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0704.0108

Reducing SAT to 2-SAT

Abstract Description of a polynomial time reduction of SAT to 2-SAT of polynomial size. http://arxiv.org/abs/0704.0108v1 Reducing SAT to 2-SAT Sergey Gubin November 4, 2018 1 Introduction Among all dimensions, 2-SAT possesses many special properties unique in the sense of computational complexity [1, 2, 3, 4, 5]. But in light of works [6, 8, 7, 9] a problem arose: either those properties are accidental or there are polynomial time reductions of SAT to 2-SAT of polynomial size. This article describes one such reduction. 2 Presenting SAT with XOR In [6] was described one of the ways to present SAT with a conjunction of XOR. Let us summarize it. Let Boolean formula f define a given SAT instance: f = c1 ∧ c2 ∧ . . . ∧ cm. (1) Clauses ci are disjunctions of literals: ci = Li1 ∨ Li2 ∨ . . . ∨ Lini , i = 1, 2, . . . , m - where ni is the number of literals in clause ci; and Lij are the literals. Using distributive laws, formula (1) can be rewritten in disjunctive form: f = d1 ∨ d2 ∨ . . . dp, p = n1n2 . . . nm. Clauses dk in this presentation are conjunctions of m literals - one literal from each clause ci, i = 1, 2, . . . , m: dk = L1k1 ∧ L2k2 ∧ . . . ∧ Lmkm , k = 1, 2, . . . , p. (2) ∗Author’s email: [email protected] It is obvious that formula (1) is satisfiable iff there are clauses without com- plimentary literals amongst conjunctive clauses (2). Disjunction of all those clauses is the disjunctive normal form of formula (1). Thus, formula (1) is satisfiable iff there are members in its disjunctive normal form. There is a generator for conjunctive clauses (2): (ξi1 ⊕ ξi2 ⊕ . . .⊕ ξini) = true, (3) - where Boolean variable ξµν indicates whether literal Lµν participates in con- junction (2). Solutions of equation (3) generate conjunctive clauses (2). Let’s call the variables ξ the indicators. To select from all solutions of equation (3) those without complimentary clauses, let’s use another Boolean equation. For each of the combination of clauses (ci, cj), 1 ≤ i < j ≤ m, let’s build a set of all couples of literals participating in the clauses: Aij = { (Liµ, Ljν) | ci = Liµ ∨ . . . ; cj = Ljν ∨ . . . }. Let Bij be a set of such couples of indicators (ξiµ, ξjν), that the literals they present are complimentary: Bij = { (ξiµ, ξjν) | (Liµ, Ljν) ∈ Aij, Liµ = L̄jν }. There are C2m sets Bij , 1 ≤ i < j ≤ m, and |Bij| ≤ min{ni, nj}. Let’s mention that some of the sets can be empty. Then, the following equa- tion will select from all solutions of equation (3) those without complimentary clauses: 1≤i<j≤m (ξ,ζ)∈Bij (ξ̄ ∨ ζ̄) = true. (4) Due to the above estimations of the number of sets Bij and of their sizes, the number of clauses in formula (4) is n = O(t2m - where t2 is the second number in the row of clauses’ sizes sorted by value: t1 = max{n1, n2, . . . , nm}, t2 = max min{ni, nj}, . . . Because satisfiability of formula (1) means that the disjunctive normal form of formula (1) has conjunctive clauses, formula (1) is satisfiable iff the following formula/equation is satisfiable: g ∧ h = true. (5) The reasons for replacing formula (1) with formula (5) are explained in [6]. The number of true-strings in truth-tables of XOR clauses of formula (3) is linear over initial input. The number of true-strings in truth-tables of disjunctive clauses of formula (4) is just 3. The number of all clauses in (5) is cubic over initial input. It can be estimated as m+ n = O(t2m Thus, application of the simplified compatibility matrices method [6] to equa- tion (5) will produce a polynomial time algorithm for SAT. But let’s return to the reduction. 3 SAT vs. 2-SAT Let’s apply the simplified method of compatibility matrices [6] to equation (5). The method consists of sequential Boolean transformations of compat- ibility matrices of equation (5). Let’s mention that after m iterations, due to the allocation of formula (4) at the end of formula (5), there will only be compatibility matrices of equation (4) left in play. They will be grouped in an upper triangular box matrix S = (Fm+µ,m+ν)1≤µ<ν≤n. (6) The matrix is displayed below: Fm+1,m+2 Fm+1,m+3 . . . Fm+1,m+n Fm+2,m+3 . . . Fm+2,m+n . . . Fm+n−1,m+n If there are no complimentary literals in different clauses of formula (1), then formula (4) is just missing. The size of matrix (6) is 0× 0. In this case, formula (1) is reducible to 1-SAT instance ω1 ∧ ω2 ∧ . . . ∧ ωm, - where ωi = ξi1 ⊕ ξi2 ⊕ . . .⊕ ξini, i = 1, 2, . . . , m. This singularity belongs to the set of all 2-SAT instances. If, during the first m iterations, a pattern of unsatisfiability arises (one of the compatibility matrices becomes filled with false entirely), then formulas (5) and (1) are both unsatisfiable [6]. This case may be thought of as a case of formula (1) being reduced to an unsatisfiable formula false. Let’s include this singularity in the set of all 2-SAT instances. Otherwise, boxes Fm+µ,m+ν in matrix (6) are what is left of the compati- bility matrices of equation (4) after the first m iterations of the method. Due to their construction [6], the boxes are 3× 3 matrices: Fm+µ,m+ν = (xij)3×3, 1 ≤ µ < ν ≤ n (7) - where xij ∈ {false, true}. The number of boxes is C n. Thus, the number of all elements in matrix (6) is e = 9C2n = O(t Let’s enumerate the elements arbitrarily: y1, y2, . . . , ye. Then, distribution of true/false in matrix (6) can be described with a 1-SAT formula/equation w = η1 ∧ η2 . . . ∧ ηe = true, (8) - where ηi are literals over a set of Boolean variables { b1, b2, . . . , be }. The literals are bi, yi = true b̄i, yi = false , i = 1, 2, . . . , e. Let’s take the following 2-SAT instance: h ∧ w. (9) Box matrix (6) is an initialization of the modified method of compatibility matrices [6] for formula (9): compatibility matrices of formula (4) are de- pleted to satisfy equation (8). Thus, continuation of the simplified method of compatibility matrices for equation (5) from its Step m+ 1 to its finish is an application of the modified method of compatibility matrices to system (9) from its Step 1 to its finish [6]. After n−2 iterations, both methods must result with the same version of satisfiability of formula (1). Thus, formulas (5) and (1) are satisfiable iff 2-SAT formula (9) is satisfiable. The number of clauses in formula (9) is e+ n = O(t22m According to [6], the time to deduce formula (9) can be safely estimated as O(t41t 4 SAT vs. 1-SAT Let’s take one step further. Applying to formula (1)/(5) either of the varia- tions of the compatibility matrices method [6] will produce a Boolean matrix. Let it be a matrix R: R = (rij)a×b. Size of the matrix depends on the method’s variation and the order of clauses in formula (1). The size can be changed if permute the clauses and repeat the method [6]. The formula (1) is satisfiable iff matrix R contains true-elements [6] (elements which are true). The existence/absence of the true-elements is the only invariant. If formula (1) is unsatisfiable, then that formula is reducible to formula “false”. Otherwise, formula (1) is reducible to a 1-SAT instance. Proof. Let’s enumerate elements of matrix R in arbitrarily order: z1, z2, . . . , zab. Let B be a set of t = ab Boolean variables: B = { bi ∈ {false, true} | i = 1, 2, . . . , t }. Then the following 1-SAT formula describes distribution of true/false in matrix R: θ1 ∧ θ2 ∧ . . . ∧ θt, (10) - where literals θi are bi, zi = true b̄i, zi = false , i = 1, 2, . . . , t. Thus, the compatibility matrices method reduces satisfiable formula (1) to 1-SAT formula (10). In its turn, formula (10) can be rewritten as SAT of any dimension by appropriate substitution of variables. If use the simplified method of compatibility matrices, then matrix R is a 3× 3 Boolean matrix [6]. Let there be two clauses shorter than 3 in formula (1). Let’s permute all clauses and make those shortest clauses to be the last ones in formula (1). Then, result of the modified method [6] will be a matrix R of size less than 3× 3. That proves the following theorem. Theorem 1. Any SAT instance is reducible to a 1-SAT instance with 9 variables or less. A SAT instance is unsatisfiable iff its 1-SAT presentation is “false” - there is not any variables in its 1-SAT presentation. 5 Conclusions Formula (1) may be thought of as a “Business Requirements”. And any appropriate computer program may be thought of as a solution of the SAT instance. Then, theorem 1 can be an explanation of the remarkable efficiency of the “natural programs”. From this point of view, the iterations of the method of compatibility matrices may be thought of as a learning/modeling of the business domain. In the artificial programming, the calculation of the compatibility matrices - a virtual business domain - could be a conclusion of the stage “Business Requirements Analysis/Mathematical Modeling”. That would improve the programs’ performance. The resulting compatibility ma- trices may be thought of as a fussy logic’s tables of rules for the domain. The whole solution of formula (1) can be achieved, with one of the fol- lowing approaches, for example. ANN approach is the applying of the com- patibility matrices method backward, starting from matrix R. An example of that can be found in [7]. DTM approach is the looping trough of the following three steps: selection of any true-element from matrix R; substi- tution of the appropriate true-assignments in formula (1); and repeating of the compatibility matrices method. The last method is an implication of the self-reducibility property of SAT [5]. In certain sense, theorem 1 may be seen as an answer to the Feasibility Thesis [2]. References [1] Stephen Cook. The complexity of theorem-proving procedures. In Con- ference Record of Third Annual ACM Symposium on Theory of Com- puting. p.151-158, 1971 [2] Stephen Cook. The P versus NP problem. http://www.claymath.org/millennium/P_vs_NP/pvsnp.pdf [3] Richard M. Karp. Reducibility Among Combinatorial Problems. In Complexity of Computer Computations, Proc. Sympos. IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y. New York: Plenum, p.85-103, 1972. [4] M.R. Garey and D.S. Johnson. Computers and Intractability, a Guide to the Theory of NP-Completeness. W.H. Freeman and Co. San Francisco, 1979. http://www.claymath.org/millennium/P_vs_NP/pvsnp.pdf [5] Lane A. Hemaspaandra, Mitsunori Ogihara. The Complexity Theory Companion. Springer-Verlag Berlin Heidelberg, 2002. [6] Sergey Gubin. A Polynomial Time Algorithm for SAT. http://www.arxiv.org/pdf/cs/0703146 [7] Sergey Gubin. A Polynomial Time Algorithm for 3-SAT. Examples of use. http://www.arxiv.org/pdf/cs/0703098 [8] Sergey Gubin. A Polynomial Time Algorithm for 3-SAT. http://www.arxiv.org/pdf/cs/0701023 [9] Sergey Gubin. A Polynomial Time Algorithm for The Traveling Sales- man Problem. http://www.arxiv.org/pdf/cs/0610042 http://www.arxiv.org/pdf/cs/0703146 http://www.arxiv.org/pdf/cs/0703098 http://www.arxiv.org/pdf/cs/0701023 http://www.arxiv.org/pdf/cs/0610042 Introduction Presenting SAT with XOR SAT vs. 2-SAT SAT vs. 1-SAT Conclusions

0704.0109

Half-metallic silicon nanowires

Half-metallic silicon nanowires E. Durgun,1, 2 D. Çakır,1, 2 N. Akman,2, 3 and S. Ciraci1, 2, ∗ Department of Physics, Bilkent University, Ankara 06800, Turkey National Nanotechnology Research Center, Bilkent University, Ankara 06800, Turkey Department of Physics, Mersin University, Mersin, Turkey (Dated: November 19, 2021) From first-principles calculations, we predict that transition metal (TM) atom doped silicon nanowires have a half-metallic ground state. They are insulators for one spin-direction, but show metallic properties for the opposite spin direction. At high coverage of TM atoms, ferromagnetic sil- icon nanowires become metallic for both spin-directions with high magnetic moment and may have also significant spin-polarization at the Fermi level. The spin-dependent electronic properties can be engineered by changing the type of dopant TM atoms, as well as the diameter of the nanowire. Present results are not only of scientific interest, but can also initiate new research on spintronic applications of silicon nanowires. PACS numbers: 73.22.-f, 68.43.Bc, 73.20.Hb, 68.43.Fg Rod-like, oxidation resistant Si nanowires (SiNW) can now be fabricated at small diameters[1] (1-7 nm) and dis- play diversity of interesting electronic properties. In par- ticular, the band gap of semiconductor SiNWs varies with their diameters. They can serve as a building material in many of electronic and optical applications like field effect transistors [2] (FETs), light emitting diodes [3], lasers [4] and interconnects. Unlike carbon nanotubes, the con- ductance of semiconductor nanowire can be tuned easily by doping during the fabrication process or by applying a gate voltage in a SiNW FET. In this letter, we report a novel spin-dependent elec- tronic property of hydrogen terminated silicon nanowires (H-SiNW): When doped by specific transition metal (TM) atoms they show half-metallic[5, 6] (HM) ground state. Namely, due to broken spin-degeneracy, energy bands En(k, ↑) and En(k, ↓) split and the nanowire re- mains to be insulator for one spin-direction of electrons, but becomes a conductor for the opposite spin-direction achieving 100% spin polarization at the Fermi level. Un- der certain circumstances, depending on the dopant and diameter, semiconductor H-SiNWs can be also either a ferromagnetic semiconductor or metal for both spin di- rections. High-spin polarization at the Fermi level can be achieved also for high TM coverage of specific SiNWs. Present results on the asymmetry of electronic states of TM doped SiNWs are remarkable and of technological in- terest since room temperature ferromagnetism is already discovered in Mn-doped SiNW[8]. Once combined with advanced silicon technology, these properties can be re- alizable and hence can make ”known silicon” again a po- tential material with promising nanoscale technological applications in spintronics, magnetism. Even though 3D ferromagnetic Heusler alloys and transition-metal oxides exhibit half-metallic properties [7], they are not yet appropriate for spintronics because of difficulties in controlling stoichiometry and the defect levels destroying the coherent spin-transport. Qian et al. have proposed HM heterostructures composed of δ- doped Mn layers in bulk Si [9]. Recently, Son et al. [10] predicted HM properties of graphene nanoribbons. Stable 1D half-metals have been also predicted for TM atom doped arm-chair single-wall carbon nanotubes [11] and linear carbon chains [12, 13]; but synthesis of these nanostructures appears to be difficult. Our results are obtained from first-principles plane wave calculations [14] (using a plane-wave basis set up to kinetic energy of 350 eV) within generalized gradient ap- proximation expressed by PW91 functional[15]. All cal- culations for paramagnetic, ferromagnetic and antiferro- magnetic states are carried out using ultra-soft pseudopo- tentials [16] and confirmed by using PAW potential[17]. All atomic positions and lattice constants are optimized by using the conjugate gradient method where total en- ergy and atomic forces are minimized. The convergence for energy is chosen as 10−5 eV between two steps, and the maximum force allowed on each atom is 0.05 eV/Å[18]. Bare SiNW(N)s (which are oriented along [001] direc- tion and have N Si atoms in their primitive unit cell) are initially cut from the ideal bulk Si crystal in rod-like forms and subsequently their atomic structures and lat- tice parameter are relaxed [19]. The optimized atomic structures are shown for N=21, 25, and 57 in Fig. 2. While bare SiNW(21) is a semiconductor, bare SiNW(25) and SiNW(57) are metallic. The average cohesive energy relative to a free Si atom (Ec) is comparable with the calculated cohesive energy of bulk crystal (4.64 eV per Si atom) and it increases with increasing N. The average co- hesive energy relative to the bulk Si crystal, E c, is small but negative as expected. Upon passivation of dangling bonds with hydrogen atoms all of these SiNWs (specified as H-SiNW) become semiconductor with a band gap EG. The binding energy of adsorbed hydrogen relative to the free H atom (Eb), as well as relative to the free H2 (E are both positive and increases with increasing N. Exten- http://arxiv.org/abs/0704.0109v1 FIG. 1: (Color online) Upper curve in each panel with numerals indicate the distribution of first, second, third, fourth etc nearest neighbor distances of SiNW(N) as cut from the ideal Si crystal, same for structure-optimized bare SiNW(N)(middle curve) and structure optimized H-SiNW(N) (bottom curve) for N=21, 57 and 81. Vertical dashed line cor- responds to the distance of Si-H bond. sive ab initio molecular dynamics calculations have been carried out at 500 K using supercells, which comprise ei- ther two or four primitive unit cells of nanowires to lift artificial limitations imposed by periodic boundary con- dition. After several iterations lasting 1 ps, the structure of all SiNW(N) and H-SiNW(N) remained stable. Even though SiNWs are cut from ideal crystal, their optimized structures deviate substantially from crystalline coordi- nation, especially for small diameters as seen in Fig.1. Upon hydrogen termination the structure is healed sub- stantially, and approaches the ideal case with increas- ing N (or increasing diameter), as expected. The cal- culated response of the wire to a uniaxial tensile force, κ = ∂ET /∂c, ranging from 172 to 394 eV/cell indicates that the strength of H-SiNW(N)s (N=21-57) is rather high. The adsorption of a single TM (TM=Fe, Ti, Co, Cr, and Mn) atom per primitive cell, denoted by n = 1, have been examined for different sites (hollow, top, bridge etc) on the surface of H-SiNW(N) for N=21, 25 and 57. In Fig. 2(c) we present only the most energetic adsorption geometry for a specific TM atom for each N, which re- sults in a HM state. These are Co-doped H-SiNW(21), Cr-doped SiNW(25) and Cr-doped SiNW(57). These nanowires have ferromagnetic ground state, since their energy difference between calculated spin-unpolarized and spin-polarized total energy, i.e. ∆Em = EsuT − is positive. We calculated ∆Em =0.04, 0.92 and 0.94 eV for H-SiNW(21)+Co, H-SiNW(25)+Cr and H- SiNW(57)+Cr, respectively [20]. Moreover, these wires have the integer number of unpaired spin in their prim- itive unit cell. In contrast to usually weak binding of TM atoms on single-wall carbon nanotubes which can lead to clustering [21], the binding energy of TM atoms (EB) on H-SiNWs is high and involve significant charge transfer from TM atom to the wire [22]. Mulliken anal- ysis indicates that the charge transfer from Co to H- FIG. 2: (Color online) Top and side views of optimized atomic structures of various SiNW(N)’s. (a) Bare SiNWs; (b) H- SiNWs; (c) single TM atom doped per primitive cell of H- SiNW (n = 1); (d) H-SiNWs covered by n TM atom corre- sponding to n > 1. Ec, E c, Eb, E b, EG, and µ, respectively denote the average cohesive energy relative to free Si atom, same relative to the bulk Si, binding energy of hydrogen atom relative to free H atom, same relative to H2 molecule, energy band gap and the net magnetic moment per primitive unit cell. Binding energies in regard to the adsorption of TM atoms, i.e. EB, E B for n = 1 and average values EB , E for n > 1 are defined in the text and in Ref[22]. The [001] direction is along the axis of SiNWs. Small, large-light and large-dark balls represent H, Si and TM atoms, respectively. Side views of atomic structure comprise two primitive unit cells of the SiNWs. Binding and cohesive energies are given in eV/atom. SiNW(21) is 0.5 electrons. The charge transfer from Cr to H-SiNW(25) and H-SiNW(57) is even higher (0.8 and 0.9 electrons, respectively). Binding energies of ad- sorbed TM atoms relative to their bulk crystals (E′B) are negative and hence indicate endothermic reaction. Due to very low vapor pressure of many metals, it is proba- bly better to use some metal-precursor to synthesize the structures predicted here. The band structures of HM nanowires are presented in Fig.3. Once a Co atom is adsorbed above the center of a hexagon of Si atoms on the surface of H-SiNW(21) the spin degeneracy is split and whole system becomes magnetic with a magnetic moment of µ=1 µB (Bohr mag- neton per primitive unit cell). Electronic energy bands become asymmetric for different spins: Bands of major- ity spins continue to be semiconducting with relatively smaller direct gap of EG=0.4 eV. In contrast, two bands of minority spins, which cross the Fermi level, become metallic. These metallic bands are composed of Co-3d and Si-3p hybridized states with higher Co contribution. The density of majority and minority spin states, namely D(E, ↑) and D(E, ↓), display a 100% spin-polarization P = [D(EF , ↑)−D(EF , ↓)]/[D(EF , ↑)+D(EF , ↓)] at EF . Cr-doped H-SiNW(25) is also HM. Indirect gap of major- ity spin bands has reduced to 0.5 eV. On the other hand, two bands constructed from Cr-3d and Si-3p hybridized states cross the Fermi level and hence attribute metal- licity to the minority spin bands. Similarly, Cr-doped H-SiNW(57) is also HM. The large direct band gap of undoped H-SiNW(57) is modified to be indirect and is reduced to 0.9 eV for majority spin bands. The mini- mum of the unoccupied conduction band occurs above but close to the Fermi level. Two bands formed by Cr-3d and Si-3p hybridized states cross the Fermi level. The net magnetic moment is 4 µB . Using PAW potential results, we estimated Curie temperature of half-metallic H-SiNW+TMs as 8, 287, and 709 K for N=21, 25, and 57, respectively. The well-known fact that density functional theory un- derestimates the band gap, EG does not concern the present HM states, since H-SiNWs are already verified to be semiconductor experimentally[1] and upon TM- doping they are predicted to remain semiconductor for one spin direction. In fact, band gaps predicted here are in fair agreement with experiment and theory. As for par- tially filled metallic bands of the opposite spin, they are properly represented. Under uniaxial compressive strain the minimum of the conduction band of majority spin states rises above the Fermi level. Conversely, it becomes semi-metallic under uniaxial tensile strain. Since conduc- tion and valence bands of both H-SiNW(21)+Co and H- SiNW(25)+Cr are away from EF , their HM behavior is robust under uniaxial strain. Also the effect of spin-orbit coupling is very small and cannot destroy HM properties [12]. The form of two metallic bands crossing the Fermi level eliminates the possibility of Peierls distortion. On FIG. 3: (Color online) Band structure and spin-dependent to- tal density of states (TDOS) for N=21, 25 and 57. Left panels: Semiconducting H-SiNW(N). Middle panels: Half-metallic H- SiNW(N)+TM. Right panels: Density of majority and mi- nority spin states of H-SiNW(N)+TM. Bands described by continuous and dotted lines are majority and minority bands. Zero of energy is set to EF . FIG. 4: (Color online) D(E, ↓), density of minority (light) and D(E, ↑), majority (dark) spin states. (a) H-SiNW(25)+Cr, n = 8; (b) H-SiNW(25)+Cr, n = 16. P and µ indicate spin- polarization and net magnetic moment (in Bohr magnetons per primitive unit cell), respectively. the other hand, HM ground state of SiNWs is not com- mon to all TM doping. For example H-SiNW(N)+Fe is consistently ferromagnetic semiconductor with different EG,↑ and EG,↓. H-SiNW(N)+Mn(Cr) can be either fer- romagnetic metal or HM depending on N. To see whether spin-dependent GGA properly repre- sents localized d-electrons and hence possible on-site re- pulsive Coulomb interaction destroys the HM, we also carried out LDA+U calculations[23]. We found that in- sulating and metallic bands of opposite spins coexist up to high values of repulsive energy (U = 4) for N=25. For N=57, HM persists until U∼1. Clearly, HM character of TM doped H-SiNW revealed in Fig.3 is robust and unique behavior. Finally, we note that HM state predicted in TM-doped H-SiNWs occurs in perfect structures; complete spin- polarization may deviate slightly from P=100% due to the finite extent of devices. Even if the exact HM charac- ter corresponding to n = 1 is disturbed for n > 1, the pos- sibility that some H-SiNWs having high spin-polarization at EF at high TM coverage can be relevant for spintronic applications. We therefore investigated electronic and magnetic structure of the above TM-doped H-SiNWs at n > 1 as described in Fig. 2(d). Figure 4 presents the calculated density of minority and majority spin states of Cr covered H-SiNWs. It is found that H-SiNW(21) covered by Co is non- magnetic for both coverage of n = 4 and 12. H-SiNW(25) is, however, ferromagnetic for different level of Cr cover- age and has high net magnetic moment. For example, n = 8 can be achieved by two different geometries; both geometries are ferromagnetic with µ=19.6 and 32.3 µB and are metallic for both spin directions. Interestingly, while P is negligible for the former geometry, the lat- ter one has P = 0.84 and hence is suitable for spin- tronic applications (See Fig. 4). Similarly, Cr covered H-SiNW(57) with n = 8 and 16 are both ferromagnetic with µ= 34.3 (P =56) and µ=54.5 µB (P =0.33), respec- tively. The latter nanostructure having magnetic mo- ment as high as 54.5 µB can be a potential nanomagnet. Clearly, not only total magnetic moment, but also the spin polarization at EF of TM covered H-SiNMs exhibits interesting variations depending on n, N and the type of In conclusion, hydrogen passivated SiNWs can exhibit half-metallic state when doped with certain TM atoms. Resulting electronic and magnetic properties depend on the type of dopant TM atom, as well as on the diam- eter of the nanowire. As a result of TM-3d and Si-3p hybridization two new bands of one type of spin direc- tion are located in the band gap, while the bands of other spin-direction remain to be semiconducting. Elec- tronic properties of these nanowires depend on the type of dopant TM atoms, as well as on diameter of the H-SiNW. When covered with more TM atoms, perfect half-metallic state of H-SiNW is disturbed, but for cer- tain cases, the spin polarization at EF continues to be high. High magnetic moment obtained at high TM coverage is another remarkable result which may lead to the fabrication of nanomagnets for various applica- tions. Briefly, functionalizing silicon nanowires with TM atoms presents us a wide range of interesting properties, such as half-metals, 1D ferromagnetic semiconductors or metals and nanomagnets. We believe that our findings hold promise for the use of silicon -a unique material of microelectronics- in nanospintronics including magne- toresistance, spin-valve and non-volatile memories. ∗ Electronic address: [email protected] [1] D. D. D. Ma et al., Science 299, 1874 (2003). [2] Y. Cui, Z. Zhong, D. Wang, W. U. Wang and C. M. Lieber, Nano Lett. 3, 149 (2003). [3] Y. Huang, X. F. Duan and C. M. Lieber, Small 1, 142 (2005). [4] X. F. Duan, Y. Huang, R. Agarwal and C. M. Lieber, Nature(London) 421, 241 (2003). [5] R.A. de Groot et al., Phys. Rev. Lett. 50, 2024 (1983). [6] W.E. Pickett and J. S. Moodera, Phys. Today 54, 39 (2001). [7] J.-H. Park et al., Nature (London) 392, 794 (1998). [8] W.H. Wu, J.C. Tsai and J.L. Chen, Appl. Phys. Lett. 90, 043121 (2007). [9] M.C. Qian et al., Phys. Rev. Lett. 96, 027211 (2006). [10] Y-W Son, M.L. Cohen and S.G. Louie, Nature 444, (2006); Phys. Rev. Lett. 97, 216803 (2006). [11] C. Yang, J. Zhao and J.P. Lu, Nano. Lett. 4, 561 (2004); Y. Yagi et al., Phys. Rev. B 69, 075414 (2004). [12] S. Dag et al., Phys. Rev. B 72, 155444 (2005). [13] E. Durgun et al., Europhys. Lett. 73, 642 (2006). [14] Numerical computations have been carried out by us- ing VASP software: G. Kresse, J. Hafner, Phys Rev. B 47, R558 (1993). Calculations of charge transfer, orbital contribution and local magnetic moments have been re- mailto:[email protected] peated by SIESTA code using local basis set, P. Ordejon, E. Artacho and J.M. Soler, Phys. Rev. B 53, R10441 (1996). [15] J. P. Perdew et al., Phys. Rev. B 46, 6671 (1992). [16] D. Vanderbilt, Phys. Rev. B 41, R7892 (1990). [17] P.E. Bloechl, Phys. Rev. B 50, 17953 (1994). [18] All structures have been treated within supercell geom- etry using the periodic boundary conditions with lattice constants of a and b ranging from 20 Å to 25 Å depending on the diameter of the SiNW and c = co (co being the optimized lattice constant of SiNW along the wire axis). Some of the calculations have been carried out in dou- ble and quadruple primitive unit cells of SiNW by taking c = 2co and c = 4co, respectively. In the self-consistent potential and total energy calculations the Brillouin zone is sampled in the k-space within Monkhorst-Pack scheme [H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976)] by (1x1x15) mesh points. [19] Numerous theoretical studies on SiNW have been pub- lished in recent years. See for example: A. K. Singh et al., Nano. Lett. 6, 920 (2006); Q. Wang et al., Phys. Rev. Lett. 95, 167200 (2005); Nano Lett. 5, 1587 (2005). [20] Spin-polarized calculations have been carried by relax- ing the magnetic moment and by starting with different initial µ values. Whether antiferromagnetic ground state exists in H-SiNW(N)+TM’s has been explored by using supercell including double primitive cells. [21] E. Durgun et al., Phys. Rev. B 67, 201401(R) (2003); J. Phys. Chem. B 108, 575 (2004). [22] Binding energy corresponding to n=1 is calculated by the following expression, EB = ET [H − SiNW (N)] + ET [TM ] − ET [H − SiNW (N) + TM ] in terms of the total energy of optimized H-SiNW(N) and TM-doped H- SiNW(N) (i.e. H-SiNW(N)+TM) and the total energy of the string of TM atoms having the same lattice parameter co of H-SiNW(N)+TM, all calculated in the same super- lattice. Hence EB can be taken as the binding energy of single isoalted TM atom, since the coupling amaong ad- sorbed TM atoms has been excluded. To calculate E′B, ET (TM) is taken as the total energy of bulk TM crystal per atom. For n¿1, ET (TM) is taken as the free TM atom energy, and hence EB includes the coupling between TM atoms. For this reason E B > 0 for H-SiNW(21)+Co at [23] S.L. Dudarev et al., Phys. Rev. B, 57, 1505 (1998).

0704.0111

Invariance and the twisted Chern character : a case study

arXiv:0704.0111v1 [math.QA] 2 Apr 2007 Invariance and the twisted Chern character : a case study Debashish Goswami Stat-Math Unit, Indian Statistical Institute 203, B. T. Road, Kolkata 700108, India. E-mail : [email protected] Abstract We give details of the proof of the remark made in [7] that the Chern characters of the canonical generators on the K homology of the quantum group SUq(2) are not invariant under the natural SUq(2) coaction. Furthermore, the conjecture made in [7] about the nontriv- iality of the twisted Chern character coming from an odd equivariant spectral triple on SUq(2) is settled in the affirmative. 1 Introduction Noncommutative geometry (NCG) (a la Connes, see [2] ) and the C∗- algebraic theory of quantum groups (see, for example, [11], [10]) are two well-developed mathematical areas which share the basic idea of ‘noncom- mutative mathematics’, namely, to view a general (noncommutative) C∗ algebra as noncommutative analogue of a topological space, equipped with additional structures resembling and generalizing those in the classical (com- mutative) situation, e.g. manifold or Lie group structure. A lot of fruitful interaction between these two areas is thus quite expected. However, such an interaction was not very common until recently, when a systematic effort by a number of mathematicians for understanding C∗-algebraic quantum groups as noncommutative manifolds in the sense of Connes triggered a rapid and interesting development to this direction. However, quite sur- prisingly, such an effort was met with a number of obstacles even in the case of the simplest non-classical quantum group, namely SUq(2) and it was not so clear for some time whether this (and other standard examples of quantum groups) could be nicely fitted into the framework of Connes’ NCG (see [6] and the discussion and references therein). The problem of finding a nontrivial equivariant spectral triple for SUq(2) was finally settled in the affirmative in the papers by Chakraborty and Pal ([4], see also [3] and [5] for subsequent development), which increased the hope for a happy mar- riage between NCG and quantum group theory. However, even in the case of SUq(2), a few puzzling questions remain to be answered. One of them is the issue of invariance of the Chern character, which we have addressed in [7] and attempted to suggest a solution through the twisted version of http://arxiv.org/abs/0704.0111v1 the entire cyclic cohomology theory, building on the ideas of [8]. In that paper, we also made an attempt to study the connection between twisted and the conventional NCG following a comment in [3]. The present article is a follow-up of [7], and we mainly concentrate on SUq(2), considering it as a test-case for comparing the twisted and conventional formulation of NCG. 2 Notation and background Let A = SUq(2) (with 0 < q < 1) denote the C∗-algebra generated by two elements α, β satisfying α∗α+ β∗β = I, αα∗ + q2ββ∗ = I, αβ − qβα = 0, αβ∗ − qβ∗α = 0, β∗β = ββ∗. We also denote the ∗-algebra generated by α and β (without taking the norm completion) by A∞. There is a Hopf ∗ algebra structure on A∞, as can be seen from, for example, [10]. We denote the canonical coproduct on A∞ by ∆. We shall also use the so-called Sweedler convention, which we briefly explain now. For a ∈ A∞, there are finitely many elements , i = 1, 2, ..., p (say), such that ∆(a) = ⊗a(2) . For notational convenience, we abbreviate this as ∆(a) = a(1)⊗a(2). For any positive integer m, let A∞m be the m-fold algebraic tensor product of A∞. There is a natural coaction of A∞ on A∞m given by ∆mA(a1 ⊗ a2 ⊗ ...⊗ am) := (a1(1) ⊗ ...am(1))⊗ (a1(2)...am(2)), using the Sweedler notation, with summation being implied. Let us recall the convolution ∗ defined in [7]. If φ : A∞m → C is an m-linear functional, and ψ : A∞ → C is a linear functional, we define their convolution φ ∗ ψ : A∞m → C by the following : (φ ∗ ψ)(a1 ⊗ ...⊗ am) := φ(a(1)1 ⊗ ...⊗ a(1)m )ψ(a 1 ...a using the Sweedler convention. We say that an m-linear functional φ is invariant if φ ∗ ψ = ψ(1)φ for every functional ψ on A∞. In [9], the K-homology K∗(A∞) has been explicitly computed. It has been shown there that K0(A∞) = K1(A∞) = Z, and the Chern charac- ters (in cyclic cohomology ) of the generators of these K-homology groups, denoted by [τev] and [τodd] respectively, are also explicitly written down. 3 Main results 3.1 Chern characters are not invariant In this subsection, we give detailed arguments for a remark made in [7] about the impossibility of having an invariant Chern character for A∞ under the conventional (non-twisted) framework of NCG. To make the notion of invariance precise, we give the following definition (motivated by a comment by G. Landi, which is gratefully acknowledged). Definition 3.1 We say that a class [φ] ∈ HCn(A∞) is invariant if there is an invariant n + 1-linear functional φ′ such that φ′ is a cyclic cocycle and φ′ ∼ φ (i.e. [φ] = [φ′]). It is easy to see that the Chern chracter [τev] cannot be invariant. Had it been so, it would follow from the uniqueness of the Haar state (say h) on SUq(2) that τev must be a scalar multiple of h. Since τev is a nonzero trace, it would imply that h is a trace too. But it is known (see [10]) that h is not a trace. However, proving that [τodd] is not invariant requires little bit of detailed arguments. We begin with the following observation. Lemma 3.2 If τ is a trace on A∞, i.e. τ ∈ HC0(A∞), then we have that (∂ξ) ∗ τ = ∂(ξ ∗ τ) for every functional ξ on A∞, where the Hochschild coboundary operator ∂ is defined by (∂ξ)(a, b) = ξ(ab)− ξ(ba). Proof : We shall use the Swedler notation. We have that for a0, a1 ∈ A∞, (∂ξ ∗ τ)(a0, a1) = (∂ξ)(a 0 ⊗ a 1 )τ(a = ξ(a 1 )τ(a 1 )− ξ(a 0 )τ(a = ξ(a 1 τ(a 1 )− ξ(a 0 )τ(a 0 ) (since τ is a trace) = (ξ ∗ τ)(a0a1)− (ξ ∗ τ)(a1a0) = ∂(ξ ∗ τ)(a0, a1). The above lemma allows us to define the multiplication ∗ at the level of cohomology classes. More precisely, for [φ] ∈ HC1(A∞) and [η] ∈ HC0(A∞), we set [φ] ∗ [η] := [φ ∗ η] ∈ HC1(A∞), which is well-defined by the Lemma 3.2. Similarly [η] ∗ [φ] and [η] ∗ [η′] (where [η′] ∈ HC0(A∞)) can be defined. We now recall from [9] that [τev] ∗ [τev] = [τev], [τev] ∗ [τodd] = [τodd] ∗ [τev] = 0. We also note that τev(1) = 1 and that τev is a trace, i.e. τev(ab) = τev(ba). Using this observation, we are now in a position to prove that the Chern character of the generator of K1(A∞) is not an invariant class. Theorem 3.3 [τodd] is not invariant. Proof : Suppose that there is φ ∼ τodd such that φ is invariant. Then we have [φ ∗ τev] = [φ] ∗ [τev] = [τodd] ∗ [τev] = 0. However, since we have φ ∗ τev = τev(1)φ = φ by the invariance of φ, it follows that [φ] = [φ ∗ τev] = 0, that is, [τodd] = 0, which is a contradiction. 3.2 Nontrivial pairing with the twisted Chern character As already mentioned in the introduction, in [7] we have made an attempt to recover the desirable property of invariance by making a departure from the conventional NCG and using the twisted entire cyclic cohomology. We briefly recall here some of the basic concepts from that paper and refer the reader to [7] and the references therein for more details of this approach. We shall use the results derived in that paper wihout always giving a specific reference. Let us give the definition of twisted entire cyclic cohomology for Banach algebras for simplicity, but note that the theory extends to locally convex algebras, which we actually need. The extension to the locally convex al- gebra case follows exactly as remarked in [1, page 370]. So, let A be a unital Banach algebra, with ‖.‖∗ denoting its Banach norm, and let σ be a continuous automorphism of A, σ(1) = 1. For n ≥ 0, let Cn be the space of continuous n + 1-linear functionals φ on A which are σ-invariant, i.e. φ(σ(a0), ..., σ(an)) = φ(a0, ..., an)∀a0, ..., an ∈ A; and Cn = {0} for n < 0. We define linear maps Tn, Nn : C n → Cn, Un : Cn → Cn−1 and Vn : C n → Cn+1 by, (Tnf)(a0, ..., an) = (−1)nf(σ(an), a0, ..., an−1), Nn = T jn, (Unf)(a0, ..., an−1) = (−1)nf(a0, ..., an−1, 1), (Vnf)(a0, ..., an+1) = (−1)n+1f(σ(an+1)a0, a1, ..., an). Let Bn = Nn−1Un(Tn − I), bn = j=0 T n+1 VnT n. Let B, b be maps on the complex C ≡ (Cn)n given by B|Cn = Bn, b|Cn = bn. It is easy to verify (similar to what is done for the untwisted case , e.g. in [2]) that B2 = 0, b2 = 0 and Bb = −bB, so that we get a bicomplex (Cn,m ≡ Cn−m) with differentials d1, d2 given by d1 = (n − m + 1)b : Cn,m → Cn+1,m, : Cn,m → Cn,m+1. Furthermore, let Ce = {(φ2n)n ∈ IN ;φ2n ∈ C2n∀n ∈ IN}, and Co = {(φ2n+1)n ∈ IN ;φ2n+1 ∈ C2n+1∀n ∈ IN}. We say that an element φ = (φ2n) of C e is a σ-twisted even entire cochain if the radius of convergence of the complex power series ‖φ2n‖z is infinity, where ‖φ2n‖ := sup‖aj‖∗≤1 |φ2n(a0, ...., a2n)|. Similarly we define σ-twisted odd entire cochains, and let Ceǫ (A, σ) (Coǫ (A, σ) respectively) denote the set of σ-twisted even (respectively odd) entire cochains. Let ∂̃ = d1 + d2 , and we have the short complex Ceǫ (A, σ) Coǫ (A, σ). We call the cohomology of this complex the σ-twisted entire cyclic cohomology of A and denote it by H∗ǫ (A, σ). Let Aσ = {a ∈ A : σ(a) = a} be the fixed point subalgebra for the automorphism σ. There is a canonical pairing < ., . >σ,ǫ: K∗(Aσ)× H∗ǫ (A, σ) → C. We shall need the pairing for the odd case, which we write down : < [u], [ψ] >≡< [u], [ψ] >σ,ǫ= (−1)n n! (2n + 1)! ψ2n+1(u −1, u, ..., u−1, u), where [u] ∈ K1(Aσ) and [ψ] ∈ H1ǫ (A, σ). Definition 3.4 Let H be a separable Hilbert space, A∞ be a ∗ subalgebra (not necessarily complete) of B(H), R be a positive (possibly unbounded) operator on H, D be a self-adjoint operator in H with compact resolvents such that the following hold : (i) [D, a] ∈ B(H) ∀a ∈ A∞, (ii) R commutes with D, (iii) For any real number s and a ∈ A∞, σs(a) := R−saRs is bounded and be- longs to A∞. Furthermore, for any positive integer n, sups∈[−n,n] ‖σs(a)‖ < Then we call the quadruple (A∞,H,D,R) an odd R-twisted spectral data. We say that the odd twisted spectral data is Θ-summable if Re−tD is trace- class for all t > 0. Let us now recall the construction of twisted Chern character from a given odd twisted spectral data (A∞,H,D,R). Let B denote the set of all A ∈ B(H) for which σs(A) := R−sARs ∈ B(H) for all real number s, [D,A] ∈ B(H) and s 7→ ‖σs(A)‖ is bounded over compact subsets of the real line. In particular, A∞ ⊆ B. We define for n ∈ IN an n+1-linear functional Fn on B by the formula Fn(A0, ..., An) = Tr(A0e ...Ane R)dt0...dtn, where Σn = {(t0, ..., tn) : ti ≥ 0, i=0 ti = 1}. Let us now equip A∞ with the locally convex topology given by the fam- ily of Banach norms ‖.‖∗,n, n = 1, 2, ..., where ‖a‖∗,n := sups∈[−n,n](‖σs(a)‖+ ‖[D,σs(a)]‖). Let A denote the completion of A∞ under this topology, and thus A is Frechet space. We can construct the (twisted) Chern character in Hoǫ (A, σ), where σ = σ1, which extends on the whole of A by continuity. Theorem 3.5 Let φo ≡ (φ2n+1)n be defined by φ2n+1(a0, ..., a2n+1) = 2iF2n+1(a0, [D, a1], ..., [D, a2n+1]), ai ∈ A. Then we have (b+B)φo = 0, hence ψo ≡ ((2n + 1)!φ2n+1)n ∈ Hoǫ (A, σ). We shall also need some results from the theory of semifinite spectral triples and the corresponding JLO cocycles and index formula, as discussed in, for example, [1]. An odd semifinite spectral triple is given by (C,N ,K,D), where K is a separable Hilbert space, N ⊆ B(K) is a von Neumann algebra with a faithful semifinite normal trace (say τ), D is a self-adjoint operator affiliated to N , C is a ∗-subalgebra of B(K) such that [D, c] ∈ B(K) for all c ∈ C. In the terminology of [1], (N ,D) is also called an odd, unbounded Breuer-Fredholm module for the norm-closure of C. It is called Θ-summable if τ(e−tD ) < ∞ for all t > 0. For a Θ-summable semifinite spectral triple, there is a canonical construction of JLO cocycle and index theorem (see [1]), which are very similar to their counterparts in the conventional framework of NCG. Let us now settle in the affirmative conjecture made in [7] about the nontriviality of the twisted Chern character of a natural twisted spectral data obtained from the equivariant spectral triple of [4]. For reader’s convenience, we briefly recall the construction of this equivariant spectral triple. Let us index the space of irreducible (co-)representations of SUq(2) by half-integers, i.e. n = 0, 1 , 1, ...; and index the orthonormal basis of the corresponding (2n + 1)2 dimensional subspace of L2(SUq(2), h) by i, j = −n, ..., n, instead of 1, 2, ..., (2n + 1). Thus, let us consider the orthonormal basis eni,j , n = , ...; i, j = −n,−n + 1, ..., n in the notation of [4]. We consider any of the equivariant spectral triples constructed by the authors of [4] and in the associated Hilbert space H = L2(SUq(2), h) define the following positive unbounded operator R : R(eni,j) = q −2i−2jeni,j , n = 0, 1 , , 1, ...; i, j = −n,−n+ 1, ..., n. Let us choose a spectral triple given by the Dirac operator D on H, defined by D(eni,j) = d(n, i)e i,j , where d(n, i) are as in (3.12) of [4], i.e. d(n, i) = 2n + 1 if n = i, d(n, i) = −(2n+ 1) otherwise. It can easily be seen that (A∞,H,D,R) is an odd R- twisted spectral data and furthermore, the fixed point subalgebra SUq(2)σ for σ(.) = R−1 ·R is the unital ∗-algebra generated by β, so it contains u = ∗β)(β−I)+I which can be chosen to be a generator of K1(SUq(2)) = Z (see [4]). It is easily seen that the map from K1(C ∗(u)) to K1(SUq(2)), induced by the inclusion map, is an isomorphism of the K1-groups (where C∗(u) denotes the unital C∗-algebra generated by u). Thus, we can consider the pairing of the twisted Chern character with K1(C ∗(u)), and in turn with K1(SUq(2)) using the isomorphism noted before. The important question raised in [7] is whether we recover the nontrivial pairing obtained in [4] in our twisted framework, and in what follows, we shall give an affirmative answer to this question. Theorem 3.6 The pairing between K1(SUq(2)σ) ∼= K1(SUq(2)) and the (twisted) Chern character of the above twisted spectral data coincides with the pairing between K1(SUq(2)) and the Chern character of the (non-twisted) spectral triple (A∞,H,D). In particular, this pairing is nontrivial. Proof : Let N be the von Neumann algebra in B(H) generated by β and f(D) for all bounded measurable functions f : R → C. Since R commutes with both β and D, it is easy to see that the functional N ∋ X 7→ τ(X) := Tr(XR) defines a faithful, normal, semifinite trace on the von Neumann algebra N . Moreover, (N ,D) is an unbounded Θ-summable Breuer-Fredholm module for the norm-closure of the unital ∗-algebra (say C) generated by β. Moreover, it follows from the fact that R commutes with D and u that the pairing of [u] with the twisted Chern character (say ψo ≡ (ψ2n+1)) coming from the twisted spectral data (A∞,H,D,R) is given by < [u], [ψo] > (−1)n n! (2n+ 1)! ψ2n+1(u −1, u, ..., u−1, u) (−1)nn! Σ2n+1 Tr(u−1e−t0D [D,u]et1D ...[D,u]et2n+1D R)dt0...dt2n+1, (−1)nn! Σ2n+1 τ(u−1e−t0D [D,u]et1D ...[D,u]et2n+1D )dt0...dt2n+1 which is nothing but the pairing between [u] ∈ K1(C) and the Breuer- Fredholm module (N ,D) mentioned before. By Theorem 10.8 of [1] and a straightforward but somewhat lengthy calculation along the lines of index computation in [4], we can show that the value of this pairing is equal to −indτ (A) ≡ −(τ(PA) − τ(QA)) for the following operator A : H0 → H0, where H0 is the closed subspace spanned by {enn,j , n = 0, 12 , ..., j = −n,−n+ 1, ..., n}, PA, QA are the orthogonal projections onto the kernel of A and the kernel of A∗ respectively and where r is a positive integer such that q2r < 1 < q2r−2 : Aenn,j = −q(n+j)(2r+1)(1−q2(n−j))r(1−q2(n−j−1)) ,j− 1 +(1−q2r(n+j)(1−q2(n−j))r)enn,j. It can be verified by computations as in [4] that Ker(A) = {0} and Ker(A∗) is the one dimensional subspace spanned by the vector ξ = n,−n, where p 1 = 1 and for n ≥ 3 1− (1− q4n−2)r (1− q4n) 12 (1− q4n−2)r 1− (1− q2)r (1 − q4) 12 (1− q2)r Clearly, since Ren−n,n = e n,−n, we have Rξ = ξ and thus −indτ (A) = ‖ξ‖2 τ(|ξ >< ξ|) = ‖ξ‖2Tr(R|ξ >< ξ|) = 1, which is the same as the value of the pairing between [u] ∈ K1(SUq(2)) and the conventional Chern character corresponding to the spectral triple constructed in [4]. ✷ Thus we see that both the conventional and twisted frameworks of NCG give essentially the same results for the example we considered, namely SUq(2). The aparent weakness of the twisted NCG arising from the fact that the twisted cyclic cohomology can be paired naturally with only the K theory of the invariant subalgebra and not of the whole algebra, does not seem to pose any essential difficulty for studying the noncommutative geometric aspects of SUq(2), since by a suitable choice of the twisting operator R as we did one could make sure that the K theory of the corresponding invariant subalgebra is isomorphic with the K theory of the whole, and also the pairing between the Chern character and the generator of the K theory in the twisted framework is equal to the similar pairing in the ordinary (non-twisted) framework of NCG. It will be important and interesting to investigate whether a similar fact remains true for a larger class of quantum groups, and we hope to pursue this in the future. References [1] A. Carey and J. Phillips, Spectral flow in Fredholm modules, eta invariants and the JLO cocycle, K-Theory31 (2004), no. 2, 135–194. [2] A. Connes, Noncommutative Geometry, Academic Press (1994). [3] A. Connes, Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2), J. Inst. Math. Jussieu 3 (2004), no. 1, 17-68. [4] P. S. Chakraborty and A. Pal, Equivariant spectral triples on the quantum SU(2) group, K-Theory 28(2003), No. 2, 107-126. [5] L. Dabrowski, G. Landi, A. Sitarz, W. van Suijlekom and J. C. Varilly, The Dirac operator on SUq(2), Commun.Math.Phys. 259 (2005) 729-759. [6] D. Goswami, Some Noncommutative Geometric Aspects of SUq(2), preprint ( math-ph/0108003). [7] D. Goswami, Twisted entire cyclic cohomology, J-L-O cocycles and equivariant spectral triples, Rev. Math. Phys. 16 (2004), no. 5, 583-602. [8] J. Kustermans, G.J. Murphy and L. Tuset, Differential Calculi over Quantum Groups and Twisted Cyclic Cocycles, J. Geom. Phys. 44 (2003), no. 4, 570–594. [9] T. Masuda, Y. Nakagami and J.Watanabe, Noncommutative Differ- ential Geometry on the Quantum SU(2), I: An Algebraic Viewpoint, K Theory 4 (1990), 157-180. [10] S. L. Woronowicz, Twisted SU(2)-group : an example of a non- commutative differential calculus, Publ. R. I. M. S. (Kyoto Univ.) 23(1987) 117-181. [11] S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), no. 4, 613–665.

0704.0112

Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces

7 Placeholder Substructures III: A Bit-String-Driven “Recipe Theory” for Infinite-Dimensional Zero-Divisor Spaces Robert P. C. de Marrais ∗ Thothic Technology Partners, P.O.Box 3083, Plymouth MA 02361 October 29, 2018 Abstract Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N- dimensional hypercomplex numbers (N a power of 2, and at least 4) can represent singularities and, as N → ∞, fractals – and thereby, scale-free net- works. Any integer > 8 and not a power of 2 generates a meta-fractal or Sky when it is interpreted as the strut constant (S) of an ensemble of octahe- dral vertex figures called Box-Kites (the fundamental ZD building blocks). Remarkably simple bit-manipulation rules or recipes provide tools for trans- forming one fractal genus into others within the context of Wolfram’s Class 4 complexity. 1 The Argument So Far In Parts I[1] and II[2], the basic facts concerning zero-divisors (ZDs) as they arise in the hypercomplex context were presented and proved. “Basic,” in the context of this monograph, means seven things. First, they emerged as a side-effect of apply- ing CDP a minimum of 4 times to the Real Number Line, doubling dimension to the Complex Plane, Quaternion 4-Space, Octonion 8-Space, and 16-D Sedenions. With each such doubling, new properties were found: as the price of sacrificing ∗Email address: [email protected] http://arxiv.org/abs/0704.0112v3 counting order, the Imaginaries made a general theory of equations and solution- spaces possible; the non-commutative nature of Quaternions mapped onto the re- alities of the manner in which forces deploy in the real world, and led to vector calculus; the non-associative nature of Octonions, meanwhile, has only come into its own with the need for necessarily unobservable quantities (because of confor- mal field-theoretical constraints)in String Theory. In the Sedenions, however, the most basic assumptions of all – well-defined notions of field and algebraic norm (and, therefore, measurement) – break down, as the phenomena correlated with their absence, zero-divisors, appear onstage (never to leave it for all higher CDP dimension-doublings). Second thing: ZDs require at least two differently-indexed imaginary units to be defined, the index being an integer larger than 0 (the CDP index of the Real Unit) and less than 2N for a given CDP-generated collection of 2N-ions. In “pure CDP,” the enormous number of alternative labeling schemes possible in any given 2N-ion level are drastically reduced by assuming that units with such indices interact by XOR-ing: the index of the product of any two is the XOR of their indices. Signing is more tricky; but, when CDP is reduced to a 2-rule construction kit, it becomes easy: for index u < G, G the Generator of the 2N-ions (i.e., the power of 2 immediately larger than the highest index of the predecessor 2N−1-ions), Rule 1 says iu · iG = +i(u+G). Rule 2 says take an associative triplet (a,b,c), assumed written in CPO (short for “cyclically positive order”: to wit, a · b = +c, b · c = +a, and c · a = +b). Consider, for instance, any (u,G,G+ u) index set. Then three more such associative triplets (henceforth, trips) can be generated by adding G to two of the three, then switching their resultants’ places in the CPO scheme. Hence, starting with the Quaternions’ (1,2,3) (which we’ll call a Rule 0 trip, as it’s inherited from a prior level of CDP induction), Rule 1 gives us the trips (1,4,5), (2,4,6), and (3,4,7), while Rule 2 yields up the other 4 trips defining the Octonions: (1,7,6), (2,5,7), and (3,6,5). Any ZD in a given level of 2N-ions will then have units with one index < G, written in lowercase, and the other index > G, written in uppercase. Such pairs, alternately called “dyads” or “Assessors,” saturate the diagonal lines of their planes, which diagonals never mutually zero-divide each other (or make DMZs, for ”divisors (or dyads) making zero”), but only make DMZs with other such diagonals, in other such Assessors. (This is, of course, the opposite situation from the projection operators of quantum mechanics, which are diagonals in the planes formed by Reals and dimensions spanned by Pauli spin operators contained within the 4- space created by the Cartesian product of two standard imaginaries.) Third thing: Such ZDs are not the only possible in CDP spaces; but they define the “primitive” variety from which ZD spaces saturating more than 1-D regions can be articulated. A not quite complete catalog of these can be found in our first monograph on the theme [3]; a critical kind which was overlooked there, involving the Reals (and hence, providing the backdrop from which to see the projection-operator kind as a degenerate type), were first discussed more recently [4]. (Ironically, these latter are the easiest sorts of composites to derive of any: place the two diagonals of a DMZ pairing with differing internal signing on axes of the same plane, and consider the diagonals they make with each other!) All the primitive ZDs in the Sedenions can be collected on the vertices of one of 7 copies of an Octahedron in the Box-Kite representation, each of whose 12 edges indicates a two-way “DMZ pathway,” evenly divided between 2 varieties. For any vertex V, and k any real scalar, indicate the diagonals this way: (V,/) = k · (iv+ iV ), while (V, \)= k · (iv− iV ). 6 edges on a Box-Kite will always have negative edge-sign (with unmarked ET cell entries: see the “sixth thing”). For vertices M and N, exactly two DMZs run along the edge joining them, written thus: (M,/) ·(N, \) = (M, \) · (N,/) = 0 The other 6 all have positive edge-sign, the diagonals of their two DMZs hav- ing same slope (and marked – with leading dashes – ET cell entries): (Z,/) ·(V, /)= (Z, \) ·(V,\)= 0 Fourth thing: The edges always cluster similarly, with two opposite faces among the 8 triangles on the Box-Kite being spanned by 3 negative edges (con- ventionally painted red in color renderings), with all other edges being positive (painted blue). One of the red triangles has its vertices’ 3 low-index units forming a trip; writing their vertex labels conventionally as A, B, C, we find there are in fact always 4 such trips cycling among them: (a,b,c), the L-trip; and the three U-trips obtained by replacing all but one of the lowercase labels in the L-trip with uppercase: (a,B,C); (A,b,C); (A,B,c). Such a 4-trip structure is called a Sail, and a Box-Kite has 4 of them: the Zigzag, with all negative edges, and the 3 Trefoils, each containing two positive edges extending from one of the Zigzag vertices to the two vertices opposite its Sailing partners. These opposite vertices are always joined by one of the 3 negative edges comprising the Vent which is the Zigzag’s opposite face. Again by convention, the vertices opposite A, B, C are written F, E, D in that order; hence, the Trefoil Sails are written (A,D,E); (F,D,B), and (F,C,E), ordered so that their lowercase renderings are equivalent to their CPO L-trips. The graphical convention is to show the Sails as filled in, while the other 4 faces, like the Vent, are left empty: they show “where the wind blows” that keeps the Box-Kite aloft. A real-world Box-Kite, meanwhile, would be held together by 3 dowels (of wood or plastic, say) spanning the joins between the only vertices left unconnected in our Octahedral rendering: the Struts linking the strut-opposite vertices (A, F); (B, E); (C, D). Fifth thing: In the Sedenions, the 7 isomorphic Box-Kites are differentiated by which Octonion index is missing from the vertices, and this index is designated by the letter S, for “signature,” “suppressed index,” or strut constant. This last designation derives from the invariant relationship obtaining in a given Box-Kite between S and the indices in the Vent and Zigzag termini (V and Z respectively) of any of the 3 Struts, which we call the “First Vizier” or VZ1. This is one of 3 rules, involving the three Sedenion indices always missing from a Box-Kite’s vertices: G, S, and their simple sum X (which is also their XOR product, since G is always to the left of the left-most bit in S). The Second Vizier tells us that the L-index of either terminus with the U-index of the other always form a trip with G, and it true as written for all 2N-ions. The Third shows the relationship between the L- and U- indices of a given Assessor, which always form a trip with X. Like the First, it is true as written only in the Sedenions, but as an unsigned statement about indices only, it is true universally. (For that reason, references to VZ1 and VZ3 hereinout will be assumed to refer to the unsigned versions.) First derived in the last section of Part I, reprised in the intro of Part II, we write them out now for the third and final time in this monograph: VZ1: v · z =V ·Z = S VZ2: Z · v =V · z = G VZ3: V · v = z ·Z = X. Rules 1 and 2, the Three Viziers, plus the standard Octonion labeling scheme derived from the simplest finite projective group, usually written as PSL(2,7), pro- vide the basis of our toolkit. This last becomes powerful due to its capacity for recursive re-use at all levels of CDP generation, not just the Octonions. The sim- plest way to see this comes from placing the unique Rule 0 trip provided by the Quaternions on the circle joining the 3 sides’ midpoints, with the Octonion Gen- erator’s index, 4, being placed in the center. Then the 3 lines leading from the Rule 0 trip’s (1, 2, 3) midpoints to their opposite angles – placed conventionally in clockwise order in the midpoints of the left, right, and bottom sides of a triangle whose apex is at 12 o’clock – are CPO trips forming the Struts, while the 3 sides themselves are the Rule 2 trips. These 3 form the L-index sets of the Trefoil Sails, while the Rule 0 trip provides the same service for the Zigzag. By a process analo- gized to tugging on a slipcover (Part I) and pushing things into the central zone of hot oil while wok-cooking (Part II), all 7 possible values of S in the Sedenions, not just the 4, can be moved into the center while keeping orientations along all 7 lines of the Triangle unchanged. Part II’s critical Roundabout Theorem tells us, moreover, that all 2N-ion ZDs, for all N > 3, are contained in Box-Kites as their minimal ensemble size. Hence, by placing the appropriate G, S, or X in the center of a PSL(2,7) triangle, with a suitable Rule 0 trip’s indices populating the circle, any and all candidate primitive ZDs can be discovered and situated. Sixth thing: The word “candidate” in the above is critical; its exploration was the focus of Part II. For, starting with N = 5 and hence G = 16 (which is to say, in the 32-D Pathions), whole Box-Kites can be suppressed (meaning, all 12 edges, and not just the Struts, no longer serve as DMZ pathways). But for all N, the full set of candidate Box-Kites are viable when S≤ 8 or equal to some higher power of 2. For all other S values, though, the phenomenon of carrybit overflow intervenes – leading, ultimately, to the “meta-fractal” behavior claimed in our abstract. To see this, we need another mode of representation, less tied to 3-D visualizing, than the Box-Kite can provide. The answer is a matrix-like method of tabulating the products of candidate ZDs with each other, called Emanation Tables or ETs. The L-indices only of all candidate ZDs are all we need indicate (the U-indices being forced once G is specified); these will saturate the list of allowed indices < G, save for the value of S whose choice, along with that of G, fixes an ET. Hence, the unique ET for given G and S will fill a square spreadsheet whose edge has length 2N−1 −2. Moreover, a cell entry (r,c) is only filled when row and column labels R and C form a DMZ, which can never be the case along an ET’s long diagonals: for the diagonal starting in the upper left corner, R xor R = 0, and the two diagonals within the same Assessor, can never zero-divide each other; for the righthand diagonal, the convention for ordering the labels (ascending counting order from the left and top, with any such label’s strut-opposite index immediately being entered in the mirror-opposite positions on the right and bottom) makes R and C strut-opposites, hence also unable to form DMZs. For the Sedenions, we get a 6 x 6 table, 12 of whose cells (those on long diagonals) are empty: the 24 filled cells, then, correspond to the two-way traffic of “edge-currents” one imagines flowing between vertices on a Box-Kite’s 12 edges. A computational corollary to the Roundabout Theorem, dubbed the Trip- Count Two-Step, is of seminal importance. It connects this most basic theorem of ETs to the most basic fact of associative triplets, indicated in the opening pages of Part I, namely: for any N, the number TripN of associative triplets is found, by simple combinatorics, to be (2N −1)(2N −2)/3! – 35 for the Sedenions, 155 for the Pathions, and so on. But, by Trip-Count Two-Step, we also know that the maximum number of Box-Kites that can fill a 2N-ion ET = TripN−2. For S a power of 2, beginning in the Pathions (for S= 25−2 = 8), the Number Hub Theorem says the upper left quadrant of the ET is an unsigned multiplication table of the 2N−2- ions in question, with the 0’s of the long diagonal (indicated Real negative units) replaced by blanks – a result effectively synonymous with the Trip-Count Two- Step. Seventh thing: We found, as Part II’s argument wound down, that the 2 classes of ETs found in the Pathions – the “normal” for S ≤ 8, filled with indices for all 7 possible Box-Kites, and the “sparse” so-called Sand Mandalas, showing only 3 Box-Kites when 8 < S < 16, were just the beginning of the story. A simple formula involving just the bit-string of s and g, where the lowercase indicates the values of S and G modulo G/2, gave the prototype of our first recipe: all and only cells with labels R or C, or content P ( = R xor C ), are filled in the ET. The 4 “missing Box-Kites” were those whose L-index trip would have been that of a Sail in the 2N−1 realm with S = s and G = g. The sequence of 7 ETs, viewed in S-increasing succession, had an obvious visual logic leading to their being dubbed a flip-book. These 7 were obviously indistinguishable from many vantages, hence formed a spectrographic band. There were 3 distinct such bands, though, each typified by a Box-Kite count common to all band-members, demonstrable in the ETs for the 64-D Chingons. Each band contained S values bracketed by multiples of 8 (either less than or equal to the higher, depending upon whether the latter was or wasn’t a power of 2). These were claimed to underwrite behaviors in all higher 2N-ion ETs, according to 3 rough patterns in need of algorithmic refining in this Part III. Corresponding to the first unfilled band, with ETs always missing 4N−4 of their candidate Box-Kites for N > 4, we spoke of recursivity, meaning the ETs for constant S and increasing N would all obey the same recipe, properly abstracted from that just cited above, empirically found among the Pathions for S > 8. The second and third behaviors, dubbed, for S ascending, (s,g)-modularity and hide/fill involution respectively, make their first showings in the Chingons, in the bands where 16 < S ≤ 24, and then where 24 < S < 32. In all such cases, we are concerned with seeing the “period-doubling” inherent in CDP and Chaotic attractors both become manifest in a repeated doubling of ET edge-size, leading to the fixed-S, N increasing analog of the fixed-N,S increasing flip-books first ob- served in the Pathions, which we call balloon-rides. Specifying and proving their workings, and combining all 3 of the above-designated behaviors into the “funda- mental theorem of zero-division algebra,” will be our goals in this final Part III. Anyone who has read this far is encouraged to bring up the graphical complement to this monograph, the 78-slide Powerpoint show presented at NKS 2006 [5], in another window. (Slides will be referenced by number in what follows.) 2 8 < S < 16,N → ∞ : Recursive Balloon Rides in the Whorfian Sky We know that any ET for the 2N-ions is a square whose edge is 2N−1 − 2 cells. How, then, can any simply recursive rule govern exporting the structure of one such box to analogous boxes for progressively higher N? The answer: include the label lines – not just the column and row headers running across the top and left margins, but their strut-opposite values, placed along the bottom and right margins, which are mirror-reversed copies of the label-lines (LLs) proper to which they are parallel. This increases the edge-size of the ET box to 2N−1. Theorem 11. For any fixed S > 8 and not a power of 2, the row and column indices comprising the Label Lines (LLs) run along the left and top borders of the 2N- ion ET ”spreadsheet” for that S. Treat them as included in the spreadsheet, as labels, by adding a row and column to the given square of cells, of edge 2N−1−2, which comprises the ET proper. Then add another row and column to include the strut-opposite values of these labels’ indices in “mirror LLs,” running along the opposite edges of a now 2N−1-edge-length box, whose four corner cells, like the long diagonals they extend, are empty. When, for such a fixed S, the ET for the 2N+1-ions is produced, the values of the 4 sets of LL indices, bounding the contained 2N-ion ET, correspond, as cell values, to actual DMZ P-values in the bigger ET, residing in the rows and columns labeled by the contained ET’s G and X (the containing ET’s g and g+S). Moreover, all cells contained in the box they bound in the containing ET have P-values (else blanks) exactly corresponding to – and including edge-sign markings of – the positionally identical cells in the 2N-ion ET: those, that is, for which the LLs act as labels. Proof. For all strut constants of interest, S < g(= G/2); hence, all labels up to and including that immediately adjoining its own strut constant (that is, the first half of them) will have indices monotonically increasing, up to and at least including the midline bound, from 1 to g − 1. When N is incremented by 1, the row and column midlines separating adjoining strut-opposites will be cut and pulled apart, making room for the labels for the 2N+1-ion ET for same S, which middle range of label indices will also monotonically increase, this time from the current 2N-ion generation’s g (and prior generation’s G), up to and at least including its own midline bound, which will be g plus the number of cells in the LL inherited from the prior generation, or g/2− 1. The LLs are therefore contained in the rows and columns headed by g and its strut opposite, g+S. To say that the immediately prior CDP generation’s ET labels are converted to the current generation’s P-values in the just-specified rows and columns is equivalent to asserting the truth of the following calculation: (g+u)+(sg) · (G+g+uopp) g + (G+g+S) −(vz) · (G+uopp) +(vz) · (sg) ·u +u − (sg) · (G+uopp) 0 only if vz = (−sg) Here, we use two binary variables, the inner-sign-setting sg, and the Vent-or- Zigzag test, based on the First Vizier. Using the two in tandem lets us handle the normal and “Type II” box-kites in the same proof. Recall (and see Appendix B of Part II for a quick refresher) that while the “Type I” is the only type we find in the Sedenions, we find that a second variety emerges in the Pathions, indistinguishable from Type I in most contexts of interest to us here: the orientation of 2 of the 3 struts will be reversed (which is why VZ1 and VZ3 are only true generally when unsigned). For a Type I, since S < g, we know by Rule 1 that we have the trip (S,g,g+S); hence, g – for all 2N-ions beyond the Pathions, where the Sand Mandalas’ g = 8 is the L-index of the Zigzag B Assessor – must be a Vent (and its strut-opposite, g+S, a Zigzag). For a Type II, however, this is necessarily so only for 1 of the 3 struts – which means, per the equation above, that sg must be reversed to obtain the same result. Said another way, we are free to assume either signing of vz means +1, so the “only if” qualifying the zero result is informative. It is u and its relationship to g+ u that is of interest here, and this formulation makes it easier to see that the products hold for arbitrary LL indices u or their strut-opposites. But for this, the term-by-term computations should seem routine: the left bottom is the Rule 1 outcome of (u,g,g+ u): obviously, any u index must be less than g. To its right, we use the trip (uopp,g,g+ uopp) → (G+ g+ uopp,g,G+uopp), whose CPO order is opposite that of the multiplication. For the top left, we use (u,S,uopp) as limned above, then augment by g, then G, leaving uopp unaffected in the first augmenting, and g + u in the second. Finally, the top right (ignoring sg and vz momentarily) is obtained this way: (u,S,uopp) → (u,g+uopp,g+S)→ (u,G+g+ s,G+g+uopp); ergo, +u. Note that we cannot eke out any information about edge-sign marks from this setup: since labels, as such, have no marks, we have nothing to go on – unlike all other cells which our recursive operations will work on. Indeed, the exact algo- rithmic determination of edge-sign marks for labels is not so trivial: as one iterates through higher N values, some segments of LL indexing will display reversals of marks found in the ascending or descending left midline column, while other seg- ments will show them unchanged – with key values at the beginnings and ends of such octaves (multiples of 8, and sums of such multiples with S mod 8) some- times being reversed or kept the same irrespective of the behavior of the terms they bound. Fortunately, such behaviors are of no real concern here – but they are, nevertheless, worth pointing out, given the easy predictability of other edge-sign marks in our recursion operations. Now for the ET box within the labels: if all values (including edge-sign marks) remain unchanged as we move from the 2N-ion ET to that for the 2N+1-ions, then one of 3 situations must obtain: the inner-box cells have labels u,v which belong to some Zigzag L-trip (u,v,w); or, on the contrary, they correspond to Vent L-indices – the first two terms in the CPO triplet (wopp,vopp,u), for instance; else, finally, one term is a Vent, the other a Zigzag (so that inner-signs of their multiplied dyads are both positive): we will write them, in CPO order, vopp and u, with third trip member wopp. Clearly, we want all the products in the containing ET to indicate DMZs only if the inner ET’s cells do similarly. This is easily arranged: for the containing ET’s cells have indices identical to those of the contained ET’s, save for the appending of g to both (and ditto for the U-indices). Case 1: If (u,v,w) form a Zigzag L-index set, then so do (g+ v,g+u,w), so markings remain unchanged; and if the (u,v) cell entry is blank in the contained, so will be that for (g+ u,g+ v) in its container. In other words, the following holds: (g+ v)+(sg) · (G+g+ vopp) (g+u) + (G+g+uopp) −(G+wopp) − (sg) ·w −w − (sg) · (G+wopp) 0 only if sg = (−1) (g+u) · (g+ v) = P : (u,v,w)→ (g+ v,g+u,w); hence, (−w). (g+ u) · (sg) · (G+ g+ vopp) = P : (u,wopp,vopp) → (g+ vopp,wopp,g+ u) → (G+wopp,G+g+ vopp,g+u); hence, (sg) · (−(G+wopp)). (G+g+uopp) · (g+ v) = P : (uopp,wopp,v)→ (g+ v,wopp,g+uopp) → (g+ v,G+g+uopp,G+wopp); hence, (−(G+wopp)). (G+g+uopp) ·(G+g+vopp) = P : Rule 2 twice to the same two terms yields the same result as the terms in the raw, hence (−w). Clearly, cycling through (u,v,w) to consider (g+ v) · (g+ w) will give the exactly analogous result, forcing two (hence three) negative inner-signs in the candidate Sail; hence, if we have DMZs at all, we have a Zigzag Sail. Case 2: The product of two Vents must have negative edge-sign, and there’s no cycling through same-inner-signed products as with the Zigzag, so we’ll just write our setup as a one-off, with upper inner-sign explicitly negative, and claim its outcome true. (g+ vopp)− (G+g+ v) (g+wopp) + (G+g+w) +(G+uopp) +u −u − (G+uopp) (g+wopp) · (g+ vopp) = P : (wopp,vopp,u) → (g+ vopp,g+wopp,u); hence, (−u). (g+wopp) · (G+g+v) = P : (wopp,v,uopp)→ (g+v,g+wopp,uopp)→ (G+ uopp,g+wopp,G+ g+ v); but inner sign of upper dyad is negative, so (−(G+ uopp)). (G+g+w) · (g+vopp) = P : (vopp,uopp,w)→ (g+w,uopp,g+vopp)→ (G+ uopp,G+g+w,g+ vopp); hence, (+(G+uopp)). (G+g+w) · (G+g+ v) = P : Rule 2 twice to the same two terms yields the same result as the terms in the raw; but inner sign of upper dyad is negative, so (+u). Case 3: The product of Vent and Zigzag displays same inner sign in both dyads; hence the following arithmetic holds: (g+u)+(G+g+uopp) (g+ vopp) + (G+g+ v) −(G+w) +wopp −wopp +(G+w) The calculations are sufficiently similar to the two prior cases as to make their writing out tedious. It is clear that, in each of our three cases, content and marking of each cell in the contained ET and the overlapping portion of the container ET are identical. � To highlight the rather magical label/content involution that occurs when N is in- or de- cremented, graphical realizations of such nested patterns, as in Slides 60-61, paint LLs (and labels proper) a sky-blue color. The bottom-most ET being overlaid in the central box has g = the maximum high-bit in S, and is dubbed the inner skybox. The degree of nesting is strictly measured by counting the number of bits B that a given skybox’s g is to the left of this strut-constant high-bit. If we partition the inner skybox into quadrants defined by the midlines, and count the number Q of quadrant-sized boxes along one or the other long diagonal, it is obvi- ous that the inner skybox itself has B = 0 and Q = 1; the nested skyboxes contain- ing it have Q = 2B. If recursion of skybox nesting be continued indefinitely – to the fractal limit, which terminology we will clarify shortly – the indices contained in filled cells of any skybox can be interpreted in B distinct ways, B → ∞, as rep- resentations of distinct ZDs with differing G and, therefore, differing U-indices. By obvious analogy to the theory of Riemann surfaces in complex analysis, each such skybox is a separate “sheet”; as with even such simple functions as the log- arithmic, the number of such sheets is infinite. We could then think of the infinite sequence of skyboxes as so many cross-sections, at constant distances, of a flash- light beam whose intensity (one over the ET’s cell count) follows Kepler’s inverse square law. Alternatively, we could ignore the sheeting and see things another Where we called fixed-N, S varying sequences of ETs flip-books, we refer to fixed-S, N varying sequences as balloon rides: the image is suggested by David Niven’s role as Phineas Fogg in the movie made of Jules Vernes’ Around the World in 80 Days: to ascend higher, David would drop a sandbag over the side of his hot- air balloon’s basket; if coming down, he would pull a cord that released some of the balloon’s steam. Each such navigational tactic is easy to envision as a bit- shift, pushing G further to the left to cross LLs into a higher skybox, else moving Figure 1: ETs for S=15, N=5,6,7 (nested skyboxes in blue)· · · and “fractal limit.” it rightward to descend. Using S = 15 as the basis of a 3-stage balloon-ride, we see how increasing N from 5 to 6 to 7 approaches the white-space complement of one of the simplest (and least efficient) plane-filling fractals, the Cesàro double sweep [6, p. 65]. The graphics were programmatically generated prior to the proving of the the- orems we’re elaborating: their empirical evidence was what informed (indeed, demanded) the theoretical apparatus. And we are not quite finished with the cur- rent task the apparatus requires of us. We need two more theorems to finish the discussion of skybox recursion. For both, suppose some skybox with B = k, k any non-negative integer, is nested in one with B = k + 1. Divide the former along midlines to frame its four quadrants, then block out the latter skybox into a 4×4 grid of same-sized window panes, partitioned by the one-cell-thick borders of its own midlines into quadrants, each of which is further subdivided by the out- side edges of the 4 one-cell-thick label lines and their extensions to the window’s frame. These extended LLs are themselves NSLs, and have R,C values of g and g+S; for S = 15, they also adjoin NSLs along their outer edges whose R,C val- ues are multiples of 8 plus S mod 8. These pane-framing pairs of NSLs we will henceforth refer to (as a windowmaker would) as muntins. It is easy to calculate that while the inner skybox has but one muntin each among its rows and columns, each further nesting has 2B+1 − 1. But we are getting ahead of ourselves, as we still have two proofs to finish. Let’s begin with Four Corners, or Theorem 12. The 4 panes in the corners of the 16-paned B = k + 1 window are identical in contents and marks to the analogously placed quadrants of the B = k skybox. Proof. Invoke the Zero-Padding Lemma with regard to the U-indices, as the labels of the boxes in the corners of the B = k+1 ET are identical to those of the same- sized quadrants in the B = k ET, all labels ≥ the latter’s g only occurring in the newly inserted region. � Remarks. For N = 6, all filled Four Corners cells indicate edges belonging to 3 Box-Kites, whose edges they in fact exhaust. These 3, not surprisingly, are the zero-padded versions of the identically L-indexed trio which span the entirety of the N = 5 ET. By calculations we’ll see shortly, however, the inner skybox, when considered as part of the N = 6 ET, has filled cells belonging to all the other 16 Box-Kites, even though the contents of these cells are identical to those in the N = 5 ET. As B increases, then, the “sheets” covering this same central region must draw upon progressively more extensive networks of interconnected Box-Kites. As we approach the fractal limit – and “the Sky is the limit” – these networks hence become scale-free. (Corollarily, for N = 7, the Four Corners’ cells exhaust all the edges of the N = 6 ET’s 19 Box-Kites, and so on.) Unlike a standard fractal, however, such a Sky merits the prefix “meta”: for each empty ET cell corresponds to a point in the usual fractal variety; and each pair of filled ET cells, having (r,c) of one = (c,r) of the other), correspond to diagonal-pairs in Assessor planes, orthogonal to all other such diagonal-pairs be- longing to the other cells. Each empty ET cell, in other words, not only corre- sponds to a point in the usual plane-confined fractal, but belongs to the comple- ment of the filled cells’ infinite number of dimensions framing the Sky’s meta- fractal. We’ve one last thing to prove here. The French Windows Theorem shows us the way the cell contents of the pairs of panes contained between the B = k+ 1 skybox’s corners are generated from those of the analogous pairings of quadrants in the B = k skybox, by adding g to L-indices. Theorem 13. For each half-square array of cells created by one or the other midline (the French windows), each cell in the half-square parallel to that adjoining the midline (one of the two shutters), but itself adjacent to the label-line delimiting the former’s bounds, has content equal to g plus that of the cell on the same line orthogonal to the midline, and at the same distance from it, as it is from the label- line. All the empty long-diagonal cells then map to g (and are marked), or g+S (and are unmarked). Filled cells in extensions of the label-lines bounding each shutter are calculated similarly, but with reversed markings; all other cells in a shutter have the same marks as their French-window counterparts. Preamble. Note that there can be (as we shall see when we speak of hide/fill involution) cells left empty for rule-based reasons other than P = R⊻C = 0 | S. The shutter-based counterparts of such French-window cells, unlike those of long- diagonal cells, remain empty. Proof. The top and left (bottom and right) shutters are equivalent: one merely switches row for column labels. Top/left and bottom/right shutter-sets are likewise equivalent by the symmetry of strut-opposites. We hence make the case for the left shutter only. But for the novelties posed by the initially blank cells and the label lines (with the only real subtleties involving markings), the proof proceeds in a manner very similar to Theorem 11: split into 3 cases, based on whether (1) the L-index trip implied by the R,C,P values is a Zigzag; (2) u,v are both Vents; or, (3) the edge signified by the cell content is the emanation of same-inner-signed dyads (that is, one is a Vent, the other a Zigzag). Case 1: Assume (u,v,w) a Zigzag L-trip in the French window’s contained skybox; the general product in its shutter is v − (G+ vopp) (g+u) + (G+g+uopp) −(G+g+wopp) +(g+w) −(g+w) +(G+g+wopp) (g+u) · v = P : (u,v,w)→ (g+w,v,g+u); hence, (−(g+w)). (g+ u) · (G+ vopp) = P : (u,wopp,vopp) → (g+wopp,g+ u,vopp) → (G+ vopp,g+u,G+g+wopp); dyads’ opposite inner signs make (G+g+wopp) pos- itive. (G+g+uopp) · v = P : (uopp,wopp,v) → (g+wopp,g+uopp,v) → (G+g+ uopp,G+g+wopp,v); hence, (−(G+g+wopp)). (G+g+uopp) · (G+ vopp) = P : (vopp,uopp,w) → (vopp,g+w,g+uopp) → (G+g+uopp,g+w,G+vopp); dyads’ opposite inner signs make (g+w) positive. Case 2: The product of two Vents must have negative edge-sign, hence nega- tive inner sign in top dyad to lower dyad’s positive. The shutter product thus looks like this: (uopp)− (G+u) (g+ vopp) + (G+g+ v) +(G+g+wopp) +(g+w) −(g+w) − (G+g+wopp) (g+vopp) ·uopp = P : (vopp,uopp,w)→ (g+w,uopp,g+vopp); hence, (−(g+ (g+ vopp) · (G+ u) = P : (vopp,u,wopp) → (g+wopp,u,g+ vopp) → (G+ u,G+ g+wopp,g+ vopp); but dyads’ inner signs are opposite, so (−(G+ g+ wopp)). (G+g+v) ·uopp = P : (uopp,wopp,v)→ (uopp,g+v,g+wopp)→ (uopp,G+ g+wopp,G+g+ v); hence, (+(G+g+wopp)). (G+g+v) ·(G+u)= P : (u,v,w)→ (u,g+w,g+v)→ (G+g+v,g+w,G+ u); but dyads’ inner signs are opposite, so (+(g+w)). Case 3: The product of Vent and Zigzag displays same inner sign in both dyads; hence the following arithmetic holds: (uopp)+(G+u) (g+ v) + (G+g+ vopp) +(G+g+w) +(g+wopp) −(g+wopp) − (G+g+w) As with the last case in Theorem 11, we omit the term-by-term calculations for this last case, as they should seem “much of a muchness” by this point. What is clear in all three cases is that index values of shutter cells have same markings as their French-window counterparts, at least for all cells which have markings in the contained skybox; but, in all cases, indices are augmented by g. The assignment of marks to the shutter-cells linked to blank cells in French windows is straightforward for Type I box-kites: since any containing skybox must have g > S, and since g+ s has g as its strut opposite, then the First Vizier tells us that any g must be a Vent. But then the R,C indices of the cell containing g must belong to a Trefoil in such a box-kite; hence, one is a Vent, the other a Zigzag, and g must be marked. Only if the R,C,P entry in the ET is necessar- ily confined to a Type II box-kites will this not necessarily be so. But Part II’s Appendix B made clear that Type II’s are generated by excluding g from their L-indices: recall that, in the Pathions, for all S ¡ 8, all and only Type II box-kites are created by placing one of the Sedenion Zigzag L-trips on the “Rule 0” circle of the PSL(2,7) triangle with 8 in the middle (and hence excluded). This is a box- kite in its own right (one of the 7 “Atlas” box-kites with S = 8); its 3 sides are “Rule 2” triplets, and generate Type II box-kites when made into zigzag L-index sets. Conversely, all Pathion box-kites containing an ’8’ in an L-index (dubbed ”strongboxes” in Appendix B) are Type I. Whether something peculiar might oc- cur for large N (where there might be multiple powers of 2 playing roles in the same box-kite) is a matter of marginal interest to present concerns, and will be left as an open question for the present. We merely note that, by a similar argument, and with the same restrictions assumed, g+S must be a Zigzag L-index, and R,C either both be likewise (hence, g+S is unmarked); or, both are Vents in a Trefoil (so g+S must be unmarked here too). The last detail – reversal of label-line markings in their g-augmented shutter- cell extensions – is demonstrated as follows, with the same caveat concerning Type II box-kites assumed to apply. Such cells house DMZs (just swap u for g+u in Theorem 11’s first setup – they form a Rule 1 trip – and compute). The LL extension on top has row-label g; that along the bottom, the strut-opposite g+S. Given trip (u,v,w), the shutter-cell index for R,C = (g,u) corresponds to French- window index for R,C = (g,g+u). But (u,g,g+u) is a Trefoil, since g is a Vent. So if u is one too, g+u isn’t; hence marks are reversed as claimed. � 3 Maximal High-Bit Singletons: (s,g)-Modularity for 16 < S ≤ 24 The Whorfian Sky, having but one high bit in its strut constant, is the simplest possible meta-fractal – the first of an infinite number of such infinite-dimensional zero-divisor-spanned spaces. We can consider the general case of such single- ton high-bit recursiveness in two different, complementary ways. First, we can supplement the just-concluded series of theorems and proofs with a calculational interlude, where we consider the iterative embeddings of the Pathion Sand Man- dalas in the infinite cascade of boxes-within-boxes that a Sky oversees. Then, we can generalize what we saw in the Pathions to consider the phenomenology of strut constants with singleton high-bits, which we take to be any bits representing a power of 2 ≥ 3 if S contains low bits (is not a multiple of 8), else a power of 2 strictly greater than 3 otherwise. Per our earlier notation, g = G/2 is the highest such singleton bit possible. We can think of its exponential increments – equiva- lent to left-shifts in bit-string terms – as the side-effects of conjoint zero-padding of N and S. This will be our second topic in this section. Maintaining our use of S = 15 as exemplary, we have already seen that NSLs come in quartets: a row and column are each headed by S mod g (henceforth, s) and g, hence 7 and 8 in the Sand Mandalas. But each recursive embedding of the current skybox in the next creates further quartets. Division down the midlines to insert the indices new to the next CDP generation induces the Sand Mandala’s adjoining strut-opposite sets of s and g lines (the pane-framing muntins) to be displaced to the borders of the four corners and shutters, with the new skybox’s g and g+ s now adjoining the old s and g to form new muntins, on the right and left respectively, while g+g/2 (the old G+g) and its strut opposite form a third muntin along the new midlines. Continuing this recursive nesting of skyboxes generates 1, 3, 7, · · ·, 2B+1 −1 row-and-column muntin pairs involving multiples of 8 and their supplementings by s, where (recalling earlier notation) B = 0 for the inner skybox, and increments by 1 with each further nesting. Put another way, we then have a muntin number µ = (2N−4 −1), or 4µ NSL’s in all. The ET for given N has (2N−1 −2) cells in each row and column. But NSLs divvy them up into boxes, so that each line is crossed by 2µ others, with the 0, 2 or 4 cells in their overlap also belonging to diagonals. The number of cells in the overlap-free segments of the lines, or ω , is then just 4µ · (2N−1 − 2− 2µ) = 24µ(µ +1): an integer number of Box-Kites. For our S = 15 case, the minimized line shuffling makes this obvious: all boxes are 6 x 6, with 2-cell-thick boundaries (the muntins separating the panes), with µ boundaries, and (µ + 1) overlap-free cells per each row or column, per each quartet of lines. The contribution from diagonals, or δ , is a little more difficult, but straight- forward in our case of interest: 4 sets of 1,2,3, · · · ,µ boxes are spanned by mov- ing along one empty long diagonal before encountering the other, with each box contributing 6, and each overlap zone between adjacent boxes adding 2. Hence, δ = 24 · (2N−3 − 1)(2N−3 − 2)/6 – a formula familiar from associative-triplet counting: it also contributes an integer number of Box-Kites. The one-liner we want, then, is this: BKN, 8<S<16 = ω +δ = (2N−4)(2N−4 −1) + (2N−3 −1)(2N−3 −2)/6 For N = 4,5,6,7,8,9,10, this formula gives 0,3,19,91,395, 1643,6699. Add 4N−4 to each – the immediate side-effect of the offing of all four Rule 0 candidate trips of the Sedenion Box-Kite exploded into the Sand Mandala that begins the recursion – and one gets “déjà vu all over again”: 1, 7, 35, 155, 651, 2667, 10795 – the full set of Box-Kites for S ≤ 8. It would be nice if such numbers showed up in unsuspected places, having nothing to do with ZDs. Such a candidate context does, in fact, present itself, in Ed Pegg’s regular MAA column on “Math Games” focusing on “Tournament Dice.” [7] He asks us, “What dice make a non-transitive four player game, so that if three dice are chosen, a fourth die in the set beats all three? How many dice are needed for a five player non-transitive game, or more?” The low solution of 3 explicitly involves PSL(2,7); the next solution of 19 entails calculations that look a lot like those involved in computing row and column headers in ETs. No solutions to the dice-selecting game beyond 19 are known. The above formulae, though, suggest the next should be 91. Here, ZDs have no apparent role save as dummies, like the infinity of complex dimensions in a Fourier-series convergence problem, tossed out the window once the solution is in hand. Can a number-theory fractal, with intrinsically structured cell content (something other, non-meta, fractals lack) be of service in this case – and, if not in this particular problem, in others like it? Now let’s consider the more general situation, where the singleton high-bit can be progressively left-shifted. Reverting to the use of the simplest case as exemplary, use S = g+1 = 9 in the Pathions, then do tandem left-shifts to pro- duce this sequence: N = 6, S = g+1 = 17; N = 7, S = g+1 = 33; · · · ; N = K, S = g+1 = 2K−2 + 1. A simple rule governs these ratchetings: in all cases, the number of filled cells = 6 · (2N−1 − 4), since there are two sets of parallel sides which are filled but for long-diagonal intersections, and two sets of g and 1 entries distributed one per row along orthogonals to the empty long diagonals. Hence, for the series just given, we have cell counts of 72, 168, · · · , 6 · (2N−1 − 4) for BKN, S = 3, 7, · · · , 2 N−3 − 1, for g < S < g + 8 = G in the Pathions, and all g < S ≤ g+8 in the Chingons, 27-ions, and general 2N-ions, in that order. Algorithmically, the situation is just as easy to see: the splitting of dyads, sending U- and L- indices to strut-opposite Assessors, while incorporating the S and G of the current CDP generation as strut-opposites in the next, continues. For S = 17 in the Chingons, there are now 2N−3 −1 = 7, not 3, Box-Kites sharing the new g = 16 (at B) and S mod g = 1 (at E) in our running example. The U- indices of the Sand Mandala Assessors for S = g+1 = 9 are now L-indices, and so on: every integer < G and 6= S gets to be an L-index of one of the 30(= 2N−1 − 2) Assessors, as 16 and S mod g = 1 appear in each of the 7 Box-Kites, with each other eligible integer appearing once only in one of the 7 ·4= 28 available L-index slots. As an aside, in all 7 cases, writing the smallest Zigzag L-index at a mandates all the Trefoil trips be “precessed” – a phenomenon also observed in the S = 8 Pathion case, as tabulated on p. 14 of [8]. For Zigzag L-index set (2,16,18), for instance, (a,d,e) = (2,3,1) instead of (1,2,3); ( f ,c,e) = (19,18,1) not (1,19,18); and ( f ,d,b) = (19,3,16). But otherwise, there are no surprises: for N = 7, there are (27−3 − 1) = 15 Box-Kites, with all 62(= 2N−1 − 2) available cells in the rows and columns linked to labels g and S mod g being filled, and so Note that this formulation obtains for any and all S > 8 where the maximum high-bit (that is, g) is included in its bitstring: for, with g at B and S mod g at E, whichever R,C label is not one of these suffices to completely determine the remaining Assessor L-indices, so that no other bits in S play a role in determining any of them. Meanwhile, cell contents P containing either g or S mod g, but created by XORing of row and column labels equal to neither, are arrayed in off- diagonal pairs, forming disjoint sets parallel or perpendicular to the two empty ones. If we write S mod g with a lower-case s, then we could call the rule in play here (s,g)-modularity. Using the vertical pipe for logical or, and recalling the special handling required by the 8-bit when S is a multiple of 8 (which we signify with the asterisk suffixed to “mod”), we can shorthand its workings this way: Theorem 14. For a 2N-ion inner skybox whose strut constant S has a singleton high-bit which is maximal (that is, equal to g = G/2 = 2N−2), the recipe for its filled cells can be condensed thus: R | C| P = g | S mod∗ g Under recursion, the recipe needs to be modified so as to include not just the inner-skybox g and S mod∗ g (henceforth, simply lowercase s), but all integer multiples k of g less than the G of the outermost skybox, plus their strut opposites k ·g+ s. Proof. The theorem merely boils down the computational arguments of prior para- graphs in this section, then applies the last section’s recursive procedures to them. The first claim of the proof is identical to what we’ve already seen for Sand Man- dalas, with zero-padding injected into the argument. The second claim merely assumes the area quadrupling based on midline splitting, with the side-effects al- ready discussed. No formal proof, then, is called for beyond these points. � Remarks. Using the computations from two paragraphs prior to the theorem’s statement, we can readily calculate the box-kite count for any skybox, no matter how deeply nested: recall the formula 6 · (2N−1 − 4) for BKN, S = 2 N−3 − 1. It then becomes a straightforward matter to calculate, as well, the limiting ratio of this count to the maximal full count possible for the ET as N → ∞, with each cell approaching a point in a standard 2-D fractal. Hence, for any S with a singleton high-bit in evidence, there exists a Sky containing all recursive redoublings of its inner skybox, and computations like those just considered can further be used to specify fractal dimensions and the like. (Such computations, however, will not concern us.) Finally, recall that, by spectrographic equivalence, all such compu- tations will lead to the same results for each S value in the same spectral band or octave. 4 Hide/Fill Involution: Further-Right High-Bits with 24 < S < 32. Recall that, in the Sand Mandala flip-book, each increment of S moved the two sets of orthogonal parallel lines one cell closer toward their opposite numbers: while S = 9 had two filled-in rows and columns forming a square missing its cor- ners, the progression culminating in S = 15 showed a cross-hairs configuration: the parallel lines of cells now abutted each other in 2-ply horizontal and vertical arrays. The same basic progression is on display in the Chingons, starting with S = 17. But now the number of strut-opposite cell pairs in each row and column is 15, not 7, so the cross-hairs pattern can’t arise until S = 31. Yet it never arises in quite the manner expected, as something quite singular transpires just after flip- ping past the ET in the middle, for S = 24. Here, rows and columns labeled 8 and 16 constrain a square of empty cells in the center · · · quickly followed by an ET which seems to continue the expected trajectory – except that almost all the non- long-diagonal cells left empty in its predecessor ETs are now inexplicably filled. More, there is a method to the “almost all” as well: for we now see not 2, but 4 rows and columns, all being blanked out while those labeled with g and S mod g are being filled in. This is an inevitable side effect of a second high-bit in S: we call this phe- nomenon, first appearing in the Chingons, hide/fill involution. There are 4, not 2, line-pairs, because S and G, modulo a lower power of 2 (because devolving upon a prior CDP generation’s g), offer twice the possibilities: for S = 25, S mod 16 is now 9, but S mod 8 can result in either 1 or 17 as well – with correlated multiples of 8 (8 proper, and 24) defining the other two pairings. All cells with R |C | P equal to one of these 4 values, but for the handful already set to “on” by the first high-bit, will now be set to “off,” while all other non-long-diagonal cells set to “off” in the Pathion Sand Mandalas are suddenly “on.” What results for each Chingon ET with 24 < S < 32 is an ensemble comprised of 23 Box-Kites. (For the flip-book, see Slides 40 – 54.) Why does this happen? The logic is as straight- forward as the effect can seem mysterious, and is akin, for good reason, to the involutory effect on trip orientation induced by Rule 2 addings of G to 2 of the trip’s 3 indices. In order to grasp it, we need only to consider another pair of abstract calcula- tion setups, of the sort we’ve seen already many times. The first is the core of the Two-Bit Theorem, which we state and prove as follows: Theorem 15. 2N-ion dyads making DMZs before augmenting S with a new high-bit no longer do so after the fact. Proof. Suppose the high-bit in the bitstring representation of S is 2K, K < (N − 1). Suppose further that, for some L-index trip (u,v,w), the Assessors U and V are DMZ’s, with their dyads having same inner signs. (This last assumption is strictly to ease calculations, and not substantive: we could, as earlier, use one or more binary variables of the sg type to cover all cases explicitly, including Type I vs. Type II box-kites. To keep things simple, we assume Type I in what follows.) We then have (u+ u ·X)(v+ v · X) = (u+U)(v+V ) = 0. But now suppose, without changing N, we add a bit somewhere further to the left to S, so that S < (2K = L) < G. The augmented strut constant now equals SL = S+L. One of our L-indices, say v, belongs to a Vent Assessor thanks to the assumed inner signing; hence, by Rule 2 and the Third Vizier, (V,v,X)→ (X +L,v,V +L). Its DMZ partner u, meanwhile, must thereby be a Zigzag L-index, which means (u,U,X)→ (u,X +L,U +L). We claim the truth of the following arithmetic: v + (V +L) u + (U +L) +(W + L) +w + w − (W +L) NOT ZERO (+w’s don’t cancel) The left bottom product is given. The product to its right is derived as follows: since u is a Zigzag L-index, the Trefoil U-trip (u,V,W) has the same orientation as (u,v,w), so that Rule 2 → (u,W +L,V +L), implying the negative result shown. The left product on the top line, though, has terms derived from a Trefoil U-trip lacking a Zigzag L-index, so that only after Rule 2 reversal are the letters arrayed in Zigzag L-trip order: (U +L,v,W +L). Ergo, +(W +L). Similarly for the top right: Rule 2 reversal “straightens out” the Trefoil U-trip, to give (U+L,V +L,w); therefore, (+w) results. If we explicitly covered further cases by using an sg variable, we would be faced with a Theorem 2 situation: one or the other product pair cancels, but not both. � Remark. The prototype for the phenomenon this theorem covers is the “explo- sion” of a Sedenion box-kite into a trio of interconnected ones in a Pathion sand mandala, with the S of the latter = the X of the former. As part of this process, 4 of the expected 7 are “hidden” box-kites (HBKs), with no DMZs along their edges. These have zigzag L-trips which are precisely the L-trips of the 4 Sedenion Sails. Here, an empirical observation which will spur more formal investigations in a sequel study: for the 3 HBKs based on trefoil L-trips, exactly 1 strut has reversed orientation (a different one in each of them), with the orientation of the triangular side whose midpoint it ends in also being reversed. For the HBK based on the zigzag L-trip, all 3 struts are reversed, so that the flow along the sides is exactly the reverse of that shown in the “Rule 0” circle. (Hence, all possible flow patterns along struts are covered, with only those entailing 0 or 2 reversals corresponding to functional box-kites: our Type I and Type II designations.) It is not hard to show that this zigzag-based HBK has another surprising property: the 8 units defined by its own zigzag’s Assessors plus X and the real unit form a ZD-free copy of the Octonions. This is also true when the analogous Type II situation is explored, al- beit for a slightly different reason: in the former case, all 3 Catamaran “twistings” take the zigzag edges to other HBKs; in the latter, though, the pair of Assessors in some other Type II box-kite reached by “twisting” – (a,B) and (A,b), say, if the edge be that joining Assessors A and B, with strut-constant copp = d – are strut opposites, and hence also bereft of ZDs. The general picture seems to mirror this concrete case, and will be studied in “Voyage by Catamaran” with this expecta- tion: the bit-twiddling logic that generates meta-fractal “Skies” also underwrites a means for jumping between ZD-free Octonion clones in an infinite number of HBKs housed in a Sky. Given recent interest in pure “E8” models giving a privi- leged place to the basis of zero-divisor theory, namely “G2” projections (viz., A. Garrett Lisi’s “An Exceptionally Simple Theory of Everything”); a parallel vogue for many-worlds approaches; and, the well-known correspondence between 8-D closest-packing patterns, the loop of the 240 unit Octonions which Coxeter dis- covered, and E8 algebras – given all this, tracking the logic of the links across such Octonionic “brambles” might prove of great interest to many researchers. Now, we still haven’t explained the flipside of this off-switch effect, to which prior CDP generation Box-Kites – appropriately zero-padded to become Box- Kites in the current generation until the new high-bit is added to the strut-constant – are subjected. How is it that previously empty cells not associated with the sec- ond high-bit’s blanked-out R, C, P values are now full? The answer is simple, and is framed in the Hat-Trick Theorem this way. Theorem 16. Cells in an ET which represent DMZ edges of some 2N-ion Box- Kites for some fixed S, and which are offed in turn upon augmenting of S by a new leftmost bit, are turned on once more if S is augmented by yet another new leftmost bit. Proof. We begin an induction based upon the simplest case (which the Chingons are the first 2N-ions to provide): consider Box-Kites with S ≤ 8. If a high-bit be appended to S, then the associated Box-Kites are offed. However, if another high-bit be affixed, these dormant Box-Kites are re-awakened – the second half of hide/fill involution. We simply assume an L-index set (u,v,w) underwriting a Sail in the ET for the pre-augmented S, with Assessors (u,U) and (v,V ). Then, we introduce a more leftified bit 2Q = M, where pre-augmented S < L < M < G, then compute the term-by term products of (u+(U +L+M)) and (v+ sg · (V +L+ M)), using the usual methods. And as these methods tell us that two applications of Rule 2 have the same effect as none in such a setup, we have no more to prove. Corollary. The induction just invoked makes it clear that strut constants equal to multiples of 8 not powers of 2 are included in the same spectral band as all other integers larger than the prior multiple. The promissory note issued in the second paragraph of Part II’s concluding section, on 64-D Spectrography, can now be deemed redeemed. In the Chingons, high-bits L and M are necessarily adjacent in the bitstring for S < G = 32; but in the general 2N-ion case, N large, zero-padding guarantees that things will work in just the same manner, with only one difference: the recursive creation of “harmonics” of relatively small-g (s,g)-modular R,C,P values will propagate to further levels, thereby effecting overall Box-Kite counts. In general terms, we have echoes of the formula given for (s,g)-modular cal- culations, but with this signal difference: there will be one such rule for each high-bit 2H in S, where residues of S modulo 2H will generate their own near- solid lines of rows and columns, be they hidden or filled. Likewise for multiples of 2H <G which are not covered by prior rules, and multiples of 2H supplemented by the bit-specific residue (regardless of whether 2H itself is available for treat- ment by this bit-specific rule). In the simplest, no-zero-padding instances, all even multiples are excluded, as they will have occurred already in prior rules for higher bits, and fills or hides, once fixed by a higher bit’s rule, cannot be overridden. Cases with some zero-padding are not so simple. Consider this two-bit in- stance, S = 73,N = 8: the fill-bit is 64, the hide-bit is just 8, so that only 9 and 64 generate NSLs of filled values; all other multiples of 8, and their supplementing by 1 (including 65) are NSLs of hidden values. Now look at a variation on this example, with the single high-bit of zero-padding removed – i. e., S = 41,N = 8. Here, the fill-bit is 32, and its multiples 64 and 96, as well as their supplements by S modulo 32 = 9, or 9 and 73 and 105, label NSLs of filled values; but all other multiples of 8, plus all multiples of 8 supplemented by 1 not equal to 9 or 73 or 105, label NSLs of hidden values. Cases with multiple fill and hide bits, with or without additional zero-padding, are obviously even more complicated to handle explicitly on a case-by-case basis, but the logic framing the rules remain simple; hence, even such messy cases are programmatically easy to handle. Hide/fill involution means, then, that the first, third, and any further odd- numbered high-bits (counting from the left) will generate “fill” rules, whereas all the even-numbered high-bits generate “hide” rules – with all cells not touched by a rule being either hidden (if the total number of high-bits B is odd) or filled (B is even). Two further examples should make the workings of this protocol more clear. First, the Chingon test case of S = 25: for (R | C | P = 9 | 16), all the ET cells are filled; however, for (R | C | P = 1 | 8 | 17 | 24), ET cells not already filled by the first rule (and, as visual inspection of Slide 48 indicates, there are only 8 cells in the entire 840-cell ET already filled by the prior rule which the current rule would like to operate on) are hidden from view. Because the 16- and 8- bits are the only high-bits, the count of same is even, meaning all remaining ET cells not covered by these 2 rules are filled. We get 23 for Box-Kite count as follows. First, the 16-bit rule gives us 7 Box- Kites, per earlier arguments; the 8-bit rule, which gives 3 filled Box-Kites in the Pathions, recursively propagates to cover 19 hidden Box-Kites in the Chingons, according to the formula produced last section. But hide/fill involution says that, of the 35 maximum possible Box-Kites in a Chingon ET, 35− 19 = 16 are now made visible. As none of these have the Pathion G = 16 as an L-index, and all the 7 Box-Kites from the 16-bit rule do, we therefore have a grand total of 7+16= 23 Box-Kites in the S = 25 ET, as claimed (and as cell-counting on the cited Slide will corroborate). The concluding Slides 76–78 present a trio of color-coded “histological slices” of the hiding and filling sequence (beginning with the blanking of the long diago- nals) for the simplest 3-high-bit case, N = 7,S = 57. Here, the first fill rule works on 25 and 32; the first hide rule, on 9, 16, 41, and 48; the second fill rule, on 1, 8, 17, 24, 33, 40, 49, and 56; and the rest of the cells, since the count of high-bits is odd, are left blank. We do not give an explicit algorithmic method here, however, for computing the number of Box-Kites contained in this 3,720-cell ET. Such recursiveness is best handled programmatically, rather than by cranking out an explicit (hence, long and tedious) formula, meant for working out by a time-consuming hand cal- culation. What we can do, instead, is conclude with a brief finale, embodying all our results in the simple “recipe theory” promised originally, and offer some reflections on future directions. 5 Fundamental Theorem of Zero-Divisor Algebra All of the prior arguments constitute steps sufficient to demonstrate the Funda- mental Theorem of Zero-Divisor Algebra. Like the role played by its Gaussian predecessor in the legitimizing of another “new kind of [complex] number the- ory,” its simultaneous simplicity and generality open out on extensive new vistas at once alien and inviting. The Theorem proper can be subdivided into a Proposi- tion concerning all integers, and a “Recipe Theory” pragmatics for preparing and “cooking” the meta-fractal entities whose existence the proposition asserts, but cannot tell us how to construct. Proposition: Any integer K > 8 not a power of 2 can uniquely be associated with a Strut Constant S of ZD ensembles, whose inner skybox resides in the 2N-ions with 2N−2 < K < 2N−1. The bitstring representation of S completely determines an infinite-dimensional analog of a standard plane-confined fractal, with each of the latter’s points associated with an empty cell in the infinite Emanation Table, with all non-empty cells comprised wholly of mutually orthogonal primitive zero- divisors, one line of same per cell. Preparation: Prepare each suitable S by producing its bitstring representation, then determining the number of high-bits it contains: if S is a multiple of 8, right- shift 4 times; otherwise, right-shift 3 times. Then count the number B of 1’s in the shortened bitstring that results. For this set {B} of B elements, construct two same-sized arrays, whose indices range from 1 to B: the array {i} which indexes the left-to-right counting order of the elements of {B}; and, the array {P} which indexes the powers of 2 of the same element in the same left-to-right order. (Example: if K = 613, the inner skybox is contained in the 211-ions; as the number is not a multiple of 8, the bistring representation 1001100101 is right- shifted thrice to yield the substring of high-bits 1001100; B = 3, and for 1 ≤ i ≤ 3, P1 = 9, P2 = 6;P3 = 5.) Cookbook Instructions: [0] For a given strut-constant S, compute the high-bit count B and bitstring arrays {i} and {P}, per preparation instructions. [1] Create a square spreadsheet-cell array, of edge-length 2I , where I ≥G/2= g of the inner skybox for S, with the Sky as the limit when I → ∞. [2] Fill in the labels along all four edges, with those running along the right (bottom) borders identical to those running along the left (top), except in reversed left-right (top-bottom) order. Refer to those along the top as col- umn numbers C, and those along the left edge, as row numbers R, setting candidate contents of any cell (r,c) to R⊻C = P. [3] Paint all cells along the long diagonals of the spreadsheet just constructed a color indicating BLANK, so that all cells with R =C (running down from upper left corner) else R⊻C = S (running down from upper right) have their P-values hidden. [4] For 1 ≤ i ≤ B, consider for painting only those cells in the spreadsheet created in [1] with R | C | P = m · 2γ | m · 2γ + σ , where γ = Pi,σ = S mod∗ 2γ , and m is any integer ≥ 0 (with m= 0 only producing a legitimate candidate for the right-hand’s second option, as an XOR of 0 indicates a long-diagonal cell). [5] If a candidate cell has already been painted by a prior application of these instructions to a prior value of i, leave it as is. Otherwise, paint it with R⊻C if i = odd, else paint it BLANK. [6] Loop to [4] after incrementing i. If i < B, proceed until this step, then reloop, reincrement, and retest for i = B. When this last condition is met, proceed to the next step. [7] If B is odd, paint all cells not already painted, BLANK; for B even, paint them with R⊻C. In these pseudocode instructions, no attention is given to edge-mark gener- ation, performance optimization, or other embellishments. Recursive expansion beyond the chosen limits of the 2N-ion starting point is also not addressed. (Just keep all painted cells as is, then redouble until the expanded size desired is at- tained; compute appropriate insertions to the label lines, then paint all new cells according to the same recipe.) What should be clear, though, is any optimization cannot fail to be qualitatively more efficient than the code in the appendix to [9], which computes on a cell-by-cell basis. For S > 8, N > 4, we’ve reached the onramp to the Metafractal Superhighway: new kinds of efficiency, synergy, con- nectedness, and so on, would seem to more than compensate for the increase in dimension. It is well-known that Chaotic attractors are built up from fractals; hence, our results make it quite thinkable to consider Chaos Theory from the vantage of pure Number · · · and hence the switch from one mode of Chaos to another as a bitstring- driven – or, put differently, a cellular automaton-type – process, of Wolfram’s Class 4 complexity. Such switching is of the utmost importance in coming to terms with the most complex finite systems known: human brains. The late Fran- cisco Varela, both a leading visionary in neurological research and its computer modeling, and a long-time follower of Madhyamika Buddhism who’d collabo- rated with the Dalai Lama in his “Tibetan Buddhists talk with brain scientists” dialogues [10], pointed to just the sorts of problems being addressed here as the next frontier. In a review essay he co-authored in 2001 just before his death [11, p. 237], we read these concluding thoughts on the theme of what lies “Beyond Synchrony” in the brain’s workings: The transient nature of coherence is central to the entire idea of large- scale synchrony, as it underscores the fact that the system does not be- have dynamically as having stable attractors [e.g., Chaos], but rather metastable patterns – a succession of self-limiting recurrent patterns. In the brain, there is no “settling down” but an ongoing change marked only by transient coordination among populations, as the attractor it- self changes owing to activity-dependent changes and modulations of synaptic connections. Varela and Jean Petitot (whose work was the focus of the intermezzo conclud- ing Part I, in which semiotically inspired context the Three Viziers were intro- duced) were long-time collaborators, as evidenced in the last volume on Naturaliz- ing Phenomenology [12] which they co-edited. It is only natural then to re-inscribe the theme of mathematizing semiotics into the current context: Petitot offers sepa- rate studies, at the “atomic” level where Greimas’ “Semiotic Square” resides; and at the large-scale and architectural, where one must place Lévi-Strauss’s “Canon- ical Law of Myths.” But the pressing problem is finding a smooth approach that lets one slide the same modeling methodology from the one scale to the other: a fractal-based “scale-free network” approach, in other words. What makes this distinct from the problem we just saw Varela consider is the focus on the structure, rather than dynamics, of transient coherence – a focus, then, in the last analysis, on a characterization of database architecture that can at once accommodate meta- chaotic transiency and structural linguists’ cascades of “double articulations.” Starting at least with C. S. Peirce over a century ago, and receiving more recent elaboration in the hands of J. M. Dunn and the research into the “Semantic Web” devolving from his work, data structures which include metadata at the same level as the data proper have led to a focus on “triadic logic,” as perhaps best exemplified in the recent work of Edward L. Robertson. [13] His exploration of a natural triadic-to-triadic query language deriving from Datalog, which he calls Trilog, is not (unlike our Skies) intrinsically recursive. But his analysis depends upon recursive arguments built atop it, and his key constructs are strongly resonant with our own (explicitly recursive) ones. We focus on just a few to make the point, with the aim of provoking interest in fusing approaches, rather than in proving any particular results. The still-standard technology of relational databases based on SQL statements (most broadly marketed under the Oracle label) was itself derived from Peirce’s triadic thinking: the creator of the relational formalism, Edgar F. “Ted” Codd, was a PhD student of Peirce editor and scholar Arthur W. Burks. Codd’s triadic “re- lations,” as Robertson notes (and as Peirce first recognized, he tells us, in 1885), are “the minimal, and thus most uniform” representations “where metadata, that is data about data, is treated uniformly with regular data.” In Codd’s hands (and in those of his market-oriented imitators in the SQL arena), metadata was “relegated to an essentially syntactic role” [13, p. 1] – a role quite appropriate to the appli- cations and technological limitations of the 1970’s, but inadequate for the huge and/or highly dynamic schemata that are increasingly proving critical in bioinfor- matics, satellite data interpretation, Google server-farm harvesting, and so on. As Robertson sums up the situation motivating his own work, Heterogeneous situations, where diverse schemata represent semanti- cally similar data, illustrate the problems which arise when one per- son’s semantics is another’s syntax – the physical “data dependence” that relational technology was designed to avoid has been replaced by a structural data dependence. Hence we see the need to [use] a simple, uniform relational representation where the data/metadata distinction is not frozen in syntax. [13, pp. 1-2] As in relational database theory and practice, the forming and exploiting of inner and outer joins between variously keyed tables of data is seminal to Robert- son’s approach as well as Codd’s. And while the RDF formalism of the Semantic Web (the representational mechanism for describing structures as well as contents of web artifacts on the World Wide Web) is likewise explicitly triadic, there has, to date, been no formal mechanism put in place for manipulating information in RDF format. Hence, “there is no natural way to restrict output of these mecha- nisms to triples, except by fiat” [13, p. 4], much less any sophisticated rule-based apparatus like Codd’s “normal forms” for querying and tabulating such data. It is no surprise, then, that Robertson’s “fundamental operation on triadic relations is a particular three-way join which takes explicit advantage of the triadic structure of its operands.” This triadic join, meanwhile, “results in another triadic relation, thus providing the closure required of an algebra.” [13, p. 6] Parsing Robertson’s compact symbolic expressions into something close to standard English, the trijoin of three triadic relations R, S, T is defined as some (a,b,c) selected from the universe of possibilities (x,y,z), such that (a,x,z) ∈ R, (x,b,y) ∈ S, and (z,y,c) ∈ T . This relation, he argues, is the most fundamental of all the operators he defines. When supplemented with a few constant relations (analogs of Tarski’s “infinite constants” embodied in the four binary relations of universality of all pairs, identity of all equal pairs, diversity of all unequal pairs, and the empty set), it can express all the standard monotonic operators (thereby excluding, among his primitives, only the relative complement). How does this compare with our ZD setup, and the workings of Skies? For one thing, Infinite constants, of a type akin to Tarski’s, are embodied in the fact that any full meta-fractal requires the use of an infinite G, which sits atop an endless cascade of singleton leftmost bits, determining for any given S an indefinite tower of ZDs. One of the core operators massaging Robertson’s triads is the flip, which fixes one component of a relation while interchanging the other two · · · but our Rule 2 is just the recursive analog of this, allowing one to move up and down towers of values with great flexibilty (allowing, as well, on and off switching effecting whole ensembles). The integer triads upon which our entire apparatus depends are a gift of nature, not dictated “by fiat,” and give us a natural basis for generating and tracking unique IDs with which to “tag” and “unpack” data (with “storage” provided free of charge by the empty spaces of our meta-fractals: the “atoms” of Semiotic Squares have four long-diagonal slots each, one per each of the “controls” Petitot’s Catastrophe Theory reading calls for, and so on.) Finally, consider two dual constructions that are the core of our own triadic number theory: if the (a,b,c) of last paragraph, for instance, be taken as a Zigzag’s L-index set, then the other trio of triples correlates quite exactly with the Zigzag U-trips. And this 3-to-1 relation, recall, exactly parallels that between the 3 Tre- foil, and 1 Zigzag, Sails defining a Box-Kite, with this very parallel forming the support for the recursion that ultimately lifts us up into a Sky. We can indeed make this comparison to Robinson’s formalism exceedingly explicit: if his X, Y, Z be considered the angular nodes of PSL(2,7) situated at the 12 o’clock apex and the right and left corners respectively, then his (a,b,c) correspond exactly to our own Rule 0 trip’s same-lettered indices! Here, we would point out that these two threads of reflection – on underwriting Chaos with cellular-automaton-tied Number Theory, and designing new kinds of database architectures – are hardly unrelated. It should be recalled that two years prior to his revolutionary 1970 paper on relational databases [14], Codd published a pioneering book on cellular automata [15]. It is also worth noting that one of the earliest technologies to be spawned by fractals arose in the arena of data compression of images, as epitomized in the work of Michael Barnsley and his Iterative Systems company. The immediate focus of the author’s own commercial efforts is on fusing meta-fractal mathematics with the context-sensitive adaptive- parsing “Meta-S” technology of business associate Quinn Tyler Jackson. [16] And as that focus, tautologically, is not mathematical per se, we pass it by and leave it, like so many other themes just touched on here, for later work. References [1] Robert P. C. de Marrais, “Placeholder Substructures I: The Road From NKS to Scale-Free Networks is Paved with Zero Divisors,” Complex Systems, 17 (2007), 125-142; arXiv:math.RA/0703745 [2] Robert P. C. de Marrais, “Placeholder Substructures II: Meta-Fractals, Made of Box-Kites, Fill Infinite-Dimensional Skies,” arXiv:0704.0026 [math.RA] [3] Robert P. C. de Marrais, “The 42 Assessors and the Box-Kites They Fly,” arXiv:math.GM/0011260 [4] Robert P. C. de Marrais, “The Marriage of Nothing and All: Zero-Divisor Box-Kites in a ‘TOE’ Sky,” in Proceedings of the 26th International Col- loquium on Group Theoretical Methods in Physics, The Graduate Center of the City University of New York, June 26-30, 2006, forthcoming from Springer–Verlag. [5] Robert P. C. de Marrais, “Placeholder Substructures: The Road from NKS to Small-World, Scale-Free Networks Is Paved with Zero-Divisors,” http:// wolframscience.com/conference/2006/ presentations/materials/demarrais.ppt (Note: the author’s surname is listed under “M,” not “D.”) [6] Benoit Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Company, San Francisco, 1983) [7] Ed Pegg, Jr., “Tournament Dice,” Math Games column for July 11, 2005, on the MAA website at http://www.maa.org/editorial/ mathgames/mathgames _07_11_05.html [8] Robert P. C. de Marrais, “The ‘Something From Nothing’ Insertion Point,” http://www.wolframscience.com/conference/2004/ presentations/ materials/rdemarrais.pdf [9] Robert P. C. de Marrais, “Presto! Digitization,” arXiv:math.RA/0603281 [10] Francisco Varela, editor, Sleeping, Dreaming, and Dying: An Exploration of Consciousness with the Dalai Lama (Wisdom Publications: Boston, 1997). [11] F. J. Varela, J.-P. Lachauz, E. Rodrigues and J. Martinerie, “The brainweb: phase synchronization and large-scale integration,” Nature Reviews Neuro- science, 2 (2001), pp. 229-239. http://arxiv.org/abs/math/0703745 http://arxiv.org/abs/0704.0026 http://arxiv.org/abs/math/0011260 http://www.maa.org/editorial/ http://www.wolframscience.com/conference/2004/ http://arxiv.org/abs/math/0603281 [12] Jean Petitot, Francisco J. Varela, Bernard Pachoud and Jean-Michel Roy, Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science (Stanford University Press: Stanford, 1999) [13] Edward L. Robertson, “An Algebra for Triadic Relations,” Technical Re- port No. 606, Computer Science Department, Indiana University, Bloom- ington IN 47404-4101, January 2005; online at http://www.cs.indiana.edu/ pub/techreports/TR606.pdf [14] E. F. Codd, The Relational Model for Database Management: Version 2 (Addison-Wesley: Reading MA, 1990) is the great visionary’s most recent and comprehensive statement. [15] E. F. Codd, Cellular Automata (Academic Press: New York, 1968) [16] Quinn Tyler Jackson, Adapting to Babel – Adaptivity and Context-Sensiti- vity in Parsing: From anbncn to RNA (Ibis Publishing: P.O. Box3083, Ply- mouth MA 02361, 2006; for purchasing information, contact Thothic Tech- nology Partners, LLC, at their website, www.thothic.com). http://www.cs.indiana.edu/ The Argument So Far 8 <S < 16, N : Recursive Balloon Rides in the Whorfian Sky Maximal High-Bit Singletons: (s,g)-Modularity for 16 < S 24 Hide/Fill Involution: Further-Right High-Bits with 24 < S < 32. Fundamental Theorem of Zero-Divisor Algebra

0704.0113

Langmuir blodgett assembly of densely aligned single walled carbon nanotubes from bulk materials

Langmuir-Blodgett Assembly of Densely Aligned Single-Walled Carbon Nanotubes from Bulk Materials Xiaolin Li, Li Zhang, Xinran Wang, Iwao Shimoyama, Xiaoming Sun, Won-Seok Seo, Hongjie Dai* Department of Chemistry, Stanford University, Stanford, CA 94305, USA. RECEIVED DATE (automatically inserted by publisher); [email protected] Single-walled carbon nanotubes (SWNTs) exhibit advanced properties desirable for high performance nanoelectronics. Important to future manufacturing of high-current, speed and density nanotube circuits is large-scale assembly of SWNTs into densely aligned forms. Despite progress in oriented synthesis and assembly including the Langmuir-Blodgett (LB) method,2-9 no method exists for producing assemblies of pristine SWNTs (free of extensive covalent modifications) with both high density and high degree of alignment of SWNTs. Here, we develop a LB method achieving monolayers of aligned non-covalently functionalized SWNTs from organic solvent with dense packing. The monolayer SWNTs are readily patterned for device integration by microfabrication, enabling the high currents (~3mA) SWNT devices with narrow channel widths. Our method is generic for different bulk materials with various diameters. Suspensions of as-grown laser-ablation and Hipco SWNTs in 1,2-dichloroethane (DCE) solutions of poly(m-phenylenevinylene -co-2,5-dioctoxy-p-phenylenevinylene) (PmPV) were prepared by sonication, ultra centrifugation and filtration (see supplementary information). The suspension contained mostly individual nanotubes (average diameter~1.3nm and ~1.8nm respectively for Hipco and laser-ablation materials, mean length ~500nm, Fig.1d and 1e) well solubilized in DCE without free unbound PmPV. PmPV is known to exhibit high binding affinity to SWNT sidewall via  stacking of its conjugated backbone (Fig.1a) and thus impart solubility of nanotubes in organic solvents.10 Indeed, we obtained homogeneous suspensions of nanotubes in PmPV solutions. However, we found that DCE was the only solvent in which PmPV bound SWNTs remained stably suspended when free unbound PmPV molecules were removed (Inset of Fig.1b). The PmPV treated SWNTs exhibited no aggregation in DCE over several months. DCE without PmPV could suspend low concentrations of SWNTs (~50X lower than with PmPV functionalization), insufficient for LB formation, especially for larger SWNTs in laser materials with lower solubility. The excitation and emission spectra of PmPV bound SWNTs (in PmPV-SWNT solution with excess PmPV removed) exhibited ~20nm and ~3nm shifts respectively relative to those of pure PmPV in DCE (Fig.1b), providing spectroscopic evidence of strong interaction between PmPV and SWNTs. No change in the spectra was observed with the highly stable PmPV-SWNT/DCE suspension for months, indicating strong binding of PmPV on SWNT without detachment in DCE. The fact that PmPV-SWNTs were not stably suspended in other solvents without excess PmPV and that addition of large amounts of these solvents (e.g., chloroform) into a PmPV-SWNT/DCE suspension causing nanotube precipitation suggested significant detachment of PmPV from nanotubes in most organic solvents. The unique stability of PmPV coating on SWNT in DCE over other solvents is not fully understood currently. Nevertheless, it is highly desirable for chemical assembly of high quality nanotubes and integrated devices since it enables non-covalently functionalized SWNTs (both large diameter laser and small diameter Hipco materials) soluble in organics in nearly pristine form, as gleaned from the characteristic UV-vis-NIR absorbance (Fig.1c) and Raman signatures of non-covalently modified SWNTs (Fig.2c). PmPV-SWNTs were spread on a water subphase from a DCE solution, compressed upon DCE vaporization to form a LB film using compression-retraction-compression cycles to reduce hysteresis (supplementary Fig.S1, S2&S3) and then vertically transferred onto a SiO2 or any other substrate (glass, plastic, etc.). Organic solutions of stably suspended SWNTs without excess free polymer are critical to high density SWNT LB film formation. Microscopy (Fig.2a&2b) and spectroscopy (Fig.2c&2d) characterization revealed high quality densely aligned SWNTs (normal to the compression and substrate pulling Hipco Figure 1. PmPV functionalized SWNTs. (a) Schematic drawings of a SWNT and two units of a PmPV chain. (b) Excitation and fluorescence spectra of pure PmPV in DCE vs. PmPV bound Hipco SWNTs in DCE. Inset: photograph of PmPV coated Hipco SWNTs suspended in DCE without excess PmPV in the solution. (c) UV-vis- NIR spectrum of PmPV suspended Hipco SWNTs with no excess PMPV. (d) & (e) Atomic force microscopy (AFM) images of Hipco and laser-ablation SWNTs randomly deposited on a substrate from solution. Insets: Diameter distributions. = OC8H17 (b) (c) 300 400 500 5.0x10 1.0x10 1.5x10 Wavelength (nm) PmPV Hipco-PmPV 600 800 1000 Wavelength (nm) Hipco-PmPV 1.0 1.5 2.03.00.5 d (nm) 1.0 1.5 2.03.00.5 d (nm) 1.0 1.5 2.0 2.50.5 d (nm) 1.0 1.5 2.0 2.50.5 d (nm) (d) (e) direction) formed uniformly over large substrates for both Hipco and laser ablation materials. Height of the film relative to tube- free regions of the substrate was <2nm under AFM, suggesting monolayer of packed SWNTs. Micro-Raman spectra of the SWNTs showed ~ cos2polarization dependence of the G band (~1590cm-1) intensity (Fig. 2d), where  is the angle between the laser polarization and the SWNT alignment direction. The peak to valley ratio of the Raman intensities was ~8 with little variation over the substrate, indicating alignment of SWNTs over large areas. Nevertheless, imperfections existed in the quasi-aligned dense SWNT assembly including voids, bending and looping of nanotubes formed during the compression process for LB film formation due to the high aspect ratio (diameter <~2nm, length ~200nm-1m) and mechanical flexibility of SWNTs. Our aligned SWNT monolayers on oxide substrates can be treated as carbon-nanotube on insulator (CNT_OI) materials for patterning and integration into potential devices, much like how Si on insulator (SOI) has been used for electronics. We used lithographic patterning techniques and oxygen plasma etching to remove unwanted nanotubes and form patterned arrays of squares or rectangles comprised of aligned SWNTs (Fig.3a and 3b). We then fabricated arrays of two-terminal devices with Ti/Au metal source (S) and drain (D) contacting massively parallel SWNTs in ~10 m wide S-D regions with channel length ~250nm (Fig.3c and 3d). Current vs. bias voltage (I-V) measurements showed that such devices made from Hipco SWNTs were more than 25 times more resistive than similar devices made from laser-ablation SWNTs, with currents reaching ~0.13mA and ~3.5mA respectively at a bias of 3 V through collective current carrying of SWNTs in parallel (Fig.3e and 3f). Further, Hipco SWNT devices exhibited higher non-linearity in the I-V characteristics than laser ablation nanotubes (Fig.3e). These results were attributed to the diameter difference between Hipco and laser-ablation materials. Hipco SWNTs were small in diameter with many tubes ≤1.2nm, giving rise to high (non-ohmic) contact resistance for both semiconducting and metallic SWNTs. Smaller SWNTs could also be more susceptible to defects and disorder, contributing to degraded current carrying ability. The LB assembly of densely aligned SWNTs can be combined with chemical separation and selective chemical reaction methods12 to afford purely metallic or semiconducting SWNTs in massive parallel configuration useful for interconnection or high speed transistor applications at large scale. The method is generic in terms of the type of nanotube materials and substrates. Acknowledgment. We thank Dr. Pasha Nikolaev for providing laser-ablation SWNTs and MARCO-MSD and Intel for support. Supporting Information Available: Experimental details are available free of charge via the internet at http://pubs.acs.org. REFERENCES 1. Guo, J., Hasan, S., Javey, A., Bosman, G. & Lundstrom, M. IEEE Trans. Nanotechnology 2005, 4, 715. 2. Zhang, Y., Chang, A. & Dai, H. J. Appl. Phys. Lett. 2001, 79, 3155. 3. Huang, S. M., Maynor, B., Cai, X. Y. & Liu, J. Adv. Mater. 2003, 15, 1651. 4. Kocabas, C., Hur, S., Gaur, A., Meitl, M. A., Shim, M. and Rogers, J. A. Small 2005, 11, 1110. 5. Han, S., Liu, X. & Zhou, C. W. J. Am. Chem. Soc. 2005, 127, 5294. 6. Gao, J., Yu, A., Itkis M. E., Bekyarova, E., Zhao, B., Niyogi, S. & Haddon, R. C. J. Am. Chem. Soc. 2004, 126, 16698. 7. Rao, S. G., Huang, L., Setyawan, W. & Hong, S. Nature 2003, 425, 36. 8. Guo, Y., Wu, J., Zhang, Y. Chem. Phys. Lett. 2002, 362, 314. 9. Krstic, V., Duesberg, G. S., Muster, J., Burghard, M., Roth, S. Chem. Mater. 1998, 10, 2338. 10. Star, A., Stoddart, J. F., Steuerman, D., Diehl, M., Boukai, A., Wong, E. W., Yang, X., Chung, S. W., Choi, H. & Heath, J. R. Angew. Chem. Int. Ed. 2001, 40, 1721. 11. Kim, W. Javey, A., Tu, R., Cao, J., Wang, Q. & Dai, H. Appl. Phys. Lett. 2005, 87, 173101. 12. Zhang, G. Y., Qi, P. F., Wang, X. R., Lu, Y. R., Li, X. L., Tu, R., Bangsaruntip, S., Mann, D., Zhang, L. & Dai, H. Science 2006, 314, Figure 2. LB monolayers of aligned SWNTs. (a) AFM image of a LB film of Hipco SWNTs on a SiO2 substrate. (b) AFM image of a LB film of laser-ablation SWNTs. (c) Raman spectra of the G line of a Hipco SWNT LB film recorded at various angles () between the polarization of laser excitation and SWNT alignment direction. Inset: Raman spectrum showing the radial breathing mode (RBM) region of the Hipco LB film at ~0. (d) G line (1590cm-1) intensity vs. angle  for the Hipco SWNT LB film in (c). The red curve is a cos2 fit. (a) (b) 0 50 100 150 5.0x10 1.0x10 1.5x10 2.0x10 Angle (deg.) ngle (deg.) Raman s hift (cm ngle (deg.) Raman s hift (cm 150 200 250 Raman shift (cm (c) (d) Figure 3. Microfabrication patterning and device integration of SWNT LB films. (a) Optical image of a patterned SWNT LB film. The squares and rectangles are regions containing densely aligned SWNTs. Other areas are SiO2 substrate regions. (b) SEM image of a region highlighted in (a) with packed SWNTs aligned vertically. (c) SEM image showing a 10-micron-wide SWNT LB film between source and drain electrodes formed in a region marked in (b). (d) AFM image of a region in (c) showing aligned SWNTs and the edges of the S and D electrodes. (e) Current vs. bias (Ids-Vds) curve of a device made of Hipco SWNTs (10m channel width and 250nm channel length). (f) Ids-Vds of a device made of laser-ablation SWNTs (10m channel width and 250nm channel length). -3 -2 -1 0 -3 -2 -1 0 -0.15 -0.10 -0.05 (e) (f) 400m 400nm 80m(b) Angle (deg.) Raman shift (cm Angle (deg.) Raman shift (cm ABSTRACT FOR WEB PUBLICATION. Single-walled carbon nanotubes (SWNTs) exhibit advanced electrical and surface properties useful for high performance nanoelectronics. Important to future manufacturing of nanotube circuits is large-scale assembly of SWNTs into aligned forms. Despite progress in assembly and oriented synthesis, pristine SWNTs in aligned and close-packed form remain elusive and needed for high-current, -speed and -density devices through collective operations of parallel SWNTs. Here, we develop a Langmuir-Blodgett (LB) method achieving monolayers of aligned SWNTs with dense packing, central to which is a non- covalent polymer functionalization by poly(m-phenylenevinylene-co-2,5-dioctoxy-p-phenylenevinylene) (PmPV) imparting high solubility and stability of SWNTs in an organic solvent 1,2-dichloroethane (DCE). Pressure cycling or ‘annealing’ during LB film compression reduces hysteresis and facilitates high-degree alignment and packing of SWNTs characterized by microscopy and polarized Raman spectroscopy. The monolayer SWNTs are readily patterned for device integration by microfabrication, enabling the highest currents (~3mA) through the narrowest regions packed with aligned SWNTs thus far.

0704.0114

Quantum Phase Transition in the Four-Spin Exchange Antiferromagnet

Quantum Phase Transition in the Four-Spin Exchange Antiferromagnet Valeri N. Kotov, Dao-Xin Yao, A. H. Castro Neto, and D. K. Campbell Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215 We study the S=1/2 Heisenberg antiferromagnet on a square lattice with nearest-neighbor and plaquette four- spin exchanges (introduced by A.W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007).) This model undergoes a quantum phase transition from a spontaneously dimerized phase to Néel order at a critical coupling. We show that as the critical point is approached from the dimerized side, the system exhibits strong fluctuations in the dimer background, reflected in the presence of a low-energy singlet mode, with a simultaneous rise in the triplet quasiparticle density. We find that both singlet and triplet modes of high density condense at the transition, signaling restoration of lattice symmetry. In our approach, which goes beyond mean-field theory in terms of the triplet excitations, the transition appears sharp; however since our method breaks down near the critical point, we argue that we cannot make a definite conclusion regarding the order of the transition. PACS numbers: 75.10.Jm, 75.30.Kz, 75.50.Ee I. INTRODUCTION Problems related to quantum criticality in quantum spin systems are of both fundamental and practical importance1. Numerous materials, such as Mott insulators, exhibit either antiferromagnetic (Néel) order or quantum disordered (spin gapped) ground state depending on the distribution of Heisen- berg exchange couplings and geometry. External perturba- tions (such as doping or frustration) can also cause quantum transitions between these phases. Systems with spin 1/2 are indeed the most interesting as they are the most susceptible to such transitions. It is well understood that the quantum transi- tion between a quantum disordered and a Néel phase is in the O(3) universality class1, where a triplet state condenses at the quantum critical point (QCP). A recent exciting development in our theoretical under- standing of QCPs originated from the proposal that if the quantum disordered (QD) phase spontaneously breaks lattice symmetries (e.g. is characterized by spontaneous dimer or- der), and the transition is of second order, then exactly at the QCP spinon deconfinement occurs, i.e. the excitations are fractionalized2. It is assumed that the Hamiltonian itself does not break the lattice symmetries (i.e. does not have “triv- ial” dimer order caused by some exchanges being stronger than the others). We use the terms “dimer order” and “va- lence bond solid (VBS) order” interchangeably. It is expected that the dimer order vanishes exactly at the point where Néel order appears, i.e. there is no coexistence between the two phases. Deconfinement thus is intimately related to disap- pearance of VBS order; indeed if the latter persisted in the Néel phase it would be impossible to isolate a spinon, as “pairing” would always take place. Spontaneous VBS or- der driven by frustration has been a common theme in quan- tum antiferromagnetism3, although its presence and the na- ture of criticality in specific models, such as the 2D square- lattice frustrated Heisenberg antiferromagnet, is still some- what controversial4. It would be particularly useful to apply unbiased numerical approaches, such as the Quantum Monte Carlo (QMC) method, to study frustrated spin models; how- ever due to the fatal “sign” problem5, frustrated Heisenberg systems are beyond the QMC reach. In a recent study, the QMC method was applied to a four- spin exchange quantum spin model without frustration, which was shown to exhibit columnar dimer VBS order and a mag- netically ordered phase with a deconfined QCP separating them6. These conclusions were later confirmed by further QMC studies7. Extensions of the model, which include for example additional (six-spin) interactions, provide additional support for a continuous QCP8. A different VBS pattern (pla- quette order) was also proposed for the four-spin exchange model9. At the same time, the nature of the quantum phase transition was challenged in Refs.[10,11], where arguments were given that the transition is in fact of (weakly) first order. It is the objective of the present work to study the Sand- vik model6, by approaching the quantum transition from the dimer VBS phase. Our approach uses as a starting point a symmetry broken state (i.e. one out of four degenerate VBS configurations), and we thus must search for signatures that the system attempts to restore the lattice symmetry at the QCP. Even though full restoration is impossible within the present framework, we find a QCP characterized by condensation of triplet modes of high density; this is in contrast to the conven- tional situation when the condensing particles are in the dilute Bose gas limit. The high density itself is due to the presence of a singlet mode that condenses at the QCP, and reflects the strong fluctuations of the background dimer order. The above effects lead to the vanishing of the VBS order parameter; at the same time our method, which accounts for the strong fluc- tuations, leads to a rather sharp phase transition. It appears that we cannot draw a definite conclusion about the order of the transition because in the vicinity of the QCP the triplon density increases uncontrollably, suggesting that other states (such a plaquette states and larger clusters) are strongly ad- mixed into the ground state. This is generally expected in a situation where the lattice symmetry is restored at the quan- tum critical point. The model under consideration is H = J 〈a,b〉 Sa.Sb −K a,b,c,d (Sa · Sb)(Sc · Sd), (1) where J > 0,K > 0, and all spins are S = 1/2. Consider the numbers 1,2,3,4 in Fig. 1. The summation in the four- spin term is over indexes (a, b) = (1, 2), (c, d) = (3, 4) and http://arxiv.org/abs/0704.0114v3 FIG. 1: (Color online) Dimer pattern in the quantum disordered (VBS) phase, K/J > (K/J)c. (a, b) = (1, 4), (c, d) = (2, 3) on a given plaquette, and then summation is made over all plaquettes12. The range of pa- rameters explored in Ref.[6] is K/J ≤ 2, and the QCP is at (K/J)c ≈ 1.85. Our coupling notation is slightly different from the one used in Refs.[6,7]; the coupling K is related to the parameter Q6,7 via K = Q/(1 + Q/(2J)), and the criti- cal point in that notation is (Q/J)c ≈ 25. The dimerization pattern is proposed to be of the “columnar” type, as shown in Fig. 1. Four such configurations exist. We will assume a con- figuration of this type, will show that it is stable at K/J ≫ 1, and will then search for an instability towards the Néel state as K/J decreases. The rest of the paper is organized as followed. In Sec- tion II we present results based on the mean-field approach in terms of the dimer (triplon) operators. In Section III we extend our treatment beyond mean-field, and even further in Section IV, where we also take into account low-energy singlet two- triplon excitations. Section V contains our conclusions. II. MEAN-FIELD TREATMENT We start by rewriting Eq. (1) in the the bond-operator representation13, where on a dimer i, the two spins forming it are expressed as: Sα1,2 = tiα±t†iαsi−iǫαβγt tiγ), and , α = x, y, z create a singlet and triplet of states. We re- fer to the triplet (S=1) quasiparticle, t† , as “triplon”. The bold indexes i, j,m, l label the dimers (see Fig. 1). Summa- tion over repeated Greek indexes is assumed, unless indicated otherwise. The hard-core constraint, s† si + t tiα = 1, must be en- forced on every site, which at the mean-field (MF) level can be done by introducing a term in the Hamiltonian, −µ (s2 + tiα − 1). Then µ and the (condensed) singlet amplitude s ≡ 〈si〉, are determined by the MF equations13. We obtain at the quadratic level, in momentum representation: tkα + −kα + h.c. where Ak = J/4− µ+ s2(ξ−k +K/2) + s 4Σ(k) , Bk = s + s4Σ(k) , = −(J/2) coskx + (J ±K/4) cosky . (3) The four-spin interaction from (1) acting between two dimers (e.g. i, j in Fig. 1) contributes to the “on-site” gap and hopping (ξ− ) via Ak, as well as to the quantum fluctuations term Bk. The part involving four dimers has been split in a mean-field fashion, leading to the Hartree-Fock self-energy −Σ(k)/K = 2Σx cos kx + 2Σy cos ky +Σxy cos kx cos ky, tmα + t t†mα〉 , (5) where i,m are neighboring dimers in the x (horizontal) direc- tion (Fig. 1), and similarly for the y and the diagonal contribu- tions. The triplon dispersion is ω(k) = , and has a minimum at the Néel ordering wave-vector kAF = (0, π) (since we work on a dimerized lattice). The ground state en- ergy is then easily computed, EGS = E0 + 〈H2〉 , (6) where E0/N = − (Js2 +Ks4) + µ(−s2 + 1) + (7) Σ2x +Σ , (8) 〈H2〉 = (ω(k)−Ak) . (9) The mean-field equations require a numerical minimization of EGS with respect to the parameters {µ, s,Σx,Σy,Σxy}. This amounts to the self-consistent Hartree-Fock approxima- tion for Σ(k). The result for the triplon gap ∆ = ω(kAF ) is presented in Fig. 2 (black curve). The MF result (K/J)c ≈ 0.6 substantially underestimates the location of the critical point, compared to the the QMC calculations, where (K/J)c ≈ 1.856,7. Interestingly, if one solves the MF equations ignoring both the hard core and the Σ(k), one finds (K/J)c = 1. Physically, in the full MF, the hard core contribution increases the gap (and hence the stabil- ity of the dimer phase) while at the same time suppressing the antiferromagnetic fluctuations (which favor the Néel state). We also note that a recent (hierarchical) MF treatment based on the plaquette ground state also underestimates very strongly the QCP location ((K/J)c ≈ 19), similarly to our result. In our view this means that both mean field approaches are not sufficient to attack the present problem, where fluctu- ations are apparently very strong. We choose to accept that the numerical QMC result gives the most accurate determina- tion of the QCP, and therefore in what follows we extend our treatment in several directions beyond mean-field theory. 0.5 1 1.5 2 2.5 Brueckner field theory (II) Brueckner +Singlet fluctuations (III) Mean-Field Theory (I) (III) FIG. 2: (Color online) Triplon excitation gap ∆ = ω(kAF ) in vari- ous approximations. The point ∆ → 0 corresponds to transition to the Néel phase. III. BEYOND MEAN-FIELD: THE DILUTE TRIPLON GAS APPROXIMATION A more accurate treatment of fluctuations is possible by taking into account the hard-core constraint beyond mean- field. One can set the singlet amplitude s = 1 in the pre- vious formulas, but introduce an infinite on-site repulsion be- tween the triplons, U tβitαi, U → ∞. As long as the triplon density (determined by the quantum fluctuations) is low, an infinite repulsion corresponds to a finite scattering am- plitude between excitations and can be calculated by resum- ming ladder diagrams for the scattering vertex14. This leads to the effective triplon-triplon vertex Γ(k, ω) which was pre- viously calculated15: Γ−1(k, ω)= ω(q) + ω(k− q)− ω u → v ω → −ω This vertex in turn affects the triplon dispersion via (what we call) the Brueckner self-energy15: ΣB(k, ω) = 4 v2qΓ(k+ q, ω − ω(q)). (11) The corresponding parameters in the quadratic Hamiltonian (2) in this case are Ak = J + 2K(1− 4nt/3) + ξ−k +Σ(k) + ΣB(k, 0), Bk = ξ +Σ(k). (12) The Bogolubov coefficients are defined in the usual way = 1/2 + Ak/(2ω(k)) = 1 + v . The various terms in Σ(k) can be expressed through them: for example Σx = + vkuk) cos kx, and so on. The density of triplons is nt = 〈t†iαtiα〉 = 3 v2k. In addition, the renormaliza- tion of the quasiparticle residue, Z−1 = 1 − ∂ΣB(k, 0)/∂ω, −k y y FIG. 3: Renormalization of quantum fluctuations by resummation of a ladder series, with (13) at the vertices. implies the replacement uk → Zkuk, vk → Zkvk in all the formulas15, and the renormalized spectrum ω(k) = An iterative numerical evaluation of the spectrum using the above equations, which amounts to solution of the Dyson equation, leads to the result shown in Fig. 2 (blue curve). The above approach appears to be well justified since the quasi- particle density nt < 0.1. The resulting critical point is still in the “weak-coupling” regime K/J < 1, with about 100% de- viation from the QMC result ((K/J)c ≈ 1.85). This suggests that the on-site triplon fluctuations are not the dominant cause for the disagreement with the QMC results; thus we proceed to include two-particle fluctuations (in the triplon language), which amounts to including dimer-dimer correlations. IV. STRONG FLUCTUATIONS IN THE SINGLET BACKGROUND: QCP BEYOND THE DILUTE TRIPLON GAS APPROXIMATION It is clear that “non-perturbative” effects are responsible for driving the QCP towards the “strong-coupling” regionK/J ∼ 2. To proceed we make two improvements to the previous low-density, weak-coupling theory. First, we take into account fluctuations in the singlet back- ground, i.e. the manifold on which the triplons are built and interact. The main effect originates from the action of the four-spin K-term from (1) on two dimers, e.g. i, j in Fig. 1. Part of this action has led to the on-site gap 2K in (12), fa- voring dimerization. However, a strong attraction between the two dimers is also present, since theK-term is symmetric with respect to the index pair exchange (1, 2)(3, 4) ↔ (1, 4)(2, 3), leading to a “plaquettization” tendency as well. In the triplon language this is manifested by formation of bound states of two triplons, due to their nearest-neighbor interactions H4,y = 〈i,j〉y,αβ tβitαj + γ2t tβitβj + γ3t tαitβj , (13) γ1 = − , γ2 = − , γ3 = − We also checked that on the perturbative (Hartree-Fock) level, the effect of this term on equations (3) and (12) was negligible (and we did not write it explicitly). An intuitive way of taking into account the effect of two- triplon bound states (with total spin S=0) on the one-triplon 2.1 2.2 2.3 2.4 2.5 FIG. 4: (Color online) (a.) Singlet bound state energy Es (black), binding energy ǫ = 2∆ − Es (blue), and the triplon gap ∆ (red). (b.) Triplon density nt. (c.) Dimer order parameters. Dashed parts of the lines represent points corresponding to rapid growth of the quasiparticle density. spectrum, is to work in the “local” approximation. This means effectively neglecting the triplon dispersion and directly eval- uating the ladder series that renormalizes the quantum fluctu- ation term Bk in (2), corresponding to emission of a pair of triplons with zero total momentum. This is illustrated graphi- cally in Fig. 3, with the result Bk = − cos kx + J +K/4 1− |γ|  cos ky +Σ(k) , γ ≡ γ1 + 3γ2 + γ3 = −J − K, (14) where γ is the effective attraction of two triplons with total S = 0, and ∆E = 2J + K (15) is the energy of two (non-interacting) triplons on adjacent sites. This calculation is justified for K/J ≫ 1 and leads to an increase of the quantum fluctuations, and from there to almost doubling of the triplon density nt (see Fig. 4 below). It contributes significantly to the shift of the QCP. We can go beyond the “local” approximation by solving the Bethe-Salpeter equation for the bound state, formed due to the attraction (13), and taking into account the full triplon dispersion. The equation for the singlet bound state energy Es(Q), corresponding to total pair momentum Q is 1 = 2γ u4q cos Es(Q)− ω(Q/2 + q)− ω(Q/2− q) . (16) Here we have, for simplicity, written only the main contribu- tion to pairing (Eq. (13)) in the limit K/J ≫ 1, and have neglected the on-site repulsion (which leads to slightly di- minished pairing), as well as small pairing due to the ex- change J from dimers in the x-direction on Fig. 1. It is easily seen that the lowest energy corresponds to Q = 0; we define from now on Es ≡ Es(Q = 0). The bind- ing energy is ǫ = 2∆ − Es, where ∆ is the one-particle gap. The bound state wave-function corresponding to Es is |Ψ〉 = α,i,j,qy iqy(i−j)t |0〉. In the “local” limit (nearest-neighbor pairing), Ψqy = 2 cos qy . Second, we have made subtle changes to the resumma- tion procedure concerning the quasiparticle renormalization Z , based on both formal and physical grounds. On the one hand it is clear that in the Brueckner approximation (Eq. (11)), where the self energy is linear in the density (ΣB ∝ nt), the dependence of the vertex Γ−1 on density is beyond the accu- racy of the calculation, meaning one can put uq = 1, vq = 0 in (10), instead of determining them self-consistently. This leads to a decreased influence of the hard-core ΣB (which fa- vors the dimer state) on the Hartree-Fock self-energy Σ(k) from (4) (which favors the Néel state). It is indeed the mutual interplay between ΣB > 0 and Σ(k) < 0, that determines the exact location of the QCP in the course of the Dyson’s equa- tion iterative solution. While in the “weak-coupling” regime K/J < 1, ΣB always dominates, in the “strong-coupling” re- gion K/J > 2, Σ(k) starts playing a significant role, since parametrically Σ ∝ Knt. It is physically consistent that in the region where singlet fluctuations in the dimer background are strong, the hard-core effect is less important, i.e. in ef- fect the kinematic hard-core constraint is “relaxed”. We also observe that in typical models with QCP driven by explicit dimerization, such as the bilayer model, the described differ- ence in approximation schemes makes a very small difference on the location of the QCP16, since those models are always in the “weak-coupling” regime, dominated by the hard-core repulsion of excitations on a non-fluctuating dimer configura- tion. The purpose of the above rather technical diversion is to emphasize that care has been taken to take into account as ac- curately as possible the effect of the (low-energy) two-particle spectrum on the one-particle triplon gap. Our results are summarized in Fig. 4 and Fig. 2 (red line) for the gap. The critical point is shifted towards (K/J)c ≈ 2.16 (in much better agreement with QMC data), with a very strong increase of the density towards Kc. This translates into a decrease of the dimer order, as measured by the two dimer order parameters that we compute from the expres- sions: Dx = |〈S3 · S4〉 − 〈S5 · S4〉| = | − 34 + nt + Dy = |〈S3 ·S4〉−〈S1 ·S4〉| = |− 34+nt− Σy|. The spins are labeled as in Fig. 1. The singlet bound state energy Es(0) also tends towards zero at the QCP, with the corresponding binding energy remaining quite large ǫ/J ≈ 1. All these effects point towards a tendency of the system to restore the lattice sym- metry, although it is certainly clear that as the critical point is approached, our approximation scheme (low density of quasi- particles) breaks down (dashed lines on figures). We should point out that the sharpness of variation near Kc is not due to divergence in any of the self-energies but is a result of rapid cancellation at high orders (i.e. iterations in the Dyson equa- tion). In fact cutting off our iterative procedure at finite order gives a smooth curve, suggesting that additional classes of di- agrams become important (although in practice their classifi- cation is an insurmountable task). The merger of singlet and triplet modes, which we find near the QCP, in principle reflects a tendency towards quasiparticle fractionalization (spinon de- confinement) and is also found in the 1D Heisenberg chain with frustration17, where spinons are always deconfined. Since we are now dealing with a situation where the density is not very small nt ≈ 0.2, it is prudent to check how the next order in the density may affect the above results. For example at second order in the density, the self-energy Σ(k) changes by amount δΣ(k), i.e. one has to add this contribution to the right hand side of Eq.(12), namely: Ak → Ak + δΣ(k), Bk → Bk + δΣ(k). (17) We have found δΣ(k) = −2K(Q2x − P 2x ) cos kx + +2K(Q2y − P 2y ) cos ky − −K(Q2xy − P 2xy) cos kx cos ky, (18) and the following definitions are used: Px = (1/3) tmα〉 = v2k cos kx, Qx = (1/3) 〈tiαtmα〉 = ukvk cos kx, (19) and similarly for the other directions, for example Py = v2k cos ky , Pxy = v2k cos kx cos ky , etc. After includ- ing these expressions in our numerical iterative procedure, we have found that the QCP is shifted by a very small amount, and the overall picture, as summarized in Fig. 4 and Fig. 2 (red line), still stands. V. CONCLUSIONS In conclusion, we have shown that the QCP between the Néel and the dimer state in the model (1) is of unconven- tional nature, in the sense that it is characterized by the pres- ence of both triplet and singlet low-energy modes. Near the QCP, whose location ((K/J)c ≈ 2.16) we find in fairly good agreement with recent QMC studies, the system exhibits: (1.) Strong rise of the triplon excitation density, due to increased quantum fluctuations, (2.) Corresponding strong decrease (and ultimately vanishing) of the dimer order at the QCP (3.) Vanishing of a singlet energy scale, related to the destruction of the dimer “columns” in Fig. 1. The above effects are all re- lated and influence strongly one another, ultimately meaning that the QCP reflects strong fluctuations and can not be de- scribed in a mean-field theory framework. These results also suggest a desire of the system to restore the lattice symmetry at the QCP, as found in the QMC studies6. At the same time all our improvements beyond mean-field theory have also resulted in a very sharp transition, which ap- pears to be first order. However in our view our approach is not capable of addressing correctly the issue of the order of the phase transition, basically because once we take the strong (inter) dimer fluctuations in to account, the triplon den- sity starts rising quickly beyond control. This is in a certain sense natural in a situation where the system wants to restore the lattice symmetry at the QCP and thus the ground state acquires strong admixture of plaquette, etc. fluctuations as the dimers begin to “disappear.” This is also manifested in the fact that our procedure is sensitive to the number of iterations in the Dyson equation; all presented results are for an “infi- nite” number of iterations, so that a fixed point is reached, but cutting off the procedure results in a smoother behavior and a shift of the QCP, which becomes iteration dependent. We have not previously encountered such volatile behavior in any other spin model with a dimer to magnetic order transi- tion. Since iterations translate into accounting of more and more fluctuations, the sensitivity of the results seems to mean that the situation starts spiraling out of control near the QCP, quite likely because classes of fluctuations become important that are not included in the dimer description, such as longer range correlations, etc. All this suggests that the triplon quasi- particle description breaks down near the QCP which indeed appears natural in a model where spinon deconfinement is ex- pected to take place at the QCP6. On the other hand, if we put aside the arguments that our approach is not reliable near the QCP, the natural conclusion would be that the transition is first order. Acknowledgments We are grateful to A. W. Sandvik, K. S. D. Beach, S. Sachdev, and O. P. Sushkov for numerous stimulating dis- cussions. A.H.C.N. was supported through NSF grant DMR- 0343790; V.N.K., D.X.Y., and D.K.C. were supported by Boston University. 1 S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999). 2 T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004); T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P.A. Fisher, Phys. Rev. B 70, 144407 (2004). 3 S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). 4 R. R. P. Singh, Z. Weihong, C. J. Hamer, and J. Oitmaa, Phys. Rev. B 60, 7278 (1999); L. Capriotti, F. Becca, A. Parola, and S. Sorella, Phys. Rev. Lett. 87, 097201 (2001); M. Mambrini, A. Läuchli, D. Poilblanc, and F. Mila, Phys. Rev. B 74, 144422 (2006). 5 P. Henelius and A. W. Sandvik, Phys. Rev. B 62, 1102 (2000). 6 A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007). 7 R. G. Melko and R. K. Kaul, Phys. Rev. Lett. 100, 017203 (2008). 8 J. Lou, A. W. Sandvik, and N. Kawashima, arXiv:0908.0740. 9 L. Isaev, G. Ortiz, and J. Dukelsky, arXiv:0903.1630. 10 A. B. Kuklov, M. Matsumoto, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett. 101, 050405 (2008). 11 F.-J. Jiang, M. Nyfeler, S. Chandrasekharan, and U.-J. Wiese, J. Stat. Mech., P02009 (2008). 12 The possibility of four-spin exchange induced dimerization has been discussed in the context of the full ring exchange, of which the interaction (1) is part; see e.g. A. Läuchli, J. C. Domenge, C. Lhuillier, P. Sindzingre, and M. Troyer, Phys. Rev. Lett. 95, 137206 (2005). 13 S. Sachdev and R. N. Bhatt, Phys. Rev. B 41, 9323 (1990). 14 A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, Mineola, NY, 2003). 15 V. N. Kotov, O. P. Sushkov, Z. Weihong, and J. Oitmaa, Phys. Rev. Lett. 80, 5790 (1998). 16 P. V. Shevchenko, A. W. Sandvik, and O. P. Sushkov, Phys. Rev. B 61, 3475 (2000). 17 W. H. Zheng, C. J. Hamer, R. R. P. Singh, S. Trebst, and H. Monien, Phys. Rev. B 63, 144411 (2001); ibid. 63, 144410 (2001). http://arxiv.org/abs/0908.0740 http://arxiv.org/abs/0903.1630

0704.0115

Smooth maps with singularities of bounded K-codimensions

Smooth maps with singularities of bounded K-codimensions ∗† Yoshifumi ANDO Abstract Let N and P be smooth manifolds of dimensions n and p respectively such that n ≧ p ≧ 2 or n < p. Let Oℓ(N, P ) denote a K-invarinat open subspace of J∞(N, P ) which consists of all regular jets and singular jets z with codimKz ≦ ℓ (including fold jets if n ≧ p). An Oℓ-regular map f : N → P refers to a smooth map such that j∞f(N) ⊂ Oℓ(N, P ). We will prove that a continuous section s of Oℓ(N,P ) over N has an Oℓ-regular map f such that s and j∞f are homotopic as sections. We next study the filtration of the group of homotopy self-equivalences of a manifold P which is constructed by the sets of Oℓ-regular homotopy self-equivalences for nonnegative integers ℓ. 1 Introduction Let N and P be smooth (C∞) manifolds of dimensions n and p respectively. Let Jk(N,P ) denote the k-jet space of the manifolds N and P with the projections πkN and π P onto N and P mapping a jet onto its source and target respectively. The canonical fiber is the k-jet space Jk(n, p) of C∞-map germs (Rn, 0) → (Rp, 0). Let K denote the contact group defined in [MaIII]. Let O(n, p) denote a K-invariant nonempty open subset of Jk(n, p) and let O(N,P ) denote an open subbundle of Jk(N,P ) associated to O(n, p). In this paper a smooth map f : N → P is called an O-regular map if jkf(N) ⊂ O(N,P ). We will study what is called the homotopy principle for O-regular maps. As for the long history of the several types of homotopy principles and their applications we refer to the Smale-Hirsch Immersion Theorem ([Sm] and [H]), the Feit k-mersion Theorem ([F]), the Phillips Submersion Theorem ([P]) and the general theorems due to Gromov ([G1]) and du Plessis ([duP1], [duP2] and [duP3]). Furthermore, we should refer to the homotopy principle on the 1-jet level for fold-maps due to Èliašberg ([E1] and [E2]) (see further references in [G2]). ∗2000 Mathematics Subject Classification. Primary 58K30; Secondary 57R45, 58A20 †Key Words and Phrases: smooth map, singularity, homotopy principle ‡This research was partially supported by Grand-in-Aid for Scientific Research (No. 16540072). http://arxiv.org/abs/0704.0115v2 Let C∞ (N,P ) denote the space consisting of all O-regular maps, N → P equipped with the C∞-topology. Let ΓO(N,P ) denote the space consisting of all continuous sections of the fiber bundle πkN |O(N,P ) : O(N,P ) → N equipped with the compact-open topology. Then there exists a continuous map jO : (N,P ) → ΓO(N,P ) defined by jO(f) = j kf . If the following property (h-P) holds, then we say in this paper that the relative homotopy principle on the existence level holds for O-regular maps. (h-P) Let C be a closed subset of N with ∂N = ∅. Let s be a section in ΓO(N,P ) which has an O-regular map g defined on a neighborhood of C to P , where jkg = s. Then there exists an O-regular map f : N → P such that s and jkf are homotopic relative to a neighborhood of C by a homotopy sλ in ΓO(N,P ) with s0 = s and s1 = j As important applications of [An7, Theorem 0.1] we will prove the following relative homotopy principles in (h-P). Here, Σn−p+1,0(n, p) refers to the space consisting of all fold jets in Jk(n, p). Theorem 1.1 Let n and p be positive integers with n ≧ p ≧ 2 or n < p. Let k be a positive integer with k ≧ n− |n− p|+ 2. Let O(n, p) denote a K-invariant open subspace of Jk(n, p) containing all regular jets such that if n ≧ p ≧ 2, then O(n, p) contains Σn−p+1,0(n, p) at least. Let N and P be connected smooth manifolds of dimensions n and p respectively with ∂N = ∅. Let C be a closed subset of N . Let s be a section in ΓO(N,P ) which has an O-regular map g defined on a neighborhood of C to P , where jkg = s. Then there exists an O-regular map f : N → P such that jkf is homotopic to s relative to a neighborhood of C as sections in ΓO(N,P ). Let ρ be an integer with ρ ≧ 1. Let W kρ denote the subset consisting of all z ∈ Jk(n, p) such that the codimension of Kz in Jk(n, p) is not less than ρ (k may be ∞). Let Okℓ (n, p) denote a K-invariant nonempty open subset of Jk(n, p)\W kℓ+1. By applying Theorem 1.1 we will prove the following theorem. Theorem 1.2 Let ℓ be a positive integer. Let k ≧ max{ℓ+1, n−|n−p|+2} or k = ∞. Let Okℓ (n, p) denote a K-invariant open subspace of J k(n, p) containing all regular jets such that if n ≧ p ≧ 2, then Okℓ (n, p) contains Σ n−p+1,0(n, p) at least. Then the relative homotopy principle in (h-P) holds for Okℓ -regular maps. It is well known that any smooth map f : N → P is homotopic to a smooth map g : N → P such that j∞x g is of finite K-codimension for any x ∈ N (see, for example, [W, Theorem 5.1]). There have been described many important applications of the homotopy principles in [G2]. We only refer to the recent applications of the relative ho- motopy principle on the existence level to the problems in topology such as the elimination of singularities and the existence of Okl -regular maps in [An1-7] and [Sa] and the relation between the stable homotopy groups of spheres and higher singularities in [An4]. Let P be a closed manifold of dimension p. Let h(P ) denote the group of all homotopy classes of homotopy equivalences of P . Let hℓ(P ) denote the subset of h(P ) which consists of all homotopy classes of maps which are homotopic to O∞l -regular homotopy equivalences. In particular, h0(P ) is the subset of all homotopy classes of maps which are homotopic to diffeomorphisms of P . In this paper we will prove that the following filtration h0(P ) ⊂ h1(P ) ⊂ · · · ⊂ hℓ(P ) ⊂ · · · ⊂ h(P ). (1.1) is never trivial in general. Theorem 1.3 For a given positive integer d, there exists a closed oriented p- manifold P and a sequence of positive integers ℓ1, ℓ2, · · · , ℓd with ℓj < ℓj+1 for 1 ≤ j < d such that h0(P ) & hℓ1(P ) & hℓ2(P ) & · · · & hℓd(P ) & h(P ). In Section 2 we will review the results on the Boardman manifolds and the fundamental properties of K-equivalence and K-determinacy which are neces- sary in this paper. In Section 3 we will recall [An7, Theorem 0.1] and apply it in the proofs of Theorems 1.1 and 1.2. In Section 4 we will study the nonexistence problem of Okl -regular maps. In Section 5 we will study the filtration in (1.1) and prove Theorem 1.3. 2 Boardman manifolds and K-orbits Throughout the paper all manifolds are Hausdorff, paracompact and smooth of class C∞. Maps are basically smooth (of class C∞) unless otherwise stated. For a Boardman symbol (simply symbol) I = (i1, · · · , ik) with n ≧ i1 ≧ · · · ≧ ik ≧ 0, let Σ I(n, p) denote the Boardman manifold of symbol I in Jk(n, p) which has been defined in [T], [L], [Bo] and [MaTB]. Let An = R[[x1, · · · , xn]] denote the formal power series of algebra on variables x1, · · · , xn. Letmn be its maximal ideal and An(k) = An/m n . Let z = j 0 f ∈ J k(n, p) where f = (f1, · · · , fp) : (Rn, 0) → (Rp, 0). We define I(z) to be the ideal in An(k) generated by the image in An(k) of the Taylor expansions of f 1, · · · , fp. It has been proved in [Bo] and [MaTB] that the Boardman symbol I(z) of z depends only on the ideal I(z) by the notion of the Jacobian extension. Let ΣI(N,P ) denote the subbundle of Jk(N,P ) overN×P associated to ΣI(n, p). Let ΣIx,y(N,P ) denote the fiber of ΣI(N,P ) over (x, y) ∈ N × P . Since codimΣi1(n, p) = (p−n+ i1)i1, the following proposition follows from [An6, Remark 2.1], which has been proved by using the results in [Bo, Section Proposition 2.1 Let I = (i1, · · · , iℓ) be a symbol such that i1 ≧ max{n− p+ 1, 1} and ΣI(n, p) is nonempty. Then we have codimΣI(n, p) ≧ (p− n+ i1)i1 + (1/2)Σ j=2 ij(ij + 1). In particular, if iℓ > 0, then we have codimΣ I(n, p) ≧ |n− p|+ ℓ. Let ΩI(n, p) denote the union of all Boardman manifolds ΣJ (N,P ) with J ≤ I in the lexicographic order. We have the following lemma (see [duP1]). Lemma 2.2 The space ΩI(n, p) is open in Jk(n, p). Let us review the K-equivalence of two smooth map germs f, g : (N, x) → (P, y), which has been introduced in [MaIII, (2.6)], by following [Mart, II, 1]. We say that the above two map germs f and g are K-equivalent if there exists a smooth map germ φ : (N, x) → GL(Rp) and a local diffeomorphism h : (N, x) → (N, x) such that f(x) = φ(x)g(h(x)). It is known that this K-equivalence is nothing but the contact equivalence introduced in [MaIII]. The contact group K is defined as a certain subgroup of the group of germs of local diffeomorphisms (N, x)× (P, y) and acts on Jkx,y(N,P ). For a k-jet z in J x,y(N,P ) let Kz denote the orbit of K through z. As is well known, Kz is an orbit of a Lie group. Hence, Kz is a submanifold of Jkx,y(N,P ). This fact is also observed from the above definition. The following lemma is important in this paper. Lemma 2.3 The Boardman manifold ΣIx,y(N,P ) in J x,y(N,P ) is invariant with respect to the action of K. Proof. Let z = jkxf and w = j xg be k-jets in J x,y(N,P ) such that two map germs f and g are K-equivalent as above. Let h∗ : Cx → Cx be the isomorphism defined by h∗(φ) = φ◦h. By the definition of K-equivalence we have h∗(I(g)) = I(f). The Thom-Boardman symbols of jkxf and j xg are determined by I(f) and I(g), and are the same by [MaTB, 2, Corollary]. This proves the assertion. Let us review the results in [MaIII], [MaIV] and [MaV] which are necessary in this paper. Let C∞(N, x) and C∞(P, y) denote the rings of smooth function germs on (N, x) and (P, y) respectively. Let mx and my denote their maximal ideals respectively. Let f : (N, x) → (P, y) be a germ of a smooth map. Let f∗ : C∞(P, y) → C∞(N, x) denote the homomorphism defined by f∗(a) = a◦f . Let θ(N)x denote the C ∞(N, x)-module of all germs at x of smooth vector fields on (N, x). We define θ(P )y similarly for y ∈ P . Let θ(f)x denote the C ∞(N, x)- module of germs at x of smooth vector fields along f , namely which consists of all smooth germs ς : (N, x) → TP such that pP ◦ ς = f . Here, pP : TP → P is the canonical projection. Then we have the homomorphisms tf : θ(N)x → θ(f)x (2.1) defined by tf(uN) = df ◦ uN for uN ∈ θ(N)x. For a singular jet z = j 0 f ∈ Jk(N,P ) there has been defined the isomorphism x,y(N,P )) −→ mxθ(f)x/m x θ(f)x (2.2) in [MaIII, (7.3)] such that Tz(Kz) corresponds to tf(mxθ(N)x)+f ∗(my)(θ(f)x) modulo mk+1x θ(f)x. We do not here explain the definition. According to [MaIII] we define d(f,K) to be dimmxθ(f)x/(tf(mxθ(N)x) + f ∗(my)(θ(f)x)), (2.3) which is equal to codimKz. 3 Proofs of Theorems 1.1 and 1.2. In this section we prove Theorems 1.1 and 1.2. Let k be a positive integer. Let W kρ = W ρ (n, p) denote the subset consisting of all z ∈ Jk(n, p) such that the codimension of Kz in Jk(n, p) is not less than ρ. The following lemma has been observed in [MaV, Section 7 and Proof of Theorem 8.1]. Lemma 3.1 Let ρ be an integer with ρ ≧ 1. Then W kρ is an algebraic subset of Jk(n, p). The order of K-determinacy is estimated by the codimension of a K-orbit as follows. Proposition 3.2 Let k be an integer with k > ρ. Let z = jkf be a singular jet in Jk(n, p)\W kρ+1. Then z is K-k-determined. Proof. It follows from [W, Theorem 1.2 (iii)] that if d =codimKz, then z is K-(d + 1)-determined. Hence, if z ∈ Jk(n, p)\W kρ+1, then d ≤ ρ and z is K-k-determined. We define the bundle homomorphism d : (πkN ) ∗(TN) −→ (πkk−1) ∗(TJk−1(N,P )), (3.1) d1 : (π ∗(TN) −→ (πkP ) ∗(TP ). Let w = jkxf ∈ J x,y(N,P ) and z = π k(w). Then we have j k−1f : (N, x) → (Jk−1(N,P ), z) and d(jk−1f) : TxN → Tz(J k−1(N,P )). We set dz(w,v) = (w, d(j k−1f)(v)) and (d1)z(w,v) = (w, df(v)). Let I ′ be a symbol of length k. Let K(ΣI ) denote the kernel subbundle of (πkN |Σ I′(N,P ))∗(TN) defined by )w = (w,Ker(dxf)). The following theorem follows from the corresponding assertion for the case k = ∞ in [B, (7.7)]. This is very important in the proof of Theorem 1.1. Theorem 3.3 If I ′ = (i1, · · · , ik−2, 0, 0) and I = (i1, · · · , ik−2,0), then we have d(K(ΣI )w) ∩ (π k−1|Σ I′(N,P ))∗(T (ΣI(N,P ))w = {0} for any w ∈ ΣI (N,P ). Let us review a general condition on O(n, p) for the relative homotopy prin- ciple on the existence level in [An7]. We say that a nonempty K-invariant open subset O(n, p) is admissible if O(n, p) consists of all regular jets and a finite number of disjoint K-invariant nonempty submanifolds V i(n, p) of codimension ρi (1 ≤ i ≤ ι) such that the following properties (H-i) to (H-v) are satisfied. (H-i) V i(n, p) consists of singular k-jets of rank ri, namely, V i(n, p) ⊂ Σn−ri(n, p). (H-ii) For each i, the set O(n, p)\{∪ιj=iV j(n, p)} is an open subset. (H-iii) For each i with ρi ≤ n, there exists a K-invariant submanifold V i(n, p)(k−1) of Jk−1(n, p) such that V i(n, p) is open in (πkk−1) −1(V i(n, p)(k−1)). (H-iv) If n ≧ p ≧ 2, then V 1(n, p) = Σn−p+1,0(n, p). Here, Σn−p+1,0(n, p) denotes the Thom-Boardman manifold in Jk(n, p), which consists of K-orbits of fold jets. Let V i(N,P ) denote the subbundle of Jk(N,P ) associated to V i(n, p). Let K(V i) be the kernel bundle in (πkN ) ∗(TN)|V i(N,P ) defined by K(V i)z = (z,Ker(dxf)). (H-v) For each i with ρi ≤ n and any z ∈ V i(N,P ), we have d(K(V i)z) ∩ (π k−1|V i(N,P ))∗(T (V i(N,P )(k−1))z = {0}. (3.2) Then we have proved the following theorem in [An7, Theorem 0.1]. Theorem 3.4 Let k ≧ n − |n − p| + 2. Let n ≧ p ≧ 2 or n < p. Let O(n, p) denote an admissible open subspace of Jk(n, p). Then the relative homotopy principle in (h-P) holds for O-regular maps. We set VI(n, p) = O(n, p) ∩ Σ I(n, p). Let J = (j1, · · · , jk) be a symbol of a singular jet with codimΣ J (n, p) ≤ n. If k ≧ n − |n − p|+ 2, we have by Proposition 2.1 that ik−1 = ik = 0. Indeed, if ik−1 > 0, then codimΣJ (n, p) ≧ |n− p|+ k − 1 ≧ n+ 1. So we set J = (j1, · · · , jk−2, 0, 0), J ∗ = (i1, · · · , ik−2,0) and VJ∗(n, p) (k−1) = πkk−1(O(n, p)) ∩ Σ J∗(n, p). Lemma 3.5 Let J = (j1, · · · , jk−2, 0, 0) and J ∗ = (j1, · · · , jk−2,0) be as above. Then VJ (n, p) is open in (π −1(VJ∗(n, p) (k−1)). Proof. It is evident that ΣJ(n, p) = (πkk−1) −1(ΣJ (n, p)) and O(n, p) ⊂ (πkk−1) −1(πkk−1(O(n, p))). So we have VJ (n, p) ⊂ (π −1(VJ∗(n, p) (k−1)). Since πkk−1 is an open map, we have that VJ (n, p) is an open subset of (π −1(VJ∗(n, p) (k−1)). Let us prove Theorem 1.1. Proof of Theorem 1.1. By Theorem 3.4 it is enough to prove that O(n, p) is admissible. Let J be a symbol of length k. By Lemma 2.3, VJ (n, p) is K- invariant. We have that (H1) O(n, p) is decomposed into a finite union of all VJ (n, p), (H2) For each symbol J , the set O(n, p) ∩ ΩJ(n, p) is an open subset of O(n, p), (H3) VJ (n, p) is open in (π −1(VJ∗(n, p) (k−1)) by lemma 3.5, (H4) If n ≧ p ≧ 2, then O(n, p) ⊃ Σn−p+1,0(n, p) by the assumption, (H5) Property (3.2) holds for VJ (n, p) by Theorem 3.3 and Lemma 3.5. Since O(n, p) satisfies the properties (H1) to (H5), we have proved Theorem We next prove Theorem 1.2. Proof of Theorem 1.2. If ℓ is finite, then it follows from Lemma 3.2 that if k > ℓ, then any k-jet z of Jk(n, p)\W kℓ+1 is K-k-determined and we have (π∞k ) −1(Okℓ (n, p)) = O ℓ (n, p). Therefore, if k ≧ max{ℓ+1, n−|n−p|+2}, then the relative homotopy principle in (h-P) holds for Okℓ -regular maps by Theorem 1.1 and also for O ℓ -regular maps. Corollary 3.6 Under the same assumption of Theorem 1.2, given a map f : N → P is homotopic to an Okℓ -regular map if and only if there exists a section s ∈ Γ (N,P ) such that πkP ◦ s is homotopic to f . Corollary 3.7 Let hℓ(P ) be as in Introduction. Then the homotopy class of a homotopy equivalence f : P → P lies in hℓ(P ) if and only if j ∞f is homotopic to a section in ΓO∞ (N,P ). Here we give two remarks. Remark 3.8 Let W∞ denote the subspace of J∞(n, p) which consists of all jets z such that any smooth map germ f with z = j∞f is not finitely determined. Let (N,P ) is the subbundle of J∞(N,P ) associated to W∞ . It has been proved (see, for example, [W, Theorem 5.1]) that W∞ is not of finite codimension in J∞(n, p). Consequently, the space of all smooth maps f : N → P with j∞f(N) ⊂ J∞(N,P )\W∞ (N,P ) is dense in C∞(N,P ). In other words if N is compact, then a smooth map f : N → P has an integer ℓ such that f is homotopic to an O∞ℓ -regular map. Remark 3.9 It is very important to study the topology of the space W kℓ+1(n, p) and obstructions for finding an Okℓ -regular map. The Thom polynomials related to W kℓ+1(n, p) have been studied in the dimensions n = p ≦ 8 in [O] and [F-R]. 4 Nonexistence theorems In this section we will discuss the nonexistence of Okℓ -regular maps f : N → P . Let W kℓ+1(N,P ) denote the subbundle of J k(N,P ) associated to W kℓ+1(n, p). By the homotopy principle for Okℓ -regular maps in Theorem 1.2, the existence of a section of Jk(N,P )\W kℓ+1(N,P ) over N is equivalent to the existence of an Okℓ -regular map. However, it is not so easy to find obstructions associated to W kℓ+1(N,P ) such as Thom polynomials of W ℓ+1(N,P ), and so we will adopt a method applied in [An1], [I-K] and [duP4] in this section. For k ≧ p+1, let Σ(n, p; k) denote the algebraic subset of all C∞-nonstable k-jets of Jk(n, p) defined in [MaV]. Note that for k′ > k, (πk −1(Σ(n, p; k)) = Σ(n, p; k′). We have proved the following proposition in [An1, Corollary 5.6]. Proposition 4.1 Let k ≧ p+ 1. If (p− n+ i)( i(i+ 1)− p+ n)− i2 ≧ n, then we have that Σi(n, p) ⊂ Σ(n, p; k). In [I-K] the following proposition has been proved, while it has not been stated explicitly and the proof has been given in the context without the details. So we give a sketchy proof. Proposition 4.2 ([I-K]) Let ℓ be a nonnegative integer and k ≧ p+ ℓ+ 1. If (p− n+ i)( i(i+ 1)− p+ n)− i2 ≧ n+ ℓ, then we have that Σi(n, p) ⊂ W kℓ+1(n, p). In particular, if n = p and i2(i−1) ≧ n+ ℓ, then we have that Σi(n, n) ⊂ W kℓ+1(n, n). Proof. Take a jet z in Σi(n, p) such that z = jk0 f . Suppose that z /∈ W ℓ+1, and hence codimKz ≦ ℓ. By [MaIV] there exists a versal unfolding F : (Rn×Rℓ, 0) → (Rp × Rℓ, 0) of f and jk (0,0) F /∈ Σ(n + ℓ, p+ ℓ; k). Here, we note that jk (0,0) of kernel rank i. By the assumption and Proposition 4.1 we have Σi(n+ ℓ, p+ ℓ) ⊂ Σ(n+ ℓ, p+ ℓ; k). This implies jk (0,0) F ∈ Σ(n + ℓ, p+ ℓ; k). This is a contradiction. Hence, z lies in W kℓ+1. We show the following proposition by applying Proposition 4.2. Proposition 4.3 Let ℓ be a nonnegative integer and k ≧ p+ℓ+1. If Σi(n, p) ⊂ W kℓ+1(n, p), then we have that for any positive integer m, Σ i(m + n,m + p) ⊂ W kℓ+1(m+ n,m+ p). Proof. Let z = jk0f ∈ Σ i(m + n,m+ p). Setting α = j10f , we identify α with the homomorphism Rm+n → Rm+p. Let Ker(α)⊥ and Im(α)⊥ be the orthogonal complement of the kernel Ker(α) and the image Im(α) of α respectively. Let L and M be subspaces of Ker(α)⊥ and Im(α) of dimension m such that α maps L onto M isomorphically. Let L⊥ and M⊥ be their orthogonal complements in Ker(α)⊥ and Im(α) respectively. Then α is decomposed as in the following exact sequence. 0 → Ker(α) → L⊕ L⊥ ⊕Ker(α) → M ⊕M⊥ ⊕ Im(α)⊥ → Im(α)⊥ → 0 Let us choose coordinates (u1, · · · , um), (um+1, · · · , um+n−i) and (um+n−i+1, · · · , um+n) of L, L⊥ and Ker(α), and coordinates (y1, · · · , ym), (ym+1, · · · , ym+n−i) and (ym+n−i+1, · · · , ym+p) of M , M⊥ and Im(α)⊥ respectively. Since α maps L onto M isomorphically, there exist the new coordinates (x1, · · · , xm+n) of R m+n such that xj = xj(u1, · · · , um+n) (1 ≤ j ≤ m) and xj = uj (m+ 1 ≤ j ≤ m+ n) and that yj ◦ f(x1, · · · , xm+n) = xj (1 ≤ j ≤ m). (4.1) Setting x = (xm+1, · · · , xm+n), we define the map g : (R n, 0) → (Rp, 0) by yj ◦ g( x) = yj ◦ f(0, · · · , 0, x) (m+ 1 ≤ j ≤ m+ p). Then f is an unfolding of g by (4.1) and g is of kernel rank i at the origin. We next prove by following the argument and the notation used in [MaIV, Section 1] that d(g,K) is equal to d(f,K). Define π : θ(f) → θ(g) by ajtf( j=m+1 j=m+1 ◦ g), where aj ∈ C ∞(Rm+n, 0), a′j ∈ C ∞(Rn, 0) and a′j( x) = aj(0, · · · , 0, x). We note tf(∂/∂xj) = (∂/∂yj) ◦ f + t=m+1(∂yt ◦ f/∂xj)(∂/∂yt) ◦ f (1 ≤ j ≤ m), tf(∂/∂xj) = t=m+1(∂yt ◦ f/∂xj)(∂/∂yt) ◦ f (m+ 1 ≤ j ≤ m+ n), (∂yt ◦ f/∂xj)(0, · · · , 0, x) = (∂yt ◦ g/∂xj)( x) (m+ 1 ≤ t ≤ m+ p). Since yt ◦ f(x1, · · · , xm+n)− yt ◦ f(0, · · · , 0, xubu(x1, · · · , xm+n), for some bj ∈ C ∞(Rm+n, 0), we have ∂yt ◦ f/∂xj − ∂yt ◦ g/∂xj = xu(∂bu/∂xj) (m+ 1 ≤ j ≤ m+ n). Hence, the assertion follows from an elementary calculation under the definition in (2.3). Since jk0 g ∈ Σ i(n, p) ⊂ W kℓ+1(n, p), we have d(g,K) ≧ ℓ+ 1. Hence, we have d(f,K) ≧ ℓ+ 1. This shows z ∈ W kℓ+1(m+ n,m+ p). This is what we want. Let ξ be a stable vector bundle over a space. Let c(Σi, ξ) denote the de- terminant of the (p− n+ i)-matrix whose (s, t)-component is the (i+ s− t)-th Stiefel-Whitney class Wi+s−t(ξ). If n − p and i are even, say n − p = 2u and i = 2v, and if ξ is orientable, then cZ(Σ i, ξ) expresses the determinant of the (v − u)-matrix whose (s, t)-component is the (v + s − t)-th Pontrjagin class Pv+s−t(ξ). Wi · · · Wn−p+1 . . . Wp−n+2i−1 · · · Wi Pv · · · Pu+1 . . . P2v−u−1 · · · Pv Let τX denote the stable tangent bundle of a manifold X . If f : N → P is a smooth map transverse to Σi(N,P ) and ξ = τN − f ∗(τP ), then c(Σ i, ξ) (resp. i, ξ)) is equal to the (resp. integer) Thom polynomial of the topological closure of (jkf)−1(Σi(N,P )) ([Po], [Ro] and see also [An1, Proposition 5.4]). If it does not vanish, then (jkf)−1(Σi(N,P )) cannot be empty by the obstruction theory in [St]. Hence, we have the following corollary of Propositions 4.2 and Corollary 4.4 Let f : M → Q be a smooth map with dimM = m + n and dimQ = m+p. Under the same assumption of Proposition 4.2. we assume that either (i) c(Σi, τM − f ∗(τQ)) does not vanish, or (ii) M and τM −f ∗(τQ) are orientable, n−p and i are even and cZ(Σ i, τM − f∗(τQ)) does not vanish. Then f is not homotopic to any Okℓ -regular map. 5 Homotopy equivalences In this section we will study the filtration in (1.1) in Introduction by applying Corollaries 3.7 and 4.4 and Remark 3.8. Let us first review what is called the Sullivan’s exact sequence in the surgery theory following [M-M] (see also [K-M], [Su] and [Br]). In what follows P is a closed and oriented n-manifold. We define the set S(P ) to be the set of all equivalence classes of homotopy equivalences f : N → P of degree 1 under the following equivalence relation. Let Nj be closed oriented n- manifolds and let fj : Nj → P be homotopy equivalences of degree 1 (j = 1, 2). We say that f1 and f2 are equivalent if there exists an h-cobordism W of N1 and N2 and a homotopy equivalence F : (W,N1 ∪ (−N2)) → (P × [0, 1], P × 0∪ (−P )× 1) of degree 1 such that F |Nj = fj (j = 1, 2). Let O(k) denote the rotation group of Rk and let Gk denote the space of all homotopy equivalence of the (k − 1)-sphere Sk−1 equipped with the compact- open topology. By considering the canonical inclusions O(k) → O(k + 1) and Gk → Gk+1, we set O = limk→∞ O(k) and G = limk→∞ Gk. Let BO and BG denote the classifying spaces for O and G. Then we have the canonical maps π(m) : BO(m) → BG(m) and π : BO → BG, which are regarded as fibrations with fibers G(m)/O(m) and G/O respectively. For a sufficiently large number m, let ηO(m) denote the universal vector bundle over BO(m) and let iG/O : G(m)/O(m) → BO(m) be the inclusion of a fiber. Set ηG/O = (iG/O) ∗ηO(m). Then ηG/O has a trivialization tG/O : ηG/O → R m as a spherical fibration. We next recall the surgery obstruction sP4q : [P,G/O] → Z only in the case of n = 4q. For [α] ∈ [P,G/O] let η = α∗(ηG/O) with the canonical bundle map α : η → ηG/O covering α and the projection πη onto P . We deform tG/O ◦ α to a map transverse to 0 ∈ Rm and let M be the inverse image of 0 with a map πη|M : M → P of degree 1. We define s 4q([α]) = (1/8)(σ(M) − σ(P )). If P is simply connected in addition, then there have been defined an injection jP : S(P ) → [P,G/O] such that if sP4q([α]) = 0, πη|M is deformed to a homotopy equivalence f : N → P of degree 1 under a certain cobordism. The following is the Sullivan’s exact sequence. 0 −→ S(P ) −→ [P,G/O] Let us recall the cobordism group Ωh−eqn of homotopy equivalences of degree 1 in [An5]. Let Nj and Pj be oriented closed n-manifolds and let fj : Nj → Pj be homotopy equivalences of degree 1 (j = 1, 2). We say that f1 and f2 are cobordant if there exists an oriented (n + 1)-manifold W , V and a homotopy equivalence F : (W,∂W ) → (V, ∂V ) of degree 1 such that ∂W = N1 ∪ (−N2), ∂V = P1 ∪ (−P2) and F |Nj = fj . The cobordism class of f : N → P is denoted by [f : N → P ]. Let Ωh−eqn denote the set which consists of all cobordism classes of homotopy equivalences of degree 1. We provide Ωh−eqn with a module structure by setting • [f1 : N1 → P1] + [f2 : N2 → P2] = [f1 ∪ f2 : N1 ∪N2 → P1 ∪ P2], • −[f : N → P ] = [f : (−N) → (−P )]. The null element is defined to be [f : N → P ] which bound a homotopy equiv- alence F : (W,∂W ) → (V, ∂V ) of degree 1 such that ∂W = N , ∂V = P and F |N = f . Even if P is not simply connected, we can find f1 : N1 → P1 with P1 being simply connected in the same cobordism class by killing π1(N) ≈ π1(P ) by usual surgery. Let cQ(Σ 2i, ηG/O) denote the image of cZ(Σ 2i, ηG/O) in H 4i2(G/O;Q). Let α = jP ([f : N → P ]) and let cP : P → BSO be a classifying map of the tangent bundle TP of P . Then it induces the homomorphism C2i : Ω H4q−4i2(G/O;Q) defined by C2i([f : N → P ]) = cQ(Σ 2i, ηG/O) ∩ α([P ]) = cQ(Σ 2i, ηG/O)⊗ 1 ∩ (α × cP )∗([P ]), under the identification H4q−4i2 (G/O;Q) = H4q−4i2(G/O;Q)⊗ 1 j=0 H4j(G/O;Q)⊗H4q−4i2−4j(BSO;Q). We have that C2i(α) = cQ(Σ 2i, ηG/O) ∩ (α)∗([P ]) = cQ(Σ 2i, ηG/O) ∩ (α ◦ f)∗([N ]) = (α ◦ f)∗((α ◦ f) ∗(cQ(Σ 2i, ηG/O)) ∩ [N ]) = (α ◦ f)∗(cZ(Σ 2i, τN − f ∗(τP )) ∩ [N ]). Furthermore, we have proved in [An5, Theorems 3.2 and 4.1] that for integers q and i with q ≧ i2 ≧ 1, 4q /(Ω 4q ∩Ker(C2i))⊗ Q = dimH4q−4i2 (BSO;Q). (5.1) The following theorem follows from (5.1), Proposition 4.2 and Corollary 4.4. Theorem 5.1 Let ℓ, q and i be integers with ℓ ≧ 0 and q ≧ i2. Let k ≧ 4q + ℓ + 1. There exists a cobordism class [f : N → P ] ∈ Ω 4q such that 2i, τN − f ∗(τP )) is not a torsion element and that if 4i 3 − 2i2 ≧ 4q + ℓ ≧ 4i2 + ℓ, then f is not cobordant in Ω 4q to any O ℓ -regular map. We can prove the following theorem using Theorem 5.1 by applying the same argument in the proof of [An5, Theorem 0.2]. However, Theorem 1.2 is very important in the following and the situation is rather different. Therefore, we give its proof. Theorem 5.2 Let ℓ, q and i be given integers with ℓ ≧ 0 and q ≧ i2. Let k ≧ 8q + ℓ + 1. If 4i3 − 2i2 ≧ 4q + ℓ ≧ 4i2 + ℓ, then there exists a closed connected oriented 8q-manifold P and a homotopy equivalence f : P → P of degree 1 such that cZ(Σ 2i, τP −f ∗(τP )) 6= 0 and that f is not cobordant in Ω to any Okℓ -regular homotopy equivalence of degree 1. Proof. It follows from Theorem 5.1 that there exists a homotopy equivalence f : N → P of degree 1 between 4q-manifolds such that cZ(Σ 2i, τN − f ∗(τP )) is not a torsion element. Let f−1 : P → N be a homotopy inverse of f . Define g : N×P → N×P by g(x, y) = (f−1(y), f(x)). We have k ≥ dimN×P + ℓ+1. If we prove that cZ(Σ 2i, τN×P − g ∗(τN×P )) does not vanish, then, by Corollary 4.4, g is not homotopic to any Okℓ -regular map. We set ξ = τN×P −g ∗(τN×P ) = τN × τP − f ∗(τP )× (f −1)∗(τN ). Then pj(ξ) = s+t=j ps(τN × τP )pt(f ∗(τP )× (f −1)∗(τN )) s+t=j s1 + s2 = s t1 + t2 = t ps1(τN )pt1(f ∗(τP ))⊗ ps2(τP )pt2((f −1)∗(τN )) modulo torsion in H∗(N ;Z) ⊗ H∗(P ;Z). The term of pj(ξ) which lies in H4j(N ;Z)⊗H0(P ;Z) is equal modulo torsion to s+t=j ps(τN )pt(f ∗(τP ))⊗ 1 = pj(τN − f ∗(τP ))⊗ 1. Hence, we have that cZ(Σ 2i, τN×P−g ∗(τN×P ) is equal to the sum of cZ(Σ 2i, τN− f∗(τP )) ⊗ 1 and the other term which lies in Σ 4i2−4j(N ;Z) ⊗ H4j(P ;Z) modulo torsion. Since cZ(Σ 2i, τN − f ∗(τP )) does not vanish, it follows that 2i, τN×P − g ∗(τN×P )) does not vanish. This completes the proof. We are now ready to prove Theorem 1.3. Proof of Theorem 1.3. In the proof k refers to a sufficiently large integer. Let i0 = 2, which is the smallest integer such that 4i 3 − 2i2 ≧ 4i2 with q = 4 and ℓ = 8. Then we have, by Theorem 5.2, a closed connected oriented 8 · 4- manifold P0 and a homotopy equivalence f0 : P0 → P0 of degree 1 such that 4, τP0 − f 0 (τP0)) 6= 0 and that f0 is not homotopic to any O 8 -regular map. By Remark 3.8 there exists an integer ℓ such that f0 is homotopic to an O regular map. Let ℓ1 be such a smallest integer. We assume the following (A-t) for an integer t ≧ 0, where ℓ0 = 8. (A-t) We have constructed integers ℓt, ℓt+1, it, a closed oriented 8·i t -manifold Pt and an O -regular homotopy equivalence ft : Pt → Pt of degree 1 such that 4i3t − 2i t ≧ 4i t + ℓt, ℓt+1 > ℓt, cZ(Σ 2it , τPt − f t (τPt)) 6= 0 and that ft is not homotopic to any Okℓt-regular map. Under the assumption (A-t) we prove (A-(t+ 1)) with ℓt+1 < ℓt+2. Let it+1 be the smallest integer among the integers i > 0 with 4i3 − 2i2 ≧ 4i2 + ℓt+1. Then it follows from Theorem 5.2 that there exist a closed connected oriented 8 · i2t+1-manifold Pt+1 and a homotopy equivalence ft+1 : Pt+1 → Pt+1 of degree 1 such that cZ(Σ 2it , τPt+1 − f t+1(τPt+1)) 6= 0 and that ft+1 is not homotopic to any Okℓt+1-regular map. It follows Remark 3.8 that there exists an integer ℓ such that ft+1 is homotopic to an O ℓ -regular map. Let ℓt+2 be the smallest integer among those integers ℓ. Hence, we have ℓt+2 > ℓt+1. This proves (A-(t+ 1)). Thus we have defined the sequences {it}, {ℓt}, closed connected oriented manifolds {Pt} of dimensions {8 · i t} and homotopy equivalences {ft} of degree 1 which satisfy the above properties. Given a positive integer d, let P = P0 × P1 × P2 × · · · × Pd, Ft = idP0 × · · · × idPt−1 × ft × idPt+1 × · · · × idPd (0 ≦ t ≦ d), and p = t=0 8 · i t . We show that Ft /∈ hℓt(P ) and Ft ∈ hℓt+1(P ). Let qt : P → Pt be the canonical projection. Then the stable tangent bundle τP is isomorphic to q∗0(τP0)⊕ q 1(τP1 )⊕ · · · ⊕ q d(τPd). Hence, τP − F t (τP ) is equal to q∗0(τP0)⊕ q 1(τP1)⊕ · · · ⊕ q d(τPd) − ((q0 ◦ Ft) ∗(τP0 )⊕ (q1 ◦ Ft) ∗(τP1)⊕ · · · ⊕ (qd ◦ Ft) ∗(τPd)) = q∗0(τP0 )⊕ q 1(τP1)⊕ · · · ⊕ q d(τPd) − (q∗0(τP0 )⊕ · · · ⊕ q t−1(τPt−1 )⊕ (ft ◦ qt) ∗(τPt)⊕ · · · ⊕ q d(τPd)) = q∗t (τPt)− (ft ◦ qt) ∗(τPt) = q∗t ((τPt)− f t (τPt)). This shows that 2it , τP − F t (τP )) = cZ(Σ 2it , q∗t ((τPt)− f t (τPt)) = q∗t (cZ(Σ 2it , τPt − f t (τPt)), which does not vanish in H2i t (P ;Z) since cZ(Σ 2it , τPt − f t (τPt)) 6= 0 and since q∗t : H t (Pt;Z) → H t (P ;Z) is injective. Furthermore, it follows from Propo- sition 4.3 that Σ2it(p, p) ⊂ W kℓ+1(p, p) and from Corollary 4.4 that Ft is not homotopic to any Okℓt -regular map. However, since ft is homotopic to an O regular map, Ft is also homotopic to an O -regular map. This proves the theorem. We prepare further results which are necessary to study the filtration in (1.1). The assertions (i) and (ii) in the following theorem have been proved in [An2, Theorem 4.8] and [An4, Theorem 4.1] respectively, which are applications of the relative homotopy principles for O-regular maps. Theorem 5.3 Let P be orientable and f : P → P be a smooth map. (i) A map f is homotopic to a fold-map if and only if τP is isomorphic to f∗(τP ). (ii) If a map f is Ω1-regular, then f is homotopic to an Ω(1,1,0)-regular map. Let V (n, p) be an algebraic set of Jk(n, p) which is invariant with respect to the actions of local diffeomorphisms of (Rn, 0) and (Rn, 0) and Let V (N,P ) be the subbundle of Jk(N,P ) associated to V (n, p). By [B-H] we have the fundamental class of V (N,P ) under the coefficient group Z/2, and have the Thom polynomial c(V (n, p), τN − f ∗(τP )) of V (N,P ). Theorem 5.4 Let V (p, p) be as above. Let P be orientable and f : P → P be a smooth map. (i) If f is a homotopy equivalence, then c(V (p, p), τP − f ∗(τP )) vanishes. (ii) cZ(W p (p, p), τP − f ∗(τP )) = 0 for p = 5, 6, 7 and 8 (8, 8), τP − f ∗(τP )) = 9P2(τP − f ∗(τP )) + 3P 1 (τP − f ∗(τP )) for p = 8. (iii) Let 2 ≦ p ≦ 8. Then there exists a section s of Okp−1(P, P ) over P with πkP ◦ s and f being homotopic if and only if cZ(W p (p, p), τP − f ∗(τP )) = 0. Proof. (i) Let S(νP ) denote the spherical normal fiber space of P . It follows from [Sp] that S(νP ) is equivalent to f ∗(S(νP )). Hence, the associated spherical spaces of τP and f ∗(τP ) are equivalent. In particular, the Stiefel-Whitney classes of τP − f ∗(τP ) vanish. (ii) If p ≦ 8, then a map f : P → P is homotopic to a smooth map with only K-simple singularities by [MaVI]. According to [F-R], the integer Thom polynomial ofW kp (p, p) is equal to the formula for p = 8 and vanish for p = 5, 6, 7 in Hp(P ;Z) ≈ Z. (iii) It follows from the relative homotopy principle for Okp−1-regular maps P → P that the primary obstruction in Hp(P ;πp−1(O p−1(p, p)) is the unique obstruction for finding the required section. By an elementary argument we πp−1(O p−1(p, p)) ≈ Hp−1(O p−1(p, p);Z) ≈ H dimWk (p,p)(W kp (p, p);Z). This shows the assertion. Finally we study the filtration in (1.1) in the case of P being orientable and p ≦ 8 by applying the homotopy principles in Theorems 1.2 and 5.3. We have hp(P ) = h(P ). Examples. Case: p ≦ 3; h0(P ) ⊂ h1(P ) = h(P ). Since P is parallelizable, TP and f∗(TP ) are trivial. So a map f : P → P is homotopic to a fold-map. We refer the reader to [Ru, 1]. Case: p = 4; h0(P ) ⊂ h1(P ) ⊂ h2(P ) = h3(P ) ⊂ h4(P ). It is known that cZ(Σ 4; τP − f ∗(τP )) = P2(τP − f ∗(τP )). If this class van- ish, then there exists a section P → Ω1(P, P ) covering f , and hence an Ω1- regular map by [F]. By Theorems 5.3 and 5.4 we obtain an Ω(1,1,0)-regular map homotopic to f . 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W. Milnor, Groups of homotopy spheres: I, Ann. Math. 77(1963), 504-537. [L] H. I. Levine, Singularities of differentiable maps, Proc. Liverpool Singu- larities Symposium, I, Springer Lecture Notes in Math. Vol. 192, 1-85, Springer-Verlag, Berlin, 1971. [M-M] I. Madsen and R. J. Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, Ann. Math. Studies 92, Princeton Univ. Press, Princeton, 1979. [Mart] J. Martinet, Déploiements versels des applications différentiables et clas- sification des applications stables, Springer Lecture Notes in Math. Vol. 535, 1-44, Spribger-Verlag, Berlin, 1976. [MaIII] J. N. Mather, Stability of C∞ mappings, III: Finitely determined map- germs, Publ. Math. Inst. Hautes Étud. Sci. 35(1968), 127-156. [MaIV] J. N. Mather, Stability of C∞ mappings, IV: Classification of stable germs by R-algebra, Publ. Math. Inst. Hautes Étud. Sci. 37(1970), 223- [MaV] J. N. Mather, Stability of C∞ mappings: V, Transversality, Adv. Math. 4(1970), 301-336. [MaTB] J. N. Mather, On Thom-Boardman singularities, Dynamical Systems, Academic Press, 1973, 233-248. [O] T. Ohmoto, Vassiliev complex for contact classes of real smooth map- germs, Res. Fac. Sci. Kagoshima Univ. 27(1994), 1-12. [Ph] A. Phillips, Submersions of open manifolds, Topology 6(1967), 171-206. [Po] I. R. Porteous, Simple singularities of maps, Proc. Liverpool Singulari- ties Symp. I, Springer Lecture Notes in Math. 192(1971), 286-307. [Ro] F. Ronga, Le calcul de la classe de cohomologie entière dual a Σk, Proc. Liverpool Singularities Symp. I, Springer Lecture Notes in Math. 192(1971), 313-315. [Ru] J. W. Rutter, Homotopy self-equivalences 1988-1999, Contemporary Math. 274(2001), 1-11. [Sa] O. Saeki, Fold maps on 4-manifolds, Comment. Math. Helv., 78(2003), 627-647. [Sm] S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. Math. 327-344, 69(1969). [Sp] M. Spivak, Spaces satisfying Poincaré duality, Topology 6(1969), 77- [St] N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951. [Su] D. Sullivan, Triangulating homotopy equivalences, Thesis, Princeton Univ., 1965. [T] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier 6(1955-56), 43-87. [W] C. T. C. Wall, Finite determinacy of smooth map germs, Bull. London Math. Soc. 13(1981), 481-539. Department of Mathematical Sciences Faculty of Science, Yamaguchi University Yamaguchi 753-8512, Japan E-mail: [email protected] Introduction Boardman manifolds and K-orbits Proofs of Theorems 1.1 and 1.2. Nonexistence theorems Homotopy equivalences

0704.0116

Stringy Jacobi fields in Morse theory

arXiv:0704.0116v2 [math-ph] 21 May 2007 Stringy Jacobi fields in Morse theory Yong Seung Cho∗ National Institute for Mathematical Sciences, 385-16 Doryong, Yuseong, Daejeon 305-340 Korea and Department of Mathematics, Ewha Womans University, Seoul 120-750 Korea Soon-Tae Hong† Department of Science Education and Research Institute for Basic Sciences, Ewha Womans University, Seoul 120-750 Korea (Dated: November 4, 2018) We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particles to discuss conjugate strings on the geodesic surface. PACS numbers: 02.40.-k, 04.20.-q, 04.90.+e, 11.25.-w, 11.40.-q Keywords: Nambu-Goto string action, geodesic surface, Jacobi field, index of geodesic surface, conjugate strings I. INTRODUCTION It is well known that string theory [1, 2] is one of the best candidates for a consistent quantum theory of grav- ity to yield a unification theory of all the four basic forces in nature. In D-brane models [2], closed strings represent gravitons propagating on a curved manifold, while open strings describe gauge bosons such as photons, or mat- ter attached on the D-branes. Moreover, because the two ends of an open string can always meet and con- nect, forming a closed string, there are no string theories without closed strings. On the other hand, the supersymmetric quantum me- chanics has been exploited by Witten [3] to discuss the Morse inequalities [4, 5, 6]. The Morse indices for pair of critical points of the symplectic action function have been also investigated based on the spectral flow of the Hes- sian of the symplectic function [7], and on the Hilbert spaces the Morse homology [8] has been considered to discuss the critical points associated with the Morse in- dex [9]. The string topology was initiated in the seminal work of Chas and Sullivan [10]. Using the Morse theoretic techniques, Cohen in Ref. [11] constructs string topology operations on the loop space of a manifold and relates the string topology operations to the counting of pseudo- holomorphic curves in the cotangent bundle. He also speculates the relation between the Gromov-Witten in- variant [12] of the cotangent bundle and the string topol- ogy of the underlying manifold. Recently, the Jacobi fields and their eigenvalues of the Sturm-Liouville oper- ator associated with the particle geodesics on a curved manifold have been investigated [13], to relate the phase factor of the partition function to the eta invariant of Atiyah [14, 15]. In this paper, we will exploit the Nambu-Goto string ∗Electronic address: [email protected] †Electronic address: [email protected] action to investigate the geodesic surface equation and the geodesic surface deviation equation associated with a Jacobi field. The index form of a geodesic surface will be also discussed for the closed strings on the curved manifold. In Section II, the string action will be introduced to investigate the geodesic surface equation in terms of the world sheet currents associated with τ and σ world sheet coordinate directions. By taking the second variation of the surface spanned by closed strings, the geodesic sur- face deviation equation will be discussed for the closed strings on the curved manifold. In Section III, exploiting the orthonormal gauge, the index form of a geodesic sur- face will be also investigated together with breaks on the string tubes. The geodesic surface deviation equation in the orthonormal gauge will be exploited to discuss the Jacobi field on the geodesic surface. II. STRINGY GEODESIC SURFACES IN MORSE THEORY In analogy of the relativistic action of a point parti- cle, the action for a string is proportional to the area of the surface spanned in spacetime manifold M by the evolution of the string. In order to define the action on the curved manifold, let (M, gab) be a n-dimensional manifold associated with the metric gab. Given gab, we can have a unique covariant derivative ∇a satisfying [6] ∇agbc = 0, ∇aω b = ∂aω b + Γbac ω c and (∇a∇b −∇b∇a)ωc = R abc ωd. (2.1) We parameterize the closed string by two world sheet coordinates τ and σ, and then we have the correspond- ing vector fields ξa = (∂/∂τ)a and ζa = (∂/∂σ)a. The Nambu-Goto string action is then given by [1, 2, 16] S = − dτdσf(τ, σ) (2.2) http://arxiv.org/abs/0704.0116v2 where the coordinates τ and σ have ranges 0 ≤ τ ≤ T and 0 ≤ σ ≤ 2π respectively and f(τ, σ) = [(ξ · ζ)2 − (ξ · ξ)(ζ · ζ)]1/2. (2.3) We now perform an infinitesimal variation of the tubes γα(τ, σ) traced by the closed string during its evolution in order to find the geodesic surface equation from the least action principle. Here we impose the restriction that the length of the string circumference is τ independent. Let the vector field ηa = (∂/∂α)a be the deviation vector which represents the displacement to an infinitesimally nearby tube, and let Σ denote the three-dimensional sub- manifold spanned by the tubes γα(τ, σ). We then may choose τ , σ and α as coordinates of Σ to yield the com- mutator relations, a = ξb∇bη a − ηb∇bξ a = 0, a = ζb∇bη a − ηb∇bζ a = 0, a = ξb∇bζ a − ζb∇bξ a = 0. (2.4) Now we find the first variation as follows [17] dτdσ ηb(ξ τ + ζ dσ P bτ ηb| τ=0 − dτ P bσηb| σ=0 , (2.5) where the world sheet currents associated with τ and σ directions are respectively given by [17] P aτ = [(ξ · ζ)ζa − (ζ · ζ)ξa], P aσ = [(ξ · ζ)ξa − (ξ · ξ)ζa]. (2.6) Using the endpoint conditions ηa(0) = ηa(T ) = 0 and pe- riodic condition ηa(σ+2π) = ηa(σ), we have the geodesic surface equation [17] ξa∇aP τ + ζ σ = 0, (2.7) and the constraint identities [17] Pτ · ζ = 0, Pτ · Pτ + ζ · ζ = 0, Pσ · ξ = 0, Pσ · Pσ + ξ · ξ = 0. (2.8) Let γα(τ, σ) denote a smooth one-parameter family of geodesic surfaces: for each α ∈ R, the tube γα is a geodesic surface parameterized by affine parameters τ and σ. For an infinitesimally nearby geodesic surface in the family, we then have the following geodesic surface deviation equation ξb∇b(η τ ) + ζ b∇b(η +R abcd (ξ bP dτ + ζ bP dσ )η c ≡ (Λη)a = 0. (2.9) For a small variation ηa, our goal is to compare S(α) with S(0) of the string. The second variation d2S/dα2(0) is then needed only when dS/dα(0) = 0. Explicitly, the second variation is given by |α=0 = − (ηc∇cP τ )(ξ a∇aηb) +(ηc∇cP a∇aηb)−R acb (ξ aP bτ + ζ aP bσ)η dσ P bτ η a∇aηb| τ=0 − dτ P bση a∇aηb| σ=0 . (2.10) Here the boundary terms vanish for the fixed endpoint and the periodic conditions, even though on the geodesic surface we have breaks which we will explain later. After some algebra using the geodesic surface deviation equa- tion, we have |α=0 = dτdσ ηa(Λη) a. (2.11) III. JACOBI FIELDS IN ORTHONORMAL GAUGE The string action and the corresponding equations of motion are invariant under reparameterization σ̃ = σ̃(τ, σ) and τ̃ = τ̃ (τ, σ). We have then gauge degrees of freedom so that we can choose the orthonormal gauge as follows [17] ξ · ζ = 0, ξ · ξ + ζ · ζ = 0, (3.1) where the plus sign in the second equation is due to the fact that ξ·ξ is timelike and ζ·ζ is spacelike. Note that the gauge fixing (3.1) for the world sheet coordinates means that the tangent vectors are orthonormal everywhere up to a local scale factor [17]. In this parameterization the world sheet currents (2.6) satisfying the constraints (2.8) are of the form P aτ = −ξ a, P aσ = ζ a. (3.2) The geodesic surface equation and the geodesic surface deviation equation read − ξa∇aξ b + ζa∇aζ b = 0, (3.3) −ξb∇b(ξ a) + ζb∇b(ζ −R abcd (ξ bξd − ζbζd)ηc = (Λη)a = 0. (3.4) We now restrict ourselves to strings on constant scalar curvature manifold such as Sn. We take an ansatz that on this manifold the string shape on the geodesic surface γ0 is the same as that on a nearby geodesic surface γα at a given time τ . We can thus construct the variation vectors ηa(τ) as vectors associated with the centers of the string of the two nearby geodesic surfaces at the given time τ . We then introduce an orthonormal basis of spatial vectors eai (i = 1, 2, ..., n−2) orthogonal to ξ a and ζa and parallelly propagated along the geodesic surface. The geodesic surface deviation equation (3.4) then yields for i, j = 1, 2, ..., n− 2 + (R iτjτ −R σjσ)η j = 0. (3.5) The value of ηi at time τ must depend linearly on the initial data ηi(0) and dη (0) at τ = 0. Since by con- struction ηi(0) = 0 for the family of geodesic surfaces, we must have ηi(τ) = Aij(τ) (0). (3.6) Inserting (3.6) into (3.5) we have the differential equation for Aij(τ) d2Aij + (R iτkτ −R σkσ)A j = 0, (3.7) with the initial conditions Aij(0) = 0, (0) = δij . (3.8) Note that in (3.7) we have the last term originated from the contribution of string property. Next we consider the second variation equation (2.10) under the above restrictions |α=0 = − (R iτjτ −R σjσ)η (3.9) We define the index form Iγ of a geodesic surface γ as the unique symmetric bilinear form Iγ : Tγ × Tγ → R such that Iγ(V, V ) = |α=0 (3.10) for V ∈ Tγ . From (3.9) we can easily find Iγ(V,W ) = −(R mτjτ −R σjσ )W . (3.11) If we have breaks 0 = τ0 < · · · < τk+1 = T , and the restriction of γ to each set [τi−1, τi] is smooth, then the tube γ is piecewise smooth. The variation vector field V of γ is always piecewise smooth. However dV/dτ will generally have a discontinuity at each break τi (1 ≤ i ≤ k). This discontinuity is measured by (τi) = (τ+i )− (τ−i ), (3.12) where the first term derives from the restrictions γ|[τi, τi+1] and the second from γ|[τi−1, τi]. If γ and V ∈ Tγ have the breaks τ1 < · · · < τk, we have ∫ τi+1 dτ = − (3.13) to yield Iγ(V,W ) = − dτdσ V m (3.14) +(R mτjτ −R σjσ )W dσ Vm∆ (τi). (3.15) Here note that if we do not have the breaks, (3.9) yields |α=0 = − dτdσ ηi + (R iτjτ −R σjσ)η (3.16) A solution ηa of the geodesic surface deviation equation (3.5) is called a Jacobi field on the geodesic surface γ. A pair of strings p, q ⊂ γ defined by the centers of the closed strings on the geodesic surface is then conjugate if there exists a Jacobi field ηa which is not identically zero but vanishes at both strings p and q. Roughly speaking, p and q are conjugate if an infinitesimally nearby geodesic surface intersects γ at both p and q. From (3.6), q will be conjugate to p if and only if there exists nontrivial initial data: dηi/dτ(0) 6= 0, for which ηi = 0 at q. This occurs if and only if detAij = 0 at q, and thus detA j = 0 is the necessary and sufficient condition for a conjugate string to p. Note that between conjugate strings, we have detAij 6= 0 and thus the inverse of A j exists. Using (3.7) we can easily see that Aik −Aij = 0. (3.17) In addition, the quantity in parenthesis of (3.17) vanishes at p, since Aij(0) = 0. Along a geodesic surface γ, we thus find Aik −Aij = 0. (3.18) If γ is a geodesic surface with no string conjugate to p between p and q, then Aij defined above will be nonsingu- lar between p and q. We can then define Y i = (A−1)ijη or ηi = AijY j . From (3.16) and (3.18), we can easily verify |α=0 = ≥ 0. (3.19) Locally γ minimizes the Nambu-Goto string action, if γ is a geodesic surface with no string conjugate to p between p and q. On the other hand, if γ is a geodesic surface but has a conjugate string r between strings p and q, then we have a non-zero Jacobi field J i along γ which vanishes at p and r. Extend J i to q by putting it zero in [r, q]. Then dJ i/dτ(r−) 6= 0, since J i is nonzero. But dJ i/dτ(r+) = 0 to yield (r) = − (r−) 6= 0. (3.20) We choose any ki ∈ Tγ such that (r) = c, (3.21) with a positive constant c. Let ηi be ηi = ǫki + ǫ−1J i where ǫ is some constant, then we have Iγ(η, η) = ǫ 2Iγ(k, k) + 2Iγ(k, J) + ǫ −2Iγ(J, J). (3.22) By taking ǫ small enough, the first term in (3.22) vanishes and the third term also vanishes due to the definition of the Jacobi field and (3.15). Substituting (3.21) into (3.15) we have Iγ(k, J) = −2πc and thus |α=0 = −4πc, (3.23) which is negative definite. From the above arguments, we conclude that given a smooth timelike tube γ connecting two strings p, q ⊂ M , the necessary and sufficient con- dition that γ locally minimizes the surface of the closed string tube between p and q over smooth one parameter variations is that γ is a geodesic surface with no string conjugate to p between p and q. It is also interesting to see that on Sn, the first non-minimal geodesic sur- face has n − 1 conjugate strings as in the case of point particle. Moreover, on the Riemannian manifold with the constant sectional curvature K, the geodesic surfaces have no conjugate strings for K < 0 or K = 0, while conjugate strings occur for K > 0 [18]. IV. CONCLUSIONS The Nambu-Goto string action has been introduced to study the geodesic surface equation in terms of the world sheet currents associated with τ and σ directions. By constructing the second variation of the surface spanned by closed strings, the geodesic surface deviation equation has been discussed for the closed strings on the curved manifold. Exploiting the orthonormal gauge, the index form of a geodesic surface has been defined together with breaks on the string tubes. The geodesic surface deviation equa- tion in this orthonormal gauge has been derived to find the Jacobi field on the geodesic surface. Given a smooth timelike tube connecting two strings on the manifold, the condition that the tube locally minimizes the sur- face of the closed string tube between the two strings over smooth one parameter variations has been also dis- cussed in terms of the conjugate strings on the geodesic surface. In the Morse theoretic approach to the string theory, one could consider the physical implications associated with geodesic surface congruences and their expansion, shear and twist. It would be also desirable if the string topology and the Gromov-Witten invariant can be in- vestigated by exploiting the Morse theoretic techniques. These works are in progress and will be reported else- where. Acknowledgments The work of YSC was supported by the Korea Re- search Council of Fundamental Science and Technol- ogy (KRCF), Grant No. C-RESEARCH-2006-11-NIMS, and the work of STH was supported by the Korea Re- search Foundation (MOEHRD), Grant No. KRF-2006- 331-C00071, and by the Korea Research Council of Fun- damental Science and Technology (KRCF), Grant No. C-RESEARCH-2006-11-NIMS. [1] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory Vol. 1 (Cambridge Univ. Press, Cambridge, 1987). [2] J. Polchinski, String Theory Vol. 1 (Cambridge Univ. Press, Cambridge, 1999). [3] E. Witten, J. Diff. Geom. 17, 661 (1982). [4] M. Morse, The Calculus of Variations in the Large (Amer. Math. Soc., New York, 1934). [5] J. Milnor, Morse Theory (Princeton Univ. Press, Prince- ton, 1963). [6] R.M. Wald, General Relativity (The Univ. of Chicago Press, Chicago, 1984). [7] A. Floer, Comm. Pure Appl. Math. 41, 393 (1988). [8] M. Schwarz, Morse Homology, Vol. 111 of Prog. Math. (Birkhäuser, Basel, 1993). [9] A. Abbondandolo, P. Majer, Comm. Pure Appl. Math. 54, 689 (2001). [10] M. Chas and D. Sullivan, String Topology, to appear in Ann. Math., math.GT/9911159. [11] P. Biran, O. Cornea and F. Lalonde, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topol- ogy Series II: Mathematics, Physics and Chemistry, Vol. 217 of NATO Sci. Series (Springer, New York, 2004). [12] D. McDuff and D. Salamon, J-holomorphic Curves and Quantum Cohomology, Vol. 6 of Univ. Lecture Series (Amer. Math. Soc., Providence, 1994). [13] S.T. Hong, J. Geom. Phys. 48, 135 (2003). [14] M.F. Atiyah, V. Patodi and I. Singer, Math. Proc. Camb. Phil. Soc. 77, 43 (1975); Math. Proc. Camb. Phil. Soc. 78, 405 (1975); Math. Proc. Camb. Phil. Soc. 79, 71 (1976). [15] E. Witten, Comm. Math. Phys. 121, 351 (1989). [16] Y. Nambu, Lecture at the Copenhagen Symposium, 1970, unpublished; T. Goto, Prog. Theor. Phys. 46, 1560 (1971). [17] J. Scherk, Rev. Mod. Phys. 47, 123 (1975); J. Govaerts, Lectures given at Escuela Avanzada de Verano en Fisica, Mexico City, Mexico (1986). [18] J. Cheeger and D. Ebin, Comparison Theorems in Rie- mannian Geometry (North-Holland, Amsterdam, 1975).

0704.0117

Lower ground state due to counter-rotating wave interaction in trapped ion system

Lower ground state due to counter-rotating wave interaction in trapped ion system T. Liu1, K.L. Wang1,2, and M. Feng3 ∗ The School of Science, Southwest University of Science and Technology, Mianyang 621010, China The Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China (Dated: November 4, 2018) We consider a single ion confined in a trap under radiation of two traveling waves of lasers. In the strong-excitation regime and without the restriction of Lamb-Dicke limit, the Hamiltonian of the system is similar to a driving Jaynes-Cummings model without rotating wave approximation (RWA). The approach we developed enables us to present a complete eigensolutions, which makes it available to compare with the solutions under the RWA. We find that, the ground state in our non-RWA solution is energically lower than the counterpart under the RWA. If we have the ion in the ground state, it is equivalent to a spin dependent force on the trapped ion. Discussion is made for the difference between the solutions with and without the RWA, and for the relevant experimental test, as well as for the possible application in quantum information processing. PACS numbers: 32.80.Lg, 42.50.-p, 03.67.-a I. INTRODUCTION Ultracold ions trapped as a line are considered as a promising system for quantum information processing [1]. Since the first quantum gate performed in the ion trap [2], there have been a series of experiments with trapped ions to achieve nonclassical states [3], simple quantum algorithm [4], and quantum communication [5]. There have been also a number of proposals to employ trapped ions for quantum computing, most of which work only in the weak excitation regime (WER), i.e., the Rabi frequency smaller than the trap frequency. While as bigger Rabi frequency would lead to faster quantum gating, some proposals [6, 7, 8] have aimed to achieve operations in the case of the Rabi frequency larger than the trap frequency, i.e., the so called strong excitation regime (SER). The difference of the WER from the SER is mathematically reflected in the employment of the rotating wave approximation (RWA), which averages out the fast oscillating terms in the interaction Hamiltonian. As the RWA is less valid with the larger Rabi frequency, the treatment for the SER was complicated, imcomplete [9], and sometimes resorted to numerics [10]. In addition, the Lamb-Dicke limit strongly restricts the application of the trapped ions due to technical challenge and the slow quantum gating. We have noticed some ideas [11, 12] to remove the Lamb-Dicke limit in designing quantum gates, which are achieved by using some complicated laser pulse sequences. In the present work, we investigate, from another research angle, the system mentioned above in SER and in the absence of the Lamb-Dicke limit. The main idea, based on an analytical approach we have developed, is to check the eigenvectors and the eigenenergies of such a system, with which we hope to obtain new insight into the system for more application. The main result in our work is a newly found ground state, energically lower than the ground state calculated by standard Jaynes-Cummings model. We will also present the analytical forms of the eigenvectors and the variance of the eigenenergies with respect to the parameters of the system, which might be used in understanding the time evolution of the system. The paper is organized as follows. In Section II we will solve the system in the absence of the RWA. Then some numerical results will be presented in comparison with the RWA solutions in Section III. We will discuss about the new results for their possible application. More extensive discussion and the conclusion are made in Section IV. Some analytical deduction details could be found in Appendix. II. THE ANALYTICAL SOLUTION OF THE SYSTEM As shown in Fig. 1, we consider a Raman Λ-type configuration, which corresponds to the actual process in NIST experiments. Like in [13], we will employ some unitary transformations to get rid of the assumption of Lamb-Dicke limit and the WER. So our solution is more general than most of the previous work [14]. For a single trapped ∗ Electronic address: [email protected] http://arxiv.org/abs/0704.0117v1 ion experiencing two off-resonant counter-propagating traveling wave lasers with frequencies ω1 and ω2, respectively, and in the case of a large detuning δ, we have an effective two-level system with the lasers driving the electric-dipole forbidden transition |g〉 ↔ |e〉 by the effective laser frequency ωL = ω1−ω2. So we have the dimensionless Hamiltonian σz + a iηx̂ + σ−e −iηx̂), (1) in the frame rotating with ωL, where ∆ = (ω0 − ωL)/ν, ω0 and ν are the resonant frequency of the two levels of the ion and the trap frequency, respectively. Ω is the dimensionless Rabi frequency in units of ν and η the Lamb-Dicke parameter. σ±,z are usual Pauli operators, and we have x̂ = a † + a for the dimensionless position operator of the ion with a† and a being operators of creation and annihilation of the phonon field, respectively. We suppose that both Ω and ν are much larger than the atomic decay rate and the phonon dissipative rate so that no dissipation is considered below. Like in [13], we first carry out some unitary transformations on Eq. (1) to avoid the expansion of the exponentials. So we have HI = UHU † = σz + a †a+ g(a† + a)σx + ǫσx + g 2, (2) where F †(η) F (η) −F †(η) F (η) with F (η) = exp [iη(a† + a)/2], g = η/2, and ǫ = −∆/2. Eq. (2) is a typical driving Jaynes-Cummings model including the counter-rotating wave terms. In contrast to the usual treatments to consider the Lamb-Dicke limit by using the RWA in a frame rotation, we remain the counter-rotating wave interaction in the third term of the right-hand side of Eq. (2) in our case. To go on our treatment, we make a further rotation with V = exp (iπσy/4), yielding = V HIV † = − σx + a †a+ g(a† + a)σz + ǫσz + g 2, (3) where we have used exp (iθσy)σx exp (−iθσy) = cos(2θ)σx + sin(2θ)σz , and exp (iθσy)σz exp (−iθσy) = cos(2θ)σz − sin(2θ)σx. For convenience of our following treatment, we rewrite Eq. (3) to be = ǫ(|e〉〈e| − |g〉〈g|)− (|e〉〈g|+ |g〉〈e|) + a†a+ g(a† + a)(|e〉〈e| − |g〉〈g|) + g2. (4) Using Schrödinger equation, and the orthogonality between |e〉 and |g〉, we suppose |〉 = |ϕ1〉|e〉+ |ϕ2〉|g〉, (5) which yields ǫ|ϕ1〉+ a†a|ϕ1〉+ g(a† + a)|ϕ1〉 − |ϕ2〉+ g2|ϕ1〉 = E|ϕ1〉, (6) − ǫ|ϕ2〉+ a†a|ϕ2〉 − g(a† + a)|ϕ2〉 − |ϕ1〉+ g2|ϕ2〉 = E|ϕ2〉. (7) To make the above equations concise, we apply the displacement operator D̂(g) = exp [g(a† − a)] on a† and a, which givesA = D̂(g)†aD̂(g) = a+g, A† = D̂(g)†a†D̂(g) = a†+g, B = D̂(−g)†aD̂(−g) = a−g, and B† = D̂(−g)†a†D̂(−g) = a† − g. So we have (A†A+ ǫ)|ϕ1〉 − |ϕ2〉 = E|ϕ1〉, (8) (B†B − ǫ)|ϕ2〉 − |ϕ1〉 = E|ϕ2〉. (9) Obvious, the new operators work in different subspaces, which leads to different evolutions regarding different internal levels |g〉 and |e〉. We will later refer to this feature to be relevant to spin-dependent force. The solution of the two equations above can be simply set as |ϕ1〉 = cn|n〉A, (10) |ϕ2〉 = dn|n〉B, (11) with N a large integer to be determined later, |n〉A = 1√ (a† + g)n|0〉A = 1√ (a† + g)nD̂(g)†|0〉 = 1√ (a† + g)n exp{−ga† − g2/2}|0〉, and |n〉B = 1√ (a† − g)n|0〉B = 1√ (a† − g)nD̂(−g)†|0〉 = 1√ (a† − g)n exp{ga† − g2/2}|0〉. Taking Eqs. (10) and (11) into Eqs. (8) and (9), respectively, and multiplying by A〈m| and B〈m|, respectively, we have, (m+ ǫ)cm − (−1)nDmndn = Ecm, (12) (m− ǫ)dm − (−1)mDmncn = Edm, (13) where we have set (−1)nDmn =A 〈m|n〉B and (−1)mDmn =B 〈m|n〉A, whose deduction can be found in Appendix. Diagonizing the relevant determinants, we may have the eigenenergies Ei and the eigenvectors regarding c n and d (n = 0, · · · , N, i = 0, · · · , N). Therefore, as long as we could find a closed subspace with ciN+1 and diN+1 approaching zero for a certain big integer N, we may have a complete eigensolution of the system. III. DISCUSSION BASED ON NUMERICS Before doing numerics, we first consider a treatment by involving the RWA. As the RWA solution could present complete eigenenergy spectra, it is interesting to make a comparison between the RWA solution and our non-RWA one. We consider a rotation in Eq. (2) with respect to exp{−i[(Ω/2)σz + a†a]t}, which results in σz + a †a+ g(aσ+ + a †σ−) + g 2, (14) where the RWA has been made by setting Ω = 1, and we have corresponding eigenenergies E±n = (n+ g 2 + 1/2)± g n+ 1. (15) So the system is degenerate in the case of η = 0 and there are two eigenenergy spectra corresponding to E±n as long as η 6= 0. Figs. 2(a) and 2(b) demonstrate two spectra, respectively, and in each figure we compare the differences between the RWA and non-RWA solutions [15]. In contrast to the two spectra in the RWA solution, the non-RWA solution includes only one spectrum. Comparing the two eigensolutions, we find that the even-number and odd-number excited levels in the non-RWA case correspond to E+n and E n of the RWA case, respectively, and the difference becomes bigger and bigger with the increase of η. It is physically understandable for these differences because the RWA solution, valid only for small η, does not work beyond the Lamb-Dicke regime. Above comparison also demonstrates the change of the ion trap system from an integrable case (i.e., with RWA validity) to the non-integrable case (i.e., without RWA validity). But besides these differences, we find an unusual result in this comparison, i.e., a new level without the counterpart in RWA solution appearing in our solution, which is lower than the ground state in RWA solution by ν + xη with x a η-dependent coefficient. In the viewpoint of physics, due to additional counter-rotating wave interaction involved, it is reasonable to have something more in our solution than the RWA case, although this does not surely lead to a new level lower than the previous ground state. Anyway, this is a good news for quantum information processing with trapped ions. As the situation in SER and beyond the Lamb-Dicke limit involves more instability, a stable confinement of the ion requires a stronger trapping condition. In this sense, our solution, with the possibility to have the ion stay in an energically lower state, gives a hope in this respect. We will come to this point again later. Since no report of the new ground state had been found either theoretically or experimentally in previous pub- lications, we suggest to check it experimentally by resonant absorption spectrum. As shown above, in the case of non-zero Lamb-Dicke parameter, the degeneracy of the neighboring level spacing is released, and the bigger the η, the larger the spacing difference between the neighboring levels. Therefore, an experimental test of the newly found ground state should be available by resonant transition between the ground and the first excited states in Fig. 2, once the SER is reached. We have noticed that the SER could be achieved by first cooling the ions within the Lamb-Dicke limit and under the WER, and then by decreasing the trap frequency by opening the trap adiabatically [6]. Since it is lower in energy than the previously recognized ground states, the new ground state we found is more stable, and thereby more suitable to store quantum information. Once the trapped ion is cooled down to the ground state in the SER, it is, as shown in Eq. (5) with n = 0, actually equivalent to the effect of a spin-dependent force on the trapped ion [16]. If we make Hadamard gate on the ion by |g〉 → (|g〉 + |e〉)/ 2 and |e〉 → (|g〉 − |e〉)/ 2, we reach a Schrödinger cat state, i.e., (1/2){[D†(g)|0〉+D†(−g)|0〉]|g〉 − [D†(g)|0〉 −D†(−g)|0〉]|e〉}. Two ions confined in a trap in above situation will yield two-qubit gates without really exciting the vibrational mode [11]. It is also the way with this spin-dependent force towards scalable quantum information processing [12]. As in SER, we may have larger Rabi frequency than in WER, the quantum gate could be in principle carried out faster in the SER. In addition, as it is convergent throughout the parameter subspace, our complete eigensolution enables us to accurately write down the state of the system at an arbitrary evolution time, provided that we have known the initial state. This would be useful for future experiments in preparing non-classical states and in designing any desired quantum gates with trapped ions in the SER and beyond the Lamb-Dicke limit. Moreover, as shown in Figs 3(a), 3(b) and 3(c), our present solution is helpful for us to understand the particular solutions in previous publication [13]. The comparison in the figures shows that the results in [13] are actually mixtures of different eigensolutions. For example, the lowest level in Fig. 2 in [13], corresponding to Ω = 2 and η = 0.2, is actually constituted at least by the third, the fourth, and the fifth excited states of the eigensolution. IV. FURTHER DISCUSSION AND CONCLUSION The observation of the counter-rotating effects is an interesting topic discussed previously. In [17], a standard method is used to study the observable effects regarding the rotating and the counter-rotating terms in the Jaynes-Cummings model, including to observe Bloch-Siegert shift [18] and quantum chaos in a cavity QED by using differently polarized lights. A recent work [19] for a two-photon Jaynes-Cummings model has also investigated the observability of the counter-rotating terms. By using perturbation theory, the authors claimed that the counter-rotating effects, although very small, can be in principle observed by measuring the energy of the atom going through the cavity. Actually, for the cavity QED system without any external source involved, it is generally thought that the counter-rotating terms only make contribution in some virtual fluctuations of the energy in the weak coupling regime. While the interference between the rotating and counter-rotating contributions could result in some phase dependent effects [20]. Anyway, if there is an external source, for example, the laser radiating a trapped ultracold ion, the counter-rotating terms will show their effects, e.g., related to heating in the case of WER [21]. In this sense, our result is somewhat amazing because the counter-rotating interaction in the SER, making entanglement between internal and vibrational states of the trapped ion, plays positive role in the ion trapping. We argue that our approach is applicable to different physical processes involving counter-rotating interaction. Since the counter-rotating terms result in energy nonconservation in single quanta processes, usual techniques cannot solve the Hamiltonian with eigenstates spanning in an open form. In this case, path-integral approach [22] and perturbation approach [20], assisted by numerical techniques were employed in the weak coupling regime of the Jaynes-Cummings model. In contrast, our method, based on the diagonalization of the coherent-state subspace, could in principle study the Jaynes-Cummings model without the RWA in any cases. We have also noticed a recent publication [23] to treat a strongly coupled two-level system to a quntum oscillator under an adiabatic approximation, in which something is similar to our work in the solution of the Hamiltonian in the absence of the RWA. But due to the different features in their system from our atomic case, the two-level splitting term, much smaller compared to other terms, can be taken as a perturbation. So the advantage of that treatment is the possibility to analytically obtain good approximate solutions. In contrast, not any approximation is used in our solution, which should be more efficient to do the relevant In summary, we have investigated the eigensolution of the system with a single trapped ion, experiencing two traveling waves of lasers, in the SER and in the absence of the Lamb-Dicke limit. We have found the ground state in the non-RWA case to be energically lower than the counterpart of the solution with RWA, which would be useful for quantum information storage and for quantum computing. The analytical forms of the eigenfunction and the complete set of the eigensolutions would be helpful for us to understand a trapped ion in the SER and with a large Lamb-Dicke parameter. We argue that our work would be applied to different systems in dealing with strong coupling problems. V. ACKNOWLEDGMENTS This work is supported in part by NNSFC No. 10474118, by Hubei Provincial Funding for Distinguished Young Scholars, and by Sichuan Provincial Funding. VI. APPENDIX We give the deduction of A〈m|n〉B and B〈m|n〉A below, A〈m|n〉B = 〈0|e−ga−g 2/2(a+ g)m(a† − g)nega †−g2/2|0〉 〈0|(a+ g)mega e−ga(a† − g)n|0〉 〈0|(a+ 2g)m(a† − 2g)n|0〉 = (−1)nDmn, Dmn = e min[m,n] (−1)−i m!n!(2g)m+n−2i (m− i)!(n− i)!i! It is easily proven following a similar step to above that B〈m|n〉A = 〈0|ega−g 2/2(a− g)m(a† + g)ne−ga †−g2/2|0〉, would finally get to (−1)mDmn. [1] Cirac J I, Zoller P 1995 Phys. Rev. Lett. 74 4091 [2] Monroe C, Meekhof D M, King B E, Itano W M, Wineland D J 1995 Phys. Rev. Lett. 75 4714 [3] Turchette Q A, Wood C S, King B E, Myatt C J, Leibfried D, Itano W M, Monroe C, Wineland D J 1998 Phys. Rev. Lett. 81 3631; Sackett C A, Kielpinski D, King B E, Langer C, Meyer V, Myatt C J, Rowe M, Turchette Q A, Itano W M, Wineland D J, Monroe C 2000 Nature 404 256 [4] Gulde S, Riebe M, Lancaster G P T, Becher C, Eschner J, Haeffner H, Schmidt-Kaler F, Chuang I L, Blatt R 2003 Nature 421 48 [5] Riebe M, Haeffner H, Roos C F, Haensel W, Benhelm J, Lancaster G P T, Koerber T W, Becher C, Schmidt-Kaler F, James D F V, Blatt R 2004 Nature 429 734; Barrett M D, Chiaverini J, Schaetz T, Britton J, Itano W M, Jost J D, Knill E, Langer C, Leibfried D, Ozeri R, Wineland D J 2004 Nature 429 737 [6] Poyators J F, Cirac J I, Blatt R, Zoller P 1996 Phys. Rev. A 54 1532; Poyatos J F, Cirac J I, Zoller P 1998 Phys. Rev. Lett. 81 1322 [7] Zheng S, Zhu X W, Feng M 2000 Phys. Rev. A 62 033807 [8] Feng M 2004 Eur. Phys. J. D 29 189 [9] Feng M, Zhu X, Fang X, Yan M, Shi L 1999 J. Phys. B 32 701; Feng M 2002 Eur. Phys. J. D 18 371 [10] Zeng H, Lin F, Wang Y, Segawa Y 1999 Phys. Rev. A 59 4589 [11] Garcia-Ripoll J J, Zoller P and Cirac J I 2003 Phys. Rev. Lett. 91 157901; [12] Duan L -M 2004 Phys. Rev. Lett. 93 100502 [13] Feng M 2001 J. Phys. B 34 451 [14] Most of the previous work in this respect were carried out by cuting off the expansion of the exponentials regarding the quantized phonon operators, which is only reasonable in the WER and within the Lamb-Dicke limit. In contrast, our treatment can be used in both the SER and the WER cases. [15] We take throughout this paper N = 40 in which the coefficients ci41 and d 41 with i = 0, 1, ..40 are negligible in the case of Ω = 1 and 2. Although with the increase of values of Ω the diagonalization space has to be enlarged, our analytical method generally works well in a wide range of parameters. [16] Haljan P C, Brickman K -A, Deslauriers L, Lee P J and Monroe C 2005 Phys. Rev. Lett. 94 153602 [17] Crisp M D 1991 Phys. Rev. A 43 2430 [18] Bloch F and Siegert A 1940 Phys. Rev. 57 522 [19] Janowicz M and Orlowski A 2004 Rep.Math. Phys. 54 71 [20] Phoenix S J D 1989 J. Mod. Optics 3 127 [21] Leibfrid D, Blatt R, Monroe C, and Wineland D J 2003 Rev. Mod. Phys. 75 281 [22] Zaheer K and Zubairy M S 1998 Phys. Rev. A 37 1628 [23] Irish E K, Gea-Banacloche J, Martin I, and Schwab K C 2005 Phys. Rev. B 72 195410 The captions of the figures Fig. 1 Schematic of a single trapped ion under radiation of two traveling wave lasers, where ω1 and ω2 are frequencies regarding the two lasers, respectively, ω0 is the resonant frequency between |g〉 and |e〉, and δ and ∆ are relevant detunings. This is a typical Raman process employed in NIST experiments, with for example Be+, for quantum computing. Fig. 2 The eigenenergy spectra with Ω = 1, where (a) and (b) correspond to two different sets of eigenenergies with respect to Lamb-Dicke parameter. In (a) the comparison is made between E+n in the RWA case (dashed-dotted curves) and En with n = even numbers in the non-RWA case (star curves for n = 0 and solid curves for others); In (b) the comparison is for E−n in the RWA case (dashed-dotted curves) to En with n = odd numbers in the non-RWA case (solid curves). Fig. 3 The eigenenergy with respect to the detuning ∆, where for convenience of comparison we have used the same parameter numbers as in [13]. For clarity, we plot the different levels with different lines. The parameter numbers are Ω = 2, and (a) η = 0.2; (b) η = 0.4; (c) η = 0.6. introduction The analytical solution of the system discussion based on numerics further discussion and conclusion acknowledgments appendix References

0704.0118

Strained single-crystal Al2O3 grown layer-by-layer on Nb (110) thin films

Strained single-crystal Al2O3 grown layer-by-layer on Nb (110) thin films Paul B. Welander and James N. Eckstein Department of Physics and Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (Dated: April 1, 2007) We report on the layer-by-layer growth of single-crystal Al2O3 thin-films on Nb (110). Single- crystal Nb films are first prepared on A-plane sapphire, followed by the evaporation of Al in an O2 background. The first stages of Al2O3 growth are layer-by-layer with hexagonal symmetry. Electron and x-ray diffraction measurements indicate the Al2O3 initially grows clamped to the Nb lattice with a tensile strain near 10%. This strain relaxes with further deposition, and beyond about 50 Å we observe the onset of island growth. Despite the asymmetric misfit between the Al2O3 film and the Nb under-layer, the observed strain is surprisingly isotropic. The present challenge of constructing solid-state quan- tum bits with long coherence times [1] has ignited new interest in Josephson junctions fabricated from single- crystal materials. It has been found that critical-current 1/f noise cannot fully account for the observed deco- herence times in junctions-based qubits [2]. However, amorphous tunnel-barrier defects can give rise to two- level charge fluctuations that destroy quantum coher- ence across the junction [3, 4]. Oh et al have recently found that tunnel-junctions from epitaxial Re/Al2O3/Al tri-layers have a significantly reduced density of two-level fluctuators [5]. The pairing of Re and Al2O3 is advantageous because of the very small misfit between the basal planes and because Re is less likely to oxidize compared with other superconducting refractory metals. However, epitaxial Re films develop domains due to basal-plane twinning, causing the surface to be rough on the length scales of a typical tunnel-junction [6]. An alternative to Al2O3 hetero-epitaxy on a close-packed metal surface is to grow on bcc (110), where such twinning is absent. To date single-crystal Al2O3 films have been grown on a number of such metals: Ta [7], Mo [8], W [9], and more recently Nb [10]. In a recent paper, Dietrich et al reported on their in- vestigations of ultra-thin epitaxial α-Al2O3 (0001) films on Nb using tunneling microscopy and spectroscopy [10]. Their films were grown on Nb (110) by evaporating Al in an O2 background near room temperature. Crystalliza- tion was achieved by annealing the sample up to 1000 ◦C. Subsequent microscopy showed the film to be atomically smooth, but spectroscopic scans found localized defect states around ±1 eV, well below the 9 eV sapphire band We report here on our findings concerning the hetero- epitaxy of Al2O3 on Nb (110) films. Unlike the previ- ous study, our Al2O3 films are grown layer-by-layer with co-deposition of Al and O at elevated substrate tem- peratures. Epitaxial bi-layers (Nb/Al2O3) and tri-layers (Nb/Al2O3/Nb) are grown by molecular beam epitaxy (MBE). Characterization techniques include in situ re- ∗This article has been submitted to Applied Physics Letters. flection high-energy electron diffraction (RHEED) and x-ray photo-electron spectroscopy (XPS), and ex situ atomic force microscopy (AFM) and x-ray diffraction (XRD). The process for growing high-quality single-crystal Nb films on sapphire is well understood [11]. Our samples start with a thick Nb base layer (2000 Å) grown on A- plane sapphire – α-Al2O3 (112̄0) – with a nominal miscut of 0.1◦. Nb (99.99%) is evaporated via e-beam bombard- ment at a rate of about 0.3 Å/s onto a substrate held near 800 ◦C. The base pressure of our chamber is about 10−11 torr, with the growth pressure around 10−9 torr. After deposition, the film is annealed above 1300 ◦C for 30 min. During growth and annealing the film surface is monitored with RHEED. Epitaxial Nb on A-plane sapphire grows in the (110) orientation with Nb [11̄1] ‖ α-Al2O3 [0001], in accordance with the well-established three-dimensional relationship [11, 12]. Nb RHEED patterns (Figure 1) reveal a two- dimensional, reconstructed film surface that takes one form after growth [13], and a second one upon annealing [14]. Annealed films also show a sharp specular spot in- FIG. 1: Nb (110) on A-plane sapphire. Top: RHEED images along the (a) [001] and (b) [11̄1] azimuths after growth at 800 ◦C, and (c) [11̄1] after annealing above 1300 ◦C. Left: 5 × 5 µm2 AFM image of annealed Nb, 10 Å height scale. Right: XRD radial scan of the Nb (110) Bragg peak. http://arxiv.org/abs/0704.0118v1 FIG. 2: Top: RHEED images from epitaxial Al2O3 on Nb (110), taken along the [11̄00] azimuth after deposition of (a) 4 Å, (b) 25 Å, and (c) 125 Å. Bottom: 5× 5 µm2 AFM scans on Al2O3 films that are 20 Å (left, 10 Å height-scale) and 100 Å (right, 50 Å) thick. dicating long-range film flatness, which is confirmed by AFM measurements. Scans show large terraces about 2000 Å wide and monolayer step-edges that align them- selves according to the substrate miscut (Figure 1). An- nealed Nb films typically have an rms surface roughness less than 2 Å. XRD measurements on these Nb films show sharp Bragg peaks and narrow rocking curves, both indicative of single-crystal growth. Figure 1 shows a radial scan (2θ-ω) of the Nb (110) Bragg peak from a 2000 Å-thick film, with intensity fringes indicating a structural coher- ence that extends over the entire film thickness. Rock- ing curves typically have a FWHM of about 0.03◦. In addition, measurements of specular and off-axis Bragg peaks demonstrate that a 2000 Å-thick annealed Nb film is strained 0.1% or less with respect to bulk. Al2O3 is deposited in situ onto similar Nb films at a substrate temperature of around 750 ◦C. Using a stan- dard effusion cell, Al (99.9995%) is evaporated at about 0.1 Å/s in an O2 (99.995%) background up to 5 × 10 torr. Under these growth conditions we estimate that the O2 flux is about 1000 times greater than that of Al [15]. After deposition the sample is cooled before turning the O2 off. Al2O3 films included in this report range in thickness from 15 to 125 Å. Chemical analysis of the Al2O3 is carried out in an XPS system adjacent to the growth chamber. Measure- ments of the Al 2p, O 1s and Nb 3d levels indicate that the Al is completely oxidized with no measurable oxida- tion of the underlying Nb. The observed energy differ- ence between the O 1s and Al 2p levels is 457.1 eV, in good agreement with what has been reported for sap- phire (456.6 eV) [16]. The Nb 3d level shows no side bands which would indicate oxide formation. RHEED of the Al2O3 thin film reveals a hexagonal FIG. 3: Strain vs. film thickness for epitaxial Al2O3 on Nb (110). (•) Strain of a 100 Å film measured during deposition. (⋄) Strain observed for a number of samples after deposition and cooling, with error bars indicating the range of strain values measured along different RHEED azimuths. C-plane-like surface in the Nishiyama-Wasserman orien- tation: α-Al2O3 (0001) [1̄100] ‖ Nb (110) [001] [17]. (Be- cause both α-Al2O3 [18] and γ-Al2O3 [19] have close- packed planes, no definitive crystal structure can be inferred. Hexagonal Miller indices will be employed for defining crystallographic orientations by convention only.) Diffraction images from various stages of growth are shown in Figure 2. Immediately after the oxide depo- sition begins the Nb diffraction pattern and specular spot disappear. After about 2 ML (4 Å) the Al2O3 diffraction pattern becomes visible. At a thickness of 25 Å, RHEED shows an elongated specular spot and well-defined first- order streaks. Up to about 50 Å the Al2O3 growth is layer-by-layer (Frank-van der Merwe mode). Beyond this thickness the 2D streaks evolve into 3D spots, indicating the growth of islands (Stranski-Krastanov mode). As the transformation from 2D to 3D growth is occurring, the measured spacing between RHEED streaks/spots increases, indicating a shrinking of the Al2O3 surface lattice. Using the RHEED from the base- layer Nb as a ruler, we find that the Al2O3 film experi- ences a tensile strain that relaxes with increasing thick- ness, as shown in Figure 3. The strain-thickness curve is determined from RHEED along the [1̄100] azimuth dur- ing Al2O3 deposition near 750 ◦C. With respect to C- plane sapphire (a = 4.759 Å), the tensile strain is nearly 10% initially and by 20 Å has fallen to about 8%. After 100 Å of deposition, the Al2O3 exhibits a tensile strain of around 3%. After deposition and cooling in O2, Al2O3 films of var- ious thicknesses show further lattice relaxation (Figure 3). On average, RHEED measurements near room tem- perature show a strain reduction of about 1% when com- pared to measurements just after Al deposition. Ther- mal contraction accounts for a significant portion of the strain change during cooling. (Both Nb and Al2O3 have expansion coefficients in this temperature range around 7-8×10−6 K−1.) However, due to the limited precision FIG. 4: XRD pole figure for an epitaxial Nb/Al2O3/Nb tri- layer grown on A-plane sapphire. Both Nb layers have a (110) surface-orientation. This scan shows the off-axis 〈110〉 Bragg peaks. The four peaks connected by the dashed rectangle are approximately four times stronger than the others. of our measurements, the presence of other strain-relief mechanisms cannot be determined. Regardless, the measured tensile strain in epitaxial Al2O3 films on Nb (110) is significant. What’s more, the strain is fairly isotropic – RHEED patterns along the {1̄100} azimuths reveal relatively small variations. The strain for each azimuth is determined by averaging oppo- site directions – eg. [1̄100] and [11̄00] – to reduce system- atic errors. The mean and range of the measured tensile strain for the three azimuths is shown in Figure 3. The strain-isotropy is surprising since the misfit along the Nb [001] or α-Al2O3 [1̄100] direction is rather large (20%), while along the Nb [11̄0] or α-Al2O3 [1̄1̄20] it is much smaller (−1.7%). Despite such an anisotropic misfit, the Al2O3 films exhibit isotropic strain. Thin Al2O3 films are also very flat. AFM imaging of a 20 Å-thick film shows an atomically flat surface with monolayer steps (c/6 = 2.165 Å) and an rms roughness of about 2 Å (Figure 2). On the other hand, the surface of a 100 Å-thick film is comprised of islands about 1000 Å wide and 50 Å in height. This agrees well with our interpretation of Al2O3 RHEED - evidence for islands in the diffraction images appeared after about 50 Å of deposition. For those samples where an epitaxial Nb over-layer is deposited in situ, the substrate is warmed back up above 700 ◦C. Under these conditions growth on C-plane sapphire would yield (111)-oriented films [11, 12]. How- ever, XRD analysis indicates that the top Nb layer is (110)-oriented with Nb [001] ‖ α-Al2O3 [1̄100], [01̄10] and [101̄0]. A pole scan of off-axis 〈110〉 Bragg peaks is shown in Figure 4, and despite the surface orientation, the Nb over-layer reproduces the hexagonal symmetry of the Al2O3 film. The top Nb film grows in three domains of roughly equal weight rotated with respect to one an- other by 120◦, with one domain aligned to the base Nb layer. This type of film structure has been observed for Nb growth on C-plane sapphire, but only under the fol- lowing conditions: evaporation above 1000 ◦C [20], post- growth annealing up to 1500 ◦C [21], and niobium sput- tering near 850 ◦C [22]. That we observe this growth structure for evaporation near 700 ◦C suggests that the surface lattice of the Al2O3 film, while hexagonal, is not identical to that of C-plane sapphire. Tunnel-junctions were fabricated from several of these epitaxial tri-layers. The I-V characteristics showed a large conductance shunting the Josephson junction. While an inhomogeneous morphology may cause such a conductance, no metallurgical pinholes were ever ob- served in our Al2O3 films. Devices with 20 Å Al2O3 lay- ers had critical current densities around 104 A/cm2 and normal state conductances near 109 S/cm2. Assuming a homogeneous barrier, the latter value gives an effective barrier height of about 1.3 eV. This is similar to the en- ergy of sub-gap states found spectroscopically by Dietrich et al [10] in epitaxial Al2O3 on Nb. Among the previous studies of Al2O3 epitaxy on bcc (110) metals, only Chen et al reported any measure of tensile strain [7]. For Al2O3 films 5-40 Å thick on Ta (110) they measured a lattice enlargement of about 9%. The agreement with our findings could be expected since the lattice constants of Ta and Nb are nearly identi- cal. One difference though is that Chen et al observed a Kurdjumov-Sachs relationship, α-Al2O3 (0001) [1̄100] ‖ Nb (110) [11̄1] [7], instead of the Nishiyama-Wasserman orientation we observe. In summary, single-crystal Nb/Al2O3 and Nb/Al2O3/Nb multi-layers were grown by MBE. Various methods of materials analysis suggest these layers are all high-quality. Our principal finding is that epitaxial Al2O3 on Nb (110) grows under uniform tensile strain, despite the anisotropic misfit. As the Al2O3 film thickness is increased the strain relaxes and the surface roughens. The over-layer Nb grows with a (110) surface orientation under growth conditions that would yield Nb (111) on C-plane sapphire. AFM and XRD analysis was carried out in the Cen- ter for Microanalysis of Materials, University of Illinois at Urbana-Champaign, which is partially supported by the U.S. Department of Energy under grant DEFG02- 91ER45439. This project was funded by the National Science Foundation through grant EIA 01-21568. [1] M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000). [2] D. J. van Harlingen, T. L. Robertson, B. L. T. Plourde, P. A. Reichardt, T. A. Crane, and J. Clarke, Phys. Rev. B 70, 064517 (2004). [3] I. Martin, L. Bulaevskii, and A. Shnirman, Phys. Rev. Lett. 95, 127002 (2005). [4] J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pap- pas, R. Simmonds, et al., Phys. Rev. Lett. 95, 210503 (2005). [5] S. Oh, K. Cicak, J. S. Kline, M. A. Sillanpää, K. D. Osborn, J. D. Whittaker, R. W. Simmonds, and D. P. Pappas, Phys. Rev. B 74, 100502 (2006). [6] S. Oh, D. A. Hite, K. Cicak, K. D. Osborn, R. W. Sim- monds, R. McDermott, K. B. Cooper, M. Steffen, J. M. Martinis, and D. P. Pappas, Thin Solid Films 389, 496 (2006). [7] P. J. Chen and D. W. Goodman, Surf. Sci. Lett. 312, L767 (1994). [8] M.-C. Wu and D. W. Goodman, J. Phys. Chem. 98, 9874 (1994). [9] J. Günster, M. Brause, T. Mayer, A. Hitzke, and V. Kempter, Nuc. Instr. and Meth. in Phys. Res. B 100, 411 (1995). [10] C. Dietrich, B. Koslowski, and P. Ziemann, J. Appl. Phys. 97, 083515 (2005). [11] S. M. Durbin, J. E. Cunningham, M. E. Mochel, and C. P. Flynn, J. Phys. F: Met. Phys. 11, L223 (1981). [12] J. Mayer, C. P. Flynn, and M. Rühle, Ultramicroscopy 33, 51 (1990). [13] C. Sürgers and H. v. Löhneysen, Appl. Phys. A 54, 350 (1992). [14] M. Ondrejcek, R. S. Appleton, W. Swiech, V. L. Petrova, and C. P. Flynn, Phys. Rev. Lett. 87, 116102 (2001). [15] K. G. Tscherich and V. von Bonin, J. Appl. Phys. 84, 4065 (1998). [16] W. M. Mullins and B. L. Averbach, Surf. Sci. 206, 29 (1988). [17] L. A. Bruce and H. Jaeger, Phil. Mag. A 38, 223 (1978). [18] W. E. Lee and K. P. D. Lagerlof, J. Elec. Micros. Tech. 2, 247 (1985). [19] F. H. Streitz and J. W. Mintmire, Phys. Rev. B 60, 773 (1999). [20] T. Wagner, J. Mater. Res. 13, 693 (1998). [21] T. Wagner, M. Lorenz, and M. Rühle, J. Mater. Res. 11, 1255 (1996). [22] H.-G. B. Ch. Dietrich and B. Koslowski, J. Appl. Phys. 94, 1478 (2003).

0704.0119

Quasi-quartet crystal electric field ground state in a tetragonal CeAg$_2$Ge$_2$ single crystal

Quasi-quartet crystal electric field ground state in a tetragonal CeAg2Ge2 single crystal A. Thamizhavel ∗, R. Kulkarni, S. K. Dhar Department of condensed matter physics and materials science, Tata institute of fundamental research, Colaba, Mumbai 400 005, India Abstract We have successfully grown the single crystals of CeAg2Ge2, for the first time, by flux method and studied the anisotropic physical properties by measuring the electrical resistivity, magnetic susceptibility and specific heat. We found that CeAg2Ge2 undergoes an antiferromagnetic transition at TN = 4.6 K. The electrical resistivity and susceptibility data reveal strong anisotropic magnetic properties. The magnetization measured at T = 2 K exhibited two metamagnetic transitions at Hm1 = 31 kOe andHm2 = 44.7 kOe, for H ‖ [100] with a saturation magnetization of 1.6 µB/Ce. The crystalline electric field (CEF) analysis of the inverse susceptibility data reveals that the ground state and the first excited states of CeAg2Ge2 are closely spaced indicating a quasi-quartet ground state. The specific heat data lend further support to the presence of closely spaced energy levels. Key words: CeAg2Ge2; CEF; quartet ground state; antiferromagnetism PACS: 81.10.-h, 71.27.+a, 71.70.Ch, 75.10.Dg, 75.50.Ee Compounds crystallizing in the ThCr2Si2 type struc- ture are the most extensively studied among the strongly correlated electron systems. A wide range of compounds crystallize in this type of tetragonal crystal structure and exhibit novel physical properties. Some of the promi- nent examples include the first heavy fermion supercon- ductor CeCu2Si2, pressure induced superconductors like CePd2Si2, CeRh2Si2, CeCu2Ge2, unconventional metam- agnetic transition in CeRu2Si2 etc. CeAg2Ge2 also crys- tallizes in the tetragonal ThCr2Si2 type crystal structure. Previous reports of CeAg2Ge2 were on polycrystalline samples and there were conflicting reports on the antifer- romagnetic ordering temperature [1,2,3]. Furthermore, the ground state properties of CeAg2Ge2 are also quite intrigu- ing. Neutron scattering experiments on a polycrystalline sample could detect only one excited state at 11 meV indi- cating that the ground state and the first excited states are closely spaced. In order to study the anisotropic physical properties and to study the crystalline electric field ground state, we have grown the single crystals of CeAg2Ge2. Single crystals of CeAg2Ge2 were grown by self flux method, using Ag:Ge (75.5: 24.5) binary alloy, which forms an eutectic at 650 ◦C, as flux. The details about the crystal ∗ Corresponding author. Tel: (81)22-2280-4556 Email address: [email protected] (A. Thamizhavel). growth process are given elsewhere [4]. Figure 1(a) shows the temperature dependence of electrical resistivity of CeAg2Ge2 for the current direction parallel to both [100] and [001] directions. There is a large anisotropy in the electrical resistivity. The electrical resistivity shows a shal- low minimum at 20 K, marginally increases with decrease in temperature down to 4.6 K. With further decrease in the temperature the electrical resistivity drops due to the reduction in the spin-disorder scattering caused by the an- tiferromagnetic ordering of the magnetic moments, as seen in the inset of Fig. 1(a). The antiferromagnetic transition can be clearly seen at 4.6 K as indicated by the arrow in the figure. Figure 1(b) shows the temperature dependence of the magnetic susceptibility along the two principle directions. As can be seen from the figure there is a large anisotropy in the susceptibility due to tetragonal crystal structure. The high temperature susceptibility does not obey the simple Curie-Weiss law; on the other hand it can be very well fit- ted to a modified Curie-Weiss law which is given by χ = , where χ0 is the temperature independent part of the magnetic susceptibility and C is the Curie constant. The value of χ0 was estimated to be 1.33 × 10 −3 and 1.41× 10−3 emu/mol forH ‖ [001] and [100], respectively such that an effective moment of 2.54 µB/Ce is obtained for Preprint submitted to Elsevier 25 October 2018 http://arxiv.org/abs/0704.0119v1 ) J || [001] [100] CeAg2Ge2 3002001000 Temperature (K) u)H || [100] [001] H || [001] [100] CeAg2Ge2 [100] [001] 6040200 Magnetic Field (kOe) CeAg2Ge2 [001] H || [100] T = 2 K Fig. 1. (a) The temperature dependence of electrical resistivity of CeAg2Ge2, inset shows the low temperature part, (b) Temperature dependence of the magnetic susceptibility together with inverse mag- netic susceptibility plot, solid lines indicate the CEF fitting and (c) Magnetization of CeAg2Ge2 measured at T = 2 K. temperatures above 100 K. In order to perform the CEF analysis of the susceptibility data, we plotted the inverse susceptibility as 1/(χ− χ0) versus T . The solid line in fig- ure 1(b) are the fitting to the inverse susceptibility with the CEF Hamiltonian given byHCEF = B where Bm and Om are CEF parameters and the Stevens operators respectively. The level splitting energies are esti- mated to be ∆1 = 5 K and ∆2 = 130 K. It may be noted that the first excited state is very close to the ground state indicating that the ground state is a quasi-quartet state. Figure 1(c) shows the field dependence of magnetization at 2 K. For H ‖ [100], the magnetization increases linearly with the field and exhibit two metamagnetic transition at Hm1 = 31 kOe and Hm2 = 44.7 kOe before it saturates at 1.6 µB/Ce at 70 kOe, indicating the easy axis of magneti- zation. On the other hand the magnetization along [001] is very small and varies linearly with field reaching a value of 0.32 µB/Ce at 50 kOe. Figure 2(a) shows the temperature dependence of the specific heat of CeAg2Ge2 together with the specific heat of a reference sample LaAg2Ge2. The antiferromagnetic or- dering is manifested by the clear jump in the specific heat at TN = 4.6 K as indicated by the arrow. The inset of Fig. 2(a) shows the Cmag/T versus T together with the 20151050 Temperature(K) H // [100] 0 kOe 20 kOe 40 kOe 50 kOe 60 kOe 80 kOe 100 kOe 120 kOe 40200 Temperature (K) R ln 2 R ln 4 Fig. 2. (a) Temperature dependence of the specific heat of CeAg2Ge2 and LaAg2Ge2. The inset shows the magnetic entropy. (b) The field dependence of the specific heat of CeAg2Ge2 for the field applied along the easy axis of magnetization, namely [100]. magnetic entropy. The entropy reaches R ln 4 not too far away from the magnetic ordering temperature leading to the conclusion that the ground state and the first excited states are closely spaced or nearly degenerate, thus corrob- orating our CEF analysis of the inverse susceptibility data. Figure 2(b) shows the field dependence of the specific heat for the field applied parallel to the easy axis of magnetiza- tion namely [100]. With the increase in the magnetic field the Néel temperature decreases and the antiferromagnetic ordering apparently vanishes at a critical field of 50 kOe indicating a possibility of a field induced quantum critical point in this compound. However, further low temperature measurements are necessary to confirm this. In summary, we have successfully grown the single crys- tals of CeAg2Ge2 by the flux method. CeAg2Ge2 orders an- tiferromagnetically at TN = 4.6 K. The CEF analysis of the inverse susceptibility data indicate the ground state and the first excited states are closely spaced. The heat capacity data support this quasi-quartet ground state. Furthermore, the heat capacity in applied magnetic fields revealed that the Néel temperature vanishes at a critical field of 50 kOe indicating a possible field induced quantum critical point in this compound. References [1] R. Rauchschwalbe et al., J. Less Common. Metals 111, (1985) [2] G. Knopp et al., J. Magn. Magn. Mater. 63 & 64, (1987) 88. [3] E. Cordruwish et al., J. Phase Equilibria 20, (1999) 407. [4] A. Thamizhavel et al., Phys. Rev. B (2007) to be published References

0704.0120

Strong Phase and $D^0-D^0bar$ mixing at BES-III

Strong Phase and D0 − D0 mixing at BES-III Xiao-Dong Cheng1,2,∗ Kang-Lin He1,† Hai-Bo Li1,‡ Yi-Fang Wang1,§ and Mao-Zhi Yang1¶ Institute of High Energy Physics, P.O.Box 918, Beijing 100049, China Department of Physics, Henan Normal University, XinXiang, Henan 453007, China (Dated: October 25, 2018) Most recently, both BaBar and Belle experiments found evidences of neutral D mixing. In this paper, we discuss the constraints on the strong phase difference in D0 → Kπ decay from the measurements of the mixing parameters, y′, yCP and x at the B factories. With CP tag technique at ψ(3770) peak, the extraction of the strong phase difference at BES-III are discussed. The sensitivity of the measurement of the mixing parameter y is estimated in BES-III experiment at ψ(3770) peak. Finally, we also make an estimate on the measurements of the mixing rate RM . PACS numbers: 13.25.Ft, 12.15.Ff, 13.20.Fc, 11.30.Er Due to the smallness of ∆C = 0 amplitude in the Standard Model (SM), D0 − D0 mixing offers a unique opportunity to probe flavor-changing interactions which may be generated by new physics. The recent measure- ments from BaBar and Belle experiments indicate that the D0 − D0 mixing may exist [1, 2]. At the B fac- tories, the decay time information can be used to ex- tract the neutral D mixing parameters. At t = 0 the only term in the amplitude is the direct doubly-Cabibbo- suppressed (DCS) mode D0 → K+π−, but for t > 0 D0 − D0 mixing may contribute through the sequence D0 → D0 → K+π− , where the second stage is Cabibbo favored (CF). The interference of this term with the DCS contribution involves the lifetime and mass differences of the neutral D mass eigenstates, as well as the final- state strong phase difference δKπ between the CF and the DCS decay amplitudes. This interference plays a key role in the measurement of the mixing parameters at time-dependent measurements. With the assumption of CPT invariance, the mass eigenstates of D0 −D0 system are |D1〉 = p|D0〉+ q|D0〉 and |D2〉 = p|D0〉−q|D0〉 with eigenvalues µ1 = m1− and µ2 = m2 − Γ2, respectively, where the m1 and Γ1 (m2 and Γ2) are the mass and width of D1 (D2). For the method of detecting D0 − D0 mixing involving the D0 → Kπ decay mentioned above, in order to separate the DCS decay from the mixing signal, one must study the time-dependent decay rate. The proper-time evolu- tion of the particle states |D0 (t)〉 and |D0 (t)〉 are given by |D0phys(t)〉 = g+(t)|D 0〉 − q g−(t)|D0〉, |D0phys(t)〉 = g+(t)|D 0〉 − p g−(t)|D0〉, (1) where (e−im2t− Γ2t ± e−im1t− 12Γ1t), (2) with definitions m ≡ m1 +m2 , ∆m ≡ m2 −m1, Γ ≡ Γ1 + Γ2 , ∆Γ ≡ Γ2 − Γ1, (3) Note the sign of ∆m and ∆Γ is to be determined by experiments. In practice, one define the following mixing parameters x ≡ ∆m , y ≡ ∆Γ . (4) The time-dependent decay amplitudes for D0 (t) → K+π− and D0 (t) → K−π+ are described as 〈K+π−|H|D0phys(t)〉 = g+(t)AK+π− − g−(t)AK+π− AK+π− [λg+(t)− g−(t)], (5) 〈K−π+|H|D0phys(t)〉 = g+(t)AK−π+ − g−(t)AK−π+ AK−π+ [λg+(t)− g−(t)], (6) where AK+π− = 〈K+π−|H|D0〉, AK+π− = 〈K+π−|H|D0〉, AK−π+ = 〈K−π+|H|D0〉, and AK−π+ = 〈K−π+|H|D0〉. Here, λ and λ are de- fined as: λ ≡ p AK+π− AK+π− , (7) λ ≡ q AK−π+ AK−π+ . (8) From Eqs. (5) and (6), one can derive the general ex- pression for the time-dependent decay rate, in agreement http://arxiv.org/abs/0704.0120v3 with [3, 4]: dΓ(D0 (t) → K+π−) dtN = |AK +π− |2 e−Γt × [(|λ|2 + 1)cosh(yΓt) + (|λ|2 − 1)cos(xΓt) + 2Re(λ)sinh(yΓt) + 2Im(λ)sin(xΓt)] (9) dΓ(D0 (t) → K−π+) dtN = |AK−π+ | e−Γt × [(|λ|2 + 1)cosh(yΓt) + (|λ|2 − 1)cos(xΓt) + 2Re(λ)sinh(yΓt) + 2Im(λ)sin(xΓt)] (10) where N is a common normalization factor. In order to simplify the above formula, we make the following defi- nition: ≡ (1 +AM )e−iβ , (11) where β is the weak phase in mixing and AM is a real- valued parameter which indicates the magnitude of CP violation in the mixing. For f = K−π+ final state, we define AK+π− AK+π− r′e−iα AK−π+ AK−π+ re−iα, (12) where r′ and α′ (r and α) are the ratio and relative phase of the DCS decay rate and the CF decay rate. Then, λ and λ can be parameterized as λ = − 1 +AM e−i(α ′−β) , (13) λ = − r(1 +AM )e −i(α+β). (14) In order to demonstrate the CP violation in decay, we define RD(1 + AD) and 1 +AD Thus, Eqs. (13) and (14) can be expressed as λ = − 1 +AD 1 +AM e−i(δ−φ) , (15) λ = − 1 +AM 1 +AD e−i(δ+φ) , (16) where δ = α′ + α is the averaged phase difference be- tween DCS and CF processes, and φ = α− α′ We can characterize the CP violation in the mixing amplitude, the decay amplitude, and the interference between amplitudes with and without mixing, by real- valued parameters AM , AD, and φ as in Ref [5, 6]. In the limit of CP conservation, AM , AD and φ are all zero. AM = 0 means no CP violation in mixing, namely, |q/p| = 1; AD = 0 means no CP violation in decay, for this case, r = r′ = RD = |AK−π+/AK−π+ |2 = |AK+π−/AK+π− |2; φ = 0 means no CP violation in the interference between decay and mixing. In experimental searches, one can define CF decay as right-sign (RS) and DCS decay or via mixing followed by a CF decay as wrong-sign (WS). Here, we define the ratio of WS to RS decays as for D0: R(t) = dΓ(D0 (t) → K+π−) dtN × e−Γ|t| × 2|AK+π− |2 , (17) and for D0: R(t) = dΓ(D0 (t) → K−π+) dtN × e−Γ|t| × 2|AK−π+ |2 , (18) Taking into account that |λ|, |λ| ≪ 1 and x, y ≪ 1, keeping terms up to order x2, y2 and RD in the ex- pressions, neglecting CP violation in mixing, decay and the interference between decay with and without mixing (AM = 0, AD = 0, and φ = 0), expanding the time- dependent for xt, yt <∼ Γ−1, combing Eqs. (9) and (10), we can write Eqs. (17) and (18) as R(t) = R(t) = RD + (Γt)2, (19) where x′ = xcosδ + ysinδ, (20) y′ = −xsinδ + ycosδ. (21) In the limit of SU(3) symmetry, AK+π− and AK+π− (AK−π+ and AK−π+) are simply related by CKM fac- tors, AK+π− = (VcdV us/VcsV ud)AK+π− [7]. In particular, AK+π− and AK+π− have the same strong phase, leading to α′ = α = 0 in Eq. (12). But the SU(3) symmetry is broken according to the recent precise measurements from the B factories, the ratio [5]: R = BR(D 0 → K+π−) BR(D0 → K+π−) , (22) is unity in the SU(3) symmetry limit. But, the world average for this ratio is Rexp = 1.21± 0.03, (23) computed from the individual measurements using the standard method of Ref. [4]. Since the SU(3) is bro- ken in D → Kπ decays at the level of 20%, in which case the strong phase δ should be non-zero. Recently, a time-dependent analysis in D → Kπ has been performed based on 384 fb−1 luminosity at Υ (4S) [1]. By assuming CP conservation, they obtained the following neutral D mixing results RD = (3.03± 0.16± 0.10)× 10−3, = (−0.22± 0.30± 0.21)× 10−3, y′ = (9.7± 4.4± 3.1)× 10−3. (24) TABLE I: Experimental results used in the paper. Only one error is quoted, we have combined in quadrature statistical and systematic contributions. Parameter BaBar (×10−3) Belle(×10−3) Technique -0.22± 0.37 [1] 0.18+0.21−0.23 [8] Kπ ′ 9.7± 5.4 [1] 0.6+4.0−3.9 [8] Kπ RD 3.03± 0.19 [1] 3.64 ± 0.17 [8] Kπ yCP - 13.1 ± 4.1 [2] K −, π+π− x - 8.0± 3.4 [9] KSπ y - 3.3± 2.8 [9] KSπ The result is inconsistent with the no-mixing hypoth- esis with a significance of 3.9 standard deviations. The results from BaBar and Belle are in agreement within 2 standard deviation on the exact analysis of y′ measure- ment by using D → Kπ as listed in Table I. As indicated in Eq. (23), the strong phase δ should be non-zero due to the SU(3) violation. One has to know the strong phase difference exactly in order to extract the direct mixing parameters, x and y as defined in Eqs. (4). However, at the B factory, it is hard to do that with a model- independent way [7, 10]. In order to extract the strong phase δ we need data near the DD threshold to do a CP tag as discussed in Ref. [7]. Here, we would like to figure out the possible physics solution of the strong phase δ by using the recent results from the B factories with differ- ent decay modes, so that we can have an idea about the sensitivity to measure the strong phase at the BES-III project. In Ref [2], Belle collaboration also reported the result of yCP = τ(D0→K+π−) τ(D0→fCP ) − 1, where fCP = K+K− and yCP = (13.1± 3.2± 2.5)× 10−3. (25) The result is about 3.2σ significant deviation from zero (non-mixing). In the limit of CP symmetry, yCP = y [11, 12]. In the decay of D0 → KSπ+π−, Belle experiment has done a Dalitz plot (DP) analysis [9], they obtained the direct mixing parameters x and y as x = (8.0± 3.4)× 10−3, y = (3.3± 2.8)× 10−3, (26) where the error includes both statistic and systematic un- certainties. Since the parameterizations of the resonances on the DP are model-dependent, the results suffer from large uncertainties from the DP model. In this analysis, they see a significance of 2.4 standard deviations from non-mixing. Here, we will use the value of x measured in the DP analysis for further discussion. As shown in Eq. (21), once y, y′ and x are known, it is straightforward to extract the strong phase difference between DCS and CF decay in D0 → Kπ decay. If taking the measured central values of x, yCP (≈ y) , and y′ as input parameters, we found two-fold solutions for tanδ as below: tanδ = 0.35± 0.63, or − 7.14± 29.13, (27) which are corresponding to (19± 32)0 and (−820 ± 30)0, respectively. At ψ(3770) peak, to extract the mixing parameter y, one can make use of rates for exclusive D0D0 combina- tion, where both the D0 final states are specified (known as double tags or DT), as well as inclusive rates, where either the D0 or D0 is identified and the other D0 de- cays generically (known as single tags or ST) [13]. With the DT tag technique [14, 15], one can fully consider the quantum correlation in C = −1 and C = +1 D0D0 pairs produced in the reaction e+e− → D0D0(nπ0) and e+e− → D0D0γ(nπ0) [13, 16, 17], respectively. For the ST, in the limit of CP conservation, the rate of D0 decays into a CP eigenstate is given as [13]: Γfη ≡ Γ(D0 → fη) = 2A2fη [1− ηy] , (28) where fη is a CP eigenstate with eigenvalue η = ±1, and Afη = |〈fη|H|D0〉| is the real-valued decay amplitude. For the DT case, Gronau et. al. [7] and Xing [18] have considered time-integrated decays into correlated pairs of states, including the effects of non-zero final state phase difference. As discussed in Ref. [7], the rate of (D0D0)C=−1 → (l±X)(fη) is described as [7]: Γl;fη ≡ Γ[(l±X)(fη)] = A2l±XA (1 + y2) ≈ A2l±XA , (29) where Al±X = |〈l±X |H|D0〉| is real-valued amplitude for semileptonic decays, here, we neglect y2 term since y ≪ For C = −1 initial D0D0 state, y can be expressed in term of the ratios of DT rates and the double ratios of ST rates to DT rates [13]: Γl;f+Γf− Γl;f−Γf+ Γl;f−Γf+ Γl;f+Γf− . (30) For a small y, its error, ∆(y), is approximately 1/ Nl±X , where Nl±X is the total number of (l ±X) events tagged with CP -even and CP -odd eigenstates. The num- ber Nl±X of CP tagged events is related to the to- tal number of D0D0 pairs N(D0D0) through Nl±X ≈ N(D0D0)[BR(D0 → l± +X)× BR(D0 → f±)× ǫtag] ≈ 1.5 × 10−3N(D0D0), here we take the branching ratio- times-efficiency factor (BR(D0 → f±)× ǫtag) for tagging CP eigenstates is about 1.1% (the total branching ratio into CP eigenstates is larger than about 5% [4]). We find ∆(y) = N(D0D0) = ±0.003. (31) If we take the central value of y from the measurement of yCP at Belle experiment [2], thus, at BES-III exper- iment [19], with 20fb−1 data at ψ(3770) peak, the sig- nificance of the measurement of y could be around 4.3 σ deviation from zero. We can also take advantage of the coherence of the D0 mesons produced at the ψ(3770) peak to extract the strong phase difference δ between DCS and CF decay am- plitudes that appears in the time-dependent mixing mea- surement in Eq. (19) [7, 13]. Because the CP properties of the final states produced in the decay of the ψ(3770) are anti-correlated [16, 17], one D0 state decaying into a final state with definite CP properties immediately iden- tifies or tags the CP properties of the other side. As discussed in Ref. [7], the process of one D0 decaying to K−π+, while the other D0 decaying to a CP eigenstate fη can be described as ΓKπ;fη ≡ Γ[(K−π+)(fη)] ≈ A2A2fη |1 + η −iδ|2 ≈ A2A2fη (1 + 2η RDcosδ), where A = |〈K−π+|H|D0〉| and Afη = |〈fη|H|D0〉| are the real-valued decay amplitudes, and we have neglected the y2 terms in Eq. (32). In order to estimate the total sample of events needed to perform a useful measurement of δ, one defined [7, 10] an asymmetry ΓKπ;f+ − ΓKπ;f− ΓKπ;f+ + ΓKπ;f− , (33) where ΓKπ;f± is defined in Eq. (32), which is the rates for the ψ(3770) → D0D0 configuration to decay into flavor eigenstates and a CP -eigenstates f±. Eq. (32) implies a small asymmetry, A = 2 RDcosδ. For a small asymme- try, a general result is that its error ∆A is approximately NK−π+ , where NK−π+ is the total number of events tagged with CP -even and CP -odd eigenstates. Thus one obtained ∆(cosδ) ≈ 1 NK−π+ . (34) The expected number NK−π+ of CP -tagged events can be connected to the total number of D0D0 pairs N(D0D0) through NK−π+ ≈ N(D0D0)BR(D0 → K−π+)×BR(D0 → f±)×ǫtag ≈ 4.2×10−4N(D0D0) [7], here, as in Ref [7], we take the branching ratio-times- efficiency factor BR(D0 → f±) × ǫtag = 1.1%. With 0 0.2 0.4 0.6 0.8 1 FIG. 1: Illustrative plot of the expected error (∆δ) of the strong phase with various central values of cosδ. The expected error of cosδ is 0.04 by ssuming 20fb−1 data at ψ(3770) peak at BES-III. The two asterisks correspond to δ = 190 and −820, respectively. the measured RD = (3.03± 0.19)× 10−3 and BR(D0 → K−π+) = 3.8% [4], one found [7] ∆(cosδ) ≈ ±444√ N(D0D0) . (35) At BESIII, about 72 × 106 D0D0 pairs can be collected with 4 years’ running. If considering both K−π+ and K+π− final states, we thus estimate that one may be able to reach an accuracy of about 0.04 for cosδ. Fig- ure 1 shows the expected error of the strong phase δ with various central values of cosδ. With the expected ∆(cosδ) = ±0.04, the sensitivity of the strong phase varies with the physical value of cosδ. For δ = 190 and −820, the expected error could be ∆(δ) = ±8.70 and ±2.90, respectively. By combing the measurements of x inD0 → KSππ and yCP from Belle, one can obtain RM = (1.18±0.6)×10−4. At the ψ(3770) peak, D0D0 pair are produced in a state that is quantum-mechanically coherent [16, 17]. This en- ables simple new method to measure D0 mixing param- eters in a way similar proposed in Ref. [7]. At BES-III, the measurement of RM can be performed unambigu- ously with the following reactions [16]: (i) e+e− → ψ(3770) → D0D0 → (K±π∓)(K±π∓), (ii) e+e− → ψ(3770) → D0D0 → (K−e+ν)(K−e+ν), (iii) e+e− → D−D∗+ → (K+π−π−)(π+ [K+e−ν]). Reaction (i) in Eq. (36) can be normalized to D0D0 → (K−π+)(K+π−), the following time-integrated ratio is obtained by neglecting CP violation: N [(K−π+)(K−π+)] N [(K−π+)(K+π−)] 2 + y2 = RM . (37) For the case of semileptonic decay, as (ii) in Eq. (36), we N(l±l±) N(l±l∓) x2 + y2 = RM , (38) The observation of reaction (i) would be definite evi- dence for the existence of D0 −D0 mixing since the final state (K±π∓)(K±π∓) can not be produced from DCS decay due to quantum statistics [16, 17]. In particular, the initial D0D0 pair is in an odd eigenstate of C which will preclude, in the absence of mixing between the D0 and D0 over time, the formation of the symmetric state required by Bose statistics if the decays are to be the same final state. This final state is also very appealing experimentally, because it involves a two-body decay of both charm mesons, with energetic charged particles in the final state that form an overconstrained system. Par- ticle identification is crucial in this measurement because if both the kaon and pion are misidentified in one of the two D-meson decays in the event, it becomes impossi- ble to discern whether mixing has occurred. At BESIII, where the data sample is expected to be 20 fb−1 inte- grated luminosity at ψ(3770) peak, the limit will be 10−4 at 95% C.L. for RM , but only if the particle identification capabilities are adequate. Reactions (ii) and (iii) offer unambiguous evidence for the mixing because the mixing is searched for in the semileptonic decays for which there are no DCS decays. Of course since the time-evolution is not measured, obser- vation of Reactions (ii) and (iii) actually would indicate the violation of the selection rule relating the change in charm to the change in leptonic charge which holds true in the standard model [16]. In Table II, the sensitivity for RM measurements in different decay modes are estimated with 4 years’ run at BEPCII. TABLE II: The sensitivity for RM measurements at BES-III with different decay modes with 4 years’ run at BESPCII 0 Mixing Reaction Events Sensitivity RS(×104) RM (×10 ψ(3770) → (K−π+)(K−π+) 10.4 1.0 ψ(3770) → (K−e+ν)(K−e+ν) 8.9 ψ(3770) → (K−e+ν)(K−µ+ν) 8.1 3.7 ψ(3770) → (K−µ+ν)(K−µ+ν) 7.3 In the limit of CP conservation, by combing the mea- surements of x in D0 → KSππ and yCP from Belle, one can obtain RM = (1.18± 0.6)× 10−4. With 20fb−1 data at BES-III, about 12 events for the precess D0D0 → (K±π∓)(K±π∓) can be produced. One can observe 3.0 events after considering the selection efficiency at BE- SIII, which could be about 25% for the four charged particles. The background contamination due to double particle misidentification is about 0.6 event with 20fb−1 data at BES-III [20]. Table III lists the expected mixing signal for Nsig = N(K ±π∓)(K±π∓), background Nbkg , and the Poisson probability P (n), where n is the possible number of observed events in experiment. In Table III, we assume the RM = 1.18× 10−4, the expected number of mixing signal events are estimated with 10fb−1 and 20fb−1, respectively. TABLE III: The expected mixing signal for Nsig = N(K±π∓)(K±π∓), background Nbkg , and the Poisson prob- ability P (n) in 10 fb−1 and 20 fb−1 at BES-III at ψ(3770) peak, respectively. Here, we take the mixing rate RM = 1.18× 10−4. 10 fb−1 (ψ(3770)) 20 fb−1 (ψ(3770)) 36 million D0D0 72 million D0D0 Nsig 1.5 3.0 Nbkg 0.3 0.6 P (n = 0) 15.7% 2.5% P (n = 1) 29.1% 9.1% P (n = 2) 26.9% 16.9% P (n = 3) 16.6% 20.9% P (n = 4) 7.7% 19.3% P (n = 5) 2.8% 14.3% P (n = 6) 0.9% 8.8% P (n = 7) 0.2% 4.7% P (n = 8) 0.1% 2.2% P (n = 9) 0.01% 0.9% In conclusion, we discuss the constraints on the strong phase difference in D0 → Kπ decay according to the most recent measurements of y′, yCP and x from B fac- tories. We estimate the sensitivity of the measurement of mixing parameter y at ψ(3770) peak in BES-III experi- ment. With 20 fb−1 data, the uncertainty ∆(y) could be 0.003. Thus, assuming y at a percent level, we can make a measurement of y at a significance of 4.3σ deviation from zero. The sensitivity of the strong phase differ- ence at BES-III are obtained by using data near the DD threshold with CP tag technique at BES-III experiment. Finally, we estimated the sensitivity of the measurements of the mixing rate RM , and find that BES-III experiment may not be able to make a significant measurement of RM with current luminosity by using coherent DD state at ψ(3770) peak. One of the authors (H. B. Li) would like to thank David Asner and Zhi-Zhong Xing for stimulating dis- cussion, Chang-Zheng Yuan for useful discussion on the statistics used in this paper, and also thank Stephen L. Olsen and Yang-Heng Zheng for commenting on this manuscript. We thank BES-III collaboration for provid- ing us many numerical results based on GEANT4 simula- tion. This work is supported in part by the National Nat- ural Science Foundation of China under contracts Nos. 10205017, 10575108,10521003, and the Knowledge Inno- vation Project of CAS under contract Nos. U-612 and U-530 (IHEP). ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] [1] B. Aubert, et. al., (BaBar Collaboration), hep-ex/0703020. [2] K. Abe et. al., (Belle Collaboration), hep-ex/0703036. [3] Y. Nir, hep-ph/0703235. [4] W. M. Yao et. al., (Partcle Data Group), J. Phys.G 33, 1(2006). [5] A. F. Falk, Y. Nir, and A. Petrov, JHEP12, 019 (1999). [6] H. B. Li and M. Z. Yang, Phys. Rev. D74, 094016(2006). [7] M. Gronau, Y. Grossman, J. L. Rosner, Phys. Lett. B508, 37 (2001). [8] L. M. Zhang et. al., (Belle Collaboration), Phys. Rev. Lett. 96, 151801 (2006). [9] M. Staric, Talk given at the 42th Renocontres De Moriond On Electroweak Interactions And Unified The- ories, March 10-17, 2007, La Thuile, Italy. [10] G. Burdman and I. Shipsey, Ann. Rev. Nucl. Part. Sci. 53, 431 (2003). [11] S. Bergmann, Y. Grossman, Z. Ligeti, Y. Nir and A. A. Petrov, Phys. Lett. B486, 418(2000). [12] D. Atwood, A. A. Petrov, Phys. Rev. D71, 054032 (2005). [13] D. M. Asner and W. M. Sun Phys. Rev. D73, 034024 (2006);D. M. Asner et. al., Int. J. Mod. Phys. A21, 5456 (2006); W. M. Sun, hep-ex/0603031, AIP Conf. Proc. 842:693-695 (2006). [14] R. M. Baltrusaitis, et. al., (MARK III Collaboration), Phys. Rev. Lett. 56, 2140(1986). [15] J. Adler, et. al., (MARK III Collaboration), Phys. Rev. Lett. 60, 89 (1988). [16] I. I. Bigi, Proceed. of the Tau-Charm Workshop, L. V. Beers (ed.), SLAC-Report-343, page 169, (1989). [17] I. Bigi, A. Sanda, Phys. Lett. B171, 320(1986). [18] Z. Z. Xing, Phys. Rev. D55, 196(1997); Z. Z. Xing, Phys. Lett. B372,317(1996). [19] BES-III Collaboration, ”The Preliminary Design Report of the BESIII Detector”, Report No. IHEP-BEPCII-SB- [20] Y. Z. Sun et. al., to appear at HEP & NP 31, 1 (2007). mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] http://arxiv.org/abs/hep-ex/0703020 http://arxiv.org/abs/hep-ex/0703036 http://arxiv.org/abs/hep-ph/0703235 http://arxiv.org/abs/hep-ex/0603031

0704.0121

Meta-Stable Brane Configuration of Product Gauge Groups

Meta-Stable Brane Configuration of Product Gauge Groups Changhyun Ahn Department of Physics, Kyungpook National University, Taegu 702-701, Korea [email protected] Abstract Starting from the N = 1 SU(Nc) × SU(N ′c) gauge theory with fundamental and bifun- damental flavors, we apply the Seiberg dual to the first gauge group and obtain the N = 1 dual gauge theory with dual matters including the gauge singlets. By analyzing the F-term equations of the superpotential, we describe the intersecting type IIA brane configuration for the meta-stable nonsupersymmetric vacua of this gauge theory. By introducing an orientifold 6-plane, we generalize to the case for N = 1 SU(Nc)×SO(N ′c) gauge theory with fundamental and bifundamental flavors. Finally, the N = 1 SU(Nc)× Sp(N ′c) gauge theory with matters is also described very briefly. http://arxiv.org/abs/0704.0121v3 1 Introduction It is well-known that the N = 1 SU(Nc) QCD with fundamental flavors has a vanishing superpotential before we deform this theory by mass term for quarks. The vanishing su- perpotential in the electric theory makes it easier to analyze its nonvanishing dual magnetic superpotential. Sometimes by tuning the various rotation angles between NS5-branes and D6-branes appropriately, even if the electric theory has nonvanishing superpotential, one can make the nonzero superpotential to vanish in the electric theory. Two procedures, deforming the electric gauge theory by adding the mass for the quarks and taking the Seiberg dual magnetic theory from the electric theory, are crucial to find out meta-stable supersymmetry breaking vacua in the context of dynamical supersymmetry breaking [1, 2]. Some models of dynamical supersymmetry breaking can be studied by gauging the subgroup of the flavor symmetry group by either field theory analysis or using the brane configuration 1. In this paper, starting from the known N = 1 supersymmetric electric gauge theories, we construct the N = 1 supersymmetric magnetic gauge theories by brane motion and linking number counting. The dual gauge group appears only on the first gauge group. Based on their particular limits of corresponding magnetic brane configurations in the sense that their electric theories do not have any superpotentials except the mass deformations for the quarks, we describe the intersecting brane configurations of type IIA string theory for the meta-stable nonsupersymmetric vacua of these gauge theories. We focus on the cases where the whole gauge group is given by a product of two gauge groups. One example can be realized by three NS5-branes with D4- and D6-branes, and the other by four NS5-branes with D4- and D6-branes. For the latter, the appropriate orientifold 6-plane should be located at the center of this brane configuration in order to have two gauge groups. Of course, it is also possible, without changing the number of gauge groups, to have the brane configuration consisting of five NS5-branes and orientifold 6-plane, at which the extra NS5-brane is located, with D4- and D6-branes, but we’ll not do this particular case in this paper. In section 2, we review the type IIA brane configuration that contains three NS5-branes, corresponding to the electric theory based on the N = 1 SU(Nc) × SU(N ′c) gauge theory [4, 5, 6] with matter contents and deform this theory by adding the mass term for the quarks. Then we construct the Seiberg dual magnetic theory which is N = 1 SU(Ñc)×SU(N ′c) gauge theory with corresponding dual matters as well as various gauge singlets, by brane motion and linking number counting. We do not touch the part of second gauge group SU(N ′c) in 1For the type IIA brane configuration description of N = 1 supersymmetric gauge theory, see the review paper [3]. this dual process. In section 3, we consider the nonsupersymmetric meta-stable minimum by looking at the magnetic brane configuration we obtained in section 2 and present the corresponding intersecting brane configuration of type IIA string theory, along the line of [7, 8, 9, 10, 11](see also [12, 13, 14]) and we describe M-theory lift of this supersymmetry breaking type IIA brane configuration. In section 4, we describe the type IIA brane configuration that contains four NS5-branes, corresponding to the electric theory based on the N = 1 SU(Nc) × SO(N ′c) gauge theory [15] with matter contents and deform this theory by adding the mass term for the quarks. Then we take the Seiberg dual magnetic theory which is N = 1 SU(Ñc) × SO(N ′c) gauge theory with corresponding dual matters as well as various gauge singlets, by brane motion and linking number counting. The part of second gauge group SO(N ′c) in this dual process is not changed under this process. In section 5, the nonsupersymmetric meta-stable minimum by looking at the magnetic brane configuration we obtained in section 4 is constructed and we present the corresponding intersecting brane configuration of type IIA string theory and describe M-theory lift of this supersymmetry breaking type IIA brane configuration, as we did in section 3. In section 6, we describe the similar application to the N = 1 SU(Nc) × Sp(N ′c) gauge theory [15] briefly and make some comments for the future directions. 2 The N = 1 supersymmetric brane configuration of SU(Nc)× SU(N ′c) gauge theory After reviewing the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(Nc)×SU(N ′c) gauge theory [4, 5, 6], we construct the Seiberg dual magnetic theory which is N = 1 SU(Ñc)× SU(N ′c) gauge theory. 2.1 Electric theory with SU(Nc)× SU(N ′c) gauge group The gauge group is given by SU(Nc)×SU(N ′c) and the matter contents [4, 5, 6] are given by • Nf chiral multiplets Q are in the fundamental representation under the SU(Nc), Nf chiral multiplets Q̃ are in the antifundamental representation under the SU(Nc) and then Q are in the representation (Nc, 1) while Q̃ are in the representation (Nc, 1) under the gauge group • N ′f chiral multiplets Q′ are in the fundamental representation under the SU(N ′c), N ′f chiral multiplets Q̃′ are in the antifundamental representation under the SU(N ′c) and then Q are in the representation (1,N′ ) while Q̃′ are in the representation (1,N′ ) under the gauge group • The flavor singlet field X is in the bifundamental representation (Nc,N′c) under the gauge group and its complex conjugate field X̃ is in the bifundamental representation (Nc,N under the gauge group In the electric theory, since there exist Nf quarks Q, Nf quarks Q̃, one bifundamental field X which will give rise to the contribution of N ′c and its complex conjugate X̃ which will give rise to the contribution of N ′c, the coefficient of the beta function of the first gauge group factor is bSU(Nc) = 3Nc −Nf −N ′c and similarly since there exist N ′f quarks Q ′, N ′f quarks Q̃ ′, one bifundamental field X which will give rise to the contribution of Nc and its complex conjugate X̃ which will give rise to the contribution of Nc, the coefficient of the beta function of the second gauge group factor is bSU(N ′c) = 3N c −N ′f −Nc. The anomaly free global symmetry is given by [SU(Nf ) × SU(N ′f )]2 × U(1)3 × U(1)R [4, 5, 6] and let us denote the strong coupling scales for SU(Nc) as Λ1 and for SU(N c) as Λ2. The theory is asymptotically free when bSU(Nc) = 3Nc − Nf − N ′c > 0 for the SU(Nc) gauge theory and when bSU(N ′c) = 3N c −N ′f −Nc > 0 for the SU(N ′c) gauge theory. The type IIA brane configuration for this theory can be described by Nc color D4-branes (01236) suspended between a middle NS5-brane (012345) and the right NS5’-brane (012389) (denoted by NS5′R-brane) along x 6 direction, together with Nf D6-branes (0123789) which are parallel to NS5′R-brane and have nonzero (45) directions. Moreover, an extra left NS5’-brane (denoted by NS5′L-brane) is located at the left hand side of a middle NS5-brane along the x6 direction and there exist N ′c color D4-branes suspended between them, with N f D6-branes which have zero (45) directions. These are shown in Figure 1 explicitly. See also [3] for the brane configuration. By realizing that the two outer NS5′L,R-branes are perpendicular to a middle NS5-brane and the fact that Nf D6-branes are parallel to NS5 R-brane and N f D6-branes are parallel to NS5′L-brane, the classical superpotential vanishes. However, one can deform this theory. Then the classical superpotential by deforming this theory by adding the mass term for the quarks Q and Q̃, along the lines of [1, 11, 10, 9, 8, 7], is given by W = mQQ̃ (2.1) and this type IIA brane configuration can be summarized as follows 2: • One middle NS5-brane with worldvolume (012345). • Two NS5’-branes with worldvolume (012389). • Nf D6-branes with worldvolume (0123789) located at the positive region in v direction. • Nc color D4-branes with worldvolume (01236). These are suspended between a middle NS5-brane and NS5′R-brane. • N ′c color D4-branes with worldvolume (01236). These are suspended between NS5′L- brane and a middle NS5-brane. Now we draw this electric brane configuration in Figure 1 and we put the coincident Nf D6-branes in the nonzero v direction. If we ignore the left NS5′L-brane, N c D4-branes and N ′f D6-branes, then this brane configuration corresponds to the standard N = 1 SQCD with the gauge group SU(Nc) with Nf massive flavors. The electric quarks Q and Q̃ correspond to strings stretching between the Nc color D4-branes with Nf D6-branes, the electric quarks Q′ and Q̃′ correspond to strings between the N ′c color D4-branes with N f D6-branes and the bifundamentals X and X̃ correspond to strings stretching between the Nc color D4-branes with N ′c color D4-branes. Figure 1: The N = 1 supersymmetric electric brane configuration of SU(Nc)×SU(N ′c) with Nf chiral multiplets Q, Nf chiral multiplets Q̃, N f chiral multiplets Q ′, N ′f chiral multiplets Q̃′, the flavor singlet bifundamental field X and its complex conjugate bifundamental field X̃ . The Nf D6-branes have nonzero v coordinates where v = m for equal massive case of quarks Q, Q̃ while Q′ and Q̃′ are massless. 2We introduce two complex coordinates v ≡ x4 + ix5 and w ≡ x8 + ix9 for simplicity. 2.2 Magnetic theory with SU(Ñc)× SU(N ′c) gauge group One can consider dualizing one of the gauge groups regarding as the other gauge group as a spectator. One takes the Seiberg dual for the first gauge group factor SU(Nc) while remaining the second gauge group factor SU(N ′c) unchanged. Also we consider the case where Λ1 >> Λ2, in other words, the dualized group’s dynamical scale is far above that of the other spectator group. Let us move a middle NS5-brane to the right all the way past the right NS5′R-brane. For example, see [12, 13, 14, 11, 10, 9, 8, 7]. After this brane motion, one arrives at the Figure 2. Note that there exists a creation of Nf D4-branes connecting Nf D6-branes and NS5 R-brane. Recall that the Nf D6-branes are perpendicular to a middle NS5-brane in Figure 1. The linking number [16] of NS5-brane from Figure 2 is L5 = − Ñc. On the other hand, the linking number of NS5-brane from Figure 1 is L5 = −Nf2 +Nc−N c. Due to the connection of N ′c D4-branes with NS5 R-brane, the presence of N c in the linking number arises. From these two relations, one obtains the number of colors of dual magnetic theory Ñc = Nf +N c −Nc. (2.2) The linking number counting looks similar to the one in [7] where there exists a contribution from O4-plane. Let us draw this magnetic brane configuration in Figure 2 and recall that we put the coincident Nf D6-branes in the nonzero v directions in the electric theory, along the lines of [12, 13, 14, 11, 10, 9, 8, 7]. The Nf created D4-branes connecting between D6-branes and NS5′R-brane can move freely in the w direction. Moreover since N c D4-branes are suspending between two equal NS5′L,R-branes located at different x 6 coordinate, these D4-branes can slide along the w direction also. If we ignore the left NS5′L-brane, N c D4-branes and N D6-branes(detaching these from Figure 2), then this brane configuration corresponds to the standard N = 1 SQCD with the magnetic gauge group SU(Ñc = Nf −Nc) with Nf massive flavors [12, 13, 14]. The dual magnetic gauge group is given by SU(Ñc) × SU(N ′c) and the matter contents are given by • Nf chiral multiplets q are in the fundamental representation under the SU(Ñc), Nf chiral multiplets q̃ are in the antifundamental representation under the SU(Ñc) and then q are in the representation (Ñc, 1) while q̃ are in the representation (Ñc, 1) under the gauge group • N ′f chiral multiplets Q′ are in the fundamental representation under the SU(N ′c), N ′f chiral multiplets Q̃′ are in the antifundamental representation under the SU(N ′c) and then Q Figure 2: The N = 1 supersymmetric magnetic brane configuration of SU(Ñc = Nf +N ′c − Nc) × SU(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, N ′f chiral multiplets Q′, N ′f chiral multiplets Q̃ ′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ as well as Nf fields F ′, its complex conjugate Nf fields F̃ ′, N f fields M and the gauge singlet Φ. There exist Nf flavor D4-branes connecting D6-branes and NS5′R-brane. are in the representation (1,N′ ) while Q̃′ are in the representation (1,N′ ) under the gauge group • The flavor singlet field Y is in the bifundamental representation (Ñc,N′c) under the gauge group and its complex conjugate field Ỹ is in the bifundamental representation (Ñc,N ) under the gauge group There are (Nf +N 2 gauge singlets in the first dual gauge group factor as follows: • Nf -fields F ′ are in the fundamental representation under the SU(N ′c), its complex con- jugate Nf -fields F̃ ′ are in the antifundamental representation under the SU(N c) and then F are in the representation (1,N′ ) under the gauge group while F̃ ′ are in the representation (1,N′ ) under the gauge group These additional Nf SU(N c) fundamentals and Nf SU(N c) antifundamentals are origi- nating from the SU(Nc) chiral mesons X̃Q and XQ̃ respectively. It is clear to see that from the Figure 2, since the Nf D6-branes are parallel to the NS5 R-brane, the newly created Nf D4-branes can slide along the plane consisting of these D6-branes and NS5′R-brane arbitrar- ily. Then strings stretching between the Nf D6-branes and N c D4-branes will give rise to these additional Nf SU(N c) fundamentals and Nf SU(N c) antifundamentals. • N2f -fields M are in the representation (1, 1) under the gauge group This corresponds to the SU(Nc) chiral meson QQ̃ and the fluctuations of the singlet M correspond to the motion of Nf flavor D4-branes along (789) directions in Figure 2. • The N ′2c -fields Φ is in the representation (1,N′2c − 1)⊕ (1, 1) under the gauge group This corresponds to the SU(Nc) chiral meson XX̃ and note that X has a representation of SU(N ′c) while X̃ has a representation N of SU(N ′c). The fluctuations of the singlet Φ correspond to the motion of N ′c D4-branes suspended two NS5 L,R-branes along the (789) directions in Figure 2. In the dual theory, since there exist Nf quarks q, Nf quarks q̃, one bifundamental field Y which will give rise to the contribution of N ′c and its complex conjugate Ỹ which will give rise to the contribution of N ′c, the coefficient of the beta function for the first gauge group factor [6] is SU( eNc) = 3Ñc −Nf −N ′c = 2Nf + 2N ′c − 3Nc where we inserted the number of colors given in (2.2) in the second equality and since there exist N ′f quarks Q ′, N ′f quarks Q̃ ′, one bifundamental field Y which will give rise to the contribution of Ñc, its complex conjugate Ỹ which will give rise to the contribution of Ñc, Nf fields F ′, its complex conjugate Nf fields F̃ ′ and the singlet Φ which will give rise to N c, the coefficient of the beta function of second gauge group factor [6] is SU(N ′c) = 3N ′c −N ′f − Ñc −Nf −N ′c = N ′c +Nc − 2Nf −N ′f . Therefore, both SU(Ñc) and SU(N c) gauge couplings are IR free by requiring the negativeness of the coefficients of beta function. One can rely on the perturbative calculations at low energy for this magnetic IR free region b SU( eNc) < 0 and b SU(N ′c) < 0. Note that the SU(N ′c) fields in the magnetic theory are different from those of the electric theory. Since bSU(N ′c)−b SU(N ′c) SU(N ′c) is more asymptotically free than SU(N mag [6]. Neglecting the SU(N ′c) dynamics, the magnetic SU(Ñc) is IR free when Nf +N Nc [6]. The dual magnetic superpotential, by adding the mass term (2.1) for Q and Q̃ in the electric theory which is equal to put a linear term in M in the dual magnetic theory, is given Wdual = Mqq̃ + Y F ′q̃ + Ỹ qF̃ ′ + ΦY Ỹ +mM (2.3) where the mesons in terms of the fields defined in the electric theory are M ≡ QQ̃, Φ ≡ XX̃, F ′ ≡ X̃Q, F̃ ′ ≡ XQ̃. By orientifolding procedure(O4-plane) into this brane configuration, the q(Q) and q̃(Q̃) are equivalent to each other, the Y (X) and Ỹ (X̃) become identical and F ′ and F̃ ′ become the same. Then the reduced superpotential is identical with the one in [7]. Here q and q̃ are fun- damental and antifundamental for the gauge group index respectively and antifundamentals for the flavor index. Then, qq̃ has rank Ñc while m has a rank Nf . Therefore, the F-term condition, the derivative the superpotential Wdual with respect to M , cannot be satisfied if the rank Nf exceeds Ñc. This is so-called rank condition and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of Y, Ỹ , F ′ and F̃ ′ to vanish. The classical moduli space of vacua can be obtained from F-term equations qq̃ +m = 0, q̃M + F̃ ′Ỹ = 0, Mq + Y F ′ = 0, F ′q̃ + Ỹ Φ = 0, q̃Y = 0, qF̃ ′ + ΦY = 0, Ỹ q = 0, Y Ỹ = 0. Then, it is easy to see that there exist three reduced equations q̃M = 0 = Mq, qq̃ +m = 0 and other F-term equations are satisfied if one takes the zero vacuum expectation values for the fields Y, Ỹ , F ′ and F̃ ′. Then the solutions can be written as follows: < q > = meφ1 eNc , < q̃ >= me−φ1 eNc 0 , < M >= 0 Φ01Nf− eNc < Y > = < Ỹ >=< F ′ >=< F̃ ′ >= 0. (2.4) Let us expand around a point on (2.4), as done in [1]. Then the remaining relevant terms of superpotential are given by W reldual = Φ0 (δϕ δϕ̃+m) + δZ δϕ q̃0 + δZ̃ q0δϕ̃ by following the same fluctuations for the various fields as in [9]: q01 eNc + (δχ+ + δχ−)1 eNc , q̃ = q̃01 eNc + (δχ+ − δχ−)1 eNc δϕ̃ δY δZ δZ̃ Φ01Nf− eNc as well as the fluctuations of Y, Ỹ , F ′ and F̃ ′. Note that there exist also three kinds of terms, the vacuum < q > multiplied by δỸ δF̃ ′, the vacuum < q̃ > multiplied by δF ′δY , and the vacuum < Φ > multiplied by δY δỸ . However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [17], one gets that m2Φ0 will contain (log 4− 1) > 0 implying that these are stable. 3 Nonsupersymmetric meta-stable brane configuration of SU(Nc)× SU(N ′c) gauge theory Now we recombine Ñc D4-branes among Nf flavor D4-branes connecting between D6-branes and NS5′R-brane with those connecting between NS5 R-brane and NS5-brane and push them in +v direction from Figure 2. After this procedure, there are no color D4-branes between NS5′R-brane and NS5-brane. For the flavor D4-branes, we are left with only (Nf − Ñc) flavor D4-branes. Then the minimal energy supersymmetry breaking brane configuration is shown in Figure 3, along the lines of [12, 13, 14, 11, 10, 9, 8, 7]. If we ignore the left NS5′L-brane, N c D4- branes and N ′f D6-branes(detaching these from Figure 3), as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU(Ñc = Nf − Nc) with Nf massive flavors [12, 13, 14]. Figure 3: The nonsupersymmetric minimal energy brane configuration of SU(Ñc = Nf + N ′c −Nc)× SU(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, N ′f chiral multiplets Q′, N ′f chiral multiplets Q̃ ′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ and various gauge singlets. The type IIA/M-theory brane construction for the N = 2 gauge theory was described by [18] and after lifting the type IIA description to M-theory, the corresponding magnetic M5-brane configuration 3 with equal mass for the quarks where the gauge group is given by 3The M5-brane lives in (0123) directions and is wrapping on a Riemann surface inside (4568910) directions. The Taub-NUT space in (45610) directions is parametrized by two complex variables v and y and the flat two dimensions in (89) directions by a complex variable w. See [14] for the relevant discussions. SU(Ñc)×SU(N ′c), in a background space of xt = vN k=1(v−ek) where this four dimensional space replaces (45610) directions, is described by t3 + (v eNc + · · · )t2 + (vN ′c + · · · )t+ vN ′f (v − ek) = 0 (3.1) where ek is the position of the D6-branes in the v direction(for equal massive case, we can write ek = m) and we have ignored the lower power terms in v in t 2 and t denoted by · · · and the scales for the gauge groups in front of the first term and the last term, for simplicity. For fixed x, the coordinate t corresponds to y. From this curve (3.1) of cubic equation for t above, the asymptotic regions for three NS5- branes can be classified by looking at the first two terms providing NS5-brane asymptotic region, next two terms providing NS5′R-brane asymptotic region and the final two terms giving NS5′L-brane asymptotic region as follows 1. v → ∞ limit implies w → 0, y ∼ v eNc + · · · NS asymptotic region. 2. w → ∞ limit implies v → m, y ∼ wNf+N ′f−N ′c + · · · NS ′L asymptotic region, v → m, y ∼ wN ′c− eNc + · · · NS ′R asymptotic region. Here the two NS5′L,R-branes are moving in the +v direction holding everything else fixed instead of moving D6-branes in the +v direction, in the spirit of [14]. The harmonic function sourced by the D6-branes can be written explicitly by summing over two contributions from the Nf and N f D6-branes and similar analysis to both solve the differential equation and find out the nonholomorphic curve can be done [14, 10, 9, 8, 7]. An instability from a new M5-brane mode arises. 4 The N = 1 supersymmetric brane configuration of SU(Nc)× SO(N ′c) gauge theory After reviewing the type IIA brane configuration corresponding to the electric theory based on the N = 1 SU(Nc) × SO(N ′c) gauge theory [15], we describe the Seiberg dual magnetic theory which is N = 1 SU(Ñc)× SO(N ′c) gauge theory. 4.1 Electric theory with SU(Nc)× SO(N ′c) gauge group The gauge group is given by SU(Nc)× SO(N ′c) and the matter contents [15](similar matter contents are found in [4]) are given by • Nf chiral multiplets Q are in the fundamental representation under the SU(Nc), Nf chiral multiplets Q̃ are in the antifundamental representation under the SU(Nc) and then Q are in the representation (Nc, 1) while Q̃ are in the representation (Nc, 1) under the gauge group • 2N ′f chiral multiplets Q′ are in the fundamental representation under the SO(N ′c) and then Q′ are in the representation (1,N′ ) under the gauge group • The flavor singlet field X is in the bifundamental representation (Nc,N′c) under the gauge group and the flavor singlet X̃ is in the bifundamental representation (Nc,N ) under the gauge group In the electric theory, since there exist Nf quarks Q, Nf quarks Q̃, one bifundamental field X which will give rise to the contribution of N ′c and its complex conjugate X̃ which will give rise to the contribution of N ′c, the coefficient of the beta function of the first gauge group factor is bSU(Nc) = 3Nc −Nf −N ′c and similarly, since there exist 2N ′f quarks Q ′, one bifundamental field X which will give rise to the contribution of Nc and its complex conjugate X̃ which will give rise to the contribution of Nc, the coefficient of the beta function of the second gauge group factor is bSO(N ′c) = 3(N c − 2)− 2N ′f − 2Nc. The anomaly free global symmetry is given by SU(Nf ) 2 × SU(2N ′f)×U(1)2 ×U(1)R and let us denote the strong coupling scales for SU(Nc) as Λ1 and for SO(N c) as Λ2, as in previous section. The theory is asymptotically free when bSU(Nc) > 0 for the SU(Nc) gauge theory and when bSO(N ′c) > 0 for the SO(N c) gauge theory. The type IIA brane configuration of N = 2 gauge theory [19] consists of four NS5-branes (012345) which have different x6 values, Nc and N c D4-branes (01236) suspended between them, 2Nf and 2N f D6-branes (0123789) and an orientifold 6 plane (0123789) of positive Ramond charge 4. According to Z2 symmetry of orientifold 6-plane(O6-plane) sitting at v = 0 and x6 = 0, the coordinates (x4, x5, x6) transform as −(x4, x5, x6), as usual. See also [3] for the discussion of O6-plane. 4There are many different brane configurations with O6-plane in the literature and some of them are present in [20, 21, 22, 23, 24]. By rotating the third and fourth NS5-branes which are located at the right hand side of O6-plane, from v direction toward −w and +w directions respectively, one obtains N = 1 theory. Their mirrors, the first and second NS5-branes which are located at the left hand side of O6-plane, can be rotated in a Z2 symmetric manner due to the presence of O6-plane simultaneously. That is, if the first NS5-brane rotates by an angle −ω in (v, w) plane, denoted by NS5−ω-brane [3], then the mirror image of this NS5-brane, the fourth NS5-brane, is rotated by an angle ω in the same plane, denoted by NS5ω-brane. If the second NS5-brane rotates by an angle θ in (v, w) plane, denoted by NS5θ-brane [3], then the mirror image of this NS5-brane, the third NS5-brane, is rotated by an angle −θ in the same plane, denoted by NS5−θ-brane. For more details, see the Figure 4 We also rotate the N ′f D6-branes which are located between the second NS5-brane and an O6-plane and make them be parallel to NS5θ-brane and denote them as D6θ-brane with zero v coordinate(the angle between the unrotated D6-branes and D6θ-branes is equal to − θ) and its mirrors N ′f D6-branes appear as D6−θ-branes between the O6-plane and third NS5-brane. There is no coupling between the adjoint field and the quarks since the rotated D6θ-branes are parallel to the rotated NS5θ-brane [5, 3]. Similarly, the Nf D6-branes which are located between the third NS5-brane and the fourth NS5-brane can be rotated and we can make them be parallel to NS5ω-brane and denote them as D6ω-branes with nonzero v coordinate(the angle between the unrotated D6-branes and D6ω-branes is equal to −ω) and its mirrors Nf D6-branes appear as D6−ω-branes between the first NS5-brane and the second NS5-brane. Moreover the Nc D4-branes are suspended between the first NS5-brane and the second NS5-brane(and its mirrors) and the N ′c D4-branes are suspended between the second NS5- brane and the third NS5-brane. For this brane setup 6, the classical superpotential is given by [15] W = −1 4 tan(ω − θ) + tan 2θ tr(XX̃)2 + trXX̃X̃X 4 sin 2θ (trXX̃)2 4Nc tan(ω − θ) . (4.1) It is easy to see that when θ approaches 0 and ω approaches π , then this superpotential vanishes. 5The angles of θ1 and θ2 in [15] are related to the angles θ and ω as follows: θ = θ1 and ω = θ2. 6For arbitrary angles θ and ω, the superpotential for the SU(Nc) sector is given by W = XφX̃ + tan(ω − θ) trφ2 where φ ia an adjoint field for SU(Nc). There is no coupling between φ and Nf quarks because D6±ω-branes are parallel to NS5±ω-branes. The superpotential for the SO(N c) sector is given by W = XφAX̃ +XφSX̃ + tan θ trφ A − 1tan θ trφ S where φA and φS are an adjoint field and a symmetric tensor for SO(N ′c) [25]. After integrating out φ, φA and φS , the whole superpotential can be written as in (4.1). Now one summarizes the supersymmetric electric brane configuration with their worldvol- umes in type IIA string theory as follows. • NS5−ω-brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with negative x6. • NS5θ-brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with negative x6. • NS5−θ-brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with positive x6. • NS5ω-brane with worldvolume by both (0123) and two spatial dimensions in (v, w) plane and with positive x6. • N ′f D6θ-branes with worldvolume by both (01237) and two spatial dimensions in (v, w) plane and with negative x6 and v = 0. • N ′f D6−θ-branes with worldvolume by both (01237) and two space dimensions in (v, w) plane and with positive x6 and v = 0. • Nf D6ω-branes with worldvolume by both (01237) and two spatial dimensions in (v, w) plane and with positive x6. Before the rotation, the distance from Nc color D4-branes in the +v direction is nonzero. • Nf D6−ω-branes with worldvolume by both (01237) and two space dimensions in (v, w) plane and with negative x6. Before the rotation, the distance from Nc color D4-branes in the −v direction is nonzero. • O6-plane with worldvolume (0123789) with v = 0 = x6. •Nc D4-branes connecting NS5−ω-brane and NS5θ-brane, with worldvolume (01236) with v = 0 = w(and its mirrors). • N ′c D4-branes connecting NS5θ-brane and NS5−θ-brane, with worldvolume (01236) with v = 0 = w. We draw the type IIA electric brane configuration in Figure 4 which was basically given in [15] already but the only difference is to put Nf D6-branes in the nonzero v direction in order to obtain nonzero masses for the quarks which are necessary to obtain the meta-stable vacua. 4.2 Magnetic theory with SU(Ñc)× SO(N ′c) gauge group One takes the Seiberg dual for the first gauge group factor SU(Nc) while remaining the second gauge group factor SO(N ′c), as in previous case. Also we consider the case where Λ1 >> Λ2, in other words, the dualized group’s dynamical scale is far above that of the other spectator group. Figure 4: The N = 1 supersymmetric electric brane configuration of SU(Nc)×SO(N ′c) with Nf chiral multiplets Q, Nf chiral multiplets Q̃, 2N f chiral multiplets Q ′, the flavor singlet bifundamental field X and its complex conjugate bifundamental field X̃ . The Nf D6ω-branes have nonzero v coordinates where v = m(and its mirrors) for equal massive case of quarks Q, Q̃ while Q′ is massless. Let us move the NS5−θ-brane to the right all the way past the right NS5ω-brane(and its mirrors to the left). After this brane motion, one arrives at the Figure 5. Note that there exists a creation of Nf D4-branes connecting Nf D6ω-branes and NS5ω-brane(and its mirrors). Recall that the Nf D6ω-branes are not parallel to the NS5−θ-brane in Figure 4(and its mirrors). The linking number of NS5−θ-brane from Figure 5 is L5 = − Ñc. On the other hand, the linking number of NS5−θ-brane from Figure 4 is L5 = −Nf2 +Nc −N c. From these, one gets the number of colors in dual magnetic theory Ñc = Nf +N c −Nc. (4.2) Let us draw this magnetic brane configuration in Figure 5 and remember that we put the coincident Nf D6ω-branes in the nonzero v directions(and its mirrors). The Nf created D4- branes connecting between D6ω-branes and NS5ω-brane can move freely in the w direction, as in previous case. Moreover, since N ′c D4-branes are suspending between two unequal NS5±ω-branes located at different x 6 coordinate, these D4-branes cannot slide along the w direction, for arbitrary rotation angles. If we are detaching all the branes except NS5ω-brane, NS5−θ-brane, D6ω-branes, Nf D4-branes and Ñc D4-branes from Figure 5, then this brane configuration corresponds to N = 1 SQCD with the magnetic gauge group SU(Ñc = Nf−Nc) with Nf massive flavors with tilted NS5-branes. The dual magnetic gauge group is given by SU(Ñc) × SO(N ′c) and the matter contents are given by Figure 5: The N = 1 supersymmetric magnetic brane configuration of SU(Ñc = Nf +N ′c − Nc) × SO(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, 2N ′f chiral multiplets Q′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ as well as Nf fields F ′, its complex conjugate Nf fields F̃ ′, N f fields M and the gauge singlet Φ. There exist Nf flavor D4-branes connecting D6ω-branes and NS5ω-brane(and its mirrors). • Nf chiral multiplets q are in the fundamental representation under the SU(Ñc), Nf chiral multiplets q̃ are in the antifundamental representation under the SU(Ñc) and then q are in the representation (Ñc, 1) while q̃ are in the representation (Ñc, 1) under the gauge group • 2N ′f chiral multiplets Q′ are in the fundamental representation under the SO(N ′c) and then Q′ are in the representation (1,N′ ) under the gauge group • The flavor singlet field Y is in the bifundamental representation (Ñc,N′c) under the gauge group and its complex conjugate field Ỹ is in the bifundamental representation (Ñc,N ) under the gauge group There are (Nf +N 2 gauge singlets in the first dual gauge group factor as follows: • Nf -fields F ′ are in the fundamental representation under the SO(N ′c), Nf -fields F̃ ′ are in the fundamental representation under the SO(N ′c) and then F ′ are in the representation (1,N′ ) under the gauge group while F̃ ′ are in the representation (1,N′ ) under the gauge group These additional 2Nf SO(N c) vectors are originating from the SU(Nc) chiral mesons X̃Q and XQ̃ respectively. It is easy to see that from the Figure 5, since the D6−ω-branes are par- allel to the NS5−ω-brane, the newly created Nf D4-branes can slide along the plane consisting of D6−ω-branes and NS5−ω-brane arbitrarily(and its mirrors). Then strings connecting the Nf D6−ω-branes and N c D4-branes will give rise to these additional 2Nf SO(N c) vectors. • N2f -fields M are in the representation (1, 1) under the gauge group This corresponds to the SU(Nc) chiral meson QQ̃ and the fluctuations of the singlet M correspond to the motion of Nf flavor D4-branes along (789) directions in Figure 5. • The N ′2c singlet Φ is in the representation (1, adj)⊕ (1, symm) under the gauge group This corresponds to the SU(Nc) chiral meson XX̃ and note that both X and X̃ have representation N′ of SO(N ′c). In general, the fluctuations of the singlet Φ correspond to the motion of N ′c D4-branes suspended two NS5±ω-branes along the (789) directions in Figure 5. In the dual theory, since there exist Nf quarks q, Nf quarks q̃, one bifundamental field Y which will give rise to the contribution of N ′c and its complex conjugate Ỹ which will give rise to the contribution of N ′c, the coefficient of the beta function of the first gauge group factor with (4.2) is SU( eNc) = 3Ñc −Nf −N ′c = 2Nf + 2N ′c − 3Nc and since there exist 2N ′f quarks Q ′, one bifundamental field Y which will give rise to the contribution of Ñc, its complex conjugate Ỹ which will give rise to the contribution of Ñc, Nf fields F ′, its complex conjugate Nf fields F̃ ′ and the singlet Φ which will give rise to N c, the coefficient of the beta function is SO(N ′c) = 3(N ′c − 2)− 2N ′f − 2Ñc − 2Nf − 2N ′c = −N ′c + 2Nc − 4Nf − 2N ′f − 6. Therefore, both SU(Ñc) and SO(N c) gauge couplings are IR free by requiring the negativeness of the coefficients of beta function. One can rely on the perturbative calculations at low energy for this magnetic IR free region b SU( eNc) < 0 and b SO(N ′c) < 0. Note that the SO(N ′c) fields in the magnetic theory are different from those of the electric theory. Since bSO(N ′c)−b SO(N ′c) SO(N ′c) is more asymptotically free than SO(N mag. Neglecting the SO(N ′c) dynamics, the magnetic SU(Ñc) is IR free when Nf +N Nc, as in previous case. The dual magnetic superpotential, by adding the mass term for Q and Q̃ in the electric theory which is equal to put a linear term in M in the dual magnetic theory, is given by 7 Wdual = (Φ2 + · · · ) +Q′ΦQ′ +Mqq̃ + Y F̃ ′q̃ + Ỹ qF ′ + ΦY Ỹ +mM (4.3) where the mesons in terms of the fields defined in the electric theory are M ≡ QQ̃, Φ ≡ XX̃, F ′ ≡ X̃Q, F̃ ′ ≡ XQ̃. 7There appears a mismatch between the number of colors from field theory analysis and those from brane motion when we take the full dual process on the two gauge group factors simultaneously [15]. By adding D4-branes to the dual brane configuration without affecting the linking number counting, this mismatch can be removed. Similar phenomena occurred in [5, 26]. Then this turned out that there exists a deformation ∆W generated by the meson Q′XX̃Q′. This is exactly the second term, Q′ΦQ′, in (4.3). In previous example, there is no such deformation term in (2.3). We abbreviated all the relevant terms and coefficients appearing in the quartic superpotential for the bifundamentals in electric theory (4.1) and denote them here by Φ2 + · · · . Here q and q̃ are fundamental and antifundamental for the gauge group index respectively and antifundamentals for the flavor index. Then, qq̃ has rank Ñc and m has a rank Nf . Therefore, the F-term condition, the derivative the superpotential Wdual with respect to M , cannot be satisfied if the rank Nf exceeds Ñc and the supersymmetry is broken. Other F-term equations are satisfied by taking the vacuum expectation values of Y, Ỹ , F ′, F̃ ′ and Q′ to vanish. The classical moduli space of vacua can be obtained from F-term equations and one gets qq̃ +m = 0, q̃M + F ′Ỹ = 0, Mq + Y F̃ ′ = 0, F̃ ′q̃ + Ỹ Φ = 0, q̃Y = 0, qF ′ + ΦY = 0, Ỹ q = 0, Q′Q′ + Y Ỹ = 0, ΦQ′ = 0. Then, it is easy to see that there exists a solution q̃M = 0 = Mq, qq̃ +m = 0. Other F-term equations are satisfied if one takes the zero vacuum expectation values for the fields Y, Ỹ , F ′, Q′ and F̃ ′. Then the solutions can be written as < q > = meφ1 eNc , < q̃ >= me−φ1 eNc 0 , < M >= 0 Φ01Nf− eNc < Y > = < Ỹ >=< F ′ >=< F̃ ′ >=< Q′ >= 0. (4.4) Let us expand around a point on (4.4), as done in [1]. Then the remaining relevant terms of superpotential are given by W reldual = Φ0 (δϕ δϕ̃+m) + δZ δϕ q̃0 + δZ̃ q0δϕ̃ by following the similar fluctuations for the various fields as in [9]. Note that there exist also four kinds of terms, the vacuum < q > multiplied by δỸ δF ′, the vacuum < q̃ > multiplied by δF̃ ′δY , the vacuum < Φ > multiplied by δY δỸ , and the vacuum < Φ > multiplied by δQ′δQ′. However, by redefining these, they do not enter the contributions for the one loop result, up to quadratic order. As done in [17], one gets that m2Φ0 will contain (log 4 − 1) > 0 implying that these are stable. 5 Nonsupersymmetric meta-stable brane configuration of SU(Nc)× SO(N ′c) gauge theory Since the electric superpotential (4.1) vanishes for θ = 0 and ω = π , the corresponding magnetic superpotential in (4.3) does not contain the terms Φ2 + · · · and it becomes Wdual = Q′ΦQ′ +Mqq̃ + Y F̃ ′q̃ + Ỹ qF ′ + ΦY Ỹ Now we recombine Ñc D4-branes among Nf flavor D4-branes connecting between D6ω=π D6-branes and NS5ω=π = NS5′R-brane with those connecting between NS5 R-brane and NS5−θ=0 = NS5R-brane(and its mirrors) and push them in +v direction from Figure 5. Of course their mirrors will move to−v direction in a Z2 symmetric manner due to the O6+-plane. After this procedure, there are no color D4-branes between NS5′R-brane and NS5R-brane. For the flavor D4-branes, we are left with only (Nf − Ñc) D4-branes(and its mirrors). Then the minimal energy supersymmetry breaking brane configuration is shown in Figure 6. If we ignore all the branes except NS5′R-brane, NS5R-brane, D6-branes, (Nf − Ñc) D4- branes and Ñc D4-branes, as observed already, then this brane configuration corresponds to the minimal energy supersymmetry breaking brane configuration for the N = 1 SQCD with the magnetic gauge group SU(Ñc) with Nf massive flavors [12, 13, 14]. Note that N D4-branes can slide w direction for this brane configuration. The type IIA/M-theory brane construction for the N = 2 gauge theory was described by [19] and after lifting the type IIA description we explained so far to M-theory, the correspond- ing magnetic M5-brane configuration with equal mass for the quarks where the gauge group is given by SU(Ñc)×SO(N ′c), in a background space of xt = (−1)Nf+N k=1(v 2− e2k) where this four dimensional space replaces (45610) directions, is characterized by t4 + (v eNc + · · · )t3 + (vN ′c + · · · )t2 + (v eNc + · · · )t+ v2N ′f+4 (v2 − e2k) = 0. From this curve of quartic equation for t above, the asymptotic regions can be classified by looking at the first two terms providing NS5R-brane asymptotic region, next two terms providing NS5′R-brane asymptotic region, next two terms providing NS5 L-brane asymptotic region, and the final two terms giving NS5L-brane asymptotic region as follows: 1. v → ∞ limit implies w → 0, y ∼ v eNc + · · · NS5R asymptotic region, w → 0, y ∼ v2Nf+2N ′f− eNc+4 + · · · NS5L asymptotic region. Figure 6: The nonsupersymmetric minimal energy brane configuration of SU(Ñc = Nf + N ′c −Nc)× SO(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, 2N ′f chiral multiplets Q′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ and gauge singlets. The N ′c D4-branes and 2(Nf − Ñc) D4-branes can slide w direction freely in a Z2 symmetric way. 2. w → ∞ limit implies v → −m, y ∼ w eNc−N ′c + · · · NS5′L asymptotic region, v → +m, y ∼ wN ′c− eNc + · · · NS5′R asymptotic region. Now the two NS5′L,R-branes are moving in the ±v direction holding everything else fixed instead of moving D6-branes in the ±v direction. Then the mirrors of D4-branes are moved appropriately. The harmonic function sourced by the D6-branes can be written explicitly by summing of three contributions from the Nf and N f D6-branes(and its mirrors) plus an O6- plane, and similar analysis to solve the differential equation and find out the nonholomorphic curve can be done [14, 10, 9, 8, 7]. In this case also, we expect an instability from a new M5-brane mode. 6 Discussions So far, we have dualized only the first gauge group factor in the gauge group SU(Nc)×SO(N ′c). What happens if we dualize the second gauge group factor SO(N ′c)?(For the case SU(Nc)× SU(N ′c), the behavior of dual for the second gauge group will be the same as when we take the dual for the first gauge group factor.) This can be done by moving the NS5θ-brane and N ′f D6θ-branes that can be located at the nonzero v coordinate for massive quarks Q ′, to the right passing through O6-plane(and their mirrors to the left). According to the linking number counting, one obtains the dual gauge group SU(Nc)×SO(Ñ ′c = 2Nc+2N ′f −N ′c+4). One can easily see that there is a creation of N ′f D4-branes connecting NS5θ-brane and D6θ-branes(and its mirrors). Then from the brane configuration, there exist the additional 2N ′f SU(Nc) quarks originating from the SO(N c) chiral mesons Q ′X ≡ F̃ ′ and Q′X̃ ≡ F ′. The deformed superpotential ∆W = Q′XX̃Q′ can be interpreted as the mass term of F ′F̃ ′. Then one can write dual magnetic superpotential in this case. However, it is not clear how the recombination of color and flavor D4-branes and splitting procedure between them in the construction of meta-stable vacua arises since there is no extra NS5-brane between two NS5±θ-branes. If there exists an extra NS5-brane at the origin of our brane configuration(then the gauge group and matter contents will change), it would be possible to construct the corresponding meta-stable brane configuration. It would be interesting to study these more in the future. As already mentioned in [8] and section 4, the matter contents in [4] are different from the ones in section 4 with the same gauge group. In other words, the theory of SU(Nc)×SO(N ′c) with X , which transform as fundamental in SU(Nc) and vector in SO(N c), a antisymmetric tensor A in SU(Nc), as well as fundamentals for SU(Nc) and vectors for SO(N c) can confine either SU(Nc) factor or SO(N c) factor. This theory can be described by the web of branes in the presence of O4−-plane and orbifold fixed points. With two NS5-branes and O4−-plane, by modding out Z3 symmetry acting on (v, w) as (v, w) → (v exp(2πi3 ), w exp( )), the resulting gauge group will be SU(Nc)×SO(Nc+4) with above matter contents [27]. Similar analysis for SU(Nc)×Sp(Nc2 −2) gauge group with opposite O4 +-plane can be done. Then in this case, the matter in SU(Nc) will be a symmetric tensor S and other matter contents are present also. It would be interesting to see whether this gauge theory and corresponding brane configuration will provide a meta-stable vacuum. Let us comment on other possibility where the gauge group is given by SU(Nc)× Sp(N ′c) and the matter contents are given by • Nf chiral multiplets Q are in the fundamental representation under the SU(Nc), Nf chiral multiplets Q̃ are in the antifundamental representation under the SU(Nc) and then Q are in the representation (Nc, 1) while Q̃ are in the representation (Nc, 1) under the gauge group • 2N ′f chiral multiplets Q′ are in the fundamental representation under the Sp(N ′c) and then Q′ are in the representation (1, 2N′ ) under the gauge group • The flavor singlet field X is in the bifundamental representation (Nc, 2N′c) under the gauge group and the flavor singlet X̃ is in the bifundamental representation (Nc, 2N ) under the gauge group One can compute the coefficients of beta functions of the each gauge group factor, as we did for previous examples. The type IIA brane configuration of an electric theory is exactly the same as the Figure 4 except the RR charge O6-plane with negative sign. The classical superpotential 8 is given by [15] W = −1 4 tan(ω − θ) + tan 2θ tr(XX̃)2 − trXX̃X̃X 4 sin 2θ (trXX̃)2 4Nc tan(ω − θ) . (6.1) In this case, when θ approaches π and ω approaches 0, then this superpotential vanishes. The dual magnetic gauge group is given by SU(Ñc = Nf + 2N c −Nc)× Sp(N ′c) with the same number of colors of dual theory as those in previous cases and the matter contents are given by • Nf chiral multiplets q are in the fundamental representation under the SU(Ñc), Nf chiral multiplets q̃ are in the antifundamental representation under the SU(Ñc) and then q are in the representation (Ñc, 1) while q̃ are in the representation (Ñc, 1) under the gauge group • 2N ′f chiral multiplets Q′ are in the fundamental representation under the Sp(N ′c) and then Q′ are in the representation (1, 2N′ ) under the gauge group • The flavor singlet field Y is in the bifundamental representation (Ñc, 2N′c) under the gauge group and its complex conjugate field Ỹ is in the bifundamental representation (Ñc, 2N under the gauge group There are (Nf + 2N 2 gauge singlets in the first dual gauge group factor • Nf -fields F ′ are in the fundamental representation under the Sp(N ′c), Nf -fields F̃ ′ are in the fundamental representation under the Sp(N ′c) and then F ′ are in the representation (1, 2N′ ) under the gauge group while F̃ ′ are in the representation (1, 2N′ ) under the gauge group • N2f -fields M are in the representation (1, 1) under the gauge group • The 4N ′2c singlet Φ is in the representation (1, adj)⊕ (1, antisymm) under the gauge group The dual magnetic superpotential for arbitrary angles is given by (4.3) with appropriate Sp(N ′c) invariant metric J . The stability analysis can be done similarly. 8The superpotential for the Sp(N ′c) sector is given by W = XφAX̃+XφSX̃+tan θ trφ S− 1tan θ trφ A where φS and φA are an adjoint field(symmetric tensor) and an antisymmetric tensor for Sp(N c) [25]. Note that there is a sign change in the second trace term of the superpotential in (6.1), compared to (4.1). After following the procedure from Figure 4 to Figure 5 with opposite RR charge for O6- plane and by taking the limit where θ → π and ω → 0, the minimal energy supersymmetry breaking brane configuration is shown in Figure 7. Figure 7: The nonsupersymmetric minimal energy brane configuration of SU(Ñc = Nf + 2N ′c −Nc)×Sp(N ′c) with Nf chiral multiplets q, Nf chiral multiplets q̃, 2N ′f chiral multiplets Q′, the flavor singlet bifundamental field Y and its complex conjugate bifundamental field Ỹ and gauge singlets. Note the RR charge of O6-plane is negative and its charge is equivalent to −4 D6-branes. The 2N ′c D4-branes and 2(Nf − Ñc) D4-branes can slide w direction freely in a Z2 symmetric way. Compared to the previous nonsupersymmetric brane configuration in Figure 6, the role of NS5-brane and NS5’-brane is interchanged to each other: undoing the Seiberg dual in the context of [13]. This kind of feature of recombination and splitting between color D4-branes and flavor D4-branes occurs in [8]. At the electric brane configuration, Nf D6-branes are perpendicular to NS5-brane and this leads to the coupling between the quarks and adjoint in the superpotential. However, the overall coefficient function including this extra terms vanishes and eventually the whole electric superpotential will vanish according to the above limit we take. From the quartic equation with the presence of opposite RR charge for O6-plane, in a background space of xt = (−1)Nf+N ′fv2N ′f−4 k=1(v 2 − e2k), t4 + (v eNc + · · · )t3 + (vN ′c + · · · )t2 + (v eNc + · · · )t+ v2N ′f−4 (v2 − e2k) = 0, the asymptotic regions can be classified as follows: 1. v → ∞ limit implies w → 0, y ∼ vN ′c− eNc · · · NS5R asymptotic region, w → 0, y ∼ v eNc−N ′c + · · · NS5L asymptotic region. 2. w → ∞ limit implies v → −m, y ∼ w2Nf+2N ′f− eNc−4 + · · · NS5′L asymptotic region, v → +m, y ∼ w eNc + · · · NS5′R asymptotic region. In [28], the SU(7)×S̃p(1) model and SU(9)×S̃p(2) model can be obtained by dualizing the SU(7)× SU(2) model with a bifundamental and two antifundamentals for SU(7) and a fun- damental for SU(2) and the SU(9)×SU(2) with a bifundamental and two antifundamentals for SU(9) and a fundamental for Sp(1) respectively(Note that Sp(1) ∼ SU(2)). The matter contents in an electric theory are different from those in previous paragraph. The matter contents in the magnetic description are given by an antisymmetric tensor and a fundamen- tal in the first gauge group as well as a bifundamental, a fundamental in the second gauge group and two antifundamentals in the first gauge group. There exists a nonzero dual mag- netic superpotential. Also the dual description the SU(7)× S̃p(1) model and SU(9)× S̃p(2) model can be constructed from the antisymmetric models of Affleck-Dine-Seiberg by gauging a maximal flavor symmetry and adding the extra matter to cancel all anomalies and extra flavor. On the other hand, the models SU(2Nc + 1)× SU(2) have its brane box model descrip- tion in [29] where the above examples correspond to Nc = 3 and Nc = 4 respectively. In particular, the case where Nc = 1(the gauge group is SU(3) × SU(2), i.e., (3, 2) model [30]) was described by brane box model with superpotential or without superpotential. Then it would be interesting to obtain the Seiberg dual for these models using brane box model and look for the possibility of having meta-stable vacua for these models. Moreover, this gauge theory was generalized to SU(2Nc + 1) × Sp(N ′c) model with a bifundamental and 2N ′c antifundamentals for SU(2Nc + 1) and a fundamental for Sp(N c) and its dual descrip- tion SU(2Nc + 1)× Sp(Ñ ′c = Nc −N ′c − 1) with a bifundamental and 2N ′c antifundamentals for SU(2Nc + 1) and a fundamental for Sp(N c) as well as two gauge singlets [28]. For the particular range of Nc, the dual theory is IR free, not asymptotically free. According to [31], SU(2Nc) with antisymmetric tensor and antifundamentals can be de- scribed by two gauge groups Sp(2Nc−4)×SU(2Nc) with bifundamental and antifundamentals for SU(2Nc). Some of the brane realization with zero superpotential was given in the brane box model in [29]. Similarly from the result of [32] by following the method of [31], the dual description for SU(2Nc +1) with antisymmetric tensor and fundamentals can be represented by two gauge group factors. This dual theory breaks the supersymmetry at the tree level. Similar discussions are present in [33]. Then it would be interesting to construct the corre- sponding Seigerg dual and see how the electric theory and its magnetic theory can be mapped into each other in the brane box model. Ther are also different directions concerning on the meta-stable vacua in different contexts and some of the relevant works are present in [34]-[43] where some of them use anti D-branes and some of them describe the type IIB theory and it would be interesting to find out how similarities if any appear and what are the differences in what sense between the present work and those works. Acknowledgments I would like to thank A. Hanany and K. Landsteiner for discussions. 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Kachru, “Gauge / gravity dual- ity and meta-stable dynamical supersymmetry breaking,” JHEP 0701, 083 (2007) [arXiv:hep-th/0610212]. http://arxiv.org/abs/hep-th/0703236 http://arxiv.org/abs/hep-th/0703135 http://arxiv.org/abs/hep-th/0703112 http://arxiv.org/abs/hep-th/0703047 http://arxiv.org/abs/hep-th/0702077 http://arxiv.org/abs/hep-th/0611303 http://arxiv.org/abs/hep-th/0611069 http://arxiv.org/abs/hep-th/0610249 http://arxiv.org/abs/hep-th/0610212 Introduction The N=1 supersymmetric brane configuration of SU(Nc) SU(Nc') gauge theory Electric theory with SU(Nc) SU(Nc') gauge group Magnetic theory with SU(N"0365Nc) SU(Nc') gauge group Nonsupersymmetric meta-stable brane configuration of SU(Nc) SU(Nc') gauge theory The N=1 supersymmetric brane configuration of SU(Nc) SO(Nc') gauge theory Electric theory with SU(Nc) SO(Nc') gauge group Magnetic theory with SU(N"0365Nc) SO(Nc') gauge group Nonsupersymmetric meta-stable brane configuration of SU(Nc) SO(Nc') gauge theory Discussions

0704.0122

Spinor dipolar Bose-Einstein condensates; Classical spin approach

Spinor dipolar Bose-Einstein condensates; Classical spin approach M. Takahashi1, Sankalpa Ghosh1,2, T. Mizushima1, K. Machida1 Department of Physics, Okayama University, Okayama 700-8530, Japan and Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India (Dated: October 26, 2018) Magnetic dipole-dipole interaction dominated Bose-Einstein condensates are discussed under spin- ful situations. We treat the spin degrees of freedom as a classical spin vector, approaching from large spin limit to obtain an effective minimal Hamiltonian; a version extended from a non-linear sigma model. By solving the Gross-Pitaevskii equation we find several novel spin textures where the mass density and spin density are strongly coupled, depending upon trap geometries due to the long-range and anisotropic natures of the dipole-dipole interaction. PACS numbers: 03.75.Mn, 03.75.Hh, 67.57.Fg Bose-Einstein condensates (BEC) with internal de- grees of freedom, the so-called spinor BEC have attract much attention experimentally and theoretically in re- cent years [1]. Spinor BEC opens up a new paradigm where the order parameter of condensates is described by a multi-component vector [2, 3]. This can be possi- ble by optically trapping cold atoms where all hyperfine states are liberated, while magnetic trapping freezes its freedom. So far 23Na (the hyperfine state F = 1), and 87Rb (F = 2) are extensively investigated. Griesmaier et al. [4] have recently succeeded in achiev- ing BEC of 52Cr atom gases whose magnetic moment per atom is 3 µB (Bohr magneton). There has been already emerging [5] several novel aspects associated with larger magnetic moment in 52Cr atom even in this magnetic trapping, where all spin moments are polarized along an external magnetic field. Namely the magnetic dipole- dipole (d-d) interaction, which is proportional to F 2 is expected to play an important role in a larger spin atom. It is natural to expect realization of BEC with still larger spin atomic species under the spinful situations by optical trapping or control the d-d interaction via the Feshbach resonance relative to other interaction channels. There has already been existing a large amount of theo- retical studies for dipolar BEC [6]. Most of them treat the polarized case where the dipolar moments are aligned along an external field. The intrinsic anisotropic or ten- sorial nature of the d-d interaction relative to the polar- ization axis manifests itself in various properties. The head-to-tail moment arrangement due to the d-d interac- tion is susceptible to a shape instability by concentrating atoms in the central region. We have seen already that tensorial and long-ranged d-d interaction is responsible for this kind of shape dependent phenomenon where the mass density is constrained by the polarization axis. In contrast the theoretical studies of the spinor dipolar BEC are scarce, and just started with several impressive works [7, 8, 9, 10]. They consider either the F = 1 spinor BEC by taking into account the d-d interaction or F = 3 for 52Cr atom gases in a realistic situation. Here one must handle a 7-component spinor with 5 different interaction channels g0, g2, g4, g6, and gd. The parameter space to hunt is large and difficult enough to find a stable con- figuration. The situation becomes further hard towards a larger F where the d-d interaction is more important and eventually dominant one among various channels. Here we investigate generic properties of the spinor dipolar BEC under an optical trapping where the d-d interaction dominates other interactions except for the s-wave repulsive channel. A proposed model Hamilto- nian is intended to capture essential properties of the spinor BEC system. We note that this long-ranged and anisotropic d-d interaction has fascinated researches for a long time, for example, Luttinger and Tisza in their seminal paper [11] theoretically discussed the stable spin configurations of a spin model on a lattice where classi- cal spins with a fixed magnitude are free to rotate on a lattice. The present paper is designed to generalize this lattice spin model to a dipolar BEC system. Here we are interested in the interplay between the spin degrees of freedom and the mass density through the d-d interac- tion. We approach this problem from atomic species with large magnetic moment. This spinor dipole BEC with the hyper fine state F (Fz = −F,−F+1, · · · , F ) is character- ized by 2F+1 components Ψα(r). In general the number of the interaction channels are F + 1. For example, the F = 1 spinor BEC [2, 3] is characterized by the scattering lengths a0 and a2, leading to the spin independent repul- sive interaction g0 = 4πh̄ 2(a0 + 2a0)/3m and the spin dependent exchange interaction g2 = 4πh̄ 2(a2 − a0)/3m. Since a0 and a2 are comparable, g2 is actually small; |g2|/g0 ∼ 1/10 for 23Na [12, 13] and ∼ 1/35 for 87Rb [14, 15]. This tendency that, except for the dominant re- pulsive part g0, other spin-dependent channels are nearly cancelled is likely to be correct for other F ’s [16]. We can take a view in this paper that instead of work- ing with ~Ψ(r) full quantum mechanical 2F + 1 compo- nents (ΨF ,ΨF−1, · · · ,Ψ−F ) with the interaction param- eters g0, g2, g4, ..., and g2F , the order parameter can be simplified to ~Ψ(ri) = ψ(ri)~S(ri) where ~S(ri) is a clas- sical vector with |~S(ri)|2 = 1. Namely we can treat it http://arxiv.org/abs/0704.0122v2 as the classical spin vector whose magnitude |ψ(ri)|2 is proportional to the local condensate density. In other words, we focus on long-wavelength and low energy tex- tured solutions of a dipolar system which will manifest the interplay between the mass and spin density degrees of freedom. We start with the following minimal model Hamilto- d3ri~Ψ †(ri)H0(ri)~Ψ(ri) d3rid rjVdd(ri, rj)|ψ(ri)|2|ψ(rj)|2,(1) H0 = − ∇2i + Vtrap(ri)− µ+ |~Ψ(ri)|2, (2) Vdd(ri, rj) = ~Si · ~Sj − 3(~Si · ~eij)(~Sj · ~eij) , (3) where ~eij ≡ (ri − rj)/rij with rij = |ri − rj |. The uni- axially symmetric trap potential is given by Vtrap(r) = mω2{γ(x2 + y2) + z2} with γ being the anisotropy pa- rameter. µ is the chemical potential. The repulsive (g > 0) and the dipole-dipole (gd) interaction are intro- duced. The classical spin vector ~Si ≡ ~S(ri) characterizes the internal degrees of freedom of the system at the site i and is denoted by spherical coordinates (ϕ(ri), θ(ri)) with |~Si|2 = 1. A dimensionless form of this Hamiltonian may be written as |∇ψ(ri)|2 + ni ∇θ(ri) + sin2 θ(ri) ∇ϕ(ri) + γ2(x2 + y2) + z2 − 2µni + gn2i d3rid ~Si · ~Sj − 3(~Si · ~eij)(~Sj · ~eij) ninj , (4) with |ψ(ri)|2 = ni. We note that the spin gradient term in the first line is a Non-linear sigma model[17]. Here it is extended to include the dipole-dipole interaction be- tween the different parts of the spin density. The energy (length) is measured by the harmonic frequency ω (har- monic length d ≡ 1/ mω) with h̄ = 1 The functional derivatives with respect to ψ∗(ri), ϕ(ri) and θ(ri) lead to the corresponding Gross-Pitaevskii equations. In this paper under a fixed repulsive interaction (g/ωd3 = 0.01) we vary the d-d interaction gd in a range of 0 ≤ gd ≤ 0.4g, beyond which the system is unsta- ble. We consider two types of the confinement: A pan- cake (γ = 0.2) and a cigar (γ = 5.0) to see the shape dependence of the d-d interaction, which is long-ranged and anisotropic. The total particle number ∼ 104. The three dimensional space is discretized into the lattice sites ∼ 2.5 × 104. Using the imaginary time (τ) evolution of Gross-Pitaevskii equations e.g. ∂ψi/∂τ = −δH/δψ∗i , we obtain stable configurations for spin and particle densi- ties by starting with various initial patterns. We start with the pancake shape (γ = 0.2). Figure 1 shows a stereographic image of the particle density and spin distributions. We call it spin current texture, where the spin direction circulates around the origin and is con- fined into the x-y plane without the third component, that is, a coplanar texture. It is seen that the particle density distribution is strongly coupled to the spin one; FIG. 1: Stereographic view of the spin current texture, dis- playing simultaneously the number and spin densities. The pancake (γ = 0.2) is distorted and at the center the number density is depleted to give a doughnut like shape. gd = 0.2g. All spins lie in the x-y plane, i.e. a coplanar spin structure, circulating around the origin O. The length of the arrow is proportional to its number density. Inset shows the schematic spin configuration on z = 0 plane. In the central region the particles are depleted over the coherent length ξd of the d-d interaction. In the present case ξd ∼ 2.0ξc (ξc is the ordinary coherent length of the FIG. 2: The r-flare texture. Left (right) column shows the cross-sectional density plots of the particle number (the corre- sponding spin structure). The circular profile in the x-y plane is spontaneously broken. gd = 0.2g, γ = 0.2. repulsive interaction). This spin current texture can be readily explained in the following way: (1) Locally, along the stream line of the spin current the head-to-tail configuration minimizes the energy. (2) Globally, the spins at A and B which are situated far apart about the origin O shown in inset of Fig. 1 are orientated anti-parallel to minimize the d-d in- teraction. (3) When the two antiparallel spins at A and B come closer towards the origin O, the kinetic energy due to the spin modulation increases. To avoid this en- ergy loss, the particle number is depleted in the central region at the cost of the harmonic potential energy. For an alternative explanation of the spin current texture we rewrite the d-d interaction as vdd(rij) ∝ Y2µ(cos θ)Σµ(ij) with Σµ(ij) being a rank 2 tensor consisting of the two spins at i and j sites, and Y2µ(cos θ) a spherical harmonics [18]. θ is the polar angle in spherical coordinates of the system. The spin current texture shown in inset of Fig. 1 picks up the phase factor e2iϕ when winding around the origin. This is coupled to Y2±2(cos θ) ∝ sin2 θ, meaning that this orbital moment dictates the number density depletion at the pancake cen- ter. The spin-orbit coupling directly manifests itself here. The total angular momentum consisting of the spin and orbit ones is a conserved quantity of the present axis- symmetric system, leading to the Einstein-de Haas effect [7]. The spin current texture is stable for the wide range of anisotropy γ: 0.01 ≤ γ ≤ 0.6, beyond which it becomes unstable. Figure 2 displays another stable configuration in a sim- ilar situation. The left (right) column shows the den- sity plots of the particle number (the corresponding spin structure). The spins are almost parallel to the x-axis, but at the outer region they bent away. We call it r-flare texture, which is a non-coplanar spin arrangement. It is clearly seen that the axis-symmetry in the x-y plane, FIG. 3: Cross sections of the particle number in Fig. 2 along the x and y-axis compared with Thomas-Fermi (TF) profile for gd = 0. The profile is elongated (compressed) along the x (y)-axis. which was originally circular, is spontaneously broken so that the circular shape is elongated along the x-axis and compressed along the y-axis. Figure 3 displays the x and y-axis cross-sections of the particle density, com- pared with the Thomas-Fermi (TF) profile for gd = 0 with the same particle number. Because of the d-d in- teraction which favors the head-to-tail arrangement, the particle number is increased at the center. The bending tendency at the circumference increases with increasing gd. Beyond a certain critical value gd ∼= 0.27g for ∼ 104 particles, the r-flare texture becomes unstable, indicating a quantum phase transition. Upon increasing the total particle number the r-flare is replaced by the spin current texture. We also note that the z-flare texture in which the polarization points to the z-axis is equally stable as we explain shortly. Let us turn to the cigar shape case elongated along the z-axis with the trap anisotropy γ = 5.0. The stable configuration we obtain is shown in Fig. 4 where the spin structure is basically a flare spin texture which is a non- coplanar spin arrangement. Namely, the bending occurs radially so that the spin texture is a three dimensional object, but keeps axis-symmetry around the z-axis. The particle density is modified from the TF profile for gd = 0, elongated along the z direction and compressed to the z- axis. This can be understood by seeing Fig. 4 (b). The up- spin density near the center exerts the d-d force so as to align the outer spins parallel to the vector connecting the center and its position, taking the head-to-tail configu- ration. This results in a non-coplanar structure, but the axis-symmetry about the z-axis is preserved. This spin texture is stable for gd ≤ 0.3g and robust for different aspect ratios: γ = 0.2 and 1.5. The bending angle of FIG. 4: (a) The z-flare spin texture in the cigar trap along the z-axis. The spins almost point to the z direction. In the outer regions they bent. The bright region in background corresponds to high number density. gd = 0.2g, γ = 5.0. (b) Schematic figure to explain this spin configuration due to d-d interaction. FIG. 5: (a) The two-z-flare spin texture under the same parameter set (gd = 0.2g, γ = 5.0.) as in Fig. 4 with different initial spin configuration. The bright region in background corresponds to high number density. (b) Schematic figure to explain this spin structure. At the z = 0 plane two oppositely aligned spins meet and the number density is depleted. the flare spin texture increases and the elongation along the z direction becomes larger as gd increases (= 0.1 and 0.2). Finally we display an example to show how the model Hamiltonian admits many subtle spin textures with com- parable energies. Figure 5 (a) shows the two-z-flares op- positely polarized stacked back to back. This configura- tion is stabilized starting with a hedgehog spin config- uration, or skyrmion at the center from which all the spins point outward from the origin. In the end the two-z-flares oppositely polarized become stable, but at the central z = 0 plane the antiparallel spins meet as seen from Fig. 5 (b). To avoid drastic changes of the spin direction, or the spin kinetic energy loss, the parti- cle density decreases there. As a result even though the harmonic potential energy is minimal there, the two-z- flare spin textures oppositely polarized are stacked back to back, but two objects are almost split. This example illustrates strong coupling between the particle number and spin densities through the d-d interaction. These spin textures can be observed directly via a novel phase-sensitive in situ detection [1] or indirectly via con- ventional absorption imaging for the number density. It is interesting to examine the vortex properties under ro- tation. For the spin current texture, the vortex entry into a system should be easy because in the central region the mass density is already depleted. We point out that the collective modes might be also intriguing because the mass density is tightly coupled with the spin degrees of freedom. These problems belong to future work. In summary, we have introduced a model Hamiltonian to capture the essential nature of dipolar spinor BEC where the spin magnitude is large enough, focusing on long wavelength and low energy textured solutions. We show several typical stable configurations by solving the Gross-Pitaevshii equation where the spin and mass den- sities are strongly coupled due to the dipole-dipole inter- action. The shape of the harmonic potential trapping is crucial to determine the spin texture. The model Hamil- tonian is a minimal extension of the Non-linear sigma model with the d-d interaction, and yet complicate and versatile enough to explore further because it is expected that there are many stable configurations with compara- ble energies. Finally the model Hamiltonian is applica- ble literally for electric dipolar systems without further approximation. We expect that BEC of hetero-nuclear molecules with permanent electric dipole moment might be realized in near future [19] where the formation of such textures may be possible. We thank Tarun K. Ghosh and W. Pogosov for useful discussions in the early stage of this research. This work of S. G. was supported by a grant of the Japan Society for the Promotion of Science. [1] See for example, L. E. Sadler et al., Nature (London) 443, 312 (2006). [2] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). [3] T. -L. Ho, Phys. Rev. Lett. 81, 742 (1998). [4] A. Griesmaier et al., Phys. Rev. Lett. 94, 160401 (2005). [5] J. Stuhleret al., Phys. Rev. Lett. 95, 150406 (2005); L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404 (2006); A. Griesmaier et al., Phys. Rev. Lett. 97, 250402 (2006); S. Giovanazzi et al., Phys. Rev. A 74, 013621 (2005). [6] See for review, M. A. Baranov et al., Phys. Scr. T 102, 74 (2002). [7] Y. Kawaguchi et al., Phys. Rev. Lett. 96, 080405 (2006), 97, 130404 (2006), and 98, 110406 (2007). [8] S. Yi and H. Pu, Phys. Rev. Lett. 97, 020401 (2006). [9] R. B. Diener and T. -L. Ho, Phys. Rev. Lett. 96, 190405 (2006). [10] R. Cheng et al., J. Phys. B 38, 2569 (2005). [11] J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946). [12] J. Stenger et al., Nature (London) 396, 345 (1998). [13] J. P. Burke et al., Phys. Rev. Lett. 81, 3355 (1998). [14] M. D. Barrett et al., Phys. Rev. Lett. 87, 010404 (2001). [15] N. N. Klausen et al., Phys. Rev. A 64, 053602 (2001). [16] For 87Rb (F = 2), the two spin dependent interactions are 80 and 50 times smaller than the spin-independent one. T. Kuwamoto et al., Phys. Rev. A 69, 063604 (2004). [17] F.D.M. Haldane, Phys. Rev. Lett. 50, 1153(1983). R. Ra- jaraman, Solitons and Instantons (North-Holland, Ams- terdam, 1989). [18] C. J. Pethick and H. Smith, in Bose-Einstein condensa- tion in dilute gases (Cambridge University Press, Cam- bridge, 2002). Chap. 5, (5.76). [19] See the special issue on ultracold polar molecules; Eur. Phys. J. D31, 149-445 (2004).

0704.0123

Nonlinear Dynamics of the Phonon Stimulated Emission in Microwave Solid-State Resonator of the Nonautonomous Phaser Generator

Nonlinear Dynamics of Phonon Stimulated Emission Nonlinear Dynamics of the Phonon Stimulated Emission in Microwave Solid- State Resonator of the Nonautonomous Phaser Generator* D.N. Makovetskii A. Usikov Institute of Radio Physics and Electronics, National Academy of Sciences of Ukraine 12, Academician Proskura St., Kharkov 61085, Ukraine The microwave phonon stimulated emission (SE) has been investigated in a solid- state resonator of a non-autonomous acoustic quantum generator (phaser). Branching and long-time refractority (absence of the reaction on the external pulses) for deterministic chaotic and regular processes of SE were observed in the experiments with weak and strong electromagnetic pumping. The clearly depined increase of the number of the independently co-existing SE states as the pumping level rise has been observed both in physical and computer experiments. This confirms the analytical estimations of the branching density in the phase space. The nature of the SE pulses refractority is closely connected with such branching and reflects the crises of strange attractors, i.e. their collisions with unstable periodic components of the higher branches. The stimulated emission (SE) of microwave phonons has been first observed in experiments on quantum hypersound amplification, when in 1960-ies it was reported for dielectric single crystals doped with paramagnetic ions of the iron group [1] - [3]. The hypersound amplification is due to inversed population of the ion spin levels, transitions between which are allowed for the spin-phonon interaction, and is, in fact, an acoustic analog of the linear maser amplification of electromagnetic fields [4], [5]. At the same time, the nature of this effect – quantum generation of microwave phonons – has remained not fully understood for a long time. There was some inertia of thinking, with analogies proposed [6] between the photon quantum generator (phaser) and electromagnetic maser generator in the same way as it had been done for amplifiers. Experimental studies of phonon SE in Ni2+:Al2O3 and Cr3+:Al2O3 crystals ([7] - [10]) have shown that phaser generation mechanism is actually much closer to the processes in optical lasers than those in maser generators. Really, due to very low velocity of hypersound (as compared with the velocity of light) the wavelength of acoustic SE in a microwave solid-state resonator is about 1-3 µm, i.e., it corresponds to the near-infrared range of electromagnetic radiation. The resonator Q-factor CAVQ is, like in typical lasers, ≈CAVQ 10 6, i.e., again it is by several orders of magnitude higher than in electromagnetic masers. As a result, observed SE power spectra, transient processes, stability of stationary modes and other properties of a phaser are very much similar to these characteristics for solid- state lasers at SCAVS TTT 21 >>>> . Here SS TT 21 , are, respectively, axial and transversal relaxation times of active (inverted) centers, CAVCAVCAV QT Ω= , where CAVΩ is the operation mode frequency of the resonator with active medium, and SESCAV Ω≈Ω≈Ω , SΩ is the quantum transition frequency for inverse population difference, SEΩ - SE carrier frequency. However, there is also a fundamental difference between phaser and optical lasers, which is related to the intensity of intrinsic quantum noises (intrinsic, or spontaneous, emission) spontJ . As for a phaser =ΩS 3 - 10 GHz, i.e., five orders of magnitude less than for a laser, the relative level of the intrinsic component in the first generator is by ~15 orders lower than in the second one (due to spont SJ ∞ Ω ). In fact, this allows us to consider a phaser as a deterministic dynamic system practically in all practically available ranges of SE intensity. Investigation of complex (including irregular) determined movements in dynamic systems is now steadily proceeding. This is related not only to the fundamental character of this problem, but also to prospects of practical application of the methods involved and results obtained in studies on the mechanisms of associative memory [11], [12], for development of computer image and symbol recognition [13], [14] etc. The multiplicative noises (which also include intrinsic radiation) affect the behavior of dynamical systems in a very non-trivial manner [15], e.g., leading to roughing of phase space topology [16] and substantial reduction of the associative memory volume. Such considerations substantially increase our interest in phaser generator-type systems, where, as distinct from optical lasers, internal noise level is low for practically all feasible combinations of controlling parameters (CP). In the present paper, we summarize the data of our experimental studies of a non-autonomic phaser generator, which were partially reported in our separate presentations [17, 18]. We also report numerical modeling of SE dynamics on the basis of a deterministic generation model that is an acoustic analog of the laser model [19]. The experiments were carried out using a ruby phaser as described earlier [7] - [10] under modulation of various CP (pumping, static magnetic field H , intensity of hypersound injJ injected into the resonator, etc.) in the frequency range =mω 30 - 3000 Hz. Fig. 1. Energy level diagram for Cr3+:Al2O3 active crystal. The Cr3+:Al2O3 spin system is formed upon splitting of the principal energy level of Cr3+ ion with orbital quantum number L = 0 and spin S = 3/2 in a trigonal crystalline field and static magnetic field H (Fig.1). Electromagnetic pumping of the Cr3+:Al2O3 spin system was carried out with a klystron of power =pumpP 12 mW at frequency PΩ = 0P PΩ + ∆Ω using a cylindrical cavity on the 011H mode. The cavity was tuned to frequency 0PΩ = 23 GHz. This corresponds to the resonance magnetic field 0H = 3,9 kOe at the angle =ϑ 54°44' between H and the third order crystallographic axis C of ruby (when conditions for symmetrical pumping are realized – see Fig.1). Detuning of the pumping source P∆Ω varied within several tens of MHz. A solid-state microwave Fabry-Perot acoustic resonator (FPAR) of Q-factor ≈CAVQ 5⋅10 5 ( ≈CAVT 10 -5 s) was placed inside the pumping cavity along its axis. FPAR is a cylinder of synthetic rose ruby with optically planar and parallel end sides – acoustic mirrors. Upon one of the FPAR acoustic mirrors, deposition a thin textured ZnO film with Al sublayer was applied by vacuum evaporation, this film being the main element of the hypersound converter. Using this film, detection was carried out of SE signals having carrier frequency =ΩSE 9,16 GHz and intensity ( )tJ . The SE signal is formed on 2 3E E↔ transition of Cr3+ ions under electromagnetic pumping of 1 3E E↔ и 2 4E E↔ transitions (diagram of energy levels for Cr 3+:Al2O3 is presented in Fig.1, spin levels iE and respective wave functions iψ are numbered in the order of increasing energy). By means of the same film, external longitudinal hypersound of frequency ≈Ω≈Ω Sinj 9 GHz and pulse intensity ≤injJ 300 mW/cm 2 was injected into the ruby crystal. All experiments were carried out at 1.8 K. The axial relaxation time on the active transition was ≈ST1 0,1 s, the transverse relaxation time ≈ST2 10 ns. The third-order axis C of ruby was coincided with FPAR geometrical axis O , and conical refraction was observed for the transverse component of the phonon SE[7]; thus, the contribution of the longitudinal component to SE intensity J was overwhelming. As the texture axis of ZnO film was also parallel to axes C and O , the injected hypersound was also purely longitudinal. It should be stressed that the momentary value ( )tJ is determined by SE processes in FPAR and is (alongside with the inverse population difference n ) a dynamic variable, while injJ is among CP. The time-averaged SE intensity value J can be different from zero only if ( )Gpumppump PP > . Here pumpP is the value of electromagnetic pumping power at frequency =ΩP 23 GHz, where phaser generation begins. The pumping parameter A (the ratio of inverted population difference at frequency ≈ΩS 9 GHz to its threshold value) is equal to unity. At low levels of pumping ( A -1 ≈ 1/30) and small periodic perturbation of the system, ( mk << 1, where mk is modulation coefficient of one of CP, e.g., pumpP , H or injJ ) regular modulation of the recorded SE signal J with period mT is always observed in experiments all over the modulation frequency range mω = 30 - 3000 Hz. This corresponds to a soft birth (emergence) of a limiting cycle of period mT on the main (zeroth) generation branch 0B . When mk is increased, the SE output signal modulation coefficient smoothly rises up to values normally not higher than 60 - 70 %. Then in a certain critical point ( )0m mk k= the value of SER increases stepwise to nearly 100 %, not acquiring, however, a pulse character and preserving the same SE period mT . If we change the sign of mdk d t , hysteresis will be observed of the ( )SE mR k dependence – the jump of SER back (decreasing) is at ( ) ( )1 0 m mk k< , indicating the existence of an above-located hard branch 1B of IR. Thus, even in the range of mk where SE is not yet of pulse character, a bistability of periodic movements is observed. Co-existence of the soft 0B and hard 1B branches with one and the same modulation period of integral intensity of the generated photon flux is realized. If now, being on the branch 1B , to increase (and not to decrease) mk , at ≈−1A 1/30 consequent doubling of the SE period will be observed according to Feigenbaum’s scenario, ending in the transition to the chaotic pulse modulation mode ( )tJ . Varying the system detuning over magnetic field 0H H H∆ = − within several Oersteds, we succeeded in finding two qualitatively different types of irregular (chaotic) modulation with pseudoperiods mT and 2 mT (pseudoperiod is the maximum interval between the neighboring pulses rounded to the nearest integer value [20]). As for the soft branch 0B , at ≈−1A 1/30 it has strictly regular character with period mT in the all range of ( )( )00 ,m mk k∈ , cutting itself off at the transition point to the hard branch 1B independently of the H∆ detuning (though specific values of ( )0mk naturally change upon variation of H∆ ). As it was shown by experiments with modulation of each of the selected CP ( pumpP , H , injJ ) the further increase in mk at low pumping level ( A -1 ≈ 1/30) leads to appearance of new hard branches 2 3, , …B B , which begin by an appropriate periodic (which is now pulse) SE mode with period mMT , where, respectively, =M 2, 3, ... , and most often end by the chaotic mode after the series of Feigenbaum’s doublings. It should be noted that coexistence region of the SE branches was typically narrow for small 1−A , and at <∆H 4 Oe no bistability of periodical movements with 1>M was observed at all. The picture becomes qualitatively different when pumping parameter A is increased up to values of ≈A 1.5 – 1.8. Here strong overlapping of periodical regions occurs on branches 2 3, , …B B , accompanied by a hysteresis under slow scanning of any of non-modulated CP (including mk , p∆Ω , etc.). Fig.2 shows a typical picture observed for coexistence of the regular SE modes. a) b) Fig. 2. Regular SE of microwave phonons (experiment): coexistence of the modes with doubled (a) and tripled (b) sequence periods of generation pulses. The SE modes with doubled (Fig.2a) and tripled (Fig.3b) sequence periods of generated phonon pulses are coexisted for common modulation period mT and a fixed set of other CP. Transitions 2 3m mT T↔ are of hysteretic character when p∆Ω is scanned within about 1 MHz. The dashed line indicated the pumping modulation period mT = 1/120 s. The amplification factor of the recording equipment in Fig.2a is four times larger than in Fig.2b. It should be underlined that the observed movement with period 2 mT (Fig.2a) is not an intermediate stage of the Feigenbaum’s scenario on the first hard branch, but is a primary SE mode for the second hard branch – just as the primary mode with period 3 mT for the third hard branch (Fig.2b). This follows from the fact that at 0mdk /dt < both for 2B and 3B SE frustration (suppression) points mk ↓ and mk ↓ were found experimentally, where after sign inversion of mdk /dt (but with other CP unchanged) the disrupted movements were not restored. It was possible to restore the modes 2 mT (Fig.2a) and 3 mT (Fig.2b) only by stepwise variation of other CP – the most convenient was to apply a hypersound monopulse injJ up to 10 mW/сm 2 at ( ) ( ) ( )2,3 2,3 2,3mm mk k k↑ ↓= + ∆ . It appeared that restoration occurred even at ( ) ( )2,3 2,3210m mk k ∆ ≈ , with further increase in injJ not leading to new modes for such low overcriticities. It follows from these experimental data that at ( )2m mk k ↑≈ the second hard branch, which is shown in Fig.2a, is also the toppest, i.e., located above the others. Similarly, at ( )3m mk k ↑≈ it is the third hard branch that is the toppest (Fig.2b). From the other side, the fact that ( ) ( )2,3 2,3m mk k↓ ↑≈ directly indicates that the periods 2 mT and 3 mT for these branches are the initial ones. In other words, SE refractority (i.e., absence of SE pulse during time R mT KT= , where 1K > ) for hard branches 2B and 3B appears already at the moment of their birth. The presence of refractority is the main qualitative difference between the mode with period 2 mT on branch 2B from the mode with the same period 2 mT (after Feigenbaum’s doubling) on branch 1B . Really, for the latter mode doubling of the period ( 2SE mT T= ) is accompanied only by changes in the amplitude ratio of the neighboring SE pulses that follow with a unit pseudoperiod 2R SE mT T T= = (no refractority). At the same time, for branch 2B we have 2R SE mT T T= = . a) b) c) d) Fig. 3. Chaotic SE of microwave phonons (experiment): one realization with constant refractority (a) and three realizations with variable refractority (b) Another type of refractority is observed at large P∆Ω , when, in particular, the effective pumping parameter is also decreasing. Fig.3 shows SE oscillograms recorded at =∆ΩH 1 Oe, mω = 120 Hz in the cases of detuning ≈∆Ω p 4 МHz (oscillogram a) and ≈∆Ω p 30 МHz (oscillograms b-d). A transition takes place from the so-called helical chaotic mode with pseudoperiod 1R mT T= (when each period of pumping modulation mT is matched by one SE pulse) to the chaotic mode with variable refractority, where pseudoperiods R mT KT= (1 ≤≤ K 5) alternate irregularly. Amplification factor of the recording equipment in Fig.3 is the same for all oscillograms a-d. Besides large refractority intervals that appear on oscillograms b-d, sharp increase in the amplitude of pulses (as compared with oscillogram a) is observed, which is an evidence of substantial expansion of the attraction range of the corresponding attractor in the phase space of the system. Subsequent measurements have shown that refractority of variable duration like shown in Fig.3 (oscillogrammes b-d) are observed for chaotic SE pulse modes even more often that purely helical chaotic movements with the unit half-period shown in Fig.3a. It is essential that the observed transitions between SE chaotic modes with different maximum half-periods be of non-hysteretic character, as distinct from the above-described hysteretic transitions between independent regular branches with different periods of movement (like those shown in Fig.2). To clear up specific mechanisms of branching of SE modes and appearance of variable refractority in chaotic realizations of pulse phonon generation, we should address the deterministic SE model in a high Q-factor resonator with active centers, for which the condition 1S CAVT T>> is also met (this model is an acoustic analog of the deterministic model for class “B” lasers [19]). In this paper, we will limit ourselves in our numerical modeling by a relatively simple case of a three-level active system. Although we used experimentally a more complex four-level system (Fig.1), which allowed us to substantially enhance the inversion. Let us consider a three-level spin system 3 2 1E E E> > , where signal transition 1 2E E↔ (S- transition) is allowed for interaction with a coherent microwave acoustic field (hypersound) of specified direction and polarization, and the pumping transition 1 3E E↔ is allowed for magneto-dipole interactions with the corresponding pumping field. The third transition (F-transition) is an idle (no-load) one, and its frequency ( )3 2 /F E EΩ ≡ − is, by definition, not equal to the frequency of S-transition ( )2 1 /S E EΩ ≡ − or its multiple values. In addition, FΩ is not an integer divisor of the P-transition frequency ( )3 1 /P E EΩ ≡ − . In the impurity paramagnetic Cr3+:Al2O3, where the ground state (orbital quantum number L = 0, spin S = 3/2 – see Fig.1) is weakly bonded with the lattice (just by spin-orbital interaction), longitudinal relaxation times 1 1 1, ,S P FT T T for all the above-mentioned transitions at low temperatures are many orders of magnitude higher than the respective latitudinal relaxation times 2 2 2, ,S P FT T T . Consequently, it is possible to choose the amplitude of microwave electromagmetic pumping field 1PH such as to make populations 1n и 3n of spin levels 1E equal, but with broadening of these levels still the same as at 1 0PH = . In other words, we assume that the following two inequalities hold simultaneously: ( )2 11; 1P P P PZ T / T Z>> << (1) or, in a somewhat different form 1 21P P P PY T Y T>> >> , (2) where PZ is saturation factor of P-transition 2 21 2 1 4P P P P P Z T T г H= , (3) and PY is interaction probability of the pumping field with the P-transition. Here Pγ is effective gyromagnetic ratio for this spin transition (accounting for direction and polarization of vector 1PH ). Besides this, phonon life time CAVT in an FPAR of high Q-factor does normally meet the requirement 1 2S CAV ST T T>> >> . (4) Similar inequalities are true for photon life times in a microwave electromagnetic pumping resonator. In addition, it is usually assumed that the pumping resonator has no intrinsic frequencies in the vicinity of FΩ . In this case, to calculate the difference in populations 2 1N n n∆ ≡ − on S-transition, one could use the balance approximation equations, which, accounting for equality 3 1n n= , can be presented in the following form: 12 1 32 1 2 2 21 2 31 1 1 1 W n W n W n Y N W n W n W n Y N = + − − ∆  = + − + ∆ , (5) where 2 21 23W W W≡ + ; 1 12 13W W W≡ + ; SY is the probability of interaction of the hypersound field with the spin S-transition; ijW are probabilities of longitudinal spin relaxation. Following [1], [7], we obtain 22 2 22 12 1S S u S SY T q U F / Z / T= ≡ , (6) where /u S uq V= Ω ; U and uV are, respectively, the hypersound amplitude and phase velocity; 12F is spin-phonon interaction factor for S-transition at specified values of the hypersound polarization and propagation direction. General expressions for 12F can be found in [1]. For the case of latitudinal hypersound propagating along the crystallographic axis O of the third or higher order (as it is in the ruby phaser at 9 Ghz [7]), we have from [1] 12 1 2Hsu F ψ ψ . (7) Here zzε is a component of elastic deformation tensor; 1ψ и 2ψ are wave functions corresponding to spin levels 1E and 2E ; Hsu is the spin-phonon interaction Hamiltonian. Using the approximation 3P Bk θΩ << (where Bk is the Boltzmann constant; θ - temperatute of the thermostat), we find from (5) ( ) ( ) 2 S P N Nd N ∆ − ∆∆ = − ∆ + , (8) where ( ) ( ) -10 4 2 2S F P E cN s W f W p W W Nθ θ θ∆ = − + − ; S ET / W= ; 1 2 3cN n n n= + + ; / 3S Bs kθ θ= Ω ; 12 21 / 3P Bp kθ θ= Ω ; 13 31 / 3F Bf kθ θ= Ω ; 23 32 ( ) ( )6 2 3 2E S F PW s W f W p Wθ θ θ= + + − + . The effective longitudinal relaxation time (relaxation time of active centers for the phaser signal channel) ( )1 ST that is referred to in (8) is not the conventional spin-lattice relaxation time 1ST used in studies of passive systems, because pumping ("hidden" in (5) due to 1 3n n= ) leads to renormalization of the longitudinal relaxation time [26]. E.g., at ,F S PW W W>> we find ( ) ( )0 S F ST W T≈ << . Further we will omit the upper index for ( )1 ST . Let us introduce a dynamical variable M , which is proportional to the average SE intensity in the solid-state resonator of a phaser generator ( )2 12 2 /S SM U Y Bρ −′ ′= Ω ≡ , (9) where 2 22 12S S uB T F Vρ′ ′= Ω ; ρ′ is the crystal density, and the line above denotes averaging over the FPAR volume. On the basis of the wave equation for hypersound in active paramagnetic medium [26], using approximations of works [4], [5], we obtain an equation for the first derivative of phaser SE intensity B M N ′= − ; (10) where ( )N N= ∆ . Averaging (8) as well, we obtain the second equation of our system 0 N NdN B M N ′= − + ; (11) The system (10) - (11) is a system of reduced movement equations for an autonomous phaser. This system, due to reduction of the pumping equations, is isomorphic to the movement equations for a two-level autonomous laser of class "В" [19]. Introducing 1St Tτ = , we will further work with a dimensionless form of these equations. For convenience, let us use the following dynamic variables: ( ) 1 2S SJ B T M Zτ ′= ≡ ; ( ) trn N Nτ = (12) and the following CP : 0 trA N N= ; 1S CAVB T T= , (13) where trN is the threshold value of the inverted population difference of spin levels corresponding to self-excitation of the phonon SE: CAV CAV S S B T T T F . (14) Let us now introduce a periodic perturbation of amplitude mk and frequency mω into the equation system (10), (11). The concrete form of the perturbation naturally depends on the choice of the modulated CP. As it has been shown theoretically [19] and follows from our experiments, the transition from modulation of one parameter to the modulation of another primarily affects the quantitative movement characteristics of the system, while on the qualitative level, behavior of the modulated system remains essentially unchanged. For convenience, we introduce the modulation into the right-hand side of equation (10) (in this case, according to [19], the smallest values of mk are needed to reach the first critical points). Then, finally, our equations for a non-linear phaser system containing a microwave solid- state resonator with inverted paramagnetic centers acquire the form ( ) ( ), cosJ m m J n BJk = Φ − , (15) ( ),n = Φ , (16) where 1St Tτ = ; 1m m STω ω= ; 1S CAVB T T= ; JΦ and nΦ are components of the unperturbed vector field Φ of our system having the following form: ( )1 ;J BJ nΦ = − ( 1)n A n JΦ = − + . The equation system (15), (16) at 0mk = has two singular points: saddle [st1]: ( )0,J n A= = ,where [st1]1Re 0Λ > ; [st1]2Re 0Λ < ; [st1]1,2Im 0Λ = ( 1,2Λ - Lyapunov’s indexes) and attractor [st2]: ( )1, 1J A n= − = , where: [ ] ( )st21,2 2AΛ = − ± ( ) ( ) 2 22 1A A B − − . (17) Taking into attention that 1B >> , at ( ) 14 1 4B A B− << − << we find that [st2] is a focus with so- called relaxation frequency: [st2]1,2 1 1Imrel S rel ST Tω ω= Λ ≡ , where we introduced, for convenience, the dimensionless frequency ( )1rel A Bω = − . As it is easy to find for our oscillator (15), (16), at small mk the branch 0B does softly emerge from this focus as a limiting cycle of period 1 mT , what has been actually observed in the experiments. The oscillator non-linearity (15), (16) begins to really manifest itself as mk increases and mT comes close to the resonance: 2m rel relT T π ω≈ ≡ . Here numerical methods are needed. Let us make some necessary estimates by introducing parameters 1ε , 2ε in the following way: 1 1 2 1 ;A B ε ε ε = + = . (18) At ( )1 1A O− = we find that relationships 1 21 2 1Bε ε −≈ ≈ << ; relT ≡ 2 relπ ω ≈ 12 S CAVT Tπ are fulfilled. At small 1A− ≈ 24 Bπ we obtain 1ε ≈ 1 2π ; 2 2 Bε π≈ ; 1rel ST T≈ . Let us now evaluate the value of ppN , which corresponds to the minimum number of required steps with time stepT for the modulation period, i.e., ( ) 1/ 2pp m step m stepN T T Tπ ω = = . Accounting for the fact that the field subsystem in our case is fast with respect to the atomic subsystem, for m relω ω≈ we obtain κ π κ > ≡ ≡ , (19) where 2 1κ > . E.g., for A - 1=1/30 and B = 3.7⋅10 3 , from (19) we find that ppN > 2⋅10 The full evolution time of the system wholeT is another important parameter in looking for solutions of our equation system. In all cases, we carried out our trial calculations for such number of modulation periods /p whole mN T T≡ as to surpass the value of 1 /S mT T ratio. This means that there should be ( )1 12pN κ πε≈ , where 1κ > 1. In the vicinity of some special points (e.g., period doubling points) effect of critical deceleration is observed – like at phase transition points in conservative systems. Here pN values were adapted to the real duration of the transient process by the trial-and-error method. The total number of points for calculations calc p ppN N N≡ ⋅ outside the critical deceleration regions does not depend on A and has a simple form p ppN N⋅ > 1 2Bκ κ . However, it should be taken into account that the problem with 1 1A− << requires larger ppN because of inverse root dependence of mT upon the overrun of the generation threshold ( pN is considered to be small while 1m rel ST T T≈ → ). This circumstance is related to the A –dependence of the normalized dissipativity nD introduced in [20] as Re ImnD ≡ Λ Λ , where Lyapunov’s indexes are taken at the stationary point in the absence of modulation. As shown in [20], the value of nD can be used for preliminary prediction of the degree of phase space stratification – when nD decreases, the number of existing attractors, generally speaking, tends to be increased at the given CP set [20]. For the attracting point [2] of our system in the autonomous mode, using (17), we obtain ( ) ( ) 11 222 1 2nD A A B A   =− − −    , (20) Formula (20) is defined at ( ) ( )NF NFA A A − +< < , where saddle-focus bifurcation points ( )±NFA are ( ) ( ){ }1/ 212 1 1NFA B B± −= ± − . For the phaser system, 1B >> (typical values are B ≈ 103-105), so ( ) 4NFA B + ≈ ; ( ) ( ) 11 4NFA B −− ≈ + . Then for 1 2A A A< < , where ( ) 1 1 4A B − >> ; 2 4A B<< , we obtain ( )2 1 . (21) As distinct from the unperturbed vector field divergence ( )0 0mkΦ ≡Φ = characterizing the contraction (i.e., the rate of phase volume contraction) ( ) ( )[st2] [st2]0div 1 1n B J AΦ = − − + = − (22) the value of nD has a non-monotonous dependence on А: d A ( ) . (23) Instead of linear increasing, which is characteristic for 0div Φ , the normalized dissipativity module is rapidly decreasing when А is changed from 1 to 2, and only then (at 2A > ) begins to slowly increase with the asymptotics nD ∞ A . Consequently, there are three qualitatively different regions of the pumping parameter А (both from physical and calculation point of view) that require separate approaches: 1. Pumping parameter values are small: 1 1A− << , accounting for the aforementioned limitation 1A− >> ( )1 4B . In this case, slight stratification (layering) of the phase space is predicted (due to large nD ) in combination with weak contraction (as 0div Φ is small); 2. Vicinity of the nD minimum: 2A ≈ . Here contraction is moderate, but maximum phase space stratification is predicted; 3. The region of high pumping parameter values, 2A >> (accounting to be limitation 4A B<< ). With increased А, a certain decrease in stratification degree is predicted (as compared with the previous case) on the background of enhanced contraction. Condition 3 is much less typical in real experiments (it is difficult to obtain such high inversion of the spin system) than conditions 1, 2, but we consider all three cases in order to obtain a full picture. Main results of our numerical modeling of SE dynamics in a solid-state resonator of the phaser generator can be summarized as follows. The total number of the observed coexisting attractors, including limiting cycles with periods mMT (where М is the external force undertone number), invariant toroids and strange attractors, is first increased with increased А (upon transition from condition 1 to condition 2), and is then decreased upon further monotonous increase of A (transition from condition 2 to condition 3) – according to formula (21). From the other side, the generated SE pulse amplitude is monotonously increasing upon increase of A - in a complete accordance with formula (23). This supports interpretation of nD and div Φ as correct characteristics of different behavior aspects of one and the same dissipative system [20]. Let us go now to concrete numerical results. Phase space stratification of our system begins under the scenario of primary saddle-knot bifurcations, which had been already known (e.g., for biochemical systems) in the beginning of 1970-ies [21], and which adequately describes the global properties of Class “B” modulated lasers [19]. Pairs of stable and unstable limiting cycles with periods mMT ( 1, 2, 3,M = … ), i.e., natural undertones mω , evolutionate (with, say, increasing mk ) from phase space singularities in all the studied range low highA A A< < ( 1 1 30lowA − = ; 30highA = ), but the "density" of the coexisting limiting cycles changes non-monotonously in this range. The simples phase space structure was observed in region 3. The calculations were carried out at A = 30; B = 3700; mω = 290. The primary hard branch 1B is developed into a series of Feigenbaum’s period doublings, becoming chaotic. This chaotic attractor at mk ≈ 0,18 has maximum refractority time 2 mT≈ , it coexists with the higher regular branch 2B with initial period 2 mT , and at mk ≈ 0.19 the chaotic attractor belonging to branch 1B is destroyed. As itself, the limiting cycle on branch 2B is not a result of the first step of the doubling series (though its secondary evolution proceeds according to the Feigenbaum’s scenario) – this cycle is a primary one, i.e., it is born just with the period 2 mT and refractority that is also equal to 2 mT . The nature of this primary undertone (as all the subsequent primary undertones – see below) is one and the same – continuous formation of independent subharmonic orbits in a dissipative system [21]. This becomes clear with further increase in the modulation amplitude: at mk = 0.44, the only attractor of the system is the limiting cycle with period 3 mT and refractority that is also equal to 3 mT (for the Feigenbaum’s series, the period of 4 mT should emerge at once, and refractority should remain equal to 1 mT ). At the same time, for this third hard branch, when mk is increased from 0.44 to 0.62, a secondary Feigenbaum’s evolution is observed, with the tripled initial period: 3 mT → 6 mT → 12 mT → 24 mT →… and the unchanged refractority time 3R mT T= . For even higher values of mk , chaotic SE mode is realized. The bifurcation sequences in the region 1 for small mk are similar to those realized in the region 3. However, upon mk increase in the region 1 we also observed much more complex dynamic properties. A very small contraction leads to the to excitation of chaotic oscillations with very large refractority time. At A -1 = 1/30; B = (1/27)⋅105 ≈ 3704; mk = 2.4⋅10 -2; mω = 10; ( )0J = 10-10; ( )0n = 1.0 we obtain maximum values of RT , up to 5 mT (Fig.4). Even higher values - RT 12 mT= - are noted when the modulation coefficient becomes as high as mk = 0.1 (at the same CP as in Fig.4). Fig. 4. Calculated time dependence of SE intensity. Refractority, as in the experiment (see Fig. 3, oscillograms b-d), varies within the limits of one to five modulation periods. Fig. 5. Calculated time dependence of inverse population difference at the same controlling parameters and initial conditions as in Fig. 4. Comparing the experimentally obtained dependences ( )J t for the mode with prolonged variable refractority (oscillograms b-d in Fig.3) with results on ( )J t obtained by calculations using our SE model in the region 1 (Fig.4), we can see their good agreement: both in experimental and calculated SE pulse sequences all refractority times are present, without exceptions - from mT to 5 mT , including the last value. Moreover, comparing Fig.4 and Fig.5, one can clearly see the qualitative difference in behavior of dynamic variables J and n, related to substantial difference in relaxation times for the field and the atomic subsystems. As a matter of fact, the above-described refractority effects are in one or another way related to this substantial difference. As 1B >> (т. е. 1S CAVT T>> ), the slow relaxation atomic subsistem (variable n) has not enough time “to be tuned" to the fast phonon processes (variable J). Therefore, when there is a periodic perturbation imposed from outside, the population difference n for certain ranges of CP does not reach over-critical values (ensuring phonon generation) in each of the modulation periods. Correspondingly, it is clearly seen from both experimental (Fig.3, oscillograms 6d) and calculated (Fig.4) pulse sequences ( )J t that larger intensity of “irradiated” SE pulse is accompanied by a longer subsequent state of refractority – the phaser is not responding to several modulation pulses. Let us underline once more that in our experiments (where B ≈ 104) such modes are more typical than SE without refractority. An essential point is that each increase in the maximum refractority time observed in our computer experiments occurred stepwise., was accompanied by an increase in maximum SE pulse amplitude and showed no hysteresis, which was also in full agreement with our real physical experiments. It is generally assumed [22] that such effects are caused by a qualitative rearrangement of the stratified phase space as a result of so-called external crises (or crises of second kind [22]), when the attraction region of a strange attractor is suddenly expanded, accompanied by an increase in the average refractority time. Each subsequent crisis gives rise to lower and lower pseudoperiods as a result of collisions of the strange attractor with unstable regular manifolds, specifically – with saddle components emerging (at primary saddle-knot bifurcations) together with branches MB [22]. Therefore, the strange attractor crises can be realized only in these regions of CP space where branching of the above-lying regular attractors has already taken place. We have noted such behavior for all SE modes with pseudoperiods up to 5 mT (physical experiment) and 12 mT (computer modeling). As further calculations have shown, when pumping power is increased, and values of A ≈ 2 are reached, the number of coexisting attractors becomes higher, enhancing stratification of the phase space, which is also in good agreement with experimental data.. The bifurcation diagrams acquire now a very complex indented structure with multiple interweaving in the CP space. As a result, the system sensitivity becomes even higher both to CP perturbations and variation of initial conditions. As a conclusion, let us stress once more that all the above-described SE processes are adequately described within the deterministic non-linear oscillator model (15), (16) under the double inequality 1 2S CAV ST T T>> >> that is typical for phaser generators. 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0704.0124

Proper J-holomorphic discs in Stein domains of dimension 2

arXiv:0704.0124v3 [math.CV] 5 Oct 2008 Proper J-holomorphic discs in Stein domains of dimension 2 Bernard Coupet*, Alexandre Sukhov** and Alexander Tumanov*** * Université de Provence, CMI, 39 rue Joliot-Curie, Marseille, Cedex, Bernard.Coupet@cmi. math-mrs.fr ** Université des Sciences et Technologies de Lille, Laboratoire Paul Painlevé, U.F.R. de Mathé- matique, 59655 Villeneuve d’Ascq, Cedex, France, [email protected] *** University of Illinois, Department of Mathematics 1409 West Green Street, Urbana, IL 61801, USA, [email protected] Abstract. We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2. MSC: 32H02, 53C15. Key words: almost complex manifold, strictly pseudoconvex domain, Morse function, Bishop disc. 1 Introduction The problem of embedding complex discs or general Riemann surfaces into complex manifolds has been well-known for a long time. The interest to the case of almost complex manifolds has grown due to a strong link with symplectic geometry (Gromov [13]). We present the following result. Theorem 1.1 Let (M,J) be an almost complex manifold of complex dimension 2 admitting a strictly plurisubharmonic exhaustion function ρ. Then for every non-critical value c of ρ, every point p ∈ Ωc = {ρ < c} and every vector v ∈ Tp(M) there exists a J-holomorphic immersion f : ID −→ Ωc, where ID ⊂ IC is the unit disc, such that f(bID) ⊂ bΩc, f(0) = p, and df0 ∂Re ζ = λv for some λ > 0. For a domainM ⊂ ICn with the standard complex structure, the result is due to Forstnerič and Globevnik [12]; there are various generalizations including embedding bordered Riemann surfaces into singular complex spaces (see [7] and references there). http://arxiv.org/abs/0704.0124v3 Recently Biolley [4] proved a similar result for an almost complex manifold M of any dimension n, but under the additional hypothesis that the defining function ρ is subcritical. The latter means that ρ does not have critical points of the maximum Morse index n. (A plurisubharmonic function can not have critical points of index higher than n.) We don’t impose such a restriction. Furthemore, Biolley [4] does not prescribe the direction of the disc. Her method is based on the Floer homology and substantially uses recent work of Viterbo [23] and Hermann [14]. Our proof is self-contained; we adapt the ideas of Forstnerič and Globevnik [12] to the almost complex case using the methods of classical complex analysis and PDE. In most work on the existence of global discs with boundaries in prescribed totally real manifolds ([2, 9, 10, 15, 17] and others) the authors use the continuity principle. By the implicit function theorem and the linearized equation they show that any given disc generates a family of nearby discs. Then the compactness argument allows for passing to the limit. In contrast, we construct the discs by solving the almost Cauchy-Riemann equation directly. Following [12], we start with a small disc passing through the given point in given direction and push the boundary of the disc in the directions complex-tangent to the level sets of the defining function ρ; it results in increasing ρ due to pseudoconvexity. This plan leads to a problem of attaching J-holomorphic discs to totally real tori in a level set of ρ. The problem is of independent interest and may occur elsewhere. It reduces in turn to the existence theorem for a boundary value problem for a quasilinear elliptic system of partial differential equations in the unit disc (Theorem 4.1). We prove it by the classical methods of the Beltrami equations and quasiconformal mappings (Ahlfors, Bers, Boyarskii, Lavrentiev, Morrey, Vekua; see [3, 21] and references there). The result can be viewed as a far reaching generalization of the Riemann mapping theorem. Since the almost Cauchy-Riemann equation is nonlinear, one can only hope to find a solution close to a current disc f . By measuring the closeness in the Lp norm, we are able in fact to construct a disc sufficiently far from f in the sup-norm. To make sure we are looking for a disc close to f , we adapt the idea of [12] of adding to f(ζ) a term with a factor of ζn (ζ ∈ ID) with big n. We develop a nonlinear version of this idea. The above procedure works well in the absence of critical points of ρ. In order to push the boundary of the disc through critical level sets, we use a method by Drinovec Drnovšek and Forstnerič [7, 11], which consists of temporarily switching to another plurisubharmonic function at each critical level set. We point out that adapting this method to the almost complex case is not a major problem because the difficulties are localized near the critical points, in which the almost complex structure can be closely approximated by the standard complex structure. Although higher dimension gives one more freedom for constructing J-holomorphic discs, we must admit that our proof of the main result goes through in dimension 2 only. The reason is that our main tool (Theorem 4.1) needs a special coordinate system in which coordinate hyperplanes z = const are J-complex, which generally can be achieved only in dimension 2. For a domain in ICn with the standard complex structure, the result is obtained in [12] by reduction to dimension 2 using sections by 2-dimensional complex hypersurfaces. Such a reduction in not possible for almost complex structures. We thank Franc Forstnerič and Josip Globevnik for helpful discussions, in particular, for pointing out at some difficulties in the problem and for the important references [7, 11]. Parts of the work were completed when the third author was visiting Université de Provence and Université des Sciences et Technologies de Lille in the spring of 2006. He thanks these universities for support and hospitality. 2 Almost complex manifolds Let (M,J) be an almost complex manifold. Denote by ID the unit disc in IC and by Jst the standard complex structure of ICn; the value of n is usually clear from the context. Let f be a smooth map from ID intoM . Recall that f is called J-holomorphic if df ◦Jst = J◦df . We also call such a map f a J-holomorphic disc or a pseudoholomorphic disc or just a holomorphic disc when a complex structure is fixed. We will often denote by ζ the standard complex coordinate on IC. A fundamental result of the analysis and geometry of almost complex structures is the Nijehnuis–Woolf theorem which states that given point p ∈ M and given tangent vector v ∈ TpM there exists a J-holomorphic disc f : ID −→M centered at p, that is, f(0) = p and such that df(0)(∂/∂Re ζ) = λv for some λ > 0. This disc f depends smoothly on the initial data (p, v) and the structure J . A short proof of this theorem is given in [19]. This result will be used several times in the present paper. It is well known that an almost complex manifold (M,J) of complex dimension n can be locally viewed as the unit ball IB in ICn equipped with an almost complex structure which is a small deformation of Jst. More precisely, let (M,J) be an almost complex manifold of complex dimension n. Then for every p ∈M , δ0 > 0, and k ≥ 0 there exist a neighborhood U of p and a smooth coordinate chart z : U −→ IB such that z(p) = 0, dz(p) ◦ J(p) ◦ dz−1(0) = Jst, and the direct image z∗(J) := dz ◦ J ◦ dz −1 satisfies the inequality ||z∗(J)− Jst||Ck(ĪB) ≤ δ0. For a proof we point out that there exists a diffeomorphism z from a neighborhood U ′ of p ∈ M onto IB such that z(p) = 0 and dz(p) ◦ J(p) ◦ dz−1(0) = Jst. For δ > 0 consider the isotropic dilation dδ : t 7→ δ −1t in ICn and the composite zδ = dδ ◦ z. Then limδ→0 ||(zδ)∗(J)−Jst||Ck(ĪB) = 0. Setting U = z δ (IB) for positive δ small enough, we obtain the desired result. As a consequence we obtain that for every point p ∈ M there exists a neighborhood U of p and a diffeomorphism z : U → IB with center at p (in the sense that z(p) = 0) such that the function |z|2 is J-plurisubharmonic on U and z∗(J) = Jst +O(|z|). Let u be a function of class C2 on M , let p ∈ M and v ∈ TpM . The Levi form of u at p evaluated on v is defined by LJ(u)(p)(v) := −d(J∗du)(v, Jv)(p). The following result is well known (see, for instance, [6]). Proposition 2.1 Let u be a real function of class C2 on M , let p ∈M and v ∈ TpM . Then LJ(u)(p)(v) = ∆(u◦f)(0) where f : rID −→M for some r > 0 is an arbitrary J-holomorphic map such that f(0) = p and df(0)(∂/∂Re ζ) = v, ζ ∈ rID. The Levi form is invariant with respect to J-biholomorphisms. More precisely, let u be a C2 real function onM , let p ∈M and v ∈ TpM . If Φ is a (J, J ′)-holomorphic diffeomorphism from (M,J) into (M ′, J ′), then LJ(u)(p)(v) = LJ (u ◦ Φ−1)(Φ(p))(dΦ(p)(v)). Finally, it follows from Proposition 2.1 that a C2 function u is J-plurisubharmonic on M if and only if LJ (u)(p)(v) ≥ 0 for all p ∈ M , v ∈ TpM . Thus, similarly to the case of the integrable structure one arrives in a natural way to the following definition: a C2 real valued function u on M is strictly J-plurisubharmonic on M if LJ(u)(p)(v) is positive for every p ∈M , v ∈ TpM\{0}. Let J be a smooth almost complex structure on a neighborhood of the origin in ICn and J(0) = Jst. Denote by z = (z1, ..., zn) the standard coordinates in IC n (in matrix computations below we view z as a column). Then a map z : ID −→ ICn is J-holomorphic if and only if it satisfies the following system of partial differential equations zζ − A(z)zζ = 0, (1) where A(z) is the complex n× n matrix defined by A(z)v = (Jst + J(z)) −1(Jst − J(z))v (2) It is easy to see that right-hand side of (2) is IC-linear in v ∈ ICn with respect to the standard structure Jst, hence A(z) is well defined. Since J(0) = Jst, we have A(0) = 0. Then in a sufficiently small neighborhood U of the origin the norm ‖ A ‖L∞(U) is also small, which implies the ellipticity of the system (1). However, we will need a more precise choice of coordinates imposing additional restric- tions on the matrix function A. The proof of the following elementary statement can be found, for instance, in [6]. Lemma 2.2 After a suitable polynomial second degree change of local coordinates near the origin z 7→ z + akjzkzj we can achieve A(0) = 0, Az(0) = 0 In these coordinates the Levi form of a given C2 function u with respect to J at the origin coincides with its Levi form with respect to Jst that is LJ(u)(0)(v) = LJst(u)(0)(v) for every vector v ∈ T0IR 3 Integral transforms in the unit disc Let Ω be a domain in IC. Let TΩ denote the Cauchy-Green transform TΩf(ζ) = f(τ)dτ ∧ dτ τ − ζ . (3) Let RΩ denote the Ahlfors-Beurling transform RΩf(ζ) = f(τ)dτ ∧ dτ (τ − ζ)2 , (4) where the integral is considered in the sense of the Cauchy principal value. We omit the index Ω if it is clear form the context. Denote by B the Bergman projection for ID. Bf(ζ) = f(τ)dτ ∧ dτ (τζ − 1)2 We need the following properties of the above operators. Proposition 3.1 (i) Let p > 2 and α = (p−2)/p. Then the linear operator T : Lp(ID) −→ Cα(IC) is bounded, in particular, T : Lp(ID) −→ L∞(ID) is compact. If f ∈ Lp(ID), then ∂ζTf = f , ζ ∈ ID, as a Sobolev derivative. (ii) Let m ≥ 0 be integer and let 0 < α < 1. Then the linear operators T : Cm,α(ID) −→ Cm+1,α(IC) and R : Cm,α(ID) −→ Cm,α(ID) are bounded. Furthermore,if f ∈ Cm,α(ID), then ∂ζTf = f and ∂ζTf = Rf , ζ ∈ ID, in the usual sense. (iii) The operator RΩ can be uniquely extended to a bounded linear operator RΩ : L p(Ω) −→ Lp(Ω) for every p > 1. If f ∈ Lp(ID), p > 1 then ∂ζTf = Rf as a Sobolev derivative. Moreover, the operator RIC is an isometry of L 2(IC), therefore ‖ RIC ‖L2(IC)= 1. (iv) The Bergman projection B : Lp(ID) −→ Ap(ID) is bounded. Here Ap(ID) denotes the space of all holomorphic functions in ID of class Lp(ID). (v) The functions p 7→‖ T ‖Lp(Ω) and p 7→‖ R ‖Lp(Ω) are logarithmically convex and in particular, continuous for p > 1. The proofs of the parts (i)–(iii) are contained in [21]. The part (iv) follows from (iii); see e. g. [8]. The part (v) follows by the classical interpolation theorem of M. Riesz–Torin (see e. g. [24]). We introduce modifications of the operators T and R for solving certain boundary value problems in the unit disc ID. For f ∈ Lp(ID) we define T0f(ζ) = Tf(ζ)− Tf(ζ ), ζ ∈ ID. (5) By Proposition 3.1 for p > 2 and α = (p− 2)/p, the linear operator T0 : L p(ID) −→ Cα(ID) is bounded, in particular, T0 : L p(ID) −→ L∞(ID) is compact. Since the function Tf is holomorphic and bounded in IC\ID, then the function ζ 7→ (Tf)(ζ ) is holomorphic in ID. Hence ∂ζT0f = ∂ζTf = f . Furthermore, for ζ ∈ bID, we have ζ = ζ , therefore by (5), ReT0f(ζ) = 0. Hence for f ∈ L p(ID), the function u = T0f solves the boundary value problem ∂ζu = f, ζ ∈ ID, Reu|bID = 0 We further define R0f := ∂ζT0f. Since ∂ζTf = Rf and ∂ζTf = f , then R0f(ζ) = ∂ζT0f(ζ) = Rf(ζ)− ∂ζTf(ζ ) = Rf(ζ) + ζ−2Rf(ζ ), (6) and we obtain a nice formula R0f = Rf +Bf, where B is the Bergman projection. By Propositions 3.1(iv) and (v), the operator R0 : Lp(ID) −→ Lp(ID) is bounded, and the map p 7→‖ R0 ‖Lp(ID) is continuous for p > 1. By Proposition 3.1(iii), R is an isometry of L2(IC). The analogue of this result for the operator R0 may have been used for the first time by Vinogradov [22]. In fact we came across [22] after proving the following Theorem 3.2 R0 is a IR-linear isometry of L 2(ID), in particular, ‖ R0 ‖L2(ID)= 1. Since we could not find a proof in the literature, for completeness we include it here. Proof : For a domain G ⊂ IC we use the inner product (f, g)G = − fgdζ ∧ dζ. We put σf(ζ) = ζ ), ψ(ζ) = ζ Then σ2 = id. By substitution ζ 7→ ζ we obtain (σf, σg)ID = (g, f)IC\ID, Rσ = ψσR, R = ψσRσ. (7) By (6) we have R0f = Rf + ψσRf. Let f ∈ L2(ID). Extend f to all of IC by putting f(ζ) = 0 for |ζ | > 1. Then ‖ R0f ‖ L2(ID)= (Rf + ψσRf,Rf + ψσRf)ID = (Rf,Rf)ID + 2Re (Rf, ψσRf)ID + (ψσRf, ψσRf)ID. Since |ψ| = 1, by (7) we obtain (ψσRf, ψσRf)ID = (σRf, σRf)ID = (Rf,Rf)IC\ID, (Rf, ψσRf)ID = (ψσRσf, ψσRf)ID = (Rσf,Rf)IC\ID = (ψσRf,Rf)IC\ID = (Rf, ψσRf)IC\ID. Then by the previous line and because R is an isometry 2Re (Rf, ψσRf)ID = Re (Rf, ψσRf)IC = Re (Rf,Rσf)IC = Re (f, σf)IC = 0. Hence ‖ R0f ‖ L2(ID)= (Rf,Rf)ID + (Rf,Rf)IC\ID =‖ Rf ‖ L2(IC)=‖ f ‖ L2(IC)=‖ f ‖ L2(ID), which proves the theorem. 4 Riemann mapping theorem for an elliptic system The Riemann mapping theorem asserts that for every simply connected domain G ⊂ IC there exists a conformal map of G onto ID. If G is smooth, then there is a diffeomorphism f : G −→ ID, which defines an almost complex structure J = f∗(Jst) in ID. Then the Riemann mapping theorem reduces to constructing a J-holomorphic map z : (ID, Jst) −→ (ID, J). The latter satisfies the Beltrami type equation ∂ζz = A(z)∂ζz, which is equivalent to the linear Beltrami equaion ∂zζ + A(z)∂zζ = 0. We consider the following more general system ∂ζz = a(z, w)∂ζz, ∂ζw = b(z, w)∂ζz, which cannot be reduced to a linear one. Here z, w are unknown functions of ζ ∈ ID and a, b are C∞ coefficients. By eliminating ζ , the system reduces to a nonhomogeneous quasilinear Beltrami type equation ∂zw + a∂zw = b, but we prefer to deal with (8) directly. The following theorem is our main technical tool for constructing pseudoholomorphic discs with boundaries in a prescribed torus. For r > 0 denote IDr := rID. Theorem 4.1 Let a, b : ID× ID1+γ −→ IC (γ > 0) be smooth functions such that a(z, 0) = b(z, 0) = 0 and |a(z, w)| ≤ a0 < 1. Then there exists C > 0 such that for every integer n ≥ 1 the system (8) admits a smooth solution (zn, wn) with the following properties: (i) |zn(ζ)| = |wn(ζ)| = 1 for |ζ | = 1. (ii) zn : ID −→ ID is a diffeomorphism with zn(0) = 0. (iii) |wn(ζ)| ≤ C|ζ | n, |wn(ζ)| < 1 + γ. Proof : Shrinking γ > 0 if necessary, we extend the functions a and b to all of IC2 preserving their properties. We will look for a solution of (8) in the form z = ζeu, w = ζnev. Then for the new unknowns u and v we have the following boundary value problem ∂ζu = A(u, v, ζ)(1 + ζ∂ζu), ζ ∈ ID ∂ζv = B(u, v, ζ)(1 + ζ∂ζu), ζ ∈ ID Re u(ζ) = Re v(ζ) = 0, |ζ | = 1 where A = aζ−1eu−u, B = bζ−neu−v. Put ∂ζu = h and choose u in the form u = T0h. Then ∂ζu = R0h, which we plug into (9). We obtain the following system of singular integral equations for u, v and h: h = A(1 + ζR0h), u = T0h, v = T0(B(1 + ζR0h)) We denote by ‖ f ‖p the L p-norm of f in ID. Since the function p 7→‖ R0 ‖p is continuous in p and ‖ R0 ‖2= 1 we choose p > 2 such that a0 ‖ R0 ‖p< 1. For given u, v ∈ L∞(ID) the map h 7→ A(1 + ζR0h) is a contraction in L p(ID) because ‖ ζA ‖∞‖ R0 ‖p< 1. Hence there exists a unique solution h = h(u, v) of the first equation of (10) satisfying ‖ h ‖p≤ ‖ A ‖p 1− a0 ‖ R0 ‖p Consider the map F : L∞(ID)× L∞(ID) −→ L∞(ID)× L∞(ID) defined by F : (u, v) 7→ (U, V ) = (T0h, T0(B(1 + ζR0h))) where h = h(u, v) is determined above. Then F is continuous (even Lipschitz) map. Let E = {(u, v) ∈ L∞(ID)× L∞(ID) :‖ u ‖∞≤ u0, ‖ v ‖∞≤ v0} We need the following Lemma 4.2 There exist u0 > 0, v0 > 0 such that E is invariant under F . Assuming the lemma, we prove the existence of the solution of (10). Indeed, since T0 : L p(ID) −→ L∞(ID) is compact for p > 2, then F : E −→ E is compact. Since E is a bounded, closed and convex, then the existence of the solution of (10) follows by Schauder’s principle. Proof of Lemma 4.2 : Since a(z, 0) = b(z, 0) = 0, we have |a(z, w)| ≤ C1|w|, |b(z, w)| ≤ C1|w|. Here and below we denote by Cj constants independent of n. We have |a| = |a(ζeu, ζnev)| ≤ C1e ‖v‖∞ |ζ |n, ‖ A ‖p=‖ aζ −1 ‖p≤ C2 ‖ ζ n−1 ‖p e ‖v‖∞ ≤ C3e ‖v‖∞n−1/p. By (11), ‖ h ‖p≤ C4e ‖v‖∞n−1/p, hence ‖ U ‖∞≤ C5e ‖v‖∞n−1/p. Similarly |B| = |b(ζeu, ζnev)ζ−neu−v| ≤ C1e ‖u‖∞ , ‖ B ‖∞≤ C1e ‖ V ‖∞≤ C7(‖ B ‖p + ‖ B ‖∞‖ h ‖p) ≤ C8e ‖u‖∞ . Let δ = n−1/p. Then ‖ U ‖∞≤ C9δe ‖v‖∞ , ‖ V ‖∞≤ C9e Consider the system u0 = C9δe v0 , v0 = C9e with the unknowns u0, v0. Then u0 = C9δe For small δ > 0 this equation has two positive roots. Let u0 = u0(δ) be the smaller root and v0 = v0(δ) = C9e u0. Now if ‖ u ‖∞≤ u0, ‖ v ‖∞≤ v0, then ‖ U ‖∞≤ C9δe ‖v‖∞ ≤ C9δe v0 ≤ u0, ‖ V ‖∞≤ C9δe ‖u‖∞ ≤ C9δe u0 ≤ v0 Hence E is invariant under F , which proves the lemma. Thus the solution of (10) in L∞(ID) exists for n big enough. Since h ∈ Lp(ID), p > 2, the second and the third equations of (10) imply that u, v ∈ Cα(ID), α = (p − 2)/p. Since ∂ζu = h ∈ L p(ID) and ∂ζu = R0h ∈ L p(ID) as Sobolev’s derivatives, then u and v are solutions of (9), hence z = ζeu and w = ζnev are solutions of (8). By the ellipticity of the system, z, w ∈ C∞(ID). The smoothness up to the boundary can be derived directly from the properties of the Beltrami equation; it also follows by the reflection principle for pseudoholomorphic discs attached to totally real manifolds (see, e.g., [18]). Since the winding number of z|bID about 0 equals 1 and ∣∂ζz/∂ζz ∣ = |a| ≤ a0 < 1 then z : ID −→ ID is a homeomorphism by the classical properties of the Beltrami equation [21], and (ii) follows. Note that u0 −→ 0, v0 −→ C9 as n −→ ∞. Since T0 : L p(ID) −→ Cα(ID) is bounded, then we have ‖ v ‖Cα(ID)≤ C10, ‖ e v ‖Cα(ID)≤ C11, and |w(ζ)| ≤ C11|ζ | n. Furthermore, since |ev| = 1 on bID, then |ev(ζ)| ≤ 1 + C11(1 − |ζ |) for |ζ | < 1. Then |w(ζ)| ≤ |ζ |n(1 + C11(1 − |ζ |) α), hence ‖ w ‖∞−→ 1 as n −→ ∞. Hence ‖ w ‖∞< 1+ γ for n big enough, and (iii) follows. This completes the proof of Theorem 4.1. 5 Pseudoholomorphic discs attached to real tori This section concerns the geometrization of Theorem 4.1. We apply Theorem 4.1 in order to obtain a crucial technical result on (approximately) attaching pseudoholomorphic discs to a given real 2-dimensional torus in (M,J). We will use this result later for pushing discs across level sets of the defining function ρ in Theorem 1.1. The tori and the discs considered in this section are not arbitrary. We study a special case which will suffice for the proof of the main result. Given a psedoholomorphic immersed disc f , we associate with f a real 2-dimensional torus Λ formed by the boundary circles of discs hζ centered at the boundary points f(ζ), ζ ∈ bID. Thus, our initial data is a pair (f,Λ). Our goal is to construct a pseudoholomorphic disc with the boundary attached to the torus Λ. First we find a suitable neighborhood of the disc f which can be parametrized by the bidisc in IC2. We transport the structure J onto this bidisc and choose the coordinates there such that the equations for J-holomorphic discs take the form used in Theorem 4.1. The theorem will provide a pseudoholomorphic disc approximately attached to Λ. 5.1 Admissible parametrizations by the bidisc and generated tori Let f : ID −→ (M,J) be a J-holomorphic disc of class C∞(ID). Suppose f is an immersion. Let γ > 0. Given ζ ∈ D consider a J-holomorphic disc hζ : (1 + γ)ID −→ M satisfying the condition hζ(0) = f(ζ) and such that the direction dhζ(0)( ∂Re τ ) is not tangent to f . Admitting some abuse of notation, we sometimes write hf(ζ) for hζ . This allows to define a C∞ map H : ID× (1 + γ)ID −→M, H(ζ, τ) = hζ(τ). Then H has the following properties: (i) For every ζ ∈ ID the map hζ := H(ζ, •) is J-holomorphic. (ii) For every ζ ∈ ID we have H(ζ, 0) = f(ζ). (iii) For every ζ ∈ ID the disc hζ is transversal to f at the point f(ζ). We assume in addition that (iv) H : ID× (1 + γ)ID −→M is locally diffeomorphic. Then Λ = H(bID×(1+γ)bID) is a real 2-dimensional torus immersed intoM . It is formed by a family of topological circles γζ = hζ((1 + γ)bID) parametrized by ζ ∈ bID. Every such a circle bounds a J-holomorphic disc hζ : (1 + γ)ID −→ M centered at f(ζ). In particular the torus Λ can be continuously deformed to the circle f(bID). If the above conditions (i) - (iv) hold we say that a mapH is an admissible parametrization of a neighborhood of f(ID) and Λ is the torus generated by H . 5.2 Ellipticity of admissible parametrizations We prove the following consequence of Theorem 4.1. Theorem 5.1 Let f : ID −→ (M,J) be a C∞ immersion J-holomorphic in ID. Suppose that there exists an admissible parametrization H of a neighborhood of f(ID) and let Λ be the generated torus. Then there exists an immersed J-holomorphic disc f̃ of class C∞(ID) centered at f(0), tangent to f at f(0) and satisfying the boundary condition f̃(bID) ⊂ H(bID× bID). We stress that the boundary of f̃ is attached to the torus H(bID×bID) and not to Λ. However since γ > 0 can be chosen arbitrarily close to 0, this leads to the following result sufficient for applications. Corollary 5.2 In the hypothesis of the former theorem for any positive integer n there exists an immersed J-holomorphic disc fn of class C∞(ID) centered at f(0), tangent to f at f(0) and such that dist(fn(bID),Λ) −→ 0 as n −→ ∞. Here dist denotes any distance compatible with the topology of M . We begin the proof of Theorem 5.1 with the remark that the discs hζ , ζ ∈ D, fill a subset V of M containing f(ID) which can be viewed as a fiber space with the base f(ID) and the generic fiber hζ((1 + γ)ID). Therefore the defined above map H : ID× (1 + γ)ID −→ V gives a natural parametrization of V by the bidisc Uγ := ID × (1 + γ)ID. Since H is locally diffeomorphic (see (iv) above) the inverse map H−1 is defined in a neighborhood of every point of V . This allows to define the almost complex structure J̃ = H∗(J) = dH−1 ◦ J ◦ dH on Uγ . The structure J̃ has a special form. Indeed, in the standard basis of IR 4 we have J̃11 J̃12 J̃21 J̃22 where J̃kj are real 2×2 matrices. We recall that in this basis the standard complex structure st of IC has the form It follows by the property (i) of H that the maps τ 7→ (c, τ) are J̃-holomorphic for every fixed c. This implies that J̃12 = 0 and J̃22 = J st . Furthermore, since the map ζ 7→ (ζ, 0) is J̃-holomorphic, we have J̃11(z, 0) = J st and J̃21(z, 0) = 0. Let now g : ID −→ Uγ be a J̃ -holomorphic map. If we set ζ = ξ+iη, the Cauchy–Riemann equations have expressing the J̃-holomorphicity of g have the form = 0 (13) Suppose now that the matrix Jst+J is invertible. Then the Cauchy–Riemann equations can be rewritten in the form gζ + A(g)gζ = 0 (14) where A is defined by (2). If we use the notation g = (z, w), then the Cauchy–Riemann equations (14) can be written in the form ∂ζz = a(z, w)∂ζz, ∂ζw = b(z, w)∂ζz identical to (8). Furthermore, since J̃(z, 0) = Jst, the conditions a(z, 0) = b(z, 0) = 0 are satisfied. Proposition 5.3 We have ‖ a ‖∞< 1. Proof : The proof consists of two steps. First we study the matrix J̃+Jst which determines the matrix A in the Cauchy–Riemann equations (14). Lemma 5.4 The matrix J̃(z, w) + Jst is non-degenerate for any (z, w) ∈ ID× (1 + γ)ID. Proof : It suffices to verify the condition det(J̃11(z, w) + J st ) 6= 0. For every fixed (z, w) the matrix J̃11(z, w) defines a complex structure on the euclidean space IR 2 so there exists a matrix P = P (z, w) such that J̃11(z, w) = PJ −1. (16) Recall that the manifold J2 of all complex structures on IR 2 can be identified with the quotient GL(2, IR)/GL(1, IC) and has two connected components: J +2 and J 2 . A structure J̃11 belongs to J 2 (resp. to J 2 ) if in the representation (16) we have detP > 0 (resp. detP < 0). Suppose now that det(PJ st ) = 0 or equivalently det(PJ st +J st P ) = 0 at some point (z, w). If we denote by pjk the entries of the matrix P , the last equality means jk=1 p jk = 0 which together with the non-degeneracy of P implies that detP < 0 so that J̃11(z, w) ∈ J 2 . On the other hand, for the point (z, 0) we have detP > 0 since J̃11(z, 0) = J st so J̃(z, 0) ∈ J 2 . But we can join the points (z, 0) and (z, w) by a real segment, so this contradiction proves lemma. Now we can conclude the proof of Proposition 5.3. It follows by Lemma 5.4 that the Cauchy–Riemann equations (13) can be written in the form (15) on ID × (1 + γ)ID. The Cauchy–Riemann equations are elliptic at every point and this condition is independent of the choice of the coordinates. The system (15) is ellipitic at a point (z, w) if and only if |a(z, w)| 6= 1. Since a(z, 0) = 0 we obtain by connectedness that |a| < 1 on ID × (1 + γ)ID, which concludes the proof. Now Theorem 5.1 follows by Theorem 4.1. 5.3 Construction of an admissible parametrization with a pre- scribed generated torus So far we studied a situation where an admissible parametrization of a neighborhood of an immersed J-holomorphic disc was given and proved the existence of discs with bound- aries close to the generated torus. In the proof of our main result, we need an admissible parametrization of a neighborhood of a J-holomorphic disc with a given generated torus. Let f : ID −→ M be an immersed J-holomorphic disc of class C∞(ID). We extend f smoothly to a neighborhood of ID. Let U be a small neighborhood of bID. For every point f(ζ), ζ ∈ U , consider a J-holomorphic disc hζ : 2ID −→ M . Suppose that the map hζ smoothly depends on ζ ∈ U . Thus we obtain a smooth map H : bID × ID −→ M, H : (ζ, τ) 7→ hζ(τ). Then Λ := H(bID × bID) is a real 2-dimensional torus. In order to construct an admissible parametrization with the generated torus Λ we need to extend the map H from the cylinder bID × ID to the bidisc ID× ID. Definition 5.5 We call the described above torus Λ admissible. We further put Xζ := Xf(ζ) = dhζ(0)( ∂Re τ ) for every ζ ∈ U . Theorem 5.6 Let f : ID −→ (M,J) be an immersed J-holomorphic disc of class C∞(ID). Let Λ be an admissible torus. Then there is a sequence of admissible tori Λn converging to Λ such that for every n there exists an immersed J-holomorphic disc fn of class C∞(ID) centered at f(0), tangent to f at f(0) and satisfying the boundary condition fn(bID) ⊂ Λn. Proof : Let Λ be an admissible torus and let X be the vector field given by Definition 5.5. In general it is impossible to extend X as a non-vanishing vector field transversal to f(ID) at every point. However, for any integer (not necessarily positive) n we can consider the discs hnζ : τ 7→ hζ(ζ nτ), where ζ ∈ bID. Their tangent vectors at the points f(ζ) are equal to Xnζ := ζ nXζ, where by multiplying a vector by a complex number ζ n we mean applying the operator (Re ζ + (Im ζ)J)n. We need the following Lemma 5.7 After a suitable choice of n the vector field Xnζ can be extended on the disc as a nonvanishing field transversal to f at every point. Proof : First we look for a global parametrization of a neighborhood of f(ID). Fix an arbitrary vector field Y transversal to f(ID) at every point. By Nijenhuis - Woolf theorem we obtain a family of J-holomorphic discs gz : w 7→ gz(w), z ∈ ID so that gz(0) = f(z) and Xf(ζ) is tangent to gz. Then the map G : (z, w) 7→ gz(w) is a local diffeomorphism from a neighborhood of ID × ID onto a neighborhood of f(ID) and G(z, 0) = f(z) so we can use the coordinates (z, w). We pull back the vector field X by G−1 and consider the vector field (G−1)∗(X) : ζ 7→ (G −1)∗(Xζ). Let m be the winding number of the w-component of the vector field (G−1)∗(X) when ζ runs along the circle bID. We set n = −m. Then the field (G−1)∗(X n) extends on the disc (ζ, 0) as a smooth vector field Z transversal to this disc at every point. Then the map G∗(Z) associates to every point of ID a vector transversal to f(ID) and so defines the desired extension X̃n of the vector field Xn. This proves the lemma. Now by the Nijenhuis - Woolf theorem there exists a map h̃ζ : ID −→ M which is J- holomorphic on ID such that h̃ζ = hζ for every ζ in a neighborhood of bID and the vector X̃ is tangent to hζ at the origin. Thus we can extend H to a function defined on ID × ID such that the map H(ζ, •) is J-holomorphic for any ζ ∈ ID. This map H is a local diffeomorphism and so determines an admissible parametrization of a neighborhood of f(ID) such that the generated torus coincides with Λ. Theorem 5.6 now follows by Theorem 5.1. 6 Pushing discs through non-critical levels In this section we explain how to push a given disc through non-critical level sets of a strictly plurisubharmonic function. Proposition 6.1 Suppose that ρ does not have critical values in the closed interval [c1, c2]. Let f : ID −→ Ωc1 be an immersed J-holomorphic disc such that f(bID) ⊂ bΩc1 . Then there exists an immersed J-holomorphic disc f̃ : ID −→ Ωc2 such that f̃(0) = f(0), df̃(0) = λdf(0) for some λ > 0 and f̃(bID) ⊂ bΩc2. For the proof we need some preparations. Let ρ be a strictly plurisubharmonic function on an almost complex manifold (M,J). For real c consider the domain Ωc = {ρ < c}. Suppose that its boundary has no critical points. Let f : ID −→ Ωc be a J-holomorphic disc of class C∞(ID) and such that f(bID) ⊂ bΩc. For every point p ∈ f(bID) consider a J-holomorphic disc hp : 2ID −→ M touching bΩc from outside such that ρ ◦ hp|2ID\{0} > c. We call the discs hp the Levi discs. The map hp can be chosen smoothly depending on p ∈ f(bID). An explicit construction of the Levi discs is given in [12]. In the almost complex case the proof is similar; the only thing which has to be justified is the existence of discs hp touching a strictly pseudoconvex level set from outside. This was recently proved by Barraud and Mazzilli [1] and Ivashkovich and Rosay [16]. In [6] the result is obtained in any dimension. For reader’s convenience we include a simple proof (see [6]). Lemma 6.2 For a point p ∈ bΩc there exists a J-holomorphic disc hp such that hp(0) = p and hp(ID\{0}) is contained in M\Ωc. Proof : We fix local coordinates z = (z1, z2) near p such that p = 0 and J(0) = Jst. Denote by ej , j = 1, 2 the vectors of the standard basis of IC 2. By an additional change of coordinates we may achieve that the map h : ζ 7→ ζe1 is J-holomorphic on ID. We can assume that the Levi form LJr (0, e1) = 1 so that r(z) = 2Re z2 + 2Re ajkzjzk + αjkzjzk + o(|z| α11 = ∆(r ◦ h)(0) = 1. Now for every δ > 0 consider the non-isotropic dilation Λδ : (z1, z2) 7→ (δ −1/2z1, δ −1z2). The J-holomorphicity of the map h implies that the direct images Jδ := (Λδ)∗(J) converge to Jst as δ −→ 0 in the Ck norm for every positive integer k on any compact subset of IC2. Similarly, the functions rδ := δ −1r ◦ Λ−1 converge to the function r0 := 2Re z2 + |z1| 2 + 2Re βz21 (for some β ∈ IC). Consider a Jst-holomorphic disc ĥ : ζ 7→ ζe1 − βζ 2e2. According to the Nijenhuis- Woolf theorem for every δ ≥ 0 small enough there exists a Jδ-holomorphic discs h δ such that the family (hδ)δ≥0 depends smoothly on the parameter δ and for every δ ≥ 0 we have hδ(ζ) = ζe1 + o(|ζ |) and h 0 = ĥ. Since (r0 ◦ h 0)(ζ) = |ζ |2, we obtain that for δ > 0 small enough that (rδ ◦ h δ)(ζ) = Aδ(ζ) + o(|ζ | 2), where Aδ is a positive definite quadratic form on IR2. Since the structures Jδ and J are biholomorphic, then the lemma follows. Thus we obtain a smooth map H : bID× ID −→M, H : (ζ, τ) 7→ hf(ζ)(τ) =: hζ(τ). For simplicity we assume here that H is a local diffeomorphism although the Levi discs hζ can intersect even for close values of ζ . We prove in a forthcoming paper that the pullback H∗(J) of J to the bidisc can be defined even if H is not a local diffeomorphism. Thus Λ := H(bID × bID) is an admissible torus and ρ|Λ ≥ c + ε for some ε > 0. We stress that ε depends only on ρ (more precisely on a constant separating the norm of the gradient of ρ from zero) and the C2-norm of J . Now Theorem 5.6 implies that there exists a disc f̃ with the same direction as f at the center and with the boundary attached to a torus arbitrarily close to Λ. Now we cut off the discs hζ by the level set {ρ = c+ε/2} and obtain a disc with boundary attached to this level set. Indeed, we have the following Lemma 6.3 Suppose that ρ ◦ f |bID ≥ c0 and c0 is a non-critical value of ρ. Then there exists a J-holomorphic disc f̃ centered at f(0) and tangent to f at the center with boundary attached to the level set {ρ = c0}. Proof : By the Hopf lemma the disc f intersects the level set {ρ = c0} transversally at every point. Therefore the open set Ω = {ζ ∈ ID : ρ ◦ f(ζ) < c0} has a smooth bound- ary. The set Ω may be disconnected, but the connected component of 0 ∈ Ω is simply connected by the maximum principle applied to the function ρ ◦ f . Now the lemma follows via reparametrization by the Riemann mapping theorem. Then we again consider the Levi discs for this level set etc. By iterating this argument a finite number of times we obtain Proposition 6.1. 7 Pushing discs through a critical level In order to push the boundary of the disc f through critical level sets of ρ, we use a method of [11, 7], which consists of temporarily switching to another plurisubharmonic function at each critical level set. We need a version of the Morse lemma for almost complex manifolds. Proposition 7.1 Let (M,J) be an almost complex manifold of complex dimension 2. Let ρ be a strictly plurisubharmonic Morse function on M . Then there exists another strictly plurisubharmonic Morse function ρ̃ close to ρ with the same critical points, such that at each critical point of Morse index k in local coordinates given by Lemma 2.2 one has ρ̃(z) = ρ̃(0) + |z1| 2 + |z2| 2 − a1Re z 1 − a2Re z 2 (17) where (i) a1 = a2 = 0 if k = 0, (ii) a1 = 2 and a2 = 0 if k = 1, (iii) a1 = a2 = 2 if k = 2. Remark. This is a weak version of the Morse lemma because we change the given function ρ instead of reducing it to a normal form. The following result must be well known. For convenience we include a proof. Lemma 7.2 Let B be a complex symmetric n×n matrix. Then there exists a unitary matrix U such that U tBU is diagonal with nonnegative elements. Proof : Using coordinate-free language, given a hermitian positive definite form H and a complex symmetric bilinear form B on a vector space V , dimIC V = n, we need u1, ..., un ∈ V such that H(ui, uj) = δij , B(ui, uj) = ciδij, ci ≥ 0. If the above holds with just ci ∈ IC, then by rotation ui 7→ σiui, |σi| = 1, we obtain ci ≥ 0. It suffices to find u1 ∈ V , H(u1, u1) = 1, such that for every x ∈ V , H(x, u1) = 0 implies B(x, u1) = 0. (18) Then the rest of ui in the H-orthogonal complement of u1 are found by induction. Given u ∈ V , by duality, there is a unique vector L(u) ∈ V such that for every x ∈ V , H(x, L(u)) = B(x, u). (19) Then L : V → V is a IR-linear (IC-antilinear) transformation. Since B is symmetric, then by (19), L is real symmetric (self-adjoint) with respect to the form ReH . Then the eigenvalues of L are real and the eigenvectors are in V (generally they are in V ⊗IR IC). Let u1 ∈ V be an eigenvector of L, that is L(u1) = λu1, for some λ ∈ IR. We normalize u1 so that H(u1, u1) = 1. Then for u = u1, (19) implies (18), and the lemma follows. Proof of Proposition 7.1 : Let p be a critical point of ρ. Introduce a coordinate system with the origin at p given by Lemma 2.2. In these coordinates the function ρ is strictly plurisubharmonic at the origin with respect to Jst. Then ρ(z) = ρ(0) + aijzizj + Re bijzizj +O(|z| where aij = aji and bij = bji. By a linear transformation we can reduce to the form aij = δij . If we now make a unitary transformation z 7→ Uz preserving |z1| 2 + |z2| 2, then the matrix B = (bij) changes to U tBU . By Lemma 7.2 the expression of ρ reduces to ρ(z) = ρ(0) + |z1| 2 + |z2| 2 − Re (a1z 1 + a2z 2) +O(|z| where aj ≥ 0, j = 1, 2. The remainder ϕ = O(|z| 3) can be removed by changing ρ to ρ̃ = ρ−ϕλ, where λ(z) = λ0(z/ε) is a smooth cut-off function with λ0 ≡ 1 in a neighborhood of the origin and λ0(z) = 0 for |z| ≥ 1, ε > 0 small enough. Since ϕ(z) = O(|z|3), then |d(ϕλ)| ≤ C|z|2, ‖ ϕλ ‖C2(IC2)≤ Cε where C > 0 is independent of ε. Since |dρ| ≥ C|z| in a neighborhood of 0 for some C > 0, then for small ε > 0 the function ρ̃ has only one critical point at the origin, is strictly plurisubharmonic and matches with ρ for |z| > ε. The coefficients aj can be reduced to the standard values 0 and 2 depending on the index k of the critical point. We need a cut-off function that falls down from 1 to 0 sufficiently slowly. Lemma 7.3 Given δ > 0 there exists a smooth non-increasing function φ with a compact support on IR+ such that (i) φ = 1 near the origin. (ii) |tφ′(t)| ≤ δ. (iii) |t2φ′′(t)| ≤ δ The lemma follows because Let bj = 0 (resp. 2) if 0 ≤ aj < 1 (resp. aj > 1). Let λ(z) = φ(|z|/ε), where φ is provided by Lemma 7.3 for sufficiently small δ. Then the function ρ̃(z) = ρ(z) + λ[(a1 − b1)Re z 1 + (a2 − b2)Re z for sufficiently small ε has all the desired properties. Proposition 7.1 is proved. Thus in what follows we assume that ρ has the properties given by Proposition 7.1. Let p be a critical point of ρ and ρ(p) = 0. Without loss of generality assume that the index k of p is equal to 1 or 2 since the disc obtained by Proposition 6.1 cannot approach a minimum of ρ. Choose a small neighborhood U of p. By (17) ρ is strictly plurisubharmonic with respect to Jst. We apply the construction of Lemma 6.7 of [11]. Consider c0 > 0 small enough such that 0 is the only critical value of ρ in the interval [−c0, 3c0]. We can assume that c0 is small enough so that the set K(c0) := {z : ρ(z) ≤ 3c0, |x ′|2 ≤ c0} is compactly contained in a neighborhood of the origin corresponding to U . Here we use the notation x′ = x1, x ′′ = x2 and |x′|2 = x21 (resp. x ′ = (x1, x2) and |x ′|2 = x21 + x 2 ) if k = 1 (resp. k = 2). We will use similar notations for the coordinates x, y and the coordinates u, v introduced below. Let E = {y′ = 0, z′′ = 0, |x′|2 ≤ c0}. (20) Then E is a totally real submanifold with boundary and dimE = k. Consider the isotropic dilations of coordinates dc0 : z 7→ w = u+ iv = c Set Jc0 = (dc0)∗(J). The structures Jc0 converge to Jst in any C m norm on compact subsets of IC2 as c0 −→ 0. Consider the function ρ̂(w) := c 0 ρ(c 0 w). This function has no critical values in [−1, 3] and its expression in the coordinates w = u+iv is the same as the expression (17) of ρ that is ρ̂(w) = 3v21 + v 2 − u 1 + u if k = 1 and ρ̂(w) = 3v21 + 3v 2 − u 1 − u if k = 2. In particular the set K = dc0(K(c0)) is given by {w : ρ̂(w) ≤ 3, |u ′|2 ≤ 1} and is a fixed compact independent of c0. It is important that the origin is a critical point of the function ρ and the local coordinates and the function ρ are given by Proposition 7.1. This allows to use the isotropic dilations in contrast with Lemma 6.2. Since the function ρ̂ is strictly plurisubharmonic with respect to Jst, we can apply the construction of [11] (Lemma 6.7 and section 6.4). We replace the function ρ̂ by a new function ϕ defined by ϕ(w) = 3v21 + v 2 − h(u 1) + u if k = 1 and ϕ(w) = 3v21 + 3v 2 − h(u 1 + u if k = 2, where h ≥ 0 is a suitable function. The construction of h depends on the parameter c0 only. In our “delated” coordinates w we apply this construction taking c0 = 1. Namely, according to [11] there exist constants 0 < τ0 < τ1 < 1 depending on the eigenvalues of ρ̂ and a function ϕ strictly plurisubharmonic on IC2 with respect to Jst satisfying the following properties: (i) ρ̂ ≤ ϕ ≤ ρ̂+ τ1, (ii) ρ̂+ τ0 ≤ ϕ on the set {|u ′|2 ≥ τ0} (iii) ϕ = ρ̂+ τ1 on {|u ′|2 ≥ 1} Since ρ̂ is strictly plurisubharmonic with respect to the structure Jc0 , the function ϕ also is strictly Jc0-plurisubharmonic on {|u ′|2 ≥ 1} in view of (iii). On the other hand the structures Jc0 converge to Jst in any C m norm on compact subsets of IC2 as c0 −→ 0. Therefore, since ϕ is strictly Jst-plurisubharmonic, it also is strictly Jc0-plurisubharmonic on K if c0 is small enough. Thus, ϕ is strictly Jc0-plurisubharmonic on {ρ̂ ≤ 3}. Now consider the function ρ̃(z) = c0ϕ(c 0 z) and set t0 = τ0c0. The function ρ̃ satisfies the following properties: (i) ρ̃ is strictly plurisubharmonic (with respect to Jst) in a neighborhood V ⊂ U of 0 and ρ̃ = ρ+ t1 on the complement of V . Here t1 > 0 is a constant. (ii) ρ̃ has no critical values on (0, 3c0) (iii) There exists t0 ∈ (0, c0) such that {ρ ≤ −c0} ∪ E ⊂ {ρ̃ ≤ 0} ⊂ {ρ ≤ −t0} ∪ E, (21) where E is defined above by (20). (iv) We have {ρ ≤ c0} ⊂ {ρ̃ ≤ 2c0} ⊂ {ρ < 3c0} (22) By Proposition 6.1 we construct an immersed J-holomorphic disc f such that −t0 < ρ ◦ f |bID < 0. The boundary of f is contained in a torus Λ formed by discs complex tangent to a level set of ρ. We will perturb the disc f slightly in order to avoid the intersection of its boundary with E. Proposition 7.4 Let f : ID −→ M be an immersed J-holomorphic disc in (M,J), where dimICM = 2. Let E be a smooth submanifold in M . Then for every m ≥ 2 there exists a J-holomorphic disc f̃ arbitrarily close to f in Cm(ID) such that f̃(0) = f(0), df̃(0) = df(0), and f̃ |bID is transverse to E. In particular, if dimIR E ≤ 2, then f̃(bID) ∩ E = ∅. Proof : By the implicit function theorem, the restriction f |bD admits infinitesimal pertur- bations in all directions. Then the proposition follows by the proof of Thom’s transversality theorem. We now assume f(bID) ∩ E = ∅. In view of the inclusion (21) we conclude that ρ̃ > 0 on f(bID). By Lemma 6.3 we cut off the disc f by a level set {ρ̃ = c} for some c > 0 to assume that now f(bID) is contained in this level set. The function ρ̃ has no critical values in (0, 3c0). By Proposition 6.1 applied to the disc f and the function ρ̃ there exists a new disc f̃ with the boundary contained in {ρ̃ > 2c0}. In view of (22) we have the inclusion {ρ̃ > 2c0} ⊂ {ρ > c0}. Now the boundary of f̃ is outside the critical level {ρ = 0} as desired, and we switch back to the original function ρ. 8 Proof of Theorem 1.1 Since the function ρ is strictly plurisubharmonic, then after a generic perturbation of ρ which does not change the given level set, we can assume that ρ is a Morse function. Let p be the given point in D. If p is not a point of minimum of ρ, we proceed as follows. Consider a small J-holomorphic disc f centered at p with the given direction v. Consider a non-critical level set ρ = c such that ρ(p) < c. Consider a foliation of a neighborhood of f by a complex one-parameter family of J-holomorphic discs hq, q ∈ f(ID) such that the boundaries of these discs are outside the sublevel set ρ < c. When q runs over the circle f(bID) these boundaries form a torus. Applying Proposition 5.1 we obtain a new disc f̃ centered at p and still in the same direction at p but with ρ ◦ f̃ |bID > 0. If p is a point of minimum for ρ, we drop this first step and directly have this situation with f̃ = f . Now the desired results follow by Proposition 6.1 combined with the above argument allowing to push boundaries of discs through critical levels. References [1] J.-F. Barraud, E. Mazzilli, Regular type of real hypersurfaces in (almost) complex manifolds, Math. 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0704.0125

Anisotropic thermo-elasticity in 2D -- Part I: A unified approach

ANISOTROPIC THERMO-ELASTICITY IN 2D PART I: A UNIFIED TREATMENT MICHAEL REISSIG AND JENS WIRTH Abstract. In this note we develop tools and techniques for the treatment of anisotropic thermo-elasticity in two space dimensions. We use a diagonalisation technique to obtain properties of the characteristic roots of the full symbol of the system in order to prove Lp–Lq decay rates for its solutions. Keywords: thermo-elasticity, a-priori estimates, anisotropic media 1. The problem under consideration Systems of thermo-elasticity are hyperbolic-parabolic or hyperbolic-hyperbolic coupled sys- tems (type-1, type-2 or type-3 models) describing the elastic and thermal behaviour of elastic, heat-conducting media. The classical type-1 model of thermo-elasticity is based on Fourier’s law, which means, that the heat flux is proportional to the gradient of the temperature. The present paper is devoted to the study of type-1 systems for homogeneous but anisotropic media in R2. There are different results in the literature for certain anisotropic media (cubic in [3]; rhombic in [4]). Our goal is to present an approach, which allows to consider (an)isotropic models in R2 from a unified point of view. We consider the type-1 system of thermo-elasticity Utt +A(D)U + γ∇θ = 0, (1.1a) θt − κ∆θ + γ∇TUt = 0. (1.1b) Here A(D) denotes the elastic operator, which is assumed to be a homogeneous second order 2×2 matrix of (pseudo) differential operators and models the elastic properties of the underlying medium. Furthermore, κ describes the conduction of heat and γ the thermo-elastic coupling of the system. We assume κ > 0 and γ 6= 0. We solve the Cauchy problem for system (1.1) with initial data U(0, ·) = U1, Ut(0, ·) = U2, θ(0, ·) = θ0, (1.2) for simplicity we assume U1, U2 ∈ S(R2,R2) and θ0 ∈ S(R2). We denote by A(ξ) the symbol of the elastic operator and we set η = ξ/|ξ| ∈ S1. Then some basic examples for our approach are given as follows. The material constants are always specified in such a way that the matrix A(η) becomes positive. Example 1.1. Cubic media in 2D are modelled by A(η) = (τ − µ)η21 + µ (λ+ µ)η1η2 (λ+ µ)η1η2 (τ − µ)η22 + µ (1.3) with constants τ, µ > 0, −2µ−τ < λ < τ . This case was treated e.g. in [3]. For the corresponding elastic system see [12]. Example 1.2. Rhombic media in 2D are modelled by A(η) = (τ1 − µ)η21 + µ (λ + µ)η1η2 (λ+ µ)η1η2 (τ2 − µ)η22 + µ (1.4) http://arxiv.org/abs/0704.0125v3 2 MICHAEL REISSIG AND JENS WIRTH with constants τ1, τ2, µ > 0 and −2µ− τ1τ2 < λ < τ1τ2. For this case we refer also to [4]. Example 1.3. Although it is not the main point of this note, we can consider isotropic media, where A(η) = µI + (λ + µ)η ⊗ η (λ+ µ)η21 + µ (λ+ µ)η1η2 (λ+ µ)η1η2 (λ+ µ)η 2 + µ (1.5) with Lamé constants µ > 0 and λ+ µ > 0. We will present a unified treatment of these cases of (in general) anisotropic thermo-elasticity. For this we assume that the homogeneous symbol A = A(ξ) = |ξ|2A(η), η = ξ/|ξ|, is given as a function A : S1 → C2×2 (1.6) subject to the conditions (A1): A is real-analytic in η ∈ S1, (A2): A(η) is self-adjoint and positive for all η ∈ S1. For some results we require that (A3): A(η) has two distinct eigenvalues, # specA(η) = 2. Under assumption (A3) the direction η ∈ S1 is called (elastically) non-degenerate. In this case we know that the elasticity equation Utt+A(D)U = 0 is strictly hyperbolic and can be diagonalised smoothly using a corresponding system of normalised eigenvectors rj(η) to the eigenvalues κj(η) of A(η). If (A3) is violated we will call the corresponding directions η ∈ S1 degenerate. For these directions we can use the one-dimensionality of S1 in connection with the analytic perturbation theory of self-adjoint matrices (cf. [6]). So we can always find locally smooth eigenvalues κj(η) and corresponding locally smooth normalised eigenvectors rj(η) of A(η). For the following we assume for simplicity that these functions extend to global smooth functions on S1. This classification of directions is not sufficient for a precise study of Lp–Lq decay estimates for solutions to the thermo-elastic system. It turns out that different microlocal directions η = ξ/|ξ| ∈ S1 from the phase space have different influence on decay estimates. But how to distinguish these directions and how to understand their influence? In general, this can be done by a refined diagonalisation procedure applied to a corresponding first order system (first order with respect to time). Applying a partial Fourier transform and chosing a suitable energy (of minimal dimension) this system reads as DtV = B(ξ)V . The properties of the matrix B(ξ) are essential for our understanding: • the notions of hyperbolic and parabolic directions depend on the behaviour of the eigen- values of B(ξ) (see (2.3), (2.5), Definition 1); • the matrix B(ξ) contains spectral data of A(ξ) together with certain coupling functions (see (2.6)) between different components of the energy. The behaviour of these coupling functions close to hyperbolic directions has an essential influence on decay rates (see Theorems 3.1, 3.5 and 3.6). It turns out that we have to exclude some exceptional values of the coupling constant γ by assuming that (see Definition 1 for the notion of hyperbolic directions) (A4): γ2 6= 2κj0(η̄)− trA(η̄) for all hyperbolic directions η̄ with respect to κj0 . Basically this implies the non-degeneracy of the 1-homogeneous part of B(ξ). In the following we will call a hyperbolic direction violating (A4) a γ-degenerate direction. Assumption (A4) is used for the treatment of small hyperbolic frequencies and plays there a similar rôle like (A3) for large frequencies. ANISOTROPIC THERMO-ELASTICITY IN 2D 3 In Section 2 we will give the transformation of the thermo-elastic system (1.1) to a system of first order and the diagonalisation procedure in detail. The proposed procedure generalises those from [14], [15], [9], [17]. The obtained results are used to represent solutions of the original system as Fourier integrals with complex phases. Based on these representations we give micro- localised decay estimates for solutions in Section 3. They and their method of proof depend • the classification of directions (to be hyperbolic or parabolic); • the order of contact between Fresnel curves (coming from the elastic part) and their tangents for hyperbolic directions; • the vanishing order of the coupling functions in hyperbolic directions. Let us formulate some of the results. The first one follows from Theorem 3.1 and Corollary 3.2. Result. Under assumptions (A1) to (A4) and if the coupling functions vanish to first order in hyperbolic directions the solutions U(t, x) and θ(t, x) to (1.1) satisfy the Lp–Lq estimate ‖DtU(t, ·)‖q + ‖ A(D)U(t, ·)‖q + ‖θ(t, ·)‖q . (1 + t)− ) (‖U1‖p,r+1 + ‖U2‖p,r + ‖θ0‖p,r) (1.7) for dual indices p ∈ (1, 2], pq = p+ q, and Sobolev regularity r > 2(1/p− 1/q). If the coupling functions vanish to higher order we have to relate their vanishing order ℓ to the order of contact γ̄ between the Fresnel curve and its tangent in the corresponding direction and for the corresponding sheet. In Theorems 3.5 and 3.6 we show that in this case the 1/2 in the exponent is changed to 1/min(2ℓ, γ̄). Our main motivation to write this paper is to provide a unified way to treat anisotropic models of thermo-elasiticity. New analytical tools presented in these notes generalise to higher dimensions and allow to treat especially models in 3D (outside degenerate directions). The two- dimensional results from [3] for cubic media and [4] for rhombic media are contained / extended; general anisotropic media can be treated. This is discussed in some detail in the second part [16] of this note. Acknowledgements. The first author thanks Prof. Wang Ya-Guang (Shanghai Jiao Tong University) for the discussion about some basic ideas of the approach presented in this paper during his stay at the TU Bergakademie Freiberg in August 2004. The stay was supported by the German-Chinese research project 446 CHV 113/170/0-2. 2. General treatment of the thermo-elastic system We use a partial Fourier transform with respect to the spatial variables to reduce the Cauchy problem for (1.1) to the system of ordinary differential equations Ûtt + |ξ|2A(η)Û + iγξθ̂ = 0, (2.1a) θ̂t + κ|ξ|2θ̂ + iγξ · Ût = 0, (2.1b) Û(0, ·) = Û1, Ût(0, ·) = Û2, θ̂(0, ·) = θ̂0 (2.1c) parameterised by the frequency variable ξ. We denote by κ1,κ2 ∈ C∞(S1) the eigenvalues of A(η) and by r1, r2 ∈ C∞(S1, S1) corre- sponding normalised eigenvectors. Both depend in a real-analytic way on η ∈ S1. In a first step we reduce (2.1) to a first order system. For this we use the diagonaliser of the elastic operator, i.e. the matrix M(η) = (r1(η)|r2(η)) build up from the normalised eigenvectors, and define U (0)(t, ξ) =MT (η)Û (t, ξ). (2.2) 4 MICHAEL REISSIG AND JENS WIRTH Then we define by the aid of D1/2(η) = diag(ω1(η), ω2(η)), ωj(η) = κj(η) ∈ C∞(S1), (2.3) the vector-valued function V (0)(t, ξ) = (Dt +D1/2(ξ))U (0)(t, ξ) (Dt −D1/2(ξ))U (0)(t, ξ) θ̂(t, ξ)  , (2.4) where as usual Dt = −i∂t. It satisfies a first order system with apparently simple structure. A short calculation yields DtV (0)(t, ξ) = B(ξ)V (0)(t, ξ) with B(ξ) = ω1(ξ) iγa1(ξ) ω2(ξ) iγa2(ξ) −ω1(ξ) iγa1(ξ) −ω2(ξ) iγa2(ξ) a1(ξ) − iγ2 a2(ξ) − a1(ξ) − iγ2 a2(ξ) iκ|ξ| , (2.5) where we used the coupling functions aj(ξ) = rj(η) · ξ. (2.6) For later use we introduce the notation B1(ξ) and B2(ξ) for the homogeneous components of B(ξ) of order 1 and 2, respectively. The coupling functions aj(η) can be understood as the co-ordinates of η with respect to the orthonormal eigenvector basis {r1(η), r2(η)}. Therefore, it holds a21(η) + a 2(η) = 1. Furthermore, they are well-defined and real-analytic functions on S In the following proposition we collect some information on the characteristic polynomial of the matrix B(ξ). Proposition 2.1. (1) trB(ξ) = iκ|ξ|2 and detB(ξ) = iκ|ξ|6 detA(η). (2) The characteristic polynomial of B(ξ) is given by det(νI −B(ξ)) =(ν − iκ|ξ|2)(ν2 − κ1(ξ))(ν2 − κ2(ξ)) − νγ2|ξ|2a21(η)(ν2 − κ2(ξ))− νγ2|ξ|2a22(η)(ν2 − κ1(ξ)). (2.7) (3) An eigenvalue ν ∈ specB(ξ), ξ 6= 0, is real if and only if ν2 = κj0 (ξ) for an index j0 = 1, 2. If the direction is non-degenerate this is equivalent to aj0(η) = 0. (4) If aj(η) 6= 0, j = 1, 2 the eigenvalues ν ∈ specB(ξ) satisfy iκ|ξ|2 a21(ξ) ν2 − κ1(ξ) a22(ξ) ν2 − κ2(ξ) . (2.8) It turns out that the property of B(ξ) to have real eigenvalues depends only on the direction η = ξ/|ξ| ∈ S1. We will introduce a notation. Definition 1. We call a direction η ∈ S1 hyperbolic if B(ξ) has a real eigenvalue and parabolic if all eigenvalues of B(ξ) have non-zero imaginary part. In hyperbolic directions we always have a pair of real eigenvalues. If η ∈ S1 is hyperbolic with ±ωj0(ξ) ∈ specB(ξ) for ξ = |ξ|η, we call η hyperbolic with respect to the index j0 (or with respect to the eigenvalue κj0(η) of A(η)) and ν±(ξ) = ±ωj0(ξ) the corresponding pair of hyperbolic eigenvalues of B(ξ). A non-degenerate direction is parabolic if and only if aj(η) 6= 0, j = 1, 2, while for non- degenerate hyperbolic directions one of the coupling functions aj0(η) vanishes. Degenerate di- rections are always hyperbolic (in 2D), see (2.7). ANISOTROPIC THERMO-ELASTICITY IN 2D 5 Example 2.1. If the medium is isotropic, A(η) = µI+(λ+µ)η⊗η, the eigenvalues of A are µ and λ+µ with corresponding eigenvectors η and η⊥. Thus all directions are hyperbolic (with respect to the second eigenvalue). In this case the matrix B(ξ) decomposes into a diagonal hyperbolic 2 × 2-block and a parabolic 3 × 3-block. This decomposition coincides with the Helmholtz decomposition as used in the standard treatment of isotropic thermo-elasticity. Example 2.2. For cubic media (where we assume in addition µ 6= τ and µ + λ 6= 0) there exist eight hyperbolic directions determined by η1η2 = 0 or η 1 = η 2 . The functions aj(η) have simple zeros at these directions. Example 2.3. Weakly coupled cubic media with λ+µ = 0, µ 6= τ , have the degenerate directions η21 = η 2 , media with µ = τ , λ + µ 6= 0, for η1η2 = 0. In both cases the coupling functions aj(η) do not vanish in these directions. If µ = τ = −λ, the elastic system decouples directly into two wave equations with propagation speed µ. In this case all directions are degenerate. Example 2.4. For rhombic media we have to distinguish between three cases. Case 1. If the material constants satisfy (λ+2µ− τ1)(λ+2µ− τ2) > 0, we are close to the cubic case and there exist eight hyperbolic directions given by η1η2 = 0 and η21(λ+ 2µ− τ1) = η22(λ+ 2µ− τ2). Case 2. If we assume on the contrary that (λ+2µ−τ1)(λ+2µ−τ2) < 0, only the four hyperbolic directions η1η2 = 0 exist. In the Cases 1 and 2 in each hyperbolic direction one of the coupling functions aj(η) vanishes to first order. Case 3. In the borderline case τ1 = λ+2µ or τ2 = λ+2µ, but τ1 6= τ2, three hyperbolic directions collapse to one. We have the four hyperbolic directions η1η2 = 0, at two of them (ηj = ±1 if τj = λ+ 2µ) the vanishing order of the coupling function is three. Rhombic media are degenerate if a) µ = τ1 (or µ = τ2) with degenerate direction (0, 1) T (or (1, 0)T ) or b) λ+µ = 0 (weakly coupled case) and (µ−τ1)(µ−τ2) > 0 with degenerate directions determined by η21(µ− τ1) = η22(µ− τ2) or c) τi = µ = −λ (exceptional case) where all directions are degenerate. Proposition 2.2. Let the direction η̄ ∈ S1 be non-degenerate and hyperbolic with respect to the index j0. Then the corresponding eigenvalues ν±(ξ) satisfy a2j0(ξ) ν2±(ξ) − κj0(ξ) = qη̄(|ξ|) = Cη̄ ∓ iDη̄|ξ| (2.9) for all non-tangential limits with real constants Cη̄, Dη̄ ∈ R, Dη̄ > 0. Furthermore, the imaginary part of the hyperbolic eigenvalue satisfies Im ν±(ξ) a2j0(η) D2η̄|ξ|2 C2η̄ + |ξ|2D2η̄ > 0, (2.10) and thus vanishes like Im ν(|ξ|η) ∼ a2j0(η) as η → η̄ for all ξ 6= 0. 6 MICHAEL REISSIG AND JENS WIRTH Proof. Let for simplicity j0 = 1. We use the characteristic polynomial of B(ξ) to deduce γ2|ξ|2a21(η) ν2± − |ξ|2κ1(η) iκ|ξ|2 γ2|ξ|2a22(η) ν2± − |ξ|2κ2(η) → 1∓ iκ|ξ| ω1(η̄) κ1(η̄)− κ2(η̄) (2.11) = 1− γ κ1(η̄)− κ2(η̄) ︸ ︷︷ ︸ γ2Cη̄ ω1(η̄) ︸ ︷︷ ︸ γ2Dη̄ |ξ| = γ2qη̄(|ξ|). (2.12) The existence of the limit is implied by ν± 6= 0 and ν2± 6= κ2(ξ) as consequence of ν±(ξ) → ±|ξ|ω1(η) as η → η̄ by Proposition 2.1. Obviously, Im qη̄(|ξ|) = ∓κ|ξ|/γ2ω1(η̄) = ∓Dη̄|ξ| is non-zero for ξ 6= 0 and considering the imaginary part of the first limit expression Im qη̄(|ξ|) = lim a21(ξ) ν2±(ξ)− κ1(ξ) = lim −2Re ν±(ξ) Im ν±(ξ) a21(ξ) |ν2±(ξ)− κ1(ξ)|2 = ∓2ω1(|ξ|η̄)|qη̄(|ξ|)|2 lim Im ν±(|ξ|η) |ξ|2a21(η) proves the second statement, limη→η̄ Im ν±(ξ) a21(η) Dη̄ |ξ| 2ω1(η̄)(C η̄+|ξ| 2D2η̄) In the case of isolated degenerate directions (others are not of interest, because then the system is decoupled) we can find a replacement for Proposition 2.1. Proposition 2.3. Let η̄ ∈ S1 be an isolated degenerate direction, κ1(η̄) = κ2(η̄). Then the corresponding hyperbolic eigenvalues ν±(ξ) satisfy ω1(ξ)− ν2±(ξ) ω1(ξ)− ω2(ξ) = a21(η̄) > 0, (2.13) and, therefore, Im ν±(ξ) ω1(ξ)− ω2(ξ) = 0, lim ω1(ξ)− Re ν±(ξ) ω1(ξ)− ω2(ξ) = a21(η̄). (2.14) Thus, if a1(η̄) 6= 0 then the eigenvalues ν±(ξ) approach ±ω1(ξ) at the contact order between ω1(ξ) and ω2(ξ) (while they approach ±ω1(ξ) with a higher order if a1(ξ) = 0). 2.1. Asymptotic expansion of the eigenvalues as |ξ| → 0. If |ξ| is small the first order part B1(ξ) dominates B2(ξ), so the properties of the eigenvalues are governed by spectral properties of B1(ξ). Proposition 2.4. (1) trB1(ξ) = 0 and detB1(ξ) = 0. (2) If the direction η ∈ S1 is parabolic the nonzero eigenvalues ν̃ of B1(η) satisfy γ−2 = a21(η) ν̃2 − κ1(η) a22(η) ν̃2 − κ2(η) (2.15) and are thus real and related to κj(η) by 0 < κ1(η) < ν̃ 1(η) < κ2(η) < ν̃ 2(η) (2.16) (if κ1(η) < κ2(η)). ANISOTROPIC THERMO-ELASTICITY IN 2D 7 (3) If η is non-degenerate and hyperbolic with respect to the index 1 we have κ1(η) = ν̃ 1(η), while for hyperbolic directions with respect to the index 2 three cases occur depending on the size of the coupling constant γ: γ2 < κ2(η̄)− κ1(η̄) : κ2(η) = ν̃22 (η), γ2 = κ2(η̄)− κ1(η̄) : ν̃21(η) = κ2(η) = ν̃22 (η), γ2 > κ2(η̄)− κ1(η̄) : ν̃21(η) = κ2(η). (2.17) (4) If the direction is degenerate, κ1(η) = κ2(η), we have the eigenvalues ± κ1(η) and κ1(η) + γ2. The existence of five distinct eigenvalues of the homogeneous principal part B1(η) for all parabolic and most hyperbolic directions allows us to calculate the full asymptotic expansion of the eigenvalues ν(ξ) of B(ξ) as |ξ| → 0. We will give only the first terms in detail, but provide the whole diagonalisation procedure. Assumption (A4) guarantees the non-degeneracy of B1(ξ). Note that, even if (A4) is violated the matrix B1(ξ) is diagonalisable (as consequence of its block structure). Proposition 2.5. As |ξ| → 0 the eigenvalues of the matrix B(ξ) behave as ν0(ξ) = iκ|ξ|2b0(η) +O(|ξ|3), (2.18a) ν±j (ξ) = ±|ξ|ν̃j(η) + iκ|ξ| 2bj(η) +O(|ξ|3) (2.18b) for all non-γ-degenerate directions, where the functions bj ∈ C∞(S1) are given by b0(η) = γ2a21(η) κ1(η) γ2a22(η) κ2(η) > 0 (2.19) bj(η) = 1 + γ2a21(η) ν̃2j + κ1(η) (ν̃2j − κ1(η))2 + γ2a22(η) ν̃2j + κ2(η) (ν̃2j − κ2(η))2 ≥ 0. (2.20) Furthermore, bj(η) > 0 if η is parabolic and bj0(η) = 0 if η is hyperbolic with respect to the index Proof. We apply a diagonalisation scheme in order to extract the spectral information for B(ξ). We assume that the eigenvalues are denoted such that κ1(η) ≤ κ2(η). Step 1. By Proposition 2.4 we know that the homogeneous first order part B1(η) has the distinct eigenvalues ν̃0 = 0 and ν̃ j (η) = ±ν̃j(η), which are ordered as κ1(η) ≤ ν̃1(η) ≤ κ2(η) ≤ ν̃2(η) (where equality holds only under the exceptions stated in Proposition 2.4). We denote corresponding normalised and bi-orthogonal left and right eigenvectors of the matrix B1(η) by ±(η) and e±j (η). If we collect them in the matrices L(η) = (0e(η)|1e+(η)|1e−(η)|2e+(η)|2e−(η)), (2.21a) R(η) = (e0(η)|e+1 (η)|e 1 (η)|e 2 (η)|e 2 (η)), (2.21b) we have L∗(η)R(η) = I and L∗(η)B1(η)R(η)) = D1(η) = diag(0, ν̃1(η),−ν̃1(η), ν̃2(η),−ν̃2(η)). (2.22) Further we get L∗(η)B2(η)R(η) = iκb∗(η)⊗ ∗b(η), (2.23) where b∗(η) and ∗b(η) are vectors collecting the last entries b0(η), b j (η) and 0b(η), jb ±(η) of the eigenvectors e0(η), e j (η) and 0e(η), je ±(η), respectively. 8 MICHAEL REISSIG AND JENS WIRTH The matrix B(0)(ξ) = L∗(η)B(ξ)R(η) = |ξ|D1(η) + |ξ|2iκb∗(η)⊗ ∗b(η) (2.24) is diagonalised modulo O(|ξ|2) as |ξ| → 0 and has a main part with distinct entries. We denote R(2)(ξ) = |ξ|2iκb∗ ⊗ ∗b. Step 2. We construct a diagonaliser of B(0)(ξ) as |ξ| → 0 of the form Nk(ξ) = I + |ξ|jN (j)(η). (2.25) For this we denote the k-homogeneous part of R(k)(ξ) by R̃(k)(ξ) and its entries by R̃ ij (η). Then we set for k = 1, 2, . . . Dk+1(η) = diag R̃(k+1)(η), (2.26) N (k)(η) = (k+1) 12 (η) d1(η)−d2(η) · · · R̃ (k+1) 15 (η) d1(η)−d5(η) (k+1) 21 (η) d2(η)−d1(η) 0 · · · R̃ (k+1) 25 (η) d2(η)−d5(η) . . . (k+1) 51 (η) d5(η)−d1(η) (k+1) 52 (η) d5(η)−d2(η) · · · 0 , (2.27) where dj(η) are the entries of D1(η). By construction we have the commutator relation [D1(η), N (k)(η)] = Dk+1(η)− R̃(k+1)(η), (2.28) such that R(k+2)(ξ) = B(0)(ξ)Nk(ξ)−Nk(ξ) |ξ|jDj(η) = R(k+1)(ξ) + |ξ|kB(0)(ξ)N (k)(η)− |ξ|kN (k)(η) |ξ|jDj(η) −Nk(ξ)|ξ|k+1Dk+1(η) = R(2)(ξ)|ξ|kN (k)(η)− |ξ|kN (k)(η) |ξ|jDj(η)− (Nk(ξ)− I)|ξ|k+1Dk+1(η) = O(|ξ|k+2). Using Nk(ξ) − I = O(|ξ|) we see that for |ξ| ≤ ck, ck sufficiently small, the matrix Nk(ξ) is invertible and N−1k (ξ)B (0)(ξ)Nk(ξ) = |ξ|jDj(η) +O(|ξ|k+2). (2.29) Thus, the entries of Dj(η) contain the asymptotic expansion of the eigenvalues, while the rows of Nk(ξ)R(η) (and L(η)N k (ξ)) give asymptotic expansions of the right (and left) eigenvectors of B(ξ). Furthermore, the construction implies that all occurring matrices are smooth functions of η ∈ S1. Step 3. We calculate the first terms explicitly. For this we need the diagonal entries of the matrix b∗ ⊗ ∗b. Therefore, we determine the left and right eigenvectors of B1(η). If we assume ANISOTROPIC THERMO-ELASTICITY IN 2D 9 that the direction η is non-degenerate we get for e0(η) = (r 1 , r 1 , r 2 , r 2 , r0) T and 0e(η) = (ℓ+1 , ℓ 1 , ℓ 2 , ℓ 2 , ℓ0) T the equations ±r±j ωj + iγajr0 = 0, ±ℓ j ωj − ajℓ0 = 0, (2.30a) 1 + a2r 2 + a1r 1 + a2r 2 = 0, a1ℓ 1 + a2ℓ 2 + a1ℓ 1 + a2ℓ 2 = 0, (2.30b) together with the normalisation condition r+1 ℓ 1 + · · · r 2 + r0ℓ0 = 1. (2.31) The first equations imply the representation ±r±j (η) = − iγaj(η) ωj(η) r0(η), ±ℓ±j (η) = iγaj(η) 2ωj(η) ℓ0(η), (2.32) the second line of equations follows from the first, while the normalisation condition yields b0(η) = r0(η) ℓ0(η) = γ2a21(η) κ1(η) γ2a22(η) κ2(η) 6= 0. (2.33) To calculate the eigenvectors we can further require r0(η) = ℓ0(η) = b0(η) > 0. Similarly, we obtain for the eigenvectors e+k (η) = (r 1 , . . . , r 2 , r0) T and ke +(η) = (ℓ+1 , . . . , ℓ 2 , ℓ0) the equations (we use the same notation as above in the hope that this will not lead to confusion here) ±r±j ωj + iγajr0 = ±ν̃kr j , ±ℓ j ωj − ajℓ0 = ±ν̃k(η)ℓ±j (2.34) together with the normalisation condition. Thus for parabolic directions we get ±r±j (η) = iγaj(η) ν̃k(η)− ωj(η) r0(η), ±ℓ±j (η) = − iγaj(η) 2(ν̃k(η)− ωj(η)) ℓ0(η), (2.35) and hence b+k (η) = r0(η) ℓ0(η) = j=1,2 γ2a2j(η) 2(ν̃k(η)− ωj(η))2 j=1,2 γ2a2j(η) 2(ν̃k(η) + ωj(η))2 1 + γ2a21(η) ν̃2k(η) + κ1(η) (ν̃2k(η) − κ1(η))2 + γ2a22(η) ν̃2k(η) + κ2(η) (ν̃2k(η)− κ2(η))2 . (2.36) For e−k (η) and ke −(η) we have to replace ν̃k(η) by −ν̃k(η) and obtain b−k (η) = b k (η) = bk(η). If the direction η is non-degenerate and hyperbolic with respect to the index j0, the entries r and ℓ±j0 are undetermined by (2.34), while the other entries of the vectors are zero. Together with the normalisation condition this determines the eigenvectors and gives bj0(η) = 0. It remains to consider degenerate directions. Then we have ν̃1 = ω1 and ν̃2 > ω1 such that for k = 1 we have ℓ0 = r0 = 0, r 1 and ℓ 1 are non-zero while r 2 = ℓ 2 = 0, especially b1(η) = 0. For k = 2 we get from the above expression for b2(η) = (2 + 2κ 2/γ2)−1 > 0. � Remark. 1. Note that for all non-degenerate hyperbolic directions η̄ ∈ S1 with respect to the index 1 the limit a21(η)b 1 (η) = κ1(η̄) κ1(η̄)− κ2(η̄) (2.37) is taken and non-zero, while for hyperbolic directions with respect to the index 2 the corre- sponding limit is non-zero only if γ2 6= κ2(η̄) − κ1(η̄), i.e. if the direction is not γ-degenerate. Near γ-degenerate directions Step 1 of the previous proof is still valid. Similar to Step 2 we 10 MICHAEL REISSIG AND JENS WIRTH can diagonalise to a (2, 2, 1) block structure. The eigenvalues of these blocks can be calculated explicitely. 2. For degenerate directions we obtain similarly b1(η) (κ1(η)− κ2(η))2 a21(η̄)a 2(η̄) 2γ2κ1(η̄) (2.38) and b1(η) vanishes to the double contact order. 2.2. Asymptotic expansion of the eigenvalues as |ξ| → ∞. If we consider large frequencies the second order part B2(ξ) dominates B1(ξ). This makes it necessary to apply a different two- step diagonalisation scheme. We follow partly ideas from [9], [14], [15] adapted to our special situation. Proposition 2.6. As |ξ| → ∞ the eigenvalues of the matrix B(ξ) behave as ν0(ξ) = iκ|ξ|2 − +O(|ξ|−1), (2.39a) ν±j (ξ) = ±|ξ|ωj(η) + a2j (η) +O(|ξ|−1). (2.39b) for all non-degenerate directions ξ/|ξ| ∈ S1. Remark. 1. Note that, while in hyperbolic directions we always have ν±j0(ξ) = ±|ξ|ωj0(η) for one index j0, in degenerate hyperbolic directions all aj(η) may be non-zero. Hence the statement of the above theorem cannot be valid in such directions in general. 2. Degenerate directions play for large frequencies a similar rôle as γ-degenerate directions play for small frequencies. Proof. The proof will be decomposed into several steps. In a first step we use the main part B2(ξ) = iκ|ξ|2 diag(0, 0, 0, 0, 1) to block-diagonalise B(ξ). In a second step we diagonalise the upper 4× 4 block for all non-degenerate directions. Step 1. For a matrix B ∈ C5×5 we denote by b-diag4,1B the block diagonal of B consisting of the upper 4 × 4 block and the lower corner entry. We construct a diagonalisation scheme to block-diagonalise B(ξ) as |ξ| → ∞. We set R(−1)(ξ) = B(ξ) − B2(ξ) = B1(ξ) and B−2(ξ) = B2(ξ) and construct recursively a diagonaliser modulo the upper 4× 4 block, Mk(ξ) = I + |ξ|−jM (j)(η). (2.40) Again we denote by R̃(k)(η) the (−k)-homogeneous part of R(k)(ξ) (which exists because it exists for R(−1)(ξ) and the existence is transfered by the construction). Then we introduce the recursive scheme Bk−2(η) = b-diag4,1 R̃(k−2)(η), (2.41) M (k)(η) = (k−2) 15 (η) (k−2) 45 (η) −R̃(k−2)51 (η) · · · −R̃ (k−2) 54 (η) 0 (2.42) for k = 1, 2, . . ., such that the commutator relation [B−2(η),M (k)(η)] = Bk−2(η)− R̃(k−2)(η), (2.43) ANISOTROPIC THERMO-ELASTICITY IN 2D 11 holds. Thus it follows R(k−1)(ξ) = B(ξ)Mk(ξ)−Mk(ξ) |ξ|−jBj(η) = O(|ξ|1−k) (2.44) and using that Mk(ξ) is invertible for |ξ| ≥ Ck, Ck sufficiently large, we obtain the block diago- nalisation M−1k (ξ)B(ξ)Mk(ξ) = |ξ|−jBj(η) +O(|ξ|1−k), (2.45) where Bj(η) = b-diag4,1 Bj(η) is (4, 1)-block diagonal. Step 2. By Step 1 we constructed Mk(ξ) such that M k (ξ)B(ξ)Mk(ξ) is (4, 1)-block diagonal modulo O(|ξ|1−k). The upper 4×4 block has already diagonal main part |ξ|−1D−1(η) = B1(ξ). If the direction η = ξ/|ξ| is non-degenerate, the diagonal entries ±ω1(η) and ±ω2(η) are mutually distinct and thus we can apply the standard diagonalisation procedure (cf. proof of Proposi- tion 2.5) in the corresponding subspace. This does not alter the lower corner entry and gives only combinations of the entries of the last column and of the last row (without changing their asymptotics). Thus we can construct a matrix Nk−1(ξ) = I + j=1 |ξ|−jN (j)(ξ), which is invertible for |ξ| > C̃k−1, C̃k−1 sufficiently large, such that N−1k−1(ξ)M k (ξ)B(ξ)Mk(ξ)Nk−1(ξ) = |ξ| 2B−2(η) + |ξ|−jDj(η) +O(|ξ|1−k) (2.46) is diagonal modulo O(|ξ|1−k). Step 3. We give the first matrices explicitly. Following Step 1 we get M (1)(η) = γa1(η) γa2(η) γa1(η) γa2(η) γa1(η) γa2(η) γa1(η) γa2(η) (2.47) together with B−1(η) = diag(ω1(η), ω2(η),−ω1(η),−ω2(η), 0) and R̃(0)(η) = B1(η)M (1)(η)−M (1)(η)B−1(η) γ2a21(η) γ2a1(η)a2(η) · · · γa1(η)ω1(η) γ2a1(η)a2(η) γ2a22(η) · · · γa2(η)ω2(η) . . . γa1(η)ω1(η) γa2(η)ω2(η) · · · −iγ , (2.48) B0(η) = b-diag4,1 R̃(0)(η) (2.49) in the first diagonalisation step. Applying a second step alters only the last row and column to O(|ξ|−1). Following Step 2 we diagonalise the upper 4× 4 block to |ξ|B−1(η) +B0(η) +O(|ξ|−1) modulo O(|ξ|−1) and the statement is proven. � Remark. If the direction η is degenerate, i.e. ω1(η) = ω2(η), we can block-diagonalise in Step 2 to (2, 2, 1)-block form. To diagonalise further we have to know that the 0-homogeneous part of these 2× 2-blocks has distinct eigenvalues. 12 MICHAEL REISSIG AND JENS WIRTH One possible treatment of degenerate directions is given in the following proposition. Note that a corresponding statement can be obtained for γ-degenerate directions as |ξ| → 0. Proposition 2.7. Let η̄ be an isolated degenerate direction. Then the corresponding hyperbolic eigenvalue satisfies in a small conical neighbourhood of η̄ ν−j0(ξ) = ω1(ξ) + ω2(ξ) (ω1(ξ)− ω2(ξ))2 iγ2(ω1(ξ)− ω2(ξ))(a21(η)− a22(η)) +O(|ξ|−1). (2.50) Proof. We follow the treatment of the previous proof to (2, 2, 1)-block-diagonalise B(ξ) modulo |ξ|−1. Now, we consider one of its 2× 2-blocks. (We use a similar notation as before in the hope that it will not lead to any confusion.) Such a block is given by B(ξ) = |ξ|B−1(η) + B0(η) +O(|ξ|−1), (2.51) where B−1(η) = diag ω1(η), ω2(η) , (2.52) B0(η) = a21(η) a1(η)a2(η) a1(η)a2(η) a . (2.53) In the direction η̄ both diagonal entries of B−1 coincide. In a small conical neighbourhood we denote the eigenvalues of |ξ|B−1(η) + B0(η) as δ+(ξ) and δ−(ξ). A simple calculation yields δ±(ξ) = ω1(ξ) + ω2(ξ) (ω1(ξ)− ω2(ξ))2 iγ2(ω1(ξ)− ω2(ξ))(a21(η)− a22(η)) (2.54) with δ−(ξ̄) = ω1(ξ̄) and δ+(ξ̄) = ω1(ξ̄) + . The hyperbolic eigenvalue corresponds to δ−(ξ). These eigenvalues are distinct in a sufficiently small neighbourhood of η̄ (and may coincide only if a1(η) = a2(η) and (ω1(ξ)−ω2(ξ))2 = γ4/(4κ2), which gives eventually two parabolic directions). Hence the perturbation theory of matrices implies ν+j0(ξ) = δ−(ξ) +O(|ξ| −1) in a sufficiently small neighbourhood of η̄ and the statement is proven. � Remark. Note, that ω1(ξ)− δ−(ξ) ω1(ξ)− ω2(ξ) = a21(η̄) (2.55) for all fixed |ξ|, which coincides with the result (2.13) of Proposition 2.3 for the eigenvalue ν(ξ). 2.3. Collecting the results. The asymptotic expansions from Propositions 2.5 and Proposi- tion 2.6 imply estimates for eigenvalues of B(ξ) and the proofs give representations of corre- sponding eigenvectors. For the application of multiplier theorems and the proof of Lp–Lq decay estimates it is essential to provide also estimates for derivatives of them. Assume that the eigenvalues under consideration are simple. From the asymptotic expansions we know that this is the case for small frequencies and also for large frequencies. For the middle part it will be sufficient to know that the hyperbolic eigenvalues are separated, which follows for sufficiently small conical neighbourhoods of these directions. ANISOTROPIC THERMO-ELASTICITY IN 2D 13 In a first step we consider derivatives of the eigenvalues. Differentiating the characteristic polynomial 0 = det(ν(ξ)I −B(ξ)) = Ik(ξ)ν(ξ) k (2.56) with respect to ξ yields by Leibniz formula ξ Ik(ξ) ξ ν(ξ) (2.57) for all multi-indices α ∈ N20. Thus we can express the highest derivative of ν(ξ)k in terms of lower ones and hence Faà di Bruno’s formula (see e.g. [5]) yields an expression Dαν(ξ) kIk(ξ)ν(ξ) k−1 = Ck,α,β Dα−βIk(ξ) Dβν(ξ) (2.58) with certain constants Ck,α,β . Because the eigenvalue has multiplicity one, the sum on the left- hand side is nonzero and therefore we can calculate the derivatives of ν(ξ) by this expression. Furthermore, it follows that for small and large frequencies the derivatives of the eigenvalue have full asymptotic expansions and thus we are allowed to differentiate the asymptotic expansions term by term. It remains to consider the corresponding eigenprojections. Recall that if ν(ξ) is a eigenvalue of multiplicity one and r(ξ) and l(ξ) are corresponding right and left eigenvectors, the correspond- ing eigenprojection is given by the dyadic product Pν(ξ) = l(ξ) ⊗ r(ξ). Thus the constructed diagonaliser matrices imply asymptotic expansions of these operators. Again we are only inter- ested whether the derivatives of these eigenprojections also possess asymptotic expansions (in order to see whether it is allowed to differentiate term by term). For this we use the representation Pν(ξ) = ν̃∈specB(ξ)\{ν} (ν̃(ξ)I −B(ξ))(ν̃(ξ)− ν(ξ))−1 (2.59) given e.g. in [4], [7]. All terms on the right-hand side have full asymptotic expansions as |ξ| → 0 and |ξ| → ∞ together with all of their derivatives. Differentiating with respect to ξ yields the same result for the eigenprojection. Thus we obtain Proposition 2.8. The asymptotic expansions from Proposition 2.5 and Proposition 2.6 may be differentiated term by term to get asymptotic expansions for the derivatives of the eigenvalues. Furthermore, the same holds true for the corresponding eigenprojections. From Proposition 2.1 we know that an eigenvalue ν(ξ) of the matrix B(ξ) is real if and only if η = ξ/|ξ| is hyperbolic. We want to combine this information with the asymptotic expansions of Proposition 2.5 and Proposition 2.6 and derive some estimates for the behaviour of the imaginary part. Proposition 2.9. Let c > 0 be a given constant. (1) Let η = ξ/|ξ| ∈ S1 be parabolic. Then the eigenvalues of B(ξ) satisfy Im ν(ξ) ≥ Cη > 0, |ξ| ≥ c, (2.60) with a constant Cη depending on the direction η and c. Furthermore, Im ν(ξ) ∼ b(η)|ξ|2, |ξ| ≤ c, (2.61) where b(η) is one of the functions from Proposition 2.5. 14 MICHAEL REISSIG AND JENS WIRTH (2) Let η̄ be non-degenerate and hyperbolic with respect to the index 1. Then ν±1 (|ξ|η̄) = ±|ξ|ω1(η̄) and ν0(ξ) and ν±2 (ξ) satisfy the statement of point 1. Furthermore, Im ν±1 (ξ) ∼ a21(η), |ξ| ≥ c, |η − η̄| ≪ 1, (2.62) Im ν±1 (ξ) ∼ |ξ|2a21(η), |ξ| ≤ c, |η − η̄| ≪ 1. (2.63) Proof. The first point follows directly from the asymptotic expansions, we concentrate on the second one. We know that the hyperbolic eigenvalues ν±1 (ξ) satisfy by Proposition 2.2 Im ν±1 (ξ) = a 1(η)N 1 (ξ) (2.64) with a smooth non-vanishing function N±1 (ξ) defined in a neighbourhood of η̄. By Proposition 2.5 and 2.6 we see that N±1 (ξ) also has full asymptotic expansions and thus N±1 (ξ) = +O(|ξ|−1), |ξ| → ∞ (2.65a) N±1 (ξ) = iκ|ξ| 2 b1(η) a21(η) +O(|ξ|3), |ξ| → 0. (2.65b) Together with (2.37) we get upper and lower bounds on N±1 (ξ) and the desired statement follows. Remark. A similar reasoning allows to replace the hyperbolic eigenvalue near degenerate di- rections by the model expression obtained in Proposition 2.7, thus ν(ξ) ∼ δ−(ξ) uniformly in a sufficiently small conical neighbourhood of η̄. 3. Decay estimates for solutions Our strategy to give decay estimates for solutions to the thermo-elastic system (1.1) is to micro-localise them. In principle we have to distinguish four different cases. On the one hand we differentiate between small and large frequencies, on the other hand between hyperbolic directions and parabolic ones. We distinguish between two cases depending on the vanishing order of the coupling functions. If the coupling functions vanish to first order at hyperbolic directions only, we rely on simple multiplier estimates. Later on we discuss coupling functions with higher vanishing order, where the decay rates are obtained by tools closely related to the treatment of the elasticity equation. 3.1. Coupling functions with simple zeros. In a first step we consider the first order system DtV = B(D)V, V (0, ·) = V0 (3.1) to Cauchy data V0 ∈ S(R2,C5). For a cut-off function χ ∈ C∞(R+) with χ(s) = 0, s ≤ ǫ, and χ(s) = 1, s ≥ 2ǫ, we consider Ppar(η) = η̄ hyperbolic χ(|η − η̄|), Phyp(η) = 1− Ppar(η). (3.2) Then Phyp(D) localises in a conical neighbourhood of the set of hyperbolic directions, while Ppar(D) localises to a compact set of parabolic directions. The asymptotic formulae and representations for the characteristic roots of the full symbol B(ξ) allow us to proof decay estimates for the solutions. ANISOTROPIC THERMO-ELASTICITY IN 2D 15 Theorem 3.1. Assume that (A1) to (A4) are satisfied and the coupling functions vanish in hyperbolic directions to first order. Then the solution V (t, x) to (3.1) satisfies the following a-priori estimates: ‖χ(D)Ppar(D)V (t, ·)‖q . e−Ct‖V0‖p,r (3.3a) ‖(1− χ(D))Ppar(D)V (t, ·)‖q . (1 + t)−( )‖V0‖p (3.3b) ‖χ(D)Phyp(D)V (t, ·)‖q . (1 + t)− )‖V0‖p,r (3.3c) ‖(1− χ(D))Phyp(D)V (t, ·)‖q . (1 + t)− )‖V0‖p (3.3d) for dual indices p ∈ (1, 2], pq = p+ q, and with Sobolev regularity r > 2(1/p− 1/q). Remark. If B(ξ) is diagonalisable for ξ 6= 0 (which is valid e.g. if B(ξ) has no double eigenvalues for ξ 6= 0) we can make the result even more precise. To each eigenvalue ν(ξ) ∈ specB(ξ) we have corresponding left and right eigenvectors and associated to them the eigenprojection Pν(D) such that Pν(D)V (t, ·) = eitν(D)Pν(D)V0. (3.4) Thus, we can single out the influence of one eigenvalue in this way. Note, that this is only of interest in the neighbourhood of hyperbolic directions and for the corresponding eigenvalue and there the assumption of diagonalisability of B(ξ) may be skipped (real eigenvalues are always simple, for small |ξ| diagonalisation works, for large |ξ| everything goes well under the assumption of non-degeneracy and on the middle part we make the neighbourhood small enough to exclude possible multiplicities). Proof. We decompose the proof into four parts corresponding to the four estimates. The micro- localised estimates are merely standard multiplier estimates. We do not use stationary phase method. Step 1. Parabolic directions, large frequencies. In this case we have uniformly in ξ ∈ suppχPpar , the estimate Im ν(ξ) ≥ C′ > 0. Taking 0 < C < C′ we obtain for these ξ ImB(ξ) = B(ξ)−B∗(ξ) in the sense of self-adjoint operators and the estimate follows from the L2–L2 estimate ‖χ(D)Ppar(D)V (t, x)‖22 = ‖χ(ξ)Ppar(ξ)V̂ (t, ξ)‖22 = 2Re χ(ξ)Ppar(ξ)V̂ (t, ξ), χ(ξ)Ppar(ξ)∂tV̂ (t, ξ) = −2 Im χ(ξ)Ppar(ξ)V̂ (t, ξ), χ(ξ)Ppar(ξ)B(ξ)V̂ (t, ξ) ≤ −2C‖χ(ξ)Ppar(ξ)V̂ (t, ξ)‖2 = −2C‖χ(D)Ppar(D)V (t, x)‖22, viewed as Hs–Hs estimate and combined with Sobolev embedding. Step 2. Parabolic directions, small frequencies. We know from Proposition 2.5 that in this case the matrix B(ξ) has only simple eigenvalues. We will make use of the representation of solutions V (t, x) = ν(ξ)∈specB(ξ) eitν(D)Pν(D)V0(x) (3.5) with corresponding eigenprojections (amplitudes) Pν(ξ). The amplitudes are uniformly bounded on the set of all occurring ξ and possess full asymptotic expansions in |ξ| as ξ → 0 together with their derivatives. Especially, by Hörmander-Mikhlin multiplier theorem [10, p. 96, Theorem 3] the operators Pν(D) are L p-bounded for 1 < p <∞. 16 MICHAEL REISSIG AND JENS WIRTH It remains to consider the model multiplier eitν(ξ). From Proposition 2.5 we know that |eitν(ξ)| . e−Ct|ξ| , |ξ| ≤ c, (3.6) with suitable constants c and C and thus the L1–L∞ estimate ‖eitν(D)f‖∞ ≤ ‖eitν(ξ)f̂‖1 ≤ ‖eitν(ξ)‖L1({|ξ|≤c})‖f̂‖∞ . ‖f‖1 e−Ct|ξ| |ξ|d|ξ| . (1 + t)−1 ‖f‖1 holds for all f ∈ L1(R2) with supp f̂ ⊆ {|ξ| ≤ c}. Riesz-Thorin interpolation [1, Chapter 4.2] with the obvious L2–L2 estimate gives the desired decay result. Step 3. Hyperbolic directions, large frequencies. We take the conical neighbourhoods small enough to exclude all multiplicities (related to the eigenvalue which becomes real in the hyperbolic direction). Then similar to (3.5) the solution is represented as V (t, x) = eitν(D)P+ν (D)V0(x) + e −itν(D)P−ν (D)V0(x) + Ṽ (t, x), (3.7) where Ṽ (t, x) corresponds to the remaining parabolic eigenvalues of B(ξ) and satisfies the es- timate from Step 1. Again we can use smoothness of P±ν (ξ) together with the existence of a full asymptotic expansion as |ξ| → ∞ to get Lp-boundedness of P±ν (D) for 1 < p < ∞ from Hörmander-Mikhlin multiplier theorem. It remains to understand the model multiplier e±itν(ξ) for the hyperbolic eigenvalue related to the hyperbolic direction η̄. Using the estimate from Proposition 2.9 we conclude for r > 2 ‖e±itν(D)f‖∞ ≤ ‖e±itν(ξ)f̂‖1 ≤ ‖e±itν(ξ)|ξ|−r‖L1(S1)‖ |ξ| r f̂‖∞ . ‖〈D〉rf‖1 |ξ|1−rd|ξ| e−C1tφ . t−1/2 ‖〈D〉rf‖1, t ≥ 1 for all f ∈ 〈D〉−rL1(R2) with supp f̂ ⊆ S1 = {|ξ| ≥ c, |η − η̄| ≤ ǫ}. Riesz-Thorin interpolation with the L2–L2 estimate gives the desired decay result. Step 4. Hyperbolic directions, small frequencies. Like for large hyperbolic frequencies we make use of the representation (3.7) to separate hyperbolic and parabolic influences. As in the previous cases the existence of full asymptotic expansions imply that the projections P±ν (D) are L bounded for 1 < p <∞. It remains to understand the model multiplier e±itν(ξ). Using Proposition 2.9 we have |e±itν(ξ)| . e−C2t|ξ| 2φ2 such that after introducing polar co-ordinates ‖e±itν(D)f‖∞ ≤ ‖e±itν(ξ)f̂‖1 ≤ ‖e±itν(ξ)‖L1(S2)‖f‖1 . ‖f‖1 e−C2tφ 2|ξ|2dφ|ξ|d|ξ| . t−1/2‖f‖1 d|ξ| . t−1/2 ‖f‖1, t ≥ 1 for all f ∈ L1(R2) with supp f̂ ⊆ S2 = {|ξ| ≤ c, |η − η̄| ≤ ǫ}. Riesz-Thorin interpolation with the L2–L2 estimate gives the desired decay result. � Remark. 1. In non-degenerate hyperbolic directions where the coupling function vanishes to order ℓ (cf. Example 2.4 case 3) we obtain by the same reasoning the weaker Lp–Lq decay rate ‖χ(D)Phyp(D)V (t, ·)‖q . (1 + t)− )‖V0‖p,r. (3.8) ANISOTROPIC THERMO-ELASTICITY IN 2D 17 It remains to understand whether this weaker decay rate is also sharp or whether an application of stationary phase method may be used to improve this. We will discuss this in Section 3.2. 2. We can extend the estimate of Theorem 3.1 to the limit case p = 1, if we include the eigen- projections Pν(ξ) into the considered model multiplier and use just their boundedness instead of Hörmander-Mikhlin multiplier theorem. This is possible, because all the multiplier estimates were based on Hölder inequalities. So far we understood properties of solutions to the transformed problem (3.1) for the vector- valued function V (t, x) given by (2.4) as V (t, x) = (Dt +D1/2(D))U (0)(t, x) (Dt −D1/2(D))U (0)(t, x) θ(t, x)  , (3.9) where U (0)(t, x) =MT (D)U(t, x) is the elastic displacement after transformation with the diag- onaliser of the elastic operator. Because M(η) is unitary and homogeneous of degree zero this diagonaliser is Lp-bounded for 1 < p <∞ with bounded inverse. Thus we have DtU(t, x) =M(D) V (t, x) (3.10) A(D)U(t, x) =M(D) 0 − 1 0 − 1 V (t, x) (3.11) such that as corollary of Theorem 3.1 we obtain Corollary 3.2. Assume (A1) to (A4) and that the coupling functions vanish of first order in all hyperbolic directions. Then the solution U(t, x) and θ(t, x) to (1.1) satisfy the a-priori estimates ‖DtU(t, ·)‖q + ‖ A(D)U(t, ·)‖q + ‖θ(t, ·)‖q . (1 + t)− ) (‖U1‖p,r+1 + ‖U2‖p,r + ‖θ0‖p,r) (3.12) for dual indices p ∈ (1, 2], pq = p+ q, and Sobolev regularity r > 2(1/p− 1/q). Remark. Including the diagonaliser M(ξ) into the model multipliers of Theorem 3.1 allows to overcome the restriction on p and to extend this statement up to p = 1. We preferred this way of presenting the results because they allow to decouple both statements. Theorem 3.1 gives deeper insight into the asymptotic behaviour of solutions than Corollary 3.2 does. 3.2. Coupling functions vanishing to higher order. We have seen that under the assump- tion that the coupling functions aj(η) ∈ C∞(S1), aj(η) = η · rj(η) related to the symbol of the elastic operator A(η) have only zeros of first order, we can deduce Lp–Lq decay estimates without relying on the method of stationary phase. Now we want to discuss how to use the method of stationary phase to deduce decay estimates in the remaining cases. From Proposition 2.2 we know that in hyperbolic directions the imaginary part Im ν±j0(ξ) of the hyperbolic eigenvalues vanishes like the square of the corresponding coupling function a2j0(η), while the real part Re ν (ξ) is essentially described by ±ωj0(ξ). First, we make this more precise and formulate an estimate for Re νj0(ξ) ± ∓ ωj0(ξ) and its derivatives. Proposition 3.3. Let η̄ ∈ S1 be hyperbolic with respect to the index j0 and aj0(η) vanish to order ℓ in η̄. Then there exists a conical neighbourhood of the direction η̄ such that for all k = 0, 1, . . . , 2ℓ− 1 the estimates ∣∂kη (Re ν (ξ) ∓ ωj0(ξ)) ∣ ≤ c(|ξ|)|η − η̄|2ℓ−k, (3.13a) ∣∂kη Im ν ∣ ≤ c(|ξ|)|η − η̄|2ℓ−k, (3.13b) 18 MICHAEL REISSIG AND JENS WIRTH hold uniformly on it, where c(|ξ|) ∼ 1 as |ξ| → ∞ and c(|ξ|) ∼ |ξ| as |ξ| → 0. This estimate may be used to transfer the micro-localised decay estimate for the elasticity equation to the thermo-elastic system on a sufficiently small conical neighbourhood of η̄. We need one further notion to prepare the main theorem of this section. Decay rates for solutions to the elasticity equation depend heavily on the order of contact between the sheets of the Fresnel surface and its tangents, cf. [13] or point 2 of the concluding remarks on page 21. Proposition 3.4. For η ∈ S1 we denote by γ̄j(η) the order of contact between the j-th sheet Sj = {ω−1j (η)η | η ∈ S 1} (3.14) of the Fresnel curve and its tangent in the point ω−1j (η)η. Then ∂kη (∂ ηωj(η) + ωj(η)) = 0, k = 0, . . . , γ̄j(η)− 2, (3.15) (if γ̄j(η) > 2) and ∂γ̄j(η)η ωj(η) + ∂ γ̄j(η)−2 η ωj(η) 6= 0. (3.16) Proof. Follows by straight-forward calculation. The curvature of the j-th Fresnel curve Sj at the point ω−1j (η)η factorises as ωj(η) + ∂ ηωj(η) and a smooth non-vanishing analytic function. � For the following we chose the conical neighbourhood of the hyperbolic direction η̄ (with respect to the index j0) small enough in order that the curvature of Sj0 vanishes at most in η̄. Theorem 3.5. Let η̄ be hyperbolic with respect to the index j0 and let aj0(η) vanish in η̄ to order ℓ > 1. Let us assume further that γ̄j0(η̄) < 2ℓ. Then ||e±itνj0 (D)f ||q . (1 + t) )||f ||p,r (3.17) for dual indices p ∈ [1, 2], pq = p + q, regularity r > 2(1/p − 1/q) and for all f with supp f̂ contained in a sufficiently small conical neighbourhood of η̄. Proof. We distinguish between two cases related to the Fourier support of f , for general f we can use linearity. In both cases we apply the method of stationary phase. Key lemma will be the Lemma of van der Corput, cf. [11, p.334], combined for large frequencies with a dyadic decomposition. It suffices to consider t ≥ 1, the estimate for t ≤ 1 follows from the uniform boundedness of the Fourier multiplier together with Sobolev embedding using the regularity imposed on the data, r > 2(1/p− 1/q). In the following we skip the index j0 of the eigenvalue. Thus ω(η) stands for ωj0(η) and γ̄(η) for γ̄j0(η) to shorten the notation. Large and medium frequencies. Assume that f̂ is supported in a sufficiently small conical neigh- bourhood of η̄ bounded away from zero, |ξ| > 1. We will use a dyadic decomposition in the radial variable. Let for this χ ∈ C∞0 (R) be chosen in such a way that χ(R) = [0, 1], suppχ ⊆ [1/2, 2] j∈Z χ(2 js) = 1 for all s ∈ R+. We set χj(s) = χ(2−js) such that suppχj ⊆ [2j−1, 2j+1]. We follow the treatment of Brenner [2] and use Besov spaces. Due to our assumptions on the Fourier support of f Besov norms are given by ||f ||Brp,q = ∥2jr||χj(|D|)f ||p ℓq(N0) (3.18) and corresponding Besov spaces are related to Sobolev and Lp-spaces by the embedding relations Hp,r →֒ Brp,2, B0q,2 →֒ Lq (3.19) ANISOTROPIC THERMO-ELASTICITY IN 2D 19 for p ∈ (1, 2] and q ∈ [2,∞). Thus it suffices to prove an Brp,2 → B0q,2 estimate for dual p and q. The case p = q = 2, r = 0 is trivial by Plancherel and the uniform boundedness of the multiplier, we concentrate on the B21,2 → B0∞,2 estimate. Therefore, we use ||e±itν(D)f ||B0 ||χj(D)e±itν(D)f ||2∞ ≤ sup Ij(t) ||f ||Br , (3.20) where Ij(t) are estimates of the dyadic components of the operator, Ij(t) = sup |F−1[eitν(ξ)ψ(ξ)χj(|ξ|)|ξ|−r]|. (3.21) Here ψ(ξ) localises to a small conical neighbourhood of η̄ bounded away from zero, |ξ| > 1, containing the support of f̂ . We assume |ψ(ξ)|+ |∂ηψ(ξ)|+ |∂|ξ|ψ(ξ)| . 1 and ψ ≡ 1 on a smaller conic set around η̄. We introduce polar co-ordinates |ξ| and φ (with φ = 0 corresponding to η̄) and set x = tz. Using that the hyperbolic directions η̄ ∈ S1 are isolated we can assume that the conical neighbourhood under consideration contains no further hyperbolic direction. For the calculation of Ij(t) we integrate over the interval [−ǫ, ǫ] for φ. This yields for all j ∈ N0 Ij(t) = sup ∫ 2j+1 eit(Re ν(ξ)+z·ξ)−t Im ν(ξ)ψ(ξ)χj(|ξ|)|ξ|1−rdφd|ξ| = 2j(2−r) sup jt(Re ν(2jξ)2−j+z·ξ)e−t Im ν(2 jξ)ψ(2jξ)χ(|ξ|)|ξ|1−rdφd|ξ| Due to Proposition 3.3 (let us restrict to the − case) we have Re ν(2jξ) = 2jω(ξ) + 2j|ξ|α(2jξ), (3.22) where ∣∂kφα(2 ∣ ≤ Ck |φ|2ℓ−k, k = 0, 1, . . . , 2ℓ− 1. (3.23) By our assumptions 2ℓ− 1 ≥ γ̄(η̄). We use this to reduce the problem to properties of the elastic eigenvalue ωj(η) and consider Ij(t) = 2 j(2−r) sup j |ξ|(ω(η)+z·η+α(2jξ))e−t Im ν(2 jξ)ψ(2jξ)dφχ(|ξ|)|ξ|1−rdξ . (3.24) Note that, |α(2jξ)| . |φ|2ℓ is small, if we choose the neighbourhood of η̄ small enough. If |ω(η)+z ·η| ≥ δ for some small δ the outer integral has no stationary points and integration by parts implies arbitrary polynomial decay. We restrict to the z ∈ R2 with |ω(η) + z · η| ≤ δ and consider only the inner integral Ij(t, |ξ|) = j|ξ|(ω(η)+z·η+α(2jξ))e−t Im ν(2 jξ)ψ(2jξ)dφ . (3.25) It can be estimated using the Lemma of van der Corput. For this we need an estimate for deriva- tives of the phase. We start by considering the unperturbed phase ω(η) + z · η. Differentiation yields ∂γ̄(η̄)η (ω(η) + z · η) = ∂γ̄(η̄)−2η (∂2ηω(η) + ω(η))− ∂γ̄(η̄)−4η (∂2ηω(η) + ω(η)) +− · · ·+− ω(η) + z · η, 2|γ̄(η̄) ∂ηω(η) + z · ηT , 26 |γ̄(η̄) (3.26) 20 MICHAEL REISSIG AND JENS WIRTH and by Proposition 3.4 the first term is non-zero for η̄ while the others are small in a neighbour- hood of η̄. Choosing δ and the neighbourhood small enough and using (3.23) this implies a lower bound on the γ̄(η̄)-th derivative of the phase, |∂γ̄(η̄)η (ω(η) + z · η + α(2jξ))| & 1 (3.27) uniformly on this neighbourhood of η̄ and independent of j. Thus we conclude by [11, p. 334] for all t ≥ 1 and uniformly in z with |ω(η) + z · η| ≤ δ Ij(t, |ξ|) ≤ Ck(t2j |ξ|)− γ̄(η̄) −t Im ν(2jξ)ψ(2jξ) . (3.28) To estimate the remaining integral we use Proposition 3.3 −t Im ν(2jξ)ψ(2jξ) ∣ dφ . e−ctφ |φ|2ℓ−1tdφ = 2 e−ctφ φ2ℓ−1ctdφ e−ctφ Integrating over ξ ∈ [1/2, 2] and choosing r ≥ 2 we obtain for j ∈ N0 Ij(t) . (1 + t) γ̄(η̄) , (3.29) where the occurring constant is independent of j and (3.20) implies the desired B21,2 → B0∞,2 estimate. Interpolation with the L2–L2 estimate gives the corresponding Brp,2 → B0q,2 estimates with r ≥ 2(1/p− 1/q) and finally the embedding relations to Sobolev and Lp-spaces the desired estimate ||eitν(D)f ||q . (1 + t)− γ̄(η̄) ||f ||p,r (3.30) for all f satisfying the required Fourier support conditions. Small frequencies. Assume now that f̂ is supported in a small conical neighbourhood of η̄ with |ξ| ≤ 2. In this case we estimate directly by the method of stationary phase. We sketch the main ideas. It is sufficient to estimate I(t) = sup eitν(ξ)ψ(ξ)χ(|ξ|) ∣ , (3.31) where the function ψ ∈ C∞(R+) localises to the small neighbourhood of η̄ with |ξ| ≤ 2. Again we require |ψ(ξ)| + |∂ηψ(ξ)| + |∂|ξ|ψ(ξ)| . 1. In correspondence to large frequencies I(t) equals I(t) = sup eit|ξ|(ω(η)+z·η+α(ξ))e−t Im ν(ξ)ψ(ξ)χ(|ξ|)dφ|ξ|d|ξ| . (3.32) We distinguish between |ω(η) + z · η| ≥ δ, where the outer integral has no stationary points, and |ω(η) + z · η| ≤ δ. In the first case we can apply one integration by parts and get t−1 using |ξ| |∂|ξ|e−t Im ν(ξ)| . |ξ|2te−t|ξ| 2φ2ℓ . 1. (3.33) In the second case we can reduce the consideration to I(t, |ξ|) = eit|ξ|(ω(η)+z·η+α(ξ))e−t Im ν(ξ)ψ(ξ)dφ (3.34) and an application of the Lemma of van der Corput. By Propositions 3.3 and 3.4 we have γ̄(η̄) φ (ω(η) + z · η + α(ξ)) 6= 0 (3.35) ANISOTROPIC THERMO-ELASTICITY IN 2D 21 for all ξ in a sufficiently small conic neighbourhood of η̄. So we obtain I(t, |ξ|) . (t|ξ|)− γ̄(η̄) −t Im ν(ξ)ψ(ξ) . (t|ξ|)− γ̄(η̄) . (3.36) Integrating with respect to |ξ| yields the estimate for I(t) I(t) ≤ I(t, |ξ|)|ξ|d|ξ| . t− γ̄(η̄) |ξ|1− γ̄(η̄) d|ξ| . (1 + t)− γ̄(η̄) , t ≥ 1 (3.37) and by Hölder inequality the L1–L∞ estimate ||eitν(D)f ||∞ ≤ I(t)||f ||1 . (1 + t)− γ̄(η̄) ||f ||1 (3.38) for all f satisfying the required Fourier support condition. By interpolation with the obvious L2–L2 estimate we get Lp–Lq estimates and combining them with the estimate of the first part proves the theorem. � If the order of contact exceeds the vanishing order of the coupling funtions, the best we can do is to use the idea of Section 3.1 and to apply standard multiplier estimates. As already remarked after the proof of Theorem 3.1 we obtain as decay rate in this case: Theorem 3.6. Let η̄ be hyperbolic with respect to the index j0 and let aj0(η) vanish in η̄ of order ℓ. Let us assume further that γ̄j0(η̄) ≥ 2ℓ, where γ̄j0(η) is defined in Proposition 3.4. Then ||e±itνj0 (D)f ||q . (1 + t)− )||f ||p,r (3.39) for dual indices p ∈ [1, 2], pq = p + q, regularity r > 2(1/p − 1/q) and for all f with supp f̂ contained in a sufficiently small conical neighbourhood of η̄. 4. Concluding remarks 1. In a second part of this note, [16], we will give concrete applications of the general treatment presented so far. From the remarks and examples we made in the previous sections it follows that we indeed cover the estimates of [3] for cubic media and [4] for rhombic media in the situations of coupling vanishing to first order. In [16] we will discuss the situation of higher order tangencies for rhombic media and give a new estimate extending the results of [4]. Furthermore, we will give concrete examples of media for all achievable decay rates in the case of a differential elastic operator A(D). 2. In general decay rates of solutions are determined by vanishing properties of the coupling functions. If the coupling functions are nonzero, decay rates are parabolic. If they vanish to sufficiently high order, decay rates are hyperbolic and determined by the elastic operator (micro- localised to this direction). In the intermediate case simple multiplier estimates are sufficient. 3. In the case of isotropic media one of the coupling functions vanishes identically. In this case one pair of eigenvalues of the matrix B(ξ) is purely real ν±(ξ) = ±(λ+µ)|ξ| and the components related to these eigenvalues solve a wave equation. Thus they satisfy the usual Strichartz type decay estimates, [2], ‖e±it(λ+µ)|D|P±(D)V0‖q . (1 + t)− )‖V0‖p,r (4.1) for p ∈ (1, 2], pq = p + q and r = 2(1/p − 1/q). More generally, if we know that one pair of eigenvalues satisfies ν±(ξ) = ±|ξ|ω(η) for all η ∈ S1 with a smooth function ω : S1 → R+, the decay rates for the corresponding components depend heavily on geometric properties of the Fresnel curve S = {ω−1(η)η | η ∈ S1 } ⊂ R2. Following [8] the estimate (4.1) is valid in this case as long as the curvature of S never vanishes. If there exist directions η where the curvature of S vanishes, the constant 1 in the exponent has to be altered to 1 , where γ̄ denotes the maximal order of contact of the curve S to its tangent, [13]. 22 MICHAEL REISSIG AND JENS WIRTH 4. The main focus of this paper was on the treatment of non-degenerate cases, thus assumptions (A1) to (A4) are required. Nevertheless, we showed how to obtain a control on the eigenvalues of the matrix B(ξ) in the exceptional cases where either (A3) or (A4) is violated. In the first case the treatment of large frequencies has to be replaced by the investigation of the expression obtained in Proposition 2.7, while in the latter one a corresponding replacement has to be made for small frequencies. 5. The results in this paper are essentially two-dimensional. For the study of anisotropic thermo- elasticity in higher space dimensions there arise two essential problems. The first is that we can not assume (A3). For example in three space dimensions and for cubic media the symbol of the elastic operator A : S2 → C3×3 has degenerate directions with multiple eigenvalues related to the crystal axes. Nevertheless, the multi-step diagonalisation scheme used in Section 2.2 can be adapted to such a case. This will be done in the sequel. The second problem is that the geometry of the set of hyperbolic directions becomes more complicated. In general we can not expect isolated hyperbolic directions, it will be necessary to consider manifolds of hyberbolic directions on S2. References [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988 [2] P. Brenner, On Lp–Lp′ estimates for the wave equation, Math. Z. 145(3):251–254, 1975. [3] J. Borkenstein, Lp–Lq Abschätzungen der linearen Thermoelastizitätsgleichungen für kubische Medien im R2, Diplomarbeit, Bonn, 1993. [4] M. S. Doll, Zur Dynamik (magneto-) thermoelastischer Systeme im R2, Dissertation, Konstanz, 2004. [5] W.P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly, Vol. 109(3): 217–234, 2002. [6] T. Kato, Perturbation Theory for linear Operators, Springer, 1980. [7] O. Liess, Decay estimates for the solutions of the system of crystal optics, Asymptot. Anal. 4:61–95, 1991. [8] R. Racke, Lectures on Nonlinear Evolution Equations – Initial Value Problems, Aspects of Mathemat- ics: E, Vol. 19, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1992. [9] M. Reissig, Y.-G. Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable, Math. Meth. Appl. Sci. 28:1359–1381, 2005. [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. [11] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 1993. [12] M. Stoth, Lp–Lq Abschätzungen für eine Klasse von Lösungen linearer Cauchy-Probleme bei anisotropen Medien, Dissertation, Bonn, 1994. [13] M. Sugimoto, Estimates for hyperbolic equations with non-convex characteristics, Math. Z. 222(4):521– 531, 1996. [14] Y.-G. Wang, Microlocal analysis in nonlinear thermoelasticity, Nonlinear Anal. 54:683–705, 2003 . [15] Y.-G. Wang, A new approach to study hyperbolic-parabolic coupled systems, in R. Picard (ed.) et al., Evolution equations. Propagation phenomena, global existence, influence of non-linearities. Based on the workshop, Warsaw, Poland, July 1-July 7, 2001, Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 60, p. 227-236, 2003. [16] J. Wirth, Anisotropic thermo-elasticity in 2D - Part II: Applications, Asymptotic Anal. ??:???– ???,????. [17] K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics, micro-local ap- proach, Akademie-Verlag, Berlin, 1997. Michael Reissig, Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany Jens Wirth, Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany current address: Department of Mathematics, Imperial College, London SW7 2AZ, UK 1. The problem under consideration 2. General treatment of the thermo-elastic system 2.1. Asymptotic expansion of the eigenvalues as ||0 2.2. Asymptotic expansion of the eigenvalues as || 2.3. Collecting the results 3. Decay estimates for solutions 3.1. Coupling functions with simple zeros 3.2. Coupling functions vanishing to higher order 4. Concluding remarks References

0704.0126

I-V characteristics of the vortex state in MgB2 thin films

I-V characteristics of the vortex state in MgB2 thin films Huan Yang,1 Ying Jia,1 Lei Shan,1 Yingzi Zhang1 and Hai-Hu Wen1∗ National Laboratory for Superconductivity, Institute of Physics and National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, P. R. China Chenggang Zhuang,2,3 Zikui Liu,4 Qi Li,2 Yi Cui2 and Xiaoxing Xi2,4 Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Department of Physics, Peking University, Beijing 100871, PR China and Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA (Dated: October 22, 2018) The current-voltage (I-V ) characteristics of various MgB2 films have been studied at different magnetic fields parallel to c-axis. At fields µ0H between 0 and 5 T, vortex liquid-glass transitions were found in the I-V isotherms. Consistently, the I-V curves measured at different temperatures show a scaling behavior in the framework of quasi-two-dimension (quasi-2D) vortex glass theory. However, at µ0H ≥ 5 T, a finite dissipation was observed down to the lowest temperature here, T = 1.7 K, and the I-V isotherms did not scale in terms of any known scaling law, of any dimensionality. We suggest that this may be caused by a mixture of σ band vortices and π band quasiparticles. Interestingly, the I-V curves at zero magnetic field can still be scaled according to the quasi-2D vortex glass formalism, indicating an equivalent effect of self-field due to persistent current and applied magnetic field. PACS numbers: 74.70.Ad, 74.25.Qt, 74.25.Sv I. INTRODUCTION Since the discovery of the two-gap superconductor MgB2 in 2001, 1 the mechanism of its superconductiv- ity and vortex dynamics has attracted considerable in- terests. The two three-dimension (3D) π bands and two quasi-two-dimension (quasi-2D) σ bands in this sim- ple binary compound seem to play an important role in the superconductivity,2 as well as the normal state properties.3,4 The two sets of bands have different en- ergy gaps, i.e., about 7 meV for the σ bands, and about 2 meV for the π bands.5,6 And the coherent length of the π bands is much larger than that of the σ bands2. Many experiments have demonstrated that the π-band superconductivity is induced from the σ-band and there is a rich evidence for both the interband and intraband scattering. Owing to the complicated nature of super- conductivity in this system, its vortex dynamics may ex- hibit some interesting or novel features. Among various experimental methods, measuring the current-voltage (I- V ) characteristics at different temperatures and magnetic fields can provide important information for understand- ing the physics of the vortex state. Up to now, the transport properties of MgB2 have been studied on both polycrystalline bulk samples7 and thin films8. In both cases, the I-V characteristics demonstrated good agree- ment with the 3D vortex glass (VG) theory. This was partially due to the limited magnetic fields in the experi- ment. In addition, it has been shown that the properties of MgB2 are very sensitive to the impurities and defects introduced in the process of sample preparation, and the vortex dynamics must be influenced, too. Therefore, it is necessary to investigate the vortex dynamics in high quality MgB2 epitaxial thin films and to reveal the in- trinsic properties of the vortex matter in this interesting multiband system. In this paper, we present the I-V characteristics of high-quality MgB2 thin films measured at various temperatures and magnetic fields. The vortex dynamics in this system is then investigated in detail. II. EXPERIMENT The high-quality MgB2 thin films studied in this work were prepared by the hybrid physical-chemical vapor de- position technique9 on (0001) 6H-SiC substrates. All the films had c-axis orientation with the thickness of about 100 nm. Fig. 1 (a) shows the θ-2θ scan of the MgB2 film, and the sharp (000l) peaks indicate the pure phase of the c-axis orientation of MgB2. In order to show the good crystallinity of the film, we present in Fig. 1(b) the same data in a semilogarithmic scale which enlarges the data in the region of small magnitude. It is clear that, besides the background noise, we can only observe the diffraction peaks from MgB2 and the SiC substrate, i.e., there is no trace of the second phase in the film. The c-axis lattice constant calculated from the MgB2 peak positions was about 3.517 Å(bulk value1: 3.524 Å). The φ scan (az- imuthal scan) shown elsewhere9 indicated well the six- fold hexagonal symmetry of the MgB2 film matching the substrate. The full width at half maximum (FWHM) of the 0002 peak taken on the film in θ-2θ scan [MgB2 0002 peak in Fig. 1(a)] and ω scan [rocking curve, shown in Fig. 1(c)] is 0.15◦ and 0.39◦, respectively. The scan- ning electron microscopy (SEM) image in Fig. 1(d) gave a rather smooth top surface view without any observable http://arxiv.org/abs/0704.0126v2 2x103 4x103 6x103 24 26 3x103 6x103 10 20 30 40 50 60 70 80 (degrees) FWHM=0.39o (0002) 500nm 2 (degrees) FIG. 1: (a) X-ray diffraction pattern of the MgB2 film on a (0001) 6H-SiC substrate in the θ-2θ scan, which shows only the 000l peaks of MgB2 in addition to substrate peaks, in- dicating a phase-pure c-axis-oriented MgB2 film. (b) The semilogarithmic plot of the θ-2θ scan. (c) The rocking curve of the 0002 MgB2 peak, which shows the FWHM of about 0.39◦. (d) The SEM image of the MgB2 film, which shows the smooth surface without obvious granularity. grain boundaries, which suggested that the film had a homogeneous quality. Ion etching was used to pattern a four-lead bridge with the effective size of 380× 20 µm2. The resistance measurements were made in an Oxford cryogenic system Maglab-Exa-12 with magnetic field up to 12 T. Magnetic field was applied along the c axis of the film for all the measurements. The temperature sta- bilization was better than 0.1% and the resolution of the voltmeter was about 10 nV. We have done all the mea- surements on several MgB2 films, and the experimental data and scaling behaviors are similar; so, in this paper, we present the data from one film. In Fig. 2, we present the resistive transitions (R-T re- lations) of a MgB2 thin film measured at various mag- netic fields in a semilogarithmic scale. The current den- sity in the measurement was about 500 A/cm , much smaller than the critical value for low temperatures, 106 A/cm 2 10. It can be determined from Fig. 2 that the sample had a superconducting transition tempera- ture of Tc = 40.05 K, with a transition width of about 0.5 K. Its normal state resistivity was about 2.45 µΩcm and the residual resistance ratio [≡ ρ(300 K)/ρ(42 K)] was about 6.4. The I-V curves were measured at various temperatures for each field, and then we got the electric 0 10 20 30 40 50 H=0 T H=1 T H=3 T H=6 T T (K) FIG. 2: Temperature dependence of resistive transitions for µ0H = 0, 1, 3, and 6 T, with the current density j = 500 A/cm field (E) and the current density (j) according to the sample dimension. The current density was swept from 5 to 105 A/cm during the I-V measurements. III. THEORETICAL MODELS In the mixed state of high-Tc superconductors with randomly distributed pointlike pinning centers, a second- order phase transition is predicted between VG state and vortex-liquid state.11 The I-V curves at different tem- peratures near the VG transition temperature Tg can be scaled onto two different branches12 by the scaling law j (T − Tg) ν(z+2−D) |T − Tg| ν(D−1) . (1) The scaling parameter z has the value of 4–7, and ν ≈ 1– 2; D denotes the dimension of the system with the value 3 for 3D and 2 for quasi-2D13; f+ and f− represent the functions for two sets of the branches above and below Tg. Above Tg, the linear resistivity is given by ρlin = dE/dj|j→0 ∝ (T − Tg) ν(z+2−D) . (2) At Tg, the electric field versus the current density curve satisfies the relationship E(j)|T=Tg ≈ j (z+1)/(D−1). (3) In 2D superconductors at µ0H = 0 T, a Berezinskii- Kosterlitz-Thouless (BKT) transition was found at a spe- cific temperature TBKT. 14 At TBKT, E ∝ j 3, which is a sign of the BKT transition. A continuous change from the BKT transition at zero field to a quasi-2D VG transi- tion, and then to a true 2D VG transition with Tg = 0 K was found in TlBaCaCuO film,15 which shows a field- induced crossover of criticalities. A 2D VG transition may exist in a true 2D system with Tg = 0 K, i.e., there is no zero-resistance state at any finite temperatures. The E-j curves can be scaled T 1+ν2D , (4) where T0 is a characteristic temperature, ν2D ≈ 2, and p ≥ 1, while g is a scaling function for all temperatures at a given magnetic field. The linear resistance is given ρlin ∝ exp[−(T0/T ) p]. (5) This 2D scaling law can be achieved in the very thin films17 or in highly anisotropic systems at high magnetic fields.18,19 IV. EXPERIMENTAL RESULTS AND DISCUSSIONS A. Quasi-two-dimension vortex-glass scaling in the low-field region (µ0H < 5 T) The E-j characteristics have been measured at various magnetic fields up to 12 T. In Fig. 3 we show the typical example at µ0H = 1 T for (a) E-j curves and (b) the cor- responding ρ-j curves in double-logarithmic scales. It is obvious that when the temperature goes below some par- ticular value (this is actually the vortex-glass transition temperature Tg according to following discussions), the resistivity falls rapidly with decreasing current density and finally reaches the zero-resistance state which is the characteristic of the so-called VG state. At the tempera- tures above Tg, the resistivity remains constant in small current limit. The current density of 500 A/cm used in ρ-T measurement shown in Fig. 2 lies in this linear resis- tivity regime from about 10−3 to 1 µΩcm. Consequently, these data sets provide the basic information on scaling if the data are describable by the VG theory. The inset in Fig. 4 shows the data of the ρlin versus (T −Tg) and the fit to Eq. (2). The data are the same as those shown in Fig. 2 for µ0H = 1 T, and the attempt Tg value is 31.4 K. In this double-logarithmic plot, the slope of the linear fitting gives just the exponent of ν(z+2−D), and the determined value is 8.08± 0.05. In order to have reasonable values for ν and z, the dimension parameterD needs to be chosen as 2, i.e., the investigated system has the property of quasi-2D, which is similar to the situation found in BiSrCaCuO.13,20 This is further supported by the VG scaling of the data at 1 T. As shown in the main frame of Fig. 4, the scaling experimental E-j curves form two universal branches corresponding to the data above and below Tg (31.4 K) with ν = 1.32 and z = 6.12. At very large current density or a temperature near the onset of superconducting transition, the free flux flow regime dominates and, hence, the data do not scale. The 101 102 103 104 105 10 nV j (A/cm2) H=1 T FIG. 3: (Color online) (a) E-j characteristics measured at fixed temperatures ranging from 30 to 36 K for µ0H = 1 T. The increments are 0.30 K in the range from 30.00 to 31.20 K, and 0.25 K in the range from 31.50 to 34.00 K respectively, and finally 35 K on the top. The dashed line shows the po- sition of Tg, and the symbols denote the segments that scale well according to the quasi-2D VG theory. The thin solid lines denote also the measured data, however, located outside the scalable range. (b) ρ-j curves corresponding to the E-j data in (a). The thick solid line in (b) denotes the voltage resolution of 10 nV. 1 2 3 4 5 6 From R(T) From E(j) 100 101 102 103 104 105 106 Quasi-2D VG Scaling for H=1 T D=2, T =31.4 K, =1.32, z=6.12 j/|T-T | (D-1) FIG. 4: (Color online) Quasi-2D VG scaling of the E-j curves measured at 1 T. The inset shows a double-logarithmic plot of the temperature dependence of the linear resistivity. The dashed line is a guide for the eyes. 3 6 9 121518 From R(T) From E(j) 10-2 10-1 100 101 102 103 104 105 10-10 Quasi-2D VG Scaling for H=3 T D=2, T =15.4 K, =1.00, z=7.70 j/(T|T-T | (D-1)) 10-1 100 101 102 103 104 105 106 10-11 Quasi-2D VG Scaling for H=3 T D=2, T =15.4 K, =1.17, z=6.58 j/|T-T | (D-1) FIG. 5: (Color online)(a) Scaling curves of the E-j data mea- sured in 3 T based on the quasi-2D VG scaling theory. The inset shows a log-log plot of the temperature dependence of the linear resistivity. (b) VG scaling with another form of scaling variable j/(T |T − Tg| ν(D−1) symbols in the figure denote the range of the data well described by the scaling law. The situation at µ0H = 3 T is similar to that at µ0H = 1 T. As shown in Fig. 5(a), the determined pa- rameters are Tg = 15.4 K, ν = 1.17, and z = 6.58. Inter- estingly, the previous work on MgB2 film 8 indicated that the 3D VG scaling theory (D = 3) is a better choice in describing the I-V characteristics in this system, though this experiment was done at magnetic fields lower than 1 T. Moreover, the I-V curves were demonstrated to scale well by using the argument of j/(T |T − Tg| ). The same conclusions were also drawn on the polycrystalline MgB2 samples7. In order to clarify this issue, we also analyzed our data using the form suggested in Ref. 8. As shown in Fig. 5(b), such a scaling with j/(T |T − Tg| ν(D−1) ) as the scaling variable is worse than that with j/ |T − Tg| ν(D−1) Most importantly, the dimension parameter D is still re- quired to be 2 instead of 3 as proposed in Refs. 7 and 8. This confusion can be easily understood in terms of the two-band superconductivity of MgB2. As we know, there are two types of bands contributing to the super- conductivity of MgB2, namely, the 3D π bands and the 101 102 103 104 105 H=6 T j (A/cm2) FIG. 6: (Color online) ρ-j data at temperatures 1.7 K and 4 K to 20 K with 2 K-step, for µ0H = 6 T. Temperature of the isotherms increases from bottom to top. 2D σ bands. Therefore, the structure of the vortex mat- ter must be affected by both of them. Although the su- perconductivity of π bands, induced possibly by that of σ bands, is much weaker, it provides a large coherence length with 3D characteristics in the low-field region. Therefore, the vortices in this system may be quasi-2D like and, at the same time, they can possess large cores characterized by the coherence length of the π band su- perfluid. In this sense, the quasi-2D scaling should be more appropriate than the 3D one. However, when a higher disorder is induced in the system, especially in the boron sites, the interband scattering gets stronger and the anisotropy decreases, which may lead to a 3D vortex scaling. In this case, a more rigid vortex line can be observed, especially at low fields.21 The good quasi-2D scaling at 1 and 3 T demonstrated here suggests that the phase transition from VG to the vortex liquid in MgB2 resembles that in the high-Tc superconductors. Together with the data shown below, we can safely conclude that a vortex glass state with zero linear resistivity can be achieved in the low field region due to the presence of the finite superfluid density from the π bands. Regarding the VG scaling22 a principal requirement is a proper deter- mination of Tg, namely the temperature with a straight logE-log j curve in the low dissipation part. The toler- ance for Tg variation is very small (about ±0.3 K). With an inappropriately chosen Tg, the scaling quality dra- matically deteriorates and, simultaneously, the values of ν and z quickly deviate from those reported above and those proposed by theory. This validates our analysis here. 10-3 10-1 101 103 A scaling attempt with quasi-2D VG for H=6 T D=2, T =0 K, =3, z=0.58 j/|T-T | (D-1) Qusi-2D VG theory From R(T) From E(j) FIG. 7: (Color online) The scaling of the E-j isotherms with quasi-2D VG model for µ0H = 6 T. The inset shows the deviation of the ρlin vs T − Tg (Tg = 0) relation from the linearity in the double logarithmic scale. B. Anomalous vortex properties in high field region As shown in Fig. 2, when the magnetic field reaches 6 T, no zero-resistance state can be observed down to the lowest temperature, here 1.7 K. Consequently, no VG transition exists above 1.7 K at this field, as shown in Fig. 6. The shape of the curve at T = 1.7 K suggests that the resistivity goes to a finite value as the current den- sity approaches zero.23 As shown in Fig. 7, the ρlin versus T − Tg seriously deviates from linearity for any possible Tg value, indicating the inapplicability of Eq. (2) in the present case. Correspondingly, the quasi-2D scaling law fails here. A natural explanation is that, with increas- ing field, the 3D supercurrent from π bands is seriously suppressed6,24 and the quasi-2D vortex structure trans- forms into a 2D-like one dominated by the σ band super- fluid. In Fig. 8, we show our attempt to apply 2D VG scaling on the data. Surprisingly, this attempt also failed, even though this model has been successfully applied to the layered superconductors with large anisotropy (or 2D property) such as Tl- and Bi-based high-Tc thin films at high magnetic fields.18,19 The most reasonable explanation for this anomaly is that the supercurrent contribution from the π bands is much easier to suppress by the magnetic field than that from the σ bands, since the gap in the π bands is several times smaller than that in the σ band. We suggest that at high magnetic fields (above 5 T), a different vortex matter state is formed, composed of quasi-particles from the π bands and vortices formed mainly by the residual superfluid from the σ band. The π-band quasiparticles diminish the long range phase coherence of the supercon- ducting phase, which leads to a finite dissipation. Once the long range superconducting phase coherence is de- stroyed by the proliferation of a large amount of these 5 10 15 20 10-4 10-3 10-2 10-1 100 101 102 103 A scaling attempt with 2D VG for H=6 T =2, p=1, T =20 K j/T1+ 2D 2D VG theory From R(T) From E(j) T (K) FIG. 8: (Color online) Attempted scaling of the data with 2D VG model [Eq. (4)] for µ0H = 6 T. The inset shows the nonlinearity of the relationship between ρlin and temperature, the solid line shows the theoretical curve of true 2D VG theory [Eq. (5)]. π-band quasiparticles, neither 3D nor quasi-2D VG scal- ing is applicable. Such a mixed state is obviously diffi- cult to be simply described by any known scaling theory. Recently, scanning tunneling microscopy studies showed that the quasiparticles of the π bands disperse over all of the superconductor, both within and outside the vortex cores25, which strongly supports our arguments. This is the basis for the explanation of the nonvanishing vortex dissipation at high magnetic fields in a zero temperature limit found recently on MgB2 thin films. C. Self-field effect at µ0H = 0 T For a 2D layered superconductor in zerofield, the above mentioned BKT transition may exist and be reflected in the I-V characteristics15. In the present MgB2 sam- ples, we have not found any evidence of this transition in low magnetic fields which would be consistent with the quasi-2D (instead of 2D) configuration of the vor- tex matter. Moreover, both the E-j curves and the ρ-j curves (as presented in Fig. 9) are similar to the situation of µ0H = 1 T. Considering the narrow transition width at zero-field, we did the measurement carefully with an increment of 0.05 K. Obviously, there is no E(j) curve which satisfies the E ∝ j3 dependence, as expected by the BKT theory. Since the current can induce self-generated vortices, it might be interesting to look at whether the quasi-2D VG model applies here. Similar in Sec. IV A, we present ρlin versus (T − Tg) in a double logarithmic plot. From this graph, we de- termined the exponent in Eq. 2 (as shown by the inset of Fig. 10(a)). A good quasi-2D scaling was obtained with parameters Tg = 39.94 K, ν = 1.12, and z = 6.61, as presented in Fig. 10(a). Using the parameters deter- ) 0H=0 T 101 102 103 104 105 j (A/cm2) 10 nV FIG. 9: (Color online) (a) E-j data at various temperatures from 39.7 K to 40.5 K with an interval of 0.05 K for µ0H = 0 T, the symbols denote the region, where the data are scaled (from 39.70 K to 40.30 K). Temperature of the isotherms increases from bottom to top. The dashed line shows the position of Tg and the symbols denote the segments, which scale well according to the quasi-2D VG theory. The thin solid lines are also the measured data lying outside the scalable range. (b) ρ-j curves corresponding to the E-j data in (a). The thick solid line in (b) denotes the voltage resolution of 10 nV. mined here, one finds a self-consistency with the value of ν(z + 2 − D), as determined in fitting the linear re- sistivity [Eq. (2)]. Both the temperature dependence of ρlin and the scaling curves at µ0H = 0 T are similar to the situation at small field µ0H = 0.1 T [shown in Fig. 10(b)] and µ0H = 0.5 T (not shown in this pa- per), except for the slight differences of the scaling pa- rameters. The scaling parameters including the ones at µ0H = 0.5 T are listed in Table I. It was proven that current and magnetic field exhibit analogous effects in suppressing superconductivity and generating quasipar- ticles in conventional superconductors.26 Similarly, the current-induced self-field may lead to a similar effect in the vortex state as an applied magnetic field. Nonethe- less, the good agreement of this simple scaling law with the zero-field data is interesting and worth studying in detail. Moreover, the values of ν and z for zero field are very close to those for µ0H = 1 and 3 T, indicating a similar vortex dynamics in the whole low-field region. H=0 T 101 102 103 104 105 j (A/cm2) 10 nV FIG. 10: (Color online) (a) Quasi-2D VG scaling of the data measured at 0 T. The inset indicates a good linearity of the temperature dependence of the linear resistivity. (b) Quasi- 2D VG scaling of the data measured at 0.1 T. The inset indi- cates a good linearity of the temperature dependence of the linear resistivity. TABLE I: Quasi-2D VG scaling parameters at different fields. µ0H (T) Tg(K) ν z 0.0 39.94 1.12 6.61 0.1 39.28 1.30 6.08 0.5 35.95 1.37 6.42 1.0 31.4 1.32 6.12 3.0 15.4 1.17 6.58 V. SUMMARY We have measured I-V curves on high-quality MgB2 films at various magnetic fields and temperatures. At magnetic fields below 5 T including the zero field, the curves scaled well according to the quasi-2D VG theory instead of the 3D model, in good agreement with the multiband superconductivity of MgB2 contributed from the strong 2D σ bands and weak 3D π bands. At the fields above 5 T, the curves did not scale according to any known VG scaling laws, accompanied by the disap- pearance of a zero-resistance state. Based on our result combined with recent tunneling experiments, a different vortex state was suggested, namely, a state where the vortices composed of the superfluid from the σ bands move through the space filled with numerous quasiparti- cles from π bands. VI. ACKNOWLEDGMENTS This work is supported by the National Science Foun- dation of China, the Ministry of Science and Tech- nology of China (973 project: 2006CB601000 and 2006CB921802), and the Knowledge Innovation Project of the Chinese Academy of Sciences (ITSNEM). The work at Penn State is supported by NSF under Grants Nos. DMR-0306746 (X.X.X.), DMR-0405502 (Q.L.), and DMR-0514592 (Z.K.L. and X.X.X.), and by ONR under grant No. N00014-00-1-0294 (X.X.X.). ∗ Electronic address: [email protected] 1 J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Nature (London) 410, 63 (2001). 2 A. Rydh, U. Welp, A. E. Koshelev, W. K. Kwok, G. W. Crabtree, R. Brusetti, L. Lyard, T. Klein, C. Marcenat, B. Kang, K. H. Kim, K. H. P. Kim, H.S. Lee, and S. I. Lee, Phys. Rev. B 70, 132503 (2004); A. E. Koshelev and A. A. Golubov, Phys. Rev. Lett. 92, 107008 (2004); ibid, Phys. Rev. B 68, 104503 (2003). 3 P. de la Mora, M. Castro, and G. Tavizon, J. Phys.: Con- dens. Matter 17, 965 (2005); I. Pallecchi, M. Monni, C. Ferdeghini, V. Ferrando, M. Putti, C. Tarantini, and E. Galleani D’Agliano, Eur. Phys. J. B 52, 171 (2006). 4 Q. Li, B. T. Liu, Y. F. Hu, J. Chen, H. Gao, L. Shan, H. H. Wen, A. V. Pogrebnyakov, J. M. Redwing, and X. X. Xi, Phys. Rev. Lett. 96, 167003 (2006). 5 H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G. Louie, Nature (London) 418, 758 (2002). 6 M. Iavarone, G. Karapetrov, A. E. Koshelev, W. K. Kwok, G. W. Crabtree, D. G. Hinks, W. N. Kang, E. M. Choi, H. J. Kim, H. J. Kim, and S. I. Lee, Phys. Rev. Lett. 89, 187002 (2002). 7 K. H. P. Kim, W. N. Kang, M. S. Kim, C. U. Jung, H. J. Kim, E. M. Choi, M. S. Park, and S. I. Lee, Physica C 370, 13 (2002). 8 S. K. Gupta, S. Sen, A. Singh, D. K. Aswal, J. V. Yakhmi, E. M. Choi, H. J. Kim, K. H. P. Kim, S. Choi, H. S. Lee, W. N. Kang, and S. I. Lee, Phys. Rev. B 66, 104525 (2002). 9 X. H. Zeng, A. V. Pogrebnyakov, A. Kotcharov, J. E. Jones, X. X. Xi, E. M. Lysczek, J.M. Redwing, S. Y. Xu, J. Lettieri, D. G. Schlom, W. Tian, X. Q. Pan, Z. K. Liu , Nature Mater. 1, 35 (2002). 10 H. H. Wen, S. L. Li, Z. W. Zhao, H. Jin, Y. M. Ni, W. N. Kang, H. J. Kim, E. M. Choi, and S. I. Lee, Phys. Rev. B 64, 134505 (2001). 11 M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989); D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B 43, 130 (1991); D. A. Huse, M. P. A. Fisher, and D. S. Fisher, Nature (London) 358, 553 (1992). 12 R. H. Koch, V. Foglietti, G. Koren, A. Gupta, and M. P. A. Fisher, Phys. Rev. Lett. 63, 1511 (1989); R. H. Koch, V. Foglietti, and M. P. A. Fisher, ibid. 64, 2586 (1990). 13 H. Yamasaki, K. Endo, S. Kosaka, M. Umeda, S. Yoshida, and K. Kajimura, Phys. Rev. B 50, 12959 (1994). 14 V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1970); J. M. Kosterlitz, and D. J. Thouless, J. Phys. C 6, 1181 (1973). 15 H. H. Wen, P. Ziemann, H. A. Radovan, and S. L. Yan, Europhys. Lett. 42, 319 (1998). 16 M. P. A. Fisher, T. A. Tokuyasu, and A. P. Young, Phys. Rev. Lett. 66, 2931 (1991); 17 C. Dekker, P. J. M. Wöltgens, R. H. Koch, B. W. Hussey, and A. Gupta, Phys. Rev. Lett. 69, 2717 (1992). 18 H. H. Wen, A. F. Th. Hoekstra, R. Griessen, S. L. Yan, L. Fang, and M. S. Si, Phys. Rev. Lett. 79, 1559 (1997). 19 H. H. Wen, H. A. Radovan, F. M. Kamm, P. Ziemann, S. L. Yan, L. Fang, and M. S. Si, Phys. Rev. Lett. 80, 3859 (1998). 20 Y. Z. Zhang, R. Deltour, J. F. de Marneffe, H. H. Wen, Y. L. Qin, C. Dong, L. Li, and Z. X. Zhao, Phys. Rev. B 62, 11373 (2000). 21 H. Jin, H. H. Wen, H. P. Yang, Z. Y. Liu, Z. A. Ren, G. C. Che, and Z. X. Zhao, App. Phys. Lett. 83, 2626 (2003). 22 D. R. Strachan, M. C. Sullivan, P. Fournier, S. P. Pai, T. Venkatesan and C. J. Lobb, Phys. Rev. Lett. 87, 067007 (2001). 23 Y. Jia, H. Yang, Y. Huang, L. Shan, C. Ren, C. G. Zhuang, Y. Cui, Q. Li, Z. K. Liu, X. X. Xi, and H. H. Wen, arXiv:cond-mat/0703637 (unpubilished). 24 R. S. Gonnelli, D. Daghero, G. A. Ummarino, V. A. Stepanov, J. Jun, S. M. Kazakov, and J. Karpinski, Phys. Rev. Lett. 89, 247004 (2002). 25 M. R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M. Kaza- kov, J. Karpinski, and Ø. Fischer, Phys. Rev. Lett. 89, 187003 (2002). 26 A. Anthore, H. Pothier, and D. Esteve, Phys. Rev. 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0704.0127

Magnetic Fingerprints of sub-100 nm Fe Nanodots

Microsoft Word - DumasLV10695BRrevised.doc Magnetic Fingerprints of sub-100 nm Fe Nanodots Randy K. Dumas,1 Chang-Peng Li,2 Igor V. Roshchin,2 Ivan K. Schuller2 and Kai Liu1,* 1Physics Department, University of California, Davis, California 95616 2Physics Department, University of California - San Diego, La Jolla, California 92093 Abstract Sub-100 nm nanomagnets not only are technologically important, but also exhibit complex magnetization reversal behaviors as their dimensions are comparable to typical magnetic domain wall widths. Here we capture magnetic “fingerprints” of 109 Fe nanodots as they undergo a single domain to vortex state transition, using a first-order reversal curve (FORC) method. As the nanodot size increases from 52 nm to 67 nm, the FORC diagrams reveal striking differences, despite only subtle changes in their major hysteresis loops. The 52 nm nanodots exhibit single domain behavior and the coercivity distribution extracted from the FORC distribution agrees well with a calculation based on the measured nanodot size distribution. The 58 and 67 nm nanodots exhibit vortex states, where the nucleation and annihilation of the vortices are manifested as butterfly-like features in the FORC distribution and confirmed by micromagnetic simulations. Furthermore, the FORC method gives quantitative measures of the magnetic phase fractions, and vortex nucleation and annihilation fields. PACS’s: 75.60.Jk, 75.60.-d, 75.70.Kw, 75.75.+a Phys. Rev. B, in press. I. Introduction Deep sub-100 nm magnetic nanoelements have been the focus of intense research interest due to their fascinating fundamental properties and potential technological applications.1-6 At such small dimensions, comparable to the typical magnetic domain wall width, properties of the nanomagnets are rich and complex. It is known that well above the domain wall width, in micron and sub-micron sized patterns, magnetization reversal often occur via a vortex state (VS).7-12 At reduced sizes, single domain (SD) static states are energetically more favorable.13 However, even in SD nanoparticles, the magnetization reversal can be quite complex, involving thermally activated incoherent processes.14 The VS-SD crossover itself is fascinating. For example, recently Jausovec et al. have proposed that a third, metastable, state exists in 97 nm permalloy nanodots, based on minor loop and remanence curve studies.15 To date, direct observation of the VS-SD crossover, especially in the deep sub-100 nm regime, has been challenging. Fundamentally, the vortex core is expected to have a nanoscale size, comparable to the exchange length. Magnetic imaging techniques face resolution limits and often are limited to remanent state and room temperature studies. Practically, collections of nanomagnets inevitably have variations in size, shape, anisotropy, etc.16 The ensemble- averaged properties obtained by collective measurements such as magnetometry no longer yield clear signatures of the nucleation / annihilation fields. Furthermore, quantitatively capturing the distributions of magnetic properties are essential to the understanding and application of magnetic and spintronic devices, which may consist of billions of nanomagnets. How to qualitatively and quantitatively investigate the properties of such nanomagnets remains a key challenge for condensed matter physics and materials science. In this study, we investigate the VS-SD crossover in deep sub-100 nm Fe nanodots. We have captured “fingerprints” of such nanodots using a first-order reversal curve (FORC) method,17-21 which circumvents the resolution, remanent state and room temperature limits by measuring the collective magnetic responses of the dots. The “fingerprints”, shown as FORC diagrams, reveal remarkably rich information about the nanodots. A qualitatively different reversal pattern is observed as the dot size is increased from 52 to 67 nm, despite only subtle differences in their major hysteresis loops. The 52 nm nanodots behave as SD particles; the 67 nm ones exhibit VS reversal; and the 58 nm ones have both SD and VS characteristics. Quantitatively, the FORC diagram shows explicitly a coercivity distribution for the SD dots, which agrees well with calculations; it yields SD and VS phase fractions in the larger dots; it also extracts unambiguously the nucleation and annihilation fields for the VS dots and distinguishes annihilations from opposite sides of the dots. II. Experimental Samples for the study are Fe nanodots fabricated using a nanoporous alumina shadow mask technique in conjunction with electron beam evaporation.22,23 This method allows for fabrication of high density nanodots (~1010 /cm2) over macroscopic areas (~1 cm2). Three different types of samples have been made on Si and MgO substrates with mean nanodot sizes of 52±8, 58±8, and 67±13 nm, and a thickness of 20 nm, 15 nm, and 20 nm, respectively. The nanodot center-to-center spacing is typically twice its diameter. The Fe nanodots thus made are polycrystalline, capped with an Al or Ag layer. A scanning electron microscopy (SEM) image of the 67 nm sample is shown in Fig. 1. A survey of the size distribution is illustrated in Fig. 1 inset. Magnetic properties have been measured using a Princeton Measurements Corp. 2900 alternating gradient and vibrating sample magnetometer (AGM/VSM), with the applied field in the plane of the nanodots. Samples have been cut down to ~ 3 × 3 mm2 pieces, which contains ~ 109 Fe nanodots each. Additionally, the FORC technique has been employed to study details of the magnetization reversal. After saturation, the magnetization M is measured starting from a reversal field HR back to positive saturation, tracing out a FORC. A family of FORC’s is measured at different HR, with equal field spacing, thus filling the interior of the major hysteresis loop [Figs. 2(a)-2(c)]. The FORC distribution is defined as a mixed second order derivative:17-21 ( ) ( ) −≡ , (1) which eliminates the purely reversible components of the magnetization. Thus any non- zero ρ corresponds to irreversible switching processes.19-21 The FORC distribution is plotted against (H, HR) coordinates on a contour map or a 3-dimensional plot. For example, along each FORC in Fig 4(a) with a specific reversal field HR, the magnetization M is measured with increasing applied field H; the corresponding FORC distribution ρ in Fig. 4(b) is represented by a horizontal line scan at that HR along H. Alternatively ρ can be plotted in coordinates of (HC, HB), where HC is the local coercive field and HB is the local interaction or bias field. This transformation is accomplished by a simple rotation of the coordinate system defined by: HB=(H+HR)/2 and HC=(H-HR)/2. Both coordinate systems are discussed in this paper. III. Results Families of the FORC’s for the 52, 58, and 67 nm nanodots are shown in Figs. 2(a)-2(c). The major hysteresis loops, delineated by the outer boundaries of the FORC’s, exhibit only subtle differences. The 52 nm nanodots show a regular major loop, with a remanence of 57 % and a coercivity of 475 Oe [Fig. 2(a)]. The 67 nm nanodots have a slight “pinching” in its loop near zero applied field, with a remanence of 27 % and a coercivity of 246 Oe [Fig. 2(c)]. The unique shape, small values of coercivity and remanence suggest that the magnetization reversal is via a VS. Indeed, the VS is confirmed by polarized neutron reflectivity measurements on similarly prepared 65 nm Fe nanodots, which find an out of plane magnetic moment corresponding to a vortex core of 15 nm.24 However, due to the relatively gradual changes in magnetization along the major loop, averaged over signals from ~ 109 Fe nanodots, it is difficult to determine the vortex nucleation and annihilation fields. In contrast, the relatively fuller major loop of the 52 nm nanodots is suggestive of a SD state. The loop of the 58 nm nanodots appears to have combined features from those of the other two samples [Fig. 2(b)]. The subtle differences seen in the major hysteresis loops manifest themselves as striking differences in the corresponding FORC distributions, shown in Figs. 2(d)-2(i). For the 52 nm nanodots, the only predominant feature is a narrow ridge along the local coercivity HC-axis with zero bias [Figs. 2(d) and 2(g)]. The ridge is peaked at HC = 525 Oe, near the major loop coercivity value of 475 Oe. This pattern is characteristic of a collection of non-interacting SD particles.25 Given that the nanodot spacing is about twice its diameter and the random in-plane easy axes, dipolar interactions are expected to be small.26 The relative spread of the FORC distribution along the HB-axis actually gives a direct measure of the interdot interactions, as we have shown in single domain magnetite nanoparticles with different separations.27 The sharp ridge shown in Fig. 2(g) is similar to that of an assembly of well–dispersed magnetite nanoparticles with little dipolar interactions. In the present case, the ridge is localized between bias field of HB ~ ±100 Oe and has a narrow FWHM (full width at half maximum) of about 136 Oe [Fig. 3(a)]. A simple calculation of the dipolar fields yields a value similar to the FWHM. As the nanodot size is increased, the FORC distribution becomes much more complex. The 67 nm sample is characterized by three main features, as shown in Figs. 2(f) and 2(i): two pronounced peaks at HC= 650 Oe and HB = ± 750 Oe, and a ridge along HB = 0, forming a butterfly-like contour plot [Fig. 2(i)]. The ridge has changed significantly from that of the 52 nm sample: a peak corresponding to the coercivity of the major loop is now virtually absent; instead a large peak at HC =1500 Oe has appeared, accompanied by two small negative regions nearby. The 58 nm sample shows a FORC pattern representative of both the 52 and 67 nm samples [Figs. 2(e) and 2(h)]. The overall distribution resembles that of the 67 nm sample, with two peaks centered at HC = 650 Oe and HB = ±400 Oe. A ridge along HB = 0, peaking at roughly the major loop coercivity, is similar to that seen in the 52 nm sample. Note that the 58 nm sample, being thinner, would tend to inhibit the formation of a VS in the smaller dots and therefore show magnetic reversal via a SD. However, significant fractions of the nanodots in the ensemble are apparently reversing via a VS. The FORC distribution also allows us to extract quantitative information about the reversal processes. Since each sample measured consists of ~ 1 billion nanodots with a distribution of sizes, a coercivity spread is contained in the FORC distribution. For the 52 nm sample, we have indeed extracted this distribution by projecting the ridge in Fig. 2(g) onto the HC-axis (HB = 0), as shown as the open circles in Fig. 3(b). The relative height gives the appropriate weight of nanodots with a given coercivity. This extracted coercivity distribution can be compared with a simple theoretical calculation. The coercivity of a SD particle undergoing reversal via a curling mode increases strongly with decreasing particle size d, according to HC −∝ , (2) where C1 and C2 are constants.28 Based on the mean nanodot size and the size distribution determined from SEM, we have calculated a coercivity distribution [solid circles in Fig. 3(b)]. A good agreement is obtained with that determined from the FORC distribution, after a rescaling of the latter by an arbitrary weight. Thus for other non- interacting single-domain particle systems with unknown size distributions, the FORC method may be used to extract that information. This is particularly important in 3D distributions of nanostructures where there is no direct image access to the individual dots, as is the case here for a 2D distribution. As we have demonstrated earlier, the FORC distribution ρ is extremely sensitive to irreversible switching.19 This is most convenient to see in the (H, HR) coordinate system (meaningful data is in H>HR), as non-zero values of ρ correspond to the degree of irreversibility along a given FORC. We have employed this capability to analyze the VS nucleation and annihilation for the 67 nm sample. The complex butterfly-like pattern of Fig. 2(i) now transforms into irreversible switching mainly along line scans 1 and 2 in Fig. 4(b), which correspond to FORC’s starting at HR= 100 Oe and -1450 Oe, respectively [marked as bold with large open circles as starting points in Fig. 4(a)]. Along line scan 1 (HR=100 Oe), when applied field H=100 Oe, vortices have already nucleated in most of the nanodots. With increasing field H, ρ becomes non-zero and increases with H and peaks at 1320 Oe. This corresponds to the annihilation of the vortices in majority of the nanodots, and eventually ρ returns to zero near positive saturation. Line scan 2 starts at HR= -1450 Oe, where the majority, but not all, of the nanodots have been negatively saturated. As H is increased, a first maximum in ρ is seen at H= -100 Oe, corresponding to the nucleation of vortices within the nanodots. Between -100 Oe < H < 1450 Oe, ρ is essentially zero, indicating reversible motion of the vortices through the nanodots. A second ρ maximum is found at H = 1450 Oe, as the vortices are annihilated. This is again followed by reversible behavior near positive saturation. Note that along line scan 1, the vortices are annihilated from the same side of the nanodot from which they first nucleated, and thus the net magnetization remains positive; along line scan 2, the vortices nucleate on one side of the dot and are annihilated from the other, and consequently the net magnetization changes sign [Fig. 4(a)]. Interestingly the annihilation field along line scan 2, 1450 Oe, is larger than that along line scan 1, 1320 Oe. It seems more difficult to drive a vortex across the nanodot and then annihilate it. The peaks in Fig. 4(b) are rather broad, which is a manifestation of vortex nucleation and annihilation field distributions. Also note that the interactions among the VS dots are expected to be negligible due to the high degree of flux closure, as confirmed by simulations.29 IV. Simulations and Discussions For comparison, micromagnetic simulations have been carried out on nearly circular nanodots with 60 nm diameters and 20 nm thicknesses.30 We have used parameters appropriate for Fe (exchange stiffness A = 2.1 × 10-11 J/m, saturation magnetization Ms = 1.7 × 106 A/m, and anisotropy constant K = 4.8 × 104 J/m3). Each polycrystalline nanodot is composed of 2 nm square cells that are 20 nm thick, where each cell is a different grain with a random easy axis. A small cut on one side of the nanodot generates two distinct annihilation fields that depend on which side of the nanodot the vortex annihilates from. This exercise models the fact that our fabricated dots are not perfectly circular.31 We have simulated FORC’s generated by two nanodots with different orientations [Fig. 4(c)]: the edge-cut in one is parallel, and in the other at a 45º angle, to the applied field. The simulated M-H curves show abrupt magnetization changes, corresponding to the nucleation, propagation, and annihilation of vortices. The corresponding FORC distribution is shown in Fig. 4(d). Peaks in the simulated FORC distribution clearly indicate the nucleation and annihilation fields of the vortices which are apparent in Fig 4(c). Along line scan 1 of Fig. 4(d), a vortex is already nucleated at HR = 100 Oe and subsequently annihilated at H = 2300 Oe (upper right corner). Along line scan 2 with HR = -2450 Oe, a vortex is nucleated at H = -100 Oe and finally annihilated at H = 2450 Oe (lower right corner). It is clear that the simulated FORC reproduces the key features of the experimentally obtained one in Fig. 4(b). Here the asymmetric dot shape is essential to obtain a different annihilation field along scan 2 than that along scan 1. We have simulated the angular dependence of nucleation and annihilation fields in such circular dot with a small cut as the cut orientation is varied in a field. We find that for most angles it is harder to annihilate a vortex from the opposite side of its nucleation site. However, for a small range of angles near 45° it is actually slightly easier to annihilate from the opposite side. It is the combination of these two behaviors that gives rise to the negative-positive-negative trio of features in the lower right portion of the FORC distribution. The presence of similar features in the experimental data shown in Fig. 4(b) thus illustrates that the FORC distribution is also sensitive to variations of dot shapes in the array. Because only two dots are simulated, the features in the FORC distribution are much sharper than the experimental data where distributions of vortex nucleation and annihilation fields are present. Including more dots in the simulation with different applied field orientations and size distributions would tend to broaden the features generated by the two dots simulated. Additionally, by selectively integrating the normalized FORC distribution20,21 corresponding to the SD phase (the aforementioned ridge centered at low coercivity values in Fig. 2), we can quantitatively determine the percentage of nanodots in SD state for each sample. The SD phase fraction is 100%, 43%, and 10% for the 52, 58, and 67 nm sample, respectively. Thus the 58 nm nanodots have a significant co-existence of both SD and VS states. However, we do not observe clear evidence of any additional metastable phase.15 V. Conclusions In summary, we have used the FORC method to “fingerprint” the rich magnetization reversal behavior in arrays of 52, 58, and 67 nm sized Fe nanodots. Distinctly different reversal mechanisms have been captured, despite only subtle differences in the major hysteresis loops. The 52 nm nanodots are in SD states. A coercivity distribution has been extracted, which agrees with calculations. The 67 nm dots reverse their magnetization via the nucleation and annihilation of vortices. Different fields are required to annihilate vortices from opposite sides of the dots. Quantitative measures of the vortex nucleation and annihilation fields have been obtained. OOMMF simulations confirm the experimental FORC distributions. The 58 nm sample shows coexistence of SD and VS reversal, without evidence of additional reversal mode. These results further demonstrate the FORC method as a simple yet powerful technique for studying magnetization reversal, due to its capability of capturing distributions of magnetic properties, sensitivity to irreversible switching, and the quantitative phase information it can extract. Acknowledgements This work has been supported by ACS (PRF-43637-AC10), AFOSR, and the Alfred P. Sloan Foundation. We thank J. E. Davies, J. Olamit, M. Winklhofer, C. R. Pike, H. G. Katzgraber, R. T. Scalettar, G. T. Zimányi, and K. L. Verosub for helpful discussions. R.K.D. acknowledges support from the Katherine Fadley Pusateri Memorial Travel Award. References * Corresponding author, email address: [email protected]. 1 C. Chappert, H. Bernas, J. Ferre, V. Kottler, J.-P. Jamet, Y. Chen, E. Cambril, T. Devolder, F. Rousseaux, V. Mathet, and H. Launois, Science 280, 1919 (1998). 2 B. Terris, L. Folks, D. Weller, J. Baglin, A. Kellock, H. Rothuizen, and P. Vettiger, Appl. Phys. Lett. 75, 403 (1999). 3 S. H. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, Science 287, 1989 (2000). 4 C. Ross, Annu. Rev. Mater. Res. 31, 203 (2001). 5 J. I. Martin, J. Nogues, K. Liu, J. L. Vicent, and I. K. Schuller, J. Magn. Magn. Mater. 256, 449 (2003). 6 F. Q. Zhu, G. W. Chern, O. Tchernyshyov, X. C. Zhu, J. G. Zhu, and C. L. Chien, Phys. Rev. Lett. 96, 027205 (2006). 7 T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). 8 R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042 (1999). 9 A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, and R. Wiesendanger, Science 298, 577 (2002). 10 K. Y. Guslienko, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, Phys. Rev. B 65, 024414 (2002). 11 H. F. Ding, A. K. Schmid, D. Li, K. Y. Guslienko, and S. D. Bader, Phys. Rev. Lett. 94, 157202 (2005). 12 J. Sort, K. S. Buchanan, V. Novosad, A. Hoffmann, G. Salazar-Alvarez, A. Bollero, M. D. Baró, B. Dieny, and J. Nogués, Phys. Rev. Lett. 97, 067201 (2006). 13 We use "single domain state" to refer to all magnetization configurations with no domain wall and a non-zero net magnetization. 14 Y. Li, P. Xiong, S. von Molnár, Y. Ohno, and H. Ohno, Phys. Rev. B 71, 214425 (2005). 15 A.-V. Jausovec, G. Xiong, and R. P. Cowburn, Appl. Phys. Lett. 88, 052501 (2006). 16 Even in nanomagnets with identical size and shape, there may exist a distribution of intrinsic anisotropy. See e.g., T. Thomson, G. Hu and B. D. Terris, Phys. Rev. Lett. 96, 257204 (2006). 17 C.R. Pike and A. Fernandez, J. Appl. Phys. 85, 6668 (1999). 18 H. G. Katzgraber, F. Pazmandi, C. R. Pike, K. Liu, R. T. Scalettar, K. L. Verosub, and G. T. Zimanyi, Phys. Rev. Lett. 89, 257202 (2002). 19 J. E. Davies, O. Hellwig, E. E. Fullerton, G. Denbeaux, J. B. Kortright, and K. Liu, Phys. Rev. B 70, 224434 (2004). 20 J. E. Davies, J. Wu, C. Leighton, and K. Liu, Phys. Rev. B 72, 134419 (2005). 21 J. Olamit, K. Liu, Z. P. Li, and I. K. Schuller, Appl. Phys. Lett. 90, 032510 (2007). 22 K. Liu, J. Nogues, C. Leighton, H. Masuda, K. Nishio, I. V. Roshchin, and I. K. Schuller, Appl. Phys. Lett. 81, 4434 (2002). 23 C. P. Li, I. V. Roshchin, X. Batlle, M. Viret, F. Ott, I. K. Schuller, J. Appl. Phys. 100, 074318 (2006). 24 I. V. Roshchin, C. P. Li, X. Battle, J. Mejia-Lopez, D. Altbir, A. H. Romero, S. Roy, S. K. Sinha, M. Fitzimmons, F. Ott, M. Viret, and I. K. Schuller, unpublished. 25 S. J. Cho, A. M. Shahin, G. J. Long, J. E. Davies, K. Liu, F. Grandjean, and S. M. Kauzlarich, Chem. Mater. 18, 960 (2006). 26 M. Grimsditch, Y. Jaccard, and I. K. Schuller, Phys. Rev. B 58, 11539 (1998). 27 J. E. Davies, J. Y. Kim, F. E. Osterloh, and K. Liu, unpublished. 28 A. Aharoni, Introduction to the Theory of Ferromagnetism, 2nd Edition (Oxford University Press, Oxford, 2000). This is an approximation for reversal via curling as the dot sizes are larger than the coherence radius. 29 J. Mejía-López, D. Altbir, A. H. Romero, X. Batlle, I. V. Roshchin, C. P. Li, and I. K. Schuller, J. Appl. Phys. 100, 104319 (2006). 30 OOMMF code. http://math.nist.gov/oommf. 31 Simulations done on perfectly circular and slightly elliptical dots show only a single annihilation field. To reproduce the two distinct annihilation fields the symmetry of the dot must be broken by some type of shape defect. Figure Captions Fig. 1. (color online). Scanning electron micrograph of the 67 nm diameter nanodot sample. Inset is a histogram showing the distribution of nanodot sizes. Fig. 2. First-order reversal curves and the corresponding distributions. Families of FORC’s (a-c), whose starting points are represented by black dots for the 52, 58, and 67 nm Fe nanodots, respectively. The corresponding FORC distributions are shown in 3-dimensional plots (d-f) and contour plots (g-i). Fig. 3. (color online). Projection of the FORC distribution ρ of the 52 nm nanodots onto (a) the HB-axis, showing weak dipolar interactions; and (b) the HC-axis (open circles), showing a coercivity distribution that agrees with a calculation based on measured size distribution (solid circles). Fig. 4. (a) A family of measured FORC’s for the 67nm diameter dots. (b) The corresponding experimental FORC distribution plotted against applied field H and reversal field HR. (c) A family of simulated FORC’s generated using the OOMMF code. Inset shows the orientations of the two dots simulated. (d) The FORC distribution calculated from the simulated FORC’s shown in (c). The two white dashed lines in (b) and (d) correspond to the two bold FORC’s whose starting points are large open circles in (a) and (c), respectively. Fig. 1, Dumas, et al. Fig. 2, Dumas, et al. Fig. 3, Dumas, et al. Fig. 4, Dumas, et al.

0704.0128

An online repository of Swift/XRT light curves of GRBs

Astronomy & Astrophysics manuscript no. 7530evans c© ESO 2018 October 26, 2018 An online repository of Swift /XRT light curves of GRBs. P.A. Evans1⋆, A.P. Beardmore1, K.L. Page1, L.G. Tyler1, J.P. Osborne1, M.R. Goad1, P.T. O’Brien1, L. Vetere2, J. Racusin2, D. Morris2, D.N. Burrows2, M. Capalbi3, M. Perri3, N. Gehrels4, and P. Romano5,6 1 Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK 2 Department of Astronomy and Astrophysics, 525 Davey Lab., Pennsylvania State University, University Park, PA 16802, USA 3 ASI Science Data Center, ASDC c/o ESRIN, via G. Galilei, 00044 Frascati, Italy 4 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA 5 INAF-Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate (LC), Italy 6 Università degli Studi di Milano, Bicocca, Piazza delle Scienze 3, I-20126, Milano, Italy Received / Accepted ABSTRACT Context. Swift data are revolutionising our understanding of Gamma Ray Bursts. Since bursts fade rapidly, it is desirable to create and disseminate accurate light curves rapidly. Aims. To provide the community with an online repository of X-ray light curves obtained with Swift. The light curves should be of the quality expected of published data, but automatically created and updated so as to be self-consistent and rapidly available. Methods. We have produced a suite of programs which automatically generates Swift/XRT light curves of GRBs. Effects of the damage to the CCD, automatic readout-mode switching and pile-up are appropriately handled, and the data are binned with variable bin durations, as necessary for a fading source. Results. The light curve repository website⋆⋆ contains light curves, hardness ratios and deep images for every GRB which Swift’s XRT has observed. When new GRBs are detected, light curves are created and updated within minutes of the data arriving at the UK Swift Science Data Centre. Key words. Gamma rays: bursts - Gamma rays: observations - Methods: data analysis - Catalogs 1. Introduction The data from the Swift satellite (Gehrels et al. 2004), and partic- ularly its X-ray Telescope (XRT, Burrows et al. 2005), are rev- olutionising our understanding of Gamma Ray Bursts (GRBs, see Zhang 2007 for a recent review). The XRT typically begins observing a GRB ∼ 100 s after the trigger, and usually follows it for several days, and occasionally for months (e.g., Grupe et al. 2007). However, creating light curves of the XRT data is a non-trivial process with many pitfalls. The UK Swift Science Data Centre is automatically generating light curves of GRBs – an example light curve is given in Fig. 1 – and making them im- mediately available online. In this paper we detail how the light curves are created, and particularly, how the complications spe- cific to these data are treated. 1.1. Aspects of light curve generation In general, creation of X-ray light curves is a relatively simple, quick task using ftools such as the xselect and lcmath pack- ages. Building Swift/XRT light curves of GRBs, however, has a number of complications which can make the task difficult and slower, as described below. ⋆ [email protected] ⋆⋆ http://www.swift.ac.uk/xrt curves 100 1000 104 105 106 Time since BAT trigger (s) Fig. 1. Swift X-ray light curve of GRB 051117a (Goad et al. 2007), created using the software described in this paper and obtained from the Swift Light Curve Repository. 1.1.1. GRBs fade The standard light curve tools, such as those mentioned above, produce light curves with uniform bin durations. Since GRBs fade by many orders of magnitude, long-duration bins are needed at late times in order to detect the source. However, GRBs show rapid variability and evolution at early times, and http://arxiv.org/abs/0704.0128v2 2 P.A. Evans et al.: An online repository of Swift/XRT light curves of GRBs. 1062×105 5×105 2×106 5×106 Time since BAT trigger (s) Static region size 1062×105 5×105 2×106 5×106 Time since BAT trigger (s) Dynamic region size Fig. 2. Late-time Swift X-ray light curves of GRB 060614 (Mangano et al. 2007), showing the need for the source region to be reduced as the data fades. Top panel: Where the source extraction region remains large at late times, the source cannot be detected after 600 ks. Bottom panel: Using a smaller source extraction region at later times suppresses the background, yielding 6 more datapoints on the light curve. short time bins are needed to resolve these features. A better ap- proach to producing GRB light curves is to bin data based on the number of counts in a bin, rather than the bin duration. This is common practice for X-ray spectroscopy, however there are no ftools available to do this for light curves. While this is our chosen means of binning GRB light curves, it is not the only option. For example, one could use the Bayesian blocks method (Scargle 1998) to determine the bin size. Another complication caused by the fading nature of GRBs is that when the burst is bright, it is best to extract data for a relatively large radius around the GRB position, to maximise the number of counts measured. When the GRB has faded, using such a large region means that the measured counts would be dominated by background counts, making it harder to detect the source, thus it is necessary to reduce the source region size as the GRB fades. This is illustrated in Fig. 2. 1.1.2. Swift data contain multiple observations and snapshots The Swift observing schedule is planned on a daily basis, and each day’s observation of a given target has its own observa- tion identification (ObsID) and event list. Thus if Swift follows a GRB for two weeks, it will produce up to fourteen event lists, all of which need to be used in light curve creation. At late times it may become necessary to combine several datasets just to detect the GRB. Also, Swift’s low-Earth orbit means that it is unable to ob- serve most targets continuously. Thus, any given ObsID may contain multiple visits to the target (‘snapshots’) which again will need to be combined (this differentiation between observa- tions – datasets with a unique ObsID – and snapshots – different on-target times within an ObsID – will be used throughout this paper). Combining snapshots/observations can result in bins on a light curve where the fractional exposure is less than 1. This must be taken into account in calculating the count rate. The standard pipeline processing of Swift data1 ensures that the sky coordinates are correctly attained for each event, how- ever the position of the GRB on the physical detector can be different each snapshot due to changes in the spacecraft attitude. This becomes a problem when one considers the effects of bad pixels and columns. 1.1.3. CCD Damage On 2005-May-27 the XRT was struck by a micrometeoroid (Abbey et al. 2005). Several of the detector columns became flooded with charge (‘hot’), and have had to be permanently screened out. Unfortunately, these lie near the centre of the CCD, so the point spread function (PSF) of a GRB often extends over these bad columns. As well as these columns there are individual ‘hot pixels’ which are screened out, and other pixels which be- come hot when the CCD temperature rises, so may be screened out in one event list, but not in the next. Exposure maps and the xrtmkarf tool can be used to correct for this, however this has to be done individually for each Swift snapshot (since the source will not be at the same detector position from one snap- shot to the next). A single day’s observation contains up to 15 snapshots, thus to do this manually is a slow, laborious task. The forthcoming xrtlccorr program should make this process eas- ier, however it will still need to be executed for each observation. 1.1.4. Automatic readout-mode switching One of the XRT’s innovative features is that it changes readout mode automatically depending on the source intensity (Hill et al. 2004). At high count-rates it operates in Windowed Timing (WT) mode, where some spatial information is sacrificed to gain time resolution (∆t = 1.8 × 10−3 s). At lower count-rates Photon Counting (PC) mode is used, yielding full spatial infor- mation, but lower time resolution (∆t = 2.5 s). The XRT also has Photodiode (PD) mode, which contains no spatial information, but has very high time resolution (∆t = 1.4× 10−4 s). This mode was designed to operate for higher count-rates than WT mode, however it was disabled following the micrometeoroid impact. Prior to this, the XRT produced very few PD mode frames be- fore switching to WT so we have limited our software to WT and PC modes. 1 http://swift.gsfc.nasa.gov/docs/swift/analysis/xrt swguide v1 2.pdf P.A. Evans et al.: An online repository of Swift/XRT light curves of GRBs. 3 For a simple, decaying GRB the earliest data are in WT mode and as the burst fades the XRT switches to PC mode. This is not always the case; the XRT can toggle between modes. GRB 060929 for example, had a count-rate of ∼ 0.1 counts s−1 and the XRT was in PC mode, when a giant flare pushed the count-rate up to ∼ 100 counts s−1 and the XRT switched into WT mode, causing a ∼ 200 s gap in the PC exposure (Fig. 3, up- per panel). Since the initial CCD frames are taken in WT mode, and the PC data both preceded and succeeded the WT data taken during the flare, there are large overlaps between the WT and PC data. On occasions, such as when Swift was observing GRB 050315 (Vaughan et al. 2006), the XRT oscillates rapidly be- tween WT and PC modes (Fig. 3, lower panel). This ‘mode switching’ occurs when the count-rate in the central window of the CCD changes rapidly. Such variation is usually due to the rapid appearance and disappearance of hot pixels at high (∼ −52◦C) CCD temperatures (the XRT is only passively cooled due to the failure of the on-board thermoelectric cooler, Kennea et al. 2005), although contamination by photons from the illumi- nated face of the Earth can also induce mode switching. Recent changes in the on-board calibration have significantly reduced the effects of hot-pixel induced mode switching, however when it does happen it complicates light curve production by causing a variable fractional exposure. Also, during mode switching the XRT does not stay in either mode long enough to collect suffi- cient data to produce a light curve bin (see Section 2.2), thus the WT and PC bins can overlap. The lower panel of Fig. 3 illus- trates these points. 1.1.5. Pile-up Pile-up occurs when two photons are incident upon on the same or adjacent CCD pixels in the same CCD frame. Thus, when the detector is read out the two photons are recorded as one event. Pile-up in the Swift XRT has been discussed by Romano et al. (2006) for WT mode and Vaughan et al. (2006) and Pagani et al. (2006) for PC mode. Their quantitative analyses show the effects of pile-up at different count rates, and we used these val- ues to determine when we consider pile-up to be a problem (see Section 2.2). This problem is not unique to Swift, but because GRBs vary by many orders of magnitude, pile-up must be identified and cor- rected in a time-resolved manner. The standard way to correct for pile-up is to use an annular source extraction region, discard- ing the data near the centre of the PSF where pile-up occurs. For constant sources, or those which vary about some roughly constant mean, it is usually safe to use this annular region at all times. This is not true for GRBs, which can span five decades in brightness; using an annulus when the burst is faint would make it almost undetectable! In the following sections we detail the algorithm used to gen- erate light curves automatically, and in particular we concentrate on how the above issues are resolved. Time since BAT trigger (s) 100 1000200 500 2000 Time since BAT trigger (s) Fig. 3. Swift X-ray light curves of two GRBs, showing the switching between readout modes. WT mode is blue, PC mode Top panel: GRB 060929. The XRT changed from PC to WT mode due to a large flare. Bottom panel: GRB 050315. The XRT was ‘mode-switching’ during the second snapshot. The lower pane shows the fractional exposure, which is highly variable due to this effect. 2. Light curve creation procedure The raw Swift/XRT data are processed at the Swift Data Center at NASA’s Goddard Space Flight Center, using the standard Swift software developed at the ASI Science Data Center (ASDC) in Italy. The processed data are then sent to the Swift quick-look archives at Goddard, the ASDC, and the UK. As soon as data for a new GRB arrive at the UK site, the light curve generation software is triggered, and light curves made available within minutes. The light curve creation procedure can be broken down into three phases. The preparation phase gathers together all of the observations of the GRB, creating summed source and back- ground event lists. The production phase converts these data into time-binned ASCII files, applying corrections for the above- mentioned problems in the process. The presentation phase then produces light curves from the ASCII files, and transfers them to the online light curve repository. 4 P.A. Evans et al.: An online repository of Swift/XRT light curves of GRBs. 2.1. Phase #1 – Preparation Phase In overview: this phase collates all of the observations, de- fines appropriate source and background regions (accounting for pileup where necessary), and ultimately produces a source event list and background event list for WT and PC mode, which are then passed to the production phase. The preparation phase begins by creating a list of ObsIDs for the GRB, and then searching the file metadata to ascertain the position of the burst, the trigger time, and the name. An im- age is then created from the first PC-mode event list and the ftool xrtcentroid is used to obtain a more accurate position. A circular source region is then defined, centred on this position, and initially 30 pixels (71”) in radius. A background region is also defined, as an annulus centred on the burst with an inner ra- dius of 60 pixels (142”) and an outer radius of 110 pixels (260”). The image is also searched for serendipitous sources close to the GRB (e.g. there is a flare star 40” from GRB 051117A; Goad et al. 2007), and if any are found to encroach on the source region, the source extraction radius is reduced to prevent contamination. The software then takes each event list in turn. The bad pixel information is obtained from the ‘BADPIX’ FITS extension and stored for use in the production phase. An image of the back- ground region is created, and the detect routine in ximage is used to identify any sources with a count-rate ≥ 3σ above the background level. For each source thus found, a circular region centred on the source with a radius equal to the source extent returned by ximage, is excluded from the background region. The event list is then broken up into individual snapshots, the mean count-rate during each snapshot is ascertained and used to determine an appropriate source region size (Table 1) and ap- pended to the event list for later use. The values in Table 1 were determined by manual analysis of many GRB observations, and reflect a compromise between minimising the background level while maximising the proportion of source counts that are de- tected. For each snapshot, the detector coordinates of the object are found using the pointxform ftool, and used to confirm that both source and background regions lie within the CCD. If the back- ground region falls off the edge of the detector it is simply shifted by an appropriate amount (ensuring that the inner ring of the an- nulus remains centred on the source). The source region must remain centred on the source in order for later count-rate correc- tions to be valid, however if this results in part of the extraction region lying outside of the exposed CCD area, the source region for this snapshot is reduced. The first part of pile-up correction is carried out at this stage. A simple, uniform time-bin light curve is created with bins of 1 s (5 s) for WT (PC) mode, and then parsed to identify times where the count rate climbs above 150 (0.6) counts s−1; such times are considered to be at risk of pile-up. For WT mode we are un- able to investigate further, since we have only one-dimensional spatial information arranged at an arbitrary (albeit known) an- gle in a two dimensional plane, and no tools currently exist to extract a PSF from such data. Instead, the centre of the source region is excluded, such that the count rate in the remaining pix- els never rises above 150 counts s−1. The number of excluded pixels is typically in the range ∼ 6–20, depending on the source brightness. For PC mode, a PSF profile is obtained for the times of interest, and the wings of this (from 25” outwards) are fitted with a King function which accurately reproduces Swift’s PSF (Moretti et al. 2005). This fit is then extrapolated back to the PSF core, and if the model exceeds the data by more than the 1-σ error on the data, the source is classified as piled up (Fig. 4). 10 100 Radius (arc sec) Fig. 4. The PSF of GRB 061121 (Page et al. 2007), during the first snapshot of PC data. The model PSF was fitted to the data more than 25” from the burst. The central 10” are clearly piled Table 1. Source extraction radii used for given count rates. R is the measured, uncorrected count rate. Count rate R (counts s−1) Source radius in pixels (arc sec) R > 0.5 30 (70.8”) 0.1 < R ≤ 0.5 25 (59.0”) 0.05 < R ≤ 0.1 20 (47.2”) 0.01 < R ≤ 0.05 15 (35.4”) 0.005 < R ≤ 0.01 12 (28.3”) 0.001 < R ≤ 0.005 9 (21.2”) 0.0005 < R ≤ 0.001 7 (16.5”) R ≤ 0.0005 5 (11.8”) The source region is then replaced with an annular region whose inner radius is that at which the model PSF and the data agree to within 1-σ of the data. Note that these annular regions are only used during the intervals for which pile-up was detected, the rest of the time a circular region is used (or a box-shaped region for WT mode). If there are several separate intervals of pile-up (e.g., pile-up lasts for several snapshots, or a flare causes the count-rate to rise into the pile-up régime), they each have their own annular region. The inner radii of the annuli (or size of the excluded region in WT mode) are stored in the event list, so that in the production phase the count-rate can be corrected for events lost by the exclusion of the central part of the PSF. The time-dependent region files thus created are used to generate source and background event lists for this snapshot. This process is performed for every snapshot in every observa- tion of the GRB, and the event lists are then combined to yield one source and one background event list for each XRT mode. Additionally, all PC-mode event lists are merged for use in the presentation phase (phase #3) 2.2. Phase #2 – Production Phase In this phase the data are first filtered so that only events with energy in the range 0.3–10 keV are included. For WT (PC) mode, only events with grades 0–2 (0–12) are accepted. Each mode is then processed separately: WT and PC mode data are not merged. The process described in this section occurs three times in parallel: once on the entire dataset, once binning only soft photons (with energies in the range 0.3–1.5 keV), and once binning only the hard photons (1.5–10 keV). The data are then binned and background subtracted. Since the source region is P.A. Evans et al.: An online repository of Swift/XRT light curves of GRBs. 5 dynamic and could change within a bin, each background pho- ton is individually scaled to the source area (the source radius used was saved in each event list during the preparation phase). A bin (i.e. a point on the light curve) is defined as the small- est possible collection of events which satisfies the following criteria: – There must be at least C counts from the source event list. – The bin must span at least 0.5 (2.51) s in WT (PC) mode. – The source must be detected at a significance of at least 3σ. – There must be no more events within the source region in this CCD frame For the energy-resolved data, both the soft and hard data must meet these criteria individually to complete a bin. C, the minimum number of counts in the source region, is a dynamic parameter. Its default value of 30 for WT mode and 20 for PC mode is valid when the source count-rate is one count per second. It scales with count rate, such that an order of magnitude change in count rate produces a factor of 1.5 change in C. This is done discretely, i.e. where 1 ≤ rate < 10, C=30 counts (WT mode), for 10 ≤ rate < 100, C = 45 counts etc. C must always be above 15 counts, so that Gaussian statistics remain valid. Note that C always refers to the number of measured counts, with no corrections applied, however ‘rate’ refers to the corrected count rate (see below). These values of C give poor signal-to-noise levels in the hardness ratio, so for the energy-resolved data we require 2C counts in each band in order to create a bin. The second criterion (the bin duration) is in place to enable reasonable sampling of the background. For the third criterion we define the detection significance as σ = N/ B, where N is the number of net counts from the source and B is the number of background counts scaled to the source area. Thus we require that a datapoint have a < 0.3% probability of being a background fluctuation before we regard it as ‘real’. The final criterion is used because the CCD is read out at dis- crete times, thus all events that occur between successive read- outs (i.e. within the same frame) have the same time stamp. Thus, if the final event in one bin and the first event in the next were from the same frame, those bins would overlap. Apart from being cosmetically unpleasant, this will also make modelling the light curves much harder, and is thus avoided. At the end of a Swift snapshot, there may be events left over which do not yet comprise a full bin. These will be appended to the last full bin from this snapshot, if there is one, other- wise they are carried over to the next snapshot. At the end of the event list, if there are still spare events, this bin is replaced with an upper limit on the count rate. This is calculated at the 3σ (i.e. 99.7%) confidence level, using the Bayesian method cham- pioned by Kraft, Burrows and Nousek (1991). As the data are binned and background subtracted, the count- rates are corrected for losses due to pile-up, dead zones on the CCD (i.e. bad pixels and bad columns) and source pho- tons which fell outside the source extraction region. This cor- rection, which is applied on an event-by-event basis, is achieved by numerically simulating the PSF for the relevant XRT mode over a radius of 150 pixels, and summing it. It is then summed again, however this time, the value of any pixel in the simu- lated PSF which corresponds to a bad pixel in the data is set to zero before the summation (the lists of bad pixels and the times for which they were bad were saved in the preparation phase). Furthermore, only the parts of the PSF which were within the data extraction region are included. Taking the ratio of the com- plete PSF to the partial PSF gives the correction factor. This method is analogous to using exposure maps and the xrtmkarf task, as is done when manually creating light curves. Alternative methods of using xrtmkarf give correction factors which differ by up to 5%; we compared our correction factors with these, and found them to lie in the middle of this distribution. In addition to these corrections, we need to ensure that the exposure time is calculated correctly: mode switching, or bins spanning multiple snapshots, will result in a bin duration which is much longer than the exposure time. This is done by using the Good Time Interval (GTI) information from the event lists: if a bin spans multiple GTIs the dead-times between GTIs are summed, and the result is subtracted from the bin duration to give the exposure time, which is used to calculate the count rate. The fractional exposure is defined as the exposure time divided by the bin duration. Finally, the data are written to ASCII files. The following information is saved for each bin: – Time in seconds (with errors). The bin time is defined as the mean photon arrival time, and the (consequentially asym- metric) errors span the entire time interval covered by the bin. Time zero is defined as the BAT trigger time. For non- Swift bursts, the trigger time given in the GCN circular which announced the GRB is used as time zero. – Source count rate (and error) in counts s−1. This is the final count rate, background subtracted and fully corrected, with a ±1-σ error. – Fractional exposure. – Background count rate (and error) in counts s−1. This is the background count rate scaled to the source region, with a ±1-σ error. – Correction factor applied to correct for to pile-up, dead zones on the CCD, and source photons falling outside of the source extraction region. – Measured counts in the source region. – Measured background counts, scaled to the source region. – Exposure time – Detection significance (σ), before corrections were applied. If an upper limit is produced, the measured counts and de- tection significance columns refer to the data which have been replaced with an upper limit. The significance of the upper limit is always 3σ. σ is always calculated before the corrections are applied, since it is a measure of how likely it is that the measured counts, not corrected counts, were caused by a fluctuation in the back- ground level. 2.2.1. Counts to flux conversion The conversion from count rates (as in our light curves) to flux requires spectral information. Since automatic spectral fitting is prone to errors (e.g due to local minima of the fit statistic), we refrain from doing this. Furthermore, accurate flux conversion needs to take into account spectral variation as the flux evolves, which is beyond the scope of this work. The GCN reports issued by the Swift team contain a mean conversion factor for a given burst. These tend to be around 5×10−11 erg cm−2 count−1(0.3–10 keV), suggesting such a value could be used as an approximate conversion. For 10 Swift bursts between GRB 070110 and GRB 070306, the mean flux conver- sion is 5.04 × 10−11 erg cm−2 count−1, with a standard deviation of 2.61 × 10−11 erg cm−2 count−1. 6 P.A. Evans et al.: An online repository of Swift/XRT light curves of GRBs. 2.3. Phase #3 – Presentation Phase The final phase parses the output of the production phase to pro- duce light curves. Three such curves are produced, and Fig. 5 shows an example of each; count-rates and the time since trig- ger are plotted logarithmically. The first is a basic light curve, simply showing count-rate against time. The second also shows the background level and fractional exposure. In WT mode when the GRB is bright, the background tends to be dominated by the < 1% of the PSF which leaks into the background region, but because of the high source count rate, this has negligible effects on the corrected count rates. The PC mode background should generally be approximately constant. If it shows large variations, the data may be contaminated by enhanced background linked to the sunlit Earth. Unfortunately, such contamination is currently unpredictable and varies both spatially and temporally; it is thus very difficult to correct for manually, and our automated pro- cessing does not currently correct for this. PC mode data points which occur during times of variable background should thus be treated with caution. We note, however, that our testing proce- dure (Section 3) does not show our light curves to be degraded when bright Earth characteristics become apparent. The third light curve produced in this phase is energy- resolved. The hard- and soft-band light curves are shown sep- arately, and the hard/soft ratio makes up the bottom panel of this plot. Also created in this phase is a deep PC mode image, using the summed PC event list created in the preparation phase. This image is split into three energy bands: 0.3–1.2 keV, 1.2–1.8 keV and 1.8–10 keV. These bands were chosen based on the spectra of the GRBs seen by Swift to date, to ensure that for a ‘typical’ burst, there will be approximately equal numbers of counts in each band. These three energy-resolved images are plotted on a logarithmic scale, and combined (using ImageMagick to pro- duce a 3-colour image (with red, green and blue being the soft, medium and hard bands respectively). This is then smoothed us- ing ImageMagick. Once created, these products are transferred to the online repository. 2.4. Immediate light curve regeneration Our light curve generation is a dynamic process: a light curve is created when the first XRT data arrive in the UKSSDC archive – typically 1.5–2 hours after the burst – and it is then up- dated whenever new data have been received and undergone the pipeline processing. Thus, a light curve should never be more than ∼ 15 minutes older than the quick-look data. If the GRB is being observed every orbit, new data can be received as often as every 96 minutes. The update procedure is identical to that described in Sections 2.1–2.3 above, except that only the new data are pro- cessed and the results appended to the existing light curve. In the case where the existing light curves ends with an upper limit, the data from this upper limit are reprocessed with the new data, hopefully enabling that limit to be replaced with a detection. 3. Testing procedure In order to confirm that our light curves are correct, we used manually created light curves for every GRB detected by the Swift XRT up to GRB 070306, which had been produced by one of us (K.L. Page). We broke each light curve up into phases of constant (power-law) decay, and compared the count-rate and 100 1000 104 105 106 107 Time since BAT trigger (s) 100 1000 104 105 106 107 Time since BAT trigger (s) 100 1000 104 105 106 107 Time since BAT trigger (s) Fig. 5. Light curve images for GRB 060729. These data have been discussed by Grupe et al. (2007). Top panel: Basic light curve. Centre panel: Detailed light curve, with background levels shown below the light curve, and fractional exposure given in the lower pane. Bottom panel: Hardness ratio. The 3 panes are (top to bot- tom) Hard: (1.5–10 keV) , Soft (0.3–1.5 keV) and the ratio (Hard/Soft). P.A. Evans et al.: An online repository of Swift/XRT light curves of GRBs. 7 Fig. 6. An example three colour image. This is GRB 060729, and the exposure time is 1.2 Ms. time at the start and end of each of these phases. We also con- firmed that the shape of the decay was the same in both auto- matic and manually created light curves. Where applicable, we also confirmed that the transition between XRT read-out modes looked the same in both sets of light curves. Once we were sat- isfied that our light curves passed this test, we also compared a random sample of 30 GRBs with those manually created by other members of the Swift/XRT team and again found good agreement. 4. Data availability and usage Our light curve repository is publicly available via the internet, http://www.swift.ac.uk/xrt curves/ Specific light curves can be accessed directly by appending their Swift target ID to this URL2. While every effort has been made to make this process com- pletely automatic, there may be cases where the light curve gen- eration fails (e.g. if the source is too faint to centroid on, or if there are multiple candidates within the BAT error circle). In this event, a member of the Swift/XRT team will manually instigate the creation procedure as soon as possible. For GRBs detected by other observatories which Swift subsequently observes, the creation procedure will not be automatically triggered, however the XRT team will trigger it manually in a timely manner. These light curves, data and images may be used by anyone. In any publication which makes use of these data, please cite this paper in the body of your publication where the light curves are presented. The suggested wording is: “For details of how these light curves were produced, see Evans et al. (2007).” 2 The target ID is the trigger number, given in the GCN notices and circulars, but padded with leading zeroes to be 8 digits long. e.g. GRB 060729 had the trigger number 221755, so its target ID is 00221755 Please also include the following paragraph in the Acknowledgements section: “This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.” 5. Acknowledgements PAE, APB, KLP, LGT and JPO acknowledge PPARC support. LV, JR, PM and DNB are supported by NASA contract NAS5- 00136. References Abbey, A.F., Carpenter, J., Read, A., et al. 2005, in ESA-SP 604, ‘The X-ray Universe 2005’, 943 Burrows, D.N., Hill, J.E., Nousek, J.A., et al. 2005, Sp. Sci. Rev, 120, 165 Gehrels N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005 Goad, M.R., Page, K.L., Godet, O., et al., 2007, A&A, in press (astro-ph/0612661) Grupe, D., Burrows, D.N., Patel, S.K., et al. 2007, ApJ, in press (astro-ph/0611240) Hill, J.E., Burrows, D.N., Nousek, J.A., et al. 2004, SPIE, 5165, 217 Kraft, R.P., Burrows, D.N., Nousek, J.A., et al. 1991, ApJ, 374, 344 Kennea, J.A., Burrows, D.N., Wells, A., et al. 2005, SPIE, 5898, 329 Moretti, A., Campana, S., Mineo, T., et al. 2005, SPIE, 5898, 360 Pagani, C., Morris, D.C., Kobayashi S., et al. 2006, ApJ, 645, 1315 Page, K.L., Willingale, R., Osborne, J.P., et al. 2007, ApJ, in press (astro- ph/0704.1609) Romano, P., Campana, S., Chincarini, G., et al. 2006, A&A, 456, 917 Scargle J.D., 1998, ApJ, 504, 405 Vaughan, S., Goad, M.R., Beardmore, A.P., et al., 2006, ApJ, 638, 920 Mangano, V., Holland, S.T., Malesani, D., et al., 2007, A&A, in press Zhang, B., 2007, ChJAA, 7, 1 http://www.swift.ac.uk/xrt_curves/ http://arxiv.org/abs/astro-ph/0612661 http://arxiv.org/abs/astro-ph/0611240 Introduction Aspects of light curve generation GRBs fade Swift data contain multiple observations and snapshots CCD Damage Automatic readout-mode switching Pile-up Light curve creation procedure Phase #1 – Preparation Phase Phase #2 – Production Phase Counts to flux conversion Phase #3 – Presentation Phase Immediate light curve regeneration Testing procedure Data availability and usage Acknowledgements

0704.0129

On the total disconnectedness of the quotient Aubry set

ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET ALFONSO SORRENTINO Abstra t. In this paper we show that the quotient Aubry set, asso iated to a su� iently smooth me hani al or symmetri al Lagrangian, is totally dis- onne ted (i.e., every onne ted omponent onsists of a single point). This result is optimal, in the sense of the regularity of the Lagrangian, as Mather's ounterexamples in [19℄ show. Moreover, we dis uss the relation between this problem and a Morse-Sard type property for (di�eren e of) riti al subsolu- tions of Hamilton-Ja obi equations. 1. Introdu tion. In Mather's studies of the dynami s of Lagrangian systems and the existen e of Arnold di�usion, it turns out that understanding ertain aspe ts of the Aubry set and, in parti ular, what is alled the quotient Aubry set, may help in the onstru - tion of orbits with interesting behavior. While in the ase of twist maps (see for instan e [3, 12℄ and referen es therein) there is a detailed stru ture theory for these sets, in more degrees of freedom quite few is known. In parti ular, it seems to be useful to know whether the quotient Aubry set is �small� in some sense of dimension (e.g., vanishing topologi al or box dimension). In [18℄ Mather showed that if the state spa e has dimension ≤ 2 (in the non- autonomous ase) or the Lagrangian is the kineti energy asso iated to a Riemann- ian metri and the state spa e has dimension ≤ 3, then the quotient Aubry set is totally dis onne ted, i.e., every onne ted omponent onsists of a single point (in a ompa t metri spa e this is equivalent to vanishing topologi al dimension). In the autonomous ase, with dimM ≤ 3, the same argument shows that this quotient is totally dis onne ted as long as the Aubry set does not interse t the zero se tion of TM (this is the ase when the ohomology lass is large enough in norm). What happens in higher dimension? Unfortunately, this is generally not true. In fa t, Burago, Ivanov and Kleiner in [6℄ provided an example that does not satisfy this property (they did not dis uss it in their work, but it follows from the results therein). More strikingly, Mather provided in [19℄ several examples of quotient Aubry sets that are not only non-totally-dis onne ted, but even isometri to losed intervals. All these examples ome from me hani al Lagrangians on TTd (i.e., the sum of the kineti energy and a potential) with d ≥ 3. In parti ular, for every ε > 0, he provided a potential U ∈ C2d−3,1−ε(Td), whose asso iated quotient Aubry set is isometri to an interval. As the author himself noti ed, it is not possible to improve the di�erentiability of these examples, due to the onstru tion arried out. The main aim of this arti le is to show that the ounterexamples provided by Mather are optimal, in the sense that for more regular me hani al Lagrangians, the asso iated quotient Aubry sets - orresponding to the zero ohomology lass - are totally dis onne ted. In parti ular, our result will also apply to slightly more general Lagrangians, satis- fying ertain onditions on the zero se tion; in this ase, we shall be able to show http://arxiv.org/abs/0704.0129v1 2 ALFONSO SORRENTINO that the quotient Aubry set, orresponding to a well spe i�ed ohomology lass, is totally dis onne ted. We shall also outline a possible approa h to generalize this result, pointing out how it is related to a Morse-Sard type problem; from this and Sard's lemma, one an easily re over Mather's result in dimension d = 2 (autonomous ase). It is important to point out, that most of this approa h has been inspired by Albert Fathi's talk [9℄, in whi h he used this relation with Sard's lemma to show a simpler way to onstru t me hani al Lagrangians on TTN , whose quotient Aubry sets are Lips hitz equivalent to any given doubling metri spa e or, equivalently, to any spa e with �nite Assouad dimension (see [13℄ for a similar onstru tion). In this ase we do not get a neat relation between their regularity and N , as in Mather's, but we an only observe that N goes to in�nity as r in reases. It would be interesting to study in depth the relation between the dimension of the quotient Aubry set, the regularity of the Lagrangian and the dimension of the state spa e. Our result may be seen as a �rst step in this dire tion. Post S riptum. Just before submitting this paper, we learnt that analogous results had been proven indipendently by Albert Fathi, Alessio Figalli and Ludovi Ri�ord, using a similar approa h (to be published). Moreover, in �A generi property of families of Lagrangian systems� (to appear on Annals of Mathemati s), Patri k Bernard and Gonzalo Contreras managed to show that generi ally, in Mañé's sense, there are at most 1+dimH1(M ;R) ergodi min- imizing measures, for ea h ohomology lass c ∈ H1(M ;R). As a orollary of this striking result, one gets that generi ally the quotient Aubry set is �nite for ea h ohomology lass and it onsists of at most 1 + dimH1(M ;R) elements. 2. The Aubry set and the quotient Aubry set. Let M be a ompa t and onne ted smooth manifold without boundary. Denote by TM its tangent bundle and T∗M the otangent one. A point of TM will be denoted by (x, v), where x ∈ M and v ∈ TxM , and a point of T∗M by (x, p), where p ∈ T∗xM is a linear form on the ve tor spa e TxM . Let us �x a Riemannian metri g on it and denote with d the indu ed metri on M ; let ‖ · ‖x be the norm indu ed by g on TxM ; we shall use the same notation for the norm indu ed on T De�nition. A fun tion L : TM −→ R is alled a Tonelli Lagrangian if: i) L ∈ C2(TM); ii) L is stri tly onvex in the �bers, i.e., the se ond partial verti al derivative (x, v) is positive de�nite, as a quadrati form, for any (x, v) ∈ TM ; iii) L is superlinear in ea h �ber, i.e., ‖v‖x→+∞ L(x, v) (this ondition is independent of the hoi e of the Riemannian metri ). Given a Lagrangian, we an de�ne the asso iated Hamiltonian, as a fun tion on the otangent bundle: H : T∗M −→ R (x, p) 7−→ sup v∈TxM {〈p, v〉x − L(x, v)} where 〈 ·, · 〉x represents the anoni al pairing between the tangent and otangent spa e. ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 3 If L is a Tonelli Lagrangian, one an easily prove that H is �nite everywhere, C2, superlinear and stri tly onvex in the �bers. Moreover, under the above assump- tions, one an de�ne a di�eomorphism between TM and T∗M , alled the Legendre transform: L : TM −→ T∗M (x, v) 7−→ (x, v) In parti ular, L is a onjugation between the two �ows (namely the Euler-Lagrange and Hamiltonian �ows) and H ◦ L(x, v) = (x, v), v − L(x, v) . Observe that if η is a 1-form on M , then we an de�ne a fun tion on the tangent spa e η̂ : TM −→ R (x, v) 7−→ 〈η(x), v〉x and onsider a new Tonelli Lagrangian Lη = L − η̂. The asso iated Hamiltonian will be Hη(x, p) = H(x, p + η). Moreover, if η is losed, then Ldt and have the same extremals and therefore the Euler-Lagrange �ows on TM asso iated to L and Lη are the same. Although the extremals are the same, this is not generally true for the minimizers. What one an say is that they stay the same when we hange the Lagrangian by an exa t 1-form. Thus, for �xed L, the minimizers depend only on the de Rham ohomology lass c = [η] ∈ H1(M ;R). From here, the interest in onsidering mod- i�ed Lagrangians, orresponding to di�erent ohomology lasses. Let us �x η, a smooth (C2 is enough for what follows) 1-form on M , and let c = [η] ∈ H1(M ;R) be its ohomology lass. As done by Mather in [16℄, it is onvenient to introdu e, for t > 0 and x, y ∈ M , the following quantity: hη,t(x, y) = inf Lη(γ(s), γ̇(s)) ds , where the in�mum is taken over all pie ewise C1 paths γ : [0, t] −→ M , su h that γ(0) = x and γ(t) = y. We de�ne the Peierls barrier as: hη(x, y) = lim inf (hη,t(x, y) + α(c)t) , where α : H1(M ;R) −→ R is Mather's α fun tion (see [15℄). It an be shown that this fun tion is onvex and that (only for the autonomous ase) the lim inf an be repla ed by lim. Observe that hη does not depend only on the ohomology lass c, but also on the hoi e of the representant; namely, if η′ = η+df , then hη′(x, y) = hη(x, y)+ f(y)− f(x). Proposition 1. The values of the map hη are �nite. Moreover, the following properties hold: i) hη is Lips hitz; ii) for ea h x ∈ M , hη(x, x) ≥ 0; iii) for ea h x, y, z ∈ M , hη(x, y) ≤ hη(x, z) + hη(z, y); iv) for ea h x, y ∈ M , hη(x, y) + hη(y, x) ≥ 0. 4 ALFONSO SORRENTINO For a proof of the above laims and more, see [16, 10, 8℄. Inspired by these prop- erties, we an de�ne δc : M ×M −→ R (x, y) 7−→ hη(x, y) + hη(y, x) (observe that this fun tion does a tually depend only on the ohomology lass). This fun tion is positive, symmetri and satis�es the triangle inequality; therefore, it is a pseudometri on AL,c = {x ∈ M : δc(x, x) = 0} . AL,c is alled the Aubry set (or proje ted Aubry set) asso iated to L and c, and δc is Mather's pseudometri . In [16℄, Mather has showed that this is a non-empty ompa t subset of M , that an be Lips hitzly lifted to a ompa t invariant subset of TM . De�nition. The quotient Aubry set (ĀL,c, δ̄c) is the metri spa e obtained by identifying two points in AL,c, if their δc-pseudodistan e is zero. We shall denote an element of this quotient by x̄ = {y ∈ AL,c : δc(x, y) = 0}. These elements (that are also alled c-stati lasses, see [8℄) provide a partition of AL,c into ompa t subsets, that an be lifted to invariant subsets of TM . They are really interesting from a dynami al systems point of view, sin e they ontain the α and ω limit sets of c-minimizing orbits (see [16, 8℄ for more details). For the sake of our proof, it is onvenient to adopt Fathi's weak KAM theory point of view (we remand the reader to [10℄ for a self- ontained presentation). De�nition. A lo ally lips hitz fun tion u : M −→ R is a subsolution ofHη(x, dxu) = k, with k ∈ R, if Hη(x, dxu) ≤ k for almost every x ∈ M . This de�nition makes sense, be ause, by Radema her's theorem, we know that dxu exists almost everywhere. It is possible to show that there exists c[η] ∈ R, su h that Hη(x, dxu) = k admits no subsolutions for k < c[η] and has subsolutions for k ≥ c[η]. The onstant c[η] is alled Mañé's riti al value and oin ides with α(c), where c = [η] (see [8℄). De�nition. u : M −→ R is a η- riti al subsolution, if Hη(x, dxu) ≤ α(c) for almost every x ∈ M . Denote by Sη the set of riti al subsolutions. This set Sη is non-empty. In fa t, Fathi showed (see [10℄) that: Proposition 2. If u : M −→ R is a η- riti al subsolution, then for every x, y ∈ M u(y)− u(x) ≤ hη(x, y) . Moreover, for any x ∈ M , the fun tion hη,x(·) := hη(x, ·) is a η- riti al subsolution. Using this result, he provided a ni e representation of hη, in terms of the η- riti al subsolutions. Corollary 1. If x ∈ AL,c and y ∈ M , hη(x, y) = sup (u(y)− u(x)) . This supremum is a tually attained. Proof. It is lear, from the proposition above, that hη(x, y) ≥ sup (u(y)− u(x)) . ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 5 Let us show the other inequality. In fa t, sin e hη,x is a η- riti al subsolution and x ∈ AL,c (i.e., hη(x, x) = 0), then hη(x, y) = hη,x(y)− hη,x(x) ≤ sup (u(y)− u(x)) . This shows that the supremum is attained. ✷ This result an be still improved. Fathi and Si onol� proved in [11℄: Theorem (Fathi, Si onol�). For any η- riti al subsolution u : M −→ R and for ea h ε > 0, there exists a C1 fun tion ũ : M −→ R su h that: i) ũ(x) = u(x) and Hη(x, dxũ) = α(c) on AL,c; ii) |ũ(x) − u(x)| < ε and Hη(x, dxũ) < α(c) on M \ AL,c. In parti ular, this implies that C1 η- riti al subsolutions are dense in Sη with the uniform topology. This result has been re ently improved by Patri k Bernard (see [5℄), showing that every η- riti al subsolution oin ides, on the Aubry set, with a C1,1 η- riti al subsolution. Denote the set of C1 η- riti al subsolutions by S1η and the set of C1,1 η- riti al subsolutions by S1,1η . Corollary 2. For x, y ∈ AL,c, the following representation holds: hη(x, y) = sup u∈S1η (u(y)− u(x)) = sup u∈S1,1η (u(y)− u(x)) . Moreover, these suprema are attained. It turns out to be onvenient, to hara terize the elements of ĀL,c (i.e., the c- quotient lasses) in terms of η- riti al subsolutions. Let us onsider the following set: Dc = {u− v : u, v ∈ Sη} (it depends only on the ohomology lass c and not on η) and denote by D1c and D1,1c , the sets orresponding, respe tively, to C1 and C1,1 η- riti al subsolutions. Proposition 3. For x, y ∈ AL,c, δc(x, y) = sup (w(y) − w(x)) = sup w∈D1c (w(y) − w(x)) = = sup w∈D1,1c (w(y)− w(x)) and this suprema are attained. Proof. From the de�nition of δc(x, y), we immediately get: δc(x, y) = hη(x, y) + hη(y, x) = = sup (u(y)− u(x)) + sup (v(x) − v(y)) = = sup u,v∈Sη [(u(y)− v(y))− (u(x)− v(x))] = = sup (w(y) − w(x)) . The other equalities follow from the density results we mentioned above. ✷ Proposition 4. If w ∈ Dc, then dxw = 0 on AL,c. Therefore AL,c ⊆ w∈D1,1c Crit(w) , where Crit(w) is the set of riti al points of w. Proof. This is an immediate onsequen e of a result by Fathi (see [10℄); namely, if u, v ∈ Sη, then they are di�erentiable on AL,c and dxu = dxv. ✷ 6 ALFONSO SORRENTINO Proposition 5. If w ∈ Dc, then it is onstant on any quotient lass of ĀL,c; namely, if x, y ∈ AL,c and δc(x, y) = 0, then w(x) = w(y). Proof. From the representation formula above, it follows that: 0 = δc(x, y) = sup w̃∈Dc (w̃(y)− w̃(x)) ≥ w(y) − w(x) 0 = δc(y, x) = sup w̃∈Dc (w̃(x)− w̃(y)) ≥ w(x) − w(y) . For any w ∈ D1c , let us de�ne the following evaluation fun tion: ϕw : (ĀL,c, δ̄c) −→ (R, | · |) x̄ 7−→ w(x) . • ϕw is well de�ned, i.e., it does not depend on the element of the lass at whi h w is evaluated; • ϕw(ĀL,c) = w(AL,c) ⊆ w(Crit(w)); • ϕw is Lips hitz, with Lips hitz onstant 1. In fa t: ϕw(x̄)− ϕw(ȳ) = w(x) − w(y) ≤ δc(x, y) = δ̄c(x̄, ȳ) ϕw(ȳ)− ϕw(x̄) = w(y)− w(x) ≤ δc(y, x) = δ̄c(ȳ, x̄) . Therefore: |ϕw(x̄)− ϕw(ȳ)| ≤ δ̄c(x̄, ȳ) . As we shall see, these fun tions play a key role in the proof of our result. 3. The main result. Our main goal is to show that, under suitable hypotheses on L, there is a well spe i�ed ohomology lass cL, for whi h (ĀL,cL , δ̄cL) is totally dis onne ted, i.e., every onne ted omponent onsists of a single point. Consider L : TM −→ R a Tonelli Lagrangian and the asso iated Legendre trans- L : TM −→ T∗M (x, v) 7−→ (x, v) Remember that T∗M , as a otangent bundle, may be equipped with a anoni al symple ti stru ture. Namely, if (U , x1, . . . , xd) is a lo al oordinate hart for M and (T∗U , x1, . . . , xd, p1, . . . , pd) the asso iated otangent oordinates, one an de�ne the 2-form dxi ∧ dpi . It is easy to show that ω is a symple ti form (i.e., it is non-degenerate and losed). In parti ular, one an he k that ω is intrinsi ally de�ned, by onsidering the 1-form on T∗U pi dxi , whi h satis�es ω = −dλ and is oordinate-indipendent; in fa t, in terms of the natural proje tion π : T∗M −→ M (x, p) 7−→ x ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 7 the form λ may be equivalently de�ned pointwise without oordinates by λ(x,p) = (dπ(x,p)) ∗p ∈ T∗(x,p)T∗M . The 1-form λ is alled the Liouville form (or the tautologi al form). Consider now the se tion of T∗M given by ΛL = L(M × {0}) = (x, 0) : x ∈ M orresponding to the 1-form ηL(x) = (x, 0) · dx = (x, 0) dxi . We would like this 1-form to be losed, that is equivalent to ask ΛL to be a Lagrangian submanifold, in order to onsider its ohomology lass cL = [ηL] ∈ H1(M ;R). Observe that this ohomology lass an be de�ned in a more intrinsi way; in fa t, onsider the proje tion π|ΛL : ΛL ⊂ T∗M −→ M ; this indu es an isomorphism between the ohomology groupsH1(M ;R) andH1(ΛL;R). The preimage of [λ|ΛL ] under this isomorphism is alled the Liouville lass of ΛL and one an easily show that it oin ides with cL. We an de�ne the set: L(M) = {L : TM −→ R : L is a Tonelli Lagrangian and ΛL is Lagrangian} . This set is non-empty and onsists of Lagrangians of the form L(x, v) = f(x) + 〈η(x), v〉x +O(‖v‖2), with f ∈ C2(M) and η a C2 losed 1-form on M . In parti ular, it in ludes the me hani al Lagrangians, i.e., Lagrangians of the form L(x, v) = ‖v‖2x + U(x) , namely the sum of the kineti energy and a potential U : M −→ R. More gen- erally, it ontains the symmetri al (or reversible) Lagrangians, i.e., Lagrangians L : TM −→ R su h that L(x, v) = L(x,−v) , for every (x, v) ∈ TM . In fa t, in the above ases, (x, 0) ≡ 0; therefore ΛL = M × {0} (the zero se tion of the otangent spa e), that is learly Lagrangian, and cL = 0. We an now state our main result: Main Theorem. Let M be a ompa t onne ted manifold of dimension d ≥ 1 and let L ∈ L(M) be a Lagrangian su h that L(x, 0) ∈ Cr(M), with r ≥ 2d − 2 and (x, 0) ∈ C2(M). Then, the quotient Aubry set (ĀL,cL , δ̄cL), orresponding to the Liouville lass of ΛL, is totally dis onne ted, i.e., every onne ted omponent onsists of a single point. This result immediately implies: Corollary 3 (Symmetri al Lagrangians). Let M be a ompa t onne ted manifold of dimension d ≥ 1 and let L(x, v) be a symmetri al Tonelli Lagrangian on TM , su h that L(x, 0) ∈ Cr(M), with r ≥ 2d−2. Then, the quotient Aubry set (ĀL,0, δ̄0) is totally dis onne ted. 8 ALFONSO SORRENTINO More spe i� ally, Corollary 4 (Me hani al Lagrangians). Let M be a ompa t onne ted manifold of dimension d ≥ 1 and let L(x, v) = 1 ‖v‖2x + U(x) be a me hani al Lagrangian on TM , su h that the potential U ∈ Cr(M), with r ≥ 2d − 2. Then, the quotient Aubry set (ĀL,0, δ̄0) is totally dis onne ted. Remark. This result is optimal, in the sense of the regularity of the potential U , for ĀL,0 to be totally dis onne ted. In fa t, Mather provided in [19℄ examples of quotient Aubry sets isometri to the unit interval, orresponding to me hani al Lagrangians L ∈ C2d−3,1−ε(TTd), for any 0 < ε < 1. Before proving the main theorem, it will be useful to show some useful results. Lemma 1. Let us onsider L ∈ L(M), su h that ∂L (x, 0) ∈ C2(M), and let H be the asso iated Hamiltonian. (1) Every onstant fun tion u ≡ const is a ηL- riti al subsolution. In parti u- lar, all ηL- riti al subsolutions are su h that dxu ≡ 0 on AL,cL. (2) For every x ∈ M , (x, 0) = (x, ηL(x)) = 0 . Proof. (1) The se ond part follows immediatly from the fa t that, if u, v ∈ SηL , then they are di�erentiable on AL,cL and dxu = dxv (see [10℄). Let us show that u ≡ const is a ηL- riti al subsolution; namely, that HηL(x, 0) ≤ α(cL) for every x ∈ M . It is su� ient to observe: • HηL(x, 0) = −L(x, 0); in fa t: HηL(x, 0) = H(x, ηL(x)) = H (x, 0) (x, 0), 0 − L(x, 0) = = −L(x, 0) . • let v be dominated by LηL + α(cL) (see [10℄, for the existen e of su h fun tions), i.e., for ea h ontinuous pie ewise C1 urve γ : [a, b] −→ M we have v(γ(b))− v(γ(a)) ≤ LηL(γ(t), γ̇(t)) dt + α(cL)(b − a) . Then, onsidering the onstant path γ(t) ≡ x, one an easily dedu e α(cL) ≥ sup (−LηL(x, 0)) = − inf LηL(x, 0) ; therefore, α(cL) ≥ −LηL(x, 0) = −L(x, 0) = HηL(x, 0) for every x ∈ M . (2) The inverse of the Legendre transform an be written in oordinates L−1 : T∗M −→ TM (x, p) 7−→ (x, p) ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 9 Therefore, (x, 0) = L−1 (L(x, 0)) = L−1 (x, 0) = L−1((x, ηL(x))) = (x, ηL(x) In parti ular, observing that for any ηL- riti al subsolution u, HηL(x, dxu) = α(cL) on AL,cL , we an easily dedu e from above that: AL,cL ⊆ {L(x, 0) = −α(cL)} = {H(x, ηL(x)) = α(cL)} α(cL) = sup (−L(x, 0)) = − inf L(x, 0) =: e0 , as denoted in [14, 7℄. Let us observe that in general e0 ≤ min c∈H1(M ;R) α(c) = −β(0) , where β : H1(M ;R) −→ R is Mather's β-fun tion, i.e., the onvex onjugate of α (in [14, 7℄, the right-hand-side quantity is referred to as stri t riti al value). Therefore, we are onsidering an extremal ase in whi h e0 = α(cL) = minα(c); it follows also quite easily that cL ∈ ∂β(0), namely, it is a subgradient of β at 0. A ru ial step in the proof of our result will be the following lemma, that an be read as a sort of relaxed version of Sard's Lemma (the proof will be mainly based on the one in [1℄). Main Lemma. Let U ∈ Cr(M), with r ≥ 2d − 2, be a non-negative fun tion, vanishing somewhere and denote A = {U(x) = 0}. If u : M −→ R is C1 and satis�es ‖dxu‖2x ≤ U(x) in an open neighborhood of A, then |u(A)| = 0 (where | · | denotes the Lebesgue measure in R). See se tion 4 for its proof. In parti ular, it implies this essential property. Corollary 5. Under the hypotheses of the main theorem, if u ∈ SηL , then |u(AL,cL)| = 0 (where | · | denotes the Lebesgue measure in R). Proof (Corollary). First of all, we an assume that u ∈ S1ηL , be ause of Fathi and Si onol�'s theorem. By Taylor's formula, it follows that there exists an open neighborhood W of AL,cL , su h that for all x ∈ W : α(cL) ≥ HηL(x, dxu) = HηL(x, 0) + (x, 0) · dxu+ (1− t)∂ (x, t dxu)(dxu) 2 dt . Let us observe the following. • From the previous lemma, one has that (x, 0) = 0 , for every x ∈ M . 10 ALFONSO SORRENTINO • From the stri t onvexity hypothesis, it follows that there exists γ > 0 su h that: (x, t dxu)(dxu) 2 ≥ γ‖dxu‖2x for all x ∈ M and 0 ≤ t ≤ 1. Therefore, for x ∈ W : α(cL) ≥ HηL(x, dxu) ≥ HηL(x, 0) + ‖dxu‖2x = = −L(x, 0) + γ ‖dxu‖2x . The assertion will follow from the previous lemma, hoosing U(x) = (α(cL) + L(x, 0)) . In fa t, U ∈ Cr, with r ≥ 2d − 2, by hypothesis; moreover, it satis�es all other onditions, be ause α(cL) = − inf L(x, 0) AL,cL ⊆ {x ∈ W : L(x, 0) = −α(cL)} = {x ∈ W : U(x) = 0} =: A . For, the previous lemma allows us to on lude that |u(AL,cL)| = 0 . Proof (Main Theorem). Suppose by ontradi tion that ĀL,cL is not totally dis- onne ted; therefore it must ontain a onne ted omponent Γ with at least two points x̄ and ȳ. In parti ular δ̄c(x̄, ȳ) = hηL(x, y) + hηL(y, x) > 0, for some x ∈ x̄ and y ∈ ȳ; therefore, we have hηL(x, y) > 0 or hηL(y, x) > 0. From the representation formula for hηL , it follows that there exists u ∈ S1,1ηL ⊆ D (sin e u = u−0, and v = 0 is a ηL- riti al subsolution), su h that |u(y)−u(x)| > 0. This implies that the set ϕu(Γ) is a onne ted set in R with at least two di�erent points, hen e it is a non degenerate interval and its Lebesgue measure is positive. ϕu(Γ) ⊆ ϕu(ĀL,cL) = u(AL,cL) and onsequently ∣ϕu(Γ) ∣ ≤ |u(AL,cL)|. This ontradi ts the previous orollary. ✷ In parti ular, this proof suggests a possible approa h to generalize the above result to more general Lagrangians and other ohomology lasses. De�nition. A C1 fun tion f : M −→ R is of Morse-Sard type if |f(Crit(f))| = 0, where Crit(f) is the set of riti al points of f and | · | denotes the Lebesgue measure in R. Proposition 6. Let M be a ompa t onne ted manifold of dimension d ≥ 1, L a Tonelli Lagrangian and c ∈ H1(M ;R). If ea h w ∈ D1,1c is of Morse-Sard type, then the quotient Aubry set (ĀL,c, δ̄c) is totally dis onne ted. This proposition and Sard's lemma (see [4℄) easily imply Mather's result in dimen- sion d ≤ 2 (autonomous ase); it su� es to noti e that Sard's lemma (in dimension d) holds for Cd−1,1 fun tions. ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 11 Corollary 6. Let M be a ompa t onne ted manifold of dimension d ≤ 2. For any L Tonelli Lagrangian and c ∈ H1(M ;R), the quotient Aubry set (ĀL,c, δ̄c) is totally dis onne ted. Remark. The main problem be omes now to understand under whi h onditions on L and c, these di�eren es of subsolutions are of Morse-Sard type. Unfortunately, one annot use the lassi al Sard's lemma, due to a la k of regularity of riti al subsolutions: in general they will be at most C1,1. In fa t, although it is always possible to smooth them up out of the Aubry set and obtain fun tions in C∞(M \ AL,c)∩C1,1(M), the presen e of the Aubry set (where the value of their di�erential is pres ribed) represents an obsta le that it is impossible to over ome. It is quite easy to onstru t examples that do not admit C2 riti al subsolutions: just onsider a ase in whi h AL,c is all the manifold and it is not a C1 graph. For instan e, this is the ase if M = T and H(x, p) = 1 )2 − sin2(πx); in fa t, there is only one riti al subsolution (up to onstants), that turns out to be a solution (AL, 2 = T), and it is given by a primitive of sin(πx)− 2 ; this is learly C1,1 but not C2. On the other hands, the above results suggest that, in order to prove the Morse-Sard property, one ould try to ontrol the omplexity of these fun tions (à la Yomdin), using the rigid stru ture provided by Hamilton-Ja obi equation and the smoothness of the Hamiltonian, rather than the regularity of the subsolutions. There are several di� ulties in pursuing this approa h in the general ase, mostly related to the nature of the Aubry set. We hope to understand these �spe ulations� more in depth in the future. 4. Proof of the Main Lemma. De�nition. Consider a fun tion f ∈ Cr(Rd). We say that f is s - �at at x0 ∈ Rd (with s ≤ r), if all its derivatives, up to the order s, vanish at x0. The proof of the main lemma is based on the following version of Kneser-Glaeser's Rough omposition theorem (see [1, 20℄). Proposition 7. Let V, W ⊂ Rd be open sets, A ⊂ V , A∗ ⊂ W losed sets. Consider U ∈ Cr(V ), with r ≥ 2, a non-negative fun tion that is s-�at on A ⊂ {U(x) = 0}, with s ≤ r − 1, and g : W −→ V a Cr−s fun tion, with g(A∗) ⊂ A. Then, for every open pre- ompa t set W1 ⊃ A∗ properly ontained in W , there exists F : Rd −→ R satisfying the following properties: i) F ∈ Cr−1(Rd); ii) F ≥ 0; iii) F (x) = U(g(x)) = 0 on A∗; iv) F is s-�at on A∗; v) {F (x) = 0} ∩W1 = A∗; vi) there exists a onstant K > 0, su h that U(g(x)) ≤ KF (x) on W1. See se tion 5 for its proof. To prove the main lemma, it will be enough to show that for every x0 ∈ M , there exists a neighborhood Ω su h that it holds. For su h a lo al result, we an assume that M = U is an open subset of Rd, with x0 ∈ U . In the sequel, we shall identify T∗U with U × Rd and for x ∈ U , we identify T∗xU = {x} × Rd. We equip U × Rd with the natural oordinates (x1, . . . , xd, p1, . . . , pd). 12 ALFONSO SORRENTINO Before pro eeding in the proof, let us point out that it is lo ally possible to repla e the norm obtained by the Riemannian metri , by a onstant norm on R Lemma 2. For ea h 0 < α < 1 and x0 ∈ M , there exists an open neighborhood Ω of x0, with Ω ⊂ U and su h that (1− α)‖p‖x0 ≤ ‖p‖x ≤ (1 + α)‖p‖x0 , for every p ∈ T∗xU ∼= Rd and ea h x ∈ Ω. Proof. By ontinuity of the Riemannian metri , the norm ‖p‖x tends uniformly to 1 on {p : ‖p‖x0 = 1}, as x tends to x0. Therefore, for x near to x0 and every p ∈ Rd \ {0}, we have: (1− α) ≤ ‖p‖x0 ≤ (1 + α). We an now prove the main result of this se tion. Proof ( Main Lemma). By hoosing lo al harts and by lemma 2, we an assume that U ∈ Cr(Ω), with Ω open set in Rd, A = {x ∈ Ω : U(x) = 0} and u : Ω −→ R is su h that ‖dxu‖2 ≤ βU(x) in Ω, where β is a positive onstant. De�ne, for 1 ≤ s ≤ r: Bs = {x ∈ A : U is s - �at at x} and observe that A = B1 := {x ∈ A : DU(x) = 0} . We shall prove the lemma by indu tion on the dimension d. Let us start with the following laim. Claim. If s ≥ 2d− 2, then |u(Bs)| = 0. Proof. Let C ⊂ Ω be a losed ube with edges parallel to the oordinate axes. We shall show that |u(Bs ∩C)| = 0. Sin e Bs an be overed by ountably many su h ubes, this will prove that |u(Bs)| = 0. Let us start observing that, by Taylor's theorem, for any x ∈ Bs ∩C and y ∈ C we U(y) = Rs(x; y), where Rs(x; y) is Taylor's remainder. Therefore, for any y ∈ C U(y) = o(‖y − x‖s) . Let λ be the length of the edge of C. Choose an integer N > 0 and subdivide C in Nd ubes Ci with edges , and order them so that, for 1 ≤ i ≤ N0 ≤ Nd, one has Ci ∩Bs 6= ∅. Hen e, Bs ∩ C = Bs ∩ Ci. Observe that for every ε > 0, there exists ν0 = ν0(ε) su h that, if N ≥ ν0, x ∈ Bs ∩ Ci and y ∈ Ci, for some 0 ≤ i ≤ N0, then U(y) ≤ ε 4β(dλ2)d ‖y − x‖s . Fix ε > 0. Choose xi ∈ Bs ∩ Ci and all yi = u(xi). De�ne, for N ≥ ν0, the following intervals in R: , yi + ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 13 Let us show that, if N is su� iently big, then u(Bs ∩ C) ⊂ i=1 Ei. In fa t, if x ∈ Bs ∩ C, then there exists 1 ≤ i ≤ N0, su h that x ∈ Bs ∩ Ci. Therefore, |u(x)− yi| = |u(x)− u(xi)| = = ‖dxu(x̃)‖ · ‖x− xi‖ ≤ βU(x̃)‖x− xi‖ ≤ 4β(dλ2)d ‖x̃− xi‖ 2 ‖x− xi‖ ≤ 2(dλ2) ‖x− xi‖ 2(dλ2) where x̃ is a point in the segment joining x and xi. Sin e by hypothesis s ≥ 2d− 2, ≥ d. Hen e, assuming that N > max{λ d, ν0}, one gets |u(x)− yi| ≤ and an dedu e the in lusion above. To prove the laim, it is now enough to observe: |u(Bs ∩ C)| ≤ |Ei| ≤ ≤ εN0 ≤ εNd 1 = ε . From the arbitrariness of ε, the assertion follows easily. ✷ This laim immediately implies that u(B2d−2) has measure zero. In parti ular, this proves the ase d = 1 (sin e in this ase 2d− 2 = 0) and allows us to start the indu tion. Suppose to have proven the result for d− 1 and show it for d. Sin e A = (B1 \B2) ∪ (B2 \B3) ∪ . . . ∪ (B2d−3 \B2d−2) ∪B2d−2 , it remains to show that |u(Bs \Bs+1)| = 0 for 1 ≤ s ≤ 2d− 3 ≤ r − 1. Claim. Every x̃ ∈ Bs \Bs+1 has a neighborhood Ṽ , su h that |u((Bs \Bs+1) ∩ Ṽ )| = 0 . Sin e Bs\Bs+1 an be overed by ountably many su h neighborhoods, this implies that u(Bs \Bs+1) has measure zero. Proof. Choose x̃ ∈ Bs \ Bs+1. By de�nition of these sets, all partial derivatives of order s of U vanish at this point, but there is one of order s + 1 that does not. Assume (without any loss of generality) that there exists a fun tion w(x) = ∂i1∂i2 . . . ∂isU(x) su h that w(x̃) = 0 but ∂1w(x̃) 6= 0 . 14 ALFONSO SORRENTINO De�ne h : Ω −→ Rd x 7−→ (w(x), x2, . . . , xd) , where x = (x1, x2, . . . , xd). Clearly, h ∈ Cr−s(Ω) and Dh(x̃) is non-singular; hen e, there is an open neighborhood V of x̃ su h that h : V −→ W is a Cr−s isomorphism (with W = h(V )). Let V1 be an open pre ompa t set, ontaining x̃ and properly ontained in V , and de�ne A = Bs ∩ V1, A∗ = h(A) and g = h−1. If we onsider W1, any open set ontaining A∗ and properly ontained in W , we an apply proposition 7 and dedu e the existen e of F : Rd −→ R satisfying properties i)-vi). De�ne Ŵ = {(x2, . . . , xd) ∈ Rd−1 : (0, x2, . . . , xd) ∈ W1} Û(x2, . . . , xd) = C F (0, x2, . . . , xd), where C is a positive onstant to be hosen su� iently big. Observe that Û ∈ Cr−1(Rd−1). Moreover, property v) of F and the fa t that A∗ = h(A) ⊆ {0} × Ŵ imply that: ∗ = {0} × B̂1 , where B̂1 = {(x2, . . . xd) ∈ Ŵ : F (0, x2, . . . , xd) = 0}. Denote  := {(x2, . . . , xd) ∈ Ŵ : Û = 0} = B̂1 and de�ne the following fun tion on Ŵ : û(x2, . . . , xd) = u(g(0, x2, . . . , xd)). We want to show that these fun tions satisfy the hypotheses for the (d − 1)- dimensional ase. In fa t: • Û ∈ Cr−1(Rd−1), with r − 1 ≥ 2d− 3 > 2(d− 1)− 2; • û ∈ C1(Ŵ ) (sin e g is in Cr−s(W ), where 1 ≤ s ≤ r − 1); • if we denote by µ = supW1 ‖dxg‖ < +∞ (sin e g is C on W1), then we have that for every point in Ŵ : ‖dû(x2, . . . , xd)‖2 ≤ ‖dxu(g(0, x2, . . . , xd))‖2‖dxg(0, x2, . . . , xd)‖2 ≤ ≤ µ2‖dxu(g(0, x2, . . . , xd))‖2 ≤ ≤ βµ2U(g(0, x2, . . . , xd)) ≤ ≤ βµ2KF (0, x2, . . . , xd) ≤ ≤ Û(x2, . . . , xd), if we hoose C > βµ2K, where K is the positive onstant appearing in proposition 7, property vi). Therefore, it follows from the indu tive hypothesis, that: |û(Â)| = 0. Sin e, u(Bs ∩ V1) ⊆ u(A) = u(g(A∗)) = u(g({0} × B̂1)) = = û(B̂1) = û(Â) , de�ning Ṽ = V1, we may on lude that |u(Bs ∩ Ṽ )| ≤ |û(Â)| = 0 . ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 15 This ompletes the proof of the Main Lemma. ✷ 5. Proof of a modified version of Kneser-Glaeser's Rough omposition theorem. Now, let us prove proposition 7. We shall mainly follow the presentation in [1℄, adapted to our needs. Proof (Proposition 7). Let us start, de�ning a family of polynomials. Supposing that g is Cr and using the s-�atness hypothesis, we have, for x ∈ A∗ and k = 0, 1, . . . , r : fk(x) = D k(U ◦ g)(x) = s<q≤k qU(g(x))Di1g(x) . . . Diqg(x) ,(1) where the se ond sum is over all the q-tuples of integers i1, . . . , iq ≥ 1 su h that i1 + . . .+ iq = k, and σk = σk(i1, . . . , iq). The ru ial observation is that (1) makes sense on A∗, even when g is Cr−s smooth (in fa t ij ≤ k − q + 1 ≤ r − s). We would like to pro eed in the fashion of Whitney's extension theorem, in order to �nd a smooth fun tion F su h that DkF = fk on A , and satisfying the stated onditions. Remark. Note that, without any loss of generality, we an assume that W is ontained in an open ball of diameter 1. The general ase will then follow from this spe ial one, by a straightforward partition of unity argument. Let us start with some te hni al lemmata. Lemma 3. For x, x′, x0 ∈ A∗ and k = 0, . . . , r, we have: i≤r−k fk+i(x) (x′ − x)i +Rk(x, x′) , |Rk(x, x′)| ‖x′ − x‖r−k −→ 0 as x, x′ −→ x0 in A∗. The proof of this lemma appears without any major modi� ation in [1℄ (on pages 36-37). De�ne, for x ∈ A∗ and y ∈ Rd P (x, y) = i=s+1 fi(x) (y − x)i and its k-th derivative Pk(x, y) = i≤r−k fi+k(x) (y − x)i . Lemma 4. For x ∈ A∗ and y ∈ W1, U(g(y)) = P (x, y) +R(x, y) , where |R(x, y)| ≤ C‖y − x‖r. 16 ALFONSO SORRENTINO Proof. The proof follows the same idea of lemma 3. By Taylor's formula for U , U(g(y)) = q=s+1 DqU(g(x)) (g(y)− g(x))q + I(g(x), g(y))(g(x) − g(y))r . Obviously, |I(g(x), g(y))(g(x) − g(y))r| ≤ C1‖y − x‖r , therefore it is su� ient to estimate the �rst term. Observe that: g(y) = g(x) + Dig(x)(y − x)i + J(x, y)(y − x)r−s . Hen e, the �rst term in the sum above be omes: q=s+1 DqU(g(x)) Dig(x)(y − x)i + J(x, y)(y − x)r−s k=s+1 ak(y − x)k + R̂(x, y) = = P (x, y) + R̂(x, y) , sin e s+1≤q≤k DqU(g(x))Di1g(x) . . . Diqg(x) = fk(x) The remainder terms onsist of: • terms ontaining (y − x)k, with k > r; • terms of the binomial produ t, ontaining J(x, y)(y − x)r−s. They are of the form: . . . (y − x)(r−s)j+ where αi ≥ 0 and αi = q − j. Sin e q ≥ s+ 1 and s ≤ r − 1, then: (r − s)j + iαi ≥ (r − s)j + = (r − s)j + q − j = = rj − sj + q − j ≥ ≥ rj − (s+ 1)j + s+ 1 = = r + r(j − 1)− (s+ 1)(j − 1) = = r + (r − s− 1)(j − 1) ≥ r . Therefore, for x ∈ A∗ and y ∈ W1 R̂(x, y) ≤ C2‖y − x‖r , and the lemma follows taking C = C1 + C2. ✷ Next step will onsist of reating a Whitney's partition. We will start by overing W1 \ A∗ with an in�nite olle tion of ubes Kj, su h that the size of ea h Kj is roughly proportional to its distan e from A∗. First, let us �x some notation. We shall write a ≺ b instead of �there exists a positive real onstant M , su h that a ≤ Mb � and a ≈ b as short for a ≺ b and b ≺ a. Let λ = 1 ; this hoi e will ome in handy later. For any losed ube K (with edges parallel to the oordinate axes), Kλ will denote the (1 + λ) - dilation of K ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 17 about its enter. Let ‖ · ‖ be the eu lidean metri on Rd and d(y) = d(y,A∗) = inf{‖y − x‖ : x ∈ A∗} . If {Kj}j is the sequen e of losed ubes de�ned below, with edges of length ej , let dj be its distan e from A , i.e., dj = d(A ,Kj) = inf{‖y − x‖ : x ∈ A∗, y ∈ Kj} . One an show the following lassi al lemma (see for instan e [1℄ for a proof). Lemma 5. There exists a sequen e of losed ubes {Kj}j with edges parallel to the oordinate axes, that satis�es the following properties: i) the interiors of the Kj 's are disjoint; ii) W1 \A∗ ⊂ iii) ej ≈ dj ; iv) ej ≈ d(y) for all y ∈ Kλj ; v) ej ≈ d(z) for all z ∈ W1 \ A∗, su h that the ball with enter z and radius d(z) interse ts Kλj ; vi) ea h point of W1 \A∗ has a neighborhood interse ting at most N of the Kλj , where N is an integer depending only on d. Now, let us onstru t a partition of unity on W1 \ A∗. Let Q be the unit ube entered at the origin. Let η be a C∞ bump fun tion de�ned on Rd su h that η(y) = 1 for y ∈ Q 0 for y 6∈ Qλ and 0 ≤ η ≤ 1. De�ne ηj(y) = η y − cj where cj is the enter of Kj and ej is the length of its edge, and onsider σ(y) = ηj(y) . Then, 1 ≤ σ(y) ≤ N for all y ∈ W1 \ A∗. Clearly, for ea h k = 0, 1, 2, . . . we have that Dkηj(y) ≺ e−kj , for all y ∈ W1 \A∗. Hen e, by properties iv) and vi) of lemma 5, we have that for ea h k = 0, 1, . . . , r: ηj(y) ≺ d(y)−k for all y ∈ W1 \A∗ Dkσ(y) ≺ d(y)−k for all y ∈ W1 \A∗ . De�ne ϕj(y) = ηj(y) These fun tions satisfy the following properties: i) ea h ϕj is C and supported on Kλj ; ii) 0 ≤ ϕj(y) ≤ 1 and j ϕj(y) = 1, for all y ∈ W1 \A∗; iii) every point of W1 \ A∗ has a neighborhood on whi h all but at most N of the ϕj 's vanish identi ally; iv) for ea h k = 0, 1, . . . , r, Dkϕj(y) ≺ d(y)−k for all y ∈ W1 \ A∗; namely, there are onstants Mk su h that D kϕj(y) ≤ Mkd(y)−k; v) there is a onstant α and points xj ∈ A∗, su h that: ‖xj − y‖ ≤ αd(y), whenever ϕj(y) 6= 0 . This follows from properties iii) and iv) of lemma 5. 18 ALFONSO SORRENTINO We an now onstru t our fun tion F . Observe that, from lemma 4: 0 ≤ U(g(y)) = P (xj , y) +R(xj , y) ≤ P (xj , y) + C‖y − xj‖r ; therefore P (xj , y) ≥ −C‖y − xj‖r. First, de�ne P̂j(y) = P (xj , y) + 2C‖y − xj‖r where C is the same onstant as in lemma 4; for what said above, P̂j(y) ≥ C‖y − xj‖r > 0 in W1 \ {xj} .(2) Hen e, onstru t F in the following way: F (y) = 0 y ∈ A∗ ϕj(y)P̂j(y) y ∈ Rd \A∗ . We laim that this satis�es all the stated properties i)-vi). In parti ular, properties ii), iii) and v) follow immediately from the de�nition of F and (2). Moreover, F ∈ C∞(Rd \ A∗). We need to show that DkF = fk (for k = 0, 1, . . . , r − 1) on ∂A∗ (namely, the boundary of A∗) and that Dr−1F is ontinuous on it. The main di� ult in the proof, is that DkF is expressed as a sum ontaining terms Dk−mϕj(y)Pm(xj , y), where ϕj(y) 6= 0. Even if y is lose to some x0 ∈ A∗, it ould be loser to A∗ and hen e the bound given by property iv) of ϕj might be ome large. One an over ome this problem by hoosing a point x∗ ∈ A∗, so that ‖x∗ − y‖ is roughly the same as d(y) and hen e, xj is lose to x Lemma 6. For every η > 0, there exists δ > 0 su h that for all y ∈ W1 \ A∗, x, x∗ ∈ A∗ and x0 ∈ ∂A∗, we have ‖Pk(x, y)− Pk(x∗, y)‖ ≤ η d(y)r−k ≤ η‖y − x0‖r−k, whenever k ≤ r and ‖y − x‖ < αd(y) ‖y − x∗‖ < αd(y) ‖y − x0‖ < δ , where α is the same onstant as in v) above. See [1℄ (on page 126) for its proof. Lemma 7. For every η > 0, there exist 0 < δ < 1 and a onstant E, su h that for all y ∈ W1 \A∗, x∗ ∈ A∗ and x0 ∈ ∂A∗, we have ‖DkF (y)− Pk(x∗, y)‖ ≤ E d(y)r−k ≤ η d(y)r−k−1, whenever k ≤ r − 1 and ‖y − x∗‖ < αd(y) ‖y − x0‖ < δ . Proof. Let Sj,k(x ∗, y) = ∂kP̂j(y)− Pk(x∗, y) . From lemma 6 (with η = ε, to be de�ned later) and the de�nition of P̂j , we get: ‖Sj,k(x∗, y)‖ ≤ ‖∂kP̂j(y)− Pk(xj , y)‖+ ‖Pk(xj , y)− Pk(x∗, y)‖ ≤ ≤ Ckd(y)r−k + εd(y)r−k = = (Ck + ε)d(y) r−k . Then, F (y)− P (x∗, y) = ϕj(y)Sj,0(x ∗, y) ON THE TOTAL DISCONNECTEDNESS OF THE QUOTIENT AUBRY SET 19 and hen e DkF (y)− Pk(x∗, y) = Dk−iϕj(y)Sj,i(x ∗, y) . Therefore, hoosing ε su� iently small: ‖DkF (y)− Pk(x∗, y)‖ ≤ ‖Dk−iϕj(y)‖ · ‖Sj,i(x∗, y)‖ ≤ Mk−id(y) −k+i(Ck + ε)d(y) r−i ≤ ≤ E d(y)r−k ≤ η d(y)r−k−1 . Lemma 8. For every η > 0, there exist 0 < δ < 1 su h that, for all y ∈ W1 \ A∗, x∗ ∈ A∗ and x0 ∈ ∂A∗, we have ‖Pk(x∗, y)− Pk(x0, y)‖ ≤ η‖y − x0‖r−k , whenever k ≤ r and ‖y − x∗‖ < αd(y) ‖y − x0‖ < δ . Proof. The proof goes as the one of lemma 6, observing that ‖x∗ − x0‖ ≤ (1 + α)‖y − x0‖ and Pk(x0, y)− Pk(x∗, y) = q≤r−k Rk+q(x ∗, x0) (y − x)q . Claim. For every x0 ∈ ∂A∗ and k = 0, 1, . . . , r − 1: F (x0) = fk(x0) . Moreover, Dr−1F is ontinuous at x0 ∈ ∂A∗. This laim follows easily from the lemmata above (see [1℄, on page 128, for more details). This proves that F ∈ Cr−1(Rd) and ompletes the proof of i) and iv). It remains to show that property vi) holds, namely that there exists a onstant K > 0, su h that U(g(x)) ≤ KF (x) on W1. Obviously, this holds at every point in A∗, for every hoi e of K (sin e both fun tions vanish there). Claim. There exists a onstant K > 0, su h that U◦g ≤ K on W1 \A∗. Proof. Sin e F > 0 on W1 \ A∗, it is su� ient to show that U◦gF is uniformly bounded by a onstant, as d(y) goes to zero. Let us start observing that, for y ∈ Kλj , P̂j(y) ≥ C‖y − xj‖r ≥ Cd(y)r ; therefore: F (y) = ϕj(y)P̂j(y) ≥ ϕj(y)Cd(y) = Cd(y)r . 20 ALFONSO SORRENTINO Moreover, if x∗ ∈ A∗ su h that d(y) = ‖y − x∗‖, lemma 4 and 7 imply: |U(g(y))− F (y)| ≤ |U(g(y))− P (x∗, y)|+ |P (x∗, y)− F (y)| ≤ ≤ Cd(y)r + Ed(y)r = (C + E)d(y)r . Hen e, U(g(y)) F (y) U(g(y))− F (y) + F (y) F (y) ≤ 1 + |U(g(y))− F (y)| F (y) ≤ 1 + (C + E)d(y) Cd(y)r ≤ 2 + E =: K̃ . This proves property vi) and on ludes the proof of the proposition. ✷ A knowledgements. I wish to thank John Mather for having introdu ed me to this area and suggested this problem. I am very grateful to him and to Albert Fathi for their interest and for several helpful dis ussions. Referen es [1℄ Ralph Abraham and Joel Robbin. Transversal mappings and �ows. 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Trans. Amer. Math. So ., 36(1):63�89, 1934. Department of Mathemati s, Prin eton University, Prin eton (NJ), 08544-1000 U.S. E-mail address: asorrent�math.prin eton.edu 1. Introduction. 2. The Aubry set and the quotient Aubry set. 3. The main result. 4. Proof of the Main Lemma. 5. Proof of a modified version of Kneser-Glaeser's Rough composition theorem. References

0704.0130

New simple modular Lie superalgebras as generalized prolongs

NEW SIMPLE MODULAR LIE SUPERALGEBRAS AS GENERALIZED PROLONGS SOFIANE BOUARROUDJ1, PAVEL GROZMAN2, DIMITRY LEITES3 Abstract. Over algebraically closed fields of characteristic p > 2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified. 1. Introduction 1.1. Setting. We use standard notations of [FH, S]; for the precise definition (algorithm) of generalized Cartan-Tanaka–Shchepochkina (CTS) complete and partial prolongations, and algorithms of their construction, see [Shch]. Hereafter K is an algebraically closed field of characteristic p > 2, unless specified. Let g′ = [g, g], and c(g) = g ⊕ center, where dim center = 1. Let n)g denote the incarnation of the Lie (super)algebra g with the n)th Cartan matrix, cf. [GL4, BGL1]. On classification of simple vectorial Lie superalgebras with polynomial coefficients (in what follows referred to as vectorial Lie superalgebras of polynomial vector fields over C, see [LSh, K3]). The works of S. Lie, Killing and È. Cartan, now classical, completed classification over C (1) simple Lie algebras of finite dimension and of polynomial vector fields. Lie algebras and Lie superalgebras over fields in characteristic p > 0, a.k.a. modular Lie (super)algebras, were first recognized and defined in topology, in the 1930s. The simple Lie algebras drew attention (over finite fields K) as a step towards classification of simple finite groups, cf. [St]. Lie superalgebras, even simple ones and even over C or R, did not draw much attention of mathematicians until their (outstanding) usefulness was observed by physicists in the 1970s. Meanwhile mathematicians kept discovering new and new examples of simple modular Lie algebras until Kostrikin and Shafarevich ([KS]) formulated a conjecture embracing all previously found examples for p > 7. Its generalization reads: select a Z-form gZ of every g of type 1) (1), take gK := gZ⊗Z K and its simple finite dimensional subquotient si(gK) (there can be several such si(gK)). Together with deformations 2) of these examples we get in this way all simple finite dimensional Lie algebras over algebraically closed fields if p > 5. If p = 5, we should add to the above list Melikyan’s examples. 1991 Mathematics Subject Classification. 17B50, 70F25. Key words and phrases. Cartan prolongation, nonholonomic manifold, Lie superalgebra. We are thankful to I. Shchepochkina for help; DL is thankful to MPIMiS, Leipzig, for financial support and most creative environment. 1)Observe that the algebra of divided powers (the analog of the polynomial algebra for p > 0) and hence all prolongs (Lie algebras of vector fields) acquire one more — shearing — parameter: N , see [S]. 2)It is not clear, actually, if the conventional notion of deformation can always be applied if p > 0 (for the arguments, see [LL]; cf. [Vi]); to give the correct (better say, universal) notion is an open problem, but in some cases it is applicable, see [BGL4]. http://arxiv.org/abs/0704.0130v1 2 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites Having built upon ca 30 years of work of several teams of researchers, and having added new ideas and lots of effort, Block, Wilson, Premet and Strade proved the generalized KSh conjecture for p > 3, see [S]. For p ≤ 5, the above KSh-procedure does not produce all simple finite dimensional Lie algebras; there are other examples. In [GL4], we returned to É. Cartan’s description of Z-graded Lie algebras as CTS prolongs, i.e., as subalgebras of vectorial Lie algebras preserving certain distributions; we thus interpreted the “mysterious” at that moment exceptional examples of simple Lie algebras for p = 3 (the Brown, Frank, Ermolaev and Skryabin algebras), further elucidated Kuznetsov’s interpretation [Ku1] of Melikyan’s algebras (as prolongs of the nonpositive part of the Lie algebra g(2) in one of its Z-gradings) and discovered three new series of simple Lie algebras. In [BjL], the same approach yielded bj, a simple super versions of g(2), and Bj(1;N |7), a simple p = 3 super Melikyan algebra. Both bj and Bj(1;N |7) are indigenous to p = 3, the case where g(2) is not simple. 1.2. Classification: Conjectures and results. Recently, Elduque considered super analogs of the exceptional simple Lie algebras; his method leads to a discovery of 10 new simple (presumably, exceptional) Lie superalgebras for p = 3. For a description of the Elduque superalgebras, see [CE, El1, CE2, El2]; for their description in terms of Cartan matrices and analogs of Chevalley relations and notations we use in what follows, see [BGL1, BGL2]. In [L], a super analog of the KSh conjecture embracing all types of simple (finite dimen- sional) Lie superalgebras is formulated based on an entirely different idea in which the CTS prolongs play the main role: F o r e v e r y s i m p l e f i n i t e d i m e n s i o n a l L i e ( s u p e r ) a l g e b r a o f t h e f o r m g(A) , t a k e i t s n o n - p o s i t i v e p a r t w i t h r e s p e c t t o a c e r t a i n s i m p l e s t Z - g r a d i n g , c o n s i d e r i t s c o m p l e t e a n d p a r t i a l p r o l o n g s a n d t a k e t h e i r s i m p l e s u b q u o t i e n t s . The new examples of simple modular Lie superalgebras (BRJ, Bj(3;N |3), Bj(3;N |5)) support this conjecture. (This is how Cartan got all simple Z-graded Lie algebras of poly- nomial growth and finite depth — the Lie algebras of type (1) — at the time when the root technique was not discovered yet.) 1.2.1. Yamaguchi’s theorem ([Y]). This theorem, reproduced in [GL4, BjL], states that for almost all simple finite dimensional Lie algebras g over C and their Z-gradings g = ⊕ of finite depth d, the CTS prolong of g≤ = ⊕ −d≤i≤0 gi is isomorphic to g, the rare exceptions being two of the four series of simple vectorial algebras (the other two series being partial prolongs). 1.2.2. Conjecture. In the following theorems, we present the results of SuperLie-assisted ([Gr]) computations of the CTS-prolongs of the non-positive parts of the simple finite di- mensional Lie algebras and Lie superalgebras g(A); we have only considered Z-grading cor- responding to each (or, for larger ranks, even certain selected) of the simplest gradings r = (r1, . . . , rrkg), where all but one coordinates of r are equal to 0 and only one — selected — is equal to 1, and where we set degX±i = ±ri for the Chevalley generators X±i of g(A), see [BGL1]. O t h e r g r a d i n g s ( a s w e l l a s a l g e b r a s g(A) o f h i g h e r r a n k s ) d o n o t y i e l d n e w s i m p l e L i e ( s u p e r ) a l g e b r a s a s p r o l o n g s o f t h e n o n - p o s i t i v e p a r t s. New simple modular Lie superalgebras 3 1.3. Theorem. The CTS prolong of the nonpositive part of g returns g in the following cases: p = 3 and g = f(4), e(6), e(7) and e(8) considered with the Z-grading with one selected root corresponding to the endpoint of the Dynkin diagram. 1.3.1. Conjecture. [The computer got stuck here, after weeks of computations] To the cases of Theorem 1.3, one can add the case for p = 5 and g = el(5) (see [BGL2]) in its Z-grading with only one odd simple root and with one selected root corresponding to any endpoint of the Dynkin diagram. 1.4. Theorem. Let p = 3. For the previously known (we found more, see Theorems 1.6, 1.7) simple finite dimensional Lie superalgebras g of rank ≤ 3 with Cartan matrix and for their simplest gradings r, the CTS prolongs (of the non-positive part of g) different from g are given in the following table elucidated below. 1.5. Melikyan superalgebras for p = 3. There are known the two constructions of the Melikyan algebra Me(5;N) = ⊕ Me(5;N)i, defined for p = 5: 1) as the CTS prolong of the triple Me0 = cvect(1; 1), Me−1 = O(1; 1)/const and the trivial module Me−2, see [S]; this construction would be a counterexample to our conjecture were there no alternative: 2) as the complete CTS prolong of the non-negative part of g(2) in its grading r = (01), with g(2) obtained now as a partial prolong, see [Ku1, GL4]. In [BjL], we have singled out Bj(1;N |7) as a p = 3 simple analog of Me(5;N) as a partial CTS prolongs of the pair (the negative part of k(1;N |7), Bj(1;N |7)0 = pgl(3)), and bj as a p = 3 simple analog of g(2) whose non-positive part is the same as that of Bj(1;N |7), i.e., bj and Bj(1;N |7) are analogs of the construction 2). The original Melikyan’s construction 1) also has its super analog for p = 3 (only in the situation described in Theorem 1.6) and it yields a new series of simple Lie superalgebras as the complete prolongs, with another simple analog of g(2) as a partial prolong. Recall ([BGL1]) that we normalize the Cartan matrix A so that Aii = 1 or 0 if the ith root is odd, whereas if the ith root is even, we set Aii = 2 or 0 in which case we write 0̄ instead of 0 in order not to confuse with the case of odd roots. 1.6. Theorem. A p = 3 analog of the construction 1) of the Melikyan algebra is given by setting g0 = ck(1; 1|1), g−1 = O(1; 1|1)/const and g−2 being the trivial module. It yields a simple super Melikyan algebra that we denote by Me(3;N |3), non-isomorphic to a superMe- likyan algebra Bj(1;N |7). The partial prolong of the non-positive part of Me(3;N |3) is a new (exceptional) simple Lie superalgebra that we denote by brj(2; 3). This brj(2; 3) has the three Cartan matrices: and 2) −1 0̄ joined by an odd reflection, and −1 0̄ . It is a super analog of the Brown algebra br(2) = brj(2; 3)0̄, its even part. The CTS prolongs for the simplest gradings r of 1)brj(2; 3) returns known simple Lie superalgebras, whereas the CTS prolong for a simplest grading r of 2)brj(2; 3) returns, as a partial prolong, a new simple Lie superalgebra we denote BRJ. Unlike br(2), the Lie superalgebra brj(2; 3) has analogs for p 6= 3, e.g., for p = 5, we get a new simple Lie superalgebra brj(2; 5) such that brj(2; 5)0̄ = sp(4) with the two Cartan matrices 1) and 2) . The CTS prolongs of brj(2; 5) for all its Cartan matrices and the simplest r return brj(2; 5). 4 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites Having got this far, it was impossible not to try to get classification of simple g(A)’s. Here is its beginning part, see [BGL5]. 1.7. Theorem. If p > 5, every finite dimensional simple Lie superalgebra with a 2 × 2 Cartan matrix is isomorphic to osp(1|4), osp(3|2), or sl(1|2). If p = 5, we should add brj(2; 5). If p = 3, we should add brj(2; 3). Remark. For details of description of the new simple Lie superalgebras of types Bj and Me and their subalgebras, in particular, presentations of brj(2; 3) and brj(2; 5), and proof of Theorem 1.7 and its generalization for higher ranks, see [BGL4, BGL5]. The new simple Lie superalgebras obtained are described in the next subsections. g Cartan matrix r prolong osp(3|2) k(1|3) k(1|3; 1) osp(3|2) k(1|3; 1) sl(1|2) vect(0|2) vect(1|1) vect(1|1) osp(1|4) k(3|1) osp(1|4) brj(2; 3) 0̄ −1 Me(3;N |3) Brj(4|3) Brj(4;N |3) Brj(3;N |4) ⊃ BRJ brj(2; 3) 0̄ −1 Brj(3;N |3) Brj(3;N |4) ⊃ BRJ brj(2; 5) brj(2; 5) brj(2; 5) brj(2; 5) brj(2; 5) New simple modular Lie superalgebras 5 g Cartan matrix r prolong sl(1|3) 0 −1 0 −1 2 −1 0 −1 2 (100) (010) (001) vect(0|3) sl(1|3) vect(2|1) 0 −1 0 −1 0 −2 0 −1 2 (100) (010) (001) vect(2|1) sl(1|3) vect(2|1) psl(2|2) any matrix (100) (010) (001) svect(1|2) psl(2|2) svect(1|2) osp(1|6) 2 −1 0 −1 2 −1 0 −1 1 (100) (010) (001) k(5|1) osp(1|6) osp(1|6) osp(3|4) 2 −1 0 −1 0 −1 0 −2 2 0 −1 0 −1 0 1 0 −1 1 (100) (010) (001) k(3|3) osp(3|4) osp(3|4) 0 −1 0 −1 2 −1 0 −1 1 (100) (010) (001) osp(3|4) osp(5|2) 2 −1 0 −1 0 1 0 −1 1 0 −1 0 −1 0 1 0 −2 2 (100) (010) (001) osp(5|2) 0 −1 0 −1 2 −1 0 −2 2 (100) (010) (001) osp(5|2) k(1|5) osp(4|2;α) α generic 2 −1 0 α 0 −1− α 0 −1 2 0 1 −1− α −1 0 −α −1− α α 0 (100) (010) (001) osp(4|2;α) osp(4|2;α) α = 0,−1 1) The simple part of 1)osp(4|2;α) is sl(2|2); for the CTS of psl(2|2), see above 2) 2)osp(4|2;α) ≃ sl(2|2); for the CTS of sl(2|2), see above 6 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites g Cartan matrix r prolong osp(2|4) 0 1 0 −1 2 −2 0 −1 2 0 1 0 −1 0 2 0 −1 2 0 −2 1 −2 0 1 −1 −1 2 (100) (010) (001) (100) (010) (001) (100) (010) (001) osp(2|4) osp(2|4) if p > 3 Bj(3;N |3) if p = 3 osp(2|4) k(3|2) osp(2|4) if p > 3 Bj(3;N |3) if p = 3 osp(2|4) osp(2|4) osp(2|4) k(3|2) g(2|3) 3) 0 0 −1 0 0 −2 −1 −2 2 (100) (010) (001) Bj(2|4) Bj(3|5) 1.8. A description of Bj(3;N |3). For g = 1)osp(2|4) and r = (0, 1, 0), we have the following realization of the non-positive part: gi the generators (even | odd) g−2 Y6 = ∂1 | Y8 = ∂4 g−1 Y2 = ∂2, Y5 = x2∂1 + x5∂4 + ∂3, | Y4 = ∂5, Y7 = 2x2∂4 + ∂6, g0 ≃ Y3 = x22∂1 + x2x5∂4 + x2∂3 + 2x5∂6, Z3 = x32∂1 + 2x3x6∂4 + x3∂2 + 2x6∂5 sl(1|1) ⊕ sl(2) ⊕K H2 = 2x1∂1 + 2x2∂2 + x4∂4 + x5∂5 + 2x6∂6, H1 = [Z1, Y1],H3 = [Z3, Y3] | Y1 = x1∂4 + 2x2∂5 + x3∂6, Z1 = 2x4∂1 + 2x5∂2 + x6∂3 The g0-module g−1 is irreducible, having one highest weight vector Y2. Let p = 3. The CTS prolong gives sdim(g1) = 4|4. The g0-module g1 has the following two lowest weight vectors: V ′1 x1x2∂4 + 2x1∂6 + 2x2 2∂5 + x2x3∂6 V ′′1 x1x2∂1 + x1x5∂4 + 2x2x4∂4 + x1∂3 + x2 2∂2 + 2x2x5∂5 + x2x6∂6 + x3x5∂6 + x4∂6 Since g1 generates the positive part of the CTS prolong, [g−1, g1] = g0, and [g−1, g−1] = g−2, the standard criteria of simplicity ensures that the CTS prolong is simple. Since none of the Z-graded Lie superalgebras over C of polynomial growth and finite depth has grading of this form (with g0 ≃ sl(1|1)⊕ sl(2)⊕ K), we conclude that this Lie superalgebra is new. We denote it by Bj(3;N |3), where N is the shearing parameter of the even indeterminates. Our calculations show that N2 = N3 = 1 always. For N1 = 1, 2, the super dimensions of the positive components of Bj(3;N |3) are given in the following tables: N = (1, 1, 1) g1 g2 g3 g4 g5 g6 – – – sdim 4|4 5|5 4|4 4|4 2|2 0|3 N = (2, 1, 1) g1 g2 g3 g4 g5 g6 · · · g11 g12 sdim 4|4 5|5 4|4 5|5 4|4 5|5 · · · 2|2 0|3 New simple modular Lie superalgebras 7 Let V ′i , V i and V i be the lowest height vectors of gi with respect to g0. For N = (1, 1, 1), these vectors are as follows: gi lowest weight vectors V ′2 x1 2∂4 + 2 x1x2∂5 + x1x3∂6 + x2x3 V ′′2 x1x2 2∂1 + x1x2x5∂4 + x1x2∂3 + 2 x1x5∂6 + x2 2x3∂3 + 2 x2 2x5∂5 + x2x3x5∂6 V ′′′2 x1 2∂1 + x2 2∂1 + x2 2x3∂2 + 2 x2 2x6∂5 + 2 x1x2∂2 + 2 x1x3∂3 + 2 x1x4∂4 +x2x3 2∂3 + x2x4∂5 + 2 x3x4∂6 + 2 x2 2x3x6∂4 + 2 x2x3x6∂6 V ′3 x1 2x2∂4 + 2 x1 2∂6 + 2 x1x2 2∂5 + x1x2x3∂6 + x2 V ′′3 x1 2x2∂1 + x1 2x5∂4 + x1 2∂3 + x1x2x3∂3 + 2 x1x2x5∂5 + x1x3x5∂6 +x2x3 2x5∂6 + x1x2x4∂4 + 2 x1x2 2∂2 + x1x2x3∂3 + 2 x1x2x6∂6 +2 x1x4∂6 + 2 x2 2∂3 + 2 x2 2x3x6∂6 + 2 x2 2x4∂5 + x2x3x4∂6 V ′4 x1 2∂1 + x1 2x2x5∂4 + x1 2x2∂3 + 2 x1 2x5∂6 + x1x2 2x3∂3 + 2 x1x2 2x5∂5 +x1x2x3x5∂6 + x2 2x5∂6 V ′′4 x1 2x4∂4 + x1 2x5∂5 + x1 2x6∂6 + x2 2x6∂6 + x1x2 2∂1 + x1x2 2x3∂2 + 2 x1x2 2x6∂5 +2 x1x3 2x5∂6 + 2 x1x2x3 2∂3 + 2 x1x2x4∂5 + x1x3x4∂6 + x2x3 2x4∂6 + 2 x1x2 2x3x6∂4 V ′5 x1 2∂2 + x1 2x4∂6 + 2 x1 2x2x3∂3 + 2 x1 2x2x4∂4 + x1 2x2x6∂6 + 2 x2 2x4∂6 +x1x2 2∂3 + x1x2 2x4∂5 + x1x2 2x3x6∂6 + 2 x1x2x3x4∂6 V ′6 x1 2x4∂1 + x1 2x5∂2 + 2 x1 2x6∂3 + x1 2x2x4∂3 + 2 x1 2x4x5∂6 +2 x1 2x2x3x5∂3 + x1 2x2x4x5∂4 + x1 2x2x5x6∂6 + x2 2x4x5∂6 +x1x2 2x5∂3 + x1x2 2x3x4∂3 + 2 x1x2 2x4x5∂5 + x1x2 2x3x5x6∂6 + x1x2x3x4x5∂6 For N = (2, 1, 1), the lowest hight vectors are as in the table above together with the following ones gi lowest weight vectors V ′′′4 x1 3∂4 + 2 x1 2x2∂5 + x1 2x3∂6 + x1x2x3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V ′11 x1 2∂2 + x1 5x4∂6 + x1 2∂3 + x1 2x4∂5 + 2 x1 5x2x3∂3 + 2 x1 5x2x4∂4 5x2x6∂6 + 2 x1 2x4∂6 + x1 2x3x6∂6 + 2 x1 4x2x3x4∂6 V ′12 x1 2x4∂1 + x1 2x5∂2 + 2 x1 2x6∂3 + x1 5x2x4∂3 + 2 x1 5x4x5∂6 + x1 2x5∂3 2x3x4∂3 + 2 x1 2x4x5∂5 + 2 x1 5x2x3x5∂3 + x1 5x2x4x5∂4 + x1 5x2x5x6∂6 2x4x5∂6 + x1 2x3x5x6∂6 + x1 4x2x3x4x5∂6 Let us investigate if Bj(3;N |3) has partial prolongs as subalgebras: (i) Denote by g′1 the g0-module generated by V 1 . We have sdim(g 1) = 2|2. The CTS par- tial prolong (g−, g0, g 1)∗ gives a graded Lie superalgebra with the property that [g−1, g1] ≃ {Y1, h1} := aff. From the description of irreducible modules over solvable Lie superalge- bras [Ssol], we see that the irreducible aff-modules are 1-dimensional. For irreducible aff- submodules g′−1 in g−1 we have two possibilities: to take g −1 = {Y4} or g′−1 = {Y7}; for both of them, g′−1 is purely odd and we can never get a simple Cartan prolong. (ii) Denote by g′′1 the g0-module generated by V 1 . We have sdim(g 1) = 2|2. The CTS partial prolong (g−, g0, g 1)∗ returns osp(2|4). 1.9. A description of Bj(2|4). We consider 3)g(2|3) with r = (1, 0, 0). In this case, sdim(g(2, 3)−) = 2|4. Since the g(2, 3)0-module action is not faithful, we consider the quo- tient algebra g0 = g(2, 3)0/ann(g−1) and embed (g(2, 3)−, g0) ⊂ vect(2|4). This realization 8 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites is given by the following table: gi the generators (even | odd) g−1 Y6 = ∂2, Y8 = ∂1 | Y11 = ∂3, Y10 = ∂4, Y4 = ∂5, Y1 = ∂6 g0 ≃ Y3 = x2∂1 + 2x4∂3 + x6∂5, Y9 = [Y2, [Y3, Y5]], Z3 = x1∂2 + 2x3∂4 + x5∂6, Z9 = [Z2, [Z3, Z5]], H2 = [Z2, Y2],H3 = [Z3, Y3] | Y2 = x1∂4 + x5∂2, Y5 = [Y2, Y3], osp(3|2) Y7 = [Y3, [Y2, Y3]], Z2 = x2∂5 + 2x4∂1, Z5 = [Z2, Z3], Z7 = [Z3, [Z2, Z3]] The g0-module g−1 is irreducible, having one lowest weight vector Y11 and one highest weight vector Y1. The CTS prolong (g−, g0)∗ gives a Lie superalgebra of superdimension 13|14. Indeed, sdim(g1) = 4|4 and sdim(g2) = 1|0. The g0-module g1 has one lowest vector: V1 = 2x1x2∂3 + x1x6∂1 + 2x2 2∂4 + x2x5∂1 + 2x2x6∂2 + x4x5∂3 + 2x4x6∂4 + x5x6∂5 The g2 is one-dimensional spanned by the following vector 2x2∂1 + x1 2x4∂3 + 2x1 2x6∂5 + x1x2 2∂2 + 2x1x2x3∂3 + x1x2x4∂4 + 2x1x2x5∂5 + x1x2x6∂6 +x1x3x6∂1 + 2x1x4x5∂1 + x1x4x6∂2 + 2x2 2x3∂4 + x2 2x5∂6 + x2x3x5∂1 + 2x2x3x6∂2 + x2x4x5∂2 +x3x4x5∂3 + 2x3x4x6∂4 + x3x5x6∂5 + 2x4x5x6∂6 Besides, if i > 2, then gi = 0 for all values of the sharing parameter N = (N1, N2). A direct computation gives [g1, g1] = g2 and [g−1, g1] = g0. SuperLie tells us that this Lie superalgebra has three ideals I1 ⊂ I2 ⊂ I3 with the same non-positive part but different positive parts: sdim(I1) = 10|14, sdim(I2) = 11|14, sdim(I3) = 12|14. The ideal I1 is just our bj, see [BjL, CE]. The partial CTS prolong with I1 returns I1 plus an outer derivation given by the vector above (of degree 2). It is clear now that Bj(2|4) is not simple. 1.10. A description of Bj(3|5). We consider 3)g(2|3) and r = (0, 1, 0). In this case, sdim(g(2, 3)−) = 3|5. Since the g(2, 3)0-module action is again not faithful, we consider the quotient module g0 = g(2, 3)0/ann(g−1) and embed (g(2, 3)−, g0) ⊂ vect(3;N |5). This realization is given by the following table: gi the generators (even | odd) g−2 Y9 = ∂1 | Y10 = ∂3, Y11 = ∂2 g−1 Y8 = ∂4, Y6 = ∂5 | Y5 = 2x4∂2 + 2x5∂3 + 2x7∂1 + ∂7, Y2 = x4∂3 − 2x6∂1 + ∂8 Y7 = x5∂2 + ∂6 g0 ≃ sl(1|2) H1 = [Z1, Y1],H3 = [Z3, Y3], Y3 = 2x3∂2 + 2x7x8∂1 + x5∂4 + 2x7∂6 + x8∂7, Z3 = 2x2∂3 + 2x6x7∂1 + x4∂5 + x6∂7 + 2x7∂8 | Y4 = [Y1, Y3], Z4 = [Z1, Z3], Y1 = 2 (2x1∂3 + 2x6x7∂2 + x6∂4 + x7∂5) Z1 = 2 x3∂1 + 2x4x5∂2 + 2x5 2∂3 + 2x5x7∂1 + 2x4∂6 + x5∂7 The g0-module g−1 is irreducible, having one highest weight vector Y2. We have sdim(g1) = 6|4. The g0-module g1 has two lowest weight vectors given by V ′1 x1x5∂2 + 2x5x6x8∂2 + x5x7x8∂3 + 2x1∂6 + 2x3∂4 + x5x7∂4 + x5x8∂5 + 2x7x8∂7 V ′′1 x6x7x8∂2 + 2x1∂4 + x7x8∂5 Now, the g0-module generated by the the vectors V 1 and V 1 is not the whole g1 but a g0- module that we denote by g′′1, of sdim = 4|4. The CTS prolong (g−, g0, g1)∗ is not simple, so New simple modular Lie superalgebras 9 consider the Lie subsuperalgebra (g−, g0, g 1)∗; the superdimensions of its positive part are adig′′1 (g′′1) g 1 adg′′1 (g 1) ad (g′′1) ad (g′′1) ad (g′′1) sdim 4|4 4|4 4|4 3|2 2|1 The lowest weight vectors of the above components are precisely {V ′2 , V ′′2 , V3, V4, V5} described bellow: adig1(g1) lowest weight vectors V ′2 x1 2∂2 + 2x1x7∂4 + 2x1x8∂5 + x1x6x8∂2 + 2x1x7x8∂3 V ′′2 2x1 2∂1 + x1x2∂2 + x1x3∂3 + x1x6∂6 + x1x7∂7 + x1x8∂8 + 2x2x7∂4 + 2x2x8∂5 +2x3x6∂4 + 2x3x7∂5 + x2x6x8∂2 + 2x2x7x8∂3 + x3x6x7∂2 + x6x7x8∂7 V3 x1 2∂4 + 2x1x7x8∂5 + 2x1x6x7x8∂2 V4 x1 2x3∂2 + 2x1 2x5∂4 + x1 2x7∂6 + 2x1 2x8∂7 + x1 2x7x8∂1 + 2x1x3x7∂4 + 2x1x3x8∂5 +x1x3x6x8∂2 + 2x1x3x7x8∂3 + x1x5x7x8∂5 + x1x5x6x7x8∂2 V5 x1 2x2∂4 + 2x1 2x3∂5 + 2x1 2x6x7∂6 + x1 2x6x8∂7 + 2x1 2x7x8∂8 + 2x1 2x6x7x8∂1 +2x1x2x7x8∂5 + 2x1x3x6x7∂4 + 2x1x3x6x8∂5 + 2x1x2x6x7x8∂2 + 2x1x3x6x7x8∂3 Since none of the known simple finite dimensional Lie superalgebra over (algebraically closed) fields of characteristic 0 or > 3 has such a non-positive part in any Z-grading, it follows that Bj(3;N |5) is new. Let us investigate if Bj(3;N |5) has subalgebras — partial prolongs. (i) Denote by g′1 the g0-module generated by V 1 . We have sdim(g 1) = 2|3. The CTS partial prolong (g−1, g0, g 1)∗ gives a graded Lie superalgebra with sdim(g 2) = 2|2 and g′i = 0 for i > 3. An easy computation shows that [g−1, g 1] = g0 and [g 1] ( g 2. Since we are investigating simple Lie superalgebra, we take the simple part of (g−1, g0, g 1)∗. This simple Lie superalgebra is isomorphic to g(2, 3)/c = bj. (ii) Denote by g′′1 the g0-module generated by V 1 . We just saw that sdim(g 1) = 4|4. The CTS partial prolong (g−1, g0, g 1)∗ gives also Bj(3|5). r = (0, 0, 1). In this case, sdim(g(2, 3)−) = 4|5. Since the g(2, 3)0-module action is not faithful, we consider the quotient algebra g0 = g(2, 3)0/ann(g−1) and embed (g(2, 3)−, g0) ⊂ vect(4;N |5). The CTS prolong returns bj := g(2, 3)/c. 1.11. A description of Me(3;N |3). 1) Our first idea was to try to repeat the above construction with a suitable super version of g(2). There is only one simple super analog of g(2), namely ag(2), but our attempts [BjL] to construct a super analog of Melikyan algebra in the above way as Kuznetsov suggested [Ku1] (reproduced in [GL4]) resulted in something quite distinct from the Melikyan algebra: The Lie superalgebras we obtained, an exceptional one bj (cf. [CE, BGL1]) and a series Bj, are indeed simple but do not resemble either g(2) or Me. 2) Our other idea is based on the following observation. The anti-symmetric form (3) (f, g) := fdg = fg′dt, on the quotient space F/const of functions (with compact support) modulo constants on the 1-dimensional manifolds, has its counterpart in 1|1-dimensional case in presence of a contact structure a n d o n l y i n t h i s c a s e as follows from the description of invariant bilinear differential operators, see [KLV]. Indeed, the Lie superalgebra k(1|1) does not distinguish 10 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites between the space of volume forms (let its generator be denoted vol) and the quotient Ω1/Fα, where α = dt+ θdθ is the contact form. For any prime p therefore, on the space g−1 := O(1;N |1)/ const of “functions (with com- pact support) in one even indeterminate u and one odd, θ modulo constants”, the superanti- symmetric bilinear form (4) (f, g) := (fdg mod Fα) = 0 − f1g1)dt, where f = f0(t) + f1(t)θ and g = g0(t) + g1(t)θ and where ′ := d , is nondegenerate. Therefore, we may expect that, for p small and N = 1, the Melikyan effect will reappear. Consider p = 5 as the most plausible. We should be careful with parities. The parity of vol is a matter of agreement, let it be even. Then the integral is an odd functional but the factorization modulo Fα makes the form (4) even. (Setting p(vol) = 1̄ we make the integral an even functional and the factorization modulo Fα makes the form (4) even again.) Since the form (4) is even, we get the following realization of k(1; 1|1) ⊂ osp(5|4) ≃ k(5; 1, ..., 1|5) by generating functions of contact vector fields on the 5|5-dimensional superspace with the contact form, where the coefficients are found from the explicit values of (p̂idq̂i − q̂2dp̂i) + ξ̂jdηj + η̂jdξ̂j − θ̂dθ̂. The coordinates on this 5|5-dimensional superspace are hatted in order not to confuse them with generating functions of k(1; 1|1): gi basis elements g−2 1̂ g−1 p̂1 = t, p̂2 = t 2, q̂1 = t 3, q̂2 = t ξ̂1 = θ, ξ̂2 = tθ, θ̂ = t 2θ, η̂2 = t 3θ, η̂1 = t We explicitly have: (t, t4) = t · t3dtN = 4t4dtN = 4 = −(t4, t); (t2, t3) = t2 · t2dtN = 6t4dtN = 1 = −(t3, t2); (t4θ, θ) = − t4 · 1dtN = −1 = (θ, t4θ); (t3θ, tθ) = − t3 · tdtN = −4 = (tθ, t3θ); (t2θ, t2θ) = − t2 · t2dtN = −6 = −1. New simple modular Lie superalgebras 11 Now, let us realize k(1; 1|1) by contact fields in hatted functions: gi basis elements g−2 1̂ g−1 p̂1 = t, p̂2 = t 2, q̂2 = 4t 3, q̂1 = t ξ1 = θ, ξ2 = tθ, θ̂ = t 2θ, η2 = 4t 3θ, η1 = t g0 1 = 2 p̂1·q̂2 + 2p̂22 + 3 ξ1η2 + 3 ξ2θ̂; t = 2 p̂1q̂1 + 4 p̂2q̂2 + 4 ξ1η1 + 2 ξ2η2; t2 = 2 p̂2q̂1 + 4 q̂ 2 + 4 ξ2η1 + θ̂η2; t 3 = 3 q̂1q̂2 + 4 θ̂η1; t 4 = q̂21 + η2η1; θ = p̂1η2 + p̂2θ̂ + q̂1ξ1 + q̂2ξ2; tθ = p̂1η1 + 2 p̂2η2 + q̂1ξ2 + 2 q̂2θ̂; t2θ = p̂2η1 + q̂1θ̂ + 2 q̂2η2; t 3θ = 4 q̂1η2 + 4 q̂2η1; t 4θ = q̂1η1 The CTS prolong gives that g1 = 0. The case where p = 3 is more interesting because it will give us the series Me(3;N |3). The non-positive part is as follows: gi basis elements g−2 1̂ g−1 p̂1 = t, q̂2 = t 2, ξ1 = θ, θ̂ = tθ, η1 = t g0 1 = p̂ 1 + 2ξ̂1η̂1; t = 2 p̂1q̂1 + 2 ξ̂1η̂1; t 2 = 2 q̂21 + 2 θ̂η̂1; θ = 2 p̂1θ̂ + q̂1ξ̂1; tθ = p̂1η̂1 + q̂1θ̂; t 2θ = q̂1η̂1 The Lie superalgebra g0 is not simple because [g−1, g1] = g0\{t2θ = q̂1η̂1}. Denote g′0 := [g−1, g1] ≃ osp(1|2). The CTS partial prolong (g−, g′0)∗ seems to be very interesting. First, our computation shows that the parameter M = (M1,M2,M3) depends only on the first parameter (relative to t). Namely, M = (M1, 1, 1). For M1 = 1, 2, the super dimensions of the positive components of Bj(3;M |3) are given in the following table: M = (1, 1, 1) g′1 g 5 – – – – – sdim 2|4 4|2 2|4 3|2 0|1 M = (2, 1, 1) g′1 g 5 · · · g′14 g′15 g′16 g′17 sdim 2|4 4|2 2|4 4|2 2|4 · · · 4|2 2|4 3|2 0|1 Here we have that [g−1, g1] = g 0 and the g 1 generates the positive part. The standard criteria for simplicity ensures that Me(3;N |3) is simple. For N = (1, a, b), the lowest weight vectors 12 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites are as follows: gi lowest weight vectors V ′1 2p1 (2)η1 + 2 p1q1θ + q1 (2)ξ1 + ξ1θη1 V ′′1 2 p1q1η1 + q1 (2)θ + η1t V ′2 2 p1q1θη1 + q1 (2)ξ1η1 + tq1 (2) + tθη1 V ′′2 2 p1 (2)q1 (2) + p1 (2)θη1 + p1q1ξ1η1 + 2q1 (2)ξ1θ + t V ′3 2 tp1 (2)η1 + tp1q1θ + 2 tq1 (2)ξ1 + tξ1θη1 V ′′3 2 p1 (2)q1 (2)η1 + 2 tp1q1η1 + 2 q1 (2)ξ1θη1 + tq1 (2)θ + t(2)η1 V ′4 2 p1 (2)q1 (2)θη1 + 2 tp1q1θη1 + tq1 (2)ξ1η1 + t (2)q1 (2) + t(2)θη1 V ′5 p1 (2)q1 (2)ξ1θη1 + t (2)p1 (2)η1 + t (2)p1q1θ + 2 t (2)q1 (2)ξ1 + 2 t (2)ξ1θη1 For N = (2, a, b), the lowest weight vectors are as above together with: gi lowest weight vectors V ′′4 2 tp1 (2)q1 (2) + tp1 (2)θη1 + tp1q1ξ1η1 + 2 tq1 (2)ξ1θ + t V ′′5 2 tp1 (2)q1 (2)η1 + 2 t (2)p1q1η1 + 2 tq1 (2)ξ1θη1 + t (2)q1 (2)θ + t(3)η1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V ′15 2 t (5)p1 (2)q1 (2)ξ1θη1 + 2t (7)p1 (2)η1 + 2 t (7)p1q1θ + t (7)q1 (2)ξ1 + t (7)ξ1θη1 2 t(6)p1 (2)q1 (2)η1 + 2 t (7)p1q1η1 + 2 t (6)q1 (2)ξ1θη1 + t (7)q1 (2)θ + t(8)η1 V ′′16 2 t (6)p1 (2)q1 (2)θη1 + 2 t (7)p1q1θη1 + t (7)q1 (2)ξ1η1 + t (8)q1 (2) + t(8)θη1 V ′′17 2t (6)p1 (2)q1 (2)ξ1θη1 + 2 t (8)p1 (2)η1 + 2 t (8)p1q1θ + t (8)q1 (2)ξ1 + t (8)ξ1θη1 Let us investigate the subalgebras of Me(3;N |3) — partial prolongs: (i) Denote by g′1 the g0-module generated by V 1 . We have sdim(g 1) = 0|1 and gi = 0 for all i > 1. The CTS partial prolong (g−, g0, g 1)∗ gives a graded Lie superalgebra with the property that [g−1, g1] ≃ osp(1|2). The partial CTS prolong (g−, osp(1|2))∗ is not simple (ii) Denote by g′′1 the g0-module generated by V 1 . We have sdim(g 1) = 3|2. The CTS partial prolong (g−, g0, g 1)∗ returns brj(2; 3). 1.12. A description of Brj(4|3). We have the following realization of the non-positive part inside vect(4|3): gi the generators (even | odd) g−4 Y8 = ∂1 | Y7 = ∂5 g−3 Y6 = ∂2 | g−2 Y4 = ∂3 | Y5 = x3∂5 + x6∂1 + ∂6 g−1 Y3 = 2x2∂1 + 2x3∂2 + ∂4 | Y2 = x2∂5 + 2x4 (2)x7∂1 + x4x6∂1 + x6x7∂5 + x4x7∂2 + x6∂2 + 2x4∂6 + 2x7∂3 + 1∂7, g0 ≃ hei(0|2) ⊕ K H1 = [Z1, Y1], H2 = 2x5∂5 + x2∂2 + 2x3∂3 + 2x4∂4 + x7∂7, | Y1 = 2x3 (2)∂5 + 2x3x6∂1 + 2x5∂1 + 2x3∂6 + x7∂4, Z1 = x4 (2)∂6 + 2x4 (2)x6∂1 + x4 (2)x7∂2 + x4x7∂3 + 2x4x6x7∂5 + x1∂5 + 2x4∂7 + 2 x6∂3 The Lie superalgebra g0 is solvable, and hence the CTS prolong (g−, g0)∗ is NOT simple since g1 does not generate the positive part. Our calculation shows that the prolong does not depend on N , i.e., N = (1, 1, 1, 1). The simple part of this prolong is brj(2; 3). The sdim New simple modular Lie superalgebras 13 of the positive parts are described as follows: g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 sdim 1|1 2|2 1|2 2|2 1|1 2|2 1|1 1|1 0|1 1|1 and the lowest weight vectors are as follows: gi lowest weight vectors V ′1 2x2x3∂5 + 2x2x6∂1 + x3x4∂6 + 2 x3x6∂2 + x3x7∂3 + x4x7∂4 + x3x4 (2)x7∂1 + 2x3x4x6∂1 +2x3x4x7∂2 + 2x3x6x7∂5 + 2x2∂6 + 2x3∂7 + 2x5∂2 + 2 x6∂4 V ′2 x4 (2)∂4 + x1x3∂5 + x1x6∂1 + x3x4 (2)∂6 + 2x3x4∂7 + 2x3x6∂3 + 2x4x6∂4 + 2x6x7∂7 +2x3x4 (2)x6∂1 + x3x4 (2)x7∂2 + x3x4x7∂3 + 2x3x4x6x7∂5 + x1∂6 + 2x5∂3 V ′′2 2x2 (2)∂1 + x3 (2)∂3 + 2x2x3∂2 + x3x4∂4 + x3x5∂5 + 2x3x7∂7 + x5x6∂1 + 2x6x7∂4 + x2∂4 + x5∂6 V ′3 2x1x2∂1 + 2x1x3∂2 + x2x3∂3 + 2 x2x4∂4 + 2x2x5∂5 + x2x6∂6 + 2x2x7∂7 + 2x3x6∂7 + x4x5∂6 +2x5x6∂2 + x5x7∂3 + x4 (2)x5x7∂1 + x3x4x6∂6 + x3x4x7∂7 + x3x6x7∂3 + 2x4x5x6∂1 + 2x4x5x7∂2 +2x4x6x7∂4 + 2x5x6x7∂5 + x3x4 (2)x6x7∂1 + 2x3x4x6x7∂2 + x1∂4 + 2x5∂7 V ′′3 2x3 (2)∂7 + x3 (2)x4∂6 + 2x3 (2)∂x6∂2 + x3 (2)x7∂3 + 2x2x3 (2)∂5 + 2x2x3∂6 + 2x2x5∂1 + x2x7∂4 +2x3x5∂2 + 2x3x6∂4 + x3 (2)x4 (2)x7∂1 + 2x3 (2)x4x6∂1 + 2x3 (2)x4x7∂2 + 2x3 (2)x6x7∂5 + 2x2x3x6∂1 +x3x4x7∂4 + x5∂4 V4 2x2 (2)∂6 + 2x2 (2)x3∂5 + 2x2 (2)x6∂1 + 2x3 (2)x4 (2)∂6 + x3 (2)x4∂7 + x3 (2)x6∂3 + 2x1x3 (2)∂5 +2x1x3∂6 + 2x1x5∂1 + x1x7∂4 + 2x2x3∂7 + 2x2x5∂2 + 2x2x6∂4 + 2 x3x5∂3 + x5x6∂6 + x5x7∂7 (2)x4 (2)x6∂1 + 2x3 (2)x4 (2)x7∂2 + 2x3 (2)x4x7∂3 + 2x1x3x6∂1 + x2x3x4∂6 + 2x2x3x6∂2 +x2x3x7∂3 + x2x4x7∂4 + x3x4 (2)x7∂4 + 2x3x4x6∂4 + x3 (2)x4x6x7∂5 + x2x3x4 (2)x7∂1 + 2 x2x3x4x6∂1 +2x2x3x4x7∂2 + 2x2x3x6x7∂5 V ′′4 x1 (2)∂1 + x4 (2)x5∂6 + 2x1x3∂3 + x1x4∂4 + x1x5∂5 + 2x1x6∂6 + x1x7∂7 + 2x4x5∂7 + 2x5x6∂3 (2)x5x6∂1 + x4 (2)x5x7∂2 + 2x4 (2)x6x7∂4 + x3x4 (2)x6∂6 + x3x4 (2)x7∂7 + 2x3x4x6∂7 +x4x5x7∂3 + x3x4 (2)x6x7∂2 + x3x4x6x7∂3 + 2x4x5x6x7∂5 V ′5 x1x2∂6 + x1x3∂7 + x1x5∂2 + x1x6∂4 + 2x2x5∂3 + x5x6∂7 + x1x2x3∂5 + x1x2x6∂1 + 2x1x3x4∂6 +x1x3x6∂2 + 2x1x3x7∂3 + 2x1x4x7∂4 + x2x4 (2)x7∂4 + x2x3x4 (2)∂6 + 2x2x3x4∂7 + 2x2x3x6∂3 +2x2x4x6∂4 + 2x2x6x7∂7 + 2x4x5x6∂6 + 2x4x5x7∂7 + 2x5x6x7∂3 + 2x4 (2)x5x6x7∂1 +2x1x3x4 (2)x7∂1 + x1x3x4x6∂1 + x1x3x4x7∂2 + x1x3x6x7∂5 + 2x2x3x4 (2)x6∂1 + x2x3x4 (2)x7∂2 +x2x3x4x7∂3 + 2x3x4x6x7∂7 + x4x5x6x7∂2 + 2x2x3x4x6x7∂5 V ′6 x1 (2)∂6 + x1 (2)x3∂5 + x1 (2)x6∂1 + 2x1x5∂3 + x4 (2)x5x6∂6 + x4 (2)x5x7∂7 + x1x4 (2)x7∂4 +x1x3x4 (2)∂6 + 2x1x3x4∂7 + 2x1x3x6∂3 + 2x1x4x6∂4 + 2x1x6x7∂7 + 2x4x5x6∂7 (2)x5x6x7∂2 + 2x1x3x4 (2)x6∂1 + x1x3x4 (2)x7∂2 + x1x3x4x7∂3 + x3x4 (2)x6x7∂7 +x4x5x6x7∂3 + 2x1x3x4x6x7∂5 V ′′6 x2 (2)x3∂3 + 2x2 (2)x4∂4 + 2x2 (2)x5∂5 + x2 (2)x6∂6 + 2x2 (2)x7∂7 + 2x1x2 (2)∂1 + x1x3 (2)∂3 +x1x2∂4 + x1x5∂6 + 2x2x5∂7 + 2x3 (2)x4 (2)x6∂6 + 2x3 (2)x4 (2)x7∂7 + x3 (2)x4x6∂7 + 2x1x2x3∂2 +x1x3x4∂4 + x1x3x5∂5 + 2x1x3x7∂7 + x1x5x6∂1 + 2x1x6x7∂4 + 2x2x3x6∂7 + x2x4x5∂6 +2x2x5x6∂2 + x2x5x7∂3 + x3x4 (2)x5∂6 + 2x3x4x5∂7 + 2x3x5x6∂3 + x5x6x7∂7 (2)x4 (2)x6x7∂2 + 2x3 (2)x4x6x7∂3 + x2x4 (2)x5x7∂1 + x2x3x4x6∂6 + x2x3x4x7∂7 + x2x3x6x7∂3 +2x2x4x5x6∂1 + 2x2x4x5x7∂2 + 2x2x4x6x7∂4 + 2x2x5x6x7∂5 + 2x3x4 (2)x5x6∂1 + x3x4 (2)x5x7∂2 +2x3x4 (2)x6x7∂4 + x3x4x5x7∂3 + x2x3x4 (2)x6x7∂1 + 2x2x3x4x6x7∂2 + 2x3x4x5x6x7∂5 14 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites V ′7 x1 (2)∂4 + 2x1 (2)x2∂1 + 2x1 (2)x3∂2 + 2x1x5∂7 + x1x2x3∂3 + 2x1x2x4∂4 + 2x1x2x5∂5 + x1x2x6∂6 +2x1x2x7∂7 + 2x1x3x6∂7 + x1x4x5∂6 + 2x1x5x6∂2 + x1x5x7∂3 + 2x2x4 (2)x5∂6 + x2x4x5∂7 +x2x5x6∂3 + x1x4 (2)x5x7∂1 + x1x3x4x6∂6 + x1x3x4x7∂7 + x1x3x6x7∂3 + 2x1x4x5x6∂1 + 2x1x4x5x7∂2 +2x1x4x6x7∂4 + 2x1x5x6x7∂5 + x2x4 (2)x5x6∂1 + 2x2x4 (2)x5x7∂2 + x2x4 (2)x6x7∂4 + 2x2x3x4 (2)x6∂6 +2x2x3x4 (2)x7∂7 + x2x3x4x6∂7 + 2x2x4x5x7∂3 + x4x5x6x7∂7 + x1x3x4 (2)x6x7∂1 + 2x1x3x4x6x7∂2 +2x2x3x4 (2)x6x7∂2 + 2x2x3x4x6x7∂3 + x2x4x5x6x7∂5 V ′8 x1 (2)x3 (2)∂5 + x1 (2)x3∂6 + x1 (2)x5∂1 + 2x1 (2)x7∂4 + 2x2 (2)x5∂3 + x1x2 (2)∂6 + x1 (2)x3x6∂1 (2)x4 (2)x7∂4 + x2 (2)x3x4 (2)∂6 + 2x2 (2)x3x4∂7 + 2x2 (2)x3x6∂3 + 2x2 (2)x4x6∂4 + 2x2 (2)x6x7∂7 +x1x2 (2)x3∂5 + x1x2 (2)x6∂1 + x1x3 (2)x4 (2)∂6 + 2x1x3 (2)x4∂7 + 2 x1x3 (2)x6∂3 + x1x2x3∂7 + x1x2x5∂2 +x1x2x6∂4 + x1x3x5∂3 + 2x1x5x6∂6 + 2x1x5x7∂7 + x2x5x6∂7 + 2x2 (2)x3x4 (2)x6∂1 + x2 (2)x3x4 (2)x7∂2 (2)x3x4x7∂3 + x3 (2)x4 (2)x6x7∂7 + 2x1x3 (2)x4 (2)x6∂1 + x1x3 (2)x4 (2)x7∂2 + x1x3 (2)x4x7∂3 +2x1x2x3x4∂6 + x1x2x3x6∂2 + 2x1x2x3x7∂3 + 2x1x2x4x7∂4 + 2x1x3x4 (2)x7∂4 + x1x3x4x6∂4 +2x2x4x5x6∂6 + 2x2x4x5x7∂7 + 2x2x5x6x7∂3 + 2x3x4 (2)x5x6∂6 + 2x3x4 (2)x5x7∂7 + x3x4x5x6∂7 (2)x3x4x6x7∂5 + 2x1x3 (2)x4x6x7∂5 + 2x1x2x3x4 (2)x7∂1 + x1x2x3x4x6∂1 + x1x2x3x4x7∂2 +x1x2x3x6x7∂5 + 2x2x4 (2)x5x6x7∂1 + 2x2x3x4x6x7∂7 + x2x4x5x6x7∂2 + 2x3x4 (2)x5x6x7∂2+ 2x3x4x5x6x7∂3 V ′9 x1 (2)x2x3∂5 + x1 (2)x2x6∂1 + 2x1 (2)x3x4 (2)x7∂1 + x1 (2)x3x4x6∂1 + x1 (2)x3x6x7∂5 + 2x1x2x3x4 (2)x6∂1 +2x1x2x3x4x6x7∂5 + 2x1x4 (2)x5x6x7∂1 + x1 (2)x3x4x7∂2 + x1 (2)x3x6∂2 + x1 (2)x5∂2 +x1x2x3x4 (2)x7∂2 + x1x4x5x6x7∂2 + x2x4 (2)x5x6x7∂2 + x1 (2)x2∂6 + 2x1 (2)x3x4∂6 + 2 x1 (2)x3x7∂3 +x1x2x3x4 (2)∂6 + x1x2x3x4x7∂3 + 2x1x2x3x6∂3 + 2x1x2x5∂3 + 2x1x4x5x6∂6 + 2x1x5x6x7∂3 +x2x4 (2)x5x6∂6 + x2x4x5x6x7∂3 + x1 (2)x3∂7 + 2x1 (2)x4x7∂4 + x1 (2)x6∂4 + 2x1x2x3x4∂7 +x1x2x4 (2)x7∂4 + 2x1x2x4x6∂4 + 2x1x2x6x7∂7 + 2x1x3x4x6x7∂7 + 2x1x4x5x7∂7 + x1x5x6∂7 +x2x3x4 (2)x6x7∂7 + x2x4 (2)x5x7∂7 + 2x2x4x5x6∂7 V ′10 x1 (2)x2 (2)∂1 + 2x1 (2)x3 (2)∂3 + 2 x1 (2)x2∂4 + 2x1 (2)x5∂6 + x1 (2)x2x3∂2 + 2x1 (2)x3x4∂4 (2)x3x5∂5 + x1 (2)x3x7∂7 + 2x1 (2)x5x6∂1 + x1 (2)x6x7∂4 + x2 (2)x4 (2)x5∂6 + 2x2 (2)x4x5∂7 (2)x5x6∂3 + 2x1x2 (2)x3∂3 + x1x2 (2)x4∂4 + x1x2 (2)x5∂5 + 2x1x2 (2)x6∂6 + x1x2 (2)x7∂7 +x1x2x5∂7 + 2x2 (2)x4 (2)x5x6∂1 + x2 (2)x4 (2)x5x7∂2 + 2x2 (2)x4 (2)x6x7∂4 + x2 (2)x3x4 (2)x6∂6 (2)x3x4 (2)x7∂7 + 2x2 (2)x3x4x6∂7 + x2 (2)x4x5x7∂3 + x1x3 (2)x4 (2)x6∂6 + x1x3 (2)x4 (2)x7∂7 +2x1x3 (2)x4x6∂7 + x1x2x3x6∂7 + 2x1x2x4x5∂6 + x1x2x5x6∂2 + 2x1x2x5x7∂3 + 2x1x3x4 (2)x5∂6 +x1x3x4x5∂7 + x1x3x5x6∂3 + 2x1x5x6x7∂7 + x2 (2)x3x4 (2)x6x7∂2 + x2 (2)x3x4x6x7∂3 (2)x4x5x6x7∂5 + x1x3 (2)x4 (2)x6x7∂2 + x1x3 (2)x4x6x7∂3 + 2x1x2x4 (2)x5x7∂1 + 2x1x2x3x4x6∂6 +2x1x2x3x4x7∂7 + 2x1x2x3x6x7∂3 + x1x2x4x5x6∂1 + x1x2x4x5x7∂2 + x1x2x4x6x7∂4 + x1x2x5x6x7∂5 +x1x3x4 (2)x5x6∂1 + 2x1x3x4 (2)x5x7∂2 + x1x3x4 (2)x6x7∂4 + 2x1x3x4x5x7∂3 + 2x2x4x5x6x7∂7 +2x3x4 (2)x5x6x7∂7 + 2x1x2x3x4 (2)x6x7∂1 + x1x2x3x4x6x7∂2 + x1x3x4x5x6x7∂5 New simple modular Lie superalgebras 15 1.13. A description ofBrj(3;N |4). We have the following realization of the non-positive part inside vect(3;N |4): gi the generators (even | odd) g−3 | Y6 = ∂4 g−2 Y5 = ∂1, Y6 = ∂2, Y7 = ∂3 | g−1 | Y2 = 2x3∂4 + ∂5, Y3 = x2∂4 + x6∂1 + ∂6 Y4 = 2x1∂4 + 2x5x7∂4 + x5∂1 + x6∂2 + 2x7∂3 + ∂7 g0 ≃ hei(2|0)⊂+ KH2 H1 = [Z1, Y1], H2 = 2x1∂1 + x3∂3 + x4∂4 + x6∂6 + 2x7∂7 Y1 = 2x5x6x7∂4 + 2x1∂2 + 2x2∂3 + 2x5x6∂1 + x6x7∂3 + 2x5∂6 + 2x6∂7, Z1 = 2x2∂1 + 2x3∂2 + x6x7∂1 + 2x6∂5 + x7∂6 | The Lie superalgebra g0 is solvable with the property that [g0, g0] = hei(2|0). The CTS prolong (g−, g0)∗ is NOT simple since g1 does not generate the positive part. Our calculation shows that the prolong does not depend on N , i.e., N = (1, 1, 1, 1). The simple part of this prolong is brj. The sdim of the positive parts are described as follows: g1 g2 g3 sdim 0|3 3|0 0|2 and the lowest weight vectors are V ′1 2x1 (2)∂4 + 2x1x5x7∂4 + x1x5∂1 + x1x6∂2 + 2x1x7∂3 + x4∂3 + x1∂7 + 2x5x6∂6 + x5x7∂7 V ′2 2x1x4∂4 + x2x5x6x7∂4 + 2x4x5x7∂4 + 2x1 (2)∂1 + x1x2∂2 + x2 (2)∂3 + x2x5x6∂1 + 2x2x6x7∂3 +x4x5∂1 + x4x6∂2 + 2x4x7∂3 + 2x1x5∂5 + x1x6∂6 + x2x5∂6 + x2x6∂7 + x4∂7 + x5x6x7∂6 V ′3 x1 (2)x2∂4 + x1x2x5x7∂4 + 2x4x5x6x7∂4 + x1 (2)x6∂1 + 2x1x2x5∂1 + 2x1x2x6∂2 + x1x2x7∂3 +2x1x4∂2 + 2 x1x5x6x7∂1 + 2x2x4∂3 + 2 x4x5x6∂1 + x4x6x7∂3 + x1 (2)∂6 + 2x1x2∂7 +x1x5x6∂5 + 2x1x5x7∂6 + x2x5x6∂6 + 2x2x5x7∂7 + 2x4x5∂6 + 2x4x6∂7 V ′′3 x1 (2)x3∂4 + x1x2 (2)∂4 + x1x3x5x7∂4 + x2 (2)x5x7∂4 + x1x2x6∂1 + 2x1x3x5∂1 + 2x1x3x6∂2 +x1x3x7∂3 + 2x1x4∂1 + 2x2 (2)x5∂1 + 2x2 (2)x6∂2 + x2 (2)x7∂3 + 2x2x4∂2 + 2x2x5x6x7∂1 +2x3x4∂3 + 2 x1 (2)∂5 + x1x2∂6 + 2x1x3∂7 + x1x6x7∂6 + 2x2 (2)∂7 + x2x5x6∂5 + 2x2x5x7∂6 +x3x5x6∂6 + 2x3x5x7∂7 + x4x5∂5 + x4x6∂6 + x4x7∂7 Let us study now the case where g′0 = der0(g−). Our calculation shows that g 0 is generated by the vectors Y1, Z1, H1, H2 above together with V = 2x3∂1 + x7∂5. The Lie algebra g solvable of sdim = 5|0. The CTS prolong (g−, g′0)∗ gives a Lie superalgebra that is not simple because g′1 does not generate the positive part. Its simple part is a new Lie superalgebra that we denote by BRJ, described as follows (here also N = (1, 1, 1): g′1 adg′1(g 1) ad (g′1) ad (g′1) ad (g′1) ad (g′1) sdim 0|6 6|0 0|5 3|0 0|3 1|0 16 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites 1.14. A description ofBrj(3;N |3). We have the following realization of the non-positive part inside vect(3;N |3): gi the generators (even | odd) g−2 Y7 = ∂1 | Y5 = ∂4, Y8 = ∂5 g−1 Y1 = ∂2, Y6 = 2x2∂1 + ∂3 | Y3 = x2∂4 + x3∂5 + 2x6∂1 + ∂6 g0 H2 = [X2, Y2], H1 = x1∂1 + x3∂3 + 2x4∂4 + 2x6∂6, X4 = [X2,X2], Y4 = [Y2, Y2] | Y2 = x 2 ∂4 + x2 x3∂5 + 2x2 x6∂1 + x1∂5 + x2∂6 + x4∂1 + x6∂3 X2 = x 3 ∂5 + 2x1∂4 + x3∂6 + x5∂1 + 2x6∂2 The Lie superalgebra g0 is isomorphic to osp(1|2)⊕ K. The CTS prolong (g−, g0)∗ is NOT simple since it gives back brj(2; 3) + an outer derivation. The sdim of the positive parts are described as follows: g1 g2 g3 sdim 2|1 1|2 0|1 and the lowest weight vectors are V ′1 2 x1 x2∂1 + x2 x4∂4 + x3 x4∂5 + 2x4 x6∂1 + x1∂3 + 2x2 x3∂3 + x2 x6∂6 + x4∂6 V ′2 x 1 ∂5 + x1 x 2 ∂4 + x1 x2 x3∂5 + 2x1 x2 x6∂1 + x1 x4∂1 + 2x 3 ∂5 + 2x 2 x5∂1 + 2x4 x5∂5 + x1 x2∂6 +x1 x6∂3 + 2x 2 x3∂6 + x 2 x6∂2 + x2 x3 x6∂3 + x2 x4∂2 + x2 x5∂3 + 2x4 x6∂6 V ′3 x 1 x2∂4 + x 1 x3∂5 + 2x 1 x6∂1 + 2x1 x2 x 3 ∂5 + 2x1 x2 x5∂1 + x2 x 3 x4∂1 + 2x2 x4 x5∂4 + 2x3 x4 x5∂5 +x4 x5 x6∂1 + x 1 ∂6 + 2x1 x2 x3∂6 + x1 x2 x6∂2 + x1 x3 x6∂3 + x1 x4∂2 + x1 x5∂3 + 2x2 x 3 x6∂3 + 2x2 x3 x4∂2 +2x2 x3 x5∂3 + x2 x5 x6∂6 + 2x3 x4 x6∂6 + 2x4 x5∂6 1.15. A description ofBrj(3;N |4). We have the following realization of the non-positive part inside vect(3;N |4): gi the generators (even | odd) g−3 | Y8 = ∂4 g−2 Y4 = ∂1, Y6 = ∂2, Y7 = ∂3 | g−1 | Y2 = x3∂4 + 2x5∂1 + ∂5, Y3 = ∂6 + x5 x6∂4 + x2∂4 + x5∂2 + 2x6∂3 Y5 = x1∂4 + x5∂3 + ∂7 g0 ≃ hei(2|0)⊂+ KH2 H1 = [Z1, Y1], H2 = 2x1∂1 + x2∂2 + x4∂4 + x5∂5 + 2x7∂7 Y1 = x1∂2 + x2∂3 + x5 x6∂3 + 2x5∂6 + 2x6∂7 Z1 = 2x5 x6 x7∂4 + 2x2∂1 + x3∂2 + x5 x6∂1 + 2x6 x7∂3 + 2x6∂5 + x7∂6 | The Lie superalgebra g0 is solvable with the property that [g0, g0] = hei(2|0). The CTS prolong (g−, g0)∗ is NOT simple since g1 does not generate the positive part. Our calculation shows that the prolong does not depend on N , i.e., N = (1, 1, 1, 1). The simple part of this prolong is 3)brj(2; 3). The sdim of the positive parts are described as follows: g1 g2 g3 sdim 0|3 3|0 0|2 New simple modular Lie superalgebras 17 and the lowest weight vectors are V ′1 2x1 x3∂4 + x 2 ∂4 + x2 x5 x6∂4 + x1 x5∂1 + x2 x5∂2 + 2 x2 x6∂3 + 2x3 x5∂3 + x4∂3 + 2x1∂5 + x2∂6 + 2 x3∂7 + 2x5 x7∂7 V ′2 2x1 x4∂4 + x 1 ∂1 + 2x1 x2∂2 + 2x 2 ∂3 + 2x2 x5 x6∂3 + 2x4 x5∂3 + 2x1 x5∂5 + x1 x7∂7 + x2 x5∂6 + x2 x6∂7 + 2x4∂7 V ′3 x1 x 3 ∂4 + 2x 2 x3∂4 + 2x2 x3 x5 x6∂4 + 2x1 x3 x5∂1 + 2x1 x4∂1 + x 2 x5∂1 + 2x2 x3 x5∂2 + x2 x3 x6∂3 + 2x2 x4∂2 +2x2 x5 x6 x7∂3 + x 3 x5∂3 + 2x3 x4∂3 + x1 x3∂5 + x1 x6 x7∂6 + 2x 2 ∂5 + 2x2 x3∂6 + 2x2 x5 x6∂5 + x2 x5 x7∂6 +2x2 x6 x7∂7 + x 3 ∂7 + x3 x5 x7∂7 + x4 x5∂5 + x4 x6∂6 + x4 x7∂7 Let us study now the case where g′0 = der0(g−). The Lie algebra g 0 is solvable of sdim = 5|0. The CTS prolong (g−, g 0)∗ gives a Lie superalgebra that is not simple because g 1 does not generate the positive part. Its simple part is a new Lie superalgebra that we had denoted by BRJ, described as follows (here also N = (1, 1, 1): g′1 adg′1(g 1) ad (g′1) ad (g′1) ad (g′1) ad (g′1) sdim 0|6 6|0 0|5 3|0 0|3 1|0 1.16. Constructing Melikyan superalgebras. Denote by F1/2 := O(1; 1) dx the space of semi-densities (weighted densities of weight 1 ). For p = 3, the CTS prolong of the triple (K,Π(F1/2), cvect(1; 1))∗ gives the whole k(1;N |3). For p = 5, let us realize the non-positive part in k(1;N |5): gi the generators g−2 1 g−1 Π(F1/2) g0 ∂1 ←→ 4 ξ1η2 + ξ2θ, x1∂1 ←→ 2 ξ1η1 + ξ2η2, x 1 ∂1 ←→ 2 ξ2η1 + 3 θη2, x 1 ∂1 ←→ 2 θη1 1 ∂1 ←→ 2 η2η1, t The CTS prolong gives that gi=0 for all i > 0. Consider now the case of (K,Π(F1/2), cvect(2; 1))∗, where p = 3. The non-positive part is realized in k(1;N |9) as follows: gi the generators g−2 1 g−1 Π(F1/2) g0 ∂1 ←→ 2 ξ1η3 + x2θ + 2 ξ3η4, x1∂1 ←→ ξ1η1 + ξ2η2 + 2 ξ4η4, x21∂1 ←→ ξ3η1 + ξ4η3 + θη2, ∂2 ←→ 2 ξ1η2 + ξ2ξ4 + ξ3θ, x2∂2 ←→ ξ1η1 + ξ3η3 + ξ4η4, x22∂2 ←→ ξ2η1 + θη3 + 2 η4η2, x1x2∂1 ←→ ξ2η1 + η4η2, x1x2∂2 ←→ ξ3η1 + 2 ξ4η3, x21x2∂1 ←→ θη1 + 2 η3η2, x21x2∂2 ←→ ξ4η1, x1x22∂1 ←→ 2 η4η1, x1x22∂2 ←→ θη1 + η3η2, x21x22∂1 ←→ η3η1, 2∂2 ←→ η2η1, t The CTS prolong (g−, g0)∗ gives a Lie superalgebra that is not simple with the property that sdim(g1) = 0|4 and gi = 0 for all i > 1. The generating functions of g1 are ξ2η2η1 + 2 ξ3η3η1 + ξ4η4η1 + θη3η2, 2 ξ4η3η1 + θη2η1, θη3η1 + η4η2η1, η3η2η1. 1.17. Defining relations of the positive parts of brj(2; 3) and brj(2; 5). For the presentations of the Lie superalgebras with Cartan matrix, see [GL1, BGL1]. The only non- trivial part of these relations are analogs of the Serre relations (both the straightforward 18 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites ones and the ones different in shape). Here they are: brj(2; 3); sdim brj(2; 3) = 10|8. 1) [[x1, x2] , [x2, [x1, x2]]] = 0, [[x2, x2] , [[x1, x2] , [x2, x2]]] = 0. 2) ad3x2(x1) = 0, [[x1, x2] , [[x1, x2] , [x1, x2]]] = 0, [[x2, [x1, x2]] , [[x1, x2] , [x2, [x1, x2]]]] = 0. 3) ad3x1(x2) = 0, [x2, [x1, [x1, x2]]]− [[x1, x2], [x1, x2]] = 0, [[x1, x2], [x2, x2]] = 0. brj(2; 5); sdim brj(2; 5) = 10|12. 1) [[x2, [x1, x2]] , [x2, [x1, x2]]] = 2 [[x1, x2] , [[x1, x2] , [x2, x2]]], [[x2, x2] , [[x1, x2] , [x2, x2]]] = 0, [[x2, [x1, x2]] , [[x1, x2] , [x2, [x1, x2]]]] = 0. 2 )ad4x2(x1) = 0, [[x2, [x1, x2]] , [x2, [x2, [x1, x2]]]] = 0, [[[x1, x2] , [x1, x2]] , [[x1, x2] , [x2, [x1, x2]]]] = 0. References [BKK] Benkart, G.; Kostrikin, A. I.; Kuznetsov, M. I. 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(NY), 141 (2007) no.4, 1390–98; math.RT/0606682 [BGL1] Bouarroudj S., Grozman P., Leites D., Cartan matrices and presentations of Elduque and Cunha simple Lie superalgebras; MPIMiS preprint 124/2006 (www.mis.mpg.de) [BGL2] Bouarroudj S., Grozman P., Leites D., Cartan matrices and presentations of the exceptional simple Elduque Lie superalgebra; MPIMiS preprint 125/2006 (www.mis.mpg.de) [BGL4] Bouarroudj S., Grozman P., Leites D., Infinitesimal deformations of the simple modular Lie super- algebras with Cartan matrices for p = 3. IN PREPARATION [BGL5] Bouarroudj S., Grozman P., Leites D., Simple modular Lie superalgebras with Cartan matrices. IN PREPARATION [C] Cartan É., Über die einfachen Transformationsgrouppen, Leipziger Berichte (1893), 395–420. Reprinted in: Œuvres complètes. Partie II. (French) [Complete works. Part II] Algèbre, systèmes différentiels et problèmes d’équivalence. [Algebra, differential systems and problems of equivalence] Second edition. 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VUZ) 48 (2004), no. 3, 13–22. http://arxiv.org/abs/math/0606682 http://personal-homepages.mis.mpg.de/clarke/Tensor-Calculations.tar.gz http://arxiv.org/abs/math/0605379 http://arxiv.org/abs/math/0608191 http://arxiv.org/abs/math/0512654 New simple modular Lie superalgebras 19 [FH] Fulton, W., Harris, J., Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp [GK] Gregory, T.; Kuznetsov, M. On depth-three graded Lie algebras of characteristic three with classical reductive null component. Comm. Algebra 32 (2004), no. 9, 3339–3371 [Gr] Grozman P., SuperLie, http://www.equaonline.com/math/SuperLie [GL1] Grozman P., Leites D., Defining relations for classical Lie superalgebras with Cartan matrix, Czech. J. Phys., Vol. 51, 2001, no. 1, 1–22; arXiv: hep-th/9702073 [GL2] Grozman P., Leites D., SuperLie and problems (to be) solved with it. 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IN PREPARATION [LSh] Leites D., Shchepochkina I., Classification of the simple Lie superalgebras of vector fields, preprint MPIM-2003-28 (http://www.mpim-bonn.mpg.de) [Ssol] Sergeev, A. Irreducible representations of solvable Lie superalgebras. Represent. Theory 3 (1999), 435–443; math.RT/9810109 [Shch] Shchepochkina I., How to realize Lie algebras by vector fields. Theor. Mat. Fiz. 147 (2006) no. 3, 821–838; arXiv: math.RT/0509472 [Sk] Skryabin, S. M. New series of simple Lie algebras of characteristic 3. (Russian. Russian summary) Mat. Sb. 183 (1992), no. 8, 3–22; translation in Russian Acad. Sci. Sb. Math. 76 (1993), no. 2, 389–406 [S] Strade, H. Simple Lie algebras over fields of positive characteristic. I. Structure theory. de Gruyter Expositions in Mathematics, 38. Walter de Gruyter & Co., Berlin, 2004. viii+540 pp. [St] Steinberg, R. Lectures on Chevalley groups. Notes prepared by John Faulkner and Robert Wilson. Yale University, New Haven, Conn., 1968. iii+277 pp. http://arxiv.org/abs/math/0202193 http://arxiv.org/abs/math/9810109 20 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites [Vi] Viviani F., Deformations of Simple Restricted Lie Algebras I, II. math.RA/0612861, math.RA/0702499; Deformations of the restricted Melikian Lie algebra,math.RA/0702594; Restricted simple Lie algebras and their infinitesimal deformations, math.RA/0702755 [WK] Weisfeiler, B. Ju.; Kac, V. G. Exponentials in Lie algebras of characteristic p. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 762–788. [Y] Yamaguchi K., Differential systems associated with simple graded Lie algebras. Progress in differential geometry, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, 1993, 413–494 1Department of Mathematics, United Arab Emirates University, Al Ain, PO. Box: 17551; [email protected], 2Equa Simulation AB, Stockholm, Sweden; [email protected], 3MPIMiS, Inselstr. 22, DE-04103 Leipzig, Germany, on leave from Department of Mathe- matics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, SE-106 91 Stockholm, Sweden; [email protected], [email protected] http://arxiv.org/abs/math/0612861 http://arxiv.org/abs/math/0702499 http://arxiv.org/abs/math/0702594 http://arxiv.org/abs/math/0702755 1. Introduction References

0704.0132

Counterflow of electrons in two isolated quantum point contacts

Counterflow of electrons in two isolated quantum point contacts V.S. Khrapai,1, 2 S. Ludwig,1 J.P. Kotthaus,1 H.P. Tranitz,3 and W. Wegscheider3 Center for NanoScience and Department für Physik, Ludwig-Maximilians-Universität, Geschwister-Scholl-Platz 1, D-80539 München, Germany Institute of Solid State Physics RAS, Chernogolovka, 142432, Russian Federation Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93040 Regensburg, Germany We study the interaction between two adjacent but electrically isolated quantum point contacts (QPCs). At high enough source-drain bias on one QPC, the drive-QPC, we detect a finite electric current in the second, unbiased, detector-QPC. The current generated at the detector-QPC always flows in the opposite direction than the current of the drive-QPC. The generated current is maximal, if the detector-QPC is tuned to a transition region between its quantized conductance plateaus and the drive-QPC is almost pinched-off. We interpret this counterflow phenomenon in terms of an asymmetric phonon-induced excitation of electrons in the leads of the detector-QPC. PACS numbers: 73.23.-b, 73.23.Ad, 73.50.Lw The state of a confined quantum system is modified by interactions with an external field (or with exter- nal sources of energy). In semiconductor nanostruc- tures the energy and quasi-momentum of electrons act- ing as probe are strongly influenced by the environment, e. g. via electron-electron or electron-phonon interaction. If driven out of equilibrium, Coulomb forces establish the local equilibrium within the electron system whereas electron-phonon interactions dominate the energy ex- change with the environment [1]. Drag experiments in semiconductor nanostructures provide a tool to study the effect of external electrons or phonons onto a probe elec- tron system. Current drag between parallel two-dimensional (2D) electron layers has been investigated in GaAs/AlGaAs bilayer systems. At small interlayer separations, ob- servations are consistent with the Coulomb drag phe- nomenon [2]. At larger separations virtual-phonon ex- change has been invoked to explain the data [3]. A neg- ative sign of a current drag between 2D and 3D electron gases in GaAs was explained by the Peltier effect [4]. At high filling factors in a perpendicular magnetic field a sign change of the longitudinal drag between parallel 2D layers was found as a function of the imbalance of the electron density in the two layers [5, 6]. Interactions between two lateral quantum wires in GaAs have been investigated in Ref. [7]. The observed frictional drag, strongly oscillating as a function of the one-dimensional (1D) subband occupation, was inter- preted in terms of Coulomb interaction between two Luttinger liquids. Recently, the observation of negative Coulomb drag between two disordered lateral 1D wires in GaAs in perpendicular magnetic fields was reported [8]. Here we report on a novel interaction effect between two neighboring quantum point contacts (QPCs), em- bedded in mutually isolated electric circuits. When a strong current is flowing through the partially transmit- ting drive-QPC, we detect a small current in the sec- ond, unbiased, detector-QPC. The detector current flows in the opposite direction of the drive current and shows a nonlinear dependence on the source-drain bias of the drive-QPC. It oscillates as a function of the detector- QPC transmission. We suggest an explanation of this counterflow phenomenon in terms of asymmetric phonon- induced excitation of ballistic electrons in the leads of the detector-QPC. Our samples are prepared on a GaAs/AlGaAs het- erostructure containing a two-dimensional electron gas 90 nm below the surface, with an electron density of nS = 2.8 × 10 11 cm−2 and a low-temperature mobility of µ = 1.4 × 106 cm2/Vs. An AFM micrograph of the split-gate nanostructure, produced with e-beam lithogra- phy, is shown in the left inset of Fig. 1. The negatively biased central gate C divides the electron system into two separate circuits, and prevents leakage currents between them. Two QPCs are defined on the upper and lower side of the central gate, respectively, by biasing gates 8 and 3. Other gates are grounded if not stated otherwise. The right inset of Fig. 1 shows a sketch of the coun- terflow experiment. We use separate electric circuits for the (upper) drive-QPC and (lower) detector-QPC. A dc bias voltage, Vdrive, is applied to the left lead of the drive- QPC, while the right lead is grounded. A current-voltage amplifier with an input voltage-offset of about 10 µV is connected to the right lead of the detector-QPC. Its left lead is always maintained at the same offset potential in order to assure zero voltage drop across the detector- QPC. In both circuits, a positive sign of the current cor- responds to electrons flowing to the left. For differential counterflow conductance measurements, the drive bias is modulated at a frequency of 21 Hz and the resulting ac current component in the detector circuit is measured with lock-in detection in the linear response regime. All measurements are performed in a dilution refrigerator at an electron temperature below 150 mK. The experimen- tal results are the same if detector and drive QPC are interchanged. First, we characterize the QPCs using a standard dif- ferential conductance measurement. Figure 1 displays the differential conductances of both QPCs in linear re- http://arxiv.org/abs/0704.0132v2 drive -0.7 -0.5 -0.3 FIG. 1: Conductance of the drive-QPC (dashed line) and the detector-QPC (solid line) in the linear response regime as a function of respective gate voltages V8 and V3. Symbols on the detector-QPC curve mark the V3 values used for coun- terflow conductance measurement presented in Fig. 2b. Left inset: AFM micrograph of the metal gates on the surface of the heterostructure (bright tone). Crossed squares mark contacted 2DEG regions. The scale bar equals 1 µm. Right inset: sketch of the counterflow measurement. The directions of currents are shown for the case of Vdrive > 0. sponse, measured as a function of the respective gate voltage V3, or V8. At low gate voltages, the QPCs are pinched-off and the conductance is close to zero. With increasing gate voltage, 1D channels successively open up [9]. For both QPCs we observe three conductance plateaus approximately quantized in units of G0 = 2e With high bias spectroscopy [10] we find the spacing between the two lowest subbands to be approximately 4 meV (3 meV) for the drive (detector) QPC. The half- width of the energy window for opening a 1D subband is ∆ ≈ 0.5 meV in both QPCs. Having characterized the QPCs, we turn to counter- flow measurements. Fig. 2a shows the dc counterflow current, Icf, through the detector-QPC and the differ- ential counterflow conductance, gcf ≡ dIcf/dVdrive, as a function of the bias on the drive-QPC. Here, the drive- QPC is tuned to nearly half a conductance quantum Gdrive = G0/2, while the detector-QPC is in the pinch-off regime (i.e. the lowest 1D subband bottom is well above the Fermi level) with Gdet ≃ 10 GΩ −1. Surprisingly, for |Vdrive| & 1 mV, a finite current is observed in the un- biased detector circuit. The direction of Icf is opposite to that of the drive-QPC current Idrive. The dc coun- terflow current is a threshold-like, nearly odd function of Vdrive. Correspondingly, the differential counterflow con- ductance is negative and a nearly even function of Vdrive. The sign of gcf expresses a phase shift of π between the applied ac modulation of Vdrive and the detected ac com- ponent of the counterflow current. Figures 2c and 2d show the absolute value of Icf for the nearly pinched-off detector as a function of the voltage on gate 8, which tunes the drive-QPC transmission. The -4 -2 0 2 4 -0.04 -0.02 -0.7 -0.6 -0.5 -0.4 drive (mV) (d) G FIG. 2: (a) - Icf and gcf for the nearly pinched-off detector- QPC as a function of Vdrive. (b) - gcf measured for a set of Gdet values marked by according symbols in Fig. 1. (c,d) - Absolute value of Icf as a function of the drive-QPC gate voltage V8, for Vdrive = ±2.25 mV (c) and Vdrive = ±4 mV(d). Also shown is the drive-QPC’s conductance in linear response (c) and its differential conductance at Vdrive = ±4 mV (d). Solid (dotted) lines correspond to Vdrive < 0 (> 0). In (a),(b) gates 7 and 9 are grounded, while in (c),(d) V7 = V9 = −0.4 V. The drive bias modulation used to measure gcf is 92 µV rms. corresponding drive-QPC differential conductance curves are also shown. For not too high Vdrive (Fig. 2c), a non- zero counterflow current is only detected in the region between pinch-off and the first conductance plateau of the drive-QPC. For higher Vdrive (Fig. 2d) Icf increases superlinearly with Vdrive at its maximum and remains finite at higher gate voltages V8. Since the source bias effects the potential distribution near the constriction, the nonlinear 1/2 conductance plateau of the drive-QPC shifts when changing Vdrive [11]. This causes the shift of the extrema on Fig. 2d as well as the asymmetry of gcf in Fig. 2a when reversing the bias. We proceed to study the counterflow effect in the -0.6 -0.5 -0.4 -0.3 -0.10 -0.05 -0.65 -0.60 -0.55 =17 kΩ =17 kΩ FIG. 3: (a)- gcf as a function of the detector-QPC gate volt- age V3. Filled symbols correspond to the gcf measured at a finite external resistance Rext = 17 kΩ, while open symbols show the corrected counterflow conductance Rext = 0 (see text). Also shown are the transmission functions Tn(1 − Tn) for the three lowest 1D subbands of the detector-QPC (dashed lines), scaled to fit the corrected data. During the gcf mea- surement the drive bias is modulated with a 230 µV rms signal about the mean value Vdrive = +2.05 mV. (b) - Normalized gcf (symbols as in (a)) and transmission function of the lowest 1D detector-QPC subband 4T0(1−T0) (dashed line) as a function of V3. The scale bar shows a gate voltage interval correspond- ing to a change of the 1D subband energy by 0.5 meV. Inset: Sketch of possible scattering processes of nonequilibrium elec- trons and holes at a partially transmitting detector-QPC. regime of a more opened detector-QPC. Figure 2b plots gcf [12] as a function of Vdrive for several values of Gdet between 0 and G0 (marked with the same symbols in Fig. 1). The qualitative appearance of gcf(Vdrive) is in- dependent of Gdet. However, the amplitude of gcf is a strongly non-monotonic function of the detector trans- mission. The counterflow conductance reaches its maxi- mum for Gdet ≈ G0/2 and decreases rapidly with further increasing Gdet. Note that the absolute value of gcf is small, corresponding to a maximal ratio of the counter- flow and drive currents |Icf/Idrive| . 10 In Fig. 3a gcf is plotted as a function of V3, controlling the detector transmission. Vdrive and V8 are adjusted for maximal gcf and kept fixed. Confirming the trend seen in Fig. 2b, the measured gcf (solid symbols) strongly oscil- lates with increasing V3 and displays three pronounced maxima before the detector-QPC is fully opened. The position of the n-th maximum (n = 0,1,2) is close to the value of V3, where Gdet/G0 ≃ n + 0.5 (Fig. 1). Here, the energy EnS of the bottom of the n-th 1D subband -4 -2 0 2 4 �������������� drive (mV) FIG. 4: Drive bias dependence of the counterflow current through the pinched-off detector-QPC for the drive-QPC formed with gate 6 (dotted line) or gate 10 (solid line). The detector-QPC conductance is about Gdet = 5 GΩ −1. The drive-QPCs are tuned to provide the maximal effect. Insets: sketches of the two counterflow measurements. The directions of currents are shown for the case of Vdrive > 0. of the detector-QPC aligns with the Fermi level of the leads EnS ≃ EF. In contrast, gcf is close to zero for fully transmitting 1D channels (Gdet/G0 ≃ n+1). The over- all magnitude of gcf decreases with increasing V3, hence Gdet. This is caused by a finite series resistance Rext of the external circuit, which results in a measured gcf lower than the case for an ideal ammeter [13]. The corrected counterflow conductance, gidealcf ≡ gcf · (1 + Rext · Gdet), corresponding to Rext = 0, is shown in Fig. 3a with open symbols. The corrected maxima are roughly equal in size and symmetric. Moreover, the shape of the n-th maxi- mum compares quite well with the corresponding func- tion of the equilibrium transmission Tn(1−Tn), extracted from the detector conductance data Tn ≡ Gdet/G0 − n (dashed lines in Fig. 3a). In Figure 3b we plot the normalized gcf and the trans- mission function 4T0(1− T0) on a logarithmic scale near the detector pinch-off. In the pinch-off regime (i.e. for T0 ≪ 1) the transmission probability of a QPC is ex- pressed as T0(E) ∝ exp([E − E S]/∆) [11]. Here E is the kinetic energy of current carrying electrons and ∆ is the half-width of the energy window for opening a 1D- subband. The energy E0S of the detector-QPC is con- trolled by gate 3 via E0S ∝ −|e|V3. This explains a nearly exponential drop of the transmission function with de- creasing V3 (Fig. 3b). In contrast, the measured gcf drops considerably slower and remains finite even where the detector-QPC is already pinched-off in equilibrium. This experimentally observed excess contribution of the normalized gcf versus T0(EF ) signals that the counter- flow current carrying electrons are excited well above the Fermi level. Converting the shift in gate voltage (see the bar in Fig. 3b) to energy, we find a characteristic excita- tion energy of E∗ ≈ 0.5 meV. This is consistent with a recently reported 1 meV bandwidth excitation provided by the drive-QPC for electrons in a nearby double-dot quantum ratchet [14]. Next we study the counterflow effect between spatially shifted QPCs. Figure 4 shows Icf through the nearly pinched-off detector-QPC as a function of the bias on the drive-QPC, which is formed either with gate 10 or gate 6, while gate 8 is now grounded (Fig. 1). Despite the shift of the drive-QPC position relative to the detector- QPC by about 300 nm, the odd drive bias dependence of the counterflow current found in Fig. 2 is practically preserved. This indicates that the excitation of electrons in one of the leads of the detector-QPC is not restricted to the close vicinity of the drive-QPC. The oscillations of the counterflow conductance gcf in Fig. 3 are reminiscent of thermopower oscillations that have been investigated on individual QPCs [15, 16]. This suggests that Icf is caused by an energetic imbalance across the detector-QPC. If the bottom of the n-th 1D- subband of the detector-QPC is well separated from the Fermi-energy in comparison to the characteristic excita- tion energy, i. e. if |EnS −EF| ≫ E ∗, this subband is either fully transmitting (Tn(E) = 1) or closed (Tn(E) = 0). In both cases electrons (holes) excited by E∗ above (below) EF are equally transmitted and gcf = 0. In contrast, if EnS ≃ EF excited electrons are more likely transmit- ted than excited holes (see inset of Fig. 3b), resulting in gcf 6= 0. The energetic imbalance across the detector-QPC we propose to be caused by phonon-based energy transfer from the drive-QPC. The excess energy of carriers in- jected across the drive-QPC is mainly relaxed by emis- sion of acoustic phonons. We consider the drive-QPC in the non-linear regime near pinch-off where µS − µD ≫ ∆ and the transmission probability is strongly energy- dependent (the source and drain leads are defined so that their chemical potentials satisfy µS > µD). In this case electrons injected into the drain lead have an excess en- ergy of about e|Vdrive| ≡ µS−µD whereas the source lead remains essentially in thermal equilibrium [17]. Hence acoustic phonons are predominantly generated in the drain lead of the drive-QPC. Because of this asymme- try electron-hole pairs are excited preferentially in the adjacent lead of the detector-QPC [18]. This gives rise to Icf directed opposite to the current through the drive- QPC (and gcf < 0). The data in Fig. 2 clearly show, that the counterflow effect is only observed in the non-linear regime of the drive-QPC. For a rough estimate we consider injected electrons with a momentum relaxation time of 60 ps limited by elastic scattering and an energy relaxation time of 1 ns [19, 20]. Assuming isotropic phonon emission we estimate an energy transfer ratio which can account for the observed value of Icf/Idrive within one order of mag- nitude. In summary, the current in a strongly biased drive- QPC generates a current flowing in the opposite direc- tion through an adjacent unbiased detector-QPC. This counterflow current is maximal in between the conduc- tance plateaus of the detector-QPC. The effect is most pronounced near pinch-off of the drive-QPC, where it behaves strongly non-linear. We interpret the results in terms of an asymmetric phonon-based energy transfer. The authors are grateful to V.T. Dolgopolov, A.W. Holleitner, C. Strunk, F. Wilhelm, I. Favero, A.V. Khaetskii, N.M. Chtchelkatchev, A.A. Shashkin, D.V. Shovkun and P. Hänggi for valuable discussions and to D. Schröer and M. Kroner for technical help. We thank the DFG via SFB 631, the BMBF via DIP-H.2.1, the Nanosystems Initiative Munich (NIM) and VSK the A. von Humboldt foundation, RFBR, RAS, and the pro- gram ”The State Support of Leading Scientific Schools” for support. [1] V. F. Gantmakher and Y. B. Levinson, in Carrier Scat- tering in Metals and Semiconductors (North-Holland, Amsterdam, 1987) [2] T.J. Gramila et al., Phys. Rev. Lett. 66, 1216 (1991) [3] T.J. Gramila et al., Phys. Rev. B 47, 12957 (1993); H. Rubel et al., Semicond. Sci. Technol. 10, 1229 (1995) [4] B. Laikhtman et al., Phys. Rev. B. 41, 9921 (1990) [5] X.G. Feng et al., Phys. Rev. Lett. 81, 3219 (1998) [6] J.G.S. Lok et al., Phys. Rev. B 63, 041305 (2001) [7] P. Debray et al., J. Phys.: Condens. Matter 13, 3389, (2001); P. Debray et al., Semicond. Sci. Technol. 17, R21, (2002) [8] M. Yamamoto et al., Science 313, 204, (2006) [9] B.J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988); D.A. Wharam et al., J. Phys. C 21, L209 (1988) [10] A. Kristensen et al., Phys. Rev. B 62, 10950 (2000) [11] L.I. Glazman, A.V. Khaetskii JETP Lett. 48 591 (1988) [12] For increasing Gdet the noises in the detector circuit in- crease, making the dc measurements very difficult. [13] The input resistance of the I-V amplifier, the ohmic con- tacts and wiring resistances result in Rext = 17 kΩ. The validity of the above formula has been checked by apply- ing an additional 47 kΩ resistor in series to Rext. [14] V.S. Khrapai et al., Phys. Rev. Lett. 97, 176803 (2006) [15] L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992); H. van Houten et al., Semicond. Sci. Technol. 7, B215 (1992) [16] A.S. Dzurak et al., J. Phys.: Condens. Matter 5, 8055, (1993) [17] A. Palevski et al., Phys. Rev. Lett. 62, 1776 (1989); for asymmetric heat production in 3D point contacts see U. Gerlach-Meyer, H.J. Queisser Phys. Rev. Lett. 51, 1904 (1983) [18] |Icf| is reduced for Vdrive < 0 (> 0) and the drive-QPC shifted to the lh (rh) side of the detector-QPC (Fig. 4). This is understood in terms of absorption of phonons in both leads of the detector-QPC. [19] B.K. Ridley, Rep. Prog. Phys 54, 169 (1991) [20] A.A. Verevkin et al., Phys. Rev. B 53, R7592 (1996)

0704.0133

PAH emission and star formation in the host of the z~2.56 Cloverleaf QSO

Accepted for publication in ApJL PAH emission and star formation in the host of the z∼2.56 Cloverleaf QSO D. Lutz1, E. Sturm1, L.J. Tacconi1, E. Valiante1, M. Schweitzer1 H. Netzer2, R. Maiolino3, P. Andreani4, O. Shemmer5, S. Veilleux6 ABSTRACT We report the first detection of the 6.2µm and 7.7µm infrared ‘PAH’ emis- sion features in the spectrum of a high redshift QSO, from the Spitzer-IRS spec- trum of the Cloverleaf lensed QSO (H1413+117, z∼2.56). The ratio of PAH features and rest frame far-infrared emission is the same as in lower luminosity star forming ultraluminous infrared galaxies and in local PG QSOs, supporting 1Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, 85741 Garching, Germany [email protected], [email protected], [email protected], [email protected], [email protected] 2School of Physics and Astronomy and the Wise Observatory, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel [email protected] 3INAF, Osservatorio Astronomico di Roma, via di Frascati 13, 00040 Monte Porzio Catone, Italy [email protected] 4ESO, Karl-Schwarzschildstraße 2, 85748 Garching, Germany [email protected] 5Department of Astronomy and Astrophysics, 525 Davey Laboratory, Pennsylvania State University, University Park, PA 16802, USA [email protected] 6Department of Astronomy, University of Maryland, College Park, MD 20742-2421, USA [email protected] http://arxiv.org/abs/0704.0133v1 – 2 – a predominantly starburst nature of the Cloverleaf’s huge far-infrared luminosity (5.4 × 1012L⊙, corrected for lensing). The Cloverleaf’s period of dominant QSO activity (LBol ∼ 7 × 10 13L⊙) is coincident with an intense (star formation rate ∼ 1000M⊙yr −1) and short (gas exhaustion time ∼ 3× 107yr) star forming event. Subject headings: galaxies: active, galaxies: starburst, infrared: galaxies 1. Introduction Redshifts ∼2.5 witness both the ‘quasar epoch’ with peak number density of luminous accreting black holes (e.g. Schmidt et al. 1995) and the peak in number of the most intense star forming events as traced by the submillimeter galaxy population (Chapman et al. 2005), suggestive of a relation of the two phenomena. Detailed evolutionary connections between massive starbursts and QSOs have been discussed for many years (e.g. Sanders et al. 1988; Norman & Scoville 1988) and form an integral part of some recent models of galaxy and merger evolution (e.g. Granato et al. 2004; Springel et al. 2005; Hopkins et al. 2006). Phases of intense star formation coincident with the active phase of the quasars are a natural postulate of such models but have been exceedingly difficult to confirm and quantify due to the effects of the powerful AGN outshining tracers of star formation at most wavelengths. Perhaps the strongest constraint on the potential significance of star formation in QSOs comes from the far-infrared part of their spectral energy distribution (SED). Indeed, this far- infrared emission has been interpreted as due to star formation (e.g. Rowan-Robinson 1995), but alternative models successfully ascribe it to AGN heated dust, by postulating a dust distribution in which relatively cold dust at large distance from the AGN has a significant covering factor, for example in a warped disk configuration (Sanders et al. 1989). Additional diagnostics are needed to break this degeneracy. CO surveys of local QSOs (e.g. Evans et al. 2001; Scoville et al. 2003; Evans et al. 2006) have produced a significant number of detections of molecular gas reservoirs that might power star formation. Depending on the adopted ‘star formation efficiency’ SFE=LFIR/LCO the detected gas masses may be sufficient or not for ascribing the QSO far-infrared emission to star formation. Optical studies have identified significant ‘post-starburst’ stellar populations in QSOs (Canalizo & Stockton 2001; Kauffmann et al. 2003). On the other hand, Ho (2005) suggested low star formation in QSOs, perhaps actively inhibited by the AGN, on the basis of observations of the [OII] 3727Å line. We have used the much less extinction sensitive mid- infrared PAH emission features to infer that in a sample of local (PG) QSOs, star formation is sufficient to power the observed far-infrared emission (Schweitzer et al. 2006). – 3 – The observational situation remains complex for high redshift QSOs. Metallicity studies of the broad-line region suggest significant enrichment by star formation (e.g. Hamann & Ferland 1999; Shemmer et al. 2004) but may not be representative for the host as a whole. Submm and mm studies of luminous radio quiet QSOs have produced significant individual detec- tions of dust emission of some QSOs, as well as statistical detection of the entire popula- tion (e.g. Omont et al. 2003; Priddey et al. 2003; Barvainis & Ivison 2002). These suggest potential starburst luminosities up to and exceeding 1013L⊙. CO studies have detected large gas reservoirs in many high-z QSOs (see summaries in Solomon & Vanden Bout 2005; Greve et al. 2005). Emission from high density molecular gas tracers has been detected in some of the brightest systems (Barvainis et al. 1997; Solomon et al. 2003; Carilli et al. 2005; Riechers et al. 2006; Garćıa-Burillo et al. 2006; Guélin et al. 2007) and may well originate in dense high pressure star forming regions, but AGN effects on chemistry and molecular line excitation could also play a role (e.g. Maloney et al. 1996). Finally, the [CII] 157µm rest wavelength fine structure line was detected in the z=6.42 quasar SDSS J114816.64+525150.3 (Maiolino et al. 2005) at a ratio to the rest frame far-infrared emission similar to the ratio in local ULIRGs, consistent with massive star formation. We have initiated a program extending the use of mid-infrared PAH emission as star formation tracer to high redshift QSOs. In this Letter, we use Spitzer mid-infrared spectra to detect and quantify star formation in one of the brightest and best studied z∼2.5 QSOs, the lensed Cloverleaf (H1413+117, Hazard et al. 1984; Magain et al. 1988). We adopt Ωm = 0.3, ΩΛ = 0.7 and H0 = 70 kms −1 Mpc−1. 2. Observations and Results We obtained low resolution (R∼ 60 − 120) mid-infrared spectra of the Cloverleaf QSO using the Spitzer infrared spectrograph IRS (Houck et al. 2004) in staring mode on July 24, 2006, at J2000 target position RA 14h15m46.27s, DEC +11d29m43.40s. The IRS aperture includes all lensed images. 30 cycles of 120sec integration time per nod position were taken in the LL1 (19.5 to 38.0 µm) and 15 cycles in the LL2 (14.0 to 21.3 µm) module, leading to effective on-source integration times of 2 and 1 hours, respectively. We use the pipeline 14.4.0 processed basic calibrated data, own deglitching and coaddition procedures, and SMART (Higdon et al. 2004) for extraction. When combining the two orders into the final spectrum, we scaled the LL2 spectrum by a factor 1.02 for best match in the overlapping region. Fig. 1 shows the IRS spectrum embedded into the infrared to radio SED of the Clover- leaf, and Fig. 2 the IRS spectrum proper, together with the location of key features in the corresponding rest wavelength range. The rest frame mid-infrared emission is dominated by – 4 – a strong continuum, approximately flat in νFν , due to dust heated by the powerful active nucleus to temperatures well above those reached in star forming regions. Superposed on this continuum are emission features, which we identify with the 6.2µm and 7.7µm aromatic ‘PAH’ emission features normally detected in star-forming galaxies over a very wide range of properties. As expected for a Type 1 AGN, there are no indications for the ice (6µm) or silicate (9.6µm) absorptions seen in heavily obscured galaxies. None of the well-known emission lines in this wavelength range is bright enough to be significantly detected in this low resolution spectrum, although we cannot exclude a contribution of [NeVI] 7.64µm to the 7.7µm feature. Adopting standard mid-infrared low resolution diagnostics (Genzel et al. 1998; Laurent et al. 2000), the weak PAH features on top of a strong continuum agree with the notion that the Cloverleaf is energetically dominated by its AGN. The detection of PAH features with several mJy peak flux density in a z∼2.6 galaxy, however, implies intense star formation, which we discuss in conjunction with other properties of the Cloverleaf. By fitting a Lorentzian superposed on a local polynomial continuum, we measure a flux of 1.52×10−21Wcm−2 for the 6.2µm feature, with a S/N of 6. The 7.7µm feature is more difficult to quantify. Schweitzer et al. (2006) have discussed PAHs as star formation indicators in local (PG) QSOs, PAHs are also detected in the average QSO spectrum of Hao et al. (2007). The AGN continuum of those QSOs shows superposed silicate emission features at & 9µm (see also Siebenmorgen et al. 2005; Hao et al. 2005). If PAH emission is additionally present, the PAH features partly ‘fill in’ the minimum in the AGN emission before the onset of the silicate feature (see Fig. 2 of Schweitzer et al. (2006)), causing a seemingly flat overall spectrum. In reality, there is simultaneous presence of AGN continuum, the 6.2-8.6µmPAH complex, silicate emission, and more PAH emission at longer wavelengths. From inspection of Fig. 2, a similar co-presence of PAH and silicate emission is observed for the Cloverleaf. We note that the presence of silicate emission in the luminous Cloverleaf Type 1 QSO, other high z Type 1 QSOs (Maiolino et al., in prep) as well as in luminous Type 2 QSOs (Sturm et al. 2006; Teplitz et al. 2006) has implications for the location of this cool silicate component (∼200K, Hao et al. 2005) in unified AGN schemes. We leave further discussion of the properties of silicates to a future paper. For the 7.7µm feature of the Cloverleaf, we adopt a flux of 6.1×10−21Wcm−2. This flux was determined by scaling a PAH template (ISO spectrum of M82, Sturm et al. 2000) to the measured Cloverleaf 6.2µm feature flux, and then fitting three lorentzians to represent the 6.2, 7.7, and 8.6 features plus a local polynomial continuum to this template. Similar Lorentzian fits were also used for local comparison objects discussed below. Brandl et al. (2006) have quantified the scatter of the 6.2 to 7.7µm flux ratio in starbursts, with 0.07 in the log this scatter indicates the modest uncertainty induced by tying the longer wavelength features to the 6.2µm one. The result of subtracting the scaled M82 template from the Cloverleaf spectrum is indicated – 5 – in Fig. 2, and shows a combination of continuum and silicate emission very similar to local QSOs. Directly measuring the 7.7µm flux by fitting a single Lorentzian plus local continuum to the Cloverleaf spectrum gives a ∼40% lower feature flux, which would be a systematic underestimate because of the complexity of the underlying continuum/silicates discussed above. 3. Intense star formation in the host of the Cloverleaf QSO The Cloverleaf SED (Fig. 1) shows strong rest frame far-infrared emission in addition to the AGN heated dust emitting in the rest frame mid-infrared. Weiß et al. (2003) decom- posed the SED into two modified blackbodies of temperature 50 and 115K, the rest frame far-infrared (40-120µm) luminosity of 5.4 × 1012L⊙ is dominated by the colder component and could largely originate in star formation. Comparison of PAH and far-infrared emission can shed new light on this question. The bolometric (rather than rest-frame far-infrared) luminosity of the Cloverleaf will still be dominated by the AGN. We estimate LBol extrap- olating from the observed rest frame 6µm continuum which for a mid-infrared spectrum with weak PAH but strong continuum will be AGN dominated (Laurent et al. 2000). Using LBol ∼ 10× νLν(6µm) based on an Elvis et al. (1994) radio-quiet QSO SED, the AGN lu- minosity is ∼ 7 × 1013L⊙. A similar estimate ∼ 5 × 10 13L⊙ is obtained from the rest frame optical (observed near-infrared; Barvainis et al. 1995) continuum, tracing the AGN ionizing continuum, and the same global SED. Schweitzer et al. (2006) have measured PAH emission in local QSOs and compared the PAH to far-infrared emission ratio to that for starbursting ULIRGs, i.e. those among a larger ULIRG sample not showing evidence for dominant AGN and not having absorption domi- nated mid-infrared spectra. Fig. 3 places the Cloverleaf on their relation between 7.7µm PAH luminosity and far-infrared luminosity. L(PAH)/L(FIR) is 0.014 for the Cloverleaf, very close to the mean value for the 12 starburst-dominated ULIRGs of < L(PAH)/L(FIR) >= 0.0130. The scatter of this relation is 0.2 in the log for these 12 comparison ULIRGs, indicating the minimum uncertainty of extrapolating from the PAH to far-infrared emission. The Clover- leaf thus extends the relation between PAH and far-infrared luminosity for the local QSOs and ULIRGs to ∼5 times larger luminosities. Its PAH emission is consistent with an ex- tremely luminous starburst of ULIRG-like physical conditions powering essentially all of the rest frame far-infrared emission. Teplitz et al. (2006) present the IRS spectrum of the lensed FIR-bright Type 2 AGN IRAS F10214+4724 at similar redshift. They report a marginal feature at 6.2µm rest wave- length which they do not interpret as PAH given the lack of a 7.7µm maximum. Given – 6 – the strength of silicate emission in this target, PAH emission may be present in the blue wing of the silicate feature without producing a maximum, and such a component may be suggested by comparing their Fig. 1 with the later onset of silicate emission in the spectra of local QSOs. The tentative 6.2µm peak in IRAS F10214+4724 has similar peak height as the Cloverleaf PAH feature, in line with our interpretation and the similar rest frame FIR fluxes of the two objects. With ∼5-10% of its total luminosity originating in the rest frame far-infrared and by star formation, the Cloverleaf is within the range of local QSOs, and not a pronounced infrared excess object. Specifically, its ratio of FIR to total luminosity and the ratio of rest frame far- infrared (60µm) to mid-infrared (6µm) continuum are about twice those of the Elvis et al. (1994) radio-quiet QSO SED. Adopting the conclusion of Schweitzer et al. (2006) that star formation already dominates the FIR emission of local PG QSOs and considering the modest FIR ‘excess’ of the Cloverleaf compared to the Elvis et al. (1994) SED then suggests only a small AGN contribution to its FIR luminosity. Other z∼2 QSOs may have lower ratios of FIR and total luminosity, and conversely larger AGN contributions to their more modest FIR emission, though. After correcting for lensing, the Cloverleaf submm flux is a factor ∼2 above the typical bright z∼2 QSOs of Priddey et al. (2003) whose rest frame B magnitudes in addition are typically brighter than the delensed Cloverleaf. Submillimeter galaxies host starbursts of similar luminosity as the Cloverleaf, at similar redshift. Lutz et al. (2005) and Valiante et al. (2007) have obtained IRS mid-infrared spec- tra of 13 SMGs with median redshift 2.8, finding mostly starburst dominated systems. A comparison can be made between PAH peak flux density and flux density at rest wavelength 222µm which is obtained with minimal extrapolation from observed SCUBA 850µm fluxes. Combining the 7.7µm feature peak of 5.1mJy (Fig. 2) with a ν3.5 extrapolation of the SCUBA flux of Barvainis & Ivison (2002) places the Cloverleaf at Log(SPAH7.7/S222µm) ∼ −1.2, near the center of the distribution of this quantity for the SMGs of Valiante et al. (2007, their Fig.4). Like the SMGs, the Cloverleaf appears to host a scaled up ULIRG-like starburst, but with superposition of a much more powerful AGN, also in comparison to the gas mass. Tracers of high density gas, in particular HCN but also HCO+ have been detected in a few high redshift QSOs including the Cloverleaf (Barvainis et al. 1997; Solomon et al. 2003; Riechers et al. 2006). Their ratio to far-infrared emission is similar to the one for Galactic dense star forming regions, and has been used to argue for dense, high pressure star forming regions dominating the far-infrared luminosity of these QSOs as well as of local ULIRGs (e.g. Solomon et al. 2003). Intense HCN emission is observed also from X-ray dominated regions close to AGN (e.g. Tacconi et al. 1994), and there is ongoing debate as to the possible contributions of chemistry and excitation in X-ray dominated regions, and other effects like – 7 – radiative pumping, to the emission of dense molecular gas tracers in ULIRGs and QSOs (Kohno 2005; Imanishi et al. 2006; Graćıa-Carpio et al. 2006). Unlike HCN, PAH emission is severely reduced in X-ray dominated regions close to AGN (Voit 1992) and provides an independent check of the effects of the AGN on the molecular gas versus the role of the host and its star formation. In a scenario where XDRs dominate the strong HCN emission and the hosts PAH emission, reproducing the consistent ratios of these quantities to rest-frame far-infrared over a wide range of far-infrared luminosities would thus require a considerable amount of finetuning. In contrast, these consistent ratios are a natural implication if all these components are dominated by ULIRG-like dense star formation. Our detection of PAH emission is strong support to a scenario in which the Cloverleaf QSO coexists with intense star formation. Applying the Kennicutt (1998) conversion from infrared luminosity to star formation rate to LFIR = 5.4× 10 12L⊙ suggests a star formation rate close to 1000 M⊙yr −1, which can be maintained for a gas exhaustion timescale of only 3×107yr, for the molecular gas mass inferred by Weiß et al. (2003). At this time resolution, the period of QSO activity coincides with what likely is the most significant star forming event in the history of the Cloverleaf host. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a con- tract with NASA. Support for this work was provided by NASA under contracts 1287653 and 1287740 (S.V.,O.S.). We thank the referee for helpful comments. 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Cloverleaf properties Quantity Value Reference Redshift z 2.55784 Weiß et al. (2003) Amplification µL 11 Venturini & Solomon (2003) F(PAH 6.2µm) 1.5× 10−21Wcm−2 this work F(PAH 7.7µm) 6.1× 10−21Wcm−2 this work L(PAH 7.7µm)a 7.6× 1010L⊙ this work L(40-120µm)a 5.4× 1012L⊙ Weiß et al. (2003) M(H2) a 3.0× 1010M⊙ Weiß et al. (2003) LBol(QSO) ∼ 7× 1013L⊙ this work, 10× νLν(6µm) aCorrected for lensing amplification 11 and to our adopted cosmology Ωm = 0.3, ΩΛ = 0.7 and H0 = 70 kms −1 Mpc−1 (DL=20.96 Gpc). – 12 – Fig. 1.— Infrared to radio spectral energy distribution for the Cloverleaf QSO. The IRS spectrum (continuous line) is supplemented by photometric data from the literature (Barvainis et al. 1995; Alloin et al. 1997; Barvainis & Lonsdale 1997; Hughes et al. 1997; Benford 1999; Rowan-Robinson 2000; Solomon et al. 2003; Weiß et al. 2003). The ISOCAM- CVF spectrum of Aussel et al. (1998) is indicated by the short dotted line. The ISO 12µm flux appears too high while the other mid-infrared data are consistent within plausible cali- bration uncertainties. – 13 – Fig. 2.— IRS spectrum of the Cloverleaf QSO. The PAH emission features as well as the expected locations of strong spectral lines in this wavelength range are marked. The dotted line shows the spectrum after the subtraction of a PAH template (spectrum of M82, see also bottom of figure), redshifted and scaled to the measured strength of the Cloverleaf 6.2µm PAH feature. Note that the noise in IRS low resolution spectra increases strongly from ∼33µm towards the long wavelength end. – 14 – Fig. 3.— Relation of 7.7µm PAH luminosity and rest frame FIR luminosity for the Cloverleaf and for local PG QSOs and starbursting ULIRGs from Schweitzer et al. (2006). Introduction Observations and Results Intense star formation in the host of the Cloverleaf QSO

0704.0134

Causal dissipative hydrodynamics for QGP fluid in 2+1 dimensions

Causal dissipative hydrodynamics for QGP fluid in 2+1 dimensions A. K. Chaudhuri∗ Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata 700 064, India (Dated: February 21, 2013) In 2nd order causal dissipative theory, space-time evolution of QGP fluid is studied in 2+1 di- mensions. Relaxation equations for shear stress tensors are solved simultaneously with the energy- momentum conservation equations. Comparison of evolution of ideal and viscous QGP fluid, ini- tialized under the same conditions, e.g. same equilibration time, energy density and velocity profile, indicate that in a viscous dynamics, energy density or temperature of the fluid evolve slowly, than in an ideal fluid. Cooling gets slower as viscosity increases. Transverse expansion also increases in a viscous dynamics. For the first time we have also studied elliptic flow of ’quarks’ in causal viscous dynamics. It is shown that elliptic flow of quarks saturates due to non-equilibrium correction to equilibrium distribution function, and can not be mimicked by an ideal hydrodynamics. PACS numbers: 47.75.+f, 25.75.-q, 25.75.Ld I. INTRODUCTION One of the most important discoveries in Relativistic Heavy ion collider (RHIC) at Brokhaven National Lab- oratory is the large elliptic flow in non-central Au+Au collisions [1, 2, 3, 4] . Elliptic flow measures the momen- tum anisotropy of produced particles and is quantified by the 2nd harmonic of the azimuthal distribution, v2(pT ) =< cos(2φ) >= dyd2pT cos(2φ)dφ dyd2pT (1.1) Elliptic flow is naturally explained in hydrodynamics. Hydrodynamic pressure is built up from rescattering of secondaries, and pressure gradients drive the subsequent collective motion. In non-central Au+Au collisions, ini- tially, the reaction zone is asymmetric (almond shaped). The pressure gradient is large in one direction and small in the other. The asymmetric pressure gradients gener- ates the elliptic flow. Naturally, in a central collision, reaction zone is symmetric and elliptic flow vanishes. Observed elliptic flow then give the strongest indication that in non-central Au+Au collisions, a collective QCD matter is produced. Whether the formed matter can be identified as the much sought after Quark-Gluon Plasma (QGP) as predicted in Lattice QCD simulations [5] is presently debatable. Ideal hydrodynamics has been partly successful in ex- plaining the observed elliptic flow, quantitatively [6]. El- liptic flow of identified particles, up to pT ∼1.5 GeV are well reproduced in ideal hydrodynamics. Ideal hydrody- namics also explains the transverse momentum spectra of identified particles (up to pT ∼ 1.5 GeV). Success of ideal hydrodynamics in explaining bulk of the data [6], together with the string theory motivated lower limit of ∗E-mail:[email protected] shear viscosity η/s ≥ 1/4π [7, 8] has led to a paradigm that in Au+Au collisions, a nearly perfect fluid is created. However, the paradigm of ”perfect fluid” produced in Au+Au collisions at RHIC need to be clarified. As in- dicated above, the ideal hydrodynamics is only partially successful and in a limited pT range (pT ≤1.5 GeV) [9]. The transverse momentum spectra of identified particles also starts to deviate form ideal fluid dynamics prediction beyond pT ≈ 1.5 GeV. Experimentally determined HBT radii are not reproduced in the ideal fluid dynamic mod- els, the famous ”HBT puzzle” [10]. It also do not repro- duce the experimental trend that elliptic flow saturates at large transverse momentum. These shortcomings of ideal fluid dynamics indicate greater importance of dis- sipative effects in the pT ranges greater than 1.5 GeV or in more peripheral collisions. Indeed, ideal fluid is a concept, never realized in nature. As suggested in string theory motivated models [7, 8], QGP viscosity could be small, η/s ≥ 1/4π, nevertheless it is non-zero. It is im- portant to study the effect of viscosity, even if small, on space-time evolution of QGP fluid and quantify its effect. This requires a numerical implementation of relativistic dissipative fluid dynamics. Furthermore, if QGP fluid is formed in heavy ion collisions, it has to be charac- terized by measuring its transport coefficients, e.g. heat conductivity, bulk and shear viscosity. Theoretically, it is possible to obtain those transport coefficients in a ki- netic theory model. However, in the present status of theory, the goal can not be achieved immediately, even more so for a strongly interacting QGP (sQGP). Alter- natively, one can use the experimental data to obtain a ”phenomenological” limit of transport coefficients of sQGP. It will also require a numerical implementation of relativistic dissipative fluid dynamics. There is another incentive to study dissipative hydrodynamics. Ideal hy- drodynamics depends on the assumption of local equilib- rium. Before local equilibrium is attained, the system has to pass through a non-equilibrium stage, where (if non- equilibrium effects are small) dissipative hydrodynamics may be applicable. Indeed, we can explore early times of fluid evolution better in a dissipative hydrodynamics. http://arxiv.org/abs/0704.0134v2 mailto:[email protected] Theory of dissipative relativistic fluid has been formu- lated quite early. The original dissipative relativistic fluid equations were given by Eckart [11] and Landau and Lif- shitz [12]. They are called 1st order theories. Formally, relativistic dissipative hydrodynamics are obtained from an expansion of entropy 4-current, in terms of dissipative fluxes. In 1st order theories, entropy 4-current contains terms linear in dissipative quantities. 1st order theory of dissipative hydrodynamics suffer from the problem of causality violation. Signal can travel faster than light. Causality violation is unwarranted in any theory, even more in a relativistic theory. The problem of causal- ity violation is removed in the Israel-Stewart’s 2nd order theory of dissipative fluid [13]. In 2nd order theory, ex- pansion of entropy 4-current contains terms 2nd order in dissipative fluxes. However, these leads to complications that dissipative fluxes are no longer function of the state variables only. They become dynamic. The space of ther- modynamic variables has to be extended to include the dissipative fluxes (e.g. heat conductivity, bulk and shear viscosity). Even though 2nd order theory was formulated some 30 years back, significant progress towards its numer- ical implementation has only been made very recently [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. At the Cyclotron Centre, Kolkata, we have developed a code ”AZHYDRO- KOLKATA” to simulate the hydrodynamic evolution of dissipative QGP fluid. Presently only dissipative ef- fect included is the shear viscosity. Some results of AZHYDRO-KOLKATA, for first order dissipative hydro- dynamics have been published earlier [19, 20, 21]. In the present paper, for the first time, we will present some results for 2nd order dissipative hydrodynamics in 2+1 dimensions. In the present paper, we will consider effect of dissipation in the QGP phase only. Effect of phase transition will be studied in a later publication. The paper is organized as follows: In section II we briefly review relativistic dissipative fluid dynamics. In section III we derive the relevant equations in 2+1 di- mension (assuming boost-invariance). Required inputs e.g. the equation of state, viscosity coefficient and initial conditions are discussed in section IV. Simulation re- sults from AZHYDRO-KOLKATA are shown in section V. In section VII we compare the transverse momentum spectra and elliptic flow of quarks in ideal and viscous dynamics. The concluding section IX summarizes our results. II. DISSIPATIVE FLUID DYNAMICS In this section, I briefly discuss the phenomenological theory of dissipative hydrodynamics. More detailed ex- position can be found in [13]. A simple fluid, in an arbitrary state, is fully speci- fied by primary variables: particle current (Nµ), energy- momentum tensor (T µν) and entropy current (Sµ) and a number of additional (unknown) variables. Primary variables satisfies the conservation laws; µ = 0, (2.1) µν = 0, (2.2) and the 2nd law of thermodynamics, µ ≥ 0. (2.3) In relativistic fluid dynamics, one defines a time-like hydrodynamic 4-velocity, uµ (normalized as u2 = 1). One also define a projector, ∆µν = gµν − uµuν , orthog- onal to the 4-velocity (∆µνuν = 0). In equilibrium, an unique 4-velocity (uµ) exists such that the particle den- sity (n), energy density (ε) and the entropy density (s) can be obtained from, Nµeq = nuµ (2.4) T µνeq = εu µuν − p∆µν (2.5) Sµeq = suµ (2.6) An equilibrium state is assumed to be fully specified by 5-parameters, (n, ε, uµ) or equivalently by the thermal potential, α = µ/T (µ being the chemical potential) and inverse 4-temperature, βµ = uµ/T . Given a equation of state, s = s(ε, n), pressure p can be obtained from the generalized thermodynamic relation, Sµeq = pβ µ − αNµeq + βλT λµeq (2.7) Using the Gibbs-Duhem relation, d(pβµ) = Nµeqdα − T λµeq dβλ, following relations can be established on the equilibrium hyper-surface Σeq(α, β dSµeq = −αdNµeq + βλdT λµeq (2.8) In a non-equilibrium system, no 4-velocity can be found such that Eqs.2.4,2.5,2.6 remain valid. Tensor de- composition leads to additional terms, Nµ = Nµeq + δN µ = nuµ + V µ (2.9) T µν = T µνeq + δT = [εuµuν − p∆µν ] + Π∆µν + πµν +(Wµuν +W νuµ) (2.10) Sµ = Sµeq + δS µ = suµ + Φµ (2.11) The new terms describe a net flow of charge V µ = ∆µνNν , heat flow, W µ = (ε + p)/nV µ + qµ (where qµ is the heat flow vector), and entropy flow Φµ. Π = µν − p is the bulk viscous pressure and πµν = (∆µσ∆ντ +∆νσ∆µτ − 1 ∆µν∆στ ]Tστ is the shear stress tensor. Hydrodynamic 4-velocity can be chosen to elimi- nate either V µ (the Eckart frame, uµ is parallel to particle flow) or the heat flow qµ (the Landau frame, uµ is par- allel to energy flow). In relativistic heavy ion collisions, central rapidity region is nearly baryon free and Lan- dau’s frame is more appropriate than the Eckart’s frame. Dissipative flows are transverse to uµ and additionally, shear stress tensor is traceless. Thus a non-equilibrium state require 1+3+5=9 additional quantities, the dissi- pative flows Π, qµ (or V µ) and πµν . In kinetic theory, Nµ and T µν are the 1st and 2nd moment of the distri- bution function. Unless the function is known a-priori, two moments do not furnish enough information to enu- merate the microscopic states required to determine Sµ, and in an arbitrary non-equilibrium state, no relation exists between, Nν , T µν and Sµ. Only in a state, close to a equilibrium one, such a relation can be established. Assuming that the equilibrium relation Eq.2.8 remains valid in a ”near equilibrium state” also, the entropy cur- rent can be generalized as, Sµ = Sµeq + dS µ = pβµ − αNµ + βλT λµ +Qµ (2.12) where Qµ is an undetermined quantity in 2nd order in deviations, δNµ = Nµ − Nµeq and δT µν = T µν − T µνeq . Detail form of Qµ is constrained by the 2nd law ∂µS 0. With the help of conservation laws and Gibbs-Duhem relation, entropy production rate can be written as, µ = −δNµ∂µα+ δT µν∂µβν + ∂µQµ (2.13) Choice of Qµ leads to 1st order or 2nd order theories of dissipative hydrodynamics. In 1st order theories the simplest choice is made, Qµ = 0, entropy current con- tains terms up to 1st order in deviations, δNµ and δT µν. Entropy production rate can be written as, µ = ΠX − qµXµ + πµνXµν (2.14) where, X = −∇.u; Xµ = ∇ − uν∂νuµ and Xµν = ∇<µuν>. The 2nd law, ∂µS µ ≥ 0 can be satisfied by postulat- ing a linear relation between the dissipative flows and thermodynamic forces, Π = −ζθ, (2.15) qµ = −λ nT ∇µ(µ/T ), (2.16) πµν = 2η∇<µuν> (2.17) where ζ, λ and η are the positive transport coefficients, bulk viscosity, heat conductivity and shear viscosity re- spectively. In 1st order theories, causality is violated. If, in a given fluid cell, at a certain time, thermodynamic forces vanish, corresponding dissipative fluxes also vanish instantly. Vi- olation of causality is unwanted in any theory, even more so in relativistic theory. Causality violation of dissipative hydrodynamics is corrected in 2nd order theories [13]. In 2nd order theories, entropy current contain terms up to 2nd order in the deviations, Qµ 6= 0. The most general Qµ containing terms up to 2nd order in deviations can be written as, Qµ = −(β0Π2−β1qνqν+β2πνλπνλ) (2.18) As before, one can cast the entropy production rate (T∂µS µ) in the form of Eq.2.14. Neglecting the terms involving dissipative flows with gradients of equilib- rium thermodynamic quantities (both are assumed to be small) and demanding that a linear relation exists be- tween the dissipative flows and thermodynamic forces, following relaxation equations for the dissipative flows can be obtained, Π = −ζ(θ + β0DΠ) (2.19) qµ = −λ ) − β1Dqµ (2.20) πµν = 2η ∇<µuν> − β2Dπµν , (2.21) where D = uµ∂µis the convective time derivative. Unlike in the 1st order theories, in 2nd order theories, dynamical equations control the dissipative flows. Even if thermo- dynamic forces vanish, dissipative flows do not vanish instantly. Before we proceed further, it may be mentioned that the parameters, α and βλ are not connected to the actual state (Nµ, T µν). The pressure p in Eq.2.12 is also not the ”actual” thermodynamics pressure, i.e. not the work done in an isentropic expansion. Chemical potential α and 4-inverse temperature βλ has meaning only for the equilibrium state. Their meaning need not be extended to non-equilibrium states also. However, it is possible to fit a fictitious ”local equilibrium” state, point by point, such that pressure p in Eq.2.12 can be identified with the thermodynamic pressure, at least up to 1st order. The conditions of fit fixes the underlying non-equilibrium phase-space distribution. III. (2+1)-DIMENSIONAL VISCOUS HYDRODYNAMICS WITH LONGITUDINAL BOOST INVARIANCE Complete dissipative hydrodynamics is a numerically challenging problem. It requires simultaneous solution of 14 partial differential equations (5 conservation equations and 9 relaxation equations for dissipative flows). We re- duce the problem to solution of 6 partial differential equa- tions (3 conservation equations and 3 relaxation equa- tions). In the following, we will study boost-invariant evolution of baryon free QGP fluid, including the dissi- pative effect due to shear viscosity only. Shear viscosity is the most important dissipative effect. For example, in a baryon free QGP, heat conduction is zero and we can disregard Eq.2.20. Bulk viscosity is also zero for the QGP fluid (point particles) and Eq.2.19 can also be neglected. Shear pressure tensor has 5 independent components but the assumption of boost invariance reduces the number of independent components to three. For a baryon free fluid, we can also disregard the conservation equation Eq.2.1. With the assumption of boost-invariance, energy- momentum conservation equation ∂µT µη = 0 become re- dundant and only three energy-momentum conservation equations are required to be solved. -10 -5 0 5 10 X (fm) -10 -5 0 5 10 _______ η/s=0.08 _______ η/s=0.135 _______ η/s=0 -10 -5 0 5 10 energy density in x-y plane, τ =2.6 fm. FIG. 1: (color online). Constant energy density contours in x-y plane at τ=2.6 fm. The black lines are for ideal fluid (η/s=0). The red and blue lines are for viscous fluid with ADS/CFT and perturbative estimate of viscosity, η/s=0.08 and 0.135. Heavy ion collisions are best described in (τ, x, y, η) coordinates, where τ = t2 − z2 is the longitudinal proper time and η = 1 ln t+z is the space-time rapid- ity. r⊥ = (x, y) are the usual cartisan coordinate in the plane, transverse to the beam direction. Relevant equations concerning this coordinate transformations are given in the appendix A. Explicit equations for energy-momentum conservation in (τ ,x,y,η) coordinates are given in the appendix B. We note that unlike in ideal fluid, in viscous fluid dynam- ics, conservation equations (see Eqs.B1-B3) contain addi- tional pressure gradients due to shear viscosity. Both T τx and T τy components of energy-momentum tensor now evolve under additional pressure gradients. The right- most term of Eq.B3 also indicate that in viscous dynam- ics, longitudinal pressure is effectively reduced (note that the πηη component is negative). Since pressure can not be negative, shear viscosity is limited by the condition, p+ τ2πηη ≥ 0. As evident from the Eqs.B1-B3, in boost-invariant dis- sipative hydrodynamics, with shear viscosity taken into account, fluid evolution depends only on seven compo- nents of the shear stress tensors. They are πττ , πτx, πτy, πxx, πyy, πxy and πηη. However, all the seven com- ponents are not independent. Tracelessness, transver- sality to uµ and the assumption of boost-invariance re- duces the independent components to three. Presently, we choose πxx πyy and πxy as the independent compo- nents. Relaxation equations for the independent compo- nents are given in the appendix C (see Eqs.C4-C6). They are solved simultaneously with the three energy momen- tum conservation equations Eqs.B1-B3, with inputs as discussed below. -10 -5 0 5 10 X (fm) -10 -5 0 5 10 _______ η/s=0.08 _______ η/s=0.135 _______ η/s=0 -10 -5 0 5 10 energy density in x-y plane, τ =8.6 fm FIG. 2: (color online). same as Fig.1 but at time τ=8.6 fm. IV. EQUATION OF STATE, VISCOSITY COEFFICIENT AND INITIAL CONDITIONS A. Equation of state One of the most important inputs of a hydrodynamic model is the equation of state. Through this input, the macroscopic hydrodynamic models make contact with the microscopic world. In the present demonstrative cal- culation we will show results for the QGP phase only. In the QGP phase, we use the simple equation of state, p = 1 ε, with energy density given as, gqgpT 4 (4.1) where gqgp = ggluon+ gquark is the degeneracy factor for QGP. ggluon = 2(helicity) × 8(color) is the degeneracy factor for gluons and gquark = 2(spin)× 3(color)× 2(q+ q̄) ×Nf is the degeneracy factor for Nf flavored quarks. For Nf ≈ 2.5, the degeneracy factor is gqgp = 42.25 B. Shear viscosity coefficient Shear viscosity coefficient (η) of QGP or sQGP is quite uncertain. In a strongly coupled QGP, shear viscosity can not be computed. Recently, using the ADS/CFT correspondence [7, 8] shear viscosity of a strongly coupled gauze theory, N=4 SUSY YM, has been evaluated, η = N2c T 3 and the entropy is given by s = π N2c T 3. Thus in the strongly coupled field theory, ADS/CFT ≈ 0.08, (4.2) Shear viscosity is quite uncertain in perturbative QCD also. At high temperature, shear viscosity, in leading log, can be written as [24, 25], η = κ g4 ln g−1 , (4.3) where g is the strong coupling constant. The leading log shear viscosity coefficient κ depend on the number of fermion flavors (Nf ). For example, for two flavored QGP, κ = 86.47 and κ = 106.7 for a three flavored QGP. With entropy density of QGP, s = π gqgpT 3. For two flavored QGP and αs ≈0.5, the ratio of viscosity over the entropy, in the perturbative regime is estimated as, ≈ 0.135, (4.4) For lower αs , perturbative estimation of η/s could be even higher. Shear viscosity can also be expressed in terms of sound attenuation length, Γs, defined as, (4.5) Γs is equivalent to mean free path and for a valid hy- drodynamic description Γs/τ << 1, i.e. mean free path is much less than the system size. Initial conditions of the fluid must be chosen carefully such that the validity condition Γs/τ << 1 remains valid initially as well as at later time also. In the present work, we have treated vis- cosity as a parameter. To explore the effect of viscosity, we have used both the ADS/CFT estimate η/s=0.08 and perturbative estimate η/s=0.135. We have also run the code with a higher value of viscosity η/s=0.2. C. Initial conditions Solution of Eqs.B1-B3 require initial conditions, the initial time τi, the transverse distribution of energy den- sity ε(x, y) and the velocities vx(x, y) and vy(x, y). Fol- lowing [6], initial transverse energy density is parame- terized geometrically. At an impact parameter ~b, trans- verse distribution of wounded nucleons NWN (x, y,~b) and -10 -5 0 5 10 X axis Title -0.78 -0.56 -0.33 -0.11 -10 -5 0 5 10 contour plot of Vx at τ=8.6 fm X axis Title -10 -5 0 5 10 X axis Title FIG. 3: (color online). contours of constant vx in x-y plane at τ=8.6 fm. The black lines are for ideal fluid (η/s=0). The red and blue lines are for viscous fluid with ADS/CFT and perturbative estimate of viscosity, η/s=0.08 and 0.135. -10 -5 0 5 10 X axis Title -0.78 -0.56 -0.33 -0.11 -10 -5 0 5 10 contour plot of Vy at τ=8.6 fm X axis Title -10 -5 0 5 10 X axis Title FIG. 4: (color online). contours of constant vy in x-y plane at τ=8.6 fm. The black lines are for ideal fluid (η/s=0). The red and blue lines are for viscous fluid with ADS/CFT and perturbative estimate of viscosity, η/s=0.08 and 0.135. of binary NN collisions NBC(x, y,~b) to are calculated in a Glauber model. A collision at impact parameter ~b is assumed to contain 25% hard scattering (proportional to number of binary collisions) and 75% soft scattering (proportional to number of wounded nucleons). Trans- verse energy density profile at impact parameter~b is then obtained as, ε(x, y,~b) = ε0(0.75×NWN (x, y,~b) + 0.25×NBC(x, y,~b)) (4.6) with central energy density ε0=30GeV/fm −3. The equi- libration time is chosen as τi=0.6 fm [6]. The initial ve- locities vx and vy are assumed to be zero initially. ______ η/s=0.08 ______ η/s=0 ______ η/s=0 Temperature in τ-x plane. y=0. X (fm)0 2 4 6 8 10 FIG. 5: (color online). Constant temperature contour in x − τ plane, for fixed y=0. The black, red and blue lines are for ideal, viscous fluid with η/s=0.08 and viscous fluid with η/s=0.135. In dissipative hydrodynamics, one requires initial con- ditions for the viscous pressures also. Due to longitudinal boost invariance of the problem, we assume that viscous pressures have attained their boost-invariant values at the time of equilibration. Boost invariant values of the three independent shear stress-tensors can be easily ob- tained from Eqs.C4-C6, σxx = σyy = θ = 1 and σxy = 0 ( at the initial time τi, u µ = (1, 0, 0, 0), Duµ = 0. The initial distribution of shear pressure tensors are then ob- tained as, πxx(x, y,~b) = 2ησxx = 2η/τi (4.7) πyy(x, y,~b) = 2ησyy = 2η/τi (4.8) πxy(x, y,~b) = 2ησxy = 0 (4.9) As stated earlier, the viscous coefficient η is obtained using the relation, η/s = const, const=0.08, 0.135 or 0.2. For these values of shear viscosity, the validity condition Γs/τ << 1 is satisfied initially. The validity condition is better satisfied at later time. V. RESULTS VI. STABILITY OF NUMERICAL SOLUTIONS A. Evolution of the viscous QGP fluid The energy-momentum conservation equations B1-B3, and the relaxation equations C2-C4 are solved simultane- ously using the code, AZHYDRO-KOLKATA, developed at the Cyclotron Centre, Kolkata. As mentioned earlier, we have solved the equations in the QGP phase only and did not consider any phase transition. In the following we will show the results for central Au+Au collisions (im- pact parameter b = 0 fm). To understand the effect of shear viscosity, with the same initial conditions, we have solved the energy-momentum conservation equations for ideal fluid and viscous fluid. As mentioned earlier, we have considered two values of viscosity, the ADS/CFT motivated value η/s=0.08 and the perturbative estimate, η/s=0.135. 0 2 4 6 8 10 X (fm) _____ η/s=0.08 _____ η/s=0.135 _____ η/s=0 Temperature in τ-x plane. y=5 fm. FIG. 6: (color online). same as fig.5 but at y=5.0 In Fig.1, we have shown the contours of constant en- ergy density in x-y plane, after an evolution of 2.6 fm. The black lines are for ideal fluid evolution. The red and blue lines are for viscous fluid with ADS/CFT (η/s=0.08) and perturbative (η/s=0.135) estimate of viscosity. Con- stant energy density contours, as depicted in Fig.1, indi- cate that with viscosity fluid cools slowly. Cooling gets slower as viscosity increases. Thus at any point in the x-y plane, energy density of viscous fluid is higher than that of an ideal fluid. At later time also, compared to an ideal fluid, viscous fluid evolve slowly. In Fig.2, contours of constant energy density at time τ=8.6 fm is shown. Here also we find than at any point energy density of viscous fluid is higher than its ideal counter part. The result is in accordance with our expectation. For dissipative fluid, equation of motion can be written as, Dε = −(ε+ p)∇µuµ + πµν∇<µuν> (6.1) Due to viscosity, evolution of energy density (or tem- perature) is slowed down. In Fig.3 and 4, we have shown the contour plot of the fluid velocity, vx and vy, after evolution of 8.6 fm. As before the black lines are for the ideal fluid evolution. The red and blue lines are for viscous fluid with η/s=0.08 and 0.135 respectively. Fluid velocities in viscous and ideal fluid differ very little. Even at late time, as shown in Fig.3 and 3, we find that for η/s=0.08-0.135, x and y component of the fluid velocity show marginal difference. However, there is an indication that in a viscous fluid, velocity grow faster than in ideal fluid. But as mentioned earlier, the difference is marginal. η/s=0.08 η/s=0.135 τ (fm) 0 2 4 6 8 10 FIG. 7: (color online). In the upper panel, temporal evolu- tion of the shear pressure tensor πxx at the fluid cell x=y=0 is shown. In the lower panel, evolution of πxy at the fluid cell x=y=5 fm is shown. The black and red lines are for ADS/CFT motivated viscosity η/s=0.08 and perturbative es- timate η/s=0.135 respectively. As seen in Fig.1-2, in viscous dynamics, QGP fluid evolves slowly. Thus life-time of the QGP phase is en- hanced in viscous dynamics. To obtain an idea about the enhanced life-time, in Fig.5, we have shown the constant temperature contours in τ − x plane , at a fixed value of y=0 fm. As seen in Fig.5, temperature evolves slowly in a viscous fluid and life-time of the QGP phase is ex- tended. For small viscosity η/s=0.08-0.135, the increase is not large. At the center of the fluid, for η/s=0.135, QGP life-time is increased approximately by 5% only. It is even less for the ADS/CFT estimate of viscosity. How- ever, enhancement of QGP life-time depends on the fluid cell position. It could be more. In Fig.6, constant tem- perature contours at y=5 fm is shown. For η/s=0.135, at x=0,y=5 fm, the QGP life-time is enhanced by ∼ 10%. We conclude that in a viscous dynamics, with moder- ate viscosity η/s=0.08-0.135, QGP life-time could be en- hanced by 5-10%. Enhanced lifetime of QGP in a viscous fluid can have significant effect on observables produced early in the collisions e.g. direct photon production or in J/ψ suppression. 1.01.21.4 0 2 4 6 8 10 0.030 0.180.21 0 2 4 6 8 10 (d) π at τ=2.6 fm(c) π at τ=0.6 fm (b) π at τ=2.6 fm(a) π at τ=0.6 fm 1.01.21.4 0 2 4 6 8 10 0.030 0.150.18 0.210.24 0 2 4 6 8 10 X (fm) FIG. 8: (color online). In panel (a) and (b), contours of constant pressure tensor πxx at initial time τi=0.6 fm and at time τ=2.6 fm is shown. In panel (c) and (d) same results for shear pressure tensor πyy is shown. B. Evolution of shear pressure tensors We have assumed that initially the shear pressure ten- sors πxx, πyy and πxy attained their longitudinal boost- invariant values. As the fluid evolve, pressure tensors also evolve. Here we investigate the evolution of shear pres- sure tensors with time. In the top panel of Fig.7 evolu- tion of shear pressure tensor πxx at the fluid cell position x=y=0 is shown. The black line is for the ADS/CFT motivated viscosity, η/s=0.08 and the red line is for the perturbative estimate of viscosity η/s=0.135. Just after the start of the evolution the shear pressure tensor πxx increases, but for a short duration and then steadily de- creases with time. By 4 fm of evolution, πxx at the center of the fluid reduces to negligibly small values. Identical behavior is seen for the shear pressure tensor πyy. In the bottom panel of Fig.7 we have shown the evolution of the third independent shear pressure tensor πxy. Ini- tially πxy is zero. As the fluid evolve, it grow in the negative direction. We find that at the centre of the fluid (x=y=0), it never grows. In Fig.7, temporal evolution of πxy at the fluid cell position x = y = 5fm is shown. From the initial zero value, πxy rapidly increases in the negative direction. It reaches its maximum around τ ≈1 fm and then decreases again. We also note that πxy never grows to large values. Compared to πxx or πyy stress ten- sor πxy is negligible. The results indicate that in a QGP fluid, viscous effect persist for a short duration (3-4 fm) only. At late time the fluid evolve essentially as an ideal fluid. The result is understandable. Shear viscosity de- pend strongly on temperature (η ∝ T 3). As the fluid cools, effect of viscosity decreases rapidly. τ (fm) 0 2 4 6 8 10 η/s=0.08 η/s=0.135 FIG. 9: Evolution of average entropy with time, for two val- ues of viscosity, the ADS/CFT motivated viscosity η/s=0.08 and perturbative estimate η/s=0.135 are shown. To show the spatial distribution of the stress tensors, in Fig.8, πxx and πyy at initial time τi=0.6 fm and after an evolution of τ = 2.6 fm are shown. As shown earlier, πxx and also πyy rapidly decreases with time. By 2 fm of evolution they are reduced by approximately by a fac- tor 6. It is also interesting to note that the initial x-y symmetric distribution of πxx and πyy quickly evolves to asymmetric distribution. With time πxx evolves faster in the x-direction than in y-direction. Similarly, πyy evolve faster in the y-direction than in the x-direction. For cen- tral collisions the asymmetric evolution of πxx and πyy counter balance each other. As shown in Fig.1 and 2, the contour plots of energy density do not show any in- dication of asymmetry even at late time. However, the asymmetric pressure tensors can have important effects on elliptic flow of observables produced early in the col- lisions, say in elliptic flow of direct photons. C. Entropy generation In a viscous fluid dynamics, entropy is generated. We can easily calculate the entropy generated during the evo- lution, πµνπµν [(πττ )2 + (πxx)2 + (πyy)2 + (τ2πηη)2 −2(πτx)2 − 2(πτy)2 + 2(πxy)2] (6.2) Evolution of spatially averaged entropy is shown in Fig.9, for the two values of viscosity coefficients η/s=0.08 and 0.135. As expected, entropy generation is more if vis- cosity is more. For both the values of viscosity, we find that entropy generation saturates after ≈ 3 fm of evolu- tion. It is expected also. As shown previously, viscous fluxes reduces to very small values after τ=3 fm. Natu- rally, entropy generation is negligible thereafter. 0.190.22 0 2 4 6 8 10 10 Temperature in τ-x plane, y= 5m. _____ viscous fluid (2nd order) _____ ideal fluid _____ viscous fluid (1st order) X (fm) FIG. 10: (color online) constant temperature contours in x − τ plane at y=5 fm. The black lines are for ideal fluid. The red and blue lines are for viscous fluid in 1st order and in 2nd order theory respectively. η/s=0.135. D. 1st order theory vs. 2nd order theory As mentioned earlier, 1st order theory of dissipative hydrodynamics is acausal, signal can travel faster than light. This is corrected in 2nd order theory, but we have to pay the price, relaxation equations for dissipa- tive fluxes are required to be solved. It is interesting to τ (fm) 0 2 4 6 8 10 12 1st order theory 2nd order theory η/s=0.135 FIG. 11: Evolution of average entropy production in a 1st order (solid line) and 2nd order (dashed line) theory. 2nd order theory generate more entropy. compare the difference we can expect in a first order the- ory and in a 2nd order theory of dissipation. In Fig. 10, we have shown the contours of constant temperature in x−τ , for a fixed y = 5fm. The black lines are for an ideal fluid. The red lines are for a viscous fluid treated in the 1st order theory. The blue lines are for viscous fluid in 2nd order theory. In 2nd order theory fluid evolve more slowly than in a first order theory. Entropy generation is also more in a 2nd order theory. In Fig.11, average en- tropy evolution with proper time is shown, both for the 1st order theory (the solid line) and the 2nd order theory. In 2nd order theory, approximately 80% more entropy is generated. VII. TRANSVERSE MOMENTUM AND ELLIPTIC FLOW OF QUARKS Presently we can not compare predictions from viscous hydrodynamics with experimental data. Hadrons are not included in the model. The initial QGP fluid evolve and cools but remain in the QGP phase, it did not undergo a phase transition to hadronic gas. However, from the momentum distribution of quarks we can get some idea about the viscous effect on particle production. Viscos- ity generates entropy, which will be reflected in enhanced multiplicity. We use the standard Cooper-Frey prescrip- tion to obtained the transverse momentum distribution of quarks. In Cooper-Frey prescription, particle distribu- tion is obtained by convoluting the one body distribution function over the freeze-out surface, µf(x, p) (7.1) where dΣµ is the freeze-out hyper-surface and f(x, p) is the one-body distribution function. Now in a viscous dy- namics, the fluid is not in equilibrium and f(x, p) can not be approximated by the equilibrium distribution func- tion, f (0)(x, p) = exp[β(uµpµ − µ)] ± 1 , (7.2) with inverse temperature β = 1/T and chemical poten- tial µ. In a highly non-equilibrium system, distribution function f(x, p) is unknown. If the system is slightly off-equilibrium, then it is possible to calculate correction to equilibrium distribution function due to (small) non- equilibrium effects. Slightly off-equilibrium distribution function can be approximated as, f(x, p) = f (0)(x, p)[1 + φ(x, p)], (7.3) φ(x, p) is the deviation from equilibrium distribution function f (0). With shear viscosity as the only dissi- pative forces, φ(x, p) can be locally approximated by a quadratic function of 4-momentum, φ(x, p) = εµνp µpν . (7.4) Without any loss of generality εµν can be written as εµν = 2(ε+ p)T 2 πµν , (7.5) completely specifying the non-equilibrium distribution function. As expected, correction factor increases with increasing viscosity. We also note that non-equilibrium correction is more on large momentum particles. The effect of viscosity is more on large momentum parti- cles. The correction factor reduces if freeze-out occurs at higher temperature. With the corrected distribution function, we can cal- culate the quark momentum spectra at freeze-out sur- face Σµ. In appendix D, relevant equations are given. The quark momentum distribution has two parts, (i) dyd2pT , obtained by convoluting the equilibrium distribu- tion function over the freeze-out surface and (ii) dN dyd2pT obtained by convoluting the correction to the equilibrium distribution function over the freeze-out surface. Since the correction factor is obtained under the assumption that non-equilibrium effects are small, φ(x, p) << 1, it necessarily imply that, dN dyd2pT << dN dyd2pT . The ratio, dNneq dNneq dyd2pT dyd2pT , (7.6) could at best be unity or less. If the ratio exceeds unity, it will imply that non-equilibrium effects are large (GeV) 0 1 2 3 4 5 η/s=0.2 η/s=0.135 η/s=0.08 Au+Au@b=6.5 fm τi=0.6fm,Sini=110fm-3,TF=160MeV FIG. 12: Ratio of quark spectra with non-equilibrium distri- bution function to that with equilibrium distribution function. and the distribution function f(x, p) can not be approx- imated as in Eq.7.3. Using AZHYDRO-KOLKATA, we have simulated a b=6.5 fm Au+Au collision. dN dyd2pT dNneq dyd2pT at freeze-out temperature TF =160 MeV are cal- culated. The ratio dN for η/s=0.08,0.135 and 0.2, are shown in Fig.12. With ADS/CFT estimate of viscosity, η/s=0.08, non-equilibrium correction to particle produc- tion become comparable to equilibrium contribution only beyond pT =5 GeV. However, with perturbative estimate, η/s=0.135, non-equilibrium correction become compara- ble to or exceeds the equilibrium contribution at pT ∼ 4 GeV. pT range is further reduced for higher viscosity η/s=0.2. Thus with perturbative estimate of viscosity (η/s = 0.135 − 0.2), hydrodynamic description remain valid upto transverse momentum pT ∼ 3.5-4 GeV. In Fig.13, we have compared the transverse momen- tum spectra of quarks in ideal hydrodynamics with that in a viscous dynamics. In Fig.13, the dotted line is the spectra obtained in ideal dynamics (η/s = 0). The pT spectra in viscous dynamics are shown by black lines. We have shown the spectra for three values of viscos- ity η/s=0.08,0.135 and 0.2. Compared to ideal dynam- ics, quarks yield in viscous dynamics increases. The in- crease is more at large pT . For low values of viscosity the increase is modest, a factor of 2 at pT =3 GeV. But yield increase by a factor or 4(10) if viscosity increases to η/s=0.135 (2). Please note that even though we have shown pT spectra upto 5 GeV, for η/s=0.2 and 0.135, hydrodynamic description fails beyond pT ∼ 3.5 and 4 We have also studied the effect of viscosity on quark elliptic flow. Effect of viscosity is very prominent on elliptic flow. In Fig.14, pT dependence of elliptic flow of quarks, in a b=6.5 fm collision is shown. The black line is v2 in ideal dynamics. In ideal dynamics, ellip- pT (GeV) 0 1 2 3 4 5 η/s=0.0(ID.Fluid) η/s=0.2 η/s=0.135 η/s=0.08 Au+Au@b=6.5 fm τi=0.6fm,Sini=110fm-3,TF=160MeV FIG. 13: Qurak transverse momentum spectra at freeze-out temperature of 160 MeV. The dotted line is the quarks spectra in ideal hydrodynamics. The solid lines (top to bottom) are in viscous dynamics with η/s=0.08,0.135 and 0.2. tic flow continually increases with pT . It well known, in contrast to experiments, where elliptic flow saturates at large pT , in ideal hydrodynamics, elliptic flow continue to increase with pT . Indeed, this is a major problem in ideal hydrodynamics. The renewed the interest in dis- sipative hydrodynamics is partly due to the inability of ideal hydrodynamics to predict the trend of elliptic flow in Au+Au collisions. In Fig.14, the blue lines are v2 in viscous dynamics with η/s=0.08,0.135 and 0.2 respec- tively. In a viscous dynamics, pT dependence of v2 is drastically changed. In contrast to ideal dynamics where v2 continue to increase with pT , in viscous dynamics, v2 continue to increase only upto pT ∼ 1.5− 2GeV . There- after v2 decreases. For perturbative estimate of viscos- ity η/s=0.135 and beyond, v2 even become negative at large pT . Veering about of v2 after pT ∼1.5-2 GeV is due to viscous effect only or more explicitly due to the non-equilibrium correction to the equilibrium distribu- tion function. This is clearly manifested from the red lines in Fig.14. The red lines are calculated ignoring the non-equilibrium corrections to the equilibrium distribu- tion function. If non-equilibrium correction is ignored, in viscous dynamics also, v2 continue to increase with pT , albeit its magnitude is reduced compared to ideal dy- namics. The result is very important. It imply that the experimental trend of elliptic flow (saturation at large pT ) could only be explained if the QGP fluid is viscous. An ideal QGP, will not be able to explain the saturation trend of the experimental data. As stated earlier, non-equilibrium correction to the equilibrium distribution function depends on the freeze- out condition. To show the effect of freeze-out condition, on v2, in Fig.15 we have shown v2 for a values of freeze- out temperature TF =160,150,140,130 and 120 MeV. As pT (GeV) 0 1 2 3 4 5 vis.fluid with Fneq vis.fluid with Feq id. fluid Au+Au@b=6.5 fm τi=0.6fm,Sini=110fm-3,TF=160MeV FIG. 14: (color online) Elliptic flow as a function of transverse momentum. The black line is v2 in ideal hydrodynamics. The blue lines are v2 in viscous dynamics with viscosity to entropy ratio η/s=0.08,0.135 and 0.2 (top to bottom) respectively, including the correction to equilibrium distribution function. The red lines are same as the blues lines but ignoring the non-equilibrium correction to the distribution function. freeze-out occur at higher and higher temperature, the veering of v2 takes place at larger and larger pT and for TF =120 MeV, the elliptic flow saturates. The result is understood easily. With decreasing freeze-out tempera- ture, the fluid evolves for longer time, the shear stress- tensor’s at the freeze-out surface is reduced and the non- equilibrium correction, proportional to shear stress ten- sors, decreases. VIII. STABILITY OF NUMERICAL SOLUTIONS IN AZHYDRO-KOLKATA Before we summarise our results, we would like to comment on the stability of numerical solutions in AZHYDRO-KOLKATA. As indicated above, with shear viscosity as the only dissipative force, boost-invariant causal hydrodynamics require simultaneous solution of six partial differential equations. Numerical solution of six partial differential equations is non-trivial and it is important to check for the numerical stability of the solutions. Analytical solutions of viscous hydrodynam- ics, even in restrictive conditions are not available, and we can not check the solutions against analytical re- sults. However, we can check for the stability of the numerical solutions. The standard procedure of check- ing the numerical stability is to change the integration step lengths and look for the difference in the solution. In Fig.16, for viscosity η/s=0.135, we have shown the constant temperature contours in x − τ plane at a fixed pT (GeV) 0 1 2 3 4 5 Au+Au@b=6.5 fm τi=0.6fm,η/s=.2,Sini=110fm-3 TF=160,150,140,130 and 120 MeV (bottom to top) FIG. 15: Dependence of elliptic flow on the freeze-out tem- perature. The solid lines (from bottom to top) are elliptic flow (v2) in viscous dynamics with TF =160,150,140,130 and 120 MeV respectively. 0.250.28 0 2 4 6 8 10 ____dx=dy=0.1,dτ=0.01 ____dx=dy=0.2,dτ=0.02Au+Au@b=0 X (fm) FIG. 16: (color online) constant temperature contours in x − τ plane at a fixed y=0 fm. The black lines are obtained with integration step lengths dx=dy=0.2 fm and dτ=0.02 fm. The blue lines are obtained with integration step lengths, dx=dy=0.1 fm and dτ=0.01 fm. Halving the step lengths do not change the evolution. The numerical solutions are stable. y=0 fm. The black and blue lines are obtained when integration step lengths are dx=dy=0.2fm,dτ=0.02 fm, and dx=dy=0.1 fm,dτ=0.01fm respectively. Evolution of QGP fluid donot alter by changing the step lengths, the solutions are stable against mesh size. IX. SUMMARY AND CONCLUSIONS In Israel-Stewart’s 2nd order theory of dissipative rel- ativistic hydrodynamics, we have studied evolution QGP fluid. In 2nd order theory, in addition to usual thermo- dynamic quantities e.g. energy density, pressure, hydro- dynamic velocities, dissipative flows are treated as ex- tended thermodynamic variables. Relaxation equations for dissipative flows are solved, simultaneously with the energy-momentum conservation equations. This greatly enhances the complexity of the problem. Altogether 14 partial differential equations are required to be solved. We simplify the problem to solution of six partial differ- ential equations by considering the evolution of baryon free QGP fluid with longitudinal boost-invariance. We also consider dissipation due to shear viscosity only, dis- regarding the bulk viscosity and the heat conduction (for a baryon free QGP fluid they do not contribute). The six partial differential equations are solved using the code AZHYDRO-KOLKATA, developed at the Cy- clotron Centre, Kolkata. To bring out the effect of viscosity, we have considered the evolution of ideal as well as viscous QGP fluid. Both ideal and viscous fluid are initialized similarly, at initial time τi=0.6 fm, the central entropy density is 110 fm Viscous dynamics require initial conditions for the shear- stress tensor components. It is assumed that at the equi- libration time, the shear stress tensors components have attained their boost-invariant values. Explicit simulation of ideal and viscous fluids confirms that energy density of a viscous fluid, evolve slowly than its ideal counterpart. Thus in a viscous fluid, lifetime of the QGP phase will be enhanced. Transverse expansion is also more in viscous dynamics. For a similar freeze-out condition freeze-out surface is extended in viscous fluid. As the fluid evolve, shear pressure tensors also evolve. Explicit simulations indicate that shear pressure tensors πxx and πyy which are initially non zero, rapidly de- creases as the fluid evolve. By 3-4 fm of evolution they reduced to very small values. The other independent shear tensor πxx is zero initially. At later time it grow in the negative direction but never grow to large value and is always order of magnitude smaller than the stress tensors (πxx and πyy). Spatial distribution of shear pres- sure tensors πxx and πyy reveal an interesting feature of viscous dynamics. Initially πxx and πyy have sym- metric distribution. As the fluid evolve, pressure tensors quickly become asymmetric, e.g. πxx evolve faster in the x-direction than in the y-direction, πyy evolve faster in y direction than in x-direction. However, in a central collision, we did not see any effect of asymmetry in the energy density distribution. In a central b=0 collision, the two opposite asymmetry cancels each other. We could not study effect of shear viscosity on par- ticle production. However, we have explored the effect of viscosity on parton momentum distribution and ellip- tic flow. We have simulated b=6.5 fm Au+Au collision. Using the Cooper-Frey prescription, transverse momen- tum spectra as well as elliptic flow of quarks at freeze-out temperature of TF =160 MeV are obtained. Viscous dy- namics flattens the quark yield at large pT . At pT =3 GeV, even a small viscosity, η/s=0.8, increase the yield by a factor of 2. The increase is even more if viscosity is large. Viscous effect is most prominent on elliptic flow. In ideal hydrodynamics, elliptic flow continue to increase with pT . But in viscous dynamics v2 veer about around pT =1.5-2 and even become negative at large pT . With appropriate choice of viscosity, freeze-out condition, el- liptic flow show saturation. The saturation effect is es- sentially due to non-equilibrium correction to the equi- librium distribution function and can not be mimicked in an ideal hydrodynamics. Only in viscous dynamics, saturation of elliptic flow can be explained. APPENDIX A: COORDINATE TRANSFORMATIONS Instead of Cartesian coordinates xµ = (t, x, y, z) we use curvilinear coordinates in longitudinal proper time and rapidity, x̄m = (τ, x, y, η): t = τ cosh η; τ = t2 − z2 (A1) z = τ sinh η; η = . (A2) The differentials dt = dτ cosh η + dη τ sinh η, (A3) dz = dτ sinh η + dη τ cosh η, (A4) and the metric tensor is easily read off from ds2 = gµνdx µdxν = dt2 − dx2 − dy2 − dz2 = ḡmndx̄ mdx̄n = dτ2 − dx2 − y2 − τ2dη2,(A5) namely ḡmn = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −τ2 , ḡmn = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1/τ2 In curvilinear coordinates we must replace the partial derivatives with respect to xµ by covariant derivatives (denoted by a semicolon) with respect to x̄m: T̄ ik;p = ∂T̄ ik + ΓipmT̄ mk + T̄ imΓkmp. The only non-vanishing Christoffel symbols are Γτηη = τ ; Γ τη = Γ ητ = 1/τ. (A7) The hydrodynamic 4-velocity uµ = γ(1, vx, vy, vz) is transformed to ūm = γ(1, vx, vy, 0), with γ⊥ = 1−v2r . From here on, we drop the bars over tensor components in x̄-coordinates for simplicity. The projector can be easily calculated, ∆µν = gµν − uµuν 1 − γ2⊥ −γ2⊥vx γ2⊥vy 0 −γ2⊥vx −1 − γ2⊥v2x −γ2⊥vxvy 0 −γ2⊥vy −γ2⊥vxvy −1 − γ2⊥v2y 0 0 0 0 1 .(A8) In (τ, x, y, η) coordinate system, the convective time derivative can be obtained as, D = u · ∂ = γ(∂τ + vx∂x + vy∂y). (A9) For future reference,we also write down the the scalar expansion rate θ = ∂·u = ∂τuτ + ∂xux + ∂yuy + (A10) APPENDIX B: ENERGY-MOMENTUM CONSERVATION With longitudinal boost-invariance the energy- momentum conservation equations Tmn;n = 0 yield ∂τ T̃ ττ + ∂x(T̃ ττvx) + ∂y(T̃ ττvy) = − (p+ τ2πηη) (B1) ∂τ T̃ τx + ∂x(T̃ τxvx) + ∂y(T̃ τxvy) = −∂x(p̃+ π̃xx − π̃τxvx) − ∂y(π̃xy − π̃τxvy) (B2) ∂τ T̃ τy + ∂x(T̃ τyvx) + ∂y(T̃ τyvy) = −∂x(π̃xy − π̃τyvx) − ∂y(p̃+ π̃yy − π̃τyvy) (B3) where Ãmn ≡ τAmn, p̃ ≡ τp, and vx ≡ T τx/T ττ , vy ≡ T τy/T ττ . The components of the energy momentum tensors, in- cluding the shear pressure tensor are, T ττ = (ε+ p)γ2⊥ − p+ πττ (B4) T τx = (ε+ p)γ2⊥vx + π τx (B5) T τy = (ε+ p)γ2⊥vy + π τy (B6) In causal dissipative hydrodynamics, energy momen- tum conservation equations are solved simultaneously with the relaxation equations. Given an equation of state, if energy density (ε) and fluid velocity (vx and vy) distributions, at any time τi are known, Eqs.B1,B2 and B3 can be integrated to obtain ε, vx and vy at the next time step τi+1. While for ideal hydrodynamics, this procedure works perfectly, viscous hydrodynamics poses a problem that shear stress-tensor components contains time derivatives, ∂τγ⊥, ∂τu x, ∂τu x etc. Thus at time step τi one needs the still unknown time derivatives. Numer- ically, time derivatives at step τi could be obtained if ve- locities at time step τi and τi+1 are known. One possible way to circumvent the problem, is to use time derivatives of the previous step, i.e. use velocities at time step τi−1 and τi to calculate the derivatives at time step τi [18]. The underlying assumption that fluid velocity changes slowly with time. In 1st order theories, this problem is circumvented by calculating the time derivatives from the ideal equation of motion , Duµ = , (B7) Dε = −(ε+ p)∇µuµ. (B8) With the help of these two equations all the time derivatives can be expressed entirely in terms of spa- tial gradients [15, 26]. 1st order theories are restricted to contain terms at most linear in dissipative quantities. Neglect of viscous terms can contribute only in 2nd or- der corrections, which are neglected in 1st order theories. While the procedure is not correct in 2nd order theory, we still use it in the present calculations. The alternative procedure of using the derivative of earlier time step is not correct either. APPENDIX C: RELAXATION EQUATIONS FOR THE VISCOUS PRESSURE TENSOR Being symmetric and traceless, the viscous pressure tensor πµν has 9 independent components. The as- sumption of boost invariance reduces this number by 3 (∇〈mu η〉 =0, m 6= η). The transversality condition mn = 0 eliminates another three components ( uη vanish and thus yield no constraint). Thus, with boost- invariance the viscous pressure tensor has only three in- dependent components. As seen in Eqs.B1,B2 and B3 in a boost-invariant evolution only seven pressure tensors πττ , πxx, πyy, πηη, πτx, πτy and πxy are of importance. Only three of these seven are independent. In an ear- lier publication [17], we have debated about the choice of the independent components and suggested use of ei- ther (πττ , πηη, ∆ = πxx − πyy) or (πττ ,πηη,πτx, πτy) (which will require solution of an additional relaxation equation) as choice of independent components. How- ever, while computing we find that the three pressure tensors πxx and πyy and πxy as independent components are computationally more convenient. The choice has the advantage that the dependent shear stress tensors can be obtained from the 3 independent stress tensors by multiplying them by fluid velocity, vx and vy (see Eqs. C7-C10). In any other choice of independent components (e.g. πττ ,πηη,∆ = πxx − πyy), the evaluation of depen- dent stress tensors requires division by fluid velocities. Since initially, fluid velocities are assumed to be zero and they grow slowly, these choices will involve division by very small numbers. Unless proper care is not taken, di- vision by small numbers can lead to unrealistically large values for the dependent stress tensors and ruin the com- putation. The relaxation equations for the independent shear stress tensors πxx, πyy and πxy, in (τ ,x,y,η) co-ordinate can be written as, xx + vx∂xπ xx + vy∂yπ xx = − 1 (πxx − 2ησxx) (C1) yy + vx∂xπ yy + vy∂yπ yy = − 1 (πyy − 2ησyy) (C2) xy + vx∂xπ xy + vy∂yπ xy = − 1 (πxy − 2ησxy) (C3) where τπ is the relaxation time, τπ = 2ηβ2 (see Eq.2.21). In ultra-relativistic limit, for a Boltzman gas, β2 can be evaluated, β2 ≈ 34p where p is the pressure [13]. In the present paper, we use this limit to obtain the relaxation time τπ. The viscous pressure tensor relaxes on a time scale τπ to 2η times the shear tensor σµν = ∇〈µ u ν〉. The xx, yy and xy components of the shear tensor σµν can be written as σxx = −∂xux − uxDux − ∆xxθ (C4) σyy = −∂yuy − uyDuy − ∆yyθ (C5) σxy = −1 y − ∂yux − uxDuy − uyDux] ∆xyθ (C6) The dependent shear stress tensors can easily be ob- tained from the independent ones as, πτx = vxπ xx + vyπ xy (C7) πτy = vxπ xy + vyπ yy (C8) πττ = v2xπ xx + v2yπ yy + 2vxvyπ xy (C9) τ2πηη = −(1 − v2x)πxx − (1 − v2y)πyy +2vxvyπ xy (C10) The expressions for the convective time derivative D and expansion scalar θ = ∂u̇, in (τ ,x,y,η) are given in Eqs. A9 and A10. APPENDIX D: PARTICLE SPECTRA With the non-equilibrium distribution function thus specified, it can be used to calculate the particle spectra from the freeze-out surface. In the standard Cooper-Frye prescription, particle distribution is obtained as, dyd2pT µf(x, p) (D1) In (τ, x, y, ηs) coordinate, the freeze-out surface is pa- rameterised as, Σµ = (τf (x, y) cosh ηs, x, y, τf (x, y) sinh ηs), (D2) and the normal vector on the hyper surface is, dΣµ = (cosh ηs,− ,−∂τf ,− sinh ηs)τfdxdydηs At the fluid position (τ, x, y, ηs) the particle 4- momenta are parameterised as, pµ = (mT cosh(ηs − Y ), px, py,mT sinh(ηs − Y )) (D4) The volume element pµdΣµ become, pµdΣµ = (mT cosh(η − Y ) − ~pT .~∇T τf )τfdxdydη (D5) Equilibrium distribution function involve the term which can be evaluated as, γ(mT cosh(η − Y ) − ~vT .~pT − µ/γ) The non-equilibrium distribution function require the sum pµpνπµν , pµpνπ µν = a1cosh 2(η − Y ) + a2cosh(η − Y ) + a3 (D7) a1 = m ττ + τ2πηη) (D8) a2 = −2mT (pxπτx + pyπτy) (D9) a3 = p xx + p2yπ yy + 2pxpyπ xy −m2T τ2πηη(D10) Inserting all the relevant formulas in Eq.D1 and inte- grating over spatial rapidity one obtains, dyd2pT dyd2pT dNneq dyd2pT (D11) with, dyd2pT (2π)3 dxdyτf [mTK1(nβ) − pT ~∇T τfK0(nβ)] (D12) dNneq dyd2pT (2π)3 dxdyτf [mT { K3(nβ) + K2(nβ) + ( + a3)K1(nβ) + K0(nβ)} −~pT .~∇T τf{ K2(nβ) + a2K1(nβ) + ( + a3)K0(nβ)}] (D13) where K0, K1, K2 and K3 are the modified Bessel func- tions. We will also show results for elliptic flow v2. It is de- fined as, dyd2pT cos(2φ)dφ dyd2pT (D14) Expanding to the 1st order, elliptic flow as a function of transverse momentum can be obtained as, v2(pT ) = v 2 (pT ) 2Nneq pT dpT dφ pT dpT dφ dφcos(2φ) d 2Nneq pT dpT dφ pT dpT dφ (D15) where v 2 is the elliptic flow calculated with the equi- librium distribution feq. [1] BRAHMS Collaboration, I. Arsene et al., Nucl. Phys. A 757, 1 (2005). [2] PHOBOS Collaboration, B. B. Back et al., Nucl. Phys. A 757, 28 (2005). [3] PHENIX Collaboration, K. Adcox et al., Nucl. Phys. A 757 (2005), in press [arXiv:nucl-ex/0410003]. [4] STAR Collaboration, J. Adams et al., Nucl. Phys. A 757 (2005), in press [arXiv:nucl-ex/0501009]. [5] Karsch F, Laermann E, Petreczky P, Stickan S and Wet- zorke I, 2001 Proccedings of NIC Symposium (Ed. H. Rollnik and D. Wolf, John von Neumann Institute for Computing, Jülich, NIC Series, vol.9, ISBN 3-00-009055- X, pp.173-82,2002.) [6] P. F. Kolb and U. 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0704.0135

A Single Trapped Ion as a Time-Dependent Harmonic Oscillator

A Single Trapped Ion as a Time-Dependent Harmonic Oscillator Nicolas C. Menicucci1, 2, ∗ and G. J. Milburn2 Department of Physics, Princeton University, Princeton, NJ 08544, USA School of Physical Sciences, The University of Queensland, Brisbane, Queensland 4072, Australia (Dated: November 4, 2018) We show how a single trapped ion may be used to test a variety of important physical mod- els realized as time-dependent harmonic oscillators. The ion itself functions as its own motional detector through laser-induced electronic transitions. Alsing et al. [Phys. Rev. Lett. 94, 220401 (2005)] proposed that an exponentially decaying trap frequency could be used to simulate (thermal) Gibbons-Hawking radiation in an expanding universe, but the Hamiltonian used was incorrect. We apply our general solution to this experimental proposal, correcting the result for a single ion and showing that while the actual spectrum is different from the Gibbons-Hawking case, it nevertheless shares an important experimental signature with this result. PACS numbers: 03.65.-w, 32.80.Pj I. INTRODUCTION The time-dependent quantum harmonic oscillator has long served as a paradigm for nonadiabatic time- dependent Hamiltonian systems and has been applied to a wide range of physical problems by choosing the mass, the frequency, or both, to be time-dependent. The ear- liest application is to squeezed state generation in quan- tum optics [1, 2, 3], in which the effect of a second-order optical nonlinearity on a single-mode field can be mod- eled by a harmonic oscillator with a frequency that is harmonically modulated at twice the bare oscillator fre- quency. It was subsequently shown that any modulation of the frequency could produce squeezing [4], and thus the same model could be used to approximately describe the generation of photons in a cavity with a time-dependent boundary [5, 6]. The model has been used in a number of quantum cosmological models. In Ref. [7], a time-dependent fre- quency has been used to explain entropy production in a quantum mini-superspace model. The model, with both mass and frequency time-dependent, has been particu- larly important in developing an understanding of how quantum fluctuations in a scalar field can drive classical metric fluctuation during inflation [8, 9]. In a cosmo- logical setting the time-dependence is not harmonic and is usually exponential. In all physical applications, of course, the model is only an approximation to the true physics, and its validity can be tested only with consid- erable difficulty, especially in the cosmological setting. Here we propose a realistic experimental context in which the time-dependent quantum harmonic oscillator can be studied directly. Many decades of effort to refine spectroscopic measure- ments for time standards now enable a single ion to be confined in three dimensions, its vibrational motion re- stricted effectively to one dimension, and the ion cooled ∗Electronic address: [email protected] to the vibrational ground state with a probability greater than 99% [10]. Laser cooling is based on the ability to couple an internal electronic transition to the vibrational motion of the ion [11]. These methods can easily be ex- tended to more than one ion and their collective normal modes of vibration [12]. Indeed so carefully can the cou- pling between the electronic and vibrational states be engineered that is is possible to realise simple quantum information processing tasks [13, 14]. We use the control of trapping potential afforded by ion traps, together with the ability to reach quantum limited motion, to propose a simple experimental test of quantum harmonic oscilla- tors with time-dependent frequencies. We also make use of the ability to make highly efficient quantum measure- ments, based on fluorescent shelving [10], to propose a practical means to test our predictions. In this paper, we calculate the excitation probability of a trapped ion in a general time-dependent potential. When beginning in the vibrational ground state of the unchirped trap and starting the chirping process adia- batically, the excitation probability is simply related to the Fourier transform of the solution of the Heisenberg equations of motion (which is also the same as the trajec- tory of the equivalent classical oscillator). We compare our result with that of Ref. [15] for the case of a single ion undergoing an exponential frequency chirp. The cited work attempts to use this experimental setup to model a massless scalar field during an inflating (i.e., de Sitter) universe, which would give a thermal excitation spectrum as a function of the detector response frequency [16]. The analysis is incorrect, however, because the wrong Hamil- tonian was used. Nevertheless, the corrected calculation presented here also gives an excitation spectrum with a thermal signature, although the particular functional form is different. II. GENERAL SOLUTION The quantum Hamiltonian for a single ion in a time- dependent harmonic trap can be well-approximated in http://arxiv.org/abs/0704.0135v2 mailto:[email protected] one dimension by ν(t)2q2 , (1) where ν(t) is time-dependent but always assumed to be much slower than the timescale of the micromo- tion [10]. For emphasis, we have indicated the explicit time-dependence of the frequency ν; we will often omit this from now on. Working in the Heisenberg picture, we get the following equations of motion for q and p: , (2) ṗ = −Mν2q . (3) Dots indicate total derivatives with respect to time. Dif- ferentiating again and plugging in these results gives 0 = q̈ + ν2q , (4) 0 = p̈− 2 ṗ+ ν2p . (5) As we shall see, only Eq. (4) is necessary for calculating excitation probabilities, so we will focus only on it. These equations are operator equations, but they are identical to the classical equations of motion for the analogous classical system. Interpreting them as such, we will la- bel the two linearly independent c-number solutions as h(t) and g(t), where the following initial conditions are satisfied: h(0) = ġ(0) = 1 and ḣ(0) = g(0) = 0 , (6) Writing q(0) = q0 and p(0) = p0, the unique solution for q to the initial value problem above is q(t) = q0h(t) + g(t) . (7) By differentiating and using the relations above, we know also that p(t) = Mq0ḣ(t) + p0ġ(t) . (8) To check our math, we can verify that [q(t), p(t)] = i~, which is fulfilled if and only if the Wronskian W (h, g) of the two solutions is one for all times—specifically, hġ − ḣg = 1 , (9) where we have assumed that [q0, p0] = i~. Moreover, if the initial state at t = 0 is symmetric with respect to phase-space rotations, then we have additional rotational freedom in choosing the initial quadratures. (This would be the case, for instance, if we start in the instantaneous ground state.) Notice that Eq. (7) can be written as the inner product of two vectors: q(t) = h(t), ν0g(t) (and similarly for Eq. (8)), where we have normalized the quadrature operators to have the same units. As an inner product, this expression is invariant under simulta- neous rotations of both vectors. Thus, if the initial state possesses rotational symmetry in the phase plane, then the rotated quadratures are equally as valid as the orig- inal ones for representing the initial state, which means that an arbitrary rotation can be applied to the second vector above without changing any measurable property of the system. This freedom can be used, for instance, to define new functions h′(t) and g′(t) that are more con- venient for calculations, where the linear transformation between them and the original ones (with prefactors as in Eq. (10)) is a rotation. We will use this freedom in the next section. One reason why ion traps have become a leading im- plementation for quantum information processing is the ability to efficiently read out the internal electronic state using a fluorescence shelving scheme [10]. As the internal state can become correlated with the vibrational motion of the ion, this scheme can be configured as a way to measure the vibrational state directly [17]. To correlate the internal electronic state with the motion of the ion, an external laser can be used to drive an electronic tran- sition between two levels |g〉 and |e〉, separated in energy by ~ωA. The interaction between an external classical laser field and the ion is described, in the dipole and rotating-wave approximation, by the interaction-picture Hamiltonian [10] HL = −i~Ω0 σ+(t)e ik cos θq(t) − σ−(t)e −ik cos θq(t) where Ω0 is the Rabi frequency for the laser-atom inter- action, ωL is the laser frequency, k is the magnitude of the wave vector ~k, which makes an angle θ with the trap axis, q(t) is given in Eq. (7), and σ±(t) = e ±i∆tσ± . (12) The electronic-state raising and lowering operators are defined as σ+ = |e〉〈g| and σ− = |g〉〈e|, respectively, and ∆ = ωA − ωL (13) is the detuning of the laser below the atomic transi- tion. We can construct a meaningful quantity that char- acterizes the “size” of q(t) based on the width of the ground-state wave packet for an oscillator with frequency ν(t), namely ~/2Mν(t). As long as this quantity is much smaller than k cos θ throughout the chirping pro- cess, then we can expand the exponentials in Eq. (11) to first order and define the interaction Hamiltonian HI be- tween the electronic states and vibrational motion (still in the interaction picture) by HI = ~Ω0k cos θq(t) e−i∆tσ− + e +i∆tσ+ . (14) where we have assumed that ωL is far off-resonance, and thus ∆ 6≃ 0. Using first-order time-dependent perturbation theory, the probability to find the ion in the excited state is P (1) = dt2 〈HI(t1)PeHI(t2)〉 = Ω20k 2 cos2 θ dt2 e −i∆(t1−t2) 〈q(t1)q(t2)〉 , where Pe = 1vib ⊗ |e〉〈e| is the projector onto the ex- cited electronic state (and the identity on the vibrational subspace). We always assume that the ion begins in the electronic ground state. If the ion also starts out in the instantaneous vibrational ground state for a static trap of frequency ν0 = ν(0) at t = 0 (which is most useful when the chirping begins in the adiabatic regime), then we can evaluate the two-time correlation function as 〈q(t1)q(t2)〉ground = h(t1)h(t2) + g(t1)g(t2) 〈q0p0〉 h(t1)g(t2)− h(t2)g(t1) h(t1)− iν0g(t1) h(t2) + iν0g(t2) f(t1)f ∗(t2) , (16) where we have used the facts that for the vibrational ground state, (p0/Mν0) = ~/2Mν0 and 〈q0p0〉 = 〈{q0, p0}+ [q0, p0]〉 = i~/2, and we have de- fined the complex function f(t) = h(t)− iν0g(t) , (17) which is the solution to Eq. (4) with initial the conditions, f(0) = 1 and ḟ(0) = −iν0. Plugging this into Eq. (15) gives, quite simply, P (1) → (Ω0η0) 2 |F̧| , (18) where dt e−i∆tf(t) , (19) and we have defined the unitless, time-dependent Lamb- Dicke parameter [10] as η(t) = ~k2 cos2 θ 2Mν(t) , (20) and η0 = η(0). Recalling that f(t) can be considered a complex c-number solution to the equations of motion for the equivalent classical Hamiltonian, Eq. (18) shows that the excitation probability is simply related to the Fourier transform of the classical trajectories when beginning in the vibrational ground state. III. EXPONENTIAL CHIRPING Recent work [15] has suggested that an exponen- tially decaying trap frequency has the same effect on the phonon modes of a string of ions as an expand- ing (i.e., de Sitter) spacetime does on a one-dimensional scalar field [18]. An inertial detector that responds to such an expanding scalar field would register a thermal bath of particles, called Gibbons-Hawking radiation [16]. Ref. [15] suggests that the acoustic analog [19] of this radiation could be seen in an ion trap, causing each ion to be excited with a thermal spectrum with temperature ~κ/2πkB, as a function of the detuning ∆, where κ is the trap-frequency decay rate. The analysis used an in- correct Hamiltonian that neglected squeezing and source terms that have no analog in the expanding scalar field model but which are present when considering trapped ions in this way, and the results are incorrect. In this sec- tion, we revisit this problem and calculate the excitation probability for a single ion in an exponentially decaying harmonic potential, as a function of the detuning ∆. We write the time-dependent frequency as [20] ν(t) = ν0e −κt . (21) This results in q̈ + ν20e −2κtq = 0 . (22) Solutions with initial conditions (6) are h(t) = , (23) g(t) = where the time dependence is carried in ν = ν(t) from Eq. (21), and Jn and Yn are Bessel functions. We could plug these directly into the formulas from the last section, but we will simplify the calculations by considering the limits of slow and long-time frequency decay, represented ν0 ≫ κ and ν0e −κT ≪ κ , (25) respectively. This allows us to do several things. First, it allows us to use the usual ground state of the unchirped trap at frequency ν0 as a good approximation to the ground state of the expanding trap at t = 0, since at that time the system is being chirped adiabatically. This is important because it allows the experiment to begin with a static potential, which is useful for cooling. Second, it allows us to simplify h(t) and g(t) using the phase-space rotation freedom discussed above. Using asymptotic ap- proximations for the Bessel functions in the coefficients, ≃ −Y1 , (26) , (27) we get h(t) ≃ sinϕY0 + cosϕJ0 , (28) ν0g(t) ≃ − cosϕY0 + sinϕJ0 . (29) where ϕ = ν0/κ − π/4. Since we are taking the initial state to be the ground state, which is symmetric with respect to phase-space rotations, we can use the freedom discussed in the previous section to undo the rotation represented by Eqs. (28) and (29) and define the simpler functions h(t) → h′(t) = , (30) g(t) → g′(t) = . (31) The primes are unnecessary due to the symmetry of the initial state, so we drop them from now on and plug di- rectly into Eq. (17): f(t) = − iJ0 , (32) where H n is a Hankel function of the first kind. The integral in Eq. (19) can be evaluated in the limits (25) using techniques similar to those used in Ref. [15]. First, define , τ = α− κt , u = eτ , and x = ∆/κ . The integral in question then becomes (neglecting the prefactor) dt e−i∆tH dt e−i∆tH α−κt) dτ e−ix(α−τ)H e−ixα dτ eixτH e−ixα du uix−1H 0 (u) . (34) Inserting a convergence factor with x → x− iǫ, and then taking the limit ǫ → 0+, we can use the formula du uix−1H 0 (u) = −2 ix Γ(ix/2) (eπx − 1)Γ(1− ix/2) to evaluate Γ(ix/2) Γ(1− ix/2) (eπx − 1)2 (eπx − 1)2 . (36) When plugging in for the dummy variables (33), this gives P (1) = (Ω0η0) 2 2πν0 (eπ∆/κ − 1)2 . (37) The calculated result from Ref. [15] for a single ion is GH = (Ω0η0) e2π∆/κ − 1 , (38) which contains a Planck factor with Gibbons- Hawking [16] temperature T = ~κ/2πkB but is different from the actual result for a single ion, given by Eq. (37). Several things should be noted about these functions. First, they both break down as ∆ → 0 because of the ap- proximation made in obtaining Eq. (14). They also fail if the time-dependent Lamb-Dicke parameter (20) ever becomes too large throughout the chirping process. Fur- thermore, most cases of interest will be ∆ ≃ ν0 (the first red sideband) and near ∆ ≃ −ν0 (the first blue side- band), which means that |∆| ≫ κ, since ν0 ≫ κ. The first red sideband represents a detector that requires the absorption of one phonon (plus one laser photon) in order to excite the atom—the usual thing we mean by “particle detector” when the particles are phonons. The first blue sideband, on the other hand, represents a detector that emits a phonon in order to excite the atom (along with absorbing one laser photon). There are a couple of ways to compare these functions. First, we can take the ratio of the two for both the red- and blue-sideband cases. In both cases, we obtain P (1) (1 + 2e−π|∆|/κ) (39) plus terms of order O(e−2π|∆|/κ). Since |∆| ≃ ν0, the prefactor is close to one, and the second term is very small (since ν0 ≫ κ). Furthermore, it is cumbersome to directly compare the measured probability to the full function (with all the prefactors). It is often easier in- stead to make measurements on both the first red side- band and the first blue sideband and then take the ratio of the two. The constant prefactors disappear in this calculation, and both functions then have the same ex- perimental signature: P (1)(∆) P (1)(−∆) GH(∆) GH(−∆) = e−2π∆/κ , (40) which is that of a thermal distribution with tempera- ture T = ~κ/2πkB, which is of the Gibbons-Hawking form [16] with the expansion rate given by κ. There- fore, although the Hamiltonian used in the calculations in Ref. [15] was missing terms, the intuition (at least for a single ion) was correct in that the actual experimental signature in this case matches that of an ion undergoing thermal motion in a static trap, where the temperature is proportional to κ. To see whether this experiment is feasible, we must ex- amine the validity of our approximations. For a typical trap, we expect that ν0 ≃ 1 MHz, and thus if we take κ ≃ 1 kHZ, we easily satisfy the first of conditions (25), namely ν0 ≫ κ. The second of these conditions gives a constraint on the modulation time T . For these param- eters we expect that T ≃ a few msec. This is compat- ible with typical cooling and readout time scales and is less than those for heating due to fluctuating patch po- tentials [10]. Thus, this is a realizable experiment with current technology. IV. CONCLUSION We have shown that a single trapped ion in a modu- lated trapping potential can serve as an experimentally accessible implementation of a quantum harmonic oscilla- tor with time-dependent frequency, including robust con- trol over state preparation, manipulation, and measure- ment. The ion itself serves both as the oscillating particle and as the local detector of vibrational motion via cou- pling to internal electronic states by an external laser. For the case of a general time-dependent trap frequency, we calculated the first-order excitation probability for the ion in terms of the solution to the classical equations of motion for the equivalent classical oscillator. We applied this general result to the case of exponential chirping and corrected the calculation in Ref. [15] for a single ion. We found that while the results from the two calculations dif- fer, the experimental signature in both cases is the same and equivalent to that of a thermal ion in a static trap. We thank Dave Kielpinski for invaluable help with the experimental details. We also thank Paul Alsing, Bill Unruh, John Preskill, Jeff Kimble, Greg Ver Steeg, and Michael Nielsen for useful discussions and suggestions. NCM extends much appreciation to the faculty and staff of the Caltech Institute for Quantum Information for their hospitality during his visit, which helped bring this work to fruition. NCM was supported by the United States Department of Defense, and GJM acknowledges support from the Australian Research Council. [1] D. Stoler, Phys. Rev. D 1, 3217 (1970). [2] H. P. Yuen, Phys. Rev. A 13, 2226 (1976). [3] J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). [4] X. Ma and W. Rhodes, Phys. Rev. A 39, 1941 (1989). [5] V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664 (1996). [6] G. T. Moore, J. Math. Phys. 11, 2679 (1970). [7] S. P. Kim and S.-W. Kim, Phys. Rev. D 51, 4254 (1995). [8] D. Polarski and A. A. Starobinsky, Classical and Quan- tum Gravity 13, 377 (1996). [9] C. Kiefer, J. Lesgourgues, D. Polarski, and A. A. Starobinsky, Classical and Quantum Gravity 15, L67 (1998). [10] D. Leibfried, R. Blatt, C. Monroe, and D.Wineland, Rev. Mod. Phys. 75, 281 (2003). [11] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. L. Gould, Phys. Rev. Lett. 75, 4011 (1995). [12] D. F. V. James, Applied Physics B: Lasers and Optics 66, 181 (1998). [13] D. Leibfried, B. De Marco, V. Meyer, D. Lucas, M. Bar- rett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, et al., Nature 422, 412 (2003). [14] F. Schmidt-Kaler, H. Häffner, M. Riebe, G. P. T. Lan- caster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, Nature 422, 408 (2003). [15] P. M. Alsing, J. P. Dowling, and G. J. Milburn, Phys. Rev. Lett. 94, 220401 (2005). [16] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). [17] S. Wallentowitz and W. Vogel, Phys. Rev. A 54, 3322 (1996). [18] A. M. de M. Carvalho, C. Furtado, and I. A. Pedrosa, Phys. Rev. D 70, 123523 (pages 6) (2004). [19] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [20] The authors of Ref. [15] consider both signs in the ex- ponential, but we will restrict ourselves to the case that allows us to begin chirping in the adiabatic limit.

0704.0136

Compounding Fields and Their Quantum Equations in the Trigintaduonion Space

Microsoft Word - Compounding Fields and Their Quantum Equations in the Trigintaduonion Space-10.doc Compounding Fields and Their Quantum Equations in the Trigintaduonion Space Zihua Weng School of Physics and Mechanical & Electrical Engineering, P. O. Box 310, Xiamen University, Xiamen 361005, China Abstract The 32-dimensional compounding fields and their quantum interplays in the trigintaduonion space can be presented by analogy with octonion and sedenion electromagnetic, gravitational, strong and weak interactions. In the trigintaduonion fields which are associated with the electromagnetic, gravitational, strong and weak interactions, the study deduces some conclusions of field source particles (quarks and leptons) and intermediate particles which are consistent with current some sorts of interaction theories. In the trigintaduonion fields which are associated with the hyper-strong and strong-weak fields, the paper draws some predicts and conclusions of the field source particles (sub-quarks) and intermediate particles. The research results show that there may exist some new particles in the nature. Keywords: sedenion space; strong interaction; weak interaction; quark; sub-quark. 1. Introduction Nowadays, there still exist some movement phenomena which can’t be explained by current sub-quark field theories. Therefore, some scientists bring forward some new field theories to explain the strange phenomena in the strong interaction and weak interaction etc. A new insight on the problem of the sub-quark movement and their interactions can be given by the concept of trigintaduonion space. According to previous research results and the ‘SpaceTime Equality Postulation’ [1-5], the eight sorts of interactions in the paper can all be described by quaternion spacetimes. Based on the conception of space verticality etc., these eight types of quaternion spacetimes can be united into the 32-dimensional trigintaduonion space. In the trigintaduonion space, the characteristics of eight sorts of interactions can be described by single trigintaduonion space uniformly. By analogy with the octonionic and sedenion fields, four sorts of trigintaduonion fields which consist of octonionic fields H-S, S-W and E-G etc., can be obtained in the paper. The paper describes the trigintaduonion fields and their quantum theory, and deduces some predicts and new conclusions which are consistent with the current sub-quark theories etc. _________ E-mail Addresses: [email protected], [email protected] 2. Compounding fields in trigintaduonion spaces Through the analysis of the different fields in the octonionic and sedenion spaces, we find that each interaction possesses its own spacetime, field and operator in accordance with the ‘SpaceTime Equality Postulation’. In the sedenion spaces, sixteen sorts of sedenion fields can be tabulated in Table 1, including their operators, spaces and fields. Table 1. The compounding fields and operators in the different sedenion spaces operator X H-S / k H-S + +A S-W / k S-W A H-S / k B E-G / k E-G +B H-S / k +S H-W / k S H-S / k space octonion space sedenion space SW-HS sedenion space EG-HS sedenion space HW-HS field H-S SW-HS EG-HS HW-HS operator X H-S / k +X S-W / k A S-W / k S-W + B E-G / k E-G +B S-W / k +S H-W / k S S-W / k space sedenion space HS-SW octonion space sedenion space EG-SW sedenion space HW-SW field HS-SW S-W EG-SW HW-SW operator X H-S / k +X E-G / k +A S-W / k A E-G / k B E-G / k E-G + +S H-W / k S E-G / k space sedenion space HS-EG sedenion space SW-EG octonion space sedenion space HW-EG field HS-EG SW-EG E-G HW-EG operator X H-S / k +X H-W / k +A S-W / k A H-W / k B E-G / k E-G +B H-W / k S H-W / k H-W + space sedenion space HS-HW sedenion space SW-HW sedenion space EG-HW octonion space field HS-HW SW-HW EG-HW H-W In the Cayley-Dickson algebra, there exists the Cayley-Dickson construction [6]. This is the process based on which the 2n-dimensional hypercomplex number is constructed from a pair of (2n-1)-dimensional hypercomplex numbers, where n is a positive integer. This is accomplished by defining the multiplication rule for the two 2n-dimensional hypercomplex numbers in terms of the four (2n-1)-dimensional hypercomplex numbers. The 2-dimensional complex numbers (n = 1), 4-dimensional quaternions (n = 2), 8-dimensional octonions (n = 3), 16-dimensional sedenions (n = 4), 32-dimensional trigintaduonions (n = 5), etc., can all be constructed from real numbers by the iterations of this process [7]. At each iteration some new basal elements, e k , are introduced with the property, e k = 1. We define the product and conjugate on the trigintaduonions, (u, v) and (x, y), in terms of the sedenions, u, v, x and y, as follows: (u, v) (x, y) = (u x y* v, y u + v x*) , (u, v)* = (u*, v) where, the mark (*) denotes the conjugate. In the trigintaduonion space, there exist different constructions of fields in the terms of different operators. By analogy with the cases in the different octonionic spaces and sedenion spaces, the operators and fields in the different trigintaduonion spaces can be written in Table 2. There exist four sorts of compounding fields in the trigintaduonion spaces. Table 2. The compounding fields and operators in the trigintaduonion space operator X H-S / k H-S +X S-W / k S-W X E-G / k E-G +X H-W / k A H-S / k H-S +A S-W / k S-W A E-G / k E-G +A H-W / k B H-S / k H-S +B S-W / k S-W B E-G / k E-G +B H-W / k S H-S / k H-S +S S-W / k S-W S E-G / k E-G +S H-W / k space T-X T-A T-B T-S field T-X T-A T-B T-S 3. Compounding field in trigintaduonion space T-X It is believed that hyper-strong field, strong-weak field, electromagnetic-gravitational field and hyper-weak field are unified, equal and interconnected. By means of the conception of the space expansion etc., four types of octonionic spaces can be combined into a trigintaduonion space T-X. In trigintaduonion space, some properties of eight sorts of interactions including strong, weak, electromagnetic and gravitational interactions etc. can be described uniformly. In the trigintaduonion space T-X, the displacement r should be extended to the new displacement R = (r + krx X ) and be consistent with the definition of momentum M. In the octonionic space H-S, the base E H-S can be written as E H-S = (1, e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 ) (1) The displacement R H-S = ( R0 , R1 , R2 , R3 , R4 , R5 , R6 , R7 ) in the octonionic space H-S is consist of the displacement r H-S = ( r0 , r1 , r2 , r3 , r4 , r5 , r6 , r7 ) and physical quantity X H-S = ( x0 , x1 , x2 , x3 , x4 , x5 , x6 , x7 ). R H-S = r H-S + k rx X H-S = R0 + e 1 R1 + e 2 R2 + e 3 R3 + e 4 R4 + e 5 R5 + e 6 R6 + e 7 R7 (2) where, R j = r j + k rx x j ; j = 0, 1, 2, 3, 4, 5, 6, 7. r0 = c H-S t H-S , r4 = c H-S T H-S . c H-S is the speed of intermediate particle in the hyper-strong field, t H-S and T H-S denote the time. The octonionic differential operator T-X1 and its conjugate operator are defined as T-X1 = 0 + e 1 1 + e 2 2 + e 3 3 + e 4 4 + e 5 5 + e 6 6 + e 7 7 (3) T-X1 = 0 e 1 1 e 2 2 e 3 3 e 4 4 e 5 5 e 6 6 e 7 7 (4) where, j = / R j . The mark (*) denotes the octonionic conjugate. In the octonionic space S-W, the base E S-W can be written as E S-W = ( e 8 , e 9 , e 10 , e 11 , e 12 , e 13 , e 14 , e 15 ) (5) The displacement R S-W = (R8 , R9 , R10 , R11 , R12 , R13 , R14 , R15 ) in the octonionic space S-W is consist of the displacement r S-W = ( r8 , r9 , r10 , r11 , r12 , r13 , r14 , r15 ) and the physical quantity X S-W = ( x8 , x9 , x10 , x11 , x12 , x13 , x14 , x15 ). R S-W = r S-W + k rx X S-W = e 8 R8 + e 9 R9 + e 10 R10 + e 11 R11 + e 12 R12 + e 13 R13 + e 14 R14 + e 15 R15 (6) where, R j = r j + k rx x j ; j = 8, 9, 10, 11, 12, 13, 14, 15. r8 = c S-W t S-W , r12 = c S-W T S-W . c S-W is the speed of intermediate particle in strong-weak field, t S-W and T S-W denote the time. The octonionic differential operator T-X2 and its conjugate operator are defined as T-X2 = e 8 8 + e 9 9 + e 10 10 + e 11 11 + e 12 12 + e 13 13 + e 14 14 + e 15 15 (7) T-X2 = e 8 8 e 9 9 e 10 10 e 11 11 e 12 12 e 13 13 e 14 14 e 15 15 (8) where, j = / R j . In the octonionic space E-G, the base E E-G can be written as E E-G = ( e 16 , e 17 , e 18 , e 19 , e 20 , e 21 , e 22 , e 23 ) (9) The displacement R E-G = ( R16 , R17 , R18 , R19 , R20 , R21 , R22 , R23 ) in the octonionic space E-G is consist of the displacement r E-G = ( r16 , r17 , r18 , r19 , r20 , r21 , r22 , r23 ) and the physical quantity XE-G = ( x16 , x17 , x18 , x19 , x20 , x21 , x22 , x23 ). R E-G = r E-G + k rx X E-G = e 16 R16 + e 17 R17 + e 18 R18 + e 19 R19 + e 20 R20 + e 21 R21 + e 22 R22 + e 23 R23 (10) where, R j = r j + k rx x j ; j = 16, 17, 18, 19, 20, 21, 22, 23. r16 = c E-G t E-G , r20 = c E-G T E-G . c E-G is the speed of intermediate particle in electromagnetic-gravitational field, t E-G and T E-G denote the time. The octonionic differential operator T-X3 and its conjugate operator are defined as T-X3 = e 16 16 + e 17 17 + e 18 18 + e 19 19 + e 20 20 + e 21 21 + e 22 22 + e 23 23 (11) T-X3 = e 16 16 e 17 17 e 18 18 e 19 19 e 20 20 e 21 21 e 22 22 e 23 23 (12) where, j = / R j . In the octonionic space H-W, the base E H-W can be written as E H-W = ( e 24 , e 25 , e 26 , e 27 , e 28 , e 29 , e 30 , e 31 ) (13) The displacement R H-W = ( R24 , R25 , R26 , R27 , R28 , R29 , R30 , R31 ) in octonionic space H-W is consist of the displacement r H-W = ( r24 , r25 , r26 , r27 , r28 , r29 , r30 , r31 ) and the physical quantity XH-W = ( x24 , x25 , x26 , x27 , x28 , x29 , x30 , x31 ). R H-W = r H-W + k rx X H-W = e 24 R24 + e 25 R25 + e 26 R26 + e 27 R27 + e 28 R28 + e 29 R29 + e 30 R30 + e 31 R31 (14) where, R j = r j + k rx x j ; j = 24, 25, 26, 27, 28, 29, 30, 31. r24 = c H-W t H-W , r28 = c H-W T H-W . c H-W is the speed of intermediate particle in hyper-weak field, t H-W and T H-W denote the time. The octonionic differential operator T-X4 and its conjugate operator are defined as, T-X4 = e 24 24 + e 25 25 + e 26 26 + e 27 27 + e 28 28 + e 29 29 + e 30 30 + e 31 31 (15) T-X4 = e 24 24 e 25 25 e 26 26 e 27 27 e 28 28 e 29 29 e 30 30 e 31 31 (16) where, j = / R j . In the trigintaduonion space T-X, the base E T-X can be written as E T-X = E T-X1 + E T-X2 + E T-X3 + E T-X4 = (1, e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8, e 9, e 10, e 11, e 12, e 13, e 14, e 15, e 16, e 17, e 18, e 19, e 20, e 21, e 22, e 23, e 24, e 25, e 26, e 27, e 28, e 29, e 30, e 31) (17) The displacement R T-X = ( R0 , R1 , R2 , R3 , R4 , R5 , R6 , R7 , R8 , R9 , R10 , R11 , R12 , R13 , R14 , R15 , R16 , R17 , R18 , R19 , R20 , R21 , R22 , R23 , R24 , R25 , R26 , R27 , R28 , R29 , R30 , R31 ) in trigintaduonion space T-X is R T-X = R T-X1 + R T-X2 + R T-X3 + R T-X4 = R0 + e 1 R1 + e 2 R2 + e 3 R3 + e 4 R4 + e 5 R5 + e 6 R6 + e 7 R7 + e 8 R8 + e 9 R9 + e 10 R10 + e 11 R11 + e 12 R12 + e 13 R13 + e 14 R14 + e 15 R15 + e 16 R16 + e 17 R17 + e 18 R18 + e 19 R19 + e 20 R20 + e 21 R21 + e 22 R22 + e 23 R23 + e 24 R24 + e 25 R25 + e 26 R26 + e 27 R27 + e 28 R28 + e 29 R29 + e 30 R30 + e 31 R31 (18) The trigintaduonion differential operator T-X and its conjugate operator are defined as T-X = T-X1 + T-X2 + T-X3 + T-X4 (19) T-X = T-X1 + T-X2 + T-X3 + T-X4 (20) In the trigintaduonion space T-X, there exists one kind of field (trigintaduonion field T-X, for short) can be obtained related to the operator (X/K + ). In the trigintaduonion field T-X, by analogy with the octonion and sedenion fields, the trigintaduonion differential operator needs to be generalized to the operator (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ). This is because the trigintaduonion field T-X includes the hyper-strong, strong-weak, electromagnetic-gravitational and hyper-weak fields. It can be predicted that the eight sorts of interactions are interconnected each other. The physical features of each subfield in the trigintaduonion field T-X meet the requirements of the equations set in the Table 3. In the trigintaduonion field T-X, the field potential A = (a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 , a13 , a14 , a15 , a16 , a17 , a18 , a19 , a20 , a21 , a22 , a23 , a24 , a25 , a26 , a27 , a28 , a29 , a30 , a31 ) is defined as A = (X/K + )* X = (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) = a0 + a1 e 1 + a2 e 2 + a3 e 3 + a4 e 4 + a5 e 5 + a6 e 6 + a7 e 7 + a8 e 8 + a9 e 9 + a10 e 10 + a11 e 11 + a12 e 12 + a13 e 13 + a14 e 14 + a15 e 15 + a16 e 16 + a17 e 17 + a18 e 18 + a19 e 19 + a20 e 20 + a21 e 21 + a22 e 22 + a23 e 23 + a24 e 24 + a25 e 25 + a26 e 26 + a27 e 27 + a28 e 28 + a29 e 29 + a30 e 30 + a31 e 31 (21) where, the mark (*) denotes the trigintaduonion conjugate. krx X = krx XT-X = k rx XH-S + rx XS-W + k rx XE-G + k rx XH-W . K = KT-X , k H-S , k S-W , k E-G , k H-W , k rx , k rx , rx and k rx are coefficients. XH-S is the physical quantity in the octonionic space H-S; XS-W is the physical quantity in octonionic space S-W; XE-G is the physical quantity in the octonionic space E-G; XH-W is the physical quantity in the octonionic space H-W. The field strength B of the trigintaduonion field T-X can be defined as B = (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) A (22) The field source and force of the trigintaduonion field T-X can be defined respectively as S = (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) * B (23) Z = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) S (24) where, the coefficient is interaction intensity of the trigintaduonion field T-X. The angular momentum of trigintaduonion field can be defined as (k rx is the coefficient) M = S (r + k rx X) (25) and the energy and power in the trigintaduonion field can be defined respectively as W = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) * M (26) N = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) W (27) Table 3. Equations set of trigintaduonion field T-X Spacetime trigintaduonion space T-X X physical quantity X = XT-X Field potential A = (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) Field strength B = (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) A Field source S = (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) Force Z = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) S Angular momentum M = S (r + k rx X) Energy W = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) Power N = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) W In the trigintaduonion space T-X, the wave functions of the quantum mechanics are the trigintaduonion equations set. The Dirac and Klein-Gordon equations of quantum mechanics are actually the wave equations set which are associated with particle’s angular momentum. In the trigintaduonion field T-X, the Dirac equation and Klein-Gordon equation can be attained respectively from the energy equation (26) and power equation (27) after substituting the operator K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) for the operator (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ). The coefficients b H-S , b S-W , b E-G and b H-W are the Plank-like constant. The U equation of the quantum mechanics can be defined as U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * M (28) The L equation of the quantum mechanics can be defined as L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U (29) The four sorts of Dirac-like equations can be obtained from the Eqs.(21), (22), (23) and (24) respectively. The D equation of quantum mechanics can be defined as D = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * X (30) The G equation of quantum mechanics can be defined as G = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) D (31) The T equation of quantum mechanics can be defined as T = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * G (32) The O equation of quantum mechanics can be defined as O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) T (33) In the trigintaduonion field T-X, the intermediate and field source particles can be obtained. We can find that the intermediate particles and other kinds of new and unknown particles may be existed in the nature. Table 4. Quantum equations set of trigintaduonion field T-X Energy quantum U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Power quantum L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U Field potential quantum D = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Field strength quantum G = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) D Field source quantum T = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Force quantum O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) T 4. Compounding field in trigintaduonion space T-A It is believed that strong-weak field, hyper-strong field, electromagnetic-gravitational field and hyper-weak field are unified, equal and interconnected. By means of the conception of the space expansion etc., four types of octonionic spaces can be combined into a trigintaduonion space T-A. In trigintaduonion space, some properties of eight sorts of interactions including strong, weak, electromagnetic and gravitational interactions etc. can be described uniformly. In the trigintaduonion space T-A, there exists one kind of field (trigintaduonion field T-A, for short) which is different to the trigintaduonion field T-X, can be obtained related to the operator (A/K + ). In the trigintaduonion space T-A, the base E T-A can be written as E T-A = E T-X (34) The displacement R T-A in trigintaduonion space T-A is R T-A = R T-X (35) The trigintaduonion differential operator T-A and its conjugate operator are defined as T-A = T-X , T-A = T-X (36) In the trigintaduonion field T-A, by analogy with the octonion and sedenion fields, the trigintaduonion differential operator needs to be generalized to the operator (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ). This is because the trigintaduonion field T-A includes hyper-strong, strong-weak, electromagnetic-gravitational and hyper-weak fields. It can be predicted that the eight sorts of interactions are interconnected each other. The physical features of each subfield in the trigintaduonion field T-A meet the requirements of the equations set in the Table 5. In the trigintaduonion field T-A, the field potential A = (a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 , a13 , a14 , a15 , a16 , a17 , a18 , a19 , a20 , a21 , a22 , a23 , a24 , a25 , a26 , a27 , a28 , a29 , a30 , a31 ) is defined as A = * X = a0 + a1 e 1 + a2 e 2 + a3 e 3 + a4 e 4 + a5 e 5 + a6 e 6 + a7 e 7 + a8 e 8 + a9 e 9 + a10 e 10 + a11 e 11 + a12 e 12 + a13 e 13 + a14 e 14 + a15 e 15 + a16 e 16 + a17 e 17 + a18 e 18 + a19 e 19 + a20 e 20 + a21 e 21 + a22 e 22 + a23 e 23 + a24 e 24 + a25 e 25 + a26 e 26 + a27 e 27 + a28 e 28 + a29 e 29 + a30 e 30 + a31 e 31 (37) where, the mark (*) denotes the trigintaduonion conjugate. X = XT-A = XT-X . The field strength B of the trigintaduonion field T-A can be defined as B = (A/K + ) A = (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) A (38) where, K= KT-A , k H-S , k S-W , k E-G and k H-W are coefficients in the trigintaduonion space. The field potentials are A H-S = a0 + a1 e 1 + a2 e 2 + a3 e 3 + a4 e 4 + a5 e 5 + a6 e 6 + a7 e 7 A S-W = a8 e 8 + a9 e 9 + a10 e 10 + a11 e 11 + a12 e 12 + a13 e 13 + a14 e 14 + a15 e 15 A E-G = a16 e 16 + a17 e 17 + a18 e 18 + a19 e 19 + a20 e 20 + a21 e 21 + a22 e 22 + a23 e 23 A H-W = a24 e 24 + a25 e 25 + a26 e 26 + a27 e 27 + a28 e 28 + a29 e 29 + a30 e 30 + a31 e 31 The field source and force of the trigintaduonion field T-A can be defined respectively as S = (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) * B (39) Z = K (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) S (40) where, the coefficient is interaction intensity of the trigintaduonion field T-A. The angular momentum of trigintaduonion field can be defined as (k rx is the coefficient) M = S (r + k rx X) (41) and the energy and power in the trigintaduonion field can be defined respectively as W = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) * M (42) N = K (X H-S / k H-S X S-W / k S-W X E-G / k E-G +X H-W / k H-W + ) W (43) In the trigintaduonion space T-A, the wave functions of the quantum mechanics are the trigintaduonion equations set. The Dirac and Klein-Gordon equations of quantum mechanics are actually the wave equations set which are associated with particle’s angular momentum. In the trigintaduonion field T-A, the Dirac equation and the Klein-Gordon equation can be attained respectively from the energy equation (42) and power equation (43) after substituting the operator K (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) for the operator (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ). The coefficients b H-S , b S-W , b E-G and b H-W are the Plank-like constant. Table 5. Equations set of trigintaduonion field T-A Spacetime trigintaduonion space T-A X physical quantity X = XT-X Field potential A = * X Field strength B = (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) A Field source S = (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) Force Z = K (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) S Angular momentum M = S (r + k rx X) Energy W = K (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) Power N = K (A H-S / k H-S A S-W / k S-W A E-G / k E-G +A H-W / k H-W + ) W The U equation of the quantum mechanics can be defined as U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * M (44) The L equation of the quantum mechanics can be defined as L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U (45) Table 6. Quantum equations set of trigintaduonion field T-A Energy quantum U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Power quantum L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U Field strength quantum G = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) A Field source quantum T = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Force quantum O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) T The three sorts of Dirac-like equations can be obtained from Eqs.(38), (39) and (40) respectively. The G equation of the quantum mechanics can be defined as G = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) A (46) The T equation of the quantum mechanics can be defined as T = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * G (47) The O equation of the quantum mechanics can be defined as O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) T (48) In the trigintaduonion field T-A, the intermediate and field source particles can be obtained. We can find that the intermediate particles and other kinds of new and unknown particles may be existed in the nature. 5. Compounding field in trigintaduonion space T-B It is believed that electromagnetic-gravitational field, strong-weak field, hyper-strong field and hyper-weak field are unified, equal and interconnected. By means of the conception of the space expansion etc., four types of octonionic spaces can be combined into a trigintaduonion space T-B. In trigintaduonion space, some properties of eight sorts of interactions including strong, weak, electromagnetic and gravitational interactions etc. can be described uniformly. In the trigintaduonion space T-B, there exists one kind of field (trigintaduonion field T-B, for short) which is different to the trigintaduonion field T-X or T-A, can be obtained related to the operator (B/K + ). In the trigintaduonion space T-B, the base E T-B can be written as E T-B = E T-X (49) The displacement R T-B in trigintaduonion space T-B is R T-B = R T-X (50) The trigintaduonion differential operator T-B and its conjugate operator are defined as T-B = T-X , T-B = T-X (51) In the trigintaduonion field T-B, by analogy with the octonion and sedenion fields, the trigintaduonion differential operator needs to be generalized to the operator (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ). This is because the trigintaduonion field T-B includes hyper-strong, strong-weak, electromagnetic-gravitational and hyper-weak fields. It can be predicted that the eight sorts of interactions are interconnected each other. The physical features of each subfield in the trigintaduonion field T-B meet the requirements of the equations set in the Table 7. In the trigintaduonion field T-B, the field potential A = (a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 , a13 , a14 , a15 , a16 , a17 , a18 , a19 , a20 , a21 , a22 , a23 , a24 , a25 , a26 , a27 , a28 , a29 , a30 , a31 ) is defined as A = * X = a0 + a1 e 1 + a2 e 2 + a3 e 3 + a4 e 4 + a5 e 5 + a6 e 6 + a7 e 7 + a8 e 8 + a9 e 9 + a10 e 10 + a11 e 11 + a12 e 12 + a13 e 13 + a14 e 14 + a15 e 15 + a16 e 16 + a17 e 17 + a18 e 18 + a19 e 19 + a20 e 20 + a21 e 21 + a22 e 22 + a23 e 23 + a24 e 24 + a25 e 25 + a26 e 26 + a27 e 27 + a28 e 28 + a29 e 29 + a30 e 30 + a31 e 31 (52) where, the mark (*) denotes the trigintaduonion conjugate. X = XT-B = XT-X . The field strength B of the trigintaduonion field T-B can be defined as B = A (53) The field source of the trigintaduonion field T-B can be defined as S = (B/K + ) * B = (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) * B (54) where, K = KT-B , k H-S , k S-W , k E-G and k H-W are coefficients in the trigintaduonion space. The coefficient is interaction intensity of trigintaduonion field T-B. The field strengths are B H-S = b0 + b1 e 1 + b2 e 2 + b3 e 3 + b4 e 4 + b5 e 5 + b6 e 6 + b7 e 7 B S-W = b8 e 8 + b9 e 9 + b10 e 10 + b11 e 11 + b12 e 12 + b13 e 13 + b14 e 14 + b15 e 15 B E-G = b16 e 16 + b17 e 17 + b18 e 18 + b19 e 19 + b20 e 20 + b21 e 21 + b22 e 22 + b23 e 23 B H-W = b24 e 24 + b25 e 25 + b26 e 26 + b27 e 27 + b28 e 28 + b29 e 29 + b30 e 30 + b31 e 31 The force of the trigintaduonion field T-B can be defined as Z = K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) S (55) The angular momentum of trigintaduonion field can be defined as (k rx is the coefficient) M = S (r + k rx X) (56) and the energy and power in the trigintaduonion field can be defined respectively as W = K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) * M (57) N = K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) W (58) In the trigintaduonion space T-B, the wave functions of the quantum mechanics are the trigintaduonion equations set. The Dirac and Klein-Gordon equations of quantum mechanics are actually the wave equations set which are associated with particle’s angular momentum. In the trigintaduonion field T-B, the Dirac equation and the Klein-Gordon equation can be attained respectively from the energy equation (57) and power equation (58) after substituting the operator K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) for the operator (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ). The coefficients b H-S , b S-W , b E-G and b H-W are the Plank-like constant. Table 7. Equations set of trigintaduonion field T-B Spacetime trigintaduonion space T-B X physical quantity X = XT-X Field potential A = * X Field strength B = A Field source S = (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) Force Z = K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) S Angular momentum M = S (r + k rx X) Energy W = K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) Power N = K (B H-S / k H-S B S-W / k S-W B E-G / k E-G +B H-W / k H-W + ) W The U equation of the quantum mechanics can be defined as U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * M (59) The L equation of the quantum mechanics can be defined as L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U (60) The two sorts of Dirac-like equations can be obtained from the field source equation (54) and force equation (55) respectively. The T equation of the quantum mechanics can be defined as T = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * B (61) The O equation of the quantum mechanics can be defined as O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) T (62) Table 8. Quantum equations set of trigintaduonion field T-B Energy quantum U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Power quantum L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U Field source quantum T = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Force quantum O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) T In the trigintaduonion field T-B, the intermediate and field source particles can be obtained. We can find that the intermediate particles and other kinds of new and unknown particles may be existed in the nature. 6. Compounding field in trigintaduonion space T-S It is believed that hyper-weak field, electromagnetic-gravitational field, strong-weak field and hyper-strong field are unified, equal and interconnected. By means of the conception of the space expansion etc., four types of octonionic spaces can be combined into a trigintaduonion space T-S. In trigintaduonion space, some properties of eight sorts of interactions including strong, weak, electromagnetic and gravitational interactions etc. can be described uniformly. In the trigintaduonion space T-S, there exists one kind of field (trigintaduonion field T-S, for short) which is different to trigintaduonion field T-X, T-A or T-B, can be obtained related to operator (S/K + ). In the trigintaduonion space T-S, the base E T-S can be written as E T-S = E T-X (63) The displacement R T-S in trigintaduonion space T-S is R T-S = R T-X (64) The trigintaduonion differential operator T-S and its conjugate operator are defined as T-S = T-X , T-S = T-X (65) In the trigintaduonion field T-S, by analogy with the octonion and sedenion fields, the trigintaduonion differential operator needs to be generalized to a new operator (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ). This is because the trigintaduonion field T-S includes the hyper-strong, strong-weak, electromagnetic-gravitational and hyper-weak fields. It can be predicted that the eight sorts of interactions are interconnected each other. The physical features of each subfield in the trigintaduonion field T-S meet the requirements of the equations set in the Table 9. In the trigintaduonion field T-S, the field potential A = (a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 , a13 , a14 , a15 , a16 , a17 , a18 , a19 , a20 , a21 , a22 , a23 , a24 , a25 , a26 , a27 , a28 , a29 , a30 , a31 ) is defined as A = * X = a0 + a1 e 1 + a2 e 2 + a3 e 3 + a4 e 4 + a5 e 5 + a6 e 6 + a7 e 7 + a8 e 8 + a9 e 9 + a10 e 10 + a11 e 11 + a12 e 12 + a13 e 13 + a14 e 14 + a15 e 15 + a16 e 16 + a17 e 17 + a18 e 18 + a19 e 19 + a20 e 20 + a21 e 21 + a22 e 22 + a23 e 23 + a24 e 24 + a25 e 25 + a26 e 26 + a27 e 27 + a28 e 28 + a29 e 29 + a30 e 30 + a31 e 31 (66) where, the mark (*) denotes the trigintaduonion conjugate. X = XT-S = XT-X . The field strength B of the trigintaduonion field T-S can be defined as B = A (67) The field source of the trigintaduonion field T-S can be defined as S = * B (68) where, the coefficient is interaction intensity of the trigintaduonion field T-S. The force of the trigintaduonion field T-S can be defined as Z = K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) S (69) where, K = KT-S , k H-S , k S-W , k E-G and k H-W are coefficients in the trigintaduonion space. And the field sources are S H-S = s0 + s1 e 1 + s2 e 2 + s3 e 3 + s4 e 4 + s5 e 5 + s6 e 6 + s7 e 7 S S-W = s8 e 8 + s9 e 9 + s10 e 10 + s11 e 11 + s12 e 12 + s13 e 13 + s14 e 14 + s15 e 15 S E-G = s16 e 16 + s17 e 17 + s18 e 18 + s19 e 19 + s20 e 20 + s21 e 21 + s22 e 22 + s23 e 23 S H-W = s24 e 24 + s25 e 25 + s26 e 26 + s27 e 27 + s28 e 28 + s29 e 29 + s30 e 30 + s31 e 31 Table 9. Equations set of trigintaduonion field T-S Spacetime trigintaduonion space T-S X physical quantity X = XT-X Field potential A = * X Field strength B = A Field source S = * B Force Z = K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) S Angular momentum M = S (r + k rx X) Energy W = K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) Power N = K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) W The angular momentum of trigintaduonion field can be defined as (k rx is the coefficient) M = S (r + k rx X) (70) and the energy and power in the trigintaduonion field can be defined respectively as W = K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) * M (71) N = K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) W (72) In the trigintaduonion space T-S, the wave functions of the quantum mechanics are the trigintaduonion equations set. The Dirac and Klein-Gordon equations of quantum mechanics are actually the wave equations set which are associated with particle’s angular momentum. In the trigintaduonion field T-S, the Dirac equation and the Klein-Gordon equation can be attained respectively from the energy equation (71) and power equation (72) after substituting the operator K (S H-S / k H-S S S-W / k S-W S E-G / k E-G +S H-W / k H-W + ) for the new operator (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ). The coefficients b H-S , b S-W , b E-G and b H-W are the Plank-like constant. The U equation of the quantum mechanics can be defined as U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) * M (73) The L equation of the quantum mechanics can be defined as L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U (74) The Dirac-like equation can be obtained from the force equation (69). The O equation of the quantum mechanics can be defined as O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) S (75) Table 10. Quantum equations set of trigintaduonion field T-S Energy quantum U = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) Power quantum L = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) U Force quantum O = (W H-S / k H-S b H-S W S-W / k S-W b S-W W E-G / k E-G b E-G +W H-W / k H-W b H-W + ) S In the trigintaduonion field T-S, the intermediate and field source particles can be obtained. We can find that the intermediate particles and other kinds of new and unknown particles may be existed in the nature. 7. Special case of compounding field in trigintaduonion space It is believed that different sorts of interactions are all unified, equal and interconnected. By means of the conception of the space expansion etc., four types of the octonionic spaces can be combined into a trigintaduonion space T-C. In the trigintaduonion space, some properties of eight sorts of interactions including the strong, weak, electromagnetic and gravitational interactions etc. can be described uniformly. In the trigintaduonion space T-C, there exists one kind of field (trigintaduonion field T-C, for short) which is the special case of the trigintaduonion fields T-X, T-A, T-B or T-S, can be obtained related to the operator . In the trigintaduonion space T-C, the base E T-C can be written as E T-C = E T-X (76) The displacement R T-C in trigintaduonion space T-C is R T-C = R T-X (77) The trigintaduonion differential operator T-C and its conjugate operator are defined as T-C = T-X , T-C = T-X (78) It can be predicted that the eight sorts of interactions are interconnected each other. The physical features of each subfield in the trigintaduonion field T-C meet the requirements of the equations set in the Table 11. In the trigintaduonion field T-C, the field potential A = (a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 , a13 , a14 , a15 , a16 , a17 , a18 , a19 , a20 , a21 , a22 , a23 , a24 , a25 , a26 , a27 , a28 , a29 , a30 , a31 ) is defined as A = * X (79) where, the mark (*) denotes the trigintaduonion conjugate. X = XT-C = XT-X . The field strength B of the trigintaduonion field T-C can be defined as B = A (80) The field source of the trigintaduonion field T-C can be defined as S = * B (81) where, the coefficient is interaction intensity of the trigintaduonion field T-C. The force of the trigintaduonion field T-C can be defined as Z = K S (82) where, K = KT-C is the coefficient in the trigintaduonion space. The angular momentum of trigintaduonion field can be defined as (k rx is the coefficient) M = S (r + k rx X) (83) and the energy and power in the trigintaduonion field can be defined respectively as W = K * M (84) N = K W (85) Table 11. Equations set of trigintaduonion field T-C Spacetime trigintaduonion space T-C X physical quantity X = XT-X Field potential A = * X Field strength B = A Field source S = * B Force Z = K S Angular momentum M = S (r + k rx X) Energy W = K * M Power N = K W In the trigintaduonion space T-C, the wave functions of the quantum mechanics are the trigintaduonion equations set. The Dirac and Klein-Gordon equations of quantum mechanics are actually the wave equations set which are associated with particle’s angular momentum M = b . The coefficient b is the Plank-like constant. In the trigintaduonion field T-C, the Dirac equation and the Klein-Gordon equation can be attained respectively from the energy equation (84) and the power equation (85). The U equation of the quantum mechanics can be defined as U = (b )* (M / b) (86) The L equation of the quantum mechanics can be defined as L = (b ) (U / b) (87) The four sorts of Dirac-like equations can be obtained from Eqs.(79), (80), (81) and (82) respectively. The D equation of the quantum mechanics can be defined as D = (b )* (X / b) (88) The G equation of the quantum mechanics can be defined as G = (b ) (D / b) (89) The T equation of the quantum mechanics can be defined as T = (b )* (G / b) (90) The O equation of the quantum mechanics can be defined as O = (b ) (T / b) (91) Table 12. Quantum equations set of trigintaduonion field T-C Energy quantum U = (b )* (M /b) Power quantum L = (b ) (U /b) Field potential quantum D = (b )* (X /b) Field strength quantum G = (b ) (D /b) Field source quantum T = (b )* (G /b) Force quantum O = (b ) (T /b) 8. Conclusions By analogy with the four sorts of octonionic fields and twelve sorts of sedenion fields, four sorts of trigintaduonion fields and their special case have been developed, including their field equations, quantum equations and some new unknown particles. In trigintaduonion field T-X, the study deduces the Dirac equation, Schrodinger equation, Klein-Gordon equation and some newfound equations of sub-quarks etc. It infers four sorts of Dirac-like equations of intermediate particles among sub-quarks etc. It predicts that there are some new particles of field sources (sub-quarks etc.) and their intermediate particles. In trigintaduonion field T-A, the paper draws the Yang-Mills equation, Dirac equation, Schrodinger equation and Klein-Gordon equation of the quarks and leptons etc. It infers three sorts of Dirac-like equations of intermediate particles among quarks and leptons. It draws some conclusions of field source particles and intermediate particles which are consistent with current electro-weak theory. It predicts that there are some new unknown particles of field sources (quarks and leptons) and their intermediate particles. In trigintaduonion field T-B, the research infers the Dirac equation, Schrodinger equation, Klein-Gordon equation and some newfound equations of electrons and masses etc. It deduces two sorts of Dirac-like equations of intermediate particles among electrons and masses etc. It draws some conclusions of field source particles and intermediate particles which are consistent with current electromagnetic and gravitational theories etc. It predicts that there are some new unknown particles of field sources (electrons and masses etc.) and their intermediate particles. In trigintaduonion field T-S, the thesis concludes the Dirac equation, Schrodinger equation and Klein-Gordon equation of the galaxies etc. It infers Dirac-like equation of intermediate particles among galaxies. It predicts that there are some new unknown particles of field sources and their intermediate particles. In the trigintaduonion field theory, we can find that the interplays among all eight sorts of interactions are much more mysterious and complicated than we found and imagined before. Acknowledgements The author thanks Shaohan Lin, Minfeng Wang, Yun Zhu, Zhimin Chen and Xu Chen for helpful discussions. This project was supported by National Natural Science Foundation of China under grant number 60677039, Science & Technology Department of Fujian Province of China under grant number 2005HZ1020 and 2006H0092, and Xiamen Science & Technology Bureau of China under grant number 3502Z20055011. References [1] Zihua Weng. Octonionic electromagnetic and gravitational interactions and dark matter. arXiv: physics /0612102. [2] Zihua Weng. Octonionic quantum interplays of dark matter and ordinary matter. arXiv: physics /0702019. [3] Zihua Weng. Octonionic strong and weak interactions and their quantum equations. arXiv: physics /0702054. [4] Zihua Weng. Octonionic hyper-strong and hyper-weak fields and their quantum equations. arXiv: physics /0702086. [5] Zihua Weng. Compounding fields and their quantum equations in the sedenion space. arXiv: physics /0703055. [6] S. Kuwata, H. Fujii, A. Nakashima. Alternativity and reciprocity in the Cayley-Dickson algebra. J. Phys. A, 39 (2006) 1633-1644. [7] Yongge Tian, Yonghui Liu. On a group of mixed-type reverse-order laws for generalized inverses of a triple matrix product with applications. J. Linear Algebra, 16 (2007) 73-89.

0704.0137

Topological defects, geometric phases, and the angular momentum of light

Topological defects, geometric phases, and the angular momentum of light S. C. Tiwari Institute of Natural Philosophy c/o 1 Kusum Kutir Mahamanapuri,Varanasi 221005, India Recent reports on the intriguing features of vector vortex bearing beams are analyzed using geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence is the origin of topological phase singularities arising in the inhomogeneous polarization patterns. A unified picture of recent results is presented based on this proposition. Angular momentum shift within the light beam (OAM) has exact equivalence with the angular momentum holonomy associated with the geometric phase consistent with our conjecture. PACS numbers: 42.25.-p, 41.20.Jb, 03.65.Vf Topological defects in continuous field theoretic frame- work are usually associated with field singularities, how- ever in analogy with crystal defects wavefront disloca- tions for scalar waves [1] and disclinations for vector waves [2] have been discussed in the literature. An im- portant advancement was the realization that topological charge was related with the orbital angular momentum (OAM) of finite sized (transverse) light beams: typically for the Laguerre-Gaussian (LG) beams helicoidal spatial structure of the wavefront with azimuthal phase exp(ilφ) gives rise to OAM per photon of lh̄ where l is the topolog- ical charge, see review [3]. Adopting the fluid dynamical paradigm topological defects in optics are termed vor- tices; singularities in the polarization patterns are called vector vortices [4]. The aim of this Letter is to present a unified picture of the underlying physics of intriguing aspects of recent reports [5, 6, 7] in terms of the transformation of topo- logical charges due to spin redirection phase (SRP) such that OAM is exchanged within the beams [8]. The role of Pancharatnam phase (PP) invoked in [4, 6, 7] is also critically examined. For the sake of clarity we briefly review the essentials of geometric phases (GP) in optics which are primarily of two types, see [8, 9] for details and original references. Rytov-Vladimirskii phase rediscovered by Chiao and Wu in 1986 (inspired by the Berry phase) arises in the wave vector or momentum space of light. The unit wave vector κ and polarization vector ǫ(κ) describe the intrinsic prop- erties of the light wave. A plane wave propagating along a slowly varying path in the real space can be mapped on to the surface of a unit sphere in the wave vector space, and under parallel transport along a curve in this space preserving the spin helicity, ǫ.κ, the polarization vector is found to be rotated after the completion of a closed cycle on the sphere. The magnitude of the rotation is given by the solid angle enclosed by the cycle, and the sign is determined by the initial polarization state. Since left circular |L > and right circular |R > polarization states acquire equal but opposite geometric phases, Berry terms this effect as geometric circular birefringence [10]. A polarized light wave propagating in a fixed direction passed through optical elements traversing a polariza- tion cycle on the Poincare sphere acquires Pancharatnam phase equal to half of the solid angle of the cycle. Berry points out that [11] Pancharatnam actually made two important contributions. One,a notion of Pancharatnam connection was introduced for the phase difference be- tween two arbitrary nonorthogonal polarizations which can be written as arg(E1 ∗.E2) for complex electric field vectors. Secondly this connection is nontransitive result- ing into the Pancharatnam phase for a geodesic triangle on the sphere. Note that a parallel transport on the Poincare sphere is made with fixed direction of propaga- tion for the occurrence of PP. In the case of space varying polarization patterns, defining a direction of propagation is not easy, however Nye [12] has used Pancharatnam connection to define propagation vector kδ as a gradient of phase difference between fields at spatial locations r1 and r2 given by dδ = Im(E∗.dE)/|E|2 (1) In [4] authors correctly use Pancharatnam connection to obtain the phase difference of light at two locations in the space varying polarization plane, however the GP involved is not Pancharatnam phase as one cannot com- plete polarization cycle without changing the wave vec- tor direction. As discussed above we have to construct appropriate wave vector space for the case of vector vor- tices. For an initial beam propagating along z-axis, at each point (r, φ) on the inhomogeneous polarization plane there will correspond a k-space, and spin helicity preserv- ing parallel transport will give SRP for closed cycles. It is known that for a linearly polarized plane wave repre- sented by |P >= e−iα|R > +eiα|L > (2) the SRP corresponding to a cycle with solid angle Ω re- sults into [10] |P t >= e −i(α+Ω)|R > +ei(α+Ω)|L > (3) For the optical vortices this equation has to be general- ized: we suggest a spatially evolving GP embodied in the solid angle as a function of (r, φ). This is one of the main contributions of this Letter leading to Eq.(4) below. For http://arxiv.org/abs/0704.0137v1 the special case in which only the azimuthal angle de- pendence matters we obtain Ω in the following way. In the transverse plane consider a point A on a circle spec- ified by φ, then the area enclosed by the great circles in k-space corresponding to this point and the reference point O specified by φ = 0 would subtend a solid an- gle which varies linearly with φ; the solid angle will vary from 0 to 4π as φ varies from 0 to 2π. Thus we obtain the generalization of Eq.(3) to |P v >= e −i(α+2φ)|R > +ei(α+2φ|L > (4) Spatially evolving SRP [13] is crucial to understand in- teresting features observerd in vector vortices; we state our second principal contribution in the form of a propo- sition. Proposition: Geometric phase induced circular bire- fringence is the origin of topological charge transforma- tion in vector vortex carrying beams, and angular mo- mentum holonomy is manifested as OAM. We demonstrate in the following that this proposition offers transparent physical mechanism to explain the re- cent reports on inhomogeneous polarization patterns. Backscattered polarization [5] : Theory and experi- ments on the backscattered light for linearly polarized light from random media have been of current interest. Interesting features for the backscattering geometry have been observed. Authors of [5] give insightful treatment of the observations invoking GP in wave vector space, and this is in agreement with our proposition. Note that backscattered light wave vector could be treated similar to the discrete transformation for reflection from a mir- ror, see discussion in [9] one can envisage an adiabatic path in a modified k-space. It may be noted that essen- tially spatially evolving SRP is used in [5]. It seems the term ’geometrical phase vortex’ introduced by them for scalar vortices appearing in space varying polarization pattern is quite revealing. The q-plate experiment [6]: An inhomogeneous anisotropic optical element called q-plate has novel addi- tion to HWP : inhomogeneity is introduced orienting the fast (or slow) optical axis making an angle of α with the x-axis in the xy-plane for a planar slab given by α(r, φ) = qφ+α0. Jones calculus applied at each point of the q-plate shows that the output beam for an incident |L > state is not only converted to |R > state but it also acquires an azimuthal phase factor of exp(2iqφ). Simi- lar to the LG beams this phase is interpreted as OAM of 2qh̄ per photon in the output wave. Experiment is carried out using nematic liquid crystal planar cell for q = 1 and the measurements on the interference pattern formed by the superposition of the output beam with the reference beam display the wave front singularities and helical modes in the output beam. We argue that in the light of our proposition SRP in- duced circular birefringence is the origin of topological phase singularity and OAM in q-plates. We picture he- licity preserving transformation in the wave vector space defined by kδ. Since the polarization variation is confined in the transverse plane for the q-plates the constraint of the spin helicity preserving process in the q-plate with the azimuthal dependence of α would lead to a spiral path for the wave vector. In q=1 plate the circular plus linear propagation along z-axis will result into a helical path and the width of the plate ensures that the input and output ends of the helix are parallel. On the unit sphere in the wave vector space this will correspond to a great circle, and the solid angle would be 2π. Since the SRP equals the solid angle for the evolving paths, our Eq.(4) above is in agreement with Eq.(3) of [6]. The im- portant observation emphasized by the authors that the incident polarization controls the sign of the orbital helic- ity or topological charge is easily explained in view of the property of the geometric birefingence in which handed- ness decides the sign of the phase. Thus both magnitude and sign of the azimuthal phase have been explained in accordance with our proposition. Tightly focused beams [7]: Analysis of the light field at the focal plane of a high numerical aperture lens for the incoming circularly polarized plane wave shows the existence of inhomogeneous polarization pattern. Pan- charatnam connection at two different points on the cir- cle around the focus shows φ-dependence of the phase difference. The field can be decomposed into |L > and |R > states, Eq.(7) in [7], and it is found that the compo- nent with the spin same as that in the object plane does not change phase while the one with opposite spin ac- quires an azimuthal phase of 2φ, i. e. topological charge 2 and OAM of 2h̄ per photon. Application of Eq. (4) immediately leads to this result in conformity with our proposition. We may remark that the construction used by the authors to derive PP, namely the geodesic trian- gle on the Poincare sphere formed by the pole, E(r, 0), and E(r, φ) cannot be completed with a fixed direction of propagation for space varying polarization pattern, and therefore it is SRP not PP that arises. Having established first part of the proposition, we dis- cuss the role of angular momentum (AM) holonomy con- jecture [8, 9]. Transfer of spin AM of light to matter was measured long ago by Beth [14], and there are many reports of OAM transfer to particles in recent years [3]. Since polarization cycle for PP requires spin exchange with optical elements, it is natural to envisage a role of AM in GP; however it would be trivial. In the AM holon- omy conjecture, we argued AM level shifts within the light beams as physical mechanism for GP. This implies exact equivalence between AM shift and GP. Indirectly the backscattered light experiments and their interpre- tation in terms of SRP supports our conjecture. In the context of the AM conservation [5] the redistribution of total AM within the beam is also indicative of AM level shifts. The q=1 plate is a special optical element in which no transfer of AM to the crystal takes place, and total AM is conserved within the light beam. We argue that spin is intrinsic, and the OAM is a manifestation of the GP with exact equivalence between them in this case. The counter-intuitive interpretation in terms of spin to OAM conversion claimed in [6] is clearly ruled out. In fact, Marrucci et al experiment offers first direct evidence in support of our conjecture. It is remarkable that the light field structure calculated for tightly focused beams shows strong resemblence with the action of q-plates on light wave, and offers another setting to test our propo- sition. To conclude the Letter we make few observations. First let us note that even without the existence of phase singu- larities it should be possible to exchange AM within the light beam accompanied with GP: as argued earlier trans- verse shifts in the beam would account for the change in OAM [9]. Secondly the interplay of evolving GP in space and time domains could be of interest. A simple rotat- ing q-plate experiment is suggested: polarized light beam after traversing the q-plate is made to pass through a ro- tating HWP. Another variant with nonintegral q for this arrangement, i.e. q-plate plus rotating HWP, is also sug- gested. Analysis of the emerging beams may delineate the role of SRP and PP as well as provide further test to AM holonomy conjecture. The physical mechanism proposed here for space vary- ing polarization pattern of light could find important application in ’all optical information processing’. The angular momentum holonomy associated with GP, and the strong evidence of its proof discussed here will have significance in the context of the controversy surround- ing ’the hidden momentum’ and Aharanov-Bohm effect. We believe present ideas also hold promise to address some fundamental questions in physics. An important recent example is that of birefringence of the vacuum in quantum electrodynamics in strong external magnetic field. Though this has been known since long, last year PVLAS experiment reported polarization rotation [15] apparently very much in excess than the expected one. This has led to a controversy on the interpretation of QED birefringence in external rotating magnetic field, see [16] for a short review. As remarked by Adler essen- tially it involves light wave propagation in a nontrivial refracive media, and he finds that to first order there should be no rotation of the polarization of light. Could there be a role of GP in this case? It would be interesting to use Pancharatnam connection to calculate the phase of propagating light, and see if evolving GP in time domain will arise due to rotating magnetic field. It is interesting to note that the magnetic field direction rotates in the plane transverse to the direction of the propagation of the light. Obviously it would give additional polariza- tion rotation. This problem is being investigated, and will be reported elewhere. The Library facility at Banaras Hindu University is acknowledged. [1] J. F. Nye and M. V. Berry, Dislocations in wave trains, Proc. R. Soc. Lond. A 336,165 (1974). [2] J. F. Nye, Polarization effects in the diffraction of electro- magnetic waves: the role of disclinations, Proc. R. Soc. Lond. A 387, 105 (1983). [3] L. Allen, M. J. Padgett, and M. Babiker, The orbital angular momentum of light, Prog. Opt. 39,291 (1999). [4] A. Niv, G. Biener, V. Kleiner, and E. Hasman, Manip- ulation of the Pancharatnam phase in vectorial vortices, Opt. Express, 14, 4208 (2006). [5] C. Schwartz and A. Dogariu, Backscattered polarization patterns, optical vortices, and the angular momentum of light, Opt. Lett. 31,1121(2006). [6] L. Marrucci, C. Manzo, and D. Paparo, Optical spin-to- orbital angular momentum conversion in inhomogeneous anisotropic media, Phys. Rev. Lett. 96,163905(2006). [7] Z. Bomzon, M. Gu, and J. Shamir, Angular momentum and geometrical phases in tight-focused circularly polar- ized plane waves, Appl. Phys. Lett. 89,241 (2006). [8] S. C. Tiwari, Geometric phase in optics: quantal or clas- sical?, J. Mod. Opt. 39,1097(1992). [9] S. C. Tiwari, Geometric phase in optics and angular mo- mentum of light, J.Mod. Opt. 51,2297(2004). [10] M. V. Berry, Quantum adiabatic anholonomy, Lectures Ferrara School on Anomalies, defects, phases..., June 1989. [11] M. V. Berry, The adiabatic phase and Pancharatnam’s phase for polarized light, J. Mod. Opt. 34, 1401 (1987). [12] J. F. Nye, Phase gradient and crystal-like geometry in electromagnetic and elastic wavefields, in Sir Charles Frank OBE, FRS:An eightieth birthday tribute (IOP,UK 1991)pp220-231. [13] S. C. Tiwari, Nature of the angular momentum of light: rotational energy and geometric phase, arxiv.org : quant-ph/0609015. [14] R. A. Beth, Direct detection of the angular momentum of light, Phys. Rev. 48, 471 (1935). [15] E. Zavattini et al, Experimental observation of optical rotation generated in vacuum by a magnetic field, Phys. Rev. Lett. 96, 110406 (2006). [16] S. L. Adler, Vacuum birefringence in a rotating magnetic field, J. Phys. A: Math. Theor. 40, F143 (2007). http://arxiv.org/abs/quant-ph/0609015

0704.0138

Circular and non-circular nearly horizon-skimming orbits in Kerr spacetimes

Circular and non-circular nearly horizon-skimming orbits in Kerr spacetimes Enrico Barausse∗ SISSA, International School for Advanced Studies and INFN, Via Beirut 2, 34014 Trieste, Italy Scott A. Hughes Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Ave., Cambridge, MA 02139 USA Luciano Rezzolla Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Potsdam, Germany and Department of Physics, Louisiana State University, Baton Rouge, LA 70803 USA (Dated: November 1, 2018) We have performed a detailed analysis of orbital motion in the vicinity of a nearly extremal Kerr black hole. For very rapidly rotating black holes — spin parameter a ≡ J/M > 0.9524M — we have found a class of very strong field eccentric orbits whose orbital angular momentum Lz increases with the orbit’s inclination with respect to the equatorial plane, while keeping latus rectum and eccentricity fixed. This behavior is in contrast with Newtonian intuition, and is in fact opposite to the “normal” behavior of black hole orbits. Such behavior was noted previously for circular orbits; since it only applies to orbits very close to the black hole, they were named “nearly horizon-skimming orbits”. Our current analysis generalizes this result, mapping out the full generic (inclined and eccentric) family of nearly horizon-skimming orbits. The earlier work on circular orbits reported that, under gravitational radiation emission, nearly horizon-skimming orbits exhibit unusual inspiral, tending to evolve to smaller orbit inclination, toward prograde equatorial configuration. Normal orbits, by contrast, always demonstrate slowly growing orbit inclination — orbits evolve toward the retrograde equatorial configuration. Using up-to-date Teukolsky-based fluxes, we have concluded that the earlier result was incorrect — all circular orbits, including nearly horizon-skimming ones, exhibit growing orbit inclination under radiative backreaction. Using kludge fluxes based on a Post-Newtonian expansion corrected with fits to circular and to equatorial Teukolsky-based fluxes, we argue that the inclination grows also for eccentric nearly horizon- skimming orbits. We also find that the inclination change is, in any case, very small. As such, we conclude that these orbits are not likely to have a clear and peculiar imprint on the gravitational waveforms expected to be measured by the space-based detector LISA. PACS numbers: 04.30.-w I. INTRODUCTION The space-based gravitational-wave detector LISA [1] will be a unique tool to probe the nature of supermassive black holes (SMBHs), making it possible to map in detail their spacetimes. This goal is expected to be achieved by observing gravitational waves emitted by compact stars or black holes with masses µ ≈ 1 − 100M⊙ spiraling into the SMBHs which reside in the center of galaxies [2] (particularly the low end of the galactic center black hole mass function, M ≈ 105− 107M⊙). Such events are known as extreme mass ratio inspirals (EMRIs). Current wisdom suggests that several tens to perhaps of order a thousand such events could be measured per year by LISA [3]. Though the distribution of spins for observed astrophysi- cal black holes is not very well understood at present, very rapid spin is certainly plausible, as accretion tends to spin- up SMBHs [4]. Most models for quasi-periodic oscillations (QPOs) suggest this is indeed the case in all low-mass x-ray binaries for which data is available [5]. On the other hand, continuum spectral fitting of some high-mass x-ray binaries indicates that modest spins (spin parameter a/M ≡ J/M2 ∼ ∗Electronic address: [email protected] 0.6− 0.8) are likewise plausible [6]. The continuum-fit tech- nique does find an extremely high spin of a/M & 0.98 for the galactic “microquasar” GRS1915+105 [7]. This argues for a wide variety of possible spins, depending on the detailed birth and growth history of a given black hole. In the mass range corresponding to black holes in galactic centers, measurements of the broad iron Kα emission line in active galactic nuclei suggest that SMBHs can be very rapidly rotating (see Ref. [8] for a recent review). For instance, in the case of MCG-6-30-15, for which highly accurate observa- tions are available, a has been found to be larger than 0.987M at 90% confidence [9]. Because gravitational waves from EM- RIs are expected to yield a very precise determination of the spins of SMBHs [10], it is interesting to investigate whether EMRIs around very rapidly rotating black holes may possess peculiar features which would be observable by LISA. Should such features exist, they would provide unambiguous infor- mation on the spin of SMBHs and thus on the mechanisms leading to their formation [11]. For extremal Kerr black holes (a = M ), the existence of a special class of “circular” orbits was pointed out long ago by Wilkins [12], who named them “horizon-skimming” or- bits. (“Circular” here means that the orbits are of constant Boyer-Lindquist coordinate radius r.) These orbits have vary- ing inclination angle with respect to the equatorial plane and have the same coordinate radius as the horizon, r = M . De- http://arxiv.org/abs/0704.0138v2 mailto:[email protected] spite this seemingly hazardous location, it can be shown that all these r = M orbits have finite separation from one another and from the event horizon [13]. Their somewhat pathological description is due to a singularity in the Boyer-Lindquist co- ordinates, which collapses a finite span of the spacetime into r = M . Besides being circular and “horizon-skimming”, these or- bits also show peculiar behavior in their relation of angular momentum to inclination. In Newtonian gravity, a generic or- bit has Lz = |L| cos ι, where ι is the inclination angle relative to the equatorial plane (going from ι = 0 for equatorial pro- grade orbits to ι = π for equatorial retrograde orbits, passing through ι = π/2 for polar orbits), and L is the orbital angular momentum vector. As a result, ∂Lz(r, ι)/∂ι < 0, and the an- gular momentum in the z-direction always decreases with in- creasing inclination if the orbit’s radius is kept constant. This intuitively reasonable decrease of Lz with ι is seen for almost all black hole orbits as well. Horizon-skimming orbits, by contrast, exhibit exactly the opposite behavior: Lz increases with inclination angle. Reference [14] asked whether the behavior ∂Lz/∂ι > 0 could be extended to a broader class of circular orbits than just those at the radius r = M for the spin value a = M . It was found that this condition is indeed more general, and extended over a range of radius from the “innermost stable circular or- bit” to r ≃ 1.8M for black holes with a > 0.9524M . Or- bits that show this property have been named “nearly horizon- skimming”. The Newtonian behavior ∂Lz(r, ι)/∂ι < 0 is recovered for all orbits at r & 1.8M [14]. A qualitative understanding of this behavior comes from recalling that very close to the black hole all physical pro- cesses become “locked” to the hole’s event horizon [15], with the orbital motion of point particles coupling to the horizon’s spin. This locking dominates the “Keplerian” tendency of an orbit to move more quickly at smaller radii, forcing an or- biting particle to slow down in the innermost orbits. Lock- ing is particularly strong for the most-bound (equatorial) or- bits; the least-bound orbits (which have the largest inclination) do not strongly lock to the black hole’s spin until they have very nearly reached the innermost orbit [14]. The property ∂Lz(r, ι)/∂ι > 0 reflects the different efficiency of nearly horizon-skimming orbits to lock with the horizon. Reference [14] argued that this behavior could have ob- servational consequences. It is well-known that the incli- nation angle of an inspiraling body generally increases due to gravitational-wave emission [16, 17]. Since dLz/dt < 0 because of the positive angular momentum carried away by the gravitational waves, and since “normal” orbits have ∂Lz/∂ι < 0, one would expect dι/dt > 0. However, if during an evolution ∂Lz/∂ι switches sign, then dι/dt might switch sign as well: An inspiraling body could evolve towards an equatorial orbit, signalling the presence of an “almost- extremal” Kerr black hole. It should be emphasized that this argument is not rigorous. In particular, one needs to consider the joint evolution of or- bital radius and inclination angle; and, one must include the dependence of these two quantities on orbital energy as well as angular momentum1. As such, dι/dt depends not only on dLz/dt and ∂Lz/∂ι, but also on dE/dt, ∂E/∂ι, ∂E/∂r and ∂Lz/∂r. In this sense, the argument made in Ref. [14] should be seen as claiming that the contribution coming from dLz/dt and ∂Lz/∂ι are simply the dominant ones. Using the nu- merical code described in [17] to compute the fluxes dLz/dt and dE/dt, it was then found that a test-particle on a circu- lar orbit passing through the nearly horizon-skimming region of a Kerr black hole with a = 0.998M (the value at which a hole’s spin tends to be buffered due to photon capture from thin disk accretion [19]) had its inclination angle decreased by δι ≈ 1◦ − 2◦ [14] in the adiabatic approximation [20]. It should be noted at this point that the rate of change of in- clination angle, dι/dt, appears as the difference of two rel- atively small and expensive to compute rates of change [cf. Eq. (3.8) of Ref. [17]]. As such, small relative errors in those rates of change can lead to large relative errors in dι/dt. Fi- nally, in Ref. [14] it was speculated that the decrease could be even larger for eccentric orbits satisfying the condition ∂Lz/∂ι > 0, possibly leading to an observable imprint on EMRI gravitational waveforms. The main purpose of this paper is to extend Ref. [14]’s anal- ysis of nearly horizon-skimming orbits to include the effect of orbital eccentricity, and to thereby test the speculation that there may be an observable imprint on EMRI waveforms of nearly horizon-skimming behavior. In doing so, we have re- visited all the calculations of Ref. [14] using a more accurate Teukolsky solver which serves as the engine for the analysis presented in Ref. [21]. We have found that the critical spin value for circular nearly horizon-skimming orbits, a > 0.9524M , also delin- eates a family of eccentric orbits for which the condition ∂Lz(p, e, ι)/∂ι > 0 holds. (More precisely, we consider vari- ation with respect to an angle θinc that is easier to work with in the extreme strong field, but that is easily related to ι.) The parameters p and e are the orbit’s latus rectum and eccentric- ity, defined precisely in Sec. II. These generic nearly horizon- skimming orbits all have p . 2M , deep in the black hole’s extreme strong field. We next study the evolution of these orbits under gravitational-wave emission in the adiabatic approximation. We first revisited the evolution of circular, nearly horizon- skimming orbits using the improved Teukolsky solver which was used for the analysis of Ref. [21]. The results of this anal- ysis were somewhat surprising: Just as for “normal” orbits, we found that orbital inclination always increases during in- spiral, even in the nearly horizon-skimming regime. This is in stark contrast to the claims of Ref. [14]. As noted above, the inclination’s rate of change depends on the difference of two expensive and difficult to compute numbers, and thus can be strongly impacted by small relative errors in those numbers. 1 In the general case, one must also include the dependence on “Carter’s constant” Q [18], the third integral of black hole orbits (described more carefully in Sec. II). For circular orbits, Q = Q(E,Lz): knowledge of E and Lz completely determines Q. A primary result of this paper is thus to retract the claim of Ref. [14] that an important dynamic signature of the nearly horizon-skimming region is a reversal in the sign of incli- nation angle evolution: The inclination always grows under gravitational radiation emission. We next extended this analysis to study the evolution of generic nearly horizon-skimming orbits. The Teukolsky code to which we have direct access can, at this point, only com- pute the radiated fluxes of energy E and angular momentum Lz; results for the evolution of the Carter constant Q are just now beginning to be understood [22], and have not yet been incorporated into this code. We instead use “kludge” expres- sions for dE/dt, dLz/dt, and dQ/dt which were inspired by Refs. [23, 24]. These expression are based on post-Newtonian flux formulas, modified in such a way that they fit strong-field radiation reaction results obtained from a Teukolsky integra- tor; see Ref. [24] for further discussion. Our analysis indicates that, just as in the circular limit, the result dι/dt > 0 holds for generic nearly horizon-skimming orbits. Furthermore, and contrary to the speculation of Ref. [14], we do not find a large amplification of dι/dt as orbits are made more eccentric. Our conclusion is that the nearly horizon-skimming regime, though an interesting curiosity of strong-field orbits of nearly extremal black holes, will not imprint any peculiar observa- tional signature on EMRI waveforms. The remainder of this paper is organized as follows. In Sec. II, we review the properties of bound stable orbits in Kerr, pro- viding expressions for the constants of motion which we will use in Sec. III to generalize nearly horizon-skimming orbits to the non-circular case. In Sec. IV, we study the evolution of the inclination angle for circular nearly horizon-skimming orbits using Teukolsky-based fluxes; in Sec. V we do the same for non-circular orbits and using kludge fluxes. We present and discuss our detailed conclusions in Sec. VI. The fits and post- Newtonian fluxes used for the kludge fluxes are presented in the Appendix. Throughout the paper we have used units in which G = c = 1. II. BOUND STABLE ORBITS IN KERR SPACETIMES The line element of a Kerr spacetime, written in Boyer- Lindquist coordinates reads [25] ds2 = − 1− 2Mr dt2 + dr2 +Σ dθ2 r2 + a2 + 2Ma2r sin2 θ sin2 θ dφ2 −4Mar sin2 θ dt dφ, (1) where Σ ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2Mr + a2. (2) Up to initial conditions, geodesics can then be labelled by four constants of motion: the mass µ of the test particle, its energy E and angular momentum Lz as measured by an observer at infinity and the Carter constant Q [18]. The presence of these four conserved quantities makes the geodesic equations sepa- rable in Boyer-Lindquist coordinates. Introducing the Carter time λ, defined by ≡ Σ , (3) the geodesic equations become = Vr(r), µ = Vt(r, θ), = Vθ(θ), µ = Vφ(r, θ) , (4) Vt(r, θ) ≡ E − a2 sin2 θ + aLz Vr(r) ≡ E̟2 − aLz )2 −∆ µ2r2 + (Lz − aE)2 +Q Vθ(θ) ≡ Q− L2z cot2 θ − a2(µ2 − E2) cos2 θ, (5c) Vφ(r, θ) ≡ Lz csc2 θ + aE , (5d) where we have defined ̟2 ≡ r2 + a2 . (6) The conserved parameters E, Lz , and Q can be remapped to other parameters that describe the geometry of the orbit. We have found it useful to describe the orbit in terms of an angle θmin — the minimum polar angle reached by the orbit — as well as the latus rectum p and the eccentricity e. In the weak- field limit, p and e correspond exactly to the latus rectum and eccentricity used to describe orbits in Newtonian gravity; in the strong field, they are essentially just a convenient remap- ping of the orbit’s apoastron and periastron: rap ≡ , rperi ≡ 1 + e . (7) Finally, in much of our analysis, it is useful to refer to z− ≡ cos2 θmin , (8) rather than to θmin directly. To map (E,Lz, Q) to (p, e, z−), use Eq. (4) to impose dr/dλ = 0 at r = rap and r = rperi, and to impose dθ/dλ = 0 at θ = θmin. (Note that for a circular orbit, rap = rperi = r0. In this case, one must apply the rules dr/dλ = 0 and d2r/dλ2 = 0 at r = r0.) Following this ap- proach, Schmidt [26] was able to derive explicit expressions for E, Lz and Q in terms of p, e and z−. We now briefly review Schmidt’s results. Let us first introduce the dimensionless quantities Ẽ ≡ E/µ , L̃z ≡ Lz/(µM) , Q̃ ≡ Q/(µM)2 , (9) ã ≡ a/M , r̃ ≡ r/M , ∆̃ ≡ ∆/M2 , (10) Figure 1: Left panel: Inclination angles θinc for which bound stable orbits exist for a black hole with spin a = 0.998M . The allowed range for θinc goes from θinc = 0 to the curve corresponding to the eccentricity under consideration, θinc = θ inc . Right panel: Same as left but for an extremal black hole, a = M . Note that in this case θmaxinc never reaches zero. and the functions f(r̃) ≡ r̃4 + ã2 r̃(r̃ + 2) + z−∆̃ , (11) g(r̃) ≡ 2 ã r̃ , (12) h(r̃) ≡ r̃(r̃ − 2) + 1− z− ∆̃ , (13) d(r̃) ≡ (r̃2 + ã2z−)∆̃ . (14) Let us further define the set of functions (f1, g1, h1, d1) ≡{ (f(r̃p), g(r̃p), h(r̃p), d(r̃p)) if e > 0 , (f(r̃0), g(r̃0), h(r̃0), d(r̃0)) if e = 0 , (f2, g2, h2, d2) ≡{ (f(r̃a), g(r̃a), h(r̃a), d(r̃a)) if e > 0 , (f ′(r̃0), g ′(r̃0), h ′(r̃0), d ′(r̃0)) if e = 0 , and the determinants κ ≡ d1h2 − d2h1 , (17) ε ≡ d1g2 − d2g1 , (18) ρ ≡ f1h2 − f2h1 , (19) η ≡ f1g2 − f2g1 , (20) σ ≡ g1h2 − g2h1 . (21) The energy of the particle can then be written κρ+ 2ǫσ − 2D σ(σǫ2 + ρǫκ− ηκ2) ρ2 + 4ησ . (22) The parameter D takes the values ±1. The angular momen- tum is a solution of the system 2 − 2g1ẼL̃z − h1L̃2z − d1 = 0 , (23) 2 − 2g2ẼL̃z − h2L̃2z − d2 = 0 . (24) By eliminating the L̃2z terms in these equations, one finds the solution L̃z = ρẼ2 − κ for the angular momentum. Using dθ/dλ = 0 at θ = θmin, the Carter constant can be written Q̃ = z− ã2(1− Ẽ2) + 1− z− . (26) Additional constraints on p, e, z− are needed for the orbits to be stable. Inspection of Eq. (4) shows that an eccentric orbit is stable only if (rperi) > 0 . (27) It is marginally stable if ∂Vr/∂r = 0 at r = rperi. Similarly, the stability condition for circular orbits is (r0) < 0 ; (28) marginally stable orbits are set by ∂2Vr/∂r 2 = 0 at r = r0. Finally, we note that one can massage the above solutions for the conserved orbital quantities of bound stable orbits to rewrite the solution for L̃z as L̃z = − g21Ẽ 2 + (f1Ẽ2 − d1)h1 . (29) From this solution, we see that it is quite natural to refer to orbits with D = 1 as prograde and to orbits with D = −1 as retrograde. Note also that Eq. (29) is a more useful form than the corresponding expression, Eq. (A4), of Ref. [21]. In that expression, the factor 1/h1 has been squared and moved inside the square root. This obscures the fact that h1 changes sign for very strong field orbits. Differences between Eq. (29) and Eq. (A4) of [21] are apparent for a & 0.835, although only for orbits close to the separatrix (i.e., the surface in the space of parameters (p, e, ι) where marginally stable bound orbits lie). III. NON-CIRCULAR NEARLY HORIZON-SKIMMING ORBITS With explicit expressions for E, Lz and Q as functions of p, e and z−, we now examine how to generalize the condi- tion ∂Lz(r, ι)/∂ι > 0, which defined circular nearly horizon- skimming orbits in Ref. [14], to encompass the non-circular case. We recall that the inclination angle ι is defined as [14] cos ι = Q+ L2z . (30) Such a definition is not always easy to handle in the case of eccentric orbits. In addition, ι does not have an obvious phys- ical interpretation (even in the circular limit), but rather was introduced essentially to generalize (at least formally) the def- inition of inclination for Schwarzschild black hole orbits. In that case, one has Q = L2x+L y and therefore Lz = |L| cos ι. A more useful definition for the inclination angle in Kerr was introduced in Ref. [21]: θinc = −D θmin , (31) where θmin is the minimum reached by θ during the orbital motion. This angle is trivially related to z− (z− = sin 2 θinc) and ranges from 0 to π/2 for prograde orbits and from π/2 to π for retrograde orbits. It is a simple numerical calculation to convert between ι and θinc; doing so shows that the differences between ι and θinc are very small, with the two coinciding for a = 0, and with a difference that is less than 2.6◦ for a = M and circular orbits with r = M . Bearing all this in mind, the condition we have adopted to generalize nearly horizon-skimming orbits is ∂Lz(p, e, θinc) ∂θinc > 0 . (32) We have found that certain parts of this calculation, particular the analysis of strong-field geodesic orbits, are best done us- ing the angle θinc; other parts are more simply done using the angle ι, particularly the “kludge” computation of fluxes de- scribed in Sec. V. (This is because the kludge fluxes are based on an extension of post-Newtonian formulas to the strong- field regime, and these formulas use ι for inclination angle.) Accordingly, we often switch back and forth between these two notions of inclination, and in fact present our final results for inclination evolution using both dι/dt and dθinc/dt. Before mapping out the region corresponding to nearly horizon-skimming orbits, it is useful to examine stable or- bits more generally in the strong field of rapidly rotating black holes. We first fix a value for a, and then discretize the space of parameters (p, e, θinc). We next identify the points in this space corresponding to bound stable geodesic orbits. Suffi- ciently close to the horizon, the bound stable orbits with spec- ified values of p and e have an inclination angle θinc ranging from 0 (equatorial orbit) to a maximum value θmaxinc . For given p and e, θmaxinc defines the separatrix between stable and unsta- ble orbits. Example separatrices are shown in Fig. 1 for a = 0.998M and a = M . This figure shows the behavior of θmaxinc as a function of the latus rectum for the different values of the ec- centricity indicated by the labels. Note that for a = 0.998M the angle θmaxinc eventually goes to zero. This is the general behavior for a < M . On the other hand, for an extremal black hole, a = M , θmaxinc never goes to zero. The orbits which re- side at r = M (the circular limit) are the “horizon-skimming orbits” identified by Wilkins [12]; the a = M separatrix has a similar shape even for eccentric orbits. As expected, we find that for given latus rectum and eccentricity the orbit with θinc = 0 is the one with the lowest energy E (and hence is the most-bound orbit), whereas the orbit with θinc = θ inc has the highest E (and is least bound). Having mapped out stable orbits in (p, e, θinc) space, we then computed the partial derivative ∂Lz(p, e, θinc)/∂θinc and identified the following three overlapping regions: • Region A: The portion of the (p, e) plane for which ∂Lz(p, e, θinc)/∂θinc > 0 for 0 ≤ θinc ≤ θmaxinc . This region is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dot-dashed line (green in the color version). • Region B: The portion of the (p, e) plane for which (Lz)most bound(p, e) is smaller than (Lz)least bound(p, e). In other words, Lz(p, e, 0) < Lz(p, e, θ inc ) (33) in Region B. Note that Region B contains Region A. It is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dotted line (red in the color version). • Region C: The portion of the (p, e) plane for which ∂Lz(p, e, θinc)/∂θinc > 0 for at least one angle θinc between 0 and θmaxinc . Region C contains Region B, and is illustrated in Fig. 2 as the area under the heavy solid line and to the left of the dashed line (blue in the color version). Figure 2: Left panel: Non-circular nearly horizon-skimming orbits for a = 0.998M . The heavy solid line indicates the separatrix between stable and unstable orbits for equatorial orbits (ι = θinc = 0). All orbits above and to the left of this line are unstable. The dot-dashed line (green in the color version) bounds the region of the (p, e)-plane where ∂Lz/∂θinc > 0 for all allowed inclination angles (“Region A”). All orbits between this line and the separatrix belong to Region A. The dotted line (red in the color version) bounds the region (Lz)most bound < (Lz)least bound (“Region B”). Note that B includes A. The dashed line (blue in the color version) bounds the region where ∂Lz/∂θinc > 0 for at least one inclination angle (“Region C”); note that C includes B. All three of these regions are candidate generalizations of the notion of nearly horizon-skimming orbits. Right panel: Same as the left panel, but for the extreme spin case, a = M . In this case the separatrix between stable and unstable equatorial orbits is given by the line p/M = 1 + e. Orbits in any of these three regions give possible general- izations of the nearly horizon-skimming circular orbits pre- sented in Ref. [14]. Notice, as illustrated in Fig. 2, that the size of these regions depends rather strongly on the spin of the black hole. All three regions disappear altogether for a < 0.9524M (in agreement with [14]); their sizes grow with a, reaching maximal extent for a = M . These regions never extend beyond p ≃ 2M . As we shall see, the difference between these three regions is not terribly important for assessing whether there is a strong signature of the nearly horizon-skimming regime on the inspi- ral dynamics. As such, it is perhaps most useful to use Region C as our definition, since it is the most inclusive. IV. EVOLUTION OF θinc: CIRCULAR ORBITS To ascertain whether nearly horizon-skimming orbits can affect an EMRI in such a way as to leave a clear imprint in the gravitational-wave signal, we have studied the time evolution of the inclination angle θinc. To this purpose we have used the so-called adiabatic approximation [20], in which the infalling body moves along a geodesic with slowly changing parame- ters. The evolution of the orbital parameters is computed us- ing the time-averaged fluxes dE/dt, dLz/dt and dQ/dt due to gravitational-wave emission (“radiation reaction”). As dis- cussed in Sec. II, E, Lz and Q can be expressed in terms of p, e, and θinc. Given rates of change of E, Lz and Q, it is then straightforward [23] to calculate dp/dt, de/dt, and dθinc/dt (or dι/dt). We should note that although perfectly well-behaved for all bound stable geodesics, the adiabatic approximation breaks down in a small region of the orbital parameters space very close to the separatrix, where the transition from an inspiral to a plunging orbit takes place [27]. However, since this region is expected to be very small2 and its impact on LISA wave- forms rather hard to detect [27], we expect our results to be at least qualitatively correct also in this region of the space of parameters. Accurate calculation of dE/dt and dLz/dt in the adiabatic approximation involves solving the Teukolsky and Sasaki- Nakamura equations [28, 29]. For generic orbits this has been done for the first time in Ref. [21]. The calculation of dQ/dt for generic orbits is more involved. A formula for dQ/dt has been recently derived [22], but has not yet been implemented (at least in a code to which we have access). On the other hand, it is well-known that a circular orbit will remain circular under radiation reaction [30, 31, 32]. This constraint means that Teukolsky-based fluxes for E and Lz 2 Its width in p/M is expected to be of the order of ∆p/M ∼ (µ/M)2/5 , where µ is the mass of the infalling body [27]. are sufficient to compute dQ/dt. Considering this limit, the rate of change dQ/dt can be expressed in terms of dE/dt and dL/dt as = −N1(p, ι) N5(p, ι) − N4(p, ι) N5(p, ι) where N1(p, ι) ≡ E(p, ι) p4 + a2 E(p, ι) p2 − 2 aM (Lz(p, ι)− aE(p, ι)) p , (35) N4(p, ι) ≡ (2M p− p2)Lz(p, ι)− 2M aE(p, ι) p , (36) N5(p, ι) ≡ (2M p− p2 − a2)/2 . (37) (These quantities are for a circular orbit of radius p.) Using this, it is simple to compute dθinc/dt (or dι/dt). This procedure was followed in Ref. [14], using the code presented in Ref. [17], to determine the evolution of ι; this analysis indicated that dι/dt < 0 for circular nearly horizon- skimming orbits. As a first step to our more general analy- sis, we have repeated this calculation but using the improved Sasaki-Nakamura-Teukolsky code presented in Ref. [21]; we focused on the case a = 0.998M . Rather to our surprise, we discovered that the fluxes dE/dt and dLz/dt computed with this more accurate code indicate that dι/dt > 0 (and dθinc/dt > 0) for all circular nearly horizon-skimming orbits — in stark contrast with what was found in Ref. [14]. As mentioned in the introduction, the rate of change of inclination angle appears as the difference of two quantities. These quantities nearly cancel (and indeed cancel exactly in the limit a = 0); as such, small relative errors in their values can lead to large relative error in the inferred in- clination evolution. Values for dE/dt, dLz/dt, dι/dt, and dθinc/dt computed using the present code are shown in Ta- ble I in the columns with the header “Teukolsky”. V. EVOLUTION OF θinc: NON-CIRCULAR ORBITS The corrected behavior of circular nearly horizon- skimming orbits has naturally led us to investigate the evo- lution of non-circular nearly horizon-skimming orbits. Since our code cannot be used to compute dQ/dt, we have resorted to a “kludge” approach, based on those described in Refs. [23, 24]. In particular, we mostly follow the procedure de- veloped by Gair & Glampedakis [24], though (as described below) importantly modified. The basic idea of the “kludge” procedure is to use the func- tional form of 2PN fluxes E, Lz and Q, but to correct the circular part of these fluxes using fits to circular Teukolsky data. As developed in Ref. [24], the fluxes are written = (1− e2)3/2 (1− e2)−3/2 (p, e, ι) (p, 0, ι) + fit circ (p, ι) , (38) = (1 − e2)3/2 (1 − e2)−3/2 (p, e, ι) (p, 0, ι) + fit circ (p, ι) , (39) = (1− e2)3/2 Q(p, e, ι) × (1− e2)−3/2 dQ/dt√ (p, e, ι)− dQ/dt√ (p, 0, ι) dQ/dt√ fit circ (p, ι) . (40) The post-Newtonian fluxes (dE/dt)2PN, (dLz/dt)2PN and (dQ/dt)2PN are given in the Appendix [particularly Eqs. (A.1), (A.2), and (A.3)]. Since for circular orbits the fluxes dE/dt, dLz/dt and dQ/dt are related through Eq. (34), only two fits to circu- lar Teukolsky data are needed. One possible choice is to fit dLz/dt and dι/dt, and then use the circularity constraint to obtain3 [24] dQ/dt√ fit circ (p, ι) = 2 tan ι fit circ Q(p, 0, ι) sin2 ι fit circ , (41) fit circ (p, ι) = − N4(p, ι) N1(p, ι) fit circ (p, ι) − N5(p, ι) N1(p, ι) Q(p, 0, ι) dQ/dt√ fit circ (p, ι) . (42) As stressed in Ref. [24], one does not expect these fluxes to work well in the strong field, both because the post-Newtonian approximation breaks down close to the black hole, and be- cause the circular Teukolsky data used for the fits in Ref. [24] was computed for 3M ≤ p ≤ 30M . As a first attempt to improve their behavior in the nearly horizon-skimming re- gion, we have made fits using circular Teukolsky data for orbits with M < p ≤ 2M . In particular, for a black hole with a = 0.998M , we computed the circular Teukolsky-based fluxes dLz/dt and dι/dt listed in Table I (columns 8 and 10). These results were fit (with error . 0.2%); see Eqs. (A.4) and (A.6) in the Appendix. 3 This choice might seem more involved than fitting directly dLz/dt and dQ/dt, but, as noted by Gair & Glampedakis, ensures more sensible re- sults for the evolution of the inclination angle. This generates more physi- cally realistic inspirals [24]. Despite using strong-field Teukolsky fluxes for our fit, we found fairly poor behavior of these rates of change, particu- larly as a function of eccentricity. To compensate for this, we introduced a kludge-type fit to correct the equatorial part of the flux, in addition to the circular part. We fit, as a function of p and e, Teukolsky-based fluxes for dE/dt and dLz/dt for orbits in the equatorial plane, and then introduce the following kludge fluxes for E and Lz: (p, e, ι) = (p, e, ι) (p, e, 0) + fit eq (p, e) (43) (p, e, ι) = (p, e, ι) (p, e, 0) + fit eq (p, e) . (44) [Note that Eq. (40) for dQ/dt is not modified by this proce- dure since dQ/dt = 0 for equatorial orbits.] Using equatorial non-circular Teukolsky data provided by Drasco [21, 33] for a = 0.998 and M < p ≤ 2M (the ι = 0 “Teukolsky” data in Tables II, III and IV), we found fits (with error . 1.5%); see Eqs. (A.9) and (A.10). Note that the fits for equatorial fluxes are significantly less accurate than the fits for circular fluxes. This appears to be due to the fact that, close to the black hole, many harmonics are needed in order for the Teukolsky-based fluxes to converge, especially for eccentric orbits (cf. Figs. 2 and 3 of Ref. [21], noting the number of radial harmonics that have significant contribution to the flux). Truncation of these sums is likely a source of some error in the fluxes themselves, making it difficult to make a fit of as high quality as we could in the circular case. These fits were then finally used in Eqs. (43) and (44) to calculate the kludge fluxes dE/dt and dLz/dt for generic or- bits. This kludge reproduces to high accuracy our fits to the Teukolsky-based fluxes for circular orbits (e = 0) or equato- rial orbits (ι = 0). Some residual error remains because the ι = 0 limit of the circular fits do not precisely equal the e = 0 limit of the equatorial fits. Table I compares our kludge to Teukolsky-based fluxes for circular orbits; the two methods agree to several digits. Tables II, III and IV compare our kludge to the generic Teukolsky- based fluxes for dE/dt and dLz/dt provided by Drasco [21, 33]. In all cases, the kludge fluxes dE/dt and dLz/dt have the correct qualitative behavior, being negative for all the orbital parameters under consideration (a = 0.998M , 1 < p/M ≤ 2, 0 ≤ e ≤ 0.5 and 0◦ ≤ ι ≤ 41◦). The relative difference between the kludge and Teukolsky fluxes is always less than 25% for e = 0 and e = 0.1 (even for orbits very close to separatrix). The accuracy remains good at larger eccentricity, though it degrades somewhat as orbits come close to the separatrix. Tables I, II, III and IV also present the kludge values of the fluxes dι/dt and dθinc/dt as computed using Eqs. (43) and (44) for dE/dt and dLz/dt, plus Eq. (40) for dQ/dt. Though certainly not the last word on inclination evolution (pending rigorous computation of dQ/dt), these rates of change proba- bly represent a better approximation than results published to date in the literature. (Indeed, prior work has often used the crude approximation dι/dt = 0 [21] to estimate dQ/dt given dE/dt and dLz/dt.) Most significantly, we find that (dι/dt)kludge > 0 and (dθinc/dt)kludge > 0 for all of the orbital parameters we con- sider. In other words, we find that dι/dt and dθinc/dt do not ever change sign. Finally, in Table V we compute the changes in θinc and ι for the inspiral with mass ratio µ/M = 10−6. In all cases, we start at p/M = 1.9. The small body then inspirals through the nearly horizon-skimming region until it reaches the sep- aratrix; at this point, the small body will fall into the large black hole on a dynamical timescale ∼ M , so we terminate the calculation. The evolution of circular orbits is computed using our fits to the circular-Teukolsky fluxes of E and Lz; for eccentric orbits we use the kludge fluxes (40), (43) and (44). As this exercise demonstrates, the change in inclination during inspiral is never larger than a few degrees. Not only is there no unique sign change in the nearly horizon-skimming region, but the magnitude of the inclination change remains puny. This leaves little room for the possibility that this class of orbits may have a clear observational imprint on the EMRI- waveforms to be detected by LISA. VI. CONCLUSIONS We have performed a detailed analysis of the orbital mo- tion near the horizon of near-extremal Kerr black holes. We have demonstrated the existence of a class of orbits, which we have named “non-circular nearly horizon-skimming orbits”, for which the angular momentum Lz increases with the or- bit’s inclination, while keeping latus rectum and eccentricity fixed. This behavior, in stark contrast to that of Newtonian orbits, generalizes earlier results for circular orbits [14]. Furthermore, to assess whether this class of orbits can pro- duce a unique imprint on EMRI waveforms (an important source for future LISA observations), we have studied, in the adiabatic approximation, the radiative evolution of incli- nation angle for a small body orbiting in the nearly horizon- skimming region. For circular orbits, we have re-examined the analysis of Ref. [14] using an improved code for comput- ing Teukolsky-based fluxes of the energy and angular momen- tum. Significantly correcting Ref. [14]’s results, we found no decrease in the orbit’s inclination angle. Inclination always increases during inspiral. We next carried out such an analysis for eccentric nearly horizon-skimming orbits. In this case, we used “kludge” fluxes to evolve the constants of motion E, Lz and Q [24]. We find that these fluxes are fairly accurate when compared with the available Teukolsky-based fluxes, indicating that they should provide at least qualitatively correct information re- garding inclination evolution. As for circular orbits, we find that the orbit’s inclination never decreases. For both circular and non-circular configurations, we find that the magnitude of the inclination change is quite paltry — only a few degrees at most. Quite generically, therefore, we found that the inclination angle of both circular and eccentric nearly horizon-skimming orbits never decreases during the inspiral. Revising the results obtained in Ref. [14], we thus conclude that such orbits are not likely to yield a peculiar, unique imprint on the EMRI- waveforms detectable by LISA. Acknowledgments It is a pleasure to thank Kostas Glampedakis for enlight- ening comments and advice, and Steve Drasco for useful discussions and for also providing the non-circular Teukol- sky data that we used in this paper. The supercomputers used in this investigation were provided by funding from the JPL Office of the Chief Information Officer. This work was supported in part by the DFG grant SFB TR/7, by NASA Grant NNG05G105G, and by NSF Grant PHY-0449884. SAH gratefully acknowledges support from the MIT Class of 1956 Career Development Fund. Appendix In this Appendix we report the expressions for the post- Newtonian fluxes and the fits to the Teukolsky data necessary to compute the kludge fluxes introduced in Sec. V. In partic- ular the 2PN fluxes are given by [24] = −32 (1− e2)3/2 g1(e)− ã g2(e) cos ι− g3(e) + π g4(e) g5(e) + ã g6(e)− sin2 ι , (A.1) = −32 (1− e2)3/2 g9(e) cos ι+ ã (ga10(e)− cos2 ιgb10(e)) − g11(e) cos ι g12(e) cos ι− g13(e) cos ι+ ã cos ι g14(e)− sin2 ι , (A.2) )7/2 √ Q sin ι (1− e2)3/2 g9(e)− ã cos ιgb10(e)− g11(e) g12(e)− g13(e) + ã g14(e)− sin2 ι , (A.3) where µ is the mass of the infalling body and where the various e-dependent coefficients are g1(e) ≡ 1 + e4 , g2(e) ≡ e6 , g3(e) ≡ g4(e) ≡ 4 + e2 , g5(e) ≡ 44711 172157 e2 , g6(e) ≡ e2 , g9(e) ≡ 1 + ga10(e) ≡ e4 , gb10(e) ≡ e4 , g11(e) ≡ g12(e) ≡ 4 + e2 , g13(e) ≡ 44711 302893 e2 , g14(e) ≡ The fits to the circular-Teukolsky data of Table I are instead given by fit circ (p, ι ) = −32 )7/2 { cos ι+ )3/2 ( cos2 ι+ 4π cos ι − 1247 cos ι cos ι −1625 sin2 ι d̃1(p/M) + d̃2(p/M) cos ι + d̃3(p/M) cos + d̃4(p/M) cos 3 ι + d̃5(p/M) cos 4 ι+ d̃6(p/M) cos 5 ι+ cos ι )3/2 ( A+B cos2 ι , (A.4) (A.5) fit circ (p, ι ) = sin2 ι√ Q(p, 0, ι) d̃1(p/M) + cos ι a7d + b + c7d )3/2] + cos2 ι d̃8(p/M) + cos ι )5/2 [ h̃1(p/M) + cos 2 ι h̃2(p/M) , (A.6) where d̃i(x) ≡ aid + bid x−1/2 + cid x−1 , i = 1, . . . , 8, h̃i(x) ≡ aih + bih x−1/2 , i = 1, 2 (A.7) and the numerical coefficients are given by a1h = −278.9387 , b1h = 84.1414 , a2h = 8.6679 , b2h = −9.2401 , A = −18.3362 , B = 24.9034 , (A.8) and by the following table i 1 2 3 4 5 6 7 8 aid 15.8363 445.4418 −2027.7797 3089.1709 −2045.2248 498.6411 −8.7220 50.8345 bid −55.6777 −1333.2461 5940.4831 −9103.4472 6113.1165 −1515.8506 −50.8950 −131.6422 cid 38.6405 1049.5637 −4513.0879 6926.3191 -4714.9633 1183.5875 251.4025 83.0834 Note that the functional form of these fits was obtained from Eqs. (57) and (58) of Ref. [24] by setting ã (i.e., q in their notation) to 1. Finally, we give expressions for the fits to the equatorial Teukolsky data of tables II, III and IV (data with ι = 0, columns with header “Teukolsky”): fit eq (p, e) = (p, e, 0)− 32 )2 (M (1− e2)3/2 g̃1(e) + g̃2(e) + g̃3(e) + g̃4(e) + g̃5(e) , (A.9) fit eq (p, e) = (p, e, 0)− 32 (1− e2)3/2 f̃1(e) + f̃2(e) + f̃3(e) + f̃4(e) + f̃5(e) , (A.10) g̃i(e) ≡ aig + big e2 + cig e4 + dig e6 , f̃i(e) ≡ aif + bif e2 + cif e4 + dif e6 , i = 1, . . . , 5 (A.11) where the numerical coefficients are given by the following table i aig b 1 6.4590 −2038.7301 6639.9843 227709.2187 5.4577 −3116.4034 4711.7065 214332.2907 2 -31.2215 10390.6778 −27505.7295 −1224376.5294 −26.6519 15958.6191 −16390.4868 −1147201.4687 3 57.1208 −19800.4891 39527.8397 2463977.3622 50.4374 -30579.3129 15749.9411 2296989.5466 4 -49.7051 16684.4629 −21714.7941 −2199231.9494 −46.7816 25968.8743 656.3460 −2038650.9838 5 16.4697 −5234.2077 2936.2391 734454.5696 15.6660 −8226.3892 −4903.9260 676553.2755 e θinc ι dθinc dθinc (deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (Teukolsky) 1.3 0 0 0 −9.108×10−2 −9.109×10−2 −2.258×10−1 −2.259×10−1 0 0 0 0 1.3 0 10.4870 11.6773 −9.328×10−2 −9.332×10−2 −2.304×10−1 −2.306×10−1 1.837×10−2 1.839×10−2 6.462×10−3 6.475×10−3 1.3 0 14.6406 16.1303 −9.588×10−2 −9.588×10−2 −2.359×10−1 −2.360×10−1 2.397×10−2 2.400×10−2 8.645×10−3 8.667×10−3 1.3 0 17.7000 19.3172 −9.875×10−2 −9.876×10−2 −2.420×10−1 −2.421×10−1 2.728×10−2 2.731×10−2 1.007×10−2 1.010×10−2 1.3 0 20.1636 21.8210 −1.019×10−1 −1.019×10−1 −2.486×10−1 −2.488×10−1 2.943×10−2 2.950×10−2 1.111×10−2 1.117×10−2 1.4 0 0 0 −8.700×10−2 −8.709×10−2 −2.311×10−1 −2.312×10−1 0 0 0 0 1.4 0 14.5992 16.0005 −9.062×10−2 −9.070×10−2 −2.386×10−1 −2.386×10−1 2.316×10−2 2.319×10−2 8.823×10−3 8.848×10−3 1.4 0 20.1756 21.7815 −9.520×10−2 −9.526×10−2 −2.482×10−1 −2.482×10−1 2.875×10−2 2.877×10−2 1.141×10−2 1.143×10−2 1.4 0 24.1503 25.7517 −1.006×10−1 −1.007×10−1 −2.595×10−1 −2.596×10−1 3.140×10−2 3.141×10−2 1.289×10−2 1.288×10−2 1.4 0 27.2489 28.7604 −1.067×10−1 −1.068×10−1 −2.725×10−1 −2.725×10−1 3.274×10−2 3.275×10−2 1.378×10−2 1.377×10−2 1.5 0 0 0 −8.009×10−2 −7.989×10−2 −2.270×10−1 −2.265×10−1 0 0 0 0 1.5 0 16.7836 18.1857 −8.401×10−2 −8.383×10−2 −2.348×10−1 −2.343×10−1 2.360×10−2 2.351×10−2 9.602×10−3 9.545×10−3 1.5 0 23.0755 24.6167 −8.917×10−2 −8.897×10−2 −2.454×10−1 −2.449×10−1 2.872×10−2 2.863×10−2 1.228×10−2 1.222×10−2 1.5 0 27.4892 28.9670 −9.537×10−2 −9.516×10−2 −2.583×10−1 −2.579×10−1 3.091×10−2 3.082×10−2 1.372×10−2 1.367×10−2 1.5 0 30.8795 32.2231 −1.025×10−1 −1.023×10−1 −2.733×10−1 −2.728×10−1 3.184×10−2 3.173×10−2 1.452×10−2 1.443×10−2 1.6 0 0 0 −7.181×10−2 −7.156×10−2 −2.168×10−1 −2.162×10−1 0 0 0 0 1.6 0 18.3669 19.7220 −7.568×10−2 −7.545×10−2 −2.242×10−1 −2.237×10−1 2.240×10−2 2.229×10−2 9.600×10−3 9.515×10−3 1.6 0 25.1720 26.6245 −8.084×10−2 −8.062×10−2 −2.346×10−1 −2.341×10−1 2.701×10−2 2.685×10−2 1.223×10−2 1.210×10−2 1.6 0 29.9014 31.2625 −8.708×10−2 −8.687×10−2 −2.474×10−1 −2.470×10−1 2.889×10−2 2.872×10−2 1.363×10−2 1.349×10−2 1.6 0 33.5053 34.7164 −9.425×10−2 −9.399×10−2 −2.622×10−1 −2.616×10−1 2.964×10−2 2.951×10−2 1.441×10−2 1.432×10−2 1.7 0 0 0 −6.332×10−2 −6.317×10−2 −2.034×10−1 −2.031×10−1 0 0 0 0 1.7 0 19.6910 20.9859 −6.702×10−2 −6.687×10−2 −2.101×10−1 −2.098×10−1 2.057×10−2 2.052×10−2 9.202×10−3 9.171×10−3 1.7 0 26.9252 28.2884 −7.197×10−2 −7.184×10−2 −2.199×10−1 −2.196×10−1 2.467×10−2 2.456×10−2 1.170×10−2 1.162×10−2 1.7 0 31.9218 33.1786 −7.794×10−2 −7.782×10−2 −2.319×10−1 −2.316×10−1 2.632×10−2 2.620×10−2 1.306×10−2 1.296×10−2 1.7 0 35.7100 36.8118 −8.475×10−2 −8.465×10−2 −2.457×10−1 −2.455×10−1 2.698×10−2 2.686×10−2 1.384×10−2 1.373×10−2 1.8 0 0 0 −5.531×10−2 −5.528×10−2 −1.888×10−1 −1.887×10−1 0 0 0 0 1.8 0 20.8804 22.1128 −5.879×10−2 −5.874×10−2 −1.948×10−1 −1.946×10−1 1.858×10−2 1.858×10−2 8.635×10−3 8.639×10−3 1.8 0 28.5007 29.7791 −6.343×10−2 −6.336×10−2 −2.036×10−1 −2.035×10−1 2.221×10−2 2.223×10−2 1.098×10−2 1.101×10−2 1.8 0 33.7400 34.9034 −6.901×10−2 −6.894×10−2 −2.146×10−1 −2.144×10−1 2.368×10−2 2.371×10−2 1.228×10−2 1.232×10−2 1.8 0 37.6985 38.7065 −7.533×10−2 −7.533×10−2 −2.271×10−1 −2.271×10−1 2.429×10−2 2.427×10−2 1.306×10−2 1.303×10−2 1.9 0 0 0 −4.809×10−2 −4.811×10−2 −1.740×10−1 −1.740×10−1 0 0 0 0 1.9 0 21.9900 23.1615 −5.132×10−2 −5.134×10−2 −1.792×10−1 −1.793×10−1 1.666×10−2 1.664×10−2 8.022×10−3 8.007×10−3 1.9 0 29.9708 31.1702 −5.562×10−2 −5.564×10−2 −1.872×10−1 −1.872×10−1 1.986×10−2 1.987×10−2 1.019×10−2 1.020×10−2 1.9 0 35.4385 36.5176 −6.078×10−2 −6.077×10−2 −1.971×10−1 −1.970×10−1 2.118×10−2 2.122×10−2 1.143×10−2 1.148×10−2 1.9 0 39.5592 40.4847 −6.659×10−2 −6.658×10−2 −2.082×10−1 −2.082×10−1 2.177×10−2 2.182×10−2 1.222×10−2 1.228×10−2 2.0 0 0 0 −4.174×10−2 −4.175×10−2 −1.598×10−1 −1.598×10−1 0 0 0 0 2.0 0 23.0471 24.1605 −4.471×10−2 −4.472×10−2 −1.643×10−1 −1.643×10−1 1.489×10−2 1.489×10−2 7.425×10−3 7.424×10−3 2.0 0 31.3715 32.4978 −4.867×10−2 −4.871×10−2 −1.713×10−1 −1.714×10−1 1.773×10−2 1.770×10−2 9.436×10−3 9.411×10−3 2.0 0 37.0583 38.0608 −5.341×10−2 −5.345×10−2 −1.801×10−1 −1.801×10−1 1.893×10−2 1.889×10−2 1.062×10−2 1.057×10−2 2.0 0 41.3358 42.1876 −5.873×10−2 −5.875×10−2 −1.900×10−1 −1.900×10−1 1.950×10−2 1.948×10−2 1.141×10−2 1.138×10−2 Table I: Teukolsky-based fluxes and kludge fluxes [computed using Eqs. (40), (43) and (44)] for circular orbits about a hole with a = 0.998M ; µ represents the mass of the infalling body. The Teukolsky-based fluxes have an accuracy of 10−6. [1] http://lisa.nasa.gov/; http://sci.esa.int/home/lisa/ [2] J. Kormendy and D. Richstone, Ann. Rev. Astron. Astrophys. 33, 581 (1995). [3] J. R. Gair, L. Barack, T. Creighton, C. Cutler, S. L. Larson, E. S. Phinney, and M. Vallisneri, Class. Quantum Grav. 21, S1595 (2004). [4] S. L. Shapiro, Astrophys. J. 620, 59 (2005). [5] L. Rezzolla, T. W. Maccarone, S. Yoshida, and O. Zanotti, Mon. Not. Roy. Astron. Soc 344, L37 (2003). [6] R. Shafee, J. E. McClintock, R. Narayan, S. W. Davis, L.-X. Li, and R. A. Remilland, Astrophys. J. 636, L113 (2006). [7] J. E. McClintock, R. Shafee, R. Narayan, R. A. Remilland, S. W. Davis, and L.-X. Li, Astrophys. J. 652, 518 (2006). [8] A. C. Fabian and G. Miniutti, G. 2005, to appear in Kerr Space- time: Rotating Black Holes in General Relativity, edited by D. L. Wiltshire, M. Visser, and S. M. Scott; astro-ph/0507409. [9] L. W. Brenneman and C. S. Reynolds, Astrophys. J. 652, 1028 (2006). [10] L. Barack and C. Cutler, Phys. Rev. D 69, 082005 (2004). [11] M. Volonteri, P. Madau, E. Quataert, and M. J. Rees, Astrophys. J. 620, 69 (2005). [12] D. C. Wilkins, Phys. Rev. D 5, 814 (1972). http://lisa.nasa.gov/ http://sci.esa.int/home/lisa/ http://arxiv.org/abs/astro-ph/0507409 e θinc ι dθinc (deg.) (deg.) (kludge) (Teukolsky) (kludge) (Teukolsky) (kludge) (kludge) 1.3 0.1 0 0 −8.804×10−2 −8.804×10−2 −2.098×10−1 −2.098×10−1 0 0 1.4 0.1 0 0 −8.728×10−2 −8.719×10−2 −2.274×10−1 −2.275×10−1 0 0 1.4 0.1 8 8.8664 −9.110×10−2 −8.736×10−2 −2.355×10−1 −2.273×10−1 4.066×10−2 2.938×10−2 1.4 0.1 16 17.4519 −1.030×10−1 −8.958×10−2 −2.602×10−1 −2.309×10−1 7.428×10−2 5.475×10−2 1.4 0.1 24 25.5784 −1.243×10−1 −9.771×10−2 −3.037×10−1 −2.415×10−1 9.663×10−2 7.316×10−2 1.5 0.1 0 0 −8.069×10−2 −8.095×10−2 −2.255×10−1 −2.260×10−1 0 0 1.5 0.1 8 8.7910 −8.323×10−2 −8.133×10−2 −2.310×10−1 −2.264×10−1 2.996×10−2 2.070×10−2 1.5 0.1 16 17.3490 −9.121×10−2 −8.395×10−2 −2.483×10−1 −2.314×10−1 5.512×10−2 3.888×10−2 1.5 0.1 24 25.5197 −1.059×10−1 −8.980×10−2 −2.792×10−1 −2.423×10−1 7.255×10−2 5.264×10−2 1.6 0.1 0 0 −7.255×10−2 −7.281×10−2 −2.161×10−1 −2.168×10−1 0 0 1.6 0.1 8 8.7195 −7.430×10−2 −7.321×10−2 −2.201×10−1 −2.173×10−1 2.258×10−2 1.502×10−2 1.6 0.1 16 17.2437 −7.986×10−2 −7.533×10−2 −2.323×10−1 −2.212×10−1 4.179×10−2 2.839×10−2 1.6 0.1 24 25.4388 −9.025×10−2 −8.040×10−2 −2.547×10−1 −2.309×10−1 5.554×10−2 3.886×10−2 1.6 0.1 32 33.2683 −1.082×10−1 −9.435×10−2 −2.920×10−1 −2.551×10−1 6.316×10−2 4.559×10−2 1.7 0.1 0 0 −6.427×10−2 −6.440×10−2 −2.036×10−1 −2.040×10−1 0 0 1.7 0.1 8 8.6555 −6.552×10−2 −6.478×10−2 −2.065×10−1 −2.045×10−1 1.742×10−2 1.124×10−2 1.7 0.1 16 17.1454 −6.953×10−2 −6.651×10−2 −2.154×10−1 −2.075×10−1 3.240×10−2 2.134×10−2 1.7 0.1 24 25.3531 −7.707×10−2 −7.052×10−2 −2.317×10−1 −2.150×10−1 4.342×10−2 2.948×10−2 1.7 0.1 32 33.2416 −9.009×10−2 −7.959×10−2 −2.590×10−1 −2.324×10−1 4.998×10−2 3.512×10−2 1.8 0.1 0 0 −5.640×10−2 −5.640×10−2 −1.897×10−1 −1.897×10−1 0 0 1.8 0.1 8 8.5991 −5.732×10−2 −5.676×10−2 −1.918×10−1 −1.902×10−1 1.371×10−2 8.640×10−3 1.8 0.1 16 17.0562 −6.028×10−2 −5.817×10−2 −1.984×10−1 −1.925×10−1 2.562×10−2 1.647×10−2 1.8 0.1 24 25.2693 −6.588×10−2 −6.139×10−2 −2.105×10−1 −1.983×10−1 3.456×10−2 2.291×10−2 1.8 0.1 32 33.2018 −7.555×10−2 −6.849×10−2 −2.307×10−1 −2.120×10−1 4.020×10−2 2.765×10−2 1.9 0.1 0 0 −4.915×10−2 −4.911×10−2 −1.753×10−1 −1.751×10−1 0 0 1.9 0.1 8 8.5494 −4.985×10−2 −4.945×10−2 −1.768×10−1 −1.755×10−1 1.097×10−2 6.791×10−3 1.9 0.1 16 16.9760 −5.208×10−2 −5.064×10−2 −1.817×10−1 −1.774×10−1 2.055×10−2 1.298×10−2 1.9 0.1 24 25.1898 −5.633×10−2 −5.328×10−2 −1.908×10−1 −1.819×10−1 2.788×10−2 1.816×10−2 1.9 0.1 32 33.1555 −6.364×10−2 −5.870×10−2 −2.059×10−1 −1.920×10−1 3.272×10−2 2.214×10−2 2.0 0.1 0 0 −4.263×10−2 −4.264×10−2 −1.607×10−1 −1.608×10−1 0 0 2.0 0.1 8 8.5057 −4.316×10−2 −4.292×10−2 −1.619×10−1 −1.611×10−1 8.862×10−3 5.424×10−3 2.0 0.1 16 16.9042 −4.488×10−2 −4.390×10−2 −1.656×10−1 −1.625×10−1 1.666×10−2 1.039×10−2 2.0 0.1 24 25.1156 −4.815×10−2 −4.604×10−2 −1.724×10−1 −1.660×10−1 2.271×10−2 1.459×10−2 2.0 0.1 32 33.1064 −5.376×10−2 −5.031×10−2 −1.838×10−1 −1.736×10−1 2.684×10−2 1.793×10−2 2.0 0.1 40 40.8954 −6.339×10−2 −6.236×10−2 −2.027×10−1 −1.967×10−1 2.917×10−2 2.036×10−2 Table II: As in Table I but for non-circular orbits; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 −3. Note that our code, as all the Teukolsky-based code that we are aware of, presently does not have the capability to compute inclination angle evolution for generic orbits. [13] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972). [14] S. A. Hughes, Phys. Rev. D 63, 064016 (2001). [15] K. S. Thorne, R. H. Price, and D. A. MacDonald, Black Holes: The Membrane Paradigm (Yale University Press, New Haven, CT, 1986). [16] F. D. Ryan, Phys. Rev. D 52, R3159 (1995). [17] S. A. Hughes, Phys. Rev. D 61, 084004 (2000). [18] B. Carter, Phys. Rev. 174, 1559 (1968). [19] K. S. Thorne, Astrophys. J. 191, 507 (1974). [20] Y. Mino, Phys. Rev. D 67, 084027 (2003) [21] S. Drasco and S. A. Hughes, Phys. Rev. D 73, 024027 (2006). [22] N. Sago, T. Tanaka, W. Hikida, and H. Nakano, Prog. Theor. Phys. 114, 509 (2005); N. Sago, T. Tanaka, W. Hikida, K. Ganz, and H. Nakano, Prog. Theor. Phys. 115, 873 (2006). [23] K. Glampedakis, S. A. Hughes, and D. Kennefick, Phys. Rev. D 66, 064005 (2002). [24] J. R. Gair and K. Glampedakis, Phys. Rev. D 73, 064037 (2006). [25] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [26] W. Schmidt, Class. Quantum Grav. 19, 2743 (2002). [27] A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000) [28] S. A. Teukolsky, Astrophys. J. 185, 635 (1973). [29] M. Sasaki and T. Nakamura, Prog. Theor. Phys. 67, 1788 (1982). [30] F. D. Ryan, Phys. Ref. D 53, 3064 (1996). [31] D. Kennefick and A. Ori, Phys. Rev. D 53, 4319 (1996). [32] Y. Mino, unpublished Ph. D. thesis, Kyoto University, 1996. [33] Data available at http://gmunu.mit.edu/sdrasco/snapshots/ http://gmunu.mit.edu/sdrasco/snapshots/ e θinc ι dθinc (deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge) 1.4 0.2 0 0 −8.636×10−2 −8.642×10−2 −2.119×10−1 −2.121×10−1 0 0 1.4 0.2 8 8.8215 −9.853×10−2 −8.240×10−2 −2.374×10−1 −2.015×10−1 1.148×10−1 9.714×10−2 1.5 0.2 0 0 −8.362×10−2 −8.349×10−2 −2.236×10−1 −2.230×10−1 0 0 1.5 0.2 8 8.7595 −9.141×10−2 −8.276×10−2 −2.410×10−1 −2.206×10−1 7.893×10−2 6.549×10−2 1.5 0.2 16 17.2957 −1.145×10−1 −8.394×10−2 −2.915×10−1 −2.215×10−1 1.466×10−1 1.230×10−1 1.5 0.2 24 25.4608 −1.524×10−1 −9.230×10−2 −3.712×10−1 −2.357×10−1 1.952×10−1 1.661×10−1 1.6 0.2 0 0 −7.596×10−2 −7.616×10−2 −2.171×10−1 −2.176×10−1 0 0 1.6 0.2 8 8.6935 −8.111×10−2 −7.641×10−2 −2.292×10−1 −2.177×10−1 5.520×10−2 4.502×10−2 1.6 0.2 16 17.1994 −9.649×10−2 −7.798×10−2 −2.647×10−1 −2.198×10−1 1.032×10−1 8.500×10−2 1.6 0.2 24 25.3891 −1.221×10−1 −8.314×10−2 −3.212×10−1 −2.288×10−1 1.388×10−1 1.160×10−1 1.7 0.2 0 0 −6.765×10−2 −6.799×10−2 −2.057×10−1 −2.068×10−1 0 0 1.7 0.2 8 8.6329 −7.116×10−2 −6.813×10−2 −2.144×10−1 −2.066×10−1 3.963×10−2 3.176×10−2 1.7 0.2 16 17.1064 −8.171×10−2 −6.995×10−2 −2.398×10−1 −2.096×10−1 7.441×10−2 6.024×10−2 1.7 0.2 24 25.3085 −9.948×10−2 −7.443×10−2 −2.806×10−1 −2.178×10−1 1.009×10−1 8.290×10−2 1.7 0.2 32 33.2037 −1.257×10−1 −8.558×10−2 −3.371×10−1 −2.366×10−1 1.175×10−1 9.806×10−2 1.8 0.2 0 0 −5.965×10−2 −5.962×10−2 −1.927×10−1 −1.926×10−1 0 0 1.8 0.2 8 8.5789 −6.211×10−2 −5.997×10−2 −1.990×10−1 −1.930×10−1 2.919×10−2 2.300×10−2 1.8 0.2 16 17.0211 −6.953×10−2 −6.147×10−2 −2.175×10−1 −1.954×10−1 5.504×10−2 4.380×10−2 1.8 0.2 24 25.2283 −8.216×10−2 −6.502×10−2 −2.474×10−1 −2.016×10−1 7.515×10−2 6.068×10−2 1.8 0.2 32 33.1656 −1.009×10−1 −7.410×10−2 −2.890×10−1 −2.190×10−1 8.839×10−2 7.258×10−2 1.9 0.2 0 0 −5.218×10−2 −5.210×10−2 −1.786×10−1 −1.783×10−1 0 0 1.9 0.2 8 8.5312 −5.394×10−2 −5.244×10−2 −1.833×10−1 −1.787×10−1 2.197×10−2 1.704×10−2 1.9 0.2 16 16.9441 −5.928×10−2 −5.373×10−2 −1.970×10−1 −1.807×10−1 4.156×10−2 3.254×10−2 1.9 0.2 24 25.1518 −6.843×10−2 −5.669×10−2 −2.192×10−1 −1.858×10−1 5.706×10−2 4.535×10−2 1.9 0.2 32 33.1207 −8.213×10−2 −6.277×10−2 −2.502×10−1 −1.966×10−1 6.767×10−2 5.475×10−2 2.0 0.2 0 0 −4.528×10−2 −4.530×10−2 −1.637×10−1 −1.638×10−1 0 0 2.0 0.2 8 8.4891 −4.657×10−2 −4.557×10−2 −1.671×10−1 −1.641×10−1 1.679×10−2 1.283×10−2 2.0 0.2 16 16.8749 −5.049×10−2 −4.664×10−2 −1.774×10−1 −1.657×10−1 3.184×10−2 2.457×10−2 2.0 0.2 24 25.0802 −5.725×10−2 −4.904×10−2 −1.941×10−1 −1.696×10−1 4.391×10−2 3.440×10−2 2.0 0.2 32 33.0730 −6.743×10−2 −5.427×10−2 −2.175×10−1 −1.793×10−1 5.243×10−2 4.184×10−2 1.5 0.3 0 0 −8.481×10−2 −8.478×10−2 −2.094×10−1 −2.094×10−1 0 0 1.5 0.3 8 8.7037 −1.006×10−1 −7.824×10−2 −2.442×10−1 −1.934×10−1 1.484×10−1 1.301×10−1 1.5 0.3 16 17.2003 −1.469×10−1 −7.811×10−2 −3.435×10−1 −1.864×10−1 2.766×10−1 2.440×10−1 1.6 0.3 0 0 −8.144×10−2 −8.123×10−2 −2.183×10−1 −2.178×10−1 0 0 1.6 0.3 8 8.6498 −9.182×10−2 −7.807×10−2 −2.426×10−1 −2.095×10−1 1.028×10−1 8.918×10−2 1.6 0.3 16 17.1246 −1.223×10−1 −8.089×10−2 −3.122×10−1 −2.144×10−1 1.928×10−1 1.683×10−1 1.6 0.3 24 25.3046 −1.716×10−1 −8.666×10−2 −4.197×10−1 −2.229×10−1 2.607×10−1 2.295×10−1 1.7 0.3 0 0 −7.362×10−2 −7.314×10−2 −2.104×10−1 −2.095×10−1 0 0 1.7 0.3 8 8.5953 −8.060×10−2 −7.224×10−2 −2.277×10−1 −2.065×10−1 7.240×10−2 6.224×10−2 1.7 0.3 16 17.0415 −1.013×10−1 −7.369×10−2 −2.774×10−1 −2.084×10−1 1.365×10−1 1.180×10−1 1.7 0.3 24 25.2339 −1.349×10−1 −7.800×10−2 −3.547×10−1 −2.153×10−1 1.861×10−1 1.622×10−1 1.8 0.3 0 0 −6.488×10−2 −6.484×10−2 −1.973×10−1 −1.972×10−1 0 0 1.8 0.3 8 8.5454 −6.970×10−2 −6.480×10−2 −2.099×10−1 −1.966×10−1 5.206×10−2 4.436×10−2 1.8 0.3 16 16.9628 −8.402×10−2 −6.671×10−2 −2.461×10−1 −1.998×10−1 9.857×10−2 8.445×10−2 1.8 0.3 24 25.1601 −1.075×10−1 −7.030×10−2 −3.026×10−1 −2.056×10−1 1.353×10−1 1.169×10−1 1.8 0.3 32 33.1047 −1.404×10−1 −8.153×10−2 −3.762×10−1 −2.255×10−1 1.600×10−1 1.394×10−1 1.9 0.3 0 0 −5.669×10−2 −5.690×10−2 −1.829×10−1 −1.832×10−1 0 0 1.9 0.3 8 8.5010 −6.010×10−2 −5.683×10−2 −1.922×10−1 −1.824×10−1 3.823×10−2 3.229×10−2 1.9 0.3 16 16.8911 −7.025×10−2 −5.818×10−2 −2.189×10−1 −1.844×10−1 7.263×10−2 6.165×10−2 1.9 0.3 24 25.0887 −8.701×10−2 −6.054×10−2 −2.609×10−1 −1.874×10−1 1.003×10−1 8.579×10−2 1.9 0.3 32 33.0624 −1.106×10−1 −6.912×10−2 −3.157×10−1 −2.034×10−1 1.195×10−1 1.032×10−1 2.0 0.3 0 0 −4.953×10−2 −4.946×10−2 −1.683×10−1 −1.683×10−1 0 0 2.0 0.3 8 8.4616 −5.199×10−2 −4.970×10−2 −1.753×10−1 −1.685×10−1 2.862×10−2 2.395×10−2 2.0 0.3 16 16.8262 −5.932×10−2 −5.079×10−2 −1.954×10−1 −1.699×10−1 5.452×10−2 4.585×10−2 2.0 0.3 24 25.0215 −7.150×10−2 −5.328×10−2 −2.269×10−1 −1.737×10−1 7.564×10−2 6.411×10−2 2.0 0.3 32 33.0172 −8.878×10−2 −6.003×10−2 −2.682×10−1 −1.864×10−1 9.077×10−2 7.771×10−2 Table III: As in Table II, but for additional values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 e θinc ι dθinc (deg.) (deg.) (kludge) (Teukolsky ) (kludge) (Teukolsky) (kludge) (kludge) 1.6 0.4 0 0 −7.766×10−2 −7.772×10−2 −1.918×10−1 −1.919×10−1 0 0 1.6 0.4 8 8.5863 −9.433×10−2 −7.645×10−2 −2.297×10−1 −1.881×10−1 1.528×10−1 1.370×10−1 1.6 0.4 16 17.0151 −1.432×10−1 −7.651×10−2 −3.382×10−1 −1.837×10−1 2.873×10−1 2.584×10−1 1.7 0.4 0 0 −7.882×10−2 −7.953×10−2 −2.097×10−1 −2.115×10−1 0 0 1.7 0.4 8 8.5426 −9.002×10−2 −7.408×10−2 −2.367×10−1 −1.978×10−1 1.087×10−1 9.656×10−2 1.7 0.4 16 16.9502 −1.229×10−1 −7.682×10−2 −3.143×10−1 −2.025×10−1 2.054×10−1 1.830×10−1 1.7 0.4 24 25.1282 −1.760×10−1 −8.090×10−2 −4.336×10−1 −2.075×10−1 2.809×10−1 2.514×10−1 1.8 0.4 0 0 −7.107×10−2 −7.007×10−2 −2.013×10−1 −1.988×10−1 0 0 1.8 0.4 8 8.4989 −7.877×10−2 −7.001×10−2 −2.209×10−1 −1.981×10−1 7.788×10−2 6.879×10−2 1.8 0.4 16 16.8817 −1.015×10−1 −7.009×10−2 −2.774×10−1 −1.965×10−1 1.478×10−1 1.309×10−1 1.8 0.4 24 25.0646 −1.383×10−1 −7.314×10−2 −3.646×10−1 −2.003×10−1 2.036×10−1 1.810×10−1 1.8 0.4 32 33.0184 −1.887×10−1 −9.193×10−2 −4.755×10−1 −2.319×10−1 2.414×10−1 2.156×10−1 1.9 0.4 0 0 −6.187×10−2 −6.267×10−2 −1.861×10−1 −1.881×10−1 0 0 1.9 0.4 8 8.4591 −6.728×10−2 −6.216×10−2 −2.006×10−1 −1.861×10−1 5.666×10−2 4.980×10−2 1.9 0.4 16 16.8173 −8.328×10−2 −6.222×10−2 −2.424×10−1 −1.844×10−1 1.079×10−1 9.506×10−2 1.9 0.4 24 25.0006 −1.094×10−1 −6.486×10−2 −3.071×10−1 −1.878×10−1 1.495×10−1 1.322×10−1 1.9 0.4 32 32.9804 −1.452×10−1 −7.884×10−2 −3.896×10−1 −2.158×10−1 1.787×10−1 1.588×10−1 2.0 0.4 0 0 −5.483×10−2 −5.457×10−2 −1.735×10−1 −1.729×10−1 0 0 2.0 0.4 8 8.4235 −5.871×10−2 −5.445×10−2 −1.844×10−1 −1.720×10−1 4.222×10−2 3.686×10−2 2.0 0.4 16 16.7586 −7.020×10−2 −5.555×10−2 −2.158×10−1 −1.733×10−1 8.064×10−2 7.057×10−2 2.0 0.4 24 24.9396 −8.902×10−2 −5.844×10−2 −2.645×10−1 −1.778×10−1 1.122×10−1 9.860×10−2 2.0 0.4 32 32.9389 −1.150×10−1 −6.536×10−2 −3.267×10−1 −1.896×10−1 1.351×10−1 1.193×10−1 1.7 0.5 0 0 −7.421×10−2 −7.401×10−2 −1.815×10−1 −1.810×10−1 0 0 1.7 0.5 8 8.4736 −8.957×10−2 −7.168×10−2 −2.173×10−1 −1.750×10−1 1.379×10−1 1.256×10−1 1.7 0.5 16 16.8300 −1.347×10−1 −6.999×10−2 −3.201×10−1 −1.676×10−1 2.611×10−1 2.378×10−1 1.8 0.5 0 0 −7.589×10−2 −7.620×10−2 −1.993×10−1 −2.000×10−1 0 0 1.8 0.5 8 8.4395 −8.644×10−2 −6.929×10−2 −2.254×10−1 −1.829×10−1 1.005×10−1 9.076×10−2 1.8 0.5 16 16.7776 −1.175×10−1 −7.210×10−2 −3.004×10−1 −1.880×10−1 1.911×10−1 1.726×10−1 1.8 0.5 24 24.9413 −1.678×10−1 −7.395×10−2 −4.158×10−1 −1.881×10−1 2.638×10−1 2.385×10−1 1.9 0.5 0 0 −6.646×10−2 −6.620×10−2 −1.855×10−1 −1.849×10−1 0 0 1.9 0.5 8 8.4059 −7.386×10−2 −6.320×10−2 −2.048×10−1 −1.768×10−1 7.312×10−2 6.579×10−2 1.9 0.5 16 16.7233 −9.572×10−2 −6.551×10−2 −2.603×10−1 −1.809×10−1 1.395×10−1 1.255×10−1 1.9 0.5 24 24.8877 −1.312×10−1 −7.087×10−2 −3.461×10−1 −1.909×10−1 1.937×10−1 1.744×10−1 1.9 0.5 32 32.8741 −1.795×10−1 −8.247×10−2 −4.544×10−1 −2.091×10−1 2.320×10−1 2.092×10−1 2.0 0.5 0 0 −5.987×10−2 −5.995×10−2 −1.761×10−1 −1.763×10−1 0 0 2.0 0.5 8 8.3750 −6.516×10−2 −5.918×10−2 −1.906×10−1 −1.738×10−1 5.456×10−2 4.882×10−2 2.0 0.5 16 16.6725 −8.081×10−2 −5.817×10−2 −2.324×10−1 −1.694×10−1 1.044×10−1 9.343×10−2 2.0 0.5 24 24.8347 −1.063×10−1 −6.254×10−2 −2.970×10−1 −1.776×10−1 1.456×10−1 1.304×10−1 2.0 0.5 32 32.8378 −1.412×10−1 −6.993×10−2 −3.787×10−1 −1.893×10−1 1.756×10−1 1.576×10−1 Table IV: As in Tables II and III, but for different values of eccentricity e; the Teukolsky-based fluxes for E and Lz have an accuracy of 10 e θinc ι ∆t/M ∆θinc ∆ι (deg.) (deg.) (deg.) (deg.) 0 0 0 1.250×106 0 0 0 5 5.355510 1.217×106 1.949×10−1 4.954×10−1 0 10 10.679331 1.118×106 3.468×10−1 8.631×10−1 0 15 15.943192 9.574×105 4.236×10−1 1.019 0 20 21.125167 7.446×105 4.109×10−1 9.440×10−1 0 25 26.211779 4.981×105 3.158×10−1 6.860×10−1 0 30 31.199048 2.528×105 1.732×10−1 3.527×10−1 0 35 36.092514 6.584×104 4.636×10−2 8.806×10−2 0.1 0 0 1.228×106 0 0 0.1 5 5.351602 1.198×106 4.517×10−1 7.766×10−1 0.1 10 10.671900 1.103×106 6.900×10−1 1.236 0.1 15 15.932962 9.426×105 7.283×10−1 1.344 0.1 20 21.113129 7.315×105 6.433×10−1 1.187 0.1 25 26.199088 4.900×105 4.780×10−1 8.547×10−1 0.1 30 31.186915 2.513×105 2.730×10−1 4.585×10−1 0.1 35 36.082095 6.589×104 8.385×10−2 1.279×10−1 0.2 0 0 1.173×106 0 0 0.2 5 5.339916 1.150×106 1.204 1.598 0.2 10 10.649670 1.064×106 1.698 2.331 0.2 15 15.902348 9.043×105 1.618 2.293 0.2 20 21.077081 6.980×105 1.324 1.900 0.2 25 26.161046 4.693×105 9.545×10−1 1.351 0.2 30 31.150481 2.486×105 5.674×10−1 7.711×10−1 0.2 35 36.050712 7.562×104 2.070×10−1 2.648×10−1 0.3 0 0 1.087×106 0 0 0.3 5 5.320559 1.069×106 2.307 2.788 0.3 10 10.612831 1.001×106 3.256 4.007 0.3 15 15.851572 8.454×105 2.984 3.741 0.3 20 21.017212 6.483×105 2.375 2.998 0.3 25 26.097732 4.408×105 1.700 2.129 0.3 30 31.089639 2.493×105 1.040 1.276 0.3 35 35.997987 1.108×105 4.626×10−1 5.569×10−1 Table V: Variation in the inclination angles ι and θinc as well as time needed to reach the separatrix for several inspirals through the nearly horizon-skimming regime. In all of these cases, the binary’s mass ratio was fixed to µ/M = 10−6, the large black hole’s spin was fixed to a = 0.998M , and the orbits were begun at p = 1.9M . The time interval ∆t is the total accumulated time it takes for the inspiralling body to reach the separatrix (at which time it rapidly plunges into the black hole). The angles ∆θinc and ∆ι are the total integrated change in these inclination angles that we compute. For the e = 0 cases, inspirals are computed using fits to the circular-Teukolsky fluxes of E and Lz ; for eccentric orbits we use the kludge fluxes (40), (43) and (44). Notice that ∆θinc and ∆ι are always positive — the inclination angle always increases during the inspiral through the nearly horizon-skimming region. The magnitude of this increase never exceeds a few degrees.

0704.0139

The Blue Straggler Population of the Globular Cluster M5

The Blue Straggler Population of the Globular Cluster M5 1 B. Lanzoni1,2, E. Dalessandro1,2, F.R. Ferraro1, C. Mancini3, G. Beccari2,4,5, R.T. Rood6, M. Mapelli7, S. Sigurdsson8 1 Dipartimento di Astronomia, Università degli Studi di Bologna, via Ranzani 1, I–40127 Bologna, Italy 2 INAF–Osservatorio Astronomico di Bologna, via Ranzani 1, I–40127 Bologna, Italy 3 Dipartimento di Astronomia e Scienza dello Spazio, Università degli Studi di Firenze, Largo Enrico Fermi 2, I– 50125 Firenze, Italy 4 Dipartimento di Scienze della Comunicazione, Università degli Studi di Teramo, Italy 5 INAF–Osservatorio Astronomico di Collurania, Via Mentore Maggini, I–64100 Teramo, Italy 6 Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802 7 S.I.S.S.A., Via Beirut 2 - 4, I–34014 Trieste, Italy 8 Astronomy Department, University of Virginia, P.O. Box 400325, Charlottesville, VA, 22904 20 March, 07 ABSTRACT By combining high-resolution HST and wide-field ground based observations, in ultraviolet and optical bands, we study the Blue Stragglers Star (BSS) popula- tion of the galactic globular cluster M5 (NGC 5904) from its very central regions up to its periphery. The BSS distribution is highly peaked in the cluster center, decreases at intermediate radii and rises again outward. Such a bimodal dis- tribution is similar to those previously observed in other globular clusters (M3, 47 Tucanae, NGC 6752). As for these clusters, dynamical simulations suggest that, while the majority of BSS in M5 could be originated by stellar collisions, a significant fraction (20-40%) of BSS generated by mass transfer processes in primordial binaries is required to reproduce the observed radial distribution. A candidate BSS has been detected beyond the cluster tidal radius. If confirmed, this could represent an interesting case of an ”evaporating” BSS. http://arxiv.org/abs/0704.0139v1 – 2 – Subject headings: Globular clusters: individual (M5); stars: evolution – binaries: general - blue stragglers 1. INTRODUCTION In globular cluster (GC) color-magnitude diagrams (CMD) blue straggler stars (BSS) appear to be brighter and bluer than the Turn-Off (TO) stars and lie along an extension of the Main Sequence. Since BSS mimic a rejuvenated stellar population with masses larger than the normal cluster stars (this is also confirmed by direct mass measurements; e.g. Shara et al. 1997), they are thought to be objects that have increased their initial mass during their evolution by means of some process. Two main scenarios have been proposed for their formation: the collisional scenario suggests that BSS are the end-products of stellar mergers induced by collisions (COL-BSS), while in the mass-transfer scenario BSS form by the mass- transfer activity between two companions in a binary system (MT-BSS), possibly up to the complete coalescence of the two stars. Hence, understanding the origin of BSS in stellar clusters provides valuable insight both on the binary evolution processes and on the effects of dynamical interactions on the (otherwise normal) stellar evolution. The relative efficiency of the two formation mechanisms is thought to depend on the en- vironment (Fusi Pecci et al. 1992; Ferraro et al. 1999a; Bellazzini et al. 2002; Ferraro et al. 2003). COL-BSS are expected to be formed preferentially in high-density environments (i.e., the GC central regions), where stellar collisions are most probable, and MT-BSS should mainly populate lower density environments (the cluster peripheries), where binary systems can more easily evolve in isolation without suffering exchanges or ionization due to gravita- tional encounters. The overall scenario is complicated by the fact that primordial binaries can also sink to the core due to mass segregation processes, and “new” binaries can be formed in the cluster centers by gravitational encounters. The two formation mechanisms are likely to be at work simultaneously in every GC (see the case of M3 as an example; Ferraro et al. 1993, 1997), but the identification of the cluster properties that mainly affect their relative efficiency is still an open issue. One possibility for distinguishing between the two types of BSS is offered by high- resolution spectroscopic studies. Anomalous chemical abundances are expected at the surface 1Based on observations with the NASA/ESA HST, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555. Also based on WFI observations collected at the European Southern Observatory, La Silla, Chile, within the observing programs 62.L-0354 and 64.L-0439. – 3 – of BSS resulting from MT activity (Sarna & de Greve 1996), while they are not predicted in case of a collisional formation (Lombardi, Rasio & Shapiro 1995). Such studies have just become feasible, and the results found in the case of 47 Tucanae (47 Tuc; Ferraro et al. 2006a) are encouraging. The detection of unexpected properties of stars along standard evolutionary sequences (e.g., variability, anomalous population fractions, or peculiar radial distributions) can help estimating the fraction of binaries within a cluster (see, e.g., Bailyn 1994; Albrow et al. 2001; Bellazzini et al. 2002; Beccari et al. 2006), but such evidence does not directly allow the determination of the relative efficiency of the two BSS formation processes. The most widely applicable tool to probe the origin of BSS is their radial distribution within the clusters (see Ferraro 2006, for a review). This has been observed to be bimodal (i.e., highly peaked in the cluster centers and peripheries, and significantly lower at inter- mediate radii) in at least 4 GCs: M3 (Ferraro et al. 1997), 47 Tuc (Ferraro et al. 2004), NGC 6752 (Sabbi et al. 2004), and M5 (Warren, Sandquist & Bolte 2006, hereafter W06). Preliminary evidence of bimodality has also been found in M55 (Zaggia, Piotto & Capaccioli 1997). Dynamical simulations suggest that the bimodal radial distributions observed in M3, 47 Tuc and NGC 6752 (Mapelli et al. 2004, 2006) result from ∼ 40− 50% of MT-BSS with the balance being COL-BSS. In this context, the case of ω Cen is atypical: the BSS radial distribution in this cluster is flat (Ferraro et al. 2006b), and mass segregation processes have not yet played a major role, thus implying that this system is populated by a vast majority of MT-BSS (Mapelli et al. 2006). These results demonstrate that detailed studies of the BSS radial distribution within GCs are very powerful tools for better understanding the complex interplay between dynamics and stellar evolution in dense stellar systems. In the present paper we extend this kind of investigation to M5 (NGC 5904). With HST- WFPC2 and -ACS ultraviolet and optical high-resolution images of the core we have been able to efficiently detect the BSS population even in the severely crowded central regions. Moreover, with wide-field optical observations performed with ESO-WFI we sampled the entire cluster extension. The combination of these two data sets allowed us to study the dynamical properties of M5, accurately redetermining its center of gravity, its surface density profile, and the BSS radial distribution over the entire cluster. The BSS population of M5 has been recently studied by W06, but we have extended the analysis to larger distances from the cluster center, and we have used Monte-Carlo dynamical simulations to interpret the observational results. – 4 – 2. OBSERVATIONS AND DATA ANALYSIS 2.1. The data sets The present study is based on a combination of two different photometric data sets: 1. The high-resolution set – It consists of a series of ultraviolet (UV) and optical images of the cluster center obtained with HST-WFPC2 (Prop. 6607, P.I. Ferraro). To efficiently resolve the stars in the highly crowded central regions, the Planetary Camera (PC, being the highest resolution instrument: 0.′′046/pixel) has been pointed approximately on the cluster center, while the three Wide Field Cameras (WF, having a lower resolution: 0.′′1/pixel) have been used to sample the surrounding regions. Observations have been performed through filter F255W (medium UV) in order to efficiently select the BSS and horizontal branch (HB) populations, and through filters F336W (approximately corresponding to an U filter) and F555W (V ) for the red giant branch (RGB) population and to guarantee a proper combination with the ground-based data set (see below). The photometric reduction of the high-resolution images was carried out using ROMAFOT (Buonanno et al. 1983), a package developed to perform accurate photometry in crowded fields and specifically optimized to handle under-sampled Point Spread Functions (PSFs; Buonanno & Iannicola 1989), as in the case of the HST-WF chips. To obtain a better coverage of the innermost regions of the cluster, we have also used a set of public HST-WFPC2 and HST-ACS observations. The HST-WFPC2 data set has been obtained through filters F439W (B) and F555W (V ) by Piotto et al. (2002), and because of the different orientation of the camera, it is complementary to ours. Additional HST-ACS data in filters F435W (B), F606W (V ), and F814W (I) have been retrieved from the ESO- STECF Science Archive, and have been used to sample the central area not covered by the WFPC2 observations. All the ACS images were properly corrected for geometric distortions and effective flux (over the pixel area) following the prescriptions of Sirianni et al. (2005). The photometric analysis was performed independently in the three drizzled images by using the aperture photometry code SExtractor (Source-Extractor; Bertin & Arnouts 1996), and adopting a fixed aperture radius of 2.5 pixels (0.125′′). The magnitude lists were finally cross-correlated in order to obtain a combined catalog. The adopted combination of the three HST data sets is sketched in Figure 1 and provided a good coverage of the cluster up to r = 115′′. 2. The wide-field set - A complementary set of wide-field B and V images was secured by using the Wide Field Imager (WFI) at the 2.2m ESO-MPI telescope during an observing run in April 2000. Thanks to the exceptional imaging capabilities of WFI (each image consists of a mosaic of 8 CCDs, for a global field of view of 34′×34′), these data cover the entire cluster – 5 – extension (see Figure 2, where the cluster is roughly centered on CCD #7). The raw WFI images were corrected for bias and flat field, and the overscan regions were trimmed using IRAF2 tools. The PSF fitting procedure was performed independently on each image using DoPhot (Schechter, Mateo & Saha 1993). All the uncertain detections, usually caused by photometric blends, stars near the CCD gaps or saturated stars, have been checked one by one using ROMAFOT (Buonanno et al. 1983). 2.2. Astrometry and center of gravity The HST+WFI catalog has been placed on the absolute astrometric system by adopting the procedure already described in Ferraro et al. (2001, 2003). The new astrometric Guide Star Catalog (GSC-II3) was used to search for astrometric standard stars in the WFI field of view (FoV), and a cross-correlation tool specifically developed at the Bologna Observatory (Montegriffo et al. 2003, private communication) has been employed to obtain an astrometric solution for each of the 8 CCDs. Several hundred GSC-II reference stars were found in each chip, thus allowing an accurate absolute positioning of the stars. Then, a few hundred stars in common between the WFI and the HST FoVs have been used as secondary standards to place the HST catalog on the same absolute astrometric system. At the end of the procedure the global uncertainties in the astrometric solution are of the order of ∼ 0.′′2, both in right ascension (α) and declination (δ). Given the absolute positions of individual stars in the innermost regions of the cluster, the center of gravity Cgrav has been determined by averaging coordinates α and δ of all stars lying in the PC FoV following the iterative procedure described in Montegriffo et al. (1995; see also Ferraro et al. 2003, 2004). In order to correct for spurious effects due to incompleteness in the very inner regions of the cluster, we considered two samples with different limiting magnitudes (m555 < 19.5 and m555 < 20), and we computed the barycenter of stars for each sample. The two estimates agree within ∼ 1′′, giving Cgrav at α(J2000) = 15h 18m 33.s53, δ(J2000) = +2o 4′ 57.′′06, with a 1σ uncertainty of 0.′′5 in both α and δ, corresponding to about 10 pixels in the PC image. This value of Cgrav is located at ∼ 4′′ south-west (∆α = −4′′, ∆δ = −0.′′9) from that previously derived by Harris (1996) on the basis of the surface brightness distribution. 2IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Associa- tion of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. 3Available at http://www-gsss.stsci.edu/Catalogs/GSC/GSC2/GSC2.htm. http://www-gsss.stsci.edu/Catalogs/GSC/GSC2/GSC2.htm – 6 – 2.3. Photometric calibration and definition of the catalogs The optical HST magnitudes (i.e., those obtained through the WFPC2 filters F439W and F555W, and through ACS filters F435W, F606W, F814W), as well as the WFI B and V magnitudes have been all calibrated on the catalog of Sandquist et al. (1996). The UV magnitudes m160 and m255 have been calibrated to the Holtzman et al. (1995) zero-points following Ferraro et al. (1997, 2001), while the U magnitude m336 has been calibrated to Dolphin (2000). In order to reduce spurious effects due to the low resolution of the ground-based obser- vations in the most crowded regions of the cluster, we use only the HST data for the inner 115′′, this value being imposed by the FoV of the WFPC2 and ACS cameras (see Figure 1). In particular, we define as HST sample the ensemble of all the stars in the WFPC2 and ACS combined catalog having r ≤ 115′′ from the center, and as WFI sample all stars detected with WFI at r > 115′′ (see Figure 2). The CMDs of the HST and WFI samples in the (V, U − V ) and (V, B − V ) planes are shown in Figure 3. 2.4. Density profile We have determined the projected density profile over the entire cluster extension, from Cgrav out to ∼ 1400 ∼ 23.′3, by direct star counts, considering only stars brighter than V = 20 (see Figure 3) in order to avoid incompleteness biases. The brightest RGB stars that are strongly saturated in the ACS data set have been excluded from the analysis, but since they are few in number, the effect on the resulting density profile is completely negligible. Following the procedure already described in Ferraro et al. (1999a, 2004), we have divided the entire HST+WFI sample in 27 concentric annuli, each centered on Cgrav and split in an adequate number of sub-sectors. The number of stars lying within each sub-sector was counted, and the star density was obtained by dividing these values by the corresponding sub-sector areas. The stellar density in each annulus was then obtained as the average of the sub-sector densities, and its standard deviation was estimated from the variance among the sub-sectors. The radial density profile thus derived is plotted in Figure 4, where we also show the best- fit mono-mass King model and the corresponding values of the core radius and concentration: rc = 27 ′′ (with a typical error of ∼ ±2′′) and c = 1.68, respectively. These values confirm that M5 has not yet experienced core collapse, and they are in good agreement with those quoted by McLaughlin & van der Marel (2005, rc = 26. ′′3 and c = 1.71), and marginally consistent with those listed by Harris (1996, rc = 25. ′′2 and c = 1.83), both derived from – 7 – the surface brightness profile. Our value of rc corresponds to ∼ 1 pc assuming the distance modulus (m−M)0 = 14.37 (d ∼ 7.5 Kpc, Ferraro et al. 1999b). 3. DEFINITION OF THE SAMPLES In order to study the BSS radial distribution and detect possible peculiarities, both the BSS and a reference population must be properly defined. Since the HST and the WFI data sets have been observed in different photometric bands, different selection boxes are needed to separate the samples in the CMDs. The adopted strategy is described in the following sections (see also Ferraro et al. 2004 for a detailed discussion of this issue). 3.1. The BSS selection At UV wavelengths BSS are among the brightest objects in a GC, and RGB stars are particularly faint. By combining these advantages with the high-resolution capability of HST, the usual problems associated with photometric blends and crowding in the high density central regions of GCs are minimized, and BSS can be most reliably recognized and separated from the other populations in the UV CMDs. For these reasons our primary criterion for the definition of the BSS sample is based on the position of stars in the (m255, m255 −U) plane. In order to avoid incompleteness bias and the possible contamination from TO and sub-giant branch stars, we have adopted a limiting magnitude m255 = 18.35, roughly corresponding to 1 magnitude brighter than the cluster TO. This is also the limiting magnitude used by W06, facilitating the comparison with their study. The resulting BSS selection box in the UV CMD is shown in Figure 5. Once selected in the UV CMD, the bulk of the BSS lying in the field in common with the optical-HST sample has been used to define the selection box and the limiting magnitude in the (B, B − V ) plane. The latter turns out to be B ≃ 17.85, and the adopted BSS selection box in the optical CMD is shown in Figure 6. The two stars lying outside the selection box (namely BSS-19 and BSS-20 in Table 1) have been identified as BSS from the (m255, m255−U) CMD. Indeed, they are typical examples of how the optical magnitudes are prone to blend/crowding problems, while the BSS selection in UV bands is much more secure and reliable. An additional BSS (BSS-47 in Table 1) lies near the edge of the ACS FoV and has only V and I observations; thus it was selected in the (V, V − I) plane (see Figure 7, where this BSS is shown together with the other 5 identified in the ACS complementary sample). With these criteria we have identified 60 BSS: 47 BSS in the HST sample (r ≤ 115′′) – 8 – and 13 in the WFI one. Their coordinates and magnitudes are listed in Table 1. Out of the 47 BSS identified in the HST sample, 41 are from the WFPC2 data set, and 6 from the ACS catalog. As shown in Figure 1 their projected distribution is quite asymmetric with the N-E sector seemingly underpopulated. The statistical significance of such an asymmetry appears even higher if only the BSS outside the core are considered. However a quantitative discussion of this topic is not warranted unless additional evidences supporting this anomalous spatial distribution are collected. One of the inner BSS (BSS-29 in Table 1) lying at 21.′′76 from the center, corresponds to the low-amplitude variable HST-V28 identified by Drissen & Shara (1998)4. In the WFI sample (r > 115′′) we find 13 BSS, with a more symmetric spatial distribution (see Figure 2). The most distant BSS (BSS-60 in Table 1, marked with an empty triangle in Fig.6) lies at ∼ 24′ from the center, i.e., beyond the cluster tidal radius. Hence, it might be an evaporating BSS previously belonging to the cluster. However, further investigations are needed before firmly assessing this issue. In order to perform a proper comparison with W06 study, we have transformed their BSS catalog in our astrometric system, and we have found that 50 BSS of their bright sample lie at r ≤ 115′′: 35 are from the HST sample, 13 from the Canada France Hawaii Telescope (CFHT) data set, and 2 from the Cerro Tololo Inter-American Observatory (CTIO) sample; in the outer regions (115′′ < r <∼ 425 ′′) 9 BSS are identified, all from the CTIO data set. By cross correlating W06 bright sample with our catalog we have found 43 BSS in common (see Table 1), 37 at r ≤ 115′′ and 6 outward. In particular, 33 BSS out of the 41 (i.e., 80% of the total) that we have identified in the WFPC2-HST sample5 are found in both catalogs, while 3 of our BSS belong 5 to their faint BSS sample (namely, BSS-27, 34, and 40, corresponding to their Core BSS 70, 79, and 76, respectively), 5 of our BSS have been missed in W06 paper, and 2 objects in their sample are classified as HB stars in our study. This is probably due to different selection criteria, and/or small differences in the measured magnitudes, caused by the different data reduction procedures and photometric analysis. For example, W06 identify the BSS on the basis of both the UV and the optical observations, while we select the BSS only in the UV plane whenever possible. Out of the other 15 BSS found at r ≤ 115′′ in the ground-based CFHT/CTIO sample of W06, 8 BSS (Core BSS 38–45 in their Table 2) clearly are false identifications. They are arranged in a very unlikely ring around a strongly saturated star, as can be seen in Figure 8, where the position in the sky of the 8 spurious BSS are overplotted on the CFHT image. Though they 4The observations presented here do not have the time coverage needed to properly search for BSS variability. 5Note that the WFPC2-HST observations used in W06 and in the present study are the same. – 9 – clearly are spurious identifications, they still define a clean sequence in the (B, B−I) CMD, nicely mimicking the BSS magnitudes and colors. As already discussed in previous papers, this once again demonstrates how automatic procedures for the search of peculiar objects are prone to errors, especially when using ground-based observations to probe very crowded stellar regions. We emphasize that all the candidate BSS listed in our Table 1 have been visually inspected evaluating the quality and the precision of the PSF fitting. This procedure significantly reduces the possibility of introducing spurious objects in the sample. Out of the remaining 7 BSS, 4 objects (namely their Core BSS 32, 30, 37 and 28) are also confirmed by our ACS observations (BSS-42, 43, 44, and 45 respectively), while 2 others (their Core BSS 27 and Ground BSS 6) are not found in the ACS data set, and the remaining one (their Ground BSS 7) is not included in our observation FoV. In turn, two BSS identified in our ACS data set (BSS 46 and 47) are missed in their sample. Concerning the BSS lying at 115′′ < r < 450′′, 6 objects (out of 9 found in both samples) are in common between the two catalogs (see Table 1), one (BSS-55) belongs to W06 faint sample (their Ground BSS 23), while the remaining 2 do not coincide. Moreover, 4 additional BSS have been identified at r > 450′′ in our study. 3.2. The reference population Since the HB sequence is bright and well separable in the UV and optical CMDs, we chose these stars as the primary representative population of normal cluster stars to be used for the comparison with the BSS data set. As with the BSS, the HB sample was first defined in the (m255, m255 −U) plane, and the corresponding selection box in (B, B − V ) has then been determined by using the stars in common between the UV and the optical samples. The resulting selection boxes in both diagrams are shown in Figures 5 and 6, and are designed to include the bulk of HB stars6. Slightly different selection boxes would include or exclude a few stars only without affecting the results. We have used WFI observations to roughly estimate the impact of possible foreground field stars contamination on the cluster population selection. As shown in the right-hand panel of Figure 6, field stars appear to define an almost vertical sequence at 0.4 < B−V < 1 in the (B, B − V ) CMD. Hence, they do not affect the BSS selection box, but marginally contaminate the reddest end of the HB. In particular, 5 objects have been found to lie within the adopted HB box in the region at r > rt sample by our observations (∼ 194 arcmin this corresponds to 0.026 spurious HB stars per arcmin2. On the basis of this, 11 field stars 6The large dispersion in the redder HB stars arises because RR Lyrae variables are included. – 10 – are expected to ”contaminate” the HB population over the sampled cluster region (r < rt). 4. THE BSS RADIAL DISTRIBUTION The radial distribution of BSS in M5 has been studied following the same procedure previously adopted for other clusters (see references in Ferraro 2006; Beccari et al. 2006). First, we have compared the BSS cumulative radial distribution to that of HB stars. A Kolmogorov-Smirnov test gives a ∼ 10−4 probability that they are extracted from the same population (see Figure 9). BSS are more centrally concentrated than HB stars at ∼ 4σ level. For a more quantitative analysis, the surveyed area has been divided into 8 concentric annuli, with radii listed in Table 2. The number of BSS (NBSS) and HB stars (NHB), as well as the fraction of sampled luminosity (Lsamp) have been measured in the 8 annuli and the obtained values are listed in Table 2. Note that HB star counts listed in the table are already decontaminated from field stars, according to the procedure described in Section 3.2 (1, 2, and 8 HB stars in the three outer annuli have been estimated to be field stars). The listed values have been used to compute the specific frequency FHBBSS ≡ NBSS/NHB, and the double normalized ratio (see Ferraro et al. 1993): Rpop = (Npop/N (Lsamp/L tot ) , (1) with pop = BSS, HB. In the present study luminosities have been calculated from the surface density profile shown in Figure 4. The surface density has been transformed into luminosity by means of a normalization factor obtained by assuming that the value obtained in the core (r ≤ 27′′) is equal to the sum of the luminosities of all the stars with V ≤ 20 lying in this region. The distance modulus quoted in Section 2.4 and a reddening E(B−V ) = 0.03 have been adopted (Ferraro et al. 1999b). The fraction of area sampled by the observations in each annulus has been carefully computed, and the sampled luminosity in each annulus has been corrected for incomplete spatial coverage (in the case of annuli 3 and 8; see Figures 1 and 2). The resulting radial trend of RHB is essentially constant with a value close to unity over the surveyed area (see Figure 10). This is just what expected on the basis of the stellar evolution theory, which predicts that the fraction of stars in any post-main sequence evolutionary stage is strictly proportional to the fraction of the sampled luminosity (Renzini & Fusi Pecci 1988). Conversely, BSS follow a completely different radial distribution. As shown in Figure 10 the specific frequency RBSS is highly peaked at the cluster center (a – 11 – factor of ∼ 3 higher than RHB in the innermost bin), decreases to a minimum 7 at r ≃ 10 rc, and rises again outward. The same behavior is clearly visible also in Figure 11, where the population ratio NBSS/NHB is plotted as a function of r/rc. Note that the region between 800′′ and rt ≃ 1290 ′′ (and thus also BSS-59, that lies at r ≃ 995.′′5) has not been considered in the analysis, since our observations provide a poor sampling of this annulus: only 35% of its area, corresponding to ∼ 0.4% of the total sampled light, is covered by the WFI pointing. However, for sake of completeness, we have plotted in Figure 12 the corresponding value of FHBBSS even for this annulus (empty circle in the upper panel): as can be seen, there is a hint for a flattening of the BSS radial distribution in the cluster outskirts. 4.1. Dynamical simulations Following the same approach as Mapelli et al. (2004, 2006), we now exploit dynamical simulations to derive some clues about the BSS formation mechanisms from their observed radial distribution. We use the Monte-Carlo simulation code originally developed by Sig- urdsson & Phinney (1995) and upgraded in Mapelli et al. (2004, 2006). In any simulation run we follow the dynamical evolution of N BSS within a background cluster, taking into account the effects of both dynamical friction and distant encounters. We identify as COL- BSS those objects having initial positions ri ∼ rc, and as MT-BSS stars initially lying at ri ≫ rc (this because stellar collisions are most probable in the central high-density regions of the cluster, while primordial binaries most likely evolve in isolation in the periphery). Within these two radial ranges, all initial positions are randomly generated following the probability distribution appropriate for a King model. The BSS initial velocities are ran- domly extracted from the cluster velocity distribution illustrated in Sigurdsson & Phinney (1995), and an additional natal kick is assigned to the COL-BSS in order to account for the recoil induced by the encounters. Each BSS has characteristic mass M and lifetime tlast. We follow their dynamical evolution in the cluster (fixed) gravitational potential for a time ti (i = 1, N), where each ti is a randomly chosen fraction of tlast. At the end of the simulation we register the final positions of BSS, and we compare their radial distribution with the observed one. We repeat the procedure until a reasonable agreement between the simulated and the observed distributions is reached; then, we infer the percentage of collisional and mass-transfer BSS from the distribution of the adopted initial positions in the simulation. For a detailed discussion of the ranges of values appropriate for these quantities and 7Note that no BSS have been found between 3.′5 and 5′. – 12 – their effects on the final results we refer to Mapelli et al. (2006). Here we only list the assumptions made in the present study: – the background cluster is approximated with a multi-mass King model, determined as the best fit to the observed profile8. The cluster central velocity dispersion is set to σ = 6.5 km s−1 (Dubath et al. 1997), and, assuming 0.5M⊙ as the average mass of the cluster stars, the central stellar density is nc = 2× 10 4 pc−3 (Pryor & Meylan 1993); – the COL-BSS are distributed with initial positions ri ≤ rc and are given a natal kick velocity of 1× σ; – initial positions ranging between 5 rc and rt (with the tidal radius rt ≃ 48 rc) have been considered for MT-BSS in different runs; – BSS masses have been fixed to M = 1.2M⊙ (Ferraro et al. 2006a), and their charac- teristic lifetime to tlast = 2 Gyr; – in each simulation run we have followed the evolution of N = 10, 000 BSS. The simulated radial distribution that best reproduces the observed one (with a reduced χ2 ≃ 0.6) is shown in Figure 11 (solid line) and is obtained by assuming that ∼ 80% of the BSS population was formed in the core through stellar collisions, while only ∼ 20% is made of MT-BSS. A higher fraction ( >∼ 40%) of MT-BSS does not correctly reproduce the steep decrease of the distribution and seriously overpredict the number of BSS at r ∼ 10 rc, where no BSS at all are found, but it nicely matches the observed upturning point at r ≃ 13 rc (see the dashed line in Figure 11). On the other hand, a population of only COL-BSS is unable to properly reproduce the external upturn of the distribution (see the dotted line in Figure 11), and 100% of MT-BSS is also totally excluded. Assuming heavier BSS (up to M = 1.5M⊙) or different lifetimes tlast (between 1 and 4 Gyr) does not significantly change these conclusions, since both these parameters mainly affect the external part of the simulated BSS distribution. Thus, an appreciable effect can be seen only in the case of a relevant upturn, and negligible variations are found in the best-fit case and when assuming 100% COL-BSS. The effect starts to be relevant in the simulations with 40% or more MT-BSS, which are however inconsistent with the observations at intermediate radii (see above). By using the simulations and the dynamical friction timescale (from, e.g., Mapelli et al. 2006), we have also computed the radius of avoidance of M5. This is defined as the char- acteristic radial distance within which all MT-BSS are expected to have already sunk to 8By adopting the same mass groups as those of Mapelli et al. (2006), the resulting value of the King dimensionless central potential is W0 = 9.7 – 13 – the cluster core, because of mass segregation processes. Assuming 12 Gyr for the age of M5 (Sandquist et al. 1996) and 1.2M⊙ for the BSS mass, we find that ravoid ≃ 10 rc. This nicely corresponds to the position of the minimum in the observed BSS radial distribution, in agreement with the findings of Mapelli et al. (2004, 2006). 5. SUMMARY AND DISCUSSION In this paper we have used a combination of HST UV and optical images of the cluster center and wide-field ground-based observations covering the entire cluster extension to de- rive the main structural parameters and to study the BSS population of the galactic globular cluster M5. The accurate determination of the cluster center of gravity from the high-resolution data gives α(J2000) = 15h 18m 33.s53, δ(J2000) = +2o 4′ 57.′′06, with a 1σ uncertainty of 0.′′5 in both α and δ. The cluster density profile, determined from direct star counts, is well fit by a King model with core radius rc = 27 ′′ and concentration c = 1.68, thus suggesting that M5 has not yet suffered the core collapse. The BSS population of M5 amounts to a total of 59 objects, with a quite asymmetric projected distribution (see Figure 1) and a high degree of segregation in the cluster center. With respect to the sampled luminosity and to HB stars, the BSS radial distribution is bimodal: highly peaked at r <∼ rc, decreasing to a minimum at r ≃ 10 rc, and rising again outward (see Figures 10 and 11). The comparison with results of W06 has revealed that 43 (out of 59) bright BSS iden- tified by these authors at r <∼ 450 ′′ are in common with our sample. Moreover, 4 additional stars classified as faint BSS in their study are in common with our BSS sample at r <∼ 450 Considering that we find 56 BSS within the same radial distance from the center, this corre- sponds to 84% matching of our catalogue. The discrepancies are explained by different data reduction procedures, photometric analysis, and adopted selection criteria, other than the spurious identification of 8 BSS by W06, due a strongly saturated star in their sample. The central peak of the RBSS distribution in our study is slightly higher (but compatible within the error bar) compared to that of W06, and we extend the analysis to larger distance from the center (out to r > 800′′), thus unveiling the external upturn and the possible flattening of the BSS distribution in the cluster outskirts. Moreover, we have compared the BSS radial distribution of M5 with that observed in other GCs studied in a similar way. In Figure 12 we plot the specific frequency FHBBSS as a function of (r/rc) for M5, M3, 47 Tuc, and NGC 6752. Such a comparison shows – 14 – that the BSS radial distributions in these clusters are only qualitatively similar, with a high concentration at the center and an upturn outward. However, significant quantitative differences are apparent: (1) the FHBBSS peak value, (2) the steepness of the decreasing branch of the distribution, (3) the radial position of the minimum (marked by arrows in the figure), and (4) the extension of the “zone of avoidance,” i.e., the intermediate region poorly populated by BSS. In particular M5 shows the smallest FHBBSS peak value: it turns out to be ∼ 0.24, versus a typical value >∼ 0.4 in all the other cases. It also shows the mildest decreasing slope: at r ≈ 2 rc the specific frequency in M5 is about a half of the peak value, while it decreases by a factor of 4 in all the other clusters. Conversely, it is interesting to note that the value reached by FHBBSS in the external regions is ∼ 50-60% of the central peak in all the studied clusters. Another difference between M5 and the other systems concerns the ratio between the radius of avoidance and the tidal radius: ravoid ≃ 0.2 rt for M5, while ravoid ∼ 0.13 rt for 47 Tuc, M3, and NGC 6752 (see Tables 1 and 2 in Mapelli et al. 2006). The dynamical simulations discussed in Section 4.1 suggest that the majority of BSS in M5 are collisional, with a content of MT-BSS ranging between 20% and 40% of the overall population. This fraction seems to be smaller than that (40-50%) derived for M3, 47 Tuc and NGC 6752 by Mapelli et al. (2006), in qualitative agreement with the smaller value of ravoid/rt estimated for M5, which indicates that the fraction of cluster currently depopulated of BSS is larger in this system than in the other cases. More in general, the results shown in Figure 11 exclude a pure collisional BSS content for M5. Our study has also revealed the presence of a candidate BSS at ∼ 24′ from the center, i.e., beyond the cluster tidal radius (see Figures 2 and 6 and BSS-59 in Table 1). If confirmed, this could represent a very interesting case of a BSS previously belonging to M5 and then evaporating from the cluster (a BSS kicked off from the core the because of dynamical interactions?). This research was supported by Agenzia Spaziale Italiana under contract ASI-INAF I/023/05/0, by the Istituto Nazionale di Astrofisica under contract PRIN/INAF 2006, and by the Ministero dell’Istruzione, dell’Università e della Ricerca. RTR is partially funded by NASA through grant number GO-10524 from the Space Telescope Science Institute. We thank the referee E. Sandquist for the careful reading of the manuscript and the useful comments and suggestions that significantly improved the presentation of the paper. REFERENCES Albrow, M. D., et al. 2001, ApJ, 559, 1060 – 15 – Bailyn, C. 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P. 1996, QJRAS, 37, 11 Schechter, P. L., Mateo, M., & Saha, A. 1993, PASP, 105, 1342 Shara, M. M., Saffer, R. A., & Livio, M. 1997, ApJ, 489, L59 Sigurdsson S., Phinney, E. S., 1995, ApJS, 99, 609 Sirianni, M., et al. 2005, PASP, 117, 1049 Warren, S. R., Sandquist, E. L., & Bolte, M., 2006, ApJ 648, 1026 (W06) Zaggia, S. R., Piotto, G., & Capaccioli, M., 1997, A&A, 327, 1004 – 17 – Fig. 1.— Map of the HST sample. The heavy solid line delimits the HST-WFPC2 FoV of our UV observations (Prop. 6607), the dashed line bounds the FoV of the optical HST- WFPC2 observations by Piotto et al. (2002), and the dotted line marks the edge of the complementary ACS data set. The derived center of gravity Cgrav is marked with a cross. BSS (heavy dots) and the concentric annuli used to study their radial distribution (cfr. Table 2) are also shown. The inner and outer annuli correspond to r = rc = 27 ′′ and r = 115′′, respectively. This preprint was prepared with the AAS LATEX macros v5.2. – 18 – Fig. 2.— Map of the WFI sample. All BSS detected in the WFI sample are marked as heavy dots, and the concentric annuli used to study their radial distribution are shown as solid lines, with the inner and outer annuli corresponding to r = 115′′ and r = 800′′, respectively (cfr. Table 2). The circle corresponding to the tidal radius (rt ≃ 21. ′5) is also shown as dashed- dotted line. The BSS lying beyond rt might represent a BSS previously belonging to M5 and now evaporating from the cluster. – 19 – – 20 – Fig. 3.— Optical CMDs of the WFPC2-HST and the WFI samples. The hatched regions indicate the magnitude limit (V ≤ 20) adopted for selecting the stars used to construct the cluster surface density profile. – 21 – Fig. 4.— Observed surface density profile (dots and error bars) and best-fit King model (solid line). The radial profile is in units of number of stars per square arcseconds. The dotted line indicates the adopted level of the background, and the model characteristic parameters (core radius rc, concentration c, dimensionless central potential W0) are marked in the figure. The lower panel shows the residuals between the observations and the fitted profile at each radial coordinate. – 22 – Fig. 5.— CMD of the ultraviolet HST sample. The adopted magnitude limit and selection box used for the definition of the BSS population are shown. The resulting fiducial BSS are marked with empty circles. The open square corresponds to the variable BSS identified by Drissen & Shara (1998). The box adopted for the selection of HB stars is also shown. – 23 – Fig. 6.— CMD of the optical HST-WFPC2 and WFI samples. The adopted BSS and HB selection boxes are shown, and all the BSS identified in these samples are marked with the empty circles. The two BSS not included in the box in the left-hand panel lie well within the selection box in the UV plane and are therefore considered as fiducial BSS. The empty triangle in the right-hand panel corresponds to the BSS identified beyond the cluster tidal radius, at r ≃ 24′. – 24 – Fig. 7.— CMD of the ACS complementary sample. The BSS selection box is shown, and the resulting fiducial BSS are marked with empty circles. – 25 – Fig. 8.— Left-hand panel: position of the 8 false BSS (marked with white circles) as derived from Table 2 of W06, overplotted to the CFHT image (units are the same as in their Figure 1). As can be seen, a heavily saturated star is responsible for the false identification. Right- hand panel: location of the 8 false BSS (empty circles) in the (B, B − I) plane, as derived from Table 2 of W06 (cfr. to their Fig. 2). – 26 – Fig. 9.— Cumulative radial distribution of BSS (solid line) and HB stars (dashed line) as a function of the projected distance from the cluster center for the combined HST+WFI sample. The two distributions differ at ∼ 4σ level. – 27 – Fig. 10.— Radial distribution of the BSS and HB double normalized ratios, as defined in equation (1), plotted as a function of the radial coordinate expressed in units of the core radius. RHB (with the size of the rectangles corresponding to the error bars computed as described in Sabbi et al. 2004) is almost constant around unity over the entire cluster extension, as expected for any normal, non-segregated cluster population. Instead, the radial trend of RBSS (dots with error bars) is completely different: highly peaked in the center (a factor of ∼ 3 higher than RHB), decreasing at intermediate radii, and rising again outward. – 28 – Fig. 11.— Observed radial distribution of the specific frequency NBSS/NHB (filled circles with error bars), as a function of r/rc. The simulated distribution that best reproduces the observed one is shown as a solid line and is obtained by assuming 80% of COL-BSS and 20% of MT-BSS. The simulated distributions obtained by assuming 40% of MT-BSS (dashed line) and 100% COL-BSS (dotted line) are also shown. – 29 – Fig. 12.— Radial distribution of the population ratio NBSS/NHB for M5, M3, 47 Tuc, and NGC 6752, plotted as a function of the radial distance from the cluster center, normalized to the core radius rc (from Mapelli et al. 2006, rc ≃ 30 ′′, 21′′, 28′′ for M3, 47 Tuc, and NGC 6752, respectively). The arrows indicate the position of the minimum of the distribution in each case. The outermost point shown for M5 (empty circle) corresponds to BSS-58, lying at r ≃ 995′′. This star has not been considered in the quantitative study of the BSS radial distribution since only a negligible fraction of the annuls between 800′′ and rt is sampled by our observations. – 30 – Table 1. The BSS population of M5 Name RA DEC m255 U B V I W06 [degree] [degree] BSS-1 229.6354506 2.0841090 16.52 16.15 15.88 15.71 - CR2 BSS-2 229.6388102 2.0849660 17.95 17.38 17.40 17.04 - CR4 BSS-3 229.6383433 2.0842640 18.21 17.63 17.64 17.32 - CR3 BSS-4 229.6416234 2.0851791 17.59 17.22 17.05 16.90 - CR5 BSS-5 229.6416518 2.0836794 16.28 15.99 15.79 15.70 - CR1 BSS-6 229.6381953 2.0810119 17.36 16.99 16.81 16.65 - CR21 BSS-7 229.6403657 2.0824062 17.40 17.07 16.97 16.76 - CR12 BSS-8 229.6412279 2.0823768 17.91 17.47 17.41 17.15 - CR13 BSS-9 229.6376256 2.0793288 17.84 17.12 16.99 16.77 - CR23 BSS-10 229.6401139 2.0794858 17.57 16.98 16.87 16.62 - CR22 BSS-11 229.6396566 2.0784944 17.51 17.20 17.12 16.92 - CR24 BSS-12 229.6432834 2.0797197 18.12 17.64 17.78 17.54 - - BSS-13 229.6384406 2.0776614 17.36 16.88 16.88 16.59 - CR25 BSS-14 229.6274500 2.0864896 18.07 17.63 17.64 17.33 - CR8 BSS-15 229.6204246 2.0879629 18.33 17.61 17.75 17.36 - CR11 BSS-16 229.6209379 2.0917858 17.80 17.28 17.26 16.98 - CR18 BSS-17 229.6264834 2.0960870 16.32 16.22 16.20 16.13 - CR20 BSS-18 229.6368731 2.0896002 16.56 16.30 16.11 16.01 - CR14 BSS-19 229.6367309 2.0917639 18.27 17.35 17.58 17.07 - CR17 BSS-20 229.6345837 2.0906438 17.88 16.81 16.96 16.43 - CR16 BSS-21 229.6382677 2.0934706 18.25 17.58 17.71 17.35 - CR19 BSS-22 229.6340227 2.0853879 17.67 17.32 17.22 17.03 - CR7 BSS-23 229.6332685 2.0875294 17.69 17.34 17.21 17.08 - CR10 BSS-24 229.6366685 2.0807168 18.23 17.78 17.67 17.37 - - BSS-25 229.6393544 2.0762832 18.11 17.79 17.72 17.50 - - BSS-26 229.6378381 2.0779999 17.86 17.52 17.43 17.27 - - BSS-27 229.6349851 2.0807202 18.17 17.51 17.74 17.30 - CR70 BSS-28 229.6397645 2.0736403 18.19 17.60 17.69 17.28 - CR33 BSS-29 229.6370495 2.0770798 16.83 16.56 16.57 17.75 - CR26 BSS-30 229.6358816 2.0747883 18.25 17.81 17.79 17.51 - CR31 BSS-31 229.6361653 2.0720147 18.29 17.77 17.81 17.47 - CR36 BSS-32 229.6339822 2.0723032 16.73 16.10 16.16 15.95 - CR35 BSS-33 229.6281392 2.0756490 17.74 17.41 17.22 17.09 - CR29 BSS-34 229.6241278 2.0750261 18.21 17.50 17.65 17.27 - CR79 BSS-35 229.6332759 2.0603761 17.48 17.17 16.95 16.86 - CR48 BSS-36 229.6270877 2.0662947 17.33 17.18 17.06 16.95 - CR47 BSS-37 229.6244175 2.0693612 16.89 16.41 16.51 15.71 - CR46 BSS-38 229.6180419 2.0724090 17.37 17.23 17.12 17.00 - CR34 – 31 – Table 1—Continued BSS-39 229.6311963 2.0857800 18.31 17.33 17.40 16.76 - - BSS-40 229.6297499 2.0664961 18.16 17.58 - 17.27 - CR76 BSS-41 229.6443367 2.0872809 - - 17.50 17.23 - CR9 BSS-42 229.6448646 2.0738335 - - 16.53 16.06 15.95 CR32 BSS-43 229.6460645 2.0748695 - - 16.64 16.44 16.66 CR30 BSS-44 229.6481631 2.0718829 - - 16.72 16.61 16.87 CR37 BSS-45 229.6433942 2.0760163 - - 17.03 16.79 16.91 CR28 BSS-46 229.6439884 2.0775670 - - 17.44 16.99 16.81 - BSS-47 229.6180420 2.0598328 - - - 17.18 17.12 - BSS-48 229.6092873 2.1680914 - - 16.85 16.68 - OR2 BSS-49 229.6723094 2.0882827 - - 16.94 16.64 - OR9 BSS-50 229.6006551 2.0814678 - - 17.00 16.74 - OR10 BSS-51 229.6669956 1.9781808 - - 17.20 16.74 - OR1 BSS-52 229.5949935 2.0469325 - - 17.69 17.46 - OR4 BSS-53 229.6706625 2.0695464 - - 17.82 17.50 - - BSS-54 229.6667908 2.1149550 - - 17.82 17.72 - - BSS-55 229.7370667 2.0323392 - - 17.80 17.42 - OR23 BSS-56 229.5476990 2.0112610 - - 16.88 16.60 - OR5 BSS-57 229.6711255 1.9415566 - - 16.98 16.64 - - BSS-58 229.4381714 2.0302088 - - 17.75 17.33 - - BSS-59 229.7408412 2.3399166 - - 17.49 17.08 - - BSS-60 229.3218200 2.3271022 - - 16.34 16.09 - - Note. — The first 41 BSS have been identified in the WFPC2 sample; BSS-42– 46 are from the complementary ACS observations; BSS-47–59 are from the WFI data-set. BSS-59 lies beyond the cluster tidal radius, at ∼ 24′ from the center. The last column list the corresponding BSS in W06 sample, with ”CR” indicating their ”Core BSS” and ”OR” their ”Outer Region BSS”. – 32 – Table 2. Number counts of BSS and HB stars ′′ re ′′ NBSS NHB L samp/L 0 27 22 94 0.14 27 50 15 94 0.16 50 115 10 135 0.26 115 150 3 46 0.09 150 210 2 52 0.10 210 300 0 45† 0.10 300 450 4 42† 0.09 450 800 2 38† 0.06 Note. — † The NHB values listed here are those corrected for field contamination (i.e., 1, 2 and 8 stars have been subtracted to the observed number counts in these three external annuli, respectively). INTRODUCTION OBSERVATIONS AND DATA ANALYSIS The data sets Astrometry and center of gravity Photometric calibration and definition of the catalogs Density profile DEFINITION OF THE SAMPLES The BSS selection The reference population THE BSS RADIAL DISTRIBUTION Dynamical simulations SUMMARY AND DISCUSSION

0704.0140

Entanglement entropy of two-dimensional Anti-de Sitter black holes

Entanglement Entropy of two-dimensional anti-de Sitter black holes Mariano Cadoni∗ Dipartimento di Fisica, Università di Cagliari, and INFN sezione di Cagliari, Cittadella Universitaria 09042 Monserrato, ITALY Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein-Hawking entropy SBH , whereas the subleading term behaves as lnSBH . This subleading term has the universal form typical for the entanglement entropy of physical systems described by effective conformal fields theories (e.g. one-dimensional statistical models at the critical point). The well-known form of the entanglement entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our result and is related with the entanglement entropy of a black hole with negative mass. Quantum entanglement is a fundamental feature of quantum systems. It is related to the existence of correlations between parts of the system. The degree of entanglement of a quantum system is measured by the entanglement entropy Sent. In quantum field theory (QFT), or more in general in many body systems, we can localize observable and unobservable degrees of freedom in spatially separated regions Q and R. Sent is then defined as the von Neumann entropy of the system when the degrees of freedom in the region R are traced over, Sent = −TrQρ̂Q ln ρ̂Q, where the trace is taken over states in the observable region Q and the reduced density matrix ρ̂Q = TrRρ̂ is obtained by tracing the density matrix ρ̂ over states in the region R. Investigation of the entanglement entropy (EE) has become relevant in many research areas. Apart from quantum information theory, the field that gave birth to the notion of entanglement entropy, it plays a crucial role in condensed matter systems, where it helps to understand quantum phases of matter (e.g spin chains and quantum liquids)[1, 2, 3, 4, 5]. Entanglement (geometric) entropy is also an useful concept for investigating general features of QFT, in particular two-dimensional conformal field theory (CFT) and the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [6, 7, 8, 9, 10, 11, 12] . Last but not least entanglement may held the key for unraveling the mystery of black hole entropy [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. We will be mainly concerned with the entanglement entropy of two-dimensional (2D) CFT and its relationship with the entropy of 2D black holes. It is an old idea that black hole entropy may be explained in terms of the EE of the quantum state of matter fields in the black hole geometry [13]. The main support to this conjecture comes from the fact that both the EE of matter fields and the Bekenstein-Hawking (BH) entropy depend on the area of the boundary region. On the other hand any attempt to explain the BH entropy as originating from quantum entanglement has to solve conceptual and technical difficulties. The usual statistical paradigm explains the BH entropy in terms of a microstate gas. This is conceptually different from the EE that measures the observer’s lack of information about the quantum state of the system in a inaccessible region of spacetime. Moreover, the EE depends both on the number of species ns of the matter fields, whose entanglement should reproduce the BH entropy, and on the value of the UV cutoff δ arising owing to the presence of a sharp boundary between the accessible and inaccessible regions of the spacetime. Conversely, the BH entropy is meant to be universal, hence independent from ns and δ. Some conceptual difficulties can be solved using Sakharov’s induced gravity approach [23, 24, 25], but the problem of the dependence on ns and δ still remains unsolved. In this letter we will show that in the case of two-dimensional AdS black hole these difficulties can be completely solved. We will derive an expression for the black hole EE that in the large black hole mass limit reproduces exactly the BH entropy. Moreover, we will show that the subleading term has the universal behavior typical for CFTs and in particular for critical phenomena. The reason of this success is related to the peculiarities of 2D AdS gravity, namely the existence of an AdS/CFT correspondence and the fact that 2D Newton constant can be considered as wholly induced by quantum fluctuations of the dual CFT. Most of the progress in understanding the EE in QFT has been achieved in the case of 2D CFT. Conformal invariance in two space-time dimension is a powerful tool that allows us to compute the EE in closed form. The entanglement entropy for the ground state of a 2D CFT originated from tracing over correlations between spacelike separated points has been calculated by Holzhey, Larsen and Wilckzek [6]. Introducing an infrared cutoff Λ the spacelike coordinate of our 2D universe will belong to C = [0,Λ[. The subsystem where measurements are performed is Q = [0,Σ[, whereas the outside region where the degrees of freedom are traced over is R = [Σ,Λ[. Because of the contribution of localized excitations arbitrarily near to the boundary the entanglement entropy diverges. Introducing http://arxiv.org/abs/0704.0140v2 an ultraviolet cutoff δ, the regularized entanglement entropy turns out to be [6] Sent = c+ c̄ , (1) where c and c̄ are the central charges of the 2D CFT. The expression (1) emphasizes the characterizing features of the entanglement entropy, namely subadditivity and invariance under the transformation which exchanges the inside and outside regions Σ → Λ− Σ. (2) Moreover, Sent is not a monotonic function of Σ, but increases and reaches its maximum for Σ = Λ/2 and then decreases as Σ increases further. This behavior has an obvious explanation. When the subsystem begins to fill most of the universe there is lesser information to be lost and the entanglement entropy decreases. Let us now consider 2D AdS black holes. As classical solutions of a 2D gravity theory they are endowed with a non-constant scalar field, the dilaton Φ. In the Schwarzschild gauge the 2D AdS black hole solutions are [26], ds2 = − dt2 + dr2, Φ = Φ0 , (3) where the length L is related to cosmological constant of the AdS spacetime (λ = 1/L2), Φ0 is the dimensionless 2D inverse Newton constant and a is an integration constants related to the black hole mass M and horizon radius rh by . (4) The thermodynamical, Bekenstein-Hawking, entropy of the black hole is [26] SBH = 2πΦ0a = 2π 2Φ0ML, (5) whereas the black hole temperature is T = a/2πL. Setting a = 0 in Eq. (3) we have the AdS black hole ground state ( in the following called AdS0) with zero mass, temperature and entropy. The AdS black hole (3) can be considered as the thermalization of the AdS0 solution at temperature a/2πL [26]. It has been shown that the 2D black hole has a dual description in terms of a CFT with central charge [27, 28, 29, 30] c = 12Φ0. (6) The dual CFT can have both the form of a 2D [29, 30] or a 1D [27, 28] conformal field theory. This AdS2/CFT2 ( or AdS2/CFT1) correspondence has been used to give a microscopical meaning to the thermodynamical entropy of 2D AdS black holes. Eq. (5) has been reproduced by counting states in the dual CFT. In Ref. [16] (see also Refs. [24, 25, 31]) it was observed that in two dimensions black hole entropy can be ascribed to quantum entanglement if 2D Newton constant is wholly induced by quantum fluctuations of matter fields. On the other hand the AdS2/CFT2 correspondence, and in particular Eq. (6), tells us that the 2D Newton constant is induced by quantum fluctuations of the dual CFT. It follows that the black hole entropy (5) should be explained as the entanglement of the vacuum of the 2D CFT of central charge given by Eq. (6) in the gravitational black hole background (3). At first sight one is tempted to use Eq. (1) to calculate the entanglement entropy of the vacuum of the dual CFT. The exterior region of the 2D black hole can be easily identified with the region Q, whereas the black hole interior has to be identified with the R region where the degrees of freedom are traced over. There are two obstacles that prevents direct application of Eq. (1). First, Eq. (1) holds for a 2D flat spacetime, whereas we are dealing with a curved 2D background. Second, the calculations leading to Eq. (1) are performed for spacelike slice Q, whereas in our case the coordinate singularities at r = rh (the horizon) and r = ∞ (the timelike asymptotic boundary of the AdS spacetime) do not allow for a global notion of spacelike coordinate (a coordinate system covering the whole black hole spacetime in which the metric is non-singular and static). Owing to these geometrical features, in the black hole case we cannot give a direct meaning to both the measures Σ and (Λ−Σ) of the subsystems Q,R. As a consequence invariance under the transformation (2) is meaningless in the black hole case. The second difficulty can be circumvented using appropriate coordinate system and regularization procedure, the first using instead of Eq. (1) the formula derived by Fiola et al. [16], which gives the EE of the vacuum of matter fields in the case of a curved gravitational background. σ= ∞σ=ε FIG. 1: Regularized euclidean instanton corresponding to the 2D AdS black hole in the coordinate system (t, σ) covering only the black hole exterior. The euclidean time is periodic. The point σ = ∞ correspond to the black hole horizon. σ = 0 corresponds to the asymptotic timelike boundary of AdS2. In the coordinate system used to define the vacuum of scalar fields in AdS2, the 2D black hole metric (3) is [26] ds2 = sinh2(aσ −dt2 + dσ2 . (7) The coordinate system (t, σ) covers only the black hole exterior. The black hole horizon corresponds to σ = ∞ where the conformal factor of the metric vanishes. The asymptotic r = ∞ timelike conformal boundary of the AdS2 spacetime is located at σ = 0, where the conformal factor diverges. The entanglement entropy of the CFT vacuum in the curved background (7) can be calculated, using the formula of Ref. [16] as the half line entanglement entropy seen by an observer in the 0 < σ < ∞ region. From the CFT point of view the AdS black hole has to be considered as the AdS0 vacuum seen by the observer using the black hole coordinates (7) [26]. Moreover, this observer sees the the AdS0 vacuum as filled with thermal radiation with negative flux [26]. It follows that the black hole entanglement entropy is given by the formula of Ref. [16] with reversed sign, ent = − ρ(σ = 0)− ln , (8) where ρ defines the conformal factor of the metric in the conformal gauge (ds2 = exp(2ρ)(−dt2+dσ2)), c is the central charge given by Eq. (6) and δ,Λ are respectively UV and IR cutoffs. Notice that in Eq. (8) we have only contributions from only one sector (e.g. right movers) of the CFT. In Ref. [29, 30] it has been shown that the 2D AdS black hole is dual to an open string with appropriate boundary conditions. These boundary conditions are such that only one sector of the CFT2 is present. The same is obviously true for the AdS2/CFT1 realization of the correspondence [27, 28]. The conformal factor of the metric (7), hence the entanglement entropy (8) blows up on the σ = 0 boundary of the AdS spacetime. The simplest regularization procedure that solves this problem is to consider a regularized boundary at σ = ǫ. Notice that ǫ plays the role of a UV cutoff for the coordinate σ, which is the natural spacelike coordinate of the dual CFT. ǫ is an IR cutoff for the coordinate r, which is the natural spacelike coordinate for the AdS2 black hole. The regularized euclidean instanton corresponding to the black hole (7) is shown in figure (1). The regularizing parameter ǫ can be set equal to the UV cutoff, δ = ǫ. Moreover, the regularized boundary is at finite proper distance from the horizon so that ǫ acts also as IR regulator, making the presence of the IR cutoff Λ in Eq. (8) redundant. It follows that the regularized EE is given by S ent = − ρ(ǫ)− ln ǫ , which using equations (7) and (4) becomes ent = . (9) As a check of the validity of our formula we note that in the case of AdS0 (rh = 0) the entanglement entropy vanishes. The AdS/CFT correspondence enable us to identify the cutoff ǫ as the UV cutoff of the CFT : ǫ ∝ L. The proportionality factor can be determined by requiring that the analytical continuation of Eq. (9) is invariant under the transformation (2) (see later). This requirement fixes ǫ = πL. With this position we get ent = . (10) This formula is our main result, it gives the entanglement entropy of the 2D AdS black hole. This entanglement entropy has the expected behavior as a function of the horizon radius rh or, equivalently, of the black hole mass M . ent becomes zero in the AdS0 ground state, rh = 0 (M = 0), whereas it grows monotonically for rh > 0 (M > 0). In order to compare the black hole EE (10) with the BH entropy (5) let us consider the limit of macroscopic black holes, that is the limit a → ∞ or equivalently rh >> L or also M >> 1/L. Expanding Eq. (10) and using Eqs. (4) and (6) we get ent = 2π 2Φ0ML− Φ0 lnLM +O(1) = SBH − 2Φ0 lnSBH +O(1). (11) We have obtained the remarkable result that the leading term in the large mass expansion of the black hole en- tanglement entropy reproduces exactly the Bekenstein-Hawking entropy. Moreover, the subleading term behaves as the logarithm of the BH entropy and describes quantum corrections to SBH . It is an universally accepted result that the quantum corrections to the BH entropy behave as lnSBH [32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. However, there is no general consensus about the value of the prefactor of this term. For the microcanonical ensemble this term has to be negative, whereas there are positive contributions coming from thermal fluctuation. Equation (11) fixes the prefactor of lnSBH in terms of the 2D Newton constant. This result contradicts some previous results supporting a Φ0-independent value of the prefactor. Our result is consistent with the approach followed in this paper, which considers 2D gravity as induced from the quantum fluctuations of a CFT with central charge 12Φ0. The first (Bekenstein-Hawking) term in Eq. (11) is the induced entanglement entropy, whereas the second term, −(c/6) ln(rh/L), is determined by the conformal symmetry. It gives the entanglement entropy (1) of a CFT in 2D flat spacetime with central charge 12Φ0 and Σ = rh in the limit Σ << Λ [6]. The subleading term in Eq. (11) represents therefore an universal behavior shared with other systems described by 2D QFTs, such as one-dimensional statistical models near to the critical point (with the black hole radius rh corresponding to the correlation length) or free scalars fields [7, 9]. Eq. (10) shows a close resemblance with the CFT entanglement entropy (1). Eqs. (10) and (1) differs in two main points: the absence in the black hole case of something corresponding to the measure of the whole space (the parameter Λ in Eq. (1)) and the appearance of hyperbolic instead of trigonometric functions. These are expected features for the entanglement entropy of a black hole. They solve the problems concerning the application of formula (1) to the black hole case. For a black hole one cannot define a measure of the whole space analogue to Λ. For static solutions the coordinate system covers only the black hole exterior. The appearance of hyperbolic instead of trigonometric functions allows for monotonic increasing of S ent (rh), eliminating the unphysical decreasing behavior of Sent(Σ) in the region Σ > Λ/2. It is interesting to see how Eq. (1) can be obtained as the analytic continuation rh → irh of our formula (10), i.e by considering an AdS black hole with negative mass. The analytically continued black hole solution is given by Eq. (3) with a2 < 0. In the conformal gauge the solution reads now ds2 = [a2/ sin2(aσ/L)](−dt2 + dσ2). The range of the spacelike coordinate, corresponding to 0 < r < ∞, is now 0 < σ < πL/2a. Regularizing the solution at σ = 0 by introducing the cutoff ǫ we get the euclidean instanton shown in Fig. (2). In terms of the 2D CFT we have to trace over the degrees of freedom outside the spacelike slice ǫ < σ < πL/2a. The related entanglement entropy can be calculated using the formula of Ref. [16] in the case of a spacelike slice with two boundary points: Sent = −c/6[ρ(ǫ)+ρ(πL/2a)− ln(δ/Λ)]. Applying this formula to the case of the black hole solution of negative mass, identifying ǫ in terms of the IR cutoff Λ, ǫ = πL2/Λ, and redefining appropriately the UV cutoff δ, we get Sent = . (12) σ= πL/2a σ=ε FIG. 2: Regularized euclidean instanton corresponding to the 2D AdS black hole with negative mass. The euclidean time is periodic. The point σ = πL/2a corresponds to the black hole singularity at r = 0. σ = 0 corresponds to the asymptotic timelike boundary of AdS2. Thus, the entanglement entropy of the 2D CFT in the curved background given by the AdS black hole of negative mass has exactly the form given by Eq. (1) with the horizon radius rh playing the role of Σ. Notice that the presence of the factor π in the argument of the sin-function is necessary if one wants invariance under the transformation (2). The requirement that equation (12) is the analytic continuation of Eq. 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0704.0141

Towards self-consistent definition of instanton liquid parameters

Towards self-consistent definition of instanton liquid parameters S.V. Molodtsov1,2, G.M. Zinovjev3 1Joint Institute for Nuclear Research, RU-141980, Dubna, Moscow region, Russia 2Institute of Theoretical and Experimental Physics, RU-117259, Moscow, Russia 3Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, UA-03680, Kiev-143, Ukraine The possibility of self-consistent determination of instanton liquid parameters is discussed to- gether with the definition of optimal pseudo-particle configurations and comparing the various pseudo-particle ensembles. The weakening of repulsive interactions between pseudo-particles is argued and estimated. The problem of finding the most effective pseudo-particle profile for instanton liquid (IL) model of the QCD vacuum [1] has already been formulated in the first papers treating the pseudo-particle superposition as the quasi-classical configuration saturating the generating functional [2] of the fol- lowing form D[A] e−S(A) , (1) where S(A) is the Yang-Mills action. Although the solution proposed in Ref. [2] was quite acceptable phenomenologically the consequent more accurate analysis discovered several imperfect conclusions putting into doubt the assertion about the instanton ensemble getting stabilization and some addi- tional mechanism should be introduced to fix such an ensemble [3]. In this note we revisit the task formulated in Ref. [2] within the self-consistent approach proposed in our previous paper [4]. We are not speculating on the detailed mechanism of stabilizing and are based on one crucial assumption which is the existence of non-zero gluon condensate in the QCD vacuum. This idea is not very orig- inal but turns out far reaching in the context of our approach. The particular form and properties of this condensate will be discussed in the following paper. Thus, as the configuration saturating the generating functional (1) we take the following super- position Aaµ(x) = Baµ(x) + Aaµ(x; γi) , (2) here Aaµ stands for the (anti-)instanton field in the singular gauge Aaµ(x; γ) = ωabη̄bµν f(y), y = x− z , (3) γi = (ρi, zi, ωi) denotes all the parameters describing the i-th (anti-)instanton, in particular, its size ρ, colour orientation ω, center position z and as usual g is the coupling constant of gauge field. The function f(y) introduces the pseudo-particle profile and will be fixed by resolving the suitable variational problem. For example, for the conventional singular instanton it looks like f(y) = . (4) http://arxiv.org/abs/0704.0141v1 In analogy with this form we consider the function f depending on y2 or, more precisely, on the variable x = at some characteristic mean pseudo-particle size ρ̄. Dealing with the anti-instanton one should make the substitution of the ’t Hooft symbol η̄ → η. It is seen from (2) we ’singled out’ one pseudo-particle of ensemble and introduced the special symbol B for its field which actually has the same form as Eq. (3). The strength tensor of this ’external’ field and the field of every separate pseudo-particle A can be written as Gaµν = G µν(B) +G µν(A) +G µν(A,B) , (5) where two first terms are given by the standard definition of field strength Gaµν(A) = ∂µA ν − ∂νAaµ + g fabcAbµAcν , (6) with the entirely antisymmetric tensor fabc. In particular, for the singular instanton of Eq. (3) it takes the form Gaµν = − η̄kαβ f(1− f) + (η̄kµβ yν − η̄kνα yµ) f ′ − f(1− f) , (7) where f ′ means the derivative over y2. The third term of Eq. (5) presents the ’mixed’ component of field strength and is Gaµν(A,B) = g f abc(BbµA ν − BbνAcµ) = g fabcωcd (Bbµ η̄dνα −Bbν η̄dµα) f. (8) It was shown in Ref. [4] that in quasi-classical regime which is of particular interest for appli- cations, the generating functional (1) could be essentially simplified if reformulated in terms of the field BA averaged over ensemble A. Performing the cluster decomposition [5] of stochastic exponent in Eq. (1) 〈exp(−S)〉ωz = exp (−1)k 〈〈Sk〉〉ωz , (9) where 〈S1〉 = 〈〈S1〉〉, 〈S1S2〉 = 〈S1〉〈S2〉 + 〈〈S1S2〉〉, . . . (the first cumulant is simply defined by averaging the action) the higher terms of effective action for the ’external’ field in IL could be presented as 〈〈S[BA]〉〉A = G(BA) G(BA) , (10) and the mass m is defined by the IL parameters developing for the standard singular pseudo-particles (4) the following form (see, also below) m2 = 9π2 n ρ̄2 N2c − 1 , (11) with n = N/V where N is the total number of pseudoparticles in the volume V and Nc is the number of colours. The small magnitude of characteristic IL parameter (packing fraction) nρ̄4 allows us at decomposing to keep the contributions of one pseudo-particle term (∼ n) only. The effective action in Eq. (10) implies a functional integration in which the vacuum stochastic fields are not destroyed by the external field. Then there is no reason to develop the detailed description of the field B driven by the symmetries of initial gauge invariant Lagrangian for the Yang-Mills fields. In practice it could be understood as an argument to do use the averaged action dealing with the field B. It means the colourless binary (and similar even) configurations only of field B survive in the effective action. In other words the decomposition B ≃ BA + · · · is used (in what follows we are not maintaining the index for the field B). Obviously, if there is any need of more detailed description including, for example, information on the fluctuations of field B one should operate with the correlation functions of higher order and the corresponding chain of the Bogolyubov equations. The selfconsistent description of pseudo-particle ensemble may not be developed based on Eq. (10) only because in such a form the pseudo-particles of zero size ρ = 0 are most advantageous. In Ref. [4] the version of variational principle was proposed which makes it possible to determine the selfconsistent solution in long wave-length approximation for the pseudo-particle ensemble (anti- instantons in the singular gauge with standard profile (4)) and external field. Here it adapts to the saturating configuration (2) also and its more optimal (than standard) profile is defined, as suggested in Ref. [2], taking into account the IL parameter change while the pseudo-particle field is present. The contribution of saturating configuration into the generating functional is evaluated as (see [2] for the denotions) Z ≃ Y = dγi e −S(B,γ) . (12) The following terms should be taken into consideration S(B, γ) = − ln d(ρi) + β Uint + U iext(B) + S(B) , (13) (the details of deducing this expression can be found in [4]). Here we remind only that to obtain it one should average over the pseudo-particle parameters and to hold the highest contributions only at summing up the pseudo-particles. If the saturating configurations are the instantons in singular gauge with the standard profile (4) the first term describing the one instanton contributions takes the form of distribution function over (anti-)instanton sizes d(ρ) = CNcΛ b ρb−5β̃2Nc , (14) where Nf , (15) β̃ = −b ln(Λρ̄), CNc ≈ 4.66 exp(−1.68Nc) π2(Nc − 1)!(Nc − 2)! If one considers the profile of Eq. (3) the change of one pseudo-particle action which has the form Si = 3 (y2f ′)2 + f 2(1− f)2 , (16) should be absorbed while calculating. Here β = 8π2/g2 is the characteristic action of single pseudo- particle (4) which is defined at the scale of average pseudo-particle size β = β(ρ̄) where β(ρ) = − lnCNc−b ln(Λρ). The coefficient b enters the corresponding equations (in particular the distribution function (14)) always with the additional factor s = . It means that in all the formula containing the one instanton contribution the following substitution b → b s . (17) should be done. The penultimate term of Eq. (13) accumulates the partial pseudo-particle contribu- tions coming from the ’mixed’ component of the strength tensor (8) and describing the interaction of pseudo-particle ensemble with the detached one, i.e. U iext(B) = Gaµν(Ai, B) G µν(Ai, B) The other terms at the characteristic IL parameters are small as it was shown in Ref. [4]. The average value of ’mixed’ component is given by the following formula 〈Gaµν(A,B) Gaµν(A,B)〉ωz = N2c − 1 I Bbµ B µ , B , (18) here I is defined by the integrated profile function of pseudo-particle Iα,β = δα,β I = f 2 , I = dx f 2 , x = In particular, for the standard form of pseudo-particle we have dx f 2 = 1 . The corresponding constant (see [4]) ζ0 = N2c−1 should be changed for the modified one ζ = λζ0 , λ = dx f 2 , in all terms describing the interaction of IL with detached pseudo-particle if the profile function f is arbitrary. Eq. (18) demonstrates that we are formally dealing with non-zero value of gluon condensate which is given by the correlation function 〈Aaµ(x; γ)Aaµ(y; γ)〉ωz = N2c − 1 |x− y| . (19) For the pseudo-particle of standard form the function F (∆) equals to F (∆) = ∆2 + 2 ∆2 + 4 ln ∣∣∣∣∣ ∆2 + 4(∆2 + 1) + ∆3 + 3∆√ ∆2 + 4−∆ ∣∣∣∣∣− − π2 (∆ 2 + 1)2 ln(1 + ∆2) + π2 ∆2 ln |∆| , with the asymptotic behaviours F (∆) → π2 − π ∆2 + π2 ∆2 ln |∆| , lim F (∆) → π The presence of this condensate (19) which leads, in particular, to the mass definition as in (11) just signifies the assumption mentioned at the beginning this note. The second term of (13) describes the repulsive interaction between the pseudo-particles of en- semble β Uint = Gaµν(Ai, Aj) G µν(Ai, Aj) γi,γj and actually presents the same contribution as Uext but being integrated with the field B of every individual pseudo-particle as β Uint = d4x m 2. It results in the change of coupling constant ξ20 = 27 π2 N2c−1 describing the pseudo-particle interaction (see [2]) for new form ξ2 = λ2 ξ20 , (similar to the change of constant ζ). And eventually the last term of Eq. (13) presents simply the Yang-Mills action of the B field S(B) = Gaµν(B) G µν(B) It is worthwhile to notice that the topological charge of the configuration (4) is retained to be equal GaµνG̃ dx f ′f(1− f) = 1 , G̃aµν = εµναβ G here εµναβ is an entirely antisymmetric tensor, ε1234 = 1. The generating functional (12) might be estimated with the approximating functional (see [2]) as Y ≥ Y1 exp(−〈S − S1〉) , (21) where dγi e −S1(B,γ)−S(B) , S1(B, γ) = − lnµ(ρi) , and µ(ρ) is an effective one particle distribution function defined by solving the variational problem. In our particular situation the average value of difference of the actions is given as follows 〈S − S1〉 = dγi [β Uint + Uext(γ, B)− ln d(ρi) + lnµ(ρi)] e lnµ(ρi) = dρ µ(ρ) ln dγ1dγ2 Uint(γ1, γ2) µ(ρ1)µ(ρ2) + ρ2ζ B2 = d4x n + ζρ2 B2 , (22) with µ0 = dρ µ(ρ). In this note we estimate the functionals in the long wave length (adiabatic) approximation, i.e. consider the IL elements to be equilibrated by the external fixed field B. After- wards, with finding the optimal IL parameters out we receive the effective action for the external field in the selfconsistent form. Eq. (22) is taken just in such a form in order to underline the inte- gration is executed over the IL elements and the parameters describing their states are the functions of external field (i.e. could finally be the functions of a coordinate x). The physical meaning of such a functional is quite transparent and implies that each separate IL element develops its characteristic screening of the attached field. Now calculating the variation of action difference 〈S − S1〉 over µ(ρ) we obtain µ(ρ) = C d(ρ) e−(nβξ 2ρ2+ζB2)ρ2 , where C is an arbitrary constant and its value is fixed by requiring the coincidence of the distribution function when the external field is switched off (B = 0) with vacuum distribution function then µ(ρ) = CNcβ̃ 2NcΛbsρbs−5 e−(nβξ 2ρ2+ζB2)ρ2 . (23) With defining the average size as dρ ρ2 µ(ρ) we come to the practical interrelation between the IL density and average size of pseudo-particles (n β ξ2 ρ2 + ζ B2) ρ2 ≃ ν , (24) where ν = bs− 42 . Apparently, the size distribution of pseudo-particles can be presented by the well-known form as µ(ρ) = CNcβ̃ 2NcΛbsρbs−5 e ρ2 . (25) Figure 1: The energy E(α) when the profile function includes a screening effect (29) with the pa- rameter λ (s = 1) only taken into consideration (lower curve) and with both parameters used (upper curve) (see the text). Eqs. (22) and (25) allow us to get the estimate of generating functional (21) in the following form D[B] e−S(B) e−E , (26) d4x n − 1− ν ζ ρ2 B2 CNc β̃ − ν ln ρ Now taking into account Eq. (24) and fixing a field B, parameters s and λ the maximum of functional (26) over the IL parameters can be calculated by solving the corresponding transcendental equation = 0) numerically. Here it is a worthwhile place to notice the presence of new factor in the denominator of 2 what is caused by the Gaussian form of the corresponding integral over ρ squared and, hence, the integration element requires the introduction of 2ρ dρ. In Ref. [2] this factor was missed. However, this fact has not generated a serious consequence because any application of these results is actually related to the choice of suitable quantity of the parameter Λ entering the observables (the pion decay constant, for example). It means we should make the proper choice of basic scale. Besides, we should also keep in mind the approximate character of IL model. Further we give the results for both versions to demonstrate the dependence of final results on the renormalized constant CNc . Searching the optimal configuration f we take the effective action in the form of nonlinear func- tional as Seff = Gaµν(B) G µν(B) + E[B] , (27) in which the IL state is described by solutions ρ̄[B, s, λ], n[B, s, λ]. In practice the following differ- ential equation should be resolved = − 1 f(1− f)(1− 2f) , (28) at fixed initial magnitude of f(x0) putting up the derivative in the initial point f ′(x0) in such a way to have the solution going to zero when x is going to infinity. Parameter β0 is introduced to fix a priori Figure 2: The IL density as the function of x = y2/ρ̄2. Three dashed curves correspond to the different profile functions. The lowest dashed line corresponds to the standard form (4). The top dashed line corresponds to the profile function with the screening factor (29) and one parameter λ (s = 1) included and the middle line presents the same function but with two parameters included. The solid line presents the selfconsistent solution of variational problem. unknown value of coupling constant in the pseudo-particle definition (3). If the profile function has been fixed the configuration should be found in the form in which the starting values of parameters s, λ and β0 coincide (within the given precision) with the parameters obtained from the solution f . Nowadays this approach looks the most optimal one among other existing possibilities not only because of the computational arguments but in view of the poor current level of understanding the interrelation between perturbative and non-perturbative contributions while calculating the effective Lagrangian. In fact, it was mentioned in Ref. [2] that in more general (realistic) formulation of this problem Eq. (28) should include the term responsible for the change of ’quantum’ constant CNc with the function f changing. In principle, it could imply that the problem of pseudo-particle ensemble stabilization is connected at the fundamental dynamics level with the anticipated smallness of the contribution and, apparently, should be addressed not so much to the description of the interacting pseudo-particles and their interactions with the perturbative fields but rather to investigation of the time hierarchy corresponding to the breakdown of quasi-stationary behaviour of the vacuum fluctuations which will certainly lead to the changes of suitable effective Lagrangian (10). In order to receive the preliminary parameter estimates we consider the simplified model with the profile function containing only one additional parameter for describing the screening effect as regards f(y) = 1 + x , x = . (29) The energy E as the function of the screening parameter α is depicted in Fig. 1. The lowest dashed curve shows the behaviour when the changes related to weakening of repulsive interaction are taken into account by switching on the parameter λ only (at s = 1). The top dashed curve was obtained with both parameters switched on. The optimal value of the screening parameter α is determined by the minimum point of function E(α). Besides, this figure demonstrates the stability of variational procedure of extracting the IL parameters. For the first calculation the values of characteristic parameters for corresponding solution were taken as α = 0.06, λ = 0.775, s = 1.0067 with the following set of the IL parameters ρ̄Λ = 0.3305, n/Λ4 = 0.919, β = 17.186. These values give for the ratio of average pseudo-particle size and average distance between pseudo-particles the quite suitable quantity ρ̄/R = 0.324. For another calculation we have treated the parameter set characterizing the solution as α = 0.02, λ = 0.888, s = 1.0015 and for the IL parameters the following values ρ̄Λ = 0.315, n/Λ4 = 0.829, β = 17.67, ρ̄/R = 0.3. In order to get more orientation we would like to mention that for the ensemble of standard pseudo-particles (α = 0, λ = 1, s = 1) the corresponding values are ρ̄Λ = 0.301, n/Λ4 = 0.769, β = 18.103, ρ̄/R = 0.282. Figure 3: The average size of IL pseudo-particles as the function of x = y2/ρ̄2. Three dashed curves correspond to different profile functions. The lowest curve corresponds to the standard form (4). The top dashed curve corresponds to the profile function with the screening factor (29) which includes one parameter λ (s = 1) and the middle line shows the same function with two parameters included. The solid curve corresponds to the selfconsistent solution of the variational problem. Now we examine the impact of correction introduced in Eq. (26) when we changed the term which has been obtained in Ref. [2]. For the first calculation with the set of solution parameters as α = 0.24, λ = 0.546, s = 1.029 we have for the IL parameters ρ̄Λ = 0.331, n/Λ4 = 1.844, β = 17.173 which lead to the ratio discussed equal to ρ̄/R = 0.386. For another calculation we have the following results α = 0.05, λ = 0.799, s = 1.0053 and ρ̄Λ = 0.291, n/Λ4 = 1.356, β = 18.483, ρ̄/R = 0.314. And for the ensemble of standard pseudo-particles (α = 0, λ = 1, s = 1) these parameters are ρ̄Λ = 0.265, n/Λ4 = 1.186, β = 19.305, ρ̄/R = 0.277. The Fig. 2 and Fig. 3 show the behaviours of IL density and average pseudo-particle size as the functions of distance x. The dashed lines on both plots correspond to the similar ensembles. The lowest curves demonstrate the behaviours for the ensembles of standard pseudo-particles (4). The top curves present the ensemble of pseudo-particles with the profile function (29) at α = 0.06 and s = 1. And the middle dashed lines correspond to the profile functions with α = 0.02 and s ∼ 1.03. Obviously, it may be concluded that including even small change of the second parameter value (s ∼ 1.03) leads to the noticeable change of ensemble characteristics (for example, the IL density) because the highest contribution to the action when the coupling constant becomes the function of ρ is essentially modified. Let us make now several comments as to the ’complete’ formulation of the problem of analyzing the equation (28). It was numerically resolved by the Runge-Kutta method. This approach combined with numerical calculation of the derivative dE at every point of consequent integration interval allows us to avoid the problems which appear when searching the minimum of complicated functional in multidimensional space. The initial data were fixed at the point x0 = = 0.1. Since the IL density value at the coordinate origin is inessential the initial form of pseudo-particle profile function is taken without any deformations as f(x0) = 1 + x0 . Then at fixed values of the parameters λ, s and β0 the coefficient c is calculated. It allows to set the slope of trajectory f ′(x0) = −cf(1− f)/x0 at initial point in such a form in order to have the solution going to zero at large distances. Afterwards we find out the values of parameters λ and s requiring the input data to coincide with the output ones within the fixed precision. The parameter values which obey the imposed constraints are the following (input values) λ = 0.69099, s = 1.049, β0 = 16.26 at c = 1.361 and λ = 0.691, s = 1.049, β0 = 16.263 (at the output of variational procedure). The solid line in Fig. 4 shows the obtained profile f as the function of x = . The differences of profiles are smoothed over if they are presented as the functions of y because the large magnitude of the screening coefficient, for example α = 0.06, is compensated by enlargening the pseudo-particle size. The dashed lines on this plot show the profile functions for the standard form (4) (top dashed line), with the screening factor (29) including one parameter only α (s = 1) (lowest dashed curve) and two parameters included (middle dashed line). Figure 4: The various profile functions. The top dashed curve corresponds to the standard form (4), the lowest dashed curve shows the function with the screening factor (29) including one parameter λ (s = 1) and the middle line presents the same function with two parameters included. The solid line corresponds to the selconsistent solution of variational problem. Another calculation (with modified Γ-function contribution) was based on the slightly different set of relevant parameters which are for the input values λ = 0.607, s = 1.0515, β0 = 17.04 at c = 1.545 and λ = 0.6066, s = 1.0515, β0 = 17.042 for the output one at the finish of variational procedure. The behaviours of IL density and average pseudo-particle size for selfconsistent solution are plotted in Fig. 2 and Fig. 3 (solid lines, respectively)1. In the Table 1 we present the IL parameters at 1It is interesting to notice that considering IL (ensemble of pseudo-particles in the singular gauge) in the field of regular pseudo-particle we obtain the IL density value in the center of regular pseudo-particle which is larger than its value at large distances what looks like the anti-screening effect. the large distances from pseudo-particle (the first line) together with the data for the ensemble of pseudo-particles with the standard profile function (the second line). The third and fourth lines of this Table 1 are devoted to the calculations with the second set of parameters (with factor 2 absent in Eq. (26)). The fourth line, in particular, presents the calculations for pseudo-particles with standard form of profile function. Table 1. Parameters of IL. ρ̄Λ n/Λ4 β ρ̄/R nρ̄4 0.381 0.743 16.263 0.354 1.582·10−2 0.331 0.769 18.103 0.282 6.277·103 0.354 1.245 17.042 0.379 1.955·10−2 0.265 1.186 19.305 0.277 5.849·10−3 It is quite obvious that the utilization of optimal pseudo-particle profile function leads to the larger pseudo-particle size but the packing fraction parameter holds, nevertheless, a small quantity which is quite suitable for the perturbative expansion. Besides, the results obtained allow us to conclude that with tuning Λ a fully satisfactory agreement our calculations of pseudo-particle size, the ensemble diluteness and gluon condensate value with their phenomenological magnitudes extracted from the other models are easily reachable. The calculations of several dimensional quantities in our approach are also very indicative. The values of the screening mass (11), average pseudo-particle size and IL density obtained for two values of Λ (200 MeV and 280 MeV) are shown in Table 2. The sequence of line meanings is identical to that in Table 1 as well as the meanings of last four lines which present the results of calculations with the second set of parameters (with factor 2 absent in Eq. (26)). Table 2. Screening mass and IL parameters Λ MeV m MeV ρ̄ GeV−1 n fm−4 200. 381 1.906 0.7496 304 1.503 0.7688 280. 533 1.361 2.88 426 1.074 2.95 200. 456 1.77 1.245 333 1.325 1.186 280. 638 1.264 4.78 466 0.946 4.56 Another interesting feature of this calculation is the weakening of pseudo-particle interaction. This effect is driven by the coefficient ξ2 (∼ λ2). Our estimates for the first set of parameters give λ = 0.691 and, hence, λ2 ∼ 0.48 and for the second set we have (λ = 0.607) and λ2 ∼ 0.37. Let us mention here that the reasonable description of instanton ensemble can be reached in the framework of two-component models [6] as well. Our calculations enable us to conclude that dealing with IL model (formulated in one-loop ap- proach) one is able to reach quite reasonable description of gluon condensate even being constrained by the values of average pseudo-particle size and other routine phenomenological parameters. More- over, the ensemble of pseudo-particles with standard profile functions turns out to be very practical because introducing the other configurations to make the similar estimates is simply unoperable. With such an approximation of the vacuum configurations the coefficient of interaction weakening develops the magnitude about λ2 ∼ 0.3 — 0.5. Including this effect leads to the enlargening of pseudo-particle size. It allows us to conclude that nowadays the instantons in the singular gauge is the only serious instrument for effective practising. The authors are sincerely grateful to A.E. Dorokhov and S.B. Gerasimov for interesting discussions and practical remarks. The financial support of the Grants INTAS-04-84-398 and NATO PDD(CP)- NUKR980668 is also acknowledged. References [1] C.G. Callan, R. Dashen, and D.J. Gross, Phys. Lett. B 66 (1977) 375; C.G. Callan, R. Dashen, and D.J. Gross, Phys. Rev. D 17, (1978) 2717. [2] D.I. Diakonov and V.Yu. Petrov, Nucl. Phys. B 245, (1984) 259. [3] I.I. Balitsky and A.V. Yung, Phys. Lett. B 168, (1986) 113; D. Förster, Phys. Lett. B 66, (1977) 279; E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. B 364, (1991) 255; T. Schäfer and E.V. Shuryak, Rev. Mod. Phys. 70, (1998) 323. [4] S.V. Molodtsov, G.M. Zinovjev, Yad. Fiz. 70, N0 6, (2007). [5] N.G. Van Kampen, Phys. Rep. 24 (1976) 171; Physica 74 (1974) 215, 239; Yu.A. Simonov, Phys. Lett. B 412 (1997) 371. [6] A.E. Dorokhov, S.V. Esaibegyan, A.E. Maximov, and S.V. Mikhailov, Eur. Phys.J C 13 (2000) 331; N.O. Agasian and S.M. Fedorov, JHEP 12 (2001) 019.

0704.0142

Some aspects of the nonperturbative renormalization of the phi^4 model

0 Some aspects of the nonperturbative renormalization of the ϕ4 model J. Kaupužs ∗ Institute of Mathematics and Computer Science, University of Latvia Raiņa bulvāris 29, LV–1459 Riga, Latvia November 4, 2018 Abstract A nonperturbative renormalization of the ϕ4 model is considered. First we inte- grate out only a single pair of conjugated modes with wave vectors ±q. Then we are looking for the RG equation which would describe the transformation of the Hamil- tonian under the integration over a shell Λ − dΛ < k < Λ, where dΛ → 0. We show that the known Wegner–Houghton equation is consistent with the assumption of a simple superposition of the integration results for ±q. The renormalized action can be expanded in powers of the ϕ4 coupling constant u in the high temperature phase at u → 0. We compare the expansion coefficients with those exactly calculated by the diagrammatic perturbative method, and find some inconsistency. It causes a question in which sense the Wegner–Houghton equation is really exact. 1 Introduction The renormalization group (RG) approach, perhaps, is the most extensively used one in numerous studies of critical phenomena [1, 2]. Particularly, the perturbative RG approach to the ϕ4 or Ginzburg–Landau model is widely known [3, 4, 5, 6]. However, the pertur- bative approach suffers from some problems [7]. Therefore it is interesting to look for a nonperturbative approach. Historically, nonperturbative RG equations have been devel- oped in parallel to the perturbative ones. These are so called exact RG equations (ERGE). The method of deriving such RG equations is close in spirit to the famous Wilson’s ap- proach, where the basic idea is to integrate out the short–wave fluctuations corresponding to the wave vectors within Λ/s < q < Λ with the upper (or ultraviolet) cutoff parameter Λ and the renormalization scale s > 1. The oldest nonperturbative equation of this kind, originally presented by Wegner and Houghton [8], uses the sharp momentum cutoff. Later, a similar equation with smooth momentum cutoff has been proposed by Polchinski [9]. The RG equations of this class are reviewed in [10]. According to the known classification [10, 11], there is another class of nonperturbative RG equations proposed by [12] and reviewed in [11]. Some relevant discussion can be found in [10], as well. Such equations describe the variation of an average effective action Γk[φ] depending on the running cutoff scale k. Here φ(x) = 〈ϕ(x)〉 is the averaged order– parameter field (for simplicity, we refer to the case of scalar field). According to [12], the averaging is performed over volume ∼ k−d such that the fluctuation degrees of freedom E–mail: [email protected] http://arxiv.org/abs/0704.0142v2 with momenta q > k are effectively integrated out. In fact, the averaging over volume ∼ k−d is the usual block–spin–averaging procedure of the real–space renormalization. At the same time, the fluctuations with q . k are suppressed by a smooth infrared cutoff. As one can judge from [11], the existence of a deterministic relation between the configuration of external source {J(x)} and that of the averaged order parameter {φ(x)} is (implicitly) assumed in the nonperturbative derivation of the RG flow equation. Namely, it is stated (see the text between (2.28) and (2.29) in [11]) that δJ(x)/δφ(y) is the inverse of δφ(x)/δJ(y), which has certain meaning as a matrix identity. To make this point clearer, let us consider a toy example ~J = A~φ, where ~J = (J(x1), J(x2), . . . , J(xN)) and ~φ = (φ(x1), φ(x2), . . . , φ(xN)) are N–component vectors and A is a matrix of size N ×N . In this case ∂J(xi)/∂φ(xj) is the element Aij of matrix A, whereas ∂φ(xi)/∂J(xj) is the element of the inverse matrix A−1. In the continuum limit N → ∞, this toy example corresponds to a linear dependence between {J(x)} and {φ(x)}. The calculation of derivative always implies the linearisation around some point, so that the matrix identity used in [11] (as a continuum limit in the above example) has a general meaning. However, it makes sense only if there exists a deterministic relation between the configurations of φ(x) and J(x) or, in a mathematical notation, if there exist mappings f : {J(x)} → {φ(x)} and f−1 : {φ(x)} → {J(x)}. On the other hand, according to the block–averaging, the values of φ(x) should be understood as the block–averages. These, of course, are not uniquely determined by the external sources, but are fluctuating quantities. So, we are quite sceptical about the exactness of such an approach of averaged effective action. The integration over fluctuation degrees with momenta q > k does not alter the be- havior of the infrared modes, directly related to the critical exponents. From this point of view, the approach based on the equations of Wegner–Houghton and Polchinski type seems to be more natural. These are widely believed to be the exact RG equations, although, in view of our currently presented results, it turns out to be questionable in which sense they are really exact. In any case, the nonperturbative RG equations cannot be solved exactly, therefore a suitable truncation is used. The convergence of several truncation schemes and of the derivative expansion has been widely studied in [13, 14, 15, 16, 17]. Here [17] refers to the specific approach of [12]. A review about all the methods of approximate solution can be found in [18]. Another problem is to test and verify the nonperturbative RG equations, comparing the results with the known exact and rigorous solutions, as well as with the results of the perturbation theory. In [15], the derivative expansion of the RG β–function has been considered, showing the agreement up to the second order between the perturbative results and those obtained from the Legendre flow equation, which also belongs to the same class of RG equations as the Wegner–Houghton and Polchinski equations. It has been stated in [13] that the critical exponent ν, extracted from the Wegner–Houghton equation in the local potential approximation, agrees with the ε–expansion up to the O(ε) order, as well as with the 1/n (1/N in the notations of [13]) expansion in the leading order. However, looking carefully on the results of [13], one should make clear that “the leading order of the 1/n expansion” in this case is no more than the zeroth order, whereas the expansion coefficient at 1/n is inconsistent with that proposed by the perturbative RG calculation at any fixed dimension d except only d = 4. The inconsistency could be understood from the point of view that the Wegner–Houghton equation has been solved approximately. Therefore it would be interesting to verify whether the problem is eliminated beyond the local potential approximation. One should also take into account that the perturbative RG theory is not rigorous and, therefore, we think that a possible inconsistency still would not prove that something is really wrong with the nonperturbative RG equation. In any case, it is a remarkable fact that correct RG eigenvalue spectrum and critical exponents are obtained in the local potential approximation at n → ∞ from the Wegner–Houghton equation [13], as well as from similar RG equations [19], in agreement with the known exact and rigorous results for the spherical model. It shows that some solutions, being not exact, nevertheless can lead to exact critical exponents. From this point of view, it seems also possible that some kind of approximations, made in the derivation of an RG equation, are not harmful for the critical exponents. We propose a simple test of the Wegner–Houghton equation: to verify the expansion of the renormalized action S of the ϕ4 model in powers of the coupling constant u at u → 0 in the high–temperature phase. Such a test is rigorous, in the sense that the natural domain of validity of the perturbation theory is considered. We think that it would be quite natural to start with such a relatively simple and straightforward test before passing to more complicated ones, considered in [13, 15, 19]. We have made this simplest test in our paper and have found that the Wegner–Houghton equation fails to give all correct expansion coefficients. We have also proposed another derivation of the Wegner–Houghton equation (Secs. 2, 3). It is helpful to clarify the origin of the mentioned inconsistency. It is also less obscure from the point of view that the used assumptions and approximations are clearly stated. As regards the derivation in [8], at least one essential step is obscure and apparently contains an implicit approximation which, in very essence, is analogous to that pointed out in our derivation. We will discuss this point in Sec. 3. 2 An elementary step of renormalization To derive a nonperturbative RG equation for the ϕ4 model, we should start with some elementary steps, as explained in this section. Consider the action S[ϕ] which depends on the configuration of the order parameter field ϕ(x) depending on coordinate x. By definition, it is related to the Hamiltonian H of the model via S = H/T , where T is the temperature measured in energy units. In general, ϕ(x) is an n–component vector with components ϕj(x) given in the Fourier representation as ϕj(x) = V k<Λ ϕj,ke ikx, where V = Ld is the volume of the system, d is the spatial dimensionality, and Λ is the upper cutoff of the wave vectors. We consider the action of the Ginzburg–Landau form. For simplicity, we include only the ϕ2 and ϕ4 terms. The action of such ϕ4 model is given by S[ϕ] = Θ(k)ϕj,kϕj,−k + uV j,l,k1,k2,k3 ϕj,k1ϕj,k2ϕl,k3ϕl,−k1−k2−k3 , (1) where Θ(k) is some function of wave vector k, e. g., Θ(k) = r0 + ck 2 like in theories of critical phenomena [4, 5, 6, 7]. In the sums we set ϕl,k = 0 for k > Λ. The renormalization group (RG) transformation implies the integration over ϕj,k for some set of wave vectors with Λ′ < k < Λ, i. e., the Kadanoff’s transformation, followed by certain rescaling procedure [4]. The action under the Kadanoff’s transformation is changed from S[ϕ] to Stra[ϕ] according to the equation e−Stra[ϕ] = e−S[ϕ] j,Λ′<k<Λ dϕj,k . (2) Alternatively, one often writes −Stra[ϕ]+AL d instead of −Stra[ϕ] to separate the constant part of the action ALd. This, however, is merely a redefinition of Stra, and for our purposes it is suitable to use (2). Note that ϕj,k = ϕ j,k+iϕ j,k is a complex number and ϕj,−k = ϕ holds (since ϕj(x) is always real), so that the integration over ϕj,k means in fact the integration over real and imaginary parts of ϕj,k for each pair of conjugated wave vectors k and −k. The Kadanoff’s transformation (2) can be split in a sequence of elementary steps S[ϕ] → Stra[ϕ] of the repeated integration given by e−Stra[ϕ] = e−S[ϕ]dϕ′j,qdϕ j,q (3) for each j and q ∈ Ω, where Ω is the subset of independent wave vectors (±q represent one independent mode) within Λ′ < q < Λ. Thus, in the first elementary step of renor- malization we have to insert the original action (1) into (3) and perform the integration for one chosen j and q ∈ Ω. In an exact treatment we must take into account that the action is already changed in the following elementary steps. For Λ′ > Λ/3, we can use the following exact decomposition of (1) S[ϕ] = A0 +A1ϕj,q +A 1ϕj,−q +A2ϕj,qϕj,−q +B2ϕ j,q +B j,−q +A4ϕ j,−q , (4) where A0 = S|ϕj,±q=0 , (5) ∂ϕj,q ϕj,±q=0 = 4uV −1 l,k1,k2 ϕj,k1ϕl,k2ϕl,−q−k1−k2 , (6) ∂ϕj,q∂ϕj,−q ϕj,±q=0 = Θ(q) + Θ(−q) + 4uV −1 (1 + 2δlj) | ϕl,k | 2 , (7) ∂ϕ2j,q ∣∣∣∣∣ ϕj,±q=0 = 2uV −1 (1 + 2δlj)ϕl,kϕl,−2q−k , (8) ∂2ϕj,q∂2ϕj,−q ϕj,±q=0 = 6uV −1 . (9) Here the sums are marked by a prime to indicate that terms containing ϕj,±q are omitted. This is simply a splitting of (1) into parts with all possible powers of ϕj,±q. The condition Λ′ > Λ/3, as well as the existence of the upper cutoff for the wave vectors, ensures that terms of the third power are absent in (4). Besides, the derivation is performed formally considering all ϕl,k as independent variables. Taking into account (4), as well as the fact that A1 = A 1+ iA 1 and B2 = B 2+ iB 2 are complex numbers, the transformed action after the first elementary renormalization step reads Stra[ϕ] = A0 − ln j,q −A ϕ′j,q + ϕ′′j,q × exp −2B′2 ϕ′j,q − ϕ′′j,q + 4B′′2ϕ × exp ϕ′j,q + ϕ′′j,q dϕ′j,qdϕ . (10) Considering only the field configurations which are relevant in the thermodynamic limit V → ∞, Eq. (10) can be simplified, omitting the terms with B2 and A4. Really, using the coordinate representation ϕl,k = V ϕl(x) e −ikxdx, we can write B2 = 2uV (1 + 2δlj) ϕ2l (x)e iqx dx − 3ϕ2j,−q  . (11) The quantity V −1 ϕ2l (x)e iqx dx is an average of ϕ2l (x) over the volume with oscillating weight factor eiqx. This quantity vanishes for relevant configurations in the thermody- namic limit: due to the oscillations, positive and negative contributions are similar in magnitude and cancel at V → ∞. Since 〈| ϕj,−q | 2〉 = 〈| ϕj,q | 2〉 is the Fourier transform of the two–point correlation function, it is bounded at V → ∞ and, hence, ϕ2j,−q also is bounded for relevant configurations giving nonvanishing contribution to the statistical averages 〈·〉 in the thermodynamic limit. Consequently, for these configurations, A2 is a quantity of order O(1), whereas V −1ϕ2j,−q and B2 vanish at V → ∞. Note, however, that the term with A4 = O in (10) cannot be neglected unless A2 is positive. One can judge that the latter condition is satisfied for the relevant field configurations due to existence of the thermodynamic limit for the RG flow. Omitting the terms with B2 and A4, the integrals in (10) can be easily calculated. It yields Stra[ϕ] = S ′[ϕ] + ∆Seltra[ϕ] , (12) where S′[ϕ] = A0 is the original action, where only the ±q modes of the j-th field com- ponent are omitted, whereas ∆Seltra[ϕ] represents the elementary variation of the action given by ∆Seltra[ϕ] = ln | A1 | . (13) According to the arguments provided above, this equation is exact for the relevant field configurations with A2 > 0 in the thermodynamic limit. The contributions to (6) and (7) provided by modes with wave vectors k, obeying two relations Λ − dΛ < k < Λ and k 6= ±q, are irrelevant in the thermodynamic limit at dΛ → 0. It can be verified by the method of analysis introduced in Sec. 2. Hence, Eq. (13) can be written as ∆Seltra[ϕ] = ln | Ã1 | + δSeltra[ϕ] , (14) where Ã1 = P ∂ϕj,q = 4uV −1 Λ−dΛ∑ l,k1,k2 ϕj,k1ϕl,k2ϕl,−q−k1−k2 , (15) Ã2 = P ∂ϕj,q∂ϕj,−q = Θ(q) + Θ(−q) + 4uV −1 Λ−dΛ∑ (1 + 2δlj) | ϕl,k | 2 , (16) and δSeltra[ϕ] is a vanishingly small correction in the considered limit. Here the operators P set to zero all ϕj,k within the shell Λ− dΛ < k < Λ (i. e., the derivatives are evaluated at zero ϕj,k for k within the shell), and the upper border Λ− dΛ for sums implies that we set ϕl,k = 0 for k > Λ− dΛ. The above replacements are meaningful, since they allow to obtain easily the Wegner–Houghton equation, as discussed in the following section. 3 Superposition hypothesis and the Wegner–Houghton equation Intuitively, it could seem very reasonable that the result of integration over Fourier modes within the shell Λ − dΛ < k < Λ at dΛ → 0 can be represented as a superposition of elementary contributions given by (14), neglecting the irrelevant corrections δSeltra[ϕ]. We will call this idea the superposition hypothesis. We remind, however, that strictly exact treatment requires a sequential integration of exp(−S[ϕ]) over a set of ϕj,q. The renormalized action changes after each such integration, and these changes influence the following steps. A problem is to estimate the discrepancy between the results of two methods: (1) the exact integration and (2) the superposition approximation. Since it is necessary to perform infinitely many integration steps in the thermodynamic limit, the problem is nontrivial and the superposition hypothesis cannot be rigorously justified. Nevertheless, the summation of elementary contributions in accordance with the su- perposition hypothesis leads to the known Wegner–Houghton equation [8]. In this case the variation of the action due to the integration over shell reads ∆Stra[ϕ] = Λ−dΛ<q<Λ Ã2(j,q) | Ã1(j,q) | Ã2(j,q) . (17) It is exactly consistent with Eq. (2.13) in [8]. The factor 1/2 appears, since only half of the wave vectors represent independent modes. Here we have indicated that the quantities Ã1 and Ã2 depend on the current j and q. They depend also on the considered field con- figuration [ϕ]. If Ã1 and Ã2 are represented by the derivatives of S[ϕ] (see (15) and (16)), then the equation is written exactly as in [8]. To avoid possible confusion, one has to make clear that the operators P in (15) and (16) influence the result, as discussed further on. It means that the equation where these operators are simply omitted, referred in the review paper [10] as the Wegner–Houghton equation, is not really the Wegner–Houghton equation. The derivation in [8] is somewhat different. Instead of performing only one elementary step of integration first, the expansion of Hamiltonian in terms of all shell variables is made there. The basic method of [8] is to show that, in the thermodynamic limit at dΛ → 0, the expansion consists of terms containing no more than two derivatives with respect to the field components. Moreover, it is assumed implicitly that only the diagonal terms with k′ = −k are important finally, when performing the summation over the wave vectors k,k′. It leads to Eq. (2.12) in [8]. The omitting of nondiagonal terms is equivalent to the superposition assumption we discussed already. Indeed, in this and only in this case the integration over the shell variables can be performed independently, as if the superposition principle were hold. Hence, essentially the same approximation is used in [8] as in our derivation, although it is not stated explicitly. Our derivation refers to the ϕ4 model, whereas in the form with derivatives the equation may have a more general validity, as supposed in [8]. Indeed, (14) remains correct for a generalized model provided that higher than second order derivatives of S[ϕ] vanish for relevant field configurations in the thermodynamic limit. It, in fact, has been assumed and shown in [8]. Based on similar arguments we have used already, the latter assumption can be justified for certain class of models, for which the action is represented by a linear combination of ϕm–kind terms with wave–vector dependent weights and vanishing sum of the wave vectors l=1 kl = 0 related to the ϕ factors. In this case we have Ã1(j,q) = ∂ϕj,q − ϕj,−q ∂ϕj,qϕj,−q , (18) Ã2(j,q) = ∂ϕj,qϕj,−q for the relevant configurations at V → ∞ and dΛ → 0. The second term in (18) appears because the derivative ∂S/∂ϕj,q contains relevant terms with ϕj,−q, which have to be removed. The influence of the operators P is seen from (15) and (18). Here we do not include the second, i. e., the rescaling step of the RG transformation. It, however, can be easily calculated for any given action, as described, e. g., in [4]. It is not relevant four our further considerations. 4 The weak coupling limit Here we consider the weak coupling limit u → 0 of the model with Θ(k) = r0 + ck a given positive r0, i. e., in the high temperature phase. In this case ∆Stra[ϕ] can be expanded in powers of u. It is the natural domain of validity of the perturbation theory, and the expansion coefficients can be calculated exactly by the known methods applying the Feynman diagram technique and the Wick’s theorem [4, 5, 11]. On the other hand, the expansion can be performed in (17). Our aim is to compare the results of both methods to check the correctness of (17), since the latter equation is based on assumptions. Let us denote by ∆S̃tra[ϕ] the variation of S[ϕ] omitting the constant (independent of the field configuration) part. Then the expansion in powers of u reads ∆S̃tra[ϕ] = ∆S1[ϕ]u+ 2 [ϕ] + ∆S 2 [ϕ] + ∆S 2 [ϕ] u2 +O , (20) where the expansion coefficient at u2 is split in three parts ∆S 2 [ϕ], ∆S 2 [ϕ], and ∆S 2 [ϕ] corresponding to the ϕ2, ϕ4, and ϕ6 contributions, respectively. The contribution of order u is related to the diagram r✐ , whereas the three second–order contributions — to the diagrams q q❦ , ❛✦q q✦❛ , and ❛✦q q✦❛ . The diagram technique represents the expansion of −S[ϕ] in terms of connected Feynman diagrams, where the coupled lines are associated with the Gaussian averages. In particular, the Fourier transformed two–point correlation function in the Gaussian approximation G0(k) = 〈ϕj,kϕj,−k〉0 = 1/[2Θ(k)] appears due to the integration over ϕ′j,k and ϕ j,k. It is represented as the coupling of lines, in such a way that each line related to the wave vector k and vector–component j is coupled with another line having the wave vector −k and the same component j. Thus, if we integrate over ϕj,k within Λ−dΛ < k < Λ in (2), then it corresponds to the coupling of lines in the same range of wave vectors in the diagram technique. According to the Wick’s theorem, one has to sum over all possible couplings, which finally yields the summation (integration) over the wave vectors obeying the constraint Λ− dΛ < k < Λ for each of the coupled lines associated with the factors G0(k). In the n–component case, it is suitable to represent the ϕ4 vertex as ❛✦q q✦❛ , where the same index j is associated with two solid lines connected to one node. The above diagrams are given by the sum of all possible couplings of the vertices ❛✦q q✦❛ , yielding the corresponding topological pictures when the dashed lines shrink to points. In this case factor n corresponds to each closed loop of solid lines, which comes from the summation over j. For a complete definition of the diagram technique, one has to mention that factors −uV −1 are related to the dashed lines, G0(k) – to the coupled solid lines, and the fields ϕj,k – to the outer uncoupled solid lines. Besides, each diagram contains a combinatorial factor. For a diagram consisting of m vertices ❛✦q q✦❛ , it is the number of all possible couplings of (numbered) lines, divided by m!. At dΛ → 0, the diagrammatic calculation for the n–component case yields ∆S1[ϕ] = (n + 2) dΛ Λ−dΛ∑ | ϕj,k | 2 (21) 2 [ϕ] = −4V Λ−dΛ∑ j,l,k1,k2,k3 ϕj,k1ϕj,k2ϕl,k3ϕl,−k1−k2−k3 (22) × [(n+ 4)Q (k1 + k2,Λ, dΛ) + 4Q (k1 + k3,Λ, dΛ)] 2 [ϕ] = −8V Λ−dΛ∑ i,j,l,k1,k2,k3,k4,k5 ϕi,k1ϕi,k2ϕj,k3ϕj,k4ϕl,k5ϕl,−k1−k2−k3−k4−k5 ×G0 (k1 + k2 + k3) F (| k1 + k2 + k3 |,Λ, dΛ) , (23) where Kd = S(d)/(2π) d, S(d) = 2πd/2/Γ(d/2) is the area of unit sphere in d dimensions, Θ(Λ) is the value of Θ(k) at k = Λ, whereas F(k,Λ, dΛ) is a cutoff function which has the value 1 within Λ− dΛ < k < Λ and zero otherwise. The quantity Q is given by Q(k,Λ, dΛ) = V −1 Λ−dΛ<q<Λ G0(q)G0(k− q)F(| k− q |,Λ, dΛ) . (24) Below we will give some details of calculation of (22), which is the most important term in our further discussion. To obtain this result, we have dechipered the ❛✦q q✦❛ diagram as a sum of three diagrams of different topologies made of vertices ❛✦q q✦❛ , i. e., q q q q✐ ✦ ✍ , and q , providing the same topological picture ❛ when shrinking the dashed lines to points. Recall that any loop made of solid lines of ❛✦q q✦❛ gives a factor n, and one needs also to compute the combinatorial factors. For the above three diagrams, the resulting factors are 4n, 16, and 16, which enter the prefactors of Q in (22). To obtain the correct sign, we recall that the diagram expansion is for −S[ϕ]. The other diagrams are calculated in a similar way. The expansion of (17) gives no contribution ∆S 2 [ϕ], and we have skipped it in the diagrammatic calculation as an irrelevant term, which vanishes faster than ∝ dΛ at dΛ → 0 in the thermodynamic limit V → ∞. The expansion of the logarithm term in (17) yields ∆S1[ϕ] exactly consistent with (21). Similarly, ∆S 2 [ϕ] is exactly consistent with (23). One has to remark that two propogators are involved in (24) and, therefore, the volume of summation region with nonvanishing cut function F shrinks as (dΛ)2 for a given nonzero wave vector k at dΛ → 0. However, there is a contribution linear in dΛ for k = 0. As a result, a contribution proportional to dΛ appears in (22). Note that the contributions (21) and (23) come from diagrams with only one coupled line. The term (22) is related to the diagram with two coupled lines. The expansion of (17) provides a different result for the corresponding part of ∆S̃tra[ϕ]: 2 [ϕ] = − d−1 dΛ Θ2(Λ) Λ−dΛ∑ j,l,k1,k2 (n+ 4 + 4δjl) | ϕj,k1 | 2 | ϕl,k2 | 2 . (25) Note that (25) comes from the ln Ã2 term in (17), and the calculation is particularly simple in this case, since the related sum in (16) is independent of q. Eq. (25) is obtained if we set Q(k,Λ, dΛ) → δk,0Q(0,Λ, dΛ) in (22) (in this case only the diagonal terms j = l are relevant when summing up the contributions with Q (k1 + k3,Λ, dΛ), as it can be shown by an analysis of relevant real–space configurations, since 〈ϕj(x)ϕl(x)〉 = 0 holds for j 6= l). It means that a subset of terms is missing in (25), as compared to (22). The following analysis will show that this discrepancy between (22) and (25) is important. It is interesting to mention that (25) is obtained also by the diagrammatic perturbation method if we first integrate out only the mode with ϕj,±q and then formally apply the superposition hypothesis, as in the derivation of the Wegner–Houghton equation. It shows that the discrepancy between (25) and (22) arises because in one case the superposition hypothesis is applied, whereas in the other case it is not used. The difference between (22) and (25) can be better seen in the coordinate representa- tion. In this case (22) reads 2 [ϕ] = − (4n+ 16) ϕ2(x1)R 2(x1 − x2)ϕ 2(x2) dx1dx2 (26) ϕj(x1)ϕl(x1)R 2(x1 − x2)ϕj(x2)ϕl(x2) dx1dx2 , where R(x) = V −1 G0(q)F(q,Λ, dΛ)e iqx (27) is the Fourier transform of G0F , and ϕ 2(x) = l (x). In three dimensions we have R(x) = (2π)2Θ(Λ) sin(Λx) (28) for any given x at dΛ → 0 and L → ∞, where L is the linear system size. The continuum approximation (28), however, is not correct for x ∼ L and therefore, probably, should not be used for the evaluation of (26). The coordinate representation of (25) is 2 [ϕ] = − d−1 dΛ Θ2(Λ) (n+ 4) ϕ2(x1)V −1 ϕ2(x2) dx1dx2 ϕ2j (x1)V −1 ϕ2j (x2) dx1dx2  . (29) Eq. (29) represents a relevant contribution at dΛ → 0, as it is proportional to dΛ. It is ob- viously not consistent with (26). In fact, the term (29) represents a mean-field interaction, which is proportional to 1/V and independent of the distance, whereas (26) corresponds to another non-local interaction given by R2(x1 − x2). Hence, the Wegner–Houghton equation (17) does not yield all correct expansion coefficients at u → 0. 5 Discussion The results of our test, stated at the end of Sec. 4, reveal some inconsistency between the Wegner–Houghton equation and the diagrammatic perturbation theory in the high temperature phase at u → 0. Since this is the natural domain of validity of the per- turbation theory, there should be no doubts that it produces correct results here, which agree with (2). So, the results of our test point to some inconsistency between the Wegner– Houghton equation and (2), which causes a question in which sense the Wegner–Houghton equation is really exact. The same can be asked about the equations of Polchinski type, since these (as it is believed) are generalizations of the Wegner–Houghton equation to the case of smooth momentum cutoff. There is no contradiction with the tests of consis- tency made in [13, 15], since our test is independent and quite different. According to our derivation of the Wegner–Houghton equation and the related discussion, it turns out that the reason of the inconsistency, likely, is the superposition approximation (defined at the beginning of Sec. 3) used in our paper and analogous approximation implicitly used in [8]. Despite of this problem, the Wegner–Houghton equation is able to reproduce the exact RG eigenvalue spectrum and critical exponents of the spherical model at n → ∞ [13]. This fact can be interpreted in such a way that the superposition approximation (or an analogous approximation) is valid to derive such nonperturbative RG equations, which can produce correct (exact) critical exponents in some limit cases, at least. From a general point of view, it concerns the fundamental question about the relation between the form of RG equation and the universal quantities. It has been verified in several known studies that the universal quantities are invariant with respect to some kind of variations in the RG equation, like changes in the shape of the momentum cutoff function. This property is known as the reparametrisation invariance [10]. Probably, the universal quantities are invariant also with respect to such a variation of the Wegner–Houghton equation, which makes it exactly consistent with (2). However, this is only a hypothesis. 6 Conclusions 1. The nonperturbative Wegner–Houghton RG equation has been rederived (Secs. 2 and 3), discussing explicitly some assumptions which are used here. In particular, our derivation assumes the superposition of small contributions provided by elemen- tary integration steps over the short–wave fluctuation modes. We consider it as an approximation. As discussed in Sec. 3, the original derivation by Wegner and Houghton includes essentially the same approximation, although not stated explic- itly. 2. According to our calculation in Sec. 4, the Wegner–Houghton equation is not com- pletely consistent with the diagrammatic perturbation theory in the limit of small ϕ4 coupling constant u in the high temperature phase. This fact, together with some other important results known from literature, is discussed in Sec. 5. Apart from critical remarks, a hypothesis has been proposed that the equations of Wegner– Houghton type, perhaps, can give exact universal quantities. References [1] D. J. Amit, Field theory, the renormalization group, and critical phenomena, World Scientific, Singapore, 1984 [2] D. Sornette, Critical Phenomena in Natural Sciences, Springer, Berlin, 2000 [3] K. G. Wilson, M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972) [4] Shang–Keng Ma, Modern Theory of Critical Phenomena, W.A. Benjamin, Inc., New York, 1976 [5] J. Zinn–Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1996 [6] H. Kleinert, V. Schulte–Frohlinde, Critical properties of φ4 theories, World Scientific, [7] J. Kaupužs, Ann. Phys. (Leipzig) 10, 299 (2001) [8] F. Wegner, A. Houghton, Phys. Rev. A 8, 401 (1973) [9] J. Polchinski, Nucl. Phys. B 231, 269 (1984) [10] C. Bagnuls, C. Bervillier, Phys. Rep. 348, 91 (2001) [11] J. Berges, N. Tetradis, C. Wetterich, Phys. Rep. 363, 223 (2002) [12] C. Wetterich, Phys. Lett. B 301, 90 (1993) [13] K.-I. Aoki, K. Morikava, W. Souma, J.-I. Sumi, H. Terao, Progress of Theoretical Physics 95, 409 (1996) [14] K.-I. Aoki, K. Morikava, W. Souma, J.-I. Sumi, H. Terao, Progress of Theoretical Physics 99, 451 (1998) [15] T. R. Morris, J. F. Tighe, Journal of High Energy Physics 9908, 007 (1999) [16] T. R. Morris, J. F. Tighe, Int. J. Mod. Phys. A 16, 2095 (2001) [17] L. Canet, B. Delamotte, D. Mouhanna, J. Vidal, Phys. Rev. B 68, 064421 (2003) [18] B. Delamotte, D. Mouhanna, M. Tissier, Phys. Rev. B 69, 134413 (2004) [19] M. D’Attanasio, T. R. Morris, Physics Letters B 409, 363 (1997) 1 Introduction 2 An elementary step of renormalization 3 Superposition hypothesis and the Wegner–Houghton equation 4 The weak coupling limit 5 Discussion 6 Conclusions

0704.0143

Instanton Liquid at Finite Temperature and Chemical Potential of Quarks

Instanton Liquid at Finite Temperature and Chemical Potential of Quarks S.V. Molodtsov1,3, G.M. Zinovjev2 1Joint Institute for Nuclear Research, Dubna, 141980 RUSSIA 2Bogolyubov Institute for Theoretical Physics, ul. Metrolohichna 14-b, Kiev, 03680 UKRAINE 3Institute of Theoretical and Experimental Physics, Moscow, 117259 RUSSIA Instanton liquid in heated and strongly interacting matter is studied using the variational princi- ple. The dependence of the instanton liquid density (gluon condensate) on the temperature and the quark chemical potential is determined under the assumption that, at finite temperatures, the dominant contribution is given by an ensemble of calorons. The respective one-loop effective quark Lagrangian is used. In current studies of strong-interacting matter under extreme conditions, primary attention is focused on a description of its phase state at given temperature and chemical potential. For def- initeness, we consider that T is the temperature of quarks and µ is the quark chemical potential (it is assumed that gluons are in thermodynamical equilibrium with quarks). However, there is no approach making it possible to describe main features of the expected phase diagram of quark-gluon matter at least qualitatively. In the present study, we argue that the instanton liquid model of the QCD vacuum [1] can shed light on some important features of a full picture. It is frequently noted that this model offers a useful tool for obtaining phenomenologically plausible estimates in spite of the fact that it is poorly justified because the typical size of an instanton is not properly fixed. As of now, this fact is considered as inessential because a connection has been revealed between limitations on the instanton size due to repulsion [2] and generation of mass of the gluon field in the framework of the quasi-classical approximation [3]. The latter mechanism is a more general property of stochastic gluon fields than the former one. We will discuss this question later. Here we assume that the problem of instanton size is solved in one of the following scenarios: self-stabilization of the saturating ensemble [2],[4], freezing of the coupling constant [5], or influence of the confining component [6]. In the present study, primary attention is focused on a plausible qualitative model describing a behavior of the gluon condensate. In the beginning, we recollect the variational principle proposed in [2] and the method of determi- nation of the size of pseudoparticles and the density of the instanton liquid and introduce notation for further considerations. In the model of instanton liquid describing the QCD vacuum, it is assumed that the leading contribution to the QCD generating functional is given by the background fields representing superposition of instantons in the singular gauge: Aaµ(x; γ) = ωabη̄bµν aν(y) , aν(y) = y2 + ρ2 , y = x− z , µ, ν = 1, 2, 3, 4 . (1) where ρ is the size, ω is the matrix of color rotation, and z is the position of the center of a pseudoparticle (in the case of anti-instanton, the ’t Hooft symbol should be replaced as follows: η̄ → η). This being so, the QCD generating functional takes the form dγi d(ρi) e −β Uint(γ) = dγi e −E(γ) , (2) http://arxiv.org/abs/0704.0143v1 E(γ) = β Uint(γ)− ln d(ρi) , where d(ρ) = β̃2Nc e−β(ρ) , (3) is the instanton size distribution [7]; dγi = dzi dωi dρi, and β(ρ) = = −b ln(C1/bNc Λρ) is the action of a single instanton, where (Λ = ΛMS = 0.92ΛP.V.) CNc , depends on the renormalization scheme and, in the case under consideration, is given by CNc ≈ 4.66 exp(−1.68Nc) π2(Nc − 1)!(Nc − 2)! , and b = 11 Nc − 2 Nf . We assume that Nf=2 here because the leading contribution to renormalization comes from hard massless gluons and quarks. The auxiliary function β̃ = −b ln(Λρ̄) , is evaluated at the scale ρ̄ defined by an average size of pseudoparticles, Uint(γ) is considered assuming pair interaction dominance. Its contribution has the form [2] dω1 dω2 dz1 dz2 Uint(γ1, γ2) = V ξ 2 ρ21 ρ where ξ2 = 27 π N2c − 1 . The factor β that appears in the exponent in formula (2) is also evaluated at the scale of an average size of pseudoparticles ρ̄. Assuming that the instanton liquid is topologically neutral, we do not introduce notation to distinguish between instantons and anti-instantons, N denotes the ovarall number of pseudoparticles in volume V . Since the interaction is independent of coordinates or orientation in color space, it is natural to calculate the generating functional Y on the basis of the effective one-particle distribution function µ(ρ), which can be determined from the solution of the variational problem dρi µ(ρ) = dγi e −E1(γ) , (4) E1(γ) = − lnµ(ρi) , where the factor V N in (4) is isolated in order that the result be expressed in terms of the respective density and convenience in interpretation of the function µ(ρ). With regard to convexity of the exponential function, the generating functional (2) for every fixed N partial contribution can be estimated using the approximating inequality Y ′ ≥ Ya = Y ′1 exp(−〈E − E1〉) , (5) where an average over approximate ensemble is implied. In the case under consideration, the average of difference 〈E − E1〉 is given by: 〈E −E1〉 = dγi [β Uint − ln d(ρi) + lnµ(ρi)] e lnµ(ρi) = dρ µ(ρ) ln dρ1dρ2 ξ 2 ρ21ρ 2 µ(ρ1)µ(ρ2) where µ0 = dρ µ(ρ). Variation of the functional 〈E −E1〉 with respect to µ(ρ) results formally in the equation µ(ρ) = e−1 d(ρ) e−nβξ 2ρ2ρ2 (where n = N/V is the density of the instanton liquid). Here an unwanted factor of e−1, emerges. It can be excluded due to the fact that the approximate functional Ya is independent of the constant factor of C that can be added to the expressionfor µ(ρ). For convenience, we set C = e, and therefore, arrive at µ(ρ) = d(ρ) e−nβξ 2ρ2ρ2 . (6) Substituting this solution to the approximate functional, we obtain V N µN0 (ρ2)2 . Defining suitable parameter ν the integral for determination µ0 can be represented in the form µ0 = Λ dρΛ CNcβ̃ 2Nc (ρΛ)b−5 e ρ2 . (7) From the comparison of which with formula (6) we obtain = βξ2nρ2 . (8) Provided that ν is known, this formula offers a relation between the average instanton size and the density of the instanton liquid. To find this relation, we consider the equation dρ ρb−3 e dρ ρb−5 e ρ2 ν−1 Γ( b−4 Γ( b−4 which gives ν = b−4 , and therefore, µ0 = Λ 4 CNcβ̃ 2Nc (ρΛ) 2 . It should be noted that the factor of two in the denominator of this expression stems from the integration measure 2ρdρ, which, in its turn, emerges in transformation to the Gaussian integral with respect to ρ squired. This factor was omited in [2]; however, this fact has no noticeable consequences. The reason is that the parameter Λ is determined from a fit to some observable, for example, to the pion decay constant. In so doing, everything is governed by a choice of scale. Moreover, it should be remembered that the instanton liquid model is merely a rough approximation. From the above, we derive an approximate expression for the functional as follows: Ya = exp [ln(n/Λ4)− 1] +N ln CNcβ̃ 2Nc(βξ2ν)−ν/2 . (9) Now we find the value of n at which the argument of the exponential approaches its maximum. To do this, we should solve the equation ln(n/Λ4) + ln CNcβ̃ 2Nc(βξ2ν)−ν/2 − n ν = 0 . (10) From the relation (8) we obtain = 0 . On the other hand, = − bρ̄ , . We represent the derivative of β with respect to the density in the form , and obtain 4β − b , . (11) Thus we derive the expressionfor the instanton liquid density n/Λ4 = CNcβ̃ 2Nc(βξ2ν)−ν/2 ν + 2 ν − 2 ν + 2 2β − ν − 2 . (12) The contribution of the derivatives of the functions β and β̃ with respect to the density was disre- garded in [2]. This contribution compensates for the above-mentioned factor of 2 though, as was noted above, this is not essential. The obtained formula for the instanton liquid density by itself does not provide a solution to the problem because it remains to solve the transcendental equation (8) in ρ̄, where the function β involves the logarithm of ρ̄. To solve this equation, it is convenient to reformulate the problem without resort to the explicit formula (12) for the instanton liquid density. By definition of the function β, the action of an isolated pseudoparticle must be positive. This gives a limitation to the maximum size of an (anti-)instanton as follows: ρ̄ΛC ≤ 1 (actually, ρ̄Λ ≤ 1). Now we can solve the transcendental equation (10), by bisection of the segment. In so doing, a stationary value of ρ̄ is determined at each step and the respective instanton liquid density is determined from equation (8). In the calculation of the generating functional, the contributions of the type (ρ̄Λ)2ν are used rather than the expression for the instanton liquid density. Now we modify the variational principle in order to extend our description to the case of finite temperatures. For this purpose, we employ calorons — solutions of the Yang–Mills equations periodic in the Euclidean time. The background field should be replaced by a superposition of calorons and anti-calorons as follows [8]: Aaµ(x, γ) = − ωab η̄bµν ∂ν lnΠ, Π = 1 + sinh(2πrT ) cosh(2πrT )− cos(2πτT ) , where T−1 is the period of the caloron, r = |x−z| is the distance from the center of the caloron z in three dimensional space, and τ = x4 − z4 – is the respective interval of ”time”. As the temperature tends to zero, such solutions go over to (anti-)instantons in the singular gauge. Yet another modifi- cation of the variational principle is the replacement of the distribution (3) in the instanton size by the function d(ρ, T ) = β̃2Nc exp[−β(ρ)− ANcT 2ρ2] , (14) where the coefficient ANc = 6 Nc − 1 π2 accounts for the additional contribution to the action of each individual pseudoparticle. It provides an approximation to a more exact expression d(ρ, T ) = d(ρ, 0) exp g2T 2 (Nc +Nf/2) 4π2ρ2 + 12 A(πρT ) [1 + (Nc −Nf)/6] , (15) constructed from the respective determinants [9]. For our purposes it is sufficient to say that the function A(πρT ) is determined by a shape of the pseudoparticle (13). This function was studied in the cited work; however, we do not use it in the present article. It should be mentioned that the expansion up to the terms of the order T 2 can be used as an approximate expression for the function A(πρT ) because, within the accuracy of the variational principle, only the terms up to order ρ2 should be kept in the argument of the exponential in formula (14). The first term in formula (15) is represented as a product of two factors; each factor was interpreted in [9]. The first factor is the square of the electric mass, that is, the temporal component of the gluon polarization tensor evaluated at the zero energy and momentum. It has the form m2el = Π44(ω = 0,p = 0) = g (Nc +Nf/2) . (16) The remaining components being equal to zero at zero energy-momentum. Therefore, the magnetic mass vanishes. Note that the one-loop quark and gluon contributions to the polarization tensor are taken into account [10], the resulting sum being rearranged in order that the quark and gluon contributions in the medium sum up to a finite value. This, formally, gives rise to a generation of the mass of the gluon field. The second factor is the integral of the square of the fourth component A4 of the field in formula (13) ∫ dy Aa4(y)A 4(y) = 4π2ρ2 . (17) It is independent of the temperature [11]. It is seen that one can take into account only one-loop contribution 12m 4 to the Lagrangian of the gluon field and neglect other corrections. It was demonstrated [3] that the term Uint describing the interaction of pseudoparticles can be brought in the form 12m 2 AaµA µ, wherem 2 = 9π2 n ρ̄2 N2c − 1 . Thus the interaction term also describes generation of the mass of the gluon field in the instanton–anti-instanton medium in quasi-classical approximation. This being so, chromoelectric and chromomagnetic fields are screened equally well provided that the instanton liquid density is not equal to zero. It was shown that screening is a consequence of stochastic character of the ensemble of gluon fields being unrelated to a specific instanton solution of the type (1) or details of the repulsion mechanism responsible for stabilization of the ensemble [2]. An application of these considerations to the (anti-)instanton solution (1) leads precisely to the formula for Uint. It turns out that, in the caloron ensemble, screening of chromomagnetic fields and the interaction term depends only weakly on the temperature. However, the anisotropy is negligible small and the interaction term coincides with that obtained for the (anti-)instanton solution. First it was found in [11], where the instanton liquid was studied at non zero temperature. The one-loop contribution of Plank gluons is proportional to Nc (see formula (16)) and does not vary as the chemical potential becomes different from zero. On the other hand, it is known that the one-loop fermion contribution in the medium can be calculated exactly. It has no dangerous singularities [12], [13]. The ”temporal” component of the polarization tensor generated by a quark of definite flavor has the form 44(k4, ω) = g dp p2 4ε2p − k2 (k2 + 2pω)2 + 4ε2pk (k2 − 2pω)2 + 4ε2pk24 − εpk4 arctan 8pω εpk4 4ε2pk 4 − 4p2ω2 + k4 where ω = |k|, k2 = ω2 + k24, εp = (m2 + p2)1/2, where m – is the quark mass, np = n−p + n+p , n−p = (e T + 1)−1, n+p = (e T + 1)−1. After summation over all components, the polarization tensor takes the form Πf(k4, ω) = g dp p2 2m2 − k2 (k2 + 2pω)2 + 4ε2pk (k2 − 2pω)2 + 4ε2pk24 . (18) It is seen that, at k4 = 0, and small values of ω, the first term (that is, unit) gives the dominant contribution to the gluon mass. The spatial components are negligibly small. In particular, at ω = 0 we obtain Πf(0, 0) = Π 44(0, 0) = g dp p2 np , (19) and at T = 0 we arrive at Πf (0, 0) = g2 (µ2 −m2)1/2µ µ+ (µ2 −m2)1/2 The ultimate expression for the electric mass has the form m2el = g2T 2 Πf (0, 0)  . (20) In this approximation, the effect of the instanton liquid is completely accounted for by the quark mass dynamically generated in the instanton medium. With such definition of mass, the formula (16) at µ = 0 and T 6= 0 should be modified. The coefficient 16 at Nf should be replaced by . However, this replacement has only a little effect; self-consistency of our calculations will be discussed below. Using the integral (17), which is also valid for the caloron solution, we derive the expression for the distribution of pseudoparticles: d(ρ;µ, T ) = d(ρ; 0, 0) e−η 2(µ,T ) ρ2 , η2 = 2 π2 Πf(0, 0)  . (21) The one-loop quark contribution to the instanton action at zero temperature, finite chemical poten- tial, and ω 6= 0 was studied in detail in [14] (see also [15], [16]). These studies make it possible to improve our description, however, we work within the approximation (21) and, moreover, we consider the limit of massless quarks. A self-consistent calculation for the quark with dynamically generated mass can be the subject of a separable study. Necessary modifications in the variational principle are as follows. It was revealed that only the distribution function d(ρ;µ, T ) of pseudoparticles changes, whereas the repulsion interaction Uint between pseudoparticles remains as before. Similar to the case of instantons, we introduce the parameter ν satisfying the relation = η2 + βξ2nρ2 , (22) instead of (8). Since the instanton liquid density is greater than zero, a new limitation on the average size of pseudoparticle emerges ρ̄Λ ≤ ν η . If this limit is smaller than the limit descussed above, then it must be the starting point for the determination of the equilibrium size of pseudoparticles by the bisection method. The derivative of the function β with respect to the density of the instanton liquid can be determined from the relation (22). The result is 4β − b+ 2η2 ρ̄2β ν−η2 ρ̄2 , (23) it should be substituted in Eq. (10) that determines the saddle-point. The integral (19) should be evaluated numerically because it cannot be calculated analytically at arbitrary temperatures even though the quark mass equals zero. Thus we are ready to determine the parameters of the instanton liquid everywhere over the µ – T plane. For simplicity, the calculations are performed at zero quark masses. We neglect the light-quark contribution to the respective determinants [17] (see also [18]). We also disregard a possible temperature molecular behavior of instanton–anti-instanton pairs [19]. The results of the calculations are shown in Fig. 1 by the lines of constant density. The instanton liquid density is plotted in Fig 2. versus the temperature (at zero chemical potential) and versus the chemical potential (at zero temperature). Though the conventional natation for the instanton liquid density at nonzero temperature is n = TN/V3, we use the label n which is more simple. At T 6= 0, and µ = 0, our results coincide with the results obtained in [11] and [9]. It sould be noted that our results are consistent with recent calculations on a lattice at finite temperatures [20], [21], where a rapid decrease of the chromoelectric components in the respective correlation functions was found. In our model, such suppression is due to the term 12m 4 in the effective action; with neglect of this factor, the chromoelectric and chromomagnetic correlators coincide. From this point of view, our calculations may seem inconsistent. We use the caloron solution (13), which is symmetric under an interchange of chromoelectric and chromomagnetic fields. However, the caloron components manifest themthelves in the observables differently because of the anisotropy of the weight function. In fact, our method of taking the gluon mass term into account is consistent only in perturbation theory. In a complete study, one must find an analogue of the solution (13) for the effective Lagrangian with the Figure 1: Lines of equal density of the instanton liquid in the temperature-chemical potential plane. Curve 1 corresponds to the density n = 0.75 n0, where n0 is the density at zero temperature and chemical potential. Also shown are the densities from (curve 2) n = 0.5 n0 to (curve 6) n = 0.1 n0. Curves 3–5 correspond to intermediate densities at intervals of 0.1. Figure 2: Instanton liquid density versus (curve 1) temperature and (curve 2) chemical potential. gluon mass generated for the chromoelectric field and gain a self-consistent description of ensemble of pseudoparticles in the long-wave approximation [3]. It is of interest that the data on correlation functions for cooled configurations [20] are fitted well by the instanton ensemble [22]. In so doing, the contribution of the terms of the second order in the instanton liquid density (∼ n2) is in excellent agreement with the effect of the standard instanton ensemble with the respective admixture of the perturbative component everywhere over the distance range chosen for a fit [23]. This agreement indicates that the confining component is absent from the lattice configurations isolated by cooling. It is surprising because lattice simulations with cooling were aimed at the searches for a long-wave confining component. However, an interpretation of lattice simulations at finite temperature presents difficulties because it is not clear what scale corresponds to the configarations used for the measurements. The magnitude of deformation of the chromoelectric component of the solution for the effective Lagrangian with the mass term is also poorly known. The scale of lattice configurations can, in principle, be estimated using the scale at which the chromoelectric field decrease since only this scale has emerged in our calculations. In conclusion we note that, though we used only a rough approximation, the most important features of the behavior of the instanton liquid density (gluon condensate) in the medium have been revealed. The lines of equal density are markedly extended along the µ axis because, according to the formula (20), the most substantial gluon component of screening vanishes at small temperatures. Typical values of T and µ at which the effects of the medium become significant are related to each other by the formula (Nc +Nf/2) (T/Λ)2 ∼ Nf (µ/Λ)2 ∼ 1, which leads to a plausible coefficient of oblongness along the µ axis 2π Tc , (at Nc = 3 and Nf = 2). A fall in density evaluated with allowance for the dynamically generated quark mass should begin at a greater value and be more steep. The reason is that, at chemical potentials less than the quark mass, the quark contribution to screening is reduced. This gives rise to formation of a plateau and concentration of the lines of equal density. The dependence of the dynamical quark mass on the momentum ω is significant at small temperatures leading to a decrease of screening approximately by a factor of two [14]. We are grateful to A.E. Dorokhov for helful discussions. This work was supported in part by grants STCU #P015c, CERN-INTAS 2000-349, NATO 2000-PST.CLG 977482. References [1] C.G. Callan, R. Dashen, and D.J. Gross, Phys. Lett. B66 (1977) 375; C.G. Callan, R. Dashen, and D.J. Gross, Phys. Rev. D17 (1978) 2717. A. Schäfer and E.V. Shuryak, Rev. Mod. Phys. 70 (1998) 323. [2] D. I. Diakonov, V. Yu. Petrov, Nucl. Phys. B245 (1984) 259. [3] S.V. Molodtsov, G.M. Zinovjev, hep-ph/0510015 [4] I.V. Musatov, A.N. Tavkhelidze and V.F. Tokarev, Theor. Math. Phys. 86, 20 (1991); A.N. Tavkhelidze and V.F. Tokarev, Fiz. Elem. Chast. Atom. Yadra 21, 1126 (1990). [5] E.V. Shuryak, Phys. Rev. D52, 5370 (1995). [6] A.E. Dorokhov, S.V. Esaibegian, A.E. Maximov and S.V. Mikhailov, Eur. Phys. J. C 13, 331 (2000). [7] G.’t Hooft, Phys.Rev.D14 (1976) 3432. [8] B.J. Harrington, H.K. Shepard, Phys. Rev. D17 (1978) 2122. [9] D.J. Gross, R.D. Pisarski, and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43. [10] E.V. Shuryak, JETP 74 (1978) 408. [11] D. I. Diakonov, A. D. Mirlin, Phys. Lett. B203 (1988) 299. [12] I.A. Akhiezer, S.V. Peletminsky, JETP 38 (1960) 1829. [13] B.A. Freedman, L.D. McLerran, Phys. Rev. D16 (1977) 1130, 1147, 1169. [14] C.A. Carvalho, Nucl. Phys. B183 (1981) 182. [15] A.A. Abrikosov (Jr), Yad. Fiz. 37 (1983) 772; V. Baluni, Phys. Lett. B106 (1981) 491. http://arxiv.org/abs/hep-ph/0510015 [16] E.V. Shuryak, Preprint INP, N0 82-03, 1982. [17] M.A. Novak, J.J.M. Verbaarschot, and I. Zahed, Nucl. Phys. B325 (1989) 581. [18] G.V. Dunne, J. Hur, Ch. Lee, H. Min, Phys. Rev. D71 (2005) 085019; G.V. Dunne, J. Hur, Ch. Lee, H. Min, Phys. Rev. Lett. 94 (2005) 072001. [19] E.-M. Ilgenfritz, E.V. Shuryak, Phys. Lett. B325 (1994) 263. [20] A. Di Giacomo, E. Meggiolaro, H. Panagopoulos, Nucl. Phys. B483 (1997) 371. [21] M. DÉlia, A. Di Giacomo and E. Meggiolaro, Phys. Rev. D67 (2003) 114504. [22] A.E. Dorokhov, S.V. Esaibegyan, and S.V. Mikhailov, Phys. Rev. D56 (1997) 4062; E.-M. Ilgenfritz, B.V. Martemyanov, S.V. Molodtsov, M. Müller-Preussker, and Yu.A. Simonov, Phys. Rev. D58 (1998) 114508. [23] E.-M. Ilgenfritz, B.V. Martemyanov, M. Müller-Preussker, Phys. Rev. D62 (2000) 096004.

0704.0144

Eternal inflation and localization on the landscape

Eternal inflation and localization on the landscape D. Podolsky1∗ and K. Enqvist1,2 1 Helsinki Institute of Physics, P.O. Box 64 (Gustaf Hällströmin katu 2), FIN-00014, University of Helsinki, Finland and 2 Department of Physical Sciences, P.O. Box 64, FIN-00014, University of Helsinki, Finland (Dated: November 4, 2018) We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential V (φ) = V0 + δV (φ), where |δV (φ)| ≪ V0 is random. We find that the diffusion of the distribution function ρ(φ, t) of the inflaton expectation value in different Hubble patches may be suppressed due to the effect analogous to the Anderson localization in disordered quantum systems. At t → ∞ only the localized part of the distribution function ρ(φ, t) survives which leads to dynamical selection principle on the landscape. The probability to measure any but a small value of the cosmological constant in a given Hubble patch on the landscape is exponentially suppressed at t → ∞. PACS numbers: 98.80.Bp,98.80.Cq,98.80.Qc String theory is believed to imply a wide landscape [1] of both metastable vacua with a positive cosmological constant and true vacua with a vanishing or a negative cosmological constant; the latter are called anti-de Sitter or AdS vacua, where space-time collapses into a singular- ity. In regions with positive cosmological constant, or in de Sitter (dS) vacua, the universe inflates, and because of the possibility of tunneling between different de Sitter vacua inflation is eternal. The problem of calculating statistical distributions of the landscape vacua is very complicated [2] and is even considered to be NP-hard [3] (the total number of vacua on the landscape is estimated to be of order 10100 ÷ 101000). Our aim is to consider how eternal in- flation proceeds on the landscape by using the mere fact that the number of vacua within the landscape is ex- tremely large, so that their distribution can have signif- icant disorder. The dynamics of eternal inflation is then described by the Fokker-Planck equations in the disor- dered effective potential.1 In that case, the landscape dynamics may have some interesting parallels in solid state physics, as we will discuss in the present paper. Eternal inflation on the landscape can be modeled as follows [5, 6]. Let us numerate vacua on the landscape by the discrete index i and define Pi(t) as the probability to measure a given (positive) value of the cosmological constant Λi in a given Hubble patch. If the rates of tunneling between the metastable minima i and j on the landscape are given by the time independent matrix Γij , then the probabilities Pi satisfy the system of “vacuum dynamics” equations [7] Ṗi = j 6=i (ΓjiPj − ΓijPi)− ΓisPi. (1) The last term in this equation corresponds to tunneling ∗On leave from Landau Institute for Theoretical Physics, 119940, Moscow, Russia. 1 An approach somewhat similar to ours was also presented in [4]. between the metastable de Sitter vacuum i and a true vacuum with a negative cosmological constant (an AdS vacuum), i.e. tunneling into a collapsing AdS space-time [8]. The collapse time tcol ∼ MP /V is much shorter than the characteristic time trec ∼ exp M4P /VdS for tun- neling back into a de Sitter metastable vacuum, so that the AdS true vacua effectively play the role of sinks for the probability current (1) describing eternal inflation on the landscape [5]. In what follows we will assume that the effect of the AdS sinks is relatively small; otherwise the landscape will be divided into almost unconnected “islands” of vacua [6], preventing the population of the whole landscape by eternal inflation. In the limit of weak tunneling only the vacua closest to each other are important. It is convenient to classify parts (islands) of the landscape according to the typical number of adjacent vacua within each part. Technically, the landscape of vacua of the string theory can be repre- sented as a graph with 10100÷101000 nodes and a number of connections between them of the same order. By an island on the landscape, we mean a subgraph relatively weakly connected to the major ”tree”. The dimension- ality of the island can then be defined as the Hausdorff dimension NH of the corresponding subgraph [17]. For example, if there are only two adjacent vacua for any vacuum in a given island, then NH = 1 for this island and we denote it as quasi-one-dimensional; a domain of vacua with NH = 2 is quasi-two-dimensional, and so on. In the quasi-one-dimensional case (neglecting the AdS sinks) the system (1) reduces to Ṗi = −Γi,i+1Pi+Γi+1,iPi+1−Γi,i−1Pi+Γi−1,iPi−1. (2) While in general Γij 6= Γji, we will take 〈Γij〉 = 〈Γji〉 on the average.2 Furthermore, suppose that the initial 2 This condition is never satisfied for the Bousso-Polchinski land- scape [9], where the adjacent vacua are those with closest values http://arxiv.org/abs/0704.0144v3 condition for Eq. (2) is Pi(0) = 1, Pj 6=i(0) = 0. (3) so that the initial state is well localized. Naively, one may expect that the distribution function Pi(t) would start to spread out according to the usual diffusion law and the system of vacua would exponentially quickly reach a “thermal” equilibrium distribution of probabilities for a given Hubble patch to be in a given dS vacuum. However, there exists a well known theorem [10] from the theory of diffusion on random lattices stating that the distribution function Pi remains localized near the initial distribution peak for a very long time, with its characteristic width behaving as 〈i2(t)〉 ∼ log t . (4) This is a surprising result when applied to eternal infla- tion where the general lore (see for example [11]) is that the initial conditions for eternal inflation will be forgot- ten almost immediately after its beginning. Instead, in what follows we will argue that the memory about the initial conditions may survive during a very long time on the quasi-one-dimensional islands of the landscape. We will model the landscape by a continuous inflaton potential V (φ) = V0 + δV (φ), (5) where V0 is constant, and δV (φ) is a random contribution such that |δV (φ)| ≪ V0, and φ is the inflaton or the or- der parameter describing the transitions. As in stochas- tic inflation [16], in different causally connected regions fluctuations have a randomly distributed amplitude and observers living in different Hubble patches see differ- ent expectation values of the inflaton. When stochastic fluctuations of the inflaton are large enough, the expec- tation value of the inflaton in a given Hubble patch is determined by the Langevin equation [16] φ̇ = − + f(t), (6) where the stochastic force f(φ, t) is Gaussian with corre- lation properties 〈f(t)f(t′)〉 = δ(t− t′). (7) From (6) one can derive the Fokker-Planck equation, which controls the evolution of the probability distribu- tion ρ(φ, t) describing how the values of φ are distributed of the effective cosmological constant. However, the spectrum of states on Bousso-Polchinski landscape is not disordered, so that the analysis based on averaging over disorder is not appli- cable. Disorder appears in more realistic multithroat models of the string theory landscape. among different Hubble patches in the multiverse. One finds [16] ∂ρ(φ, t) . (8) The general solution to Eq. (8) is given by ρ = e 4π2δV (φ) cnψn(φ)e 0 (t−t0) 4π2 , (9) where ψn and En are respectively the eigenfunctions and the eigenvalues of the effective Hamiltonian Ĥ = − +W (φ). (10) W (φ) = is a functional of the scalar field potential V (φ). It is often denoted as the superpotential due to its “super- symmetric” form: the Hamiltonian (10) can be rewrit- ten as Ĥ = Q̂†Q̂, where Q̂ = −∂/∂φ + v′(φ) with v(φ) = 4π2δV (φ)/(3H40 ). The eigenfunctions of the Hamiltonian (10) satisfy the Schrödinger equation + (En −W (φ))ψn = 0, (12) and its solutions have the following well known features [16]: 1. The eigenvalues of the Hamiltonian (10) are all pos- itive definite. 2. The contributions from eigenfunctions of excited states ψn>0(φ) to the solution Eq. (9) become ex- ponentially quickly damped with time. However, if one is interested in what happens at time scales ∆t . 1/En, the first n eigenfunctions should be taken into account. In particular, if the spectrum of the Hamiltonian (10) is very dense, as in the case of the string theory landscape, knowing the ground state alone is not enough for complete understand- ing dynamics of eternal inflation. We now recall that the potential V (φ) is a random func- tion of the inflaton field and has extremely large number of minima. This allows us to draw several conclusions about the form of the eigenfunctions ψn(φ) using the for- mal analogy between Eq. (12) and the time-independent Schrödinger equation describing the motion of carriers in disordered quantum systems such as semiconductors with impurities. The physical quantities in disordered systems can be calculated by averaging over the random potential of the impurities.3 A famous consequence of the random potential gener- ated by impurities in crystalline materials is the strong suppression of the conductivity, known as Anderson lo- calization [12, 13]. This effect is essential in dimensions lower than 3 and completely defines the kinetics of carri- ers in one-dimensional systems. There, impurities create a random potential for Bloch waves with the correlation properties 〈u(r)u(r′)〉 = δ(r − r′), 〈u(r)〉 = 0, (13) where τ is the mean free path for electrons and ν is the density of states per one spin degree of freedom of the electron gas at the Fermi surface. As a consequence, in the one-dimensional case all eigenstates of the electron hamiltonian become localized with ψn(r) ∼ exp |r − rn| at t → ∞, where rn are the positions of localization centers, and the localization length L is of the order of the mean free path lτ = 〈v〉τ . As a result, the probability density ρ(R, t) to find electron at the point R at time t asymptotically approaches the limit ρ(R) ∼ exp(−R/L) for R ≫ L, or ρ(R) ∼ Const for R ≪ L at t → ∞. The one-dimensional Anderson localization takes place for an arbitrarily weak disorder and arbitrary correlation properties of the random potential u(r) [13]. Also, in a two-dimensional case all the electron eigen- states in a random potential remain localized. However, the localization length grows exponentially with energy, the rate of growth being related to the strength of the dis- order. In three-dimensional case, the localization prop- erties of eigenstates are defined by the Ioffe-Regel-Mott criterion: if the corresponding eigenvalue of the Hamil- tonian of electrons En satisfies the condition En < Eg where Eg is so called mobility edge, then the eigenstate is localized. The mobility edge Eg is a function of the strength of the disorder. In higher dimensional cases the situation is unknown. Let us now return to the discussion of eternal infla- tion described by the Fokker-Planck equation (8). Since the localization is the property of the eigenfunctions of the time-independent hamiltonian (10), it is also a nat- ural consequence of the effective randomness of the po- tential of the string theory landscape.4 The diffusion 3 Observe that the typical number of these impurities varies be- tween 1012 to 1017 per cm3 while the number of vacua on the string theory landscape is 10100 ÷ 101000. 4 The Anderson localization on the landscape of string theory was discussed before in [14] in the context of the Wheeler-deWitt equation in the minisuperspace. The possibility to have the An- derson localization on the landscape was also mentioned in [15]. of the probability distribution (4) is suppressed due to the localization of the eigenfunctions ψn(φ) contributing to the overall solution (9). This counteracts the general wisdom that eternal inflation rapidly washes out any in- formation of the initial conditions. Indeed,in the quasi- one-dimensional case all the wave functions ψn(φ) are localized, i.e., for a particular realization of disorder they behave as ψn(φ) ∼ exp |φ− φn| . (15) where φn define the ”localization centers” as in the Eq. (14), and L is the localization length which is of the same order of magnitude as the “mean free path” related to the strength of the disorder in the superpotential W (φ). Let us now discuss how eternal inflation proceeds on islands where the typical number of adjacent vacua is larger than two. In the quasi-two-dimensional case the network of vacua within a given island is described by a composite index ~i = (i, j). The distribution function ρ for finding a given value of the cosmological constant in a given Hubble patch is a two-dimensional matrix. Again, all the eigenstates of the corresponding tunneling hamil- tonian Ĥ are localized. However, since the localization length grows exponentially with energy, the distribution function effectively spreads out almost linearly with 〈~i2(t)〉 ∼ t 1 + c1 logα t + · · · , (16) where α > 0 are constants depending on the correlation properties of the disorder on the landscape [18]. The low energy eigenstates (namely, the states with E < Eg where Eg is the mobility edge) are localized with a rela- tively small localization length. In the quasi-higher-dimensional cases the distribution function spreads out according to the linear diffusion law at intermediate times. Again, there exists a mobility edge Eg such that the eigenstates of the tunneling Hamilto- nian with energies E < Eg are localized. These low energy eigenstates define the asymptotics of the distri- bution function ρ at t≫ E−1g . (17) The value of the mobility edge Eg strongly depends on the dimensionality of the island and the strength of the disorder, and the higher is the dimensionality, the lower is the mobility edge [17]. Localization of the low energy eigenstates in two- and higher-dimensional cases introduces an effective dynam- ical selection principle for different vacua on the land- scape (5): in the asymptotic future, not all of them will be populated, but only those near the localization centers φn, and the probability to populate other minima will be suppressed exponentially according to the Eq. (15). It is interesting to note that in condensed matter sys- tems the localization centers are typically located near the points where the effective potential has its deepest minima [13]. In the case of eternal inflation, it means that the probability to measure any but very low value of the cosmological constant in a given Hubble patch will be exponentially suppressed in the asymptotic future [17]. Finally, we discuss the effect of sinks on the dynamics of tunneling between the vacua. On the string theory landscape, dS metastable vacua are typically realized by uplifting stable AdS vacua (as, for example, in the well known KKLT model [19]). The probability to tunnel from the uplifted dS state i back into the AdS vacuum is related to the value of gravitino mass m3/2 in the dS state [8] and given by tAdS ∼ Γ is ∼ exp Const.M2P 3/2,i . (18) The gravitino mass after uplifting [20] has the order of magnitude m3/2,i ∼ |VAdS,i| 1/2/MP . Since at long time scales VAdS,i can also be regarded as a random quantity, our analysis of the general solution of “vacuum dynam- ics” equations (1) does not have to be modified in any essential way [17]. In addition to AdS sinks, Hubble patches where eternal inflation has ended (stochastic fluctuations of the infla- ton expectation value became smaller than the effect of classical force) also effectively play a role of sinks for the probability current described by the Eq. (8). In par- ticular, the Hubble patch we live in is one of such sinks. Related to the effect of sinks, there exists a time scale tend for eternal inflation on the landscape (5) such that the unitarity of the evolution of the probability distribution ρ breaks down at t ≫ tend [17]. Our discussion remains valid if t ≪ tend. It is unclear whether the probability distribution ρ has achieved the late time asymptotics in the corner of the landscape we live in. In summary, we have argued that eternal inflation on the landscape may lead to a strong localization of the inflaton distribution function among different Hub- ble patches. This is a consequence of the high density of the vacua, which effectively implies a random poten- tial for the order parameter responsible for inflation. We found that the inflaton motion is analogous to the mo- tion of carriers in disordered quantum systems, and there exists an analogue of the Anderson localization for eter- nal inflation on the landscape. Physically, this means that not all the vacua on the landscape are populated by eternal inflation in the asymptotic future, but only those near the localization centers of the inflaton effective po- tential. They are located near the deepest minima of the potential, which implies that the probability to measure any but very low value of the cosmological constant in a given Hubble patch is exponentially suppressed at late times. Acknowledgements The authors belong to the Marie Curie Research Train- ing Network HPRN-CT-2006-035863. D.P. is thankful to I. Burmistrov, N. Jokela, J. Majumder, M. Skvortsov, K. 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0704.0145

Singularity Resolution in Isotropic Loop Quantum Cosmology: Recent Developments

IMSc/2007/03/2 Singularity Resolution in Isotropic Loop Quantum Cosmology: Recent Developments Ghanashyam Date1, ∗ 1The Institute of Mathematical Sciences, CIT Campus, Chennai-600 113, INDIA Abstract Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue has been discussed in terms of physical quantities (expectation values of Dirac observables) and there is also an “improved” quantization of the Hamiltonian constraint. These developments are briefly discussed. This is an expanded version of the review talk given at the 24th IAGRG meeting in February 2007. PACS numbers: 04.60.Pp,98.80.Jk,98.80.Bp ∗Electronic address: [email protected] http://arxiv.org/abs/0704.0145v1 mailto:[email protected] I. COSMOLOGY, QUANTUM COSMOLOGY, LOOP QUANTUM COSMOLOGY Our current understanding of large scale properties of the universe is summarised by the so called Λ−CDM Big Bang model – homogeneous and isotropic, spatially flat space- time geometry with a positive cosmological constant and cold dark matter. Impressive as it is, the model is based on an Einsteinian description of space-time geometry which has the Big Bang singularity. The existence of cosmological singularities is in fact much more general. There are homogeneous but anisotropic solution space-times which are singular and even in the inhomogeneous context there is a general solution which is singular [1]. The singularity theorems give a very general argument for the existence of initial singularity for an everywhere expanding universe with normal matter content, the singularity being characterized as incompleteness of causal geodesic in the past. Secondly, in conjunction with an inflationary scenario, one imagines the origin of the smaller scale structure to be attributed to quantum mechanical fluctuations of matter and geometry. On account of both the features, a role for the quantum nature of matter and geometry is indicated. Quantum mechanical models for cosmological context were in fact constructed, albeit formally. For the homogeneous and isotropic sector, the geometry is described by just the scale factor and the extrinsic curvature of the homogeneous spatial slices. In the gravitational sector, a quantum mechanical wave function is a function of the scale factor i.e. a function on the (mini-) superspace of gravity. The scale factor being positive, the minisuperspace has a boundary and wave functions need to satisfy a suitable boundary condition. Furthermore, the singularity was not resolved in that the Wheeler-De Witt equation (or the Hamiltonian constraint) which is a differential equation with respect to the scale factor, had singular coefficient due to the matter density diverging near the boundary. Thus quantization per se does not necessarily give a satisfactory replacement of the Big Bang singularity. Meanwhile, over the past 20 years, a background independent, non-perturbative quan- tum theory of gravity is being constructed starting from a (gauge-) connection formulation of classical general relativity [2]. The background independence provided strong constraints on the construction of the quantum theory already at the kinematical level (i.e. before imposi- tion of the constraints) and in particular revealed a discrete and non-commutative nature of quantum (three dimensional Riemannian) geometry. The full theory is still quite unwieldy. Martin Bojowald took to step of restricting to homogeneous geometries and quantizing such models in a loopy way. Being inherited from the connection formulation, the geometry is described in terms of densitized triad which for the homogeneous and isotropic context is described by p ∼ sgn(p)a2 which can also take negative values (encoding the orientation of the triad). This means that the classical singularity (at p = 0) now lies in the interior of the superspace. Classically, the singularity prevents any relation between the two regions of positive and negative values of p. Quantum mechanically however, the wave functions in these two regions, could be related. One question that becomes relevant in a quantum theory is that if a wave function, specified for one orientation and satisfying the Hamilto- nian constraint, can be unambiguously extended to the other orientation while continuing to satisfy the Hamiltonian constraint. Second main implication of loop quantization is the necessity of using holonomies – exponentials of connection variable c – as well defined op- erators. This makes the Hamiltonian constraint a difference equation on the one hand and also requires an indirect definition for inverse triad (and inverse volume) operators entering in the definition of the matter Hamiltonian or densities and pressures. Quite interestingly, the Hamiltonian constraint equation turns out to be non-singular (i.e. deterministic) and in the effective classical approximation suggests interesting phenomenological implication quite naturally. These two features in fact made LQC an attractive field. We will briefly summarise the results prior to 2005 and then turn to more recent devel- opments. An extensive review of LQC is available in [3]. For simplicity and definiteness, we will focus on the spatially flat isotropic models. II. SUMMARY OF PRE 2005 LQC Classical model: Using coordinates adapted to the spatially homogeneous slicing of the space-time, the metric and the extrinsic curvature are given by, ds2 := −dt2 + a2(t) (dr2 + r2dΩ2 . (1) Starting from the usual Einstein-Hilbert action and scalar matter for definiteness, one can get to the Hamiltonian as, |detgµν | φ̇2 − V (φ) (−aȧ2) + a3φ̇2 − V (φ)a3 pa = − aȧ , pφ = V0a 3φ̇ , V0 := d3x ; H(a, pa, φ, pφ) = Hgrav +Hmatter + a3V0V (φ) Hmatter Thus, H = 0 ↔ Friedmann Equation. For the spatially flat model, one has to choose a fiducial cell whose fiducial volume is denoted by V0. In the connection formulation, instead of the metric one uses the densitized triad i.e. instead of the scale factor a one has p̃, |p̃| := a2/4 while the connection variable is related to the extrinsic curvature as: c̃ := γȧ/2 (the spin connections is absent for the flat model). Their Poisson bracket is given by {c̃, p̃} = (8πGγ)/(3V0). The arbitrary fiducial volume can be absorbed away by defining c := V 0 c̃, p := V 0 p̃. Here, γ is the Barbero-Immirzi pa- rameter which is dimensionless and is determined from the Black hole entropy computations to be approximately 0.23 [4]. From now on we put 8πG := κ. The classical Hamiltonian is then given by, γ−2c2 |p|−3/2p2φ + |p|3/2V (φ) . (5) For future comparison, we now take the potential for the scalar field, V (φ) to be zero as well. One can obtain the Hamilton’s equations of motion and solve them easily. On the con- strained surface (H = 0), eliminating c in favour of p and pφ, one has, c = ± γ , ṗ = ± |pφ||p|−1/2 . φ̇ = pφ|p|−3/2 , ṗφ = 0 , (6) |p| ⇒ p(φ) = p∗e± (φ−φ∗) (7) Since φ is a monotonic function of the synchronous time t, it can be taken as a new “time” variable. The solution is determined by p(φ) which is (i) independent of the constant pφ and (ii) passes through p = 0 as φ → ±∞ (expanding/contracting solutions). It is immediate that, along these curves, p(φ), the energy density and the extrinsic curvature diverge as p → 0. Furthermore, the divergence of the density implies that φ(t) is incomplete i.e. t ranges over a semi-infinite interval as φ ranges over the full real line 1. Thus a singularity is signalled by a solution p(φ) passing through p = 0 in finite synchronous time (or equivalently by the density diverging somewhere along the solution). A natural way to ensure that all solutions are non-singular is to ensure that either of the two terms in the Hamiltonian constraint are bounded. Question is: does a quantum theory replace the Big Bang singularity by something non-singular?. There are at least two ways to explore this question. One can imagine computing cor- rections to the Hamiltonian constraint such that individual terms in the effective constraint are bounded. Alternatively and more satisfactorily, one should be able to define suitable observables whose expectation values will generate the analogue of p(φ) curves along which physical quantities such as energy density, remain bounded. The former method was used pre-2005 because it could be used for more general models (non-zero potential, anisotropy etc). The latter has been elaborated in 2006, for the special case of vanishing potential. Both methods imply that classical singularity is resolved in LQC but not in Wheeler-De Witt quantum cosmology. We will first discuss the issue in terms of effective Hamiltonian because it is easier and then discuss it in terms of the expectation values. In the standard Schrodinger quantization, one can introduce wave functions of p, φ and quantize the Hamiltonian operator by c→ i~κγ/3∂p , pφ → −i~∂φ, in equation (5). With a choice of operator ordering, ĤΨ(p, φ) = 0 leads to the Wheeler-De Witt partial differential equation which has singular coefficients. The background independent quantization of Loop Quantum Gravity however suggest a different quantization of the isotropic model. One should look for a Hilbert space on which only exponentials of c (holonomies of the connection) are well defined operators and not ĉ. Such a Hilbert space consists of almost periodic functions of c which implies that the triad operator has every real number as a proper eigenvalue: p̂|µ〉 := 1 γℓ2Pµ|µ〉, ∀µ ∈ R , ℓ2P := κ~. This has a major implication: inverses of positive powers of triad operators do not exist [5]. These have to be defined by using alternative classical expressions and promoting them to quantum operators. This can be done with at least one parameter worth of freedom, eg. |p|−1 = 8πGγl {c, |p|l} ]1/(1−l) , l ∈ (0, 1) . (8) 1 For the FRW metric, integral curves of ∂t are time-like geodesics and hence incompleteness with respect to t is synonymous with geodesic incompleteness. Only positive powers of |p| appear now. However, this still cannot be used for quantization since there is no ĉ operator. One must use holonomies: hj(c) := e iτi , where τi are anti-hermitian generators of SU(2) in the jth representation satisfying Trj(τiτj) = −13j(j + 1)(2j + 1)δij, Λ i is a unit vector specifying a direction in the Lie algebra of SU(2) and µ0 is the coordinate length of the loop used in defining the holonomy. Using the holonomies, |p|−1 = (8πGµ0γl) j(j + 1)(2j + 1) TrjΛ · τ hj{h−1j , |p|l} , (9) which can be promoted to an operator. Two parameters, µ0 ∈ R and j ∈ N/2, have crept in and we have a three parameter family of inverse triad operators. The definitions are: |̂p|−1 |µ〉 = (Fl(q)) 1−l |µ〉 , q := µ Fl(q) := l + 2 (q + 1)l+2 − |q − 1|l+2 l + 1 (q + 1)l+1 − sgn(q − 1)|q − 1|l+1 Fl(q ≫ 1) ≈ Fl(q ≈ 0) ≈ l + 1 . (11) All these operators obviously commute with p̂ and their eigenvalues are bounded above. This implies that the matter densities (and also intrinsic curvatures for more general homogeneous models), remain bounded at the classically singular region. Most of the phenomenological novelties are consequences of this particular feature predominantly anchored in the matter sector. We have thus two scales: p0 := P and 2jp0 := µ0(2j)ℓ P. The regime |p| ≪ p0 is termed the deep quantum regime, p≫ 2jp0 is termed the classical regime and p0 . |p| . 2jp0 is termed the semiclassical regime. The modifications due to the inverse triad defined above are strong in the semiclassical and the deep quantum regimes. For j = 1/2 the semiclassical regime is absent. Note that such scales are not available for the Schrodinger quantization. The necessity of using holonomies also imparts a non-trivial structure for the gravitational Hamiltonian. The expression obtained is: Hgrav = − 8πGγ3µ30 ǫijkTr hihjh j hk{h−1k , V } In the above, we have used j = 1/2 representation for the holonomies and V denotes the volume function. In the limit µ0 → 0 one gets back the classical expression. While promoting this expression to operators, there is a choice of factor ordering involved and many are possible. We will present two choices of ordering: the non-symmetric one which keeps the holonomies on the left as used in the existing choice for the full theory, and the particular symmetric one used in [6]. Ĥnon−symgrav = γ3µ30ℓ sin2µ0c V̂ cos − cosµ0c V̂ sin Ĥsymgrav = 24i(sgn(p)) γ3µ30ℓ sinµ0c V̂ cos − cos V̂ sin sinµ0c (14) At the quantum level, µ0 cannot be taken to zero since ĉ operator does not exist. The action of the Hamiltonian operators on |µ〉 is obtained as, Ĥnon−symgrav |µ〉 = (Vµ+µ0 − Vµ−µ0) (|µ+ 4µ0〉 − 2|µ〉+ |µ− 4µ0〉) (15) Ĥsymgrav|µ〉 = [|Vµ+3µ0 − Vµ+µ0 | |µ+ 4µ0〉+ |Vµ−µ0 − Vµ−3µ0 | |µ− 4µ0〉 − {|Vµ+3µ0 − Vµ+µ0 |+ |Vµ−µ0 − Vµ−3µ0 |} |µ〉] (16) where Vµ := ( γℓ2P|µ|)3/2 denotes the eigenvalue of V̂ . Denoting quantum wave function by Ψ(µ, φ) the Wheeler-De Witt equation now becomes a difference equation. For the non- symmetric one we get, A(µ+ 4µ0)ψ(µ+ 4µ0, φ)− 2A(µ)ψ(µ, φ) + A(µ− 4µ0)ψ(µ− 4µ0, φ) 3ℓ2PHmatter(µ)ψ(µ, φ) (17) where, A(µ) := Vµ+µ0 − Vµ−µ0 and vanishes for µ = 0. For the symmetric operator one gets, f+(µ)ψ(µ+ 4µ0, φ) + f0(µ)ψ(µ, φ) + f−(µ)ψ(µ− 4µ0, φ) 3ℓ2PHmatter(µ)ψ(µ, φ) where, (18) f+(µ) := |Vµ+3µ0 − Vµ+µ0 | , f−(µ) := f+(µ− 4µ0) , f0 := − f+(µ)− f−(µ) . Notice that f+(−2µ0) = 0 = f−(2µ0), but f0(µ) is never zero. The absolute values have entered due to the sgn(p) factor. These are effectively second order difference equations and the Ψ(µ, φ) are determined by specifying Ψ for two consecutive values of µ eg for µ = µ̂+ 4µ0, µ̂ for some µ̂. Since the highest (lowest) order coefficients vanishes for some µ, then the corresponding component Ψ(µ, φ) is undetermined by the equation. Potentially this could introduce an arbitrariness in extending the Ψ specified by data in the classical regime (eg µ ≫ 2j) to the negative µ. For the non-symmetric case, the highest (lowest) A coefficients vanish for their argument equal to zero thus leaving the corresponding ψ component undetermined. However, this un- determined component is decoupled from the others. Thus apart from admitting the trivial solution ψ(µ, φ) := Φ(φ)δµ,0, ∀µ, all other non-trivial solutions are completely determined by giving two consecutive components: ψ(µ̂, φ), ψ(µ̂+ 4µ0, φ). For the symmetric case, due to these properties of the f±,0(µ), it looks as if the difference equation is non-deterministic if µ = 2µ0 + 4µ0n, n ∈ Z. This is because for µ = −2µ0, ψ(2µ0, φ) is undetermined by the lower order ψ’s and this coefficient enters in the deter- mination of ψ(2µ0, φ). However, the symmetric operator also commutes with the parity operator: (Πψ)(µ, φ) := ψ(−µ, φ). Consequently, ψ(2µ0, φ) is determined by ψ(−2µ0, φ). Thus, we can restrict to µ = 2µ0 + 4kµ0, k ≥ 0 where the equation is deterministic. In both cases then, the space of solutions of the constraint equation, is completely de- termined by giving appropriate data for large |µ| i.e. in the classical regime. Such a de- terministic nature of the constraint equation has been taken as a necessary condition for non-singularity at the quantum level 2. As such this could be viewed as a criterion to limit the choice of factor ordering. By introducing an interpolating, slowly varying smooth function, Ψ(p(µ) := 1 γℓ2P), and keeping only the first non-vanishing terms, one deduces the Wheeler-De Witt differential equation (with a modified matter Hamiltonian) from the above difference equation. Making a WKB approximation, one infers an effective Hamiltonian which matches with the classical Hamiltonian for large volume (µ≫ µ0) and small extrinsic curvature (derivative of the WKB phase is small). There are terms of o(~0) which contain arbitrary powers of the first derivative of the phase which can all be summed up. The resulting effective Hamiltonian now contains modifications of the classical gravitational Hamiltonian, apart from the modifications in the matter Hamiltonian due to the inverse powers of the triad. The largest possible domain of validity of effective Hamiltonian so deduced must have |p| & p0 [7, 8]. An effective Hamiltonian can alternatively obtained by computing expectation values of 2 For contrast, if one just symmetrizes the non-symmetric operator (without the sgn factor), one gets a difference equation which is non-deterministic. the Hamiltonian operator in semiclassical states peaked in classical regimes [9]. The leading order effective Hamiltonian that one obtains is (spatially flat case): non−sym eff = − B+(p)sin 2(µ0c) + A(p)− 1 B+(p) +Hmatter ; B+(p) := A(p+ 4p0) + A(p− 4p0) , A(p) := (|p+ p0|3/2 − |p− p0|3/2) , (19) γℓ2Pµ , p0 := γℓ2Pµ0 . For the symmetric operator, the effective Hamiltonian is the same as above except that B+(p) → f+(p) + f−(p) and 2A(p) → f+(p) + f−(p). The second bracket in the square bracket, is the quantum geometry potential which is negative and higher order in ℓP but is important in the small volume regime and plays a role in the genericness of bounce deduced from the effective Hamiltonian [10]. This term is absent in effective Hamiltonian deduced from the symmetric constraint. The matter Hamiltonian will typically have the eigenvalues of powers of inverse triad operator which depend on the ambiguity parameters j, l. We already see that the quantum modifications are such that both the matter and the gravitational parts in the effective Hamiltonian, are rendered bounded and effective dynamics must be non-singular. For large values of the triad, p ≫ p0, B+(p) ∼ 6p0 p − o(p−3/2) while A(p) ∼ 3p0 o(p−3/2). In this regime, the effective Hamiltonians deduced from both symmetric and non- symmetric ordering are the same. The classical Hamiltonian is obtained for µ0 → 0. From this, one can obtain the equations of motion and by computing the left hand side of the Friedmann equation, infer the effective energy density. For p≫ p0 one obtains 3, := ρeff = Hmatter 1− 8πGµ Hmatter , p := a2/4 . (20) The effective density is quadratic in the classical density, ρcl := Hmatterp −3/2. This modi- fication is due to the quantum correction in the gravitational Hamiltonian (due to the sin2 feature). This is over and above the corrections hidden in the matter Hamiltonian (due to the “inverse volume” modifications). As noted before, we have two scales: p0 controlled by µ0 in the gravitational part and 2p0j in the matter part. For large j it is possible that we 3 For p in the semiclassical regime, one should include the contribution of the quantum geometry potential present in the non-symmetric ordering, especially for examining the bounce possibility [8]. can have p0 ≪ p ≪ 2p0j in which case the above expressions will hold with j dependent corrections in the matter Hamiltonian. In this semiclassical regime, the corrections from sin2 term are smaller in comparison to those from inverse volume. If p ≫ 2p0j then the matter Hamiltonian is also the classical expression. For j = 1/2, there is only the p ≫ p0 regime and ρcl is genuinely the classical density. Let us quickly note a comparison of the two quantizations as reflected in the corresponding effective Hamiltonians, particularly with regards to the extrema of p(t). For this, we will assume same ambiguity parameters (j, l) in the matter Hamiltonian, (1/2)p2φF jl and explore the regime p & p0. The two effective Hamiltonians differ significantly in the semiclassical regime due to the quantum geometry potential. The equations of motion imply that pφ is a constant of motion, φ varies monotonically with t and on the constraint surface, we can eliminate c in favour of p and pφ. Let us focus on p(t) and in particular consider its possible extrema. It is immediate that ṗ = 0 implies sin(2µ0c) = 0. This leads to two possibilities: (A) sin(µ0c) = 0 or (B) cos(µ0c) = 0. A local minimum signifies a bounce while a local maximum signifies a re-collapse. The value of p at an extremum, p∗, gets determined in terms of the constant pφ. The bounce/re- collapse nature of an extremum depends upon whether p∗ is in the classical regime or in the semiclassical regime and also on the case (A) or (B). Note that for the case (A) to hold, it is necessary that the quantum geometry potential is present. Thus, for the symmetric ordering, case (A) cannot be realised – it will imply pφ = 0. An extremum determined by case (A): It is a bounce if p∗ is in in the semiclassical regime; p∗ varies inversely with pφ while the corresponding density varies directly. p∗ being limited to the semiclassical regime implies that pφ is also bounded both above and below, for such an extremum to occur. It turns out that p∗ can be in the classical regime, provided pφ ∼ ℓ2P. Thus, the non-symmetric constraint, at the effective level, can accommodate a bounce only in the semiclassical regime and with large densities. An extremum determined by case (B): It is bounce if p∗ is in the classical regime; p∗ varies directly with pφ and the corresponding density varies inversely. p∗ being limited to the classical regime implies that pφ must be bounded below but can be arbitrarily large and thus the density can be arbitrarily small. This is quite unreasonable and has been sited as one of the reasons for considering the “improved” quantization (more on this later). If p∗ is in the semiclassical regime, it has to be a re-collapse with pφ ∼ ℓ2P. In the early works, one worked with the non-symmetric constraint operator and the sin2 corrections were not incorporated (i.e. µ0c≪ π/2 was assumed) and the phenomenological implications were entirely due to the modified matter Hamiltonian. These already implied genericness of inflation and genericness of bounce. These results were discussed at the previous IAGRG meeting in Jaipur. To summarize: LQC differs from the earlier quantum cosmology in three basic ways (a) the basic variables are different and in particular the classical singularity is in the inte- rior of the mini-superspace; (b) the quantization is very different, being motivated by the background independent quantization employed in LQG; (c) there is a “parent” quantum theory (LQG) which is pretty much well defined at the kinematical level, unlike the metric variables based Wheeler-De Witt theory. The loop quantization has fundamentally distinct implications: its discrete nature of quantum geometry leads to bounded energy densities and bounded extrinsic and intrinsic curvatures (for the anisotropic models). These two features are construed as “resolving the classical singularity”. Quite un-expectedly, the ef- fective dynamics incorporating quantum corrections is also singularity-free (via a bounce), accommodates an inflationary phase rather naturally and is well behaved with regards to perturbations. Although there are many ambiguity parameters, these results are robust with respect to their values. III. POST 2004 ISOTROPIC LQC Despite many attractive features of LQC, many points need to be addressed further: • LQC being a constrained theory, it would be more appropriate if singularity resolution is formulated and demonstrated in terms of physical expectation values of physical (Dirac) operators i.e. in terms of “gauge invariant quantities”. This can be done at present with self-adjoint constraint i.e. a symmetric ordering and for free, massless scalar matter. • There are at least three distinct ambiguity parameters: µ0 related to the fiducial length of the loop used in writing the holonomies; j entering in the choice of SU(2) representation which is chosen to be 1/2 in the gravitational sector and some large value in the matter sector; l entering in writing the inverse powers in terms of Poisson brackets. The first one was thought to be determined by the area gap from the full theory. The j = 1/2 in the gravitational Hamiltonian seems needed to avoid high order difference equation and larger j values are hinted to be problematic in the study of a three dimensional model [11]. Given this, the choice of a high value of j in the matter Hamiltonian seems unnatural4. For phenomenology however the higher values allowing for a larger semiclassical regime are preferred. The l does not play as significant a role. • The bounce scale and density at the bounce, implied by the effective Hamiltonian (from symmetric ordering), is dependent on the parameters of the matter Hamiltonian and can be arranged such that the bounce density is arbitrarily small. This is a highly undesirable feature. Furthermore, the largest possible domain of validity of WKB approximation is given by the turning points (eg the bounce scale). However, the approximation could break down even before reaching the turning point. An independent check on the domain of validity of effective Hamiltonian is thus desirable. • A systematic derivation of LQC from LQG is expected to tighten the ambiguity pa- rameters. However, such a derivation is not yet available. A. Physical quantities and Singularity Resolution When the Hamiltonian is a constraint, at the classical level itself, the notion of dynamics in terms of the ‘time translations’ generated by the Hamiltonian is devoid of any physical meaning. Furthermore, at the quantum level when one attempts to impose the constraint as Ĥ|Ψ〉 = 0, typically one finds that there are no solutions in the Hilbert space on which Ĥ is defined - the solutions are generically distributional. One then has to consider the space of all distributional solutions, define a new physical inner product to turn it into a Hilbert space (the physical Hilbert space), define operators on the space of solutions (which must thus act invariantly) which are self-adjoint (physical operators) and compute expectation values, uncertainties etc of these operators to make physical predictions. Clearly, the space of solutions depends on the quantization of the constraint and there is an arbitrariness in the choice of physical inner product. This is usually chosen so that a complete set of Dirac 4 For an alternative view on using large values of j, see reference [12]. observables (as deduced from the classical theory) are self-adjoint. This is greatly simplified if the constraint has a separable form with respect to some degree of freedom 5. For LQC (and also for the Wheeler-De Witt quantum cosmology), such a simplification is available for a free, massless scalar matter: Hmatter(φ, pφ) := p2φ|p|−3/2. Let us sketch the steps schematically, focusing on the spatially flat model for simplicity [6, 13]. 1. Fundamental constraint equation: The classical constraint equations is: |p|+ 8πG p2φ |p|−3/2 = 0 = Cgrav + Cmatter ; (21) The corresponding quantum equation for the wave function, Ψ(p, φ) is: 8πGp̂2φΨ(p, φ) = [B̃(p)] −1ĈgravΨ(p, φ) , [B̃(p)] is eigenvalue of |̂p|−3/2 ; (22) Putting p̂φ = −i~∂φ, p := µ and B̃(p) := ( )−3/2B(µ), the equation can be written in a separated form as, ∂2Ψ(µ, φ) = [B(µ)]−1 ℓ−1P Ĉgrav Ψ(µ, φ) := − Θ̂(µ)Ψ(µ, φ). (23) The Θ̂ operator for different quantizations is different. For Schrodinger quantization (Wheeler-De Witt), with a particular factor ordering suggested by the continuum limit of the difference equation, the operator Θ̂(µ) is given by, Θ̂Sch(µ)Ψ(µ, φ) = − |µ|3/2∂µ µ ∂µΨ(µ, φ) (24) while for LQC, with symmetric ordering, it is given by, Θ̂LQC(µ)Ψ(µ, φ) = −[B(µ)]−1 C+(µ)Ψ(µ+ 4µ0, φ) + C 0(µ)Ψ(µ, φ)+ C−(µ)Ψ(µ− 4µ0, φ) C+(µ) := ∣∣ |µ+ 3µ0|3/2 − |µ+ µ0|3/2 ∣∣ , (25) C−(µ) := C+(µ− 4µ0) , C0(µ) := − C+(µ)− C−(µ) . Note that in the Schrodinger quantization, the BSch(µ) = |µ|−3/2 diverges at µ = 0 while in LQC, BLQC(µ) vanishes for all allowed choices of ambiguity parameters. In both cases, B(µ) ∼ |µ|−3/2 as |µ| → ∞. 5 A general abstract procedure using group averaging is also available. 2. Inner product and General solution: The operator Θ̂ turns out to be a self-adjoint, positive definite operator on the space of functions Ψ(µ, φ) for each fixed φ with an inner product scaled by B(µ). That is, for the Schrodinger quantization, it is an operator on L2(R, BSch(µ)dµ) while for LQC it is an operator on L2(RBohr, BBohr(µ)dµBohr). Because of this, the operator has a complete set of eigenvectors: Θ̂ek(µ) = ω 2(k)ek(µ), k ∈ R, 〈ek|ek′〉 = δ(k, k′), and the general solution of the fundamental constraint equation can be expressed as Ψ(µ, φ) = dk Ψ̃+(k)ek(µ)e iωφ + Ψ̃−(k)ēk(µ)e −iωφ . (26) The orthonormality relations among the ek(µ) are in the corresponding Hilbert spaces. Different quantizations differ in the form of the eigenfunctions, possibly the spectrum itself and of course ω(k). In general, these solutions are not normalizable in L2(RBohr× R, dµBohr × dµ), i.e. these are distributional. 3. Choice of Dirac observables: Since the classical kinematical phase space is 4 dimensional and we have a single first class constraint, the phase space of physical states (reduced phase space) is two dimensional and we need two functions to coordinatize this space. We should thus look for two (classical) Dirac observables: functions on the kinematical phase space whose Poisson bracket with the Hamiltonian constraint vanishes on the constraint surface. It is easy to see that pφ is a Dirac observable. For the second one, we choose a one parameter family of functions µ(φ) satisfying {µ(φ), C(µ, c, φ, pφ)} ≈ 0. The corresponding quantum definitions, with the operators acting on the solutions, are: p̂φΨ(µ, φ) := −i~∂φΨ(µ, φ) , (27) ̂|µ|φ0Ψ(µ, φ) := ei Θ̂(φ−φ0)|µ|Ψ+(µ, φ0) + e−i Θ̂(φ−φ0)|µ|Ψ−(µ, φ0) (28) On an initial datum, Ψ(µ, φ0), these operators act as, ̂|µ|φ0Ψ(µ, φ0) = |µ|Ψ(µ, φ0) , p̂φΨ(µ, φ0) = ~ Θ̂Ψ(µ, φ0) . (29) 4. Physical inner product: It follows that the Dirac operators defined on the space of solutions are self-adjoint if we define a physical inner product on the space of solutions as: 〈Ψ|Ψ′〉phys := “ dµB(µ)” Ψ̄(µ, φ)Ψ′(µ, φ) . (30) Thus the eigenvalues of the inverse volume operator crucially enter the definition of the physical inner product. For Schrodinger quantization, the integral is really an integral while for LQC it is actually a sum over µ taking values in a lattice. The inner product is independent of the choice of φ0. A complete set of physical operators and physical inner product has now been specified and physical questions can be phrased in terms of (physical) expectation values of functions of these operators. 5. Semiclassical states: To discuss semiclassical regime, typically one defines semiclassical states: physical states such that a chosen set of self-adjoint operators have specified expectation values with uncertainties bounded by specified tolerances. A natural choice of operators for us are the two Dirac operators defined above. It is easy to construct semiclassical states with respect to these operators. For example, a state peaked around, pφ = p and |µ|φ0 = µ∗ is given by (in Schrodinger quantization for instance), Ψsemi(µ, φ0) := (k−k∗)2 2σ2 ek(µ)e iω(φ0−φ ∗) (31) k∗ = − 3/2κ~−1p∗φ , φ ∗ = φ0 +− 3/2κℓn|µ∗| . (32) For LQC, the ek(µ) functions are different and the physical expectation values are to be evaluated using the physical inner product defined in the LQC context. 6. Evolution of physical quantities: Since one knows the general solution of the constraint equation, Ψ(µ, φ), given Ψ(µ, φ0), one can compute the physical expectation values in the semiclassical so- lution, Ψsemi(µ, φ) and track the position of the peak as a function of φ as well as the uncertainties as a function of φ. 7. Resolution of Big Bang Singularity: A classical solution is obtained as a curve in (µ, φ) plane, different curves being labelled by the points (µ∗, φ∗) in the plane. The curves are independent of the constant value of p∗φ These curves are already given in (7). Quantum mechanically, we first select a semiclassical solution, Ψsemi(p ∗ : φ) in which the expectation values of the Dirac operators, at φ = φ0, are p φ and µ ∗ re- spectively. These values serve as labels for the semiclassical solution. The former one continues to be p∗φ for all φ whereas 〈 ̂|µ|φ0〉(φ) =: |µ|p∗φ,µ∗(φ), determines a curve in the (µ, φ) plane. In general one expects this curve to be different from the classical curve in the region of small µ (small volume). The result of the computations is that Schrodinger quantization, the curve |µ|p∗ ,µ∗(φ), does approach the µ = 0 axis asymptotically. However for LQC, the curve bounces away from the µ = 0 axis. In this sense – and now inferred in terms of physical quantities – the Big Bang singularity is resolved in LQC. It also turns out that for large enough values of p∗φ, the quantum trajectories constructed by the above procedure are well approximated by the trajectories by the effective Hamiltonian. All these statements are for semiclassical solutions which are peaked at large µ∗ at late times. Two further features are noteworthy as they corroborate the suggestions from the effective Hamiltonian analysis. First one is revealed by computing expectation value of the matter density operator, ρmatter := ̂(p∗φ) 2|p|−3, at the bounce value of |p|. It turns out that this value is sensitive to the value of p∗φ and can be made arbitrarily small by choosing p φ to be large. Physically this is unsatisfactory as quantum effects are not expected to be significant for matter density very small compared to the Planck density. This is traced to the quantization of the gravitational Hamiltonian, in particular to the step which introduces the ambiguity parameter µ0. A novel solution proposed in the “improved quantization”, removes this undesirable feature. The second one refers to the role of quantum modifications in the gravitational Hamil- tonian compared to those in the matter Hamiltonian (the inverse volume modification or B(µ)). The former is much more significant than the latter. So much so, that even if one uses the B(µ) from the Schrodinger quantization (i.e. switch-off the inverse volume mod- ifications), one still obtains the bounce. So bounce is seen as the consequence of Θ̂ being different and as far as qualitative singularity resolution is concerned, the inverse volume modifications are un-important. As the effective picture (for symmetric constraint) showed, the bounce occurs in the classical region (for j = 1/2) where the inverse volume corrections can be neglected. For an exact model which seeks to understand as to why the bounces are seen, please see [14]. B. Improved Quantization The undesirable features of the bounce coming from the classical region, can be seen readily using the effective Hamiltonian, as remarked earlier. To see the effects of modifi- cations from the gravitational Hamiltonian, choose j = 1/2 and consider the Friedmann equation derived from the effective Hamiltonian (20), with matter Hamiltonian given by Hmatter = p2φ|p|−3/2. The positivity of the effective density implies that p ≥ p∗ with p∗ determined by vanishing of the effective energy density: ρ∗ := ρcl(p∗) = ( 8πGµ20γ −1. This leads to |p∗| = 4πGµ20γ |pφ| and ρ∗ = 8πGµ20γ )−3/2|pφ|−1. One sees that for large |pφ|, the bounce scale |p∗| can be large and the maximum density – density at bounce – could be small. Thus, within the model, there exist a possibility of seeing quantum effects (bounce) even when neither the energy density nor the bounce scale are comparable to the corre- sponding Planck quantities and this is an undesirable feature of the model. This feature is independent of factor ordering as long as the bounce occurs in the classical regime. One may notice that if we replace µ0 → µ̄(p) := ∆/|p| where ∆ is a constant, then the effective density vanishes when ρcl equals the critical value ρcrit := ( 8πG∆γ2 )−1, which is independent of matter Hamiltonian. The bounce scale p∗ is determined by ρ∗ = ρcrit which gives |p∗| = ( 2ρcrit )1/3. Now although the bounce scale can again be large depending upon pφ, the density at bounce is always the universal value determined by ∆. This is a rather nice feature in that quantum geometry effects are revealed when matter density (which couples to gravity) reaches a universal, critical value regardless of the dynamical variables describing matter. For a suitable choice of ∆ one can ensure that a bounce always happen when the energy density becomes comparable to the Planck density. In this manner, one can retain the good feature (bounce) even for j = 1/2 thus “effectively fixing” an ambiguity parameter and also trade another ambiguity parameter µ0 for ∆. This is precisely what is achieved by the “improved quantization” of the gravitational Hamiltonian [15]. The place where the quantization procedure is modified is when one expresses the cur- vature in terms of the holonomies along a loop around a “plaquette”. One shrinks the plaquette in the limiting procedure. One now makes an important departure: the plaque- tte should be shrunk only till the physical area (as distinct from a fiducial one) reaches its minimum possible value which is given by the area gap in the known spectrum of area operator in quantum geometry: ∆ = 2 3πγG~. Since the plaquette is a square of fiducial length µ0, its physical area is µ 0|p| and this should set be to ∆. Since |p| is a dynamical variable, µ0 cannot be a constant and is to be thought of a function on the phase space, µ̄(p) := ∆/|p|. It turns out that even with such a change which makes the curvature to be a function of both connection and triad, the form of both the gravitational constraint and inverse volume operator appearing in the matter Hamiltonian, remains the same with just doing the replacement, µ0 → µ̄ defined above, in the holonomies. The expressions simplify by using eigenfunctions of the volume operator V̂ := ˆ|p| , instead of those of the triad. The relevant expressions are: v := Ksgn(µ)|µ|3/2 , K := 2 ; (33) V̂ |v〉 = )3/2 ℓ3P |v||v〉 , (34) Ψ(v) := Ψ(v + k) , (35) |̂p|−1/2 j=1/2,l=3/4 Ψ(v) = )−1/2 K1/3|v|1/3 ∣∣|v + 1|1/3 − |v − 1|1/3 ∣∣Ψ(v) (36) B(v) = ∣∣|v + 1|1/3 − |v − 1|1/3 ∣∣3 (37) Θ̂ImprovedΨ(v, φ) = −[B(v)]−1 C+(v)Ψ(v + 4, φ) + C0(v)Ψ(v, φ)+ C−(v)Ψ(v − 4, φ) , (38) C+(v) := |v + 2| | |v + 1| − |v + 3|| , (39) C−(v) := C+(v − 4) , C0(v) := − C+(v)− C−(v) . (40) Thus the main changes in the quantization of the Hamiltonian constraint are: (1) replace µ0 → µ̄ := ∆/|p| in the holonomies; (2) choose symmetric ordering for the gravitational constraint; and (3) choose j = 1/2 in both gravitational Hamiltonian and the matter Hamil- tonian (in the definition of inverse powers of triad operator). The “improvement” refers to the first point. This model is singularity free at the level of the fundamental constraint equation (even though the leading coefficients of the difference equation do vanish, because the the parity symmetry again saves the day); the densities continue to be bounded above – and now with a bound independent of matter parameters; the effective picture continues to be singularity free and with undesirable features removed and the classical Big Bang being replaced by a quantum bounce is established in terms of physical quantities. C. Close Isotropic Model While close model seems phenomenologically disfavoured, it provides further testing ground for quantization of the Hamiltonian constraint. Because of the intrinsic (spatial) curvature, the plaquettes used in expressing the Fij in terms of holonomies, are not bounded by just four edges – a fifth one is necessary. This was attempted and was found to lead to an “unstable” quantization. This difficulty was bypassed by using the holonomies of the extrinsic curvature instead of the gauge connection which is permissible in the homogeneous context. The corresponding, non-symmetric constraint and its difference equation was anal- ysed for the massless scalar matter. Green and Unruh, found that solutions of the difference equation was always diverging (at least for one orientation) for large volumes. Further, the divergence seemed to set in just where one expected a re-collapse from the classical theory. In the absence of physical inner product and physical interpretation of the solutions, it was concluded that this version of LQC for close model is unlikely to accommodate classical re-collapse even though it avoided the Big Bang/Big Crunch singularities. Recently, this model has been revisited [16]. One went back to using the gauge connec- tion and the fifth edge difficulty was circumvented by using both the left-invariant and the right-invariant vector fields to define the plaquette. In addition, the symmetric ordering was chosen and finally the µ0 → µ̄ improvement was also incorporated. Without the improve- ment, there were still the problems of getting bounce for low energy density and also not getting a reasonable re-collapse (either re-collapse is absent or the scale is marginally larger than the bounce scale). With the improvement, the bounces and re-collapses are neatly accommodated and one gets a cyclic evolution. In this case also, the scalar field serves as a good clock variable as it continues to be monotonic with the synchronous time. I have focussed on the singularity resolution issue in this talk. Other developments have also taken place in the past couple of years. I will just list these giving references. 1. Effective models and their properties: The effective picture was shown to be non- singular and since this is based on the usual framework of GR, it follows that energy conditions must be violated (and indeed they are thanks to the inverse volume modifi- cations). This raised questions regarding stability of matter and causal propagation of perturbations. Golam Hossain showed that despite the energy conditions violations, neither of the above pathologies result [17]. Minimally coupled scalar has been used in elaborating inflationary scenarios. However non-minimally coupled scalars are also conceivable models. The singularity resolution and inflationary scenarios continue to hold also in this case. Furthermore sufficient e-foldings are also admissible [18]. In the improved quantization, one sets the ambiguity parameter j = 1/2 and shifts the dominant effects to the the gravitational Hamiltonian. All the previous phenomeno- logical implications however were driven by the inverse volume modifications in the matter sector. Consequently, it is necessary to check if and how the phenomenology works with the improved quantization. This has been explored in [19]. Using the effective dynamics for the homogeneous mode, density perturbations were explored and power spectra were computed with the required small amplitude [20, 21]. As many of the phenomenology oriented questions have been explored using effective Hamiltonian which incorporate quantum corrections from various sources (gravity, matter etc). This motivates a some what systematic approach to constructing effective approximations. This has been initiated in [22]. 2. Anisotropic models: The anisotropic models provide further testing grounds for loop quantization. At the difference equation level, the non-singularity has been checked also for these models in the non-symmetric scheme. For the vacuum Bianchi I model, there is no place for the inverse volume type corrections to appear at an effective Hamiltonian level and the effective dynamics would continue to be singular. However, once the gravitational corrections (sin2) are incorporated, the effective dynamics again is non-singular and one can obtain the non-singular version of the (singular) Kasner solution [23]. More recently, the Bianchi I model with a free, massless scalar is also analysed in the improved quantization [24]. A perturbative treatment of anisotropies has been explored in [25]. 3. Inhomogeneities: Inhomogeneities are a fact of nature although these are small in the early universe. This suggests a perturbative approach to incorporate inhomogeneities. On the one hand one can study their evolution in the homogeneous, isotropic back- ground (cosmological perturbation theory). One can also begin with a (simplified) inhomogeneous model and try to see how a homogeneous approximation can become viable. The work on the former has already begun. For the latter part, Bojowald has discussed a simplified lattice model to draw some lessons for the homogeneous models. In particular he has given an alternative argument for the µ0 → µ̄ modification which does not appeal to the area operator [12]. IV. OPEN ISSUES AND OUT LOOK In summary, over the past two years, we have seen how to phrase and understand the fate of Big Bang singularity in a quantum framework. Firstly, with the help of a minimally coupled, free, massless scalar which serves as a good clock variable in the isotropic context, one can define physical inner product, a complete set of Dirac observables and their physical matrix elements. At present this can be done only for self-adjoint Hamiltonian constraint. Using these, one can construct trajectories in the (p, pφ) plane which are followed by the peak of a semiclassical state as well as the uncertainties in the Dirac observables. It so happens that these trajectories do not pass through the zero volume – Big Bang is replaced by a Bounce. For close isotropic model, the Big Crunch is also replaced by a bounce while retaining classically understood re-collapse. In conjunction with the µ̄ improvement, the gravitational Hamiltonian can be given the the main role in generating the bounce. A corresponding treatment in Schrodinger quantization (Wheeler-De Witt theory), does not generate a bounce nor does it render the density, curvatures bounded. Thus, quantum representation plays a significant role in the singularity resolution. Secondly, the improved quantization motivated by the regulation of the Fij invoking the area operator from the full theory (or by the argument from the inhomogeneous lattice model), also leads the bounce to be “triggered” when the energy density reaches a critical value (∼ 0.82ρPlanck) which is independent of the values of the dynamical variables. Close model also gives the same critical value. While the improvement is demonstrated to be viable in the isotropic context, the proce- dure differs from that followed in the full theory. One may either view this as something special to the mini-superspace model(s) or view it as providing hints for newer approaches in the full theory. A general criteria for “non-singularity” is not in sight yet and so also a systematic deriva- tion of the mini-superspace model(s) from a larger, full theory. Acknowledgements: I would like to thank Parampreet Singh for discussions regarding the improved quantization as well as for running his codes for exploring bounce in the semiclassical regime. Thanks are due to Martin Bojowald for comments on an earlier draft. [1] Belinskii V A, Khalatnikov I M, and Lifschitz E M, 1982, A general solution of the Einstein equations with a time singularity, Adv. Phys., 13, 639–667. [2] Rovelli C, 2004, Quantum Gravity, Cambridge University Press, Cambridge, UK, New York, USA ; Thiemann T, 2001, Introduction to Modern Canonical Quantum General Relativity, [gr-qc/0110034]; Ashtekar A and Lewandowski J, 2004, Background Independent Quantum Gravity: A Status Report, Class. Quant. Grav., 21, R53, gr-qc/0404018; [3] Bojowald M, 2005, Loop Quantum Cosmology, Living Rev. Relativity, 8, 11, http://www.livingreview.org/lrr-2005-11, [gr-qc/0601085]. [4] Domagala M and Lewandowski J, 2004, Black hole entropy from Quantum Geometry, Class. Quant. Grav., 21, 5233-5244, [gr-qc/0407051]; Krzysztof A. Meissne K A, 2004, Black hole entropy in Loop Quantum Gravity, Class. Quant. Grav., 21, 5245-5252, [gr-qc/0407052]; [5] Ashtekar A, Bojowald M and Lewandowski J, 2003, Mathematical structure of loop quantum cosmology, Adv. Theor. Math. Phys., 7, 233-268, [gr-qc/0304074]. [6] Ashtekar A, Pawlowski T and Singh P, 2006, Quantum Nature of the Big Bang, Phys. Rev. Lett., 96, 141301, [gr-qc/0602086]. [7] Bojowald M, 2001, The Semiclassical Limit of Loop Quantum Cosmology, Class. Quant. Grav., 18, L109-L116, [gr-qc/0105113]. [8] Date G and Hossain G M, 2004, Effective Hamiltonian for Isotropic Loop Quantum Cos- mology, Class. Quant. 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[11] Vandersloot K, 2005, On the Hamiltonian Constraint of Loop Quantum Cosmology, Phys. Rev., D 71, 103506, [gr-qc/0502082]; Perez A, 2006, On the regularization ambiguities in loop quantum gravity, Phys. Rev., D 73, 044007, [gr-qc/0509118]. [12] Bojowald M, 2006, Loop quantum cosmology and inhomogeneities Gen. Rel. Grav., 38, 1771- 1795, [gr-qc/0609034]. [13] Ashtekar A, Pawlowski T and Singh P, 2006, Quantum Nature of the Big Bang: An Analytical and Numerical Investigation, Phys. Rev. D, 73, 124038, [gr-qc/0604013]. [14] Bojowald M, 2006, Large scale effective theory for cosmological bounces, gr-qc/0608100. [15] Ashtekar A, Pawlowski T and Singh P, 2006, Quantum Nature of the Big Bang: Improved dynamics, Phys. Rev. D, 74, 084003, [gr-qc/0607039]. [16] Ashtekar A, Pawlowski T, Singh P and Vandersloot K, 2006, Loop quantum cosmology of k=1 FRW models [gr-qc/0612104]. [17] Hossain G M, 2005, On Energy Conditions and Stability in Effective Loop Quantum Cosmol- ogy, Class. Quant. Grav., 22, 2653, [gr-qc/0503065]. [18] Bojowald M and Kagan M, 2006, Singularities in Isotropic Non-Minimal Scalar Field Models, Class. Quant. Grav., 23, 4983-4990, [gr-qc/0604105]; Bojowald M and Kagan M, 2006, Loop cosmological implications of a non-minimally coupled scalar field, Phys. Rev. D, 74, 044033, [gr-qc/0606082]. [19] Singh P, Vandersloot K and Vereshchagin G V, 2006, Non-Singular Bouncing Universes in Loop Quantum Cosmology, Phys. Rev. D, 74, 043510 [gr-qc/0606032]. [20] Hossain G M, 2005, Primordial Density Perturbation in Effective Loop Quantum Cosmology, Class. Quant. Grav., 22, 2511, [gr-qc/0411012]. [21] Calcagni G and Cortes M, 2007, Inflationary scalar spectrum in loop quantum cosmology, Class. Quantum Grav., 24, 829, [gr-qc/0607059]. [22] Bojowald M and Skirzewski A, 2006, Effective Equations of Motion for Quantum Systems Rev. Math. Phys., 18, 713-746, [math-ph/0511043]. http://cgpg.gravity.psu.edu/archives/thesis/index.shtml http://arxiv.org/abs/gr-qc/0407074 http://arxiv.org/abs/gr-qc/0502082 http://arxiv.org/abs/gr-qc/0509118 http://arxiv.org/abs/gr-qc/0609034 http://arxiv.org/abs/gr-qc/0604013 http://arxiv.org/abs/gr-qc/0608100 http://arxiv.org/abs/gr-qc/0607039 http://arxiv.org/abs/gr-qc/0612104 http://arxiv.org/abs/gr-qc/0503065 http://arxiv.org/abs/gr-qc/0604105 http://arxiv.org/abs/gr-qc/0606082 http://arxiv.org/abs/gr-qc/0606032 http://arxiv.org/abs/gr-qc/0411012 http://arxiv.org/abs/gr-qc/0607059 http://arxiv.org/abs/math-ph/0511043 [23] Date G, 2005, Absence of the Kasner singularity in the effective dynamics from loop quantum cosmology, Phys. Rev. D, 72, 067301 [gr-qc/0505002]. [24] Chiou D, 2006, Loop Quantum Cosmology in Bianchi Type I Models: Analytical Investigation, [gr-qc/0609029]. [25] Bojowald M, Hernndez H. H, Kagan M, Singh P and Skirzewski A, 2006, Hamiltonian cosmo- logical perturbation theory with loop quantum gravity corrections Phys. Rev. D, 74, 123512, [gr-qc/0609057]; Bojowald M, Hernndez H. H, Kagan M, and Skirzewski A, 2006, Effective constraints of loop quantum gravity, Phys. Rev. D, 74, [gr-qc/0611112]; Bojowald M, Hern- ndez H. H, Kagan M, Singh P and Skirzewski A, 2006, Formation and Evolution of Structure in Loop Cosmology Phys. Rev. Lett., 98, 031301, [astro-ph/0611685]. http://arxiv.org/abs/gr-qc/0505002 http://arxiv.org/abs/gr-qc/0609029 http://arxiv.org/abs/gr-qc/0609057 http://arxiv.org/abs/gr-qc/0611112 http://arxiv.org/abs/astro-ph/0611685 Cosmology, quantum cosmology, loop quantum cosmology Summary of pre 2005 LQC Post 2004 Isotropic LQC Physical quantities and Singularity Resolution Improved Quantization Close Isotropic Model Open Issues and Out look References

0704.0146

Vortices in Bose-Einstein Condensates: Theory

arXiv:0704.0146v1 [cond-mat.other] 2 Apr 2007 Vortices in Bose-Einstein Condensates: Theory N. G. Parker1, B. Jackson2, A. M. Martin1, and C. S. Adams3 1 School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia. [email protected],[email protected] 2 School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom. [email protected] 3 Department of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom. [email protected] 1 Quantized vortices Vortices are pervasive in nature, representing the breakdown of laminar fluid flow and hence playing a key role in turbulence. The fluid rotation associated with a vortex can be parameterized by the circulation Γ = dr · v(r) about the vortex, where v(r) is the fluid velocity field. While classical vortices can take any value of circulation, superfluids are irrotational, and any rotation or angular momentum is constrained to occur through vortices with quan- tized circulation. Quantized vortices also play a key role in the dissipation of transport in superfluids. In BECs quantized vortices have been observed in several forms, including single vortices [1, 2], vortex lattices [3, 4, 5, 6] (see also Chap. VII), and vortex pairs and rings [7, 8, 9]. The recent observation of quantized vortices in a fermionic gas was taken as a clear signature of the underlying condensation and superfluidity of fermion pairs [10]. In addition to BECs, quantized vortices also occur in superfluid Helium [11, 12], nonlinear optics, and type-II superconductors [13]. 1.1 Theoretical Framework Quantization of circulation Quantized vortices represent phase defects in the superfluid topology of the system. Under the Madelung transformation, the macroscopic condensate ‘wavefunction’ ψ(r, t) can be expressed in terms of a fluid density n(r, t) and a macroscopic phase S(r, t) via ψ(r) = n(r, t) exp[iS(r, t)]. In order that the wavefunction remains single-valued, the change in phase around any closed contour C must be an integer multiple of 2π, ∇S · dl = 2πq, (1) http://arXiv.org/abs/0704.0146v1 2 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams where q is an integer. The gradient of the phase S defines the superfluid velocity via v(r, t) = (h̄/m)∇S(r, t). This implies that the circulation about the contour C is given by, v · dl = q . (2) In other words, the circulation of fluid is quantized in units of (h/m). The circulating fluid velocity about a vortex is given by v(r, θ) = qh̄/(mr)θ̂, where r is the radius from the core and θ̂ is the azimuthal unit vector. Theoretical model The Gross-Pitaevskii equation (GPE) provides an excellent description of BECs at the mean-field level in the limit of ultra-cold temperature [14]. It supports quantized vortices, and has been shown to give a good description of the static properties and dynamics of vortices [14, 15]. Dilute BECs require a confining potential, formed by magnetic or optical fields, which typically varies quadratically with position. We will assume an axially-symmetric harmonic trap of the form V = 1 m(ω2rr 2 + ω2zz 2), where ωr and ωz are the radial and axial trap frequencies respectively. Excitation spectra of BEC states can be obtained using the Bogoliubov equations, and specify the stability of station- ary solutions of the GPE. For example, the presence of the so-called anomalous modes of a vortex in a trapped BEC are indicative of their thermodynamic instability. The GPE can also give a qualitative, and sometimes quantitative, understanding of vortices in superfluid Helium [11, 12]. Although this Chapter deals primarily with vortices in repulsively-inte- racting BECs, vortices in attractively-interacting BECs have also received theoretical interest. The presence of a vortex in a trapped BEC with attractive interactions is less energetically favorable than for repulsive interactions [16]. Indeed, a harmonically-confined attractive BEC with angular momentum is expected to exhibit a center-of-mass motion rather than a vortex [17]. The use of anharmonic confinement can however support metastable vortices, as well as regimes of center-of-mass motion and instability [18, 19, 20]. Various approximations have been made to incorporate thermal effects into the GPE to describe vortices at finite temperature (see also Chap. XI). The Popov approximation self-consistently couples the condensate to a normal gas component using the Bogoliubov-de-Gennes formalism [21] (cf. Chap. I Sec. 5.2). Other approaches involve the addition of thermal/quantum noise to the system, such as the stochastic GPE method [22, 23, 24] and the classical field/truncated Wigner methods [25, 26, 27, 28]. Thermal effects can also be simulated by adding a phenomenological dissipation term to the GPE [29]. Basic properties of vortices In a homogeneous system, a quantized vortex has the 2D form, Vortices in Bose-Einstein Condensates: Theory 3 ψ(r, θ) = nv(r) exp(iqθ). (3) The vortex density profile nv(r) has no analytic solution, although approx- imate solutions exist [30]. Vortex solutions can be obtained numerically by propagating the GPE in imaginary time (t→ −it) [31], whereby the GPE con- verges to the lowest energy state of the system (providing it is stable). By en- forcing the phase distribution of Eq. (3), a vortex solution is generated. Figure 1 shows the solution for a q = 1 vortex at the center of a harmonically-confined BEC. The vortex consists of a node of zero density with a width characterized by the condensate healing length ξ = h̄/ mn0g, where g = 4πh̄ 2a/m (with a the s-wave scattering length) and n0 is the peak density in the absence of the vortex. For typical BEC parameters [3], ξ ∼ 0.2 µm. For a q = 1 vortex at the center of an axially-symmetric potential, each particle carries h̄ of angular momentum. However, if the vortex is off-center, the angular momentum per particle becomes a function of position [15]. 1.2 Vortex structures Increasing the vortex charge widens the core due to centrifugal effects. In harmonically-confined condensates a multiply-quantized vortex with q > 1 is energetically unfavorable compared to a configuration of singly-charged vor- tices [32, 33]. Hence, a rotating BEC generally contains an array of singly- charged vortices in the form of a triangular Abrikosov lattice [3, 4, 5, 6, 34] (see also Chap. VII), similar to those found in rotating superfluid helium [11]. A q > 1 vortex can decay by splitting into singly-quantized vortices via a dynamical instability [35, 36], but is stable for some interaction strengths [37]. Multiply-charged vortices are also predicted to be stabilized by a suitable localized pinning potential [38] or the addition of quartic confinement [33]. Two-dimensional vortex-antivortex pairs (i.e. two vortices with equal but opposite circulation) and 3D vortex rings arise in the dissipation of superflow, and represent solutions to the homogeneous GPE in the moving frame [39, 40], with their motion being self-induced by the velocity field of the vortex lines. When the vortex lines are so close that they begin to overlap, these states are no longer stable and evolves into a rarefaction pulse [39]. Having more than one spin component in the BECs (cf. Chap. IX) pro- vides an additional topology to vortex structures. Coreless vortices and vortex ‘molecules’ in coupled two-component BECs have been probed experimentally [41] and theoretically [42]. More exotic vortex structures such as skyrmion ex- citations [43] and half-quantum vortex rings [44] have also been proposed. 2 Nucleation of vortices Vortices can be generated by rotation, a moving obstacle, or phase imprinting methods. Below we discuss each method in turn. 4 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams 2.1 Rotation As discussed in the previous section, a BEC can only rotate through the existence of quantized vortex lines. Vortex nucleation occurs only when the rotation frequency Ω of the container exceeds a critical value Ωc [15, 32, 46]. Consider a condensate in an axially-symmetric trap which is rotating about the z-axis at frequency Ω. In the Thomas-Fermi limit, the presence of a vortex becomes energetically favorable when Ω exceeds a critical value given by [47], 0.67R . (4) This is derived by integrating the kinetic energy density mn(r)v(r)2/2 of the vortex velocity field in the radial plane. The lower and upper limits of the integration are set by the healing length ξ and the BEC Thomas-Fermi radius R, respectively. Note that Ωc < ωr for repulsive interactions, while Ωc > ωr for attractive interactions [16]. In a non-rotating BEC the presence of a vortex raises the energy of the system, indicating thermodynamic instability [48]. In experiments, vortices are formed only when the trap is rotated at a much higher frequency than Ωc [3, 4, 5], demonstrating that the energetic criterion is a necessary, but not sufficient, condition for vortex nucleation. There must also be a dynamic route for vorticity to be introduced into the condensate, and hence Eq. (4) provides only a lower bound for the critical frequency. The nucleation of vortices in rotating trapped BECs appears to be linked to instabilities of collective excitations. Numerical simulations based on the GPE have shown that once the amplitude of these excitations become sufficiently large, vortices are nucleated that subsequently penetrate the high-density bulk of the condensate [23, 27, 29, 49, 50]. One way to induce instability is to resonantly excite a surface mode by adding a rotating deformation to the trap potential. In the limit of small perturbations, this resonance occurs close to a rotation frequency Ωr = ωℓ/ℓ, where ωℓ is the frequency of a surface mode with multipolarity ℓ. In the Thomas-Fermi limit, the surface modes satisfy ωℓ = ℓωr [51], so Ωr = ℓ. For example, an elliptically-deformed trap, which excites the ℓ = 2 quadrupole mode, would nucleate vortices when rotated at Ωr ≈ ωr/ This value has been confirmed in both experiments [3, 4, 5] and numerical simulations [23, 27, 29, 49, 50]. Higher multipolarities were resonantly excited in the experiment of Ref. [6], finding vortex formation at frequencies close to the expected values, Ω = ωr/ ℓ, and lending further support to this picture. A similar route to vortex nucleation is revealed by considering stationary states of the BEC in a rotating elliptical trap, which can be obtained in the Thomas-Fermi limit by solving hydrodynamic equations [52]. At low rotation rates only one solution is found; however at higher rotations (Ω > ωr/ bifurcation occurs and up to three solutions are present. Above the bifurcation point one or more of the solutions become dynamically unstable [53], leading Vortices in Bose-Einstein Condensates: Theory 5 to vortex formation [54]. Madison et al. [55] followed these stationary states experimentally by adiabatically introducing trap ellipticity and rotation, and observed vortex nucleation in the expected region. Surface mode instabilities can also be induced at finite temperature by the presence of a rotating noncondensed “thermal” cloud. Such instabilities occur when the thermal cloud rotation rate satisfies Ω > ωℓ/ℓ [56]. Since all modes can potentially be excited in this way, the criterion for instability and hence vortex nucleation becomes Ωc > min(ωℓ/ℓ), analogous to the Landau criterion. Note that such a minimum exists at Ωc > 0 since the Thomas-Fermi result ωℓ = ℓωr becomes less accurate for high ℓ [57]. This mechanism may have been important in the experiment of Haljan et al. [34], where a vortex lattice was formed by cooling a rotating thermal cloud to below Tc. 2.2 Nucleation by a moving object Vortices can also be nucleated in BECs by a moving localized potential. This problem was originally studied using the GPE for 2D uniform condensate flow around a circular hard-walled potential [58, 59], with vortex-antivortex pairs being nucleated when the flow velocity exceeded a critical value. In trapped BECs a similar situation can be realized using the optical dipole force from a laser, giving rise to a localized repulsive Gaussian potential. Under linear motion of such a potential, numerical simulations revealed vortex pair formation when the potential is moved at a velocity above a critical value [60]. The experiments of [61, 62] oscillated a repulsive laser beam in an elongated condensate. Although vortices were not observed directly, the measurement of condensate heating and drag above a critical velocity was consistent with the nucleation of vortices [63]. An alternative approach is to move the laser beam potential in a circular path around the trap center [64]. By “stirring” the condensate in this way one or more vortices can be created. This technique was used in the experiment of Ref. [6], where vortices were generated even at low stirring frequencies. 2.3 Other mechanisms and structures A variety of other schemes for vortex creation have been suggested. One of the most important is that by Williams and Holland [65], who proposed a combination of rotation and coupling between two hyperfine levels to create a two-component condensate, one of which is in a vortex state. The non- vortex component can then either be retained or removed with a resonant laser pulse. This scheme was used by the first experiment to obtain vortices in BEC [1]. A related method, using topological phase imprinting, has been used to experimentally generate multiply-quantized vortices [66]. Apart from the vortex lines considered so far, vortex rings have also been the subject of interest. Rings are the decay product of dynamically unstable dark solitary waves in 3D geometries [7, 8, 67, 68]. Vortex rings also form 6 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams in the quantum reflection of BECs from surface potentials [69], the unstable motion of BECs through an optical lattice [70], the dragging of a 3D object through a BEC [71], and the collapse of ultrasound bubbles in BECs [72]. The controlled generation of vortex rings [73] and multiple/bound vortex ring structures [74] have been analyzed theoretically. A finite temperature state of a quasi-2D BEC, characterized by the ther- mal activation of vortex-antivortex pairs, has been simulated using classical field simulations [75]. This effect is thought to be linked to the Berezinskii- Kosterlitz-Thouless phase transition of 2D superfluids, recently observed ex- perimentally in ultracold gases [76]. Similar simulations in a 3D system have also demonstrated the thermal creation of vortices [77, 78]. 3 Dynamics of vortices The study of vortex dynamics has long been an important topic in both clas- sical [79] and quantum [12] hydrodynamics. Helmholtz’s theorem for uniform, inviscid fluids, which is also applicable to quantized vortices in superfluids near zero temperature, states that the vortex will follow the motion of the background fluid. So, for example, in a superfluid with uniform flow velocity vs, a single straight vortex line will move with velocity vL, such that it is stationary in the frame of the superfluid. Vortices similarly follow the “background flow” originating from circulat- ing fluid around a vortex core. Hence vortex motion can be induced by the presence of other vortices, or by other parts of the same vortex line when it is curved. Most generally, the superfluid velocity vi due to vortices at a partic- ular point r is given by the Biot-Savart law [12], in analogy with the similar equation in electromagnetism, (s − r) × ds |s− r|3 ; (5) where s(ζ, t) is a curve representing the vortex line with ζ the arc length. Equation (5) suffers from a divergence at r = s, so in calculations of vortex dynamics this must be treated carefully [80]. Equation (5) also assumes that the vortex core size is small compared to the distance between vortices. In particular, it breaks down when vortices cross during collisions, where recon- nection events can occur. These reconnections can either be included manually [81], or by solving the full GPE [82]. The latter method also has the advantage of including sound emission due to vortex motion or reconnections [83, 84]. In a system with multiple vortices, motion of one vortex is induced by the circulating fluid flow around other vortices, and vice-versa [11]. This means that, for example, a pair of vortices of equal but opposite charge will move linearly and parallel to each other with a velocity inversely proportional to the distance between them. Two or more vortices of equal charge, meanwhile, Vortices in Bose-Einstein Condensates: Theory 7 will rotate around each other, giving rise to a rotating vortex lattice as will be discussed in Chap. VII. When a vortex line is curved, circulating fluid from one part of the line can induce motion in another. This effect can give rise to helical waves on the vortex, known as Kelvin modes [85]. It also has interesting consequences for a vortex ring, which will travel in a direction perpendicular to the plane of the ring, with a self-induced velocity that decreases with in- creasing radius. Classically, this is most familiar in the motion of smoke rings, though similar behavior has also been observed in superfluid helium [86]. This simple picture is complicated in the presence of density inhomo- geneities or confining walls. In a harmonically-trapped BEC the density is a function of position, and therefore the energy, E, of a vortex will also de- pend on its position within the condensate. To simplify matters, let us con- sider a quasi-2D situation, where the condensate is pancake-shaped and the vortex line is straight. In this case, the energy of the vortex depends on its displacement r from the condensate center [87], and a displaced vortex feels a force proportional to ∇E. This is equivalent to a Magnus force on the vortex [88, 89, 90] and to compensate the vortex moves in a direction perpendicular to the force, leading it to precess around the center of the condensate along a line of constant energy. This precession of a single vortex has been observed experimentally [2], with a frequency in agreement with theoretical predictions. In more 3D situations, such as spherical or cigar-shaped condensates, the vor- tex can bend [91, 92, 93, 94] leading to more complicated motion [15]. Kelvin modes [95, 96] and vortex ring dynamics [88] are also modified by the density inhomogeneity in the trap. In the presence of a hard-wall potential, a new constraint is imposed such that the fluid velocity normal to the wall must be zero, vs ·n̂ = 0. The resulting problem of vortex motion is usually solved mathematically [79] by invoking an “image vortex” on the other side of the wall (i.e. in the region where there is no fluid present), at a position such that its normal flow cancels that of the real vortex at the barrier. The motion of the real vortex is then simply equal to the induced velocity from the image vortex circulation. 4 Stability of vortices 4.1 Thermal instabilities At finite temperatures the above discussion is modified by the thermal oc- cupation of excited modes of the system, which gives rise to a noncondensed normal fluid in addition to the superfluid. A vortex core moving relative to the normal fluid scatters thermal excitations, and will therefore feel a frictional force leading to dissipation. This mutual friction force can be written as [11], fD = −nsΓ{αs′ × [ s′ × (vn − vL)] + α′s′ × (vn − vL)}, (6) where ns is the background superfluid density, s ′ is the derivative of s with respect to arc length ζ, α and α′ are temperature dependent parameters, 8 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams while vL and vn are the velocities of the vortex line and normal fluid respec- tively. The mutual friction therefore has two components perpendicular to the relative velocity vn − vL. To consider an example discussed in the last section, an off-center vortex in a trapped BEC at zero temperature will precess such that its energy remains constant. In the presence of a non-condensed component, however, dissipation will lead to a loss of energy. Since the vortex is topological it cannot simply vanish, so this lost energy is manifested as a radial drift of the vortex towards lower densities. In Eq. (6) the α term is responsible for this radial motion, while α′ changes the precession frequency. The vortex disappears at the edge of the condensate, where it is thought to decay into elementary excitations [97]. Calculations based upon the stochastic GPE have shown that thermal fluctuations lead to an uncertainty in the position of the vortex, such that even a central vortex will experience thermal dissipation and have a finite lifetime [24]. This thermodynamic lifetime is predicted to be of the order of seconds [97], which is consistent with experiments [1, 3, 94]. 4.2 Hydrodynamic instabilities Experiments indicate that the crystallization of vortex lattices is temperature- independent [5, 98]. Similarly, vortex tangles in turbulent states of superfluid Helium have been observed to decay at ultracold temperature, where thermal dissipation is virtually nonexistent [99]. These results highlight the occurrence of zero temperature dissipation mechanisms, as listed below. Instability to acceleration The topology of a 2D homogeneous superfluid can be mapped on to a (2+1)D electrodynamic system, with vortices and phonons playing the role of charges and photons respectively [100]. Just as an accelerating electron radiates ac- cording to the Larmor acceleration squared law, a superfluid vortex is inher- ently unstable to acceleration and radiates sound waves. Vortex acceleration can be induced by the presence of an inhomogeneous background density, such as in a trapped BEC. Sound emission from a vortex in a BEC can be probed by considering a trap of the form [45], Vext = V0 1 − exp mω2rr 2. (7) This consists of a gaussian dimple trap with depth V0 and harmonic frequency component ωd, embedded in an ambient harmonic trap of frequency ωr. A 2D description is sufficient to describe this effect. This set-up can be realized with a quasi-2D BEC by focussing a far-off-resonant red-detuned laser beam in the center of a magnetic trap. The vortex is initially confined in the inner region, where it precesses due to the inhomogeneous density. Since sound excitations Vortices in Bose-Einstein Condensates: Theory 9 Fig. 1. Profile of a singly-quantized (q = 1) vortex at the center of a harmonically- confined BEC: (a) condensate density along the y = 0 axis (solid line) and the corresponding density profile in the absence of the vortex (dashed line). (b) 2D density and (c) phase profile of the vortex state. These profiles are calculated nu- merically by propagating the 2D GPE in imaginary time subject to an azimuthal 2π phase variation around the trap center. −6 −4 −2 0 2 4 6 x (ξ) (i) (ii) Fig. 2. Vortex path in the dimple trap geometry of Eq. (7) with ωd = 0.28(c/ξ). Deep V0 = 10µ dimple (dotted line): mean radius is constant, but modulated by the sound field. Shallow V0 = 0.6µ dimple and homogeneous outer region ωr = 0 (dot- ted line): vortex spirals outwards. Outer plots: Sound excitations (with amplitude ∼ 0.01n0) radiated in the V0 = 0.6µ system at times indicated. Top: Far-field distri- bution [−90, 90]ξ×[−90, 90]ξ. Bottom: Near-field distribution [−25, 25]ξ×[−25, 25]ξ, with an illustration of the dipolar radiation pattern. Copyright (2004) by the Amer- ican Physical Society [45]. have an energy of the order of the chemical potential µ, the depth of the dimple relative to µ leads to two distinct regimes of vortex-sound interactions. V0 ≫ µ: The vortex effectively sees an infinite harmonic trap - it precesses and radiates sound but there is no net decay due to complete sound reabsorp- tion. However, a collective mode of the background fluid is excited, inducing slight modulations in the vortex path (dotted line in Fig 2). V0 < µ: Sound waves are radiated by the precessing vortex. Assuming ωr = 0, the sound waves propagate to infinity without reinteracting with the vortex. 10 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams The ensuing decay causes the vortex to drift to lower densities, resulting in a spiral motion (solid line in Fig. 2), similar to the effect of thermal dissipation. The sound waves are emitted in a dipolar radiation pattern, perpendicularly to the instantaneous direction of motion (subplots in Fig. 2), with a typical amplitude of order 0.01n0 and wavelength λ ∼ 2πc/ωV [15], where c is the speed of sound and ωV is the vortex precession frequency. The power radiated from a vortex can be expressed in the form [45, 101, 102], P = βmN , (8) where a is the vortex acceleration, N is the total number of atoms, and β is a dimensionless coefficient. Using classical hydrodynamics [101] and by map- ping the superfluid hydrodynamic equations onto Maxwell’s electrodynamic equations [102], it has been predicted that β = π2/2 under the assumptions of a homogeneous 2D fluid, a point vortex, and perfect circular motion. Full numerical simulations of the GPE based on a realistic experimental scenario have derived a coefficient of β ∼ 6.3 ± 0.9 (one standard deviation), with the variation due to a weak dependence on the geometry of the system [45]. When ωr 6= 0, the sound eventually reinteracts with the vortex, slowing but not preventing the vortex decay. By varying V0 it is possible to control vortex decay, and in suitably engineered traps this decay mechanism is expected to dominate over thermal dissipation [45]. Vortex acceleration (and sound emission) can also be induced by the pres- ence of other vortices. A co-rotating pair of two vortices of equal charge has been shown to decay continuously via quadrupolar sound emission, both an- alytically [103] and numerically [104]. Three-body vortex interactions in the form of a vortex-antivortex pair incident on a single vortex have also been sim- ulated numerically, with the interaction inducing acceleration in the vortices with an associated emission of sound waves [104]. Simulations of vortex lattice formation in a rotating elliptical trap show that vortices are initially nucleated in a turbulent disordered state, before relaxing into an ordered lattice [50]. This relaxation process is associated with an exchange of energy from the sound field to the vortices due to these vortex-sound interactions. This agrees with the experimental observation that vortex lattice formation is insensitive to temperature [5, 98]. Kelvin wave radiation and vortex reconnections In 3D a Kelvin wave excitation will induce acceleration in the elements of the vortex line, and therefore local sound emission. Indeed, simulations of the GPE in 3D have shown that Kelvin waves excitations on a vortex ring lead to a decrease in the ring size, indicating the underlying radiation process [84]. Kelvin wave excitations can be generated from a vortex line reconnection [83, 84] and the interaction of a vortex with a rarefaction pulse [105]. Vortices in Bose-Einstein Condensates: Theory 11 Vortex lines which cross each other can undergo dislocations and reconnec- tions [106], which induce a considerable burst of sound emission [83]. Although they have yet to be probed experimentally in BECs, vortex reconnections are hence thought to play a key role in the dissipation of vortex tangles in Helium II at ultra-low temperatures [11]. 5 Dipolar BECs A BEC has recently been formed of chromium atoms [107], which feature a large dipole moment. This opens the door to studying of the effect of long- range dipolar interactions in BECs. 5.1 The Modified Gross-Pitaevskii Equation The interaction potential Udd(r) between two dipoles separated by r, and aligned by an external field along the unit vector ê is given by, Udd(r) = êiêj (δij − 3r̂ir̂j) . (9) For low energy scattering of two atoms with dipoles induced by a static electric field E = Eê, the coupling constant Cdd = E 2α2/ǫ0 [108, 109], where α is the static dipole polarizability of the atoms and ǫ0 is the permittivity of free space. Alternatively, if the atoms have permanent magnetic dipoles, dm, aligned in an external magnetic field B = Bê, one has Cdd = µ0d m [110], where µ0 is the permeability of free space. Such dipolar interactions give rise to a mean-field potential Φdd(r) = d3rUdd (r − r′) |ψ (r′) |2, (10) which can be incorporated into the GPE to give, ih̄ψt = ∇2 + g|ψ|2 + Φdd + V ψ. (11) For an axially-symmetric quasi-2D geometry (ωz ≫ ωr) rotating about the z -axis, the ground state wavefunction of a single vortex has been solved numer- ically [111]. Considering 105 chromium atoms and ωr = 2π × 100Hz, several solutions were obtained depending on the strength of the s-wave interactions and the alignment of the dipoles relative to the trap. For the case of axially-polarized dipoles the most striking results arise for attractive s-wave interactions g < 0. Here the BEC density is axially symmetric and oscillates in the vicinity of the vortex core. Similar density oscillations have been observed in numerical studies of other non-local inter- action potentials, employed to investigate the interparticle interactions in 4He 12 N. G. Parker, B. Jackson, A. M. Martin, and C. S. Adams [112, 113, 114, 115], with an interpretation that relates to the roton structure in a superfluid [115]. For the case of transversely-polarized dipoles, where the polarizing field is co-rotating with the BEC, and repulsive s-wave interactions (g > 0), the BEC becomes elongated along the axis of polarization [116] and as a consequence the vortex core is anisotropic. 5.2 Vortex Energy Assuming a dipolar BEC in the TF limit (cf. Sec. 5.1 in Chap. I), the en- ergetic cost of a vortex, aligned along the axis of polarization (z-axis), has been derived using a variational ansatz for the vortex core [117], and thereby the critical rotation frequency Ωc at which the presence of a vortex becomes energetically favorable has been calculated. For an oblate trap (ωr < ωz), dipolar interactions decrease Ωc, while for prolate traps (ωr > ωz) the pres- ence of dipolar interactions increases Ωc. A formula resembling Eq. (4) for the critical frequency of a conventional BEC can be used to explain these results, with R being the modified TF radius of the dipolar BEC. Indeed, using the TF radius of a vortex-free dipolar BEC [118, 119] and the conven- tional s-wave healing length ξ, it was found that Eq. (4) closely matches the results from the energy cost calculation. Deviations become significant when the dipolar interactions dominate over s-wave interactions. In this regime the s-wave healing length ξ is no longer the relevant length scale of the system, and the equivalent dipolar length scale ξd = Cddm/(12πh̄ ) will characterize the vortex core size. For g > 0 and in the absence of dipolar interactions, the rotation frequency at which the vortex-free BEC becomes dynamically unstable, Ωdyn, is always greater than the critical frequency for vortex stabilization Ωc. However in the presence of dipolar interactions, Ωdyn can become less than Ωc, leading to an intriguing regime in which the dipolar BEC is dynamically unstable but vortices will not enter [117, 120]. As with attractive condensates [17], the angular momentum may then be manifested as center of mass oscillations. 6 Analogs of Gravitational Physics in BECs There is growing interest in pursuing analogs of gravitational physics in con- densed matter systems [121], such as BECs. The rationale behind such models can be traced back to the work of Unruh [122, 123], who noted the analogy between sound propagation in an inhomogeneous background flow and field propagation in curved space-time. This link applies in the TF limit of BECs where the speed of sound is directly analogous to the speed of light in the corresponding gravitational system [124]. This has led to proposals for exper- iments to probe effects such as Hawking radiation [125, 126] and superradiance [127]. For Hawking radiation it is preferable to avoid the generation of vortices [121, 128], and as such will not be discussed here. However, the phenomena Vortices in Bose-Einstein Condensates: Theory 13 of superradiance in BECs, which can be considered as stimulated Hawking radiation, relies on the presence of a vortex [129, 130, 131, 132], which is analogous to a rotating black hole. Below we outline the derivation of how the propagation of sound in a BEC can be considered to be analogous to field propagation [121]. From the GPE it is possible to derive the continuity equation for an irrotational fluid flow with phase S(r, t) and density n(r, t), and a Hamilton-Jacobi equation whose gradient leads to the Euler equation. Linearizing these equations with respect to the background it is found that ′ = − 1 ∇S · ∇S′ − gn′ + h̄ , (12) ′ = − 1 ∇ · (n∇S′) − 1 ∇ · (n′∇S) , (13) where n′ and S′ are the perturbed values of the density n and phase S respec- tively. Neglecting the quantum pressure ∇2-terms, the above equations can be rewritten as a covariant differential equation describing the propagation of phase oscillations in a BEC. This is directly analogous to the propagation of a minimally coupled massless scalar field in an effective Lorentzian geometry which is determined by the background velocity, density and speed of sound in the BEC. Hence, the propagation of sound in a BEC can be used as an analogy for the propagation of electromagnetic fields in the corresponding space-time. Of course one has to be aware that this direct analogy is only valid in the TF regime, which breaks down on scales of the order of a healing length, i.e. the theory is only valid on large length scales, as is general relativity. 6.1 Superradiance Superradiance in BECs relies on sound waves incident on a vortex structure and is characterized by the reflected sound energy exceeding the incident energy. This has been studied using Eqs. (12) and (13) for monochromatic sound waves of frequency ωs and angular wave number qs incident upon a vortex [129] and a ‘draining vortex” (a vortex with outcoupling at its center) [130, 131, 132]. For the vortex case, a vortex velocity field v(r, θ) = (β/r)θ̂ and a density profile ansatz was assumed. Superradiance then occurs when βqs > Ac∞, where A is related to the vortex density ansatz and c∞ is the speed of sound at infinity [129]. Interestingly, this condition is frequency independent. For the case of a draining vortex, an event horizon occurs at a distance a from the vortex core, where the fluid circulates at frequency Ω. Assuming a homogeneous density n and a velocity profile v(r, θ) = −car̂ +Ωa2θ̂ where c is the homogeneous speed of sound, superradiance occurs when 0 < ωs < qsΩ [130, 131, 132]. The increase in energy of the outgoing sound is due to an extraction of energy from the vortex and as such it is expected to lead to slowing of the 14 N. G. Parker, B. Jackson, A. M. 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0704.0147

A POVM view of the ensemble approach to polarization optics

arXiv:0704.0147v2 [physics.optics] 20 Jun 2007 A POVM view of the ensemble approach to polarization optics Sudha,1 A.V. Gopala Rao,2 A. R. Usha Devi,3 and A.K. Rajagopal4 1Department of P.G. Studies in Physics, Kuvempu University, Shankaraghatta-577 451, India∗ 2Department of Studies in Physics, Manasagangothri, University of Mysore, Mysore 570 006, India 3 Department of Physics, Jnanabharathi Campus, Bangalore University, Bangalore-560 056, India 4 Department of Computer Science and Center for Quantum Studies, George Mason University, Fairfax, VA 22030, USA and Inspire Institute Inc., McLean, VA 22101, USA Statistical ensemble formalism of Kim, Mandel and Wolf (J. Opt. Soc. Am. A 4, 433 (1987)) offers a realistic model for characterizing the effect of stochastic non-image forming optical media on the state of polarization of transmitted light. With suitable choice of the Jones ensemble, various Mueller transforma- tions - some of which have been unknown so far - are deduced. It is observed that the ensemble approach is formally identical to the positive operator val- ued measures (POVM) on the quantum density matrix. This observation, in combination with the recent suggestion by Ahnert and Payne (Phys. Rev. A 71, 012330, (2005)) - in the context of generalized quantum measurement on single photon polarization states - that linear optics elements can be employed in setting up all possible POVMs, enables us to propose a way of realizing different types of Mueller devices. c© 2018 Optical Society of America OCIS codes: 030.0030, 230.0230. ∗Corresponding author: [email protected] http://arxiv.org/abs/0704.0147v2 1. Introduction The intensity and polarization of a beam of light passing through an isolated optical de- vice undergoes a linear transformation. But this is an ideal situation because, in general, the optical system is embedded in some media such as atmosphere or other ambient mate- rial, which further modifies the polarization properties of the light beam passing through it. A statistical ensemble model describing random linear optical media was formulated two decades ago by Kim, Mandel and Wolf [1], but is not examined in any detail in the literature, to the best of our knowledge. The purpose of the present paper is to pursue this avenue in a new way arising from the realization of a relationship, presented here, with the positive operator valued measures (POVM) of quantum measurement theory. This is because the transformation of the polarization states of a light beam propagating through an ensemble of deterministic optical devices exhibits a structural similarity with the POVM transformation of quantum density matrices. This connection motivates, in view of the re- cent interest in the implementations of POVMs on single photon density matrix employing linear optics elements [2], identification of experimental schemes to realize various kinds of Muller transformations. The properties of the transformation of the polarization states of light form a much studied topic in literature [3 – 17]. Thus the power of the ensemble approach becomes evident in elucidating the known optical devices as well as some hitherto unknown types [17], which had remained only a mathematical possibility. The contents of this paper are organized as follows. In Sec. 2, a concise formulation of the Jones and Mueller matrix theory, along with a summary of main results of Gopala Rao et al. [17] is given. Based on the approach of Kim, Mandel and Wolf [1] suitable Jones en- sembles, corresponding to various types of Mueller transformations are identified in Sec. 3. In Sec. 4, a structural equivalence between Jones ensemble and POVMs of quantum mea- surement theory is established. Following the linear optics scheme of Ahnert and Payne [2] for the implementation of POVMs on single photon density matrix, experimental setup for realizing Mueller matrices of types I and II are suggested in Sec. 5. The final section has some concluding remarks. 2. Brief summary of known results on the Jones and the Mueller formalism. Following the standard procedure, let E1 and E2, defined here as a column matrix E = , denote two components of the transverse electric field vector associated with a light beam. The coherency matrix (or the polarization matrix) of the light beam is a positive semidefinite 2x2 hermitian matrix defined by, C = 〈E⊗E†〉. (1) Expressing this in terms of the standard Pauli matrices σ1 =  , σ3 =  and the unit matrix σ0 = , we have siσi = s0 + s3 s1 − is2 s1 + is2 s0 − s3  (2) The physical significance of the quantities arising here are s0 = Tr (Cσ0) = Intensity of the beam si = Tr (Cσi) = Components of Polarization vector ~s of the beam Thus the coherency matrix completely specifies the physical properties of the light beam. The four-vector S =  defined by Eq. (3) is the well known Stokes vector, which represents the state of polarization of the light beam. Because C is hermitian, the Stokes vector is real. The positive semidefiniteness of C implies that the Stokes vector must satisfy the properties s0 > 0, s 0 − |~s|2 ≥ 0 (4) A 2x2 complex matrix J, called the Jones matrix, represents the so-called deterministic optical device [18] or medium. When a light beam represented by E passes through such a medium, the transformed light beam is given by E′ = JE. Correspondingly, the coherency matrix C transforms as ′ = JCJ† (5) (Here J† is the hermitian conjugate of J.) Alternatively, instead of the 2 × 2 matrix transformation of the coherency matrix, as given by Eq. (5), a transformation ′ = MS (6) of the four componental Stokes column S through a real 4x4 matrix M, called the Mueller matrix, is found be more useful [18]. Using Eq. (3) and Eq. (5 we have, s′i = Tr(C ′σi) = Tr(JCJ † σi) = Tr(J†σiJσj)sj which leads to the well-known relationship [1] Mij = Tr(J†σiJσj) between the elements of a Jones matrix and that of corresponding Mueller matrix. But in the case where medium cannot be represented by a Jones matrix, it is not possible to characterize the change in the state of polarization of the light beam through Eq. (5). In such a situation, Mueller formalism provides a general approach for the polarization transformation of the light beam. The Mueller matrix M is said to be non-deterministic when it has no corresponding Jones characterization. Mathematically, a Mueller device can be represented by any 4 × 4 matrix such that the Stokes parameters of the outgoing light beam satisfy the physical constraint Eq. (4). In other words, a Mueller matrix is any 4×4 real matrix that transforms a Stokes vector into another Stokes vector. There are many aspects of the relationships between these two formulations of the polarization optics and a complete characterization of Mueller matrices has been the subject matter of Ref. [1, 3-17]. It was Gopala Rao et al. [17] who presented a complete set of necessary and sufficient conditions for any 4x4 real matrix to be a Mueller matrix. In so doing, they found that there are two algebraic types of Mueller matrices called type I and type II; and it has been shown [17] that only a subset of the type-I Mueller matrices - called deterministic or pure Mueller matrices - have corresponding Jones characterization. All the known polarizing optical devices such as retarders, polarizers, analyzers, optical rotators are pure Mueller type and are well understood. Mueller matrices of the Type II variety are yet to be physically realized and have remained as mere mathematical possibility. For the sake of completeness, we present here the characterization as well as categorization of these two types of Mueller matrices as is given in Ref. [17]. This will enable us to show that both Type I and II Mueller devices are realizable in an unified manner in terms of the proposed ensemble approach [1]. I. A 4× 4 real matrix M is called a type-I Mueller matrix iff (i) M00 ≥ 0 (ii) The G-eigenvalues ρ0, ρ1, ρ2, ρ3 of the matrix N=M̃GM are all real. (Here, M̃ stands for the transpose of M; G-eigenvalues are the eigenvalues of the matrix GN, with G = diag(1, −1, −1, −1)). iii) The largest G-eigenvalue ρ0 possesses a time-like G-eigenvector and the G-eigenspace of N contains one time-like and three space-like G-eigenvectors. II. A 4× 4 real matrix M is called a type-II Mueller matrix iff (i) M00 > 0. (ii) The G-eigenvalues ρ0, ρ1, ρ2, ρ3 of N=M̃GM are all real. (iii) The largest G-eigenvalue ρ0 possesses a null G-eigenvector and the G-eigenspace of N contains one null and two space-like G-eigenvectors. (iv) If X0 = e0 + e1 is the null G-eigenvector of N such that e0 is a time-like vector with positive zeroth component, e1 is a space-like vector G-orthogonal to e0 then ẽ0Ne0 > 0. Despite the knowledge of these new category of Mueller matrices [15, 17], not much attention is paid for realizing the corresponding devices. An experimental arrangement involving a parallel combination of deterministic (pure Mueller) optical devices is proposed in Ref. [17] for realizing type-II Mueller devices. The physical situations, where the beam of light is subjected to the influence of a medium such as atmosphere was addressed in Ref. [1]. In the next section, we discuss this ensemble approach for random optical media, proposed by Kim, Mandel and Wolf [1] . 3. Mueller matrices as ensemble of Jones devices Kim et. al. [1] associate a set of probabilities {pe, pe = 1} to describe the stochastic medium. Then a Jones device Je associated with each element e of the ensemble gives a corresponding coherency matrix C′e = JeCJ e. The ensemble averaged coherency matrix Cav = pe(JeCJ e) (7) then describes the effects of the medium on the beam of light. In a similar fashion, the corresponding ensemble of Mueller matrices {Me} associated with the ensemble of Jones matrices {Je} is constructed and its ensemble averaged Mueller matrix is similarly formed as Mav = e peMe. Since a linear combination of Mueller matrices with non-negative coefficients is also a Mueller matrix, the ensemble averaged Mueller matrix Mav is a Mueller matrix 1. We now turn to the question of constructing an appropriate ensemble designed to describe a given physical situation. The simplest example of an ensemble is one where the elements are chosen entirely randomly, i.e., the system is described by a chaotic ensemble where the probabilities are all equal, pe = , where n denotes the number of elements in the ensemble. The coherency matrix Cav of the light beam passing through such a chaotic assembly is just an arithmetic average of the coherency matrices C′e = JeCJ e and hence Cav = e (8) More general models can be constructed depending on the medium for the propagation of the beam of light. For example, one may employ various types of filters or solid state systems through which the light passes; the assignment of the Jones matrices and the corresponding probabilities will then differ depending on the weights placed on these elements. Restricting ourselves to an ensemble consisting of only two Jones devices which occur with equal probability p1 = 1/2, p2 = 1/2, we have found out that the resultant Mueller matrices can either be deterministic or non-deterministic. We give in the foregoing (see Table I) some examples of Mueller matrices corresponding to different choices of Jones matrices in an ensemble Je, e = 1, 2, for some representative cases. This will also serve to show the 1 This is because, each Mueller matrix M transforms an initial Stokes vector into a final Stokes vector and a linear combination of Stokes vectors with non-negative coefficients p is again a Stokes vector. generality of the ensemble procedure in capturing the physical realizations for the Mueller devices discussed in Ref. [17]. Table 1. Mueller matrices resulting from 2-element Jones ensemble. J1 J2 M = p1M1 + p2M2, Type of M p1 = p2 = 1. 1√ 1 1− i 1 + i −1 1 1− i 1 + i −1 3 0 0 0 0 −1 2 2 0 2 −1 2 0 2 2 −1 Pure Mueller 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 Type-I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 Type-I 4. 1√ 3 0 0 0 0 −1 0 2 0 0 −1 0 0 2 0 −1 Type-I 1 1− i 1 + i −1 5 0 0 0 0 −1 2 4 0 2 −3 2 0 4 2 −1 Type-I 1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Type-II 2 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 Type-II In Table I, the Jones matrices chosen are so as to give pure Mueller (deterministic) and non-deterministic type-I, type-II matrices respectively. We observe that an assembly of Jones matrices can result in a pure Mueller matrix if and only if all elements of the assembly correspond to the same optical device. This is because, with all Je’s are same, a transformation of the form Cav = e pe(JeCJ e) is equivalent to a transformation of the Stokes vector S through a Mueller matrix Mav = peMe = Mpure. When the medium is represented by a pure Mueller matrix, the outgoing light beam will have the same degree of polarization as the incoming light beam. In fact, pure Mueller matrix is the simplest among type-I Mueller matrices. Not all type-I Mueller matrices preserve the degree of polarization of the incident light beam. To see this, note that the type-I matrix of example 2 (see Table I) converts any incident light beam into a linearly polarized light beam; the other three type-I matrices (examples 3 to 5) transform completely polarized light beams into partially polarized light beams. Similarly, type-II Mueller matrices do not, in general, preserve the degree of polarization of the incident light beam. It may be seen that the type-II Mueller matrix of example 7 is a depolarizer matrix, since it converts any incident light beam into an unpolarized light beam. Though one cannot a priori state which choices of Jones matrices result in type-I or type- II, it is interesting to observe that all types of Mueller matrices result - even in 2-element ensembles. It is not difficult to conclude that an ensemble, with more Jones devices and with different weight factors, can give rise to a variety of Mueller matrices of all possible algebraic types. It would certainly be interesting to physically realize such systems. In the following section, a connection between the ensemble approach for optical devices and the POVMs of quantum measurement theory is established. 4. A connection to Positive Operator Valued Measures We will now show that the phenomenology of the ensemble construction of Kim, Mandel and Wolf [1] described above has a fundamental theoretical underpinning, if we make a formal identification of the coherency matrix with the density matrix description of the subsystem of a composite quantum system. The coherency matrix defined by Eqs. (1) and (2) resembles a quantum density matrix in that both describe a physical system by a hermitian, trace-class, and positive semi-definite matrix. While the quantum density matrix has unit trace, the coherency matrix has intensity of the beam as the value of the trace. The Jones matrix transformation is a general transformation of the coherency matrix, which preserves its hermiticity and positive semi-definiteness - but changes the values of the elements of the coherency matrix. The most general transformation of the density matrix ρ, which preserves its hermiticity, positive semi-definiteness and also the unit trace is the positive operator valued measures (POVM) [19]: iVi = I (9) where Vi’s are general matrices and I is the unit element in the Hilbert space. More generally, one could relax the condition of preservation of the unit trace of the density matrix by examining the possibility of a contracting transformation, where the unit matrix condition on the POVM operators is replaced by an inequality. This mathematical theorem has a physical basis in the Kraus operator formalism [19] when we consider the Hamiltonian description of a composite interacting system A, B described by a density matrix ρ(A, B) and deduce the subsystem density matrix of A given by, ρ(A) = TrB ρ(A, B). In this case, the Kraus operators are the explicit expressions of the POVM operators and contain the effects of interaction between the systems A and B in the description of the subsystem A. It is thus clear that the phenomenology of Ref. [1] has a correspondence with the Kraus formulation and the POVM theory. In order to make this association complete, we compare Eq. (9) with the expression given by Eq. (7). Apart from a phase factor, the Kraus operators {Vi}, associated with POVMs, may be related to the Jones assembly {Ji}, chosen in the form iVi = iJi (10) In the construction of the Table I presented earlier, a simple model was proposed where all probabilities were chosen to be equal and the condition on the sum over the Jones matrix combinations was set equal to unit matrix. In such cases, the intensity of the beam gets reduced by 1/n and the polarization properties of the beam gets changed as was described earlier. With this identification, we have provided here an important interpretation and meaning to the phenomenology of the ensemble approach of Kim et al.[1]. Recently Ahnert and Payne [2] proposed an experimental scheme to implement all possible POVMs on single photon polarization states using linear optical elements. In view of the connection between the ensemble formalism for Jones and Mueller matrices with the POVMs, a possible experimental realization of the two types of Mueller matrices is suggested in the next section. 5. Possible experimental realization of types I and II Mueller matrices. We first observe that the density matrix of a single photon polarization state, ρ = ρHH |H〉 〈H|+ ρHV |H〉 〈V |+ ρ∗HV |V 〉 〈H|+ ρV V |V 〉 〈V | (11) is nothing but the coherency matrix of the photon [20] 〈â†H âH〉 〈â H âV 〉 〈â†V âH〉 〈â V âV 〉  , (12) where âH and âV are the creation operators of the polarization states of the single photon; {|H〉, |V 〉} denote the transverse orthogonal polarization states of photon. This is seen explicitly by noting that the average values of the Stokes operators are obtained as, s0 = 〈Ŝ0〉 = 〈(â†H âH + â V âV )〉 = ρHH + ρV V = Tr(ρ), s1 = 〈Ŝ1〉 = 〈(â†H âV + â V âH)〉 = ρHV + ρ∗HV = Tr(ρ σ1), s2 = 〈Ŝ2〉 = i 〈(â†V âH − â H âV )〉 = i (ρHV − ρ∗HV ) = Tr(ρ σ2), s3 = 〈Ŝ3〉 = 〈(â†H âH − â V âV )〉 = ρHH − ρV V = Tr(ρ σ3). (13) Hence the proposed setup [2], involving only linear optics elements such as polarizing beam splitters, rotators and phase shifters, that promises to implement all possible POVMs on a single photon polarization state leads to all possible ensemble realizations for the Mueller matrices. More specifically, this provides a general experimental scheme to realize varieties of Mueller matrices - including the hitherto unreported type-II Mueller matrices. We briefly describe the scheme proposed in Ref. [2] and illustrate, by way of examples, how it leads to both type-I and type-II Mueller matrices. In Ref. [2], a module corresponds to an arrangement having polarization beam splitters, polarization rotators, phase shifters and unitary operators. For an n element POVM, a setup involving n − 1 modules are needed. That means, a single module is enough for a 2 element POVM; a setup involving two modules is required for a 3 element POVM and so on. We describe two, three element POVMs by specifying the optical elements in the respective modules and by specifying the corresponding Kraus operators in terms of these elements. For any two operator POVM, the Kraus operators V1, V2 are given by V1 = U and V2 = U ′′D2U. Here U, U ′, U′′ are the three unitary operators in a single module. Denoting θ, φ as the angles of rotation of the two variable polarization rotators and γ, ξ, the angles of the two variable phase shifters in the module, the diagonal matrices D1, D2 are given by, eiγ cos θ 0 0 cos φ  , D2 = eiξ sin θ 0 0 sin φ  (14) The POVM elements F1 = V 1V1 = U 1D1U, F2 = V 2V2 = U 2D2U (15) satisfy the condition i=1,2 Fi = F1 + F2 = I. For any three operator POVM, the Kraus operators are given by V1 = U IDIUI, V2 = U IIDIIUIIU V3 = U IIUIIU Here, the diagonal D matrices are eiγI cos θI 0 0 cosφI  , D′I = eiξI sin θI 0 0 sinφI  (17) DII = eiγII cos θII 0 0 cos φII  , D′II = eiξII sin θII 0 0 sinφII  (18) (θI, φI), (γI, ξI) are respectively the pair of angles corresponding to variable polarization rotators and variable phase shifters in the first module. Similarly, (θII, φII), (γII, ξII) are the pairs of angles corresponding to variable polarization rotators and variable phase shifters respectively in the second module. UI, U I are the unitary operators used in the first module and UII, U II, U II are the unitary operators used in the second module. (Notice that all the unitary operators in the above schemes are arbitrary and a particular choice of the associated unitary operators gives rise to a different experimental arrangement). The extension of this scheme to n operator POVM involving n-1 modules is quite similar and is given in [2]. We had identified, in Sec. 3, that an ensemble average of Jones devices will lead to all possible types of Mueller matrices, some examples of which are given in Table 1. We now show that the experimental set up proposed in Ref. [2] can also be used to realize varieties of Mueller devices. To substantiate our claim, we identify here the linear optical elements needed in the single module set up of Ahnert and Payne [2], which lead to the physical realization of two typical Mueller matrices given in Table 1. To obtain the type-I Mueller matrix M= 1  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1  of example 3 (see Table I), we use U = I, U′ =  and U′′ =  as the required unitary Jones devices and both the variable polarization rotators are set with their rotation angles θ=φ = π/4. There is no need of phase shifter devices in this case i.e, γ=ξ=0. Similarly for the type-II Mueller matrix M= 1  2 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0  of example 7, we find that U = I, U′ = 1√  andU′′ = 1√  are the required unitary Jones devices. The rotation angles of the variable polarization rotators are, as in the earlier case, θ=φ = π/4 and there is no need of phase shifter devices i.e., γ=ξ=0. Notice that in both the above examples the unitary operators U′, U′′ correspond to linear and circular retarders [18]. These two examples illustrate that the experimental set up given in Ref. [2] may be utilized to realize the required non-determinisitc Mueller devices. In fact Mueller matri- ces corresponding to an ensemble with more than two Jones devices may also be realized by employing larger number of modules as given in the experimental scheme proposed by Ref. [2]. 6. Conclusion We have established here a connection between the phenomenological ensemble approach [1] for the coherency matrix and the POVM transformation of quantum density matrix. This opens up a fresh avenue to physically realize types I and II of the Mueller matrix classification of Ref. [17]. We have also given experimental setup to implement Mueller transformations corresponding to ensemble average of Jones devices by employing the POVM scheme on the single photon density matrix suggested in Ref. [2], in the context of quantum measurement theory. It is gratifying to note that two decades after the introduction of the ensemble approach, which had remained obscure and only received passing reference in textbooks such as [20], its value is revealed in this paper through its connection with the new developments in quantum measurement theory. We plan on exploring further the POVM transformation in the description of quantum polarization optics. References 1. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media”, J. Opt. Soc. Am. A 4, 433–437 (1987). 2. S. E. Ahnert and M. C. Payne, “General implementation of all possible positive- operator-value measurements of single photon polarization states”, Phys. Rev. A 71, 012330-33, (2005). 3. R. Barakat, “Bilinear constraints between elements of the 4× 4 Mueller-Jones transfer matrix of polarization theory”, Opt. Commun. 38,159–161 (1981). 4. R. Simon, “The connection between Mueller and Jones matrices of Polarization Op- tics”, Opt. Commun. 42, 293–297 (1982). 5. A. B. Kostinski, B. James, and W. M. Boerner, “Optimal reception of partially polar- ized waves” J. Opt. Soc. Am. A 5, 58–64 (1988). 6. A. B. Kostinski, Depolarization criterion for incoherent scattering” Appl. Optics 31, 3506–3508 (1992). 7. J. J. Gil, and E. Bernabeau, “A depolarization criterion in Mueller matrices” Optica Acta, 32, 259–261 (1985). 8. R. Simon,“ Mueller matrices and depolarization criteria” J. Mod. Optics 34, 569–575 (1987). 9. R. Simon, “Non-depolarizing systems and degree of polarization” Opt. Commun. 77, 349–354 (1990) 10. M. Sanjay Kumar, and R. Simon, “Characterization of Mueller matrices in Polarizatio Optics”, Optics Commun. 88, 464–470 (1992). 11. R. Sridhar and R. Simon, “Normal form for Mueller matrices in Polarization Optics” J. Mod. Optics 41, 1903–1915 (1994). 12. D. G. M. Anderson, and R. Barakat,“Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix” J. Opt. Soc. Am. A 11, 2305–2319 (1994). 13. C. V. M. van der Mee, and J. W. Hovenier, “Structure of matrices transforming Stokes parameters”, J. Math. Phys. 33, 3574–3584 (1992). 14. C. R. Givens, and A. B. Kostinski, “A simple necessary and sufficient criterion on physically realizable Mueller matrices”, J. Mod. Opt. 40, 471–481 (1993). 15. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes pa- rameters”, J. Math. Phys. 34, 5072–5088 (1993). 16. S. R. Cloude, “Group Theory and Polarization algebra”, Optik 75, 26–36 (1986). 17. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix” J. Mod. Optics, 45, 955–987 (1998). 18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized light, (North Holland Publishing Co., Amsterdam, 1977) 19. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2002). 20. L. Mandel and E. Wolf, Quantum Coherence and quantum optics, (Cambridge Univer- sity Press, Cambridge, 1995).

0704.0148

Reexamination of spin decoherence in semiconductor quantum dots from equation-of-motion approach

Reexamination of spin decoherence in semiconductor quantum dots from equation-of-motion approach J. H. Jiang,1, 2 Y. Y. Wang,2 and M. W. Wu1, 2, ∗ Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui, 230026, China Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China (Dated: November 2, 2018) The longitudinal and transversal spin decoherence times, T1 and T2, in semiconductor quantum dots are investigated from equation-of-motion approach for different magnetic fields, quantum dot sizes, and temperatures. Various mechanisms, such as the hyperfine interaction with the surrounding nuclei, the Dresselhaus spin-orbit coupling together with the electron–bulk-phonon interaction, the g-factor fluctuations, the direct spin-phonon coupling due to the phonon-induced strain, and the coaction of the electron–bulk/surface-phonon interaction together with the hyperfine interaction are included. The relative contributions from these spin decoherence mechanisms are compared in detail. In our calculation, the spin-orbit coupling is included in each mechanism and is shown to have marked effect in most cases. The equation-of-motion approach is applied in studying both the spin relaxation time T1 and the spin dephasing time T2, either in Markovian or in non-Markovian limit. When many levels are involved at finite temperature, we demonstrate how to obtain the spin relaxation time from the Fermi Golden rule in the limit of weak spin-orbit coupling. However, at high temperature and/or for large spin-orbit coupling, one has to use the equation-of-motion approach when many levels are involved. Moreover, spin dephasing can be much more efficient than spin relaxation at high temperature, though the two only differs by a factor of two at low temperature. PACS numbers: 72.25.Rb, 73.21.La,71.70.Ej I. INTRODUCTION One of the most important issues in the growing field of spintronics is quantum information processing in quantum dots (QDs) using electron spin.1,2,3,4,5 A main obstacle is that the electron spin is unavoidably coupled to the environment (such as, the lattice) which leads to considerable spin decoherence (including lon- gitudinal and transversal spin decoherences).6,7 Vari- ous mechanisms, such as, the hyperfine interaction with the surrounding nuclei,8,9 the Dresselhaus/Rashba spin- orbit coupling (SOC)10,11 together with the electron- phonon interaction, g-factor fluctuations,12 the direct spin-phonon coupling due to the phonon-induced strain,9 and the coaction of the hyperfine interaction and the electron-phonon interaction can lead to the spin deco- herence. There are quite a lot of theoretical works on spin decoherence in QD. Specifically, Khaetskii and Nazarov analyzed the spin-flip transition rate using a perturbative approach due to the SOC together with the electron-phonon interaction, g-factor fluctuations, the di- rect spin-phonon coupling due to the phonon-induced strain qualitatively.13,14,15 After that, the longitudinal spin decoherence time T1 due to the Dresslhaus and/or the Rashba SOC together with the electron-phonon in- teraction were studied quantitatively in Refs. 16,17,18,19, 20,21,22,23,24,25,26. Among these works, Cheng et al.18 developed an exact diagonalization method and showed that due to the strong SOC, the previous perturbation method14,15,16 is inadequate in describing T1. Further- more, they also showed that, the perturbation method previously used missed an important second-order en- ergy correction and would yield qualitatively wrong re- sults if the energy correction is correctly included and only the lowest few states are kept as those in Refs. 14,15,16. These results were later confirmed by Deste- fani and Ulloa.21 The contribution of the coaction of the hyperfine interaction and the electron-phonon interac- tion to longitudinal spin decoherence was calculated in Refs. 27 and 28. In contrast to the longitudinal spin de- coherence time, there are relatively fewer works on the transversal spin decoherence time, T2, also referred to as the spin dephasing time (while the longitudinal spin de- coherence time is referred to as the spin relaxation time for short). The spin dephasing time due to the Dressel- haus and/or the Rashba SOC together with the electron- phonon interaction was studied by Semenov and Kim29 and by Golovach et al..20 The contributions of the hyper- fine interaction and the g-factor fluctuation were studied in Refs. 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44 and in Ref. 45 respectively. However, a quantitative calcula- tion of electron spin decoherence induced by the direct spin-phonon coupling due to phonon-induced strain in QDs is still missing. This is one of the issues we are going to present in this paper. In brief, the spin re- laxation/dephasing due to various mechanisms has been studied previously in many theoretical works. However, almost all of these works only focus individually on one mechanism. Khaetskii and Nazarov discussed the ef- fects of different mechanisms on the spin relaxation time. Nevertheless, their results are only qualitative and there is no comparison of the relative importance of the dif- http://arxiv.org/abs/0704.0148v5 ferent mechanisms.13,14,15 Recently, Semenov and Kim discussed various mechanisms contributed to the spin dephasing,46 where they gave a “phase diagram” to in- dicate the most important spin dephasing mechanism in Si QD where the SOC is not important. However, the SOC is very important in GaAs QDs. To fully under- stand the microscopic mechanisms of spin relaxation and dephasing, and to achieve control over the spin coherence in QDs,47,48,49 one needs to gain insight into the relative importance of each mechanism to T1 and T2 under vari- ous conditions. This is one of the main purposes of this paper. Another issue we are going to address relates to differ- ent approaches used in the study of the spin relaxation time. The Fermi-Golden-rule approach, which is widely used in the literature, can be used in calculation of the re- laxation time τi→f between any initial state |i〉 and final state |f〉.12,13,14,15,16,17,18,19,21,23,24,25,27,28,50,51,52 How- ever, the problem is that when the process of the spin relaxation relates to many states, (e.g., when tempera- ture is high, the electron can distribute over many states), one should find a proper way to average over the relax- ation times (τi→f ) of the involved processes to give the total spin relaxation time (T1). What makes it difficult in GaAs QDs, is that all the states are impure spin states with different expectation values of spin. In the existing literature, spin relaxation time is given by the average of the relaxation times of processes from the initial state |i〉 to the final state |f〉 (with opposite majority spin of |i〉) weighted by the distribution of the initial states fi, 18,51,52 i.e., T−11 = i→f . (1) This is a good approximation in the limit of small SOC as each state only carries a small amount of minority spin. However, when the SOC is very strong which hap- pens at high levels, it is difficult to find the proper way to perform the average. We will show that Eq. (1) is not adequate any more. Thus, to investigate both T1 and T2 at finite temperature for arbitrary strength of SOC, we develop an equation-of-motion approach for the many-level system via projection operator technique56 in the Born approximation. With the rotating wave ap- proximation, we obtain a formal solution to the equation of motion. By assuming a proper initial distribution, we can calculate the evolution of the expectation value of spin. We thus obtain the spin relaxation/dephasing time by the 1/e decay of the expectation value of spin operator 〈Sz〉 or |〈S+〉| (to its equilibrium value), with S+ ≡ Sx+ iSy. With this approach, we are able to study spin relaxation/dephasing for various temperature, SOC strength, and magnetic field. For quantum information processing based on electron spin in QDs, the quantum phase coherence is very im- portant. Thus, the spin dephasing time is a more rel- evant quantity. There are two kinds of spin dephasing times: the ensemble spin dephasing time T ∗2 and the irreversible spin dephasing time T2. For a direct mea- surement of an ensemble of QDs58 or an average over many measurements at different times where the config- urations of the environment have been changed,59,60,61 it gives the ensemble spin dephasing time T ∗2 . The ir- reversible spin dephasing time T2 can be obtained by spin echo measurement.60,61 A widely discussed source which leads to both T ∗2 and T2 is the hyperfine interac- tion between the electron spin and the nuclear spins of the lattice. As it has been found that T ∗2 is around 10 ns, which is too short and makes a practical quantum infor- mation processing difficult in electron spin based qubits in QDs. Thus a spin echo technique is needed to remove the free induction decay and to elongate the spin dephas- ing time. Fortunately, this technique has been achieved first by Petta et al. for two electron triplet-singlet system and then by Koppens et al. for a single electron spin sys- tem. The achieved spin dephasing time is ∼ 1 µs, which is much longer than T ∗2 . We therefore discuss only the ir- reversible spin dephasing time T2 throughout the paper, i.e., we do not consider the free induction decay in the hyperfine-interaction-induced spin dephasing. It is further noticed that Golovach et al. have shown that the spin dephasing time T2 is two times the spin relaxation time T1. 20 However, as temperature increases, this relation does not hold. Semenov and Kim on the other hand reported that the spin dephasing time is much smaller than the spin relaxation time.29 In this paper, we calculate the temperature dependence of the ratio of the spin relaxation time to the spin dephasing time and analyze the underlying physics. This paper is organized as follows: In Sec. II, we present our model and formalism of the equation-of- motion approach. We also briefly introduce all the spin decoherence mechanisms considered in our calculations. In Sec. III we present our numerical results to indicate the contribution of each spin decoherence mechanism to spin relaxation/dephasing time under various conditions based on the equation-of-motion approach. Then we study the problem of how to obtain the spin relaxation time from the Fermi Golden rule when many levels are involved in Sec. IV. The temperature dependence of T1 and T2 is investigated in Sec. V. We conclude in Sec. VI. II. MODEL AND FORMALISM A. Model and Hamiltonian We consider a QD system, where the QD is confined by a parabolic potential Vc(x, y) = m∗ω20(x 2 + y2) in the quantum well plane. The width of the quantum well is a. The external magnetic field B is along z direction, except in Sec. IV. The total Hamiltonian of the system of electron together with the lattice is: HT = He +HL +HeL , (2) where He, HL, HeL are the Hamiltonians of the elec- tron, the lattice and their interaction, respectively. The electron Hamiltonian is given by + Vc(r) +HZ +HSO (3) where P = −i~∇+ e A with A = (B⊥/2)(−y,x) (B⊥ is the magnetic field along z direction), HZ = gµBB · σ is the Zeeman energy with µB the Bohr magneton, and HSO is the Hamiltonian of SOC. In GaAs, when the quantum well width is small or the gate-voltage along the growth direction is small, the Rashba SOC is unimportant.53 Therefore, only the Dresselhaus term10 contributes to HSO. When the quantum well width is smaller than the QD radius, the dominant term in the Dresselhaus SOC reads Hso = 〈P 2z 〉λ0(−Pxσx + Pyσy) , (4) with γ0 denoting the Dresselhaus coefficient, λ0 being the quantum well subband index of the lowest one and 〈P 2z 〉λ ≡ −~2 ψ∗zλ(z)∂ 2/∂z2ψzλ(z)dz. The Hamiltonian of the lattice consists of two parts HL = Hph +Hnuclei, where Hph = qη ~ωqηa qηaqη (a †/a is the phonon cre- ation/annihilation operator) describes the vibration of the lattice and Hnuclei = j γIB · Ij (γI is the gyro- magnetic ratios of the nuclei and Ij is the spin of the j- th nucleus) describes the precession of the nuclear spins of the lattice in the external magnetic field. We focus on the spin dynamics due to hyperfine interaction at a time scale much smaller than the nuclear dipole-dipole correlation time (10−4 s in GaAs33,40), where the nuclear dipole-dipole interaction can be ignored. Under this ap- proximation, the equation of motion for the reduced elec- tron system can be obtained which only depends on the initial distribution of the nuclear spin bath.33 The in- teraction between the electron and the lattice also has two parts HeL = HeI + He−ph, where HeI is the hy- perfine interaction between the electron and nuclei and He−ph represents the electron-phonon interaction which is further composed of the electron–bulk-phonon (BP) interaction Hep, the direct spin-phonon coupling due to the phonon-induced strain Hstrain and phonon-induced g-factor fluctuation Hg. B. Equation-of-motion approach The equations of motion can describe both the co- herent and the dissipative dynamics of the electron sys- tem. When the quasi-particles of the bath relax much faster than the electron system, the Markovian approx- imation can be made; otherwise the kinetics is the non- Markovian. For electron-phonon coupling, due to the fast relaxation of the phonon bath and the weak electron- phonon scattering, the kinetics of the electron is Marko- vian. Nevertheless, as the nuclear spin bath relaxes much slower than the electron spin, the kinetics due to the coupling with nuclei is of non-Markovian type.30,32,33 It is further noted that there is also a contribution from the coaction of the electron-phonon and electron-nuclei couplings, which is a fourth order coupling to the bath. For this contribution, the decoherence of spin is mainly controlled by the electron-phonon scattering while the hyperfine (Overhauser) field54 acts as a static magnetic field. Thus, this fourth order coupling is also Markovian. Finally, since the electron orbit relaxation is much faster than the electron spin relaxation,55 we always assume a thermo-equilibrium initial distribution of the orbital de- grees of freedom. Generally, the interaction between the electron and the quasi-particle of the bath is weak. Therefore the first Born approximation is adequate in the treatment of the interaction. Under this approximation, the equation of motion for the electron system coupled to the lattice envi- ronment can be obtained with the help of the projection operator technique.56 We then assume a sudden approxi- mation so that the initial distribution of the whole system is ρ(t = 0) = ρe(0) ⊗ ρL(0), where ρe, ρL is the density matrix of the system and bath respectively. This approx- imation corresponds to a sudden injection of the electron into the quantum dot, which is reasonable for genuine experimental setup.33 As the initial distribution of the the lattice ρL(0) commutates with the Hamiltonian of the lattice HL, the equation of motion can be written as dρe(t) = − i [He +TrL(HeLρ L(0)), ρe(t)] dτTrL[HeL, U0(τ)(P̂ [HeL, ρe(t− τ) ⊗ρL(0)])U †0 (τ)] , (5) where ρe(t) is the density operator of the electron sys- tem at time t, TrL stands for the trace over the lattice degree of freedom, and U0(τ) = e −i(HL+He)τ is time- evolution operator without HeL. P̂ = 1̂ − ρL(0) ⊗ TrL is the projection operator. The initial distribution of the phonon system is chosen to be the thermo-equilibrium distribution.20 It has been shown by previous theoretical studies that the initial state of the nuclear spin bath is crucial to the spin dephasing and relaxation.30,32,33 Al- though it may take a long time (e.g., seconds) for the nuclear spin system to relax to its thermo-equilibrium state, one can still assume that its initial state is the thermo-equilibrium one. This assumption corresponds to the genuine case of enough long waiting time during every individual measurement. For a typical setup at above 10 mK and with about 10 T external magnetic field, the thermo-equilibrium distribution is a distribu- tion with equal probability on every state. For these ini- tial distributions of phonons and nuclear spins, the term TrL(HeLρ L(0)) is zero. Thus, P̂ [HeL, ρe(t−τ)⊗ρL(0)] = [HeL, ρe(t−τ)⊗ρL(0)] . (6) The equation of motion is then simplified to, dρe(t) = − i [He, ρ e(t)] − 1 dτTrL[HeL, [H eL(−τ), Ue0 (t)ρ Ie(t− τ)Ue0 (t)ρL(0)]] , (7) where HIeL and ρ Ie are the corresponding operators (HeL and ρe ) in the interaction picture, and Ue0 (t) = e −iHet is the time-evolution operator of He. It should be fur- ther noted that the first Born approximation can not fully account for the non-Markovian dynamics due to the hyperfine interaction with nuclear spins.33,57 Only when the Zeeman splitting is much larger than the fluctuating Overhauser shift, the first Born approximation is ade- quate. For GaAs QDs, this requires B ≫ 3.5 T.33 In this paper, we focus on the study of spin dephasing for the high magnetic field regime of B > 3.5 T under the first Born approximation, where the second Born approxima- tion only affects the long-time behavior.33 Later we will argue that this correction of long time dynamics changes the spin dephasing time very little. 1. Markovian kinetics The kinetics due to the coupling with phonons can be investigated within the Markovian approximation, where the equation of motion reduces to, dρe(t) = − i [He, ρ e(t)]− 1 dτTrph[He−ph, [HIe−ph(−τ), ρe(t)⊗ ρph(0)]] . (8) Here Trph is the trace over phonon degrees of freedom and ρph(0) is the initial distribution of the phonon bath. Within the basis of the eigen-states of the electron Hamil- tonian, {|ℓ〉}, the above equation reads, ρeℓ1ℓ2= −i (εℓ1 − εℓ2) ρeℓ1ℓ2 Trp(H I e−ph ρeℓ4ℓ2 ⊗ ρ −HI e−phℓ1ℓ3 ρ ⊗ ρpeqH ) +H.c. . (9) Here H = 〈ℓ1|He−ph|ℓ3〉 and HI e−phℓ1ℓ3 = 〈ℓ1|HIe−ph(−τ)|ℓ3〉. A general form of the electron- phonon interaction reads He−ph = Φqη(aqη + a −qη)Xqη(r,σ) . (10) Here, η represents the phonon branch index; Φqη is the matrix element of the electron-phonon interaction; aqη is the phonon annihilation operator; Xqη(r,σ) denotes a function of electron position and spin. Substituting this into Eq. (9), we obtain, after integration within the Markovian approximation,49 ρeℓ1ℓ2 = i (εℓ1 − εℓ2) ρeℓ1ℓ2 |Φqη|2{Xqηℓ1ℓ3X ρeℓ4ℓ2 ×Cqη(εℓ4 − εℓ3)−X ρeℓ3ℓ4 ×Cqη(εℓ3 − εℓ1)} +H.c. in which X = 〈ℓ1|Xqη(r,σ)|ℓ2〉, and Cqη(∆ε) = n̄(ωqη)δ(∆ε+ωqη)+[n̄(ωqη)+1]δ(∆ε−ωqη). Here n̄(ωqη) represents the Bose distribution function. Equation (11) can be written in a more compact form ρeℓ1ℓ2 = − Λℓ1ℓ2ℓ3ℓ4ρ , (12) which is a linear differential equation. This equation can be solved by diagonalizing Λ. Given an initial distribu- tion ρeℓ1ℓ2(0), the density matrix ρ (t) and the expec- tation value of any physical quantity 〈O〉t = Tr(Ôρe(t)) at time t can be obtained:49 〈O〉t = Tr(Ôρe) ℓ1···ℓ6 〈ℓ2|Ô|ℓ1〉P(ℓ1ℓ2)(ℓ3ℓ4) × e−Γ(ℓ3ℓ4)tP−1 (ℓ3ℓ4)(ℓ5ℓ6) ρeℓ5ℓ6(0) (13) with Γ = P−1ΛP being the diagonal matrix and P repre- senting the transformation matrix. To study spin dynam- ics, we calculate 〈Sz〉t (|〈S+〉t|) and define the spin relax- ation (dephasing) time as the time when 〈Sz〉t (|〈S+〉t|) decays to 1/e of its initial value (to its equilibrium value). 2. Non-Markovian kinetics Experiments have already shown that for a large en- semble of quantum dots or for an ensemble of many mea- surements on the same quantum dot at different times, the spin dephasing time due to hyperfine interaction is quite short, ∼ 10 ns.58,59,60,61 This rapid spin dephas- ing is caused by the ensemble broadening of the preces- sion frequency due to the hyperfine fields.40 When the external magnetic field is much larger than the random Overhauser field, the rotation due to the Overhauser field perpendicular to the magnetic field is blocked. Only the broadening of the Overhauser field parallel to the magnetic field contribute to the spin dephasing. To de- scribe this free induction decay for this high magnetic field case, we write the hyperfine interaction into two parts: HeI = h · S = HeI1 +HeI2. Here h = (hx, hy, hz) and S = (Sx, Sy, Sz) are the Overhauser field and the electron spin respectively. HeI1 = hzSz and HeI2 = (h+S− + h−S+) with h± = hx ± ihy. The longitudi- nal part HeI1 is responsible for the free induction de- cay, while the transversal part HeI2 is responsible for high order irreversible decay. As the rapid free induction decay can be removed by spin echo,60,61 elongating the spin dephasing time to ∼ 1 µs which is more favorable for quantum computation and quantum information pro- cessing, we then discuss only the irreversible decay. We first classify the states of the nuclear spin system with its polarization. Then we reconstruct the states within the same class to make it spatially uniform. These uniformly polarized pure states, |n〉’s, are eigen-states of hz. They also form a complete-orthogonal basis of the nuclear spin system. A formal expression of |n〉 is33 |n〉 = m1···mN αnm1···mN |I,mj〉 . (14) Here |I,mj〉 denotes the eigen-state of the z-component of the j-th nuclear spin Ijz with the eigenvalue ~mj. N denotes the number of the nuclei. The equation of motion for the case with initial nuclear spin state ρns1 (0) = |n〉〈n| is given by33 dρe(t) [He +Trns(HeIρ 1 (0)), ρ e(t)] dτTrns[HeI2, U 0 (τ) ×[HeI2, ρe(t− τ)⊗ ρns1 (0)]UeI0 (τ)] . (15) As in traditional projection operator technique, the dy- namics of the nuclear spin subsystem is incorporated self-consistently in the last term.33,56 Here Trns is the trace over nuclear spin degrees of freedom. UeI0 (τ) = exp[−iτ(He+HI +HeI1)]. The Overhauser field is given by h = j Av0Ijδ(r − Rj), where the constants A and v0 are given later. Ij and Rj are the spin and posi- tion of j-th nucleus respectively. As mentioned above, the initial state of the nuclear spin bath is chosen to be a state with equal probability of each state, therefore ρns(0) = n 1/Nw|n〉〈n|, with Nw = n 1 being the number of states of the basis {|n〉}. To quantify the ir- reversible decay, we calculate the time evolution of S for every case with initial nuclear spin state |n〉. We then sum over n and obtain ||〈S+〉t|| = |〈S(n)+ 〉t|. (16) It is noted that the summation is performed after the absolute value of 〈S(n)+ 〉t. Therefore, the destructive in- terference due to the difference in precession frequency ωzn, which originates from the longitudinal part of the hyperfine interaction (HeI1), is removed. We thus use 1/e decay of the envelope of ||〈S+〉t|| to describe the irre- versible spin dephasing time T2. Similar description has been used in the irreversible spin dephasing in semicon- ductor quantum wells62 and the irreversible inter-band optical dephasing in semiconductors.63,64 Expanding Eq. (15) in the basis of {|n〉}, one obtains, ρeℓ1ℓ2= − (εℓ1δℓ1ℓ3 +H nℓ1;nℓ3 )ρeℓ3ℓ2 −ρeℓ1ℓ3(εℓ3δℓ3ℓ2 +H nℓ3;nℓ2 [HeI2nℓ1;n1ℓ3H I eI2 n1ℓ3;nℓ4 ρeℓ4ℓ2(t− τ) −HI eI2nℓ1;n1ℓ3ρ (t− τ)HeI2n1ℓ4;nℓ2 ] +H.c. . (17) Here HeI2nℓ1;n1ℓ3 = 〈nℓ1|HeI2|n1ℓ3〉 and H I eI2 nℓ1;n1ℓ3 〈nℓ1|HIeI2(−τ)|n1ℓ3〉. For simplicity, we neglect the terms concerning different orbital wavefunctions which are much smaller. For small spin mixing, assuming an equilibrium distribution in orbital degree of freedom, un- der rotating wave approximation, and trace over the or- bital degree of freedom, we finally arrive at 〈S(n)+ 〉t = iωzn〈S + 〉t − fk([h+]knn′ × [h−]kn′n + [h−]knn′ [h+]kn′n) × exp[iτ(ωkn − ωkn′)]}〈S(n)+ 〉t−τ . (18) Here ωzn = k fk(Ezk/~ + ωkn) with Ezk representing the electron Zeeman splitting of the k-th orbital level. [hi]knn′ = 〈n|〈k|hi|k〉|n′〉 (i = ±, z). ωkn = [hz]knn + ǫnz with ǫnz denoting the nuclear Zeeman splitting which is very small and can be neglected. By solving the above equation, we obtain |〈S(n)+ 〉t| for a given |n〉. We then sum over n and determine the irreversible spin dephasing time T2 as 1/e decay of the envelop of ||〈S+〉t||. By noting that only the polarization of nuclear spin state |n〉 determines the evolution of |〈S(n)+ 〉t|, the summation over n is then reduced to summation over polarization which becomes a integration for large N . This integration can be handled numerically. In the limiting case of zero SOC and very low tem- perature, only the lowest two Zeeman sublevels are con- cerned. The equation for 〈S+〉t with initial nuclear spin state ρns1 (0) = |n〉〈n| reduce to 〈S+〉t = iωzn〈S+〉t − ([h+]nn′ × [h−]n′n + [h−]nn′ [h+]n′n) exp[iτ(ωn − ωn′)]}〈S+〉t−τ = iωz〈S+〉t − dτΣ(τ)〈S+〉t−τ . (19) In this equation ωzn = (gµBB + [hz]nn′)/~, [hξ]nn′ = 〈n|〈ψ1|hξ|ψ1〉|n′〉 (ξ = ±, z and ψ1 is the orbital quantum number of the ground state), and ωn = [hz]nn. Similar equation has been obtained by Coish and Loss,33 and later by Deng and Hu35 at very low temperature such that only the lowest two Zeeman sublevels are considered. Coish and Loss also presented an efficient way to evaluate Σ(τ) in terms of their Laplace transformations, Σ(s) = dτe−sτΣ(τ). They gave, Σ(s) = ([h+]nn′ [h−]n′n + [h−]nn′ [h+]n′n)/(s− iδωnn′) , (20) with δωnn′ = (ωn − ωn′). With the help of this tech- nique, we are able to investigate the spin dephasing due to the hyperfine interaction. C. Spin decoherence mechanisms In this subsection we briefly summarize all the spin de- coherence mechanisms. It is noted that the SOC modifies all the mechanisms. This is because the SOC modifies the Zeeman splitting18 and the spin-resolved eigen-states of the electron Hamiltonian, it hence greatly changes the effect of the electron-BP scattering.18 These two modifi- cations, especially the modification of the Zeeman split- ting, also change the effect of other mechanisms, such as, the direct spin-phonon coupling due to the phonon- induced strain, the g-factor fluctuation, the coaction of the electron-phonon interaction and the hyperfine inter- action. In the literature, except for the electron-BP scat- tering, the effects from the SOC are neglected except for the work by Woods et al.16 in which the spin relaxation time between the two Zeeman sub-levels of the lowest electronic state due to the phonon-induced strain is in- vestigated. However, the perturbation method they used does not include the important second-order energy cor- rection. In our investigation, the effects of the SOC are included in all the mechanisms and we will show that they lead to marked effects in most cases. 1. SOC together with electron-phonon scattering As the SOC mixes different spins, the electron-BP scat- tering can induce spin relaxation and dephasing. The electron-BP coupling is given by Hep = Mqη(aqη + a −qη)e iq·r , (21) where Mqη is the matrix element of the electron-phonon interaction. In the general form of the electron phonon interaction He−ph, Φqη = Mqη and Xqη(r,σ) = e iq·r. |Mqsl|2 = ~Ξ2q/2ρvslV for the electron-BP coupling due to the deformation potential. For the piezoelectric cou- pling, |Mqpl|2 = (32~π2e2e214/κ2ρvslV )[(3qxqyqz)2/q7] for the longitudinal phonon mode and j=1,2 |Mqptj |2 = [32~π2e2e214/(κ 2ρvstq 5V )][q2xq y + q z + q (3qxqyqz) 2/q2] for the two transverse modes. Here Ξ stands for the acoustic deformation potential; ρ is the GaAs volume density; V is the volume of the lattice; e14 is the piezoelectric constant and κ denotes the static dielectric constant. The acoustic phonon spectra ωqql = vslq for the longitudinal mode and ωqpt = vstq for the transverse mode with vsl and vst representing the corresponding sound velocities. Besides the electron-BP scattering, electron also cou- ples to vibrations of the confining potential, i.e., the surface-phonons,28 δV (r) = − 2ρωqηV (aqη + a −qη)ǫqη · ∇rVc(r) , in which ǫqη is the polarization vector of a phonon mode with wave-vector q in branch η. However, this contri- bution is much smaller than the electron-BP coupling. Compared to the coupling due to the deformation poten- tial for example, the ratio of the two coupling strengths is ≈ ~ω0/Ξql0 , where l0 is the characteristic length of the quantum dot and ~ω0 is the orbital level splitting. The phonon wave-vector q is determined by the energy differ- ence between the final and initial states of the transition. Typically phonon transitions between Zeeman sublevels and different orbital levels, ql0 ranges from 0.1 to 10. Bearing in mind that ~ω0 is about 1 meV while Ξ = 7 eV in GaAs, ~ω0/Ξql0 is about 10 −3. The piezoelectric cou- pling is of the same order as the deformation potential. Therefore spin decoherence due to the electron–surface- phonon coupling is negligible. 2. Direct spin-phonon coupling due to phonon-induced strain The direct spin-phonon coupling due to the phonon- induced strain is given by65 Hstrain = s(p) · σ , (23) where hsx = −Dpx(ǫyy − ǫzz), hsy = −Dpy(ǫzz − ǫxx) and hsz = −Dpz(ǫxx− ǫyy) with p = (px, py, pz) = −i~∇ and D being the material strain constant. ǫij (i, j = x, y, z) can be expressed by the phonon creation and annihilation operators: ǫij = qη=l,t1,t2 2ρωqηV (aq,η + a −q,η)(ξiηqj + ξjηqi)e iq·r , (24) in which ξil = qi/q for the longitudinal phonon mode and (ξxt1 , ξyt1 , ξzt1) = (qxqz, qyqz,−q2‖)/qq‖, (ξxt2 , ξyt2 , ξzt2) = (qy,−qx, 0)/q‖ for the two trans- verse phonon modes with q‖ = q2x + q y . There- fore, in the general form of electron-phonon interaction He−ph, Φqη = −iD ~/(32ρωqηV ) and Xqη(r,σ) = ijk ǫijk(ξjηqj − ξkηqk)pieiq·rσi with ǫijk denoting the Levi-Civita tensor. 3. g-factor fluctuation The spin-lattice interaction via phonon modulation of the g-factor is given by12 ijkl=x,y,z AijklµBBiσjǫkl , (25) where ǫkl is given in Eq. (24) and Aijkl is a tensor determined by the material. Therefore in He−ph, Φqη = i ~/(32ρωqηV ) and Xqη(r,σ) = i,j,k,l Ai,j,k,lµBBi(ξkηqk − ξlηql)σjeiq·r. Due to the axial symmetry with respect to the z-axis, and keep- ing in mind that the external magnetic field is along the z direction, the only finite element of Hg is Hg = [(A33−A31)ǫzz+A31 i ǫii]~µBBσz/2 with A33 = Azzzz , A31 = Azzxx and A66 = Axyxy. A33 + 2A31 = 0. 4. Hyperfine interaction The hyperfine interaction between the electron and nu- clear spins is66 HeI(r) = Av0S · Ijδ(r−Rj) , (26) where S = ~σ/2 and Ij are the electron and nucleus spins respectively, v0 = a 0 is the volume of the unit cell with a0 representing the crystal lattice parameter, and r (Rj) denotes the position of the electron (the j-th nu- cleus). A = 4µ0µBµI/(3Iv0) is the hyperfine coupling constant with µ0, µB and µI representing the perme- ability of vacuum, the Bohr magneton and the nuclear magneton separately. As the Zeeman splitting of the electron is much larger (three orders of magnitude larger) than that of the nu- cleus spin, to conserve the energy for the spin relax- ation processes, there must be phonon-assisted transi- tions when considering the spin-flip processes. Tak- ing into account directly the BP induced motion of nu- clei spin of the lattice leads to a new spin relaxation mechanism:28 eI−ph(r) = − Av0S · Ij(u(R0j) ·∇r)δ(r−Rj) , (27) where u(R0j) = ~/(2ρωqηv0)(aqη + a qη)ǫqηe iq·R0 is the lattice displacement vector. Therefore using the notation of Eq. (10), Φ = ~/(2ρV ωqη) and Xqη = j Av0S·Ij∇rδ(r−Rj). The second-order process of the surface phonon and the BP together with the hyperfine interaction also leads to spin relaxation: eI−ph(r) = |ℓ2〉 m 6=ℓ1 〈ℓ2|δVc(r)|m〉〈m|HeI (r)|ℓ1〉 εℓ1 − εm m 6=ℓ2 〈ℓ2|HeI(r)|m〉〈m|δVc(r)|ℓ1〉 εℓ2 − εm 〈ℓ1| , (28) eI−ph = |ℓ2〉 m 6=ℓ1 〈ℓ2|Hep|m〉〈m|HeI(r)|ℓ1〉 εℓ1 − εm m 6=ℓ2 〈ℓ2|HeI(r)|m〉〈m|Hep|ℓ1〉 εℓ2 − εm 〈ℓ1| , (29) in which |ℓ1〉 and |ℓ2〉 are the eigen states ofHe. By using the notations in He−ph, Φqη = ~/(2ρωqηv0) and Xqη = |ℓ2〉ǫqη · m 6=ℓ1 εℓ1 − εm 〈ℓ2|[He,P]|m〉 × 〈m|S · Ijδ(r−Rj)|ℓ1〉+ m 6=ℓ2 εℓ2 − εm 〈m|[He,P]|ℓ1〉 Av0〈ℓ2|S · Ijδ(r−Rj)|m〉 〈ℓ1| (30) for V eI−ph. Similarly Φqη =Mqη and Xqη = |ℓ2〉 m 6=ℓ1 〈ℓ2|eiq·r|m〉 εℓ1 − εm Av0〈m|S · Ij × δ(r−Rj)|ℓ1〉+ m 6=ℓ2 εℓ2 − εm 〈m|eiq·r|ℓ1〉 Av0〈ℓ2|S · Ijδ(r−Rj)|m〉 〈ℓ1| (31) for V eI−ph. Again as the contribution from the surface phonon is much smaller than that of the BP, V eI−ph can be neglected. It is noted that, the direct spin-phonon coupling due to the phonon-induced strain together with the hyperfine interaction gives a fourth-order scattering and hence induces a spin relaxation/dephasing. The in- teraction is eI−ph = |ℓ2〉 m 6=ℓ1 〈ℓ2|Hzstrain|m〉〈m|HeI(r)|ℓ1〉 εℓ1 − ǫm m 6=ℓ2 〈ℓ2|HeI(r)|m〉〈m|Hzstrain|ℓ1〉 ǫℓ2 − ǫm 〈ℓ1| , (32) with Hzstrain = h sσz/2 only changing the electron energy but conserving the spin polarization. It can be written hzs = − 2ρωq,ηV (ξyηqy−ξzηqz)qzeiq·r . (33) Comparing this to the electron-BP interaction Eq. (21), the ratio is ≈ ~Dq/Ξ, which is about 10−3. Therefore, the second-order term of the direct spin-phonon coupling due to the phonon-induced strain together with the hy- perfine interaction is very small and can be neglected. Also the coaction of the g-factor fluctuation and the hy- perfine interaction is very small compared to that of the electron-BP interaction jointly with the hyperfine inter- action as µBB/Ξ is around 10 −5 when B = 1 T. There- fore it can also be neglected. In the following, we only retain the first and the third order terms V eI−ph and eI−ph in calculating the spin relaxation time. The spin dephasing time induced by the hyperfine in- teraction can be calculated from the non-Markovian ki- netic Eq. (18), for unpolarized initial nuclear spin state |n0〉, resulting in 〈S(n0)+ 〉t ∝ dr|ψk(r)|4 cos( |ψk(r)|2t) , where fk is the thermo-equilibrium distribution of the orbital degree of freedom. When only the lowest two Zeeman sublevels are considered, assuming a simple form of the wavefunction, |Ψ(r)|2 = 1 exp(−r2 /d20) with d‖/az representing the QD diameter/quantum well width, and r‖ = x 2 + y2, the integration can be carried 〈S(n0)+ 〉t ∝ cos(t/t0)− 1 (t/t0)2 sin(t/t0) . (35) Here, t0 = (2πazd ‖)/(Av0) determines the spin dephas- ing time. Note that t0 is proportional to the factor azd where az/d is the characteristic length/area of the QD along z direction / in the quantum well plane. By solving Eq. (18) for various n, and summing over n, we obtain ||〈S+〉t|| = n |〈S + 〉t|. We then define the time when the envelop of ||〈S+〉t|| decays to 1/e of its initial value as the spin dephasing time T2. As mentioned above the hyperfine interaction can not transfer an energy of the order of the Zeeman splitting, thus the hyperfine inter- action alone can not lead to any spin relaxation.43 In the above discussion, the nuclear spin dipole-dipole interaction is neglected. Recently, more careful exami- nations based on quantum cluster expansion method or pair correlationmethod have been performed.41,42,43,47 In these works, the nuclear spin dipole-dipole interaction is also included. This interaction together with the hyper- fine mediated nuclear spin-spin interaction is the origin of the fluctuation of the nuclear spin bath. To the lowest order, the fluctuation is dominated by nuclear spin pair flips.41,42,43,47 This fluctuation provides the source of the electron spin dephasing, as the electron spin is coupled to the nuclear spin system via hyperfine interaction. Our method used here includes only the hyperfine interaction to the second order in scattering. However, it is found that the dipole-dipole-interaction–induced spin dephas- ing is much weaker than the hyperfine interaction for a QD with a = 2.8 nm and d0 = 27 nm until the par- allel magnetic field is larger than ∼ 20 T.42 Therefore, for the situation in this paper, the nuclear dipole-dipole- interaction–induced spin dephasing can be ignored.67 III. SPIN DECOHERENCE DUE TO VARIOUS MECHANISMS Following the equation-of-motion approach developed in Sec. II, we perform a numerical calculation of the spin relaxation and dephasing times in GaAs QDs. Two mag- netic field configurations are considered: i.e., the mag- netic fields perpendicular and parallel to the well plane (along x-axis). The temperature is taken to be T = 4 K unless otherwise specified. For all the cases we con- sidered in this manuscript, the orbital level splitting is larger than an energy corresponding to 40 K. Therefore, the lowest Zeeman sublevels are mainly responsible for the spin decoherence. When calculating T1, the initial distribution is taken to be in the spin majority down state of the eigen-state of the Hamiltonian He with a Maxwell-Boltzmann distribution fk = C exp[−ǫk/(kBT )] for different orbital levels (C is the normalization con- stant). For the calculation of T2, we assign the same distribution between different orbital levels, but with a superposition of the two spin states within the same or- bital level. The parameters used in the calculation are listed in Table I.8,68,69 TABLE I: Parameters used in the calculation ρ 5.3× 103 kg/m3 κ 12.9 vst 2.48 × 10 3 m/s g −0.44 vsl 5.29 × 10 3 m/s Ξ 7.0 eV e14 1.41 × 10 9 V/m m∗ 0.067m0 A 90 µeV A33 19.6 γ0 27.5 Å 3·eV I 3 D 1.59 × 104 m/s a0 5.6534 Å A. Spin Relaxation Time T1 We now study the spin relaxation time and show how it changes with the well width a, the magnetic field B and the effective diameter d0 = ~π/m∗ω0. We also compare the relative contributions from each relaxation mechanism. 1. Well width dependence In Fig. 1(a) and (b), the spin relaxation times induced by different mechanisms are plotted as function of the width of the quantum well in which the QD is confined for perpendicular magnetic field B⊥ = 0.5 T and parallel magnetic field B‖ = 0.5 T respectively. We first concen- trate on the perpendicular magnetic field case. In Fig. 1(a), the calculation indicates that the spin relaxation due to each mechanism decreases with the increase of well g-factor strain eI−ph eI−ph B⊥ = 0.5 T a (nm) 1098765432 10−10 B‖ = 0.5 T a (nm) 1098765432 10−10 FIG. 1: (Color online) T−11 induced by different mechanisms vs. the well width for (a): perpendicular magnetic field B⊥ = 0.5 T with (solid curves) and without (dashed curves) the SOC; (b) parallel magnetic field B‖ = 0.5 T with the SOC. The effective diameter d0 = 20 nm, and temperature T = 4 K. Curves with � — T−11 induced by the electron-BP scattering together with the SOC; Curves with • — T−11 induced by the second-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with N — T 1 induced by the first-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with H — T 1 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−11 induced by the g-factor fluctuation. width. Particularly the electron-BP scattering mecha- nism decreases much faster than the other mechanisms. It is indicated in the figures that when the well width is small (smaller than 7 nm in the present case), the spin re- laxation time is determined by the electron-BP scattering together with the SOC. However, for wider well widths, the direct spin-phonon coupling due to phonon-induced strain and the first-order process of hyperfine interac- tion combined with the electron-BP scattering becomes more important. The decrease of spin relaxation due to each mechanism is mainly caused by the decrease of the SOC which is proportional to a−2. The SOC has two effects which are crucial. First, in the second order per- turbation the SOC contributes a finite correction to the Zeeman splitting which determines the absorbed/emitted phonon frequency and wave-vector.18 Second, it leads to spin mixing. The decrease of the SOC thus leads to the decrease of Zeeman splitting and spin mixing. The for- mer leads to small phonon wave-vector and small phonon absorption/emission efficiency.18 Therefore the electron- BP mechanism decreases rapidly with increasing a. On the other hand, the other two largest mechanisms can flip spin without the help of the SOC. The spin relaxations due to these two mechanisms decrease in a relatively mild way. It is further confirmed that without SOC they de- creases in a much milder way with increasing a (dashed curves in Fig. 1). It is also noted that the spin relaxation rate due to the g-factor fluctuation is at least six orders of magnitude smaller than that due to the leading spin decoherence mechanisms and can therefore be neglected. It is noted that in the calculation, the SOC is always included as it has large effect on the eigen-energy and eigen-wavefunction of the electron.18 We also show the spin relaxation times induced by the hyperfine interac- tions (V eI−ph and V eI−ph) and the direct spin-phonon coupling due to the phonon-induced strain but without the SOC as in the literature.27,28,45 It can be seen clearly that the spin relaxation that includes the SOC is much larger than that without the SOC. For example, the spin relaxation induced by the second-order process of the hy- perfine interaction together with the BP (V eI−ph) is at least one order of magnitude larger when the SOC is in- cluded than that when the SOC is neglected. This is because when the SOC is neglected, 〈m|HeI(r)|ℓ1〉 and 〈ℓ2|HeI(r)|m〉 in Eq. (29) are small as the matrix el- ements of HeI(r) between different orbital energy lev- els are very small. However, when the SOC is taken into account, the spin-up and -down levels with differ- ent orbital quantum numbers are mixed and therefore |ℓ〉 and |m〉 include the components with the same or- bital quantum number. Consequently the matrix ele- ments of 〈m|HeI(r)|ℓ1〉 and 〈ℓ2|HeI(r)|m〉 become much larger. Therefore, spin relaxation induced by this mech- anism depends crucially on the SOC. It is emphasized from the above discussion that the SOC should be included in each spin relaxation mecha- nism. In the following calculations it is always included unless otherwise specified. In particular in reference to the mechanism of electron-BP interaction, we always con- sider it together with the SOC. We further discuss the parallel magnetic field case. In Fig. 1(b) the spin relaxation times due to different mech- anisms are plotted as function of the quantum well width for same parameters as Fig. 1(a), but with a parallel mag- netic field B‖ = 0.5 T. It is noted that the spin relaxation rate due to each mechanism becomes much smaller for small a compared with the perpendicular case. Another feature is that the spin relaxation due to each mecha- nism decrease in a much slower rate with increasing a. The electron-BP mechanism is dominant even at a = 10 nm but decrease faster than other mechanisms with a. It is expected to be less effective than the V eI−ph mecha- nism or V eI−ph mechanism or the direct spin-phonon cou- pling due to phonon-induced strain mechanism for large enough a. The g-factor fluctuation mechanism is negli- gible again. These features can be explained as follows. For parallel magnetic field the contribution of the SOC to Zeeman splitting is much less than in the perpendicular magnetic field geometry.21 Moreover, this contribution is negative which makes Zeeman splitting smaller.21 There- fore, the phonon absorption/emission efficiency becomes much smaller for small a, i.e., large SOC. When a in- creases, the Zeeman splitting increases. However, the spin mixing decreases. The former effect is weak, and only cancels part of the latter, thus the spin relaxation due to each mechanism decrease slowly with a. g-factor strain eI−ph eI−ph a = 5 nm B⊥ (T) 543210 a = 10 nm B⊥ (T) 543210 FIG. 2: (Color online) T−11 induced by different mechanisms vs. the perpendicular magnetic field B⊥ for d0 = 20 nm and (a) a = 5 nm and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by the electron-BP scattering; Curves with • — T−11 induced by the second-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with N — T−11 induced by the first-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with H — T−11 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−11 induced by the g-factor fluctuation. g-factor strain eI−ph eI−ph a = 5 nm B‖ (T) 543210 10−10 a = 10 nm B‖ (T) 543210 FIG. 3: (Color online) T−11 induced by different mechanisms vs. the parallel magnetic field B‖ for d0 = 20 nm and (a) a = 5 nm and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by the electron-BP scattering; Curves with • — T−11 induced by the second-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with N — T−11 induced by the first-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with H — T−11 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−11 induced by the g-factor fluctuation. 2. Magnetic Field Dependence We first study the perpendicular-magnetic-field case. The magnetic field dependence of T1 for two different well widths are shown in Fig. 2(a) and Fig. 2(b). In the calculation, d0 = 20 nm. It can be seen that the ef- fect of each mechanism increases with the magnetic field. Particularly the electron-BP mechanism increases much faster than other ones and becomes dominant at high magnetic fields. For small well width (5 nm in Fig. 2a), the spin relaxation induced by the electron-BP scattering is dominant except at very low magnetic fields (0.1 T in the figure) where contributions from the first-order pro- cess of hyperfine interaction together with the electron- BP scattering and the direct spin-phonon coupling due to phonon-induced strain also contribute. It is interesting to see that when a is increased to 10 nm, the electron-BP scattering is the largest spin relaxation mechanism only at high magnetic fields (>1.1 T). For 0.4 T < B⊥ < 1.1 T (B⊥ < 0.4 T), the direct spin-phonon coupling due to the phonon-induced strain (the first order hyperfine interaction together with the BP ) becomes the largest relaxation mechanism. It is also noted that there is no single mechanism which dominates the whole spin relax- ation. Two or three mechanisms are jointly responsible for the spin relaxation. It is indicated that the spin relax- ations induced by different mechanisms all increase with B⊥. This can be understood from a perturbation theory: when the magnetic field is small the spin relaxation be- tween two Zeeman split states for each mechanism is pro- portional to n̄(∆E)(∆E)m (∆E is the Zeeman splitting) with m = 7 for electron-BP scattering due to the de- formation potential18,25 and for the second-order process of the hyperfine interaction together with the electron- BP scattering due to the deformation potential V eI−ph; m = 5 for electron-BP scattering due to the piezoelec- tric coupling15,18,25 and for the second-order process of the hyperfine interaction together with the electron-BP scattering due to the piezoelectric coupling V eI−ph; 27 and m = 5 for the direct spin-phonon coupling due to phonon- induced strain;15 m = 1 for the first-order process of the hyperfine interaction together with the BP V eI−ph. The spin relaxation induced by the g-factor fluctuation is pro- portional to n̄(∆E)(∆E)5B2⊥. For most of the cases stud- ied, ∆E is smaller than kBT , hence n̄(∆E) ∼ kBT/∆E, and n̄(∆E)(∆E)m ∼ (∆E)m−1. m > 1 hold for all mech- anism except the V eI−ph mechanism, therefore the spin relaxation due to these mechanisms increases with in- creasing B⊥. However, from Eq. (27) one can see that it has a term with ∇r, which indicates that the effect of this mechanism is proportional to 1/d0. As the vector potential of the magnetic field increases the confinement of the QD and gives rise to smaller effective diameter d0, this mechanism also increases with the magnetic field in the perpendicular magnetic field geometry. We then study the case with the magnetic field par- allel to the quantum well plane. In Fig. 3 the spin re- laxation induced by different mechanisms are plotted as function of the parallel magnetic field B‖ for two dif- ferent well widths. In the calculation, d0 = 20 nm. It can be seen that, similar to the case with perpendicular magnetic field, the effects of most mechanisms increase with the magnetic field. Also the electron-BP mechanism increases much faster than the other ones and becomes dominant at high magnetic fields. However, without the orbital effect of the magnetic field in the present con- figuration, the effect of V eI−ph changes very little with the magnetic field. For both small (5 nm in Fig. 3(a)) and large (10 nm in Fig. 3(b)) well widths, the electron- BP scattering is dominant except at very low magnetic field (0.1 T in the figure), where the first-order process of the hyperfine interaction together with the electron-BP interaction V eI−ph also contributes. a = 5 nm d0 (nm) 3025201510 g-factor strain eI−ph eI−ph a = 10 nm d0 (nm) 3025201510 10−10 FIG. 4: (Color online) T−11 induced by different mechanisms vs. the effective diameter d0 for B⊥ = 0.5 T and (a) a = 5 nm and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by the electron-BP scattering; Curves with • – T−11 induced by the second-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with N — T 1 induced by the first-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with H — T 1 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−11 induced by the g-factor fluctuation. 3. Diameter Dependence We now turn to the investigation of the diameter de- pendence of the spin relaxation. We first concentrate on the perpendicular magnetic field geometry. The spin re- laxation rate due to each mechanism is shown in Fig. 4a for a small (a = 5 nm) and Fig. 4b for a large (a = 10 nm) well widths respectively with a fixed perpendicu- lar magnetic field B⊥ = 0.5 T. In the figure, the spin relaxation rate due each mechanism except V eI−ph in- creases with the effective diameter. Specifically, the ef- fect of the electron-BP mechanism increases very fast, while the effect of the direct spin-phonon coupling due to phonon-induced strain mechanism increases very mildly. The V eI−ph decreases with d0 slowly. Other mechanisms are unimportant. The electron-BP mechanism eventu- ally dominates spin relaxation when the diameter is large a = 5 nm d0 (nm) 3025201510 10−10 g-factor strain eI−ph eI−ph a = 10 nm d0 (nm) 3025201510 FIG. 5: (Color online) T−11 induced by different mechanisms vs. the effect diameter d0 with B‖ = 0.5 T and (a) a = 5 nm and (b) 10 nm. T = 4 K. Curves with � — T−11 induced by the electron-BP scattering; Curves with • — T−11 induced by the second-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with N — T 1 induced by the first-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with H — T 1 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−11 induced by the g-factor fluctuation. enough. The threshold increases from 12 nm to 26 nm when the well width increases from 5 nm to 10 nm. For small diameter the V eI−ph and the direct spin-phonon coupling due to phonon-induced strain mechanism dom- inate the spin relaxation. The increase/decrease of the spin relaxation due to these mechanisms can be under- stood from the following. The effect of the SOC on the Zeeman splitting is proportional to d20 for small magnetic field.18 The increase of d0 thus leads to a increase of Zee- man splitting, therefore the efficiency of the phonon ab- sorption/emission increases. Another effect is that the in- crease of d0 will increase the phonon absorption/emission efficiency due to the increase of the form factor.18 Thus the spin relaxation increases. Moreover, the spin mix- ing is also proportional to d0 also. 18 This leads to much faster increasing of the effect of the electron-BP mecha- nism and the V eI−ph mechanism. However, the spin re- laxation due to V eI−ph decreases with the diameter. This is because V eI−ph contains a term ∇r [Eq. (27)] which decreases with the increase of d0. Physically speaking, the decrease of the effect of V eI−ph is due to the fact that the spin mixing due to the hyperfine interaction de- creases with the increase of the number of nuclei within the dot N as the random Overhauser field is proportional to 1/ N . The spin relaxation induced by the g-factor is also negligible here for both small and large well width. We then turn to the parallel magnetic field case. In the calculation, B‖ = 0.5 T. The results are shown for both small well width (a = 5 nm in Fig. 5(a)) and large well width (a = 10 nm in Fig. 5(b)) respectively. Simi- lar to the perpendicular magnetic field case, the effect of every mechanism except the V eI−ph mechanism increases with increasing diameter. The effect of the electron-BP mechanism increases fastest and becomes dominant for d0 > 12 nm for both small and large well width. For d0 < 12 nm for the two cases the first-order process of the V eI−ph mechanism becomes dominant. The ef- fect of the V eI−ph mechanism become larger than that of the direct spin-phonon coupling due to phonon-induced strain mechanism. However, these two mechanism are still unimportant and becomes more and more unimpor- tant for larger d0. Here, the spin relaxation induced by the g-factor is negligible. 4. Comparison with Experiment In this subsection, we apply our analysis to experiment data in Ref. 7. We first show that our calculation is in good agreement with the experimental results. Then we compare contributions from different mechanisms to spin relaxation as function of the magnetic field. In the cal- culation we choose the quantum dot diameter d0 = 56 nm (~ω0 = 1.1 meV as in experiment). The quantum well is taken to be an infinite-depth well with a = 13 nm. The Dresselhaus SOC parameter γ0〈k2z〉 is taken to be 4.5 meV·Åand the Rashba SOC parameter is 3.3 meV·Å. T = 0 K as kBT ≪ gµBB in the experiment. The magnetic field is applied parallel to the well plane in [110]-direction. The Dresselhaus cubic term is also taken into consideration. All these parameters are the same with (or close to) those used in Ref. 24 in which a cal- culation based on the electron-BP scattering mechanism agrees well with the experimental results. For this mech- anism, we reproduce their results. The spin relaxation time measured by the experiments (black dots with er- ror bar in the figure) almost coincide with the calculated spin relaxation time due to the electron-BP scattering mechanism (curves with � in the figure).71 It is noted from the figure that other mechanisms are unimportant for small magnetic field. However, for large magnetic field the effect of the direct-spin phonon coupling due to phonon-induced strain becomes comparable with that of the electron-BP mechanism. At B‖ = 10 T, the two differs by a factor of ∼ 5. g-factor strain eI−ph eI−ph B‖ (T) 15129630 10−10 10−12 FIG. 6: (Color online) T−11 induced by different mechanisms vs. the parallel magnetic field B‖ in the [110] direction for d0 = 56 nm and a = 13 nm with both the Rashba and Dres- selhaus SOCs. T = 0 K. The black dots with error bar is the experimental results in Ref. 7. Curves with � — T−11 induced by the electron-BP scattering; Curves with • — T−11 induced by the second-order process of the hyperfine interaction to- gether with the BP (V eI−ph); Curves with N — T 1 induced by the first-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with H — T 1 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−11 induced by the g-factor fluctuation. B. Spin Dephasing Time T2 In this subsection, we investigate the spin dephasing time for different well widths, magnetic fields and QD diameters. As in the previous subsection, the contribu- tions of the different mechanisms to spin dephasing are compared.70 To justify the first Born approximation in studying the hyperfine interaction induced spin dephas- ing, we focus mainly on the high magnetic field regime of B > 3.5 T. A typical magnetic field is 4 T. We also demonstrate via extrapolation that in the low magnetic field regime spin dephasing is dominated by the hyperfine interaction. 1. Well Width Dependence In Fig. 7 the well width dependence of the spin dephas- ing induced by different mechanisms is presented under the perpendicular (a) and parallel (b) magnetic fields. In the calculations B⊥ = 4 T/B‖ = 4 T and d0 = 20 nm. It can be seen in both figures that the spin dephasing due to each mechanism decreases with a. Moreover, the spin de- phasing due to the electron-BP scattering decreases much faster than that due to the hyperfine interaction. These features can be understood as following. The spin de- phasing due to electron-BP scattering depends crucially on the SOC. As the SOC is proportional to a−2, the spin dephasing decreases fast with a. For the hyperfine inter- action, from Eq. (35) one can deduce that the decay rate of ||〈S+〉t|| is mainly determined by the factor 1/(azd2‖) (here az = a), which thus decreases with a, but in a very mild way. The fast decrease of the electron-BP mecha- nism makes it eventually unimportant. For the present perpendicular-magnetic-field case the threshold is around 2 nm. For parallel magnetic field it is even smaller. A higher temperature may enhance the electron-BP mech- anism (see discussion in Sec. V) and make it more im- portant than the hyperfine mechanism. It is noted that other mechanisms contribute very little to the spin de- phasing. Thus, in the following discussion, we do not con- sider these mechanisms. Comparing Figs 7(a) and (b), one finds that a main difference is that the electron-BP mechanism is less effective for the parallel-magnetic-field case. As has been discussed in the previous subsection, the spin mixing and the Zeeman splitting in the parallel filed case is smaller than those in the perpendicular field case. Therefore, the electron-BP mechanism is weakened markedly. Similar to Fig. 1, the SOC is always included in the computation as it has large effect on the eigen-energy and eigen-wavefunction of the electrons. The spin dephasings calculated without the SOC for the hyperfine interaction, the direct spin-phonon coupling due to phonon-induced strain and the g-factor fluctuation are also shown in Fig. 7(a) as dashed curves. It can be seen from the figure that for the spin dephasings induced by the direct spin- phonon coupling due to phonon-induced strain and by the g-factor fluctuation, the contributions with the SOC are much larger than those without. This is because when the SOC is included, the fluctuation of the effective field induced by both mechanisms becomes much stronger and more scattering channels are opened. However, what should be emphasized is that the spin dephasings induced by the hyperfine interaction with and without the SOC are nearly the same (the solid and the dashed curves nearly coincide). That is because the change of the wave- function Ψ(r) due to the SOC is very small (less than 1 % in our condition) and therefore the factor 1/(azd ‖) is almost unchanged when the SOC is neglected. Thus the spin dephasing rate is almost unchanged. In the inset of Fig. 7(a), the time evolution of ||〈S+〉t|| induced by the hyperfine interaction is shown, with a = 2 nm. It can be seen that ||〈S+〉t|| decays very fast and de- creases to less than 10 % of its initial value within the first two oscillating periods. Therefore, T2 is determined by the first two or three periods of ||〈S+〉t||. Thus the correction of the long time dynamics due to higher or- der scattering33 contributes little to the spin dephasing time. For quantum computation and quantum informa- tion processing, the initial, e.g., 1 % decay of ||〈S+〉t|| may be more important than the 1/e decay.42,43 Indeed, the spin dephasing time defined by the exponential fitting of 1 % decay is short than that defined by the 1/e decay. However, the two differs less than 5 times. For a rough B⊥ = 4 T t (µs) 6543210 a (nm) 1098765432 eI−ph eI−ph g-factor strain hyperfine B‖ = 4 T a (nm) 1098765432 10−10 10−15 10−20 10−25 FIG. 7: (Color online) T−12 induced by different mechanisms vs. the well width for d0 = 20 nm. T = 4 K. (a): B⊥ = 4 T with (solid curves) and without (dashed curves) the SOC; (b): B‖ = 4 T only with the SOC. Curve with � — T 2 induced by the electron-BP interaction; Curves with • — T−12 induced by the hyperfine interaction; Curves with H — T−12 induced by the direct spin-phonon coupling due to phonon-induced strain; Curves with � — T−12 induced by g-factor fluctuation; N — T−12 induced by the second-order process of the hyper- fine interaction together with the BP (V eI−ph); Curves with � — T−12 induced by the first-order process of the hyperfine in- teraction together with the BP (V eI−ph). The time evolution of ||〈S+〉t|| induced by the hyperfine interaction with a = 2 nm is shown in the inset of (a). comparison of contributions from different mechanisms to spin dephasing where only the order-of-magnitude differ- ence is concerned (see Figs. 7-9), this difference due to the definition does not jeopardize our conclusions. 2. Magnetic Field Dependence We then investigate the magnetic field dependence of the spin dephasing induced by the electron-BP scatter- ing and by the hyperfine interaction for two different well widths (a = 3 nm and a = 5 nm) with both perpendicular and parallel magnetic field. From Fig. 8(a) and (b) one hyperfine B⊥ (T) 87.576.565.554.543.5 B‖ (T) 87.576.565.554.543.5 FIG. 8: (Color online) T−12 induced by the electron-BP scat- tering and the hyperfine interaction vs. (a): the perpendic- ular magnetic field B⊥ ; (b): the parallel magnetic field B‖ for a = 3 nm (solid curves) and 5 nm (dashed curves). T = 4 K, and d0 = 20 nm. Curves with � — T 2 induced by the electron-BP interaction; Curves with • — T−12 induced by the hyperfine interaction. can see that the spin dephasing due to the electron-BP scattering increases with magnetic field, whereas that due to the hyperfine interaction decreases with magnetic field. Thus, the electron-BP mechanism eventually dominates the spin dephasing for high enough magnetic field. The threshold is Bc⊥ = 4 T / B = 7 T for a = 3 nm with per- pendicular/parallel magnetic field. For larger well width, e.g., a = 5 nm with parallel magnetic field or perpendicu- lar magnetic field, the threshold magnetic fields increase to larger than 8 T. The different magnetic field depen- dences above can be understood as following. Besides spin relaxation, the spin-flip scattering also contributes to spin dephasing.20 As has been demonstrated in Sec. IIIA, the electron-BP scattering induced spin-flip tran- sition rate increases with the magnetic field. Therefore the spin dephasing rate increases with the magnetic field also. In contrast, spin dephasing induced by the hyper- fine interaction decreases with the magnetic field. This is because when the magnetic field becomes larger, the fluctuation of the effective magnetic field due to the sur- rounding nuclei becomes insignificant compared. There- fore, the hyperfine-interaction-induced spin dephasing is reduced. Similar results have been obtained by Deng and Hu.44 hyperfine B⊥ = 4 T d0 (nm) 3025201510 B‖ = 4 T (b) d0 (nm) 3025201510 FIG. 9: (Color online) T−12 induced by the electron-BP scat- tering and the hyperfine interaction vs. the effective diameter d0 T = 4 K. (a): B⊥ = 4 T ; (b): B‖ = 4 T for a = 3 nm (solid curves) and 5 nm (dashed curves). Curves with � — T−12 induced by the electron-BP interaction; Curves with • — T−12 induced by the hyperfine interaction. 3. Diameter Dependence In Fig. 9 the spin dephasing times induced by the electron-BP scattering and the hyperfine interaction are plotted as function of the diameter d0 for a small (a = 3 nm) and a large (a = 5 nm) well widths. In the cal- culation, B⊥ = 4 T in Fig. 9(a) and B‖ = 4 T in (b). It is noted that the effect of the electron-BP mechanism increases rapidly with d0, whereas the effect of the hy- perfine mechanism decreases slowly. Consequently, the electron-BP mechanism eventually dominates the spin dephasing for large enough d0. The threshold is d 0 = 19 (27) nm for a = 3 (5) nm case with the perpendicular magnetic field and dc0 = 26 (30) nm for a = 3 (5) nm case under the parallel magnetic field. As has been dis- cussed in Sec. IIIA, both the effect of the SOC and the ef- ficiency of the phonon absorption/emission increase with d0. Therefore, the spin dephasing due to the electron-BP mechanism increases rapidly with d0. 18,21 The decrease of the effect of the hyperfine interaction is due to the de- crease of the factor 1/(azd ) [Eq. 35] with the diameter IV. SPIN RELAXATION TIMES FROM FERMI GOLDEN RULE AND FROM EQUATION OF MOTION In this section, we will try to find a proper method to average over the transition rates from the Fermi Golden rule, τ−1i→f , to give the spin relaxation time T1. In the limit of small SOC, we rederive Eq. (1) from the equation of motion. We further show that Eq. (1) fails for large SOC where a full calculation from the equation of motion is needed. T = 12 K 10987654321 γ = γ0 T (K) 4035302520151050 FIG. 10: (Color online) Spin relaxation time T1 calculated from the equation-of-motion approach (�) v.s. that obtained from Eq. (1) (•) as function of (a): the strength of the SOC for T=12 K; (b): the temperature for γ = γ0. The well width a = 5 nm, perpendicular magnetic field B⊥ = 0.5 T, QD diameter d0 = 30 nm. The ratio of the two R is also plotted in the figure. Note the scale of T−11 is at the right hand side of the frame. We first rederive Eq. (1) for small SOC from the equa- tion of motion. In QDs, the orbital level splitting is usually much larger than the Zeeman splitting. Each Zeeman sublevel has two states: one with majority up- spin, the other with majority down-spin. We call the former “minus state” (as it corresponds to a lower en- ergy) while the latter “plus state”. For small SOC, the spin mixing is small. Thus we neglect the much smaller contribution from the off-diagonal terms of the density matrix to Sz. Therefore Sz(t) = z fi±(t) where i± denotes the plus/minus state of the i-th or- bital state. For small SOC, the spin relaxation is much slower than the orbital relaxation.25,55 This im- plies that the time takes to establish equilibrium within the plus/minus states is much smaller than the spin relax- ation time. Thus we can assume a equilibrium (Maxwell- Boltzmann) distribution between the plus/minus states at any time. The distribution function is therefore given by fi±(t) = N±(t) exp(−εi±/kBT )/Z±. Here N±(t) = i fi±(t) is the total probability of the plus/minus states with N+(t) + N−(t) = 1 for single electron in QD and i exp(−εi±/kBT ) is the partition function for the plus/minus state. At equilibrium, N± = N ± . The equation for Sz(t) is hence, Sz(t) = [Sz(t)− Seqz ] Si±z exp(−εi±/kBT )/Z± δN±(t) , (36) with δN±(t) = N±(t) − Neq± . As the orbital level splitting is usually much larger than the Zeeman split- ting, the factor exp(−εi±/kBT )/Z± can be approximated by exp(−εi0/kBT )/Z0 with εi0 = 12 (εi+ + εi−) and i exp[−εi0/kBT ]. Further using the particle- conservation relation ± δN±(t) = 0, one has Sz(t) = [ (Si+z −Si−z ) exp(−εi0/kBT )/Z0] δN+(t) . As Sz(t) − Seqz = [δN+(t)/Z0] Si−z ) exp(−εi0/kBT ), one finds that the spin relax- ation time is nothing but the relaxation time of N+. The next step is to derive the equation of d δN+(t), which is given in our previous work:49 δN+(t) = δfi+(t) [τ−1i+→f−δfi+(t)− τ i−→f+δfi−(t)] [τ−1i+→f− + τ i−→f+] e−εi0/kBT δN+(t) .(38) Thus spin relaxation time is given by, (τ−1i+→f− + τ i−→f+) e−εi0/kBT . (39) Furthermore, substituting e−εi0/kBT /Z0 by f exp(−εi±/kBT )/Z±, we have (τ−1i+→f−f i+ + τ i−→f+f i−) . (40) This is exactly Eq. (1). T1/T2 T (K) 10−10 10−11 10−12 10−13 2520151050 FIG. 11: (Color online) Spin relaxation time T1, spin dephas- ing time T2 and T1/T2 against temperature T . B⊥ = 4 T, a = 5 nm and d0 = 30 nm. Note the scale of T1 and T2 is at the right hand side of the frame. For large SOC, or large spin mixing due to anticrossing of different spin states,19,25 the spin relaxation rate be- comes comparable with the orbital relaxation rate. Fur- thermore, the decay of the off-diagonal term of the den- sity matrix should contribute to the decay of Sz. There- fore, the above analysis does not hold. In this case, it is difficult to obtain such a formula, and a full calculation from the equation-of-motion is needed. In Fig. 10(a), we show that (for T = 12 K, a = 5 nm, B⊥ = 0.5 T, d0 = 30 nm) the spin relaxation times T1 calculated from equation-of-motion approach and that obtained from Eq. (40). Here, for simplicity and without loss of generality, we consider only the electron-BP scat- tering mechanism. The discrepancy of T1 obtained from the two approaches increases with γ. At γ = 10γ0, the ratio of the two becomes as large as ∼ 3. In Fig. 10(b), we plot the spin relaxation times obtained via the two approaches as function of temperature for γ = γ0 with other parameters remaining unchanged. It is noted that the discrepancy of T1 obtained from the two approaches increases with temperature. For high temperature, the higher levels are involved in the spin dynamics where the SOC becomes larger. At 40 K, the discrepancy is as large as 60 %. The ratio increases very slowly for T < 20 K where only the lowest two Zeeman sublevels are involved in the dynamics. g-factor strain eI−ph eI−ph B⊥ = 0.5 T T (K) 2520151050 10−10 10−15 B⊥ = 0.9 T T (K) 2520151050 FIG. 12: (Color online) Spin relaxation time T1 against tem- perature T for (a): B⊥ = 0.5 T; (b): B⊥ = 0.9 T. a = 10 nm and d0 = 20 nm. Curves with � — T 1 induced by the electron-BP scattering together with the SOC; Curves with • — T−11 induced by the second-order process of the hyperfine interaction together with the BP (V eI−ph); Curves with N — T−11 induced by the first-order process of the hyperfine inter- action together with the BP (V eI−ph); Curves with H — T induced by the direct spin-phonon coupling due to phonon- induced strain; Curves with � — T−11 induced by the g-factor fluctuation. V. TEMPERATURE DEPENDENCE OF SPIN RELAXATION TIME T1 AND SPIN DEPHASING TIME T2 We first study the relative magnitude of the spin re- laxation time T1 and the spin dephasing time T2. We consider a QD with d0 = 30 nm and a = 5 nm at B⊥ = 4 T where the largest contribution to both spin relaxation and dephasing comes from the electron-BP scattering (see Fig. 4(a) and Fig. 9(a), we have checked that the electron-BP scattering mechanism is dominant through- out the temperature range). From Fig. 11, one finds that when the temperature is low (T < 5 K in the figure), T2 = 2T1, which is in agreement with the discussion in Ref. 20. However, T1/T2 increases very quickly with T and for T = 20 K, T1/T2 ∼ 2 × 102. This is understood from the fact that when T is low, the electron mostly dis- tributes in the lowest two Zeeman sublevels. For small SOC, Golovach et al. have shown via perturbation the- ory that phonon induces only the spin-flip noise in the leading order. Consequently, T2 = 2T1. 20 When the tem- perature becomes comparable with the orbital level split- ting ~ω0, the distribution over the upper orbital levels is not negligible any more. As mentioned previously, the SOC contributes a non-trivial part to the Zeeman split- ting. Specifically, the second order energy correction due to the SOC contributes to the Zeeman splitting. The energy correction for different orbital levels is generally unequal (always larger for higher levels). When the elec- tron is scattered by phonons randomly from one orbital state to another one with the same major spin polariza- tion, the frequency of its precession around z direction changes. Continuous scattering leads to random fluctu- ation of the precession frequency and thus leads to spin dephasing.29,46 Note that this fluctuation only leads to a phase randomization of S+, but not flips the z compo- nent spin Sz, i.e., not leads to spin relaxation. There- fore, the spin dephasing becomes stronger than the spin relaxation for high temperatures. Moreover, this effect increases with temperature rapidly as the distribution over higher levels and the phonon numbers both increase with temperature. We further study the temperature dependence of spin relaxation for lower magnetic field and larger quantum well width where other mechanisms may be more im- portant than the electron-BP mechanism. In Fig. 12(a), the spin relaxation time is plotted as function of tem- perature for B⊥ = 0.5 T, a = 10 nm and d0 = 20 nm. It is seen from the figure that the direct spin-phonon coupling due to phonon-induced strain mechanism dom- inates the spin relaxation throughout the temperature range. It is also noted that for T ≤ 4 K the spin relax- ation rates induced by different mechanisms all increase with temperature according to the phonon number fac- tor 2n̄(Ez1) + 1 with Ez1 being the Zeeman splitting of the lowest Zeeman sublevels. However, for T > 4 K, the spin relaxation rates induced by the direct spin-phonon coupling due to phonon-induced strain and the electron- BP interaction increase rapidly with temperature, while the spin relaxation rates induced by V eI−ph and V eI−ph increase mildly according to 2n̄(Ez1) + 1 throughout the temperature range. These features can be understood as what follows. For T ≤ 4 K, the distribution over the high levels is negligible. Only the lowest two Zeeman sublevels involve in the spin dynamics. The spin relaxation rates thus increase with 2n̄(Ez1) + 1 and the relative impor- tance of each mechanism does not change. Therefore, our previous analysis on comparison of relative importance of different spin decoherence mechanisms at 4 K holds true for the range 0 ≤ T ≤ 4 K. When the temperature gets higher, the contribution from higher levels becomes more important. Although the distribution at the higher lev- els is still very small, for the direct spin-phonon coupling mechanism, the transition rates between the higher lev- els and that between higher levels and the lowest two sublevels are very large. For the electron-BP mechanism the transition rates between the higher levels are very large due to the large SOC in these levels. Therefore, the contribution from the higher levels becomes larger than that from the lowest two sublevels. Consequently, the in- crease of temperature leads to rapid increase of the spin relaxation rates. However, for the two hyperfine mech- anisms: the V eI−ph and the V eI−ph, the spin relaxation rates does not change much when the higher levels are involved. They thus increase by the phonon number fac- In Fig. 12(b) we show the temperature dependence of the spin relaxation time for the same condition but with B = 0.9 T. It is noted that the spin relaxation rate due to the electron-BP mechanism catches up with that in- duced by the direct spin-phonon coupling due to phonon- induced strain at T = 9 K and becomes larger for higher temperature. This indicates that the temperature depen- dence of the two mechanisms are quite different. In Fig. 13 we show the spin dephasing induced by electron-BP scattering and the hyperfine interaction as function of temperature for B⊥ = 4 T, a = 10 nm and d0 = 20 nm. We choose the conditions so that the spin dephasing is dominated by the hyperfine interaction at low temperature. However, the effect of the electron-BP mechanism increases with temperature quickly while that of the hyperfine interaction remains nearly unchanged. The fast increase of the effect from the electron-BP scat- tering is due to three factors: 1) the increase of the phonon number; 2) the increase of scattering channels; and 3) the increase of the SOC induced spin mixing in higher levels. On the other hand, from Eq. 35, one can deduce that the spin dephasing rate of the hyperfine in- teraction depends mainly on the factor 1/(azd ) with is the characteristic length/area along the z di- rection / in the quantum well plane. For higher levels, the d2‖ is larger, but only about a factor smaller than 10. Thus the effect of the hyperfine interaction increases very slowly with temperature. It should be noted that in the above discussion, we neglected the two-phonon scattering mechanism,15,46,50 which may be important at high temperature. The con- tribution of this mechanism should be calculated via the equation-of-motion approach developed in this paper, and compared with the contribution of other mechanisms showed here. VI. CONCLUSION In conclusion, we have investigated the longitudinal and transversal spin decoherence times T1 and T2, called spin relaxation time and spin dephasing time, in differ- ent conditions in GaAs QDs from the equation-of-motion approach. Various mechanisms, including the electron- BP scattering, the hyperfine interaction, the direct spin- hyperfine T (K) 2520151050 FIG. 13: (Color online) Spin relaxation time T1 against tem- perature T . B⊥ = 4 T, a = 10 nm and d0 = 20 nm. Curves with � — T−12 induced by the electron-BP scattering together with the SOC; Curves with • — T−12 induced by the hyperfine interaction phonon coupling due to phonon-induced strain and the g-factor fluctuation are considered. Their relative im- portance is compared. There is no doubt that for spin decoherence induced by electron-BP scattering, the SOC must be included. However, for spin decoherence induced by the hyperfine interaction, the direct spin-phonon cou- pling due to phonon-induced strain, g-factor fluctua- tion, and hyperfine interaction combined with electron- phonon scattering, the SOC is neglected in the existing literature.27,28,45 Our calculations have shown that, as the SOC has marked effect on the eigen-energy and the eigen-wavefunction of the electron, the spin decoherence induced by these mechanisms with the SOC is larger than that without it. Especially, the decoherence from the second-order process of hyperfine interaction combined with the electron-BP interaction increases at least one order of magnitude when the SOC is included. Our cal- culations show that, with the SOC, in some conditions some of these mechanisms (except g-factor fluctuation mechanism) can even dominate the spin decoherence. There is no single mechanism which dominates spin re- laxation or spin dephasing in all parameter regimes. The relative importance of each mechanism varies with the well width, magnetic field and QD diameter. In particu- lar, the electron-BP scattering mechanism has the largest contribution to spin relaxation and spin dephasing for small well width and/or high magnetic field and/or large QD diameter. However, for other parameters the hyper- fine interaction, the first-order process of the hyperfine interaction combined with electron-BP scattering, and the direct spin-phonon coupling due to phonon-induced strain can be more important. It is noted that the g- factor fluctuation always has very little contribution to spin relaxation and spin dephasing which can thus be neglected all the time. For spin dephasing, the electron- BP scattering mechanism and the hyperfine interaction mechanism are more important than other mechanisms for magnetic field higher than 3.5 T. For this regime, other mechanisms can thus be neglected. It is also shown that spin dephasing induced by the electron-BP mecha- nism increases rapidly with temperature. Extrapolated from our calculation, the hyperfine interaction mecha- nism is believed to be dominant for small magnetic field. We also discussed the problem of finding a proper method to average over the transition rates τ−1i→f obtained from the Fermi Golden rule, to give the spin relaxation time T1 at finite temperature. For small SOC, we red- erived the formula for T1 at finite temperature used in the existing literature18,51,52 from the equation of mo- tion. We further demonstrated that this formula is in- adequate at high temperature and/or for large SOC. For such cases, a full calculation from the equation-of-motion approach is needed. The equation-of-motion approach provides an easy and powerful way to calculate the spin decoherence at any temperature and SOC. We also studied the temperature dependence of spin re- laxation T1 and dephasing T2. 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Skierbiszewski, A. Zduniak, E. Litwin- Staszewska, D. Bertho, F. Kobbi, J. L. Robert, G. E. Pikus, F. G. Pikus, S. V. Iordanskii, V. Mosser, K. Zekentes, and Yu. B. Lyanda-Geller, Phys. Rev. B 53, 3912 (1996). 70 It should be mentioned that one effect is not included : when electron is scattered by phonon from one orbital state to another, it feels a difference in the spin precession fre- quency since the strength of longitudinal (along the exter- nal magnetic field) component of Overhauser field differs with orbital states. This effect randomizes the spin preces- sion phase and leads to a pure spin dephasing. However, this effect is negligible in our manuscript. 71 The deviation of our calculation from the experiment data at T = 14 T is due to the fact that we do not include the cyclotron effect along the-z direction. For B & 10 T, the cyclotron orbit length is smaller than the quantum well width, which makes our model unrealistic. http://arxiv.org/abs/0711.0479

0704.0149

Construction of initial data for 3+1 numerical relativity

Construction of initial data for 3+1 numerical relativity Eric Gourgoulhon Laboratoire Univers et Théories, UMR 8102 du C.N.R.S., Observatoire de Paris, Université Paris 7 - Denis Diderot, F-92195 Meudon Cedex, France E-mail: [email protected] Abstract. This lecture is devoted to the problem of computing initial data for the Cauchy problem of 3+1 general relativity. The main task is to solve the constraint equations. The conformal technique, introduced by Lichnerowicz and enhanced by York, is presented. Two standard methods, the conformal transverse- traceless one and the conformal thin sandwich, are discussed and illustrated by some simple examples. Finally a short review regarding initial data for binary systems (black holes and neutron stars) is given. Submitted to: Journal of Physics: Conference Series, for the Proceedings of the VII Mexican School on Gravitation and Mathematical Physics, held in Playa del Carmen, Quintana Roo, Mexico (November 26 - December 2, 2006) 1. Introduction The 3+1 formalism is the basis of most modern numerical relativity and has lead, along with alternative approaches [82], to the recent successes in the binary black hole merger problem [6, 7, 99, 25, 26, 27, 28] (see [24, 69, 86] for a review). Thanks to the 3+1 formalism, the resolution of Einstein equation amounts to solving a Cauchy problem, namely to evolve “forward in time” some initial data. However this is a Cauchy problem with constraints. This makes the set up of initial data a non trivial task, because these data must fulfill the constraints. In this lecture, we present the most wide spread methods to deal with this problem. Notice that we do not discuss the numerical techniques employed to solve the constraints (see e.g. Choptuik’s lecture for finite differences [32] and Grandclément and Novak’s review for spectral methods [58]). Standard reviews about the initial data problem are the articles by York [106] and Choquet-Bruhat and York [36]. Recent reviews are the articles by Cook [37], Pfeiffer [79] and Bartnik and Isenberg [10]. 2. The initial data problem 2.1. 3+1 decomposition of Einstein equation In this lecture, we consider a spacetime (M, g), where M is a four-dimensional smooth manifold and g a Lorentzian metric on M. We assume that (M, g) is globally http://arxiv.org/abs/0704.0149v2 Construction of initial data for 3+1 numerical relativity 2 hyperbolic, i.e. that M can be foliated by a family (Σt)t∈R of spacelike hypersurfaces. We denote by γ the (Riemannian) metric induced by g on each hypersurface Σt and K the extrinsic curvature of Σt, with the same sign convention as that used in the numerical relativity community, i.e. for any pair of vector fields (u,v) tangent to Σt, g(u,∇vn) = −K(u,v), where n is the future directed unit normal to Σt and ∇ is the Levi-Civita connection associated with g. The 3+1 decomposition of Einstein equation with respect to the foliation (Σt)t∈R leads to three sets of equations: (i) the evolution equations of the Cauchy problem (full projection of Einstein equation onto Σt), (ii) the Hamiltonian constraint (full projection of Einstein equation along the normal n), (iii) the momentum constraint (mixed projection: once onto Σt, once along n). The latter two sets of equations do not contain any second derivative of the metric with respect to t. They are written‡ R+K2 −KijKij = 16πE (Hamiltonian constraint), (1) DjKij −DiK = 8πpi (momentum constraint), (2) where R is the Ricci scalar (also called scalar curvature) associated with the 3-metric γ, K is the trace of K with respect to γ: K = γijKij , D stands for the Levi-Civita connection associated with the 3-metric γ, and E and pi are respectively the energy density and linear momentum of matter, both measured by the observer of 4-velocity n (Eulerian observer). In terms of the matter energy-momentum tensor T they are expressed as E = Tµνn µnν and pi = −Tµνnµγνi. (3) Notice that Eqs. (1)-(2) involve a single hypersurface Σ0, not a foliation (Σt)t∈R. In particular, neither the lapse function nor the shift vector appear in these equations. 2.2. Constructing initial data In order to get valid initial data for the Cauchy problem, one must find solutions to the constraints (1) and (2). Actually one may distinguish two problems: • The mathematical problem: given some hypersurface Σ0, find a Riemannian metric γ, a symmetric bilinear form K and some matter distribution (E,p) on Σ0 such that the Hamiltonian constraint (1) and the momentum constraint (2) are satisfied. In addition, the matter distribution (E,p) may have some constraints from its own. We shall not discuss them here. • The astrophysical problem: make sure that the solution to the constraint equations has something to do with the physical system that one wish to study. Facing the constraint equations (1) and (2), a naive way to proceed would be to choose freely the metric γ, thereby fixing the connection D and the scalar curvature R, and to solve Eqs. (1)-(2) for K. Indeed, for fixed γ, E, and p, Eqs. (1)-(2) form a quasi-linear system of first order for the components Kij . However, as discussed by Choquet-Bruhat [45], this approach is not satisfactory because we have only four equations for six unknowns Kij and there is no natural prescription for choosing arbitrarily two among the six components Kij . In 1944, Lichnerowicz [70] has shown that a much more satisfactory split of the initial data (γ,K) between freely choosable parts and parts obtained by solving ‡ we are using the standard convention for indices, namely Greek indices run in {0, 1, 2, 3}, whereas Latin ones run in {1, 2, 3} Construction of initial data for 3+1 numerical relativity 3 Eqs. (1)-(2) is provided by a conformal decomposition of the metric γ. Lichnerowicz method has been extended by Choquet-Bruhat (1956, 1971) [45, 33], by York and Ó Murchadha (1972, 1974, 1979) [103, 104, 76, 106] and more recently by York and Pfeiffer (1999, 2003) [107, 80]. Actually, conformal decompositions are by far the most widely spread techniques to get initial data for the 3+1 Cauchy problem. Alternative methods exist, such as the quasi-spherical ansatz introduced by Bartnik in 1993 [8] or a procedure developed by Corvino (2000) [39] and by Isenberg, Mazzeo and Pollack (2002) [63] for gluing together known solutions of the constraints, thereby producing new ones. Here we shall limit ourselves to the conformal methods. 2.3. Conformal decomposition of the constraints In the conformal approach initiated by Lichnerowicz [70], one introduces a conformal metric γ̃ and a conformal factor Ψ such that the (physical) metric γ induced by the spacetime metric on the hypersurface Σt is γij = Ψ 4γ̃ij . (4) We could fix some degree of freedom by demanding that det γ̃ij = 1. This would imply Ψ = (det γij) 1/12. However, in this case γ̃ and Ψ would be tensor densities. Moreover the condition det γ̃ij = 1 has a meaning only for Cartesian-like coordinates. In order to deal with tensor fields and to allow for any type of coordinates, we proceed differently and introduce a background Riemannian metric f on Σt. If the topology of Σt allows it, we shall demand that f is flat. Then we replace the condition det γ̃ij = 1 by det γ̃ij = det fij . This fixes det γij det fij )1/12 . (5) Ψ is then a genuine scalar field on Σt (as a quotient of two determinants). Consequently γ̃ is a tensor field and not a tensor density. Associated with the above conformal transformation, there are two decomposi- tions of the traceless part Aij of the extrinsic curvature, the latter being defined by Kij =: Aij + Kγij . (6) These two decompositions are Aij =: Ψ−10Âij , (7) Aij =: Ψ−4Ãij . (8) The choice −10 for the exponent of Ψ in Eq. (7) is motivated by the following identity, valid for any symmetric and traceless tensor field, ij = Ψ−10D̃j Ψ10Aij , (9) where D̃j denotes the covariant derivative associated with the conformal metric γ̃. This choice is well adapted to the momentum constraint, because the latter involves the divergence of K. The alternative choice, i.e. Eq. (8), is motivated by time evolution considerations, as we shall discuss below. For the time being, we limit ourselves to the decomposition (7), having in mind to simplify the writing of the momentum constraint. Construction of initial data for 3+1 numerical relativity 4 By means of the decompositions (4), (6) and (7), the Hamiltonian constraint (1) and the momentum constraint (2) are rewritten as (see Ref. [51] for details) D̃iD̃ iΨ− 1 Âij ij Ψ−7 + 2πẼΨ−3 − 1 K2Ψ5 = 0, (10) D̃j ij − 2 Ψ6D̃iK = 8πp̃i, (11) where R̃ is the Ricci scalar associated with the conformal metric γ̃ and we have introduced the rescaled matter quantities Ẽ := Ψ8E and p̃i := Ψ10pi. (12) Equation (10) is known as Lichnerowicz equation, or sometimes Lichnerowicz-York equation. The definition of p̃i is such that there is no Ψ factor in the right-hand side of Eq. (11). On the contrary the power 8 in the definition of Ẽ is not the only possible choice. As we shall see in § 3.4, it is chosen (i) to guarantee a negative power of Ψ in the Ẽ term in Eq. (10), resulting in some uniqueness property of the solution and (ii) to allow for an easy implementation of the dominant energy condition. 3. Conformal transverse-traceless method 3.1. Longitudinal/transverse decomposition of Âij In order to solve the system (10)-(11), York (1973,1979) [104, 105, 106] has decomposed Âij into a longitudinal part and a transverse one, setting Âij = (L̃X)ij +  TT, (13) where  TT is both traceless and transverse (i.e. divergence-free) with respect to the metric γ̃: γ̃ij TT = 0 and D̃j TT = 0, (14) and (L̃X)ij is the conformal Killing operator associated with the metric γ̃ and acting on the vector field X: (L̃X)ij := D̃iXj + D̃jX i − 2 k γ̃ij . (15) (L̃X)ij is by construction traceless: γ̃ij(L̃X) ij = 0 (16) (it must be so because in Eq. (13) both Âij and  TT are traceless). The kernel of L̃ is made of the conformal Killing vectors of the metric γ̃, i.e. the generators of the conformal isometries (see e.g. Ref. [51] for more details). The symmetric tensor (L̃X)ij is called the longitudinal part of Âij , whereas  TT is called the transverse part. Given Âij , the vector X is determined by taking the divergence of Eq. (13): taking into account property (14), we get D̃j(L̃X) ij = D̃j ij . (17) The second order operator D̃j(L̃X) ij acting on the vector X is the conformal vector Laplacian ∆̃L: ∆̃L X i := D̃j(L̃X) ij = D̃jD̃ jX i + D̃iD̃jX j + R̃i jX j , (18) Construction of initial data for 3+1 numerical relativity 5 where the second equality follows from the Ricci identity applied to the connection D̃, R̃ij being the associated Ricci tensor. The operator ∆̃L is elliptic and its kernel is, in practice, reduced to the conformal Killing vectors of γ̃, if any. We rewrite Eq. (17) as ∆̃L X i = D̃j ij . (19) The existence and uniqueness of the longitudinal/transverse decomposition (13) depend on the existence and uniqueness of solutions X to Eq. (19). We shall consider two cases: • Σ0 is a closed manifold, i.e. is compact without boundary; • (Σ0,γ) is an asymptotically flat manifold, i.e. is such that the background metric f is flat (except possibly on a compact sub-domain B of Σt) and there exists a coordinate system (xi) = (x, y, z) on Σt such that outside B, the components of f are fij = diag(1, 1, 1) (“Cartesian-type coordinates”) and the variable x2 + y2 + z2 can take arbitrarily large values on Σt; then when r → +∞, the components of γ and K with respect to the coordinates (xi) satisfy γij = fij +O(r −1) and = O(r−2), (20) Kij = O(r −2) and = O(r−3). (21) In the case of a closed manifold, one can show (see Appendix B of Ref. [51] for details) that solutions to Eq. (19) exist provided that the source D̃j ij is orthogonal to all conformal Killing vectors of γ̃, in the sense that ∀C ∈ ker L̃, γ̃ijC iD̃k γ̃ d3x = 0. (22) But the above property is easy to verify: using the fact that the source is a pure divergence and that Σ0 is closed, we may integrate the left-hand side by parts and get, for any vector field C, γ̃ijC i D̃k γ̃ d3x = −1 γ̃ij γ̃kl(L̃C) ikÂjl γ̃ d3x. (23) Then, obviously, when C is a conformal Killing vector, the right-hand side of the above equation vanishes. So there exists a solution to Eq. (19) and this solution is unique up to the addition of a conformal Killing vector. However, given a solution X, for any conformal Killing vector C, the solution X +C yields to the same value of L̃X, since C is by definition in the kernel of L̃. Therefore we conclude that the decomposition (13) of Âij is unique, although the vector X may not be if (Σ0, γ̃) admits some conformal isometries. In the case of an asymptotically flat manifold, the existence and uniqueness is guaranteed by a theorem proved by Cantor in 1979 [30] (see also Appendix B of Ref. [87] as well as Refs. [35, 51]). This theorem requires the decay condition ∂2γ̃ij ∂xk∂xl = O(r−3) (24) in addition to the asymptotic flatness conditions (20). This guarantees that R̃ij = O(r −3). (25) Then all conditions are fulfilled to conclude that Eq. (19) admits a unique solution X which vanishes at infinity. To summarize, for all considered cases (asymptotic flatness and closed manifold), any symmetric and traceless tensor Âij (decaying as O(r−2) in the asymptotically flat case) admits a unique longitudinal/transverse decomposition of the form (13). Construction of initial data for 3+1 numerical relativity 6 3.2. Conformal transverse-traceless form of the constraints Inserting the longitudinal/transverse decomposition (13) into the constraint equations (10) and (11) and making use of Eq. (19) yields to the system D̃iD̃ iΨ− 1 (L̃X)ij +  (L̃X)ij +  + 2πẼΨ−3 − 1 K2Ψ5 = 0, (26) ∆̃L X i − 2 Ψ6D̃iK = 8πp̃i, (27) where (L̃X)ij := γ̃ikγ̃jl(L̃X) kl and ÂTTij := γ̃ikγ̃jl TT. (28) With the constraint equations written as (26) and (27), we see clearly which part of the initial data on Σ0 can be freely chosen and which part is “constrained”: • free data: – conformal metric γ̃; – symmetric traceless and transverse tensor  TT (traceless and transverse are meant with respect to γ̃: γ̃ij TT = 0 and D̃j TT = 0); – scalar field K; – conformal matter variables: (Ẽ, p̃i); • constrained data (or “determined data”): – conformal factor Ψ, obeying the non-linear elliptic equation (26) (Lichnerowicz equation) – vector X, obeying the linear elliptic equation (27) . Accordingly the general strategy to get valid initial data for the Cauchy problem is to choose (γ̃ij ,  TT,K, Ẽ, p̃ i) on Σ0 and solve the system (26)-(27) to get Ψ and X Then one constructs γij = Ψ 4γ̃ij (29) Kij = Ψ−10 (L̃X)ij +  Ψ−4Kγ̃ij (30) E = Ψ−8Ẽ (31) pi = Ψ−10p̃i (32) and obtains a set (γ,K, E,p) which satisfies the constraint equations (1)-(2). This method has been proposed by York (1979) [106] and is naturally called the conformal transverse traceless (CTT ) method. 3.3. Decoupling on hypersurfaces of constant mean curvature Equations (26) and (27) are coupled, but we notice that if, among the free data, we choose K to be a constant field on Σ0, K = const, (33) then they decouple partially : condition (33) implies D̃iK = 0, so that the momentum constraint (27) becomes independent of Ψ: ∆̃L X i = 8πp̃i (K = const). (34) Construction of initial data for 3+1 numerical relativity 7 The condition (33) on the extrinsic curvature of Σ0 defines what is called a constant mean curvature (CMC ) hypersurface. Indeed let us recall that K is nothing but (minus three times) the mean curvature of (Σ0,γ) embedded in (M, g). A maximal hypersurface, having K = 0, is of course a special case of a CMC hypersurface. On a CMC hypersurface, the task of obtaining initial data is greatly simplified: one has first to solve the linear elliptic equation (34) to get X and plug the solution into Eq. (26) to form an equation for Ψ. Equation (34) is the conformal vector Poisson equation discussed above (Eq. (19), with D̃j ij replaced by 8πp̃i). We know then that it is solvable for the two cases of interest mentioned in Sec. 3.1: closed or asymptotically flat manifold. Moreover, the solutions X are such that the value of L̃X is unique. 3.4. Lichnerowicz equation Taking into account the CMC decoupling, the difficult problem is to solve Eq. (26) for Ψ. This equation is elliptic and highly non-linear§. It has been first studied by Lichnerowicz [70, 71] in the case K = 0 (Σ0 maximal) and Ẽ = 0 (vacuum). Lichnerowicz has shown that given the value of Ψ at the boundary of a bounded domain of Σ0 (Dirichlet problem), there exists at most one solution to Eq. (26). Besides, he showed the existence of a solution provided that Âij ij is not too large. These early results have been much improved since then. In particular Cantor [29] has shown that in the asymptotically flat case, still with K = 0 and Ẽ = 0, Eq. (26) is solvable if and only if the metric γ̃ is conformal to a metric with vanishing scalar curvature (one says then that γ̃ belongs to the positive Yamabe class) (see also Ref. [74]). In the case of closed manifolds, the complete analysis of the CMC case has been achieved by Isenberg (1995) [62]. For more details and further references, we recommend the review articles by Choquet-Bruhat and York [36] and Bartnik and Isenberg [10]. Here we shall simply repeat the argument of York [107] to justify the rescaling (12) of E. This rescaling is indeed related to the uniqueness of solutions to the Lichnerowicz equation. Consider a solution Ψ0 to Eq. (26) in the case K = 0, to which we restrict ourselves. Another solution close to Ψ0 can be written Ψ = Ψ0 + ǫ, with |ǫ| ≪ Ψ0: D̃iD̃ i(Ψ0 + ǫ)− (Ψ0 + ǫ) + Âij ij (Ψ0 + ǫ) −7 + 2πẼ(Ψ0 + ǫ) −3 = 0. (35) Expanding to the first order in ǫ/Ψ0 leads to the following linear equation for ǫ: D̃iD̃ iǫ− αǫ = 0, (36) Âij ijΨ−80 + 6πẼΨ 0 . (37) Now, if α ≥ 0, one can show, by means of the maximum principle, that the solution of (36) which vanishes at spatial infinity is necessarily ǫ = 0 (see Ref. [34] or § B.1 of Ref. [35]). We therefore conclude that the solution Ψ0 to Eq. (26) is unique (at least locally) in this case. On the contrary, if α < 0, non trivial oscillatory solutions of Eq. (36) exist, making the solution Ψ0 not unique. The key point is that the scaling (12) of E yields the term +6πẼΨ−40 in Eq. (37), which contributes to make α positive. If we had not rescaled E, i.e. had considered the original Hamiltonian constraint, the § although it is quasi-linear in the technical sense, i.e. linear with respect to the highest-order derivatives Construction of initial data for 3+1 numerical relativity 8 contribution to α would have been instead −10πEΨ40, i.e. would have been negative. Actually, any rescaling Ẽ = ΨsE with s > 5 would have work to make α positive. The choice s = 8 in Eq. (12) is motivated by the fact that if the conformal data (Ẽ, p̃i) obey the “conformal” dominant energy condition γ̃ij p̃ip̃j, (38) then, via the scaling (12) of pi, the reconstructed physical data (E, pi) will automatically obey the dominant energy condition γijpipj. (39) 4. Conformally flat initial data by the CTT method 4.1. Momentarily static initial data In this section we search for asymptotically flat initial data (Σ0,γ,K) by the CTT method exposed above. As a purpose of illustration, we shall start by the simplest case one may think of, namely choose the freely specifiable data (γ̃ij ,  TT,K, Ẽ, p̃ to be a flat metric: γ̃ij = fij , (40) a vanishing transverse-traceless part of the extrinsic curvature: TT = 0, (41) a vanishing mean curvature (maximal hypersurface) K = 0, (42) and a vacuum spacetime: Ẽ = 0, p̃i = 0. (43) Then D̃i = Di, where D denotes the Levi-Civita connection associated with f , R̃ = 0 (f is flat) and the constraint equations (26)-(27) reduce to (LX)ij(LX) ij Ψ−7 = 0 (44) i = 0, (45) where ∆ and ∆L are respectively the scalar Laplacian and the conformal vector Laplacian associated with the flat metric f : ∆ := DiDi and ∆LX i := DjDjX i + DiDjXj . (46) Equations (44)-(45) must be solved with the boundary conditions Ψ = 1 when r → ∞ (47) X = 0 when r → ∞, (48) which follow from the asymptotic flatness requirement. The solution depends on the topology of Σ0, since the latter may introduce some inner boundary conditions in addition to (47)-(48). Let us start with the simplest case: Σ0 = R 3. Then the unique solution of Eq. (45) subject to the boundary condition (48) is X = 0. (49) Construction of initial data for 3+1 numerical relativity 9 Figure 1. Hypersurface Σ0 as R 3 minus a ball, displayed via an embedding diagram based on the metric γ̃, which coincides with the Euclidean metric on 3. Hence Σ0 appears to be flat. The unit normal of the inner boundary S with respect to the metric γ̃ is s̃. Notice that D̃ · s̃ > 0. Consequently (LX)ij = 0, so that Eq. (44) reduces to Laplace equation for Ψ: ∆Ψ = 0. (50) With the boundary condition (47), there is a unique regular solution on R3: Ψ = 1. (51) The initial data reconstructed from Eqs. (29)-(30) is then γ = f (52) K = 0. (53) These data correspond to a spacelike hyperplane of Minkowski spacetime. Geometrically the condition K = 0 is that of a totally geodesic hypersurface [i.e. all the geodesics of (Σt,γ) are geodesics of (M, g)]. Physically data with K = 0 are said to be momentarily static or time symmetric. Indeed, if we consider a foliation with unit lapse around Σ0 (geodesic slicing), the following relation holds: Ln g = −2K, where Ln denotes the Lie derivative along the unit normal n. So if K = 0, Ln g = 0. This means that, locally (i.e. on Σ0), n is a spacetime Killing vector. This vector being timelike, the configuration is then stationary. Moreover, the Killing vector n being orthogonal to some hypersurface (i.e. Σ0), the stationary configuration is called static. Of course, this staticity properties holds a priori only on Σ0 since there is no guarantee that the time development of Cauchy data with K = 0 at t = 0 maintains K = 0 at t > 0. Hence the qualifier ‘momentarily’ in the expression ‘momentarily static’ for data with K = 0. 4.2. Slice of Schwarzschild spacetime To get something less trivial than a slice of Minkowski spacetime, let us consider a slightly more complicated topology for Σ0, namely R 3 minus a ball (cf. Fig. 1). The sphere S delimiting the ball is then the inner boundary of Σ0 and we must provide boundary conditions for Ψ and X on S to solve Eqs. (44)-(45). For simplicity, let us choose = 0. (54) Altogether with the outer boundary condition (48), this leads to X being identically zero as the unique solution of Eq. (45). So, again, the Hamiltonian constraint reduces to Laplace equation ∆Ψ = 0. (55) Construction of initial data for 3+1 numerical relativity 10 Figure 2. Same hypersurface Σ0 as in Fig. 1 but displayed via an embedding diagram based on the metric γ instead of γ̃. The unit normal of the inner boundary S with respect to that metric is s. Notice that D · s = 0, which means that S is a minimal surface of (Σ0,γ). If we choose the boundary condition Ψ| = 1, then the unique solution is Ψ = 1 and we are back to the previous example (slice of Minkowski spacetime). In order to have something non trivial, i.e. to ensure that the metric γ will not be flat, let us demand that γ admits a closed minimal surface, that we will choose to be S. This will necessarily translate as a boundary condition for Ψ since all the information on the metric is encoded in Ψ (let us recall that from the choice (40), γ = Ψ4f ). S is a minimal surface of (Σ0,γ) iff its mean curvature vanishes, or equivalently if its unit normal s is divergence-free (cf. Fig. 2): = 0. (56) This is the analog of ∇ · n = 0 for maximal hypersurfaces, the change from minimal to maximal being due to the change of metric signature, from the Riemannian to the Lorentzian one. Expressed in term of the connection D̃ = D (recall that in the present case γ̃ = f), condition (56) is equivalent to Di(Ψ6si) = 0. (57) Let us rewrite this expression in terms of the unit vector s̃ normal to S with respect to the metric γ̃ (cf. Fig. 1); we have s̃ = Ψ−2s, (58) since γ̃(s̃, s̃) = Ψ−4γ̃(s, s) = γ(s, s) = 1. Thus Eq. (57) becomes Di(Ψ4s̃i) fΨ4s̃i = 0. (59) Let us introduce on Σ0 a coordinate system of spherical type, (x i) = (r, θ, ϕ), such that (i) fij = diag(1, r 2, r2 sin2 θ) and (ii) S is the sphere r = a, where a is some positive constant. Since in these coordinates f = r2 sin θ and s̃i = (1, 0, 0), the minimal surface condition (59) is written as = 0, (60) Construction of initial data for 3+1 numerical relativity 11 Figure 3. Extended hypersurface Σ′ obtained by gluing a copy of Σ0 at the minimal surface S; it defines an Einstein-Rosen bridge between two asymptotically flat regions. = 0 (61) This is a boundary condition of mixed Newmann/Dirichlet type for Ψ. The unique solution of the Laplace equation (55) which satisfies boundary conditions (47) and (61) is Ψ = 1 + . (62) The parameter a is then easily related to the ADM mass m of the hypersurface Σ0. Indeed for a conformally flat 3-metric (and more generally in the quasi-isotropic gauge, cf. Chap. 7 of Ref. [51]), the ADM mass m is given by the flux of the gradient of the conformal factor at spatial infinity: m = − 1 r=const r2 sin θ dθ dϕ = − 1 = 2a. (63) Hence a = m/2 and we may write Ψ = 1 + . (64) Therefore, in terms of the coordinates (r, θ, ϕ), the obtained initial data (γ,K) are γij = diag(1, r2, r2 sin θ) (65) Kij = 0. (66) So, as above, the initial data are momentarily static. Actually, we recognize on (65)- (66) a slice t = const of Schwarzschild spacetime in isotropic coordinates. The isotropic coordinates (r, θ, ϕ) covering the manifold Σ0 are such that the range of r is [m/2,+∞). But thanks to the minimal character of the inner boundary S, we can extend (Σ0,γ) to a larger Riemannian manifold (Σ′0,γ ′) with γ′|Σ0 = γ and γ′ smooth at S. This is made possible by gluing a copy of Σ0 at S (cf. Fig. 3). Construction of initial data for 3+1 numerical relativity 12 Figure 4. Extended hypersurface Σ′ depicted in the Kruskal-Szekeres representation of Schwarzschild spacetime. R stands for Schwarzschild radial coordinate and r for the isotropic radial coordinate. R = 0 is the singularity and R = 2m the event horizon. Σ′ is nothing but a hypersurface t = const, where t is the Schwarzschild time coordinate. In this diagram, these hypersurfaces are straight lines and the Einstein-Rosen bridge S is reduced to a point. The topology of Σ′0 is S 2×R and the range of r in Σ′0 is (0,+∞). The extended metric γ ′ keeps exactly the same form as (65): γ′ij dx i dxj = dr2 + r2dθ2 + r2 sin2 θdϕ2 . (67) By the change of variable r 7→ r′ = m it is easily shown that the region r → 0 does not correspond to some “center” but is actually a second asymptotically flat region (the lower one in Fig. 3). Moreover the transformation (68), with θ and ϕ kept fixed, is an isometry of γ ′. It maps a point p of Σ0 to the point located at the vertical of p in Fig. 3. The minimal sphere S is invariant under this isometry. The region around S is called an Einstein-Rosen bridge. (Σ′0,γ ′) is still a slice of Schwarzschild spacetime. It connects two asymptotically flat regions without entering below the event horizon, as shown in the Kruskal-Szekeres diagram of Fig. 4. 4.3. Bowen-York initial data Let us select the same simple free data as above, namely γ̃ij = fij ,  TT = 0, K = 0, Ẽ = 0 and p̃ i = 0. (69) For the hypersurface Σ0, instead of R 3 minus a ball, we choose R3 minus a point: Σ0 = R 3\{O}. (70) The removed point O is called a puncture [21]. The topology of Σ0 is S 2×R; it differs from the topology considered in Sec. 4.1 (R3 minus a ball); actually it is the same topology as that of the extended manifold Σ′0 (cf. Fig. 3). Construction of initial data for 3+1 numerical relativity 13 Thanks to the choice (69), the system to be solved is still (44)-(45). If we choose the trivial solution X = 0 for Eq. (45), we are back to the slice of Schwarzschild spacetime considered in Sec. 4.1, except that now Σ0 is the extended manifold previously denoted Σ′0. Bowen and York [20] have obtained a simple non-trivial solution to the momentum constraint (45) (see also Ref. [15]). Given a Cartesian coordinate system (xi) = (x, y, z) on Σ0 (i.e. a coordinate system such that fij = diag(1, 1, 1)) with respect to which the coordinates of the puncture O are (0, 0, 0), this solution writes X i = − 1 7f ijPj + k, (71) where r := x2 + y2 + z2, ǫ k is the Levi-Civita alternating tensor associated with the flat metric f and (Pi, Sj) = (P1, P2, P3, S1, S2, S3) are six real numbers, which constitute the six parameters of the Bowen-York solution. Notice that since r 6= 0 on Σ0, the Bowen-York solution is a regular and smooth solution on the entire Σ0. The conformal traceless extrinsic curvature corresponding to the solution (71) is deduced from formula (13), which in the present case reduces to Âij = (LX)ij ; one Âij = xiP j + xjP i − f ij − x ǫiklSkx lxj + ǫ , (72) where P i := f ijPj . The tensor  ij given by Eq. (72) is called the Bowen-York extrinsic curvature. Notice that the Pi part of  ij decays asymptotically as O(r−2), whereas the Si part decays as O(r Remark : Actually the expression of Âij given in the original Bowen-York article [20] contains an additional term with respect to Eq. (72), but the role of this extra term is only to ensure that the solution is isometric through an inversion across some sphere. We are not interested by such a property here, so we have dropped this term. Therefore, strictly speaking, we should name expression (72) the simplified Bowen-York extrinsic curvature. The Bowen-York extrinsic curvature provides an analytical solution of the momentum constraint (45) but there remains to solve the Hamiltonian constraint (44) for Ψ, with the asymptotic flatness boundary condition Ψ = 1 when r → ∞. Since X 6= 0, Eq. (44) is no longer a simple Laplace equation, as in Sec. 4.1, but a non-linear elliptic equation. There is no hope to get any analytical solution and one must solve Eq. (44) numerically to get Ψ and reconstruct the full initial data (γ,K) via Eqs. (29)-(30). The parameters Pi of the Bowen-York solution are nothing but the three components of the ADM linear momentum of the hypersurface Σ0 Similarly, the parameters Si of the Bowen-York solution are nothing but the three components of the angular momentum of the hypersurface Σ0, the latter being defined relatively to the quasi-isotropic gauge, in the absence of any axial symmetry (see e.g. [51]). Remark : The Bowen-York solution with P i = 0 and Si = 0 reduces to the momentarily static solution found in Sec. 4.1, i.e. is a slice t = const of the Schwarzschild spacetime (t being the Schwarzschild time coordinate). However Bowen-York initial data with P i = 0 and Si 6= 0 do not constitute a slice of Kerr spacetime. Indeed, it has been shown [47] that there does not exist any foliation of Kerr spacetime by hypersurfaces which (i) are axisymmetric, (ii) smoothly Construction of initial data for 3+1 numerical relativity 14 reduce in the non-rotating limit to the hypersurfaces of constant Schwarzschild time and (iii) are conformally flat, i.e. have induced metric γ̃ = f , as the Bowen-York hypersurfaces have. This means that a Bowen-York solution with Si 6= 0 does represent initial data for a rotating black hole, but this black hole is not stationary: it is “surrounded” by gravitational radiation, as demonstrated by the time development of these initial data [22, 49]. 5. Conformal thin sandwich method 5.1. The original conformal thin sandwich method An alternative to the conformal transverse-traceless method for computing initial data has been introduced by York in 1999 [107]. The starting point is the identity K = − 1 LNnγ = − γ, (73) where N is the lapse function and β is the shift vector associated with some 3+1 coordinates (t, xi). The traceless part of Eq. (73) leads to Ãij = γ̃ij − 2 k γ̃ij , (74) where Ãij is defined by Eq. (8). Noticing that − Lβ γ̃ij = (L̃β)ij + k, (75) and introducing the short-hand notation γ̃ij , (76) we can rewrite Eq. (74) as Ãij = + (L̃β)ij . (77) The relation between Ãij and Âij is [cf. Eqs. (7)-(8)] Âij = Ψ6Ãij . (78) Accordingly, Eq. (77) yields Âij = + (L̃β)ij , (79) where we have introduced the conformal lapse Ñ := Ψ−6N. (80) Equation (79) constitutes a decomposition of Âij alternative to the longitudi- nal/transverse decomposition (13). Instead of expressing Âij in terms of a vector X and a TT tensor  TT, it expresses it in terms of the shift vector β, the time derivative of the conformal metric, ˙̃γ , and the conformal lapse Ñ . The Hamiltonian constraint, written as the Lichnerowicz equation (10), takes the same form as before: D̃iD̃ iΨ− R̃ Âij ij Ψ−7 + 2πẼΨ−3 − K Ψ5 = 0, (81) Construction of initial data for 3+1 numerical relativity 15 except that now Âij is to be understood as the combination (79) of βi, ˙̃γ and Ñ . On the other side, the momentum constraint (11) becomes, once expression (79) is substituted for Âij , (L̃β)ij + D̃j Ψ6D̃iK = 16πp̃i. (82) In view of the system (81)-(82), the method to compute initial data consists in choosing freely γ̃ij , ˙̃γ , K, Ñ , Ẽ and p̃i on Σ0 and solving (81)-(82) to get Ψ and β This method is called conformal thin sandwich (CTS ), because one input is the time derivative ˙̃γ , which can be obtained from the value of the conformal metric on two neighbouring hypersurfaces Σt and Σt+δt (“thin sandwich” view point). Remark : The term “thin sandwich” originates from a previous method devised in the early sixties by Wheeler and his collaborators [4, 101]. Contrary to the methods exposed here, the thin sandwich method was not based on a conformal decomposition: it considered the constraint equations (1)-(2) as a system to be solved for the lapse N and the shift vector β, given the metric γ and its time derivative. The extrinsic curvature which appears in (1)-(2) was then considered as the function of γ, ∂γ/∂t, N and β given by Eq. (73). However, this method does not work in general [9]. On the contrary the conformal thin sandwich method introduced by York [107] and exposed above was shown to work [35]. As for the conformal transverse-traceless method treated in Sec. 3, on CMC hypersurfaces, Eq. (82) decouples from Eq. (81) and becomes an elliptic linear equation for β. 5.2. Extended conformal thin sandwich method An input of the above method is the conformal lapse Ñ . Considering the astrophysical problem stated in Sec. 2.2, it is not clear how to pick a relevant value for Ñ . Instead of choosing an arbitrary value, Pfeiffer and York [80] have suggested to compute Ñ from the Einstein equation giving the time derivative of the trace K of the extrinsic curvature, i.e. K = −Ψ−4 D̃iD̃ iN + 2D̃i lnΨ D̃ 4π(E + S) + Ãijà , (83) where S is the trace of the matter stress tensor as measured by the Eulerian observer: S = γµνTµν . This amounts to add this equation to the initial data system. More precisely, Pfeiffer and York [80] suggested to combine Eq. (83) with the Hamiltonian constraint to get an equation involving the quantity NΨ = ÑΨ7 and containing no scalar products of gradients as the D̃i lnΨD̃ iN term in Eq. (83), thanks to the identity D̃iD̃ iN + 2D̃i lnΨ D̃ iN = Ψ−1 D̃iD̃ i(NΨ) +ND̃iD̃ . (84) Expressing the left-hand side of the above equation in terms of Eq. (83) and substituting D̃iD̃ iΨ in the right-hand side by its expression deduced from Eq. (81), Construction of initial data for 3+1 numerical relativity 16 we get D̃iD̃ i(ÑΨ7)− (ÑΨ7) K2Ψ4 + Âij ijΨ−8 + 2π(Ẽ + 2S̃)Ψ−4 K̇ − βiD̃iK Ψ5 = 0, (85) where we have used the short-hand notation K̇ := and have set S̃ := Ψ8S. (87) Adding Eq. (85) to Eqs. (81) and (82), the initial data system becomes D̃iD̃ iΨ− R̃ Âij ij Ψ−7 + 2πẼΨ−3 − K Ψ5 = 0 (88) (L̃β)ij + D̃j Ψ6D̃iK = 16πp̃i (89) D̃iD̃ i(ÑΨ7)− (ÑΨ7) K2Ψ4 + Âij ijΨ−8 + 2π(Ẽ + 2S̃)Ψ−4 K̇ − βiD̃iK Ψ5 = 0, (90) where Âij is the function of Ñ , βi, γ̃ij and ˙̃γ defined by Eq. (79). Equations (88)-(90) constitute the extended conformal thin sandwich (XCTS ) system for the initial data problem. The free data are the conformal metric γ̃, its coordinate time derivative ˙̃γ, the extrinsic curvature trace K, its coordinate time derivative K̇, and the rescaled matter variables Ẽ, S̃ and p̃i. The constrained data are the conformal factor Ψ, the conformal lapse Ñ and the shift vector β. Remark : The XCTS system (88)-(90) is a coupled system. Contrary to the CTT system (26)-(27), the assumption of constant mean curvature, and in particular of maximal slicing, does not allow to decouple it. 5.3. XCTS at work: static black hole example Let us illustrate the extended conformal thin sandwich method on a simple example. Take for the hypersurface Σ0 the punctured manifold considered in Sec. 4.3, namely Σ0 = R 3\{O}. (91) For the free data, let us perform the simplest choice: γ̃ij = fij , ˙̃γ = 0, K = 0, K̇ = 0, Ẽ = 0, S̃ = 0, and p̃i = 0, (92) i.e. we are searching for vacuum initial data on a maximal and conformally flat hypersurface with all the freely specifiable time derivatives set to zero. Thanks to (92), the XCTS system (88)-(90) reduces to Âij ij Ψ−7 = 0 (93) (Lβ)ij = 0 (94) ∆(ÑΨ7)− Âij ijΨ−1Ñ = 0. (95) Construction of initial data for 3+1 numerical relativity 17 Aiming at finding the simplest solution, we notice that β = 0 (96) is a solution of Eq. (94). Together with ˙̃γ = 0, it leads to [cf. Eq. (79)] Âij = 0. (97) The system (93)-(95) reduces then further: ∆Ψ = 0 (98) ∆(ÑΨ7) = 0. (99) Hence we have only two Laplace equations to solve. Moreover Eq. (98) decouples from Eq. (99). For simplicity, let us assume spherical symmetry around the puncture O. We introduce an adapted spherical coordinate system (xi) = (r, θ, ϕ) on Σ0. The puncture O is then at r = 0. The simplest non-trivial solution of (98) which obeys the asymptotic flatness condition Ψ → 1 as r → +∞ is Ψ = 1 + , (100) where as in Sec. 4.1, the constant m is the ADM mass of Σ0 [cf. Eq. (63)]. Notice that since r = 0 is excluded from Σ0, Ψ is a perfectly regular solution on the entire manifold Σ0. Let us recall that the Riemannian manifold (Σ0,γ) corresponding to this value of Ψ via γ = Ψ4f is the Riemannian manifold denoted (Σ′0,γ) in Sec. 4.1 and depicted in Fig. 3. In particular it has two asymptotically flat ends: r → +∞ and r → 0 (the puncture). As for Eq. (98), the simplest solution of Eq. (99) obeying the asymptotic flatness requirement ÑΨ7 → 1 as r → +∞ is ÑΨ7 = 1 + , (101) where a is some constant. Let us determine a from the value of the lapse function at the second asymptotically flat end r → 0. The lapse being related to Ñ via Eq. (80), Eq. (101) is equivalent to Ψ−1 = r + a r +m/2 . (102) Hence . (103) There are two natural choices for limr→0 N . The first one is N = 1, (104) yielding a = m/2. Then, from Eq. (102) N = 1 everywhere on Σ0. This value of N corresponds to a geodesic slicing. The second choice is N = −1. (105) This choice is compatible with asymptotic flatness: it simply means that the coordinate time t is running “backward” near the asymptotic flat end r → 0. This contradicts the assumption N > 0 in the standard definition of the lapse function. However, we shall generalize here the definition of the lapse to allow for negative values: whereas the unit vector n is always future-oriented, the scalar field t is allowed to decrease towards the future. Such a situation has already been encountered for the Construction of initial data for 3+1 numerical relativity 18 part of the slices t = const located on the left side of Fig. 4. Once reported into Eq. (103), the choice (105) yields a = −m/2, so that . (106) Gathering relations (96), (100) and (106), we arrive at the following expression of the spacetime metric components: gµνdx µdxν = − 1 + m dt2 + dr2 + r2(dθ2 + sin2 θdϕ2) . (107) We recognize the line element of Schwarzschild spacetime in isotropic coordinates. Hence we recover the same initial data as in Sec. 4.1 and depicted in Figs. 3 and 4. The bonus is that we have the complete expression of the metric g on Σ0, and not only the induced metric γ. Remark : The choices (104) and (105) for the asymptotic value of the lapse both lead to a momentarily static initial slice in Schwarzschild spacetime. The difference is that the time development corresponding to choice (104) (geodesic slicing) will depend on t, whereas the time development corresponding to choice (105) will not, since in the latter case t coincides with the standard Schwarzschild time coordinate, which makes ∂t a Killing vector. 5.4. Uniqueness of solutions Recently, Pfeiffer and York [81] have exhibited a choice of vacuum free data (γ̃ij , ˙̃γ ,K, K̇) for which the solution (Ψ, Ñ , βi) to the XCTS system (88)-(90) is not unique (actually two solutions are found). The conformal metric γ̃ is the flat metric plus a linearized quadrupolar gravitational wave, as obtained by Teukolsky [92], with a tunable amplitude. ˙̃γ corresponds to the time derivative of this wave, and both K and K̇ are chosen to zero. On the contrary, for the same free data, with K̇ = 0 substituted by Ñ = 1, Pfeiffer and York have shown that the original conformal thin sandwich method as described in Sec. 5.1 leads to a unique solution (or no solution at all if the amplitude of the wave is two large). Baumgarte, Ó Murchadha and Pfeiffer [14] have argued that the lack of uniqueness for the XCTS system may be due to the term − (ÑΨ7)7 Âij ijΨ−8 = − 7 Ψ6γ̃ikγ̃jl + (L̃β)ij + (L̃β)kl (ÑΨ7)−1 (108) in Eq. (90). Indeed, if we proceed as for the analysis of Lichnerowicz equation in Sec. 3.4, we notice that this term, with the minus sign and the negative power of (ÑΨ7)−1, makes the linearization of Eq. (90) of the type D̃iD̃ iǫ+αǫ = σ, with α > 0. This “wrong” sign of α prevents the application of the maximum principle to guarantee the uniqueness of the solution. The non-uniqueness of solution of the XCTS system for certain choice of free data has been confirmed by Walsh [100] by means of bifurcation theory. 5.5. Comparing CTT, CTS and XCTS The conformal transverse traceless (CTT) method exposed in Sec. 3 and the (extended) conformal thin sandwich (XCTS) method considered here differ by the choice of free data: whereas both methods use the conformal metric γ̃ and the trace Construction of initial data for 3+1 numerical relativity 19 of the extrinsic curvature K as free data, CTT employs in addition  TT, whereas for CTS (resp. XCTS) the additional free data is ˙̃γ , as well as Ñ (resp. K̇). Since  TT is directly related to the extrinsic curvature and the latter is linked to the canonical momentum of the gravitational field in the Hamiltonian formulation of general relativity, the CTT method can be considered as the approach to the initial data problem in the Hamiltonian representation. On the other side, ˙̃γ being the “velocity” of γ̃ij , the (X)CTS method constitutes the approach in the Lagrangian representation [108]. Remark : The (X)CTS method assumes that the conformal metric is unimodular: det(γ̃ij) = f (since Eq. (79) follows from this assumption), whereas the CTT method can be applied with any conformal metric. The advantage of CTT is that its mathematical theory is well developed, yielding existence and uniqueness theorems, at least for constant mean curvature (CMC) slices. The mathematical theory of CTS is very close to CTT. In particular, the momentum constraint decouples from the Hamiltonian constraint on CMC slices. On the contrary, XCTS has a much more involved mathematical structure. In particular the CMC condition does not yield to any decoupling. The advantage of XCTS is then to be better suited to the description of quasi-stationary spacetimes, since ˙̃γ = 0 and K̇ = 0 are necessary conditions for ∂t to be a Killing vector. This makes XCTS the method to be used in order to prepare initial data in quasi-equilibrium. For instance, it has been shown [57, 43] that XCTS yields orbiting binary black hole configurations in much better agreement with post-Newtonian computations than the CTT treatment based on a superposition of two Bowen-York solutions. Indeed, except when they are very close and about to merge, the orbits of binary black holes evolve very slowly, so that it is a very good approximation to consider that the system is in quasi-equilibrium. XCTS takes this fully into account, while CTT relies on a technical simplification (Bowen-York analytical solution of the momentum constraint), with no direct relation to the quasi-equilibrium state. A detailed comparison of CTT and XCTS for a single spinning or boosted black hole has been performed by Laguna [68]. 6. Initial data for binary systems A major topic of contemporary numerical relativity is the computation of the merger of a binary system of black holes [24] or neutron stars [84], for such systems are among the most promising sources of gravitational radiation for the interferometric detectors either groundbased (LIGO, VIRGO, GEO600, TAMA) or in space (LISA). The problem of preparing initial data for these systems has therefore received a lot of attention in the past decade. 6.1. Helical symmetry Due to the gravitational-radiation reaction, a relativistic binary system has an inspiral motion, leading to the merger of the two components. However, when the two bodies are sufficiently far apart, one may approximate the spiraling orbits by closed ones. Moreover, it is well known that gravitational radiation circularizes the orbits very efficiently, at least for comparable mass systems [18]. We may then consider that the motion is described by a sequence of closed circular orbits. Construction of initial data for 3+1 numerical relativity 20 Figure 5. Action of the helical symmetry group, with Killing vector ℓ. χτ (P ) is the displacement of the point P by the member of the symmetry group of parameter τ . N and β are respectively the lapse function and the shift vector associated with coordinates adapted to the symmetry, i.e. coordinates (t, xi) such that ∂t = ℓ. The geometrical translation of this physical assumption is that the spacetime (M, g) is endowed with some symmetry, called helical symmetry. Indeed exactly circular orbits imply the existence of a one-parameter symmetry group such that the associated Killing vector ℓ obeys the following properties [46]: (i) ℓ is timelike near the system, (ii) far from it, ℓ is spacelike but there exists a smaller number T > 0 such that the separation between any point P and its image χT (P ) under the symmetry group is timelike (cf. Fig. 5). ℓ is called a helical Killing vector, its field lines in a spacetime diagram being helices (cf. Fig. 5). Helical symmetry is exact in theories of gravity where gravitational radiation does not exist, namely: • in Newtonian gravity, • in post-Newtonian gravity, up to the second order, • in the Isenberg-Wilson-Mathews (IWM) approximation to general relativity, based on the assumptions γ̃ = f and K = 0 [61, 102]. Moreover helical symmetry can be exact in full general relativity for a non- axisymmetric system (such as a binary) with standing gravitational waves [44]. But notice that a spacetime with helical symmetry and standing gravitational waves cannot be asymptotically flat [48]. To treat helically symmetric spacetimes, it is natural to choose coordinates (t, xi) that are adapted to the symmetry, i.e. such that ∂t = ℓ. (109) Then all the fields are independent of the coordinate t. In particular, = 0 and K̇ = 0. (110) Construction of initial data for 3+1 numerical relativity 21 If we employ the XCTS formalism to compute initial data, we therefore get some definite prescription for the free data ˙̃γ and K̇. On the contrary, the requirements (110) do not have any immediate translation in the CTT formalism. Remark : Helical symmetry can also be useful to treat binary black holes outside the scope of the 3+1 formalism, as shown by Klein [67], who developed a quotient space formalism to reduce the problem to a three dimensional SL(2,R)/SO(1, 1) sigma model. Taking into account (110) and choosing maximal slicing (K = 0), the XCTS system (88)-(90) becomes D̃iD̃ Âij ij Ψ−7 + 2πẼΨ−3 = 0 (111) (L̃β)ij − 16πp̃i = 0 (112) D̃iD̃ i(ÑΨ7)− (ÑΨ7) Âij ijΨ−8 + 2π(Ẽ + 2S̃)Ψ−4 = 0, (113) where [cf. Eq. (79)] Âij = (L̃β)ij . (114) 6.2. Helical symmetry and IWM approximation If we choose, as part of the free data, the conformal metric to be flat, γ̃ij = fij , (115) then the helically symmetric XCTS system (111)-(113) reduces to Âij ij Ψ−7 + 2πẼΨ−3 = 0 (116) ∆βi + DiDjβj − (Lβ)ijDj ln Ñ = 16πÑp̃i (117) ∆(ÑΨ7)− (ÑΨ7) Âij ijΨ−8 + 2π(Ẽ + 2S̃)Ψ−4 = 0, (118) where Âij = (Lβ)ij (119) and D is the connection associated with the flat metric f , ∆ := DiDi is the flat Laplacian [Eq. (46)], and (Lβ)ij := Diβj +Djβi− 2 Dkβk f ij [Eq. (15) with D̃i = Di]. We remark that the system (116)-(118) is identical to the system defining the Isenberg-Wilson-Mathews approximation to general relativity [61, 102] (see e.g. Sec. 6.6 of Ref. [51]). This means that, within helical symmetry, the XCTS system with the choice K = 0 and γ̃ = f is equivalent to the IWM system. Remark : Contrary to IWM, XCTS is not some approximation to general relativity: it provides exact initial data. The only thing that may be questioned is the astrophysical relevance of the XCTS data with γ̃ = f . Construction of initial data for 3+1 numerical relativity 22 6.3. Initial data for orbiting binary black holes The concept of helical symmetry for generating orbiting binary black hole initial data has been introduced in 2002 by Gourgoulhon, Grandclément and Bonazzola [52, 57]. The system of equations that these authors have derived is equivalent to the XCTS system with γ̃ = f , their work being previous to the formulation of the XCTS method by Pfeiffer and York (2003) [80]. Since then other groups have combined XCTS with helical symmetry to compute binary black hole initial data [38, 1, 2, 31]. Since all these studies are using a flat conformal metric [choice (115)], the PDE system to be solved is (116)-(118), with the additional simplification Ẽ = 0 and p̃i = 0 (vacuum). The initial data manifold Σ0 is chosen to be R 3 minus two balls: Σ0 = R 3\(B1 ∪ B2). (120) In addition to the asymptotic flatness conditions, some boundary conditions must be provided on the surfaces S1 and S2 of B1 and B2. One choose boundary conditions corresponding to a non-expanding horizon, since this concept characterizes black holes in equilibrium. We shall not detail these boundary conditions here; they can be found in Refs. [38, 40, 41, 54, 65]. The condition of non-expanding horizon provides 3 among the 5 required boundary conditions [for the 5 components (Ψ, Ñ , βi)]. The two remaining boundary conditions are given by (i) the choice of the foliation (choice of the value of N at S1 and S2) and (ii) the choice of the rotation state of each black hole (“individual spin”), as explained in Ref. [31]. Numerical codes for solving the above system have been constructed by • Grandclément, Gourgoulhon and Bonazzola (2002) [57] for corotating binary black holes; • Cook, Pfeiffer, Caudill and Grigsby (2004, 2006) [38, 31] for corotating and irrotational binary black holes; • Ansorg (2005, 2007) [1, 2] for corotating binary black holes. Detailed comparisons with post-Newtonian initial data (either from the standard post- Newtonian formalism [17] or from the Effective One-Body approach [23, 42]) have revealed a very good agreement, as shown in Refs. [43, 31]. An alternative to (120) for the initial data manifold would be to consider the twice-punctured R3: Σ0 = R 3\{O1, O2}, (121) where O1 and O2 are two points of R 3. This would constitute some extension to the two bodies case of the punctured initial data discussed in Sec. 5.3. However, as shown by Hannam, Evans, Cook and Baumgarte in 2003 [60], it is not possible to find a solution of the helically symmetric XCTS system with a regular lapse in this case‖. For this reason, initial data based on the puncture manifold (121) are computed within the CTT framework discussed in Sec. 3. As already mentioned, there is no natural way to implement helical symmetry in this framework. One instead selects the free data  TT to vanish identically, as in the single black hole case treated in Secs. 4.1 and 4.3. Then Âij = (L̃X)ij . (122) ‖ see however Ref. [59] for some attempt to circumvent this Construction of initial data for 3+1 numerical relativity 23 The vector X must obey Eq. (45), which arises from the momentum constraint. Since this equation is linear, one may choose for X a linear superposition of two Bowen-York solutions (Sec. 4.3): X = X(P (1),S(1)) +X(P (2),S(2)), (123) where X(P (a),S(a)) (a = 1, 2) is the Bowen-York solution (71) centered on Oa. This method has been first implemented by Baumgarte in 2000 [11]. It has been since then used by Baker, Campanelli, Lousto and Takashi (2002) [5] and Ansorg, Brügmann and Tichy (2004) [3]. The initial data hence obtained are closed from helically symmetric XCTS initial data at large separation but deviate significantly from them, as well as from post-Newtonian initial data, when the two black holes are very close. This means that the Bowen-York extrinsic curvature is bad for close binary systems in quasi-equilibrium (see discussion in Ref. [43]). Remark : Despite of this, CTT Bowen-York configurations have been used as initial data for the recent binary black hole inspiral and merger computations by Baker et al. [6, 7, 99] and Campanelli et al. [25, 26, 27, 28]. Fortunately, these initial data had a relative large separation, so that they differed only slightly from the helically symmetric XCTS ones. Instead of choosing somewhat arbitrarily the free data of the CTT and XCTS methods, notably setting γ̃ = f , one may deduce them from post-Newtonian results. This has been done for the binary black hole problem by Tichy, Brügmann, Campanelli and Diener (2003) [94], who have used the CTT method with the free data (γ̃ij ,  given by the second order post-Newtonian (2PN) metric. This work has been improved recently by Kelly, Tichy, Campanelli and Whiting (2007) [66]. In the same spirit, Nissanke (2006) [75] has provided 2PN free data for both the CTT and XCTS methods. 6.4. Initial data for orbiting binary neutron stars For computing initial data corresponding to orbiting binary neutron stars, one must solve equations for the fluid motion in addition to the Einstein constraints. Basically this amounts to solving ∇νT µν = 0 in the context of helical symmetry. One can then show that a first integral of motion exists in two cases: (i) the stars are corotating, i.e. the fluid 4-velocity is colinear to the helical Killing vector (rigid motion), (ii) the stars are irrotational, i.e. the fluid vorticity vanishes. The most straightforward way to get the first integral of motion is by means of the Carter-Lichnerowicz formulation of relativistic hydrodynamics, as shown in Sec. 7 of Ref. [50]. Other derivations have been obtained in 1998 by Teukolsky [93] and Shibata [83]. From the astrophysical point of view, the irrotational motion is much more interesting than the corotating one, because the viscosity of neutron star matter is far too low to ensure the synchronization of the stellar spins with the orbital motion. On the other side, the irrotational state is a very good approximation for neutron stars that are not millisecond rotators. Indeed, for these stars the spin frequency is much lower than the orbital frequency at the late stages of the inspiral and thus can be neglected. The first initial data for binary neutron stars on circular orbits have been computed by Baumgarte, Cook, Scheel, Shapiro and Teukolsky in 1997 [12, 13] in the corotating case, and by Bonazzola, Gourgoulhon and Marck in 1999 [19] in the irrotational case. These results were based on a polytropic equation of state. Since then configurations in the irrotational regime have been obtained Construction of initial data for 3+1 numerical relativity 24 • for a polytropic equation of state [73, 96, 97, 53, 90, 91] (the configurations obtained in Ref. [91] have been used as initial data by Shibata [84] to compute the merger of binary neutron stars); • for nuclear matter equations of state issued from recent nuclear physics computations [16, 77]; • for strange quark matter [78, 72]. All these computation are based on a flat conformal metric [choice (115)], by solving the helically symmetric XCTS system (116)-(118), supplemented by an elliptic equation for the velocity potential. Only very recently, configurations based on a non flat conformal metric have been obtained by Uryu, Limousin, Friedman, Gourgoulhon and Shibata [98]. The conformal metric is then deduced from a waveless approximation developed by Shibata, Uryu and Friedman [85] and which goes beyond the IWM approximation. 6.5. Initial data for black hole - neutron star binaries Let us mention briefly that initial data for a mixed binary system, i.e. a system composed of a black hole and a neutron star, have been obtained very recently by Grandclément [55] and Taniguchi, Baumgarte, Faber and Shapiro [88, 89]. Codes aiming at computing such systems have also been presented by Ansorg [2] and Tsokaros and Uryu [95]. Acknowledgments I warmly thank the organizers of the VII Mexican school, namely Miguel Alcubierre, Hugo Garcia-Compean and Luis Urena, for their support and the success of the school. I also express my gratitude to Marcelo Salgado for his help and many discussions and to Nicolas Vasset for the careful reading of the manuscript. References [1] M. Ansorg : Double-domain spectral method for black hole excision data, Phys. Rev. D 72, 024018 (2005). [2] M. Ansorg: Multi-Domain Spectral Method for Initial Data of Arbitrary Binaries in General Relativity, Class. Quantum Grav. 24, S1 (2007). [3] M. Ansorg, B. Brügmann and W. Tichy : Single-domain spectral method for black hole puncture data, Phys. Rev. D 70, 064011 (2004). [4] R.F. Baierlein, D.H Sharp and J.A. Wheeler : Three-Dimensional Geometry as Carrier of Information about Time, Phys. Rev. 126, 1864 (1962). [5] J.G. Baker, M. Campanelli, C.O. Lousto and R. 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Introduction The initial data problem 3+1 decomposition of Einstein equation Constructing initial data Conformal decomposition of the constraints Conformal transverse-traceless method Longitudinal/transverse decomposition of "705EAij Conformal transverse-traceless form of the constraints Decoupling on hypersurfaces of constant mean curvature Lichnerowicz equation Conformally flat initial data by the CTT method Momentarily static initial data Slice of Schwarzschild spacetime Bowen-York initial data Conformal thin sandwich method The original conformal thin sandwich method Extended conformal thin sandwich method XCTS at work: static black hole example Uniqueness of solutions Comparing CTT, CTS and XCTS Initial data for binary systems Helical symmetry Helical symmetry and IWM approximation Initial data for orbiting binary black holes Initial data for orbiting binary neutron stars Initial data for black hole - neutron star binaries

0704.0150

Magnetism and Thermodynamics of Spin-1/2 Heisenberg Diamond Chains in a Magnetic Field

Magnetism and Thermodynamics of Spin-1/2 Heisenberg Diamond Chains in a Magnetic Field Bo Gu and Gang Su∗ College of Physical Sciences, Graduate University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China The magnetic and thermodynamic properties of spin-1/2 Heisenberg diamond chains are investi- gated in three different cases: (a) J1, J2, J3 > 0 (frustrated); (b) J1, J3 < 0, J2 > 0 (frustrated); and (c) J1, J2 > 0, J3 < 0 (non-frustrated), where the bond coupling Ji (i = 1, 2, 3) > 0 stands for an antiferromagnetic (AF) interaction, and < 0 for a ferromagnetic (F) interaction. The density matrix renormalization group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer matrix renormalization group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor S(q) shows peaks at wavevectors q = aπ/3 (a = 0, 1, 2, 3, 4, 5) for different couplings in a zero magnetic field, which, however in the magnetic fields where the magnetization plateau with m = 1/6 pertains, exhibits the peaks only at q = 0, 2π/3 and 4π/3, which are found to be couplings-independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the XXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat and magnetization for the compound Cu3(CO3)2(OH)2 are fairly compared with our TMRG results. PACS numbers: 75.10.Jm, 75.40.Cx I. INTRODUCTION Low-dimensional quantum spin systems with compet- ing interactions have become an intriguing subject in the last decades. Among many achievements in this area, the phenomenon of the topological quantization of mag- netization has attracted much attention both theoreti- cally and experimentally. A general necessary condi- tion for the appearance of the magnetization plateaus has been proposed by Oshikawa, Yamanaka and Affleck (OYA) [1], stating that for the Heisenberg antiferromag- netic (AF) spin chain with a single-ion anisotropy, the magnetization curve may have plateaus at which the magnetization per site m is topologically quantized by n(S − m) = integer, where S is the spin, and n is the period of the ground state determined by the explicit spatial structure of the Hamiltonian. As one of fasci- nating models which potentially possesses the magne- tization plateaus, the Heisenberg diamond chain, con- sisting of diamond-shaped topological unit along the chain, as shown in Fig. 1, has also gained much atten- tion both experimentally and theoretically (e.g. Refs. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). It has been observed that the compounds, A3Cu3(PO4)4 with A = Ca, Sr[2], and Bi4Cu3V2O14[3] can be nicely modeled by the Heisenberg diamond chain. Another spin-1/2 compound Cu3Cl6(H2O)2·2H8C4SO2 was initially regarded as a model substance for the spin-1/2 diamond chain [4], but a later experimental research reveals that this compound should be described by a double chain model with very weak bond alterna- tions, and the lattice of the compound is found to be Cu2Cl4·H8C4SO2 [5]. Recently, Kikuchi et al.[6] have reported the experimental results on a spin-1/2 com- pound Cu3(CO3)2(OH)2, where local Cu 2+ ions with spin S = 1/2 are arranged along the chain direction, and the diamond-shaped units consist of a one-dimensional (1D) lattice. The 1/3 magnetization plateau and the double peaks in the magnetic susceptibility as well as the specific heat as functions of temperature have been observed experimentally[6, 7], which has been discussed in terms of the spin-1/2 Heisenberg diamond chain with AF couplings J1, J2 and J3 > 0. On the theoretical aspect, the frustrated diamond spin chain with AF interactions J1, J2 and J3 > 0 was studied by a few groups. The first diamond spin chain was ex- plored under a symmetrical condition J1 = J3[8]. Owing to the competition of AF interactions, the phase diagram in the ground state of the spin-1/2 frustrated diamond chain was found to contain different phases, in which the magnetization plateaus at m = 1/6 as well as 1/3 are predicted[9, 10, 11, 12, 13]. Another frustrated diamond chain with ferromagnetic (F) interactions J1, J3 < 0 and AF interaction J2 > 0 was also investigated theoretically, which can be experimentally realized if all angles of the exchange coupling bonds are arranged to be around 90◦, a region where it is usually hard to determine safely the coupling constants and even about their signs[14, 15]. Despite of these works, the investigations on the Heisen- http://arxiv.org/abs/0704.0150v1 berg diamond spin chain with various competing inter- actions are still sparse. Motivated by the recent experimental observation on the azurite compound Cu3(CO3)2(OH)2[6, 7], we shall explore systematically the magnetic and thermodynamic properties of the spin-1/2 Heisenberg diamond chain with various competing interactions in a magnetic field, and attempt to fit into the experimental observation on the azurite in a consistent manner. The density matrix renor- malization group (DMRG) as well as the transfer ma- trix renormalization group (TMRG) techniques will be invoked to study the ground-state properties and ther- modynamics of the model under interest, respectively. The local magnetic moments, spin correlation functions, and static structure factors will be discussed for three cases at zero temperature. It is found that the static structure factor S(q) shows peaks in zero magnetic field at wavevectors q = aπ/3 (a = 0, 1, 2, 3, 4, 5) for different couplings, while in the magnetic fields where the mag- netization plateau with m = 1/6 remains, the peaks ap- pear only at wavevectors q = 0, 2π/3 and 4π/3, which are found to be couplings-independent. These informa- tion could be useful for further neutron studies. The double-peak structures of the susceptibility and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the com- petition of the couplings. It is uncovered that the XXZ anisotropy of F and AF couplings leads the system with- out frustration (see below) to display quite different be- haviors. In addition, the experimental data of the sus- ceptibility, specific heat and magnetization for the com- pound Cu3(CO3)2(OH)2 are fairly compared with our TMRG results. The rest of this paper is outlined as follows. In Sec. II, we shall introduce the model Hamiltonian for the spin- 1/2 Heisenberg diamond chain with three couplings J1, J2 and J3, where three particular cases are identified. In Sec. III, the magnetic and thermodynamic properties of a frustrated diamond chain with AF interactions J1, J2 and J3 > 0 will be discussed. In Sec. IV, the physi- cal properties of another frustrated diamond chain with F interactions J1, J3 < 0 and AF interaction J2 > 0 will be considered. In Sec. V, the magnetism and ther- modynamics of a non-frustrated diamond chain with AF interactions J1, J2 > 0 and F interaction J3 < 0 will be explored, and a comparison to the experimental data on the azurite compound will be made. Finally, a brief summary and discussion will be presented in Sec. VI. FIG. 1: (Color online) Sketch of the Heisenberg diamond chain. The bond interactions are denoted by J1, J2, and J3. Three cases will be considered: (a) J1, J2, J3 > 0 (a frustrated diamond chain); (b) J1, J3 < 0, J2 > 0 (a frustrated diamond chain with competing interactions); and (c) J1, J2 > 0, J3 < 0 (a diamond chain without frustration). Note that Ji > 0 stands for an antiferromagnetic interaction while Ji < 0 for a ferromagnetic interaction, where i = 1, 2, 3. II. MODEL The Hamiltonian of the spin-1/2 Heisenberg diamond chain reads (J1S3i−2 · S3i−1 + J2S3i−1 · S3i +J3S3i−2 · S3i + J3S3i−1 · S3i+1 +J1S3i · S3i+1)−H · Sj , (1) where Sj is the spin operator at the jth site, L is the to- tal number of spins in the diamond chain, Ji (i = 1, 2, 3) stands for exchange interactions, H is the external mag- netic field, gµB = 1 and kB = 1. Ji > 0 represents the AF coupling while Ji < 0 the F interaction. There are three different cases particularly interesting, as displayed in Fig. 1, which will be considered in the present paper: (a) a frustrated diamond chain with J1, J2, J3 > 0; (b) a frustrated diamond chain with competing interactions J1, J3 < 0, J2 > 0; and (c) a diamond chain without frustration with J1, J2 > 0, J3 < 0. It should be re- marked that in the case (c) of this model, since the two end points of the J2 bond represent the two different lat- tice sites, it is possible that J1 and J3 can be different, even their signs. The magnetic properties and thermodynamics for the aforementioned three spin-1/2 Heisenberg diamond chains in the ground states and at finite temperatures will be investigated by means of the DMRG and TMRG methods, respectively. As the DMRG and TMRG tech- niques were detailed in two nice reviews[16, 17], we shall not repeat the technical details for concise. In the ground-state calculations, the total number of spins in the diamond chain is taken at least as L = 120. At finite temperatures, the thermodynamic properties presented below are calculated down to temperature T = 0.05 (in units of |J1|) in the thermodynamic limit. In our calcu- lations, the number of kept optimal states is taken as 81; the width of the imaginary time slice is taken as ε = 0.1; the Trotter-Suzuki error is less than 10−3; and the trun- cation error is smaller than 10−6. III. A FRUSTRATED HEISENBERG DIAMOND CHAIN (J1, J2, J3 > 0) A. Local Magnetic Moment and Spin Correlation Function Figure 2(a) manifests the magnetization process of a frustrated spin-1/2 Heisenberg diamond chain with the couplings satisfying J1 : J2 : J3 = 1 : 2 : 2 at zero tem- perature. The plateau of magnetization per site m = 1/6 is observed. According to the OYA necessary condition [1], the m = 1/6 plateau of a spin-1/2 Heisenberg chain corresponds to the period of the ground state n = 3. Beyond the magnetization plateau region, the magnetic curve goes up quickly with increasing the magnetic field H . Above the upper critical field, the magnetic curve shows a s-like shape. To further look at how this magneti- zation plateau appears, the spatial dependence of the av- eraged local magnetic moment 〈Szj 〉 in the ground states under different external fields is presented, as shown in Fig. 2(b). It is seen that in the absence of external field, the expectation value 〈Szj 〉 changes its sign at every three sites within a very small range of (−10−3, 10−3) because of quantum fluctuations, resulting in the magne- tization per site m = j=1〈Szj 〉/L = 0. 〈Szj 〉 increases with increasing the magnetic field, and oscillates with increasing j, whose unit of three spins is gradually di- vided into a pair and a single, as displayed in Fig. 2(c). At the field H/J1 = 1.5, as demonstrated in Fig. 2(d), the behavior of 〈Szj 〉 falls into a perfect sequence such as {..., (Sa, Sa, Sb), ...} with Sa = 0.345 and Sb = −0.190, giving rise to the magnetization per site m = 1/6. In ad- dition, such a sequence remains with increasing the mag- netic field till H/J1 = 2.5, implying that the m = 1/6 plateau appears in the range of H/J1 = 1.5 ∼ 2.5, as manifested in Fig. 2(a). When the field is promoted fur- ther, the sequence changes into a waved succession with smaller swing of (Sa − Sb), as shown in Fig. 2(e), which corresponds to the fact that the plateau state ofm = 1/6 is destroyed, and gives rise to a s-like shape of M(H). It is noting that when the plateau state of m = 1/6 is de- stroyed, the increase of m at first is mainly attributed to a rapid lift of Sb, and later, the double Sa start to flimsily increase till Sa = Sb = 0.5 at the saturated field. The physical picture for the above results could be un- derstood as follows. For the m = 1/6 plateau state at 0 1 2 3 4 5 0 40 80 0 40 80 T = 0 = 1 : 2 : 2, -1E-3 T = 0, H = 0 (d) H/J = 1.5, 2.5 (e) H/J FIG. 2: (Color online) For a spin-1/2 frustrated Heisenberg diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : 2, (a) the magnetization per site m as a function of magnetic field H in the ground states; and the spatial dependence of the averaged local magnetic moment 〈Szj 〉 in the ground states with external field (b) H/J1 = 0, (c) 1, (d) 1.5 and 2.5, and (e) 4. J1 : J2 : J3 = 1 : 2 : 2, we note that if an approximate wave function defined by[13] (2| ↑3i−2↑3i−1↓3i〉 ± | ↑3i−2↓3i−1↑3i〉 ± | ↓3i−2↑3i−1↑3i〉), (i = 1, ..., L/3) (2) where ↑j (↓j) denotes spin up (down) on site j, is applied, one may obtain 〈ψi|Sz3i−2|ψi〉 = 1/3, 〈ψi|Sz3i−1|ψi〉 = 1/3, 〈ψi|Sz3i|ψi〉 = −1/6, giving rise to a sequence {..., (13 , ), ...}, andm = (1 )/3 = 1/6, which is in agree- ment with our DMRG results {..., (0.345, 0.345, −0.190), ...}. This observation shows that the ground state of this plateau state might be described by trimerized states. Let Hc1 and Hc2 be the lower and upper critical mag- netic field at which the magnetization plateau appears and is destructed, respectively. For Hc1 ≤ H ≤ Hc2, the magnetization m = mp = 1/6, namely, the system falls into the magnetization plateau state. For 0 ≤ H ≤ Hc1 and J1 : J2 : J3 = 1 : 2 : 2, the magnetization curve shows the following behavior m(H) = mp( )[1+α1(1− )−α2(1− )2/3], (3) where Hc1/J1 = 1.44, the parameters α1 = 2/3 and α2 = 1. Obviously, when H = 0, m = 0; H = Hc1 , m = mp. A fair comparison of Eq. (3) to the DMRG results is presented in Fig. 3(a). For Hc2 ≤ H ≤ Hs and J1 : J2 : J3 = 1 : 2 : 2, where Hs is the saturated magnetic field, the magnetization curve has the form of m(H) = mp + (H −Hc2){kc + (Hs −H)[ (H −Hc2)1/3 (Hs −H)1/3 ]}, (4) where kc = (ms −mp)/(Hs −Hc2) with ms the satura- tion magnetization, and β1, β2 the parameters. One may see that when H = Hc2 , m = mp; H = Hs, m = ms. A nice fitting to the DMRG result gives the parameters Hc2/J1 = 3.15, ms = 1/2, Hs/J1 = 4.55, β1 = 0.143, β2 = 0.178 and kc = 0.238, as shown in Fig. 3(b). It should be remarked that from the phenomenological Eqs.(3) and (4), we find that, away from the plateau region, the magnetic field dependence of the magnetiza- tion of this model differs from those of Haldane-type spin chains where m(H) ∼ (H −Hc1)1/2. To explore further the magnetic properties of the frus- trated spin-1/2 Heisenberg diamond chain in the ground states with the couplings J1 : J2 : J3 = 1 : 2 : 2 at dif- ferent external fields, let us look at the static structure factor S(q) which is defined as S(q) = eiqj〈Szj Sz0 〉, (5) where q is the wave vector, and 〈Szj Sz0 〉 is the spin corre- lation function in the ground state. As demonstrated in Fig. 4(a), in the absence of the external field, S(q) shows three peaks: two at q = π/3, 5π/3, and one at q = π, which is quite different from that of the spin S = 1/2 Heisenberg AF chain, where S(q) only diverges at q = π. As indicated by Eq. (5), the peaks of S(q) reflect the periods of the spin correlation function 〈Szj Sz0 〉, i.e., the peaks at q = π/3 (5π/3) and π reflect the periods of 6 and 2 for 〈Szj Sz0 〉, respectively. As shown in Fig. 4(b), in the absence of the external field, 〈Szj Sz0 〉 changes sign every three sites, which corresponds really to the peri- ods of 6 and 2. With increasing the magnetic field, the small peak of S(q) at q = π becomes a round valley while the peak at q = π/3 (5π/3) continuously shifts towards q = 2π/3 (4π/3) with the height enhanced, indicating the corruption of the periods of 6 and 2 but the emer- gence of the new period ∈ (3, 6) for 〈Szj Sz0 〉, as shown in Fig. 4(c). At the field H/J1 = 1.5, two peaks shift to q = 2π/3 and 4π/3 respectively, and merge into the peaks already existing there, showing the existence of 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 DMRG =1, J =2, J Eq. (4) = 1/6, k = 0.238, = 3.15, H = 4.55, = 0.143, = 0.178 DMRG Eq.(3) =1, J =2, J = 1/6, H = 1.44, =2/3, FIG. 3: (Color online) For a spin-1/2 frustrated Heisenberg diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : 2, the DMRG results of the magnetization per site m as a function of magnetic field H away from the plateau state can be fairly fitted by Eqs. (3) and (4), (a) for 0 ≤ H ≤ Hc1 , and (b) for Hc2 ≤ H ≤ Hs. the period 3 for 〈Szj Sz0〉, as clearly displayed in Fig. 4(d). The valley and peaks of S(q) keep intact in the plateau state at m = 1/6. When the plateau state is destroyed at the field H/J1 = 4, the peaks at q = 2π/3 and 4π/3 are depressed dramatically while the peaks at q = π/3, 5π/3 and π appear again with very small heights, reveal- ing the absence of the period 3 and the slight presence of the period 2 and 6, as shown in Fig. 4(e). At the field H/J1 = 4.8, all peaks disappear and become zero, ex- cept for the peak at q = 0, which is the saturated state. Therefore, the static structure factor S(q) shows differ- ent characteristics in different magnetic fields[18]. On the other hand, it is known that S(q) also reflects the low- lying excitations of the system. It is thus reasonable to expect that the low-lying excitations of the frustrated di- amond chain will behave differently in different magnetic fields. To investigate the zero-field static structure factor S(q) in the ground state for the frustrated spin-1/2 diamond chains with various AF couplings, the four cases with J1 = 1, J3 > 0, and J2 = 0.5, 1, 2, and 4 are shown in Figs. 5(a)-5(d), respectively. For J2 = 0.5, as shown in Fig. 5(a), S(q) displays a sharp peak at q = π when J3 < 0.5, and three peaks at q = 0, 2π/3 and 4π/3 when J3 > 0.5. It is shown from the ground state phase -0.01 -0.01 0 40 80 0 40 80 (a) J1 = 1, = 2, = 2, T = 0 =1.5, T = 0, H = 0 (c) H/J (d) H/J = 1.5, 2.5 FIG. 4: (Color online) For a spin-1/2 frustrated Heisenberg diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : 2, (a) the static structure factor S(q) in the ground states under different external fields; and the spatial dependence of the spin correlation function 〈SzjS 0 〉 in the ground states under external field (b) H/J1 = 0, (c) 1, (d) 1.5 and 2.5, and (e) 4. diagram [10] that the system is in the spin fluid (SF) phase when J1 = 1, J2 = 0.5, J3 < 0.5, and enters into the ferrimagnetic (FRI) phase when J1 = 1, J2 = 0.5, J3 > 0.5. For J2 = 1, as indicated in Fig. 5(b), the incommensurate peaks exist, such as the case of J3 = 0.8, where the system is in the dimerized (D) phase [10]. For J2 = 2, as manifested in Fig. 5(c), S(q) has a sharp peak at q = π and two ignorable peaks at π/3 and 5π/3 when J3 < 1; three sharp peaks at q = 0, 2π/3 and 4π/3 at J3 = 1; a round valley (J3 = 1.5) or a small peak (J3 = 4) at q = π and two mediate peaks at π/3 and 5π/3 when J3 > 1. The system with J1 = 1 and J2 = 2 is in the D phase when 1 < J3 < 2.8, and in the SF phase when J3 < 1 or J3 > 2.8 [10]. For J2 = 4 revealed in Fig. 5(d), the situations are similar to that of Fig. 5(c), but here only the SF phase exists for the system with J1 = 1, J2 = 4 and J3 > 0 [10]. It turns out that even in the same phase, such as the SF phase, the zero- field static structure factor S(q) could display different characteristics for different AF couplings. In fact, we note that the exotic peak of S(q) has been experimentally observed in the diamond-typed compound Sr3Cu3(PO4)4 [19]. 0 1 2 (a) T = 0, H = 0, = 1, = 0.5 (b) J1 = 1, = 1J3=0.3 (c) J1 = 1, = 2J3=0.5 (d) J1 = 1, FIG. 5: (Color online) The zero-field static structure factor S(q) in the ground states for the spin-1/2 frustrated Heisen- berg diamond chains with length L = 120, J1 = 1, J3 > 0 and J2 taken as (a) 0.5, (b) 1, (c) 2 and (d) 4. If the spin correlation function for the spin-S chain can be expressed as 〈Szj Sz0 〉 = α(−1)je−jβ , where α and β are two parameters, its static structure factor will take the form of S(q) = S(S + 1) − α(cos q + e cos q + coshβ , (6) which can recover exactly the S(q) of the spin-S AKLT chain S(q) = S+1 1−cos q 1+cos q+2/S(S+2) [20] with α = (S + 1)2/3 and β = ln(1+2/S). Eq.(6) has a peak at wavevec- tor q = π. By noting that the zero-field static struc- ture factor S(q) for the frustrated diamond chains dis- plays peaks at wavevectors q = aπ/3 (a = 0, 1, 2, 3, 4, 5) for different AF couplings, the spin correlation func- tion 〈Szj Sz0 〉 could be reasonably divided into six modes 〈Sz6m+lSz0 〉 = cl + αle−(6m+l)β or 〈Sz6m+lSz0 〉 = αl(6m + l)−β with j = 6m+ l and l = 1, 2, ..., 6, whose contribu- tions to the static structure factor should be considered separately[21]. Thus, the static structure factor for the present systems could be mimicked by a superposition of six modes, which leads to S(q) = e(6−l)β cos(lq)− e−lβ cos[(6 − l)q] cosh(6β)− cos(6q) cos(lq)− cos[(6− l)q] 1− cos(6q) , (7) S(q) = 2(6m+ l)−β cos[(6m+ 1)q] + , (8) 0 1 2 0 1 2 DMRG T= 0, H= 0, = 0.5, = 0.3 Eq.(8) 1,3,5 = -0.17, 2,4,6 = 0.17, = 1.4 DMRG T=0, H=0, =0.5, Eq.(7) = -0.15, = -0.01, = 0.16, = 0.27, = -0.0007, = -0.0013, = 0.0117 Eq.(8) = -0.11, = 0.04, = -0.21, = 1.3 DMRG T= 0, H= 0, = 0.5 DMRG Eq.(8) = -0.04, = -0.16, = -0.14, = 1.1 T=0, H=0, J FIG. 6: (Color online) The DMRG results of the zero-field static structure factor as a function of wavevector for the spin- 1/2 frustrated Heisenberg diamond chains are fitted: (a) J1 = 1, J2 = 0.5, J3 = 0.3 by Eq. (8); (b) J1 = 1, J2 = 0.5, J3 = 4 by Eq. (7); (c) J1 = 1, J2 = 4, J3 = 0.5, and (d) J1 = 1, J2 = 4, J3 = 4 by Eq. (8). respectively, depending on which phase the system falls into, where αl, cl and β are couplings-dependent param- eters. As presented in Fig. 6, the DMRG results of the static structure factor as a function of wavevector are fitted by Eqs. (7) and (8) for the spin-1/2 frustrated Heisen- berg diamond chains with various AF couplings in zero magnetic field. It can be found that the characteristic peaks can be well fitted by Eqs. (7) and (8), with only a slightly quantitative deviation, showing that the main features of the static structure factor for the present sys- tems can be reproduced by a linear superposition of six modes. The fitting results give six different modes in general, as shown in Fig. 4(b). To further understand the above-mentioned behav- iors of the zero-field static structure factor, S(q), we have applied the Jordan-Wigner (JW) transformation to study the low-lying excitations of the spin-1/2 frus- trated Heisenberg diamond chain with various AF cou- plings (see Appendix A for derivations). It can be seen that the zero-field low-lying fermionic excitation ε(k) be- haves differently for different AF couplings, as shown in Figs. 7(a)-(d). Obviously, these low-lying excitations are responsible for the DMRG calculated behaviors of S(q), where the positions of minimums of ε(k) for different AF couplings, as indicated by arrows in Figs. 7(a)-(c), are exactly consistent with the locations of the peaks of zero- field static structure factor S(q) shown in Figs. 6(a)-(c), respectively, although there is a somewhat deviation for Fig. 7(d) and Fig. 6(d). It also shows that the six 0 1 2 0 1 2 T=0, H=0, = 1, = 0.5, = 0.3 (b) T=0, H=0, = 1, = 0.5, (c) T=0, H=0, = 1, = 4, = 0.5 (d) T=0, H=0, = 1, = 4, FIG. 7: (Color online) The zero-field low-lying fermionic ex- citation as a function of wavevector for the spin-1/2 frustrated Heisenberg diamond chain with (a) J1 = 1, J2 = 0.5, J3 = 0.3, (b) J1 = 1, J2 = 0.5, J3 = 4, (c) J1 = 1, J2 = 4, J3 = 0.5, and (d) J1 = 1, J2 = 4, J3 = 4. The arrows indicate the locations of minimums of ε(k). modes suggested by Eqs. (7) and (8) is closely related to the low-lying excitations of the system. B. Magnetization, Susceptibility and Specific Heat The magnetization process for the spin-1/2 frustrated diamond chain with J1 = 1, J3 > 0, and J2 = 0.5 and 2 is shown in Fig. 8(a) and 8(b), respectively, where temperature is fixed as T/J1 = 0.05. It is found that the magnetization exhibits different behaviors for differ- ent AF couplings: a plateau at m = 1/6 is observed, in agreement with the ground state phase diagram [12, 13]; for J2 = 0.5, as shown in Fig. 8(a), the larger J3 is, the larger the width of the plateau at m = 1/6 becomes; for J2 = 2, as presented in Fig. 8(b), the width of the plateau at m = 1/6 becomes larger with increasing J3 < 1, and then turns smaller with increasing J3 > 1; for J3 < 1, the larger J2, the larger the width of the plateau atm = 1/6; for J3 = 2, the larger J2, the smaller the width of the plateau at m = 1/6. The saturated field is obviously promoted with increasing AF J3 and J2. Figures 8(c) and 8(d) give the susceptibility χ as a function of temperature T for the spin-1/2 frustrated diamond chain with J1 = 1, J3 > 0 and J2 = 0.5 and 2, respectively, while the external field is taken as H/J1 = 0.01. For J2 = 0.5, as shown in Fig. 8(c), the low temperature part of χ(T ) keeps finite when J3 < 0.5, and becomes divergent when J3 > 0.5. As clearly man- ifested in the inset of Fig. 8(c), J3 = 0.5 is the critical value, which is consistent with the behaviors of static structure factor S(q) in Fig. 5(a). For J2 = 2, as shown in Fig. 8(d), an unobvious double-peak structure at low temperature is observed at small and large J3 such as 0.2 and 3. The temperature dependence of the specific heat C with J1 = 1, J3 > 0 and J2 = 0.5 and 2 is shown in Fig. 8(e) and 8(f), respectively, while the external field is fixed as H/J1 = 0.01. For J2 = 0.5, as given in Fig. 8(e), a double-peak structure of C(T ) is observed at low temperature for small and large J3 such as 0.3 and 1. The case with J2 = 2 shown in Fig. 8(f) exhibits the similar characteristics. Thus, the thermodynamics of the system demonstrate different behaviors for different AF couplings. As manifested in Fig. 5, the low-lying excita- tions behave differently for different AF couplings. The double-peak structure of the susceptibility as well as the specific heat could be attributed to the excited gaps in the low-lying excitation spectrum[22]. IV. A FRUSTRATED DIAMOND CHAIN WITH COMPETING INTERACTIONS (J1, J3 < 0, J2 > 0) A. Local Magnetic Moment and Spin Correlation Function Figure 9(a) shows the magnetization process of a frus- trated spin-1/2 Heisenberg diamond chain in the ground states with the couplings J1 : J2 : J3 = −1 : 4 : −0.5. The plateau of magnetization per site m = 1/6 is clearly obtained. To understand the occurrence of the magne- tization plateau, the spatial dependence of the averaged local magnetic moment 〈Szj 〉 in the ground states at dif- ferent external fields is calculated. It is seen that in the absence of the magnetic field, as presented in Fig. 9(b), the expectation values of 〈Szj 〉 change sign every three sites within a very small range of (−2× 10−4, 2× 10−4), resulting in the magnetization per site m = 0. With increasing the field, the expectation values of 〈Szj 〉 in- crease, whose unit of three spins is gradually divided into a pair and a single, as displayed in Fig. 9(c). At the field H/|J1| = 0.05, as illustrated in Fig. 9(d), the behavior of 〈Szj 〉 shows a perfect sequence such as {..., (Sa, Sb, Sb), ...} with Sa = 0.496 and Sb = 0.002, giving rise to the mag- netization per site m = 1/6. In addition, the sequence is fixed with increasing the field until H/|J1| = 3.2, cor- responding to the plateau of m = 1/6. As the field is enhanced further, the double Sb begin to rise, and the sequence becomes a waved series with smaller swing of (Sa − Sb) as revealed in Fig. 9(e), which corresponds to the plateau state at m = 1/6 that is destroyed. It is noting that the increase of m is mainly attributed to the promotion of double Sb, as Sa is already saturated until Sa = Sb = 0.5 at the saturated field. As discussed above, the physical picture of them = 1/6 plateau state at J1 : J2 : J3 = −1 : 4 : −0.5 can be 0 2 4 0 2 4 0.0 0.4 0.8 0 1 2 0.0 0.2 0.4 0.6 0 1 2 0 1 2 =0.2 1 2 = 1, J = 0.5, = 0.05 1 2J3 = 0.2 = 1, J = 2, = 0.05 = 1, J = 0.5, H/J =0.01 =1, J = 0.01 = 0.2 =0.5, = 0.01 3 = 3 =1, J = 0.01 = 0.2 FIG. 8: (Color online) For the spin-1/2 frustrated Heisenberg diamond chains with J1 = 1 and J3 > 0, the magnetization process m(H) at temperature T/J1 = 0.05 with (a) J2 = 0.5 and (b) J2 = 2; the susceptibility χ(T ) at field H/J1 = 0.01 with (c) J2 = 0.5 and (d) J2 = 2; the specific heat C(T ) at field H/J1 = 0.01 with (e) J2 = 0.5 and (f) J2 = 2. understood by the following approximate wave function (| ↑3i−2↑3i−1↓3i〉 ± | ↑3i−2↓3i−1↑3i〉). By use of this wave function, we have 〈ψi|Sz3i−2|ψi〉 = 1/2, 〈ψi|Sz3i−1|ψi〉 = 0, 〈ψi|Sz3i|ψi〉 = 0, leading to a sequence of {..., (1/2, 0, 0), ...}, and m = (1/2 + 0 + 0)/3 = 1/6. This is in agreement with our DMRG re- sults {..., (0.496, 0.002, 0.002), ...}. The static structure factor S(q) of the frustrated spin- 1/2 Heisenberg diamond chain with the competing cou- plings J1 : J2 : J3 = −1 : 4 : −0.5 in the ground states is considered in different external fields. At zero field, shown in Fig. 10(a), S(q) has three peaks at q = π/3, 5π/3 and π with mediate heights, which reflects the pe- riods of 6 and 2 for 〈Szj Sz0 〉, respectively. As shown in 0 1 2 3 0 40 80 0 40 80 T = 0 = -1:4:-0.5, -2E-3 T = 0, H = 0 | = 0.025 (d) H/|J | = 0.05, 3.2 (e) H/|J | = 3.27 FIG. 9: (Color online) For a spin-1/2 frustrated Heisenberg diamond chain with fixed couplings J1 : J2 : J3 = −1 : 4 : −0.5, (a) the magnetization per site m as a function of mag- netic field H in the ground states; and the spatial dependence of the averaged local magnetic moment 〈Szj 〉 in the ground states with external field (b) H/|J1| = 0, (c) 0.025, (d) 0.05 and 3.2, and (e) 3.27. Fig. 10(b), in the absence of the external field, 〈Szj Sz0 〉 changes sign every three sites, corresponding to the pe- riods of 6 and 2. With increasing the field, the peak at q = π becomes flat with height depressed forwardly, while the peak at q = π/3 (5π/3) is divided into two peaks shifting oppositely from q = π/3 (5π/3) with height de- creased, indicating the corruption of the periods of 6 and 2 and the emergence of new periods for 〈Szj Sz0 〉, as shown in Fig. 10(c). At the field H/|J1| = 0.05, two shift- ing peaks have respectively reached q = 2π/3 and 4π/3, and are merged with the existing peaks, showing the oc- currence of period 3 for 〈Szj Sz0 〉, as clearly displayed in Fig. 10(d). The flat and peaks keep constant during the plateau state at m = 1/6. When the plateau state is destroyed at field H/J1 = 3.27, the peaks at q = 2π/3 and 4π/3 are depressed sharply, revealing the decay of the period 3, as shown in Fig. 10(e). At the saturated field of H/J1 = 3.3, all peaks disappear and become flat with the value zero, which is the saturated state. So, the static structure factor S(q) shows different characteristics in different magnetic fields. Similar to the discussions in -0.02 0 40 80 0 40 80 = -1, = 4, = -0.5, T = 0 0.05, 3.2 | = 3.3 T = 0, H = 0 | = 0.025 | = 0.05, 3.2 | = 3.27 FIG. 10: (Color online) For a spin-1/2 frustrated Heisenberg diamond chain with fixed couplings J1 : J2 : J3 = −1 : 4 : −0.5, (a) the static structure factor S(q) in the ground states with different external fields; and the spatial dependence of the spin correlation function 〈Szj S 0 〉 in the ground states with external field (b) H/|J1| = 0, (c) 0.025, (d) 0.05 and 3.2, and (e) 3.27. Fig. 4, the low-lying excitations of this frustrated di- amond chain would also behave differently in different magnetic fields. To further investigate the zero-field static structure factor S(q) in the ground state for the frustrated spin-1/2 diamond chains with various J1, J3 < 0 and J2 > 0, two cases with J1 = −1, J3 < 0 and J2 = 1 and 4 are illus- trated in Fig. 11(a) and 11(b), respectively. For J2 = 1, S(q) shows a round peak at q = π, two sharp peaks at π/3 and 5π/3 when |J3| < 1, and a very sharp peak at q = 0 when |J3| > 1. For J2 = 4, S(q) shows three peaks at q = π/3, π and 5π/3 as |J3| < 1, and a very sharp peak at q = π and nearly ignorable peaks at π/3 and 5π/3 as |J3| > 1. In general, the zero-field static structure factor S(q) shows different characteristics with different competing couplings, whose exotic characteristics could be experimentally observed in the related diamond-typed compounds. As shown in Fig. 12, the DMRG results of the static structure factor as a function of wavevector are fitted by Eq.(7) for the spin-1/2 frustrated Heisenberg diamond chains with J1, J3 < 0 and J2 > 0. It can be found 0 1 2 (b) J = -1, J = 4J3 = -16 (a) T = 0, H = 0, J = -1, J = -0.1 FIG. 11: (Color online) The zero-field static structure factor S(q) in the ground states for the spin-1/2 frustrated Heisen- gberg diamond chains with length L = 120, J1 = −1, J3 < 0 and J2 taken as (a) 1; (b) 4. that the characteristic behaviors can be nicely fitted by Eq.(7), with only a slightly quantitative deviation, show- ing that the main features of the static structure factor for the present systems can be captured by a superposi- tion of six modes. It is consistent with the fact that the spin correlation function for the present systems has six different modes, as manifested in Fig. 10(b). Similar to the case (a) with all AF couplings in the last section, the characteristics of zero-field S(q) for the spin-1/2 frustrated Heisenberg diamond chain with J1, J3 < 0 and J2 > 0 can be further undersood in terms of the low-lying excitations of the system (see Appendix A). By means of the JW transformation, the zero-field low-lying fermionic excitation ε(k) of the present case is calculated, as shown in Figs. 13(a)-(b). It is found that the zero-field low-lying fermionic excitation ε(k) differs for different couplings, but the positions of minimums of ε(k) for different couplings, as indicated by arrows in Figs. 13(a) and (b), appear to be the same. One may see that these positions coincide exactly with the locations of peaks of the zero-field static structure factor S(q) mani- fested in Figs. 12(a) and (b), respectively, showing that our fitting equation is qualitatively consistent with the low-lying excitations of the system. 0 1 2 (b) DMRG T = 0, H = 0, = -1, = 4, = -0.5 Eq. (7) = -0.07, = 0.3, c 1,2,...,6 = 0.0 = -0.07, = -0.21, (a) DMRG T = 0, H = 0, = -1, = -0.1 Eq. (7) = -0.03, = -0.14, = -0.12, = 0.19, 1,2,...,6 = 0.0 FIG. 12: (Color online) The DMRG results of the zero-field static structure factor as a function of wavevector are fitted by Eq. (7) for the spin-1/2 frustrated Heisenberg diamond chains with (a) J1 = −1, J2 = 1, J3 = −0.1, and (b) J1 = −1, J2 = 4, J3 = −0.5. 0 1 2 -3.35 -3.30 -3.25 (b) T=0, H=0, = -1, = 4, = -0.5 (a) T=0, H=0, = -1, = 1, = -0.1 FIG. 13: (Color online) The zero-field low-lying fermionic excitation as a function of wavevector for the spin-1/2 frus- trated Heisenberg diamond chain with (a) J1 = −1, J2 = 1, J3 = −0.1, and (b) J1 = −1, J2 = 4, J3 = −0.5. The arrows indicate the locations of minimums of ε(k). B. Magnetization, Susceptibility and Specific Heat Figures 14(a) and 14(b) show the magnetization pro- cess for the spin-1/2 frustrated diamond chain at a finite temperature T/|J1| = 0.05 with J1 = −1, J3 < 0, and J2 = 1 and 4, respectively. It is shown that the magneti- zation exhibits different behaviors for different J1, J3 < 0 and J2 > 0. A plateau at m = 1/6 is observed at small |J3|; with a fixed J2, the larger |J3|, the smaller the width of the plateau at m = 1/6, and the plateau disappears when |J3| exceeds the critical value; for a fixed J3, the larger J2, the wider the width of the plateau at m = 1/6; the saturation field is obviously depressed with the in- crease of |J3| at a fixed J2, and is enhanced with the increase of J2 at a fixed J3. Figures 14(c) and 14(d) manifest the susceptibility χ as a function of temperature T for the spin-1/2 frus- trated diamond chain with J1 = −1, J3 < 0 and J2 = 1 and 4, respectively, where the external field is taken as H/|J1| = 0.01. For J2 = 1, the low temperature part of χ(T ) keeps finite when |J3| < 1, and becomes divergent when |J3| > 1. As clearly revealed in the inset of Fig. 14(c), J3 = −1 is the critical value, which is in agreement with the behaviors of static structure factor S(q) shown in Fig. 11(a). For J2 = 4, a clear double-peak structure of χ(T ) is obtained at |J3| = 8. The temperature depen- dence of the specific heat C with J1 = −1, J3 < 0 and J2 = 1 and 4 is shown in Figs. 14(e) and 14(f), respec- tively, where the external field is fixed as H/|J1| = 0.01. For J2 = 1, a double-peak structure of C(T ) is observed for the case of |J3| = 0.5. The case with J2 = 4 shown in Fig. 14(f) exhibits the similar characteristics. It is also found that, owing to the competitions among J1, J3 and J2, the thermodynamics demonstrate rich behaviors at different couplings. As reflected in Fig. 11, the low-lying excitations behave differently with various F interactions J1, J3 and AF interaction J2, while the excitation gaps could induce the double-peak structure in the suscepti- bility as well as in the specific heat[22]. V. A DIAMOND CHAIN WITHOUT FRUSTRATION (J1, J2 > 0, J3 < 0) A. Local Magnetic Moment and Spin Correlation Function Figure 15(a) shows the magnetization process of a non-frustrated spin-1/2 Heisenberg diamond chain in the ground states with the couplings satisfying J1 : J2 : J3 = 1 : 2 : −0.5. The plateau of magnetization per site m = 1/6 is observed. The appearance of the magneti- zation plateau can be understood from the spatial de- pendence of the averaged local magnetic moment 〈Szj 〉 in the ground states under different external fields. In ab- sence of the external field, the expectation values of 〈Szj 〉 change sign every one site with a waved swing within a very small range of (−2× 10−4, 2× 10−4), giving rise to 0.0 0.4 0.8 0 1 2 3 4 0 1 2 0 1 2 0.0 0.6 1.2 0 1 2 0 1 2 -1 -0.5 J3 = -0.1 = -1, J = 1, | = 0.05 = -8 = -1, J = 4, | = 0.05 -0.5-4 = -1.5 = -1, J = 1, |=0.01 = -1 =-1, J | = 0.01 = -1 =-1, J | = 0.01 = -1 =-1, J | = 0.01 = -0.1 FIG. 14: (Color online) For the spin-1/2 frustrated Heisen- berg diamond chains with J1 = −1 and J3 < 0, the mag- netization process m(H) at temperature T/J1 = 0.05 with (a) J2 = 1 and (b) J2 = 4; the susceptibility χ(T ) at field H/J1 = 0.01 with (c) J2 = 1 and (d) J2 = 4; the specific heat C(T ) at field H/J1 = 0.01 with (e) J2 = 1 and (f) J2 = 4. the magnetization per site m = 0. Under a finite field, every three successive spins have gradually cooperated into a pair and a single, as shown in Fig. 15(c). At H/J1 = 0.9, as given in Fig. 15(d), 〈Szj 〉 shows a perfect sequence such as {..., (Sa, Sb, Sb), ...} with Sa = 0.393 and Sb = 0.053, resulting in the magnetization per site m = 1/6. Moreover, the sequence is fixed with the field increased until H/J1 = 1.8, corresponding to the plateau state ofm = 1/6. As the field is increased further, double Sb begin to increase, and the sequence becomes a waved succession with a smaller swing of (Sa − Sb), as mani- fested in Fig. 15(e), which corresponds to the fact that the plateau state at m = 1/6 is destroyed. It is observed that the increase of m at first is mainly attributed to the speedy boost of double Sb, and later, Sa starts to increase weakly until Sa = Sb = 0.5 at the saturated field. 0 1 2 3 0 40 80 0 40 80 T = 0 = 1:2:-0.5, -2E-4 (b) T = 0, H = 0 = 0.7 (d) H/J = 0.9, 1.8 = 2.5 FIG. 15: (Color online) For a spin-1/2 non-frustrated Heisen- berg diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : −0.5, (a) the magnetization per site m as a function of magnetic field H in the ground states; and the spatial de- pendence of the averaged local magnetic moment 〈Szj 〉 in the ground states with external field (b) H/J1 = 0, (c) 0.7, (d) 0.9 and 1.8, and (e) 2.5. For this non-frustrated case with couplings J1 : J2 : J3 = 1 : 2 : −0.5, the obtained perfect sequence of {..., (0.393, 0.053, 0.053), ...} for the m = 1/6 plateau state could be understood by the following approximate trimerized wave function (2| ↑3i−2↑3i−1↓3i〉 ± 2| ↑3i−2↓3i−1↑3i〉 ± | ↓3i−2↑3i−1↑3i〉). (9) According to this function, we find 〈ψi|Sz3i−2|ψi〉 = 7/18, 〈ψi|Sz3i−1|ψi〉 = 1/18, 〈ψi|Sz3i|ψi〉 = 1/18, giving rise to a sequence of {..., (7/18, 1/18, 1/18), ...}. It turns out that m = (7/18 + 1/18 + 1/18)/3 = 1/6. This observation implies that the ground state of the plateau state can also be described by the trimerized states. The static structure factor S(q) of the non-frustrated spin-1/2 Heisenberg diamond chain in the ground states with the couplings J1 : J2 : J3 = 1 : 2 : −0.5 is probed in different external fields. As shown in Fig. 16(a), in absence of the external field, S(q) shows a sharp peak at q = π, similar to the behaviors of the S = 1/2 Heisen- berg AF chain, which reflects the period of 2 for 〈Szj Sz0 〉. 0 1 2 -0.02 0 40 80 0 40 80 (a) J1 = 1, = 2, = -0.5, T = 0 0.9, 1.8 T = 0, H = 0 = 0.7 (d) H/J = 0.9, 1.8 = 2.5 FIG. 16: (Color online) For a spin-1/2 non-frustrated Heisen- berg diamond chain with fixed couplings J1 : J2 : J3 = 1 : 2 : −0.5, (a) the static structure factor S(q) in the ground states with different external fields; and the spatial dependence of the spin correlation function 〈Szj S 0 〉 in the ground states with external field (b) H/J1 = 0, (c) 0.7, (d) 0.9 and 1.8, and (e) As displayed in Fig. 16(b), at zero external field, 〈Szj Sz0 〉 changes sign every one lattice site, corresponding to the period of 2. When the field is increased, the peak at q = π becomes a flat with the height depressed greatly, while two new peaks with small heights at q = 2π/3 and 4π/3 appear, indicating the corruption of the period of 2 and the emergence of the new period of 3 for 〈Szj Sz0 〉, as demonstrated in Fig. 16(c). At the field H/J1 = 0.9, the peaks at q = 0, 2π/3 and 4π/3 become sharper. The flat and peaks of S(q) keep unchanged during the plateau state at m = 1/6. When the plateau state is de- stroyed at the field H/J1 = 2.5, the peaks at q = 2π/3 and 4π/3 are suppressed dramatically, revealing the de- cay of the period 3, as shown in Fig. 16(e). At the field H/J1 = 2.8, all peaks disappear, and become a flat with the value zero, except for the peak at q = 0, which is the saturated state. It can be stated that the static struc- ture factor S(q) shows various characteristics in different magnetic fields. Similar to what discussed in Figs. 4 and 10, the low-lying excitations of this non-frustrated diamond chain would also behave differently in different magnetic fields. The zero-field static structure factor S(q) in the ground state for the present system displays a peak at q = π with different couplings, but whose static correlation function 〈Szj Sz0 〉 varies with the couplings. As illustrated in Fig. 17, only are the values of 〈Szj Sz0 〉 larger than zero pre- sented for convenience. In order to gain deep insight into physics, for a comparison we also include the static correlation function for the S = 1/2 Heisenberg antifer- romagnetic (HAF) chain in Fig. 17(a), whose asymptotic behavior has the form of [23, 24] 〈Sj · S0〉 ∝ (−1)j (2π)3/2 . (10) Eq. (10) is depicted as solid lines in Figs. 17. To take the finite-size effect into account, the length of the diamond chain is taken as L = 90, 120 and 160, respectively. As revealed in Fig. 17(a), the DMRG result of the S = 1/2 HAF chain with an infinite length agrees well with the solid line. Fig. 17(b) shows that the static correlation function for the spin-1/2 diamond chain with frustrated couplings J1 : J2 : J3 = 1 : 1.2 : 0.5 decays faster than that of the HAF chain. Compared with Figs. 17(c)-(f), it can be found that all the static correlation functions for the spin-1/2 non-frustrated diamond chain with AF interactions J1, J2 and F interaction J3 fall more slowly than that of the S = 1/2 HAF chain; for fixed AF inter- actions J1 and J2, the static correlation functions drop more leisurely with increasing the F interaction |J3|; for fixed AF interaction J1 and F interaction J3, the static correlation functions decrease more rapidly with increas- ing the AF interaction J2. B. Magnetization, Susceptibility and Specific Heat Figures 18(a) and 18(b) show the magnetization pro- cess for the spin-1/2 non-frustrated diamond chain at a finite temperature T/J1 = 0.05 with J1 = 1, J3 < 0, and J2 = 0.5 and 2, respectively. It is found that the magneti- zation behaves differently with different AF interactions J1, J2 and F interaction J3. A plateau at m = 1/6 is ob- tained at small |J3|; for fixed J1 and J2, the larger |J3|, the narrower the width of the plateau at m = 1/6, and after |J3| exceeds a critical value, the plateau at m = 1/6 is eventually smeared out; for fixed J1 and J3, the larger J2, the wider the width of the plateau at m = 1/6; the saturated field is obviously unchanged with changing the F interaction J3. The coupling-dependence of the spin- 1/2 non-frustrated diamond chain with AF interactions J1, J2 and F interaction J3 is similar to that of trimerized F-F-AF chains[25]. Figures 18(c) and 18(d) present the susceptibility χ as a function of temperature T for the spin-1/2 frus- trated diamond chain with J1 = 1, J3 < 0 and J2 = 0.5 and 2, respectively, where the external field is taken as H/J1 = 0.01. A double-peak structure of χ(T ) is ob- served at small F interaction |J3| and disappears at large |J3|. The temperature dependence of the specific heat C with J1 = 1, J3 < 0 and J2 = 0.5 and 2 is shown in Figs. 0 50 100 150 0 50 100 150 (a) J =1:1:0 (HAF) Eq. (10) L=90 (DMRG) L=120 (DMRG) L=160 (DMRG) T = 0, H = 0 = 1:1.2:0.5 (frustrated) = 1:2:-0.5 = 1:2:-2 = 1:0.5:-0.5 = 1:0.5:-2 FIG. 17: (Color online) The zero-field static correlation func- tion 〈Szj S 0 〉 versus site j in the ground state for a spin-1/2 diamond chain with different lengths and various couplings. The couplings ratio J1 : J2 : J3 is taken as (a) 1 : 1 : 0 (HAF), (b) 1 : 1.2 : 0.5 (frustrated), (c) 1 : 2 : −0.5, (d) 1 : 2 : −2, (e) 1 : 0.5 : −0.5, (f) 1 : 0.5 : −2. The length is taken as L = 90, 120 and 160, respectively. 18(e) and 18(f), respectively, where H/J1 = 0.01. It is seen that, when J2 is small, C(T ) exhibits only a single peak; when J2 is large, a double-peak structure of C(T ) is observed. In the latter case, the double-peak structure is more obvious for small F interaction |J3|, and tend to disappear at large |J3|. Therefore, the thermodynamics demonstrate various behaviors with different AF interac- tions J1, J2 and F interaction J3. C. Effect of Anisotropy of Bond Interactions Some magnetic materials show different behaviors un- der longitudinal and transverse magnetic fields, showing that the anisotropy plays an important role in the physi- cal properties of the system. First, let us investigate the XXZ anisotropy of the AF interaction J2 on the proper- ties of the spin-1/2 non-frustrated diamond chain with the couplings J1 : J2z : J3 = 1 : 2 : −0.5 for vari- ous anisotropy parameter defined by γ2 = J2x/J2z = J2y/J2z, where the z axis is presumed to be perpendic- ular to the chain direction. For γ2 ≥ 1, the magneti- zation m(H), susceptibility χ(T ) and specific heat C(T ) are presented in Figs. 19(a), (b) and (c), respectively. 0.0 0.8 1.6 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 0 1 2 = 0.05, = 1, J = 0.5 = -2 = 1, J = 0.05, = -2 = 1, J = 0.5 = 0.01, = -2 -0.5-0.2 = 1, J = 0.01, = -0.2 = 1, J = 0.5 = 0.01, = -2 -0.2 = 0.01, = 1, J -0.2 -0.5 = -2 FIG. 18: (Color online) For the spin-1/2 non-frustrated Heisenberg diamond chains with J1 = 1 and J3 < 0, the mag- netization process m(H) at temperature T/J1 = 0.05 with (a) J2 = 0.5 and (b) J2 = 2; the susceptibility χ(T ) at field H/J1 = 0.01 with (c) J2 = 0.5 and (d) J2 = 2; the specific heat C(T ) at field H/J1 = 0.01 with (e) J2 = 0.5 and (f) J2 = 2. With increasing γ2, it is found that when the magnetic field H is along the z direction, the width of the mag- netization plateau at m = 1/6 as well as the saturation field are enlarged, while those are more increased for H along the x direction than along the z direction; the peak of the susceptibility χ(T ) for H along the z direction at lower temperature side is promoted, and the second round peak at high temperature side is depressed with a little shift, while χ(T ) for H along the x direction shows the similar varying trend; the peak of the specific heat C(T ) for H along the z direction at lower temperature side leaves almost unchanged, and the second round peak at high temperature side moves towards the higher tem- perature side, while C(T ) for H along the x direction coincides with those for H along the z direction. For 0 < γ2 < 1, the anisotropy just shows very reverse effect on the thermodynamic properties in comparison to what we discussed above. Now let us discuss the effect of the XXZ anisotropy of J3 < 0 on the magnetic and thermodynamic properties of the spin-1/2 non-frustrated diamond chain with the couplings J1 : J2 : J3z = 1 : 2 : −0.5. Recall that as the J2 bond connects two different lattice sites, as shown in Fig. 1, J1 and J3 can be different, even their signs. Define a parameter γ3 to characterize the anisotropy as γ3 = J3x/J3z = J3y/J3z, where the z axis is perpendicu- lar to the chain direction. For γ3 ≥ 1, the magnetization m(H), susceptibility χ(T ) and specific heat C(T ) are de- picted in Figs. 19(d), (e) and (f), respectively. With increasing γ3, it is seen that the width of the plateau at m = 1/6 for H along the z direction becomes slightly wider, while it goes smaller for H along the x direc- tion; the saturation field is not changed with γ3 along both directions; the peak of χ(T ) for H along the z di- rection at lower temperature side is promoted, and the second round peak at higher temperature side is slightly depressed, while the situations along the x direction are just reverse, namely, the peak at lower temperature side is depressed, and the second peak at higher tempera- ture side is slightly promoted; the peak of C(T ) for H along the z direction at lower temperature side leaves al- most unchanged, and the second round peak at higher temperature side moves slightly to the higher tempera- ture side, while C(T ) along the x direction coincides with that along the z direction. For 0 < γ3 < 1, the situation just becomes reverse in comparison to what we discussed above. D. Comparison to Experimental Results Recently, Kikuchi et al. [6] have performed a nice measurement on a spin-1/2 diamond-chain compound Cu3(CO3)2(OH)2, i.e., azurite. They have observed the 1/3 magnetization plateau, unambiguously confirming the previous theoretical prediction. The two broad peaks both in the magnetic susceptibility and the specific heat are observed. We note that in Ref. [6], the experimen- tal data at finite temperatures are fitted by the zero- temperature theoretical results obtained by the exact di- agonalization and DMRG methods, while the result of the high temperature series expansion fails to fit the low-temperature behavior of the susceptibility. In ac- cordance with our preceding discussions, by using the TMRG method, we have attempted to re-analyse the ex- perimental data presented in Ref. [6] to fit the experi- ments for the whole available temperature region. Our fitting results for the temperature dependence of the susceptibility χ of the compound Cu3(CO3)2(OH)2 are presented in Fig. 20(a). For a comparison, we have also included the TMRG result calculated by us- ing the parameters given in Ref. [6]. Obviously, our TMRG results with J1 : J2 : J3z = 1 : 1.9 : −0.3 and J3x/J3z = J3y/J3z = 1.7 fit very well the experimental 0 1 2 3 4 0 1 2 3 4 0 1 2 3 0 1 2 3 0 1 2 0 1 2 = 0.05 =1:2:-0.5 =2, z =2, x = 0.05 =1:2:-0.5 =2, z =2, x = 0.01 =2, z =2, x = 0.01 =2, z =2, x = 0.01 = 0.01 FIG. 19: (Color online) For the spin-1/2 non-frustrated dia- mond chain with the couplings satisfying J1 : J2z : J3 = 1 : 2 : −0.5 for various anisotropy γ2 = J2x/J2z ≥ 1: (a) the magnetization process m(H) at temperature T/J1 = 0.05; (b) the susceptibility χ(T at field H/J1 = 0.01; and (c) the specific heat C(T ) at field H/J1 = 0.01. For the spin-1/2 non-frustrated diamond chain with couplings J1 : J2 : J3z = 1 : 2 : −0.5 for various anisotropy γ3 = J3x/J3z ≥ 1: (d) the magnetization process m(H) at temperature T/J1 = 0.05; (e) the susceptibility χ(T ) at field H/J1 = 0.01; and (f) the specific heat C(T ) at field H/J1 = 0.01. data of χ, and the two round peaks at low temperatures are nicely reproduced, while the result with J1 : J2 : J3 = 1 : 1.25 : 0.45 obtained in Ref. [6] cannot fit the low-temperature behavior of χ [26]. On the other hand, the fitting results for the temperature dependence of the specific heat C(T ) of the compound Cu3(CO3)2(OH)2 are shown in Fig. 20(b). The lattice contribution, which is included in the raw experimental data in Ref.[6], is subtracted according to C(T ) = CExp(T ) − αT 3, where α is a parameter. Obviously, our TMRG result with the same set of parameters J1 : J2 : J3z = 1 : 1.9 : −0.3 and J3x/J3z = J3y/J3z = 1.7 fits also remarkably well the experimental data of C(T ), and the two round peaks at low temperatures are nicely reproduced, while the re- sult with J1 : J2 : J3 = 1 : 1.25 : 0.45 given in Ref. [6] cannot fit the low-temperature behavior of C(T ), even qualitatively. In addition, the sharp peak of C(T ) ex- perimentally observed at temperature around 2K cannot be reproduced by both sets of the coupling parameters, which might be a three-dimensional long-range ordering due to interchain interactions. The fitting results for the magnetization m(H) of the compound Cu3(CO3)2(OH)2 are shown in Fig. 20(c). We would like to point out that the quantitative fitting by our above parameters to the width of the plateau is not so good, but the quali- tative behavior is quite consistent with the experiments both in the transverse and longitudinal magnetic fields, say, H c1 > H c1, H c2 < H c2, and the saturation field is fixed along both directions, suggesting that our fitting parameters capture the main characteristics. It is worth pointing out that if the anisotropy ratio is increased up to J3x/J3z = J3y/J3z = 2.5 with the same couplings J1 : J2 : J3z = 1 : 1.9 : −0.3, the width of the 1/3 plateau for H ‖ b will be decreased to about one-half of that for H ⊥ b. Therefore, our calculations show that (i) the best cou- plings obtained by fitting the experimental data of the susceptibility for the azurite could be J1 : J2 : J3z = 1 : 1.9 : −0.3 with the anisotropic ratio for the ferromag- netic interaction J3x/J3z = J3y/J3z = 1.7, where z ⊥ b; (ii) the compound may not be a spin frustrated magnet; (iii) the double peaks of the susceptibility and the specific heat are not caused by the spin frustration effect, but by the two kind of gapless and gapful excitations owing to the competition of the AF and F interactions. One might argue that for this diamond chain com- pound, from the point of the lattice distance it is unlikely that J1 is AF without XXZ anisotropy while J3 is F with strong XXZ anisotropy. We may offer another possibil- ity to support our findings, namely, the case of J1 and J3 with opposite signs is not excluded from the lattice structure of the compound. A linear relationship exists between the exchange energy and the metal-ligand-metal bridge angle: the coupling energy, positive (ferromag- netic) at angles near 90o, becomes increasingly smaller (more antiferromagnetic) as the angle increases[27]. As the ferromagnetic coupling J3 is determined by fitting the experimental low-temperature behaviors of χ(T ) and C(T ), this fitting coupling parameters should not be im- possible if one considers the angle of J1 bridge to keep the antiferromagnetic coupling while the angle of J3 bridge to induce the ferromagnetic coupling. On the other hand, we note that there is another compound with Cu ions, Cu2(abpt)(SO4)2(H2O)·H2O, whose g factors in XY plane are different from that in z direction [28]. Besides, someone might argue that the condition J2 ≫ J1, |J3| is necessary to explain the double peak behavior of the dia- mond chain. In fact, such an argument is not necessarily true, as manifested in Fig. 18(c), where the double peaks of χ(T ) at low temperatures can also be produced with 0 10 20 30 40 0 10 20 30 40 0.020 0.025 0.030 H (T) (c) H b (Exp.) H//b (Exp.) (TMRG) (Ref.[12]) H b (TMRG) (present work) H//b (TMRG) (present work) T (K) (b) CExp- T =0.0005 =0.00055 =0.0006 (TMRG) (Ref.[12]) (TMRG) (present work) H b (Exp.) (TMRG) =1:1.25:0.45, H b (TMRG) H//b (TMRG) (a) H//b (Exp.) =19K (Ref.[12]) =1:1.9:-0.3, =23K, J (present work) FIG. 20: (Color online) A comparison of experimental re- sults for (a) the magnetic susceptibility, (b) the specific heat and (c) the magnetization process for the spin-1/2 diamond compound Cu3(CO3)2(OH)2 with the TMRG results. The experimental data are taken from Ref. [6]. See the context for details. the parameters J1 : J2 : J3 = 1 : 0.5 : −0.1. In other words, the double-peak behavior of the diamond chain may not depend on whether J2 ≫ J1, J3 or not, but may be strongly dependent on the competition of AF and F interactions, as discussed above. VI. SUMMARY AND DISCUSSION In this paper, we have numerically studied the mag- netic and thermodynamic properties of spin-1/2 Heisen- berg diamond chains with three different cases (a) J1, J2, J3 > 0 (frustrated), (b) J1, J3 < 0, J2 > 0 (frus- trated), and (c) J1, J2 > 0, J3 < 0 (non-frustrated) by means of the DMRG and TMRG methods. In the ground states, the local magnetic moment, spin correlation func- tion, and static structure factor are explored. The static structure factor S(q) at zero field shows peaks at wave vector q = 0, π/3, 2π/3, π, 4π/3 and 5π/3 for different couplings, in which the peaks at q = 0, 2π/3 and 4π/3 in the magnetization plateau state with m = 1/6 are ob- served to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is seen that the six modes are closely related to the low-lying excitations of the sys- tem. At finite temperatures, the magnetization, suscep- tibility and specific heat are calculated, which show var- ious behaviors for different couplings. The double-peak structure of the susceptibility and specific heat can be procured, whose positions and heights are found to be dependent on competing couplings. It has been shown that the XXZ anisotropy of F and AF couplings can have remarkable effect on the physical behaviors of the sys- tem. In addition, the experimental susceptibility, specific heat and magnetization of the diamond chain compound Cu3(CO3)2(OH)2[6] can be nicely fitted by our TMRG results. For the spin-1/2 frustrated Heisenberg diamond chains with AF couplings J1, J2 and J3, the magnetization plateau at m = 1/6 in the ground state coincides with a perfect fixed sequence of the averaged local magnetic moment such as {..., (Sa, Sa, Sb), ...} with 2Sa+Sb = 1/2, which might be described by trimerized states. On the other hand, the static structure factor S(q) shows peaks at wave vectors q = 0, π/3 (5π/3), and 2π/3 (4π/3) for different external fields and different AF couplings. We note that the similar behavior of S(q) has been experimentally observed in diamond-typed compound Sr3Cu3(PO4)4 [19]. In addition, the DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes. It is observed that the six modes are closely related to the low-lying excitations of the present case. At finite temperatures, the magnetizationm(H), susceptibility χ(T ) and specific heat C(T ) demonstrate different behaviors at different AF couplings, say, the magnetization plateau atm = 1/6 is observed whose width is found to be dependent on the couplings; the double peak structure is observed for the susceptibility χ(T ) and specific heat C(T ) as a function of temperature, and the heights and positions of the peaks are found to be dependent on the AF couplings. For the spin-1/2 frustrated Heisenberg diamond chains with F couplings J1, J3 and AF coupling J2, the mag- netization plateau at m = 1/6 in the ground state cor- responds to a perfect fixed sequence of the averaged lo- cal magnetic moment such as {..., (Sa, Sb, Sb), ...} with Sa+2Sb = 1/2, which could be understood by trimerized states. The static structure factor S(q) shows peaks also at wave vectors q = 0, π/3 (5π/3), and 2π/3 (4π/3) for different external fields and different F couplings J1, J3 and AF coupling J2, which is expected to be experimen- tally observed in the related diamond-type compound. In addition, the DMRG results of the zero-field static structure factor can be nicely fitted by a linear superpo- sition of six modes with the fitting equations mentioned above. The six modes are closely related to the low- lying excitations of the system. At finite temperatures, the magnetization m(H), susceptibility χ(T ) and spe- cific heat C(T ) demonstrate various behaviors for differ- ent couplings, namely, the magnetization plateau at m = 1/6 is observed whose width is found to depend on the couplings; the double-peak structure is also observed for the susceptibility χ(T ) and specific heat C(T ), and the heights and positions of the peaks are found dependent on F couplings J1, J3 and AF coupling J2. For the spin-1/2 non-frustrated Heisenberg diamond chains with AF couplings J1, J2 and F coupling J3, the magnetization plateau at m = 1/6 in the ground state coincides with a perfect fixed sequence of the averaged local magnetic moment such as {..., (Sa, Sb, Sb), ...} with Sa+2Sb = 1/2, which could be understood by trimerized states. The static structure factor S(q) is observed to ex- hibit the peaks at wave vectors q = 0 and 2π/3 (4π/3) for different external fields and different AF couplings J1, J2 and F coupling J3, which could be experimen- tally detected in the related diamond-type compound. In addition, it is found that the zero-field spin correlation function 〈Szj Sz0 〉 is similar to that of the S = 1/2 Heisen- berg AF chain. At finite temperatures, the magnetiza- tion m(H), susceptibility χ(T ) and specific heat C(T ) are found to reveal different behaviors for different cou- plings, i.e., the magnetization plateau at m = 1/6 is ob- tained, whose width is found to depend on the couplings; the double-peak structure is observed for the tempera- ture dependence of the susceptibility χ(T ) and specific heat C(T ), where the heights and positions of the peaks depend on different AF couplings J1, J2 and F coupling The effect of the anisotropy of the AF and F inter- actions on the physical properties of the non-frustrated Heisenberg diamond chain is also investigated. For the case of the couplings satisfying J1 : J2z : J3 = 1 : 2 : −0.5, when the anisotropic ratio γ2 = J2x/J2z = J2y/J2z 6= 1, it is found that the width of the plateau at m = 1/6, the saturation field, and the susceptibility χ(T ) show the same tendency, but quantitatively different, un- der the external field H along the z and x directions, while the specific heat C(T ) for H along the z direction coincides with that along the x direction. For the case of the couplings satisfying J1 : J2 : J3z = 1 : 2 : −0.5, when the anisotropic ratio γ3 = J3x/J3z = J3y/J3z 6= 1, it is seen that the width of the plateau at m = 1/6, the saturation field, and the susceptibility χ(T ) exhibit the opposite trends for H along the z and x directions, while the specific heat C(T ) for H along the z direction also coincides with that along the x direction. For all the three cases, plateau states of m = 1/6 are observed during the magnetization, whose static struc- ture factor S(q) shows peaks at wavevectors q = 0, 2π/3 and 4π/3. But in absence of the magnetic field, the static structure factor S(q) in the ground state displays peaks at q = 0, π/3, 2π/3, π, 4π/3, and 5π/3 for the frustrated case with J1, J2, J3 > 0; peaks at q = 0, π/3, π, and 5π/3 for the frustrated case with J1, J3 < 0, J2 > 0; and a peak at q = π for the non-frustrated case with J1, J2 > 0, J3 < 0. In addition, the DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where the fitting equa- tion is proposed. At finite temperatures, the double-peak structure of the susceptibility and specific heat against temperature can be obtained for all the three cases. It is found that the susceptibility shows ferrimagnetic char- acteristics for the two frustrated cases with some cou- plings, while no ferrimagnetic behaviors are observed for the non-frustrated case. The compound Cu3(CO3)2(OH)2 is regarded as a model substance for the spin-1/2 Heisenberg diamond chain. The 1/3 magnetization plateau and the two broad peaks both in the magnetic susceptibility and the spe- cific heat have been observed experimentally[6]. Our TMRG calculations with J1 : J2 : J3z = 1 : 1.9 : −0.3 and J3x/J3z = J3y/J3z = 1.7 capture well the main characteristics of the experimental susceptibility, spe- cific heat and magnetization, indicating that the com- pound Cu3(CO3)2(OH)2 may not be a spin frustrated magnet[26]. APPENDIX A: LOW-LYING EXCITATIONS OF SPIN-1/2 FRUSTRATED HEISENBERG DIAMOND CHAINS In this Appendix, the low-lying excitations of the spin- 1/2 frustrated Heisenberg diamond chain are investigated by means of the Jordan-Wigner (JW) transformation. The Hamiltonian of the system reads (J1S3i−2 · S3i−1 + J2S3i−1 · S3i + J3S3i−2 · S3i +J3S3i−1 · S3i+1 + J1S3i · S3i+1)−H Szj ,(A1) where 3N is the total number of spins in the diamond chain, Ji > 0 (i = 1, 2, 3) represent the AF coupling while Ji < 0 the F interaction, and H is the external magnetic field along the z direction. In accordance with the spin configuration of the diamond chain, we start from the Jordan-Wigner (JW) transformation with spin- less fermions S+j = a j exp[iπ a+mam], Szj = a j aj − , (A2) where j = 1, · · · , 3N . Because the period of the present system is 3, three kinds of fermions in moment space can be introduced through the Fourier transformations a3i−2 = eik(3i−2)a1k, a3i−1 = eik(3i−1)a2k, a3i = eik(3i)a3k. (A3) Ignoring the interactions between fermions, the Hamilto- nian takes the form of H = E0 + [(ω1a 1ka1k + ω2a 2ka2k + ω3a 3ka3k) +(γ1a1ka 2k + γ2a2ka 3k + γ3a3ka 1k + h.c.)],(A4) where E0 = (2J1+J2+2J3−6H), ω1=−(J1+J3)−H , ω2=− 12 (J1+J2+J3)−H , ω3=ω2, γ1=(J1e ik+J3e −i2k)/2, γ2=(J2e ik)/2, and γ3=(J3e ik + J1e −ik)/2. Via the Bogoliubov transformation a1k = u11(k)α1k + u12(k)α2k + u13(k)α3k, a2k = u21(k)α1k + u22(k)α2k + u23(k)α3k, a3k = u31(k)α1k + u32(k)α2k + u33(k)α3k, (A5) the Hamiltonian can be diagonalized as H = Eg + ikαik. (A6) The coefficients of the Bogoliubov transformation can be found through equations of motion i~ȧik = [aik, H ]: ω1 γ1 γ3 γ∗1 ω2 γ2 γ∗3 γ  = ǫik  . (A7) For a given k, the eigenvalues εik and eigenvectors (u1i, u2i, u3i) can be numerically calculated by the driver ZGEEV.f of the LAPACK, which is available on the website[29]. Figs.7 show the zero-field low-lying fermionic excitation ε(k) for the frustrated diamond chain with different AF coupling, while Figs.13 present the zero-field low-lying fermionic excitation ε(k) for the frustrated diamond chain with J1, J3 < 0 and J2 > 0. Acknowledgments We are grateful to Prof. D. P. Arovas for useful com- munication. This work is supported in part by the Na- tional Science Fund for Distinguished Young Scholars of China (Grant No. 10625419), the National Science Foundation of China (Grant Nos. 90403036, 20490210, 10247002), and by the MOST of China (Grant No. 2006CB601102). [] ∗Corresponding author. E-mail: [email protected] [1] M.Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett. 78, 1984 (1997). [2] M. Drillon, E. Coronado, M. Belaiche and R. L. Carlin, J. Appl. Phys. 63, 3551 (1988); M. Drillon, M. Belaiche, P. Legoll, J. Aride, A. Boukhari and A. Moqine, J. Magn. Magn. Mater. 128, 83 (1993). [3] H. Sakurai, K. Yoshimura, K. Kosuge, N. Tsujii, H. Abe, H. Kitazawa, G. Kido, H. Michor and G. Hilscher, J. Phys. Soc. Japan 71, 1161 (2002). [4] M. Ishii, H. Tanaka, M. Mori, H. Uekusa, Y. Ohashi, K. Tatani, Y. Narumi, and K. Kindo, J. Phys. Soc. Jpn. 69, 340 (2000). [5] M. Fujisawa, J. Yamaura, H. Tanaka, H. Kageyama, Y. Narumi, and K. Kindo, J. Phys. Soc. Jpn. 72, 694 (2003). [6] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Klehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta, Phys. Rev. Lett. 94, 227201 (2005). [7] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, and T. Ide- hara, Physica B 329, 967 (2003). [8] K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Con- dens. Matter 8, 6405 (1996). [9] K. Okamoto, T. Tonegawa, Y. Takahashi, and M. Kaburagi, J. Phys.: Condens. Matter 11, 10485 (1999). [10] T. Tonegawa, K. Okamoto, T. Hikihara, Y. Takahashi, and M. Kaburagi, J. Phys. Soc. Jpn. 69, 332 (2000). [11] K. Sano and K. Takano, J. Phys. Sco. Jpn. 69, 2710 (2000). [12] T. Tonegawa, K. Okamoto, T. Hikihara, Y. Takahashi, and M. Kaburagi, J. Phys. Chem. Solids 62, 125 (2001). [13] K. Okamoto, T. Tonegawa, and M. Kaburagi, J. Phys. Condens. Matter 15, 5979 (2003). [14] A. Honecker and A. Lauchli, Phys. Rev. B 63, 174407 (2001). [15] D. D. Swank and R. D.Willett, Inorganica Chimica Acta, 8, 143 (1974). [16] S. White; T. Xiang and X. Wang, Density-Matrix Renor- malization, Lecture Notes in Physics, Vol. 528, edited by I. Peschel, X. Wang, M. Kaulke and K. Hallberg (Springer-Verlag, New York, 1999). [17] U. Schollwock, Rev. Mod. Phys. 77, 259 (2005). [18] We have checked that in the present situation, the static structure factor S(q) calculated from 〈Szj S 0 〉 coincides with that from 〈(Szj − 〈S j 〉)(S 0 − 〈S 0 〉)〉. [19] Y. Ajiro, T. Asano, K. Nakaya, M. Mekata, K. Ohoyama, Y. Yamaguchi, Y. Koike, Y. Morii, K. Kamishima, H. A. Katori, and T. Goto, J. Phys. Soc. Jpn. 70, Suppl. A, 186 (2001). [20] D. P. Arovas, A. Auerbach, and F. D. M. Haldane, Phys. Rev. Lett. 60, 531 (1988). [21] The system under interest involves possibly ferrimag- netic, dimerized and spin liquid phases, leading to the spin-spin correlation functions exhibit different behav- iors including exponential or power-law decaying. The constant cl in Eq. (7) characterizes the long-rang order in ferrimagnetic phase. [22] B. Gu, G. Su, and S. Gao, Phys. Rev. B 73, 134427 (2006). [23] I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, J. Phys. A: Math. Gen. 22, 511 (1989). [24] R. P. Singh, M. E. Fisher, and R. Shanker, Phys. Rev. B 39, 2562 (1989). [25] B. Gu, G. Su, and S. Gao, J. Phys.:Condens. Matter 17, 6081 (2005). [26] B. Gu and G. Su, Phys. Rev. Lett. 97, 089701 (2006). [27] J. C. Livermore, R. D. Willett, R. M. Gaura, and C. P. Landee, Inorg. Chem. 21, 1403 (1982). [28] P. J. van Koningsbruggen, D. Gatteschi, R. A. G. de Graaff, J. G. Haasnoot, J. Reedijk, and C. Zanchini, In- org. Chem. 34, 5175 (1995). [29] http://www.netlib.org/lapack/ http://www.netlib.org/lapack/

0704.0151

Extraction of physical laws from joint experimental data

EPJ manuscript No. (will be inserted by the editor) Extraction of physical laws from joint experimental data Igor Grabec Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, PP 394, 1001 Ljubljana, Slovenia, Tel: +386 01 4771 605, Fax: +386 01 4253 135, E-mail: [email protected] Received: date / Revised version: date Abstract. The extraction of a physical law y = yo(x) from joint experimental data about x and y is treated. The joint, the marginal and the conditional probability density functions (PDF) are expressed by given data over an estimator whose kernel is the instrument scattering function. As an optimal estimator of yo(x) the conditional average is proposed. The analysis of its properties is based upon a new definition of prediction quality. The joint experimental information and the redundancy of joint measurements are expressed by the relative entropy. With the number of experiments the redundancy on average increases, while the experimental information converges to a certain limit value. The difference between this limit value and the experimental information at a finite number of data represents the discrepancy between the experimentally determined and the true properties of the phenomenon. The sum of the discrepancy measure and the redundancy is utilized as a cost function. By its minimum a reasonable number of data for the extraction of the law yo(x) is specified. The mutual information is defined by the marginal and the conditional PDFs of the variables. The ratio between mutual information and marginal information is used to indicate which variable is the independent one. The properties of the introduced statistics are demonstrated on deterministically and randomly related variables. PACS. 06.20.DK Measurement and error theory – 02.50.+s Probability theory, stochastic processes, and statistics – 89.70.+c Information science http://arxiv.org/abs/0704.0151v1 2 Igor Grabec: Extraction of physical laws from joint experimental data 1 Introduction The progress of natural sciences depends on advancement in the fields of experimental techniques and modeling of relations between experimental data in terms of physical laws.[1,2] By utilizing computers a revolution appeared in the acquisition of experimental data while modeling still awaits a corresponding progress. For this purpose the modeling process should be generally described in terms of operations that could be autonomously performed by a computer. A step in this direction was taken recently by a nonparametric statistical modeling of the probability dis- tribution of measured data.[3] The nonparametric model- ing requires no a priori assumptions about the probability density function (PDF) of measured data and therefore provides for a fairly general and autonomous experimen- tal modeling of physical laws by a computer.[1,4] More- over, the inaccuracy of measurement caused by stochastic influences can be properly accounted for in the nonpara- metric modeling that further leads to the expression of ex- perimental information, redundancy of repeated measure- ments and model cost function in terms of entropy of infor- mation. These variables have already been applied when formulating an optimal nonparametric modeling of PDF, in the most simple case of a one–dimensional variable.[3] However, more frequently than modeling of a PDF the problem is to extract a physical law from joint data about various variables and to analyze its properties. Therefore, the aim of this article is to propose a general statistical approach also to the solution of this problem. As an optimal statistical estimator of an experimen- tal physical law we propose the conditional average (CA) that is determined by the conditional PDF.[1] This esti- mator represents a nonparametric regression whose struc- ture is case independent; hence it can be generally pro- grammed and autonomously determined by a computer. Due to these convenient properties, we consider CA as a basis for the autonomous extraction of experimental phys- ical laws in data acquisition systems. The fundamental steps of the proposed approach to extraction of experimental physical laws from given data are explained in the second section. We first define the estimators of the joint, the marginal and the conditional PDFs and derive from them the conditional average as an optimal estimator of a physical law that is hidden in joint data. In order to estimate the number of data ap- propriate for the extraction of a physical law, we further introduce the statistics that characterize the information provided by joint measurements. In the third section of the article the properties of the CA estimator and the other introduced statistics are demonstrated on cases of deterministically and randomly related data. 2 Statistics of joint measurements 2.1 Uncertainty of experimental observation Without loss of generality we consider a phenomenon that can be quantitatively characterized by two scalar valued variables x and y comprising a vector z = (x, y). We fur- ther assume that the phenomenon can be experimentally Igor Grabec: Extraction of physical laws from joint experimental data 3 explored by repetition of joint measurements on a two– channel instrument having equal spans Sx = (−L,L), Sy = (−L,L). Their Cartesian product Sxy = Sx ⊗ Sy determines the joint span. We treat a measurement of a joint datum as a process in which the measured object generates the instrument output z = (x, y). The basic properties of the instrument and measurement procedure can be characterized by a calibration based on a set of objects {wkl = (uk, vl); k = 1, . . . l = 1, . . .} that repre- sent joint physical units. Using these units, a scale net can be determined in the joint span Sxy of the instrument. In order to simplify the notation, we further omit the indices of units. A common property of measurements is that the out- put of the instrument fluctuates even when calibration is repeated.[1,2] We describe this property by the joint PDF ψ(z|w), which characterizes the scattering of the in- strument output at a given joint unit w. For the sake of simplicity, we consider an instrument whose channels can be calibrated mutually independently. In this case the instrument scattering function is expressed by the prod- uct of scattering functions corresponding to both channels ψ(z|w) = ψ(x|u)ψ(y|v). Their mean values u, v, and stan- dard deviations σx, σy represent an element of the instru- ment scale and the scattering of instrument output at the joint calibration. These values can be estimated statisti- cally by the sample mean and variance of both components measured during repeated calibration by a joint unit w. The standard deviation σ characterizes the uncertainty of the measurement procedure performed on a unit.[1,2] We further consider the most frequent case in which the output scattering does not depend on the channel index and the position w = (u, v) on the joint scale. In this case it can be expressed as a function of the difference z − w = (x − u, y − v) and a common standard devia- tion σ = σx = σy as ψ(z|w) = ψ(z −w, σ). We consider scattering of instrument output during calibration as a consequence of random disturbances in the measurement system. When these disturbances are caused by contribu- tions from mutually independent sources, the central limit theorem of the probability theory leads us to the Gaussian scattering function ψ(z−w, σ) = g(x−u, σ)g(y−v, σ), in which the scattering of a single component is determined ψ(x|u) = g(x− u, σ) = 1√ − (x− u) . (1) 2.2 Estimation of probability density functions Let us consider a single measurement which yields a joint datum z1 = (x1, y1). We assume that this joint datum appears at the outputs of instrument channels, since it is the most probable at a given state z of the observed phe- nomenon and the instrument during measurement. There- fore, we utilize the measured datum z1 as the center of the probability distribution ψ(z− z1, σ) = ψ(x− x1, σ)ψ(y − y1, σ) that represents the corresponding state. Consider next a series of N repeated measurements which yield the basic data set {zi; i = 1, . . . , N}. In ac- cordance with the above–given interpretation of measured data we adapt to them the distributions {ψ(z−zi, σ); i = 1, . . . , N}. If the data z1, . . . , zN are spaced more than σ 4 Igor Grabec: Extraction of physical laws from joint experimental data apart, we assume that their scattering is caused by varia- tion of the state z in repeated measurements and generally consider z as a random vector variable. Its joint PDF is determined by the statistical average over distributions {ψ(z− zi, σ); i = 1, . . . , N} as: fN (z) = ψ(z− zi, σ). (2) This function represents an experimental model of PDF and resembles Parzen’s kernel estimator, which is often used in statistical modeling of PDFs.[5,4] However, in Parzen’s modeling the kernel width σ plays the role of a smooth- ing parameter whose value decreases with the number of data N , which is not consistent with the general proper- ties of measurements. In opposition to this, we consider σ as an instrumental parameter that is determined by the inaccuracy of measurement.[3,4] In the majority of experi- mental observations σ is a constant during measurements, and hence need not be further indicated in the scattering function ψ. From the joint PDF f(z) = f(x, y) the marginal PDF f(x) of a component x is obtained by integration over the other component, for example: f(x) = f(x, y)dy (3) The conditional PDF of the variable y at a given condition x is then defined by the ratio of the joint PDF and the marginal PDF of the condition: f(y|x) = f(x, y) Using the experimental model of joint PDF (2) we obtain for the marginal and conditional PDFs the following kernel estimators: fN(x) = ψ(x− xi, σ) (5) fN (y|x) = i=1 ψ(x− xi, σ)ψ(y − yi, σ) i=1 ψ(x− xi, σ) 2.3 Estimation of a physical law It is often observed that the joint PDF resembles a crest along some line y = ŷ(x). We consider ŷ(x) as an estimator of a hidden physical law y = yo(x) that provides for a prediction of a value y from the given value x. If we repeat joint measurements, and consider only those that yield the value x, we can generally observe that corresponding values of the variable y are scattered, at least due to the stochastic character of the measurements. As an optimal predictor of the variable y at the given value x, we consider the value ŷ that yields the minimum of the mean square prediction error D at a given x: D = E[(ŷ − y)2|x] = min(ŷ) (7) The minimum takes place when dD/dŷ = 0. The solu- tion of this equation yields as the optimal predictor ŷ the conditional average ŷ(x) = E[y|x] = y f(y|x)dy (8) By using Eq. 6 for the conditional probability, we obtain for CA the superposition ŷN (x) = i=1 yiψ(x− xi, σ) i=1 ψ(x− xi, σ) yiCi(x) (9) The coefficients Ci(x) = ψ(x− xi, σ) i=1 ψ(x − xi, σ) Igor Grabec: Extraction of physical laws from joint experimental data 5 represent a normalized measure of similarity between the given value x and sample values xi and satisfy the condi- tions: Ci(x) = 1 , (11) 0 ≤ Ci(x) ≤ 1. (12) The more similar given value x is to a datum xi, the larger the coefficient Ci(x) is and the contribution of the corre- sponding term yiCi(x) to the sum in Eq.(9). The pre- diction of the value ŷN (x), which best corresponds to the given value x, thus resembles the associative recall of mem- orized items in the brains of intelligent beings, and there- fore could be treated as a basis for the development of computerized autonomous modelers of physical laws and related machine intelligence.[1] The predictor Eq. (9) is completely determined by the set of measured data {z − zi; i = 1, . . . , N} and the in- strument scattering function ψ. The predictor is not based on any a priori assumption about the functional relation between the variables x and y, as is done for example when a physical law is described by some regression func- tion in which parameters are adapted to given data. The conditional average Eq. (9) can thus be treated as a non- parametric regression, although the scattering functions ψ(z−zi, σ) still depend on the parameters zi, σ. However, these parameters, as well as the form of the function ψ, are totally specified by measurements. They represent a property of the observed phenomenon and not an assumed auxiliary of the modeling. Since the form of the CA pre- dictor does not depend on a specific phenomenon under consideration, it could be considered as a generally ap- plicable basis for statistical modeling of physical laws in terms of experimental data in an autonomous computer. It is convenient that Eq. (9) can be simply generalized to a multi–dimensional case by substituting the condition and the estimated variable by the corresponding vectors.[1] Moreover, it is convenient that the ordering into depen- dent and independent variables is done automatically by a specification of the condition. 2.3.1 Description of predictor quality We can interpret a phenomenon which is characterized by the vector z = (x, y) as a process that maps the vari- able x to the variable y. When the variables x and y are stochastic, we most generally describe this mapping by the joint PDF f(x, y). Similarly, we can interpret the predic- tion of the variable ŷ(x) from the given value x as a pro- cess that runs in parallel with the observed phenomenon. This process is also generally characterized by the PDF f(x, ŷ), while the relation between the variables y and ŷ is characterized by the PDF f(y, ŷ). The better the pre- dictor is, the more the distribution f(y, ŷ) is concentrated along the line y = ŷ(x). For a good predictor we generally expect that the prediction error Er = y − ŷ is close to 0. Since both variables are considered as stochastic ones, we expect that the first and second moments of the pre- diction error E[y − ŷ], E[(y − ŷ)2] are small, while for an exact prediction E[y − ŷ] = 0, and E[(y − ŷ)2] = 0. The second moment of the error is equal to E[(y − ŷ)2] = Var(y)+Var(ŷ)−2Cov(y, ŷ)+(my−mŷ)2, wheremy = E[y] andmŷ = E[ŷ] denote mean values. If the variables y and ŷ 6 Igor Grabec: Extraction of physical laws from joint experimental data are statistically independent and have equal mean values, the covariance vanishes: Cov(y, ŷ) = 0, and my −mŷ = 0, so that E[(y − ŷ)2] = Var(y) + Var(ŷ). Based upon this property we introduce a relative statistic called the pre- dictor quality with the formula Q = 1− E[(y − ŷ) Var(y) + Var(ŷ) 2Cov(y, ŷ) Var(y) + Var(ŷ) − (my −mŷ) Var(y) + Var(ŷ) Its value equals 1 for an exact prediction: ŷ = y, while it equals 0, if the variables y, ŷ are statistically independent and have equal mean values. If the mean values differ: my −mŷ 6= 0, the quality Q can also be negative. When the predictor is determined by the conditional average (8), we obtain for its mean value mŷ = E[ŷ] = ŷf(x)dx = yf(y|x)f(x)dxdy yf(y, x)dxdy = E[y] = my. (14) Since in this case my −mŷ = 0, we further get 2Cov(y, ŷ) Var(y) + Var(ŷ) Similarly we get for the covariance Cov(y, ŷ) = (y −my)(ŷ(x) −mŷ(x)])f(y, x)dxdy (ŷ(x)−mŷ(x))(y −my)f(y|x)dyf(x)dx (ŷ(x)−mŷ(x))2f(x)dx = Var(ŷ), (16) so that the expected quality of the CA predictor is 2Var(ŷ) Var(y) + Var(ŷ) . (17) In the case when the relation between both components of the vector z is determined by some physical law yo(x), and only the measurement procedure introduces an additive noise ν with zero mean E[ν] = 0, and variance E[ν2] = σ2, we can express the variable y as y = yo(x) + ν. In this case the following equations: E[(y − ŷ)2] = σ2, Var(y) = Var(ŷ) + σ2 hold, and we get for the expected predictor quality the expression: 2Var(ŷ) 2Var(ŷ) + σ2 . (18) For Var(ŷ) ≫ σ2/2 we have Q ≈ 1, while for Var(ŷ) ≪ σ2/2 we have Q ≈ 0. In the last case ŷ ≈ constant, while y fluctuates around this constant, and consequently the prediction quality is low. Since generally Var(y) ≥ Var(ŷ) and Var(ŷ) ≥ 0, we obtain from Eq. (17) the inequality 0 ≤ Q ≤ 1. It describes a mean property, which need not be fulfilled exactly if the conditional average is statistically estimated from a finite number of samples N ; but we can expect that it holds ever more with an increasing N . However, we can gen- erally expect that with an increasing N , the statistically estimated CA ever better represents the underlying physi- cal law y = yo(x). However, with an increasing N , the cost of experiments increases, and consequently there generally appears the question: ”How to specify a number of sam- ples N that is reasonable for the experimental estimation of a hidden law yo(x)?” 2.4 Experimental information In order to answer the last question, we proceed with the description of the indeterminacy of the vector variable z in terms of the entropy of information. Following the def- initions given for a scalar random variable in the previous Igor Grabec: Extraction of physical laws from joint experimental data 7 article,[3] we first describe the indeterminacy of the com- ponent x. For this purpose we introduce a uniform refer- ence PDF ρ(x) = 1/(2L) that hypothetically corresponds to the most indeterminate noninformative observation of variable x; or to equivalently prepared initial states of the instrument before executing the experiments in a series of observations. By using this reference and the marginal PDF f(x), we first define the indeterminacy of a continu- ous random variable by the negative value of the relative entropy[6,7] Hx = − f(x) log (f(x) dx. (19) Using the expressions for the reference, instrumental scat- tering function, and experimentally estimated PDF, we obtain the expressions for the uncertainty Hu of calibra- tion performed on a unit u, the uncertainty Hx of the component x, experimental information Ix provided by N measurements of x, and the redundancy Rx of these measurements as follows [3]: Hu = − ψ(x, u) log(ψ(x, u)) dx − log(2L), Hx = − fN (x) log(fN (x)) dx − log(2L), Ix(N) = Hx −Hu, Rx(N) = log(N)− Ix(N), (20) Similar equations are obtained for the component y by substituting x→ y. In order to describe the uncertainty of the random vec- tor z, we utilize the reference PDF that is uniform inside the joint span Sxy: ρ(z) = ρ(x)ρ(y) = 1/(2L) 2, and van- ishes elsewhere. By analogy with the scalar variable we define the indeterminacy of the random vector z by the negative value of the relative entropy:[6] Hxy = − f(z) log (f(z) dxdy. (21) In the case of a uniform reference PDF we obtain Hxy = − f(z) log(f(z)) dxdy − 2 log(2L). (22) With this formula we then express the uncertainty of the joint instrument calibration as ψ(z,w) log(ψ(z,w)) dxdy − 2 log(2L). For σ ≪ L we obtain from the Gaussian scattering func- tion ψ(z, zi) = g(x− xi, σ)g(y − yi, σ) the approximation ≈ log + log + 1, (24) The uncertainty of calibration depends on the ratio be- tween the scattering width 2σ and the instrument span 2L in both directions. The number 2 log(σ/L) determines the lowest possible uncertainty of measurement on the given two–channel instrument, as achieved at its joint calibra- tion. The indeterminacy of the random vector z, which char- acterizes the scattering of experimental data, is defined by the estimated joint PDF as Hxy = − fN (z) log(fN (z)) dxdy − 2 log(2L) (25) and is generally greater than the uncertainty of calibra- tion described by H . Since H denotes the lowest possi- ble indeterminacy of observation carried out over a given instrument, we define the joint experimental information 8 Igor Grabec: Extraction of physical laws from joint experimental data Ixy about vector z = (x, z) by the difference Ixy(N) = Hxy −Hw fN (z) log(fN (z)) dxdy ψ(z,w) log(ψ(z,w)) dxdy. (26) Most properties of the uncertainty and information apper- taining to a random vector are similar to those in the case of a scalar variable. For example, the reference density ρ(z) can be arbitrarily selected since it is excluded from the specification of the experimental information.[3] Further- more, the joint experimental information Ixy(1) provided by a single measurement is zero. For a measurement which yields multiple samples z1, . . . , zN that are mutually sep- arated by several σ in both directions, the distributions ψ(z, z1) = g(x− xi, σ)g(y− yi, σ) are nonoverlapping and the first integral on the right of Eq. 26 can be approxi- mated as ψ(z, zi) log ψ(z, zi) ≈ log(N)− ψ(z, z1) logψ(z, z1) dxdy (27) so that we get Ixy(N) ≈ log(N). If the distributions ψ(z, zi) are overlapping but not concentrated at a single point, the inequality 0 ≤ Ixy(N) ≤ log(N) holds generally. Similarly as the entropy of information for a discrete random vari- able, the experimental information describes how much information is provided by N experiments performed by an instrument that is not infinitely accurate.[6] In accor- dance with these properties the experimental information describes the complexity of experimental data in units of information entropy, which are here nats. When the distributions ψ(z, zi) are nonoverlapping, N repeated experiments yield the maximal possible informa- tion log(N). However, with an increasing number N , ever more overlapping of distributions ψ(z, zi) takes place, and therefore the experimental information Ixy(N) increases more slowly than log(N). Consequently, the repetition of joint measurements becomes on average ever more redun- dant with an increasing number N . The difference Rxy(N) = log(N)− Ixy(N) . (28) thus represents the redundancy of repeated joint measure- ments in N experiments. Since the overlapping of distri- butions ψ(z, zi) increases with an increasing number of ex- periments, the experimental information on average tends to a constant value Ixy(∞), and along with this, the re- dundancy increases with N . The number Kxy(N) = e Ixy(N) (29) describes how many nonoverlapping distributions are needed to represent the experimental observation. With an in- creasing N , the number Kxy(N) tends to a fixed value Kxy(∞) that can be well estimated already from a finite number of experiments. We could conjecture thatKxy(∞) approximately determines a reasonable number of experi- ments that provide sufficient data for an acceptable mod- eling of the joint PDF. However, it is still better to de- termine such a number from a properly introduced cost function of the experimental observation. With this aim we consider the difference Dxy(N) = Ixy(∞)− Ixy(N) as the measure of the discrepancy between the experimen- Igor Grabec: Extraction of physical laws from joint experimental data 9 tally observed and the true properties of the phenomenon. An information cost function is then comprised of the re- dundancy and the discrepancy measure: Cxy(N) = Rxy(N) +Dxy(N). (30) Since the redundancy on average increases, while the dis- crepancy measure decreases with the number of measure- ments N , we expect that the cost function Cxy(N) ex- hibits a minimum at a certain number No, which could be considered as an optimal one for the experimental model- ing of a phenomenon. From the definition of redundancy and the discrepancy measure we further obtain Cxy(N) = Rxy(N)+Dxy(N) = log(N)−2Ixy(N)+Ixy(∞). Since the last term is a constant for a given phenomenon, it is not essential for the determination of No, and can be omitted from the definition of the cost function. This yields a more simple version Cxy(N) = log(N)− 2Ixy(N), (31) which is more convenient for application since it does not include the limit value Ixy(∞). In a previous article [3] we have proposed a cost function that is comprised from the redundancy and the information measure of the dis- crepancy between the hypothetical and experimentally ob- served PDFs. However, such a definition is less convenient than the present one, although the values of No deter- mined from both cost functions do not differ essentially. Numerical investigations also show that the optimal num- ber No approximately corresponds to Kxy(∞) = eIxy(∞) if the distribution of the data points is approximately uni- form. Although the experimental information of a vector vari- able and its scalar components exhibits similar properties, their values generally do not coincide since the overlapping of distributions ψ(z, zi) generally differs from that of dis- tributions ψ(x, xi) or ψ(y, yi). Therefore, the experimen- tal information provided by joint measurements generally differs from that provided by measurements of single com- ponents. 2.5 Mutual information and determination of one variable by the other In order to describe the information corresponding to the relation between variables x, y we introduce conditional entropy. At a given value x we express the entropy per- taining to the variable y by the conditional PDF as Hy|x = − f(y|x) log (f(y|x) dy (32) If we express in Eq. (21) the joint PDF by the conditional one f(z) = f(y|x)f(x) we obtain the following equation: Hxy = Hy|x +Hx (33) in which Hy|x denotes the average conditional entropy of information Hy|x = − Hy|xf(x) dx. (34) When we exchange the meaning of the variables we get Hxy = Hx|y +Hy. (35) Based on these equations and Eq. (26) we obtain the fol- lowing relation between the joint and the conditional in- 10 Igor Grabec: Extraction of physical laws from joint experimental data formation Ixy = Hx|y +Hy −Hu −Hv = Iy|x + Ix = Ix|y + Iy (36) where the conditional information is defined by Ix|y = Hx|y −Hu or Iy|x = Hy|x −Hv. (37) When the components of the vector z are statistically independent, the joint PDF is equal to the product of marginal probabilities and the joint information is given by the sum Ixy = Ix + Iy, which represents the maxi- mal possible information that could be provided by joint measurements. However, when x and y are not statisti- cally independent, the joint information is less than the maximal possible one: Ixy < Ix + Iy. The difference Im = Ix + Iy − Ixy = Ix − Ix|y = Iy − Iy|x. (38) can be interpreted as the experimental information that a measurement of one variable provides about another one and is consequently called the mutual information.[6,8,9,10] In accordance with the previous interpretation of the re- dundancy, it follows from the last two terms in Eq. (38) that the mutual information also describes how redun- dant on average is a measurement of the variable y at a given x or vice versa. In accordance with the definition of the redundancy of a certain number N of measurements Rx(N) = log(N) − Ix, we further define also the mutual redundancy of N joint measurements Rm(N) = log(N)− Im(N) . (39) If we then take into account all the definitions of the re- dundancies and types of information, we obtain the for- mula: Rxy(N) = Rx(N) +Ry(N)−Rm(N) (40) It should be pointed out that redundanciesRxy(N), Rx(N), Ry(N), and Rm(N) generally increase with N , while the corresponding experimental information tends to fixed val- ues that correspond to the amount of data needed for pre- senting related variables. In order to describe quantitatively how well determined the value of the variable y by the value of x is on aver- age, we propose a relative measure of determination by the ratio Dy|x = . (41) If Dy|x > Dx|y, the value of the variable x better deter- mines the value of y than vice versa. In this case the vari- able x could be considered as more fundamental for the description of the phenomenon, and consequently as an independent one. In the case of functional dependence de- scribed by a physical law y = yo(x), the relative measure of determination is Dy|x = 1, while for the statistically independent variables x and y it is Dy|x = 0. The entropy of information is generally decreased if the distribution of scattered experimental data at a given x is compressed to the estimated physical law ŷ(x). The corresponding information gain is in drastic contrast to the information loss that is caused by the noise in a mea- surement system.[11] Igor Grabec: Extraction of physical laws from joint experimental data 11 3 Illustration of statistics 3.1 Data with a hidden law The purpose of this section is to demonstrate graphically the basic properties of the statistics introduced above. For this purpose it is most convenient to generate data nu- merically since in this case the relation between the vari- ables x and y, as well as the properties of the scatter- ing function ψ(z), can be simply set. For our demonstra- tion we arbitrarily selected a third order polynomial law yo(x) = [x(x − 5)(x + 10)]/100 and the Gaussian scatter- ing function with standard deviation σ = 0.2. To simulate the basic data set {xi, yi; i = 1, . . . , N}, we first calcu- lated 50 sample values xi by summing two random terms obtained from a generator with a uniform distribution in the interval [−8,+8] and from a Gaussian generator hav- ing the mean value 0 and standard deviation σ = 0.2. The corresponding sample values yi were then calculated as a sum of terms obtained from the selected law yo(xi) and the same random Gaussian generator with a different seed. The generated data {xi, yi; i = 1, . . . , 50} were used as centers of scattering function when estimating the joint PDF based on Eq. (2). An example of such PDF is shown in Fig. 1, while the corresponding joint data of the basic set are shown by points in the top curve of Fig. 2 together with the underlying law yo(x). The conditional average predictor, which corresponds to the presented example, was modeled by inserting data from the basic data set into Eq. (9). To demonstrate its performance, we additionally generated a test data set by N=50, σ=0.2 Fig. 1. The joint PDF f(x, y) utilized to demonstrate the properties of the conditional average predictor. −10 −8 −6 −4 −2 0 2 4 6 8 10 TESTING OF CA PREDICTOR σ = 0.2 N=50 Q = 0.977 Fig. 2. Testing of CA predictor. Curves representing the un- derlying law and given data yo, y – (top), test and predicted data yt, yp – (middle), and prediction error Er = yp − yt – (bottom) are displaced in vertical direction for a better visu- alization. the same procedure as in the case of the basic data set, but with different seeds of all the random generators. Using the values xi,t of the test set, we then predicted the cor- responding values ŷi by the modeled CA predictor. With this procedure we simulated a situation that is normally 12 Igor Grabec: Extraction of physical laws from joint experimental data met when a natural law is modeled and tested based upon experimental data. The test and predicted data are shown by the middle two curves in Fig. 2. From both data sets the prediction error Er = ŷ − yt was calculated that is presented by the bottom curve (..*..) in Fig. 2. The curve representing the predicted data (–o–) is smoother than the curve representing the original test data (..·..). This prop- erty is a consequence of smoothing caused by estimating the conditional mean value from various data included in the modeled CA predictor. In spite of this smoothing, it is obvious that the characteristic properties of the relation between the variables x and y is approximately extracted from the given data by the CA predictor. This further means that the properties of the hidden law y = yo(x) can be approximately described in the region where measured data appear based on a finite number of joint samples. The quality of estimation of the hidden law yo(x) de- pends on the values and number N of statistical samples utilized in Eq. (9) in the modeling of CA and its testing. To demonstrate this property, we repeated the complete pro- cedure three times, using various statistical data sets with increasing N and determined the dependence of predic- tor quality Q on N . The result is presented in Fig. 3. The quality statistically fluctuates with the increasing N , but the fluctuations are ever less pronounced, so that quality determined from different data sets converges to a com- mon limit value at a large N . In our example with σ = 0.2 the limit value is approximately Q = 0.98. With increas- ing N , the curves corresponding to different data sets join approximately at NCA ≈ 30. At a higher N the fluctua- 0 5 10 15 20 25 30 35 40 45 50 PREDICTOR QUALITY σ = 0.2 Fig. 3. Dependence of predictor quality Q on number of sam- ples N determined by various statistical data sets. tions of Q are ever less expressive. We could conjecture that about 30 data values are needed to model the CA predictor in the presented case approximately. The smaller the scattering width σ is, the higher gen- erally the limit value of the predictor quality is, but on average Q is still less than 1 if 1/σ and N are finite. This property is in tune with the well–known fact that it is impossible to determine exactly the law y = yo(x) from joint data that are measured by an instrument which is subject to output scattering due to inherent stochastic disturbances. The properties of the statistics that are formulated based upon the entropy of information are demonstrated for the case with σ = 0.2 in Fig. 4. It shows the depen- dence of experimental information Ixy, mutual informa- tion Im, redundancy Rxy, and cost function Cxy on the number of samples N for three different sample sets. In the same figure the maximal possible information, which Igor Grabec: Extraction of physical laws from joint experimental data 13 0 10 20 30 40 50 60 70 80 90 100 log(N) Ixy Im Rxy Cxy σ=0.2 Fig. 4. Dependence of log(N), experimental information Ixy, mutual information Im, redundancy Rxy, and cost function Cxy on the number of samples N determined by various sta- tistical data sets. corresponds to the ideal case with no scattering, is also presented by the curve log(N), since it represents the ba- sis for defining the redundancy. Similarly as in the one– dimensional case [3], the experimental information Ixy in the two–dimensional case also converges with increasing N to a fixed value. In the presented case the limit value is Ixy(∞) ≈ 3.2, which yields the number K∞ ≈ 25. This number is approximately equal to the ratio of standard deviation of variable x and the scattering width σ and describes how many uniformly distributed samples are needed to represent the PDF of the data.[3] Due to the convergence of experimental information to a fixed value, the curve Ixy(N) starts to deviate from log(N) with the in- creasingN . Consequently the redundancyRxy = log(N)− Ixy(N) starts to increase, which further leads to the min- imum of the cost function Cxy(N) = log(N) − 2Ixy(N). 0 10 20 30 40 50 60 70 80 90 100 log(N) Ixy Ix Iy Im σ=0.2 Fig. 5. Dependence of log(N), experimental information Ixy, marginal informations Ix, Iy, and mutual information Im on the number of samples N . The minimum is not well pronounced due to statistical variations, but it takes place at approximately No ≈ 30. Not surprisingly, the optimal number No approximately corresponds to K∞ and also to NCA. Similarly as the joint experimental information Ixy, the marginal experimental information Ix, Iy also converges to fixed values with increasing N .[3] These statistics are presented in Fig. 5 for the same data generator as applied in the case of Fig. 4. The sample values of variable x take place in a larger interval than those of variable y. Hence there is less overlapping of scattering functions comprising the marginal PDF of x and consequently Ix is larger than Iy. It is also characteristic that Ixy is larger than Ix since the data points in the joint span Sxy are more separated than in the marginal span Sx. Since the mutual informa- tion Im is defined as Im = Ix + Iy − Ixy, its properties depend on both the marginal and the joint information, 14 Igor Grabec: Extraction of physical laws from joint experimental data 0 10 20 30 40 50 60 70 80 90 100 log(N) Ixy Rxy Cxy σ=0.1 σ=0.4 σ=0.4 σ=0.4 σ=0.1 σ=0.1 Fig. 6. Dependence of log(N), experimental information Ixy, redundancy Rxy, and cost function Cxy on the number of sam- plesN determined from various data sets and scattering widths and consequently Im converges more quickly to the limit value than the experimental information Ixy. To demonstrate the influence of scattering width on the presented statistics the calculations were repeated with σ = 0.1 and 0.4. The results are presented in Fig. 6. For the sake of clear presentation, the curves representing the mutual information Im are omitted. As could be expected, the limit value of Ixy increases with decreasing σ. This property is consistent with the well–known fact that more information can be obtained by experimental observation when using an instrument of higher accuracy that corre- sponds to a lesser scattering width. In opposition to this, the redundancy of measurement decreases, and along with it, the optimal number No increases with the decreasing scattering width. 0 10 20 30 40 50 60 70 80 90 100 Dx|yσ=0.2 Fig. 7. Dependence of relative measure of determination Dy|x – (top lines) and Dx|y – (bottom lines) on the number of sam- ples N determined from various statistical data sets. From the calculated mutual and marginal information, the relative measures of determinationDy|x andDx|y were further determined using various statistical data sets. The results are presented in Fig. 7 for the case of scattering width σ = 0.2. When the number of data N surpasses the interval around the optimal number No, statistical varia- tions of Dy|x and Dx|y become less pronounced and their values settle close to limit ones. The limit value Dx|y is essentially lower than Dy|x. This is the consequence of the fact that in our case the variable y is uniquely determined by the underlying law yo(x) based upon the variable x, but not vice versa. In our case, there are three values of the variable x corresponding to a value of y in a certain inter- val. Consequently, y is better determined by a given x than vice versa, which further yields Dy|x > Dx|y. Hence the relative measure of determination indicates that variable x Igor Grabec: Extraction of physical laws from joint experimental data 15 N=500, σ=0.2 random data Fig. 8. The joint PDF f(x, y) of N = 500 statistically inde- pendent random data with σ = 0.2. could be considered more fundamental for the description of the relation between the variables x and y. 3.2 Data without a hidden law To support the last conclusion let us examine an exam- ple in which the sample values of the variables x and y were calculated by two statistically independent ran- dom generators. The corresponding joint PDF is shown in Fig. 8, while the properties of the other statistics are demonstrated by Figs. 9, 10 and 11. The properties of the presented statistics could be un- derstood, if the overlapping of scattering functions com- prising the estimator of the joint PDF is examined. In the previous case with the underlying law yo(x), the joint data are distributed along the corresponding line where −8 ≤ x ≤ +8, while in the last case, they take place in the square region −8 ≤ x ≤ +8,−8 ≤ y ≤ +8. Conse- quently, the number of samples with nonoverlapping scat- tering functions in the last case is approximately L/σ = 16 0 50 100 150 200 250 300 350 400 450 500 log(N) Ixy Im Rxy Cxy σ=0.2 random data Fig. 9. Dependence of log(N), experimental information Ixy, redundancy Rxy, and cost function Cxy on the number of sam- ples N determined by various statistical data sets and scatter- ing widths σ. 0 50 100 150 200 250 300 350 400 450 500 log(N) Ixy Ix Iy Im σ=0.2 random data Fig. 10. Dependence of log(N), experimental information Ixy, marginal informations Ix, Iy, and mutual information Im on the number of samples N in the case of statistically indepen- dent random variables x, y. 16 Igor Grabec: Extraction of physical laws from joint experimental data 0 50 100 150 200 250 300 350 400 450 500 σ=0.2 random data Fig. 11. Dependence of relative measure of determinationDy|x – (top lines) and Dx|y – (bottom lines) on the number of ran- dom samples N in the case of statistically independent random data with σ = 0.2. times larger than in the previous case. In the last case we can therefore expect the optimal number of samples in the interval around Nro ≈ 16No = 480. Since in the last case a larger region is covered by the joint PDF, the overlapping of scattering functions is less probable than previously, and therefore, the joint experimental informa- tion Ixy deviates less quickly from the line log(N) with the increasingN . Therefore, the redundancy increases less quickly and the minimum of the cost function takes place at a much higher number of Nro = 480, which corre- sponds well to our estimation. Since in the last case the experimental information Ixy converges less quickly to the limit value than the marginal information Ix, Iy, the mu- tual information Im first increases and later decreases to its limit value. Related to this is the approach of rela- tive measures of determination Dy|x, Dx|y to much lower limit values as in the previous case. Since the marginal information Ix, Iy is approximately equal, the curves rep- resenting Dy|x, Dx|y join with increasing N , and there is no argument to consider any variable as a more funda- mental one for the description of the phenomenon under examination. This conclusion is consistent with the fact that the centers of the scattering functions are determined by two statistically independent random generators. How- ever, the limit values of the statistics Dy|x, Dx|y are not equal to zero since the region −8 ≤ x ≤ +8,−8 ≤ y ≤ +8 where the data appear is limited, while the characteristic region −σ ≤ x ≤ +σ,−σ ≤ y ≤ +σ covered by the joint scattering function does not vanish. 4 Conclusions Following the procedures proposed in the previous article [3], we have shown how the joint PDF of a vector variable z = (x, y) can be estimated nonparametrically based upon measured data. For this purpose the inaccuracy of joint measurements was considered by including the scattering function in the estimator. It is essential that the properties of the scattering function need not be a priori specified, but could be determined experimentally based upon cali- bration procedure. The joint PDF was then transformed into the conditional PDF that provides for an extraction of the law yo(x) that relates the measured variables x, y. For this purpose the estimation by the conditional average yo(x) ≈ E[y|x] is proposed. The quality of the prediction by the conditional average is described in terms of the es- timation error and the variance of the measured data. It is outstanding that the quality exhibits a convergence to Igor Grabec: Extraction of physical laws from joint experimental data 17 some limit value that represents the measure of applicabil- ity of the proposed approach. Examination of the quality convergence makes it feasible to estimate an appropriate number of joint data needed for the modeling of the law. It is important that the conditional average makes feasi- ble a nonparametric autonomous extraction of underlying law from the measured data. Using the joint PDF estimator we have also defined the experimental information, the redundancy of measure- ment and the cost function of experimental exploration. It is characteristic that experimental information converges with an increasing number of joint samples to a certain limit value which characterizes the number of nonoverlap- ping scattering distributions in the estimator of the joint PDF. The most essential terms of the cost function are the experimental information and the redundancy. Dur- ing cost minimization the experimental information pro- vides for a proper adaptation of the joint PDF model to the experimental data, while the redundancy prevents an excessive growth of the number of experiments. By the position of the cost function minimum we introduced the optimal number of the data that is needed to represent the phenomenon under exploration. This number roughly cor- responds to the ratio between the magnitude of the charac- teristic region where joint data appear and the magnitude of the characteristic region covered by the joint scattering function. It also corresponds to the appropriate number estimated from the quality of prediction by the conditional average. Based upon the experimental information corre- sponding to the joint and marginal PDFs, the mutual in- formation has been introduced and further utilized in the definition of the relative measure of determination of one variable by another. This statistic provides an argument for considering one variable as a fundamental one for the description of the phenomenon. In this article we graphically present the properties of the proposed statistics by two characteristic examples that represent data related by a certain law and statistically independent random data. The exhibited properties agree well with the expectations given by experimental science. The problems related to the extraction of laws represent- ing relations such as y2 + x2 = 1 and the expression of physical laws by differential equations or analytical mod- eling were not considered. For this purpose the statistical methods are developed in the fields of pattern recognition, system identification and artificial intelligence. Acknowledgment The research was supported by the Ministry of Science and Technology of Slovenia and EU COST. References 1. I. Grabec and W. Sachse, Synergetics of Measurement, Pre- diction and Control (Springer-Verlag, Berlin, 1997). 2. J. C. G. Lesurf, Information and Measurement (Institute of Physics Publishing, Bristol, 2002) 3. I. Grabec, Experimental modeling of physical laws, Eur. Phys. J., B, 22 129-135 (2001) 4. R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (J. Wiley and Sons, New York, 1973), Ch. 4. 5. E. Parzen, Ann. Math. Stat., 35 1065-1076 (1962). 18 Igor Grabec: Extraction of physical laws from joint experimental data 6. T. M. Cover and J. A. Thomas Elements of Information Theory (John Wiley & Sons, New York, 1991). 7. A. N. Kolmogorov, IEEE Trans. Inf. Theory, IT-2 102-108 (1956). 8. B. S. Clarke, A. R. Barron, IEEE Trans. Inf. Theory, 36 (6) 453-471 (1990) 9. D. Haussler, M. Opper, Annals of Statistics, 25 (6) 2451- 2492 (1997) 10. D. Haussler, IEEE Trans. Inform. Theory, 43 (4) 1276- 1280 (1997) 11. C. E. Shannon, Bell. Syst. Tech. J., 27 379-423 (1948). Introduction Statistics of joint measurements Illustration of statistics Conclusions

0704.0152

Kinetic equation for finite systems of fermions with pairing

arXiv:0704.0152v2 [nucl-th] 18 Dec 2007 Kinetic equation for finite systems of fermions with pairing V. I. Abrosimov a, D. M. Brink b, A. Dellafiore c,∗, F. Matera c,d aInstitute for Nuclear Research, 03028 Kiev, Ukraine bOxford University, Oxford, U.K. cIstituto Nazionale di Fisica Nucleare, Sezione di Firenze dDipartimento di Fisica, Università degli Studi di Firenze, via Sansone 1, I 50019 Sesto F.no (Firenze), Italy Abstract The solutions of the Wigner-transformed time-dependent Hartree–Fock–Bogoliubov equations are studied in the constant-∆ approximation. This approximation is known to violate particle-number conservation. As a consequence, the density fluc- tuation and the longitudinal response function given by this approximation contain spurious contributions. A simple prescription for restoring both local and global particle-number conservation is proposed. Explicit expressions for the eigenfrequen- cies of the correlated systems and for the density response function are derived and it is shown that the semiclassical analogous of the quantum single–particle spectrum has an excitation gap of 2∆, in agreement with the quantum result. The collective response is studied for a simplified form of the residual interaction. Key words: Pairing, Vlasov equation PACS: 21.10.Pc, 03.65.Sq 1 Introduction The problem of extending the Vlasov equation to systems in which pairing correlations play an important role has been tackled some time ago by Di Toro and Kolomietz [1] in a nuclear physics context and, more recently, by Urban and Schuck [2] for trapped fermion droplets. These last authors derived ∗ Corresponding author Email address: [email protected] (A. Dellafiore). Preprint submitted to Elsevier 16 November 2021 http://arxiv.org/abs/0704.0152v2 the TDHFB equations for the Wigner transform of the normal density matrix ρ and of the pair correlation function κ (plus their time-reversal conjugates) and used them to study the dynamics of a spin-saturated trapped Fermi gas. In the time-dependent theory one obtains a system of four coupled differential equations for ρ, κ, and their conjugates [2] and, if one wants an analytical so- lution, some approximation must be introduced. Here we try to find a solution of the equations of motion derived by Urban and Schuck in the approximation in which the pairing field ∆(r,p, t) is treated as a constant. It is well known that such an approximation violates both particle-number-conservation and gauge invariance (see e.g. sect. 8-5 of [3] and [4]), nonetheless we study it because of its simplicity, with the aim of correcting the final results for its shortcomings. Moreover, the constant-∆ approximation is not satisfactory for describing long wavelength pairing modes in a large system. Such modes have frequencies which are much less than the pairing frequency ∆/~ and for their study it is essential to use a self consistent theory where the gap ∆ is related to the pair density κ through the pairing interaction. The phases of ∆ and κ are particularly important because they describe the collective superfluid currents. On the other hand nuclei are small systems. Shell gaps are large compared with ∆, or equivalently giant resonance frequencies are large com- pared with the pairing frequency. The constant-∆ approximation is much more reasonable in such systems. In Sect. 2, the basic equations are recalled and reformulated in terms of the even and odd components of the normal density ρ. In Sect. 3, the static limit is studied by following the approach of [5] and the constant-∆ approximation is introduced. In Sect. 4, the simplified dynamic equations resulting from the constant-∆ approxmation are derived and their solutions are determined in lin- ear approximation. In Sect. 5, these solutions are studied in a one-dimensional model and the problem of particle-number conservation is examined in detail. By studying the energy-weighted sum rule (in the Appendix), we find that the constant-∆ approximation introduces some spurious strength into the density response of the system. A simple prescription, based on the continuity equa- tion, is proposed in order to eliminate the spurious strength. The resulting strength function gives the same energy-weighted sum rule as for the uncor- related systems. In Sect. 6, the general solution found in Sect. 4 is re-written for spherical systems, where the angular integrations can be performed ex- plicitly, leading to expressions containing only radial integrations. In Sect. 7, the collective response function of spherical nuclei is derived for a simple multipole-multipole residual interaction. In Sect. 8, the quadrupole and oc- tupole channels, that are the ones most affected by the pairing correlations are shown explicitly. Finally, in Sect. 9 conclusions are drawn. 2 Basic equations We assume that our system is saturated both in spin and isospin space and do not distinguish between neutrons and protons, so we can use directly the equations of motion of Urban and Schuck. We start from the equations of motion derived in Ref. [2] for the Wigner- transformed density matrices ρ = ρ(r,p, t) and κ = κ(r,p, t), with the warning that the sign of κ that we are using agrees with that of Ref. [1], hence it is opposite to that of [2]. Moreover we find convenient to use the odd and even combinations of the normal density introduced in [2]: ρev = [ρ(r,p, t) + ρ(r,−p, t)] , (1) ρod = [ρ(r,p, t)− ρ(r,−p, t)] . (2) Thus, the equations of motion given by Eqs.(15a...d) of Ref. [2] read i~∂tρev = i~{h, ρod} − 2iIm[∆∗(r,p, t)κ] (3) i~∂tρod = i~{h, ρev}+ i~Re{∆∗(r,p, t), κ} (4) i~∂tκ=2(h− µ)κ−∆(r,p, t)(2ρev − 1) + i~{∆(r,p, t), ρod} . (5) Here h is the Wigner-transformed Hartree–Fock hamiltonian h(r,p, t), while ∆(r,p, t) is the Wigner-transformed pairing field. Since the time-dependent part of κ is complex, κ = κr+iκi, the last equation gives two separate equations for the real and imaginary parts of κ. Moreover, from the supplementary normalization condition ([6], p. 252) R2 = R (6) satisfied by the generalized density matrix R, the two following independent equations are obtained: ρodκ+ i {ρev, κ}=0 , (7) ρev(ρev − 1) + ρ2od + κκ∗=0 . (8) We shall use the equations of motion (3–5), together with these equations, as our starting point, but first we notice that, in the limit of no pairing, both ∆ and κ vanish, the third equation of motion reduces to a trivial identity, while the first two give the Vlasov equation for normal systems, expressed in terms of the even and odd components of ρ: ∂tρev = {h, ρod} , (9) ∂tρod = {h, ρev} . (10) A solution of the linearized Vlasov equation for normal systems (i. e. without pairing) has been obtained in Ref. [7] and our aim here is to study the changes introduced by the pairing interaction in the solution of [7]. Moreover, before studying the time-dependent problem, it is useful to look at the static limit. 3 Static limit In this section we follow the approach of Ref. [5]. At equilibrium we have ρev = ρ0(r,p) , (11) ρod =0 , (12) h=h0(r,p) , (13) κ=κ0(r,p) (14) ∆0 =∆0(r,p) (15) and equations (3–5) give 0=−2iIm(∆∗0κ0) , (16) 0= i~{h0, ρ0}+ i~Re{∆∗0, κ0} , (17) 0=2(h0 − µ)κ0 −∆0(2ρ0 − 1) , (18) while Eqs. (7, 8) give {ρ0, κ0}=0 , (19) ρ0(ρ0 − 1) + |κ0|2=0 . (20) Equation (16) is satisfied if we assume that ∆0 and κ0 are real quantities, while Eqs. (18) and (20), taken as a system, have the solution:[5] ρ0(r,p) = h0(r,p)− µ E(r,p) κ0(r,p) = − ∆0(r,p) 2E(r,p) , (22) with the quasiparticle energy E(r,p) = ∆20(r,p) + (h0(r,p)− µ)2 . (23) It can be easily checked that Eqs. (21, 22) satisfy also Eqs. (17) and (19), that {h0, ρ0}+ {∆0, κ0} = 0 (24) {ρ0, κ0} = 0 . (25) The (semi)classical equilibrium phase-space distribution is closely related to ρ0(r,p): f0(r,p) = (2π~)3 ρ0(r,p) (26) and the statistical factor 4 takes into account the fact that there are two kinds of fermions. The parametrer µ is determined by the condition drdpf0(r,p) , (27) where A is the number of particles. This integral should keep the same value also out of equilibrium (global particle-number conservation). 3.1 Constant-∆ approximation In a fully self-consistent approach, the pairing field ∆(r,p, t) is related to κ(r,p, t), however here we introduce an approximation and replace the pairing field of the HFB theory with the phenomenological pairing gap of nuclei, hence in all our equations we put ∆(r,p, t) ≈ ∆0(r,p) ≈ ∆ = const. , (28) with ∆ ≈ 1MeV. In the constant-∆ approximation the equilibrium distributions become ρ0(ǫ) = 1− ǫ− µ , (29) κ0(ǫ) =− 2E(ǫ) and the quasiparticle energy E(ǫ) = ∆2 + (ǫ− µ)2 , (31) ǫ = h0(r,p) = + V0(r) the particle energy in the equilibrium mean field. In the following we shall use the relation: κ0(ǫ) = E2(ǫ) dρ0(ǫ) . (32) 4 Dynamic equations Always in the approximation where ∆ is constant and real, the time-dependent equations (3–5) become i~∂tρev = i~{h, ρod} − 2i∆Im(κ) (33) i~∂tρod = i~{h, ρev} (34) i~∂tκ=2(h− µ)κ−∆(2ρev − 1) . (35) This is the simplified set of equations that we want to study here. The sum of the first two equations gives an equation that is similar to the Vlasov equation of normal systems, only with the extra term −2i∆Im(κ). This extra term couples the equation of motion of ρ with that of κ, thus, instead of a single differential equation (Vlasov equation), now we have a system of two coupled differential equations (for ρ and κi). Our aim here is that of determining the effects of pairing on the linear response of nuclei, thus we assume that our system is initially at equilibrium, with densities given by Eqs. (29,30), and that at time t = 0 a weak external field of the kind δV ext(r, t) = βδ(t)Q(r) (36) is applied to it. This simple time-dependence is sufficient to determine the linear response of the system. In a self-consistent approach, we should take into account also the changes of the mean field surrounding each particle induced by the external force and consider a perturbing hamiltonian of the δh = δV ext + δV int , (37) however we start with the zero-order approximation δh = δV ext (38) and will consider collective effects in a second stage. Since we want to solve Eqs. (33–35) in linear approximation, we consider small fluctuations of the time-dependent quantities about their equilibrium values and neglect terms that are of second order in the fluctuations. Hence, in Eqs. (33–35) we put: h=h0 + δh , (39) ρev = ρ0 + δρev , (40) ρod = δρod , (41) κ=κ0 + δκ = κ0 + δκr + iδκi . (42) Then, the linearized form of Eqs. (33–35) is i~∂tδρev = i~{h0, δρod} − 2i∆δκi (43) i~∂tδρod= i~{h0, δρev}+ i~{δh, ρ0} (44) −~∂tδκi=2(ǫ− µ)δκr + 2κ0δh− 2∆δρev , (45) ~∂tδκr =2(ǫ− µ)δκi . (46) Taking the sum of the first two equations gives i~∂tδρ(r,p, t) = (47) i~{h0, δρ(r,p, t)}+ i~{δh(r,p, t), ρ0} − 2i∆δκi(r,p, t) , which can be regarded as an extension of the linearized Vlasov equation stud- ied in [7]. In order to make the comparison with [7] easier, from now on we change the normalization of the phase-space densities and define f(r,p, t)= (2π~)3 ρ(r,p, t) (48) χ(r,p, t)= (2π~)3 κ(r,p, t) , (49) moreover, we put F (ǫ) = (2π~)3 ρ0(ǫ) . (50) In terms of the new functions Eqs. (47) and (45) read i~∂tδf(r,p, t) = i~{h0, δf(r,p, t)}+ i~{δh(r,p, t), f0} − 2i∆δχi(r,p, t) , (51) −~∂tδχi(r,p, t) = 2(ǫ− µ)δχr(r,p, t) + 2χ0δh(r,p, t) − 2∆δfev(r,p, t) . (52) The function fev is given by the obvious extension of Eq. (1). In order to get a closed system of equations, we still need an extra equation for δχr(r,p, t). This can be obtained from the linearized form of the supplementary condition (8) that reads δρev(2ρ0 − 1) = −2κ0δκr , (53) δκr(r,p, t) = 1− 2ρ0(ǫ) δρev(r,p, t) = − δρev(r,p, t) . (54) The last expression has been obtained with the help of Eq. (18). In terms of the new functions f and χ, the last equation reads δχr(r,p, t) = − δfev(r,p, t) . (55) Equations (51, 52) and (55) are the set of coupled equations for the phase- space densities that we have to solve. Replacing Eq. (55) into Eq. (52), and using Eq. (30), gives the following system of coupled differential equations: ∂tδf(r,p, t)= {h0, δf}+ {δh, f0} − 2 δχi(r,p, t) , (56) ∂tδχi(r,p, t)= E2(ǫ) [δf(r,p, t) + δf(r,−p, t)]− 2 δh(r,p, t) . (57) Taking the Fourier transform in time, gives −iωδf(r,p, ω)= {h0, δf}+ {δh, f0} − 2 δχi(r,p, ω) , (58) −iωδχi(r,p, ω))= E2(ǫ) [δf(r,p, ω) + δf(r,−p, ω)] − 2χ0 δh(r,p, ω) , (59) or (for ω 6= 0) −iωδf(r,p, ω) + {δf, h0}=−iωd2 [δf(r,p, ω) + δf(r,−p, ω)] +F ′(ǫ)[{δh, h0}+ iωd2δh] , (60) (Ω(ǫ) Ω(ǫ) = 2 . (62) This frequency plays a crucial role in our approach, its minimum value is 2∆/~. In Eq. (60) we have used the relation {f0, δh} = F ′(ǫ){ho, δh} as well as Eq. (32). By comparing Eq. (60) with the analogous equation for normal systems −iωδf(r,p, ω) + {δf, h0} = F ′(h0){δh, h0} , (63) we can see that the only effect of pairing in the constant-∆ approximation is that of adding the terms proportional to d2. The normal Vlasov equation (63) can be solved in a very compact way by using the method of action-angle variables [8], [7]. In that approach one expands δhn(I)e in·Φ , (64) where I and Φ are the action and angle variables, respectively. Moreover δf(r,p, ω) = δfn(I, ω)e in·Φ (65) {δf, h0} = i(n · ~ω)δfn(I, ω)ein·Φ , (66) where the vector ~ω has components . (67) Then Eq.(63) gives δfn(I, ω) = (n · ~ω) (n · ~ω)− (ω + iε) F ′(ǫ)δhn(I) . (68) The (zero-order) eigenfrequencies of the (normal) physical system are ωn = n · ~ω . (69) Here want to use the same method to solve the more complicated equation (60). Since that equation contains also the function δf(r,−p, ω), we need also the analogous equation for this other quantity: −iωδf(r,−p, ω)− {δf, h0}r,p=−iωd2 [δf(r,p, ω) + δf(r,−p, ω)] +F ′(ǫ)[−{δh, h0}r,p + iωd2δh] . (70) By expanding δf(r,p, ω) and δf(r,−p, ω) as δf(r,±p, ω) = δf±n (I, ω)e in·Φ , (71) Eqs. (60) and (70) give [−ω(1− d ) + ωn]δf n + ω δf−n = F ′(ǫ)[ωn + ωd 2]δhn , (72) ]δf+n + [−ω(1− )− ωn]δf−n = F ′(ǫ)[−ωn + ωd2]δhn , (73) which is a system of two coupled algebraic equations for the coefficients δf+n and δf−n . Its solution is δf+n = ω̄2n + ωωn ω̄2n − ω2 F ′(ǫ)δhn , (74) δf−n = ω̄2n − ωωn ω̄2n − ω2 F ′(ǫ)δhn , (75) where ω̄2n = ω n + Ω 2(ǫ) (76) are the (squared) eigenfrequencies of the correlated system. These eigenfre- quencies are in agreement with the enegy spectrum of a superfluid infinite ho- mogeneous Fermi gas (see e. g. Sect. 39 of [10]) and they lead to a low-energy gap of 2∆ in the excitation spectrum of the correlated systems. However, as anticipated, we expect problems with particle-number conservation. These problems are better discussed in one dimension, where formulae are simpler. 5 One-dimensional systems and particle-number conservation In one dimension, Eq.(60) reads −iωδf(x, p, ω) + ẋ∂xδf − dV0(x) ∂pδf = (77) −iωd2 1 [δf(x, p, ω) + δf(x,−p, ω)] + F ′(ǫ)( p ∂xδh + iωd 2δh) . In zero-order approximation δh(x, p, ω) = βQ(x), moreover, in one dimension F ′(ǫ) = . (78) The vectors ~ω and Φ have only one component: ω0(ǫ) = T (ǫ) Φ(x) = ω0τ(x) , (80) τ(x) = v(ǫ, x′) , (81) v(ǫ, x) = [ǫ− V0(x)] . (82) The time T (ǫ) is the period of the bound motion of particles with enegy ǫ in the equilibrium potential well V0(x): T = 2τ(x2). The points x1,2 are the classical turning points for the same particles. Instead of the action variable I(ǫ) = 1 dxp(ǫ, x), it is more convenient to use the particle energy ǫ as constant of motion. As pointed out in [7], the range of values of τ can be extended to the whole interval (0, T ), by defining τ(x) = v(ǫ, x′) when τ > T . With this extension, the angle variable Φ(x) takes values between 0 and 2π, as it should. In one dimension, Eqs. (74, 75) give δf±n (ǫ, ω) = ω̄2n ± ω ωn ω̄2n − ω2 F ′(ǫ)δhn , (84) δhn = βQn v(ǫ, x) e−iωnτ(x) v(ǫ, x) cos[ωnτ(x)] , (85) The frequencies ωn are the eigenfrequencies of the uncorrelated system: ωn = nω0 , (86) while ω̄n are the new eigenfrequencies modified by the pairing correlations: ω̄n = ± ω2n + Ω 2(ǫ) . (87) Note that, since δh−n = δhn, then δf n = δf By using the solutions (84), we can also obtain an expansion for the even and odd parts of δf : δfev(x, ǫ, ω) = An(ω) cosnω0τ(x) , (88) δfod(x, ǫ, ω) = Bn(ω) sinnω0τ(x) , (89) An(ω) = ω̄2n − ω2 F ′(ǫ)δh′n , (90) Bn(ω) = iω ω̄2n − ω2 F ′(ǫ)δh′n (91) δh′n =2δhn , n 6= 0 , (92) = δhn , n = 0 . (93) Note that, while Bn=0(ω) = 0, we have An=0(ω) 6= 0, and this fact leads to an unphysical fluctuation of the number of particles, induced by the applied external field. These fluctuations are given by δA(ω) = dxδ̺(x, ω) , (94) where δ̺(x, ω) is the density fluctuation at point x: δ̺(x, ω) = dp δf(x, p, ω) = 2 v(ǫ, x) δfev(x, ǫ, ω) . (95) Equation (88) gives δA(ω) = 2 An(ω) dτ cos nω0τ . (96) Since the integrals ∫ T/2 0 dτ cosnω0τ vanish when n 6= 0, the term with n = 0 is the only one contributing to this sum, thus givig an unphysical fluctuation of the number of particles. This problem could be solved simply by excluding the mode n = 0 from the sum in Eq. (88), however this would not be sufficient to solve all problems with particle-number conservation, since we can easily check that the solutions (88, 89) do not satisfy the continuity equation iω̺(x, ω) = ∂xj(x, ω) . (97) The density fluctuation involves only the even part of δf , while the current density j(x, ω) involves only the odd part: j(x, ω) = δf(x, p, ω) = 2 dǫδfod(x, ǫ, ω) . (98) The fact that the continuity equation is violated is a very serious shortcom- ing of the constant-∆ approximation. However, since we have seen that this approximation leads to very simple equations and to rather satisfactory ex- pressions for the eigenfrequencies of the correlated systems, we still use it, but with the following prescription: when calculating the longitudinal response function, the density fluctuations should be evaluated by using Eq. (97), instead of Eq, (95). Then, the density fluctuations (95) should be replaced by δ ¯̺(x, ω) = dǫ∂xδfod(x, ǫ, ω) . (99) In practice we are proposing to evaluate the longitudinal response function in terms of the transverse response function. It is well known that also the more familiar BCS approximation gives a more accurate description of the transverse response (see e.g. sect. 8-5 of [3]). In the Appendix we show that the longitudinal response function resulting from the present prescription satisfies the same energy-weighted sum rule as the uncorrelated response function. This would not necessarily happen if, instead of changing only the even part of δf , we had modified also its odd part. It is interesting to see how the solutions (84) are changed by our prescription. By using Eq. (89) forfod, Eq. (99) gives δ ¯̺(x, ω) = 2 v(ǫ, x) δf̄ev(x, ǫ, ω) , (100) δf̄ev(x, ǫ, ω) = Ān(ω) cosnω0τ(x) , (101) Ān(ω) = Bn(ω) , (102) (note that Ān=0(ω) = 0). Then δf̄(x,±p, ω) = δf̄ev(x, ǫ, ω)± δfod(x, ǫ, ω) (103) and Eq. (84) is replaced by δf̄±n (ǫ, ω) = ω2n ± ω ωn ω̄2n − ω2 F ′(ǫ)δhn . (104) By comparing this expression to Eq. (84), we can see that the fluctuations of the phase-space density given by the constant-∆ approximation contain an extra contribution that we identify as spurious: δf±n (ǫ, ω) = δf̄ n (ǫ, ω) + δf n (ǫ, ω) , (105) δf spurn (ǫ, ω) = Ω2(ǫ) ω̄2n − ω2 F ′(ǫ)δhn . (106) The spurious character of δf spurn is suggested also by sum-rule arguments (see Appendix). The term f spurn (ǫ, ω) contributes to all modes of the density strength function: the contribution to the mode n = 0 gives a fluctuation of the particle-number integral (global paticle-number violation), while the other modes give a spurious contribution to the density strength function, increas- ing the sum rule and violating the continuity equation (local particle-number violation). Note that the spurious contribution (106) affects only the even part of the pase-space density, not the odd part. 6 Spherical Systems The method of action-angle variables gives a very compact solution of the lin- earized Vlasov equation both in the uncorrelated and correlated cases, however it may be useful to make a connection between the results given by this method and the more explicit treatment of spherical nuclei given in [7]. For uncorre- lated system this has been done in [11]. Here we follow that approach in order to derive useful expressions for correlated spherical systems. The components of the vecor n are (n1, n2, n3), the first point to notice is that, because of the degeneracy associated with any central-force field, the vector ~ω has only two non vanishing components: ~ω = (0, ωϕ(ǫ, λ), ω0(ǫ, λ)) . (107) With λ we denote the magnitude of the particle angular momentum. According to Eq. (69), the eigenfrequencies of the uncorrelated system are [7] ωn = ωn3,n2(ǫ, λ) = n3ω0 + n2ωϕ , (108) while Eq. (76) gives the correlated eigenfrequencies ω̄n = ω̄n3,n2 = ± ω2n3,n2 + Ω 2(ǫ) . (109) In three dimensions, the Fourier coefficients analogous to (85) are Qn(I) = (2π)3 dΦe−in·ΦQ(r) . (110) The external field Q(r) can be expanded in partial waves as Q(r) = QL(r)YLM(r̂) , (111) giving Qn(I) = Q(LM)n , (112) with [11] Q(LM)n = dLMN(β ′)δM,n1δN,n2Q . (113) By using this last equation (and changing n3 → n), the expansion (71) be- comes δf(r,±p, ω) = (114) δfL±nN (ǫ, λ, ω)e iφnN (r) DLMN(α, β ′, γ) δfL±nN (ǫ, λ, ω) = ω̄2nN ± ωωnN ω̄2nN − ω2 βF ′(ǫ)QLnN (115) and QLnN the semiclassical limit of the radial matrix elements: QLnN = vr(r) e−iφnN (r)QL(r) , vr(r) cos[φnN(r)]QL(r) . (116) Here T is the period of radial motion, vr(r) the radial velocity vr(r) = ǫ− V0(r)− (117) and the phases φnN(r) are given by φnN(r) = ωnNτ(r)−Nγ(r) , (118) where τ(r) = vr(r′) (119) γ(r) = vr(r′) . (120) The frequencies ω0 and ωϕ are given by τ(r2) , (121) γ(r2) τ(r2) . (122) The Wigner rotation matrix elements in Eq. (114) are given by [9] DLMN(α, β ′, γ) = e−iMαdLMN(β ′)e−iNγ , (123) where (α, β ′, γ) are the Euler angles introduced in [7]. On the basis of the discussion in Sect. (5), we expect that the solution (114) will contain some spurious strength introduced by the constant-∆ approxima- tion. In order to eliminate the spurious contributions, we should replace the coefficients (115) with δf̄L±nN (ǫ, λ, ω) = ω2nN ± ωωnN ω̄2nN − ω2 βF ′(ǫ)QLnN . (124) These modified coefficients allow us to obtain the modified zero-order propa- gator D̄0L(r, r ′, ω) = dǫF ′(ǫ) d̄LnN(r, r ω − ω̄nN + iε , (125) d̄LnN(r, r ′) = (126) (4π)2 2L+ 1 |YLN( (−2ωnN )(ωnN )cosφnN(r) r2vr(r) cosφnN(r r′2vr(r′) and the corresponding response and strength functions: R̄0L(ω)= drdr′r2QL(r)D̄ L(r, r ′, ω)r′2QL(r ′) , (127) S̄0L(ω)=− ImR̄0L(ω) . (128) For multipole response: QL(r) = r For normal systems, the zero-order propagator D0L(r, r ′, ω) is given by Eqs. (125) and(126) where ω̄nN is replaced by ωnN and F ′(ǫ) is proportional to a δ-function[7]. 7 Collective response Up to now, we have been concerned only with the zero-order approximation, which corresponds to the single-particle approximation of the quantum ap- proach. In this approximation, the perturbing part of the hamiltonian is given only by the external field, while a more consistent approach would require taking into account also the mean-field fluctuation induced by the external force, so that δh = δV ext(r, ω) + δV int(r, ω) . (129) In the Hartree approximation, δV int(r, ω) = dr′v(r− r′)δ̺(r′, ω) . (130) where v(r− r′)is the (long-range) interaction between constituents. For consistency, we take δV int(r, ω) = dr′v(r− r′)δ ¯̺(r′, ω) , (131) then the collective propagator for correlated systems satisfies the same kind of integral equation as for normal systems [7]: D̄L(r, r ′, ω) = (132) D̄0L(r, r ′, ω) + dyy2D̄0L(r, x, ω)vL(x, y)D̄L(y, r ′, ω) . Here vL(x, y) is the partial-wave component of the interaction between par- ticles. We assume that this interaction can be approximated by a separable form of the kind vL(x, y) = κLx LyL , (133) where κL is a parameter that determines the strength of the interaction. Then, the integral equation (132) gives an algebraic equation for the collective cor- related response function R̄L(ω) = drr2rL dr′r′2r′LD̄L(r, r ′, ω) leading to the expression R̄L(ω) = R̄0L(ω) 1− κLR̄0L(ω) . (134) 8 Results Here we compare the multipole strength functions given by our simplified model of pairing correlations with that of the corresponding uncorrelated sys- tem. This comparison is made for the quadrupole and octupole strength func- tions, since these channels are the ones that are most affected. The static nuclear mean field is approximated with a spherical cavity of radius R = 1.2A 3 fm and the A nucleons are treated on the same footing, i. e., we do not distinguish between neutrons and protons. Moreover, we chose A = 208 for ease of comparison with previous calculations of uncorrelated response functions [12,13]. Shell effects are not included in our semiclassical picture and the results shown below should be considered as an indication of the qualitative effects to be expected in heavy nuclei. For the uncorrelated calculations, the Fermi energy is determined by the parametrization chosen for the radius as ǫF ≈ 33.33 MeV, while for the correlated case, the parameter µ is determined by the condition (27); with the value of ∆ = 1 MeV used here, the value of µ is practically coincident with that of ǫF , so we have used µ = ǫF = 33.33 MeV in the calculations below. Moreover, the small parameter ε appearing in Eq. (125) has been given the value ε = 0.1 MeV. This value is chosen to simplify the evaluation of the response function by smoothing out discontinuities in its dependence on ω. In the evaluation of the collective response, the value of parameters κL is the same as in [12,13], that is: κ2 = −1 × 10−3 MeV/fm4 and κ3 = −2 × 10−5 MeV/fm6. 8.1 Quadrupole response Figure 1 show the longitudinal quadrupole strength function evaluated in the zero-order approximation (corresponding to the quantum single-particle ap- proximation). The dashed curve shows the uncorrelated response evaluated according to the theory of [7], while the full curve shows the result of the present correlated calculation. As we can see the effect of pairing correlations on this zero-order strength function is rather small, however, since pairing af- fects also the real part of the zero-order response function,in Fig. 2 we plot 0 5 10 15 20 25 30 ÑΩ @MeVD Fig. 1. Quadrupole stength function in zero-order approximation. The dashed curve gives the response of a normal system of A = 208 nucleons contained in a spherical cavity, while the solid curve includes the effects of pairing correlations in constant-∆ approximation. 0 5 10 15 20 25 30 ÑΩ @MeVD Fig. 2. Collective quadrupole strength function showing the giant quadrupole res- onance. The solid curve involves also pair correlations, the dashed curve has no pairing. also the collective strength function given by Eq. (134). Again, the effect of pairing correlations is very small, in agreement with the results of [1]. 0 1 2 3 4 5 ÑΩ @MeVD 0.0025 0.005 0.0075 0.0125 0.015 0.0175 Fig. 3. Same as Fig.1, at low excitation energy. The main difference between the uncorrelated and correlated responses occurs at small excitation energy, Fig. 3 shows a detail of Fig. 1 at low excitation energy. The correlated strength function displays a gap of about 2 ∆ , the very small strength extending below 2 MeV is entirely due to the finite value of the small parameter ε used in the numerical evaluation of the propagator (125). 8.2 Octupole response Figures 4 and 5 show the zero-order and collective octupole strength functions, both correlated and uncorrelated. As we can see, in this case too the effect is rather small. 0 5 10 15 20 25 30 ÑΩ @MeVD Fig. 4. Zero-order octupole strength function. The solid curve involves pair correla- tions, the dashed curve has no pairing. 0 5 10 15 20 25 30 ÑΩ @MeVD Fig. 5. Collective octupole strength function. The solid curve involves pair correla- tions, the dashed curve has no pairing. 9 Conclusions The solutions of the semiclassical time-dependent Hartree–Fock–Bogoliubov equations have been studied in a simplified model in which the pairing field ∆(r,p, t) is treated as a constant phenomenological parameter. Such an ap- proximation is known to violate some important constraints, like global (particle- number integral) and local (continuity equation) particle-number conserva- tion. In a linearized approach , we have shown that the global particle-number violation is related only to one particular mode of the density fluctuations, while the violation of the continuity equation gives a spurious contribution to all modes of the density response. Both global and local particle-number conservation can be restored by introducing a new density fluctuation that is related to the current density by the continuity equation. This prescription changes the strength associated with the various eigenmodes of the density fluctuations, but not the eigenfrequencies of the system. We have shown in a one-dimensional model that the energy-weighted sum rule calculated accord- ing to this prescription has exactly the same value as for normal, uncorrelated systems, thus we conclude that our prescription eliminates all the spurious strength introduced by the constant-∆ approximation. In a simplified model of nuclei, the effects of pairing correlations on the isoscalar strength functions has been studied in detail for the quadrupole and octupole channels, in the region of giant resonances. In both cases the effects of pairing are rather small. More sizable effects are found at lower excitation energy, in the region of surface modes which have not been included in the present model, but will certainly be more affected by pairing correlations. Appendix In this Appendix we show that the correlated zero-order response function given by the modified density fluctuation (99) satisfies the same energy-weighted sum rule (EWSR) as the uncorrelated response function. We assume that par- ticles move in one-dimensional square-well potential, so that formulae become simpler because the particle velocity does not depend on position: v(ǫ, x) = v(ǫ) = Uncorrelated sum rule The uncorrelated propagator [7] D0(x, x′, ω) = (135) dǫF ′(ǫ) −2nω0 cos[nω0τ(x)] v(ǫ, x) ω − nω0(ǫ) + iε cos[nω0τ(x v(ǫ, x′) gives the uncorrelated strength function S0(ω) = − 1 dxdx′Q(x)D0(x, x′, ω)Q(x′) and the first moment dωωS0(ω) dǫF ′(ǫ) (−2nω0(ǫ) T (ǫ) )(T (ǫ) Q2nnω0(ǫ). (136) Correlated sum rule The modified density fluctuation (99) allows us to evaluate the correlated propagator D̄0 through the relation δ ¯̺(x, ω) = β dx′D̄0(x, x′, ω)Q(x′) , (137) giving D̄0(x, x′, ω) = (138) dǫF ′(ǫ) −2nω0 )cosnω0τ(x) v(ǫ, x) ω − ω̄n + iε cosnω0τ(x v(ǫ, x′) and the correlated first moment M̄1 = 2 dǫF ′(ǫ) (−2nω0(ǫ) T (ǫ) )(T (ǫ) (nω0(ǫ) ω̄n(ǫ) ω̄n(ǫ) . (139) The only difference between this expression and Eq. (136) is in the form of F ′(ǫ), which is proportional to a δ-function in (136), while it is smoother in the correlated case, however, if the parameter µ is determined by the one- dimensional version of Eq. (27), then it can be easly found that, for a square- well mean field, M̄1 = M1 . (140) The detailed argument goes as follows: both for correlated and uncorrelated systems, the number of particles is given by dxdpF (ǫ) = dǫT (ǫ)F (ǫ) , (141) with F (ǫ) = 4 θ(ǫF − ǫ) for uncorrelated fermions and F (ǫ) = 42π~ρ0(ǫ) in the correlated case, while the moments (136, 139) are given by dǫF ′(ǫ)G(ǫ) , (142) G(ǫ) = − (2π)2 T (ǫ) Q2n (143) in both cases, F ′(ǫ) obviously differs in the two cases. Integrating by parts the last expression in (141), gives A = J(ǫ)F (ǫ) F ′(ǫ)J(ǫ) , (144) J(ǫ) = dǫT (ǫ) . (145) For a square-well potential of size L: T (ǫ) = , (146) J(ǫ) = 2mL 2 ǫ . (147) Since J(ǫ)F (ǫ) = lim J(ǫ)F (ǫ) = 0 , (148) both for the correlated and uncorrelated distributions, we have A = − dǫF ′(ǫ)J(ǫ) . (149) The explicit expressions of J(ǫ) and T (ǫ) give A=−2L dǫF ′(ǫ) ǫ , (150) (2π)2√ dǫF ′(ǫ) ǫ (151) for both distributions. From these relations follows that, for a square-well mean field, the relation (140) is exact. If we had used the density fluctuation (95), instead of (99), to evaluate the correlated propagator, we would have obtained a different value of the first moment because the additional term (106) in the phase-space density gives an extra contribution to the density response function and hence to the EWSR. Because of the fundamental character of the EWSR, as well as of the continuity equation, we conclude that this term is a spurious contribution generated by the constant-∆ approximation. References [1] M. Di Toro and V.M. Kolomietz, Zeit. Phys. A-Atomic Nuclei 328 (1987) 285 [2] M. Urban and P. Schuck, Phys. Rev. A 73 (2006) 013621 [3] J.R. Schrieffer, Theory of superconductivity, (W.A. Benjamin, Inc., New York, 1964) [4] R. Combescot, M. Yu. Kagan, S. Stringari, Phys. Rev. A 74, (2006) 042717 [5] R. Bengtsson and P. Schuck, Phys. Lett. 89B (1980) 321 [6] P. Ring and P. Schuck, The Nuclear Many-Body Problem, (Springer, New York, 1980) [7] D.M. Brink, A. Dellafiore, M. Di Toro, Nucl. Phys. A456 (1986) 205 [8] V.L. Polyachenko, I.G. Schuckman, Sov. Astron. 25 (1981) 533 [9] D.M. Brink and G.R. Satchler, Angular Momentum (Oxford University Press, Oxford,U.K., 1968), p. 147 [10] E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2 (Pergamon Press, Oxford, 1980) [11] A. Dellafiore, F. Matera, D.M. Brink, Phys. Rev. A 51 (1995) 914 [12] V.I. Abrosimov, A. Dellafiore, F. Matera, Nucl. Phys. A697 (2002) 748 [13] V.I. Abrosimov, O.I. Davidovskaya, A. Dellafiore, F. Matera, Nucl. Phys. A727 (2003) 748220

0704.0153

Reciprocal Symmetry and Classical Discrete Oscillator Incorporating Half-Integral Energy Levels

Microsoft Word - Class. oscl incorp. half-inegral.31.03.doc Reciprocal Symmetry and Classical Discrete Oscillator Incorporating Hal-Integral Energy Levels Mushfiq Ahmad Department of Physics, Rajshahi University, Rajshahi, Bangladesh E-mail: [email protected] Abstract Classical oscillator differential equation is replaced by the corresponding (finite time) difference equation. The equation is, then, symmetrized so that it remains invariant under the change d -d, where d is the smallest span of time. This symmetric equation has solutions, which come in reciprocally related pairs. One member of a pair agrees with the classical solution and the other is an oscillating solution and does not converge to a limit as d 0. This solution contributes to oscillator energy a term which is a multiple of half- integers. 1. Introduction The differential equation = (1.1) has a unique solution. The corresponding finite difference equation has more solutions1. When the function represents a harmonic oscillator, different solutions will contribute to oscillator energy in different ways. We intend to study these contributions and compare them to the corresponding quantum mechanical values. 2. Oscillator Finite Difference Equation Classical simple harmonic oscillator function f (with angular speed w) satisfies differential equation (1.1) To exploit its symmetry properties we replace the above differential equation by the corresponding symmetric finite difference equation2 ± = giW ),( δ (2.1) where ),(),( ),( −−+ = ±±± twgtwg (2.2) The above difference quotient has the following symmetry under the change δδ −→ ),(),( δδ tD Dg ±± = (2.3) We require that at least one of the solutions, +g , of (2.1) should go over to (1.1) in the limit 0→δ ⎯⎯→⎯= 0)(),( δδ = (2.4) With fg ⎯⎯→⎯ →+ 0δ and wW ⎯⎯→⎯ →0δ (2.5) 3. Reciprocal Symmetry Let ±g be of the form δ/)( tag ±± ±= so that ),()1(),( / twgtwg t −+ −= δ (3.1) Consider equation (2.1) ),(),( −−+ ++ twgtwg ),(. twgiW += (3.2) Using (3.1) we find that −g also satisfies the equation. This establishes reciprocal symmetry of (2.1), that the equation remains invariant under transformation (3.1). 4. Reciprocal symmetric Solutions (2.1) has a pair of solutions ( )iwt ±=± exp)1( sin(1 sin(1 )2/( / (4.1) +g and −g satisfy (2.1) with δ )sin(w W = (4.2) We may write ).exp()exp( exp tiwiwti g ++ =⎟ (4.3) ).exp()exp( exp tiwiwti g −− =−⎟ (4.4) where wywnw +=+= ++ δπ /)2( (4.5) wywnw −=−+= −− δπ /)12( (4.6) 5. Classical and Half-Integral Energy Levels The energy of the oscillator is proportional to 22222 }/)2{(2}/)2{(2)( wwnnwwyyw ++=++= +++ δπδπ (5.1) 22222 }/)2/1{(4}/)12{(2)( wwnnwwyyw ++++=++= −+− δπδπ (5.2) For n=0 (5.1) gives the classical value. The middle term of (5.2) is a product of half- integers and w. To this extent it corresponds to quantum mechanical value. 4. Conclusion We have replaced oscillator differential equation by the corresponding symmetric discrete equation (2.1). This has brought to surface important parts of oscillator function, which were lost in the conventional solution. These parts contain discrete – integral and half integral -- energy levels. 1 Mushfiq Ahmad. Reciprocal Symmetry and Equivalence between Relativistic and Quantum Mechanical Concepts. http://www.arxiv.org/abs/math-ph/0611024 2 Mushfiq AhmadReciprocal Symmetric and Origin of Quantum Statistics. http://www.arxiv.org/abs/physics/0703194

0704.0154

Hadrons in Medium -- Theory confronts experiment

Untitled Hadrons in Medium – Theory confronts experiment Fabian Eichstaedt, Stefan Leupold, Ulrich Mosel∗) and Pascal Muehlich Institut fuer Theoretische Physik, Universitaet Giessen, Giessen, Germany In this talk we briefly summarize our theoretical understanding of in-medium selfenergies of hadrons. With the special case of the ω meson we demonstrate that earlier calculations that predicted a significant lowering of the mass in medium are based on an incorrect treat- ment of the model Lagrangian; more consistent calculations lead to a significant broadening, but hardly any mass shift. We stress that the experimental reconstruction of hadron spectral functions from measured decay products always requires knowledge of the decay branching ratios which may also be strongly mass-dependent. It also requires a quantitatively reliable treatment of final state interactions which has to be part of any reliable theory. §1. Introduction The study of in-medium properties of hadrons has attracted quite some inter- est among experimentalists and theorists alike because of a possible connection with chiral symmetry restoration in hot and/or dense matter. Experiments using ultrarel- ativistic heavy ions reach not only very high densities, but connected with that also very high temperatures. In their dynamical evolution they run through various – physically quite different – states, from an initial high-nonequilibrium stage through a very hot stage of – possibly - a new state of matter (QGP) to an equilibrated ’clas- sical’ hadronic stage at moderate densities and temperatures. Any observed signal necessarily represents a time-integral over all these physically quite distinct states of matter. On the contrary, in experiments with microscopic probes on cold nuclei one tests interactions with nuclear matter in a well-known state, close to cold equilib- rium. Even though the density probed is always smaller than the nuclear saturation density, the expected signals are as large as those from ultrarelativistic heavy-ion collisions.1), 2) In this talk we discuss as an example the theoretical situation concerning the ω meson in medium and use it to point out various essential points both in the theoretical framework as well as in the interpretation of data (for further refs see the reviews in3)–5)). §2. In-medium Properties: Theory The interest in in-medium properties arose suddenly in the early 90’s when sev- eral authors6), 7) predicted a close connection between in-medium masses and chiral symmetry restoration in hot and/or dense matter. This seemed to establish a direct link between nuclear properties on one hand and QCD symmetries on the other. Later on it was realized that the connection between the chiral condensates of QCD and hadronic spectral functions is not as direct as originally envisaged. The only ∗) Speaker, e-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.0154v1 2 U. Mosel strict connection is given by QCD sum rules which restrict only an integral over the hadronic spectral function by the values of the quark and gluon condensates which themselves are known only for the lowest twist configurations. Indeed a simple, but more realistic analysis of QCD sum rules showed that these do not make precise predictions for hadron masses or widths, but can only serve to constrain hadronic spectral functions.8)–11) Thus hadronic models are needed for a more specific pre- diction of hadronic properties in medium. For example, in the past a lively discussion has been going on about a pos- sible mass shift of the ω-meson in a nuclear medium. While there seems to be a general agreement that the ω acquires a certain width of the order of 40-60 MeV in the medium, the mass shift is not so commonly agreed on. While some groups have predicted a dropping mass,12)–14) there have also been suggestions for a rising mass15)–18) or even a structure with several peaks.19), 20) In this context a recent experiment by the CBELSA/TAPS collaboration is of particular interest, since it is the first indication of a downward shift of the mass of the ω-meson in a nuclear medium.21) Since Klingl et al.13) were among the first to predict such a downward shift it is worthwhile to look into their approach again. The central quantity that contains all the information about the properties of an ω meson in medium is the spectral function Amed(q) = − q2 − (m0ω) 2 −Πvac(q)−Πmed(q) , (2.1) with the bare mass m0ω of the ω. The vacuum part of the ω selfenergy Πvac is dominated by the decay ω → π+π0π−.22) For the calculation of the in-medium part one can employ the low-density-theorem13), 19), 20) which states that at sufficiently small density of the nuclear medium one can expand the selfenergy in orders of the density ρ Πmed(ν, ~q = 0; ρ) = −ρT (ν) , (2.2) where T (ν) is the ω-nucleon forward-scattering amplitude. We note that a priori it is not clear up to which densities this low-density-theorem is reliable.23) To obtain the imaginary part of the forward scattering amplitude via Cutkosky’s Cutting Rules Klingl et. al12), 13) used an effective Lagrangian that combined chiral SU(3) dynamics with VMD. The ω selfenergy was evaluated on tree-level which needs as input the inelastic reactions ωN → πN (1π channel) and ωN → 2πN (2π channel) to determine the effective coupling constants. The amplitude ωN → πN is more or less fixed by the measurable and measured back reaction.25) This is in contrast to the reaction ωN ↔ ρN which – in the calculations of ref.12), 13) – is not constrained by any data and which dominates the 2π channel. Furthermore, Klingl et. al12), 13) employed a heavy baryon approximation (HBA) to drop some of the tree-level diagrams generated by their Lagrangian. All the calculations were made for isospin-symmetric nuclear matter at temperature T = 0. The scattered ω was taken to have ~q = 0 relative to the nuclear medium. We have repeated these calculations without, however, invoking the HBA.∗) For ∗) For further details of the present calculations we refer to ref.26) Hadrons in Medium 3 the 2π channel which decides about the in-medium mass shift of the ω in the calcu- lations of ref.12), 13) we find considerable differences – up to one order of magnitude in the imaginary part of the selfenergy – when comparing calculations using the full model with those using the HBA.26) We thus have to conclude already at this point that the HBA is unjustified for the processes considered here and leads to grossly incorrect results. We show our resulting in-medium spectral function of the ω (where HBA was not employed) in figure 1. Note that in the medium the peak is shifted to 544 MeV which is due to the large effects of a relativistic, full treatment of the imaginary and real parts of the amplitudes obtained in the present model. This has to be compared with the results obtained by Klingl et al.12) Since Klingl et al. find an in-medium peak at about 620 MeV it is obvious that in the relativistic calculation the physical picture changes drastically. It is also obvious that the correct treatment of the same 0.001 0.01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω [GeV] vacuum Fig. 1. Spectral function of the ω meson in the vacuum and at normal nuclear density. Lagrangian as used in ref.12) on tree-level leads to an unrealistic lowering of the ω spectral function. It is, therefore, worthwhile to look into another method to calculate the ω self- energy that takes experimental constraints as much as possible into account and – in contrast to the tree-level calculations of ref.12) – respects unitarity. A first study in this direction has been performed by Lutz et al.19) who solved the Bethe-Salpeter equation with local interaction kernels. These authors found a rather complex spec- tral function with a second peak at lower energies due to a coupling to nucleon resonances with masses of about ≈ 1500 MeV. We have recently used a large-scale K-matrix analysis of all available γN and πN data27)–29) that does respect unitarity and thus constrains the essential 2π channel by the inelasticities in the 1π chan- nel.20) By consistently using the low-density-approximation we have obtained the result shown in Fig. 2. Fig. 2 clearly exhibits a broadened ω spectral function with only a small (upwards) shift of the peak mass. In agreement with the calculations of Lutz et al.,19) although with less strength, it also exhibits a second peak at masses around 550 MeV that is due to a coupling to a N*(1535)-nucleon hole configuration. Such a resonance-hole coupling is known to play also a major role in the determi- 4 U. Mosel 0.5 0.6 0.7 0.8 0.9 1.0 [GeV] Fig. 2. ω spectral function for an ω meson at rest, i.e. q0 = q2 (from ref.20)). The appropriately normalized data points correspond to the reaction e+e− → ω → 3π in vacuum. Shown are results for densities ρ = 0, ρ = ρ0 = 0.16 fm −3 (solid) and ρ = 2ρ0 (dashed). nation of the ρ meson spectral function;23), 24) in the context of QCD sum rules it has been examined in ref.17) It is obviously quite sensitive to the detailed coupling strength of this resonance to the ωN channel which energetically opens up only at much higher masses. As mentioned earlier, there is general consensus among different theories, that the on-shell width of the ω meson in medium reaches values of about 50 MeV at saturation density. To illustrate this point we show in Fig. 3 the width as a function of omega momentum relative to the nuclear matter restframe both for the transverse and the longitudinal polarization degree of freedom. It is clearly seen that the 0.0 0.1 0.2 0.3 0.4 0.5 0.6 |q| [GeV] Fig. 3. On-shell width of the ω in nuclear matter at nuclear matter density ρ0 (from ref. 20)). The open (solid) points give the width for the transverse (longitudinal) degree of freedom. transverse width increases strongly as a function of momentum. At values of about Hadrons in Medium 5 500 MeV, i.e. the region, where CBELSA/TAPS measures, the transverse width has already increased to about 125 MeV and even the polarization averaged width amounts to 100 MeV. §3. Spectral Functions and Observables Apart from invariant mass measurements, there is another possibility to exper- imentally constrain the in-medium broadening of the ω-meson. The total width plotted in Fig. 3 is the sum of elastic and inelastic widths. In general, the inelastic width alone is determined by the imaginary part of the selfenergy and the latter determines the amount of reabsorption of ω mesons in the medium. In a Glauber approximation the cross section for ω production on a nucleus reads dσγ+A→ω+X d3x ρ(~x) dσγ+N→ω+X ℑΠ(p, ρ(~x ′)) (3.1) The ratio of this cross section on the nucleus to that on the nucleon then deter- mines the nuclear transmission T which depends on the imaginary part of the omega selfenergy ℑΠ T (A) ≈ d3x ρ(~x) exp ℑΠ(p, ρ(~x ′)) . (3.2) Using in addition the low-density-approximation ℑΠ(p, ~x) = −pρ(~x)σinelωN (3 one obtains the usual Glauber result T (A) = d3x ρ(~x) exp dz′ ρ(~x ′)σinelωN . (3.4) We show the calculated transmission T in Fig. 4 together with the data obtained by CBELSA/TAPS. The measured cross section dependence on massnumber A is reproduced very well30) if the inelastic ωN cross section is increased by 25% over the usually used parametrization. This may indicate a problem with the usually used cross section, or - more interesting - it may indicate a breakdown of the low-density- approximation. It is, furthermore, important to realize that the spectral functions themselves are not observable. What can be observed are the decay products of the meson under study. It is thus obvious that even in vacuum the invariant mass distribution of the decaying resonance (V → X1X2), reconstructed from the four-momenta of the decay products (X1,X2), involves a product of spectral function and partial decay width into the channel being studied dRV→X1X2 ∼ A(q2)× ΓV→X1X2(q Γtot(q2) . (3.5) Since in general the branching ratio also depends on the invariant mass of the decay- ing resonance this dependence may distort the observed invariant mass distribution 6 U. Mosel 0 50 100 150 200 Fig. 4. Transparency of nuclei for ω production. Calculations and preliminary data are normalized to 12C. Dashed lines reflect error estimates obtained from the spread of the data. Data are from CBELSA/TAPS.31) compared with the spectral function itself. This effect is obviously the more impor- tant the broader the decaying resonance is and the stronger the widths depend on While these branching ratios are usually well known in vacuum there is con- siderable uncertainty about their value in the nuclear medium. This uncertainty is connected with the lack of knowledge about the in-medium vertex corrections, i.e. the change of coupling constants with density. Even if we assume that these quan- tities stay the same, then at least the total width appearing in the denominator of the branching ratio has to be changed, consistent with the change of the width in the spectral function. This point has only rarely been discussed so far, but it has far-reaching consequences. For example, for the ρ meson the partial decay width into the dilepton channel goes like Γρ→e+e− ∼ , (3.6) where the first factor on the rhs originates in the photon propagator and the last factor M comes from phase-space. On the other hand, the total decay width of the ρ meson in vacuum is given by (neglecting the pion masses for simplicity) Γtot ≈ Γρ→ππ ∼ M , (3.7) so that the branching ratio in vacuum goes like Γρ→e+e− . (3.8) This strong M -dependence distorts the spectral function, in particular, for a broad resonance such as the ρ meson. This effect is contained and clearly seen in theoretical simulations of the total dilepton yield from nuclear reactions (see, e. g., Figs. 8− 10 Hadrons in Medium 7 in32)); it leads to a considerable shift of strength in the dilepton spectrum towards lower masses. For the semileptonic decay channel π0γ that has been exploited in the CBELSA TAPS experiment again a strong mass-dependence of the branching ratio shows up because just at the resonance the decay channel ω → ρπ opens up. In both of these cases the in-medium broadening changes the total widths in the denominator of the branching ratios even if the partial decay width stays the same as in vacuum. Such an in-medium broadening of the total width, which should be the same as in the spectral function, will tend to weaken the M -dependence of the total width and thus the branching ratio as a whole. In medium another complication arises: the spectral function no longer depends on the invariant mass alone, but – due to a breaking of Lorentz-invariance because of the presence of the nuclear medium – in addition also on the three-momentum of the hadron being probed. Again, this p-dependence of the vector meson selfenergy has only rarely been taken into account (see, however, refs.20), 24), 33)). In addition, final state interactions do affect hadronic decay channels. A quantitatively reliable treatment of these FSI thus has to be integral part of any trustable theory that aims at describing these data. §4. Conclusions QCD sum rules establish a very useful link between the chiral condensates, both in vacuum and in medium, but their connection to hadronic spectral functions is indirect. The latter can thus only be constrained by the QCDSR, but not be fixed; for a detailed determination hadronic models are needed. We have pointed out in this talk that the low-density-approximation nearly always used in these studies does not answer the question up to which densities it is applicable. First studies23) have shown that this may be different from particle to particle. While the in-medium properties of all vector mesons ρ, ω, and φ are the subject of intensive experimental and theoretical research, in this talk we have concentrated on the ω meson for which recent experiments indicate a lowering of the mass by about 60 MeV in photon-produced experiments on nuclei. A tree-level calculation, based on an effective Lagrangian, that predicted such a lowering, has been shown to be incorrect because of the heavy-baryon approximation used in that calculation. A correct tree-level calculation with the same Lagrangian gives strong contributions from the ω → 2πN channel, which, however, is unconstrained by any data; in effect, the spectral function is softened by an unreasonable pole mass shift. This problem might partially be based on the fact that all the inelastic processes ωN → πN and ωN → 2πN are only treated at tree-level. Here an improved calculation is needed, which incorporates coupled-channels and rescattering, e.g. a Bethe-Salpeter19) or a K-matrix approach.27), 34), 35) We have indeed shown that a better calculation that again starts out from an effective Lagrangian and takes unitarity, channel-coupling and rescattering into ac- count yields a significantly different in-medium spectral function in which the pole mass hardly changes, but a broadening of about 60 MeV at nuclear saturation density takes place, which increases with momentum, primarily in the transverse channel. 8 U. Mosel Finally, we have pointed out that any measurement of the spectral function necessarily involves also a branching ratio into the channel being studied. The ex- perimental in-medium signal thus contains changes of both the spectral function and the branching ratio. Acknowledgements The authors acknowledge discussions with Norbert Kaiser and Wolfram Weise. They have also benefitted a lot from discussions with Vitaly Shklyar. This work has been supported by DFG through the SFB/TR16 ”Subnuclear Structure of Matter”. References 1) U. Mosel, in: QCD Phase Transitions, Proc. Int. Workshop Hirschegg 1997, GSI Darm- stadt, p. 201, arXiv:nucl-th/9702046. 2) U. Mosel, in: Hadrons in Dense Matter, Proc. Int. Workshop Hirschegg 2000, GSI Darm- stadt, p. 11, arXiv:nucl-th/0002020. 3) T. Falter, J. Lehr, U. Mosel, P. Muehlich and M. Post, Prog. Part. Nucl. Phys. 53, 25 (2004). 4) L. Alvarez-Ruso, T. Falter, U. Mosel and P. Muehlich, Prog. Part. Nucl. Phys. 55, 71 (2005). 5) R. Rapp and J. Wambach, Adv. Nucl. Phys. 25, 1 (2000). 6) G. E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991). 7) T. Hatsuda and S. H. Lee, Phys. Rev. C 46, 34 (1992). 8) S. Leupold, W. Peters and U. Mosel, Nucl. Phys. A 628 (1998) 311 9) S. Leupold and U. Mosel, Phys. Rev. C 58, 2939 (1998). 10) S. Leupold, Phys. Rev. C 64, 015202 (2001). 11) S. Leupold and M. Post, Nucl. Phys. A 747, 425 (2005). 12) F. Klingl, T. Waas and W. Weise, Nucl. Phys. A 650, 299 (1999). 13) F. Klingl, N. Kaiser and W. Weise, Nucl. Phys. A 624, 527 (1997). 14) T. Renk, R. A. Schneider and W. Weise, Phys. Rev. C 66, 014902 (2002). 15) A. K. Dutt-Mazumder, R. Hofmann and M. Pospelov, Phys. Rev. C 63, 015204 (2001). 16) M. Post and U. Mosel, Nucl. Phys. A 699, 169 (2002). 17) B. Steinmueller and S. Leupold, Nucl. Phys. A 778, 195 (2006). 18) S. Zschocke, O. P. Pavlenko and B. Kampfer, Phys. Lett. B 562, 57 (2003). 19) M. F. M. Lutz, G. Wolf and B. Friman, Nucl. Phys. A 706, 431 (2002) [Erratum-ibid. A 765, 431 (2006)]. 20) P. Muehlich, V. Shklyar, S. Leupold, U. Mosel and M. Post, Nucl. Phys. A 780, 187 (2006). 21) D. Trnka et al. [CBELSA/TAPS Collaboration], Phys. Rev. Lett. 94, 192303 (2005). 22) F. Klingl, N. Kaiser and W. Weise, Z. Phys. A 356, 193 (1996). 23) M. Post, S. Leupold and U. Mosel, Nucl. Phys. A 741, 81 (2004). 24) W. Peters, M. Post, H. Lenske, S. Leupold and U. Mosel, Nucl. Phys. A 632, 109 (1998). 25) B. Friman, arXiv:nucl-th/9801053. 26) F. Eichstaedt, Diploma Thesis, Institut fuer Theoretische Physik, JLU Giessen, 2006, http://theorie.physik.uni-giessen.de/documents/diplom/eichstaedt.pdf . 27) G. Penner and U. Mosel, Phys. Rev. C 66, 055211 (2002). 28) G. Penner and U. Mosel, Phys. Rev. C 66, 055212 (2002). 29) V. Shklyar, H. Lenske, U. Mosel and G. Penner Phys.Rev. C72, 015210 (2005). 30) P. Muehlich and U. Mosel, Nucl.Phys.A773,156 (2006). 31) M. Kotulla, nucl-ex/0609012. 32) M. Effenberger, E. L. Bratkovskaya and U. Mosel, Phys. Rev. C 60, 044614 (1999). 33) M. Post, S. Leupold and U. Mosel, Nucl. Phys. A 689, 753 (2001). 34) T. Feuster and U. Mosel, Phys. Rev. C 59, 460 (1999). 35) V. Shklyar, H. Lenske, U. Mosel and G. Penner, Phys. Rev. C 71, 055206 (2005) [Erratum- ibid. C 72, 019903 (2005)]. 0 50 100 150 200

0704.0155

A computer program for fast non-LTE analysis of interstellar line spectra

Astronomy & Astrophysics manuscript no. aa6820 c© ESO 2013 February 12, 2013 A computer program for fast non-LTE analysis of interstellar line spectra With diagnostic plots to interpret observed line intensity ratios F. F. S. van der Tak1,2, J. H. Black3, F. L. Schöier4, D. J. Jansen5, and E. F. van Dishoeck5 1 Netherlands Institute for Space Research (SRON), Landleven 12, 9747 AD Groningen, The Netherlands e-mail: [email protected] 2 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany 3 Onsala Space Observatory, Chalmers University of Technology, 43992 Onsala, Sweden 4 Stockholm Observatory, AlbaNova University Center, 10691 Stockholm, Sweden 5 Leiden University Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands Received 27 November 2006; accepted 27 March 2007 ABSTRACT Aims. The large quantity and high quality of modern radio and infrared line observations require efficient modeling techniques to infer physical and chemical parameters such as temperature, density, and molecular abundances. Methods. We present a computer program to calculate the intensities of atomic and molecular lines produced in a uniform medium, based on statistical equilibrium calculations involving collisional and radiative processes and including radiation from background sources. Optical depth effects are treated with an escape probability method. The program is available on the World Wide Web at http://www.sron.rug.nl/∼vdtak/radex/index.shtml . The program makes use of molecular data files maintained in the Leiden Atomic and Molecular Database (LAMDA), which will continue to be improved and expanded. Results. The performance of the program is compared with more approximate and with more sophisticated methods. An Appendix provides diagnostic plots to estimate physical parameters from line intensity ratios of commonly observed molecules. Conclusions. This program should form an important tool in analyzing observations from current and future radio and infrared telescopes. Key words. Radiative transfer – Methods: numerical – Radio lines – Infrared lines – Submillimeter 1. Introduction Observations of spectral lines at radio, (sub)millimeter and infrared wavelengths are a powerful tool to in- vestigate the physical and chemical conditions in the dilute gas of astronomical sources where thermody- namic equilibrium is a poor approximation (e.g., Genzel 1991; Black 2000). To extract astrophysical parame- ters from the data, the excitation and optical depth of the lines need to be estimated, for which various meth- ods may be used, depending on the available observa- tions (Van Dishoeck & Hogerheijde 1999; Van der Tak 2005). If only one or two lines of a molecule1 have been ob- served, the excitation must be deduced from observa- tions of other species or from theoretical considerations. An example is the assumption that the excitation tem- perature equals the kinetic temperature, a case known as Local Thermodynamic Equilibrium (LTE) which holds at high densities. If many lines have been observed, a popular method is the ‘rotation diagram’, also called ‘Boltzmann plot’ or ‘population diagram’ (e.g., Blake et al. 1987; Helmich et al. 1994; Goldsmith & Langer 1999). This method describes the excitation by a single tempera- ture, obtained by a fit to the line intensities as a func- tion of upper level energy. Provided that beam sizes are similar and optical depths are low, or that appropriate corrections are made, this method yields estimates of the excitation temperature and column density of the molecule. The excitation temperature approaches the ki- netic temperature in the high-density limit, but generally depends on both kinetic temperature and volume den- sity. Spectral line surveys are often analyzed with ro- tation diagrams, although more advanced methods are also used (Helmich & van Dishoeck 1997; Comito et al. 2005). More sophisticated methods retain the assumption of a local excitation, but solve for the balance of ex- citation and de-excitation rates from and to a given http://arxiv.org/abs/0704.0155v1 http://www.sron.rug.nl/~vdtak/radex/index.shtml 2 Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra probability method and the Large Velocity Gradient (LVG) method (Sobolev 1960; De Jong et al. 1975; Goldreich & Scoville 1976). These ‘intermediate-level’ methods require knowledge of molecular collisional data, whereas the previous ‘basic-level’ methods only required spectroscopic and dipole moment information. This extra requirement limits the use of these methods to some extent, because collisional data do not exist for all astrophysically relevant species. The advantage is that column density, kinetic temperature and volume density can be constrained, if accurate collision rates are known. As with rotation diagrams, this method can be used to compute synthetic spectra to be compared with data with a χ2 statistic (Jansen 1995; Leurini et al. 2004). The most advanced methods drop the local approxi- mation and solve for the intensities (or the radiative rates) as functions of depth into the cloud, as well as of velocity. Such methods are usually of the Accelerated Lambda Iteration (ALI) or Monte Carlo (MC) type, al- though hybrids also exist. The performance and con- vergence of such programs have recently been tested by Van Zadelhoff et al. (2002). Using such programs one can constrain temperature, density, and velocity gradients within sources (e.g., Van der Tak et al. 1999; Tafalla et al. 2002; Jakob et al. 2007), and, if enough observations are available, even molecular abundance profiles (e.g., Van der Tak et al. 2000a; Schöier et al. 2002; Maret et al. 2005), especially when coupled to chemical networks (e.g., Doty et al. 2004; Evans et al. 2005; Goicoechea et al. 2006). This paper presents the public version of a radiative transfer code at the ‘intermediate’ level. The assump- tion of a homogeneous medium limits the number of free parameters and makes the program a useful tool in rapidly analyzing a large set of observational data, in order to provide constraints on physical conditions, such as density and kinetic temperature (Jansen 1995). The program can be used for any molecule for which collisional rate coefficients are available. The input for- mat for spectroscopic and collisional data is that of the LAMDA database (Schöier et al. 2005)2 where an on- line calculator for molecular line intensities3, based on our program, can also be found4. The paper is set up as follows. Section 2 describes the radiative transfer formalism and introduces our notation of the key quantities. Section 3 describes the formalism which the program actually uses, and discusses its im- plementation. Section 4 compares the results of the pro- gram to those of other programs. The paper concludes in § 5 with suggested future directions of astrophysical radiative transfer modeling. 2 http://www.strw.leidenuniv.nl/∼moldata 3 In this paper, the ‘strength’ of a line is an intrinsic quantity 2. Radiative transfer and molecular excitation This section summarizes the formalism to analyze molecular line observations which our program adopts. For more detailed discussions of radiative transfer see, e.g., Cannon (1985) or Rybicki & Lightman (1979). 2.1. Basic formalism Describing the transfer of radiation requires a quantity which is conserved along its path as long as no local absorption or emission processes take place, and which includes the direction of travel. The quantity that sat- isfies this requirement is the specific intensity Iν, de- fined as the amount of energy passing through a surface normal to the path, per unit time, surface, bandwidth (measured here in frequency units), and solid angle. The transfer equation for radiation propagating a distance ds can then be written as = jν − αν Iν, (1) where jν and αν are the local emission and extinction coefficients, respectively. The two terms on the right- hand side may be combined into the source function, defined by S ν ≡ . (2) Writing the transport equation in its integral form and defining the optical depth, dτν ≡ αν ds, measured along the ray5 one arrives at Iν = Iν (0)e −τν + S ν(τ −(τν−τ′ν) dτ′ν, (3) where Iν is the radiation emerging from the medium and Iν(0) is the ‘background’ radiation entering the medium. The above equations hold both for continuum radiation, which is emitted over a large bandwidth, and for spectral lines, which arise when the local emission and absorp- tion properties change drastically over a very small fre- quency interval, due to the presence of molecules. From this point the discussion will focus on bound-bound transitions within a multi-level molecule consisting of N levels with spontaneous downward rates Aul, Einstein coefficients for stimulated transitions Bul and Blu, and collisional rates Cul and Clu, between upper levels u and lower levels ℓ. The rate of collision is equal to Cul = ncolγul, (4) where ncol is the number density of the collision partner (in cm−3) and γul is the downward collisional rate coeffi- cient (in cm3 s−1). The rate coefficient is the Maxwellian average of the collision cross section, σ, γul = 8kTkin )−1/2 ( kTkin σEe−E/kTkin dE, (5) http://www.strw.leidenuniv.nl/~moldata http://www.sron.rug.nl/~vdtak/radex.php Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra 3 where E is the collision energy, k is the Boltzmann con- stant, Tkin is the kinetic temperature, and µ is the re- duced mass of the system. The upward rates are ob- tained through detailed balance γlu = γul e−hν/kTkin , (6) where gi is the statistical weight of level i. The local emission in transition u→l with laboratory frequency νul, can be expressed as nu Aul φν, (7) where nu is the number density of molecules in level u and φν is the frequency-dependent line emission profile. The absorption coefficient reads (nl Blu ϕν − nu Bul χν) , (8) where φν and χν are the line profiles for absorption and stimulated emission (counted as negative extinction), respectively. From here on we assume complete angular and fre- quency redistribution of the emitted photons, so that φν=ϕν=χν, which is strictly only valid when collisional excitation dominates. This assumption allows the source function to be written as S νul = nuAul nlBlu − nuBul , (9) where we have used the Einstein relations. It is com- mon to introduce an excitation temperature Tex defined through the Boltzmann equation exp [−(Eu − El)/kTex] , (10) where Ei is the energy of level i, such that S νul = Bν(Tex), the specific intensity of a blackbody radiating at Tex. In the interstellar medium, the dominant line broadening mechanism is Doppler broadening. Except in very cold and dark cloud cores, observed line widths are much larger than expected from the kinetic temperature: this effect is commonly ascribed to random macroscopic gas motions or ‘turbulence’. The result is a Gaussian line profile ν − νul − v · n , (11) where νD is the Doppler width, v is the velocity vector of the moving gas at the position of the scattering, n is a unit vector in the direction of the propagating beam of radiation, and c is the speed of light. The Doppler width is the 1/e half-width of the profile, equal to ∆V/2 where ∆V is its full width at half-maximum. If the level populations ni are known, the radiative trans- fer equation can be solved exactly. In particular, under circumstellar media, the density is too low to attain LTE, but statistical equilibrium (SE) can often be assumed: = 0 = n jP ji − ni Pi j = Fi − niDi , (12) where Pi j, the destruction rate coefficient of level i, and its formation rate coefficient P ji are given by Pi j = Ai j + Bi j J̄ν + Ci j (i > j) Bi j J̄ν +Ci j (i < j). In Eq. 13, Bi j J̄ν = Bi j Jν φ(ν) dν (14) is the number of induced radiative (de-)excitations from state i to state j per second per particle in state i, and Iν dΩ (15) is the specific intensity Iν integrated over solid angle dΩ and averaged over all directions. The SE equations thus include the effects of non-local radiation. This discussion assumes that the state-specific rates of formationFi [cm3 s−1] and destructionDi [s−1] are zero to ensure that the radiative transfer is solved indepen- dently of assumptions about chemical processes. In gen- eral, formation and destruction processes should be in- cluded explicitly to be able to deal with situations in which the chemical time scales are very short or the radiative lifetimes very long. For example, the forma- tion temperature (in Fi) affects the rotational excitation of C3 (Roueff et al. 2002) and the vibrational excitation of H2 (Black & van Dishoeck 1987; Burton et al. 1990; Takahashi & Uehara 2001), systems for which line ra- diation only occurs as slow electric quadrupole transi- tions. The rotational excitation of reactive ions like CO+ (Fuente et al. 2000; Black 1998) is also sensitive to Fi and Di because the rates of reactions with H and H2 rival the inelastic collision excitation rates. Similar con- siderations apply to the excitation of H+3 in the Sgr A region close to the Galactic Center (Van der Tak 2006), where electron impact excitation competes with disso- ciative recombination. 2.2. Molecular line cooling Once the radiative transfer problem has been solved and the level populations are known, the cooling (or heating) from molecular line emission can be estimated. Since the level populations contain all the information of the radiative transfer, a general expression for the cooling is obtained from considering all possible collisional tran- sitions (nlγlu − nuγul)hνul, (16) 4 Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra are in detailed balance at the kinetic temperature; there- fore it is possible for net heating to occur (Λ < 0) in cases where the crucial level populations have Tex>Tkin, owing to strong radiative excitation in a hot external ra- diation field. 2.3. Escape probability The difficulty in solving radiative transfer problems is the interdependence of the molecular level populations and the local radiation field, requiring iterative solution methods. In particular, for inhomogeneous or geometri- cally complex objects, extensive calculations with many grid points are required. However, if only the global properties of an interstellar cloud are of interest, the cal- culation can be greatly simplified through the introduc- tion of a geometrically averaged escape probability β, the probability that a photon will escape the medium from where it was created. This probability depends only on the optical depth τ and is related to the inten- sity within the medium, ignoring background radiation and any local continuum, through Jνul = S νul (1 − β). (17) Several authors have developed detailed relations be- tween β and τ for specific geometrical assumptions. Our program offers the user a choice of three such ex- pressions. The first is the expression derived for an ex- panding spherical shell, the so-called Sobolev or large velocity gradient (LVG) approximation (Sobolev 1960; Castor 1970; Elitzur 1992, p. 42-44). This method is also widely applied for moderate velocity gradients, to mimic turbulent motions. Our program uses the formula by Mihalas (1978) and De Jong et al. (1980) for this ge- ometry: βLVG = dτ′ = 1 − e−τ . (18) Second, in the case of a static, spherically symmet- ric and homogeneous medium the escape probability is (Osterbrock & Ferland 2006, Appendix 2)6 βsphere = . (19) Third, for a plane-parallel ‘slab’ geometry, applicable for instance to shocks, βslab = 1 − e−3τ is derived (De Jong et al. 1975). Figure 1 plots the be- haviour of β as a function of τ for these three cases; for more detailed comparisons see Stutzki & Winnewisser (1985) and Ossenkopf et al. (2001). Users of our pro- gram can select either expression for their calculations. The on-line version of the program uses the formula for the uniform sphere, Eq. (19). 0 1 2 3 4 5 6 7 8 9 10 Optical depth τ Sphere Fig. 1. Escape probability β as a function of optical depth τ for three different geometries: uniform sphere (solid line), expanding sphere (dotted line) and plane- parallel slab (dashed line). 3. The program RADEX is a non-LTE radiative transfer code, written originally by J. H. Black, that uses the escape proba- bility formulation assuming an isothermal and homo- geneous medium without large-scale velocity fields. With the current increase of observational possibilities in mind, we have developed a version of this program which is suitable for public use. A guide for using the code in practice is provided in Appendix A and on-line7; Appendix B describes the adopted coding style. This section focuses on the implementation of the formalism of § 2 in the program. 3.1. Basic capabilities For a homogeneous medium with no global velocity field, the optical depth at line centre can be expressed using Eqs. (2, 7, 9, 11), as AulNmol 1.064∆V , (21) where Nmol is the total column density,∆V the full width at half-maximum of the line profile in velocity units, and xi the fractional population of level i. The formal- ism is analogous to the LVG method, with the global n/(dV/dR) replaced by the local N/∆V , as in microtur- bulent codes (Leung & Liszt 1976). The program itera- tively solves the statistical equilibrium equations start- ing from optically thin statistical equilibrium (§ 3.4) for the initial level populations. The program can handle up to seven collision partners simultaneously. In dense molecular clouds, H2 is the main collision partner for most species, but in some cases, separate cross sections may exist for collisions with the ortho and para forms of H2, and electron col- lisions may be important for ionic species. In diffuse molecular clouds and PDRs, excitation by atomic H be- comes important, particularly for fine structure lines, while for comets, H2O is the main collision partner. We Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra 5 refer to Flower (1989) for the basic theory of molecular collisions, and to Dubernet (2005) for an update of the latest results. The output of the program is the background-subtracted line intensity in units of the equivalent radiation temper- ature in the Rayleigh-Jeans limit. The background sub- traction follows traditional cm- and mm-wave spectro- scopic observations where the differences between on- source and off-source measurements are recorded, such Iemν − I . (22) The radiation peak temperature TR can be directly com- pared to the observed antenna temperature corrected for the optical efficiency of the telescope. However, it should be emphasized that RADEX contains no informa- tion about the geometry or length scale and that it is assumed that the source fills the antenna beam. If the source is expected to be smaller than the observational beam, computed line fluxes must be corrected before comparing to observed fluxes. In other types of observations, the continuum may not be subtracted from the data. In (sub-)millimeter and THz observations, for example with ESA’s fu- ture Herschel space observatory, the dust continuum of many sources will be much stronger than any in- strumental error, and baseline subtraction may not be needed. The same is true for interferometer data, where the instrumental passband is well characterized. 3.2. Background radiation field The average Galactic background (interstellar radia- tion field, ISRF) adopted in RADEX consists of several components. The main contribution is the cosmic mi- crowave background (CMB) whose absolute tempera- ture is taken to be TCBR = 2.725±0.001K based on the full COBE data set as analyzed by Fixsen & Mather (2002). This model of the microwave background rep- resents the broadband continuum only and does not in- clude the strong emission lines, several of which contain significant power in the far-infrared and (sub-)millime- ter part of the spectrum (see, e.g., Fixsen et al. 1999). The ultraviolet/visible/near-infrared part of the spec- trum is based on the model of average Galactic starlight in the solar neighborhood of Mathis et al. (1983). The far-infrared and (sub-)millimeter part of the spectrum is based on the single-temperature fit to the Galactic ther- mal dust emission of Wright et al. (1991). At frequen- cies below 10 cm−1 (30 GHz), there is a background contribution from non-thermal radiation in the Galaxy. A tabulation of this spectrum in ASCII format is avail- able on-line8, and a graphical representation is shown in Black (1994). One subtle aspect of the calculation is the distinction between the background seen by the observer and the background seen by the molecules. The continuum con- tribution to the rate equations may be composed of (1) an external component which arises outside the emitting region and (2) an internal continuum that arises within the emitting region. The CMB and ISRF are examples of external continuum components; dust emission from the line-emitting region is an example of an internal continuum. While an external continuum always fills the entire sky, an internal continuum may only fill a frac- tion of it, for example in the case of a circumstellar disk. With this distinction in mind, the internal intensity be- comes Jintν = β[Bν(TCBR)+ηI ν ]+(1−β)[Bν(Tex)+θ(1−η)I where Iuserν is the continuous spectrum defined by the user. The factor η, is the fraction of local continuum which arises outside the line emitting region, and the factor θ is the fraction of local sky filled by the internal continuum. 3.3. Chemical formation and destruction rates The equations of statistical equilibrium (12) include source and sink terms. By default, RADEX sets the destruction rates equal to the same small value, Di ≡ D = 10−15 s−1, appropriate for cosmic-ray ionization plus cosmic-ray induced photodissociation (Prasad & Tarafdar 1983; Gredel et al. 1989). The cor- responding formation rates are Fi = 10−24ntotalgi exp(−Ei/kTform)/Q(Tform) (24) where ntotal is the sum of the densities of all collision partners, Tform is a formation temperature (default value 300 K), and Q(T ) = gi exp(−Ei/kTform) (25) is the partition function. These assumptions imply a nominal fractional abundance of every molecule ntotal ntotalD = 10−9 . (26) The value of the nominal abundance is inconsequential because the results in RADEX depend on Nmol/∆V , but not on the fractional abundance. For most molecules currently in the associated database (LAMDA) and for the most commonly encountered interstellar conditions, these choices will not affect the observable excitation. The formation and destruction rates are computed in a subroutine that can be modified by the user to provide a more realistic description of chemical processes. For example, users may treat the combined ortho/para forms of molecules by introducing a realistic Tform, especially in cases where no o/p interchange processes is likely to be effective. Other cases of potential interest include the photodissociation of large molecules into smaller molecules, or the evaporation of icy grain mantles into the gas phase. Our formulation in terms of a volume rate of formation is chosen to be independent of the de- tails of the formation process. In general, formation and destruction processes are important for molecules that http://www.oso.chalmers.se/~jblack/RESEARCH/ 6 Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra 3.4. Calculation The input parameters of RADEX and its output are de- scribed in Appendix A. Calculations with RADEX pro- ceed as follows. A first guess of the populations of the molecular energy levels is produced by solving statisti- cal equilibrium in the optically thin case. The only ra- diation taken into account is the unshielded background radiation field; internally produced radiation is not yet available. The solution for the level populations allows calculation of the optical depths of all the lines, which are then used to re-calculate the molecular excitation. The new calculation treats the background radiation in the same manner as the internally produced radiation. The program iteratively finds a consistent solution for the level populations and the radiation field. When the optical depths of the lines with τ > 10−2 are stable from one iteration to the next to a given tolerance (default 10−6), the program writes output and stops. 3.5. Results There are several ways in which RADEX can be used to analyze molecular line observations. In most of these applications, the modeled quantity is the velocity- integrated line intensity, as the excitation is assumed to be independent of velocity. As a consequence, self- absorbed lines cannot be modeled satisfactorily with RADEX. In the simplest case, the temperature and den- sity are known from other observations and only the col- umn density of the molecule under consideration needs to be varied to get the best agreement with the observed line intensity. If the H2 column density is known from other observations, for example from an optically thin CO isotopic line, the ratio of the two column densi- ties gives the molecular abundance, averaged over the source. The RADEX distribution contains a Python script to automate this procedure which is further described in Appendix D. Another often-used application of RADEX is to deter- mine temperatures and densities from the observed in- tensity ratios of lines of the same molecule. If the abun- dance of the molecule is constant throughout the source, the ratios should give source-averaged physical con- ditions independent of the specific chemistry of the molecule. Appendix C presents illustrative plots of line ratios for commonly observed molecules and lines in the optically thin case. For higher optical depths, the qualitative trends remain the same but there are quanti- tative differences. RADEX can readily be used to generate similar plots for moderately thick cases. Again, Python scripts are made available to automate this procedure (Appendix D). To illustrate the use of RADEX on actual observations, we take the observations of the HCO+ 1–0 and 3–2 lines to- ward a relatively simple source, the photon-dominated region IC 63 (Jansen et al. 1994). The observed 1–0/3–2 ratio corrected for beam dilution is 5.5±1.5. The kinetic temperature of the source is constrained from CO ob- servations to be ∼50 K. Fig. C.3 shows that the inferred Fig. 2. Comparison of the predicted line strengths for the 10 lowest rotational transitions of CO for a homo- geneous isothermal sphere, with nH2 = 10 5 cm−3 and Tkin = 50 K, using different methods. Upper panel: The total optical depth through the sphere at line centre τ and excitation temperature Tex as a function of the up- per rotational level J involved in the transition. Middle panel: The radiation temperature TR obtained for each transition using RADEX with different prescriptions of the escape probability β and compared with the result from the Monte-Carlo code (MCC) of Schöier (2000). Also shown are the results for optically thin emission in LTE. Lower panel: TR obtained from RADEX compared with the results from MCC, δTR = T R − TR. inferred column density from the absolute intensities is 8 × 1012 cm−2, which, together with the overall H2 col- umn density of 5×1021 cm−2, gives an HCO+ abundance with respect to H2 of 1.6 × 10−9. A slightly more complicated situation arises for the Orion Bar PDR (Hogerheijde et al. 1995). For this source, both HCO+ 1–0, 3–2 and 4–3 lines have been observed. The 1–0/3–2 ratio gives an order of magni- tude lower density than the 3–2/4–3 ratio. This differ- Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra 7 density tracers such as CO are usually much less sensi- tive to density. One possible solution is a clumpy PDR model in which the 1–0 line is mostly produced in the low-density interclump gas containing 90% of the mate- rial and the 4–3 line in the high-density clumps. Within this clumpy model, a single column density fits all three lines and an accurate abundance can be derived. Note that this technique of adding the results of two models is only applicable at low optical depths. If only two lines of a molecule have been observed, the line ratio can be used as indicator of tempera- ture or density, depending on molecule and transition (Appendix C). A single line ratio is never enough to constrain both temperature and density, though. For multi-line observations, a comparison of data and models in terms of χ2 is preferred. See for example Van der Tak et al. (2000b), Schöier et al. (2002), and Leurini et al. (2004) for details of such calculations. 3.6. Limitations of the program The current version of the program does not in- clude a contribution from continuous (dust or free-free) opacity to the escape probability, as for example in Takahashi et al. (1983). Continuum radiation from dust is generally negligible at long wavelengths ( >∼ 1 mm) but becomes important for regions with very high col- umn densities (such as protoplanetary disks) and at far-infrared and shorter wavelengths ( <∼ 100µm). Free- free radiation may become important for the calcula- tion of atomic fine structure lines from H II regions; other programs such as CLOUDY (Ferland 2003)9 may be more suitable for this purpose. The absence of con- tinuous opacity limits the applicability of the program particularly in situations where infrared pumping is im- portant, either directly through rotational transitions or via vibrational transitions (Carroll & Goldsmith 1981; Hauschildt et al. 1993). Another limitation of the program is that only one molecule is treated at a time, so that the effects of line overlap are not taken into account. Such overlaps may occur both at radio and at infrared wavelengths (e.g., Expósito et al. 2006). In special cases, overlap between lines of the same molecule may influence their excita- tion, for example the hyperfine components of HCN or + (Daniel et al. 2006). For certain molecules under certain physical conditions (especially low density and/or strong radiation field), population inversions occur, which cause negative op- tical depth and hence nonlinear amplification of the incoming radiation (Elitzur 1992). This phenomenon, known as ‘maser’ action, requires non-local treatment of the radiative transfer, in particular a fine sampling of directions, for which RADEX is not set up. Generally, the escape probability approximation is justified until the masers saturate, which occurs at τ ≈ −1. In practice, the computed intensities of lines with τ <∼ − 0.1 are not as accurate as those of other lines, and the intensities of lines with τ <∼ − 1 should be disregarded altogether. If Fig. 3. Excitation temperature of the HCO+ 1–0 tran- sition as a function of radius for the model cloud of § 4.1.2, calculated with RADEX assuming static spher- ical, expanding spherical, and slab geometry (dashed / dash-dotted / dotted lines), with a multi-zone es- cape probability program (long/short dashes) and with a Monte Carlo code (solid lines). The panels are for dif- ferent column densities, hence optical depths. Note the different vertical scales. of non-maser lines may also be affected. While special- ized programs should be used to calculate the intensities of maser lines (e.g., Spaans & van Langevelde 1992; Gray & Field 1995; Yates et al. 1997), RADEX may well be used to predict which lines of a molecule may display http://www.nublado.org/ 8 Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra which are pumped by infrared radiation (Leurini et al. 2004). 4. Comparison with other methods This section shows a comparison of RADEX with other programs, first for the case of constant physical con- ditions (§ 4.1) and second for variable conditions (§ 4.2). Comparison is with the analytical rotation di- agram method and with Monte Carlo methods, which have been benchmarked to high accuracy, both for the case of HCO+ (Van Zadelhoff et al. 2002)10 and of H2O (Van der Tak et al. 2005) 11. Throughout this sec- tion, molecular data have been taken from the LAMDA database (Schöier et al. 2005). 4.1. Homogeneous models 4.1.1. The case of CO To test the RADEX code, we have compared its output both to an optically thin LTE analysis (rotation diagram method) and a full radiative transfer analysis using a Monte-Carlo method (Schöier 2000). The test problem consists of a spherically symmetric cloud with a con- stant density, n(H2), of 1 × 105 cm−3 within a radius of 100 AU. In this example only the CO emission is treated using a fractional abundance of 1 × 10−4 relative to H2 yielding a central CO column density of NCO = 3 × 1016 cm−2 and an average value of N=2×1016 cm−2. The kinetic temperature is set to 50 K, the background tem- perature to 2.73 K, and the line width to ∆V=1.0 km s−1. Fig. 2 presents the results of the calculations for the ten lowest rotational transitions. The excitation temper- atures of the lines vary from being close to thermalized for transitions involving low J-levels, to sub-thermally excited for the higher-lying lines. The optical depth in the lines is moderate (∼ 1 − 2) to low. It is seen that the expressions of the escape probability for the uniform sphere and the expanding sphere give almost identical solutions which are close to that obtained from the full radiative transfer (MCC in Fig. 2). The slab geometry gives slightly higher intensities, in particular for high- lying lines. The optically thin approximation, where the gas is assumed to be in LTE at 50 K, produces much larger discrepancies, up to a factor of∼ 2, and only gives the correct intensity for the J = 1 → 0 line, where the LTE conditions are met. 4.1.2. The case of HCO+ To further verify the performance of the RADEX program, we have compared its results to that of another program that does not use the local approximation: the Monte Carlo program RATRAN (Hogerheijde & van der Tak 2000). We also compare the results to those from the multi-zone escape proba- bility program by Poelman & Spaans (2005). The test case is a cloud with n(H2) = 1×104 cm−3, Tkin=10 K, Tbg=2.73 K, and a line width of ∆V=1.0 km s −1, equiv- alent to bD=0.6 km s −1. The pure rotational emission spectrum of HCO+ was calculated for column densi- ties of 1012, 1013, 1014 and 1015 cm−2, which for RADEX were given directly as input parameters. For the multi- zone programs, a cloud radius of 1018 cm was specified along with abundances of 10−10–10−7, distributed over 50 cells. Figure 3 shows the calculated excitation temperature of the HCO+ 1–0 transition as a function of radius for these physical conditions. For N(HCO+) <∼ 10 12 cm−2, the excitation is independent of radius and the calcu- lations for the various geometries agree to ≈10%. The dependence of the excitation on radius and on geom- etry increases with increasing column density, and for N(HCO+) >∼ 10 15 cm−2, the curvature of the Tex distri- bution becomes too large to ignore. The corresponding line optical depth is ≈100, with ≈20% spread between the various estimates (Fig. 4). The curvature arises be- cause at the cloud center, photon trapping thermalizes the excitation, while at the edge, the emission can es- cape the cloud (Bernes 1979). We do not recommend to use RADEX at line optical depths >∼ 100, because the cal- culated excitation temperature may not be representa- tive of the emitting region. However, even if some lines are highly optically thick, RADEX may well be used to analyze other lines which are optically thin. For exam- ple for H2O, the ground state lines often have τ ∼ 1000, but RADEX is well capable of computing intensities for higher-lying transitions which are not as optically thick. At low optical depth, variations in Tex translate directly into changes in emergent line intensity. Thus, differ- ences as large as 20% in calculated line flux can arise depending on the choice of escape probability descrip- tion, even for moderately thick cases. At high optical depth, the direct connection between Tex and line flux is lost because of the dependence on the adopted veloc- ity field. The assumption in the program that the opti- cal depth is independent of velocity breaks down in this case. In this limit, the peak line temperature TR gives the value of Tex at the τ = 1 surface of the cloud in this specific transition. The results shown in this section do not translate easily to other HCO+ lines such as J=3→2, because the ex- citation is governed by several competing effects. The optical depth of the J=3→2 line may be higher or lower than that of the J=1→0 line, depending on temperature and density. Observers are encouraged to use RADEX to study the excitation of their lines as a function of these parameters, and also consider geometric variations. 4.2. Observations of a Young Stellar Object To compare a typical RADEX analysis with other meth- ods for a situation which varying physical conditions, we choose a molecule for which many lines can be observed: para-formaldehyde, p-H2CO. Figure 5 shows observations of p-H2CO (sub-)millimeter emission lines http://www.strw.leidenuniv.nl/astrochem/ http://www.sron.rug.nl/~vdtak/H2O/ Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra 9 Fig. 4. Optical depth of the HCO+ 1–0 transition for the model of § 4.1.2, calculated with RADEX assuming static spherical, expanding spherical, and slab geometry (filled triangles/filled squares/open triangles), with the multi-zone escape probability code (open circles) and with the Monte Carlo code (filled circles). Fig. 5. Line strengths of p-H2CO observed towards the embedded low-mass protostar IRAS 16293–2422 (squares with error bars), modeled assuming LTE (solid line), using RADEX (triangles), and using a Monte Carlo program assuming a constant abundance (solid circles) and an abundance varying with radius (open circles). Fig. 6. Distributions of the χ2 parameter correspond- ing to the models in Figure 5. The RADEX results are for n(H2) = 10 6 cm−3 as found by Van Dishoeck et al. (1995). Van Dishoeck et al. (1995). The data are analyzed us- ing three methods: assuming LTE (with a rotation dia- gram), assuming SE (using RADEX), and using a Monte Carlo program. The free parameters for the LTE fit are the excitation temperature Tex and the column density Figure 6 shows the distributions of the χ2 parameter for the LTE and SE fits, calculated in the standard way (see, e.g., Van der Tak et al. 2000b; Schöier et al. 2002) as- suming a 20% uncertainty for all observed points except the line with the highest upper level energy where a 30% uncertainty was used. As seen from the figure, the non- LTE method gives a better fit to the data, as quantified by the lower minimum χ2 value. This result is not nec- essarily surprising given that more free parameters are available. A more important difference is that the esti- mates of temperature and column density between the two methods are substantially different, in particular the temperature (50 vs 150 K). Since the non-LTE method involves fewer assumptions about the physical state of the cloud, its results are to be preferred. These results illustrate that rotation diagrams may give misleading results when determining physical proper- ties of interstellar gas clouds (cf. Johnstone et al. 2003 for the case of CH3OH). Figure 5 also demonstrates that temperatures and column densities derived from rota- tion diagrams tend to depend on which lines happen to have been observed (cf. the HCO+ case in § 4.1.2). From other data, IRAS 16293–2422 is actually known to have a gradient in temperature and density throughout its en- velope, which cannot be modelled properly with either technique. For such situation, a full Monte Carlo radia- tive transfer method is needed in which both the physi- cal conditions and the abundances can vary with radius (Fig. 5, circles). Nevertheless, the column densities and abundances inferred with RADEX using the physical con- ditions inferred from the line ratios differ by only a fac- tor of a few from those found with the more sophis- ticated analysis, at least for the particular zone of the source to which those conditions apply (Schöier et al. 2002). 5. Conclusions We have presented a computer program to analyze spec- tral line observations at radio and infrared wavelengths, based on the escape probability approximation. The pro- gram can be used for any molecule for which collisional data exist; such input data are available in the required format from the LAMDA database. The program can be used for optical depths from ≈–0.1 to ≈100. The limited number of free parameters makes RADEX very useful to rapidly analyze large datasets. As an example, observed line intensity ratios may be com- pared with the plots in Appendix C to estimate density and kinetic temperature. Ratios of other lines and other molecules may be easily computed using the Python scripts included in the RADEX distribution. The program may also be used to create synthetic spectra. This capa- bility will be important to model the THz line surveys from the HIFI instrument onboard the Herschel space observatory. In the future, we plan to incorporate a multi-zone es- cape probability formalism (Poelman & Spaans 2005; Elitzur & Asensio Ramos 2006) which will enable 10 Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra depths, the calculation may also start from LTE condi- tions rather than from optically thin statistical equilib- rium. Robust convergence may be achieved by starting from either initial condition and requiring the two an- swers to be equal. For the modeling of crowded spec- tra, the effects of line overlap will also need to be con- sidered, for instance in the ‘all or nothing’ approach (Cesaroni & Walmsley 1991). Such spectra will be rou- tinely observed with the superb resolution and sensitiv- ity of ALMA. Our program is free for anybody to use for science, pro- vided that appropriate reference is made to this paper. For any other purpose such as to incorporate the pro- gram into other packages which may be distributed to the public, prior agreement with the authors is needed. Acknowledgements. The authors wish to thank Huib Jan van Langevelde for his efforts in documenting RADEX, and Erik Deul for computing support at Leiden Observatory. JHB and FLS acknowledge the Swedish Research Council for financial support. FvdT and EvD thank the Netherlands Organization for Scientific Research (NWO) and the Netherlands Research School for Astronomy (NOVA). 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D. 1997, MNRAS, 285, 303 Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 1 Online Material Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 2 Appendix A: Program input and output A.1. Program input The input parameters to RADEX are the following: 1. The name of the molecular data file to be used. 2. The name of the file to write the output to. 3. The frequency range for the output file [GHz]. All transitions from the molecular data file are always taken into account in the calculation, but often it is practical to write only a limited set of lines to the output. 4. The kinetic temperature of the cloud [K]. 5. The number of collision partners to be used. Most users will want H2 as only collision partner, but in more specialized cases, additional collisions with H or electrons may for instance play a role. See the molecular datafiles for details. For some species (CO, atoms) separate collision data for ortho and para H2 exist; the program then uses the thermal or- tho/para ratio unless the user specifies otherwise. 6. The name (case-insensitive) and the density [cm−3] of each collision partner. Possibilities are H2, p-H2, o-H2, electrons, atomic H, He, and H 7. The temperature of the background radiation field – If >0, a black body at this temperature is used. Most users will adopt the cosmic microwave background at TCMB=2.725(1+z) K for a galaxy at redshift z. – If =0, the average interstellar radiation field (ISRF) is used, taken from Black (1994) with modifications described in § 3.2. This spectrum is not adjustable by a scale factor because it con- sists of several components that are not expected to scale linearly with respect to each other. – If <0, a user-defined radiation field is used, spec- ified by values of frequency [cm−1], intensity [Jy nsr−1], and dilution factor [dimensionless]. Spline interpolation and extrapolation are ap- plied to this table. The intensity need not be specified at all frequencies of the line list, but a warning message will appear if extrapolation (rather than interpolation) is required. 8. The column density of the molecule [cm−2]. 9. The FWHM line width [km s−1]. A.2. Program output The output file written by RADEX first replicates the in- put parameters, and then lists the following quantities for each spectral line within the specified frequency range. 1. Quantum numbers, upper state energy [K], fre- quency [GHz], and wavelength [µm]. These num- bers are just copied from the molecular data file, which usually comes from the LAMDA database. Frequencies from this database are generally of line catalogs such as CDMS (Müller et al. 2001)12 should be consulted. 2. The excitation temperature [K] as defined in Eq. (10). In general, different lines have different excitation temperatures. Lines are thermalized if Tex=Tkin; in LTE, all lines are thermalized. 3. The line optical depth, defined as the optical depth of the equivalent rectangular line shape (φν = 1/∆ν). 4. The line intensity, defined as the Rayleigh-Jeans equivalent temperature TR [K]. 5. The line flux, defined as the velocity-integrated intensity, both in units of K km s−1 (common in radio astronomy) and of erg cm−2 s−1 (common in infrared astronomy). The line flux is calcu- lated as 1.0645TR∆V, where the factor 1.0645 =√ ln 2) converts the adopted rectangular line profile into a Gaussian profile with an FWHM of ∆V. The integrated profile is useful to estimate the total emission in the line, but it has limited mean- ing at high optical depths, because the change of optical depth over the line profile is not taken into account. Proper modeling of optically thick lines re- quires programs that resolve the source both spec- trally and spatially (see § 4.1.2 for further discus- sion). Auxiliary output files can be generated, for example to display the adopted continuum spectrum. Appendix B: Coding standards The original version of RADEX was written in such a way as to minimize the use of machine memory which was expensive until a decade ago. Nowadays, clarity and easy maintenance are more important requirements, which is why the source code has been re-written fol- lowing the rules below. We hope that these rules will be useful for the development of other ‘open source’ as- tronomical software. For further guidelines on scientific programming we recommend the Software Carpentry13 on-line course. 1. All the action is in subroutines; the sole purpose of the main program is to show the structure of the pro- gram. The subroutines are grouped into several files for a better overview; compilation instructions for automated builds on a variety of platforms are in a Makefile. 2. The program text is interspersed with comments at a ratio of ≈1:1. In particular, each subroutine starts with a description of its contents, its input and out- put, where incoming calls come from, and which calls go out. Then the properties of each variable are described: contents, units and type. 3. Variables and subroutines have descriptive names with a length of 5–10 letters. Names of integer vari- ables start with the letters i..n; names of floating- point variables (always of double precision) with a..h or o..z. There are no specific namings for vari- ables of character or logical type, as for example http://cdms.de http://www.swc.scipy.org/ Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 3 in CLOUDY. Names are always based on the English language. We do not use upper-, lower-, or camel- case to distinguish types of identifiers, as some pro- grams do. 4. Loops are marked by indenting the program text. The loop variables are always called ilev, iline .., never just i. 5. Subroutines start with a check whether the input pa- rameters have reasonable values. Such checks force soft landings if necessary, and avoid runtime errors. 6. Statements with calculations use spaces around the =, + and – symbols, but not around others. Calculations that consist of multiple steps are split over as many program lines. Multiple assignments in a row are aligned at the = sign. 7. The program text avoids “magic numbers” both in calculations and in definitions. Numbers that are of- ten used such as Planck’s constant are defined at one central place in the program. Similarly, often-used variables such as physical parameters are stored in shared memory rather than passed on via subroutine calls. Appendix C: Diagnostic plots of molecular line ratios We have used the RADEX program and the LAMDA database to calculate line ratios of several commonly observed molecules for a range of kinetic tempera- tures and H2 densities. The plots in this Appendix may be used by observers to estimate physical conditions from their data. Line ratios have the advantage of be- ing less sensitive to calibration errors than absolute line strengths, especially when the two lines have been mea- sured with the same telescope, receiver and spectrome- The calculations assume a column density of 1012 cm−2, a line width of 1.0 km s−1, and use a 2.73 K blackbody as background radiation field. Under these conditions, the lines are optically thin, so that the line ratios do not depend on column density. The calculations also assume that the emission in both lines fills the telescope beams equally, which may be the case if the lines are close in frequency. However, lines are generally measured in beams of different sizes, and the observations need to be corrected to account for this effect, if the source is known to be compact. Linear molecules such as CO (Fig. C.1) are tracers of density at low densities, when collisions compete with radiative decay. At higher densities, the excitation becomes thermalized and the line ratios are sensitive to temperature. For a given molecule, moving up the J−ladder means probing higher temperatures and densi- ties. Note that for the column densities of typical dense interstellar clouds, the CO lines are optically thick, and observations of 13CO or even rarer isotopologues must be used to probe physical conditions. The critical densities of molecular lines scale as µ2ν3, where µ is the permanent dipole moment of the molecule and ν is the frequency of the line. Indeed, the CS molecule (Fig. C.2) has a larger dipole moment than CO, and its line ratios are mainly probes of the den- sity. The small frequency spacing between the lines of CS makes this molecule very useful to probe density structure (e.g., Van der Tak et al. 2000b). The HCO+ and HCN molecules (Fig. C.3) display similar trends to CS, although their line spacing is not as small. Non-linear molecules such as H2CO (Figs. C.4 and C.5) have the advantage that both temperature and density may be probed within the same frequency range. Ratios of lines from different J−states tend to be density tracers (left panels), while ratios of lines from the same J−state but different K−states are mostly temperature probes (right panels). The lines of H2CO are often quite strong, making this molecule a favourite tracer of temperature and density (Mangum & Wootten 1993). Other asym- metric molecules have also been used, such as H2CS (Blake et al. 1994) and CH3OH (Leurini et al. 2004) al- though abundance variations from source to source or even within sources often complicate the interpretation (Johnstone et al. 2003). Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 4 Appendix D: The Python scripts The RADEX distribution comes with two scripts, radex line.py and radex grid.py, to automate standard modeling procedures. The scripts are written in Python and are run from the Unix shell command line after manual editing of parameters. The first script, radex line.py, calculates the column density of a molecule from an observed line intensity, given estimates of kinetic temperature and H2 volume density. The input parameters are: 1. The kinetic temperature [K] 2. The number density of H2 molecules [cm 3. The temperature of the background radiation field [K], usually 2.73 (CMB). 4. The name of the molecule (or molecular data file) 5. The frequency of the line [GHz] 6. The observed line intensity [K] 7. The observed line width [km s−1] Furthermore, two numerical parameters have good de- fault values but will need to be changed occasionally: 1. The free spectral range around the line (default 10%): this number must be smaller for molecules with many line close in frequency, such as CH3OH. The program uses this parameter to find the ob- served line from the list of lines in the molecular model. 2. The required accuracy (default 10%): The default corresponds to the calibration uncertainty of most telescopes. The script iterates on column density until the observed and modeled line fluxes agree to within the desired ac- curacy. The best-fit column density is directly written to the screen. The file radex.out gives details of the best-fit model. The second script, radex grid.py, runs a series of RADEX models to estimate the kinetic temperature and/or the volume density from an observed line ratio. The user needs to set the following input parameters: 1. The grid boundaries: minimum and maximum ki- netic temperature [K] and minimum and maximum H2 volume density [cm 2. The temperature of the background radiation field [K], usually 2.73 (CMB). 3. The molecular column density [cm−2]. For the illustrative plots in Appendix C, a low value (N=1012 cm−2) was used, so that the line ratios are independent of column density (optically thin limit). However, in modeling specific observations, it is worth varying this parameter to assess the sensitivity of the line ratio to column density. 4. The observed line width [km s−1], usually an aver- age of the widths of the two lines. 1. The number of grid points along the temperature and density axes. 2. The free spectral range around the line (see above) The name of the molecule and a list of observed line ratios and names of associated output files are given at the start of the main program. The script produces a file radex.out which is a tabular listing of temperature, log density, and line ratio. This results may be plotted with the user’s favourite plotting program. Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 5 Fig. C.1. Line ratios of CO in the optically thin limit as a function of kinetic temperature and H2 density. Contours are spaced linearly and some contours are la- beled for easy identification. Fig. C.2. Line ratios of CS in the optically thin limit as a function of kinetic temperature and H2 density. Contours are spaced linearly and some contours are la- beled for easy identification. Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 6 Fig. C.3. Line ratios of HCO+ and HCN in the optically thin limit as a function of kinetic temperature and H2 density. Contours are spaced linearly and some contours are labeled for easy identification. Fig. C.4. Line ratios of o-H2CO in the optically thin limit as a function of kinetic temperature and H2 den- sity. Contours are spaced linearly and some contours are labeled for easy identification. Van der Tak et al.: Fast non-LTE analysis of interstellar line spectra, Online Material p 7 Fig. C.5. Line ratios of p-H2CO in the optically thin limit as a function of kinetic temperature and H2 den- sity. Contours are spaced linearly and some contours are labeled for easy identification. Introduction Radiative transfer and molecular excitation Basic formalism Molecular line cooling Escape probability The program Basic capabilities Background radiation field Chemical formation and destruction rates Calculation Results Limitations of the program Comparison with other methods Homogeneous models The case of CO The case of HCO+ Observations of a Young Stellar Object Conclusions Program input and output Program input Program output Coding standards Diagnostic plots of molecular line ratios The Python scripts

0704.0157

Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids

arXiv:0704.0157v1 [cond-mat.stat-mech] 2 Apr 2007 7 Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids M. López de Haro1, S. B. Yuste2 and A. Santos3 1 Centro de Investigación en Enerǵıa, Universidad Nacional Autónoma de México (U.N.A.M.), Temixco, Morelos 62580, Mexico [email protected] 2 Departamento de F́ısica, Universidad de Extremadura, E-06071 Badajoz, Spain [email protected] 3 Departamento de F́ısica, Universidad de Extremadura, E-06071 Badajoz, Spain [email protected] An overview of some analytical approaches to the computation of the struc- tural and thermodynamic properties of single component and multicompo- nent hard-sphere fluids is provided. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus–Yevick theory. Approximate ex- pressions for the contact values of the radial distribution functions and the corresponding analytical equations of state are also discussed. Extensions of this methodology to related systems, such as sticky hard spheres and square- well fluids, as well as its use in connection with the perturbation theory of fluids are briefly addressed. 1 Introduction In the statistical thermodynamic approach to the theory of simple liquids, there is a close connection between the thermodynamic and structural prop- erties [1–4]. These properties depend on the intermolecular potential of the system, which is generally assumed to be well represented by pair interactions. The simplest model pair potential is that of a hard-core fluid (rods, disks, spheres, hyperspheres) in which attractive forces are completely neglected. In fact, it is a model that has been most studied and has rendered some analytical results, although up to this day no general (exact) explicit expression for the equation of state is available, except for the one-dimensional case. Something similar applies to the structural properties. An interesting feature concerning the thermodynamic properties is that in hard-core systems the equation of state depends only on the contact values of the radial distribution functions. In the absence of a completely analytical approach, the most popular methods http://arxiv.org/abs/0704.0157v1 2 M. López de Haro, S. B. Yuste and A. Santos to deal with both kinds of properties of these systems are integral equation theories and computer simulations. It is well known that in real gases and liquids at high temperatures the state and thermodynamic properties are determined almost entirely by the repulsive forces among molecules. At lower temperatures, attractive forces become significant, but even in this case they affect very little the configu- ration of the system at moderate and high densities. These facts are taken into account in the application of the perturbation theory of fluids, where hard-core fluids are used as the reference systems in the computation of the thermodynamic and structural properties of real fluids. However, successful results using perturbation theory are rather limited due to the fact that, as mentioned above, there are in general no exact (analytical) expressions for the thermodynamic and structural properties of the reference systems which are in principle required in the calculations. On the other hand, in the realm of soft condensed matter the use of the hard-sphere model in connection, for instance, with sterically stabilized colloidal systems is quite common. This is due to the fact that nowadays it is possible to prepare (almost) monodisperse spherical colloidal particles with short-ranged harshly repulsive interparticle forces that may be well described theoretically with the hard-sphere potential. This chapter presents an overview of the efforts we have made over the last few years to compute the thermodynamic and structural properties of hard-core systems using relatively simple (approximate) analytical methods. It is structured as follows. In Section 2 we describe our proposals to derive the contact values of the radial distribution functions of a multicomponent mixture (with an arbitrary size distribution, either discrete or continuous) of d-dimensional hard spheres from the use of some consistency conditions and the knowledge of the contact value of the radial distribution function of the corresponding single component system. In turn, these contact values lead to equations of state both for additive and non-additive hard spheres. Some con- sequences of such equations of state, in particular the demixing transition, are briefly analyzed. This is followed in Section 3 by the description of the Ratio- nal Function Approximation method to obtain analytical expressions for the structural quantities of three-dimensional single component and multicompo- nent fluids. The only required inputs in this approach are the contact values of the radial distribution functions and so the connection with the work of the previous section follows naturally. Structural properties of related systems, like sticky hard spheres or square-well fluids, that may also be tackled with the same philosophy are also discussed in Section 4. Section 5 provides an account of the reformulation of the perturbation theory of liquids using the results of the Rational Function Approximation method for a single compo- nent hard-sphere fluid and its illustration in the case of the Lennard–Jones fluid. In the final section, we provide some perspectives of the achievements obtained so far and of the challenges that remain ahead. Alternative Approaches to Hard-Sphere Liquids 3 2 Contact Values and Equations of State for Mixtures As stated in the Introduction, a nice feature of hard-core fluids is that the expressions of all their thermodynamic properties in terms of the radial distri- bution functions (RDF) are particularly simple. In fact, for these systems the internal energy reduces to that of the ideal gas and in the pressure equation it is only the contact values rather than the full RDF which appear explicitly. In this section we present our approach to the derivation of the contact values of hard-core fluid mixtures in d dimensions. 2.1 Additive Systems in d Dimensions If σij denotes the distance of separation at contact between the centers of two interacting fluid particles, one of species i and the other of species j, the mixture is said to be additive if σij is just the arithmetic mean of the hard-core diameters of each species. Otherwise, the system is non-additive. We deal in this subsection and in Subsection 2.2 with additive systems, while non-additive hard-core mixtures will be treated in Subsection 2.3. Definitions Let us consider an additive mixture of hard spheres (HS) in d dimensions with an arbitrary number N of components. In fact, our discussion will remain valid for N → ∞, i.e., for polydisperse mixtures with a continuous distribution of sizes. The additive hard core of the interaction between a sphere of species i and a sphere of species j is σij = (σi + σj), where the diameter of a sphere of species i is σii = σi. Let the number density of the mixture be ρ and the mole fraction of species i be xi = ρi/ρ, where ρi is the number density of species i. From these quantities one can define the packing fraction η = vdρMd, where vd = (π/4) d/2/Γ (1 + d/2) is the volume of a d-dimensional sphere of unit diameter and Mn ≡ 〈σn〉 = i (1) denotes the nth moment of the diameter distribution. In a HS mixture, the knowledge of the contact values gij(σij) of the RDF gij(r), where r is the distance, is important for a number of reasons. For example, the availability of gij(σij) is sufficient to get the equation of state (EOS) of the mixture via the virial expression Z(η) = 1 + i,j=1 xixjσ ijgij(σij), (2) 4 M. López de Haro, S. B. Yuste and A. Santos where Z = p/ρkBT is the compressibility factor of the mixture, p being the pressure, kB the Boltzmann constant, and T the absolute temperature. The exact form of gij(σij) as functions of the packing fraction η, the set of diameters {σk}, and the set of mole fractions {xk} is only known in the one-dimensional case, where one simply has [5] gij(σij) = , (d = 1). (3) Consequently, for d ≥ 2 one has to resort to approximate theories or empirical expressions. For hard-disk mixtures, an accurate expression is provided by Jenkins and Mancini’s (JM) approximation [6, 7], gJMij (σij) = (1− η)2 σiσjM1 σijM2 , (d = 2). (4) The associated compressibility factor is ZJM(η) = 1 + η/8 (1 − η)2 , (d = 2). (5) In the case of three-dimensional systems, some important analytical expres- sions for the contact values and the corresponding compressibility factor also exist. For instance, the expressions which follow from the solution of the Percus–Yevick (PY) equation of additive HS mixtures by Lebowitz [8] are gPYij (σij) = (1− η)2 σiσjM2 σijM3 , (d = 3), (6) ZPY(η) = (1− η)2 (1 − η)2 , (d = 3). (7) Also analytical are the results obtained from the Scaled Particle Theory (SPT) [9–12], gSPTij (σij) = (1 − η)2 σiσjM2 σijM3 (1 − η)3 σiσjM2 σijM3 , (d = 3), ZSPT(η) = (1− η)2 (1− η)3 , (d = 3). (9) Neither the PY nor the SPT lead to particularly accurate values and so Boubĺık [13] and, independently, Grundke and Henderson [14] and Lee and Levesque [15] proposed an interpolation between the PY and the SPT contact values, that we will refer to as the BGHLL values: gBGHLLij (σij) = (1− η)2 σiσjM2 σijM3 (1− η)3 σiσjM2 σijM3 , (d = 3). Alternative Approaches to Hard-Sphere Liquids 5 This leads through Eq. (2) to the widely used and rather accurate Boubĺık– Mansoori–Carnahan–Starling–Leland (BMCSL) EOS [13,16] for HS mixtures: ZBMCSL(η) = (1− η)2 η2(3− η) (1 − η)3 , (d = 3). (11) Refinements of the BGHLL values have been subsequently introduced, among others, by Henderson et al. [17–22], Matyushov and Ladanyi [23], and Barrio and Solana [24] to eliminate some drawbacks of the BMCSL EOS in the so- called colloidal limit of binary HS mixtures. On a different path, but also having to do with the colloidal limit, Viduna and Smith [25] have proposed a method to obtain contact values of the RDF of HS mixtures from a given EOS. However, none of these proposals may be easily generalized so as to be valid for any dimensionality and any number of components. Therefore, if one wants to have a more general framework able to deal with arbitrary d and N an alternative strategy is called for. Universality Ansatz In order to follow our alternative strategy, it is useful to make use of exact limit results that can help one in the construction of approximate expressions for gij(σij). Let us consider first the limit in which one of the species, say i, is made of point particles, i.e., σi → 0. In that case, gii(σi) takes the ideal gas value, except that one has to take into account that the available volume fraction is 1− η. Thus, gii(σi) = . (12) An even simpler situation occurs when all the species have the same size, {σk} → σ, so that the system becomes equivalent to a single component system. Therefore, {σk}→σ gij(σij) = gs, (13) where gs is the contact value of the RDF of the single component fluid at the same packing fraction η as that of the mixture. Table 1 lists some of the most widely used proposals for the contact value gs and the associated compressibility factor Zs = 1 + 2 d−1ηgs (14) in the case of the single component HS fluid. Equations (12) and (13) represent the simplest and most basic conditions that gij(σij) must satisfy. There is a number of other less trivial consistency conditions [11, 17, 19, 20, 23, 24,32–34], some of which will be used later on. In order to proceed, in line with a property shared by earlier proposals [see, in particular, Eqs. (4), (6), (8), and (10)], we assume that, at a given packing fraction η, the dependence of gij(σij) on the parameters {σk} and {xk} takes place only through the scaled quantity 6 M. López de Haro, S. B. Yuste and A. Santos Table 1. Some expressions of gs and Zs for the single component HS fluid. In the SHY proposal, ηcp = ( 3/6)π is the crystalline close-packing fraction for hard disks. In the LM proposal, b3 and b4 are the (reduced) third and fourth virial coefficients, ζ(η) = 1.2973(59)−0.062(13)η/ηcp for d = 4, and ζ(η) = 1.074(16)+0.163(45)η/ηcp for d = 5, where the values of the close-packing fractions are ηcp = π 2/16 ≃ 0.617 and ηcp = π 2/30 ≃ 0.465 for d = 4 and d = 5, respectively. d gs Zs Label Ref. 1− 7η/16 (1− η)2 1 + η2/8 (1− η)2 H [26] 1− η(2ηcp − 1)/2η2cp 1− 2η + η2(ηcp − 1)/2η2cp 1− 2η + η2(ηcp − 1)/2η2cp SHY [27] 2 gHs − 27(1− η)4 ZHs − 26(1− η)4 L [28] 1 + η/2 (1− η)2 1 + 2η + 3η2 (1− η)2 PY [29] 1− η/2 + η2/4 (1− η)3 1 + η + η2 (1− η)3 SPT [9] 1− η/2 (1− η)3 1 + η + η2 − η3 (1− η)3 CS [30] 1 + [21−db3 − ζ(η)b4/b3]η 1− ζ(η)(b4/b3)η + [ζ(η)− 1] 21−db4η2 1 + 2d−1ηgLMs LM [31] zij ≡ . (15) More specifically, we assume gij(σij) = G(η, zij), (16) where the function G(η, z) is universal in the sense that it is a common func- tion for all the pairs (i, j), regardless of the composition and number of compo- nents of the mixture. Of course, the function G(η, z) is in principle different for each dimensionality d. To clarify the implications of this universality ansatz, let us imagine two mixtures M and M′ having the same packing fraction η but strongly differing in the set of mole fractions, the sizes of the parti- cles, and even the number of components. Suppose now that there exists a pair (i, j) in mixture M and another pair (i′, j′) in mixture M′ such that zij = zi′j′ . Then, according to Eq. (16), the contact value of the RDF for the pair (i, j) in mixture M is the same as that for the pair (i′, j′) in mixture M′, i.e., gij(σij) = gi′j′ (σi′j′ ). In order to ascribe a physical meaning to the parameter zij , note that the ratio Md−1/Md can be understood as a “typ- ical” inverse diameter (or curvature) of the particles of the mixture. Thus, z−1ij = (σ−1i + σ j )/(Md−1/Md) represents the arithmetic mean curvature, in units of Md−1/Md, of a particle of species i and a particle of species j. Alternative Approaches to Hard-Sphere Liquids 7 Once the ansatz (16) is adopted, one may use the limits in (12) and (13) to get G(η, z) at z = 0 and z = 1, respectively. Since zii → 0 in the limit σi → 0, insertion of Eq. (12) into (16) yields G(η, 0) = 1 ≡ G0(η). (17) Next, if all the diameters are equal, zij → 1, so that Eq. (13) implies that G(η, 1) = gs. (18) Linear Approximation As the simplest approximation [35], one may assume a linear dependence of G on z that satisfies the basic requirements (17) and (18), namely G(η, z) = 1 z. (19) Inserting this into Eq. (16), one has ge1ij (σij) = . (20) Here, the label “e1” is meant to indicate that (i) the contact values used are an extension of the single component contact value gs and that (ii) G(η, z) is a linear polynomial in z. This notation will become handy below. Although the proposal (20) is rather crude and does not produce especially accurate results for gij(σij) when d ≥ 3, it nevertheless leads to an EOS that exhibits an excellent agreement with simulations in 2, 3, 4, and 5 dimensions, provided that an accurate gs is used as input [35–39]. This EOS may be written as Ze1(η) = 1 + 2d−1(Ω0 −Ω1) + [Zs(η)− 1]Ω1, (21) where the coefficients Ωm depend only on the composition of the mixture and are defined by Ωm = 2 −(d−m) Mmd−1 Mm+1d Mn+mMd−n. (22) In particular, for d = 2 and d = 3, Ze1(η) = Zs(η) − , (d = 2), (23) 8 M. López de Haro, S. B. Yuste and A. Santos Ze1(η) = Zs(η)− , (d = 3). (24) As an extra asset, from Eq. (21) one may write the virial coefficients of the mixture Bn, defined by Z = 1 + Bn+1ρ n, (25) in terms of the (reduced) virial coefficients of the single component fluid bn defined by Zs = 1 + bn+1η n. (26) The result is Bn = v Ω1bn + 2 d−1(Ω0 −Ω1) . (27) In the case of binary mixtures, these coefficients are in very good agreement with the available exact and simulation results [35,37], except when the mix- ture involves components of very disparate sizes, especially for high dimen- sionalities. One may perform a slight modification such that this deficiency is avoided and thus get a modified EOS [37, 40]. For d = 2 and d = 3 it reads Z(η) = Zs(η) + x1 1− η2 1− η2 − Zs(η) σ2 − σ1 1− η1 1− η1 − Zs(η) σ1 − σ2 , (d = 2, 3), where ηi = vdρiσ i is the partial volume packing fraction due to species i. In contrast to most of the approaches (PY, SPT, BMCSL, e1, . . . ), the proposal (28) expresses Z(η) in terms not only of Zs(η) but also involves Zs and Zs . Equation (28) should in principle be useful in particular for binary mixtures involving components of very disparate sizes. However, it is slightly less accurate than the one given in Eq. (21) for ordinary mixtures [37]. Quadratic Approximation In order to improve the proposal contained in Eq. (20), in addition to the consistency requirements (12) and (13), one may consider the condition stem- ming from a binary mixture in which one of the species (say i = 1) is much Alternative Approaches to Hard-Sphere Liquids 9 larger than the other one (i.e., σ1/σ2 → ∞), but occupies a negligible volume (i.e., x1(σ1/σ2) d → 0). In that case, a sphere of species 1 is felt as a wall by particles of species 2, so that [17, 20, 41] σ1/σ2→∞ x1(σ1/σ2)d→0 g12(σ12)− 2d−1ηg22(σ2) = 1. (29) Hence, in the limit considered in Eq. (29), we have z22 → 1, z12 → 2. Conse- quently, under the universality ansatz (16), one may rewrite Eq. (29) as G(η, 2) = 1 + 2d−1ηG(η, 1). (30) Thus, Eqs. (17), (18), and (30) provide complete information on the function G at z = 0, z = 1, and z = 2, respectively, in terms of the contact value gs of the single component RDF. The simplest functional form of G that complies with the above consistency conditions is a quadratic function of z [42]: G(η, z) = G0(η) + G1(η)z + G2(η)z2, (31) where the coefficients G1(η) and G2(η) are explicitly given by G1(η) = (2− 2d−2η)gs − 2− η/2 1− η , (32) G2(η) = 1− η/2 − (1− 2d−2η)gs. (33) Therefore, the explicit expression for the contact values is ge2ij (σij) = (2− 2d−2η)gs − 2− η/2 1− η/2 − (1− 2d−2η)gs . (34) Following the same criterion as the one used in connection with Eq. (20), the label “e2” is meant to indicate that (i) the resulting contact values represent an extension of the single component contact value gs and that (ii) G(η, z) is a quadratic polynomial in z. Of course, the quadratic form (31) is not the only choice compatible with conditions (17), (18), and (30). For instance, a rational function was also considered in Ref. [42]. However, although it is rather accurate, it does not lead to a closed form for the EOS. In contrast, when Eq. (34) is inserted into Eq. (2), one gets a closed expression for the compressibility factor in terms of the packing fraction η and the first few moments Mn, n ≤ d. The result is Ze2(η) = 1 + 2 d−2 η 1− η [2(Ω0 − 2Ω1 +Ω2) + (Ω1 −Ω2)η] + [Zs(η)− 1] 2Ω1 −Ω2 + 2d−2(Ω2 −Ω1)η , (35) 10 M. López de Haro, S. B. Yuste and A. Santos where the quantities Ωm are defined in Eq. (22). Quite interestingly, in the two-dimensional case Eq. (35) reduces to Eq. (23), i.e., Ze1(η) = Ze2(η), (d = 2). (36) This illustrates the fact that two different proposals for the contact values gij(σij) can yield the same EOS when inserted into Eq. (2). On the other hand, for three-dimensional mixtures Eq. (35) becomes Ze2(η) = 1− η + M Zs(η)− , (d = 3), which differs from Eq. (24). In fact, Ze1(η)− Ze2(η) = 1 + η − (1 − 2η)Zs(η) , (d = 3). Specific Examples In this subsection, rather than carrying out an exhaustive comparison with the wealth of results available in the literature, we will consider only a few representative examples. In particular, for d = 3, we will restrict ourselves to a comparison with classical proposals (say BGHLL, PY, and SPT for the contact values). The comparison with more recent ones may be found in Refs. [35, 42, 43]. Thus far the development has been rather general since gs remains free in Eqs. (20) and (34). In order to get specific results, it is necessary to fix gs [cf. Table 1]. In the one-dimensional case, one has gs = 1/(1− η) and so one gets the exact result (3) after substitution into Eq. (20). Similarly Eqs. (32) and (33) lead to G1 = G2 = 0 and so we recover again the exact result. If in the two-dimensional case we take Henderson’s value [26] gs = g then the linear approximation (20) reduces to the JM approximation, Eq. (4). This equivalence can be symbolically represented as geH1ij = g ij , where the label “eH1” refers to the extension of Henderson’s single component value in the linear approximation. While gJMij is very accurate, even better results are provided by the quadratic form (34), especially if Luding’s value [28] gs = g is used [44]. In the three-dimensional case, Eq. (20) is of the form of the solution of the PY equation [8]. In fact, insertion of gs = g s leads to Eq. (6), i.e., g gPYij . Similarly, if the SPT expression [9] gs = g s is used for the single component contact value in the quadratic approximation (34), we reobtain the SPT expression for the mixture, Eq. (8). In other words, geSPT2ij = g On the other hand, if the much more accurate CS [30] expression gs = g used as input, we arrive at the following expression: Alternative Approaches to Hard-Sphere Liquids 11 geCS2ij = η(1 − η/3) (1 − η)2 σiσjM2 σijM3 η2(1− η/2) (1− η)3 σiσjM2 σijM3 , (d = 3), which is different from the BGHLL one, Eq. (10), improves the latter for zij > 1, and leads to similar results for zij < 1, as comparison with computer simulations shows [42]. The four approximations (6), (8), (10), and (39) are consistent with conditions (12) and (13), but only the SPT and eCS2 are also consistent with condition (29). It should also be noted that if one considers a binary mixture in the infinite solute dilution limit, namely x1 → 0, so that z12 → 2/(1 + σ2/σ1), Eq. (39) yields the same result for g12(σ12) as the one proposed by Matyushov and Ladanyi [23] for this quantity on the basis of exact geometrical relations. However, the extension that the same authors propose when there is a non-vanishing solute concentration, i.e., for x1 6= 0, is different from Eq. (39). Equation (34) can also be used in the case of hyperspheres (d ≥ 4) [42]. In particular, a very good agreement with available computer simulations [38] is obtained for d = 4 and d = 5 by using Luban and Michels [31] value gs = g 0.0 0.1 0.2 0.3 0.4 0.5 -0.02 eCS1 eCS2 Fig. 1. Deviation of the compressibility factor from the BMCSL value, as a function of the packing fraction η for an equimolar three-dimensional binary mixture with σ2/σ1 = 0.6. The open (Ref. [18]) and closed (Ref. [45]) circles are simulation data. The lines are the PY EOS (– · · –), the SPT EOS (– · – ·), the eCS1 EOS (· · · ), and the eCS2 EOS (– – –). 12 M. López de Haro, S. B. Yuste and A. Santos 0.00 0.05 0.10 0.15 0.20 0.25 4D, / =1/2 4D, / =1/3 5D, / =2/5 Fig. 2. Compressibility factor for three equimolar mixtures in 4D and 5D systems. Lines are the eLM1 predictions, while symbols are simulation data [38]. Now we turn to the compressibility factors (21) and (35), which are ob- tained from the contact values (20) and (34), respectively. Since they depend on the details of the composition through the d first moments, they are mean- ingful even for continuous polydisperse mixtures. As said above, in the two-dimensional case both Eqs. (21) and (35) reduce to Eq. (23), which yield very accurate results when a good Zs is used as input [39, 42, 44]. For three-dimensional mixtures, insertion of Zs = Z Eqs. (24) and (37) yields ZeCS1(η) = ZBMCSL(η) + (1 − η)3M23 M1M3 −M22 , (d = 3), (40) ZeCS2(η) = ZBMCSL(η)− (1 − η)2M23 M1M3 −M22 , (d = 3), (41) where ZBMCSL(η) is given by Eq. (11). Note that ZeCS1(η) > ZBMCSL(η) > ZeCS2(η). Since simulation data indicate that the BMCSL EOS tends to un- derestimate the compressibility factor, it turns out that, as illustrated in Fig. 1 for an equimolar binary mixture with σ2/σ1 = 0.6, the performance of ZeCS1 is, paradoxically, better than that of ZeCS2 [42], despite the fact that the un- derlying linear approximation for the contact values is much less accurate than the quadratic approximation. This shows that a rather crude approximation such as Eq. (20) may lead to an extremely good EOS [35, 37–39], which, as Alternative Approaches to Hard-Sphere Liquids 13 clearly seen in Fig. 1, represents a substantial improvement over the classical proposals. Interestingly, the EOS corresponding to ZeCS1 has recently been independently derived as the second order approximation of the Fundamental Measure Theory for the HS fluid by Hansen-Goos and Roth [46]. In the case of d = 4 and d = 5, use of Zs(η) = Z s (η) in Eq. (21) produces a simple extended EOS of a mixture of hard additive hyperspheres in these dimensionalities. The accuracy of these two EOS for hard hypersphere mixtures in the fluid region has been confirmed by simulation data [38] for a wide range of compositions and size ratios. In Fig. 2, this accuracy is explicitly exhibited in the case of three equimolar mixtures, two in 4D and one in 5D. 2.2 A More Consistent Approximation for Three-Dimensional Additive Mixtures Up to this point, we have considered an arbitrary dimensionality d and have constructed, under the universality assumption (16), the acurate quadratic approximation (34), which fulfills the consistency conditions (12), (13), and (29). However, there exist extra consistency conditions that are not necessarily satisfied by (34). In particular, when the mixture is in contact with a hard wall, the state of equilibrium imposes that the pressure evaluated near the wall by considering the impacts with the wall must be the same as the pressure in the bulk evaluated from the particle-particle collisions. This consistency condition is especially important if one is interested in deriving accurate expressions for the contact values of the particle-wall correlation functions. Since a hard wall can be seen as a sphere of infinite diameter, the contact value gwj of the correlation function of a sphere of diameter σj with the wall can be obtained from gij(σij) as gwj = lim gij(σij). (42) Note that gwj provides the ratio between the density of particles of species j adjacent to the wall and the density of those particles far away from the wall. The sum rule connecting the pressure of the fluid and the above contact values is [47] Zw(η) = xjgwj, (43) where the subscript w in Zw has been used to emphasize that Eq. (43) repre- sents a route alternative to the virial one, Eq. (2), to get the EOS of the HS mixture. The condition Z = Zw is equivalent to (29) in the special case where one has a single fluid in the presence of the wall. However, in the general case of a mixture plus a wall, the condition Z = Zw is stronger than Eq. (29). In the two-dimensional case, it turns out that the quadratic approximation (34) already satisfies the requirement Z = Zw, regardless of the density and composition of the mixture [44]. However, this is not the case for d ≥ 3. 14 M. López de Haro, S. B. Yuste and A. Santos Our problem now consists of computing gij(σij) and the associated gwj for the HS mixture in the presence of a hard wall, so that the condition Z = Zw is satisfied for an arbitrary mixture [43]. Due to the mathematical complexity of the problem, here we will restrict ourselves to three-dimensional systems (d = 3). Similarly to what we did in the preceding subsection, we consider a class of approximations of the universal type (16), so that conditions (12) and (13) lead again to Eqs. (17) and (18), respectively. Notice that Eq. (16) implies in particular that gwj = G(η, zwj), zwj = 2σj . (44) Assuming that z = 0 is a regular point and taking into account condition (17), G(η, z) can be expanded in a power series in z: G(η, z) = G0(η) + Gn(η)zn. (45) After simple algebra, using the ansatz (16) and Eq. (45) in Eqs. (2) (with d = 3) and (43) one gets Z = G0 + 3η G0 + 4η Mn+13 i,j=1 xixjσ ij , (46) Zw = G0 + Mn. (47) Notice that if the series (45) is truncated after a given order n ≥ 3, Zw is given by the first n moments of the size distribution only. On the other hand, Z still involves an infinite number of moments if the truncation is made after n ≥ 4 due to the presence of terms like i,j xixjσ 4/σij , i,j xixjσ 5/σ2ij , . . . . Therefore, if we want the consistency condition Z = Zw to be satisfied for any discrete or continuous polydisperse mixture, either the whole infinite series (45) needs to be considered or it must be truncated after n = 3. The latter is of course the simplest possibility and thus we make the approximation G(η, z) = G0(η) + G1(η)z + G2(η)z2 + G3(η)z3. (48) As a consequence, Z and Zw depend functionally on the size distribution of the mixture only through the first three moments (which is in the spirit of Rosenfeld’s Fundamental Measure Theory [48]). Using the approximation (48) in Eqs. (46) and (47) we are led to Z = G0 + η (3G0 + 2G1) + 2 (G1 + 2G2 + 2G3) , (49) Alternative Approaches to Hard-Sphere Liquids 15 Zw = G0 + 2 G1 + 4 (G2 + 2G3) . (50) Thus far, the dependence of both Z and Zw on the momentsM1,M2, andM3 is explicit and we only lack the packing-fraction dependence of G1, G2, and G3. From Eqs. (49) and (50) it follows that the difference between Z and Zw is given by Z−Zw = [3ηG0 − 2(1− η)G1]+2 [ηG1 − 2(1− η)G2 − 2(2− η)G3] . Therefore, Z = Zw for any dispersity provided that G1(η) = 2 (1− η)2 , (52) G2(η) = 4 (1− η)3 − 2− η G3(η), (53) where use has been made of the definition of G0, Eq. (17). To close the problem, we use the equal size limit given in Eq. (18), which yields G0+G1+G2+G3 = gs. After a little algebra we are led to G2(η) = (2− η)gs − 2 + η2/4 (1− η)2 , (54) G3(η) = (1− η) gSPTs − gs . (55) This completes the derivation of our improved approximation, which we will call “e3”, following the same criterion as the one used to call “e1” and “e2” to the approximations (20) and (34), respectively. In Eq. (55), gSPTs is the SPT contact value for a single fluid, whose expression appears in Table 1. From Eq. (55) it is obvious that the choice gs = g s makes our e3 approximation to become the e2 approximation, both reducing to the SPT for mixtures, Eq. (8). This means that the SPT is fully internally consistent with the require- ment Z = Zw, although it has the shortcoming of not being too accurate in the single component case. The e3 proposal, on the other hand, satisfies the condition Z = Zw and has the flexibility of accommodating any desired gs. For the sake of concreteness, let us write explicitly the contact values in the e3 aproximation: ge3ij (σij) = 2 (1− η)2 (2 − η)gs − 2 + η2/4 (1− η)2 + (1− η) gSPTs − gs , (56) 16 M. López de Haro, S. B. Yuste and A. Santos ge3wj = 1− η + (1− η)2 σj + 4 (2 − η)gs − 2 + η2/4 (1− η)2 +8(1− η) gSPTs − gs . (57) With the above results the compressibility factor may be finally written in terms of Zs as Ze3(η) = (1− η) (1− η)2 Zs(η)− . (58) A few comments are in order at this stage. First, from Eq. (49) we can observe that, for the class of approximations (48), the compressibility factor Z does not depend on the individual values of the coefficients G2 and G3, but only on their sum. As a consequence, two different approximations of the form (48) sharing the same density dependence of G1 and G2 + G3 also share the same virial EOS. For instance, if one makes the choice gs = g then ZePY3 = ZPY, even though g ij (σij) 6= gPYij (σij). Furthermore, if one makes the more accurate choice gs = g s , then ZeCS3 = ZBMCSL, but again geCS3ij (σij) 6= gBGHLLij (σij). The eCS3 contact values are geCS3ij (σij) = 2 (1− η)2 η2(1 + η) 4(1− η)3 4(1− η)2 , (59) geCS3wj = (1− η)2 η2(1 + η) (1− η)3 (1 − η)2 . (60) In Figs. 3 and 4 we display the performance of the contact values as given by Eqs. (59) and (60), respectively, by comparison with results of computer simulations for both discrete and polydisperse mixtures. In both figures we have also included the results that follow from the classical proposals as well as those of the eCS1 and eCS2 approximations. It is clear that for the wall- particle contact values the eCS3 approximation yields the best performance, while for the particle-particle contact values both the eCS2 and eCS3 are of comparable accuracy. A further feature to be pointed out is that the practical collapse on a common curve of the simulation data in Figs. 3 and 4 provide a posteriori support for the universality ansatz made in Eq. (16). As mentioned earlier, there exist extra consistency conditions (see for in- stance Ref. [12]) that one might use as well within our approach. Assuming Alternative Approaches to Hard-Sphere Liquids 17 0.0 0.5 1.0 1.5 2.0 eCS1 eCS2 eCS3 Fig. 3. Plot of the difference gij(σij)− gBGHLLij (σij) as a function of the parameter zij = (σiσj/σij)M2/M3 for hard spheres (d = 3) at a packing fraction η = 0.49. The symbols are simulation data for the single fluid (circle, Ref. [36]), three binary mixtures (squares, Ref. [49]) with σ2/σ1 = 0.3 and x1 = 0.0625, 0.125, and 0.25, and a ternary mixture (triangles, Ref. [50]) with σ2/σ1 = , σ3/σ1 = , and x1 = 0.1, x2 = 0.2. The lines are the PY approximation (– · · –), the SPT approximation (– · – ·), the eCS1 approximation (· · · ), the eCS2 approximation (– – –), and the eCS3 approximation (—). that the ansatz (16) still holds, some of these conditions are related to the derivatives of G with respect to z, namely ∂G(η, z) 2(1− η)2 , (61) ∂2G(η, z) gPYs − , (62) ∂3G(η, z) = 0. (63) Interestingly enough, as shown by Eq. (52), condition (61) is already satisfied by our e3 approximation without having to be imposed. On the other hand, condition (63) implies G3 = 0 in the e3 scheme and thus it is only satisfied if gs = g s , in which case we recover the SPT. Condition (62) is not fulfilled either by the SPT or by the e3 approximation (except for a particular ex- pression of gs which is otherwise not very accurate). Thus, fulfilling the extra 18 M. López de Haro, S. B. Yuste and A. Santos 0.0 0.5 1.0 1.5 2.0 2.5 eCS1 eCS2 eCS3 Fig. 4. Plot of the difference gwj − gBGHLLwj as a function of the parameter zwj/2 = σjM2/M3 for hard spheres (d = 3) at a packing fraction η = 0.4. The symbols are simulation data for a polydisperse mixture with a narrow top-hat distribution (open squares, Ref. [51]), a polydisperse mixture with a wide top-hat distribution (open circles, Ref. [51]), a polydisperse mixture with a Schulz distribution (open triangles, Ref. [51]), and a binary mixture (closed circles, Ref. [52]). The lines are the PY approximation (– · · –), the SPT approximation (– · – ·), the eCS1 approximation (· · · ), the eCS2 approximation (– – –), and the eCS3 approximation (—). conditions (62) and (63) with a free gs requires either considering a higher order polynomial in z (in which case the consistency condition Z = Zw can- not be satisfied for arbitrary mixtures, as discussed before) or not using the universality ansatz at all. In the first case, we have checked that a quartic or even a quintic polynomial does not improve matters, whereas giving up the universality assumption increases significantly the number of parameters to be determined and seems not to be adequate in view of the behavior observed in the simulation data. An additional comment has to do with the restriction to d = 3 in this subsection. As noted before, the approximation e1 reduces to the exact result (3) for d = 1. For d = 2, the approximation e2 already fulfills the condition Z = Zw and so there is no real need to go further in that case. Since we have needed the approximation e3 to satisfy Z = Zw for d = 3, it is tempting to speculate that a polynomial form for G(z) of degree d could be found to be consistent with the condition Z = Zw for d ≥ 4. However, a detailed analysis shows that this is not the case for an arbitrary mixture, since the Alternative Approaches to Hard-Sphere Liquids 19 number of conditions exceeds the number of unknowns, unless the universality assumption is partially relaxed. As a final comment, let us stress that, although the discussion in this section has referred, for the sake of simplicity, to discrete mixtures, all the dependence on the details of the composition occurs through a finite number of moments, so that the results remain meaningful even for continuous poly- disperse mixtures [53]. In that case, instead of a set of mole fractions {xi} and a set of diameters {σi}, one has to deal with a distribution function w(σ) such that w(σ)dσ is the fraction of particles with a diameter comprised between σ and σ + dσ. Therefore, the moments (1) are now defined as dσ σnw(σ), (64) and with such a change the results we have derived for discrete mixtures also hold for polydisperse systems. 2.3 Non-Additive Systems Non-additive hard-core mixtures, where the distance of closest approach be- tween particles of different species is no longer the arithmetic mean of the diameters of both particles, have received much less attention than additive mixtures, in spite of their in principle more versatility to deal with interesting aspects occurring in real systems (such as fluid-fluid phase separation) and of their potential use as reference systems in perturbation calculations on the thermodynamic and structural properties of, say, Lennard–Jones mixtures. Nevertheless, the study of non-additive systems goes back fifty years [54–56] and is still a rapidly developing and challenging problem. As mentioned in the paper by Ballone et al. [57], where the relevant references may be found, experimental work on alloys, aqueous electrolyte solutions, and molten salts suggests that hetero-coordination and homo- coordination may be interpreted in terms of excluded volume effects due to non-additivity of the repulsive part of the intermolecular potential. In particu- lar, positive non-additivity leads naturally to demixing in HS mixtures, so that some of the experimental findings of phase separation in the above mentioned (real) systems may be accounted for by using a model of a binary mixture of (positive) non-additive HS. On the other hand, negative non-additivity seems to account well for chemical short-range order in amorphous and liquid binary mixtures with preferred hetero-coordination [58]. Some Preliminary Definitions Let us consider an N -component mixture of non-additive HS in d dimensions. In this case, σij = (σi + σj)(1 + ∆ij), where ∆ij ≥ −1 is a symmetric matrix with zero diagonal elements (∆ii = 0) that characterizes the degree 20 M. López de Haro, S. B. Yuste and A. Santos of non-additivity of the interactions. If ∆ij > 0 the non-additivity character of the ij interaction is said to be positive, while it is negative if ∆ij < 0. In the case of a binary mixture (N = 2), the only non-additivity parameter is ∆ ≡ ∆12 = ∆21. The virial EOS (2) remains being valid in the non-additive case. The contact values gij(σij) can be expanded in a power series in density gij(σij) = 1 + vdρ xkck;ij + (vdρ) k,ℓ=1 xkxℓckℓ;ij +O(ρ3). (65) The coefficients ck;ij , ckℓ;ij , . . . are independent of the composition of the mix- ture, but they are in general complicated nonlinear functions of the diameters σij , σik, σjk, σkℓ, . . . . Insertion of the expansion (65) into Eq. (2) yields the virial expansion of Z, namely Z(ρ) = 1 + Bn(vdρ) = 1 + vdρ i,j=1 Bijxixj + (vdρ) i,j,k=1 Bijkxixjxk +(vdρ) i,j,k,ℓ=1 Bijkℓxixjxkxℓ +O(ρ4). (66) Note that, for further convenience, we have introduced the coefficients Bn ≡ −(n−1) d Bn, where Bn are the usual virial coefficients [cf. Eq. (25)]. The composition-independent second, third, and fourth (barred) virial coefficients are given by Bij = 2 d−1σdij , (67) Bijk = ck;ijσ ij + cj;ikσ ik + ci;jkσ , (68) Bijkℓ = ckℓ;ijσ ij + cjℓ;ikσ ik + ciℓ;jkσ jk + cjk,iℓσ iℓ + cik,jℓσ +cij;kℓσ . (69) A Simple Proposal for the Equation of State of d-Dimensional Non-Additive Mixtures Our goal now is to generalize the e1 proposal given by Eq. (20) to the non- additive case [59]. We will not try to extend the e2 and e3 proposals, Eqs. (34) and (56), because of two reasons. First, given the inherent complexity of non- additive systems, we want to keep the approach as simple as possible. Second, Alternative Approaches to Hard-Sphere Liquids 21 we are more interested in the EOS than in the contact values themselves and, as mentioned earlier, the e1 proposal provides excellent EOS, at least in the additive case, despite the simplicity of the corresponding contact values. As the simplest possible extension, we impose again the point particle and equal size consistency conditions, Eqs. (12) and (13), and thus keep in this case also the ansatz (16) and the linear structure of Eq. (19). However, instead of using Eq. (15), we determine the parameters zij as to reproduce Eq. (65) to first order in the density. The result is readily found to be [59] zij = )−1(∑ k xkck;ij . (70) Here b2 = 2 d−1 and b3 are the second and third virial coefficients for the single component fluid, as defined by Eq. (26). The proposal of Eq. (19) sup- plemented by Eq. (70) is, by construction, accurate for densities low enough as to justify the truncated approximation gij(σij) ≈ 1 + vdρ k xkck;ij . On the other hand, the limitations of this truncated expansion for moderate and large densities may be compensated by the use of gs. When Eqs. (16), (19), and (70) are inserted into Eq. (2) one gets Z(η) = 1 + b3MdB2 − b2B3 (b3 − b2)M2d + [Zs(η) − 1] B3 −MdB2 (b3 − b2)M2d . (71) Equation (71) is the sought generalization of Eq. (21) to non-additive hard- core systems. As in the additive case, the the density dependence in the EOS of the mixture is rather simple: Z(η)− 1 is expressed as a linear combination of η/(1 − η) and Zs(η) − 1, with coefficients such that the second and third virial coefficients are reproduced. Again, Eq. (71) is bound to be accurate for sufficiently low densities, while the limitations of the truncated expansion for moderate and large densities are compensated by the use of the EOS of the pure fluid. The exact second virial coefficient B2 is known from Eq. (67). In principle, one should use the exact coefficients ck;ij to compute B3. However, to the best of our knowledge they are only known for d ≤ 3. Since our objective is to have a proposal which is explicit for any d, we can make use of a reasonable approximation for them [59], as described below. An Approximate Proposal for ck;ij The values of the coefficients ck;ij are exactly known for d = 1 and d = 3 and from these results one may approximate them in d dimensions as [59] ck;ij = σ k;ij + σd−1k;ij σi;jkσj;ik, (72) where we have called 22 M. López de Haro, S. B. Yuste and A. Santos σk;ij ≡ σik + σjk − σij (73) and it is understood that σk;ij ≥ 0 for all sets ijk. Clearly, σi;ij = σi. For a binary mixture Eq. (72) yields c1;11 = (b3/b2)σ c2;11 = (2σ12 − σ1)d + (b3/b2 − 1)σ1(2σ12 − σ1)d−1, c1;12 = σ 1 + (b3/b2 − 1) (2σ12 − σ1)σd1/σ12. Of course, Eqs. (72) and (74) reduce to the exact results for d = 1 (b2 = b3 = 1) and for d = 3 (b2 = 4, b3 = 10). The quantities σk;ij may be given a simple geometrical interpretation. Assume that we have three spheres of species i, j, and k aligned in the sequence ikj. In such a case, the distance of closest approach between the centers of spheres i and j is σik + σjk. If the sphere of species k were not there, that distance would of course be σij . Therefore σk;ij as given by Eq. (73) represents a kind of effective diameter of sphere k, as seen from the point of view of the interaction between spheres i and j. Inserting Eq. (72) into Eq. (70), one gets zij = )−1(∑ k xkσ k xkσ k;ij σi;jkσj;ik Mdσij . (75) It can be easily checked that in the additive case (σk;ij → σk), Eq. (75) reduces to Eq. (15). Equations (72) and (74) are restricted to the situation σk;ij ≥ 0 for any choice of i, j, and k, i.e., 2σ12 ≥ max(σ1, σ2) in the binary case. This excludes the possibility of dealing with mixtures with extremely high negative non- additivity in which one sphere of species k might “fit in” between two spheres of species i and j in contact. Since for d = 3 and N = 2 the coefficients ck;ij are also known for such mixtures [60], we may extend our proposal to deal with these cases: c1;11 = (b3/b2)σ c2;11 = σ̂ 2 + (b3/b2 − 1)σ1σ̂d−12 , c1;12 = (2σ12 − σ̂2)d + (b3/b2 − 1) σ̂2σd1/σ12, where we have defined σ̂2 = max (2σ12 − σ1, 0) . (77) With such an extension, we recover the exact values of ck;ij for a binary mixture of hard spheres (d = 3), even if σ1 > 2σ12 or σ2 > 2σ12. The EOS (71) becomes explicit when B3 is obtained from Eq. (68) by using the approximation (72). The resulting virial coefficient is the exact one for d = 1 and d = 3. For hard disks (d = 2), it turns out that the approximate third virial coefficient is practically indistinguishable from the exact one [59]. Alternative Approaches to Hard-Sphere Liquids 23 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Fig. 5. Plot of the compressibility factor versus the non-additivity parameter ∆ for a symmetric binary mixture of non-additive hard spheres (d = 3) at η = π/30 and two different compositions. The solid lines are our proposal, Eq. (71), with Zs = Z while the dashed lines are Hamad’s proposal (Refs. [61–63]). The symbols are results from Monte Carlo simulations (Refs. [64,65]). When the approximate B3 is used, Eq. (71) reduces to Eq. (21) in the additive case. From the comparison with simulation results, both for the compressibility factor and higher order virial coefficients, we find that the EOS (71) does a good job for non-additive mixtures, thus representing a reasonable compro- mise between simplicity and accuracy, provided that Zs is accurate enough. This is illustrated in Fig. 5, where the proposal (71) with Zs = Z s and a similar proposal by Hamad [61–63] are compared with simulation data [64,65] for some three-dimensional symmetric mixtures. A more extensive compari- son [59] shows that Eq. (71) seems to work better (especially as the density is increased) in the case of positive non-additivities, at least for d = 1, d = 2, and d = 3, but its performance is also reasonably good in highly asymmetric mixtures, even for negative ∆. Of course the full assessment of this proposal is still pending since it involves many facets (non-additivity parameters, size ratios, density, and composition). Without this full assessment and given its rather satisfactory performance so far, going beyond the approximation given by Eq. (19) (taking similar steps to the ones described in Subsections 2.1 and 2.2 for additive systems) does not seem to be necessary at this stage, although it is in principle feasible. 24 M. López de Haro, S. B. Yuste and A. Santos 2.4 Demixing Demixing is a common phase transition in fluid mixtures usually originated on the asymmetry of the interactions (e.g., their strength and/or range) be- tween the different components in the mixture. In the case of athermal systems such as HS mixtures in d dimensions, if fluid-fluid separation occurs, it would represent a neat example of an entropy-driven phase transition, i.e., a phase separation based only on the size asymmetry of the components. The exis- tence of demixing in binary additive three dimensional HS mixtures has been studied theoretically since decades, and the issue is still controversial. In this subsection we will present our results following different but related routes that attempt to clarify some aspects of this problem. Binary Mixtures of Additive d-Dimensional Spheres (d = 3, d = 4 and d = 5) Now we look at the possible instability of a binary fluid mixture of HS of diameters σ1 and σ2 (σ1 > σ2) in d dimensions by looking at the Helmholtz free energy per unit volume, f , which is given by = −1 + xi ln Z(η′)− 1 , (78) where λi is the thermal de Broglie wavelength of species i. We locate the spinodals through the condition f11f22−f212 = 0, with fij ≡ ∂2f/∂ρi∂ρj . Due to the spinodal instability, the mixture separates into two phases of different composition. The coexistence conditions are determined through the equality of the pressure p and the two chemical potentials µ1 and µ2 in both phases (µi = ∂f/∂ρi), leading to binodal (or coexistence) curves. We begin with the case d = 3. It is well known that the BMCSL EOS, Eq. (11), does not lead to demixing. However, other EOS for HS mixtures have been shown to predict demixing [41, 66], including the EOS that is obtained by truncating the virial series after a certain number of terms [67, 68]. In particular, it turns out that both Z = ZeCS1, Eq. (40), and Z = ZeCS2, Eq. (41), lead to demixing for certain values of the parameter γ ≡ σ2/σ1 that measures the size asymmetry. The critical values of the pressure, the composition, and the packing fraction are presented in Table 2 for a few values of γ. As discussed earlier, the eCS1 EOS and, to a lesser extent, the eCS2 EOS are both in reasonably good agreement with the available simulation results for the compressibility factor [18, 36, 45] and lead to the exact second and third virial coefficients but differ in the predictions for Bn with n ≥ 4. The scatter in the values for the critical constants shown in Table 2 is evident and so there is no indication as to whether one should prefer one equation over the other in connection with this problem. Notice, for instance, that the eCS2 Alternative Approaches to Hard-Sphere Liquids 25 Table 2. Critical constants pcσ 1/kBT , x1c, and ηc for different γ-values as obtained from the two extended CS equations (40) and (41). eCS1 eCS2 γ pcσ 1/kBT x1c ηc pcσ 1/kBT x1c ηc 0.05 3599 0.0093 0.822 1096 0.0004 0.204 0.1 1307 0.0203 0.757 832.0 0.0008 0.290 0.2 653.4 0.0537 0.725 — — — 0.3 581.9 0.0998 0.738 — — — 0.4 663.4 0.1532 0.766 — — — does not predict demixing for γ ≥ 0.2, while both the values of the critical pressures and packing fractions for which it occurs according to the eCS1 EOS suggest that the transition might be metastable with respect to a fluid-solid transition. Now we turn to the cases d = 4 and d = 5. Here we use the extended Luban–Michels equation (eLM1) described in Subsection 2.1 [see Eq. (21) and Table 1]. As seen in Fig. 6, the location of the critical point tends to go down and to the right in the η2 vs η1 plane as γ decreases for d = 4 [69]. On the other hand, while it also tends to go down as γ decreases if d = 5, its behavior in the η2 vs η1 plane is rather more erratic in this case. Also, the value of the critical pressure pc (in units of kBT/σ 1) is not a monotonic function of γ; its minimum value lies between γ = 1/3 and γ = 1/2 when d = 4, and it is around γ = 3/5 for d = 5. This non-monotonic behavior is also observed for three-dimensional HS [66, 68]. It is conceivable that the demixing transition in binary mixtures of hard hyperspheres in four and five dimensions described above may be metastable with respect to a fluid-solid transition, as it may also be the case of 3D HS. In fact, the value of the pressure at the freezing transition for the single component fluid is [31] pfσ d/kBT ≃ 12.7 (d = 3), 11.5 (d = 4), and 12.2 (d = 5), i.e., pfσ d/kBT does not change appreciably with the dimensionality but is clearly very small in comparison with the critical pressures pcσ 1/kBT we obtain for the mixture; for instance, pcσ 1/kBT ≃ 600 (d = 3, γ = 3/10), 300 (d = 4, γ = 1/3) and 123 (d = 5, γ = 3/5). However, one should also bear in mind that, if the concentration x1 of the bigger spheres decreases, the value of the pressure at which the solid-fluid transition in the mixture occurs in 3D is also considerably increased with respect to pf [cf. Fig. 6 of Ref. [66]]. Thus, for concentrations x1 ≃ 0.01 corresponding to the critical point of the fluid-fluid transition, the maximum pressure of the fluid phase greatly exceeds pf. If a similar trend with composition also holds in 4D and 5D, and given that the critical pressures become smaller as the dimensionality d is increased, it is not clear whether the competition between the fluid-solid and the fluid-fluid transitions in these dimensionalities will always be won by the former. The point clearly deserves further investigation. 26 M. López de Haro, S. B. Yuste and A. Santos 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1400 2/ 1=3/4 4 1/k 0.0 0.1 0.2 0.3 0.4 5 1/k Fig. 6. Spinodal curves (upper panels: lines) and binodal curves (upper panels: open symbols; lower panels: lines) in a 4D system (left panels) and in a 5D system (right panels). The closed symbols are the critical consolute points. An interesting feature must be mentioned. There is a remarkable similarity between the binodal curves represented in the pσdi –η1 and in the µi–η1 planes [69]. By eliminating η1 as if it were a parameter, one can represent the binodal curves in a µi vs pσ i plane. Provided the origin of the chemical potentials is such as to make λi = σi, the binodals in the µi–pσ i plane practically collapse into a single curve (which is in fact almost a straight line) for each dimensionality (d = 3, d = 4, and d = 5) [69]. A closer analysis of this Alternative Approaches to Hard-Sphere Liquids 27 phenomenon shows, however, that it is mainly due to the influence on µi of terms which are quantitatively dominant but otherwise irrelevant to the coexistence conditions. Binary Mixtures of Non-Additive Hard Hyperspheres in the Limit of High Dimensionality Let us now consider a binary mixture of non-additive HS of diameters σ1 and σ2 in d dimensions. Thus in this case σ12 ≡ 12 (σ1 + σ2)(1 +∆) where as before ∆ may be either positive or negative. Further assume (something that will become exact in the limit d → ∞ [70]) that the EOS of the mixture is described by the second virial coefficient only, namely p = ρkBT [1 +B2(x1)ρ] , (79) where, according to Eq. (67), B2(x1) = vd2 1 + x 2 + 2x1x2σ . (80) The Helmholtz free energy per unit volume is given by f/ρkBT = −1 +∑2 i=1 xi ln +B2ρ, where Eq. (78) has been used. The Gibbs free energy per particle is g = (f + p)/ρ = xi ln + 2B2(x1)ρ, (81) where without loss of generality we have set kBT = 1. Given a size ratio γ, a value of ∆, and a dimensionality d, the consolute critical point (x1c, pc) is the solution to ∂2g/∂x21 ∂3g/∂x31 = 0, provided of course it exists. Then, one can get the critical density ρc from Eq. (79). We now introduce the scaled quantities [71] p̃ ≡ 2d−1vdd−2pσd1/kBT, u ≡ d−1B2ρ. (82) Consequently, Eqs. (79) and (81) can be rewritten as p̃ = u u+ d−1 /B̃2, (83) xi ln (xiΛi) + ln Adu/B̃2 + 2du, (84) where B̃2 ≡ B2/2d−1vdσd1 , Λi ≡ (λi/σ1)d, and Ad ≡ d/2d−1vd. Next we take the limit d → ∞ and assume that the volume ratio γ̃ ≡ γd is kept fixed and that there is a (slight) non-additivity ∆ = d−2∆̃ such that the scaled non- additivity parameter ∆̃ is also kept fixed in this limit. Thus, the second virial coefficient can be approximated by 28 M. López de Haro, S. B. Yuste and A. Santos B̃2 = B̃ 2 +B̃ −1+O(d−2), B̃(0)2 = x1 + x2γ̃ 2 = x1x2γ̃ 1/2J, J ≡ 1 (ln γ̃) + 2∆̃. (86) Let us remark that, in order to find a consolute critical point, it is essential to keep the term of order d−1 if ∆̃ ≤ 0. The EOS (83) can then be inverted to yield u = u(0)+u(1)d−1+O(d−2), u(0) = 2 , u (1) = −1 1− u(0) B̃ In turn, the Gibbs free energy (84) becomes g = g(0)d+ g(1) +O(d−1), g(0) = 2u(0), g(1) = i=1 xi ln (xiΛi) + ln (0)/B̃ + 2u(1), while the chemical potentials µ1 = g+x2 (∂g/∂x1)p and µ2 = g−x1 (∂g/∂x1)p are given by µi = µ i d+ µ i +O(d−1), µ 1 = 2p̃ 1 = ln Adx1Λ1 p̃/B̃ 2 + (x2/x1)(γ̃p̃) 1/2B̃ 2 /B̃ where µ2 is obtained from µ1 by the changes x1 ↔ x2, Λ1 → Λ2/γ̃, γ̃ → 1/γ̃, p̃→ p̃γ̃, B̃2 → B̃2/γ̃. The coordinates of the critical point are readily found to be x1c = γ̃3/4 1 + γ̃3/4 , p̃c = 1 + γ̃1/4 4γ̃J2 . (90) Note that x1c is independent of ∆̃. The coexistence curve, which has to be obtained numerically, follows from the conditions µ i (xA, p̃) = µ i (xB , p̃) (i = 1, 2) where x1 = xA and x1 = xB are the mole fractions of the co- existing phases. Once the critical consolute point has been identified in the pressure/concentration plane, we can obtain the critical density. The domi- nant behaviors of B̃2 and u at the critical point are 2 (x1c) = 1− γ̃1/4 + γ̃1/2 )2 , u 1 + γ̃1/4 1− γ̃1/4 + γ̃1/2 . (91) Hence, the critical density readily follows after substitution in the scaling relation given in Eq. (82). It is also convenient to consider the scaled version η̃ ≡ d−12dη of the packing fraction η = vdρσd1 (x1 + x2γ̃). At the critical point, it takes the nice expression Alternative Approaches to Hard-Sphere Liquids 29 0.00 0.04 0.08 0.12 0.16 p =-0.1 =0.1 =0.01 Fig. 7. Binodal curves in the planes ep vs x1 and eη vs x1 corresponding to eγ = 0.01 and e∆ = −0.1, e∆ = 0, and e∆ = 0.1. η̃c = γ̃1/8 + γ̃−1/8 . (92) The previous results clearly indicate that a demixing transition is possible not only for additive or positively non-additive mixtures but even for negative non-additivities. The only requirement is J > 0, i.e., ∆̃ > − 1 (ln γ̃) equivalently, ∆ > − 1 (ln γ) . Figure 7 shows the binodal curves corresponding to γ̃ = 0.01 and ∆̃ = −0.1 (negative non-additivity), ∆̃ = 0 (additivity), and ∆̃ = 0.1 (positive non-additivity). While the high dimensionality limit has allowed us to address the prob- lem in a mathematically simple and clear-cut way, the possibility of demixing with negative non-additivity is not an artifact of that limit. As said before, demixing is known to occur for positive non-additive binary mixtures of HS in three dimensions and there is compelling evidence on the existence of this phenomenon in the additive case, at least in the metastable fluid region. Even though in a three-dimensional mixture the EOS is certainly more complicated than Eq. (79) and the demixing transition that we have just discussed for neg- ative non-additivity is possibly metastable with respect to the freezing transi- tion, the main effects at work (namely the competition between depletion due to size asymmetry and hetero-coordination due to negative non-additivity) are also present. In fact, it is interesting to point out that Roth et al. [72], using the approximation of an effective single component fluid with pair inter- 30 M. López de Haro, S. B. Yuste and A. Santos actions to describe a binary mixture of non-additive 3D HS and employing an empirical rule based on the effective second virial coefficient, have also sug- gested that demixing is possible for small negative non-additivity and high size asymmetry. Our exact results lend support to this suggestion and con- firm that, in some cases, the limit d→ ∞ highlights features already present in real systems. 3 The Rational Function Approximation (RFA) Method for the Structure of Hard-Sphere Fluids The RDF g(r) and its close relative the (static) structure factor S(q) are the basic quantities used to discuss the structure of a single component fluid [1–4]. The latter quantity is defined as S(q) = 1 + ρh̃(q), (93) where h̃(q) = dr e−iq·rh(r) (94) is the Fourier transform of the total correlation function h(r) ≡ g(r)−1, i being the imaginary unit. An important related quantity is the direct correlation function c(r), which is defined in Fourier space through the Ornstein–Zernike (OZ) relation [1–4] c̃(q) = h̃(q) 1 + ρh̃(q) , (95) where c̃(q) is the Fourier transform of c(r) The usual approach to obtain g(r) is through one of the integral equa- tion theories, where the OZ equation is complemented by a closure relation between c(r) and h(r) [1]. However, apart from requiring in general hard nu- merical labor, a disappointing aspect is that the substitution of the (necessar- ily) approximate values of g(r) obtained from them in the (exact) statistical mechanical formulae may lead to the thermodynamic inconsistency problem. The two basic routes to obtain the EOS of a single component fluid of HS are the virial route, Eq. (14), and the compressibility route χs ≡ kBT = [1− ρc̃(0)]−1 = S(0) = 1 + 2ddησ−d dr rd−1h(r). (96) Thermodynamic consistency implies that χ−1s (η) = [ηZs(η)], (97) Alternative Approaches to Hard-Sphere Liquids 31 but, in general, this condition is not satisfied by an approximate RDF. In the case of a HS mixture, the virial route is given by Eq. (2), while the compressibility route is indicated below [cf. Eq. (145)]. In this section we describe the RFA method, which is an alternative to the integral equation approach and in particular leads by construction to thermodynamic consistency. 3.1 The Single Component HS Fluid We begin with the case of a single component fluid of HS of diameter σ. The following presentation is equivalent to the one given in Refs. [73, 74], where all details can be found, but more suitable than the former for direct generalization to the case of mixtures. The starting point will be the Laplace transform G(s) = dr e−srrg(r) (98) and the auxiliary function Ψ(s) defined through G(s) = [ρ+ esσΨ(s)] . (99) The choice of G(s) as the Laplace transform of rg(r) and the definition of Ψ(s) from Eq. (99) are suggested by the exact form of g(r) to first order in density [73]. Since g(r) = 0 for r < σ while g(σ+) = finite, one has g(r) = Θ(r − σ) g(σ+) + g′(σ+)(r − σ) + · · · , (100) where g′(r) ≡ dg(r)/dr. This property imposes a constraint on the large s behavior of G(s), namely eσssG(s) = σg(σ+) + g(σ+) + σg′(σ+) s−1 +O(s−2). (101) Therefore, lims→∞ e sσsG(s) = σg(σ+) = finite or, equivalently, s−2Ψ(s) = 2πσg(σ+) = finite. (102) On the other hand, according to Eq. (96) with d = 3, χs = 1− 24ησ−3 lim dr e−srr [g(r) − 1] = 1− 24ησ−3 lim G(s)− s−2 . (103) Since the (reduced) isothermal compressibility χs is also finite, one has∫∞ dr r2 [g(r) − 1] = finite, so that the weaker condition dr r [g(r)− 1] = lims→0[G(s)− s−2] = finite must hold. This in turn implies 32 M. López de Haro, S. B. Yuste and A. Santos Ψ(s) = −ρ+ ρσs− ρσ2s2+ ρσ3 + ρσ3 + σs4+O(s5). (104) First-Order Approximation (PY Solution) An interesting aspect to be remarked is that the minimal input we have just described on the physical requirements related to the structure and thermo- dynamics of the system is enough to determine the small and large s limits of Ψ(s), Eqs. (102) and (104), respectively. While infinite choices for Ψ(s) would comply with such limits, a particularly simple form is a rational function. In particular, the rational function having the least number of coefficients to be determined is Ψ(s) = E(0) + E(1)s+ E(2)s2 + E(3)s3 L(0) + L(1)s , (105) where one of the coefficients can be given an arbitrary non-zero value. We choose E(3) = 1. With such a choice and in view of Eq. (104), one finds E(0) = −ρL(0), E(1) = −ρ(L(1) − σL(0)), E(2) = ρ(σL(1) − 1 σ2L(0)), and L(0) = 2π 1 + 2η (1− η)2 , (106) L(1) = 2πσ 1 + η/2 (1− η)2 . (107) Upon substitution of these results into Eqs. (99) and (105), we get G(s) = L(0) + L(1)s ϕ2(σs)σ3L(0) + ϕ1(σs)σ2L(1) ] , (108) where ϕn(x) ≡ x−(n+1) (−x)m − e−x . (109) In particular, ϕ0(x) = 1− e−x , ϕ1(x) = 1− x− e−x , ϕ2(x) = 1− x+ x2/2− e−x (110) Note that limx→0 ϕn(x) = (−1)n/(n+ 1)!. It is remarkable that Eq. (108), which has been derived here as the sim- plest rational form for Ψ(s) complying with the requirements (102) and (104), coincides with the solution to the PY closure, c(r) = 0 for r > σ, of the OZ equation [29]. Application of Eq. (102) yields the PY contact value gPYs and compressibility factor ZPYs shown in Table 1. Analogously, Eq. (103) yields χPYs = (1− η)4 (1 + 2η)2 . (111) It can be easily checked that the thermodynamic relation (97) is not satisfied by the PY theory. Alternative Approaches to Hard-Sphere Liquids 33 Second-Order Approximation In the spirit of the RFA, the simplest extension of the rational approximation (105) involves two new terms, namely αs4 in the numerator and L(2)s2 in the denominator, both of them necessary in order to satisfy Eq. (102). Such an addition leads to Ψ(s) = E(0) + E(1)s+ E(2)s2 + E(3)s3 + αs4 L(0) + L(1)s+ L(2)s2 . (112) Applying Eq. (104), it is possible to express E(0), E(1), E(2), E(3), L(0), and L(1) in terms of α and L(2). This leads to G(s) = L(0) + L(1)s+ L(2)s2 1 + αs− ρ ϕ2(σs)σ3L(0) + ϕ1(σs)σ2L(1) + ϕ0(σs)σL(2) (113) where L(0) = 2π 1 + 2η (1− η)2 , (114) L(1) = 2πσ 1 + 1 (1 − η)2 1 + 2η α− 3ηL . (115) Thus far, irrespective of the values of the coefficients L(2) and α, the condi- tions lims→∞ e sσsG(s) = finite and lims→0[G(s) − s−2] = finite are satisfied. Of course, if L(2) = α = 0, one recovers the PY approximation. More gen- erally, we may determine these coefficients by prescribing the compressibility factor Zs (or equivalently the contact value gs) and then, in order to ensure thermodynamic consistency, compute from it the isothermal compressibility χs by means of Eq. (97). From Eqs. (102) and (103) one gets L(2) = 2πασgs, (116) 1− 12η 1 + 2 αL(2) . (117) Clearly, upon substitution of Eqs. (114) and (116) into Eq. (117) a quadratic algebraic equation for α is obtained. The physical root is α = − 12η(1 + 2η)E4 (1− η)2 + 36η [1 + η − Zs(1− η)]E4 , (118) where Zs − 13 Zs − 13 Zs − ZPYs )]1/2} . (119) The other root must be discarded because it corresponds to a negative value of α, which, according to Eq. (116), yields a negative value of L(2). This would 34 M. López de Haro, S. B. Yuste and A. Santos imply the existence of a positive real value of s at which G(s) = 0 [73, 74], which is not compatible with a positive definite RDF. However, according to the form of Eq. (119) it may well happen that, once Zs has been chosen, there exists a certain packing fraction ηg above which α is no longer positive. This may be interpreted as an indication that, at the packing fraction ηg where α vanishes, the system ceases to be a fluid and a glass transition in the HS fluid occurs [74–76]. Expanding (113) in powers of s and using Eq. (101) one can obtain the derivatives of the RDF at r = σ+ [77]. In particular, the first derivative is g′(σ+) = L(1) − L(2) , (120) which may have some use in connection with perturbation theory [15]. It is worthwhile to point out that the structure implied by Eq. (113) coin- cides in this single component case with the solution of the Generalized Mean Spherical Approximation (GMSA) [78], where the OZ relation is solved under the ansatz that the direct correlation function has a Yukawa form outside the core. For a given Zs, once G(s) has been determined, inverse Laplace trans- formation yields rg(r). First, note that Eq. (99) can be formally rewritten G(s) = − ρn−1 [−Ψ(s)]−n e−nsσ. (121) Thus, the RDF is then given by g (r) = ρn−1ψn (r − nσ)Θ (r − nσ) , (122) with Θ (x) denoting the Heaviside step function and ψn (r) = −L−1 s [−Ψ (s)]−n , (123) L−1 denoting the inverse Laplace transform. Explicitly, using the residue the- orem, ψn (r) = − (n−m)!(m− 1)! rn−m, (124) where a(i)mn = lim s [−Ψ (s) /(s− si)]−n , (125) si (i = 1, . . . , 4) being the poles of 1/Ψ(s), i.e., the roots of E (0) + E(1)s + E(2)s2 + E(3)s3 + αs4 = 0. Explicit expressions of g(r) up to the second coordination shell σ ≤ r ≤ 3σ can be found in Ref. [79]. Alternative Approaches to Hard-Sphere Liquids 35 On the other hand, the static structure factor S(q) [cf. Eq. (93)] and the Fourier transform h̃(q) may be related to G(s) by noting that h̃(q) = dr r sin(qr)h(r) = −2π G(s)−G(−s) . (126) Therefore, the basic structural quantities of the single component HS fluid, namely the RDF and the static structure factor, may be analytically deter- mined within the RFA method once the compressibility factor Zs, or equiva- lently the contact value gs, is specified. In Fig. 8 we compare simulation data of g(r) for a density ρσ3 = 0.9 [80] with the RFA prediction and a recent approach by Trokhymchuk et al. [81], where Zs = Z s [cf. Table 1] and the associated compressibility χCSs = (1− η)4 1 + 4η + 4η2 − 4η3 + η4 (127) are taken in both cases. Both theories are rather accurate, but the RFA cap- tures better the maxima and minima of g(r) [82]. It is also possible to obtain within the RFA method the direct correlation function c(r). Using Eqs. (95) and (126), and applying the residue theorem, one gets, after some algebra, 1 2 3 4 1 2 3 4 Fig. 8. Radial distribution function of a single component HS fluid for ρσ3 = 0.9. The solid lines represent simulation data [80]. The dashed lines represent the results of the approach of Ref. [81], while the dotted lines refer to those of the RFA method. The inset shows the oscillations of g(r) in more detail. 36 M. López de Haro, S. B. Yuste and A. Santos c(r) = +K0 +K1r +K3r Θ(1 − r) +K (128) where 12αηL(2)/π + 1− 12α(1 + 2α)η/(1− η), (129) 4α2(1 − η)4κ6 2 [1 + 2(1 + 3α)η]± [2 + η + 2α(1 + 2η)]κ +(1− η) κ2 − η (12 + (κ± 6)κ) L(2)/π 12η [1 + 2(1 + 3α)η] ±6η [3η − 2α(1− 4η)]κ− 6η(1 + 2α)(1− η)κ2 − (1 − η)2κ3(ακ∓ 1) +6η(1− η) κ2 − η (12 + (κ± 6)κ) L(2)/π , (130) K−1 = − κ +K−e −κ +K0 +K1 +K3 , (131) K0 = − 1 + 2 (1 + 3α) η − 6η (1− η)L(2)/π ακ (1− η)2 , (132) 2α2κ2 (1− η)4 [2 + η + 2α(1 + 2η)] 2 − 4 (1− η) [1 + η ×(7 + η + 6α (2 + η))]L(2)/π + 12η (2 + η) (1− η)2L(2) ,(133) K0, (134) K = − (K+ +K− +K−1) . (135) In Eqs. (129)–(135) we have taken σ = 1 as the length unit. Note that Eq. (135) guarantees that c(0) = finite, while Eq. (131) yields c(σ+) − c(σ−) = L(2)/2πα = g(σ+). The latter equation proves the continuity of the indirect correlation function γ(r) ≡ h(r) − c(r) at r = σ. With the above results, Eqs. (122) and (128), one may immediately write the function γ(r). Finally, we note that the bridge function B(r) is linked to γ(r) and to the cavity (or background) function y(r) ≡ eφ(r)/kBT g(r), where φ(r) is the interaction potential, through B(r) = ln y(r) − γ(r), (136) and so, within the RFA method, the bridge function is also completely speci- fied analytically for r > σ once Zs is prescribed. If one wants to have B(r) also for 0 ≤ r ≤ σ, then an expression for the cavity function is required in that region. Here we propose such an expression using a limited number of constraints. First, since the cavity function and its first derivative are continuous at r = σ, we have Alternative Approaches to Hard-Sphere Liquids 37 0.0 0.2 0.4 0.6 0.8 1.0 1000 3=0.3 3=0.5 3=0.7 Fig. 9. Cavity function of a single component HS fluid in the overlap region for ρσ3 = 0.3, 0.5, and 0.7. The solid lines represent our proposal (140) with Zs = Z while the symbols represent Monte Carlo simulation results [84]. y(1) = gs, y′(1) − 1, (137) where Eqs. (116) and (120) have been used and again σ = 1 has been taken. Next, we consider the following exact zero-separation theorems [83]: ln y(0) = Zs(η)− 1 + ′)− 1 , (138) y′(0) = −6ηy(1). (139) The four conditions (137)–(139) can be enforced by assuming a cubic poly- nomial form for ln y(r) inside the core, namely y(r) = exp Y0 + Y1r + Y2r 2 + Y3r , (0 ≤ r ≤ 1), (140) where Y0 = Zs(η)− 1 + ′)− 1 , (141) Y1 = −6ηy(1), (142) 38 M. López de Haro, S. B. Yuste and A. Santos 0 2 4 6 8 10 RFA, =0.3 RFA, =0.49 Fig. 10. Parametric plot of the bridge function B(r) versus the indirect correlation function γ(r). The dashed line refers to the RFA for η = 0.3, while the solid line refers to the RFA for η = 0.49. In each case, the branch of the curve to the right of the circle corresponds to r ≤ 1, while that to the left corresponds to r ≥ 1. For comparison, the PY closure B(r) = ln[1 + γ(r)]− γ(r) is also plotted (dash-dotted line). Y2 = 3 ln y(1)− y′(1) − 3Y0 − 2Y1, (143) Y3 = −2 ln y(1) + y′(1) + 2Y0 + Y1. (144) The proposal (140) is compared with available Monte Carlo data [84] in Fig. 9, where an excellent agreement can be observed. Once the cavity function y(r) provided by the RFA method is comple- mented by (140), the bridge function B(r) can be obtained at any distance. Figure 10 presents a parametric plot of the bridge function versus the indirect correlation function as given by the RFA method for two different packing fractions, as well as the result associated with the PY closure. The fact that one gets a smooth curve means that within the RFA the oscillations in γ(r) are highly correlated to those of B(r). Further, the effective closure relation in the RFA turns out to be density dependent, in contrast with what occurs for the PY theory. Note that the absolute value |B(r)| for a given value of γ(r) is smaller in the RFA than the PY value and that the RFA and PY curves become paradoxically closer for larger densities. Since the PY theory is known Alternative Approaches to Hard-Sphere Liquids 39 to yield rather poor values of the cavity function inside the core [85, 86], it seems likely that the present differences may represent yet another manifes- tation of the superiority of the RFA method, a point that certainly deserves to be further explored. 3.2 The Multicomponent HS Fluid The method outlined in the preceding subsection will be now extended to an N -component mixture of additive HS. Note that in a multicomponent system the isothermal compressibility χ is given by χ−1 = T,{xj} T,{xj} = 1− ρ i,j=1 xixj c̃ij(0), (145) where c̃ij(q) is the Fourier transform of the direct correlation function cij(r), which is defined by the OZ equation h̃ij(q) = c̃ij(q) + ρkh̃ik(q)c̃kj(q), (146) where hij(r) ≡ gij(r) − 1. Equations (145) and (146) are the multicompo- nent extensions of Eqs. (96) and (95), respectively. Introducing the quantities ĥij(q) ≡ ρiρj h̃ij(q) and ĉij(q) ≡ ρiρj c̃ij(q), the OZ relation (146) be- comes, in matrix notation, ĉ(q) = ĥ(q) · [I+ ĥ(q)]−1, (147) where I is the N ×N identity matrix. Thus, Eq. (145) can be rewritten as χ−1 = i,j=1 xixj [δij − ĉij(0)] = i,j=1 I+ ĥ(0) . (148) Similarly to what we did in the single component case, we introduce the Laplace transforms of rgij(r): Gij(s) = dr e−srrgij(r). (149) The counterparts of Eqs. (100) and (101) are gij(r) = Θ(r − σij) gij(σ ij) + g ij)(r − σij) + · · · , (150) 40 M. López de Haro, S. B. Yuste and A. Santos eσijssGij(s) = σijgij(σ ij) + gij(σ ij) + σijg s−1 +O(s−2). (151) Moreover, the condition of a finite compressibility implies that h̃ij(0) = finite. As a consequence, for small s, s2Gij(s) = 1 +H 3 + · · · (152) with H ij = finite and H ij = −h̃ij(0)/4π = finite, where dr (−r)nrhij(r). (153) We are now in the position to generalize the approximation (113) to the N -component case [87]. While such a generalization may be approached in a variety of ways, two motivations are apparent. On the one hand, we want to recover the PY result [8] as a particular case in much the same fashion as in the single component system. On the other hand, we want to maintain the development as simple as possible. Taking all of this into account, we propose Gij(s) = e−σijs L(s) · [(1 + αs)I− A(s)]−1 , (154) where L(s) and A(s) are the matrices Lij(s) = L ij + L ij s+ L 2, (155) Aij(s) = ρi ϕ2(σis)σ ij + ϕ1(σis)σ ij + ϕ0(σis)σiL , (156) the functions ϕn(x) being defined by Eq. (109). We note that, by construc- tion, Eq. (154) complies with the requirement lims→∞ e σijssGij(s) = finite. Further, in view of Eq. (152), the coefficients of s0 and s in the power series expansion of s2Gij(s) must be 1 and 0, respectively. This yields 2N 2 condi- tions that allow us to express L(0) and L(1) in terms of L(2) and α. The solution is [87] ij = ϑ1 + ϑ2σj + 2ϑ2α− ϑ1 ρkσkL kj , (157) ij = ϑ1σij + ϑ2σiσj + (ϑ1 + ϑ2σi)α− ρkσkL kj , (158) where ϑ1 ≡ 2π/(1− η) and ϑ2 ≡ 6π(M2/M3)η/(1− η)2. In parallel with the development of the single component case, L(2) and α can be chosen arbitrarily. Again, the choice L ij = α = 0 gives the PY solution [8, 88]. Since we want to go beyond this approximation, we will determine those coefficients by taking prescribed values for gij(σij), which in turn, via Eq. (2), give the EOS of the mixture. This also leads to the required value of Alternative Approaches to Hard-Sphere Liquids 41 χ−1 = ∂(ρZ)/∂ρ, thus making the theory thermodynamically consistent. In particular, according to Eq. (151), ij = 2πασijgij(σ ij). (159) The condition related to χ is more involved. Making use of Eq. (152), one can get h̃ij(0) = −4πH(1)ij in terms of L(2) and α and then insert it into Eq. (148). Finally, elimination of L ij in favor of α from Eq. (159) produces an algebraic equation of degree 2N , whose physical root is determined by the requirement that Gij(s) is positive definite for positive real s. It turns out that the physical solution corresponds to the smallest of the real roots. Once α is known, upon substitution into Eqs. (154), (157), (158), and (159), the scheme is complete. Also, using Eq. (151), one can easily derive the result g′ij(σ ij) = 2πασij ij − L . (160) It is straightforward to check that the results of the preceding subsection are recovered by setting σi = σ, regardless of the values of the mole factions. Once Gij(s) has been determined, inverse Laplace transformation directly yields rgij(r). Although in principle this can be done analytically, it is more practical to use one of the efficient methods discussed by Abate and Whitt [89] to numerically invert Laplace transforms [90]. In Fig. 11 we present a comparison between the results of the RFA method with the PY theory and simulation data [50] for the RDF of a ternary mixture. In the case of the RFA, we have used the eCS2 contact values and the cor- responding isothermal compressibility. The improvement of the RFA over the PY prediction, particularly in the region near contact, is noticeable. Although the RFA accounts nicely for the observed oscillations, it seems to somewhat overestimate the depth of the first minimum. Explicit knowledge of Gij(s) also allows us to determine the Fourier trans- form h̃ij(q) through the relation h̃ij(q) = −2π Gij(s)−Gij(−s) . (161) The structure factor Sij(q) may be expressed in terms of h̃ij(q) as [4] Sij(q) = xiδij + ρxixj h̃ij(q). (162) In the particular case of a binary mixture, rather than the individual structure factors Sij(q), it is some combination of them which may be easily associated with fluctuations of the thermodynamic variables [91, 92]. Specifically, the quantities [4] Snn(q) = S11(q) + S22(q) + 2S12(q), (163) 42 M. López de Haro, S. B. Yuste and A. Santos 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 10.750.50.250 10.750.50.250 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 Fig. 11. Radial distribution functions gij(r) for a ternary mixture with diameters σ1 = 1, σ2 = 2, and σ3 = 3 at a packing fraction η = 0.49 with mole fractions x1 = 0.7, x2 = 0.2, and x3 = 0.1. The circles are simulation results [50], the solid lines are the RFA predictions, and the dotted lines are the PY predictions. Snc(q) = x2S11(q)− x1S22(q) + (x2 − x1)S12(q), (164) Scc(q) = x 2S11(q) + x 1S22(q)− 2x1x2S12(q) (165) are sometimes required. After replacement of ĥij(q) = ρiρj h̃ij(q) in Eq. (147), one easily gets c̃ij(q). Subsequent inverse Fourier transformation yields cij(r). The result gives cij(r) for r > σij as the superposition of N Yukawas [93], namely Alternative Approaches to Hard-Sphere Liquids 43 cij(r) = e−κℓr , (166) where q = ±iκℓ with ℓ = 1, . . . , N are the zeros of det I+ ĥ(q) and the amplitudes K ij are obtained by applying the residue theorem as q→iκℓ c̃ij(q)(q − iκℓ). (167) The indirect correlation functions γij(r) ≡ hij(r) − cij(r) readily follow from the previous results for the RDF and direct correlation functions. Finally, in this case the bridge functions Bij(r) for r > σij are linked to gij(r) and cij(r) through Bij(r) = ln gij(r) − γij(r) (168) and so once more we have a full set of analytical results for the structural properties of a multicomponent fluid mixture of HS once the contact values gij(σij) are specified. 4 Other Related Systems The philosophy behind the RFA method to derive the structural properties of three-dimensional HS systems can be adapted to deal with other related systems. The main common features of the RFA can be summarized as follows. First, one chooses to represent the RDF in Laplace space. Next, using as a guide the low-density form of the Laplace transform, an auxiliary function is defined which is approximated by a rational or a rational-like form. Finally, the coefficients are determined by imposing some basic consistency conditions. In this section we consider the cases of sticky-hard-sphere, square-well, and hard-disk fluids. In the two former cases the RFA program is followed quite literally, while in the latter case it is done more indirectly through the RFA method as applied to hard rods (d = 1) and hard spheres (d = 3). 4.1 Sticky Hard Spheres The sticky-hard-sphere (SHS) fluid model has received a lot of attention since it was first introduced by Baxter in 1968 [94] and later extended to multi- component mixtures by Perram and Smith [95] and, independently, by Bar- boy [96]. In this model, the molecular interaction may be defined via square- well (SW) potentials of infinite depth and vanishing width, thus embodying the two essential characteristics of real molecular interactions, namely a harsh repulsion and an attractive part. In spite of their known shortcomings [97], an important feature of SHS systems is that they allow for an exact solution of the 44 M. López de Haro, S. B. Yuste and A. Santos OZ equation in the PY approximation [94,95]. Furthermore, they are thought to be appropriate for describing structural properties of colloidal systems, mi- celles, and microemulsions, as well as some aspects of gas-liquid equilibrium, ionic fluids and mixtures, solvent mediated forces, adsorption phenomena, polydisperse systems, and fluids containing chainlike molecules [98–102]. Let us consider an N -component mixture of spherical particles interacting according to the SW potential φij(r) = ∞, r < σij , −ǫij , σij < r < Rij , 0, r > Rij . (169) As in the case of additive HS, σij = (σi + σj)/2 is the distance between the centers of a sphere of species i and a sphere of species j at contact. In addition, ǫij is the well depth and Rij − σij indicates the well width. We now take the SHS limit [94], namely Rij → σij , ǫij → ∞, τij ≡ Rij − σij e−ǫij/kBT = finite, (170) where the τij are monotonically increasing functions of the temperature T and their inverses measure the degree of “adhesiveness” of the interacting spheres i and j. Even without strictly taking the mathematical limits (170), short-range SW fluids can be well described in practice by the SHS model [103]. The virial EOS for the SHS mixture is given by Z = 1 + i,j=1 dr ryij(r) e−φij(r)/kBT = 1 + i,j=1 xixjσ ijyij(σij) 12τij y′ij(σij) yij(σij) , (171) where yij(r) ≡ gij(r)eφij(r)/kBT is the cavity function and y′ij(r) = dyij(r)/dr. Since yij(r) must be continuous, it follows that gij(r) = yij(r) Θ(r − σij) + 12τij δ(r − σij) . (172) The case of a HS system is recovered by taking the limit of vanishing adhe- siveness τ−1ij → 0, in which case Eq. (171) reduces to the three-dimensional version of Eq. (2). On the other hand, the compressibility EOS, Eq. (145), is valid for any interaction potential, including SHS. As in the case of HS, it is convenient to define the Laplace transform (149). The condition yij(σij) = finite translates into the following large s behavior of Gij(s): eσijsGij(s) = σ ijyij(σij) 12τij + σ−1ij s +O(s−2), (173) Alternative Approaches to Hard-Sphere Liquids 45 which differs from (151): while eσijsGij(s) ∼ s−1 for HS, eσijsGij(s) ∼ s0 for SHS. However, the small s behavior is still given by Eq. (152), as a consequence of the condition χ−1 = finite. The RFA proposal for SHS mixtures [104] keeps the form (154), except that now Lij(s) = L ij + L ij s+ L 2 + L 3, (174) Aij(s) = ρi ϕ2(σis)σ ij + ϕ1(σis)σ ij + ϕ0(σis)σiL ij − e −σisL (175) instead of Eqs. (155) and (156). By construction, Eqs. (154), (174), and (175) comply with the requirement lims→∞ e σijsGij(s) = finite. Further, in view of Eq. (152), the coefficients of s0 and s in the power series expansion of s2Gij(s) must be 1 and 0, respectively. This yields 2N 2 conditions that allow us to express L(0) and L(1) in terms of L(2), L(3), and α as [104] ij = ϑ1+ϑ2σj+2ϑ2α−ϑ1 kj − L ρkσkL kj , (176) ij = ϑ1σij + ϑ2σiσj + (ϑ1 + ϑ2σi)α− kj − L (ϑ1 + ϑ2σi) ρkσkL kj , (177) where ϑ1 and ϑ2 are defined below Eq. (158). We have the freedom to choose (3) and α, but L(2) is constrained by the condition (173), i.e., the ratio between the first and second terms in the expansion of eσijsGij(s) for large s must be exactly equal to σij/12τij. First-Order Approximation (PY Solution) The simplest approximation consists of making α = 0. In view of the condition eσijsGij(s) ∼ s0 for large s, this implies L(3)ij = 0. In that case, the large s behavior that follows from Eq. (154) is 2πeσijsGij(s) = L (2) · D s−1 +O(s−2), (178) where Dij ≡ ρi σ2i L ij − σiL ij + L . (179) Comparison with Eq. (173) yields yij(σij) = πσ2ij ij , (180) 46 M. López de Haro, S. B. Yuste and A. Santos 12τijL ik Dkj . (181) Taking into account Eqs. (176) and (177) (with L ij = L ji and of course also with α = 0 and L(3) = 0), Eq. (181) becomes a closed equation for L(2): 12τijL = ϑ1σij+ ϑ2σiσj− ki σj + L kj σi (182) The physical root L(2) of Eq. (182) is the one vanishing in the HS limit τij → ∞. Once known, Eq. (180) gives the contact values. This first-order approximation obtained from the RFA method turns out to coincide with the exact solution of the PY theory for SHS [95]. Second-Order Approximation As in the case of HS mixtures, a more flexible proposal is obtained by keeping α (and, consequently, L ij ) different from zero. In that case, instead of Eq. (178), one has 2πeσijsGij(s) = +O(s−2). (183) This implies πσ2ij αyij(σij), (184) 12τijL . (185) If we fix yij(σij), Eqs. (176), (177), (184), and (185) allow one to express L (1), L(2), and L(3) as linear functions of α. Thus, only the scalar parameter α remains to be fixed, analogously to what happens in the HS case. As done in the latter case, one possibility is to choose α in order to reproduce the isothermal compressibility χ given by Eq. (148). To do so, one needs to find the coefficients H ij appearing in Eq. (152). The result is [104] (0) = C(0) · I− A(0) , (186) (1) = C(1) · I− A(0) , (187) where Alternative Approaches to Hard-Sphere Liquids 47 αδkj −A(1)kj δkj −A(0)kj (188) σ2ik +H αδkj −A(1)kj σ3ik + σikH δkj −A(0)kj , (189) ij = (−1) σn+3i (n+ 3)! σn+2i (n+ 2)! σn+1i (n+ 1)! (190) Equation (187) gives H(1) in terms of α: H ij = Pij(α)/[Q(α)] 2, where Pij(α) denotes a polynomial in α of degree 2N and Q(α) denotes a polynomial of degree N . It turns out then that, seen as a function of α, χ is the ratio of two polynomials of degree 2N . Given a value of χ, one may solve for α. The physical solution, which has to fulfill the requirement that Gij(s) is positive definite for positive real s, corresponds to the smallest positive real root. Once α is known, the scheme is complete: Eq. (184) gives L(3), then L(2) is obtained from Eq. (185), and finally L(1) and L(0) are given by Eqs. (176) and (177), respectively. Explicit knowledge of Gij(s) through Eqs. (154), (174), and (175) allows one to determine the Fourier transform h̃ij(q) and the struc- ture factor Sij(q) through Eqs. (161) and (162), respectively. Finally, inverse Laplace transformation of Gij(s) yields gij(r) [90]. Single Component SHS Fluids The special case of single component SHS fluids [105, 106] can be obtained from the multicomponent one by taking σij = σ and τij = τ . Thus, the Laplace transform of rg(r) in the RFA is G(s) = L(0) + L(1)s+ L(2)s2 + L(3)s3 1 + αs− ρ ϕ2(s)L(0) + ϕ1(s)L(1) + ϕ0(s)L(2) − e−sL(3) (191) where we have taken σ = 1. Equations (176) and (177) become L(0) = 2π 1 + 2η (1− η)2 − L(2) (1 − η)2 (1 − 4η)L(3), (192) L(1) = 2π 1 + 1 (1− η)2 1 + 2η α− 3ηL(2) − 18η (1− η)2 L(3). (193) The choice α = L(3) = 0 makes Eq. (191) coincide with the exact solution to the PY approximation for SHS [94], where L(2) is the physical root (i.e., the one vanishing in the limit τ → ∞) of the quadratic equation [see Eq. (182)] 48 M. López de Haro, S. B. Yuste and A. Santos 12τL(2) = 2π 1 + 2η (1− η)2 − 12η L(2) + ηL(2) . (194) We can go beyond the PY approximation by prescribing a contact value y(1), so that, according to Eqs. (184) and (185), L(3) = y(1), (195) L(2) = 12τ + L(3). (196) By prescribing the isothermal compressibility χ, the parameter α can be ob- tained as the physical solution (namely, the one remaining finite in the limit τ → ∞) of a quadratic equation [106]. Thus, given an EOS for the SHS fluid, one can get the thermodynamically consistent values of y(1) and χ and determine from them all the coefficients appearing in Eq. (191). Figure 12 shows the cavity function for η = 0.164 and τ = 0.13 as obtained from Monte Carlo simulations [101] and as predicted by the PY and RFA theories, the latter making use of the EOS recently proposed by Miller and Frenkel [102]. It can be observed that the RFA is not only more accurate than the PY approximation near r = 1 but also near r = 2. On the other hand, none of these two approximations account for the singularities (delta-peaks and/or discontinuities) of y(r) at r = 8/3, 5/3, 3, 2, . . . [100, 101]. 1.0 1.5 2.0 2.5 =0.164, =0.13 MC Fig. 12. Cavity function of a single component SHS fluid for η = 0.164 and τ = 0.13. The solid line represents simulation data [101]. The dotted and dashed lines represent the PY and RFA approaches, respectively. Alternative Approaches to Hard-Sphere Liquids 49 4.2 Single Component Square-Well Fluids Now we consider again the SW interaction potential (169) but for a single fluid, i.e., σij = σ, ǫij = ǫ, Rij = R. Since no exact solution of the PY theory for the SW potential is known, the application of the RFA method is more challenging in this case than for HS and SHS fluids. As in the cases of HS and SHS, the key quantity is the Laplace transform of rg(r) defined by Eq. (98). It is again convenient to introduce the auxiliary function Ψ(s) through Eq. (99). As before, the conditions g(r) = finite and χ = finite imply Eqs. (102) and (104), respectively. However, the important difference between HS and SHS fluids is that in the latter case G(s) must reflect the fact that g(r) is discontinuous at r = R as a consequence of the discontinuity of the potential φ(r) and the continuity of the cavity function y(r). This implies that G(s), and hence Ψ(s), must contain the exponential term e−(R−σ)s. This manifests itself in the low-density limit, where the con- dition limρ→0 y(r) = 1 yields Ψ(s) = (1 + s)− e−(R−1)s(e1/T∗ − 1)(1 +Rs) , (197) where T ∗ ≡ kBT/ǫ and we have taken σ = 1. In the spirit of the RFA method, the simplest form that complies with Eq. (102) and is consistent with Eq. (197) is [107] Ψ(s) = −12η + E1s+ E2s2 + E3s3 1 +Q0 +Q1s− e−(R−1)s (Q0 +Q2s) , (198) where the coefficients Q0, Q1, Q2, E1, E2, and E3 are functions of η, T ∗, and R. The condition (104) allows one to express the parameters Q1, E1, E2, and E3 as linear functions of Q0 and Q2 [107, 108]: 1 + 2η + 2η(R3 − 1)Q2 − (R − 1)2(R2 + 2R+ 3)Q0 +Q2 − (R − 1)Q0, (199) 1 + 2η 3− 4(R3 − 1)Q2 + (R− 1)2(R2 + 2R+ 3)Q0 , (200) 1 + 2η {1− η − 2(R− 1) [1− 2ηR(R+ 1)]Q2 +(R− 1)2 (1− η(R + 1)2 , (201) 1 + 2η (1 − η)2 + 6η(R− 1) R+ 1− 2ηR2 −η(R − 1)2[4 + 2R− η(3R2 + 2R+ 1)]Q0 . (202) 50 M. López de Haro, S. B. Yuste and A. Santos From Eq. (102), we have g(1+) = . (203) The complete RDF is given by Eq. (122), where now Eq. (198) must be used in Eq. (123). In particular, ψ1(r) and ψ2(r) are ψ1(r) = ψ10(r)Θ(r) + ψ11(r + 1−R)Θ(r + 1−R), (204) ψ2(r) = ψ20(r)Θ(r)+ψ21(r+1−R)Θ(r+1−R)+ψ22(r+2−2R)Θ(r+2−2R), (205) where ψ1k(r) = 2π W1k(si) E′(si) six, (206) ψ2k(r) = −4π2 rW2k(si) +W 2k(si)−W2k(si) E′′(si) E′(si) [E′(si)]2 . (207) Here, si are the three distinct roots of E(s) ≡ −12η+E1s+E2s2 +E3s3 and W10(s) ≡ 1 +Q0 +Q1s, W11(s) ≡ −(Q0 +Q2s). (208) W20(s) ≡ s[W10(s)]2, W21(s) ≡ 2sW10(s)W11(s), W22(s) ≡ s[W11(s)]2. (209) To close the proposal, we need to determine the parameters Q0 and Q2 by imposing two new conditions. An obvious condition is the continuity of the cavity function at r = R, what implies g(R+) = e1/T g(R−). (210) This yields ( 1− e−1/T ψ10(R− 1) = −ψ11(0) = 2π . (211) As an extra condition, we could enforce the continuity of the first derivative y′(r) at r = R [109]. However, this complicates the problem too much without any relevant gain in accuracy. In principle, it might be possible to impose consistency with a given EOS, via either the virial route, the compressibility route, or the energy route. But this is not practical since no simple EOS for SW fluids is at our disposal for wide values of density, temperature, and range. As a compromise between simplicity and accuracy, we fix the parameter Q0 at its exact zero-density limit value, namely Q0 = e 1/T∗ − 1 [107]. Therefore, Eq. (211) becomes a transcendental equation for Q2 that needs to be solved numerically. For narrow SW potentials, however, it is possible to replace the exact condition (210) by a simpler one allowing Q2 to be obtained analytically [108], which is especially useful for determining the thermodynamic properties [108, 110]. Alternative Approaches to Hard-Sphere Liquids 51 R=1.5, 3=0.4, T*=1.5 1.0 1.5 2.0 2.5 3.0 2.0 R=2, 3=0.4, T*=3 R=1.05, 3=0.8, T*=0.5 Fig. 13. Radial distribution function of a single component SW fluid for R = 1.05, ρσ3 = 0.8, and T ∗ = 0.5 (top panel), for R = 1.5, ρσ3 = 0.4, and T ∗ = 1.5 (middle panel), and for R = 2.0, ρσ3 = 0.4, and T ∗ = 3.0 (bottom panel). The circles represent simulation data [111] and the solid lines refer to the results obtained from the RFA method. It can be proven that the RFA proposal (198) reduces to the exact solutions of the PY equation [29, 94] in the HS limit, i.e., ǫ → 0 or R → 1, and in the SHS limit, i.e., ǫ→ ∞ and R → 1 with (R− 1)e1/T∗ = finite [107, 108]. Comparison with computer simulations [107, 108, 110, 111] shows that the RFA for SW fluids is rather accurate at any fluid density if the potential well is sufficiently narrow (say R ≤ 1.2), as well as for any width if the density is small enough (say ρσ3 ≤ 0.4). However, as the width and/or the density increase, the RFA predictions worsen, especially at low temperatures. As an 52 M. López de Haro, S. B. Yuste and A. Santos illustration, Fig. 13 compares the RDF provided by the RFA with Monte Carlo data [111] for three representative cases. 4.3 Hard Disks As is well known, the PY theory is exactly solvable for HS fluids with an odd number of dimensions [112–114]. In particular, in the case of hard rods (d = 1), the PY theory provides the exact RDF g(r) or, equivalently, the exact cavity function y(r) outside the hard core (i.e., for r > σ). However, it does not reproduce the exact y(r) in the overlapping region (i.e., for r < σ) [85]. The full exact one-dimensional cavity function is [85] yHR(r|η) = e−(r−1)η/(1−η) ηn−1e−(r−n)η/(1−η) (1 − η)n(n− 1)! (r − n)n−1Θ(r − n), (212) where the subscript HR stands for hard rods and, as usual, σ = 1 has been taken. Consequently, one has gHR(1 +|η) = 1 dr rhHR(r|η) ≡ H(0)HR(η) = − η2. (213) When d is even, the PY equation is not analytically solvable for the HS interaction. In particular, in the important case of hard disks (d = 2), one must resort to numerical solutions of the PY equation [1, 115]. Alternatively, a simple heuristic approach has proven to yield reasonably good results [116]. Such an approach is based on the näıve assumption that the structure and spatial correlations of a hard-disk fluid share some features with those of a hard-rod and a hard-sphere fluid. This fuzzy idea becomes a more specific one by means of the following simple model [116]: gHD(r|η) = ν(η)gHR(r|ω1(η)η) + [1− ν(η)]gHS(r|ω3(η)η). (214) Here, the subscript HD stands for hard disks (d = 2) and the subscript HS stands for hard spheres (d = 3). The parameter ν(η) is a density-dependent mixing parameter, while ω1(η)η and ω3(η)η are the packing fractions in one and three dimensions, respectively, which are “equivalent” to the packing fraction η in two dimensions. In Eq. (214), it is natural to take for gHR(r|η) the exact solution, Eq. (212). As for gHR(r|η), one might use the RFA recipe described in Section 3. However, in order to keep the model (214) as simple as possible, it is sufficient for practical purposes to take the PY solution, Eq. (108). In the latter approximation, gHS(1 +|η) = 1 + η/2 (1− η)2 dr rhHS(r|η) ≡ H(0)HS (η) = − 10− 2η + η2 20(1 + 2η) (215) Alternative Approaches to Hard-Sphere Liquids 53 In order to close the model (214), we still need to determine the parameters ν(η), ω1(η), and ω3(η). To that end, we first impose the condition that Eq. (214) must be consistent with a prescribed contact value gHD(1 +|η) or, equiv- alently, with a prescribed compressibility factor ZHD(η) = 1 + 2ηgHD(1 +|η), with independence of the choice of the mixing parameter ν(η). In other words, gHD(1 +|η) = gHR(1+|ω1(η)η) = gHS(1+|ω3(η)η). (216) Making use of Eqs. (213) and (215), this yields ω1(η) = gHD(1 +|η)− 1 ηgHD(1+|η) , ω3(η) = 4gHD(1 +|η) + 1− 24gHD(1+|η) + 1 4ηgHD(1+|η) (217) Once ω1(η) and ω3(η) are known, we can determine ν(η) by imposing that the model (214) reproduces the isothermal compressibility χHD(η) thermody- namically consistent with the prescribed ZHD(η) [cf. Eq. (97)]. From Eqs. (96) and (214) one has χHD(η) = 1 + 8η dr r {ν(η)hHR(r|ω1(η)η) + [1− ν(η)] hHS(r|ω3(η)η)} , (218) so that ν(η) = [χHD(η)− 1] /8η −H(0)HS (ω3(η)η) HR(ω1(η)η) −H HS (ω3(η)η) , (219) where H HR(η) and H HS (η) are given by Eqs. (213) and (215), respectively. Once a sensible EOS for hard disks is chosen [see, for instance, Table 1], Eqs. (217) and (219) provide the parameters of the model (214). The results show that the scaling factor ω1(η) is a decreasing function, while ω3(η) is an increasing function [116]. As for the mixing parameter ν(η), it is hardly dependent of density and takes values around ν(η) ≃ 0.35–0.40. Comparison of the interpolation model (214) with computer simulation re- sults shows a surprisingly good agreement, despite the crudeness of the model and the absence of empirical fitting parameters, especially at low and mod- erate densities [116]. The discrepancies become important only for distances beyond the location of the second peak and for densities close to the stability threshold. 5 Perturbation Theory When one wants to deal with realistic intermolecular interactions, the prob- lem of deriving the thermodynamic and structural properties of the system becomes rather formidable. Thus, perturbation theories of liquids have been devised since the mid twentieth century. In the case of single component flu- ids, the use of an accurate and well characterized RDF for the HS fluid in a 54 M. López de Haro, S. B. Yuste and A. Santos perturbation theory opens up the possibility of deriving a closed theoretical scheme for the determination of the thermodynamic and structural proper- ties of more realistic models, such as the Lennard–Jones (LJ) fluid. In this section, we will consider this model system, which captures the basic physical properties of real non-polar fluids, to illustrate the procedure. In the application of the perturbation theory of liquids, the stepping stone has been the use of the HS RDF obtained from the solution to the PY equa- tion. Unfortunately, the absence of thermodynamic consistency present in the PY approximation (as well as in other integral equation theories) may clearly contaminate the results derived from its use within a perturbative treatment. In what follows we will reanalyze the different theoretical schemes for the ther- modynamics of LJ fluids that have been constructed with perturbation theory, taking as the reference system the HS fluid. This includes the consideration of the RDF as obtained with the RFA method, which embodies thermody- namic consistency, as well as the proposal of a unifying framework in which all schemes fit in. With our development, we will be able to present a formula- tion which lends itself to relatively easy numerical calculations while retaining the merits that analytical results provide, namely a detailed knowledge and control of all the approximations involved. Let us consider a three-dimensional fluid system defined by a pair inter- action potential φ(r). The virial and energy EOS express the compressibility factor Z and the excess part of the Helmholtz free energy per unit volume f ex, respectively, in terms of the RDF of the system as Z = 1− 2 ∂φ(r) g(r)r3, (220) = 2πρβ dr φ(r)g(r)r2 , (221) where β ≡ 1/kBT . Let us now assume that φ(r) is split into a known (ref- erence) part φ0(r) and a perturbation part φ1(r). The usual perturbative expansion for the Helmholtz free energy to first order in β leads to [2] + 2πρβ dr φ1(r)g0(r)r , (222) where f0 and g0(r) are the free energy and the RDF of the reference system, respectively. The LJ potential is φLJ(r) = 4ǫ r−12 − r−6 , (223) where ǫ is the depth of the well and, for simplicity, we have taken the distance at which the potential vanishes as the length unit, i.e., φLJ(r = 1) = 0. For this potential the reference system may be forced to be a HS system, i.e., one can set Alternative Approaches to Hard-Sphere Liquids 55 φ0(r) = φHS(r) = ∞, r ≤ σ0, 0, r > σ0, (224) where σ0 is a conveniently chosen effective HS diameter. In this case the Helmholtz free energy to this order is approximated by ≈ fHS + 2πρβ dr φLJ(r)gHS(r/σ0)r 2. (225) Note that Eq. (225) may be rewritten in terms of the Laplace transform G(s) of (r/σ0)gHS(r/σ0) as ≈ fHS + 2πρβσ30 ds ΦLJ(s)G(s), (226) where ΦLJ(s) satisfies rφLJ(r) = σ0 ds e−rs/σ0ΦLJ(s), (227) so that ΦLJ(s) = 4ǫσ (s/σ0) − (s/σ0) . (228) Irrespective of the value of the diameter σ0 of the reference system, the right hand side of Eq. (226) represents always an upper bound for the value of the free energy of the real system. Therefore, it is natural to determine σ0 so as to provide the least upper bound. This is precisely the variational scheme of Mansoori and Canfield [117,118] and Rasaiah and Stell [119], usually referred to as MC/RS, and originally implemented with the PY theory for G(s), Eq. (108). In our case, however, we will considerG(s) as given by the RFA method, Eq. (113). Therefore, at fixed ρ and β, the effective diameter σ0 in the MC/RS scheme is obtained from the conditions {∫ η0 ZHS(η)− 1 + 48βǫσ−20 dsG(s|η0) (s/σ0) − (s/σ0) = 0, (229) {∫ η0 ZHS(η)− 1 + 48βǫσ−20 dsG(s|η0) (s/σ0) − (s/σ0) > 0. (230) In these equations, use has been made of the thermodynamic relationship between the free energy and the compressibility factor, Eq. (78). Moreover, we have called η0 ≡ (π/6)ρσ30 and have made explicit with the notation G(s|η0) the fact that the HS RDF depends on the packing fraction η0. 56 M. López de Haro, S. B. Yuste and A. Santos Even if the reference system is not forced to be a HS fluid, one can still use Eq. (226) provided an adequate choice for σ0 is made such that the expansion involved in the right hand side of Eq. (222) yields the right hand side of Eq. (226) to order β2. This is the idea of the Barker and Henderson [120] first order perturbation scheme (BH1), where the effective HS diameter is 1− e−βφLJ(r) . (231) The same ideas may be carried out to higher order in the perturbation expansion. The inclusion of the second order term in the expansion yields the so-called macroscopic compressibility approximation [2] for the free energy, namely + 2πρβ dr φ1(r)g0(r)r −πρβ2χ0 dr φ21(r)g0(r)r , (232) where χ0 is the (reduced) isothermal compressibility of the reference system [121]. To implement a particular perturbation scheme in this approximation un- der a unifying framework that eventually leads to easy numerical evaluation, two further assumptions may prove convenient. First, the perturbation poten- tial φ1(r) ≡ φLJ(r)−φ0(r) may be split into two parts using some “molecular size” parameter ξ ≥ σ0 such that φ1(r) = φ1a(r), 0 ≤ r ≤ ξ, φ1b(r), r > ξ. (233) Next, a choice for the RDF for the reference system is done in the form g0 (r) ≈ θ(r)yHS(r/σ0), (234) where yHS is the cavity (background) correlation function of the HS system and θ(r) is a step function defined by θ(r) = θa(r), 0 ≤ r ≤ ξ, θb(r), r > ξ, (235) in which the functions θa(r) and θb(r) depend on the scheme. With these assumptions the integrals involved in Eq. (232) may be rewrit- ten as dr φn1 (r)g0(r)r dr φn1a(r)θa(r)yHS(r/σ0)r dr φn1a(r)θa(r)gHS(r/σ0)r dr φn1b(r)θb(r)gHS(r/σ0)r 2, (236) Alternative Approaches to Hard-Sphere Liquids 57 with n = 1, 2 and where the fact that yHS(r/σ0) = gHS(r/σ0) when r > σ0 has been used. Decomposing the last integral as and applying the same step as in Eq. (226), Eq. (236) becomes In = σ ds Φnb(s)G(s) + dr φn1a(r)θa(r)yHS(r/σ0)r dr [φn1a(r)θa(r) − φn1b(r)θb(r)] gHS(r/σ0)r2, (237) where the functions Φ1b(s) and Φ2b(s) are defined by the relation rφn1b(r)θb(r) = σ0 ds e−rs/σ0Φnb(s). (238) In the Barker–Henderson second order perturbation scheme (BH2), one takes θa(r) = 0, θb(r) = 1, ξ = σ0, φ1a(r) = 0, φ1b(r) = 4ǫ r−12 − r−6 (239) and σ0 is computed according to Eq. (231). This choice ensures that + 2πρβ dr φ1(r)gHS(r/σ0)r −πρβ2χHS dr φ21(r)gHS(r/σ0)r . (240) On the other hand, if one chooses θa(r) = exp [−β (φLJ(r) + ǫ)] , θb(r) = 1, ξ = 21/6, (241) φ1a(r) = −ǫ, φ1b(r) = 4ǫ r−12 − r−6 , (242) the scheme leads to the Weeks–Chandler–Andersen (WCA) theory [122] if one determines the HS diameter through the condition χ0 = χHS [123], which in turn implies dr r2e−βφ0(r)yHS(r/σ0) = ∫ 21/6 dr r2gHS(r/σ0) 1− e−βφ0(r) . (243) To close the scheme, the HS cavity function has to be provided in the range 0 ≤ r ≤ σ0. Fortunately, relatively simple expressions for yHS(r/σ0) are available in the literature [124–126], apart from our own proposal, Eq. (140). Note that θb(r) and φ1b(r), and thus also Φnb(s), are the same functions in the BH2 and WCA schemes. It is convenient, in order to have all the quantities needed to evaluate fLJ in these schemes, to provide explicit expressions for Φ1b(s) and Φ2b(s). These are given by [cf. Eq. (228)] 58 M. López de Haro, S. B. Yuste and A. Santos Φ1b(s) = ΦLJ(s), (244) Φ2b(s) = 16ǫ 2σ−20 (s/σ0) − 2(s/σ0) (s/σ0) . (245) Up to this point, we have embodied the most popular perturbation schemes within a unified framework that requires as input only the EOS of the HS fluid in order to compute the Helmholtz free energy of the LJ system and leads to relatively easy numerical computations. It should be clear that a variety of other possible schemes, requiring the same little input, fit in our unified framework, which is based on the RFA method for gHS(r/σ0) and G(s). Once fLJ has been determined, the compressibility factor of the LJ fluid at a given order of the perturbation expansion readily follows from Eqs. (222) or (232) through the thermodynamic relation ZLJ = ρ . (246) Taking into account that the HS fluid presents a fluid-solid transition at a freezing packing fraction ηf ≃ 0.494 [127] and a solid-fluid transition at a melting packing fraction ηm ≃ 0.54 [127], the fluid-solid and solid-fluid coexistence lines for the LJ system may be computed from the values (ρ, T ) determined from the conditions (π/6)ρσ30(ρ, T ) = ηf and (π/6)ρσ 0(ρ, T ) = ηm, respectively, with the effective diameter σ0(ρ, T ) obtained using any of the perturbative schemes. Similarly, admitting that there is a glass transition in the HS fluid at the packing fraction ηg ≃ 0.56 [128], one can now determine the location of the liquid-glass transition line for the LJ fluid in the (ρ, T ) plane from the simple relationship (π/6)ρσ30(ρ, T ) = ηg. With a proper choice for ZHS, it has been shown [76,129,130] that the critical point, the structure, and the phase diagram (including a glass transition) of the LJ fluid may be adequately described with this approach. 6 Perspectives In this chapter we have given a self-contained account of a simple (mostly analytical) framework for the study of the thermodynamic and structural properties of hard-core systems. Whenever possible, the developments have attempted to cater for mixtures with an arbitrary number of components (in- cluding polydisperse systems) and arbitrary dimensionality. We started con- sidering the contact values of the RDF because they enter directly into the EOS and are required as input in the RFA method to compute the structural properties. With the aid of consistency conditions, we were able to devise var- ious approximate proposals which, when used in conjunction with a sensible choice for the contact value of the RDF of the single component fluid (re- quired in the formulation but otherwise chosen at will), have been shown to Alternative Approaches to Hard-Sphere Liquids 59 be in reasonably good agreement with simulation results and lead to accurate EOS both for additive and non-additive mixtures. Some aspects of the results that follow from the use of these EOS were illustrated by looking at demixing problems in these mixtures, including the far from intuitive case of a binary mixture of non-additive hard spheres in infinite dimensionality. After that, restricting ourselves to three-dimensional systems, we described the RFA method as applied to a single component hard-sphere fluid and to a multicomponent mixture of HS. Using this approach, we have been able to obtain explicit analytical results for the RDF, the direct correlation function, the static structure factor, and the bridge function, in the end requiring as input only the contact value of the RDF of the single component HS fluid (or equivalently its compressibility factor). One of the nice assets of the RFA approach is that it eliminates the thermodynamic consistency problem which is present in most of the integral equation formulations for the computation of structural quantities. Once again, when a sensible choice for the single component EOS is made, we have shown, through the comparison between the results of the RFA approach and simulation data for some illustrative cases, the very good performance of our development. Also, the use of the RFA approach in connection with some other related systems (sticky hard spheres, square-well fluids, and hard disks) has been addressed. The final part of the chapter concerns the use of HS results for more realistic intermolecular potentials in the perturbation theory of liquids. In this instance we have been able to provide a unifying scheme in which the most popular perturbation theory formulations may be expressed and which was devised to allow for easy computations. We illustrated this for a LJ fluid but it should be clear that a similar approach might be followed for other fluids and in fact it has recently been done in connection with the glass transition of hard-core Yukawa fluids [131]. Finally, it should be clear that there are many facets of the equilibrium and structural properties of hard-core systems that may be studied with a simi- lar approach but that up to now have not been considered. For instance, the generalizations of the RFA approach for systems such as hard hyperspheres, non-additive hard spheres, square-well mixtures, penetrable spheres [132], or the Jagla potential [133] appear as interesting challenges. Similarly, the ex- tension of the perturbation theory scheme to the case of LJ mixtures seems a worthwhile task. We hope to address some of these problems in the future and would be very much rewarded if some others were taken up by researchers who might find these developments also a valuable tool for their work. References 1. J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, 587 (1976). 2. D. A. McQuarrie, Statistical Mechanics (Harper & Row, N. Y., 1976). 3. H. L. Friedman, A Course in Statistical Mechanics (Prentice Hall, Englewood Cliffs, 1985). 60 M. López de Haro, S. B. Yuste and A. Santos 4. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, (Academic Press, London, 1986). 5. J. L. Lebowitz and D. Zomick, J. Chem. Phys. 54, 3335 (1971). 6. 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0704.0158

Complexities of Human Promoter Sequences

APS/xxx Complexities of Human Promoter Sequences Fangcui Zhao1,∗ Huijie Yang2,3,† and Binghong Wang4 College of Life Science and Bioengineering, Beijing University of Technology, Beijing 100022, China Department of Physics, National University of Singapore, Science Drive 2, Singapore 117543 School of Management, University of Shanghai for Science and Technology, and Shanghai Institute for Systematic Science, Shanghai 200093, China Department of Modern Physics, University of Science and Technology of China, Anhui Hefei 230026, China (Dated: October 26, 2018) By means of the diffusion entropy approach, we detect the scale-invariance characteristics embed- ded in the 4737 human promoter sequences. The exponent for the scale-invariance is in a wide range of [0.3, 0.9], which centered at δc = 0.66. The distribution of the exponent can be separated into left and right branches with respect to the maximum. The left and right branches are asymmetric and can be fitted exactly with Gaussian form with different widths, respectively. PACS numbers: 82.39.Pj, 05.45.Tp I. INTRODUCTION Understanding gene regulation is one of the most excit- ing topics in molecular genetics [1]. Promoter sequences are crucial in gene regulation. The analysis of these re- gions is the first step towards complex models of regula- tory networks. A promoter is a combination of different regions with different functions [2, 3, 4, 5]. Surrounding the tran- scription start site is the minimal sequence for initiat- ing transcription, called core promoter. It interacts with RNA polymerase II and basal transcription factors. Few hundred base pairs upstream of the core promoter are the gene-specific regulatory elements, which are recog- nized by transcription factors to determine the efficiency and specificity of promoter activity. Far distant from the transcription start site there are enhancers and dis- tal promoter elements which can considerably affect the rate of transcription. Multiple binding sites contribute to the functioning of a promoter, with their position and context of occurrence playing an important role. Large- scale studies show that repeats participate in the regula- tion of numerous human and mouse genes [6]. Hence, the promoter’s biological function is a cooperative process of different regions such as the core promoter, the gene- specific regulatory elements, the enhancers/silencers, the insulators, the CpG islands and so forth. But how they cooperate with each other is still a problem to be inves- tigated carefully. The structures of DNA sequences determine their bio- logical functions [7]. Recent years witness an avalanche of finding nontrivial structure characteristics embedded in DNA sequences. Detailed works show that the non- coding sequences carry long-range correlations [8, 9, 10]. The size distributions of coding sequences and non- coding sequences obey Gaussian or exponential and ∗Electronic address: [email protected] †Electronic address: [email protected]; Corresponding author power-law [11, 12], respectively. Theoretical model-based simulations [13, 14, 15, 16] tell us that the parts of the promoters where the RNA transcription has started are more active than a random portion of the DNA. By means of the nonlinear modeling method it is found that along the putative promoter regions of human sequences there are some segments much more predictable com- pared with other segments [17]. All the evidences suggest that the nontrivial structure characteristics of a promoter determine its biological functions. The statistical prop- erties of a promoter may shed light on the cooperative process of different regions. Experimental knowledge of the precise 5’ ends of cD- NAs should facilitate the identification and characteri- zation of regulatory sequence elements in proximal pro- moters [18]. Using the oligocapping method, Suzuki et al. identify the transcriptional start sites from cDNA libraries enriched in full-length cDNA sequences. The identified transcriptional start sites are available at the Database, http://dbtss.hgc.jp/. [19]. Consequently, Leonardo et al. have used this data set and aligned the full-length cDNAs to the human genome, thereby extracting putative promoter regions (PPRs) [20]. Us- ing the known transcriptional start sites from over 5700 different human full-length cDNAs, a set of 4737 distinct PPRs are extracted from the human genome. Each PPR consists nucleotides from −2000 to +1000bp, relative to the corresponding transcriptional start site. They have also counted eight-letter words within the PPRs, using z-scores and other related statistics to evaluate the over- and under- representations. In this paper, by means of the concept of diffusion entropy (DE) we try to detect the scale-invariant char- acteristics in these putative promoter regions. II. DIFFUSION ENTROPY ANALYSIS The diffusion entropy (DE) method is firstly designed to capture the scale-invariance embedded in time series [21, 22, 23]. To keep the description as self-contained as http://arxiv.org/abs/0704.0158v1 mailto:[email protected] mailto:[email protected] http://dbtss.hgc.jp/ possible, we review briefly the procedures. We consider a PPR denoted with Y = (y1, y2, · · · , y3001), where ys is the element at the position s and ys = A, T,C or G. Replacing A, T and C,G with −1 and +1, respectively, the original PPR is mapped to a time series X = (x1, x2, · · · , x3001). There is not a trend in this series, i.e., X is stationary. Connecting the starting and the end of X , we can ob- tain a set of delay-register vectors, which reads, T1(t) = (x1, x2, · · · , xt) T2(t) = (x2, x3, · · · , xt+1) T3001(t) = (x3001, x1, · · · , xt−1) Regarding each vector as a trajectory of a particle in duration of t time units, all the vectors can be described as a diffusion process of a system containing 3001 parti- cles. The initial state of the system is T1(0) T2(0) T3001(0) Accordingly, at each time step t we can calculate dis- placements of all the particles. The probability distribu- tion function (PDF) of the displacements can be approx- imated with p(m, t) ∼ Km/3001, where m = −t,−t + 1, · · · , t and Km is the number of the particles whose displacements are m. It can represent the state of the system at time t. As a tenet of complexity theory [24, 25], complexity is related with the concept of scaling invariance. For the constructed diffusion process, the scaling invariance is defined as, p(m, t) ≈ , (2) where δ is the scaling exponent and can be regarded as a quantitative description of the PPR’s complexity. If the elements in the PPR are positioned randomly, the result- ing PDF obeys a Gaussian form and δ = 0.5. Complexity of the PPR is expected to generate a departure from this ordinary condition, that is, δ 6= 0.5. The value of δ can tell us the pattern characteristics of a PPR. The departure from the ordinary condition can be described with a preferential effect. Let the element is A, T (or C,G), the preferential probability for the fol- lowing element’s being A, T (or C,G) is Wpre. A positive preferential effect, i.e, Wpre > 0.5, leads to the value of δ larger than 0.5. While a negative preferential effect, i.e, Wpre < 0.5, can induce the value of δ smaller than 0.5. 101 102 103 PPR-1 Calculate =0.662 101 102 103 PPR-1000 Calculate =0.760 101 102 103 PPR-2000 Calculate =0.703 101 102 103 PPR-3000 Calculate =0.500 FIG. 1: (Color online) Typical DE results. The results for the PPRs numbered 1, 1000, 2000 and 3000 are presented. In considerable wide regions of t , the curves of DE can be fitted almost exactly with the linear relation in Eq.(4). Hence, a large value of δ implies that A, T or C,G accu- mulate strongly in a scale-invariance way, respectively. However, correct evaluation of the scaling exponent is a nontrivial problem. In literature, variance-based method is used to detect the scale-invariance. But the obtained Hurst exponent Hmay be different from the real δ, that is, generally we haveH 6= δ. And for some conditions, the variance is divergent, which leads the invalidation of the variance-method at all. To overcome these shortages, the Shannon entropy for the diffusion can be used, which reads, S(t) = − p(m, t) ln p(m, t) This diffusion-based entropy is called diffusion entropy (DE). A simple computation leads the relation between the scaling invariance defined in Eq.2 and the DE as, S(t) = A+ δ ln t, (4) where A is a constant depends on the PDF. Detailed works show that DE is a reliable method to search the correct value of δ, regardless the form of the PDF [26, 27, 28, 29]. The complexity in the PDF can be catalogued into two levels [30], the primary one due to the extension of the probability to all the possible displacements m, and the secondary one due to the internal structures. Conse- quently, we should consider also the corresponding shuf- fling sequences as comparison. 0 200 400 600 800 1000 1200 1400 1600 FIG. 2: Distribution of the maximum interval ∆t in which one can find scale-invariant characteristics. Keeping the standard deviation of the fitting result in the range of ≤ 0.05 , we can find the maximum intervals ∆t for all the PPRs. The distribution tells us that generally the scale-invariance can be found over two to three decades of the scale t . 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 600 Calculate Gauss fit Gauss fit left=0.67 wleft=0.17 right=0.65 wright=0.10 FIG. 3: (Color online) The complex index δ distributes in a wide range of [0.3, 0.9]. The distribution can be sepa- rated into two branches with respect to the center δc = 0.66. The two branches are asymmetric and obey exactly the Gau- usian function, respectively. The widths and centers of the left and right branches are (wleft, xleftc ) = (0.17, 0.67), (wright, xrightc ) = (0.10, 0.65). The centers coincide with each other, wleft ≈ wright ≈ δc = 0.66. The right branch dis- tributes in a significant narrow region. III. RESULTS AND DISCUSSIONS The DEs for all the 4737 PPRs are calculated. As a typical example, Fig.1 presents the DE results for the PPRs numbered 1, 1000, 2000 and 3000. In considerable wide regions of t, the curves of DE can be fitted almost exactly with the linear relation in Eq.4. For each PPR, there exists an interval, t0 ∼ t0+∆t, in which the PDF behaves scale-invariance. Keeping simul- taneously the standard deviation and the error of the scaling exponent for the fitting result in the range of ≤ 0.05 and ≤ 0.03, we can find the maximum intervals ∆t for all the PPRs. In the fitting procedure, the confi- dence level is set to be 95%. The distribution of ∆t, as shown in Fig.2, tells us that generally the scale-invariance can be found over two to three decades of the scale t. The concept of DE is based upon statistical theory, that is, t0 should be large enough so that the statistical as- sumptions are valid. To cite an example, we consider a random series, whose elements obey a homogenous dis- tribution in [0, 1]. Only the length of the delay-register vectors, t, in Eq.(1) is large enough, the corresponding PDF for the displacements, i.e, the summation value of each delay-register vector, approaches the Gaussian dis- tribution. Consequently, t0 is not a valuable parameter. The values of t0 for different PPRs are not presented. The resulting scaling exponent δ ± 0.03 distributes in a wide range of [0.3, 0.9]. The distribution can be separated into two branches with respect to the center δc = 0.66. The two branches are asymmet- ric and can be fitted exactly with the Gauusian func- tion, respectively. The widths and centers of the left and right branches are (wleft, xleftc ) = (0.17, 0.67), (wright, xrightc ) = (0.10, 0.65). That is to say, the centers coincide with each other, wleft ≈ wright ≈ δc = 0.66. Comparatively, the right branch distributes in a signifi- cant narrow region. The PPRs are shuffled also. For each PPR, the shuf- fling result is obtained by averaging over ten shuffling samples. The scaling exponents are almost same, i.e., δshuffling = 0.5±0.03. The detected scale-invariant char- acteristics are internal-structure-related. How to understand the asymmetric characteristic of the distribution of the complexity index δ is an in- teresting problem. In literature, some statistical char- acteristics of DNA sequences are captured with evo- lution models, such as the long-range correlations and the over- and under-representation of strings and so on [31, 32, 33]. From the perspective of evolution, per- haps the distribution characteristics may favor a stochas- tic evolution model. The initial sequences have same complexity δinitial = δc = 0.66. With the evolution processes the sequences diffuse along two directions, in- creasing complexity and decreasing complexity, i.e, the index δ increases and decreases, respectively. The diffu- sion coefficients for the two directions are significantly different, denoted with Dleft 6= Dright. Based upon the widths of the two branches we can estimate that, Dleft Dright = δleft δright = 1.7. It should be noted that, the complexity is regarded as the departure from the ordinary condition, δ = 0.5. In the totally 4737 values of δ, only a small portion of them are less than 0.5. Ac- cordingly, the PPRs may be catalogued into two classes, the PPRs with high complexity and the PPRs with low complexity. The former class evolves averagely with a slow speed while the later one with a high speed. In summary, by means of the DE method, we calculate the complexities of the 4737 PPRs. The distribution of the complexity index includes two asymmetric branches, which obey Gaussian form with different widths, respec- tively. A stochastic evolution model may provide us a comprehensive understand of these characteristics. IV. ACKNOWLEDGEMENTS This work is funded by the National Natural Sci- ence Foundation of China under Grant Nos. 70571074, 10635040 and 70471033, by the National Basic Re- search Program of China (973 Program) under grant No.2006CB705500), by the President Funding of Chinese Academy of Science, and by the Specialized Research Fund for the Doctoral Program of Higher Education of China. One of the authors (H. Yang) would like to thank Prof. Y. Zhuo for stimulating discussions. [1] Ohler,U. and Niemann,H. (2001) Identification and anal- ysis of eukaryotic promoters: recent computational ap- proaches. Trends Genet., 17, 56-60. [2] Werner,T. (1999) Models for prediction and recognition of eukaryotic promoters. Mammalian Genome, 10, 168- [3] Pedersen,A.G., Baldi,P., Chauvin,Y., Brunak,S. (1999) The biology of eukaryotic promoter prediction - a review. Comput. Chem., 23, 191-207. [4] Zhang,M.Q. (2002) Computational methods for promoter recognition. In: Jiang T, Xu Y, Zhang,M.Q., editors. Current topics in computational molecular biology. Cam- bridge, Massachusetts: MIT Press; p. 249-268. [5] Narang,V., Sung,W.-K., Mittal A. (2005) Computational modeling of oligonucleotides positional densities for hu- man promoter prediction. Art. Intel. Med., 35, 107-119. [6] Rosenberg, N. and Jolicoeur, P. (1997) Retroviral patho- genesis. In Retroviruses (Coffin, J.M. et al., eds), pp. 475–586, Cold Spring Harbor Press. [7] Buldyrev,S.V., Goldberger,A.L., Havlin,S., Man- tegna,R.N., Matsa,M.E., Peng,C.-K., Simons, M. and Stanley,H.E. (1995) Long-range correlation properties of coding and noncoding DNA sequences: GenBank analysis. Phys. Rev. E 51, 5084-5091. [8] Peng,C.K., Buldyrev,S., Goldberger,A., Havlin, S., Sciortino,F., Simons,M. and Stanley,H.E.(1992) Long- range correlations in nucleotide sequences. Nature 356, 168-171. [9] Li,W., Kaneko,K. (1992) Long-range correlations and partial 1/f −α spectrum in a noncoding DNA sequence. Europhys. Lett. 17, 655. [10] Yang,H., Zhao,F., Zhuo,Y., et al. (2002) Analysis of DNA chains by means of factorial moments. Phys. Lett. A 292,349-356. [11] Provata, A., Almirantis,Y. (1997) Scaling properties of coding and non-coding DNA sequences. Physica A 247, [12] Provata,A., Almirantis,Y. (2000) Fractal cantor patterns in the sequence structure of DNA, Fractals 8 ,15-27. [13] Yang,H., Zhuo,Y., Wu,X. (1994) Investigation of ther- mal denaturation of DNA molecules based upon non- equilibrium transport approach. J. Phys. A 27, 6147- 6156. [14] Salerno,M. (1991) Discrete model for DNA-promoter dy- namics. Phys. Rev. A 44, 5292-5297. [15] Lennholm,E., Homquist,M. (2003) Revisiting Salerno’s sine-Gordon model of DNA: active regions and robust- ness. Physica D 177, 233-241. [16] Kalosakas,G., Rasmussen,K.O. and Bishop,A.R. (2004) Sequence-specific thermal fluctuations identify start sites for DNA transcription. Europhys. Lett. 68, 127-133. [17] Yang,H., Zhao,F., Gu,J. and Wang,B. (2006) Nonlin- ear modeling approach to human promoter sequences. J. Theo. Bio. 241, 765-773. [18] Trinklein, N.D., Aldred, S.J., Saldanha, A.J., My- ers, R.M., 2003. Identification and functional analysis of human transcriptional promoters. Genome Res. 13, 308C312. [19] Suzuki,Y., Yamashita,R., Nakai,K. and Sugano,S. (2002) DBTSS:DataBase of human transcriptional start sites and full-length cDNAs. Nucleic Acids Res., 30, 328-331. [20] Leonardo,M.–R., John,L.S., Gavin,C.K. and David,L. (2004) Statistical analysis of over- represented words in human promoter sequences. Nucleic Acids Res., 32, 949-958. See also, ftp://ftp.ncbi.nlm.nih.gov/pub/marino/published/ hs promoters/fasta/. [21] Grigolini,P., Palatella,L. and Raffaelli,G. (2001) Complex Geometry, Patterns, and Scaling in Nature and Society. Fractals 9, 439-449. [22] Scafetta,N., Hamilton,P. and Grigolini,P. (2001) The thermodynamics of social processes: the teen birth phe- nomenon. Fractals 9, 193-208. [23] Scafetta,N. and Grigolini,P. (2002) Scaling detection in time series: Diffusion entropy analysis. Phys. Rev. E 66, 036130. [24] Bar-Yam, Y. (1997) Dynamics of Complex Systems. Addison-Wesley, Reading, MA. [25] Mandelbrot,B. B. (1988) Fractal Geometry of Nature. W.H. Freeman, San Francisco, CA. [26] Scafetta,N. and West, B. J. (2003) Solar flare intermit- tency and the earth’s temperature anomalies. Phys. Rev. Lett. 90, 248701. [27] Scafetta,N., Latora,V. and Grigolini,P. (2002) Levy scal- ing: The diffusion entropy analysis applied to DNA se- quences. Phys. Rev. E 66, 031906. [28] Yang,H., Zhao,F., Zhang,W. and Li,Z. (2005) Diffusion entropy approach to complexity for a Hodgkin–Huxley neuron. Physica A 347, 704-710. [29] Yang,H., Zhao,F., Qi,L. and Hu,B. (2004) Temporal se- ries analysis approach to spectra of complex networks. Phys. Rev. E 69, 066104. [30] Pipek,J. and Varga,I. (1992) Universal classification scheme for the spatial-localization properties of one- ftp://ftp.ncbi.nlm.nih.gov/pub/marino/published/ particle states in finite, d-dimensional systems. Phys. Rev. A 46,3148-3163. [31] Hsieh,L.-C., Luo,L., Ji F. and Lee,H.C. (2003) Minimal model for genome evolution and growth. Phys. Rev. Lett. 90, 018101. [32] Kloster,M. (2005) Analysis of evolution through compet- itive selection. Phys. Rev. Lett. 95, 168701. [33] Messer,P.W., Arndt,P.F. and Lassig,M. (2005) Solvable sequence evolution models and genomic correlations. Phys. Rev. Lett. 94, 138103.

0704.0159

Evidence for an excitonic insulator phase in 1T-TiSe$_{2}$

APS/123-QED Evidence for an excitonic insulator phase in 1T -TiSe2 H. Cercellier,∗ C. Monney, F. Clerc, C. Battaglia, L. Despont, M. G. Garnier, H. Beck, and P. Aebi Institut de Physique, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland L. Patthey Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen, Switzerland H. Berger Institut de Physique de la Matière Complexe, EPFL, CH-1015 Lausanne, Switzerland (Dated: October 22, 2018) We present a new high-resolution angle-resolved photoemission study of 1T -TiSe2 in both, its room-temperature, normal phase and its low-temperature, charge-density wave phase. At low tem- perature the photoemission spectra are strongly modified, with large band renormalisations at high- symmetry points of the Brillouin zone and a very large transfer of spectral weight to backfolded bands. A theoretical calculation of the spectral function for an excitonic insulator phase reproduces the experimental features with very good agreement. This gives strong evidence in favour of the excitonic insulator scenario as a driving force for the charge-density wave transition in 1T -TiSe2. PACS numbers: Transition-metal dichalcogenides (TMDC’s) are lay- ered compounds exhibiting a variety of interesting phys- ical properties, mainly due to their reduced dimension- ality [1]. One of the most frequent characteristics is a ground state exhibiting a charge-density wave (CDW), with its origin arising from a particular topology of the Fermi surface and/or a strong electron-phonon coupling [2]. Among the TMDC’s 1T -TiSe2 shows a commensu- rate 2×2×2 structural distortion below 202 K, accom- panied by the softening of a zone boundary phonon and with changes in the transport properties [3, 4]. In spite of many experimental and theoretical studies, the driv- ing force for the transition remains controversial. Sev- eral angle-resolved photoelectron spectroscopy (ARPES) studies suggested either the onset of an excitonic insula- tor phase [5, 6] or a band Jahn-Teller effect [7]. Further- more, TiSe2 has recently attracted strong interest due to the observation of superconductivity when intercalated with Cu [8]. In systems showing exotic properties, such as Kondo systems for example [9], the calculation of the spectral function has often been a necessary and deci- sive step for the interpretation of the ARPES data and the determination of the ground state of the systems. In the case of 1T -TiSe2, such a calculation for an excitonic insulator phase lacked so far. In this letter we present a high-resolution ARPES study of 1T -TiSe2, together with theoretical calculations of the excitonic insulator phase spectral function for this compound. We find that the experimental ARPES spec- tra show strong band renormalisations with a very large transfer of spectral weight into backfolded bands in the low-temperature phase. The spectral function calculated for the excitonic insulator phase is in strikingly good Electronic address: [email protected] agreement with the experiments, giving strong evidence for the excitonic origin of the transition. The excitonic insulator model was first introduced in the sixties, for a semi-conductor or a semi-metal with a very small indirect gap EG [10, 11, 12, 13]. Thermal ex- citations lead to the formation of holes in the valence band and electrons in the conduction band. For low free carrier densities, the weak screening of the electron- hole Coulomb interaction leads to the formation of sta- ble electron-hole bound states, called excitons. If the exciton binding energy EB is larger than the gap energy EG, the system becomes unstable upon formation of exci- tons. This instability can drive a transition to a coherent ground state of condensed excitons, with a periodicity given by the spanning vector w that connects the va- lence band maximum to the conduction band minimum. In the particular case of TiSe2, there are three vectors (wi, i = 1, 2, 3) connecting the Se 4p-derived valence band maximum at the Γ point to the three symmetry- equivalent Ti 3d-derived conduction band minima at the L points of the Brillouin zone (BZ) (see inset of fig. 1b)). Our calculations are based on the BCS-like model of Jérome, Rice and Kohn [12], adapted for multiple wi. The band dispersions for the normal phase have been chosen of the form ǫv(k) = ǫ v + ~ k2x + k + tv cos( c(k,wi) = ǫ c + ~ ( (kx − wix) (ky − wiy) +tc cos (2π(kz − wiz) for the valence (ǫv) and the three conduction (ǫ c) bands respectively, with c the lattice parameter perpendicular to the surface in the normal (1×1×1) phase, tv and tc the amplitudes of the respective dispersions perpendicular to the surface and mv, mc the effective masses. http://arxiv.org/abs/0704.0159v1 mailto:[email protected] The parameters for equations 1 were derived from pho- ton energy dependent ARPES measurements carried out at the Swiss Light Source on the SIS beamline, using a Scienta SES-2002 spectrometer with an overall energy resolution better than 10 meV, and an angular resolution better than 0.5◦. The fit to the data gives for the Se 4p valence band maximum -20 ± 10 meV, and for the Ti 3d conduction band a minimum -40 ± 5 meV [14]. From our measurements we then find a semimetallic band structure with a negative gap (i.e. an overlap) EG=-20 ± 15 meV for the normal phase of TiSe2, in agreement with the lit- erature [15]. The dispersions deduced from the ARPES data are shown in fig. 1a) (dashed lines). Within this model the one-electron Green’s functions of the valence and the conduction bands were calculated for the excitonic insulator phase. For the valence band, one obtains Gv(k, z) = z − ǫv(k)− |∆|2(k,wi) z − ǫc(k+wi) . (2) This is a generalized form of the equations of Ref. [12] for an arbitrary number of wi. The order parameter ∆ is related to the number of excitons in the condensed state at a given temperature. For the conduction band, there is a system of equations describing the Green’s functions Gic corresponding to each spanning vector vector wi: z − ǫic(k+wi) c(k+wi, z) = 1 +∆ ∗(k,wi) ∆(k,wj)G c(k+wj, z) z − ǫv(k) This model and the derivation of the Green’s functions will be further described elsewhere [16]. The spectral function calculated along several high- symmetry directions of the BZ is shown in fig. 1a) for an order parameter ∆=0.05 eV. Its value has been chosen for best agreement with experiment. The color scale shows the spectral weight carried by each band. For presen- tation purposes the δ-like peaks of the spectral function have been broadened by adding a constant 30 meV imagi- nary part to the self-energy. In the normal phase (dashed lines), as previously described we consider a semimetal with a 20 meV overlap, with bands carrying unity spec- tral weight. In the excitonic phase, the band structure is strongly modified. The first observation is the appear- ance of new bands (labeled C1, V2 and C3), backfolded with the spanning vector w = ΓL. The C1, V2 and C3 branches are the backfolded replicas of branches C2, V3 and C4 respectively. In this new phase the Γ and L points are now equivalent, which means that the excitonic state has a 2×2×2 periodicity of purely electronic origin, as ex- pected from theoretical considerations [10, 12]. Another effect of exciton condensation is the opening of a gap in the excitation spectrum. This results in a flattening of the valence band near Γ in the ΓM direction (V1 branch) and in the AΓ direction (V3 branch), and also an upward bend of the conduction band near L and M (C2 and C4 Γ Μ Α ΓL FIG. 1: : a) Spectral function of the excitonic insulator in a 1T structure calculated for a 20 meV overlap and an order parameter ∆=0.05 eV. The V1-V3 (resp. C1-C4) branches refer to the valence (resp. conduction) band. Dashed lines correspond to the normal phase (∆=0). The path in recipro- cal space is shown in red in the inset. b) Spectral weight of the different bands. Inset : bulk Brillouin zone of 1T -TiSe2. branches). It is interesting to notice that in the vicinity of these two points, the conduction band is split (arrows). This results from the backfolding of the L points onto each other, according to the new periodicity of the exci- tonic state [17]. The spectral weight carried by the bands is shown in fig. 1b). The largest variations occur near the Γ, L and M points, where the band extrema in the normal phase are close enough for excitons to be created. Away from these points, the spectral weight decreases in the backfolded bands (C1, V2, C3) and increases in the others. The intensity of the V1 branch, for example, de- creases by a factor of 2 when approaching Γ, whereas the backfolded C1 branch shows the opposite behaviour. Such a large transfer of spectral weight into the back- folded bands is a very uncommon and striking feature. Indeed, in most compounds with competing potentials (CDW systems, vicinal surfaces,...), the backfolded bands carry an extremely small spectral weight [18, 19, 20]. In these systems the backfolding results mainly from the in- fluence of the modified lattice on the electron gas, and the weight transfer is related to the strength of the new crystal potential component. Here, the case of the exci- tonic insulator is completely different, as the backfolding is an intrinsic property of the excitonic state. The large Τ=250 Κ Τ=250 Κ Τ=65 Κ Τ=65 Κ FIG. 2: : ARPES spectra of 1T -TiSe2 for a) the normal and b) the low temperature phase. Thick dotted lines are parabolic fits to the bands in the normal phase and thin dotted lines are guides to the eye for the CDW phase. Fine lines follow the dispersion of the 4p sidebands (see text). transfer of spectral weight is then a purely electronic ef- fect, and turns out to be a characteristic feature of the excitonic insulator phase. Fig. 2 shows ARPES spectra recorded at a photon energy hν=31 eV as a function of temperature. At this photon energy, the normal emission spectra correspond to states located close to the Γ point. For the sake of sim- plicity the description is in terms of the surface BZ high- symmetry points Γ̄ and M̄ . The 250 K spectra exhibit the three Se 4p-derived bands at Γ̄ and the Ti 3d-derived band at M̄ widely described in the literature [5, 6, 7]. The thick dotted lines (white) are fits by equation 1, giv- ing for the topmost 4p band a maximum energy of -20 ± 10 meV, and for the Ti 3d a minimum energy of -40 ± 5 meV. The small overlap EG=-20 ± 15 meV in the normal phase is consistent with the excitonic insulator scenario, as the exciton binding energy is expected to be close to that value. [5, 6]. The position of both band maxima in the occupied states is most probably due to a slight Ti overdoping of our samples [3]. In our case, a transition temperature of 180 ± 10 K was found from dif- ferent ARPES and scanning tunneling microscopy mea- surements, indicating a Ti doping of less than 1 %. On the 250 K spectrum at Γ̄, the intensity is low near normal emission. This reduced intensity and the residual inten- sity at M̄ around 150 meV binding energy (arrows) may arise from exciton fluctuations (see reduction of spectral Τ=250 Κ Τ=250 Κ Τ=65 Κ Τ=65 Κ ∆=0.05 eV FIG. 3: : Theoretical spectral function of 1T -TiSe2, calcu- lated along the path given by the free electron final state approximation shown in the inset. a) normal state and b) low temperature phase (see text). weight near Γ in the V1 branch in fig. 1b). Matrix ele- ments do not appear to play a role as the intensity vari- ation only depends very slightly on photon energy and polarization. In the 65 K data (fig. 2b)), the topmost 4p band flattens near Γ̄ and shifts to higher binding en- ergy by about 100 meV (thin white, dotted line). This shift is accompanied by a larger decrease of the spectral weight near the top of the band. The two other bands (fine black lines) are only slightly shifted and do not ap- pear to participate in the transition. In the M̄ spectrum strong backfolded valence bands can be seen, and the conduction band bends upwards, leading to a maximum intensity located about 0.25 Å−1 from M̄ (thin white dot- ted line). This observation is in agreement with Kidd et al. [6], although in their case the conduction band was unoccupied in the normal phase. The calculated spectral functions corresponding to the data of fig. 2 are shown in fig. 3, using the free-electron fi- nal state approximation with a 10 eV inner potential and a 4.6 eV work function (see inset). The effect of tempera- ture was taken into account via the order parameter and the Fermi function. Only the topmost valence band was considered, as the other two are practically not influenced by the transition (see above, fig. 2). The behavior of this band is extremely well reproduced by the calculation. In the 65 K calculation the valence band is flattened near Γ̄, and the spectral weight at this point is reduced to 44 %, close to the experimental value of 35 %. The agreement E-EF (eV)Model ARPES −0.025 FIG. 4: : Near-EF constant energy cuts in the vicinity of the Γ point. The theoretical data correspond to fig. 3b and the ARPES data are taken from the low-temperature data of fig. is very satisfying, considering that the calculation takes into account only the lowest excitonic state. The exper- imental features appear broader than in the calculation, but at finite temperatures one may expect the existence of excitons with non-zero momentum, leading to a spread of spectral weight away from the high-symmetry points. In the near-M̄ spectral function, the backfolded va- lence band is strongly present in the 65 K calculation, with comparable spectral weight as at Γ̄ and as the con- duction band at M̄ . The conduction band maximum intensity is located away from M̄ as in the experiment. The small perpendicular dispersion of the free-electron final state causes an asymmetry of the intensity of the conduction band on each side of M̄ , which is also visible in fig. 2. In our calculation, as opposed to the ARPES spectra, the conduction band is unoccupied and only the occupied tail of the peaks is visible. This difference may be simply due to the final state approximation used in the calculation, a slight shift of the chemical potential due to the transition, or to atomic displacements that would shift the conduction band [6, 7, 21]. Such atomic displacements, in terms of a band Jahn-Teller effect, were suggested as a driving force for the transition. However, the key point is that, although the lattice distortion may shift the conduction band, the very small atomic displace- ments (≈ 0.02 Å [3]) in 1T -TiSe2 are expected to lead to a negligable spectral weight in the backfolded bands [20]. As an example, 1T -TaS2, another CDW compound known for very large atomic displacements [22] (of or- der > 0.1 Å) introduces hardly detectable backfolding of spectral weight in ARPES. Clearly, an electronic origin is necessary for obtaining such strong backfolding in the presence of such small atomic displacements. Therefore, our results allow to rule out a Jahn-Teller effect as the driving force for the transition of TiSe2. Furthermore, the ARPES spectra also show evidence for the backfolded conduction band at the Γ̄ point. Fig. 4 shows constant energy cuts around the Fermi energy, taken from the data of fig. 2b and 3b (arrows). In the ARPES data two slightly dispersive peaks, reproduced in the calculation, clearly cross the Fermi level. These features turn out to be the populated tail of the back- folded conduction band, whose centroid is located just above the Fermi level. To our knowledge no evidence for the backfolding of the conduction band had been put forward so far. In summary, by comparing ARPES spectra of 1T - TiSe2 to theoretical predictions for an excitonic insula- tor, we have shown that the superperiodicity of the ex- citonic state with respect to the lattice results in a very large transfer of spectral weight into backfolded bands. This effect, clearly evidenced by photoemission, turns out to be a characteristic feature of the excitonic insula- tor phase, thus giving strong evidence for the existence of this phase in 1T -TiSe2 and its prominent role in the CDW transition. Skillfull technical assistance was provided by the work- shop and electric engineering team. This work was sup- ported by the Fonds National Suisse pour la Recherche Scientifique through Div. II and MaNEP. [1] J. A. Wilson et al., Adv. Phys. 24, 117 (1975). [2] F. Clerc et al., Phys. Rev. B 74, 155114 (2006). [3] F. J. Di Salvo et al., Phys. Rev. B 14, 4321 (1976). [4] M. Holt et al., Phys. Rev. Lett. 86, 3799 (2001). [5] T. Pillo et al., Phys. Rev. B 61, 16213 (2000). [6] T. E. Kidd et al., Phys. Rev. Lett. 88, 226402 (2002). [7] K. Rossnagel et al., Phys. Rev. B 65, 235101 (2002). [8] E. Morosan et al., Nature Physics 2, 544 (2006). [9] D. Malterre et al., Adv. Phys. 45, 299 (1996). [10] W. Kohn, Phys. Rev. Lett. 19, 439 (1967). [11] B. I. Halperin and T. M. Rice, Rev. Mod. Phys. 40, 755 (1968). [12] D. Jérome et al., Phys. Rev. 158, 462 (1967). [13] F. X. Bronold and H. Fehske, Phys. Rev. B 74, 165107 (2006). [14] The fit parameters are : ǫ0v=-0.08±0.005 eV, mv=- 0.23±0.02 me, where me is the free electron mass, tv=0.06±0.005 eV ; ǫ c=-0.01±0.0025 eV, m c=5.5±0.2 me, m c=2.2±0.1 me, tc=0.03±0.0025 eV [15] O. Anderson et al., Phys. Rev. Lett. 55, 2188 (1985). [16] C. Monney et al., to be published [17] J. A. Wilson et al., Phys. Rev. B 18, 2866 (1978). [18] C. Didiot et al., Phys. Rev. B 74, 081404(R) (2006). [19] C. Battaglia et al., Phys. Rev. B 72, 195114 (2005). [20] J. Voit et al., Science 290, 501 (2000). [21] M. H. Whangbo and E. Canadell, J. Am. Chem. Soc. 114, 9587 (1992). [22] A. Spijkerman et al., Phys. Rev. B 56, 13757 (1997).

0704.0160

Oxygen-rich droplets and the enrichment of the ISM

Oxygen-rich droplets and the enrichment of the ISM Grażyna Stasińska1, Guillermo Tenorio-Tagle2, Mónica Rodŕıguez2, William J. Henney3 1 LUTH, Observatoire de Paris-Meudon, 5 Place Jules Jansen, 92195 Meudon, France 2 Instituto Nacional de Astrof́ısica Óptica y Electrónica, AP 51, 72000, Puebla, Mexico 3Centro de Radioastronomı́a y Astrof́ısica, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 3-72, 58090 Morelia, Mexico We argue that the discrepancies observed in HII regions between abundances derived from optical recombination lines (ORLs) and collisionally excited lines (CELs) might well be the signature of a scenario of the enrichment of the interstellar medium (ISM) proposed by Tenorio- Tagle (1996). In this scenario, the fresh oxygen released during massive supernova explosions is confined within the hot superbubbles as long as supernovae continue to explode. Only after the last massive supernova explosion, the metal-rich gas starts cooling down and falls on the galaxy within metal-rich droplets. Full mixing of these metal-rich droplets and the ISM occurs during photoionization by the next generations of massive stars. During this process, the metal-rich droplets give rise to strong recombination lines of the metals, leading to the observed ORL-CEL discrepancy. (The full version of this work is submitted to Astronomy and Astrophysics.) 1. Introduction There is no doubt that galaxies suffer chemical enrichment during their lives (see e.g. Cid Fernandes et al. 2006 for a recent systematic approach using a large data base of galaxies from the Sloan Digital Survey Data Release 5 – Adelman-Mac Carthy J.K. et al., 2007). The main source of oxygen production has since long been identified as due to supernovae from massive stars (type II supernovae). Yet, the exact process by which chemical enrichment proceeds is poorly known (see a review by Scalo & Elmegreen 2004). Ten years ago, Tenorio-Tagle (1996, hereafter T-T96) proposed a scenario in which the metal-enhanced ejecta from supernovae follow a long excursion in galactic haloes before falling down on the galaxies in the form of oxygen-rich droplets. In the present work (the full version of which has been submitted to Astronomy & Astrophysics, Stasińska et al. 2007), we suggest that the discrepancy between the oxygen abundances derived from optical recombination lines (ORLs) and from collisionally ex- cited lines (CELs) in HII regions (see e.g. Garćıa-Rojas et al. 2006 and references therein) might well be the signature of those oxygen-rich droplets. In fact, Tsamis et al. (2003) and Péquignot & Tsamis (2005) already suggested that the ORL/CEL discrepancy in HII regions is the result of inhomegeneities in the chemical composition in these objects. Our aim is to explicit the link between the ORL/CEL discrepancy and the T-T96 sce- nario, and to check whether what is known of the oxygen yields allows one to explain the ORL/CEL discrepancy in a quantitative way. 2. The Tenorio-Tagle (1996) scenario Figures 1–5 present the T-T96 scenario in cartoon format. http://arxiv.org/abs/0704.0160v1 2 Stasińska et al.: Oxygen-rich droplets and the enrichment of the ISM Figure 1. Sketch of the T-T96 scenario: At time t=0, a burst of star formation occurs and a giant HII region forms. Figure 2. Sketch of the T-T96 scenario: During the next ∼ 40 Myr, supernovae explode, creating a hot superbubble confined within a large expanding supershell that bursts into the galactic halo. The superbubble contains the matter from the oxygen-rich supernova ejecta mixed with the matter from the stellar winds and with the matter thermally evaporated from the surrounding supershell. 3. The ORL-CEL discrepancy in the context of the T-T96 scenario The details of the physical arguments concerning the amount of oxygen available in the droplets, the mixing processes, as well as the simulation of the ORL-CEL discrepancy with a multizone photoionization model are described in Stasińska et al. (2007). Here, we simply give the most important conclusions. Photoionization of the oxygen-rich droplets predicted by the T-T96 scenario can repro- duce the observed abundance discrepancy factors (ADFs, i.e. the ratios of abundances obtained from ORLs and from CELs) derived for Galactic and extragalactic HII regions. The recombination lines arising from the highly metallic droplets thus show mixing at work. Stasińska et al.: Oxygen-rich droplets and the enrichment of the ISM 3 Figure 3. Sketch of the T-T96 scenario: after the last supernova has exploded, the gas in the superbubble begins to cool down. Loci of higher densities cool down quicker. Due to a sequence of fast repressurizing shocks, this leads to the formation of metal-rich cloudlets. The cooling timescale is of the order of 100 Myr. Figure 4. Sketch of the T-T96 scenario: The now cold metal-rich cloudlets fall unto the galactic disk. They are further fragmented into metal-rich droplets by Raighleigh-Taylor instabilities. This metal-rich rain affects a region whose extension is of the order of kiloparsecs, i.e. much larger than the size of the initial HII region We find that, if our scenario holds, the recombination lines strongly overestimate the metallicities of the fully mixed HII regions. The collisionally excited lines may also over- estimate them, although in much smaller proportion. In absence of any recipe to correct for these biases, we recommend to discard objects showing large ADFs to probe the chemical evolution of galaxies. 4 Stasińska et al.: Oxygen-rich droplets and the enrichment of the ISM Figure 5. Sketch of the T-T96 scenario: When a next generation of massive stars form, they photoionize the surrounding interstellar medium, including the metal-rich droplets. It is only after the droplets have been photoionized that their matter is intimately mixed with the matter from the ISM, and that proper chemical enrichment has occured. The whole process since the explosion of the supernovae that provided fresh oxygen has taken at least 100 Myr. To proceed further with this question of inhomogeneities, one needs as many observa- tional constraints as possible. On the theoretical side, one needs more robust estimates of the integrated stellar yields as well as a better knowledge of the impact of massive stars on the ISM and of the role of turbulence. All these issues are relevant to our understanding of the metal enrichment of the Universe. REFERENCES Adelman-Mac Carthy J.K. et al., 2007, in preparation Cid Fernandes, R., Vala Asari, N., Sodré Jr. L., Stasińska, G., Mateus, A., Torres-Papaqui, J.P., Schnoell, W., 2006, MNRAS in press (astro-ph/0610815) Garćıa-Rojas, J., Esteban, C., Peimbert, M., Costado, M. T., Rodŕıguez, M., Peimbert, A., & Ruiz, M. T. 2006, MNRAS, 368, 253 Scalo, J., & Elmegreen, B. G. 2004, ARA&A, 42, 275 Stasińska, G., Tenorio-Tagle, G., Rodŕıguez, M., Henney, W.J., 2007, A&A submitted Tenorio-Tagle, G. 1996, AJ, 111, 1641 (T-T96) Tsamis, Y. G., Barlow, M. J., Liu, X.-W., Danziger, I. J., & Storey, P. J. 2003, MNRAS, 338, Tsamis, Y. G., & Péquignot, D. 2005, MNRAS, 364, 687 http://arxiv.org/abs/astro-ph/0610815 Introduction The Tenorio-Tagle (1996) scenario The ORL-CEL discrepancy in the context of the T-T96 scenario

0704.0161

Soft modes and NTE in Zn(CN)2 from Raman spectroscopy and first principles calculations

Microsoft Word - word_Zn_CN__high_pressure Soft modes and NTE in Zn(CN)2 from Raman spectroscopy and first principles calculations T. R. Ravindran*, A. K. Arora, Sharat Chandra, M. C. Valsakumar and N. V. Chandra Shekar Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India We have studied Zn(CN)2 at high pressure using Raman spectroscopy, and report Gruneisen parameters of the soft phonons. The phonon frequencies and eigen vectors obtained from ab-initio calculations are used for the assignment of the observed phonon spectra. Out of the eleven zone-centre optical modes, six modes exhibit negative Gruneisen parameter. The calculations suggest that the soft phonons correspond to the librational and translational modes of C≡N rigid unit, with librational modes contributing more to thermal expansion. A rapid disordering of the lattice is found above 1.6 GPa from X-ray diffraction. PACS numbers: 62.50.+p, 63.20.Dj, 78.30.–j, 78.20.Bh *Corresponding author: Email: [email protected] Interest in materials that exhibit negative thermal expansion (NTE) was renewed after the report [1] of high and isotropic NTE in Zr(WO4)2 over a wide temperature range, leading to extensive work and several reviews on the subject [2-5]. The structure of Zr(WO4)2 and several other NTE materials consist of corner sharing tetrahedral and octahedral units. From Raman spectroscopic investigations on Zr(WO4)2 as a function of pressure and temperature, the phonons responsible for NTE have been identified, and it has been shown that in addition to the librational (rigid-unit) mode at 5 meV, several other phonons of much higher energy also contribute significantly to the NTE in this material [6-9]. Based on structural analysis transverse displacements of the shared oxygen atoms and consequent rotation of polyhedra [1] was suggested as the cause of NTE in Zr(WO4)2. In the context of corner linked structures, Zn(CN)2 is remarkable, as it has C≡N as the linking species between tetrahedral units instead of a single atom and exhibits twice as much coefficient of NTE (-17x10-6 K-1) [10] as that of Zr(WO4)2. The structure of Zn(CN)2 consists of three-dimensional, inter-penetrating, tetrahedral frameworks of Zn-CN-Zn chains [11]. Two different cubic structures, mP 34 (CN- ordered) [11] and mPn3 (CN-disordered) [10] have been reported to fit well to the diffraction patterns. In the ‘ordered structure’ the CN ions lying along the body diagonal are orientationally ordered such that they form ZnC4 and ZnN4 coordination tetrahedra around alternate cations. On the other hand, in the ‘disordered structure’, C and N atoms are randomly flipped so as to occupy the sites with equal probability. It was shown recently from factor group analysis in conjunction with Raman and IR spectroscopic measurements that the structure is indeed disordered [12]. From a topological model treating ZnN4/ZnC4 as rigid units the structure was argued to support a large number of low frequency rigid unit phonon modes (ωph< 2 THz, ≈70 cm-1) that contribute to NTE [13]. On the other hand, it was shown recently by spectroscopic measurements [12] that the lowest energy optical mode in Zn(CN)2 is an IR mode at 178 cm-1. It has been shown by atomic pair distribution function analysis of X- ray diffraction data and suitable modelling of the displacements of C/N away from the body-diagonal that this displacement increases as a function of temperature [14]. This is in any case expected from the increased amplitude of atomic vibrations as temperature is increased. However, there is no report of the role of different phonons to thermal expansion. Since phonon modes and their Gruneisen parameters are directly responsible for thermal expansion in a material, it becomes vitally important to study them. Here we report the first study of phonons in Zn(CN)2 at high pressure using Raman spectroscopy and ab-initio calculations. High pressure X-ray diffraction measurements are also carried out for obtaining the bulk modulus to calculate Gruneisen parameters. From high pressure Raman measurements soft phonons are identified. In addition, first-principles ab-initio density functional calculations are performed at different volumes and phonon dispersion curves obtained using frozen phonon approximation with SIESTA code [15]. The phonon eigen vectors are used for the assignment of phonon modes. The thermal expansion coefficient is calculated from Gruneisen parameters of all the phonons and compared with the reported value. Zn(CN)2 (>99.5%) was obtained from Alfa Aesar. X-ray diffraction pattern of this powder sample showed no observable impurity phases. A small piece of sample of lateral dimensions ~100 μm was loaded into a gasketed, Mao-Bell type diamond anvil cell. Raman spectra were recorded at different pressures in the backscattering geometry using the 488-nm line of an argon ion laser. Methanol-ethanol (4:1) mixture was used as pressure transmitting medium. Ruby fluorescence was used to measure pressure. Scattered light from the sample was analyzed by a SPEX double monochromator, and detected with a cooled photomultiplier tube operated in the photon counting mode. Scanning of the spectra and data acquisition were carried out using a PSoC (Programmable System on Chip) hardware controlled by LabVIEW® 7.1 program [16]. The spectral range covered was 10-2400 cm-1 that also includes the C≡N stretch mode around 2220 cm-1. High pressure X-ray diffraction (HPXRD) was carried out in an angle dispersive mode using Guinier diffractometer [17]. The incident Mo Kα1 radiation is obtained from a Rigaku 18 kW rotating anode x-ray generator. Ab-initio calculations were carried out in the framework of the density functional theory using the Perdew–Burke–Ernzerhof generalized gradient approximation for exchange and correlation [18]. A 3×3×3 supercell of Zn(CN)2 unit cell was used for determining the relaxed atomic configuration, and phonon frequencies calculated using the SIESTA code. The calculations were performed using a Monkhorst-Pack grid of 8×8×8 k-points with a shift of 0.5. The energy cut off was 350 Rydbergs and a double zeta plus polarization (DZP) basis set was used. Standard norm-conserving, fully relativistic Troullier-Martins TM2 pseudopotentials were used. The computations were performed in a 16-node linux cluster. The 30 degrees of freedom arising from the 10 atoms in the cubic unit cell of Zn(CN)2 result in 3 acoustic and 27 optical branches. Out of the three structural units viz., C≡N ion, ZnC4 tetrahedron and ZnN4 tetrahedron, C≡N is the most strongly bound unit and hence taken as a ‘rigid molecular unit’. The 6 degrees of freedom corresponding to the linear molecular ion C≡N can be divided into 1-internal (stretching vibration), 3 rigid-translations and 2 rigid rotational degrees of freedom. The ‘disordered’ structure of zinc cyanide has the following irreducible representations of optical phonons [12]: .OptΓ = A1g + Eg + F1g + 3F2g + A2u + Eu + 2F1u + F2u Out of these, the A1g, Eg and F2g modes are Raman active and F1u mode is IR active. The remaining four modes are optically inactive. Figure 1 shows the Raman spectra of Zn(CN)2 at several pressures including ambient. There are three Raman modes clearly seen at 2221, 342, and 200 cm-1. Out of these, the asymmetric peak about 342 cm-1 is actually a doublet that can be resolved into 339 and 343 cm-1 [12]. The linewidth of all three modes increase and their intensities reduce at high pressures. While the C≡N stretch mode about 2220 cm-1 hardens as pressure is increased, the other two modes are seen to soften. The modes at 342 cm-1 and 200 cm-1 are too weak to follow above 1 GPa. Figure 2 depicts the phonon frequency (ω) vs. pressure (P) for the three modes observed by Raman spectroscopy. Most measurements were carried out under hydrostatic conditions using (methanol + ethanol) as pressure transmitting medium. One set of measurements in which no medium was used (open circles in Fig. 2) resulted in a weaker P dependence for the 2220 cm-1 mode up to a pressure of 1.5 GPa and a negative coefficient above 2 GPa. The reason for this change of slope from positive to negative could be a structural transition occurring under non-hydrostatic pressure. However, high pressure X-ray diffraction measurements (discussed later) under hydrostatic pressure have not indicated any structural phase transition between 1.5 and 2 GPa or at any pressure up to 5.2 GPa, the highest pressure up to which measurements were made. X-ray diffraction patterns were recorded at several pressures up to 5.2 GPa. Only three reflections, viz., (110), (211) and (321) could be observed. As pressure is increased, the intensity of all lines reduces drastically. At a pressure as low as 0.2 GPa the intensities of the peaks reduce by about 50%. Above 0.6 GPa the (321) line disappears and above 1.6 GPa, only the (110) line is present, which continues up to the highest pressure of 5.2 GPa, indicating possible disordering of C≡N has taken place above 1.6 GPa. Such a partial/sublattice amorphization has been reported earlier also in other compounds [19, 20]. Lattice parameters at several pressures were obtained from the three lines using a disordered cubic space group ( mPn3 ) structure. The unit cell volume obtained as a function of pressure was fitted to Murnaghan equation of state and resulted in a bulk modulus B0=25±11 GPa. The large error in B0 is due to the scatter in the XRD data and also the small number of reflections that were used to calculate the lattice parameters. With this input of B0 the mode Gruneisen parameters (γi = B0ωi−1 ∂ωi/∂P) of the three Raman modes could be calculated (Table 1, last column). In the absence of γi values of other vibrational modes, the thermal expansion coefficient has been calculated using simulation data as detailed in the next paragraph. It is not straight forward to incorporate random disorder in ab-initio calculations. When such a disorder is introduced by randomly flipping half the C≡N species in the supercell, it is found that for this disordered structure of cubic Zn(CN)2 - when the system is allowed to relax - the ground state energy does not converge to a stable configuration but evolves into a tetragonal structure (space group nmP 24 ) with c-parameter ~0.5% larger than the a- and b-parameters. Upon further relaxation, the structure slowly becomes triclinic. Additionally, the inter-atomic forces are large and do not converge to small values. On the other hand, for the ‘ordered structure’ the forces converged to values less than 10-6 eV/Å due to geometrical considerations. Hence the ordered structure of Zn(CN)2 is used for computational purposes. It should be pointed out that the values of the vibrational frequencies obtained from either of the space groups are not expected to be different from each other, since the same kind of atomic motions are involved in the vibrational modes. The number of zone-centre optical phonon modes is also the same in either space group. The total energy of the system was computed in the relaxed configuration for different volumes of the cell up to V/V0=0.844. The energy vs. volume data was fitted to Murnaghan equation and the bulk modulus obtained is 88 GPa. A similar result (90 GPa) is obtained when WIEN2K is used to calculate the bulk modulus. Phonon dispersion curves at different volumes were calculated using the frozen-phonon method using the VIBRA module in the SIESTA package. Eigen frequencies for the various modes were obtained by diagonalizing the dynamical matrix. The phonon dispersion curves obtained at ambient volume from simulations are shown in Figure 3. Eigenvectors were viewed using the Visual Molecular Dynamics (VMD) package [21]. The highest compression corresponds to a pressure of 8.3 GPa. From the pressure dependence of the various zone centre optical phonons (inset in Fig. 2) the mode Gruneisen parameters were obtained [Table I]. Using Einstein’s specific heat Ci = R [xi2exp(xi)]/[exp(xi)-1]2, where xi=ħωi/kBT, for the various modes the total specific heat CV was obtained. Here R is the universal gas constant. Thermal expansion coefficient α=(γavCV)/(3VmB0), (where γav=½∑piCiγi)/CV, pi are the degeneracies of the respective ωi phonon branches at the Brillouin zone centre, Vm is the molar volume and B0 taken as 88 GPa) is calculated to be -22×10-6 K-1, in good agreement with the reported value. In view of the non-availability of polarized Raman measurements on oriented single crystals of Zn(CN)2, the observed modes were assigned (Table I) based on eigen vectors of calculated phonons. The CN stretching mode at 2200 cm-1 can be assigned to A1g. In the internal mode region A1g and F2g modes arise due to correlation splitting [12] and are often degenerate. It is noteworthy that six out of the eleven optical modes exhibit negative Gruneisen parameters. Furthermore, all the modes of energy lower than 360 cm- 1 have negative γi. Figure 4 shows the displacement vectors of different atoms for the phonons that exhibit large negative γi. The 143 cm-1 mode corresponds to translational motion of CN ions whereas the other three modes involve librations of CN ions about the axis joining Zn-Zn’ atoms. The difference in the values of the calculated and the observed values of γi could partly arise from the different values of B0 used. However, this does not affect the calculation of α, since B0 gets cancelled in the definition of α. Further, the total Gruneisen parameter (∑piγi) for the C≡N librational modes is -57 whereas for the translational modes this value is -41. As mentioned earlier, using a topological model the network structure of Zn(CN)2 has been argued to have a large number of low-frequency rigid units modes of ZnC4/ZnN4 in analogy with Zr(WO4)2. On the other hand, in the present lattice dynamical calculations the lowest frequency mode turns out to be a CN-translational mode. This is because the topological model treated ZnC4/ZnN4 as rigid units, whereas actually only the strongly bound CN ions should be considered as rigid units. Recent atomic pair distribution function analysis shows that the displacements of C/N away from the line joining Zn…Zn’ increases as a function of temperature. Though this appears physically reasonable, this displacement, when extrapolated to 0 K, remains as large as 0.42 Å (Fig.9 of Ref.[14]) suggesting inconsistency between the Rietweld refined structure and that obtained from PDF analysis. Furthermore, the reason for Zn…Zn’ distance (which is directly related to the lattice parameters) estimated from PDF analysis being different from that obtained from XRD analysis [14] remains unclear. On the other hand, the present phonon calculations and Raman measurements at high pressure provide the first insight into the relative role of the different phonons in causing negative thermal expansion in Zn(CN)2. In conclusion, we have identified the optical phonons responsible for NTE in Zn(CN)2 from high pressure Raman spectroscopic studies and from first principles density functional simulation studies at different volumes. Gruneisen parameters of all the vibrational modes were obtained from simulations. A large number of phonon modes in Zn(CN)2 are soft, and all contribution to NTE arises from C≡N librational and translational modes. The value of thermal expansion coefficient α calculated from the Gruneisen parameters is in good agreement with experimental value. X-ray diffraction investigations suggest growth of disorder at high pressure. References: 1. J. S. O. Evans, T. A. Mary, T. Vogt, M. A. Subramanian, and A. W. Sleight, Chem. Mater. 8, 2809 (1996). 2. M. G. Tucker, A. L. Goodwin, M. T. Dove, D. A. Keen, S. A. Wells, and J. S. O. Evans, Phys. Rev. Lett. 95, 255501 (2005) 3. A. W. Sleight, Curr. Opin. in Solid State Mater. Sci. 3, 128 (1998). 4. S. K. Sikka, J. Phys.: Condens. Matter 16, S1033 (2004). 5. G. D. Barrera, J. A. O. Bruno, T. H. K. Barron, and N. L. Allan, J. Phys.: Condens. Matter 17, R217 (2005). 6. T. R. Ravindran, A. K. Arora, and T. A. Mary, Phys. Rev. Lett. 84, 3879 (2000). 7. T. R. Ravindran, A. K. Arora, and T. A. Mary, Phys. Rev. Lett. 86, 4977 (2001) 8. T. R. Ravindran, A. K. Arora, and T. A. Mary, J. Phys.: Condens. Matter 13, 11573 (2001). 9. T. R. Ravindran, A. K. Arora, and T. A. Mary, Phys. Rev. B 67, 064301 (2003). 10. D. J. Williams, D. E. Partin, F. J. Lincoln, J. Kouvetakis, and M. O’Keefe, J. Solid State Chem. 134, 164 (1997) 11. B. F. Hoskins and R. Robson, J. Am. Chem. Soc. 112, 1546 (1990). 12. T. R. Ravindran, A. K. Arora, and T. N. Sairam, J. Raman Spectrosc. 38, 283 (2007). 13. A. L. Goodwin, and C. K. Kepert, Phys. Rev. B 71, R140301 (2005). 14. K. W. Chapman, P. J. Chupas, and C. J. Kepert, J. Am. Chem. Soc. 127, 15630 (2005) 15. J. M. Soler, E. Artacho, J. D. Gale, A. Garcia, J. Junquera, P. Ordejon and D. Sanchez-Portal, J. Phys.: Condens. Matter 14, 2745 (2002). 16. J. Jayapandian, R. Kesavamoorthy and A. K. Arora, J. Instrum. Soc. India (2007) in press. 17. P. Ch. Sahu, M. Yousuf, N. V. C. Shekar, N. Subramanian, and K. G. Rajan, Rev. Sci. Instrum. 66, 2599 (1995). 18. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 19. A. K. Arora, R. Nithya, T. Yagi, N. Miyajima and T.A. Mary, Solid State Commun. 129, 9 (2004) 20. J.B. Parise, J.S. Loveday, R.J. Nelmes, H. Kagi, Phys. Rev. Lett. 83, 328 (1999). 21. W. Humphrey, A. Dalke and K. Schulten, J. Mol. Graphics 14, 33 (1996) Table I Modes (deg. of freedom) metry Calc. freq. (cm-1) Obs. freq. (cm-1) Calc. Obs. γi Zn-trans. F2g 388 343 (R) 0.45 - F1u 143 178 (IR) -14.3 - Eu 255 Inactive -1.5 - F2g 352 216 (R) -0.13 -0.50 (15) A2u 564 Inactive 1.1 - CN-trans. (12) F1u 596 461 (IR) 1.4 - F1g 288 Inactive -8.0 - Eg 357 339 (R) -6.2 -0.54 CN-libr. F2u 326 Inactive -7 - F2g 2232 2218(IR) 1.5 0.14(1) CN-int. (4) A1g 2245 2221 (R) 1.5 - Figure and table captions: Table 1. Calculated and observed phonon frequencies in Zn(CN)2, their classification, mode assignments and Gruneisen parameters. Observed IR frequencies are from [12]. Figure 1. Raman spectra of Zn(CN)2 at several pressures. Spectra are scaled and shifted for clarity. The modes at 342 and 200 cm-1 could not be followed above 1 GPa due to weak intensities. Figure 2. Mode frequency vs. Pressure for the observed Raman modes in Zn(CN)2. Open symbols: results without pressure medium. The inset shows ω vs. P for all the eleven modes obtained from the phonon calculations. Though data were generated up to 8.3 GPa, the trend after the first three pressures (shown here) is non-linear, and hence not considered for obtaining γi. Figure 3. Phonon dispersion curves obtained from First Principles density functional simulations on a 3×3×3 supercell of Zn(CN)2. Note that the acoustic phonon branch interacts with the lowest energy optical phonon branch at 143 cm-1. Both the branches change character due to the non-crossing rule. Figure 4. Atomic displacements of vibrational modes corresponding to (a) 143 cm-1, (b) 288 cm-1, (c) 326 cm-1 and (d) 357 cm-1. The arrows fixed to the atoms are proportional to the amplitude of atomic motion. In the 326 cm-1 mode neighbouring Zn atoms also move (opposite direction) 200 250 300 350 2200 2250 CN-translation CN stretch CN libration x0.6x1.6 0.8 GPa 2.4 GPa 1.2 GPa 0.1 GPa Raman shift (cm-1) Figure 1. Ravindran et al Figure 2. Ravindran et al. 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 2 2 0 0 2 3 0 0 Figure 3. Ravindran et al. Figure 4. Ravindran et al.

0704.0162

Estimation of experimental data redundancy and related statistics

EPJ manuscript No. (will be inserted by the editor) Estimation of experimental data redundancy and related statistics Igor Grabec Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, PP 394, 1001 Ljubljana, Slovenia, Tel: +386 01 4771 605, Fax: +386 01 4253 135, E-mail: [email protected] Received: date / Revised version: date Abstract. Redundancy of experimental data is the basic statistic from which the complexity of a natural phenomenon and the proper number of experiments needed for its exploration can be estimated. The redundancy is expressed by the entropy of information pertaining to the probability density function of experimental variables. Since the calculation of entropy is inconvenient due to integration over a range of variables, an approximate expression for redundancy is derived that includes only a sum over the set of experimental data about these variables. The approximation makes feasible an efficient estimation of the redundancy of data along with the related experimental information and information cost function. From the experimental information the complexity of the phenomenon can be simply estimated, while the proper number of experiments needed for its exploration can be determined from the minimum of the cost function. The performance of the approximate estimation of these statistics is demonstrated on two–dimensional normally distributed random data. PACS. 06.20.DK Measurement and error theory – 02.50.+s Probability theory, stochastic processes, and statistics – 89.70.+c Information science 1 Introduction The basic task of experimental physical exploration of nat- ural phenomena is to provide quantitative data on mea- sured variables and, from them extract physical laws [1]. Related to this task, experimenters must decide how many experiments to perform in order to provide proper exper- http://arxiv.org/abs/0704.0162v1 2 Igor Grabec: Estimation of experimental data redundancy and related statistics imental data. We know that it is reasonable to repeat ex- periments as long as they yield essentially new data, and to stop repetition when the data become redundant. In order to describe this concept objectively, we have intro- duced in previous articles [2,3] two statistics called exper- imental information I and redundancy R of experimental data based on the entropy of information [4]. Their differ- ence C = R−I can be interpreted as the information cost function of the experimental exploration. From the cost function minimum, the proper number N◦ of experiments can be determined in an objective way. The entropy of in- formation is defined by the integral of a nonlinear function of the probability density function of experimental data, and consequently its calculation is numerically demand- ing. This property represents a serious obstacle, especially when treating multivariate data. Therefore, our aim is to show how this obstacle can be effectively avoided by es- timating data redundancy without integration. For this purpose we first briefly repeat the route to the definition of redundancy [2,3] and subsequently show how the inte- gral in the corresponding expression can be approximated. The performance of the derived approximate method of calculation is demonstrated using two–dimensional nor- mally distributed random data. 2 Redundancy of experimental data Let us consider a phenomenon characterized by N mea- surements of a variable x using an instrument with span Sx = (−L,L). Properties of the instrument are specified by calibration on a unit u. The probability density func- tion (PDF) of the instrument’s output scattering during calibration is described by the scattering function ψ(x, u). When the scattering is caused by mutually independent disturbances in the experimental system, the scattering function is Gaussian [1,4] : ψ(x, u) = g(x− xi, σ) = (x− u)2 . (1) We apply this function in our further treatment. The mean value u and standard deviation σ can be estimated statis- tically by repetition of calibration. Let xi denote the most probable instrument output in the i–th experiment. Using ψ(x, xi) we describe the properties of the explored phenomenon during the i–th experiment. Similarly, the properties in a series of N re- peated experiments, which yield the basic data set {xi; i = 1, . . . , N}, are described by the experimentally estimated fN (x) = ψ(x, xi). (2) In addition, we introduce a uniform reference PDF ρ(x) = 1/(2L) indicating that all outcomes of the experiment are hypothetically equally probable before executing the ex- periments. Based upon functions fN (x) and ρ(x) we describe the indeterminacy of variable x by the negative value of the relative entropy [5,6,7]: Hx = − f(x) log (fN (x) dx. (3) Similarly, we describe the uncertainty Hu of calibration performed on a unit u by: Hu = − ψ(x, u) log (ψ(x, u) dx. (4) Igor Grabec: Estimation of experimental data redundancy and related statistics 3 Using the difference of these statistics we define the ex- perimental information: I = Hx −Hu f(x) log(fN (x)) dx ψ(x, u) log(ψ(x, u)) dx. (5) Using Eq. 2 in this expression we get: I = log(N) − 1 ψ(x, xi) log ψ(x, xj) ψ(x, u) log ψ(x, u) dx. (6) If we express the logarithm in the second term as: ψ(x, xj) = logψ(x, xi) + log ψ(x, xj) ψ(x, xi) we obtain: I = log(N) + ψ(x, xi) log ψ(x, xi) ψ(x, xi) log ψ(x, xj) ψ(x, xi) ψ(x, u) log ψ(x, u) dx. (8) The second and the fourth term on the right side of this equation yield 0 and we get: I = log(N)− 1 ψ(x, xi) log ψ(x, xj) ψ(x, xi) With the last term we introduce the statistic called redun- dancy of data: ψ(x, xi) log ψ(x, xj) ψ(x, xi) dx (10) with which we get the basic relation: I = log(N)−R (11) If |xi − xj | ≫ σ for all pairs i#j, there is no overlapping of functions ψ(x, xi), ψ(x, xj); therefore, the sum in the logarithm is ∼ 0, and consequently the redundancy is R ∼ 0. In the opposite case, when |xi − xj | ≪ σ, it follows that ψ(x, xi) ∼ ψ(x, xj). Due to good overlapping in this case, the corresponding term in the expression of R yields log(2)/N and R > 0. This property indicates that experimental information is increasing with increasing N as I ∼ log(N) if the ac- quired data are well separated with respect to σ. However, with an increasing number of data, they are ever more densely distributed, which results in an increasing overlap- ping of distributions that causes increasing redundancy of measurements. Although the expression in Eq. 10 for re- dundancy R is rather cumbersome due to the included integral, we expect that R could be estimated without integration by the simpler function of distances between data points. For this purpose we next consider the prop- erties of the scattering function ψ(x, xi). If the Gaussian function ψ(x, xi) = g(x−xi, σ) is con- sidered as an approximation of the delta function δ(x−xi), and the logarithm as a slowly changing function, the inte- gration in Eq. 10 can be carried out, which yields for the redundancy the first order approximate expression with- out the integral: ψ(xi, xj) ψ(xi, xi) If we take into account Eq. 1, we get for the redundancy the following approximate expression that depends only 4 Igor Grabec: Estimation of experimental data redundancy and related statistics on standard functions of distances between data points: − (xi − xj) However, this first order approximation is rather rough because the distribution ψ(xi, xj) has the width σ > 0 and the logarithm in Eq. 10 includes the fraction of functions ψ(x, xj)/ψ(x, xi). To proceed to a better approximation, we have examined the case of just two data points, since it mainly determines the property of the redundancy. In this case the integration of the first three terms in a Taylor series expansion of the logarithm yields the second approx- imation: − (xi − xj) , (14) which is obtained from the previous one by merely chang- ing 2σ2 → 4σ2. This property indicates that a still better approximation could be obtained by properly adapting 2σ2 in Eq. 13. For this purpose we have proceeded with numerical investigations which have shown that a nearly optimal approximation is obtained if 2σ2 in Eq. 13 is re- placed by ∼ 5.1σ2: − (xi − xj) 5.1σ2 . (15) Numerical investigations have further shown that this for- mula also yields good results in cases with many data points. Since the integral is excluded from Eq. 15, the redun- dancy R can be estimated from Eq. 15 with essentially less computational effort than from Eq. 10. This advantage is especially outstanding in a multivariate case where the redundancy is defined by multiple integrals, while in the approximate formula in Eq. 15 only the term (xi − xj) in the exponential function has to be replaced by the norm of corresponding vectors. Due to this advantage, it is also reasonable to estimate approximately the experimental in- formation using the basic formula I = log(N) − R. The experimental information I converges with the increasing number of data N to a certain limit value from which the complexity of the phenomenon under investigation can be estimated using the formula K ≈ exp(IN→∞) introduced previously [2,3]. The complexity K indicates how many non–overlapping scattering distributions are needed in the estimator Eq. 2 to describe the PDF of the observed phe- nomenon. The information cost function is the difference of the redundancy and experimental information: C = R − I. During minimization of this cost, the experimental infor- mation provides for a proper adaptation of the PDF es- timator to the experimental data, while the redundancy prevents excessive growth of the number of data points. By the position of the cost function minimum we introduce the proper number No of the data and the corresponding experiments that are needed to judiciously represent the phenomenon under exploration. By inserting the expres- sion I = log(N) − R into C = R − I, we obtain for the information cost function the formula: C = 2R− log(N). (16) Therefore the proper number No can also be determined from the approximately estimated redundancy Ro. This number roughly corresponds to the ratio between the mag- nitude of the characteristic region where experimental data Igor Grabec: Estimation of experimental data redundancy and related statistics 5 appear and the magnitude of the characteristic region cov- ered by the scattering function [2,3]. 3 Numerical examples To demonstrate the properties of the approximations R1, R2, Ro let us first consider the case of just two data points separated by a distance x1 − x2. Fig. 1 shows the dependence of redundancy R on relative distance d = (x1−x2)/σ as determined by the integral in Eq. 10 and ap- proximations in Eqs. 13,14,15. Improvement achieved by subsequent steps of approximation and a fairly good agree- ment between approximation Ro and R calculated by the integral is evident. However, in a case with more data points we can generally expect slightly worse agreement due to overlapping of more than two scattering functions in the sum of the approximation formula in Eq. 15. The performance in such a case is demonstrated in the next example. In order to provide for reproduction of the demon- strated example, we consider a two–dimensional Gaussian random phenomenon with zero mean value. The stan- dard deviation of both components is equal to s = 2.5, while their covariance is zero. The data generated by a standard Gaussian generator are represented in the two- dimensional span (−10,+10)⊗ (−10,+10) using the scat- tering width σ = 0.5. In such a case we can theoretically predict that the proper number of data samples should be No ≈ (s/σ)2 = 25. For the demonstration, a set of Nmax = 100 two- dimensional data samples {(xi, yi); i = 1 . . .Nmax} was 0 1 2 3 4 5 6 Fig. 1. Dependence of redundancy R on relative distance d = (x1−x2)/σ between data points as determined by the integral in Eq. 10, and approximations in Eqs. 13,14,15. 0.005 0.015 0.025 0.035 Fig. 2. PDF determined by 100 data points xi, yi. generated. The corresponding probability density function was estimated using Eq. 2 adapted to the two–dimensional case with statistically independent components: fN (x, y) = ψ(x, xi)ψ(y, yi). (17) The resulting PDF with N = 100 is graphically repre- sented in Fig. 2. 6 Igor Grabec: Estimation of experimental data redundancy and related statistics 0 10 20 30 40 50 60 70 80 90 100 REDUNDANCY VERSUS NUMBER OF DATA Fig. 3. Dependence of redundancy R on number N of data points as determined by the integral in Eq. 10 – (R), and ap- proximation in Eq. 15 – (Ro) adapted to the two–dimensional case. From the generated data the redundancy was calcu- lated using Eqs. 10 and 15 adapted to the two–dimensional case. The dependence of redundancy R on the number N of accounted data points is shown in Fig. 3. Fairly good agreement between both statistics is again evident. Approximately estimated redundancy was further uti- lized in the calculation of statistics I and C. They are shown as functions of the number of data points N in Fig. 4 together with R(N) and log(N). Agreement with the same statistics calculated more exactly by integration can be established by comparing this figure with Fig. 4. In both cases we obtain for the proper number the value No = 28. This value depends on the statistical properties of the data set used in its calculation; a statistical esti- mation from 100 different data sets yields the estimate No ≈ 25±13 which agrees well with the theoretically pre- 0 10 20 30 40 50 60 70 80 90 100 INFORMATION STATISTICS VERSUS NUMBER OF DATA log( N) Fig. 4. Dependence of information statistics on the number N of data points as approximately determined from Eq. 15. The minimum of the cost function occurs at N = 28. 0 10 20 30 40 50 60 70 80 90 100 log( N) Fig. 5. Dependence of information statistics on the number N of data points as determined based on integration. dicted value No = 25. Similarly as in the one–dimensional case [2], it turns out that the function fNo(x, y) is only a rough estimator of the hypothetical PDF. This property is a consequence of the fact that experimental information I and redundancy R have equal weights in the cost function C = R− I. Igor Grabec: Estimation of experimental data redundancy and related statistics 7 Figs. 4 and 5 indicate that experimental information I converges with increasing N to a certain limit value from which the complexity of the phenomenon under investiga- tion can be approximately estimated as K ≈ exp INmax . In our case we get the estimate K ≈ 21. The number of non–overlapping scattering distributions that represent the PDF of the observed phenomenon is thus slightly smaller than the proper number No of experiments needed for its exploration. 4 Conclusions From the statistics introduced in the previous articles [2,3] based on information entropy, we have here derived an ap- proximate formula for the calculation of redundancy R of experimental data. It is important that this formula does not include the integral by which the information entropy is defined. This makes feasible a simplified and fairly good estimation of redundancy and, with it, the related exper- imental information and cost function. The advantage of the approximate calculation becomes outstanding in mul- tivariate cases because multiple integration is not needed there. A serious obstacle for the application of the con- cept of experimental information and redundancy of data can thus be avoided. Efficient estimation of the experi- mental information and cost function, and with them the determined complexity of the phenomenon and the proper number of experiments needed for its exploration, could be considered valuable in planning experimental work. In addition, the complexityK or the proper numberNo could be applied in the field of neural networks [1,8] to deter- mine the appropriate number of cells needed to deal with a certain phenomenon. Acknowledgment This work was supported by the Ministry of Higher Edu- cation, Science and Technology of the Republic of Slovenia and EU – COST. References 1. I. Grabec and W. Sachse, Synergetics of Measurement, Pre- diction and Control (Springer-Verlag, Berlin, 1997). 2. I. Grabec, Experimental modeling of physical laws, Eur. Phys. J., B, 22 129-135 (2001) 3. I. Grabec, Extraction of physical laws from joint exper- imental data, Eur. Phys. J., B, 48 279-289 (2005) (DOI: 10.1140/epjb/e2005-00391-0) 4. J. C. G. Lesurf, Information and Measurement (Institute of Physics Publishing, Bristol, 2002) 5. T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, New York, 1991). 6. A. N. Kolmogorov, IEEE Trans. Inf. Theory, IT-2 102-108 (1956). 7. D. J. C. MacKay, Information Theory, Inference, and Learn- ing Algorithms (Cambridge University Press, Cambridge, UK, 2003) 8. S. Haykin, Neural Networks, (Prentice Hall International, Inc., Upper Saddle River, New Jersey, 1999) Introduction Redundancy of experimental data Numerical examples Conclusions

0704.0163

Effective potentials for quasicrystals from ab-initio data

Effective potentials for quasicrystals from ab-initio data Peter Brommer∗and Franz Gähler Institut für Theoretische und Angewandte Physik Universität Stuttgart November 2, 2018 Abstract Classical effective potentials are indispensable for any large-scale atomistic sim- ulations, and the relevance of simulation results crucially depends on the quality of the potentials used. For complex alloys like quasicrystals, however, realistic effective potentials are practically inexistent. We report here on our efforts to develop effec- tive potentials especially for quasicrystalline alloy systems. We use the so-called force matching method, in which the potential parameters are adapted so as to optimally reproduce the forces and energies in a set of suitably chosen reference configurations. These reference data are calculated with ab-initio methods. As a first application, EAM potentials for decagonal Al-Ni-Co, icosahedral Ca-Cd, and both icosahedral and decagonal Mg-Zn quasicrystals have been constructed. The influence of the po- tential range and degree of specialisation on the accuracy and other properties is discussed and compared. Keywords: force matching; quasicrystal; effective potential; molecular dynamics; ab initio 1 Introduction Large-scale molecular dynamics simulations are possible only with classical effective po- tentials, which reduce the quantum-mechanical interactions of electrons and nuclei to an effective interaction between the atom cores. The computational task is thereby greatly simplified. Whereas ab-initio simulations are limited to a few hundred atoms at most, classical simulations can be done routinely with multi-million atom systems. For many purposes, such system sizes are indispensable. For example, fracture studies of quasicrys- tals require samples with several million atoms at least [1]. Diffusion studies, on the other ∗e-mail: [email protected] hand, can be done with a few thousand atoms (or even less), but require very large simu- lated times of the order of nanoseconds [2], which also makes them infeasible for ab-initio simulations. While physically justified effective potentials have been constructed for many elemen- tary solids, such potentials are rare for complex intermetallic alloys. For this reason, molec- ular dynamics simulations of these materials have often been done with simple model po- tentials, resulting in rather limited reliability and predictability. In order to make progress, better potentials are needed to accurately simulate complex materials. The force matching method [3] provides a way to construct physically reasonable po- tentials also for more complex solids, where a larger variety of local environments has to be described correctly, and many potential parameters need to be determined. The idea is to compute forces and energies from first principles for a suitable selection of small refer- ence systems, and to fit the potential parameters so that they optimally reproduce these reference data. Hereafter, potentials generated in this way will be referred to as fitted potentials. Thus, the force matching method allows to make use of the results of ab-initio simulations also for large-scale classical simulations, thereby bridging the gap between the sample sizes supported by these two methods. 2 Force Matching As we intend to construct potentials for complex intermetallic alloys, we have to assume a functional form which is suitable for metals. A good choice are EAM (Embedded Atom Method) potentials [4], also known as glue potentials [5]. Such potentials have been used very successfully for many metals, and are still efficient to compute, even though they include many-body terms. In contrast, pure pair potentials show a number of deficiencies when it comes to describe metals [5]. The functional form of EAM potentials is given by i,j<i φkikj(rij) + Uki(ni), with ni = j 6=i ρkj(rij), (1) where φkikj is a pair potential term depending on the two atom types kl. Uki describes the embedding term that represents an additional energy for each atom. This energy is a function of a local density ni determined by contributions ρkj of the neighbouring atoms. It is tempting to view this as embedding each atom into the electron sea provided by its neighbours. Such an interpretation is not really meaningful, however. The potential (1) is invariant under a family of “gauge” transformations [5], by which one can move contributions from the embedding term to the pair term, and vice versa, so that it makes little sense to give any of them an individual physical interpretation. In order to allow for maximal flexibility, and to avoid any bias, the potential functions in (1) are represented by tabulated values and spline interpolation, the tabulated values acting as potential parameters. This makes it unnecessary to guess the right analytic form beforehand. The sampling points can be chosen freely, which is useful for functions which vary rapidly in one region, but only slowly in another region. The forces and energies in the reference structures are computed with VASP, the Vi- enna Ab-Initio Simulation Package [6, 7], using the Projector Augmented Wave (PAW) method [8, 9]. Like all plane wave based ab-initio codes, VASP requires periodic bound- ary conditions. For quasicrystals, this means that periodic approximants have to be used as reference structures. As ab-initio methods are limited to a few hundred atoms, those approximants must be rather small. For the systems studied so far, this was not a major problem, as the relevant local environments in the quasicrystal all occur also in reasonably small approximants. Icosahedral quasicrystals with F-type lattice may be more problem- atic in this respect. For these, small approximants are rare, and the force matching method requires a sufficient variety of reference structures. Given the reference data (forces, energies, and stresses in the reference structures), the potential parameters (in our case: up to about 120 EAM potential sampling points for spline interpolation) then are optimised in a non-linear least square fit, so that the fitted potential reproduces the reference data as well as possible. The target function to be minimised is a weighted sum of the squared deviations between the reference data, denoted by the subscript 0 below, and the corresponding data computed from the fitted effective potential. It is of the form Z = ZF + ZC, with (2) α=x,y,z (fjα − f0,jα) f 20,j + εj , and ZC = (Ak − A0,k) A20,k + εk , (3) where ZF represents the contributions of the forces f j, and ZC those of some collective quantities like total stresses and energies, but also additional constraintsAk on the potential one would like to impose. The denominators of the fractions ensure that the target function measures the relative deviations from the reference data, except for really tiny quantities, where the εl prevent extremely small denominators. The Wl are the weights of the different terms. It proves useful for the fitting to give the total stresses and the cohesion energies a higher weight, although in principle they should be reproduced correctly already from the forces. We developed a programme named potfit, which optimises the potential parameters to a set of reference data. It consists of two largely independent parts. The first part imple- ments a particular parametrised potential model. It takes a list of potential parameters and computes from it the target function, i.e., the deviations of the forces, energies, and stresses from the reference data. Wrapped around this part is a second, potential indepen- dent part, which implements a least square minimisation module, using a combination of a deterministic conjugate gradient algorithm [10] and a stochastic simulated annealing algo- rithm [11]. This part knows nothing about the details of the potential, and only deals with a list of potential parameters. The programme architecture thus makes it easy to replace the potential dependent part by a different one, e.g., one which implements a different potential model, or a different way to parametrise it. 3 Results and Applications We generated several fitted potentials for decagonal Al-Ni-Co and icosahedral Ca-Cd qua- sicrystals, as well as Mg-Zn potentials suitable for both icosahedral and decagonal phases. In a first step, classical molecular dynamics simulations with simple model potentials were used to create reference configurations from small approximants (80–250 atoms). These in- cluded samples at different temperatures, but also samples which were scaled and strained in different ways. The approximants were carefully selected, so that all relevant local envi- ronments are represented. For those reference structures, the forces, stresses and energies were computed with ab-initio methods, and a first version of the fitted effective potential given by sampling points with cubic spline interpolation was fitted to the reference data. In a second step, molecular dynamics simulations with the newly determined potential were used to create new reference structures, which are better representatives of the struc- tures actually appearing in that system. The new reference structures complemented and partially replaced the previous ones, and the fitting procedure was repeated. This second iteration resulted in a significantly better fit to the reference data. In order to test the transferability of the fitted potentials, further samples similar to the reference structures were created, and their ab-initio forces and energies were compared to those determined by the classical potentials. The deviations were of the same order as the deviations found in the potential fit, which shows that the fitted potentials transfer well to similar struc- tures. For Al-Ni-Co, a force-matched potential is displayed in figure 1. Fitted potentials for Ca-Cd and Mg-Zn are not displayed here for space constraints, but are available from the authors. The potentials developed for decagonal Al-Ni-Co quasicrystals are intended to be used in high-temperature diffusion simulations [2]. It is therefore important that they describe high temperature states well, which is achieved by selecting the reference structures ac- cordingly. By using high temperature reference structures, the fitted potential is especially trained to such situations. As part of the potential validation, the melting temperature was determined by slowly heating the sample at constant pressure, and the elastic constants of decagonal Al-Ni-Co were determined. We actually have constructed two potential variants: Variant A gives excellent values for the elastic constants (Table 1), but produces a melting temperature which is somewhat too high. Conversely, variant B shows larger deviations in the elastic constants, but gives a very reasonable value of the melting temperature of about 1300 K. It is a general experience that with an effective potential it is often not possible to reproduce all desired quantities equally well at the same time. In complex intermetallic systems there are many competing candidates for the ground state structure. This is the case also for complex crystalline systems. In principle, the ground state of these can be determined directly by ab-initio simulations, but for large unit cells this is extremely time-consuming, or even impossible. Classical potentials can be used to select the most promising candidates, and to pre-relax them, so that the time for ab-initio relaxation can be dramatically reduced. Potentials used for this purpose must be able to discriminate energy differences of the order of a meV/atom. This has been largely achieved with fitted potentials for the Mg-Zn and Ca-Cd systems, by using mainly near 1 2 3 4 5 6 7 8 distance [ Å ] Al−Al Al−Co Al−Ni 1 2 3 4 5 6 7 8 distance [ Å ] Co−Co Co−Ni Ni−Ni −0.10 −0.05 1 2 3 4 5 6 7 8tr distance [ Å ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (total) local "electron density" Figure 1: Potential functions for decagonal Al-Ni-Co ground state structures as reference structures. Also, for this application it is important to choose a small εj in equation (3), so that small forces are also reproduced accurately. The so constructed Ca-Cd potentials have been used successfully for structure optimisations [13]. 4 Discussion and Conclusion The selection of the reference structures used for the potential fit largely determines the capabilities of the resulting potential. For a precise determination of the ground state, low temperature structures should be dominant in the reference structures, and it must be assured that even small forces and energy differences are reproduced accurately. For high temperature simulations, on the other hand, typical high temperature structures must be predominant in the reference structures. This opens the possibility to design specialised potentials for certain purposes by a suitable selection of reference structures. It should be kept in mind, however, that a fitted potential can only deal with situations it has been trained to. For instance, one should not expect a fitted potential to handle surfaces correctly, if it was trained only with bulk systems. Clearly, there is always a trade-off Table 1: Elastic constants of decagonal Al-Ni-Co [GPa] c11 c33 c44 c66 a c12 c13 Exp. [12] 234 232 70 88 57 67 Pot. A 230 231 55 70 91 91 Pot. B 197 187 49 58 86 84 aIn decagonal QC: c66 = 12 (c11 − c12) between the transferability and the accuracy of a fitted potential. A potential can be made more versatile by training it with many different kinds of structures, but the more versatile it becomes, the less accurate it will be on average. Conversely, very accurate fitted potentials will probably have limited transferability. For practical applications, the range of a potential is also an important issue, as it enters in the third power in the computational effort of molecular dynamics. Allowing for a larger potential range results in greater flexibility of the potential, which might improve its accuracy, but this comes at the price of a slower simulation. We therefore need a compromise between speed and accuracy. The potential range should only be increased as long as this can improve the potential quality. In a first step, our fitted potentials were constructed with a fairly generous range of about 7Å. It turned out, however, that especially the transfer function ρi did not make effective use of this range, and was essentially zero beyond 5Å. In a second fit we therefore restricted the range of ρi to 5Å, without significant loss of accuracy. This is one of the advantages of using tabulated functions: The system itself chooses the optimal functions, including the optimal range. Force Matching has proven to be a versatile method to construct physically reasonable, accurate effective potentials even for structures as complicated as quasicrystals and their approximants. Our potfit programme makes it easy to apply this method to different sys- tems, and is also easy to adapt for the support of further potential models. The potentials constructed so far have successfully been used in high temperature diffusion simulations of decagonal Al-Ni-Co [2], and in structure optimisation of approximants in the Zn-Mg and Ca-Cd systems. Further fruitful applications of the fitted potentials can certainly be expected, and we hope to apply our methods also to other complex alloy systems, where reliable potentials are still lacking. Acknowledgement This work was funded by the Deutsche Forschungsgemeinschaft through Sonderforschungs- bereich 382. Special thanks go to Marek Mihalkovič for supplying approximants and feed- back in the Ca-Cd and Mg-Zn systems, and to Hans-Rainer Trebin for supervising the thesis work of the first author. References [1] F. Rösch, Ch. Rudhart, J. Roth, H.-R. Trebin, and P. Gumbsch, Phys. Rev. B 72, 014128 (2005). [2] S. Hocker, F. Gähler, and P. Brommer, Phil. Mag. 86, 1051 (2006). [3] F. Ercolessi and J. B. Adams, Europhys. Lett. 26, 583 (1994). [4] M. S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984). [5] F. Ercolessi, M. Parrinello, and E. Tosatti, Phil. Mag. A 58, 213 (1988). [6] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). [7] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). [8] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). [9] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [10] M. J. D. Powell, Comp. J. 7, 303 (1965). [11] A. Corana, M. Marchesi, C. Martini, and S. Ridella, ACM Trans. Math. Soft. 13, 262 (1987). [12] M. A. Chernikov, H. R. Ott, A. Bianchi, A. Migliori, and T. W. Darling, Phys. Rev. Lett. 80, 321 (1998). [13] M. Mihalkovič and M. Widom, Phil. Mag. 86, 519 (2006). Introduction Force Matching Results and Applications Discussion and Conclusion

0704.0164

On smooth foliations with Morse singularities

On smooth foliations with Morse singularities Lilia Rosati Università di Firenze, Dipartimento di Matematica “U. Dini”, viale Morgagni 67/A, 50134 Firenze e-mail: [email protected] Abstract Let M be a smooth manifold and let F be a codimension one, C∞ foliation on M , with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb [Reeb] states that a manifold admitting a foliation with exactly two center- type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper [Ee-Kui] classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) the- orems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F , we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S1, we are able to extend Haefliger’s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result. Acknoledgements I am very grateful to prof. Bruno Scárdua for proposing me such an interesting subject and for his valuable advice. My hearthy good thanks to prof. Graziano Gentili for his suggestions on the writing of this article. 1 Foliations and Morse Foliations Definition 1.1 A codimension k, foliated manifold (M,F) is a manifold M with a differentiable structure, given by an atlas {(Ui, φi)}i∈I , satisfying the following properties: (1) φi(Ui) = B n−k × Bk; (2) in Ui ∩ Uj 6= ∅, we have φj ◦ φ i (x, y) = (fij(x, y), gij(y)), where {fij} and {gij} are families of, respectively, submersions and diffeomorphisms, defined on natural domains. Given a local chart (foliated chart) (U, φ), ∀x ∈ Bn−k and y ∈ Bk, the set φ−1(·, y) is a plaque and the set φ−1(x, ·) is a transverse section. The existence of a foliated manifold (M,F) determines a partition of M into subsets, the leaves, defined by means of an equivalence relation, each endowed of an intrinsic manifold structure. Let x ∈ M ; we denote by Fx or Lx the leaf of F through x. With the intrinsic manifold structure, Fx turns to be an immersed (but not embedded, in general) submanifold of M . In an equivalent way, a foliated manifold (M,F) is a manifold M with a collection of couples {(Ui, gi)}i∈I , http://arxiv.org/abs/0704.0164v1 where {Ui}i∈I is an open covering of M , gi : Ui → B k is a submersion, ∀i ∈ I , and the gi’s satisfy the cocycle relations, gi = gij ◦ gj , gii = id, for suitable diffeomorphisms gij : B k → Bk, defined when Ui ∩ Uj 6= ∅. Each Ui is said a foliation box, and gi a distinguished map. The functions γij = dgij are the transition maps [Stee] of a bundle NF ⊂ TM , normal to the foliation. More completely, there exists a G-structure on M [Law], which is a reduction of the structure group GL(n, R) of the tangent bundle to the subgroup of the matrices , where A ∈ GL(n− k, R) and C ∈ GL(k, R). A codimension one, C∞ foliation of a smooth manifold M , with isolated singularities, is a pair F = (F∗, Sing(F)), where Sing(F) ⊂ M is a discrete subset and F∗ is a codimension one, C∞ foliation (in the ordinary sense) of M∗ = M \Sing(F). The leaves of F are the leaves of F∗ and Sing(F) is the singular set of F . A point p is a Morse singularity if there is a C∞ function, fp : Up ⊂ M → R, defined in a neighborhood Up of p, with a (single) non-degenerate critical point at p and such that fp is a local first integral of the foliation, i.e. the leaves of the restriction F|Up are the connected components of the level hypersurfaces of fp in Up\{p}. A Morse singularity p, of index l, is a saddle, if 0 < l < n (where n = dimM ), and a center, if l = 0, n. We say that the foliation F has a saddle-connection when there exists a leaf accumulated by at least two distinct saddle-points. A Morse foliation is a foliation with isolated singularities, whose singular set consists of Morse singularities, and which has no saddle-connections. In this way if a Morse foliation has a (global) first integral, it is given by a Morse function. Of course, the first basic example of a Morse foliation is indeed a foliation defined by a Morse function on M . A less evident example is given by the foliation depicted in figure 2. In the literature, the orientability of a codimension k (regular) foliation is determined by the orientability of the (n− k)-plane field tangent to the foliation, x → TxFx. Similarly transverse orientability is determined by the orientability of a complementary k-plane field. A singular, codimension one foliation, F , is transversely orientable [Cam-Sc] if it is given by the natural (n − 1)-plane field associated to a one-form, ω ∈ Λ1(M), which is integrable in the sense of Frobenius. In this case, choosing a Riemannian metric on M , we may find a global vector field transverse to the foliation, X = grad(ω), ωX ≥ 0, and ωxXx = 0 if and only if x is a singularity for the foliation (ω(x) = 0). A transversely orientable, singular foliation F of M is a transversely orientable (regular) foliation F∗ of M∗ in the sense of the classical definition. Viceversa, if F∗ is transversely orientable, in general, F is not. Thanks to the Morse Lemma [Mil 1], Morse foliations reduce to few representative cases. On the other hand, Morse foliations describe a large class among transverseley orientable foliations. To see this, let F be a foliation defined by an integrable one-form, ω ∈ Λ1(M), with isolated singularies. We proceed with a local analysis; using a local chart around each singularity, we may suppose ω ∈ Λ1( Rn), ω(0) = 0, and 0 is the only singularity of ω. We have ω(x) = hi(x)dx i and, in a neighborhood of 0 ∈ Rn, we may write ω(x) = ω1(x) +O(|x| 2), where ω1 is the linear part of ω, defined by ω1(x) = idxj , aij = ∂h i(x)/∂xj . We recall that the integrability of ω implies the integrability of ω1 and that the singularity 0 ∈ R n is said to be non degenerate if and only if (aij) ∈ R(n) is non degenerate; in this latter case (aij) is symmetric: it is the hessian matrix of some real function f , defining the linearized foliation (ω1 = df ). We have {transverseley orientable foliations, with Morse singularities} = {foliations, defined by non degenerate linear one-forms} ⊂ {foliations, defined by non degenerate one-forms}. Let (σ, τ) be the space σ of integrable one-forms in Rn, with a singularity at the origin, endowed with the C1-Whitney topology, τ . If ω, ω′ ∈ σ, we say ω equivalent ω′ (ω ∼ ω′) if there exists a diffeomorphism φ : Rn → Rn, φ(0) = 0, which sends leaves of ω into leaves of ω′. Moreover, we say ω is structurally stable, if there exists a neighborhood V of ω in (σ, τ) such that ω′ ∼ ω, ∀ω′ ∈ V . Theorem 1.2 (Wagneur)[Wag] The one-form ω ∈ σ is structurally stable, if and only if the index of 0 ∈ Sing(ω) is neither 2 nor n− 2. Let us denote by S the space of foliations defined by non degenerate one-forms with singularities, whose index is neither 2 nor n− 2. If S1 ⊂ S is the subset of foliations defined by linear one-forms, then we have: Corollary 1.3 There exists a surjective map, s : S1 → S/∼. PSfrag replacementsF1 L0 L1 Figure 1: F1,F2 foliations on RP Hol(L,F1) = {e}, Hol(L0,F1) = {e, g0}, g20 = e, Hol(L1,F2) = {e, g1}, g1 orientation reversing diffeomorphism, Hol(L2,F2) = {e, g2}, g2 generator of unilateral holonomy. PSfrag replacements Figure 2: A singular foliation of the sphere S2, which does not admit a first integral. With the same spirit, a singular foliation on S3 may be given. 2 Holonomy and Reeb Stability Theorems It is well known that a basic tool in the study of foliations is the holonomy of a leaf (in the sense of Ehresmann). If L is a leaf of a codimension k foliation (M,F), the holonomy Hol(L,F) = Φ(π1(L)), is the image of a representation, Φ : π1(L) → Germ( R k, 0), of the fundamental group of L into the germs of diffeomorphisms of Rk, fixing the origin. Let x ∈ L and Σx be a section transverse to L at x; with abuse of notation, we will write that a diffeomorphism g : Dom(g) ⊂ Σx → Σx, fixing the origin, is an element of the holonomy group. For codimension one foliations (k = 1), we may have: (i) Hol(L,F) = {e}, (ii) Hol(L,F) = {e, g}, with g2 = e, g 6= e, (iii) Hol(L,F) = {e, g}, where gn 6= e, ∀n, and g is a (orientation preserving or reversing) diffeomorphism. In particular, among orientation preserving diffeomorphisms, we might find a g : Σx → Σx, such that g is the identity on one component of Σx \ {x} and it is not the identity on the other; in this case, we say that L has unilateral holonomy (see figure 1 for some examples). We recall Reeb Stability Theorems (cfr., for example, [Cam-LN] or [Mor-Sc]). Theorem 2.1 (Reeb Local Stability) Let F be a C1, codimension k foliation of a manifold M andF a compact leaf with finite holonomy group. There exists a neighborhood U of F , saturated in F (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a retraction π : U → F such that, for every leaf F ′ ⊂ U , π|F ′ : F ′ → F is a covering with a finite number of sheets and, for each y ∈ F , π−1(y) is homeomorphic to a disk of dimension k and is transverse to F . The neighborhood U can be taken to be arbitrarily small. The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf with finite holonomy, the space of leaves is Hausdorff. Under certain conditions the Reeb Local Stability Theorem may replace the Poincaré Bendixon Theorem [Pal-deM] in higher dimensions. This is the case of codimension one, singular foliations (Mn,F), with n ≥ 3, and some center-type singularity in Sing(F). Theorem 2.2 (Reeb Global Stability) Let F be a C1, codimension one foliation of a closed manifold, M . If F contains a compact leaf F with finite fundamental group, then all the leaves of F are compact, with finite fundamental group. If F is transversely orientable, then every leaf of F is diffeomorphic to F ; M is the total space of a fibration f : M → S1 over S1, with fibre F , and F is the fibre foliation, {f−1(θ)|θ ∈ S1}. This theorem holds true even when F is a foliation of a manifold with boundary, which is, a priori, tangent on certain components of the boundary and transverse on other components [God]. In this setting, let H l = {(x1, . . . , xl) ∈ Rl|xl ≥ 0}. Taking into account definition 1.1, we say that a foliation of a manifold with boundary is tangent, respectively transverse to the boundary, if there exists a differentiable atlas {(Ui, φi)}i∈I , such that property (1) of the above mentioned definition holds for domains Ui such that Ui ∩ ∂M = ∅, while φi(Ui) = B n−k × H k, respectively, φi(Ui) = H n−k × Bk for domains such that Ui ∩ ∂M 6= ∅. Moreover, we ask that the change of coordinates has still the form described in property (2). Recall that F|∂M is a regular codimension k − 1 (respectively, k) foliation of the (n− 1)-dimensional boundary. After this, it is immediate to write the definition for foliations which are tangent on certain components of the boundary and transverse PSfrag replacements b Figure 3: n = 2: a singular foliation with center- type singularities, having no first integral. PSfrag replacements Ws(q)) Wu(q) Figure 4: A trivial couple center-saddle (p, q) (Theorem 3.5, case (i)). on others. Observe that, for foliations tangent to the boundary, we have to replace S1 with [0, 1] in the second statement of the Reeb Theorem 2.2 (see Lemma 5.6). We say that a component of Sing(F) is weakly stable if it admits a neighborhood, U , such that F|U is a foliation with all leaves compact. The problem of global stability for a foliation with weakly stable singular components may be reduced to the case of foliations of manifolds with boundary, tangent to the boundary. It is enough to cut off an invariant neighborhood of each singular component. Holonomy is related to transverse orientability by the following: Proposition 2.3 Let L be a leaf of a codimension one (Morse) foliation (M,F). If Hol(L,F) = {e, g}, where g2 = e, g 6= e, then F is non-transversely orientable. Moreover, if π : M → M/F is the projection onto the space of leaves, then ∂(M/F) 6= ∅ and π(L) ∈ ∂(M/F). Proof. We choose x ∈ L and a segment Σx, transverse to the foliation at x. Then g : Σx → Σx turns out to be g(y) = −y. Let y → Ny a 1-plane field complementary to the tangent plane field y → TyFy . Suppose we may choose a vector field y → X(y) such that Ny = span{X(y)}. Then it shoud be X(x) = −X(x) = (dg)x(X(x)), a contraddiction. Consider the space of leaves near L; this space is the quotient of Σx with respect to the equivalence relation ∼ which identifies points on Σx of the same leaf. Then Σx/∼ is a segment of type (z, x] or [x, z), where π−1(x) = L. At last we recall a classical result due to Reeb. Theorem 2.4 (Reeb Sphere Theorem) [Reeb] A transversely orientable Morse foliation on a closed manifold, M , of dimension n ≥ 3, having only centers as singularities, is homeomorphic to the n-sphere. This result is proved by showing that the foliation considered must be given by a Morse function with only two singular points, and therefore thesis follows by Morse theory. Notice that the theorem still holds true for n = 2, with a different proof. In particular, the foliation need not to be given by a function (see figure 3). 3 Arrangements of singularities In section 4 we will study the elimination of singularities for Morse foliations. To this aim we will describe here how to identify special “couples” of singularities and we will study the topology of the neighbouring leaves. Definition 3.1 Let n = dimM,n ≥ 2. We define the set C(F) ⊂ M as the union of center-type singularities and leaves diffeomorphic to Sn−1 (with trivial holonomy if n = 2) and for a center singularity, p, we denote by Cp(F) the connected component of C(F) that contains p. Proposition 3.2 Let F be a Morse foliation on a manifold M . We have: (1) C(F) and Cp(F) are open in M . (2) Cp(F) ∩ Cq(F) 6= ∅ if and only if Cp(F) = Cq(F). Cp(F) = M if and only if ∂ Cp(F) = ∅. In this case the singularities of F are centers and the leaves are all diffeomorphic to Sn−1. (3) If q ∈ Sing(F) ∩ ∂ Cp(F), then q must be a saddle; in this case ∂ Cp(F) ∩ Sing(F) = {q}. Moreover, for n ≥ 3 and F transversely orientable, ∂ Cp(F) 6= ∅ if and only if ∂ Cp(F) ∩ Sing(F) 6= ∅. In these hypotheses, ∂ Cp(F) contains at least one separatrix of the saddle q. (4) ∂ Cp(F) \ {q} is closed in M \ {q}. PSfrag replacements Figure 5: A saddle q of index 1 (n− 1), accumu- lating one center p (Theorem 3.5, case (ii)). PSfrag replacements Figure 6: A saddle q of index 1 (n− 1), accumu- lating one center p (Theorem 3.5, case (iii)). Figure 7: Two saddles in trivial coupling for the foliation defined by the function fǫ = − ǫy + z , (ǫ > 0). PSfrag replacements L2p q no intersection legenda Figure 8: A dead branch of a trivial couple of sad- dles for a foliated manifold (Mn,F), n ≥ 3. Proof. (1) C(F) is open by the Reeb Local Stability Theorem 2.1. (3) If non-empty, ∂ Cp(F) ∩ Sing(F) consists of a single saddle q, as there are no saddle connections. The second part follows by the Reeb Global Stability Theorem for manifolds with boundary and the third by the Morse Lemma. (4) By the Transverse Uniformity Theorem (see, for example, [Cam-LN]), it follows that the intrinsic topology of ∂ Cp(F) \ {q} coincides with its natural topology, as induced by M \ {q}. We recall the following (cfr., for example [Mor-Sc]): Lemma 3.3 (Holonomy Lemma) Let F be a codimension one, transversely orientable foliation on M , let A be a leaf of F and K be a compact and path-connected set. If g : K → A is a C1 map homotopic to a constant in A, then g has a normal extension i.e. there exist ǫ > 0 and a C1 map G : K × [0, ǫ] → M such that Gt(x) = G x(t) = G(x, t) has the following properties: (i) G0(K) = g, (ii) Gt(K) ⊂ A(t) for some leaf A(t) of F with A(0) = A, (iii) ∀x ∈ K the curve Gx([0, ǫ]) is normal to F . For the case of center-saddle pairings we prove the following descriptions of the separatrix: Theorem 3.4 Let F be a C∞, codimension one, transversely orientable, Morse foliation of a compact n- manifold, M , n ≥ 3. Let q be a saddle of index l /∈ {1, n− 1}, accumulating to one center p. Let L ⊂ Cp(F) be a spherical leaf intersecting a neighborhood U of q, defined by the Morse Lemma. Then ∂ Cp(F) \ {q} has a single connected component (see figure 13) and is homeomorphic to Sn−1/Sl−1. If F is a leaf such that U \ Cp(F) 6= ∅, then F is homeomorphic to Bl×Sn−l−1∪φ B l×Sn−l−1, where φ is a diffeomorphism of the boundary (for example, we may have F ≃ Sl × Sn−l−1, but also F ≃ Sn−1, for l = n/2). Proof. Let ω ∈ Λ1(M) be a one-form defining the transversely orientable foliation. We choose a riemannian metric on M and we consider the transverse vector field Xx = grad(ω)x. We suppose ||X || = 1. In U , we have X = h · grad(f) for some real function h > 0 defined on U . Further, we may suppose that ∂U follows the orbits of X in a neighborhood of ∂ Cp(F). The Morse Lemma gives a local description of the foliation near its singularities; in particular the local topology of a leaf near a saddle of index l is given by the connected components of the level sets of the function f(x) = −x21 − · · · − x l + x l+1 + · · · + x n. If, for c ≥ 0, we write f −1(c) = {(x1, . . . , xn) ∈ R n|x21 + · · · + x2l + c = x l+1 + · · · + x n}, it is easy to see that f −1(0) is homeomorphic to a cone over Sl−1 × Sn−l−1 and f−1(c) ≃ Bl × Sn−l−1 (c > 0). Similarly, we obtain f−1(c) ≃ Bn−l × Sl−1 for c < 0. Therefore, by our hypothesis on l, the level sets are connected; in particular the separatrix S ⊃ f−1(0) is unique and ∂ Cp(F) = S ∪ {q}; moreover U is splitted by f −1(0) in two different components. A priori, a leaf may intersect more than one component. As F is transversely orientable, the holonomy is an orientation preserving diffeomorphism, and then a leaf may intersect only non adiacent components; then this is not the case, in our hypotheses. Let L be a spherical leaf ⊂ Cp(F) enough near q. Then L ∩ U 6= ∅ and it is not restrictive to suppose it is given by f−1(c) for some c < 0. We define the compact set K = Sn−1 \ Bn−l × Sl−1 ≃ L \ U . As n ≥ 3, the composition K // L \ U // L is homotopic to a constant in its leaf. By the proof of the Holonomy Lemma 3.3, L \ U projects diffeomorphically onto A(ǫ) = ∂ Cp(F), by means of the constant-speed vector field, X . Together with the Morse Lemma, this gives a piecewise description of ∂ Cp(F), which is obtained by piecing pieces toghether. It comes out ∂ Cp(F) ≃ S n−1/Sl−1, a set with the homotopy type of Sn−1 ∨ Sl (where ∨ is the wedge sum), simply connected in our hypotheses. Consequently, the map K×{ǫ} → ∂ Cp(F), obtained with the extension, admits, on turn, a normal extension. This completes the piecewise description of F . In case of presence of a saddle of index 1 or n− 1, we have: Theorem 3.5 Let F be a C∞, codimension one, transversely orientable, Morse foliation of a compact n- manifold, M , n ≥ 3. Let q be a saddle of index 1 or n − 1 accumulating to one center p. Let L ⊂ Cp(F) be a spherical leaf intersecting a neighborhood U of q, defined by the Morse Lemma. We may have: (i) ∂ Cp(F) contains a single separatrix of the saddle (see figure 4) and is homeomorphic to Sn−1; (ii) ∂ Cp(F) contains both separatrices S1 and S2 of the saddle (see figure 5) and is homeomorphic to S n−1/Sn−2 ≃ Sn−1 ∨Sn−1. If this is the case, there exist two leaves Fi (i = 1, 2), such that Fi and L intersect different components of U \ Si and we have that Fi is homeomorphic to S n−1 (i = 1, 2); (iii) q is a self-connected saddle (see figure 6) and ∂ Cp(F) is homeomorphic to S n−1/S0. In this case we will refer to the couple Cp(F),F| Cp(F ) a singular Reeb component. Moreover, U \ ∂ Cp(F) has three connected components and L intersects two of them. If F is a leaf intersecting the third component of U \ ∂ Cp(F), then F is homeomorphic to S 1 × Sn−2, or to R× Sn−2. Proof. The proof is quite similar to the proof of the previous theorem. Nevertheless we give a brief sketch here. The three cases arise from the fact that q has two local separatrices, S1 and S2, but not necessarily ∂ Cp(F) contains both of them. When this is the case, we may have that S1 and S2 belong to distinct leaves, or to the same leaf (in this case all spherical leaves contained in Cp(F) intersect two different components of U \ (S1 ∪ S2) ). Using the Morse lemma, we construct the set K for the application of the Holonomy Lemma 3.3. We have, respectively: K = Bn−1, K = K1 ⊔K2 = S 0 ×Bn−1 (we apply twice the Holonomy Lemma), K = B1 × Sn−2. In the first two cases, as K is simply connected, the map K → L, to be extended, is clearly homotopic to a constant in its leaf. Then L \U projects onto ∂Cp(F) and on neighbour leaves. This completes the piecewise description in case (i) and (ii). In the third case, piecing pieces together after a first application of the Holonomy Lemma, we obtain ∂ Cp(F) ≃ Sn−1/S0 and ∂ Cp(F) \ {q} ≃ B 1 × Sn−2, simply connected for n 6= 3. With a second application of the Holonomy Lemma (n 6= 3), K projects diffeomorphically onto any neighbour leaf, F . The same also happens for n = 3, because a curve γ : S1 → ∂ Cp(F), as the one depicted in figure 6, is never a generator of the holonomy, which is locally trivial (a consequence of the Morse lemma). Nevertheless, there are essentially two ways to piece pieces together. We may have F ≃ S1 × Sn−2 or F ≃ R× Sn−2. The last result gives the motivation for a new concept. Definition 3.6 In a codimension one singular foliation F it may happen that, for some leaf L and q ∈ Sing(F), the set L ∪ {q} is arcwise connected. Let C = {q ∈ Sing(F)|L ∪ {q} is arcwise connected}. If for some leaf L the set C 6= ∅, we define the corresponding singular leaf [Wag] S(L) = L ∪ C. In particular, if F is a transversely orientable Morse foliation, each singular leaf is given by S(L) = L∪{q}, for a single saddle-type singularity q, either selfconnected or not. In the case of a transversely orientable Morse foliation F on M (n = dimM ≥ 3), given a saddle q and a separatrix L of q, we may define a sort of holonomy map of the singular leaf S(L). This is done in the following way. As the foliation is Morse, in a neighborhoodU ⊂ M of q there exists a (Morse) local first integral f : U → R, with f(q) = 0. Keeping into account the structure of the level sets of the Morse function f (see Theorem 3.4 and Theorem 3.5) we observe that there are at most three connected components in U \ S(L) = U \ {f−1(0)} (notice that the number of components depends on the Morse index of q). Let γ : [0, 1] → S(L) be a C1 path through the singularity q. At first, we consider the case γ([0, 1]) ⊂ U , q = γ(t) for some 0 < t < 1. For a point x ∈ M \ Sing(F), let Σx be a transverse section at x. The set Σx \ {x} is the union of two connected components, Σ x and Σ x that we will denote by semi-transverse sections at x. For x = γ(0) ∈ S(L) we have f(x) = 0 and we can choose semi-transverse sections at x in a way that f(Σ+x ) > 0 and f(Σ x ) < 0. We repeat the construction for y = γ(1), obtaining four semi-transverse sections, which are contained in (at most) three connected components of U \S(L). As a consequence, at least two of them are in the same component. By our choices, this happens for Σ−x and Σ y (but we cannot exclude it happens also for Σ+x and Σ y ). We define the semi-holonomy map h − : Σ− ∪γ(0) → Σ− ∪γ(1) by setting h−(γ(0)) = γ(1) and h−(z) = h(z) for z ∈ Σ− , where h : Σ− is a classic holonomy map (i.e. such that for a leaf F , it is h(F ∩ Σ− ) = F ∩ Σ− ). In the same way, if it is the case, we define h+. Consider now any curve γ : [0, 1] → S(L). As F is transversely orientable, the choice of a semi-transverse section for the curve γ([0, 1]) ∩ U , may be extended continuously on the rest of the curve, γ([0, 1]) \ U ; with this remark, we use classic holonomy outside U . To complete the definition, it is enough to say what a semi- transverse section at the saddle q is. In this way we allow q ∈ γ(∂[0, 1]). To this aim, we use the orbits of the transverse vector field, grad(f). By the property of gradient vector fields, there exist points t, v such that α(t) = ω(v) = q. Let Σ+q (Σ q ) be the negative (positive) semi-orbit through t (v). Each of Σ q and Σ transverse to the foliation and such that Σ+q ∩ Σ q = {q}, is a semi-transverse section at the saddle q. In this way, the semi-holonomy of a singular leaf Hol+(S(L),F) is a representation of the fundamental group π1(S(L)) into the germs of diffeomorphisms of R≥0 fixing the origin, Germ( R≥0, 0). Now we consider the (most interesting) case of a selfconnected separatrix S(L) = ∂ Cp(F), with ∂ Cp(F) satisfying the description of Theorem 3.5, case (iii). The singular leaf ∂ Cp(F), homeomorphic to S n−1/S0, has the homotopy type of Sn−1 ∨ S1. We have Hol+(∂ Cp(F),F) = {e, h γ }, where γ is the non trivial generator of the homotopy, and h−γ is a map with domain contained in the complement ∁ Cp(F). The two options h−γ = e, h γ 6= e give an explanation of the two possible results about the topology of the leaves near the selfconnected separatrix. 4 Realization and elimination of pairings of singularities Let us describe one of the key points in our work, i.e. the elimination procedure, which allows us to delete pairs of singularities in certain configurations, and, this way, to lead us back to simple situations as in the Reeb Sphere Theorem (2.4). We need the following notion [Cam-Sc]: Definition 4.1 Let F be a codimension one foliation with isolated singularities on a manifold Mn. By a dead branch of F we mean a region R ⊂ M diffeomorphic to the product Bn−1 × B1, whose boundary, ∂R ≈ Bn−1×S0∪Sn−2×B1, is the union of two invariant components (pieces of leaves of F , not necessarily distinct leaves in F ) and, respectively, of transverse sections, Σ ≈ {t} × B1, t ∈ Sn−2. Let Σi, i = 1, 2 be two transverse sections. Observe that the holonomy from Σ1 → Σ2 is always trivial, in the sense of the Transverse Uniformity Theorem [Cam-LN], even if Σi ∩ S(L) 6= ∅ for some singular leaf S(L). In this case we refer to the holonomy of the singular leaf, in the sense above. A first result includes known situations. Proposition 4.2 Given a foliated manifold (Mn,F), with F Morse and transversely orientable, with Sing(F) ∋ p, q, where p is a center and q ∈ ∂ Cp(F) is a saddle of index 1 or n− 1, there exists a new foliated manifold (M, F̃), such that: (i) F̃ and F agree outside a suitable region R of M , which contains the singularities p, q; (ii) F̃ is nonsingular in a neighborhood of R. Proof. We are in the situations described by Theorem 3.5. If we are in case (i), the couple (p, q) may be eliminated with the technique of the dead branch, as illustrated in [Cam-Sc]. If we are in case (ii), we observe that the two leaves Fi, i = 1, 2 bound a region, A, homeomorphic to an anulus, S n−1 × [0, 1]. We may now replace the singular foliation F|A with the trivial foliation F̃|A, given by S n−1 × {t}, t ∈ [0, 1]. If we are in case (iii), we may replace the singular Reeb component with a regular one, in the spirit of [Cam-Sc]. Even in this case, we may think the replacing takes place with the aid of a new sort of dead branch, the dead branch PSfrag replacements Figure 9: On the left: the height function on the plane V defines a foliation of the torus; on the right: a possible description of the foliation. PSfrag replacements ∂ Cp(F) Figure 10: On the left, a dead branch for the self- connected saddle q of figure 9; on the right, the foliation obtained after the elimination of the two couples of singularities. of the selfconnected saddle, that we describe with the picture of figure 10, for the case of the foliation of the torus of figure 9, defined by the height Morse function [Mil 1]. Observe that the couples (p, q) and (r, s) of this foliation may be also seen as an example of the coupling described in Theorem 3.5, case (ii). In this case the elimination technique and the results are completely different (see figure 11). Definition 4.3 If the couple (p, q) satisfies the description of Theorem 3.5, case (i) (and therefore may be elim- inated with the technique of the dead branch), we will say that (p, q) is a trivial couple. A new result is the construction of saddle-saddle situations: Proposition 4.4 Given a foliation F on an n-manifold Mn, there exists a new foliation F̃ on M , with Sing(F̃) = Sing(F) ∪ {p, q}, where p and q are a couple of saddles of consecutive indices, connecting transversely (i.e. such that the stable manifold of p, Ws(p), intersects transversely the unstable manifold of q, Wu(q)). Proof. We choose the domain of (any) foliated chart, (U, φ). Observe that R′ = U (≃ φ(U)) is a dead branch for a foliation F ǫ′ , given (up to diffeomorphisms) by the submersion fǫ = −x 1/2− · · · − x k−1/2 + (x ǫxk) + xk+1/2+ · · ·+ x n/2, for some ǫ = ǫ ′ < 0. We consider F ǫ′′ , given by taking ǫ = ǫ ′′ > 0 in fǫ, which presents a couple of saddles of consecutive indices, and we choose a dead branch R′′ around them. We also choose a homeomorphism between R′ and R′′ which sends invariant sets of F ǫ′ into invariant sets of F ǫ′′ in a neighborhood of the boundary. With a surgery, we may replace F ǫ′ with F ǫ′′ . The converse of the above poposition is preceded by the following Remark 4.5 Given a foliation F on Mn with two complementary saddle singularities p, q ∈ Sing(F), having a strong stable connection γ, there exist a neighborhood U of p, q and γ in Mn, a δ ∈ R+ and a coordinate system φ : U → Rn taking p onto (0, . . . , φk = −δ, . . . , 0), q onto (0, . . . , φk = δ, . . . , 0), γ onto the xk-axis, {xl = 0}l 6=k, and such that: (i) the stable manifold of p is tangent to φ −1({xl = 0}l>k) at p, (ii) the unstable manifold of q is tangent to φ−1({xl = 0}l<k) at q (we are led to the situation considered in [Mil 2], A first cancelation theorem). So using the chart φ : U → Rn we may assume that we are on a dead branch of Rn and the foliation F|U is defined by fǫ, for ǫ = δ 2. In this way the vector field grad(fǫ) defines a transverse orientation in U . For a suitable µ > 0, the points r1 = (0, . . . , φ k = −δ − µ, . . . , 0) and r2 = (0, . . . , φ k = δ + µ, . . . , 0) are such that the modification takes place in a region of U delimited by Lri , i = 1, 2. Proposition 4.6 Given a foliation F on Mn with a couple of saddles p, q of complementary indices, having a strong stable connection, there exists a dead branch of the couple of saddles, R ⊂ M and we can obtain a foliation F̃ on M such that: (i) F̃ and F agree on M \ R; (ii) F̃ is nonsingular in a neighborhood of R; indeed F̃ |R is conjugated to a trivial fibration; (iii) the holonomy of F̃ is conjugate to the holonomy of F in the following sense: given any leaf L of F such that L ∩ (M \R) 6= ∅, then the corresponding leaf L̃ of F̃ is such that Hol(L̃, F̃) is conjugate to Hol(L,F). Example 4.7 (Trivial Coupling of Saddles) Let M = Sn, n ≥ 3. For l = 1, . . . , n − 2 we may find a Morse foliation of M = Sn, invariant for the splitting Sn = Bn−l × Sl ∪φ S n−l−1 × Bl+1, where φ is a diffeomorphism of the boundary. In fact, by theorem 3.4 or 3.5, case (iii), Bn−l × Sl admits a foliation with one center and one saddle of index l. Similarly, Sn−l−1 × Bl+1 admits a foliation with a saddle of index n− l− 1, actually a saddle of index l+ 1, after the attachment. We may eliminate the trivial couple of saddles and we are led to the well-known foliation of Sn, with a couple of centers and spherical leaves. Remark 4.8 The elimination of saddles of consecutive indices is actually a generalization of the elimination of couples center-saddle, (p, q) with q ∈ ∂ Cp(F). Indeed, we may eliminate (p, q) only when the saddle q has index 1 or n−1. This means the singularities of the couple must have consecutive indices and, as q ∈ ∂ Cp(F), there exists an orbit of the transverse vector field having p as α-limit (backward) and q as ω-limit (forward), or viceversa. Such an orbit is a strong stable connection. 5 Reeb-type theorems We shall now describe how to apply our techniques to obtain some generalizations of the Reeb Sphere Theorem (2.4) for the case of Morse foliations admitting both centers and saddles. A first generalization is based on the following notion: Definition 5.1 We say that an isolated singularity, p, of a C∞, codimension one foliation F on M is a stable singularity, if there exists a neighborhood U of p in M and a C∞ function, f : U → R, defining the foliation in U , such that f(p) = 0 and f−1(a) is compact, for |a| small. The following characterization of stable singularities can be found in [Cam-Sc]. Lemma 5.2 An isolated singularity p of a function f : U ⊂ Rn → R defines a stable singularity for df , if and only if there exists a neighborhood V ⊂ U of p, such that, ∀x ∈ V , we have either ω(x) = {p} or α(x) = {p}, where ω(x) (respectively α(x)) is the ω-limit (respectively α-limit) of the orbit of the vector field grad(f) through the point x. In particular it follows the well-known: Lemma 5.3 If a function f : U ⊂ Rn → R has an isolated local maximum or minimum at p ∈ U then p is a stable singularity for df . The converse is also true: Lemma 5.4 If p is a stable singularity, defined by the function f , then p is a point of local maximum or minimum for f . Proof. It follows immediately by Lemma 5.2 and by the fact that f is monotonous, strictly increasing, along the orbits of grad(f). With this notion, we obtain Lemma 5.5 Let F be a codimension one, singular foliation on a manifold Mn. In a neighborhood of a stable singularity, the leaves of F are diffeomorphic to spheres. Proof. Let p ∈ Sing(F) be a stable singularity. By Lemma 5.4, we may suppose p is a minimum (otherwise we use −f ). Using a local chart around p, we may suppose we are on Rn and we may write the Taylor- Lagrange expansion around p for an approximation of the function f : U → R at the second order. We have f(p + h) = f(p) + 1/2〈h,H(p + θh)h〉, where H is the Hessian of f and 0 < θ < 1. It follows 〈h,H(p+ θh)h〉 ≥ 0 in U . Then f is convex and hence the sublevels, f−1(c), are also convex. We consider the flow φ : D(φ) ⊂ R × U → U of the vector field grad(f). By the properties of gradient vector fields, in our hypothesis, D(φ) ⊃ (−∞, 0] × U and ∀x ∈ U there exists the α-limit, α(x) = p. For any x ∈ f−1(c), the tangent space, Txf −1(c), to the sublevels of f does not contain the radial direction, −→px. This is obvious otherwise, for the convexity of f−1(c), the singularity p should lie on the sublevel f−1(c), a contraddiction because, in this case, p should be a saddle. Equivalently, the orbits of the vector field grad(f) are transverse to spheres centered at p. An application of the implicit function theorem shows the existence of a smooth function x → tx, that assigns to each point x ∈ f −1(c) the (negative) time at which φ(t, x) intersects Sn−1(p, ǫ), where ǫ is small enough to have Bn(p, ǫ) ( R(f−1(c)), the compact region bounded by f−1(c) . The diffeomorphism between the leaf f−1(c) and the sphere Sn−1(p, ǫ) is given by the composition x → φ(tx, x). The lemma is proved. Lemma 5.6 Let F be a codimension one, transversely orientable foliation of M , with all leaves closed, π : M → M/F the projection onto the space of leaves. Then we may choose a foliated atlas on M and a differentiable structure on M/F , such that M/F is a codimension one compact manifold, locally diffeomor- phic to the space of plaques, and π is a C∞ map. Proof. At first we notice that the space of leaves M/F (with the quotient topology) is a one-dimensional Hau- sorff topological space, as a consequence of the Reeb Local Stability Theorem 2.1. As all leaves are closed and with no holonomy, we may choose a foliated atlas {(Ui, φi)} such that, for each leaf L ∈ F , L∩Ui consists, at most, of a single plaque. Let π : M → M/F be the projection onto the space of leaves and πi : Ui → R the projection onto the space of plaques. With abuse of notation, we may write πi = p2 ◦ φi, where p2 is the pro- jection on the second component. As there is a 1-1 correspondence between the quotient spaces π|Ui(Ui) and πi(Ui), then, are homeomorphic. Let V ⊂ M/F be open. The set π −1(V ) is an invariant open set. We may find a local chart (Ui, φi) such that π(Ui) = V . We say that (V, πi ◦ (π|Ui ) −1) is a chart for the differentiable atlas with the required property. To see this, it is enough to prove that, if (V, πj ◦(π|Uj ) −1) is another chart with the same domain, V , there exists a diffeomorphism between the two images of V , i.e. between πi◦(π|Ui) −1(V ) and πj ◦ (π|Uj ) −1(V ). This is not obvious when Ui ∩ Uj = ∅. Indeed, the searched diffeomorphism exists, and it is given by the Transverse Uniformity Theorem [Cam-LN]. Observe that, in coordinates, π coincides with the projection on the second factor. Lemma 5.7 Let n ≥ 2. A weakly stable singularity for a foliation (Mn,F) is a stable singularity. Proof. Let p be a weakly stable singularity, U a neighborhood of p with all leaves compact. We need a local first integral near p. As a consequence of the Reeb Local Stability Theorem 2.1, we can find an (invari- ant) open neighborhood V ⊂ U of p, whose leaves have all trivial holonomy. The set V \ {p} is open in M∗ = M \ Sing(F). Let F∗ = F \ Sing(F); the projection π∗ : M∗ → M∗/F∗ is an open map (see, for example [Cam-LN]). As a consequence of Lemma 5.6, the connected (as n ≥ 2) and open set π∗(V \ {p}) is a 1-dimensional manifold with boundary, i.e. it turns out to be an interval, for example (0, 1). Now, we extend smoothly π∗ to a map π on U . In particular, let W ( V be a neighborhood of p. If (for example) π∗(W \ {p}) = (0, b) for some b < 1, we set π(p) = 0. Thesis follows by lemma 5.3. Theorem 5.8 Let Mn be a closed n-dimensional manifold, n ≥ 3. Suppose that M supports a C∞, codimen- sion one, transversely orientable foliation, F , with non-empty singular set, whose elements are, all, weakly stable singularities. Then M is homeomorphic to the sphere, Sn. Proof. By hypothesis, ∀p ∈ Sing(F), p is a weakly stable singularity. Then it is a stable singularity. By lemma 5.5, in an invariant neighborhood Up of p, the leaves are diffeomorphic to spheres. Now we can proceed as in the proof of the Reeb Sphere Theorem 2.4. Theorem 5.9 (Classification of codimension one foliations with all leaves compact) Let F be a (possibly singular, with isolated singularities) codimension one foliation of M , with all leaves compact. Then all pos- sible singularities are stable. If F is (non) transversely orientable, the space of leaves is (homeomorphic to [0, 1]) diffeomorphic to [0, 1] or S1. In particular, this latter case ocurs if and only if ∂M,Sing(F) = ∅. In all the other cases, denoting by π : M → [0, 1] the projection onto the space of leaves, it is Hol(π−1(x),F) = {e}, ∀x ∈ (0, 1). Moreover, if x = 0, 1, we may have: (i) π−1(x) ⊂ ∂M 6= ∅ and Hol(π−1(x),F) = {e}; (ii) π−1(x) is a (stable) singularity; (iii) Hol(π−1(x),F) = {e, g}, g 6= e, g2 = e (in this case, ∀y ∈ (0, 1), the leaf π−1(y) is a two-sheeted covering of π−1(x). Proof. If F is transversely orientable, by the Reeb Global Stability Theorem 2.2 and Lemma 5.6, the space of leaves is either diffeomorphic to S1 or to [0, 1]. In particular, M/F ≈ S1 if and only if M is closed and F non singular. When this is not the case, M/F ≈ [0, 1], and there are exactly two points (∂[0, 1]) which come from a singular point and/or from a leaf of the boundary. If F is non transversely orientable, there is at least one leaf with (finite) non trivial holonomy, which corre- sponds a boundary point in M/F to (by Proposition 2.3). By the proof of Lemma 5.6, the projection is not differentiable and the space of leaves M/F , a Hausdorff topological 1-dimensional space, turns out to be an orbifold (see [Thu]). We pass to the transversely orientable double covering, p : (M̃, F̃) → (M,F). The fo- liation F̃ , pull-back of F , has all leaves compact, and singular set empty or with stable components; therefore we apply the first part of the classification to M̃/F̃ . Both if M̃/F̃ is diffeomorphic to S1 or to [0, 1], M/F is homeomorphic to [0, 1], but (clearly) with different orbifold structures. Before going on with our main generalization of the Reeb Sphere Theorem 2.4, which extends a similar result of Camacho and Scárdua [Cam-Sc] concerning the case n = 3, we need to recall another result, that we are going to generalize. As we know, the Reeb Sphere Theorem, in its original statement, consideres the effects (on the topology of a manifold M ) determined by the existence, on M , of a real valued function with exactly two non-degenerate singular points. A very similar problem was studied by Eells and Kuiper [Ee-Kui]. They considered manifolds admitting a real valued function with exactly three non-degenerate singular points.They obtained very interest- ing results. Among other things, it sticks out the obstruction they found about the dimension of M , which must be even and assume one of the values n = 2m = 2, 4, 8, 16. Moreover, the homotopy type of the manifold turns out to vary among a finite number of cases, including (or reducing to, if n = 2, 4) the homotopy tupe of the projective plane over the real, complex, quaternion or Cayley numbers. Definition 5.10 In view of the results of Eells and Kuiper [Ee-Kui], if a manifold M admits a real-valued func- Figure 11: Elimination technique applied in case (ii) (Theorem 3.5) for the foliation of figure 9. PSfrag replacements q Figure 12: A foliation of RP2 with three singular points. tion with exactly three non-degenerate singular points, we will say that M is an Eells-Kuiper manifold. We have (see [Cam-Sc] for the case n = 3): Theorem 5.11 (Center-Saddle Theorem) Let Mn be an n-dimensional manifold, with n ≥ 2 such that (M,F) is a foliated manifold, by means of a transversely orientable, codimension-one, Morse, C∞ folia- tion F . Moreover F is assumed to be without holonomy if n = 2. Let Sing(F) be the singular set of F , with #Sing(F) = k + l, where k, l are the numbers of, respectively, centers and saddles. If we have k ≥ l + 1, then there are two possibilities: (1) k = l + 2 and M is homeomorphic to an n-dimensional sphere; (2) k = l + 1 and M is an Eells-Kuiper manifold. Proof. If l = 0, assertion is proved by the Reeb Sphere Theorem 2.4. Let l ≥ 1; we prove our thesis by induction on the number l of saddles. We set F l = F . So let l = 1 and F1 = F . By hypothesis, in the set Sing(F) there exist at least two centers, p1, p2, with p1 6= p2, and one saddle q. We have necessarily q ∈ ∂ Cp1(F) ∩ ∂ Cp2(F). In fact, if this is not the case and, for example q /∈ ∂ Cp1(F), then (keeping into account that for n = 2, the foliation F is assumed to be without holonomy) ∂ Cp1 = ∅ and M = Cp1(F). A contraddiction. Let i(q) the Morse index of the saddle q. For n ≥ 3 we apply the results of Theorems 3.4 and 3.5 to the couples (p1, q) and (p2, q). In particular, by Theorem 3.5, (iii), it follows that the saddle q cannot be selfconnected. We now have the following two possi- bilities: (a) i(q) = 1, n− 1 and (p1, q) or (and) (p2, q) is a trivial couple, (b) i(q) 6= 1, n− 1 and there are no trivial couples. For n = 2, we have necessarily i(q) = 1 and, in our hypotheses, q is always selfconnected. With few changes, we adapt Theorem 3.5, to this case, obtaining ∂ Cp(F) ≃ S 1 or ∂ Cp(F) ≃ S 1 ∨ S1; in this latter case we will say that the saddle q is selfconnected with respect to p. We obtain: (a’) (p1, q) or (and) (p2, q) is a trivial couple; (b’) q is selfconnected both with respect to p1 and to p2. In cases (a) and (a’) we proceed with the elimination of a trivial couple, as stated in Proposition 4.2, and then we obtain the foliated manifold (M,F0), with no saddle-type and some center-type singularities. We apply the Reeb Sphere Theorem 2.4 and obtain #Sing(F) = 2 and M ≃ Sn. In case (b) (n ≥ 3), as a consequence of Theorem 3.4, we necessarily have i(q) = n/2 (and therefore n must be even!). Moreover Cp1(F) ≈ Cp2(F) and M = Cp1(F) ∪φ Cp2(F) may be thought as two copies of the same (singular) manifold glued together along the boundary, by means of the diffeomorphism φ. In case (b’) (n = 2), we obtain the same result as above, i.e. Cp1(F) ≈ Cp2(F) and M = Cp1(F)∪φ Cp2(F). We notice that case (b’) occurs when the set Cpi(F) ≃ B 2/S0 is obtained by identifying two points of the boundary in a way that reverses the orientation. In cases (b) and (b’), it turns out that #Sing(F1) = 3. Moreover, F1 has a first integral, which is given by the projection of M onto the space of (possibly singular) leaves. In fact, by Lemma 5.6, the space of leaves is diffeomorphic to a closed interval of R. In this way M turns out to be an Eells-Kuiper manifold. This ends the case l = 1. Let l > 1 (and #Sing(F) > 3). As above, in Sing(F) there exist at least one saddle q and two (distinct) cen- ters, p1, p2 such that q ∈ ∂ Cp1(F)∩ ∂ Cp2(F); we are led to the same possibilities (a), (b) for n ≥ 3 and (a)’, (b)’ for n = 2. Anyway (b) and (b’) cannot occur, otherwise M = Cp1(F) ∪φ Cp2(F) and #Sing(F) = 3, a contraddiction. Then we may proceed with the elimination of a trivial couple. In this way we obtain the foliated manifold (M,F l−1), which we apply the inductive hypothesis to. The theorem is proved, observing that, a posteriori, case (1) holds if k = l + 2 and case (2) if k = l + 1. 6 Haefliger-type theorems In this paragraph, we investigate the existence of leaves of singular foliations with unilateral holonomy. Keep- ing into account the results of the previous paragraph, for Morse foliations, we may state or exclude such an occurrence, according to the following theorem: Theorem 6.1 Let F be a C∞, codimension one, Morse foliation on a compact manifold Mn, n ≥ 3, assumed to be transversely orientable, but not necessarily closed. Let k be the number of centers and l the number of saddles. We have the following possibilities: (i) if k ≥ l + 1, then all leaves are closed in M \ Sing(F); in particular, if ∂M 6= ∅ or k ≥ l + 2 each regular (singular) leaf of F , is diffeomorphic (homeomorphic) to a sphere (in the second option, it is diffeomorphic to a sphere with a pinch at one point); (ii) if k = l there are two possibilities: all leaves are closed in M \Sing(F), or there exists some compact (regular or singular) leaf with unilateral holonomy. Example 6.2 The foliation of example 4.7 is an occurrence of theorem 6.1, case (ii) with all leaves closed. The Reeb foliation of S3 and each foliation we may obtain from it, with the introduction of l = k trivial couples center-saddle, are examples of theorem 6.1, case (ii), with a leaf with unilateral holonomy. Now we consider other possibilities for Sing(F). Definition 6.3 Let F be a C∞, codimension one foliation on a compact manifold Mn, n ≥ 3, with singular set Sing(F) 6= ∅. We say that Sing(F) is regular if its connected components are either isolated points or smoothly embedded curves, diffeomorphic to S1. We extend the definition of stability to regular components, by saying that a connected component Γ ⊂ Sing(F) is (weakly) stable, if there exists a neighborhood of Γ, where the foliation has all leaves compact (notice that we can repeat the proof of Lemma 5.7 and obtain that a weakly stable component is a stable component). In the case Sing(F) is regular, with stable isolated singularities, when n ≥ 3 we may exclude a Haefliger- type result, as a consequence of Lemma 5.5 and the Reeb Global Stability Theorem for manifolds with bound- ary. Then we study the case Sing(F) regular, with stable components, all diffeomorphic to S1. Let J be a set such that for all j ∈ J , the curve γj : S 1 → M , is a smooth embedding and Γj := γj(S 1) ⊂ Sing(F) is stable. Then J is a finite set. This is obvious, otherwise ∀j ∈ J , we may select a point xj ∈ Γj and obtain that the set {xj}j∈J has an accumulation point. But this is not possible because the singular components are separated. We may regard a singular component Γj , as a degenerate leaf, in the sense that we may associate to it, a single point of the space of leaves. We need the following definition Definition 6.4 Let F be a C∞, codimension one foliation on a compact manifold M . Let D2 be the closed 2-disc and g : D2 → M be a C∞ map. We say that p ∈ D2 is a tangency point of g with F if (dg)p( R Tg(p)Fg(p). We recall a proposition which Haefliger’s theorem (cfr. the book [Cam-LN]) is based upon. Proposition 6.5 Let A : D2 → M be a C∞ map, such that the restriction A|∂D2 is transverse to F , i.e. ∀x ∈ ∂D2, (dA)x(Tx(∂D 2)) + TA(x)FA(x) = TA(x)M . Then, for every ǫ > 0 and every integer r ≥ 2, there exists a C∞ map, g : D2 → M , ǫ-near A in the Cr-topology, satisfying the following properties: (i) g|∂D2 is transverse to F . (ii) For every point p ∈ D 2 of tangency of g with F , there exists a foliation box U of F with g(p) ∈ U and a distinguished map π : U → R such that p is a non-degenerate singularity of π ◦ g : g−1(U) → R. In particular there are only a finite number of tangency points of g with F , since they are isolated, and they are contained in the open disc D2 = {z ∈ R2 : ||z|| < 1}. (iii) If T = {p1, . . . , pt} is the set of tangency points of g with F , then g(pi) and g(pj) are contained in distinct leaves of F , for every i 6= j. In particular, the singular foliation g∗(F) has no saddle connections. We are now able to prove a similar result, in the case of existence of singular components. Proposition 6.6 Let F be a codimension one, C∞ foliation on a compact manifold Mn, n ≥ 3, with regular singular set, Sing(F) = ∪j∈JΓj 6= ∅, where Γj are all stable components diffeomorphic to S 1 and J is finite. Let A : D2 → M be a C∞ map, such that the restriction A|∂D2 is transverse to F . Then, for every ǫ > 0 and every integer r ≥ 2, there exists a C∞ map, g : D2 → M , ǫ-near A in the Cr-topology, satisfying properties (i) and (iii) of proposition 6.5, while (ii) is changed in: (ii’) for every point p ∈ D2 of tangency of g with F , we have two cases: (1) if Lg(p) is a regular leaf of F , there exists a foliation box, U of F , with g(p) ∈ U , and a distinguished map, π : U → R, satisfying properties as in (ii) of Proposition 6.5; (2) if Lg(p) is a degenerate leaf of F , there exists a neighborhood, U of p, and a singular submersion, π : U → R, satisfying properties as in (ii) Proposition 6.5. Proof. We start by recalling the idea of the classical proof. We choose a finite covering of A(D2) by foliation boxes {Qi} i=1. In each Qi the foliation is defined by a distinguished map, the submersion πi : Qi → R. We choose an atlas, {(Qi, φi)} i=1, such that the last component of φi : Qi → R n is πi, i.e. φi = (φ i , φ i , . . . , φ i , πi). We construct the finite cover of D {Wi = A −1(Qi)} i=1; the expression of A in coordinates is A|Wi = (A i , . . . , A i , πi ◦A). We may choose covers of D2, {Ui} i=1, {Vi} i=1, such that Ui ⊂ Vi ⊂ Vi ⊂ Wi, i = 1, . . . , r; then we proceed by induction on the number i. Starting with i = 1 and setting g0 = A, we apply a result ([Cam-LN], Cap. VI, §2, Lemma 1, pag. 120) and we modify gi−1 in a new function gi, in a way that gi(Wi) ⊂ Qi and πi ◦ gi : Wi → R is Morse on the subset Ui ⊂ Wi. At last we set g = gr. In the present case, essentially, it is enough to choose a set of couples, {(Uk, πk)}k∈K , where {Uk}k∈K is an open covering of M , πk : Uk → R, for k ∈ K , is a (possibly singular) submersion and, if Uk ∩ Ul 6= ∅ for a couple of indices k, l ∈ K , there exists a diffeomorphism plk : πk(Uk ∩ Ul) → πl(Uk ∩ Ul), such that πl = plk ◦ πk. By hypothesis, there exists the set of couples {(Ui, πi)}i∈I , where {Ui}i∈I , is an open covering of M \ Sing(F), and, for i ∈ I , the map πi : Ui → R, is a distinguished map, defining the foliated manifold (M \ Sing(F),F∗). Let y ∈ Sing(F), then y ∈ Γj , for some j ∈ J . As y ∈ M , there exists a neighborhood C ∋ y, homeomorphic to an n-ball. Let h : C → Bn be such a homeomorphism. As the map γj : S 1 → Γj is a smooth embedding, we may suppose that, locally, Γj is sent in a diameter of the ball Bn, i.e. h(C ∩ Γj) = {x2 = · · · = xn = 0}. For each singular point z = h −1(b, 0, . . . , 0), the set D = h−1(b, x2, . . . , xn), homeomorphic to a small (n−1)-ball, is transverse to the foliation at z. Moreover, if z1 6= z2, then D1 ∩D2 = ∅. The restriction F|D is a singular foliation with an isolated stable singularity at z. By lemma 5.5, the leaves of F|D are diffeomorphic to (n− 2)-spheres. It turns out that y has a neighborhood homeomorphic to the product (−1, 1)×Bn−1, where the foliation is the image of the singular trivial foliation of (−1, 1)×Bn−1, given by (−1, 1)×Sn−2×{t}, t ∈ (0, 1), with singular set (−1, 1)×{0}. Let πy : Uy → [0, 1) be the projection. If, for a couple of singular points y, w ∈ Sing(F), we have Uy ∩Uw 6= ∅, we may suppose they belong to the same connected component, Γj . We have πw ◦ π y (0) = 0 and, as a consequence of lemma 5.6, there exists a diffeomorphism between πy(Uy ∩ Uw \ Γj) and πw(Uy ∩ Uw \ Γj). The same happens if Uy ∩Ui 6= ∅ for some Ui ⊂ M \Sing(F). It comes out that πy is singular on Uy ∩Sing(F) and non-singular on Uy \ Sing(F), i.e. (dπy)z = 0 ⇔ z ∈ Uy ∩ Sing(F). At the end, we set K = I ∪ Sing(F). Let g : D2 → M be a map. Then g defines the foliation g∗(F), pull-back of F , on D2. Observe that if Sing(F) = ∅, then Sing(g∗(F)) = {tangency points of g with F}, but in the present case, as Sing(F) 6= ∅, we have Sing(g∗(F)) = {tangency points of g with F} ∪ g∗(Sing(F)). Either if p is a point of tangency of g with F or if p ∈ g∗(Sing(F)), we have d(πk)p = 0. With this remark, we may follow the classical proof. As a consequence of proposition 6.6, we have: Theorem 6.7 (Haefliger’s theorem for singular foliations) Let F be a codimension one, C2, possibly singular foliation of an n-manifold M , with Sing(F), (empty or) regular and with stable components diffeomorphic to S1. Suppose there exists a closed curve transverse to F , homotopic to a point. Then there exists a leaf with unilateral holonomy. 7 Novikov-type theorems We end this article with a result based on the original Novikov’s Compact Leaf Theorem and on the notion of stable singular set. To this aim, we premise the following remark. Novikov’s statement establishes the existence of a compact leaf for foliations on 3-manifolds with finite fundamental group. This result actually proves the existence of an invariant submanifold, say N ⊂ M , with boundary, such that F|N contains open leaves whose universal covering is the plane. Moreover these leaves accumulate to the compact leaf of the boundary. In what follows, a submanifold with the above properties will be called a Novikov component. In particular a Novikov component may be a Reeb component, i.e. a solid torus endowed with its Reeb foliation. We recall that two PSfrag replacements ∂ Cp(F) Figure 13: p − q is not a trivial coupling when 1 < l < n− 1, where l is the index of the saddle PSfrag replacements ST1 Figure 14: A singular foliation of S3, with no van- ishing cycles. Reeb components, glued together along the boundary by means of a diffeomorphism which sends meridians in parallels and viceversa, give the classical example of the Reeb foliation of S3. If F is a Morse foliation of a 3-manifold, as all saddles have index 1 or 2, we are always in conditions of proposition 4.2 and then we are reduced to consider just two (opposite) cases: (i) all singularities are centers, (ii) all singularities are saddles. In case (i), by the proof of the Reeb Sphere Theorem 2.4, we know that all leaves are compact; in case (ii), all leaves may be open and dense, as it is shown by an example of a foliation of S3 with Morse singularities and no compact leaves [Ros-Rou]. As in the previous paragraph, we study the case in which Sing(F) is regular with stable components,Γj , j ∈ J , where J is a finite set. We have: Theorem 7.1 Let F be a C∞, codimension one foliation on a closed 3-manifold M3. Suppose Sing(F) is (empty or) regular, with stable components. Then we have two possibilities: (i) all leaves of F are compact; (ii) F has a Novikov component. Proof. If Sing(F) = ∅, thesis (case (ii)) follows by Novikov theorem. Let Sing(F) 6= ∅. We may suppose that F is transversely orientable (otherwise we pass to the transversely orientable double covering). If Sing(F) contains an isolated singularity, as we know, we are in case (i). Then we suppose Sing(F) contains no isolated singularity, i.e. Sing(F) = Γj . Set D(F) = {Γj, j ∈ J} ∪ { compact leaves with trivial holonomy}. By the Reeb Local Stability Theorem 2.1, D(F) is open. We may have ∂D(F) = ∅, and then we are in case (i), or ∂D(F) 6= ∅, and in this case it contains a leaf with unilateral holonomy, F . It is clear that F bounds a Novikov component, and then we are in case (ii); in fact, from one side, F is accumulated by open leaves. If F ′ is one accumulating leaf, then its universal covering is p : R2 → F ′. Suppose, by contraddiction, that the universal covering of F ′ is p : S2 → F ′. By the Reeb Global Stability Theorem for manifolds with boundary, all leaves are compact, diffeomorphic to p(S2). This concludes the proof since F must have infinite fundamental group. The last result may be reread in terms of the existence of closed curves, transverse to the foliation. We have: Lemma 7.2 LetF be a codimension one, C∞ foliation on a closed 3-manifoldM , with singular set, Sing(F) 6= ∅, regular, with stable components. Then F is a foliation with all leaves compact if and only if there exist no closed transversals. Proof. (Sufficiency) If the foliation admits an open (in M \ Sing(F)) leaf, L, it is well known that we may find a closed curve, intersecting L, transverse to the foliation. Viceversa (necessity), let F be a foliation with all leaves compact. If necessary, we pass to the transversely orientable double covering p : (M̃, F̃) → (M,F). In this way, we apply Lemma 5.6 and obtain, as Sing(F̃) 6= ∅, that the projection onto the space of leaves is a (global) C∞ first integral of F̃ , f : M̃ → [0, 1] ⊂ R. Suppose, by contraddiction, that there exists a C1 closed transversal to the foliation F , the curve γ : S1 → M . The lifting of γ2 is a closed curve, Γ : S1 → M̃ , transverse to F̃ . The set f(Γ(S1)) is compact and then has maximum and minimum, m1,m2 ∈ R. A contrad- diction, because Γ cannot be transverse to the leaves {f−1(m1)}, {f −1(m2)}. With this result, we may rephrase the previous theorem. Corollary 7.3 Let F be a codimension one, C∞ foliation on a 3-manifold M , such that Sing(F) is regular with stable components. Then (i) there are no closed transversals, or equivalently, F is a foliation by compact leaves, (ii) there exists a closed transversal, or equivalently, F has a Novikov component. Remark 7.4 In the situation we are considering, we cannot state a singular version of Auxiliary Theorem I (see, for example [Mor-Sc]). In fact, even though a singular version of Haefliger Theorem is given, the existence of a closed curve transverse the foliation, homotopic to a constant, does not lead, in general, to the existence of a vanishing cycle, as it is shown by the following counterexample. Example 7.5 We consider the foliation of S3 given by a Reeb component, ST1, glued (through a diffeomor- phism of the boundary which interchanges meridians with parallels) to a solid torus ST2 = S 1 × D2 = T 2 × (0, 1) ∪ S1. The torus ST2 is endowed with the singular trivial foliation F|ST2 = T 2 × {t}, for t ∈ (0, 1), where Sing(F|ST2) = S 1 = Sing(F). As a closed transversal to the foliation, we consider the curve γ : S1 → ST1 ⊂ S 3, drawed in figure 14. Let f : D2 → S3 be an extension of γ; the extension f is assumed to be in general position with respect to F , as a consequence of proposition 6.5. As γ(S1) is linked to the singular component S1 ⊂ ST2, then f(D2) ∩ Sing(F) 6= ∅. As a consequence, we find a decreasing sequence of cycles, {βn}, (the closed curves of the picture) which does not admit a cycle, β∞, such that βn > β∞, for all n. In fact the “limit” of the sequence is not a cycle, but the point f(D2) ∩ Sing(F). Example 7.6 The different situations of Theorem 7.1 or Corollary 7.3 may be exemplified as follows. It is easy to see that S3 admits a singular foliation with all leaves compact (diffeomorphic to T 2) and two singular (stable) components linked together, diffeomorphic to S1. In fact one can verify that S3 is the union of two solid tori, ST1 and ST2, glued together along the boundary, both endowed with a singular trivial foliation. We construct another foliation on S3, modifying the previous one. We set S̃T1 = S 1 × {0} ∪ T 2 × (0, 1/2]. In this way, ST1 = S̃T1 ∪ T 2 × (1/2, 1]. We now modify the foliation in ST1 \ S̃T1, by replacing the trivial foliation with a foliation with cylindric leaves accumulating to the two components of the boundary. References [Cam-LN] C. Camacho, A. Lins Neto: Geometric theory of foliations, Boston, Birkhauser, 1985 [Cam-Sc] C. Camacho, B. Scárdua: On codimension one foliations with Morse singularities on three- manifolds, Topology and its Applications 154 (2007) 1032-1040. [Ee-Kui] J. Eells, N.H. Kuiper: Manifolds which are like projective planes, Pub. Math. de l’I.H.E.S., 14, 1962. [God] C. Godbillon: Feuilletages, etudies geometriques, Basel, Birkhauser, 1991 [Law] H.B. Lawson, jr.: Foliations, Bull. Amer. Math. Soc., Vol. 80, N. 3, May 1974. [Mil 1] J. Milnor: Morse theory, Princeton, NJ, Princeton University Press, 1963. [Mil 2] J. Milnor: Lectures on the h-cobordism theorem, Princeton, NJ, Princeton University Press, 1965. [Mor-Sc] C.A. Morales, B. Scárdua: Geometry and Topology of foliated manifolds. [Nov] S.P. Novikov: Topology of foliations. Trudy Moskov. Mat. Obshch. 14 (1965), 248-278. [Pal-deM] J. Palis, jr., W. de Melo: Geometric theory of dinamical systems: an introduction, New-York, Springer,1982. [Reeb] G. Reeb: Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique. CRAS 222 (1946), 847-849. [Ros-Rou] H. Rosemberg, R. Roussarie: Some remarks on stability of foliations, J. Diff. Geom. 10, 1975, 207-219. [Stee] N. Steenrod: The topology of fiber bundles, Princeton, NJ, Princeton University Press, 1951 [Thu] W.P. Thurston: Three-dimensional geometry and topology, Princeton, NJ, Princeton University Press, 1997. [Wag] E. Wagneur: Formes de Pfaff à singularités non dégénérées, Annales de l’institut Fourier, tome 28 n. 3 (1978), p. 165-176. Foliations and Morse Foliations Holonomy and Reeb Stability Theorems Arrangements of singularities Realization and elimination of pairings of singularities Reeb-type theorems Haefliger-type theorems Novikov-type theorems

0704.0165

Frobenius-Schur indicators for semisimple Lie algebras

FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS MOHAMMAD ABU-HAMED AND SHLOMO GELAKI 1. Introduction Classically Frobenius-Schur indicators were defined for irreducible representa- tions of finite groups over the field of complex numbers. The interest in doing so came from the second indicator which determines whether an irreducible represen- tation is real, complex or quaternionic. Namely, a classical theorem of Frobenius and Schur asserts that an irreducible representation is real, complex or quaternionic if and only if its second indicator is 1, 0 or −1, respectively (see e.g. [S]). However, no representation-theoretic interpretation of the higher indicators is known. Recently, Frobenius-Schur indicators of irreducible representations of complex semisimple finite dimensional (quasi-)Hopf algebras H were defined by Linchenko and Montgomery [LM] and Mason and Ng [MN] (see also [KSZ]), generalizing the definition in the group case. The values of the mth indicator are cyclotomic integers in Qm. Moreover, an analog of the Frobenius-Schur theorem on the second indicator was proved, and in general it has been shown that the indicators carry rich information on H , as well as on its representation category (see also [NS2]). In fact, one can generalize the definition of Frobenius-Schur indicators to simple objects of any semisimple tensor categories which admit a pivotal structure (= tensor isomorphism id →∗∗), thus showing in particular that the indicators are categorical invariants (see e.g. [FGSV], [NS1]). The category of finite dimensional representations of a finite dimensional complex semisimple Lie algebra is a pivotal semisimple tensor category, and hence one can define the Frobenius-Schur indicators of its simple objects. The second indicator was already defined and known to be nonzero if and only if the simple representation is self-dual, and 1 or −1 if and only if the representation is orthogonal or symplectic, respectively. Furthermore, Tits gave an explicit formula for it in representation- theoretic terms (see Section 3). The purpose of this paper is to study Frobenius-Schur indicators (of all degrees) for semisimple Lie algebras. More specifically to find a closed formula for the indicators in representation-theoretic terms and deduce its asymptotical behavior. In particular we obtain that the indicators take integer values. The organization of the paper is as follows. Section 2 is devoted to preliminaries. We recall some basic definitions and facts from Lie theory which we need (e.g. the Weyl integration formula). Next we define the mth Frobenius-Schur indicator of the representation categories of finite dimensional complex semisimple Lie algebras. Date: February 11, 2007. http://arxiv.org/abs/0704.0165v1 2 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI In section 3 we recall the properties of the second indicator. For the benefit of the reader we also give a proof of Tits’ theorem. Section 4 is dedicated to the proof of our main results. In 4.1 we prove the formula for the mth Frobenius-Schur indicator νm, m ≥ 2, which is given by the following theorem. Theorem 1.1. Let g be a finite dimensional complex semisimple Lie algebra. Let V (λ) be an irreducible representation of g with highest weight λ, W the Weyl group of g, ρ the half sum of positive roots, and V (λ) ρ−σ·ρ the weight space of the weight ρ−σ·ρ where m ≥ 2 is an integer. Then the mth Frobenius-Schur indicator νm(V (λ)) of V (λ) is given by νm(V (λ)) = sn(σ) dimV (λ) ρ− σ · ρ Our proof of Theorem 1.1 is analytic. Namely, we work with the equivalent representation category of the associated simply connected Lie group and use the Weyl integration formula to obtain our formula. Next, in 4.2 we prove the following corollary of Theorem 1.1. Corollary 1.2. For large enough m, νm(V (λ)) = dimV (λ)[0] (which is not zero if and only if λ belongs to the root lattice). In particular for the classical Lie algebras sl(n,C), so(2n,C), so(2n + 1,C) and sp(2n,C), νm(V (λ)) = dimV (λ)[0] for m greater or equal to 2n− 1, 4n− 5, 4n− 3 and 2n+ 1, respectively. Finally in 4.3 we use our formula and Kostant’s theorem to compute explicitly the Frobenius-Schur indicators for the representation category of sl(3,C). More specifically, we prove: Theorem 1.3. Let V (a, b) be an irreducible representation of sl(3,C). Then (1) ν2(V (a, b)) = 1 if a = b, and ν2(V (a, b)) = 0 if a 6= b. (2) ν3(V (a, b)) = 1 +min{a, b}. (3) For m > 3 we have, νm(V (a, b)) = 1 + min{a, b} if (a, b) is in the root lattice and νm(V (a, b)) = 0 otherwise. Acknowledgments. This research was supported by the Israel Science Foundation (grant No. 125/05). 2. Preliminaries Throughout let g be a finite dimensional complex semisimple Lie algebra of rank r, ( , ) its Killing form, h a Cartan subalgebra (CSA) of g, Φ the root system corresponding to h, ∆ a fixed base, {h1, ..., hr} the corresponding coroot system, and W the Weyl group. Let λ ∈ h∗ be a dominant integral weight (i.e. λ(hi) is a nonnegative integer for all i), V (λ) the finite dimensional irreducible representation of g with highest weight λ and Π(λ) the set of integral weights occurring in V (λ); it is a finite set which is invariant under the action of the Weyl group. For µ ∈ Π(λ), let mλ(µ) = dimV (λ)[µ] be the multiplicity of µ in V (λ). Recall that the multiplicities are invariant under the Weyl group action. Let ρ = α (half sum of positive roots); it is a strongly dominant integral weight. FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 3 Let us recall Kostant’s theorem on the multiplicities of weights (for a proof see [Hu]). Let µ ∈ h∗ and define p(µ) to be the number of sets of non-negative integers {kα|α ≻ 0} for which µ = kαα (p is called the Kostant’s partition function). Of course, p(µ) = 0 if µ is not in the root lattice. Theorem 2.1. (Kostant) Let λ be a dominant weight and µ ∈ Π(λ). Then the multiplicities of V (λ) are given by the formula mλ(µ) = sn(σ)p(σ(λ + ρ)− µ− ρ). Let gc be the compact real form of g, and G the corresponding simply connected compact matrix Lie group with Lie algebra gc. It is known that Rep(g), Rep(gc) and Rep(G) are equivalent symmetric tensor categories. Let t be a CSA of gc; it corresponds to a maximal torus T of G. Then h = t⊕ it. It is known that α(h) is purely imaginary for all h ∈ t and α ∈ Φ. If t∗ denotes the space of real-valued linear functionals on t, then the roots are contained in it∗ ⊂ h∗. It is then convenient to introduce the real roots, which are simply 1 times the ordinary roots, the real coroots hα which are the elements of t corresponding to the elements 2α (α,α) where α is a real root, and the real weights of an irreducible representation of G. An element µ of t∗ is said to be integral if µ(hα) ∈ Z for each real coroot hα. The real weights of any finite dimensional representation of g are integral. (See [Ha].) The Weyl denominator is the function Aρ : T −→ C given by Aρ(t) = Aρ(e sn(ω)ei(ω·ρ)(h). Theorem 2.2. (Weyl integration formula) Let G be a simply connected compact Lie group. Let f be a continuous class function on G, dg the normalized Haar measure on G, and dt the normalized Haar measure on T . Then f(g)dg = f(t)|Aρ(t)| Let us now define the Frobenius-Schur indicators of an irreducible representation of g. Definition 2.3. Let V be an irreducible representation of g and m ≥ 2 be an inte- ger. The mth Frobenius-Schur indicator of V is the number νm(V ) = tr(c|(V ⊗m)g), where c is the cyclic automorphism of V ⊗m given by v1⊗· · ·⊗vm 7→ vm⊗· · ·⊗vm−1. Remark 2.4. In fact, as we mentioned in the introduction, the indicators can be defined categorically. Applying the categorical definition to Rep(g) yields the above definition, while applying it to Rep(G) yields tr(c|(V ⊗m)G). Since the indicators of V regarded as a g-module coincide with the indicators of V regarded as a G-module we have νm(V ) = tr(c|(V ⊗m)G). 3. Tits’ theorem on the second indicator Theorem 3.1. (See [B]) Let G be a compact Lie group. Let V be an irreducible complex representation of G, and set ǫV = χ(g2)dg. Then V is self dual if and only if ǫV 6= 0. Furthermore, suppose V is self dual and let B be a (unique up to 4 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI scalar) G-invariant non-degenerate bilinear form on V . Then B is either symmetric or skew-symmetric, and it is such if and only if ǫV = 1,−1, respectively. Remark 3.2. In Proposition 4.4 we will prove that ǫV = ν2(V ) as defined above. Historically ν2(V ) was defined by ǫV . Example 3.3. Let us use Theorem 1.1 to calculate νm(V ) in the representation category of sl(2,C). Let sl(2,C) = sp{h, x, y}, where h = , x = . The root system is Φ = {α,−α}, where α(h) = 2. The Weyl group is W = {1, σα} , ρ = α and σα(ρ) = − α. Let V (n) = ⊕nj=0V [n − 2j] be the irreducible representation of highest weight λ(h) = n with its weight space decomposition. By Theorem 1.1, νm(V (n)) = dimV (n)[0]− dimV (n) Let m = 2. By the formula above, if n is odd, then dimV (n)[0] = 0 and dimV (n)[α ] = 1. Hence ν2(V (n)) = −1. Similarly, if n is even, ν2(V (n)) = 1. Consequently ν2(V (n)) = (−1) n = (−1)λ(h). For m ≥ 3, is not an integer and hence νm(V (n)) = dimV (n)[0]. Therefore we have νm(V (n)) = 1 if n is even 0 if n is odd Let g = h (⊕α∈Φgα) be the root space decomposition of g and ∆ = {α1, ..., αr} a fixed base. Fix a standard set of generators for g: xi ∈ gαi , yi ∈ g−αi so that [xi, yi] = hi. Let ρ̌ := 1/2 α∈Φ+ hα be the half sum of positive coroots. Proposition 3.4. Let E := x1 + ...+ xr and H := 2ρ̌. Then there exist constants a1, ..., ar such that the subalgebra P generated by H,E, F := a1y1 + ... + aryr is isomorphic to sl(2,C). The Lie subalgebra P ⊆ g is called a principal sl(2,C)-subalgebra of g (see [K] or [D]). Lemma 3.5. Let V = V (λ) be an irreducible representation of g. Let P be a principal sl(2,C)-subalgebra of g. Consider V as a P-module. Then its highest weight is λ(H), and it contains the irreducible sl(2,C)-representation V (λ(H)) with multiplicity one. Proof. Let v+ be a highest weight vector of V considered as a g-module. Then obviously we have Hv+ = λ(H)v+ and Ev+ = 0. Hence v+ is a highest weight vector with weight λ(H) for V considered as a P -module. Therefore we can write V = V (λ(H)) V (nj). Now it remains to show that λ(H) > nj for any j. Let V = V [λ] ⊕ V [µ] be the weight space decomposition of V as a g-module. It is also a weight space decomposition of V considered as a P -module, so V [µ] is a weight space of P with weight µ(H). Recall that µ = λ− j=1 kjαj where kj ∈ Z Note that λ(H) > µ(H) if and only if λ(H) > λ(H)− j=1 kjαj(H) if and only if j=1 kjαj(2ρ̆) > 0 if and only if j=1 kj(αj , 2ρ) > 0. But 2ρ is strongly dominant, i.e., (αj , 2ρ) > 0 for all 1 ≤ j ≤ r. The proof is complete. � FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 5 Let ω0 ∈ W be the unique element sending ∆ to −∆. Theorem 3.6. (Tits) Let V = V (λ) be a finite dimensional irreducible represen- tation of g. If λ+ ω0λ 6= 0 then ν2(V ) = 0. Otherwise, ν2(V ) = (−1) λ(2ρ̌). Proof. It is known that the dual of V (λ) is V (−ω0λ), so if V (λ) is not self dual (i.e., λ+ ω0λ 6= 0) then ν2(V ) = 0. Suppose that V is self dual as a g-module. Then V admits a non-degenerate g-invariant bilinear form, and we have to decide if it is symmetric or skew sym- metric. To do so, consider the principal sl(2,C)-subalgebra P as in Lemma 3.5. The restriction of V to P has a unique copy of the largest representation of P occurring in V , with highest weight λ(2ρ̌). We already proved that this represen- tation has indicator (−1)λ(2ρ̆). Now we can use Theorem 3.1 to prove that V has a symmetric (skew-symmetric) g-invariant form if and only if it has a symmetric (skew-symmetric) P -invariant form. The first direction is obvious. Conversely, sup- pose that V has a symmetric P -invariant form and suppose on the contrary that V admits a skew-symmetric g-invariant form. Then if we restrict the bilinear g-form to P we get that V has a skew-symmetric P -invariant form which is a contradiction. Similar considerations are applied when V has a skew-symmetric P -invariant form. We conclude that ν2(V ) = (−1) λ(2ρ̆). � 4. The Main results 4.1. Proof of Theorem 1.1. Let G be the associated simply connected compact Lie group. From now on we will consider V (λ) as a G-module. For convenience set V = V (λ), N = V (λ)⊗m, and let π : G −→ GL(V ) be the irreducible represen- tation. The following lemma is easily derived from linear algebra. Lemma 4.1. Let T ∈ End(V ) be a projection, W = ImT and S ∈ End(V ) an operator preserving W. Then tr|W (S) = tr|V (S ◦ T ). Proof. Fix a basis A = {w1, ..., wk} for W , and let à = {w1, ..., wk, wk+1, ..., wn} be a completion to a basis for V . Let C = [S|W ]A be the matrix representing S|W with respect to the basis A. Since T |W = idW and S(W ) ⊆ W we find out that , [S] , and hence [S] . The lemma follows easily now. � Proposition 4.2. We have, νm(V ) = tr|NG(c) = tr|V (c ◦ π ⊗m(g))dg. Proof. We follow the lines of the proof of the first formula for Frobenius-Schur indicators in the Hopf case, given in Section 2.3 of [KSZ]. Set τ = π⊗m. Consider the operator τ(g)dg : N −→ N . Let us first show that the image of this operator is NG. Indeed, by the invariance of the Haar measure, τ(g)vdg = τ(hg)vdg = τ(g)vdg for all h ∈ G and v ∈ N . Hence τ(g)dg ⊆ NG. Conversely, suppose that u ∈ NG, then τ(g)udg = udg = u dg = u. Hence NG ⊆ Im τ(g)dg and we are done. 6 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI In fact, the above shows also that the operator τ(g)dg is a projection onto Finally, c ∈ Aut(NG), so by Lemma 4.1, tr|NG(c) = tr|N τ(g)dg tr|N (c ◦ τ(g))dg, as claimed. � The following lemma is a particular case of a lemma in Section 2.3 of [KSZ] and its proof replicates the proof of that lemma. Lemma 4.3. Let f1, ..., fm ∈ End(V ). Then, tr|V ⊗m(c ◦ (f1 ⊗ ...⊗ fm)) = tr|V (f1 ◦ ... ◦ fm). Proof. Let v1, ..., vn be a basis of V with dual basis v 1 , ..., v n. For l = 1, ...,m, fl is presented by the matrix i,j=1 , where alij = (v i , fl(vj)). Therefore, tr i=1(v i , fl(vi)). We now have tr|V ⊗m(c ◦ (f1 ⊗ ...⊗ fm)) = i1,...,im=1 (v∗i1 ⊗ v ⊗ ...⊗ v∗im , c(f1(vi1)⊗ f2(vi2)⊗ ...⊗ fm(vim))) = i1,...,im=1 (v∗i1 , f2(vi2)) · · · (v , fm(vim ))(v , f1(vi1 )) = i1,...,im=1 a2i1,i2a i2,i3 · · · amim−1,ima im,i1 = tr|V (f2 ◦ f3 ◦ · · · ◦ fm ◦ f1) = tr|V (f1 ◦ f2 ◦ · · · ◦ fm), as desired. � Consequently we have the following proposition which is analogous to the finite group case. Proposition 4.4. Let χ be the irreducible character of V . Then νm(V ) = χ(gm)dg. Proof. We follow the lines of the proof of the first formula for Frobenius-Schur indicators in the Hopf case, given in Section 2.3 of [KSZ]. It follows immediately from Proposition 4.2 and Lemma 4.3 that νm(V ) = tr|N (c ◦ π ⊗m(g))dg = tr|N (c ◦ (π(g) ⊗ ...⊗ π(g))dg = tr|V (π(g) ◦ .. ◦ π(g))dg = χ(gm)dg. FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 7 Recall the integral real elements which are those elements µ of t∗ for which 2(µ,α) (α,α) is an integer for any simple real root α. For each real integral element µ, there is a function µ̃ on T given by µ̃(eh) = eiµ(h) for all h in t. Functions of this form are called torus characters and they have the following property. Lemma 4.5. µ̃(t)dt = eiµ(h)deh = 1 µ = 0, 0 otherwise. Proof. Suppose that µ 6= 0, then there exists t0 ∈ t such that µ̃(t0) 6= 1. Therefore µ̃(t)dt = µ̃(t0t)dt = µ̃(t0) µ̃(t)dt, hence µ̃(t)dt = 0. � Let χ be the character of V . Before we begin the proof of Theorem 1.1, recall that if t = eh ∈ T then for all t ∈ T , (1) χ(t) = χ(eh) = µ∈Π(V ) dim(V [µ])eiµ(h). We can now prove our main result. Proof of Theorem 1.1: By Proposition 4.4 and the Weyl integration formula we have, (2) νm(V ) = χ(gm)dg = χ(tm)|Aρ(t)| On the other hand, (3) χ(tm) = χ(emh) = µ∈Π(V ) dim(V [µ])eimµ(h). Hence by (2) and (3) we have, (4) νm(V ) = µ∈Π(V ) dimV [µ] eimµ(h)|Aρ(e h)|2deh. Now let us calculate the last integral. We have eimµ(h)|Aρ(e h)|2deh = eimµ(h)Aρ(e h)Aρ(eh)de eimµ(h) ω∈W sn(ω)e i(ω·ρ)(h) τ∈W sn(τ)e −i(τ ·ρ)(h) deh = ω,τ∈W sn(ωτ) ei(mµ+ω·ρ−τ ·ρ)(h)deh. But from Lemma 4.5 we have ei(mµ+ω·ρ−τ ·ρ)(h)deh = 1 if mµ+ ω · ρ− τ · ρ = 0 0 otherwise. 8 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI Hence (4) becomes, νm(V ) = ω,τ∈W τ·ρ−ω·ρ sn(ωτ)dimV [µ] ω,τ∈W sn(ωτ)dimV τ · ρ− ω · ρ Since dimV [ζ] = dimV [τ · ζ] for all ζ ∈ Π(V ) and τ ∈ W , we can write, νm(V ) = ω,τ∈W sn(ωτ)dimV ρ− τ−1ω · ρ Now if we fix ω ∈ W , substitute σ = τ−1ω and use the fact that sn(ωτ) = sn(τ−1ω), we get sn(ωτ)dimV ρ− τ−1ω · ρ sn(τ−1ω)dimV ρ− τ−1ω · ρ sn(σ)dimV ρ− σ · ρ Consequently, νm(V ) = ω,σ∈W sn(σ)dimV ρ− σ · ρ sn(σ)dimV ρ− σ · ρ as desired. � It may be interesting to state the following immediate consequence of Theorem 1.1 and Theorem 3.6. Corollary 4.6. Let V (λ) be an irreducible self dual representation of g, then sn(σ)dimV (λ) ρ− σ · ρ = (−1)λ(2ρ̌). If V (λ) is not self dual, the sum equals 0. 4.2. Proof of Corollary 1.2. Since ρ is strongly dominant, σ · ρ = ρ only when σ = 1. Write νm(V ) = dimV [0] + σ 6=1 sn(σ)dimV ρ− σ · ρ We wish to show that for large enough m, ρ−σ·ρ is not a weight of V when σ 6= 1. Indeed, suppose that σ 6= 1. Recall that ρ− σ · ρ is an integral element, hence if we fix some coroot hα, we have the following set of integers: Uα = {(ρ− σ · ρ)(hα)|σ ∈ W , σ 6= 1}. Therefore if we take mα = 1 + uα, where uα is the maximal element of Uα, then ρ−σ·ρ /∈ Π(V ). Hence dimV ρ−σ·ρ = 0 for all σ 6= 1, and therefore νm(V ) = dimV [0], for all m ≥ mα. � Note that by the procedure of the above proof, m =: min{mα|α ∈ ∆} is a better bound. Let us now give an explicit such lower bound. FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 9 Lemma 4.7. If ω ∈ W then ω · ρ = ρ− −1(α)∈Φ− In particular, sα(ρ) = ρ− α for α ∈ ∆. Proof. Evidently, ω · ρ is half sum of the set {ω(α)|α ∈ Φ+}. Like Φ+, this is a set of exactly half of the roots, containing each root or its negative but not both. More precisely, this set is obtained from Φ+ by replacing each α ∈ Φ+ such that ω−1 · α ∈ Φ− by its negative. Now, ω · ρ = ρ− −1(α)∈Φ− is evident , and sα(ρ) = ρ− α is a special case since one shows that if α ∈ ∆ and β ∈ Φ+, then either β = α or sα(β) ∈ Φ Proposition 4.8. Let V be an irreducible representation of G. Then νm(V ) = dimV [0] for all m ≥ M := minα∈∆{ |β(hα)|+ 1}. Proof. Let h = hα be a simple coroot. For all 1 6= ω ∈ W we have, |(ρ− ω · ρ)(h)| = β∈Φ+, −1(β)∈Φ− β∈Φ+, −1(β)∈Φ− |β(h)| ≤ |β(h)|. Therefore if we choose m = β∈Φ+ |β(h)|+ 1 then ρ−ω·ρ (h) /∈ Z, namely, ρ−ω·ρ not a weight. Consequently, σ 6=1 sn(σ)dimV ρ− σ · ρ = 0, and we are done. � Let us calculate the bound M defined in Proposition 4.8 for sl(n,C). Let the Cartan subalgebra be the set of diagonal matrices in sl(n,C). Let the set of positive roots be Φ+ = {βi,j|1 ≤ i < j ≤ n}, where βi,j(diag(a1, . . . , an)) = ai − aj . The subset ∆ = {βi,i+1|1 ≤ i ≤ n− 1} is a base. With respect to this base the simple coroots are {hi|1 ≤ i ≤ n− 1}, where hi is the matrix with 1 in the (i, i) position, −1 in the (i+1, i+1) position and 0 elsewhere. Then, by an elementary calculation, we get that for any simple coroot h, 1≤i<j≤n |βi,j(h)|+ 1 = 2n− 1. Consequently we obtain that M = 2n− 1. Let us calculate the bound M defined in Proposition 4.8 for so(2n+ 1,C). Let the Cartan subalgebra be the set of diagonal matrices in so(2n + 1,C). Let the set of positive roots be Φ+ = {βi ± βj |1 ≤ i < j ≤ n} ∪ {βi|1 ≤ i ≤ n}, where βi(hj) = δij . The subset ∆ = {βi − βi+1, βn|1 ≤ i ≤ n − 1} is a base. With respect to this base the simple coroots are {hi − hi+1, 2hn|1 ≤ i ≤ n − 1}, where hi is the matrix with 1 in the (i, i) position, −1 in the (n+ i, n+ i) position and 0 10 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI elsewhere. Then, by an elementary calculation, we get that for any simple coroot h := hk − hk+1, 1 ≤ k ≤ n− 1, the sum β∈Φ+ |β(h)|+ 1 equals 1≤i<j≤n |(βi + βj)(h)|+ 1≤i<j≤n |(βi − βj)(h)| + |βi(h)|+ 1 = 4n− 3, while for the simple coroot h := 2hn it equals 4n− 1. Consequently we obtain that M = 4n− 3. Applying similar arguments to the other classical simple Lie algebras yields the following result. Proposition 4.9. The bound M for sl(n,C), so(2n,C), so(2n+1,C) and sp(2n,C) is equal to 2n− 1, 4n− 5, 4n− 3 and 2n+ 1, respectively. 4.3. The proof of Theorem 1.3. Let h be the CSA of sl(3,C) generated by the two elements h1 = diag(1,−1, 0) and h2 = diag(0, 1,−1). We will identify any functional α on h with the pair (α(h1), α(h2)). Under this identification the six roots of sl(3,C) are α1 = (2,−1), α2 = (−1, 2), α1 + α2 = (1, 1), −α1 = (−2, 1), −α2 = (1,−2) and −α1 − α2 = (−1,−1). The roots α1 = (2,−1), α2 = (−1, 2) form a base and the corresponding simple coroots are h1, h2, respectively. Recall that if V = V (λ) is an irreducible representation of sl(3,C) of highest weight λ, then λ is of the form (a, b) with a and b non-negative integers. Recall thatW ∼= S3 and it acts on h by σ·diag(d1, d2, d3) = diag(dσ(1), dσ(2), dσ(3)). Therefore, (12) · α1 = −α1, (12) · α2 = α1 + α2; (13) · α1 = −α2, (13) · α2 = −α1; (23) · α1 = α1 + α2, (23) · α2 = −α2; (123) · α1 = −α1 − α2, (123) · α2 = α1; and (132) · α1 = α2, (123) · α2 = −α1 − α2. The half sum of positive roots is ρ = 1 (2α1 + 2α2) = α1 + α2. We have, ρ− (12)ρ = (2,−1), ρ− (13)ρ = (2, 2), ρ− (23)ρ = (−1, 2), ρ− (123)ρ = (0, 3), and ρ− (132)ρ = (3, 0). Let m = 2. Considering our formula, we cancel all the summands which include roots that one of their two components is not divisible by 2. Consequently we get ν2(V ) = dimV [(0, 0)]− dimV [(1, 1)]. Recall that an irreducible representation V (a, b) is self dual if and only if a = b. Since λ = (s, s) = sα1 + sα2, (λ, 2ρ̌) = (λ, 2h1 + 2h2) = (λ, 2h1) + (λ, 2h2) = 4s, it follows from Tits’ theorem that ν2(V (a, b)) = 0 a 6= b, 1 otherwise. Similar considerations for m ≥ 3 yield, ν3(V ) = dimV [(0, 0)] + dimV [(1, 0)] + dimV [(0, 1)] νm≥4(V ) = dimV [(0, 0)]. In particular, if λ does not belong to the root lattice, νm≥4(V ) = 0. We now calculate dimV [(0, 0)], dimV [(1, 0)] and dimV [(0, 1)]. Recall that for η ∈ h∗, p(η) ≥ 1 if and only if η belongs to the root lattice and η ≻ 0. If η = kα1 + lα2 with nonnegative integers k and l, then p(η) = 1 +min{k, l}. Write λ = kα1 + lα2 where k and l are real numbers and identify it with the pair (λ(h1), λ(h2)) = (a, b) = (2k − l, 2l− k). FROBENIUS-SCHUR INDICATORS FOR SEMISIMPLE LIE ALGEBRAS 11 Note that (0, 1) = 1 α2 and (1, 0) = α2. Therefore by Kostant’s formula (see Theorem 2.1), dimV [(0, 0)] = sn(ω)p((k + 1)ω · α1 + (l + 1)ω · α2 − α1 − α2), dimV [(0, 1)] = sn(ω)p (k + 1)ω · α1 + (l + 1)ω · α2 − dimV [(1, 0)] = sn(ω)p (k + 1)ω · α1 + (l + 1)ω · α2 − It is straightforward to verify that in each of the three cases the surviving terms correspond to ω = 1, (12), (23). For example, in the first case calculating (k + 1)ω · α1 + (l + 1)ω · α2 − α1 − α2 for ω = (12), (13), (23), (123), (132), yields (l − k − 1)α1 + lα2, −(l + 2)α1 − (k + 2)α2 (hence p = 0), kα1 + (k − l − 1)α2, (l−k−1)α1− (k+2)α2 (hence p = 0), and −(l+2)α1+(k− l−1)α2 (hence p = 0), respectively. Therefore we have that dimV [(0, 0)] equals b+ 2a 2b+ a b− a− 3 2b+ a b+ 2a −b+ a− 3 dimV [(0, 1)] equals b+ 2a− 1 2b+ a− 2 b− a− 4 2b+ a− 2 b+ 2a− 1 −b+ a− 5 and dimV [(1, 0)] equals b+ 2a− 2 2b+ a− 1 b− a− 5 2b+ a− 1 b+ 2a− 2 −b+ a− 4 Now, modulo 3, exactly one of the following holds: 1) b+2a = 0 and 2b+a = 0 (in this case λ belongs to the root lattice), 2) b+2a = 1 and 2b+a = 2 and 3) b+2a = 2 and 2b + a = 1. Hence by the above and elementary calculations, we obtain that in the first case dimV [(0, 1)] = dimV [(1, 0)] = 0, in the second case dimV [(0, 0)] = dimV [(1, 0)] = 0 and in the third case dimV [(0, 0)] = dimV [(0, 1)] = 0. Therefore, in the first case ν3(V (a, b)) equals 1 +min b+ 2a 2b+ a 1 +min b− a− 3 2b+ a 1 +min b+ 2a −b+ a− 3 12 MOHAMMAD ABU-HAMED AND SHLOMO GELAKI in the second case it equals 1 +min b+ 2a− 1 2b+ a− 2 1 +min b− a− 4 2b+ a− 2 1 +min b+ 2a− 1 −b+ a− 5 and in the third case it equals 1 +min b+ 2a− 2 2b+ a− 1 1 +min b− a− 5 2b+ a− 1 1 +min b+ 2a− 2 −b+ a− 4 Finally, it is easy to check that in each case the sum equals 1 + min{a, b}, as claimed. This completes the proof of the theorem. � References [B] D. Bump, Lie Groups, Springer-Verlag NY, LLC, (2004). [D] E. Dynkin, Semisimple subalgebras of semisimple Lie algebras (Russian) Mat.Sbornik N.S. 30 (27) (1952) 349-462, English: AMS Translations 6 (1957), 111-244. [FGSV] J. Fucs, C. Ganchev, K. Szlachányi, and P. Vescernyes, S4-symmetry of 6j-sympols and Frobenius-Schur indicators in rigid monoidal C∗-categories. J.Math Phys. 40 (1999), 408-426. [Ha] B. Hall, Lie groups, Lie algebras and representations, Springer-Verlag, Berlin-Heidelberg- New York, (2006). [Hu] J. Humphreys, Introdution to Lie algebras and representation theory, Springer-Verlag, Berlin-Heidelberg-New York, (1972). [K] B. Kostant, The principal three dimensional subgroup and betti numbers of complex simple Lie group, Amer.J.Math. 81 (1959), 973-1032. [KSZ] Y. Kashina, Y. Sommerhaeuser, and Y. Zhu, On higher Frobenius-Schur indicators, Memoirs of the AMS 181, no 855 (2006). [LM] V. Linchenko and S. Montgomery, A Frobenius-Schur theorem for Hopf algebras, Algebr. Represent. Theory 3 (2000), no. 4, 347-355, Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. [MN] G. Mason and S-H. Ng, Central invariants and Frobenius-Schur indicators for semi- simple qusi-Hopf algebras, Adv. Math. 190 (2005), 161-195. [NS1] S-H. Ng and P. Schauenburg, Higher Frobenius-Schur indicators for pivotal categories, preprint arXiv:math,QA/0503167. [NS2] S-H. Ng and P. Schauenburg, Central invariants and higher indicators for semisimple quasi-Hopf algebras, Transactions of the AMS, to appear, arXiv:math,QA/0508140. [S] J-P. Serre, Linear Representation of Finite Groups, Springer-Verlag, New York, (1977). Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail address: [email protected], [email protected] Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail address: [email protected] 1. Introduction 2. Preliminaries 3. Tits' theorem on the second indicator 4. The Main results 4.1. Proof of Theorem ?? 4.2. Proof of Corollary ?? 4.3. The proof of Theorem ?? References

0704.0166

Supersymmetry breaking metastable vacua in runaway quiver gauge theories

IFT-UAM/CSIC-07-14 CERN-PH-TH/2007-063 Supersymmetry breaking metastable vacua in runaway quiver gauge theories I. Garćıa-Etxebarria, F. Saad, A.M.Uranga PH-TH Division, CERN CH-1211 Geneva 23, Switzerland Instituto de F́ısica Teórica C-XVI, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain Abstract In this paper we consider quiver gauge theories with fractional branes whose infrared dy- namics removes the classical supersymmetric vacua (DSB branes). We show that addition of flavors to these theories (via additional non-compact branes) leads to local meta-stable supersymmetry breaking minima, closely related to those of SQCD with massive flavors. We simplify the study of the one-loop lifting of the accidental classical flat directions by di- rect computation of the pseudomoduli masses via Feynman diagrams. This new approach allows to obtain analytic results for all these theories. This work extends the results for the dP1 theory in hep-th/0607218. The new approach allows to generalize the computa- tion to general examples of DSB branes, and for arbitrary values of the superpotential couplings. http://arxiv.org/abs/0704.0166v2 http://arxiv.org/abs/hep-th/0607218 1 Introduction Systems of D-branes at singularities provide a very interesting setup to realize and study diverse non-perturbative gauge dynamics phenomena in string theory. In the context of N = 1 supersymmetric gauge field theories, systems of D3-branes at Calabi-Yau singularities lead to interesting families of tractable 4d strongly coupled conformal field theories, which extend the AdS/CFT correspondence [1, 2, 3] to theories with reduced (super)symmetry [4, 5, 6] and enable non-trivial precision tests of the correspondence (see for instance [7, 8]). Addition of fractional branes leads to families of non-conformal gauge theories, with intricate RG flows involving cascades of Seiberg dualities [9, 10, 11, 12, 13], and strong dynamics effects in the infrared. For instance, fractional branes associated to complex deformations of the singular geometry (denoted deformation fractional branes in [12]), correspond to supersym- metric confinement of one or several gauge factors in the gauge theory [9, 12]. The generic case of fractional branes associated to obstructed complex deformations (de- noted DSB branes in [12]), corresponds to gauge theories developing a non-perturbative Affleck-Dine-Seiberg superpotential, which removes the classical supersymmetric vacua [14, 15, 16]. As shown in [15] (see also [17, 18]), assuming canonical Kahler potential leads to a runaway potential for the theory, along a baryonic direction. A natural suggestion to stop this runaway has been proposed for the particular example of the dP1 theory (the theory on fractional branes at the complex cone over dP1) in [19]. It was shown that, upon the addition of D7-branes to the configuration (which intro- duce massive flavors), the theory develops a meta-stable minimum (closely related to the Intriligator-Seiberg-Shih (ISS) model [20]), parametrically long-lived against decay to the runaway regime (see [21] for an alternative suggestion to stop the runaway, in compact models). In this paper we show that the appearance of meta-stable minima in gauge theories on DSB fractional branes, in the presence of additional massless flavors, is much more general (and possibly valid in full generality). We use the tools of [15] to introduce D7-branes on general toric singularities, and give masses to the corresponding flavors. Since quiver gauge theories are rather involved, we develop new techniques to efficiently analyze the one-loop stability of the meta-stable minima, via the direct computation of Feynman diagrams. These tools can be used to argue that the results plausibly hold for general systems of DSB fractional branes at toric singularities. It is very satisfactory to verify the correspondence between the existence of meta-stable vacua and the geometric property of having obstructed complex deformations. The present work thus enlarges the class of string models realizing dynamical su- persymmetry breaking in meta-stable vacua (see [22, 23, 24, 25, 26] for other proposed realizations, and [27, 28, 29] for models of dynamical supersymmetry breaking in ori- entifold theories). Although we will not discuss it in the present paper, these results can be applied to the construction of models of gauge mediation in string theory as in [30] (based on the additional tools in [31]), in analogy with [32]. This is another motivation for the present work. The paper is organized as follows. In Section 2 we review the ISS model, evaluating one-loop pseudomoduli masses directly in terms of Feynman diagrams. In Section 3 we study the theory of DSB branes at the dP1 and dP2 singularities upon the addition of flavors, and we find that metastable vacua exist for these theories. In Section 4 we extend this analysis to the general case of DSB branes at toric singularities with massive flavors, and we illustrate the results by showing the existence of metastable vacua for DSB branes at some well known families of toric singularities. Finally, the Appendix provides some technical details that we have omitted from the main text in order to improve the legibility. 2 The ISS model revisited In this Section we review the ISS meta-stable minima in SQCD, and propose that the analysis of the relevant piece of the one-loop potential (the quadratic terms around the maximal symmetry point) is most simply carried out by direct evaluation of Feynman diagrams. This new tool will be most useful in the study of the more involved examples of quiver gauge theories. 2.1 The ISS metastable minimum The ISS model [20] (see also [33] for a review of these and other models) is given by N = 1 SU(Nc) theory with Nf flavors, with small masses Welectric = mTrφφ̃, (2.1) where φ and φ̃ are the quarks of the theory. The number of colors and flavors are chosen so as to be in the free magnetic phase: Nc + 1 ≤ Nf < Nc. (2.2) This condition guarantees that the Seiberg dual is infrared free. This Seiberg dual is the SU(N) theory (with N = Nf −Nc) with Nf flavors of dual quarks q and q̃ and the meson M . The dual superpotential is given by rewriting (2.1) in terms of the mesons and adding the usual coupling between the meson and the dual quarks: Wmagnetic = h (Tr q̃Mq − µ 2TrM), (2.3) where h and µ can be expressed in terms of the parameters m and Λ, and some (unknown) information about the dual Kähler metric1. It was also argued in [20] that it is possible to study the supersymmetry breaking minimum in the origin of (dual) field space without taking into account the gauge dynamics (their main effect in this discussion consists of restoring supersymmetry dynamically far in field space). In the following we will assume that this is always the case, and we will forget completely about the gauge dynamics of the dual. Once we forget about gauge dynamics, studying the vacua of the dual theory be- comes a matter of solving the F-term equations coming from the superpotential (2.3). The mesonic F-term equation reads: − FMij = hq̃i · qj − hµ2δij = 0, (2.4) where i and j are flavor indices and the dot denotes color contraction. This has no solution, since the identity matrix δij has rank Nf while q̃ i · qj has rank N = Nf −Nc. Thus this theory breaks supersymmetry spontaneously at tree level. This mechanism for F-term supersymmetry breaking is called the rank condition. The classical scalar potential has a continuous set of minima, but the one-loop potential lifts all of the non-Goldstone directions, which are usually called pseudomod- uli. The usual approach to study the one-loop stabilization is the computation of the complete one-loop effective potential over all pseudomoduli space via the Coleman- Weinberg formula [34]: M4B log −M4F log . (2.5) This approach has the advantage that it allows the determination of the one-loop minimum, without a priori information about its location, and moreover it provides the full potential around it, including higher terms. However, it has the disadvantage 1The exact expressions can be found in (5.7) in [20], but we will not need them for our analysis. We just take all masses in the electric description to be small enough for the analysis of the metastable vacuum to be reliable. of requiring the diagonalization of the mass matrix, which very often does not admit a closed expression, e.g. for the theories we are interested in. In fact, we would like to point out that to determine the existence of a meta-stable minimum there exists a computationally much simpler approach. In our situation, we have a good ansatz for the location of the one-loop minimum, and are interested just in the one-loop pseudomoduli masses around such point. This information can be directly obtained by computing the one-loop masses via the relevant Feynman diagrams. This technique is extremely economical, and provides results in closed form in full generality, e.g. for general values of the couplings, etc. The correctness of the original ansatz for the vacuum can eventually be confirmed by the results of the computation (namely positive one-loop squared masses, and negligible tadpoles for the classically massive fields 2). Hence, our strategy to study the one-loop stabilization in this paper is as follows: • First we choose an ansatz for the classical minimum to become the one-loop vacuum. It is natural to propose a point of maximal enhanced symmetry (in particular, close to the origin in the space of vevs for M there exist and R- symmetry, whose breaking by gauge interactions (via anomalies) is negligible in that region). Hence the natural candidate for the one-loop minimum is q = q̃T = , (2.6) with the rest of the fields set to 0. This initial ansatz for the one-loop minimum is eventually confirmed by the positive square masses at one-loop resulting from the computations described below. In our more general discussion of meta-stable minima in runaway quiver gauge theories, our ansatz for the one-loop minimum is a direct generalization of the above (and is similarly eventually confirmed by the one-loop mass computation). • Then we expand the field linearly around this vacuum, and identify the set of classically massless fields. We refer to these as pseudomoduli (with some abuse of language, since there could be massless fields which are not classically flat directions due to higher potential terms) 2Since supersymmetry is spontaneously broken the effective potential will get renormalized by quantum effects, and thus classically massive fields might shift slightly. This appears as a one loop tadpole which can be encoded as a small shift of µ. This will enter in the two loop computation of the pseudomoduli masses, which are beyond the scope of the present paper. • As a final step we compute one-loop masses for these pseudomoduli by evaluating their two-point functions via conventional Feynman diagrams, as explained in more detail in appendix A.1 and illustrated below in several examples. The ISS model is a simple example where this technique can be illustrated. Con- sidering the above ansatz for the vacuum, we expand the fields around this point as: µ+ 1√ (ξ+ + ξ−) (ρ+ + ρ−) , q̃T = µ+ 1√ (ξ+ − ξ−) (ρ+ − ρ−) , M = Z̃T Φ (2.7) where we have taken linear combinations of the fields in such a way that the bosonic mass matrix is diagonal. This will also be convenient in section 2.2, where we discuss the Goldstone bosons in greater detail. We now expand the superpotential (2.3) to get 2µξ+Y + µZρ+ + µZρ− + µρ+Z̃ − µρ−Z̃ ρ2+Φ− ρ2−Φ− µ2Φ+ . . . , (2.8) where we have not displayed terms of order three or higher in the fluctuations, unless they contain Φ, since they are irrelevant for the one loop computation we will perform. Note also that we have set h = 1 and we have removed the trace (the matricial structure is easy to restore later on, here we just set Nf = 2 for simplicity). The massless bosonic fluctuations are given by Re ρ+, Im ρ−, Φ and ξ−. The first two together with Im ξ− are Goldstone bosons, as explained in section 2.2. Thus the pseudomoduli we are interested in are given by Φ and Re ξ−. Let us focus on Φ (the case of Re ξ− admits a similar discussion). In this case the relevant terms in the superpotential simplify further, and just the following superpotential contributes: W = µZ (ρ+ + ρ−) + µZ̃ (ρ+ − ρ−) + ρ2+Φ− ρ2−Φ− µ2Φ+ . . . , which we recognize, up to a field redefinition, as the symmetric model of appendix A.2. We can thus directly read the result δm2Φ = |h|4µ2 (log 4− 1). (2.9) This matches the value given in [20], which was found using the Coleman-Weinberg potential. 2.2 The Goldstone bosons One aspect of our technique that merits some additional explanation concerns the Goldstone bosons. The one-loop computation of the masses for the fluctuations associ- ated to the symmetries broken by the vacuum, using just the interactions described in appendix A.1, leads to a non-vanishing result. This puzzle is however easily solved by realizing that certain (classically massive) fields have a one-loop tadpole. This leads to a new contribution to the one-loop Goldstone two-point amplitude, given by the dia- gram in Figure 1. Adding this contribution the total one-loop mass for the Goldstone bosons is indeed vanishing, as expected. This tadpole does not affect the computation of the one-loop pseudomoduli masses (except for Re ξ+, but its mass remains positive) as it is straightforward to check. Re ξ+ Figure 1: Schematic tadpole contribution to the Im ξ− two point function. Both bosons and fermions run in the loop. The structure of this cancellation can be understood by using the derivation of the Goldstone theorem for the 1PI effective potential, as we now discuss. The proof can be found in slightly more detail, together with other proofs, in [35]. Let us denote by V the 1PI effective potential. Invariance of the action under a given symmetry implies ∆φi = 0, (2.10) where we denote by ∆φi the variation of the field φi under the symmetry, which will in general be a function of all the fields in the theory. Taking the derivative of this equation with respect to some other field φk δφiδφk ∆φi + · δ∆φi = 0. (2.11) Let us consider how this applies to our case. At tree level, there is no tadpole and the above equation (truncated at tree level) states that for each symmetry generator broken by the vacuum, the value of ∆φi gives a nonvanishing eigenvector of the mass matrix with zero eigenvalue. This is the classical version of the Goldstone theorem, which allows the identification of the Goldstone bosons of the theory. For instance, in the ISS model in the previous section (for Nf = 2), there are three global symmetry generators broken at the minimum described around (2.6). The SU(2) × U(1) symmetry of the potential gets broken down to a U(1)′, which can be understood as a combination of the original U(1) and the tz generator of SU(2). The Goldstone bosons can be taken to be the ones associated to the three generators of SU(2), and correspond (for µ real) to Im ξ−, Im ρ− and Re ρ+, in the parametrization of the fields given by equation (2.7). Even in the absence of tree-level tadpoles, there could still be a one-loop tadpole. When this happens, there should also be a non-trivial contribution to the mass term for the Goldstone bosons in the one-loop 1PI potential, related to the tadpole by the one-loop version of (2.11). This relation guarantees that the mass term in the physical (i.e. Wilsonian) effective potential, which includes the 1PI contribution, plus those of the diagram in Figure 1, vanishes, as we described above. In fact, in the ISS example, there is a non-vanishing one-loop tadpole for the real part of ξ+ (and no tadpole for other fields). The calculation of the tadpole at one loop is straightforward, and we will only present here the result iM = −i|h| (4π)2 (2 log 2). (2.12) The 1PI one-loop contribution to the Goldstone boson mass is also simple to calculate, giving the result iM = −i|h| (4π)2 (log 2). (2.13) Using the variations of the relevant fields under the symmetry generator, e.g. for tz, ∆Re ξ+ = −Im ξ− (2.14) ∆Im ξ− = Re ξ+ + 2µ. (2.15) we find that the (2.11) is satisfied at one-loop. δφiδφk ∆φi + · δ∆φi = m2Im ξ− · 2µ+ (Re ξ+tadpole) · (−1) = 0. (2.16) A very similar discussion applies to tx and ty. The above discussion of Goldstone bosons can be similarly carried out in all ex- amples of this paper. Hence, it will be enough to carry out the computation of the 1PI diagrams discussed in appendix A.1, and verify that they lead to positive squared masses for all classically massless fields (with Goldstone bosons rendered massless by the additional diagrams involving the tadpole). 3 Meta-stable vacua in quiver gauge theories with DSB branes In this section we show the existence of a meta-stable vacuum in a few examples of gauge theories on DSB branes, upon the addition of massive flavors. As already discussed in [19], the choice of fractional branes of DSB kind is crucial in the result. The reason is that in order to have the ISS structure, and in particular supersymmetry breaking by the rank condition, one needs a node such that its Seiberg dual satisfies Nf > N , with N = Nf − Nc with Nc, Nf the number of colors, flavors of that gauge factor. Denoting Nf,0, Nf,1 the number of massless and massive flavors (namely flavors arising from bi-fundamentals of the original D3-brane quiver, or introduced by the D7- branes), the condition is equivalent to Nf,0 < Nc. This is precisely the condition that an ADS superpotential is generated, and is the prototypical behavior of DSB branes [14, 15, 16, 18]. Another important general comment, also discussed in [19], is that theories on DSB branes generically contain one or more chiral multiplets which do not appear in the superpotential. Being decoupled, such fields remain as accidental flat directions at one-loop, so that the one-loop minimum is not isolated. The proper treatment of these flat directions is beyond the reach of present tools, so they remain an open question. However, it is plausible that they do not induce a runaway behavior to infinity, since they parametrize a direction orthogonal to the fields parametrizing the runaway of DSB fractional branes. 3.1 The complex cone over dP1 In this section we describe the most familiar example of quiver gauge theory with DSB fractional branes, the dP1 theory. In this theory, a non-perturbative superpotential removes the classical supersymmetric vacua [14, 15, 16]. Assuming canonical Kähler potential the theory has a runaway behavior [15, 17]. In this section, we revisit with our techniques the result in [19] that the addition of massive flavors can induce the ap- pearance of meta-stable supersymmetry breaking minima, long-lived against tunneling to the runaway regime. As we show in coming sections, this behavior is prototypical and extends to many other theories with DSB fractional branes. The example is also representative of the computations for a general quiver coming from a brane at a toric singularity, and illustrates the usefulness of the direct Feynman diagram evaluation of one-loop masses. Consider the dP1 theory, realized on a set ofM fractional D3-branes at the complex cone over dP1. In order to introduce additional flavors, we introduce sets of Nf,1 D7-branes wrapping non-compact 4-cycles on the geometry and passing through the singular point. We refer the reader to [19], and also to later sections, for more details on the construction of the theory, and in particular on the introduction of the D7-branes. Its quiver is shown in Figure 2, and its superpotential is W = λ(X23X31Y12 −X23Y31X12) + λ′(Q3iQ̃i2X23 +Q2jQ̃j1X12 +Q1kQ̃k3X31) + m3Q3iQ̃k3δik +m2Q2jQ̃i2δji +m1Q1kQ̃j1δkj , (3.1) where the subindices denote the groups under which the field is charged. The first line is the superpotential of the theory of fractional brane, the second line describes 77-73-37 couplings between the flavor branes and the fractional brane, and the last line gives the flavor masses. Note that there is a massless field, denoted Z12 in [19], that does not appear in the superpotential. This is one of the decoupled fields mentioned above, and we leave its treatment as an open question. SU(3M) SU(2M) SU(M) PSfrag replacements Q3i Q2j Q̃j1 Figure 2: Extended quiver diagram for a dP1 theory with flavors, from [19]. We are interested in gauge factors in the free magnetic phase. This is the case for the SU(3M) gauge factor in the regime M + 1 ≤ Nf,1 < M. (3.2) To apply Seiberg duality on node 3, we introduce the dual mesons: M21 = X23X31 ; Nk1 = Q̃k3X31 M ′21 = X23Y31 ; N Q̃k3Y31 N2i = X23Q3i ; Φki = Q̃k3Q3i (3.3) and we also replace the electric quarks Q3i, Q̃k3, X23, X31, Y31 by their magnetic duals Q̃i3, Q3k, X32, X13, Y13. The magnetic superpotential is given by rewriting the confined fields in terms of the mesons and adding the coupling between the mesons and the dual quarks, W = h (M21X13X32 + M 21Y13X32 + N2iQ̃i3X32 + Nk1X13Q3k + N k1Y13Q3k + ΦkiQ̃i3Q3k ) + hµ0 (M21Y12 − M ′21X12 ) + µ′Q1kNk1 + µ′N2iQ̃i2 − hµ 2TrΦ + λ′Q2jQ̃j1X12 + m2Q2iQ̃i2 + m1Q1iQ̃i1. (3.4) This is the theory we want to study. In order to simplify the treatment of this example we will disregard any subleading terms in mi/µ ′, and effectively integrate out Nk1 and N2i by substituting them by 0. This is not necessary, and indeed the computations in the next sections are exact. We do it here in order to compare results with [19]. As in the ISS model, this theory breaks supersymmetry via the rank condition. The fields Q̃i3, Q3k and Φki are the analogs of q, q̃ and M in the ISS case discussed above. This motivates a vacuum ansatz analogous to (2.6) and the following linear expansion: φ00 φ01 φ10 φ11 ; Q̃i3 = µeθ +Q3,1 Q̃3,2 ; QT3i = µe−θ +Q3,1 Q̃k1 = Q̃1,1 ; Q2j = Q2,11 x Q2,21 x ; M21 = M21,1 M21,2 Y13 = (Y13) ; X X12,1 X12,2 ; XT32 = X32,1 X32,2 Y T12 = Y12,1 Y12,2 ; N ′k1 = N ′k1,1 ; M ′21 = M ′21,1 M ′21,2 X13 = (X13) . (3.5) Note that we have chosen to introduce the nonlinear expansion in θ in order to re- produce the results found in the literature in their exact form3. Note also that for the sake of clarity we have not been explicit about the ranks of the different matrices. They can be easily worked out (or for this case, looked up in [19]), and we will restrict ourselves to the 2 flavor case where the matrix structure is trivial. As a last remark, we are not being explicit either about the definitions of the different couplings in terms of the electric theory. This can be done easily (and as in the ISS case they involve 3A linear expansion would lead to identical conclusions concerning the existence of the meta-stable vacua, but to one-loop masses not directly amenable to comparison with results in the literature. an unknown coefficient in the Kähler potential), but in any event, the existence of the meta-stable vacua can be established for general values of the coefficients in the superpotential. Hence we skip this more detailed but not very relevant discussion. The next step consists in expanding the superpotential and identifying the massless fields. We get the following quadratic contributions to the superpotential: Wmass = 2hµφ00Q̃3,1 + hµφ01Q̃3,2 + hµφ10Q3,2 + hµ0M21,1Y12,1 + hµ0M21,2Y12,2 − λ′M ′21,1X12,1 − λ′M ′21,2X12,2 + hµN ′k1,1Y13 − h1µQ̃1,1X13 − h2µQ2,11X32,1 − h2µQ2,21X32,2. (3.6) The fields massless at tree level are x, x′, y, z, φ11, θ, Q3,2 and Q̃3,2. Three of these are Goldstone bosons as described in the previous section. For real µ they are Im θ, Re (Q̃3,2 +Q3,2) and Im (Q̃3,2 −Q3,2). We now show that all other classically massless fields get masses at one loop (with positive squared masses). As a first step towards finding the one-loop correction, notice that the supersym- metry breaking mechanism is extremely similar to the one in the ISS model before, in particular it comes only from the following couplings in the superpotential: Wrank = hQ3,2Q̃3,2φ11 − hµ2φ11 + . . . (3.7) This breaks the spectrum degeneracy in the multiplets Q3,2 and Q̃3,2 at tree level, so we refer to them as the fields with broken supersymmetry. Let us compute now the correction for the mass of x, for example. For the one-loop computation we just need the cubic terms involving one pseudomodulus and at least one of the broken supersymmetry fields, and any quadratic term involving fields present in the previous set of couplings. From the complete expansion one finds the following supersymmetry breaking sector: Wsymm. = hφ11Q3,2Q̃3,2 + hµφ01Q̃3,2 + hµφ10Q3,2 − hµ2φ11. (3.8) The only cubic term involving the pseudomodulus x and the broken supersymmetry fields is Wcubic = −h2 x Q̃3,2X32,1, (3.9) and there is a quadratic term involving the field X32,1 Wmass coupling = −h2µQ2,11X32,1. (3.10) Assembling the three previous equations, the resulting superpotential corresponds to the asymmetric model in appendix A.2, so we can directly obtain the one-loop mass for x: δm2x = |h|4µ2C |h2|2 . (3.11) Proceeding in a similar way, the one-loop masses for φ11, x ′, y and z are: δm2φ11 = |h|4µ2(log 4− 1) δm2x′ = |h|4µ2C |h2|2 δm2y = |h|4µ2C |h1|2 δm2z = |h|4µ2(log 4− 1). (3.12) There is just one pseudomodulus left, Re θ, which is qualitatively different to the others. With similar reasoning, one concludes that it is necessary to study a superpo- tential of the form W = h(Xφ1φ2 + µe θφ1φ3 + µe −θφ2φ4 − µ2X). (3.13) Due to the non-linear parametrization, the expansion in θ shows that there is a term quadratic in θ which contributes to the one-loop mass via a vertex with two bosons and two fermions, the relevant diagram is shown in Figure 16d. The result is a vanishing mass for Im θ, as expected for a Goldstone boson (the one-loop tadpole vanishes in this case), and a non-vanishing mass for Re θ δm2Re θ = |h|4µ4(log 4− 1). (3.14) We conclude by mentioning that all squared masses are positive, thus confirming that the proposed point in field space is the one-loop minimum. As shown in [19], this minimum is parametrically long-lived against tunneling to the runaway regime. 3.2 Additional examples: The dP2 case Let us apply these techniques to consider new examples. In this section we consider a DSB fractional brane in the complex cone over dP2, which provides another quiver theory with runaway behavior [15]. The quiver diagram for dP2 is given in Figure 3, with superpotential W = X34X45X53 −X53Y31X15 −X34X42Y23 + Y23X31X15X52 + X42X23Y31X14 −X23X31X14X45X52 (3.15) Figure 3: Quiver diagram for the dP2 theory. We consider a set of M DSB fractional branes, corresponding to choosing ranks (M, 0,M, 0, 2M) for the corresponding gauge factors. The resulting quiver is shown in Figure 4, with superpotential W = −λX53Y31X15 (3.16) U(2M) U(M)U(M) Figure 4: Quiver diagram for the dP2 theory with M DSB fractional branes. Following [19] and appendix B, one can introduce D7-branes leading to D3-D7 open strings providing (possibly massive) flavors for all gauge factors, and having cubic couplings with diverse D3-D3 bifundamental chiral multiplets. We obtain the quiver in Figure 5. Adding the cubic 33-37-73 coupling superpotential, and the flavor masses, the complete superpotential reads Wtotal = −λX53Y31X15 − λ′(Q1iQ̃i3Y31 +Q3jQ̃j5X53 +Q5kQ̃k1X15) + m1Q1iQ̃k1 +m2Q3jQ̃i3 +m5Q5kQ̃j5 (3.17) where 1, 2, 3 are the gauge group indices and i, j, k are the flavor indices. We consider the U(2M) node in the free magnetic phase, namely M + 1 ≤ Nf,1 < 2M (3.18) U(M) U(M) U(2M) PSfrag replacements Q1i Qi3 Figure 5: Quiver for the dP2 theory with M fractional branes and flavors. After Seiberg Duality the dual gauge factor is SU(N) withN = Nf,1−M and dynamical scale Λ. To get the matter content in the dual, we replace the microscopic flavors Q5k, Q̃j5, X53, X15 by the dual flavors Q̃k5, Q5j , X35, X51 respectively. We also have the mesons related to the fields in the electric theory by M1k = X15Q5K ; Ñj3 = Q̃j5X53 M13 = X15X53 ; Φ̃jk = Q̃j5Q5k (3.19) There is a cubic superpotential coupling the mesons and the dual flavors Wmes. = h (M1kQ̃k5X51 + M13X35X51 + Ñj3X35Q5j + Φ̃jkQ̃k5Q5j ) (3.20) where h = Λ/Λ̂ with Λ̂ given by Λ 3Nc−Nf elect Λ 3(Nf−Nc)−Nf = Λ̂Nf , where Λelect is the dynamical scale of the electric theory. Writing the classical superpotential terms of the new fields gives Wclas. = −hµ0M13Y31 + λ′Q1iQ̃i3Y31 + µ′ Ñj3Q3j + µ′M1kQ̃k1 + m1Q1iQ̃k1 + m3Q3jQ̃i3 − hµ 2TrΦ (3.21) where µ0 = λΛ, µ ′ = λ′Λ, and µ 2 = −m5Λ̂. So the complete superpotential in the Seiberg dual is Wdual = −hµ0M13Y31 + λ′Q1iQ̃i3Y31 + µ′ Ñj3Q3j + µ′M1kQ̃k1 + m1Q1iQ̃k1 + m3Q3jQ̃i3 − hµ 2TrΦ + h (M1kQ̃k5X51 + M13X35X51 + Ñj3X35Q5j + Φ̃jkQ̃k5Q5j ) (3.22) This superpotential has a sector completely analogous to the ISS model, triggering supersymmetry breaking by the rank condition. This suggests the following ansatz for the point to become the one-loop vacuum Q5k = Q̃ , (3.23) with all other vevs set to zero. Following our technique as explained above, we expand fields at linear order around this point. Focusing on Nf,1 = 2 and Nc = 1 for simplicity (the general case can be easily recovered), we have Q̃k5 = µ+ δQ̃5,1 δQ̃5,2 ; Q5k = (µ+ δQ5,1 ; δQ5,2) ; Φ = δΦ0,0 δΦ0,1 δΦ1,0 δΦ1,1 Q̃k1 = δQ̃1,1 δQ̃1,2 ; Q1i = (δQ1,1 ; δQ1,2) ; Q̃i3 = δQ̃3,1 δQ̃3,2 ; Q3j = (δQ3,1 ; δQ3,2) Ñj3 = δÑ3,1 δÑ3,2 ; M1k = (δM1,1 ; δM1,2) ; M13 = δM13 ; Y31 = δY31 ; X51 = δX51 X35 = δX35 (3.24) Inserting this into equation (3.22) gives Wdual = −hµ0 δM13δY31 + λ′ δQ1,1δQ̃3,1δY31 + λ′ δQ1,2δQ̃3,2δY31 + µ′ δÑ3,1δQ3,1 + µ ′ δÑ3,2δQ3,2 + µ ′ δM1,1δQ̃1,1 + µ ′ δM1,2δQ̃1,2 + m1δQ1,1δQ̃1,1 + m1δQ1,2δQ̃1,2 + m3δQ3,1δQ̃3,1 + m3δQ3,2δQ̃3,2 − hµ 2δΦ11 + h (µδM1,1δX51 + δM1,1δQ̃5,1δX51 + δM1,2δQ̃5,2δX51 + δM13δX35δX51 + µδX35δÑ3,1 + δX35δÑ3,1δQ5,1 + δX35δÑ3,2δQ5,2 + µδQ̃5,1δΦ00 + µδQ5,1δΦ00 + δQ5,1δQ̃5,1δΦ00 + µδΦ01δQ̃5,2 + δQ5,1δΦ01δQ̃5,2 + µδΦ10δQ5,2 + δQ̃5,1δΦ10δQ5,2 + δQ̃5,2δΦ11δQ5,2). We now need to identify the pseudomoduli, in other words the massless fluctuations at tree level. We focus then just on the quadratic terms in the superpotential Wmass = −hµ0 δM13δY31 + µ′ δÑ3,1δQ3,1 +m3δQ3,1δQ̃3,1 + hµδX35δÑ3,1 + µ′ δÑ3,2δQ3,2 +m3δQ3,2δQ̃3,2 + µ′ δM1,1δQ̃1,1 + m1δQ1,1δQ̃1,1 + hµδM1,1δX51 + µ′ δM1,2δQ̃1,2 + m1δQ1,2δQ̃1,2 + hµδQ̃5,1δΦ00 + hµδQ5,1δΦ00 + hµδΦ01δQ̃5,2 + µδΦ10δQ5,2. (3.25) We have displayed the superpotential so that fields mixing at the quadratic level appear in the same line. In order to identify the pseudomoduli we have to diagonalize4 these fields. Note that the structure of the mass terms corresponds to the one in appendix C, in particular around equation (C.9). From the analysis performed there we know that upon diagonalization, fields mixing in groups of four (i.e., three mixing terms in the superpotential, for example the δM1,1, δQ̃1,1, δQ1,1, δX51 mixing) get nonzero masses, while fields mixing in groups of three (two mixing terms in the superpotential, for example δM1,2, δQ̃1,2 and δQ1,2) give rise to two massive perturbations and a massless one, a pseudomodulus. We then just need to study the fate of the pseudomoduli. From the analysis in appendix C, the pseudomoduli coming from the mixing terms are Y1 = m3δÑ3,2 − µ′δQ̃3,2 , Y2 = m1δM1,2 − µ′δQ1,2 , Y3 = hµ(δQ5,1 − δQ̃5,1) . (3.26) In order to continue the analysis, one just needs to change basis to the diagonal fields and notice that the one loop contributions to the pseudomoduli are described again by the asymmetric model of appendix A.2, so they receive positive definite contributions. The exact analytic expressions can be easily found with the help of some computer algebra program, but we omit them here since they are quite unwieldy. 4 The general case In the previous section we showed that several examples of quiver gauge theories on DSB fractional branes have metastable vacua once additional flavors are included. In this section we generalize the arguments for general DSB branes. We will show how to add D7–branes in a specific manner so as to generate the appropriate cubic flavor couplings and mass terms. Once this is achieved, we describe the structure of the Seiberg dual theory. The results of our analysis show that, with the specified configuration of D7–branes, the determination of metastability is greatly simplified and only involves looking at the original superpotential. Thus, although we do not prove that DSB branes on arbitrary singularities generate metastable vacua, we show how one can determine the existence of metastability in a very simple and systematic 4As a technical remark, let us note that it is possible to set all the mass terms to be real by an appropriate redefinition of the fields, so we are diagonalizing a real symmetric matrix. manner. Using this analysis we show further examples of metastable vacua on systems of DSB branes. 4.1 The general argument 4.1.1 Construction of the flavored theories Consider a general quiver gauge theory arising from branes at singularities. As we have argued previously, we focus on DSB branes, so that there is a gauge factor satisfying Nf,0 < Nc, which can lead to supersymmetry breaking by the rank condition in its Seiberg dual. To make the general analysis more concrete, let us consider a quiver like that in Figure 6, which is characteristic enough, and let us assume that the gauge factor to be dualized corresponds to node 2. In what follows we analyze the structure of the fields and couplings in the Seiberg dual, and reduce the problem of studying the meta-stability of the theory with flavors to analyzing the structure of the theory in the absence of flavors. PSfrag replacements X21 Y21 Y32 Z32 X43 Y43 Figure 6: Quiver diagram used to illustrate general results. It does not correspond to any geometry in particular. The first step is the introduction of flavors in the theory. As discussed in [19], for any bi-fundamental Xab of the D3-brane quiver gauge theory there exist a supersymmetric D7-brane leading to flavors Qbi, Q̃ia in the fundamental (antifundamental) of the b (ath) gauge factor. There is also a cubic coupling XabQbiQ̃ia. Let us now specify a concrete set of D7-branes to introduce flavors in our quiver gauge theory. Consider a superpotential coupling of the D3-brane quiver gauge theory, involving fields charged under the node to be dualized. This corresponds to a loop in the quiver, involving node 2, for instance X32X21X14Y43 in Figure 6. For any bi-fundamental chiral multiplet in this coupling, we introduce a set of Nf,1 of the corresponding D7-brane. This leads to a set of flavors for the different gauge factors, in a way consistent with anomaly cancellation, such as that shown in Figure 7. The description of this system of D7- branes in terms of dimer diagrams is carried out in Appendix B. The cubic couplings described above lead to the superpotential terms5 Wflavor = λ ′ (X32Q2bQb3 + X21Q1aQa2 + X14Q4dQd1 + Y43Q3cQc4 ) (4.1) Finally, we introduce mass terms for all flavors of all involved gauge factors: Wmass = m2Qa2Q2b + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a (4.2) These mass terms break the flavor group into a diagonal subgroup. PSfrag replacements X21 Y21 Y32 Z32 X43 Y43 Q2b Qb3 Q4dQd1 Figure 7: Quiver diagram with flavors. White nodes denote flavor groups. 4.1.2 Seiberg duality and one-loop masses We consider introducing a number of massive flavors such that node 2 is in the free magnetic phase, and consider its Seiberg dual. The only relevant fields in this case are those charged under gauge factor 2, as shown if Figure 8. The Seiberg dual gives us Figure 9 where the M ’s are mesons with indices in the gauge groups, R’s and S’s are 5Here we assume the same coupling, but the conclusions hold for arbitrary non-zero couplings. PSfrag replacements X21 Y21 Y32 Z32Qa2 Figure 8: Relevant part of quiver before Seiberg duality. PSfrag replacements X̃12 Ỹ12 Ỹ23Z̃23 M1, . . . ,M6 Figure 9: Relevant part of the quiver after Seiberg duality on node 2. mesons with only one index in the flavor group, and Xab is a meson with both indices in the flavor groups. The original cubic superpotential and flavor mass superpotentials become Wflavor dual = λ ′ (S13bQb3 + R a1Q1a + X14Q4dQd1 + Y43Q3cQc4 ) Wmass dual = m2Xab + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a (4.3) In addition we have the extra meson superpotential Wmesons = h (XabQ̃b2Q̃2a + R a1X̃12Q̃2a + R a1Ỹ12Q̃2a + S 3bQ̃b2X̃23 + S 3bQ̃b2Ỹ23 + S33bQ̃b2Z̃23 + M 31X̃12X̃23 + M 31X̃12Ỹ23 + M 31X̃12Z̃23 + M431Ỹ12X̃23 + M 31Ỹ12Ỹ23 + M 31Ỹ12Z̃23 ). (4.4) The crucial point is that we always obtain terms of the kind underlined above, namely a piece of the superpotential reading m2Xab + hXabQ̃b2Q̃2a. This leads to tree level supersymmetry breaking by the rank condition, as announced. Moreover the superpo- tential fits in the structure of the generalized asymmetric O’Raifeartaigh model studied in appendix A.2, with Xab, Q̃b2, Q̃2a corresponding to X , φ1, φ2 respectively. The mul- tiplets Q̃b2 and Q̃2a are split at tree level, and Xab is massive at 1-loop. From our study of the generalized asymmetric case, any field which has a cubic coupling to the supersymmetry breaking fields Q̃b2 or Q̃2a is one-loop massive as well. Using the gen- eral structure of Wmesons, a little thought shows that all dual quarks with no flavor index (e.g. X̃ , Ỹ ) and all mesons with one flavor index (e.g. R or S) couple to the supersymmetry breaking fields. Thus they all get one-loop masses (with positive squared mass). Finally, the flavors of other gauge factors (e.g. Qb3) are massive at tree level from Wmass. The bottom line is that the only fields which do not get mass from these interac- tions are the mesons with no flavor index, and the bi-fundamentals which do not get dualized (uncharged under node 2). All these fields are related to the theory in the absence of extra flavors, so they can be already stabilized at tree-level from the original superpotential. So, the criteria for a metastable vacua is that the original theory, in the absence of flavors leads, after dualization of the node with Nf < Nc, to masses for all these fields (or more mildly that they correspond to directions stabilized by mass terms, or perhaps higher order superpotential terms). For example, if we apply this criteria to the dP2 case studied previously, the original superpotential for the fractional DSB brane is W = −λX53Y31X15 (4.5) so after dualization we get W = −λM13Y31 (4.6) which makes these fields massive. Hence this fractional brane, after adding the D7- branes in the appropriate configuration, will generate a metastable vacua will all moduli stabilized. The argument is completely general, and leads to an enormous simplification in the study of the theories. In the next section we describe several examples. A more rigorous and elaborate proof is provided in the appendix where we take into account the matricial structure, and show that all fields, except for Goldstone bosons, get positive squared masses at tree-level or at one-loop. 4.2 Additional examples 4.2.1 The dP3 case Let us consider the complex cone over dP3, and introduce fractional DSB branes of the kind considered in [15]. The quiver is shown in Figure 10 and the superpotential is W = X13X35X51 (4.7) Node 1 has Nf < Nc so upon addition of massive flavors and dualization will lead to supersymmetry breaking by the rank condition. Following the procedure of the previous section, we add Nf,1 flavors coupling to the bi-fundamentals X13, X35 and X51. Node 1 is in the free magnetic phase for P +1 ≤ Nf,1 < 32P + . Dualizing node 1, the above superpotential becomes W = X35M53 (4.8) where M53 is the meson X51X13. So, following the results of the previous section, we can conclude that this DSB fractional brane generates a metastable vacua with all pseudomoduli lifted. 4.2.2 Phase 1 of PdP4 Let us consider the PdP4 theory, and introduce the DSB fractional brane of the kind considered in [15]. The quiver is shown in Figure 11 . The superpotential is W = −X25X51X12 (4.9) U(P+1) Figure 10: Quiver diagram for the dP3 theory with a DSB fractional brane. U(M+P) Figure 11: Quiver diagram for the dP4 theory with a DSB fractional branes. Node 1 has Nf < Nc and will lead to supersymmetry breaking by the rank condition in the dual. Following the procedure of the previous section, we add Nf,1 flavors coupling to the bi-fundamentals X12, X25 and X51. Node 1 is in the free magnetic phase for P + 2 ≤M +Nf,1 < 32(M + P ). Dualizing node 1, the above superpotential becomes W = X25M52, where M53 is the meson X51X12. Again we conclude that this DSB fractional brane generates a metastable vacua with all pseudomoduli lifted. 4.2.3 The Y p,q family Consider D3-branes at the real cones over the Y p,q Sasaki-Einstein manifolds [36, 37, 38, 39], whose field theory were determined in [8]. The theory admits a fractional brane [13] of DSB kind, which namely breaks supersymmetry and lead to runaway behavior [15, 18]. The analysis of metastability upon addition of massive flavors for arbitrary Y p,q’s is much more involved than previous examples. Already the description of the field theory on the fractional brane is complicated. Even for the simpler cases of Y p,q and Y p,p−1 the superpotential contains many terms. In this section we do not provide a general proof of metastability, but rather consider the more modest aim of showing that all directions related to the runaway behavior in the absence of flavors are stabilized by the addition of flavors. We expect that this will guarantee full metastability, since the fields not involved in our analysis parametrize directions orthogonal to the runaway at infinity. The dimer for Y p,q is shown in Figure 12 and consists of a column of n hexagons and 2m quadrilaterals which are just halved hexagons [18]. The labels (n,m) are related to (p, q) by n = 2q ; m = p− q (4.10) • The Y p,1 case The dimer for the theory on the DSB fractional brane in the Y p,1 case is shown in Figure 13, a periodic array of a column of two full hexagons, followed by p− 1 cut hexagons (the shaded quadrilateral has Nc = 0). As shown in [18], the top quadrilateral which has Nf < Nc, and induces the ADS superpotential triggering the runaway. The relevant part of the dimer is shown in Figure 14, where V1 and V2 are the fields that run to infinity [18]. This node will lead to supersymmetry breaking by the rank condition in the dual. It is in the free magnetic phase for M + 1 ≤ Nf,1 < pM + M2 . The piece Figure 12: The generic dimer for Y p,q, from [18]. of the superpotential involving the V1 and V2 terms is W = Y U2V2 − Y U1V1. (4.11) In the dual theory, the dual superpotential makes the fields massive. Hence, the theory has a metastable vacua where the runaway fields are stabilized. Figure 13: The dimer for Y p,1. (p−1)M (p−2)M (2p−1)M (p−1)M (p+1)M Figure 14: Top part of the dimer for Y p,1. The hexagons are labeled by the ranks of the respective gauge groups • The Y p,p−1 case The analysis for Y p,p−1 is similar but in this case it is the bottom quadrilateral which has the highest rank and thus gives the ADS superpotential [18]. The relevant part of the dimer is shown in Figure 15, and the runaway direction is described by the fields V1 and V2. Upon addition of Nf,1 flavors, the relevant node in the in the free magnetic phase for M + 1 ≤ Nf,1 < pM + M2 Considering the superpotential, it is straightforward to show that the runaway fields become massive. Complementing this with our analysis in previous section, we conclude that the theory has a metastable vacua where the runaway fields are stabilized. We have thus shown that we can obtain metastable vacua for fractional branes at cones over the Y p,1 and Y p,p−1 geometries. Although there is no obvious generalization for arbitrary Y p,q’s, our results strongly suggest that the existence of metastable vacua extends to the complete family. 5 Conclusions and outlook The present work introduces techniques and computations which suggest that the ex- istence of metastable supersymmetry breaking vacua is a general property of quiver gauge theories on DSB fractional branes, namely fractional branes associated to ob- structed complex deformations. It is very satisfactory to verify the correlation between a non-trivial dynamical property in gauge theories and a geometric property in their (p−1)M (p−1)M (2p−1)M (p−2)M (2p−2)M (2p−2)M Figure 15: Bottom part of the dimer for Y p,p−1. The hexagons are labeled by the ranks of the respective gauge groups string theory realization. The existence of such correlation fits nicely with the remark- able properties of gauge theories on D-branes at singularities, and the gauge/gravity correspondence for fractional branes. Beyond the fact that our arguments do not constitute a general proof, our analysis has left a number of interesting open questions. In fact, as we have mentioned, all theories on DSB fractional branes contain one or several fields which do not appear in the superpotential. We expect the presence of these fields to have a direct physical interpretation, which has not been uncovered hitherto. It would be interesting to find a natural explanation for them. Finally, a possible extension of our results concerns D-branes at orientifold singular- ities, which can lead to supersymmetry breaking and runaway as in [27]. Interestingly, in this case the field theory analysis is more challenging, since they would require Seiberg dualities of gauge factors with matter in two-index tensors. It is very possible that the string theory realization, and the geometry of the singularity provide a much more powerful tool to study the system. Overall, we expect other surprises and interesting relations to come up from further study of D-branes at singularities. Acknowledgments We thank S. Franco for useful discussions. A.U. thanks M. González for encouragement and support. This work has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and by the Comunidad de Madrid under project HEPHACOS P-ESP-00346. The research by I.G.-E. is supported by the Gobierno Vasco PhD fellowship program. The research of F.S is supported by the Ministerio de Educación y Ciencia through an FPU grant. I.G.-E. and F.S. thank the CERN Theory Division for hospitality during the completion of this work. A Technical details about the calculation via Feyn- man diagrams A.1 The basic amplitudes In the main text we are interested in computing two point functions for the pseudo- moduli at one loop, and in section 2.2 also tadpole diagrams. There are just a few kinds of diagrams entering in the calculation, which we will present now for the two-point function, see Figure 16. The (real) bosonic fields are denoted by φi and the (Weyl) fermions by ψi. The pseudomodulus we are interested in is denoted by ϕ. c) d) a) b) ϕ ϕ ϕ ϕ ϕ ϕ Figure 16: Feynman diagrams contributing to the one-loop two point function. The dashed line denotes bosons and the solid one fermions. Bosonic contributions These come from two terms in the Lagrangian. First there is a diagram coming from terms of the form (Figure 16b): L = . . .+ λϕ2φ2 − 1 m2φ2, (A.1) giving an amplitude (we will be using dimensional regularization) iM = −2iλ (4π)2 − γ + 1 + log 4π − logm2 . (A.2) The other contribution comes from the diagram in Figure 16a: L = . . .+ λϕφ1φ2 − 2, (A.3) which contributes to the two point function with an amplitude: iM = iλ (4π)2 − γ + log 4π − dx log∆ , (A.4) where here and in the following we denote ∆ ≡ xm21 + (1− x)m22. Fermionic contributions The relevant vertices here are again of two possible kinds, one of which is nonrenor- malizable. The cubic interaction comes from terms in the Lagrangian given by the diagram in Figure 16c: L = . . .+ ϕ(aψ1ψ2 + a∗ψ̄1ψ̄2) + 1 + ψ̄ 2 + ψ̄ 2). (A.5) We are assuming real masses for the fermions here, in the configurations we study this can always be achieved by an appropriate field redefinition. The contribution from such vertices is given by: −2im1m2 (4π)2 (a2 + (a2)∗) − γ + log 4π − log∆ − 8i|a| (4π)2 − γ + log 4π + 1 − log∆ . (A.6) The other fermionic contribution, which one does not need as long as one is dealing with renormalizable interactions only (but we will need in the main text when analyzing the pseudomodulus θ), is given by terms in the Lagrangian of the form (Figure 16d): L = . . .+ λϕ2(ψ2 + ψ̄2) + 1 m(ψ2 + ψ̄2), (A.7) which contributes to the total amplitude with: iM = 8λmi (4π)2 − γ + 1 + log 4π − logm2 . (A.8) A.2 The basic superpotentials The previous amplitudes are the basic ingredients entering the computation, but in general the number of diagrams contributing to the two point amplitudes is quite big, so calculating all the contributions by hand can get quite involved in particular examples6. Happily, one finds that complicated models (such as dP1 or dP2, studied in the main text) reduce to performing the analysis for only two different superpotentials, which we analyze in this section. The symmetric case We want to study in this section a superpotential of the form: W = h(Xφ1φ2 + µφ1φ3 + µφ2φ4 − µ2X). (A.9) 6The authors wrote the computer program in http://cern.ch/inaki/pm.tar.gz which helped greatly in the process of computing the given amplitudes for the relevant models. http://cern.ch/inaki/pm.tar.gz This model is a close cousin of the basic O’Raifeartaigh model. We are interested in the one loop contribution to the two point function of X , which is massless at tree level. From the (F-term) bosonic potential one obtains the following terms entering the one loop computation: |hXφ2|2 + |h|2µ(Xφ2φ∗3 +X∗φ∗2φ3) + |h|2µ(Xφ1φ∗4 +X∗φ∗1φ4) + |h|2µ2(φ1φ2 + φ∗1φ∗2) + |h|2µ2|φi|2 (A.10) In order to do the computation it is useful to diagonalize the mass matrix by introducing φ+ and φ− such that: (φ+ + iφ−) φ2 = (φ+ − iφ−) (A.11) and φa, φb such that: φ∗3 = (φa + iφb) φ (φa − iφb). (A.12) With these redefinitions the bosonic scalar potential decouples into identical φ+ and φ− sectors, giving two decoupled copies of: V = |h|2|X|2|φ+|2 + |h|2µ2(|φ+|2 + |φa|2) +|h|2µ(Xφ+φa +X∗φ∗+φ∗a)− |h|2µ2 φ2+ + (φ . (A.13) Calculating the amplitude consists simply of constructing the (very few) two point diagrams from the potential above and plugging the formulas above for each diagram (the fermionic part is even simpler in this case). The final answer is that in this model the one loop correction to the mass squared of X is given by: δm2X = |h4|µ2 (log 4− 1). (A.14) The generalized asymmetric case The next case is slightly more complicated, but will suffice to analyze completely all the models we encounter. We will be interested in the one loop contribution to the mass of the pseudomoduli Y in a theory with superpotential: W = h(Xφ1φ2 + µφ1φ3 + µφ2φ4 − µ2X) + k(rY φ1φ5 + µφ5φ7), (A.15) with k and r arbitrary complex numbers. The procedure is straightforward as above, so we will just quote the result. We obtain an amplitude given by: iM = −i (4π)2 |h2rµ|2C , (A.16) where we have defined C(t) as: C(t) = t log 4− t t− 1 log t . (A.17) Note that this is a positive definite function, meaning that the one loop correction to the mass is always positive, and the pseudomoduli get stabilized for any (nonzero) value of the parameters. Also note that the limit of vanishing t with |r|2t fixed (i.e., vanishing masses for φ5 and φ7, but nonvanishing coupling of Y to the supersymmetry breaking sector) gives a nonvanishing contribution to the mass of Y . B D7–branes in the Riemann surface The gauge theory of D3-branes at toric singularities can be encoded in a dimer diagram [40, 41, 42, 43, 44]. This corresponds to a bi-partite tiling of T 2, where faces corre- spond to gauge groups, edges correspond to bi-fundamentals, and nodes correspond to superpotential terms. As an example, the dimer diagram of D3–branes on the cone over dP2 is shown in Figure 17. As shown in [43], D3–branes on a toric singularity are mirror to D6–branes on intersecting 3-cycles in a geometry given by a fibration of a Riemann surface Σ with punctures. This Riemann surface is just a thickening of the web diagram of the toric singularity [45, 46, 47], with punctures associated to external legs of the web diagram. The mirror D6-branes wrap non-trivial 1-cycles on this Rie- mann surface, with their intersections giving rise to bi-fundamental chiral multiplets, and superpotential terms arising from closed discs bounded by the D6-branes. In [19], it was shown that D7–branes passing through the singular point can be described in the mirror Riemann surface Σ by non-compact 1-cycles which come from infinity at one puncture and go to infinity at another. Figure 18 shows the 1-cycles corresponding to some D3- and D7-branes in the Riemann surface in the geometry mirror to the complex cone over dP2. A D7-brane leads to flavors for the two D3-brane gauge factors whose 1-cycles are intersected by the D7-brane 1-cycle, and there is a cubic coupling among the three fields (related to the disk bounded by the three 1-cycles in the Riemann surface). Figure 17: Dimer diagram for D3–branes at a dP2 singularity. Figure 18: Riemann surface in the geometry mirror to the complex cone over dP2, shown as a tiling of a T 2 with punctures (denoted by capital letters). The figure shows the non- compact 1-cycles extending between punctures, corresponding to D7-branes, and a piece of the 1-cycles that correspond to the mirror of the D3-branes. U(M) U(M) U(2M) PSfrag replacements Q1i Qi3 Figure 19: Quiver for the dP2 theory with M fractional branes and flavors. As stated in Section 4, given a gauge theory of D3-branes at a toric singularity, we introduce flavors for some of the gauge factors in a specific way. We pick a term in the superpotential, and we introduce flavors for all the involved gauge factors, and coupling to all the involved bifundamental multiplets. For example, the quiver with flavors for the dP2 theory is shown in Figure 19. On the Riemann surface, this procedure amounts to picking a node and introducing D7-branes crossing all the edges ending on the node, see Figure 18. In this example we obtain the superpotential terms Wflavor = λ ′(Q1iQ̃i3Y31 +Q3jQ̃j5X53 +Q5kQ̃k1X15) (B.1) In addition we introduce mass terms Wmass = m1Q1iQ̃k1 +m2Q3jQ̃i3 +m5Q5kQ̃j5 (B.2) This procedure is completely general and applies to all gauge theories for branes at toric singularities7. C Detailed proof of Section 4 Recall that in Section 4 we considered the illustrative example of the gauge theory given by the quiver in Figure 20. Since node 2 is the one we wish to dualize, the only relevant part of the diagram is shown in Figure 21. We show the Seiberg dual in Figure 22. The above choice of D7–branes, which we showed in appendix B can be applied to arbitrary toric singularities, gives us the superpotential terms Wflavor = λ ′ (X32Q2bQb3 + X21Q1aQa2 + X14Q4dQd1 + Y43Q3cQc4 ) Wmass = m2Qa2Q2b + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a (C.1) Taking the Seiberg dual of node 2 gives Wflavor dual = λ ′ (S13bQb3 + R a1Q1a + X14Q4dQd1 + Y43Q3cQc4 ) Wmass dual = m2Xab + m3Qb3Q3c + m4Qc4Q4d + m1Qd1Q1a Wmesons = h (XabQ̃b2Q̃2a + R1a1X̃12Q̃2a + R a1Ỹ12Q̃2a 7This procedure does not apply if the superpotential (regarded as a loop in the quiver) passes twice through the node which is eventually dualized in the derivation of the metastable vacua. However we have found no example of this for any DSB fractional branes. PSfrag replacements X21 Y21 Y32 Z32 X43 Y43 Q2b Qb3 Q4dQd1 Figure 20: Quiver diagram with flavors. White nodes denote flavor groups PSfrag replacements X21 Y21 Y32 Z32Qa2 Figure 21: Relevant part of quiver before Seiberg duality. PSfrag replacements X̃12 Ỹ12 Ỹ23Z̃23 M1, . . . ,M6 Figure 22: Relevant part of the quiver after Seiberg duality on node 2. + S13bQ̃b2X̃23 + S 3bQ̃b2Ỹ23 + S 3bQ̃b2Z̃23 + M131X̃12X̃23 + M 31X̃12Ỹ23 + M 31X̃12Z̃23 + M431Ỹ12X̃23 + M 31Ỹ12Ỹ23 + M 31Ỹ12Z̃23 ) (C.2) where we have not included the original superpotential. The crucial point is that the underlined terms appear for any quiver gauge theory with flavors introduced as described in appendix B. As described in the main text, supersymmetry is broken by the rank condition due to the F-term of the dual meson associated to the massive flavors. Our vacuum ansatz is (we take Nf = 2 and Nc = 1 for simplicity; this does not affect our conclusions) Q̃b2 = ; Q̃2a = (µ1Nc ; 0) (C.3) with all other vevs set to zero. We parametrize the perturbations around this minimum Q̃b2 = µ+ φ1 ; Q̃2a = (µ+ φ3 ; φ4) ; Xab = X00 X01 X10 X11 (C.4) and the underlined terms give hXabQ̃b2Q̃2a − hµ2Xab = hX11 φ2 φ4 − hµ2X11 + hµ φ2X01 + hµ φ4X10 + hµ φ1X00 + hµ φ3X00 + hφ1 φ3X00 + hφ2 φ3X01 + hφ1 φ4X10 (C.5) It is important to note that all the fields in (C.4) will have quadratic couplings only in the underlined term (C.5). Thus, one can safely study this term, and the conclusions are independent of the other terms in the superpotential. Diagonalizing (C.5) gives hXabQ̃b2Q̃2a − hµ2Xab = hX11 φ2 φ4 − hµ2X11 + hµ φ2X01 + hµ φ4X10 2hµ φ+X00 + φ2+X00 − φ2−X00 (ξ+ − ξ−)φ2X01 + (ξ+ + ξ−)φ4X10 (C.6) where (φ1 + φ3) ; ξ− = (φ1 − φ3) (C.7) This term is similar to the generalized asymmetric case studied in appendix A.2 with X11 → X ; φ4 → φ1 ; φ2 → φ2 ; X10 → φ3 ; X01 → φ4 (C.8) So here X11 is the linear term that breaks supersymmetry, and φ2, φ4 are the broken supersymmetry fields. In (C.6), the only massless fields at tree-level are X11 and ξ−. Comparing to the ISS case in Section 2.1 shows that Im ξ− is a Goldstone boson and X11, Re ξ− get mass at tree-level. As for φ2 and φ4, setting ρ+ = (φ2 + φ4) and (φ2−φ4) gives us Re(ρ+) and Im (ρ−) massless and the rest massive. Following the discussion in Section 2.1, Re(ρ+) and Im (ρ−) are just the Goldstone bosons of the broken SU(Nf ) symmetry 8. We have thus shown that the dualized flavors (e.g. Q̃b2, Q̃2a) and the meson with two flavor indices (e.g. Xab) get mass at tree-level or at 1-loop unless they are Goldstone bosons. Now, we need to verify that this is the case for the remaining fields. PSfrag replacements Q4dQd1 X̃12 Ỹ12 Ỹ23Z̃23 M1..M6 Figure 23: Quiver after Seiberg duality on node 2. The Seiberg dual of the original quiver diagram is shown in Figure 23. The dual- ized bi-fundamentals come in two classes. The first are the ones that initially (before dualizing) had cubic flavor couplings, there will always be only two of those (e.g. X̃12, X̃23). The second are those that did not initially have cubic couplings to flavors, there is an arbitrary number of those (e.g. Ỹ12, Ỹ23, Z̃23). Figure 24 shows the relevant part of the quiver for the first class. Recalling the superpotential terms (C.2), there are several possible sources of tree-level masses. For instance, these can arise in Wflavor dual and Wmass dual. Also, remembering our assignation of vevs in (C.3), tree-level masses can also arise in Wmesons from cubic couplings involving the broken supersymmetry fields (e.g. Q̃b2, Q̃2a). The first class of bi-fundamentals (e.g. X̃12, X̃23) only appear in Wmesons coupled to their respective mesons (e.g. R 1, S1). In turn these mesons will ap- 8In the case where the flavor group is SU(2), these Goldstone bosons are associated to the gener- ators tx and ty. PSfrag replacements M1, . . . ,M6 Figure 24: Relevant part of dual quiver for first class of bi-fundamentals. pear in quadratic terms in Wflavor dual coupled to flavors (e.g. S 3bQb3 and R a1Q1a), and these flavors each appear in one term in Wmass. Thus there are two sets of three terms which are coupled at tree-level and which always couple in the same way. Consider for instance the term λ′ S13bQb3 + m3Qb3Q3c + hS 3bQ̃b2X̃23 = λ ′ (S1 S2) +m1(C1 C2) + h (S1 S2) µ+ φ1 = λ′(S1B1 + S2B2) +m1(B1C1 +B2C2) + hµS1 X̃23 + hS1 φ1 X̃23 + hS2 φ2 X̃23 (C.9) where Si, Bi, Ci and X̃23 are the perturbations around the minimum. Diagonalizing (which can be done analytically for any values of the couplings), we get that all terms except one get tree-level masses, the massless field being: Y = m1S2 − λ′C2 (C.10) This massless field has a cubic coupling to φ2 X̃23 and gets mass at 1-loop since φ2 is a broken supersymmetry field, as described in appendix A.2. Figure 25 shows the relevant part of the quiver for the second class of bi-fundamentals (i.e. those that are dualized but do not have cubic flavor couplings). These fields and their mesons only appear in one term, so will always couple in the same way. 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Gauntlett, D. Martelli, J. Sparks and D. Waldram, Adv. Theor. Math. Phys. 8, 711 (2004) [arXiv:hep-th/0403002]. [38] J. P. Gauntlett, D. Martelli, J. F. Sparks and D. Waldram, Adv. Theor. Math. Phys. 8, 987 (2006) [arXiv:hep-th/0403038]. http://arxiv.org/abs/hep-th/0607183 http://arxiv.org/abs/hep-th/0610212 http://arxiv.org/abs/hep-th/0607218 http://arxiv.org/abs/hep-th/0608157 http://arxiv.org/abs/hep-th/0703236 http://arxiv.org/abs/hep-th/9806080 http://arxiv.org/abs/hep-th/0703047 http://arxiv.org/abs/hep-th/0703112 http://arxiv.org/abs/hep-th/0605166 http://arxiv.org/abs/hep-th/0603108 http://arxiv.org/abs/hep-th/0512170 http://arxiv.org/abs/hep-ph/0702069 http://arxiv.org/abs/hep-th/0402153 http://arxiv.org/abs/hep-th/0403002 http://arxiv.org/abs/hep-th/0403038 [39] D. Martelli and J. Sparks, Commun. Math. Phys. 262, 51 (2006) [arXiv:hep-th/0411238]. [40] A. Hanany and K. D. Kennaway, arXiv:hep-th/0503149. [41] S. Franco, A. Hanany, K. D. Kennaway, D. Vegh and B. Wecht, arXiv:hep-th/0504110. [42] A. Hanany and D. Vegh, arXiv:hep-th/0511063. [43] B. Feng, Y. H. He, K. D. Kennaway and C. Vafa, arXiv:hep-th/0511287. [44] S. Franco and D. Vegh, arXiv:hep-th/0601063. [45] O. Aharony and A. Hanany, Nucl. Phys. B 504, 239 (1997) [arXiv:hep-th/9704170]. [46] O. Aharony, A. Hanany and B. Kol, JHEP 9801, 002 (1998) [arXiv:hep-th/9710116]. [47] N. C. Leung and C. Vafa, Adv. Theor. Math. Phys. 2, 91 (1998) [arXiv:hep-th/9711013]. http://arxiv.org/abs/hep-th/0411238 http://arxiv.org/abs/hep-th/0503149 http://arxiv.org/abs/hep-th/0504110 http://arxiv.org/abs/hep-th/0511063 http://arxiv.org/abs/hep-th/0511287 http://arxiv.org/abs/hep-th/0601063 http://arxiv.org/abs/hep-th/9704170 http://arxiv.org/abs/hep-th/9710116 http://arxiv.org/abs/hep-th/9711013 Introduction The ISS model revisited The ISS metastable minimum The Goldstone bosons Meta-stable vacua in quiver gauge theories with DSB branes The complex cone over dP1 Additional examples: The dP2 case The general case The general argument Construction of the flavored theories Seiberg duality and one-loop masses Additional examples The dP3 case Phase 1 of PdP4 The Yp,q family Conclusions and outlook Technical details about the calculation via Feynman diagrams The basic amplitudes The basic superpotentials D7–branes in the Riemann surface Detailed proof of Section 4

0704.0167

Low Energy Aspects of Heavy Meson Decays

Low Energy Aspects of Heavy Meson Decays .∗ Jan O. Eeg Department of Physics, University of Oslo, P.O.Box 1048 Blindern, N-0316 Oslo, Norway I discuss low energy aspects of heavy meson decays, where there is at least one heavy meson in the final state. Examples are B − B mixing, B → DD, B → Dη′, and B → Dγ. The analysis is performed in the heavy quark limit within heavy-light chiral perturbation theory. Coefficients of 1/Nc suppressed chiral Lagrangian terms (beyond factorization) have been estimated by means of a heavy-light chiral quark model. PACS numbers: PACS numbers 13.20.Hw , 12.39.St , 12.39.Fe , 12.39.Hg 1. Introduction In this paper we consider non-leptonic “heavy meson to heavy meson(s)” transitions, for instance B − B-mixing [1], B → DD̄ [2] and with only one D-meson in the final state, like B → Dη′ [3] and B → γ D∗ [4, 5, 6]. The methods [7] used to describe heavy to light tansitions like B → ππ and B → Kπ are not suited for the decays we consider. We use heavy-light chiral perturbation theory (HLχPT). Lagrangian terms corresponding to factorization are then determined to zeroth order in 1/mQ, where mQ is the mass of the heavy quark (b or c). For B−B-mixing we have also calculated 1/mb corrections [1]. Colour suppressed 1/Nc terms beyond factorization can be written down, but their coefficients are unknown. However, these coefficients can be cal- culated within a heavy-light chiral quark model (HLχQM) [8] based on the heavy quark effective theory (HQEFT) [9] and HLχPT [10]. The 1/Nc suppressed non-factorizable terms calculated in this way will typically be proportional to a model dependent gluon condensate [1, 2, 3, 6, 8, 11]. Presented at the Euridice meeting in Kazimierz, Poland, 24-27th of august 2006 http://arxiv.org/abs/0704.0167v1 2 KazProc printed on November 4, 2018 2. Quark Lagrangians for non-leptonic decays The effective non-leptonic Lagrangian at quark level has the form [12]: Ci(µ) Q̂i(µ) , (1) where the Wilson coefficients Ci contain GF and KM factors. Typically, the operators are four quark operators being the product of two currents: Q̂i = j W (q1 → q2) j µ (q3 → q4) , (2) where j W (qi → qj) = (qj)L γ µ (qi)L, and some of the quarks qi,j are heavy. To leading order in 1/Nc, matrix elements of Q̂i factorize in products of matrix elements of currents. Non-factorizable 1/Nc suppressed terms are obtained from “coloured quark operators”. Using Fierz transformations δijδln = δinδlj + 2 t lj , (3) where ta are colour matrices, we may rewrite the operator Q̂i as Q̂Fi = W (q1 → q4) j µ (q3 → q2) + 2 j W (q1 → q4) a jWµ (q3 → q2)a , (4) where j W (qi → qj)a = (qj)L γµ ta (qi)L is a left-handed coloured current. The quark operators in Q̂Fi give 1/Nc suppressed terms. 3. Heavy-light chiral perturbation theory The QCD Lagrangian involving light and heavy quarks is: LQuark = ±Q v iv ·DQ(±)v +O(m−1Q ) + q̄iγ ·Dq + ... (5) where Q v are the quark fields for a heavy quark and a heavy anti-quark with velocity v, q is the light quark triplet, and iDµ = i∂µ− eqAµ− gstaAaµ. The bosonized Lagrangian have the following form, consistent with the un- derlying symmetry [10]: Lχ(Bos) = ∓Tr a (iv · Dfa)H − gATr f γµγ5A + ...(6) where the covariant derivative is iDµ ≡ δaf (i∂µ− eHAµ)−Vµfa ; a, f being SU(3) flavour indices. The axial coupling is gA ≃ 0.6. Furthermore, Vµ(orAµ) = ± (ξ†∂µξ ± ξ∂µξ†) , (7) KazProc printed on November 4, 2018 3 where ξ = exp(iΠ/f), and Π is a 3 by 3 matrix containing the light mesons (π,Kη), and the heavy (1−, 0−) doublet field (Pµ, P5) is H(±) = P±(P µ − iP (±)5 γ5) , P± = (1± γ · v)/2 , (8) where superscripts (±) means meson and anti-meson respectively. To bosonize the non-leptonic quark Lagrangian, we need to bosonize the currents. Then the b, c, and c quarks are treated within HQEFT, which means the replace- ments b → Q(+)vb , c → Q vc , and c → Q v̄ . Then the bosonization of currents within HQEFT for decay of a heavy B-meson will be: µQ(+)vb −→ ξ†γµLH ≡ Jµb , (9) where L is the left-handed projector in Dirac space, and αH = fH for H = B,D before pQCD and chiral corrections are added. Here, H represents the heavy meson (doublet) containing a b-quark. For creation of a heavy anti-meson B or D, the corresponding currents J and J c̄ are given by (9) with H b replaced by H b and H c , repectively. For the B → D transition we have µ LQ(+)vc −→ −ζ(ω)Tr ≡ Jµb→c , (10) where ζ(ω) is the Isgur-Wise function, and ω = vb · vc. For creation of DD pair we have the same expression for the current J cc̄ with H replaced c , and ζ(ω) replaced by ζ(−λ), where λ = v̄ · vc . In addition there are 1/mQ corrections for Q = b, c. The low velocity limit is ω → 1 . For B → DD and B → D∗γ one has ω ≃ 1.3 , and ω ≃ 1.6 , respectively. 3.1. Factorized lagrangians for non-leptonic processes For B −B mixing, the factorized bosonized Lagrangian is LB = CB J b (Jb̄) µ , (11) where CB is a short distance Wilson coefficient (containing (GF ) 2), which is taken at µ = Λχ ≃ 1 GeV, and the currents are given by (9). For processes obtained from two different four quark operators for b → cc̄q (q = d, s), we find the factorized Lagrangian corresponding to Fig. 1: LSpecFact = (C2 + b→c (Jc̄)µ , (12) 4 KazProc printed on November 4, 2018 B0 D+ Fig. 1. Factorized contribution for B0 → D+D−s through the spectator mechanism, which does not exist for decay mode B0 → D+s D−s . Fig. 2. Factorized contribution for B0 → D+s D−s through the annihilation mecha- nism, which give zero contributions if both D+s and D s are pseudoscalars. where Ci = GFVcbV cq ai, and [13] a1 ≃ −0.35 − 0.07i, a2 ≃ 1.29 + 0.08i. We have considered the process B0d → D s . Note that there is no factorized contribution to this process if both D-mesons in the final state are pseudoscalars! But the factorized contribution to B0 → D+D−s will be the starting point for chiral loop contributions to the process B0 → D+s D−s . The factorizable term from annihilation is shown in Fig. 2, and is: LAnnFact = (C1 + cc̄ (Jb)µ . (13) Because (C1 + C2/Nc) is a non-favourable combination of the Wilson coef- ficients, this term will give a small non-zero contribution if at least one of the mesons in the final state is a vector. 3.2. Possible 1/Nc suppressed tree level terms For B − B̄ mixing, we have for instance the 1/Nc suppressed term ξ†σµαLH ξ†σµαRH . (14) KazProc printed on November 4, 2018 5 π0, η8 π0, η8 π0, η8 π0, η8 Fig. 3. Chiral corrections to B −B mixing, i.e the bag parameter BBq for q = d, s. The black boxes are weak vertices. B0 B∗0 Fig. 4. Two classes of non-factorizable chiral loops for B0 → D+s D−s based on the factorizable amplitude proportional to the IW function ∼ ζ(ω). For B → DD̄, we have for instance the terms ξ†σµαLH c γαLH c̄ γµ , (15) ξ†σµαLH c γαLH (v̄ − vc)µ . (16) One needs a framework to estimate the coefficients of such terms. We use the HLχQM, which will pick a certain linear combination of 1/Nc terms. 3.3. Chiral loops for non-leptonic processes Within HLχPT, the leading chiral corrections are proportional to χ(M) ≡ ( )2 ln( ) , (17) where mM is the appropriate light meson mass and Λχ is the chiral symme- try breaking scale, which is also the matching scale within our framework. For B − B mixing there are chiral loops obtained from (6) and (11) shown in Fig. 3. These have to be added to the factorized contribution. 6 KazProc printed on November 4, 2018 Fig. 5. The HLχQM ansatz: Vertex for quark meson interaction For the process B0d → D s we obtain a chiral loop amplitude cor- responding to Fig. 4. This amplitude is complex and depend on ω and λ defined previously. It has been recently shown [5] that (0+, 1+) states in loops should also be added to the result. 4. The heavy-light chiral quark model The Lagrangian for HLχQM [8] contains the Lagrangian (5): LHLχQM = LHQET + LχQM + LInt , (18) where LHQET is the heavy quark part of (5), and the light quark part is LχQM = χ [γµ(iDµ + Vµ + γ5Aµ)−m]χ . (19) Here χL = ξ †qL and χR = ξqR are flavour rotated light quark fields, and m is the light constituent mass. The bosonization of the (heavy-light) quark sector is performed via the ansatz: LInt = −GH v Qv +Qv H . (20) The coupling GH is determined by bosonization through the loop diagrams in Fig 6. The bosonization lead to relations between the model depen- dent parameters GH , m, and 〈 αsπ G 2 〉, and the quadratic-, linear, and logarithmic- divergent integrals I1, I3/2, I1, and the physical quantities fπ, 〈 qq 〉, gA and fH (H = B,D). For example, the relation obtained for iden- tifying the kinetic term is: − iG2HNc (I3/2 + 2mI2 + i(8 − 3π) 384Ncm3 G2 〉) = 1 , (21) where we have used the prescription: αβ → 4π2〈 G2 〉 1 (gµαgνβ − gµβgνα) . (22) The parameters are fitted in strong sector, with 〈 αs G2 〉 = [(0.315 ± 0.020) GeV]4 , and GH 2 = 2m ρ , where ρ ≃ 1. For details , see [8]. KazProc printed on November 4, 2018 7 Fig. 6. Diagrams generating the strong chiral lagrangian at mesonic level. The kinetic term and and the axial vector term ∼ gA. Fig. 7. Non-factorizable contribution to B −B mixing; Γ ≡ ta γµ L 5. 1/N terms from HLχQM To obtain the 1/Nc terms for B − B mixing in Fig. 7 , we need the bosonization of colored current in the quark operators of eq. (4): a γαQ(+)vb GH gs GaµνTr ξ†γαLH b Σµν , (23) Σµν = σµν − 2πf [σµν , γ · vb]+ . (24) This coloured current is also used for B → DD in Fig. 8, for B → Dη′ in Fig. 9, and for B → γD∗ in Fig. 10 In addition there are more complicated bosonizations of coloured currents as indicated in Fig. 8. For B → Dη′ and B → γD∗ decays there are two different four quark operators, both for b → cūq and b → c̄uq, respectively. At µ = 1 GeV they have Wilson coefficients a2 ≃ 1.17 , a1 ≃ −0.37 (up to prefactors GF and 8 KazProc printed on November 4, 2018 Fig. 8. Non-factorizable 1/Nc contribution for B0 → D+s D−s through the annihila- tion mechanism with additional soft gluon emision. Fig. 9. Diagram for B → Dη′ within HLχQM . Γ = γµ(1− γ5) KM-factors). For B → Dη′, we must also attach a propagating gluon to the η′gg∗-vertex. Note that for B0 → γD0∗, the 1/Nc suppressed mechanism in Fig. 10 dominates, unlike B0 → γD0∗. Factorized contributions are proportional to either the favourable contribution af = a2 + a1/Nc ≃ 1.05 or the non-favourable contribution anf = a1 + a2/Nc ≃ 0.02. 5.1. 1/mc correction terms For the B → D transition we have the 1/mc suppressed terms: c + Z1γ c γα + Z2H c γ · vb , (25) where the Zi’s are calculable within HLχQM. The relative size of 1/mc corrections are typically of order 20− 30%. 6. Results 6.1. B −B mixing The result for the B(ag) parameter in B −B-mixing has the form [1] B̂Bq = 1− δBG 32π2f2 , (26) KazProc printed on November 4, 2018 9 B D B D Fig. 10. Non-factorizable contributions to B → γD∗ from the coloured operators similar to the K − K-mixing case [11]. From perturbative QCD we have b̃ ≃ 1.56 at µ = Λχ = 1 GeV. From calculations within the HLχQM we obtain, δBG = 0.5±0.1 and τb = (0.26±0.04)GeV, and from chiral corrections τχ,s = (−0.10 ± 0.04)GeV2, and τχ,d = (−0.02 ± 0.01)GeV2 . We obtained B̂Bd = 1.51± 0.09 B̂Bs = 1.40 ± 0.16 , (27) in agreement with lattice results. 6.2. B → DD decays Keeping the chiral logs and the 1/Nc terms from the gluon condensate, we find the branching ratios in the “leading approximation”. For decays of B̄0d (∼ VcbV ∗cd) and B̄0s (∼ VcbV ∗cs) we obtain branching ratios of order few ×10−4 and ×10−3, respectively Then we have to add counterterms ∼ ms for chiral loops. These may be estimated in HLχQM. 6.3. B → Dη′ and B → γD∗ decays The result corresponding to Fig. 9 is: Br(B → Dη′) ≃ 2× 10−4 . (28) The partial branching ratios from the mechanism in Fig. 10 are [6] Br(B0d → γ D ∗0)G ≃ 1× 10−5 ; Br(B0s → γ D∗0)G ≃ 6× 10−7 . (29) The corresponding factorizable contribibutions are roughly two orders of magnitude smaller. Note that the process B0d → γ D∗0 has substantial meson exchanges (would be chiral loops for ω → 1), and is different. 7. Conclusions Our low energy framework is well suited to B −B mixing, and to some extent to B → DD. Work continues to include (0+, 1+), states, countert- erms, and 1/mc terms. Note that the amplitude for B → D+s D−s is zero 10 KazProc printed on November 4, 2018 in the factorized limit. For processes like B → Dη′ and B → Dγ we can give order of magnitude estimates when factorization give zero or small amplitudes. * * * JOE is supported in part by the Norwegian research council and by the European Union RTN network, Contract No. HPRN-CT-2002-00311 (EURIDICE). He thanks his collaborators : A. Hiorth, S. Fajfer, A. Polosa, A. Prapotnik Brdnik, J.A. Macdonald Sørensen, and J. Zupan REFERENCES [1] A. Hiorth and J. O. Eeg, Eur. Phys. J. direct C30, 006 (2003) (see also references therein). [2] J.O. Eeg, S. Fajfer , and A. Hiorth, Phys.Lett. B570, 46-52 (2003); J. O. Eeg, S. Fajfer, and A. Prapotnik Eur. Phys. J. C42, 29-36 (2005). See also: J.O. Eeg, S. Fajfer, J. Zupan, Phys. Rev. D 64, 034010 (2001). [3] J. O. Eeg, A. Hiorth, A. D. Polosa, Phys. Rev. D 65, 054030 (2002). 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B 412, 201 (1994). http://arxiv.org/abs/hep-ph/0606065 http://arxiv.org/abs/hep-ph/0605078 Introduction Quark Lagrangians for non-leptonic decays Heavy-light chiral perturbation theory Factorized lagrangians for non-leptonic processes Possible 1/Nc suppressed tree level terms Chiral loops for non-leptonic processes The heavy-light chiral quark model 1/Nc terms from HLQM 1/mc correction terms Results B- B mixing B D D decays B D ' and B D* decays Conclusions

0704.0168

Radiative losses and cut-offs of energetic particles at relativistic shocks

Mon. Not. R. Astron. Soc. 000, 1–10 (????) Printed 4 November 2018 (MN LATEX style file v2.2) Radiative losses and cut-offs of energetic particles at relativistic shocks Paul Dempsey⋆ and Peter Duffy⋆ UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland. Accepted 2007 March 28. Received 2007 March 21; in original form 2007 January 24 ABSTRACT We investigate the acceleration and simultaneous radiative losses of electrons in the vicinity of relativistic shocks. Particles undergo pitch angle diffusion, gaining energy as they cross the shock by the Fermi mechanism and also emitting synchrotron radiation in the ambient magnetic field. A semi-analytic approach is developed which allows us to consider the behaviour of the shape of the spectral cut-off and the variation of that cut-off with the particle pitch angle. The implications for the synchrotron emission of relativistic jets, such as those in gamma ray burst sources and blazars, are discussed. Key words: relativistic shock acceleration, radiative losses. 1 INTRODUCTION The role of radiative losses in determining the spectra from non-thermal sources has been well understood in the non- relativistic shock limit since the work of Webb et al. (1984) and Heavens & Meisenheimer (1987). Their results were in broad agreement with the natural expectation that there would be a cut-off in the spectrum, at the shock, and at a momentum where acceleration and loss timescales are equal, with the shape of this cut-off depending critically on the momentum dependence of the particle scattering. Subse- quently, as the particles are advected downstream, and are no longer efficiently accelerated by the shock, the spectra steepens at momenta where the particles have had sufficient time to cool. At a strong, nonrelativistic shock the differen- tial number density of particles at energies where radiative cooling is unimportant is a power law with N(E) ∝ E−0.5 with a corresponding intensity of Iν ∝ ν−0.5 for the emitted synchrotron radiation. At higher momenta, where cooling becomes important, the spectrum steepens so that the ra- diation, beyond a break frequency νb, is Iν ∝ ν−1 up to a critical frequency, νc corresponding to cut-off of the particle spectrum. The position of νb depends on position away from the shock; decreasing downstream as the particles have more time to cool. The observed emission is therefore dependent on the spatial resolution with which the source is observed as discussed in Heavens & Meisenheimer (1987). The results in the existing literature refer only to non-relativistic flows and are of great use in analysing the spectra from super- novae and the jets of some active galaxies (AGN). How- ever, a number of objects of astrophysical importance, such as AGN jets, microquasars and gamma-ray bursts, contain ⋆ E-mail: [email protected] ; [email protected] flows which have bulk relativistic motion and the purpose of this paper is to examine the breaks, cut-offs and emission for such sources. While the first order Fermi process at relativistic shocks contains the same basic physics as in the nonrelativistic case, i.e. scattering leading to multiple shock crossings competing with a finite chance of escape downstream, the anisotropy of the particle distribution complicates the analysis consid- erably (Kirk & Schneider (1987), Heavens & Drury (1988) and Kirk et al. (2000)). The inclusion of self-consistent syn- chrotron losses will, as in the nonrelativistic limit, modify the spectrum at high momenta but we would also expect pitch angle effects to become apparent in the position of the cut-off and the emission itself. In order to motivate our treatment of this problem we first discuss the nonrelativistic shock limit in section 2, including the emission from a spa- tially integrated source. Section 3 then presents the analysis of synchrotron losses at relativistic shocks with particular emphasis on the shape of the momentum cut-off. We con- clude with a discussion in section 4. 2 NONRELATIVISTIC SHOCKS The effect of synchrotron losses on the energetic particle dis- tribution in the presence of nonrelativistic shocks is demon- strated rigorously in Webb et al. (1984). However a simpler approach is described in Heavens & Meisenheimer (1987) provided synchrotron losses are not considered important at the injection energies. We will follow this approach here, although we shall introduce a slightly different definition of the cut-off momentum. In the presense of a magnetic field charged particles emit synchrotron radiation with an energy loss rate given c© ???? RAS http://arxiv.org/abs/0704.0168v1 2 Paul Dempsey and Peter Duffy = −asB2p2 = −λp2 (1) where as is a positive constant. The radiative loss timescale is therefore tloss = 1/(λp). In the steady state, and in the presence of a nonrelativistic flow u, energetic particles obey a transport equation describing advection, diffusion, adiabatic compression and radiative losses, = 0. (2) In the presence of a nonrelativistic shock front where the upstream flow speed is u− and that downstream is u+ the acceleration timescale is tacc = u− − u+ . (3) At momenta for which tacc ≪ tloss the phase space den- sity will be a simple power law with f ∝ p−s where s = 3u−/(u− − u+). 2.1 Momentum Cut-off The spectrum will steepen at momentum p∗ where tacc(p ∗) = tloss(p ∗). In the case of momentum independent diffusion this gives u− − u+ . (4) In the case of a relativistic shock this result no longer strictly holds since the acceleration timescale defined above is only valid for nonrelativistic flows. Nevertheless, we will use this definition of p∗ throughout the paper for the sake of comparison. However, we require a general definition of the cut-off momentum that can be applied in the relativistic limit. An obvious alternative is to define the momentum at which the local spectral index, ∂ ln f/∂ ln p, becomes s + 1 but, as we shall see, it is necessary to perform a Laplace transform of the transport equation to proceed with this problem and it is more straightforward to define the cut-off in terms of spectral steepening of the Laplace transformed spectrum. In order to motivate such a definition we solve the nonrel- ativistic shock acceleration problem in the presence of syn- chrotron losses by first making the substitutions W ≡ p4f and y ≡ 1/p so that the transport equation, either upstream or downstream of the shock where adiabatic losses are zero, becomes (λW ) = 0. (5) Taking the Laplace transform with respect to y Ŵ (k, z) = W (y, z) exp(−yk)dy (6) and using the fact that losses prevent any particles achieving infinite energy, i.e. W (0, z) = 0, the transformed transport equation is + λkŴ = 0. (7) in the case of a momentum independent diffusion coefficient. Since the distribution function must be bounded infinitely far upstream and downstream, the solution becomes Ŵ± = A±(k) exp 1 + ω±k where we have introduced 4λ±κ± . (9) The isotropic and anisotropic parts of the particle dis- tribution function must match up at the shock giving, f−(p, 0) = f+(p, 0) (10) . (11) Multiplying the isotropic boundary condition by p4, making the substitutions as above and taking the Laplace transform with respect to 1/p gives Ŵ−(k, z = 0) = Ŵ+(k, z = 0) (12) which in turn gives A−(k) = A+(k) ≡ A(k). (13) The flux continuity condition (11) becomes A(k) = A0k 1 + ω−k − (s− 3) 1 + ω+k 1 + ω−k 1 + ω+k In the absence of synchrotron losses, ω± = 0, we have Ŵ±(k) ∝ k3−s which, upon inversion, gives f(p) ∝ p−s as expected. We can therefore define a function, Q̂, by Ŵ = k3−sQ̂. Recalling that k is the Laplace transformed variable of inverse momentum we define the cut-off momentum, pcut, to occur at the point where ∂ ln Q̂ ∂ ln k k=pcut = −1. (15) As an illustrative example, consider a power law distribution with a sharp maximum momentum, f(p) ∝ p−sH(pmax − p) with H the Heaviside function. In this case we have W = y y − 1 ⇒ Ŵ = e− pmax Γ(s− 3)k3−s where Γ is the Gamma function. With Q̂ ∝ exp(−k/pmax) we then have pcut = pmax as required physically in this sim- ple case. Returning to the solution of the shock problem we have from equation 14 = exp 1 + ω−k − (s− 3) 1 + ω+k 1 + ω−k 1 + ω+k . (17) Defining Ω = ω+/ω− (18) (s2 + 2 s2Ω+ Ω2s2 − 6Ω2s+ 2Ω s+ 9Ω2 − 8Ω ) × (s− 3) (19) c© ???? RAS, MNRAS 000, 1–10 Radiative losses and cut-offs of energetic particles at relativistic shocks 3 1e−04 0.01 1 100 10000 Figure 1. The cut-off momentum, pcut as a function of Ω for fixed equilibrium momentum, p∗ = 1, and spectral index, s = 4. Salzer Sum 0.001 0.01 0.01 0.1 1 10 Figure 2. Laplace inversion for s = 4 and Ω = 1. Using just M = 6 in the Salzer summation the inversion has already converged. The first approximation M = 1 is exactly the Laplace function Ŵk. We can see how fast the Salzer summation Post-Widder inversion converges as M = 2 is a very reasonably approximation to the actual solution. gives pcut = s4(2 + 2Ω)− s3(5 + 11Ω) + s2(5 + 8Ω− 2χ) (s2(1− Ω) + 6Ωs − 9Ω)2 s(33Ω + χ)− 36Ω (s2(1− Ω) + 6Ωs − 9Ω)2 . (20) This is always greater than p∗, s+ (s− 3)Ω as can be seen from figure 1. The minimum value for pcut occurs for Ω = 1, and is given by pcut = 2(s−1) The Laplace inversion (see Appendix for details) for s = 4 and Ω = 1 is shown in figure 2. Using just M = 6 in the Salzer summation the inversion has already converged. The first approximation M = 1 is exactly the Laplace function Ŵk. We can see how fast the Salzer summation Post-Widder inversion converges as M = 2 is a good approximation to the actual solution. Figure 3 shows how the particle distribution varies with Ω; if Ω 6= 1 the cut-off is broader. While p∗ is independent of Ω, pcut increases as the distribution broadens. The data in figure 3 can be fitted by an exponential tail Ω=3.2Ε7 Ω=1 Ω=0 Ω=3.2Ε7 log p flog p4 0.001 0.01 0.1 1 10 Figure 3. Particle Distributions of Various Ω Ω p∗ pcut β 0 1 1.25 2.25 1 1 1 2.25 9 1 1.4 2 16 1 1.53 1.8 25 1 1.63 1.75 ∞ 1 2 1.5 Table 1. Parameters for fitting particle spectra to the distribution of the form − (p/pcut)β where β ∼ 2. Table 1 shows how β varies with Ω for a shock of natural spectral index s = 4, with β attaining its maximum value of 2.25, i.e. the cut-off is sharpest, when Ω = 1. When Ω ≫ 1 particles can diffuse in the upstream without losing any energy, allowing a greater spread in momentum above pcut. 2.2 The Integrated Distribution Function and Synchrotron Spectra When the source cannot be fully resolved observationally, we must include the contribution from all particles within some distance z′ of the shock in calculating the spatially integrated emission. In the case of steady emission from a jet pointing towards us, or a completely unresolved source, z′ is essentially the source size in the optically thin limit. For log p log p4 f z’=0z’=1z’=10z’=50 0.01 0.01 0.1 1 10 Figure 4. The spatial variation of the particle distribution for Ω = 1. z′ = c© ???? RAS, MNRAS 000, 1–10 4 Paul Dempsey and Peter Duffy log p log p4 f 0.001 0.01 0.01 0.1 1 10 Figure 5. The spatially integrated particle distribution for Ω = 1 from z′ = 0 to z′ = 50. Note that as well as the particle cut-off there is a spectral break earlier were they spectrum softens from p−4 to p−5. This is analogous to synchrotron ageing. simplicity we assume that the magnetic field downstream of the shock is constant although the model can be generalised for more complex cases. The integrated Laplace distribution function is Ŵ dz k3−s Q̂ 1 + ω+k 1− exp 1 + ω+k . (23) When z′ is very small the result is k3−s with a cut-off at high k as expected. As z′ tends to infinity at low k we have k2−s so the spectrum is steepened , with the same high k cut-off. For finite values of z′ the spectrum starts as k3−s before turning into k2−s and finally cutting off. We shall see later that this result also holds in real momentum space. Figure 4 shows how the cut-off tends to lower momenta as we go further downstream. However what is most often observed a result of the integrated distribution is shown in figure 5. While the cut-off momentum is the same as at the shock, the distribution changes from an initial p−4 to a p−5 spectrum at some critical momentum, pb, which de- pends on z′. Here we will consider only synchrotron emission from an ordered magnetic field (parallel to the flow). Let 4πνm3ec where ν is the frequency, q is the charge on the electron, me is the electron mass and B is the magnetic field strength. Then given a spatially integrated particle dis- tribution f ∝ p−sg(p, µ) the total power emitted per unit frequency is (Rybicki & Lightman 1986) Ptot(ω) ∝ 1− µ2 1− µ2 !(s−5)/2 1− µ2 AF (x) dx (24) where F is the first synchrotron function F (x) ≡ x (y) dy. (25) In the case of non-relativistic diffusive shock accelera- Emission at ShockνΙ(ν) 0.001 0.01 0.001 0.01 0.1 1 10 100 Figure 6. A telescope with very high resolution may be able to observe synchrotron radiation at the shock. The only spectral feature here is the cut-off at νcut. Before νcut the spectrum has shape I(ν) ∼ ν−1/2. For this result Ω = 1. bνΙ(ν) z’=50 0.001 0.01 0.001 0.01 0.1 1 10 100 Figure 7. Spatially integrated emission for Ω = 1. The black dashed curve shows the synchrotron spectrum of an unresolved object containing a strong shock. The only feature is the spectral cut-off at νcut. Before νcut the spectrum has shape I(ν) ∼ ν The solid curve illustrates the synchrotron spectrum of a partially resolved source with emission from the shock (z′ = 0) to some downstream distance (z′ = 50). tion f downstream of a shock is assumed to be isotropic in which case g is independent of µ. Figure 6 shows the emission at the shock. The only fea- ture here is the cut-off hump, which is of course related to the particle cut-off, before which Iν ∼ ν−(s−3)/2. Figure 7 shows a more realistic plot, that of emission from an ex- tended region. If the region is infinite in size then the cut-off remains the sole feature but the spectrum before the cut-off is different Iν ∼ ν−(s−2)/2. If the region has finite size then a second feature, the spectral break νb, appears. Before the break the spectrum goes as Iν ∼ ν−(s−3)/2 while after it it is Iν ∼ ν−(s−2)/2. Again this is related to the momentum break pb we see in the particle distribution in figure 5. 3 RELATIVISTIC SHOCK ACCELERATION WITH LOSSES In the case of a relativistic shock, the particle transport equation describing advection, pitch angle diffusion and losses becomes Γ (u+ µ) + λg(µ) ∂(p4f) where µ is the cosine of the pitch angle of the particle and the flow velocity is constant upstream and downstream of the shock. λ = UB and g(µ) = 1 − µ2 for syn- chrotron losses in an ordered magnetic field, λ = 4σTUB/3 and g(µ) = 1 for synchrotron losses in a tangled magnetic field, or λ = 4σTUrad/3 and g(µ) = 1 for inverse Compton c© ???? RAS, MNRAS 000, 1–10 Radiative losses and cut-offs of energetic particles at relativistic shocks 5 losses. Equation (26) holds separately upstream and down- stream with the conditions that the distribution is isotropic infinity far downstream, there are no particles infinitely far upstream and the distribution is continuous at the shock. Al- though we will derive equations for general momentum inde- pendent pitch-angle diffusion and an arbitrary magnetic field alignment, the figures and results produced throughout the rest of this paper are for isotropic diffusion Dµµ = D(1−µ2) in an ordered (longitudinal) magnetic field with λ/D = 0.1. Guided by the treatment of the nonrelativistic case we set W = p4f and y = 1/p so that Γ(u+ µ) D(µ)(1− µ2)∂W − λg(µ)∂W . (27) Taking the Laplace Transform with respect to y and assum- ing W (0, µ, z) = 0 Γ(u+ µ) D(µ)(1− µ2)∂Ŵ − λg(µ)kŴ . (28) With the spatial and pitch angle variables separable we look for solutions of the form Ŵ (k, µ, z) = ai(k)Xi(k, z)Qi(k, µ) (29) putting this back into the reduced transport equation we get Γ(u+ µ) Qi = (DQi)Xi (30) where we have defined the differential operator D via DΦ = ∂ D(µ)(1− µ2)∂Φ − λg(µ)kΦ. (31) Separating X and Q we get the usual = Λi(k) = Qi(u+ µ) DQi (32) ⇒ Xi(k, z) = exp Λi(k)z and we have an equation for Q(k, µ) DQi − Λi(k)Qi(u+ µ) = 0. (34) Expanding out the differential operator we get D(µ)(1− µ2)∂Qi − (Λi(u+ µ) + kλg(µ))Qi = 0 which has regular singularities at µ = ±1 and so it should be possible to find solutions for Qi on [−1, 1] for all k ∈ C. 3.1 Determining the Eigenfunctions We know that along the real axis, k = x ∈ R, each Qi satisfies D(µ)(1− µ2)∂Qi − (Λi(u+ µ) + xλg(µ))Qi = 0. We define an inner product by: 〈ζ, ξ〉 = (u+ µ)ζ ξdµ. (37) u =.3; u =.076 u =.5; u =.129 u =.7; u =.189+ −Λ ( )k 1e−04 0.001 0.01 0.001 0.01 0.1 1 10 Figure 8. The zeroth order downstream eigenvalue for shock speeds u− = .3, .5, .7. Along the x-axis we have plotted the log- arithm of k+ while along the y-axis we have the logarithm of −Λ0(k+). When k+ = 0 we have Λ0(0) = 0. It can be shown that the Qi(x) are orthogonal and either real or purely imaginary, and the Λi(x) are real and distinct. We can normalise the eigenfunctions such that 〈Qi, Qj〉 = δi,j (38) or considering them as real 〈Qi, Qj〉 = δi,j(1/2 − i) / |1/2− i| ≡ ηi,j . (39) Then we have (see Appendix for details) j 6=i Λj − Λi g(µ)QiQjdµ Qjηj,j (40) g(µ)QiQidµ ηi,i. (41) We solve equation 36 at x = 0 using the Prüfer transforma- tion as in Kirk et al. (2000). We then use equations 40 and 41 to find Qi(x, µ) and Λi(x) for x > 0 using Runge-Kutta methods. Figures 8, 9, 10 and 11 show the zeroth downstream eigenvalues and eigenfunctions for shocks speeds of .3, .5 and .7. This eigenfunction is the dominant component in the downstream distribution function at the shock of such mildly relativistic shocks, where we are close to isotropy. Further downstream, where the contribution of higher eigenfunctions are more strongly damped, so the anisotropy for some z > 0 is essentially that of the zeroth eigenfunction. While Λ0(k+) is initially zero, note from figure 8 that it decreases linearly until a certain point which, as we will see later, is close to the cut-off momentum. This will play a major role in the integrated distribution function and emission. Figures 9, 10 and 11 show how anisotropy arises in the zeroth order eigenfunction which is isotropic for k = 0. Given that k is related to the momentum these figures show that, since this is the dominant eigenfunction, the anisotropy will increase with increasing energy. c© ???? RAS, MNRAS 000, 1–10 6 Paul Dempsey and Peter Duffy Figure 9. The zeroth order downstream eigenfunction for shock speed u− = .3. Along the x-axis we have plotted the logarithm of k+ while along the y-axis we have µ+. Up the z-axis we have plotted Q0(k+, µ+), which also defines the grayscale. Note the anisotropy increases with k+. Figure 10. The zeroth order downstream eigenfunction for shock speed u− = .5. Along the x-axis we have plotted the logarithm of k+ while along the y-axis we have µ+. Up the z-axis we have plotted Q0(k+, µ+), which also defines the grayscale. Note the anisotropy increases with k+. 3.2 Shock matching conditions Starting from Ŵ (k, µ, z) = ai(k)Xi(k, z)Qi(k, µ) (42) ai(k) exp Λi(k)z Qi(k, µ)) (43) we note that upstream (z < 0) we have ai = 0 for all i such that Λi 6 0 and that downstream (z > 0) we have ai = 0 for all i such that Λi > 0. The distribution function Figure 11. The zeroth order downstream eigenfunction for shock speed u− = .7. Along the x-axis we have plotted the logarithm of k+ while along the y-axis we have µ+. Up the z-axis we have plotted Q0(k+, µ+), which also defines the greyscale. Note the anisotropy increases with k+. is continuous at the shock, f−(y−, µ−, 0) = f+(y+, µ+, 0) (44) with (y−, µ−) related to (y+, µ+) by a Lorentz transforma- tion of velocity urel = (u− − u+)/(1− u−u+), y− = Γrely+(1 + urelµ−). (45) In terms of W the matching condition becomes rel(1 + urelµ−) W−(y−, µ−, 0) = W+(y+, µ+, 0) (46) and we now need to express this in terms of Ŵ , the Laplace transform with respect to y. Taking k−y− = k+y+, multi- plying the matching condition for W by exp(−k+y+) and integrating over y+ gives rel(1 + urelµ−) Ŵ−(k−, µ−, 0) = Ŵ+(k+, µ+, 0). (47) Guided by the discussion for the nonrelativistic case, we use the expansion i (k±, µ±) (48) so that the matching condition for the Laplace transformed spectrum at the shock reduces to rel(1 + urelµ−) i (k−)Q i (k−, µ−) = i (k+)Q i (k+, µ+). (49) In order to solve for the particle spectrum, we multiply by (u++µ+)Q j (k+, µ+) j > 0 and integrate over µ+. Then for a fixed k− we have i (k−) (1 + urelµ−) i (k−, µ−)× (u+ + µ+)Q j (k+, µ+)dµ+ = 0. (50) Defining a matrix S with elements i,j = (1 + urelµ−) i (k−, µ−)(u+ + µ+)Q j (k+, µ+)dµ+ we need to find the spectral index s, such that detS = 0. The Laplace inversion is then carried out numerically (see Appendix for details). As motivated by the nonrelativistic case, we define the cut-off to be the point at which d(lnR) d(ln k) = −1 (52) where R = biQi. Figures 12, 13 and 14 plot d(lnR)/d(ln k) at the shock against k as measured down- stream. The results are summarised in table 2. Figures 12, 13 and 14 show how the cut-off momentum becomes increasing anisotropic as the shock speed increases. The distribution can be fitted approximately by f ≈ p−s exp Γrelpcut(µ+) where β is typically 2. However it is difficult justify the use of the factor Γrel in general as our results are only for mildly relativistic shocks. This fit justifies our definition of pcut instead of using the equilibrium momentum p ∗. Figure c© ???? RAS, MNRAS 000, 1–10 Radiative losses and cut-offs of energetic particles at relativistic shocks 7 d(ln k) d(ln R) 0.001 0.01 0.1 1 10 100 Figure 12. Plotted along the x-axis we have the logarithm of momentum k+ while along the y-axis we have d(lnR+)/d(ln k+) for u− = .3 and R+ = i (k+)Q i (k+, µ+). Note the cut-off depends on µ+ d(ln k) d(ln R) 0.001 0.01 0.1 1 10 100 Figure 13. Plotted along the x-axis we have the logarithm of momentum k+ while along the y-axis we have d(lnR+)/d(ln k+) for u− = .5 and R+ = i (k+)Q i (k+, µ+). Note the cut-off depends on µ+ 17 illustrates this approximation for a .7c shock. β seems to be pitch angle dependent varying between 1.75 and 2.2, but typically 2. In fact for the .3c and .5c shock cases β showed much less variation about 2. For the shock speeds we have chosen, with the Juttner-Synge equation of state, the spectral indices in the absence of losses are close to 4. Figures 15, 16 and 17 illustrated a feature that was not present in the non-relativistic case. The pitch angle de- pendence of the cut-off momentum leads to a difference in the isotropy levels between particles above and below some d(ln k) d(ln R) 0.001 0.01 0.1 1 10 100 Figure 14. Plotted along the x-axis we have the logarithm of momentum k+ while along the y-axis we have d(lnR+)/d(ln k+) for u− = .7 and R+ = i (k+)Q i (k+, µ+). Note the cut-off depends on µ+ u− .3 .5 .7 u+ .076 .129 .189 Γrel 1.027 1.089 1.23 p∗ .404 1.143 2.26 Non-Rel pcut .621 1.79 3.719 pcut(µ+ = −1.0) .541 1.42 2.566 pcut(µ+ = −0.5) .595 1.71 3.612 pcut(µ+ = 0) .64 1.929 4.375 pcut(µ+ = 0.5) .682 2.138 5.105 pcut(µ+ = 1) .741 2.533 6.773 Table 2. Summary of Cut-Off Momenta aniso p pcut−off 0.01 0.001 0.01 0.1 1 10 Figure 15. The downstream function evaluated at the shock for a shock speed of .3. Along the x-axis we have plotted the logarithm of momentum p+ while along the y-axis we have the logarithm of W = p4f . critical momentum paniso. Indeed there is a clear pattern of greater levels of anisotropy at high energies as the shock speed increases, despite the fact that the results presented here are only for mildly relativistic shocks. 3.3 The Spatially Integrated Distribution While the method we follow in this paper finds the upstream particle distribution directly, it is easy to find the down- aniso cut−off 0.01 0.001 0.01 0.1 1 10 Figure 16. The downstream function evaluated at the shock for a shock speed of .5. Along the x-axis we have plotted the logarithm of momentum p+ while along the y-axis we have the logarithm of W = p4f . c© ???? RAS, MNRAS 000, 1–10 8 Paul Dempsey and Peter Duffy cut−off aniso W µ=−1 0.01 0.001 0.01 0.1 1 10 100 .185 exp(−(p/(2.566 sqrt(1.23)))^1.75) .185 exp(−(p/(4.375 sqrt(1.23)))^1.80) .185 exp(−(p/(6.773 sqrt(1.23)))^2.20) Figure 17. The downstream function evaluated at the shock for a shock speed of .7. Along the x-axis we have plotted the logarithm of momentum p+ while along the y-axis we have the logarithm of W = p4f . The lines are data while the points are the best fit described in the text. break aniso cut−off 0.01 1000 0.001 0.01 0.1 1 10 Figure 18. The downstream function integrated for a shock speed of .3 between z′ = 0 and z′ = 100 where z′ = Dz/Γ+. Along the x-axis we have plotted the logarithm of momentum p+ while along the y-axis we have the logarithm of W = p4f . stream distribution by using the matching condition, as dis- cussed in the previous section. The downstream distribution is, in many respects, more important physically as it will be responsible form most of the spatial integrated emission. As it can be difficult to spatially resolve observational data from non-thermal emitters, we must consider the emission from an extended region of space. Our eigenfunction expan- sion allows us to do this quite easily. The spatially averaged distribution from a downstream region [z0, z1] in terms of Laplace variables is R[z0,z1](k+, µ+) = ai(k) Λ+i (k+)z1 − exp Λ+i (k+)z0 i (µ+, k+) Λ+i (k+) . (54) In the case of a source which is completely spatially unre- solved this reduces to R[0,∞](k+, µ+) = −Γ+ ai(k) Q+i (µ+, k+) Λ+i (k+) . (55) Of course the optical depth of the emitting region will also have an effect on the spectrum of unresolved sources by re- ducing z1. cut−off aniso break 0.01 0.001 0.01 0.1 1 10 Figure 19. The downstream function integrated for a shock speed of .5 between z′ = 0 and z′ = 100 where z′ = Dz/Γ+. Along the x-axis we have plotted the logarithm of momentum p+ while along the y-axis we have the logarithm of W = p4f . cut−off aniso break 0.01 0.001 0.01 0.1 1 10 Figure 20. The downstream function integrated for a shock speed of .7 between z′ = 0 and z′ = 100 where z′ = Dz/Γ+. Along the x-axis we have plotted the logarithm of momentum p+ while along the y-axis we have the logarithm of W = p4f . Using the same numerical Laplace inversion as in the non-relativistic case we have calculated the distribution functions and synchrotron emission. Figures 18, 19 and 20 show the spatially integrated distribution functions for a fi- nite emission region. Now there are there features: a momen- tum break, pb, due to spatial effect; an anisotropic break, µ=−.5 νΙ(ν) 1e−06 1e−05 1e−04 0.001 0.01 1e−07 1e−06 1e−05 1e−04 0.001 0.01 0.1 1 10 100 1000 Figure 21. Synchrotron emission from the particle distribution shown in 18 measured in the downstream medium. In plotting our µ = ±1 we used µ = ±.9999 as there is no emission from an ordered field along µ = ±1. c© ???? RAS, MNRAS 000, 1–10 Radiative losses and cut-offs of energetic particles at relativistic shocks 9 µ=−.5 νΙ(ν) 1e−06 1e−05 1e−04 0.001 0.01 1e−06 1e−04 0.01 1 100 10000 Figure 22. Synchrotron emission from the particle distribution shown in 19 measured in the downstream medium. In plotting our µ = ±1 we used µ = ±.9999 as there is no emission from an ordered field along µ = ±1. µ=−.5 νΙ(ν) 1e−06 1e−05 1e−04 0.001 0.01 1e−06 1e−04 0.01 1 100 10000 Figure 23. Synchrotron emission from the particle distribution shown in 20 measured in the downstream medium. In plotting our µ = ±1 we used µ = ±.9999 as there is no emission from an ordered field along µ = ±1. paniso, due to relativistic effects; and a cut-off, pcut, due to energy losses. Given that the magnetic field is constant throughout this region it is trivial to produce the associ- ated synchrotron emission plots of figures 21, 22 and 23. It should be noted that in the emission plots Iν is measured in the downstream frame, but since Iν/ν 3 is a Lorentz invariant the transformation is trivial. The synchrotron emission also includes the same three features we observed in the particle distribution; namely a break frequency beyond which the ef- fect of synchrotron cooling becomes important, a frequency at which pitch-angle or anisotropic effects play a role and an upper cut-off beyond which there is virtually no emission. 4 DISCUSSION Particle acceleration and self-consistent synchrotron radia- tion have been considered previously by Kirk et al. (1998) using a zonal model. They were successful in explaining the radio to X-ray spectrum of Mkn 501. However such zonal models typically depend on isotropic particle distributions. We have shown, however, that for particles near the high energy cut-off this is not true even for mildly relativistic flows. The computational resources available restricted our results to be below .7c. However even for the mildly rela- tivistic shock velocities we see a clear pattern of high energy anisotropy emerging resulting in synchrotron emission which is also anisotropic. This could be extremely important in the modelling of the inverse Compton hump in γ-rays observed in TeV Blazars (Aharonian et al. 2006). As a second implica- tion of the particle anisotropy, in the presence of losses, the idealised situation, of a two sided strongly polarised iden- tical jet system can be considered. Each jet contains only forward external shocks, and the jet which is directed to- wards the observer is inclined at an angle θ = cos−1(−µ) to the line of sight (magnetic field direction same as that of shock). Then we will observe the emission from particles in the jet directed towards us which have pitch angle µ and from particles in the jet directed away from us which have pitch angle −µ. While at low energies the only difference between the observed emission of the two jets will be as a result of the effects of beaming, at energies near the syn- chrotron cut-off the details of the acceleration mechanism will amplify this difference, depending on viewing angle. Although the work in this paper is limited to an ide- alised form of diffusion, and mildly relativistic shocks, it illustrates previous unexamined features which could be im- portant in the modelling of relativistic, γ-ray sources such as microquasars, blazars and GRBs. We have parameterised the exponential shape of the distribution cut-off and identi- fied new pitch angle dependent features between break and cut-off frequencies. Further work is needed to examine both momentum dependent scattering and high Lorentz factor flows. ACKNOWLEDGMENTS Paul Dempsey would like to thank the Irish Research Coun- cil for Science, Engineering and Technology for their finan- cial support. He would also like to thank Cosmogrid for ac- cess to their computational facilities. We are grateful for discussions with Felix Aharonian. Peter Duffy would like to thank the Dublin Institute for Advanced Studies for their hospitality during the completion of this work. We would like to thank the referee for comments that improved the quality of this paper. REFERENCES Abate J., Valkó P.P., 2004, International Journal for Nu- merical Methods in Engineering, 60, 979 Aharonian F., et al. 2006, A&A, 455, 461 Boas M.L., 1983, Mathematical Methods in the Physical Sciences, 2nd Ed., John Wiley & Sons Heavens A.F., Drury L.O’C., 1988, MNRAS, 235, 997 Heavens A.F., Meisenheimer K., 1987, MNRAS, 225, 335 Kirk J.G., Guthmann A.W., Gallant Y.A., Achterberg A., 2000, ApJ, 542, 235 Kirk, J. G., Rieger, F. M., & Mastichiadis, A., 1998, A&A, 333, 452 Kirk J.G., Schneider P., 1987, ApJ, 323, L87 Rybicki, G. B., & Lightman, A. P. 1986, Radiative Pro- cesses in Astrophysics. Webb G.M., Drury L.O’C., Biermann P., 1983, A&A, 137, c© ???? RAS, MNRAS 000, 1–10 10 Paul Dempsey and Peter Duffy Valkó P.P., Abate J., 2004, Computers and Mathematics with Application, 48, 629 Widder D. V., 1932, PNAS, 18, 181 This paper has been typeset from a TEX/ LATEX file prepared by the author. APPENDIX A: INVERSE LAPLACE TRANSFORMS While Heavens & Meisenheimer (1987) invert the Laplace transform analytically for particular cases here we use nu- merical methods as we will need to when dealing with rela- tivistic flows. Formally the inverse Laplace transform is the Bromwich integral, which is a complex integral given by: f(t) = L−1 [F (s)] = 1 Z γ+i∞ F (s) ds (A1) where γ is to the right of every singularity of F (s). If the sin- gularity of F (s) all ly in the left half of the complex plane γ can be set to 0 and this reduces to the inverse Fourier transform, which is easy to do. However for complicated or numerical Laplace functions the Bromwich integral is ex- tremely difficult to solve. The four main numerical inversion techniques are Fourier Series Expansion, Talbot’s method, Weeks method and methods based on the Post-Widder for- mula. However some of these methods converge rather slowly and a lot of work has gone into creating acceleration meth- ods. Numerical Laplace inversion is a area of active research and the choice of inversion technique is as much an art as a science at the moment. In this paper the Post-Widder based method was chosen and only these methods shall by described below. Let F (s) be the Laplace transform of f(t) then Widder (1932) showed that fn(t) → f(t) where fn(t) = (−1)n ((n+ 1)/t). (A2) The advantage of this method in our case in that we see that the Laplace transform of the solution times the Laplace coordinate is the zeroth order approximation to the actual solution. W0(y) = Ŵ (1/y) ⇒ W0(p) = pŴ (p). When dealing with numerical results however it is easier to use the Gaver-Stehfest algorithm (Abate & Valkó 2004). It is an algorithm based on the Post-Widder method with the Gaver approximants, {fn(t) : n > 0}, defined as fn(t) ≡ (n+ 1) ln(2) 2(n+ 1) f̂ ((n+ 1 + k) ln(2)/t) . (A3) However the convergence for both these methods is slow. A test of methods for accelerating this convergence can be found in Valkó & Abate (2004) and two are found to be quite good: the non-linear Wynn’s Rho Algorithm and the linear Salzer summation. Again a choice has to be made and here we present only Salzer summation: f(t,M) → f(t) where f(t,M) = Wkfk−1(t) (A4) Wk = (−1)k+M . (A5) The Post-Widder method based on differentiation was im- plemented in Maple with the Salzer acceleration. It was used to produce the results in the non-relativistic limit as we have an analytic form of the Laplace function to work with. The Salzer accelerated Gaver-Stehfest algorithm was imple- mented in C/C++ code for use with the numerical output from the relativistic approach discussed above. APPENDIX B: DERIVING THE EIGENSYSTEM DIFFERENTIAL EQUATIONS The solutions, Qi, to equation 36 D(µ)(1− µ2)∂Qi − xλg(µ)Qi = Λi(u+ µ)Qi (B1) for real x, are orthogonal, with weight u+ µ and have real, distinct eigenvalues Λi (Boas 1983). Taking the derivative of this equation with respect to x gives D(µ)(1− µ2) ∂ − xλg(µ)∂Qi = Λi(u+ µ) (u+ µ) + λg(µ) Qi. (B2) Since solution to Sturm Liouville equations form an orthog- onal basis, we can write qmQm (B3) which gives D(µ)(1− µ2) ∂ qmxλg(µ)Qm qmΛm(u+ µ)Qm qmΛi(u+ µ)Qm + (u+ µ) + λg(µ) Multiplying by Q∗j and integrating over µ gives qm(Λm − Λi)〈Qj, Qm〉 〈Qj , Qi〉+ λ g(µ)QiQ j dµ. (B5) Taking j = i implies equation 41 and j 6= i implies equation c© ???? RAS, MNRAS 000, 1–10 Introduction Nonrelativistic Shocks Momentum Cut-off The Integrated Distribution Function and Synchrotron Spectra Relativistic Shock Acceleration with Losses Determining the Eigenfunctions Shock matching conditions The Spatially Integrated Distribution Discussion Inverse Laplace Transforms Deriving the eigensystem differential equations

0704.0169

Very strong and slowly varying magnetic fields as source of axions

arXiv:0704.0169v1 [hep-ph] 2 Apr 2007 VERY STRONG AND SLOWLY VARYING MAGNETIC FIELD AS SOURCE OF AXIONS Giorgio CALUCCI* Dipartimento di Fisica Teorica dell’Università di Trieste, Trieste, I 34014 Italy INFN, Sezione di Trieste, Italy Abstract The investigation on the production of particles in slowly varying but extremely intense magnetic field in extended to the case of axions. The motivation is, as for some previously considered cases, the possibility that such kind of magnetic field may exist around very compact astrophysical objects. * E-mail: [email protected] http://arxiv.org/abs/0704.0169v1 1. Statement of the problem A magnetic field of huge strength can give rise to real particles even if its rate of variation is very small: this posibility could be of some interest from a pure theoretical point of view, but it gains more physical relevance if one accepts that such kind of field configurations may be present around some very compact astrophysical objects[1-3]. In this case the time vari- ation is related to the evolution of the source, by collapse, rotations or else, and it is therefore very slow, in comparison with the times typical of elementary-particle processes. We can call the former time the macro- scopic time and the latter the microscopic one. The production of light particles in these processes has been analyzed in some detail in some pre- vious papers[4],with the suggestion that it is one of the mechanisms at work in the phenomenon of gamma-ray bursts[5]; the typical microscopic time is related to the electron mass since photons are produced through real or virtual intermediate states of e− − e+-pairs. The lightest particles that could be produced are massive neutrinos, but the magnetic-moment coupling induced by the standard electroweak interactions is extremely small. There is, at least in the theoretical realm, another very light parti- cle, that is the axion[6]: owing to its dynamical characteristics it must be coupled also to the electromagnetic field[7], even more its electomag- netic coupling is being actively studied from an experimental side[8] and the possibility of detecting such particles as coming from nonterrestrial sources has already been foreseen[9]. It is immediately seen that the pro- duction of axions by a varying magnetic field must be realized through a mechanism different from the previously considered one, in fact the axions are coupled[7] only to the pseudoscalar density E ·B so the presence of an electric field is necessary as a starting point, but a nonstatic magnetic field creates always an electric field and even though the rate of variation is small, the very large magnetic strength makes the electric field not a tiny one. In the present paper the coupling of the axions field with a given E ·B density is written in standard second-quantized formalism, then the effect of time variation of that density on the axion vacuum is determined and the consequent production is calculated. The result depends both on the spatial shape and on the time variation of the magnetic field: in accordance with the prevailing astrophysical hypotheses[1-3] the magnetic field is seen as a bundle of lines of force which may safely be considered straight in comparison with the microscopic scale. The time-variation could affect both the shape and the strength of the fields, both are effective in the production process. The calculation procedure is not the standard adiabatic approximation [10] as used in previous investigations [4], but the feature that one has to deal with a two-scale problem is still fully relevant. 2. General form of the production probability The starting point is a second-quantized axion field in presence of given, classical, magnetic and electric fields. The axion field φ(x) is cou- pled to the pseudoscalar density * G(x) = E(x) · B(x) and the coupling constant, of dimension of length, is here indicated by C. We assume that the interaction lasts from an initial time to until a final time t. So we have for the axion field the expression: φ(x) = φo(x) + χ(x) = φo(x) + C ∆R(x− y)G(y)d4y (1) Here ∆R(x − y) is the standard retarded Green function, the source is a c−number, so the same holds for χ. The field φ is free before to, where it has the standard expansion: φo(x) = (2π)3/2 ao(k)e ik·r−iωt + a†o(k)e −ik·r+iωt , (2) then it acquires a contribution from χ, this term has the following actual expression: χ(x) = (2π)3/2 e−iωkt eiωkτgk(τ)dτe ik·r − c.c. (3.1) gk(t) = (2π)3/2 G(r, t)e−ik·r (3.2) The reality condition for G, that gives g∗k = g−k have been used, together with the initial condition χ̇(to) = 0. The separation into positive and negative frequencies in χ(x) is unambiguous until the typical frequencies of G are small. Having found the time evolution of the field the total production of axions is calculated in Heisenberg description of motion, i.e. we take an initial state ao(k)|◦ >= 0 ∀k, i.e. the vacuum in the absence of inter- action then we express the mean particle number as the time dependent N (k, t) =< ◦|a†(k, t) a(k, t)|◦ > * the same kind of coupling is possible also for the neutral pion, but in this case other channels of production are present [11] Since all the effect of the interaction is a c−number shift on the opera- tors a(k, t) = ao(k) + b(k) the calculation is easy, in particular when the interaction no longer acts we get Nf (k) = |bf (k)|2 (4) The expression of bf (k) can be read off from eq.s (2,3.1,3.2), it is: bf (k) = eiωkτgk(τ)dτ (5) 3. Detailed calculations As anticipated in the introduction a more definite model of the mag- netic field can by a field of uniform direction (at a given time) with some transverse shape, both the direction and the shape may vary in time, one possible restriction is the conservation of the total flux. More explicitly these conditions are realized by giving: n ∧ r 1− w(µr⊥) B = − F n ∂⊥w(µr⊥) . (6) The unit vector n gives the instantaneous direction of the magnetic field, r2 − n · r2 and ∂⊥ the corresponding derivative, F is the total magnetic flux; the parameter µ defines the size of the field in the transverse directions, w is taken to be cylindrically symmetric and, obviously it must go to zero at infinity, the requirement w(0) = 1 avoids singularities in E. Since E = −Ȧ and A ·B = 0 only the terms coming from the time variation of the direction of A contributes to G = E ·B so we get: )2J · r (1− w)∂⊥w Here J gives the angular velocity of n i.e. J = n ∧ ṅ. In this configuration the Fourier transform of the source is gk(t) = (2π)3/2 C(2F )2δ(n · k)J · kS(k⊥/µ) (8.1) S(k⊥/µ) = (µ/k⊥) J1(ρk⊥/µ) (1− w(ρ))w′(ρ) dρ/ρ (8.2) Here J1 is the Bessel function of order one and w ′ indicates the derivative with respect to the argument. It is useful to remember that, owing to the presence of the factor δ(n · k) in the expression of S we can substitute k2⊥ simply with k2. We now remember that the model of magnetic field we have at hand is such that it is uniform along one direction, but this direction is contin- uously varying, so a most significant quantity is obtained by an angular integration dΩkNf (k) (9) So we need a quantity like dΩkgk(τ)g ′) which contains a singularity due to the presence, in the domain of integration, of a δ-square term which arises for τ = τ ′. This is clearly due to the unphysical assumption that at every time there is a direction in which the pseudoscalar density G is absolutely uniform. Since we are integrating over the direction at the end the effect of the singularity is mild enough, it results in a logarithmic divergence, however this fact must be explicitly death with, considering a finite extension of the fields. A more careful treatment is mathemetically heavier, so it is presented in an Appendix. The final result is given by ≈ C2(2F )4 k Ψ[S(k/µ)]2 2 ln 4kL . (10) If we give a definite transverse shape to the fields we get, evidently, a definite answer. In the situation where the fields change direction but not intensities, i.e. we keep µ constant, we may give, tentatively the form w(ρ) = e−ρ . Then the expression for the function S(k/µ) is[12]: S(k/µ) = − exp . (11) The limit k → 0 of this expression is finite, so the whole production goes to zero only owing to the phase-space factor k2 in front of the expression in eq.(10). This result, as it appears from the whole derivation, can be valid only for axion masses and so energies which are definitely larger than the typical frequencies of the astrophysical phenomena, no resonant dynamics is included. 4. Some conclusions The rate of production of axions by a slowly varying but very strong magnetic field has been calculated. The conditions are such that the as- trophysical frequencies are very much lower than the proper frequencies of the axion field. In fact with an axion mass mA of the order of 1eV[13] the typical frequencies of the field shall exceed 1015 Hz and so no resonance conditions appear realistic. The relevant parameter turns out to be the dimension of the spatial inhomogeneity, in the present model 1/µ, here also is seems very reasonable to assume that µ << mA. It is also possible to give an expression for the total number of produced particles. A = ζ C 2F 4µ3 Ψ lnLµ (12) Some of the factors owe their origin to the general form of the interaction, eq(1) and to dimensional requirament, it appers clear the role of the total rotation of the field (Ψ) in determining the overall production; so when the rotation is uniform the rate is proportional to the angular velocity. The numerical factor is more model dependent, in the chosen case it is ζ = 8 3− 2− 2] = 0.704 . . .. The transformation from the incoming Heisenberg field to the outgo- ing field can be implemented by the simple unitary operator U = exp d3k[a(k)b∗(k) − a†(k)b(k)] Ua(k)U† = a(k) + b(k) . The actual form of the evolution operator gives the further information that the axion are produced with a Poissonian distribution of multiplicity, strictly speaking this is true for a production in a totally defined state, in operations like the one leading to eq. (9) this particular form can be blurred. A very short mention on this problem was made at the XXXVI In- ternational Symposium on Multiparticle Dynamics, Paraty, R.J. - Brazil, September 2006 Appendix We want to calculate dΩkgk(τ)g ′), with the functions g given by eq.(8.1) and taking care of the finite extension of the magnetic field. This is implemented by substituting the δ−functions as: δ(n · k) → Lπ−1/2 exp[−L2(n · k)2] (A.1) The calculation is performed in the particular case in which the rotation takes place in a constant plane, so J‖J′. The integration over the angles is performed in Cartesian coordinates, it is useful to introduce the unit vector of the three-momentum direction k = kv, then dΩk = 2δ(v 2 − 1)d3v and through standard although lengthy calculations the representation is obtained: dΩkδ(n · k)J · kδ(n′ · k)J′ · k → JJ ′× dλ exp[iλ(w2 − 1)]w2dw (n ∧ n′)2 − 2iλ/(Lk)2 − λ2/(Lk)4 ]−1/2 (A.2) In the limit L→ ∞ it results I = JJ ′/|n ∧ n′| which can be obtained in a simpler way. Now we must integrate over τ and τ ′, times an oscillating factor eiωk(τ−τ ′). In the conditions that have been chosen the motion takes place in a plane, so n is characterized by a unique angle ψ and n′ by ψ′ hence the integration over time amounts at an angular integration, in fact (n ∧ n′)2 = (sin(ψ − ψ′))2 and moreover J = ψ̇ and J ′ = ψ̇′. So we must integrate I in dψ , dψ′ from 0 to some final angle Ψf . Defin- ing γ = ψ − ψ′ we see that the integrand shows, in the limit L → ∞, a singularity for γ = 0, so we perform the integration from −1 π to 1 because the domain which do not include zero has no singular behavior, the oscillating factor is approximated with its value on the singular point τ = τ ′, so that the exponential factor reduces to 1. Then the integral is a complete elliptic integral,[12] which can be conveniently expressed in term of the hypergeometric function. In fact the integration in dγ is: sin2 γ +Q with Q = − 2iλ (Lk)2 (Lk)4 (A.3) The result of the integration is and in the limit L→ ∞, which gives Q→ 0 we get; 2 ln 2 + lnLk Since the dominant term in the limit is independent of the auxiliary pa- rameter λ the rest of the integration in eq. (A.2) is straightforward and it gives I = Ψ 2 ln 4kL (A.4) A comment: what is excluded is the possibility that the magnetic field should perform more than a complete rotation, this would destroy the correspondence ψ = ψ′ ↔ τ = τ ′. References 1. C. Thompson, R.C. Duncan, Astrophys.J. 408 (1993) 194 2. H. Hanami, Astrophys.J. 491 (1997) 687 3. C. Kouveliotou, R.C. Duncan, C. Thompson, Sci.Am. 288,2 (2003) 4. G. Calucci, Mod.Phys.Lett. A14 (1999) 1183; A. DiPiazza, G.Calucci, Phys.Rev.D 65 (2002) 125019; A. DiPiazza, G.Calucci, Phys.Rev.D 66 (2002) 123006; A. DiPiazza, Eur. Phys. J.C 36 (2004) 25 5. T. Piran, Phys.Rep.314 (1999) 375; J. van Paradijs, C. Kouveliotou, R.A.M.J. Wijers, Annu.Rev. As- tron.Astrophys. 38 (2000) 279: P. Mészáros, Annu.Rev.Astron.Astrophys. 40 (2002)137; T. Piran, Rev.Mod.Phys. 76 (2004) 1143 R. Ruffini, Nuovo Cimento B 119 (2004) 785 6. R.D.Peccei, H.R.Quinn, Phys.Rev.Lett. 38 (1977) 1440; R.D.Peccei, H.R.Quinn, Phys.Rev.D 16 (1977) 1791; F. Wilczek, Phys.Rev.Lett. 40 (1978) 279; S. Weinberg, Phys.Rev.Lett. 40 (1978) 223 7. L. Maiani, R.Petronzio, E.Zavattini Phys.Lett.B 175 (1986) 359 8. E. Zavattini, G. Zavattini, G. Ruoso, E. Polacco, E. Milotti, M. Karuza, U. Gastaldi, G. DiDomenico, F. DellaValle, R. Cimino, S. Carusotto, G.Cantatore, M. Bregant Phys.Rev.Lett. 96 (2006) 110406 S. Lamoreaux Nature 41 (2006) 31 9. The CERN Axion solar telescope (C.E.Asleth et al.) Nucl.Phys.B (proc.suppl.) 110 (2002) 85 10. A.B. Migdal, V. Krainov Approximation methods in quantum mechan- ics (Ch. 2) (Benjamin, NewYork 1969) D.R.Bates ed. Quantum theory, vol 1 (Ch. 8)Academic press, London 11. A. DiPiazza, G.Calucci, Mod.Phys.Lett. A20 (2005), 117 12. M.Abramowitz and I.A.Stegun, Handbook of mathematical functions (Dover, 1964). 13. S. Hannestad, A. Mirizzi,G. Raffelt, J. Cosm. Astrop. Phys. 07002 (2005)

0704.0170

Symmetry disquisition on the TiOX phase diagram

Symmetry disquisition on the TiOX phase diagram Daniele Fausti,∗ Tom T. A. Lummen, Cosmina Angelescu, Roberto Macovez, Javier Luzon, Ria Broer, Petra Rudolf, and Paul H.M. van Loosdrecht† Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands. Natalia Tristan and Bernd Büchner IFW Dresden, D-01171 Dresden, Germany Sander van Smaalen Laboratory of Crystallography, University of Bayreuth, 95440 Bayreuth, Germany Angela Möller, Gerd Meyer, and Timo Taetz Institut für Anorganische Chemie, Universität zu Köln, 50937 Köln, Germany (Dated: November 2, 2018) Abstract The sequence of phase transitions and the symmetry of in particular the low temperature incom- mensurate and spin-Peierls phases of the quasi one-dimensional inorganic spin-Peierls system TiOX (TiOBr and TiOCl) have been studied using inelastic light scattering experiments. The anomalous first-order character of the transition to the spin-Peierls phase is found to be a consequence of the different symmetries of the incommensurate and spin-Peierls (P21/m) phases. The pressure dependence of the lowest transition temperature strongly suggests that magnetic interchain interactions play an important role in the formation of the spin-Peierls and the incom- mensurate phases. Finally, a comparison of Raman data on VOCl to the TiOX spectra shows that the high energy scattering observed previously has a phononic origin. PACS numbers: 68.18.Jk Phase transitions 63.20.-e Phonons in crystal lattices 75.30.Et Exchange and superexchange interactions 75.30.Kz Magnetic phase boundaries (including magnetic transitions, metamagnetism, etc.) 78.30.-j Infrared and Raman spectra http://arxiv.org/abs/0704.0170v2 I. INTRODUCTION The properties of low-dimensional spin systems are one of the key topics of contemporary condensed matter physics. Above all, the transition metal oxides with highly anisotropic interactions and low-dimensional structural elements provide a fascinating playground to study novel phenomena, arising from their low-dimensional nature and from the interplay between lattice, orbital, spin and charge degrees of freedom. In particular, low-dimensional quantum spin (S=1/2) systems have been widely discussed in recent years. Among them, layered systems based on a 3d9 electronic configuration were extensively studied in view of the possible relevance of quantum magnetism to high temperature superconductivity1,2. Though they received less attention, also spin=1/2 systems based on early transition metal oxides with electronic configuration 3d1, such as titanium oxyhalides (TiOX, with X=Br or Cl), exhibit a variety of interesting properties3,4. The attention originally devoted to the layered quasi two-dimensional 3d1 antiferromagnets arose from considering them as the electron analog to the high-Tc cuprates 5. Only recently TiOX emerged in a totally new light, namely as a one-dimensional antiferromagnet and as the second example of an inorganic spin-Peierls compound (the first being CuGeO3) The TiO bilayers constituting the TiOX lattice are candidates for various exotic electronic configurations, such as orbital ordered3, spin-Peierls6 and resonating-valence-bond states8. In the case of the TiOX family the degeneracy of the d orbitals is completely removed by the crystal field splitting, so that the only d−electron present, mainly localized on the Ti site, occupies a nondegenerate energy orbital3. As a consequence of the shape of the occupied orbital (which has lobes oriented in the b− and c−directions, where c is perpendicular to the layers), the exchange interaction between the spins on different Ti ions arises mainly from direct exchange within the TiO bilayers, along the b crystallographic direction3. This, in spite of the two-dimensional structural character, gives the magnetic system of the TiOX family its peculiar quasi one-dimensional properties6. Magnetic susceptibility6 and ESR3 measurements at high temperature are in reasonably good agreement with an antiferromag- netic, one-dimensional spin-1/2 Heisenberg chain model. At low temperature (Tc1) TiOX shows a first-order phase transition to a dimerised nonmagnetic state, discussed in terms of a spin Peierls state6,9,10. Between this low temperature spin Peierls phase (SP) and the one-dimensional antiferromagnet in the high temperature phase (HT), various experimental evidence4,11,12,13 showed the existence of an intermediate phase, whose nature and origin is still debated. The temperature region of the intermediate phase is different for the two compounds considered in this work, for TiOBr Tc1 = 28 K and Tc2 = 48 K while for TiOCl Tc1 = 67 K and Tc2 = 91 K. To summarize the properties so far reported, the intermediate phase (Tc1 < Tc2) exhibits a gapped magnetic excitation spectrum 4, anomalous broadening of the phonon modes in Raman and IR spectra9,13, and features of a periodicity incom- mensurate with the lattice14,15,16,17. Moreover, the presence of a pressure induced metal to insulator transition has been recently suggested for TiOCl18. Due to this complex phase behavior, both TiOCl and TiOBr have been extensively discussed in recent literature, and various questions still remain open: there is no agreement on the crystal symmetry of the spin Peierls phase, the nature and symmetry of the incommensurate phase is not clear and the anomalous first-order character of the transition to the spin Peierls state is not explained. Optical methods like Raman spectroscopy are powerful experimental tools for revealing the characteristic energy scales associated with the development of broken symmetry ground states, driven by magnetic and structural phase transitions. Indeed, information on the nature of the magnetic ground state, lattice distortion, and interplay of magnetic and lattice degrees of freedom can be obtained by studying in detail the magnetic excitations and the phonon spectrum as a function of temperature. The present paper reports on a vibrational Raman study of TiOCl and TiOBr, a study of the symmetry properties of the three phases and gives coherent view of the anomalous first order character of the transition to the spin Peierls phase. Through pressure-dependence measurements of the magnetic susceptibility, the role of magnon-phonon coupling in determining the complex phase diagram of TiOX is discussed. Finally, via a comparison with the isostructural compound VOCl, the previously reported13,19 high energy scattering is revisited, ruling out a possible interpretation in terms of magnon excitations. II. EXPERIMENT Single crystals of TiOCl, TiOBr, and VOCl have been grown by a chemical vapor trans- port technique. The crystallinity was checked by X-ray diffraction12. Typical crystal di- mensions are a few mm2 in the ab−plane and 10-100 µm along the c−axis, the stacking direction15. The sample was mounted in an optical flow cryostat, with a temperature sta- bilization better than 0.1 K in the range from 2.6 K to 300 K. The Raman measurements were performed using a triple grating micro-Raman spectrometer (Jobin Yvon, T64000), equipped with a liquid nitrogen cooled CCD detector (resolution 2 cm−1 for the considered frequency interval). The experiments were performed with a 532 nm Nd:YVO4 laser. The power density on the sample was kept below 500 W/cm2 to avoid sample degradation and to minimize heating effects. The polarization was controlled on both the incoming and outgoing beam, giving access to all the polarizations schemes allowed by the back-scattering configuration. Due to the macroscopic morphology of the samples (thin sheets with natural surfaces parallel to the ab−planes) the polarization analysis was performed mainly with the incoming beam parallel to the c−axis (c(aa)c̄, c(ab)c̄ and c(bb)c̄, in Porto notation). Some measurements were per- formed with the incoming light polarized along the c−axis, where the k−vector of the light was parallel to the ab−plane and the polarization of the outgoing light was not controlled. These measurements will be labeled as x(c⋆)x̄. The magnetization measurements were performed in a Quantum Design Magnetic Prop- erty Measurement System. The pressure cell used is specifically designed for measurement of the DC-magnetization in order to minimize the cell’s magnetic response. The cell was calibrated using the lead superconducting transition as a reference, and the cell’s signal (measured at atmospheric pressure) was subtracted from the data. III. RESULTS AND DISCUSSION The discussion will start with a comparison of Raman experiments on TiOCl and TiOBr in the high temperature phase, showing the consistency with the reported structure. After- wards, through the analysis of Raman spectra the crystal symmetry in the low temperature phases will be discussed, and in the final part a comparison with the isostructural VOCl will be helpful to shed some light on the origin of the anomalous high energy scattering reported for TiOCl and TiOBr13,19. A. High Temperature Phase The crystal structure of TiOX in the high temperature (HT) phase consists of buckled Ti-O bilayers separated by layers of X ions. The HT structure is orthorhombic with space group Pmmn. The full representation20 of the vibrational modes in this space group is: Γtot = 3Ag + 2B1u + 3B2g + 2B2u + 3B3g + 2B3u. (1) Among these, the modes with symmetry B1u, B2u, and B3u are infrared active in the polar- izations along the c, b, and a crystallographic axes9, respectively. The modes with symmetry Ag, B2g, and B3g are expected to be Raman active: The Ag modes in the polarization (aa), (bb), and (cc); the B2g modes in (ac) and the B3g ones in (bc). Fig.1 shows the room tem- 100 200 300 400 100 200 300 400 Energy (cm-1) c(bb)c TiOBr (T=100 K) c(aa)c TiOCl (T=300 K) FIG. 1: (Color online) Polarized Raman spectra (Ag) of TiOCl and TiOBr in the high temperature phase, showing the three Ag modes. Left panel: (bb) polarization; right panel: (aa) polarization. perature Raman measurements in different polarizations for TiOCl and TiOBr, and Fig.2 displays the characteristic Raman spectra for the three different phases of TiOBr, the spec- tra are taken at 100 (a), 30 (b) and 3K (c). At room temperature three Raman active modes are clearly observed in both compounds for the c(aa)c̄ and c(bb)c̄ polarizations (Fig.1), while none are observed in the c(ab)c̄ polarization. These results are in good agreement with the group theoretical analysis. The additional weakly active modes observed at 219 cm−1 for TiOCl and at 217 cm−1 for TiOBr are ascribed to a leak from a different polarization. This is confirmed by the measurements with the optical axis parallel to the ab-planes (x(c⋆)x̄) on TiOBr, where an intense mode is observed at the same frequency (as shown in the inset of Fig.2(a)). In addition to these expected modes, TiOCl displays a broad peak in the c(bb)c̄ polarization, centered at around 160 cm−1 at 300K; a similar feature is observed in TiOBr as a broad background in the low frequency region at 100K. As discussed for TiOCl13, these modes are thought to be due to pre-transitional fluctuations. Upon decreasing the tempera- ture, this ”peaked” background first softens, resulting in a broad mode at Tc2 (see Fig.2(b)), and then locks at Tc1 into an intense sharp mode at 94.5 cm −1 for TiOBr (Fig.2(c)) and at 131.5 cm−1 for TiOCl. The frequency of all the vibrational modes observed for TiOCl and TiOBr in their high temperature phase are summarized in Table I. Here, the infrared active modes are taken from the literature7,9 and for the Raman modes the temperatures chosen for the two compounds are 300K for TiOCl and 100K for TiOBr. The observed Raman frequencies agree well with 100 200 300 400 500 600 c(aa)c c(ab)c Energy (cm-1) c(bb)c T=3 K T=30 K c(aa)c c(ab)c c(bb)c 200 250 550 600 Energy (cm-1) TiOBr - Pol x(c*)x T=100 K c(aa)c c(ab)c c(bb)c FIG. 2: (Color online) Polarization analysis of the Raman spectra in the three phases of TiOBr, taken at 3 (a), 30 (b) and 100K (c). The spectra of TiOCl show the same main features and closely resemble those of TiOBr. Table IV reports the frequencies of the TiOCl modes. The inset shows the TiOBr spectrum in the x(c∗)x̄ polarization (see text). previous reports13. The calculated values reported in Table I are obtained with a spring- model calculation based on phenomenological longitudinal and transversal spring constants (see Appendix). The spring constants used were optimized using the TiOBr experimental frequencies (except for the ones of the B3g modes due to their uncertain symmetry) and kept constant for the other compounds. The frequencies for the other two compounds are obtained by merely changing the appropriate atomic masses and are in good agreement with the experimental values. The relative atomic displacements for each mode of Ag symmetry are shown in Table II. The scaling ratio for the lowest frequency mode (mode 1) between the two compounds is in good agreement with the calculation of the atomic displacements. The low frequency mode is mostly related to Br/Cl movement and, indeed, the ratio νT iOCl/νT iOBr = 1.42 is similar to the mass ratio MCl. The other modes (2 and 3) involve mainly Ti or O displacements, and their frequencies scale with a lower ratio, as can be expected. B. Low Temperature Phases Although the symmetry of the low temperature phases has been studied by X-ray crys- tallography, there is no agreement concerning the symmetry of the SP phase; different works TABLE I: (a)Vibrational modes for the high temperature phase in TiOCl, TiOBr and VOCl. The calculated values are obtained with a spring model. The mode reported in italics in Table I are measured in the x(c⋆)x̄ polarization they could therefore have either B2g or B3g symmetry (see experimental details). (a) TiOBr TiOCl VOCl Exp. Cal. Exp. Cal. Exp. Cal. Ag (σaa, σbb, σcc) 142.7 141 203 209.1 201 208.8 329.8 328.2 364.8 331.2 384.9 321.5 389.9 403.8 430.9 405.2 408.9 405.2 B2g(σac) 105.5 157.1 156.7 328.5 330.5 320.5 478.2 478.2 478.2 B3u(IR, a) 77 a 75.7 104b 94.4 93.7 417a 428.5 438b 428.5 425.2 B3g(σbc) 60 86.4 129.4 129.4 216 336.8 219 c 336.8 327.2 598 586.3 586.3 585.6 B2u(IR, b) 131 a 129.1 176b 160.8 159.5 275a 271.8 294b 272.1 269.8 B1u(IR, c) 155.7 194.1 192.4 304.8 301.1 303.5 aValue taken from Ref.7. bValue taken from Ref.9. cValue obtained considering the leakage in the σyy polarization. TABLE II: The ratio between the frequency of the Ag Raman active modes measured in TiOBr and TiOCl is related to the atomic displacements of the different modes as calculated for TiOBr (all the eigenvectors are fully c−polarized, the values are normalized to the largest displacement). (b) Mode ν(TiOBr) νCl/νBr Ti O Br 1 142.7 1.42 0.107 0.068 1 2 329.8 1.11 1 0.003 0.107 3 389.9 1.11 0.04 1 0.071 proposed two different space groups, P21/m 14,15,16 and Pmm221. The possible symmetry changes that a dimerisation of Ti ions in the b−direction can cause are considered in order to track down the space group of the TiOX crystals in the low temperature phases. Assuming that the low temperature phases belong to a subgroup of the high temperature orthorhombic space group Pmmn, there are different candidate space groups for the low temperature phases. Note that the assumption is certainly correct for the intermediate phase, because the transition at Tc2 is of second-order implying a symme- try reduction, while it is not necessarily correct for the low temperature phase, being the transition at Tc1 is of first-order. FIG. 3: (Color online) Comparison of the possible low temperature symmetries. The low temper- ature structures reported are discussed, considering a dimerisation of the unit cell due to Ti-Ti coupling and assuming a reduction of the crystal symmetry. The red rectangle denotes the unit cell of the orthorhombic HT structure. Structure (a) is monoclinic with its unique axis parallel to the orthorhombic c−axis (space group P2/c), (b) shows the suggested monoclinic structure for the SP phase (P21/m), and (c) depicts the alternative orthorhombic symmetry proposed for the low T phase Pmm2. Fig.3 shows a sketch of the three possible low temperature symmetries considered, and Table III reports a summary of the characteristic of the unit cell together with the number of phonons expected to be active for the different space groups. Depending on the relative po- sition of the neighboring dimerised Ti pairs, the symmetry elements lost in the dimerisation are different and the possible space groups in the SP phase are P2/c (Table III(a)), P21/m (b) or Pmm2 (c). The first two are monoclinic groups with their unique axis perpendicular to the TiO plane (along the c−axis of the orthorhombic phase), and lying in the TiO plane (‖ to the a−axis of the orthorhombic phase), respectively. The third candidate (Fig.3(c)) has orthorhombic symmetry. The group theory analysis based on the two space groups suggested for the SP phase (P21/m 14 and Pmm221) shows that the number of modes expected to be Raman active is TABLE III: Comparison between the possible low temperature space group. (a) Space group P2/c Unique axis ⊥ to TiO plane, C42h 4TiOBr per unit cell Γ = 7Ag + 6Au + 9Bg + 11Bu 7Ag Raman active σxx, σyy, σzz, σxy 11Bg Raman active σxz, σyz 6Au and 9Bu IR active (b) Space group P21/m Unique axis in the TiO plane, C22h 4 TiOBr per unit cell Γ = 12Ag + 5Au + 6Bg + 10Bu 12Ag Raman active σxx, σyy, σzz, σxy 6Bg Raman active σxz, σyz 5Au and 10Bu IR active (c) Space group Pmm2 4 TiOBr per unit cell Γ = 11A1 +A2 + 4B1 + 5B2 11A1 Raman active σxx, σyy, σzz A2 Raman active σxy 4B1 and 5B2 Raman active in σxz and σyz different in the two cases (Table III(b) and (c)). In particular, the 12 fully symmetric vibra- tional modes (Ag), in the P21/m space group, are expected to be active in the σxx, σyy, σzz and σxy polarizations, and 6Bg modes are expected to be active in the cross polarizations (σxz and σyz). Note that in this notation, z refers to the unique axis of the monoclinic cell, so σyz corresponds to c(ab)c for the HT orthorhombic phase. For Pmm2 the 11 A1 vibrational modes are expected to be active in the σxx, σyy, σzz polarizations, and only one mode of symmetry A2 is expected to be active in the cross polarization (σxy or c(ab)c). The experiments, reported in Table IV for both compounds and in Fig.2 for TiOBr only, show that 10 modes are active in the c(aa)c and c(bb)c in the SP phase (Fig.2(c)), and, more importantly, two modes are active in the cross polarization c(ab)c. This is not compatible with the expectation for Pmm2. Hence the comparison between the experiments and the group theoretical analysis clearly shows that of the two low temperature structures reported in X-ray crystallography15,21, only the P21/m is compatible with the present results. As discussed in the introduction, the presence of three phases in different temperature intervals for TiOX is now well established even though the nature of the intermediate phase is still largely debated7,12,15. The temperature dependence of the Raman active modes for TiOBr between 3 and 50 K, is depicted in Fig.4. In the spin-Peierls phase, as discussed above, the reduction of the crystal symmetry16 increases the number of Raman active modes. Increasing the temperature above Tc1 a different behavior for the various low temperature phonons is observed. As shown in Fig.4, some of the modes disappear suddenly at Tc1 (labeled LT ), some stay invariant up to the HT phase (RT ) and some others undergo a sudden broadening at Tc1 and slowly disappear upon approaching Tc2 (IT ). The polarization analysis of the Raman modes in the temperature region Tc1 < T < Tc2 shows that the number TABLE IV: Vibrational modes of the low temperature phases. spin Peierls phase (a) TiOBr Ag(σxx, σyy) 94.5 102.7 142.4 167 219 276.5 330 351 392 411∗ Ag(σxy) 175,6 506.5 TiOCl Ag(σxx, σyy) 131.5 145.8 203.5 211.5 296.5 305.3 322.6 365.1 387.5 431∗ Ag(σxy) 178.5 524.3 Intermediate phase (b) TiOBr (30K) Ag(σxx, σyy) 94.5 142 221.5 277 328.5 344.5 390.4 TiOCl (75K) Ag(σxx, σyy) 132.8 206.2 302 317.2 364.8 380 420.6 ∗ The broad line shape of this feature suggests it may originate from a two-phonon process. 100 150 200 250 300 350 400 450 10 20 30 40 50 TITRTITITRTLT Energy (cm-1) 27.5 K 32.5 K 37.5 K 42.5 K : broadens above T : high-T mode : dissappears above T 351cm-1 328cm-1 275cm-1 220cm-1 94cm-1 Temperature(K) FIG. 4: (Color online) The temperature dependence of the Raman spectrum of TiOBr is depicted (an offset is added for clarity). The 3 modes present at all temperatures are denoted by the label RT . The modes characteristic of the low temperature phase (disappearing at Tc1 = 28 K) are labelled LT , and the anomalous modes observed in both the low temperature and the intermediate phase are labelled IT . The right panel (b) shows the behavior of the frequency of IT modes, plotted renormalized to their frequency at 45 K. It is clear that the low-frequency modes shift to higher energy while the high-frequency modes shift to lower frequency. of active modes in the intermediate phase is different from that in both the HT and the SP phases. The fact that at T = Tc1 some of the modes disappear suddenly while some others do not disappear, strongly suggests that the crystal symmetry in the intermediate phase is different from both other phases, and indeed confirms the first-order nature of the transition at Tc1. In the X-ray structure determination15, the intermediate incommensurate phase is dis- cussed in two ways. Firstly, starting from the HT orthorhombic (Pmmn) and the SP mon- oclinic space group (P21/m - unique axis in the TiO planes, ‖ to a), the modulation vector required to explain the observed incommensurate peaks is two-dimensional for both space groups. Secondly, starting from another monoclinic space group, with unique axis perpen- dicular to the TiO bilayers (P2/c), the modulation vector required is one-dimensional. The latter average symmetry is considered (in the commensurate variety) in Fig.3(a) and Table III(a). In the IP, seven modes are observed in the σxx, σyy and σzz geometry on both compounds (see Table IV(b)), and none in the σxy geometry. This appears to be compatible with all the space groups considered, and also with the monoclinic group with unique axis perpendicular to the TiO planes (Table III(a)). Even though from the evidence it is not possible to rule out any of the other symmetries discussed, the conjecture that in the intermediate incom- mensurate phase the average crystal symmetry is already reduced, supports the description of the intermediate phase as a monoclinic group with a one-dimensional modulation15, and moreover it explains the anomalous first-order character of the spin-Peierls transition at Tc1. The diagram shown in Fig.5 aims to visualize that the space group in the spin-Peierls state FIG. 5: (Color online) The average crystal symmetry of the intermediate phase is proposed to be monoclinic with the unique axis parallel to the c−axis of the orthorhombic phase. Hence the low temperature space group is not a subgroup of the intermediate phase, and the transition to the spin-Peierls phase is consequently of first order. (P21/m) is a subgroup of the high temperature Pmmn group, but not a subgroup of any of the possible intermediate phase space groups suggested (possible P2/c). This requires the phase transition at Tc1 to be of first order, instead of having the conventional spin-Peierls second-order character. Let us return to Fig.4(b) to discuss another intriguing vibrational feature of the interme- diate phase. Among the modes characterizing the intermediate phase (IT ), the ones at low frequency shift to higher energy approaching Tc2, while the ones at high frequency move to lower energy, seemingly converging to a central frequency (≃300 cm−1 for both TiOCl and TiOBr). This seems to indicate an interaction of the phonons with some excitation around 300 cm−1. Most likely this is in fact arising from a strong, thermally activated coupling of the lattice with the magnetic excitations, and is consistent with the pseudo-spin gap observed in NMR experiments4,22 of ≈430 K (≃300 cm−1). C. Magnetic Interactions As discussed in the introduction, due to the shape of the singly occupied 3d orbital, the main magnetic exchange interaction between the spins on the Ti ions is along the crystallo- graphic b−direction. This, however, is not the only effective magnetic interaction. In fact, FIG. 6: (Color online) (a) Magnetization as a function of temperature measured with fields 1 T and 5 T (the magnetization measured at 1 T is multiplied by a factor of 5 to evidence the linearity). The inset shows the main magnetic interactions (see text). (b) Pressure dependence of Tc1. The transition temperature for transition to the spin-Peierls phase increases with increasing pressure. The inset shows the magnetization versus the temperature after subtracting the background signal coming from the pressure cell. one also expects a superexchange interaction between nearest and next-nearest neighbor chains (J2 and J3 in the insert of Fig.6(a)) 23. The situation of TiOX is made more interest- ing by the frustrated geometry of the interchain interaction, where the magnetic coupling J2 between adjacent chains is frustrated and the exchange energies can not be simultaneously minimized. Table V reports the exchange interaction values for the three possible magnetic interactions calculated for TiOBr. These magnetic interactions were computed with a DFT Broken symmetry approach24 using an atom cluster including the two interacting atoms and all the surrounding ligand atoms, in addition the first shell of Ti3+ ions was replaced by Al3+ ions and also included in the cluster. The calculations were performed with the Gaussian03 package25 using the hybrid exchange-correlation functional B3LYP26 and the 6-3111G* basisset. TABLE V: Calculated Exchange interactions in TiOBr TiOBr J1 = −250 K J2 = −46.99 K J3 = 11.96 K Although the computed value for the magnetic interaction along the b−axis is half of the value obtained from the magnetic susceptibility fitted with a Bonner-Fisher curve accounting for a one-dimensional Heisenberg chain, it is possible to extract some conclusions from the ab-initio computations. The most interesting outcome of the results is that in addition to the magnetic interaction along the b−axis, there is a relevant interchain interaction (J1/J2 = 5.3) in TiOBr. Firstly, this explains the substantial deviation of the Bonner-Fisher fit from the magnetic susceptibility even at temperature higher than Tc2. Secondly, the presence of an interchain interaction, together with the inherent frustrated geometry of the bilayer structure, was already proposed in literature12 in order to explain the intermediate phase and its structural incommensurability. The two competing exchange interactions J1 and J2 have different origins: the first arises from direct exchange between Ti ions, while the second is mostly due to the superexchange interaction through the oxygen ions23. Thus, the two exchange constants are expected to depend differently on the structural changes induced by hydrostatic pressure, J1 should increase with hydrostatic pressure (increases strongly with decreasing the distance between the Ti ions), while J2 is presumably weakly affected due only to small changes in the Ti– O–Ti angle (the compressibility estimated from the lattice dynamics simulation is similar along the a and b crystallographic directions). The stability of the fully dimerized state is reduced by the presence of an interchain coupling, so that Tc1 is expected to be correlated to J1/J2. Pressure dependent magnetic experiments have been performed to monitor the change of Tc1 upon increasing hydrostatic pressure. The main results, shown in Fig.6, indeed is consistent with this expectation: Tc1 increases linearly with pressure; unfortunately it is not possible to address the behavior of Tc2 from the present measurements. D. Electronic Excitations and Comparison with VOCl The nature of the complex phase diagram of TiOX was originally tentatively ascribed to the interplay of spin, lattice and orbital degrees of freedom7. Only recently, infrared spec- troscopy supported by cluster calculations excluded a ground state degeneracy of the Ti d orbitals for TiOCl, hence suggesting that orbital fluctuations can not play an important role in the formation of the anomalous incommensurate phase27,28. Since the agreement between the previous cluster calculations and the experimental results is not quantitative, the energy of the lowest 3d excited level is not accurately known, not allowing to discard the possibility of an almost degenerate ground state. For this reason a more formal cluster calculation has been performed using an embedded cluster approach. In this approach a TiO2Cl4 cluster was treated explicitly with a CASSCF/CASPT2 quantum chemistry calculation. This clus- ter was surrounded by eight Ti3+ TIP potentials in order to account for the electrostatic interaction of the cluster atoms with the shell of the first neighboring atoms. Finally, the cluster is embedded in a distribution of punctual charges fitting the Madelung’s potential produced by the rest of the crystal inside the cluster region. The calculations were per- formed using the MOLCAS quantum chemistry package29 with a triple quality basis set; for the Ti atom polarization functions were also included. The calculations reported in Table VI, confirmed the previously reported result27 for both TiOCl and TiOBr. The first excited state dxy is at 0.29-0.3 eV (> 3000 K) for both compounds, therefore the orbital degrees of freedom are completely quenched at temperatures close to the phase transition. A comparison with the isostructural compound VOCl has been carried out to confirm that the phase transitions of the TiOX compounds are intimately related to the unpaired S=1/2 spin of the Ti ions. The V3+ ions have a 3d2 electronic configuration. Each ion carries two unpaired electrons in the external d shell, and has a total spin of 1. The crystal TABLE VI: Crystal field splitting of 3d1 Ti3+ in TiOCl and TiOBr (eV). TiOCl TiOBr xy 0.29-0.29 0.29-0.30 xz 0.66-0.68 0.65-0.67 yz 1.59-1.68 1.48-1.43 x2 − r2 2.30-2.37 2.21-2.29 field environment of V3+ ions in VOCl is similar to that of Ti3+ in TiOX, suggesting that the splitting of the degenerate d orbital could be comparable. The electrons occupy the two lowest t2g orbitals, of dy2−z2 (responsible for the main exchange interaction in TiOX) and dxy symmetry respectively. Where the lobes of the latter point roughly towards the Ti ions of the nearest chain (Table VI). It is therefore reasonable to expect that the occupa- tion of the dxy orbital in VOCl leads to a substantial direct exchange interaction between ions in different chains in VOCl and thus favors a two-dimensional antiferromagnetic order. Indeed, the magnetic susceptibility is isotropic at high temperatures and well described by a quadratic two-dimensional Heisenberg model, and at TN = 80 K VOCl undergoes a phase transition to a two-dimensional antiferromagnet30. 200 300 400 1000 1500 20 K 40 K 60 K 78 K 98 K 116 K Frequency (cm-1) VOCl (3 K) TiOCl (3 K) TiOBr (10 K) FIG. 7: (Color online) Raman scattering features of VOCl. (a) High energy scattering of TiOCl/Br and VOCl, and (b) temperature dependence of the vibrational scattering features of VOCl. No symmetry changes are observed at TN = 80 K. The space group of VOCl at room temperature is the same as that of TiOX in the high temperature phase (Pmmn), and, as discussed in the previous section, three Ag modes are expected to be Raman active. As shown in Fig.7(b), three phonons are observed throughout the full temperature range (3 − 300 K), and no changes are observed at TN . The modes observed are consistent with the prediction of lattice dynamics calculations (Table I). In the energy region from 600 to 1500 cm−1, both TiOBr and TiOCl show a similar highly structured broad scattering continuum, as already reported in literature13,19. The fact that the energy range of the anomalous feature is consistent with the magnetic exchange constant in TiOCl (J=660 K) suggested at first an interpretation in terms of two-magnon Raman scattering13. Later it was shown that the exchange constant estimated for TiOBr is considerably smaller (J=406 K) with respect to that of TiOCl while the high energy scattering stays roughly at the same frequency. Even though the authors of ref.19 still assigned the scattering continuum to magnon processes, it seems clear taht the considerably smaller exchange interaction in the Br compound (J=406 K) falsifies this interpretation and that magnon scattering is not at the origin of the high energy scattering of the two compounds. Furthermore, the cluster calculation (Table VI) clearly shows that no excited crystal field state is present in the energy interval considered, ruling out a possible orbital origin for the continuum. These observations are further strengthened by the observation of a similar continuum scattering in VOCl (see fig. 7(a)) which has a different magnetic and electronic nature. Therefore, the high energy scattering has most likely a vibrational origin. The lattice dynamics calculations, confirmed by the experiments, show that a ”high” energy mode (≃600 cm−1) of symmetry B3g (Table I) is expected to be Raman active in the σyz polarization. Looking back at Fig.2, the inset shows the measurements performed with the optical axis parallel to the TiOX plane, where the expected mode is observed at 598 cm−1. The two phonon process related to this last intense mode is in the energy range of the anomalous scattering feature and has symmetry Ag (B3g ⊗ B3g). The nature of the anomalies observed is therefore tentatively ascribed to a multiple-phonon process. Further detailed investigations of lattice dynamics are needed to clarify this issue. IV. CONCLUSION The symmetry of the different phases has been discussed on the basis of inelastic light scattering experiments. The high temperature Raman experiments are in good agreement with the prediction of the group theoretical analysis (apart from one broad mode which is ascribed to pre-transitional fluctuations). Comparing group theoretical analysis with the polarized Raman spectra clarifies the symmetry of the spin-Peierls phase and shows that the average symmetry of the incommensurate phase is different from both the high temper- ature and the SP phases. The conjecture that the intermediate phase is compatible with a different monoclinic symmetry (unique axis perpendicular to the TiO planes) could explain the anomalous first-order character of the transition to the spin-Peierls phase. Moreover, an anomalous behavior of the phonons characterizing the intermediate phase is interpreted as evidencing an important spin-lattice coupling. The susceptibility measurements of TiOBr show that Tc1 increases with pressure, which is ascribed to the different pressure dependence of intrachain and interchain interactions. Finally, we compared the TiOX compounds with the ”isostructural” VOCl. The presence of the same anomalous high energy scattering fea- ture in all the compounds suggests that this feature has a vibrational origin rather than a magnetic or electronic one. Acknowledgements The authors are grateful to Maxim Mostovoy, Michiel van der Vegte, Paul de Boeij, Daniel Khomskii, Iberio Moreira and Markus Grüninger for valuable and insightful discussions. This work was partially supported by the Stichting voor Fundamenteel Onderzoek der Materie [FOM, financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)], and by the German Science Foundation (DFG). V. APPENDIX: DETAILS OF THE SPRING MODEL CALCULATION The spring model calculation reported in the paper, was carried out using the software for lattice-dynamical calculation UNISOFT31 (release 3.05). In the calculations the Born- von Karman model was used; here the force constants are treated as model parameters and they are not interpreted in terms of a special interatomic potential. Only short range interactions between nearest neighbor ions are taken into account. Considering the forces to be central forces, the number of parameters is reduced to two for each atomic interaction: the longitudinal and transversal forces respectively defined as L = d2V (r̄i,j) and T = 1 dV (r̄i,j) A custom made program was interfaced with UNISOFT to optimize the elastic constants. Our program proceeded scanning the n dimensional space (n = number of parameters) with a discrete grid, to minimize the squared difference between the calculated phonon frequencies and the measured experimental frequencies for TiOBr, taken from both Raman and infrared spectroscopy. The phonon frequencies of TiOCl and VOCl were obtained using the elastic constants optimized for TiOBr and substituting the appropriate ionic masses. The optimized force constants between different atoms are reported in N/m in the following Table. TABLE VII: Elastic constants used in the spring model calculation. The label numbers refer to Fig. ??, while the letters refer to the different inequivalent positions of the ions in the crystal. Number Ions Longitudinal (L) (N/m) Transversal (T) (N/m) 1 Ti(a)-Ti(b) 18.5 32.7 2 Ti(a)-O(a) 18.5 11.1 3 Ti(a)-O(b) 53.1 9.5 4 Ti(a)-X(a) 29.0 4.4 5 O(a)-O(b) 20.6 7.3 6 X(a)-O(a) 18.5 3.5 7 X(a)-X(b) 11.7 0.7 ∗ Electronic address: [email protected] † Electronic address: [email protected] 1 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). 2 E. Dagotto, Rep. Prog. Phys. 62, 1525 (1999). 3 V. Kataev, J. Baier, A. Möller, L. Jongen, G. Meyer, , and A. Freimuth, Phys. Rev. B 68, 140405 (2003). 4 T. Imai and F. C. Choub, cond-mat 0301425 (2003), URL http://xxx.lanl.gov/abs/cond-mat/0301425. 5 C. H. Maule, J. N. Tothill, P. Strange, and J. A. Wilson, J. Phys. C 21, 2153 (1988). 6 A. Seidel, C. A. Marianetti, F. C. Chou, G. Ceder, and P. A. Lee, Phys. Rev. B 67, 020405 (2003). 7 G. Caimi, L. Degiorgi, P. Lemmens, and F. C. Chou, J. Phys. Cond. Mat. 16, 5583 (2004). 8 R. J. Beynon and J. A. Wilson, J. Phys. Cond. 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Introduction Experiment Results and Discussion High Temperature Phase Low Temperature Phases Magnetic Interactions Electronic Excitations and Comparison with VOCl Conclusion Appendix: Details of the spring model calculation References

0704.0171

Discovery of a point-like very-high-energy gamma-ray source in Monoceros

Astronomy & Astrophysics manuscript no. 7299 c© ESO 2018 May 31, 2018 Discovery of a point-like very-high-energy γ-ray source in Monoceros F. A. Aharonian1,13, A.G. Akhperjanian2, A.R. Bazer-Bachi3 , B. Behera14, M. Beilicke4 , W. Benbow1, D. Berge1 ⋆, K. Bernlöhr1,5, C. Boisson6, O. Bolz1, V. Borrel3, I. Braun1, E. Brion7, A.M. Brown8, R. Bühler1, I. Büsching9, T. Boutelier17, S. Carrigan1, P.M. Chadwick8, L.-M. Chounet10, G. Coignet11, R. Cornils4, L. Costamante1,23, B. Degrange10, H.J. Dickinson8, A. Djannati-Atäı12 , W. Domainko1, L.O’C. Drury13, G. Dubus10, K. Egberts1, D. Emmanoulopoulos14, P. Espigat12, C. Farnier15, F. Feinstein15, A. Fiasson15, A. Förster1, G. Fontaine10, Seb. Funk5, S. Funk1, M. Füßling5, Y.A. Gallant15, B. Giebels10, J.F. Glicenstein7, B. Glück16, P. Goret7, C. Hadjichristidis8, D. Hauser1, M. Hauser14, G. Heinzelmann4, G. Henri17, G. Hermann1, J.A. Hinton1,14 ⋆⋆, A. Hoffmann18, W. Hofmann1, M. Holleran9, S. Hoppe1, D. Horns18, A. Jacholkowska15, O.C. de Jager9, E. Kendziorra18, M. Kerschhaggl5, B. Khélifi10,1, Nu. Komin15, K. Kosack1, G. Lamanna11, I.J. Latham8, R. Le Gallou8, A. Lemière12, M. Lemoine-Goumard10, T. Lohse5, J.M. Martin6, O. Martineau-Huynh19, A. Marcowith3,15, C. Masterson1,23, G. Maurin12, T.J.L. McComb8, E. Moulin15,7, M. de Naurois19, D. Nedbal20, S.J. Nolan8, A. Noutsos8, J-P. Olive3, K.J. Orford8, J.L. Osborne8, M. Panter1, G. Pedaletti14, G. Pelletier17, P.-O. Petrucci17, S. Pita12, G. Pühlhofer14, M. Punch12, S. Ranchon11, B.C. Raubenheimer9, M. Raue4, S.M. Rayner8, O. Reimer ⋆⋆⋆, J. Ripken4, L. Rob20, L. Rolland7, S. Rosier-Lees11, G. Rowell1 †, J. Ruppel21, V. Sahakian2, A. Santangelo18, L. Saugé17, S. Schlenker5, R. Schlickeiser21, R. Schröder21, U. Schwanke5, S. Schwarzburg18, S. Schwemmer14, A. Shalchi21, H. Sol6, D. Spangler8, R. Steenkamp22, C. Stegmann16, G. Superina10, P.H. Tam14, J.-P. Tavernet19, R. Terrier12, M. Tluczykont10,23 ‡, C. van Eldik1, G. Vasileiadis15, C. Venter9, J.P. Vialle11, P. Vincent19, H.J. Völk1, S.J. Wagner14, M. Ward8, Y. Moriguchi24, and Y. Fukui24,25 (Affiliations can be found after the references) Preprint online version: May 31, 2018 ABSTRACT Aims. The complex Monoceros Loop SNR/Rosette Nebula region contains several potential sources of very-high-energy (VHE) γ- ray emission and two as yet unidentified high-energy EGRET sources. Sensitive VHE observations are required to probe acceleration processes in this region. Methods. The H.E.S.S. telescope array has been used to search for very high-energy γ-ray sources in this region. CO data from the NANTEN telescope were used to map the molecular clouds in the region, which could act as target material for γ-ray production via hadronic interactions. Results. We announce the discovery of a new γ-ray source, HESSJ0632+058, located close to the rim of the Monoceros SNR. This source is unresolved by H.E.S.S. and has no clear counterpart at other wavelengths but is possibly associated with the weak X-ray source 1RXSJ063258.3+054857, the Be-star MWC148 and/or the lower energy γ-ray source 3EG J0634+0521. No evidence for an associated molecular cloud was found in the CO data. Key words. gamma rays: observations Send offprint requests to: [email protected], [email protected] ⋆ now at CERN, Geneva, Switzerland ⋆⋆ now at School of Physics & Astronomy, University of Leeds, Leeds LS2 9JT, UK ⋆⋆⋆ now at Stanford University, HEPL & KIPAC, Stanford, CA 94305-4085, USA † now at School of Chemistry & Physics, University of Adelaide, Adelaide 5005, Australia ‡ now at DESY Zeuthen 1. Introduction Shell-type supernova remnants (SNRs) have been identified as particle accelerators via their very-high-energy (VHE; E > 100 GeV) γ-ray and non-thermal X-ray emission (see e.g. Aharonian et al. (2006a) and Koyama et al. (1997)). It has been suggested that interactions of particles acceler- ated in SNR with nearby molecular clouds should produce detectable γ-ray emission (Aharonian et al. 1994). For this reason the well-known Monoceros Loop SNR (G 205.5+0.5, distance∼1.6 kpc (Graham et al. 1982; Leahy et al. 1986)), with its apparent interaction with the Rosette Nebula (a young stellar cluster/molecular cloud complex, distance http://arxiv.org/abs/0704.0171v1 2 F. A. Aharonian et al.: A point-like γ-ray source in Monoceros 1.4± 0.1 kpc (Hensberge et al. 2000)) is a prime target for observations with VHE γ-ray instruments. For the case of hadronic cosmic rays (CRs) interact- ing in the interstellar medium to produce pions and hence γ-rays via π0 decay, a spatial correlation between γ-ray emission and tracers of interstellar gas is expected. Such a correlation was used to infer the presence of a population of recently accelerated CR hadrons in the Galactic Centre re- gion (Aharonian et al. 2006b). This discovery highlights the importance of accurate mapping of available target material for the interpretation of TeV γ-ray emission. The NANTEN 4 m diameter sub-mm telescope at Las Campanas observa- tory, Chile, has been conducting a 12CO (J=1→0) survey of the Galactic plane since 1996 (Mizuno & Fukui 2004). The Monoceros region is covered by this survey and the NANTEN data are used here to trace the target material for interactions of accelerated hadrons. 2. H.E.S.S. Observations and Results The observations described here took place between March 2004 and March 2006 and comprise 13.5 hours of data after data quality selection and dead-time correction. The data were taken over a wide range of zenith angles from 29 to 59 degrees, leading to a mean energy threshold of 400 GeV with so-called standard cuts used here for spectral analysis and 750 GeV with the hard cuts used here for the source search and position fitting. These cuts are described in de- tail in Aharonian et al. (2006c). A search in this region for point-like emission was made using a 0.11◦ On source region and a ring of mean radius 0.5◦ for Off source background estimation (see Berge et al. (2006) for details). Fig. 1 shows the resulting significance map, together with CO data from NANTEN, radio con- tours and the positions of all Be-stars in this region. The peak significance in the field is 7.1σ. The number of sta- tistical trials associated with a search of the entire field of view, in 0.01◦ steps along both axes, is≈ 105. The measured peak significance corresponds to 5.3σ after accounting for these trials. A completely independent analysis based on a fit of camera images to a shower model (Model Analysis de- scribed in de Naurois (2006)), yields a significance of 7.3σ (5.6σ post-trials). The best fit position of the new source is 6h32m58.3s, +5◦48′20′′ (RA/Dec. J2000) with 28′′ statistical errors on each axis, and is hence identified as HESSJ0632+057. Systematic errors are estimated at 20′′ on each axis. There is no evidence for intrinsic extension of the source and we derive a limit on the rms size of the emission region of 2′ (at 95% confidence), under the assumption that the source follows a Gaussian profile. This source size upper limit is shown as a dashed circle in the bottom panel of Fig. 1. Fig. 2 demonstrates the point-like nature of the source. The an- gular distribution of excess γ-ray-like events with respect to the best fit position is shown together with the expected distribution for a point-like source. The reconstructed energy spectrum of the source is con- sistent with a power-law: dN/dE = k(E/1TeV)−Γ with photon index Γ = 2.53± 0.26stat ± 0.20sys and a flux nor- malisation k = 9.1±1.7stat±3.0sys×10 −13 cm−2s−1TeV−1. Fig. 3 shows the H.E.S.S. spectrum together with that for the unidentified EGRET source 3EGJ0634+0521 (dis- cussed below) and an upper limit derived for TeV emis- sion from 3EGJ0634+0521 using the HEGRA telescope s00m30h06s00m35h06 s00m30h06s00m35h06 G 205.5+0.5 Rosette Nebula HESS J0632+057 SAX J0635.2+0533 3EG 0634+0521 5.6407 5.7507 5.8607 5.9707 s30m32h06s00m33h06s30m33h06 s30m32h06s00m33h06s30m33h06 H.E.S.S. Fig. 1. Top: the Monoceros SNR / Rosette Nebula re- gion. The grey-scale shows velocity integrated (0-30 km s−1) 12CO (J=1→0) emission from the NANTEN Galactic Plane Survey (white areas have highest flux). Yellow con- tours show 4 and 6 σ levels for the statistical significance of a point-like γ-ray excess. Radio observations at 8.35 GHz from Langston et al. (2000) are overlaid as cyan contours, and illustrate the extent of the Rosette Nebula. The nomi- nal Green (2004) Catalogue position/size of the Monoceros SNR is shown as an (incomplete) dashed circle. 95% and 99% confidence regions for the position of the EGRET source 3EG0634+0521 are shown as dotted green contours. The binary pulsar SAXJ0635.2+0533 is marked with a square and Be-stars with pink stars. Bottom: an expanded view of the centre of the top panel showing H.E.S.S. sig- nificance as a colour scale. The rms size limit derived for the TeV emission is shown as a dashed circle. The unidenti- fied X-ray source 1RXSJ063258.3+054857 is marked with a triangle and the Be-star MCW 148 with a star. F. A. Aharonian et al.: A point-like γ-ray source in Monoceros 3 (square degrees)2θ 0 0.005 0.01 0.015 0.02 0.025 Fig. 2. Distribution of excess (candidate γ-ray) events as a function of squared angular distance from the best fit posi- tion of HESS J0632+057 (points), compared to the expecta- tion for this dataset from Monte-Carlo simulations (smooth curve). array (Aharonian et al. 2004), converted from an integral to a differential flux using the spectral shape measured by H.E.S.S. We find no evidence for flux variability of HESS J0632+057 within our dataset. However, we note that due to the weakness of the source and sparse sam- pling of the light-curve, intrinsic variability of the source is not strongly constrained. The bulk of the available data was taken in two short periods in December 2004 (P1, 4.7 hours) and November/December 2005 (P2, 6.2 hours). The integral fluxes (above 1 TeV) in these two periods were: 6.3±1.8×10−13 cm−2 s−1 (P1) and 6.4±1.5×10−13 cm−2 s−1 (P2). Energy (GeV) -110 1 10 210 310 410 -1010 HEGRA HESS J0623+057 3EG J0634+0521 Fig. 3. Reconstructed VHE γ-ray spectrum of HESS J0632+057 compared to the HE γ-ray source 3EGJ0634+0521. An upper limit derived for 3EGJ0634+0521 at TeV energies using the HEGRA instrument is also shown. Amongst the candidate VHE sources in this field is the 34 ms binary pulsar SAXJ0635.2+0533. There is no signif- icant γ-ray emission at the position of this object and we derive a 99% confidence upper limit on the integral flux, F (> 1TeV), of 2.6 × 10−13 cm−2 s−1, assuming an E−2 type spectrum. 3. Possible Associations of HESS J0632+057 The new VHE source HESS J0632+057 lies in a complex region and several associations with objects known at other wavelengths seem plausible. We therefore consider each of these potential counterparts in turn. The Monoceros Loop SNR is rather old in comparison to the known VHE γ-ray shell-type SNRs RXJ1713.7−3946 (Aharonian et al. 2006a), RXJ0852.0−4622 (Aharonian et al. 2005b) and Cas- A (Aharonian et al. 2001). All these objects have estimated ages less than ∼ 2000 years, in contrast the Monoceros Loop SNR has an age of ∼ 3 × 104 years (Leahy et al. 1986). This supernova remnant therefore appears to be in a different evolutionary phase (late Sedov or Radiative) compared to these known VHE sources. However, CR acceleration may occur even at this later evolutionary stage (see for example Yamazaki et al. (2006)). The principal challenge for a scenario involving the Monoceros Loop is to explain the very localised VHE emission at only one point on the SNR limb. The interaction of the SNR with a compact molecular cloud is one possible solution. In this scenario (and indeed any π0 decay scenario) for the observed γ-ray emission, a correlation is expected between the TeV emission and the distribution of target material. An unresolved molecular cloud listed in a CO survey at 115 GHz (Oliver et al. 1996) lies rather close to HESS J0623+057, at l = 205.75 b = −1.31. The distance estimate for this cloud (1.6 kpc) is consistent with that for the Monoceros SNR, making it a potential target for hadrons accelerated in the SNR. However, as can be seen clearly in the NANTEN data in Fig. 1, the intensity peak of this cloud is significantly shifted to the East of the H.E.S.S. source. We find no evidence in the NANTEN data for any clouds along the line of sight to the H.E.S.S. source. 3EGJ0634+0521 is an unidentified EGRET source (Hartman et al. 1999) with positional uncertainties such that HESS J0632+057 lies close to the 99% confidence con- tour. Given that this source is flagged as possibly extended or confused, a positional coincidence of these two objects seems plausible. Furthermore, the reported third EGRET catalogue flux above 100 MeV ((25.5± 5.1)× 10−8 photons cm−2 s−1 with a photon index of 2.03 ± 0.26, see Fig. 3), is consistent with an extrapolation of the H.E.S.S. spec- trum. A global fit of the two spectra gives a photon index of 2.41±0.06. 1RXSJ063258.3+054857 is a faint ROSAT source (Voges et al. 2000) which lies 36′′ from the H.E.S.S. source with a positional uncertainty of 21′′ (see Fig. 1 bottom). Given the uncertainties on the positions of both objects this X-ray source can certainly be considered a potential counterpart of HESS J0632+057. The chance probability of the coincidence of a ROSAT Faint Source Catalogue source within the H.E.S.S. error circle is estimated as 0.1% by scaling the total number of sources in the field of view. The ROSAT source is rather weak, with only 4 counts detected above 0.9 keV, spectral comparison is therefore rather diffi- cult. In the scenario where the γ-ray emission is interpreted as inverse Compton emission from a population of energetic electrons, the ROSAT source could be naturally ascribed to the synchrotron emission of the same electron population. However, the low level of the X-ray emission (∼ 10−13 erg cm−2 s−1) in comparison with the TeV flux (∼ 10−12 erg 4 F. A. Aharonian et al.: A point-like γ-ray source in Monoceros cm−2 s−1) implies a very low magnetic field (≪ 3µG) un- less a strong radiation source exists in the neighbourhood of the emission region and/or the X-ray emission suffers from substantial absorption. Observations at > 4 keV are required to resolve this absorption issue. In a π0 decay sce- nario for the γ-ray source, secondary electron production via muon decay is expected along with γ-ray emission. The synchrotron emission of these secondary electrons would in general produce a weaker X-ray source than the IC scenario, probably compatible with the measured ROSAT flux. MWC148 (HD 259440) is a massive emission-line star of spectral type B0pe which lies within the H.E.S.S. error circle. The chance probability of this coincidence is hard to assess, as there was no a-priori selection of stellar objects as potential γ-ray sources. However, given the presence of only 3 Be-type stars in the field of view of the H.E.S.S. observation (see Fig. 1) and the solid angle of the H.E.S.S. error circle, the naive chance probability of the associa- tion is 10−4. Stars of this spectral type have winds with typical velocities and mass loss rates of 1000 km s−1 and 10−7M⊙/year, respectively. Plausible acceleration sites are in strong internal or external shocks of the stellar wind. We estimate that an efficiency of 1-10% in the conversion of the kinetic energy of the wind into γ-ray emission would be required to explain the H.E.S.S. flux (assuming this star lies at the distance of the Rosette Nebula). However, as no associations of similar stars with point-like γ-ray sources were found in the H.E.S.S. survey of the inner Galaxy, this scenario seems rather unlikely. A related possibility is that MWC148 is part of a binary system with an, as yet undetected, compact companion. Such a system might then resemble the known VHE γ-ray source PSRB1259-63/SS2883 (Aharonian et al. 2005a). Further multi-wavelength observations are required to con- firm or refute this scenario. Acknowledgements. The support of the Namibian authorities and of the University of Namibia in facilitating the construction and op- eration of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Particle Physics and Astronomy Research Council (PPARC), the IPNP of the Charles University, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and oper- ation of the equipment. The NANTEN project is financially supported from JSPS (Japan Society for the Promotion of Science) Core-to-Core Program, MEXT Grant-in-Aid for Scientific Research on Priority Areas, and SORST-JST (Solution Oriented Research for Science and Technology: Japan Science and Technology Agency). We would also like to thank Stan Owocki and James Urquhart for very useful dis- cussions. References Aharonian, F., Akhperjanian, A., Barrio, J., et al. 2001, A&A, 370, Aharonian, F., Akhperjanian, A. G., Aye, K.-M., et al. 2005a, A&A, 442, 1 Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006a, A&A, 449, 223 Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2006b, Nature, 439, 695 Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. 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Brainerd, 59 Oliver, R. J., Masheder, M. R. W., & Thaddeus, P. 1996, A&A, 315, Voges, W., Aschenbach, B., Boller, T., et al. 2000, IAU Circ., 7432, 1 Yamazaki, R., Kohri, K., Bamba, A., et al. 2006, MNRAS, 371, 1975 1 Max-Planck-Institut für Kernphysik, P.O. Box 103980, D 69029 Heidelberg, Germany 2 Yerevan Physics Institute, 2 Alikhanian Brothers St., 375036 Yerevan, Armenia 3 Centre d’Etude Spatiale des Rayonnements, CNRS/UPS, 9 av. du Colonel Roche, BP 4346, F-31029 Toulouse Cedex 4, France 4 Universität Hamburg, Institut für Experimentalphysik, Luruper Chaussee 149, D 22761 Hamburg, Germany 5 Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D 12489 Berlin, Germany 6 LUTH, UMR 8102 du CNRS, Observatoire de Paris, Section de Meudon, F-92195 Meudon Cedex, France 7 DAPNIA/DSM/CEA, CE Saclay, F-91191 Gif-sur-Yvette, Cedex, France 8 University of Durham, Department of Physics, South Road, Durham DH1 3LE, U.K. 9 Unit for Space Physics, North-West University, Potchefstroom 2520, South Africa 10 Laboratoire Leprince-Ringuet, IN2P3/CNRS, Ecole Polytechnique, F-91128 Palaiseau, France 11 Laboratoire d’Annecy-le-Vieux de Physique des Particules, IN2P3/CNRS, 9 Chemin de Bellevue - BP 110 F-74941 Annecy-le-Vieux Cedex, France 12 APC, 11 Place Marcelin Berthelot, F-75231 Paris Cedex 05, France UMR 7164 (CNRS, Université Paris VII, CEA, Observatoire de Paris) 13 Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland 14 Landessternwarte, Universität Heidelberg, Königstuhl, D 69117 Heidelberg, Germany 15 Laboratoire de Physique Théorique et Astroparticules, IN2P3/CNRS, Université Montpellier II, CC 70, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France 16 Universität Erlangen-Nürnberg, Physikalisches Institut, Erwin-Rommel-Str. 1, D 91058 Erlangen, Germany 17 Laboratoire d’Astrophysique de Grenoble, INSU/CNRS, Université Joseph Fourier, BP 53, F-38041 Grenoble Cedex 9, France 18 Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, D 72076 Tübingen, Germany 19 Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Universités Paris VI & VII, 4 Place Jussieu, F-75252 Paris Cedex 5, France 20 Institute of Particle and Nuclear Physics, Charles University, V Holesovickach 2, 180 00 Prague 8, Czech Republic 21 Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und Astrophysik, Ruhr-Universität Bochum, D 44780 F. A. Aharonian et al.: A point-like γ-ray source in Monoceros 5 Bochum, Germany 22 University of Namibia, Private Bag 13301, Windhoek, Namibia 23 European Associated Laboratory for Gamma-Ray Astronomy, jointly supported by CNRS and MPG 24 Department of Astrophysics, Nagoya University, Chikusa- ku, Nagoya 464-8602, Japan 25 Nagoya University Southern Observatories, Nagoya 464- 8602, Japan Introduction H.E.S.S. Observations and Results Possible Associations of HESSJ0632+057

0704.0172

Thermal entanglement of qubit pairs on the Shastry-Sutherland lattice

Thermal entanglement of qubit pairs on the Shastry-Sutherland lattice S. El Shawish J. Stefan Institute, Ljubljana, Slovenia A. Ramšak and J. Bonča Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia and J. Stefan Institute, Ljubljana, Slovenia (Dated: 2 April 2007) We show that temperature and magnetic field properties of the entanglement between spins on the two- dimensional Shastry-Sutherland lattice can be qualitatively described by analytical results for a qubit tetramer. Exact diagonalization of clusters with up to 20 sites reveals that the regime of fully entangled neighboring pairs coincides with the regime of finite spin gap in the spectrum. Additionally, the results for the regime of vanishing spin gap are discussed and related to the Heisenberg limit of the model. PACS numbers: 75.10.Jm, 03.65.Yz, 03.67.Mn I. INTRODUCTION In any physical system with subsystems in interaction, indi- vidual parts of the system are to some extent entangled, even if they are far apart, as realized already at the beginning of modern quantum mechanics sixty years ago. Today it has be- come appreciated that the ability to establish entanglement be- tween quantum particles in a controlled manner is a crucial in- gredient of any quantum information processing system1. On the other hand, it turned out that the analysis of appropriately quantified entanglement between parts of the system can also be a very useful tool in the study of many body phenomena, as is, e.g., the behavior of correlated systems in the vicinity of crossovers between various regimes or even points of quantum phase transition2. Quantum entanglement of two distinguishable particles in a pure state can be quantified through von Neuman entropy3,4,5. Entanglement between two spin- 1 particles – qubit pair – can be considered a physical resource, an essential ingredient of algorithms suitable for quantum computation. For a pair of subsystems A and B, each occupied by a single electron, an appropriate entanglement measure is the entanglement of for- mation, which can be quantified from the Wootters formula6. In general, electron-qubits have the potential for even richer variety of entanglement measure choices due to both their charge and spin degrees of freedom. When entanglement is quantified in systems of indistinguishable particles, the mea- sure must account for the effect of exchange and it must ade- quately deal with multiple occupancy states7,8,9,10,11,12. A typ- ical example is the analysis of entanglement in lattice fermion models (the Hubbard model, e.g.) where double occupancy plays an essential role11. In realistic hardware designed for quantum information processing, several criteria for qubits must be fulfilled13: the existence of multiple identifiable qubits, the ability to initial- ize and manipulate qubits, small decoherence, and the ability to measure qubits, i.e., to determine the outcome of compu- tation. It seems that among several proposals for experimen- tal realizations of such quantum information processing sys- tems the criteria for scalable qubits can be met in solid state structures consisting of coupled quantum dots14,15. Due to the ability to precisely control the number of electrons in such structures16, the entanglement has become experimentally ac- cessible quantity. In particular, recent experiments on semi- conductor double quantum dot devices have shown the evi- dence of spin entangled states in GaAs based heterostuctures17 and it was shown that vertical-lateral double quantum dots may be useful for achieving two-electron spin entanglement18. It was also demonstrated recently that in double quantum dot systems coherent qubit manipulation and projective readout is possible19. Qubit pairs to be used for quantum information processing must be to a high degree isolated from their environment, oth- erwise small decoherence requirement from the DiVincenzo’s checklist can not be fulfilled. The entanglement, e.g., between two antiferromagnetically coupled spins in contact with ther- mal bath, is decreased at elevated temperatures and external magnetic field20,21,22, and will inevitably vanish at some fi- nite temperature23. Entanglement of a pair of electrons that are confined in a double quantum dot is collapsed due to the Kondo effect at low temperatures and for a very weak tunnel- ing to the leads. At temperatures below the Kondo temper- ature a spin-singlet state is formed between a confined elec- tron and conduction electrons in the leads24. For other open systems there are many possible sources of decoherence or phase-breaking, for example coupling to phonon degrees of freedom25. The main purpose of the present paper is to analyze the ro- bustness of the entanglement of spin qubit pairs in a planar lattice of spins (qubits) with respect to frustration in magnetic couplings, elevated temperatures as well as due to increasing external magnetic field. The paper is organized as follows. Sec. II introduces the model for two coupled qubit pairs – qubit tetramer – and presents exact results for temperature and magnetic field dependence of the entanglement between near- est and next-nearest-neighboring spins in a tetrahedron topol- ogy. In Sec. III the model is extended to infinite lattice of qubit pairs described by the Shastry-Sutherland model26. This model is convenient firstly, because of the existence of sta- ble spin-singlet pairs in the ground state in the limit of weak coupling between the qubit pairs, and secondly, due to a rel- atively good understanding of the physics of the model in the http://arxiv.org/abs/0704.0172v1 thermodynamic limit. Entanglement properties of the Shastry- Sutherland model were so far not considered quantitatively. Neverteless, several results concerning the role of entangle- ment at a phase transition in other low-dimensional spin lattice systems2,27,28,29,30,31,32, as well as in fermionic systems33,34,35 have been reported recently. Near a quantum phase transi- tion in some cases entanglement even proves to be more ef- ficient precursor of the transition compared to standard spin- spin correlations35,36. In Sec. IV we discuss entanglement be- tween nearest neighbors in the Heisenberg model, represent- ing a limiting case of the Shastry-Sutherland model. Results are summarized in Sec. V and some technical details are given in Appendix A. II. THERMAL ENTANGLEMENT OF A QUBIT TETRAMER IN MAGNETIC FIELD Consider first a double quantum dot composed of two adja- cent quantum dots weakly coupled via a controllable electron- hopping integral. By adjusting a global back-gate voltage, precisely two electrons can be confined to the dots. The inter- dot tunneling matrix element t determines the effective anti- ferromagnetic (AFM) superexchange interaction J ∼ 4t2/U , where U is the scale of Coulomb interaction between two electrons confined on the same dot. There are several possi- ble configurations of coupling between such double quantum dots. One of the simplest specific designs is shown schemat- ically in Fig. 1(a): four qubits at vertices of a tetrahedron. In addition to the coupling A-B, by appropriate arrangements of gate electrodes the tunneling between A-C and A-D can as well be switched on. We consider here the case where J/U ≪ 1, thus double oc- cupancy of individual dot is negligible and appropriate Hilbert space is spanned by two dimers (qubit pairs): spins at sites A-B and C-D are coupled by effective AFM Heisenberg mag- netic exchange J and at sites A-C, B-C, A-D, B-D by J ′. The corresponding hamiltonian of such a pair of dimers is given as H4 = J(SA · SB + SC · SD) + (1) + 2J ′(SA · SC + SB · SC + SA · SD + SB · SD)− − B(SzA + SzB + SzC + SzD), where Si = σi is spin operator corresponding to the site i and B is external homogeneous magnetic field in the direction of the z-axis. Factor 2 in Eq. (1) is introduced for convenience – such a parameterization represents the simplest case of finite Shastry-Sutherland lattice with periodic boundary conditions studied in Sec. III. A. Concurrence We focus here on the entanglement properties of two cou- pled qubit dimers. The entanglement of a pair of spin qubits A and B may be defined through concurrence3, C = 2|α↑↑α↓↓ − α↑↓α↓↑|, if the system is in a pure state |ΨAB〉 = ss′ αss′ |s〉A|s′〉B, where |s〉i corresponds to the basis | ↑ 〉i, Figure 1: (Color online) (a) Two coupled qubit pairs (dimers) in tetra- hedral topology. (b) Shastry-Sutherland lattice as realized, e.g., in the SrCu2(BO3)2 compound. | ↓ 〉i. Concurrence varies from C = 0 for an unentan- gled state (for example | ↑ 〉A| ↑ 〉B) to C = 1 for com- pletely entangled Bell states3 1√ (| ↑ 〉A| ↑ 〉B ± | ↓ 〉A| ↓ 〉B) or 1√ (|↑ 〉A|↓ 〉B ± |↓ 〉A|↑ 〉B). For finite inter-pair coupling J ′ 6= 0 or at elevated tempera- tures the A-B pair can not be described by a pure state. In the case of mixed states describing the subsystem A-B the concur- rence may be calculated from the reduced density matrix ρAB given in the standard basis |s〉i|s′〉j6. Concurrence can be fur- ther expressed in terms of spin-spin correlation functions2,27, where for systems that are axially symmetric in the spin space the concurrence may conveniently be given in a simple closed form37, which for the thermal equilibrium case simplifies fur- ther, CAB = 2max(0, |〈S+AS 〉〈P ↓ 〉). (2) Here S+i = (S † = Sxi + ıS i is the spin raising operator for dot i and P ↑i = (1 + 2Szi ), P (1 − 2Szi ) are the projection operators onto the state | ↑ 〉i or | ↓ 〉i, respectively. We consider the concurrence at fixed temperature, therefore the expectation values in the concurrence formula Eq. (2) are evaluated as 〈O〉 = 〈n|O|n〉e−βEn , (3) whereZ = −βEn is the partition function, β = 1/T , and {|n〉} is a complete set of states of the system. Note that due to the equilibrium and symmetries of the system, several spin- spin correlation functions vanish, 〈S+ 〉 = 0, for example. Figure 2: (Color online) (a) Zero-temperature concurrence CAB as a function of J ′/J and B/J . Different regimes are characterized by particular ground state functions |φn〉 defined in Appendix A. (b) T/J = 0.1 results for CAB. (c) Next nearest concurrence CAC for T = 0, and (d) for T/J = 0.1. Dashed lines separate CAB(C) > 0 from CAB(C) = 0. In vanishing magnetic field, where the SU(2) symmetry is restored, the concurrence formula Eq. (2) simplifies further and is completely determined by only one38 spin invariant 〈SA · SB〉, CAB = max(0,−2〈SA · SB〉 − ). (4) The concurrence may be expected to be significant whenever enhanced spin-spin correlations indicate A-B singlet forma- tion. B. Analytical results There are several known results related to the model Eq. (1). In the special case of J ′ = 0, for example, the tetramer con- sists of two decoupled spin dimers with concurrence CAB (or the corresponding thermal entanglement) as derived in Refs. 20,21. Entanglement of a qubit pair described by the related XXZ Heisenberg model with Dzyaloshinskii-Moriya anisotropic interaction can be also obtained analytically22. Hamiltonian H4 with additional four-spin exchange interac- tion but in the absence of magnetic field was considered re- cently in the various limiting cases39. Tetramer model Eq. (1) considered here is exactly solvable and in Appendix A we present the corresponding eigenvec- tors and eigenenergies. The concurrence CAB is for this case determined from Eq. (2) with 〉 = 1 − e3j/2/2− ej/2 eb + e−b + e−j/2+4j /6 + e−j/2+2j eb + e−b + e−j/2−2j eb/4 + 1/3 + e−b/4 , (5) where j = βJ , j′ = βJ ′, b = βB, and with 〈P ↑↓ 〉 = 1 + e−j/2+4j + e−j/2+2j 1 + e±b + e−j/2−2j 1/6 + e±b/2 + e±2b . (6) Z = e3j/2 + 2ej/2 eb + 1 + e−b + e−j/2+4j + e−j/2+2j eb + 1 + e−b + e−j/2−2j e2b + eb + 1 + e−b + e−2b is the partition function. Alternatively, one can define and analyze also the entangle- ment between spins at sites A and C and the corresponding concurrence CAC can be expressed from Eq. (2) by applying Figure 3: (Color online) (a) Temperature and magnetic field depen- dence of CAB for J ′/J = 0.4 and (b) J ′ = J . Dashed lines separate CAB > 0 from CAB = 0. additional correlators with replaced B→C, − e−j/2+4j − e−j/2+2j eb + e−b + e−j/2−2j eb/4 + 1/3 + e−b/4 , (8) 〈P ↑↓ 〉 = 1 e3j/2/4 + ej/2 1/2 + e±b + e−j/2+4j /12 + e−j/2+2j e±b/2 + e−j/2−2j 1/6 + e±b/2 + e±2b . (9) The line 2J ′ = J represents a particularly interesting spe- cial case where two dimers are coupled symmetrically form- ing a regular tetrahedron. An important property of this sys- tem is the (geometrical) frustration of, e.g., qubits C-A-B. Such a frustration is the driving force of the quantum phase transition found in the Shastry-Sutherland model and is the reason for similarity of the results for two coupled dimers and a large planar lattice studied in the next Section. C. Examples In the low temperature limit the concurrence is determined by the ground state properties while transitions between var- ious regimes are determined solely by crossings of eigenen- ergies, which depend on two parameters (J ′/J,B/J). There are 5 distinct regimes for CAB shown in Fig. 2(a): (i) com- pletely entangled dimers (singlets A-B and C-D, state |φ1〉 from Appendix A), CAB = 1; (ii) for B > J and smaller J ′/J the concurrence is zero because the energy of the state consisted of a product of fully polarized A-B and C-D triplets, |φ12〉, is the lowest energy in this regime; (iii) concurrence is zero also for J ′ > J/2 and low B/J , with the ground state |φ2〉. There are two regimes corresponding to 12 step in CAB where the ground state is either (iv) any linear combina- tion of degenerate states |φ6,7〉, i.e., simultaneous A-B singlet (triplet) and C-D triplet (singlet) for J ′ < J/2, or (v) state |φ5〉 at J ′ > J/2 and larger B. Qubits A-C are due to special topology never fully entangled, and the corresponding CAC is presented in Fig. 2(c). In the limit of J ′ ≫ J the tetramer corresponds to a Heisenberg model ring consisted of 4 spins and in this case qubit A is due to tetramer symmetry equally entangled to both neighbors (C and D), thus CAC = At elevated temperatures the concurrence is smeared out as shown in Figs. 2(b,d). Note the dip separating the two dif- ferent regimes with CAB = , seen also in the CAC = case. This dip clearly separates different regimes discussed in the previous T = 0 limit and signals a proximity of a disen- tangled excited state. For sufficiently high temperatures van- ishing concurrence is expected23. The critical temperature Tc denoted by a dashed line is set by the magnetic exchange scale J , since at higher temperatures local singlets are broken irre- spectively of the magnetic field. A rather unexpected result is shown in Fig. 3(a) where at B & 2J and low temperatures the concurrence slightly in- creases with increasing temperature due to the contribution of excited A-B singlet components that are absent in the ground state. Similar behavior is found for J ′ ∼ 0 around B ∼ J , which is equivalent to the case of a single qubit dimer20,21 (not shown here). There is no distinctive feature in tempera- ture and magnetic field dependence of CAB when J ′ > J/2 and a typical results is shown in Fig. 3(b) for J ′ > J . III. PLANAR ARRAY OF QUBIT PAIRS: THE SHASTRY-SUTHERLAND LATTICE A. Preliminaries The central point of this paper is the analysis of pair en- tanglement for the case of a larger number of coupled qubit pairs. In the following it will be shown that the results corre- sponding to tetramers considered in the previous Section can be very helpful for better understanding pair-entanglement of N > 4 qubits. There are several possible generalizations of coupled dimers and one of the simplest in two dimensions is the Shastry-Sutherland lattice shown in Fig. 1(b). Neighbor- ing sites A-B are connected with exchange interaction J and next-neighbors with J ′. The corresponding hamiltonian for N/2 dimers (N sites) is given with HN = J Si · Sj + J ′ Si · Sj −B Szi . (10) Periodic boundary conditions are used. For the special case N = 4 the model reduces to Eq. (1) where due to periodic boundary conditions sites A-C (and other equivalent pairs) are doubly connected, therefore a factor of 2 in Eq. (1), as men- tioned in Sec. II. The Shastry-Sutherland model (SSM) was initially pro- posed as a toy model possessing an exact dimerized eigenstate known as a valence bond crystal26. Recently, the model has experienced a sudden revival of interest by the discovery of the two-dimensional spin-liquid compound SrCu2(BO3)2 40,41 since it is believed that magnetic properties of this compound are reasonably well described by the SSM42. In fact, several generalizations of the SSM have been introduced to account better for recent high-resolution measurements revealing the magnetic fine structure of SrCu2(BO3)2 42,43,44,45. Soon af- ter the discovery of the SrCu2(BO3)2 system, the SSM thus became a focal point of theoretical investigations in the field of frustrated AFM spin systems, particularly low-dimensional quantum spin systems where quantum fluctuations lead to magnetically disordered ground states (spin liquids) with a spin gap in the excitation spectrum. The SSM is a two-dimensional frustrated antiferromagnet with a unique spin-rotation invariant exchange topology that leads in the limit J ≫ J ′ to an exact gapped dimerized ground state with localized spin singlets on the dimer bonds (dimer phase). In the opposite limit, J ≪ J ′, the model becomes ordinary AFM Heisenberg model with a long-range Néel or- der and a gapless spectrum (Néel phase). While two of the phases are known, there are still open questions regarding the existence and the nature of the intermediate phases. Several possible scenarios have been proposed, e.g.: either a direct transition between the two states occurs at the quantum criti- cal point near J ′/J ∼ 0.746,47, or a transition via an interme- diate phase that exists somewhere in the range of J ′/J > 0.6 and J ′/J < 0.948. Although different theoretical approaches have been applied, a true nature of the intermediate phase (if any) has still not been settled. As will be evident later on, our exact-diagonalization results support the first scenario. The SSM phase diagram reveals interesting behavior also for varying external magnetic field. In particular, experiments on SrCu2(BO3)2 in strong magnetic fields show formation of magnetization plateaus41,49, which are believed to be a con- sequence of repulsive interaction between almost localized spin triplets. Several theoretical approaches support the idea that most of these plateaus are readily explained within the (bare) SSM46,50,51. Recent variational treatment based on en- tangled spin pairs revealed new insight into various phases of the SSM48. Although extensively studied, the zero-temperature phase diagram of the SSM remains elusive. This lack of reliable solutions is even more pronounced when considering thermal fluctuations in SSM as only few methods allow for the inclu- 0 0.2 0.4 0.6 0.8 1 T/J=0.1-2 <S > - 1/2. -2 <S > - 1/2 AFM limit dimer limit Figure 4: (Color online) Results for the Shastry-Sutherland lattice with N = 20 sites and periodic boundary conditions. Presented are renormalized spin-spin correlation functions −2〈SA · SB,C〉 − 12 as a function of J ′/J and for various temperatures. Asterisk indicates critical J ′c which roughly separates the dimer and Néel phase. sion of finite temperatures in frustrated spin systems. In this respect, the calculation of thermal entanglement between the spin pairs would also provide a new insight into the complex- ity of the SSM. B. Numerical method We use the low-temperature Lanczos method52 (LTLM), an extension of the finite-temperature Lanczos method53 (FTLM) for the calculation of static correlation functions at low tem- peratures. Both methods are nonperturbative, based on the Lanczos procedure of exact diagonalization and random sam- pling over different initial wave functions. A main advan- tage of LTLM is that it accurately connects zero- and finite- temperature regimes with rather small numerical effort in comparison to FTLM. On the other hand, while FTLM is lim- ited in reaching arbitrary low temperatures on finite systems, it proves to be computationally more efficient at higher tem- peratures. A combination of both methods therefore provides reliable results in a wide temperature regime with moderate computational effort. We note that FTLM was in the past suc- cessfully used in obtaining thermodynamic as well as dynamic properties of different models with correlated electrons as are: the t-J model,53 the Hubbard model,54 as well as the SSM model.43,45 In comparison with the conventional Quantum Monte Carlo (QMC) methods LTLM possesses the following advantages: (i) it does not suffer from the minus-sign problem that usually hampers QMC calculations of many-electron as well as frus- trated spin systems, (ii) the method continuously connects the zero- and finite-temperature regimes, (iii) it incorporates as well as takes the advantage of the symmetries of the prob- lem, and (iv) it yields results of dynamic properties in the real time in contrast to QMC calculations where imaginary- time Green’s function is obtained. The LTLM (FTLM) is on the other hand limited to small lattices which usually leads to sizable finite-size effects. To account for these, we ap- Figure 5: (Color online) (a) Zero-temperature concurrence CAB for a 20-site cluster for various J ′/J and B/J . Shaded area represents the regime of fully entangled dimers, CAB = 1. (b) The corresponding results for T/J = 0.1. (c) Next nearest concurrence CAC for T = 0, and (d) for T/J = 0.1. Note qualitative and even quantitative similarity with the tetramer results, Fig. 2. Dashed lines separate CAB(C) > 0 from CAB(C) = 0. plied LTLM to different square lattices with N = 8, 16 and 20 sites using periodic boundary conditions (we note that next-larger system, N = 32, was too large to be handled numerically). Another drawback of the LTLM (FTLM) is the difficulty of the Lanczos procedure to resolve degener- ate eigenstates that emerge also in the SSM. In practice, this manifests itself in severe statistical fluctuations of the cal- culated amplitude for T → 0 since in this regime only a few (degenerate) eigenstates contribute to thermal average. The simplest way to overcome this is to take a larger num- ber of random samples R ≫ 1, which, however, requires a longer CPU time. We have, in this regard, also included a small portion of anisotropy in the SSM (in the form of the anisotropic interdimer Dzyaloshinskii-Moriya interaction {AC}(S C − S C), D z/J ∼ 0.01), which slightly splits the doubly degenerate single-triplet levels. In this way, R ∼ 30 per Sz sector was enough for all calculated curves to converge within ∼ 1% for T/J < 1. Here, the number of Lanczos iterations M = 100 was used along with the full reorthogonalization of Lanczos vectors at each step. C. Entanglement Entanglement in the absence of magnetic field is most prominently reflected in spin-spin correlation functions, e.g., 〈SA ·SB〉 and 〈SA ·SC〉. In zero temperature limit due to quan- tum phase transition at J ′c these correlations change sign. In Fig. 4 are presented renormalized spin-spin correlation func- tions (for positive values identical to concurrence) as a func- tion of J ′/J : (i) CAB > 0 in dimer phase and (ii) CAC > 0 in the Néel phase. Critical J ′c is indicated by asterisk. The re- sults for N = 16 are qualitatively and quantitatively similar to the N = 20 case presented here. At finite temperatures spin correlations are smeared out as shown in Fig. 4 for various T . Limiting Heisenberg case, J ′ → ∞, is discussed in more de- tail in the next Section. J ′ = 0 case corresponds to the single dimer limit21 and Sec. II. Complete phase diagram of the SSM at T = 0 but with fi- nite magnetic field can be classified in terms of concurrence instead of spin correlations. In Fig. 5(a)CAB is presented as a function of (J ′/J,B/J) as in the case of a single tetramer, Fig. 2(a). Presented results correspond to the N = 20 case, while N = 16 system exhibits very similar structure (not shown here). N = 8 and N = 4 cases are qualitatively sim- ilar, the main difference being the value of critical J ′c which increases with N . Remarkable similarity between all these cases can be interpreted by local physics in the regime of fi- nite spin gap, J ′ < J ′c. Qubit pairs are there completely entan- gled, CAB = 1, and CAB ∼ 12 for magnetic field larger than the spin gap, but B < J+2J ′. For even larger B concurrence approaches zero, similar to the N = 4 case. Concurrence is Figure 6: (Color online) Temperature and magnetic field dependence of CAB for J ′/J = 0.4 and N = 20. Note the similarity with the corresponding tetramer results, Fig. 3(a). Dashed lines separate CAB > 0 from CAB = 0. zero also for J ′ > J ′c, except along the B ∼ 4J ′ line where weak finite concurrence could be the finite size effect. Similar results are found also for N = 16, 8 cases, and are most pro- nounced in the N = 4 case. At finite temperature the structure of concurrence is smeared out [Fig. 5(b)] similar to Fig. 2(b). Concurrence CAC corresponding to next-nearest neighbors is, complementary to CAB, increased in the Néel phase of the diagram, Fig. 5(c). The similarity with N = 4, Fig. 2(c) is somewhat surprising because in this regime long-range cor- relations corresponding to the gapless spectrum of AFM-like physics are expected to change also short range correlations. The only quantitative difference compared to N = 4 is the maximum value of CAC ∼ 0.3 instead of 0.5 (beside the crit- ical value J ′c discussed in the previous paragraph). Concur- rence is very small for B > J + 2J ′. At finite temperatures fine fluctuations in the concurrence structure are smeared out, Fig. 5(d). Temperature and magnetic field dependence of CAB in the dimer phase is presented in Fig. 6 for fixed J ′/J = 0.4. Sim- ilarity with the corresponding N = 4 tetramer case, Fig. 3(a), is astonishing and is again the consequence of local physics in the presence of a finite spin gap. Finite size effects (in com- parison with N = 16 and N = 8 cases) are very small (not shown). Dashed line represents the borderline of the CAB = 0 region: critical Tc ≈ 0.75J valid for B/J . 3, that is in this regime nearly independent of B, is slightly larger than in the single tetramer case where its insensitivity to B is even more pronounced. IV. HEISENBERG LIMIT The concurrence corresponding to next-nearest neighbors in SSM, CAC, is non zero in the Néel phase for J ′ > J ′c. Typical result for concurrence in this regime (for fixed J ′/J = 1) in terms of temperature and magnetic field is presented in Fig. 7(a). At zero temperature the concurrence is zero for B > 4J ′ [compare with Fig. 2(c) and Fig. 5(c)]. Figure 7: (Color online) (a) Next nearest neighbor concurrence CAC for J ′ = J . (b) Heisenberg lattice result as a special case of the SSM, J = 0. Shaded region represents CAC = 0. In the line shaded region (low finite temperature and large magnetic field) our numer- ical results set only the upper limit CAC < 5 · 10 −4. Dashed lines separate CAC > 0 from CAC = 0. In the limit J = 0 the model simplifies to the AFM Heisen- berg model on a square lattice of N sites, HAC = J Si · Sj −B Szi . (11) Several results for this model have already been presented for very small clusters55,56,57, however the temperature and mag- netic field dependence of the concurrence for systems with sufficiently large number of states and approaching thermo- dynamic limit has not been presented so far. In Fig. 7(b) we further presented temperature and magnetic field dependence of concurrence for the Heisenberg model for N = 20 (results for N = 16 are quantitatively similar, but not shown here). Temperature and magnetic field dependence of CAC exhibits peculiar semi-island shape where at fixed value of B the concurrence increases with increasing temperature. This effect is to some extent seen in all cases and is the con- sequence of exciting local singlet states, which do not appear in the ground state. At T → 0 finite steps with increasing B correspond to gradual transition from the singlet ground state to totally polarized state with total spin S = 10 and vanish- ing concurrence. This is in more detail presented in Fig. 8(a) for various N = 4, 8, 16, 20. At B = 0 and for N = 20 we get CAC = 0.19. It is interesting to compare this results 0 0.05 0.1 /2N=8 1/6+2N -3/2C Figure 8: (Color online) (a) Zero-temperature concurrence CAC in the Heisenberg limit as a function of B/J ′ and for various N = 4, 8, 16, 20. Sections with different total spin values are addition- ally labeled. (b) Finite-size scaling of concurrence in the absence of magnetic field. Full line represents the fit corresponding to Ref. 58, CAC ≈ + 2N−3/2. with the known finite-size analysis scaling for the ground state energy of the Heisenberg model58. The same scaling gives CAC ≈ 16 + 2N −3/2. Our finite-size scaling, Fig. 8(b), is in perfect agreement with this result for N → ∞ at T = 0 and B = 0. In the opposite limit of high magnetic fields, the vanishing concurrence CAC = 0, is observed for B above the critical value Bc = 4J ′ for all system sizes shown in Fig. 8(a). This result can be deduced also analytically. Since in a fully po- larized state CAC = 0, this Bc actually denotes a transition from S1 = N/2 − 1 to S0 = N/2 ferromagnetic ground state with energy E0 = N(J ′ − B)/2. The energy of the one-magnon excitation above the ferromagnetic ground state is given by the spin wave theory, which is in this case exact, as E1 = E0 −J ′(2− cos kxa− cos kya)+B, where (kx, ky) is the magnon wave vector and a denotes the lattice spacing. Ev- idently, a transition to a fully polarized state occurs precisely at Bc = 4J ′ at (π/a, π/a) point in the one-magnon Brillouin zone. V. SUMMARY The aim of this paper was to analyze and understand how concurrence (and related entanglement) of qubit pairs (dimers) is affected by their mutual magnetic interactions. In particular, we were interested in a planar array of qubit dimers described by the Shastry-Sutherland model. This model is suitable due to very robust ground state composed of entan- gled qubit pairs which breaks down by increasing the in- terdimer coupling. It is interesting to study both, the en- tanglement between nearest and between next-nearest spins (qubits) at finite temperature and magnetic field. The results are based on numerical calculations using low-temperature Lanczos methods on lattices of 4, 8, 16 and 20 sites with pe- riodic boundary conditions. A comprehensive analysis of concurrence for various pa- rameters revealed two general conclusions: (1) For a weak coupling between qubit dimers, J ′ < J ′c, qubit pairs are locally entangled in accordance with the local nature of the dimer phase. This is due to a finite singlet-triplet gap (spin gap) in the excitation spectrum that is a consequence of strong geometrical frustration in magnetic couplings. The regime of fully entangled neighbors perfectly coincides with the regime of finite spin gap as presented in Fig. 9. Calcu- lated lines for various system sizes N in Fig. 9(a) denote re- gions (shaded for N = 20) in the (J ′/J,B/J) plane where CAB = 1 at T = 0. In the lower panel [Fig. 9(b)] the lines represent the energy gap E1 − EGS between the first excited state with energy E1 with total spin projection S z = 1 and the ground state with energy EGS and total spin projection Sz = 0, calculated for B = 0. For J ′ < J ′c (full lines) E1 − EGS corresponds to the value of the spin gap. With an increasing magnetic field the spin gap closes (shaded region for N = 20) and eventually vanishes at the CAB = 1 border line. Shaded regions in Figs. 9(a),(b) therefore coincide. Note also that the results for N = 16 and 20 sites differ mainly in J ′c. As a consequence of finite spin gap and local character of correlations it is an interesting observation that even N = 4 results as a function of temperature and magnetic field quali- tatively correctly reproduce N = 20 results in the regime of J ′ < J ′c. The main quantitative difference is in a renormal- ized value of J ′c = J/2 for N = 4, as is evident from the comparison of Figs.2,3 and Figs.5,6. This similarity of the re- sults appears very useful due to the fact that concurrence for tetrahedron-like systems (N = 4) is given analytically (Sec. (2) In the opposite, strong interdimer coupling regime, J ′ > J ′c, the excitation spectrum is gapless and the concurrence be- tween next-nearest qubits, CAC, exhibits a similar behavior as in the antiferromagnetic Heisenberg model J/J ′ → 0. Our B = 0 results coincide with the known result extrapolated to the thermodynamic limit CAC ≈ 16 . In finite magnetic field and T = 0 the concurrence vanishes at Bc = 4J ′ when the system becomes fully polarized (ground state with the total spin S = N/2). However, at elevated temperatures the con- currence increases due to excited singlet states and eventually drops to zero at temperatures above Tc ≈ J ′. We can conclude with the observation that our analysis of concurrence and related entanglement between qubit pairs was also found to be a very useful measure for classify- ing various phases of the Shastry-Sutherland model. As our numerical method is based on relatively small clusters, we were unable to unambiguously determine possible interme- diate phases of the model in the regime J ′ ∼ J ′c, but we be- lieve that concurrence will prove to be a useful probe for the classification of various phases also in this regime using alter- native approaches. However, we were able to sweep through all other dominant regimes of the parameters including finite temperature and magnetic field. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 singlet - triplet gap Figure 9: (Color online) (a) Zero-temperature CAB = 1 region in the plane (J ′/J, B/J) for various N . (b) The corresponding spin gap at B = 0 (the energy of the lowest total Sz = 1 state relative to the ground state energy). VI. ACKNOWLEDGMENTS The authors acknowledge J. Mravlje for useful discussions and the support from the Slovenian Research Agency under Contract No. P1-0044. Appendix A: EIGENENERGIES AND EIGENVECTORS FOR PERIODICALLY COUPLED TWO QUBIT DIMERS Consider two qubit dimers coupled into a tetramer and de- scribed with the Hamiltonian Eq. (1) and Fig. 1(a). The model is exactly solvable in the separate {S, Sz} subspaces corre- sponding to different values of the total spin S and its z com- ponent Sz . Following the abbreviations for singlet and triplet states on nearest-neighbor (dimer) sites i and j, |sij〉 = |↑i↓j − ↓i↑j〉, |t0ij〉 = |↑i↓j + ↓i↑j〉, |t+ij〉 = |↑i↑j〉, |t−ij〉 = |↓i↓j〉, (A1) the resulting eigenstates |φk〉 and eigenenergies Ek corre- sponding to the hamiltonian Eq. (1) are: S = 0, Sz = 0 : |φ1〉 = |sAB〉|sCD〉, E1 = −3J/2, (A2) |φ2〉 = − |t0AB〉|t0CD〉+ |t+AB〉|t 〉+ |t− E2 = J/2− 4J ′. (A3) S = 1, Sz = −1 : |φ3〉 = |sAB〉|t−CD〉, |φ4〉 = |t−AB〉|sCD〉, E3,4 = −J/2−B, (A4) |φ5〉 = |t0AB〉|t 〉 − |t− 〉|t0CD〉 E5 = J/2− 2J ′ −B. (A5) S = 1, Sz = 0 : |φ6〉 = |sAB〉|t0CD〉, |φ7〉 = |t0AB〉|sCD〉, E6,7 = −J/2, (A6) |φ8〉 = 〉 − |t− E8 = J/2− 2J ′. (A7) S = 1, Sz = 1 : |φ9〉 = |sAB〉|t+CD〉, |φ10〉 = |t+AB〉|sCD〉, E9,10 = −J/2 +B, (A8) |φ11〉 = − |t0AB〉|t+CD〉+ |t 〉|t0CD〉 E11 = J/2− 2J ′ +B. (A9) S = 2, Sz = −2 : |φ12〉 = |t−AB〉|t E12 = J/2 + 2J ′ − 2B. (A10) S = 2, Sz = −1 : |φ13〉 = |t0AB〉|t−CD〉+ |t 〉|t0CD〉 E13 = J/2 + 2J ′ −B. (A11) S = 2, Sz = 0 : |φ14〉 = 2|t0AB〉|t0CD〉+ |t+AB〉|t 〉+ |t− E14 = J/2 + 2J ′. (A12) S = 2, Sz = 1 : |φ15〉 = |t0AB〉|t+CD〉+ |t 〉|t0CD〉 E15 = J/2 + 2J ′ +B. (A13) S = 2, Sz = 2 : |φ16〉 = |t+AB〉|t E16 = J/2 + 2J ′ + 2B. (A14) 1 M. A. Nielsen and I. A. Chuang, Quantum Information and Quantum Computation (Cambridge University Press, Cambridge, 2001). 2 A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002). 3 C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996); C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W.K. Wootters, ibid. 54, 3824 (1996). 4 S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997). 5 V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. 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0704.0173

Bonding of H in O vacancies of ZnO

Bonding of H in O vacancies of ZnO H. Takenaka and D.J. Singh Materials Science and Technology Division and Center for Radiation Detection Materials and Systems, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032 (Dated: October 25, 2018) We investigate the bonding of H in O vacancies in ZnO using density functional calculations. We find that H is anionic and does not form multicenter bonds with Zn in this compound. PACS numbers: 71.20.Ps,71.55.Gs ZnO is of importance as an extremely fast inorganic scintillator material when doped with Ga or In. It is useful in alpha particle detection, e.g. for devices such as deuterium-tritium neutron generators used in radiography.1,2,3,4,5,6 In this application, H treatment has been shown to improve properties. ZnO has also at- tracted much recent attention motivated by potential applications as an oxide electronic material,7,8,9,10 and in optoelectronic and lighting applications.11,12,13,14,15 H has been implicated as playing an important role in the electronic properties for ZnO for those applications as well.16,17,18 From a fundamental point of view, the be- havior, and especially bonding of H, is of great inter- est; H plays an exceptionally important role in chemistry, and shows unique bonding characteristics. For example, it readily forms compounds where it behaves as a halo- gen ion and forms structures similar to fluorides, such as rutile, perovskite, rocksalt, etc.,19,20,21 and at the same time readily occurs a cation in other chemical environ- ments. Polar covalent bonds involving H and hydrogen bonds are central to much of organic chemistry as well the properties of important substances such as water.22 Thus the recent report by Janotti and Van de Walle (JV) that H forms a new type of strong multicenter bond in O vacancies in ZnO is of wide ranging interest.23 In this paper, we present standard local density ap- proximation (LDA) calculations of the electronic proper- ties and structure of H containing O vacancies in ZnO. We do not find the multicenter covalent bonds claimed by JV, and instead characterize the behavior of H as quite conventional in that it occurs as an anion on the anion site in a polar crystalline environment. Our calculations were done within the standard local density approximation using the general potential lin- earized augmented planewave (LAPW) method, includ- ing local orbitals.24,25 Specifically, we constructed a 72 atom 3x3x2 wurtzite supercells of ZnO, with one O atom removed and replaced by H. The calculations were done using the bulk lattice parameters of ZnO, but the inter- nal coordinates of all atoms in the supercell were fully relaxed. No symmetry was assumed in the relaxations. The LAPW method is an all electron method that makes no shape approximations to either the potential or charge density. It divides space into non-overlapping atom cen- tered spheres and an interstitial region. The method then employs accurate basis sets appropriate for each region.24 In the present calculations, LAPW sphere radii of 2.0 a0, 1.6 a0 and 1.2 a0 were used for Zn, O, and H respec- tively, along with a basis set consisting of more than 8500 LAPW functions and local orbitals. Convergence tests were done with a larger basis set of approximately 12000 functions, but no significant changes were found. The relaxations were done without any imposed symmetry, with a 2x2x2 special k-point zone sampling. A sampling using only the Γ point was found to yield slightly different quantitative results, due to the limited size of our super- cell, but would lead to the same conclusions. The calcu- lated value of the internal parameter is u=0.119, which agrees almost exactly with the experimental value. The densities of states used to analyze the electronic proper- ties were obtained using the linear tetrahedron method based on eigenvalues and wavefunctions at 36 k-points in the half zone (k and -k are connected by time reversal). In our relaxed structure for a neutral cell, we find that H occurs in a slightly asymmetric position, with three Zn neighbors at 2.03 Å, and one Zn neighbor (the one along the c-axis direction) at 2.17 Å. For the singly charged cell, we obtain a very similar result, specifically three Zn neighbors at 2.02 Å, and the apical Zn at 2.21 Å. In the following, we focus on the neutral cell except as noted. Fig. 1 shows the projection of the electronic density of states onto the H LAPW sphere, of radius 1.2 a0. The Fermi energy for our neutral cell lies at a position one electron per cell into the conduction bands, correspond- ing to the valence difference of one between O and H. As may be seen, there are two prominent peaks in the H component of the density of states, one, denoted “B”, at ∼ -8 eV with respect to the Fermi level (-6 eV with respect to the valence band maximum), and the other, denoted “A”, high in the conduction bands at ∼ 6 eV. JV identified these peaks, “B” and “A”, respectively, as the bonding and antibonding combinations of metal and H orbitals giving rise to the multicenter bond. In addition, there is significant H s character distributed over the va- lence bands, especially near the valence band maximum. We note that the very large bonding-antibonding split- ting of 14 eV implied by the assignment of JV indicates extremely strong covalent bonds, which is somewhat sur- prising considering the Zn-H distances. In any case, such a large covalent gap would imply that the bonding and antibonding states should have mixed character. In other words, the bonding state should be of roughly half H s character, while the remaining H 1s character should oc- cur in the unoccupied antibonding level, so that the oc- http://arxiv.org/abs/0704.0173v1 -8 -6 -4 -2 0 2 4 6 8 E(eV) H pH in O vacancy FIG. 1: Projection of the electronic density of states onto the s and p components inside the H LAPW sphere, radius 1.2 a0, for the 72 atom neutral cell. The two peaks identified by JV as bonding and antibonding combinations are indicated by “B” and “A”, respectively. The Fermi level lies in the conduction bands. The position of the valence band maximum is denoted by “VBM” (note that the LDA strongly underestimates the 3.3 eV band gap of ZnO). cupancy of the H 1s orbital should be roughly 1 e, and certainly significantly less than 2 e. To analyze the bonding further it is convenient to com- pare the charge density with an ionic model, as was done for some alanates.30,31 As mentioned, H is known to en- ter some solids as an anion, including tetragonal MgH2. ZnH2 also exists though it is not as well characterized. Furthermore, the simplest hydride, LiH, is of this ioni- cally bonded type and includes H− anions coordinated by six metal atoms.33,34 In these hydrogen anion based materials, the negative H ion is stabilized by the Ewald field. In fact, the importance of the Ewald field is one of the essential differences between chemistry in solid state and the chemistry of molecules. The long range Coulomb interaction stabilizes ionic bonding for species that would generally be largely covalent in small molecules, and in particular stabilizes anions such as O2− and H−, which are common in solid state chemistry but much less so in gas phase molecules. This stabilization by the Ewald field is reflected in the variability of the effective size of H in crystal structure data for anionic hydrides.27,28,29 In view of the common occurrence of H as an anion in many metal hydrides, it would not be surprising if H− were sta- bilized by the Ewald field of an anion vacancy in a polar crystal such as ZnO. Thus we consider an ionic model, based on the charge density of a H− ion stabilized by the Ewald field, as simulated by a Watson sphere,35 as in Ref. 31. For such a H− ion, 0.525 e out of 2.0 e, i.e. ∼ 26% of the charge, is inside a radius of 1.2 a0, so the majority of the charge is outside. Because of the small sphere radius used for H in our calculations the amount of charge in- -8 -6 -4 -2 0 2 4 6 8 E(eV) FIG. 2: Integration of the projection of the H s projected density of states as in Fig. 1 normalized according to the fraction of charge inside a 1.2 a0 sphere for a Watson sphere stabilized H− anion (see text). side the sphere is only weakly dependent on the Watson sphere radius, which reflects the environment. For a non- spin polarized neutral H in free space as described in the LDA, 0.378 e (38%) would be inside a radius of 1.2 a0, showing that there is a strong dependence on the charge state, though not precise proportionality. Fig. 2 shows the integral of the H s character as a function of energy normalized by the fraction of the H− charge inside a 1.2 a0 sphere (0.525/2). Over the valence band region the p contribution is less than 2%, and the d contribution is less than 0.2%. The conduction bands, which are more Zn sp derived, show a larger proportion of H p character, as may be seen in Fig. 1. Thus the charge inside the sphere, which comes from the occupied valence bands, is mainly due to H s states, and not from orbitals on neighboring atoms. Using the ionic model for H−, i.e. incorporating the factor of 0.525/2 as the fraction of charge inside the H LAPW sphere, and integrating, one finds that the peak “B” contains ∼ 0.8 H s electrons. Integrating over the remaining valence bands brings the H s count to 2.0 elec- trons, i.e. what is expected for H−. This leads to an interpretation of the electronic structure, where the peak “B” comes from the H 1s state. This hybridizes with valence band states, which have mixed Zn d and O p character. The second peak “A”, 14 eV higher, is then the H s resonance. This is a very reasonable position for the resonance of H−. In particular, the H−− resonance of atomic H− is at ∼ 14.5 eV.36,37,38 JV emphasized the shape of the charge density associated with the states in the peak “B” and argued for bonding based partly on real space images of this charge. As mentioned, in our pro- jected density of states we find that this peak contains 0.8 s electrons (i.e. 40% H s character), which would be consistent with a bonding orbital. However, the hy- bridization is with other occupied states, and when the integration is done over all the valence bands, we find 2 s electrons, consistent with H−. We emphasize that mix- ing of occupied states does not contribute to the energy, and that such hybridizations do not constitute bonds. Our calculated binding energy relative H2 and a relaxed neutral supercell with an O vacancy is 87 kJ/mol H.39 This may be an overestimate due to LDA errors,40 but in any case is much smaller than the binding that would be suggested by a 14 eV bonding-antibonding splitting. We also calculated the positron wavefunction and life- times for ZnO with an O vacancy and with the H contain- ing O vacancy. This was done using the LAPW method in the full inverted self-consistent Coulomb potential plus the correlation and enhancement factors of Boronski and Nieminen41 as calculated from the full charge density. We obtain a bulk positron lifetime for ZnO of 144 ps, which is at the lower end of the experimental range. Reported experimental values are 151 ps (Ref. 42), 170 ps (Ref. 43), 141 ps - 155 ps (Ref. 44), and 182 ps (Ref. 45). Sig- nificantly, positrons, which are positively charged, tend to localize in voids and in sites that are favorable for cations, and localize weakly if at all in anion sites, due to the unfavorable Coulomb potential. We do not find positron localization at the O vacancy in our ZnO super- cell, indicating that the O is indeed an anion as expected, nor do we find positron localization or a significant life- time increase in the cell with a H containing O vacancy. We also find no significant change in lifetime for H in an O vacancy within a charged supercell with one electron removed. In contrast, we obtain a bound positron state for Zn vacancies, both with and without H, reflecting the fact that Zn is on a cation site. The calculated lifetime in a supercell with a Zn vacancy is 212 ps, while with a H filled Zn vacancy we obtain 175 ps (in this case H bonds to a single adjacent O to form a hydroxyl like unit with H-O bond length of 1.01 Å).46 To summarize, we have performed density functional calculations for ZnO supercells with both empty and H filled O vacancies. 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0704.0174

Reparametrization Invariance, the controversial extraction of $\alpha$ from $B\to\pi\pi$ and New Physics

IFIC/07-17 FTUV-07-0402 Reparametrization Invariance, the controversial extraction of α from B → ππ and New Physics Francisco J. Botella a, Miguel Nebot b a Departament de F́ısica Teòrica and IFIC, Universitat de València-CSIC, E-46100, Burjassot, Spain b Centro de F́ısica Teórica de Part́ıculas (CFTP), Instituto Superior Técnico, P-1049-001, Lisboa, Portugal Abstract The extraction of the weak phase α from B → ππ decays has been controversial from a statistical point of view, as the frequentist vs. bayesian confrontation shows. We analyse several relevant questions which have not deserved full attention and pervade the extraction of α. Reparametrization Invariance proves appropriate to understand those issues. We show that some Standard Model inspired parametriza- tions can be senseless or inadequate if they go beyond the minimal Gronau and London assumptions: the single weak phase α just in the ∆I = 3/2 amplitudes, the isospin relations and experimental data. Beside those analyses, we extract α through the use of several ade- quate parametrizations, showing that there is no relevant discrepancy between frequentist and bayesian results. The most relevant informa- tion, in terms of α, is the exclusion of values around α ∼ π/4; this result is valid in the presence of arbitrary New Physics contributions to the ∆I = 1/2 piece. http://arxiv.org/abs/0704.0174v1 1 Introduction The extraction of the CP violating phase α [1] has lead to some recent con- troversy confronting the results and statistical methods of two different col- laborations: the frequentist approach advocated in references [2, 4] and the bayesian approach employed in reference [3]. In reference [2] J. Charles et al. presented an important criticism to the bayesian methods used by the UTfit collaboration in order to extract the angle α of the unitarity triangle b–d from ππ and ρρ data. The criticism relies heavily on the statistical treatment of data: frequentist vs. bayesian. The answer of the UTfit collaboration [3] rises some interesting points, both on the interpretation of the results and on the importance of the physical assumptions on the hadronic amplitudes. The authors of [2] have recently answered to this UTfit reply in [4]. The aim of the present work is to clarify several issues central to an adequate understanding of the physics at stake. We also want to call the attention on the importance of reparametrization invariance (RpI) in the sense intro- duced by F.J.B. and J. Silva in reference [5] to do so. We will not enter the polemic arena of statistical confrontation. With regard to this, we will instead illustrate the compatibility of results obtained in both approaches as long as things are done properly; notwithstanding, we will not ignore some “obscure” aspects of both approaches that are somehow swept under the rug as the statistical confrontation rages on, they illustrate that rather than sticking to one approach and deprecating the other it may be wiser to learn lessons from both. This work is organized as follows. We start section 2 with a short re- minder on reparametrization invariance and its implications, then we use the exclusion or inclusion of B → π0π0 data together with RpI to clarify the ori- gin of our knowledge on α. In section 3 we study critically Standard Model inspired parametrizations. We devote section 4 to a detailed analysis of the impact on the results of allowed ranges for some parameters. The lessons from previous sections set up the stage for an adequate extraction of α, to which section 5 is dedicated, especially in the presence of New Physics (NP) in loops. Several appendices deal with aspects left out of the main flow of the discussion. 2 Reparametrization invariance and B → ππ 2.1 Weak Phases We start this section with a short reminder of the findings presented in reference [5] concerning the parametrization of decay amplitudes and the election of weak phases. A generic parametrization of the decay amplitude of a B meson to a given final state and the CP-conjugate amplitude is the following1: A = M1 e +iφ1 eiδ1 +M2 e +iφ2 eiδ2 , Ā = M1 e −iφ1 eiδ1 +M2 e −iφ2 eiδ2 , (1) where φj are CP-odd weak phases, δj are CP-even strong phases and Mj the magnitudes of the different contributions. The first property to consider is the full generality, as long as φ1 − φ2 6= 0 mod [π], of Eq. (1), i.e. any additional contribution M3e ±iφ3eiδ3 can be recast into the previous form as e±iφ3 = sin(φ3 − φ2) sin(φ1 − φ2) e±iφ1 + sin(φ3 − φ1) sin(φ2 − φ1) e±iφ2 , (2) and thus A′ = A +M3e +iφ3eiδ3 = M ′1e +iφ1eiδ 1 +M ′2e +iφ2eiδ Ā′ = Ā +M3e −iφ3eiδ3 = M ′1e −iφ1eiδ 1 +M ′2e −iφ2eiδ 2 , (3) M ′1e 1 = M1e iδ1 +M3e sin(φ3 − φ2) sin(φ1 − φ2) M ′2e 2 = M2e iδ2 +M3e sin(φ3 − φ1) sin(φ2 − φ1) . (4) We can also use Eq. (2) to change our basic set {φ1, φ2} of weak phases to any other arbitrary set of weak phases {ϕ1, ϕ2}, as long as ϕ1−ϕ2 6= 0 mod [π]: A = M1 e+iϕ1 ei∆1 +M2 e+iϕ2 ei∆2 , Ā = M1 e−iϕ1 ei∆1 +M2 e−iϕ2 ei∆2 , (5) where M1ei∆1 = M1eiδ1 sin(φ1 − ϕ2) sin(ϕ1 − ϕ2) sin(φ2 − ϕ2) sin(ϕ1 − ϕ2) M2ei∆2 = M1eiδ1 sin(φ1 − ϕ1) sin(ϕ2 − ϕ1) sin(φ2 − ϕ1) sin(ϕ2 − ϕ1) . (6) 1If the final state is ±1 CP eigenstate, Ā should include an additional ±1 factor. This change in the basic set of chosen weak phases should have no physical implications, hence the name reparametrization invariance. We remind two main consequences of RpI in the absence of hadronic inputs. For an extensive discussion see [5]: 1. Consider two basic sets of weak phases {φ1, φ2} and {φ1, ϕ2} with φ2 6= ϕ2; if an algorithm allows us to write φ2 as a function of physical observables then, owing to the functional similarity of equation (1) and (5), we would extract ϕ2 with exactly the same function, leading to φ2 = ϕ2, in contradiction with the assumptions; then, a priori, the weak phases in the parametrization of the decay amplitudes have no physical meaning, or cannot be extracted without hadronic input. 2. If, experimentally, the direct CP asymmetry C = (|A|2 − |Ā|2)/(|A|2+ |Ā|2) is C = 0, then the decay amplitudes can be expressed in terms of a single weak phase, which could be sensibly extracted, up to discrete am- bigüities, through the indirect CP asymmetry S = 2 Im(ĀA∗)/(|A|2 + |Ā|2). Additionally, if the theoretical description of the decay ampli- tudes only involves a single weak phase from a basic Lagrangian, then it can be identified with the phase measured through S. As we will see, this two results apply respectively to the π+π− and π+π0 channels. Essentially, the first one will be operative in the ∆I = 1/2 piece and the second one in the ∆I = 3/2. 2.2 Removing π0π0 information To make our point transparent we will start by studying the extraction – in fact the non-extraction – of α from ππ data when B → π0π0 experimental information is removed. Let us start with a widely used [2, 3], Standard Model inspired, parametrization of the decay amplitudes: A+− ≡ A(B0d → π+π−) = e−iαT+− + P ,√ 2A+0 ≡ 2A(B+ → π+π0) = e−iα(T+− + T 00) , 2A00 ≡ 2A(B0d → π0π0) ≡ 2A+0 −A+− = e−iαT 00 − P , Ā+− ≡ A(B̄0d → π+π−) = e+iαT+− + P ,√ 2Ā+0 ≡ 2A(B− → π−π0) = e+iα(T+− + T 00) , 2Ā00 ≡ 2A(B̄0d → π0π0) ≡ 2Ā+0 − Ā+− = e+iαT 00 − P . (7) When π0π0 experimental information is removed we have two decoupled de- cays: 1. π+π0 data, i.e. the average branching ratio B+0 and the direct CP asymmetry C+0, provide, respectively, |T+− + T 00| and a consistency check C+0 = 0; α is irrelevant there. 2. π+π− data, i.e. B+−, C+− and the mixing induced CP asymmetry S+−, give information on α decoupled from π+π0, on |T+−|, |P | and the relative (strong) phase δPT+− between T +− and P . With three observables and four parameters everybody knows or suspects that one cannot really extract α: we have C+− 6= 0, as reminded in section 2.1, α cannot be extracted from B → π+π− in this limited case. One can try, nevertheless, to obtain a probability distribution function (PDF) for α as in reference [2]. This PDF, obtained in an analysis with three observables and four unknowns, has obviously a strong dependence in the priors, as in figure 2 of [2]. Even worse, reparametrization invariance [5] tells us that A+−, Ā+− can also be written as A+− = e −iα′T ′+− + P ′, Ā+− = e +iα′T ′+− + P ′ , (8) where α′ is any weak phase – known or unknown, α′ 6= 0, π –. In this scenario the conclusion is clear: any information one would get for α would also be valid for any α′ and thus it cannot be assigned to α. This solves the puzzle raised in the MA and RI parametrizations within figure 4 of reference [2]: those PDFs cannot be attributable to α. Just with that data alone we cannot extract α′ – whatever it is –, as we have emphasized in 2.1. To illustrate this issue we compute the PDFs of figure 1 in the following parametrization: A+− ≡ A(B0d → π+π−) = e−iα T+− + P , 2A+0 ≡ 2 A(B+ → π+π0) = e−iα(T+− + T 00) , 2A00 ≡ 2 A(B0d → π0π0) ≡ 2A+0 − A+− , Ā+− ≡ A(B̄0d → π+π−) = e+iα T+− + P , 2Ā+0 ≡ 2 A(B− → π−π0) = e+iα(T+− + T 00) , 2Ā00 ≡ 2 A(B̄0d → π0π0) ≡ 2Ā+0 − Ā+− . Notice that just with α = α′, Eq. (9) recovers the parametrization in Eq. (7). The phase of T+− is set to zero (i.e. all strong phases are relative to arg(T+−)) and flat priors are used for all the parameters2, that is, moduli |T+−|, |T 00|, 2The allowed ranges for the different moduli and the sensitivity to them in this and other cases will be addressed later, for instance, for this example, they are all limited to lie in the range [0; 10]× 10−3 ps−1/2. |P | and phases δP = arg(P ), δ0 = arg(T 00), α and α′. Results in other parametrizations, being equally illustrative, are relegated to appendix C. 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α′ PDF 0 25 50 75 100 125 150 175 α = α′ (c) Joint (α′, α) PDF 25 50 75 100 125 150 175 (d) α = α′ PDF Figure 1: PDFs of α and α′ from B → ππ without π0π0 data. The lesson of this example is rather obvious: the set of observables being insensitive to α, its PDF is uninformative (just the flat prior in this case); the PDF in figure 1(d), erroneously identified with α, is nothing else than α′ itself, whatever it could be. 2.3 Including back π0π0 information When we incorporateB → π0π0 data to the isospin construction, |A00| (|Ā00|) gives the angle among A+0 (Ā+0) and A+− (Ā+−); using then the known phase difference between A+− and Ā+−, the angle among A+0 and Ā+0 is obtained. This is just the isospin analysis giving α. Knowing α, i.e. with α fixed, A+− = e −iαT+− + P would have full meaning and {B+−, C+−, S+−} would fix the three hadronic parameters. Unfortunately the isospin analysis as explained above yields allowed values for α spanning a wide range. The degeneracy of solutions together with the experimental errors do not fix α, just exclude some region. In this situation {B+−, C+−, S+−} do not really fix the hadronic parameters and, consequently, they tend to generate a spurious PDF for α as we have seen. The final “α” is thus a sort of convolution of the α obtained from the isospin analysis and the spurious one “extracted” purely from π+π− data. This is illustrated with the PDFs of figure 2, making use of the parametrization in Eq. (9). 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α′ PDF 0 25 50 75 100 125 150 175 α = α′ (c) Joint (α′, α) PDF 25 50 75 100 125 150 175 (d) α = α′ PDF Figure 2: PDFs of α and α′ from B → ππ. To stress the importance of this issue we repeat the previous example while arbitrarily reducing all experimental uncertainties by a common factor of 5. The PDFs corresponding to this fake scenario are displayed in figure 3. The results shown in figures 1, 2 and 3 deserve some comment: 1. Figures 1(b) and 2(b) are almost identical; in the former we were not using B → π0π0 information while in the later we were doing so. This similarity is a dramatic illustration of the spurious nature of the “ex- tracted” α′. 2. Figure 2(d) is the cut of the joint PDF in figure 2(c) along the line α = α′. Therefore the so called MA extraction of α is a sort of convolution 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α′ PDF 25 50 75 100 125 150 175 (c) α = α′ PDF Figure 3: PDFs of α and α′ from B → ππ with experimental uncertainties reduced by a factor of 5. of the Gronau-London α – figure 2(a) – and the spurious one. 3. This α′ PDF basically allows any value of α′ except the neighborhoods of 0 and π, which are a priori forbidden by S+−, C+− 6= 0: obviously there is no way to produce CP violation in the π+π− channel without two weak phases in the amplitude that controls it. The exclusion of α′ = 0, π is the only physical information one can extract in the SM from the PDF of α′. 4. The deep in the α distributions around α ∼ π/4, which is transmitted to the α = α′ PDF, is senseful. The exclusion of α ∼ 0, π is also physical inside the SM. Nevertheless, how strongly these 0, π regions are excluded is highly sensitive to the allowed ranges for |T+−|, |T 00| and |P | – see section 4 –. As we move away form the α = 0, π points, the final PDF of α would be more influenced by the spurious α′ distribution. One can see that in the shape of the α distribution for α < 25◦ or α > 75◦. 5. As uncertainties are reduced, even with α ≡ α′, the valid ranges for the “real” α emerge, despite the α′ distribution. That is, as experimen- tal uncertainties are reduced, the α′ “pollution” of α through α ≡ α′ becomes increasingly ineffective, as it should, and just transmits the physical exclusion of α = 0, π inside the SM. The main lesson from the previous example is: α is obtained from purely ∆I = 3/2 amplitudes, without additional hadronic input. Including it in ∆I = 1/2 pieces, as reparametrization invariance shows, pollutes the legiti- mate extraction with information that one cannot claim is concerning α. 3 Standard Model inspired parametrizations As stated above, following the consequences of reparametrization invariance, the really legitimate sources of our knowledge on α are A+0, Ā+0. We have referred to the parametrization in Eq. (7) as a “SM inspired parametrization” of the amplitudes and we have discussed how the inclusion of α in A+−, Ā+− is dangerous with present uncertainties. Nevertheless, it is clear that the exclusion of α ∼ 0, π inside the SM is a valid physical consequence that comes from having α in A+− and Ā+−. To further illustrate the importance and the subtlety of this issue let us consider in detail what can be interpreted as a “SM inspired parametrization”. Once we take into account reparametrization invariance, we only need3 to focus on A+− and Ā+−: 1. RpI allows us to write {A+−, Ā+−} in terms of any pair of weak phases {φ1, φ2} (as long as φ1 − φ2 6= 0 mod [π]), nothing enforces the use of {0, α}. 2. SM compliance of any parametrization only requires that the vanishing of all the SM phases leads to no CP violation, once again nothing singles out or requires the use of {0, α}. Consequently, as we have at our disposal other SM phases that we can choose to parametrize A+−, Ā+−, namely 4 γ, β, χ, χ′, instead of A+− = e −iαT+−+P and Ā+− = e iαT+− + P , we can for example write, on equal footing, A+− = M1e iδ1e−iχ +M2e iδ2e−iβ, Ā+− = M1e iδ1e+iχ +M2e iδ2e+iβ , (10) A+− = e −iχT+− + P, Ā+− = e +iχT+− + P . (11) Within the SM χ ∼ O(λ2), had we used this last parametrization (Eq. (11)), we would have found extreme compatibility problems5 that would be absent with another SM inspired parametrization: this is a dramatic illustration of the consequences of RpI mentioned in section 2.1. In other words, pre- tending that one obtains information on SM “theoretical” phases just by parametrizing A+− and Ā+− with them is in general senseless. In this case we would have obtained that figure 2(b) is the PDF of the phase χ, the one that appears in Bs–B̄s mixing [8, 9, 10, 11, 12, 13]. 3A+0 and Ā+0 can be parametrized with a single weak phase, identifiable with α, A00 and Ā00 will follow from the isospin relations. 4γ = arg(−VudVcbV ∗ubV ∗cd), β = arg(−VcdVtbV ∗cbV ∗td), χ = arg(−VcbVtsV ∗csV ∗tb) and χ′ = arg(−VusVcdV ∗udV ∗cs) [6]. 5Just look, for example, to the O(λ2) ∼ 2 − 3◦ region of the different α′ PDFs in the plots of previous sections [7]. 4 Physics and parametrical problems In section 2 we mentioned that the exclusion of the “dangerous” α′ near 0 and π depended on the allowed ranges for the parameters |T ij| and |P |. Figure 4 shows the PDFs of α, α′ and α = α′ for four different sets of allowed ranges of |T ij| and |P |. On the one hand, the PDFs of α in figures 4(a), 4(d), 4(g) and 4(j) are quite similar. On the other hand, the PDFs of α′ in figures 4(b), 4(e), 4(h) and 4(k) are completely different: the “dangerous” α′, especially in the regions close to 0,π, is sensitive to the applied bounds. This is automatically transmitted to the α = α′ PDF and it is in this way that the region with “α” close to 0,π is suppressed (even wipped out as in figures 4(c) and 4(i)) through the cuts on the spurious α′, induced by the cuts on |T ij| and |P |. One could think that this is particular to the bayesian statistical approach, figure 5 shows the frequentist confidence level curves for α computed under the same parametric restrictions. As we use the parametrization of Eq. (7), they correspond to the α = α′ plots in Figure 4. It is rather clear that without regard to the statistical approach, limiting the values of |T ij| and |P | has observable effects in the extraction of α. Note that figure 5(c) differs from figure 5(a) not by a cut but by a change in the shape, even if it is not a dramatic change. The authors of reference [2] pointed out that there is some peculiar limit with α → 0 together with P/T+−, T 00/T+− → −1, |T+−| → ∞ – using the parametrization of Eq. (7) – that keeps all the observables “in place”: it is in fact a question of having α′ → 0 rather than α → 0. This peculiar limit is useful to understand the α ∼ 0, π exclusion above mentioned. To obtain parameter configurations with high likelihood when α(′) approaches 0 or π, the required values of |T ij| and |P | are increasingly large. Imposing bounds on |T ij| and |P | automatically limits how close to 0, π one can push the weak phase while producing likely branching ratios and asymmetries. The use of the parametrization in Eq. (9) shows how this works for the dangerous α′ and is then transmitted to α. 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α′ PDF 25 50 75 100 125 150 175 (c) α = α′ PDF Allowed ranges: |T ij| ∈ [0; 10]× 10−3 ps−1/2, |P | ∈ [0; 2.5]× 10−3 ps−1/2 25 50 75 100 125 150 175 (d) α PDF 25 50 75 100 125 150 175 (e) α′ PDF 25 50 75 100 125 150 175 (f) α = α′ PDF Allowed ranges: |T ij| ∈ [0; 10]× 10−3 ps−1/2, |P | ∈ [0; 10]× 10−3 ps−1/2 25 50 75 100 125 150 175 (g) α PDF 25 50 75 100 125 150 175 (h) α′ PDF 25 50 75 100 125 150 175 (i) α = α′ PDF Allowed ranges: |T ij| ∈ [0; 5]× 10−3 ps−1/2, |P | ∈ [0; 1.25]× 10−3 ps−1/2 25 50 75 100 125 150 175 (j) α PDF 25 50 75 100 125 150 175 (k) α′ PDF 25 50 75 100 125 150 175 (l) α = α′ PDF Allowed ranges: |T ij| ∈ [0; 25]× 10−3 ps−1/2, |P | ∈ [0; 25]× 10−3 ps−1/2 Figure 4: PDFs obtained using the parametrization in Eq. (9) and different allowed ranges for |T ij| and |P |. 25 50 75 100 125 150 175 (a) |T ij| < 10, |P | < 2.5 25 50 75 100 125 150 175 (b) |T ij | < 10, |P | < 10 25 50 75 100 125 150 175 (c) |T ij | < 5, |P | < 1.25 25 50 75 100 125 150 175 (d) |T ij | < 25, |P | < 25 Figure 5: α CL; as usual |T ij| and |P | in units of 10−3 ps−1/2. 5 The extraction of α from B → ππ and New Physics Recently the UTfit collaboration has proposed to add information on the moduli of the amplitudes in order to extract α inside the SM. In particular, to add reasonable QCD based cuts on the moduli of T ij and P . Even if we agree with this procedure, we must stress that the resulting PDF of α – see figures 4(c) or 4(i) – in the non zero region mixes ∆I = 3/2 information with spurious ∆I = 1/2 information. In this case it does not seem dramatic, but it can be so in the B → ρρ case – see [2] –. In addition, if one is trying to make a general fit of the SM it is more natural to use the ∆I = 3/2 piece of B → ππ to get reliable bounds on α and once α is fixed by the general unitarity triangle analysis, use the ∆I = 1/2 piece of B → ππ to obtain better information on the hadronic parameters. In fact, the UTfit collaboration presents results along this line in [3]. This implies our recommendation of using α in the A+0 amplitude and another phase in A+− or in the ∆I = 1/2 piece. After confronting the SM à la CKM with data, the most important ob- jective in overconstraining the unitarity triangle is in fact to look for New Physics (NP) [14, 15, 16, 8, 9, 10, 11, 12, 13]. When there is NP – just in the mixings or also in the ∆I = 1/2 decay amplitudes6 – it is not appropriate to use a SM inspired parametrization. In the limit where all SM phases go to zero, C+− and S+− can still be reproduced by NP loops. So, if we want to interpret the α PDF as7 ᾱ we have to use a different CP-violating phase in the ∆I = 1/2 piece or in A+−. Parametrizations that fulfill these requirements are the so-called PLD, ES, the ’τ ’ parametrization in [2] and even our SM-like parametrization with α′ in Eq. (9) despite having one more parameter. A similar one, which additionally factorizes an overall scale of the amplitudes, is the following, that we call ’1i’: A+− ≡ e−iαT3/2(T + iP ), 2A00 ≡ e−iαT3/2(1− T − iP ),√ 2A+0 ≡ e−iαT3/2, 2Ā+0 ≡ e+iαT3/2, Ā+− ≡ e+iαT3/2(T − iP ), 2Ā00 ≡ e+iαT3/2(1− T + iP ). Notice that a global weak phase in A+− is irrelevant in C +− and amounts to a global shift of arg(Ā+−A In this section we will “extract” α in a bayesian approach making use of different parametrizations; we will show the consistency of all those results 6With great accuracy – up to small electroweak penguins – this case corresponds to having NP everywhere except in tree level amplitudes. 7Where ᾱ = π − β̄ − γ, β̄ = β − φd and the NP phase in B0d–B̄0d mixing is defined by = r2de −i2φd [Md ]SM . and then compare to frequentist results. From a fundamental point of view, as stressed in previous sections, we are not willing to use information beside assuming the triangular isospin relations, the single “tree level” weak phase of the ∆I = 3/2 piece and experimental results themselves. Reparametrization invariance and the presence of a single weak phase, α, in the ∆I = 3/2 amplitudes A+0 and Ā+0 imply that all the results to be presented in this section will be valid in the presence of New Physics in loops. Figure 6 shows the PDF of α in three different cases: the ’PLD’ [17] and ’1i’ (Eq. (12)) parametrizations, and the explicit extraction (as in [17] or [5]). Corresponding 68%, 90% and 95% probability regions are displayed in table 1, together with the frequentist 68%, 90% and 95% CL regions (in the fol- lowing, frequentist calculations are carried with the ’PLD’ parametrization). These regions are represented in figure 7. Despite some small differences in the 68% regions, somehow expectable as they are more sensitive to details, the results are consistent, they coincide rather well. B → ππ data are still too uncertain to really provide important constraints on α, the only relevant feature being the exclusion of the α ∼ π/4 region, which could be understood (see section C.1 in appendix C) in terms of the smallness of B00. 25 50 75 100 125 150 175 (a) PLD parametrization 25 50 75 100 125 150 175 (b) 1i parametrization 25 50 75 100 125 150 175 (c) Explicit extraction Figure 6: α PDFs. 68% 90% 95% PLD [0; 5]◦ ∪ [85; 101]◦∪ [0; 8]◦ ∪ [82; 107]◦∪ [0; 9]◦ ∪ [82; 110]◦ [121; 150]◦ ∪ [168; 180]◦ [114; 157]◦ ∪ [162; 180]◦ ∪[113; 180]◦ 1i [95; 174]◦ [0; 1]◦ ∪ [89; 180]◦ [0; 5]◦ ∪ [85; 180]◦ [2; 8]◦ ∪ [82; 88]◦∪ [0; 9]◦ ∪ [81; 91]◦ Explicit [100; 120]◦ ∪ [125; 145]◦∪ [95; 175]◦ ∪ [179; 180]◦ [0; 10]◦ ∪ [80; 180]◦ [150; 170]◦ CL [0; 7]◦ ∪ [83; 104]◦ [0; 12]◦ ∪ [78; 180]◦ [0; 14]◦ ∪ [76; 180]◦ [115; 154]◦ ∪ [166; 180]◦ Table 1: α regions within [0; 180◦]. 25 50 75 100 125 150 175 Figure 7: α regions (the ordering, top to bottom, is in each case: ’PLD’ pa- rametrization, ’1i’ parametrization, Explicit extraction and frequentist anal- ysis). Conclusions To our knowledge the discrepancies between frequentist and bayesian ap- proaches using the so-called MA and RI parametrizations with Eq. (7) have not been previously understood. We explain that with present experimen- tal uncertainties it is extremely unsecure to introduce the phase α in the ∆I = 1/2 piece. To a great extent a spurious PDF of α tends to be gener- ated. The Gronau and London analysis is critically based on the appearance of one weak phase in the ∆I = 3/2 piece (C+− = 0). Introducing α in the ∆I = 1/2 piece – or A+− – (C +− 6= 0) brings this “second” α to the category of ’not observable’ even if one is using a Standard Model inspired parametri- zation. This difficulty is operative in the so-called MA and RI parametriza- tions. The introduction of α in the ∆I = 1/2 piece and some QCD-based bounds on the amplitudes allows – as done by the UTfit collaboration – to eliminate the solutions around α ∼ 0, π inside the SM. The PDF can still be partially contaminated with the spurious α distribution. In B → ππ it is not dramatic but it could be so in other channels. This last procedure cannot be applied to an analysis with NP in loops. Therefore, we strongly recommend to use parametrizations where α is just included in the ∆I = 3/2 piece. We partially agree with the UTfit collaboration that, in spite of the differences among the frequentist and bayesian methods, both approaches give similar results if one uses parametrizations with a clear physical meaning. In this sense the most relevant result is the exclusion of the region ᾱ ∼ 25◦ − 75◦. Acknowledgments This research has been supported by European FEDER, Spanish MEC under grant FPA 2005-01678, Generalitat Valenciana under GVACOMP 2007-172, by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects PDCT/FP/63912/2005, PDCT/FP/63914/2005, CFTP-FCT UNIT 777, and by the Marie Curie RTN MRTN-CT-2006-035505. M.N. acknowl- edges financial support from FCT. The authors thank J. Bernabéu and P. Paradisi for reading the manuscript and useful comments. A Inputs and numerical methods Along this work we use the set of experimental measurements [18, 19, 20, 21, 22,23,24,25,26,27,28], combined by the Heavy Flavour Averaging Group [29], in table 2. B+−ππ B 5.2± 0.2 1.31± 0.21 5.7± 0.4 C+−ππ S −0.39± 0.07 −0.59± 0.09 −0.37± 0.32 Table 2: Experimental results, branching ratios are multiplied by 10−6. In terms of B → ππ amplitudes, Bij = τBi+j |Aij|2 + |Āij |2 , C ij = |Aij|2 − |Āij|2 |Aij|2 + |Āij|2 , Sij = 2 Im(ĀijAij |Aij|2 + |Āij |2 All frequentist CL computations are performed by: (1) minimizing χ2 with respect to all parameters except the one of interest which is fixed (in this case α), (2) computing the corresponding CL through an incomplete Γ function. All bayesian PDFs are computed using especially adapted Markov Chain MonteCarlo techniques. B Experimental results and isospin relations The isospin relations A+− + 2A00 = 2A+0 , Ā+− + 2Ā00 = 2Ā+0 , (14) define two triangles in the complex plane whose relative orientation fixes α. The sizes of the different sides follow from Eq. (14). |A+−| 2|A00| 2|A+0| |Ā+−| 2|Ā00| 2|Ā+0| 1.441 1.040 2.634 2.176 1.533 2.634 Table 3: Numerical values of the sides of the isospin triangles computed with experimental central values, to be multiplied by 10−3 ps−1/2. This allows the reconstruction, up to a number of discrete ambigüities - namely up to eight -, of both triangles. Central values of present measure- ments yield the values of the sides in table 3. One straightforward question is mandatory: do those would-be triangles “close”? The answer is in the negative because |A+−|+ 2|A00| = 2.481 ≯ 2.634 = 2|A+0| , |Ā+−|+ 2|Ā00| = 3.709 > 2.634 = 2|Ā+0| . In fact, for those central values, the first triangle is not a triangle [30]. In terms of likelihood, the closest configuration to that situation, the most likely one, is having the first triangle flat, a feature which naturally explains the reduced – by a factor of two, from eight to four – degeneracy of α “solutions”. That is, while for old data the almost flatness of this same isospin triangle yielded eight different solutions distributed in four almost-degenerate pairs, those pairs are now degenerate and rather than exact solutions for the central values of the observables they produce best-fitting points. Consequently, the use of explicit solution constructions requires the rejec- tion of the joint regions of experimental input incompatible with the isospin relations Eqs. (14). For old data, this meant rejecting some 48.2% of allowed experimental input (weighting each observable with a gaussian with mean and standard deviation given by the corresponding central value and uncer- tainty), for the new data set this rejection rate is 70.9%. In the bayesian and frequentist treatments the isospin relations are assumed valid and all the subsequent analyses are “normalized” to that assumption. C Removing B → π0π0 information C.1 Explicit extraction of α This appendix is devoted to some complementary results extending what is presented in section 2.2. The first issue we will address is the explicit extrac- tion8 of α when B → π0π0 information is removed, that is, no knowledge of B00 and C00. The explicit extraction of α assumes the isospin relations in Eqs. (14) so to start with, the ignorance on B00 is not ”just plain ignorance” (whatever this could stand for) as it will operatively mean that for any ex- perimental set of results {B+−, B+0, C+−, S+−}, B00 and C00 should be such 8Beside the explicit formula for α in terms of the available observables presented in reference [17] we also make use of the extraction of α explained in [5]; the results are com- pletely equivalent, however the later does not make any use of a particular parametrization of the amplitudes and is easily interpreted in terms of the isospin construction. that both would-be isospin triangles are in fact isospin triangles. C00 is obvi- ously restricted to be in the range [−1; 1]; what about B00? One could argue that if there is no information on B → π0π0 it should be smaller than a given bound or one can just let it be as large as allowed by other data and isospin constraints. This rather trivial fact is apparently at the origin of the discrep- ancy in the results presented in references [2,3] for the explicit extraction of α “without” B → π0π0 information: figure 8 shows two PDFs of α. They are obtained by generating known experimental sets {B+−, B+0, C+−, S+−} according to gaussian distributions with central values and standard devia- tions given by the quoted measurements and uncertainties (C+− and S+− are also restricted to be within [−1; 1]), then C00 and B00 are generated through flat distributions, C00 in the range [−1; 1] and B00 in a range [0;B00Max]. Sets {B+−, B+0, C+−, S+−, B00, C00} which fulfill the isospin relations Eqs. (14) are retained and used to extract α. The PDFs of α represented in figure 8 only differ in the value of B00Max, Fig. 8(a) was obtained with B Max equal to two times the present measurement while Fig. 8(b) was obtained with B00Max equal to twenty times the present measurement. On the one hand, the PDF in figure 8(a) coincides with the one presented in figure 4 ’ES’ of reference [2]; on the other hand the PDF in figure 8(b) agrees, more or less, with figure 4 of reference [3]. It is now clear that the difference among both may be just due to the numerical procedure. Figure 8(b) shows that the removal of B → π0π0 information leads to a loss of knowledge on α. Ironically, there is a lesson in this example: numerics apart, the smallness of B00 is responsible for the exclusion of values α ∼ π/4. 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α PDF Figure 8: Explicit extraction without B → π0π0; lighter curves correspond to the different individual contributions related by the discrete ambigüities. C.2 Parametrizations To complete the picture we now proceed to repeat the extraction of α when B → π0π0 information is removed in several parametrizations. We will make use of the ’PLD’ parametrization [17], of the ’1i’ parametrization with fixed weak phases in {A+−, Ā+−} (Eq. (12)) and, finally, of the parametrization in Eq. (9) but in this case, apart from α and α′, instead of moduli and phases we will use real and imaginary parts of T+−, P and T 00 (the RI parametrization in reference [2]). The PDFs of α obtained for the first two parametrizations are shown in figure 9, they are eloquent: no knowledge on α. 25 50 75 100 125 150 175 (a) α PDF, PLD parametrization 25 50 75 100 125 150 175 (b) α PDF, 1i parametrization Figure 9: Extraction without B → π0π0. For the RI parametrization we show the PDFs of α, α′ and the one obtained by setting α = α′ in figure 10. Once again it is clear that there is no information on α and that inappropriately insisting on including it in {A+−, Ā+−} produces the senseless result of figure 10(c). 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α′ PDF 25 50 75 100 125 150 175 (c) α = α′ PDF Figure 10: Extraction without B → π0π0, RI parametrization. The conclusion of this appendix is straightforward: just dealing with a reduced scenario in which B → π0π0 information is removed, a proper understanding of the subtleties involved in the parametrization of B → ππ amplitudes avoids peculiar results as for instance the ’MA’ and ’RI’ ones included in figure 4 of reference [2]. We have shown here that starting with a flat prior for α consistently gives highly non-informative posteriors in several sensible parametrizations. D Using the RI parametrization In section 2.3 we used the parametrization in Eq. (9) to obtain figure 2 with flat |T+−|, |P |, |T 00|, arg(P ), arg(T 00), α and α′ priors. For completness we also show – figure 11 – the PDFs of α, α′ and α = α′ in case one uses flat |T+−|, Re [P ], Im [P ], Re [T 00], Im [T 00], α and α′ priors. Beside the effect of the spurious α′ in the PDF of α = α′, we can also appreciate the influence of the change in the priors: the integration domain is the same as in figure 2 but the integration measure is now different. The main effect is the relative enhancement of the contributions from regions with large parameters, including the contributions from the α′ → 0 driven region. 25 50 75 100 125 150 175 (a) α PDF 25 50 75 100 125 150 175 (b) α′ PDF 25 50 75 100 125 150 175 (c) α = α′ PDF Figure 11: α extraction, RI parametrization. E One short statistical comment Leaving completely aside philosophical aspects of probability, both frequen- tist and bayesian approaches start with a common likelihood function. Each approach reduces the information provided by the likelihood function in a different manner. Consequently, they do not yield strictly coincident results: • Bayesian posteriors obviously depend on the priors, for example the allowed ranges or the shape. As we have seen, we obtain different posteriors with different priors. However, as long as one is using sen- sible parametrizations and reasonable priors, we end up finding rather compatible results. • Frequentist CL curves do depend on the parametrization, to be precise, they depend on the allowed ranges for the parameters; once sensible parametrizations and adequate ranges are used, CL curves obtained with them are identical. The α → 0 limit in the SM inspired parame- trization of Eq. (7) illustrates this issue. Beside those well known issues, we may find troublesome that: 1. Most probable values in the bayesian PDFs do not coincide with the analytical solutions for α. 2. Intimately related to this aspect, bayesian PDFs seem unable to dis- tinguish among degenerate solutions. We remind that these statements concern one dimensional PDFs of α. Fre- quentist one dimensional CL curves distinguish α solutions because they are obtained through best fitting points for fixed α. Bayesian PDFs do not dis- tinguish them as the uncertainties produce distributions for the degenerate solutions which overlap and add up in the complete PDF. One can still have a hint of the proximity of different solutions from this kind of overlap, but this is not the point here. For reduced experimental uncertainties, bayesian PDFs would not overlap and would distinguish among those different solu- tions. This could be sufficient to think that, per se, there is no discriminating advantage in using one or the other approach. With present uncertainties, bayesian analyses seem incapable of pinning down the right location of the solutions in α and telling us something about their degeneracy. It is not a fundamental problem of bayesian methods as reduced uncertainties would overcome these “difficulties”. If it is not a fundamental problem, could we somehow overcome these “difficulties” with present uncertainties? The an- swer is in the positive as the problem only arises because we are insisting in the reduction of the available experimental information to obtain one- dimensional PDFs of α; let us take a look to the joint PDFs in figure 12. These are the joint PDFs of (δ, α) and (αeff , α) obtained with the ’PLD’ parametrization. They are quite illustrative, one can see the different solu- tions in α concentrated around the values of α dictated by the analytical expectations. The pretended fundamental drawbacks of bayesian methods to adequately place and distinguish the solutions are just a consequence of pushing too far, for the present level of experimental uncertainty in the re- sults, the statistical “reduction of information process”. 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D73, 071501 (2006), hep-ph/0602060. http://arxiv.org/abs/hep-ph/0406263 http://arxiv.org/abs/hep-ex/0703016 http://arxiv.org/abs/hep-ex/0608003 http://arxiv.org/abs/hep-ex/0501071 http://arxiv.org/abs/hep-ex/0412037 http://arxiv.org/abs/hep-ex/0608035 http://arxiv.org/abs/hep-ex/0502035 http://arxiv.org/abs/hep-ex/0408101 http://arxiv.org/abs/hep-ex/0401029 http://arxiv.org/abs/hep-ex/0311061 http://arxiv.org/abs/hep-ex/0308040 http://arxiv.org/abs/hep-ex/0301032 http://www.slac.stanford.edu/xorg/hfag/ http://arxiv.org/abs/hep-ph/0602060 Introduction Reparametrization invariance and bold0mu mumu BBBBBB Weak Phases Removing bold0mu mumu 000000000000 information Including back bold0mu mumu 000000000000 information Standard Model inspired parametrizations Physics and parametrical problems The extraction of bold0mu mumu from bold0mu mumu BBBBBB and New Physics Inputs and numerical methods Experimental results and isospin relations Removing bold0mu mumu B00B00B00B00B00B00 information Explicit extraction of bold0mu mumu Parametrizations Using the RI parametrization One short statistical comment

0704.0175

Solar System Constraints on Gauss-Bonnet Mediated Dark Energy

Solar system constraints on Gauss-Bonnet mediated dark energy Luca Amendola1, Christos Charmousis2 and Stephen C Davis3 1 INAF/Osservatorio Astronomico di Roma, Viale Frascati 33, 00040 Monte Porzio Catone (Roma), Italy 2 LPT, Université Paris–Sud, Bâtiment 210, 91405 Orsay CEDEX, France 3 Lorentz Institute, Postbus 9506, 2300 RA Leiden, The Netherlands E-mail: [email protected], [email protected] and [email protected] Abstract. Although the Gauss-Bonnet term is a topological invariant for general relativity, it couples naturally to a quintessence scalar field, modifying gravity at solar system scales. We determine the solar system constraints due to this term by evaluating the post-Newtonian metric for a distributional source. We find a mass dependent, 1/r7 correction to the Newtonian potential, and also deviations from the Einstein gravity prediction for light-bending. We constrain the parameters of the theory using planetary orbits, the Cassini spacecraft data, and a laboratory test of Newton’s law, always finding extremely tight bounds on the energy associated to the Gauss-Bonnet term. We discuss the relevance of these constraints to late-time cosmological acceleration. Keywords: dark energy theory, gravity, string theory and cosmology 1. Introduction Supernovae measurements [1] indicate that our universe has entered a phase of late- time acceleration. One can question the magnitude of the acceleration and its equation of state, although given the concordance of different cosmological data, acceleration seems a robust observation (although see [2] for criticisms). Commonly, in order to explain this phenomenon one postulates the existence of a minute cosmological constant Λ ∼ 10−12 eV4. This fits the data well and is the most economic explanation in terms of parameter(s). However such a tiny value is extremely unnatural from a particle physics point of view [3]. Given the theoretical problems of a cosmological constant, one hopes that the intriguing phenomenon of acceleration is a window to new observable physics. This could be in the matter sector, in the form of dark energy [4, 5], or in the gravity sector, in the form of a large distance modification of Einstein gravity [6, 7, 8, 9]. Scalar field driven dark energy, or quintessence [4] is one of the most popular of the former possibilities. However these models have important drawbacks, such as the fine tuning of the mass of the quintessence field (which has to be smaller than the actual Hubble parameter, H0 ∼ 10−33 eV), and stability of radiative corrections from http://arxiv.org/abs/0704.0175v2 Solar system constraints on Gauss-Bonnet mediated dark energy 2 the matter sector [10] (see however [11]). Modified gravity models have the potential to avoid these problems, and can give a more profound explanation of the acceleration. However, these are far more difficult to obtain since Einstein’s theory is experimentally well established [12], and the required modifications happen at very low (classical) energy scales which are (supposed to be) theoretically well understood. Furthermore, many apparently successful modified gravity models suffer from instabilities or are incompatible with gravity experiments. For example the self-accelerating solutions of DGP [8] suffer from perturbative ghosts [13], and f(R) gravity theories [9] can conflict with solar system measurements and present instabilities [14]. In this paper we will consider observational constraints on a class of gravity theories which feature both dark energy and modified gravity. Specifically, we will examine solar system and laboratory constraints resulting from the response of gravity to a quintessence-like scalar field, which couples to quadratic order curvature terms such as the Gauss-Bonnet term. Such couplings arise naturally [15], and modify gravity at local and cosmological scales [15, 16]. Although the Gauss-Bonnet invariant shares many of the properties of the Einstein-Hilbert term, the resulting theory can have substantially different features, see for example [17]. It is a promising candidate for a consistent explanation of cosmological acceleration, but as we will show, can also produce undesirable effects at solar system scales. In particular, we will determine constraints from deviations in planetary orbits around the sun, the frequency shift of signals from the Cassini probe, and table- top experiments. In contrast to some previous efforts in the field [18], we will not suppose a priori the order of the Gauss-Bonnet correction or the scalar field potential. Instead we will calculate leading-order gravity corrections for each of them, and obtain constraints on the relevant coupling constants (checking they fall within the validity of our perturbative expansion). Hence our analysis will apply for large couplings, which as we will see, are in accord with Gauss-Bonnet driven effective dark energy models. In this way we will show such models generally produce significant deviations from general relativity at local scales. We also include higher-order scalar field kinetic terms, although for the solutions we consider they turn out to be subdominant. In the next section we will present the theory in question and calculate the corrections to a post-Newtonian metric for a distributional point mass source. In section 3, we derive constraints from planetary motion, the Cassini probe, and a table- top experiment. For the Cassini constraint, we have to explicitly derive the predicted frequency shift for our theory, as it does not fall within the usual Parametrised Post- Newtonian (PPN) analysis. We discuss the implications of our results in section 4. 2. Quadratic Curvature Gravity We will consider a theory with the second-order gravitational Lagrangian R− (∇φ)2 − 2V (φ) Solar system constraints on Gauss-Bonnet mediated dark energy 3 ξ1(φ)LGB + ξ2(φ)Gµν∇µφ∇νφ+ ξ3(φ)(∇φ)2∇2φ+ ξ4(φ)(∇φ)4 , (1) which includes the Gauss-Bonnet term LGB = R2 − 4RµνRµν + RµνρσRµνρσ. Note for example that such a Lagrangian with given ξ’s arises naturally from higher dimensional compactification of a pure gravitational theory [15]. On its own, in four dimensions, the Gauss-Bonnet term does not contribute to the gravitational field equations. However we emphasise that when coupled to a scalar field (as above), it has a non-trivial effect. Throughout this paper we take the dimensionless couplings ξi and their derivatives to be O(1). There is then only one scale for the higher curvature part of the action, given by the parameter α, with dimensions of length squared. Similarly we assume that all derivatives of the potential V are of O(V ), which in our conventions has dimensions of inverse length squared. These two simplifying assumptions will hold for a wide range of theories, including those in which ξi and V arise from a toroidal compactification of a higher dimensional space [15]. On the other hand it is perfectly conceivable that they do not apply for our universe, in which case the corresponding gravity theories will not be covered by the analysis in this article. Using the post-Newtonian limit, the metric for the solar system can be written [12] ds2 = −(1− h00)(c dt)2 + (δij + hij)dxidxj +O(ǫ3/2) . (2) with h00, hij = O(ǫ). The dimensionless parameter ǫ is the typical gravitational strength, given by ǫ = Gm/(rc2) where m is the typical mass scale and r the typical length scale (see below). For the solar system ǫ is at most 10−5, while for cosmology, or close to the event horizon of a black hole, it is of order unity. The scale of planetary velocities v, is of order ǫ1/2, and so the h0i components of the metric are O(ǫ 3/2), as are ∂th00 and ∂thij . In what follows, we will take φ = φ0 +O(ǫ). For the linearised approximation we are using, we can adopt a post-Newtonian gauge in which the off-diagonal components of hij are zero. We can then write hij = −2Ψδij , h00 = −2Φ , (3) and so c2Φ is the Newtonian potential. In this paper we will consider the leading-order corrections in ǫ without assumptions on the magnitude of V and α. To leading order in ǫ, the Einstein equations take the nice compact form, ρm − V − 2αξ′1D(Φ + Ψ, φ) + O(ǫ2, αǫ3/r2, V ǫr2) (4) 2ξ′1D(Ψ, φ) + D(φ, φ) +O(ǫ2, αǫ3/r2, V ǫr2) (5) where primes denote ∂/∂φ, and V , ξ′1, etc. are evaluated at φ = φ0. The matter energy density in the solar system is ρm, and G0 is its bare coupling strength (without quadratic gravity corrections). Other components of the energy-momentum tensor are higher order in ǫ. The scalar field equation is ∆φ = V ′ − α [4ξ′1D(Φ,Ψ) + ξ2D(Φ−Ψ, φ) + ξ3D(φ, φ)] + O(ǫ2, αǫ3/r2, V ǫr2) . (6) Solar system constraints on Gauss-Bonnet mediated dark energy 4 We have defined the operators X,ii , D(X, Y ) = X,ijY,ij −∆X∆Y . (7) with i, j = 1, 2, 3 where to leading order, the Gauss-Bonnet term is then LGB = 8D(Φ,Ψ). For standard Einstein gravity (V = α = 0), the solution of the above equations is Φ = Ψ = −Um , φ = φ0 , (8) where ρm(~x ′, t) |~x− ~x′| . (9) We will now study solutions which are close to the post-Newtonian limit of general relativity, and take Φ = −Um + δΦ , Ψ = −Um + δΨ , φ = φ0 + δφ , (10) where δφ, etc. are the leading-order α- and V -dependent corrections. Note that the Laplacian carries a distribution and therefore we have to be careful with the implementation of the D operator. We see that δφ is O(V, αǫ2), and so, to leading order, we have ∆ δφ = V ′ − 4αξ′1D(Um, Um) . (11) Having calculated δφ, we obtain ∆ δΦ = −V + 4αξ′1D(Um, δφ) (12) ∆ δΨ = + 2αξ′1D(Um, δφ) . (13) In the case of a spherical distributional source ρm = mδ (3)(x), . (14) In accordance to our estimations for ǫ the solar system Newtonian potentials are Um . 10 −5, and the velocities satisfy v2 . Um. For planets we have Um . 10 (with the maximum attained by Mercury). With the aid of the relation D(r−n, r−m) = 2nm n+m+ 2 ∆r−(n+m+2) (15) the above expressions evaluate, at leading order, to φ = φ0 + r2V ′ − 2ξ′1 α(G0m) Φ = −G0m − 64(ξ α2(G0m) Ψ = −G0m − 32(ξ α2(G0m) . (18) Solar system constraints on Gauss-Bonnet mediated dark energy 5 We find that there are now non-standard corrections to the Newtonian potential which do not follow the usual parametrised expansion, in agreement with [19], but not [18] (which uses different assumptions on the form of the theory). First of all note that the Gauss-Bonnet coupling α couples to the running of the dark energy potential V ′, giving a 1/r contribution to the modified Newtonian potential (17). We absorb this into the gravitational coupling, G = G0 . (19) The corresponding term in (18) gives a constant contribution to the effective γ PPN parameter. The r2V terms in (17), (18) are typical of a theory with a cosmological constant, whereas the final, 1/r7 terms are the leading pure Gauss-Bonnet correction, which is enhanced at small distances. If we take the usual expression for the PPN parameter γ = Ψ/Φ, we see that it is r dependent. In using the Cassini constraint on γ we must be careful to calculate the frequency shift from scratch. For the above derivation we have assumed δφ ≪ Um, which implies V ≪ Um/r2 and α ≪ r2/Um. This will hold in the solar system if V ≪ 10−36m−2 and α ≪ 1023m2 (everywhere) 1029m2 (planets only) in geometrised units. Note that strictly speaking there is also a lower bound on our coupling constants, if the above analysis is to be valid. Indeed, if we were to find corrections of order ǫ2 ∼ 10−14, then it would imply that higher-order corrections from general relativity were just as important as the ones appearing in (17), (18). 3. Constraints 3.1. Planetary motion Deviations from the usual Newtonian potential will affect planetary motions, which provides a way of bounding them. This idea has been used to bound dark matter in the solar system [20], and also the value of the cosmological constant [21]. We will apply the same arguments to our theory. From the above gravitational potential (17), we obtain the Newtonian acceleration gacc(r) = −c2 64(αξ′1) Gmeff where rg ≡ Gm/c2 is gravitational radius of the mass m. The above expression gives the effective mass meff felt by a body at distance r. If the test body is a planet with semi-major axis a, we can use this formula at r ≈ a. Its mean motion n ≡ Gm/a3 will then be changed by δn = (n/2)(δmeff/m). By evaluating the statistical errors of the mean motions of the planets, δn = −(3n/2)δa/a, we can derive a bound on δmeff and hence our deviations from general relativity δmeff 64(αξ′1) . (22) Solar system constraints on Gauss-Bonnet mediated dark energy 6 The values of a for the planets are determined using Kepler’s third law, with a constant sun’s mass m⊙. Constraints on δΦ then follow from the errors δa, in the measure of a. These can be found in [22], and are also listed in the appendix for convenience. Given their different r-dependence, the two corrections to δmeff are unlikely to cancel. We will therefore bound them separately, giving constraints on α and V . The strongest bound on the combination ξ′1α comes from Mercury, with . 1.8× 10−12 . (23) Neglecting the cosmological constant term, and using a ≈ 5.8×107 km and rg ≈ 1.5 km, we find |ξ′1α| . (3a5δa) ≈ 3.8× 1022m2 . (24) We see that this is within range of validity (20) for our perturbative treatment of gravity. In cosmology, the density fraction corresponding to the Gauss-Bonnet term is [15] ΩGB = 4ξ . (25) If this is to play the role of dark energy in our universe, it needs to take, along with the contribution of the potential, a value around 0.7 at cosmological length scales (and for redshift z ∼ 1). If we wish to accurately apply the bound on α (24) to cosmological scales, details of the dynamical evolution of φ will be required. These will depend on the form of V and the ξi, and are expected to involve complex numerical analysis, all of which is beyond the scope of this work. Here we will instead assume that the cosmological value of φ is also φ0, which, while crude, will allow us to estimate the significance of the above result. Given the hierarchy between cosmological and solar system scales it is natural to question this assumption but we will make it here, and discuss it in more detail in the concluding section. Making the further, and less controversial, assumption that dφ/dt ≈ H , we obtain a very stringent constraint on ΩGB: |ΩGB| ≈ 4|ξ′1α|H20 . 8.8× 10−30 . (26) Hence we see that solar system constraints on Gauss-Bonnet fraction of the dark energy are potentially very significant, despite the fact that the Gauss-bonnet term is quadratic in curvature. Since we are assuming that all the ξi are of the same order, the above bound also applies to the dark energy fractions arising from the final three terms in (1). Clearly there are effective dark energy models for which the analysis leading to the above bound (26) does not apply. However any successful model will require a huge variation of ξ1 between local and cosmological scales, or a very substantial violation of one of our other assumptions. Solar system constraints on Gauss-Bonnet mediated dark energy 7 For comparison, we apply similar arguments to obtain a constraint on the potential. The strongest bound comes from the motion of Mars [21], and is |V | . 9rgδa ≈ 1.2× 10−40m−2 . (27) This suggests ΩV = V/(3H 0) . 7.3×1011, which is vastly weaker than the corresponding cosmological constraint (ΩV . 1). Hence planetary orbits tell us little of significance about dark energy arising from a potential, in sharp contrast to the situation for Gauss- Bonnet dark energy. 3.2. Cassini spacecraft The most stringent constraint on the PPN parameter γ was obtained from the Cassini spacecraft in 2002 while on its way to Saturn. The signals between the spacecraft and the earth pass close to the sun, whose gravitational field produces a time delay. The smallest value of r on the light ray’s path defines the impact parameter b. A small impact parameter maximises the light delay. During that year’s superior solar conjunction the spacecraft was re = 8.43AU = 1.26 × 1012m away from the sun, and the impact parameter dropped as low as bmin = 1.6R⊙. A PPN analysis of the system produced the strong constraint δγ ≡ γ − 1 = (2.1± 2.3)× 10−5 . (28) Given that our theory is not PPN we have to undertake the calculation from scratch. The above constraint comes from considering a round trip, in which the light travels from earth, grazes the sun’s ‘surface’, reaches the spacecraft, and then returns by the same route. We take the path of the photon to be the straight line between the earth and the spacecraft, ~x = (x, b, 0) with x varying from −xe to x⊕. For a round trip (there and back), the additional time delay for a light ray due to the gravitational field of the sun is then c∆t = 2 h00(r) + hxx(r) dx = −2 (Φ + Ψ)|r=√x2+b2 dx . (29) For the solution (17) and (18), this evaluates to c∆t = 4rg 1− 2αξ a3⊕ + r + b2(a⊕ + re) 1024(αξ′1) , (30) where we have assumed x⊕ ≈ a⊕ ≫ b, and similarly for the spacecraft. Rather than directly measure ∆t, the Cassini experiment actually found the frequency shift in the signal [23] ygr = . (31) The results obtained were ygr = − 10−5 s (2 + δγ) . (32) Solar system constraints on Gauss-Bonnet mediated dark energy 8 If gravitation were to be described by the standard PPN formalism, then δγ would be the possible deviation of the PPN parameter γ from the general relativity value of 1. From (30) we obtain ygr = − b2V (a⊕ + re) 1536(αξ′1) 4αξ′1V . (33) Requiring that the corrections are within the errors (28) of (32), implies |ξ′1α| . . 1.6× 1020m2 . (34) This suggests the dark energy bound |ΩGB| . 3.6× 10−32 , (35) although obtaining this bound from solar system data requires major assumptions about the cosmological behaviour of φ, as we will point out in section 4. The data obtained by the spacecraft were actually for a range of impact parameters b, but we have just used the most conservative value b = bmin = 1.6R⊙. The above constraint is even stronger than (24), which was obtained for planetary motion. This is because the experiment involved smaller r, and so the possible Gauss-Bonnet effects were larger. Taking the above expression for ygr (33) at face value, we can also constrain the potential to be |V | . 10−22m2 and the cross-term |αξ′1V ′| . 10−5. However these are of little interest as they are much weaker than the planetary motion constraints (24), (27), and also the former is far outside the range of validity (20) of our analysis. 3.3. A table-top experiment Laboratory experiments can also be used to obtain bounds on deviations from Newton’s law. For illustration we will consider the table-top experiment described in [24]. It consists of a 60 cm copper bar, suspended at its midpoint by a tungsten wire. Two 7.3 kg masses are placed on carts far (105 cm) from the bar, and another mass of m ≈ 43 g is placed near (5 cm) to the side of bar. Moving the masses to the opposite sides of the bar changes in the torque felt by it. The experiment measures the torques N105 and −N5 produced respectively by the far and near masses. The masses and distances are chosen so that the two torques roughly cancel. The ratio R = N105/N5 is then determined, and compared with the theoretical value. The deviation from the Newtonian result is Rexpt RNewton − 1 = (1.2± 7)× 10−4 . (36) In fact, to help reduce errors, additional measurements were taken. To account for the gravitational field of the carts that the far masses sit on, the experiment was repeated with only the carts and a m′ ≈ 3 g near mass. The measured torque was then subtracted from the result for the loaded carts. The Gauss-Bonnet corrections to the Newton potential (17) will alter the torques produced by all four masses, as well as the carts. Furthermore, since δΦ is non-linear Solar system constraints on Gauss-Bonnet mediated dark energy 9 in mass, there will be further corrections coming from cross terms. The expressions derived in section 2 are just for the gravitational field of a single mass, and so will not fully describe the above table-top experiment. However, we find that the contribution from the mass m will dominate the other corrections, and so we can get a good estimate of the Gauss-Bonnet contribution to the ratio R by just considering m. The torque experienced by the copper bar, due to a point mass at ~X = (X, Y, Z) is d3x (~x ∧ ~F )z = ρCu yX − xY r=| ~X−~x| , (37) where ρCu is the bar’s density. A full list of parameters for the experiment is given in table I of [24]. The bar’s dimensions are 60 cm × 1.5 cm × 0.65 cm. Working in coordinates with the origin at the centre of the bar, the mass m = 43.58 g is at ~X = (24.42,−4.77,−0.03) cm. Treating m as a point mass, Newtonian gravity implies a torque of N5 ≈ (8.2 cm2)GmρCu is produced. The Gauss-Bonnet correction is δN5 = ρCu 64G3m3(αξ′1) yX − xY | ~X − ~x|9 ≈ −(0.025 cm−4) (Gm)3(αξ′1) . (38) To be consistent with the bound (36), we require δN5/N5 to be within the range of δR. This implies |αξ′1| . (18 cm3) . 1.3× 1022m2 , (39) which is comparable to the planetary constraint (24). Extrapolating it to cosmological scales gives |ΩGB| . 3.1× 10−30 . (40) There are of course many more recent laboratory tests of gravity, and we expect that stronger constraints can be obtained from them. Table-top experiments frequently involve multiple gravitational sources, or gravitational fields which cannot reasonably be treated as point masses. A more detailed calculation than the one presented in section 2 will then be required. For example, the gravitational field inside a sphere or cylinder will not receive corrections of the form (17), and so any experiment involving a test mass moving in such a field requires a different analysis. 4. Discussion We have shown that significant constraints on Gauss-Bonnet gravity can be derived from both solar system measurements and table-top laboratory experiments (note that further constraints arise when imposing theoretical constraints like absence of superluminal or ghost modes, see [25]). The fact that the corrections to Einstein gravity are second order in curvature suggests they will automatically be small. However this does not take into account the fact that the dimensionfull coupling of the Gauss-Bonnet term must be large if it is to have any hope of producing effective dark energy. Additional constraints will come from the perihelion precession of Mercury, although the linearised Solar system constraints on Gauss-Bonnet mediated dark energy 10 analysis we have used is inadequate to determine this, and higher-order (in ǫ) effects will need to be calculated. Performing an extrapolation of our results to cosmological scales suggests that the density fraction ΩGB will be far too small to explain the accelerated expansion of our universe. This agrees with the conclusions of [19]. Hence if Gauss-Bonnet gravity is to be a viable dark energy candidate, one needs to find a loophole in the above arguments. This is not too difficult, and we will now turn to this question. In particular, we have assumed no spatial or temporal evolution of the field φ between cosmological and solar system scales, even though the supernova measurements correspond to a higher redshift and a far different typical distance scale. A varying φ would of course imply that different values of ξi, and their derivatives, would be perceived by supernovas and the planets. It is interesting to note that the size of the bound we have found (26) is of order the square of the ratio of the solar system and the cosmological horizon scales, s = (1AUH0) 2 ∼ 10−30. Therefore one could reasonably argue that the small number appearing in (26) could in fact be due to the hierarchy scale, s, rather than a very stringent constraint on ΩGB. This could perhaps be concretely realised with something similar to the chameleon effect [26] giving some constraint on the running of the quintessence theory. One other possibility is that the baryons (which make up the solar system) and dark matter (which is dominant at cosmological scales) have different couplings to φ [27]. Again, this would alter the relation between local and cosmological constraints. Alternatively, it may be that our assumptions on the form of the theory should be changed. The scalar field could be coupled directly to the Einstein-Hilbert term, as in Brans-Dicke gravity. Additionally, the couplings ξi and their derivatives could be of different orders. The same could be true of the potential. In particular, if φ were to have a significant mass, this would suppress the quadratic curvature effects, as they operate via the scalar field. This would be similar to the situation in scalar-tensor gravity with a potential, where the strong constraints on the theory can be avoided by giving the scalar a large mass (which, however, would inhibit acceleration). Finally, the behaviour of the scalar field could be radically different. We took it to be O(ǫ), like the metric perturbations. However since our constraints are on the metric, and not φ, this need not be true. Furthermore, since the theory is quadratic, there may well be alternative solutions of the field equations, and not just the one we studied. Hence to obtain a viable Gauss-Bonnet dark energy model, which is compatible with solar system constraints, at least one of the above assumptions must be broken. For many of the above ideas the higher-order scalar kinetic terms will play a significant role. This then opens up the possibility that the higher-gravity corrections will cancel each other, further weakening the constraints. We hope to address some of these issues in the near future. Solar system constraints on Gauss-Bonnet mediated dark energy 11 Acknowledgments CC thanks Martin Bucher, Gilles Esposito-Farese and Lorenzo Sorbo for discussions. SCD thanks the Netherlands Organisation for Scientific Research (NWO) for financial support. Appendix For the benefit of readers without an astronomical background, we list relevant solar system parameters. The values for δa come from table 4 of [22]. We take the Hubble constant to be H0 = 70 kms −1 Mpc−1. 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D 62, 043511 (2000) [astro-ph/9908023] http://arxiv.org/abs/hep-th/0604086 http://arxiv.org/abs/hep-th/0503191 http://arxiv.org/abs/astro-ph/0307338 http://arxiv.org/abs/astro-ph/0307285 http://arxiv.org/abs/astro-ph/0603703 http://arxiv.org/abs/hep-th/0506137 http://arxiv.org/abs/astro-ph/0606078 http://arxiv.org/abs/hep-th/0609155 http://arxiv.org/abs/hep-th/0702002 http://arxiv.org/abs/hep-th/0608178 http://arxiv.org/abs/hep-th/0206089 http://arxiv.org/abs/hep-th/0309083 http://arxiv.org/abs/gr-qc/0612065 http://arxiv.org/abs/gr-qc/0306018 http://arxiv.org/abs/gr-qc/0409081 http://arxiv.org/abs/astro-ph/0602438 http://arxiv.org/abs/hep-th/0604201 http://arxiv.org/abs/astro-ph/0408415 http://arxiv.org/abs/astro-ph/9908023 Introduction Quadratic Curvature Gravity Constraints Planetary motion Cassini spacecraft A table-top experiment Discussion

0704.0176

Switching mechanism of photochromic diarylethene derivatives molecular junctions

Swit hing me hanism of photo hromi diarylethene derivatives mole ular jun tions Jing Huang, Qunxiang Li, Hao Ren, Haibin Su, Q.W.Shi, and Jinlong Yang Hefei National Laboratory for Physi al S ien es at Mi ros ale, University of S ien e and Te hnology of China, Hefei, Anhui 230026, People's Republi of China Division of Materials S ien e, Nanyang Te hnologi al University, 50 Nanyang Avenue, 639798, Singapore (Dated: November 4, 2018) Abstra t The ele troni transport properties and swit hing me hanism of single photo hromi diarylethene derivatives sandwi hed between two gold surfa es with losed and open on�gurations are inves- tigated by a fully self- onsistent nonequilibrium Green's fun tion method ombined with density fun tional theory. The al ulated transmission spe tra of two on�gurations are strikingly distin - tive. The open form la ks any signi� ant transmission peak within a wide energy window, while the losed stru ture has two signi� ant transmission peaks on the both sides of the Fermi level. The ele troni transport properties of the mole ular jun tion with losed stru ture under a small bias voltage are mainly determined by the tail of the transmission peak ontributed unusually by the perturbed lowest perturbed uno upied mole ular orbital. The al ulated on-o� ratio of urrents between the losed and open on�gurations is about two orders of magnitude, whi h reprodu es the essential features of the experimental measured results. Moreover, we �nd that the swit hing behavior within a wide bias voltage window is extremely robust to both substituting F or S for H or O and varying end an horing atoms from S to Se and Te. PACS numbers: 73.63.-b, 85.65.+h, 82.37.Vb http://arxiv.org/abs/0704.0176v1 I. INTRODUCTION A riti al mission of the mole ular ele troni s is to develop innovative devi es at single mole ular s ale. The representative mole ular wires, re ti�ers, swit hes, and transistors have been intensively studied in the past years. Obviously, a single mole ular swit h holds great promise sin e the swit h is a ru ial element of any modern design of memory and logi appli ations. Now various s hemes have been proposed to realize mole ular swit hing pro ess in luding relative motion of mole ule internal stru ture, 3,4,5,6,7 hange of mole ule harge states, and bond �u tuation between the mole ule and their ele tri al onta ts. Re ently, an alternative routine has been suggested to design swit hes based on single stably existing mole ule whi h an reversibly transform between two ondu tive states in response to external triggers. 11,12,13,14 Among various triggers, light is a very attra tive external stim- ulus be ause of the ease of addressability, fast response times, and ompatibility with a wide range of ondensed phases. The swit hing properties through the so- alled photo hromi mole ules have been ar- ried out by several experimental and theoreti al groups. 15,16,17,18,19,20,21,22 In parti ular, the dithienyl y lopentene (DTC) derivatives (as the entral swit hing unit) hold great promise as arti� ial photoele troni swit hing mole ules be ause of their reversible photo-indu ed transformations that modulate ele tri al ondu tivity and their ex eptional thermal sta- bility and fatigue resistan e. 15,16,17,18,19 Using the me hani ally ontrollable break-jun tion te hnique, Dulić et al.16 designed mole ular swit hes based on DTC mole ules (1,2-bis[5′- -a etylsulfanylthien-2 - yl)-2 -methylthien-3 -yl℄ y lopentene) with two thiophene link- ers, however, whi h operates only one-way, i.e. from ondu ting to the insulating state under visible light with λ=546 nm with resistan e hange at 2-3 orders of magnitude. Interestingly, He et al. found that the transition an be opti al two-way for DTC mole ules where H atoms in y lopentene are substituted by six F atoms (�uorined-DTC). The single-mole ule resistan e in the open form is about 130 times larger than that of in the losed stru ture measured by using s anning tunneling mi ros opy with a gold tip. In a parallel study, Kat- sonis et al. used aromati (meta-phenyl) linkers and observed that the light- ontrolled swit hing of single DTC mole ules onne ted to gold nanoparti les was reversible. Very re ently, to improve the poor stability of su h kind of onjugated mole ules with thiols on both ends, Tanigu hi et al. developed an inter onne t method in solution for diarylethene photo hromi mole ular swit hes that an ameliorate ele trode-mole ule binding, mole u- lar orientation, and devi e fun tions. In their experiment, one light- ontrolled swit hing mole ule onsists of a entral �uorined-DTC mole ule, diaryls on two sides and two thiol groups at both ends. The orresponding losed and open forms are shown Figure 1(a). Consequently, the urrent through the mole ular jun tion with losed stru ture is about 20 times larger than that of the open form measured by STM. To gain better understanding of these experimental observations, several theoreti al work have been arried out. 20,21,22 Li et al. performed quantum mole ular dynami s and density fun tional theory (DFT) al ulations on the ele troni stru tures and transport properties through several photo hromi mole ules with several di�erent spa ers sandwi hed between gold onta ts. They hose dithienylethene (DTE) derivatives to model the experimental measured mole ules and predi ted an about 30 times ondu tion enhan ement when on- verting the open form into a losed one by opti al te hnique. Subsequently, two resear h groups independently investigated the swit hing properties of DTC mole ular jun tions, and found that the transmission peak originates from the highest o upied mole ular orbital (HOMO) of the losed form lying near the ele trode Fermi level. 21,22 In their al ulations, the ele trodes are simulated with luster models and the e�e ts on the transport properties oming from six H hydrogen atoms substituted by F atom are not onsidered. Till now, to our best knowledge, there is no theoreti al study about the swit hing me hanism through exa tly the same measured mole ules (�uorined-DTC with two diaryls on two sides, named diarylethene in Tanigu hi et al.'s experiment). In this paper, we employ the non-equilibrium green's fun tion te hnique (NEGF) ombined with DFT method to address ele troni trans- port and swit hing behavior of diarylethene based mole ular jun tion. Moreover, we examine the robustness of this type of swit hing devi e against various hemi al substitution (where six F atoms in the peripheral of y lopentene and S atoms in thienyl are substituted by H and O atoms, respe tively) and alternations of an horing atoms. II. COMPUTATIONAL MODEL AND METHOD The omputational model system is s hemati ally illustrated in Fig. 1(b). The mole ules with open and losed on�gurations are sandwi hed between two gold ele trodes through S-Au bonds. The Au (111) surfa e is represented by a (4×4) ell with periodi boundary on- ditions. Sin e the hollow site on�guration is energeti ally preferable by 0.2 and 0.6 eV than the bridge and atop sites, respe tively, the diarylethene mole ule onne ts to sulfur atoms whi h are lo ated at hollow sites of two Au (111) surfa es. Both ele trodes are repeated by three layers (A, B, and C). The whole system is arranged as (BCA)-(BC-mole ule-CBA)- (CBA), whi h an be divided into three regions in luding the left lead (BCA), the s attering region, and the right lead (CBA). The s attering regions in lude a diarylethene mole ule, two surfa e layers of the left (BC), and three surfa e layers of the right lead (CBA), where all the s reening e�e ts are in luded into the onta t region, within whi h the harge-density matrix is solved self- onsistently with the NEGF method. The ele troni transport properties are studied by the NEGF ombined with DFT al- ulations, whi h are implemented in ATK pa kage. This methodology has been adopted to explain various experimental results su essfully. 26,27 In our al ulations, Ceperley-Alder lo al-density approximation is used. Core ele trons are modeled with Troullier-Matrins nonlo al pseudopotential, and valen e ele trons are expanded in a SIESTA lo alized basis 29,30 A energy uto� of 150 Ry for the grid integration is set to present the a urate harge density. The optimized ele trode-ele trode distan e is 39.5 Å for the losed on�guration whi h is 0.7 Å longer than that of the open one. All atomi positions are relaxed and the orresponding gold-sulfur distan e is 2.5 Å, whi h is lose to the typi al theoreti al values. In addition, we �nd that the geometri hanges of two diarylethene mole ule sandwi hed between two Au(111) surfa es are negligible omparing to the orresponding free mole ules. III. RESULTS AND DISCUSSION A. Ele troni stru tures of free diarylethene mole ules with losed and open stru - tures Atomi positions of two free diarylethene mole ules with losed and open stru tures are optimized by Gaussion03 pa kage at general gradient approximation level. In the ground ele troni states, both optimized on�gurations are featured by out-of-plane distortions. The entral dihedral angle is 60 degrees for the open form, while only about 8 degrees for the losed one. This distortion leads the distan e between arbon and arbon bond (the bond an be broken by photon) lose to 4.0 Å in the open ase ompared to 1.5 Å for the losed on�guration. These important geometri parameters are onsistent with the previous DFT predi tions for bisbenzothienylethene mole ules. Experimental studies have demonstrated that the mole ule an transform reversibly between the losed and open forms by shining ultraviolet and visible lights, respe tively. Drawing from the hemi al intuition, one would expe t that the ele troni stru tures have distin tive hara teristi s due to the signi� ant geometri di�eren e between losed and open stru tures. For example, it is lear that both single and double bonds appearing in the entral swit hing unit get almost swapped within the losed and open on�gurations as shown in Fig. 1(a). The number of double bonds is 9 in the open form in ontrast to 8 in the losed one. Thus, the energy of HOMO in the open form is expe ted to be lower than that in the losed one. In deed, the energies of the HOMO and lowest uno upied mole ular orbital (LUMO) of the losed form are -4.6 and -3.3 eV, respe tively, whereas the HOMO and LUMO energies of the open one are -4.9 and -2.7 eV. The frontier orbital lo alizes primarily on ea h onjugated unit of the mole ule or on the entral swit hing unit for the diarylethene mole ule with open on�guration. The mole ule in the losed form belongs to a onjugated system, whose HOMO and LOMO orbitals are essentially delo alizated π orbitals extending over the entire mole ule. More interesting, when six F atoms in the peripheral of y lopentene are substituted by H atoms, we �nd that the HOMO and LUMO energies of this modi�ed mole ule shift dramati ally to -4.2 and -2.5 eV for the losed form, and to -4.7 and -1.6 eV for the open one respe tively. These remarkable di�eren es of the geometries and ele troni stru tures are expe ted to a�e t signi� antly transport properties. B. Transport properties of diarylethene mole ular jun tions The urrents through the mole ular jun tion with losed and open on�gurations in the bias voltage range [-1.0, 1.0V℄ are al ulated by the Landauer-Bütiker formalism. It should be pointed out that at ea h bias voltage, the urrent is determined self- onsistently under the nonequilibrium ondition. The al ulated I-V urves are presented in Figure 2. The triangles linking with bla k solid lines are for the diarylethene mole ular jun tion, while the ir les linking with short red dotted lines stand for the jun tion where six F atoms in the peripheral of y lopentene are substituted by H atoms. The �lled (empty) symbols orrespond to the losed (open) stru tures. Our al ulations apture the key features of the experimental results. The urrent through the losed form is remarkably higher than that of the open one. When the diarylethene mole ule in the jun tion hanges from a losed on�guration to the open one, the mole ular wire is predi ted to swit h from the on (low resistan e) state to the o� (high resistan e) state. The urrent enhan ement is quanti�ed by the on-o� ratio of urrent de�ned as R(V ) = Iclosed(V )/Iopen(V ). For example, the urrent of the losed form at 1.0 V is about 4.5 µA, whi h is about 500 times larger than that of the open ase. Su h a large on-o� ratio in this given range of bias voltage an be readily measured and is desirable for the real appli ation. Note that the predi ted on-o� ratio at 1.0 V is larger by about one order of magnitude ompared to experiment. We think one possible reason for this dis repan y is the limitation of the omputational method. It is well known that the al ulated value of the urrent through mole ular jun tion using NEGF ombined with DFT is larger about 1-2 orders of magnitude than that of these experimental measured result. 26,31 Other two possible reasons are environment e�e t and geometry dif- feren e. Firstly, solvent e�e t is not onsidered in presented al ulations. Se ondly, in our omputational model, diarylethene mole ules are dire tly bound to gold ele trodes through Au-S bonds in va uum. In the experimental setup, the entral swit hing mole ules bind to the long orientation ontrol mole ules (polyrotaxane), whi h onne t to the interfa e ontrol mole ules (4-iodobenzenethiol) an hored with gold nanoele trodes in solution (the distan e between two ele trodes is about 30 nm). Note that the slight geometri distortion due to the mole ule-ele trode intera tion an result in a slight asymmetry in the al ulated I-V urves at small bias voltage range as shown in the inset below right of Fig. 2 in small s ale for larity. To understand the dramati di�eren e in ondu tivities of the losed and open on�gu- rations, we ompute the energy dependen e of total zero-bias voltage transmission spe tra shown in Figure 3, where the Fermi level (EF ) is set to be zero for larity. Generally speaking, the ondu tan e of the mole ular jun tion is determined by the number of the eigen hannel, the properties of the perturbed frontier orbitals of the mole ule due to the presen e of the gold ele trodes and the alignment of the metal Fermi level within the per- turbed HOMO-LUMO gap. Applying an e�e tive s heme named mole ular proje ted self- onsistent Hamiltonian (MPSH) method, the orbital energies and eigenstates (referred as perturbed MOs) of the MPSH are obtained and plotted in Fig. 3. The energy positions of these perturbed MOs relative to the EF are denoted in Figs. 3(a) and 3(b) with red short verti al lines, whi h mat h ni ely with the transmission peaks. The spatial distributions of the perturbed-HOMOs and -LUMOs are presented in Fig. 3 lo ating on the right and left sides of the EF , respe tively. Both al ulated ondu tan es are very small at zero bias. It is 4.2×10 G0 (G0=2e /h) for the losed on�guration at the EF , and only 5.4×10 for the open one whi h is about 800 times smaller than the former one. The diarylethene mole ule with a losed stru ture has two broad and strong transmission peaks lo ating at -0.8 and 0.5 eV, respe tively. For the open form, note that the la k of any signi� ant peaks in between -1.5 and 1.7 eV learly elu idates its lower ondu tivity. More importantly, the transmission spe tra display extraordinarily dis repant hara ter- isti s. It is lear that for the diarylethene mole ular jun tion with losed stru ture, the signi� ant transmission peaks lo ating below and above the EF (about -0.8 and 0.5 eV) are mainly ontributed by the perturbed-HOMO and -LUMO, respe tively. Notably, the perturbed HOMOs and LUMOs of the losed on�guration in Fig. 3(a) are delo alized π- onjugated orbitals, whi h provide good hannels for ele tron tunneling through the mole u- lar jun tion and lead to two signi� ant transmission peaks. Very interestingly, the transport properties are predominated by the tail of the perturbed LUMO ontributed transmission peak at small bias voltage (for example, less than 1.0 V), sin e the transmission oming from the perturbed LUMO is just 0.5 eV away from EF , whi h is 0.3 eV loser than that of the perturbed HOMO. Note that this �nding is di�erent from the mi ros opi pi tures of other existing mole ular jun tions based on photo hromi DTE and DTC swit hing mole ules, 21,22 azobenzene, and quintuple bond [PhCrCrPh℄ mole ules, whose transport properties are prevailed by the transmission peak ontributed by the perturbed HOMO. Yet ontradi torily, the spatial pro�les shown in Fig.3 (b) of the perturbed LUMO strongly lo alizes at the entral swit hing unit with open on�guration. This leads to no appre iable transmission peaks in the wide bias window (i.e. from -1.5 to 1.7 eV). The sig- ni� ant transmission peak at -1.7 eV originates from the perturbed HOMO and HOMO-1 (both are π orbitals) for the open stru ture, however, it is lo ated too far away from the EF . Here, omparing to the losed ase, we note that the position of the perturbed HOMO for the open on�guration is buried deeply below the EF , whi h is onsistent with the previous theoreti al results of DTC mole ules. 21,22 These theoreti al �ndings ensure us to on lude that the sharp ontrast of the alignment of the perturbed orbital energies with respe t to the ele trode Fermi level and the shape of these perturbed frontier mole ular orbitals are the essential auses for the striking ontrast in transport properties through diarylethene mole ular jun tions with losed and open on�gurations. It should be pointed out that the number of transmission paths an not a ount for the dramati di�eren e in ondu tivities of the losed and open on�gurations sin e the eigen hannel analysis indi ates that there is a single eigen hannel for both ases within a wide window (i.e. [-1.5, 1.5 eV℄). A ording to the features mentioned above of the al ulated zero-bias transmission spe tra (Fig. 3), One an spe ulate that this type of mole ular swit h an operate robustly in a pretty wide range of bias voltages with fairly large on -o� ratio. Additional urrents through the diarylethene mole ular jun tions with two di�erent on�gurations at -2.0, -1.5, 1.5 and 2.0 V are also al ulated. The on-o� ratios of urrent are predi ted to be about two orders of magnitude. This suggests that the bias voltage window of this kind of mole ular swit h (in Tanigu hi et al.'s experiments ) with reasonably large on-o� ratio is surprisingly wider than that of other photo-sensitive mole ules. 13,21 Experimentalist found that diarylethene mole ular swit h is reversible when the mole ules are sandwi hed through aromati linkers. 18,23 Theoreti al al ulations argued that whether it an be swit hed reversibly or not depending on the mole ule-ele trode hybridization. The weak intera tion between mole ule and ele trode is required to fa ilitate the desired reversible transition. A ording to these �ndings, the reversible transition between the open and losed on�gurations in this diarylethtene derivatives based mole ular jun tion is highly possible, sin e the mole ules are sandwi hed with phenyl linkers and the mole ule-ele trode hybridization is weak. C. Substituting e�e t on diarylethene mole ule Previous theoreti al al ulations fo us on the end linking groups, no attempts so far have been made to examine the side substituting e�e t on transport properties through the diarylethene derivations. It is important to investigate the ondu tan e of the mole ular jun tion, where six F atoms in the peripheral of y lopentene are substituted by H atoms. The al ulated transmission spe tra for the H-substituted mole ular jun tion with the losed and open forms at zero bias voltage are shown by bla k solid lines in Figures. 4(a) and 4(b), and two orresponding I-V urves are presented in Fig. 2 with �lled and empty ir les (linked by red dotted lines), respe tively. The urrent through the H-substituted mole ular jun tion with losed on�guration is about half of that of the y lopentene with six F atoms in the peripheral. The reasons are summarized in the following three points. (1) The repla ement of H with F on the swit hing unit results in the variation of band gaps. The energy gap of the H-substituted diarylethene mole ule is about 1.7 eV, while the gap of diarylethene (F) mole ule is about 1.3 eV. (2) The alignment of the Fermi level is di�erent for two systems. For the jun tion with the H-substituted mole ule, the peak oming from the perturbed HOMO lo ates at -0.7 eV, whi h is loser to the Fermi level than the perturbed LUMO transmission peak (at 1.0 eV). This result is onsistent with these previous theoreti al studies on other DTC and DTE derivations. 21,22 However, the Fermi level lies lose to the transmission ontributed by the perturbed LUMO for the diarylethene (F) mole ular jun tion, as shown in Fig. 3. (3) The transport properties under small bias voltage are mainly determined by the tail of the transmission peak oming from the perturbed LUMO for the losed diarylethene (F) mole ular jun tion. However, the ondu tivity of the losed H-substituted one is ontrolled the tail of transmission peak ontributed by the perturbed HOMO. Nonetheless, the light- ontrolled swit hing feature is undoubtedly retained. Experimental and theoreti al results revealed that the visible adsorption spe tra hanged when two S atoms of the swit hing unit were substituted by O atoms. Thus, the trans- mission spe tra of the mole ular jun tion shown in Fig.1 (b) where two S atoms in entral swit hing unit are repla ed by O atoms are also al ulated here, as shown in Fig. 4 with red dotted lines. Clearly, the swit hing behavior does not depend sensitively on the O- substituent. However, it should be pointed out that the positions of signi� ant transmission peaks obviously shift when ompared to Fig. 3. Parti ularly, the transmission peaks oming from the perturbed HOMO and LUMO lo ates at -0.7 and 0.8 eV, respe tively. Again, the tail of perturbed HOMO transmission peak ontributes largely to the low bias ele troni ondu tan e. D. E�e t of varing end an horing atoms In general, the transport properties of mole ular jun tions depend nontrivially on the end linking atoms. 36,37 Now we turn to explore the e�e t of alternating end an horing atoms. The al ulated transmission spe tra at zero bias voltage are shown in Figure 5. The bla k solid and red dotted lines stand for end Se- and Te-an hored ases, respe tively. It is lear that the main hara teristi s of the transmission spe tra are maintained and the losed stru ture is undoubtedly more ondu tive. For the end Se-an hored ase, the energies of perturbed MO are quite lose to the data presented in Fig. 3 of the end S-an hored one. Interestingly, learly observable hanges have been shown for the end Te-an hored ase. The transmission peaks originating from the perturbed HOMO and LUMO for the mole ular jun tion onne ting to gold ele trodes through Te atoms lo ates at -1.0 (-0.8 for S-an hored one) and 0.3 eV (0.5 for S-an hored one), respe tively. The very interesting �nding of this study is that the swit hing behavior of diarylethene derivatives based mole ular jun tions is robust to vary end an horing atoms from S to Se and Te. To examine the sensitivity of results shown in Fig. 3 to small hange of the ele trode- ele trode distan e, we ompute the zero-bias transmission spe tra of diarylethene swit hes with the losed and open stru tures as elongating and shortening ele trode-ele trode separa- tion up to 0.3 Å. We �nd that the transmission spe tra experien e little hange, and trans- port properties of this kind of diarylethene mole ular jun tion is not dete tably sensitive to the ele trode-ele trode distan e. It indi ates that this kind of light- ontrolled swit hing based on diarylethene derivatives is stable as a mole ular swit hing devi e. Note that the transport behavior is des ribed by the ele tron elasti s attering theory in our al ulations. The e�e t arising from the ele troni vibration and the a ompanying heat dissipation on the al ulated on-o� ratio an be negle ted be ause of the remarkable di�eren e of the I-V urves. IV. CONCLUSION In summary, we investigate the transport properties of the diarylethene with losed and open stru tures using the NEGF ombined the DFT method. The zero-bias transmission fun tion of two di�erent forms is strikingly distin tive. The open form la ks any signi�- ant transmission peak within a wide energy window, while the losed stru ture has two signi� ant transmission peaks on the both sides of the Fermi level. The ele troni trans- port properties of the mole ular jun tion with losed stru ture under a small bias voltage are mainly determined by the tail of the transmission peak ontributed unusually by the perturbed lowest perturbed uno upied mole ular orbital. The al ulated on-o� ratio of ur- rents between the losed and open on�gurations is about two orders of magnitude, whi h reprodu es the essential features of the experimental measured results. Moreover, although the alignments of the perturbed mole ular orbitals's energies with respe t to the ele trode's Fermi level are not exa tly the same, we �nd that the swit hing behavior within a wide bias voltage window is extremely robust to both substituting F or S for H or O and varying end an horing atoms from S to Se and Te. ACKNOWLEDGMENTS This work was partially supported by the National Natural S ien e Foundation of China under Grants 10674121, 10574119, 50121202, and 20533030, by National Key Basi Resear h Program under Grant No. 2006CB922004, by the USTC-HP HPC proje t, and by the SCCAS and Shanghai Super omputer Center.Work at NTU is supported in part by A*STAR SERC grant (No. 0521170032). Corresponding author. E-mail: liqun�ust .edu. n Corresponding author. E-mail: jlyang�ust .edu. n A. Aviram and M. A. Ratner, Mole ular Ele troni s: S ien e and Te hnology (The New York A ademy of S ien es, New York, 1999); A. Nitzan and M. A. Ratner, S ien e 300, 1384 (2003). C. Joa him, J. K. Gimzewski, and A. Aviram, Nature 408, 541 (2000). B. Y. Choi, S. J. Kahng, S. Kim, H. Kim, H. W. Kim, Y. J. Song, J. Ihm, and Y. Kuk, Phys. Rev. Lett. 96, 156106 (2006). J. Henzl, M. Mehlhorn, H. Gawronski, K. H. Rieder, and K. Morgenstern, Angew. Chem. Int. Ed. 45, 603 (2006). J. Chen, M. A. Reed, A. M. Rawlett, and J. M. 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A. Perpète, Chem. Phys. Lett. 429, 147 (2006). S. H. Ke, H. U. Baranger, and W. T. Yang, J. Am. Chem. So . 126, 15897 (2004). Y. Q. Xue, and M. A. Ratner, Phys. Rev. B 69, 085403 (2004). J. K. Viljas, J. C. Cuevas, F. Pauly, and M. Häfner, Phys. Rev. B 72, 245415 (2005). Figure 1: (Color online) (a) The diarylethene derivative in losed and open on�gurations. (b) A s hemati of the swit hing jun tion. Diarylethene mole ules are sandwi hed between two Au (111) surfa es, and two S an horing atoms are lo ated at the hollow site. The verti al blue line denotes the interfa e between the s attering region and the left or right gold ele trode. Figure 2: (Color online) The al ulated urrent-voltage hara teristi s of the diarylethene and its derivative mole ular jun tions with two di�erent on�gurations. The triangles linking with bla k solid lines are for the diarylethene mole ular jun tion, while the ir les linking with short red dotted lines stand for the jun tion where six F atoms in the peripheral of y lopentene are substituted by H atoms. The �lled (empty) symbols orrespond to the losed (open) stru tures. The inset below right is the I-V urve for the open stru tures (with F and H atoms in the peripheral, respe tively) in small s ale for larity. Figure 3: (Color online) The zero-bias voltage transmission spe tra versus the energy E-EF of diarylethene mole ular jun tions with the losed (a) and open (b) on�gurations. Here, EF is the Fermi level of ele trodes. The red short verti al lines stand for the positions of MPSH mole ular energy levels. The spatial distributions of the perturbed HOMOs and LUMOs are inserted in the �gure, and pla ed at the right and left sides of the EF , respe tively. Figure 4: (Color online) The al ulated transmission spe tra versus the energy E-EF at zero-bias voltage for diarylethene mole ular jun tions with losed (a) and open (b) forms, respe tively. One ase is that six F atoms in the peripheral of entral y lopentene are substituted by H atoms (with bla k solid lines); the other is that two S atoms are repla ed by O atoms (with red dotted lines). The red short verti al lines stand for the positions of MPSH mole ular energy levels. Figure 5: (Color online) The al ulated transmission spe tra for diarylethene mole ular jun tions with losed (a) and open (b) stru tures. The bla k solid and red dotted lines stand for the end an horing Se and Te atoms, respe tively. Here, the red short verti al lines stand for the positions of MPSH mole ular energy levels. Fig.1 of Huang et al. Fig.2 of Huang et al. Fig.3 of Huang et al. Fig.4 of Huang et al. Fig.5 of Huang et al. INTRODUCTION COMPUTATIONAL MODEL AND METHOD RESULTS AND DISCUSSION Electronic structures of free diarylethene molecules with closed and open structures Transport properties of diarylethene molecular junctions Substituting effect on diarylethene molecule Effect of varing end anchoring atoms CONCLUSION ACKNOWLEDGMENTS References

0704.0177

Robust manipulation of electron spin coherence in an ensemble of singly charged quantum dots

Robust manipulation of electron spin coherence in an ensemble of singly charged quantum dots A. Greilich, M. Wiemann, F. G. G. Hernandez † , D. R. Yakovlev § , I. A. Yugova ‡ , and M. Bayer Experimentelle Physik II, Universität Dortmund, D-44221 Dortmund, Germany A. Shabaev ⋆ and Al. L. Efros Naval Research Laboratory, Washington, DC 20375, USA D. Reuter and A. D. Wieck Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (Dated: November 4, 2018, robustcontrol-03-27-07-fin.tex) Using the recently reported mode locking effect [1] we demonstrate a highly robust control of electron spin coherence in an ensemble of (In,Ga)As quantum dots during the single spin coherence time. The spin precession in a transverse magnetic field can be fully controlled up to 25 K by the parameters of the exciting pulsed laser protocol such as the pulse train sequence, leading to adjustable quantum beat bursts in Faraday rotation. Flipping of the electron spin precession phase was demonstrated by inverting the polarization within a pulse doublet sequence. PACS numbers: 72.25.Dc, 72.25.Rb, 78.47.+p, 78.55.Cr The spin of an electron in a quantum dot (QD) is an attractive quantum bit candidate [2, 3, 4, 5] due to its favorable coherence properties [1, 6, 7, 8]. As the inter- action strength is rather small for direct spin manipula- tion, the idea to swap spin into charge has been furbished [6, 9, 10]. For example, the electron may be converted into a charged exciton by optical injection of an electron- hole pair [10], depending on the residual electron’s spin orientation, leading to distinctive polarization selection rules. The fundamental quantity regarding spin coherence is the transverse relaxation time T2 . In a QD ensemble, this time is masked by dephasing, mostly caused by dot- to-dot variations of the spin dynamics. The dephasing time does not exceed 10 ns, much shorter than T2 . This leads to the general believe that manipulations ought to -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 = 1.86 ns 3.26 ns 3.66 ns 3.76 ns 3.86 ns = 4.26 ns B = 6 T, T = 6 K 4.92 ns 5.22 ns 5.42 ns 5.62 ns 5.92 ns Time (ns) FIG. 1: Faraday rotation traces measured as function of delay between probe and first pump pulse at time zero. A second pump pulse was applied, delayed relative to the first one by TD , indicated at each trace. The top left trace gives the FR without second pump. be performed on a single spin. Measurement of a single electron spin polarization, however, also results in de- phasing due to temporal sampling of varying nuclear spin configurations [11, 12], as statistically significant mea- surements on a single QD may require multiple repetition of the experiment. The dephasing can be overcome by spin-echo techniques, which give a single electron spin co- herence time on the scale of micro-seconds [8]. This long coherence time derived by spin-echo is result of a refo- cusing of the electron spin and possibly the nuclear spin configuration [11], and it is viewed as an upper bound on the free-induction decay of spin coherence [11, 13]. Recently, however, we have shown that mode locking of electron spin coherence allows one to overcome the en- semble dephasing [14] and to measure the single electron spin relaxation time T2 without applying spin-echo re- focusing [1]. For monitoring the coherence, pump-probe Faraday rotation (FR) measurements [15] on a QD en- semble were used: after optical alignment of the spins normal to an external magnetic field the electron spins precess about this field. Due to precession frequency variations the ensemble phase coherence is quickly lost. However, a periodic train of circularly polarized pulses emitted by a mode-locked laser synchronizes those spin precession modes, for which the precession frequency is a multiple of the laser repetition rate. This synchroniza- tion leads to constructive interference (CI) of these modes in the ensemble spin polarization before arrival of each pump pulse (see Fig. 1, upper left trace). The limit for spin mode locking is set by the single electron spin co- herence time which can last up to a few microseconds [1] reaching the low bound on echo-like decays [16]. Here we develop a detailed understanding of the de- gree of control which can be reached for the electron spin coherence in an ensemble of singly charged QDs by ex- ploiting the mode locking. For this purpose trains of excitation pump pulse doublets were designed to vary http://arxiv.org/abs/0704.0177v1 the phase synchronization condition (PSC) for electron spin precession frequencies. The PSC selects a QD sub- set, whose contribution to the ensemble spin polarization shows a well controlled phase recovery. Variation of the pulse separation results in tunable patterns of quantum oscillation bursts in time-resolved FR, in good agreement with our calculation, which rely on a newly developed theoretical model. This tailoring of electron spin coher- ence is very robust, as the spin mode locking is stable up to 25 K. For higher temperatures the coherence ampli- tude decreases due to phonon-assisted scattering of holes during the laser pulse excitation by which the spin co- herence is created. The studied self-assembled (In,Ga)As/GaAs QDs were fabricated by molecular beam epitaxy on a (001)-oriented GaAs substrate. The sample contains 20 QD layers with a layer dot density of about 10 10 cm −2 , separated by 60 nm wide barriers [17]. For average occupation by a sin- gle electron per dot, the structures were n -modulation doped 20 nm below each layer with a Si-dopant density matching roughly the dot density. The sample was held in the insert of an optical magneto-cryostat, allowing temperature variation from T = 6 to 50 K. FR with picosecond time resolution was used for study- ing the spin dynamics: Thereby spin polarization along the growth direction ( z -axis) is generated by a circu- larly polarized pump pulse hitting the sample along z , and its precession in a transverse magnetic field B ≤ 7 T along the x -axis is tested by the rotation of the linear polarization of a probe pulse. For optical excitation, a Ti- sapphire laser was used emitting pulses with a duration of ∼ 1.5 ps (full width at half maximum of ∼ 1 meV) at 75.6 MHz repetition rate (corresponding to a repetition period TR = 13.2 ns). The laser energy was tuned into resonance with the QD ground state transition and the laser pulses were split into pump and probe. The pump beam was split further into two pulses with variable de- lay TD in between. The circular polarization of the two pumps could be controlled individually. For detecting the rotation angle of the probe beam linear polarization, a homodyne technique was used. Figure 1 shows FR traces excited by the two-pulse train with a repetition period TR = 13.2 ns, in which both pulses have the same intensity and polarization, and the delay between these pulses TD was varied be- tween ∼ TR/7 and ∼ TR/2 . The FR pattern varies strongly for the case when the delay time TD is commen- surate with the repetition period TR : TD = TR/i with i = 2, 3, ... , and for the case TD 6= TR/i . For commen- surability TD = TR/i , the FR signal shows strong peri- odic bursts of quantum oscillations only at times equal to multiples of TD , as seen in the left panel of Fig. 1 for TD = 1.86 ns≈ TR/7 . Commensurability is also given to a good approximation for delays TD = TR/4 ≈ 3.26 ns and TD = TR/3 ≈ 4.26 ns. For incommensurability of TD and TR , TD 6= TR/i , the FR signal shows bursts of quantum oscillations be- tween the two pulses of each pump doublet, in addition to the bursts outside of the doublet. For example, one can see a single burst in the mid between the pumps for TD = 3.76 and 5.22 ns. Two bursts, each equidis- tant from the closest pump and also equidistant from one another, appear at TD = 4.92 and 5.62 ns. Three equidistant bursts occur at TD = 5.92 ns. Note also that the FR amplitude before the second pump arrival is al- ways significantly larger than before the first pump for any TD . Although the time dependencies of the FR signal look very different for commensurate and incommensurate TD and TR , in both cases they can be fully controlled by designing the synchronization of electron spin pre- cession modes in order to reach CI of their contribu- tions to the FR signal [1]. A train of circularly polar- ized pump pulse singlets synchronizes those spin preces- sions for which the precession frequency satisfies the PSC [1, 18]: ωe = 2πN/TR . Then the electron spin under- goes an integer number, N ≫ 1 , of full 2π rotations in the interval TR between the pump pulses. For a train of pump pulse doublets the PSC has to be extended to account for the intervals TD and TR − TD in the laser excitation protocol ωe = 2πNK/TD = 2πNL/(TR − TD) , (1) where K and L are integers. On first glance this con- dition imposes severe limitations on the TD values, for which synchronization is obtained: TD = [K/(K + L)]TR , (2) which for TD < TR/2 leads to K < L . When Eq. (2) is satisfied, the contribution of synchronized precession modes to the average electron spin polarization Sz(t) is proportional to −0.5 cos[N(2πKt/TD)] . Summing over all relevant oscillations leads to CI of their contributions with a period TD/K in time [1]. The rest of QDs does not contribute to Sz(t) at times longer than the ensem- ble dephasing time. The PSC Eq. (1) explains the posi- tion of all bursts in the FR signal for commensurate and incommensurate ratios of TD and TR . For commensu- rability, K ≡ 1 and TD = TR/(1 + L) according to Eq. (2). In this case CIs should occur with period TD as seen in Fig. 1 for TD = 1.86 ns (L = 6 ). For incommensurability of TD and TR the number of FR bursts between the two pulses within a pump doublet and the delays at which they appear can be tailored. There should be just one burst between the pulses, when K ≡ 2 , because then the CI must have a period TD/2 . A single burst is seen in Fig. 1 for TD = 3.76 and 5.22ns. The corresponding ratios TD/TR are 0.285 and 0.395, respectively. At the same time Eq.(2) gives a ratio TD/TR = 2/(L+ 2) , which is equal to 0.285 and 0.4 for L=5 and 3, respectively, in good accord with experiment. Next, two FR bursts are seen for TD = 4.92 and 5.62ns, corresponding to TD/TR ≈ 0.372 and 0.426. The corresponding CI period TD/3 is reached for K ≡ 3 . Then from Eq.(2) TD/TR = 3/(L + 3) , giving 0.375 -1 0 1 2 3 4 5 6 305 310 315 305 310 315 TD=2TR/7 Time (ns) TD=TR/3 t = TD t = 0 TD=TR/3 Precession frequency (GHz) t = 0 t = TD TD=2TR/7 FIG. 2: (a,b): Spectra of electron spin precession modes, −Sz(t) , which are phase synchronized by the two-pulse train calculated for TD = TR/3 and TD = 2TR/7 at the moments of first ( t = 0 ) and second ( t = TD ) pulse arrival (red). Single-pump spectra are shown in blue. (c): FR traces cal- culated for two ratios of TD/TR . Laser pulse area Θ = π . TR = 13.2 ns. Electron g -factor | ge |= 0.57 and its disper- sion ∆g = 0.005 . B = 6 T. and 0.429 for L = 5 and 4, respectively. Finally, the FR signal with TD = 5.92 ns ( TD/TR ≈ 0.448) shows three FR bursts between the two pumps. The CI period TD/4 is obtained for K ≡ 4 , for which Eq.(2) gives TD/TR = 4/(L + 4) ≈ 0.444 with L = 5 . Obviously good general agreement between experiment and theory is established, highlighting the high flexibility of the laser protocol. In turn, this understanding can be used to in- duce FR bursts at wanted delays TD/K , so that at these times further coherent manipulation of all electron spins involved in the burst is facilitated. However, the question arises how accurate condition Eq. (2) for the TD/TR ratio must be fulfilled to reach phase synchronization. Formally, one can find for any arbitrary TD/TR large K and L values such that Eq. (2) is satisfied with high accuracy. But the above analysis shows, that only the smallest of all available L leads to PSC matching. Experimentally, the facilities to address this point are limited, as the largest TD for which FR signal can be measured are delays around 5 ns between the two pumps. For larger delays the FR bursts shift out of the scanning range. For short TD , on the other hand, the bursts are overlapping with the FR signal from the pump pulses. To answer this question, we have modeled the FR sig- nal for commensurate and incommensurate ratios of TD and TR . Figure 2 shows the results together with spec- tra of synchronized spin precession modes (SSPM) at the moment of the first and second pump pulse arrival. The SSPM were calculated similar those induced by a single pulse train [1]. Figure 2(a) gives the SSPM for com- mensurate TD = TR/3 superimposed on the SSPM cre- ated by a single pulse train with the same TR . Panel (c) shows the FR signal created by such a two pulse train. The SSPM for the considered strong excitation are considerably broadened and contain modes for which ωe = 2πM/TD = 2π3M/TR with integer M , which co- incide with each third mode created by a single pulse train. However, the SSPM given by ωe = 2πN/TR , which do not satisfy the PSC for a two pulse train, are not completely suppressed, because the train synchro- nizes the electron spin precession in some frequency range around the PSC. One sees also, that at t = 0 the two pulse train leads to a significant alignment of electron spins opposite to the direction of spins satisfying the PSC. This ”negative” alignment decreases the CI mag- nitude and therefore the FR signal before the first pulse arrival, and is also responsible for a significantly larger magnitude of the FR signal before the second pulse ar- rival [see Figs. 1 and 2 (c)]. For incommensurate ratios of TD and TR the SSPM become much more complex. Still we are able to recover the modes which satisfy the PSC at the pulse arrival times. In Fig. 2 (b) we show the SSPM at t = 0 and t = TD for TD = 2TR/7 (K = 2 , L = 5 ), where the arrows indicate the frequencies which satisfy the PSC for the two pulse train. Only a small number of such modes fall within the average distribution of electron spin precession modes, because the distance between the PSC modes is proportional to 2πK/TD = 2π(K +L)/TR . The diluted spectra of PSC modes for incommensurability decrease the magnitude of the FR bursts between the pump pulses, in accord with experiment. This shows, that although any ratio of TD/TR can be satisfied by large K and L , the FR signal between the pulses should be negligibly small in this case. Consequently, not any ratio of TD/TR leads to pronounced FR bursts. To obtain further insight into the tailoring of electron spin coherence, which can be reached by a two-pulse train, we have turned from co- to counter-circularly po- larized pumps. The delay between pumps TD was fixed at TR/6 ≈ 2.2 ns. The time dependencies of the corre- sponding FR signals are similar, as shown in Fig. 3. Be- sides the two FR bursts directly connected to the pump pulses, one sees a burst +1 due to CI of spin synchro- nized modes. The insets in Fig. 3 (a) show closeups of the different FR bursts. The sign, κ , of the FR am- plitude for the counter-circular configuration undergoes 2TD -periodic changes in time relative to the co-circular case, as seen in Fig. 3 (b), which demonstrates optical switching of the electron spin precession phase by π in an ensemble of QDs. The observed effect of sign reversal is well described by our model. Let us consider first a two-pulse train with delay time TD = TR/2 for which the two pumps -1 0 1 2 3 4 5 0 1 2 3 4 5 co, counter, Time (ns) pump 1 pump 2 +1 burst - (b) Time (ns) B = 3 T T = 6 K (c) FIG. 3: (a): Faraday rotation traces in the co-circularly (blue traces) or counter-circularly (red traces) polarized two pump pulse experiments measured for TD = 2.2 ns and B = 6T. Insets give close-ups showing the relative sign, κ , of the FR amplitude between the two traces. κ is plotted in (b) vs time. (c): Effect of temperature on the FR amplitude in two- pump-pulse experiment. TD = 1.88 ns. are counter-circularly polarized. In this case an electron spin can be synchronized only if at the moment of pulse arrival it has an orientation opposite to the orientation at the previous pulse. This leads to the PSC ωe = 2π(N + 1/2)/TD . The contribution of such precession modes to the electron spin polarization is proportional to cos[2π(N+1/2)t/TD)] = cos(2πNt/TD) cos(πt/TD)− sin(2πNt/TD) sin(πt/TD) . Summing these contribu- tions, only the first term gives a CI, whose modulus has period TD , while the sign of cos(πt/TD) changes with period 2TD . Only each third of the precession frequen- cies can be synchronized by a counter-circularly polarized two pulse train when the delay time is TR/6 as in our ex- periment. However, the corresponding PSC has the same dependence on TD . The CI modulus also has period TD and its sign changes with period 2TD . The relative sign of the FR amplitude for the counter- and co-circularly case, κ = sgn{cos[πt/TD]} , is in accord with the exper- imental dependence in Fig. 3 (b). The CIs of the electron spin contributions can be seen only as long as the coherence of the electron spins is main- tained. In this respect the temperature stability of the CI is especially important. Fig. 3 (c) shows FR traces in a two-pump-pulse configuration with TD = 1.88 ns at different temperatures. For both positive and negative delays, the FR amplitude at a fixed delay is about con- stant for temperatures up to 25 K, irrespective of slight variations which might arise from changes in the phase synchronization of QD subsets. Above 30 K a sharp drop occurs, which can be explained by thermally activated destruction of the spin coherence. The electron spin coherence in charged QDs is initi- ated by generation of a superposition of an electron and a charged exciton state by resonant pump pulses [17, 19]. The simultaneous decrease of the FR magnitude before each pump pulse and afterwards (when the CI signal is controlled by the excitation pulse) suggests that the co- herence at elevated temperatures is lost already during its generation. The 30K temperature threshold corresponds to an activation energy of ∼ 2.5 meV. This energy may be assigned only to the splitting between the two lowest confined hole levels, because the electron level splitting dominates the 20 meV splitting between p- and s-shell emission in photoluminescence and is much larger than 2.5 meV. The decoherence of the hole spin results from two phonon scattering, which is thermally activated and should occur on a sub-picosecond time scale, i.e. within the laser pulse [20]. The fast decoherence of the hole spin at T > 30 K suppresses formation of the electron-trion superposition state. ps-pulses as used here are therefore not sufficiently short for initialization of the superposi- tion and creation of a long-lived electron spin coherence. In summary, we have demonstrated that the mode- locking effect allows a far-reaching control of electron spin coherence in QD ensembles during the spin coher- ence time of microseconds [1]. Two-pulse train mode- locking selects QD subsets which give a non-dephasing contribution to the ensemble spin precession. The tech- nique shows remarkable stability with respect to temper- ature increase up to 25 K, a property which is important for utilizing it in quantum information processing. The robustness of this control technique is provided by the dispersion of the spin precession frequencies in the QD ensemble. Acknowledgments. This work was supported by the BMBF program nanoquit, the DARPA program QuIST, the ONR, the DFG (FOR485) and FAPESP. [ † ] on leave from the Instituto de Fisica Gleb Wataghin, Campinas, SP , Brazil. [ § ] also at Ioffe Physico-Technical Institute, 194021, St. Pe- tersburg, Russia. [ ‡ ] also at Institute of Physics, St. Petersburg State Univer- sity, 198504, St. Petersburg, Russia. [ ⋆ ] also at School of Computational Sciences, George Mason University, Fairfax VA 22030. [1] A. Greilich et al., Science 313, 341 (2006). [2] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [3] A. Imamoglu et al., Phys. Rev. Lett. 83, 4204 (1999). [4] Semiconductor Spintronics and Quantum Computation, ed. by D. D. Awschalom, D. Loss, and N. Samarth (Springer-Verlag, Heidelberg 2002). [5] S. M. Clark et al., cond-mat/0610152. [6] J. M. Elzerman et al., Nature 430, 431 (2004). [7] M. Kroutvar et al., Nature 432, 81 (2004). [8] J. R. Petta et al., Science 309, 2180 (2005). [9] see, for example, R. Hanson et al., Phys. Rev. Lett. 94, 196802 (2005). [10] see, for example, T. Calarco et al., Phys. Rev. A 68, 012310 (2003); P. Chen et al., Phys. Rev. B 69, 075320 (2004). [11] W. Yao, R. Liu, and L. J. Sham, Phys. Rev. B 74, 195301 (2006). [12] R. Liu, S. E. Economou, L. J. Sham, and D. G. Steel, Phys. Rev. B 75, 085322 (2007). [13] W. A. Coish et al., Phys. Stat. Sol. (b) 243, 3658 (2006). [14] For a general treatment on suppression of phase noise see A. G. Kofman and G. Kurizki, Phys. Rev. Lett. 93, 130406 (2004). [15] J. M. Kikkawa and D. D. Awschalom, Science 287, 473 (2000). [16] R. Hanson et al., cond-mat/0610433. [17] A. Greilich et al., Phys. Rev. Lett. 95, 227401 (2006). [18] A. Shabaev et al., Phys. Rev. B 68, 201305(R) (2003). [19] T. A. Kennedy et al., Phys. Rev. B 73, 045307 (2006). [20] T. Takagahara, Phys. Rev. B 62, 16840 (2000). http://arxiv.org/abs/cond-mat/0610152 http://arxiv.org/abs/cond-mat/0610433

0704.0178

Equation of state for dense hydrogen and plasma phase transition

cpp header will be provided by the publisher Equation of state for dense hydrogen and plasma phase transition B. Holst∗1, N. Nettelmann 1, and R. Redmer 1 1 Universität Rostock, Institut für Physik, D-18051 Rostock, Germany Received date?, revised date?, accepted date? Published online date? Key words equation of state, dense hydrogen, phase transitions PACS 51.30+i, 52.25.Jm, 52.25.Kn, 52.35.Tc We calculate the equation of state of dense hydrogen within the chemical picture. Fluid variational theory is generalized for a multi-component system of molecules, atoms, electrons, and protons. Chemical equilibrium is supposed for the reactions dissociation and ionization. We identify the region of thermodynamic instability which is related to the plasma phase transition. The reflectivity is calculated along the Hugoniot curve and compared with experimental results. The equation-of-state data is used to calculate the pressure and temperature profiles for the interior of Jupiter. 1 Introduction The equation of state (EOS) of hydrogen and helium at high pressures is of great relevance for models of the interior of giant planets and other astrophysical objects as well as for inertial confinement fusion experiments. For detailed calculations accurate knowledge of the EOS over a wide range of densities and temperatures is needed. Especially, in the range of warm dense matter with high densities characteristic for condensed matter and at temperatures of a few eV the EOS is crucial for modelling giant planets. This region is challenging for many-particle theory because strong correlations dominate the physical behavior. Progress in shock-wave experimental technique has allowed to study this region only recently. To probe the EOS, experimental investigations were performed statically with diamond anvil cells or dynami- cally by using shock waves, see [1] for a recent review. The experimental data indicate that a nonmetal-to-metal transition occurs at about 1 Mbar which is identified by a strong increase of the conductivity [2] and reflectiv- ity [3]. Some theoretical models yield a thermodynamic instability in this transition region, the plasma phase transition (PPT) [4, 5, 6, 7, 8], which would strongly affect models for planetary interiors and the evolution of giant planets [9, 10, 11]. After a long period of controversial discussions, new results of shock wave experiments on deuterium support the existence of such a PPT [12]. This fundamental problem of high-pressure physics will also be studied with the FAIR facility at GSI Darmstadt within the LAPLAS project, see [13, 14]. In this paper we present new results for the EOS of dense hydrogen within the chemical picture. We treat the reactions pressure dissociation and ionization self-consistently via respective mass action laws. We identify the region of thermodynamic instability and calculate the phase diagram as well as the reflectivity in order to verify the corresponding nonmetal-to-metal transition. The EOS data is used to model the interior of Jupiter within a three-layer model. The agreement with astrophysical constraints such as the core mass and the fraction of heavier elements can serve as an additional test of the theoretical EOS. 2 Equation of state for dense hydrogen Warm dense hydrogen is considered as a partially ionized plasma in the chemical picture. A mixture of a neutral component (atoms and molecules) and a plasma component (electrons and protons) is in chemical equilibrium ∗ Corresponding author: e-mail: [email protected], Phone: +49 381 498 6919, Fax: +49 381 498 6912 cpp data will be provided by the publisher http://arxiv.org/abs/0704.0178v1 2 EOS for dense hydrogen: EOS for dense hydrogen with respect to dissociation and ionization. The EOS is derived from an expression for the free energy of the neutral (F0) and charged particles (F±), see [15, 16]: F (T, V,N) = F0 + F± + Fpol. (1) The first two terms consist of ideal and interaction contributions and can be written as F0 = F 0 and F± = +F int . Fpol contains interaction terms between charged and neutral components caused by polarization [17]. Applying fluid variational theory (FVT), the EOS is determined by calculating the free energy F int0 (T, V,N) via the Gibbs-Bogolyubov inequality [18]. This method has been generalized to two-component systems with a reaction [19, 20, 21] so that also molecular systems at high pressure can be treated where pressure dissociation occurs, e.g. H2 ⇀↽ 2H for hydrogen. In chemical equilibrium, µH2 = 2µH is fulfilled, and the number of atoms and molecules can be determinded self-consistently via the chemical potentials µc = (∂F/∂Nc)T . The effective interactions between the neutral species are modeled by exp-6 potentials, and the free energy of a multi- component reference system of hard spheres has to be known; for details, see [19, 20, 22]. The charged component is treated by using efficient Padé approximations for the free energy developed by Chabrier and Potekhin [23]. The coupling with the neutral component occurs via the ionization equilibrium, H⇀↽e+p. In chemical equilibrium, the relation µH = µe + µp determines the degree of ionization. Since atoms and molecules are particles of finite size there is an additional interaction between the charged component and the neutral fluid. According to the concept of reduced volume, point-like particles cannot pene- trate into the volume occupied by atoms and molecules. This leads to a correction in the description of the ideal gas of the charged component [24, 25] so that the ideal free energy of protons and electrons F id is dependent on the reduced volume V ∗ = V · (1− η), (T, V ∗, N) = N±kBT · f , (2) where η is the ratio of the volume which cannot be penetrated by point-like particles to the total volume. It is derived from hard sphere diameters obtained within the FVT self-consistently. The free energy density f id,∗ given by Fermi integrals which take into account quantum effects. In order to avoid an intersection of pressure isotherms, which is important for modelling planetary interiors, a minimum diameter dmin has been introduced. It was determined starting at low temperatures where it remains almost constant up to 15.000 K, then it increases up to 20.000 K and remains constant again for higher temperatures, see Fig. 1. These values are in the range of the results for the diameter of the hydrogen atom derived from the confined atom model [26]. 5000 10000 15000 20000 T [K] Fig. 1 Minimum diameter for expanded particles (atoms, molecules) introduced within the reduced volume concept. Consequently, the reduced volume concept changes the chemical potential of each component drastically at higher densities and results in pressure ionization. This is due to the fact that additional terms appear in the chemical potential, which is the particle number derivative of the free energy, and thermodynamic functions of degenerate plasmas are very sensitive to changes in density. This current model FVT+ includes all interaction contributions to the chemical potentials, thus being a gener- alization of earlier work [22] where only ideal plasma contributions have been treated (FVT+ cpp header will be provided by the publisher 3 mass density ρ [g/cm 20000 K mass density ρ [g/cm 50000 K Fig. 2 Composition of dense hydrogen for 20.000 K (left) and 50.000 K (right). In Fig. 2 the composition of hydrogen derived from the present approach is shown for two temperatures. Hydrogen is an atomic gas at low temperatures (left) and low densities. With increasing densities molecules are formed due to the mass action law. Pressure dissociation and ionization can be observed in the high-density region. The nonideality corrections to the free energy force a transition from a molecular fluid to a fully ionized plasma. At higher temperatures (right) the formation of molecules is suppressed and pressure ionization becomes the dominating process. At low densities and high temperatures a fully ionized plasma is produced due to thermal ionization. We show pressure isotherms over a wide range of temperatures and densities in Fig. 3. At low densities the system behaves like a neutral fluid. Between densities of 10−3 g/cm3 and 10−1 g/cm3 nonideality corrections to the free energy of atoms and molecules lead to a nonlinear behavior of the isotherms. For still higher densities a phase transition occurs which is treated by a Maxwell construction. The thermodynamic instability vanishes with increasing temperatures, and the critical point is located at 16.800 K, 0.35 g/cm3, and 45 GPa. ρ [g/cm 100000 K 50000 K 30000 K 20000 K 15000 K 5000 K Fig. 3 Pressure isotherms for dense hydrogen. The critical point and the related coexistence line are shown in Fig. 4 and compared with results of other EOS. The critical point itself lies within the range of other predictions, whereas the coexistence line is lower than most of the other results. For a comparison of data concerning the PPT, see Table 1. New shock-wave experiments [12] imply that a PPT occurs in deuterium at densities of 1.5 g/cm3 and a coexistence pressure of about 1 megabar. Each of these values is twice as high as evaluated in the recent model. 4 EOS for dense hydrogen: EOS for dense hydrogen 0 5000 10000 15000 20000 T [K] Fig. 4 Phase diagram for dense hydrogen. Present results of the FVT+ (red) are compared with other predictions for the PPT: SC [4, 5], RK [27], MH [28], ER [29], SBT [30], RRN [31], BEF [32], MCPB [33]. Tc pc ρc Method Authors Reference (103 K) (GPa) (g/cm3) 12.6 95 0.95 PIP Ebeling/Sändig (1973) [34] 19 24 0.14 PIP Robnik/Kundt (1983) [27] 16.5 22.8 0.13 PIP Ebeling/Richert (1985) [29] 16.5 95 0.43 PIP Haronska et al. (1987) [35] 15 64.6 0.36 PIP Saumon/Chabrier (1991) [4] 15.3 61.4 0.35 PIP Saumon/Chabrier (1992) [5] 14.9 72.3 0.29 PIP Schlanges et al. (1995) [30] 16.5 57 0.42 PIP Reinholz et al. (1995) [31] 11 55 0.25 PIMC Magro et al. (1996) [33] 20.9 0.3 0.002 Kitamura/Ichimaru (1998) [36] 16.8 45 0.35 PIP present FVT+ Table 1 Theoretical results for the critical point of the hypothetical plasma phase transition (PPT) in hydrogen which was predicted by Zeldovich and Landau [37] and Norman and Starostin [38]. 3 Conductivity and reflectivity The PPT is an instability driven by the nonmetal-to-metal transition (pressure ionization). We calculate the electrical conductivity as well as the reflectivity by applying the COMPTRA04 program package [39, 40] in order to locate this transition in the density-temperature plane. Optical properties are calculated within the Drude model. The reflectivity R(ω) is given in the long-wavelength limit via the dielectric function ε(ω) which is determined by a dynamic collision frequency ν(ω) or, alternatively, by the dynamic conductivity σ(ω) [41]: R(ω) = ε(ω)− 1 ε(ω)− 1 , (3) ε(ω) = 1− ω [ω + iν(ω)] = 1 + σ(ω), (4) σ(ω) = σ(0) . (5) ωpl = nee2/(ε0me) is the plasma frequency of the electrons. The reflectivity was determined along the Hugoniot curve and is compared with experimental results [3] and those of the earlier model FVT+ [42] in Fig. 5. The results of the current model show a much better agreement with the experiment. The characteristic and abrupt rise with increasing pressure was reproduced more accurately. cpp header will be provided by the publisher 5 This drastic increase appears due to pressure ionization in the vicinity of the criotical point of the PPT. As a result, the reflectivity advances from very low values to metallic-like ones almost instantly. 10 100 1000 P [GPa] 808 nm 1064 nm Celliers et al. 2000 Fig. 5 Reflectivity of dense hydrogen within the models FVT+ and FVT+ along the Hugoniot curve in comparison with experiments [3]. 4 Planetary interiors Modelling the interiors of giant planets and comparison with their observational parameters offers an alternative tool besides laboratory experiments of probing the EOS of the components the planets are predominantly made of. Giant planets such as Jupiter and Saturn consist mainly of hydrogen and, in decreasing order, of helium, water and rocks, covering a wide range of pressures and temperatures. Independently from the H-EOS used for modelling, the simplest interior structure that is compatible with the observational constraints requires at least three homogenous layers with a transition from a cold molecular fluid in the outer envelope to a pressure ionized plasma in the deep interior and a dense solid core of ices and rocks. A solid core may be explained as a result of the formation process and the seperation into two fluid envelopes with different particle abundances by an existence of a PPT as provided by the FVT+ EOS. The constraining observational parameters are the total mass of the planet M , its equatorial radius Req , the temperature T at the outer boundary, the average helium content Ȳ , the period of rotation ω and the gravitational moments J2, J4, J6. From measurements of the luminosity it has been argued [43] that the temperature profile should be adiabatic. For a given EOS, the interior profiles of pressure P and density ρ are calculated by integration of the equation of hydrostatic equilibrium ρ(r, θ) ∇~rP (r, θ) = ∇~r ρ(r, θ) |~r − ~r′| ω2r2 sin θ2 along an isentrope defined by the outer boundary. The first term on the right hand side of eq. (6) is the gravitational potential and the second term the centrifugal potential assuming axialsymmetric rotation. We apply the theory of figures [44] up to third order to solve this equation and to calculate the gravitational moments. They are defined as the coefficients of the expansion of the gravitational potential into Legendre polynomials, taken at the outer boundary. Being integrals of the density distribution weighted by some power of the radius, they are very sensitive with respect to the amount and distribution of helium and heavier elements within the planet. In accordance with previous calculations [45, 46], mixtures of hydrogen with helium and heavier elements have been derived from the EOS of the pure materials via the additive volume rule. It states that the entropy of mixing can be neglected. 6 EOS for dense hydrogen: EOS for dense hydrogen Assuming a three-layer structure, we present results for Jupiter for the profiles of temperature, density, and pressure along the radius in Fig. 6 using two different H-EOS, the standard Sesame table 5251 for hydrogen [47] and the FVT+ model presented above. 0.0001 0.001 0.01 1000 10000 100000 0 2 4 6 8 10 12 radius [RE] profiles inside Jupiter density pressure temperature Sesame Fig. 6 Profiles of temperature, density, pressure along the radius within Jupiter using two different H-EOS, FVT+ (solid) and Sesame 5251 (dashed). The profiles of temperature appear very similar, meaning a small uncertainty about the real profiles. Contrary, the density and pressure profiles exhibit more differences and require some explanation. In the fluid part of Jupiter, the presence of a PPT leads to a jump in density between the envelopes. Since the gravitational moments as integrals over the density have to be the same for both H-EOS, the density profile of a H-EOS with PPT has to be smaller in the outer envelope and larger in the inner envelope. The different size and composition of the core for these specific H-EOS are a consequence of their different compressibility in the regime of pressure ionization at about 1 Mbar, where the gravitational moments are most sensitive to the density distribution. In case of a stiff H-EOS like Sesame, a larger amount of heavy elements is needed in the two fluid envelopes to compensate for the smaller hydrogen density at a given pressure. As a result, this material is added to the well-known density-pressure relation of degenerate electrons in the deep interior, leaving less material for the core. Thus, in case of the Sesame-EOS, the amount of heavy elements becomes with 10% very large and an unlikely solution with a very small core of light material (e.g. water) can be found. In case of the FVT+ EOS which is more compressible than the Sesame EOS at about 1 Mbar, the helium content is below the value of 27.5% for the protosolar cloud in order to reproduce the lowest gravitational moment J2. Furthermore, the next gravitational moment J4 cannot be reproduced correctly because the transition to the metallic envelope occurs already at about 90% of the radius and, thus, at too low densities. For opposite reasons, both the Sesame and FVT+ EOS applied in a three-layer model of Jupiter are not compatible with all of the observational constraints. While Sesame is probably too stiff, the FVT+ model is likely too soft in the WDM region at about 1 Mbar. 5 Conclusions In this paper, we have extended the earlier chemical model FVT+ to calculate the EOS of dense hydrogen. The current model FVT+ includes nonideality corrections to the free energy of each commponent of the partially ionized plasma. We have shown results for the composition and the thermodynamic properties of dense hydrogen. The PPT was located in the phase diagram, its critical point coincides with earlier results. Furthermore, we have determined optical properties such as reflectivity and conductivity, within linear response theory using the program package COMPTRA04. The calculated reflectivity along the experimental Hugoniot curve shows a good agreement with the experiments. However, application of the FVT+ EOS to the interior structure of Jupiter indicates that the behavior at about 1 Mbar is probably too soft. The same conclusion can be drawn from a cpp header will be provided by the publisher 7 comparison with shock-wave experiments that indicate the existence of a PPT [12]. FVT+ predicts the PPT at too low pressures as well as at too low densities. Further efforts to solve this problem, especially concerning the reduced volume concept, are necessary. Acknowledgements We thank P. M. Celliers, W. Ebeling, V. E. Fortov, V. K. Gryaznov, W.-D. Kraeft, and G. Röpke for stimulating discussions. This work was supported by the DFG within the SFB 652 Strongly Correlated Matter in Radiation Fields and the GRK 567 Strongly Correlated Many Particle Systems. References [1] W. J. Nellis, Rep. Prog. Phys. 69, 1479 (2006). [2] S. T. Weir, A. C. Mitchell, W. J. Nellis, Phys. Rev. Lett. 76, 1860 (1996). [3] P. M. Celliers et al., Phys. Rev. Lett. 84, 5564 (2000). [4] D. Saumon and G. Chabrier, Phys. Rev. A 44, 5122 (1991). [5] D. Saumon and G. Chabrier, Phys. Rev. A 46, 2084 (1992). [6] W. Ebeling and G. 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Nettelmann, and B. Holst, AIP Conf. Proc. 845, 127 (2006). [43] W. B. Hubbard, Astrophys. J. 152, 745 (1968). [44] V. N. Zharkov and V. P. Trubytsin, Physics of planetary Interiors, in: Astronomy and Astrophysics Series (Pachart, Tucson/AZ, 1978) [45] G. Chabrier, D. Saumon, W. B. Hubbard, and J. I. Lunine, Astrophys. J. 391, 817 (1992). [46] D. Saumon and T. Guillot, Astrophys. J. 609, 1170 (2004). [47] Sesame table 5251 (1982), derived from Sesame table 5263, G. Kerley, Report LA-4776 (1972). http://www.mpg.uni-rostock.de/sp/pages/comptra Introduction Equation of state for dense hydrogen Conductivity and reflectivity Planetary interiors Conclusions References

0704.0179

Experimental nonclassicality of single-photon-added thermal light states

Experimental nonclassicality of single-photon-added thermal light states Alessandro Zavatta,1, 2, ∗ Valentina Parigi,2,3 and Marco Bellini1, 3, † 1Istituto Nazionale di Ottica Applicata (CNR), L.go E. Fermi, 6, I-50125, Florence, Italy 2Department of Physics, University of Florence, I-50019 Sesto Fiorentino, Florence, Italy 3LENS, Via Nello Carrara 1, 50019 Sesto Fiorentino, Florence, Italy (Dated: October 29, 2018) We report the experimental realization and tomographic analysis of novel quantum light states obtained by exciting a classical thermal field by a single photon. Such states, although completely incoherent, possess a tunable degree of quantumness which is here exploited to put to a stringent experimental test some of the criteria proposed for the proof and the measurement of state non- classicality. The quantum character of the states is also given in quantum information terms by evaluating the amount of entanglement that they can produce. PACS numbers: 42.50.Dv, 03.65.Wj INTRODUCTION The definition and the measurement of the nonclassi- cality of a quantum light state is a hot and widely dis- cussed topic in the physics community; nonclassical light is the starting point for generating even more nonclas- sical states [1, 2] or producing the entanglement which is essential to implement quantum information protocols with continuous variables [3, 4]. A quantum state is said to be nonclassical when it cannot be written as a mixture of coherent states. In terms of the Glauber-Sudarshan P representation [5, 6], the P function of a nonclassi- cal state is highly singular or not positive, i.e. it cannot be interpreted as a classical probability distribution. In general however, since the P function can be badly be- haved, it cannot be connected to any observable quan- tity. In recent years, a nonclassicality criterion based on the measurable quadrature distributions obtained from homodyne detection has been proposed by Richter and Vogel [7]. Moreover, a variety of nonclassical states has recently been characterized by means of the negative- ness of their Wigner function [8, 9, 10, 11], this however being just a sufficient and not necessary condition for nonclassicality [12]. It is still an open question which is the universal way to experimentally characterize the nonclassicality of a quantum state. A conceptually simple way to generate a quantum light state with a varying degree of nonclassicality consists in adding a single photon to any completely classical one. This is quite different from photon subtraction which, on the other hand, produces a nonclassical state only when starting from an already nonclassical one [13, 14]. In this Letter we report the generation and the analy- sis of single-photon-added thermal states (SPATSs), i.e., completely classical states excited by a single photon, ∗Electronic address: [email protected] †Electronic address: [email protected] first described by Agarwal and Tara in 1992 [15]. We use the techniques of conditioned parametric amplifica- tion recently demonstrated by our group [10, 11] to gen- erate such states, and we employ ultrafast pulsed ho- modyne detection and quantum tomography to investi- gate their character. The peculiar nonclassical behavior of SPATSs has recently triggered an interesting debate [7, 16] and has been described in several theoretical pa- pers [14, 15, 16, 17, 18]; their experimental generation has already been proposed, although with more complex schemes [14, 18, 19], but never realized. Thanks to their adjustable degree of quantumness, these states are an ideal benchmark to test the different experimental crite- ria of nonclassicality recently proposed, and to investi- gate the possibility of multi-photon entanglement gener- ation. The nonclassicality of SPATSs is here analyzed by reconstructing their negative-valued Wigner functions, by using the quadrature-based Richter-Vogel (RV) crite- rion, and finally comparing these with two other methods based on quantum tomography. In particular, we show that the so-called entanglement potential [20] is a sensi- tive measurement of nonclassicality, and that it provides quantitative data about the possible use of the states for quantum information applications in terms of the entan- glement that they would generate once sent to a 50-50 beam-splitter. EXPERIMENTAL The main source of our apparatus is a mode-locked Ti:Sa laser which emits 1.5 ps pulses with a repetition rate of 82 MHz. The pulse train is frequency-doubled to 393 nm by second harmonic generation in a LBO crystal. The spatially-cleaned UV beam then serves as a pump for a type-I BBO crystal which generates spontaneous para- metric down-conversion (SPDC) at the same wavelength of the laser source. Pairs of SPDC photons are emitted in two distinct spatial channels called signal and idler. Along the idler channel the photons are strongly filtered http://arxiv.org/abs/0704.0179v1 mailto:[email protected] mailto:[email protected] in the spectral and spatial domain by means of etalon cavities and by a single-mode fiber which is directly con- nected to a single-photon-counting module (further de- tails are given in [9, 11]). The signal field is mixed with a strong local oscillator (LO, an attenuated portion of the main laser source) by means of a 50% beam-splitter (BS). The BS outputs are detected by two photodiodes connected to a wide-bandwidth amplifier which provides the difference (homodyne) signal between the two pho- tocurrents on a pulse-to-pulse basis [21]. Whenever a single photon is detected in the idler channel, an homo- dyne measurement is performed on the correlated spatio- temporal mode of the signal channel by storing the corre- sponding electrical signal (proportional to the quadrature operator value) on a digital scope. FIG. 1: (color online) Experimental setup. HR (HT) is a high reflectivity (transmittivity) beam splitter; SPCM is a single- photon-counting module; all other symbols are defined in the text. The mode-cleaning fiber used to inject the thermal state coming from the rotating ground glass disk (RD) into the parametric crystal is not shown here for clarity. When no field is injected in the SPDC crystal, con- ditioned single-photon Fock states are generated from spontaneous emission in the signal channel [8, 9]. We have recently shown that, if the SPDC crystal is injected with a coherent state, stimulated emission comes into play and single-photon excitation of such a pure state is obtained [10, 11]. However, a coherent state is still at the border between the quantum and the classical regimes; it is therefore extremely interesting to use a truly clas- sical state, like the thermal one, as the input, and to observe its degaussification [13]. In order to avoid the technical problems connected to the handling of a true high-temperature thermal source, we use pseudo-thermal one, obtained by inserting a rotating ground glass disk (RD) in a portion of the laser beam (see Fig.1). By cou- pling a fraction (much smaller than the typical speckle size) of the randomly scattered light into a single-mode fiber, at the output we obtain a clean spatial mode with random amplitude and phase yielding the photon distri- bution typical of a thermal source [22] which is then used to inject the parametric amplifier. PROPERTIES OF SPATSS In order to describe the state generated in our exper- iment, we give a general treatment of photon addition based on conditioned parametric amplification. By first- order perturbation theory, the output of the parametric amplifier when a pure state |ϕm〉 is injected along the signal channel is given by |ψm〉 = [1 + (gâ†sâ i − g ∗âsâi)] |ϕm〉s |0〉i , (1) where g accounts for the coupling and the amplitude of the pump and â, ↠are the usual noncommuting annihi- lation and creation operators. For a generic signal input, the output state of the parametric amplifier can be writ- ten as ρ̂out = Pm |ψm〉 〈ψm| (2) where the input mixed state is ρ̂s = Pm |ϕm〉 〈ϕm| and Pm is the probability for the state |ϕm〉. If we condition the preparation of the signal state to single-photon de- tection on the idler channel, we obtain the prepared state ρ̂ = Tri(ρ̂out |1〉i 〈1|i) = |g| 2â†sρ̂sâs. (3) When the input state ρ̂s is a thermal state with mean photon number n̄, we obtain that the single-photon- added thermal state is described by the following density operator expressed in the Fock base: n̄(n̄+ 1) 1 + n̄ n |n〉 〈n|. (4) The lack of the vacuum term and the rescaling of higher excited terms is evident in this expression. The P phase- space representation can be easily calculated and is given by (see also [15]) P (α) = [(1 + n̄)|α|2 − n̄]e−|α| 2/n̄, (5) while the corresponding Wigner function reads as W (α) = |2α|2(1 + n̄)− (1 + 2n̄) (1 + 2n̄)3 e−2|α| 2/(1+2n̄) (6) where α = x + iy. SPATSs have a well-behaved P func- tion which is always negative around α = 0; this feature is also present in the Wigner function and assures their nonclassicality, however both P (0) andW (0) tend to zero in the limit of n̄→ ∞. DATA ANALYSIS AND DISCUSSION After the acquisition of about 105 quadrature values with random phases, we have performed the reconstruc- tion of the diagonal density matrix elements using the maximum likelihood estimation [23]. This method gives the density matrix that most likely represents the mea- sured homodyne data. Firstly, we build the likelihood function contracted for a density matrix truncated to 25 diagonal elements (with the constraints of Hermiticity, positivity and normalization), then the function is max- imized by an iterative procedure [24, 25] and the errors on the reconstructed density matrix elements are evalu- ated using the Fisher information [25]. The results are shown in Fig. 2, together with the corresponding recon- structed [11] Wigner functions for two different temper- FIG. 2: (color online) Experimentally reconstructed diagonal density matrix elements (reconstruction errors of statistical origin are of the order of 1%) and Wigner functions for ther- mal states (left) and SPATSs (right): a) n̄ = 0.08; b) n̄ = 1.15. Filled circles indicate the density matrix elements calculated for thermal states and SPATSs with the expected efficiencies. atures of the injected thermal state. Since in the low- gain regime the count rate in the idler channel is given by 〈n̂〉 = Tr(ρ̂outâ†i âi) = |g|2(1 + n̄), the mean photon number values n̄ reported in Fig. 2 and in the following are obtained from the ratio between the trigger count rates when the thermal injection is present and when it is blocked (see Ref. [11] and references therein). The finite experimental efficiency in the preparation and homodyne detection of SPATSs is fully accounted for by a loss mechanism which can be modeled by the trans- mission of the ideal state ρ̂ of Eq.(4) through a beam splitter of trasmittivity η coupling vacuum into the de- tection mode, such that the detected state ρ̂η is finally found as: ρ̂η = TrR{Uη(ρ̂ |0〉 〈0|)U †η} (7) where Uη is the beam splitter operator acting on two in- put modes containing the state ρ̂ and the vacuum, and the states of the reflected mode (indicated by R) are traced out. In the case of finite efficiency the expression for the Wigner function thus results: Wη(α) = 1 + 2η[n̄+ 2(1 + n̄)|α|2 − 2n̄η − 1] (1 + 2n̄η)3 −2|α|2 1+2n̄η . It should be noted that the value of experimental ef- ficiency which best fits the data is the same (η = 0.62) as that obtained for single-photon Fock states (i.e., with- out injection), and implies that only a portion of vac- uum due to losses enters the mode during the generation of SPATS. Thanks to a very low rate of dark counts in the trigger detector, the portion of the injected thermal state which survives the conditional preparation proce- dure and contributes to degradation of the SPATSs is in fact completely negligible. However, since the nonclassi- cal features of the state get weaker for large n̄, a limited efficiency (η < 1) has the effect of progressively hiding them among unwanted vacuum components. Indeed, the measured negativity of the Wigner func- tion at the origin (see Fig.3a and b) rapidly gets smaller as the mean photon number of the input thermal state is increased. With the current level of efficiency and recon- struction accuracy we are able to prove the nonclassical- ity of all the generated states (up to n̄ = 1.15), but one may expect to experimentally detect negativity above the reconstruction noise, and thus prove state nonclassical- ity, up to about n̄ ≈ 1.5 (also see Fig.6a). It should be noted that, even for a single-photon Fock state, the Wigner function loses its negativity for efficiencies lower than 50%, so that surpassing this experimental threshold is an essential requisite in order to use this nonclassicality criterion. After having experimentally proved the nonclassical- ity of the states for all the investigated values of n̄, it is interesting to verify the nonclassical character of the measured SPATSs also using different criteria. The first one has been recently proposed by Richter and Vogel [7] and is based on the characteristic func- tion G(k, θ) = 〈eikx̂(θ)〉 of the quadratures (i.e., the Fourier transform of the quadrature distribution), where x̂(θ) = (âe−iθ + â†eiθ)/2 is the phase-dependent quadra- ture operator. At the first-order, the criterion defines a phase-independent state as nonclassical if there is a value of k such that |G(k, θ)| ≡ |G(k)| > Ggr(k), where -2 -1 0 1 2 0.0 0.4 0.8 1.2 -0.16 -0.12 -0.08 -0.04 0.08 0.34 0.70 1.15 Classical limit FIG. 3: (color online) a) Sections of the experimentally re- constructed Wigner functions for SPATSs with different n̄; b) Experimental values for the minimum of the Wigner func- tion W (0) as a function of n̄ for SPATSs (solid squares) and for single-photon Fock states (empty circles) obtained by blocking the injection; the values calculated from Eq.(8) for η = 0.62 (solid curves) are in very good agreement with experimental data and clearly show the appropriateness of the model. Negativity of the Wigner function is a sufficient condition for affirming the nonclassical character of the state. Ggr(k) is the characteristic function for the vacuum mea- sured when the signal beam is blocked before homodyne detection. In other words, the evidence of structures nar- rower than those associated to vacuum in the quadrature distribution is a sufficient condition to define a nonclas- sical state [12]. However, it has been shown that non- classical states exist (as pointed out by Diósi [16] for a vacuum-lacking thermal state [17], which is very similar to SPATSs) which fail to fulfil such inequality; when this happens, the first-order Richter-Vogel (RV) criterion has to be extended to higher orders: the second-order RV inequality reads as 2G2(k/2)Ggr(k/ 2)−G(k) > Ggr(k). (9) It is evident that, as higher orders are investigated, the increasing sensitivity to experimental and statistical noise may soon become unmanageable. The measured |G(k)| and left hand side of Eq. (9) are plotted in Fig. 4a) and b), together with the Ggr(k) char- acteristic function, also obtained from the experimental quadrature distribution of vacuum. While the detected 0 2 4 6 8 0 2 4 6 8 1.0a) Ggr(k) 0.53 0.70 0.90 1.15 4 6 8 Ggr(k) 0.08 0.34 FIG. 4: (color online) Experimental characteristic functions involved in the RV nonclassicality criterion for the detected SPATSs: a) first order; b) second order (the inset shows a magnified view of the region where the state with n̄ = 0.53 is just slightly fulfilling the criterion). SPATSs satisfy the nonclassical first-order RV criterion only for the two lowest values of n̄, it is necessary to extend the criterion to the second order to just barely show nonclassicality at large values of k for n̄ = 0.53 (see the inset of Fig.4b, where the shaded region indicates the error area of the experimental Ggr(k)). At higher temperatures, no sign of nonclassical be- havior is experimentally evident with this approach, al- though the Wigner function of the corresponding states still clearly exhibits a measurable negativity (see Fig.3). It should be noted that the second-order RV criterion for the ideal state of Eq. (4) is expected to prove the nonclas- sicality of SPATSs up to n̄ ≈ 0.6 [7]; however, when the limited experimental efficiency and the statistical noise is taken into account, it will start to fail even earlier. The tomographic reconstruction of the state that was earlier used for the nonclassicality test based on the neg- ativity of the Wigner function, can also be exploited to test alternative criteria: for example by reconstructing the photon-number distribution ρn = 〈n| ρ̂meas |n〉 and then looking for strong modulations in neighboring pho- ton probabilities by the following relationship [26, 27] B(n) ≡ (n+ 2)ρnρn+2 − (n+ 1)ρ2n+1 < 0, (10) introduced by Klyshko in 1996, which is known to hold for nonclassical states. In the ideal situation of unit ef- ficiency SPATSs should always give B(0) < 0 due to the absence of the vacuum term ρ0, in agreement with Ref. [17]. The experimental results obtained for B(0) by using the reconstructed density matrix ρ̂meas are pre- sented in Fig.5a) together with those calculated for the state described by ρ̂η (see Eq.(7)) with η = 0.62. The agreement between the experimental data and the ex- pected ones is again very satisfactory, showing that our model state ρ̂η well represents the experimental one. Our current efficiency should in principle allow us to find neg- ative values of B(0) even for much larger values of n̄; however, if one takes the current reconstruction errors due to statistical noise into account, the maximum n̄ for which the corresponding SPATS can be safely declared nonclassical is of the order of 2. It should be noted that, differently from the Wigner function approach, here the nonclassicality can be proved even for experimental effi- ciencies much lower than 50%, as far as the mean photon number of the thermal state is not too high (see Fig.6b). Finally, it is particularly interesting to measure the en- tanglement potential (EP) of our states as recently pro- posed by Asboth et al. [20]. This measurement is based on the fact that, when a nonclassical state is mixed with vacuum on a 50-50 beam splitter, some amount of entan- glement (depending on the nonclassicality of the input state) appears between the BS outputs. No entangle- ment can be produced by a classical initial state. For a given single-mode density operator ρ̂, one calculates the entanglement of the bipartite state at the BS out- puts ρ̂′ = UBS(ρ̂|0〉〈0|)U †BS by means of the logarithmic negativity EN (ρ̂ ′) based on the Peres separability cri- terion and defined in [28], where UBS is the 50-50 BS transformation. The computed entanglement potentials for the reconstructed SPATS density matrices ρ̂meas are shown in Fig. 5b) together with those expected at the experimentally-evaluated efficiency (i.e., obtained from ρ̂η with η = 0.62). The EP is definitely greater than zero (by more than 13σ) for all the detected states, thus con- firming that they are indeed nonclassical, in agreement with the findings obtained by the measurement of B(0) and W (0). As a comparison, the EP would be equal to unity for a pure single-photon Fock state, while it would reduce to 0.43 for a single-photon state mixed with vac- uum ρ̂ = (1− η) |0〉 〈0|+ η |1〉 〈1| with η = 0.62. To summarize, the three tomographic approaches to test nonclassicality have all been able to experimentally prove it for all the generated states (i.e., SPATSs with 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 Classical limit Classical limit FIG. 5: (color online) a) Experimental data (squares) and calculated values (solid curve) of B(0) as a function of n̄; negative values indicate nonclassicality of the state. b) The same as above for the entanglement potential (EP) of the SPATSs; here nonclassicality is demonstrated by EP values greater than zero. an average number of photons in the seed thermal state up to n̄ = 1.15) for a global experimental efficiency of η = 0.62. In order to gain a better view of the range of values for n̄ and for the global experimental efficiency η which allow to prove the nonclassical character of single- photon-added thermal states under realistic experimen- tal conditions, we have calculated the indicators W (0), B(0), and EP from the model state described by ρ̂η. The results are shown in Fig.6: the contour plots define the regions of parameters where the detected state is classi- cal (white areas), where it would result nonclassical if the reconstruction errors coming from statistical noise could be neglected (grey areas) and, finally, where it is defi- nitely nonclassical even with the current level of noise (black areas). From such plots it is evident that, as al- ready noted, the Wigner function negativity only works for sufficiently high efficiencies, while both B(0) and EP are able to detect nonclassical behavior even for η < 50%. In particular, the entanglement potential is clearly seen to be the most powerful criterion, at least for these par- ticular states, and to allow for an experimental proof of a) b) W(0) EPB(0) FIG. 6: Calculated regions of nonclassical behavior of SPATSs as a function of n̄ and η according to: a) the negativity of the Wigner function at the origin W (0); b) the Klyshko criterion B(0); c) the entanglement potential EP. White areas indicate classical behavior; grey areas indicate where a potentially nonclassical character is not measurable due to experimental reconstruction noise (estimated as the average error on the experimentally reconstructed parameters); black areas indicate regions where the nonclassical character is measurable given the current statistical uncertainties. nonclassicality for all combinations of n̄ and η, as long as reconstruction errors can be neglected. Also considering the current experimental parameters, EP should show the quantum character of SPATSs even for n̄ > 3, thus demonstrating its higher immunity to noise. Although at a different degree, all three indicators are however very sensitive to the presence of reconstruction noise of statistical origin which may completely mask the nonclassical character of the states, even for relatively low values of n̄ or for low efficiencies. In order to unambigu- ously prove the quantum character of higher-temperature SPATSs in these circumstances the only possibility is to reduce the “grey zone” by significantly increasing the number of quadrature measurements. CONCLUSIONS In conclusion, we have generated a completely incoher- ent light state possessing an adjustable degree of quan- tumness which has been used to experimentally test and compare different criteria of nonclassicality. Although the direct analysis of quadrature distributions, done fol- lowing the criterion proposed by Richter and Vogel, has been able to show the nonclassical character of some of the states with lower mean photon numbers, quantum to- mography, with the reconstruction of the density matrix and the Wigner function from the homodyne data, has allowed us to unambiguously show the nonclassical char- acter of all the generated states: three different criteria, the negativity of the Wigner function, the Klyshko crite- rion and the entanglement potential, have been used with varying degree of effectiveness in revealing nonclassical- ity. Besides being a useful tool for the measurement of nonclassicality through the definition of the entanglement potential, the combination of nonclassical field states - such as those generated here - with a beam-splitter, can be viewed as a simple entangling device generating multi- photon states with varying degree of purity and entangle- ment and allowing the future investigation of continuous- variable mixed entangled states [29]. ACKNOWLEDGMENTS The authors gratefully acknowledge Koji Usami for giving the initial stimulus for this work and Milena D’Angelo and Girish Agarwal for useful discussions and comments. This work was partially supported by Ente Cassa di Risparmio di Firenze and MIUR, under the PRIN initiative and FIRB contract RBNE01KZ94. [1] A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, Phys. Rev. A 70, 020101(R) (2004). [2] H. Jeong, A. P. Lund, and T. C. Ralph, Phys. Rev. A 72, 013801 (2005). [3] M. S. Kim, W. Son, V. Bužek, and P. L. Knight, Phys. Rev. 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Jones, J. Haight, and C. T. Lee, Quantum Semi- class. Opt. 9, 411 (1997). [19] M. Dakna, L. Knöll, and D.-G. Welsch, Eur. Phys. J. D 3, 295 (1998). [20] J. K. Asboth, J. Calsamiglia, and H. Ritsch, Phys. Rev. Lett. 94, 173602 (2005). [21] A. Zavatta, M. Bellini, P. L. Ramazza, F. Marin, and F. T. Arecchi, J. Opt. Soc. Am. B 19, 1189 (2002). [22] F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965). [23] K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Phys. Rev. A 61, 010304 (1999). [24] A. I. Lvovsky, J. Opt. B: Quantum Semiclass. Opt. 6, 556 (2004). [25] Z. Hradil, D. Mogilevtsev, and J. Rehacek, Phys. Rev. Lett. 96, 230401 (2006). [26] D. N. Klyshko, Phys. Lett. A 231, 7 (1996). [27] G. M. D’Ariano, M. F. Sacchi, and P. Kumar, Phys. Rev. A 59, 826 (1999). [28] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002). [29] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998).

0704.0180

Neutron Skin and Giant Resonances in Nuclei

Neutron Skin and Giant Resonances in Nuclei Vadim Rodin Institute for Theoretical Physics, University of Tübingen Auf der Morgenstelle 14, D-72076 Tübingen, Germany November 1, 2018 Abstract Some aspects, both experimental and theoretical, of extracting the neutron skin ∆R from properties of isovector giant resonances are discussed. Existing proposals are critically reviewed. The method relying on the energy difference between the GTR and IAS is shown to lack sensitivity to ∆R. A simple explanation of the linear relation between the symmetry energy and the neutron skin is also presented. 1 Introduction Accurate experimental data on the neutron skin in neutron rich nuclei would allow to further constrain model parameters involved in the calculations of the nuclear symmetry energy [1]. The latter plays a central role in a variety of nuclear phenomena. The value a4 ≈ 30 MeV of the nuclear symmetry energy S(ρ0) = a4+ (ρ− ρ0)+ . . . at nuclear saturation density ρ0 ≈ 0.17 fm −3 seems reasonably well established. On the other hand, the density dependence of the symmetry energy can vary substantially with the many-body approximations employed. Several authors have pointed out [2, 3] a strong correlation between the neutron skin, ∆R = 〈r2〉n− 〈r2〉p = Rn−Rp, and the symmetry energy of neutron matter near saturation density. In the framework of a mean field approach Furnstahl [3] demonstrated that in heavy nuclei there exists an almost linear empirical correlation between theoretical predictions in terms of various mean field approaches to S(ρ) (i.e., a bulk property) and the neutron skin, ∆R (a property of finite nuclei). This observation has contributed to a renewed interest in an accurate determination of the neutron skin in neutron rich nuclei. Besides, a precise value of the neutron skin is required as an input in several processes of physical interest, e.g. the analysis of energy shifts in deeply bound pionic atoms [4], and in the analysis of atomic parity violation experiments (weak charge) [5]. It is worth to stress that to experimentally determine the skin in heavy nuclei is extremely challenging as ∆R is just about few percents of the nuclear radius. The present contribution is partially based upon the results published previously in [6]. 2 Relationship between the symmetry energy and ∆R Brown [2] and Furnstahl [3] have pointed out that within the framework of mean field models there exists an almost linear empirical correlation between theoretical predictions for both a4 and its density http://arxiv.org/abs/0704.0180v1 dependence, p0, and the neutron skin ∆R in heavy nuclei. This observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here, following the analysis of [6], this question is addressed from a point of view of the Landau-Migdal approach. Let us consider a simple mean-field model with the Hamiltonian consisting of the single-particle mean field part Ĥ0 and the residual particle-hole interaction Ĥp−h: Ĥ = Ĥ0 + Ĥp−h, Ĥph = (F ′ +G′~σa~σb)~τa~τbδ(~ra − ~rb), (1) Ĥ0 = (Ta + U(xa)), U(x) = U0(x) + U1(x) + UC(x), (2) U0(x) = U0(r) + Uso(x); U1(x) = Spot(r)τ (3); UC(x) = UC(r)(1− τ (3)). (3) Here, U0(x) is the phenomenological isoscalar part of the mean field potential U(x) (x = {~r, ~σ, ~τ}), U0(r) and Uso(x) are the central and spin-orbit parts, respectively; F ′ and G′ are the phenomenological Landau-Migdal parameters. The isovector part U1(x) and the Coulomb mean field UC(x) are both calculated consistently in the Hartree approximation, Spot(r) is the symmetry potential (r-dependent symmetry energy in finite nuclei). The model Hamiltonian Ĥ (1) preserves the isospin symmetry within the RPA if a selfconsistency relation between the symmetry potential and the Landau-Migdal parameter F ′ is fulfilled: Spot(r) = 2F ′n(−)(r), (4) where n(−)(r) = nn(r)− np(r) is the neutron excess density. Thus, in this model the depth of the sym- metry potential is controlled by the Landau-Migdal parameter F ′ (analogous role plays the parameter g2ρ in relativistic mean field models). Spot(r) is obtained from Eq.(4) by an iterative procedure; the resulting dependence of ∆R on the dimensionless parameter f ′ = F ′/(300 MeV fm3) shown in fig. 1 indeed illustrates that ∆R depends almost linearly on f ′. Then with the use of the Migdal relation (1 + 2f ′) [7] relating the symmetry energy and f ′, a similar, almost linear, correlation between a4 and ∆R is obtained. To get more insight in the role of f ′ we consider small variations δF ′. Neglecting the varia- tion of n(−)(r) with respect to δF ′, the corresponding linear variation of the symmetry potential is δSpot(r) = 2δF ′n(−)(r). Then in the first order perturbation theory, such a variation of Spot causes the following variation of the ground-state wave function |δ0〉 = δF ′ 〈s|N̂(−)|0〉 E0−Es |s〉, with “s” labeling the eigenstates of the nuclear Hamiltonian and a single-particle operator N̂ (−) defined as N̂ (−) = n(−)(ra)τ a . Consequently, the variation of the expectation value 〈0|V̂ (−)|0〉 = NR2n − ZR of another single-particle operator V̂ (−) = a can be written as Rpδ(∆R) = δF Re〈0|N̂ (−)|s〉〈s|V̂ (−)|0〉 E0 −Es . (5) In practice the sum in Eq. (5) is exhausted mainly by the isovector monopole resonance (IMR) which high excitation energy (about 24 MeV in 208Pb) justifies the perturbative consideration. Eq. (5) is able to reproduce directly calculated δ(∆R) shown in Fig. 1 with the accuracy of about 10%. As a result, a simple microscopic interpretation of the linear correlation between ∆R and Landau parameter F ′ is obtained. 0.6 0.8 1 1.2 1.4 Figure 1: Neutron skin in 208Pb versus the Landau-Migdal parameter f ′. 3 Extracting neutron skin from properties of isovector giant resonances Parity violating electron scattering off nuclei is probably the least model dependent approach to probe the neutron distribution [8]. The weak electron-nucleus potential is Ṽ (r) = V (r) + γ5A(r), where the axial potential A(r) = GF ρW (r). The weak charge is mainly determined by neutrons ρW (r) = (1 − 4 sin2 θW )ρp(r)− ρn(r), with sin 2 θW ≈ 0.23. In a scattering experiment using polarized electrons one can determine the cross section asymmetry [8] which comes from the interference between the A and V contributions. Using the measured neutron form factor at small finite value of Q2 and the existing information on the charge distribution one can uniquely extract the neutron skin. Some slight model dependence comes from the need to assume a certain radial dependence for the neutron density, to extract Rn from a finite Q 2 form factor. However, the best claimed accuracy of the experimental determination of neutron radii would be on the level of 1%, that translates to relatively large uncertainty of 20-30% in the neutron skin. On such accuracy level, some indirect experimental probes of ∆R still can be competitive. A variety of experimental approaches have been employed to obtain indirect information on ∆R. To some extent all the analysis contain a certain model dependence, which in many cases is difficult to estimate quantitatively. For choosing an indirect probe it is very important to address the question how sensitive is the proposed physical quantity with respect to a variation of ∆R in a single nucleus. The higher is the sensitivity, the better is the choice of the correlation for the indirect deducing ∆R from the measured values. It is not intended here to give a comprehensive review of the existing methods. In particular, the results from the analysis of the antiprotonic atoms, elastic proton and neutron scattering reactions, and the pygmy dipole resonance are completely left out. Here, special emphasis will be put on proposals to provide accurate information on the neutron skin from properties of isovector giant resonances. 3.1 Spin-dipole Giant Resonance In [9] it has been proposed to utilize the excitation probability of the spin-dipole resonance in charge exchange reactions for determining the neutron skin. The method has been applied to obtain information on the variation of the neutron skin in the Sn isotopes [9]. For the relevant operator, a [~σa ⊗ ~ra]JM , (J = 0, 1, 2) the summed ∆L = 1 strength is S(−) − S(+) = C(NR2n − ZR p). (6) Here S(−) and S(+) are the spin-dipole total strengths in β(−) and β(+) channels, respectively; C is the factor depending on the normalization of the spin-dipole operator (in the definition of Ref. [2] C = 1/4π, we use here C = 1). Because S(+) could not be measured experimentally, the model-dependent energy- weighted sum rule was invoked in the analysis of [9] to eliminate S(+). However, the used analytical representation for the sum rule was oversimplified and led in some cases, e.g. for 208Pb, to absurdly negative S(+). In [10] another way was proposed, namely, to use for the analysis the ratio S(+)/S(−) calculated within the pn-RPA. The parameterization of the RPA calculation results for tin isotopes in the form S(+)/S(−) = 0.388− 0.012(N − Z) was used later in [11] to reanalyze the experimental data and led to a marked change in the extracted ∆R’s. Let us now assess the experimental accuracy for S(−) needed to determine the neutron skin to a given accuracy. Putting S(+) = 0 (that seems to be a very good approximation for 208Pb) and one has S(−) = (N − Z)R2p + 2NRp∆R. (7) The ratio of the second term on the rhs to the first one in case of 208Pb is 2N∆R/((N − Z)Rp) ≈ 5.7∆R/Rp. Therefore, for Rp = 5.5 fm and ∆R = 0.2 fm the second term is only 25% of the first one and one needs 5% accuracy in S(−) to determine ∆R with 20% accuracy. Because the SD strength is spread out and probably has a considerable strength at low-energy, the results for the ∆R can be only considered as qualitative with a relatively large uncertainty (up to 30-50%). 3.2 Isobaric analogue state The dominant contribution to the energy weighted sum rule (EWSR) for Fermi excitations by the operator T (−) = a comes from the Coulomb mean field (EWSR)F = UC(r)n (−)(r)d3r, (8) The Coulomb mean field UC(r) resembles very much that of the uniformly charged sphere, being inside a nucleus a quadratic function: UC(r) = (3 − (r/Rc) 2), r ≤ Rc. It turns out that if one extends such a quadratic dependence also to the outer region r > Rc (instead of proportionality to Rc/r), it gives numerically just a very small deviation in (EWSR)F (less than 0.5%, due to the fact, that both the difference and its first derivative go to zero at r = Rc and n (−)(r) is exponentially decreasing for r > Rc). Using such an approximation, one gets: (EWSR)F ≈ (N − Z)∆C 3(N − Z)R2c with ∆C = , and S(−) given in Eq.(7). Since the IAS exhausts almost 100% of the NEWSR and EWSR, one may hope to extract S(−) from the IAS energy. However, the term depending on S(−) contributes only about 20% to (EWSR)F , and as a result, the part of S(−) depending on ∆R contributes only about 4% to (EWSR)F (in 208Pb). ¿From the experimental side, the IAS energy can be determined with unprecendently high accuracy, better than 0.1%. Also, from the experimentally known charge density distribution the Coulomb mean field UC(r) can be calculated rather accurately, and hence one can determine the small difference between Eqs.(9) and (8). But at the level of 1% accuracy several theoretical effects discarded in Eq.(8) come into play that makes such an accurate description of the IAS energy very difficult (the Nolen-Schiffer anomaly). Also in [12] it was stated that the Coulomb displacement energies (CDE) are sensitive to ∆R. A gross estimate ∆R = 0.80(5)(N −Z)/A fm was obtained from a four-parameter fit of the experimental Rp and observed mirror CDE’s. The authors claimed 127 keV to be the rms error of the fit, but they assumed the nuclear wave functions calculated within the Nuclear Shell Model to be isospin pure. Thus, the important effect of the Coulomb mixing of the IAS and the IMR was not taken into account, which is known to decrease the IAS energy by a few percents. Therefore, the Nolen-Schiffer anomaly does not seem to have been resolved yet. 3.3 Is the energy spacing between GTR and IAS a good candidate for determining the neutron skin in isotopic chains? In a recent paper [13] a proposal has been put forward to use the isotopic dependence of the energy spacing, ∆E, between the Gamow-Teller resonance (GTR) and the IAS as a tool for determining the evolution of the neutron skin in nuclei along an isotopic chain. Here, we would like to present some physical arguments which question the physical relevance of this method. The authors of [13] have used the fact that both functions, ∆R and ∆E, are monotonic functions (increasing and decreasing, respectively) of the neutron excess (N−Z) to state that “isotopic dependence of the energy spacings between the GTR and IAS provides direct information on the evolution of neutron skin-thickness along the Sn isotopic chain”. Arguing in such a way one can find a correlation between any two monotonic functions of a single physical parameter and plot them as a function of one another like is done in Fig. 2 of [13] 1. However, it does not imply automatically a real physical correlation between the functions which are determined also by many other model parameters which are kept fixed while performing calculations (the calculations in [13] have been performed within the relativistic mean field (RMF) and relativistic QRPA (RQRPA) approaches). Again, the relevant question to be addressed is how sensitive is one physical quantity with respect to a variation of another in a single nucleus? In other words, one has to evaluate what variation of ∆E is produced by varying ∆R in a single nucleus. Imaging an extreme situation (which is actually not far from reality) that ∆E were not sensitive to ∆R at all, one would get by varying ∆R a family of different calculated dependences (like shown in the upper panel of Fig. 2 of [13]) which would give no clue about the real dependence seen in nature. Thus, it is quite important to understand the physical reasons which cause the energy splitting between the GTR and the IAS. It is well-known that if the nuclear Hamiltonian possessed Wigner SU(4) symmetry then the GTR and the IAS would be degenerate, ∆E = 0. In such a case any variation of ∆R, not violating the symmetry, would not affect ∆E at all. However, it is also known that the spin-isospin SU(4) symmetry is broken in nuclei. Hence, ∆E is determined by those terms in 1note that, to avoid confusion in comparing the measured and calculated dependences, the authors should have plotted the experimental points in the upper panel as the function of the measured ∆R rather than calculated ∆R and should have added the horizontal error bars to them reflecting the experimental uncertainty in ∆R shown in the lower panel. the nuclear Hamiltonian which violate the symmetry. Their qualitative and semi-quantitative estimates in terms of the energy weighted sum rules for the Gamow-Teller (EWSRGT ) and the Fermi (EWSRF ) excitations have been already known for more than 20 years (see, e.g., [14]). The analysis of these authors as well as a quantitative analysis performed recently in [15] has shown that there are three basic sources in the Hamiltonian which violate SU(4) symmetry and contribute to the difference of the sum rules: spin-orbit mean field and both particle-particle and particle-hole residual charge-exchange interactions. One sees that none of the sources explicitly refers to the symmetry potential, to which ∆R is especially sensitive. An estimate of ∆E as ∆E = EWSRGT − EWSRF N − Z can be calculated according to [15] in the Sn isotopes. From the sources violating SU(4) symmetry, spin-orbit mean field represents the major one and contributes about 5 MeV to the splitting. The contribution of the particle-hole interaction is negative and about 1–2 MeV in the absolute value. The contribution of the particle-particle interaction is rather difficult to evaluate (due to uncertainty in the strength of the spin-dependent particle-particle interaction) but it seems to be of minor importance (very probably no more than 0.5 MeV, especially for large (N − Z)) and can safely be neglected. Now let us turn to the discussion of the sensitivity of the contributions to the variation of ∆R. We could reproduce the corresponding analytical expressions from [15] explicitly, but it is enough for our purpose just to mention that the dominating contribution to ∆E from the spin-orbit mean field is given by its expectation value in the ground state and is determined basically only by the unfilled spin-orbit doublets. This expectation value is completely insensitive to the variation of ∆R. Within the Landau-Migdal approach described above, the particle-hole contribution ∆Eph = 2(G′ − F ′) N − Z (n(−)(r))2d3r (10) is given by the product of the volume integral of the neutron excess density squared and the difference of the p-h strengths G′ and F ′ [15]. In the SU(4)-symmetric limit one has G′ = F ′ and ∆Eph = 0 explicitly. Still, in this limit one has a freedom to choose different F ′ that produces a variation in ∆R, similar to shown in fig 1. Therefore, as already mentioned, one can get no clue about the actual ∆R from ∆E = 0 in the SU(4)-symmetric limit. In a realistic situation G′ 6= F ′ (f ′ = 1.0 and g′ = 0.8 were taken in [15]), but ∆Eph depends only on the difference G′ − F ′. One usually fixes G′ in order to reproduce the GTR energy in some nuclei (the authors of [13] have followed this way, too) and possible information from ∆E about the absolute value of F ′ is lost. Furthermore, one can a priori think that a degree of violation of the SU(4) symmetry should be a sort of a fundamental property of the residual interaction.Therefore, the difference G′ − F ′ should stay more stable in different models as compared to some possible variation of F ′ producing different ∆R. Considering value of G′ − F ′ fixed, one can employ a simple model varying only ρn(r) to see how a change of ∆R affects ∆Eph via variation of the neutron excess density n (−)(r). A small variation of ρn(r) can be approximately represented as δρn(r) = − (3ρn(r) +R dρn(r) ), where δRn is a change of the rms neutron radius Rn, R is the nuclear radius (with R n ≈ 0.6R 2). Assuming the proton and neutron densities be constant inside a nucleus, the final estimate is δ∆Eph N + Z − 2γN N − Z ), where γ = n(−)(R)/n(−)(0). Thus, in Sn isotopes with the experimental charge radii about Rp =4.6 fm a rather significant variation of ∆R about 0.1 fm, that is of the order of magnitude of δR, would cause δ∆Eph = 0.3 and δ∆Eph = 0.15 for 112Sn and 132Sn, respectively (γ = 0.5), that corresponds to the absolute change about 0.3 MeV in ∆E, to be compared with the experimental uncertainties in ∆E of the same order. It is clear that to draw any conclusion about ∆R from the measured ∆E would be premature. Even if the experimental errors in ∆E were exactly zero, the accuracy of the theoretical model itself would be hardly believed to be of the necessary level. For instance, apart from the obvious uncertainties in the isotopic dependence of the spin-orbit potential, the GTR does not exhaust 100% of the corresponding sum rules and the shell-structure effects such as configurational and isospin splitting of the GTR can have some effect on the calculated GTR energy. It is also noteworthy that, in spite of the claimed self-consistency of the calculations, the slope of the calculated isotopic dependence of the IAS energy is about 3 times larger than the experimental one (see inset in Fig. 1 of [13]). Note, that the isospin self-consistent continuum-QRPA calculations of [15] were able to nicely reproduce the slope (while overall underestimated the IAS energy by about 0.5 MeV, the well-known Nolen-Schiffer anomaly). To conclude, we believe that the suggested in [13] method to deduce the neutron skin from the energy spacing between GTR and IAS is rather questionable in its origin and does not fairly provide “direct information on the evolution of neutron skin-thickness”. 4 Some implications of ∆R In several processes of physical interest knowledge of ∆R plays a crucial role and in fact a more accurate value could lead to more stringent tests: (i) The pion polarization operator [4] (the s-wave optical potential) in a heavy nucleus Π(ω, ρp, ρn) = −T+(ω)ρ−T−(ω)(ρn−ρp) has mainly an isovector character (T +(mπ) ∼ 0). Parameterizing the densities by Fermi shapes for the case of 208Pb the main nuclear model dependence in the analysis comes from the uncertainty in the value of ∆R multiplying T−. (ii) The parity violation in atoms is dominated by Z−boson exchange between the electrons and the neutrons [5]. Taking the proton distribution as a reference there is a small so-called neutron skin (ns) correction to the parity non-conserving amplitude, δEnspnc, for, say, a 6s1/2 → 7s1/2 transition, which is related to ∆R as (independent of the electronic structure) δEnspnc (αZ)2 . (11) In 133Cs it amounts to a δE/E ≈ −(0.1−0.4)% depending on whether the non-relativistic or relativistic estimates for ∆R are used [5]. The corresponding uncertainty in the weak charge QW is −(0.2− 0.8)σ. (iii) The pressure in a neutron star matter can be expressed as in terms of symmetry energy and its density dependence P (ρ, x) = ρ2 ∂E(ρ, x) = ρ2[E ′(ρ, 1/2) + S ′(ρ)(1− 2x)2 + . . .]. (12) By using beta equilibrium in a neutron star, µe = µn − µp = − ∂E(ρ,x) , and the result for the electron chemical potential, µe = 3/4h̄cx(3π 2ρx)1/3, one finds the proton fraction at saturation density, ρ0, to be quite small, x0 ∼ 0.04. Hence, the pressure at saturation density can be approximated as P (ρ0) = ρs(1− 2x0)(ρ0S ′(ρ0)(1− 2x0) + S(ρ0)x0) ∼ ρ ′(ρ0). (13) At higher densities the proton fraction increases; this increase is more rapid in case of larger p0 [1]. While for the pressure at higher densities contributions from other nuclear quantities like compressibility will play a role in it was argued that that there is a correlation of the neutron star radius and the pressure which does not depend on the EoS at the highest densities. Numerically the correlation can be expressed in the form of a power law, RM ∼ C(ρ,M)( P (ρ) MeVfm−3 )0.25 km, where C(ρ = 1.5ρs,M = 1.4Msolar) ∼ 7. This shows that a determination of a neutron star radius would provide some constraint on the symmetry properties of nuclear matter. 5 Conclusion In this contribution we discuss some aspects of extracting the neutron skin from properties of isovector giant resonances and critically review existing proposals. The theoretical method relying on the energy difference between the GTR and IAS is shown to lack sensitivity to ∆R. It is also shown that the phe- nomenological, almost linear, relationship between the symmetry energy and the neutron skin in finite nuclei, observed in mean field calculations, can be understood in terms the Landau-Migdal approach. Acknowledgments The work is supported in part by the Deutsche Forschungsgemeinschaft (grant FA67/28-2) and by the EU ILIAS project (contract RII3-CT-2004-506222). The author would like to thank Profs. L. Dieperink and M. Urin for useful discussions. References [1] C.J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86 (2001) 5647 ; Phys. Rev. C 66 (2002) 055803 . [2] B.A. Brown, Phys. Rev. Lett. 85 (2000) 5296 . [3] R. J. Furnstahl, Nucl. Phys. A 706 (2002) 85 . [4] E.E. Kolomeitsev, N. Kaiser and W. Weise, Phys. Rev. Lett. 90 (2003) 092501 . [5] S. J. Pollock and M. C. Welliver, Phys. Lett. B 464 (1998) 177 . [6] A.E.L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier, V. Rodin, Phys. Rev. C 68 (2003) 064307 . [7] A.B. Migdal, Theory of finite Fermi-systems and properties of atomic nuclei (Moscow, Nauka, 1983) (in Russian). [8] C. J. Horowitz, S. J. Pollock, P. A. Souder and R. Michaels, Phys. Rev. C 63 (2001) 025501 ; http://hallaweb.jlab.org/parity/prex. [9] A. Krasznohorkay et al., Phys. Rev. Lett. 82 (1999) 3216 . [10] V. Rodin, M. Urin, KVI annual report (2000). [11] M. Csatlós et al., Acta Phys. Polonica B33 (2002) 331 . [12] J. Duflo and A. P. Zuker, Phys. Rev. C 66 (2002) 051304 . [13] D. Vretenar, N. Paar, T. Nikšić, P. Ring, Phys. Rev. Lett. 91 (2003) 262502 . [14] Yu.V. Gaponov, Yu.S. Lyutostansky, V.G. Aleksankin, JETP Lett. 34 (1981) 386 ; T. Suzuki, Phys. Lett. B 104 (1981) 92 ; K. Nakayama, A. Pio Galeao, F. Krmpotic, Phys. Lett. B 114 (1982) 217 . [15] V.A. Rodin and M.H. Urin, Phys. At. Nuclei 66 (2003) 2128 , nucl-th/0201065. http://hallaweb.jlab.org/parity/prex http://arxiv.org/abs/nucl-th/0201065 Introduction Relationship between the symmetry energy and R Extracting neutron skin from properties of isovector giant resonances Spin-dipole Giant Resonance Isobaric analogue state Is the energy spacing between GTR and IAS a good candidate for determining the neutron skin in isotopic chains? Some implications of R Conclusion

0704.0181

Genetic Optimization of Photonic Bandgap Structures

Genetic Optimization of Photonic Bandgap Structures Joel Goh, Ilya Fushman, Dirk Englund, Jelena Vučković Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA [email protected]; [email protected]; [email protected]; [email protected] Abstract: We investigate the use of a Genetic Algorithm (GA) to design a set of photonic crystals (PCs) in one and two dimensions. Our flexible design methodology allows us to optimize PC structures which are optimized for specific objectives. In this paper, we report the results of several such GA-based PC optimizations. We show that the GA performs well even in very complex design spaces, and therefore has great potential for use as a robust design tool in present and future applications. © 2018 Optical Society of America OCIS codes: (130) Integrated optics; (130.2790) Guided waves; (130.3210) Integrated optics devices; (140) Lasers and laser optics; (140.3410) Laser resonators; (140.5960) Semiconductor lasers; (230) Optical devices; (230.5750) Resonators; (230.6080) Sources; (250) Optoelectron- ics; (250.5300) Photonic integrated circuits; (260) Physical optics; (260.5740) Resonance; References and links 1. Sajeev John. Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett., 58(23):2486–2489, Jun 1987. 2. Eli Yablonovitch. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett., 58(20):2059–2062, May 1987. 3. Dirk Englund, David Fattal, Edo Waks, Glenn Solomon, Bingyang Zhang, Toshihiro Nakaoka, Yasuhiko Arakawa, Yoshihisa Yamamoto, and Jelena Vučković. Controlling the spontaneous emission rate of single quan- tum dots in a two-dimensional photonic crystal. Phys. Rev. Lett., 95(013904), July 2005. 4. Misha Boroditsky, Rutger Vrijen, Thomas Krauss, Roberto Coccioli, Raj Bhat, and Eli Yablonovitch. Control of spontaneous emission in photonic crystals. Proceedings of SPIE - The International Society for Optical Engineering, 3621:190–197, 1999. 5. Hatice Altug and Jelena Vučković. Experimental demonstration of the slow group velocity of light in two- dimensional coupled photonic crystal microcavity arrays. Applied Phys. Lett., 86(111102), March 2005. 6. Yurii A. Vlasov, Martin O’Boyle, Hendrik F. Hamann, and Sharee J. McNab. Active control of slow light on a chip with photonic crystal waveguides. Nature, 438:65–69, November 2005. 7. Hatice Altug and Jelena Vučković. Photonic crystal nanocavity array laser. Opt. Express, 13:8819 – 8828, October 2005. 8. Bong-Shik Song, Susumu Noda, Takashi Asano, and Yoshihiro Akahane. Ultra-high-Q photonic double- heterostructure nanocavity. Nature Materials, 4:207–210, 2005. 9. Jelena Vučković, Marko Lončar, Hideo Mabuchi, and Axel Scherer. Design of photonic crystal microcavities for cavity QED. Phys. Rev. E, 65(1):016608, Dec 2001. 10. Dirk Englund, Ilya Fushman, and Jelena Vučković. General recipe for designing photonic crystal cavities. Opt. Express, 13:5961–5975, August 2005. 11. David A.B. Miller Yang Jiao, Shanhui Fan. Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator. Opt. Lett., 30:141–143, 2005. 12. Stefan Preble, Hod Lipson, and Michal Lipson. Two-dimensional photonic crystals designed by evolutionary algorithms. Applied Phys. Lett., 86(061111), 2005. 13. Robert P. Drupp, Jeremy A. Bossard, Douglas H. Werner, and Theresa S. Mayer. Single-layer multiband infrared metallodielectric photonic crystals designed by genetic algorithm optimization. Applied Phys. Lett., 86, Feb 2005. 14. J. H. Holland. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence. Univ. of Michigan Press, 1975. http://arxiv.org/abs/0704.0181v1 15. D. E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley, 1989. 16. L. Davis. Genetic Algorithms and Simulated Annealing. Morgan Kaufmann, 1987. 17. Linfang Shen, Zhuo Ye, and Sailing He. Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm. Phys. Rev. B, 68(035109), 2003. 18. E. Kerrinckx, L. Bigot, M. Douay, and Y. Quiquempois. Photonic crystal fiber design by means of a genetic algorithm. Opt. Express, 12, May 2004. 19. Steven G. Johnson and J. D. Joannopoulos. Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis. Opt. Express, 8(3):173–190, 2001. 20. Yoshihiro Akahane, Takashi Asano, Bong-Shik Song, and Susumu Noda. High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature, 425:944–947, 2003. 21. P. Lalanne, S. Mias, and J. Hugonin. Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities. Opt. Express, 12:458–467, Feb 2004. 22. A. Yariv and P. Yeh. Optical Waves in Crystals: Propagation and Control of Laser Radiation. John Wiley and Sons Inc, 2002. 1. Introduction Photonic crystals (PCs) describe a class of semiconductor structures which exhibit a periodic variation of refractive index in 1, 2, or 3 dimensions. As a result of this periodic variation, PCs possess a photonic band gap – a range of frequencies in which the propagation of light is forbidden [1, 2]. This is the analog of the electronic bandgap in traditional semiconductors. This unique characteristic of PCs enables them to be used to effectively manipulate light. PCs have already been used for applications such as modifying the spontaneous emission rate of emitters [3, 4], slowing down the group velocity of light [5, 6], and designing highly efficient nanoscale lasers [7]. Given that Photonic Crystals find applications in a myriad of areas, we proceed to investigate the question: What is the best possible PC design for a given application? Traditionally, the design of optimal PC structures has been largely done by either trial-and-error, iterative searches through a design space, by physical intuition, or some combination of the above methods [8, 9]. However, such methods of design have their limitations, and recent developments in PC design optimization have instead taken on a more systematic and algorithmic nature [10, 11, 12, 13]. In this work, we report the results of a Genetic Algorithm to optimize the design of a set of one and two-dimensional PC structures. We show that the Genetic Algorithm can effectively optimize PC structures for any given design objective, and is thus a highly robust and useful design tool. 2. Genetic Algorithms Genetic Algorithms (also known as Evolutionary Algorithms) are a class of optimization algo- rithms that apply principles of natural evolution to optimize a given objective [14, 15, 16]. In the genetic optimization of a problem, different solutions to the problem are picked (usually randomly), and a measure of fitness is assigned to each solution. On a given generation of the design, a set of operations, analogous to mutation and reproduction in natural selection, are per- formed on these solutions to create a new generation of solutions, which should theoretically be “fitter” than their parents. This process is repeated until the algorithm terminates, typically after a pre-defined number of generations, or after a particularly “fit” solution is found, or more generally, when a generation of solutions meets some pre-defined convergence criterion. 3. Implementation Genetic Algorithms have already been used in PC design - to find non-intuitive large-bandgap designs [12, 17] and for designing PC fibers [18]. In our work, we performed the genetic opti- mization by varying the sizes of circular holes in a triangular lattice. This approach was chosen because the search space is conveniently constrained in this paradigm, and the optimized struc- tures can be easily fabricated, if desired. A freely available software package [19] was used to simulate the designed structures. In addition, we used the following parameters for the implementation of our Genetic Algo- rithm: Chromosome Encoding. We used a direct-chromosome encoding, where the various opti- mization parameters were stored in a vector. For the current simulations, for simplicity, we only varied the radii of cylindrical holes in a triangular lattice. Our implementation can be easily modified to include other optimization parameters as well, such as the po- sitions of the various holes, or the refractive index of the dielectric material. Selection. We used fitness-proportionate selection (also known as roulette-wheel selection), to choose parent chromosomes for mating. In this selection scheme, a chromosome is selected with a probability Pi that is proportional to its fitness fi, as shown in Eq. (1). Mating. After a pair of parent chromosomes vparent,1 and vparent,2 were selected, they were mated to produce a child chromosome vchild by taking a random convex combination of the parent vectors, as in Eq. (2). λ ∼ U(0,1) ~vchild = λ~vparent,1 +(1−λ )~vparent,2 (2) Mutation. Mutation was used to introduce diversity in the population. We used two types of mutation in our simulations, a random-point crossover and a gaussian mutation. 1) Random-point crossover: For an original chromosome vector ~vorig of length N, we select a random index, k, from 0 to N as the crossover point, and swap the two halves of ~vorig to produce the mutated vector,~vmut , as represented in Eq. (3). ~vorig = (v1,v2, . . . ,vN) k ∼ U{0,1,2, ....,N} ~vmut = (vk+1,vk+2, . . . ,vN ,v1,v2, . . . ,vk−1) T (3) 2) Gaussian mutation: To mutate a chromosome vector by Gaussian mutation, we define each element of ~vmut to be independent and identically distributed Gaussian Random Variables with mean ~vorig and a standard deviation proportional to the corresponding elements of ~vorig. This searches the space in the vicinity of the original chromosome vector~vorig. vmuti ∼ N vorigi ,σ , i ∈ {0,1,2, ....,N} (4) σ2 is a algorithm-specific variance, and can be tuned to change the extent of parameter- space exploration due to mutation. Cloning. To ensure that the maximum fitness of the population was would never decrease, we copied (cloned) the top few chromosomes with the highest fitness in each generation and inserted them into the next generation. 4. Simulation Results 4.1. Optimizing Planar Photonic Cavity Cavities 4.1.1. Q-factor Maximization One problem of interest in PC design is the inverse problem, where one tries to find a dielectric structure to confine a given (target) electromagnetic mode. Here we consider the inverse design problem of optimizing a linear-defect cavity in a planar photonic crystal cavity. The Q-factor is a common figure of merit measuring how well a cavity can confine a given mode, and can be approximated (assuming no material absorption) by the following expression: Qtotal where Q|| represents the Q-factor in the direction parallel to the slab, and Q⊥ represents the Q-factor perpendicular to the slab. Q⊥ is usually the limiting factor for Qtotal. As was shown previously [10, 20], the vertical mode confinement, which occurs through total internal reflection (TIR), can be improved if the mode has minimal k-space components inside the light cone. In the subsequent sections, we report the results where we employed our GA to minimize the light cone radiation of such cavities. We used one-dimensional photonic crystals as approx- imations to these cavities [21], and simulated these cavities using the standard Transfer Matrix method for the E-field [22]. 4.1.2. Matching to a Target Function In [10] it was noted that minimization of light cone radiation could be performed via mode- matching to a target function which already possessed such a property. We therefore used a fitness function that was equal (up to a normalizing factor) to the reciprocal of the mean-squared difference between our simulated mode and a target mode (see Eq (6)). For this simulation, our chromosome encoded the thicknesses of the dielectric slabs in the structure, and was a vector of length 10. We used 100 chromosomes in each generation and allowed them to evolve for 80 generations. f itness ∝ | fsim(x)− ftarget (x)| We used target modes that were sinusoidal functions multiplied by sinc and sinc-squared envelope respectively. Such target modes have theoretically no radiation at or near the Gamma point and are therefore ideal candidates as target functions. The results, shown in Fig. 1, clearly feature a suppression of k-vector components at low spatial-frequencies. Matching using the the sinc-squared envelope target function appeared to produce a better match. From the k- space plots, the GA evidently had difficulty matching the sharp edges for the sinc-envelope target mode. 4.1.3. Direct Minimization of Light Cone Radiation In the preceding subsection, we observed that when we formulated our objective as a matching problem, in the case of the sinc-envelope, the GA sacrificed the desired low spatial-frequency suppression in an effort to match the overall shape of the function. The preceding formulation therefore poses an implicit constraint on our optimization. By reformulating the optimization problem, we were able to effectively remove this constraint, and obtain a better result. 0 100 200 300 400 500 −3 −2 −1 0 1 2 3 k (a/λ) 0 100 200 300 400 500 −3 −2 −1 0 1 2 3 k (a/λ) Fig. 1: Top-left: Real-space mode profile after optimizing for closest-match to a sinc-envelope target mode. Top-right: k-space mode profile of optimized simulated mode and a sinc-envelope target mode. Bottom: Real-space and k-space mode profiles for matching against a sinc2- envelope target mode. Our reformulation directly minimized the k-vector components in the light cone, by min- imizing the integrated square-magnitude of the simulated E-field mode in k-space inside the light cone. The fitness function that we used is given as in Eq (7), where V represents the set of k-vectors within the light cone. f itness = |F(k)|2dk The final, evolved structure, together with the corresponding real-space and k-space mode profiles are shown in Fig 2. The k-space mode profile features a strong suppression of radiation at low frequencies, to a greater extent as compared to the optimized fields from the preced- ing simulations. By relaxing our constraint and performing a direct optimization, our GA has designed a structure that achieves better light cone suppression than before. Our direct opti- mization paradigm has exploited the extreme generality of the GA, which simply requires that a fitness function be defined, with little further constraint thereafter. 4.2. Maximal Gap at any k-vector Point Moving on to the more generic case of 2D photonic crystals, we will proceed to show the results of simulations for maximizing the TE bandgap at any point in k-space for a 2-Dimensional PC structure with a triangular lattice of air holes. This could be useful for PC design applications where the target mode to be confined is centered around a particular point in k-space [10]. By maximizing the bandgap at that k-space point, we would effectively design a better mirror for a mode resonating along this k-space direction. 0 50 100 150 200 250 300 350 400 450 Optimized Mode − real space −3 −2 −1 0 1 2 3 k (a/λ) Optimized Mode − k space Fig. 2: Top: Real-space mode profile of optimized resonant E-field mode. Bottom: Correspond- ing k-space mode profile of optimized mode We used a supercell which was three periods wide in each dimension and varied the radii of the nine holes in total, and we encoded the chromosome as a vector of these nine holes. We used a population size of 60 chromosomes for each generation, and allowed the optimization to run for a total of 100 generations. To evaluate the fitness of each chromosome, we used the eigensolver in Ref [19] to calculate the gap-to-midgap ratio at the K-point of the band diagram. We then scaled the calculated ratio exponentially to tune the selection pressure of the optimization. Figure 3 shows the variation of the gap-to-midgap ratio of our structures as the algorithm progressed. Our Genetic Algorithm performs as expected, and we get a general increase of fitness as the algorithm progresses. All the four runs do not show any significant increase in fitness af- ter Generation 80, at which point they have maximum fitnesses (i.e. ratio of their bandgap to midgap value) of around 72%. All the optimized structures after the run have similar dielectric structures and band diagrams. The dielectric structures and a sample band diagram is shown in Figure 4. 4.3. Optimal dual PC structures As a more complex example, let us consider two similar PC designs, (1) a triangular lattice of air holes in a dielectric slab, and (2) a triangular lattice of dielectric rods in air. Structure (1) possesses a bandgap for TE light, but no bandgap for TM light, while structure (2) possesses a bandgap for TM light, but not for TE light. Our objective is to use the Genetic Algorithm to find a PC design in which the TE eigenmode for structure (1) and the TM eigenmode for structure (2) are most similar. Maxwell’s equations can be cast as eigenproblems for the Electric or Magnetic fields, and our approach could be po- tentially useful in future PC design, because solving the inverse problem is analytically simpler (at least intuitively) for the eigenproblem involving the E-field. We used a 3x3 supercell for the optimization, and we minimize the mean-square difference of the z-components of the electric and magnetic fields of the dual structures at the K-point of the band diagram. We recognize a priori that a trivial solution, which we wish to avoid, is 0 10 20 30 40 50 60 70 80 90 100 Generation number Fig. 3: Fitness (gap-to-midgap ratio at K-point of the band diagram) of maximally-fit struc- ture of each generation for 100 generations. The maximum fitness is a monotonically non- decreasing function due to cloning. A general increase in fitness arises as a result of various genetic operations (selection, mating, mutation). a structure that has a uniform refractive index (either dielectric or air) throughout, and so we prevent the genetic algorithm from obtaining this by restricting our mutation to only a Gaussian mutation (see Eq. 4). This preferentially searches the locality of points, and is a necessary trade- off for obtaining a reasonable solution. This illustrates the versatility of the Genetic approach - the extent of the search can be easily modified by a simple change of algorithm parameters. Fig. 5 shows the optimal dual structures with the corresponding simulated fields. 5. Conclusion From the results above, we have shown that our Genetic Algorithm is able to effectively opti- mize PC designs to meet specific design criteria. Furthermore, by our choice of encoding, we could easily impose constraints upon the design space to ensure that every design searched by the algorithm could be realistically fabricated. Between different optimizations, all that needed to be changed was the measure of how well a given structure complied with our design crite- rion - the ”fitness function” in Genetic Algorithm parlance. Our Genetic Algorithm is therefore highly robust and can be easily modified to optimize any user-defined objective function. (a) Run 1 (b) Run 2 (c) Run 3 (d) Run 4 (e) Band Diagram - optimized (f) Band Diagram - uniform holes, r/a = 0.3 Fig. 4: Dielectric structures (a-d), showing the optimal PC structures predicted by 4 runs our Genetic Algorithm. The unit cell for each structure is depicted by the yellow bounding box. A sample band diagram (for Run 3) is shown in (e). The optimized TE-bandgap, calculated as the ratio of the size of the gap to the midgap value, was found to be ≃ 72%. The TE-bandgap for a triangular lattice with uniform air holes (r/a = 0.3) is shown in (f ) for reference. (a) Band 1, E-field (b) Band 1, H-field (c) Band 2, E-field (d) Band 2, H-field (e) Band 3, E-field (f) Band 3, H-field (g) Band 4, E-field (h) Band 4, H-field Fig. 5: Genetic Algorithm prediction of PC structures that have optimally matched E and H fields, for the lowest 4 bands, at the K point. The E-fields are shown for structure with dielectric rods, and have a TM bandgap, while the H-fields are shown for structures with air holes, and have a TE bandgap. The shown fields are in the direction aligned with the rods. The fields for the lowest 3 bands are very well matched, but begin to deviate significantly from each other at band 4. Introduction Genetic Algorithms Implementation Simulation Results Optimizing Planar Photonic Cavity Cavities Q-factor Maximization Matching to a Target Function Direct Minimization of Light Cone Radiation Maximal Gap at any k-vector Point Optimal dual PC structures Conclusion

0704.0182

Huge magneto-crystalline anisotropy of x-ray linear dichroism observed on Co/FeMn bilayers

Huge magneto-crystalline anisotropy of x-ray linear dichroism observed on Co/FeMn bilayers W. Kuch∗ Freie Universität Berlin, Institut für Experimentalphysik, Arnimallee 14, D-14195 Berlin, Germany F. Offi,† L. I. Chelaru,‡ J. Wang,§ K. Fukumoto,¶ M. Kotsugi,∗∗ and J. Kirschner Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany J. Kuneš Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany. and Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Praha 6, Czech Republic. (Dated: 23.03.2007) We present an x-ray spectromicroscopic investigation of single-crystalline magnetic FeMn/Co bilayers on Cu(001), using X-ray magnetic circular (XMCD) and linear (XMLD) dichroism at the Co and Fe L3 absorption edges in combination with photoelectron emission microscopy (PEEM). Using the magnetic coupling between the ferromagnetic Co layer and the antiferromagnetic FeMn layer we are able to produce magnetic domains with two different crystallographic orientations of the magnetic easy axis within the same sample at the same time. We find a huge difference in the XMLD contrast between the two types of magnetic domains, which we discuss in terms of intrinsic magneto-crystalline anisotropy of XMLD of the Co layer. We also demonstrate that due to the high sensitivity of the method, the small number of induced ferromagnetic Fe moments at the FeMn–Co interface is sufficient to obtain magnetic contrast from XMLD in a metallic system. PACS numbers: 75.70.Ak, 68.37.-d, 75.50.Ee I. INTRODUCTION The recent interest in the magnetic coupling between antiferromagnetic (AF) and ferromagnetic (FM) mate- rials is motivated by the quest for fundamental insight into the phenomenon of exchange bias.1 This effect, the discovery of which dates back to the 1950’s,2 manifests itself in a shift of the magnetization curve along the field axis. Nowadays the exchange bias effect is employed in a variety of devices, such as sensors or hard disk read heads, based on magnetic thin films.3,4 Only few methods can be used to study the spin struc- ture of ultrathin antiferromagnetic films. While neutron diffraction and Mössbauer spectroscopy have been suc- cessfully employed to explore the spin structures of many bulk antiferromagnets already decades ago, both meth- ods suffer from a lack of signal if films of a few atomic layers are to be investigated. X-ray magnetic linear dichroism (XMLD) in the soft x-ray absorption, on the other hand, is a method with sub-monolayer sensitivity. XMLD refers to the difference between x-ray absorption spectra for the plane of x-ray polarization aligned parallel and perpendicular to the atomic moments.5 By symme- try the XMLD signal does not depend on the orientation of magnetic moments but only on their axial alignment, and is thus suitable for the investigation of ferromagnetic as well as collinear antiferromagnetic spin structures, an- tiferromagnetic thin films in particular. Linear dichroic signal is also encountered in case of structural, not mag- netic, reduction of the symmetry, as is commonly the case in the direction along the film normal due to the presence of interfaces.6,7 This can be eliminated if mea- surements are compared in which only the direction of the spin axis is varied, while the lattice geometry is fixed. In the following we will use the acronym “XMLD” for this situation only. Unlike x-ray magnetic circular dichroism (XMCD), which in 3d metallic systems essentially measures inte- gral quantities, namely the spin and orbital magnetic moments, the size and shape of XMLD depend also on the details of the electronic structure. Although in- tegral sum rules have been put forward for XMLD,8 which relate the integral over the XMLD signal to the magneto-crystalline-anisotropy energy, this integral is usually much smaller than the amplitude of the plus– minus feature in the XMLD spectrum. Theoretical cal- culations predict that the latter may vary significantly with the crystallographic orientation of the magnetic moments.9 This is what we call the magneto-crystalline anisotropy of XMLD. In this paper we study the magneto-crystalline anisotropy of XMLD of a thin Co layer. We use the AF– FM coupling between the FeMn and Co layers to manipu- late the orientation of the Co moments, namely the obser- vation that Co moments in a Co/FeMn bilayer align along 〈110〉 directions when in contact with a magnetically dis- ordered (paramagnetic) FeMn layer (above its Néel tem- perature), while they prefer 〈100〉 directions when the FeMn layer magnetically orders (antiferromagnetic).10,11 We take advantage of the fact that the Néel temperature depends on the thickness of the FeMn layer.12 Growing wedge-shaped samples we are thus able to study both 〈110〉- and 〈100〉-oriented domains at the same time. We use a photoelectron emission microscope (PEEM), as described in Refs. 13,14,15 for the microscopic laterally resolved detection of the x-ray absorption cross section of the FeMn/Co bilayers. In combination with x-ray mag- netic circular dichroism (XMCD) in the soft x-ray absorp- tion as a magnetic contrast mechanism, PEEM is rou- tinely used for the element-resolved observation of mag- netic domain patterns in multilayered structures.13,16,17 XMLD can equally serve as the magnetic contrast mech- anism for PEEM if linearly polarized x-rays are used. Here, a combined XMCD and XMLD spectromicroscopic investigation of single-crystalline FeMn/Co bilayers on Cu(001) is presented. We find that the XMLD signal of the FM Co layer exhibits a strikingly different behav- ior when in contact with a paramagnetic and an anti- ferromagnetically ordered FeMn layer, while the XMCD contrast does not differ appreciably. We compare our ob- servations to ab initio calculations of the L3 XMLD in bulk fcc Co for different crystallographic orientations of the magnetic moments. Investigation of the influence of spin and electronic structure on the XMLD requires single-crystalline sam- ples with well characterized AF–FM interfaces. Be- cause of the small lattice mismatch (0.4%),18 Fe50Mn50 films (FeMn in the following) on a Cu(001) single crys- tal are ideal candidates for such investigations. Epitax- ial, virtually unstrained FeMn films can be grown in a layer-by-layer mode by thermal deposition on Cu(001) at room temperature.12 This provides an opportunity to study the magnetic properties of an AF/FM system in single crystalline FeMn/Co and Co/FeMn bilayers on Cu(001).10,12,19 Scanning tunneling microscopy revealed atomically smooth interfaces with islands or vacancies of single atomic height.20 Based on XMCD-PEEM in- vestigations of FM/FeMn/FM trilayers and on XMLD spectroscopy experiments of Co/FeMn bilayers, we con- cluded previously that a non-collinear three-dimensional spin structure is present in the ultrathin FeMn layers, possibly similar to the so-called 3Q spin structure present in bulk FeMn.21 Combination of the Kerr magnetometry and XMCD-PEEM imaging showed that the magnetic coupling across the interface is mediated by step edges of single atom height, while atomically flat areas do not contribute.22 II. EXPERIMENT All experiments were performed in-situ in an ultrahigh vacuum system with a base pressure below 10−8 Pa. The disk-shaped Cu(001) single crystal was cleaned by cycles of 1 keV argon ion bombardment at 300 K and subse- quent annealing at 873 K for 15 minutes. The surface exhibited a sharp (1× 1) low energy electron diffraction pattern. No contaminations were detectable by Auger electron spectroscopy (AES). The films were grown by thermal evaporation on the clean substrate at room temperature in zero external magnetic field. Fe and Co were evaporated by electron bombardment of high purity wires (99.99% purity) of 2 mm diameter, while a rod (99.5% purity) of 4 mm di- ameter was used for Mn. FexMn1−x films of equiatomic composition (x = 0.50 ± 0.02) were obtained by simul- taneous evaporation of Fe and Mn from two different sources. During the deposition the pressure in the cham- ber was kept below 5 × 10−8 Pa. A typical evaporation rate was 1 ML per minute. The composition of the FeMn films was estimated from the evaporation rates of the two sources, determined by medium energy electron diffrac- tion (MEED), and cross-checked by Auger electron spec- troscopy peak ratios. No indication of segregation of Cu into or on top of the FeMn layers was found. The thick- ness of the films was determined by MEED, which shows pronounced layer-by-layer oscillations.12 The FeMn layer was grown in the form of small wedges of 200 µm width, using the method described in Ref. 23. The experiments were performed at the UE56/2- PGM1 helical undulator beamline of the Berlin syn- chrotron radiation facility BESSY, which can be set to deliver circularly polarized radiation of either helicity with a degree of circular polarization of about 80%, or linear vertical or horizontal polarization of > 97%.24 The set-up of the electrostatic PEEM was identical to that de- scribed in Refs. 13,14,15. The light was incident at an angle of 30◦ with respect to the sample surface. Rotation of the sample about the surface normal allowed to take images for different x-ray azimuthal angles of incidence. Parameters were set to a lateral resolution of 350 nm, and a field of view of 60 µm. The XMCD images represent a grayscale-coded ab- sorption asymmetry for opposite helicities of the circu- larly polarized x-rays at the L3 absorption maximum (777.5 eV), AXMCD = I+ − I− I+ + I− , (1) i.e., the difference of absorption images acquired with opposite helicities divided by their sum. For the quan- titative analysis, background images acquired at lower photon energy (5 eV below the L3 maximum) were sub- tracted. For XMLD, the maximum contrast was determined from a series of images acquired with 0.2 eV photon en- ergy step around the maximum of the Co L3 absorption peak of a 6 ML Co/Cu(001) film, using p-polarized light. Maximum contrast was found between images taken at photon energies E1 = 776.5 eV and E2 = 777.9 eV. Since the acquisition time necessary to observe XMLD contrast at the Fe L3 edge in FeMn/Co/Cu(001) bilayers was of the order of hours, no such photon energy sweeps were undertaken for the Fe L3 edge (maximum at 707.5 eV); instead, the same relative photon energies as determined for the Co L3 edge were tentatively used (E1 = 706.5 eV FIG. 1: Magnetic domain images of FeMn/Co/Cu(001) struc- ture obtained at the L3 edges of (a), (b) Co, (c), (d) Fe, and (e) Mn. The thickness of the FeMn layer, increasing from the top to the bottom of each image, is shown on the ver- tical axis. Crystallographic orientation of the Cu substrate and azimuthal angle of incidence are shown in panels (a) and (b), respectively. The left and right columns represent the XMCD and XMLD contrast, respectively. Arrows indicate the domain magnetization. and E2 = 707.9 eV). Images at the two photon energies were taken using s and p polarized x-rays. Because of the 30◦ incidence, XMLD from in-plane magnetization is larger by a factor of 4/3 for s polarized excitation. Tak- ing into account the opposite sign of the effect for the two polarizations we used the following formula for the XMLD contrast AXMLD = Is(E2)− Is(E1) Is(E2) + Is(E1) Ip(E2)− Ip(E1) Ip(E2) + Ip(E1) The images thus reproduce quantitatively the difference between images taken at the higher photon energy minus images taken at the lower photon energy for s polarized x-rays, divided by the sum of these images. III. RESULTS AND DISCUSSION This section is divided into three parts. First, we demonstrate the performance of the spectromicroscopic domain imaging and present the contrast obtained on the L3 edges of Co, Fe and Mn. Next, we study the depen- dence of the XMCD and XMLD contrast at the Co L3 edge on the azimuthal angle of incidence. Finally, we discuss the quantitative difference between the XMLD signal obtained from 〈110〉 and 〈100〉 domains. In Fig. 1 typical XMCD and XMLD images obtained at the L3 edge of Co are shown together with those ob- tained at the L3 edges of Fe and Mn. A wedge-shaped FeMn/12 ML Co bilayer was used. The Co XMCD im- age (a) shows micron-sized magnetic domains. While the domain magnetization lies along 〈110〉 directions in the upper half of the image, only domains with magnetiza- tion along 〈100〉 can be seen the lower half. This be- havior was observed previously,10,11 and is related to the fact that for less than about 10 ML FeMn thickness is paramagnetic at room temperature, while thicker FeMn layers develop AF order.12 Identical domain patterns with strongly reduced XMCD contrast are observed at the Fe (c) and Mn (e) L3 edges. The contrast arises due to the induced moments in the FeMn layer.19 The Mn image is a negative of the Fe and Co images, indicative of an antiparallel orientation of the Mn moments with respect to the magnetization direction of the FM Co layer. Images of the same spot obtained with XMLD as mag- netic contrast mechanism are shown in the right column of Fig. 1. The magnetic domain pattern is clearly visible at the Co L3 edge (b). Note that domains with oppo- site magnetization direction cannot be distinguished by XMLD (compare the right lower parts of images (a) and (b)). The XMLD contrast at the Fe L3 edge is much weaker than at the Co L3 edge. Only after averaging over images with about 170 minutes total acquisition time we were able to recognize at least the magnetic domains in the top part of the image. The images of Fig. 1 are presented on quite different grayscale ranges: While the full contrast from saturated white to saturated black in the Co XMCD image (a) is 20%, it is amplified to 8% for the Fe (c) and 1.5% for Mn (e) XMCD images, and amounts to 3% in the Co XMLD image (b), and only 0.7% in the Fe XMLD image (d). Our previously published XMLD spectra,21 in which the XMLD signal at the Fe L3 edge is below the noise level, supported a non-collinear antiferromagnetic ar- rangement of Fe spins in the AF FeMn layer. In this case only the small induced ferromagnetic moment in the FeMn layer, the XMCD signal of which corresponds to about 30% of the Fe atoms in the interface atomic layer,19 leads to an XMLD signal. Although XMLD has been successfully applied in the past to image antifer- romagnetic domains in PEEM,25,26,27,28 no attempt was made on metallic antiferromagnets, and it is commonly believed that the reduced crystal field splitting of the electronic states in metals9,29,30 is prohibiting the use of XMLD for magnetic imaging. Fig. 1 (d), however, shows that it is possible to image the XMLD signal, even of the comparably low number of the induced moments, by PEEM. Quantitative estimates support the interpretation of the contrast observed in Fig. 1 (d): In the top part of the image, the Fe XMLD is about a factor of 6 weaker than the Co XMLD at the same position. This is about the same ratio as between the respective XMCD contrasts in panels (a) and (c) at about 6 ML FeMn thickness. This size of induced ferromagnetic alignment is consistent with our earlier investigation of FeMn/Co bilayers.19 Further- more, the Fe XMLD originating from induced moments at the interface decreases with increasing total FeMn thickness, so that the expected Fe XMLD signal would be within the noise of the measurement of the spectra of Ref. 21, which were taken for a 15 ML FeMn film. Fig. 2 shows a series of magnetic domain images of the Co layer from a sample in which a 0–25 ML wedge of FeMn was deposited on top of a continuous film of 6 ML Co/Cu(001). The left column shows the XMCD contrast at the Co L3 absorption maximum. The right column shows images of the same spot of the sample, acquired with linear polarization of the x-rays. As in Fig. 1, the FeMn thickness increases from the top to the bottom of the images. Panels (a) through (g) show images obtained for different azimuthal angles of incidence, indicated by an arrow at the right hand side of each panel. Note that the field of view slightly shifted due to readjustment of the sample. The azimuth angle was read from a dial at the sample holder with an accuracy of 1◦. 0◦ corresponds to the nominal [010] direction of the Cu substrate; how- ever, as will be outlined below, the angular dependence of the magnetic contrast indicates that the real [010] di- rection was at −2◦ azimuth angle. This deviation from the nominal direction is within the accuracy with which the substrate could be oriented upon mounting to the sample holder. The data have been taken in the sequence from (a) to (g). Typical acquisition times were 4 minutes per he- licity for the circular polarization, and 20 minutes per polarization direction and photon energy for the linear polarization. Including the necessary sample manipu- lations, the time to obtain the data of Fig. 2 totalled 28 hours. The time evolution of the domain pattern is clearly visible in the (a)–(g) series from the shift of the transition line between paramagnetic and antiferromag- netic FeMn towards higher thicknesses. We attribute this to progressing contamination and possibly oxidation by residual gas of the surface of the FeMn layer. The Co layer, however, which is investigated here, is protected against contamination by the FeMn overlayer. The right column of Fig. 2 shows images taken with linear x-ray polarization. The upper parts of the im- ages clearly reveal identical domain pattern as seen with XMCD. The behavior of the XMLD contrast follows the geometric expectations including the reversal of contrast between panels (a) and (c), and near vanishing of the contrast in panel (b) for the azimuthal angle of incidence close to 45◦ with respect to the magnetization. The most prominent observation, and the main result of this work, is the strong suppression of the XMLD contrast visible in the bottom parts of the images which correspond to domains with 〈100〉 direction of magnetization. In the following we discuss the angular dependence of XMCD and XMLD contrast in detail. In Fig. 3 we show the Co XMCD contrast as a function of the azimuthal FIG. 2: Magnetic domain images of a FeMn/Co bilayer on Cu(001), acquired at the Co L3 edge with circular polarization (left column) and linear polarization (right column). Rows (a)–(g) correspond to different azimuthal angles of incidence, as labeled at the right hand side and indicated by arrows. Local magnetization directions are indicated by arrows. The FeMn thickness increases from top to bottom in each image, from 7.2 to 14.7 in (a), gradually shifting to from 9.4 to 16.9 ML in (g). angle of incidence obtained from the data of Fig. 2. Fig. 3 (a) shows the contrast of the domains in the upper part of the images, where the FeMn is paramagnetic and Co domains are magnetized along 〈110〉 directions. Panel (b) presents the contrast from the lower part of the images, where the FeMn is antiferromagnetic and Co domains [110] [110] [110] 6040200-20 x-ray incidence azimuth ϕ (deg) (b) [010] [100] [010] FIG. 3: Angular dependence of the Co L3 XMCD contrast: (a) the XMCD signal of three domains with magnetization directions [11̄0], [1̄10], and [110] represented by solid trian- gles, circles, and squares, respectively (the crosses at −20◦ are measurements from a Co/Cu(001) reference film without FeMn layer), (b) the XMCD signal of three domains with magnetization directions [01̄0], [1̄00], and [010] represented by open triangles, circles, and squares, respectively. The solid lines are the result of simultaneous sin(ϕ) fits to the data. -0.01 6040200-20 x-ray incidence azimuth ϕ (deg) <110> <100> FIG. 4: Angular dependence of the Co L3 XMLD contrast. The contrast between Co magnetic domains with mutually perpendicular magnetization direction along 〈110〉 directions is represented by solid symbols, the contrast between Co do- mains with perpendicular magnetization direction along 〈100〉 directions is represented by open symbols. The solid lines are the result of simultaneous sin(2ϕ) fits to the data. align along 〈100〉 directions. The lines correspond to a sine fit. The fit reveals a −2◦ phase shift, which can be attributed to the inaccuracy of the sample mounting. The angular dependence of the XMCD contrast of the individual domains confirms very nicely the assignment of magnetization directions in Fig. 2. A small vertical offset may be attributed to instrumental asymmetries, as for example different intensities of the two helicities. Importantly, the amplitude of the XMCD contrast is only about 5% lower in panel (b) than in panel (a). The latter is equal to the contrast of Co domains in a Co/Cu(001) reference sample without FeMn layer, which is indicated in Fig. 3 (a) by crosses at −20◦ incidence azimuth. Fig. 4 shows the angular dependence of the Co XMLD contrast from data of Fig. 2. Solid and open symbols rep- resent the contrast between domains with mutually per- -0.03 -0.02 -0.01 784782780778776774772770 photon energy (eV) Dhesi et al., Co/Cu(001)vic 4° Kuch et al., Co/FeMn/Cu(001) Co L3 FIG. 5: Comparison of different experimental Co XMLD data from literature. (a): Polarization-averaged absorption at the Co L3 edge, (b): Difference between absorption for parallel and perpendicular x-ray polarization. Markers and thin lines: Fig. 1 of Ref. 30, for 6 ML Co on 4◦ miscut Cu(001), magnetization along 〈110〉. Thick lines: Data from Fig. 4 of Ref. 21, for 15 ML FeMn/6 ML Co/Cu(001), magnetization along 〈100〉, scaled to the same absorption maximum. The data of Ref. 30 have been shifted in energy by −1.34 eV for overlap of the absorption curves in (a). pendicular magnetization direction in the regions where the FeMn layer is paramagnetic and antiferromagnetic, respectively. The solid lines are the result of sine fits for which the phases were fixed at +43◦ and −2◦, respec- tively, using the result of the fits of Fig. 3. Again, the data confirm the assignment of the magnetization axes in Fig. 2. Based on the fits we are able to quantify the sup- pression of the XMLD contrast in 〈100〉 domains as com- pared to the 〈110〉 domains to be a factor of 0.28. Such a large effect of magnetization direction on any physi- cal quantity is rather unusual in 3d metals, which ex- hibit only a weak spin–orbit coupling. In order to prove that we see a genuine magneto-crystalline anisotropy we next discuss the role of spin non-collinearity and make comparison to other experimental data and theoretical calculations. Besides changing the easy axis direction in the Co layer, magnetic ordering of the FeMn may also induce a small non-collinearity of the Co moments. Such a non-collinear fanning out of the FM moments in Fe/MnF2 bilayers as a consequence of the AF–FM cou- pling was recently suggested on the basis of Mössbauer spectroscopy.31 We consider now if a similar scenario can explain our XMLD data. A distribution of the Co spins around a mean direction would lead to a reduction of both the XMCD and XMLD signals compared to the fully aligned case. While the reduction of the XMCD signal is proportional to the reduction of the net mo- ment, XMLD, due to its different angular dependence, is more sensitive and depends also on the distribution of the fanning angles. In the extreme case of moments oriented -0.02 -0.01 2520151050-5 energy (eV) [110] [100] FIG. 6: Calculated XMLD in the fcc Co for magneti- zation along two different crystallographic directions. (a): Polarization-averaged absorption spectrum for [100] magneti- zation. (b): XMLD difference for magnetization along [100] (solid line) and along [110] (line and symbols). The data for [100] magnetization are reproduced from Ref. 9. The energy scale is relative to the absorption edge. at 45◦ with respect to the net magnetization, the XMLD would be reduced to zero, while the XMCD would still be at 71% of its maximum value. Using the reduction factor of 0.95 of the net magnetization, obtained from XMCD contrast, and assuming both binary (moments point at a fixed angle on either side of the net magnetization) and normal distributions (Gaussian distribution of angles) of the fanning angle we arrive at a reduction factor of about 0.81 for the XMLD contrast. Therefore fanning of the Co moments due to the interaction with the FeMn layer, if at all present, can explain only a small fraction of the observed effect, which amounts to the reduction factor of 0.28. Since the stability of the instrument is not sufficient to acquire series of microspectroscopic images for different photon energies, we use published data to compare exper- imental Co XMLD spectra for magnetization along [100] and [110] directions. The [100] data are taken from Ref. 21, in which a 15 ML FeMn/6 ML Co bilayer on Cu(001) was measured. The spectra for Co magnetized along [110] direction are taken from the work of Dhesi et al., in which 6 ML Co on Cu(001), miscut by 4◦, was measured.30 The two spectra at the Co L3 edge, rescaled to the same ab- sorption maximum and shifted to the common position of the absorption edge, are compared in Fig. 5. Both spectra had been measured under similar conditions.32 Note that the peak-to-peak ratio of the XMLD spectra of Fig. 5 (b) cannot be compared directly to the ratio of the XMLD asymmetry magnitudes of Fig. 4, which corresponds to the contrast between XMLD signal at the energies marked by arrows in Fig. 5.33 Although smaller than the asymmetry anisotropy of 3.6 from Fig. 4, the ra- tio of peak-to-peak XMLD of 2.3 between the two curves obtained from Fig. 5 still indicates substantial magneto- crystalline anisotropy of the XMLD signal. As mentioned in the Introduction, such a large anisotropy is unusual in 3d metals since the spin–orbit coupling is rather weak, e.g. the calculated magneto- crystalline anisotropy energy in bulk fcc Co is only 2 µeV per atom.34 Also the XMCD spectrum, which depends essentially only on integral quantities, namely spin and orbital moments, exhibits a very small anisotropy.9 As pointed out by one of us and P. M. Oppeneer, the XMLD signal in metallic Co depends only weakly on the small valence band spin-orbit coupling. The major contribu- tion to XMLD comes from the exchange splitting of the 2p levels (≈ 1 eV).9 The magneto-crystalline anisotropy then arises from the fact that different final 3d states are probed for different orientations of the sample magneti- zation. To assess the feasibility of our experimental data, we used the calculated XMLD spectrum of Ref. 9 for the [100] direction (what is referred to in Ref. 9 as “full cal- culation”) and augmented these with equal calculations for the [110] magnetization on the same system (see Fig. 6). In the calculations performed on bulk fcc Co a siz- able anisotropy of XMLD is found, however, the [100] exhibits larger XMLD magnitude contrary to the exper- iment. Before dismissing these results as a disagreement a few remarks are in order. First, the calculations were done on bulk material while the experiment is performed on a thin layer sandwiched by other materials, therefore a good quantitative agreement is unlikely. Second, we can- not judge the calculated anisotropy based on the present data only. Note that due to a slight mutual shift of the calculated spectra, the [100] contrast at the maximum amplitude of the [110] XMLD would be rather small. Such a shift is not present in Fig. 5, where [100] and [110] spectra obtained on slightly different samples are compared. Third, a possible non-collinearity of Co spins due to the presence of the AF FeMn layer would lead to local moments pointing neither along [110] nor fully along [100]. Taking these uncertainties into account we draw a modest, nevertheless non-trivial, conclusion that the the- ory does not prohibit a magneto-crystalline anisotropy of XMLD as large as observed in our experiment. IV. CONCLUSIONS We have presented a spectromicroscopic PEEM inves- tigation of the magnetic domain pattern on Co/FeMn bilayers using XMCD and XMLD as the contrast mech- anism. The sensitivity of the method allows to visualize even the tiny XMLD signal of the induced ferromagnetic moments in the FeMn layer. We have found a factor of 3.6 difference in the XMLD contrast between the Co L3 signal from 〈110〉 and 〈100〉 domains in a single sam- ple. We argue that this huge difference is mainly due to an intrinsic magneto-crystalline anisotropy of XMLD of the Co layer. Comparison of experimental XMLD spec- tra obtained from different samples published previously and ab initio calculations on bulk fcc Co suggest that such an anisotropy is indeed possible. Acknowledgments We thank B. Zada and W. Mahler for technical assis- tance, and S. S. Dhesi for providing the data from Ref. 30. Financial support by the German Minister for Education and Research (BMBF) under grant No. 05 SL8EF19 is gratefully acknowledged. J. K. acknowledges the support by an Alexander von Humboldt Research Fellowship. ∗ Electronic address: [email protected]; URL: http: //www.physik.fu-berlin.de/~ag-kuch † Present address: CNISM and Dipartimento di Fisica, Uni- versità Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy. ‡ Present address: Universität Duisburg–Essen, Institut für Experimentelle Physik, Lotharstraße 1, D-47057 Duisburg, Germany. § Present address: Department of Physics and HKU-CAS Joint Lab on New Materials, The University of Hong Kong, Hong Kong, China. ¶ Present address: SPring-8, 1–1–1 Kouto, Sayo-cho, Sayo- gun, Hyogo 679-5198, Japan. ∗∗ Present address: Hiroshima Synchrotron Radiation Cen- ter, 2–313 Kagamiyama, Higashi-Hiroshima, 739-8526 Hi- roshima, Japan. 1 J. Nogués and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999). 2 W. H. Meiklejohn and C. P. 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Introduction Experiment Results and discussion Conclusions Acknowledgments References

0704.0183

Temperature Dependence of the Tensile Properties of Single Walled Carbon Nanotubes: O(N) Tight Binding MD Simulation

Carbon nanotubes (CNTs) have attracted attention for application to nanodevices due to their unique mechanical and electronic properties TEMPERATURE DEPENDENCE OF THE TENSILE PROPERTIES OF SINGLE WALLED CARBON NANOTUBES: O(N) TIGHT BINDING MD SIMULATION GÜLAY DERELİ * , BANU SÜNGÜ Department of Physics, Yildiz Technical University, 34210 Istanbul, Turkey Abstract This paper examines the effect of temperature on the structural stability and mechanical properties of 20 layered (10,10) single walled carbon nanotubes (SWCNTs) under tensile loading using an O(N) tight-binding molecular dynamics (TBMD) simulation method. We observed that (10,10) tube can sustain its structural stability for the strain values of 0.23 in elongation and 0.06 in compression at 300K. Bond breaking strain value decreases with increasing temperature under stretching but not under compression. The elastic limit, Young’s modulus, tensile strength and Poisson ratio are calculated as 0.10, 0.395 TPa, 83.23 GPa, 0.285, respectively, at 300K. In the temperature range from 300K to 900K; Young’s modulus and the tensile strengths are decreasing with increasing temperature while the Poisson ratio is increasing. At higher temperatures, Young’s modulus starts to increase while the Poisson ratio and tensile strength decrease. In the temperature range from 1200K to 1800K, the SWCNT is already deformed and softened. Applying strain on these deformed and softened SWCNTs do not follow the same pattern as in the temperature range of 300K to 900K. PACS numbers: 61.46.Fg, 62.25.+g, 62.20 Dc, 62.20 Fe Keywords: Single Wall Carbon Nanotubes, Order N Tight-binding Molecular Dynamics, tensile properties. *Corresponding author: [email protected] mailto:[email protected] 1. Introduction Tensile properties of SWCNTs have been widely investigated by experimental and theoretical techniques. Experimentally, the Young’s modulus of SWCNTs are measured as ranging from 0.9 to 1.9 TPa in [1]. In the SEM measurements of [2], SWCNT ropes broke at the strain values of 5.3 % or lower and the determined mean values of breaking strength and Young’s modulus are 30 GPa and 1002 GPa, respectively. AFM and SGM measurements [3] show that SWCNTs can sustain elongations as great as 30% without breaking. On the other hand, the ab initio simulation study of SWCNTs [4] showed that Young’s modulus and Poisson ratio values of the tubes are ranging from 0.5 TPa to 1.1 TPa and from 0.11 to 0.19, respectively. The Young modulus and Poisson ratio of armchair nanotubes is given as 0.764 TPa and 0.32 respectively in [5]. Young modulus of (10,0); (8,4) and (10,10) tubes are calculated as 1.47 TPa; 1.10 TPa and 0.726 TPa, respectively in [6]. The results of [7] proposed that the structural failure should occur at 16% for zigzag and above 24% for armchair tubes. An empirical force-constant model of [8] gave the Young’s modulus between 0.971 TPa- 0.975 TPa and Poisson ratio between 0.277 - 0.280. An empirical pair potential simulations of [9] gave the Young’s modulus between 1.11 TPa -1.258 TPa and the Poisson ratio between 0.132-0.151. Continuum shell model of [10], calculated the elastic modulus as 0.94 TPa.and the maximum stress and failure strain values as 70 GPa, 88 GPa; 11%, 15% for (17,0) and (10,10) tubes, respectively. Finite element method [11] determined the strength of CNTs between 77 GPa to 101 GPa and Poisson ratio between 0.31-0.35. Analytical model in [12] found the tensile strength as 126.2 GPa of armchair tubes to be stronger than that (94.56 GPa) of zigzag tubes and the failure strains are 23.1% for armchair and 15.6-17.5% for zigzag tubes. MD simulations of [13-17] determined Young’s modulus between 0.311 to 1.017 TPa for SWCNT. We found the Young’s modulus, tensile strength and the Poisson ratio as 0.311 TPa, 4.92 GPA and 0.287 for (10,10) tubes in [16]. C.Goze et al. [18] calculated the Young’s modulus as 0.423 TPa and Poisson ratio as 0.256 for (10,10) tube. Nonlinear elastic properties of SWCNTs under axial tension and compression were studied by T.Xiao et al. [19,20] using MD simulations with the second-generation Brenner potential. They showed that the energy change of the nanotubes are a cubic function of the tensile strains, both in tension and under compression. The maximum elongation strains are 15% and 17% for zigzag and armchair tubes, respectively. Also the maximum compression strain decreases with increasing tube diameter, and it is almost 4% for (10,10) tube. M.Sammalkorpi et al. [21] studied the effects of vacancy-related defects on the mechanical characteristics of SWCNTs by employing MD simulations and continuum theory. They calculated the Young’s modulus for perfect SWCNTs as 0.7 TPa. They showed that at 10K temperature, the critical strains of (5,5) and (10,10) tubes are 26% and 27%, respectively; also tensile strength is 120 GPa. On the other hand, for (9,9) and (17,0) tubes, the critical strains are found as 22% and 21%, respectively, and tensile strength is 110 GPa. Y.Wang et al. [22] investigated the compression deformation of SWCNTs by MD simulations using the Tersoff-Brenner potential to describe the interactions of carbon atoms. They determined that the SWCNTs whose diameters range from 0.5 nm to 1.7 nm and length ranges from 7 nm to 19 nm, the Young’s modulus range from 1.25 TPa to 1.48 TPa. S.H.Yeak et al. [23] used MD and TBMD method to examine the mechanical properties of SWCNTs under axial tension and compression. Their results showed that the Young’s modulus of the tubes are around 0.53 TPa; the maximum strain under axial tension is 20% for (12,12) and (7,7) tubes and also under this strain rate, the tensile stresses are 100 GPa and 90 GPa, respectively. Many elastic characteristics like the Young’s modulus show a wide variations (0.3 TPa- 1.48 TPa) in all reported results in literature. These results are obtained at room temperature or without the mention of the temperature. The following reasons may be given for the variety of results: i) Young’s modulus depends on the tube diameter and the chirality ii) different values are used for the wall thickness iii) different procedures are applied to represent the strain iv) accuracy of the applied methods (first principle methods in comparison with emprical model potentials) SWCNTs will be locally subject to abrupt temperature increases in electronics circuits and the temperature increase affects their structural stability and the mechanical properties. MD simulation studies on the mechanical properties of the SWCNTs at various temperatures under tensile loading simulations can be followed in [24-28]. M.B.Nardelli et al. [24] showed that all tubes are brittle at high strains and low temperatures, while at low strains and high temperatures armchair nanotubes can be completely or partially ductile. In zigzag tubes ductile behavior is expected for tubes with n<14 while larger tubes are completely brittle. N.R.Raravikar et al. [25] showed between 0-800K temperature range radial Young’s modulus of nanotubes decreases with increasing temperature and its slope is -7.5x10-5 (1/K) . C.Wei et al. [26,27] studied the tensile yielding of SWCNTs and MWCNTs under continuous stretching using MD simulations and a transition state theory based model. They showed that the yield strain decreases at higher temperatures and at slower strain rates. The tensile yield strain of SWCNT has linear dependence on the temperature and has a logarithmic dependence on the strain rate. The slope of the linear dependence increases with temperature. From their results it is shown that the yield strain of (10,0) tube decreased from 18% to 5% for the temperature range increasing from 300K to 2400K and for the different strain rates. Another MD simulation study was performed by Y.-R.Jeng et al. [28] investigated the effect of temperature and vacancy defects on tensile deformation of (10,0); (8,3); (6,6) tubes of similar radii. Their Young’s modulus and Poisson ratio values range from 0.92 to 1.03 TPa and 0.36- 0.32, respectively. Their simulations also demonstrate that the values of the majority of the considered mechanical properties decrease with increasing temperature and increasing vacancy percentage. In this study, the effect of temperature increase on the structural stability and mechanical properties of (10,10) armchair SWCNT under tensile loading is investigated by using O(N) tight-binding molecular dynamics (TBMD) simulations. Extensive literature survey is given in order to show the importance of our present study. The armchair 20 layers (10,10) SWCNT is chosen in the present work because it is one of the most synthesized nanotube in the experiments. For the first time we questioned how the strain energy of these nanotubes changes for positive and negative strain values at high temperatures. Along with the high temperature stress-strain curves for the first time we displayed the bond-breaking strain values through total energy graphs. Mechanical properties (Young’s modulus, Poisson ratio, tensile strength and elastic limit) of this nanotube are reported at high temperatures. 2. Method Traditional TB theory solves the Schrödinger equation by direct matrix diagonalization, which results in cubic scaling with respect to the number of atoms O(N3). The O(N) methods, on the other hand, make the approximation that only the local environment contributes to the bonding and hence the bond energy of each atom. In this case the run time would be in linear scaling with respect to the number of atoms. G.Dereli et al. [29,30] have improved and succesfully applied the O(N) TBMD technique to SWCNTs. In this work, using the same technique, we performed SWCNT simulations depending on conditions of temperature and unaxial strain. The electronic structure of the simulated system is calculated by a TB Hamiltonian so that the quantum mechanical many body nature of the interatomic forces is taken into account. Within a semi-empirical TB, the matrix elements of the Hamiltonian are evaluated by fitting a suitable database. TB hopping integrals, repulsive potential and scaling law is fixed in the program [31,32]. Application of the technique to SWCNTs can be seen in our previous studies [16,29,30]. An armchair (10,10) SWCNT consisting of 400 atoms with 20 layers is simulated. Periodic boundary condition is applied along the tube axis. Velocity Verlet algorithm along with the canonical ensemble molecular dynamics (NVT) is used. Our simulation procedure is as follows: i.) The tube is simulated at a specified temperature during a 3000 MD steps of run with a time step of 1 fs. This eliminates the possibility of the system to be trapped in a metastable state. We wait for the total energy per atom to reach the equilibrium state. ii.) Next, uniaxial strain is applied to the tubes. We further simulated the deformed tube structure (the under uniaxial strain) for another 2000 MD steps. In our study, while the nanotube is axially elongated or contracted, reduction or enlargement of the radial dimension is observed. Strain is obtained from 00 /)( LLL −=ε , where and 0L L are the tube lengths before and after the strain, respectively. We applied the elongation and compression and calculated the average total energy per atom. Following this procedure we examined the structural stability, total energy per atom, stress-strain curves, elastic limit, Young’s modulus, tensile strength, Poisson ratio of the (10,10) tube as a function of temperature. The stress is determined from the resulting force acting on the tube per cross sectional area under stretching. The cross sectional area of the tube, is defined by S RRS δπ2= , where R and Rδ are the radius and the wall thickness of the tube, respectively. We have used 3.4 Å for wall thickness. Mechanical properties are calculated from the stress-strain curves. Elastic limit is obtained from the linear regions of the stress-strain curves. Young’s modulus, which shows the resistivity of a material to a change in its length, is determined from the slope of the stress-strain curve at studied temperatures. The tensile strength can be defined as the maximum stress which may be applied to the tube without perturbing its stability. Poisson ratio which is a measure of the radial reduction or expansion of a material under tensile loading can be defined as ⎟⎟ where R and are the tube radius at the strainoR ε and before the strain, respectively. 3. Results and Discussion In Figure 1, we present the total energy per atom of the (10,10) SWCNT as a function of strain. Several strain values are applied. The positive values of strain corresponds to elongation and the negative values to compression. We obtained the total energy per atom vs strain curves in the temperature range between 300K-1800K in steps of 300K. Total energy per atom increases as we increase the temperature. An asymmetric pattern is observed in these curves. Repulsive forces are dominant in the case of compression. SWCNT does not have a high strength for compression as much as for elongation. (10,10) SWCNT is stable up to 0.06 strain in compression in the temperature range between 300K-1500K and 0.03 at 1800K. In elongation (10,10) SWCNT is stable up to 0.23 strain at 300K. As we increase the temperature the tube is stable up to 0.15 in elongation until 1800K. At 1800K we can only apply the strain of 0.08 in elongation before bond breakings. Figure 2a shows the variation of the total energy per atom during simulations for the strain values of 0.23 in elongation and 0.06 in compression at 300K. This figure indicates that the tube can sustain its structural stability up to these strain values. Beyond these, bond-breakings between the carbon atoms are observed at the strain values of 0.24 in elongation and 0.07 in compression as given in Figure 2b. In Figure 2b. sharp peaks represent the disintegrations of atoms from the tube. Next, bond-breaking strains are studied with increasing temperature. In Figure 3, we show the bond-breaking strain values with respect to temperature: as the temperature increases, disintegration of atoms from their places is possible at lower strain values due to the thermal motion of atoms. But this is not the case for compression as can be seen in Figure 3. Some examples of the variation of the total energy as a function of MD Steps under uniaxial strain values at various temperatures are given in Figure 4 and Figure 5. Figure 4a and Figure 5a shows that the tube can sustain its structural stability for strain values of 0.14 in elongation and 0.06 in compression at 900K; 0.08 in elongation and 0.03 in compression at 1800K, respectively. Beyond these points, bond-breakings between carbon atoms are observed at the strain values of 0.15 in elongation and 0.07 in compression, at 900 K (Figure 4b) and 0.09 in elongation and 0.04 in compression, at 1800K (Figure 5b). The stress-strain curves of the tube are given in Figure 6. at studied temperatures. Our results show that the temperature have a significant influence on the stress-strain behaviour of the tubes. The stress-strain curves are in the order of increasing temperatures between 300K- 900K. Stress value is increasing with increasing temperature . On the other hand between 1200K-1800K the stress value decreases with increasing temperature. This is due to the smaller energy difference under tensile loading with respect to 300K-900K temperature range. This result can also be followed in the total energy changes observed in Figures 2b , 4b and 5b. Table1. gives a summary of the variations of the mechanical properties of (10,10) SWCNT with temperature. As given in Table1, elastic limit has the same value (0.10) in the 300K- 900K temperature range. It drops to 0.09 in the 1200K-1500K temperature range and to 0.08 at 1800K. Young’s modulus, Poisson ratio and the tensile strength of the tube have been found to be sensitive to the temperature (Table 1.). Our calculated value at 300K is 0.401 TPa. It decreases to 0.370 TPa at 600K and to 0.352 TPa at 900K. In this temperature range Young’s modulus decreases 12 %. After 1200K as we increase the temperature to 1800K there is 3% increase in the Young’s modulus. We determined the tensile strength of (10,10) tube as 83.23 GPa at 300K. There is an abrupt decrease in tensile strength as we increase the temperature to 900K. Between 900K-1500K temperature range tensile strength does not change appreciably. At 1800K, it drops to 43.78 GPa. We specified the Poisson ratio at 300K as 0.3. Between 300K-900K temperature range Poisson ratio increases to 0.339 (12.5 %). This corresponds to the increase in the radial reduction. As we increase the temperature to 1200K its value drops to 0.315 and at 1800K to 0.289. We can conclude that for 20 layer (10,10) SWCNT in the 300K-900K temperature range : Young’s modulus, the tensile strengths are decreasing with increasing temperature while the Poisson ratio is increasing. At higher temperatures, Young’s modulus and the tensile strengths start to increase while the Poisson ratio decreases. In the 1200K-1800K temperature range, the SWCNT is already deformed and softened. Applying strain on these deformed and softened SWCNT do not follow the same pattern of 300K- 900K temperature range. 4. Conclusion This paper reports for the first time the effect of temperature on the stress-strain curves, Young’s modulus, tensile strength, Poisson ratio and elastic limit of (10,10) SWCNT. Total energy per atom of the (10,10) tube increases with axial strain under elongation. We propose that SWCNTs do not have a high strength for compression as much as for elongation. This is due to the dominant behavior of repulsive forces in compression. At room temperature, the bond breaking strain values of the tube are 0.24 in elongation and 0.07 in compression. We showed that as the temperature increases, the disintegration of atoms from their places is possible at lower strain values (0.09 at 1800K) in elongation due to the thermal motion of atoms. But this is not the case for compression. For 20 layers SWCNT bond-breaking negative strain values are temperature independent between 300K-1500K temperature range. Bond breakings occurs at 0.07 compression in this temperature range. When we increase the number of layers to 50, bond-breaking negative strain value decrease from 0.07 to 0.05 and remains the same in this temperature range. However, this is not a robust property for negative strains. When we decrease on the other hand the layer size to 10; bond-breaking negative strain values vary with increasing temperature. We note that for short tubes the critical strain values for compressive deformations are dependent on the size of the employed supercell and therefore they are an artifact of the calculation. In literature, various critical strain values were mentioned for the tube deformations. Our room temperature critical strain values are in aggrement with the experimental results of [3] and the computational results of [7,10,12]. MD simulations of [21] determined the critical strain value of (10,10) tube as 0.27 at 10K. To our knowledge the only reported temperature simulation study on tensile property comes from the MD simulation results of [26]. They showed that the yield strain of (10,0) tube decreases from 0.18 to 0.05 for the temperature range increasing from 300K to 2400K. Our results follow the same trend such that the bond breaking strain values decrease with increasing temperature. In [20] the maximum compression strain of (10,10) tube is given as 0.04 using Brenner potential without the mention of temperature, we obtained this value at 1800K. We obtained the stress-strain curves in the temperature range between 300K-1800K. Our results show that the temperature have a significant influence on the stress-strain behavior of the tubes. (10,10) tube is brittle between 300K-900K and soft after 1200K. The elastic limit decreased from 0.10 to 0.08 with increasing temperature. There is a wide range of values given in literature for Young’s modulus of SWCNTs due to the accuracy of the method and the choice of the wall thickness of the tube. The experimental results are in the range from 0.9 TPa to 1.9 TPa [1,2], ab initio results are in the range from 0.5 TPa to 1.47 TPa [4-6], empirical results are in the range from 0.971 TPa to 0.975 TPa [8] and from 1.11 TPa to 1.258 TPa [9], and also MD simulation results are in the range from 0.311 TPa to 1.48 TPa [13-28]. Our calculated value at 300K is 0.401 TPa is consistent with [4, 18,23]. We determined the tensile strength of (10,10) tube as 83.23 GPa at 300K and it decreases with increasing temperature. Maximum stress value of (10,10) tube is reported as 88 GPa in [10], 77 to 101 GPa in [11]. At 300K, we calculated the Poisson ratio of (10,10) tube as 0.3. This is in accord with the ab initio [5]; empirical [8,11], tight binding [18] results. M.B.Nardelli et al. [24] showed that all tubes are brittle at high strains and low temperatures, while at low strains and high temperatures armchair nanotubes can be completely or partially ductile. Our findings agree with this (10,10) armchair SWCNT is brittle at low temperatures and ductile at higher temperatures. Contrary to [28] our extensive temperature study has shown that Young’s modulus changes with temperature. 5. Comments Carbon nanotubes have the highest tensile strength of any material yet measured, with labs producing them at a tensile strength of 63 GPa, still well below their theoretical limit of 300 GPa. Carbon nanotubes are one of the strongest and stiffest materials known, in terms of their tensile stress and Young’s modulus. This strength results from the covalent sp2 bonds formed between the individual carbon atoms. Our simulation study using the interactions between electrons and ions also predicts a similar tensile strength and also shows that when exposed to heat they still keep their tensile strength around this value until very high temperatures like 1800K. CNTs are not nearly as strong under compression. Because of their hollow structure and high aspect ratio, they tend to undergo buckling when placed under compressive stress. The elastic limit is the maximum stress a material can undergo at which all strain are recoverable. (i.e., the material will return to its original size after removal of the stress). At stress levels below the elastic limit the material is said to be elastic.Once the material exceeds this limit, it is said to have undergone plastic deformation (also known as permanent deformation). When the stress is removed, some permanent strain will remain, and the material will be a different size. Our study shows that when the nanotube is exposed to heat this property does not change appreciably until 1800K. Through our tight-binding molecular dynamics simulation study we reported the high temperature positive/negative bond- breaking strain values and stress-strain curves of (10,10) SWCNTs . As far as we are aware, the strain energy values corresponding to positive/negative strain values at different temperatures are given here for the first time. We hope this extensive study of high temperature mechanical properties will be useful for aerospace applications of CNTs. Acknowledgement The research reported here is supported through the Yildiz Technical University Research Fund Project No: 24-01-01-04. The calculations are performed at the Carbon Nanotubes Simulation Laboratory at the Department of Physics, Yildiz Technical University, Istanbul, Turkey. References [1] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J.Treacy, Phys. Rev. B 58, 14013 (1998). [2] M.F. Yu, B.S. Files, S. Arepalli, R.S. Ruoff, Phys. Rev. Lett. 84, 5552 (2000). [3] D. Bozovic, M. Bockrath, J.H. Hafner, C.M. Leiber, H. Park, M. Tinkham, Phys. Rev. B 67, 033407 (2003). [4] D.Sanchez-Portal, E. Artacho, J.M. Soler, A. Rubio, P. Ordejon, Phys. Rev. B 59, 12678 (1999). [5] G. Zhou, W. Duan, B. Gu, Chem. Phys. Lett. 333, 344 (2001). [6] A. Pullen, G.L. Zhao, D. Bagayoko, L.Yang, Phys. Rev. B 71, 205410 (2005). [7] T.Dumitrica, T.Belytschko, B.I.Yakobson, Journal of Chem. Phys. 118, 9485 (2003). [8] J.P. Lu, Phys. Rev. Lett. 79, 1297 (1997). [9] S. Gupta, K.Dharamvir, V.K.Jindal, Phys. Rev. B 72, 165428 (2005). [10] T. Natsuki, M. Endo, Carbon 42, 2147 (2004). [11] X. Sun, W. Zhao, Mater. Sci. And Engineering A 390, 366 (2005). [12] J.R. Xiao, B.A. Gama, J.W. Gillespie Jr., Int. Journal of Solids and Structures 42, 3075 (2005). [13] T. Ozaki, Y. Iwasa, T. Mitani, Phys. Rev. Lett 84, 1712 (2000). [14] B. Ni, S.B. Sinnott, P.T. Mikulski, J.A. Harrison, Phys. Rev. Lett. 88, 205505 (2002). [15] L.G. Zhou, S.Q. Shi, Comp. Mater. Sci. 23, 166 (2002). [16] G. Dereli, C. Özdogan, Phys. Rev. B 67, 035416 (2003). [17] S. Ogata, Y. Shibutani, Phys. Rev. B 68, 165409 (2003). [18] C.Goze, L.Vaccarini, L. Henrard, P. Bernier, E. Hernandez, A.Rubio, Synthetic Metals, 103, 2500 (1999). [19] T.Xiao, K.Liao, Phys. Rev. B 66, 153407 (2002). [20] T.Xiao, X.Xu, Journal of Appl. Phys. 95, 8145 (2004). [21] M.Sammalkorpi, A.Krasheninnikov, A.Kuronen, K.Nordlund, K.Kaski, Phys. Rev. B 71, 169906(E) (2005). [22] Y.Wang, X.Wang, X.Ni, H.Wu, Comput. Mater. Sci. 32, 141 (2005). [23] S.H.Yeak, T.Y.Ng, K.M.Liew, Phys. Rev. B 72, 165401 (2005). [24] M.B. Nardelli, B.I. Yakobson, J. Bernholc, Phys. Rev. Lett. 81, 4656 (1998). [25] N.R. Raravikar, P. Keblinski, A.M. Rao, M.S. Dresselhaus, L.S. Schadler, P.M. Ajayan, Phys. Rev. B 66, 235424 (2002). [26] C. Wei, K. Cho, D. Srivastava, Phys. Rev. B 67, 115407 (2003). [27] C. Wei, K. Cho, D. Srivastava, Appl. Phys. Lett. 82, 2512 (2003). [28] Y.R. Jeng, P.C. Tsai, T.H. Fang, Journal of Physics and Chemistry of Solids 65, 1849 (2004). [29] C. Özdoğan, G. Dereli, T. Çağın, Comp. Phys. Comm. 148, 188 (2002). [30] G. Dereli, C. Özdogan, Phys. Rev. B 67, 035415 (2003). [31] L.Colombo, Comput. Mater. Sci. 12, 278 (1998). [32] C.H. Xu,C.Z. Wang, C.T. Chan, K.M. Ho, J. Phys.:Cond. Matt. 4, 6047 (1992). Temperature (K) Elastic Limit Young’s Modulus (TPa) Tensile Strength (GPa) Poisson Ratio 300 0.10 0.401 83.23 0.300 600 0.10 0.370 69.78 0.332 900 0.10 0.352 67.62 0.339 1200 0.09 0.360 67.33 0.315 1500 0.09 0.356 68.14 0.320 1800 0.08 0.365 43.78 0.289 Table 1. High Temperature Mechanical Properties of (10,10) SWCNT Figure Captions Figure 1. “Color online” Total energy per atom curves as a function of strain at different temperatures (negative strain values correspond to compression). Figure 2 a. “Color online” (10,10) SWCNT is stable for the strains of 0.23 and -0.06 at 300K. Figure 2 b. “Color online” Bond- breakings are observed between the carbon atoms for the strains of 0.24 and -0.07 at 300K. System is not in equilibrium. Figure 3. “Color online” Bond- breaking strain variations as a function of temperature for a) tension, b) compression. Figure 4 a. “Color online” (10,10) SWCNT is stable for the strains of 0.14 and -0.06 at 900K. Figure 4 b. “Color online” Bond- breakings are observed between the carbon atoms for the strains of 0.15 and -0.07 at 900 K. System is not in equilibrium. Figure 5 a. “Color online” (10,10) SWCNT is stable for the strains of 0.08 and -0.03 at 1800K. Figure 5 b. “Color online” Bond- breakings are observed between the carbon atoms for the strains of 0.09 and -0.04 at 1800K. System is not in equilibrium. Figure 6. “Color online” The stress-strain curves of (10,10) SWCNT at different temperatures.

0704.0184

Gamma-ray emitting AGN and GLAST

arXiv:0704.0184v1 [astro-ph] 2 Apr 2007 Gamma-ray emitting AGN and GLAST P. Padovani European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching bei München, Germany Abstract. I describe the different classes of Active Galactic Nuclei (AGN) and the basic tenets of unified schemes. I then review the properties of the extragalactic sources detected in the GeV and TeV bands, showing that the vast majority of them belong to the very rare blazar class. I further discuss the kind of AGN GLAST is likely to detect, making some predictions going from the obvious to the likely, all the way to the less probable. Keywords: active galactic nuclei, radio sources, gamma-ray sources PACS: 98.54.Cm, 98.54.Gr, 98.70.Dk, 98.70.Rz THE ACTIVE GALACTIC NUCLEI ZOO Active Galactic Nuclei (AGN) are extragalactic sources, in some cases clearly associated with nuclei of galaxies (although generally the host galaxy light is swamped by the nucleus), whose emission is dominated by non-stellar processes in some waveband(s). Based on a variety of observations, we believe that the inner parts of AGN are not spherically symmetric and therefore that emission processes are highly anisotropic [4, 28]. The current AGN paradigm includes a central engine, almost certainly a massive black hole, surrounded by an accretion disk and by fast-moving clouds, which under the influence of the strong gravitational field emit Doppler-broadened lines. More distant clouds emit narrower lines. Absorbing material in some flattened configuration (usually idealized as a torus) obscures the central parts, so that for transverse lines of sight only the narrow-line emitting clouds are seen and the source is classified as a so-called "Type 2" AGN. The near-infrared to soft-X-ray nuclear continuum and broad-lines, including the UV bump typical of classical quasars, are visible only when viewed face-on, in which case the object is classified as a "Type 1" AGN. In radio-loud objects, which constitute ≈ 10% of all AGN, we have the additional presence of a relativistic jet, likely perpendicular to the disk (see Fig. 1 of [28]). This axisymmetric model of AGN implies widely different observational properties (and therefore classifications) at different aspect angles. Hence the need for "Unified Schemes" which look at intrinsic, isotropic properties, to unify fundamentally identical (but apparently different) classes of AGN. Seyfert 2 galaxies are though to be the "parent" population of, and have been "unified" with, Seyfert 1 galaxies, whilst low-luminosity (Fanaroff-Riley type I [FR I][7]) and high-luminosity (Fanaroff-Riley type II [FR II] ) radio galaxies have been unified with BL Lacs and radio quasars respectively [28]. In other words, BL Lacs are thought to be FR I radio galaxies with their jets at relatively small ( < 15− 20◦) angles w.r.t. the line of sight. Similarly, we believe flat-spectrum radio quasars (FSRQ) to be FR II radio galaxies oriented at small ( < 15◦) angles, while steep-spectrum radio quasars (SSRQ) should be at angles in between those of FSRQ and FR II’s (15◦ <∼ θ <∼ 40◦; a spectral index value αr = 0.5 at a few GHz [where fν ∝ ν−α ] is usually taken as the dividing line between FSRQ and SSRQ). BL Lacs and FSRQ, that is radio-loud AGN with their jets practically oriented towards the observer, make up the blazar class. Blazars, as I show below, play a very important role in γ-ray astronomy and it is therefore worth expanding on their properties. Blazars Blazars are the most extreme variety of AGN. Their signal properties include irregular, rapid variability, high polarization, core-dominant radio morphology (and therefore flat [αr <∼ 0.5] radio spectra), apparent superluminal motion, and a smooth, broad, non-thermal continuum extending from the radio up to the γ-rays [28]. Blazar properties are consistent with relativistic beaming, that is bulk relativistic motion of the emitting plasma at small angles to the line of sight, which gives rise to strong amplification and collimation in the observer’s frame. Adopting the usual definition of the relativistic Doppler factor δ = [Γ(1−β cosθ )]−1, Γ = (1−β 2)−1/2 being the Lorentz factor, β = v/c being the Gamma-ray emitting AGN and GLAST November 4, 2018 1 http://arxiv.org/abs/0704.0184v1 0 10 20 30 40 50 60 70 80 90 FIGURE 1. The dependence of the Doppler factor on viewing angle. Different curves correspond to different Lorentz factors Γ. The expanded scale on the inset shows the angles for which δ = 1. ratio between jet speed and the speed of light, and θ the angle w.r.t. the line of sight, and applying simple relativistic transformations, it turns out that the observed luminosity at a given frequency is related to the emitted luminosity in the rest frame of the source via Lobs = δ pLem with p ∼ 2−3. For θ ∼ 0◦, δ ∼ 2Γ (Fig. 1) and the observed luminosity can be amplified by factors 400 – 10,000 (for Γ ∼ 10 and p ∼ 2−3, which are typical values). That is, for jets pointing almost towards us the emitted luminosity can be overestimated by up to four orders of magnitude. For more typical angles θ ∼ 1/Γ, δ ∼ Γ and the amplification is ∼ 100− 1,000. In a nut-shell, blazars can be defined as sites of very high energy phenomena, with bulk Lorentz factors up to Γ ≈ 30 [6] (corresponding to velocities ∼ 0.9994c) and photon energies reaching the TeV range (see below). Given their peculiar orientation, blazars are very rare. Assuming that the maximum angle w.r.t. the line of sight an AGN jet can have for a source to be called a blazar is ∼ 15◦, only ∼ 3% of all radio-loud AGN, and therefore ≈ 0.3% of all AGN, are blazars. For a ∼ 1− 10% fraction of galaxies hosting an AGN, this implies that only 1 out of ≈ 3,000− 30,000 galaxies is a blazar! Blazar spectral energy distributions (SEDs) are usually explained in terms of synchrotron and inverse Compton emission, the former dominating at lower energies, the latter being relevant at higher energies. Blazars have a large range in synchrotron peak frequency, νpeak, which is the frequency at which the synchrotron energy output is maximum (i.e., the frequency of the peak in a ν − ν fν plot). Although the νpeak distribution appears now to be continuous, it is still useful to divide blazars into low-energy peaked (LBL), with νpeak in the IR/optical bands, and high-energy peaked (HBL) sources, with νpeak in the UV/X-ray bands [21]. The location of the synchrotron peaks suggests in fact a different origin for the X-ray emission of the two classes. Namely, an extension of the synchrotron emission responsible for the lower energy continuum in HBL, which display steep (αx ∼ 1.5) X-ray spectra [29], and inverse Compton (IC) emission in LBL, which have harder (αx ∼ 1) spectra [20]. This distinction applies almost only to BL Lacs, as most known FSRQ are of the low-energy peak type and, therefore, with the X-ray band dominated by Gamma-ray emitting AGN and GLAST November 4, 2018 2 inverse Compton emission. Very few “HFSRQ” (as these sources have been labelled), i.e., FSRQ with high (UV/X-ray energies) νpeak are in fact known. Moreover, νpeak for all these sources (apart from one) appears to be ∼ 10−100 times smaller than the values reached by BL Lacs (see [19] for a review). THE GEV AND TEV SKIES Before moving on to GLAST we need to assess the present status of the γ-ray sky. I do this first at GeV and then TeV energies. The third EGRET catalogue [15] includes 271 sources (E > 100 MeV), out of which 95 were identified as extragalactic (including 28 lower confidence sources). Further work [17, 24, 25], which provided more identifications, allows us to say that EGRET has detected at least ∼ 130 extragalactic sources (since a large fraction of sources is still unidentified), all of them AGN apart from the Large Magellanic Cloud. Furthermore, all the AGN are radio-loud and ∼ 97% of them are blazars, with the remaining sources including a handful of radio galaxies (e.g., Centaurus A, NGC 6251). Most of the blazars are FSRQ, in a ratio ∼ 3/1 with BL Lacs. Finally, ∼ 80% of the BL Lacs are LBL and the few HBL are all local (z < 0.12). As all of the FSRQ are also of the LBL type, ∼ 93% of EGRET detected blazars are of the low-energy peak type. The situation at TeV energies is at first order similar to that in the GeV band, with some significant differences. All confirmed extragalactic TeV sources are radio-loud AGN and include 16 BL Lacs and one radio galaxy (M87) (a starburst galaxy is also a possible TeV source) [18, 3]. That is, the blazar fraction is ∼ 94%. Unlike the GeV band, however, no FSRQ is detected and all but one BL Lacs are HBL. This is due to the fact that in HBL the very high-energy flux is higher than in LBL, as both peaks of the two humps in their SED are shifted to higher frequencies. The fact that the GeV and TeV skies are dominated by blazars seems to be at odds with these sources being extremely rare (see previous section). The explanation has to be found in the peculiar properties of the blazar class and rests on the fact that blazars are characterized by: 1. high-energy particles, which can produce GeV and TeV photons; 2. relativistic beaming, to avoid photon-photon collision and amplify the flux; 3. strong non-thermal (jet) component. Point 1 is obvious. We know that in some blazars synchrotron emission reaches at least the X-ray range, which reveals the presence of high-energy electrons which can produce γ-rays via inverse Compton emission (although other processes can also be important: e.g., [5]). Point 2 is vital, as otherwise in sources as compact as blazars all GeV photons, for example, would be absorbed through photon-photon collisions with target photons in the X-ray band (see, e.g., [16]). Beaming means that the intrinsic radiation density is much smaller than the observed one and therefore γ- ray photons manage to escape from the source. The flux amplification in the observer’s frame makes also the sources more easily detectable. Point 3 is also very important. γ-ray emission is clearly non-thermal (although we still do not know for sure which processes are responsible for it) and therefore related to the jet component. The stronger the jet component, the stronger the γ-ray flux. GLAST AND AGN We can know ask which (and how many) AGN GLAST will detect. This I describe in the following, in decreasing order of "obviousness". Blazars Given that blazars are well know γ-ray sources, GLAST will certainly detect many flat-spectrum radio quasars and BL Lacs. How many exactly depends on a variety of factors. These include blazar evolution and intrinsic number density (which can to some extent be estimated from deep surveys in other bands), their duty cycle in the γ-ray band (as we know that EGRET was detecting mostly sources in outburst), and their SED (see below). Finally, any prediction will have not to violate the extragalactic γ-ray background. Gamma-ray emitting AGN and GLAST November 4, 2018 3 To get an order of magnitude estimate, I make the following simple assumptions: a) EGRET has detected 130 blazars, which is likely to be a lower limit given the still unidentified sources; b) the number counts are Euclidean, that is N(> S) ∝ S−1.5, where S is the flux density; this is a very likely upper limit as we know that, after the initial steep rise, number counts of extragalactic sources tend to flatten out at lower fluxes; c) GLAST is 30 times more sensitive than EGRET. The total number of blazars GLAST will detect over the whole sky is then < 20,000. This corresponds to <∼ 0.5 objects/deg 2, which, interestingly enough, is the surface density of blazars down to ∼ 50 mJy at 5 GHz in the Deep X-ray Radio Blazar Survey (DXRBS) [22]. Note also that by means of Monte Carlo simulations a value around 5,000 has been predicted all-sky (extrapolating from the high Galactic latitude value of [10]; see also [11]). As discussed above, EGRET has detected very few blazars of the high-energy peak type (HBL). This is because the EGRET band was sampling the "valley" between the two (synchrotron and IC) humps in their SED. A look at the SED of some of the TeV detected HBL [1, 2, 27] shows that many, if not all, of them should be easily detected by GLAST. Radiogalaxies Unified schemes predict that the "parent" population of blazars is made up of radio-galaxies, a much more numerous class (by a factor ≈ 30 for a dividing angle between the two classes ∼ 15◦). However, at large angles w.r.t. the line of sight, jet emission is not only not-amplified but actually de-amplified. Fig. 1 shows that for typical Lorentz factors δ < 1 for viewing angles > 20− 30◦. This implies that radio-galaxies on average are weaker sources (by factors ≈ 1,000) than blazars, in all bands. And indeed, the handful of GeV/TeV-detected radio-galaxies are all local (z < 0.02). Large scale, that is kpc-scale jet emission, as opposed to the small, pc-scale, one, is also unlikely to be relevant in the γ-ray band for the bulk of radio-galaxies [26, 23]. However, the radio-galaxy cause might not be totally lost. It has been proposed that blazar jets are structured or decelerated. The first scenario [9], which ties in with Very Long Baseline Interferometry (VLBI) observations of limb brightening [12], suggests the presence of a fast spine surrounded by a slower external layer. In the other case [8], which tries to reconcile the low δ values from VLBI observations of TeV BL Lacs with the high values inferred from SED modeling of the same sources, the jet is supposed to decelerate from a Lorentz factor Γ ∼ 20 down to Γ ∼ 5 over a length of ∼ 0.1 pc. In both instances the presence of the two velocity fields implies that each of the two components sees an enhanced radiation field produced by the other. The net result is that IC emission gets boosted and therefore the GeV flux is higher than that predicted in the simpler case of an homogeneous jet (at the price of having a larger number of free parameters). Assuming that the γ-ray/radio flux ratio observed for the three GeV/TeV-detected radio-galaxies sources is typical, at least 10 3CR radio-galaxies should to be detected by GLAST [9]. Note that some Broad Line Radio Galaxies (BLRG), which are Type 1 sources in which the jet is at angles intermediate between those of blazars and radio-galaxies, are also likely to be detected by GLAST [13, 14]. Radio-Quiet AGN The large majority of AGN are of the radio-quiet type, that is they are characterized by very weak radio emission, on average ∼ 1,000 times fainter than in radio-loud sources. Radio-quiet does not mean radio-silent, however, and the nature of radio emission in these sources is still debated. Two extreme options ascribe it either to processes related to star-formation (synchrotron emission from relativistic plasma ejected from supernovae) or to a scaled down version of the non-thermal processes associated with energy generation and collimation present in radio-loud AGN. In the latter case, one would expect also radio-quiet AGN to be (faint) γ-ray sources. Assuming their GeV flux to scale roughly as the radio flux this would be, on average, a factor ≈ 30 below the GLAST detection limit. Detection might be possible, however, for the (few) high core radio flux radio-quiet AGN. Even a negative detection, supported by detailed calculations, could prove very valuable in constraining the nature of radio-emission in these sources. Gamma-ray emitting AGN and GLAST November 4, 2018 4 SUMMARY The main conclusions are as follows: 1. Blazars, even though they make up a small minority of AGN, dominate the γ-ray sky; 2. GLAST will certainly detect "many thousand" blazars, with the exact number being somewhat model dependent; 3. GLAST will most likely detect "many" high-energy peaked blazars, which have so far escaped detection at GeV energies due to the fact that EGRET was sampling the "valley" between the two (synchrotron and IC) humps in their spectral energy distribution; 4. GLAST will possibly detect a "fair" number of radio-galaxies; 5. GLAST might also detect some radio-quiet AGN, depending on the nature of their radio emission. In any case, GLAST will constrain (radio-loud) AGN physics and populations, as described very well at this conference! ACKNOWLEDGMENTS It is a pleasure to thank Paolo Giommi for useful discussions and Annalisa Celotti for reading the manuscript. REFERENCES 1. J. Albert, et al., The Astrophysical Journal 648, L105–L108 (2006). 2. J. Albert, et al., The Astrophysical Journal 654, L199–L122 (2007). 3. J. Albert, et al., The Astrophysical Journal in press (2007) (arXiv:astro-ph/0703084). 4. R. Antonucci, Annual Review of Astronomy and Astrophysics 31, 473–521 (1993). 5. A. Celotti, these proceedings (2007). 6. M. H. Cohen, et al., The Astrophysical Journal in press (2007) (arXiv:astro-ph/0611642). 7. B. L. Fanaroff, and J. M. Riley, Monthly Notices of the Royal Astronomical Society 167, 31p–36p (1974). 8. M. Georganopoulos, E. S. Perlman, and D. Kazanas, The Astrophysical Journal 643, L33–L36 (2005). 9. G. Ghisellini, F. Tavecchio, and M. Chiaberge, Astronomy & Astrophysics 432, 401–410 (2005). 10. P. Giommi, and S. Colafrancesco, "Non-thermal Cosmic Backgrounds and prospects for future high-energy observations of blazars" in Gamma-Wave 2005 in press (2007) (arXiv:astro-ph/0602243). 11. P. Giommi, these proceedings (2007). 12. M. Giroletti, et al., The Astrophysical Journal 600, 127–140 (2004). 13. P. Grandi, and G. Palumbo, The Astrophysical Journal in press (2007) (arXiv:astro-ph/0611342). 14. P. Grandi, and G. Palumbo, these proceedings (2007). 15. R. C. Hartman, et al., The Astrophysical Journal Supplement Series 123, 79–202 (1999). 16. L. Maraschi, G. Ghisellini, and A. Celotti, The Astrophysical Journal 397, L5–L9, (1992). 17. J. R. Mattox, R. C. Hartman, and O. Reimer, The Astrophysical Journal Supplement Series 135, 155–175 (2001). 18. D. Mazin, these proceedings (2007). 19. P. Padovani, "Blazar Sequence: Validity and Predictions" in The Multi-messenger approach to high energy gamma-ray sources in press (2007) (arXiv:astro-ph/0610545). 20. P. Padovani, L. Costamante, P. Giommi, G. Ghisellini, A. Celotti, and A. Wolter, Monthly Notices of the Royal Astronomical Society 347, 1282–1293 (2004). 21. P. Padovani, and P. Giommi, The Astrophysical Journal 444, 567–581 (1995). 22. P. Padovani, P. Giommi, H. Landt, and E. S. Perlman, The Astrophysical Journal in press (2007) (arXiv:astro-ph/0702740). 23. R. Sambruna, these proceedings (2007). 24. D. Sowards-Emmerd, R. W. Romani, and P. F. Michelson, The Astrophysical Journal 590, 109–122 (2003). 25. D. Sowards-Emmerd, R. W. Romani, P. F. Michelson, and J. S. Ulvestad, The Astrophysical Journal 609, 564–575 (2004). 26. Ł. Stawarz, M. Sikora, and M. Ostrowski, The Astrophysical Journal 597, 186–201 (2003). 27. F. Tavecchio, et al., The Astrophysical Journal 554, 725–733 (2001). 28. C. M. Urry, and P. Padovani, Publications of the Astronomical Society of the Pacific 107, 803–845 (1995). 29. A. Wolter, et al., Astronomy & Astrophysics 335, 899–911 (1998). Gamma-ray emitting AGN and GLAST November 4, 2018 5

0704.0185

Potfit: effective potentials from ab-initio data

Potfit: effective potentials from ab-initio data Peter Brommer and Franz Gähler Institut für Theoretische und Angewandte Physik (ITAP), Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany E-mail: [email protected] Abstract. We present a program called potfit which generates an effective atomic interaction potential by matching it to a set of reference data computed in first-principles calculations. It thus allows to perform large-scale atomistic simulations of materials with physically justified potentials. We describe the fundamental principles behind the program, emphasizing its flexibility in adapting to different systems and potential models, while also discussing its limitations. The program has been used successfully in creating effective potentials for a number of complex intermetallic alloys, notably quasicrystals. Modelling Simulation Mater. Sci. Eng. 15 (2007), 295–304 online at http://stacks.iop.org/ms/15/295 doi: 10.1088/0965-0393/15/3/008 PACS numbers: 02.60.Pn, 02.70.Ns, 07.05.Tp, 61.44.Br 1. Introduction Classical effective potentials reduce the quantum-mechanical interactions of electrons and nuclei in a solid to an effective interaction between atom cores. This greatly reduces the computational effort in molecular dynamics (MD) simulations. Whereas first principles simulations are limited to a few hundred atoms at most, classical MD calculations with many millions of atoms are routinely performed. Such system sizes are possible, because molecular dynamics with short-range interactions scales linearly with the number of atoms. Moreover, it can easily be parallelized using a geometrical domain decomposition scheme [1, 2], thereby achieving linear scaling also in the number of CPUs. The study of many problems in materials science and nanotechnology indeed requires simulations of systems with millions of atoms. Quite generally, this is the case whenever long-range mechanical stresses are involved. Examples of such problems are the study of fracture propagation [3], nano-indentation, or the motion and pinning of dislocations. Other problems may be simulated with more moderate numbers of atoms, but require very long simulated times, of the order of nanoseconds, an example of which is the study of atomic diffusion [4]. In either case, if large systems and/or long time scales are required, classical effective potentials are the only way to make molecular dynamics simulations possible. The reliability and predictive power of classical MD simulations depend cruicially on the quality of the effective potentials employed. In the case of elementary solids, such potentials are usually obtained by adjusting a few potential parameters to http://arxiv.org/abs/0704.0185v2 mailto:[email protected] http://stacks.iop.org/ms/15/295 http://dx.doi.org/10.1088/0965-0393/15/3/008 Potfit: effective potentials from ab-initio data 2 optimally reproduce a set of reference data, which typically includes a number of experimental values like lattice constants, cohesive energies, or elastic constants, sometimes supplemented with ab-initio cohesive energies and stresses [5, 6]. In the case of more complex systems with a large variety of local environments and many potential parameters to be determined, such an approach cannot help, however; there is simply not enough reference data available. The force matching method [7] provides a way to construct physically justified potentials even under such circumstances. The idea is to compute forces and energies from first principles for a suitable selection of small reference systems and to adjust the parameters of the potential to optimally reproduce them. For that purpose, we developed a program called potfit‡. By separating the process of optimization from the form of the potential, potfit allows for maximal flexibility in the choice of potential model and parametrization. The underlying algorithms are described in section 2. Section 3 focuses on the implementation of the algorithms, followed by details on employing potfit in section 4. We discuss advantages and limitations of the force matching method and our implementation in section 5, and present our conclusions in the final section 6. 2. Algorithms As mentioned above, potfit consists of two separate parts. The first one implements a particular parametrized potential model and calculates from a set of potential parameters ξi the target function that quantifies the deviations of the forces, stresses and energies from the reference values. Wrapped around is a second, potential independent part which implements a least squares minimization module. As this part is completely independent of the potential model and just deals with the list of parameters ξi, it is fairly straightforward to change the parametrization of the potential (tabulated or analytic), or even to switch to a different potential model. 2.1. Optimization From a mathematical point of view, force matching is a basic optimization problem: There is a set of parameters ξi, a set of values bk(ξi) depending on them, and a set of reference values b0,k which the bk have to match. This leads to the well- known method of least squares, where one tries to minimize the sum of squares of the deviations between the bk and the b0,k. In our case, the reference values can either be the components of the force vector ~f0,j acting on each individual atom j, or global data A0,k like stresses, energies, or certain external constraints. We found it helpful to measure the relative rather than the absolute deviations from the reference data, except for very small reference values. The least squares target function thus becomes Z = ZF + ZC, (1) with ZF = α=x,y,z (fjα − f0,jα) 0,j + εj , (2) and ZC = (Ak −A0,k) , (3) ‡ http://www.itap.physik.uni-stuttgart.de/%7Eimd/potfit http://www.itap.physik.uni-stuttgart.de/%7Eimd/potfit Potfit: effective potentials from ab-initio data 3 where ZF represents the contributions of the forces, and ZC that of the global data. The (small and positive) εℓ impose a lower bound on the denominators, thereby avoiding a too accurate fitting of small quantities which are actually not known to such a precision. The Wℓ are the weights of the different terms. It proves useful for the fitting to give the total stresses and the cohesion energies an increased weight, although in principle they should be reproduced correctly already from the forces. Even if all forces are matched with a small deviation only, those deviations can add up in an unfortunate way when determining stresses, thus leading to potentials giving wrong elastic constants. Including global quantities in the fit with a sufficiently high weight supresses such undesired behaviour of the fitting process. As the evaluation of the highly nonlinear target function (1) is computationally rather expensive, a careful choice of the minimization method has to be made. We chose a combination of a conjugate-gradient-like deterministic algorithm [8] and a stochastic simulated annealing algorithm [9]. For the deterministic algorithm we take the one described by Powell [8], which takes advantage of the form of the target function (which is a sum of squares). By re-using data obtained in previous function calls it arrives at the minimum faster than standard least squares algorithms. It also does not require any knowledge of the gradient of the target function. The algorithm first determines the gradient matrix at the starting point in the high-dimensional parameter space by finite differences. The gradient matrix is assumed to be slowly varying around the starting point. A new optimal search direction towards the minimum is determined by the method of conjugate gradients. Then, the target function is minimized along this direction. This operation is called line minimization. When the minimum is found, the direction unit vector replaces one of the basis vectors spanning the parameter space. The gradient matrix is updated only with respect to this new direction, using the finite differences calculated in the line minimization. In this way, no finite differences have to be calculated explicitly except in the very first step. The line minimization is performed by Brent’s algorithm [10] in an implementation taken from the GNU Scientific Library [11]. The algorithm is restarted (including a calculation of the full gradient matrix) when either a step has been too large to maintain the assumption of a constant gradient matrix, the basis vectors spanning the parameter space become almost linearly dependent, or the linear equation involved in Powell’s algorithm cannot be solved with satisfactory numerical precision. The other minimization method implemented is a simulated annealing[12] algorithm proposed by Corana [9]. While the deterministic algorithm mentioned above will always find the closest local minimum, simulated annealing samples a larger part of the parameter space and thus has a chance to end up in a better minimum. The price to pay is a computational burdon which can be several orders of magnitude larger. For the basic Monte Carlo move, we chose adding Gaussian-shaped bumps to the potential functions. The bump heights are normally distributed around zero, with a standard deviation adjusted so that on average half of the Monte Carlo steps are accepted. This assures optimal progress: Neither are too many calculations wasted because the changes are too large to be accepted, nor are the steps too small to make rapid progress. Potfit: effective potentials from ab-initio data 4 2.2. Potential models and parametrizations The simplest effective potential is a pair potential, which only depends on interatomic distances. It takes the form i,j<i φsisj (rij), (4) where rij is the distance between atoms i and j, and φsisj is a potential function depending on the two atom types si and sj . This function can either be given in analytic form, using a small number of free parameters, like for a Lennard-Jones potential, or in tabulated form together with an interpolation scheme for distances between the tabulation points. Whereas the parameters of an analytic potential can often be given a physical meaning, such an interpretation is usually not possible for tabulated potentials. On the other hand, an inappropriate form of an analytic potential may severely constrain the optimization, leading to a poor fit. For this reason, we chose the functions φ to be defined by tabulated values and spline interpolation, thus avoiding any bias introduced by an analytic potential. This choice results in a relatively high number of potential parameters, compared to an analytic description of the potentials. This is not too big a problem, however. Force matching provides enough reference data to fit even a large number of parameters. The potential functions φ only need to be defined at pair distances r between a minimal distance rmin and a cutoff radius rcut, where the function should go to zero smoothly. We found pair potentials to be insufficient for the simulation of complex metallic alloys. More suited are EAM (Embedded Atom Method [13, 5]) potentials, also known as glue potentials [14], which have many advantages over pair potentials in the description of metals [14]. EAM potentials include a many-body term depending on a local density ni: i,j<i φsisj (rij) + Usi(ni) with ni = j 6=i ρsj (rij). (5) ni is a sum of contributions from the neighbours through a transfer function ρsj , and Usi is the embedding function that yields the energy associated with placing atom i at a density ni. Again, all functions are specified by their values at a number of sampling points. The parameters ξi specifying a tabulated potential are naturally the values at the sampling points. Due to the nature of spline interpolation, either the gradient or the curvature at the exterior sampling points of each function can also be chosen freely. Depending on the type of potential one can keep the gradients fixed, or adapt them dynamically by adding them to the set of parameters ξi. The EAM potential described by (5) has two gauge degrees of freedom, i.e., two sets of parameter changes which do not alter the physics of the potential: ρs(r) → κρs(r), Usi(ni) → Usi( φsisj (r) → φsisj (r) + λsiρsj (r) + λsjρsi(r), Usi(ni) → Usi(ni)− λsini. According to (6), the units of the density ni can be chosen arbitrarily. We use this degree of freedom to set the units such that the densities ni computed for the reference Potfit: effective potentials from ab-initio data 5 configurations are contained in the interval (−1; 1], but not in any significantly smaller interval. The transformation (7) states that certain energy contributions can be moved freely between the pair and the embedding term. An embedding function U which is linear in the density n can be gauged away completely. This also makes any separate interpretation of the pair potential part and the embedding term void; the two must only be judged together. The latter degeneracy is usually lifted by choosing the gradients of the Ui(ni) to vanish at the average density for each atom type. potfit also uses this convention when exporting potentials for plotting and MD simulation. As the average density might change during minimization, potfit internally uses a slightly different gauge: It requires that the gradient vanishes at the center of the domain of the respective embedding function. potfit can perform the transformations (6,7) periodically on its own, thus eliminiating the need to fix the gauge by an additional term in the target function (1). Unfortunately, for tabulated functions the transformations cannot be performed exactly due to the nature of spline interpolation. A change of gauge therefore can lead to an increase of the target function, which is why we suppress such gauge transformations in the very late stages of a minimization. 3. Implementation potfit is implemented in ANSI C. While the user may specify most options in a parameter file read when running the program, some fundamental choices must be made at compile time, like for example the potential model used, or whether to allow for automatic gauge transformations in EAM potentials. This is a compromise between convenience and computation speed. Compile time options can be selected by passing them to the make command, and thus do not require any changes of the source files. For solving the linear equations in Powell’s minimization algorithm, potfit makes use of routines from the LAPACK library [15], which must be installed separately, probably together with the BLAS library [16] LAPACK is based on. 3.1. Parallelization and optimization The program spends almost all CPU time in calculating the forces for a given potential; finding a new potential to be tested against the reference data takes only a tiny fraction of that time. Thus, the only way to improve performance is to reduce the total time needed for the force computations, either by minimizing their number, or by making each force computation faster. Powell’s algorithm leaves only little room for further reduction of the number of force evaluations. One could for instance adjust the precision required in a line minimization. If the tolerance is too small, time is wasted in refining a minimum beyond need, whereas an insufficent precision may stop too far from the minimum, thus requiring more steps in total. The choice of this tolerance was made empirically. Much more time can be saved by parallelizing the calculation of forces, energies, and stresses for a given potential. This is done in a straightforward way: As the forces, energies, and stresses of the different reference configurations can be computed independently, we simply distribute the reference configurations on several processes. Before the force computation, the potential parameters are distributed to all processes, and afterwards the computed forces, energies and stresses are collected. The communication is performed using the standard Message Passing Interface (MPI [17]). Potfit: effective potentials from ab-initio data 6 This simple parallelization scheme works well as long as the number of configurations per process does not drop below 10 to 15. Otherwise, the communication overhead starts to show up, and load balancing problems may appear. A shared memory OpenMP parallelization also exists, but produces inferior results. In force matching, the reference configurations stay fixed. Therefore, all distances between atoms remain fixed, and potfit can use neighbour lists, which need to be computed only once at startup. In fact, for each neighbour pair all data required for spline interpolation are pre-computed, allowing for a fast lookup of the tabulated functions. This data needs to be recomputed only when the tabulation points of a function are changed. 3.2. Input and output files Tabulated potential functions can be specified with equidistant or with arbitrary tabulation points. For equidistant tabulation points, the boundaries of the domain and the number of sampling points of each function are read from the potential file, followed by a list of function values at the sampling points and the gradients at the domain boundaries. In the case of free tabulation points, only their number is specified at the beginning of the potential file, followed by a list of argument-value pairs and again the gradients of the potential functions at the domain boundaries. Reference configuration files contain the number of atoms, the box vectors, the cohesive energy, and the stresses on the unit cell, followed by a list of atoms, with atom species, position and reference force for each atom. Such reference configuration files can simply be concatenated. potfit was designed to cooperate closely with the first-principles code VASP [18, 19] and with IMD [20], our own classical MD code. VASP, which is a plane wave code implementing ultrasoft pseudopotentials and the Projector-Augmented Wave (PAW) method [21, 22], is used to compute the reference data for the force matching, whereas the resulting potentials are intended to be used with IMD. For this reason, potfit provides import and export filters for potentials and configurations to communicate with these programs. These filters are implemented as scripts, which can easily be modified to interface with other programs. 4. Results and validation As a first test, potfit should be able to recover a classical potential from reference data computed with that potential. For this test, we used snapshots from several molecular dynamics runs as reference structures, first for a Lennard-Jones fcc solid, then for a complex Ni-Al alloy simulated with EAM potentials [23]. In order to ensure that all reference data presented to potfit is consistent, the potentials were approximated by cubic spline polynomials, in the same way as potfit represents the potentials. With such reference data and starting with vanishing potential functions, potfit could in both cases perfectly recover the potentials. This test therefore demonstrates the correctness of the program. One should keep in mind, however, that reference data from ab-inito computations often cannot be reproduced perfectly by any classical potential. Our primary research interest are quasicrystals [24] and other complex metal alloys, for which good potentials are hardly available. potfit has been developed in order to generate effective potentials for such complex metal alloys, which feature large (or even infinite) unit cells, several atom species, and a wide variety of different local Potfit: effective potentials from ab-initio data 7 environments. So far, force matching had been used mainly to determine potentials for monoatomic metals and a small selection of relatively simple binary alloys. As a first application beyond simple alloys, we have developed potentials for the quasicrystalline and nearby crystalline phases in the systems Al-Ni-Co, Ca-Cd, and Mg-Zn. Due to the complexity of the structures and also due to the choice of tabulated potential functions, a relatively large number of potential parameters is required. This is especially true for ternary EAM potentials, which comprise 12 tabulated functions, with 10–15 tabulation points each. Correspondingly, a relatively large amount of reference data is required. A computationally efficient implementation of the force matching method is therefore essential. It turned out that potfit scales well under those circumstances and is up to its task. Although the potentials to be generated are intended for (aperiodic) quasicrystals and crystals with large unit cells, all reference structures have to be periodic crystals with unit cell sizes suitable for the ab-initio computation of the reference data. On the other hand, the reference structures should approximate the quasicrystal in the sense, that all their unit cells together accommodate all relevant structural motifs. To do so, they must be large enough. For instance, the quasicrystalline and related crystalline phases of Ca-Cd and Mg-Zn consist of packings of large icosahedral clusters in different arrangements. Reference structures must be able to accomodate such clusters. A further constraint is, that the unit cell diameter must be larger than the range of potentials. We found that reference structures with 80–200 atoms represent a good compromise between these requirements. Starting from a selection of basic reference structures, further ones were obtained by taking snapshots of MD simulations with model potentials at various temperatures and pressures. Also samples which were strained in different ways were included. For all these reference structures, the ab-initio forces, stresses and energies werde determined with VASP, and a potential was fitted to reproduce these data. As reference energy, the cohesive energy was used, i.e., the energies of the constituent atoms was subtracted from the VASP energies. Instead of absolute cohesive energies one can also use the energy relative to some reference structure. Once a first version of the fitted potential was available, the MD snapshots were replaced or complemented with better ones obtained with the new potential, and the procedure was iterated. As expected, no potential could be found which would reproduce the reference data exactly. During the optimization, the target function (1) does not converge to zero, which indicates that quantum mechanical reality (taking density functional theory as reality) is not represented perfectly by the potential model used. The forces computed from the optimal potential typically differ by about 10% from the reference forces, which seems acceptable. For the energies and stresses a much higher agreement could be reached. Cohesion energy differences for instance can be reproduced with an accuracy better than 1%. The generated potentials were then used in molecular dynamics simulations to determine various material properties, such as the melting temperature and the elastic constants, for which values consistent with experiment were obtained. The Ca-Cd potentials were especially tuned towards ground-state like structures, whose energies are reproduced with high accuracy, in agreement with ab-initio results. Details of these applications can be found in [4, 25, 26, 27]. Probably the best tested EAM potential constructed with potfit was obtained by Rösch, Trebin and Gumbsch [28]. This potential is intended for the simulation of crack propagation in the C15 Laves Phase of NbCr2, and has undergone a broad validation. These authors calculated the Potfit: effective potentials from ab-initio data 8 lattice constant, the elastic constants and the melting temperature and compared these values to experimental and ab-initio results with reasonable success. They also studied relaxation of surface atoms, surface energy and the crack propagation in NbCr2. According to the authors [28], the force-matched potentials created with potfit clearly outperform previously published potentials. But this example also shows [28] that a large number of fitting-validation cycles are usually required, before a usable and satisfactory potential is obtained. This makes force matching a time-consuming and tedious process. 5. Discussion 5.1. Transferability It should be kept in mind that force-matched potentials will only work well in situations they have been trained to. Therefore, all local environments that might occur in the simulation should also be present in the set of reference configurations. Otherwise the results may not be reliable. Using a very broad selection of reference configurations will make the potential more transferable, making it usable for many different situations, e.g. for different phases of a given alloy. On the other hand, giving up some transferability may lead to a higher precision in special situations. By carefully constraining the variety of reference structures one may generate a potential that is much more precise in a specific situation than a general purpose potential, which was trained on a broader set of reference structures. The latter potential, on the other hand, will be more versatile, but less accurate on average. Finding sufficiently many suitable reference structures might not always be trivial. For certain complex structure like quasicrystals, there may be only very few (if any) approximating periodic structures with small enough unit cells. 5.2. Optimal number and location of sampling points Each reference database has an optimum number of parameters it can support. Using too few parameters, the potential functions lack flexibility. On the other hand, exceeding this number may lead to overfitting beyond the limit of the potential model. potfit cannot determine that optimal number automatically, but there is a simple strategy the user can employ. The set of reference configurations is split in two subsets, one of which is used for fitting and the other for testing the potential. If the root-mean-square (rms) deviation of the test set significantly exceeds that of the fitting set, the database is probably overfitted [29]. By starting with a relatively low number of parameters, that is increased as long as the rms of the testing stage decreases, one can arrive at the optimal number of parameters [30]. This strategy also helps in dealing with oscillatory artefacts of the spline interpolation: If the sampling points are not spaced too densely, and there is enough data to support each tabulation point, artificial wiggles are suppressed. potfit provides the frequency with which each tabulation interval is accessed during an evaluation of the target function (1). With this information, sampling point density can be reduced for distances that do not appear frequently enough in the reference configurations. Potfit: effective potentials from ab-initio data 9 5.3. Number of atom types and choice of reference structures The most obvious impact of an increasing number of atom types is the corresponding increase in the number of potential parameters. For instance, an EAM potential for n atom types requires n(n + 1)/2 + 2n tabulated functions, each with 10 to 15 tabulation points. For a ternary system, this already amounts to the order of 150 potential parameters. Whereas such a number of parameters can still be handled, an increasing number of atom types leads to yet another problem, which is more serious. To see this, it must be kept in mind that any potential function depending on the interatomic distance must be determined for the entire argument range between rmin and rcut. If tabulated functions are used, for each tabulation interval there must be distances actually occuring in the reference structures, for otherwise there are potential parameters which do not affect the target function, and which conseqently cannot be determined in the fit. The requirement that all distances for all combinations of atom types actually occur in the reference structures becomes especially problematic if the atoms of one type form only a small minority, in which case some distances between such atoms might be completely absent in all reasonable reference structures. If the number of atom types is large, there is unfortunately always at least one element which is a minority constituent. In such situations it might be unavoidable to use a much broader selection of reference structures with varying stoichiometry, instead of a fixed stoichiometry with a minority constituent. It might even be necessary to include energetically less favourable configurations to provide a complete set of reference data. Another solution would be to use a non-local (or less local) parametrisation of the potential functions, like a superposition of broad gaussians or functions given by analytic formulae. Changing one parameter can then affect the function over a broader range of arguments, making it again possible to fit the function even if only sparse information on it is provided by the reference data. Potentials represented in this way would also not suffer from the wiggle artefacts of spline interpolation described above. 5.4. Experimental values as reference data potfit does currently not use experimental data during force matching. The potentials are determined exclusively from ab-initio data, which means they cannot exceed the accuracy of the first principles calculations. While it is possible, in principle, to support also the comparison to experimental values, we decided against such an addition. For once, available experimental values can often not be calculated directly from the potentials, so determining them would considerably slow down the target function (1) evaluation. Secondly, experimental values often also depend on the exact structure of the system, which in most cases is not completely known beforehand for complex structures, for instance due to fractional occupancies in the experimentally determined structure model. A better way to use experimental data is to test whether the newly generated potentials lead to structures that under MD simulation show the behaviour known from experiment. 6. Conclusion Large scale molecular dynamics simulations are possible only with classical effective potentials, but for many complex systems physically justified potentials do not exist so far. Our program potfit allows the generation of effective potentials even for complex Potfit: effective potentials from ab-initio data 10 binary and ternary intermetallics, adjusting them to ab-initio determined reference data using the force matching method. Potentials for several complex intermetallic compounds have been generated, and were successfully used in molecular dynamics studies of various properties [4, 25, 26, 28]. It should be emphasized, however, that constructing potentials is still tedious and time-consuming. Potentials have to be thoroughly tested against quantities not included in the fit. In this process, candiate potentials often need to be rejected or refined. Many iterations of the fitting-validation cycle are usually required. It takes experience and skill to decide when a potential is finished and ready to be used for production, and for which conditions and systems it is suitable. potfit is only a tool that assists in this process. Flexibility and easy extensibility was one of the main design goals of potfit. While at present only pair and EAM potentials with tabulated potential functions are implemented in potfit, it would be easy to complement these by other potential models, or to add support for differently represented potential functions. Acknowledgments This work was funded by the Deutsche Forschungsgemeinschaft through Collaborative Research Centre (SFB) 382, project C14. Special thanks go to Stephen Hocker and Frohmut Rösch for fruitful discussion and feedback, and to Hans-Rainer Trebin for supervising the thesis work of the first author. References [1] Allen M P and Tildesley D J 1987 Computer Simulation of Liquids, Oxford Science Publications, (Oxford: Clarendon) [2] Beazley D M, Lohmdahl P S, Grønbech-Jensen N, Giles R and Tamayou P 1995 Parallel algorithms for short-range molecular dynamics vol III of Annual Reviews of Computational Physics (Singapore: World Scientific) pp 119–175 ISBN 981–02–2427–3 [3] Rösch F, Rudhart C, Roth J, Trebin H R and Gumbsch P 2005 Phys. Rev. B 72 014128 [4] Hocker S, Gähler F and Brommer P 2006 Phil. Mag. 86(6–8) 1051–1057 [5] Daw M S, Foiles S M and Baskes M I 1993 Mater. Sci. Rep. 9(7–8) 251–310 [6] Chantasiriwan S and Milstein F 1996 Phys. Rev. B 53(21) 14080–14088 [7] Ercolessi F and Adams J B 1994 Europhys. Lett. 26(8) 583–588 [8] Powell M J D 1965 Comp. J. 7(4) 303–307 [9] Corana A, Marchesi M, Martini C and Ridella S 1987 ACM Trans. Math. Soft. 13(3) 262–280 [10] Brent R P 1973 Algorithms for minimization without derivatives Prentice-Hall series in automatic computation (Englewood Cliffs, NJ: Prentice-Hall) ISBN 0–13–022335–2 [11] Galassi M, Davies J, Theiler J, Gough B, Jungman G, Booth M and Rossi F 2005 GNU Scientific Library Reference Manual - Revised Second Edition (Bristol: Network Theory Ltd) ISBN 0– 9541617–3–4 [12] Kirkpatrick S, Gelatt C D and Vecci M P 1983 Science 220(4598) 671–680 [13] Daw M S and Baskes M I 1984 Phys. Rev. B 29(12) 6443–6453 [14] Ercolessi F, Parrinello M and Tosatti E 1988 Phil. Mag. A 58(1) 213–226 [15] Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A and Sorensen D 1999 LAPACK Users’s Guide, Third Edition (Philadelphia, PA: Society for Industrial and Applied Mathematics) ISBN 0–89871–447–8 [16] Lawson C L, Hanson R J, Kincaid D R and Krogh F T 1979 ACM Trans. Math. Soft. 5(3) 308–323 [17] Gropp W, Lusk E and Skjellum A 1999 Using MPI - 2nd Edition (Cambridge, MA: MIT Press) ISBN 0–262–57132–3 [18] Kresse G and Hafner J 1993 Phys. Rev. B 47(1) 558–561 [19] Kresse G and Furthmüller J 1996 Phys. Rev. B 54(16) 11169–11186 [20] Stadler J, Mikulla R and Trebin H R 1997 Int. J. Mod. Phys. C 8(5) 1131–1140 [21] Blöchl P E 1994 Phys. Rev. B 50(24) 17953–17979 http://dx.doi.org/10.1103/PhysRevB.72.014128 http://dx.doi.org/10.1080/14786430500259734 http://dx.doi.org/10.1016/0920-2307(93)90001-U http://dx.doi.org/10.1103/PhysRevB.53.14080 http://dx.doi.org/10.1145/29380.29864 http://dx.doi.org/10.1126/science.220.4598.671 http://dx.doi.org/10.1103/PhysRevB.29.6443 http://dx.doi.org/10.1145/355841.355847 http://dx.doi.org/10.1103/PhysRevB.47.558 http://dx.doi.org/10.1103/PhysRevB.54.11169 http://dx.doi.org/10.1142/S0129183197000990 http://dx.doi.org/10.1103/PhysRevB.50.17953 Potfit: effective potentials from ab-initio data 11 [22] Kresse G and Joubert D 1999 Phys. Rev. B 59(3) 1758–1775 [23] Ludwig M and Gumbsch P 1995 Modelling Simul. Mater. Sci. Eng. 3(4) 533–542 [24] Trebin H R, ed 2003 Quasicrystals. Structure and Physical Properties (Weinheim: Wiley-VCH) [25] Brommer P and Gähler F 2006 Phil. Mag. 86(6–8) 753–758 [26] Mihalkovič M and Widom M 2006 Phil. Mag. 86(3–5) 519–527 [27] Brommer P, Gähler F and Mihalkovič M 2007 Phil. Mag. (at press) [28] Rösch F, Trebin H R and Gumbsch P 2006 Int. Journal Fracture 139(3–4) 517–526 [29] Robertson I J, Heine V and Payne M C 1993 Phys. Rev. Lett. 70(13) 1944–1947 [30] Mishin Y, Farkas D, Mehl M J and Papaconstantopoulos D A 1999 Phys. Rev. B 59(5) 3393–3407 http://dx.doi.org/10.1103/PhysRevB.59.1758 http://dx.doi.org/10.1088/0965-0393/3/4/008 http://dx.doi.org/10.1080/14786430500333349 http://dx.doi.org/10.1080/14786430500333356 http://dx.doi.org/10.1007/s10704-006-0065-8 http://dx.doi.org/10.1103/PhysRevLett.70.1944 http://dx.doi.org/10.1103/PhysRevB.59.3393 Introduction Algorithms Optimization Potential models and parametrizations Implementation Parallelization and optimization Input and output files Results and validation Discussion Transferability Optimal number and location of sampling points Number of atom types and choice of reference structures Experimental values as reference data Conclusion

0704.0186

Dark energy and neutrino model in SUSY -- Remarks on active and sterile neutrinos mixing --

October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka International Journal of Modern Physics E c© World Scientific Publishing Company Dark energy and neutrino model in SUSY – Remarks on active and sterile neutrinos mixing – Ryo Takahashi∗ Graduate School of Science and Technology, Niigata University, 950-2181 Niigata, Japan [email protected] Morimitsu Tanimoto Department of Physics, Niigata University, 950-2181 Niigata, Japan [email protected] We consider a Mass Varying Neutrinos (MaVaNs) model in supersymmetric theory. The model includes effects of supersymmetry breaking transmitted by the gravitational interaction from the hidden sector, in which supersymmetry was broken, to the dark energy sector. Then evolutions of the neutrino mass and the equation of state parameter of the dark energy are presented in the model. It is remarked that only the mass of a sterile neutrino is variable in the case of the vanishing mixing between the left-handed and a sterile neutrino on cosmological time scale. The finite mixing makes the mass of the left-handed neutrino variable. 1. Introduction Cosmological observations have provided the strong evidence that the Universe is flat and its energy density is dominated by the dark energy component whose negative pressure causes the cosmic expansion to accelerate.1 In order to clarify the origin of the dark energy, one has tried to understand the connection of the dark energy with particle physics. In the Mass Varying Neutrinos (MaVaNs) scenario proposed by Fardon, Nelson and Weiner, relic neutrinos could form a negative pressure fluid and cause the present cosmic acceleration.2 In the model, an unknown scalar field, which is called “acceleron”, is introduced and neutrinos are assumed to interact through a new scalar force. The acceleron sits at the instantaneous minimum of its potential, and the cosmic expansion only modulates this minimum through changes in the number density of neutrinos. Therefore, the neutrino mass is given by the acceleron, in other words, it depends on the number density of neutrinos and changes with the expansion of the Universe. The equation of state parameterw and the energy density of the dark energy also evolve with the neutrino mass. Those evolutions depend ∗talked at the International Workshop on Neutrino Masses and Mixings, University of Shizuoka, Shizuoka, Japan, December 17-19, 2006 http://arxiv.org/abs/0704.0186v1 October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka 2 Ryo Takahashi on the form of a scalar potential and the relation between the acceleron and the neutrino mass strongly. Some examples of the potential have been considered.3 The idea of the variable neutrino mass was considered at first in a model of neutrino dark matter and was discussed for neutrino clouds.4 Interacting dark en- ergy scalar with neutrinos was considered in the model of a sterile neutrino.5 The coupling to the left-handed neutrino and its implication on the neutrino mass limit from baryogenesis was discussed.6 In the context of the MaVaNs scenario, there have been a lot of works. 7,8,9,10,11,12 In this talk, we present a MaVaNs model including the supersymmetry breaking effect mediated by the gravity. Then we show evolutions of the neutrino mass and the equation of state parameter in the model. 2. MaVaNs Model in Supersymmetric Theory We discuss the Mass Varying Neutrinos scenario in supersymmetric theory and present a model. We assume a chiral superfield A in dark sector. A is assumed to be a singlet under the gauge group of the standard model. It is difficult to construct a viable MaVaNs model without fine-tunings in some parameters when one assumes one chiral superfield in dark sector, which couples to only the left-handed lepton doublet superfield. 8 Therefore, we assume that the superfield A couples to both the left- handed lepton doublet superfield L and the right-handed neutrino superfield R. For simplicity, we consider the MaVaNs scenario in one generation of neutrinos.a In such framework, we suppose the following superpotential, AA+mDLA+MDLR+ RR, (1) where λ is a coupling constant of O(1) and MA, MD, MR and mD are mass pa- rameters.b The scalar and the spinor component of A are represented by φ and ψ, respectively. The scalar component corresponds to the acceleron which cause the present cosmic acceleration. The spinor component is a sterile neutrino. The third term of the right-hand side in Eq. (1) is derived from the Yukawa coupling such as yLAH with y < H >= mD, where H is the Higgs doublet. In the MaVaNs scenario, the dark energy is assumed to be composed of the neutrinos and the scalar potential for the acceleron. Therefore, the energy density of the dark energy is given as ρDE = ρν + V (φ). (2) Since only the acceleron potential contributes to the dark energy, we assume the vanishing vacuum expectation values of sleptons, and thus we find the following aThree generations model of this scenario has presented in non supersymmetric theory.9 bOther supersymmetric model so called “hybrid model” has been proposed.10 October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 3 effective scalar potential, V (φ) = |φ|4 +M2A|φ| 2 +m2D|φ| 2. (3) We can write down a lagrangian density from Eq. (1), L = λφψψ +MAψψ +mDνLψ +MDνLνR +MRνRνR + h.c.. (4) It is noticed that the lepton number conservation in the dark sector is violated because this lagrangian includes both MAψψ and mDνLψ. After integrating out the right-handed neutrino, the effective neutrino mass matrix is given by mD MA + λφ , (5) in the basis of (νL, ψ), where c ≡ −M D/MR and we assume λφ≪ MD ≪MR. The first term of the (1, 1) element of this matrix corresponds to the usual term given by the seesaw mechanism in the absence of the acceleron. We obtain masses of the left-handed and a sterile neutrino as follows, mνL = c+MA + λ < φ > [c− (MA + λ < φ >)]2 + 4m , (6) c+MA + λ < φ > [c− (MA + λ < φ >)]2 + 4m . (7) It is remarked that only the mass of a sterile neutrino is variable in the case of the vanishing mixing (mD = 0) between the left-handed and a sterile neutrino on cosmological time scale. The finite mixing (mD 6= 0) makes the mass of the left- handed neutrino variable. We will consider these two cases of mD = 0 and mD 6= 0 later. In the MaVaNs scenario, there are two constraints on the scalar potential. The first one comes from cosmological observations. It is that the magnitude of the present dark energy density is about 0.74ρc. ρc is the critical density. Thus, the first constraint turns to V (φ0) = 0.74ρc − ρ ν , (8) where “0” means the present value. The second one is the stationary condition: ∂V (φ) = 0. (9) In this scenario, the neutrino mass is represented by a function of the acceleron; mν = f(φ). Since the energy density of the neutrino varies on cosmological times scale, the vacuum expectation value of the acceleron also varies. This property makes the neutrino mass variable. If ∂mν/∂φ 6= 0, Eq. (9) is equivalent to ∂V (φ(mν)) = 0. (10) October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka 4 Ryo Takahashi Eq. (10) is rewritten by using the cosmic temperature T : ∂V (φ) = −T 3 ∂F (ξ) , (11) where ξ ≡ mν/T , ρν = T 4F (ξ) and F (ξ) ≡ y2 + ξ2 ey + 1 . (12) We can get the time evolution of the neutrino mass from Eq. (11). Since the sta- tionary condition should be always satisfied in the evolution of the Universe, this one at the present epoch is the second constraint on the scalar potential: ∂V (φ) mν=m0ν = −T 3 ∂F (ξ) mν=m0ν ,T=T0 . (13) In addition to two constraints for the potential, we also have two relations between the vacuum expectation value of the acceleron and the neutrino masses at the present epoch: m0νL = c+MA + λ < φ > [c− (MA + λ < φ >0)]2 + 4m , (14) m0ψ = c+MA + λ < φ > [c− (MA + λ < φ >0)]2 + 4m . (15) Next, let us consider the dynamics of the acceleron field. In order that the acceleron does not vary significantly on distance of inter-neutrino spacing, the ac- celeron mass at the present epoch must be less than O(10−4eV) 2. Here and below, we fix the present acceleron mass as m0φ = 10 −4 eV. (16) Once we adjust parameters which satisfy five equations (8) and (13)∼(16), we can have evolutions of the neutrino masses by using the Eq. (11). The dark energy is characterized by the evolution of the equation of state pa- rameter w. The equation of state is derived from the energy conservation law and the stationary condition Eq. (11): w + 1 = [4− h(ξ)]ρν , (17) where h(ξ) ≡ ∂F (ξ) F (ξ) . (18) It seems that w in this scenario depend on the neutrino mass and the cosmic temperature. This means that w varies with the evolution of the Universe unlike the cosmological constant. October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 5 In the last of this section, we comment on the hydrodynamical stability of the dark energy in the MaVaNs scenario. The speed of sound squared in the neutrino- acceleron fluid is given by c2s = ẇρDE + wρ̇DE , (19) where pDE is the pressure of the dark energy. Recently, it was argued that when neutrinos are non-relativistic, this speed of sound squared becomes negative in this scenario.11 The emergence of an imaginary speed of sound means that the MaVaNs scenario with non-relativistic neutrinos is unstable, and thus the fluid in this sce- nario cannot acts as the dark energy. However, finite temperature effects provide a positive contribution to the speed of sound squared and avoid this instability. 12 Then, a model should satisfy the following condition, 5aT 2 25aT 20 (z + 1) > 0, (20) where z is the redshift parameter, z ≡ (T/T0)− 1, and ≃ 6.47. (21) The first and the second term of left hand side in Eq. (20) are negative and positive contributions to the speed of sound squared, respectively. We find that a model which leads to small ∂mν/∂z is favored. A model with a small power-law scalar potential; V (φ) = Λ4(φ/φ0)k, k ≪ 1, and a constant dominant neutrino mass; mν = C + f(φ), f(φ) ≪ C, leads to small ∂mν/∂z. c Actually, some models have been presented.9 3. Effect of supersymmetry breaking Let us consider effect of supersymmetry breaking in the dark sector. We assume a superfield X , which breaks supersymmetry, in the hidden sector, and the chiral superfield A in the dark sector is assumed to interact with the hidden sector only through the gravity. This framework is shown graphically in Fig. 1. Once supersym- metry is broken at TeV scale, its effect is transmitted to the dark sector through the following operators: X† +X A†A, (22) where Mpℓ is the Planck mass. Then, the scale of soft terms FX(TeV 2)/Mpℓ ∼ O(10−3-10−2eV) is expected. In the “acceleressence” scenario, this scale is identi- fied with the dark energy scale.14 We consider only one superfield which breaks cA model with the masses of the left-handed neutrinos given by the see-saw mechanism is unstable even if it has a small power-law scalar potential.13 October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka 6 Ryo Takahashi Fig. 1. The illustration of interactions among three sectors. The dark sector couples to the left- handed neutrino through a new scalar force in the MaVaNs scenario. The dark sector is also assumed to be related with the hidden sector only through the gravity. supersymmetry for simplicity. If one extends the hidden sector, one can consider a different mediation mechanism between the standard model and the hidden sector from one between the dark and the hidden sector. In this framework, taking supersymmetry breaking effect into account, the scalar potential is given by V (φ) = |φ|4 − (φ3 + h.c.) +M2A|φ| 2 +m2D|φ| 2 −m2|φ|2 + V0, (23) where κ and m are supersymmetry breaking parameters, and V0 is a constant determined by the condition that the cosmological constant is vanishing at the true minimum of the acceleron potential. We consider two types of the neutrino mass matrix in this scalar potential. They are the cases of the vanishing and the finite mixing between the left-handed and a sterile neutrino. 3.1. Case of the Vanishing Mixing When the mixing between the left-handed and a sterile neutrino is vanishing,mD = 0 in the neutrino mass matrix (5). Then we have the masses of the left-handed and a sterile neutrino as mνL = c, (24) mψ = MA + λ < φ > . (25) In this case, we find that only the mass of a sterile neutrino is variable on cosmo- logical time scale due to the second term of the right hand side in Eq. (25). Let us adjust parameters which satisfy Eqs. (8) and (13)∼(16). In Eq. (8), the scalar potential Eq. (23) is used. Putting typical values for four parameters by hand as follows: λ = 1, mD = 0, m = 2× 10−2 eV, m0ψ = 10 −2 eV, (26) October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 7 we have < φ >0≃ −1.31× 10−5 eV, c = 2× 10−2 eV, MA ≃ 10 −2 eV, m ≃ 10−2 eV, κ ≃ 4.34× 10−3 eV. (27) We need fine-tuning between MA and m in order to satisfy the constraint on the present accerelon mass of Eq. (16). We show evolutions of the mass of a sterile neutrino and the equation of state parameter in Figs. 2, 3 and 4. The behavior of the mass of a neutrino near the present epoch is shown in Fig. 3. We find that the mass of a sterile neutrino have varied slowly in this epoch. This means that the first term of the left hand side in Eq. (20), which is a negative contribution to the speed of sound squared, is tiny. We can also check the positive speed of sound squared in a numerical calculation. Therefore, the neutrino-acceleron fluid is hydrodynamically stable and acts as the dark energy. 3.2. Case of the Finite Mixing Next, we consider the case of the finite mixing between the left-handed and a sterile neutrino (mD 6= 0). In this case, the left-handed and a sterile neutrino mass are given by mνL = c+MA + λ < φ > [c− (MA + λ < φ >)]2 + 4m , (28) c+MA + λ < φ > [c− (MA + λ < φ >)]2 + 4m . (29) We find that both masses of the left-handed and a sterile neutrino are variable on cosmological time scale due to the term of the acceleron dependence. Taking typical values for four parameters as λ = 1, mD = 10 −3 eV, m0νL = 2× 10 −2 eV, m0ψ = 10 −2 eV, (30) we have < φ >0≃ −1.31× 10−5 eV, c ≃ 1.99× 10−2 eV, MA ≃ 1.01× 10 −2 eV, m ≃ 1.02× 10−2 eV, κ ≃ 4.34× 10−3 eV. (31) where we required that the mixing between the active and a sterile neutrino is tiny. In our model, the small present value of the acceleron is needed to satisfy the constraints on the scalar potential in Eqs. (8) and (13). Values of parameters in (31) are almost same as the case of the vanishing mixing (27). However, the mass of the left-handed neutrino is variable unlike the vanishing mixing case. The time evolution of the left-handed neutrino mass is shown in Fig. 5. The mixing does not affect the evolution of a sterile neutrino mass and the equation of state parameter, which are shown in Figs. 6, 7. Since the variation in the mass of the left-handed neutrino is not vanishing but extremely small, the model can also avoid the instability of speed of sound. October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka 8 Ryo Takahashi Finally, we comment on the smallness of the evolution of the neutrino mass at the present epoch. In our model, the mass of the left-handed and a sterile neutrino include the constant part. A variable part is a function of the acceleron. In the present epoch, the constant part dominates the neutrino mass because the present value of the acceleron should be small. This smallness of the value of the acceleron is required from the cosmological observation and the stationary condition in Eqs. (8) and (13). 4. Summary We presented a supersymmetric MaVaNs model including effects of the supersym- metry breaking mediated by the gravity. Evolutions of the neutrino mass and the equation of state parameter have been calculated in the model. Our model has a chiral superfield in the dark sector, whose scalar component causes the present cosmic acceleration, and the right-handed neutrino superfield. In our framework, supersymmetry is broken in the hidden sector at TeV scale and the effect is assumed to be transmitted to the dark sector only through the gravity. Then, the scale of soft parameters of O(10−3-10−2)(eV) is expected. We considered two types of model. 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Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 20) October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 11 Fig. 4. Evolution of w (0 ≤ z ≤ 50) Fig. 5. Evolution of the mass of the left-handed neutrino (0 ≤ z ≤ 2000) October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka 12 Ryo Takahashi Fig. 6. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 2000) Fig. 7. Evolution of w (0 ≤ z ≤ 50) Introduction MaVaNs Model in Supersymmetric Theory Effect of supersymmetry breaking Case of the Vanishing Mixing Case of the Finite Mixing Summary

0704.0187

The transverse proximity effect in spectral hardness on the line of sight towards HE 2347-4342

Astronomy & Astrophysics manuscript no. 7585 c© ESO 2018 October 27, 2018 The transverse proximity effect in spectral hardness on the line of sight towards HE 2347−4342 ⋆ G. Worseck1, C. Fechner2,3, L. Wisotzki1, and A. Dall’Aglio1 1 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany 2 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany 3 Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany Received 2 April 2007 / Accepted 31 July 2007 ABSTRACT We report the discovery of 14 quasars in the vicinity of HE 2347−4342, one of the two quasars whose intergalactic He ii forest has been resolved with FUSE. By analysing the H i and the He ii opacity variations separately, no transverse proximity effect is detected near three foreground quasars of HE 2347−4342: QSO J23503−4328 (z = 2.282, ϑ = 3.′59), QSO J23500−4319 (z = 2.302, ϑ = 8.′77) and QSO J23495−4338 (z = 2.690, ϑ = 16.′28). This is primarily due to line contamination and overdensities probably created by large-scale structure. By comparing the H i absorption and the corresponding He ii absorption, we estimated the fluctuating spectral shape of the extragalactic UV radiation field along this line of sight. We find that the UV spectral shape near HE 2347−4342 and in the projected vicinity of the three foreground quasars is statistically harder than expected from UV background models dominated by quasars. In addition, we find three highly ionised metal line systems near the quasars. However, they do not yield further constraints on the shape of the ionising field. We conclude that the foreground quasars show a transverse proximity effect that is detectable as a local hardening of the UV radiation field, although the evidence is strongest for QSO J23495−4338. Thus, the relative spectral hardness traces the proximity effect also in overdense regions prohibiting the traditional detection in the H i forest. Furthermore, we emphasise that softening of quasar radiation by radiative transfer in the intergalactic medium is important to understand the observed spectral shape variations. From the transverse proximity effect of QSO J23495−4338 we obtain a lower limit on the quasar lifetime of ∼ 25 Myr. Key words. quasars: general – quasars: absorption lines – intergalactic medium – diffuse radiation 1. Introduction After reionisation the intergalactic medium (IGM) is kept highly photoionised by the metagalactic UV radiation field generated by the overall population of quasars and star-forming galax- ies (e.g. Haardt & Madau 1996; Fardal et al. 1998; Bianchi et al. 2001; Sokasian et al. 2003). The intensity and spectral shape of the UV background determines the ionisation state of the observ- able elements in the IGM. In particular, the remaining fraction of intergalactic neutral hydrogen and singly ionised helium is responsible for the Lyα forest of H i and He ii. On lines of sight passing near quasars the IGM will be sta- tistically more ionised due to the local enhancement of the UV flux that should result in a statistically higher IGM transmission (’void’) in the QSO’s vicinity (Fardal & Shull 1993; Croft 2004; McDonald et al. 2005). This so-called proximity effect has been found with high statistical significance on lines of sight towards luminous quasars (e.g. Bajtlik et al. 1988; Giallongo et al. 1996; Scott et al. 2000). On the other hand, a transverse proximity ef- fect created by foreground ionising sources nearby the line of sight has not been clearly detected in the H i forest, except the recent detection at z = 5.70 by Gallerani et al. (2007). While two large H i voids have been claimed to be due to the transverse Send offprint requests to: G. Worseck, e-mail: [email protected] ⋆ Based on observations collected at the European Southern Observatory, Chile (Proposals 070.A-0425 and 074.A-0273). Data col- lected under Proposals 068.A-0194, 070.A-0376 and 116.A-0106 was obtained from the ESO Science Archive. proximity effect by Dobrzycki & Bechtold (1991a, however see Dobrzycki & Bechtold 1991b) and Srianand (1997), other studies find at best marginal evidence (Fernández-Soto et al. 1995; Liske & Williger 2001), and most attempts resulted in non-detections (Crotts 1989; Møller & Kjærgaard 1992; Crotts & Fang 1998; Schirber et al. 2004; Croft 2004). This has led to explanations involving the systematic effects of anisotropic radiation, quasar variability (Schirber et al. 2004), intrinsic overdensities (Loeb & Eisenstein 1995; Rollinde et al. 2005; Hennawi & Prochaska 2007; Guimarães et al. 2007) and finite quasar lifetimes (Croft 2004). Intergalactic He ii Lyα absorption (λrest = 303.7822 Å) can be studied only towards the few quasars at z > 2 whose far UV flux is not extinguished by intervening Lyman limit systems (Picard & Jakobsen 1993; Jakobsen 1998). Of the six quasars successfully observed so far, the lines of sight to- wards HE 2347−4342 (z = 2.885) and HS 1700+6416 (z = 2.736) probe the post-reionisation era of He ii with an emerg- ing He ii forest that has been resolved with FUSE (Kriss et al. 2001; Shull et al. 2004; Zheng et al. 2004; Fechner et al. 2006; Fechner & Reimers 2007a). In a highly ionised IGM a comparison of the H i with the corresponding He ii absorption yields an estimate of the spectral shape of the UV radiation field due to the different ionisation thresholds of both species. The amount of He ii compared to H i gives a measure of the spectral softness, generally expressed via the column density ratio η = NHe ii/NH i. Typically, η <∼ 100 indi- cates a hard radiation field generated by the surrounding quasar http://arxiv.org/abs/0704.0187v2 2 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 population, whereas η >∼ 100 requires a significant contribu- tion of star-forming galaxies or heavily softened quasar radiation (e.g. Haardt & Madau 1996; Fardal et al. 1998; Haardt & Madau 2001). The recent FUSE observations of the He ii Lyα forest re- vealed large η fluctuations (1 <∼ η <∼ 1000) on small scales of 0.001 <∼ ∆z <∼ 0.03 with a median η ≃ 80–100. Apart from scatter due to the low-quality He ii data at S/N ∼ 5 (Fechner et al. 2006; Liu et al. 2006) and possible systematic er- rors due to the generally assumed line broadening mechanism (Fechner & Reimers 2007a), several physical reasons for these η variations have been proposed. A combination of local den- sity variations (Miralda-Escudé et al. 2000), radiative transfer effects (Maselli & Ferrara 2005; Tittley & Meiksin 2006) and lo- cal differences in the properties of quasars may be responsible for the fluctuations. In particular, at any given point in the IGM at z > 2 only a few quasars with a range of spectral indices (Telfer et al. 2002; Scott et al. 2004) contribute to the UV back- ground at hν ≥ 54.4 eV (Bolton et al. 2006). Already low-resolution He ii spectra obtained with HST in- dicate a fluctuating radiation field, which has been interpreted as the onset of He ii reionisation in Strömgren spheres around hard He ii photoionising sources along or near the line of sight (Reimers et al. 1997; Heap et al. 2000; Smette et al. 2002). Jakobsen et al. (2003) found a quasar coinciding with the promi- nent He ii void at z = 3.05 towards Q 0302−003, thereby pre- senting the first clear case of a transverse proximity effect. In Worseck & Wisotzki (2006), hereafter Paper I, we revealed the transverse proximity effect as a systematic increase in spectral hardness around all four known foreground quasars along this line of sight. This suggests that a hard radiation field is a sensi- tive probe of the transverse proximity effect even if there is no associated void in the H i forest, either because of the weakness of the effect, or because of large-scale structure. Along the line of sight towards HE 2347−4342 several He ii voids have been claimed to be due to nearby unknown AGN (Smette et al. 2002). Likewise, some forest regions with a de- tected hard radiation field may correspond to proximity ef- fect zones of putative foreground quasars (Fechner & Reimers 2007a). Here we report on results from a slitless spectroscopic quasar survey in the vicinity of HE 2347−4342 and on spectral shape fluctuations of the UV radiation field probably caused by foreground quasars towards the sightline of HE 2347−4342. The paper is structured as follows. Sect. 2 presents the observations and the supplementary data employed for the paper. Although we do not detect any transverse proximity effect in the H i for- est (Sect. 3), the fluctuating UV spectral shape along the line of sight indicates a hard radiation field in the projected vicinity of the foreground quasars (Sect. 4). In Sect. 5 we study three nearby metal line systems which could further constrain the ion- ising field. We interpret the statistically significant excesses of hard radiation as being due to the transverse proximity effect (Sect. 6). We present our conclusions in Sect. 7. Throughout the paper we adopt a flat cosmological model with Ωm = 0.3, ΩΛ = 0.7 and H0 = 70 km s −1 Mpc−1. 2. Observations and data reduction 2.1. Search for QSO candidates near HE 2347−4342 In October 2002 we observed a 25′ × 33′ field centered on HE 2347−4342 (z = 2.885) with the ESO Wide Field Imager (WFI, Baade et al. 1999) at the ESO/MPI 2.2 m Telescope (La Silla) in its slitless spectroscopic mode (Wisotzki et al. 2001) as part of a survey for faint quasars in the vicinity of established high-redshift quasars. A short summary of the survey is given in Paper I; a detailed description will follow in a separate paper. A semi-automated search for emission line objects among the slitless spectra of the ∼ 1400 detected objects in the field resulted in 10 prime quasar candidates. 2.2. Spectroscopic follow-up Follow-up spectroscopy of these 10 quasar candidates was ob- tained with the Focal Reducer/Low Dispersion Spectrograph 2 (FORS2, Appenzeller et al. 1998) on ESO VLT UT1/Antu in Visitor Mode on November 17 and 19, 2004 under variable see- ing but clear conditions. The spectra were taken either with the 300V grism or the 600B grism and a 1′′ slit kept at the parallactic angle, resulting in a spectral resolution of ∼ 10 Å FWHM and ∼ 4.5 Å FWHM, respectively. No order separa- tion filter was employed, leading to possible order overlap at λ > 6600 Å in the spectra taken with the 300V grism. Exposure times were adjusted to yield S/N ∼ 20 in the quasar con- tinuum. The spectra were calibrated in wavelength against the FORS2 He/Ne/Ar/HgCd arc lamps and spectrophotometrically calibrated against the HST standard stars Feige 110 and GD 108. Data reduction was performed with standard IRAF tasks us- ing the optimal extraction algorithm by Horne (1986). Figure 1 shows the spectra of the quasars together with 4 quasars from an- other survey (Sect. 2.3). Table 1 summarises our spectroscopic follow-up observations. 2.3. Additional quasars We checked the ESO Science Archive for additional quasars in the vicinity of HE 2347−4342 and found several unpublished quasars from a deeper slitless spectroscopic survey using the ESO VLT, the results of which (on the field of Q 0302−003) are described in Jakobsen et al. (2003). We obtained their follow-up spectra of quasars surrounding HE 2347−4342 from the archive and publish them here in agreement with P. Jakobsen. In the course of their survey FORS1 spectra of 10 candidates were taken with the 300V grism crossed with the GG435 order separa- tion filter and a 1′′ slit, calibrated against the standards LTT 7987 and GD 50. Seven of their candidates are actually quasars, of which 3 were also found independently by our survey. The re- maining 4 quasars are beyond our redshift-dependent magnitude limit. The FORS1 spectra of the 4 additional quasars are dis- played in Fig. 1 and listed separately in Table 1. According to the quasar catalogue by Véron-Cetty & Véron (2006) there are no other previously known quasars within a radius < 30′ around HE 2347−4342. 2.4. Redshifts and magnitudes Redshifts of the 14 quasars were determined by taking every de- tectable emission line into account. Line peaks were measured by eye and errors were estimated taking into account the S/N of the lines, line asymmetries and the presence of absorption systems. The quasar redshifts were derived by weighting the measurements of detected lines. Since high-ionisation lines suf- fer from systematic blueshifts with respect to the systemic red- shift (Gaskell 1982; Tytler & Fan 1992; McIntosh et al. 1999), a higher weight was given to low-ionisation lines. Obviously blueshifted lines were discarded. Redshift errors were estimated G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 3 Fig. 1. VLT/FORS spectra of quasars in the vicinity of HE 2347−4342. The spectra are shown in black together with their 1σ noise arrays (green lines). The small inserts show the corresponding discovery spectra from our slitless survey in the same units. from the redshift differences of the remaining lines and their es- timated errors. The 14 discovered quasars lie in the broad redshift range 0.720 ≤ z ≤ 3.542. Fig. 2 shows their angular separations with respect to HE 2347−4342. We find three background quasars to HE 2347−4342 and we identify a pair of bright quasars at z ≃ 1.763 separated by 7.′8. Three foreground quasars (labelled A–C in Table 1 and Fig. 2) are located in the redshift range to study the transverse proximity effect. Table 2 provides the redshift measurements for the detected emission lines in their spectra. The redshift of QSO J23503−4328 was based on Lyα and C iv. The measurement of the Mg ii is uncertain because of the decline of the resolving power of the 300V grism towards the red, but yields a slightly higher redshift than the adopted one. For QSO J23500−4319 we measured a consistent redshift from the C iv and the C iii] line. The redshift measurement of QSO J23495−4338 was difficult due to several metal absorp- tion line systems of which only two Mg ii systems at z = 0.921 and z = 1.518 could be identified. In particular, Fe ii absorp- tion from the z = 0.912 system hampered a redshift measure- 4 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 Table 1. Quasars observed near the line of sight of HE 2347−4342. The first 10 listed quasars have been found in our survey, the remaining 4 quasars result from the previously unpublished survey by P. Jakobsen. Quasar magnitudes are B and V magnitudes for our survey and Jakobsen’s survey, respectively. Object α (J2000) δ (J2000) z Magnitude Night Grism Exposure Airmass Seeing Abbr. QSO J23510−4336 23h51m05.s50 −43◦36′57.′′2 0.720 ± 0.002 20.74 ± 0.27 19 Nov 2004 300V 1200 s 1.30 1.′′3 QSO J23507−4319 23h50m44.s97 −43◦19′26.′′0 0.850 ± 0.003 19.90 ± 0.07 17 Nov 2004 600B 360 s 1.28 0.′′7 QSO J23507−4326 23h50m45.s39 −43◦26′37.′′0 1.635 ± 0.003 21.05 ± 0.14 17 Nov 2004 300V 200 s 1.23 1.′′0 QSO J23509−4330 23h50m54.s80 −43◦30′42.′′2 1.762 ± 0.004 18.23 ± 0.03 17 Nov 2004 600B 300 s 1.08 0.′′7 QSO J23502−4334 23h50m16.s18 −43◦34′14.′′7 1.763 ± 0.003 18.95 ± 0.04 17 Nov 2004 300V 60 s 1.18 0.′′7 QSO J23503−4328 23h50m21.s55 −43◦28′43.′′7 2.282 ± 0.003 20.66 ± 0.11 17 Nov 2004 300V 400 s 1.20 0.′′7 A QSO J23495−4338 23h49m34.s53 −43◦38′08.′′7 2.690 ± 0.006 20.21 ± 0.17 19 Nov 2004 300V 360 s 1.13 1.′′2 C QSO J23511−4319 23h51m09.s44 −43◦19′41.′′6 3.020 ± 0.004 21.00 ± 0.14 17 Nov 2004 600B 1000 s 1.09 1.′′1 QSO J23514−4339 23h51m25.s54 −43◦39′02.′′9 3.240 ± 0.004 21.57 ± 0.29 17 Nov 2004 300V 1400 s 1.14 1.′′2 QSO J23503−4317 23h50m21.s94 −43◦17′30.′′0 3.542 ± 0.005 21.94 ± 0.62 19 Nov 2004 300V 1800 s 1.23 1.′′2 600B 1800 s 1.33 1.′′2 QSO J23515−4324 23h51m33.s05 −43◦24′45.′′2 1.278 ± 0.002 20.82 ± 0.14 06 Oct 2002 300V 900 s 1.24 0.′′7 QSO J23512−4332 23h51m15.s18 −43◦32′34.′′3 1.369 ± 0.001 21.52 ± 0.25 06 Oct 2002 300V 900 s 1.18 0.′′7 QSO J23508−4335 23h50m52.s91 −43◦35′06.′′8 1.778 ± 0.002 22.01 ± 0.32 06 Oct 2002 300V 900 s 1.11 0.′′9 QSO J23500−4319 23h50m00.s28 −43◦19′46.′′1 2.302 ± 0.002 22.61 ± 0.83 06 Oct 2002 300V 900 s 2.37 0.′′8 B Table 2. Detected emission lines and redshifts of QSOs A–C. Object Emission line λobs [Å] z QSO J23503−4328 Lyα 3989 ± 4 2.281 ± 0.003 N v 4070 ± 8 2.282 ± 0.006 Si iv+O iv] 4585 ± 8 2.276 ± 0.006 C iv 5082 ± 4 2.281 ± 0.003 C iii] 6253 ± 7 2.276 ± 0.004 Mg ii 9196 ± 12 2.286 ± 0.004 2.282 ± 0.003 QSO J23500−4319 Si iv+O iv] 4613 ± 6 2.296 ± 0.004 C iv 5115 ± 3 2.302 ± 0.002 C iii] 6305 ± 2 2.303 ± 0.001 2.302 ± 0.002 QSO J23495−4338 Lyα 4513 ± 10 2.712 ± 0.008 O i+Si ii 4823 ± 10 2.694 ± 0.008 C ii 4930 ± 10 2.692 ± 0.007 Si iv+O iv] 5135 ± 15 2.669 ± 0.011 C iv 5691 ± 10 2.674 ± 0.006 C iii] 7028 ± 10 2.682 ± 0.005 2.690 ± 0.006 ment of the Lyα line. The C iv and the C iii] lines show unidenti- fied absorption features. Thus, the redshift of QSO J23495−4338 is heavily weighted towards the very noisy low-ionisation lines O i+Si ii and C ii. However, redshift uncertainties of the fore- ground quasars do not significantly affect our results. Apparent magnitudes were derived from target aquisition images photometrically calibrated against the standard star fields PG 2213−006 or Mark A (Landolt 1992). Unfortunately the aquisition exposures of the faintest quasars were too short to de- termine their magnitudes accurately. Magnitudes derived from integration of the spectra are consistent with the photometric ones after correcting for slit losses. We note that QSO J23507−4326 is variable. This quasar has been detected in both slitless surveys and had V ≃ 20.3 in October 2001, V ≃ 20.7 in October 2002 and V ≃ 21.0 in November 2004. We were able to discover this quasar in its bright phase while missing the slightly fainter quasar QSO J23515−4324 detected only in the survey by P. Jakobsen. 2.5. Optical spectra of HE 2347−4342 From the ESO Science Archive we retrieved the optical spectra of HE 2347−4342 taken with UVES at VLT UT2/Kueyen in the Large Programme “The Cosmic Evolution of the Intergalactic Medium” (Bergeron et al. 2004). Data reduction was performed Fig. 2. Distribution of separation angles ϑ vs. redshift z of the quasars from Table 1 with respect to HE 2347−4342. Symbol size indicates apparent optical magnitude. using the UVES pipeline provided by ESO (Ballester et al. 2000). The vacuum-barycentric corrected co-added spectra yield a S/N ∼ 100 in the Lyα forest at R ∼ 45000. The spectrum was normalised in the covered wavelength range 3000 <∼ λ <∼ 10000 Å using a cubic spline interpolation algorithm. 2.6. Far-UV spectra of HE 2347−4342 HE 2347−4342 is one of the two high-redshift quasars observed successfully in the He ii Lyα forest below 303.7822 Å rest frame wavelength with the Far Ultraviolet Spectroscopic Explorer (FUSE) at a resolution of R ∼ 20000, although at a S/N <∼ 5 (Kriss et al. 2001; Zheng et al. 2004). G. Kriss and W. Zheng kindly provided the reduced FUSE spec- trum of HE 2347−4342 described in Zheng et al. (2004). We adopted their flux normalisation with a power law fλ = 3.3 × 10−15 λ/1000Å )−2.4 erg cm−2 s−1 Å−1 reddened by the Cardelli et al. (1989) extinction curve assuming E(B − V) = 0.014 (Schlegel et al. 1998). 3. The Lyα forest near the foreground quasars Aiming to detect the transverse proximity effect as an underden- sity (’void’) in the Lyα forest towards HE 2347−4342 we ex- amined the forest regions in the projected vicinity of the three foreground quasars labelled A–C in Table 1. The H i forest of G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 5 Fig. 3. The Lyα forest of HE 2347−4342 in the vicinity of the foreground quasars A–C from Table 1. The upper panels show the normalised optical spectrum of HE 2347−4342 including Lyβ and metal lines (red) and the H i Lyα transmission obtained from the line list by T.-S. Kim (black). The binned blue line shows the mean H i Lyα transmission in ∆z = 0.005 bins towards HE 2347−4342, whereas the dashed green line indicates the expected mean transmission 〈T 〉exp. The lower panels display the corresponding He ii transmission from the FUSE spectrum. [See the online edition of the Journal for a colour version of this figure.] HE 2347−4342 has been analysed in several studies, e.g. by Zheng et al. (2004) and Fechner & Reimers (2007a), hereafter called Z04 and FR07, respectively. Since the line list from FR07 is limited to z > 2.29, T.-S. Kim (priv. comm.) kindly provided an independent line list including the lower redshift Lyα forest (z > 1.79). Both line lists agree very well in their overlapping redshift range 2.29 < z < 2.89. Figure 3 displays the H i and the He ii forest regions near the foreground quasars A–C. The H i Lyα forest is contaminated by metals. In particular at z < 2.332 there is severe contam- ination due to the O vi absorption of the associated system of HE 2347−4342 (Fechner et al. 2004). Because the strong O vi absorption overlaps with the projected positions of QSO A and QSO B it is very difficult to obtain a well-determined H i line sample in this region. Furthermore, there is Lyβ absorption of H i and He ii at z < 2.294. We also overplot in Fig. 3 the mean H i Lyα transmission in ∆z = 0.005 bins obtained from T.-S. Kim’s line list and the generally expected mean transmission over sev- eral lines of sight 〈T 〉exp = e−τ eff with τ eff = 0.0032 (1 + z) (Kim et al. 2002). We do not detect a significant void near the three foreground quasars, neither in the H i forest nor in the He ii forest. In the vicinity of QSO A and QSO B, even a careful decontamina- tion of the optical spectrum does not reveal a significant H i underdensity. Instead, the transmission is fluctuating around the mean. Due to the poor quality of the FUSE data in this region (S/N <∼ 2) and the He ii Lyβ absorption from higher redshifts, a simple search for He ii voids near QSO A and QSO B is impos- sible. In the vicinity of QSO C the H i Lyα absorption is slightly higher than on average. There is a small void at z ≃ 2.702 that can be identified in the forests of both species. The probability of chance occurrence of such small underdensities is high, so link- ing this void to QSO C seems unjustified. However, note that the He ii absorption in the vicinity of QSO C (z ∼ 2.69) is lower than at z ∼ 2.71 in spite of the same H i absorption. This points to fluctuations in the spectral shape of the ionising radiation near the quasar (Sect. 4.3). Given the luminosities and distances of our foreground quasars to the sightline of HE 2347−4342, could we expect to detect the transverse proximity effect as voids in the H i forest? As in Paper I, we modelled the impact of the foreground quasars on the line of sight towards HE 2347−4342 with the parameter ω (z) = fνLL , j 4πJν (z) 1 + z′j )−α j+1 1 + z j αJν + 3 α j + 3 z j, 0 z j, z which is the ratio between the summed photoionisation rates of n quasars at redshifts z j with rest frame Lyman limit fluxes fνLL , j, penetrating the absorber at redshift z and the overall UV back- ground with Lyman limit intensity Jν. dL(z j, 0) is the luminosity distance of QSO j, and dL(z j, z) is its luminosity distance as seen at the absorber; the redshift of the quasar as seen at the absorber is z′j (Liske 2000). A value ω ≫ 1 predicts a highly significant proximity effect. We assumed a constant UV background at 1 ryd of Jν = 7×10−22 erg cm−2 s−1 Hz−1 sr−1 (Scott et al. 2000) with a power- law shape Jν ∝ ν −αJν and αJν = 1.8. The quasar Lyman limit fluxes were estimated from the spectra by fitting a power law fν ∝ ν −α to the quasar continuum redward of the Lyα emis- sion line, excluding the emission lines. The spectra were scaled to yield the measured photometric magnitudes. Table 3 lists the resulting spectral indices, the H i Lyman limit fluxes, and the transverse distances. The combined effects of QSOs A and B result in a peak ωmax ≃ 0.89, while QSO C yields ωmax ≃ 0.11. So we expect only a weak signature of the transverse proximity effect that can be easily diluted by small-scale transmission fluctuations around 〈T 〉exp. Thus, the apparent lack of a transverse proximity effect in the H i forest is no surprise. We can also roughly estimate the amplitude of the proximity effect in the He ii forest. Extrapolating the power laws (QSOs and background) above 4 ryd at η = 50 (Haardt & Madau 1996, hereafter HM96) we get ωmax ≃ 20 near QSO A and ωmax ≃ 2 near QSO C. A softer background would result in higher values of ω, whereas absorption of ionising photons in the He ii forest would decrease ω. However, due to the arising He ii Lyβ forest and the low S/N in the FUSE data near QSOs A and B, even high ω values do not necessarily result in a visible He ii void. 6 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 Table 3. Rest frame Lyman limit fluxes of foreground QSOs. A power law fν ∝ ν −α is fitted to the QSO continua and fνLL is the extrapolated H i Lyman limit flux in the QSO rest frame. d⊥(z) denotes the transverse proper distance to the line of sight towards HE 2347−4342. QSO Abbr. z α fνLL [µJy] d⊥(z) [Mpc] QSO J23503−4328 A 2.282 0.21 16 1.76 QSO J23500−4319 B 2.302 0.84 1 4.33 QSO J23495−4338 C 2.690 0.24 29 7.75 In the direct vicinity of QSO C the He ii data is not saturated, but shows no clear void structure either. We will show in the following sections that the spectral shape of the radiation field is a more sensitive indicator of the transverse proximity effect than the detection of voids in the forests. 4. The fluctuating shape of the UV radiation field towards HE 2347−4342 4.1. Diagnostics If both hydrogen and helium are highly ionised in the IGM with roughly primordial abundances, the column density ra- tio η = NHe ii/NH i indicates the softness of the UV radiation field impinging on the absorbers. Theoretically, η can be de- rived numerically via photoionisation models of the IGM with an adopted population of ionising sources. At the redshifts of interest, 50 <∼ η <∼ 100 is predicted for a UV background gen- erated by quasars (HM96; Fardal et al. 1998), whereas higher values indicate a contribution of star-forming galaxies (e.g. Haardt & Madau 2001, hereafter HM01). The He ii forest has been resolved with FUSE towards HE 2347−4342 and HS 1700+6416, allowing a direct estimation of η by fitting the absorption lines (Kriss et al. 2001; Zheng et al. 2004; Fechner et al. 2006, FR07). Due to the low S/N and the strong line blending in the He ii forest the He ii lines have to be fitted with absorber redshifts and Doppler parameters fixed from the fitting of the H i data of much higher quality. Generally, pure non-thermal line broadening (bHe ii = bH i) is assumed (how- ever, see FR07 and Sect. 6 below). The He ii forest towards HE 2347−4342 was fitted independently by Z04 and FR07. In the following, we rely on the line fitting results from FR07, which at any rate are consistent with those obtained by Z04 in the redshift ranges near the quasars. All current studies indicate that η is strongly fluctuating on very small scales in the range 1 <∼ η <∼ 1000. The median column density ratio towards HE 2347−4342 is η ≃ 62 (Z04), whereas Fechner et al. (2006) find a higher value of η ≃ 85 towards HS 1700+6416. Both studies find evidence for an evolution of η towards smaller values at lower redshifts. However, only part of the scatter in η is due to redshift evolution and statistical er- rors, so the spectral shape of the UV radiation field has to fluc- tuate (FR07). Although the analyses of both available lines of sight give consistent results, cosmic variance may bias the de- rived median η and its evolution. This is of particular interest for our study, since we want to reveal local excesses of low η near the quasars with respect to the median (Sect. 4.3). Clearly, more lines of sight with He ii absorption would be required to yield tighter constraints on the redshift evolution of η. The detailed results of visual line fitting may be subjective and may depend on the used fitting software. In particular, am- biguities in the decomposition of blended H i lines can affect the derived η values (Fechner & Reimers 2007b). Therefore we also analyse the UV spectral shape variations using the ratio of the effective optical depths τeff,He ii τeff,H i . (2) As introduced in Paper I, this parameter is a resolution- independent estimator of the spectral shape of the UV radiation field with small (high) R values indicating hard (soft) radiation on a certain redshift scale ∆z. Shull et al. (2004) followed a sim- ilar approach by taking η ≃ 4τHe ii/τH i for a restricted τ range on scales of ∆z = 1.6× 10−4 and ∆z = 6.6× 10−4. However, this scaling relation between τ and η is only valid at the centre of an absorption line (Miralda-Escudé 1993). The column density ratio is defined per absorption line and not as a continuous quan- tity, whereas R can be defined on any scale. While R and η are correlated (see below), there is no simple conversion between R and η and the correlation will depend on the adopted redshift scale of R. 4.2. Fluctuations in R and η along the line of sight We obtained R(z) by binning both normalised Lyα forest spec- tra of H i and He ii into aligned redshift bins of ∆z = 0.005 in the range 2.3325 < z < 2.8975 and computed R = ln 〈THe ii〉/ ln 〈TH i〉 with the mean transmission 〈THe ii〉 and 〈TH i〉. The choice of the redshift binning scale was motivated by the typical scale of η fluctuations 0.001 <∼ ∆z <∼ 0.03 (Kriss et al. 2001, FR07). We adopted the binning procedure by Telfer et al. (2002) in order to deal with original flux bins that only partly overlap with the new bins. The errors were computed accord- ingly. Due to the high absorption and the low S/N of the He ii data we occasionally encountered unphysical values 〈THe ii〉 ≤ 0. These were replaced by their errors, yielding lower limits on R. We mostly neglected the usually small metal contamination in the computation of 〈TH i〉 in the Lyα forest because the errors in R are dominated by the low S/N and the more uncertain contin- uum level of the He ii spectrum. The FUSE data in the redshift bins at z = 2.375, 2.380, 2.730, 2.735, 2.845 and 2.850 are con- taminated by galactic H2 absorption, so no R measurement on the full scale of ∆z = 0.005 can be performed there. At 2.29 <∼ zLyα <∼ 2.33 the H i Lyα forest is severely con- taminated by O vi from the associated system of HE 2347−4342 (Fechner et al. 2004). Furthermore, the Lyβ forest of both species emerges at zLyα < 2.294. Because this excess absorp- tion would bias the direct estimation of R in the spectra, we tried to decontaminate the forests at z < 2.332. 〈TH i〉 was computed from the H i Lyα forest reconstructed from the line list by T.- S. Kim (Sect. 3). The corresponding 〈THe ii〉 was obtained after dividing the FUSE data by the simulated Lyβ absorption of the lines at higher redshift. Since the decontamination depends on the validity of the He ii line parameters as well as on the com- pleteness of the H i line list in the complex region contaminated by O vi, the derived R values at z < 2.332 have to be regarded as rough estimates. The resulting R(z) is shown in the upper panel of Fig. 4. The optical depth ratio strongly fluctuates around its median value R ≃ 4.8 obtained for uncontaminated redshift bins, indicating spectral fluctuations in the UV radiation field. We also show in Fig. 4 the median η(z) on the same redshift bins based on the line fitting results in FR07. Also the median η strongly fluctuates with a slight trend of an increase with redshift (Z04). Clearly, the data is inconsistent with a spatially uniform UV background, but the median η ≃ 70 of the line sample is consistent with G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 7 Fig. 4. The fluctuating spectral shape of the UV background towards HE 2347−4342. The up- per panel shows the ratio of effective optical depths R vs. redshift z in ∆z = 0.005 bins. Data points at z < 2.332 (crosses) have been decontaminated from O vi and Lyβ absorp- tion (see text). Foreground quasars are marked with letters and vertical dotted lines as well as HE 2347−4342 (star symbol). The green dashed line indicates the median R ≃ 4.8 ob- tained at z > 2.332 in uncontaminated bins. The lower panel shows the median η from FR07 in the same redshift bins. The red dashed line in- dicates the median η ≃ 70 of the line sample. quasar-dominated models of the UV background. A comparison of R(z) and η(z) reveals that both quantities are correlated. The Spearman rank order correlation coefficient is rS = 0.67 with a probability of no correlation PS = 6 × 10 There is a scatter in the relation between R and η, which is due to noise in the He ii data and due to the fact that R is a spec- tral softness indicator that is smoothed in redshift. Therefore, in addition to the UV spectral shape, R will depend on the den- sity fluctuations of the Lyα forest on the adopted scale. In or- der to estimate the scatter in R due to these density fluctua- tions, we simulated H i and He ii Lyα forest spectra. We gen- erated 100 H i forests with the same overall redshift evolution of eff,H i = 0.0032 (1 + z) 3.37 (Kim et al. 2002) based on the empir- ical line distribution functions in redshift z, column density NH i and Doppler parameter bH i (e.g. Kim et al. 2001). We modelled each forest as a composition of lines with Voigt profiles using the approximation by Tepper-Garcı́a (2006). The spectral resolution (R ∼ 42000) and quality (S/N ∼ 100) closely matches the opti- cal data of HE 2347−4342. The corresponding He ii forests were generated at FUSE resolution with a S/N = 4 for four constant values of η = 10, 20, 50 and 100. We assumed pure non-thermal broadening of the lines. Then we computed R at 2 ≤ z ≤ 3 on our adopted scale ∆z = 0.005, yielding 20000 R measurements for each considered η. For convenience we took out the general red- shift dependence of τeff,H i via dividing by the expected effective optical depth τ eff,H i, so τeff,H i eff,H i is a measure of H i overdensity (D > 1) or underdensity (D < 1). In Fig. 5 we show the relation R(D) obtained from the Monte Carlo simulations and compare it to the distribution observed towards HE 2347−4342. The simulated R(D) can be fitted rea- sonably with a 3rd order polynomial in logarithmic space, yield- ing a general decrease of R with D for every η. The root-mean- square scatter increases from 0.13 dex for η = 10 to 0.18 dex for η = 100. At D >∼ 3 the R(D) distribution flattens due to saturation of high-column density absorbers on the flat part of the curve of growth. The flattening causes substantial overlap between the simulated R distributions at D >∼ 5, making R increasingly in- sensitive to the underlying η. However, at D <∼ 3 hard radiation Fig. 5. Dependence of R on D = τeff,H i/τ eff,H i for different sim- ulated values of η. The black lines indicate the polynomial fits to the simulated distributions in logarithmic space. Red filled circles represent the measured R(D) towards HE 2347−4342 in uncontaminated bins at z > 2.332. The horizontal dotted line de- notes R = 2. [See the online edition of the Journal for a colour version of this figure.] and soft radiation can be reasonably well distinguished. We also overplot the measured R(D) towards HE 2347−4342 in Fig. 5. The observed distribution is inconsistent with a constant η, but the majority of values falls into the modelled η range. While many high R values indicate η > 100, values with R <∼ 2 corre- spond to η <∼ 20 at D <∼ 3. Thus, the very low R values always in- dicate a hard radiation field up to moderate overdensities. As we will see in the next section, the saturation effect probably does not play a role in relating a hard radiation field to the nearby quasars. 4.3. The UV radiation field near the quasars We now investigate in greater detail the spectral shape of the UV radiation field near the four quasars with available data on R and η: the background quasar HE 2347−4342 and the foreground QSOs A–C. Due to the small number of comparison values de- rived from only two lines of sight, we will adopt η = 100 as a characteristic value for the overall UV background at z > 2.6 8 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 (HE 2347−4342, QSO C) and a value of η = 50 at z ∼ 2.3 (QSOs A and B). The former value is close to the median η = 102 obtained by Fechner et al. (2006) at 2.58 < z < 2.75 towards HS 1700+6416, whereas the latter η value accounts for the prob- able evolution of η with redshift. Furthermore, we will compare the η values in the vicinity of the quasars to models of the UV background. 4.3.1. HE 2347−4342 A close inspection of Fig. 4 reveals a strongly fluctuating radi- ation field near HE 2347−4342 with some very small, but also high R values. Also the column density ratio shows large fluc- tuations (1 <∼ η <∼ 1000) with six η <∼ 10 absorbers out of the 20 absorbers at z > 2.86. These strong variations of the spec- tral shape are likely due to radiative transfer effects in the asso- ciated absorption system causing an apparent lack of the prox- imity effect of HE 2347−4342 (Reimers et al. 1997). The high He ii column densities of the associated system may soften the quasar radiation with increasing distance and Fig. 4 supports this interpretation. Due to the probable strong softening of the hard quasar radiation on small scales, the relative spectral hardness near HE 2347−4342 is only revealed by individual low η val- ues instead of robust median values. However, also the highly ionised metal species of the associated system (Fechner et al. 2004) favour the presence of hard QSO radiation. Thus we conclude that despite the lack of a radiation-induced void near HE 2347−4342, its impact onto the IGM can be detected via the relative spectral hardness of the UV radiation field. The three R < 2 values near HE 2347−4342 have D < 3, so they are prob- ably not affected by saturation. 4.3.2. QSOs A and B If our decontamination of the Lyα forests near the two z ∼ 2.3 QSOs A and B is correct, R should reflect UV spectral shape variations also in that region. Indeed, the redshift bin at z = 2.280 next to QSO A (z = 2.282) is a local R minimum with R ≃ 1.5. At z = 2.270 we find R ≃ 0.8. At the redshift of QSO B (z = 2.302) the radiation field is quite soft, but we note a low R ∼ 1 at z = 2.310. We obtain D < 3 for the four R <∼ 2 values near QSO A and QSO B, so saturation is not relevant, and the low R values correspond to low η values. The measured η values in this redshift region are presented in Fig. 6. The error bars are only indicative, since blended line components are not independent and the He ii column densities are derived with constraints from the H i forest. Lower limits on η result from features detected in He ii but not in H i. Due to ambi- guities in the line profile decomposition at the H i detection limit and the present low quality of the He ii data it is hard to judge the reality of most of these added components (Fechner & Reimers 2007b). Nevertheless, since η for adjacent lines may be not inde- pendent due to line blending, we must include the lower limits in the analysis. At z < 2.294 the fitting of He ii lines becomes unreliable due to the arising Lyβ forest. Therefore, no direct es- timates of η can be obtained in the immediate vicinity of QSO A. Furthermore, the H i line sample may be incomplete or the line parameters may be not well constrained due to blending with the O vi of the associated system of HE 2347−4342. Considering these caveats, the median η ≃ 19 obtained for the values at z < 2.332 shown in Fig. 6 is only an estimate. Nevertheless, this is much lower than the typical values η ∼ 50 found at z ∼ 2.3 towards HS 1700+6416 (Fechner et al. 2006). Fig. 6. Column density ratio η vs. redshift z in the vicinity of QSO A and QSO B. The long (short) dashed line indicates the median η ≃ 19 in this redshift range (η = 50 for a UV back- ground generated by quasars). At z < 2.294 the He ii Lyβ forest sets in. Moreover, it is also lower than at slightly higher redshifts to- wards HE 2347−4342. For instance, the median η increases to η = 79 in the redshift range 2.35 ≤ z ≤ 2.40. This is inconsis- tent with the smooth redshift evolution of η on large scales in- ferred by Z04 and Fechner et al. (2006) for both available sight- lines. Thus, we infer an excess of hard radiation in the vicinity of QSO A and QSO B. The most extreme η values are located in the projected vicinity of QSO B, with 6 lines reaching η < 1. If es- timated correctly, these low η values require local hard sources and cannot be generated by the diffuse UV background. Both foreground quasars could be responsible for the hard radiation field because of similar light travel times to the probably affected absorbers (tA ≃ 2tB). 4.3.3. QSO C Since metal contamination of the H i forest is small in the pro- jected vicinity of QSO C (Fig. 3), the UV spectral shape is bet- ter constrained here than near QSO A and QSO B. From Fig. 4 we note a local R minimum (R ≃ 1.3) that exactly coincides with the redshift of QSO C (z = 2.690). At higher redshifts R rises, possibly indicating a softer ionising field. However, at 2.63 <∼ z <∼ 2.695 the optical depth ratio is continuously below the median with R < 2 in five redshift bins. Due to the H i over- densities near QSO C, all R < 2 values have D > 1, but only the bin at z = 2.635 has D ≃ 4, so the remaining ones may still indicate low column density ratios η. Figure 7 displays the η values from FR07 in the redshift range 2.63 < z < 2.73 in the projected vicinity of QSO C. For comparison, we also indicate η = 100 that is consistent with the median η = 102 towards HS 1700+6416 in this red- shift range (Fechner et al. 2006). While the data generally shows strong fluctuations around the median over the whole covered redshift range (Z04; Fechner et al. 2006), there is an apparent excess of small η values near QSO C indicating a predominantly hard radiation field. From the data, a median η ≃ 46 is obtained at 2.63 < z < 2.73 including the lower limits on η. The median η near QSO C is lower than the median η towards HS 1700+6416 by a factor of two and also slightly lower than the η obtained for spatially uniform UV backgrounds generated by quasars. The relative agreement of the median η near QSO C and hard ver- sions of quasar UV background models may result from the soft- ening of the quasar radiation by the IGM at the large proper dis- tances d >∼ 7.75 Mpc considered here (Table 3). This will be G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 9 Fig. 7. Column density ratio η vs. redshift z in the vicinity of QSO C. The short dashed line denotes η = 100 that is consistent with the median η ≃ 102 obtained in the range 2.58 < z < 2.75 towards HS 1700+6416 (Fechner et al. 2006). The long dashed line in- dicates the median η ≃ 46 obtained for the shown η values (2.63 < z < 2.73). further explored in Sect. 6.2. The larger contrast between the median η near QSO C and the median η towards HS 1700+6416 yields stronger evidence for a local hardening of the UV radia- tion near QSO C. However, this comparison value derived from the single additional line of sight tracing this redshift range may be biased itself. Near QSO C the column density ratio still fluctuates and is not homogeneously low as naively expected. We also note an apparent offset of the low η region near QSO C towards lower redshift due to fewer absorbers with low η at z > 2.69. While some of the fluctuations can be explained by uncertainties to recover η reliably from the present data, the very low η ≤ 10 values (≃ 24% of the data in Fig. 7) are likely intrinsically low. These η values are in conflict with a homogeneous diffuse UV background, and are likely affected by a local hard source. In Sect. 6.4 we will estimate the error budget of η by Monte Carlo simulations. If QSO C creates a fluctuation in the spectral shape of the UV background, the distance between the quasar and the line of sight implies a light travel time of t = 25 Myr. The low (high) redshift end of the region shown in Fig. 7 corresponds to a light travel time of 64 Myr (44 Myr). Since these light travel times are comparable, we argue that it is important to consider not only the immediate projected vicinity of QSO C to be affected by the proximity effect (see also Fig. 10 below). In summary, both spectral shape indicators R and η indicate a predominantly hard UV radiation field near all four known quasars in this field. Many η values in the projected vicinity of the quasars indicate a harder radiation than expected even for model UV backgrounds of quasars alone. This points to a transverse proximity effect detectable via the relative spectral hardness. However, there are other locations along the line of sight with an inferred hard radiation field, but without an asso- ciated quasar, most notably the regions at z ∼ 2.48 and z ∼ 2.53 (Fig. 4). Before discussing these in detail (Sect. 6.3), we search for additional evidence for hard radiation near the foreground quasars by analysing nearby metal line systems. 5. Constraints from metal line systems Observed metal line systems provide an additional tool to con- strain the spectral shape of the ionising radiation. Since pho- toionisation modelling depends on several free parameters, ap- propriate systems should preferably show many different ionic species. Fechner et al. (2004) analysed the associated metal line system of HE 2347−4342 and found evidence for a hard quasar spectral energy distribution at the absorbers with highest veloc- ities that are probably closest to the quasar. Their large He ii column densities probably shield the other absorbers which are better modelled with a softer radiation field. The results by Table 4. Measured column densities of the metal line system at z = 2.2753. Several components of H i remain unresolved. # v [km s−1] H i C iv Nv 1 −106.2 13.25 ± 0.59 12.70 ± 0.27 2 −94.3 13.634 ± 0.005 13.26 ± 0.57 12.40 ± 0.42 3 −45.8 12.76 ± 0.39 12.43 ± 0.14 4 −32.0 13.319 ± 0.015 12.80 ± 0.40 12.88 ± 0.06 5 0.0 13.232 ± 0.019 13.17 ± 0.03 12.94 ± 0.02 6 44.8 12.604 ± 0.020 12.51 ± 0.09 12.21 ± 0.05 7 91.5 13.042 ± 0.007 12.79 ± 0.05 11.98 ± 0.11 Fechner et al. (2004) are consistent with the more direct hard- ness estimators R and η near HE 2347−4342 (Sect. 4.3.1). In the spectrum of HE 2347−4342 an intervening metal line system is detected at z = 2.7119 which is close to the redshift of QSO C (∆z = 0.022) at a proper distance of d ≃ 10.0 Mpc. At z = 2.2753 there is another system showing multiple com- ponents of C iv and N v as well as only weak H i absorption (NH i < 10 13.7 cm−2). The presence of N v and weak H i features with associated metal absorption are characteristic of intrinsic absorption systems exposed to hard radiation. Due to the small proper distance to QSO A (d ≃ 3.1 Mpc) this system is proba- bly illuminated by the radiation of the close-by quasar. A third suitable metal line system at z = 2.3132 is closer to QSO B (d ≃ 6.1 Mpc) than to QSO A (d ≃ 12.0 Mpc). But since QSO A is much brighter than QSO B (Table 3), the metal line system at z = 2.3132 might be affected by both quasars. Due to their small relative velocities with respect to the quasars of < 3000 km s−1 the systems are likely associated to the quasars (e.g. Weymann et al. 1981). In order to construct CLOUDY models (Ferland et al. 1998, version 05.07) we assumed a single-phase medium, i.e. all ob- served ions arise from the same gas phase, as well as a solar abundance pattern (Asplund et al. 2005) at a constant metallicity throughout the system. Furthermore, we assumed pure photoion- isation and neglected a possible contribution of collisional ion- isation. The absorbers were modelled as distinct, plane-parallel slabs of constant density testing different ionising spectra. 5.1. The system at z = 2.275 near QSO A The system at z = 2.2753 shows seven components of C iv and N v along with unsaturated features of H i (Fig. 8). The ab- sorber densities are constrained by the C iv/N v ratio. For the HM01 background scaled to yield log Jb = −21.15 at the H i Lyman limit (Scott et al. 2000), we derive densities in the range 10−4.38 to 10−3.35 cm−3 at a metallicity of ∼ 0.6 solar. The es- 10 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 Fig. 8. Metal line system at z = 2.2753 towards HE 2347−4342. The displayed profiles assume the rescaled HM01 background. Zero velocity corresponds to z = 2.2753. timated absorber sizes are ∼ 5 kpc or even smaller, where the sizes are computed according to NH = nH l with the absorbing path length l. With an additional contribution by QSO A, mod- elled as a power law with α = 0.21 and H i Lyman limit intensity log Jq = −21.9 at the location of the absorber, we obtain an even higher metallicity of ∼ 11 times solar. Densities in the range 10−2.95 to 10−2.01 cm−3 are found leading to very small absorbers of . 10 pc. Both models lead to unusually high metallicities and very small absorber sizes. However, Schaye et al. (2007) recently re- ported on a large population of compact high-metallicity ab- sorbers. Using the HM01 background they found typical sizes of ∼ 100 pc and densities of 10−3.5 cm−3 for absorbers with nearly solar or even super-solar metallicities. In fact, this system is part of the sample by Schaye et al. (2007). Since the system exhibits only a few different species, it is impossible to discriminate between the soft and the hard radi- ation model. In principle, both models lead to a consistent de- scription of the observed metal lines. The soft HM01 UV back- ground yields η ∼ 170 for the modelled absorbers, whereas the model including the hard radiation of QSO A leads to η ∼ 10. Recall that the He ii forest cannot be used to measure η directly due to blending with Lyβ features and very low S/N. 5.2. The systems near QSO B and QSO C The systems at z = 2.3132 and z = 2.7119 are located near QSO B and QSO C, respectively. Only few ions are observed and some of them may even be blended. Therefore, no significant conclusions based on CLOUDY models can be drawn. Using the column density estimates we find that the system at z = 2.7119 close to QSO C exhibiting C iv and O vi can be described con- sistently with a HM01+QSO background. Models assuming a quasar flux of log Jq & −22.5 seen by the absorber yield η . 40, consistent with the direct measurements from the He ii forest. However, the metal transitions alone do not provide strong con- straints. The system at z = 2.3132 shows C iv in six components along with Si iv and Si iii. The Lyman series of this system suf- fers from severe blending preventing a reliable estimation of the H i column density. Therefore metallicities and absorber sizes cannot be estimated. Adopting our column density estimates we infer that this system can be reasonably modelled with or with- out a specific quasar contribution. 6. Discussion 6.1. The transverse proximity effect in spectral hardness Fourteen quasars have been found in the vicinity of HE 2347−4342 of which three are located in the usable part of the H i Lyα forest towards HE 2347−4342. No H i underdensity is detected near these foreground quasars even when correcting for contamination by the O vi absorption from the associated system of HE 2347−4342. An estimate of the predicted effect confirms that even if existing, the classical proximity effect is probably too weak to be detected on this line of sight due to the high UV background at 1 ryd and small-scale variance in the H i transmission (Sect. 3). However, the analysis of the spectral shape of the UV radia- tion field near the foreground quasars yields a markedly different result. The spectral shape is fluctuating, but it is predominantly hard near HE 2347−4342 and the known foreground quasars. Close to QSO C, both estimators R and η are consistent with a significantly harder radiation field than on average. There is a sharp R minimum located precisely at the redshift of the quasar, but embedded in a broader region of low R values statistically consistent with a hard radiation field of η <∼ 10 (Fig. 4). The column density ratio η is also lower than on average and indi- cates a harder radiation field than obtained for quasar-dominated models of the UV background (Fig. 7). Because of line blend- ing, only one of the three metal line systems detected near the foreground quasars can be used to estimate the shape of the ionising field. The metal line system at z = 2.275 can be de- scribed reasonably by the HM01 background with or without a local ionising component by QSO A. The He ii forest does not provide independent constraints for this absorber. Line blending prevents an unambiguous detection of O vi at z = 2.712, leaving the shape of the ionising field poorly constrained without tak- ing into account the He ii forest. Thus, the systems show highly ionised metal species, but our attempts to identify a local quasar radiation component towards them remain inconclusive. The most probable sources for the hard radiation field at z ∼ 2.30 and z ∼ 2.69 towards HE 2347−4342 are the nearby foreground quasars. In particular, the absorbers with η <∼ 10 have to be located in the vicinity of an AGN, since the filtering of quasar radiation over large distances results in η >∼ 50. Also star- forming galaxies close to the line of sight cannot yield the low η values, since they are unable to produce significant numbers of photons at hν > 54.4 eV (Leitherer et al. 1999; Smith et al. 2002; Schaerer 2003). We conclude that there is evidence for a transverse proximity effect of QSO C detectable via the relative spectral hardness. There are also indications that QSO A and QSO B show the same effect, although contamination adds un- certainty to the spectral shape variations in their projected vicin- Given these incidences of a hard radiation field near the quasars, how do these results relate to those of Paper I, in which we investigated the line of sight towards Q 0302−003? Both lines of sight show He ii absorption and on both lines of sight we find evidence for a predominantly hard radiation field near the quasars in the background and the foreground. However, the decrease of η near quasars towards Q 0302−003 appears to be much smoother than towards HE 2347−4342. There are several reasons for the lack of small-scale spec- tral shape variations on the line of sight to Q 0302−003. First, the low-resolution STIS spectrum of Q 0302−003 does not re- solve the He ii lines and limits the visible scale of fluctuations to ∆z & 0.006 (Paper I). Much smaller scales can be probed in the resolved He ii forest of HE 2347−4342, but the fitting of G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 11 blended noisy He ii features may result in artifical η variations. We will discuss the uncertainties of η below (Sect. 6.4). Second, Q 0302−003 (z = 3.285) probes higher redshifts, where the He ii fraction in the IGM is significantly higher and the inferred radi- ation field is very soft (η ∼ 350 in the Gunn-Peterson trough). Therefore, the impact of a hard source on the spectral shape is likely to be more pronounced than at lower redshifts after the end of He ii reionisation, where η of the UV background gradu- ally decreases. 6.2. The decrease of η near QSO C We now investigate quantitatively whether the foreground quasars are capable of creating a hardness fluctuation on the sightline towards HE 2347−4342. Unfortunately, since only one quasar is located in an uncontaminated region of the Lyα forests, we can present sufficient evidence only for QSO C. For the other two quasars the data is too sparse and contamination adds uncer- tainty to the derived η, but in principle QSO A should also show a strong effect, because its Lyman limit flux penetrating the line of sight is ∼ 8 times higher than the one of QSO C. Heap et al. (2000) and Smette et al. (2002) presented simple models of the decrease of η in front of a quasar taking into ac- count the absorption of ionising photons by the IGM. In a highly photoionised IGM with helium mass fraction Y ≃ 0.24 and tem- perature T ≃ 2 × 104 K we have 4 (1 − Y) αHe ii ΓHe ii ≃ 0.42 ΓHe ii , (4) where Γi and αi are the photoionisation rate and the radiative recombination coefficient for species i (Fardal et al. 1998). The photoionisation rate is Γi = Γi,b + Γi,q with a contribution of the background and the quasar. The contribution of the quasar to the photoionisation rate of species i at the jth absorber in front of it (z j > z j+1) is Γi,q(z j) = σi fν,i h(1 + zq) 1 + zq 1 + z j )−α+1 ( dL(zq, 0) dL(zq, z j) x−α−4exp Ni,kσix 1 + zk 1 + z j dx, (5) with the photoionisation cross section at the Lyman limit σi, the observed Lyman limit flux fν,i and x = ν/νi with the Lyman limit frequency νi. Extrapolating the power law continuum flux to the He ii Lyman limit yields fν,He ii = fν,H i4 −α. With the spectral in- dex α from Table 3 we obtain ηmin ≃ 2.3 for QSO C. We simulated η(z) for a set of 1000 Monte Carlo Lyα for- est spectra generated with the procedure discussed in Sect. 4.2. We assumed ΓH i,b = 1.75 × 10 −12 s−1 corresponding to the UV background from Sect. 3 and ηb = 100, which agrees with the median η towards HS 1700+6416 in the redshift range under consideration (Fechner et al. 2006). The intervening absorbers successively block the quasar flux. Especially, every absorber with log NH i > 15.8 will truncate the quasar flux at hν > 4 ryd due to a He ii Lyman limit system, leading to an abrupt softening of the radiation field. Figure 9 presents the simulated decrease of the median η approaching QSO C assuming a constant quasar luminosity, isotropic radiation and an infinite quasar lifetime together with the upper and lower percentiles of the η distribution obtained in bins of proper distance ∆d = 2 Mpc. The spread in the simulated η is due to line-of-sight differences in the absorber Fig. 9. Column density ratio η vs. proper distance d. The black line shows the modelled decrease of the median η approaching QSO C with respect to the ambient ηb = 100 (dashed line). Green lines mark the upper and lower quartiles of the simu- lated η distribution in bins of ∆d = 2 Mpc. QSO C is located at 7.75 Mpc. Filled circles show the median η from FR07 in con- centric rings of ∆d = 2 Mpc around the quasar. Error bars are the quartile distances to the median. The arrow marks the metal line system at z = 2.7122 at d = 12.03 Mpc. [See the online edition of the Journal for a colour version of this figure.] properties. Since we consider the transverse proximity effect, we are limited to a proper distance d >∼ 7.75 Mpc (Table 3). The model agrees reasonably with the median η of the data ob- tained in concentric rings around the quasar. As expected, in- dividual η values strongly deviate from this simple model due to the assumptions of the quasar properties (constant luminosity and spectral index, isotropic radiation) and due to the unknown real distribution of absorbers in transverse direction. Recently, Hennawi & Prochaska (2007) found evidence for excess small- scale clustering of high-column density systems in transverse direction to quasar sightlines. In Sect. 4.3.3 we found indica- tions that the η distribution around QSO C is not symmetric, which could be due to such anisotropic shielding. However, this does not imply an intrinsic anisotropy due to the unknown mat- ter distribution around the quasar and the large uncertainties in individual η values. Moreover, the line-of-sight variance at a constant η = 100 is too small to explain the large observed spread of the η values. Clearly, a self-consistent explanation of the small-scale η fluctuations would require hydrodynami- cal simulations of cosmological radiative transfer in order to in- vestigate possible shielding effects and the statistical distribu- tion of η values near quasars. While there is recent progress in case of the UV background (Sokasian et al. 2003; Croft 2004; Maselli & Ferrara 2005; Bolton et al. 2006), a proper treatment of three-dimensional radiative transfer in the IGM around a quasar is still in its infancy. However, our simplified approach suggests that QSO C is capable of changing the spectral shape of the UV radiation field by the right order of magnitude to ex- plain the low η values in its vicinity. Also a variation in the sizes and the centres of the bins chosen for Fig. 9 does not drastically change the indicated excess of low η at d <∼ 14 Mpc. Figure 9 also shows very clearly that the sphere of influence for the trans- verse proximity effect is not limited to the immediate vicinity of the quasar. Figure 10 shows a two-dimensional cut in comoving space near QSO C in the plane spanned by both lines of sight. The minimum separation of both lines of sight corresponds to a light travel time of ≃ 25.2 Myr, but the lifetime of QSO C could be >∼ 40 Myr due to the low η values at larger distances. The fluctua- 12 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 Fig. 10. Transverse comoving separation ∆r⊥ vs. line-of-sight comoving separation ∆r‖ with respect to QSO C. Black (green) points denote absorbers with η < 100 (η ≥ 100) on the line of sight towards HE 2347−4342 (curved line) with indicated red- shifts. The blue arrow points to the metal line system at z = 2.712. The half circles show the distance travelled by light emit- ted at the indicated times prior to our observation. The minimum light travel time between the two lines of sight is 25.19 Myr. [See the online edition of the Journal for a colour version of this fig- ure.] tions of the UV spectral shape could be explained by shadowing of the hard QSO radiation by unknown intervening structures between both lines of sight. 6.3. Other regions with an inferred hard UV radiation field In Fig. 4 we note two additional regions at z ∼ 2.48 and z ∼ 2.53 where R is prominently small and where there is no nearby quasar. Also the fitted η(z) shows very low values apparently unrelated to a known foreground quasar. Figure 11 displays the redshift distribution of the η ≤ 10 subsample. The low η values are clustered with two peaks near the foreground quasars, but also at z ∼ 2.40, z ∼ 2.48 and z ∼ 2.53. At the first glance the ex- istence of such regions seems to undermine the relation between the foreground quasars and a low η in their vicinity. However, there are several plausible explanations for the remaining low η values: 1. Unknown quasars: We can conclude from Paper I that the quasars responsible for hardness fluctuations may be very faint (like Q 0302-D113 in Paper I) or may reside at large distances (Q 0301−005 in Paper I). QSO C is located near the edge of our survey area centred on HE 2347−4342, so other quasars capable of influencing the UV spectral shape might be located outside the field of view. Moreover, in or- der to sample the full quasar luminosity domain (MB ≤ −23) at z ∼ 2.5 our survey is still too shallow by ∼ 1 mag- nitude. Therefore, a larger and/or deeper survey around HE 2347−4342 is desirable. 2. Quasar lifetime: Assuming that quasars are long-lived and radiate isotropically, every statistically significant low η fluc- tuation should be due to a nearby quasar. On the other hand, short-lived quasars will not be correlated with a hard radia- tion field due to the light travel time from the quasar to the background line of sight. Quasar lifetimes are poorly con- strained by observations to 1 <∼ tq <∼ 100 Myr (Martini 2004). This could be short enough to create relic light echoes from extinct quasars. The comoving space density of quasars with Fig. 11. Redshift distribution of low-η absorbers. The open (hashed) histogram shows all (NH i ≤ 10 14 cm−2) absorbers with η ≤ 10. Letters and dotted lines mark foreground quasars. The horizontal dashed line denotes the estimated average number of absorbers scattered from η = 80 to η ≤ 10 (≃ 0.64 per bin). MB < −23 at z ≃ 2.5 is ≃ 3.7×10 −6 Mpc−3 (Wolf et al. 2003) resulting in an average proper separation of ∼ 18.5 Mpc be- tween two lines of sight. This translates into a light travel time of ∼ 60 Myr which is of the same order as the quasar lifetime. So it is quite possible that some quasars have al- ready turned off, but their hard radiation is still present. 3. Obscured quasars: Anisotropic emission of type I quasars may lead to redshift offsets between regions with an inferred hard radiation field and quasars close to the line of sight. In the extreme case the putative quasar radiates in transverse di- rection, but is obscured on our line of sight (type II quasar). The space density of type II AGN at z > 2 is very uncer- tain due to the challenging optical follow-up that limits the survey completeness (e.g. Barger et al. 2003; Szokoly et al. 2004; Krumpe et al. 2007). Thus, the fraction of obscured AGN at high redshift is highly debated (Akylas et al. 2006; Treister & Urry 2006), but may well equal that of type I AGN in the luminosity range of interest (Ueda et al. 2003). We believe that a combination of the above effects is respon- sible for the loose correlation between low η values and active quasars. In particular, at z ∼ 2.4 we infer a hard radiation field in a H i void (FR07), which may have been created by a luminous quasar that is unlikely to be missed by our survey (V <∼ 22). In Fig. 11 we also indicate the error level due to inaccurate line fitting and noise in the He ii data (dashed line) obtained from simulated data (see below). The low number of η values scattered from a simulated η = 80 to η ≤ 10 implies that the overdensities of such small η values are statistically significant. Constraining the sample to lines with log(NH i) < 14 due to a possible bias caused by thermal broadening does not remove the significant clusters of lines with small η. 6.4. Uncertainties in the spectral hardness Our findings are likely to be affected by random errors and possi- bly also by systematic errors mostly related to the He ii data. The poor quality of the FUSE spectrum of HE 2347−4342 (S/N <∼ 5) contributes to the fluctuations in η even if the η value was con- stant (Fechner et al. 2006, see also below). The optical depth ratio R should be less affected by noise, since it is an average over a broader redshift range ∆z = 0.005. The low S/N and the generally high absorption at η ≫ 1 provide uncertainty for the continuum determination in the He ii spectrum. The extrapolated reddened power law is certainly an approximation. Although the η fitting results from FR07 are broadly consis- tent with those of Z04 and agree well in the regions near the fore- G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 13 ground quasars, there are substantial differences in some redshift ranges. This is probably due to the combined effects of low He ii data quality, different data analysis software and ambiguities in the deblending of lines. At present, η cannot be reliably deter- mined at individual absorbers unless metal transitions provide further constraints. In order to assess the random scatter in η due to the low S/N He ii data and ambiguities in the line deblending of both species, we again used Monte Carlo simulations. Ten H i Lyα forest spec- tra were generated in the range 2 < z < 3 via the Monte Carlo procedure outlined in Sect. 4.2. The resolution R ∼ 42000 and S/N = 100 closely resembles the optical data of HE 2347−4342. We also generated the corresponding He ii forests at FUSE res- olution and S/N = 4. We assumed pure non-thermal line broad- ening and η = 80. Voigt profiles were automatically fitted to the H i spectra using AUTOVP 1 (Davé et al. 1997). The He ii spectra were then automatically fitted with redshifts z and non- thermal Doppler parameters bH i fixed from the fitted H i line lists, yielding 7565 simulated η values. On average the recov- ered η is slightly higher than the simulated one (median η ≃ 89) with a large spread (0 < η <∼ 8000), but only 285 lines have η ≤ 10. Thus, we estimate a probability P ≃ 3.8% that η is scattered randomly from η = 80 to η ≤ 10 if the assumption of non-thermal broadening is correct. Note that this probability is likely an upper limit due to the fact that only H i Lyα was used to obtain the line parameters, which results in large error bars for saturated lines on the flat part of the curve of growth. In the real data, these errors were avoided by fitting unsaturated higher orders of the Lyman series wherever possible. In the line sample by FR07, 94 out of the 526 absorbers have η ≤ 10, whereas our simulation implies that only ∼ 20 are ex- pected to be randomly scattered to η ≤ 10 if η was constant. Thus, the major part of the scatter of η in the data is due to real fluctuations in the UV spectral shape. The majority of the low η ≤ 10 values is inconsistent with η ≥ 80, so they indicate a hard radiation field in spite of the low S/N in the He ii data. Yet, due to the large intrinsic scatter obtained from the simulations, individual η values hardly trace the variations of the UV spectral shape. Local spatial averages should be more reliable (FR07). Since the transverse proximity effect zones always extend over some redshift range, this requirement is fulfilled and on average we reveal a harder radiation field than expected. Concerning the high tail of the simulated distribution at η = 80, ∼ 15% of the lines are returned with η >∼ 200. This may indicate that a fraction of the observed high η values is still consistent with a substantially harder radiation field, underlining that single η values poorly constrain the spectral shape. Possibly, some η values are systematically too low due to the assumption of non-thermal broadening (bHe ii = bH i) when fitting the He ii forest. FR07 found that this leads to underes- timated η values at NH i >∼ 10 13 cm−2 if the lines are in fact thermally broadened (bHe ii = 0.5bH i). Non-thermal broaden- ing is caused by turbulent gas motions or the differential Hubble flow, with the latter affecting in particular the low-column den- sity forest. Thermal broadening becomes important in collapsed structures at high column densities. In simulations of the Lyα forest, non-thermal broadening has been found to dominate (Zhang et al. 1995, 1998; Hernquist et al. 1996; Weinberg et al. 1997; Bolton et al. 2006; Liu et al. 2006). This has been con- firmed observationally for the low-column density forest (Z04; Rauch et al. 2005). On the other hand, eight out of eleven ab- sorbers with NH i > 10 14 cm−2 in the vicinity of QSO C have 1 http://ursa.as.arizona.edu/˜rad/autovp.tar η ≤ 10 (Fig. 7). Although the column density ratio of these absorbers could be underestimated due to an unknown contri- bution of thermal broadening, the statistical evidence for a hard radiation field is based on the vast majority of low-column den- sity lines. The median η obtained in this region does not in- crease significantly after excluding the suspected lines (∼ 53 vs. ∼ 40). This is still much lower than the median η ∼ 100 towards HS 1700+6416 in this redshift range (Fechner et al. 2006). Therefore, it is unlikely that our results are biased due to the assumed line broadening. 7. Conclusions Traditionally, the transverse proximity effect of a quasar has been claimed to be detectable as a radiation-induced void in the H i Lyα forest. But due to several systematic effects like quasar variability, finite quasar lifetime, intrinsic overdensities around quasars, or anisotropic radiation, most searches yielded negative results (e.g. Schirber et al. 2004; Croft 2004). In this paper, we have analysed the fluctuating spectral shape of the UV background in the projected vicinity of the three foreground quasars QSO J23503−4328, QSO J23500−4319 and QSO J23495−4338 (dubbed QSO A, B and C) on the line of sight towards HE 2347−4342 (z = 2.885). By comparing the H i absorption and the corresponding He ii absorption, we have presented evidence for a statistical excess of hard UV radiation near the foreground quasars. However, due to contamination of the forests near QSO A (z = 2.282) and QSO B (z = 2.302), the evidence is strongest for QSO C (z = 2.690). We interpret these indicators for an excess of hard radiation near the fore- ground quasars as a manifestation of the transverse proximity effect. A simple model indicates that the foreground quasars are capable of generating the observed hard radiation over the ob- served distances of several Mpc. Furthermore, we tried to model the ionising radiation field of three metal line systems close to the foreground quasars. Two of those are strongly affected by line blending and do not allow for reliable photoionisation mod- els. The remaining system can be modelled reasonably with or without a contribution by a local quasar. Future larger samples of highly ionised unblended metal systems near foreground quasars may provide evidence for local hardness fluctuations. In Worseck & Wisotzki (2006) we revealed the transverse proximity effect as a systematic local hardness fluctuation around four foreground quasars near Q 0302−003 and pointed out that the relative UV spectral hardness is a sensitive phys- ical indicator of the proximity effect over distances of several Mpc. In this study we are able to confirm this on a second line of sight. Evidently, small-scale transmission fluctuations in the H i forest can dilute the small predicted signature of the effect. However, the hard spectral shape of the UV radiation field still indicates the transverse proximity effect despite the H i density fluctuations. Thus, we confirm our previous result that the spec- tral hardness breaks the density degeneracy that affects the tradi- tional searches for the proximity effect. Moreover, the predicted transverse proximity effect of the quasars in the H i forest is weak due to the high UV background at 1 ryd. Still the UV spectral shape is able to discriminate local UV sources independent of the amplitude of the UV background. Bolton et al. (2006) find that the large UV spectral shape fluctuations in the IGM are likely due to the small number of quasars contributing to the He ii ionisation rate at any given point, whereas the H i ionisation rate is rather homogeneous due to the probable contribution of star-forming galaxies (e.g. Bianchi et al. 2001; Sokasian et al. 2003; Shapley et al. 2006). 14 G. Worseck et al.: The transverse proximity effect in spectral hardness towards HE 2347−4342 Our findings confirm the picture that AGN create the hard part of the intergalactic UV radiation field. If the quasar is active long enough, its hard radiation field can be observed penetrating a background line of sight. It is also likely that light echoes from already extinguished quasars are responsible for some locations of hard radiation without an associated quasar. The transverse proximity effect of QSO C implies a minimum quasar lifetime of ∼ 25 Myr (probably even ∼ 40 Myr), providing additional constraints to more indirect estimates (e.g. Martini 2004, and references therein). However, the UV radiation field near the foreground quasars is not homogeneously hard as naively expected, but still shows fluctuations. Apart from substantial measurement uncertainties, the unknown density structure around the quasar could shield the ionising radiation in some directions, maybe even preferentially in transverse direction to the line of sight (Hennawi & Prochaska 2007). Thus, radiative transfer effects may become important to explain a fluctuating UV spectral shape in the presence of a nearby quasar. Large-scale simulations of cosmological radia- tive transfer with discrete ionising sources are required to adress these issues in detail. Moreover, the He ii forest has been resolved so far only to- wards two quasars at a very low S/N <∼ 5. While the low data quality primarily creates uncertainties in the spectral shape on small spatial scales, large scales could be affected by cosmic variance. Thus, the general redshift evolution of the UV spec- tral shape is not well known and estimates obtained from single lines of sight may well be biased by local sources. Acknowledgements. We thank the staff of the ESO observatories La Silla and Paranal for their professional assistance in obtaining the optical data discussed in this paper. We are grateful to Peter Jakobsen for agreeing to publish the quasars from his survey. We thank Gerard Kriss and Wei Zheng for providing the re- duced FUSE spectrum of HE 2347−4342. Tae-Sun Kim kindly supplied an addi- tional line list of HE 2347−4342. GW and ADA acknowledge support by a HWP grant from the state of Brandenburg, Germany. CF is supported by the Deutsche Forschungsgemeinschaft under RE 353/49-1. We thank the anonymous referee for helpful comments. 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A., Deharveng, J.-M., et al. 2004, ApJ, 605, 631 List of Objects ‘HE 2347−4342’ on page 1 ‘HS 1700+6416’ on page 1 ‘Q 0302−003’ on page 2 ‘QSO J23510−4336’ on page 4 ‘QSO J23507−4319’ on page 4 ‘QSO J23507−4326’ on page 4 ‘QSO J23509−4330’ on page 4 ‘QSO J23502−4334’ on page 4 ‘QSO J23503−4328’ on page 4 ‘QSO J23495−4338’ on page 4 ‘QSO J23511−4319’ on page 4 ‘QSO J23514−4339’ on page 4 ‘QSO J23503−4317’ on page 4 ‘QSO J23515−4324’ on page 4 ‘QSO J23512−4332’ on page 4 ‘QSO J23508−4335’ on page 4 ‘QSO J23500−4319’ on page 4 ‘Q 0302-D113’ on page 12 ‘Q 0301−005’ on page 12 Introduction Observations and data reduction Search for QSO candidates near HE 2347-4342 Spectroscopic follow-up Additional quasars Redshifts and magnitudes Optical spectra of HE 2347-4342 Far-UV spectra of HE 2347-4342 The Ly forest near the foreground quasars The fluctuating shape of the UV radiation field towards HE 2347-4342 Diagnostics Fluctuations in R and along the line of sight The UV radiation field near the quasars HE 2347-4342 QSOs A and B QSO C Constraints from metal line systems The system at z=2.275 near QSO A The systems near QSO B and QSO C Discussion The transverse proximity effect in spectral hardness The decrease of near QSO C Other regions with an inferred hard UV radiation field Uncertainties in the spectral hardness Conclusions

0704.0188

Biased random walks on combs

Biased random walks on random combs Tanya M Elliott and John F Wheater Department of Physics, University of Oxford Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK E-mail: [email protected], [email protected] Abstract. We develop rigorous, analytic techniques to study the behaviour of biased random walks on combs. This enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine. Two specific examples of random comb ensembles are discussed; the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths. We also analyze transport properties along the spine for these probability measures. PACS numbers: 05.40.Fb, 04.60.Nc, 05.45.Df 1. Introduction The behaviour of random walks on random combs is of interest from a number of points of view. Condensed matter physicists have studied such structures because they serve as a model for diffusion in more complicated fractals and percolation clusters [1, 2, 3, 4]. In the context of quantum gravity, random combs are a tractable example of a random manifold ensemble and understanding their geometric properties can provide insight into higher dimensional problems [5, 6, 7]. Most of the literature concerns approximate analytical techniques and numerical solutions, although there are exact calculations of leading order behaviour in some cases [8]. To this end, it is desirable to have rigorous methods for determining the geometric quantities of interest and that is the purpose of this paper. One such quantity is the dimensionality of the ensemble. On a sufficiently smooth manifold all definitions of dimension will agree, but for fractal geometries like random combs this is not necessarily true. The spectral dimension is defined to be ds provided the ensemble average probability of a random walker being back at the origin at time t, takes the asymptotic form t−ds/2. This concept of dimension does not in general agree with the Hausdorff dimension dH , which is defined when the expectation value of the volume enclosed within a geodesic distance R from a marked point scales like RdH as R → ∞. http://arxiv.org/abs/0704.0188v2 Biased random walks on random combs 2 We know that for diffusion on regular structures the mean square displacement at large times is proportional to t, but for a fractal substrate there is anomalous diffusion and the mean square displacement behaves like t2/dw , where dw represents the fractal dimension of the walk and depends sensitively on the nature of the random structure. Biased random walks on combs have also been studied in connection with disordered materials, since such a system is a paradigm for diffusion on fractal structures in the presence of an applied field [9, 10]. As we discuss later there are several different bias regimes. Topological bias, where at every vertex in the comb there is an increased probability of moving away from the origin was first studied for a random comb with a power-law distribution of tooth lengths in [11]. Other works have discussed the effects of bias away from the origin only in the teeth [12] and only in the spine [13]. The effect of going into the teeth can be viewed as creating a waiting time for the walk along the spine; the distribution of the waiting time depends on both the bias and the length of the teeth and the outcome is the result of subtle interplay between the two. In [14] some new, rigorous techniques were developed to study random walks on combs. This enabled an exact, but very simple calculation of the spectral dimension of random combs. The principal idea is to split both random combs and random walks into subsets that give either strictly controllable or exponentially decaying contributions to the calculation of physical characteristics. These methods were later reinforced to prove that the spectral dimension of generic infinite tree ensembles is 4/3 [15, 16]. In this paper we use and extend the techniques of [14] to deal with biased walks on combs. Some of our results are new; some qualify statements made in the literature; and some merely confirm results already derived by other, usually less rigorous, methods. The random combs, the bias scenario, some useful generating functions and the critical exponents are defined in the next section. In Section 3 we introduce some deterministic combs, discuss general properties of the generating functions and establish bounds that will be instrumental when studying random ensembles. Section 4 looks at regions of bias where the large time behaviour is independent of the comb ensemble or simply dependent on the expectation value of the first return generating function in the teeth. In Section 5 we compute the spectral dimension in regions of bias where it is influenced by the probability measure on the teeth. Two specific cases are considered: the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths. Section 6 examines transport properties along the spine for these same probability measures and in the final section we review the main results, compare with the literature and discuss their significance. Some exact calculations and proofs omitted from the main text are outlined in the appendices. 2. Definitions Wherever possible we use the definitions and notation of [14]; we repeat them here for the reader’s convenience but mostly refer back to [14] for proofs and derived properties. Biased random walks on random combs 3 2.1. Random combs LetN∞ denote the nonnegative integers regarded as a graph so that n has the neighbours n ± 1 except for 0 which only has 1 as a neighbour. Let Nℓ be the integers 0, 1, . . . , ℓ regarded as a graph so that each integer n ∈ Nℓ has two neighbours n± 1 except for 0 and ℓ which only have one neighbour, 1 and ℓ− 1, respectively. A comb C is an infinite Figure 1. A comb. rooted tree-graph with a special subgraph S called the spine which is isomorphic to N∞ with the root, which we denote r, at n = 0. At each vertex of S, except the root r, there is attached by their endpoint 0 one of the graphs Nℓ or N∞. The linear graphs attached to the spine are called the teeth of the comb, see figure 1. We will denote by Tn the tooth attached to the vertex n on S, and by Ck the comb obtained by removing the links (0, 1), . . . , (k − 1, k), the teeth T1, . . . , Tk and relabelling the remaining vertices on the spine in the obvious way. An arbitrary comb is specified by a list of its teeth {T1, . . .} and |Tk| denotes the length of the tooth. Note that we have excluded the possibility of a tooth of zero length. This is for technical convenience in what follows and can be relaxed [17]. In this paper we are interested in random combs for which the length ℓ of each tooth is identically and independently distributed with probability µℓ. This induces a probability measure µ on the positive integers and expectation values with respect to this measure will be denoted 〈·〉µ. In particular we will consider the two measures µAℓ = p, ℓ = ∞, 1− p, ℓ = 1, 0, otherwise; µBℓ = , a > 1. (1) Biased random walks on random combs 4 However, the results proved for µB apply to any measure with the same behaviour at large ℓ and we note in passing that the methods used here will work for any distribution that is reasonably smooth, for example the exponential distribution. The measure µB has been discussed quite extensively in the literature but µA has not. 2.2. Biased random walks We regard time as integer valued and consider a walker who makes one step on the graph for each unit time interval. If the walker is at the root or at the end-point of a tooth then she leaves with probability 1. If at any other vertex the probabilities are parametrized by two numbers ǫ1 and ǫ2 as shown in figure 2a and the allowed range of these parameters is shown in figure 2b. For walks in the teeth there is bias away from or towards the spine depending on whether ǫ2 is positive or negative; similarly a walk on the spine is biased away from or towards the root depending on whether ǫ1 is positive or negative. When there is no bias we say that the walk is ‘critical’; the fully critical case ǫ1 = ǫ2 = 0 was covered in [14]. The notation b− = 1− ǫ1 − ǫ2, b+ = 1 + ǫ1 − ǫ2, bT = 1 + 2ǫ2, (2) will be used where applicable since these combinations appear often in our analysis. We denote by B,B′, B1, B2 etc constants which depend on ǫ1 and ǫ2 and may vary from line to line but are positive and finite on the relevant range; other constants will be denoted c, c′ etc. The generating function for the probability pC(t) that the walker on C is back at the root at time t having left it at t = 0 is defined by QC(x) = (1− x)t/2pC(t). (3) Letting ω be a walk on C starting at r, ω(t) the vertex where the walker is to be found at time t, and ρω(t) the probability for the walker to step from ω(t) to ω(t+1), we have QC(x) = ω:r→r (1− x) |ω|−1 ρω(t). (4) A similar relation gives the generating function for probabilities for first return to the root, PC(x), except that the trivial walk of duration 0 is excluded. The two functions are related by QC(x) = 1− PC(x) , (5) and it is straightforward to show that PC(x) satisfies the recurrence relation PC(x) = (1− x)b− 3− b+PC1(x)− bTPT1(x) . (6) Biased random walks on random combs 5 PSfrag replacements (1 + 2ǫ2) (1− 2ǫ2)1 (1 + 2ǫ2) (1− ǫ1 − ǫ2) 13(1 + ǫ1 − ǫ2) PSfrag replacements (1 + 2ǫ2) (1− 2ǫ2) (1 + 2ǫ2) (1− ǫ1 − ǫ2) (1 + ǫ1 − ǫ2) ǫ1 Figure 2. Bias parameterisation. Note that PC(x) and QC(x) depend upon ǫ1 and ǫ2; to avoid clutter we will normally suppress this dependence but if necessary it will appear as superscripts. It is important for what follows that QC is a convex function of PC which is itself a convex function of PT1 , ...PTk , PCk for any k > 0. For an ensemble of combs, we will denote the expectation values of the generating functions for return and first return probabilities as Q(x) = 〈QC(x)〉µ P (x) = 〈PC(x)〉µ . (7) We will say that g(x) ∼ f(x) if there exist positive constants c, c′, σ, σ′ and x0 such c f(x) exp −σ(log |f(x)|)1/a < g(x) < c′ f(x)| log f(x)|σ′ (8) for 0 < x ≤ x0. The tactic of this paper is to prove bounds of this form for the generating functions; in almost all cases our results are in fact a little stronger having σ = σ′ = 0 Biased random walks on random combs 6 when we will say that g(x) ≈ f(x). The random walk on C is recurrent if PC(0) = 1 in which case we define the exponent β through 1− PC(x) ∼ xβ . (9) If β is an integer then we expect logarithmic corrections and define β̃ if 1− PC(x) ≈ xβ | log x|−β̃. (10) It follows that QC(x) diverges as x → 0 and we define α by QC(x) ∼ x−α, (11) and if α is an integer, α̃ when QC(x) ≈ x−α| log x|α̃. (12) If PC(0) < 1 then the random walk is non-recurrent, or transient, and QC(x) is finite as x → 0. Then if, as x → 0, the first k − 1 derivatives of QC(x) are finite but the kth derivative diverges we define the exponent αk by C (x) ∼ x −αk , (13) and if αk is an integer, α̃k when C (x) ≈ x −αk | log x|α̃k . (14) In considering the ensemble of combs µ, we define all these exponents in exactly the same way simply replacing PC(x) with 〈PC(x)〉µ and so on. Note that for a single recurrent comb β = α but in an ensemble this is no longer necessarily the case; applying Jensen’s inequality to (5) we see that β ≤ α. If Q(k)(x) ∼ x−αk then it is straightforward to show that Rk(λ) = tk 〈pC(t)〉µ ∼ λ −αk . (15) It follows that if the sequence decays uniformly at large t, which we do not prove, then it falls off as tαk−1−k. Thus we define ds = 2(1 + k − αk). Similarly if Q(x) ≈ | log x|α̃ then R(λ) ≈ | log λ|α̃ and, again assuming uniformity, p(t) falls off as t−1 | log t|α̃−1. 2.3. Two-point functions Let p1C(t;n) denote the probability that the walker on C, having left r at t = 0 and not subsequently returned there, is at point n on the spine at time t. The corresponding generating function, which we will call the two-point function, is defined by GC(x;n) = (1− x)t/2p1C(t;n). (16) Letting ω be a walk on C starting at r and ending at n without returning to r we have GC(x;n) = ω:r→n (1− x) |ω|−1 ρω(t). (17) Biased random walks on random combs 7 Following the discussion in section 2.2 of [14] this leads us to the representation GC(x;n) = b+(1− x)n/2 PCk(x). (18) 2.4. The Heat kernel Let KC(t;n, ℓ) denote the probability that the walker on C, having left r at t = 0, is at point ℓ in tooth Tn at time t. KC(t;n, ℓ) satisfies the diffusion equation on C so we call it the heat kernel. The probability that the walker has travelled a distance n along the spine at time t is given by KC(t;n) = KC(t;n, ℓ), (19) and has generating function HC(x;n) = (1− x)t/2KC(t;n). (20) HC(x;n) can be written as HC(x;n) = GC(x;n) 1− PC(x) D|Tn|(x), (21) where Dℓ(x) = 1 + GNℓ(x; k), (22) and we define H(x;n) = 〈HC(x;n)〉µ . (23) Note that, because KC(t;n) is a probability, H(x;n) = . (24) The exponent dk is defined through the moments in n nk H(x;n) ≈ x−1−dk , (25) and in the case dk = 0 the exponent d̃k is defined when nk H(x;n) ≈ x−1 | log x|d̃k . (26) If ǫ1 ≥ 0 one can show that on any comb 〈n〉ω:|ω|=t is a non-decreasing sequence and thus that there is some constant T0 such that for T > T0 | log T | 〈n〉ω:|ω|=T + 〈n〉ω:|ω|=T+1 < c T d1, d1 6= 0 c | log T |d̃1 < 〈n〉ω:|ω|=T + 〈n〉ω:|ω|=T+1 < c | log T |d̃1, d1 = 0. Biased random walks on random combs 8 If ǫ1 < 0 (for which we always have d̃1 = 0) then we have only the weaker result that for T > T0 | log T | )1+d1 〈n〉ω:|ω|=t < c T 1+d1 . (28) 3. Basic properties 3.1. Results for simple regular combs The relation (6) can be used to compute the generating functions for a number of simple regular graphs which will be important in our subsequent analysis [14]. (i) An infinitely long tooth, N∞: P∞(x) = 2 if ǫ2 = 0; 1− 2|ǫ2| 4|ǫ2| (1− 2ǫ2) +O(x2) otherwise. (ii) A tooth of length ℓ, Nℓ: Pℓ(x) = P∞(x) 1 +XY 1−ℓ 1 +XY −ℓ where bT (1− P∞(x)) 2− bT (1 + P∞(x)) , Y = 2− bTP∞(x) bTP∞(x) . (31) (iii) The comb ♯ given by {Tk = N1, ∀k} has all teeth of length 1, and P♯(x) = 1− B1x 2 +O(x) if ǫ1 = 0; 1− ǫ2 − |ǫ1| − x B2 +O(x2) otherwise. Note that ♯ is non-recurrent if ǫ1 > 0. It is also convenient to define ℓ♯ to be {T1 = Nℓ, C1 = ♯}. (iv) The comb ∗ given by {Tk = N∞, ∀k} has all teeth of length ∞ and is non-recurrent for ǫ2 > 0, P∗(x) = 1 + ǫ2 − 4ǫ2 + ǫ − x B1√ 4ǫ2 + ǫ +O(x2). (33) Otherwise P∗(x) = 1− |ǫ1| 1 + ǫ1 2 +O(x) if ǫ2 = 0, ǫ1 6= 0; 1− ǫ2 − |ǫ1| +O(x2) if ǫ2 < 0, ǫ1 6= 0; 1− B4x 2 +O(x) if ǫ2 < 0, ǫ1 = 0. Biased random walks on random combs 9 (v) The comb ♭ℓ given by {Tk = Nℓ, ∀k} has all teeth of length ℓ and P♭ℓ(x) = 1− |ǫ1| 1 + ǫ1 (ℓ+ 1 + |ǫ1|)x+O(x2ℓ2) if ǫ2 = 0, ǫ1 6= 0; 1− ǫ2 − |ǫ1| |ǫ1ǫ2| +O(xY −ℓ) if ǫ2 < 0, ǫ1 6= 0; 2 +O(x 2Y −ℓ) if ǫ2 < 0, ǫ1 = 0; where, as x → 0, Y → 1 + 2|ǫ2| 1− 2|ǫ2| . (36) When ǫ2 > 0 let ℓ̄ = ⌊| log x|/ log Y ⌋, where ⌊z⌋ denotes the integer below z. For ℓ > 2ℓ̄ the teeth are long enough that P♭ℓ(x) behaves like (33). For ℓ̄ < ℓ ≤ 2ℓ̄, P♭ℓ(x) is non-recurrent with the leading power of x being fractional. For ℓ ≤ ℓ̄ P♭ℓ<ℓ̄(x) = 1− ǫ2 − |ǫ1| |ǫ1ǫ2| +O(x) if ǫ1 6= 0; ℓ +O(x ℓ, xY −ℓ) if ǫ1 = 0, where the notation O(a, b) means O(max (a, b)). 3.2. General properties of the generating functions The generating functions for any comb satisfy three simple properties which can be derived from (6): (i) Monotonicity The value of PC(x) decreases monotonically if the length of a tooth is increased. (ii) Rearrangement If the comb C ′ is created from C by swapping the adjacent teeth Tk and Tk+1 then PC′(x) > PC(x) if |Tk+1| < |Tk|. (iii) Inheritance If walks on Ck or Tk are non-recurrent for finite k then walks on C are non-recurrent. The proof of the first two follows that given in [14] for the special case ǫ2 = ǫ1 = 0. The third can be shown by assuming that either PC1(0) < 1 or PT1(0) < 1; it then follows immediately from (6) that PC(0) < 1 and the result follows by induction. 3.3. Useful elementary bounds By monotonicity GC(x;n) is always bounded above by G♯(x;n) from which we get GC(x;n) < exp(−nΛǫ1,ǫ2(x)), (38) Biased random walks on random combs 10 where Λǫ1,ǫ2(x) = 2 + ǫ2 if ǫ1 > 0, 2 + ǫ2 1− ǫ2 if ǫ1 = 0, if ǫ1 < 0. Now let P C (x) denote the contribution to PC(x) from walks that reach beyond n = N on the spine. It is straightforward to show using the arguments of section 2.5 of [14] C (x) ≤ ǫ1,ǫ2 C (x;N)G −ǫ1,ǫ2 C (x;N). (40) Combining this with (38) we obtain the useful bound C (x) ≤ exp(−N(Λǫ1,ǫ2(x) + Λ−ǫ1,ǫ2(x))). (41) Now consider the ensemble µ′ of combs C for which: Tk = N1, k = 1..K − 1, TK = Nℓ; at k > K teeth are short, Tk = N1, with probability 1 − p or long, Tk = Nℓ, with probability p; and the nth tooth is short, Tn = N1. Then using the representation (18) GC(x;n) can be bounded above by noting that if Tk+1 = Nℓ then PCk < Pℓ♯, otherwise PCk < P♯. This gives GC(x, n) ≤ (1− x)−n/2 Pℓ♯(x) n−K−kP♯(x) k+K , (42) and hence 〈GC(x, n)〉µ′ = n−K−1 n−K−1 pn−K−1−k(1− p)kGC(x, n) P♯(x) KPℓ♯(x) (1− x)n/2 ((1− p)P♯(x) + pPℓ♯(x))n−K−1 . (43) 4. Results independent of the comb ensemble µ In this section we show that in some regions of ǫ1,2 the behaviour at large time is essentially independent of the comb ensemble, or else simply dependent upon 〈PT (x)〉µ. The leading, and where different, the leading non-analytic, behaviour of 〈PT (x)〉µ as x → 0 for the measures studied here is given in table 1. The results for µA are trivial, as are those for any measure when ǫ2 < 0, while the case µ B and ǫ2 = 0 can be derived using the techniques in [14]. The calculation for µB and ǫ2 > 0 is somewhat subtle and is included in Appendix A. Biased random walks on random combs 11 Table 1. Leading and leading non-analytic behaviour of 1− 〈PT 〉µ in various cases. ensemble ǫ2 < 0 ǫ2 = 0 ǫ2 > 0 µA Bx Bx 2 B +B′x µB, a < 2 Bx Bxa/2 B(| log x|a−1)−1 µB, a = 2k Bx Bx+ . . . B′xk| log x| B(| log x|a−1)−1 µB, a > 2, a 6= 2k Bx Bx+ . . . B′xa/2 B(| log x|a−1)−1 4.1. ds when ǫ2 < 0 First we show that for any comb ensemble 0 if ǫ1 < 0 and ǫ2 < 0; 1 if ǫ1 = 0 and ǫ2 < 0. By monotonicity we have that for any comb C P∗(x) ≤ PC(x) ≤ P♯(x). (45) Taking expectation values and using (32) and (34) it follows that for ǫ2 < 0 P (x) = 〈PC(x)〉µ = 1−B1x 2 +O(x) if ǫ1 = 0, 1− ǫ2 − |ǫ1| +O(x2) otherwise. Similarly Q∗(x) ≤ QC(x) ≤ Q♯(x) (47) and so Q(x) = 〈QC(x)〉µ = +O(1) if ǫ1 = 0, B2|ǫ1| +O(1) if ǫ1 < 0, and (44) follows. 4.2. ds when ǫ1 > 0 When ǫ1 > 0 all combs are non-recurrent and so we must examine the derivatives of Q(x). Differentiating (5) and (6) gives C (x) = QC(x) C (x), (49) C (x) = −PC(x) PC(x) (1− x)b− (x) + b+P . (50) Biased random walks on random combs 12 By monotonicity (50) can be bounded above and below by replacing PC with P∗ and P♯ respectively. Taking the expectation value and using translation invariance to note that 〈PC〉µ = 〈PC1〉µ shows that, if T (x) diverges as x → 0, then Q(1)(x) ∼ B T (x) T (x) , 1). (51) As can be seen from table 1, in some cases 〈PT (x)〉µ is analytic, or only higher derivatives diverge. For the measures considered here it can be shown that if 〈PT (x)〉µ is analytic at x = 0 then so is Q(x). If on the other hand 〈PT (x)〉µ is not analytic but the k’th derivative diverges then Q(k)(x) = B T (x) T (x) , 1). (52) The proof is a straightforward but tedious generalization of (49) and (50) and is relegated to Appendix B. If a derivative of Q(x) diverges then ds can be read off using (14) and (52). Otherwise if all finite order derivatives are finite then pC(t) decays at large t faster than any power and ds is not defined. 4.3. dk when ǫ2 < 0 or ǫ1 < 0 We show that for any comb ensemble d̃k = 0, dk = 0 if ǫ1 < 0, k/2 if ǫ1 = 0 and ǫ2 < 0, k if ǫ1 > 0 and ǫ2 < 0. It is trivial to show that 1 ≤ Dℓ ≤ , ǫ2 < 0, (54) and then by monotonicity we get G∗(x;n) 1− P∗(x) ≤ H(x;n) ≤ B G♯(x;n) 1− P♯(x) . (55) Combining this with (32) and (34) yields the results for ǫ2 < 0. To deal with ǫ1 < 0 and ǫ2 ≥ 0 note that monotonicity gives D|Tn|(x) 1− PC(x) G∗(x;n) ≤ H(x;n) ≤ D|Tn|(x) 1− PC(x) G♯(x;n). (56) Using the lower bound and (18), (24) and (33) we get after summing over n D|Tn|(x) 1− PC(x) 4ǫ2 + ǫ 1 − ǫ1 − 2ǫ2 H(x;n) ≤ . (57) Inserting this into the upper bound of (56) gives H(x;n) ≤ 4ǫ2 + ǫ 1 − ǫ1 − 2ǫ2 G♯(x;n). (58) Biased random walks on random combs 13 It is a trivial consequence of (24) that nkH(x;n) > , k > 0, (59) and the results then follow by using (38). 5. The spectral dimension when ǫ2 ≥ 0 and ǫ1 ≤ 0 Here and in some of the sections to follow we will need to sum over the location of the first long tooth to determine the spectral dimension. Most generally we call a tooth long when it has length ≥ ℓ and short when it has length < ℓ. Consider combs for which the first L− 1 teeth are short but the Lth tooth is long; the probability for this is p(1− p)L−1, where p is the probability of a tooth being long. Denoting by ℓL a comb having the first long tooth at vertex L gives Q(x) = 〈QℓL(x)〉µ p(1− p) L−1. (60) QℓL(x) is bounded above by the comb in which all teeth at n ≥ L + 1 are short, and below by the comb in which all teeth at n ≥ L+ 1 are infinite, Q{Tn<L=Nℓ′ ,ℓ′≤ℓ;Tn≥L=N∞}(x) < QℓL(x) < Q{Tn6=L=N1;TL=Nℓ}(x). (61) 5.1. µA – Infinite teeth at random locations 5.1.1. ǫ2 = 0, ǫ1 < 0 We first show that the exponent β = – so it is unchanged from the comb ∗. This result follows from the inequalities 1− pBx +O(x) ≤ P (x) ≤ 1− pB′x 2 +O(x). (62) The lower bound is obtained by applying Jensen’s inequality to (6). To get the upper bound we average over the first tooth and then by monotonicity we obtain P (x) ≤ pPℓ♯(x) + (1− p)P♯(x), (63) with ℓ = ∞ and using (6) and (32) gives the bound required. The spectral dimension is given by 1 if p ≥ 2|ǫ1|(1 + |ǫ1|)−1, log(1− p) 1−|ǫ1| 1+|ǫ1| ) otherwise. This result follows from estimating the sum in (60) using the bounds in (61) with ℓ = ∞ and short teeth being N1. PC(x) for these bounding combs is computed in Appendix C and using (C.3) we get upper and lower bounds on Q∞L(x) of the form Bx+B′x 1−|ǫ1| 1+|ǫ1| . (65) Biased random walks on random combs 14 5.1.2. ǫ2 > 0, ǫ1 < 0 The probability that C is non-recurrent is at least p, the probability that T1 = N∞, and hence P (0) < 1. (66) In fact it follows from the lemma of Appendix B that P (k)(x) is finite for all finite k so the exponent β is undefined. The spectral dimension is given by 2 log(1− p) 1−|ǫ1|−ǫ2 1+|ǫ1|−ǫ2 ) . (67) To show this we start by estimating Q(x) in exactly the same way as in 5.1.1 except that the behaviour of the limiting combs is now given by (C.5) so that there are upper and lower bounds on Q∞L(x) of the form Bx+B′ 1−|ǫ1|−ǫ2 1+|ǫ1|−ǫ2 . (68) When p ≤ 1 − b+/b− this sum diverges at x = 0 and it is then straightforward to obtain (67). For larger p the sum is convergent at x = 0 so we next examine Q(1)(x) = . Note that −P (1)C ≥ 13b−; then letting Z be a very large integer and using Hölder’s inequality QC(x) ≤ −Q(1)(x) ≤ QC(x) 2+1/Z −P (1)C (x) . (69) By the lemma of Appendix B the second factor in the upper bound is finite as x → 0 so we need an estimate of 〈Q2C〉µ. This is provided by (68) modified by squaring the denominator; when p ≤ 1 − (b+/b−)2 this sum diverges at x = 0 and once again we obtain (67). For still larger p both Q and Q(1) are finite at x = 0 and we examine the second and higher derivatives. This uses (B.4), (−1)kP (k)C ≥ bk−b 2k−1, Hölder’s inequality and the lemma; the term with the highest power of QC dominates and the result is always (67). ‡ 5.1.3. ǫ2 > 0, ǫ1 = 0 By the same argument as in 5.1.2 we find P (0) < 1, so β is again undefined. An upper bound on Q(x) may be obtained as in 5.1.1 using (C.9) to get Q∞L(x) ≤ (L+ (1− ǫ2)/4ǫ2) (70) which means the upper bound of (60) is finite. A proof that all derivatives of Q(x) are finite is given in Appendix B.2, so pC(t) decays faster than any power at large t. ‡ Strictly speaking when 1− (b+/b−)k < p ≤ 1− (b+/b−)k+1/Z the upper bounds diverge so our proof does not work for these arbitrarily small intervals. Biased random walks on random combs 15 5.2. µB – Teeth of random length In this subsection we are concerned with random combs that have a distribution of tooth lengths. The general strategy for determining quantities of interest is to identify teeth that are long enough to affect the critical behaviour of the biased random walk and consider the probability with which they occur. It will be useful to define the function λ(δ, η, ζ) = ⌊ δ| log x|η − ζ(a− 1) log | log x| log Y ⌋, (71) which will be used to denote a tooth length, and the function (ℓ) = (a− 1)ℓa−1 1 +O(ℓ−1) , (72) which is the probability that a tooth has length greater than ℓ− 1. 5.2.1. ǫ2 = 0, ǫ1 < 0 We first show that if a < 2, 1 otherwise. The proof follows the lines described in section 5.1.1 with a slight modification for the upper bound on P (x). Note that, from (30), teeth of length ℓ > ⌊x− 12 ⌋ have PT (x) ≤ 1− Bx 2 . We then proceed as in (63) but with ℓ = ⌊x− 12 ⌋+ 1. The exponent β is non-trivial if a < 2 but, as we now show, ds = 0 for all a > 1 so mean field theory does not apply when a < 2. This result follows from the inequalities x| log x| ≤ Q(x) ≤ B . (74) The upper bound is a consequence of Q(x) < Q♯(x). To obtain the lower bound consider the combs for which at least the first N teeth are all shorter than ℓ0. Then using monotonicity and (41) Q(x) ≥ (1− p>(ℓ0)) 1− P♭ℓ0(x) +O(exp(−N(Λǫ1,ǫ2(x) + Λ−ǫ1,ǫ2(x)))) . (75) Setting ℓ0 = λ(1, (a− 1)−1, 0), N = ⌊2(Λǫ1,ǫ2 +Λ−ǫ1,ǫ2)−1| log x|⌋+ 1 and using (35) the result follows for small enough x. 5.2.2. ǫ2 > 0, ǫ1 < 0 The exponent β = 0 but there are computable logarithmic corrections and we find that | log x|a−1 ≤ P (x) ≤ 1− B | log x|a−1 . (76) The lower bound follows from applying Jensen’s inequality to (6). For the upper bound note that teeth of length ℓ > λ(1, 1, 0) have PT < B. Again proceed as in (63) with ℓ = λ(1, 1, 0) + 1. Biased random walks on random combs 16 Table 2. 〈Dℓ〉µ in various cases. ensemble ǫ2 < 0 ǫ2 = 0 ǫ2 > 0 µA B +O(x) Bx− 2 +O(1) Bx−1 +O(1) µB, a ≥ 2 B +O(x) B +O(x) B(x| log x|a−1)−1 +O(1) µB, a < 2 B +O(x) Bxa/2−1 +O(1) B(x| log x|a−1)−1 +O(1) The spectral dimension is ds = 0 showing again that mean field theory does not apply. This follows from the inequalities B′ exp −B′′| log x|1/a ≤ Q ≤ B , (77) for small enough x. The upper bound is a consequence of Q(x) < Q♯(x) and the lower bound follows from (75) by setting ℓ0 = λ(1, 1/a, 0), N = ⌊2(Λǫ1,ǫ2+Λ−ǫ1,ǫ2)−1| log x|⌋+1 and using (37). 5.2.3. ǫ2 > 0, ǫ1 = 0 The exponent β = 0, but there are logarithmic corrections which follow from the inequalities | log x|(a−1)/2 ≤ P (x) ≤ 1− B | log x|(a−1)/2 . (78) The lower bound comes from applying Jensen’s inequality to the recurrence relation (6). The upper bound is obtained by requiring unitarity of the heat kernel and its proof is relegated to Appendix D. The spectral dimension and logarithmic exponent are given by ds = 2, α̃ = a− 1, (79) which shows that mean field theory does not apply. This result follows from B′ | log x|a−1 < Q(x) < B | log x|a−1 (80) for small enough x which is obtained by a modified version of the argument in 5.1.1. First let ℓ0 = λ(1, 1, ζ), so that Pℓ0(x) = 1− | log x|ζ(a−1) | log x|2ζ(a−1) . (81) To obtain (80) we use (60) and (61) with p = p>(ℓ0), ℓ = ℓ0 and for the lower bound set Tn<L = Nℓ0. Then using the bounds in (C.8) with ζ = 1 and (C.7) with ζ = 2 and estimating the sums gives the result. 6. Heat Kernel when ǫ1 ≥ 0, ǫ2 ≥ 0 These calculations require 〈Dℓ〉µ in the various cases which are tabulated in table 2 for convenience. Biased random walks on random combs 17 6.1. µA – Infinite teeth at random locations We show that 0 if ǫ2 > 0 and ǫ1 ≥ 0, k/2 if ǫ2 = 0 and ǫ1 > 0. These results follow from (85), (86) and (87) below. Noting that for ǫ1 > 0 all combs have 1 − B−1− − < PC(x) < 1 − B−1+ and using monotonicity gives D|Tn|(x) G∗(x;n) ≤ H(x;n) ≤ D|Tn|(x) GC′(x;n) 1− PC′(x) , (83) D|Tn|(x) 〈GC′(x;n)〉µ , (84) where C ′ is constructed from C by forcing Tn = N1. If ǫ2 > 0 then using (43) with K = 0, ℓ = ∞ gives the upper bound H(x;n) < exp(−B′n). (85) If ǫ2 = 0 then exactly the same calculation gives H(x;n) < exp(−B′x 2n) (86) and evaluating the left hand side of (83) gives a lower bound of the same form. If ǫ2 > 0 and ǫ1 = 0 it is necessary to sum over the location of the first infinite tooth. Using (C.9), (43) and introducing C ′ as in (83) gives H(x;n) < exp(−B′n). (87) 6.2. µB – Teeth of random length We show that 0 if ǫ2 > 0 and ǫ1 ≥ 0; ka/2 if ǫ2 = 0, ǫ1 > 0 and a < 2; k if ǫ2 = 0, ǫ1 > 0 and a ≥ 2. These results follow from H(x;n) < x| log x|a−1 exp(−B′n/| log x|a−1) if ǫ2 > 0 and ǫ1 ≥ 0; x1−a/2 exp(−B′nxa/2) if ǫ2 = 0, ǫ1 > 0 and a < 2; B exp(−nB′x) if ǫ2 = 0, ǫ1 > 0 and a ≥ 2, when x is small enough and lower bounds of the same form. The upper bounds are obtained by proceeding as in subsection 6.1: for ǫ1 > 0 and ǫ2 > 0 setting ℓ = λ(1, 1, 0) + 1 and for ǫ1 > 0 and ǫ2 = 0 setting ℓ = ⌊x− 2 ⌋ + 1. For ǫ1 = 0 and ǫ2 > 0 we start with the upper bound of (83); let ℓ1 = λ(1, 1, 2), p1 = p>(ℓ1) and ℓ2 = λ(2, 1, 0), p2 = p>(ℓ2). The latter shall be called long teeth and we denote by Biased random walks on random combs 18 (ℓ2K♯) the comb with a single long tooth at vertex K. We now sum over the location of the first long tooth using (18), (43) and (C.8) and taking account of the fact that the first long tooth may be before or after the nth tooth H(x;n) ≤ D|Tn|(x) p2(1− p1)K−1 1− P(ℓ2K♯)(0) P(ℓ2K♯)m × ((1− p1)P♯ + (p1 − p2)Pℓ1♯ + p2Pℓ2♯) n−K−1 θ(ℓ2 − |Tn|)D|Tn|(x) p2(1− p1)K−1 1− P(ℓ2K♯)(0) P(ℓ2K♯)m . (90) In the first sum we use the value given in Table 2 for D|Tn|(x) . In the second sum θ(ℓ2 − |Tn|)D|Tn|(x) = B(x| log x|2(a−1))−1+O(1) for |Tn| < ℓ2 and the result follows. To obtain the lower bounds when ǫ1 > 0 we note that H(x;n) ≥ B− D|Tn|(x)GC(x;n) D|Tn|(x) 〈GC(x;n)〉µ (91) where the measure µ is defined by µℓ = µℓ, for teeth Tk, k 6= n, 〈Dℓ〉µ , for tooth Tn. (92) Using the decomposition (18) and Jensen’s inequality 〈GC(x;n)〉µ ≥ 3(1− x)n/2 exp(−Sn), (93) where − PCk+1(x) 1− PTk+1(x) . (94) Now applying Jensen’s inequality with the measure µ to (6) shows that the lower bounds satisfy a recursion formula of exactly the same form as discussed in Appendix C. So from (C.1) we find that Sn ≤ n − P (x) + 〈1− PT (x)〉µ (〈PT (x)〉µ − 〈PT (x)〉µ) + P (x)(1− A(x)) A(x)k−1(P̄ (x)− A(x)P (x))/(P̄ (x)− P (x))− 1 , (95) where P (x) = (1− x)b− 3− bT 〈PT (x)〉µ − b+P (x) P̄ (x) = (1− x)b− 3− bT 〈PT (x)〉µ − b+P (x) A(x) = (1− x)b− P (x)2 b+ . (96) Biased random walks on random combs 19 For ǫ1 > 0 it is straightforward to check that A(x) > c > 1 and that the sum in (95) is bounded above by an n independent constant. Lower bounds of the form of (89) then follow by inserting the appropriate 〈PT 〉µ in (96) and (95). When ǫ1 = 0 H(x;n) ≥ D|Tn|(x) GC′(x;n) 1− PC′(x) , (97) where C ′ is constructed from C by setting Tk≥n = N∞. Choosing ℓ0 = λ(1, 1, 2) and using (18) and (C.7) gives H(x;n) ≥ D|Tn|(x) (1− p>(ℓ0))n−13(1− x)−n/2P∗(x)2 P♭ℓ0(x)− 1k−1 1− P♭ℓ0(x) + 1n−1 D|Tn|(x) (1− p>(ℓ0))n−13(1− x)−n/2P∗(x)2 1− P♭ℓ0(x) + 1n−1 (n− 3)2(1− P♭ℓ0(x)) 2P♭ℓ0(x)− 1 , (98) for n ≥ 4 which gives the result. 7. Results and discussion Figure 3 outlines the results that we have computed for µA. These are new and show that the most interesting regime is actually when the bias along the spine is towards the origin, a circumstance which has not been studied much in the literature. When ǫ1 ≥ 0 and ǫ2 > 0 the walker disappears rapidly, never to return, and p(t) decays faster than any power. When ǫ1 < 0 the bias along the spine is keeping the walker close to the origin but if there are any infinite teeth present the walker can spend a lot of time in the teeth; the conflict between these effects leads to a non-trivial ds. The fact that d1 = 0 whenever ǫ2 > 0 shows that the walker never gets far down the spine; if she disappears then it is up a tooth that she is lost. The Hausdorff dimension for µA is dH = 2, regardless of bias and so we have here several examples of violation of the bound 2dH/(1 + dH) ≤ ds ≤ dH , which applies for unbiased diffusion [19]. Figure 4 shows our results for µB as well as the results for the unbiased case studied in [14]. This length distribution has been studied quite extensively in the literature but usually under the assumption that ǫ2 ≥ 0. As can be seen the interesting behaviour displayed by µA when ǫ1 < 0 does not occur here – essentially because very long teeth are not common enough. We believe that with more work α̃ when ǫ1 < 0, ǫ2 ≥ 0 can be found using our methods, but as this will not give further physical insight we leave the calculations to elsewhere [17]. The case ǫ1 > 0, ǫ2 > 0 (often called topological bias) was originally studied using mean field theory, which gave the mean square displacement 〈n2(t)〉 ∼ (log t)2(a−1), (99) and this is in fact correct since the walker spends much of the time in the teeth. However the claim in [3] that (99) holds for ǫ2 > 0 regardless of ǫ1 is false. The mean field method gives the correct result when ǫ1 = 0 only because the walk on the spine is ignored, which Biased random walks on random combs 20 amounts to using PT (x) for PC(x) in (21) and naively applying Jensen’s inequality. The case ǫ1 > 0, ǫ2 = 0 was studied by Pottier [13] who computed the leading contribution exactly, but without complete control over the sub-leading terms; she also calculated the leading behaviour 〈n2〉 − 〈n〉2 which we have not. Of course our results for ds and d1 agree with hers. The Hausdorff dimension for µ B is dH = 3 − a when a < 2 and dH = 1 when a ≥ 2 and so again we see that, as expected, a biasing field intensifies the difference between the purely geometric definition of dimension and that which is related to particle propagation. The results for ǫ2 < 0 are intuitively obvious and, as we have proved, apply for any model with identically and independently distributed tooth lengths. The walker never gets far into the tooth and therefore combs have long time behaviour characteristic of the spine alone. This paper has given a comprehensive treatment of biased random walks on combs using rigorous techniques – namely recursion relations for generating functions combined with unitarity and monotonicity arguments. It serves to put in context many previous results as well as present new ones. In the unbiased case [14] and in some bias regimes mean field theory is sufficient to compute the leading order behaviour because the walker either does not reach the ends of the longest teeth or does not travel far enough down the spine for variations from average to be important. But, as is illustrated in many examples here, a full treatment is needed when such fluctuations cannot be ignored. Finally, while the results are of interest in themselves, an important point of the paper was to demonstrate that rigorous analytic methods can be used to treat biased diffusion on random geometric structures and it is to be hoped that these tools can be extended to higher dimensional problems. Acknowledgments We would like to thank Bergfinnur Durhuus and Thordur Jonsson for valuable discussions. This work is supported in part by Marie Curie grant MRTN-CT-2004- 005616 and by UK PPARC grant PP/D00036X/1. T.E. would like to acknowledge an ORS award and a Julia Mann Graduate Scholarship from St Hilda’s College, Oxford. Appendix A. Calculation of 〈PT (x)〉µB for ǫ2 > 0 First we rewrite (30) as Pℓ(x) = P∞(x)Y − (Y − 1)P∞(x)X−1 X−1 + Y −ℓ , (A.1) so that 〈PT (x)〉µB = P∞(x)Y − (Y − 1)P∞(x)X X−1 + Y −ℓ (A.2) Biased random walks on random combs 21 PSfrag replacements (1 + 2ǫ2) (1− 2ǫ2) (1 + 2ǫ2) (1− ǫ1 − ǫ2) (1 + ǫ1 − ǫ2) ds = 0 dk = 0 dk = 0dk = 0 dk = 0 dk = 0 ds = 1 ds = 1 ds = 3 dk = k ds n.d. ds n.d. ds n.d. ds = 2 log(1− p) Ω−1 ds = log(1− p) Ω−1 p < p∗ p ≥ p∗ Figure 3. Results for µA where Ω = log 1−|ǫ1|−ǫ2 1+|ǫ1|−ǫ2 and p∗ = 2|ǫ1|(1 + |ǫ1| − ǫ2)−1. The logarithmic exponents α̃ and d̃k are always zero for µ PSfrag replacements (1 + 2ǫ2) (1− 2ǫ2) (1 + 2ǫ2) (1− ǫ1 − ǫ2) (1 + ǫ1 − ǫ2) ds = 0 ds = 0 ds = 0 dk = 0 dk = 0 dk = 0 dk = 0 dk = 0 ds = 1 ds = 3 dk = k dk = k ds n.d. ds = 2 log(1− p) Ω−1 ds = log(1− p) Ω−1 p < p∗ p ≥ p∗ ds = 2 ds = 2 ds = 2 + a α̃ ≤ 0 α̃k = 1 α̃k = 0 α̃ = a− 1 α̃ = −a d̃k = 0 d̃k = 0 d̃k = 0 d̃k = k(a− 1) d̃k = k(a− 1) ≤ α̃ ≤ 0 if a < 2 if a < 2 if a < 2 if a ≥ 2 if a ≥ 2 if a ≥ 2 if a = 2k if a 6= 2k Figure 4. Results for µB . When ǫ2 < 0 the logarithmic exponents α̃ and d̃k are always zero. Biased random walks on random combs 22 X−1 + Y −ℓ X−1 + Y −ℓ ≡ S. (A.3) Since for ǫ2 > 0, Y > 1 we let log Y = ρ and write | log x| X−1 + e−ρℓ | log x| X−1 + e−ρℓ , (A.4) where σ is an arbitrary constant < 1. This is bounded above by taking ℓ in the exponential to be its value at the top of each sum to give S ≤ Ca X−1 + xσ | log x| ℓ−a + | log x| | log x|a−1 . (A.5) Noting that as x → 0, X−1 → Bx we get a lower bound on 〈Pℓ(x)〉µB of 〈PT (x)〉µB ≥ 1− | log x|a−1 , (A.6) for small enough x. An equivalent upper bound is calculated in the same manner by ignoring the first term in (A.4) and setting σ = 1, which leads to the result quoted in table 1. A similar procedure leads to bounds of the form B/x| log x|a on T (x) which we also need, at small enough x. Appendix B. Proof of results for non-recurrent regime First we define a structure of ordered lists of ordered integers. Let S denote an ordered list of hS integers [n1, n2, . . . nhS ], n1 ≥ n2 ≥ . . . ≥ nhS ≥ 1, hS ≥ 1, [ ], hS = 0. (B.1) Define |S| = ni, hS ≥ 1, 0, hS = 0, (B.2) and let SN denote the set of all distinct lists S with |S| = N . Within SN the lists S and S ′ are ordered by letting j = min(i : ni 6= n′i) and then setting S > S ′ if nj > n′j . Finally if S ∈ SN and S ′ ∈ SN ′ with N > N ′ then S > S ′. It is convenient to denote by S + 1 the lowest list above S, and by S ∪ S ′ the list obtained by concatenating S and S ′ and then ordering as above. Biased random walks on random combs 23 Now define H(S; f(x)) = (−1)|S| f (ni)(x), (B.3) and for the empty list H([ ]; f(x)) = 1. We need the following lemma, which is proved in Appendix B.1: Lemma (i) If 〈H(S;PT (x))〉µ is finite as x → 0 for all S ≤ S̄ then 〈H(S;PC(x))〉µ is finite as x → 0 for all S ≤ S̄ and ǫ1 6= 0. (ii) If the conditions of part (i) apply and, as x → 0, H(S̄ + 1, PT (x)) diverges as x−γ , γ > 0, then H(S̄ + 1, PC(x)) also diverges as x−γ. Differentiating (5) k times gives C (x) = C (x) (1− PC(x))2 + (−1)k S∈Sk/[k] C(S)H(S;PC(x)) (1− PC(x))hS+1 (B.4) where C(S) is a combinatorial coefficient. It is straightforward to check for any S that 〈H(S;PT (x))〉µA is analytic for ǫ2 6= 0, and that 〈H(S;PT (x))〉µB is analytic when ǫ2 < 0. When ǫ2 = 0 H(S;Pℓ(x))|x=0 = cSℓ2|S|−hS 1 +O(l−2) (B.5) from which 〈H(S;Pℓ(x))〉µB is divergent for S = [⌈a/2⌉], and with smaller degree for [⌈a/2⌉ − 1, 1] if 2k < a ≤ 2k + 1, k ∈ Z, but always convergent for any inferior S. The results given in section 4.2 then follow from noting that P∗(x) < PC(x) < P♯(0) < 1 and using the lemma. Appendix B.1. Proof of lemma To prove the lemma note that H(S; f + g) = S′∪S′′=S H(S ′; f)H(S ′′; g) (B.6) and differentiate (6) n times to get (−1)nP (n)C (x) = (1− x)F C (x) + nF (n−1) C (x) (B.7) where C (x) = PC(x) PC(x)b+ (1− x)b− S′∪S′′=S )hS′′ H(S ′;PC1(x))H(S ′′;PT1(x)). (B.8) Biased random walks on random combs 24 It is then straightforward to generalise this formula to H(S, PC(x)) = R+ (PC(x))hS S′∈S|S| C(S, S ′) PC(x)b+ (1− x)b− S′′∪S′′′=S′ )hS′′′ H(S ′′;PC1(x))H(S ′′′;PT1(x)), (B.9) where the leading terms are written out explicitly and R contains contributions depending only on lists inferior to S |S|. Every term on the right hand side is positive so it can be bounded above by using PC(x) < P♯(0) and then the expectation value taken; moving the S ′′ = S term to the left hand side gives 〈H(S, PC(0))〉µ P♯(0) R+ (P♯(0))hS S′∈S|S| C(S, S ′) P♯(0)b+ S′′∪S′′′=S′ S′′ 6=S )hS′′′ 〈H(S ′′;PC1(0))〉µ 〈H(S ′′′;PT (0))〉µ . (B.10) Part (i) is true for S̄ = [1] so the lemma then follows immediately by induction on S. To prove part (ii) use part (i) to isolate the potentially divergent terms in (B.9) leaving H(S̄, PC(x)) = PC(x) (1− x)b− H(S̄;PC1(x)) + H(S̄;PT1(x)) + finite terms. (B.11) For small enough x, PC(x) (1− x)b− < 1, ∀C (B.12) and part (ii) follows upon taking expectation values. Appendix B.2. ǫ1 = 0, ǫ2 > 0 We will show that H(S, PC(x)) (1− PC(x))hS+1 (B.13) is finite at x = 0, which together with (B.4) gives the result. Using (61) and (C.9) gives C H(S, PC(x)) , (B.14) Biased random walks on random combs 25 where nC is the location of the first infinite tooth of C. Applying (B.9) iteratively we find that the right hand side is bounded above by terms of the form 〈H(S ′, PT (x))〉µA . (B.15) The maximum value of K occurring is hS + 1 + ΦS where ΦS is the number of strings inferior to S. As remarked before 〈H(S ′, PT (x))〉µA is analytic and is trivially finite which completes the proof. Appendix C. Calculation of PC(x) for some useful combs Let the comb C have Tk = Nℓ, k < L and arbitrary TL and CL. Then following the method of Appendix A of [14] we find P ǫ1ǫ2C (x) = P (1−A)(P ǫ1ǫ2CL−1(x)− P AL−1(P ǫ1ǫ2CL−1(x)−AP (x))− (P ǫ1ǫ2CL−1(x)− P (C.1) where (1− x)b− (P ǫ1ǫ2 (x))2b+ . (C.2) Setting ǫ2 = 0, ǫ1 < 0, ℓ = 1, TL = N∞ and CL = ♯ we find after some algebra that P ǫ10C (x) = P ♯ (x) 1 + A−L +O(x) (C.3) and, as x → 0, 1 + |ǫ1| 1− |ǫ1| . (C.4) Repeating the exercise but with CL = ∗ yields a similar result. If instead we set ǫ2 > 0, ǫ1 < 0, ℓ = 1, TL = N∞ and CL = ♯ we find P ǫ1ǫ2C (x) = P ♯ (x) 2ǫ2(A− 1)A−L ǫ1 − 2ǫ2(1− A−L) (1 +O(x)) (C.5) and, as x → 0, A → 1 + |ǫ1| − ǫ2 1− |ǫ1| − ǫ2 . (C.6) Again, repeating the exercise but with CL = ∗ yields a similar result. With ǫ2 > 0, ǫ1 = 0, and C = {Tk<L = Nℓ, Tk≥L = N∞} we find that P 0ǫ2C (x) > P (x)− 1 , L > 2, (C.7) (it is good enough to use P∗(x) for k = 2); and for C = {Tk 6=L = N1, TL = Nℓ}, x < x0, P 0ǫ2C (x) < P ♯ (x) 1− 1 AL−1−1  (C.8) where A = (1− x)(P 0ǫ2♯ (x))−2 and B is a positive constant depending on x0, A and ǫ2. Finally for ǫ2 > 0, ǫ1 = 0, and C = {Tk 6=L = N1, TL = N∞} we find that P 0ǫ2C (0) = 1− L+ (1− ǫ2)/4ǫ2 . (C.9) Biased random walks on random combs 26 Appendix D. Upper bound on P (x) when ǫ1 = 0, ǫ2 > 0 We start by writing H(x;n) = D|Tn|(x) GC(x;n) 1− PC(x) , (D.1) where the measure µ̄ is defined in (92). Applying Jensen’s inequality with this measure to (6) results in a recursion formula of the same form as discussed in Appendix C and it is easy to verify that 〈PCk(x)〉µ ≥ 〈PCk(x)〉µ to give GC(x;n) 1− PC(x) GC(x;n) 1− PC(x) b+(1− x)n/2 −n 〈1− PC(x)〉µ 〈1− PC(x)〉µ , (D.2) where in the last line we have again used Jensen’s inequality when averaging over the ensemble. Applying this result to (D.1), summing over n, and using (24) we obtain the inequality D|Tn|(x) 〈1− PC(x)〉2µ . (D.3) Using the value for 〈Dℓ(x)〉µ given in table 2 and rearranging gives the upper bound on P (x) quoted in 5.2.3. References [1] G. H. Weiss and S. Havlin, Some properties of a random walk on a comb structure, Physica 134A (1986) 474-484 [2] S. Revathi, V. Balakrishnan, S. Lakshmibala and K. P. N. Murthy, Validity of the mean-field approximation for diffusion on a random comb, Phys. Rev. E 54 (1996) 2298-2302 [3] D. ben-Avraham and S. Havlin, Diffusion and reactions in fractals and disordered systems, Cambridge University Press, Cambridge (2000) [4] S. Havlin, J. E. Kiefer and G. H. Weiss, Anomalous diffusion on a random comblike structure, Phys. Rev. A 36 (1987) 1403-1408 [5] J. Ambjørn, B. Durhuus and T. Jonsson, Quantum geometry: a statistical field theory approach, Cambridge University Press, Cambridge (1997) [6] J. Ambjørn and Y. Watabiki, Scaling in quantum gravity, Nucl. Phys. B445 (1995) 129-144, hep-th/9501049 [7] J. Ambjørn, J. Jurkiewicz and R. Loll, Spectral dimension of the universe, Phys. Rev. Lett. 95 (2005) 171301, hep-th/0505113 [8] C. Aslangul, P. Chvosta and N. Pottier, Analytic study of a model of diffusion on a random comblike structure, Physica A 203 (1994) 533-565 [9] S. Havlin, A. Bunde, H. E. Stanley and D. Movsholvitz, Diffusion on percolation clusters with a bias in topological space: non-universal behaviour, J. Phys. A 19 (1986) L693-L698 [10] V. Balakrishnan and C. Van den Broeck, Transport properties on a random comb, Physica A 217 (1995) 1-21 [11] S. Havlin, A. Bunde, Y. Glaser and H. E. Stanley, Diffusion with a topological bias on random structures with a power-law distribution of dangling ends, Phys. Rev. A 34 (1986) 3492-3495 http://arxiv.org/abs/hep-th/9501049 http://arxiv.org/abs/hep-th/0505113 Biased random walks on random combs 27 [12] N. Pottier, Diffusion on random comblike structures: field-induced trapping effects, Physica A 216 (1995) 1-19 [13] N. Pottier, Analytic study of a model of biased diffusion on a random comblike structure, Physica A 208 (1994) 91-123 [14] B. Durhuus, T. Jonsson and J. F. Wheater, Random walks on combs, J. Phys. A 39 (2006) 1009- 1038, hep-th/0509191 [15] B. Durhuus, T. Jonsson and J. F. Wheater, The spectral dimension of generic trees, math-ph/0607020 [16] B. Durhuus, T. Jonsson and J. F. Wheater, On the spectral dimension of generic trees, DMTCS proc. AG (2006), 183-192 [17] T.M. Elliott, Oxford University D.Phil Thesis, in preparation. [18] W. Feller, An introduction to probability theory and its applications, Vol.2, Wiley, London (1968) [19] A. Grigoryan and T. Coulhon, Pointwise estimates for transition probabilities of random walks in infinite graphs, in: Trends in mathematics: Fractals in Graz 2001, Ed. P. Grabner and W. Woess. Birkhäueser (2002) http://arxiv.org/abs/hep-th/0509191 http://arxiv.org/abs/math-ph/0607020 Introduction Definitions Random combs Biased random walks Two-point functions The Heat kernel Basic properties Results for simple regular combs General properties of the generating functions Useful elementary bounds Results independent of the comb ensemble ds when 2<0 ds when 1>0 dk when 2<0 or 1<0 The spectral dimension when 20 and 10 A – Infinite teeth at random locations 2= 0, 1< 0 2> 0, 1< 0 2> 0, 1= 0 B – Teeth of random length 2= 0, 1< 0 2> 0, 1 <0 2> 0, 1 =0 Heat Kernel when 1 0, 2 0 A – Infinite teeth at random locations B – Teeth of random length Results and discussion Calculation of "426830A PT(x) "526930B B for 2>0 Proof of results for non-recurrent regime Proof of lemma 1=0, 2>0 Calculation of PC(x) for some useful combs Upper bound on P(x) when 1=0, 2>0

0704.0189

Monoid generalizations of the Richard Thompson groups

Monoid generalizations of the Richard Thompson groups J.C. Birget ∗ 24 Jan. 2016 Abstract The groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1 have connections with circuit complexity (studied in another paper). Here we prove thatMk,1 and Invk,1 are congruence-simple for all k. Their Green relations J and D are characterized: Mk,1 and Invk,1 are J -0-simple, and they have k− 1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra Ok. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets.1 1 Thompson-Higman monoids Since their introduction by Richard J. Thompson in the mid 1960s [29, 26, 30], the Thompson groups have had a great impact on infinite group theory. Graham Higman generalized the Thompson groups to an infinite family [20]. These groups and some of their subgroups have appeared in many contexts and have been widely studied; see for example [12, 8, 15, 10, 17, 18, 9, 11, 23]. The definition of the Thompson-Higman groups lends itself easily to generalizations to inverse monoids and to more general monoids. These monoids are also generalizations of the finite symmetric monoids (of all functions on a set), and this leads to connections with circuit complexity; more details on this appear in [2, 3, 5]. By definition the Thompson-Higman group Gk,1 consists of all maximally extended isomorphisms between finitely generated essential right ideals of A∗, where A is an alphabet of cardinality k. The multiplication is defined to be composition followed by maximal extension: for any ϕ,ψ ∈ Gk,1, we have ϕ · ψ = max(ϕ ◦ ψ). Every element ϕ ∈ Gk,1 can also be given by a bijection ϕ : P → Q where P,Q ⊂ A∗ are two finite maximal prefix codes over A; this bijection can be described concretely by a finite function table. For a detailed definition according to this approach, see [4] (which is also similar to [28], but with a different terminology); moreover, Subsection 1.1 gives all the needed definitions. It is natural to generalize the maximally extended isomorphisms between finitely generated essential right ideals of A∗ to homomorphisms, and to drop the requirement that the right ideals be essential. It will turn out that this generalization leads to interesting monoids, or inverse monoids, which we call Thompson-Higman monoids. Our generalization of the Thompson-Higman groups to monoids will also generalize the embedding of these groups into the Cuntz algebras [4, 27], which provides an additional ∗ Earlier versions of this paper appeared in http://arxiv.org/abs/0704.0189 (v1 in April 2007, and v2 in April 2008), and also appeared in reference [1]. Supported by NSF grant CCR-0310793. Changes in this version: Section 4 has been thoroughly revised, and errors have been corrected; however, the main results of Section 4 do not change. The main changes are in Theorem 4.5, Definition 4.5A (the concept of a normal right-ideal morphism), and the final proof of Theorem 4.13. Sections 1, 2, and 3 are unchanged, except for the proof of Theorem 2.3, which was incomplete; a complete proof was published in the Appendix of reference [6], and is also given here. http://arxiv.org/abs/0704.0189v3 http://arxiv.org/abs/0704.0189 motivation for our definition. Moreover, since these homomorphisms are close to being arbitrary finite string transformations, there is a connection between these monoids and combinational boolean circuits; the study of the connection between Thompson-Higman groups and circuits was started in [5, 3] and will be developed more generally for monoids in [2]; the present paper lays some of the foundations for [2]. 1.1 Definition of the Thompson-Higman groups and monoids Before defining the Thompson-Higman monoids we need some basic definitions, that are similar to the introductory material that is needed for defining the Thompson-Higman groups Gk,1; we follow [4] (which is similar to [28]). We use an alphabet A of cardinality |A| = k, and we list its elements as A = {a1, . . . , ak}. Let A∗ denote the set of all finite words over A (i.e., all finite sequences of elements of A); this includes the empty word ε. The length of w ∈ A∗ is denoted by |w|; let An denote the set of words of length n. For two words u, v ∈ A∗ we denote their concatenation by uv or by u · v; for sets B,C ⊆ A∗ the concatenation is BC = {uv : u ∈ B, v ∈ C}. A right ideal of A∗ is a subset R ⊆ A∗ such that RA∗ ⊆ R. A generating set of a right ideal R is a set C such that R is the intersection of all right ideals that contain C; equivalently, R = CA∗. A right ideal R is called essential iff R has a non-empty intersections with every right ideal of A∗. For words u, v ∈ A∗, we say that u is a prefix of v iff there exists z ∈ A∗ such that uz = v. A prefix code is a subset C ⊆ A∗ such that no element of C is a prefix of another element of C. A prefix code is maximal iff it is not a strict subset of another prefix code. One can prove that a right ideal R has a unique minimal (under inclusion) generating set, and that this minimal generating set is a prefix code; this prefix code is maximal iff R is an essential right ideal. For right ideals R′ ⊆ R ⊆ A∗ we say that R′ is essential in R iff R′ intersects all right subideals of R in a non-empty way. Tree interpretation: The free monoid A∗ can be pictured by its right Cayley graph, which is the rooted infinite regular k-ary tree with vertex set A∗ and edge set {(v, va) : v ∈ A∗, a ∈ A}. We simply call this the tree of A∗. It is a directed tree, with all paths moving away from the root ε (the empty word); by “path” we will always mean a directed path. A word v is a prefix of a word w iff v is is an ancestor of w in the tree. A set P is a prefix code iff no two elements of P are on the same path. A set R is a right ideal iff any path that starts in R has all its vertices in R. The prefix code that generates R consists of the elements of R that are maximal (within R) in the prefix order, i.e., closest to the root ε. A finitely generated right ideal R is essential iff every infinite path of the tree eventually reaches R (and then stays in it from there on). Similarly, a finite prefix code P is maximal iff any infinite path starting at the root eventually intersects P . For two finitely generated right ideals R′ ⊂ R, R′ is essential in R iff any infinite path starting in R eventually reaches R′ (and then stays in R′ from there on). In other words for finitely generated right ideals R′ ⊆ R, R′ is essential in R iff R′ and R have the same “ends”. For the prefix tree of A∗ we can consider also the “boundary” Aω (i.e., all infinite words), a.k.a. the ends of the tree. In Thompson’s original definition [29, 30], G2,1 was given by a total action on {0, 1}ω . In [4] this total action was extended to a partial action on A∗∪Aω; the partial action on A∗ ∪Aω is uniquely determined by the total action on Aω; it is also uniquely determined by the partial action on A∗. Here, as in [4], we only use the partial action on A∗. Definition 1.1 A right ideal homomorphism of A∗ is a total function ϕ : R1 → A∗ such that R1 is a right ideal of A∗, and for all x1 ∈ R1 and all w ∈ A∗: ϕ(x1w) = ϕ(x1) w. For any partial function f : A∗ → A∗, let Dom(f) denote the domain and let Im(f) denote the image (range) of f . For a right ideal homomorphism ϕ : R1 → A∗ it is easy to see that the image Im(ϕ) is also right ideal of A∗, which is finitely generated (as a right ideal) if the domain R1 = Dom(ϕ) is finitely generated. A right ideal homomorphism ϕ : R1 → R2, where R1 = Dom(ϕ) and R2 = Im(ϕ), can be described by a total surjective function P1 → S2, with P1, S2 ⊂ A∗; here P1 is the prefix code (not necessarily maximal) that generates R1 as a right ideal, and S2 is a set (not necessarily a prefix code) that generates R2 as a right ideal; so R1 = P1A ∗ and R2 = S2A ∗. The function P1 → S2 corresponding to ϕ : R1 → R2 is called the table of ϕ. The prefix code P1 is called the domain code of ϕ and we write P1 = domC(ϕ). When S2 is a prefix code we call S2 the image code of ϕ and we write S2 = imC(ϕ). We denote the table size of ϕ (i.e., the cardinality of domC(ϕ)) by ‖ϕ‖. Definition 1.2 An injective right ideal homomorphism is called a right ideal isomorphism. A right ideal homomorphism ϕ : R1 → R2 is called total iff the domain right ideal R1 is essential. And ϕ is called surjective iff the image right ideal R2 is essential. The table P1 → P2 of a right ideal isomorphism ϕ is a bijection between prefix codes (that are not necessarily maximal). The table P1 → S2 of a total right ideal homomorphism is a function from a maximal prefix code to a set, and the table P1 → S2 of a surjective right ideal homomorphism is a function from a prefix code to a set that generates an essential right ideal. The word “total” is justified by the fact that if a homomorphism ϕ is total (and if domC(ϕ) is finite) then ϕ(w) is defined for every word that is long enough (e.g., when |w| is longer than the longest word in the domain code P1); equivalently, ϕ is defined from some point onward on every infinite path in the tree of A ∗ starting at the root. Definition 1.3 An essential restriction of a right ideal homomorphism ϕ : R1 → A∗ is a right ideal homomorphism Φ : R′1 → A∗ such that R′1 is essential in R1, and such that for all x′1 ∈ R′1: ϕ(x′1) = Φ(x′1). We say that ϕ is an essential extension of Φ iff Φ is an essential restriction of ϕ. Note that if Φ is an essential restriction of ϕ then R′2 = Im(Φ) will automatically be essential in R2 = Im(ϕ). Indeed, if I is any non-empty right subideal of R1 then I∩R′1 6= ∅, hence ∅ 6= Φ(I∩R′1) ⊆ Φ(I) ∩ Φ(R′1) = Φ(I) ∩ R′2; moreover, any right subideal J of R2 is of the form J = Φ(I) where I = Φ−1(J) is a right subideal of R1; hence, for any right subideal J of R2, ∅ 6= J ∩R′2. Proposition 1.4 (1) Let ϕ,Φ be homomorphisms between finitely generated right ideals of A∗, where A = {a1, . . . , ak}. Then Φ is an essential restriction of ϕ iff Φ can be obtained from ϕ by starting from the table of ϕ and applying a finite number of restriction steps of the following form: Replace (x, y) in a table by {(xa1, ya1), . . . , (xak, yak)}. (2) Every homomorphism between finitely generated right ideals of A∗ has a unique maximal essential extension. Proof. (1) Consider a homomorphism between finitely generated right ideals ϕ : R1 → R2, let P1 be the finite prefix code that generates the right ideal R1, and let S2 = ϕ(P1), so S2 generates the right ideal R2. If x ∈ P1 and y = ϕ(x) ∈ S2 then (since ϕ is a right ideal homomorphism), yai = ϕ(xai) for i = 1, . . . , k. Then R1 − {x} is a right ideal which is essential in R1, and R1 − {x} is generated by (P1 − {x}) ∪ {xa1, . . . , xak}. Indeed, in the tree of A∗ every downward directed path starting at vertex x goes through one of the vertices xai. Thus, removing (x, y) from the graph of ϕ is an essential restriction; for the table of ϕ, the effect is to replace the entry (x, y) by the set of entries {(xa1, ya1), . . . , (xak, yak)}. If finitely many restriction steps of the above type are carried out, the result is again an essential restriction of ϕ. Conversely, let us show that if Φ is an essential restriction of ϕ then Φ can be obtained by a finite number of replacement steps of the form “replace (x, y) by {(xa1, ya1), . . . , (xak, yak)} in the table”. Using the tree of A∗ we have: If R and R′ are right ideals of A∗ generated by the finite prefix codes P , respectively P ′, and if R′ is essential in R then every infinite path from P intersects P ′. It follows from this characterization of essentiality and from the finiteness of P1 and P 1 that R1 − R′1 is finite. Hence ϕ and Φ differ only in finitely many places, i.e., one can transform ϕ into Φ in a finite number of restriction steps. So, the restriction Φ of ϕ is obtained by removing a finite number of pairs (x, y) from ϕ; however, not every such removal leads to a right ideal homomorphism or an essential restriction of ϕ. If (x0, y0) is removed from ϕ then x0 is removed from R1 (since ϕ is a function). Also, since R 1 is a right ideal, when x0 is removed then all prefixes of x0 (equivalently, all ancestor vertices of x0 in the tree of A have to be removed. So we have the following removal rule (still assuming that domain and image right ideals are finitely generated): If Φ is an essential restriction of ϕ then ϕ can be transformed into Φ by removing a finite set of strings from R1, with the following restriction: If a string x0 is removed then all prefixes of x0 are also removed from R1; moreover, x0 is removed from R1 iff (x0, ϕ(x0)) is removed from ϕ. As a converse of this rule, we claim that if the transformation from ϕ to Φ is done according to this rule, then Φ is an essential restriction of ϕ. Indeed, Φ will be a right ideal homomorphism: if Φ(x1) is defined then Φ(x1z) will also be defined (if it were not, the prefix x1 of x1z would have been removed), and Φ(x1z) = ϕ(x1z) = ϕ(x1) z = Φ(x1) z. Moreover, Dom(Φ) = R 1 will be essential in R1: every directed path starting at R1 eventually meets R 1 because only finitely many words were removed from R1 to form R 1. Hence by the tree characterization of essentiality, R 1 is essential in R1. In summary, if Φ is an essential restriction of ϕ then Φ is obtained from ϕ by a finite sequence of steps, each of which removes one pair (x, ϕ(x)). In Dom(ϕ) the string x is removed. The domain code becomes (P1 −{x}) ∪ {xa1, . . . , xak}, since {xa1, . . . , xak} is the set of children of x in the tree of A∗. This means that in the table of ϕ, the pair (x, ϕ(x)) is replaced by {(xa1, ϕ(x) a1), . . . , (xak, ϕ(x) ak)}. (2) Uniqueness of the maximal essential extension: By (1) above, essential extensions are obtained by the set of rewrite rules of the form {(xa1, ya1), . . . , (xak, yak)} → (x, y), applied to tables. This rewriting system is locally confluent (because different rules have non-overlapping left sides) and ter- minating (because they decrease the length); hence maximal essential extensions exist and are unique. Proposition 1.4 yields another tree interpretation of essential restriction: Assume first that a total order a1 < a2 < . . . < ak has been chosen for the alphabet A; this means that the tree of A ∗ is now an oriented rooted tree, i.e., the children of each vertex v have a total order (namely, va1 < va2 < . . . < vak). The rule “replace (x, y) in the table by {(xa1, ya1), . . . , (xak, yak)}” has the following tree interpretation: Replace x and y = ϕ(x) by the children of x, respectively of y, matched according to the order of the children. Important remark: As we saw, every right ideal homomorphism can be described by a table P → S where P is a prefix code and S is a set. But we also have: Every right ideal homomorphism ϕ has an essential restriction ϕ′ whose table P ′ → Q′ is such that both P ′ and Q′ are prefix codes; moreover, Q′ can be chosen to be a subset of An for some n ≤ max{|s| : s ∈ S}. Example (with alphabet A = {a, b}): has an essential restriction aa ab b aa ab aa . Theorem 4.5B gives a tighter result with poly- nomial bounds. Definition 1.5 The Thompson-Higman partial function monoid Mk,1 consists of all maximal essen- tial extensions of homomorphisms between finitely generated right ideals of A∗. The multiplication is composition followed by maximal essential extension. In order to prove associativity of the multiplication of Mk,1 we define the following and we prove a few Lemmas. Definition 1.6 By RIk we denote the monoid of all right ideal homomorphisms between finitely generated right ideals of A∗, with function composition as multiplication. We consider the equivalence relation ≡ defined for ϕ1, ϕ2 ∈ RI k by: ϕ1 ≡ ϕ2 iff max(ϕ1) = max(ϕ2). It is easy to prove that RI k is closed under composition. Moreover, by existence and uniqueness of the maximal essential extension (Prop. 1.4(2)) each ≡-equivalence class contains exactly one element of Mk,1. We want to prove: Proposition 1.7 The equivalence relation ≡ is a monoid congruence on RI k, and Mk,1 is isomorphic (as a monoid) to RI k/≡. Hence, Mk,1 is associative. First some Lemmas. Lemma 1.8 If R′i ⊆ Ri (i = 1, 2) are finitely generated right ideals with R′i essential in Ri, then R′1 ∩R′2 is essential in R1 ∩R2. Proof. We use the tree characterization of essentiality. Any infinite path p in R1 ∩ R2 is also in Ri (i = 1, 2), hence p eventually enters into R′i. Thus p eventually meets R 1 and R 2, i.e., p meets R 1∩R′2. Lemma 1.9 All ϕ1, ϕ2 ∈ RI k have restrictions Φ1,Φ2 ∈ RI k (not necessarily essential restrictions) such that: • Φ2 ◦ Φ1 = ϕ2 ◦ ϕ1, and • Dom(Φ2) = Im(Φ1) = Dom(ϕ2) ∩ Im(ϕ1). Proof. Let R = Dom(ϕ2) ∩ Im(ϕ1). This is a right ideal which is finitely generated since Dom(ϕ2) and Im(ϕ1) are finitely generated (see Lemma 3.3 of [4]). Now we restrict ϕ1 to Φ1 in such a way that Im(Φ1) = R and Dom(Φ1) = ϕ 1 (R), and we restrict ϕ2 to Φ2 in such a way that Dom(Φ2) = R and Im(Φ2) = ϕ2(R). Then Φ2 ◦Φ1(.) and ϕ2 ◦ϕ1(.) agree on ϕ−11 (R); moreover, Dom(Φ2 ◦Φ1) = ϕ 1 (R). Since ϕ2 ◦ ϕ1(x) is only defined when ϕ1(x) ∈ R, we have Φ2 ◦ Φ1 = ϕ2 ◦ ϕ1. Also, by the definition of R we have Dom(Φ2) = Im(Φ1). ✷ Lemma 1.10 For all ϕ1, ϕ2 ∈ RI k we have: max(ϕ2 ◦ ϕ1) = max(max(ϕ2) ◦ ϕ1) = max(ϕ2 ◦max(ϕ1)). Proof. We only prove the first equality; the proof of the second one is similar. By Lemma 1.9 we can restrict ϕ1 and ϕ2 to ϕ 1, respectively ϕ 2, so that ϕ 2 ◦ ϕ′1 = ϕ2 ◦ ϕ1, and Dom(ϕ′2) = Im(ϕ′1) = Dom(ϕ2) ∩ Im(ϕ1); let R′ = Dom(ϕ2) ∩ Im(ϕ1). Similarly we can restrict ϕ1 and max(ϕ2) to ϕ 1 , respectively ϕ 2 , so that ϕ 2 ◦ ϕ′′1 = max(ϕ2) ◦ ϕ1, and Dom(ϕ′′2) = Im(ϕ 1) = Dom(max(ϕ2)) ∩ Im(ϕ1); let R′′ = Dom(max(ϕ2)) ∩ Im(ϕ1). Obviously, R′ ⊆ R′′ (since ϕ2 is a restriction of max(ϕ2)). Moreover, R′ is essential in R′′, by Lemma 1.8; indeed, Dom(ϕ2) is essential in Dom(max(ϕ2)) since max(ϕ2) is an essential extension of ϕ2. Since R ′ is essential in R′′, ϕ2 ◦ϕ1 is an essential restriction of max(ϕ2)◦ϕ1. Hence by uniqueness of the maximal essential extension, max(max(ϕ2) ◦ ϕ1) = max(ϕ2 ◦max(ϕ1)). ✷ Proof of Prop. 1.7: If ϕ2 ≡ ψ2 then, by definition, max(ϕ2) = max(ψ2), hence by Lemma 1.10: max(ϕ2 ◦ ϕ) = max(max(ϕ2) ◦ ϕ) = max(max(ψ2) ◦ ϕ) = max(ψ2 ◦ ϕ), for all ϕ ∈ RI k. Thus (by the definition of ≡), ϕ2 ◦ ϕ ≡ ψ2 ◦ ϕ, so ≡ is a right congruence. Similarly one proves that ≡ is a left congruence. Thus, RI k/≡ is a monoid. Since every ≡-equivalence class contains exactly one element of Mk,1 there is a one-to-one cor- respondence between RI k/≡ and Mk,1. Moreover, the map ϕ ∈ RI k 7−→ max(ϕ) ∈ Mk,1 is a homomorphism, by Lemma 1.10 and by the definition of multiplication in Mk,1. Hence RI k/≡ is isomorphic to Mk,1. ✷ 1.2 Other Thompson-Higman monoids We now introduce a few more families of Thompson-Higman monoids, whose definition comes about naturally in analogy with Mk,1. Definition 1.11 The Thompson-Higman total function monoid totMk,1 and the Thompson-Higman surjective function monoid surMk,1 consist of maximal essential extensions of homomorphisms between finitely generated right ideals of A∗ where the domain, respectively, the image ideal, is an essential right ideal. The Thompson-Higman inverse monoid Invk,1 consists of all maximal essential extensions of iso- morphisms between finitely generated (not necessarily essential) right ideals of A∗. Every element ϕ ∈ totMk,1 can be described by a function P → Q, called the table of ϕ, where P,Q ⊂ A∗ with P a finite maximal prefix code over A. A similar description applies to surMk,1 but now with Q a finite maximal prefix code. Every ϕ ∈ Invk,1 can be described by a bijection P → Q where P,Q ⊂ A∗ are two finite prefix codes (not necessarily maximal). It is easy to prove that essential extension and restriction of right ideal homomorphisms, as well as composition of such homomorphisms, preserve injectiveness, totality, and surjectiveness. Thus totMk,1, surMk,1, and Invk,1 are submonoids of Mk,1. We also consider the intersection totMk,1 ∩ surMk,1, i.e., the monoid of all maximal essential extensions of homomorphisms between finitely generated essential right ideals of A∗; we denote this monoid by totsurMk,1. The monoids Mk,1, totMk,1, surMk,1, and totsurMk,1 are regular monoids. (A monoid M is regular iff for every m ∈ M there exists x ∈ M such that mxm = m.) The monoid Invk,1 is an inverse monoid. (A monoid M is inverse iff for every m ∈ M there exists one and only one x ∈M such that mxm = m and x = xmx.) We consider the submonoids totInv k,1 and surInvk,1 of Invk,1, described by bijections P → Q where P,Q ⊂ A∗ are two finite prefix codes with P , respectively Q maximal. The (unique) inverses of elements in totInv k,1 are in surInvk,1, and vice versa, so these submonoids of Invk,1 are not regular monoids. We have totInv k,1 ∩ surInvk,1 = Gk,1 (the Thompson-Higman group). It is easy to see that for all n > 0, Mk,1 contains the symmetric monoids PF kn of all partial functions on kn elements, represented by all elements of Mk,1 with a table P → Q where P,Q ⊆ An. Hence Mk,1 contains all finite monoids. Similarly, totMk,1 contains the symmetric monoids Fkn of all total functions on kn elements. And Invk,1 contains Ikn (the finite symmetric inverse monoid of all injective partial functions on An). 1.3 Cuntz algebras and Thompson-Higman monoids All the monoids, inverse monoids, and groups, defined above, are submonoids of the multiplicative part of the Cuntz algebra Ok. The Cuntz algebra Ok, introduced by Dixmier [16] (for k = 2) and Cuntz [14], is a k-generated star-algebra (over the field of complex numbers) with identity element 1 and zero 0, given by the following finite presentation. The generating set is A = {a1, . . . , ak}. Since this is defined as a star- algebra, we automatically have the star-inverses {a1, . . . , ak}; for clarity we use overlines rather than stars. Relations of the presentation: aiai = 1, for i = 1, . . . , k; aiaj = 0, when i 6= j, 1 ≤ i, j ≤ k; a1a1 + . . .+ akak = 1. It is easy to verify that this defines a star-algebra. The Cuntz algebras are actually C∗-algebras with many remarkable properties (proved in [14]), but here we only need them as star-algebras, without their norm and Cauchy completion. In [4] and independently in [27] it was proved that the Thompson-Higman group Gk,1 is the subgroup of Ok consisting of the elements that have an expression of the form x∈P f(x) x where we require the following: P and Q range over all finite maximal prefix codes over the alphabet {a1, . . . , ak}, and f is any bijection P → Q. Another proof is given in [22]. More generally we also have: Theorem 1.12 The Thompson-Higman monoid Mk,1 is a submonoid of the multiplicative part of the Cuntz algebra Ok. Proof outline. The Thompson-Higman partial function monoid Mk,1 is the set of all elements of Ok that have an expression of the form x∈P f(x) x where P ⊂ A∗ ranges over all finite prefix codes, and f ranges over functions P → A∗. The details of the proof are very similar to the proofs in [4, 27]; the definition of essential restriction (and extension) and Proposition 1.4 insure that the same proof goes through. ✷ The embeddability into the Cuntz algebra is a further justification of the definitional choices that we made for the Thompson-Higman monoid Mk,1. 2 Structure and simplicity of the Thompson-Higman monoids We give some structural properties of the Thompson-Higman monoids; in particular, we show that Mk,1 and Invk,1 are simple for all k. 2.1 Group of units, J-relation, simplicity By definition, the group of units of a monoid M is the set of invertible elements (i.e., the elements u ∈M for which there exists x ∈M such that xu = ux = 1, where 1 is the identity element of M). Proposition 2.1 The Thompson-Higman group Gk,1 is the group of units of the monoids Mk,1, totMk,1, surMk,1, totsurMk,1, and Invk,1. Proof. It is obvious that the groups of units of the above monoids contain Gk,1. Conversely, we want to show that that if ϕ ∈ Mk,1 (and in particular, if ϕ is in one of the other monoids) and if ϕ has a left inverse and a right inverse, then ϕ ∈ Gk,1. First, it follows that ϕ is injective, i.e., ϕ ∈ Invk,1. Indeed, existence of a left inverse implies that for some α ∈Mk,1 we have α ϕ = 1; hence, if ϕ(x1) = ϕ(x2) then x1 = α ϕ(x1) = α ϕ(x2) = x2. Next, we show that domC(ϕ) is a maximal prefix code, hence ϕ ∈ totInv k,1. Indeed, we can again consider α ∈Mk,1 such that α ϕ = 1. For any essential restriction of 1 the domain code is a maximal prefix code, hence domC(α ◦ ϕ) is maximal (where ◦ denotes functional composition). Moreover, domC(α ◦ϕ) is also contained in the domain code of some restriction of ϕ, since ϕ(x) must be defined when α ◦ ϕ(x) is defined. Hence domC(ϕ′), for some restriction ϕ′ of ϕ, is a maximal prefix code; it follows that domC(ϕ) is a maximal prefix code. If we apply the reasoning of the previous paragraph to ϕ−1 (which exists since we saw that ϕ is injective), we conclude that domC(ϕ−1) = imC(ϕ) is a maximal prefix code. Thus, ϕ ∈ surInvk,1. We proved that if ϕ has a left inverse and a right inverse then ϕ ∈ totInv k,1 ∩ surInv k,1. Since totInv k,1 ∩ surInvk,1 = Gk,1 we conclude that ϕ ∈ Gk,1. ✷ We now characterize some of the Green relations of Mk,1 and of Invk,1, and we prove simplicity. By definition, two elements x, y of a monoid M are J-related (denoted x ≡J y) iff x and y belong to exactly the same ideals of M . More generally, the J-preorder of M is defined as follows: x ≤J y iff x belongs to every ideal that y belongs to. It is easy to see that x ≡J y iff x ≤J y and y ≤J x; moreover, x ≤J y iff there exist α, β ∈ M such that x = αyβ. A monoid M is called J-simple iff M has only one J-class (or equivalently, M has only one ideal, namely M itself). A monoid M is called 0-J-simple iff M has exactly two J-classes, one of which consist of just a zero element (equivalently, M has only two ideals, one of which is a zero element, and the other is M itself). See [13, 19] for more information on the J-relation. Cuntz [14] proved that the multiplicative part of the C∗-algebra Ok is a 0-J-simple monoid, and that as an algebra Ok is simple. We will now prove similar results for the Thompson-Higman monoids. Proposition 2.2 The inverse monoid Invk,1 and the monoid Mk,1 are 0-J-simple. The monoid totMk,1 is J-simple. Proof. Let ϕ ∈ Mk,1 (or ∈ Invk,1). When ϕ is not the empty map there are x0, y0 ∈ A∗ such that y0 = ϕ(x0). Let us define α, β ∈ Invk,1 by the tables α = {(ε 7→ x0)} and β = {(y0 7→ ε)}. Recall that ε denotes the empty word. Then β ϕα(.) = {(ε 7→ ε)} = 1. So, every non-zero element of Mk,1 (and of Invk,1) is in the same J-class as the identity element. In the case of totMk,1 we can take α = {(ε 7→ x0)} as before (since the domain code of α is {ε}, which is a maximal prefix code), and we take β′ : Q 7→ {ε} (i.e., the map that sends every element of Q to ε), where Q is any finite maximal prefix code containing y0. Then again, β ′ ϕα(.) = {(ε 7→ ε)} = 1. Thompson proved that V (= G2,1) is a simple group; Higman proved more generally that when k is even then Gk,1 is simple, and when k is odd then Gk,1 contains a simple normal subgroup of index 2. We will show next that in the monoid case we have simplicity for all k (not only when k is even). For a monoid M , “simple”, or more precisely, “congruence-simple” is defined to mean that the only congruences on M are the trivial congruences (i.e., the equality relation, and the congruence that lumps all elements of M into one congruence class). Theorem 2.3 The Thompson-Higman monoids Invk,1 and Mk,1 are congruence-simple for all k. Proof. Let ≡ be any congruence on Mk,1 that is not the equality relation. We will show that then the whole monoid is congruent to the empty map 0. We will make use of 0-J -simplicity. Case 0: Assume that Φ ≡ 0 for some element Φ 6= 0 of Mk,1. Then for all α, β ∈ Mk,1 we have obviously αΦβ ≡ 0. Moreover, by 0-J -simplicity of Mk,1 we have Mk,1 = {αΦβ : α, β ∈ Mk,1} since Φ 6= 0. Hence in this case all elements of Mk,1 are congruent to 0. For the remainder we suppose that ϕ ≡ ψ and ϕ 6= ψ, for some elements ϕ,ψ of Mk,1 − {0}. For a right ideal R ⊆ A∗ generated by a prefix code P we call PAω the set of ends of R. We call two right ideals R1, R2 essentially equal iff R1 and R2 have the same ends, and we denote this by R1 =ess R2. This is equivalent to the following property: Every right ideal that intersects R1 also intersects R2, and vice versa (see [6] and [7]). Case 1: Dom(ϕ) 6=ess Dom(ψ). Then there exists x0 ∈ A∗ such that x0A∗ ⊆ Dom(ϕ), but Dom(ψ) ∩ x0A∗ = ∅; or, vice versa, there exists x0 ∈ A∗ such that x0A∗ ⊆ Dom(ψ), but Dom(ϕ) ∩ x0A∗ = ∅. Let us assume the former. Letting β = (x0 7→ x0), we have ϕβ(.) = (x0 7→ ϕ(x0)). We also have ψ β(.) = 0, since ∗ ∩Dom(ψ) = ∅. So, ϕβ ≡ ψ β = 0, but ϕβ 6= 0. Hence case 0, applied to Φ = ϕβ, implies that the entire monoid Mk,1 is congruent to 0. Case 2.1: Im(ϕ) 6=ess Im(ψ) and Dom(ϕ) =ess Dom(ψ). Then there exists y0 ∈ A∗ such that y0A∗ ⊆ Im(ϕ), but Im(ψ) ∩ y0A∗ = ∅; or, vice versa, ∗ ⊆ Im(ψ), but Im(ϕ) ∩ y0A∗ = ∅. Let us assume the former. Let x0 ∈ A∗ be such that y0 = ϕ(x0). Then (y0 7→ y0) ◦ ϕ ◦ (x0 7→ x0) = (x0 7→ y0). On the other hand, (y0 7→ y0) ◦ ψ ◦ (x0 7→ x0) = 0. Indeed, if x0A∗ ∩ Dom(ψ) = ∅ then for all w ∈ A∗ : ψ ◦ (x0 7→ x0)(x0w) = ψ(x0w) = ∅. And if x0A∗ ∩ Dom(ψ) 6= ∅ then for those w ∈ A∗ such that x0w ∈ Dom(ψ) we have (y0 7→ y0) ◦ψ ◦ (x0 7→ x0)(x0w) = (y0 7→ y0)(ψ(x0w)) = ∅, since Im(ψ) ∩ y0A∗ = ∅. Now case 0 applies to 0 6= Φ = (y0 7→ y0) ◦ ϕ ◦ (x0 7→ x0) ≡ 0; hence all elements of Mk,1 are congruent to 0. Case 2.2: Im(ϕ) =ess Im(ψ) and Dom(ϕ) =ess Dom(ψ). Then after restricting ϕ and ψ to Dom(ϕ) ∩ Dom(ψ) (=ess Dom(ϕ) =ess Dom(ψ)), we have: domC(ϕ) = domC(ψ), and there exist x0 ∈ domC(ϕ) = domC(ψ) and y0 ∈ Im(ϕ), y1 ∈ Im(ψ) such that ϕ(x0) = y0 6= y1 = ψ(x0). We have two sub-cases. Case 2.2.1: y0 and y1 are not prefix-comparable. Then (y0 7→ y0) ◦ ϕ ◦ (x0 7→ x0) = (x0 7→ y0). On the other hand, (y0 7→ y0) ◦ ψ ◦ (x0 7→ x0)(x0w) = (y0 7→ y0)(y1w) = ∅ for all w ∈ A∗ (since y0 and y1 are not prefix-comparable). So (y0 7→ y0) ◦ψ ◦ (x0 7→ x0) = 0. Hence case 0 applies to 0 6= Φ = (y0 7→ y0) ◦ ϕ ◦ (x0 7→ x0) ≡ 0. Case 2.2.2: y0 is a prefix of y1, and y0 6= y1. (The case where y0 is a prefix of y1 is similar.) Then y1 = y0au1 for some a ∈ A, u1 ∈ A∗. Letting b ∈ A−{a}, and y2 = y0b, we obtain a string y2 that is not prefix-comparable with y1. Now, (y2 7→ y2)◦ϕ◦(x0 7→ x0)(x0v2) = (y2 7→ y2)(y0v2) = y2. But for all w ∈ A∗, (y2 7→ y2) ◦ψ ◦ (x0 7→ x0)(x0w) = (y2 7→ y2)(y1w) = ∅, since y2 and y1 are not prefix-comparable. Thus, case 0 applies to 0 6= Φ = (y2 7→ y2) ◦ ϕ ◦ (x0 7→ x0) ≡ 0. The same proof works for Invk,1 since all the multipliers used in the proof (of the form (u 7→ v) for some u, v ∈ A∗) belong to Invk,1. ✷ 2.2 D-relation Besides the J-relation and the J-preorder, based on ideals, there are the R- and L−relations and R- and L−preorders, based on right (or left) ideals. Two elements x, y ∈ M are R-related (denoted x ≡R y) iff x and y belong to exactly the same right ideals of M . The R-preorder is defined as follows: x ≤R y iff x belongs to every right ideal that y belongs to. It is easy to see that x ≡R iff x ≤R y and y ≤R x; also, x ≤R y iff there exists α ∈ M such that x = yα. In a similar way one defines ≡L and ≤L. Finally, there is the D-relation of M , which is defined as follows: x ≡D y iff there exists s ∈ M such that x ≡R s ≡L y; this is easily seen to be equivalent to saying that there exists t ∈M such that x ≡L t ≡R y. For more information on these definitions see for example [13, 19]. The D-relation of Mk,1 and Invk,1 has an interesting characterization, as we shall prove next. We will represent all elements of Mk,1 by tables of the from ϕ : P → Q, where both P and Q are finite prefix codes over A (with |A| = k). For such a table we also write P = domC(ϕ) (the domain code of ϕ) and Q = imC(ϕ) (the image code of ϕ). In general, tables of elements of Mk,1 have the form P → S, where P is a finite prefix code and S is a finite set; but by using essential restrictions, if necessary, every element of Mk,1 can be given a table P → Q, where both P and Q are finite prefix codes. Note the following invariants with respect to essential restrictions: Proposition 2.4 Let ϕ1 : P1 → Q1 be a table for an element of Mk,1, where P1, Q1 ⊂ A∗ are finite prefix codes. Let ϕ2 : P2 → Q2 be another finite table for the same element of Mk,1, obtained from the table ϕ1 by an essential restriction. Then P2, Q2 ⊂ A∗ are finite prefix codes and we have |P1| ≡ |P2| mod (k − 1) and |Q1| ≡ |Q2| mod (k − 1). These modular congruences also hold for essential extensions, provided that we only extend to tables in which the image is a prefix code. Proof. An essential restriction consists of a finite sequence of essential restriction steps; an essential restriction step consists of replacing a table entry (x, y) of ϕ1 by {(xa1, ya1), . . . , (xak, yak)} (according to Proposition 1.4). For a finite prefix code Q ⊂ A∗, and q ∈ Q, the finite set (Q−{q})∪{qa1, . . . , qak} is also a prefix code, as is easy to prove. In this process, the cardinalities change as follows: |P1| becomes |P1|−1+k and |Q1| becomes |Q1|−1+k. Indeed (looking at Q1 for example), first an element y is removed from Q1, then the k elements {ya1, . . . , yak} are added. The elements yai that are added are all different from the elements that are already present in Q1−{y}; in fact, more strongly, yai and the elements of Q1 − {y} are not prefixes of each other. ✷ As a consequence of Prop. 2.4 it makes sense, for any ϕ ∈ Mk,1, to talk about |domC(ϕ)| and |imC(ϕ)| as elements of Zk−1, independently of the representation of ϕ by a right-ideal homomorphism. Theorem 2.5 For any non-zero elements ϕ,ψ of Mk,1 (or of Invk,1) the D-relation is characterized as follows: ϕ ≡D ψ iff |imC(ϕ)| ≡ |imC(ψ)| mod (k − 1). Hence, Mk,1 and Invk,1 have k− 1 non-zero D-classes. In particular, M2,1 and Inv2,1 are 0-D-simple (also called 0-bisimple). The proof of Theorem 2.5 uses several Lemmas. Lemma 2.6 ([5] Lemma 6.1; Arxiv version of [5] Lemma 9.9). For every finite alphabet A and every integer i ≥ 0 there exists a maximal prefix code of cardinality 1+(|A|−1) i. And every finite maximal prefix code over A has cardinality 1 + (|A| − 1) i, for some integer i ≥ 0. It follows that when |A| = 2, there are finite prefix codes over A of every finite cardinality. ✷ As a consequence of this Lemma we have for all ϕ ∈ Gk,1: ‖ϕ‖ ≡ 1 mod (k− 1). Thus, except for the Thompson group V (when k = 2), there is a constraint on the table size of the elements of the group. In the following idQ denotes the element of Invk,1 given by the table {(x 7→ x) : x ∈ Q} where Q ⊂ A∗ is any finite prefix code. Lemma 2.7 (1) For any ϕ ∈ Mk,1 (or ∈ Invk,1) with table P → Q (where P,Q are finite prefix codes) we have: ϕ ≡R idQ. (2) If S, T are finite prefix codes with |S| = |T | then idS ≡D idT . (3) If ϕ1 : P1 → Q1 and ϕ2 : P2 → Q2 are such that |Q1| = |Q2| then ϕ1 ≡D ϕ2. Proof. (1) Let P ′ ⊆ P be a set of representatives modulo ϕ (i.e., we form P ′ by choosing one element in every set ϕ−1ϕ(x) as x ranges over P ). So, |P ′| = |Q|. Let α ∈ Invk,1 be given by a table Q→ P ′; the exact map does not matter, as long as α is bijective. Then ϕ ◦ α(.) is a permutation of Q, and ϕ ◦ α ≡R ϕ ◦ α ◦ (ϕ ◦ α)−1 = idQ. Now, ϕ ≥R ϕ ◦ α ≥R ϕ ◦ α ◦ (ϕ ◦ α)−1 ◦ ϕ = idQ ◦ ϕ = ϕ, hence ϕ ≡R ϕ ◦ α (≡R idQ). (2) Let α : S → T be a bijection (which exists since |S| = |T |); so α represents an element of Invk,1. Then α = α ◦ idS(.) and idS = α−1 ◦ α(.); hence, α ≡L idS . Also, α = idT ◦ α(.) and idT = α ◦ α−1(.); hence, α ≡R idT . Thus, idS ≡L α ≡R idT . (3) If |Q1| = |Q2| then idQ1 ≡D idQ2 by (2). Moreover, ϕ1 ≡D idQ1 and ϕ2 ≡D idQ2 by (1). The result follows by transitivity of ≡D. ✷ Lemma 2.8 (1) For any m ≥ k let i be the residue of m modulo k − 1 in the range 2 ≤ i ≤ k, and let us write m = i+ (k − 1)j, for some j ≥ 0. Then there exists a prefix code Qi,j of cardinality |Qi,j| = m, such that idQi,j is an essential restriction of id{a1,...,ai}. Hence, idQi,j = id{a1,...,ai} as elements of Invk,1. (2) In Mk,1 and in Invk,1 we have id{a1} ≡D id{a1,...,ak} = 1. Proof. (1) For any m ≥ k there exist i, j ≥ 0 such that 1 ≤ i ≤ k and m = i+ (k− 1)j. We consider the prefix code Qi,j = {a2, . . . , ai} ∪ r=1 a 1(A− {a1}) ∪ a It is easy to see that Qi,j is a prefix code, which is maximal iff i = k; see Fig. 1 below. Clearly, |Qi,j| = i + (k − 1)j. Since Qi,j contains aj1A, we can perform an essential extension of idQi,j by replacing the table entries {(aj1a1, a 1a1), (a 1a2, a 1a2), . . . , (a 1ak, a 1ak)} by (a 1). This replaces Qi,j by Qi,j−1. So, idQi,j can be essentially extended to idQi,j−1 . By repeating this we find that idQi,j is the same element (in Mk,1 and in Invk,1) as idQi,0 = id{a1,...,ai}. (2) By essential restriction, id{a1} = id{a1a1,a1a2,...,a1ak}, in Mk,1 and in Invk,1. And by Lemma 2.7(2), id{a1a1,a1a2,...,a1ak} ≡D id{a1,...,ak}; the latter, by essential extension, is 1. ✷ {a2, . . . , ai} ❍❍. . .✁✁ a1 (A− {a1}) ❍❍. . .✁✁ ar1 (A− {a1}) ❍❍. . .✁✁ 1 (A− {a1}) ❍❍. . .✁✁ ✁✁ ❆❆. . . Fig. 1: The prefix tree of Qi,j. Lemma 2.9 For all ϕ,ψ ∈ Invk,1: If ϕ ≥L(Mk,1) ψ, where ≥L(Mk,1) is the L-preorder of Mk,1, then ϕ ≥L(Ik,1) ψ, where ≥L(Ik,1) is the L-preorder of Invk,1. The same holds with ≥L replaced by ≡L, ≥R, ≡R, ≡D, ≥J and ≡J . Proof. If ψ = αϕ for some α ∈ Mk,1 then let us define α′ by α′ = α idIm(ϕ). Then we have: ψ ϕ−1 = α ϕ ϕ−1 = α idIm(ϕ) = α ′, hence α′ ∈ Invk,1 (since ϕ,ψ ∈ Invk,1). Moreover, α′ ϕ = α idIm(ϕ) ϕ = αϕ = ψ. ✷ So far our Lemmas imply that in Mk,1 and in Invk,1, every non-zero element is ≡D to one of the k − 1 elements id{a1,...,ai}, for i = 1, . . . , k − 1. Moreover the Lemmas show that if two elements of Mk,1 (or of Invk,1) are given by tables ϕ1 : P1 → Q1 and ϕ2 : P2 → Q2, where P1, Q1, P2 and Q2 are finite prefix codes, then we have: If |Q1| ≡ |Q2| mod (k − 1) then ϕ1 ≡D ϕ2. We still need to prove the converse of this. It is sufficient to prove the converse for Invk,1, by Lemma 2.9 and because every element of Mk,1 is ≡D to an element of Invk,1 (namely id{a1,...,ai}). Lemma 2.10 Let ϕ,ψ ∈ Invk,1. If ϕ ≡D ψ in Invk,1, then ‖ϕ‖ ≡ ‖ψ‖ mod (k − 1). Proof. (1) We first prove that if ϕ ≡L ψ then |domC(ϕ)| ≡ |domC(ψ)| mod (k − 1). By definition, ϕ ≡L ψ iff ϕ = β ψ and ψ = αϕ for some α, β ∈ Invk,1. By Lemma 1.9 there are restrictions β′ and ψ′ of β, respectively ψ, and an essential restriction Φ of ϕ such that: Φ = β′ ◦ ψ′, and Dom(β′) = Im(ψ′). It follows that Dom(Φ) ⊆ Dom(ψ′), since if ψ′(x) is not defined then Φ(x) = β′ ◦ ψ′(x) is not defined either. Similarly, there is an essential restriction Ψ of ψ and a restriction ϕ′ of ϕ and such that Dom(Ψ) ⊆ Dom(ϕ′). Thus, the restriction of both ϕ and ψ to the intersection Dom(Φ)∩Dom(Ψ) yields restrictions ϕ′′, respectively ψ′′ such that Dom(ϕ′′) = Dom(ψ′′). Claim: ϕ′′ and ψ′′ are essential restrictions of ϕ, respectively ψ. Indeed, every right ideal R of A∗ that intersects Dom(ψ) also intersects Dom(Ψ) (since Ψ is an essential restriction of ψ). Since Dom(Ψ) ⊆ Dom(ϕ′) ⊆ Dom(ϕ), it follows that R also intersects Dom(ϕ). Moreover, since Φ is an essential restriction of ϕ, R also intersects Dom(Φ). Thus, Dom(Φ) is essential in Dom(ψ). Since Dom(Ψ) is also essential in Dom(ψ), it follows that Dom(Φ) ∩Dom(Ψ) is essential in Dom(ψ); indeed, in general, the intersection of two right ideals R1, R2 that are essential in a right ideal R3, is essential in R3 (this is a special case of Lemma 1.8). This means that ψ ′′ is an essential restriction of ψ. Similarly, one proves that ϕ′′ is an essential restriction of ϕ. [This proves the Claim.] So, ϕ′′ and ψ′′ are essential restrictions such that Dom(ϕ′′) = Dom(ψ′′). Hence, domC(ϕ′′) = domC(ψ′′); Proposition 2.4 then implies that |domC(ϕ)| ≡ |domC(ϕ′′)| = |domC(ψ′′)| ≡ |domC(ψ)| mod (k − 1). (2) Next, let us prove that if ϕ ≡R ψ then |imC(ϕ)| ≡ |imC(ψ)| mod (k−1). In Invk,1 we have ϕ ≡R ψ iff ϕ−1 ≡L ψ−1. Also, imC(ϕ) = domC(ϕ−1). Hence, (2) follows from (1). The Lemma now follows from (1) and (2), since for elements of Invk,1, |imC(ϕ)| = |domC(ϕ)| = ‖ϕ‖, and since the D-relation is the composite of the L-relation and the R-relation. ✷ Proof of Theorem 2.5. We saw already (in the observations before Lemma 2.10 and in the preceding Lemmas) that for ϕ1 : P1 → Q1 and ϕ2 : P2 → Q2 (where P1, Q1, P2 and Q2 are non-empty finite prefix codes) we have: If |Q1| ≡ |Q2| mod (k − 1) then ϕ1 ≡D ϕ2. In particular, when |Q1| ≡ i mod (k − 1) then ϕ1 ≡D id{a1,...,ai}. It follows from Lemma 2.10 that the elements id{a1,...,ai} (for i = 1, . . . , k − 1) are all in different D-classes. ✷ So far we have characterized the D- and J-relations of Mk,1 and Invk,1. We leave the general study of the Green relations of Mk,1, Invk,1, and the other Thompson-Higman monoids for future work. The main result of this paper, to be proved next, is that the Thompson-Higman monoids Mk,1 and Invk,1 are finitely generated and that their word problem over any finite generating set is in P. 3 Finite generating sets We will show that Invk,1 and Mk,1 are finitely generated. An application of the latter fact is that a finite generating set of Mk,1 can be used to build combinational circuits for finite boolean functions that do not have fixed-length inputs or outputs. In engineering, non-fixed length inputs or outputs make sense, for example, if the inputs or outputs are handled sequentially, and if the possible input strings form a prefix code. First we need some more definitions about prefix codes. The prefix tree of a prefix code P ⊂ A∗ is, by definition, a tree whose vertex set is the set of all the prefixes of the elements of P , and whose edge set is {(x, xa) : a ∈ A, xa is a prefix of some element of P}. The tree is rooted, with root ε (the empty word). Thus, the prefix tree of P is a subtree of the tree of A∗. The set of leaves of the prefix tree of P is P itself. The vertices that are not leaves are called internal vertices. We will say more briefly an “internal vertex of P” instead of internal vertex of the prefix tree of P . An internal vertex has between 1 and k children; an internal vertex is called saturated iff it has k children. One can prove easily that a prefix code P is maximal iff every internal vertex of the prefix tree of P is saturated. Hence, every prefix code P can be embedded in a maximal prefix code (which is finite when P is finite), obtained by saturating the prefix tree of P . Moreover we have: Lemma 3.1 For any two finite non-maximal prefix codes P1, P2 ⊂ A∗ there are finite maximal prefix codes P ′1, P 2 ⊂ A∗ such that P1 ⊂ P ′1, P2 ⊂ P ′2, and |P ′1| = |P ′2|. Proof. First we saturate P1 and P2 to obtain two maximal prefix codes P 1 and P 2 such that P1 ⊂ P ′′1 , and P2 ⊂ P ′′2 . If |P ′′1 | 6= |P ′′2 | (e.g., if |P ′′1 | < |P ′′2 |) then |P ′′1 | and |P ′′2 | differ by a multiple of k − 1 (by Prop. 2.4). So, in order to make |P ′′1 | equal to |P ′′2 | we repeat the following (until |P ′′1 | = |P ′′2 |): consider a leaf of the prefix tree of P ′′1 that does not belong to P1, and attach k children at that leaf; now this leaf is no longer a leaf, and the net increase in the number of leaves is k − 1. ✷ Lemma 3.2 Let P and Q be finite prefix codes of A∗ with |P | = |Q|. If P and Q are both maximal prefix codes, or if both are non-maximal, then there is an element of Gk,1 that maps P onto Q. On the other hand, if one of P and Q is maximal and the other one is not maximal, then there is no element of Gk,1 that maps P onto Q. Proof. When P and Q are both maximal then any one-to-one correspondence between P and Q is an element of Gk,1. When P and Q are both non-maximal, we use Lemma 3.1 above to find two maximal prefix codes P ′ and Q′ such that P ⊂ P ′, Q ⊂ Q′, and |P ′| = |Q′|. Consider now any bijection from P ′ onto Q′ that is also a bijection from P onto Q. This is an element of Gk,1. When P is maximal and Q is non-maximal, then every element ϕ ∈Mk,1 that maps P onto Q will satisfy domC(ϕ) = P ; since ϕ is onto Q, we have imC(ϕ) = Q. Hence, ϕ 6∈ Gk,1 since imC(ϕ) is a non-maximal prefix code. A similar reasoning shows that no element of Gk,1 maps P onto Q if P is non-maximal and Q is maximal. ✷ Notation: For u, v ∈ A∗, the element of Invk,1 with one-element domain code {u} and one-element image code {v} is denoted by (u 7→ v). When (u 7→ v) is composed with itself j times the resulting element of Invk,1 is denoted by (u 7→ v)j . Lemma 3.3 (1) For all j > 0: (a1 7→ a1a1)j = (a1 7→ aj+11 ). (2) Let S = {aj1a1, a 1a2, . . . , a 1ai}, for some 1 ≤ i ≤ k−1, 0 ≤ j. Then idS is generated by the k+1 elements {(a1 7→ a1a1), (a1a1 7→ a1)} ∪ {id{a1a1, a1a2, ..., a1ai} : 1 ≤ i ≤ k − 1}. (3) For all j ≥ 2: (ε 7→ aj1)(.) = (a1 7→ a1a1)j−1 · (ε 7→ a1)(.). Proof. (1) We prove by induction that (a1 7→ a1a1)j = (a1 7→ a1aj1) for all j ≥ 1. Indeed, (a1 7→ a1a1)j+1(.) = (a1 7→ a1a1) · (a1 7→ a1aj1)(.), and by essential restriction this is 1 a1w (w ∈ Aj − {a a1a1a 1 a1a1w · (a1 7→ a1aj1)(.) = (a1 7→ a1a1a 1)(.). (2) For S = {aj1a1, a 1a2, . . . , a 1ai} we have idS = a1a1 a1a2 . . . a1ai 1a1 a 1a2 . . . a 1a1 a 1a2 . . . a a1a1 a1a2 . . . a1ai a1a1 a1a2 . . . a1ai 1a1 a 1a2 . . . a a1a1 a1a2 . . . a1ai a1ai+1 . . . a1ak 1a1 a 1a2 . . . a 1ai a 1ai+1 . . . a · id{a1a1, a1a2, ..., a1ai}(.) = (a1 7→ aj1)· id{a1a1, a1a2, ..., a1ai} = (a1 7→ a1a1) j−1· id{a1a1, a1a2, ..., a1ai}. The map id{a1a1} is redundant as a generator since (a1a1 7→ a1a1) = (a1a1 7→ a1) (a1 7→ a1a1)(.). (3) By (1) we have (ε 7→ aj1) = (a1 7→ a 1) · (ε 7→ a1)(.), and (a1 7→ a 1) = (a1 7→ a1a1)j−1. ✷ Theorem 3.4 The inverse monoid Invk,1 is finitely generated. Proof. Our strategy for finding a finite generating set for Invk,1 is as follows: We will use the fact that the Thompson-Higman group Gk,1 is finitely generated. Hence, if ϕ ∈ Invk,1, g1, g2 ∈ Gk,1, and if g2ϕg1 can be expressed as a product p over a fixed finite set of elements of Invk,1, then it follows that ϕ = g−12 p g 1 can also be expressed as a product over a fixed finite set of elements of Invk,1. We assume that a finite generating set for Gk,1 has been chosen. For any element ϕ ∈ Invk,1 with domain code domC(ϕ) = P and image code imC(ϕ) = Q, we distinguish four cases, depending on the maximality or non-maximality of P and Q. (1) If P and Q are both maximal prefix codes then ϕ ∈ Gk,1, and we can express ϕ over a finite fixed generating set of Gk,1. (2) Assume P and Q are both non-maximal prefix codes. By Lemma 3.1 there are finite maximal prefix codes P ′, Q′ such that P ⊂ P ′, Q ⊂ Q′, and |P ′| = |Q′|; and by Lemma 2.6, |P ′| = |Q′| = 1+(k−1)N for someN ≥ 0. Consider the following maximal prefix code C, of cardinality |P ′| = |Q′| = 1+(k−1)N : r=0 a 1(A− {a1}) ∪ a The maximal prefix code C is none other than the code Qi,j when i = k and j = N − 1 (introduced in the proof of Lemma 2.8, Fig. 1). The elements g1 : C → P ′ and g2 : Q′ → C of Gk,1 can be chosen so that ψ = g2ϕg1(.) is a partial identity with domC(ψ) = imC(ψ) ⊂ C consisting of the |P | first elements of C in the dictionary order. So, ψ is the identity map restricted to these |P | first elements of C, and ψ is undefined on the rest of C. To describe domC(ψ) = imC(ψ) in more detail, let us write |P | = i+ (k − 1) ℓ, for some i, ℓ with 1 ≤ i < k and 0 ≤ ℓ ≤ N − 1. Then domC(ψ) = imC(ψ) = aN−11 A ∪ r=j+1 a 1(A− {a1}) ∪ a 1 {a2, . . . , ai}. where j = N − 1− ℓ. Since ψ = iddomC(ψ), we claim: By essential maximal extension ψ = idS (as elements of Invk,1), where S = {aj1a1, a 1a2, . . . , a 1ai}, with i, j as in the description of domC(ψ) = imC(ψ) above, i.e., 1 < i < k, N−1 ≥ j = N−1− ℓ ≥ 0, and |P | = i+ (k − 1) ℓ. Indeed, if |P | < k then S is just domC(ψ), with i = |P |, and ℓ = 0 (hence j = N − 1). If |P | ≥ k then the maximum essential extension of ψ will replace the 1 + (k − 1) ℓ elements aN−11 A ∪ r=N−j+1 a 1(A− {a1}) by the single element a N−ℓ+1 1 = a 1 . What remains is the set S = {aj+11 } ∪ a 1 {a2, . . . , ai}. Finally, by Lemma 3.3, idS (where S = {aj1a1, a 1a2, . . . , a 1ai}) can be generated by the k + 1 elements {(a1 7→ a1a1), (a1a1 7→ a1)} ∪ {id{a1a1, a1a2, ..., a1ai} : 1 ≤ i ≤ k − 1}. (3) Assume P is a maximal prefix code and Q is non-maximal. Let Q′ be the finite maximal prefix code obtained by saturating the prefix tree of Q. Then Q ⊂ Q′, |Q′| = 1+(k−1)N ′, and |P | = 1+(k−1)N for some N ′ > N ≥ 0. We consider the maximal prefix codes C and C ′ as defined in the proof of (2), using N ′ for defining C ′. We can choose g1 : C → P and g2 : Q′ → C ′ in Gk,1 so that ψ = g2ϕg1(.) is the dictionary-order preserving map that maps C to the first |C| elements of C ′. So we have domC(ψ) = C, and imC(ψ) = S0 , where S0 ⊂ C ′ consist of the |C| first elements of C ′, in dictionary order. Since |C| = 1 + (k − 1)N , we can describe S0 in more detail by ⋃N ′−2 r=N ′−N a 1(A− {a1}) ∪ a N ′−1 Next, by essential maximal extension we now obtain ψ = (ε 7→ aN Indeed, we saw that |P | = 1+ (k− 1)N . If |P | = 1 then P = {ε}, and ψ = (ε 7→ aN ′1 ). If |P | ≥ k then maximum essential extension of ψ will replace all the elements of C by the single element ε, and it will replace all the elements of S0 by the single element a N ′−N Finally, by Lemma 3.3, (ε 7→ aN ′−N1 ) is generated by the two elements (ε 7→ a1) and (a1 7→ a1a1). (4) The case where P is a non-maximal maximal prefix code and Q is maximal can be derived from case (3) by taking the inverses of the elements from case (3). ✷ Theorem 3.5 The monoid Mk,1 is finitely generated. Proof. Let ϕ : P → Q be the table of any element of Mk,1, mapping P onto Q, where P,Q ⊂ A∗ are finite prefix codes. The map described by the table is total and surjective, so if |P | = |Q| (and in particular, if ϕ is the empty map) then ϕ ∈ Invk,1, hence ϕ can be expressed over the finite generating set of Invk,1. In the rest of the proof we assume |P | > |Q|. The main observation is the following. Claim. ϕ can be written as the composition of finitely many elements ϕi ∈Mk,1 with tables Pi → Qi such that 0 ≤ |Pi| − |Qi| ≤ 1. Proof of the Claim: We use induction on |P | − |Q|. There is nothing to prove when |P | − |Q| ≤ 1, so we assume now that |P | − |Q| ≥ 2. If ϕ(x1) = ϕ(x2) = ϕ(x3) = y1 for some x1, x2, x3 ∈ P (all three being different) and y1 ∈ Q, then we can write ϕ as a composition ϕ(.) = ψ2 ◦ψ1(.), as follows. The map ψ1 : P −→ P −{x1} is defined by ψ1(x1) = ψ1(x2) = x2, and acts as the identity everywhere else on P . The map ψ2 : P −{x1} −→ Q is defined by ψ2(x2) = ψ2(x3) = y1, and acts in the same way as ϕ everywhere else on P −{x1}. Then for ψ1 we have |P | − |P − {x1}| < |P | − |Q|, and for ψ2 we have |P − {x1}| − |Q| < |P | − |Q|. If ϕ(x1) = ϕ(x2) = y1 and ϕ(x3) = ϕ(x4) = y2 for some x1, x2, x3, x4 ∈ P (all four being different) and y1, y2 ∈ Q (y1 6= y2), then we can write ϕ as a composition ϕ(.) = ψ2 ◦ψ1(.), as follows. First the map ψ1 : P −→ P −{x1} is defined by ψ1(x1) = ψ1(x2) = x2, and acts as the identity everywhere else on P . Second, the map ψ2 : P −{x1} −→ Q is defined by ψ2(x2) = y1 and ψ2(x3) = ψ2(x4) = y2, and acts like ϕ everywhere else on P − {x1}. Again, for ψ1 we have |P | − |P − {x1}| < |P | − |Q| and for ψ2 we have |P − {x1}| − |Q| < |P | − |Q|. [End, proof of the Claim.] Because of the Claim we now only need to consider elements ϕ ∈ Mk,1 with tables P → Q such that the prefix codes P,Q satisfy |P | = |Q|+ 1. We denote P = {p1, . . . , pn} and Q = {q1, . . . , qn−1}, with ϕ(pj) = qj for 1 ≤ j ≤ n− 1, and ϕ(pn−1) = ϕ(pn) = qn−1. We define the following prefix code C with |C| = |P |: • if |P | = i ≤ k then C = {a1, . . . , ai}; note that i ≥ 2, since |P | > |Q| > 0; • if |P | > k then C = {a2, . . . , ai} ∪ r=1 a 1(A− {a1}) ∪ a where i, j are such that |P | = i + (k − 1)j, 2 ≤ i ≤ k, and 1 ≤ j (see Fig. 1). Let us write C in increasing dictionary order as C = {c1, . . . , cn}. The last element of C in the dictionary order is thus cn = ai. We now write ϕ(.) = ψ3 ψ2 ψ1(.) where ψ1, ψ2, ψ3 are as follows: • ψ1 : P −→ C is bijective and is defined by pj 7→ cj for 1 ≤ j ≤ n; • ψ2 : C −→ C − {ai} is the identity map on {c1, . . . , cn−1}, and ψ2(cn) = cn−1. • ψ3 : C − {ai} −→ Q is bijective and is defined by cj 7→ qj for 1 ≤ j ≤ n− 1. It follows that ψ1 and ψ3 can be expressed over the finite generating set of Invk,1. On the other hand, ψ2 has a maximum essential extension, as follows. • If 2 ≤ |P | = i ≤ k then a1 . . . ai−2 ai−1 ai a1 . . . ai−2 ai−1 ai−1 id{a1, ..., ai−1} ai • If |P | = i+ (k − 1)j > k and if i > 2 then, after maximal essential extension, ψ2 also becomes max(ψ2) = id{a1, ..., ai−1} ai • If |P | = i+ (k − 1)j > k and if i = 2 then, after essential extensions, max(ψ2) = a1a1 . . . a1ak−2 a1ak−1 a1ak a2 a1a1 . . . a1ak−2 a1ak−1 a1ak a1ak ida1A a2 a1 a2 a1 a1ak In summary, we have factored ϕ over a finite set of generators of Invk,1 and k additional generators in Mk,1. ✷ Factorization algorithm: The proofs of Theorems 3.4 and 3.5 are constructive; they provide algo- rithms that, given ϕ ∈ Invk,1 or ∈ Mk,1, output a factorization of ϕ over the finite generating set of Invk,1, respectively Mk,1. In [20] (p. 49) Higman introduces a four-element generating set for G2,1; a special property of these generators is that their domain codes and their image codes only contain words of length ≤ 2, and ∣ |γ(x)| − |x| ∣ ≤ 1 for every generator γ and every x ∈ domC(γ). The generators in the finite generating set of Mk,1 that we introduced above also have those properties. Thus we obtain: Corollary 3.6 The monoid M2,1 has a finite generating set such that all the generators have the following property: The domain codes and the image codes only contain words of length ≤ 2, and ∣ |γ(x)| − |x| ∣ ≤ 1 for every generator γ and every x ∈ domC(γ). ✷ For reference we list an explicit finite generating set forM2,1. It consists, first, of the Higman generators of G2,1 ([20] p. 49): Not = , (01 ↔ 1) = 00 01 1 00 1 01 , (0 ↔ 10) = 0 10 11 10 0 11 , and τ1,2 = 00 01 10 11 00 10 01 11 the additional generators for Inv2,1: (ε→ 0), (0 → ε), (0 → 00), (00 → 0); the additional generators for M2,1: , and 00 01 1 00 01 01 Observe that Higman’s generators of Gk,1 (in [20] p. 27) have domain and image codes with at most 3 internal vertices. We observe that the additional generators that we introduced for Invk,1 and Mk,1 have domain and image codes have at most 2 internal vertices. The following problem remains open: Are Invk,1 and Mk,1 finitely presented? 4 The word problem of the Thompson-Higman monoids We saw that the Thompson-Higman monoid Mk,1 is finitely generated. We want to show now that the word problem of Mk,1 over any finite generating set can be decided in deterministic polynomial time, i.e., it belongs to the complexity class P. 2 In [4] it was shown that the word problem of the Thompson-Higman group Gk,1 over any finite generating set is in P. In fact, it is in the parallel complexity class AC1 [4], and it is co-context-free [25]. In [5] it was shown that the word problem of the Thompson-Higman group Gk,1 over the infinite generating set Γk,1∪{τi,i+1 : i > 0} is coNP-complete, where Γk,1 is any finite generating set of Gk,1; the position transposition τi,i+1 ∈ Gk,1 has domC(τi,i+1) = imC(τi,i+1) = Ai+1, and is defined by uαβ 7→ uβα for all letters α, β ∈ A and all words u ∈ Ai−1. We will see below that the word problem of Mk,1 over Γk,1 ∪ {τi,i+1 : i > 0} is also coNP-complete, where Γk,1 is any finite generating set of Mk,1. 4.1 The image code formula Our proof in [4] that the word problem of the Thompson-Higman group Gk,1 (over any finite generating set) is in P, was based on the following fact (the table size formula): ∀ϕ,ψ ∈ Gk,1: ‖ψ ◦ ϕ‖ ≤ ‖ψ‖+ ‖ϕ‖. Here ‖ϕ‖ denotes the table size of ϕ, i.e., the cardinality of domC(ϕ). See Proposition 3.5, Theorem 3.8, and Proposition 4.2 in [4]. In Mk,1 the above formula does not hold in general, as the following example shows. We give some definitions and notation first. Definition 4.1 For any finite set S ⊆ A∗ we denote the length of the longest word in S by ℓ(S). The cardinality of S is denoted by |S|. The table of a right-ideal morphism ϕ is the set {(x, ϕ(x)) : x ∈ domC(ϕ)}. 2 This section has been revised in depth, to correct errors. Proposition 4.2 For every n > 0 there exists Φn = ϕ 2 ϕ1 ∈ M2,1 (for some ϕ1, ϕ2 ∈ M2,1) with the following properties: The table sizes are ‖Φn‖ = 2n, and ‖ϕ2‖ = ‖ϕ1‖ = 2. So, ‖Φn‖ is exponentially larger than (n− 1) · ‖ϕ2‖+ ‖ϕ1‖. Hence the table size formula does not hold in M2,1. The word lengths of ϕ1, ϕ2, and Φn (over the finite generating set Γ of M2,1 from Section 3 in [1]) satisfy |ϕ1|Γ = 1, |ϕ2|Γ ≤ 2, and |Φn|Γ < 2n. So the table size of Φn is exponentially larger than its word length: ‖Φn‖ > |Φn|Γ . Proof. Consider ϕ1, ϕ2 ∈ M2,1 given by the tables ϕ1 = {(0 7→ 0), (1 7→ 0)}, and ϕ2 = {(00 7→ 0), (01 7→ 0)}. One verifies that Φn = ϕn−12 ◦ ϕ1(.) sends every bitstring of length n to the word 0; its domain code is {0, 1}n, its image code is {0}, and it is its maximum essential extension. Thus, ‖ϕn−12 ◦ ϕ1‖ = 2n, whereas (n − 1) · ‖ϕ2‖+ ‖ϕ1‖ = 2n. Also, ϕ2(.) = (0 7→ 0, 1 7→ 0) · (0 7→ ε), so |ϕ1|Γ = 1, |ϕ2|Γ ≤ 2, and |Φn|Γ ≤ 2n− 1; hence ‖Φn‖ > 2|Φn|Γ/2. ✷ We will use the following facts that are easy to prove: If R ⊂ A∗ is a right ideal and ϕ is a right-ideal morphism then ϕ(R) and ϕ−1(R) are right ideals. The intersection and the union of right ideals are right ideals. We also need the following result. (Lemma 3.3 of [4]) If P,Q, S ⊆ A∗ are such that PA∗ ∩QA∗ = SA∗, and if S is a prefix code then S ⊆ P ∪Q. ✷ Lemma 4.3 Let θ be a right-ideal morphism, and assume SA∗ ⊆ Dom(θ), where S ⊂ A∗ is a finite prefix code. Then there is a finite prefix code R ⊂ A∗ such that θ(SA∗) = RA∗ and R ⊆ θ(S). Proof. Since θ is a right-ideal morphism we have θ(SA∗) = θ(S) A∗. Since θ(S) might not be a prefix code we take R = {r ∈ θ(S) : r is minimal (shortest) in the prefix order within θ(S)}. Then R is a prefix code that has the required properties. ✷ Lemma 4.4 3 For any right-ideal morphism θ and any prefix code Z ⊂ A∗, θ−1(Z) is a prefix code. In particular, θ−1(imC(θ)) is a prefix code, and θ−1(imC(θ)) ⊆ domC(θ). There exist right-ideal morphisms θ with finite domain code, such that θ−1(imC(θ)) 6= domC(θ). Proof. First, θ−1(Z) is a prefix code. Indeed, if we had x1 = x2u for some x1, x2 ∈ θ−1(Z) with u non-empty, then θ(x1) = θ(x2) u, with θ(x1), θ(x2) ∈ Z. This would contradict the assumption that Z is a prefix code. Second, let Q = imC(θ); then θ−1(Q)A∗ ⊆ θ−1(QA∗). Indeed, if x ∈ θ−1(Q), then x = pw for some p ∈ domC(θ) and w ∈ A∗. Hence, θ(x) = θ(p)w, and θ(x) ∈ Q. Since θ(p)w ∈ Q and θ(p) ∈ QA∗, we have θ(p)w = θ(p) (since Q is a prefix code). So w is empty, hence x = pw = p ∈ domC(θ). Example: Let A = {0, 1}, and let θ be the right-ideal morphism defined by domC(θ) = {01, 1}, imC(θ) = {ε}, and θ(01) = 0, θ(1) = ε. Then, θ−1(imC(θ)) = {1} 6= domC(θ). ✷ The following generalizes the “table size formula” of Gk,1 to the monoid Mk,1. Theorem 4.5 (Generalized image code formulas). 4 Let ϕi be right-ideal morphism with finite domain codes, for i = 1, 2, . . . , n. Then ∣imC(ϕn ◦ . . . ◦ ϕ1) ∣ ≤ |imC(ϕ1)| + i=2 |ϕi(domC(ϕi))|, (2) ℓ domC(ϕn ◦ . . . ◦ ϕ1) i=1 ℓ(domC(ϕi)), 3 This Lemma was incorrect in the earlier versions of this paper and in [1]. 4 This Theorem was incorrect in the previous versions and in [1]; this is a corrected (and expanded) version. (3) ℓ ϕn . . . ϕ1(domC(ϕn ◦ . . . ◦ ϕ1)) i=1 ℓ(ϕi(domC(ϕi))), (4) ℓ imC(ϕn ◦ . . . ◦ ϕ1) ≤ ℓ(imC(ϕ1)) + i=2 ℓ(ϕi(domC(ϕi))), ∣ϕn . . . ϕ1(domC(ϕn ◦ . . . ◦ ϕ1)) i=1 |(ϕi(domC(ϕi))|, and ϕn . . . ϕ1(domC(ϕn ◦ . . . ◦ ϕ1)) ⊆ i=1 ϕn . . . ϕi(domC(ϕi)). Proof. Let Pi = domC(ϕi) and Qi = imC(ϕi). (1) The proof is similar to the proof of Proposition 3.5 in [4]. We have Dom(ϕ2 ◦ϕ1) = ϕ−11 (Q1A∗ ∩ ∗) and Im(ϕ2◦ϕ1) = ϕ2(Q1A∗∩P2A∗). So the following maps are total and onto on the indicated sets: ϕ−11 (Q1A ∗ ∩ P2A∗) ϕ1−→ Q1A∗ ∩ P2A∗ ϕ2−→ ϕ2(Q1A∗ ∩ P2A∗). By Lemma 3.3 of [4] (quoted above) we have Q1A ∗ ∩ P2A∗ = SA∗ for some finite prefix code S with S ⊆ Q1 ∪ P2. Moreover, by Lemma 4.3 we have ϕ2(SA∗) = R2A∗ for some finite prefix code R2 such that R2 ⊆ ϕ2(S). Now, since S ⊆ Q1 ∪ P2 we have R2 ⊆ ϕ2(S) ⊆ ϕ2(Q1) ∪ ϕ2(P2). Thus, |imC(ϕ2 ◦ ϕ1)| = |R2| ≤ |ϕ2(P2)|+ |ϕ2(Q1)|. Since |ϕ2(Q1)| ≤ |Q1|, we have |R2| ≤ |ϕ2(P2)|+ |Q1|. By induction for n > 2, |imC(ϕn ◦ ϕn−1 ◦ . . . ◦ϕ1)| ≤ |ϕn(domC(ϕn))| + |imC(ϕn−1 ◦ . . . ◦ϕ1)| ≤ |ϕn(domC(ϕn))| + i=2 |ϕi(domC(ϕi))| + |imC(ϕ1)| . (2) We prove the formula when n = 2; the general formula then follows immediately by induction. Let x ∈ domC(ϕ2 ◦ ϕ1); then ϕ1(x) is defined, hence x = p1u for some p1 ∈ P1, u ∈ A∗. And ϕ2 is defined on ϕ1(x) = ϕ1(p1)u, so ϕ1(x) ∈ P2A∗ = Dom(ϕ2). Hence there exist p2 ∈ P2 and v ∈ A∗ such (⋆) ϕ1(p1)u = p2v ∈ ϕ1(P1)A∗ ∩ P2A∗. It follows that u and v are suffix-comparable. Claim. The words u and v in (⋆) satisfy: u = ε, or v = ε. Proof of the Claim: Since u and v are suffix-comparable, let us first consider the case where v is a suffix of u, i.e., u = tv for some t ∈ A∗. Then ϕ1(x) = ϕ1(p1) tv = p2v, hence ϕ1(p1) t = p2, hence ϕ2 is defined on ϕ1(p1) t = p2. So, ϕ2 ◦ ϕ1 is defined on p1t, i.e., p1t ∈ domC(ϕ2 ◦ ϕ1). But we also have x = p1tv ∈ domC(ϕ2 ◦ ϕ1). Since domC(ϕ2 ◦ ϕ1) is a prefix code, it follows that v = ε. Let us next consider the other case, namely where u is a suffix of v, i.e., v = su for some s ∈ A∗. Then ϕ1(x) = ϕ1(p1)u = p2su, hence ϕ1(p1) = p2s, hence ϕ2 is defined on ϕ1(p1) = p2s, hence p1 ∈ domC(ϕ2 ◦ ϕ1). But we also have x = p1u ∈ domC(ϕ2 ◦ ϕ1). Since domC(ϕ2 ◦ ϕ1) is a prefix code, it follows that u = ε. [This proves the Claim.] Now for x ∈ domC(ϕ2 ◦ ϕ1) we have x = p1u, and ϕ1(p1)u = p2v, hence |x| = |p1| + |u| and |ϕ1(p1)|+ |u| = |p2|+ |v|. By the Claim, either |u| = 0 or |v| = 0. If |u| = 0 then |x| = |p1| ≤ ℓ(domC(ϕ1)). If |v| = 0 then |x| = |p1|+ |u| = |p1|+ |p2|+ |v| − |ϕ1(p1)| = |p1|+ |p2| − |ϕ1(p1)| ≤ |p1|+ |p2| ≤ ℓ(domC(ϕ1)) + ℓ(domC(ϕ2)). (3) As in the proof of (2) we only need to consider n = 2. Let x ∈ domC(ϕ2ϕ1), hence ϕ2ϕ1(x) ∈ ϕ2ϕ1(domC(ϕ2ϕ1)). By (⋆) (and with the notation of the proof of (2)) we have ϕ2ϕ1(x) = ϕ2(ϕ1(p1)u) = ϕ2(p2) v ∈ ϕ2(ϕ1(P1)A∗ ∩ P2A∗) = Im(ϕ2ϕ1). By the reasoning of the proof of (2), we have two cases: If |u| = 0 then |v| = |ϕ1(p1)| + |u| − |p2| = |ϕ1(p1)| − |p2| ≤ |ϕ1(p1)|. Hence, |ϕ2ϕ1(x)| = |ϕ2(p2)|+ |v| ≤ |ϕ2(p2)|+ |ϕ1(p1)| ≤ ℓ(ϕ2(domC(ϕ2)) + ℓ(ϕ1(domC(ϕ1)). If |v| = 0 then ϕ2ϕ1(x) = ϕ2(p2), hence |ϕ2ϕ1(x)| = |ϕ2(p2)| ≤ ℓ(ϕ2(domC(ϕ2)). (4) We first consider the case n = 2. As we saw in the proof of (1), imC(ϕ2ϕ1) = R2 whereR2 ⊆ ϕ2(S), and where S is a prefix code such that S ⊆ Q1 ∩ P2. Hence R2 ⊆ ϕ2(Q1) ∪ ϕ2(P2). Hence for any z ∈ R2, either z ∈ ϕ2(P2) or z ∈ ϕ2(Q1). If z ∈ ϕ2(P2) then |z| ≤ ℓ(ϕ2(P2)). If z ∈ ϕ2(Q1), then z = ϕ2(q1) for some q1 ∈ Q1 ∩ P2A∗, so q1 = p2u for some p2 ∈ P2 and u ∈ A∗. We have q1 ∈ P2A∗ (= Im(ϕ2)), so q1 ∈ Im(ϕ2). Now |z| = |ϕ2(p2)| + |u|, and |u| = |q1| − |p2| ≤ |q1| ≤ ℓ(imC(ϕ1)). Thus, |z| ≤ |ϕ2(p2)|+ ℓ(imC(ϕ1)) ≤ ℓ(ϕ2(domC(ϕ2))) + ℓ(imC(ϕ1)). The formula for n > 2 now follows by induction in the same way as in the proof of (1). (5) We first prove the formula for n = 2. As we saw in the proof of (2), if x ∈ domC(ϕ2ϕ1) then there exist u, v ∈ A∗, p1 ∈ P1, p2 ∈ P2, such that x = p1u and ϕ1(x) = ϕ1(p1)u = p2v. Moreover, by the Claim in (2) we have u = ε or v = ε. Also, ϕ2ϕ1(x) = ϕ2(ϕ1(p1)u) = ϕ2(p2) v. If v = ε then ϕ2ϕ1(x) = ϕ2(p2) ∈ ϕ2(domC(ϕ2)). If u = ε then ϕ2ϕ1(x) = ϕ2ϕ1(p1) ∈ ϕ2ϕ1(domC(ϕ1)). Thus we proved the following fact: ϕ2ϕ1(domC(ϕ2ϕ1)) ⊆ ϕ2(domC(ϕ2)) ∪ ϕ2ϕ1(domC(ϕ1)). Now, since |ϕ2ϕ1(domC(ϕ1))| ≤ |ϕ1(domC(ϕ1))|, the fact implies that |ϕ2ϕ1(domC(ϕ2ϕ1))| ≤ |ϕ2(domC(ϕ2))| + |ϕ1(domC(ϕ1))|. By induction we immediately obtain ∣ϕn . . . ϕ1(domC(ϕn ◦ . . . ◦ ϕ1)) i=1 |(ϕi(domC(ϕi))|, and ϕn . . . ϕ1(domC(ϕn ◦ . . . ◦ ϕ1)) ⊆ ϕn(domC(ϕn)) ∪ ϕnϕn−1(domC(ϕn−1)) ∪ . . . . . . . . . ∪ ϕn . . . ϕi(domC(ϕi)) ∪ . . . . . . . . . ∪ ϕn . . . ϕi . . . ϕ1(domC(ϕ1)). ✷ Remarks. Obviously, Dom(ϕ2ϕ1) ⊆ Dom(ϕ1); however, in infinitely many cases (in “most” cases), domC(ϕ2ϕ1) 6⊆ domC(ϕ1). Instead, we have the more complicated formula of Theorem 4.5(5). By Prop. 4.2, we cannot have a formula for |domC(ϕn . . . ϕ1)| of a similar nature as the formulas in Theorem 4.5. The following class of right-ideal morphisms plays an important role here (as well as in Section 5 of [7], where it was introduced). 5 Definition 4.5A (Normal). A right-ideal morphism ϕ is called normal iff ϕ(domC(ϕ)) = imC(ϕ). By Lemma 5.7 of [7] we also have: ϕ is normal iff ϕ−1(imC(ϕ)) = domC(ϕ). In other words, ϕ is normal iff ϕ is entirely determined by the way it maps domC(ϕ) onto imC(ϕ). For example, every injective right-ideal morphism is normal (by Lemma 5.1 in [7]). The finite generating set Γ of Mk,1, constructed in Section 3, consist entirely of normal right-ideal morphisms. On the other hand, the composition of two normal right-ideal morphisms does not always result in a normal morphism, as is shown by the following example: domC(f) = {0, 1} and f(0) = 0, f(1) = 10; domC(g) = {0, 1} and g(0) = g(1) = 0; so f and g are normal. But domC(gf) = {0, 1} and gf(0) = 0, gf(1) = 00; so gf is not normal (for more details, see Prop. 5.8 in [7]). The next result (Theorem 4.5B) shows that every element of Mk,1 can be represented by a normal right-ideal morphism. So one can say informally that “from the point of view of Mk,1, all right-ideal morphisms are normal”. For proving this we need some definitions. We always assume |A| ≥ 2. Definitions and notation. If x1, x2 ∈ A∗ are such that x1 is a prefix of x2, i.e., x2 ∈ x1A∗, we denote this by x1 ≤pref x2. For Z ⊆ A∗, the set of prefixes of Z is pref(Z) = {v ∈ A∗ : v ≤pref z for some z ∈ Z}. 5 Def. 4.5A, Theorem 4.5B, and Cor. 4.5C are new in this version. For a set X ⊆ A∗ and a word v ∈ A∗, v−1X denotes the set {s ∈ A∗ : vs ∈ X}. The tree of A∗ has root ε, vertex set A∗, and edge set {(w,wa) : w ∈ A∗, a ∈ A}. A subtree of the tree of A∗ has as root any string r ∈ A∗, and as vertex set any subset V ⊆ rA∗, such that the following holds for all v ∈ V and u ∈ A∗: r ≤pref u ≤pref v implies u ∈ V . The following is a slight generalization of the classical notion of a prefix tree. Definition (Prefix tree). Let Z ⊆ A∗, and let q ∈ A∗. The prefix tree T (q, Z) is the subtree of the tree of A∗ with root q and vertex set Vq,Z = {v ∈ A∗ : q ≤pref v, and v ≤pref z for some z ∈ Z}. Remark. Let L be the set of leaves of T (q, Z); then L and q−1L are prefix codes. Definition (Saturated tree). A subtree T of the tree of A∗ is saturated iff for every vertex v of T we have: v has no child in T (i.e., v is a leaf), or v has |A| children in T . Definition (Tree saturation). Let T be a subtree of the tree of A∗, with root q, set of vertices V , and set of leaves L. The saturation of T is the smallest (under inclusion) saturated subtree of the tree of A∗ with root q, that contains T . In other words, if T is just {q}, it is its own saturation; otherwise the saturation has root q and has vertex set V ∪ (V − L) ·A. We denote the saturation of T by sT . Remark. (1) The prefix tree T (q, Z) and its saturation have the same depth (i.e., length of a longest path from the root). Every leaf of T (q, Z) is also a leaf of sT (q, Z), but unless T (q, Z) is already saturated, sT (q, Z) has more leaves than T (q, Z). The non-leaf vertices of T (q, Z) and sT (q, Z) are the same. (2) The number of leaves in the saturated tree sT (q, Z) is < |Vq,Z | · |A|. (3) Let L be the leaf set of the saturated tree sT (q, Z); if Z is finite then q−1L is a maximal prefix code. Theorem 4.5B (Equivalent normal morphism). For every right-ideal morphism ϕ with finite domain code there exists a normal right-ideal morphism ϕ0 with finite domain code, such that ϕ = ϕ0 in Mk,1. Moreover, |imC(ϕ0)| = |ϕ0(domC(ϕ0))| ≤ |A| · (ℓ(ϕ(P )) + 1) · |ϕ(P )|, |domC(ϕ0)| ≤ |P | · |A| · (ℓ(ϕ(P )) + 1) · |ϕ(P )|, ℓ(imC(ϕ0)) = ℓ(ϕ0(domC(ϕ0))) = ℓ(ϕ(domC(ϕ))), ℓ(domC(ϕ0)) ≤ ℓ(domC(ϕ)) + ℓ(ϕ(domC(ϕ))). Proof. Let P = domC(ϕ), Q = imC(ϕ), P0 = domC(ϕ0), Q0 = imC(ϕ0). For each p ∈ P , let ϕ(p)Wϕ(p) be the the set of leaves of the saturated tree sT ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ . By Remark (3) above, Wϕ(p) is a finite maximal prefix code. Now we define ϕ0 as follows: ϕ0 is the restriction of ϕ to p∈P pWϕ(p)A Let us verify that ϕ0 has the required properties. Since P andWϕ(p) are finite prefix codes, p∈P pWϕ(p) is a finite prefix code. So, domC(ϕ0) = p∈P pWϕ(p). Since each Wϕ(p) is a maximal prefix code, the right ideal p∈P pWϕ(p)A ∗ is essential in the right ideal PA∗; hence ϕ and ϕ0 are equal as elements of Mk,1. Finally, let us show that ϕ0(domC(ϕ0)) is a prefix code. We have ϕ0(domC(ϕ0)) = p∈P ϕ(p) Wϕ(p), which is the set of leaves of the union of the saturated prefix trees sT ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ , for p ranging over P . For each p ∈ P , the leaves of sT ϕ(p), ϕ(P )∩ϕ(p)A∗ form the prefix code ϕ(p)Wϕ(p). For p1 6= p2 in P , if ϕ(p1) is a prefix of ϕ(p2) then the leaves of sT ϕ(p2), ϕ(P ) ∩ ϕ(p2)A∗ are a subset of the leaves of sT ϕ(p1), ϕ(P ) ∩ ϕ(p1)A∗ , so the union of these two leaf sets is just the leaf set of sT ϕ(p1), ϕ(P ) ∩ ϕ(p1)A∗ ; a similar thing happens if ϕ(p2) is a prefix of ϕ(p1). So in p∈P ϕ(p) Wϕ(p) we can ignore elements p of P for which ϕ(p) is a strict prefix of another element of ϕ(P ). If ϕ(p1) and ϕ(p2) are not prefix-comparable, then the leaves of sT ϕ(pi), ϕ(P )∩ϕ(pi)A∗ ϕ(pi) as a prefix, so these two trees have leaf sets that are two-by-two prefix-incomparable (namely the sets ϕ(p1)Wϕ(p1) and ϕ(p2)Wϕ(p2)). The union of prefix codes that are two-by-two prefix-incomparable forms a prefix code; hence, p∈P ϕ(p) Wϕ(p) is a prefix code. Now, since ϕ0(domC(ϕ0)) is a prefix code it follows that imC(ϕ0) = ϕ0(domC(ϕ0)), so ϕ0 is normal. This proves the first part of the theorem. Let us prove the formulas. We saw that imC(ϕ0) = ϕ0(domC(ϕ0)) = p∈P ϕ(p) Wϕ(p), and ϕ(p) Wϕ(p) is the leaf set of the saturated tree sT ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ . By the definition of prefix trees, the vertices of all the (non-saturated) trees T ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ are subsets of pref(ϕ(P )). By Remark (2) above, the number of leaves in a saturated tree sT ϕ(p), ϕ(P )∩ϕ(p)A∗ is at most |A| times the number of vertices of the non-saturated tree. Hence, |imC(ϕ0)| ≤ |A|·|pref(ϕ(P ))|. Moreover, for any finite Z ⊂ A∗, |pref(Z)| ≤ (1 + ℓ(Z)) · |Z|, hence, |imC(ϕ0)| ≤ |A| · (ℓ(ϕ(P )) + 1) · |ϕ(P )|. We have domC(ϕ0) = p∈P pWϕ(p), and ϕ(p)Wϕ(p) is the leaf set of sT ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ Hence by the same reasoning as for |imC(ϕ0)|: |Wϕ(p)| = |ϕ(p)Wϕ(p)| ≤ |A| · (ℓ(ϕ(P )) + 1) · |ϕ(P )|. Hence, |domC(ϕ0)| ≤ p∈P |Wϕ(p)| ≤ p∈P |A|·(ℓ(ϕ(P ))+1)·|ϕ(P )| ≤ |P |·|A|·(ℓ(ϕ(P ))+1)·|ϕ(P )|. We have imC(ϕ0) = p∈P ϕ(p)Wϕ(p), and ϕ(p)Wϕ(p) is the leaf set of sT ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ Hence, ℓ(imC(ϕ0)) ≤ ℓ(ϕ(P )); indeed, tree saturation does not increase the depth of a tree, and the depth of T ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ is ≤ ℓ(ϕ(P )). We have domC(ϕ0) = p∈P pWϕ(p). And ℓ(Wϕ(p)) ≤ ℓ(ϕ(p)Wϕ(p)) ≤ ℓ(ϕ(P )), since ϕ(p)Wϕ(p) is the leaf set of sT ϕ(p), ϕ(P ) ∩ ϕ(p)A∗ . Hence, for every x ∈ domC(ϕ0) we have x ∈ pWϕ(p) for some p ∈ P , so |x| ≤ |p|+ ℓ(Wϕ(p)). Therefore, ℓ(domC(ϕ0)) ≤ ℓ(P ) + ℓ(ϕ(P )). ✷ Theorem 4.5B tells us that as far as Mk,1 is concerned, all right-ideal morphisms are normal. Corollary 4.5C (Image code formula). Let ϕi be a right-ideal morphism (for i = 1, . . . , n), and let Φ = ϕn ◦ . . . ◦ ϕ1. (1) If ϕi is normal for 2 ≤ i ≤ n, then |imC(Φ)| ≤ i=1 |imC(ϕi)| . (2) If all ϕi are normal (for 1 ≤ i ≤ n), then ℓ(domC(Φ) ∪ imC(Φ)) ≤ i=1 ℓ(domC(ϕi) ∪ imC(ϕi)) . Proof. (1) follows immediately from Theorem 4.5(1), and (2) follows from 4.5(2) and 4.5(4). ✷ Counter-examples: (1) The following shows that the image code formula of Corollary 4.5C(1) is wrong in some examples when ϕ2 is not normal (but ϕ1 is normal). Let A = {0, 1}, n ≥ 2, and ϕ1 = {(01, 00), (00, 01), (10, 1011), (11, 1100)}, and ϕ2 = {(00u0, 000u1) : u ∈ {0, 1}n−1} ∪ {(01v0, 001v1) : v ∈ {0, 1}n−1} ∪ {(10, 000), (11, 001)}. So, imC(ϕ1) = {00, 01, 1011, 1100}, and imC(ϕ2) = {000, 001}, hence |imC(ϕ1)|+ |imC(ϕ2)| = 6. Note that the right-ideal morphisms ϕ1 and ϕ2 are in maximally extended form. 6 The concept of normal morphism and Theorem 4.5 enable us to rehabilitate the image code formula (which was incorrect as stated in Theorem 4.5 of [1], but which is correct when one adds the hypothesis that the morphisms ϕi are normal). Now, ϕ2 ◦ϕ1 : 01u0 7→ 00u0 7→ 000u1 and ϕ2 ◦ϕ1 : 00v0 7→ 01v0 7→ 001v1, for all u, v ∈ {0, 1}n−1; and ϕ2 ◦ ϕ1 : 10 7→ 1011 7→ 00011, ϕ2 ◦ ϕ1 : 11 7→ 1100 7→ 00100. Note that ϕ2 ◦ ϕ1 is in maximally extended form. Then imC(ϕ2 ◦ϕ1) = {00011, 00100} ∪ 000 {00, 01, 11} {0, 1}n−2 ∪ 001 {00, 01, 11} {0, 1}n−2 . Thus when n ≥ 2: 2 + 6 · 2n−2 = |imC(ϕ2 ◦ ϕ1)| 6≤ |imC(ϕ1)|+ |imC(ϕ2)| = 6. ✷ (2) The following shows that the formula of Corollary 4.5C(2) is wrong in some examples when ϕ2 is not normal (but ϕ1 is normal). We abbreviate ℓ(domC(ϕ) ∪ imC(ϕ)) by ℓ(ϕ). Let A = {0, 1}, n ≥ 2, ϕ1 = {(0, 0n)}, and ϕ2 = {(0, 0n+1), (1, 0)}. So, ℓ(ϕ1) = n, and ℓ(ϕ2) = 1 since imC(ϕ2) = {0}. Now, ϕ2 ◦ ϕ1 = {(0, 02n)}. Thus when n ≥ 2: 2n = ℓ(ϕ2 ◦ ϕ1) 6≤ ℓ(ϕ2) + ℓ(ϕ1) = n+ 1. ✷ For elements of Invk,1 the image code has the same size as the domain code, which is also the table size. Moreover, injective right-ideal morphisms are normal, thus Corollary 4.5C implies: Corollary 4.6 For all injective right-ideal morphisms ϕ,ψ: ‖ψ ◦ ϕ‖ ≤ ‖ψ‖+ ‖ϕ‖. ✷ In other words, the table size formula holds for Invk,1. Another immediate consequence of Theorem 4.5 is the following. Corollary 4.7 Let ϕi be normal right-ideal morphisms for i = 1, . . . , n, and let c1, c2 be positive constants. (1) If |imC(ϕi)| ≤ c1 for all i then |imC(ϕn ◦ . . . ◦ ϕ1)| ≤ c1 n. (2) If ℓ(imC(ϕi)) ≤ c2 for all i then ℓ(imC(ϕn ◦ . . . ◦ ϕ1)) ≤ c2 n. ✷ The position transposition τi,j (with 0 < i < j) is, by definition, the partial permutation of A which transposes the letters at positions i and j; τi,j is undefined on words of length < j. More precisely, we have domC(τi,j) = imC(τi,j) = A j, and uαvβ 7→ uβvα for all letters α, β ∈ A and all words u ∈ Ai−1 and v ∈ Aj−i−1. In this form, τi,j is equal to its maximum essential extension. Corollary 4.8 The word-length of τi,j over any finite generating set of Mk,1 is exponential. Proof. We have |imC(τi,j)| = kj . The Corollary follows then from Corollary 4.7(1). ✷ 4.2 Some algorithmic problems about right-ideal morphisms We consider several problems about right-ideal morphisms of A∗ and show that they have deterministic polynomial-time algorithms. We also show that the word problem of Mk,1 over Γk,1∪ {τi,i+1 : 0 < i} is coNP-complete, where Γk,1 is any finite generating set of Mk,1. We saw that Γk,1 can be chosen so as to consist of normal right-ideal morphisms. Lemma 4.9 There are deterministic polynomial time algorithms for the following problems. Input: Two finite prefix codes P1, P2 ⊂ A∗, given explicitly by lists of words. Output 1: The finite prefix code Π ⊂ A∗ such that ΠA∗ = P1A∗ ∩P2A∗, where Π is produced explicitly as a list of words. Question 2: Is P1A ∗ ∩ P2A∗ essential in P1A∗ (or in P2A∗, or in both)? Proof. We saw already that Π exists and Π ⊆ P1 ∪ P2; see Lemma 3.3 of [4] (quoted before Lemma 4.3 above). Algorithm for Output 1: Since Π ⊆ P1 ∪ P2, we just need to search for the elements of Π within P1 ∪ P2. For each x ∈ P1 we check whether x also belongs to P2A∗ (by checking whether any element of P2 is a prefix of x). Since P1 and P2 are explicitly given as lists, this takes polynomial time. Similarly, for each x ∈ P2 we check whether x also belongs to P1A∗. Thus, we have computed the set Π1 = (P1 ∩ P2A∗) ∪ (P2 ∩ P1A∗). Now, Π is obtained from Π1 by eliminating every word that has another word of Π1 as a prefix. Since Π1 is explicitly listed, this takes just polynomial time. Algorithm for Question 2: We first compute Π by the previous algorithm. Next, we check whether every p1 ∈ P1 is a prefix of some r ∈ Π; since P1 and Π are given by explicit lists, this takes just polynomial time. For P2 it is similar. ✷ Lemma 4.10 The following input-output problem has a deterministic polynomial-time algorithm. • Input: A finite set S ⊂ A∗, and m right-ideal morphisms ψj for j = 1, . . . ,m, where S is given by an explicit list of words, and each ψj is given explicitly by the list of pairs of words {(x, ψj(x)) : x ∈ domC(ψj)}. • Output: The finite set ψm . . . ψ1(S), given explicitly by a list of words. Proof. Let Ψ = ψm ◦ . . . ◦ψ1 ◦ idS. Then ψm . . . ψ1(S) = Ψ(domC(Ψ)). By Theorem 4.5(3) and (5), ℓ(Ψ(domC(Ψ))) ≤ ℓ(S)+ i=1 ℓ(ψi(domC(ψi))) and |Ψ(domC(Ψ))| ≤ |S|+ i=1 |ψi(domC(ψi))|. So the size of ψm . . . ψ1(S), in terms of the number of words and their lengths, is polynomially bounded by the size of the input. We now compute ψm . . . ψ1(S) by applying ψj to ψj−1 . . . ψ1(S) for increasing j. Since the sizes of the sets remain polynomially bounded, this algorithm takes polynomial time. ✷ Corollary 4.11 The following input-output problems have deterministic polynomial-time algorithms. • Input: A list of n right-ideal morphisms ϕi for i = 1, . . . , n, given explicitly by finite tables. • Output 1: A finite set, as an explicit list of words, that contains ϕn . . . ϕ1(domC(ϕn . . . ϕ1)). • Output 2: The finite set imC(ϕn . . . ϕ1), as an explicit list of words. Proof. (1) By Theorem 4.5(5) we have ϕn . . . ϕ1(domC(ϕn . . . ϕ1)) ⊆ i=1 ϕn . . . ϕi(domC(ϕi)). By Lemma 4.10, each set ϕn . . . ϕi(domC(ϕi)), as well as their union, is computable in polynomial time (as an explicit list of words). (2) Let Φ = ϕn . . . ϕ1, Pi = domC(ϕi), and Qi = imC(ϕi). As in the proof of Theorem 4.5(1), Dom(ϕ2 ◦ ϕ1) = ϕ−11 (Q1A∗ ∩ P2A∗), Im(ϕ2 ◦ ϕ1) = ϕ2(Q1A∗ ∩ P2A∗), and the maps ϕ 1 (Q1A ϕ1−→ Q1A∗ ∩P2A∗ ϕ2−→ ϕ2(Q1A∗∩P2A∗) are total and onto. By Lemma 3.3 of [4] (mentioned before Theorem 4.5) we have Q1A ∗ ∩ P2A∗ = S1A∗ for some finite prefix code S1 with S1 ⊆ Q1 ∪ P2. Moreover, by Lemma 4.3, ϕ2(S1A ∗) = R2A ∗, where imC(ϕ2ϕ1) = R2 ⊆ ϕ2(S1). By induction, for j ≥ 2 suppose imC(ϕj . . . ϕ1) = Rj ⊆ ϕj(Sj−1), where Rj and Sj−1 are finite prefix codes such that Sj−1 ⊆ Rj−1 ∪ Pj , Sj−1A∗ = Rj−1A∗ ∩ PjA∗, RjA∗ = Im(ϕj . . . ϕ1) = ϕj(Sj−1A ∗), and the maps ϕ−1j (RjA ∗ ∩Pj+1A∗) ϕj−→ RjA∗ ∩Pj+1A∗ ϕj+1−→ ϕj+1(RjA∗ ∩Pj+1A∗) are total and onto. Then by Lemma 3.3 of [4] we again have RjA ∗ ∩Pj+1A∗ = SjA∗ for some finite prefix code Sj with Sj ⊆ Rj∪Pj+1; and by Lemma 4.3, ϕj+1(SjA∗) = Rj+1A∗ for some finite prefix code Rj+1 such that imC(ϕj+1ϕj . . . ϕ1) = Rj+1 ⊆ ϕj+1(Sj). Applying Theorem 4.5 to Ri = imC(ϕi . . . ϕ1) for any i ≥ 2 we have |Ri| ≤ |ϕi(Pi)|+ . . . + |ϕ2(P2)|+ |imC(ϕ1)|, and ℓ(Ri) ≤ ℓ(ϕi(Pi)) + . . . + ℓ(ϕ2(P2)) + ℓ(imC(ϕ1)). Since Sj ⊆ Pj ∪Rj−1, we have |Sj | ≤ |Pj |+ |Rj−1| ≤ |Pj |+ |ϕj−1(Pj−1)|+ . . . + |ϕ2(P2)|+ |imC(ϕ1)|, and ℓ(Sj) ≤ ℓ(Pj)+ ℓ(Rj−1) ≤ ℓ(Pj)+ ℓ(ϕj−1(Pj−1))+ . . . + ℓ(ϕ2(P2))+ ℓ(imC(ϕ1)). Thus, the size of each Ri and Sj is less than the input size; by input size we mean the total length of all the words in the input lists. By Lemma 4.9, the prefix code Sj is computed from Rj and Pj+1, as an explicit list, in time ≤ Tj(|Pj |+ ℓ(Pj) + |Rj−1|+ ℓ(Rj−1)), for some polynomial Tj(.). And Rj+1 is computed from Sj by applying ϕj+1 to Sj, and then keeping the elements that do not have a prefix in ϕj+1(Sj). Computing ϕj+1(Sj) takes at most quadratic time, and finding the prefix code in ϕj+1(Sj) also takes at most quadratic time. In the end we obtain Rn = imC(ϕn . . . ϕ1) as an explicit list of words. ✷ When we consider the word problem of Mk,1 over a finite generating set, we measure the input size by the length of input word (with each generator having length 1). But for the word problem of Mk,1 over the infinite generating set Γk,1 ∪ {τi−1,i : i > 1} we count the length of the position transpositions τi−1,i as i, in the definition of the input size of the word problem. Indeed, at least log2 i bits are needed to describe the subscript i of τi−1,i. Moreover, in the connection between Mk,1 (over Γk,1∪{τi−1,i : i > 1}) and circuits, τi−1,i is interpreted as the wire-crossing operation of wire number i and wire number i− 1; this suggests that viewing the size of τi−1,i as i is more natural than log2 i. In any case, we will see next that the word problem of Mk,1 over Γk,1 ∪ {τi−1,i : i > 1} is coNP-complete, even if the size of τi−1,i is more generously measured as i; this is a stronger result than if log2 i were used. Theorem 4.12 (coNP-complete word problem). The word problem of Mk,1 over the infinite generating set Γk,1 ∪ {τi−1,i : i > 1} is coNP-complete, where Γk,1 is any finite generating set of Mk,1. Proof. In [5] (see also [3]) it was shown that the word problem of the Thompson-Higman group Gk,1 over ΓGk,1 ∪ {τi−1,i : i > 1} is coNP-complete, where ΓGk,1 is any finite generating set of Gk,1. Hence, since the elements of the finite set ΓGk,1 can be expressed by a finite set of words over Γk,1, it follows that the word problem of Mk,1 over Γk,1 ∪ {τi−1,i : i > 1} is coNP-hard. We will prove now that the word problem of Mk,1 over Γk,1 ∪ {τi−1,i : i > 1} belongs to coNP. The input of the problem consists of two words (ρm, . . . , ρ1) and (σn, . . . , σ1) over Γk,1 ∪ {τi−1,i : i > 1}. The input size is the weighted length of the words (ρm, . . . , ρ1) and (σn, . . . , σ1), where each generator in Γk,1 has weight 1, and each generator of the form τi−1,i has weight i. For every right-ideal morphism ϕ we abbreviate ℓ(domC(ϕ) ∪ϕ(domC(ϕ))) by ℓ(ϕ); recall that for a finite set X ⊂ A∗, ℓ(X) denotes the length of a longest word in X. Since Γk,1 is finite there is a constant c > 0 such that c ≥ ℓ(γ) for all γ ∈ Γk,1; also, for each τi−1,i we have ℓ(τi−1,i) = i. By Theorem 4.5, the table of σn ◦ . . . ◦ σ1 (and more generally, the table of σj ◦ . . . ◦ σ1 for any j with n ≥ j ≥ 1) contains only words of length ≤ j=1 ℓ(σj), and similarly for ρm ◦ . . . ◦ρ1 (and for ρi ◦ . . . ◦ρ1, m ≥ i ≥ 1). So all the words in the tables for any σj ◦ . . . ◦σ1 and any ρi ◦ . . . ◦ρ1 have lengths that are linearly bounded by the size of the input (ρm, . . . , ρ1), (σn, . . . , σ1) Claim. Let N = max{ i=1 ℓ(ρi), j=1 ℓ(σj)}. Then ρm · . . . · ρ1 6= σn · . . . · σ1 in Mk,1 iff there exists x ∈ AN such that ρm ◦ . . . ◦ ρ1(x) 6= σn ◦ . . . ◦ σ1(x). Proof of the Claim: As we saw above, the tables of ρm ◦ . . . ◦ ρ1 and σn ◦ . . . ◦ σ1 only contain words of length ≤ N . Thus, restricting ρm ◦ . . . ◦ ρ1 and σn ◦ . . . ◦ σ1 to ANA∗ is an essential restriction, and the resulting tables have domain codes in AN . Therefore, ρm · . . . · ρ1 and σn · . . . · σ1 are equal (as elements of Mk,1) iff ρm ◦ . . . ◦ ρ1 and σn ◦ . . . ◦ σ1 are equal on AN . [End, Proof of Claim] The number N in the Claim is immediately obtained form the input. Based on the Claim, we obtain a nondeterministic polynomial-time algorithm which decides (nondeterministically) whether there exists x ∈ AN such that ρm ◦ . . . ◦ ρ1(x) 6= σn ◦ . . . ◦ σ1(x), as follows: The algorithm guesses x ∈ AN , computes ρm ◦ . . . ◦ ρ1(x) and σn ◦ . . . ◦ σ1(x), and checks that they are different words (∈ A∗) or that one is undefined and the other is a word. Applying Theorem 4.5 to ρm ◦ . . . ◦ ρ1 ◦ idAN and to σn ◦ . . . ◦ σ1 ◦ idAN shows that |ρm ◦ . . . ◦ ρ1(x)| ≤ 2N and |σn ◦ . . . ◦ σ1(x)| ≤ 2N ; here |ρm ◦ . . . ◦ ρ1(x)| denotes the length of the word ρm ◦ . . . ◦ ρ1(x) ∈ A∗, and similarly for σn ◦ . . . ◦ σ1(x). Also by Theorem 4.5, all intermediate results (as we successively apply ρi for i = 1, . . . ,m, or σj for j = 1, . . . , n) are words of length ≤ 2N . These successive words are computed by applying the table of ρi or σj (when ρi or σj belong to Γk,1), or by directly applying the position permutation τh,h−1 (if ρi or σj is τh,h−1). Thus, the output ρm ◦ . . . ◦ ρ1(x) (and similarly, σn ◦ . . . ◦ σ1(x)) can be computed in polynomial time. ✷ 4.3 The word problem of Mk,1 is in P We now move ahead with the the proof of our main result. Theorem 4.13 (Word problem in P). The word problem of the Thompson-Higman monoids Mk,1, over any finite generating set, can be decided in deterministic polynomial time. We assume that a fixed finite generating set Γk,1 of Mk,1 has been chosen. The input consists of two sequences (ρm, . . . , ρ1) and (σn, . . . , σ1) over Γk,1, and the input size is m+ n; since Γk,1 is finite and fixed, it does not matter whether we choose m+ n as input size, or the sum of the lengths of all the words in the tables of the elements of Γk,1. We want to decide in deterministic polynomial time whether, as elements of Mk,1, the products ρm · . . . · ρ1 and σn · . . . · σ1 are equal. Overview of the proof: • We compute the finite sets imC(ρm ◦ . . . ◦ ρ1), imC(σn ◦ . . . ◦ σ1) ⊂ A∗, explicitly described by lists of words. By Corollary 4.11 (Output 2) this can be done in polynomial time, and these sets have polynomial size. (Note however that by Proposition 4.2, the table sizes of ρm ◦ . . . ◦ ρ1 or σn ◦ . . . ◦ σ1 could be exponential in m or n.) • We check whether Im(ρm ◦ . . . ◦ ρ1) ∩ Im(σn ◦ . . . ◦ σ1) is essential in Im(ρm ◦ . . . ◦ ρ1) and in Im(σn ◦ . . . ◦ σ1). By Lemma 4.9 (Question 2) this can be done in polynomial time. If the answer is “no” then ρm · . . . · ρ1 6= σn · . . . · σ1 in Mk,1, since they don’t have a common maximum essential extension. Otherwise, the computation continues. • We compute the finite prefix code Π ⊂ A∗ such that ΠA∗ = Im(ρm ◦ . . . ◦ ρ1) ∩ Im(σn ◦ . . . ◦ σ1). By Lemma 4.9 (Output 1) this can be done in polynomial time, and Π has polynomial size. Hence, the table of idΠA∗ can be computed in polynomial time. • We restrict ρm ◦ . . .◦ρ1 and σn ◦ . . . ◦σ1 in such a way that their images are in ΠA∗. In other words, we replace them by ρ = idΠA∗ ◦ρm ◦ . . .◦ρ1, respectively σ = idΠA∗ ◦σn ◦ . . .◦σ1. Since ΠA∗ is essential in Im(ρm ◦ . . . ◦ ρ1) and in Im(σn ◦ . . . ◦ σ1), we have ρ = ρm · . . . · ρ1 in Mk,1, and σ = σn · . . . · σ1 in Mk,1. So, ρm · . . . · ρ1 = σn · . . . · σ1 in Mk,1 iff ρ = σ in Mk,1. • We compute finite sets R1, R2 ⊂ A∗, such that ρ(domC(ρ)) ⊆ R1 and σ(domC(σ)) ⊆ R2. Since ρ(domC(ρ)) ∪ σ(domC(σ)) ⊆ ΠA∗, we can pick R1, R2 so that R1 ∪ R2 ⊆ ΠA∗. By Corollary 4.11 (Output 1), the sets R1, R2 can be computed as explicit lists in polynomial time. Let R = R1 ∪R2. • We note that ρ = σ in Mk,1 iff for all r ∈ ρ(domC(ρ)) ∪ σ(domC(σ)): ρ−1(r) = σ−1(r). This holds iff for all r ∈ R: ρ−1(r) = σ−1(r). • For every r ∈ R we construct a deterministic finite automaton (DFA) accepting the finite set ρ−1(r) ⊂ A∗, and a DFA accepting the finite set σ−1(r) ⊂ A∗. By Corollary 4.15 this can be done in polynomial time, and the DFAs have polynomial size. (The finite sets ρ−1(r) and σ−1(r) themselves could have exponential size.) Note that domC(ρ) ⊆ ρ−1(ρ(domC(ρ))) ⊆ ρ−1(R), and similarly for σ. Note that usually, domC(ρ) 6⊆ ρ−1(imC(ρ)) (since ρ is not normal in general), and similarly for σ; so we need to use ρ(domC(ρ)), and not just imC(ρ). • For every r ∈ R we check whether the DFA for ρ−1(r) and the DFA for σ−1(r) are equivalent. By classical automata theory, equivalence of DFAs can be checked in polynomial time. [End of Overview.] Automata – notation and facts: In the following, DFA stands for deterministic finite automaton. The language accepted by a DFA A is denoted by L(A). A DFA is a structure (S,A, δ, s0, F ) where S is the set of states, A is the input alphabet, s0 ∈ S is the start state, F ⊆ S is the set of accept states, and δ : S ×A→ S is the next-state function; in general, δ is a partial function (by “function” we always mean partial function). We extend the definition of δ to a function S×A∗ → S by defining δ(s,w) to be the state that the DFA reaches from s after reading w (for any w ∈ A∗ and s ∈ S). See [21, 24] for background on finite automata. A DFA is called acyclic iff its underlying directed graph has no directed cycle. It is easy to prove that a language L ⊆ A∗ is finite iff L is accepted by an acyclic DFA. Moreover, L is a finite prefix code iff L is accepted by an acyclic DFA that has a single accept state (take the prefix tree of the prefix code, with the leaves as accept states, then glue all the leaves together into a single accept state). By the size of a DFA A we mean the number of states, |S|; we denote this by size(A). For a finite set P ⊆ A∗ we denote the length of the longest words in P by ℓ(P ), and we define the total length of P by Σ(P ) = x∈P |x|; obviously, Σ(P ) ≤ |P | · ℓ(P ). For a language L ⊆ A∗ and a partial function Φ : A∗ → A∗, we define the inverse image of L under Φ by Φ−1(L) = {x ∈ A∗ : Φ(x) ∈ L}. For L ⊆ A∗ we denote the set of all strict prefixes of the words in L by spref(L); precisely, spref(L) = {x ∈ A∗ : (∃w ∈ L)[x ≤pref w and x 6= w ]}. The reason why we use acyclic DFAs to describe finite sets is that a finite set can be exponentially larger than the number of states of a DFA that accepts it; e.g., An is accepted by an acyclic DFA with n+1 states. This conciseness plays a crucial role in our polynomial-time algorithm for the word problem of Mk,1. Lemma 4.14 Let A be an acyclic DFA with a single accept state. Let ϕ be a normal right-ideal morphism, with domC(ϕ) 6= {ε} and imC(ϕ) 6= {ε}. Then ϕ−1(L(A)) is accepted by a one-accept-state acyclic DFA ϕ−1(A) whose number of states is size(ϕ−1(A)) < size(A) + Σ(domC(ϕ)). The transition table of the DFA ϕ−1(A) can be constructed deterministically in polynomial time, based on the transition table of A and the table of ϕ. Proof. If ϕ−1(L(A)) = ∅ then size(ϕ−1(A)) = 0, so the result is trivial. Let us assume now that ϕ−1(L(A)) 6= ∅. Let A = (S,A, δ, s0, {sA}) where sA is the single accept state; sA has no out-going edges (they would be useless). For any set X ⊆ A∗ and any state s ∈ S we denote {δ(s, x) : x ∈ X} by δ(s,X). Let P = domC(ϕ) and Q = imC(ϕ). Since A is acyclic, its state set S can be partitioned into δ(s0, spref(Q)) and δ(s0, QA ∗). Since Q 6= {ε}, the block δ(s0, spref(Q)) contains s0, so the block is non-empty. The block δ(s0, QA ∗) is non-empty because of the assumption ϕ−1(L(A)) 6= ∅, which implies L(A) ∩QA∗ 6= ∅. Since L(A) is a prefix code and ϕ is a right-ideal morphism, ϕ−1(L(A)) is a prefix code. To accept ϕ−1(L(A)) we define an acyclic DFA, called ϕ−1(A), as follows: • State set of ϕ−1(A): spref(P ) ∪ δ(s0, QA∗); start state: ε, i.e., the root of the prefix tree of P (since P 6= {ε}, ε ∈ spref(P )); accept state: the accept state sA of A. • State-transition function δ1 of ϕ−1(A): For every r ∈ spref(P ) and a ∈ A such that ra ∈ spref(P ): δ1(r, a) = ra. For every r ∈ spref(P ) and a ∈ A such that ra ∈ P : δ1(r, a) = δ(s0, ϕ(ra)). For every s ∈ δ(s0, QA∗): δ1(s, a) = δ(s, a). It follows immediately from this definition that for all p ∈ P : δ1(ε, p) = δ(s0, ϕ(p)). The construction of ϕ−1(A) assumes that ϕ maps P onto Q, i.e., it uses the assumption that ϕ is normal. As usual, “function” means partial function, so δ(., .) and δ1(., .) need not be defined on every state-letter pair. The DFA ϕ−1(A) can be pictured as being constructed as follows: The DFA has two parts. The first part is the prefix tree of P , but with the leaves left out (and with edges to leaves left dangling). The second part is the DFA A restricted to the state subset δ(s0, QA∗). The two parts are glued together by connecting any dangling edge, originally pointing to a leaf p ∈ P , to the state δ(s0, ϕ(p)) ∈ δ(s0, QA∗). The description of ϕ−1(A) constitutes a deterministic polynomial time algorithm for constructing the transition table of ϕ−1(A), based on the transition table of A and on the table of ϕ. By the construction, the number of states of ϕ−1(A) is < size(A) + Σ(P ) We will prove now that the DFA ϕ−1(A) accepts exactly ϕ−1(L(A)); i.e., ϕ−1(L(A)) = L(ϕ−1(A)). [⊆] Consider any y ∈ L(A) such that ϕ−1(y) 6= ∅. We want to show that ϕ−1(A) accepts all the words in ϕ−1(y). Since ϕ−1(y) 6= ∅ we have y ∈ Im(ϕ), hence y = qw for some strings q ∈ Q = imC(ϕ) and w ∈ A∗. Since Q is a prefix code, q and w are uniquely determined by y. Moreover, since y ∈ L(A) it follows that y has an accepting path in A of the form q−→ δ(s0, q) w−→ sA. For every x ∈ ϕ−1(y) we have x ∈ Dom(ϕ) = PA∗, hence x = pv for some strings p ∈ P and v ∈ A∗. So ϕ(x) = ϕ(p) v. We also have ϕ(x) = y = qw, hence ϕ(p) and q are prefix-comparable. Therefore, ϕ(p) = q, since Q is a prefix code and since ϕ(p) ∈ Q (by normality of ϕ); hence v = w. Thus every x ∈ ϕ−1(y) has the form pw for some string p ∈ ϕ−1(q). Now in ϕ−1(A) there is the following accepting path on input x = pw ∈ ϕ−1(y): p−→ δ1(ε, p) = δ(s0, ϕ(p)) w−→ sA. Thus ϕ−1(A) accepts x = pw = pv. [⊇] Suppose ϕ−1(A) accepts x. Then, because of the prefix tree of P at the beginning of ϕ−1(A), x has the form x = pw for some strings p ∈ P and w ∈ A∗. The accepting path in ϕ−1(A) on input pw has the form p−→ δ1(ε, p) = δ(s0, ϕ(p)) w−→ sA. Also, ϕ(x) = qw where q = ϕ(p) ∈ Q (here we use normality of ϕ). Hence A has the following computation path on input qw: q−→ δ(s0, q) = δ(s0, ϕ(p)) w−→ sA. So, ϕ(x) = ϕ(p)w = qw ∈ L(A). Hence, x ∈ ϕ−1(qw) ⊆ ϕ−1(L(A)). Thus L(ϕ−1(A)) ⊆ ϕ−1(L(A)). Corollary 4.15 Let A be an acyclic DFA with a single accept state. For i = 1, . . . , n, let Pi, Qi ⊂ A∗ be finite prefix codes, and let ϕi : PiA ∗ → QiA∗ be normal right-ideal morphisms. We assume that Pi 6= {ε} and Qi 6= {ε}. Then (ϕn ◦ . . .◦ϕ1)−1(L(A)) is accepted by an acyclic DFA with size < size(A)+ i=1 Σ(Pi), with one accept state. The transition table of this DFA can be constructed deterministically in polynomial time, based on the transition table of A and the tables of ϕi (for i = 1, . . . , n). Proof. We assume that (ϕn ◦ . . . ◦ ϕ1)−1(L(A)) 6= ∅ (since the empty set is accepted by a DFA of size 0). We use induction on n. For n = 1 the Corollary is just Lemma 4.14. Let n ≥ 1, assume the Corollary holds for n normal morphisms, and consider one more normal right- ideal morphism ϕ0 : P0A ∗ → Q0A∗, where P0, Q0 ⊂ A∗ are finite prefix codes with P0 6= {ε} 6= Q0. And assume (ϕn ◦ . . . ◦ ϕ1 ◦ ϕ0)−1(L(A)) 6= ∅. Since (ϕn ◦ . . . ◦ ϕ1◦ ϕ0)−1(L(A)) = ϕ−10 ◦ (ϕn ◦ . . . ◦ ϕ1)−1(L(A)), let us apply Lemma 4.14 to ϕ0 and the acyclic DFA (ϕn ◦ . . . ◦ ϕ1)−1(A). We have ε 6∈ Dom(ϕn . . . ϕ1ϕ0); indeed, Pi 6= {ε} is equivalent to ε 6∈ Dom(ϕi); moreover we have ε 6∈ Dom(ϕ0), and Dom(ϕn . . . ϕ1ϕ0) ⊆ Dom(ϕ0). Similarly, Qi 6= {ε} is equivalent to ε 6∈ Im(ϕi); and ε 6∈ Im(ϕn) implies ε 6∈ Im(ϕn . . . ϕ1ϕ0). The conclusion of Lemma 4.14 is then that (ϕn◦. . .◦ϕ1 ◦ ϕ0)−1(L(A)) is accepted by an acyclic DFA (ϕn◦. . .◦ϕ1◦ϕ0)−1(A) whose size is < size((ϕn◦. . .◦ϕ1)−1(A)) + Σ(P0) < size(A)+ i=1 Σ(Pi)+Σ(P0) = size(A) + i=0Σ(Pi). ✷ Proof of Theorem 4.13: Let (ρm, . . . , ρ1) and (σn, . . . , σ1) be two sequences of generators from the finite generating set Γk,1. The elements of Γk,1 can be chosen so that the assumptions of Corollary 4.15 hold; see Section 3 of [1], where such a generating set is given. We want to decide in deterministic polynomial time whether the products ρm · . . . · ρ1 and σn · . . . · σ1 are the same, as elements of Mk,1. First, by Corollary 4.11 (Output 2) we can compute the sets imC(ρm◦. . .◦ρ1) and imC(σn◦. . .◦σ1), explicitly described by lists of words, in polynomial time. By Lemma 4.9 (Question 2) we can check in polynomial time whether the right ideal Im(ρm◦. . .◦ρ1) ∩ Im(σn◦. . .◦σ1) is essential in Im(ρm◦. . .◦ρ1) and in Im(σn ◦ . . . ◦ σ1). If it is not essential we immediately conclude that ρm · . . . · ρ1 6= σn · . . . · σ1. On the other hand, if it is essential, Lemma 4.9 (Output 1) lets us compute a generating set Π for the right ideal Im(ρm ◦ . . . ◦ ρ1) ∩ Im(σn ◦ . . . ◦ σ1), in deterministic polynomial time; the generating set Π is a finite prefix code, given explicitly by a list of words. By Corollary 4.7 and because Π ⊆ imC(ρm ◦ . . .◦ρ1) ∪ imC(σn ◦ . . .◦σ1), Π has linearly bounded cardinality and the length of the longest words in Π is linearly bounded in terms of n+m. We restrict ρm ◦ . . . ◦ ρ1 and σn ◦ . . . ◦ σ1 in such a way that their images are ΠA∗; i.e., we replace them by ρ = idΠA∗ ◦ ρm ◦ . . . ◦ ρ1, respectively σ = idΠA∗ ◦ σn ◦ . . . ◦ σ1. So, Im(ρ) = ΠA∗ = Im(σ). Also, since ΠA∗ is essential in Im(ρm ◦ . . .◦ρ1) and in Im(σn ◦ . . .◦σ1) we have: ρ is equal to ρm · . . . ·ρ1 in Mk,1, and σ is equal to σn · . . . · σ1 in Mk,1. So for deciding the word problem it is enough to check whether ρ = σ in Mk,1. By the next Claim, the sets ρ(domC(ρ)) and σ(domC(σ)) play a crucial role. However, instead of directly computing ρ(domC(ρ)) and σ(domC(σ)), we compute finite sets R1, R2 ⊂ A∗ such that ρ(domC(ρ)) ⊆ R1 and σ(domC(σ)) ⊆ R2 . Moreover, since ρ(domC(ρ))∪σ(domC(σ)) ⊆ ΠA∗, we can pick R1, R2 so that R1 ∪R2 ⊆ ΠA∗. By Corollary 4.11 (Output 1), the sets R1, R2 can be computed in polynomial time as explicit lists of words. Let R = R1 ∪R2. Claim. ρ = σ in Mk,1 iff ρ −1(r) = σ−1(r) for every r ∈ ρ(domC(ρ)) ∪ σ(domC(σ)). The latter is equivalent to ρ−1(r) = σ−1(r) for every r ∈ R. Proof of the Claim. If ρ = σ in Mk,1 then ρ −1(r) = σ−1(r) for every r ∈ ΠA∗ = Im(ρ) = Im(σ). Hence this holds in particular for all r ∈ ρ(domC(ρ)) ∪ σ(domC(σ)) and for all r ∈ R, since ρ(domC(ρ)) ∪ σ(domC(σ)) ⊆ R ⊂ ΠA∗. Conversely, if ρ−1(r) = σ−1(r) for every r ∈ ρ(domC(ρ)) ∪ σ(domC(σ)), then for all x ∈ ρ−1(r) = σ−1(r): ρ(x) = r = σ(x). Since domC(ρ) ⊆ ρ−1(ρ(domC(ρ))) and domC(σ) ⊆ σ−1(σ(domC(σ))), it follows that ρ and σ are equal on domC(ρ) ∪ domC(σ), and it follows that domC(ρ) = domC(σ). Hence ρ and σ are equal as right-ideal morphisms, and hence as elements of Mk,1. [This proves the Claim.] Recall that |R| and ℓ(R), and hence Σ(R), are polynomially bounded in terms of the input size. To check for each r ∈ R whether ρ−1(r) = σ−1(r), we apply Corollary 4.15, which constructs an acyclic DFA Aρ for ρ−1(r) from a DFA for {r}; this is done deterministically in polynomial time. Similarly, an acyclic DFA Aσ for σ−1(r) is constructed. Thus, ρ−1(r) = σ−1(r) iff Aρ and Aσ accept the same language. Checking whether Aρ and Aσ accept the same language is an instance of the equivalence problem for DFAs that are given explicitly by transition tables. It is well known (see e.g., [21], or [24] p. 103) that the equivalence problem for DFAs is decidable deterministically in polynomial time. This proves Theorem 4.13. ✷ Acknowledgement. I would like to thank John Meakin for many discussions over the years concern- ing the Thompson groups and generalizations to inverse monoids. References [1] J.C. Birget, “Monoid generalizations of the Richard Thompson groups”, J. of Pure and Applied Algebra, 213(2) (Feb. 2009) 264-278. (Preprint: http://arxiv.org/abs/0704.0189, v1 April 2007, v2 April 2008.) [2] J.C. Birget, “One-way permutations, computational asymmetry and distortion”, J. of Algebra 320(11) (Dec. 2008) 4030-4062. [3] J.C. Birget, “Factorizations of the Thompson-Higman groups, and circuit complexity”, International J. of Algebra and Computation, 18.2 (March 2008) 285-320. [4] J.C. Birget, “The groups of Richard Thompson and complexity”, International J. of Algebra and Compu- tation 14(5,6) (Dec. 2004) 569-626. [5] J.C. Birget, “Circuits, coNP-completeness, and the groups of Richard Thompson”, International J. of Algebra and Computation, 16(1) (Feb. 2006) 35-90. [6] J.C. Birget, “Bernoulli measure on strings, and Thompson-Higman monoids”, Semigroup Forum 83.1 (Aug. 2011) 1-32. [7] J.C. Birget, “Polynomial-time right-ideal morphisms and congruences”. Preprint: http://arxiv.org/abs/1511.02056 (Nov. 2015) [8] M. Brin, C. Squier, “Groups of piecewise linear homeomorphisms of the real line”, Inventiones Mathemat- icae 79 (1985) 485-498. [9] M. Brin, “The Chameleon Groups of Richard J. Thompson: Automorphisms and Dynamics”, Publications Math. de l’IHES 84 (1997) 5-33. [10] K. Brown, R. Geoghegan, “An infinite-dimensional torsion-free FP∞ group”, Inventiones Mathematicae 77 (1984) 367-381. [11] J. Burillo, S. Cleary, M. Stein, J. Taback, “Combinatorial and metric properties of Thompson’s group T ”, Trans. Amer. Math. Soc. 361(2) (2009) 631-652. Preprint: http://arxiv.org/pdf/math/0503670 [12] J. W. Cannon, W. J. Floyd, W. R. Parry, “Introductory notes on Richard Thompson’s groups”, L’Enseignement Mathématique 42 (1996) 215-256. [13] A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. 1 (Mathematical Survey, No 7 (I)) American Mathematical Society, Providence (1961). [14] J. Cuntz, “Simple C∗-algebras”, Communications in Mathematical Physics 57 (1977) 173-185. [15] P. Dehornoy, “Geometric presentations for Thompson’s groups”, J. of Pure and Applied Algebra 203 (2005) 1-44. [16] J. Dixmier, “Traces sur les C∗-algèbres II”, Bulletin des Sciences Mathématiques 88 (1964) 39-57. [17] E. Ghys, V. Sergiescu, “Sur un groupe remarquable de difféomorphismes du cercle”, Commentarii Mathe- matici Helvetici 62(2) (1987) 185-239. [18] V. Guba, M.V. Sapir, “Diagram groups”, Memoirs American Math. Soc., 130 no. 620 (1997), viii+117 pages. [19] P.A. Grillet, Semigroups, An Introduction to the Structure Theory, Marcel Dekker, New York (1995). http://arxiv.org/abs/0704.0189 http://arxiv.org/abs/1511.02056 http://arxiv.org/pdf/math/0503670 [20] G. Higman, “Finitely presented infinite simple groups”, Notes on Pure Mathematics 8, The Australian National University, Canberra (1974). [21] J. Hopcroft, J. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley (1979). [22] B. Hughes, “Trees, Ultrametrics, and Noncommutative Geometry”, http://arxiv.org/pdf/math/0605131v2.pdf [23] M.V. Lawson, “Orthogonal completions of the polycyclic monoids”, Communications in Algebra, 35 (2007) 1651-1660. [24] H. Lewis, Ch. Papadimitriou, Elements of the Theory of Computation, 2nd ed., Prentice Hall (1998). [25] J. Lehnert, P. Schweitzer, “The co-word problem for the Higman-Thompson group is context-free”, Bulletin of the London Mathematical Society, 39 (April 2007) 235-241. [26] R. McKenzie, R. J. Thompson, “An elementary construction of unsolvable word problems in group theory”, in Word Problems, (W. W. Boone, F. B. Cannonito, R. C. Lyndon, editors), North-Holland (1973) pp. 457-478. [27] V.V. Nekrashevych, “Cuntz-Pimsner algebras of group actions”, J. Operator Theory 52(2) (2004) 223-249. [28] Elizabeth A. Scott, “A construction which can be used to produce finitely presented infinite simple groups”, J. of Algebra 90 (1984) 294-322. [29] Richard J. Thompson, Manuscript (1960s). [30] Richard J. Thompson, “Embeddings into finitely generated simple groups which preserve the word prob- lem”, in Word Problems II, (S. Adian, W. Boone, G. Higman, editors), North-Holland (1980) pp. 401-441. J.C. Birget Dept. of Computer Science Rutgers University – Camden Camden, NJ 08102 [email protected] http://arxiv.org/pdf/math/0605131v2.pdf 1 Thompson-Higman monoids 1.1 Definition of the Thompson-Higman groups and monoids 1.2 Other Thompson-Higman monoids 1.3 Cuntz algebras and Thompson-Higman monoids 2 Structure and simplicity of the Thompson-Higman monoids 2.1 Group of units, J-relation, simplicity 2.2 D-relation 3 Finite generating sets 4 The word problem of the Thompson-Higman monoids 4.1 The image code formula 4.2 Some algorithmic problems about right-ideal morphisms 4.3 The word problem of Mk,1 is in P

0704.0190

The Reliability on the Direction of the Incident Neutrino for the Fully Contained Events and Partially Contained Events due to QEL in the Super-Kamiokande

EPJ manuscript No. (will be inserted by the editor) The Reliability on the Direction of the Incident Neutrino for the Fully Contained Events and Partially Contained Events due to QEL in the Super-Kamiokande E. Konishi1, Y. Minorikawa2, V.I. Galkin3, M. Ishiwata4, I. Nakamura4, N. Takahashi1, M. Kato5 and A. Misaki6 1 Graduate School of Science and Technology, Hirosaki University, Hirosaki, 036-8561, Japan 2 Department of Science, School of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502, Japan 3 Department of Physics, Moscow State University, Moscow, 119992, Russia 4 Department of Physics, Saitama University, Saitama, 338-8570, Japan 5 Kyowa Interface Science Co.,Ltd., Saitama, 351-0033, Japan 6 Advanced Research Institute for Science and Engineering, Waseda University, Tokyo, 169-0092, Japan e-mail: [email protected] Received: / Revised version: March 30, 2007 Abstract. In the SK analysis of the neutrino events for Fully Contained Events and Partially Contained Events on their zenith angle distribution, it is assumed that the zenith angle of the incident neutrino is the same as that of the detected charged lepton. In the present paper, we examine the validity of the SK assumption on the direction of the incident neutrinos. Concretely speaking, we analyze muon-like events due to QEL. For the purpose, we develop Time Sequential Monte Carlo Simulation to extract the conclusion on the validity of the SK assumption. In our Time Sequential Simulation, we simulate every physical process concerned as exactly as possible without any approximation. From the comparison between the zenith angle distributon of the emitted muons under the SK assumption on the direction and the corresponding one obtained under our Time Sequential Simulation, it is concluded that the measurement of the direction of the incident neutrino for the neutrino events occurring inside the detector in the SK analysis turns out to be unreliable, which holds irrespective of the existence and/or non-existence of the neutrino oscillation. PACS. Superkamiokande, QEL, Fully Contained Event, Time Sequential Monte Carlo Simulation 1 Introduction Superkamiokande have been analyzing Fully Contained Events and Partially Contained Events which are gen- erated inside the detector, and Upward Through Going Events and Stopping Events which are generated outside the detector, for the studies on the neutrino oscillation in atmospheric neutrinos. The report of oscillations between muon and tau neutrinos for atmospheric neutrinos de- tected with SuperKamiokande (SK, hereafter) is claimed to be robustly established for the following reasons: (1) The discrimination between electrons and muons in the SK energy range, say, several hundred MeV to several GeV, has been proved to be almost perfect, as demon- strated by calibration using accelerator beams [1] 1. 1 The SK discrimination procedure between muon and elec- tron is constructed on the average value theory. In our opinion, discrimination procedure should be examined, taking into ac- count the stochastic characters of the physical processes in the neutrino events concerned. If we take this effect into account, then, for example, we give uncertainties of 3◦ to 14◦ in the in- (2) The analysis for the electron-like events and the muon- like events which give the single-ring structure in Fully Contained Events and Partially Contained Events with their zenith angle distribution, based on the well es- tablished discrimination procedure mentioned in (1), reveals a significant deficit of muon-like events but the expected level of electron-like events. It is, thus, concluded that muon neutrinos oscillate into tau neu- trinos which cannot be detected due to the small ge- ometry of SK. As the most new one, the SK collab- oration published their comprehensive paper[2]. The analysis of SK data presently yields sin22θ > 0.92 and 1.5× 10−3eV2 < ∆m2 < 3.4× 10−3eV2 at 90% confi- dence level. (3) The analysis of Upward Through Going Events and Stopping Events, in which the neutrino interactions occur outside the detector, leads to similar results to cident direction of the charged lepton and uncertainties of 2m to 7m in the vertex point of the events. See, our papers [2]. However, SK give 1.8◦ to 3.0◦ and 0.3m for the same physical quantities. See, accompanied two papers. http://arxiv.org/abs/0704.0190v1 2 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK (2). The charged leptons which are produced in these categories are regarded as being exclusively muons, be- cause electrons have negligible probabilities to produce such events as they lose energy very rapidly in the sor- rounding rock. Thus for these events the discrimina- tion procedure described in (1) is not required, and, therefore, the analysis here is independent of the anal- ysis in (2). For these events, however, the SK group obtains the same parameters for neutrino oscillations as in (2) 2. (4) Now, SK assert that they have found the oscillatory signature in atomospheric neutrinos from L/E anal- ysis, which should be the ultimate evidence for the exsistence of the neutrino oscillation [3] Our critical examination on the L/E analysis by SK will be pub- lished elsewhere. As for item(3), we have clarified that SK hardly dis- criminate electron( neutrino) from muon( neutrino ) in the SK manner and, instead, propose more rigorous and suitable procedure with theoretical background for the dis- crimination between them in the preceeding two papers. Among the neutrino events both occurring inside and out- side the detector, the most robust evidence for the neu- trino oscillation, if exists, should have been obtained from the analysis of both electron-like events and muon-like events in Fully Contained Events. Because (i) all necces- sary informations for the physical interpretation are in- cluded in Fully Contained Events due to their character and (ii), furthermore, both electron-like events and muon-like events give the single structure image free from arbitrary inter- pretation with the proper electron/muon discrimination procedure. SK treat neutrino events whose energies cover from several hundred MeV to several GeV, if the neutrino events occur inside the detector. In this energy region, Quasi Elastic Scattering(QEL) [4] is dominant compared with other physical processes, such as one-pion produc- tion [5], coherent pion production [6] and deep inelastic scattering [7]. Events due to other processes, except QEL, are not free from ambiguities due to multi-ring structure of the images. Therefore, SK should have analyzed the muon- like events and the electron-like events with the single ring image in Fully Contained Events exculu- sively where QEL is dominant, without utilizing poorer quality events, if SK pursue to obtain the clear cut conclusion on the neutrino oscillation 3. 2 It seems strange that the experimental data with different qualities give similarly precise results, because Fully Contained Events whose information is totally inside the detector are of higher experimental qualities compared with those of both Up- ward Through Going Events and Stopping Events. 3 In their analysis, they really add Partially Contained Events to the experimental data as those with the same quality under the assumption that they belong to muon-like events in Fully Contained Events to raise the statistics higher. However, such the assumption lacks in theoretical background. Further- more, SK utilize to add multi-ring structure events which are Therefore, it is essential for us to examine single ring structure events among Fully Contained Events due to QEL which have the least ambiguities among the neutrino events concerned to obtain clear cut conclusion as for the neutrino oscillation. Here, the main concern of the present paper is devoted to the detailed analysis of the muon-like events from QEL, focusing on the direction of the incident neutrino. Situation around the corresponding electron-like event is the same as in the muon-like event. The exami- nation on the separation of Fully Contained Events from Partially Contained Events will be discussed in subsequent papers. Here, it should be emphasized that the direction of the incident neutrino is assumed to be the same as that of the emitted charged lepton, i.e., the (anti-)muon or (anti-)electron, in the SK analysis of both Fully Contained Events and Partially Contained Events [8,9]. The SK De- tector Simulation is to be constructed without any con- tradiction with the SK assumption on the direction. From the point of orthodoxical Monte Carlo Simula- tion, it seems to be unnatural for SK to impose such the assumption that the direction of the incident neutrino is the same as that of the emitted lepton ( hereafter, we call this assumption simply ”the SK assumption on the di- rection”) upon their Detector Simulation. Obviously, one need not any assumption on the relation between the di- rection of the incident neutrino and that of the emitted lepton in any sense, if we develop the Monte Carlo Simu- lation in a rigorous manner, which will be shown later in the present paper. In order to avoid any misunderstanding toward the SK assumption on the direction we reproduce this assumption from the original SK paper: ”However, the direction of the neutrino must be estimated from the reconstructed direction of the products of the neu- trino interaction. In water Cherenkov detectors, the direc- tion of an observed lepton is assumed to be the direction of the neutrino. Fig.11 and Fig.12 show the estimated corre- lation angle between neutrinos and leptons as a function of lepton momentum. At energies below 400 MeV/c, the lep- ton direction has little correlation with the neutrino direc- tion. The correlation angle becomes smaller with increas- ing lepton momentum. Therefore, the zenith angle depen- dence of the flux as a consequence of neutrino oscillation is largely washed out below 400 MeV/c lepton momentum. With increasing momentum, the effect can be seen more clearly. ” [8] 4. On the other hand, Ishitsuka states in his Ph.D thesis which is exclusively devoted into the L/E analysis of the caused by one-pion roduction, coherent pion production and deep inelastic scattering. However, the discrimination among the multi-ring structures is not so easy, which may lead the worse estimation of energies as well as directions of the events concerned. 4 It could be understood from this statement that SK justify the validity of this assumption above 400 MeV/c. However, it is not correct, because SK put ”to be proved ” as the proposition. See, page 101 in their paper [8]. E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 3 atmospheric neutrino from Super Kamiokande as follows: ” 8.4 Reconstruction of Lν Flight length of neutrino is determined from the neu- trino incident zenith angle, although the energy and the flavor are also involved. First, the direction of neutrino is estimated for each sample by a different way. Then, the neutrino flight lenght is calclulated from the zenith angle of the reconstructed direction. 8.4.1 Reconstruction of Neutrino Direction FC Single-ring Sample The direction of neutrino for FC single-ring sample is simply assumed to be the same as the reconstructed direc- tion of muon. Zenith angle of neutrino is reconstructed as follows: cosΘrecν = cosΘµ (8.17) ,where cosΘrecν and cosΘµ are cosine of the reconstructed zenith angle of muon and neutrino, respectively. ” [9] 5. In our understanding, SK Monte Carlo Simulation is named usually as the Detector Simulation. It is, however, noticed that the effect of the azimuthal angles of the emit- ted leptons in QEL could not be taken into account in their Simulation. As will be shown in later (see Section 3), this effect greatly influences over the final zenith angle distri- bution of the emitted leptons. Also, the back scattering due to QEL can not be neglected for the rigorous deter- mination of the direction of the incident neutrino, but this effect could not be treated in the SK Detector Simulation, which is beyond the application limitaition 6. On the other hand, we could take into account these effects correctly in our Monte Carlo Simulation which is named as Time Sequential Simulation. In the present paper, we carry out the full Monte Carlo Time Sequential Simulation as exactly as possible, with- out the SK assumption on the direction to clarify the prob- lematic issue raised by SK. We carry out simulation which starts from the opposite side of the Earth to the SK de- tector. A neutrino sampled from the atmospheric neutrino energy spectrum at the opposite side of the Earth tra- verses through the medium with different densities in the interior of the Earth and penetrates finally into the SK 5 It should be noticed that the SK assumption on the di- rection may hold on the following possible two cases: [1] The scattering angle of the emitted lepton is so small that the ef- fect of the scattering angle could be neglected really. However, in the present case, it could not be true from Fig. 1 and Fig. 2 and Table 1. [2] One may assert that the assumption could not hold on individual case, but it could hold statistically af- ter accumulation of large amount of the data. However, such assertion should be verified. We verify such assumption could not hold. See, Fig. 11 and Fig. 12 in the present paper. 6 SK have never clarified not only the details, but also the principle and its validity on their Monte Carlo Simulation. We hope disclosure of their Detector Simulation for open and fair scientific discussion. detector where the neutrino interactions occur. The emit- ted energy of the individual lepton thus produced and its direction are simulated exactly based on the probability function of the cross sections concerned. We finally show the zenith angle distribution of the emitted leptons as well as that of the incident neutrinos are quite different from corresponding ones of the SK. This indicates that the SK assumption on the direction coud not be a reliable estimator as for the determination of the direction of the incident neutrino (See, section 5). 2 Cross Sections of Quasi Elastic Scattering in the Neutrino Reaction and the Scattering Angle of Charged Leptons. We examine the following reactions due to the charged current interaction (c.c.) from QEL. νe + n −→ p+ e− νµ + n −→ p+ µ− ν̄e + p −→ n+ e+ (1) ν̄µ + p −→ n+ µ+ The differential cross section for QEL is given as fol- lows [6]. dσℓ(ℓ̄)(Eν(ν̄)) G2F cos ν(ν̄) A(Q2)±B(Q2) C(Q2) where A(Q2) = + f1f2 + g21 B(Q2) = (f1 + f2)g1Q C(Q2) = f21 + f + g21 The signs + and − refer to νµ(e) and ν̄µ(e) for charged current (c.c.) interactions, respectively. The Q2 denotes the four momentum transfer between the incident neu- trino and the charged lepton. Details of other symbols are given in [4]. The relation among Q2, Eν(ν̄), the energy of the in- cident neutrino, Eℓ, the energy of the emitted charged lepton (muon or electron or their anti-particles) and θs, the scattering angle of the emitted lepton, is given as Q2 = 2Eν(ν̄)Eℓ(1− cosθs). (3) 4 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK Fig. 1. Relation between the energy of the muon and its scattering angle for different incident muon neutrino energies, 0.5 GeV, 1 GeV, 2 GeV, 5 GeV, 10 GeV and 100 GeV. Also, the energy of the emitted lepton is given by Eℓ = Eν(ν̄) − . (4) Now, let us examine the magnitude of the scattering angle of the emitted lepton in a quantitative way, as this plays a decisive role in determining the accuracy of the direction of the incident neutrino, which is directly related to the reliability of the zenith angle distribution of both Fully Contained Events and Partially Contained Events in By using Eqs. (2) to (4), we obtain the distribution function for the scattering angle of the emitted leptons and the related quantities by a Monte Carlo method. The procedure for determining the scattering angle for a given energy of the incident neutrino is described in the Ap- pendix A. Fig. 1 shows this relation for muon, from which we can easily understand that the scattering angle θs of the emitted lepton ( muon here ) cannot be neglected. For a quantitative examination of the scattering angle, we construct the distribution function for θs of the emit- ted lepton from Eqs. (2) to (4) by using a Monte Carlo method. Fig. 2 gives the distribution function for θs of the muon produced in the muon neutrino interaction. It can be seen that the muons produced from lower energy neutrinos are scattered over wider angles and that a considerable part of them are scattered even in backward directions. Simi- lar results are obtained for anti-muon neutrinos, electron neutrinos and anti-electron neutrinos. Also, in a similar manner, we obtain not only the dis- tribution function for the scattering angle of the charged leptons, but also their average values < θs > and their standard deviations σs. Table 1 shows them for muon neu- trinos, anti-muon neutrinos, electron neutrinos and anti- electron neutrinos. In the SK analysis, it is assumed that the scattering angle of the charged particle is zero [8,9]. Fig. 2. Distribution functions for the scattering angle of the muon for muon-neutrino with incident energies, 0.5 GeV, 1.0 GeV and 2 GeV. Each curve is obtained by the Monte Carlo method (one million sampling per each curve). 3 Influence of Azimuthal Angle of Quasi Elastic Scattering over the Zenith Angle of both the Fully Contained Events and Partially Contained Events In the present section, we examine the effect of the az- imuthal angles of the emitted leptons over their own zenith angles for given zenith angles of the incident neutrinos 7. For three typical cases (vertical, horizontal and diag- onal), Fig. 3 gives a schematic representation of the re- lationship between, θν(ν̄), the zenith angle of the incident neutrino, and (θs, φ) a pair of scattering angle of the emit- ted lepton and its azimutal angle. From Fig. 3(a), it can been seen that the zenith angle θµ(µ̄) of the emitted lepton is not influenced by its φ in the vertical incidence of the neutrinos (θν(ν̄) = 0 o), as it must be. From Fig. 3(b), however, it is obvious that the influence of φ of the emitted leptons on their own zenith angle is the strongest in the case of horizontal incidence of the neutrino (θν(ν̄) = 90 o). Namely, one half of the emitted leptons are recognized as upward going, while the other half is classified as downward going ones. The di- agonal case ( θν(ν̄) = 43 o) is intermediate between the vertical and the horizontal. In the following, we examine the cases for vertical, horizontal and diagonal incidence of the neutrino with different energies, say, Eν(ν̄) = 0.5 GeV, Eν(ν̄) = 1 GeV and Eν(ν̄) = 5 GeV. The detailed procedure for the Monte Carlo simulation is described in the Appendix A. 7 Throughout this paper, we measure the zenith angles of the emitted leptons from the upward vertical direction of the incident neutrino. Consequently, notice that the sign of our direction is oposite to that of the SK ( our cos θν(ν̄) = - cos θν(ν̄) in SK) E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 5 Table 1. The average values < θs > for scattering angle of the emitted charged leptons and their standard deviations σs for various primary neutrino energies Eν(ν̄). Eν(ν̄) (GeV) angle νµ(µ̄) ν̄µ(µ̄) νe ν̄e (degree) 0.2 < θs > 89.86 67.29 89.74 67.47 σs 38.63 36.39 38.65 36.45 0.5 < θs > 72.17 50.71 72.12 50.78 σs 37.08 32.79 37.08 32.82 1 < θs > 48.44 36.00 48.42 36.01 σs 32.07 27.05 32.06 27.05 2 < θs > 25.84 20.20 25.84 20.20 σs 21.40 17.04 21.40 17.04 5 < θs > 8.84 7.87 8.84 7.87 σs 8.01 7.33 8.01 7.33 10 < θs > 4.14 3.82 4.14 3.82 σs 3.71 3.22 3.71 3.22 100 < θs > 0.38 0.39 0.38 0.39 σs 0.23 0.24 0.23 0.24 Fig. 3. Schematic view of the zenith angles of the charged muons for diffrent zenith angles of the incident neutrinos, fo- cusing on their azimuthal angles. 3.1 Dependence of the spreads of the zenith angle for the emitted leptons on the energies of the emitted leptons for different incident directions with different energies We give the scatter plots between the fractional energies of the emitted muons and their zenith angle for a definite zenith angles of the incident neutrino with different ener- gies in Figs. 4 to 6. In Fig. 4, we give the scatter plots for vertically incident neutrino with different energies 0.5 GeV, 1 GeV and 5 GeV . In this case, the relations between the emitted energies of the muon and and their zenith an- gles are unique, which comes from the definition of the zenith angle of the emitted lepton. However, the densities (frequencies of event number) along each curve is differ- ent in position to position and depend on the energies of the incident neutrinos. Generally speaking,densities along curves become higher toward cos θµ(µ̄) = 1. In this case, cos θµ(µ̄) is never influenced by the azimuthal angel in the scattering by the definition 8. Fig. 5 tells us that the horizontally incident neutrinos give the most widely spread of the zenith angle distribu- tion of the emitted lepton influenced by the azimuthal an- gle. The more lower incident neutrino energies, the more wider spreads of the emitted leptons. The diagonally in- cident neutrinos give the intermediate distribution of the emitted leptons between those of vertically incident neu- trinos and horizontally incident neutrinos. 8 The zenith angles of the particles concerned are measured from the vertical direction. 6 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK (a) (b) (c) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=0.5GeV cosθν=1(θν=0°) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=1GeV cosθν=1(θν=0°) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=5GeV cosθν=1(θν=0°) Fig. 4. The scatter plots between the fractional energies of the produced muons and their zenith angles for vertically incident muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case. (a) (b) (c) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=0.5GeV cosθν=0(θν=90°) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=1GeV cosθν=0(θν=90°) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=5GeV cosθν=0(θν=90°) Fig. 5. The scatter plots between the fractional energies of the produced muons and their zenith angles for horizontally incident muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case. (a) (b) (c) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=0.5GeV cosθν=0.731(θν=43°) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=1GeV cosθν=0.731(θν=43°) 0 0.2 0.4 0.6 0.8 1 Eµ / Eν Eν=5GeV cosθν=0.731(θν=43°) Fig. 6. The scatter plots between the fractional energies of the produced muons and their zenith angles for diagonally incident muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case. E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 7 (a) (b) (c) −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=0.5GeV cosθν=1(θν=0°) average=0.262 s.d.=0.547 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=1GeV cosθν=1(θν=0°) average=0.590 s.d.=0.439 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=5GeV cosθν=1(θν=0°) average=0.978 s.d.=0.067 Fig. 7. Zenith angle distribution of the muon for the vertically incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption. (a) (b) (c) −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=0.5GeV cosθν=0(θν=90°) average=0.003 s.d.=0.564 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=1GeV cosθν=0(θν=90°) average=0.001 s.d.=0.480 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=5GeV cosθν=0(θν=90°) average=0.006 s.d.=0.141 Fig. 8. Zenith angle distribution of the muon for the horizontally incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption. (a) (b) (c) −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=0.5GeV cosθν=0.731(θν=43°) average=0.189 s.d.=0.556 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=1GeV cosθν=0.731(θν=43°) average=0.432 s.d.=0.463 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 cosθµ muon neutrino Eν=5GeV cosθν=0.731(θν=43°) average=0.715 s.d.=0.103 Fig. 9. Zenith angle distribution of the muon for the diagonally incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption. 8 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 3.2 Zenith angle distribution of the emitted lepton for the different incidence of the neutrinos with different energies In Figs. 7 to 9, we give the zenith angle distributions of the emitted muons for the given direction of the incident neutrinos with different energies of the neutrino. These figures are obtained through summation on the energies of the emitted muons for their definite zenith angles in Figs. 4 to 6. In Figs. 7(a) to 7(c), we give the zenith angle distri- bution of the emitted muon for the case of vertically inci- dent neutrinos with different energies, say, Eν = 0.5 GeV, Eν = 1 GeV and Eν = 5 GeV. Comparing the case for 0.5 GeV with that for 5 GeV, we understand the big contrast between them as for the zenith angle distribution. The scattering angle of the emit- ted muon for 5 GeV neutrino is relatively small (See, Table 1) that the emitted muons keep roughly the same direction as their original neutrino. In this case, the effect of their azimuthal angle on the zenith angle is also small. However, in the case for 0.5 GeV which is the dominant energy for Fully Contained Events in the Superkamiokande, there is even a possibility for the emitted muon to be emitted in the backward direction due to the large angle scattering, the effect of which is enhanced by their azimuthal angle. The most frequent occurrence in the backward scatter- ing of the emitted muon appear in the horizontally inci- dent neutrino as shown in Figs. 8(a) to 8(c). In this case, the zenith angle distribution of the emitted muon should be symmetrical to the horizontal direction. Comparing the case for 5 GeV with those both for ∼0.5 GeV and for ∼1 GeV, even 1 GeV incident neutrinos lose almost the orig- inal sense of the incidence if we measure it by the zenith angle of the emitted muon. Figs. 9(a) to 9(c) for the di- agonally incident neutrino tell us that the situation for diagonal cases lies between the case for the vertically in- cident neutrino and that for horizontally incident one. 4 Zenith Angle Distribution of Fully Contained Events and Partially Contained Events for a Given Zenith Angle of the Incident Neutrino, Taking Their Energy Spectrum into Account In the previous sections, we discuss the relation between the zenith angle distribution of the incident neutrino with a single energy and that of the emited muons produced by the neutrino for the different incident direction. In order to apply our motivation around the uncertainty of the SK as- sumption on the direction for Fully Contained Events and Partially Contained Events, we must consider the effect of the energy spectrum of the incident neutrino. The Monte Carlo simulation procedure for this purpose are given in the Appendix B. In Fig. 10, we give the zenith angle distributions of the sum of µ+(µ̄) and µ− for a given zenith angle of ν̄µ̄ and νµ, taking into account primary neutrino energy spectrum at Kamioka site. In Table 2, the average values for cosθµ+µ̄ and their standard deviation for different incidences of the incident neutrinos with different energies are presented 9. In the SK case, their average values are given by cosθν(ν̄) themselves by definition and, consequently, the standard deviations are zero under the assumption, because the SK assumption is of the delta function for the incidence direc- tion. They are shown in the bottom line of Table 2. In the second line from the bottom in this table, we give the av- erage values and their standard deviations for cos θν+ν̄ ob- tained under the inclusion of the energy spectrum for pri- mary neutrinos. Thus, we found these values correspond to those for incident neutrino with the effective single energy between 0.5 GeV and 1 GeV. If we compare the average energies and the standard deviations for the inclusion of incident neutrino energy spectrum with those under the SK assumption, it is easily understood that SK assump- tion does not represent real zenith angle distribution of the emitted muon. 5 Relation between the Zenth angle Distribution of the Incident Neutrinos and that of the emitted leptons Now, we extend the results for the definite zenith angle obtained in the previous section to the case in which we consider the zenih angle distribution of the incident neu- trinos totally. Here, we examine the real correlation between cos θν and cos θµ, by peforming the exact Monte Carlo simula- tion. The detail for the simulation procedure is given the Appendix C. In Fig. 11 we classsify the correlation between cos θν and cos θµ according to the different energy range of the incident muon neutrinos. It should be noticed that the SK assumption on cos θν = cos θµ is roughly hold only for Eν ≥ 5 GeV, but the widths in cos θµ for the definite cos θν near cos θν =0 (for horizontally incident neutrino ) are much larger than those near cos θν =1 ( for the vertically incident neutrino). Of course, this is due to the effect of the azimutal angle in QEL which could not be derived by the SK simulation (DETECTOR SIMULATION ). Such tendencies become more remarkable in Eν ≤ 5 GeV and in these energies the SK assumption on the direction does not hold any more. In Fig. 12, we classify the correlation between cos θν and cos θµ according to the different energy range of Eµ. The similar argument on Fig. 11 can be done on the case of Fig.12 9 Notice that the difference in the corresponding quantites between the case for single energy and the case for the energy spectrum. The formers are given in the µ−,while th latter is given in µ− and µ+. However, such the difference does not change the essential recognition. E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 9 Table 2. Average values and their standerd deviations in cosθµ+µ̄ for the zenith angle distributions of the muons with different primary energies of the insident neutrinos. Vertical Diagonal Horizontal cosθν+ν̄ = 1 (0 ◦) cosθν+ν̄ = 0.731 (43 ◦) cosθν+ν̄ = 0 (90 Eν+ν̄(GeV) cos θµ+µ̄ σcos θµ+µ̄ cos θµ+µ̄ σcos θµ+µ̄ cos θµ+µ̄ σcos θµ+µ̄ 0.5 0.262 0.547 0.189 0.556 -0.003 0.564 1.0 0.590 0.439 0.432 0.463 0.001 0.480 2.0 0.581 0.250 0.623 0.290 0.001 0.325 5.0 0.978 0.067 0.715 0.103 0.006 0.141 Spectrum∗ 0.468 0.531 0.339 0.519 -0.005 0.500 SK∗∗ 1.00 0.00 0.731 0.000 0.000 0.000 (a) (b) (c) −1 −0.5 0 0.5 1 cosθµ+µ− µ+ and µ− cosθν+ν − =1(θν+ν −=0°) Avg.=0.81 S.D.=0.30 −1 −0.5 0 0.5 1 µ − cosθµ+µ − µ+ and µ− cosθν+ν −=0(θν+ν −=90°) Avg.=0.00 S.D.=0.34 −1 −0.5 0 0.5 1 µ − cosθµ+µ − µ+ and µ− cosθν+ν −=0.73(θν+ν −=43°) Avg.=0.60 S.D.=0.33 Fig. 10. Zenith angle distribution of µ− and µ+ for ν and ν̄ for the incident neutrinos with the vertical, horizontal and diagonal directions, respectively. The overall neutrino spectrum at Kamioka site is taken into account. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption. 0 0.2 0.4 0.6 0.8 1 cosθν Eµ < 0.5 GeV 0.5 < Eµ < 1 GeV Eµ > 1 GeV 0 0.2 0.4 0.6 0.8 1 cosθν 1 < Eµ < 2 GeV 2 < Eµ < 5 GeV Eµ > 5 GeV Fig. 12. Correlation diagrams between cos θν and cos θµ for different muon energy ranges. 10 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 0 0.2 0.4 0.6 0.8 1 cosθν 1 < Eν < 2 GeV 2 < Eν < 5 GeV Eν > 5 GeV Fig. 11. Correlation Diagram between cos θν and cos θµ for different neutrino energy regions. Thus, it could be surely concluded from Fig. 11 and Fig. 12 that the SK assumption on the direction never holds as a good estimator for the determination of the directions of the incident neutrinos. In order to obtain the zenith angle distribution of the emitted leptons for that of the incident neutrinos, we di- vide the cosine of the zenith angle of the incident neutrino into twenty regular intervals from cos θν = 0 to cos θν = 1. For the given interval of cos θν , we carry out the exact Monte Carlo simulation, the detail of which is give in the Appendix D and obtain the cosine of the zenith angle of the emitted leptons, taking account of the geometry for surronding the SK detector. Thus, for each interval of cos θν , we obtain the corre- sponding zenith angle distribution of the emitted leptons. Then, we sum up these corresponding ones over all zenth angles of the incident neutrinos and we finally obtain the relation between the zenith angle distribution for the in- cident neutrinos and that for the emitted leptons. In a similar manner, we could obtain between cos θν̄ and cos θµ̄ for anti-neutrinos. The situation for anti-neutrinos is essentially same as that for neutrinos. Here, we examine the zenith angle distribution of the muons from both upward neutrinos and downward ones in the case that neutrino oscillation does not exist. By performing the procedures described in Appendix C, a pair of sampling ( cos θν+ν̄ , Eν+ν̄ ) gives a pair of ( cos θµ+µ̄, Eµ+µ̄ ). In Fig. 13, we give the zenith angle dis- tribution of the upward neutrinos ( the sum of νµ and ν̄µ ) which is constructed from the energy spectra for different cos θν+ν̄ . (see, Honda et. al. [10] and Appendix B) Upward neutrinos may produce even downward lep- tons due to both the backscattering effect and the effect of azimuthal angle on larger forward scattering for the in- teraction concerned (see, Figure 3 and Figures 4 to 6 in −1 −0.5 0 0.5 1 cosθν+ν − µ+ ,µ− (pµ>0.4 GeV/c) upward neutrinos no oscillation from upward neutrinos cosθµ+µ −, Fig. 13. The relation between the zenith angle distribution of the incident neutrino and corresponding ones of the emitted lepton the text). As the result of it, the zenith angle distribution of the emitted muons for the upward neutrino may leak in the downward direction. From Figure 13, it is very clear that the shape of the zenith angle distribution for the in- cident neutrinos is quite different from that of the emitted muons produced by these neutrinos. If the SK assumption on the direction statistically holds, the zenith angle dis- tribution for the emitted muons should coincide totally with that of the incident neutrinos. In other words, one may say that the zenith angle distribution for the emitted muons should be understood as that of the incident neu- trino under the SK assumption on the direction. However, the muon spectrum is distinctively different from the real (computational) incident neutrino spectrum as shown in the figure. Thus we conclude that SK assumption on the direction leads to the wrong conclusion on the neutrino oscillation. The further examination on the experimental data obtained by SK will be carried out in the subsequent papers. It is, further, noticed that upward neutrino energy spectrum in the figure biggest near cos θν+ν̄ = 0 and the smallest near cos θν+ν̄ = 1, which reflects from the en- hancement of the primary incident neutrino energy spec- trum from the inclined direction and is independent on the neutrino oscillation, while in SK opinion, such tendency may be favor of the existence of neutrino oscillation. 6 Discussions and Conclusion In order to extract the definite conclusion on the neutrino oscillation from the experiment by cosmic ray neutrinos whose intensity as well as interaction with the substance are both very weak, first of all, one should analyze the E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 11 most clear cut and ambiguity free events. Among neutrino events analyzed by SK, the most clear cut events are Single Ring Events, such as electron-like events and muon-like events in Fully Contained Events which are generated by QEL. These events are provided with simplicity due to single ring and all possible measureable physical quantities are confined in the detector. Furthermore, QEL is the most dominant source for neutrino events which are generated in the SK detector. This is the reason why we examine the QEL events excul- sively in present and subsequent papers. If the neutrino oscillation really exists, the most clear cut evidence surely appears in the analysis of single ring events due to QEL in Fully Contained Events and one does not need the analysis of any other type of events, such as single ring events in Partially Contained Events, multi- ring events in either Fully Contained Events or Partially Contained Evetns, all of which include inevitably ambigu- ities for the interpretation and show merely sub-evidences compared with that from the single ring events due to QEL in Fully Contained Events. SK analyze the zenith angle distribution of the inci- dent neutrinos under the asssumption that the direction of the incident neutrino is the same as that of the emitted lepton. We conclude that this assumption is supplemented by their Monte Carlo Simulation named as Detector Sim- ulation 10. In the present paper, we adopt Time Sequential Simu- lation which starts from the incident neutrino energy spec- trum on the opposite side of the Earth to the SK detec- tor and simulate all posible physical processes which are connected with the zenith angle distribution of the in- cident neutrinos according to their probability functions concerned for the examination on the validity of the SK assumption on the direction. Concretely speaking, we take the following treatment, (i) the stochastic treatments of the scattering angle of the emitted lepton in QEL, including the scattering on the backward as well as the azimuthal angle, which could not be treated in Detector Simulation, (ii) the stochastic treat- ment on the zenith angle distribution of the emitted lep- ton, considering the incident neutrino energy spectrum, (iii) the stochastic treatment on the detection of the QEL events inside the SK detector. Furthermore, the discrimination between Fully Con- tained Events and Partially Contained Events is only pos- sible in the Time Sequential Simulation, because the events concerned may be classified into different categories by chance, Fully Contained Events and Partially Contained Events due to different occurring points and different di- rections. 10 The SK Detector Simulation for obtaing the zenith angle distribtuion of the incident neutrino as for the neutrino oscilla- tiom has never been disclosed in their papers, even in the Ph.D thesis. Consequently, this is only our onjecture as for utiliza- tion of the SK Detector Simulation . A clear thing is only that SK impose the proposittion that the direction of the incident neutrino is the same as that of the emitted lepton upon the neutrino oscillation analysis. The conclusions thus obtained are as follows: (1) The zenith angle distributions of the emitted lepton in QEL for the incident neutrino with both the def- inite zenith angle and the definite energy are widely spread, particularly, into even the backward region due to partly pure backscattering and partly the combina- tion of the azimuthal angle with the slant direction of the incident neutrinos. However, for every incident neutrino with a definte zenith angle, SK give the same definite zenith angle to the emitted lepton. Already in this stage, the SK assumption on the direction does not hold. (2) Taking account of the incident neutrino energy spec- trum and simulating all physical processes concerned, we obtain the zenith angle distribution of the emitted leptons for the incident neutrino with a definite zenith angle. It is proved that the SK assumption on the di- rection does not hold again. (3) The correlation diagrams between cos θν and cos θµ show that SK assumption does noty hold well even for higher energies of the incident neutrinos, and it is shown that the correlation between them become weaker in more inclined incident neutrinos due to the effect of the azimuthal angle in QEL. (4) Taking into account the detection efficiency for the events concerned in the simulation for upward neu- trinos and anti-neutrinos, we obtain the zenith angle distribution of the leptons ( muons plus anti-muons ). According to the SK Assumption on the direction, the zenith angle distribution is the same as that of the incident neutrinos. However, the original zenith angle distribution of incident neutrino is found to be quite different from that derived from that of leptons. This is the final conclusion that SK have not measured the di- rection of the incident neutrinos reliably, which is quite independent on either the existence or non-existence of the neutrino oscillation. (5) The SK assume that the Partially Contained Events exclusively belong to the muon-like event. However, such the assumption lacks in theoretical background. Electron events can also contribute to the Partially Contained Events under some geometrical condition, for example, partly coming from the transformation by Eq.(A.5). The quantitative examination on the Par- tially Contained Events among the electron-like event will be published elsewhere. In subsequent papers, we will give the relation between the zenith angle distributions of the incident neutrinos and the corresponding muons in the cases with and with- out neutrino oscillation, including downward neutrino and will examine whether it is possible to or not to detect the neutrino oscillation by using atmospheric neutrino. 12 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK In the following Appendices we give the concrete Monte Carlo Simulations, namely, the details of our Time Se- quential Simulation. A Appendix: Monte Carlo Procedure for the Decision of Emitted Energies of the Leptons and Their Direction Cosines Here, we give the Monte Carlo Simulation procedure for obtaining the energy and its direction cosines, (lr,mr, nr), of the emitted lepton in QEL for a given energy and its direction cosines, (l,m, n), of the incident neutrino. The relation among Q2, Eν+ν̄ , the energy of the inci- dent neutrino, Eℓ, the energy of the emitted lepton (muon or electron or their anti-particles) and θs, the scattering angle of the emitted lepton, is given as Q2 = 2Eν(ν̄)Eℓ(ℓ̄)(1− cosθs). (A·1) Also, the energy of the emitted lepton is given by Eℓ(ℓ̄) = Eν(ν̄) − . (A·2) Procedure 1 We decide Q2 from the probability function for the differ- ential cross section with a given Eν(ν̄) (Eq. (2) in the text) by using the uniform random number, ξ, between (0,1) in the following Pℓ(ℓ̄)(Eν(ν̄), Q 2)dQ2, (A·3) where Pℓ(ℓ̄)(Eν(ν̄), Q dσℓ(ℓ̄)(Eν(ν̄), Q dσℓ(ℓ̄)(Eν(ν̄), Q (A·4) From Eq. (A·1), we obtain Q2 in histograms together with the corresponding theoretical curve in Fig. 14. The agree- ment between the sampling data and the theoretical curve is excellent, which shows the validity of the utlized proce- dure in Eq. (A·3) is right. Procedure 2 We obtain Eℓ(ℓ̄) from Eq. (A·2) for the given Eν(ν̄) and Q2 thus decided in the Procedure 1. Procedure 3 We obtain cos θs, cosine of the the scattering angle of the emitted lepton, for Eℓ(ℓ̄) thus decided in the Procedure 2 from Eq. (A·1) . Procedure 4 We decide φ, the azimuthal angle of the scattering lepton, 0 0.2 0.4 0.6 0.8 1 1 GeV 2 GeV Eν=0.5 GeV Fig. 14. The reappearance of the probability function for QEL cross section. Histograms are sampling results, while the curves concerned are theoretical ones for given incident energies. which is obtained from φ = 2πξ. (A·5) Here, ξ is a uniform random number (0, 1). As explained schematically in the text(see Fig. 3 in the text), we must take account of the effect due to the az- imuthal angle φ in the QEL to obtain the zenith angle distribution of both Fully Contained Events and Partially Contained Events correctly. Procedure 5 The relation between direction cosines of the incident neu- trinos, (ℓν(ν̄),mν(ν̄), nν(ν̄)), and those of the corresponding emitted lepton, (ℓr,mr, nr), for a certain θs and φ is given ℓ2 +m2 ℓ2 +m2 ℓν(ν̄) ℓ2 +m2 ℓ2 +m2 mν(ν̄) ℓ2 +m2 0 nν(ν̄) sinθscosφ sinθssinφ cosθs, (A·6) where nν(ν̄) = cosθν(ν̄), and nr = cosθℓ. Here, θℓ is the zenith angle of the emitted lepton. The Monte Carlo procedure for the determination of θℓ of the emitted lepton for the parent (anti-)neutrino with given θν(ν̄) and Eν(ν̄) involves the following steps: We obtain (ℓr,mr, nr) by using Eq. (A·6). The nr is the cosine of the zenith angle of the emitted lepton which E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 13 Fig. 15. The relation between the direction cosine of the incident neutrino and that of the emitted charged lepton. should be contrasted to nν , that of the incident neutrino. Repeating the procedures 1 to 5 just mentioned above, we obtain the zenith angle distribution of the emitted leptons for a given zenth angle of the incident neutrino with a def- inite energy. In the SK analysis, instead of Eq. (A·6), they assume nr = nν(ν̄) uniquely for Eµ(µ̄) ≥ 400 MeV. B Appendix: Monte Carlo Procedure to Obtain the Zenith Angle of the Emitted Lepton for a Given Zentith Angle of the Incident Neutrino The present simulation procedure for a given zenith an- gle of the incident neutrino starts from the atmospheric neutrino spectrum at the opposite site of the Earth to the SK detector. We define, Nint(Eν(ν̄), t, cosθν(ν̄)), the in- teraction neutrino spectrum at the depth t from the SK detector in the following way Nint(Eν(ν̄), t, cosθν(ν̄)) = Nsp(Eν(ν̄), cos θν(ν̄))× λ1(Eν(ν̄), t1, ρ1) × · · · × λn(Eν(ν̄), tn, ρn) (B·1) Here, Nsp(Eν(ν̄), cos θν(ν̄)) is the atmospheric (anti-) neutrino spectrum for the zenith angle at the opposite surface of the Earth. Here λi(Eν(ν̄), ti, ρi) denotes the mean free path due to the neutrino(anti neutrino) with the energy Eν(ν̄) from QEL at the distance, ti, from the opposite surface of the Earth inside whose density is ρi. The procedures of the Monte Carlo Simulation for the incident neutrino(anti neutrino) with a given energy,Eν(ν̄), whose incident direction is expressde by (l,m, n) are as fol- lows. Procedure A For the given zenith angle of the incident neutrino, θν(ν̄), we formulate, Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), the produc- tion function for the neutrino flux to produce leptons at the Kamioka site in the following Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) = σℓ(ℓ̄)(Eν(ν̄))Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄), (B·2) where σℓ(ℓ̄)(Eν(ν̄)) = dσℓ(ℓ̄)(Eν(ν̄), Q dQ2. (B·3) Each differential cross section above is given in Eq. (2) in the text. Utilizing, ξ, the uniform random number between (0,1), we determine Eν(ν̄), the energy of the incident neutrino in the following sampling procedure ∫ Eν(ν̄) Eν(ν̄),min Pd(Eν(ν̄), t, cos θν(ν̄)(ν̄))dEν(ν̄), (B·4) where Pd(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) ∫ Eν(ν̄),max Eν(ν̄),min Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) . (B·5) 14 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK In our Monte Carlo procedure, the reproduction of, Pd(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), the nor- malized differential neutrino interaction probability func- tion, is confirmed in the same way as in Eq. (A·4). Procedure B For the (anti-)neutrino concerned with the energy ofEν(ν̄), we sample Q2 utlizing ξ3, the uniform random number be- tween (0,1). The Procedure B is exactly the same as in the Procedure 1 in the Appendix A. Procedure C We decide, θs, the scattering angle of the emitted lepton for given Eν(ν̄) and Q 2. The procedure C is exactly the same as in the combination of Procedures 2 and 3 in the Appendix A. Procedure D We randomly sample the azimuthal angle of the charged lepton concerned. The Procedure D is exactly the same as in the Procedure 4 in the Appendix A. Procedure E We decide the direction cosine of the charged lepton con- cerned. The Procedure E is exactly the same as in the Procedure 5 in the Appendix A. We repeat Procedures A to E until we reach the de- sired trial number. C Appendix: Correlation between the Zenith Angles of the Incident Neutrinos and Those of the Emitted Leptons Procedure A By using, Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), which is defined in Eq. (B·2), we define the spectrum for cos θν(ν̄) in the following. I(cos θν(ν̄))d(cos θν(ν̄)) = d(cos θν(ν̄)) ∫ Eν(ν̄),max Eν(ν̄),min Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄).(C·1) By using Eq. (C·2) and ξ, a sampled uniform random num- ber between (0,1), then we could determine cos θν(ν̄) from the following equation ∫ cos θν(ν̄) Pn(cos θν(ν̄))d(cos θν(ν̄)), (C·2) where Pn(cos θν(ν̄)) = I(cos θν(ν̄)) I(cos θν(ν̄))d(cos θν(ν̄)). (C·3) Procedure B For the sampled d(cos θν(ν̄)) in the Procedure A, we sam- ple Eν(ν̄) from Eq.(C·4) by using ξ, the uniform randum number between (0,1) ∫ Eν(ν̄) Eν(ν̄),min Ppro(Eν(ν̄), cos θν(ν̄))dEν(ν̄), (C·4) where Ppro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) = Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) ∫ Eν(ν̄),max Eν(ν̄),min Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) . (C·5) Procedure C For the sampled Eν(ν̄) in the Procedure B, we sample Eµ(µ̄) from Eqs. (A·2) and (A·3). For the sampled Eν(ν̄) and Eµ(µ̄), we determine cos θs, the scattering angle of the muon uniquely from Eq. (A·1). Procedure D We determine, φ, the azimuthal angle of the scattering lep- ton from Eq. (A·5) by using ξ, an uniform randum number between (0,1). Procedure E We obtain cos θµ(µ̄) from Eq. (A·6). As the result, we ob- tain a pair of (cos θν(ν̄), cos θµ(µ̄)) through Procedures A to E. Repeating the Procedures A to E, we finally the cor- relation between the zenith angle of the incident neutrino and that of the emitted muon. E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 15 D Appendix: Detection of the Neutrino Events in the SK Detector and Their Interaction Points The plane ABCD is always directed vertically to the di- rection of the incident neutrino with a given zenith an- gle, which is shown in Fig. 16. The rectangular ABCDE- FGH encloses the SK detector whose radius and height is denoted by R and H, respectively. The width and the height of the plane ABCD for a given zenith angle, θν(ν̄), is given as, R and Rcos θν(ν̄) + H sin θν ,respectively, which are shown in Fig. 16-c. Now, let us estimate the ratio of the number of the neu- trino events inside the SK detector to that in the rectan- gular ABCDEFGH. As the number of the neutrino events inside some material is proportional to the number of the nucleons in the material concerned. The number of the nucleons inside the SK detector (ρ = 1) is given as Nsk = NavogaR 2H, (D·1) whereNavoga denotes the Avogadro number, and the num- ber of the nucleons in the exterior of the SK detector inside ABCDEFGH is given as Fig 16-a Fig 16-b Injection Point (X,Y) Neutrino Injection Point Fig 16-c Fig 16-d Fig. 16. Sampling procedure for neutrino evens injected into the detector Nextr(cos θν(ν̄) ) = ρNavoga R2H + R(H2+R2) sin θν(ν̄) cos θν(ν̄) , (D·2) where ρ is the density of the rock which surrounds the SK detector. Then, the total number of the target in the rectangu- larABCDEFGH is given as Ntot(cos θν(ν̄)) = Nsk +Nextr(cos θν(ν̄)). (D·3) Here, we take 2.65, as ρ (standard rock). Then, Rtheor, the ratio of the number of the neutrino events in the SK detector to that in the rectangular ABCDE- FGH is given as Rtheor(cos θν(ν̄)) = Nsk/Ntot(cos θν(ν̄)). (D·4) We obtain Rtheor for different values of cos θν(ν̄) given in the Table 3. Here, we simulate neutrino events occured in the rectangu- lar ABCDEFH, by using the atmospheric neutrino beam which falls down on the plane ABCD. Thus, Nsmaple, the sampling number of the (anti-)neutrino events inside the rectangular ABCDEFG for a given cos θν(ν̄) is given as Nsample(cos θν(ν̄)) = Ntot(cos θν(ν̄))× ∫ Eν(ν̄),max Eν(ν̄),min ℓ(ℓ̄) (Eν(ν̄))Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄) (D·5) where σℓ(ℓ̄)(Eν(ν̄)) is the total cross section for (anti-)neutrino due to QEL, and Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄) is the dif- ferential nutrino energy spectrum for the definite zenith angle, θν(ν̄), in the plane ABCD. The injection points of the neutrinos in the plane ABCD are distributed over the plane randomly and uniformely and the injection points are determined from a pair of the uniform random num- bers between (0,1). They penetrate into the rectangular ABCDEFGH from the injection point in the plane ABCD and some of them may penetrate into the SK detector or may not, which depend on their injection point. In the neutrino events which penetrate into the SK detectorr, their geometrical total track length, Ttrack, are devided into three parts Ttrack = Tb + Tsk + Ta, (D·6) where Tb denotes the track length from the plane ABCD to the entrance point of the SK detector, Tsk denotes the track length inside the SK detector, and Ta denotes the track length from the escaping point of the SK detector to the exit point of the rectangular ABCDEF, and thus 16 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK Table 3. Occurrence probabilities of the neutrino events in- side the SK detector for different cos θν ’s. Comparison between Rtheor and Rmonte. The sampling numbers for the Monte Carlo Simulation are, 1000, 10000, 100000, respectively. cos θν Rtheor Rmonte Sampling Number 1000 10000 100000 0.000 0.58002 0.576 0.5750 0.57979 0.100 0.41717 0.425 0.4185 0.41742 0.200 0.32792 0.353 0.3252 0.32657 0.300 0.27324 0.282 0.2731 0.27163 0.400 0.23778 0.223 0.2329 0.23582 0.500 0.21491 0.206 0.2063 0.21203 0.600 0.20117 0.197 0.1946 0.19882 0.700 0.19587 0.193 0.1925 0.19428 0.800 0.20117 0.198 0.2002 0.20001 0.900 0.22843 0.230 0.2248 0.22803 1.00 0.58002 0.557 0.5744 0.57936 Ttrack denotes the geometrical length of the neutrino con- cerned in the rectangular ABCDEFGH. By the definition, the neutrinos concerned with Ttrack interact surely somewhere along the Ttrack. Here, we are interested only in the interaction point which ocuurs along Tsk. We could determine the interaction point in the Tsk in the following. We define the following quantities for the purpose. Tweight = Tsk + ρ(Tb + Ta), (D·7) ρav = Tweight/Ttrack, (D·8) ξρ = ρav/ρ, (D·9) ξsk = Tsk/Tweight. (D·10) The flow chart for the choice of the neutrino events in the SK detector and the determination of the interaction points inside the SK detector is given in Fig. 17. Thus, we obtain neutrino events whose occurrence point is decided in the SK detector in the following. xf = x0, (D·11) yf = y0 + ξTsk sin θν(ν̄) (D·12) zf = z0 + ξTsk cos θν(ν̄). (D·13) If we carry out the Monte Carlo Simulation, following the flow chart in Fig. 17, then, we obtain Nevent, the num- ber of the neutrino events generasted in the SK detector. The ratio of the selected events to the total trial is given Rmonte(cos θν(ν̄)) = Nevent(cos θν(ν̄))/Nsample(cos θν(ν̄)). (D·14) Comparison between Rtheor and Rmonte in Table 3 shows that our Monte Carlo procedure is valid. References 1. Kasuga, S. et al., Phys. Lett. B374 (1996) 238. N Nsample N=N+1 N=N+1 Entry Determination of Point in the plane ABCD by using 1 and 2 Judgement on the Event’s Entering Determination of Ta,Tb and Tsk Determination of Tweight av, and sk Determination of e interaction point of e events inside SK Fig. 17. Flow Chart for the determination of the interaction points of the neutrino events inside the detector 2. Ashie,Y. et al., Phys. Rev. D 71 (2005) 112005. 3. Ashie,Y. et al., Phys. Rev. Lett.93 (2004) 101801. 4. Renton, P., Electro-weak Interaction, Cambridge University Press (1990). See p. 405. 5. D.Rein and L.M.Sehgal, Ann. of Phys. 133 (1981) 1780. 6. D.Rein and L.M.Sehgal Nucl. Phys. B84 (1983) 29. 7. R.H.Gandhi et. al. Astropart. Phys. 5 (1996) 81. 8. Kajita, T. and Totsuka, Y. Rev. Mod. Phys., 73 (2001) 85. See p. 101. 9. Ishitsuka, M., Ph.D thesis, University of Tokyo (2004). See p. 138. 10. Honda, M., et al., Phys. Rev. D 52 (1996) 4985 Introduction Cross Sections of Quasi Elastic Scattering in the Neutrino Reaction and the Scattering Angle of Charged Leptons. Influence of Azimuthal Angle of Quasi Elastic Scattering over the Zenith Angle of both the Fully Contained Events and Partially Contained Events Zenith Angle Distribution of Fully Contained Events and Partially Contained Events for a Given Zenith Angle of the Incident Neutrino, Taking Their Energy Spectrum into Account Relation between the Zenth angle Distribution of the Incident Neutrinos and that of the emitted leptons Discussions and Conclusion Appendix: Monte Carlo Procedure for the Decision of Emitted Energies of the Leptons and Their Direction Cosines Appendix: Monte Carlo Procedure to Obtain the Zenith Angle of the Emitted Lepton for a Given Zentith Angle of the Incident Neutrino Appendix: Correlation between the Zenith Angles of the Incident Neutrinos and Those of the Emitted Leptons Appendix: Detection of the Neutrino Events in the SK Detector and Their Interaction Points

0704.0191

Intricate Knots in Proteins: Function and Evolution

plcb-02-09-10 1074..1079 Intricate Knots in Proteins: Function and Evolution Peter Virnau , Leonid A. Mirny , Mehran Kardar 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States of America, 2 Harvard–MIT Division of Health Sciences and Technology, Cambridge, Massachusetts, United States of America Our investigation of knotted structures in the Protein Data Bank reveals the most complicated knot discovered to date. We suggest that the occurrence of this knot in a human ubiquitin hydrolase might be related to the role of the enzyme in protein degradation. While knots are usually preserved among homologues, we also identify an exception in a transcarbamylase. This allows us to exemplify the function of knots in proteins and to suggest how they may have been created. Citation: Virnau P, Mirny LA, Kardar M (2006) Intricate knots in proteins: Function and evolution. PLoS Comput Biol 2(9): e122. DOI: 10.1371/journal.pcbi.0020122 Introduction Although knots are abundant and complex in globular homopolymers [1–3], they are rare and simple in proteins [4– 8]. Sixteen methyltransferases in bacteria and viruses can be combined into the a/b knot superfamily [9], and several isozymes of carbonic anhydrase (I, II, IV, V) are known to be knotted. Apart from these two folds, only a few insular knots have been reported [5,6,10,11], some of which were derived from incomplete structures [6,11]. For the most part, knotted proteins contain simple trefoil knots (31) that can be represented by three essential crossings in a projection onto a plane (see Figure 1, left). Only three proteins were identified with four projected crossings (41, Figure 1, middle). In this report we provide the first comprehensive review of knots in proteins, which considers all entries in the Protein Data Bank (http://www.pdb.org) [12], and not just a subset. This allows us to examine knots in homologous proteins. Our analysis reveals several new knots, all in enzymes. In particular, we discovered the most complicated knot found to date (52) in humanubiquitin hydrolase (Figure 1, right), and suggest that its entangled topology protects it against being pulled into the proteasome. We also noticed that knots are usually preserved among structural homologues. Sequence similarity appears to be a strong indicator for thepreservationof topology, although differences between knotted and unknotted structures are sometimes subtle. Interestingly, we have also identified a novel knot in a transcarbamylase that is not present in homologues of known structure. We show that the presence of this knot alters the functionality of the protein, and suggest how the knot may have been created in the first place. Mathematically, knots are rigorously defined in closed loops [13]. Fortunately, both the N- and C-termini of open proteins are typically accessible from the surface and can be connected unambiguously: we reduce the protein to its Ca- backbone, and draw two lines outward starting at the termini in the direction of the connection line between the center of mass of the backbone and the respective ends [5]. The lines are joined by a big loop, and the structure is topologically classified by the determination of its Alexander polynomial [1,13]. Applying this method to the Protein Data Bank in the version of January 3, 2006, we found 273 knotted structures in the 32,853 entries that contain proteins (Table S1). Knots formed by disulfide [14,15] or hydrogen bonds [7] were not included in the study. Results For further analysis, we considered 36 proteins that contain knots as defined by rather stringent criteria discussed in the Materials and Methods section. These proteins can be classified into six distinct families (Table 1). Four of these families incorporate a deeply knotted section, which persists when 25 amino acids are cut off from either terminus. Interestingly, all knotted proteins thus identified are en- zymes. Our investigation affirms that all members of the carbonic anhydrase fold (including the previously undeter- mined isozymes III, VII, and XIV) are knotted. In addition, we identify a novel trefoil in two bacterial transcarbamylase-like proteins (AOTCase in Xanthomonas campestris and SOTCase in Bacteroides fragilis) [16,17]. UCH-L3—The most complex protein knot. One of our most intriguing discoveries is a fairly intricate knot with five projected crossings (52) in ubiquitin hydrolase (UCH-L3 [18]; see Figure 1, right). This knot is the first of its kind and, apart from carbonic anhydrases, the only identified in a human protein. Human UCH-L3 also has a yeast homologue [6,19] with a sequence identity of 32% [20]. Amino acids 63 to 77 are unstructured, and if we connect the unstructured region by an arc that is present in the human structure, we obtain the same knot with five crossings. What may be the function of this knot? In eukaryotes, proteins get labeled for Editor: Robert B. Russell, European Molecular Biology Laboratory, Germany Received April 3, 2006; Accepted July 28, 2006; Published September 15, 2006 A previous version of this article appeared as an Early Online Release on July 28, 2006 (DOI: 10.1371/journal.pcbi.0020122.eor). DOI: 10.1371/journal.pcbi.0020122 Copyright: � 2006 Virnau et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abbreviations: AOTCase, N-acetylornithine transcarbamylase; SOTCase, N-succi- nylornithine transcarbamylase; UCH-L3, ubiquitin hydrolase * To whom correspondence should be addressed. E-mail: [email protected] PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221074 degradation by ubiquitin conjugation. UCH-L3 performs deconjugation of ubiquitin, thus rescuing proteins from degradation. The close association of the enzyme with ubiquitin should make it a prime target for degradation at the proteasome. We suggest that the knotted structure of UCH-L3 makes it resistant to degradation. In fact, the first step of protein degradation was shown to be ATP-dependent protein unfolding by threading through a narrow pore (;13 Å in diameter) of a proteasome [21,22]. Such threading into the degradation chamber depends on how easily a protein unfolds, with more stable proteins being released back into solution [23] and unstable ones being degraded. If ATP- dependent unfolding proceeds by pulling the C-terminus into a narrow pore [21], then a knot can sterically preclude such translocation, hence preventing protein unfolding and degradation. While arceabacterial proteasome PAN was shown to process proteins from its C- to N-terminus [21], it cannot be ruled out that some eukaryotic proteasomes process proteins in the N- to C-direction, thus requiring protection of both termini. Unfolding of a knotted protein by pulling may require a long time for global unfolding and untangling of the knot. Unknotted proteins, in contrast, have been shown to become unstable if a few residues are removed from their termini [24], suggesting that threading a few (5–10) residues into a proteasomal pore would be sufficient to unravel an unknotted structure. At both termini, UCH-L3 contains loops entangled into the knot protecting both ends against unfolding if pulled. It should also be noted that both N- and C-termini are stabilized by a number of hydrophobic interactions with the rest of the protein. The C-terminus is Figure 1. Examples of the Three Different Types of Knots Found in Proteins Colors change continuously from red (first residue) to blue (last residue). A reduced representation of the structure, based on the algorithm described in [1,6,36], is shown in the lower row. (Left) The trefoil knot (31) in the YBEA methyltransferase from E. coli (pdb code 1ns5; unpublished data) reveals three essential crossings in a projection onto a plane. (Middle) The figure-eight knot (41) in the Class II ketol-acid reductoisomerase from Spinacia oleracea (pdb code 1yve [26]) features four crossings. (Only the knotted section of the protein is shown.) (Right) The knot 52 in ubiquitin hydrolase UCH-L3 (pdb code 1xd3 [18]) reveals five crossings. Pictures were generated with Visual Molecular Dynamics (http://www.ks.uiuc.edu/Research/vmd) [43]. DOI: 10.1371/journal.pcbi.0020122.g001 PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221075 Synopsis Several protein structures incorporate a rather unusual structural feature: a knot in the polypeptide backbone. These knots are extremely rare, but their occurrence is likely connected to protein function in as yet unexplored fashion. The authors’ analysis of the complete Protein Data Bank reveals several new knots that, along with previously discovered ones, may shed light on such con- nections. In particular, they identify the most complex knot discovered to date in a human protein, and suggest that its entangled topology protects it against unfolding and degradation. Knots in proteins are typically preserved across species and sometimes even across kingdoms. However, there is also one example of a knot in a protein that is not present in a closely related structure. The emergence of this particular knot is accompanied by a shift in the enzymatic function of the protein. It is suggested that the simple insertion of a short DNA fragment into the gene may suffice to cause this alteration of structure and function. Intricate Knots in Proteins particularly stable—residues 223 to 229 are hydrophobic and form numerous contacts at 5 Å with the rest of the structure. We would like to stress that this hypothesis needs to be tested by experiments. Different proteins may also provide different levels of protection against degradation, depending on structural details, the depth of the knot, and its complex- ity. Recently, a knot in the red/far-red light photoreceptor phytochrome A in Deinococcus radiodurans was identified [11] (see Materials and Methods). Although sequence similarity suggests that the knot may also be present in plant homologues, we cannot be certain. In plants, the red- absorbing form is rather stable (half-life of 1 wk), but the far-red–absorbing form is degraded upon photoconversion by the proteasome with a half-life of 1–2 h in seedlings (and somewhat longer in adult plants) [25]. Evolutionary aspects. As expected, homologous structures tend to retain topological features. The trefoil knot in carbonic anhydrase can be found in isozymes ranging from bacteria and algae to humans (Table 1). Class II ketol-acid reductoisomerase comprises a figure-eight knot present in Escherichia coli [10] and spinach [26] (see Figure 1, middle), and S-adenosylmethione synthetase contains a deep trefoil knot in E. coli [5,27] and rat [28]. It appears that particular knots have indeed been preserved throughout evolution, which suggests a crucial role for knots in protein enzymatic activity and binding. Table 1. List of Knotted PDB Entries (January 2006) Protein Knot Family Protein Species PDB Code Length Knot Knotted Core a/b knot YbeA-like E. coli 1ns5 153 31 69–121 (32) T. maritime 1o6d 147 31 68–117 (30) S. aureus 1vh0 157 31 73–126 (31) B. subtilis 1to0 148 31 64–116 (32) tRNA(m1G37)-methyltransferase TrmD H. influenza 1uaj 241 31 93–138 (92) E. coli 1p9p 235 31 90–130 (89) SpoU-like RNA 29-O ribose mtf. T. thermophilus 1v2x 191 31 96–140 (51) H. influenza 1j85 156 31 77–114 (42) T. thermophilus 1ipa 258 31 185–229 (29) E. coli 1gz0 242 31 172–214 (28) A. aeolicus 1zjr 197 31 95–139 (58) S. viridochromog. 1x7p 265 31 192–234 (31) YggJ C-terminal domain-like H. influenza 1nxz 246 31 165–216 (30) B. subtilis 1vhk 235 31 158–208 (27) T. thermophilus 1v6z 227 31 103–202 (25) Hypothetical protein MTH1 (MT0001) A. M. Thermoautotr. 1k3r 262 31 48–234 (28) Carbonic anhydrases Carbonic anhydrase N. gonorrhoeae 1kop 223 31 36–223 (0) Carbonic anhydrase I H. sapiens 1hcb 258 31 29–256 (2) Carbonic anhydrase II H. sapiens 1lug 259 31 30–256 (3) Bos Taurus 1v9e 259 31 32–256 (3) Dunaliella salina 1y7w 274 31 37–270 (4) Carbonic anhydrase III Rattus norv. 1flj 259 31 30–256 (3) H. sapiens 1z93 263 31 28–254 (9) Carbonic anhydrase IV H.sapiens 1znc 262 31 32–261 (1) Mus musculus 2znc 249 31 32–246 (3) Carbonic anhydrase V Mus musculus 1keq 238 31 7–234 (4) Carbonic anhydrase VII H. sapiens 1jd0 260 31 28–257 (3) Carbonic anhydrase XIV Mus Musculus 1rj6 259 31 29–257 (2) Miscellaneous Ubiquitin hydrolase UCH-L3 H. sapiens 1xd3A 229 52 12–172 (11) S. cerevisiae (synth.) 1cmxA 214 31 9–208 (6) S-adenosylmethionine synthetase E. coli 1fug 383 31 33–260 (32) Rattus norv. 1qm4 368 31 30–253 (29) Class II ketol-acid reductoisomerase Spinacia oleracea 1yve 513 41 239–451 (62) E. coli 1yrl 487 41 220–435 (52) Transcarbamylase-like B. fragilis 1js1 324 31 169–267 (57) X. campestris 1yh1 334 31 171–272 (62) ‘‘Protein’’ describes the name or the family of the knotted structure. ‘‘Species’’ refers to the scientific name of the organism from which the protein was taken for structure determination. ‘‘PDB code’’ gives one example Protein Data Bank entry for each knotted protein: additional structures of the same protein can be found using the SCOP classification tool [9]. ‘‘Length’’ describes the number of Ca-backbone atoms in the structure. ‘‘Knot’’ refers to the knot type which was discovered in the protein: 31, trefoil; 41, figure-eight knot; 52, 2nd knot with five crossings according to standard knot tables [13]. The core of a knot is the minimum configuration which stays knotted after a series of deletions from each terminus; in brackets we indicate how many amino acids can be removed from either side before the structure becomes unknotted (see Materials and Methods). Structure is fragmented and becomes knotted when missing sections are joined by straight lines. The size of the knotted core refers to the thus-connected structure. The knot is also present in at least one fragment. Structure is fragmented and only knotted when missing sections are joined by straight lines. Fragments are unknotted. 1v6z is currently not classified according to SCOP (version 1.69). Sequence similarity suggests that it is part of the a/b knot fold. 1v6zB contains a shallow composite knot (31#41), which turns into a regular trefoil when two amino acids are cut from the N-terminus. (The random closure [see Materials and Methods] determines a trefoil right away.) 1uch contains the same structure as 1xd3. If the missing section in the center of this structure is joined by a straight line, it becomes knotted (52). In the yeast homologue (1cmx), amino acids 63 to 77 are unstructured, and if we replace the missing parts by a straight line, we obtain a trefoil knot that has been identified before [6]. If we connect the unstructured region by an arc present in the human structure, we obtain the same knot with five crossings. eVisual inspection reveals that the calculated size of the knotted core is too small. DOI: 10.1371/journal.pcbi.0020122.t001 PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221076 Intricate Knots in Proteins UCH-L3 in human and yeast share only 33% [29] of their sequences, but contain the same 5-fold knot as far as we can tell from the incomplete structure in yeast. It is not only likely that all species in between have the same knot—the link between sequence and structure may also be used to predict candidates for knots among isozymes or related proteins for which the structure is unknown. For example, UCH-L4 in mouse has 96% sequence identity with human UCH-L3. The similarity with UCH-L6 in chicken is 86%, and with UCH-L1 about 55%. Indeed, a reexamination of the most recent Protein Data Bank entries revealed that UCH-L1 contains the same 52 knot as UCH-L3. (See the Update section—the structure was not yet part of the January Protein Data Bank release on which this paper is based.) Unfortunately, the method is not foolproof because differences between knotted and unknotted structures are sometime subtle. As we will demonstrate in the next paragraph, a more reliable estimate has to consider the conservation of major elements of the knot, like loops and threads. AOTCase—How a protein knot can alter enzymatic activity. Somewhat surprisingly, we also identified a pair of homo- logues for which topology is not preserved. N-acetylornithine transcarbamylase (AOTCase [17]) is essential for the arginine biosynthesis in several major pathogens. In other bacteria, animals, and humans, a homologous enzyme (OTCase) processes L-ornithine instead [30]. Both proteins have two active sites. The first one binds carbamyl phosphate to the enzyme. The second site binds acetylornithine in AOTCases and L-ornithine in OTCases, enabling a reaction with carbamyl phosphate to form acetylcitrulline or citrulline, respectively [17, 31]. AOTCase in X. campestris has 41% sequence identity with OTCase from Pyrococcus furiosus [32] and 29% with human OTCase [31]. As demonstrated in Figure 2, AOTCase contains a deep trefoil knot which is not present in OTCase (Figure 2, right) and which modifies the second active site. The knot consists of a rigid proline-rich loop (residues 178–185), through which residues 252 to 256 are threaded and affixed. As elaborated in [17], the reaction product N-acetylcitrulline strongly interacts with the loop and with Lys252. Access to subsequent residues is, however, restricted by the knot. L- norvaline in Figure 2 (right) is very similar to L-ornithine but lacks the N-e atom of the latter to prevent a reaction with carbamyl phosphate. As the knot is not present in OTCase, the ligand has complete access to the dangling residues 263– 268 and strongly interacts with them [31]. This leads to a rotation of the carboxyl-group by roughly 1108 around the Ca–Cb bond [17]. This example demonstrates how the presence of a knot can modify active sites and alter the enzymatic activity of a protein—in this case, from processing L-ornithine to N- acetyl-L-ornithine. It is also easy to imagine how this alteration happened: a short insertion extends the loop and modifies the folding pathway of the protein. Discussion Nature appears to disfavour entanglements, and evolution has developed mechanisms to avoid knots. Human DNA wraps around histone proteins, and the rigidity of DNA allows it to form a spool when it is fed into a viral capsid. One end also stays in the loading channel and prevents subsequent equilibration [33]. Knotted proteins are rare, although the reason is far less well understood. Can the absence of entanglement be explained in terms of particular statistical ensembles, or is there an evolutionary bias? And how do these structures actually fold? Knots are ubiquitous in globular homopolymers [1–3,8], but rare in coil-like phases [1,34–36]. It is likely that even a flexible polymer will at least initially remain unknotted after a Figure 2. Structures of Transcarbamylase from X. campestris with a Trefoil Knot and from Human without a Knot (Left) Knotted section (residues 171–278) of N-acetylornithine transcarbamylase from X. campestris with reaction product N-acetylcitrulline (pdb code 1yh1 [17]) and interacting side chains. (Right) Corresponding (unknotted) section (residues 189–286) in human ornithine transcarbamylase (pdb code 1c9y [31]) with inhibitor L-norvaline and carbamyl phosphate. Colors change continuously from red (first residue in the section) to blue (last residue in the section). The two proteins have an overall sequence identity of 29% [41]. Pictures were generated with VMD [43]. DOI: 10.1371/journal.pcbi.0020122.g002 PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221077 Intricate Knots in Proteins collapse from a swollen state. In proteins, the free energy landscape is considerably more complex, which may allow most proteins to stay unknotted. The secondary structure and the stiffness of the protein backbone may shift the length scale at which knots typically appear, too [8]. If knotted proteins are in fact more difficult to degrade, it might also be disadvanta- geous for most proteins to be knotted in the first place. Unfortunately, few experimental papers address folding and biophysical aspects of knots in proteins. In recent work [37], Jackson and Mallam reversibly unfolded and folded a knotted methyltransferase in vitro, indicating that chaper- ones are not a necessary prerequisite. In a subsequent study [38], the authors provide an extensive kinetic analysis of the folding pathway. In conclusion, we would like to express our hope that this report will inspire more experiments in this small but nevertheless fascinating field. Materials and Methods To determine whether a structure is knotted, we reduce the protein to its backbone, and draw two lines outward starting at the termini in the direction of the connection line between the center of mass of the backbone and the respective ends. These two lines are joined by a big loop, and the structure is classified by the determination of its Alexander polynomial [1,13]. To determine the size of the knotted core, we delete successively amino acids from the N-terminus until the protein becomes unknotted [1,6]. The proce- dure is repeated at the C-terminus starting with the last structure that contained the original knot. For each deletion, the outward pointing line through the new termini is parallel to the respective lines computed for the full structure. The thus determined size should, however, only be regarded as a guideline. A better estimate can be achieved by looking at the structure. In Table 1 we include knotted structures with no missing amino acids in the center of the protein. (A list of potentially knotted structures with missing amino acids can be found in Table S3.) Technically, the numbering of the residues in the mmcif file has to be subsequent, and no two amino acids are allowed to be more than 6 Å apart. In addition, the knot has to persist when two amino acids are cut from either terminus. We have further excluded structures for which unknotted counterexamples exist (e.g., only one nuclear magnetic resonance structure among many is knotted or another structure of the same protein is unknotted). If a structure is fragmented, the knot has to appear in one fragment and in the resulting structure obtained from connecting missing sections by straight lines. Other knotted structures are only considered when at least one additional member of the same structural family [9] contains a knot according to the criteria above. The enforcement of these rules leads to the exclusion of the bluetongue virus core protein [6] (41) and photoreceptor phyto- chrome A in D. radiodurans [11] (31), which have been previously identified as being knotted. Both structures are fragmented and become knotted only when a few missing fragments are connected by straight lines. In the viral core protein, the dangling C-terminus threads through a loose loop and becomes knotted in one out of two cases. On the other hand, the photoreceptor phytochrome A appears to contain a true knot. Notably, our analysis suggests that the thus connected structure of phytochrome A contains a figure-eight knot instead of a trefoil as reported in [11]. Moreover, we excluded a structure of the Autographa California nuclear polyhedrosis virus, which contains a knot according to our criteria. However, the N-terminus is buried inside the protein and the knot only exists because of our specific connection to the outside. To further validate our criteria, we implemented an alternative method [4,8,39] that relies on the statistical analysis of multiple random closures. We arbitrarily chose two points on a sphere (which has to be larger than the protein) and connected each with one terminus. The two points can be joined unambiguously, and the resulting loop was analyzed by calculating the Alexander polynomial. We repeated the procedure 1,000 times, and defined the knot as the majority type. Applying this analysis, we discovered 241 knotted structures in the Protein Data Bank. All 241 structures are also present in the 273 structures (Table S1) that were identified by our method, and the knot type is the same. Themissing32 structures (Table S2) aremostly shallow knots and were already rejected according to our extended criteria. The random closure also correctly discards rare structures with buried termini. In conclusion, the method used in this paper is considerably faster but requires a slightly increased inspection effort. Our observations agree with [8], which provides an extensive comparison of closures applied to proteins. A complete listing of knotted Protein Data Bank structures is given in the Supporting Information. Update. Recently, the structure of human UCH-L1 was solved and released [40]. The protein shares 55% sequence identity with UCH-L3 [41], and it contains the same 5-fold knot. UCH-L1 is highly abundant in the brain, comprising up to 2% of the total brain protein [42]. The structure of UCH-L1 was not yet part of the January Protein Data Bank edition on which the rest of this study is based. We also noticed several new structures of knotted transcarbamylase-like proteins. Supporting Information Table S1. List of Knotted Protein Data Bank Entries Found at DOI: 10.1371/journal.pcbi.0020122.st001 (79 KB DOC). Table S2. List of Knotted Entries from Table S1 That Become UnknottedWhenEnds AreConnected by theRandomClosureMethod Found at DOI: 10.1371/journal.pcbi.0020122.st002 (28 KB DOC). Table S3. List of Structures That Become Knotted When Missing Sections Are Joined by Straight Lines Found at DOI: 10.1371/journal.pcbi.0020122.st003 (35 KB DOC). Accession Numbers The Protein Data Bank (http://www.pdb.org) accession numbers for the structures discussed in this paper are human UCH-L3 (1xd3), UCH-L3 yeast homologue (1cmx), human UCH-L1 (2etl), photo- receptor phytochrome A in D. radiodurans (1ztu), class II ketol-acid reductoisomerase in E. coli (1yrl), class II ketol-acid reductoisomerase in spinach (1yve), S-adenosylmethione synthetase in E. coli (1fug), S- adenosylmethione synthetase in rat (1qm4), AOTCase from X. campestris (1yh1), SOTCase from B. fragilis (1js1), OTCase from P. furiosus (1a1s), OTCase from human (1c9y), bluetongue virus core protein (2btv), and baculovirus P35 protein in Autographa California nuclear polyhedrosis virus (1p35). Acknowledgments Upon completion of this work we became aware of a related study [8], which independently identified the knots in UCH-L3 and SOTCase in a re-examination of protein knots. 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0704.0192

Star Formation in Galaxies with Large Lower Surface Brightness Disks

arXiv:0704.0192v1 [astro-ph] 2 Apr 2007 Submitted to The Astrophysical Journal Star Formation in Galaxies with Large Lower Surface Brightness Disks K. O’Neil NRAO, PO Box 2, Green Bank, WV 24944 [email protected] M. S. Oey University of Michigan, Astronomy Department, 830 Dennison Building Ann Arbor, MI 48109-1042 [email protected] G. Bothun University of Oregon, Physics Department, 1371 E 13th Avenue, Eugene, OR 97403 [email protected] ABSTRACT We present B, R, and Hα imaging data of 19 large disk galaxies whose prop- erties are intermediate between classical low surface brightness galaxies and or- dinary high surface brightness galaxies. We use data taken from the Lowell 1.8m Perkins telescope to determine the galaxies’ overall morphology, color, and star formation properties. Morphologically, the galaxies range from Sb through Irr and include galaxies with and without nuclear bars. The colors of the galaxies vary from B−R = 0.3 – 1.9, and most show at least a slight bluing of the colors with increasing radius. The Hα images of these galaxies show an average star for- mation rate lower than is found for similar samples with higher surface brightness disks. Additionally, the galaxies studied have both higher gas mass-to-luminosity and diffuse Hα emission than is found in higher surface brightness samples. Subject headings: galaxies: evolution; galaxies: colors; galaxies: luminosities; galaxies: ISM; galaxies: photometry; galaxies: spiral http://arxiv.org/abs/0704.0192v1 – 2 – 1. Introduction Large low surface brightness galaxies are galaxies with disk central surface brightnesses statistically far from the Freeman (1970) value of µB(0) = 21.65 ± 0.3 mag arcsec −2, and whose properties are significantly removed from the dwarf galaxy category (e.g. MB < −18, MHI > 10 9M⊙). Studies of large LSB galaxies have discovered a number of intriguing facts: large LSB galaxies, in contrast to dwarf LSB galaxies, can exhibit molecular gas (Das, et al. 2006; O’Neil & Schinnerer 2004; O’Neil, Schinnerer, & Hofner 2003; O’Neil, Hofner, & Schinnerer 2000); the gas mass-to-luminosity ratios of large LSB galaxies are typically higher than for similar high surface brightness counterparts by a factor of 2 or more (O’Neil, et al. 2004); and, like dwarf LSB galaxies, large LSB systems are typically dark-matter dominated (Pickering, et al. 1997; McGaugh, Rubin, & de Blok 2001). These properties, added to their typically low metallicities (de Naray, McGaugh, & de Blok 2004; Gerritsen & de Blok 1999), lead to the inference that even large LSB galaxies are under-evolved compared to their high surface brightness (HSB) counterparts. Once their typically low gas surface densities (MHI ≤ 10 21 cm−2) (Pickering, et al. 1997) and low baryonic-to-dark matter ratios (Gurovich, et al. 2004; McGaugh, et al. 2000) are taken into account, the question becomes less why LSB galaxies are under-evolved than how they can form stars at all (O’Neil, Bothun, & Schombert 2000, and references therein). Yet large LSB galaxies have the same total luminosity within them as ordinary Hubble sequence spirals (O’Neil, et al. 2004; Impey & Bothun 1997; Pickering, et al. 1997; Sprayberry, et al. 1995). On average then, star formation cannot be too inefficient in these large LSB galaxies in spite of their unevolved characteristics, else their integrated light would be significantly less then in their HSB counterparts. In an effort to better understand this enigmatic group of galaxies and their evolutionary status, we recently conducted a 21-cm survey to discover a larger nearby sample of such objects (O’Neil, van Driel, & Schneider 2006; O’Neil, et al. 2004). We succeeded in identi- fying about 25 candidates within the redshift range 0.04 < z < 0.08, whose combined HI and optical properties suggest them to be large LSB galaxies. We obtained B, R, and Hα imaging of 19 of these galaxies at Lowell Observatory to confirm whether these candidates are indeed LSB galaxies, and to obtain a dataset of their fundamental parameters. These observations are presented here, and interestingly, none of the galaxies ultimately turned out to be LSB galaxies by the strict conventional definition; we discuss this result below in § 4. However, these galaxies still represent a sample whose surface brightnesses are below average, and whose properties are intermediate between those of the bona fide massive LSB galaxies, and ordinary HSB galaxies. In this work, we quantify and parameterize the fundamental properties of this sample of large, “lower surface brightness” galaxies. – 3 – 2. Galaxy Sample There are three ways that disk galaxy surface brightness can be measured or quantified – using a surface brightness profile and fitting an exponential disk to derive the central surface brightness; measuring an average surface brightness within a given isophotal diameter; and measuring the surface brightness of the isophote at the 1/2 light radius point (the effective surface brightness). The latter two definitions suffer from the fact that the bulge light is included in the surface brightness estimates, resulting in their prediction of the disk surface brightness to be less accurate. As a result, the typical operational definition of an LSB galaxy uses the first definition, and defines an LSB galaxy as one whose whose observed disk central surface brightness is µB(0) ≥23.0 mag arcsec −2. For reference, the Freeman value of µB(0)=21.65 +/- 0.30 mag arcsec −2 defines the distribution of central surface brightness, in the blue band, for Hubble sequence spirals. Regardless of the definition, without pre-existing high quality optical imaging of galaxies, it is difficult to unambiguously identify a sample of disk galaxies that will turn out to be LSB. With only catalog data available, one is driven to use the average surface brightness and identify potential LSB galaxies as those whose average surface brightness is below some threshold level. All of the galaxies in this sample were identified as LSB by Bothun, et al. (1985) using the magnitude and diameter values found in the Uppsala General Catalog (Nilson 1973), and employing the general equation 〈µB〉 = mpg +5log(D)+8.63. Here, mpg is the photographic magnitude of the galaxies, D is the diameter in arcminutes, and the constant, 8.63, is derived from the conversion from arcminutes to arcseconds (8.89) and the conversion from mpg to mB (-0.26, as used by Bothun, et al.) Bothun, et al. (1985) then made a cut-off to the galaxies in their sample, requiring 〈µB〉 >24.0 mag arcsec −2 to look for galaxies with lower surface brightness disks, with the majority of the galaxies chosen having 〈µB〉 >25.0 mag arcsec (The inclusion of a number of galaxies with 〈µB〉=24-25 mag arcsec −2 was due to the 0.5mag errors given in the UGC.) The Bothun, et al. (1985) sample was further pared down by our desire to image large LSB galaxies. That is, we wished to avoid the dwarf galaxy category entirely. To do this, we required the galaxies to have MHI > 10 9 M⊙, W20 > 200 km s −1, and/or MB < −19. These criteria are sufficiently removed from the dwarf galaxy category to guarantee no overlap between our sample and that category exists. – 4 – 3. Observations & Data Reduction Galaxies’ integrated broad-band colors represent a convolution of the mean age of the stellar population, metallicity, and recent star formation rate; while measurements of Hα luminosity provide a direct measure of the current star formation rate (SFR). With these combined observations, is is possible to parameterize the current SFR relative to the overall star formation history. As a result, these observations are widely used in many surveys that target fundamental galaxy parameters, for example, SINGG (Meurer, et al. 2006) and 11HUGS (Kennicutt, et al. 2004), and others (e.g. Gavazzi, et al. 2006; Koopman & Kenney 2006; Helmboldt, et al. 2005). 19 galaxies were observed on 7-10 June, 2002 and 5-8 October, 2003 using the Lowell 1.8m Perkins telescope. The filter set used included Johnson B and R as well as three Hα filters from a private set (R. Walterbos) with center frequency/bandwidths of 6650/75, 6720/35, 6760/75 Å. A 1065x1024 pixel Loral SN1259 CCD camera was used, giving a 3.3′ field of view and resolution of 0.196′′/pixel. Seeing in June, 2002, ranged from 1.8′′ - 2.4′′ and from 1.4′′ - 2.2′′ for the October, 2003, observations. At least 3 frames, each shifted slightly in position, were obtained for each object through each filter and were median filtered to reduce the effect from cosmic rays, bad pixels, etc. All initial data reduction (bias and flat field removal, image alignment, etc) was done within IRAF. The R band images were scaled and used as the continuum images for data reduction purposes. Corrections to the measured fluxes were made in the following way. Atmospheric extinc- tion was obtained using the observational airmass and the atmospheric extinction coefficients for Kitt Peak which are distributed with IRAF. Galactic extinction was corrected using the values for E(B−V) obtained from NED, the reddening law of Seaton (1979) as parameter- ized by Howarth (1983) (A(λ) = X(λ)E(B−V )) and assuming the case B recombination of Osterbrock (1989) with RV=3.1 (O’Donnell 1994) (X(6563Å)=2.468). Contamination from [NII] emission in the Hα images was corrected using the relationship derived by Jansen, et al. (2000) and re-confirmed by Helmboldt, et al. (2004): [NII] = [−0.13± 0.035]MR + [−3.2 ± 0.90] , where MR is the absolute magnitude in the R band. Hα extinction was determined using the equation found in Helmboldt, et al. (2004): log (Hα)int = [−0.12± 0.048]MR + [−2.5± 0.96] which was found through a linear least squares fitting to the A(Hα)int determined using all galaxies in his sample with a measured Hβ flux. For this calculation, Helmboldt, et al. – 5 – (2004) used the Hα to Hβ ratio measured by Jansen, et al. (2000), an assumed intrinsic ratio of Hα =2.85 (Case B recombination and T=104 K (Osterbrock 1989)), the extinction curve of O’Donnell (1994), and RV=3.1 No correction for internal extinction due to inclination was made for the B and R bands. It should be noted, though, that in a number of plots inclination corrections were made to the B and R colors and central surface brightnesses, as noted in the Figure captions. The corrections used in these cases are: µ(0)λcorr = µ(0) λ − 2.5Cλlog(b/a) (1) mλcorr = m λ −Aλ (2) Aλ = −2.5log 1 + e−τ λsec(i) + (1− 2f) 1− e−τ λsec(i) τλsec(i) Here, CR,B=1 (Verheijen 1997); (b/a) is the ratio of the minor to major axis; f = 0.1 and τR,B=0.40, 0.81 (Tully, et al. 1998; Verheijen 1997). Finally, a correction was applied to account for the effect of stellar absorption in the Balmer line of Fcor = Fobs , (4) where Fcor is the corrected and Fobs is the observed Hα flux, We is the measured equivalent width and Wa is the equivalent width of the Balmer absorption lines. As we do not have measurements for Wa, we estimated Wa to be 3±1 Å, based off the values found in Oey & Kennicutt (1993); Roennback & Bergvall (1995); McCall, Rybski, & Shields (1985). Note that this effect is potentially stronger in the diffuse gas than in the H II regions due to the older stellar population likely lying in the diffuse gas. As a result we may still be underestimating the total Hα flux in the diffuse gas within the galaxies. However, as the diffuse gas fractions for these galaxies are extremely high (see Section 6, below), it is unlikely that this effect is high. Global parameters and radial profiles for the galaxies were determined primarily using the routines available in IRAF (notably ellipse) and the results are given in Tables 1 and 2. Galaxy images, surface brightness profiles, and color profiles are given in Figures 1 – 3. In all cases the inclination and position angle for the galaxies were determined from the best fit values from the B & R frames. These best fit values were then used for the ellipse fitting in all four images (B, R, Hα and continuum with Hα subtracted), a practice which insures the color profiles are obtained accurately and are not affected by, e.g. misaligned ellipses. The same apertures were also used for all four images, with the apertures found through allowing ellipse to range from 1 pixel (0.196′′) until the mean value in the ellipse reaches the sky value, – 6 – increasing geometrically by a factor of 1.2. Sky values were found through determining the mean value in more than 100 5×5 sq. pixel boxes in each frame. The error found for the sky was incorporated into all magnitude and surface brightness errors, which also include errors from the determination of the zeropoint and the errors from the N II contribution to the Hα (in the case of the Hα magnitudes). The B and R surface brightness profiles of all galaxies were fit using two methods. First, the inner regions of the galaxies’ surface brightness profiles was fit using the de Vaucouleurs r1/4 profile Σ(r) = Σeffexp −7.669[(r/reff )1/4−1] → µ(r) = µeff + 8.327 , (5) and the outer regions were fit by the exponential disk profiles Σ(r) = Σ0exp ( rα) → µ(r) = µ0 + 1.086 . (6) Additionally, we attempted to fit a disk profile (6) to both the inner and outer regions of the galaxies’, to determine if a two-disk fit would better match the data (Broeils & Courteau 1997; de Jong 1996). Roughly one-fourth of the galaxies (5/19) were best fit (in the χ2-sense) by the standard bulge+disk model. Another 47% of the galaxies were best fit by the two-disk model. Of the remaining galaxies, 21% (4 galaxies) were best fit by a single disk, and one galaxy (UGC 11840) could not be fit by any profile. The results from the fits are shown in Table 3 and Figure 2, and an asterisk (*) is placed next to the best fit model. Note that in a few cases (e.g. UGC 00189) only one model is listed in the Table. This is due to the fact that in these cases the fitting using the other model proved to be completely unrealistic. Finally, it should be noted that in all cases the same best-fit model was used for both the B and R data. The color profiles were similarly fit (using an an inverse error weighting) with a line to both the inner and outer galaxy regions (Figure 3). Here, though, the “boundary radius” was simply taken from the surface brightness profile fits, with the “boundary radius” being defined as the radius where the inner and outer surface brightness fits crossed. If only one (or no) fit was made to the surface brightness profile, then only one color profile was fit. In a number of cases the difference in slope between the inner and outer galaxy regions was less than the least-squares error for the fit. In these cases again only one line was fit for the color profiles. The HIIphot program (Thilker, Braun, & Walterbos 2000) was used both to determine the shape and number of H II regions for each galaxy and also to determine the Hα flux for each of these regions. The fluxes from the Hα, Hα-subtracted continuum, B, and R images – 7 – were measured in identical corresponding apertures, which are the H II region boundaries defined by HIIphot. While HIIphot applies an interpolation algorithm across these apertures to estimate the diffuse background in the Hα frames, we determined the background in other bands from the median flux in an annulus around each H II region aperture. Results from the analysis of the H II regions are given in Table 4, and sample H II regions are shown in Figure 4. Errors for the Hα, SFR, and EW measurements are derived from the error values reported with HIIphot. Errors for the B and R magnitudes, and colors, are derived from the total sky and zeropoint errors, as well as the error in positioning of the HII regions. The diffuse fraction errors are derived both from the total Hα flux errors and also include errors in determining the total flux within the HII regions and for the entire galaxy. Finally, it should be mentioned that the equivalent width (EW) was calculated simply as the ratio of the Hα flux to Hα-subtracted continuum flux for a given region (or the whole galaxy). The large distances to the observed galaxies (40 - 100 Mpc) results in many of the H II region being blended together. As a result, any luminosity function derived for these objects would be necessarily skewed towards larger HII regions (see Oey, et al. 2006). This can be seen in the analysis done by Thilker, Braun, & Walterbos (2000) wherein the dependence of the luminosity function found for M51 was examined. There one can clearly see the increase in the number of high luminosity regions and subsequent reduction in the number of low luminosity regions as the galaxy is ’moved’ to increasing distances. Examining their results also shows that while the distribution of H II region luminosities changes with distance, the total luminosity of the H II regions, as found by HIIphot, does not change significantly as the galaxy moves from 10 Mpc to 45 Mpc. As a result, while determining luminosity functions for the galaxies in this paper is not feasible due to the distances involved, derivations such as the diffuse fraction are unaffected by distance. This fact is also supported by the SINGG survey results (Oey, et al. 2006). 4. Surface Brightness The distribution of central surface brightnesses found for the galaxies observed is shown in Figure 5. As is plain from that Figure, the mean measured central surface brightness for this sample, falls short of the definitions discussed in Section 2. Indeed only 4 galaxies in our sample meet the operational definition of LSB galaxies as having µB(0) ≥ 23 mag arcsec If we return to the Freeman value, however, we see that the operational definition of LSB galaxies is 4.5σ from the value for Hubble sequence spirals, making it statistically extreme. For the sample defined here, half have central surface brightnesses at least two sigma above the Freeman value, a definition only 2.5% of the Freeman sample meets. As a result, while – 8 – the sample does not meet the operation criteria for LSB galaxies, we clearly do have a sample with lower central surface brightnesses that would be found in the average Hubble sequence galaxies. It should be pointed out here that the main scientific focus of Bothun, et al. (1985) was not oriented toward producing a representative sample of LSB galaxies as detected on photographic surveys (that focus did not occur until Schombert & Bothun 1988), but rather toward identifying cataloged galaxies for 21-cm based redshift determinations. The galaxies were chosen to have surface brightnesses that were too low for reliable optical spectroscopy (assuming emission lines were not present). This was done as a test of the potentially large problem of bias in on going optical redshift surveys in the time (see Bothun, et al. 1986). In fact, the operational criteria for selecting the galaxies that were observed at Arecibo 20 years ago, lay in the knowledge that these cataloged galaxies were never going to be even attempted in the optical redshift surveys of the time and this raised the very real possibility of biased redshift distributions and an erroneous mapping of large scale structure. In the original redshift measurements of Bothun, et al. (1985) a significant number of candidate LSB galaxies were not detected at 21-cm within the observational redshift window (approximately 0-12,000 km/s). Many of those non-detections would later turn out to be intrinsically large galaxies located at redshifts beyond 12,000 km/s (see O’Neil, et al. 2004). As we are interested here in the Hα properties of galaxies with large, relatively LSB disks, these initial non-detections comprise the bulk of our sample. Surface photometry of this sample not only provides detailed information regarding the galaxies’ surface brightness and color distributions, but it also probes the efficacy of the Bothun, et al. (1985) average surface brightness criteria for selecting LSB disks. Here, we used the magnitudes and diameters obtained in this study (Table 1) with two different equations for determining a galaxy’s average surface brightness within the D25 radius. The first equation used is that of Bothun, et al. (1985) 〈µ25〉 = m25 + 5log(D25) + 3.63 (7) and the second is a modified version of the above equation from Bottinelli, et al. (1995) which takes the galaxies’ inclination into account: 〈µ25〉 = m25 + 5log(D25) + 3.63− 2.5log kR−2C + (1− k)R(0.4C/K)−1 . (8) In both equations, m25 and D25 are the magnitude and diameter (in units of 0.1 ′) at the µ=25.0 mag arcsec−2 isophote, R is the axis ratio (a/b), and C is defined as (logD/logR) and is fixed at 0.04 (Bottinelli, et al. 1995). Finally, k (the ratio of the bulge-to-disk lumi- nosity) and K (a measure of how the apparent diameter changes with surface brightness at – 9 – a given axis ratio) are dependent on the revised de Vaucouleurs morphological type (T) as follows (Simien & de Vaucouleurs 1986; Fouqué & Paturel 1985): T=1 → k=0.41; T=2 → k=0.32; T=3 → k=0.24; T=4 → k=0.16; T=5 → k=0.09; T=6 → k=0.05; T=7 → k=0.02; T≥8 → k=0.0; K = 0.12− 0.007T if T < 0; K = 0.094 if T ≥ 0. The values for k at T≥8 are extrapolated from fitting the Simien & de Vaucouleurs (1986) values. The results of equations 7 and 8, plotted against the galaxies’ central surface brightness both uncorrected and corrected for inclination, are shown in Figures 6 and 7, respectively. The difference between the two plots is small, with neither equation doing an excellent job in predicting when a disk’s central surface brightness will be low. The two equations (Bothun, et al. (1985) and Bottinelli, et al. (1995)) have roughly the same fit (in the χ2 sense), which at first appears surprising. It is likely that uncertainties in the inclination measurements and morphological classification of the galaxies have increased the scatter in the Bottinelli, et al. (1995) equation, increasing the scatter in an otherwise more accurate equation. As a result, while the Bottinelli, et al. (1995) may indeed be the most accurate, the simpler equation is equally as good to use in most circumstances as it involves fewer assumptions. The second fact that is readily apparent in looking at Figures 6 and 7 is that with the new measurements of magnitude and diameter, none of the galaxies in our sample meet the criterion laid out by Bothun, et al. (1985) for an LSB galaxy. That is that none of the galaxies in this sample have 〈µ25〉 >25 mag arcsec −2. As Bothun, et al. (1985) listed all of these objects as having 〈µ25〉 >25 mag arcsec −2 using the magnitudes and diameters provided by the original UGC measurements, this shows that the UGC measurements indeed predicted fainter magnitudes/larger values for D25 than is found with more sophisticated measurement techniques. Additionally, it is good to note that the trends shown in Figures 6 and 7 indicate that any galaxy which met the 〈µ25〉 >25 mag arcsec −2 criteria would be highly likely to also have µ(0) >23 mag arcsec−2. In these days of digital sky surveys it is difficult to appreciate the immense undertaking that defines the UGC catalog. Anyone who has looked at the Nilson selected galaxies on the Palomar Observatory Sky Survey (POSS) plates with a magnifying eyepiece really has to marvel that Nilson’s eye saw objects at least one arcminute in diameter. It is thus not surprising that, at the ragged end of that catalog, many of the listed UGC diameters are systematically high. Cornell, et al. (1987) made a detailed diameter comparison between diameters as obtained from high quality CCD surface photometry and the estimates made by Nilson (1973). They compared the diameter at the 25.0 mag arcsec−2 isophote in CCD – 10 – B images to the tabulated diameter in the UGC. The study, based on approximately 250 galaxies, identified two sources of systematic error (neither of which are surprising). First, galaxies with reported diameters less than 2′ typically had D25,B as measured by the CCD images that were 15-25% smaller. Second, Cornell, et al. (1987) found a systematic bias as a function of surface brightness in the sense that lower surface brightness galaxies had a higher number of overestimated diameters in the UGC than higher surface brightness galaxies. It should also be noted that the majority of the galaxies in this study lie at low Galactic latitude. This seems to be a perverse consequence that there is a large collection of galaxies between 7,000 – 10,000 km s−1 (where the diameter criterion in the UGC yields a relatively large physical size) located at relatively low galactic latitude. Nominal corrections for galactic extinction made by Bothun, et al. (1985) turned out to underestimate the extinction as shown by later published extinction maps. In some cases, the differences were as large as one magnitude. The combination of these facts with the very uncertain magnitudes of many of these galaxies (see Bothun & Cornell 1990), it is not surprising that the measured average surface brightness could easily be 1-1.5 magnitudes higher than the average surface brightness that has been estimated from the UGC catalog parameters (roughly 40% of this comes from systematic magnitude errors and 60% from the diameter errors discovered by Cornell, et al. (1987)). 5. Morphology & Color All of the galaxies observed have large sizes (3αB = 10 – 54 kpc), bright central bulges, and well defined spiral structure (Figure 1). In most cases the galaxies can be described as late-type systems (Sbc and later). There are, though, a number of exceptions to this rule. Three of the galaxies, UGC 00023, UGC 07598, and UGC 11355 (Sb, Sc, and Sb galaxies, respectively) have clear nuclear bars. UGC 08311, classified as an Sbc galaxy, is clearly in the late stages of merging with another system. In this case the LSB classification of the galaxy is likely bogus, as the apparently LSB disk is likely just the remnant the merging process and will disappear as the galaxy compacts after the merging process. UGC 8904 is given a morphological type of S? with both NED and HYPERLEDA, yet the faint spiral arms surrounding it indicate its should be properly classified as an Sbc system. UGC 12021 is, like UGC 00023, listed as an Sb galaxy. Finally, UGC 11068 has a faint nuclear ring which is most readily visible in the B image. The differences between the galaxies becomes more apparent when the Hα images are examined. Hodge & Kennicutt (1983) classify the radial distribution of H II regions in spiral galaxies into three broad categories – galaxies with H II region surface densities which – 11 – decrease with increasing radius, galaxies with oscillating H II region surface densities, and galaxies with ring-like H II density distributions. To these categories we would add a fourth, to include those galaxies with no detectable H II regions. The first category of Hodge & Kennicutt (1983) is also the most common, as it includes all galaxies with generally decreasing radial densities of Hα. In the Hodge & Kennicutt (1983) sample this category is dominated by Sc – Sm galaxies but contains all Hubble types. In our sample, this category includes both galaxies with and without significant Hα emission in the spiral arm regions. This group includes UGC 00023, UGC 00189, UGC 02588, UGC 02796, UGC 03119, UGC 03308, UGC 07598, and UGC 12021. Interestingly, of the galaxies listed above, 4/8 are Sb/Sbc galaxies and 3/8 are Sc-Sm galaxies. (The last galaxy, UGC 02588, is an irregular galaxy.) The second category of Hodge & Kennicutt (1983), galaxies with oscillating densities, is dominated in their sample of Sb galaxies. Only a few of the galaxies in this sample fall into this category, 80% of which are also Sb/Sc galaxies. These are UGC 02299, UGC 08311, UGC 08904, UGC 11355, and UGC 11396. These galaxies all have a concentration of star formation seen in the nuclear regions and then clumps of star formation spread through the spiral arms, typically accompanied by diffuse Hα also spread throughout the arms. The third category of Hodge & Kennicutt (1983) is dominated by early-type galaxies, of which we have none in our sample. Nonetheless we have three galaxies which fall into this category – UGC 08644, UGC 10894, and UGC 11617. All three have H II regions spread throughout their disks, with no central concentration near the galaxies’ nuclei. In fact, the three brightest star forming regions within UGC 08644 all lie with the spiral arms, and are visible in all three filters. In contrast, both UGC 11617 and UGC 10894 have no bright H II regions, but instead have a large number of diffuse H II regions, with the brightest (as listed in Table 2) receiving that designation simply due to its size. The fourth category of galaxies contains UGC 01362, UGC 11068, and UGC 11840, none of which have detectable Hα. In the case of UGC 01362 and UGC 11840 this is not too surprising as the galaxies are dominated by a bright nucleus, and their surrounding spiral arms are extremely faint in both R and B. As a result, any Hα which may exist in the galaxies’ disks is too diffuse to be detected. UGC 11068, though, has both a well defined nucleus and a clear spiral structure extending out to a radius of ∼13 kpc (3α). Yet no Hα can be detected in this galaxy. This may mean that UGC 11068 is in a transition state for its star formation, with no ongoing star formation yet with enough recent activity that the spiral arms remain well defined. Perhaps the most intriguing galaxy of our sample is UGC 11355. This galaxy was – 12 – placed in Category 2, above, as it has a bright nucleus and clumpy disk in the Hα image. The B and R band images of UGC 11355 show a galaxy with a simple Sbc morphology. The Hα image, though, shows a distinct star forming ring. The ring is at a very different inclination from the rest of the galaxy (i=49◦ for the ring and 73◦ for the galaxy as a whole), and lies approximately 2.6 kpc in radius from the center of the galaxy, measured along the major axis. As the B and R images show no indication of a ring morphology this indicates unusually strong star formation in the ring. It is also useful to note the presence of a bar in UGC 11355 – shown more clearly in Figure 8. The fact that the inclination of the ring is significantly different from that of the rest of the galaxy suggests the ring a tidal effect due to an interaction, such as a small satellite galaxy being cannibalized by UGC 11355, or the influence of CGCG 143-026, 14.9′ and 68 km s−1 away. It is interesting to note that the Hα morphology of the galaxies does not appear to correlate with the galaxies’ color profile (Figure 3). The galaxy with the steepest slope in the color profile is UGC 08644 which has only a few H II regions in its outer arms. The other galaxies with steep color profiles are UGC 00023 and UGC 8904, which have a bright knot of star formation in the nucleus and faint Hα spread throughout their arms, and UGC 11840 and UGC 11068 both of which have no detectable Hα. The galaxies with the shallowest slopes similarly show no correlation between their color profiles and morphology. This suggests that the current star formation in these galaxies is largely independent of the past star-formation history, although this result should be confirmed with better, extinction-corrected, data. 6. Star Formation Figures 10 – 17 compare the properties of the H II regions and emission of our galaxy sample. Where possible, measurements from other samples of late-type galaxies are also shown (Kennicutt & Kent 1983; Jansen, et al. 2000; Helmboldt, et al. 2005; Oey, et al. 2006). Examining the figures it is clear that the overall properties of our sample are similar to those of other late-type (Sbc-Sc) galaxies. That is, the values for the individual H II region luminosities are similar to those reported by Helmboldt, et al. (2005) and Kennicutt & Kent (1983) (Figure 10) while the global Hα equivalent width (EW) and global star formation rates match those seen by all three comparison samples (Figures 11, 12). We should note that as discussed in § 3 our sample suffers from having many of the H II regions blended together as a result of the distance to our galaxy samples. As a result, it is highly likely that in the comparisons of the luminosities for the galaxies’ individual H II regions the luminosities (Figure 10) from our sample are artificially higher then those in the other sample, potentially by a factor of 3 or more. This fact does not alter the results of this – 13 – section, but it is the likely explanation for the slightly higher than average values found for L3 in Figure 10. To examine the total amount of gas found within the H II regions compared with that found in the diffuse Hα gas, we need to determine the galaxies’ Hα diffuse fraction, defined here as the ratio of Hα flux not found within the defined H II regions to the total Hα flux found for the entire galaxy. Examining Figures 13 and 14, as well as Tables 2 and 4, reveals an interesting fact – while the global SFR for these galaxies is fairly typical (0.3 – 5 M⊙/yr), the combined SFR from the galaxies’ H II regions is a factor of 2 – 10 smaller. That is, on average the majority of the Hα emission and thus the majority of the star formation in the observed galaxies comes not from the bright knots of star formation but instead from the galaxies’ diffuse Hα gas. This is in contrast to the behavior seen from typical HSB galaxies, as evidenced by the data of Oey, et al. (2006) in Figure 13. We note that blending and angular resolution effects appear to be relatively unimportant in estimating the fraction of diffuse Hα emission. Oey, et al. (2006) demonstrate this by showing no systematic changes in measured diffuse fractions as a function of distance up to almost 80 Mpc, and inclination angle, for their sample of 100+ SINGG survey galaxies. While at first glance the higher diffuse Hα fractions found for these galaxies seems surprising, recent GALEX results of the outer edges of M83, a region whose environment closely resembles that of the disks of massive LSB galaxies, also show considerable star formation outside the H II regions in that part of the galaxy (Thilker, et al. 2005). Similarly, Helmboldt, et al. (2005) found a slight trend with lower surface brightness galaxies having higher diffuse fractions than their higher surface brightness counterparts. The fact that these galaxies have higher Hα diffuse gas fractions raises an interesting question. Typically diffuse gas is believed to be ionized by OB stars lying within density- bounded H II regions. The problem of transporting the ionizing photons from these regions to the diffuse gas is extreme in these cases, as there would need to be a very large number of density-bound H II regions leaking ionizing photons to ionize the quantity of diffuse gas seen here. (See the more detailed discussion in Hoopes, Walterbos, & Bothun 2001, which also discusses shock heating from stellar winds and SNe as ionization sources.) An alternative suggestion is that field OB stars are also ionizing the diffuse gas, as was suggested by Hoopes, Walterbos, & Bothun (2001). This would imply a different stellar population within and without the H II regions, as it would likely be the later OB types (B0–O9) which either escape the H II regions or survive the regions’ destruction. We note Oey, King, & Parker (2004) predict a modest increase in the fraction of field massive stars in galaxies with the lowest absolute star-formation rates. Scheduled GALEX observations of a subset of our observed galaxies may shed light on the underlying stellar population in the galaxies’ diffuse – 14 – stellar disks. Finally, it is elucidating to look for any trends between the global and regional properties of the galaxies and their SFR and Hα content. Figure 15 plots the galaxies’ central surface brightness (in both B and R) against the galaxies’ total star formation rate. While the error bars make defining any trend difficult, there certainly appears to be a decrease in the global SFR with decreasing central surface brightness, similar to the trends seen in other studies (e.g. van den Hock, et al. 2000; Gerritsen & de Blok 1999). Figure 16 shows the galaxies’ gas-to-luminosity ratios plotted against both their global equivalent width and diffuse Hα fraction. In both cases, no trend can be seen with our data, although the small number of points available make any diagnosis difficult. Combined with the other datasets, though, we can see a general trend toward higher equivalent widths with increasing MHI/LB, but surprisingly no trend between gas fraction and the galaxies’ diffuse Hα fraction is visible. This lack of correlation is also seen by Oey, et al. (2006). The last trend which can be seen is a rough correlation between the galaxies’ global color and star formation rate (Figure 11), with redder galaxies having higher SFR, a fact which may be a reddening effect. The individual Hα regions, however, show no such trend (Figure 17). 7. Conclusion The sample of 19 galaxies observed for this project were chosen to be large galaxies with low surface brightness disks. The surface brightness measurements for this sample were obtained originally through the UGC measurements through determining the galaxies’ average surface brightness within the µ=25 mag arcsec−2 isophote. The relation employed to determine the galaxies’ average surface brightness (Equation 7) has shown itself it be a good predictor of a galaxy’s central surface brightness. But for a wide variety of reasons the UGC measurements were not sufficient to insure the galaxies contained within this catalog have true LSB disks, underscoring the difficulty in designing targeted searches for large LSB galaxies. Nonetheless, the sample of galaxies observed for this project have lower surface bright- nesses than is found for a typical sample of large high surface brightness galaxies. In most other aspects the galaxies appear fairly ‘normal’, with colors typically B−R=0.3−0.9, mor- phological types ranging from Sb – Irr, and color gradients which typically grow bluer toward the outer radius. However, the galaxies have both higher gas mass-to-luminosity fractions and diffuse Hα fractions than is found in higher surface brightness samples. This raises two questions. First, if the SFR for these galaxies has been similar to their higher sur- face brightness counterparts through the galaxies’ life, why do the lower surface brightness – 15 – galaxies have higher gas mass-to-luminosity ratios? Second, why do these galaxies have a higher fraction of ionizing photons outside the density-bounded H II regions then their higher surface brightness counterparts? The answer to the first question posed above likely comes from the difference between the studied galaxies’ current and historical SFR. As these galaxies have on average and lower metallicities (de Naray, McGaugh, & de Blok 2004; Gerritsen & de Blok 1999) than their higher surface brightness counterparts, it is likely that the galaxies’ SFR has not remained constant throughout the their lifetimes. Indeed the simplest explanation for the current similar SFRs and higher gas mass-to-luminosity ratios for the studied galaxies than for their higher surface brightness counterparts is that the galaxies’ past SFR was significantly different than is currently seen. In fact, the measured properties would be expected if the galaxies in this study have episodic star formation histories, with significant time (1-3 Gyr) lapsing between SF bursts, as has been conjectured for LSB galaxies in the past (e.g. Gerritsen & de Blok 1999). Such a star formation history would help promote significant changes in the galaxies’ mean surface brightness and allow an individual large disk galaxy to appear as either (a) a relatively normal Hubble sequence spiral, (b) a large, lower surface brightness disk, or (c) perhaps even a lower surface brightness disk if the time between episodes is sufficiently large, depending on the elapsed time since the last SF burst. 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Global Properties of Galaxies – B & R Measurements Galaxy RAa Deca Vela Typea mb Mb D25 c 〈µ〉d mb Mb D25 c 〈µ〉d rb B−Rb ie [J2000] [J2000] km s−1 [mag] [Mag] [′′] mag/′′2 [mag] [Mag] [′′] mag/′′2 [′′] [◦] UGC 00023 00 04 13.0 10 47 25 7787 3 14.4 (0.1) -20.7 (0.1) 71 23.4 13.0 (0.1) -22.1 (0.1) 86.7 22.4 40 1.4 (0.2) 52 UGC 00189 00 19 57.5 15 05 32 7649 7 15.0 (0.2) -20.1 (0.2) 84 24.4 13.8 (0.1) -21.3 (0.1) 116.1 23.8 57 1.2 (0.2) 67 UGC 01362 01 52 50.7 14 45 52 7918 8.8 16.9 (0.2) -18.2 (0.2) 30 24.3 15.6 (0.1) -19.5 (0.1) 42.1 23.5 23 1.3 (0.3) 0 UGC 02299 02 49 07.8 11 07 09 10253 8 15.4 (0.2) -20.3 (0.2) 59 24.0 14.5 (0.4) -21.2 (0.4) 65.1 23.3 33 0.9 (0.4) 32 UGC 02588 03 12 26.5 14 24 27 10093 9.9 15.8 (0.2) -19.9 (0.2) 39 23.6 14.7 (0.1) -20.9 (0.1) 50.1 23.0 28 1.1 (0.2) 28 UGC 02796 03 36 52.5 13 24 24 9076 4 14.8 (0.2) -20.6 (0.2) 66 23.6 13.3 (0.1) -22.1 (0.1) 94.1 22.7 28 1.5 (0.2) 57 UGC 03119 04 39 07.7 11 31 50 7851 4 14.3 (0.2) -20.8 (0.2) 71 23.7 12.4 (0.1) -22.7 (0.1) ‡ 24.1 40 1.9 (0.2) 72 UGC 03308 05 26 01.8 08 57 25 8517 6 14.3 (0.3) -21.0 (0.3) 89 23.8 14.0 (0.2) -21.3 (0.2) 88.1 23.5 48 0.3 (0.4) 28 UGC 07598 12 28 30.9 32 32 52 9041 5.9 15.3 (0.1) -20.1 (0.1) 46 23.5 14.8 (0.1) -20.6 (0.1) 66.8 22.8 33 0.5 (0.2) 22 UGC 08311 13 13 50.8 23 15 16 3451 4.1 15.5 (0.1) -17.9 (0.1) 50 23.8 14.8 (0.1) -18.5 (0.1) 62.0 23.5 33 0.7 (0.2) 26 UGC 08644 13 40 01.4 07 22 00 6983 8 16.1 (0.2) -18.8 (0.2) 43 24.2 15.3 (0.2) -19.6 (0.2) 49.4 23.5 28 0.8 (0.3) 30 UGC 08904 13 58 51.1 26 06 24 9773 3.6 15.9 (0.1) -19.7 (0.1) 43 23.8 14.9 (0.1) -20.7 (0.1) 55.5 23.4 33 1.0 (0.2) 48 UGC 10894 17 33 03.8 27 34 29 6890 4 16.0 (0.3) -18.8 (0.3) 48 24.1 14.9 (0.3) -19.9 (0.3) ‡ 23.0 28 1.1 (0.4) 57 UGC 11068 17 58 05.0 28 14 38 4127 3.2 15.0 (0.2) -18.7 (0.2) 64 24.0 13.8 (0.1) -19.9 (0.1) 89.4 23.4 57 1.2 (0.2) 0 UGC 11355 18 47 57.0 22 56 33 4360 3.5 13.9 (0.3) -19.9 (0.3) 173 24.8 12.5 (0.1) -21.3 (0.1) ‡ 23.7 82 1.4 (0.3) 73 UGC 11396 19 03 49.5 24 21 28 4441 3.5 14.8 (0.3) -19.1 (0.3) 114 24.4 13.8 (0.2) -20.1 (0.2) 78.7 23.0 33 1.0 (0.4) 59 UGC 11617 20 43 39.3 14 17 52 5119 6.1 14.9 (0.2) -19.2 (0.2) 75 24.1 13.9 (0.2) -20.3 (0.2) 86.1 23.3 40 1.1 (0.3) 58 UGC 11840 21 53 18.0 04 14 50 7986 4 16.3 (0.1) -18.9 (0.1) 24 22.9 15.3 (0.1) -19.9 (0.1) 23.3 21.9 11 1.0 (0.2) 40 UGC 12021 22 24 11.6 06 00 12 4472 3 14.7 (0.2) -19.2 (0.2) 113 24.7 13.6 (0.1) -20.2 (0.1) 130.4 23.9 57 1.1 (0.2) 63 Note. — Errors are given in parenthesis. aRA, Dec, velocity and type information obtained from NED, the NASA Extragalacitc Database. Galaxy types are defined in de Vaucouleurs, et al. (1991). bMagnitudes and colors were obtained at the maximum usable radius, r. Corrections applied and error estimates are described in Section 3. cD25 are the diameters for the 25 mag arcsec −2 isophotes. dAverage surface brightness, as defined by Equation 7 in Section 4. eInclinations are simply the major to minor axis ratio of the galaxies, found through isophote fitting. ‡Isophotes did not reach 25 mag arcsec−2. – 20 – Table 2. Global Properties of Galaxies – Hα Galaxy Hα flux ×10−13a EWb SFRc rd erg cm−2 s−1 UGC 00023 5 (1) 22 (8) 4 (1) 28 UGC 00189 4 (1) 63 (33) 4 (1) 23 UGC 01362 · · · · · · · · · · · · UGC 02299 0.41 (0.09) 38 (13) 0.7 (0.2) 6 UGC 02588 0.7 (0.3) 32 (21) 1.1 (0.5) 13 UGC 02796 2.1 (0.7) 12 (4) 2.7 (0.9) 13 UGC 03119 7 (4) 20 (12) 6 (6) 28 UGC 03308 1.0 (0.3) 22 (8) 1.1 (0.3) 11 UGC 07598 1.3 (0.3) 60. (18) 1.8 (0.4) 19 UGC 08311 3.1 (0.6) 91 (37) 0.6 (0.1) 19 UGC 08644 · · · · · · · · · · · · UGC 08904 0.8 (0.2) 40 (13) 1.1 (0.3) 16 UGC 10894 0.4 (1) 34 (78) 0.4 (0.1) 9 UGC 11068 · · · · · · · · · · · · UGC 11355 8 (2) 30 (9) 2.7 (0.6) 28 UGC 11396 2.4 (0.8) 400 (2000) 0.8 (0.2) 16 UGC 11617 3.1 (0.7) 30 (2) 0.3 (0.2) 19 UGC 11840 · · · · · · · · · · · · UGC 12021 4 (1) 63 (32) 1.3 (0.4) 23 UGC 12289 · · · · · · · · · · · · Note. — Derivation of quantities are described in Section 3. Errors are given in parenthesis. aTotal Hα flux found within the radius centered on the (optical) center of the galaxy and extending to the radius given in the last column. Errors were determined in the same manners as for magnitudes, and are given in Section 3. bThe equivalent width was calculated simply as the ratio of the total Hα flux to total Hα-subtracted continuum flux. cSFR = 1.26×1041ergs−1 ; from Kennicutt, Tamblyn, & Congdon (1994). dRadius at which the isophotal signal-to-noise went below 1σ. – 21 – Table 3. Fitted Galaxy Properties inner outer Galaxy Fit Filterb µeff/µ0 c Reff/α e αf Boundaryg Fith Type a mag arcsec−2 mag arcsec−2 [′′] [′′] Error ∗ UGC 00023 Two Disk B 19.06 (0.33) 1.71 (0.02) 21.17 (0.21) 11.05 (0.06) 6.55 0.82 ∗ UGC 00023 Two Disk R 17.30 (0.27) 1.72 (0.02) 19.58 (0.16) 9.58 (0.03) 6.93 0.53 UGC 00023 Bulge/Disk B 21.48 (1.98) 4.06 (0.23) 21.47 (0.51) 11.96 (0.11) 8.94 0.89 UGC 00023 Bulge/Disk R 19.20 (1.56) 3.07 (0.14) 19.82 (0.35) 9.96 (0.05) 8.79 0.58 ∗ UGC 00189 Two Disk B 20.99 (0.08) 7.75 (0.06) 24.05 (1.85) 36.90 (2.11) 28.34 0.33 ∗ UGC 00189 Two Disk R 19.53 (0.09) 6.13 (0.05) 21.66 (0.67) 18.84 (0.22) 24.92 0.97 ∗ UGC 01362 One Disk B · · · · · · 23.06 (0.33) 7.77 (0.10) · · · 1.78 ∗ UGC 01362 One Disk R · · · · · · 21.84 (0.15) 7.85 (0.04) · · · 0.46 ∗ UGC 02299 Two Disk B 20.85 (0.10) 2.00 (0.16) 22.43 (1.76) 11.24 (0.04) 6.48 1.88 ∗ UGC 02299 Two Disk R 19.84 (0.28) 2.21 (0.03) 21.46 (0.38) 10.50 (0.10) 7.23 0.99 UGC 02299 Bulge/Disk B 25.10 (2.85) 17.35 (1.95) 23.23 (2.18) 12.62 (0.27) 17.01 2.08 UGC 02299 Bulge/Disk R 24.00 (1.75) 17.76 (1.24) 22.30 (1.43) 10.37 (0.19) 0.00 1.18 UGC 02588 One Disk B · · · · · · 21.50 (0.08) 5.47 (0.02) · · · 1.40 UGC 02588 One Disk R · · · · · · 20.41 (0.07) 5.79 (0.02) · · · 1.12 ∗ UGC 02588 Bulge/Disk B 25.75 (2.43) 16.16 (1.75) 22.61 (0.97) 8.78 (0.08) 6.94 1.18 ∗ UGC 02588 Bulge/Disk R 23.77 (2.50) 7.86 (0.75) 21.09 (0.48) 7.41 (0.04) 4.57 0.82 ∗ UGC 02796 Two Disk B 19.07 (0.34) 1.89 (0.03) 21.02 (0.60) 8.75 (0.14) 6.90 0.48 ∗ UGC 02796 Two Disk R 17.52 (0.27) 2.19 (0.03) 19.66 (0.41) 8.69 (0.08) 8.59 0.68 UGC 02796 Bulge/Disk B 23.07 (2.01) 13.80 (1.10) 22.54 (4.11) 8.92 (0.45) † 0.71 UGC 02796 Bulge/Disk R 21.01 (0.90) 10.26 (0.31) 20.49 (1.06) 6.76 (0.22) † 0.21 UGC 03119 One Disk B · · · · · · 19.82 (0.07) 10.35 (0.03) · · · 0.75 UGC 03119 One Disk R · · · · · · 17.92 (0.05) 9.48 (0.01) · · · 0.49 ∗ UGC 03119 Bulge/Disk B 22.05 (9.17) 3.54 (0.91) 20.03 (0.31) 11.51 (0.08) 3.50 0.61 ∗ UGC 03119 Bulge/Disk R 21.07 (5.81) 6.53 (1.29) 18.16 (0.24) 10.10 (0.02) 3.71 0.28 ∗ UGC 03308 Two Disk B 19.83 (2.25) 0.87 (0.04) 21.70 (0.18) 15.47 (0.16) 3.47 0.35 ∗ UGC 03308 Two Disk R 19.06 (1.32) 0.92 (0.03) 20.99 (0.13) 11.60 (0.06) 3.58 0.25 UGC 03308 Bulge/Disk B 18.36 (10.1) 0.43 (0.08) 21.71 (0.23) 15.66 (0.18) 3.62 0.37 UGC 03308 Bulge/Disk R 18.70 (5.77) 0.67 (0.08) 21.03 (0.18) 11.90 (0.08) 3.85 0.20 UGC 07598 One Disk B · · · · · · 21.69 (0.07) 7.96 (0.02) · · · 0.75 UGC 07598 One Disk R · · · · · · 19.82 (0.08) 6.26 (0.02) · · · 1.12 ∗ UGC 07598 Bulge/Disk B 16.02 (6.98) 0.21 (0.03) 21.84 (0.13) 8.60 (0.03) 3.25 0.52 ∗ UGC 07598 Bulge/Disk R 15.78 (3.90) 0.40 (0.03) 20.18 (0.21) 7.34 (0.04) 4.06 0.49 ∗ UGC 08311 Two Disk B 20.82 (0.09) 2.94 (0.02) 23.29 (0.66) 14.91 (0.26) 12.50 0.48 – 22 – Table 3—Continued inner outer Galaxy Fit Filterb µeff /µ0 c Reff/α e αf Boundaryg Fith Type a mag arcsec−2 mag arcsec−2 [′′] [′′] Error ∗ UGC 08311 Two Disk R 20.28 (0.08) 3.22 (0.03) 22.73 (0.71) 15.81 (0.32) 13.64 0.39 ∗ UGC 08644 One Disk B · · · · · · 22.56 (0.14) 8.79 (0.06) · · · 0.55 ∗ UGC 08644 One Disk R · · · · · · 21.19 (0.15) 6.40 (0.04) · · · 0.78 ∗ UGC 08904 Two Disk B 20.43 (0.16) 2.56 (0.02) 22.94 (0.71) 10.07 (0.15) 11.22 0.69 ∗ UGC 08904 Two Disk R 18.90 (0.16) 2.20 (0.01) 22.05 (0.49) 9.67 (0.10) 11.19 0.35 UGC 08904 Bulge Only B 23.99 (0.27) 12.55 (0.09) · · · · · · · · · 1.17 UGC 08904 Bulge Only R 21.83 (0.25) 6.95 (0.04) · · · · · · · · · 0.48 ∗ UGC 10894 One Disk B · · · · · · 21.65 (0.10) 7.90 (0.04) · · · 0.44 ∗ UGC 10894 One Disk R · · · · · · 20.26 (0.12) 6.69 (0.04) · · · 0.42 UGC 11068 Two Disk B 20.15 (0.65) 1.15 (0.02) 22.46 (0.13) 13.65 (0.07) 4.87 0.31 UGC 11068 Two Disk R 18.67 (0.54) 1.25 (0.02) 21.02 (0.14) 11.26 (0.05) 5.23 0.54 ∗ UGC 11068 Bulge/Disk B 20.36 (0.18) 1.00 (0.00) 22.53 (0.11) 14.10 (0.06) 5.31 0.38 ∗ UGC 11068 Bulge/Disk R 19.89 (2.52) 1.66 (0.10) 21.18 (0.25) 11.97 (0.07) 6.23 0.42 ∗ UGC 11355 Two Disk B 19.20 (0.15) 3.34 (0.02) 21.61 (0.18) 32.30 (0.24) 14.23 0.86 ∗ UGC 11355 Two Disk R 17.30 (0.16) 3.01 (0.02) 19.83 (0.13) 24.07 (0.10) 13.02 0.64 UGC 11355 Bulge/Disk B 23.29 (0.95) 19.99 (0.72) 22.13 (0.56) 38.81 (0.57) 26.27 1.13 UGC 11355 Bulge/Disk R 20.70 (0.94) 10.96 (0.37) 20.19 (0.35) 26.30 (0.18) 26.30 0.93 ∗ UGC 11396 Two Disk B 20.34 (1.82) 1.14 (0.05) 22.08 (0.27) 22.66 (0.39) 4.46 0.95 ∗ UGC 11396 Two Disk R 19.77 (2.17) 1.23 (0.08) 20.20 (0.17) 13.04 (0.10) 2.80 1.69 UGC 11396 Bulge/Disk B 20.52 (9.40) 0.97 (0.19) 22.10 (0.35) 23.03 (0.45) 4.75 0.93 UGC 11396 Bulge/Disk R 20.09 (3.23) 1.09 (0.32) 20.21 (0.23) 13.03 (0.11) 3.03 1.69 ∗ UGC 11617 One Disk B · · · · · · 21.30 (0.07) 12.83 (0.05) · · · 0.69 ∗ UGC 11617 One Disk R · · · · · · 20.08 (0.08) 10.99 (0.04) · · · 0.46 ∗ UGC 11840 No Fit B · · · · · · · · · · · · · · · · · · ∗ UGC 11840 No Fit R · · · · · · · · · · · · · · · · · · UGC 12021 One Disk B · · · · · · 20.73 (0.06) 11.05 (0.02) · · · 1.63 UGC 12021 One Disk R · · · · · · 19.41 (0.06) 10.26 (0.02) · · · 1.58 ∗ UGC 12021 Bulge/Disk B 26.49 (1.28) 23.19 (0.00) 20.88 (0.16) 11.59 (0.04) 2.37 1.54 ∗ UGC 12021 Bulge/Disk R 23.85 (0.32) 23.19 (0.00) 19.96 (0.18) 11.60 (0.03) 6.23 1.01 Note. — Derivation of quantities is described in Section 3. Errors are given in parenthesis. aThe type of fit made to the surface brightenss profile – bulge+disk, two exponential disks, or one exponential disk. – 23 – bOptical filter for the data described within that row. cEffective surface brightness (R1/4 bulge fit) or central surface brightness (exponential disk fit) for the inner disk fit. See Equations 5 and 6. dEffective radius (R1/4 bulge fit) or scale length (exponential disk fit) for the inner disk fit. See Equations 5 and 6. eCentral surface brightness for the outer exponential disk fit. fScale length for the outer exponential disk fit. gBoundary between the inner and outer fits, defined by where the fitted lines cross. hχ2 error for the fits. ∗Best fit – used for all further analysis. †Fitted lines for the bulge and disk components do not cross. Table 4. Properties of Hα Regions Galaxy Regiona Hα Flux ×10−15 Hα Luminosity×1038 SFRb EWc Bd Rd B−Rd Diffusee erg cm−2 s−1 erg s−1 M⊙ yr [mag] [mag] [mag] [%] UGC 00023 1 70 (14) 760 (150) 0.6 (0.1) 23 (1) 16.9 (0.1) 15.2 (0.1) 1.7 (0.2) 0.82 (0.04) UGC 00189 1 10 (2) 110 (20) 0.09 (0.02) 30 (2) 19.9 (0.6) 18.3 (0.5) 1.7 (0.7) 0.98 (0.01) UGC 01362 0 · · · · · · · · · · · · · · · · · · · · · · · · UGC 02299 1 12 (2) 240 (50) 0.19 (0.04) 20 (1) 21 (2) 21 (2) 0. (3) · · · UGC 02299 2 15 (3) 260 (50) 0.21 (0.04) 14 (1) 18.2 (0.2) 17.1 (0.2) 1.1 (0.3) · · · UGC 02299 TOTAL 22 (3) 490 (70) 0.39 (0.06) · · · · · · · · · · · · 0.45 (0.06) UGC 02588 0 · · · · · · · · · · · · · · · · · · · · · · · · UGC 02796 1 74.8 (15) 900 (200) 0.71 (0.1) 10 (1) 16.5 (0.2) 14.8 (0.4) 1.7 (0.4) 0.67 (0.01) UGC 03119 1 151 (30) 1600 (300) 1.3 (0.3) 16 (1) 18.8 (0.1) 16.7 (0.1) 2 (0.1) 0.77 (0.02) UGC 03308 0 · · · · · · · · · · · · · · · · · · · · · · · · UGC 07598 1 6 (1) 80 (17) 0.07 (0.01) 13 (1) 22.2 (0.7) 21.5 (0.7) 0.6 (1) · · · UGC 07598 2 4.7 (0.9) 60 (12) 0.05 (0.01) 12 (1) 22 (1) 20.2 (0.9) 2 (1) · · · UGC 07598 3 2.4 (0.5) 32 (6) 0.03 (0.01) 13 (1) 23 (1) 21 (1) 2 (2) · · · UGC 07598 4 6 (1) 80 (20) 0.06 (0.01) 12 (1) 21.1 (0.8) 19.6 (0.7) 2 (1) · · · UGC 07598 5 3.3 (0.7) 43 (9) 0.03 (0.01) 13 (1) 23 (2) 22 (2) 2 (3) · · · UGC 07598 6 6 (1) 80 (20) 0.06 (0.01) 11 (1) 21 (3) 22 (3) -1 (4) · · · UGC 07598 7 4.1 (0.8) 50 (10) 0.04 (0.01) 11 (1) 22 (1) 21 (2) 1 (4) · · · UGC 07598 8 1.3 (0.3) 15 (3) 0.012 (0.003) 9 (1) 25 (3) 24 (3) 1 (3) · · · UGC 07598 9 32 (6) 430 (90) 0.34 (0.07) 13 (1) 18 (0.1) 16 (0.1) 1.9 (3) · · · UGC 07598 10 1.4 (0.3) 20. (4) 0.015 (0.003) 16 (1) 21.8 (0.5) 20.4 (0.5) 1.3 (0.5) · · · UGC 07598 11 10 (2) 130 (30) 0.11 (0.02) 11 (1) 21.4 (0.5) 20 (0.6) 1.4 (0.8) · · · UGC 07598 12 4.8 (1) 60 (10) 0.05 (0.01) 11 (1) 22 (1) 20.2 (0.9) 2 (1) · · · UGC 07598 13 4.4 (0.9) 60 (10) 0.05 (0.01) 12 (1) 21.8 (0.8) 20.1 (0.7) 2 (1) · · · UGC 07598 14 5 (1) 70 (10) 0.05 (0.01) 11 (1) 22.7 (1) 21 (1) 1 (2) · · · UGC 07598 TOTAL 69 (6) 1200 (100) 0.96 (0.08) · · · · · · · · · · · · 0.5 (0.1) UGC 08311 1 7 (1) 14 (3) 0.011 (0.002) 20. (1) 21.3 (0.7) 21 (0.9) 0. (1) · · · UGC 08311 2 100 (20) 240 (50) 0.19 (0.04) 53 (1) 18.8 (0.4) 18.4 (0.5) 0.3 (0.6) · · · UGC 08311 3 7 (1) 16 (3) 0.012 (0.003) 18 (1) 20.8 (0.1) 20. (1) 1 (1) · · · UGC 08311 4 17 (3) 38 (8) 0.03 (0.01) 29 (1) 0.9 (0.1) 20.5 (0.9) -19.6 (0.9) · · · UGC 08311 5 8 (2) 18 (4) 0.015 (0.003) 25 (1) 22 (2) 22 (2) 1 (3) · · · UGC 08311 6 140 (30) 320 (60) 0.26 (0.05) 38 (1) 16.9 (0.1) 16.3 (0.1) 0.6 (0.1) · · · UGC 08311 7 15 (3) 32 (6) 0.03 (0.01) 19 (1) 18.6 (0.1) 18.1 (0.2) 0.6 (0.2) · · · UGC 08311 TOTAL 270 (30) 680 (80) 0.54 (0.07) · · · · · · · · · · · · 0.2 (0.1) UGC 08644 1 5.1 (1) 44 (9) 0.04 (0.01) 17 (1) 21 (0.1) 20. (1) 1 (1) · · · Table 4—Continued Galaxy Regiona Hα Flux ×10−15 Hα Luminosity×1038 SFRb EWc Bd Rd B−Rd Diffusee erg cm−2 s−1 erg s−1 M⊙ yr [mag] [mag] [mag] [%] UGC 08644 2 4.8 (1) 42 (8) 0.03 (0.01) 20. (1) 22 (0.5) 21.1 (0.6) 0.9 (0.8) · · · UGC 08644 TOTAL 8.3 (1) 90 (10) 0.07 (0.01) · · · · · · · · · · · · · · · UGC 08904 1 3.4 (0.7) 47 (9) 0.04 (0.01) 10. (1) 22.4 (0.8) 22 (1) 1 (1) · · · UGC 08904 2 0.7 (0.1) 12 (2) 0.01 (0.009) 23 (1) 23.6 (0.5) 23.0 (0.6) 0.7 (0.8) · · · UGC 08904 3 1.2 (0.3) 20. (4) 0.02 (0.02) 16 (1) 24.3 (0.7) 24 (1) 1 (1) · · · UGC 08904 4 0.8 (0.2) 12 (2) 0.01 (0.009) 10. (1) 23.8 (0.4) 24 (1) 0. (1) · · · UGC 08904 5 0.7 (0.1) 9 (2) 0.01 (0.007) 10. (1) 24.5 (0.9) 24 (1) 1 (2) · · · UGC 08904 6 32 (6) 600 (100) 0.45 (0.09) 25 (1) 17.7 (0.1) 16.3 (0.1) 1.4 (0.1) · · · UGC 08904 TOTAL 33 (6) 700 (100) 0.53 (0.09) · · · · · · · · · · · · 0.6 (0.1) UGC 10894 1 1.2 (0.2) 11 (2) 0.009 (0.002) 39 (3) 24 (2) 23 (1) 1 (2) · · · UGC 10894 2 1.4 (0.3) 13 (3) 0.010 (0.002) 37 (2) 22.1 (0.5) 21.3 (0.6) 0.8 (0.8) · · · UGC 10894 3 13 (3) 110 (20) 0.09 (0.02) 27 (1) 18.9 (0.2) 17.2 (0.1) 1.8 (0.2) · · · UGC 10894 4 1.9 (0.4) 18 (4) 0.014 (0.003) 31 (2) 23 (1) 22 (1) 1 (2) · · · UGC 10894 5 1.8 (0.4) 16 (3) 0.013 (0.003) 35 (2) 22.8 (0.7) 21.9 (0.7) 1 (1) · · · UGC 10894 6 2.8 (0.6) 26 (5) 0.020 (0.004) 36 (2) 22.4 (0.7) 22 (1) 0. (1) · · · UGC 10894 7 3.9 (0.8) 35 (7) 0.03 (0.01) 31 (2) 22.6 (0.8) 21.6 (0.8) 1 (1) · · · UGC 10894 TOTAL 23 (2) 230 (25) 0.18 (0.02) · · · · · · · · · · · · 0.6 (0.3) UGC 11068 0 · · · · · · · · · · · · · · · · · · · · · · · · UGC 11355 1 180 (40) 600 (100) 0.48 (0.1) 17 (1) 19.7 (0.4) 18.2 (0.4) 1.5 (0.6) 0.87 (0.08) UGC 11396 1 25 (5) 90 (20) 0.07 (0.01) 21 (1) 22 (1) 18.1 (0.2) 4 (1) 0.91 (0.01) UGC 11617 1 9 (2) 40 (8) 0.07 (0.01) 12 (1) 21 (1) 21 (0.6) 0. (1) · · · UGC 11617 2 21 (4) 90 (20) 0.04 (0.01) 13 (1) 21 (2) 20. (2) 1 (3) · · · UGC 11617 3 13 (3) 50 (10) 0.04 (0.01) 13 (1) 22 (2) 20. (3) 1 (3) · · · UGC 11617 4 12 (2) 47 (9) 0.09 (0.01) 11 (1) 21 (2) 21 (1) 1 (3) · · · UGC 11617 5 26 (5) 110 (20) 0.03 (0.02) 14 (1) 21.5 (0.6) 21 (4) 1 (4) · · · UGC 11617 6 10 (2) 42 (8) 0.03 (0.01) 12 (1) 21.5 (0.2) 20.8 (0.7) 0.7 (0.7) · · · UGC 11617 TOTAL 69 (6) 390 (30) 0.31 (0.03) · · · · · · · · · · · · 0.8 (0.1) UGC 11840 0 · · · · · · · · · · · · · · · · · · · · · · · · UGC 12021 1 13 (3) 50 (10) 0.04 (0.01) 31 (2) 19.2 (0.4) 17.8 (0.3) 1.4 (0.5) 0.97 (0.01) Note. — Derivation of quantities is described in Section 3. Errors are given in parenthesis. aInternal numbering scheme for the HII regions found by HIIphot. bSFR for the region, defined as SFR = 1.26×1041erg s−1 cEquivalent width was calculated simply as the ratio of the total Hα flux to total Hα-subtracted continuum flux. dTotal B and R magnitudes and colors within the HII regions eDiffuse fraction found for the galaxy, defined defined here as the ratio of the fraction of Hα flux not found within the defined Hα regions to the total Hα flux found for the entire galaxy. †Due to both a (masked) star near the center of this galaxy and a (masked) CCD flaw (bad column) which also runs through the center of the galaxy, a number of H II regions which should have been identified by HIIphot were not, artificially rasing the diffuse fraction on this galaxy, possibly by as much as 20-30%. – 27 – Fig. 1.— Grey scale images of the observed galaxies. Figure available through the published AJ paper or online at http://www.gb.nrao.edu/∼koneil. Fig. 2.— Surface brightness profiles for all galaxies observed. The dash-dotted lines show the inner fit, the dashed lines show the outer fit, and the solid lines show the combined fits. Both the B (blue - bottom) and R (red - top) profiles are shown. Figure available through the published AJ paper or online at http://www.gb.nrao.edu/∼koneil. Fig. 3.— Color profiles for all the galaxies observed. Here the inner fits (when made) are shown by a dashed line and the outer fits are shown by a solid line. Figure available through the published AJ paper or online at http://www.gb.nrao.edu/∼koneil. – 28 – UGC 00189 UGC 10894 UGC 07598 UGC 08904 Fig. 4.— Example images showing the H II regions found by HIIphot for the galaxies UGC 00189, UGC 10894, UGC 07598, and UGC 08904. The H II regions are outlines in white. In the case of UGC 10894 two regions which were masked due to the presence of stars can also be seen, outlined by the square white boxes. – 29 – 18 20 22 24 µB,R(0) [mag arcsec 19 20 21 22 23 24 25 µB,R(0) [mag arcsec Fig. 5.— (Left) Histogram showing the distribution of central surface brightnesses for the observed galaxies. The (red) dashed line shows the R-band data and the (blue) solid line shows the B-band data. (Right) Plot of the observed central surface brightness against scale length of the outer disk. Here, the R-band data is demarcated by (red) open circles while the B-band data uses (blue) filled circles. 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 <µB,R> [mag arcsec 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 <µB,R> [mag arcsec Fig. 6.— Plots comparing the measured central surface brightnesses with the average surface brightness for the galaxies, as defined by Equation 7 (left) and Equation 8 (right) and using the magnitude and diameter values found herein. The R-band data is demarcated by (red) open circles while the B-band data uses (blue) filled circles. – 30 – 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 <µB,R> [mag arcsec 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 <µB,R> [mag arcsec Fig. 7.— Plots comparing the measured central surface brightnesses, corrected for inclina- tion, with the average surface brightness for the galaxies, as defined by Equation 7 (left) and Equation 8 (right) and using the magnitude and diameter values found herein. The R-band data is demarcated by (red) open circles while the B-band data uses (blue) filled circles. Fig. 8.— Images of UGC 11355 with the stretch altered to show the galaxy’s nuclear bar (left - R-band image) and star forming ring (right - Hα image). In both images the ellipse shows the shape and size of the star forming ring. The images are 1.0′ across. – 31 – 0 1 2 3 4 MHI/LR,B global [MO · /LO · ] Fig. 9.— Gas mass to B and R-band luminosity ratios plotted against the global star formation rate for the galaxies. The (blue) filled circles are for the B-band data and the (red) open circles are for the R-band data. – 32 – −14 −16 −18 −20 −22 MB, global S0 − Sa Sab − Sb Sbc−Sc Sm−Im Fig. 10.— Total B magnitude plotted against the average luminosity of the brightest three Hα regions. (If less than three regions were found, the average of all H II regions was used.) The filled (red) symbols are the data from our observations; the filled (blue) symbols are from Helmboldt, et al. (2005); and the open (black) symbols are from Kennicutt & Kent (1983). – 33 – −0.5 0.0 0.5 1.0 1.5 2.0 2.5 (B − R)global Our Data Helmboldt, et.al 2004 Jansen, et.al 2000 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 (B−R)global Our Data Jansen, et.al 2000 Fig. 11.— Global color versus equivalent width (left) and star formation rate (right). To insure any trends (or lack) remain the same, the data from this paper is shown both without inclination correction (black) and with (gray). Note that inclination corrections are described in Section 3. As the global SFR was not available for the Helmboldt, et al. (2005) data, it is not shown on the right. −17 −18 −19 −20 −21 −22 −23 MB,global Our Data Kennicutt & Kent 1983 Helmbolt, et.al 2004 −16 −17 −18 −19 −20 −21 −22 MB, global Our data Kennicutt 1983 Jansen et al. 2000 Fig. 12.— Total B magnitude plotted against the global equivalent width (left) and star formation rate (right). As the global SFR was not available for the Helmboldt, et al. (2005) data, it is not shown on the right. – 34 – 36 37 38 39 40 41 log(LHalpha, eff/area) [erg s −1 kpc−2] Our Data Oey, et.al Fig. 13.— Luminosity surface brightness (Luminosity/area) plotted against the diffuse Hα fraction for our sample and that of Oey, et al. (2006). – 35 – 0.0 0.5 1.0 1.5 2.0 SFRtotal,region [MO · /yr] 0 20 40 60 80 100 120 140 E.Wglobal [Å] Fig. 14.— A comparison of regional and global star formation rate and equivalent width for the studied galaxies. On the left is a plot of the global SFR against the total SFR found for the individual H II regions, with a line demarcating the point where the global and regional SFR are equal. On the right is a plot of the global EW against the average EW for the individual H II regions. – 36 – 19 20 21 22 23 24 25 µB,R(0) [mag arcsec Fig. 15.— Central surface brightness versus global star formation rate for the observed galaxies. The (red) open circles are from the R band data, while the (blue) filled circles are for the B data. – 37 – 0.001 0.010 0.100 1.000 10.000 MHI/LB global [MO · /LO · ] Our Data Kennicutt & Kent 1983 Helmbolt, et.al 2004 0.1 1.0 10.0 MHI/LR global [MO · /LO · ] Our Data Oey, et.al (2006) Helmboldt, et.al (2005) Fig. 16.— Gas mass to luminosity ratios plotted against global equivalent widths (left) and diffuse Hα fractions (right). On the left, the (black) circles are our data, the (blue) triangles are from Kennicutt & Kent (1983) and the (red) diamonds are from Helmboldt, et al. (2004). On the right, the (black) circles are again our data, while the (blue) asterisks are from Oey, et al. (2006). – 38 – −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (B − R)region Fig. 17.— This plot shows the regional colors versus star formation rates for the observed galaxies.

0704.0193

Domain Wall Dynamics near a Quantum Critical Point

Domain Wall Dynamics near a Quantum Critical Point Shengjun Yuan and Hans De Raedt Department of Applied Physics, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands Seiji Miyashita Department of Physics, Graduate School of Science, University of Tokyo, Bunkyo-ku,Tokyo 113-0033, Japan and CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan (Dated: November 4, 2018) We study the real-time domain-wall dynamics near a quantum critical point of the one-dimensional anisotropic ferromagnetic spin 1/2 chain. By numerical simulation, we find the domain wall is dynamically stable in the Heisenberg-Ising model. Near the quantum critical point, the width of the domain wall diverges as (∆− 1) PACS numbers: 75.10.Jm, 75.40.Gb, 75.60.Ch, 75.40.Mg. 75.75.+a I. INTRODUCTION Recent progress in synthesizing materials that contain ferromagnetic chains1,2,3,4 provides new opportunities to study the quantum dynamics of atomic-size domain walls (DW). On the atomic level, a DW is a structure that is stable with respect to (quantum) fluctuations, separating two regions with opposite magnetization. Such a struc- ture was observed in the one-dimensional CoCl2 · 2H2O chain5,6. In an earlier paper7, we studied the propagation of spin waves in ferromagnetic quantum spin chains that sup- port DWs. We demonstrated that DWs are very stable against perturbations, and that the longitudinal compo- nent of the spin wave speeds up when it passes through a DW while the transverse component is almost completely reflected. In this paper, we focus on the dynamic stability of the DW in the Heisenberg-Ising ferromagnetic chain. It is known that the ground state of this model in the subspace of total magnetization zero supports DW structures8,9. However, if we let the system evolve in time from an initial state with a DW structure and this initial state is not an eigenstate, it must contain some excited states. Therefore, the question whether the DW structure will survive in the stationary (long-time) regime is nontrivial. The question how the DW structure dynamically sur- vives in the stationary (long-time) region is an interesting problem. In particular, we focus on the stability of the DW with respect to the dynamical (quantum) fluctua- tions as we approach the quantum critical point (from Heisenberg-Ising like to Heisenberg). We show that the critical quantum dynamics of DWs can be described well in terms of conventional power laws. The behavior of quantum systems at or near a quantum critical point is of contemporary interest10. We also show that the DW profiles rapidly become very stable as we move away from the quantum critical point. II. MODEL The Hamiltonian of the system is given by8,9,11,12,13 H = −J (SxnS n+1 + S n+1 +∆S n+1), (1) where N indicates the total number of spins in the spin chain, and the exchange integrals J and J∆ determine the strength of the interaction between the x, y and z components of spin 1/2 operators Sn = (S n , S Here we only consider the system with the ferromagnetic (J > 0) nearest exchange interaction. It is well known that |∆| = 1 is a quantum critical point of the Hamil- tonian in Eq. (1), that is, the analytical expressions of the ground state energy for 1 < ∆ and −1 < ∆ < 1 are different and singular at the points ∆ = ±112. In Ref.8,9 Gochev constructed a stable state with DW structure in both the classical and quantum treatments of the Hamiltonian (1). In the classical treatment, Gochev replaces the spin operators in Eq. (1) by classical vectors of length s Szn = s cos θn, S n = s sin θn cosϕn, S n = s sin θn sinϕn, and then uses the conditions δE/δθ = 0 and ϕn = const. to find the ground state. In the ground state, the mag- netization per site is given by9 Szn = s tanh(n− n0)σ, Sxn = s cosϕ sech(n− n0)σ, Syn = s sinϕ sech(n− n0)σ, where σ = ln[∆ + ∆2 − 1], (3) ϕ is an arbitrary constant, and n0 is a constant fixing the position of the DW. The corresponding energy is EDW = 2s 2J∆tanhσ. (4) http://arxiv.org/abs/0704.0193v1 In the quantum mechanical treatment, Gochev first con- structs the eigenfunction of a bound state of k magnons9 |ψk〉 = An Bm1m2...mkS S−m2 ...S |0〉 , (5) where Bm1m2...mk = vmii ,mi < mi+1, (6) vi = cosh(i − 1)σ/ cosh(iσ), (7) A−2 = v2ii /(1− v i ), (8) and the corresponding energy is given by9 J∆tanhσ tanh kσ. (9) Then he demonstrated that for the infinite chain, the linear superposition |φn0〉 = A |ψN0+i〉 , where n0 = N0 + α, |α| ≤ 1/2, N0 → ∞, (11) A−2 = , (12) is the quantum analog of the classical domain wall, in which 〈Szn〉 , 〈S n〉 , 〈S n〉 are given in Eq. (2), and the en- ergy coincides with Eq. (4). Gochev’s work confirmed the existence of the DW structures in the one-dimensional ferromagnetic quantum spin 1/2 chain. In the infinite chain, the exact quantum analog of classical DW is represented by |φn0〉. In the fi- nite chain, the DW structure exists as a bound k-magnon state |ψk〉. The main difference between these two states is the distribution of magnetization in the XY plane. In the infinite chain, the change of the magnetization oc- curs in three dimensions, according to Eq. (2), but in the finite chain 〈Sxn〉 = 〈S n〉 = 0 for all spins. Now we consider 〈Szn〉 of the bound state |ψk〉 in the case that the number of flipped spin is half of the total spins, i.e., k = N/2 and N is an even number. Even though the formal expression for |ψk〉 is known, the ex- pression for 〈Szn〉 in this state (for finite and infinite chains) is not known. For finite N , the ground state in the subspace of total magnetizationM = 0 can, in princi- ple, be calculated from Eq. (5). However, this requires a numerical procedure and we loose the attractive features FIG. 1: The magnetization 〈Szn〉 in the ground state of the subspace of total magnetization M = 0, generated by the power method. The parameters are: (a) ∆ = 1.05, (b) ∆ = 1.1, (c) ∆ = 1.2, (d) ∆ = 2. The total number of spins in the spin chain is N = 20. It is clear that there is a DW at the centre of the spin chain. Furthermore there is no structure in the XY plane, that is, 〈Sxn〉 = 〈S n〉 = 0. FIG. 2: Top picture (a): Initial spin configuration at time t/τ = 0; Bottom pictures (b,c,d,e,f,g,h,i): Spin configuration at time t/τ = 100; Bottom left pictures (b,c,d,e): DW struc- tures disappear or are not stable. The parameters are: (b) ∆ = 0 (XY model), (c) ∆ = 0.5 (Heisenberg-XY model), (d) ∆ = 1 (Heisenberg model), (e) ∆ = 1.05 (Heisenberg-Ising model); Bottom right pictures (f,g,h,i): DW structures are dynamically stable in the Heisenberg-Ising model. The pa- rameters are: (f) ∆ = 1.1, (g) ∆ = 1.2, (h) ∆ = 2, (i) ∆ = 20. The total number of spins in the spin chain is N = 20. of the analytical approach. Indeed, it is more efficient to use a numerical method and compute directly the ground state in the subspace of total magnetization M = 0. In Fig. 1, we show some representative results as obtained by the power method14 for a chain of N = 20 spins. In all cases, the domain wall is well-defined. Obviously, be- TABLE I: The energy E = J∆/2 of the initial state |Φ〉 (see Fig. 2(a)) and the ground state E g in the M = 0 subspace, both relative to the ground state energy of the ferromagnet. ∆ E E 1.05 0.53 0.16 0.16 0.16 0.16 1.1 0.55 0.23 0.23 0.23 0.23 2 1.00 0.87 0.87 0.87 0.87 5 2.50 2.45 2.45 2.45 2.45 cause we are considering the system in the ground state, the magnetization profile will not change during the time evolution. To inject a DW in the spin chain, we take the state |Φ〉 with the left half of the spins up and the other half down as the initial state (see Fig. 2(a) for N = 20). The state |Φ〉 corresponds to the state with the largest weight in the bound state |ψk〉 with k = N/2, because |Bm1m2...mk | reaches the maximum if mi = i for all i = 1, 2, .., N/2 (note |vi| < 1). It is clear that |Φ〉 is not an eigenstate of the Hamiltonian in Eq. (1). The energy of |Φ〉, relative to the ferromagnetic ground state, is J∆/2, and its spread (〈Φ|H2|Φ〉−〈Φ|H |Φ〉2)1/2 = J/2. In Table I, we list some representative values of the energy in the initial state (see Fig. 2(a)) and in the ground state of subspace M = 0 (see Fig. 1). A priori, there is no reason why the DW of the ini- tial state |Φ〉 should relax to a DW profile that is dy- namically stable. For ∆ ≃ 1, the difference between en- ergy of the initial state and the ground state energies for N = 16, 18, 20, 22 is relatively large and the relative spread in energy (1/∆) is large also, suggesting that near the quantum critical point, the initial state may contain a significant amount of excited states. Therefore, it is not evident that a DW structure will survive in the long- time regime. In fact, from the numbers in Table I, one cannot predict whether or not the DW will be stable. For instance, for ∆ = 1.05 and N = 16, 18, 20, the DW is not dynamically stable whereas for N = 22 it is stable but the energies (see first line in Table I) do not give a clue as to why this should be the case. On the other hand, by solving the time dependent Schrödinger equa- tion (TDSE), it is easy to see if the DW is dynamically stable or not. III. DYNAMICALLY STABLE DOMAIN WALLS We solve the TDSE of the whole system with the Hamiltonian in Eq. (1) and study the time-evolution of the magnetization at each lattice site. The numeri- cal solution of the TDSE is performed by the Chebyshev polynomial algorithm, which is known to yield extremely accurate independent of the time step used15,16,17,18. We adopt open boundary conditions, not periodic bound- ary conditions, because the periodic boundary condition would introduce two DWs in the initial state. In this pa- per, we display the results at time intervals of τ = π/5J , and use units such that ~ = 1 and J = 1. The initial state of the system is shown in Fig. 2(a). The spins in the left part (n = 1 to 10) of the spin chain are all ”spin-up” and the rest (n = 11 to 20) are all ”spin-down”. Here ”spin-up” or ”spin-down” correspond to the eigenstates of the single spin 1/2 operator Szn. Whether the DW at the centre of the spin chain is stable or unstable depends on the value of ∆. In Fig. 2(b,c,d,e,f,g,h, and i), we show the states of the system as obtained by letting the system evolve over a fairly long time (t = 500J/π). It is clear that the DW totally disap- pears for 0 ≤ ∆ ≤ 1, that is, in the XY, Heisenberg-XY and Heisenberg spin 1/2 chain, the DW structures are not stable. For the Heisenberg-Ising model (∆ > 1), the DW remains stable when t ≥ 500J/π (see Ref.7), and its structure is more sharp and clear if ∆ is larger, so we will concentrate on the cases ∆ > 1. One may note that the values of ∆ in Fig. 2(e,f,g,h) are the same as in Fig. 1(a,b,c,d), but that the distributions of the magnetiza- tion are similar but not the same. This is because the energy is conserved during the time evolution and the system, which starts from the initial state shown in Fig. 2(a), will never relax to the ground state of the subspace with the total magnetization M = 0. In order to get a quantitative expression of the width of DW, we first introduce the quantity Szn (t1, t2; ∆) (n = 1, 2, ..., N) as the time average of the expectation value 〈Szn (t)〉 of nth spin: Szn (t1, t2; ∆) ≡ 〈Szn (t)〉 dt t2 − t1 . (13) We take the average in Eq. (13) over a long period dur- ing which the DW is dynamically stable. In Fig. 3, we show some results of Szn (t1, t2; ∆) for the Heisenberg- Ising model, where we take t1 = 101τ , t2 = 200τ and various ∆. We find that each curve in Fig. 3 is symmet- ric about the line n = (N + 1) /2, and can be fitted well by the function Szn (t1, t2; ∆) = a∆ tanh n− (N + 1)/2 . (14) The values of ∆ we used and the corresponding values of a∆, b∆ are shown in Table II. As we mentioned earlier, Gochev9 constructed an eigenstate of the one- dimensional anisotropic ferromagnetic spin 1/2 chain in which the mean values Szn, S n and S n coincide with the stable DW structure in the classical spin chain, that is 〈Szn〉 = tanh(n− n0)σ, (15) where n0 is the position of the DW (in our notation, this is (N + 1) /2). The fitted form of Szn (t1, t2; ∆) in Eq. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1.05 1.06 FIG. 3: (Color online) Szn (t1, t2; ∆) as a function of n for different ∆. Here t1 = 101τ , t2 = 200τ . We show the data for ∆ = 1.05, 1.06, 1.1, 1.2, 1.3, 1.5, 2, 5 and 20 only. The total number of spins in the spin chain is N = 20. (14) is similar to Eq. (15). From Table II, it is clear that as ∆ increases, |a∆| converges to 1/2, in agreement with Eq. (15). From the comparison of b∆ and 1/σ in Fig. 4, it is clear that the dependence on ∆ is qualitatively similar but not the same. This is due to the fact that Gochev’s solution is for a DW in the ground state whereas we obtain the DW by relaxation of the state shown in Fig. 2(a). We want to emphasize that the meaning of Szn (t1, t2; ∆) in Eq. (14) is different from 〈S n〉 in Eq. (15). The former describes the mean value of 〈Szn (t)〉 in a state with dynamical fluctuations, while the latter describes the distribution of 〈Szn〉 in an exact eigenstate without dynamical fluctuations. Next we introduce a definition of the DW width. From Eq. (14), we can find n1 and n2 which satisfy Szn1 (t1, t2; ∆) = 1/4, Szn2 (t1, t2; ∆) = −1/4, (16) that is, when Szn (t1, t2; ∆) equals half of its maximum value (1/2). Here n1 and n2 are not necessarily integer numbers. Now we can define the DW width W as the distance between n1 and n2: W = |n1 − n2| . (17) Clearly, the width of the DW becomes ill-defined if it approaches the size of the chain. On the other hand, the computational resources (mainly memory), required to solve the TDSE, grow exponentially with the number of spins in the chain. These two factors severely limit the minimum difference between ∆ and the quantum critical point (∆ = 1) that yields meaningful results for the width of the DW. Indeed, for fixed N , ∆ has to be larger than TABLE II: The values of ∆ we used in our simulations and the corresponding a∆, b∆ fitted by Eq. (14) for a spin chain of N = 20 spins. ∆ a∆ b∆ ∆ a∆ b∆ 1.05 −0.263 3.659 1.8 −0.493 0.524 1.06 −0.330 3.171 1.9 −0.494 0.488 1.07 −0.377 2.850 2 −0.495 0.460 1.08 −0.406 2.673 2.1 −0.495 0.436 1.09 −0.424 2.534 2.2 −0.496 0.416 1.1 −0.435 2.396 2.5 −0.497 0.370 1.15 −0.462 1.996 3 −0.498 0.322 1.2 −0.471 1.626 4 −0.499 0.270 1.25 −0.476 1.330 5 −0.499 0.240 1.3 −0.479 1.142 6 −0.500 0.220 1.35 −0.481 0.959 7 −0.500 0.206 1.4 −0.483 0.869 8 −0.500 0.195 1.45 −0.485 0.770 9 −0.500 0.187 1.5 −0.487 0.719 10 −0.500 0.179 1.6 −0.489 0.629 15 −0.500 0.156 1.7 −0.491 0.568 20 −0.500 0.141 1 3 5 7 9 11 13 15 17 19 21 FIG. 4: Comparison of b∆ and 1/σ as a function of ∆. The total number of spins in the spin chain is N = 20. the ”effective” critical value for the finite system in order for the DW width to be smaller than the system size. Although the system sizes that are amenable to numeri- cal simulation are rather small for present-day ”classical statistical mechanics” standards, it is nevertheless possi- ble to extract from these simulations useful information about the quantum critical behavior of the dynamically stable DW. In Fig. 5, we plot W as a function of ∆ (1.06 ≤ ∆ ≤ 20). By trial and error, we find that all the data can be 1 3 5 7 9 11 13 15 17 19 21 FIG. 5: The DW width as a function of ∆ in a spin chain of N = 20 spins. The black dots are the sim- ulation data and the solid line is given by W (∆) = AN/ ln ∆ − ǫN + (∆− ǫN ) + BN with ǫN = 0.046 ± 0.001, AN = 2.16 ± 0.06 and BN = −0.485 ± 0.068. TABLE III: The values of ǫN , AN and BN in Eq. (18) for a spin chain of N = 16, 18, 20, 22 and 24 spins. For the fits, we used all the data for ∆ ≤ 5. N ǫN AN BN 16 0.065 ± 0.001 2.08 ± 0.10 −0.493 ± 0.142 18 0.052 ± 0.002 2.07 ± 0.11 −0.450 ± 0.152 20 0.045 ± 0.002 2.22 ± 0.09 −0.556 ± 0.133 22 0.040 ± 0.001 2.36 ± 0.08 −0.689 ± 0.140 24 0.033 ± 0.001 2.34 ± 0.06 −0.681 ± 0.127 fitted very well by the function W (∆) = ∆− ǫN + (∆− ǫN) } +BN , where ǫN , AN and BN are fitting parameters. As shown in Fig. 6, all the data for N = 16, 18, 22, 24 and ∆ ≤ 5 fit very well to Eq. (18). The results of these fits are collected in Table III. To analyze the finite-size dependence in more detail, we adopt the standard finite-size scaling hypothesis19. We assume that in the infinite system, the DW width plays the role of the correlation length, that is, we assume that W (∆) ∼W0(∆− 1) −ν , (19) where ν is a critical exponent. Finite-size scaling predicts that the effective critical value ∆∗N = 1 + ǫN where ǫN is proportional to N−1/ν . Taking ν = 1/2, Fig. 7 shows that ∆∗N converges to one as N increases. As a check on the fitting procedure, we apply it to the data obtained by solving for the ground state in the M = 0 subspace. In view of Eq. (13) and (14), we may expect that Eq. (18) fits the data very well and, as shown in Fig. 8, this is indeed the case. If we fit the data to W (∆) =W0 (∆−∆ . (20) without assuming a priori value C, we find that C de- pends on the range of ∆ that was used in the fit, as shown in Fig. 9. Remarkably, we find that C ≈ 0.57 if we fit the data for a large range of ∆’s and that C ap- proaches 1/2 if we restrict the value of ∆ to the vicinity of the critical point. IV. THE STABILITY OF DOMAIN WALLS To describe the stability of the DW structure, we in- troduce δn (∆) (n = 1, 2, ..., N): δn (∆) = [Szn (t1, t2; ∆)] − Szn (t1, t2; ∆) , (21) where [Szn (t1, t2; ∆)] 〈Szn (t)〉 t2 − t1 . (22) In order to show the physical meaning of δn, we write 〈Szn (t)〉 as 〈Szn (t)〉 ≡ Cn +Ωn (t) , (23) where Cn is a constant and Ωn (t) is a time-dependent term. Then Eq. (21) becomes δn (∆) = Ω2n (t) dt t2 − t1 Ωn (t) dt t2 − t1 . (24) It is clear that if 〈Szn (t)〉 is a constant in the time interval [t1, t2], then δn (∆) = 0. In general, since the initial state is not an eigenstate of the Hamiltonian Eq. (1), the magnetization of each spin will fluctuate and Ωn (t) 6= 0. If, after long time, the system relaxes to a stationary state that contains a DW, the magnetization of each spin will fluctuate around its stationary value Cn. The fluctuations are given by Ωn (t). If |Ωn (t)| is large, the difference between the actual magnetization profile at time t and the stationary profile Cn may be large. From Eq. (24), it is clear that δn (∆) is a measure of the deviation of 〈Szn (t)〉 from its stationary value Cn, averaged over the time interval [t1, t2]. Thus, δn (∆) gives direct information about the dynamics stability of the Figure 10 shows the distribution δn (∆) for different values of ∆. We only show some typical results, as in Fig. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 16 N=22 1 2 3 4 5 FIG. 6: The DW width as a function of ∆ in a spin chain of N = 16, 18, 22, and 24 spins. The black dots are the simulation data and the solid line in each panel is given by Eq. (18). 3. As expected, the distribution of δn (∆) is symmetric about the centre of the spin chain (n = 10.5). We first consider how δn (∆) changes with ∆ for fixed n. From Fig.10, we conclude: 1) For the spins which are not located at the DW cen- tre, i.e., n 6= 10, 11, δn (∆) decreases if ∆ becomes larger. This means that the quantum fluctuations of these spins become smaller if we increase the value of ∆. This is reasonable because with increasing ∆, the initial state approaches an eigenstate of the Hamiltonian for which δn (∆) = 0 (Ising limit). 2) For the spins at the DW centre, i.e., n = 10, 11, when ∆ becomes larger and larger, δn (∆) first increases and then decreases. Qualitatively, this can be understood in the following way. When ∆ is close to 1, the magneti- zation at the DW centre disappears very fast and remains zero. However, if ∆ >> 1, the magnetization at the DW centre will retain its initial direction, hence the behavior of the spin at the DW centre will qualitatively change as ∆ moves away from the critical point ∆ = 1. In Fig. 11, we plot δ10 (∆) (= δ11 (∆)) as a function of ∆. It is clear that δ10 (∆) first increases as ∆ increases, reaches its maximum at ∆ = 1.3, and then decreases as ∆ be- comes larger. Now we consider the n-dependence of δn (∆) for fixed ∆. Since δn (∆) is a symmetric function of n, we may consider only one side of the whole chain, e.g., the spins with n = 1, 2, ..., N/2. From Fig.10, according to the value of ∆, there are three different regions: 1) 1.05 ≤ ∆ ≤ 1.3: starting from the boundary (n = 1), δn (∆) first decreases, then increases, and finally decreases again as n approaches the DW centre (n = 10). As we discussed already, the fluctuation of the magneti- zation at the DW centre is small when ∆ is close to 1. The spin at the boundary only interacts with one nearest spin, so it has more freedom to fluctuate. For the others, because of the influence of the DW structure (or bound- ary), the fluctuations of the spins which are near the DW (or near the boundary) are larger compared to those of a spin located in the middle of a polarized region. Thus δn (∆) is larger if the spin is located near the DW or near a boundary. 0 0.001 0.002 0.003 0.004 0.005 FIG. 7: Fit of ∆∗N to ∆ ∗ + λ ·N−2 with ∆∗ = 1.009± 0.002, and λ = 14.253 ± 0.660. 1 2 3 4 5 FIG. 8: The DW width as a function of ∆ (1.06 ≤ ∆ ≤ 20) in the ground state of subspace M = 0 in a spin chain of N = 20 spins. The black dots are the simulation data and the solid line is given by Eq. (18), with ǫN = 0.010 ± 0.001, AN = 1.87± 0.04 and BN = −0.550± 0.079. 2) 1.3 ≤ ∆ ≤ 5: δn (∆) reaches its maximum at the DW centre. The reason for this is that in this regime the magnetizations of all spins retain their initial direction, therefore the spins that are far from the centre fluctuate little. 3) 5 < ∆: in this regime (Ising limit), the initial state is very close to the eigenstate, and the fluctuations are small, even for the spins at the DW. V. SUMMARY In the presence of Ising-like anisotropy, DWs in a fer- romagnetic spin 1/2 chain are dynamically stable over 1 3 5 7 9 11 13 15 17 19 21 FIG. 9: The exponent C as a function of ∆max in a spin chain of N = 20 spins. The exponent C obtained by fitting the DW width to Eq. (20), with ∆∗N=20 as obtained from the fit shown in Fig. 7, for ∆ in the range [1.06,∆max]. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1.05 1.06 FIG. 10: (Color online) δn (∆) as a function of n for different ∆. Here t1 = 101τ , t2 = 200τ . We only show the data for ∆ = 1.05, 1.06, 1.1, 1.2, 1.3, 1.5, 2, 5 and 20. The total number of spins in the spin chain is N = 20. extended periods of time. The profiles of the magnetiza- tion of the DW are different from the profile in the ground state in the subspace of total magnetization M = 0. As the system becomes more isotropic, approaching the quantum critical point, the width of the DW increases as a power law, with an exponent equal to 1/2. 1 3 5 7 9 11 13 15 17 19 21 FIG. 11: δ10 (∆) as a function of ∆. Here t1 = 101τ , t2 = 200τ . The total number of spins in the spin chain is N = 20. 1 T. Kajiwara, M. Nakano, Y. Kaneko, S. Takaishi, T. Ito, M. Yamashita, A. Igashira-Kamiyama, H. Nojiri, Y. Ono and N. Kojima, J. Am. Chem. Soc. 127 10150 (2005). 2 M. Mito, H. Deguchi, T. Tajiri, S. Takagi, M. Yamashita and H. Miyasaka, Phys. Rev. B 72, 144421 (2005). 3 H. Kageyama, K. Yoshimura, K. Kosuge, M. Azuma, M. Takano, H. Mitamura and T. Goto, J. Phys. Soc. Jpn. 66, 3996 (1997). 4 A. Maignana, C. Michel, A.C. Masset, C. Martin and B. Raveau, Eur. Phys. J. B 15, 657 (2000). 5 J. Torrance and M. Tinkham, Phys. Rev. 187, 587 (1969). 6 D. Nicoli and M. Tinkham, Phys. Rev. B 9, 3126 (1974). 7 S. Yuan, H. De Raedt, and S. Miyashita, J. Phys. Soc. Jpn., 75, 084703 (2006). 8 I.G. Gochev, JETP Lett. 26, 127 (1977). 9 I.G. Gochev, Sov. Phys. JETP 58, 115 (1983). 10 S. Sachdev, Quantum Phase Transitions, (Cambridge Uni- versity Press, Cambridge, 1999). 11 H.J. Mikeska, S. Miyashita and G.H. Ristow, J. Phys.: Condens. Matter 3, 2985 (1991). 12 J. des Cloizeaux and M. Gaudin, J. Math. Phys. 7, 1384 (1966). 13 D.C. Mattis, The Theory of Magnetism I, Solid State Sci- ence Series 17 (Springer, Berlin 1981). 14 J.H. Wilkinson, The Algebraic Eigenvalue Problem, (Ox- ford University Press, Oxford, 1999). 15 H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81, 3967 (1984). 16 C. Leforestier, R.H. Bisseling, C. Cerjan, M.D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero and R. Kosloff, J. Comp. Phys. 94, 59 (1991). 17 T. Iitaka, S. Nomura, H. Hirayama, X. Zhao, Y. Aoyagi and T. Sugano, Phys. Rev. E56, 1222 (1997). 18 V.V. Dobrovitski and H.A. De Raedt, Phys. Rev. E67 , 056702 (2003). 19 D.P. Landau and K. Binder, AGuide to Monte Carlo Simu- lations in Statistical Physics, (Cambridge University Press, Cambridge, 2000).

0704.0194

Quantum mechanical approach to decoherence and relaxation generated by fluctuating environment

Quantum mechanical approach to decoherence and relaxation generated by fluctuating environment S.A. Gurvitz∗ Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel and Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, NM 87545, USA D. Mozyrsky Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: November 4, 2018) We consider an electrostatic qubit, interacting with fluctuating charge of a single electron tran- sistor (SET) in the framework of an exactly solvable model. The SET plays role of an environment affecting the qubits’ parameters in a controllable way. We derive the rate equations describing the dynamics of the entire system for an arbitrary qubit-SET coupling. Solving these equations we obtain decoherence and relaxation rates of the qubit, as well as the spectral density of qubit param- eters’ fluctuations. We found that in a weak coupling regime decoherence and relaxation rates are directly related to the spectral density taken at either zero or Rabi frequency, depending on which qubit parameter is fluctuating. In the latter case our result coincides with that of the spin-boson model in the weak coupling limit, despite different origin of the fluctuations. We show that this relation holds also in the presence of weak back-action of the qubit on the environment. In case of strong back-action such a simple relationship no longer holds, even if qubit-SET coupling is small. It does not hold also in the strong coupling regime, even in the absence of the back-action. In addition, we found that our model predicts localization of the qubit in the strong-coupling regime, resembling that in the spin-boson model. PACS numbers: 03.65.Yz, 05.60.Gg, 73.23.-b, 73.23.Hk I. INTRODUCTION The influence of environment on a single quantum sys- tem is the issue of crucial importance in quantum in- formation science. It is mainly associated with decoher- ence, or dephasing, which transforms any pure state of a quantum system into a statistical mixture. Despite a large body of theoretical work devoted to decoherence, its mechanism has not been clarified enough. For instance, how decoherence is related to environmental noise, in particular in the presence of back-action of the system on the environment (quantum measurements). More- over, decoherence is often intermixed with relaxation. Al- though each of them represents an irreversible process, decoherence and relaxation affect quantum systems in quite different ways. In order to establish a relation between the fluctua- tion spectrum and decoherence and relaxation rates one needs a model that describes the effects of decoherence and relaxation in a consistent quantum mechanical way. An obvious candidate is the spin-boson model1,2 which represents the environment as a bath of harmonic oscil- lators at equilibrium, where the fluctuations obey Gaus- sian statistics3. Despite its apparent simplicity, the spin- boson model cannot be solved exactly2. Also, it is hard to manipulate the fluctuation spectrum in the framework of this model. In addition, mesoscopic structures may couple only to a few isolated fluctuators, like spins, lo- cal currents, background charge fluctuations, etc. This would require models of the environment, different from Electrodes FIG. 1: Electrostatic qubit, realized by an electron trapped in a coupled-dot system (a), and its schematic representation by a double-well (b). Ω0 denotes the coupling between the two dots. the spin-boson model (see for instace4,5,6,7,8,9,10,11,12). In general, the environment can be out of equilibrium, like a steady-state fluctuating current, interacting with the qubit13,14,15,16. This for instance, takes place in the con- tinuous measurement (monitoring) of quantum systems17 and in the “control dephasing” experiments18,19,20. All these types on non-Gaussian and non-equilibrium envi- ronments attracted recently a great deal of attention21. In this paper we consider an electrostatic qubit, which can be viewed as a generic example of two-state systems. It is realized by an electron trapped in coupled quantum dots22,23,24, Fig. 1. Here E1 and E2 denote energies of the electron states in each of the dots and Ω0 is a cou- pling between these states. It is reasonable to assume that the decoherence of a qubit is associated with fluctu- http://arxiv.org/abs/0704.0194v3 Electrostatic Qubit Single Electron Transistor (a) (b) FIG. 2: Qubit near Single Electron Transistor. Here El,r and E0 denote the energy levels in the left (right) reservoirs and in the quantum dot, respectively, and µL,R are the correspond- ing chemical potentials. The electric current I generates fluc- tuations of the electrostatic opening between two dots (a), or it fluctuates the energy level of the nearest dot (b). ations of the qubit parameters, E1,2 and Ω0, generated by the environment. Indeed, a stochastic averaging of the Schrödinger equation over these fluctuations param- eters results in the qubit’s decoherence, which transfers any qubit state into a statistical mixture25,26. In general, one can expect that the fluctuating environment should result in the qubit’s relaxation, as well, as for instance in the phenomenological Redfield’s description of relaxation in the magnetic resonance27. As a quantum mechanical model of the environment we consider a Single Electron Transistor (SET) capac- itively coupled to the qubit, e.g., Fig. 2. Such setup has been contemplated in numerous solid state quantum computing architectures where SET plays role of a read- out device16,17,28,29 and contains most of the generic fea- tures of a fluctuating non-equilibrium environment. The discreteness of the electron charge creates fluctuations in the electrostatic field near the SET. If the electro- static qubit is placed near the SET, this fluctuating field should affect the qubit behavior as shown in Fig. 2. It can produce fluctuations of the tunneling coupling be- tween the dots (off-diagonal coupling) by narrowing the electrostatic opening connecting these dots, as in Fig. 2a, or make the energy levels of the dots fluctuate, as shown schematically in Fig. 2b. Note that while in some regimes the SET operates as a measuring device16,17, in other regimes it corresponds purely to a source of noise. In- deed, if the energy level E0, Fig. 2, is deeply inside the voltage bias – the case we consider in the begin- ning, the SET current is not modulated by the qubit electron. In this case the SET represents only the fluc- tuating environment affecting the qubit behavior (“pure environment”30). A similar model of the fluctuating environment has been studied mostly for small bias (linear response) or for the environment in an equilibrium. Here, however, we consider strongly non-equilibrium case where the bias voltage applied on the SET (V = µL − µR) is much larger than the levels widths and the coupling between the SET and the qubit. In this limit our model can be solved exactly for both weak and strong coupling (but is still smaller than the bias voltage). This constitutes an essential advantage with regard to perturbative treat- ments of similar models. For instance, the results of our model can be compared in different regimes with phe- nomenological descriptions used in the literature. Such a comparison would allow us to determine the regions where these phenomenological models are valid. Since our model is very simple in treatment, the deco- herence and relaxation rates can be extracted from the exact solution analytically, as well as the time-correlator of the electric charge inside the SET. This would make it possible to establish a relation between the frequency- dependent fluctuation spectrum of the environment and the decoherence and the relaxation rates of the qubit, and to determine how far this relation can be extended. We expect that such a relation should not depend on a source of fluctuations. This point can be verified by a comparison with a similar results obtained for equi- librium environment in the framework of the spin-boson model1,2. It is also important to understand how the decoher- ence and relaxation rates depend on the frequency of the environmental fluctuations. This problem has been investigated in many phenomenological approaches for “classical” environments at equilibrium. Yet, there still exists an ambiguity in the literature related to this point for non-equilibrium environment. For instance, it was found by Levinson that the decoherence rate, generated by fluctuations of the energy level in a single quantum dot is proportional to the spectral density of fluctuations at zero frequency31. The same result, but for a double-dot system has been obtained by Rabenstein et al.32. On the other hand, it follows from the Redfied’s approach that the corresponding decoherence rate is proportional to the spectral density at the frequency of the qubit’s oscillations (the Rabi frequency)27. Since our model is the exactly solvable one, we can resolve this ambiguity and establish the appropriate physical conditions that can result in different relations of decoherence rate to the environmental fluctuations. The most important results of our study are related to the situation when back-action of the qubit on the envi- ronment takes place. This problem did not receive such a considerable amount of attention in the literature as, for example, the case of “inert” environment. This is in spite of a fact that the back-action always takes place in the presence of measurement. There are many questions related to the effects of a back-action. For instance, what would be a relation between decoherence (relaxation) of the qubit and the noise spectrum of the environment? Or, how decoherence is affected by a strong response of environment? We believe that our model appears to be more suitable for studying these and other problems re- lated to the back-action than most of the other existing approaches. The plan of this paper is as follows: Sect. II presents a phenomenological description of decoherence and relax- ation in the framework of Bloch equations, applied to the electrostatic qubit. Sect. III contains description of the model and the quantum rate-equation formalism, used for its solution. Detailed quantum-mechanical derivation of these equations for a specific example is presented in Appendix A. Sect. IV deals with a configuration where the SET can generate only decoherence of the qubit. We consider separately the situations when SET produces fluctuations of the tunneling coupling (Rabi frequency) or of the energy levels. The results are compared with the SET fluctuation spectrum, evaluated in Appendix B. Sect. V deals with a configuration where the SET gener- ates both decoherence and relaxation of the qubit. Sect. VI is summary. II. DECOHERENCE AND RELAXATION OF A QUBIT In this section we describe in a general phenomeno- logical framework the effect of decoherence and relax- ation on the qubit behavior. Although the results are known, there still exists some confusion in the literature in this issue. We therefore need to define precisely these quantities and demonstrate how the corresponding deco- herenece and relaxation rates can be extracted from the qubit density matrix. Let us consider an electrostatic qubit, realized by an electron trapped in coupled quantum dots, Fig. 1. This system is described by the following tunneling Hamilto- Hqb = E1a 1a1 + E2a 2a2 − Ω0(a 2a1 + a 1a2) (1) where a 1,2, a1,2 are the creation and annihilation opera- tors of the electron in the first or in the second dot. For simplicity we consider electrons as spinless fermions. In addition, we assume that a 1a1 + a 2a2 = 1, so that only one electron is present in the double-dot. The electron wave function can be written as |Ψ(t)〉 = b(1)(t)a 1 + b (2)(t)a 〉 (2) where b(1,2)(t) are the probability amplitudes for find- ing the electron in the first or second well, obtained from the Schrödinger equation i∂t|Ψ(t)〉 = Hqb|Ψ(t)〉 (we adopt the units where ~ = 1 and the electron charge e = 1). The corresponding density matrix, σjj′ (t) = b(j)(t)b(j ′)∗(t), with j, j′ = {1, 2}, is obtained from the equation i∂t σ = [H,σ]. This can be written explicitly as σ̇11 = iΩ0(σ21 − σ12) (3a) σ̇12 = −iǫσ12 + iΩ0(1− 2σ11) , (3b) where σ22(t) = 1 − σ11(t), σ21(t) = σ∗12(t) and ǫ = E1 − E2. Solving these equations one easily finds that the electron oscillates between the two dots (Rabi oscil- lations) with frequency ωR = 4Ω20 + ǫ 2. For instance, for the initial conditions σ11(0) = 1 and σ12(0) = 1, the probability of finding the electron in the second dot is σ22(t) = 2(Ω0/ωR) 2(1− cosωRt). This result shows that for ǫ≫ Ω0 the amplitude of the Rabi oscillations is small, so the electron remains localized in its initial state. The situation is different when the qubit interacts with the environment. In this case the (reduced) density ma- trix of the qubit σ(t) is obtained by tracing out the en- vironment variables from the total density matrix. The question is how to modify Eqs. (3), written for an isolated qubit, in order to obtain the reduced density matrix of the qubit, σ(t). In general one expects that the environ- ment could affect the qubit in two different ways. First, it can destroy the off-diagonal elements of the qubit density matrix. This process is usually referred to as decoherence (or dephasing). It can be accounted for phenomenolog- ically by introducing an additional (damping) term in Eq. (3b), σ̇12 = −iǫσ12 + iΩ0(1− 2σ11)− σ12 (4) where Γd is the decoherence rate. As a result the qubit density-matrix σ(t) becomes a statistical mixture in the stationary limit, t→∞−→ 1/2 0 0 1/2 . (5) This happens for any initial conditions and even for large level displacement, ǫ ≫ Ω0,Γd (provided that Ω0 6= 0). Note that the statistical mixture (5) is proportional to the unity matrix and therefore it remains the same in any basis. Secondly, the environment can put the qubit in its ground state, for instance via photon or phonon emis- sion. This process is usually referred to as relaxation. For a symmetric qubit we would have t→∞−→ 1/2 1/2 1/2 1/2 . (6) In contrast with decoherence, Eq. (5), the relaxation pro- cess puts the qubit into a pure state. That implies that the corresponding density matrix can be always written as δ1iδ1j in a certain basis (the basis of the qubit eigen- states). This is in fact the essential difference between decoherence and relaxation. With respect to elimination of the off-diagonal density matrix elements, note that re- laxation would eliminate these terms only in the qubit’s eigenstates basis. In contrast, decoherence eliminates the off-diagonal density matrix element in any basis (Eq. (5)). In fact, if the environment has some energy, it can put the qubit into an exited state. However, if the qubit is finally in a pure state, such excitation process generated by the environment affects the qubit in the same way as relaxation: it eliminates the off-diagonal density matrix elements only in a certain qubit’s basis. Therefore exci- tation of the qubit can be described phenomenologically on the same footing as relaxation. It is often claimed that decoherence is associated with an absence of energy transfer between the system and the environment, in contrast with relaxation (excitation). This distinction is not generally valid. For instance, if the initial qubit state corresponds to the electron in the state |E2〉, Fig. 1, the final state after decoherence corresponds to an equal distribution between the two dots, 〈E〉 = (E1 +E2)/2. In the case of E1 ≫ E2, this process would require a large energy transfer between the qubit and the environment. Therefore decoherence can be consistently defines as a process leading to a statistical mixture, where all states of the system have equal probabilities (as in Eq. (5)). The relaxation (excitation) process can be described most simply by diagonalizing the qubit Hamiltonian, Eqs. (1), to obtain Hqb = E+a +a+ + E−a −a−, where the operators a± are obtained by the corresponding ro- tation of the operators a1,2 30. Here E+ and E− are the ground (symmetric) and excited (antisymmetric) state energies. Then the relaxation process can be described phenomenologically in the new qubit basis |±〉 = a†±|0¯〉 σ̇−−(t) = −Γrσ−−(t) (7a) σ̇+−(t) = i(E− − E+)σ+−(t)− σ+−(t) , (7b) where σ++(t) = 1 − σ−−(t), σ−+(t) = σ∗+−(t) and Γr is the relaxation rate. In order to add decoherence, we return to the orig- inal qubit basis |1, 2〉 = a†1,2|0¯〉 and add the damping term to the equation for the off-diagonal matrix elements, Eq. (4). We arrive at the quantum rate equation describ- ing the qubit’s behavior in the presence of both decoher- ence and relaxation30,33, σ̇11 = iΩ0(σ21 − σ12)− Γr (σ12 + σ21)− (2σ11 − 1) + Γr σ̇12 = −iǫσ12 + iΩ0 + Γr (1− 2σ11) + Γr σ12 − κ2(σ12 + σ21) σ12 , (8b) where ǫ̃ = (ǫ2 + 4Ω20) 1/2 and κ = Ω0/ǫ̃. In fact, these equations can be derived in the framework of a particu- lar model, representing an electrostatic qubit interacting with the point-contact detector and the environment, de- scribed by the Lee model Hamiltonian33. Equations (8) can be rewritten in a simpler form by mapping the qubit density matrix σ = {σ11, σ12, σ21} to a “polarization” vector S(t) via σ(t) = [1 + τ · S(t)]/2, where τx,y,z are the Pauli matrices. For instance, one obtains for the symmetric case, ǫ = 0, Ṡz = − Sz − 2Ω0 Sy (9a) Ṡy = 2Ω0 Sz − Γd + Γr Sy (9b) Ṡx = − Γd + 2Γr (Sx − S̄x) (9c) where S̄x = Sx(t→∞) = 2Γr/(Γd+2Γr). One finds that Eqs. (9) have a form of the Bloch equations for spin- precession in the magnetic field27, where the effect of environment is accounted for by two relaxation times for the different spin components: the longitudinal T1 and the transverse T2, related to Γd and 2Γr as T−11 = Γd + 2Γr , and T−12 = Γd + Γr , (10) The corresponding damping rates, the so-called “depolar- ization” (Γ1 = 1/T1) and the “dephasing” (Γ2 = 1/T2) are used for phenomenological description of two-level systems34. However, neither Γ1 nor Γ2 taken alone would drive the qubit density matrix into a statistical mixture Eq. (5) or into a pure state Eq. (6). In contrast, our definition of decoherence and relax- ation (excitation) is associated with two opposite effects of the environment on the qubit: the first drives it into a statistical mixture, whereas the second drives it into a pure state. We expect therefore that such a natural distinction between decoherence and relaxation would be more useful for finding a relation between these quantities and the environmental behavior than other alternative definitions of these quantities existing in the literature. In general, the two rates, Γd,r, introduced in phe- nomenological equations (8), (9), are consistent with our definitions of decoherence and relaxation. The only ex- ception is the case of Γr = 0 and Ω0 = 0, where are no transitions between the qubit’s states even in the pres- ence of the environment (“static” qubit). One easily finds from Eqs. (3a), (4) that σ12(t) → 0 for t → ∞, whereas the diagonal density-matrix elements of the qubit remain unchanged (so-called “pure dephasing”5,34): t→∞−→ σ11(0) 0 0 σ22(0) . (11) Thus, if the initial probabilities of finding the qubit in each of its states are not equal, σ11(0) 6= σ22(0), then the final qubit state is neither a mixture nor a pure state, but a combination of the both. It implies that Γd in Eqs. (8) would also generate relaxation (excitation) of the qubit. Note that in this case the off-diagonal density-matrix el- ements, absent in Eq.(11), would reappear in a different basis. This implies that the “pure dephasing”5,34 occurs only in a particular basis. Let us evaluate the probability of finding the electron in the first dot, σ11(t). Solving Eqs. (9) for the initial conditions σ11(0) = 1, σ12(0) = 0, we find σ11(t) = e−Γrt/2 −e−t + C2e where e± = (Γd ± Ω̃), Ω̃ = Γ2d − 64Ω20 and C1,2 = 1±(Γd/Ω̃). Solving the same equations in the limit of t→ ∞, we find that the steady-state qubit density matrix is t→∞−→ 1/2 Γr/(Γd + 2Γr) Γr/(Γd + 2Γr) 1/2 . (13) Thus the off-diagonal elements of the density matrix can provide us with a ratio of relaxation to decoherence rates33. III. DESCRIPTION OF THE MODEL Consider the setup shown in Fig. 2. The entire system can be described by the following tunneling Hamiltonian, represented by a sum of the qubit and SET Hamiltonians and the interaction term, H = Hqb +HSET +Hint. Here Hqb is given by Eq. (1) and describes the qubit. The sec- ond term, HSET, describes the single-electron transistor. It can be written as HSET = l cl + rcr + E0c l c0 +Ωrc rc0 +H.c.) , (14) where c l,r and cl,r are the creation and annihilation elec- tron operators in the state El,r of the right or left reser- voir; c 0 and c0 are those for the level E0 inside the quan- tum dot; and Ωl,r are the couplings between the level E0 and the level El,r in the left (right) reservoir. In order to avoid too lengthy formulaes, our summation indices l, r indicate simultaneously the left and the right leads of the SET, where the corresponding summation is carried out. As follows from the Hamiltonian (14), the quantum dot of the SET contains only one level (E0). This assumption has been implied only for the sake of simplicity for our presentation, although our approach is well suited for a case of n levels inside the SET, E0c 0c0 → n Enc and even when the interaction between these levels is in- cluded (providing that the latter is much less or much larger than the bias V )35,36. We also assumed a weak energy dependence of the couplings Ωl,r ≃ ΩL,R. The interaction between the qubit and the SET, Hint, depends on a position of the SET with respect to the qubit. If the SET is placed near the middle of the qubit, Fig. 2a, then the tunneling coupling between two dots of the qubit in Eq. (1) decreases, Ω0 → Ω0− δΩ0, whenever the quantum dot of the SET is occupied by an electron. This is due to the electron’s repulsive field. In this case the interaction term can be written as Hint = δΩ c 0c0(a 1a2 + a 2a1) . (15) On the other hand, in the configuration shown in Fig. 2b where the SET is placed near one of the dots of the qubit, the electron repulsive field displaces the qubit energy lev- els by ∆E = U . The interaction terms in this case can be written as Hint = U a 0c0 . (16) Consider the initial state where all the levels in the left and the right reservoirs are filled with electrons up to the Fermi levels µL,R respectively. This state will be called the “vacuum” state |0 〉. The wave function for the entire system can be written as |Ψ(t)〉 = b(1)(t)a 0l (t)a 0cl + rl (t)a rcl + l<l′,r 0rll′(t)a rclcl′ + · · · +b(2)(t)a 0l (t)a 0cl + rl (t)a rcl + l<l′,r 0rll′(t)a rclcl′ + . . . 〉, (17) where b(j)(t), b α (t) are the probability amplitudes to find the entire system in the state described by the cor- responding creation and annihilation operators. These amplitudes are obtained from the Schrödinger equation i|Ψ̇(t)〉 = H |Ψ(t)〉, supplemented with the initial condi- tion b(1)(0) = p1, b (2)(0) = p2, and b α (0) = 0, where p1,2 are the amplitudes of the initial qubit state. Note that Eq. (17) implies a fixed electron number (N) in the reservoirs. At the first sight it would lead to deple- tion of the left reservoir of electrons over the time. Yet in the limit of N →∞ (infinite reservoirs) the dynamics of an entire system reaches its steady state before such a depletion takes place37,38. The behavior of the qubit and the SET is given by the reduced density matrix, σss′ (t). It is obtained from the entire system’s density matrix |Ψ(t)〉〈Ψ(t)| by trac- ing out the (continuum) reservoir states. The space of such a reduced density matrix consists of four discrete states s, s′ = a, b, c, d, shown schematically in Fig. 3 for the setup of Fig. 2a. The corresponding density-matrix elements are directly related to the amplitudes b(t), for instance, σaa(t) = |b(1)(t)|2 + |b(1)lr (t)| l<l′,r<r′ |b(1)rr′ll′(t)| 2 + · · · (18a) σdd(t) = |b(2)0l (t)| l<l′,r |b(2)0rll′(t)| l<l′<l′′,r<r′ |b(2)0rr′ll′(t)| 2 + · · · (18b) σbd(t) = 0l (t)b 0l (t) + l<l′,r 0rll′(t)b 0rll′(t) + l<l′<l′′,r<r′ 0rr′ll′ (t)b 0rr′ll′(t) + · · · . (18c) In was shown in37,38 that the trace over the reservoir states in the system’s density matrix can be performed in the large bias limit (strong non-equilibrium limit) V = µL − µR ≫ Γ,Ω0, U (19) where the level (levels) of the SET carrying the current are far away from the chemical potentials, and Γ is the width of the level E0. In this derivation we assumed only weak energy dependence of the transition ampli- tudes Ωl,r ≡ ΩL,R and the density of the reservoir states, ρ(El,r) = ρL,R. As a result we arrive at Bloch-type rate equations for the reduced density matrix without any additional assumptions. The general form of these equa- tions is36,38 σ̇jj′ = i(Ej′ − Ej)σjj′ + i σjkΩ̃k→j′ − Ω̃j→kσkj′ P2πρ(σjkΩk→k′Ωk′→j′ + σkj′Ωk→k′Ωk′→j) P2πρ (Ωk→jΩk′→j′ +Ωk→j′Ωk′→j)σkk′ (20) Here Ωk→k′ denotes the single-electron hopping ampli- tude that generates the k → k′ transition. We distinguish between the amplitudes Ω̃ describing single-electron hop- ping between isolated states and Ω describing transitions between isolated and continuum states. The latter can generate transitions between the isolated states of the system, but only indirectly, via two consecutive jumps of an electron, into and out of the continuum reservoir states (with the density of states ρ). These transitions are rep- resented by the third and the fourth terms of Eq. (20). The third term describes the transitions (k → k′ → j) or (k → k′ → j′), which cannot change the number of electrons in the collector. The fourth term describes the transitions (k → j and k′ → j′) or (k → j′ and k′ → j) which increase the number of electrons in the collector by one. These two terms of Eq. (20) are analogues of the “loss” (negative) and the “gain” (positive) terms in the classical rate equations, respectively. The factor P2 = ±1 in front of these terms is due anti-commutation of the fermions, so that P2 = −1 whenever the loss or the gain terms in Eq. (20) proceed through a two-fermion state of the dot. Otherwise P2 = 1. Note that the reduction of the time-dependent Schrödinger equation, i|Ψ̇(t)〉 = H |Ψ(t)〉, to Eqs. (20) is performed in the limit of large bias without explicit use of any Markov-type or weak coupling approxima- tions. The accuracy of these equations is respectively max(Γ,Ω0, U, T )/|µL,R−Ej |. A detailed example of this derivation is presented in Appendix A for the case of res- onant tunneling through a single level. The derivation there and in Refs.37,38 were performed by assuming zero temperature in the leads, T = 0. Yet, this assumption is not important in the case of large bias, providing the levels carrying the current are far away from the Fermi energies, |µL,R − Ej | ≫ T . IV. NO BACK-ACTION ON THE ENVIRONMENT A. Fluctuation of the tunneling coupling Now we apply Eqs. (20) to investigate the qubit’s be- havior in the configurations shown in Fig. 2. First we consider the SET placed near the middle of the qubit, Figs. 2a,3. In this case the electron current through the SET will influence the coupling between two dots of the (d)(b)(a) (c) E E1 2 Ω0 Ω0’ ’ FIG. 3: The available discrete states of the entire system cor- responding to the setup of Fig. 2a. ΓL,R denote the tunneling rates to the corresponding reservoirs and Ω′0 = Ω0 − δΩ. qubit, making it fluctuate between the values Ω0 and Ω′0 = Ω0 − δΩ. The corresponding rate equations can be written straightforwardly from Eqs. (20). One finds, σ̇aa = −ΓLσaa + ΓRσbb − iΩ0(σac − σca), (21a) σ̇bb = −ΓRσbb + ΓLσaa − iΩ′0(σbd − σdb), (21b) σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (21c) σ̇dd = −ΓRσdd + ΓLσcc − iΩ′0(σdb − σbd), (21d) σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)− ΓLσac + ΓRσbd, (21e) σ̇bd = −iǫ0σbd − iΩ′0(σbb − σdd)− ΓRσbd + ΓLσac, (21f) where ΓL,R = 2π|ΩL,R|2ρL,R are the tunneling rates from the reservoirs and ǫ0 = E1 − E2. These equations display explicitly the time evolution of the SET and the qubit. The evolution of the for- mer is driven by the first two terms in Eqs. (21a)-(21d). They generate charge-fluctuations inside the quantum dot of the SET (the transitions a←→b and c←→d), described by the “classical” Boltzmann-type dynamics. The qubit’s evolution is described by the Bloch-type terms (c.f. Eqs. (3)), generating the qubit transitions (a←→c and b←→d). Thus Eqs. (21) are quite general, since they described fluctuations of the tunneling cou- pling driven by the Boltzmann-type dynamics. The resulting time evolution of the qubit is given by the qubit (reduced) density matrix: σ11(t) = σaa(t) + σbb(t) , (22a) σ12(t) = σac(t) + σbd(t) , (22b) and σ22(t) = 1− σ11(t). Similarly, the charge fluctuations of SET are deter- mined by the probability of finding the SET occupied, P1(t) = σbb(t) + σdd(t) . (23) It is given by the equation Ṗ1(t) = ΓL − ΓP1(t) , (24) obtained straightforwardly from Eqs. (21). Here Γ = ΓL +ΓR is the total width. The same equation for P1(t) can be obtained if the qubit is decoupled from the SET (δΩ = 0). Thus there is no back-action of the qubit on the charge fluctuations inside the SET in the limit of large bias voltage. Consider first the stationary limit, t → ∞, where Ṗ1(t)→ 0 and σ̇(t)→ 0. It follows from Eq. (24) that the probability of finding the SET occupied in this limit is P̄1 = ΓL/Γ. This implies that the fluctuations of the cou- pling Ω0, induced by the SET, would take place around the average value Ω = Ω0 − P̄1 δΩ. With respect to the qubit in the stationary limit, one easily obtains from Eqs. (21) that the qubit density ma- trix always becomes the statistical mixture (5), when t → ∞. This takes place for any initial conditions and any values of the qubit and the SET parameters. There- fore the effect of the fluctuating charge inside the SET does not lead to relaxation of the qubit, but rather to its decoherence. It is important to note, however, that for the aligned qubit, ǫ = 0, the decoherence due to fluctuations of the tunneling coupling Ω0 is not complete. Indeed, it follows from Eqs. (21) that d/dt[Re σ12(t)] = 0. The reason is that the corresponding operator, a 1a2 + a 2a1 commutes with the total Hamiltonian H = Hqb + HSET + Hint, Eqs. (1), (14) and (15), for E1 = E2. As a result, Re σ12(t) = Re σ12(0). In order to determine the decoherence rate analytically, we perform a Laplace transform on the density matrix, σ̃(E) = σ(t) exp(−iEt)dE. Then solving Eq. (21) we can determine the decoherence rate from the locations of the poles of σ̃(E) in the complex E-plane. Consider for instance the case of ǫ0 = 0 and the symmetric SET, ΓL = ΓR = Γ/2. One finds from Eqs. (21) and (22a) that σ̃11(E) = i(E − 2Ω + iΓ) 4(E − 2Ω + iΓ/2)2 + Γ2 − (2 δΩ)2 i(E + 2Ω+ iΓ) 4(E + 2Ω+ iΓ/2)2 + Γ2 − (2 δΩ)2 . (25) Upon performing the inverse Laplace transform, σ11(t) = ∞+i0∫ −∞+i0 σ̃11(E) e −iEt dE , (26) and closing the integration contour around the poles of the integrand, we obtain for Γ > 2δΩ and t≫ 1/Γ σ11(t)− (1/2) ∝ e−(Γ− Γ2−4δΩ2)t/2 sin(2Ω t). (27) Comparing this result with Eq. (12) we find that the decoherence rate is Γd = 2 Γ2 − 4δΩ2 Γ≫δΩ−→ (2δΩ)2/Γ . (28) For ǫ0 6= 0 and ǫ0,Γ ≪ Ω the decoherece rate Γd is multiplied by an additional factor [1− (ǫ0/2Ω)2]. In a general case, ΓL 6= ΓR, we obtain in the same limit (ΓL,R ≫ δΩ) for the decoherence rate: (4 δΩ)2 (ΓL + ΓR)3 It is interesting to compare this result with the fluctu- ation spectrum of the charge inside the SET, Eq. (B8), Appendix B. We find Γd = 2 (δωR) 2 SQ(0) , (30) where ωR = 4Ω2 + ǫ20 is the Rabi frequency. The latter represents the energy splitting in the diagonalized qubit Hamiltonian. Thus δωR corresponds to the amplitude of energy level fluctuations in a single dot. Although Eq. (30) has been obtained for small fluc- tuations δωR, it might be approximately correct even if δωR is of the order of Γ. It is demonstrated in Fig. 4, where we compare σ11(t) and σ12(t), obtained from Eqs. (21) and (22) (solid line) with those from Eqs. (3a) and (4) (dashed line) for the decoherence rate Γd given by Eq. (30). The initial conditions correspond to σ11(0) = 1 and σ12(0) = 0 (respectively, σaa(0) = ΓR/Γ and σbb(0) = ΓR/Γ). In the case of aligned qubit, however, Re σ12(t) = Re σ12(0), as was explained above. On the other hand, one always obtains from (3a) and (4) that Re [σ12(t → ∞)] = 0. Therefore the phenomenological Bloch equa- tions are not applicable for evaluation of Re [σ12(t)], even in the weak coupling limit (besides the case of Re [σ12(t = 0)] = 0). In the large coupling regime (δΩ≫ Γ) the phenomeno- logical Bloch equations, Eqs. (3a) and (4), cannot be used, as well. Consider for simplicity the case of ǫ = 0 and ΓL,R = Γ/2. Then one finds from Eq. (27) that the damping oscillations between the two dots take place at two different frequencies, 2Ω ± (δΩ)2 − (Γ/2)2, in- stead of the one frequency, ωR = 2Ω, given the Bloch equations. Moreover, Eq. (30) does not reproduce the decoherence (damping) rate in this limit. Indeed, one obtains from Eq. (28) that the decoherence rate Γd = 2Γ for δΩ > Γ/2, so Γd does not depend on the coupling (δΩ) at all. B. Fluctuation of the energy level Consider the SET placed near one of the qubit dots, as shown in Fig. 2b. In this case the qubit-SET inter- action term is given by Eq. (16). As a result the energy level E1 will fluctuate under the influence of the fluctua- tions of the electron charge inside the SET. The available discrete states of the entire system are shown in Fig. 5. Using Eqs. (20) we can write the rate equations, similar 10 20 30 40 50 (a) (b) 10 20 30 40 50 Re 12 Im 12 11 σ ( ) σ ( ) σ ( ) FIG. 4: The occupation probability of the first dot of the qubit for ǫ = 2Ω, ΓL = Ω, ΓR = 2Ω and δΩ = 0.5Ω. The solid line is the exact result, whereas the dashed line is obtained from the Bloch-type rate equations with the decoherence rate given by Eq. (30). (d)(b)(a) (c) ’’’ ’ Γ R ΓL ΓR E0 E0 0Ω0Ω0 E2 E2 E +U1Ω FIG. 5: The available discrete states of the entire system for the configuration shown in Fig. 2b. Here U is the repulsion energy between the electrons. to Eqs. (21), σ̇aa = −Γ′Lσaa + Γ′Rσbb − iΩ0(σac − σca), (31a) σ̇bb = −Γ′Rσbb + Γ′Lσaa − iΩ0(σbd − σdb), (31b) σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (31c) σ̇dd = −ΓRσdd + ΓLσcc − iΩ0(σdb − σbd), (31d) σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)− ΓL + Γ Rσbd, (31e) σ̇bd = −i(ǫ0 + U)σbd − iΩ0(σbb − σdd)− ΓR + Γ Lσac , (31f) where Γ′L,R are the tunneling rate at the energy E0+U Let us assume that Γ′L,R = ΓL,R. Then it follows from Eqs. (31) that the behavior of the charge inside the SET is not affected by the qubit, the same as in the previous case of the Rabi frequency fluctuations. Also the qubit density matrix becomes the mixture (5) in the stationary state for any values of the qubit and the SET parameters. Hence, there is no qubit relaxation in this case either (except for the static qubit, Ω0 = 0, and σ11(0) 6= σ22(0), Eq. (11)). Since according to Eq. (24), the probability of finding an electron inside the SET in the stationary state is P̄1 = ΓL/Γ, the energy level E1 of the qubit is shifted by P̄1U . Therefore it is useful to define the “renormalized” level displacement, ǫ = ǫ0 + P̄1U . As in the previous case we use the Laplace transform, σ(t)→ σ̃(E), in order to determine the decoherence rate analytically. In the case of ΓL = ΓR = Γ/2 and ǫ = 0 we obtain from Eqs. (31) σ̃11(E) = 32(E + iΓ)Ω20 U2 − 4E(E + iΓ) . (32) The position of the pole in the second term of this expres- sion determines the decoherence rate. In contrast with Eq. (25), however, the exact analytical expression for the decoherence rate (Γd) is complicated, since it is given by a cubic equation. We therefore evaluate Γd in a different way, by substituting E = ±2Ω0 − iγ in the second term of Eq. (32) and then expanding the latter in powers of γ by keeping only the first two terms of this expansion. The decoherence rate Γd is related to γ by Γd = 4γ, as follows from Eq. (12). Then we obtain: 2(Γ2 + 4Ω20) for U ≪ (Ω20 + ΓΩ0)1/2 64ΓΩ20 U2 + 16Ω20 for U ≫ (Ω20 + ΓΩ0)1/2 In general, if ΓL 6= ΓR, one finds from Eqs. (31) that Γd = 2U 2ΓLΓR/[Γ(Γ 2 + 4Ω20)] for U ≪ (Ω20 + ΓΩ0)1/2. The same as in the previous case, Eq. (30), the deco- herence rate in a weak coupling limit is related to the fluctuation spectrum of the SET, SQ(ω), Eq. (B8), but now taken at a different frequency, ω = 2Ω0. The lat- ter corresponds to the level splitting of the diagonalized qubit’s Hamiltonian, ωR. Thus, Γd = U 2 SQ(ωR) , (34) which can be applied also for ǫ 6= 0. This is illustrated by Fig. 6 which shows σ11(t) obtained from Eqs. (31) and (22) (solid line) with Eqs. (3a) and (4) (dashed line) for the decoherence rate Γd given by Eq. (34). As in the previous case, shown in Fig. 4, the initial conditions correspond to σ11(0) = 1 and σ12(0) = 0 (respectively, σaa(0) = ΓR/Γ and σbb(0) = ΓR/Γ). One finds from Fig. 6 that Eq. (34) can be used for an estimation of Γd even for U ∼ Γ,Ω0. In contrast with the tunneling-coupling fluctuations, Eq. (30), where the decoherence rate is given by SQ(0), the fluctuations of the qubit’s energy level generate the decoherence rate, determined by the fluctuation spec- trum at Rabi frequency, SQ(ωR), Eq. (34). A similar distinction between the decoherence rates generated by different components of the fluctuating field, exists in a phenomenological description of magnetic resonance27. One can understand this distinction by diagonalizing the qubit’s Hamiltonian. In this case the Rabi frequency, ωR, becomes the level splitting of the qubit’s states |±〉 = (|1〉 ± |2〉)/ 2 (for ǫ = 0). So in this basis, the tunneling-coupling fluctuations correspond to simultane- ous fluctuations of the energy levels in the both dots. 10 20 30 40 50 (a) (b) 10 20 30 40 50 Re 12 Im 12 11 σ ( ) σ ( ) σ ( ) FIG. 6: The probability of finding the electron in the first dot of the qubit for ǫ = 2Ω0, ΓL = Ω0, ΓR = 2Ω0 and U = 0.5Ω0. The solid line is the exact result, whereas the dashed line is obtained from the Bloch-type rate equations with the decoherence rate given by Eq. (34). Since these fluctuations are “in phase”, we could expect that the corresponding dephasing rate is determined by spectral density at zero frequency. In fact, it looks like as fluctuations of a single dot state, considered by Levin- son in a weak coupling limit31. On the other hand by fluctuating the energy level in one of the dots only, one can anticipate that the corresponding dephasing rate is determined by the fluctuation spectrum at the Rabi fre- quency, ωR, Eq. (34), which is a frequency of the inter-dot transitions. Since ωR can be controlled by the qubit’s levels dis- placement, ǫ, the relation (34) can be implied by using qubit for a measurement of the shot-noise spectrum of the environment18,19,40. For instance, it can be done by at- taching a qubit to reservoirs at different chemical poten- tials. The corresponding resonant current which would flow through the qubit in this case, can be evaluated via a simple analytical expression13 that includes explicitly the decoherence rate, Eq. (34). Thus by measuring this current for different level displacement of the qubit (ǫ0), one can extract the spectral density of the fluctuating environment acting on the qubit18. Although Eq. (34) for the decoherence rate has been obtained by using a particular mechanism for fluctuations of the qubit’s energy levels, we suggest that this mecha- nism is quite general. Indeed, the rate equations (31) can describe any fluctuating media near a qubit, driven by the Boltzmann type of equations. Therefore it is rather nat- ural to assume that Eq. (34) would be valid for any type of such (classical) environment in weak coupling limit. This implies that the decoherence rate is always deter- mined via the spectral density of a fluctuating qubit’s level, whereas the nature of a particular medium inducing these fluctuations would be irrelevant. In order to sub- stantiate this point it is important to compare Eq. (34) with the corresponding decoherence rate induced by the thermal environment in the framework of the spin-boson model. In a weak damping limit this model predicts1,2 T−11 = T 2 = (q 0/2)S(ωR) , where q0 is a coupling of the medium with the qubit levels (q0 corresponds to U in our case) and S(ω) is a spectral density. Using Eq. (10) one finds that this result coincides with Eq. (34). 0 5 10 15 20 0 5 10 15 20 t t1111 U/ =100ΩU/ =10 σ ( ) σ ( ) (a) (b) FIG. 7: The probability of finding the electron in the first dot of the qubit for ǫ = 0, ΓL = ΓR = Ω0 and U , as given by Eqs. (31) (solid line) and from the Bloch-type equations (dashed line) with the decoherence rate given by Eq. (33). C. Strong-coupling limit and localization Let us consider the limit of U ≫ (Ω20 + ΓΩ0)1/2. Our rate equation (31) are perfectly valid in this region, pro- viding only that E0 + U is deeply inside of the potential bias, Eq. (19). We find from Eq. (33) that the deco- herence rate is not directly related to the spectrum of fluctuations in strong coupling limit. In addition, the ef- fective frequency of the qubit’s Rabi oscillations (ω decreases in this limit. Indeed, by using Eqs. (32), (26), one finds that the main contribution to σ11(t), is coming from a pole of σ̃11(E), which lies on the imaginary axis. This implies that the effective frequency of Rabi oscilla- tions strongly decreases when U ≫ (Ω20 + ΓΩ0)1/2. In addition, the decoherence rate Γd → 0 in the same limit, Eq. (33). As a result, the electron would localize in the initial qubit state, Fig. 7. The results displayed in this figure show that the solu- tion of the Bloch-type rate equations, with the decoher- ence rate given by Eq. (33), represents damped oscilla- tions (dashed line). It is very far from the exact result (solid line), obtained from Eqs. (31) and corresponding to the electron localization in the first dot. The latter is a result of an effective decrease of the Rabi frequency for large U that slows down electron transitions between the dots. Thus such an environment-induced localization is different from the Zeno-type effect (unlike an assumption of Ref.12). Indeed, the Zeno effect takes place whenever the decoherence rate is much larger then the coupling be- tween the qubit’s states13,33. However, the decoherence rate in the strong coupling limit is much smaller then the coupling Ω0 . In fact, the localization shown in Fig. 7 is rather similar to that in the spin-boson model1,2. It shows that in spite of their defferences, both models trace the same physics of the back-action of the environment (SET) on the qubit. V. BACK-ACTION OF THE QUBIT ON THE ENVIRONMENT A. Weak back-action effect Now we investigate a weak dependence of the width’s ΓL,R on the energy U , Fig. 5. We keep only the linear term, Γ′L,R = ΓL,R+αL,RU , by assuming that U is small. (A similar model has been considered in28,41). In con- trast with the previous examples, where the widths have not been dependent on the energy, the qubit’s oscillation would affect the SET current and its charge correlator. A more interesting case corresponds to αL 6= αR. Let us take for simplicity αL = 0 and αR = α 6= 0. Similarly to the previous case we introduce the “renor- malized” level displacement, ǫ = ǫ0 − (ΓL/Γ)U , where ǫ = 0 corresponds to the aligned qubit. Solving Eqs. (31) in the steady-state limit, σ̄ = σ(t → ∞), and keeping only the first term in expansion in powers of U , we find for the reduced density matrix of the qubit, Eqs. (22): − α ǫ αΩ0(1 + c αU) αΩ0(1 + c αU) , (35) where c = (αǫ − 2Γ)/(4ΓRΓ). It follows from Eqs. (35) that the qubit’s density matrix in the steady-state is no longer a mixture, Eq. (5) . Indeed, the probability to occupy the lowest level is always larger than 1/2 and σ̄12 6= 0. This implies that relaxation takes place to- gether with decoherence. The ratio of the relaxation and decoherence rates is given by the off-diagonal terms of the reduced density matrix of the qubit. For ǫ = 0 one finds from Eq. (13) that Γd/Γr = σ̄ 12 − 2. In order to find a relation between the decoherence and relaxation rates, Γd,r, and the fluctuation spectrum of the qubit energy level, SQ(ω), we first evaluate the total damping rate of the qubit’s oscillations (γ). Using Eq. (12) we find that this quantity is related to the deco- herence and relaxation rates by γ = (Γd + 2Γr)/4. The same as in the previous case the rate γ is determined by poles of Laplace transformed density matrix σ(t)→ σ̃(E) in the complex E-plane. Consider for simplicity the case of ǫ = 0 and ΓL = ΓR = Γ/2. Performing the Laplace transform of Eqs. (31) we look for the poles of σ11(E) at E = ±2Ω0 − iγ by assuming that γ is small. We obtain Γd + 2Γr = 2(Γ2 + 4Ω20) Γ− αU Γ 2 − 4Ω20 2 (Γ2 + 4Ω20) for U ≪ Ω0. Now we evaluate the correlator of the charge inside the SET, SQ(ω) which induces the energy-level fluctuations of the qubit. Using Eqs. (31) and (B6) we find, SQ(ω) = 2 (Γ2 + ω2) − αU Γ 2 − ω2 4 (Γ2 + ω2) for αU ≪ Γ. Therefore in the limit of U ≪ Ω0 and αU ≪ Γ the total damping rate of the qubit’s oscillations is directly related to the spectral density of the fluctuations spectrum taken at the Rabi frequency, Γd + 2Γr = U 2SQ(2Ω0). (38) This represents a generalization of Eq. (34) for the case of a weak back-action of qubit oscillations on the spectral density of the environment. As a result, the qubit dis- plays relaxation together with decoherence. It is remark- able that the total qubit’s damping rate is still given by the fluctuation spectrum of the SET (environment) mod- ulated by the qubit. Note that Eq. (38) can be applied only if the modulation of the tunneling rate through the SET (tunneling current) is small αU ≪ Γ, in addition to a weak distortion of the qubit (U ≪ Ω0). In the case of strong back-action of the qubit on the environment the decorerence and relaxation rates of the qubit are not directly related to the fluctuation spectrum of the environment, even if the distortion of the qubit is small. This point is illustrated by the following example. B. Strong back-action Until now we considered the case where E0 + U ≪ µL, so that the interacting electron of the SET remains deeply inside the voltage bias. If however, the interaction U between the qubit and the SET is such that E0 + U ≫ µL, the qubit’s oscillation would strongly affect the fluctuation of charge inside the SET. Indeed, the current through the SET is blocked whenever the level E1 of the qubit is occupied, Fig. 8. In fact, this case can be treated with small modification of the rate equations (31), if only µL − E0 ≫ Γ and E0 + U − µL ≫ Γ, where E0 is a level of the SET carrying the current. The corresponding quantum rate equations describing the system are obtained directly from Eqs. (20). As- suming that the widths ΓL,R are energy independent we find16 σ̇aa = (ΓL + ΓR)σbb − iΩ0(σac − σca), (39a) σ̇bb = −(ΓR + ΓL)σbb − iΩ0(σbd − σdb), (39b) σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (39c) σ̇dd = −ΓRσdd + ΓLσcc − iΩ0(σdb − σbd), (39d) σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)− + ΓRσbd, (39e) σ̇bd = −i(ǫ0 + U)σbd − iΩ0(σbb − σdd) σbd . (39f) Solving Eqs. (39) in the stationary limit, σ̄ = σ(t → ∞) and introducing the “renormalized” level displace- ment, ǫ = ǫ0 −UΓL/(2Γ), we obtain for the qubit’s den- (d)(b)(a) (c) E +U1 FIG. 8: The available discrete states of the entire system when the electron-electron repulsive interaction U breaks off the current through the SET. sity matrix, Eqs. (22) in the steady state: σ̄11 = − 8ǫU 16ǫ2 + 8Uǫ+ 48Ω20 + 9(U 2 + Γ2) ,(40a) σ̄12 = 12UΩ0 16ǫ2 + 8Uǫ+ 48Ω20 + 9 (U 2 + Γ2) , (40b) where for simplicity we considered the symmetric case, ΓL = ΓR = Γ/2. It follows from Eqs. (40) that similarly to the previous example, the qubit’s density matrix is no longer a mixture (5). The relaxation takes place together with decoherence in this case too. Let us consider weak distortion of the qubit by the SET, U < Ω0. Although the values of U are restricted from below (U ≫ Γ+µL−E0), this limit can be achieved if the level E0 is close to the Fermi energy, providing only that µL − E0 ≫ Γ, and Γ≪ U . Now we evaluate σ11(t) with the rate equations (39) and then compare it with the same quantity obtained from the Bloch equations, Eq. (12), where Γd,r are given by Eqs. (34)and (13). The corresponding charge-correlator, SQ(ωR), is evaluated by Eqs. (B6) and (39). As an example, we take symmetric qubit with aligned levels, ǫ = 0, ΓL = ΓR = 0.05Ω0 and U = 0.5Ω0. The decoherence and relaxation rates, corre- sponding to these parameters are respectively: Γd/Ω0 = 0.0038 and Γr/Ω0 = 0.00059. The results are presented in Fig. 9a. The solid line shows σ11(t), obtained from the rate equations (39), where the dashed line is the same quantity obtained from Eq. (12). We find that Eq. (34) (or (38)) underestimates the actual damping rate of σ11(t) by an order of mag- nitude). This lies in a sharp contrast with the previous case, where the energy level of the SET is not distorted by the qubit, Γ′L,R = ΓL,R, Fig. 5. Indeed, in this case σ11(t) obtained Eq. (12) with Γd given by Eq. (34) and Γr = 0, agrees very well with that obtained from the rate equations (31), as shown in Fig. 9b. Such an example clearly illustrates that the decoher- ence is not related to the fluctuation spectrum of the environment, whenever the environment is strongly af- fected by the qubit, even if the coupling with a qubit is small. This is a typical case of measurement, corre- sponding to a noticeable response of the environment to the qubit’s state (a “signal”). 20 40 60 80 100 20 40 60 80 100 t t11 11σ ( ) σ ( ) FIG. 9: (a) The probability of finding the electron in the first dot of the qubit for ǫ = 0, ΓL = ΓR = 0.05Ω0 and U = 0.5Ω0. The solid line is obtained from Eqs. (39), whereas the dashed line corresponds to the Eq. (12) with Γd given by Eq. (34); (b) the same for the case, shown in Fig. 5, where the solid line corresponds to Eqs. (31). VI. SUMMARY In this paper we propose a simple model describing a qubit interacting with fluctuating environment. The latter is represented by a single electron transistor (SET) in close proximity of the qubit. Then the fluctuations of the charge inside the SET generate fluctuating field acting on the qubit. In the limit of large bias voltage, the Schrödinger equation for the entire system is reduced to the Bloch-type rate equations. The resulting equations are very simple, so that one can easily analyze the limits of weak and strong coupling of the qubit with the SET. We considered separately two different cases: (a) there is no back-action of the qubit on the SET behavior, so that the latter represents a “pure environment”; and (b) the SET behavior depends on the qubit’s state. In the latter case the SET can “measure” the qubit. The setup corresponding to the “pure environment” is realized when the energy level of the SET carrying the current lies deeply inside the potential bias. The second (measure- ment) regime of the SET is realized when the tunnel- ing widths of the SET are energy dependent, or when the energy level of the SET carrying the current is close enough to the Fermi level of the corresponding reservoir. Then the electron-electron interaction between the qubit and the SET modulates the electron current through the In the case of the “pure environment” (“no- measurement” regime) we investigate separately two dif- ferent configurations of the qubit with respect to the SET. In the first one the SET produces fluctuations of the off-diagonal coupling (Rabi frequency) between two qubit’s states. In the second configuration the SET pro- duces fluctuations of the qubit’s energy levels. In the both cases we find no relaxation of the qubit, despite the energy transfer between the qubit and the SET can take place. As a result the qubit always turns asymptot- ically to the statistical mixture. We also found that in both cases the decoherence rate of the qubit in the weak coupling limit is given by the spectral density of the cor- responding fluctuating parameter. The difference is that in the case of the off-diagonal coupling fluctuations the spectral density is taken at zero frequency, whereas in the case of the energy level fluctuations it is taken at the Rabi-frequency. In the case of the strong coupling limit, however, the decoherence rate is not related to the fluctuation spec- trum. Moreover we found that the electron in the qubit is localized in this limit due to an effective decrease of the off-diagonal coupling. This phenomenon may resem- ble the localization in the spin-boson model in the strong coupling limit. If the charge correlator and the total SET current are affected by the qubit (back-action effect), we found that the off-diagonal density-matrix elements of the qubit sur- vive in the steady-state limit and therefore the relax- ation rate is not zero. We concentrated on the case of weak coupling, when the Coulomb repulsion between the qubit and the SET is smaller then the Rabi frequency. The back-action of the qubit on the SET, however, can be weak or strong. In the first case we found that the total damping rate of the qubit due to decoherence and relaxation is again given by the spectral density of the SET charge fluctuations, modulated by the qubit. This relation, however, is not working if the back-action is strong. Indeed, we found that the damping rate of the qubit in this case is larger by an order of magnitude than that given by the spectral density of the corresponding fluctuating parameter. This looks like that in the strong back-action of the qubit on the SET the major component of decoherence is not coming from the fluctuation spectrum of the qubit’s parameters only, but also from the measurement “sig- nal” of the SET. On the first sight it could agree with an analysis of Ref.30, suggesting that the decoherence rate contains two components, generated by a measure- ment and by a “pure environment” (environmental fluc- tuations). The latter therefore represents an unavoid- able decoherence, generated by any environment. Yet, in a weak coupling regime such a separation seems not working. In this case the damping (decoherence) rate is totally determined by the environment fluctuations, even so modulated by the qubit. Although our model deals with a particular setup, it bears the main physics of a fluctuating environment, act- ing on a qubit. Indeed, the Bloch-type rate equations, which we used in our analysis have a pronounced phys- ical meaning: they relate the variation of qubit param- eters with a nearby fluctuating field described by rate equations. A particular mechanism, generated this field should not be relevant for an evaluations of the deco- herence and relaxation rates, but only its fluctuation spectrum. Indeed, in the weak coupling limit our re- sult for the decorence rate coincides with that obtained in a framework of the spin-boson model. Thus our model can be considered as a generic one. Its main advantage is that it can be easily extended to multiple coupled qubits. Such an analysis would allow to determine how decoher- ence scales with number of qubits42, which is extremely important for a realization of quantum computations. In addition, our model can be extended to a more complicated fluctuating environments, such as containing characteristic frequencies in its spectrum. It would for- mally correspond to a replacement of the SET in Fig. 2 by a double-dot (DD) coupled to the reservoirs43. All these situations, however, must be a subject of a sepa- rate investigation. VII. ACKNOWLEDGEMENT One of us (S.G.) thanks T. Brandes and C. Emary for helpful discussions and important suggestions. S.G is also grateful to the Max Planck Institute for the Physics of Complex Systems, Dresden, Germany, and to NTT Ba- sic Research Laboratories, Atsugi-shi, Kanagawa, Japan, for kind hospitality. APPENDIX A: QUANTUM-MECHANICAL DERIVATION OF RATE EQUATIONS FOR QUANTUM TRANSPORT Consider the resonant tunneling through the SET, shown schematically in Fig. 10. The entire system is described by the Hamiltonian HSET, given by Eq. (14). The wave function can be written in the same way as Eq. (17), where the variables related to the qubit are omitted, |Ψ(t)〉 = b(t) + b0l(t)c 0cl + brl(t)c l<l′,r b0rll′(t)c rclcl′ + · · · 〉. (A1) Substituting |Ψ(t)〉 into the time-dependent Schrödinger equation, i∂t|Ψ(t)〉 = HSET|Ψ(t)〉, and performing the Laplace transform, b̃(E) = exp(iEt) b(t)dt, we obtain the following infinite set of algebraic equations for the FIG. 10: Resonant tunneling through a single dot. µL,R are the Fermi energies in the collector and emitter, respectively. amplitudes b̃(E): Eb̃(E) − Ωlb̃0l(E) = i (A2a) (E + El − E0)b̃0l(E)− Ωlb̃(E) Ωr b̃lr(E) = 0 (A2b) (E + El − Er)b̃lr(E)− Ωr b̃0l(E) Ωl′ b̃0ll′r(E) = 0 (A2c) (E + El + El′ − E0 − Er)b̃0ll′r(E) − Ωl′ b̃lr(E) + Ωlb̃l′r(E)− Ωr′ b̃ll′rr′(E) = 0 (A2d) · · · · · · · · · (The r.h.s of Eq. (A2a) reflects the initial condition.) Let us replace the amplitude b̃ in the term Ωb̃ of each of the equations (A2) by its expression obtained from the subsequent equation. For example, substituting b̃0l(E) from Eq. (A2b) into Eq. (A2a) we obtain E + El − E0 b̃(E) E + El − E0 b̃lr(E) = i. (A3) Since the states in the reservoirs are very dense (contin- uum), one can replace the sums over l and r by integrals, for instance ρL(El) dEl , where ρL(El) is the density of states in the emitter, and Ωl,r → ΩL,R(El,r). Consider the first term Ω2L(El) E + El − E0 ρL(El)dEl (A4) where Λ is the cut-off parameter. Assuming weak en- ergy dependence of the couplings ΩL,R and the density of states ρL,R, we find in the limit of high bias, µL = Λ→∞ S1 = −iπΩ2L(E0 − E)ρL(E0 − E) = −i . (A5) Consider now the second sum in Eq. (A3). ρR(Er)dEr ΩL(El)ΩR(Er)b̃lr(E,El, Er) E + El − E0 ρL(El)dEl , (A6) where we replaced b̃lr(E) by b̃(E,El, Er) and took µL = Λ, µR = −Λ. In contrast with the first term of Eq. (A3), the amplitude b̃ is not factorized out the integral (A6). We refer to this type of terms as “cross-terms”. Fortu- nately, all “cross-terms” vanish in the limit of large bias, Λ → ∞. This greatly simplifies the problem and is very crucial for a transformation of the Schrödinger to the rate equations. The reason is that the poles of the integrand in the El(Er)-variable in the “cross-terms” are on the same side of the integration contour. One can find it by using a perturbation series the amplitudes b̃ in powers of Ω. For instance, from iterations of Eqs. (A2) one finds b̃(E,El, Er) = iΩLΩR E(E + El − Er)(E + El − E0) + · · · The higher order powers of Ω have the same structure. Since E → E + iǫ in the Laplace transform, all poles of the amplitude b̃(E,El, Er) in the El-variable are below the real axis. In this case, substituting Eq. (A7) into Eq. (A6) we find (E + iǫ)(E + E0 − E1 + iǫ)2(E + E0 − Er + iǫ) + · · · dEl = 0 , (A8) Thus, S2 → 0 in the limit of µL →∞, µR → −∞. Applying analogous considerations to the other equa- tions of the system (A2), we finally arrive at the following set of equations: (E + iΓL/2)b̃(E) = i (A9a) (E + El − E0 + iΓR/2)b̃0l(E) − Ωlb̃(E) = 0 (A9b) (E + El − Er + iΓL/2)b̃lr(E) − Ωrb̃0l(E) = 0 (A9c) (E + El + El′ − E0 − Er + iΓR/2)b̃0ll′r(E) − Ωl′ b̃lr(E) + Ωlb̃l′r(E) = 0 (A9d) · · · · · · · · · Eqs. (A9) can be transformed directly to the reduced density matrix σ (n,n′) jj′ (t), where j = 0, 1 denote the state of the SET with an unoccupied or occupied dot and n de- notes the number of electrons which have arrived at the collector by time t. In fact, as follows from our derivation, the diagonal density-matrix elements, j = j′and n = n′, form a closed system in the case of resonant tunneling through one level, Fig. 10. The off-diagonal elements, j 6= j′, appear in the equation of motion whenever more than one discrete level of the system carry the transport (see Eq. (20). Therefore we concentrate below on the di- agonal density-matrix elements only, σ 00 (t) ≡ σ (n,n) 00 (t) and σ 11 (t) ≡ σ (n,n) 11 (t). Applying the inverse Laplace transform on finds 00 (t) = l...,r... dEdE′ b̃l · · · r · · · (E)b̃∗l · · · r · · · (E′)ei(E ′−E)t (A10a) 11 (t) = l...,r... dEdE′ 0l · · · r · · · (E)b̃∗ 0l · · · r · · · (E′)ei(E ′−E)t (A10b) Consider, for instance, the term σ 11 (t) = l |b0l(t)|2. Multiplying Eq. (A9b) by b̃∗0l(E ′) and then subtracting the complex conjugated equation with the interchange E ↔ E′ we obtain dEdE′ (E′ − E − iΓR)b̃0l(E)b̃∗0l(E′) − 2Im Ωlb̃0l(E)b̃ ∗(E′) ′−E)t = 0 (A11) Using Eq. (A10b) one easily finds that the first inte- gral in Eq. (A11) equals to −i[σ̇(0)11 (t)+ΓRσ 11 (t)]. Next, substituting b̃0l(E) = Ωlb̃(E) E + El − E0 + iΓR/2 (A12) from Eq. (A9b) into the second term of Eq. (A11), and replacing a sum by an integral, one can perform the El- integration in the large bias limit, µL → ∞, µR → −∞. Then using again Eq. (A10b) one reduces the second term of Eq. (A11) to iΓLσ 00 (t). Finally, Eq. (A11) reads 11 (t) = ΓLσ 00 (t)− ΓRσ 11 (t). The same algebra can be applied for all other am- plitudes b̃α(t). For instance, by using Eq. (A10a) one easily finds that Eq. (A9c) is converted to the following rate equation σ̇ 00 (t) = −ΓLσ 00 (t) + ΓRσ 11 (t). With respect to the states involving more than one electron (hole) in the reservoirs (the amplitudes like b̃0ll′r(E) and so on), the corresponding equations contain the Pauli ex- change terms. By converting these equations into those for the density matrix using our procedure, one finds the “cross terms”, like Ωlb̃l′r(E)Ωl′ b̃ ′), generated by Eq. (A9d). Yet, these terms vanish after an integration over El(r) in the large bias limit, as the second term in Eq. (A3). The rest of the algebra remains the same, as described above. Finally we arrive at the following infinite system of the chain equations for the diagonal elements, σ 00 and σ 11 , of the density matrix, 00 (t) = −ΓLσ 00 (t) , (A13a) 11 (t) = ΓLσ 00 (t)− ΓRσ 11 (t) , (A13b) 00 (t) = −ΓLσ 00 (t) + ΓRσ 11 (t) , (A13c) 11 (t) = ΓLσ 00 (t)− ΓRσ 11 (t) , (A13d) · · · · · · · · · Summing over n in Eqs. (A13) we find for the reduced density matrix of the SET, σ(t) = (n)(t), the fol- lowing “classical” rate equations, σ̇00(t) = −ΓLσ00(t) + ΓRσ11(t) (A14a) σ̇11(t) = ΓLσ00(t)− ΓRσ11(t) (A14b) These equations represent a particular case of our general quantum rate equations (20), which are derived using the above described technique37,38. APPENDIX B: CORRELATOR OF ELECTRIC CHARGE INSIDE THE SET. The charge correlator inside the SET is given by SQ(ω) = S̄Q(ω) + S̄Q(−ω), where S̄Q(ω) = 〈δQ̂(0)δQ̂(t)〉eiωtdt . (B1) Here δQ̂(t) = c 0(t)c0(t) − q̄ and q̄ = P̄1 = P1(t → ∞) is the average charge inside the dot. Since the initial state, t = 0 in Eq. (B1) corresponds to the steady state, one can represent the time-correlator as 〈δQ̂(0)δQ̂(t)〉 = q=0,1 Pq(0)(q − q̄)(〈Qq(t)〉 − q̄) , (B2) where Pq(0) is the probability of finding the charge q = 0, 1 inside the quantum dot in the steady state, such that P1(0) = q̄ and P0(0) = 1 − q̄, and 〈Qq(t)〉 = P (q)1 (t) is the average charge in the dot at time t, starting with the initial condition P 1 (0) = q. Substituting Eq. (B2) into Eq. (B1) we finally obtain S̄Q(ω) = q̄(1 − q̄)[P̃ (1)1 (ω)− P̃ 1 (ω)] , (B3) where P̃ 1 (ω) is a Laplace transform of P 1 (t). These quantities are obtained directly from the rate equations, such that q̄ = σ̄bb + σ̄dd and P̃ 1 (ω) = σ̃ bb (ω) + dd (ω), where σ̄ = σ(t → ∞) and σ̃(q)(ω) is the Laplace transform σ(q)(t) with the initial conditions corresponding to the occupied (q = 1) or unoccupied (q = 0) SET. In order to find these quantities it is use- ful to rewrite the rate equations in the matrix form, σ̇(t) = Mσ(t), representing σ(t) as the eight-vector, σ = {σaa, σbb, σcc, σdd, σac, σca, σbd, σdb} and M as the corresponding 8× 8-matrix. Applying the Laplace trans- form we find the following matrix equation, (i ω I +M)σ̃(q)(ω) = −σ(q)(0) , (B4) where I is the unit matrix and σ(q)(0) is the initial con- dition for the density-matrix obtained by projecting the total wave function (17) on occupied (q = 1) and unoc- cupied (q = 0) states of the SET in the limit of t → ∞, σ(1)(0) = N1{0, σ̄bb, 0, σ̄dd, 0, 0, σ̄bd, σ̄db} , (B5a) σ(0)(0) = N0{σ̄aa, 0, σ̄cc, 0, σ̄ac, σ̄ca, 0, 0} , (B5b) and N1 = 1/q̄ and N0 = 1/(1− q̄) are the corresponding normalization factors. Finally one obtains: SQ(ω) = 2q̄(1− q̄)Re [σ̃(1)bb (ω) + σ̃ dd (ω) − σ̃(0)bb (ω)− σ̃ dd (ω)]. (B6) In the case shown in Fig. 2 one finds from Eqs. (21) or Eqs. 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0704.0195

Group-theoretical properties of nilpotent modular categories

GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK To Yuri Ivanovich Manin on his 70th birthday Abstract. We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted dou- bles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical in the sense of [ENO]. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs given in [Dr]). 1. introduction In this paper we work over an algebraically closed field k of characteristic 0. By a fusion category we mean a k-linear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects, finite dimensional spaces of morphisms, and such that the unit object 1 of C is simple. We refer the reader to [ENO] for a general theory of such categories. A fusion category is pointed if all its simple objects are invertible. A pointed fusion category is equivalent to VecωG, i.e., the category of G-graded vector spaces with the associativity constraint given by some cocycle ω ∈ Z3(G, k×) (here G is a finite group). 1.1. Main results. Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition into a tensor product of braided fusion categories whose Frobenius-Perron dimen- sions are powers of distinct primes. The notion of nilpotent fusion category was introduced in [GN]; we recall it in Subsection 2.2. Let us mention that the representation category Rep(G) of a finite group G is nilpotent if and only if G is nilpotent. It is also known that fusion categories of prime power Frobenius-Perron dimension are nilpotent [ENO]. On the other hand, VecωG is nilpotent for any G and ω. Therefore it is not true that any nilpotent fusion category is a tensor product of fusion categories of prime power dimensions. Theorem 1.2. A modular category C with integral dimensions of simple objects is nilpotent if and only if there exists a pointed modular category M such that C⊠M Date: March 31, 2007. http://arxiv.org/abs/0704.0195v2 2 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK is equivalent, as a braided tensor category, to the center of a fusion category of the form VecωG for a finite nilpotent group G. We emphasize here that in general the equivalence in Theorem 1.2 does not respect the spherical structures (equivalently, twists) of the categories involved and thus is not an equivalence of modular categories. Fortunately, this is not a very serious complication since the spherical structures on C are easy to classify: it is well known that they are in bijection with the objects X ∈ C such that X⊗X = 1, see [RT]. The categoryM in Theorem 1.2 is not uniquely determined by C. However, there are canonical ways to choose M. In particular, one can always make a canonical “minimal” choice for M such that dim(M) = αp with αp ∈ {0, 1, 2} for odd p and α2 ∈ {0, 1, 2, 3}, see Remark 6.11. Theorem 1.3. A modular category C is braided equivalent to the center of a fusion category of the form VecωG with G being a finite p-group if and only if it has the following properties: (i) the Frobenius-Perron dimension of C is p2n for some n ∈ Z+, (ii) the dimension of every simple object of C is an integer, (iii) the multiplicative central charge of C is 1. See Subsection 2.6 for the definition of multiplicative central charge. In order to avoid confusion we note that our definition of multiplicative central charge is different from the definition of central charge of a modular functor from [BK, 5.7.10]; in fact, the central charge from [BK] equals to the square of our central charge. Remark 1.4. If p 6= 2 then it is easy to see that (i) implies (ii) (see, e.g., [GN]). 1.2. Interpretation in terms of group-theoretical fusion categories and semisimple quasi-Hopf algebras. The notion of group-theoretical fusion cate- gory was introduced in [ENO, O1]. Group-theoretical categories form a large class of well-understood fusion categories which can be explicitly constructed from finite group data (which justifies the name). For example, as far as we know, all currently known semisimple Hopf algebras have group-theoretical representation categories (however, there are semisimple quasi-Hopf algebras whose representation categories are not group-theoretical, see [ENO]). Theorem 1.5. Let C be a fusion category such that all objects of C have integer dimension and such that its center Z(C) is nilpotent. Then C is group-theoretical. Remark 1.6. A consequence of this theorem is the following statement: every semisimple (quasi-)Hopf algebra of prime power dimension is group-theoretical in the sense of [ENO, Definition 8.40]. This provides a partial answer to a question asked in [ENO]. 1.3. Idea of the proof. We describe here the main steps in the proof of The- orem 1.3. First we characterize centers of pointed fusion categories in terms of Lagrangian subcategories and show that a modular category C is equivalent to the representation category of a twisted group double if and only if it has a Lagrangian (i.e., maximal isotropic) subcategory of dimension dim(C). This result is remi- niscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs [Dr, Section 2]. GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 3 Thus we need to show that a category satisfying the assumptions of Theorem 1.3 contains a Lagrangian subcategory. The proof is inspired by the following result for nilpotent metric Lie algebras (i.e, Lie algebras with an invariant non-degenerate scalar product) which can be derived from [KaO]: if g is a nilpotent metric Lie algebra of even dimension then g contains an abelian ideal k, which is Lagrangian (i.e., such that k⊥ = k). The relevance of metric Lie algebras to our considerations is explained by the fact that they appear in [Dr] as classical limits of quasi-Hopf algebras. In fact, our proof is a “categorification” of the proof of the above result. Thus we need some categorical versions of linear algebra constructions involved in this proof. Remarkably, the categorical counterparts exist for all notions required. For example the notion of orthogonal complement in a metric Lie algebra is replaced by the notion of centralizer in a modular tensor category introduced by M. Müger [Mu2]. 1.4. Organization of the paper. Section 2 is devoted to preliminaries on fusion categories, which include nilpotent fusion categories, (pre)modular categories, cen- tralizers, Gauss sums and central charge, and Deligne’s classification of symmetric fusion categories. In Section 3 we define the notions of isotropic and Lagrangian subcategories of a premodular category C, generalizing the corresponding notions for a metric group (which is, by definition, a finite abelian group with a quadratic form). We then recall a construction, due to A. Bruguières [Br] and M. Müger [Mu1], which associates to a premodular category C the “quotient” by its centralizer, called a modularization. We prove in Theorem 3.4 an invariance property of the central charge with respect to the modularization. This result will be crucial in the proof of Theorem 6.5. We also study properties of subcategories of modular categories and explain in Proposition 3.9 how one can use maximal isotropic subcategories of a modular category C to canonically measure a failure of C to be hyperbolic (i.e., to contain a Lagrangian subcategory). In Section 4 we characterize hyperbolic modular categories. More precisely, we show in Theorem 4.5 that for a modular category C there is a bijection between La- grangian subcategories of C and braided tensor equivalences C ∼−→ Z(VecωG) (where G is a finite group, ω ∈ Z3(G,K×), and Z(VecωG) is the center of VecωG). Note that the category Z(VecωG) is equivalent to Rep(Dω(G)) - the representation category of the twisted double of G [DPR]. We then prove in Theorem 4.8 that if C is a modular category such that dim(C) = n2, n ∈ Z+, the central charge of C equals 1, and C contains a symmetric subcate- gory of dimension n, then either C is equivalent to the representation category of a twisted double of a finite group or C contains an object with non-integer dimension. We also give a criterion for a modular category C to be group-theoretical. Namely, we show in Corollary 4.13 that C is group-theoretical if and only if there is an isotropic subcategory E ⊂ C such that (E ′)ad ⊆ E . In Section 5 we study pointed modular p-categories. We give a complete list of such categories which do not contain non-trivial isotropic subcategories and analyze the values of their central charges. We then prove in Proposition 5.3 that a nonde- generate metric p-group (G, q) with central charge 1 such that |G| = p2n, n ∈ Z+, contains a Lagrangian subgroup. 4 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK Section 6 is devoted to nilpotent modular categories. There we give proofs of our main results stated in 1.1 above. They are contained in Theorem 6.5, Theorem 6.6, Corollary 6.7, Theorem 6.10, and Theorem 6.12. 1.5. Acknowledgments. The research of V. Drinfeld was supported by NSF grant DMS-0401164. The research of D. Nikshych was supported by the NSF grant DMS- 0200202 and the NSA grant H98230-07-1-0081. The research of V. Ostrik was sup- ported by NSF grant DMS-0602263. S. Gelaki is grateful to the departments of mathematics at the University of New Hampshire and MIT for their warm hospi- tality during his Sabbatical. The authors are grateful to Pavel Etingof for useful discussions. 2. Preliminaries Throughout the paper we work over an algebraically closed field k of character- istic 0. All categories considered in this paper are finite, abelian, semisimple, and k-linear. 2.1. Fusion categories. For a fusion category C let O(C) denote the set of iso- morphism classes of simple objects. Let C be a fusion category. Its Grothendieck ring K0(C) is the free Z-module generated by the isomorphism classes of simple objects of C with the multiplication coming from the tensor product in C. The Frobenius-Perron dimensions of objects in C (respectively, FPdim(C)) are defined as the Frobenius-Perron dimensions of their images in the based ring K0(C) (respectively, as FPdim(K0(C))), see [ENO, 8.1]. For a semisimple quasi-Hopf algebra H one has FPdim(X) = dimk(X) for all X in Rep(H), and so FPdim(Rep(H)) = dimk(H). A fusion category is pointed if all its simple objects are invertible. By a fusion subcategory of a fusion category C we understand a full tensor subcat- egory of C. An example of a fusion subcategory is the maximal pointed subcategory Cpt generated by the invertible objects of C. A fusion category C is pseudo-unitary if its categorical dimension dim(C) coin- cides with its Frobenius-Perron dimension, see [ENO] for details. In this case C admits a canonical spherical structure (a tensor isomorphism between the iden- tity functor of C and the second duality functor) with respect to which categori- cal dimensions of objects coincide with their Frobenius-Perron dimensions [ENO, Proposition 8.23]. The fact important for us in this paper is that a fusion category of an integer Frobenius-Perron dimension is automatically pseudo-unitary [ENO, Proposition 8.24]. Let C and D be fusion categories. Recall that for a tensor functor F : C → D its image F (C) is the fusion subcategory of D generated by all simple objects Y in D such that Y ⊆ F (X) for some simple X in C. The functor F is called surjective if F (C) = D. 2.2. Nilpotent fusion categories. For a fusion category C let Cad be the trivial component in the universal grading of C (see [GN]). Equivalently, Cad is the smallest fusion subcategory of C which contains all the objects X ⊗X∗, X ∈ O(C). For a fusion category C we define C(0) = C, C(1) = Cad, and C(n) = (C(n−1))ad for every integer n ≥ 1. The non-increasing sequence of fusion subcategories of C (1) C = C(0) ⊇ C(1) ⊇ · · · ⊇ C(n) ⊇ · · · GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 5 is called the upper central series of C. We say that a fusion category C is nilpotent if every non-trivial subcategory of C has a non-trivial group grading, see [GN]. Equiv- alently, C is nilpotent if its upper central series converges to Vec (the category of finite dimensional k−vector spaces), i.e., C(n) = Vec for some n. The smallest such n is called the nilpotency class of C. If C is nilpotent then every fusion subcategory E ⊂ C is nilpotent, and if F : C → D is a surjective tensor functor, then D is nilpotent (see [GN]). Example 2.1. (1) Let G be a finite group and C = Rep(G). Then C is nilpo- tent if and only if G is nilpotent. (2) Pointed categories are precisely the nilpotent fusion categories of nilpotency class 1. A typical example of a pointed category is VecωG, the category of finite dimensional vector spaces graded by a finite group G with the associativity constraint determined by ω ∈ Z3(G, k×). In this paper we are especially interested in the following class of nilpotent fusion categories. Example 2.2. Let p be a prime number. Any category of dimension pn, n ∈ Z, is nilpotent by [ENO, Theorem 8.28]. For representation categories of semisimple Hopf algebras of dimension pn this follows from a result of A. Masuoka [Ma1]. By [GN], a nilpotent fusion category comes from a sequence of gradings, in particular it has an integer Frobenius-Perron dimension. It follows from results of [ENO] that a nilpotent fusion category C is pseudounitary. 2.3. Premodular categories and modular categories. Recall that a braided tensor category C is a tensor category equipped with a natural isomorphism c : ⊗ ∼= ⊗rev satisfying the hexagon diagrams [JS]. Let cXY : X ⊗ Y ∼= Y ⊗X with X,Y ∈ C denote the components of c. A balancing transformation, or a twist, on a braided category C is a natural automorphism θ : idC → idC satisfying θ1 = id1 and (2) θX⊗Y = (θX ⊗ θY )cY XcXY . A braided fusion category C is called premodular, or ribbon, if it has a twist θ satisfying θ∗X = θX∗ for all objects X ∈ C. The S-matrix of a premodular category C is S = {sXY }X,Y ∈O(C), where sXY is the quantum trace of cYXcXY , see [T]. Equivalently, the S-matrix can be defined as follows. For all X,Y, Z ∈ O(C) let NZXY be the multiplicity of Z in X ⊗ Y . For every object X let d(X) denote its quantum dimension. Then (3) sXY = θ Z∈O(C) NZXY θZd(Z). The categorical dimension of C is defined by (4) dim(C) = X∈O(C) d(X)2. One has dim(C) 6= 0 [ENO, Theorem 2.3]. Note 2.3. Below we consider only fusion categories with integer Frobenius-Perron dimensions of objects. Any such category C is pseudo-unitary (see 2.1). In par- ticular, if C is braided then it has a canonical twist, which we will always assume chosen. 6 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK A premodular category C is called modular if the S−matrix is invertible. Example 2.4. For any fusion category C its center Z(C) is defined as the category whose objects are pairs (X, cX,−), where X is an object of C and cX,− is a natural family of isomorphisms cX,V : X ⊗ V ∼= V ⊗ X for all objects V in C satisfying certain compatibility conditions (see e.g., [Kass]). It is known that the center of a pseudounitary category is modular. 2.4. Pointed modular categories and metric groups. Let G be a finite abelian group. Pointed premodular categories C with the group of simple objects isomorphic to G (up to a braided equivalence) are in the natural bijection with quadratic forms on G with values in the multiplicative group k∗ of the base field. Here a quadratic form q : G → k∗ is a map such that q(g−1) = q(g) and b(g, h) := q(gh) q(g)q(h) a symmetric bilinear form, i.e., b(g1g2, h) = b(g1, h)b(g2, h) for all g1, g2, h ∈ G. Namely, for g ∈ G the value of q(g) is the braiding automorphism of g ⊗ g (here by abuse of notation g denotes the object of C corresponding to g ∈ G). See [Q, Proposition 2.5.1] for a proof that if two categories C1 and C2 produce the same quadratic form then they are braided equivalent (Quinn proves less canonical but equivalent statement). We will denote the category corresponding to a group G with quadratic form q by C(G, q) and call the pair (G, q) a metric group. The category C(G, q) is pseudounitary and hence has a spherical structure such that dimensions of all simple objects equal to 1; hence the categories C(G, q) always have a canonical ribbon structure. The category C(G, q) is modular if and only if the bilinear form b(g, h) associated with q is non-degenerate (in this case we will say that the corresponding metric group is non-degenerate). 2.5. Centralizers. Let K be a fusion subcategory of a braided fusion category C. In [Mu1, Mu2] M. Müger introduced the centralizer K′ of K, which is the fusion subcategory of C consisting of all the objects Y satisfying (5) cY XcXY = idX⊗Y for all objects X ∈ K. If (5) holds we will say that objects X and Y centralize each other. In the case of a ribbon category C, condition (5) is equivalent to sXY = d(X)d(Y ), see [Mu2, Proposition 2.5]. Note that in the case of a pointed modular category the centralizer corresponds to the orthogonal complement. The subcategory C′ of C is called the transparent subcategory of C in [Br, Mu1]. For any fusion subcategory K ⊆ C of a braided fusion category C let Kco be the commutator of K [GN], i.e., the fusion subcategory of C spanned by all simple objects X ∈ C such that X ⊗ X∗ ∈ K. For example, if C = Rep(G), G a finite group, then any fusion subcategory K of C is of the form K = Rep(G/N) for some normal subgroup N of G, and Kco = Rep(G/[G,N ]) (see [GN]). It follows from the definitions that (Kco)ad ⊆ K ⊆ (Kad)co. Let K be a fusion subcategory of a pseudounitary modular category C. It was shown in [GN] that (6) (Kad)′ = (K′)co. It was shown in [Mu2, Theorem 3.2] that for a fusion subcategory K of a modular category C one has K′′ = K and (7) dim(K) dim(K′) = dim(C). GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 7 The subcategory K is symmetric if and only if K ⊆ K′. It is modular if and only if K ∩ K′ = Vec, in which case K′ is also modular and there is a braided equivalence C ∼= K ⊠K′. Let C be a modular category. Then by [GN, Corollary 6.9], Cpt = (Cad)′. 2.6. Gauss sums and central charge in modular categories. Let C be a modular category. For any subcategory K of C the Gauss sums of K are defined by (8) τ±(K) = X∈O(K) θ±1X d(X) Below we summarize some basic properties of twists and Gauss sums (see e.g., [BK, Section 3.1] for proofs). Each θX , X ∈ O(C), is a root of unity (this statement is known as Vafa’s theo- rem). The Gauss sums are multiplicative with respect to tensor product of modular categories, i.e., if C1, C2 are modular categories then (9) τ±(C1 ⊠ C2) = τ±(C1)τ±(C2). We also have that (10) τ+(C)τ−(C) = dim(C). When k = C the multiplicative central charge ξ(C) is defined by (11) ξ(C) = τ dim(C) where dim(C) is the positive root. If dim(C) is a square of an integer, then Formula (11) makes sense even if k 6= C. By Vafa’s theorem, ξ(C) is a root of unity. Example 2.5. The center Z(C) of any fusion category C (see Example 2.4) is a modular category with central charge 1 [Mu4, Theorem 1.2]. 2.7. Symmetric fusion categories. The structure of symmetric fusion categories is known, thanks to Deligne’s work [De]. Namely, let G be a finite group and let z ∈ G be a central element such that z2 = 1. Consider the category Rep(G) with its standard symmetric braiding σX,Y . Then the map σ X,Y = (1 + z|X + z|Y − z|Xz|Y )σX,Y is also a symmetric braiding on the category Rep(G) (the meaning of the factor 1 (1 + z|X + z|Y − z|Xz|Y ) is the following: if z|X or z|Y equals 1, then this factor is 1; if z|X = z|Y = −1 then this factor is (−1)). We will denote by Rep(G, z) the category Rep(G) with the commutativity constraint defined above. Theorem 2.6. ([De]) Any symmetric fusion category is equivalent (as a braided tensor category) to Rep(G, z) for uniquely defined G and z. The categorical dimen- sion of X ∈ Rep(G, z) equals Tr(z|X) and dim(C) = FPdim(C) = |G|. Now assume that the category Rep(G, z) is endowed with a twist θ such that the dimension of any object is non-negative. It follows immediately from the theorem that θX = z|X . We have Corollary 2.7. Let C be a symmetric fusion category with the canonical spherical structure (see 2.3). (i) If dim(C) is odd then θX = idX for any X ∈ C. 8 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK (ii) In general either θX = idX for any X ∈ C, or C contains a fusion subcat- egory C1 ⊂ C such that FPdim(C1) = 12FPdim(C) and θX = idX for any X ∈ C1. Proof. As for (i), it is clear that z = 1. For (ii) one takes C1 = Rep(G/〈z〉) ⊂ Rep(G). � 3. Isotropic subcategories and Bruguières-Müger modularization 3.1. Modularization. Definition 3.1. Let C be a premodular category with braiding c and twist θ. A fusion subcategory E of C is called isotropic if θ restricts to the identity on E , i.e., if θX = idX for all X ∈ E . An isotropic subcategory E is called Lagrangian if E = E ′. The category C is called hyperbolic if it has a Lagrangian subcategory and anisotropic if it has no non-trivial isotropic subcategories. Remark 3.2. (a) When C = C(G, q) is a pointed modular category defined in Example 2.4 then isotropic and Lagrangian subcategories of C correspond to isotropic and Lagrangian subgroups of (G, q), respectively. We discuss properties of pointed modular categories in Section 5. (b) Let G be a finite group and let ω ∈ Z3(G, k×). Consider the pointed fusion category VecωG. Its center C = Z(VecωG) is a modular category. It contains a Lagrangian subcategory E ∼= Rep(G) formed by all objects in C which are sent to multiples of the unit object of VecωG by the forgetful functor Z(VecωG) → Vec (c) It follows from the balancing axiom (2) that an isotropic subcategory E ⊆ C is always symmetric. Conversely, if E is symmetric and dim(E) is odd then E is isotropic, see 2.7. In particular, if dim(C) is odd then any symmetric subcategory of C is isotropic. (d) Recall that we assume that C is endowed with a canonical spherical struc- ture, see 2.3. Any isotropic subcategory E ⊂ C is equivalent, as a symmetric category, to Rep(G) for a canonically defined group G with its standard braiding and identical twist, see 2.7. In particular, if E is Lagrangian then dim(C) = dim(E)2 is a square of an integer. Let C be a premodular category such that its centralizer C′ is isotropic and dimensions of all objects X ∈ C′ are non-negative. Let us recall a construction, due to A. Bruguières [Br] and M. Müger [Mu1], which associates to C a modular category C̄ and a surjective braided tensor functor C → C̄. Let G(C) be the unique (up to an isomorphism) group such that the category C′ is equivalent, as a premodular category, to Rep(G(C)) with its standard symmetric braiding and identity twist. Let A be the algebra of functions on G(C). The group G(C) acts on A via left translations and so A is a commutative algebra in C′ and hence in C. Consider the category C̄ := CA of right A-modules in the category C (see, e.g., [KiO, 1.2]). It was shown in [Br, KiO, Mu1] that C̄ is a braided fusion category and that the “free module” functor (12) F : C → C̄, X 7→ X ⊗A is surjective and has a canonical structure of a braided tensor functor. One can define a twist φ on C̄ in such a way that φY = θX for all Y ∈ O(C̄) and X ∈ GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 9 O(C) for which HomC(X,Y ) 6= 0. It follows that the category C̄ is modular, see [Br, Mu1, KiO] for details. We will call the category C̄ a modularization of C. Let d and d̄ denote the dimension functions in C and C̄, respectively. For any object X in C̄ one has (13) d̄(X) = cf. [KiO, Theorem 3.5], [Br, Proposition 3.7]. Remark 3.3. Let E be an isotropic subcategory of a modular category C. Then dim(Ē ′) = dim(C)/ dim(E)2 (see e.g. [KiO]). 3.2. Invariance of the central charge. In this subsection we prove an invariance property of the central charge with respect to modularization, which will be crucial in the sequel. Theorem 3.4. Let C be a modular category and let E be an isotropic subcategory of C. Let F : E ′ → Ē ′ be the canonical braided tensor functor from E ′ to its modularization. Then ξ(Ē ′) = ξ(C). Proof. Let A be the canonical commutative algebra in E . We have dim(E) = d(A). By definition, Ē ′ is the category of left A-modules in E ′. Let us compute the Gauss sums of Ē ′: dim(E)τ±(Ē ′) = dim(E) Y ∈O(Ē′) φ±1Y d̄(Y ) Y ∈O(Ē′) φ±1Y d(Y )d̄(Y ) Y ∈O(Ē′) X∈O(C) dimk HomC(X, Y )d(X)  d̄(Y ) X∈O(C) θ±1X d(X) Y ∈O(Ē′) dimk HomC(X, Y )d̄(Y ) X∈O(C) θ±1X d(X) Y ∈O(Ē′) dimk HomĒ′(X ⊗A, Y )d̄(Y ) X∈O(C) θ±1X d(X)d̄(F (X)) = τ ±(C), where we used the relation (13) and the fact that F is an adjoint of the forgetful functor from Ē ′ to E ′. Combining this with the equation dim(Ē ′) = dim(C)/ dim(E)2 (see Remark 3.3) we obtain the result. � 3.3. Maximal isotropic subcategories. Let C be a modular category and let L be an isotropic subcategory of C which is maximal among isotropic subcategories of C. Below we will show that the braided equivalence class of the modular category L̄′ (the modularization of L′ by L) is independent of the choice of L. Let C be a fusion category and let A and B be its fusion subcategories such that X ⊗ Y ∼= Y ⊗ X for all X ∈ O(A) and Y ∈ O(B). Let A ∨ B denote the fusion 10 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK subcategory of C generated by A and B, i.e., consisting of all subobjects of X ⊗ Y , where X ∈ O(A) and Y ∈ O(B). Recall that the regular element of K0(C)⊗Z C is X∈O(C) d(X)X . It is defined up to a scalar multiple by the property that Y ⊗RC = d(Y )RC for all Y ∈ O(C) [ENO]. Lemma 3.5. Let C, A, B be as above. Then dim(A ∨ B) = dim(A) dim(B) dim(A∩B) . Proof. It is easy to see that (14) RA ⊗RB = aRA∨B, where the scalar a is equal to the multiplicity of the unit object 1 in RA ⊗ RB, which is the same as the multiplicity of 1 in Z∈O(A∩B) d(Z) 2Z ⊗ Z∗. Hence, a = dim(A ∩ B). Taking dimensions of both sides of (14) we get the result. � Let L(C) denote the lattice of fusion subcategories of a fusion category C. For any two subcategories A and B their meet is their intersection and their joint is the category A ∨ B. Lemma 3.6. Let C be a fusion category such that X ⊗ Y ∼= Y ⊗X for all objects X,Y in C. For all A, B, D ∈ L(C) such that D ⊆ A the following modular law holds true: (15) A ∩ (B ∨D) = (A ∩ B) ∨ D. Proof. A classical theorem of Dedekind in lattice theory states that (15) is equiv- alent to the following statement: for all A, B, D ∈ L(C) such that D ⊆ A, if A∩ B = D ∩ B and A∨ B = D ∨ B then A = D (see e.g., [MMT]). Let us prove the latter property. Take a simple object X ∈ A. Then X ∈ A ∨ B = D ∨ B so there are simple objects D ∈ D and B ∈ B such that X is contained in D ⊗ B. Therefore, B is contained in D∗ ⊗ X and so B ∈ A. So B ∈ A ∩ B = D ∩ B ⊆ D. Hence X ∈ D, as required. � Remark 3.7. When C = Rep(G) is the representation category of a finite group G, Lemma 3.6 gives a well-known property of the lattice of normal subgroups of G. The next lemma gives an analogue of a diamond isomorphism for the “quotients by isotropic subcategories.” Lemma 3.8. Let C be a modular category, let D be an isotropic subcategory of C and let B be a subcategory of D′. Let A, A0 be the canonical commutative algebras in D and D ∩ B, respectively. Then the category BA0 of A0-modules in B and the category (D ∨ B)A of A- modules in D ∨ B are equivalent as braided tensor categories. Proof. Note that dim(BA0) = dim(B) dim(D ∩ B) dim(D ∨ B) dim(D) = dim((D ∨ B)A) by Lemma 3.5. Define a functorH : BA0 → (D∨B)A byH(X) = X⊗A0A, X ∈ BA0 . ThenH has a natural structure of a braided tensor functor. Note that for X = Y ⊗A0, Y ∈ B we have H(X) = Y ⊗A, i.e., the composition of H with the free A0-module functor is the free A-module functor. The latter functor is surjective and, hence, so is H . Since a surjective functor between categories of equal dimension is necessarily an equivalence (see [ENO, 5.7] or [EO, Proposition 2.20]) the result follows. � GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 11 Proposition 3.9. Let C be a modular category and let L1, L2 be maximal among isotropic subcategories of C. Then the modularization L̄′1 and L̄′2 are equivalent as braided fusion categories. Proof. Let D = L1 and B = L′1∩L′2. By maximality of L1,L2 we have L′1∩L2 ⊆ L1 and L1 ∩ L′2 ⊆ L2. Therefore, D ∩ B = L1 ∩ L2 and D ∨ B = L′1 ∩ (L1 ∨ L′2) = L′1 by Lemma 3.6. Let A0 be the canonical commutative algebra in L1 ∩ L2. Applying Lemma 3.8 we see that L̄′1 is equivalent to the category (L′1 ∩L′2)A0 of A0-modules in L′1 ∩L′2. The proposition now follows by interchanging L1 and L2. � Remark 3.10. (i) We can call the modular category L̄′1 constructed in the proof of Proposition 3.9 “the” canonical modularization corresponding to C (it measures the failure of C to be hyperbolic). The above proof gives a concrete equivalence L̄′1 ∼= L̄′2. But given another maximal isotropic subcategory L3 ⊂ C the composition of equivalences L̄′1 ∼= L̄′2 and L̄′2 ∼= L̄′3 is not in general equal to the equivalence L̄′1 ∼= L̄′3. This is why we put “the” above in quotation marks. (ii) For a maximal isotropic subcategory L ⊂ C the corresponding modular- ization does not have to be anisotropic, in contrast with the situation for metric groups. Examples illustrating this phenomenon are, e.g., the cen- ters of non-group theoretical Tambara-Yamagami categories considered in [ENO, Remark 8.48]. 4. Reconstruction of a twisted group double from a Lagrangian subcategory 4.1. C-algebras. Let us recall the following definition from [KiO]. Definition 4.1. Let C be a ribbon fusion category. A C−algebra is a commutative algebra A in C such that dimHom(1, A) = 1, the pairing A⊗A → A → 1 given by the multiplication of A is non-degenerate, θA = idA and dim(A) 6= 0. Let C be a modular category, let A be a C−algebra, and let CA be the fusion category of right A−modules with the tensor product ⊗A. The free module functor F : C → CA, X 7→ X ⊗A has an obvious structure of a central functor. By this we mean that there is a natural family of isomorphisms F (X)⊗AY ∼= Y ⊗AF (X), X ∈ C, Y ∈ CA, satisfying an obvious multiplication compatibility, see e.g. [Be, 2.1]. Indeed, we have F (X) = X ⊗ A, and hence F (X) ⊗A Y = X ⊗ Y . Similarly, Y ⊗A F (X) = Y ⊗ X . These two objects are isomorphic via the braiding of C (one can check that the braiding gives an isomorphism of A-modules using the commutativity of A). Thus, the functor F extends to a functor F̃ : C → Z(CA) in such a way that F is the composition of F̃ and the forgetful functor Z(CA) → CA. Proposition 4.2. The functor F̃ : C → Z(CA) is injective (that is fully faithful). Proof. Consider CA as a module category over C via F and over Z(CA) via F̃ . We will prove the dual statement (see [ENO, Proposition 5.3]), namely that the functor T : CA ⊠ CopA → C∗CA dual to F̃ is surjective (here and below the superscript op refers to the tensor category with the opposite tensor product). Recall (see e.g. [O1]) that the category C∗CA is identified with the category of A−bimodules. An 12 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK explicit description of the functor T is the following: by definition, any M ∈ CA is a right A−module. Using the braiding and its inverse one can define on M two structures of a left A−module: A ⊗ M A,M−→ M ⊗ A → M . Both structures make M into an A−bimodule, and we will denote the two results by M+ and M−, respectively. Then we have T (M ⊠ N) = M+ ⊗A N−. In particular we see that the functor C ⊠ Cop F⊠F−→ CA ⊠ CopA T−→ C∗CA coincides with the functor C ⊠ Cop ≃ Z(C) ≃ Z(C∗CA) → C CA (see [O2]). Since the functor Z(C CA) → C CA is surjective (see [EO, 3.39]) we see that the functor T is surjective. The proposition is proved. � Remark 4.3. Note that since C and Z(CA) are modular we have a factorization Z(CA) = C ⊠D, where D is the centralizer of C in Z(CA). One observes that D is identified with the category of “dyslectic” A−modules Rep0(A), see [KiO, P]. Corollary 4.4. Assume that dim(A) = dim(C). Then the functors F̃ : C → Z(CA) and T : CA ⊠ CopA → C∗CA are tensor equivalences. Proof. We have already seen that dim(CA) = dim(C)dim(A) . Hence, dim(Z(CA)) = dim(C)2 dim(A)2 = dim(C). Since F̃ is an injective functor between categories of equal dimension, it is necessarily an equivalence by [EO, Proposition 2.19]. Hence the dual functor T is also an equivalence. � 4.2. Hyperbolic modular categories as twisted group doubles. We are now ready to state and prove our first main result which relates hyperbolic modular categories and twisted doubles of finite groups. Let C be a modular category. Consider the set of all triples (G,ω, F ), where G is a finite group, ω ∈ Z3(G, k×), and F : C ∼−→ Z(VecωG) is a braided tensor equivalence. Let us say that two triples (G1, ω1, F1) and (G2, ω2, F2) are equivalent if there exists a tensor equivalence ι : Vecω1G1 ∼−→ Vecω2G2 such that F2◦F2 = ι◦F1◦F1, where Fi : Z(VecωiGi) → Vec , i = 1, 2, are the canonical forgetful functors. Let E(C) be the set of all equivalences classes of triples (G,ω, F ). Let Lagr(C) be the set of all Lagrangian subcategories of C. Theorem 4.5. For any modular category C there is a natural bijection f : E(C) ∼−→ Lagr(C). Proof. The map f is defined as follows. Note that each braided tensor equivalence F : C ∼−→ Z(VecωG) gives rise to the Lagrangian subcategory f(G,ω, F ) of C formed by all objects sent to multiples of the unit object 1 under the forgetful functor Z(VecωG) → VecωG. This subcategory is clearly the same for all equivalent choices of (G,ω, F ). Conversely, given a Lagrangian subcategory E ⊆ C it follows from Deligne’s theorem [De] that E = Rep(G) for a unique (up to isomorphism) finite group G. Let A = Fun(G) ∈ Rep(G) = E ⊂ C. It is clear that A is a C−algebra and dim(A) = dim(E) = dim(C). Then by Corollary 4.4, the functor F̃ : C → Z(CA) is an equivalence. Finally, let us show that CA is pointed and K0(CA) = ZG. Note that there are |G| non-isomorphic structures Ag, g ∈ G, of an invertible A-bimodule on A, since the category of A-bimodules in E is equivalent to VecG. For each Ag there is a GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 13 pair X,Y of simple objects in CA such that T (X ⊠ Y ) = Ag. Taking the forgetful functor to CA we obtain Y = X∗ and X is invertible. Hence, for each g ∈ G there is a unique invertible Xg ∈ CA such that T (Xg ⊠X∗g ) = Ag, and therefore g 7→ Xg is an isomorphism of K0 rings. Thus, CA ∼= VecωG for some ω ∈ Z3(G, k×). We set h(E) to be the class of the equivalence F̃ : C ∼−→ Z(CA). Let show that the above constructions f and h are inverses of each other. Let E be a Lagrangian subcategory of C and let A be the algebra defined in the previous paragraph. The forgetful functor from C ∼= Z(CA) to CA is the free module functor, and so f(h(E)) consists of all objects X in C such that X ⊗ A is a multiple of A. Since A is the regular object of E , it follows that f(h(E)) = E and f ◦ h = id. Proving that h ◦ f = id amounts to a verification of the following fact. Let G be a finite group, let ω ∈ Z3(G, k×), and let A = Fun(G) be the canonical algebra in Rep(G) ⊂ Z(VecωG). Then the category of A-modules in Z(VecωG) is equivalent to VecωG and the functor of taking the free A-module coincides with the forgetful functor from Z(VecωG) to Vec G. This is straightforward and is left to the reader. � Remark 4.6. Our reconstruction of the representation category of a twisted group double from a Lagrangian subcategory can be viewed as a categorical analogue of the following reconstruction of the double of a quasi-Lie bialgebra from a Manin pair (i.e., a pair consisting of a metric Lie algebra and its Lagrangian subalgebra) in the theory of quantum groups [Dr, Section 2]. Let g be a finite-dimensional metric Lie algebra (i.e., a Lie algebra on which a nondegenerate invariant symmetric bilinear form is given). Let l be a Lagrangian subalgebra of g. Then l has a structure of a quasi-Lie bialgebra and there is an isomorphism between g and the double D(l) of l. The correspondence between Lagrangian subalgebras of g and doubles isomorphic to g is bijective, see [Dr, Section 2] for details. Remark 4.7. Given a hyperbolic modular category C there is no canonical way to assign to it a pair (G, ω) such that C ∼= Z(VecωG) as a braided fusion category. Indeed, it follows from [EG1] that there exist non-isomorphic finite groups G1, G2 such that Z(VecG1) ∼= Z(VecG2) as braided fusion categories. (See also [N].) Theorem 4.8. Let C be a modular category such that dim(C) = n2, n ∈ Z+, and such that ξ(C) = 1. Assume that C contains a symmetric subcategory V such that dim(V) = n. Then either C is the center of a pointed category or it contains an object with non-integer dimension. Proof. Assume that V is not isotropic. Then V contains an isotropic subcategory K such that dim(K) = 1 dim(V) (this follows from Deligne’s description of symmetric categories, see 2.7). Hence the category K̄′ (modularization of K′) has dimension 4 and central charge 1. It follows from the explicit classification given in Example 5.1 (b),(d) that the category K̄′ contains an isotropic subcategory of dimension 2; clearly this subcategory is equivalent to Rep(Z/2Z). Let A1 = Fun(Z/2Z) be the commutative algebra of dimension 2 in this subcategory. Let I : K̄′ → K′ be the right adjoint functor to the modularization functor F : K′ → K̄′. We claim that the object A := I(A1) has a canonical structure of a C−algebra. Indeed, we have a canonical morphism in Hom(F (A), A1) = Hom(A, I(A1)) = Hom(A,A) ∋ id. Using this one can construct a multiplication on A via Hom(A1⊗ A1, A1) → Hom(F (A)⊗F (A), A1) = Hom(F (A⊗A), A1) = Hom(A⊗A,A). Since 14 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK the functor F is braided it follows from the commutativity of A1 that A is commu- tative. Other conditions from Definition 4.1 are also easy to check. In particular dim(A) = dim(K) dim(A1) = dim(V) = dim(C). We also note that the category RepK′(A) contains precisely two simple objects (actually, the functor M 7→ I(M) is an equivalence of categories between RepK̄′(A1) and RepK′(A)); we will call these two objects 1 (for A itself considered as an A−module) and δ. Clearly δ⊗A δ = 1. By Corollary 4.4, we have an equivalence C∗CA = CA⊠C A . Moreover, the forgetful functor C∗CA → CA corresponds to the tensor product functor CA ⊠ C A → CA. Now consider the subcategory (K′)∗CA ⊂ C CA (in other words A−bimodules in K ′); the forgetful functor above restricts to S : (K′)∗CA → RepK′(A). Let M ∈ (K′)∗CA be a simple object. We claim that there are three possibilities: 1) S(M) = 1, 2) S(M) = δ or 3) S(M) = 1⊕ δ. Indeed, M = X ⊠ Y ∈ CA ⊠ CopA and S(M) = X ⊗ Y for some simple X,Y ∈ CA. Since 1 and δ are invertible the result is clear. Now, notice that if there exists M as in case 3) then we have X = Y ∗ and dim(X) = dim(Y ) = 2. Thus the category CA contains an object with non-integer dimension, which implies that the category C contains an object with non-integer dimension (see e.g. [ENO, Corollary 8.36]), and the theorem is proved in this case. Hence we will assume that for any M ∈ (K′)∗CA only 1) or 2) holds. This implies that all objects of (K′)∗CA are invertible. Note that dim((K ′)∗CA) = dim(K dim(C) and hence we have precisely 2 dim(C) simple objects. Consider all objects M ∈ (K′)∗CA such that S(M) = 1; it is easy to see that there are precisely dim(C) of those (indeed, X⊠Y 7→ X⊠ (Y ⊗A δ) gives a bijection between simple bimodules M with S(M) = 1 and simple bimodules M with S(M) = δ). Let G be the group of isomorphism classes of all objects M ∈ (K′)∗CA with S(M) = 1 (thus |G| = dim(C)). Any object of this type is of the form Xg ⊠ (Xg)∗ for some invertible Xg ∈ CA. Thus we already constructed dim(C) invertible simple objects in CA. Since dim(CA) = dim(C) the objects Xg exhaust all simple objects in CA. By Corollary 4.4, we are done. � 4.3. A criterion for a modular category to be group-theoretical. Let C be a modular category. It is known that the entries of the S-matrix of C are cyclotomic integers [CG, dBG]. Hence, we may identify them with complex numbers. In particular, the notions of complex conjugation and absolute value of the elements of the S-matrix make sense. Remark 4.9. Let K ⊆ C be a fusion subcategory. Recall from [GN] that (Kad)′ is spanned by simple objects Y such that |sXY | = dXdY for all simple X in K. In this case the ratio b(X,Y ) := sXY /(dXdY ) is a root of unity. Furthermore, for all simple X ∈ K, Y1, Y2 ∈ K′ad and any simple subobject Z of Y1 ⊗ Y2 we have (16) b(X,Y1)b(X,Y2) = b(X,Z), as explained in [Mu2]. Lemma 4.10. Let C be a modular category and let K ⊆ C be a fusion subcategory such that K ⊆ (Kad)′. (1) There is a grading K = ⊕g∈G Kg such that K1 = K′ ∩ K. (2) There is a non-degenerate symmetric bilinear form b on G such that b(g, h) = sXY /(dXdY ) for all X ∈ Kg and Y ∈ Kh. GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 15 (3) If K′ ∩ K is isotropic then there is a non-degenerate quadratic form q on G such that q(g) = θX for all X ∈ Kg. In this case b is the bilinear form corresponding to q. Proof. Since Kad ⊆ K′ ∩ K ⊆ K the assertion (1) follows from [GN]. Let b(X,Y ) = sXY /(dXdY ) for all simple X,Y ∈ K. Clearly, b is symmetric and b(X,Y ) = 1 for all simple X in K if and only if Y ∈ K′ ∩ K = K1. To prove (2) it suffices to check that b depends only on h ∈ G such that Y ∈ Kh (then the G-linear property follows from (16)). Let Y1, Y2 be simple objects in Kh. Then Y1 ⊗ Y ∗2 ∈ K′ ∩ K and so b(X,Y1)b(X,Y ∗2 ) = 1, whence b(X,Y1) = b(X,Y2), as desired. Finally, (3) is a direct consequence of our discussion in Section 3.1. � For a subcategory K ⊆ C satisfying the hypothesis of Lemma 4.10 let (GK, bK) be the corresponding abelian grading group and bilinear form. Note that if such K is considered as a subcategory of Crev then the corresponding bilinear form is (GK, b Theorem 4.11. Let C be a modular category. Then symmetric subcategories of Z(C) ∼= C ⊠ Crev of dimension dim(C) are in bijection with triples (L, R, ι), where L ⊆ C, R ⊆ Crev are symmetric subcategories such that (L′)ad ⊆ L, (R′)ad ⊆ R, and ι : (GL′ , bL′) ∼= (GR′ , bR′) is an isomorphism of bilinear forms. Namely, any such subcategory is of the form (17) DL,R,ι = ⊕g∈G Lg ⊠Rι(g). Proof. Let X1 ⊠ Y1 and X2 ⊠ Y2 be two simple objects of C⊠ Crev. They centralize each other if and only if |sX1X2 | = dX1dX2 ,(18) |sY1Y2 | = dY1dY2 , and(19) sX1X2 dX1dX2 sY1Y2 dY1dY2 = 1.(20) Let D be a symmetric subcategory of C ⊠ Crev and let L (respectively, R) be the centralizers of fusion subcategories of C (respectively, Crev) formed by left (re- spectively, right) tensor factors of simple objects in D. By conditions (18), (19), and Remark 4.9 we must have L′ad ⊆ L and R′ad ⊆ R. Hence, Lemma 4.10 gives gradings L′ = ⊕g∈GL (L′)g with (L′)1 = L′ ∩ L and R′ = ⊕g∈GR (R′)g with (R′)1 = R′ ∩ R. The condition (20) gives an isomorphism of bilinear forms ι : (GL′ , bL′) ∼= (GR′ , bR′) which is well-defined be the property that whenever X ∈ (L′)g and Y ∈ R′ are simple objects such that X ⊠ Y ∈ D then Y ∈ (R′)ι(g). Note that (21) D ⊆ ⊕g∈G Lg ⊠Rι(g), and hence dim(D) ≤ dim(L ∩ L′) dim(R∩R′)|GL′ | = dim(L′) dim(R∩R′). The same inequality holds with L and R interchanged. Therefore, dim(C)2 = dim(D)2 ≤ dim(L′) dim(L′ ∩ L) dim(R′) dim(R∩R′) ≤ dim(C)2. 16 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK Here the first inequality becomes equality if and only if the inclusion in (21) is an equality and the second inequality becomes equality if and only if L′ ∩ L = L and R′ ∩R = R, i.e., when L and R are symmetric. � Remark 4.12. The subcategory DL,R,ι constructed in Theorem 4.11 is Lagrangian if and only if L and R are isotropic subcategories of C and ι is an isomorphism of metric groups. Corollary 4.13. Let C be a modular category. The following conditions are equiv- alent: (i) C is group-theoretical. (ii) There is a finite group G and a 3-cocycle ω ∈ Z3(G, k×) such that Z(C) ∼= Z(V ecωG) as a braided fusion category. (iii) C ⊠ Crev contains a Lagrangian subcategory. (iv) There is an isotropic subcategory E ⊂ C such that (E ′)ad ⊆ E. Proof. The equivalence (i)⇔(ii) is a consequence of [ENO], (ii)⇔(iii) follows from Theorem 4.5, and (iii)⇔(iv) follows from taking E = R = L and ι = idG Theorem 4.11, cf. Remark 4.12. � Combining the above criterion with Theorem 4.8 we obtain the following useful characterization of group-theoretical modular categories. Corollary 4.14. A modular category C is group-theoretical if and only if simple objects of C have integral dimension and there is a symmetric subcategory L ⊂ C such that (L′)ad ⊆ L. 5. Pointed modular categories In this section we analyze the structure of pointed modular categories, their central charges, and Lagrangian subgroups. Recall that such categories canonically correspond to metric groups [Q]. Let G = Z/nZ. The corresponding braided categories of the form C(Z/nZ, q) are completely classified by numbers σ = q(1) such that σn = 1 (n is odd) or σ2n = 1 (n is even). Then the braiding of objects corresponding to 0 ≤ a, b < n is the multiplication by σab and the twist of the object a is the multiplication by σa [Q]). We will denote the category corresponding to σ by C(Z/nZ, σ). Example 5.1. (a) Let G = Z/2Z. There are 4 possible values of σ: ±1,±i. The categories C(Z/2Z,±i) are modular with central charge 1±i√ and the categories C(Z/2Z,±1) are symmetric. The category C(Z/2Z, 1) is isotropic and the category C(Z/2Z,−1) is not. (b) Let G = Z/4Z. The twist of the object 2 ∈ Z/4Z is σ4 = ±1. If this twist is -1 then σ is a primitive 8th root of 1 and the corresponding category is modular; its Gauss sum is 1 + σ + σ4 + σ9 = 2σ and the central charge is σ. Note that if σ4 = 1 then the category C(Z/4Z, σ) contains a nontrivial isotropic subcategory. (c) Let G = Z/2kZ with k ≥ 3. Since the twist of the object 2k−1 is σ22k−2 = 1, the category C(Z/2kZ, σ) always contains a nontrivial isotropic subcategory. (d) Let G = Z/2Z × Z/2Z. There are five modular categories with this group. We give for each of them the list of values of q on nontrivial elements of G: (1) C(Z/2Z× Z/2Z, i): the values of q are i, i,−1, and the central charge is i. GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 17 (2) C(Z/2Z× Z/2Z,−i): the values of q are −i,−i,−1, and the central charge is −i. (3) C(Z/2Z×Z/2Z,−1): the values of q are −1,−1,−1, and the central charge is −1. (4) C(Z/2Z× Z/2Z, 1): the values of q are i,−i, 1, and the central charge 1. (5) The double of Z/2Z: the values of q are 1, 1,−1, and the central charge 1. In this list, each category of central charge 1 contains a nontrivial isotropic subcat- egory while the others contain a nontrivial symmetric (but not isotropic) subcate- gory. (e) Let G = Z/2Z × Z/4Z. Assume that the category C(G, q) does not contain a nontrivial isotropic subcategory. Then C(G, q) is equivalent to C(Z/4Z, σ) ⊠ C(Z/2Z,±i) where σ is a primitive 8th root of 1. The possible central charges are ±1 and ±i. (f) Let G = Z/2Z × Z/2Z × Z/2Z. Assume that the category C(G, q) does not contain a nontrivial isotropic subcategory. Then C(G, q) is equivalent to C(Z/2Z× Z/2Z, σ)⊠ C(Z/2Z, σ′), where σ′ = ±i and σ 6= 1,−σ′. Example 5.2. Let p be an odd prime. (a) Let G = Z/pZ. The category C(Z/pZ, σ) is modular for σ 6= 1 and is isotropic for σ = 1. The central charge of the modular category C(Z/pZ, σ) is ±1 for p = 1 mod 4 and ±i for p = 3 mod 4. (b) Let G = Z/pZ × Z/pZ. There are two modular pointed categories with underlying group G. One has central charge 1 (and is equivalent to the center of Z/pZ), and the other one has central charge -1. Recall that for a metric group (G, q) its Gauss sum is τ±(G, q) = a∈G q(a) A subgroup H of G is called isotropic if q|H = 1. An isotropic subgroup is called Lagrangian if H⊥ = H . The following proposition is well known. Proposition 5.3. Let (G, q) be a non-degenerate metric group such that |G| = p2n where p is a prime number and n ∈ Z+. Suppose that τ±(G, q) = |G| (i.e., the central charge of G is 1). Then G contains a Lagrangian subgroup. Proof. It suffices to prove that G contains a non-trivial isotropic subgroup H , then one can pass to H⊥/H and use induction. Assume that p is odd. Assume that G contains a direct summand Z/pkZ with k > 1. Then the subgroup Z/pZ ⊂ Z/pkZ is isotropic, since otherwise it is a non- degenerate metric subgroup of G and hence can be factored. Thus we are reduced to the case when G is a direct sum of k copies of Z/pZ. When k > 2, the quadratic form on G is isotropic (by the Chevalley - Waring theorem). Thus we are reduced to the case k = 2, which is easy (see Example 5.2 (b)). Assume now that p = 2. Again assume that G contains a direct summand Z/2kZ with k > 1. Again the subgroup Z/2Z ⊂ Z/2kZ is inside its orthogonal complement; moreover it is isotropic if k ≥ 3. If k = 2 and the subgroup Z/2Z ⊂ Z/4Z is not isotropic then the subgroup Z/4Z is a non-degenerate metric subgroup and hence factors out; let G = G1 ⊕ Z/4Z be the corresponding decomposition of G. If G1 contains Z/2Z such that Z/2Z ⊆ Z/2Z⊥ then we are done: if this subgroup is not isotropic then the diagonal subgroup Z/2Z ⊂ Z/2Z ⊕ Z/2Z ⊂ G1 ⊕ Z/4Z is isotropic. Thus G1 is a sum of Z/2Z’s and each summand is non-degenerate. But note that the central charge of a non-degenerate metric group Z/4Z is a primitive 18 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK eighth root of 1 (see Example 5.1 (b)) which is also the central charge of a non- degenerate metric Z/2Z (see Example 5.1 (a)). This implies that the number of Z/2Z summands in G1 is odd which is impossible since the order of G is a square. Thus we are reduced to the case when G is a sum of k copies of Z/2Z. In this case all possible values of the quadratic form q are ±1,±i and since τ+(G, q) = 2k/2, there is at least one non-identity a ∈ G with q(a) = 1. So the subgroup generated by a is isotropic. The proposition is proved. � 6. Nilpotent modular categories In this section we prove our main results, stated in 1.1, and derive a few corol- laries. Recall the definitions of Kad and Kco from 2.5. Proposition 6.1. Let C be a nilpotent modular category. Then for any maximal symmetric subcategory K of C one has (K′)ad ⊆ K. Equivalently, there is a grading of K′ such that K is the trivial component: (22) K′ = ⊕g∈GK′g, K′1 = K. Proof. The two conditions are equivalent since by [GN] the adjoint subcategory is the trivial component of the universal grading. Let K be a symmetric subcategory of C, i.e., such that K ⊆ K′. Assume that (K′)ad is not contained in K. It suffices to show that K is not maximal. Let E = (Kco ∩ (K′)ad) ∨ K. Clearly, K ⊆ E ⊆ K′. We have E ′ = ((Kco ∩ (K′)ad) ∨ K)′ = K′ ∩ ((Kco)′ ∨ ((K′)ad)′) = K′ ∩ ((K′)ad ∨ Kco) = (K′ ∩ Kco) ∨ (K′)ad, where we used the modular law of the lattice L(C) from Lemma 3.6. Since K ⊆ K′ ∩ Kco and Kco ∩ (K′)ad ⊆ (K′)ad we see that E ⊆ E ′, i.e., E is symmetric. Let n be the largest positive integer such that (K′)(n) 6⊆ K. Such n exists by our assumption and the nilpotency of K′. We claim that (K′)(n) ⊆ Kco. Indeed, (K′)(n) ⊆ ((K′)(n+1))co ⊆ Kco since D ⊆ (Dad)co for every subcategory D ⊆ C. Therefore, Kco ∩ (K′)(n) = (K′)(n) is not contained in K and K ( (Kco ∩ (K′)(n)) ∨K ⊆ (Kco ∩ (K′)ad) ∨ K = E , which completes the proof. � Recall that in a fusion category whose dimension is an odd integer the dimensions of all objects are automatically integers [GN, Corollary 3.11]. Corollary 6.2. A nilpotent modular category C with integral dimensions of simple objects is group-theoretical. Proof. This follows immediately from Corollary 4.14 and Proposition 6.1. � GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 19 Remark 6.3. It follows from Corollary 4.13 that a nilpotent modular category C with integral dimensions of simple objects contains an isotropic subcategory E such that (E ′)ad ⊆ E . The corresponding grading (23) E ′ = ⊕h∈H E ′h, E ′1 = E , gives rise to a non-degenerate quadratic form q on H defined by q(h) = θV for any non-zero V ∈ Ch. We have a braided equivalence Ē ′ ∼= C(H, q). We may assume that E is maximal among isotropic subcategories of C. In this case, Proposition 3.9 implies that the isomorphism class of the above metric group (H, q) does not depend on the choice of the maximal isotropic subcategory E . Corollary 6.4. The central charge of a modular nilpotent category with integer dimensions of objects is always an 8th root of 1. Moreover, the central charge of a modular p−category is ±1 if p = 1mod 4 and ±1, ±i if p = 3mod 4. The central charge of a modular p−category of dimension p2k, k ∈ Z+ with odd p is ±1. Proof. By Remark 6.3 and Theorem 3.4 the central charge always equals the central charge of some pointed category, so the first claim follows from Examples 5.1-5.2. The second and third claims follow from Example 5.2. � Theorem 6.5. Let C be a modular category with integral dimensions of simple objects. Then C is nilpotent if and only if there exists a pointed modular category M such that C⊠M is equivalent (as a braided fusion category) to Z(VecωG), where G is a nilpotent group. Proof. Note that for a nilpotent group G the category Z(VecωG) is a tensor product of modular p-categories and, hence, is nilpotent. So if C ⊠M ∼= Z(VecωG) then C is nilpotent (as a subcategory of a nilpotent category). Let us prove the converse implication. Pick an isotropic subcategory E ⊂ C such that (E ′)ad ⊆ E (such a subcategory exists by Remark 6.3). There is a metric group (H, q) such that Ē ′ ∼= C(H, q). Let E ′ = ⊕h∈H E ′h , where E1 = E be the corresponding grading from (23). Let M be the reversed category of Ē ′ (i.e., with the opposite braiding and twist). Then M ∼= C(H, q−1) and ξ(M) = ξ(C(H, q))−1 = ξ(C)−1 by Theorem 3.4. The modular category Cnew = C ⊠ M is nilpotent and ξ(Cnew) = 1. The cate- gory Enew := ⊕h∈H Eh ⊠ h is a Lagrangian subcategory of Cnew and the required statement follows from Theorem 4.5. � Let p be a prime number. Theorem 6.6. A modular category C is equivalent to the center of a fusion category of the form VecωG with G being a p-group if and only if it has the following properties: (i) the Frobenius-Perron dimension of C is p2n for some n ∈ Z+, (ii) the dimension of every simple object of C is an integer, (iii) the multiplicative central charge of C is 1. Proof. It is clear that for any finite p-group G and ω ∈ Z3(G, k×) the modular category Z(VecωG) satisfies properties (i) and (ii). The central charge of Z(Vec equals 1 by [Mu4, Theorem 1.2]. Let us prove the converse. Suppose that C satisfies conditions (i), (ii), and (iii). Let E be an isotropic subcategory of C such that (E ′)ad ⊆ E (such an E exists by Remark 6.3). There is a grading E ′ = ⊕h∈H E ′h with E ′1 = E and θ being constant 20 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK on each E ′h, h ∈ H . Note that H is a metric p-group whose order is a square. By Proposition 5.3 it contains a Lagrangian subgroup H0, whence ⊕h∈H0 E ′h is a Lagrangian subcategory of C. Thus, C ∼= VecωG for some G and ω by Theorem 4.5. Since |G|2 = dim(Vec dim(C) it follows that G is a p-group. � Finally, we apply our results to show that certain fusion categories (more pre- cisely, representation categories of certain semisimple quasi-Hopf algebras) are group- theoretical and to obtain a categorical analogue of the Sylow decomposition of nilpotent groups. Corollary 6.7. Let C be a fusion category with integral dimensions of simple objects and such that Z(C) is nilpotent. Then C is group-theoretical. Proof. By Corollary 6.2 the category Z(C) is group-theoretical. Hence, C ⊠ Crev is group theoretical (as a dual category of Z(C), see [ENO]). Therefore, C is group- theoretical (as a fusion subcategory of C ⊠ Crev). � Corollary 6.8. Let C be a fusion category of dimension pn, n ∈ Z+, such that all objects of C have integer dimension (this is automatic if p > 2). Then C is group-theoretical. In other words, semisimple quasi-Hopf algebras of dimension pn are group- theoretical. Remark 6.9. Semisimple Hopf algebras of dimension pn were studied by several authors, see e.g., [EG2], [Kash], [Ma1], [Ma2], [MW], [Z]. From Corollary 6.2 we obtain the following Sylow decomposition. Theorem 6.10. Let C be a braided nilpotent fusion category such that all objects of C have integer dimension. Then C is group-theoretical and has a decomposition into a tensor product of braided fusion categories of prime power dimension. If the factors are chosen in such a way that their dimensions are relatively prime, then such a decomposition is unique up to a permutation of factors. Proof. It was shown in [GN, Theorem 6.11] that the center of a braided nilpotent fusion category is nilpotent. Hence, Z(C) is group-theoretical by Corollary 6.2. Since C is equivalent to a subcategory of Z(C), it is group-theoretical by [ENO, Proposition 8.44]. This means that there is a group G and ω ∈ Z3(G, k∗) such that C is dual to VecωG with respect to some indecomposable module category. The group G is necessarily nilpotent since Rep(G) ⊆ Z(VecωG) ∼= Z(C). Hence, G is isomorphic to a direct product of its Sylow p-subgroups, G = G1 × · · · ×Gn, and so VecωG is equivalent to a tensor product of p-categories. It follows from [ENO, Proposition 8.55] that the dual category C is also a product of fusion p-categories, as desired. Now suppose that C is decomposed into factors of prime power Frobenius-Perron dimension, C ≃ ⊠pCp. It is easy to see that the objects from Cp ⊂ C are characterized by the following property: (24) X ∈ Cp if and only if there exists k ∈ Z+ such that Hom(1, X⊗ ) 6= 0. This shows that the decomposition in question is unique. � GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 21 Remark 6.11. Let C be a nilpotent modular category with integral dimensions of simple objects. We already mentioned in the introduction that the choice of a tensor complement M satisfying C ⊠ M ∼= VecωG is not unique. In the proof of Theorem 6.5 such M can be chosen canonically as the category opposite to the canonical modularization corresponding to a maximal isotropic subcategory of C, see Proposition 3.9. Another canonical way is to choose an M of minimal possible dimension. This is done as follows. By Theorem 6.10, we have C = ⊠p Cp and M = ⊠p Mp, where Cp,Mp are modular p-categories. By Theorem 6.6, Mp has to be chosen in such a way that dim(Cp) dim(Mp) is a square and ξ(Mp) = ξ(Mp)−1. It follows from Examples 5.1, 5.2 and Corollary 6.4 that there is a unique such choice of Mp with minimal dim(Mp), in which case dim(Mp) ∈ {1, p, p2} for odd p and dim(M2) ∈ {1, 2, 4, 8}. Theorem 6.12. Let C be a braided nilpotent fusion category. Then C has a unique decomposition into a tensor product of braided fusion categories of prime power dimension. Proof. According to Theorem 6.10 the result is true if the dimensions of simple objects of C are integers. In general, define subcategories Cp ⊂ C by condition (24) above. For a simple object X ∈ C it is known (see [GN]) that FPdim(X) = N ∈ N. Thus X⊠X ∈ C⊠C has an integer dimension. The category C⊠C contains a fusion subcategory (C⊠ C)int consisting of all objects with integer dimension, see [GN]. We can apply Theorem 6.10 to the category (C ⊠ C)int and obtain a unique decomposition X = ⊗pXp with Xp ∈ Cp. The theorem is proved. � Corollary 6.13. Let C be a braided nilpotent fusion category. Assume that X ∈ C is simple and its dimension is not integer. Then FPdim(X) ∈ Proof. This follows immediately from Theorem 6.12 since if a category of prime power Frobenius-Perron dimension pk contains an object of a non-integer dimension then p = 2, see [ENO]. � Example 6.14. It is easy to see that the Tambara-Yamagami categories from [TY] are nilpotent and indecomposable into a tensor product. Thus Theorem 6.12 implies that if such a category admits a braiding, then its dimension should be a power of 2 (since the dimension of a Tambara-Yamagami category is always divisible by 2). A stronger result is contained in [S]. References [Be] R. Bezrukavnikov, On tensor categories attached to cells in affine Weyl groups, Represen- tation theory of algebraic groups and quantum groups, 69–90, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo, 2004. [Br] A. Bruguières, Catégories prémodulaires, modularization et invariants des variétés de dimension 3, Mathematische Annalen, 316 (2000), no. 2, 215-236. [BD] M. Boyarchenko, V. Drinfeld, A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic, math.RT/0609769. [BK] B. Bakalov, A. Kirillov Jr., Lectures on Tensor categories and modular functors, AMS, (2001). [CG] A. Coste, T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys. Lett. 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[EO] P. Etingof, V. Ostrik, Finite tensor categories, Moscow Math. Journal 4 (2004), 627-654. [GN] S. Gelaki, D. Nikshych, Nilpotent fusion categories, math.QA/0610726. [JS] A. Joyal, R. Street, Braided tensor categories, Adv. Math., 102, 20-78 (1993). [Kash] Y. Kashina, Classification of semisimple Hopf algebras of dimension 16, J. Algebra 232 (2000), no. 2, 617–663. [Kass] C. Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer, New York. [KaO] I. Kath, M. Olbrich Metric Lie algebras and quadratic extensions, Transform. Groups 11 (2006), no. 1, 87–131. [KiO] A. Kirillov Jr., V. Ostrik, On q-analog of McKay correspondence and ADE classification of ŝl2 conformal field theories, Adv. Math. 171 (2002), no. 2, 183–227. [Ma1] A. Masuoka, The pn theorem for semi-simple Hopf algebras, Proc. AMS 124 (1996), 735- [Ma2] A. Masuoka, Self-dual Hopf algebras of dimension p3 obtained by extension, J. Algebra 178 (1995), 791–806. [Mu1] M. 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Res. Notices 1 (1994), 53–59. http://arxiv.org/abs/math/0610726 http://arxiv.org/abs/math/0605530 http://arxiv.org/abs/math/0011037 GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 23 V.D.: Department of Mathematics, University of Chicago, Chicago, IL 60637, USA E-mail address: [email protected] S.G.: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail address: [email protected] D.N.: Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA E-mail address: [email protected] V.O.: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA E-mail address: [email protected] 1. introduction 1.1. Main results 1.2. Interpretation in terms of group-theoretical fusion categories and semisimple quasi-Hopf algebras. 1.3. Idea of the proof 1.4. Organization of the paper 1.5. Acknowledgments 2. Preliminaries 2.1. Fusion categories 2.2. Nilpotent fusion categories 2.3. Premodular categories and modular categories 2.4. Pointed modular categories and metric groups 2.5. Centralizers 2.6. Gauss sums and central charge in modular categories 2.7. Symmetric fusion categories 3. Isotropic subcategories and Bruguières-Müger modularization 3.1. Modularization 3.2. Invariance of the central charge 3.3. Maximal isotropic subcategories 4. Reconstruction of a twisted group double from a Lagrangian subcategory 4.1. C-algebras 4.2. Hyperbolic modular categories as twisted group doubles 4.3. A criterion for a modular category to be group-theoretical 5. Pointed modular categories 6. Nilpotent modular categories References

0704.0196

Remarks on N_c dependence of decays of exotic baryons

TPJU-03/2007 Remarks on N dependence of decays of exotic baryons Karolina Pieściuk ∗) and Micha l Prasza lowicz ∗∗) M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-049 Kraków, Poland We calculate the Nc dependence of the decay widths of exotic eikosiheptaplet within the framework of Chral Quark Soliton Model. We also discuss generalizations of regular baryon representations for arbitrary Nc. §1. Introduction One of the most puzzling results of the chiral quark-soliton model (χQSM) for exotic baryons consists in a very small hadronic decay width,1) governed by the decay constant G10. While the small mass of exotic states is rather generic for all chiral models1)–3) the smallness of the decay width appears as a subtle cancelation of three different terms that contribute to G10. Decay width in solitonic models 4) is calculated in terms of a matrix element M of the collective axial current operator corresponding to the emission of a pseudoscalar meson ϕ1) – see Ref. 5) for criticism of this approach: Ô(8)ϕ = 3 ϕi −G1 dibcD ϕb Ŝc − ϕ8 Ŝi × piϕ. (1.1) For notation see Ref. 1). Constants G0,1,2 are constructed from the so called moments of inertia that are calculable in χQSM. The decay width is given as ΓB→B′+ϕ = M M ′ M2 = 1 M M ′ A2. (1.2) The “bar” over the amplitude squared denotes averaging over initial and summing over final spin (and, if explicitly indicated, over isospin). For B(10) → B′(8) + ϕ for spin ”up” and ~pϕ = (0, 0, pϕ) we have 81/2, B ∣ Ô(8)ϕ ∣101/2, B 3G10√ × pϕ (1.3) G10 = G0 −G1 − G2. (1.4) ∗) e-mail address: [email protected] ∗∗) e-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.0196v2 2 K. Pieściuk and M. Prasza lowicz In order to have an estimate of the width (1.2) the authors of Ref. 1) calculated G10 in the nonrelativistic limit6) of χQSM and got G10 ≡ 0. It has been shown that this cancelation between terms that scale differently with Nc (G0 ∼ N c , G1,2 ∼ N is in fact consistent with large Nc counting, 7) since G10 = G0 − Nc + 1 G2 (1.5) where the Nc dependence comes from the SU(3) Clebsch-Gordan coefficients calcu- lated for large Nc. In the nonrelativistic limit (NRL): G0 = −(Nc + 2)G, G1 = −4G, G2 = −2G, G ∼ N1/2c . (1.6) In this paper we ask whether the similar cancelation takes place for the decays of 27 of spin 1/2 and 3/2. We also discuss the possible modifications of the Nc depen- dence of the decay width due to the different choice of the large Nc generalizations of regular SU(3) multiplets. §2. Baryons in large N limit Soliton is usually quantized as quantum mechanical symmetric top with two moments of inertia I1,2: B = Mcl + S(S + 1) + C2(R) − S(S + 1) − B . (2 Here S denotes baryon spin, C2(R) the Casimir operator for the SU(3) representation R = (p, q): C2(R) = p2 + q2 + pq + 3(p + q) (2.2) and quantities δ B denote matrix elements of the SU(3) breaking hamiltonian: Ĥ ′ = σ + αD 88 ) + βY + 8A ĴA. (2 Model parameters that can be found in Ref. 8) α = −Nc (σ + β), β = −ms , γ = 2ms , σ = mu + md scale with Nc in the following way: i1,2 = 3I1,2/Nc where i1,2 ∼ O(N0c ), σ, β, γ ∼ O(msN0c ). (2.4) Here ΣπN is pion-nucleon sigma term and mq denote current quark masses. Numer- ically σ > |β| , |γ|. So far we have specified explicit Nc dependence (2.4) that follows from the fact that model parameters are given in terms of the quark loop. Another type of the Remarks on Nc dependence 3 Nc dependence comes from the constraint 9) that selects SU(3)flavor representations R = (p, q) containing states with hypercharge YR = Nc/3. Therefore for arbitrary Nc ordinary baryon representations have to be extended and one has to specify which states correspond to the physical ones. Usual choice10) ”8” = (1, (Nc − 1)/2) , ”10” = (3, (Nc − 3)/2) , ”10” = (0, (Nc + 3)/2) , (2.5) depicted in Fig. 1 corresponds – in the quark language – to the case when each time when Nc is increased by 2, a spin-isospin singlet (but charged) 3 diquark is added, as depicted in Fig. 2. Fig. 1. Standard generalization of SU(3) flavor baryon representations for arbitrary Nc Fig. 2. Adding 3 diquarks to regular SU(3) baryon representations 8, 10 and 10 corresponds to the representation set of Fig.1. Extension (2.5) leads to (1.5). It implies that mass differences between centers of multiplets scale differently with Nc: ∆10−8 = ∼ O(1/Nc), ∆10−8 = Nc + 3 ∼ O(1). (2.6) The fact that ∆10−8 6= 0 in large Nc limit triggered recently discussion on the validity of the semiclassical quantization for exotic states.11) Since in the chiral limit the momentum pϕ of the outgoing meson scales according to (2.6), overall Nc dependence of the decay width is strongly affected by its third power (1.2): ΓB→B′+ϕ ∼ O(A2)O(p3ϕ). (2.7) Phenomenologically, however, scaling (2.6) is not sustained. Indeed, meson mo- menta in ∆ and Θ decays are almost identical (assuming M Θ ≃ 1540 MeV): pπ ≃ 225 MeV, pK ≃ 268 MeV. (2.8) Unfortunately, going off SU(3)flavor limit does not help. Explicitly: δ(8) = (Nc − 3) (Nc − 2)α + 32γ Nc + 7 3(Nc + 2)α− 12(2Nc + 9)γ (Nc + 3)(Nc + 7) 4 K. Pieściuk and M. Prasza lowicz (6α + (Nc + 6)γ) (Nc + 3)(Nc + 7) − I(I + 1) = 3σ + 2β − σY + . . . (2.9) δ(10) = (Nc − 3)(Nc + 4) (Nc + 1)(Nc + 9) Nc − 3 5(Nc − 3) 2(Nc + 1)(Nc + 9) 3(Nc − 1)α − 52(Nc + 3)γ (Nc + 1)(Nc + 9) Y = 3σ + 2β − σY + . . . (2.10) δ(10) = Nc(Nc − 3) (Nc + 3)(Nc + 9) Nc − 3 β − 3(Nc − 3) 2(Nc + 3)(Nc + 9) 6Ncα− 9γ 2(Nc + 3)(Nc + 9) Y = 5σ + 4β − σY + . . . (2.11) where . . . denote terms O(1/Nc), Y and I denote physical hypercharge and isospin. Interestingly in all cases in the large Nc limt, ms splittings are proportional to the hypercharge differences only. In this limit Σ−Λ splitting in the octet is zero and this degeneracy is lifted in the next order at O(1/Nc). This explains the smallness of Σ − Λ mass difference. Additionally δ(8)N ≃ δ ∆ up to higher order terms O(1/N2c ), however δ Θ − δ N ≃ σ + 2β > 0. This implies that α + β − 3 γ → 3 + σ + 2β + O(1/Nc), γ → O(1/Nc). (2.12) The first equation shows that the Θ − N 6= 0 in the large Nc limit even if ms corrections are included. We will come back to this problem in the last section. §3. Decay constants of twentysevenplet for large N In this section we shall consider decays of eikosiheptaplet (27-plet) ”27” = (2, (Nc + 1)/2) (3.1) that can have either spin 1/2 or 3/2, the latter being lighter. Mass differences read ∆273/2−8 = Nc + 1 ∼ O(1), ∆271/2−8 = Nc + 7 ∼ O(1), ∆273/2−10 = Nc + 1 ∼ O(1), ∆271/2−10 = − Nc + 7 ∼ O(1), ∆273/2−10 = ∼ O(1/Nc), ∆271/2−10 = ∼ O(1/Nc). (3.2) Matrix elements for the decays of eikosiheptaplet (with S3 = 1/2) read: A(B273/2 → B 8 + ϕ) = 3 8 ”8” 8(Nc + 5) 9(Nc + 3)(Nc + 9) ×G27, Remarks on Nc dependence 5 A(B273/2 → B 10 + ϕ) = −3 8 ”10” (Nc − 1)(Nc + 7) 9(Nc + 1)(Nc + 3)(Nc + 9) × F27, A(B273/2 → B + ϕ) = 3 8 ”10” 2(Nc + 1)(Nc + 7) 3(Nc + 3)(Nc + 9) × E27, (3.3) Decay Large Nc NRL Scaling in NRL 273/2 → 81/2 G27 = G0 − Nc−14 G1 = −3G N 273/2 → 103/2 F27 = G0 − Nc−14 G1 − G2 = 0 0 273/2 → 101/2 E27 = G0 + G1 = −(Nc + 6)G N For S = 1/2 and S3 = 1/2 we have: A(B271/2 → B 8 + ϕ) = −3 8 ”8” (Nc + 1)(Nc + 5) 9(Nc + 3)(Nc + 7)(Nc + 9) ×H27, A(B271/2 → B 10 + ϕ) = −3 8 ”10” 8(Nc − 1) 9(Nc + 3)(Nc + 9) ×G′27, A(B271/2 → B + ϕ) = 3 8 ”10” Nc + 4 9(Nc + 3)(Nc + 9) ×H ′27, (3.4) Decay Large Nc NRL Scaling in NRL 271/2 → 81/2 H27 = G0−Nc+54 G1 + G2 = 0 0 271/2 → 103/2 G′27 = G0−Nc+54 G1 = 3G N 271/2 → 101/2 H ′27 = G0 + 2Nc+52Nc+8G1 + 2Nc+8 G2 = − (Nc+3)(Nc+7)Nc+4 G N In order to calculate the Nc behavior of the width we have to know the Nc dependence of the flavor Clebsch-Gordan coefficients that depend on the states in- volved. For the decays into 8 and 10 the only possible channels are Θ27 → N(∆)+K, and the pertinent Clebsches do not depend on Nc. For the decays into 10 we have Θ27 → Θ10 + π that scales like O(1) and Θ27 → N10 +K that scales like O(1/ The resulting scaling of ΓΘ27→B′+ϕ calculated from Eq.(2 .7) reads as follows: 6 K. Pieściuk and M. Prasza lowicz decay of Nc scaling decay of Nc scaling Θ273/2 exact NRL Θ271/2 exact NRL → N8 + K O(1) O(1/N2c ) → N8 + K O(1) 0 → ∆10 + K O(1) 0 → ∆10 + K O(1) O(1/N2c ) → N10 + K O(1/N3c ) O(1/N3c ) → N10 + K O(1/N3c ) O(1/N3c ) → Θ10 + π O(1/N2c ) O(1/N2c ) → Θ10 + π O(1/N2c ) O(1/N2c ) Interestingly, we see that whenever the exact scaling is O(1), the nonrelativistic cancelation (exact or partial) lowers the power of Nc, whereas in the case when the width has good behavior for large Nc, there is no NRL cancelation. §4. Alternative choices for large N multiplets So far we have only considered the ”standard” generalization (2.5) of baryonic SU(3)flavor representations for large Nc. This choice is based on the requirement that generalized baryonic states have physical spin, isospin and strangeness, however their hypercharge and charge are not physical.10) Moreover, the generalization of the octet is not selfadjoint and antidecuplet is not complex conjugate of decuplet. Some years ago it has been proposed to consider alternative schemes.12) Fig. 3. Generalization of SU(3) flavor representations in which octet is selfadjoint Fig. 4. Adding triquarks to regular SU(3) baryon representations 8, 10 and 10 corresponds to the representation set of Fig.3. If we require the generalized octet to be self-adjoint we are led to the following set of representations ”8” = (Nc/3, Nc/3) , ”10” = ((Nc + 6)/3, (Nc − 3)/3) , ”10” = ”10”∗ (4.1) that are depicted in Figs. 3 and 4. This means that we enlarge Nc in steps of 3 adding each time a uds triquark. Generalized states have physical isospin, hypercharge (and charge), but unphysical strangeness and spin that is of the order of Nc. With this Remarks on Nc dependence 7 choice both ∆10−8, ∆10−8 6= 0 in large Nc limit: ∆10−8 = (Nc/6 − 1) /I1, ∆10−8 = (Nc/6 − 1) /I2. (4.2) With this power counting we can calculate large Nc approximation of the meson momenta in the decays of ∆ and Θ: ∆ → N pπ = (M∆ −MN )2 −m2π = 256 MeV, Θ → N pK = (MΘ −MN )2 −m2K = 339 MeV (4.3) that are much closer to the physical values (2.8) than (2.6). Fig. 5. Generalization of SU(3) flavor representations in which decuplet is fully symmetric (0, q). Fig. 6. Adding sextet diquarks to regular SU(3) baryon representations 8, 10 and 10 corresponds to the representation set of Fig.5. Finally let us mention a third possibility in which we require generalized decuplet to be a completely symmetric SU(3)flavor representation for arbitrary Nc. This leads to (see Figs. 5 and 6): ”8” = (Nc − 2, 1) ”10” = (Nc, 0) ”10” = (Nc − 3, 3) . (4.4) Interestingly this choice has a smooth limit to the one flavor case. In the quark language it amounts to adding a symmetric diquark to the original SU(3)flavor rep- resentation when increasing Nc in steps of 2. As seen from Fig. 5 physical states are situated at the bottom of infinite representations (4.4) and therefore have unphysical strangeness, charge (hypercharge) and also spin. The mass splittings for this choice read ∆10−8 = Nc/ 2I1, ∆10−8 = 3/ 2I2. (4 Here the generalized decuplet remains split from the ”8”, while ∆10−8 → 0 for large Nc. The phase space factor for Θ decay is therefore suppressed with respect to the one of ∆. §5. Summary In this short note we have shown that very small width of exotic baryons – if they exist – cannot be explained by the standard Nc counting alone. Certain degree 8 K. Pieściuk and M. Prasza lowicz of nonrelativisticity is needed to ensure cancelations between different terms in the decay constants. This phenomenon observed firstly for antidecuplet, is also operative for the decays of eikosiheptaplet. We have shown that in χQSM in the nonrelativistic limit all decays are suppressed for large Nc. Exact cancelations occur for Θ273/2 → ∆10 + K and Θ271/2 → N8 + K, leading Nc terms cancel for Θ273/2 → N8 + K and Θ271/2 → ∆10 + K. For 27 → 10 there are no cancelations, but the phase space is N−3c suppressed. We have also briefly discussed nonstandard generalizations of regular baryon representations for arbitrary Nc. For Nc > 3 bayons are no longer composed from 3 quarks and therefore they form large SU(3)flavor representations that reduce to octet, decuplet and antidecuplet for Nc = 3. The standard way to generalize regular baryon representations is to add antisymmetric antitriplet diqaurk when Nc is increased in intervals of 2. This choice fulfils many reasonable requirements; most importantly for SU(2)flavor these representations form regular isospin multiplets. However, repre- sentations (2.5) do not obey conjugation relations characteristic for regular represen- tations. Therefore we have proposed generalization (4.1) that satisfies conjugation relations. Most important drawback of (4.1) is that spin S ∼ Nc that contradicts semiclassical quantization. Nevertheless as a result meson momenta emitted in 10 and 10 decays scale in the same way with Nc (4.3), consistently with ”experimental” values (2.8), whereas for (2.5) the scaling is different (2.6). Acknowledgements One of us (MP) is grateful to the organizers of the Yukawa International Sym- posium (YKIS2006) for hospitality during this very successful workshop. References 1) D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359 (1997) 305 [arXiv:hep-ph/9703373]. 2) L.C Biedenharn and Y. Dothan, Monopolar Harmonics in SU(3)F as eigenstates of the Skyrme-Witten model for baryons, E. Gotsman and G. Tauber (eds.), From SU(3) to gravity, p. 15-34. 3) M. Prasza lowicz, talk at Workshop on Skyrmions and Anomalies, M. Jeżabek and M. Prasza lowicz eds., World Scientific 1987, page 112 and Phys. Lett. B 575 (2003) 234 [hep-ph/0308114]. 4) G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552; 5) H. Weigel, arXiv:hep-ph/0703072. 6) M. Prasza lowicz, A. Blotz and K. Goeke, Phys. Lett. B 354 (1995) 415 [hep-ph/9505328]; M. Prasza lowicz, T. Watabe and K. Goeke, Nucl. Phys. A 647 (1999) 49 [hep-ph/9806431]. 7) M. Prasza lowicz, Phys. Lett. B 583, 96 (2004) [arXiv:hep-ph/0311230]. 8) A. Blotz, D. Diakonov, K. Goeke, N. W. Park, V. Petrov and P. V. Pobylitsa, Nucl. Phys. A 555, 765 (1993). 9) E. Guadagnini, Nucl. Phys. B 236 (1984) 35; P.O. Mazur, M. Nowak and M. Prasza lowicz, Phys. Lett. B 147(1984) 137; S. Jain and S.R. Wadia, Nucl. Phys. B 258 (1985) 713. 10) G. Karl, J. Patera and S. Perantonis, Phys. Lett. B 172 (1986) 49; J. Bijnens, H. Sonoda and M. Wise, Can. J. Phys. 64 (1986) 1. Z. Duliński and M. Prasza lowicz, Acta Phys. Pol. B 18 (1988) 1157. 11) P. V. Pobylitsa, Phys. Rev. D 69, 074030 (2004) [arXiv:hep-ph/0310221]. T. D. Cohen, Phys. Rev. D 70, 014011 (2004) [arXiv:hep-ph/0312191]. 12) Z. Duliński, Acta. Phys. Pol. B 19 (1988) 891. http://arxiv.org/abs/hep-ph/9703373 http://arxiv.org/abs/hep-ph/0308114 http://arxiv.org/abs/hep-ph/0703072 http://arxiv.org/abs/hep-ph/9505328 http://arxiv.org/abs/hep-ph/9806431 http://arxiv.org/abs/hep-ph/0311230 http://arxiv.org/abs/hep-ph/0310221 http://arxiv.org/abs/hep-ph/0312191 Introduction Baryons in large Nc limit Decay constants of twentysevenplet for large Nc Alternative choices for large Nc multiplets Summary

0704.0197

Analysis of random Boolean networks using the average sensitivity

arXiv:0704.0197v1 [nlin.CG] 2 Apr 2007 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- Analysis of random Boolean networks using the average sensitivity Steffen Schober∗ and Martin Bossert Institute of Telecommunications and Applied Information Theory, Ulm University Albert-Einstein-Allee 43, 89081 Ulm, Germany November 4, 2018 Abstract In this work we consider random Boolean networks that provide a general model for genetic regulatory networks. We extend the analysis of James Lynch who was able to proof Kauffman’s conjecture that in the ordered phase of random networks, the number of ineffective and freezing gates is large, where as in the disordered phase their number is small. Lynch proved the conjecture only for networks with connectivity two and non-uniform probabilities for the Boolean functions. We show how to apply the proof to networks with arbitrary connectivity K and to random networks with biased Boolean functions. It turns out that in these cases Lynch’s parameter λ is equivalent to the expectation of average sensitivity of the Boolean functions used to construct the network. Hence we can apply a known theorem for the expectation of the average sensitivity. In order to prove the results for networks with biased functions, we deduct the expectation of the average sensitivity when only functions with specific connectivity and specific bias are chosen at random. Keywords: Random Boolean networks, phase transition, average sensitivity PACS numbers: 02.10.Eb, 05.45.+b, 87.10.+e 1 Introduction In 1969 Stuart Kauffman started to study random Boolean networks as simple models of genetic regulatory networks [1]. Random Boolean networks that con- sists of a set of Boolean gates that are capable of storing a single Boolean value. At discrete time steps these gates store a new value according to an initially chosen random Boolean function, which receives its inputs from random chosen gates. We will give a more formal definition later. Kauffman made numerical ∗Corresponding author. E-Mail: [email protected] http://arxiv.org/abs/0704.0197v1 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- studies of random networks, where the functions are chosen from the set of all Boolean functions with K arguments (the so called NK-Networks). He recog- nised that if K ≤ 2, the random networks exhibit a remarkable form of ordered behaviour: The limit cycles are small, the number of ineffective gates, which are gates that can be perturbed without changing the asymptotic behaviour, and the number of freezing gates that stop changing their state is large. In contrast if K ≥ 3, the networks do not exhibit this kind of ordered behaviour (see [1, 2]). The first analytical proof for this phase transition was given by Derrida and Pomeau (see [3]) by studying the evolution of the Hamming distance of random chosen initial states by means of so called annealed approximation. The first proof for the number of freezing and ineffective gates was given by James Lynch (see [4], although slightly weaker results appeared earlier [5, 6]). Depending on a parameter λ, that depends on the probabilities of the Boolean functions, he showed that if λ ≤ 1 almost all gates are ineffective and freezing, otherwise not. Although his analysis is very general, until now it was only applied to networks with connectivity 2 and non-uniform probabilities for the Boolean function: if the probability of choosing a constant function is larger or equal the probability of choosing a non-constant non-canalizing function (namely the XOR- or the inverted XOR-function), λ is less or equal to one. But it turns out that in some cases λ is equal to the expectation of the average sensitivity. Therefore we will first study the average sensitivity in Section 3. Afterwards it will be shown in Section 4 how to use the results from the previous section to apply Lynch’s analysis to classical NK-Networks and biased random Boolean networks 1. But first we will give some basic definition used throughout the paper in Section 2. 2 Basic Definitions In the following F2 = {0, 1} denotes the Galois field of two elements, where addition, denoted by ⊕, is defined modulo 2. The set of vectors of length K over F2 will be denoted by F 2 . If x is a vector from F 2 , its ith component will be denoted by xi. With u (i) ∈ FK2 we will denote the unit vector which has all components zero except component i which is one. The Hamming weight of x ∈ FK2 is defined as wH(x) = |{i | xi 6= 0, i = 1, . . . ,K}| and the Hamming distance of x,y ∈ FK2 as dH(x,y) = wH(x⊕ y). A Boolean function is a mapping f : FK2 → F2. A function f may be represented by its truth table tf , that is, a vector in F 2 , where each component of the truth table gives the value of f for one of the 2K possible arguments. To fix an order on the components of the truth table, suppose that its ith component equals the value of the corresponding function, given the binary representation (to K bits) of i as an argument. 1a definition will be given later DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 3 Average Sensitivity In this section we will focus on the average sensitivity. The average sensitivity is a known complexity measure for Boolean functions, see for example [7]2. It was already used to study Boolean and random Boolean networks for example in [8, 9]. Definition 1. Let f denote a Boolean function FK2 → F2 and u (i) a unit vector. 1. The sensitivity sf(w) is defined as: sf (w) = i | f(w) 6= f(w⊕ u(i)), i = 1, . . . ,K 2. The average sensitivity sf is defined as the average of sf (w) over all w ∈ sf = 2 sf (w) Now consider the random variable FK : Ω → FK, where FK denotes the set a all 22 Boolean function with K arguments. The probability measure is given by P (FK = f) = K . The expected value of the average sensitivity of this random variable is denoted by EFK (sf ), and is given by EFK (sf ) = P (FK = f)sf The expected value was already derived in [10], and is given by: Theorem 1 (Bernasconi [10]). Let the random variable FK be defined as above, then EFK (sf ) = P (FK = f)sf = We will now concentrate on biased Boolean functions. The bias of a Boolean function f : FK2 → F2 is defined as the number of 1 in the functions truth table divided by 2K . To define the bias of a random Boolean function two definitions are possible. First we can assumes that the truth tables of the Boolean functions are produced by independent Bernoulli trials with probability p for a one (This should be called mean bias, used for example in [3, 8] ). Therefore consider the random variable FK,p. The probability of choosing a function f is given by P (FK,p = f) = p wH(tf )(1− p)2 K−wH(tf ) For p = 1/2 this is equivalent to the definition of FK . 2here it is called critical complexity DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- As a second possibility, we can only choose functions which have bias p whereas to all other functions we assign probability 0 (we will call this fixed bias). Therefore consider the random variables F fixedK,p : Ω → FK. Denote the truth table of a function f by tf . Further denote the set of all Boolean functions f with K arguments and wH(tf ) = p2 K with FK,p. The probability for a certain function chosen according F fixedK,p is given by P (F fixedK,p = f) = |FK,p| if f ∈ FK,p 0 if f /∈ FK,p Both definitions ensure that the expectation to get a one is equal to p if the input of a function is chosen at random (with respect to uniform distribution). But it will turn out that these two different methods of creating biased Boolean functions, have a major impact on the average sensitivity. The expectation of the average sensitivity of FK,p was derived in [8]: Theorem 2 ([8]). Let the random variable FK,p be defined as above: EFK,p(sf ) = 2Kp(1− p) For the random variable F fixedK,p we will now proof the following theorem: Theorem 3. Let the random variable F fixed be defined as above: fixed (sf ) = 2K+1Kp(1− p) (2K − 1) Proof. To find EF fixed (sf ) we will first consider the random variable FK,t : Ω → FK where t ∈ {0, 1, · · · , 2 K} and the probability of a function is given by P (FK,t = f) = if wH(tf ) = t 0 else Consider the Boolean functions as functions into R by identifying 0, 1 ∈ F2 with 0, 1 ∈ R. Then we get or the function f : sf = 2 i | f(w) 6= f(u(i) ⊕w), i = 1, . . . ,K = 2−K (f(w)− f(w⊕ u(i)))2 = 2−K (f(w) + f(w⊕ u(i))− 2f(w)f(w⊕ u(i))). DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- where u(i) again denotes the unit vector with ith component set to 1. Hence by the linearity of the expectation EFK,t(sf ) = 2 EFK,t(f(w)) + EFK,t(f(w ⊕ u (i))) − 2EFK,t(f(w)f(w ⊕ u (i))) Now we form a matrix with the truth tables of all functions with Hamming weight t as column vectors: c(1), c(2), · · · , c(( where c(i) ∈ F2 M has exactly columns and 2K rows. Each entry Mi,j in the ith row and jth column equals the value of function fj given the binary representation of i as input. Hence EFK,t(f(w)) is determined by the number of 1 in the row associated with w divided by the length of the row. Consider an arbitrary row i. This row has a one at position j if the corresponding column c(j) has a one at position i. But there are column vectors with a 1 at position i. It follows: ∀w ∈ FK2 : EFK,t(f(w)) = . (2) As this holds for all w, we have ∀w,u(i) ∈ FK2 : EFK,t(f(w⊕ u (i))) = . (3) To find an expression for EF fixed (f(w)f(w⊕u(i))) we consider two arbitrary rows l,m (l 6= m). Define the following sum: γl,m = (Kt ) Ml,iMm,i. Obviously Ml,iMm,i = 1 only if we have a 1 in both rows at position i. This means for the column vectors c(i) of M , we have c m = 1. But there are exactly such column vectors in M . Therefore we have ∀l,m, l 6= m : γl,m = 2K − 2 As w 6= w ⊕ u(i) for all w,u(i) it follows: EFK,t(f(w)f(w ⊕ u (i))) = t(t− 1) 2K(2K − 1) . (4) DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- Hence substituting Equations (2), (3) and (4) into Equation (1) leads to EFK,t(sf ) = K(2K − t)t 2K−1(2K − 1) Finally the claimed expression for EF fixed (sf ) can be obtained from the above equation by a substitution of t: t → p2K . It should be noted, that the Theorems 1 and 2 can be proved using in a similar way. Also worth noting is the fact, that if the functions are chosen according FK , F fixed K,p or FK,p the expectation of the sensitivity of a fixed vector w (namely the expectation of sf (w)) is independent of w (see Equation (1),(2), (3) and (4)). Hence the following lemma holds Lemma 1. If F = FK , F fixed or FK,p, then ∀w,v ∈ FK2 : EF (sf (w)) = EF (sf (v)) Before proceeding to the next section, it should be noted, that using the same arguments as in the proof of Theorem 3, we can also prove the expectation of average sensitivity of order l , defined as s(l)(f) = 2−K x ∈ FK2 |wH(x) = l and f(w) 6= f(w⊕ x) In this case, instead of summing up all unit vectors in Equation (1), we sum up all vectors of Hamming weight l. As the equations (2) and (4) hold for all w ∈ FK2 we conclude that E(s(l)(F fixedK,p )) = 2K+1p(1− p) (2K − 1) and by similar arguments E(s(l)(FK,p)) = 2p(1− p) respectively E(s(l)(FK)) = 4 Extending Lynch’s analysis As already mentioned James Lynch gave a very general analysis of randomly constructed Boolean networks (see [4]). Before stating his results we give a formal definition for Boolean networks A Boolean network B is a 4-tuple DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- (V,E, F̃ ,x) where V = {1, ..., N} is a set of natural numbers, E is a set of labeled edges on V , F̃ = {f1, ..., fN} is a ordered set of Boolean functions such that for each v ∈ V the number of arguments of fv is the in-degree of v in E, these edges are labeled with 1, ..., in-degree(v), and x = (x1, . . . xn) ∈ F Suppose that a vertex i has Ki in-edges from vertices vi,1, . . . , vi,Ki . For y ∈ F we define B(y) = f1(yv1,1 , . . . , yv1,K1 ), . . . , fN(yvN,1 , . . . , yvN,KN ) The state of B at time 0 is called the initial state x, so we define B0(x) = x. For time t ≥ 1 the state is inductively defined as Bt(x) = B(Bt−1(x)). Hence we can in interpret V as set of gates, E and F̃ describes their functional dependence and x is the networks initial state. Assume some ordering f1, f2, ... on the set of all Boolean functions F , where each function fi depends on Ki arguments. Further a random variable F : Ω → F with probabilities pi = P (F = fi) such that i=i pi = 1 and i=1 piK ∞. Now a random Boolean network consisting of N gates is constructed as follows: For each gate a Boolean function is chosen independently, where the probability of choosing fi is given by pi. Suppose a function f was chosen that has K arguments, these arguments are chosen at random from all equally likely possibilities. At last an initial state is chosen at random from the set on all equally likely states. If the Boolean functions are chosen according to our previously defined random variable FK we will call this networks NK-Networks with connectivity K. If the functions are chosen according to F fixedK,p or FK,p we will call this networks biased random Boolean networks with connectivity K and fixed bias p respectively mean bias p. Let us now state Lynch’s results. His analysis depends on a parameter R ∋ λ ≥ 0 depending only on the functions and their probabilities. We will define λ later in Definition 3. First we have to state Lynch’s definition of freezing and ineffective gates: Definition 2 (Lynch [4] Definition 1 Item 2 and 5). Let x ∈ FN2 and v ∈ V . 1. Gate v freezes to y ∈ FN2 in t steps on input x if B v (x) = y for all t ′ ≥ t. 2. Let u(i) ∈ Fn2 . A gate v is t-ineffective at input x ∈ FK2 if B t(x) = Bt(x⊕ u(v)). Now we will state the main result. Theorem 4 (Lynch [4] Theorem 4 and 6). Let α, β be positive constants satisfying 2α log δ+2β < 1 and α log δ < β where δ = E(Ki). 1. There is a constant r such that for all x ∈ FN2 P (v is ineffective in α log N steps) = r When λ ≤ 1, r = 1 and when λ > 1 , r < 1. DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 2. There is a constant r such that for all x ∈ FN2 P (v is freezing in α log N steps) = r When λ ≤ 1, r = 1 and when λ > 1 , r < 1. 3 The above theorem shows that if λ ≤ 1 almost all gates are freezing and ineffective and otherwise not. The next corollary gives us more information what happens if λ > 1: Corollary 1 (Lynch [4] Corollary 3 and Corollary 6). Let λ > 1. For almost all random Boolean networks 1. if gate v is not α logN -ineffective, there is a positive constant W such that for t ≤ α logN , the number of gates affected by v at time t is asymptotic to Wλt, 2. if gate v is not freezing in α logN steps , there is a positive constant W such that for t ≤ α logN , the number of gates that affect v at time t is asymptotic to Wλt. Now we will state the definition of λ for Boolean networks: Definition 3 (Lynch [4], Definition 4). Let f be a Boolean function of K ar- guments. For i ∈ {1, . . . ,K}, we say that argument i directly affects f on input w ∈ FK2 if f(w) 6= f(w ⊕ u (i)). Now put γ(f,w) as the number of i’s that directly affect f on input w. Given a constant a ∈ [0, 1], we define γ(fi,w)a wH(w)(1− a)Ki−wH(w). Obviously γ(f,w) is identical to sf (w) which will be used instead in the further discussion. The constant a is the probability that a random gate is one (at infinite time) given that all gates at time 0 have probability 0.5 of being one. (see [4, Definiton 2]). Assume that we choose the functions according a random variable F which should be either FK , F fixed K,p or FK,p. The functions are chosen out the set FK , we denote a function’s probability with pf . It follows that awH(w)(1− a)K−wH(w) pfsf (w) (5) awH(w)(1− a)K−wH(w)E(sF (w)) (6) = E(sF (w)) ai(1− a)K−i (7) = E(sF (w)) = EF (sf ) (8) 3Please note that we here state a slightly weaker result than in the original analysis. DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- E(sF (w)) denotes the expectation of the sensitivity for a fixed w, Equation (7) follows from Lemma 1. Therefore, together with Theorem 1 and Theorem 3 we proved the following: Theorem 5 (Biased random Boolean networks). For random Boolean networks, 1. the functions are chosen according random variable FK,p, it follows that λ = 2Kp(1− p), 2. the functions are chosen according random variable F fixed , it follows that 2K+1Kp(1− p) 2K − 1 As a special case of the above theorem we get (or by using Theorem 1) Theorem 6 (NK-Networks). In random Boolean networks, where the functions are chosen according to the random variable FK 5 Discussion The results about NK-Networks are consistent with experimental results. In fact if K ≤ 2 almost all networks almost all gates are freezing and almost all gates are ineffective and otherwise not (see [2]). Obviously, the border between the ordered and disordered phase is given by λ = 1. The resulting phase diagram for biased random Boolean networks, where the functions are chosen according to F fixedK,p and FK,p is shown in Figure 1. It it interesting to note that if the functions are chosen with fixed bias, then also Boolean networks with connectivity K = 2 can become unstable. This conclusion can be drawn from Lynch’s original result already. As mentioned in the introduction, he showed for K = 2, that λ > 1 if the probability of choosing a non-constant non-canalizing function, namely the XOR or the inverted XOR function, is larger than the probability of choosing a constant function. For example if the bias is 0.5, the probability of choosing a constant function is zero, whereas both XOR and inverted XOR function have probability greater zero, hence λ > 1. It is interesting to compare our results with previous results obtained first by Derrida and Pomeau using the so called annealed approximation (see [3]). In their annealed model the functions and connections are chosen at random at each time step. Considering two instances of the same annealed network starting in two randomly chosen initial states s1(0), s2(0) they show that dH(s1(t), s2(t)) DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- 1 3 4 52 Figure 1: Phase diagram for biased random networks: Functions chosen accord- ing FK,p (dashed) and F fixed K,p (solid) where c = 1 if 2Kp(1− p) ≤ 1 and c ≤ 1 otherwise. It is remarkable that the two models behave similar, but it is unclear whether this holds in general. 6 Acknowledgement We would like to thank our colleges Georg Schmidt and Stephan Stiglmayr for proofreading and Uwe Schoening for useful hints. References [1] S. Kauffman, Metabolic stability and epigenesis in randomly constructed nets, Journal of Theoretical Biology 22 (1969) 437–467. [2] S. Kauffman, The large scale structure and dynamics of genetic control circuits: an ensemble approach, Journal of Theoretical Biology 44 (1974) 167–190. [3] B. Derrida, Y. Pomeau, Random networks of automata - a simple annealed approximation, Europhysics Letters 2 (1986) 45–49. [4] J. F. Lynch, Dynamics of random boolean networks, in: Conference on Mathematical Biology and Dynamical Systems, University of Texas at Tyler, 2005. DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- [5] J. F. Lynch, On the threshold of chaos in random boolean cellular au- tomata, Random Structures and Algorithms (1995) 236–260. [6] J. F. Lynch, Critical points for random boolean networks, Physica D: Non- linear Phenomena 172 (1-4) (2002) 49–64. [7] I. Wegener, The Complexity of Boolean Functions, Wiley-Teubner Series in Computer Science, John Wiley, B.G. Teubner, 1987. [8] I. Shmulevich, S. A. Kauffman, Activities and sensitivities in boolean net- work models, Physical Review Letters 93 (4). [9] S. Kauffman, C. Peterson, B. Samuelsson, C. Troeln, Genetic networks with canalyzing boolean rules are always stable, Proceedings of the Nationial Academy of Science 101 (49). [10] A. Bernasconi, Mathematical techniques for the analysis of boolean func- tions, Ph.D. thesis, Dipartimento di Informatica, Universita di Pisa (March 1998).

0704.0198

Theory of polariton mediated Raman scattering in microcavities

Theory of polariton mediated Raman scattering in microcavities L. M. León Hilario, A. Bruchhausen, A. M. Lobos, and A. A. Aligia Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Enerǵıa Atómica, 8400 S. C. de Bariloche, Argentina (Dated: November 4, 2018) Abstract We calculate the intensity of the polariton mediated inelastic light scattering in semiconductor microcavities. We treat the exciton-photon coupling nonperturbatively and incorporate lifetime effects in both excitons and photons, and a coupling of the photons to the electron-hole continuum. Taking the matrix elements as fitting parameters, the results are in excellent agreement with measured Raman intensities due to optical phonons resonant with the upper polariton branches in II-VI microcavities with embedded CdTe quantum wells. PACS numbers: 71.36.+c, 78.30.Fs, 78.30.-j http://arxiv.org/abs/0704.0198v1 Planar semiconductors microcavities (MC’s) have attracted much attention in the last decade as they provide a novel means to study, enhance and control the interaction between light and matter [1, 2, 3, 4, 5, 6, 7]. When the MC mode (cavity-photon) is tuned in near resonance with the embedded quantum-well (QW) exciton transitions, and the damping processes involved are weak in comparison to the photon-matter interaction, the eigenstates of the system become mixed exciton-photon states, cavity-polaritons, which are in part light and in part matter bosonic quasi-particles [1, 2, 3]. Examples of interesting new physics are the recent evidence of a Bose-Einstein condensation of polaritons in CdTe MC’s [5, 6], and the construction of devices which increase the interaction of sound and light, opening the possibility of realizing a coherent monochromatic source of acoustic phonons [7]. Raman scattering due to longitudinal optical (LO) phonons, being a coherent process is intrinsically connected with the cavity-polariton. The physics of strongly coupled photons and excitons, the polariton–phonon interaction, and the polariton–external-photon coupling are clearly displayed [8, 9, 10, 11, 12]. In particular resonant Raman scattering (RRS) experiments, in which the wave-length of the incoming radiation is tuned in a way such that, after the emission of a LO phonon, the energy of the outgoing radiation coincides with that of the cavity polariton branches, have proven to be suited to sense the dynamics of the coupled modes, and to obtain information about the dephasing of the resonant polaritonic state [8, 9, 10, 11, 12]. Unfortunately, due to the large remaining luminesence, RRS experiments in resonance with the lower polariton branch have not yet been achieved in intrinsic II-VI MC’s [13]. Therefore all reported experiments in these kind of MC’s, with embedded CdTe QW’s, consider the case of the scattered photons in outgoing resonance with the upper polariton branch (for two branch-systems: coupling of one exciton mode and the cavity photon) or with the middle polariton branch (for tree branch-systems: coupling of two exciton modes and the cavity photon) [9, 10, 14]. The measured intensities in these two systems were analyzed on the basis of a model in which one (two) exciton states |e〉 are mixed with a photon state |f〉 with the same in-plane wave vector k, leading to a 2x2 (3x3) matrix. Essentially, the Raman intensity is proportional to I ∼ TiTs|〈Pi|H ′|Ps〉| 2, (1) where Ti describe the probability of conversion of an incident photon |fi〉 into the polariton state |Pi〉, Ts has an analogous meaning for the scattered polariton |Ps〉 and the outgoing photon |fs〉, and H ′ is the interaction between electrons and the LO phonons. At the conditions of resonance with the outgoing polariton, Ti is very weakly dependent on laser energy or detuning (difference between photon and exciton energies), while Ts is proportional to the photon strength of the scattered polariton |〈fs|Ps〉| 2. Similarly, in the simplest case of only one exciton (2x2 matrix) one expects that the matrix element entering Eq. (1) is proportional to the exciton part of the scattered polariton |〈Pi|H ′|Ps〉| 2 ∼ |〈es|Ps〉| 2. Thus if the wave function is |Ps〉 = α|fs〉+ β|es〉, one has I ∼ |〈fs|Ps〉| 2|〈es|Ps〉| 2 = |α|2|β|2. (2) This model predicts a Raman intensity which is symmetric with detuning and is maximum at zero detuning. In other words, the intensity is maximum for detunings such that the scattered polariton is more easily coupled to the external photons, but at the same time when the polariton is more easily coupled to the optical phonons, which requires a large matter (exciton) component of the polariton. This result is in qualitative agreement with experiment [10]. However, for positive detuning the experimental results fall bellow the values predicted by Eq. (2) (see Fig. 1). This is ascribed to the effects of the electron-hole continuum above the exciton energy, which are not included in the model [10]. For another sample in which two excitons are involved, the above analysis can be ex- tended straightforwardly and the intensity depends on the amplitudes of a 3x3 matrix and matrix element of H ′ involving both excitons [10]. However, comparison with experiments at resonance with the middle polariton branch, shows a poorer agreement than in the previous case (Fig. 6 of Ref. 10). In addition, a loss of coherence between the scattering of both excitons with the LO phonon was assumed, which is hard to justify. Some improvement has been obtained recently when damping effects are introduced phenomenologically as imagi- nary parts of the photon and exciton energies, but still a complete loss of coherence resulted from the fit [14, 15]. In this paper we include the states of the electron-hole continuum, and the damping effects in a more rigorous way. Using some matrix elements as free parameters, we can describe accurately the Raman intensities for both samples studied in Ref. 10. In the experiments with CdTe QW’s inside II-VI MC’s, the light incides perpendicular to the (x, y) plane of the QW’s and is collected in the same direction z. Therefore the in-plane wave vector K = 0, the polarization of the electric field should lie in the (x, y) plane, and the excitons which couple with the light should have the same symmetry as the electric field (one of the two Γ5 states of heavy hole excitons [16]). Thus, to lighten the notation we suppress wave vector and polarization indices. The basic ingredients of the theory are two or three strongly coupled boson modes, one for the light MC eigenmode with boson creation operator f †, another one for the 1s exciton (e1) and if is necessary, the 2s exciton (e2) is also included. We assume that each of these boson states mixes with a continuum of bosonic excitations which broadens its spectral density. In addition, we include the electron- hole continuum above the exciton states, described by bosonic operators c , ck, where k is the difference between electron and hole momentum in the (x, y) plane (the sum is K = 0 because it is conserved). The Hamiltonian reads: H = Eff iei + if +H.c.) + prp + pf +H.c.) iqdiq + (Viqd iqei +H.c.) + f +H.c.). (3) The first three terms describe the strong coupling between the MC photon and the exciton(s) already included in previous approaches [10, 15]. The fourth and fifth terms describe a continuum of radiative modes and its coupling to the MC light eigenmode. Their main effect is to broaden the spectral density of the latter even in the absence of light-matter interaction. The following two terms have a similar effect for the exciton mode(s). The detailed structure of the states described by the d iq operators is not important in what follows. They might describe combined excitations due to scattering with acoustical phonons. The last two terms correspond to the energy of the electron hole excitations and their coupling to the MC light mode. In Eq. (3) we are making the usual approximation of neglecting the internal fermionic structure of the excitons and electron-hole operators and taking them as free bosons. This is an excellent approximation for the conditions of the experiment. We also neglect terms which do not conserve the number of bosons. Their effect is small for the energies of interest [17]. These approximations allow us formally to diagonalize the Hamiltonian by a Bogoliubov transformation. The diagonalized Hamiltonian has the form H = νpν , where the boson operators p†ν correspond to generalized polariton operators and are linear combinations of all creation operators entering Eq. (3). Denoting the latter for brevity as b then p†ν = j . In practice, instead of calculating the Eν and Aνj , it is more convenient to work with retarded Green’s functions Gjl(ω) = 〈〈bj; b l 〉〉ω and their equations of motion ω〈〈bj; b l 〉〉ω = δjl + 〈〈[bj, H ]; b l 〉〉ω. (4) As we show below, the RRS intensity can be expressed in terms of spectral densities derived from these Green’s functions, which in turn can be calculated using Eq. (4). Using Fermi’s golden rule, the probability per unit time for a transition from a polariton state |i〉 = p |0〉 to states |s〉 = p†νa †|0〉, where a† creates a LO phonon is |〈i|H ′|s〉|2ρ(ω), where ρ(ω) = δ(ω − Eν) = − ImGνν(ω + i0 +) (5) is the density of final states. As argued above, we neglect the dependence of Ti on frequency and take Ts = |Aνf | 2, the weight of photons in the scattered eigenstates. In addition, H ′ should be proportional to the matter (exciton) part of the scattered polariton states. Then, the Raman intensity is proportional to: WTs ∝ I = |Aνe1 + αAνe2| 2|Aνf | 2ρ(ω). (6) Here we are neglecting the contribution of the electron-hole continuum to H ′, and α is the ratio of matrix elements of the exciton-LO phonon interaction between 2s and 1s excitons. If the 2s excitons are unimportant, α = 0. Using Eqs. (4) it can be shown that ρjl(ω) = − [Gjl(ω + i0 +)−Gjl(ω − i0 −)] = AνjĀνlρ(ω). (7) From here and |Aνj| 2 = 1, it follows that ρ(ω) = ρjj(ω). Replacing in Eq. (6) we obtain: I(ω) = ρff [ρe1,e1 + |α| 2ρe2,e2 + 2Re(αρe2,e1)]∑ ρjj(ω) . (8) In practice, when ω is chosen such that the resonance condition for the outgoing polariton is fulfilled, we can neglect the contribution of the continuum states in the denominator of Eq. (8). In particular, if the contribution of the 2s exciton can be neglected (as in sample A of Ref. 10) I(ω) = ρff (ω)ρe1,e1(ω) ρff(ω) + ρe1,e1(ω) . (9) The Green’s functions are calculated from the equations of motion (4). In the final expres- sions, the continuum states enter through the following sums: Sf (ω) = ω + i0+ − ǫp , Si(ω) = |Viq| ω + i0+ − ǫiq , S ′f(ω) = ω + i0+ − ǫk For the first two we assume that the results are imaginary constants that we take as param- eters: Sf(ω) = −iδf , Sj(ω) = −iδj (11) This is the result expected for constant density of states and matrix elements. Our results seem to indicate that this assumption is valid for the upper and middle polariton branches. For the lower branch at small k it has been shown that the line width due to the interaction of polaritons with acoustic phonons depends on detuning Ef −E1 and k‖, being smaller for small wave vector [18, 19]. The electron-hole continuum begins at the energy of the gap and corresponds to vertical transitions in which the light promotes a valence electron with 2D wave vector k to the conduction band with the same wave vector. In the effective-mass approximation, the energy is quadratic with k and this leads to a constant density of states beginning at the gap. The matrix element Vk is proportional to Mk = 〈kv|pE |kc〉, where pE is the momentum operator in the direction of the electric field, and |kv〉, |kc〉 are the wave functions for valence and conduction electrons with wave vector k. Taking these wave functions as plane waves, one has Mk ∼ kE , the wave vector in the direction of the electric field. Then |Vk| 2 ∼ k2E. Adding the contributions of all directions of k one has |Vk| 2 ∼ |k|2 ∼ ǫk (linear with energy for small energy). This leads to S ′f = R(ω)− iA(ω −B)Θ(ω −B), (12) where B is the bottom of the electron-hole continuum (the energy of the semiconductor gap) and A is a dimensionless parameter that controls the magnitude of the interaction. The real part R(ω) can be absorbed in a renormalization of the photon energy and is unimportant in what follows. The imaginary part is a correction to the photon width for energies above the bottom of the continuum. Using the theory outlined above, we calculated the intensity of RRS corresponding to the samples A and B measured in Refs. 10, 14 and compared them with the experimental results. Sample A corresponds to the simplest case. Two polariton branches are seen and therefore only the 1s exciton plays a significant role. The binding energy of this exciton B−E1 is not well known. The Rabi splitting 2V1 = 19 meV. The width of the Raman scan as a function of frequency for zero detuning is of the order of w = 0.1 meV (see Fig. 3 of Ref. 10). This implies the relation 2w2 = δ2f + δ in our theory. In any case the results are weakly sensitive to w. Therefore, we have three free parameters in our theory in addition to a multiplicative constant: B −E1, the ratio of widths δf/δ1 and the slope A. -15 -10 -5 0 5 10 15 Ef - E1 (meV) FIG. 1: Raman intensity as a function of detuning for sample A. Solid squares: experimental results [10]. Solid line: theory (Eq. 9) for B − E1 = 14 meV, δf = δ1 = 0.1 meV, and A = 0.031. Dashed line: result for a 2x2 matrix (Eq. 2). The comparison between the experimental and the theoretical intensities is shown in Fig. 1 for a set of parameters that lead to a close agreement with experiment. The condition of resonance is established choosing the energy ω for which the intensity given by Eq. (9) has its second relative maximum (corresponding to the upper polariton branch). The dashed line corresponds to the case in which only the first three terms (with i = 1) in the Hamiltonian, Eq. (3) are included. In this case, the intensity is given in terms of the solution of a 2x2 matrix [Eq. (2)] and was used in Ref. 10 to interpret the data. This simple expression gives a Raman intensity which is an even function of detuning. When the full model is considered, the Raman intensity falls more rapidly for large detuning Ef − E1 as a consequence of the hybridization of the photon with the electron-hole continuum. When the energy of the polariton increases beyond B entering the electron-hole continuum (corresponding to the kink in Fig. 1), the Raman peak broadens and loses intensity. The kink can be smoothed if the effect of the infinite excitonic levels below the continuum is included in the model (leading to an S ′f with continuous first derivative), but this is beyond the scope of this work. If the ratio δf/δ1 is enlarged, the Raman intensity increases for negative detuning with respect to its value for positive detuning. In the experiments with sample B three polariton branches are observed [14] and the 2s exciton plays a role. Experimentally, it is known that the binding energy for the two excitons are B − E1 =17 meV and B − E2 =2 meV. From the observed Rabi splitting one has 2V1 = 13 meV, 2V2 = 2.5 meV. In comparison with the previous case, we have the additional parameter α (the ratio of exciton-LO phonon matrix elements). In addition, to be able to describe well the intensity for low energies of the middle polariton (left part of the curve shown in Fig. 2), we need to assume a small linear dependence of E1 with the position of the incident laser spot in the sample. This dependence is also inferred form the observed luminescence spectrum [14]. In our model, this corresponds to a dependence of E1 with Ef : E1 = E + z(Ef − E ) (13) In Fig. 2 we show the intensity at the second maximum of I(ω) (corresponding to the middle polariton branch) as a function of the energy of this maximum. We also show in the figure experimental results taken at lower laser excitation and a slightly higher temperature (4.5 K) than those reported in Ref. 10. The slope which better describes the data is z = 0.14. This value is close to z = 0.155 which was obtained from a fit of the maxima of luminescence spectrum of the lower and middle polaritons. We have taken the same value for A as in Fig. 1. The agreement between theory and experiment is remarkable. As for sample A, the values of δi that result from the fit are reasonable in comparison with calculated values [18]. 1,645 1,650 1,655 1,660 1,665 Middle polariton energy (eV) FIG. 2: Raman intensity as a function of the middle polariton energy. Solid squares: experimental results [14]. Solid line: theory for δf = 0.2 meV, δ1 = 0.1 meV, δ2 = 0.12 meV, α = −0.45 and z = 0.14. A is the same as in Fig. 1 . In summary, we have proposed a theory to calculate Raman intensity for excitation of longitudinal optical phonons in microcavities, in which different matrix elements are incorporated as parameters of the model. The most important advance in comparison with previous simplified theories [10, 15] is the inclusion of the strong coupling of the electron- hole continuum with the microcavity photon. Inclusion of this coupling is essential when the energy of the polariton is near the bottom of the conduction band (at the right of Fig. 1). We also have included the effects of damping of excitons and photons, coupling them with a continuum of bosonic excitations. Simpler approaches have included the spectral widths δf and δi of photons and excitons as imaginary parts of the respective energies, leading to non-hermitian matrices. Taking some of the parameters of the model as free (δf/δi, B − E1 and A for sample A, δf , δi, z and α for sample B), we obtain excellent fits of the observed Raman intensities. The resulting values of the parameters agree with previous estimates, if they are available. We are not aware of previous estimates for A and α. As an important improvement to previous approaches [10, 15] for the case of sample B, we do not have to assume a partial loss of coherence between 1s and 2s excitons in their scattering with the LO phonon. Further progress in the understanding of the interaction of excitons with light and phonons requires microscopic calculations of the parameters δf , δi, A and α, and the effects of the temperature on them. However, taking into account the difficulties in calculating these parameters accurately, the present results are encouraging and suggest that the main physical ingredients are included in our model. We thank A. Fainstein for useful discussions. This work was supported by PIP 5254 of CONICET and PICT 03-13829 of ANPCyT. [1] Kavokin A and Malpuech G 2003 Cavity Polaritons (Elsevier, Amsterdam) [2] Special issue on microcavities 2003 Semicond. Sci. Technol. 18 10 S279-S434 [3] Special issue on Photon-mediated phenomena in semiconductor nanostructures J. Phys. Con- dens. Matter 18 35 S3549-S3768 [4] Skolnick M S, Fisher T and Whittaker D M 1998 Semicond. Sci. Technol 13 645 [5] Kasprzak J, Richard M, Kundermann M, Baas A, Jeambrun P, Keeling J M J, Marchetti F M, Szymanska M H, André R, Staehli J L, Savona V, Littlewood P B, Deveaud B and Le Si Dang 2006 Nature 443 409 [6] Deng H, Press D, Götzinger S, Solomon G S, Hey R, Ploog K H and Yamamoto Y 2006 Phys. Rev. Lett. 97 146402 [7] Trigo M, Bruchhausen A, Fainstein A, Jusserand B and Thierry-Mieg V 2002 Phys. Rev. Lett. 89 227402 [8] Fainstein A, Jusserand B and Thierry-Mieg V 1997 Phys. Rev. Lett. 78 1576 [9] Fainstein A, Jusserand B and André R 1998 Phys. Rev. B 57 R9439 [10] Bruchhausen A, Fainstein A, Jusserand B and André R 2003 Phys. Rev. B 68 205326 [11] Tribe W R, Baxter D, Skolnick M S, Mowbray D J, Fisher T A and Roberts J S 1997 Phys. Rev. B 56 12 429 [12] Stevenson R M, Astratov V N, Skolnick M S, Roberts J S and Hill G 2003 Phys. Rev. B 67 081301(R) [13] RRS experiments in resonance with the lower polariton branch have only been reported in III-VI samples with doped Bragg reflectors (see refs. [11] and [12] for details). [14] Fainstein A and Jusserand B 2006 in Light Scattering in Solids vol. 9 Cardona M and Merlin R editors (Springer, Berlin) [15] Bruchhausen A, Fainstein A and Jusserand B 2005 Physics of semiconductors CP772 p.1117 Menéndez J and Van de Walle C G editors (American Institute of Physics) [16] Jorda S, Rössler U and Broido D 1993 Phys. Rev. B 48 1669 [17] Jorda S 1994 Phys. Rev. B 50 2283 [18] Savona V and Piermarocchi C 1997 Phys. Status Solidi A 164 45 [19] Cassabois G, Triques A L C, Bogani F, Delalande C, Roussignol Ph and Piermarocchi C 2000 Phys. Rev. B 61 1696 References

0704.0199

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS AND GENERALISED NON-CROSSING PARTITIONS C. KRATTENTHALER† AND T. W. MÜLLER Abstract. Given a finite irreducible Coxeter group W , a positive integer d, and types T1, T2, . . . , Td (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = σ1σ2 · · ·σd of a Coxeter element c of W , such that σi is a Coxeter element in a subgroup of type Ti in W , i = 1, 2, . . . , d, and such that the factorisation is “minimal” in the sense that the sum of the ranks of the Ti’s, i = 1, 2, . . . , d, equals the rank of W . For the exceptional types, these decomposition numbers have been computed by the first author in [“Topics in Dis- crete Mathematics,” M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126] and [Séminaire Lotharingien Combin. 54 (2006), Article B54l]. The type An decomposition numbers have been computed by Goulden and Jackson in [Europ. J. Combin. 13 (1992), 357–365], albeit using a somewhat different language. We explain how to extract the type Bn decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22–56] on map enumeration. Our formula for the type Dn decomposition numbers is new. These results are then used to determine, for a fixed positive integer l and fixed integers r1 ≤ r2 ≤ · · · ≤ rl, the number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl in Armstrong’s generalised non-crossing partitions poset, where the poset rank of πi equals ri, and where the “block structure” of π1 is prescribed. We demonstrate that this result implies all known enumerative re- sults on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non- crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type Dn generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong’s F = M Conjecture in type Dn, thus completing a computational proof of the F = M Conjecture for all types. It also allows to address another conjecture of Armstrong on maximal intervals containing a random multichain in the generalised non-crossing partitions poset. 1. Introduction The introduction of non-crossing partitions for finite reflection groups (finite Coxeter groups) by Bessis [8] and Brady and Watt [15] marks the creation of a new, exciting subject of combinatorial theory, namely the study of these new combinatorial objects which possess numerous beautiful properties, and seem to relate to several other ob- jects of combinatorics and algebra, most notably to the cluster complex of Fomin and 2000 Mathematics Subject Classification. Primary 05E15; Secondary 05A05 05A10 05A15 05A18 06A07 20F55 33C05. Key words and phrases. root systems, reflection groups, Coxeter groups, generalised non-crossing partitions, annular non-crossing partitions, chain enumeration, Möbius function, M -triangle, gener- alised cluster complex, face numbers, F -triangle, Chu–Vandermonde summation. †Research partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. http://arxiv.org/abs/0704.0199v3 2 C. KRATTENTHALER AND T. W. MÜLLER Zelevinsky [21] (cf. [2, 3, 5, 4, 8, 9, 14, 15, 16, 17, 20]). They reduce to the classical non-crossing partitions of Kreweras [30] for the irreducible reflection groups of type An (i.e., the symmetric groups), and to Reiner’s [32] type Bn non-crossing partitions for the irreducible reflections groups of type Bn. (They differ, however, from the type Dn non-crossing partitions of [32].) The subject has been enriched by Armstrong through introduction of his generalised non-crossing partitions for reflection groups in [1]. In the symmetric group case, these reduce to the m-divisible non-crossing partitions of Edelman [18], while they produce new combinatorial objects already for the reflection groups of type Bn. Again, these generalised non-crossing partitions possess numerous beautiful properties, and seem to relate to several other objects of combinatorics and algebra, most notably to the generalised cluster complex of Fomin and Reading [19] (cf. [1, 6, 7, 20, 27, 28, 29, 36, 37, 38]). From a technical point of view, the main subject matter of the present paper is the computation of the number of certain factorisations of the Coxeter element of a reflection group. These decomposition numbers, as we shall call them from now on (see Section 2 for the precise definition), arose in [27, 28], where it was shown that they play a crucial role in the computation of enumerative invariants of (generalised) non-crossing partitions. Moreover, in these two papers the decomposition numbers for the exceptional reflection groups have been computed, and it was pointed out that the decomposition numbers in type An (i.e., the decomposition numbers for the symmetric groups) had been earlier computed by Goulden and Jackson in [23]. Here, we explain how the decomposition numbers in type Bn can be extracted from results of Bóna, Bousquet, Labelle and Leroux [12] on the enumeration of certain planar maps, and we find formulae for the decomposition numbers in type Dn, thus completing the project of computing the decomposition numbers for all the irreducible reflection groups. The main goal of the present paper, however, is to access the enumerative theory of the generalised non-crossing partitions of Armstrong via these decomposition num- bers. Indeed, one finds numerous enumerative results on ordinary and generalised non-crossing partitions in the literature (cf. [1, 2, 4, 8, 9, 18, 30, 32, 37]): results on the total number of (generalised) non-crossing partitions of a given size, of those with a fixed number of blocks, of those with a given block structure, results on the number of (multi-)chains of a given length in a given poset of (generalised) non-crossing partitions, results on rank-selected chain enumeration (that is, results on the number of chains in which the ranks of the elements of the chains have been fixed), etc. We show that not only can all these results be rederived from our decomposition numbers, we are also able to find several new enumerative results. In this regard, the most general type of result that we find is formulae for the number of (multi-)chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the poset of non-crossing partitions of type An, Bn, respectively Dn, in which the block structure of π1 is fixed as well as the ranks of π2, . . . , πl−1. Even the corresponding result in type An, for the non-crossing partitions of Kreweras, is new. Furthermore, from the result in type Dn, by a suitable summation, we are able to find a formula for the rank-selected chain enumeration in the poset of generalised non-crossing partitions of type Dn, thus generalising the earlier formula of Athanasiadis and Reiner [4] for the rank-selected chain enumeration of “ordinary” non-crossing partitions of type Dn. In conjunction with the results from [27, 28], this generalisation in turn allows us to complete a computational case-by-case proof of Armstrong’s “F = M Conjecture” [1, Conjecture 5.3.2] predicting a surprising relationship between a certain face count in the DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 3 generalised cluster complex of Fomin and Reading and the Möbius function in the poset of generalised non-crossing partitions of Armstrong. (A case-free proof had been found earlier by Tzanaki in [38].) Our results allow us also to address another conjecture of Armstrong [1, Conj. 3.5.13] on maximal intervals containing a random multichain in the poset of generalised non-crossing partitions. We show that the conjecture is indeed true for types An and Bn, but that it fails for type Dn (and we suspect that it will also fail for most of the exceptional types). We remark that a totally different approach to the enumerative theory of (gener- alised) non-crossing partitions is proposed in [29]. This approach is, however, com- pletely combinatorial and avoids, in particular, reflection groups. It is, therefore, not capable of computing our decomposition numbers nor anything else which is intrinsic to the combinatorics of reflection groups. A similar remark applies to [31, Theorem 4.1], where a remarkable uniform recurrence is found for rank-selected chain enumeration in the generalised non-crossing partitions of any type. It could be used, for example, for verifying our result in Corollary 19 on the rank-selected chain enumeration in the generalised non-crossing partitions of type Dn, but it is not capable of computing our decomposition numbers nor of verifying results with restrictions on block structure. Our paper is organised as follows. In the next section we define the decomposition numbers for finite reflection groups from [27, 28], the central objects in our paper, together with a combinatorial variant, which depends on combinatorial realisations of non-crossing partitions, which we also explain in the same section. This is followed by an intermediate section in which we collect together some auxiliary results that will be needed later on. In Section 4, we recall Goulden and Jackson’s formula [23] for the full rank decomposition numbers of type An, together with the formula from [28, Theorem 10] that it implies for the decomposition numbers of type An of arbitrary rank. The purpose of Section 5 is to explain how formulae for the decomposition numbers of type Bn can be extracted from results of Bóna, Bousquet, Labelle and Leroux in [12]. The type Dn decomposition numbers are computed in Section 6. The approach that we follow is, essentially, the approach of Goulden and Jackson in [23]: we translate the counting problem into the problem of enumerating certain maps. This problem is then solved by a combinatorial decomposition of these maps, translating the decomposition into a system of equations for corresponding generating functions, and finally solving this system with the help of the multidimensional Lagrange inversion formula of Good. Sections 7–11 form the “applications part” of the paper. In the preparatory Section 7, we recall the definition of the generalised non-crossing partitions of Armstrong, and we explain the combinatorial realisations of the generalised non-crossing partitions for the types An, Bn, and Dn from [1] and [29]. The bulk of the applications is contained in Section 8, where we present three theorems, Theorems 11, 13, and 15, on the number of factorisations of a Coxeter element of type An, Bn, respectively Dn, with less stringent restrictions on the factors than for the decomposition numbers. These theorems result from our formulae for the (combinatorial) decomposition numbers upon appropriate summations. Subsequently, it is shown that the corresponding formulae imply all known enumeration results on non-crossing partitions and generalised non-crossing partitions, plus several new ones, see Corollaries 12, 14, 16–19 and the accompanying remarks. Section 9 presents the announced computational proof of the F = M (ex-)Conjecture for type Dn, based on our formula in Corollary 19 for the rank-selected chain enumeration in the poset of generalised non-crossing partitions of type Dn, while Section 10 addresses 4 C. KRATTENTHALER AND T. W. MÜLLER Conjecture 3.5.13 from [1], showing that it does not hold in general since it fails in type Dn. In the final Section 11 we point out that the decomposition numbers do not only allow one to derive enumerative results for the generalised non-crossing partitions of the classical types, they also provide all the means for doing this for the exceptional types. For the convenience of the reader, we list the values of the decomposition numbers for the exceptional types that have been computed in [27, 28] in an appendix. In concluding the introduction, we want to attract the reader’s attention to the fact that many of the formulae presented here are very combinatorial in nature (see Sections 4, 5, 8). This raises the natural question as to whether it is possible to find combinatorial proofs for them. Indeed, a combinatorial (and, in fact, almost bijective) proof of the formula of Goulden and Jackson, presented here in Theorem 5, has been given by Bousquet, Chauve and Schaeffer in [13]. Moreover, most of the proofs for the known enumeration results on (generalised) non-crossing partitions presented in [1, 2, 4, 18, 32] are combinatorial. On the other hand, to our knowledge so far nobody has given a combinatorial proof for Theorem 7, the formula for the decomposition numbers of type Bn, essentially due to Bóna, Bousquet, Labelle and Leroux [12], although we believe that this should be possible by modifying the ideas from [13]. There are also other formulae in our paper (see e.g. Corollaries 12 and 14, Eqs. (6.1) and (8.33)) which seem amenable to combinatorial proofs. However, to find combinatorial proofs for our type Dn results (cf. in particular Theorem 9.(ii) and Corollaries 16–19) seems rather hopeless to us. 2. Decomposition numbers for finite Coxeter groups In this section, we introduce the decomposition numbers from [27, 28], which are (Coxeter) group-theoretical in nature, plus combinatorial variants for Coxeter groups of types Bn and Dn, which will be important in combinatorial applications. These variants depend on the combinatorial realisation of these Coxeter groups, which we also explain here. Let Φ be a finite root system of rank n. (We refer the reader to [24] for all terminology on root systems.) For an element α ∈ Φ, let tα denote the corresponding reflection in the central hyperplane perpendicular to α. Let W = W (Φ) be the group generated by these reflections. As is well-known (cf. e.g. [24, Sec. 6.4]), any such reflection group is at the same time a finite Coxeter group, and all finite Coxeter groups can be realised in this way. By definition, any element w of W can be represented as a product w = t1t2 · · · tℓ, where the ti’s are reflections. We call the minimal number of reflections which is needed for such a product representation the absolute length of w, and we denote it by ℓT (w). We then define the absolute order on W , denoted by ≤T , via u ≤T w if and only if ℓT (w) = ℓT (u) + ℓT (u −1w). As is well-known and easy to see, this is equivalent to the statement that every shortest representation of u by reflections occurs as an initial segment in some shortest product representation of w by reflections. Now, for a finite root system Φ of rank n, types T1, T2, . . . , Td (in the sense of the classification of finite Coxeter groups), and a Coxeter element c, the decomposition number NΦ(T1, T2, . . . , Td) is defined as the number of “minimal” products c1c2 · · · cd less than or equal to c in absolute order, “minimal” meaning that ℓT (c1) + ℓT (c2) + · · ·+ ℓT (cd) = ℓT (c1c2 · · · cd), such that, for i = 1, 2, . . . , d, the type of ci as a parabolic DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 5 Coxeter element is Ti. (Here, the term “parabolic Coxeter element” means a Coxeter element in some parabolic subgroup. The reader should recall that it follows from [8, Lemma 1.4.3] that any element ci is indeed a Coxeter element in a parabolic subgroup of W = W (Φ). By definition, the type of ci is the type of this parabolic subgroup. The reader should also note that, because of the rewriting c1c2 · · · cd = ci(c i c1ci)(c i c2ci) · · · (c i ci−1ci)ci+1 · · · cd, (2.1) any ci in such a minimal product c1c2 · · · cd ≤T c is itself ≤T c.) It is easy to see that the decomposition numbers are independent of the choice of the Coxeter element c. (This follows from the well-known fact that any two Coxeter elements are conjugate to each other; cf. [24, Sec. 3.16].) The decomposition numbers satisfy several linear relations between themselves. First of all, the number NΦ(T1, T2, . . . , Td) is independent of the order of the types T1, T2, . . . , Td; that is, we have NΦ(Tσ(1), Tσ(2), . . . , Tσ(d)) = NΦ(T1, T2, . . . , Td) (2.2) for every permutation σ of {1, 2, . . . , d}. This is, in fact, a consequence of the rewriting (2.1). Furthermore, by the definition of these numbers, those of “lower rank” can be computed from those of “full rank.” To be precise, we have NΦ(T1, T2, . . . , Td) = NΦ(T1, T2, . . . , Td, T ), (2.3) where the sum is taken over all types T of rank n − rkT1 − rkT2 − · · · − rkTd (with rkT denoting the rank of the root system Ψ of type T , and n still denoting the rank of the fixed root system Φ; for later use we record that ℓT (w0) = rkT0 (2.4) for any parabolic Coxeter element w0 of type T0). The decomposition numbers for the exceptional types have been computed in [27, 28]. For the benefit of the reader, we reproduce these numbers in the appendix. The decomposition numbers for type An are given in Section 4, the ones for type Bn are computed in Section 5, while the ones for type Dn are computed in Section 6. Next we introduce variants of the above decomposition numbers for the types Bn and Dn, which depend on the combinatorial realisation of the Coxeter groups of these types. As is well-known, the reflection group W (An) can be realised as the symmetric group Sn+1 on {1, 2, . . . , n+1}. The reflection groups W (Bn) and W (Dn), on the other hand, can be realised as subgroups of the symmetric group on 2n elements. (See e.g. [11, Sections 8.1 and 8.2].) Namely, the reflection group W (Bn) can be realised as the subgroup of the group of all permutations π of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄} satisfying the property π(̄i) = π(i). (2.5) (Here, and in what follows, ¯̄i is identified with i for all i.) In this realisation, there is an analogue of the disjoint cycle decomposition of permutations. Namely, every 6 C. KRATTENTHALER AND T. W. MÜLLER π ∈ W (Bn) can be decomposed as π = κ1κ2 · · ·κs, (2.6) where, for i = 1, 2, . . . , s, κi is of one of two possible types of “cycles”: a type A cycle, by which we mean a permutation of the form ((a1, a2, . . . , ak)) := (a1, a2, . . . , ak) (a1, a2, . . . , ak), (2.7) or a type B cycle, by which we mean a permutation of the form [a1, a2, . . . , ak] := (a1, a2, . . . , ak, a1, a2, . . . , ak), (2.8) a1, a2, . . . , ak ∈ {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}. (Here we adopt notation from [15].) In both cases, we call k the length of the “cycle.” The decomposition (2.6) is unique up to a reordering of the κi’s. We call a type A cycle of length k of combinatorial type Ak−1, while we call a type B cycle of length k of combinatorial type Bk, k = 1, 2, . . . . The reader should observe that, when regarded as a parabolic Coxeter element, for k ≥ 2 a type A cycle of length k has type Ak−1, while a type B cycle of length k has type Bk. However, a type B cycle of length 1, that is, a permutation of the form (i, ī), has type A1 when regarded as a parabolic Coxeter element, while we say that it has combinatorial type B1. (The reader should recall that, in the classification of finite Coxeter groups, the type B1 does not occur, respectively, that sometimes B1 is identified with A1. Here, when we speak of “combinatorial type,” then we do distinguish between A1 and B1. For example, the “cycles” ((1, 2)) = (1, 2) (1̄, 2̄) or ((1̄, 2)) = (1̄, 2) (1, 2̄) have combinatorial type A1, whereas the cycles [1] = (1, 1̄) or [2] = (2, 2̄) have combinatorial type B1.) As Coxeter element for W (Bn), we choose c = (1, 2, . . . , n, 1̄, 2̄, . . . , n̄) = [1, 2, . . . , n]. Now, given combinatorial types T1, T2, . . . , Td, each of which being a product of Ak’s and Bk’s, k = 1, 2, . . . , the combinatorial decomposition number N (T1, T2, . . . , Td) is defined as the number of minimal products c1c2 · · · cd less than or equal to c in absolute order, where “minimal” has the same meaning as above, such that for i = 1, 2, . . . , d the combinatorial type of ci is Ti. Because of (2.1), the combinatorial decomposition numbers N combBn (T1, T2, . . . , Td) satisfy also (2.2) and (2.3). The reflection group W (Dn) can be realised as the subgroup of the group of all permutations π of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄} satisfying (2.5) and the property that an even number of elements from {1, 2, . . . , n} is mapped to an element of negative sign. (Here, the elements 1, 2, . . . , n are considered to have sign +, while the elements 1̄, 2̄, . . . , n̄ are considered to have sign −.) Since W (Dn) is a subgroup of W (Bn), and since the above realisation of W (Dn) is contained as a subset in the realisation of W (Bn) that we just described, any π ∈ W (Dn) can be decomposed as in (2.6), where, for i = 1, 2, . . . , d, κi is either a type A or a type B cycle. Requiring that π is in the subgroup W (Dn) of W (Bn) is equivalent to requiring that there is an even number of type B cycles in the decomposition (2.6). Again, the decomposition (2.6) for π ∈ W (Dn) is unique up to a reordering of the κi’s. As Coxeter element, we choose c = (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1) (n, n̄) = [1, 2, . . . , n− 1] [n]. DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 7 We shall be entirely concerned with elements π of W (Dn) which are less than or equal to c. It is not difficult to see (and it is shown in [4, Sec. 3]) that the unique factorisation of any such element π has either 0 or 2 type B cycles, and in the latter case one of the type B cycles is [n] = (n, n̄). In this latter case, in abuse of terminology, we call the product of these two type B cycles, [a1, a2, . . . , ak−1] [n] say, a “cycle” of combinatorial type Dk. More generally, we shall say for any product of two disjoint type B cycles of the form [a1, a2, . . . , ak−1] [ak] (2.9) that it is a “cycle” of combinatorial type Dk. The reader should observe that, when regarded as parabolic Coxeter element, for k ≥ 4 an element of the form (2.9) has type Dk. However, if k = 3, it has type A3 when regarded as parabolic Coxeter element, while we say that it has combinatorial type D3, and, if k = 2, it has type A 1 when regarded as parabolic Coxeter element, while we say that it has combinatorial type D2. (The reader should recall that, in the classification of finite Coxeter groups, the types D3 and D2 do not occur, respectively, that sometimes D3 is identified with A3, D2 being identified with A 1. Here, when we speak of “combinatorial type,” then we do distinguish between D3 and A3, and between D2 and A Now, given combinatorial types T1, T2, . . . , Td, each of which being a product of Ak’s and Dk’s, k = 1, 2, . . . , the combinatorial decomposition number N (T1, T2, . . . , Td) is defined as the number of minimal products c1c2 · · · cd less than or equal to c in absolute order, where “minimal” has the same meaning as above, such that for i = 1, 2, . . . , d the combinatorial type of ci is Ti. Because of (2.1), the combinatorial decomposition numbers N combDn (T1, T2, . . . , Td) satisfy also (2.2) and (2.3). 3. Auxiliary results In our computations in the proof of Theorem 9, leading to the determination of the decomposition numbers of type Dn, we need to apply the Lagrange–Good inversion formula [22] (see also [26, Sec. 5] and the references cited therein). We recall it here for the convenience of the reader. In doing so, we use standard multi-index notation. Name- ly, given a positive integer d, and vectors z = (z1, z2, . . . , zd) and n = (n1, n2, . . . , nd), we write zn for zn11 z 2 · · · z d . Furthermore, in abuse of notation, given a formal power series f in d variables, f(z) stands for f(z1, z2, . . . , zd). Moreover, given d formal power series f1, f2, . . . , fd in d variables, f n(z) is short for fn11 (z1, z2, . . . , zd)f 2 (z1, z2, . . . , zd) · · · f d (z1, z2, . . . , zd). Finally, if m = (m1, m2, . . . , md) is another vector, then m + n is short for (m1 + n1, m2 + n2, . . . , md + nd). Notation such as m − n has to be interpreted in a similar Theorem 1 (Lagrange–Good inversion). Let d be a positive integer, and let f1(z), f2(z), . . . , fd(z) be formal power series in z = (z1, z2, . . . , zd) with the property that, for all i, fi(z) is of the form zi/ϕi(z) for some formal power series ϕi(z) with ϕi(0, 0, . . . , 0) 6= 0. Then, if we expand a formal power series g(z) in terms of powers of the fi(z), g(z) = n(z), (3.1) 8 C. KRATTENTHALER AND T. W. MÜLLER the coefficients γn are given by g(z)f−n−e(z) det 1≤i,j≤d where e = (1, 1, . . . , 1), where the sum in (3.1) runs over all d-tuples n of non-negative integers, and where 〈zm〉h(z) denotes the coefficient of zm in the formal Laurent series h(z). Next, we prove a determinant lemma and a corollary, both of which will also be used in the proof of Theorem 9. Lemma 2. Let d be a positive integer, and let X1, X2, . . . , Xd, Y2, Y3, . . . , Yd be indeter- minates. Then 1≤i,j≤d 1− χ(1 6= j) , i = 1 1− χ(i 6= j) , i ≥ 2 Y2Y3 · · ·Yd X1X2 · · ·Xd , (3.2) where χ(S) = 1 if S is true and χ(S) = 0 otherwise. Proof. By using multilinearity in the rows, we rewrite the determinant on the left-hand side of (3.2) as X1X2 · · ·Xd 1≤i,j≤d X1 − χ(1 6= j)Yj, i = 1 Xi − χ(i 6= j)Yi, i ≥ 2 Next, we subtract the first column from all other columns. As a result, we obtain the determinant X1X2 · · ·Xd 1≤i,j≤d X1, i = j = 1 −Yj, i = 1 and j ≥ 2 Xi − Yi, i ≥ 2 and j = 1 χ(i = j)Yi, i, j ≥ 2  . Now we add rows 2, 3, . . . , d to the first row. After that, our determinant becomes lower triangular, with the entry in the first row and column equal to i=1Xi − i=2 Yi, and the diagonal entry in row i, i ≥ 2, equal to Yi. Hence, we obtain the claimed result. � Corollary 3. Let d and r be positive integers, 1 ≤ r ≤ d, and let X1, X2, . . . , Xd, Y and Z be indeterminates. Then, with notation as in Lemma 2, 1≤i,j≤d 1− χ(r 6= j) Z , i = r 1− χ(i 6= j) Y , i 6= r Y d−2 i=1Xi + (Y − Z)Xr − (d− 1)Y Z X1X2 · · ·Xd . (3.3) Proof. We write the diagonal entry in the r-th row of the determinant in (3.3) as Xr + Y − Z Y − Z DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 9 and then use linearity of the determinant in the r-th row to decompose the determinant Xr + Y − Z Y − Z where D1 is the determinant in (3.2) with Xr replaced by Xr+Y −Z, and with Yi = Y for all i, and where D2 is the determinant in (3.2) with d replaced by d−1, with Yi = Y for all i, and with Xi replaced by Xi−1 for i = r+1, r+2, . . . , d. Hence, using Lemma 2, we deduce that the determinant in (3.3) is equal to Y d−1 i=1Xi + Y − Z − (d− 1)Y X1X2 · · ·Xd (Y − Z)Y d−2 i=1Xi −Xr − (d− 2)Y X1X2 · · ·Xd Little simplification then leads to (3.3). � We end this section with a summation lemma, which we shall need in Sections 5 and 6 in order to compute the Bn, respectively Dn, decomposition numbers of arbitrary rank from those of full rank, and in Section 8 to derive enumerative results for (generalised) non-crossing partitions from our formulae for the decomposition numbers. Lemma 4. Let M and r be non-negative integers. Then m1+2m2+···+rmr=r m1, m2, . . . , mr M + r − 1 , (3.4) where the multinomial coefficient is defined by m1, m2, . . . , mr m1!m2! · · ·mr! (M −m1 −m2 − · · · −mr)! Proof. The identity results directly by comparing coefficients of zr on both sides of the identity (1 + z + z2 + z3 + · · · )M = (1− z)−M . 4. Decomposition numbers for type A As was already pointed out in [28, Sec. 10], the decomposition numbers for type An have already been computed by Goulden and Jackson in [23, Theorem 3.2], albeit using a somewhat different language. (The condition on the sum l(α1) + l(α2) + · · ·+ l(αm) is misstated throughout the latter paper. It should be replaced by l(α1) + l(α2) + · · ·+ l(αm) = (m− 1)n+ 1.) In our terminology, their result reads as follows. Theorem 5. Let T1, T2, . . . , Td be types with rkT1 + rkT2 + · · ·+ rkTd = n, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d. NAn(T1, T2, . . . , Td) = (n + 1) n− rkTi + 1 n− rkTi + 1 1 , m 2 , . . . , m , (4.1) where the multinomial coefficient is defined as in Lemma 4. 10 C. KRATTENTHALER AND T. W. MÜLLER Here we have used Stembridge’s [35] notation for the decomposition of types into a product of irreducibles; for example, the equation T = A32 ∗ A5 means that the root system of type T decomposes into the orthogonal product of 3 copies of root systems of type A2 and one copy of the root system of type A5. It was shown in [28, Theorem 10] that, upon applying the summation formula in Lemma 4 to the result in Theorem 5 in a suitable manner, one obtains a compact formula for all type An decomposition numbers. Theorem 6. Let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d. NAn(T1, T2, . . . , Td) = (n+ 1) rkT1 + rkT2 + · · ·+ rkTd + 1 n− rkTi + 1 n− rkTi + 1 1 , m 2 , . . . , m , (4.2) where the multinomial coefficient is defined as in Lemma 4. All other decomposition numbers NAn(T1, T2, . . . , Td) are zero. 5. Decomposition numbers for type B In this section we compute the decomposition numbers in type Bn. We show that one can extract the corresponding formulae from results of Bóna, Bousquet, Labelle and Leroux [12] on the enumeration of certain planar maps, which they call m-ary cacti. While reading the statement of the theorem, the reader should recall from Section 2 the distinction between group-theoretic and combinatorial decomposition numbers. Theorem 7. (i) If T1, T2, . . . , Td are types with rkT1 + rkT2 + · · ·+ rkTd = n, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , j − 1, j + 1, . . . , d, Tj = Bα ∗ A 1 ∗ A 2 ∗ · · · ∗ A for some α ≥ 1, then N combBn (T1, T2, . . . , Td) = n n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi n− rkTi 1 , m 2 , . . . , m (5.1) where the multinomial coefficient is defined as in Lemma 4. For α ≥ 2, the number NBn(T1, T2, . . . , Td) is given by the same formula. (ii) If T1, T2, . . . , Td are types with rkT1 + rkT2 + · · ·+ rkTd = n, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d, DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 11 NBn(T1, T2, . . . , Td) = n n− rkTi n− rkTi 1 , m 2 , . . . , m )) d∑ 1 (n− rkTj) 0 + 1 (5.2) where m 0 = n− rkTj − (iii) All other decomposition numbers NBn(T1, T2, . . . , Td) and N (T1, T2, . . . , Td) with rkT1 + rkT2 + · · ·+ rkTd = n are zero. Proof. Determining the decomposition numbers NBn(T1, T2, . . . , Td) = NBn(Td, . . . , T2, T1) (recall (2.2)), respectively N combBn (T1, T2, . . . , Td) = N (Td, . . . , T2, T1), amounts to counting all possible factorisations [1, 2, . . . , n] = σd · · ·σ2σ1, (5.3) where σi has type Ti as a parabolic Coxeter element, respectively has combinatorial type Ti. The reader should observe that the factorisation (5.3) is minimal, in the sense n = ℓT [1, 2, . . . , n] = ℓT (σ1) + ℓT (σ2) + · · ·+ ℓT (σd), since ℓT (σi) = rkTi, and since, by our assumption, the sum of the ranks of the Ti’s equals n. A further observation is that, in a factorisation (5.3), there must be at least one factor σi which contains a type B cycle in its (type B) disjoint cycle decomposition, because the sign of [1, 2, . . . , n] as an element of the group S2n of all permutations of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄} is −1, while the sign of any type A cycle is +1. We first prove Claim (iii). Let us assume, by contradiction, that there is a minimal decomposition (5.3) in which, altogether, we find at least two type B cycles in the (type B) disjoint cycle decompositions of the σi’s. In that case, (5.3) has the form [1, 2, . . . , n] = u1κ1u2κ2u3, (5.4) where κ1 and κ2 are two type B cycles, and u1, u2, u3 are the factors in between. Moreover, the factorisation (5.4) is minimal, meaning that n = ℓT (u1) + ℓT (κ1) + ℓT (u2) + ℓT (κ2) + ℓT (u3). (5.5) We may rewrite (5.4) as [1, 2, . . . , n] = κ1κ2(κ 1 u1κ1κ2)(κ 2 u2κ2)u3, or, setting u′1 = κ 1 u1κ1κ2 and u 2 = κ 2 u2κ2, as [1, 2, . . . , n] = κ1κ2u 2u3. (5.6) This factorisation is still minimal since u′1 is conjugate to u1 and u 2 is conjugate to u2. At this point, we observe that κ1 must be a cycle of the form (2.8) with a1 < a2 < · · · < ak < a1 < a2 < · · · < ak in the order 1 < 2 < · · · < n < 1̄ < 2̄ < · · · < n̄, because otherwise κ1 6≤T [1, 2, . . . , n], which would contradict (5.6). A similar argument 12 C. KRATTENTHALER AND T. W. MÜLLER Figure 1. The 3-cactus corresponding to the factorisation (5.7) applies to κ2. Now, if κ1 and κ2 are not disjoint, then it is easy to see that ℓT (κ1κ2) < ℓT (κ1) + ℓT (κ2), whence n = ℓT ([1, 2, . . . , n]) = ℓT (κ1κ2u ≤ ℓT (κ1κ2) + ℓT (u 1) + ℓT (u 2) + ℓT (u3) ≤ ℓT (κ1κ2) + ℓT (u1) + ℓT (u2) + ℓT (u3) < ℓT (κ1) + ℓT (κ2) + ℓT (u1) + ℓT (u2) + ℓT (u3), a contradiction to (5.5). If, on the other hand, κ1 and κ2 are disjoint, then we can find i, j ∈ {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}, such that i < j < κ1(i) < κ2(j) (in the above order of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}). In other words, if we represent κ1 and κ2 in the obvious way in a cyclic diagram (cf. [32, Sec. 2]), then they cross each other. However, in that case we have κ1κ2 6≤T [1, 2, . . . , n], contradicting the fact that (5.6) is a minimal factorisation. (This is one of the con- sequences of Biane’s group-theoretic characterisation [10, Theorem 1] of non-crossing partitions.) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 13 We turn now to Claims (i) and (ii). In what follows, we shall show that the formulae (5.1) and (5.2) follow from results of Bóna, Bousquet, Labelle and Leroux [12] on the enumeration of m-ary cacti with a rotational symmetry. In order to explain this, we must first define a bijection between minimal factorisations (5.3) and certain planar maps. By a map, we mean a connected graph embedded in the plane such that edges do not intersect except in vertices. The maps which are of relevance here are maps in which faces different from the outer face intersect only in vertices, and are coloured with colours from {1, 2, . . . , d}. Such maps will be referred to as d-cacti from now on.1 Examples of 3-cacti can be found in Figures 1 and 2. In the figures, the faces different from the outer face are the shaded ones. Their colours are indicated by the numbers 1, 2, respectively 3, placed in the centre of the faces. Figure 1 shows a 3-cactus in which the vertices are labelled, while Figure 2 shows one in which the vertices are not labelled. (That one of the vertices in Figure 2 is marked by a bold dot should be ignored for the moment.) In what follows, we need the concept of the rotator around a vertex v in a d- cactus, which, by definition, is the cyclic list of colours of faces encountered in a clockwise journey around v. If, while travelling around v, we encounter the colours b1, b2, . . . , bk, in this order, then we will write (b1, b2, . . . , bk) O for the rotator, meaning that (b1, b2, . . . , bk) O = (b2, . . . , bk, b1) O, etc. For example, the rotator of all the vertices in the map in Figure 1 is (1, 2, 3)O. We illustrate the bijection between minimal factorisations (5.3) and d-cacti with an example. Take n = 10 and d = 3, and consider the factorisation [1, 2, . . . , 10] = σ3σ2σ1, (5.7) where σ3 = ((7, 8)), σ2 = [2, 6, 8] ((1, 9̄, 10)) ((4, 5)), and σ1 = ((1, 8̄)) ((2, 3, 5)). For each cycle (a1, a2, . . . , ak) (sic!) of σi, we create a k-gon coloured i, and label its vertices a1, a2, . . . , ak in clockwise order. (The warning “sic!” is there to avoid misunderstand- ings: for each type A “cycle” ((b1, b2, . . . , bk)) we create two k-gons, the vertices of one being labelled b1, b2, . . . , bk, and the vertices of the other being labelled b1, b2, . . . , bk, while for each type B “cycle” [b1, b2, . . . , bk] we create one 2k-gon with vertices labelled b1, b2, . . . , bk, b1, b2, . . . , bk.) We glue these polygons into a d-cactus, the faces of which are these polygons plus the outer face, by identifying equally labelled vertices such that the rotator of each vertex is (1, 2, . . . , d). Figure 1 shows the outcome of this procedure for the factorisation (5.7). The fact that the result of the procedure can be realised as a d-cactus follows from Euler’s formula. Namely, the number of faces corresponding to the polygons is 1 + k (the 1 coming from the polygon corresponding to the type B cycle), the number of edges is 2α+2 k (k+1), and the number of vertices is 2n. Hence, if we include the outer face, the number of vertices minus the number of edges 1We warn the reader that our terminology deviates from the one in [12, 23]. We follow loosely the conventions in [25]. To be precise, our d-cacti in which the rotator around every vertex is (1, 2, . . . , d)O are dual to the coloured d-cacti in [23], respectively d-ary cacti in [12], in the following sense: one is obtained from the other by “interchanging” the roles of vertices and faces, that is, given a d-cactus in our sense, one obtains a d-cactus in the sense of Goulden and Jackson by shrinking faces to vertices and blowing up vertices of degree δ to faces with δ vertices, keeping the incidence relations between faces and vertices. Another minor difference is that colours are arranged in counter-clockwise order in [12, 23], while we arrange colours in clockwise order. 14 C. KRATTENTHALER AND T. W. MÜLLER Figure 2. A rotation-symmetric 3-cactus with a marked vertex plus the number of faces is 2n− 2α− 2 k (k + 1) + 2 k + 2 = 2n+ 2− 2α− 2 = 2n+ 2− 2 rkT1 − 2 rkT2 − · · · − 2 rkTd = 2, (5.8) according to our assumption concerning the sum of the ranks of the types Ti. We may further simplify this geometric representation of a minimal factorisation (5.3) by deleting all vertex labels and marking the vertex which had label 1. If this simplification is applied to the 3-cactus in Figure 1, we obtain the 3-cactus in Figure 2. Indeed, the knowledge of which vertex carries label 1 allows us to reconstruct all other vertex labels as follows: starting from the vertex labelled 1, we travel clockwise along the boundary of the face coloured 1 until we reach the next vertex (that is, we traverse only a single edge); from there, we travel clockwise along the boundary of the face coloured 2 until we reach the next vertex; etc., until we have travelled along an edge DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 15 bounding a face of colour d. The vertex that we have reached must carry label 2. Etc. Clearly, if drawn appropriately into the plane, a d-cactus resulting from an application of the above procedure to a minimal factorisation (5.3) is symmetric with respect to a rotation by 180◦, the centre of the rotation being the centre of the regular 2α-gon corresponding to the unique type B cycle of σj ; cf. Figure 2. In what follows, we shall abbreviate this property as rotation-symmetric. In summary, under the assumptions of Claim (i), the decomposition number N combBn (T1, T2, . . . , Td), respectively, if α ≥ 2, the decomposition number NBn(T1, T2, . . . , Td) also, equals the number of all rotation-symmetric d-cacti on 2n vertices in which one vertex is marked and all vertices have rotator (1, 2, . . . , d)O, with exactly m k pairs of faces of colour i having k + 1 vertices, arranged symmetrically around a central face of colour j with 2α vertices. Aside from the marking of one vertex, equivalent objects are counted in [12, Theo- rem 25]. In our language, modulo the “dualisation” described in Footnote 1, and upon replacing m by d, the objects which are counted in the cited theorem are d-cacti in which all vertices have rotator (1, 2, . . . , d)O, and which are invariant under a rotation (not necessarily by 180◦). To be precise, from the proof of [12, (81)] (not given in full detail in [12]) it can be extracted that the number of d-cacti on 2n vertices, in which all vertices have rotator (1, 2, . . . , d)O, which are invariant under a rotation by (360/s)◦, s being maximal with this property, and which have exactly 2m k faces of colour i having k + 1 vertices arranged around a central face of colour j with 2α vertices, equals (2n)d−2s ′µ(t/s) 2(n− rkTj)/t 1 /t, 2m 2 /t, . . . , 2m i 6=j 2(n− rkTi) 2(n− rkTi)/t 1 /t, 2m 2 /t, . . . , 2m , (5.9) where the sum extends over all t with s | t, t | 2α, and t | 2m k for all i = 1, 2, . . . , d and k = 1, 2, . . . , n. Here, µ(·) is the Möbius function from number theory.2 In presenting the formula in the above form, we have also used the observation that, for all i (including i = j !), the number of type A cycles of σi is n− rkTi. As we said above, the d-cacti that we want to enumerate have one marked vertex, whereas the d-cacti counted by (5.9) have no marked vertex. However, given a d-cactus counted by (5.9), we have exactly 2n/s inequivalent ways of marking a vertex. Hence, recalling that the d-cacti that we want to count are invariant under a rotation by 180◦, we must multiply the expression (5.9) by 2n/s, and then sum the result over all even s. Since, by definition of the Möbius function, we have 2|s|t µ(t/s) = s′| t µ(t/2s′) = 1 if t 0 otherwise, the result of this summation is exactly the right-hand side of (5.1). 2Formula (81) in [12] does not distinguish the colour or the size of the central face (that is, in the language of [12]: the colour or the degree of the central vertex), therefore it is in fact a sum over all possible colours and sizes, represented there by the summations over i and h, respectively. 16 C. KRATTENTHALER AND T. W. MÜLLER Finally, we prove Claim (ii). From what we already know, in a minimal factorisation (5.3) exactly one of the factors on the right-hand side must contain a type B cycle of length 1 in its (type B) disjoint cycle decomposition, σj say. As a parabolic Coxeter element, a type B cycle of length 1 has type A1. Since all considerations in the proof of Claim (i) are also valid for α = 1, we may use Formula (5.1) with α = 1, and with 1 replaced by m 1 − 1, to count the number of these factorisations, to obtain n− rkTj 1 − 1, m 2 , . . . , m i 6=j n− rkTi n− rkTi 1 , m 2 , . . . , m This has to be summed over j = 1, 2, . . . , d. The result is exactly (5.2). The proof of the theorem is now complete. � Combining the previous theorem with the summation formula of Lemma 4, we can now derive compact formulae for all type Bn decomposition numbers. Theorem 8. (i) Let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , j − 1, j + 1, . . . , d, Tj = Bα ∗ A 1 ∗ A 2 ∗ · · · ∗ A for some α ≥ 1. Then N combBn (T1, T2, . . . , Td) = n rkT1 + rkT2 + · · ·+ rkTd n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi n− rkTi 1 , m 2 , . . . , m , (5.10) where the multinomial coefficient is defined as in Lemma 4. For α ≥ 2, the number NBn(T1, T2, . . . , Td) is given by the same formula. (ii) Let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d. N combBn (T1, T2, . . . , Td) rkT1 + rkT2 + · · ·+ rkTd )( d∏ n− rkTi n− rkTi 1 , m 2 , . . . , m , (5.11) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 17 whereas NBn(T1, T2, . . . , Td) = nd−1 rkT1 + rkT2 + · · ·+ rkTd )( d∏ n− rkTi n− rkTi 1 , m 2 , . . . , m n− rkT1 − rkT2 − · · · − rkTd + 1 (n− rkTj) 0 + 1 , (5.12) with m 0 = n− rkTj − (iii) All other decomposition numbers NBn(T1, T2, . . . , Td) and N (T1, T2, . . . , Td) are zero. Proof. If we write r for n− rkT1− rkT2−· · ·− rkTd, then for Φ = Bn the relation (2.3) becomes NBn(T1, T2, . . . , Td) = T :rkT=r NBn(T1, T2, . . . , Td, T ), (5.13) with the same relation holding for N combBn in place of NBn . In order to prove (5.10), we let T = Am11 ∗A 2 ∗ · · · ∗A n and use (5.1) in (5.13), to obtain N combBn (T1, T2, . . . , Td) = m1+2m2+···+nmn=r m1, m2, . . . , mn n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi n− rkTi 1 , m 2 , . . . , m If we use (3.4) with M = n− r, we arrive at our claim after little simplification. In order to prove (5.11), we let T = Bα ∗ A 1 ∗ A 2 ∗ · · · ∗ A n in (5.13). The important point to be observed here is that, in contrast to the previous argument, in the present case T must have a factor Bα. Subsequently, use of (5.1) in (5.13) yields N combBn (T1, T2, . . . , Td) = m1+2m2+···+nmn=r−α m1, m2, . . . , mn n− rkTi n− rkTi 1 , m 2 , . . . , m . (5.14) Now we use (3.4) with r replaced by r − α and M = n − r, and subsequently the elementary summation formula n− α− 1 r − α n− α− 1 n− r − 1 r − 1 . (5.15) Then, after little rewriting, we arrive at our claim. To establish (5.12), we must recall that the group-theoretic type A1 does not distin- guish between a type A cycle ((i, j)) = (i, j) (̄i, j̄) and a type B cycle [i] = (i, ī). Hence, to obtain NBn(T1, T2, . . . , Td) in the case that no Ti contains a Bα for α ≥ 2, we must 18 C. KRATTENTHALER AND T. W. MÜLLER add the expression (5.11) and the expressions (5.10) with m 1 replaced by m 1 −1 over j = 1, 2, . . . , d. As is not difficult to see, this sum is indeed equal to (5.12). � 6. Decomposition numbers for type D In this section we compute the decomposition numbers for the type Dn. Theorem 9 gives the formulae for the full rank decomposition numbers, while Theorem 10 presents the implied formulae for the decomposition numbers of arbitrary rank. To our knowl- edge, these are new results, which did not appear earlier in the literature on map enumeration or on the connection coefficients in the symmetric group or other Coxeter groups. Nevertheless, the proof of Theorem 9 is entirely in the spirit of the fundamental paper [23], in that the problem of counting factorisations is translated into a problem of map enumeration, which is then solved by a generating function approach that requires the use of the Lagrange–Good formula for coefficient extraction. We begin with the result concerning the full rank decomposition numbers in type Dn. While reading the statement of the theorem below, the reader should again recall from Section 2 the distinction between group-theoretic and combinatorial decomposition numbers. Theorem 9. (i) If T1, T2, . . . , Td are types with rkT1 + rkT2 + · · ·+ rkTd = n, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , j − 1, j + 1, . . . , d, Tj = Dα ∗ A 2 ∗ · · · ∗ A for some α ≥ 2, then N combDn (T1, T2, . . . , Td) = (n− 1) n− rkTj 1 , m 2 , . . . , m i 6=j n− rk Ti − 1 n− rkTi − 1 1 , m 2 , . . . , m , (6.1) where the multinomial coefficient is defined as in Lemma 4. For α ≥ 4, the number NDn(T1, T2, . . . , Td) is given by the same formula. (ii) If T1, T2, . . . , Td are types with rkT1 + rkT2 + · · ·+ rkTd = n, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d, DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 19 N combDn (T1, T2, . . . , Td) = (n− 1)d−1 n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m −2(d− 1)(n− 1) n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m  , (6.2) while NDn(T1, T2, . . . , Td) = (n− 1)d−1 i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m n− rkTj 1 , m 2 , . . . , m n− rkTj 1 , m 2 , m 3 − 1, m 4 , . . . , m n− rkTj 1 − 2, m 2 , . . . , m −2(d− 1)(n− 1) n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m  . (6.3) (iii) All other decomposition numbers NDn(T1, T2, . . . , Td) and N (T1, T2, . . . , Td) with rkT1 + rkT2 + · · ·+ rkTd = n are zero. Remark. These formulae must be correctly interpreted when Ti contains no Dα and rkTi = n− 1. In that case, because of n− 1 = rkTi = m 1 + 2m 2 + · · ·+ nm n , there must be an ℓ, 1 ≤ ℓ ≤ n− 1, with m ℓ ≥ 1. We then interpret the term n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m n− rkTi − 1 n− rk Ti − 1 1 , m 2 , . . . , m n− rkTi − 2 1 , . . . , m ℓ − 1, . . . , m where the multinomial coefficient is zero whenever −1 = n− rkTi − 2 < m 1 + · · ·+ (m ℓ − 1) + · · ·+m except when all of m 1 , . . . , m ℓ − 1, . . . , m n are zero. Explicitly, one must read n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m 20 C. KRATTENTHALER AND T. W. MÜLLER if rkTi = n− 1 but Ti 6= An−1, and n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m if Ti = An−1. Proof of Theorem 9. Determining the decomposition number NDn(T1, T2, . . . , Td) = NDn(Td, . . . , T2, T1) (recall (2.2)), respectively N combDn (T1, T2, . . . , Td) = N (Td, . . . , T2, T1), amounts to counting all possible factorisations (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1) (n, n̄) = σd · · ·σ2σ1, (6.4) where σi has type Ti as a parabolic Coxeter element, respectively has combinatorial type Ti. Here also, the factorisation (6.4) is minimal in the sense that n = ℓT (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1) (n, n̄) = ℓT (σ1) + ℓT (σ2) + · · ·+ ℓT (σd), since ℓT (σi) = rkTi, and since, by our assumption, the sum of the ranks of the Ti’s equals n. We first prove Claim (iii). Let us assume, for contradiction, that there is a minimal factorisation (6.4), in which, altogether, we find at least two type B cycles of length ≥ 2 in the (type B) disjoint cycle decompositions of the σi’s. It can then be shown by arguments similar to those in the proof of Claim (iii) in Theorem 7 that this leads to a contradiction. Hence, “at worst,” we may find a type B cycle of length 1, (a, ā) say, and another type B cycle, κ say. Both of them must be contained in the disjoint cycle decomposition of one of the σi’s since all the σi’s are elements of W (Dn). Given that κ has length α − 1, the product of both, (a, ā) κ, is of combinatorial type Dα, α ≥ 2, whereas, as a parabolic Coxeter element, it is of type Dα only if α ≥ 4. If α = 3, then it is a parabolic Coxeter element of type A3, and if α = 2 it is of type A 1. Thus, we are actually in the cases to which Claims (i) and (ii) apply. To prove Claim (i), we continue this line of argument. By a variation of the conjuga- tion argument (5.4)–(5.6), we may assume that these two type B cycles are contained in σd, σd = (a, ā) κ σ d say, where, as above, (a, ā) is the type B cycle of length 1 and κ is the other type B cycle, and where σ′d is free of type B cycles. In that case, (6.4) takes the form c = (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1) (n, n̄) = (a, ā) κ σ′d · · ·σ1. (6.5) If a 6= n, κ 6= (n, n̄), and if κ does not fix n, then (a, ā)κ 6≤T c, a contradiction. Likewise, if a 6= n, κ = [b1, b2, . . . , bk] with n /∈ {b1, b2, . . . , bk}, then (a, ā) κ 6≤T [1, 2, . . . , n − 1], again a contradiction. Hence, we may assume that a = n, whence (a, ā) κ = κ (n, n̄) forms a parabolic Coxeter element of type Dα, given that κ has length α − 1. We are then in the position to determine all possible factorisations of the form (6.5), which reduces to (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1) = [1, 2, . . . , n− 1] = κσ′d · · ·σ1. (6.6) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 21 This is now a minimal type B factorisation of the form (5.3) with n replaced by n− 1. We may therefore use Formula (5.1) with n replaced by n− 1, and with rkTj replaced by rkTj − 1. These substitutions lead exactly to (6.1). Finally, we turn to Claim (ii). First we discuss two degenerate cases which come from the identifications D3 ∼ A3, respectively D2 ∼ A 1, and which only occur for NDn(T1, T2, . . . , Td) (but not for the combinatorial decomposition numbers N (T1, T2, . . . , Td)). It may happen that one of the factors in (6.4), let us say, without loss of generality, σd, contains a type B cycle of length 1 and one of length 2 in its disjoint cycle decomposition; that is, σd may contain (n, n̄) [a, b] = (n, n̄) (a, b, ā, b̄) = [a, b] [b, n] [b, n̄]. As a parabolic Coxeter element, this is of type A3. By the reduction (6.5)–(6.6), we may count the number of these possibilities by Formula (5.1) with n replaced by n− 1, rkTj replaced by rkTj−1, and m 3 replaced by m 3 −1. This explains the second term in the factor in big parentheses on the right-hand side of (6.3). On the other hand, it may happen that one of the factors in (6.4), let us say again, without loss of generality, σd, contains two type B cycles of length 1 in its disjoint cycle decomposition; that is, σd may contain (n, n̄) (a, ā). As a parabolic Coxeter element, this is of type A 1. By the reduction (6.5)–(6.6), we may count the number of these possibilities by Formula (5.1) with n replaced by n − 1, rkTj replaced by rkTj − 1, and m 1 replaced by m 1 − 2. This explains the third term in the factor in big parentheses on the right-hand side of (6.3). From now on we may assume that none of the σi’s contains a type B cycle in its (type B) disjoint cycle decomposition. To determine the number of minimal factorisations (6.4) in this case, we construct again a bijection between these factorisations and certain maps. In what follows, we will still use the concept of a rotator, introduced in the proof of Theorem 7. We apply again the procedure described in that proof. That is, for each (ordinary) cycle (a1, a2, . . . , ak) of σi, we create a k-gon coloured i, label its vertices a1, a2, . . . , ak in clockwise order, and glue these polygons into a map by identifying equally labelled vertices such that the rotator of each vertex is (1, 2, . . . , d). However, this map can be embedded in the plane only if we allow the creation of an inner face corresponding to the cycle (n, n̄) on the left-hand side of (6.4) (the outer face corresponding to the large cycle (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1)). Moreover, this inner face must be bounded by 2d edges. We call such a map, in which all faces except the outer face and an inner face intersect only in vertices, and are coloured with colours from {1, 2, . . . , d}, and in which the inner face is bounded by 2d edges, a d-atoll. For example, if we take n = 10 and d = 3, and consider the factorisation (1, 2, . . . , 9, 1̄, 2̄, . . . , 9̄) (10, 10) = σ3σ2σ1, (6.7) where σ3 = ((1, 4, 10, 7̄)), σ2 = ((1, 3)) ((4, 6, 10)) ((7, 8, 9)), and σ1 = ((1, 2)) ((4, 5)), and apply this procedure, we obtain the 3-atoll in Figure 3. In the figure, the faces corresponding to cycles are shaded. As in Figures 1 and 2, the outer face is not shaded. Here, there is in addition an inner face which is not shaded, the face formed by the vertices 4, 10, 4̄, 10. Again, the colours of the shaded faces are indicated by the numbers 1, 2, respectively 3, placed in the centre of the faces. 22 C. KRATTENTHALER AND T. W. MÜLLER 3 2 1 3 Figure 3. The 3-atoll corresponding to the factorisation (6.7) Unsurprisingly, the fact that the result of the procedure can be realised as a d-atoll follows again from Euler’s formula. More precisely, the number of faces corresponding to the polygons is 2 k , the number of edges is 2 k (k + 1), and the number of vertices is 2n. Hence, if we include the outer face and the inner face, the number of vertices minus the number of edges plus the number of faces is 2n− 2 k (k + 1) + 2 k + 2 = 2n+ 2− 2 = 2n+ 2− 2 rkT1 − 2 rkT2 − · · · − 2 rkTd = 2, (6.8) according to our assumption concerning the sum of the ranks of the types Ti. Again, we may further simplify this geometric representation of a minimal factorisa- tion (6.4) by deleting all vertex labels, marking the vertex which had label 1 with •, and marking the vertex that had label n with �. If this simplification is applied to the 3-atoll in Figure 3, we obtain the 3-atoll in Figure 4. Clearly, if drawn appropriately into the plane, a d-atoll resulting from an application of the above procedure to a mini- mal factorisation (6.4) is symmetric with respect to a rotation by 180◦, the centre of the rotation being the centre of the inner face; cf. Figure 4. As earlier, we shall abbreviate DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 23 3 2 1 3 Figure 4. A rotation-symmetric 3-atoll with two marked vertices this property as rotation-symmetric. In fact, there is not much freedom for the choice of the vertex marked by � once a vertex has been marked by •. Clearly, if we run through the vertex labelling process described in the proof of Theorem 7, labelling 1 the vertex which is marked by •, we shall reconstruct the labels 1, 2, . . . , n − 1, 1̄, 2̄, . . . , n− 1. This leaves only 2 vertices incident to the inner face unlabelled, one of which will have to carry the mark �. In summary, under the assumptions of Claim (ii), the number of minimal factori- sations (6.4), in which none of the σi’s contains a type B cycle in its disjoint cycle decomposition, equals twice the number of all rotation-symmetric d-atolls on 2n ver- tices, in which one vertex is marked by •, all vertices have rotator (1, 2, . . . , d)O, and with exactly m k pairs of faces of colour i having k + 1 vertices, arranged symmetri- cally around the inner face (which is not coloured). Let us denote the number of these d-atolls by N ′Dn(T1, T2, . . . , Td). We must now enumerate these d-atolls. First of all, introducing a figure of speech, we shall refer to coloured faces of a d-atoll which share an edge with the inner face but not with the outer face as faces “inside the d-atoll,” and all others as faces “outside the d-atoll.” For example, in Figure 4 we find two faces inside the 3-atoll, namely the two loop faces attached to the vertices labelled 10, respectively 10, in Figure 3. Since, in a d-atoll, the inner face is bounded by exactly 2d edges, inside the d-atoll, we find only 24 C. KRATTENTHALER AND T. W. MÜLLER coloured faces containing exactly one vertex. Next, we travel counter-clockwise around the inner face and record the coloured faces sharing an edge with both the inner and outer faces. Thus we obtain a list of the form F1, F2, . . . , Fℓ, Fℓ+1, . . . , F2ℓ, where, except possibly for the marking, Fh+ℓ is an identical copy of Fh, h = 1, 2, . . . , ℓ. In Figure 4, this list contains four faces, F̃1, F̃2, F̃3, F̃4, where F̃1 and F̃3 are the two quadrangles of colour 3, and where F̃2 and F̃4 are the two triangles of colour 2 connecting the two quadrangles. Continuing the general argument, let the colour of Fh be ih. Inside the d-atoll, because of the rotator condition, there must be {ih+1−ih−1}d faces (containing just one vertex) incident to the common vertex of Fh and Fh+1 coloured {ih+1}d, . . . , {ih+1−1}d, where, by definition, {x}d := x, if 0 ≤ x ≤ d x+ d, if x < 0 x− d, if x > d, and where ih+ℓ = ih, h = 1, 2, . . . , ℓ. Here, if {ih + 1}d > {ih+1 − 1}d, the sequence of colours {ih + 1}d, . . . , {ih+1 − 1}d must be interpreted “cyclically,” that is, as {ih + 1}d, {ih+1}d+1, . . . , d, 1, 2, . . . , {ih+1−1}d. As we observed above, the number of edges bounding the inner face is 2d. On the other hand, using the notation just introduced, this number also equals {ih+1 − ih}d = 2 (ih+1 − ih) + d · χ(ih+1 < ih) χ(ih+1 < ih). Hence, there is precisely one h for which ih+1 < ih. Without loss of generality, we may assume that h = ℓ, so that i1 < i2 < · · · < iℓ. The ascending colouring of the faces F1, F2, . . . , Fℓ breaks the (rotation) symmetry of the d-atoll. Therefore, we may first enumerate d-atolls without any marking, and multiply the result by the number of all possible markings, which is n − 1. More precisely, let N ′′Dn(T1, T2, . . . , Td) denote the number of all rotation-symmetric d-atolls on 2n vertices, in which all vertices have rotator (1, 2, . . . , d)O, and with exactly m pairs of faces of colour i having k+1 vertices, arranged symmetrically around the inner face (which is not coloured). Then, NDn(T1, T2, . . . , Td) = 2N (T1, T2, . . . , Td) = 2(n− 1)N ′′Dn(T1, T2, . . . , Td). (6.9) We use a generating function approach to determine N ′′Dn(T1, T2, . . . , Td), which re- quires a combinatorial decomposition of our objects. Let G(z) be the generating func- G(z) = w(A), (6.10) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 25 where A is the set of all rotation-symmetric d-atolls, in which all vertices have rotator (1, 2, . . . , d)O, and where w(A) = #(faces of A with colour i) #(faces of A with colour i and k vertices) i,k . Here, z = (z1, z2, . . . , zd), with the zi’s, i = 1, 2, . . . , d, and the pi,k, i = 1, 2, . . . , d, k = 1, 2, . . . , being indeterminates. Clearly, in view of the bijection between minimal factorisations (6.4) and d-atolls described earlier, and by (6.9), we have NDn(T1, T2, . . . , Td) = 2(n− 1) i,k+1 G(z), (6.11) where c = (c1, c2, . . . , cd), with ci equal to the number of type A cycles of σi; that is, k , i = 1, 2, . . . , d. Here, and in the sequel, we use the multi-index notation introduced at the beginning of Section 3. For later use, we observe that, for all i, ci is related to rkTi via ci = n− rkTi. (6.12) Now, let A be a d-atoll in A such that the faces which share an edge with both the inner and outer faces are F1, F2, . . . , Fℓ, Fℓ+1, . . . , F2ℓ, where Fh+ℓ is an identical copy of Fh, where the colour of Fh is ih, h = 1, 2, . . . , ℓ, and with i1 < i2 < · · · < iℓ. We decompose A by separating from each other the polygons which touch in vertices of the inner face. The decomposition in the case of our example in Figure 4 is shown in Figure 5. Ignoring identical copies which are there due to the rotation symmetry, we obtain a list K1, L , . . . , L , . . . , C d , C 1 , . . . , C K2, L , . . . , L , . . . , C d , C 1 , . . . , C , . . . Kℓ, L , . . . , L d , L 1 , . . . , L , . . . , C , (6.13) where Kh is the d-cactus containing the face Fh, and, hence, a d-cactus in which all but two neighbouring vertices have rotator (1, 2, . . . , d)O, the latter two vertices being incident to just one face, which is of colour ih, where L j is a face of colour j with just one vertex, and where C j is a d-cactus in which all but one vertex have rotator (1, 2, . . . , d)O, the distinguished vertex being incident to just one face, which is of colour j, h = 1, 2, . . . , ℓ and j = 1, 2, . . . , d. With this notation, our example in Figure 5 is one in which ℓ = 2, i1 = 2, i2 = 3. The d-cacti Kh can be further decomposed. Namely, assuming that the face Fh is a k-gon (of colour ih), let C1, C2, . . . , Ck−2 be the d-cacti incident to this k-gon, read in clockwise order, starting with the d-cactus to the left of the two distinguished vertices. Figure 6 illustrates this further decomposition of the d-cactus K2 from Fig- ure 5. After removal of Fh, we are left with the ordered collection C1, C2, . . . , Ck−2 of d-cacti, each of which having the property that the rotator of all but one vertex is (1, 2, . . . , d)O, the exceptional vertex having rotator (1, . . . , ih − 1, ih + 1, . . . , d) O. By separating from each other the polygons of colours 1, . . . , ih − 1, ih + 1, . . . , d which 26 C. KRATTENTHALER AND T. W. MÜLLER Figure 5. The decomposition of the 3-atoll in Figure 4 touch in the exceptional vertex, each d-cactus Ci in turn can be decomposed into d- cacti Ci,1, . . . , Ci,ih−1, Ci,ih+1, . . . , Ci,d with Ci,j ∈ Cj for all k, where Cj denotes the set of all d-cacti in which all but one vertex have rotator (1, 2, . . . , d)O, the distinguished vertex being incident to just one face, which is of colour j. Let ωj(z) denote the generating function for the d-cacti in Cj , that is, ωj(z) = w(C). (6.14) Furthermore, for i = 1, 2, . . . , d, define the formal power series Pi(u) in one variable u Pi(u) = pi,ku Then, by the decomposition (6.13) and the further decomposition of the Kh’s that we just described, the contribution of the above d-atolls to the generating function (6.10) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 27 Figure 6. The decomposition of K2 in Figure 5 zijωij (z) ω1(z) · · ·ωd(z) ω1(z) · · ·ωd(z) ωij(z) − pij ,1 j=1 zjpj,1∏ℓ j=1 zijpij ,1 (ω1(z) · · ·ωd(z)) j=1 ωij(z) zjpj,1 ωj(z) ω1(z)···ωd(z) ωij (z) pij ,1 the term in the first line corresponding to the contribution of the Kj ’s, the first term in the second line corresponding to the contribution of the L k ’s, and the second term in the second line corresponding to the contribution of the C k ’s. These expressions must be summed over ℓ = 2, 3, . . . , d and all possible choices of 1 ≤ i1 < i2 < · · · < iℓ ≤ d to 28 C. KRATTENTHALER AND T. W. MÜLLER obtain the desired generating function G(z), that is, G(z) = zjpj,1 ωj(z) 1≤i1<i2<···<iℓ≤d ω1(z)···ωd(z) ωij (z) pij , 1 zjpj,1 ωj(z) ω1(z)···ωd(z) ωj(z) ω1(z)···ωd(z) ωj(z) (6.15) Here we have used the elementary identity 1≤i1<i2<···<iℓ≤d Xi1Xi2 · · ·Xiℓ = (1 +X1)(1 +X2) · · · (1 +Xd). Before we are able to proceed, we must find functional equations for the generating functions ωj(z), j = 1, 2, . . . , d. Given a d-cactus C in Cj such that the distinguished vertex is incident to a k-gon (of colour j), we decompose it in a manner analogous to the decomposition of Kh above. To be more precise, let C1, C2, . . . , Ck−1 be the d-cacti incident to this k-gon, read in clockwise order, starting with the d-cactus to the left of the distinguished vertex. After removal of the k-gon, we are left with the ordered collection C1, C2, . . . , Ck−1 of d-cacti, each of which having the property that the rotator of all but one vertex is (1, 2, . . . , d)O, the exceptional vertex having rotator (1, . . . , j − 1, j + 1, . . . , d)O. By separating from each other the polygons of colours 1, . . . , j− 1, j+1, . . . , d which touch in the exceptional vertex, each d-cactus Ci in turn can be decomposed into d-cacti Ci,1, . . . , Ci,j−1, Ci,j+1, . . . , Ci,d with Ci,k ∈ Ck for all k. The upshot of these combinatorial considerations is that ωj(z) = zjPj(ω1(z) · · ·ωd(z)/ωj(z)), j = 1, 2, . . . , d, or, equivalently, ωj(z) Pj(ω1(z) · · ·ωd(z)/ωj(z)) , j = 1, 2, . . . , d. Using this relation, the expression (6.15) for G(z) may now be further simplified, and we obtain G(z) = 1− ω1(z)···ωd(z) ωj(z) ω1(z)···ωd(z) ωj(z) + (d− 1) ω1(z)···ωd(z) ωj(z) This is substituted in (6.11), to obtain NDn(T1, T2, . . . , Td) = −2(n− 1) i,k+1 ω1(z)···ωd(z) ωj(z) ω1(z)···ωd(z) ωj(z) + 2(n− 1)(d− 1) i,k+1 ω1(z)···ωd(z) ωj(z) ) . (6.16) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 29 Now the problem is set up for application of the Lagrange–Good inversion formula. Let fi(z) = zi/Pi(z1 · · · zd/zi), i = 1, 2, . . . , d. If we substitute fi(z) in place of zi, i = 1, 2, . . . , d, in (6.16), and apply Theorem 1 with g(z) = z1···zd z1···zd respectively g(z) = z1···zd we obtain that NDn(T1, T2, . . . , Td) = −2(n− 1) i,k+1 fj(z)pj,1 −c(z) det 1≤i,k≤d + 2(n− 1)(d− 1) i,k+1 −c(z) det 1≤i,k≤d , (6.17) where 0 stands for the vector (0, 0, . . . , 0). We treat the two terms on the right-hand side of (6.17) separately. We begin with the second term: i,k+1 −c(z) det 1≤i,k≤d i,k+1 1≤i,k≤d    z1 · · · zd , i = k −P ci−2i z1 · · · zd ×P ′i z1 · · · zd z1 · · · zd , i 6= k    i,k+1 × det 1≤i,k≤d z1 · · · zd , i = k ci − 1 i (u) ) ∣∣∣∣∣ u=z1···zd/zi , i 6= k 30 C. KRATTENTHALER AND T. W. MÜLLER Reading coefficients, we obtain ci − 1 1 , m 2 , . . . , m 1≤i,k≤d 1, i = k ci − 1 , i 6= k ci − 1 1 , m 2 , . . . , m 1≤i,k≤d 1− χ(i 6= k) ci − 1 the second line being due to (6.12). Now we can apply Lemma 2 with Xi = ci − 1 and Yi = n− 1, i = 1, 2, . . . , d. The term (ci − 1)− (d− 1)(n− 1) (n− rkTi − 1)− (d− 1)(n− 1) on the right-hand side of (3.2) simplifies to −1 due to our assumption concerning the sum of the ranks of the types Ti. Hence, if we use the relation (6.12) once more, the second term on the right-hand side of (6.17) is seen to equal −2(d− 1)(n− 1)d n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m This explains the second term in the factor in big parentheses in (6.2) and the fourth term in the factor in big parentheses on the right-hand side of (6.3). Finally, we come to the first term on the right-hand side of (6.17). We have i,k+1 fj(z)pj,1 −c(z) det 1≤i,k≤d i,k+1 i 6=j  det1≤i,k≤d    ci−1+χ(i=j) z1 · · · zd , i = k ci−2+χ(i=j) z1 · · · zd ×P ′i z1 · · · zd z1 · · · zd , i 6= k    i,k+1 i 6=j × det 1≤i,k≤d ci−1+χ(i=j) z1 · · · zd , i = k ci − 1 + χ(i = j) ci−1+χ(i=j) i (u) ) ∣∣∣∣∣ u=z1···zd/zi , i 6= k DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 31 Reading coefficients, we obtain ci − 1 + χ(i = j) 1 , m 2 , . . . , m 1≤i,k≤d 1, i = k ci − 1 + χ(i = j) , i 6= k ci − 1 + χ(i = j) 1 , m 2 , . . . , m 1≤i,k≤d 1− χ(j 6= k) n , i = j 1− χ(i 6= k) n−1 , i 6= j the second line being due to (6.12). Now we can apply Corollary 3 with r = j, Xi = ci − 1, i = 1, . . . , j − 1, j + 1, . . . , d, Xj = cj, Y = n− 1, and Z = n. The term Xi + (Y − Z)Xj − (d− 1)Y Z = n (ci − 1) − cj − (d− 1)(n− 1)n (n− rkTi − 1) + n− cj − (d− 1)(n− 1)n on the right-hand side of (3.2) simplifies to −cj due to our assumption concerning the sum of the ranks of the types Ti. Hence, if we use the relation (6.12) once more, the second term on the right-hand side of (6.17) is seen to equal the sum over j = 1, 2, . . . , d 2(n− 1)d−1 n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m This explains the first terms in the factors in big parentheses on the right-hand sides of (6.2) and (6.3). The proof of the theorem is complete. � Combining the previous theorem with the summation formula of Lemma 4, we can now derive compact formulae for all type Dn decomposition numbers. Theorem 10. (i) Let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , j − 1, j + 1, . . . , d, Tj = Dα ∗ A 2 ∗ · · · ∗ A for some α ≥ 2. Then N combDn (T1, T2, . . . , Td) = (n− 1) rkT1 + rkT2 + · · ·+ rkTd − 1 n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m , (6.18) where the multinomial coefficient is defined as in Lemma 4. For α ≥ 4, the number NDn(T1, T2, . . . , Td) is given by the same formula. 32 C. KRATTENTHALER AND T. W. MÜLLER (ii) Let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d. N combDn (T1, T2, . . . , Td) = (n− 1) rkT1 + rkT2 + · · ·+ rkTd − 1 n− rkTj 1 , m 2 , . . . , m )( d∏ i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m ℓ=1 rkTℓ n− 1− ℓ=1 rkTℓ ℓ=1 rkTℓ − 2(d− 2)(n− 1) n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m  , (6.19) whereas NDn(T1, T2, . . . , Td) = (n− 1) rk T1 + rkT2 + · · ·+ rkTd − 1 i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m n− rkTj 1 , m 2 , . . . , m n− rkTj 1 , m 2 , m 3 − 1, m 4 , . . . , m n− rkTj 1 − 2, m 2 , . . . , m ℓ=1 rkTℓ n− 1− ℓ=1 rkTℓ ℓ=1 rkTℓ − 2(d− 2)(n− 1) n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m  . (6.20) (iii) All other decomposition numbers NDn(T1, T2, . . . , Td) and N (T1, T2, . . . , Td) are zero. Remark. The caveats on interpretations of the formulae in Theorem 9 for critical choices of the parameters (cf. the Remark after the statement of that theorem) apply also to the formulae of Theorem 10. DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 33 Proof. We proceed in a manner similar to the proof of Theorem 8. If we write r for n− rkT1 − rkT2 − · · · − rkTd and set Φ = Dn, then the relation (2.3) becomes NDn(T1, T2, . . . , Td) = T :rkT=r NDn(T1, T2, . . . , Td, T ), (6.21) with the same relation holding for N combDn in place of NDn . In order to prove (6.18), we let T = Am11 ∗A 2 ∗ · · · ∗A n and use (6.1) in (6.21), to obtain N combDn (T1, T2, . . . , Td) = m1+2m2+···+nmn=r (n− 1)d n− r − 1 n− r − 1 m1, m2, . . . , mn n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m If we use (3.4) with M = n− r − 1, we arrive at our claim after little simplification. Next we prove (6.19). In contrast to the previous argument, here the summation on the right-hand side of (6.21) must be taken over all types T of the form T = Dα ∗ A 1 ∗ A 2 ∗ · · · ∗ A n , α ≥ 2, as well as of the form T = A 1 ∗ A 2 ∗ · · · ∗ A For the sum over the former types, we have to substitute (6.1) in (6.21), to get m1+2m2+···+nmn=r−α (n− 1)d m1, m2, . . . , mn n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m . (6.22) On the other hand, for the sum over the latter types, we have to substitute (6.2) in (6.21), to get m1+2m2+···+nmn=r (n− 1)d m1, m2, . . . , mn n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m m1+2m2+···+nmn=r (n− 1)d n− r − 1 n− r − 1 m1, m2, . . . , mn n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m −2(d− 1)(n− 1) n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m  . (6.23) 34 C. KRATTENTHALER AND T. W. MÜLLER We simplify (6.22) by using (3.4) with r replaced by r − α and M = n − r, and by subsequently applying the elementary summation formula n− α− 1 r − α n− α− 1 n− r − 1 r − 2 . (6.24) The expression which we obtain in this way explains the fraction in the third line of (6.19) multiplied by the expression in the last line. On the other hand, we simplify the sums in (6.23) by using (3.4) with M = n− r, respectively M = n− r − 1. Thus, the expression (6.23) becomes 2(n− 1)d n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m +(n−1)d−1 n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m − 2(d− 1)(n− 1) n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m which explains the expression in the second line of (6.19) and the second expression in the third line of (6.19) multiplied by the expression in the last line. The proof of (6.20) is analogous, using (6.3) instead of (6.2). We leave the details to the reader. � 7. Generalised non-crossing partitions In this section we recall the definition of Armstrong’s [1] generalised non-crossing partitions poset, and its combinatorial realisation from [1] and [29] for the types An, Bn, and Dn. Let again Φ be a finite root system of rank n, and letW = W (Φ) be the corresponding reflection group. We define first the non-crossing partition lattice NC(Φ) (cf. [8, 15]). Let c be a Coxeter element in W . Then NC(Φ) is defined to be the restriction of the partial order ≤T from Section 2 to the set of all elements which are less than or equal to c in this partial order. This definition makes sense since any two Coxeter elements in W are conjugate to each other; the induced inner automorphism then restricts to an isomorphism of the posets corresponding to the two Coxeter elements. It can be shown thatNC(Φ) is in fact a lattice (see [16] for a uniform proof), and moreover self-dual (this is obvious from the definition). Clearly, the minimal element in NC(Φ) is the identity element in W , which we denote by ε, and the maximal element in NC(Φ) is the chosen Coxeter element c. The term “non-crossing partition lattice” is used because NC(An) is isomorphic to the lattice of non-crossing partitions of {1, 2, . . . , n + 1}, originally introduced by Kreweras [30] (see also [20] and below), and since also NC(Bn) and NC(Dn) can be realised as lattices of non-crossing partitions (see [4, 32] and below). In addition to a fixed root system, the definition of Armstrong’s generalised non- crossing partitions requires a fixed positive integer m. The poset of m-divisible non- crossing partitions associated to the root system Φ has as ground-set the following subset DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 35 of (NC(Φ))m+1: NCm(Φ) = (w0;w1, . . . , wm) : w0w1 · · ·wm = c and ℓT (w0) + ℓT (w1) + · · ·+ ℓT (wm) = ℓT (c) . (7.1) The order relation is defined by (u0; u1, . . . , um) ≤ (w0;w1, . . . , wm) if and only if ui ≥T wi, 1 ≤ i ≤ m. (According to this definition, u0 and w0 need not be related in any way. However, it follows from [1, Lemma 3.4.7] that, in fact, u0 ≤T w0.) The poset NC m(Φ) is graded by the rank function (w0;w1, . . . , wm) = ℓT (w0). (7.2) Thus, there is a unique maximal element, namely (c; ε, . . . , ε), where ε stands for the identity element in W , but, for m > 1, there are many different minimal elements. In particular, NCm(Φ) has no least element if m > 1; hence, NCm(Φ) is not a lattice for m > 1. (It is, however, a graded join-semilattice, see [1, Theorem 3.4.4].) In what follows, we shall use the notions “generalised non-crossing partitions” and “m-divisible non-crossing partitions” interchangeably, where the latter notion will be employed particularly in contexts in which we want to underline the presence of the parameter m. In the remainder of this section, we explain combinatorial realisations of the m- divisible non-crossing partitions of types An−1, Bn, and Dn. In order to be able to do so, we need to recall the definition of Kreweras’ non-crossing partitions of {1, 2, . . . , N}, his “partitions non croisées d’un cycle” of [30]. We place N vertices around a cycle, and label them 1, 2, . . . , N in clockwise order. The circular representation of a partition of the set {1, 2, . . . , N} is the geometric object which arises by representing each block {i1, i2, . . . , ik} of the partition, where i1 < i2 < · · · < ik, by the polygon consisting of the vertices labelled i1, i2, . . . , ik and edges which connect these vertices in clockwise order. A partition of {1, 2, . . . , N} is called non-crossing if any two edges in its circular representation are disjoint. Figure 7 shows the non-crossing partition {{1, 2, 21}, {3, 19, 20}, {4, 5, 6}, {7, 17, 18}, {8, 9, 10, 11, 12, 13, 14, 15, 16}} of {1, 2, . . . , 21}. There is a natural partial order on Kreweras’ non-crossing partitions defined by refinement: a partition π1 is less than or equal to the partition π2 if every block of π1 is contained in some block of π2. If Φ = An−1, the m-divisible non-crossing partitions are in bijection with Kreweras- type non-crossing partitions of the set {1, 2, . . . , mn}, in which all the block sizes are divisible by m. We denote the latter set of non-crossing partitions by ÑCm(An−1). It has been first considered by Edelman in [18]. In fact, Figure 7 shows an example of a 3-divisible non-crossing partition of type A20. Given an element (w0;w1, . . . , wm) ∈ NC m(An−1), the bijection, ▽ say, from [1, Theorem 4.3.8] works by “blowing up” w1, w2, . . . , wm, thereby “interleaving” them, and then “gluing” them together by an operation which is called Kreweras complement in [1]. More precisely, for i = 1, 2, . . . , m, let τm,i be the transformation which maps a permutation w ∈ Sn to a permutation τm,i(w) ∈ Smn by letting (τm,i(w))(mk + i−m) = mw(k) + i−m, k = 1, 2, . . . , n, 36 C. KRATTENTHALER AND T. W. MÜLLER Figure 7. Combinatorial realisation of a 3-divisible non-crossing parti- tion of type A6 and (τm,i(w))(l) = l for all l 6≡ i (mod m). At this point, the reader should recall from Section 2 that W (An−1) is the symmetric group Sn, and that the standard choice of a Coxeter element in W (An−1) = Sn is c = (1, 2, . . . , n). With this choice of Coxeter element, the announced bijection maps (w0;w1, . . . , wm) ∈ NC m(An−1) to ▽mAn−1(w0;w1, . . . , wm) = (1, 2, . . . , mn) (τm,1(w1)) −1 (τm,2(w2)) −1 · · · (τm,m(wm)) We refer the reader to [1, Sec. 4.3.2] for the details. For example, let n = 7, m = 3, w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6). Then (w0;w1, w2, w3) is mapped to ▽3A6(w0;w1, w2, w3) = (1, 2, . . . , 21) (7, 16) (2, 20) (18, 6, 3) = (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16). (7.3) Figure 7 shows the graphical representation of (7.3) on the circle, in which we represent a cycle (i1, i2, . . . , ik) as a polygon consisting of the vertices labelled i1, i2, . . . , ik and edges which connect these vertices in clockwise order. It is shown in [1, Theorem 4.3.8] that ▽mAn−1 is in fact an isomorphism between the posets NCm(An−1) and ÑC m(An−1). Furthermore, it is proved in [1, Theorem 4.3.13] DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 37 Figure 8. Combinatorial realisation of a 3-divisible non-crossing parti- tion of type B5 ci(w0) = bi(▽ (w0;w1, . . . , wm)), i = 1, 2, . . . , n, (7.4) where ci(w0) denotes the number of cycles of length i of w0 and bi(π) denotes the number of blocks of size mi in the non-crossing partition π. If Φ = Bn, the m-divisible non-crossing partitions are in bijection with Kreweras-type non-crossing partitions π of the set {1, 2, . . . , mn, 1̄, 2̄, . . . , mn}, in which all the block sizes are divisible by m, and which have the property that if B is a block of π then also B := {x̄ : x ∈ B} is a block of π. (Here, as earlier, we adopt the convention that ¯̄x = x for all x.) We denote the latter set of non-crossing partitions by ÑCm(Bn). A block B with B = B is called a zero block . A non-crossing partition in ÑCm(Bn) can only have at most one zero block. Figures 8 and 9 give examples of 3-divisible non-crossing partitions of type B5. Figure 8 shows one without a zero block, while Figure 9 shows one with a zero block. Clearly, the condition that B is a block of the partition if and only if B is a block translates into the condition that the geometric realisation of the partition is invariant under rotation by 180◦. 38 C. KRATTENTHALER AND T. W. MÜLLER Figure 9. A 3-divisible non-crossing partition of type B5 with zero block Given an element (w0;w1, . . . , wm) ∈ NC m(Bn), the bijection, ▽ say, from [1, Theorem 4.5.6] works in the same way as for NCm(An−1). That is, recalling from Section 2 that W (Bn) can be combinatorially realised as a subgroup of the group of permutations of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}, and that, in this realisation, the standard choice of a Coxeter element is c = [1, 2, . . . , n] = (1, 2, . . . , n, 1̄, 2̄, . . . , n̄), the announced bijection maps (w0;w1, . . . , wm) ∈ NC m(Bn) to ▽mBn(w0;w1, . . . , wm) = [1, 2, . . . , mn] (τ̄m,1(w1)) −1 (τ̄m,2(w2)) −1 · · · (τ̄m,m(wm)) where τ̄m,i is the obvious extension of the above transformations τm,i: namely we let (τ̄m,i(w))(mk + i−m) = mw(k) + i−m, k = 1, 2, . . . , n, 1̄, 2̄, . . . , n̄, and (τ̄m,i(w))(l) = l and (τ̄m,i(w))(l̄) = l̄ for all l 6≡ i (mod m), where mk̄ + i − m is identified with mk + i−m for all k and i. We refer the reader to [1, Sec. 4.5] for the details. For example, let n = 5, m = 3, w0 = ((2, 4)), w1 = [1] = (1, 1̄), w2 = ((1, 4)), and w3 = ((2, 3)) ((4, 5)). Then (w0;w1, w2, w3) is mapped to ▽3B5(w0;w1, w2, w3) = [1, 2, . . . , 15] [1] ((2, 11)) ((6, 9)) ((12, 15)) = ((1, 2̄, 12)) ((3, 4, 5, 6, 10, 11)) ((7, 8, 9)) ((13, 14, 15)). (7.5) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 39 Figure 8 shows the graphical representation of (7.5). It is shown in [1, Theorem 4.5.6] that ▽mBn is in fact an isomorphism between the posets NCm(Bn) and ÑC m(Bn). Furthermore, it is proved in [1, proof of The- orem 4.3.13] that ci(w0) = bi(▽ (w0;w1, . . . , wm)), i = 1, 2, . . . , n, (7.6) where ci(w0) denotes the number of type A cycles (recall the corresponding terminology from Section 4) of length i of w0 and bi(π) denotes one half of the number of non-zero blocks of size mi in the non-crossing partition π. (Recall that non-zero blocks come in “symmetric” pairs.) Consequently, under the bijection ▽mBn , the element w0 contains a type B cycle of length ℓ if and only if ▽mBn(w0;w1, . . . , wm) contains a zero block of size The m-divisible non-crossing partitions of type Dn cannot be realised as certain “partitions non croisées d’un cycle,” but as non-crossing partitions on an annulus with 2m(n − 1) vertices on the outer cycle and 2m vertices on the inner cycle, the vertices on the outer cycle being labelled by 1, 2, . . . , mn − m, 1̄, 2̄, . . . , mn−m in clockwise order, and the vertices of the inner cycle being labelled by mn − m + 1, . . . , mn − 1, mn,mn−m+ 1, . . . , mn− 1, mn in counter-clockwise order. Given a partition π of {1, 2, . . . , mn, 1̄, 2̄, . . . , mn}, we represent it on this annulus in a manner analogous to Kreweras’ graphical representation of his partitions; namely, we represent each block of π by connecting the vertices labelled by the elements of the block by curves in clockwise order, the important additional requirement being here that the curves must be drawn in the interior of the annulus. If it is possible to draw the curves in such a way that no two curves intersect, then the partition is called a non-crossing partition on the (2m(n − 1), 2m)-annulus. Figure 10 shows a non-crossing partition on the (15, 6)- annulus. With this definition, them-divisible non-crossing partitions of typeDn are in bijection with non-crossing partitions π on the (2m(n − 1), 2m)-annulus, in which successive elements of a block (successive in the circular order in the graphical representation of the block) are in successive congruence classes modulo m, which have the property that if B is a block of π then also B := {x̄ : x ∈ B} is a block of π, and which satisfy an additional restriction concerning their zero block. Here again, a zero block is a block B with B = B. The announced additional restriction says that a zero block can only occur if it contains all the vertices of the inner cycle, that is, mn − m + 1, . . . , mn − 1, mn,mn−m+ 1, . . . , mn− 1, mn, and at least two further elements from the outer cycle. We denote this set of non-crossing partitions on the (2m(n − 1), 2m)-annulus by ÑCm(Dn). A non-crossing partition in ÑC m(Dn) can only have at most one zero block. Figures 10 and 11 give examples of 3-divisible non-crossing partitions of type D6, Figure 10 one without a zero block, while Figure 11 one with a zero block. Again, it is clear that the condition that B is a block of the partition if and only if B is a block translates into the condition that the geometric realisation of the partition is invariant under rotation by 180◦. In order to clearly sort out the differences to the earlier combinatorial realisations of m-divisible non-crossing partitions of types An−1 and Bn, we stress that for type Dn there are three major features which are not present for the former types: (1) here we consider non-crossing partitions on an annulus; (2) it is not sufficient to impose the 40 C. KRATTENTHALER AND T. W. MÜLLER 17 18 Figure 10. Combinatorial realisation of a 3-divisible non-crossing par- tition of type D6 condition that the size of every block is divisible by m: the condition on successive elements of a block is stronger; (3) there is the above additional restriction on the zero block (which is not present in type Bn). Given an element (w0;w1, . . . , wm) ∈ NC m(Dn), the bijection, ▽ say, from [29] works as follows. Recalling from Section 2 that W (Dn) can be combinatorially realised as a subgroup of the group of permutations of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}, and that, in this realisation, the standard choice of a Coxeter element is c = [1, 2, . . . , n − 1] [n] = (1, 2, . . . , n− 1, 1̄, 2̄, . . . , n− 1) (n, n̄), the announced bijection maps (w0;w1, . . . , wm) ∈ NCm(Dn) to ▽mDn (w0;w1, . . . , wm) = [1, 2, . . . , m(n− 1)] [mn−m+ 1, . . . , mn− 1, mn] ◦ (τ̄m,1(w1)) −1 (τ̄m,2(w2)) −1 · · · (τ̄m,m(wm)) where τ̄m,i is defined as above. We refer the reader to [29] for the details. For example, let n = 6, m = 3, w0 = ((2, 4̄)), w1 = ((2, 6̄)) ((4, 5)), w2 = ((1, 5̄)) ((2, 3)), and DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 41 17 18 Figure 11. A 3-divisible non-crossing partition of type D6 with zero block w3 = ((3, 6)). Then (w0;w1, w2, w3) is mapped to ▽3D6(w0;w1, w2, w3) = [1, 2, . . . , 15] [16, 17, 18]((4, 16)) ((10, 13)) ((2, 14)) ((5, 8)) ((9, 18)) = ((1, 2, 15)) ((3, 4, 17, 18, 10, 14)) ((5, 9, 16)) ((6, 7, 8)) ((11, 12, 13)). (7.7) Figure 10 shows the graphical representation of (7.7). It is shown in [29] that ▽mDn is in fact an isomorphism between the posets NC m(Dn) and ÑCm(Dn). Furthermore, it is proved in [29] that ci(w0) = bi(▽ (w0;w1, . . . , wm)), i = 1, 2, . . . , n, (7.8) where ci(w0) denotes the number of type A cycles of length i of w0 and bi(π) denotes one half of the number of non-zero blocks of size mi in the non-crossing partition π. (Recall that non-zero blocks come in “symmetric” pairs.) Consequently, under the bijection ▽mDn, the element w0 contains a type D cycle of length ℓ if and only if ▽mDn(w0;w1, . . . , wm) contains a zero block of size mℓ. 42 C. KRATTENTHALER AND T. W. MÜLLER 8. Decomposition numbers with free factors, and enumeration in the poset of generalised non-crossing partitions This section is devoted to applying our formulae from Sections 4–6 for the decompo- sition numbers of the types An, Bn, and Dn to the enumerative theory of generalised non-crossing partitions for these types. Theorems 11–15 present formulae for the num- ber of minimal factorisations of Coxeter elements in types An, Bn, and Dn, respectively, where we do not prescribe the types of all the factors as for the decomposition numbers, but just for some of them, while we impose rank sum conditions on other factors. Im- mediate corollaries are formulae for the number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1, l being given, in the posets ÑCm(An−1), ÑC m(Bn), and ÑC m(Dn), where the poset rank of πi equals ri, and where the block structure of π1 is prescribed, see Corollaries 12, 14, and 16. These results in turn imply all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations, see the remarks accom- panying the corollaries. They also imply two further new results on chain enumeration in ÑCm(Dn), see Corollaries 18 and 19. We want to stress that, since ÑC m(Φ) and NCm(Φ) are isomorphic as posets for Φ = An−1, Bn, Dn, Corollaries 12, 14, 16, 17, 18 imply obvious results for NCm(Φ) in place of ÑCm(Φ), Φ = An−1, Bn, Dn, via (7.4), (7.6), respectively (7.8). We begin with our results for type An. The next theorem generalises Theorem 6, which can be obtained from the former as the special case in which l = 1 and m1 = 1. Theorem 11. For a positive integer d, let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d, and let l, m1, m2, . . . , ml, s1, s2, . . . , sl be given non-negative integers with rkT1 + rk T2 + · · ·+ rkTd + s1 + s2 + · · ·+ sl = n. Then the number of factorisations c = σ1σ2 · · ·σdσ 2 · · ·σ 2 · · ·σ · · ·σ 2 · · ·σ , (8.1) where c is a Coxeter element in W (An), such that the type of σi is Ti, i = 1, 2, . . . , d, and such that ℓT (σ 1 ) + ℓT (σ 2 ) + · · ·+ ℓT (σ ) = si, i = 1, 2, . . . , l, (8.2) is given by (n+ 1)d−1 n− rkTi + 1 n− rkTi + 1 1 , m 2 , . . . , m m1(n+ 1) m2(n+ 1) · · · ml(n+ 1) , (8.3) where the multinomial coefficient is defined as in Lemma 4. Proof. In the factorisation (8.1), we first fix also the types of the σ i ’s. For i = 1, 2, . . . , mj and j = 1, 2, . . . , l, let the type of σ i = A (i,j) 1 ∗ A (i,j) 2 ∗ · · · ∗ A (i,j) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 43 We know that the number of these factorisations is given by (4.1) with d replaced by d+m1+m2+· · ·+ml and the appropriate interpretations of them i ’s. Next we fix non- negative integers r i and sum the expression (4.1) over all possible types T i of rank i , i = 1, 2, . . . , mj, j = 1, 2, . . . , l. The corresponding summations are completely analogous to the summation in the proof of Theorem 6. As a result, we obtain (n+ 1)d−1 n− rkTi + 1 n− rkTi + 1 1 , m 2 , . . . , m · · · n + 1 · · · × · · · × n + 1 n + 1 · · · for the number of factorisations under consideration. In view of (8.2) and (2.4), to obtain the final result, we must sum these expressions over all non-negative integers 1 , . . . , r ml satisfying the equations 1 + r 2 + · · ·+ r = sj, j = 1, 2, . . . , l. (8.4) This is easily done by means of the multivariate version of the Chu–Vandermonde summation. The formula in (8.3) follows. � In view of the combinatorial realisation of m-divisible non-crossing partitions of type An−1 which we described in Section 7, the special case d = 1 of the above theorem has the following enumerative consequence. Corollary 12. Let l be a positive integer, and let s1, s2, . . . , sl be non-negative integers with s1 + s2+ · · ·+ sl = n− 1. The number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the poset ÑCm(An−1), with the property that rk(πi) = s1 + s2 + · · ·+ si, i = 1, 2, . . . , l− 1, and that the number of blocks of size mi of π1 is bi, i = 1, 2, . . . , n, is given by b1 + b2 + · · ·+ bn b1 + b2 + · · ·+ bn b1, b2, . . . , bn · · · , (8.5) provided that b1 + 2b2 + · · ·+ nbn ≤ n, and is 0 otherwise. Remark. The conditions in the statement of the corollary imply that s1 + b1 + b2 + · · ·+ bn = n. (8.6) Proof. Let π1 ≤ π2 ≤ · · · ≤ πl−1 (8.7) be a multi-chain in ÑCm(An−1). Suppose that, under the bijection ▽ , the element πj corresponds to the tuple (w 1 , . . . , w m ), j = 1, 2, . . . , l− 1. The inequalities in (8.7) imply that w 1 , w 2 , . . . , w m can be factored in the form i = u i · · ·u i , i = 1, 2, . . . , m, where u i = w (l−1) i and, more generally, i = u (j+1) (j+2) i · · ·u i , i = 1, 2, . . . , m, j = 1, 2, . . . , l − 1. (8.8) 44 C. KRATTENTHALER AND T. W. MÜLLER For later use, we record that c = w 1 · · ·w (j+1) (j+2) 1 · · ·u (j+1) (j+2) 2 · · ·u · · · u(j+1)m u (j+2) m · · ·u . (8.9) Now, by (7.4), the block structure conditions on π1 in the statement of the corollary translate into the condition that the type of w Ab21 ∗ A 2 ∗ · · · ∗ A n−1. (8.10) On the other hand, using (7.2), we see that the rank conditions in the statement of the corollary mean that ℓT (w 0 ) = s1 + s2 + · · ·+ sj, j = 1, 2, . . . , l − 1. In combination with (8.9), this yields the conditions ℓT (u 1 ) + ℓT (u 2 ) + · · ·+ ℓT (u m ) = sj , j = 2, 3, . . . , l. (8.11) Thus, we want to count the number of factorisations c = w 1 · · ·u 2 · · ·u · · · u(2)m u m · · ·u , (8.12) where the type of w 0 is given in (8.10), and where the “rank conditions” (8.11) are satisfied. So, in view of (2.4), we are in the situation of Theorem 11 with n replaced by n− 1, d = 1, l replaced by l − 1, si replaced by si+1, i = 1, 2, . . . , l − 1, T1 the type in (8.10), m1 = m2 = · · · = ml−1 = m, except that the factors are not exactly in the order as in (8.1). However, by (2.2) we know that the order of factors is without relevance. Therefore we just have to apply Theorem 11 with the above specialisations. If we also take into account (8.6), then we arrive immediately at (8.5). � This result is new even for m = 1, that is, for the poset of Kreweras’ non-crossing par- titions of {1, 2, . . . , n}. It implies all known results on Kreweras’ non-crossing partitions and the m-divisible non-crossing partitions of Edelman. Namely, for l = 2 it reduces to Armstrong’s result [1, Theorem 4.4.4 with ℓ = 1] on the number of m-divisible non-crossing partitions in ÑCm(An−1) with a given block structure, which itself con- tains Kreweras’ result [30, Theorem 4] on his non-crossing partitions with a given block structure as a special case. If we sum the expression (8.5) over all s2, s3, . . . , sl with s2 + s3 + · · · + sl = n − 1 − s1, then we obtain that the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in Edelman’s poset ÑC m(An−1) of m-divisible non-crossing partitions of {1, 2, . . . , mn} in which π1 has bi blocks of size mi equals b1 + b2 + · · ·+ bn b1 + b2 + · · ·+ bn b1, b2, . . . , bn (l − 1)mn n− s1 − 1 b1 + b2 + · · ·+ bn b1 + b2 + · · ·+ bn b1, b2, . . . , bn (l − 1)mn b1 + b2 + · · ·+ bn − 1 , (8.13) provided that b1 + 2b2 + · · · + nbn ≤ n, a result originally due to Armstrong [1, Theorem 4.4.4]. On the other hand, if we sum the expression (8.5) over all possible b1, b2, . . . , bn, that is, b2 + 2b3 + · · ·+ (n− 1)bn = s1, use of Lemma 4 with M = n− s1 and r = s1 yields that the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in Edelman’s DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 45 poset ÑCm(An−1) ∼= NC m(An−1) where πi is of rank s1+s2+ · · ·+si, i = 1, 2, . . . , l−1, equals · · · , (8.14) a result originally due to Edelman [18, Theorem 4.2]. Clearly, this formula contains at the same time a formula for the number of all m-divisible non-crossing partitions of {1, 2, . . . , mn} with a given number of blocks upon setting l = 2 (cf. [18, Lemma 4.1]), as well as that it implies that the total number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the poset of these partitions is (l − 1)mn+ n (8.15) upon summing (8.14) over all non-negative integers s1, s2, . . . , sl with s1+s2+ · · ·+sl = n− 1 by means of the multivariate Chu–Vandermonde summation, thus recovering the formula [18, Cor. 4.4] for the zeta polynomial of the poset of m-divisible non-crossing partitions of type An−1. As special case l = 2, we recover the well-known fact that the total number of m-divisible non-crossing partitions of {1, 2, . . . , mn} is 1 (m+1)n We continue with our results for type Bn. We formulate the theorem below on factorisations inW (Bn) only with restrictions on the combinatorial type of some factors. An analogous result with group-theoretical type instead could be easily derived as well. We omit this here because, for the combinatorial applications that we have in mind, combinatorial type suffices. We remark that the theorem generalises Theorem 8, which can be obtained from the former as the special case in which l = 1 and m1 = 1. Theorem 13. (i) For a positive integer d, let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , j − 1, j + 1, . . . , d, Tj = Bα ∗ A 1 ∗ A 2 ∗ · · · ∗ A for some α ≥ 1, and let l, m1, m2, . . . , ml, s1, s2, . . . , sl be given non-negative integers rkT1 + rkT2 + · · ·+ rkTd + s1 + s2 + · · ·+ sl = n. (8.16) Then the number of factorisations c = σ1σ2 · · ·σdσ 2 · · ·σ 2 · · ·σ · · ·σ 2 · · ·σ , (8.17) where c is a Coxeter element in W (Bn), such that the combinatorial type of σi is Ti, i = 1, 2, . . . , d, and such that ℓT (σ 1 ) + ℓT (σ 2 ) + · · ·+ ℓT (σ ) = si, i = 1, 2, . . . , l, (8.18) is given by n− rkTj 1 , m 2 , . . . , m )( d∏ i 6=j n− rkTi n− rkTi 1 , m 2 , . . . , m · · · , (8.19) 46 C. KRATTENTHALER AND T. W. MÜLLER where the multinomial coefficient is defined as in Lemma 4. (ii) For a positive integer d, let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d, and let l, m1, m2, . . . , ml, s1, s2, . . . , sl be given non-negative integers. Then the number of factorisations (8.17) which satisfy (8.18) plus the condition that the combinatorial type of σi is Ti, i = 1, 2, . . . , d, is given by nd−1(n− rkT1 − rkT2 − · · · − rkTd) n− rkTi n− rkTi 1 , m 2 , . . . , m · · · . (8.20) Proof. We start with the proof of item (i). In the factorisation (8.17), we first fix also the types of the σ i ’s. For i = 1, 2, . . . , mj and j = 1, 2, . . . , l let the type of σ i = A (i,j) 1 ∗ A (i,j) 2 ∗ · · · ∗ A (i,j) We know that the number of these factorisations is given by (5.1) with d replaced by d+m1+m2+· · ·+ml and the appropriate interpretations of them i ’s. Next we fix non- negative integers r i and sum the expression (5.1) over all possible types T i of rank i , i = 1, 2, . . . , mj, j = 1, 2, . . . , l. The corresponding summations are completely analogous to the first summation in the proof of Theorem 8. As a result, we obtain n− rkTj 1 , m 2 , . . . , m )( d∏ i 6=j n− rkTi n− rkTi 1 , m 2 , . . . , m · · · · · · × · · · × · · · for the number of factorisations under consideration. In view of (8.18) and (2.4), to obtain the final result, we must sum these expressions over all non-negative integers 1 , . . . , r ml satisfying the equations 1 + r 2 + · · ·+ r = sj, j = 1, 2, . . . , l. This is easily done by means of the multivariate version of the Chu–Vandermonde summation. The formula in (8.19) follows. The proof of item (ii) is completely analogous, we must, however, cope with the complication that the type B cycle, which, according to Theorem 7, must occur in the disjoint cycle decomposition of exactly one of the factors on the right-hand side of (8.17), can occur in any of the σ i ’s. So, let us fix the types of the σ i ’s to i = A (i,j) 1 ∗ A (i,j) 2 ∗ · · · ∗ A (i,j) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 47 i = 1, 2, . . . , mj, j = 1, 2, . . . , l, except for (i, j) = (p, q), where we require that the type T (q)p = Bα ∗ A 1 ∗ A 2 ∗ · · · ∗ A Again, we know that the number of these factorisations is given by (5.1) with d replaced by d +m1 +m2 + · · ·+ml and the appropriate interpretations of the m i ’s. Now we fix non-negative integers r i and sum the expression (5.1) over all possible types T of rank r i , i = 1, 2, . . . , mj , j = 1, 2, . . . , l. Again, the corresponding summations are completely analogous to the summations in the proof of Theorem 8. In particular, the summation over all possible types T p of rank r p is essentially the summation on the right-hand side of (5.14) with d replaced by d + m1 + m2 + · · · + ml and r replaced p . If we use what we know from the proof of Theorem 8, then the result of the summations is found to be n− rkTi n− rkTi 1 , m 2 , . . . , m · · · × · · · × · · · r(q)p · · · × · · · × · · · . (8.21) The reader should note that the term r in this expression results from the sum- mation over all types T p of rank r p (compare (5.15) with r replaced by r p ; we have( ). Using (8.16), (8.18) and (2.4), we see that the sum of all r p over p = 1, 2, . . . , mq and q = 1, 2, . . . , l must be n− rkT1 − rkT2 − · · · − rkTd. Hence, the sum of the expressions (8.21) over all (p, q) equals nd−1(n− rkT1 − rkT2 − · · · − rkTd) n− rkTi n− rkTi 1 , m 2 , . . . , m · · · × · · · × · · · × · · · × · · · Finally, we must sum these expressions over all non-negative integers r 1 , . . . , r ml sat- isfying the equations 1 + r 2 + · · ·+ r = sj, j = 1, 2, . . . , l. Once again, this is easily done by means of the multivariate version of the Chu– Vandermonde summation. As a result, we obtain the formula in (8.20). � In view of the combinatorial realisation of m-divisible non-crossing partitions of type Bn which we described in Section 7, the special case d = 1 of the above theorem has the following enumerative consequence. 48 C. KRATTENTHALER AND T. W. MÜLLER Corollary 14. Let l be a positive integer, and let s1, s2, . . . , sl be non-negative integers with s1 + s2 + · · · + sl = n. The number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the poset ÑCm(Bn) with the property that rk(πi) = s1+ s2+ · · ·+ si, i = 1, 2, . . . , l−1, and that the number of non-zero blocks of π1 of size mi is 2bi, i = 1, 2, . . . , n, is given by( b1 + b2 + · · ·+ bn b1, b2, . . . , bn · · · , (8.22) provided that b1 + 2b2 + · · ·+ nbn ≤ n, and is 0 otherwise. Remark. The conditions in the statement of the corollary imply that s1 + b1 + b2 + · · ·+ bn = n. (8.23) The reader should recall from Section 7, that non-zero blocks of elements π of ÑCm(Bn) occur in pairs since, with a block B of π, also B is a block of π. Proof. The arguments are completely analogous to those of the proof of Corollary 12. The conclusion here is that we need Theorem 13 with d = 1, l replaced by l − 1, si replaced by si+1, i = 1, 2, . . . , l − 1, m1 = m2 = · · · = ml−1 = m, and T1 of the type Bn−b1−2b2−···−nbn ∗ A 1 ∗ A 2 ∗ · · · ∗ A in the case that b1 + 2b2 + · · ·+ nbn < n (which enforces the existence of a zero block of size 2(n− b1 − 2b2 − · · · − nbn) in π1), respectively Ab21 ∗ A 2 ∗ · · · ∗ A if not. So, depending on the case in which we are, we have to apply (8.19), respectively (8.20). However, for d = 1 these two formulae become identical. More precisely, under the above specialisations, they reduce to n− rkT1 b2, b3, . . . , bn · · · If we also take into account (8.23), then we arrive immediately at (8.22). � This result is new even for m = 1, that is, for the poset of Reiner’s type Bn non-crossing partitions. It implies all known results on these non-crossing partitions and their extension to m-divisible type Bn non-crossing partitions due to Armstrong. Namely, for l = 2 it reduces to Armstrong’s result [1, Theorem 4.5.11 with ℓ = 1] on the number of elements of ÑCm(Bn) with a given block structure, which itself con- tains Athanasiadis’ result [2, Theorem 2.3] on Reiner’s type Bn non-crossing partitions with a given block structure as a special case. If we sum the expression (8.22) over all s2, s3, . . . , sl with s2 + s3 + · · · + sl = n − s1, then we obtain that the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in ÑC m(Bn) in which π1 has 2bi non-zero blocks of size mi equals b1 + b2 + · · ·+ bn b1, b2, . . . , bn (l − 1)mn n− s1 b1 + b2 + · · ·+ bn b1, b2, . . . , bn (l − 1)mn b1 + b2 + · · ·+ bn (8.24) provided that b1 + 2b2 + · · · + nbn ≤ n, a result originally due to Armstrong [1, The- orem 4.5.11]. On the other hand, if we sum the expression (8.22) over all possible b1, b2, . . . , bn, that is, over b2 + 2b3 + · · · + (n − 1)bn ≤ s1, use of Lemma 4 with M = n − s1 and r = s1 − α (where α stands for the difference between s1 and DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 49 b2+2b3+ · · ·+(n−1)bn) yields that the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in ÑCm(Bn) ∼= NC m(Bn) where πi is of rank s1+s2+ · · ·+si, i = 1, 2, . . . , l−1, equals n− α− 1 s1 − α · · · · · · , (8.25) another result due to Armstrong [1, Theorem 4.5.7]. Clearly, this formula contains at the same time a formula for the number of all elements of ÑCm(Bn) ∼= NC m(Bn) with a given number of blocks (equivalently, a given rank) upon setting l = 2 (cf. [1, Theorem 4.5.8]), as well as that it implies that the total number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in ÑC m(Bn) ∼= NC m(Bn) is (l − 1)mn+ n (8.26) upon summing (8.25) over all non-negative integers s1, s2, . . . , sl with s1+s2+ · · ·+sl = n by means of the multivariate Chu–Vandermonde summation, thus recovering the formula [1, Theorem 3.6.9] for the zeta polynomial of the poset of generalised non- crossing partitions in the case of type Bn. As special case l = 2, we recover the fact that the cardinality of ÑCm(Bn) ∼= NC m(Bn) is (m+1)n (cf. [1, Theorem 3.5.3]). The final set of results in this section concerns the type Dn. We start with Theo- rem 15, the result on factorisations in W (Dn) which is analogous to Theorems 11 and 13. Similar to Theorem 13, we formulate the theorem only with restrictions on the combinatorial type of some factors. An analogous result with group-theoretical type instead could be easily derived as well. We refrain from doing this here because, again, for the combinatorial applications that we have in mind, combinatorial type suffices. We remark that the theorem generalises Theorem 10, which can be obtained from the former as the special case in which l = 1 and m1 = 1. Theorem 15. (i) For a positive integer d, let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , j − 1, j + 1, . . . , d, Tj = Dα ∗ A 2 ∗ · · · ∗ A for some α ≥ 2, and let l, m1, m2, . . . , ml, s1, s2, . . . , sl be given non-negative integers rkT1 + rk T2 + · · ·+ rkTd + s1 + s2 + · · ·+ sl = n. Then the number of factorisations c = σ1σ2 · · ·σdσ 2 · · ·σ 2 · · ·σ · · ·σ 2 · · ·σ , (8.27) where c is a Coxeter element in W (Dn), such that the combinatorial type of σi is Ti, i = 1, 2, . . . , d, and such that ℓT (σ 1 ) + ℓT (σ 2 ) + · · ·+ ℓT (σ ) = si, i = 1, 2, . . . , l, (8.28) 50 C. KRATTENTHALER AND T. W. MÜLLER is given by (n− 1)d−1 n− rkTj 1 , m 2 , . . . , m )( d∏ i 6=j n− rkTi − 1 n− rk Ti − 1 1 , m 2 , . . . , m m1(n− 1) m2(n− 1) · · · ml(n− 1) , (8.29) the multinomial coefficient being defined as in Lemma 4. (ii) For a positive integer d, let the types T1, T2, . . . , Td be given, where Ti = A 1 ∗ A 2 ∗ · · · ∗ A n , i = 1, 2, . . . , d, and let l, m1, m2, . . . , ml, s1, s2, . . . , sl be given non-negative integers. Then the number of factorisations (8.27) which satisfy (8.28) as well as the condition that the combina- torial type of σi is Ti, i = 1, 2, . . . , d, is given by 2(n− 1)d−1 n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m m1(n− 1) m2(n− 1) · · · ml(n− 1) + (n− 1)d n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m m1(n− 1) · · · mj(n− 1)− 1 sj − 2 · · · ml(n− 1) − 2(d− 1)(n− 1)d n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m m1(n− 1) m2(n− 1) · · · ml(n− 1) . (8.30) Proof. The proof of item (i) is completely analogous to the proof of item (i) in The- orem 13. Making reference to that proof, the only difference is that, instead of the expression (5.1), we must use (6.1) with d replaced by d + m1 + m2 + · · · + ml and the appropriate interpretations of the m i ’s. The summations over types T i with fixed rank r i are carried out by using (3.4) with M = n − r − 1. Subsequently, the summations over the r i ’s satisfying (8.4) are done by the multivariate version of the Chu–Vandermonde summation. We leave it to the reader to fill in the details to finally arrive at (8.29). Similarly, the proof of item (ii) is analogous to the proof of item (ii) in Theorem 13. However, we must cope with the complication that there may or may not be a type D cycle in the disjoint cycle decomposition of one of the σ i ’s on the right-hand side of DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 51 (8.27). In the case that there is no type B cycle, we fix the types of the σ i ’s to i = A (i,j) 1 ∗ A (i,j) 2 ∗ · · · ∗ A (i,j) i = 1, 2, . . . , mj , j = 1, 2, . . . , l, and sum the expression (6.2) with d replaced by d + m1 +m2 + · · ·+ml and the appropriate interpretations of the m i ’s over all possible types T i with rank r i , i = 1, 2, . . . , mj , j = 1, 2, . . . , l. This yields the expression 2(n− 1)d−1 n− rkTj 1 , m 2 , . . . , m i 6=j n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m · · · × · · · × · · · + 2(n− 1)d n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m · · · × · · · × · · · − 2(d− 1)(n− 1)d−1 n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m · · · × · · · × · · · . (8.31) In the case that there appears, however, a type B cycle in σ p , say, we adopt the same set-up as above, except that we restrict σ p to types of the form T (q)p = Dα ∗ A 2 ∗ · · · ∗ A Subsequently, we sum the expression (6.1) with d replaced by d+m1 +m2 + · · ·+ml and the appropriate interpretations of the m i ’s over all possible types T i of rank r This time, we obtain (n− 1)d n− rkTi − 1 n− rkTi − 1 1 , m 2 , . . . , m · · · × · · · × · · · n− α− 1 p − α · · · × · · · × · · · . (8.32) The sum over α can be evaluated by means of the elementary summation formula n− α− 1 r − α n− α− 1 n− r − 1 r − 2 52 C. KRATTENTHALER AND T. W. MÜLLER Finally, we must sum the expressions (8.31) and (8.32) over all non-negative integers 1 , . . . , r ml satisfying the equations 1 + r 2 + · · ·+ r = sj, j = 1, 2, . . . , l. Once again, this is easily done by means of the multivariate version of the Chu– Vandermonde summation. After some simplification, we obtain the formula in (8.30). In view of the combinatorial realisation of m-divisible non-crossing partitions of type Dn which we described in Section 7, the special case d = 1 of the above theorem has the following enumerative consequence. Corollary 16. Let l be a positive integer, and let s1, s2, . . . , sl be non-negative integers with s1 + s2 + · · · + sl = n. The number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the poset ÑCm(Dn) with the property that rk(πi) = s1+ s2+ · · ·+ si, i = 1, 2, . . . , l−1, and that the number of non-zero blocks of π1 of size mi is 2bi, i = 1, 2, . . . , n, is given by( b1 + b2 + · · ·+ bn b1, b2, . . . , bn m(n− 1) · · · m(n− 1) , (8.33) if b1 + 2b2 + · · ·+ nbn < n− 1, and b1 + b2 + · · ·+ bn b1, b2, . . . , bn m(n− 1) · · · m(n− 1) m(n− 1) b1 + b2 + · · ·+ bn − 1 b1 + b2 + · · ·+ bn − 1 b1 − 1, b2, . . . , bn m(n− 1) · · · m(n− 1)− 1 sj − 2 · · · m(n− 1) , (8.34) if b1 + 2b2 + · · ·+ nbn = n. Remark. The conditions in the statement of the corollary imply that s1 + b1 + b2 + · · ·+ bn = n. (8.35) The reader should recall from Section 7, that non-zero blocks of elements π of ÑCm(Dn) occur in pairs since, with a block B of π, also B is a block of π. The condition b1 + 2b2 + · · ·+ nbn < n− 1, which is required for Formula (8.33) to hold, implies that π1 must contain a zero block of size 2(n − b1 − 2b2 − · · · − nbn), while the equality b1 + 2b2 + · · · + nbn = n, which is required for Formula (8.34) to hold, implies that π1 contains no zero block. The extra condition on zero blocks that are imposed on elements of ÑCm(Dn) implies that b1 + 2b2 + · · ·+ nbn cannot be equal to n− 1. Proof. Again, the arguments are completely analogous to those of the proof of Corol- lary 12. Here we need Theorem 15 with d = 1, l replaced by l − 1, si replaced by si+1, i = 1, 2, . . . , l − 1, m1 = m2 = · · · = ml−1 = m, and T1 of the type Dn−b1−2b2−···−nbn ∗A 1 ∗ A 2 ∗ · · · ∗ A in the case that b1 + 2b2 + · · ·+ nbn < n− 1, respectively Ab21 ∗ A 2 ∗ · · · ∗ A DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 53 if not. So, depending on the case in which we are, we have to apply (8.29), respectively (8.30). If we also take into account (8.35), then we arrive at the claimed result after little manipulation. Since we have done similar calculations already several times, the details are left to the reader. � This result is new even for m = 1, that is, for the poset of type Dn non-crossing partitions of Athanasiadis and Reiner [4], and of Bessis and Corran [9]. Not only does it imply all known results on these non-crossing partitions and their extension to m- divisible type Dn non-crossing partitions due to Armstrong, it allows us as well to solve several open enumeration problems on the m-divisible type Dn non-crossing partitions. We state these new results separately in the corollaries below. To begin with, if we set l = 2 in Corollary 16, then we obtain the following extension to ÑCm(Dn) of Athanasiadis and Reiner’s result [4, Theorem 1.3] on the number of type Dn non-crossing partitions with a given block structure. Corollary 17. The number of all elements of ÑCm(Dn) which have 2bi non-zero blocks of size mi equals ( b1 + b2 + · · ·+ bn b1, b2, . . . , bn m(n− 1) b1 + b2 + · · ·+ bn (8.36) if b1 + 2b2 + · · ·+ nbn < n− 1, and b1 + b2 + · · ·+ bn b1, b2, . . . , bn m(n− 1) b1 + b2 + · · ·+ bn b1 + b2 + · · ·+ bn − 1 b1 − 1, b2, . . . , bn m(n− 1) b1 + b2 + · · ·+ bn − 1 (8.37) if b1 + 2b2 + · · ·+ nbn = n. On the other hand, if we sum the expression (8.33), respectively (8.34), over all s2, s3, . . . , sl with s2+s3+ · · ·+sl = n−s1, then we obtain the following generalisation. Corollary 18. The number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in ÑC m(Dn) in which π1 has 2bi non-zero blocks of size mi equals( b1 + b2 + · · ·+ bn b1, b2, . . . , bn (l − 1)m(n− 1) b1 + b2 + · · ·+ bn , (8.38) if b1 + 2b2 + · · ·+ nbn < n− 1, and b1 + b2 + · · ·+ bn b1, b2, . . . , bn (l − 1)m(n− 1) b1 + b2 + · · ·+ bn b1 + b2 + · · ·+ bn − 1 b1 − 1, b2, . . . , bn (l − 1)m(n− 1) b1 + b2 + · · ·+ bn − 1 (8.39) if b1 + 2b2 + · · ·+ nbn = n. Next we sum the expressions (8.33) and (8.34) over all possible b1, b2, . . . , bn, that is, we sum (8.33) over b2+2b3+ · · ·+(n−1)bn < s1−1, and we sum the expression (8.34) over b2+2b3+ · · ·+(n−1)bn = s1. With the help of Lemma 4 and the simple binomial summation (6.24), these sums can indeed be evaluated. In this manner, we obtain the following result on rank-selected chain enumeration in ÑCm(Dn). 54 C. KRATTENTHALER AND T. W. MÜLLER Corollary 19. The number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in ÑC m(Dn) ∼= NCm(Dn) where πi is of rank s1 + s2 + · · ·+ si, i = 1, 2, . . . , l − 1, equals m(n− 1) · · · m(n− 1) m(n− 1) · · · m(n− 1)− 1 sj − 2 · · · m(n− 1) s1 − 2 m(n− 1) · · · m(n− 1) . (8.40) This formula extends Athanasiadis and Reiner’s formula [4, Theorem 1.2(ii)] from NC(Dn) to ÑC m(Dn). Setting l = 2, we obtain a formula for the number of all elements in ÑCm(Dn) ∼= NC m(Dn) with a given number of blocks (equivalently, of given rank); cf. [1, Theorem 4.6.3]. Next, summing (8.40) over all non-negative integers s1, s2, . . . , sl with s1+s2+ · · ·+sl = n by means of the multivariate Chu–Vandermonde summation, we find that the total number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in ÑCm(Dn) ∼= NC m(Dn) is given by ((l − 1)m+ 1)(n− 1) ((l − 1)m+ 1)(n− 1) 2(l − 1)m(n− 1) + n ((l − 1)m+ 1)(n− 1) , (8.41) thus recovering the formula [1, Theorem 3.6.9] for the zeta polynomial of the poset of generalised non-crossing partitions for the type Dn. The special case l = 2 of (8.41) gives the well-known fact that the cardinality of ÑCm(Dn) ∼= NC m(Dn) is 2m(n−1)+n (m+1)(n−1) (cf. [1, Theorem 3.5.3]). In the following section, Corollary 19 will enable us to provide a new proof of Arm- strong’s F = M (Ex-)Conjecture in type Dn. 9. Proof of the F = M Conjecture for type D Armstrong’s F = M (Ex-)Conjecture [1, Conjecture 5.3.2], which extends an earlier conjecture of Chapoton [17], relates the “F -triangle” of the generalised cluster complex of Fomin and Reading [19] to the “M-triangle” of Armstrong’s generalised non-crossing partitions. The F -triangle is a certain refined face count in the generalised cluster complex. We do not give the definition here and, instead, refer the reader to [1, 27], because it will not be important in what follows. It suffices to know that, again fixing a finite root system Φ of rank n and a positive integer m, the F -triangle FmΦ (x, y) for the generalised cluster complex ∆m(Φ) is a polynomial in x and y, and that it was computed in [27] for all types. What we need here is that it was shown in [27, Sec. 11, DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 55 Prop. D] that (1− xy)nFmDn x(1 + y) 1− xy 1− xy r,s≥0 m(n− 1) m(n− 1) + s− r − 1 m(n− 1) m(n− 1) + s− r − 1 m(n− 1)− 1 r − 2 m(n− 1) + s− r − 1 m(n− 1) m(n− 1) + s− r − 2 s− r − 2 . (9.1) The “M-triangle” of NCm(Φ) is the polynomial defined by MmΦ (x, y) = u,w∈NCm(Φ) µ(u, w) xrkuyrkw, where µ(u, w) is the Möbius function in NCm(Φ). It is called “triangle” because the Möbius function µ(u, w) vanishes unless u ≤ w, and, thus, the only coefficients in the polynomial which may be non-zero are the coefficients of xkyl with 0 ≤ k ≤ l ≤ n. An equivalent object is the dual M-triangle, which is defined by (MmΦ ) ∗(x, y) = u,w∈(NCm(Φ))∗ µ∗(u, w) xrk ∗ wyrk where (NCm(Φ))∗ denotes the poset dual to NCm(Φ) (i.e., the poset which arises from NCm(Φ) by reversing all order relations), where µ∗ denotes the Möbius function in (NCm(Φ))∗, and where rk∗ denotes the rank function in (NCm(Φ))∗. It is equivalent since, obviously, we have (MmΦ ) ∗(x, y) = (xy)nMmΦ (1/x, 1/y). (9.2) Given this notation, Armstrong’s F = M (Ex-)Conjecture [1, Conjecture 5.3.2] reads as follows. Conjecture FM. For any finite root system Φ of rank n, we have FmΦ (x, y) = y 1 + y y − x y − x Equivalently, (1− xy)nFmΦ x(1 + y) 1− xy 1− xy u,w∈(NCm(Φ))∗ µ∗(u, w) (−x)rk ∗ w(−y)rk ∗ u. (9.3) So, Equation (9.1) provides an expression for the left-hand side of (9.3) for Φ = Dn. With our result on rank-selected chain enumeration in NCm(Dn) given in Corollary 19, we are now able to calculate the right-hand side of (9.3) directly. As we mentioned already in the Introduction, together with the results from [27, 28], this completes a 56 C. KRATTENTHALER AND T. W. MÜLLER computational case-by-case proof of Conjecture FM. A case-free proof had been found earlier by Tzanaki in [38]. The only ingredient that we need for the proof is the well-known link between chain enumeration and the Möbius function. (The reader should consult [33, Sec. 3.11] for more information on this topic.) Given a poset P and two elements u and w, u ≤ w, in the poset, the zeta polynomial of the interval [u, w], denoted by Z(u, w; z), is the number of (multi)chains from u to w of length z. (It can be shown that this is indeed a polynomial in z.) Then the Möbius function of u and w is equal to µ(u, w) = Z(u, w;−1). Proof of Conjecture FM in type Dn. We now compute the right-hand side of (9.3), that u,w∈(NCm(Dn))∗ µ∗(u, w)(−x)rk ∗ w(−y)rk In order to compute the coefficient of xsyr in this expression, (−1)r+s u,w∈(NCm(Dn))∗ with rk∗ u=r and rk∗ w=s µ∗(u, w), we compute the sum of all corresponding zeta polynomials (in the variable z), multiplied by (−1)r+s, (−1)r+s u,w∈(NCm(Dn))∗ with rk∗ u=r and rk∗ w=s Z(u, w; z), and then put z = −1. For computing this sum of zeta polynomials, we must set l = z+2, n−s1 = s, sl = r, s2 + s3 + · · · + sl−1 = s − r in (8.40), and then sum the resulting expression over all possible s2, s3, . . . , sl−1. (The reader should keep in mind that the roles of s1, s2, . . . , sl in Corollary 19 have to be reversed, since we are aiming at computing zeta polynomials in the poset dual to NCm(Dn).) By using the Chu–Vandermonde summation, one obtains m(n− 1) zm(n− 1) m(n− 1)− 1 r − 2 zm(n − 1) m(n− 1) zm(n − 1)− 1 s− r − 2 m(n− 1) zm(n − 1) If we put z = −1 in this expression and multiply it by (−1)r+s, then we obtain exactly the coefficient of xsyr in (9.1). � 10. A conjecture of Armstrong on maximal intervals containing a random multichain Given a finite root system of rank n, Conjecture 3.5.13 in [1] says the following: If we choose an l-multichain uniformly at random from the set π1 ≤ π2 ≤ · · · ≤ πl : πi ∈ NC m(Φ), i = 1, . . . , l, and rk(π1) = i , (10.1) DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 57 then the expected number of maximal intervals in NCm(Φ) containing this multichain Narm(Φ, n− i) Nar1(Φ, n− i) , (10.2) where Narm(Φ, i) is the i-th Fuß–Narayana number associated to NCm(Φ), that is, the number of elements of NCm(Φ) of rank i. In particular, this expected value is independent of l. We show in this section that, for types An and Bn, the conjecture follows easily from Edelman’s (8.14) respectively Armstrong’s (8.25) (presumably, this fact constituted the evidence for setting up the conjecture), while an analogous computation using our new result (8.40) demonstrates that it fails for type Dn. At the end of this section, we comment on what we think happens for the exceptional types. The computation of the expected value in the above conjecture can be approached in the following way. One first observes that a maximal interval in NCm(Φ) is an interval between an element π0 of rank 0 and the global maximum (c; ε, . . . , ε). Therefore, to compute the proposed expected value, we may count the number of chains π0 ≤ π1 ≤ π2 ≤ · · · ≤ πl, rk(π0) = 0 and rk(π1) = i, (10.3) and divide this number by the total number of all chains in (10.1). Clearly, in types An, Bn, and Dn, this kind of chain enumeration can be easily accessed by (8.14), (8.25), and (8.40), respectively. We begin with type An. By (8.14), the number of chains (10.3) equals s2+···+sl+1=n−i n + 1 m(n+ 1) m(n+ 1) · · · m(n + 1) m(n + 1) ml(n + 1) while the number of chains in (10.1) equals s2+···+sl+1=n−i n + 1 m(n+ 1) · · · m(n + 1) n + 1 ml(n + 1) In both cases, we used the multivariate Chu–Vandermonde summation to evaluate the sums over s2, . . . , sl+1. The quotient of the two numbers is n + 1 m(n+ 1) n + 1 m(n + 1) n + 1 which by (8.14) with n replaced by n + 1, l = 2, s1 = n − i, and s2 = i agrees indeed with (10.2) for Φ = An. 58 C. KRATTENTHALER AND T. W. MÜLLER For type Bn, there is an analogous computation using (8.25), the details of which we leave to the reader. The result is that the desired expected value equals which by (8.25) with l = 2, s1 = n− i, and s2 = i agrees indeed with (10.2) for Φ = Bn. The analogous computation for type Dn uses (8.40). The number of chains (10.3) equals s2+···+sl+1=n−i m(n− 1) m(n− 1) · · · m(n− 1) s2+···+sl+1=n−i m(n− 1)− 1 m(n− 1) · · · m(n− 1) s2+···+sl+1=n−i m(n− 1) m(n− 1) · · · m(n− 1)− 1 sj − 2 · · · m(n− 1) m(n− 1) ml(n− 1) m(n− 1)− 1 ml(n− 1) +m(l − 1) m(n− 1) ml(n− 1)− 1 n− i− 2 , (10.4) while the number of chains in (10.1) equals s2+···+sl+1=n−i m(n− 1) · · · m(n− 1) s2+···+sl+1=n−i m(n− 1) · · · m(n− 1)− 1 sj − 2 · · · m(n− 1) s2+···+sl+1=n−i m(n− 1) · · · m(n− 1) ml(n− 1) ml(n− 1)− 1 n− i− 2 ml(n− 1) (10.5) The quotient of (10.4) and (10.5) gives the desired expected value. It is, however, not independent of l, and therefore Armstrong’s conjecture does not hold for Φ = Dn. In the case that Φ is of exceptional type, then, as we outline in the next section, the knowledge of the corresponding decomposition numbers (see the Appendix) allows one to access the rank selected chain enumeration. Using this, the approach for computing the expected value proposed by Armstrong that we used above for the classical types DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 59 can be carried through as well for the exceptional types. We have not done this, but we expect that, similarly to the case of Dn, for most exceptional types the expected value will depend on l, so that Armstrong’s conjecture will probably also fail in these cases. 11. Chain enumeration in the poset of generalised non-crossing partitions for the exceptional types Although it is not the main topic of our paper, we want to briefly demonstrate in this section that the knowledge of the decomposition numbers also enables one to do refined enumeration in the generalised non-crossing partition posets NCm(Φ) for exceptional root systems Φ (of rank n). We restrict the following considerations to the rank-selected chain enumeration. This means that we want to count the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in NC m(Φ), where πi is of rank s1 + s2 + · · · + si, i = 1, 2, . . . , l − 1. Let us denote this number by RΦ(s1, s2, . . . , sl), with sl = n− s1 − s2 − · · · − sl. Now, the considerations at the beginning of the proof of Corollary 12, leading to the factorisation (8.12) with rank constraints on the factors, are also valid for NCm(Φ) instead of NCm(An−1), that is, they are independent of the underlying root system. Hence, to determine the number RΦ(s1, s2, . . . , sl), we have to count all possible factorisations c = w 1 · · ·u 2 · · ·u · · · u(2)m u m · · ·u under the rank constraints (8.11) and ℓT (w 0 ) = s1, where c is a Coxeter element in W (Φ). As we remarked in the proof of Corollary 12, equivalently we may count all factorisations c = w 2 · · ·u 2 · · ·u · · · 2 · · ·u (11.1) which satisfy (8.11) and ℓT (w 0 ) = s1. We can now obtain an explicit expression by fixing first the types of w 0 and all the u i ’s. Under these constraints, the number of factorisations (11.1) is just the corresponding decomposition number. Subsequently, we sum the resulting expressions over all possible types. Before we are able to state the formula which we obtain in this way, we need to recall some standard integer partition notation (cf. e.g. [34, Sec. 7.2]). An integer partition λ (with n parts) is an n-tuple λ = (λ1, λ2, . . . , λn) of integers satisfying λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0. It is called an integer partition of N , written in symbolic notation as λ ⊢ N , if λ1 + λ2 + · · ·+ λn = N . The number of parts (components) of λ of size i is denoted by mi(λ). Then, making again use of the notation for the multinomial coefficient introduced in Lemma 4, the expression for RΦ(s1, s2, . . . , sl) which we obtain in the way described above is ′ NΦ(T 0 , T 1 , T 2 , . . . , T m1(λ(j)), m2(λ(j)), . . . , mn(λ(j)) , (11.2) where ∑ ′ is taken over all integer partitions λ(2), λ(3), . . . , λ(l) satisfying λ(2) ⊢ s2, λ(3) ⊢ s3, . . . , λ (l) ⊢ sl, over all types T 0 with rk(T 0 ) = s1, and over all types T with rk(T i ) = λ i , i = 1, 2, . . . , n, j = 2, 3, . . . , l. By way of example, using this formula and the values of the decomposition num- bers NE8(. . . ) given in Appendix A.7 (and a computer), we obtain that the number 60 C. KRATTENTHALER AND T. W. MÜLLER RE8(4, 2, 1, 1) of all chains π1 ≤ π2 ≤ π3 in NC m(E8), where π1 is of rank 4, π2 is of rank 6, and π3 is of rank 7, is given by 75m3 (8055m− 1141) (which, by the independence (2.2) of decomposition numbers from the order of the types, is also equal to RE8(4, 1, 2, 1) and RE8(4, 1, 1, 2)), while the number RE8(2, 4, 1, 1) of all chains π1 ≤ π2 ≤ π3 in NC m(E8), where π1 is of rank 2, π2 is of rank 6, and π3 is of rank 7, is given by 75m3 (73125m3 − 58950m2 + 15635m− 2154) (which is also equal to RE8(2, 1, 4, 1) and RE8(2, 1, 1, 4)). Acknowledgements The authors thank the anonymous referee for a very careful reading of the original manuscript. Appendix A. The decomposition numbers for the exceptional types A.1. The decomposition numbers for type I2(a) [27, Sec. 13]. We have NI2(a)(I2(a)) = 1, NI2(a)(A1, A1) = a, NI2(a)(A1) = a, NI2(a)(∅) = 1, all other num- bers NI2(a)(T1, T2, . . . , Td) being zero. A.2. The decomposition numbers for type H3 [27, Sec. 14]. We have NH3(H3) = 1, NH3(A 1, A1) = 5, NH3(A2, A1) = 5, NH3(I2(5), A1) = 5, NH3(A1, A1, A1) = 50, plus the assignments implied by (2.2) and (2.3), all other numbers NH3(T1, T2, . . . , Td) being zero. A.3. The decomposition numbers for type H4 [27, Sec. 15]. We have NH4(H4) = 1, NH4(A1 ∗ A2, A1) = 15, NH4(A3, A1) = 15, NH4(H3, A1) = 15, NH4(A1 ∗ I2(5), A1) = 15, NH4(A 1) = 30, NH4(A 1, A2) = 30, NH4(A 1, I2(5)) = 15, NH4(A2, A2) = 5, NH4(A2, I2(5)) = 15, NH4(I2(5), I2(5)) = 3, NH4(A 1, A1, A1) = 225, NH4(A2, A1, A1) = 150, NH4(I2(5), A1, A1) = 90, NH4(A1, A1, A1, A1) = 1350, plus the assignments implied by (2.2) and (2.3), all other numbers NH4(T1, T2, . . . , Td) being zero. A.4. The decomposition numbers for type F4 [27, Sec. 16]. We have NF4(F4) = 1, NF4(A1 ∗ A2, A1) = 12, NF4(B3, A1) = 12, NF4(A 1) = 12, NF4(A 1, B2) = 12, NF4(A2, A2) = 16, NF4(B2, B2) = 3, NF4(A 1, A1, A1) = 72, NF4(A2, A1, A1) = 48, NF4(B2, A1, A1) = 36, NF4(A1, A1, A1, A1) = 432, plus the assignments implied by (2.2) and (2.3), all other numbers NF4(T1, T2, . . . , Td) being zero. DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 61 A.5. The decomposition numbers for type E6 [27, Sec. 17]. We have NE6(E6) = 1, NE6(A1 ∗ A 2, A1) = 6, NE6(A1 ∗ A4, A1) = 12, NE6(A5, A1) = 6, NE6(D5, A1) = 12, NE6(A 1 ∗ A2, A2) = 36, NE6(A 2, A2) = 8, NE6(A1 ∗ A3, A2) = 24, NE6(A4, A2) = 24, NE6(D4, A2) = 4, NE6(A 1 ∗ A2, A 1) = 18, NE6(A1 ∗ A3, A 1) = 36, NE6(A4, A 1) = 36, NE6(D4, A 1) = 18, NE6(A 1) = 12, NE6(A1 ∗ A2, A 1) = 24, NE6(A1 ∗ A2, A1 ∗ A2) = 48, NE6(A3, A 1) = 36, NE6(A3, A1 ∗ A2) = 72, NE6(A3, A3) = 27, NE6(A A2, A1, A1) = 144, NE6(A 2, A1, A1) = 24, NE6(A1∗A3, A1, A1) = 144, NE6(A4, A1, A1) = 144, NE6(D4, A1, A1) = 48, NE6(A 1, A1) = 180, NE6(A 1, A2, A1) = 168, NE6(A1 ∗ A2, A 1, A1) = 360, NE6(A1 ∗ A2, A2, A1) = 336, NE6(A3, A 1, A1) = 378, NE6(A3, A2, A1) = 180, NE6(A 1) = 432, NE6(A2, A 1) = 504, NE6(A2, A2, A 1) = 288, NE6(A2, A2, A2) = 160, NE6(A 1, A1, A1) = 2376, NE6(A2, A 1, A1, A1) = 1872, NE6(A2, A2, A1, A1) = 1056, NE6(A 1, A1, A1, A1) = 864, NE6(A1 ∗ A2, A1, A1, A1) = 1728, NE6(A3, A1, A1, A1) = 1296, NE6(A 1, A1, A1, A1, A1) = 10368, NE6(A2, A1, A1, A1, A1) = 6912, NE6(A1, A1, A1, A1, A1, A1) = 41472, plus the assignments implied by (2.2) and (2.3), all other numbers NE6(T1, T2, . . . , Td) being zero. A.6. The decomposition numbers for type E7 [28, Sec. 6]. We have NE7(E7) = 1, NE7(E6, A1) = 9, NE7(D6, A1) = 9, NE7(A6, A1) = 9, NE7(A1 ∗ D5, A1) = 9, NE7(A1∗A5, A1) = 9, NE7(A2∗D4, A1) = 0, NE7(A2∗A4, A1) = 9, NE7(A 1∗D4, A1) = 0, NE7(A 1∗A4, A1) = 0, NE7(A 3, A1) = 0, NE7(A1∗A2∗A3, A1) = 9, NE7(A 1∗A3, A1) = 0, NE7(A 2, A1) = 0, NE7(A 1 ∗ A 2, A1) = 0, NE7(A 1 ∗ A2, A1) = 0, NE7(A 1, A1) = 0, NE7(D5, A2) = 18, NE7(A5, A2) = 30, NE7(A1 ∗ A4, A2) = 54, NE7(A1 ∗ D4, A2) = 9, NE7(A2 ∗ A3, A2) = 36, NE7(A 1 ∗ A3, A2) = 36, NE7(A1 ∗ A 2, A2) = 36, NE7(A A2, A2) = 12, NE7(A 1, A2) = 0, NE7(D5, A 1) = 54, NE7(A5, A 1) = 63, NE7(A1 ∗ D4, A 1) = 27, NE7(A1 ∗ A4, A 1) = 81, NE7(A2 ∗ A3, A 1) = 27, NE7(A 1 ∗ A3, A 1) = 27, NE7(A1 ∗ A 1) = 27, NE7(A 1 ∗ A2, A 1) = 9, NE7(A 1) = 0, NE7(D5, A1, A1) = 162, NE7(A5, A1, A1) = 216, NE7(A1 ∗ D4, A1, A1) = 81, NE7(A1 ∗ A4, A1, A1) = 324, NE7(A2 ∗ A3, A1, A1) = 162, NE7(A 1 ∗ A3, A1, A1) = 162, NE7(A1 ∗ A 2, A1, A1) = 162, NE7(A 1 ∗ A2, A1, A1) = 54, NE7(A 1, A1, A1) = 0, NE7(D4, A3) = 9, NE7(A4, A3) = 54, NE7(A1 ∗ A3, A3) = 135, NE7(A 2, A3) = 54, NE7(A 1 ∗ A2, A3) = 162, NE7(A 1, A3) = 27, NE7(D4, A1∗A2) = 45, NE7(A4, A1∗A2) = 162, NE7(A1∗A3, A1∗A2) = 243, NE7(A 2, A1∗ A2) = 54, NE7(A 1 ∗ A2, A1 ∗ A2) = 162, NE7(A 1, A1 ∗ A2) = 27, NE7(D4, A 1) = 30, NE7(A4, A 1) = 99, NE7(A1 ∗ A3, A 1) = 126, NE7(A 1) = 18, NE7(A 1 ∗ A2, A 1) = 54, NE7(A 1) = 9, NE7(D4, A2, A1) = 81, NE7(A4, A2, A1) = 378, NE7(A1 ∗A3, A2, A1) = 783, NE7(A 2, A2, A1) = 270, NE7(A 1 ∗ A2, A2, A1) = 810, NE7(A 1, A2, A1) = 135, NE7(D4, A 1, A1) = 243, NE7(A4, A 1, A1) = 891, NE7(A1 ∗ A3, A 1, A1) = 1377, NE7(A A21, A1) = 324, NE7(A 1 ∗ A2, A 1, A1) = 972, NE7(A 1, A1) = 162, NE7(D4, A1, A1, A1) = 729, NE7(A4, A1, A1, A1) = 2916, NE7(A1 ∗ A3, A1, A1, A1) = 5103, NE7(A 2, A1, A1, A1) = 1458, NE7(A 1 ∗ A2, A1, A1, A1) = 4374, NE7(A 1, A1, A1, A1) = 729, NE7(A3, A3, A1) = 486, NE7(A3, A1 ∗ A2, A1) = 1458, NE7(A3, A 1, A1) = 891, NE7(A1 ∗ A2, A1 ∗ A2, A1) = 2430, NE7(A1∗A2, A 1, A1) = 1215, NE7(A 1, A1) = 540, NE7(A3, A2, A2) = 432, NE7(A1 ∗ A2, A2, A2) = 1188, NE7(A 1, A2, A2) = 711, NE7(A3, A2, A 1) = 1053, NE7(A1∗A2, A2, A 1) = 2349, NE7(A 1, A2, A 1) = 1323, NE7(A3, A 1) = 2430, NE7(A1∗ A2, A 1) = 3402, NE7(A 1) = 1539, NE7(A3, A2, A1, A1) = 3402, NE7(A1 ∗ A2, A2, A1, A1) = 8262, NE7(A 1, A2, A1, A1) = 4779, NE7(A3, A 1, A1, A1) = 8019, NE7(A1 ∗ A2, A 1, A1, A1) = 13851, NE7(A 1, A1, A1) = 7047, NE7(A3, A1, A1, A1, A1) = 26244, NE7(A1 ∗ A2, A1, A1, A1, A1) = 52488, NE7(A 1, A1, A1, A1, A1) = 28431, 62 C. KRATTENTHALER AND T. W. MÜLLER NE7(A2, A2, A2, A1) = 2916, NE7(A2, A2, A 1, A1) = 6561, NE7(A2, A 1, A1) = 13122, NE7(A 1, A1) = 19683, NE7(A2, A2, A1, A1, A1) = 21870, NE7(A2, A 1, A1, A1, A1) = 45927, NE7(A 1, A1, A1, A1) = 78732, NE7(A2, A1, A1, A1, A1, A1) = 157464, NE7(A 1, A1, A1, A1, A1, A1) = 295245, NE7(A1, A1, A1, A1, A1, A1, A1) = 1062882, plus the assignments implied by (2.2) and (2.3), all other numbers NE7(T1, T2, . . . , Td) being zero. A.7. The decomposition numbers for type E8 [28, Sec. 7]. We have NE8(E8) = 1, NE8(E7, A1) = 15, NE8(D7, A1) = 15, NE8(A7, A1) = 15, NE8(A1 ∗ E6, A1) = 15, NE8(A1∗D6, A1) = 0, NE8(A1∗A6, A1) = 15, NE8(A2∗D5, A1) = 15, NE8(A2∗A5, A1) = 0, NE8(A 1∗D5, A1) = 0, NE8(A 1∗A5, A1) = 0, NE8(A3∗D4, A1) = 0, NE8(A3∗A4, A1) = 15, NE8(A1∗A2∗D4, A1) = 0, NE8(A1∗A2∗A4, A1) = 15, NE8(A 1∗D4, A1) = 0, NE8(A A4, A1) = 0, NE8(A1 ∗ A 3, A1) = 0, NE8(A 2 ∗ A3, A1) = 0, NE8(A 1 ∗ A2 ∗ A3, A1) = 0, NE8(A 1∗A3, A1) = 0, NE8(A1∗A 2, A1) = 0, NE8(A 2, A1) = 0, NE8(A 1∗A2, A1) = 0, NE8(A 1, A1) = 0, NE8(E6, A2) = 20, NE8(D6, A2) = 15, NE8(A6, A2) = 60, NE8(A1 ∗ D5, A2) = 60, NE8(A1 ∗ A5, A2) = 60, NE8(A2 ∗D4, A2) = 20, NE8(A2 ∗ A4, A2) = 90, NE8(A 3, A2) = 45, NE8(A 1∗D4, A2) = 0, NE8(A 1∗A4, A2) = 90, NE8(A1∗A2∗A3, A2) = 90, NE8(A 1∗A3, A2) = 0, NE8(A 2, A2) = 0, NE8(A 2, A2) = 45, NE8(A 1∗A2, A2) = 0, NE8(A 1, A2) = 0, NE8(E6, A 1) = 45, NE8(D6, A 1) = 90, NE8(A6, A 1) = 135, NE8(A1 ∗ D5, A 1) = 135, NE8(A1 ∗A5, A 1) = 135, NE8(A2 ∗D4, A 1) = 45, NE8(A2 ∗A4, A 1) = 90, NE8(A 1) = 45, NE8(A 1 ∗ D4, A 1) = 0, NE8(A 1 ∗ A4, A 1) = 90, NE8(A1 ∗ A2 ∗ A3, A 1) = 90, NE8(A 1 ∗ A3, A 1) = 0, NE8(A 1) = 0, NE8(A 1 ∗ A 1) = 45, NE8(A 1 ∗A2, A 1) = 0, NE8(A 1) = 0, NE8(E6, A1, A1) = 150, NE8(D6, A1, A1) = 225, NE8(A6, A1, A1) = 450, NE8(A1 ∗ D5, A1, A1) = 450, NE8(A1 ∗ A5, A1, A1) = 450, NE8(A2 ∗ D4, A1, A1) = 150, NE8(A2 ∗ A4, A1, A1) = 450, NE8(A 3, A1, A1) = 225, NE8(A 1 ∗ D4, A1, A1) = 0, NE8(A 1 ∗ A4, A1, A1) = 450, NE8(A1 ∗ A2 ∗ A3, A1, A1) = 450, NE8(A 1 ∗ A3, A1, A1) = 0, NE8(A 2, A1, A1) = 0, NE8(A 1 ∗ A 2, A1, A1) = 225, NE8(A 1 ∗ A2, A1, A1) = 0, NE8(A 1, A1, A1) = 0, NE8(D5, A3) = 45, NE8(A5, A3) = 90, NE8(A1 ∗ A4, A3) = 315, NE8(A1 ∗ D4, A3) = 45, NE8(A2 ∗ A3, A3) = 270, NE8(A A3, A3) = 270, NE8(A1 ∗ A 2, A3) = 225, NE8(A 1 ∗ A2, A3) = 225, NE8(A 1, A3) = 0, NE8(D5, A1 ∗A2) = 195, NE8(A5, A1 ∗A2) = 390, NE8(A1 ∗A4, A1 ∗A2) = 690, NE8(A1 ∗ D4, A1 ∗A2) = 195, NE8(A2 ∗A3, A1 ∗A2) = 495, NE8(A 1 ∗A3, A1 ∗A2) = 495, NE8(A1 ∗ A22, A1 ∗A2) = 300, NE8(A 1 ∗A2, A1 ∗A2) = 300, NE8(A 1, A1 ∗A2) = 0, NE8(D5, A 150, NE8(A5, A 1) = 300, NE8(A1 ∗ A4, A 1) = 375, NE8(A1 ∗ D4, A 1) = 150, NE8(A2 ∗ A3, A 1) = 225, NE8(A 1 ∗ A3, A 1) = 225, NE8(A1 ∗ A 1) = 75, NE8(A 1 ∗ A2, A 1) = 75, NE8(A 1) = 0,NE8(D5, A2, A1) = 375,NE8(A5, A2, A1) = 750, NE8(A1∗A4, A2, A1) = 1950, NE8(A1 ∗D4, A2, A1) = 375, NE8(A2 ∗A3, A2, A1) = 1575, NE8(A 1 ∗A3, A2, A1) = 1575, NE8(A1 ∗ A 2, A2, A1) = 1200, NE8(A 1 ∗ A2, A2, A1) = 1200, NE8(A 1, A2, A1) = 0, NE8(D5, A 1, A1) = 1125, NE8(A5, A 1, A1) = 2250, NE8(A1 ∗ A4, A 1, A1) = 3825, NE8(A1∗D4, A 1, A1) = 1125, NE8(A2∗A3, A 1, A1) = 2700, NE8(A 1∗A3, A 1, A1) = 2700, NE8(A1 ∗ A 1, A1) = 1575, NE8(A 1 ∗ A2, A 1, A1) = 1575, NE8(A 1, A1) = 0, NE8(D5, A1, A1, A1) = 3375, NE8(A5, A1, A1, A1) = 6750, NE8(A1 ∗ A4, A1, A1, A1) = 13500, NE8(A1 ∗ D4, A1, A1, A1) = 3375, NE8(A2 ∗ A3, A1, A1, A1) = 10125, NE8(A A3, A1, A1, A1) = 10125, NE8(A1 ∗ A 2, A1, A1, A1) = 6750, NE8(A 1 ∗ A2, A1, A1, A1) = 6750, NE8(A 1, A1, A1, A1) = 0, NE8(D4, D4) = 5, NE8(D4, A4) = 15, NE8(A4, A4) = 138, NE8(D4, A1 ∗ A3) = 105, NE8(A4, A1 ∗ A3) = 390, NE8(A1 ∗ A3, A1 ∗ A3) = 1155, NE8(D4, A 2) = 35, NE8(A4, A 2) = 180, NE8(A1 ∗ A3, A 2) = 360, NE8(A 2) = 95, DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS 63 NE8(D4, A 1 ∗ A2) = 135, NE8(A4, A 1 ∗ A2) = 630, NE8(A1 ∗ A3, A 1 ∗ A2) = 1035, NE8(A 1 ∗A2) = 270, NE8(A 1 ∗A2, A 1 ∗A2) = 495, NE8(D4, A 1) = 30, NE8(A4, A 165, NE8(A1 ∗A3, A 1) = 255, NE8(A 1) = 60, NE8(A 1 ∗A2, A 1) = 135, NE8(A 30, NE8(D4, A3, A1) = 225, NE8(A4, A3, A1) = 1215, NE8(A1 ∗ A3, A3, A1) = 4050, NE8(A 2, A3, A1) = 1575, NE8(A 1∗A2, A3, A1) = 5400, NE8(A 1, A3, A1) = 1350, NE8(D4, A1 ∗ A2, A1) = 975, NE8(A4, A1 ∗ A2, A1) = 4590, NE8(A1 ∗ A3, A1 ∗ A2, A1) = 10800, NE8(A 2, A1 ∗A2, A1) = 3450, NE8(A 1 ∗A2, A1 ∗A2, A1) = 9900, NE8(A 1, A1 ∗A2, A1) = 2475, NE8(D4, A 1, A1) = 750, NE8(A4, A 1, A1) = 3375, NE8(A1 ∗ A3, A 1, A1) = 6750, NE8(A 1, A1) = 1875, NE8(A 1∗A2, A 1, A1) = 4500, NE8(A 1, A1) = 1125, NE8(D4, A2, A2) = 175, NE8(A4, A2, A2) = 1140, NE8(A1∗A3, A2, A2) = 3300, NE8(A 2, A2, A2) = 1300, NE8(A 1 ∗ A2, A2, A2) = 4500, NE8(A 1, A2, A2) = 1125, NE8(D4, A2, A 1) = 675, NE8(A4, A2, A 1) = 3015, NE8(A1∗A3, A2, A 1) = 8550, NE8(A 2, A2, A 1) = 2925, NE8(A A2, A2, A 1) = 9000, NE8(A 1, A2, A 1) = 2250, NE8(D4, A 1) = 1800, NE8(A4, A A21) = 8640, NE8(A1 ∗ A3, A 1) = 17550, NE8(A 1) = 5175, NE8(A 1 ∗ A2, A A21) = 13500, NE8(A 1) = 3375, NE8(D4, A2, A1, A1) = 1875, NE8(A4, A2, A1, A1) = 9450, NE8(A1 ∗ A3, A2, A1, A1) = 27000, NE8(A 2, A2, A1, A1) = 9750, NE8(A A2, A2, A1, A1) = 31500, NE8(A 1, A2, A1, A1) = 7875, NE8(D4, A 1, A1, A1) = 5625, NE8(A4, A 1, A1, A1) = 26325, NE8(A1 ∗ A3, A 1, A1, A1) = 60750, NE8(A 1, A1, A1) = 19125, NE8(A 1 ∗A2, A 1, A1, A1) = 54000, NE8(A 1, A1, A1) = 13500, NE8(D4, A1, A1, A1, A1) = 16875, NE8(A4, A1, A1, A1, A1) = 81000, NE8(A1 ∗ A3, A1, A1, A1, A1) = 202500, NE8(A 2, A1, A1, A1, A1) = 67500, NE8(A 1 ∗ A2, A1, A1, A1, A1) = 202500, NE8(A 1, A1, A1, A1, A1) = 50625, NE8(A3, A3, A2) = 1350, NE8(A3, A1 ∗A2, A2) = 5175, NE8(A3, A 1, A2) = 3825, NE8(A1 ∗ A2, A1 ∗ A2, A2) = 15000, NE8(A1 ∗ A2, A 1, A2) = 9825, NE8(A 1, A2) = 6000, NE8(A3, A3, A 1) = 4050, NE8(A3, A1 ∗ A2, A 1) = 13500, NE8(A3, A 1) = 9450, NE8(A1 ∗ A2, A1 ∗ A2, A 1) = 30825, NE8(A1 ∗ A2, A 17325,NE8(A 1) = 7875,NE8(A3, A3, A1, A1) = 12150,NE8(A3, A1∗A2, A1, A1) = 42525, NE8(A3, A 1, A1, A1) = 30375, NE8(A1 ∗ A2, A1 ∗ A2, A1, A1) = 106650, NE8(A1 ∗ A2, A 1, A1, A1) = 64125, NE8(A 1, A1, A1) = 33750, NE8(A3, A2, A2, A1) = 10575, NE8(A3, A2, A 1, A1) = 29700, NE8(A3, A 1, A1) = 76950, NE8(A1 ∗ A2, A2, A2, A1) = 35700, NE8(A1∗A2, A2, A 1, A1) = 84825, NE8(A1∗A2, A 1, A1) = 171450, NE8(A 1, A2, A2, A1) = 25125, NE8(A 1, A2, A 1, A1) = 55125, NE8(A 1, A1) = 94500, NE8(A3, A2, A1, A1, A1) = 91125, NE8(A3, A 1, A1, A1, A1) = 243000, NE8(A1 ∗ A2, A2, A1, A1, A1) = 276750,NE8(A1∗A2, A 1, A1, A1, A1) = 597375, NE8(A 1, A2, A1, A1, A1) = 185625, NE8(A 1, A1, A1, A1) = 354375, NE8(A3, A1, A1, A1, A1, A1) = 759375, NE8(A1 ∗ A2, A1, A1, A1, A1, A1) = 2025000, NE8(A 1, A1, A1, A1, A1, A1) = 1265625, NE8(A2, A2, A2, A2) = 9350, NE8(A2, A2, A2, A 1) = 24975, NE8(A2, A2, A 1) = 64350, NE8(A2, A A21, A 1) = 143100, NE8(A 1) = 261225, NE8(A2, A2, A2, A1, A1) = 78000, NE8(A2, A2, A 1, A1, A1) = 203625, NE8(A2, A 1, A1, A1) = 479250, NE8(A A1, A1) = 951750, NE8(A2, A2, A1, A1, A1, A1) = 641250, NE8(A2, A 1, A1, A1, A1, A1) = 1569375, NE8(A 1, A1, A1, A1, A1) = 3341250, NE8(A2, A1, A1, A1, A1, A1, A1) = 5062500, NE8(A 1, A1, A1, A1, A1, A1, A1) = 11390625, NE8(A1, A1, A1, A1, A1, A1, A1, A1) = 37968750, plus the assignments implied by (2.2) and (2.3), all other numbers NE8(T1, T2, . . . , Td) being zero. 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Stembridge, coxeter, Maple package for working with root systems and finite Coxeter groups; available at http://www.math.lsa.umich.edu/~jrs. [36] E. Tzanaki, Combinatorics of generalized cluster complexes and hyperplane arrangements, Ph.D. thesis, University of Crete, Iraklio, 2007. [37] E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, J. Combin. Theory Ser. A 113 (2006), 1189–1198. [38] E. Tzanaki, Faces of generalized cluster complexes and noncrossing partitions, SIAM J. Discrete Math. 22 (2008), 15–30. Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria. WWW: http://www.mat.univie.ac.at/~kratt. School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom. WWW: http://www.maths.qmw.ac.uk/~twm/. http://www.math.lsa.umich.edu/~jrs http://www.mat.univie.ac.at/~kratt http://www.maths.qmw.ac.uk/~twm/ 1. Introduction 2. Decomposition numbers for finite Coxeter groups 3. Auxiliary results 4. Decomposition numbers for type A 5. Decomposition numbers for type B 6. Decomposition numbers for type D 7. Generalised non-crossing partitions 8. Decomposition numbers with free factors, and enumeration in the poset of generalised non-crossing partitions 9. Proof of the F=M Conjecture for type D 10. A conjecture of Armstrong on maximal intervals containing a random multichain 11. Chain enumeration in the poset of generalised non-crossing partitions for the exceptional types Acknowledgements Appendix A. The decomposition numbers for the exceptional types A.1. The decomposition numbers for type I2(a) [Sec. 13]KratCB A.2. The decomposition numbers for type H3 [Sec. 14]KratCB A.3. The decomposition numbers for type H4 [Sec. 15]KratCB A.4. The decomposition numbers for type F4 [Sec. 16]KratCB A.5. The decomposition numbers for type E6 [Sec. 17]KratCB A.6. The decomposition numbers for type E7 [Sec. 6]KratCF A.7. The decomposition numbers for type E8 [Sec. 7]KratCF References

0704.0200

Electromagnetic polarizabilities and the excited states of the nucleon

Electromagnetic polarizabilities and the excited states of the nucleon Martin Schumacher [email protected] Zweites Physikalisches Institut der Universität Göttingen, Friedrich-Hund-Platz 1 D-37077 Göttingen, Germany Abstract The electromagnetic polarizabilities of the nucleon are shown to be essentially composed of the nonresonant αp(E0+) = +3.2, αn(E0+) = +4.1, the t-channel α p,n = −βtp,n = +7.6 and the resonant βp,n(P33(1232)) = +8.3 contributions (in units of 10 −4fm3). The remaining deviations from the experimental data ∆αp = 1.2 ± 0.6, ∆βp = 1.2 ∓ 0.6, ∆αn = 0.8 ± 1.7 and ∆βn = 2.0 ∓ 1.8 are contributed by a larger number of resonant and nonresonant processes with cancellations between the contributions. This result confirms that dominant contributions to the electric and magnetic polarizabilities may be represented in terms of two- photon coupling to the σ meson having the predicted mass mσ = 666 MeV and two-photon width Γγγ = 2.6 keV. PACS. 11.55.Fv Dispersion relations – 11.55.Hx Sum rules – 13.60.Fz Elastic and Compton scattering – 14.70.Bh Photons 1 Introduction Recently, it has been shown [1] that the t-channel components αt and βt of the electric (α) and magnetic (β) polarizabilities of the nucleon can be understood as a property of the constituent quarks. The constituent quarks couple to π and σ fields and, mediated by these fields, they couple to two photons. The coupling of two photons with perpendicular linear polarization to the π0 meson provides the main contribution, γtπ, to the backward-angle spin-polarizability γπ. Similarly, the coupling of two photons with parallel linear polarization provides the main contribution, (α− β)t, to the difference (α−β) of the electric and the magnetic polarizabilities. The quantitative prediction (α − β)tp,n = 15.2 (in units of 10−4fm3) makes use of the fact that the mass of the particle of the σ field is predicted by the quark-level Nambu–Jona-Lasinio (NJL) model to be mσ = 666 MeV and its two-photon width to be Γγγ = 2.6 keV [1]. The foregoing paragraph describes the résumé of a long and partial controversial history of research. The scalar-isoscalar t-channel was introduced [2] in analogy to the pseudoscalar t-channel [3]. But differing from the π0-pole contribution [3] to the scattering amplitude, the meaning and importance of the scalar-isoscalar t-channel [2] was less well known, mainly because the σ meson was not considered as a normal particle. One important step forward was the formulation of the BEFT sum rule [4], relating the s-channel part of the difference of the electric and magnetic polarizabilities, (α − β)s, to the multipole content of the total photoabsorption cross section using a fixed-θ dispersion relation at θ = π, and by relating the t-channel part (α− β)t to a dispersion relation for t with the imaginary part of the amplitude taken from the reactions γγ → ππ and NN̄ → ππ via a unitarity relation. Furthermore, the scalar-isoscalar phase δ00(t) was taken from the reaction ππ → ππ. One of the first evaluations of the BEFT sum-rule showed that for pointlike uncorrelated pions the large value of (α − β)t = +17.51 is obtained [5] which in other calculations has been reduced by very different factors (see [6] for an overview) when the ππ correlation and the pion internal structure is taken into account. The largest reduction amounting to a factor of 2 has been obtained in the latest of this early series of http://arxiv.org/abs/0704.0200v1 calculations [7]. This unsatisfactory situation has recently been clarified by showing [1, 8] that the arithmetic average of the most recent calculations of Drechsel et al. [9] and Levchuk et al. (see [6]), (α − β)tp,n = 15.3 ± 1.3, leads to a very good agreement with the experimental result and with a parameter-free calculation based on the quark-level NJL model or dynamical linear σ model (LσM) [1,10,11], leading to (α− β)tp,n = 15.2. After the size and the dynamics of the t-channel contribution to the electromagnetic polar- izabilities has been well understood, it appears of interest to get a similar understanding for the s-channel contribution. Especially, the question has to be answered what the individual contri- butions of the resonant excited states of the nucleon to the electric and magnetic polarizabilities are and how the contributions of the “pion cloud” to the electric and magnetic polarizabilities may be specified. To the author’s knowledge such an investigation has not been carried out before. 2 Electromagnetic polarizabilities obtained from the forward- angle sum-rule for (α + β) and the backward-angle sum-rule for (α− β) The appropriate tool for the present investigation is to simultaneously apply the forward-angle sum-rule for (α + β) and the backward-angle sum-rule for (α − β). This leads to the following relations α = αs + αt, (1) [A(ω)σ(ω,E1,M2, · · · ) +B(ω)σ(ω,M1, E2, · · · )] dω , (2) 5αe gπNN 12π2 m2σ fπ = 7.6, (3) β = βs + βt, (4) [A(ω)σ(ω,M1, E2, · · · ) +B(ω)σ(ω,E1,M2, · · · )] dω , (5) βt = − 5αe gπNN 12π2 m2σ fπ = −7.6, (6) ω0 = mπ + , (7) A(ω) = , (8) B(ω) = . (9) In (1) – (9) ω is the photon energy in the lab-system, mπ the pion mass and m the nucleon mass. The quantities αs, βs are the s-channel electric and magnetic polarizabilities, and αt, βt the t-channel electric and magnetic polarizabilities, respectively. The multipole content of the photoabsorption cross section enters through σ(ω,E1,M2, · · · ) = σ(ω,E1) + σ(ω,M2) + · · · , (10) σ(ω,M1, E2, · · · ) = σ(ω,M1) + σ(ω,E2) + · · · , (11) i.e. through the sums of cross sections with change and without change of parity during the electromagnetic transition, respectively1. The multipoles belonging to parity-change are favored for the electric polarizability αs whereas the multipoles belonging to parity-nonchange are fa- vored for the magnetic polarizability βs. The coefficients A(ω) and B(ω) in Eqs. (2), (5), (8) and (9) multiplying the cross sections of the parity-favored and parity-nonfavored multipoles, respectively, are A ∼ +1.07 and B ∼ −0.07 at the pion photoproduction threshold. They in- crease with photon energy, as expected for relativistic correction factors. Using A(ω) and B(ω) it is easy to prove that (α + β) ≡ (α + β)s is given by the Baldin or Baldin-Lapidus (BL) [14] sum rule, whereas (α− β)s is given by the s-channel part of the BEFT [4] sum rule. For the t-channel parts, αt and βt, we use the predictions obtained from the σ-meson pole representation2 with properties as predicted by the quark-level Nambu–Jona-Lasinio model [1, 8]. The quantities entering into this prediction are αe = e 2/4π = 1/137.04, the pion-nucleon coupling constant, gπNN = 13.169±0.057, the pion decay constant, fπ = (92.42±0.26) MeV, and the σ-meson mass, mσ = 666.0 MeV [1, 10, 11]. For convenience we summarize the arguments leading to the relations (3) and (6). The flavor wave-functions of the π0 and the σ meson are given by |π0〉 = (−uū+ dd̄), |σ〉 = (uū+ dd̄). (12) This leads to the decay matrix elements M(σ → γγ) = −5 M(π0 → γγ) = 5 . (13) Using the NJL model or the dynamical LσM with dimensional regularization we arrive at [1,11] mclσ = 4πf clπ√ , (14) where mclσ and f π = 89.8 MeV are the σ meson mass and the π decay constant in the chiral limit (cl) and Nc = 3 the number of colors. Then the mass of the σ meson is given by (mclσ ) 2 +m2π = 666 MeV. (15) Inserting this into (α− β)t = gσNNM(σ → γγ) 2πm2σ and using fσNN = fπNN and (α+ β) t = 0 we arrive at (3) and (6). 1It should be noted that this separation into cross sections for separate multipoles is possible in the presently used fixed-θ dispersion theory applied at θ = 0 and θ = π, whereas in the corresponding formulas based on fixed-t dispersion theory [12] terms containing mixed products of CGLN [13] amplitudes occur (for a discussion see [6]). 2This σ-meson pole in the complex t-plane of the Compton scattering amplitude A(s, t) is not the same, but has relations with the σ-meson pole introduced to parameterize the ππ scattering amplitude. These relations have been discussed in detail in [1,6]. 3 Components of electromagnetic polarizabilities from analyses of total photoabsorption and meson photoproduction data In the following we use different photoabsorption data to get information on partial contributions to αs and βs. Analyses of total photoabsorption cross sections have been carried out in [15]. These analyses give a very good overview over the resonant and nonresonant contributions to the electromagnetic polarizabilities. Further information is taken from the PDG2006 [16], the GWSES [17] and the Mainz [18,19] analyses of meson photoproduction data. 3.1 Components of electromagnetic polarizabilities from analysis of the total photoabsorption cross-section of the proton In the following we wish to study the contributions of nucleon resonances and nonresonant excited states to the s-channel electromagnetic polarizabilities. Only the resonances P33(1232), P11(1440), D13(1520), S11(1535) and F15(1680) have to be taken into account. The contributions of the resonances S11(1650), D15(1675) and higher lying resonances are negligible. For this analysis we use the Walker [15,20] parameterization of nucleon resonances I = Ir W 2r ΓΓγ (W 2 −W 2r )2 +W 2r Γ2 , (17) Γ = Γr )2l+1( q2r +X q2 +X2 , (18) Γγ = Γr k2r +X k2 +X2 . (19) s = 2ωm+m2, ω = photon energy in the lab system, (20) W 2 = s (21) k = |k| = , |k| = photon momentum in the c.m. system, (22) q = |q| = E2π −m2π; Eπ = s−m2 +m2π , |q| = π momentum in the c.m. system, (23) jγ , multipole angular momentum of the photon, (24) l, single π angular momentum. (25) The damping constants X are X = 160 MeV for the P33(1232) resonance and X = 350 MeV else. For the proton, parameters are given in [15] for the relevant resonant states, leading to the results given in lines 3 – 5 of Table 1. The sum αp+βp of nonresonant contributions in line 7 of Table 1 is in agreement with the corresponding number calculated from the nonresonant cross section given in [15] if the nonresonant cross section data are extrapolated to about 3.5 GeV. This shows that with the predicted t-channel contributions given in line 6 there is consistency between the experimental electromagnetic polarizabilities and the predictions. 3.2 Components of electromagnetic polarizabilities from analyses of meson photoproduction for the proton and the neutron From isospin considerations it has been derived [21] that the amplitudes for meson photopro- duction are composed of A(1/2) and A(3/2), referring to final states of definite isospin (1 Table 1: Partial contributions to the electromagnetic polarizabilities based on the analysis of the total photoabsorption cross section [15]. The t-channel parts in line 6 are the predictions based on the σ-meson pole representation (see section 2). Line 7 contains the differences between the numbers in line 2 and the sums of numbers given in lines 3–6. The experimental data are normalized to (α+ β)p = 13.9 ± 0.3 (see [6]). 1 αp βp 2 experiment 12.0 ± 0.6 1.9 ∓ 0.6 3 P33(1232) M1, E2 −1.1 +8.3 4 P11(1440) M1 −0.1 +0.3 5 D13(1520) E1,M2 +1.2 −0.3 6 S11(1535) E1 +0.1 −0.0 5 F15(1680) E2,M3 −0.1 +0.4 6 t-channel +7.6 −7.6 7 nonresonant +4.4 +0.8 Furthermore, there is an amplitude A(0) which may be related to “recoil” effects [21]. This latter amplitude makes a contribution to I = 1/2 only. Therefore, the amplitudes (1/2) = A(0) + A(1/2), nA (1/2) = A(0) − A(1/2) (26) may be introduced. Furthermore, with A(+) = (A(1/2) + 2A(3/2)), A(−) = (A(1/2) −A(3/2)), (27) the physical amplitudes may be expressed by the isospin combinations (see e.g. [19, 22]) A(γp → nπ+) = 2(A(−) +A(0)) = (1/2) − 1 A(3/2)), (28) A(γp → pπ0) = A(+) +A(0) =p A(1/2) + A(3/2), (29) A(γn → pπ−) = − 2(A(−) −A(0)) = (1/2) + A(3/2)), (30) A(γn → nπ0) = A(+) −A(0) = −nA(1/2) + A(3/2). (31) The relation for the cross section of 1π photoproduction is given by σ1π = 2π (l + 1)2 (l + 2)(|El+|2 + |M(l+1)−|2) + l(|Ml+|2 + |E(l+1)−|2) , (32) ∆σ1π = 2π (l + 1)2(−1)l (l + 2)(|El+|2 − |M(l+1)−|2) + l(|Ml+|2 − |E(l+1)−|2) ,(33) ∆σ1π = σ1π(E1,M2, · · · )− σ1π(M1, E2, · · · ). (34) The peak cross section Ir introduced in (17) is given by Ir = 2π 2J + 1 2J0 + 1 , (35) where J and J0 are the spins of the excited state and the ground state, respectively, Γγ the photon width and Γ the total width of the resonance. The photon width Γγ may be expressed through the resonance couplings A1/2 and A3/2 by the relation [16] (2J + 1)MR |A1/2|2 + |A3/2|2 , (36) where MN and MR are the nucleon and resonant masses. Combining (35) and (36) we arrive |A1/2|2 + |A3/2|2 . (37) Using (37) the quantity Ir can be calculated from the resonance couplings A1/2 and A3/2 given by the PDG [16], by GWSES [17] and Mainz [19]. The results obtained for the electromagnetic polarizabilities obtained from the data given in [19] are given in lines 3 – 7 of Table 2. Table 2: Partial contributions to the electromagnetic polarizabilities. The resonant contributions in lines 3–7 are obtained from the analysis of Drechsel et al. [19]. The t-channel parts in line 8 are the predictions based on the σ-meson pole representation (see section 2). The predicted contribution due to the E0+ amplitude in line 9 is based on the analyses given in [17–19]. Line 10 contains the differences between the numbers in line 2 and the sums of numbers given in lines 3–9. The experimental data are normalized to (α+ β)p = 13.9 ± 0.3 and (α + β)n = 15.2 ± 0.5 (see [6]). 1 αp βp αn βn 2 experiment 12.0 ± 0.6 1.9∓ 0.6 12.5 ± 1.7 2.7 ∓ 1.8 3 P33(1232) M (3/2) 1+ , E (3/2) 1+ −1.1 +8.3 −1.1 +8.3 4 P11(1440) p,nM (1/2) 1− −0.0 +0.2 −0.0 +0.1 5 D13(1520) p,nE (1/2) 2− , p,nM (1/2) 2− +0.6 −0.2 +0.5 −0.1 6 S11(1535) p,nE (1/2) 0+ +0.1 −0.0 +0.1 −0.0 7 F15(1680) p,nE (1/2) 3− , p,nM (1/2) 3− −0.1 +0.3 −0.0 +0.0 8 t-channel +7.6 −7.6 +7.6 −7.6 9 E0+ (empirical) +3.2 −0.3 +4.1 −0.4 10 background +1.7 +1.2 +1.3 +2.4 The main contributions to the nonresonant parts of the electromagnetic polarizabilities are expected from the E0+ amplitude which has to be taken from analyses of meson photoproduction data. Multipole analyses of pion photoproduction based on fixed-t dispersion relations and unitarity are given by Hanstein et al. [18] in a convenient form. Cross sections separated into resonant and nonresonant parts are provided for the reactions γp → π+n and γn → π−p up to energies of 500 MeV and extrapolations of the nonresonant parts are straightforward using the data contained in [19] and [17]. In principle there is a problem in disentangling resonant and nonresonant contributions because of interference effects. The interference of the amplitudes (1/2) 0+ with the S11(1535) and S11(1650) resonances, however, does not lead to problems in determining the nonresonant E0+ contributions because of the smallness of the resonant parts. The results for the electromagnetic polarizabilities obtained from these empirical E0+ data are contained in line 9 of Table 2. 3It should be noted that the quantity Ir of (37) contains the branching correction Γ/Γπ as required. Up to this point the electromagnetic polarizabilities find an explanation in the numbers given in lines 3 – 9 of Table 2, with the exception of the small contributions given in line 10 which deserve a further investigation. These non-E0+ parts of the nonresonant contributions are partly due to the M (3/2) 1− , p,nM (1/2) 1+ and p,nE (1/2) 1+ amplitudes which interfere with the corresponding resonant amplitudes p,nM (1/2) 1− (P11(1440)), M (3/2) 1+ (P33(1232)) and E (3/2) 1+ (P33(1232)), respec- tively [19]. Only the nonresonant parts of the M1− and M1+ amplitudes are expected to be to some extent important in comparison with dominant E0+ amplitude. Therefore we restrict the present discussion to the M1− and M1+ amplitudes. Using the data given in [19] we arrive at the estimates αnonres.p (M1−) = −0.0, βnonres.p (M1−) = +0.2, αnonres.n (M1−) = −0.1, βnonres.p (M1−) = +0.4, αnonres.p (M1+) = −0.0, βnonres.p (M1+) = +0.3, αnonres.n (M1+) = −0.1, βnonres.p (M1+) = +0.6. The conclusion we have to draw from this is that it is not possible to relate the numbers given in line 10 of Table 2 to known photoproduction processes, unless the two-pion channels are taken into account (see e.g. [23]). The ππN final states can be characterized either as quasi two- body states such as π∆ and ρN , or as a ππN component in which both pions are in S waves. Furthermore, in the Regge regime above ≈ 2000 MeV also f2(1270), a2(1320) and Pomeron t-channel exchanges play a role. The π∆ contribution has been analyzed in terms of a ∆ Kroll- Ruderman term and a ∆ pion-pole term [24]. Using data from this analysis [24] we arrive at (αp,n + βp,n) ≈ 1.0 for this partial ππ channel. The nonresonant cross section above ≈ 2000 MeV makes a contribution of about (αp,n + βp,n) ≈ 0.7. 4 Discussion 4.1 Discussion of the s-channel contribution For a long time there have been attempts to understand the electromagnetic polarizabilities predominantly in terms of properties of the “pion cloud” of the nucleon. Among these attempts CHPT in its original relativistic form [25] is among the most prominent ones. It has been shown by L’vov [26] that the results obtained for the electromagnetic polarizabilities through the evaluation of chiral loops [25] can be reproduced via dispersion theory when the Born approximation of the electric-dipole CGLN amplitude E0+ is taken into account. The results obtained in this way are shown in lines 2 and 3 of Table 3. Table 3: Predictions for the “meson cloud” contribution to the electromagnetic polarizabilities in different approaches. 1 method αp βp αn βn reference 2 CHPT +7.4 −2.0 +10.1 −1.2 Bernard [25] 3 piona) Born +7.3 −1.8 +9.8 −0.9 L’vov [26] 4 E0+ Born +7.5 −1.4 +9.9 −1.8 present a) The use of fixed-t dispersion theory requires the consideration of interference terms of the E0+ amplitude with other amplitudes. It is of interest to use also the present approach based on forward and backward dispersion relations for studies of this type. For this purpose use may be made of the Born approximation (see [22] p. 286, [27] p. 35) given in the form EBorn0+ (γN → π±N) = ± (−)Born 0+ ± E (0)Born , (38) (−)Born 1− v2 1 + v , (39) with v = |q|/ q2 +m2π being the velocity of the pion in the c.m. system. The expression given in (39) corresponds to the static approximation discussed in detail in [22, 27]. Because of the relation σE0+(γn → π−p) σE0+(γp → π+n) ≃ 1.3 (40) (see [22] p. 276) the recoil terms E (0)Born 0+ may be replaced by multiplying E (−)Born 0+ with (1 + )−1/2 and (1+mπ )+1/2 in order to get the results for the proton and neutron, respectively. The relation given in (40) is well justified at threshold but its approximate validity extends to higher energies [18,19,27]. The pseudovector coupling constant f in (39) is given by f = gπNN (mπ/2m) with gπNN = 13.169 ± 0.057. There is a remarkable agreement between the numbers given in Table 3 but these numbers are larger by a factor ∼ 2.4 than the corresponding numbers in line 9 of Table 2. Two reasons for the deviation of the empirical E0+ amplitude from the Born approximation have been discussed in [19]. The first reason is that the pseudovector (PV) coupling is not valid at high photon energies but has to be replaced by some average of the PV and pseudoscalar the (PS) coupling. The second reason are ρ and ω meson t-channel exchanges which are not taken into account in the Born approximation. In Table 2 (see also Table 1) we see that the different resonant contributions to the electric polarizabilities cancel each other, so that the electric polarizabilities are mainly due to the t-channel part αtp,n (∼60%) given in line 8 and a smaller nonresonant part α(E0+) (∼30%) given in line 9. For the magnetic polarizabilities there is an almost complete cancellation of the P11(1440), D13(1520) and F15(1680) contributions, so that the main remaining contributions are due to the P33(1232) resonance, canceled to a large extent by the t-channel contribution β t. The nonresonant background given in line 10 of Table 2 amounts to about 10% of the experimental electric polarizabilities and to about 70% of the experimental magnetic polarizabilities. This means that precise predictions of these contributions are highly desirable, especially for the magnetic polarizabilities. Unfortunately, the non-E0+ parts of the nonresonant photoabsorption cross sections are dominated by two-pion channels where the information on the multipole content is scarce. 4.2 Discussion of the t-channel contribution In [1] it has been shown that there are two independent, but apparently equivalent and comple- mentary options to calculate the scalar-isoscalar t-channel contribution to the electromagnetic polarizabilities of the nucleon. Option 1 makes use of the properties of the σ-meson as predicted by the quark-level NJL model and in this respect is of course model dependent. The quark-level NJL model predicts a definite σ-mesons mass, viz. mσ = 666 MeV, through a parameter-free relation of mσ to the pion decay constant fπ. The result (α − β)t = 15.2 is in an excellent agreement with the experimental result. The agreement between a prediction and an experimental result cannot be used as an argument for the validity of the prediction without further support. This support is provided by dispersion theory applied to the measured properties of the σ meson as showing up in particle reactions with two pions in the intermediate state (Option 2). Option 2 first takes into consideration that the σ meson has been observed in many data analyses [16] as a pole on the second sheet of the isoscalar S wave of ππ scattering. This pole describes part of the resonant structure of the σ meson without being a complete description. This latter property of the pole follows from the fact that the 90◦ crossing of the scalar-isoscalar phase δ00(s) is located at much higher energies than predicted by the structure of the pole. The analyses of Colangelo et al. [28] and Caprini et al. [29] led to s(pole) = (470± 30) − i(295 ± 20) MeV s(δS = 90 ◦) = (844± 13) MeV [28], (41) Mσ = 441 −8 MeV, Γσ = 544 −25 MeV [29]. (42) The numbers contained in (41) and (42) are extremely valuable in characterizing the properties of the σ meson as a real particle but they can only qualitatively be compared with the mass mσ = 666 MeV of the virtual σ meson, because in the latter case there is no open decay channel. This means that there is no contradiction between the existence of the broad mass distribution for the real σ meson and a precisely determined mass of the virtual σ meson. Furthermore, the numbers contained in (41) and (42) are of no direct relevance for the prediction of (α − β)t. First of all it certainly would lead only to a qualitative estimate for (α − β)t if the parameters of the σ-meson pole in (41) and (42) would be used instead of mσ = 666 MeV. Furthermore, such an insufficient attempt is not necessary because the BEFT [4] sum rule provides a precise relation between (α − β)t and the properties of the real σ meson. In the BEFT sum rule the imaginary part of the t-channel Compton scattering amplitude is given by an unitarity relation where the two reaction γγ → σ → ππ and NN̄ → σ → ππ are exploited. In these reactions the resonant structure of the σ meson enters via the experimentally determined scalar-isoscalar phase δ00(s) which is considerably different from the corresponding quantity predicted by the poles shown in (41) and (42). The real part of the t-channel Compton scattering amplitude is obtained via a dispersion relation. The present status of the evaluation of the BEFT sum rule (α−β)tnp = 15.3±1.3 is in good agreement with the experimental result as well as the prediction based on the quark-level NJL model. 5 Conclusion The good agreement of the result based on the BEFT sum rule with the experimental result as well as the prediction based on the quark level NJL model may be understood as a strong argument that the two predictions of (α−β)t are equivalent. This implies that in addition to the poles in (41) and (42) also the mass mσ = 666 MeV of the virtual σ meson is an experimentally verified property of the σ meson. Acknowledgment The author is indebted to Deutsche Forschungsgemeinschaft for the support of this work through the projects SCHU222 and 436RUS113/510. He thanks M.I. Levchuk, A.I. L’vov and A.I. Mil- stein for a long term cooperation which contributed to the motivation for the present investiga- tion. References [1] M. Schumacher, Eur. Phys. J. A 30, 413 (2006); DOI 10.1140/epja/i2006-10103-0 [hep-ph/0609040]. [2] A.C. Hearn, E. Leader, Phys. Rev. 126, 789 (1962); R. Köberle, Phys. Rev. 166, 1558 (1968). [3] E.E. Low, Phys. Rev. 120, 582 (1960) (and reference therein); M. Jacob, J. Mathews, Phys. Rev. 117, 854 (1960). [4] J. Bernabeu, T.E.O. Ericson, C. Ferro Fontan, Phys. Lett. 49 B, 381 (1974); J. Bernabeu, B. Tarrach, Phys. Lett 69 B, 484 (1977). [5] I. Guiasu, E.E. Radescu, Phys. Rev. D 14, 1335 (1976); Phys. Lett. 62 B, 193 (1976). [6] M. Schumacher, Prog. Part. Nucl. Phys. 55, 567 (2005) [hep-ph/0501167]. [7] B.R. Holstein, A.M. Nathan, Phys. Rev. D 49, 6101 (1994). [8] M.I. Levchuk, A.I. L’vov, A.I. Milstein, M. Schumacher, Proceedings of the Workshop NSTAR2005, 12–15 October 2005, Tallahassee, Florida, edited by S. Capstick, V. Crede, P. Eugenio (World Scientific 2006) 389 [hep-ph/0511193]. [9] D. Drechsel et al., Phys. Rep. 378, 99 (2003); Phys. Rev. C 61, 015204 (1999). [10] T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 221 (1994). [11] R. Delbourgo, M. Scadron, Mod. Phys. Lett. A 10, 251 (1995) [hep-ph/9910242]; Int. J. Mod. Phys. A 13, 657 (1998) [hep-ph/9807504]. [12] A.I. L’vov, V.A. Petrun’kin, M. Schumacher, Phys. Rev. C 55, 359 (1997). [13] G.F. Chew, M.L. Goldberger, F.E. Low, Y. Nambu, Phys. Rev. 106, 1345 (1957). [14] A.M. Baldin, Nucl. Phys. 18, 310 (1960); L.I. Lapidus, Zh. Eksp. Teor. Fiz. 43, 1358 (1962) [Sov. Phys. JETP 16, 964 (1963)]. [15] T.A. Armstrong et al., Phys. Rev. D 5, 1640 (1972); Nucl. Phys. B 41, 445 (1972). [16] W.-M. Yao et al., (Particle Data Group) J. Phys. G 33, 1 (2006) [URL: http://pdg.lbl.gov]. [17] R.A. Arndt, et al. Phys. Rev. C 66, 055213 (2002). [18] O. Hanstein, D. Drechsel, L. Tiator, Nucl. Phys. A 632, 561 (1998). [19] D. Drechsel, O. Hanstein, S.S. Kamalov, L. Tiator, Nucl. Phys. A 645, 145 (1999). [20] R.L. Walker, Phys. Rev. 182, 1729 (1969). [21] K.M. Watson, Phys. Rev. 95, 228 (1954). [22] T. Ericson, W. Weise, Pions and Nuclei, International Series of Monographs on Physics 74, Oxford Science Publications (1988). [23] D. Drechsel, L. Tiator, J. Phys. G: Nucl. Part. Phys. 18, 449 (1992). [24] J.A. Gómez Tejedor, E. Oset, Nucl. Phys. A 571, 667 (1994); 600, 413 (1996). http://arxiv.org/abs/hep-ph/0609040 http://arxiv.org/abs/hep-ph/0501167 http://arxiv.org/abs/hep-ph/0511193 http://arxiv.org/abs/hep-ph/9910242 http://arxiv.org/abs/hep-ph/9807504 http://pdg.lbl.gov [25] V. Bernard, N. Kaiser, U.-G. Meissner, Phys. Rev. Lett. 67, 1515 (1991); Nucl. Phys. B 373, 346 (1992). [26] A.I. L’vov, Phys. Lett. B 304, 29 (1993). [27] A. Donnachie, in: High Energy Physics, Edited by E.H.S. Burhop V, 1 Academic Press (1972) [28] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603, 125 (2001). [29] I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. 96, 132001 (2006). Introduction Electromagnetic polarizabilities obtained from the forward-angle sum-rule for (+) and the backward-angle sum-rule for (-) Components of electromagnetic polarizabilities from analyses of total photoabsorption and meson photoproduction data Components of electromagnetic polarizabilities from analysis of the total photoabsorption cross-section of the proton Components of electromagnetic polarizabilities from analyses of meson photoproduction for the proton and the neutron Discussion Discussion of the s-channel contribution Discussion of the t-channel contribution Conclusion

0704.0201

Hecke-Clifford algebras and spin Hecke algebras I: the classical affine type

HECKE-CLIFFORD ALGEBRAS AND SPIN HECKE ALGEBRAS I: THE CLASSICAL AFFINE TYPE TA KHONGSAP AND WEIQIANG WANG Abstract. Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, for- mulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras. 1. Introduction 1.1. The Hecke algebras associated to finite and affine Weyl groups are ubiquitous in diverse areas, including representation theories over finite fields, infinite fields of prime characteristic, p-adic fields, and Kazhdan- Lusztig theory for category O. Lusztig [Lu1, Lu2] introduced the graded Hecke algebras, also known as the degenerate affine Hecke algebras, associ- ated to a finite Weyl groupW , and provided a geometric realization in terms of equivariant homology. The degenerate affine Hecke algebra of type A has also been defined earlier by Drinfeld [Dr] in connections with Yangians, and it has recently played an important role in modular representations of the symmetric group (cf. Kleshchev [Kle]). In [W1], the second author introduced the degenerate spin affine Hecke algebra of type A, and related it to the degenerate affine Hecke-Clifford algebra introduced by Nazarov in his study of the representations of the spin symmetric group [Naz]. A quantum version of the spin affine Hecke algebra of type A has been subsequently constructed in [W2], and was shown to be related to the q-analogue of the affine Hecke-Clifford algebra (of type A) defined by Jones and Nazarov [JN]. 1.2. The goal of this paper is to provide canonical constructions of the degenerate affine Hecke-Clifford algebras and degenerate spin affine Hecke algebras for all classical finite Weyl groups, which goes beyond the type A case, and then establish some basic properties of these algebras. The notion of spin Hecke algebras is arguably more fundamental while the notion of the Hecke-Clifford algebras is crucial for finding the right formulation of the spin Hecke algebras. We also construct the degenerate covering affine Hecke algebras which connect to both the degenerate spin affine Hecke algebras and the degenerate affine Hecke algebras of Lusztig. http://arxiv.org/abs/0704.0201v3 2 TA KHONGSAP AND WEIQIANG WANG 1.3. Let us describe our constructions in some detail. The Schur multiplier for each finite Weyl group W has been computed by Ihara and Yokonuma [IY] (see [Kar]). We start with a distinguished double cover W̃ for any finite Weyl group W : 1 −→ Z2 −→ W̃ −→W −→ 1. (1.1) Denote Z2 = {1, z}. Assume that W is generated by s1, . . . , sn subject to the relations (sisj) mij = 1. The quotient CW− := CW̃/〈z + 1〉 is then generated by t1, . . . , tn subject to the relations (titj) mij = 1 for mij odd, and (titj) mij = −1 for mij even. In the symmetric group case, this double cover goes back to I. Schur [Sch]. Note that W acts as automorphisms on the Clifford algebra CW associated to the reflection representation h of W . We establish a (super)algebra isomorphism Φfin : CW ⋊CW ≃−→ CW ⊗ CW−, extending an isomorphism in the symmetric group case (due to Sergeev [Ser] and Yamaguchi [Yam] independently) to all Weyl groups. That is, the superalgebras CW ⋊ CW and CW − are Morita super-equivalent in the terminology of [W2]. The double cover W̃ also appeared in Morris [Mo]. We formulate the notion of degenerate affine Hecke-Clifford algebras HcW and spin affine Hecke algebras H− , with unequal parameters in type B case, associated to Weyl groups W of type D and B. The algebra HcW (and respectively H− ) contain CW ⋊CW (and respectively CW −) as subalgebras. We establish the PBW basis properties for these algebras: ∼= C[h∗]⊗ CW ⊗CW, H−W ∼= C[h ∗]⊗ CW− where C[h∗] denotes the polynomial algebra and C[h∗] denotes a noncommu- tative skew-polynomial algebra. We describe explicitly the centers for both HcW and H . The two Hecke algebras HcW and H are related by a Morita super-equivalence, i,e. a (super)algebra isomorphism Φ : HcW ≃−→ CW ⊗ H−W which extends the isomorphism Φfin. Such an isomorphism holds also for W of type A [W1]. We generalize the construction in [Naz] of the intertwiners in the affine Hecke-Clifford algebras HcW of type A to all classical Weyl groups W . We also generalize the construction of the intertwiners in [W1] for H− of type A to all classical Weyl groups W . We further establish the basic properties of these intertwiners in both HcW and H . These intertwiners are expected to play a fundamental role in the future development of the representation theory of these algebras, as it is indicated by the work of Lusztig, Cherednik and others in the setup of the usual affine Hecke algebras. We further introduce a notion of degenerate covering affine Hecke algebras H∼W associated to the double cover W̃ of the Weyl group W of classical type. The algebra H∼W contains a central element z of order 2 such that the THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 3 quotient of H∼W by the ideal 〈z+1〉 is identified with H and its quotient by the ideal 〈z−1〉 is identified with Lusztig’s degenerate affine Hecke algebras associated toW . In this sense, our covering affine Hecke algebra is a natural affine generalization of the central extension (1.1). A quantum version of the covering affine Hecke algebra of type A was constructed in [W2]. The results in this paper remain valid over any algebraically closed field of characteristic p 6= 2 (and in addition p 6= 3 for type G2). In fact, most of the constructions can be made valid over the ring Z[1 ] (occasionally we need to adjoint 1.4. This paper and [W1] raise many questions, including a geometric re- alization of the algebras HcW or H in the sense of Lusztig [Lu1, Lu2], the classification of the simple modules (cf. [Lu3]), the development of the rep- resentation theory, an extension to the exceptional Weyl groups, and so on. We remark that the modular representations of HcW in the type A case in- cluding the modular representations of the spin symmetric group have been developed by Brundan and Kleshchev [BK] (also cf. [Kle]). In a sequel [KW] to this paper, we will extend the constructions in this paper to the setup of rational double affine Hecke algebras (see Etingof- Ginzburg [EG]), generalizing and improving a main construction initiated in [W1] for the spin symmetric group. We also hope to quantize these degenerate spin Hecke algebras, reversing the history of developments from quantum to degeneration for the usual Hecke algebras. 1.5. The paper is organized as follows. In Section 2, we describe the distin- guished covering groups of the Weyl groups, and establish the isomorphism theorem in the finite-dimensional case. We introduce in Section 3 the degen- erate affine Hecke-Clifford algebras of type D and B, and in Section 4 the corresponding degenerate spin affine Hecke algebras. We then extend the isomorphism Φfin to an isomorphism relating these affine Hecke algebras, establish the PBW properties, and describe the centers of HcW and H Section 5, we formulate the notion of degenerate covering affine Hecke al- gebras, and establish the connections to the degenerate spin affine Hecke algebras and usual affine Hecke algebras. Acknowledgements. W.W. is partially supported by an NSF grant. 2. Spin Weyl groups and Clifford algebras 2.1. The Weyl groups. Let W be an (irreducible) finite Weyl group with the following presentation: 〈s1, . . . , sn|(sisj)mij = 1, mii = 1, mij = mji ∈ Z≥2, for i 6= j〉 (2.1) For a Weyl group W , the integers mij take values in {1, 2, 3, 4, 6}, and they are specified by the following Coxeter-Dynkin diagrams whose vertices correspond to the generators of W . By convention, we only mark the edge connecting i, j with mij ≥ 4. We have mij = 3 for i 6= j connected by an unmarked edge, and mij = 2 if i, j are not connected by an edge. 4 TA KHONGSAP AND WEIQIANG WANG An ◦ ◦ . . . ◦ ◦ 1 2 n− 1 n Bn(n ≥ 2) ◦ ◦ . . . ◦ ◦ 1 2 n− 1 n Dn(n ≥ 4) ◦ ◦ · · · ◦ ◦ 1 2 n− 3 En=6,7,8 ◦ ◦ ◦ . . . ◦ ◦ 1 3 4 n− 1 n F4 ◦ ◦ ◦ ◦ 1 2 3 4 G2 ◦ ◦ 2.2. A distinguished double covering of Weyl groups. The Schur mul- tipliers for finite Weyl groups W (and actually for all finite Coxeter groups) have been computed by Ihara and Yokonuma [IY] (also cf. [Kar]). The explicit generators and relations for the corresponding covering groups of W can be found in Karpilovsky [Kar, Table 7.1]. We shall be concerned about a distinguished double covering W̃ of W : 1 −→ Z2 −→ W̃ −→W −→ 1. We denote by Z2 = {1, z}, and by t̃i a fixed preimage of the generators si of W for each i. The group W̃ is generated by z, t̃1, . . . , t̃n with relations z2 = 1, (t̃it̃j) mij = 1, if mij = 1, 3 z, if mij = 2, 4, 6. THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 5 The quotient algebra CW− := CW̃/〈z+1〉 of CW̃ by the ideal generated by z+1 will be called the spin Weyl group algebra associated to W . Denote by ti ∈ CW− the image of t̃i. The spin Weyl group algebra CW− has the following uniform presentation: CW− is the algebra generated by ti, 1 ≤ i ≤ n, subject to the relations (titj) mij = (−1)mij+1 ≡ 1, if mij = 1, 3 −1, if mij = 2, 4, 6. (2.2) Note that dimCW− = |W |. The algebra CW− has a natural superalgebra (i.e. Z2-graded) structure by letting each ti be odd. By definition, the quotient by the ideal 〈z − 1〉 of the group algebra CW̃ is isomorphic to CW . Example 2.1. Let W be the Weyl group of type An, Bn, or Dn, which will be assumed in later sections. Then the spin Weyl group algebra CW− is generated by t1, . . . , tn with the labeling as in the Coxeter-Dynkin diagrams and the explicit relations summarized in the following table. Type of W Defining Relations for CW− i = 1, titi+1ti = ti+1titi+1, (titj) 2 = −1 if |i− j| > 1 t1, . . . , tn−1 satisfy the relations for CW n = 1, (titn) 2 = −1 if i 6= n− 1, n, (tn−1tn) 4 = −1 t1, . . . , tn−1 satisfy the relations for CW n = 1, (titn) 2 = −1 if i 6= n− 2, n, tn−2tntn−2 = tntn−2tn 2.3. The Clifford algebra CW . Denote by h the reflection representation of the Weyl groupW (i.e. a Cartan subalgebra of the corresponding complex Lie algebra g). In the case of type An−1, we will always choose to work with the Cartan subalgebra h of gln instead of sln in this paper. Note that h carries a W -invariant nondegenerate bilinear form (−,−), which gives rise to an identification h∗ ∼= h and also a bilinear form on h∗ which will be again denoted by (−,−). We identify h∗ with a suitable sub- space of CN and then describe the simple roots {αi} for g using a standard orthonormal basis {ei} of CN . It follows that (αi, αj) = −2 cos(π/mij). Denote by CW the Clifford algebra associated to (h, (−,−)), which is re- garded as a subalgebra of the Clifford algebra CN associated to (C N , (−,−)). We shall denote by ci the generator in CN corresponding to 2ei and denote by βi the generator of CW corresponding to the simple root αi normalized with β2i = 1. In particular, CN is generated by c1, . . . , cN subject to the relations c2i = 1, cicj = −cjci if i 6= j. (2.3) 6 TA KHONGSAP AND WEIQIANG WANG The explicit generators for CW are listed in the following table. Note that CW is naturally a superalgebra with each βi being odd. Type of W N Generators for CW An−1 n βi = (ci − ci+1), 1 ≤ i ≤ n− 1 Bn n βi = (ci − ci+1), 1 ≤ i ≤ n− 1, βn = cn Dn n βi = (ci − ci+1), 1 ≤ i ≤ n− 1, βn = 1√ (cn−1 + cn) E8 8 β1 = (c1 + c8 − c2 − c3 − c4 − c5 − c6 − c7) (c1 + c2), βi = (ci−1 + ci−2), 3 ≤ i ≤ 8 E7 8 the subset of βi in E8, 1 ≤ i ≤ 7 E6 8 the subset of βi in E8, 1 ≤ i ≤ 6 F4 4 β1 = (c1 − c2), β2 = 1√2(c2 − c3) β3 = c3, β4 = (c4 − c1 − c2 − c3) G2 3 β1 = (c1 − c2), β2 = 1√6(−2c1 + c2 + c3) The action of W on h and h∗ preserves the bilinear form (−,−) and thus W acts as automorphisms of the algebra CW . This gives rise to a semi-direct product CW ⋊CW . Moreover, the algebra CW ⋊CW naturally inherits the superalgebra structure by letting elements inW be even and each βi be odd. 2.4. The basic spin supermodule. The following theorem is due to Mor- ris [Mo] in full generality, and it goes back to I. Schur [Sch] (cf. [Joz]) in the type A, namely the symmetric group case. It can be checked case by case using the explicit formulas of βi in the Table of Section 2.3. Theorem 2.2. Let W be a finite Weyl group. Then, there exists a surjective superalgebra homomorphism CW− Ω−→ CW which sends ti to βi for each i. Remark 2.3. In [Mo], W is viewed as a subgroup of the orthogonal Lie group which preserves (h, (−,−)). The preimage of W in the spin group which covers the orthogonal group provides the double cover W̃ of W , where the Atiyah-Bott-Shapiro construction of the spin group in terms of the Clifford algebra CW was used to describe this double cover of W . The superalgebra CW has a unique (up to isomorphism) simple super- module (i.e. Z2-graded module). By pulling it back via the homomorphism Ω : CW− → CW , we obtain a distinguished CW−-supermodule, called the basic spin supermodule. This is a natural generalization of the classical construction for CS−n due to Schur [Sch] (see [Joz]). 2.5. A superalgebra isomorphism. Given two superalgebras A and B, we view the tensor product of superalgebras A ⊗ B as a superalgebra with multiplication defined by (a⊗ b)(a′ ⊗ b′) = (−1)|b||a′|(aa′ ⊗ bb′) (a, a′ ∈ A, b, b′ ∈ B) (2.4) THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 7 where |b| denotes the Z2-degree of b, etc. Also, we shall use short-hand notation ab for (a⊗ b) ∈ A ⊗ B, a = a⊗ 1, and b = 1⊗ b. We have the following Morita super-equivalence in the sense of [W2] be- tween the superalgebras CW ⋊CW and CW Theorem 2.4. We have an isomorphism of superalgebras: Φ : CW ⋊CW ≃−→ CW ⊗ CW− which extends the identity map on CW and sends si 7→ − −1βiti. The inverse map Ψ is the extension of the identity map on CW which sends ti 7→ −1βisi. We first prepare some lemmas. Lemma 2.5. We have (Φ(si)Φ(sj)) mij = 1. Proof. Theorem 2.2 says that (titj) mij = (βiβj) mij = ±1. Thanks to the identities βjti = −tiβj and Φ(si) = − −1βiti, we have (Φ(si)Φ(sj)) mij = (−βitiβjtj)mij = (βiβjtitj) mij = (βiβj) mij (titj) mij = 1. Lemma 2.6. We have βjΦ(si) = Φ(si) si(βj) for all i, j. Proof. Note that (βi, βi) = 2β i = 2, and hence βjβi = −βiβj + (βj , βi) = −βiβj + 2(βj , βi) (βi, βi) β2i = −βisi(βj). Thus, we have βjΦ(si) = − −1βjβiti −1tiβjβi = −1tiβisi(βj) = Φ(si) si(βj). Proof of Theorem 2.4. The algebra CW ⋊CW is generated by βi and si for all i. Lemmas 2.5 and 2.6 imply that Φ is a (super) algebra homomorphism. Clearly Φ is surjective, and thus an isomorphism by a dimension counting argument. Clearly, Ψ and Φ are inverses of each other. � Remark 2.7. The type A case of Theorem 2.4 was due to Sergeev and Ya- maguchi independently [Ser, Yam], and it played a fundamental role in clarifying the earlier observation in the literature (cf. [Joz, St]) that the representation theories of CS−n and Cn ⋊CSn are essentially the same. In the remainder of the paper, W is always assumed to be one of the classical Weyl groups of type A,B, or D. 8 TA KHONGSAP AND WEIQIANG WANG 3. Degenerate affine Hecke-Clifford algebras In this section, we introduce the degenerate affine Hecke-Clifford algebras of type D and B, and establish some basic properties. The degenerate affine Hecke-Clifford algebra associated to the symmetric group Sn was introduced earlier by Nazarov under the terminology of the affine Sergeev algebra [Naz]. 3.1. The algebra HcW of type An−1. Definition 3.1. [Naz] Let u ∈ C, and W =WAn−1 = Sn be the Weyl group of type An−1. The degenerate affine Hecke-Clifford algebra of type An−1, denoted by HcW or H , is the algebra generated by x1, . . . , xn, c1, . . . , cn, and Sn subject to the relation (2.3) and the following relations: xixj = xjxi (∀i, j) (3.1) xici = −cixi, xicj = cjxi (i 6= j) (3.2) σci = cσiσ (1 ≤ i ≤ n, σ ∈ Sn) (3.3) xi+1si − sixi = u(1− ci+1ci) (3.4) xjsi = sixj (j 6= i, i+ 1) (3.5) Remark 3.2. Alternatively, we may view u as a formal parameter and the algebra HcW as a C(u)-algebra. Similar remarks apply to various algebras introduced in this paper. Our convention c2i = 1 differs from Nazarov’s which sets c2i = −1. The symmetric group Sn acts as the automorphisms on the symmetric algebra C[h∗] ∼= C[x1, . . . , xn] by permutation. We shall denote this action by f 7→ fσ for σ ∈ Sn, f ∈ C[x1, . . . , xn]. Proposition 3.3. Let W = WAn−1. Given f ∈ C[x1, . . . , xn] and 1 ≤ i ≤ n− 1, the following identity holds in HcW : sif = f sisi + u f − f si xi+1 − xi cici+1f − f sicici+1 xi+1 + xi It is understood here and in similar expressions below that A In this sense, both numerators on the right-hand side of the above formula are (left-)divisible by the corresponding denominators. Proof. By the definition of HcW , we have that six j = x j si for any k if j 6= i, i + 1. So it suffices to check the identity for f = xki xli+1. We will proceed by induction. THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 9 First, consider f = xki , i.e. l = 0. For k = 1, this follows from (3.4). Now assume that the statement is true for k. Then xki+1si + u (xki − xki+1) xi+1 − xi (cici+1x i − xki+1cici+1) xi+1 + xi = xki+1 (xi+1si − u(1− ci+1ci)) (xki − xki+1) xi+1 − xi xi + u (cici+1x i − xki+1cici+1) xi+1 + xi = xk+1i+1 si + u (xk+1 − xk+1 i+1 ) xi+1 − xi (cici+1x − xk+1 i+1 cici+1) xi+1 + xi where the last equality is obtained by using (3.2) and (3.4) repeatedly. An induction on l will complete the proof of the proposition for the mono- mial f = xki x i+1. The case l = 0 is established above. Assume the formula is true for f = xki x i+1. Then using sixi+1 = xisi+u(1+ci+1ci), we compute i+1 = i+1si + u (xki x i+1 − xlixki+1) xi+1 − xi (cici+1x i+1 − xlixki+1cici+1) xi+1 + xi · xi+1 = xlix i+1(xisi + u(1 + ci+1ci)) (xki x i+1 − xlix i+1 ) xi+1 − xi (cici+1x i+1 + x i+1 cici+1) xi+1 + xi = xl+1i x i+1si + u (xki x i+1 − x xi+1 − xi (cici+1x i+1 − x xki+1cici+1) xi+1 + xi This completes the proof of the proposition. � The algebra HcW contains C[h ∗],Cn, and CW as subalgebras. We shall denote xα = xa11 · · · xann for α = (a1, . . . , an) ∈ Zn+, cǫ = c 1 · · · cǫnn for ǫ = (ǫ1, . . . , ǫn) ∈ Zn2 . Below we give a new proof of the PBW basis theorem for HcW (which has been established by different methods in [Naz, Kle]), using in effect the induced HcW -module Ind 1 from the trivial W -module 1. This induced module is of independent interest. This approach will then be used for type D and B. Theorem 3.4. LetW =WAn−1 . The multiplication of subalgebras C[h ∗],Cn, and CW induces a vector space isomorphism C[h∗]⊗ Cn ⊗ CW ≃−→ HcW . 10 TA KHONGSAP AND WEIQIANG WANG Equivalently, {xαcǫw|α ∈ Zn+, ǫ ∈ Zn2 , w ∈ W} forms a linear basis for HcW (called a PBW basis). Proof. Note that IND := C[x1, . . . , xn] ⊗ Cn admits an algebra structure by (2.3), (3.1) and (3.2). By the explicit defining relations of HcW , we can verify that the algebra HcW acts on IND by letting xi and ci act by left multiplication, and si ∈ Sn act by si.(fc ǫ) = f sicsiǫ + f − f si xi+1 − xi cici+1f − f sicici+1 xi+1 + xi For α = (a1, . . . , an), we denote |α| = a1 + · · · + an. Define a Lexico- graphic ordering < on the monomials xα, α ∈ Zn+, (or respectively on Zn+), by declaring xα < xα , (or respectively α < α′), if |α| < |α′|, or if |α| = |α′| then there exists an 1 ≤ i ≤ n such that ai < a′i and aj = a′j for each j < i. Note that the algebra HcW is spanned by the elements of the form x αcǫw. It remains to show that these elements are linearly independent. Suppose that S := xαcǫw = 0 for a finite sum over α, ǫ, w and that some coefficient a 6= 0; we fix one such ǫ. Now consider the action S on an element of the form x 2 · · · xNnn for N1 ≫ N2 ≫ · · · ≫ Nn ≫ 0. Let w̃ be such that (x 2 · · · xNnn )w̃ is maximal among all possible w with aαǫw 6= 0 for some α. Let α̃ be the largest element among all α with 6= 0. Then among all monomials in S(xN11 x 2 · · · xNnn ), the monomial xα̃(x 2 · · · xNnn )w̃cǫ appears as a maximal term with coefficient ±aα̃ǫw̃. It follows from S = 0 that aα̃ǫw̃ = 0. This is a contradiction, and hence the elements xαcǫw are linearly independent. � Remark 3.5. By the PBW Theorem 3.4, the HcW -module IND introduced in the above proof can be identified with the HcW -module induced from the trivial CW -module. The same remark applies below to type D and B. 3.2. The algebra HcW of type Dn. Let W = WDn be the Weyl group of type Dn. It is generated by s1, . . . , sn, subject to the following relations: s2i = 1 (i ≤ n− 1) (3.6) sisi+1si = si+1sisi+1 (i ≤ n− 2) (3.7) sisj = sjsi (|i− j| > 1, i, j 6= n) (3.8) sisn = snsi (i 6= n− 2) (3.9) sn−2snsn−2 = snsn−2sn, s n = 1. (3.10) In particular, Sn is generated by s1, . . . , sn−1 subject to the relations (3.6– 3.8) above. Definition 3.6. Let u ∈ C, and letW =WDn . The degenerate affine Hecke- Clifford algebra of type Dn, denoted by H W or H , is the algebra generated by xi, ci, si, 1 ≤ i ≤ n, subject to the relations (3.1–3.5), (3.6–3.10), and the THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 11 following additional relations: sncn = −cn−1sn snci = cisn (i 6= n− 1, n) snxn + xn−1sn = −u(1 + cn−1cn) (3.11) snxi = xisn (i 6= n− 1, n). Proposition 3.7. The algebra HcDn admits anti-involutions τ1, τ2 defined by τ1 : si 7→ si, cj 7→ cj , xj 7→ xj , (1 ≤ i ≤ n); τ2 : si 7→ si, cj 7→ −cj, xj 7→ xj , (1 ≤ i ≤ n). Also, the algebra HcDn admits an involution σ which fixes all generators si, xi, ci except the following 4 generators: σ : sn 7→ sn−1, sn−1 7→ sn, xn 7→ −xn, cn 7→ −cn. Proof. We leave the easy verifications on τ1, τ2 to the reader. It remains to check that σ preserves the defining relations. Almost all the relations are obvious except (3.4) and (3.11). We see that σ preserves (3.4) as follows: for i ≤ n− 2, σ(xi+1si − sixi) = xi+1si − sixi = u(1− ci+1ci) = σ(u(1 − ci+1ci)); σ(xnsn−1 − sn−1xn−1) = −xnsn − snxn−1 = u(1 + cncn−1) = σ(u(1− cncn−1)). Also, σ preserves (3.11) since σ(snxn + xn−1sn) = −sn−1xn + xn−1sn−1 = −u(1− cn−1cn) = σ(−u(1 + cn−1cn)). Hence, σ is an automorphism of HcDn . Clearly σ 2 = 1. � The natural action of Sn on C[h ∗] = C[x1, . . . , xn] is extended to an action of WDn by letting xsnn = −xn−1, x n−1 = −xn, x = xi (i 6= n− 1, n). Proposition 3.8. Let W = WDn , 1 ≤ i ≤ n − 1, and f ∈ C[x1, . . . , xn]. Then the following identities hold in HcW : (1) sif = f sisi + u f − f si xi+1 − xi cici+1f − f sicici+1 xi+1 + xi (2) snf = f snsn − u f − f sn xn + xn−1 cn−1cnf − f sncn−1cn xn − xn−1 Proof. Formula (1) has been established by induction as in type An−1. For- mula (2) can be verified by a similar induction. � 12 TA KHONGSAP AND WEIQIANG WANG 3.3. The algebra HcW of type Bn. Let W = WBn be the Weyl group of type Bn, which is generated by s1, . . . , sn, subject to the defining relation for Sn on s1, . . . , sn−1 and the following additional relations: sisn = snsi (1 ≤ i ≤ n− 2) (3.12) (sn−1sn) 4 = 1, s2n = 1. (3.13) We note that the simple reflections s1, . . . , sn belongs to two different conjugacy classes in WBn , with s1, . . . , sn−1 in one and sn in the other. Definition 3.9. Let u, v ∈ C, and let W = WBn . The degenerate affine Hecke-Clifford algebra of type Bn, denoted by H W or H , is the algebra generated by xi, ci, si, 1 ≤ i ≤ n, subject to the relations (3.1–3.5), (3.6–3.8), (3.12–3.13), and the following additional relations: sncn = −cnsn snci = cisn (i 6= n) snxn + xnsn = − snxi = xisn (i 6= n). The factor 2 above is inserted for the convenience later in relation to the spin affine Hecke algebras. When it is necessary to indicate u, v, we will write HcW (u, v) for H W . For any a ∈ C\{0}, we have an isomorphism of superalgebras ψ : HcW (au, av) → HcW (u, v) given by dilations xi 7→ axi for 1 ≤ i ≤ n, while fixing each si, ci. The action of Sn on C[h ∗] = C[x1, . . . , xn] can be extended to an action of WBn by letting xsnn = −xn, x i = xi, (i 6= n). Proposition 3.10. Let W = WBn. Given f ∈ C[x1, . . . , xn] and 1 ≤ i ≤ n− 1, the following identities hold in HcW : (1) sif = f sisi + u f − f si xi+1 − xi cici+1f − f sicici+1 xi+1 + xi (2) snf = f snsn − f − f sn Proof. The proof is similar to type A and D, and will be omitted. � 3.4. PBW basis for HcW . Note that H W contains C[h ∗],Cn,CW as subal- gebras. We have the following PBW basis theorem for HcW . Theorem 3.11. Let W =WDn or W = WBn. The multiplication of subal- gebras C[h∗],Cn, and CW induces a vector space isomorphism C[h∗]⊗ Cn ⊗ CW −→ HcW . Equivalently, the elements {xαcǫw|α ∈ Zn+, ǫ ∈ Zn2 , w ∈ W} form a linear basis for HcW (called a PBW basis). THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 13 Proof. For W = WDn , we can verify by a direct lengthy computation that the HcAn−1-action on IND = C[x1, . . . , xn]⊗Cn (see the proof of Theorem 3.4) naturally extends to an action of HcDn , where (compare Proposition 3.8) sn acts by sn.(fc ǫ) = f sncsnǫ − f − f sn xn + xn−1 − ucn−1cnf − f sncn−1cn xn − xn−1 Similarly, for W = WBn , the H -action on IND extends to an action of HcBn , where (compare Proposition 3.10) sn acts by sn.(fc ǫ) = f sncsnǫ − f − f sn It is easy to show that, for either W , the elements xαcǫw span HcW . It remains to show that they are linearly independent. We shall treat theWBn case in detail and skip the analogous WDn case. To that end, we shall refer to the argument in the proof of Theorem 3.4 with suitable modification as follows. The w̃ = ((η1, . . . , ηn), σ) ∈ WBn = {±1}n ⋊ Sn may now not be unique, but the σ and the α̃ are uniquely determined. Then, by the same argument on the vanishing of a maximal term, we obtain that w̃ aα̃ǫw̃x 2 · · · xNnn )w̃ = 0, and hence, (η1,...,ηn) aα̃ǫw̃(−1) i=1 ηiNi = 0. By choosing N1, . . . , Nn with different parities (2 n choices) and solving the 2n linear equations, we see that all aα̃ǫw̃ = 0. This can also be seen more explicitly by induction on n. By choosing Nn to be even and odd, we deduce that for a fixed ηn, (η1,...,ηn−1)∈{±1}n−1 aα̃ǫw̃(−1) i=1 ηiNi = 0, which is the equation for (n− 1) xi’s and the induction applies. � 3.5. The even center for HcW . The even center of a superalgebra A, de- noted by Z(A), is the subalgebra of even central elements of A. Proposition 3.12. Let W = WDn or W = WBn . The even center Z(H of HcW is isomorphic to C[x 1, . . . , x Proof. We first show that every W -invariant polynomial f in x21, . . . , x central in HcW . Indeed, f commutes with each ci by (3.2) and clearly f commutes with each xi. By Proposition 3.8 for type Dn or Proposition 3.10 for type Bn, sif = fsi for each i. Since H W is generated by ci, xi and si for all i, f is central in HcW and C[x 1, . . . , x W ⊆ Z(HcW ). On the other hand, take an even central element C = α,ǫ,w xαcǫw in HcW . We claim that w = 1 whenever aα,ǫ,w 6= 0. Otherwise, let 1 6= w0 ∈ W be maximal with respect to the Bruhat ordering in W such that a α,ǫ,w0 6= 0. Then x i 6= xi for some i. By Proposition 3.8 for typeDn or Proposition 3.10 for type Bn, x iC − Cx2i is equal to α,ǫ aα,ǫ,w0x α(x2i − (x 2)cǫw0 plus a 14 TA KHONGSAP AND WEIQIANG WANG linear combination of monomials not involving w0, hence nonzero. This contradicts with the fact that C is central. So we can write C = xαcǫ. Since xiC = Cxi for each i, then (3.2) forces C to be in C[x1, . . . , xn]. Now by (3.2) and ciC = Cci for each i we have that C ∈ C[x21, . . . , x2n]. Since siC = Csi for each i, we then deduce from Proposition 3.8 for type Dn or Proposition 3.10 for type Bn that C ∈ C[x21, . . . , x2n]W . This completes the proof of the proposition. � 3.6. The intertwiners in HcW . In this subsection, we will define the inter- twiners in the degenerate affine Hecke-Clifford algebras HcW . The following intertwiners φi ∈ HcW (with u = 1) for W = WAn−1 were introduced by Nazarov [Naz] (also cf. [Kle]), where 1 ≤ i ≤ n− 1: φi = (x i+1 − x2i )si − u(xi+1 + xi)− u(xi+1 − xi)cici+1. (3.14) A direct computation using (3.4) provides another equivalent formula for φi: φi = si(x i − x2i+1) + u(xi+1 + xi) + u(xi+1 − xi)cici+1. We define the intertwiners φi ∈ HcW for W = WDn (1 ≤ i ≤ n) by the same formula (3.14) for 1 ≤ i ≤ n− 1 and in addition by letting φn ≡ φDn = (x2n − x2n−1)sn + u(xn − xn−1)− u(xn + xn−1)cn−1cn. (3.15) We define the intertwiners φi ∈ HcW for W = WBn (1 ≤ i ≤ n) by the same formula (3.14) for 1 ≤ i ≤ n− 1 and in addition by letting φn ≡ φBn = 2x2nsn + 2vxn. (3.16) The following generalizes the type An−1 results of Nazarov [Naz]. Theorem 3.13. Let W be either WAn−1 , WDn, or WBn. The intertwiners φi (with 1 ≤ i ≤ n− 1 for type An−1 and 1 ≤ i ≤ n for the other two types) satisfy the following properties: (1) φ2i = 2u 2(x2i+1 + x i )− (x2i+1 − x2i )2 (1 ≤ i ≤ n− 1,∀W ); (2) φ2n = 2u 2(x2n + x n−1)− (x2n − x2n−1)2, for type Dn; (3) φ2n = 4x n − 2v2x2n, for type Bn; (4) φif = f siφi (∀f ∈ C[x1, . . . , xn],∀i,∀W ); (5) φicj = c j φi (1 ≤ j ≤ n,∀i,∀W ); (6) φiφjφi · · ·︸ ︷︷ ︸ = φjφiφj · · ·︸ ︷︷ ︸ Proof. Part (1) follows by a straightforward computation and can also be found in [Naz] (with u = 1). Part (2) follows from (1) by applying the involution σ defined in Proposition 3.7. Part (3) and (5) follow by a direct verification. Part (4) for WAn−1 follows from clearing the denominators in the formula in Proposition 3.3 and then rewriting in terms of φi as defined in (3.14). Similarly, (4) for WDn and WBn follows by rewriting the formulas given in Proposition 3.8 in type D and Proposition 3.10 in type B, respectively. THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 15 It remains to prove (6) which is less trivial. Recall that mij︷ ︸︸ ︷ sisjsi · · · = mij︷ ︸︸ ︷ sjsisj · · ·, (denoting this element by w). Let IND be the subalgebra of HcW generated by C[x1, . . . , xn] and Cn. Denote by ≤ the Bruhat ordering on W . Then we can write φiφjφi · · · = fw + pu,wu for some f ∈ C[x1, . . . , xn], and pu,w ∈ IND. We may rewrite φiφjφi · · · = fw + r′u,wφu where φu := φaφb · · · for any subword u = sasb · · · of w = sisjsi · · · , and r′u,w is in some suitable localization of IND with the central element∏ 1≤k≤n x 1≤i<j≤n(x i − x2j) ∈ IND being invertible. Note that such a localization is a free module over the corresponding localized ring of C[x1, . . . , xn]. We can then write φjφiφj · · · = fw + r′′u,wφu with the same coefficient of w as for φiφjφi · · ·, according to Lemma 3.14. The difference ∆ := (φiφjφi · · · − φjφiφj · · ·) is of the form ru,wφu for some ru,w. Observe by (4) that ∆p = p w∆ for any p ∈ C[x1, . . . , xn]. Then we have pwru,wφu = p w∆ = ∆p = ru,wφup = ru,wp In other words, (pw − pu)ru,w = 0 for all p ∈ C[x1, . . . , xn] for each given u < w. This implies that ru,w = 0 for each u, and ∆ = 0. This completes the proof of (6) modulo Lemma 3.14 below. � Lemma 3.14. The following identity holds: i · · ·︸ ︷︷ ︸ = φ0jφ j · · ·︸ ︷︷ ︸ where φ0i denotes the specialization φi|u=0 of φ at u = 0 (or rather φBn |v=0 when i = n in the type Bn case.) 16 TA KHONGSAP AND WEIQIANG WANG Proof. Let W =WBn . For 1 ≤ i ≤ n− 1, mi,i+1 = 3. So we have i = (x i+1 − x2i )si(x2i+2 − x2i+1)si+1(x2i+1 − x2i )si = (x2i+1 − x2i )(x2i+2 − x2i )(x2i+2 − x2i+1)sisi+1si = (x2i+2 − x2i+1)(x2i+2 − x2i )(x2i+1 − x2i )si+1sisi+1 = (x2i+2 − x2i+1)si+1(x2i+1 − x2i )si(x2i+2 − x2i+1)si+1 = φ0i+1φ Note that mij = 2 for j 6= i, i+ 1; clearly, in this case, φ0iφ0j = φ0jφ0i . Noting that mn−1,n = 4, we have φ0n−1φ n = 4(x n − x2n−1)sn−1x2nsn(x2n − x2n−1)sn−1x2nsn = 4(x2n − x2n−1)x2n−1(x2n−1 − x2n)x2nsn−1snsn−1sn = 4x2n(x n − x2n−1)x2n−1(x2n−1 − x2n)snsn−1snsn−1 = 4x2nsn(x n − x2n−1)sn−1x2nsn(x2n − x2n−1)sn−1 = φ0nφ This completes the proof for type Bn. The similar proofs for types An−1 and Dn are skipped. � Theorem 3.13 implies that for every w ∈ W we have a well-defined el- ement φw ∈ HcW given by φw = φi1 · · ·φim where w = si1 · · · sim is any reduced expression for w. These elements φw should play an important role for the representation theory of the algebras HcW . It will be very interest- ing to classify the simple modules of HcW and to find a possible geometric realization. This was carried out by Lusztig [Lu1, Lu2, Lu3] for the usual degenerate affine Hecke algebra case. 4. Degenerate spin affine Hecke algebras In this section we will introduce the degenerate spin affine Hecke algebra when W is the Weyl group of types Dn or Bn, and then establish the connections with the corresponding degenerate affine Hecke-Clifford algebras HcW . See [W1] for the type A case. 4.1. The skew-polynomial algebra. We shall denote by C[b1, . . . , bn] the C-algebra generated by b1, . . . , bn subject to the relations bibj + bjbi = 0 (i 6= j). This is naturally a superalgebra by letting each bi be odd. We will refer to this as the skew-polynomial algebra in n variables. This algebra has a linear basis given by bα := bk11 · · · bknn for α = (k1, . . . , kn) ∈ Zn+, and it contains a polynomial subalgebra C[b21, . . . , b THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 17 4.2. The algebra H− of type Dn. Recall that the spin Weyl group CW associated to a Weyl group W is generated by t1, . . . , tn subject to the rela- tions as specified in Example 2.1. Definition 4.1. Let u ∈ C and let W = WDn . The degenerate spin affine Hecke algebra of type Dn, denoted by H or H− , is the algebra generated by C[b1, . . . , bn] and CW − subject to the following relations: tibi + bi+1ti = u (1 ≤ i ≤ n− 1) tibj = −bjti (j 6= i, i+ 1, 1 ≤ i ≤ n− 1) tnbn + bn−1tn = u tnbi = −bitn (i 6= n− 1, n). The algebra H− is naturally a superalgebra by letting each ti and bi be odd generators. It contains the type An−1 degenerate spin affine Hecke algebra H− (generated by b1, . . . , bn, t1, . . . , tn−1) as a subalgebra. Proposition 4.2. The algebra H− admits anti-involutions τ1, τ2 defined by τ1 : ti 7→ −ti, bi 7→ −bi (1 ≤ i ≤ n); τ2 : ti 7→ ti, bi 7→ bi (1 ≤ i ≤ n). Also, the algebra H− admits an involution σ which swaps tn−1 and tn while fixing all the remaining generators ti, bi. Proof. Note that we use the same symbols τ1, τ2, σ to denote the (anti-) involutions for H− and HcDn in Proposition 3.7, as those on H are the restrictions from those on HcDn via the isomorphism in Theorem 4.4 below. The proposition is thus established via the isomorphism in Theorem 4.4, or follows by a direct computation as in the proof of Proposition 3.7. � 4.3. The algebra H− of type Bn. Definition 4.3. Let u, v ∈ C, and W = WBn . The degenerate spin affine Hecke algebra of type Bn, denoted by H or H− , is the algebra generated by C[b1, . . . , bn] and CW − subject to the following relations: tibi + bi+1ti = u (1 ≤ i ≤ n− 1) tibj = −bjti (j 6= i, i+ 1, 1 ≤ i ≤ n− 1) tnbn + bntn = v tnbi = −bitn (i 6= n). Sometimes, we will write H− (u, v) or H− (u, v) for H− or H− to indicate the dependence on the parameters u, v. 18 TA KHONGSAP AND WEIQIANG WANG 4.4. A superalgebra isomorphism. Theorem 4.4. Let W =WDn or W =WBn . Then, (1) there exists an isomorphism of superalgebras Φ : HcW−→Cn ⊗ H−W which extends the isomorphism Φ : Cn ⋊ CW −→ Cn ⊗ CW− (in Theorem 2.4) and sends xi 7−→ −2cibi for each i; (2) the inverse Ψ : Cn⊗H−W−→HcW extends Ψ : Cn⊗CW− −→ Cn⋊CW (in Theorem 2.4) and sends bi 7−→ cixi for each i. Theorem 4.4 also holds for WAn−1 (see [W1]). Proof. We only need to show that Φ preserves the defining relations in HcW which involve xi’s. Let W = WDn . Here, we will verify two such relations below. The verification of the remaining relations is simpler and will be skipped. For 1 ≤ i ≤ n− 1, we have Φ(xi+1si − sixi) = ci+1bi+1(ci − ci+1)ti − (ci − ci+1)ticibi = (1− ci+1ci)bi+1ti + (1− ci+1ci)tibi = u(1− ci+1ci), Φ(snxn + xn−1sn) = (cn−1 + cn)tncnbn + cn−1bn−1(cn−1 + cn)tn = −(1 + cn−1cn)tnbn − (1 + cn−1cn)bn−1tn = −u(1 + cn−1cn). Now let W = WBn . For 1 ≤ i ≤ n − 1, as in the proof in type Dn, we have Φ(xi+1si − sixi) = u(1− ci+1ci). Moreover, we have Φ(snxn + xnsn) = cntncnbn + cnbncntn 2cntncnbn + 2cnbncntn 2(tnbn + bncn) = − Φ(snxj) = cntncjbj = 2cntncjbj 2cjcntnbj = 2cjbjcntn = Φ(xjsn), for j 6= n. Thus Φ is a homomorphism of (super)algebras. Similarly, we check that Ψ is a superalgebra homomorphism. Observe that Φ and Ψ are inverses on generators and hence they are indeed (inverse) isomorphisms. � THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 19 4.5. PBW basis for H− . Note that H− contains the skew-polynomial algebra C[b1, . . . , bn] and the spin Weyl group algebra CW − as subalgebras. We have the following PBW basis theorem for H− Theorem 4.5. Let W = WDn or W = WBn. The multiplication of the subalgebras CW− and C[b1, . . . , bn] induces a vector space isomorphism C[b1, . . . , bn]⊗ CW− ≃−→ H− Theorem 4.5 also holds for WAn−1 (see [W1]). Proof. It follows from the definition that H− is spanned by the elements of the form bασ where σ runs over a basis for CW− and α ∈ Zn+. By Theo- rem 4.4, we have an isomorphism ψ : Cn⊗H−W−→HcW . Observe that the im- age ψ(bασ) are linearly independent in HcW by the PBW basis Theorem 3.11 for HcW . Hence the elements b ασ are linearly independent in H− 4.6. The even center for H− Proposition 4.6. Let W =WDn or W = WBn . The even center of H isomorphic to C[b21, . . . , b Proof. By the isomorphism Φ : HcW → Cn⊗H (see Theorems 4.4) and the description of the center Z(HcW ) (see Proposition 3.12), we have Z(Cn ⊗ H−W ) = Φ(Z(H W )) = Φ(C[x 1, . . . , x W ) = C[b21, . . . , b Thus, C[b21, . . . , b W ⊆ Z(H− Now let C ∈ Z(H− ). Since C is even, C commutes with Cn and thus com- mutes with the algebra Cn ⊗ H−W . Then Ψ(C) ∈ Z(H W ) = C[x 1, . . . , x and thus, C = ΦΨ(C) ∈ Φ(C[x21, . . . , x2n]W ) = C[b21, . . . , b2n]W . � In light of the isomorphism Theorem 4.4, the problem of classifying the simple modules of the spin affine Hecke algebra H− is equivalent to the classification problem for the affine Hecke-Clifford algebra HcW . It remains to be seen whether it is more convenient to find the geometric realization of instead of HcW . 4.7. The intertwiners in H− . The intertwiners Ii ∈ H−W (1 ≤ i ≤ n− 1) for W =WAn−1 were introduced in [W1] (with u = 1): Ii = (b i+1 − b2i )ti − u(bi+1 − bi). (4.1) The commutation relations in Definition 4.1 gives us another equivalent expression for Ii: Ii = ti(b i − b2i+1) + u(bi+1 − bi). We define the intertwiners Ii ∈ H−W for W = WDn (1 ≤ i ≤ n) by the same formula (4.1) for 1 ≤ i ≤ n− 1 and in addition by letting In ≡ IDn = (b2n − b2n−1)tn − u(bn − bn−1). (4.2) 20 TA KHONGSAP AND WEIQIANG WANG Also, we define the intertwiners Ii ∈ H−W for W = WBn (1 ≤ i ≤ n) by the same formula (4.1) for 1 ≤ i ≤ n− 1 and in addition by letting In ≡ IBn = 2b2ntn − vbn. (4.3) Proposition 4.7. The following identities hold in H− , for W = WAn−1 , WBn, or WDn: (1) Iibi = −bi+1Ii,Iibi+1 = −biIi, and Iibj = −bjIi (j 6= i, i + 1), for 1 ≤ i ≤ n− 1, 1 ≤ j ≤ n, and any W ; In addition, (2) Inbn−1 = −bnIn,Inbn = −bn−1In, and Inbi = −biIn (i 6= n− 1, n), for type Dn; (3) Inbn = −bnIn, and Inbi = −biIn (i 6= n), for type Bn. Proof. (1) We first prove the case when j = i: Iibi = (b i+1 − b2i )tibi − u(bi+1 − bi)bi = (b2i+1 − b2i )(−bi+1ti + u)− u(bi+1bi − b2i ) = −bi+1 (b2i+1 − b2i )ti − u(bi+1 − bi) = −bi+1Ii. The proof for Iibi+1 = −biIi is similar and thus skipped. For j 6= i, i+ 1, we have tibj = −bjti, and hence Iibj = −bjIi. (2) We prove only the first identity. The proofs of the remaining two identities are similar and will be skipped. Inbn−1 = (b n − b2n−1)tnbn−1 − u(bn − bn−1)bn−1 = (b2n − b2n−1)(−bntn + u)− u(bnbn−1 − b2n−1) = −bn (b2n − b2n−1)tn − u(bn − bn−1) = −bnIn. The proof of (3) is analogous to (2), and is thus skipped. � Recall the superalgebra isomorphism Φ : HcW−→Cn ⊗H defined in Sec- tion 4 and the elements βi ∈ Cn defined in Section 2. Theorem 4.8. Let W be either WAn−1 , WDn, or WBn . The isomorphism Φ : HcW −→ Cn ⊗H sends φi 7→ −2 −1βiIi for each i. More explicitly, Φ sends φi 7−→ − −2(ci − ci+1)⊗ Ii (1 ≤ i ≤ n− 1); φn 7−→ − −2(cn−1 + cn)⊗ In for type Dn; φn 7−→ −2 −1cn ⊗ In for type Bn. THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 21 Proof. Recall that the isomorphism Φ sends si 7→ − −1βiti, xi 7→ −2cibi for each i. So, for 1 ≤ i ≤ n− 1, we have the following Φ(φi) = Φ (x2i+1 − x2i )si − u(xi+1 + xi)− u(xi+1 − xi)cici+1 −2(ci − ci+1)(b2i+1 − b2i )ti − u −2(ci+1bi+1 − cibi) −2(ci+1bi+1 − cibi)cici+1 −2(ci − ci+1) (b2i+1 − b2i )ti − u(bi+1 − bi) −2(ci − ci+1)⊗ Ii. Next for φn ∈ HcDn , we have Φ(φn) = Φ (x2n − x2n−1)sn + u(xn − xn−1)− u(xn + xn−1)cn−1cn −2(cn + cn−1)(b2n − b2n−1)tn + u −2(cnbn − cn−1bn−1) −2(cnbn − cn−1bn−1)cn−1cn −2(cn + cn−1) (b2n − b2n−1)tn − u(bn − bn−1) −2(cn−1 + cn)⊗ In. We skip the computation for φn ∈ HcBn which is very similar but less complicated. � Proposition 4.9. The following identities hold in H− , for W = WAn−1 , WBn, or WDn: (1) I2i = u 2(b2i+1 + b i )− (b2i+1 − b2i )2, for 1 ≤ i ≤ n− 1 and every type of (2) I2n = u 2(b2n + b n−1)− (b2n − b2n−1)2, for type Dn. (3) I2n = 4b n − v2b2n, for type Bn. Proof. It follows from the counterparts in Theorem 3.13 via the explicit correspondences under the isomorphism Φ (see Theorem 4.8). It can of course also be proved by a direct computation. � Proposition 4.10. For W =WAn−1, WBn , or WDn , we have IiIjIi · · ·︸ ︷︷ ︸ = (−1)mij+1 IjIiIj · · ·︸ ︷︷ ︸ Proof. By Theorem 2.2, we have βiβjβi · · ·︸ ︷︷ ︸ = (−1)mij+1 βjβiβj · · ·︸ ︷︷ ︸ Now the statement follows from the above equation and Theorem 3.13 (6) via the correspondence of the intertwiners under the isomorphism Φ (see Theorem 4.8). � Remark 4.11. Proposition 4.7, Theorem 4.8, and Proposition 4.10 for H− can be found in [W1]. 22 TA KHONGSAP AND WEIQIANG WANG 5. Degenerate covering affine Hecke algebras In this section, the degenerate covering affine Hecke algebras associated to the double covers W̃ of classical Weyl groups W are introduced. It has as its natural quotients the usual degenerate affine Hecke algebras HW [Dr, Lu1, Lu2] and the spin degenerate affine Hecke algebras H− introduced by the authors. Recall the distinguished double cover W̃ of a Weyl group W from Sec- tion 2.2. 5.1. The algebra H∼W of type An−1. Definition 5.1. Let W = WAn−1 , and let u ∈ C. The degenerate covering affine Hecke algebra of type An−1, denoted by H W or H , is the algebra generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n−1, subject to the relations for W̃ and the additional relations: zx̃i = x̃iz, z is central of order 2 (5.1) x̃ix̃j = zx̃jx̃i (i 6= j) (5.2) t̃ix̃j = zx̃j t̃i (j 6= i, i + 1) (5.3) t̃ix̃i+1 = zx̃it̃i + u. (5.4) Clearly H∼W contains CW̃ as a subalgebra. 5.2. The algebra H∼W of type Dn. Definition 5.2. Let W = WDn , and let u ∈ C. The degenerate covering affine Hecke algebra of type Dn, denoted by H W or H , is the algebra generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n, subject to the relations (5.1–5.4) and the following additional relations: t̃nx̃i = zx̃it̃n (i 6= n− 1, n) t̃nx̃n = −x̃n−1t̃n + u. 5.3. The algebra H∼W of type Bn. Definition 5.3. Let W = WBn , and let u, v ∈ C. The degenerate covering affine Hecke algebra of type Bn, denoted by H W or H , is the algebra generated by x̃1, . . . , x̃n and z, t̃1, . . . , t̃n, subject to the relations (5.1–5.4) and the following additional relations: t̃nx̃i = zx̃it̃n (i 6= n) t̃nx̃n = −x̃nt̃n + v. 5.4. PBW basis for H∼W . Proposition 5.4. Let W = WAn−1 ,WDn , or WBn . Then the quotient of the covering affine Hecke algebra H∼W by the ideal 〈z − 1〉 (respectively, by the ideal 〈z+1〉) is isomorphic to the usual degenerate affine Hecke algebras HW (respectively, the spin degenerate affine Hecke algebras H THE CLASSICAL SPIN AFFINE HECKE ALGEBRAS 23 Proof. Follows by the definitions in terms of generators and relations of all the algebras involved. � Theorem 5.5. Let W = WAn−1 ,WDn , or WBn. Then the elements x̃ where α ∈ Zn+ and w̃ ∈ W̃ , form a basis for H∼W (called a PBW basis). Proof. By the defining relations, it is easy to see that the elements x̃αw̃ form a spanning set for H∼W . So it remains to show that they are linearly independent. For each element t ∈ W , denote the two preimages in W̃ of t by {t̃, zt̃}. Now suppose that aα,t̃x̃ αt̃+ bα,t̃zx̃ Let I+ and I− be the ideals of H∼W generated by z−1 and z+1 respectively. Then by Proposition 5.4, H∼W /I + ∼= HW and H∼W /I− ∼= H . Consider the projections: Υ+ : H W −→ H∼W /I+, Υ− : H∼W −→ H∼W /I−. By abuse of notation, denote the image of x̃α in HW by x α. Observe that 0 = Υ+ (aα,t̃x̃ αt̃+ bα,t̃x̃ αzt̃) (aα,t̃ + bα,t̃)x αt ∈ HW . Since it is known [Lu1] that {xαt|α ∈ Zn+ and t ∈ W} form a basis for the usual degenerate affine Hecke algebra HW , aα,t̃ = −bα,t̃ for all α and t. Similarly, denoting the image in CW− of t̃ by t̄, we have 0 = Υ− (aα,t̃x̃ αt̃+ bα,t̃x̃ αzt̃) (aα,t̃ − bα,t̃)x αt̄ ∈ H− Since {xαt̄} is a basis for the spin degenerate affine Hecke algebra H− have aα,t̃ = bα,t̃ for all α and t. Hence, aα,t̃ = bα,t̃ = 0, and the linear independence is proved. � References [BK] J. Brundan, A Kleshchev, Hecke-Clifford superalgebras, crystals of type A 2l and modular branching rules for Ŝn, Represent. Theory 5 (2001), 317–403. [Dr] V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 58–60. [EG] P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348. [IY] S. Ihara and T. Yokonuma, On the second cohomology groups (Schur multipliers) of finite reflection groups, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 11 (1965), 155–171. [JN] A. Jones and M. Nazarov, Affine Sergeev algebra and q-analogues of the Young symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc. 78 (1999), 481–512. [Joz] T. Józefiak, A class of projective representations of hyperoctahedral groups and Schur Q-functions, Topics in Algebra, Banach Center Publ., 26, Part 2, PWN- Polish Scientific Publishers, Warsaw (1990), 317–326. [Kar] G. Karpilovsky, The Schur multiplier, London Math. Soc. Monagraphs, New Series 2, Oxford University Press, 1987. 24 TA KHONGSAP AND WEIQIANG WANG [Kle] A. Kleshchev, Linear and projective representations of symmetric groups, Cam- bridge Tracts in Mathematics 163, Cambridge University Press, 2005. [KW] T. Khongsap and W. Wang, Hecke-Clifford algebras and spin Hecke algebras II: the rational double affine type, Preprint 2007. [Lu1] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635. [Lu2] ———, Cuspidal local systems and graded Hecke algebras I, Publ. IHES 67 (1988), 145–202. [Lu3] ———, Cuspidal local systems and graded Hecke algebras III, Represent. Theory 6 (2002), 202–242. [Mo] A. Morris, Projective representations of reflection groups, Proc. London Math. Soc 32 (1976), 403–420. [Naz] M. Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. in Math. 127 (1997), 190–257. [Sch] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155– [Ser] A. Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434. [St] J. Stembridge, The projective representations of the hyperoctahedral group, J. Al- gebra 145 (1992), 396–453. [W1] W. Wang, Double affine Hecke algebras for the spin symmetric group, preprint 2006, math.RT/0608074. [W2] ———, Spin Hecke algebras of finite and affine types, Adv. in Math. 212 (2007), 723–748. [Yam] M. Yamaguchi, A duality of the twisted group algebra of the symmetric group and a Lie superalgebra, J. Algebra 222 (1999), 301–327. Department of Math., University of Virginia, Charlottesville, VA 22904 E-mail address: [email protected] (Khongsap); [email protected] (Wang) http://arxiv.org/abs/math/0608074 1. Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 2. Spin Weyl groups and Clifford algebras 2.1. The Weyl groups 2.2. A distinguished double covering of Weyl groups 2.3. The Clifford algebra CW 2.4. The basic spin supermodule 2.5. A superalgebra isomorphism 3. Degenerate affine Hecke-Clifford algebras 3.1. The algebra HcW of type An-1 3.2. The algebra HcW of type Dn 3.3. The algebra HcW of type Bn 3.4. PBW basis for HcW 3.5. The even center for HcW 3.6. The intertwiners in HcW 4. Degenerate spin affine Hecke algebras 4.1. The skew-polynomial algebra 4.2. The algebra H-W of type Dn 4.3. The algebra H-W of type Bn 4.4. A superalgebra isomorphism 4.5. PBW basis for H-W 4.6. The even center for H-W 4.7. The intertwiners in H-W 5. Degenerate covering affine Hecke algebras 5.1. The algebra HW of type An-1 5.2. The algebra HW of type Dn 5.3. The algebra HW of type Bn 5.4. PBW basis for HW References

0704.0202

Towards Minimal Resources of Measurement-based Quantum Computation

Towards Minimal Resources of Measurement-based Quantum Computation Simon Perdrix PPS, CNRS - Université Paris 7 E-mail: [email protected] Abstract. We improve the upper bound on the minimal resources required for measurement- based quantum computation [4, 3, 6]. Minimizing the resources required for this model is a key issue for experimental realization of a quantum computer based on projective measurements. This new upper bound allows also to reply in the negative to the open question presented in [5] about the existence of a trade-off between observable and ancillary qubits in measurement- based quantum computation. 1. Introduction The discovery of new models of quantum computation (QC), such as the one-way quantum computer [7] and the projective measurement-based model [4], have opened up new experimental avenues toward the realisation of a quantum computer in laboratories. At the same time they have challenged the traditional view about the nature of quantum computation. Since the introduction of the quantum Turing machine by Deutsch [1], unitary transformations plays a central rôle in QC. However, it is known that the action of unitary gates can be simulated using the process of quantum teleportation based on projective measurements-only [4]. Characterizing the minimal resources that are sufficient for this model to be universal, is a key issue. Resources of measurement-based quantum computations are composed of two ingredients: (i) a universal family of observables, which describes the measurements that can be performed (ii) the number of ancillary qubits used to simulate any unitary transformation. Successive improvements of the upper bounds on these minimal resources have been made by Leung and others [2, 3]. These bounds have been significantly reduced when the state transfer (which is a restricted form of teleportation) has been introduced: one two-qubit observable (Z ⊗ X) and three one-qubit observables (X , Z and (X + Y )/ 2), associated with only one ancillary qubit, are sufficient for simulating any unitary-based QC [6]. Are these resources minimal ? In [5], a sub-family of observables (Z ⊗X , Z, and (X − Y )/ is proved to be universal, however two ancillary qubits are used to make this sub-family universal. These two results lead to an open question : is there a trade-off between observables and ancillary qubits in measurement-based QC ? In this paper, we reply in the negative to this http://arxiv.org/abs/0704.0202v1 Towards Minimal Resources of Measurement-based Quantum Computation 2 open question by proving that the sub-family {Z ⊗ X,Z, (X − Y )/ 2} is universal using only one ancillary qubit, improving the upper bound on the minimal resources required for measurement-based QC. 2. Measurement-based QC The simulation of a given unitary transformation U by means of projective measurements can be decomposed into: • (Step of simulation) First, U is probabilistically simulated up to a Pauli operator, leading to σU , where σ is either identity or a Pauli operator σx, σy, or σz . • (Correction) Then, a corrective strategy consisting in combinig conditionally steps of simulation is used to obtain a non-probabilistic simulation of U . Teleportation can be realized by two successive Bell measurements (figure 1), where a Bell measurement is a projective measurement in the basis of the Bell states {|Bij〉}i,j∈{0,1}, where |Bij〉 = 1√ (σiz ⊗ σjx)(|00〉+ |11〉). A step of simulation of U is obtained by replacing the second measurement by a measurement in the basis {(U † ⊗ Id)|Bij〉}i,j∈{0,1} (figure 2). Figure 1. Bell measurement-based teleportation ΦUσ ΦUσ Figure 2. Simulation of U up to a Pauli operator If a step of simulation is represented as a probabilistic black box (figure 3, left), there exists a corrective strategy (figure 3, right) which leads to a full simulation of U . This strategy consists in conditionally composing steps of simulation of U , but also of each Pauli operator. A similar step of simulation and strategy are given for the two-qubit unitary transformation ΛX (Controlled-X) in [4]. Notice that this simulation uses four ancillary qubits. As a consequence, since any unitary transformation can be decomposed into ΛX and one-qubit unitary transformations, any unitary transformation can be simulated by means of projective measurements only. More precisely, for any circuit C of size n – with basis ΛX and all one-qubit unitary transformations – and for any ǫ > 0, O(n log(n/ǫ)) projective measurements are enough to simulate C with probability greater than 1− ǫ. Towards Minimal Resources of Measurement-based Quantum Computation 3 Φ Φσy σz Figure 3. Left: step of simulation abstracted into a probabilistic black box representation – Rigth: conditional composition of steps of simulation. Approximative universality, based on a finite family of projective measurements, can also be considered. Leung [3] has shown that a family composed of five observables F0 = {Z,X ⊗ X,Z ⊗ Z,X ⊗ Z, 1√ (X − Y ) ⊗ X} is approximatively universal, using four ancillary qubits. It means that for any unitary transformation U , any ǫ > 0 and any δ > 0, there exists a conditional composition of projective measurements from F0 leading to the simulation of a unitary transformation Ũ with probability greater than 1− ǫ and such that ||U − Ũ || < δ. Figure 4. State transfer ΦUσVb X Φ U ZU ΦUσVVXVb Figure 5. Step of simulation based on state transfer In order to decrease the number of two-qubit measurements – four inF0 – and the number of ancillary, an new scheme called state transfer has been introduced [6]. The state transfer (figure 4) replaces the teleportation scheme for realizing a step of simulation. Composed of one two-qubit measurements, two one-qubit measurements, and using only one ancillary qubit, the state transfer can be used to simulate any one-qubit unitary transformation up to a Pauli operator (figure 5). For instance, simulations ofH andHT – see section 3 for definitions of H and T – are represented in figure 6. Moreover a scheme composed of two two-qubit measurements, two one-qubit measurements, and using only one ancillary qubit can be used to simulated ΛX up to a Pauli operator (figure 7). Since {H, T,ΛX} is a universal family of unitary transformations, the family F1 = {Z ⊗ X,X,Z, 1√ (X − Y )} of observables is approximatively universal, using one ancillary qubit [6]. This result improves the result by Leung, since only one two-qubit measurement and one ancillary qubit are used, instead of four two-qubit measurements and four ancillary qubits. Moreover, one can prove that at least one Towards Minimal Resources of Measurement-based Quantum Computation 4 two-qubit measurement and one ancillary qubit are required for approximative universality. Thus, it turns out that the upper bound on the minimal resources for measurement-based QC differs form the lower bound, on the number of one-qubit measurements only. b Z X Z X−Y/ 2Φ b Z X Figure 6. Simulation of H and HT up to a Pauli operator. σ Λ X Figure 7. Simulation of ΛX up to a Pauli operator. In [5], it has been shown that the number of these one-qubit measurements can be decreased, since the family F2 = {Z ⊗ X,Z, 1√ (X − Y )}, composed of one two-qubit and only two one-qubit measurements, is also approximatively universal, using two ancillary qubit. The proof is based on the simulation of X-measurements by means of Z and Z ⊗ X measurements (figure 8). This result leads to a possible trade-off between the number of one-qubit measurements and the number of ancillary qubits required for approximative universality. Figure 8. X-measurement simulation In this paper, we meanly prove that the family F2 is approximatively universal, using only one ancillary qubit. Thus, the upper bound on the minimal resources required for approximative universality is improved, and moreover we answer the open question of the trade-off between observables and ancillary qubits. Notice that we prove that the trade-off conjectured in [5] does not exist, but another trade-off between observables and ancillary qubits may exist since the bounds on the minimal resources for measurement-based quantum computation are not tight. 3. Universal family of unitary transformations There exist several universal families of unitary transformations, for instance {H, T,ΛX} is one of them: Towards Minimal Resources of Measurement-based Quantum Computation 5 H = 1√ , T = , ΛX = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 We prove that the family {HT, σy,ΛZ} is also approximatively universal: Theorem 1 U = {HT, σy,ΛZ} is approximatively universal. The proof is based on the following properties. Let R (α) be the rotation of the Bloch sphere about the axis n through an angle α. Proposition 1 If n = (a, b, c) is a real unit vector, then for any α, R (α) = cos(α/2)I − i sin(α/2)(aσx + bσy + cσz). Proposition 2 For a given vector n of the Bloch sphere, if θ is an irrational multiple of π, then for any α and any ǫ > 0, there exists k such that (α)− R (θ)k)|| < ǫ/3 Proposition 3 If n and m are non parallel vectors of the Bloch sphere, then for any one- qubit unitary transformation U , there exists α, β, γ, δ such that: U = eiαR Proposition 4 (Włodarski [8]) If α is not an integer multiple of π/4 and cos β = cos2 α, then either α or β is an irrational multiple of π. Proof of theorem 1: First we prove that any 1-qubit unitary transformation can be approximated by HT and σyHT . Consider the unitary transformations U1 = T , U2 = HTH , U3 = σyHTHσy. Notice that T is, up to an unimportant global phase, a rotation by π/4 radians around z axis on the Block sphere: U1 = T = e −iπ/8(cos(π/8)I − i sin(π/8)σz) U2 = HTH = e −iπ/8(cos(π/8)I − i sin(π/8)σx) U3 = σyHTHσy = e −iπ/8(cos(π/8)I + i sin(π/8)σx) Composing U1 and U2 gives, up to a global phase: U2U1 = (cos(π/8)I − i sin(π/8)σx)(cos(π/8)I − i sin(π/8)σz) = cos2(π/8)I − i[cos(π/8)(σx + σz)− sin(π/8)σy] sin(π/8) Towards Minimal Resources of Measurement-based Quantum Computation 6 According to proposition 1, U2U1 is a rotation of the Bloch sphere about an axis along n = (cos(π/8), − sin(π/8), cos(π/8)) and through an angle θ defined as a solution of cos(θ/2) = cos2(π/8). Since π/8 is not an integer multiple of π/4 but a rational multiple of π, according to proposition 4, a such θ is an irrational multiple of π. This irrationality implies that for any angle α, the rotation around n about angle α can be approximated to arbitrary accuracy by repeating rotations around n about angle θ (see proposition 3). For any α and any ǫ > 0, there exists k such that (α)− R (θ)k)|| < ǫ/3 Moreover, composing U1 and U3 gives, up to a global phase: U3U1 = (cos(π/8)I + i sin(π/8)σx)(cos(π/8)I − i sin(π/8)σz) = cos2(π/8)I − i[cos(π/8)(−σx + σz) + sin(π/8)σy] sin(π/8) U3U1 is a rotation of the Bloch sphere about an axis alongm = (− cos(π/8), sin(π/8), cos(π/8)) and through the angle θ. Thus, for any α and any ǫ > 0, there exists k such that (α)− R (θ)k)|| < ǫ/3 Since n and m are non-parallel, any one-qubit unitary transformation U , according to proposition 2, can be decomposed into rotations around n and m : There exist α, β, γ, δ such U = eiαR Finally, for any U and ǫ > 0, there exist k1, k2, k3 such that ||U − R (θ)k1R (θ)k2R (θ)k3 || < ǫ Thus, any one-qubit unitary transformation can be approximated by means of U2U1, and U3U1. Since U2U1 = (HT )(HT ) and U3U1 = σyHTHσyT = −(σyHT )(σyHT ), the family {HT, σy} approximates any one-qubit unitary transformation. With the additional ΛZ gate, the family U is approximatively universal. � 4. Universal family of projective measurements In [5], the family of observables F2 = {Z ⊗ X,Z, X−Y√ } is proved to be approximatively universal using two ancillary qubits. We prove that this family requires only one ancillary qubit to be universal: Theorem 2 F2 = {Z ⊗X,Z, X−Y√ } is approximatively universal, using one ancillary qubit only. The proof consists in simulating the unitary transformations of the universal family U . First, one can notice that HT can be simulated up to a Pauli operator, using measurements of F2, as it is depicted in figure 6. So, the universality is reduced to the ability to simulate ΛZ and the Pauli operators – Pauli operators are needed to simulated σy ∈ F , but also to perform the corrections required by the corrective strategy (figure 3). Towards Minimal Resources of Measurement-based Quantum Computation 7 Lemma 5 For a given 2-qubit register a, b and one ancillary qubit c, the sequence of measurements according to Zc, Za⊗Xc, Zc⊗Xb, and Zb (see figure 9) simulates ΛZ(Id⊗H) on qubits a, b, up to a Pauli operator. The resulting state is located on qubits a and c. Z(Id H)Z ZΦ Φc Figure 9. Simulation of ΛZ(Id⊗H) Proof: One can show that, if the state of the register a, b is |Φ〉 before the sequence of measurements, the state of the register a, b after the measurements is σΛZ(Id⊗H)|Φ〉, where σ = σs1z ⊗ σs3x σs2+s4z and si’s are the classical outcomes of the sequence of measurements. � In order to simulate Pauli operators, a new scheme, different from the state transfer, is introduced. Lemma 6 For a given qubit b and one ancillary qubit a, the sequence of measurements Za, Xa ⊗ Zb, and Za (figure 10) simulates, on qubit b, the application of σz with probability 1/2 and Id with probability 1/2. Figure 10. Simulation of σ Proof: Let |Φ〉 = α|0〉+ β|1〉 be the state of qubit b. After the first measurement, the state of the register a, b is |ψ1〉 = (σs1x ⊗ Id)|0〉 ⊗ |Φ〉 where s1 ∈ {0, 1} is the classical outcome of the measurement. Since 〈ψ1|X ⊗ Z|ψ1〉 = 0, the state of the register a, b is: |ψ2〉 = (σs1x ⊗ Id)(Id+ (−1)s2X ⊗ Z)|0〉 ⊗ |Φ〉 (σs1x σ z ⊗ Id)(|0〉 ⊗ |Φ〉+ |1〉 ⊗ (σz|Φ〉) Let s3 ∈ {0, 1} be the outcome of the last measurement, on qubit a. If s1 = s3 then state of the qubit b is |Φ〉, and σz|Φ〉 otherwise. One can prove that these two possibilities occur with equal probabilities. � Lemma 7 For a given qubit b and one ancillary qubit a, the sequence of measurements , Za ⊗ Xb, and (figure 11) simulates, on qubit b, the application of σx with probability 1/2 and Id with probability 1/2. Towards Minimal Resources of Measurement-based Quantum Computation 8 (X−Y)/ 2 (X−Y)/ 2 Figure 11. Simulation of σ The proof of lemma 7 is similar to the proof of lemma 6. Proof of theorem 2: First notice that the family of unitary transformations U ′ = {HT, σy,ΛZ(I ⊗ H)} is approximatively universal since U = {HT, σy,ΛZ} is universal. HT and ΛZ(I⊗H) can be simulated up to a Pauli operator (lemmas 5). The universality of the family of observables F2 = {Z ⊗ X,Z, X−Y√ } is reduced to the ability to simulate any Pauli operators. Lemma 7 (resp. lemma 6), shows that σx (σz) can be simulated with probability 1/2, moreover if the simulation fails, the resulting state is same as the original one. Thus, this simulation can be repeated until a full simulation of σx (σz). Finally, σy = iσzσx can be simulated, up to a global phase, by combining simulations of σx and σz. Thus, F2 = {Z ⊗X,Z, X−Y√ } is approximatively universal using only one ancillary qubit. � 5. Conclusion We have proved a new upper bound on the minimal resources required for measurement- based QC: one two-qubit, and two one-qubit observables are universal, using one ancillary qubit only. This new upper bound has experimental applications, but allows also to prove that the trade-off between observables and ancillary qubits, conjectured in [5], does not exist. This new upper bound is not tight since the lower bound on the minimal resources for this model is one two-qubit observable and one ancillary qubit. References [1] D. Deutsch. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A, 400:97–117, 1985. [2] S. A. Fenner and Y. Zhang. Universal quantum computation with two- and three-qubit projective measurements, 2001. [3] D. W. Leung. Quantum computation by measurements. IJQI, 2:33, 2004. [4] M. A. Nielsen. Universal quantum computation using only projective measurement, quantum memory, and preparation of the 0 state. Phys. Rev. A, 308:96–100, 2003. [5] S. Perdrix. Qubit vs observable resouce trade-offs in measurement-based quantum computation. In proceedings of Quantum communication measurement and computing, 2004. [6] S. Perdrix. State transfer instead of teleportation in measurement-based quantum computation. International Journal of Quantum Information, 3(1):219–223, 2005. [7] R. Raussendorf, D. E. Browne, and H. J. Briegel. The one-way quantum computer - a non-network model of quantum computation. Journal of Modern Optics, 49:1299, 2002. [8] L. Wlodarski. On the equation cosα1+cosα2 cosα3+cosα4 = 0. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 1969. Introduction Measurement-based QC Universal family of unitary transformations Universal family of projective measurements Conclusion

0704.0203

A Spitzer census of the IC 348 nebula

A Spitzer census of the IC 348 nebula TO APPEAR IN THE ASTRONOMICAL JOURNAL Preprint typeset using LATEX style emulateapj v. 08/22/09 A SPITZER CENSUS OF THE IC 348 NEBULA AUGUST A. MUENCH1 , CHARLES J. LADA1 , K. L. LUHMAN2,3 , JAMES MUZEROLLE4 & ERICK YOUNG4 Submitted November 20, 2006; Accepted March 30, 2007; Version October 25, 2018 ABSTRACT Spitzer mid-infrared surveys enable accurate census of young stellar objects by sampling large spatial scales, revealing very embedded protostars and detecting low luminosity objects. Taking advantage of these capabilities, we present a Spitzer based census of the IC 348 nebula and embedded star cluster, covering a 2.5 pc region and comparable in extent to the Orion nebula. Our Spitzer census supplemented with ground based spectra has added 42 class II T-Tauri sources to the cluster membership and identified ∼ 20 class 0/I protostars. The population of IC 348 likely exceeds 400 sources after accounting statistically for unidentified diskless members. Our Spitzer census of IC 348 reveals a population of class I protostars that is anti-correlated spatially with the class II/III T-Tauri members, which comprise the centrally condensed cluster around a B star. The protostars are instead found mostly at the cluster periphery about∼ 1 pc from the B star and spread out along a filamentary ridge. We further find that the star formation rate in this protostellar ridge is consistent with that rate which built the older exposed cluster while the presence of fifteen cold, starless, millimeter cores intermingled with this protostellar population indicates that the IC 348 nebula has yet to finish forming stars. Moreover, we show that the IC 348 cluster is of order 3-5 crossing times old, and, as evidenced by its smooth radial profile and confirmed mass segregation, is likely relaxed. While it seems apparent that the current cluster configuration is the result of dynamical evolution and its primordial structure has been erased, our finding of a filamentary ridge of class I protostars supports a model where embedded clusters are built up from numerous smaller sub-clusters. Finally, the results of our Spitzer census indicate that the supposition that star formation must progress rapidly in a dark cloud should not preclude these observations that show it can be relatively long lived. Subject headings: infrared: stars — circumstellar matter — open clusters and associations: individual (IC 348) 1. INTRODUCTION The IC 348 nebula on the northeastern corner of the Perseus Molecular Cloud (Barnard 1915) has been known to har- bor pre-main sequence T-Tauri stars since they were revealed through a slitless Hα grism survey by Herbig (1954). Slit- less Hα grism surveys were once the most powerful tool for searching for young stars (c.f., Herbig & Bell 1988), while the subsequent development of infrared bolometers permit- ted better census of the darker regions of molecular clouds, including very young protostars which are young stars that still retain infalling envelopes. Such infrared observations in IC 348 by Strom et al. (1974) led, for example, to the discov- ery of an optically invisible bright 2µm source about 1pc from the clustering of Hα members. Strom’s IR source was the first such hint that the stars forming in the IC 348 nebula might not all have the same age. Modern tools for identifying young stars include X-ray surveys, which parse young stellar ob- jects (YSOs) using energetic emissions from their rotationally enhanced, magnetic activity, and wide-field infrared imaging surveys, which identify YSOs using the signature in the star’s broadband spectral energy distribution (SED) of thermal re- processing of the star’s light by an optically thick circumstel- lar disk. To date roughly 300 young stars have been identi- fied in the IC 348 nebula from X-ray (e.g., Preibisch & Zin- necker 2001, 2004), optical (e.g., Trullols & Jordi 1997; Her- 1 Smithsonian Astrophysical Observatory. 60 Garden Street, Mail Stop 72. Cambridge, MA. 02138 USA; [email protected], [email protected] 2 Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative Agreement no. NCC 5-538 with the National Aeronautics and Space Administration, Office of Space Science, Planetary Astronomy Program. 3 Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA; [email protected]. 4 Steward Observatory, University of Arizona Tucson, AZ 85712; [email protected], [email protected] big 1998), near-infrared (Lada & Lada 1995; Muench et al. 2003, hereafter, LL95 and M03 respectively) and spectro- scopic surveys (Luhman et al. 1998b; Luhman 1999; Luhman et al. 2003b, 2005a). These known members are clustered at the center of the nebula and have a median age of∼ 2−3 My (Luhman et al. 2003b); we examined the disk properties of these members in Lada et al. (2006, hereafter, Paper 1), For this paper we undertook a mid-infrared survey of the IC 348 nebula with the Spitzer Space Telescope (Werner et al. 2004) to make a more complete membership census over a large cluster area. Statistical studies of the surface density of stars around IC 348 (Tej et al. 2002; Cambrésy et al. 2006) anticipated the discovery of more cluster members, but they could not identify individual members and could give no in- formation about their evolutionary status. The classification of a young star as protostellar (class I) or more evolved class II sources with optically thick disks (see Adams et al. 1987, etc) is best accomplished using its broadband spectral en- ergy distribution. Thus, we have identified and classified ap- proximately 60 new cluster members, including ∼ 20 pro- tostellar objects, by constructing each source’s broad band (0.5−70µm) SED and through spectroscopic follow up. Our census has expanded the confirmed boundaries of IC 348 to a physical size comparable to that well studied portion of the Orion Nebula Cluster (Hillenbrand & Hartmann 1998). Paper I contains all details of the data processing except for the far-infrared Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004) observations (see §2.3.2)5. Can- didate members were selected initially using spectral indices to identify the presence of infrared excess in their composite SEDs (§2.1). Ground-based spectra, including new observa- tions presented in this paper, support the membership status 5 The Spitzer data obtained for this paper were taken from AORs 3955200, 3651584, 4315904. 2 Muench et al. FIG. 1.— A) Spitzer m3.6 − m5.8 color vs m5.8 magnitude diagram; B) Spitzer α3−8µm spectral index vs m5.8 magnitude diagram. Symbol types differentiate four band (filled circles) and three band (open squares) IRAC detections. Upperlimits on m5.8 are shown with arrows (19 sources). Subsequent sorting of these sources into respective YSO classes was restricted to that sample with m5.8 < 15 (or fainter sources which were detected in the 3.6, 4.5, and 8.0µm bands). Vertical dashed lines in panel (B) correspond to α values used to segregate different YSO classes (e.g., class I, II; see §2.1 and Paper I). The reddening law is from Indebetouw et al. (2005) or as derived in §A. for nearly all of the class II candidates and many of the proto- stars. Our census of very low luminosity protostars required the removal of an overwhelming population of extragalactic sources that masquerade as young stars (§2.3). In §3 we ex- amine the nature of the IC 348 protostellar and class II mem- bers by comparing their positions to recent dense gas and dust maps from the COMPLETE6 project (Ridge et al. 2006), by analyzing their physical separations, and by placing them on the Hertzsprung-Russell (HR) diagram. We discuss briefly the implications of the cluster’s inferred structure and star form- ing history and examine the timescales for dynamical evolu- tion of the central star cluster, pointing out their relevance for the timescale for dark cloud and circumstellar disk evolution (§4). Appendices include a discussion of the effects of red- dening on the 3 − 8µm portion of a stellar or star+disk SED (§A), spectra of ∼ 40 new members (§B) and a photometric catalog of candidate apparently diskless (class III) members selected from X-ray surveys of this region (§C). 2. SPITZER CENSUS 2.1. SED selected young stellar objects Studying the previously known members of IC 348 in Paper I, we showed that the power-law fit of the 3 − 8µm portion of the young stars’ SEDs as observed by Spitzer, provided a good diagnostic of these members’ disk properties. We were able to empirically separate members with optically thick T- 6 The COordinated Molecular Probe Line Extinction Thermal Emission Survey of Star Forming Regions, http://cfa-www.harvard.edu/ COMPLETE/. Tauri disks (hereafter class II sources) from those with little (termed anemic) or no apparent disk excess at these wave- lengths7. In this section we describe how we used this SED parameter to identify new young stellar objects (YSOs) from our entire Spitzer catalog of IC 348, including new embedded protostars that were not studied in Paper I. In searching for new members using disk excess it is also important to avoid selecting reddened background stars; as we discuss in Ap- pendix A, the 3 − 8µm SED slope is fairly insensitive to ex- tinction, which allows us to be confident in the quality of our initial member selection. To fit a power law to the 3−8µm SED we required sources in our catalog to be detected in at least three of the four Spitzer IRAC (InfraRed Array Camera; Fazio et al. 2004) bands; this restricted our search to a 26.8′ by 28.5′ region of the GTO IRAC maps centered at 03:44:20.518, +32:10:34.87 with a PA of 81◦. Note, this entire region was also surveyed with MIPS. The resulting ∼ 2.5 pc region enclosed both the AV -limited completeness census of Luhman et al. (2003b) and the 20′ FLAMINGOS8 region studied by Muench et al. (2003)9. In this region there are 906 sources detected in three IRAC bands, including 282 of the 300 known mem- bers studied in Paper I. Of these 906 candidates, 648 were detected in all four IRAC bands. Only 19 of the 906 sources 7 Hereafter, we use the term “class III” to describe all members having SEDs indicative of “anemic” inner disks or simple photospheres; see Paper I. 8 The FLoridA Multi-object Imaging Near-IR Grism Observational Spec- trometer. See http://flamingos.astro.ufl.edu/. 9 These survey regions are compared in Figure 5. http://cfa-www.harvard.edu/COMPLETE/ http://cfa-www.harvard.edu/COMPLETE/ http://flamingos.astro.ufl.edu/ Spitzer census of IC 348 3 FIG. 2.— Distribution function of α3−8µm for sources with m5.8 < 15.0. The open histogram corresponds to Spitzer sources detected in the four IRAC bands; the hatched histogram corresponds to the distribution function of three IRAC band detections. The distribution functions were normalized by the total number of candidates. Two vertical lines (α = −1.8 and α = −0.5) separate the candidates into three YSO classes. For each source we overplotted the 1σ uncertainty in the α3−8µm fit versus its α3−8µm on the x-axis; again, open and filled symbols differentiate 3 and 4 band detections, respectively. lacked 5.8µm detections while 238 sources detected from 3.6 to 5.8µm lacked 8.0µm detections10. To better con- strain the candidates’ SEDs we derived 95% upper limits for all sources lacking either 5.8 or 8.0µm detections. We constructed the m3.6 − m5.8 versus m5.8 color- magnitude diagram (CMD, Figure 1a) for these 906 candi- dates to further refine our selection criteria. Upper-limits for the 19 sources lacking 5.8µm flux measurements are dis- played with arrows. Two loci are clearly evident in the CMD: one of nearly colorless stars and the second redder locus we expect to consist primarily of T-Tauri stars with disks. A strong 8.0µm magnitude cutoff for colorless stars is evident at m5.8 ∼ 14; most the 5.8µm upperlimits are m5.8 > 14.5. In Figure 1b we have replaced the m3.6 − m5.8 color with the power-law fit to the observed slope of the IRAC portion of these sources’ SEDs (α3−8µm)11; this clearly reinforces the existence of two intrinsic loci in the CMD. The two loci are more distinct when plotting m5.8 as a function of α3−8µm be- cause we have assumed the correct underlying shape of the objects’ SEDs; i.e., whether it is the Raleigh Jeans portion of a star’s photospheric SED or the thermal infrared SED of a pas- sive, re-radiating optically thick disk. Further, these power- law fits are less sensitive to uncorrelated photometric uncer- tainties than colors, which are of course ratios between only 10 One spectroscopically confirmed member, #396 (M5.25), did not have photometry at 3.6µm due to a nearby bright star; o Persi was saturated at 3.6 and 4.5µm; there were nine sources detected only at 5.8µm. Otherwise all the sources in our field were detected in bands 3.6 and 4.5µm. We found no sources detected only at 8.0µm. 11 Upper limits were not used in these calculations, although they were useful for filtering sources; see §2.1. FIG. 3.— Spitzer IRAC color-color diagrams for all 3 band IRAC detec- tions with m5.8 < 15. Sources are color coded by candidate YSO class defined by α3−8µm: class I (α > −0.5), solid grey filled circles; class II (−0.5 > α > −1.8), light grey filled circles; other (class III, “anemic” disk or non-member) objects are open circles. A) m4.5 − m5.8 vs m3.6 − m4.5. This pane includes sources not detected in 8.0µm band; B) m4.5 − m8.0 vs m3.6−m4.5. This panel includes sources lacking 5.8µm detections. Note, we found that IC 348 sources in the far upper left are contaminated by shocked emission from Herbig-Haro objects. The reddening law is from Indebetouw et al. (2005). two wavelengths. Nonetheless, we further filtered our sam- ple of candidate members based on photometric quality. We imposed an empirical flux limit of m5.8 < 15 based on the increased spread in the value of α3−8µm for fainter colorless stars; we did, however include fainter 5.8µm IRAC sources 4 Muench et al. if they were also detected at 8.0µm12. We also required the detections to have photometric errors of less than 0.2 magni- tudes. After applying these photometric constraints we had 657 candidates in our IC 348 Spitzer region. Figure 2 displays the distribution function of α3−8µm as a histogram for these candidates. The first narrow peak in the α3−8µm distribu- tion function at -2.8 reflects the narrowly constrained value of α3−8µm for stellar photospheres; photospheric α3−8µm has very little spectral type dependence (Paper I). A second peak at α3−8µm = −1.3 corresponds to class II T-Tauri stars with optically thick disks and a third peak corresponds to sources with flat or rising mid-IR SEDs. Using our empirical bound- ary between anemic and class II disks (α3−8µm > −1.8; Paper I; shown in Figure 2 ) we identified 192 candidate YSOs in our IC 348 region. Our tally of IC 348 YSOs is 20% larger than the total number of IC 348 YSOs (158) identified by Jørgensen et al. (2006). While we are using slightly lower luminosity limits than Jørgensen et al., the statistics of their Legacy survey come from a different and 70% larger cluster area, correspond to a different definition of the spectral index and include class III (by their definition) sources; thus, we do not further discuss the statistics of this Legacy project. Fi- nally, we did not search for new members with “anemic” type disks (−2.6 < α3−8µm < −1.8; Paper I) since a search for sources with very small excesses can be hampered by poor photometry, in this case due to the nebula (see the scatter in the power-law fit sigma overplotted Figure 2). We subdivided the α3−8µm > −1.8 YSO sample into two classes based on the shape of the α3−8µm distribution function: thick disk class II sources in the peak,−1.8 < α3−8µm < −0.5 and class I “protostellar” candidates with α3−8µm > −0.5. Flat spectrum sources, considered to be protostars in a later stage of envelope dispersal or with highly flared disks, can have slightly falling mid-infrared SED slopes, 0.3 > α > −0.3 (Lada 1987). A distinction between highly flared class II disks and emission from disk +remnant envelope may re- quire data at wavelengths longer than 10µm. There are a total of 136 candidate class II sources in our IC 348 Spitzer re- gion and 56 red class I candidates. For comparison to other Spitzer studies of YSOs in clusters that use color-color clas- sification techniques (e.g., Megeath et al. 2004), we plot two such diagrams in Figure 3. Together these encompassed all 3 band IRAC detected sources; protostellar and class II sources are color-coded on these plots. Sources parsed by α3−8µm are well segregated in the color-color diagram except where pho- tometric errors in a single color yield some scattering. 2.2. Class II census results 2.2.1. Membership In this section we explore the membership status of the 136 class II candidates identified in §2.1, revealing that the vast majority of them are confirmed spectroscopicallly as mem- bers. Seventy six of our 136 α3−8µm selected class II ob- jects were cataloged previously as members of IC 348 (Her- big 1998; Luhman et al. 1998b; Luhman 1999; Luhman et al. 12 We will use the m5.8 magnitude to parse the sources in our subsequent analysis for four reasons: 1) it is clearly more sensitive than the 8.0µm chan- nel; 2) when combined with a standard extinction law it will be the primary IRAC bandpass for emergent flux from heavily reddened sources (Whitney et al. (2004); Appendix §A and Figure 22) due at least in part to silicate absorption in the 8.0µm bandpass; 3) it is somewhat less contaminated by PAH emission than the 8.0µm bandpass frequently evident in the SEDs of non-cluster sources. FIG. 4.— Additional class II candidates considered to be IC 348 members but lacking spectral types. Again, sources are sorted according to their 5.8µm magnitude, which is listed in parenthesis below each source ID. Plotting sym- bols, line thickness and line color alternate from SED to SED for clarity. 2003b, 2005a). For this paper we obtained optical and near- infrared spectroscopy of 34 more class II sources; these ob- servations are detailed in Appendix B and in Table 1 we list new members with spectral types. From the remaining 26 class II candidates, we identified an additional eight sources whose SEDs suggest they are high quality candidates (55308, 10031, 1287, 1379, 22865, 753; see Figure 4 and Table 2). The four latter objects are very faint (H > 16; see also Figure 6), and if they are cluster members rather than background sources (e.g. galaxies) then they are almost certainly brown dwarfs given their low luminosities. Three sources classified initially as class II sources using IRAC data were reclassi- fied as protostellar (§2.3) based on their MIPS SEDs. The remaining sources were class II contaminants, consisting of either HH knots (2) or false excesses sources detected in only 3 bands and contaminated by nebular emission (13). We con- clude that the technique of using α3−8µm as a discriminator of class II YSOs is successful for roughly 90% of the initial class II sample (118 members from 136 candidates). Figure 5 compares the locations of our new class II sources to previous deep IR/spectroscopic census. New class II sources roughly correlate spatially with previously known members although all but 8 lie outside the Luhman et al. (2003b) AV < 4; M > 0.03M� completeness region. In- side that survey region five new class II members are deeply embedded (AV > 4) in a dark molecular gas cloud at the cluster’s southwestern boundary, while three are very faint, likely lying below that survey’s 0.03M� limit (e.g. # 1379) and lack spectroscopic followup. Most (27) of the new class II members fall within the boundaries of the Muench et al. (2003) near-infrared survey and confirmed cluster members can now be found as far as 2 pc from the cluster core. Com- pared to 40 ± 6 unidentified K < 13 members predicted by Cambrésy et al. (2006) we found 23 new class II and 4 class I with K < 13 while our disk based SED selection cri- teria could not have revealed new class III members, which outnumber class II members by a factor of two. If we con- sider that the surface density excess seen in the Cambrésy Spitzer census of IC 348 5 TABLE 1 SPECTROSCOPY OF IC 348 Spitzer EXCESS SOURCES IDa α(J2000) δ(J2000) f pb Spectral Type Membershipc Class 70 03 43 58.55 32 17 27.7 cfht M3.5(IR),M3.75(op) AV ,H2O,ex,e,Li,NaK II 117 03 43 59.08 32 14 21.3 2m M3-M4(IR) AV ,H2O,e,ex II 132 03 44 27.25 32 14 21.0 cfht M3.5(IR,op) AV ,H2O,ex,NaK II 162 03 43 48.81 32 15 51.7 cfht M4.5(IR) AV ,H2O,ex II 179 03 44 34.99 32 15 31.1 cfht M3.5(IR,op) AV ,H2O,ex,NaK II 199 03 43 57.22 32 01 33.9 wfpc M6.5(IR) AV ,H2O,ex II 215 03 44 28.95 32 01 37.9 cfht M3.25(IR) AV ,H2O,ex II 231 03 44 31.12 32 18 48.5 cfht M3.25(IR) AV ,H2O,ex II 234 03 44 45.22 32 01 20.0 cfht M5.75(IR) AV ,H2O,ex I 245 03 43 45.17 32 03 58.7 cfht ?(IR) ex I 265 03 44 34.69 32 16 00.0 cfht M3.5(IR) AV ,ex II 280 03 44 15.23 32 19 42.1 cfht M4.75(IR,op) AV ,H2O,ex,NaK II 321 03 44 22.94 32 14 40.5 cfht M5.5(IR) AV ,H2O,ex II 327 03 44 06.00 32 15 32.3 cfht M6.5(IR) H2O,ex II 364 03 44 43.03 32 15 59.8 cfht M4.75(IR,op) AV ,H2O,ex,NaK II 368 03 44 25.70 32 15 49.3 cfht M5.5(IR) AV ,H2O,ex II 406 03 43 46.44 32 11 06.1 cfht M6.5(IR),M5.75(op) AV ,H2O,ex,NaK II 643 03 44 58.55 31 58 27.3 cfht M6.5(IR) AV ,H2O,ex II 723 03 43 28.47 32 05 05.9 cfht M4(IR) AV ,H2O,e,ex II 904 03 45 13.81 32 12 10.1 cfht M3.5(IR) AV ,H2O,ex I 1679 03 44 52.07 31 58 25.5 cfht M3.5(IR) AV ,H2O,ex II 1683 03 44 15.84 31 59 36.9 cfht M5.5(IR),M5.25(op) AV ,H2O,ex,e,NaK II 1707 03 43 47.63 32 09 02.7 cfht M7(IR) H2O,ex II 1761 03 45 13.07 32 20 05.3 2m M5(IR) AV ,H2O,ex II 1833 03 44 27.21 32 20 28.7 cfht M5.25(IR),M5(op) AV ,H2O,ex,NaK II 1843 03 43 50.57 32 03 17.7 cfht M8.75(IR) AV ,H2O,ex II 1872 03 44 43.31 32 01 31.6 2m ?(IR) e,ex I 1881 03 44 33.79 31 58 30.3 cfht M4.5(IR),M3.75(op) AV ,H2O,ex,e,NaK II 1889 03 44 21.35 31 59 32.7 2m ?(IR) e,ex I 1890 03 43 23.57 32 12 25.9 cfht M4.5(op) AV ,NaK II 1905 03 43 28.22 32 01 59.2 cfht >M0(IR),M1.75(op) AV ,H2O,ex,e,Li II 1916 03 44 05.78 32 00 28.5 2m ?(IR) ex I 1923 03 44 00.47 32 04 32.7 2m M5(IR) AV ,H2O,ex II 1925 03 44 05.78 32 00 01.3 cfht M5.5(IR) AV ,H2O,ex II 1933 03 45 16.35 32 06 19.9 cfht ?(IR),K5(op) AV ,ex,e II 10120 03 45 17.83 32 12 05.9 cfht M3.75(op) e,NaK,AV ,ex II 10176 03 43 15.82 32 10 45.6 cfht M4.5(IR) AV ,H2O,ex II 10219 03 45 35.63 31 59 54.4 cfht M4.5(IR,op) AV ,H2O,ex,NaK,e II 10305 03 45 22.15 32 05 45.1 cfht M8(IR) AV ,H2O,ex II 22232 03 44 21.86 32 17 27.3 cfht M5(IR),M4.75(op) AV ,H2O,ex,e,NaK II 30003 03 43 59.17 32 02 51.3 wfpc M6(IR) AV ,H2O,ex I a The running number identifiers used in this work corresponds to and extends that system used in Luhman et al. (1998b); Luhman (1999); Luhman et al. (2003b, 2005b,a); Muzerolle et al. (2006); Lada et al. (2006). b f p is a flag on the source’s position indicating the origin of that astrometry: Muench et al. (2m: 2003, ; FLAMINGOS); Luhman et al. (cfht: 2003b); Luhman et al. (wfpc: 2005b); irac: IRAC mosaics, this paper. c Membership in IC 348 is indicated by AV & 1 and a position above the main sequence for the distance of IC 348 (“AV ”), excess emission in the IRACMIPS data (“ex”), the shape of the gravity- sensitive steam bands (“H2O”), Na I and K I strengths intermediate between those of dwarfs and giants (“NaK”), strong Li absorption (“Li”) or emission in the Balmer, Paschen or Brackett lines of hydrogen (“e”). et al. 2MASS map of IC 348 extends well beyond the borders of our Spitzer survey then we would conclude that Cambrésy et al. has underestimated somewhat the true population size at larger radii. A simple ratio of 2MASS excess to Spitzer survey areas suggests a correction factor of 3-4. Section 4.1 includes further discussion of the total cluster population size inferred from our Spitzer survey statistics. 2.2.2. Completeness We explored the completeness of our class II membership as affected by the selection requirements we used when iden- tifying new candidates and by the depth of our spectroscopic observations. Intrinsically, our Spitzer census is very sensi- tive to faint sources while insensitive to the effects of dust extinction. For example, the m5.8 magnitude limit in the Spitzer color-magnitude diagram of Figure 1a corresponds to the ability to detect a diskless 10 My 20MJup brown dwarf (K ∼ 15.6; K − 5.8 ∼ 0.6; Muench et al. 2003, ; see their Figure 12) at multiple Spitzer wavelengths.; further, we could easily detect a 3 My brown dwarf seen through through ∼ 40 visual magnitudes of extinction. Our first selection require- ment, requiring detection at three bands short-ward of 8µm, would have included 80% of the known H < 16 IC 348 mem- bers examined in Paper I; those missing were primarily class III members (those whose SEDs lack disk excess signatures). We were more concerned the application of two photometric constraints, m5.8 < 15 and merr < 0.2 mag and how these filters might affect the completeness of our census. Figure 6a is an H−K/H CMD for all potential class II can- didates. This includes the 136 class II candidates and those 6 Muench et al. FIG. 5.— Comparison of our Spitzer survey region (dashed box) to the Luhman et al. (2003b) AV < 4, M > 0.03M� complete census region (dot-dashed box) and the Muench et al. (2003) FLAMINGOS near-IR survey (dotted box). Filled dark circles are new class II sources with spectral types; unfilled circles are class II sources identified from their SEDs (§2.2) but lack optical/near-IR spectra; light filled circles are previously known members. sources excluded by our photometric constraints; each cat- egory is plotted with different symbols. Using the H band magnitude as our proxy for the mass+age+extinction limits of this study, the ratio of the photometrically filtered to the unfiltered Class II H band LFs gives an estimate of our in- completeness due to these quality filters (Figure 6b). Sources photometrically filtered from our catalog correspond to about 10 − 20% of the sample over a range of H magnitude, due probably to the variable intensity of the nebular background. Our class II census is probably more complete than suggested (> 80% complete for H < 16) because some fraction of the photometrically filtered class II candidates would have been rejected as non-members. Most of our high quality candidate members lacking spectra are faint. Dividing the H LF of our spectroscopic sample by the H LF of the unfiltered class II candidates yields a similar completeness limit: > 80% com- plete for H < 16. 2.3. Protostellar census The results of scrutinizing the 56 α3−8µm selected class I candidates are given in this section. Figure 7 displays the spectral energy distributions of the 15 brightest class I proto- stellar (α3−8µm > −0.5) candidates, including MIPS photom- etry out to 70µm. Sources are sorted on decreasing 5.8µm flux; all have m5.8 < 12.5, which should reduce the chance that they might be galaxies (Jørgensen et al. 2006). All of these sources are clearly protostars from their SEDs; previ- ous speculation on the nature of some of these objects based on the association of such red Spitzer point sources with HH objects (Walawender et al. 2006) appears to be confirmed. There is an interesting apparent correlation of SED shape with 5.8µm flux. As has been shown for protostars in Tau- rus (Kenyon & Hartmann 1995), the most luminous IC 348 protostars are exclusively flat spectrum sources, while source SEDs longward of 10µm become progressively steeper with decreasing source luminosity. Moreover, the location of the TABLE 2 Spitzer EXCESS SOURCES WITHOUT SPECTRAL TYPESa ID α(J2000) δ(J2000) f p Class 753 03 44 57.617 32 06 31.25 cfht II 1287 03 44 56.904 32 20 35.86 cfht II 1379 03 44 52.010 31 59 21.92 cfht II 1401 03 44 54.690 32 04 40.28 cfht I 1517 03 43 20.029 32 12 19.38 cfht Ib 1898 03 44 43.893 32 01 37.37 2m 0/I 4011 03 44 06.914 32 01 55.35 cfht I 10031 03 44 59.979 32 22 32.83 2m II 21799 03 43 51.586 32 12 39.92 cfht Ib 22865 03 45 17.647 32 07 55.33 cfht II 22903 03 45 19.053 32 13 54.85 cfht Ib 40150 03 43 56.162 32 03 06.11 irac I 40182 03 45 03.838 32 00 23.54 irac II 52590 03 44 20.384 32 01 58.45 irac I 52648 03 44 34.487 31 57 59.60 irac I 54299 03 43 44.284 32 03 42.41 irac II 54361 03 43 51.026 32 03 07.74 irac I 54362 03 43 50.948 32 03 26.24 irac I 54419 03 43 59.400 32 00 35.40 irac I 54459 03 44 02.415 32 02 04.46 irac I 54460 03 44 02.622 32 01 59.58 irac I 55308 03 45 13.497 32 24 34.68 irac II 55400 03 44 02.376 32 01 40.01 irac I 57025 03 43 56.890 32 03 03.40 m24m 0 HH-211 03 43 56.770 32 00 49.90 m70m 0 a Column descriptions same as in Table 1. b These class I sources are located away from any molecular material and may be background sources with SEDs that mimic circumstellar disks. TABLE 3 NON-MEMBERSa IDa α(J2000) δ(J2000) f p Spectra 398 03 43 43.28 32 13 47.3 cfht op 424 03 43 43.11 32 17 47.7 cfht op 1920 03 43 23.55 32 09 07.8 cfht op 22898 03 45 18.713 32 05 31.0 cfht IR 40163 03 44 39.994 32 01 33.5 irac IR 52827 03 45 14.012 32 06 53.0 irac IR 52839 03 45 13.199 32 10 01.9 irac IR a The optical/near-IR spectra of these sources indicate they are galaxies (§2.3.1) or field stars (§B.3). The wavelength regime of the spectral observation is given (op/IR). Column descriptions same as in Table 1. flat spectrum sources in Figure 1b mirrors another fact shown by the Kenyon & Hartmann Taurus study, namely, that flat spectrum protostars are intrinsically more luminous than class II sources. If we were to “deredden” our flat spectrum proto- stars along the reddening vector in Figure 1b then we would find them to be 2-3+ magnitudes brighter than essentially all other IC 348 members. This indicates to us that flat spectrum protostars have a star+disk+envelope structure distinct from class II sources and likely correspond to a different evolution- ary phase. For comparison to the fainter steeper class I proto- stars we plot the steep slope of the 24→ 70µm MIPS SED of #57025, which lacks detection in IRAC bands (it is placed on this plot using the 5.8µm upperlimit) and which corresponds to a previously known class 0 source that drives the HH-797 jet (§2.3.2). Five of these bright protostars had existing spectroscopy to which we have added seven new spectra (see also Ap- Spitzer census of IC 348 7 FIG. 6.— Completeness of the IC 348 class II census. A) H − K vs K color-magnitude diagram plotted for all class II candidates independent of Spitzer photometric quality. Symbols correspond to: class II (−0.5 > α3−8µm > −1.8) members with previously known spectral types (light filled circles); class II members with new spectra (dark filled circles); candidate class II members w/o spectral types (crosses); potential class II candidates filtered from this study due faint or poor Spitzer photometry (open circles); and those class II candidates rejected as members based upon further SED analysis (asterisks). Isochrones and the hydrogen burning limit are shown for 1 and 10 Myr from Baraffe et al. (1998); B) The completeness fraction from the ratio of H band luminosity functions. Solid line: the ratio of photometrically filtered class II candidates to all class II candidates; Dashed line: the ratio of class II sources with spectral types to all class II candidates. See text. pendix B). Seven of these twelve IC 348 protostars have M type spectra, ranging from M0 for the luminous IR source first identified by Strom et al. (1974, (our source #13)) to the newly typed faint M6 source #30003, which is enshrouded in a scattered light cavity that can been seen in HST/optical (Luhman et al. 2005b), near-IR (M03) and Spitzer 4.5µm im- ages. Spectral types were not measurable for the other five sources because no absorption features were detected in the infrared. New featureless infrared spectra of four of these sources are shown in Figure 8; a spectrum of the fifth, #51, appeared in Luhman et al. (1998b). Based on their mid-IR SEDs, the featureless nature of their near-IR spectra is prob- ably due to veiling by continuum emission from circumstel- lar material (Casali & Matthews 1992; Greene & Lada 1996) though these spectra do not exclude the possibility that they are embedded, early type (thus hotter) YSOs. Much hotter YSOs (corresponding to A or B type) are excluded because we do not observed the characteristics of massive protostars, namely, very large bolometric luminosities (> 100 L/L�), hydrogen absorption lines and/or evidence of embedded H II regions. Given the presence of hydrogen emission lines in a number of the objects and their proximity to other class I ob- jects and mm cores, it is very likely they are low mass mem- bers of IC 348 rather than massive members or background sources. Three additional bright class I candidates are phys- ically associated with molecular cloud cores (54460, 54459, 54362) but lack spectra. All three of these sources have MIPS SEDs consistent with significant reprocessing of their emer- gent flux by cold envelopes, and Tafalla et al. (2006) recently identified a molecular outflow associated with #54362. We tabulated all of these bright sources as protostellar members of IC 348; again, sources with spectra are listed in Table 1; those without are in Table 2. Additionally, we reclassified three α3−8µm selected class II sources as protostellar based upon their 24 − 70µm SEDs. Sources #1898, 54361 and #55419, which all appeared as nebulous blobs in the near-IR images of M03, appear as point sources in Spitzer data, have class II IRAC SED slopes yet have sharply rising MIPS SEDs (See Figure 7). Source #1898 is infact the brightest far-IR source in IC 348 (fluxes of ∼ 10 Jy @70µm and ∼ 60 Jy @160µm) and is almost cer- tainly a newly identified protostellar member (see also §3.3.3; Figure 18a). The slope of its 70/160µm SED is 1.2 compared to 1.1 and 2.4, respectively for the HH-797 and HH-211 class 0 sources. Were it not for its detection in scattered light in the near-IR and that a strong molecular outflow has not (yet) been found, its far-IR SED would suggest that it is also a class 0 source. Source #54361 is a point source from 3.6 to 24µm but is blended with #54362 at 70µm; the 70µm emission is elongated N-S, peaks right between #54361 and #54362 and cannot be ascribed confidently to either. This source also ap- pears to lie along the axis of the #54362 molecular outflow found recently by Tafalla et al. (2006), who suggested that #54361 may be a bright 24µm but unresolved knot of shocked gas instead of a young embedded star. Source #55419 also ap- pears to be blended with parts of the HH-211 outflow and is detected at 70µm. Additional spectroscopic data may clarify these latter 2 candidates’ true nature; in this work we have included them as candidate class I sources. 2.3.1. Low luminosity protostellar candidates Finally, we noted an interesting trend in Figure 1b: most of the class II candidates are bright, while most of the class I candidates are very faint. Although this low luminosity range has a high likelihood of galaxy contamination, it is important 8 Muench et al. FIG. 7.— Bright IC 348 protostars. Most (15) of these sources were initially selected using α3−8µm > −0.5 and three were added where the IRAC SED is contaminated by shock or scattered light emission (e.g., 54419 or 1898) and/or the MIPS SED appears protostellar (#54361). The steep 24 − 70µm SED of source #57025, the apparent class 0 driving source for HH-797, is shown for comparison. Sources are ordered by decreasing 5.8µm flux. Plotting symbols, line thickness and line color alternate from SED to SED for clarity. FIG. 8.— SpeX near-IR spectra of candidate members of IC 348 that show no detectable photospheric absorption features; these data have a resolution of R = 100 and are normalized at 2µm. We consider sources 245, 1872, 1889 and 1916, to be protostellar candidates based on their broadband SEDs. Source #1933, while featureless in the IR, is an accreting (Hα ∼55Å) K5 class II (α3−8µm = −0.57)member, which we were able to type using optical spectra (§B.3; see Figures 28 and 27)). to investigate these faint candidates to search for low lumi- nosity young stars that would be missed by the flux limits suggested by Jørgensen et al. (2006) and co-workers. We be- gan our exploration of these sources by plotting in Figure 9 the SEDs of the 41 faint α3−8µm > −0.5 protostellar candi- dates, sorting them by 5.8µm magnitude (or its upper limit). The ensemble population is clearly dominated by a class of objects with non-power law SEDs, which was a fact previ- ously evident in the poor quality of many of the class I SED power-law fits (Figure 2). Many have stellar-like continuum out to 5 microns with sharply inflected and rising SEDs be- yond. Such a SED feature can be ascribed to PAH emission at 6 and 8 microns, which appear in galaxies and evolved stars (Jura et al. 2006). To substantiate this point, we obtained Keck NIRC (Matthews & Soifer 1994) H K spectra of 4 of these red low luminosity sources. Two of these targets have monotoni- cally rising Spitzer SEDs, while 2 have sharp 8µm inflections. These spectra, which were obtained on 23 November 2004, are shown in Figure 10. The fact that these sources are not very red (especially compared to those spectra in Figure 8) indicates they are not class I objects, and the lack of steam indicates they are not brown dwarfs. They are probably all galaxies. Source #52839 is almost certainly a galaxy based on its emission lines, which do not correspond to rest-frame wavelengths of any lines typical of young stars. We chose to exclude all sources with PAH or similar fea- tures from our census of faint YSOs. To identify the best YSO candidates out of these faint sources and exclude PAH rich sources, we compared the monochromatic flux ratios 4.5/3.6µm and 8.0/5.8µm of these faint candidate YSOs to these flux ratios for the brighter protostars none of which show obvious 6− 8µm PAH emission (Figure 11). Flat spec- trum sources are located at (1,1) and sources with strong sil- icate absorption fall into the upper left quadrant. We traced a box around the locations of the brighter protostellar candi- dates in this diagram and chose the 11 faint candidates within it as additional protostellar candidates. This box excluded the four Keck sources whose spectra are clearly not those of YSOs. Some of these 11 low-luminosity class I candidates are more likely to be young stars than others. Two sources in particular have stellar (or sub-stellar) spectral features (#622, M6; #746, M5) and two sources are close companions to bright class I sources (#55400 and 40150). Source #4011 lies in the center of narrow dark lane/shadow clearly seen in the Spitzer census of IC 348 9 FIG. 9.— Observed spectral energy distributions of faint (m5.8 > 12.5) candidate protostellar (α3−8µm > −0.5) objects ordered by decreasing 5.8µm flux. Clearly a mix of source types is present at these faint magnitudes and PAH rich sources, which are identified by the strongly inflected 8µm SED point (e.g., #40010), are excluded from our study (§2.3.1). The near-IR spectra of monotonically increasing SEDs like #52839 indicate these are also likely extra-galactic interlopers (Figure 10). Plotting symbols, line thickness and line color alternate from SED to SED for clarity. Sources distinguished as good candidate low- luminosity protostars are marked with (*) next to their id (see text). infrared images of Muench et al. (2003). This strongly sug- gests it is a young star-disk system seen nearly edge-on, which is reinforced by the presence of a jet (HH799) that was ob- served and associated with this source by Walawender et al. (2006). However, the presence of an edge-on disk could cause a class II member to appear as a class I source (Chiang & Goldreich 1999); thus, the exact evolutionary stage of these young stars is unclear. Edge-on geometries also cause sources to appear subluminous on the HR diagram due to the fact the optical/near-IR flux is likely scattered light, which leads to low measured values of extinction, while the mid-IR flux is still quenched by the disk extinction. Both #622 and #746 are, for example, subluminous on the HR diagram. Note, if the dust in the disk is grey, the reddening vector(s) in Figure 1 are vertical and the basic IRAC SED classification remains nearly unchanged 13. Although these faint class I candidates are spatially cor- related near dark cores, others lie far from the molecular cloud, including candidates #1517 and 21799 to the NW. These are likely extragalactic contaminants despite their con- vincing SEDs; for completeness, all of the protostellar candi- dates lacking spectral confirmation are given Table 2. In total we find the accuracy of α3−8µm for uniquely selecting class I sources is quite low (less than ∼ 50%). While the application of additional selection criteria (flux limits) like those used by Jørgensen et al. (2006) and we used in Figure 11 can improve 13 See also the SED dependence of nearly-edge on disks and dust settling in D’Alessio et al. (1999). FIG. 10.— Keck NIRC spectroscopy of candidate low-luminosity proto- stars (α3−8µm > −0.5). Despite their rising mid-IR SEDs (Figure 9), these sources are not intrinsically red nor do they show absorption features typical of brown dwarfs. They are likely galaxies; source #52839 displays emission lines that do not correspond to typical transitions observed in young stars. 10 Muench et al. FIG. 11.— Parsing faint YSOs from 6− 8µm PAH emission sources using IRAC flux ratios. Flat spectrum flux ratios are marked with light dashed lines. The locations of bright protostars in this diagram were used to select the best candidate low luminosity protostars. The selection box is traced by heavy dashed lines and includes sources in the upper left quadrant. This acts to exclude sources with obvious PAH emission and include sources with silicate absorption. The SEDs of these 11 candidates are tagged with (*) in Figure 9. Sources with 5.8µm upperlimits are marked with arrows. the accuracy of a class I census, the reality is that galaxies and PAH sources masquerading as protostars dominate the statis- tics even for this nearby young cluster and follow up spec- troscopy is clearly needed to confirm low luminosity proto- stellar candidates. 2.3.2. MIPS survey of dark cloud cores near IC 348 To identify the most embedded protostars we examined our 24, 70 and 160µm MIPS images of our IRAC survey region and cross-correlated our Spitzer source list with a composite catalog (Table 6) of millimeter (mm) and sub-mm dark cloud cores near to the IC 348 nebula. Our dark core list was cat- aloged from and contains cross-references to a number of re- cent mm-wave studies of the Perseus Molecular Cloud; it is similar to but encompasses a larger area than one presented in Walawender et al. (2006). Hatchell et al. (2005) performed a Submillimetre Common-User Bolometer Array (SCUBA, Holland et al. 1999) survey that identified 15 unique dust con- tinuum peaks near the IC 348 nebula, lying mostly in a molec- ular ridge south of the cluster center. While also surveying the entire Perseus cloud at 1.1mm, Enoch et al. (2006) found 21 compact sources within our Spitzer survey region. Finally, Kirk et al. (2006) produced a archival based SCUBA mosaic of the entire Perseus cloud for the COMPLETE project and these data are publicly available on their website. For source extraction Kirk et al. used a single conservative threshold for identifying sources and recovered only some of the Hatchell et al. SCUBA sources. Yet all of the Hatchell et al. sources and most of the 1.1mm bolometer objects are clearly detected in the COMPLETE SCUBA images. Our MIPS observations of these cores were obtained with the camera in scan mode operating at medium scan rate and covering a total area of 30’ by 30’ common to all three de- tector arrays. The map consisted of 12 scan legs; half-array cross-scan offsets were employed to ensure full sky coverage at 160µm and on side “A” of the 70µm array. The total ef- fective exposure time per pixel was 80 seconds at 24µm, 40 seconds at 70µm, and 8 seconds at 160µm. The data were reduced and mosaicked using the MIPS instrument team Data Analysis Tool (Gordon et al. 2005). Coaddition and mosaick- ing of individual frames included applying distortion correc- tions and cosmic ray rejection. The 70 and 160µm frames were further processed by applying a time filter on each scan leg in order to ameliorate time-dependent transient effects such as source and stimulator latency and readout-dependent drifts. We used IRAF and the DAOPHOT package to perform point-source photometry; specifically, at 24µm, we employed PSF fitting with an empirical PSF with a 5.6” fit radius and 15-22.5” sky annulus. For the 70 and 160µm data we used aperture photometry with beamsizes of 9” and 30” and sky annuli of 9-20” and 32-56”, respectively. We applied aper- ture corrections at all wavelengths as derived from STinyTim PSF models (Engelbracht 2006). No color corrections were applied. Typical measurement uncertainties are ∼ 5 − 10% at 24 µm and 10-20% at 70 and 160 µm (though there may be somewhat larger systematic uncertainties at 160 µm be- cause of uncorrectable saturation effects). The sensitivity at the latter two channels is limited by the very bright thermal emission from the molecular cloud environs, and varies sig- nificantly with spatial position. Only four sources are confi- dently detected at 160µm. In these 26 IC 348 dark cores we found only two MIPS sources which lacked detections shortward of 8µm and, thus, were not already identified as YSOs using α3−8µm. These two MIPS-only sources corresponded to the previously iden- tified driving sources of two outflows traced by Herbig-Haro objects: HH-211 (McCaughrean et al. 1994) and HH-797 (Walawender et al. 2005). The HH-211 source appears only at 70 micron, which is the position we recorded in Table 6. The HH-797 jet was originally detected in molecular hydro- gen by McCaughrean et al. (1994) and Eislöffel et al. (2003). Eislöffel et al. discovered the 1.2 mm counterpart to the HH- 797 driving source, naming it IC348-mm while Tafalla et al. (2006) identified a strong molecular outflow correlating with the HH objects. The apparent driving source appears first at 24µm and corresponds to source #57025 in our numbering system; the position we tabulated corresponds to the 24µm source. Both of these sources have been previously char- acterized as class 0 sources (Eislöffel et al. 2003; Froebrich 2005)14. Including these class 0 sources #57025 and HH-211, we tally 20 bright protostellar members of IC 348 as well as 11 fainter candidates. Five other dark cloud cores contained sources we classi- fied as protostellar based upon their 3 − 8µm SEDs. These protostars are in systems of 1-3 bright members and we are confident of their association with these cores (also §3.3 and Table 6). Thus, 19 of our composite list of 26 mm sources in our IC 348 Spitzer region appear to be starless. To permit fu- ture SED analysis we tabulated all the relevant photometry for these starless cores. Foremost we derived 95% Spitzer upper- limits in the three MIPS bandpasses (Table 7). Since not all these sources were photometered in Kirk et al. (2006) we also derived aperture 850µm fluxes (or their upper limits) for all 26 cloud cores in the SCUBA mosaic15. Given the crowded nature of these sources and the varying background emission, we tabulated SCUBA fluxes at different aperture beamsizes 14 The definition of embedded protostars was expanded by Andre et al. (1993) to include so called “class 0” sources, whose original definition in- cluded: 1) little or no flux shortward of 10µm, 2) a spectral energy distribu- tion peaking in the sub-mm regime and characterized by a single black body temperature, and the somewhat less observable but more physical criteria 3) Menv > M∗. Their detection only at λ > 20µm appear to support this original definition. 15 A sub-region SCUBA map of the IC 348 region was provided by J. Di Francesco, private communication; it had a pixel resolution of 3′′ compared to the 6′′ COMPLETE map. Spitzer census of IC 348 11 TABLE 4 Spitzer 3− 24µM DATA FOR NEW IC 348 MEMBERS ID Magnitudes Uncertaintiesa 3.6µm 4.5µm 5.8µm 8.0µm 24µm 3.6µm 4.5µm 5.8µm 8.0µm 24µm 70 9.93 9.58 9.28 8.59 5.40 0.02 0.01 0.02 0.03 0.03 117 10.87 10.27 9.99 9.27 6.40 0.01 0.02 0.03 0.06 0.03 132 11.06 10.64 10.16 9.61 6.25 0.01 0.05 0.01 0.04 0.03 162 11.26 10.86 10.32 9.51 6.81 0.02 0.02 0.03 0.03 0.03 179 11.35 10.98 10.47 9.60 7.19 0.02 0.05 0.03 0.05 0.03 199 11.96 11.59 11.16 10.54 7.79 0.03 0.05 0.06 0.03 0.05 215 11.08 10.73 10.28 9.62 5.91 0.01 0.02 0.03 0.02 0.03 231 11.82 11.28 10.89 10.02 6.66 0.01 0.06 0.05 0.02 0.03 234 11.57 10.94 10.30 9.46 6.13 0.02 0.03 0.05 0.03 0.03 245 10.05 9.02 8.15 7.18 2.40 0.01 0.03 0.03 0.01 0.03 265 11.00 10.43 9.85 9.15 4.59 0.02 0.06 0.04 0.02 0.03 280 12.22 11.99 11.69 11.07 5.87 0.01 0.01 0.05 0.04 -9.00 321 12.70 12.45 12.12 11.40 5.61 0.01 0.03 0.07 0.08 -9.00 327 12.58 12.24 11.79 11.11 8.60 0.01 0.02 0.05 0.05 0.04 364 12.04 11.66 11.04 10.16 6.46 0.01 0.04 0.03 0.09 0.03 368 12.74 12.32 12.01 11.35 7.11 0.01 0.02 0.08 0.08 0.06 406 13.15 12.81 12.58 12.00 9.09 0.01 0.02 0.06 0.14 0.04 643 13.19 12.87 12.25 11.54 8.86 0.01 0.02 0.03 0.04 0.04 723 12.18 11.67 11.42 10.62 7.44 0.01 0.03 0.04 0.05 0.03 753 16.03 15.45 14.88 13.25 6.83 0.12 0.12 0.15 -9.00 -9.00 904 12.91 11.72 10.66 9.39 5.22 0.05 0.01 0.05 0.02 0.03 1287 13.40 12.82 12.26 11.82 7.98 0.02 0.10 0.07 0.05 0.04 1379 15.02 14.60 14.12 13.65 8.35 0.03 0.04 0.04 0.10 -9.00 1401 15.07 14.43 13.53 12.89 8.13 0.02 0.04 0.06 0.13 0.13 1517 15.35 14.32 13.31 12.54 9.04 0.04 0.03 0.11 0.08 0.04 1679 11.30 11.07 10.74 10.28 6.47 0.01 0.02 0.03 0.03 0.03 1683 12.31 12.04 11.62 10.85 7.55 0.02 0.02 0.07 0.04 0.03 1707 13.04 12.59 12.29 11.50 8.68 0.02 0.02 0.06 0.04 0.04 1761 12.77 12.54 12.13 11.60 9.04 0.01 0.03 0.05 0.04 0.05 1833 12.10 11.79 11.52 10.92 8.03 0.01 0.02 0.04 0.04 0.03 1843 13.88 13.36 12.76 11.97 5.04 0.03 0.05 0.08 0.06 -9.00 1872 7.78 6.67 5.84 4.92 1.26 0.01 0.00 0.05 0.02 0.03 1881 10.99 10.75 10.50 9.97 6.44 0.02 0.03 0.06 0.04 0.03 1889 9.76 8.78 8.06 7.26 3.48 0.02 0.03 0.03 0.03 0.03 1890 12.11 11.86 11.55 11.10 8.63 0.01 0.03 0.04 0.04 0.04 1898 12.46 11.28 10.72 10.70 4.57 0.12 0.11 0.17 0.29 0.04 1905 9.30 8.98 8.78 8.25 5.75 0.02 0.03 0.04 0.03 0.03 1916 10.87 9.75 8.97 8.61 6.40 0.03 0.02 0.02 0.04 0.03 1923 13.11 12.54 12.00 11.42 7.62 0.02 0.02 0.06 0.05 0.04 1925 12.66 12.09 11.50 10.87 7.52 0.02 0.03 0.05 0.04 0.03 1933 8.09 7.42 6.96 6.06 3.26 0.01 0.01 0.03 0.02 0.03 4011 14.61 14.00 13.30 12.28 7.60 0.04 0.03 0.10 0.06 0.03 10031 12.59 12.39 12.01 11.30 8.26 0.01 0.04 0.03 0.03 0.04 10120 11.99 11.71 11.41 11.00 8.04 0.02 0.02 0.04 0.04 0.04 10176 13.36 13.01 12.63 12.10 7.59 0.02 0.03 0.06 0.04 -9.00 10219 11.28 10.96 10.69 9.87 6.98 0.01 0.01 0.05 0.01 0.03 10305 14.33 13.82 13.55 12.89 7.30 0.03 0.02 0.16 0.15 -9.00 21799 16.51 15.79 14.71 13.64 8.37 0.06 0.13 0.20 0.20 -9.00 22232 12.05 11.69 11.34 10.38 7.67 0.02 0.02 0.10 0.03 0.04 22865 15.40 15.03 14.48 13.60 9.86 0.02 0.05 0.05 -9.00 -9.00 22903 15.37 14.48 13.24 12.41 9.03 0.04 0.03 0.08 0.03 0.04 30003 13.99 13.08 12.36 11.11 8.44 0.06 0.03 0.10 0.07 0.12 40150 14.39 13.73 12.98 11.91 9.14 0.03 0.04 0.04 0.07 0.06 40182 14.37 13.72 13.29 12.75 9.34 0.12 0.03 0.04 0.13 -9.00 52590 16.26 15.13 14.12 13.20 6.61 0.05 0.04 0.05 0.18 -9.00 52648 16.84 15.61 15.08 13.76 8.47 0.04 0.07 0.09 0.17 -9.00 54299 13.59 12.85 12.32 11.82 5.66 0.02 0.02 0.05 0.10 -9.00 54361 10.88 9.55 9.17 8.97 5.69 0.03 0.02 0.06 0.03 0.03 54362 14.25 12.87 11.87 10.91 5.12 0.07 0.08 0.05 0.04 0.03 54419 12.84 10.46 10.70 10.50 5.49 0.09 0.05 0.08 0.06 0.04 54459 14.31 12.54 11.41 10.09 4.60 0.14 0.06 0.07 0.04 0.03 54460 13.52 12.08 11.04 10.16 5.32 0.04 0.06 0.07 0.04 0.03 55308 12.01 11.64 11.51 10.81 7.87 0.02 0.02 0.04 0.03 0.03 55400 15.22 14.17 13.27 12.37 8.13 0.10 0.04 0.13 0.07 0.06 57025 · · · · · · · · · · · · 7.10 · · · · · · · · · · · · 0.04 HH-211 · · · · · · · · · · · · 7.24 · · · · · · · · · · · · -9.00 a The listed magnitude is an upper limit if the listed uncertainty is given as -9. 12 Muench et al. TABLE 5 FAR-IR & SUBMM FLUX DENSITIES FOR IC 348 PROTOSTARS ID MIPS 70µm MIPS 160µm SCUBA 850µmc Comments/ fluxa uncb flux unc f20′′ f40′′ S40′′ Blend ID 13 2.601 0.368 14.492 -9.000 0.096 · · · · · · MMP-10 51 3.944 0.453 19.126 -9.000 0.051 0.097 0.012 75 5.078 -9.000 · · · · · · 0.009 · · · · · · nebula 234 · · · · · · · · · · · · 0.110 · · · · · · 245 0.945 0.262 · · · · · · 0.049 · · · · · · 276 1.778 -9.000 · · · · · · 0.031 · · · · · · 435 · · · · · · · · · · · · 0.006 · · · · · · nebula 622 · · · · · · · · · · · · 0.014 · · · · · · nebula 746 · · · · · · · · · · · · 0.012 · · · · · · 904 1.144 -9.000 · · · · · · 0.037 · · · · · · 1401 · · · · · · · · · · · · 0.007 · · · · · · 1517 · · · · · · · · · · · · · · · · · · · · · Off SCUBA. 1872 9.583 0.705 61.196 2.650 0.249 0.633 0.624 1898 1889 0.730 0.191 14.902 -9.000 0.058 0.193 0.205 1898 9.583 0.705 61.196 2.650 0.249 0.633 0.624 1872 1916 0.248 -9.000 13.134 -9.000 0.034 0.116 0.055 4011 · · · · · · · · · · · · 0.110 · · · · · · 21799 · · · · · · · · · · · · · · · · · · · · · Off SCUBA. 22903 · · · · · · · · · · · · · · · · · · · · · Off SCUBA. 30003 1.292 -9.000 · · · · · · 0.186 · · · · · · 40150 · · · · · · · · · · · · 0.327 · · · · · · 57025 52590 · · · · · · · · · · · · 0.005 · · · · · · 52648 · · · · · · · · · · · · 0.051 · · · · · · 54361 2.320 0.337 26.142 -9.000 0.218 0.671 0.653 54362 54362 2.002 0.330 26.142 -9.000 0.218 0.671 0.653 54361 54419 0.452 0.207 · · · · · · 0.042 · · · · · · 54459 0.894 0.241 · · · · · · 0.160 · · · · · · 54460 54460 0.894 0.241 · · · · · · 0.147 · · · · · · 54459 55400 · · · · · · · · · · · · 0.085 · · · · · · 57025 3.428 0.426 19.839 3.508 0.408 0.985 1.201 40150 HH211 2.854 0.388 48.655 4.803 0.618 1.695 1.617 a All flux densities are in Janskys. b Sources with uncertainties equal to -9 correspond to 95% upperlimits. c Aperture flux derivation same as Table 7. Sources with SCUBA fluxes only in a 20′′ beam are 95% upperlimits. FIG. 12.— Spatial distribution of young stars in the IC 348 nebula. Our Spitzer survey is marked in each panel with a dashed box. A) class II vs III member surface density maps. The class III distribution is traced by filled grey contours, while the class II spatial distribution corresponds to red unfilled contours. In both cases contours start at 15 stars per square parsec and increase by 15 stars per square parsec. Subclusters reported by LL95 are plotted as crosses but do not correspond to actual clusterings of members except in 2-3 cases. B) class I protostars (§2.3) compared to IC 348 ensemble membership (light filled symbols). Bright protostars (filled diamonds), as well as many faint (m5.8 > 12.5) filtered candidates (open diamonds), are highly concentrated in a ridge ∼ 1 pc SW of the IC 348 core and anti-correlated with the central concentration of class II/III members. See also Figure 17a. Spitzer census of IC 348 13 TABLE 6 MERGED CATALOG OF MILLIMETER CORES IN IC 348 ID α(J2000) δ(J2000) f p Other IDs Associated Comments (a) (b) Protostars (c) MMS-01 3:44:43.7 32:01:32.3 3 H05-14, Bolo116, 1898, Peak on K034471+32015, 1872, 1872/1898 SMM-07 234 MMS-02 3:44:21.4 31:59:20.3 4 Bolo113, 1889 Spitzer src. offset N. SMM-14 MMS-03 3:44:12.8 32:01:37.0 4 SMM-17 51 No Bolocam src. MMS-04 3:44:05.0 32:00:27.7 2 Bolo109 1916 SCUBA peak 25′′ N. MMS-05 3:43:56.5 32:00:50.0 1 H05-12, Bolo103, HH-211 K034393+32008 SMM-01 MMS-06 3:43:57.2 32:03:01.8 3 H05-13, Bolo104, 57025, Spitzer src. K034395+32030, 40150 offset SW. IC348-mm, HH-797 SMM-02 MMS-07 3:43:50.8 32:03:24.0 1 H05-15, Bolo102, 54362, Peak on 54362. K034383+32034 54361 SMM-03 MMP-01 3:45:16.8 32:04:46.4 4 Bolo119 Starless Near IRAS 03422+3156 MMP-02 3:44:56.0 32:00:31.3 2 Bolo118 Starless No SCUBA pt. src. MMP-03 3:44:48.8 32:00:29.5 2 H05-25, Bolo117 Starless 24µm abs. SMM-12 MMP-04 3:44:36.8 31:58:49.0 1 H05-19, Bolo115, Starless 24µm abs. K034460+31587 SMM-11 MMP-05 3:44:14.1 31:57:57.0 4 Bolo111 Starless 24µm abs. SMM-15 MMP-06 3:44:06.0 32:02:14.0 4 H05-22, Bolo110, Starless SCUBA peak; K034410+32022 24µm abs. SMM-09 MMP-07 3:44:05.4 32:01:50.0 4 H05-20, Bolo110 Starless SCUBA peak; 24µm abs. MMP-08 3:44:02.8 32:02:30.5 3 H05-18, Bolo107, Starless SCUBA peak; K034405+32024 24µm abs. SMM-06 MMP-09 3:44:02.3 32:02:48.0 1 H05-21, Bolo107 Starless SCUBA peak MMP-10 3:44:01.3 32:02:00.8 3 H05-16, Bolo106, Starless 160µm src? K034401+32019 SMM-05 MMP-11 3:44:02.3 32:04:57.3 2 Bolo108 Starless No SCUBA pt. src. MMP-12 3:43:57.7 32:04:01.6 3 H05-17, Bolo105, Starless K034395+32040 SMM-08 MMP-13 3:43:45.6 32:01:45.1 2 Bolo101 Starless 24µm abs. MMP-14 3:43:43.7 32:02:53.0 4 H05-26, Bolo100, Starless SCUBA peak; K034373+32028 24µm abs. SMM-04 MMP-15 3:43:42.5 32:03:23.0 1 H05-24, Bolo100 Starless SCUBA peak; 24µm abs. MMP-16 3:43:38.0 32:03:09.0 4 H05-23, Bolo099, Starless 24µm abs. K034363+32031 SMM-10 MMP-17 3:44:23.1 32:10:01.1 4 Bolo114 Starless MMP-18 3:44:15.5 32:09:13.1 4 Bolo112 Starless MMP-19 3:43:45.8 32:03:10.4 3 Bolo100, Starless SCUBA Peak; K34346+32032 24µm abs. (a) Origin of positions (#): (1) Hatchell et al. (2005); (2) Enoch et al. (2006); (3) Kirk et al. (2006); (4) Closed Contour SCUBA, this paper. (b) Origin of acronyms []: [H]: Hatchell et al. (2005); [Bolo]: Enoch et al. (2006); [K]: Kirk et al. (2006); [SMM]: Walawender et al. (2006) c Comments include the existence of a 24µm absorption feature and whether we agreed with Hatchell et al. (2005) that a distinct SCUBA peak is present. These two criteria frequently agreed. and corrected for the varying nebular emission by subtracting a sky or pedestal value. In general the central positions of these aperture fluxes and upperlimits (and listed in Table 6) come from the better resolution SCUBA data. In some cases the actual closed contour peaks in the SCUBA survey were much better correlated to what appear to be absorption fea- tures in the 24µm nebulosity or to individual Spitzer sources than those positions previously published. If the published positions appeared to us to be inaccurate then we used either the location of the closed contour SCUBA peak or the minima of the 24µm absorption features. 3. ANALYSIS 3.1. Spatial distribution of members From our Spitzer census we have identified 42 new class II members of IC 348 and a population of ∼ 30 candidate class 14 Muench et al. FIG. 13.— Comparison of gas, dust and stars in the IC 348 nebula. North is up and East is left in all three panels. A) Locations of class II (red circles) and class I protostars (yellow crosses) compared to FCRAO 13CO data (COMPLETE dataset Ridge et al. 2006); B) false color IRAC & MIPS image of IC 348: red (24µm); green (8µm) and blue (4.5µm); C) class 0/I protostars (yellow crosses) are compared to the merged list of millimeter cores (Table 6) against the MIPS 24µm image (inverse greyscale ). TABLE 7 FAR-IR/SUBMM FLUX DENSITIES AND UPPERLIMITS FOR IC 348 STARLESS CORES ID Spitzer MIPS(a) SCUBA 850µm(b) 24µm 70µm 160µm f20′′ f40′′ S40′′ MMP-01 0.0007 0.209 6.234 0.072 0.227 0.198 MMP-02(c) 0.0008 0.393 9.285 0.034 0.115 0.119 MMP-03 0.0017 0.360 17.199 0.095 0.284 0.283 MMP-04 0.0009 0.162 18.773 0.107 0.287 0.303 MMP-05 0.0010 0.471 12.846 0.066 0.218 0.191 MMP-06 0.0016 1.726 23.118 0.150 0.486 0.402 MMP-07 0.0020 0.783 31.037 0.119 0.399 0.361 MMP-08 0.0012 0.631 22.191 0.137 0.462 0.510 MMP-09 0.0026 0.562 22.305 0.117 0.393 0.502 MMP-10 0.0190 2.326 14.492 0.164 0.536 0.592 MMP-11(c) 0.0328 2.251 20.301 0.042 0.158 0.131 MMP-12 0.0024 1.431 20.300 0.139 0.459 0.450 MMP-13 0.0005 0.521 17.214 0.087 0.298 0.156 MMP-14 0.0010 0.424 17.826 0.092 0.291 0.382 MMP-15 0.0036 0.350 12.689 0.112 0.393 0.368 MMP-16 0.0013 0.504 23.390 0.124 0.394 0.320 MMP-17 0.0300 9.158 66.411 0.050 0.164 0.165 MMP-18 0.0024 2.776 24.274 0.070 0.212 0.151 MMP-19 0.0060 · · · · · · 0.097 0.377 0.401 (a) All Spitzer flux density upperlimits are given in Jy. Central source fluxes for cores with protostars are given in Table 5. (b) Aperture flux in a 20 or 40′′ beam on the COMPLETE SCUBA 850µm Perseus image. We corrected for the non-uniform nebular emission, which includes a pedestal flux contribution or bowl-outs due to sky chopping, by subtracting a “sky” based on the mode of the pixel values in an annulus from 120 − 140′′. For comparison the last column labeled “S40′′” is the simple sum of the pixels in a 40 aperture without correction for the non-uniform background emission. The conversion from Jy/beam to Jy was 0.0802. (c) No SCUBA source is evident at the 1.1mm bolometer position; SCUBA flux given should be considered an upperlimit in that aper- ture. 0/I protostars of which we are confident in the membership and evolutionary status of ∼ 20. This section explores the spatial distribution of the cluster’s class I, II and III members. Figures 12a compares the surface density maps of all (new and old) class II and III members; these maps were created by convolving the members’ positions with a 0.2 pc box filter (∼ 2′). Note, class III source statistics are formally complete only in the region bounded by the Luhman et al. (2003b) sur- vey16. Interestingly, the locations of the class II and class III surface density peaks are essentially identical; we derive the same result when we directly calculated the median spa- tial centroids for each population17. This class II/III surface density peak corresponds approximately to the location of the B5 star HD 281159 at the center of the nebula and the con- centration of members surrounding this peak in Figure 12a represents the centrally condensed IC 348 cluster core (Her- big 1998; Muench et al. 2003). Using a near-IR survey to derive the surface density distribution toward IC 348, LL95 found that the cluster appeared to be constructed of this core, their IC 348a, and eight smaller sub-clusters. At that time they did not have access to the refined cluster membership provided by subsequent surveys. Overplotting all nine of the LL95 sub-cluster centroids on our membership filtered map reveals that only two, or maybe three of them (a, b and pos- sibly e) represent significant cluster substructure; the rest are apparently background surface density fluctuations likely due to counting statistics and/or patchy line of sight extinction. On the other hand, Figure 12b reveals that the class 0/I pro- tostars have an entirely different spatial distribution. While there are a few class I sources projected toward the cluster’s class II/III center, most were found at the periphery, wrapping around the cluster from the east to the southwest. While many are widely spaced, a large concentration of IC 348 protostars lies∼ 1 pc S.W. of the nominal cluster center and where there is no corresponding surface density enhancement of class II or III members. Strom’s IC 348 IR source lies near the center of this region, which is also the apex of most of the Herbig-Haro jets found near IC 348, including HH-211, HH-797 and many new jets recently identified by Walawender et al. (2006). 3.2. Comparison of gas, mid-IR dust emission and young stars 16 Section 4.1 discusses our incompleteness for class III members in more detail; as described in Appendix C we searched X-ray catalogs for additional class III sources, finding 27. These candidates were included to the class III source list when creating Figure 12a, although their addition or removal have little impact on our subsequent conclusions about the cluster’s structure. 17 The median spatial centroids we derived are 03:44:30.053 +32:08:33.86 for class II members and 03:44:32.809 32:09:6.00 for class III; both J2000. Spitzer census of IC 348 15 FIG. 14.— Variation in reddening of members vs declination. There are increases in the dispersion of AV at the declination of the cluster core (DEC.=32.2◦) and along the southern protostellar ridge. There is a more sig- nificant segregation of reddened (25 > AV > 4) class II/III members other foreground cluster members. Values of AV are from SED fitting (see Paper I, §3.4 and Table 8). We further examined the spatial distribution of disk bearing IC 348 members (class 0/I/II) by comparing their locations to maps of the dust and gas emission from the associated Perseus molecular cloud. The most useful sets of such dust and gas maps come from the publicly available COMPLETE project, which were published by Ridge et al. (2006). Figure 13a com- pares the locations of the young stars with disks to the Perseus COMPLETE integrated 13CO gas map. Nearly all the disk bearing members of IC 348 are projected against dense molecular gas. The class II members are more concentrated centrally near a gas filament that stretches from the southwest to the northeast. Even though our survey area is large class II sources are not distributed uniformly; there are few class II members found to the northwest or southeast of this filament (hereafter termed the central filament). Only those LL95 subclusters that are projected against the central filament are confirmed by our analysis of known members (IC 348a, b to the south and perhaps e to the north). Un- confirmed LL95 sub-clusters are located off the central fila- ment and where the reddening of background stars is probably small and patchy. Moreover, the stars in the IC 348a cluster core appear to be associated physically with the central fila- ment. Plotting the AV of individual members versus declina- tion in Figure 14, we see that the bulk of the cluster is infront of the central filament and they have fairly constant and low reddennings – AV < 4. Near the cluster core (δ ∼ 32.15◦), however, AV varies much more, reaching fairly large extinc- tions (AV > 10) and indicating that the cluster’s core remains at least partially embedded in the central filament18. Finally, there is no evidence that class II members avoid the cluster center or prefer the cluster halo as was suggested previously in 2µm studies (LL95, M03). Protostars are projected against, and embedded presumably within another molecular CO filament that stretches west-to- east along the southern edge of the nebula. We term this 18 Further evidence for the semi-embedded nature of cluster core is found in the reddened nebulosity surrounding it in the near-IR color image M03 (Figure 1; print edition). the southern filament. Note that the integrated CO emission is somewhat misleading in this respect: the apparent 13CO bridge (see Figure 13a) connecting the central and southern filaments is at a completely different (strongly blue-shifted) radial velocity compared to either filament (Borkin et al. 2005); therefo, the central and southern filaments are infact distinct. These distinct filaments, however, share a common radial velocity to within 0.45 km/sec (vlsr,cl ∼ 8.15; vlsr,s ∼ 8.6; Borkin et al. 2005); thus, they are physically related. Fur- ther, there is significantly more contrast in the reddenings of foreground and embedded members along this southern fila- ment than in the cluster core. In Figure 14 we find that most members lie in front the southern filament (δ < 32.1◦) with small reddenings, AV ∼ 1 − 2 but the reddened members very embedded with AV > 5, ranging up to 25 magnitudes. This segregation of members by AV might be evidence of dis- tinct cluster populations that orginate in the two distinct gas filaments. Figure 13b reveals additional details about the IC 348 neb- ula, using the 4.5, 8.0, and 24µm images to trace blue (scat- tered light and/or shocked hydrogen), green (scattered light and/or PAH) and red (24 micron dust emission). The opti- cal portion of the nebula appears here as a blue-green cav- ity that surrounds the centrally condensed class II/III cluster core, providing more evidence that the central filament lies mostly but not far behind these stars. At the cluster’s periph- ery, on the other hand, the molecular gas contours in panel (a) are closely mirrored by emission or scattering traced by the Spitzer 8 and 24µm data in panel (b). Northwest of the cluster center, for example, one can see the way the Spitzer dust emission traces around the perimeter of a large low den- sity CO clump facing the central B star. We also find that the bright red nearly circular (r=0.13pc) 24µm emission mi- cron ring that surrounds the central B5 type binary has corre- sponding low level (30mJy/beam) SCUBA 850µm emission that forms small clumps around this ring. This red ring also correlates with a ring of red near-IR (K band) dust emission in the color Figure 1 of M03. Unfortunately, SCUBA submm maps do not include another less red r=0.1pc cavity that sur- rounds a pair of A type stars (#3 & 14) to the NE. In this color image the class I protostars are found in strings of bright red 24 micron sources behind the southern emission wall and in the dark cloud core at the heart of the southern filament. Comparing Figure 13(a) to panel (c) one can better see how MIPS 24µm emission etches out the edges of the southern molecular filament, wrapping around and into a 13CO cavity on the ridge’s southern edge. Many of the bumps and wig- gles in the CO gas contours have counterparts in the 24µm dust emission and the dark molecular ridge does not appear sharp edged as it would were it a foreground cloud. It is in- stead enveloped in and therefore immediately adjacent to (and we believe slightly behind) the nebula surrounding the cen- tral B star. Panel (c) includes symbols marking the locations of class I protostars and sub/millimeter dark cores (Table 6). While 24µm dust emission closely follows the surface of the cloud as traced by molecular line data, the mid-IR dust emis- sion does not closely follow the contours of the SCUBA dust emission. This could be due on the one hand to the spatially chopped nature of the SCUBA data, which acts to remove larger scale and spatially smoother sub-mm emission struc- tures. On the other hand the SCUBA (starless) cores are fre- quently seen in silhouette as 24µm absorption features (§3.3) against low level scattered mid-IR light that permeates this 16 Muench et al. FIG. 15.— Detailed view of the IC 348 SW protostellar ridge with Spitzer. The false color image was created using 8, 24 and 70µm Spitzer images. Contours (Jy/beam) from the COMPLETE SCUBA 850µm map (starting at 57 mJy/beam, ∼ 10σ ) are overlaid on this image. Note the five cases where the SCUBA data peak up on IC 348 protostars (also Figure 18). Near the eastern edge of the ridge there is the 70µm source #1898 paired with the 24µm source #1872 both lying under a strong SCUBA core. The color bar scale corresponds to MJy/str in the 70µm image. FIG. 16.— Detailed view of the IC 348 SW protostellar ridge with COMPLETE SCUBA (Kirk et al. 2006). Ten logarithmically spaced contours from 100 to 1000 mJy/beam are overlaid for emphasis. MM cores from Table 6 are shown as open (yellow) circles. Spitzer protostars (§2.3) are plotted as small (red) circles. The color bar scale corresponds to Jy/beam in the SCUBA 850µm image. filament. The source of that scattered light is not clear. In projection, two starless cores are seen much closer to the cen- tral B star and its clustering, but have neither emission nor absorption features in Spitzer data; at least one (MMP-18) is associated with an 1M� N2H+ core identified by Tafalla et al. (2006) . They appear to be hidden from the illuminating source of the mid-IR dust emission by the densest part of the central gas filament in which they reside and which reaches a peak reddening of AV ∼ 15 magnitudes in the extinction maps presented in M03. 3.3. The protostars of the southern filament 3.3.1. Spitzer & SCUBA correlations In this section we focus on the southern molecular ridge, containing most of the class 0/I protostellar objects; Figures 15 and 16 compare Spitzer and SCUBA images of this region. We used a sub-region of the COMPLETE SCUBA Perseus map created by Kirk et al. (2006) to compare the cloud’s dust continuum to our detected mid-IR point sources. In the IC 348 region of the Perseus SCUBA data we measured a mean and rms of 8.6 and 4.8mJy/beam, which were used to plot loga- rithmically spaced contours starting at 57 mJy/beam (10σ). As discussed in the previous section there are a large num- ber of starless SCUBA cores (Table 6) whose 850µm con- tours correspond precisely to dark 24µm absorption features. These SCUBA emission/Spitzer dark cores appear across this protostellar ridge and indicate to us that the registration er- ror for these SCUBA and Spitzer IC 348 comparisons is no more than 1-2 SCUBA pixels (3 − 6′′). We conclude simi- larly that the removal of large-scale (> 120′′) structures from the SCUBA map by Kirk et al. has had little affect on the spatial correlations of Spitzer and SCUBA point sources we discuss below. In this southern molecular ridge there are 23 identified pro- tostars and a comparable number (22) of MM cores of which 15 are starless. Low level dusty filaments stretch across the region, threading the various star forming sites; there are not, however, spatially distinct regions of star forming versus star- less cores. Unlike the spatially anti-correlated distributions of class I and class II sources (Figure 13a), starless and star forming cores are intermingled and the empty SCUBA cores are typically no further from the B star at the cluster cen- Spitzer census of IC 348 17 ter than are the protostars. While in Figure 15 the three strongest 70 micron point sources shine through associated SCUBA core peaks and correspond to class 0 sources, the fact is that most of the protostars are only peripherally associated with SCUBA cores (Figure 16). In all but 6 cases, the clos- est SCUBA peak is more that 3000 AU (10′′) from a proto- star and we conclude that these cores are neither the original “infalling” envelope nor “common” envelopes encompassing a set of protostars (Looney et al. 2000). The intermingled SCUBA cores are instead probabe sites of future star forma- tion. Moreover, six flat-spectrum protostars appear to be com- pletely disassociated from the dust continuum, having neither SCUBA nor 1.1 mm detections. If the remnant envelopes are small (. 1000 AU) or if we are seeing the envelope pole on, the integrated dust continuum might not have enough contrast to be detected in larger beam size of the sub-mm observa- tions19. It is interesting to note that most of the protostars in or adjacent to SCUBA cores appear in systems of only 1-3 members (at the Spitzer MIPS resolution limit of∼ 1000 AU). Protostars distant from mm cores actually appear essentially solitary (down to separations of ∼ 400 AU based on the near- IR data) and are typically separated from other protostars by > 20000 AU. 3.3.2. Clustering Following our studies of protostars in Orion (Lada et al. 2000, 2004), we convolved the position of sources in the SW ridge with a box kernel to create surface density maps and to identify and characterize any embedded sub-clusterings of protostars. Unlike Orion where the embedded subclus- ters have R < 0.05 pc, convolution with kernels less than 0.2 pc (130′′) produced no significant clumping in the pro- tostellar ridge; the youngest IC 348 sources are much more spread out. Figure 17a displays the surface density maps for class II, class I and MM objects in the ridge convolved using an 0.2 pc kernel. There is an apparent protostellar clustering whose peak coincidentally coincides with Strom’s IR source and reaches ∼ 200 stars · pc−2 , which is more than an order of magnitude lower then we found for the embedded subclus- ters in the molecular gas behind the Orion nebula. The class I and MM sources are correlated except for a small group of starless cores on the western edge of the ridge. These two sets of sources are, however, anti-correlated with the LL95 IC 348b subcluster of class II sources, which splits the ridge in half and achieves a nearly identical peak surface density. This class II subgroup is obvious in Figure 15 as the central group of bright 24µm sources lacking SCUBA emission. We also applied a nearest neighbor analysis to these ridge sources. Teixeira et al. (2006) examined the nearest neigh- bor distribution of bright class I sources in the embedded Spokes subcluster of NGC 2264, finding a preferred spac- ing of 27′′ or 0.1 pc (d=900pc). A nearest neighbor analy- sis for the class II, class I and MM cores in the SW ridge reveals no resolved, preferred spacings (Figure 17b) but in- 19 These sources are reminiscent of and may be similar to those nearby “peculiar” class I Taurus sources detected by but unresolved with single dish 1.3mm data in Motte & André (2001, see also discussion in White et al. (2006)). Whether these non-detections (or those unresolved Taurus detec- tions) rule out the existence of an envelope (so removing the protostellar moniker) can only be firmly determined using observations of the silicate fea- ture at 9.6µm coupled with detailed SED modeling (e.g. Eisner et al. 2005). Indeed such SED modeling by Eisner et al. of one these Motte & André (2001) “peculiar” class I objects, L1489, nonetheless prefers a disk+envelope structure; thus, for now we retained the SED based protostellar classification. FIG. 17.— Clustering in the IC 348 SW Ridge. A) Surface density map for class II, class I and MM cores convolved with an 130′′ (∼ 0.2pc) square kernel. Contours start at 3 objects/box (∼ 75 stars · pc−2) and increase in steps of 1 object per contour. The locations of the class I protostars are overplotted as filled circles. Note the correlated class I/MM core distribution and the anti-correlated class I/II distributions. B) Nearest neighbor analysis for objects (22 mm cores; 23 protostars; 33 class II YSOs) in the SW ridge. All three distributions rise to the resolution limit of the surveys, which is smaller than the peak of the randomized distribution. The turnover in the MM core spacings appears to be due to the effective resolution (∼ 15′′) of the SCUBA data. The unresolved class I peak for r < 20′′ corresponds to the small 2-3 member systems illustrated in Figure 18. stead all rise to the resolution limit. Unlike the Spokes, the spacings of protostars are mostly flat except for a peak at or below 20′′ (< 0.03pc; < 6000AU); this unresolved peak is sharpened by including starless cores as neighbors to the pro- tostars. Visually inspecting Figures 18(a-c) reveals the nature of this difference with NGC 2264. These small spacings come from a few protostellar systems of 1-3 members with small ∼ 1000 − 6000 AU separations, although the majority of the class I sources are essentially solitary and widely spaced. The class II spacing distribution also rises down to the resolution limit. 3.3.3. Near-Infrared Images In this section we use deep near-infrared images from M03 to illustrate some of these small 1-3 member protostellar sub- clusters. Figure 18(a-c) show three closeup views of the pro- tostellar ridge, progressing from east to west. The eastern MMS-1 core, illustrated in Figure 18a, contains three proto- stars, including the brightest far-IR source in the entire clus- ter. Although the SCUBA core peaks right between proto- stars #1898 and #1872, our comparison of the near-IR, Spitzer 24µm and 70µm images leads us to conclude that the 70µm source (and thus also the 160µm source) peaks up on the east- ern red K band knot (source #1898) rather than on the west- ern protostar, #1872, which is where the 24µm source peaks (scrutinize the color version of Figure 15). Source #234 could be either the tertiary member of a hierarchal triple or an en- tirely separate clump fragment. Unlike MMS-1 in Figure 18a, panel (b) shows how most of the protostars are unassociated with individual SCUBA cores. 18 Muench et al. FIG. 18.— Detailed near-infrared views of protostars in the IC 348 SW molecular ridge. Images from Muench et al. (2003)(their Figure 1; print edition); a 0.1pc yardstick (D = 320 pc) is shown in each panel; protostars are circled and labeled. SCUBA contours are the same as in Figure 15. Panel A) MMS-1 encloses three protostars but peaks on the red 70µm source #1898, which is seen as a scattered light cavity in the near-infrared. Source 1898 is separated by 4000 AU from the dominate 24µm source #1872, which has a featureless spectrum (Figure 8). Note how the near-IR dust emission traces the SCUBA dust continuum. Panel B) MMS-3 encloses source #51, and MMS-4, detected at 1.1mm by Enoch et al. (2006), encloses #1916, which has a featureless spectrum (Figure 8). Sources #276, 4011, 54459, 54460 and 55400 cannot be firmly associated with any dust continuum peaks, though they are all seen in scattered light. Panel C) SCUBA core MMS-7 peaks on # 53462, MMS-6 is offset slightly (1500au) to the NNE from the class 0 source, #57025 (IC348-mm) which is thought to drive HH-797 (seen in red continuum arching to the NNW). Embedded in scattered light, #245 also falls outside the dust continuum contours. The edge-on source #4011 and the trio of 54459/54460/55400 are simply adjacent to starless SCUBA cores. All of the pro- tostars in Figure 18b are seen in scattered light, including the rather solitary flat spectrum protostars #51 and #276 as the flat spectrum protostar # 245 panel (c). On the other hand, two other very good SCUBA/Spitzer/near-IR correla- tions are illustrated in panel (c). The class 0 #57025 and pro- tostar #54362 both appear almost precisely at their respective SCUBA closed contour peaks (within 1000 AU of MMS-6 & MMS-7, respectively). These comparisons reinforce our ar- guement that most of these SCUBA cores are infact starless. 3.4. Inferred cluster properties Considering the expanded borders of the IC 348 cluster traced by our Spitzer census, it is useful to ask how the ad- dition of new cluster members over a large physical scale might have modified global cluster properties such as the me- dian age or stellar initial mass function (IMF). In this sec- tion we derived bolometric luminosities for the new and old members and compared them to theoretical isochrones on the Hertzsprung-Russell (HR) diagram to answer this question. In this exercise all the sources were placed on the HR diagram by dereddening a single passband flux, using the AV derived from SED fitting (see Paper I), and applying a bolometric cor- rection (BC), which is tabulated as a function of effective tem- perature and taken from our previous studies. Other fixed val- ues or assumptions included a value of Mbol,� = 4.75, the use of a subgiant spectral type to Te f f scale from Luhman (1999), and a distance of 320 pc, which is the value we have assumed in all of our previous studies of IC 348 members. One subclass spectral type uncertainties were assumed and were propagated into the L/L� uncertainty, which was the quadratic sum of the 1σ photometric error, the χ2 AV fit un- certainty and the variation in BC as a function of Te f f . The χ2 AV fit uncertainty dominates the error budget of L/L� for each star. We actually derived L/L� at all passbands from V to K and found that these derivations are extremely self- consistent in the near-IR with essentially no variation between L/L� derived from the J or H bands though there was some evidence for K band excess producing slightly higher bolo- metric luminosities (typically, however,< 0.2 dex). Figure 19 presents HR diagrams for sets of members parsed spatially or according to their disk properties. Isochrones and evolution- ary sequences were taken from Baraffe et al. (1998)20. Figure 20 presents the inferred cluster properties based on these HR diagrams and theoretical tracks. Although they lie preferentially at the edges of previous spectroscopic census, the new, primarily class II sources iden- tified in our Spitzer census fall in the same basic locations on the HR diagram as previous members (Figure 19a); specifi- cally, they have a very similar spread in L/L� at fixed Te f f . This spread in L/L� at a fixed Te f f should represent a range of radii for stars of approximately the same stellar mass and should correspond to the spread in the birth times for contract- ing pre-main sequence stars. This L/L� spread is, however, convolved with a distribution of uncertainties, which in this case we find to be dominated by uncertainties in extinction es- timates, and the age of a particular star should be viewed with caution. The ensemble of cluster members ages may yield some clues about the cluster’s star forming history, so we quantified this luminosity (age) spread by counting sources between logarithmically spaced isochrones and plotting them in Figure 20a. In this way, for example, we can show that the addition of new class II sources does not appear to modify the star forming history inferred previously for IC 348. A spatially distinct population of protostars spread along the cluster’s periphery clearly suggests that star formation in IC 348 is not necessarily coeval and that the location of star formation may have varied with time across the nebula. We tested the hypothesis of spatial variations in the SFH for IC 348 members by examining radial variations of the clus- ter loci on the HR diagram (Figures 19cd) and the inferred SFHs (Figure 20bc). Radial variations of the apparent ages 20 No single set of Baraffe et al. models fit the locations of the GG Tau quadruple or the IC 348 locus on the HR diagram (most recently see Luh- man et al. 2003b, and references therein). As prescribed previously, we use a mixed set of Baraffe et al. (1998) models with different convective properties for different mass ranges: a mixing length parameter 1/Hp=1 for M < 0.6M� and 1/Hp=1.9 forM > 0.6M�. Thus, by design, our set of isochrones will yield a constant inferred mean age as a function of Te f f (M�). Spitzer census of IC 348 19 FIG. 19.— Hertzsprung-Russell diagrams. Only sources with spectral types and subject to our census constraints (three IRAC band detections) are included. A) New Spitzer members compared to the pre-existing IC 348 population. New members do not differ in their luminosity spread on the HR diagram but are preferentially later types. B) Those sources on the nebula’s southern edge (δ < 32.07) and projected toward the protostars are compared to the ensemble population. The sources along the southern protostellar ridge appear more luminous on average than the cluster ensemble. C) Radial dependence of class II sources in the HR diagram. D) Radial dependence of class III sources in the HR diagram. Isochrones plotted correspond to ages of 1, 2, 3, 5, 10, 100 Myr as ordered by decreasing luminosity. Evolutionary tracks plotted correspond to stars of 0.03, 0.072, 0.1, 0.13, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0 and 1.3M�as ordered by increasing Te f f . of IC 348 members were reported by Herbig (1998) yet our class II Spitzer survey is spatially complete over a much larger area than his Hα based survey. We divided the population into a r < 4′ core and a r > 4′ halo, which is approx- imately the same radial distinction used by or discussed in LL95, Herbig (1998) and M03; these two samples correspond to roughly equal proportions of cluster membership (40% and 60%, specifically). We find no significant radial differences in the spread of L/L� on the HR diagram or in the extracted SFHs of the spatially complete class II populations (Figure 20b); although spatially complete only in the core, we found no radial variation in the class III SFHs (Figure 20c) either. Infact, the class II and class III age distributions are essen- tially indistinguishable, displaying a peak at 2.5 My and an age spread of 4 My, which we derived using the half dpower points of the cluster ensemble age distribution. Even if star formation were a function of time and location in the nebula, the common heritage of stars inside and outside the cluster core means that the core is either a distinct and long lived star formation site or the merger of many smaller briefer star formation events whose initial spatial distribution no longer appears terribly obvious. Although our new members are preferentially cool stars and thus low mass (< 0.3M�; Figures 20ad) this does not appear to be the result of a bias in our survey. It is instead a conse- quence of the IMF, which peaks in IC 348 for low mass stars (Luhman et al. 2003b; Muench et al. 2003) coupled with an apparent radial variation in this mass function which skews to lower mass stars at large radii where most of our new class II sources are found. Besides the unclassifiable protostars we found only 1 new early type class II K star (#1933); the so- lar mass members at larger radii are either already included in our census and/or perhaps diskless. We found that the spa- tially complete class II population of IC 348 displays a modest radial variation in the distribution of effective temperatures, which we use as a proxy for mass. The hotter, higher mass sources are more concentrated in the cluster’s core and cooler lower mass stars prefer the cluster halo (Figure 20b). This result using the HR diagram supports the luminosity function analysis of M03 which first identified radial MF variations in this cluster. Further, the M03 IMF analysis is not biased for or against the presence of disk excess so that despite the incon- clusive HR diagram results for the spatially incomplete class III population (Figure 20c) we conclude that this MF radial skew is real. 4. DISCUSSION 4.1. Total young star population of the IC 348 nebula We have added a substantial contingent of new young stars to the membership of IC 348, bringing the total known mem- bership to 363 sources. This is larger than anticipated statis- tically by Cambrésy et al. (2006) using 2MASS all sky data. We now perform an estimate of the total young star popu- lation in IC 348, accounting statistically for undocumented 20 Muench et al. FIG. 20.— Distributions of IC 348 member properties as a function of cluster structure and disk evolutionary phase. Panels A-C plot the star forming histories and panels D-F plot the distribution of effective temperatures which we use a proxy for the mass function. All distribution functions were normalized by (individual) population size for these comparisons and panels A-C were also divided by the bin width in Myr and thus have units of fractional stars / Myr. Note that any star lying above the 1 Myr isochrone was placed into the first log age bin. TABLE 8 DERIVED PROPERTIES FOR IC 348 MEMBERS. ID Te f f AV a L/L�b SED Paramsc Sp.T. (K) Best Fit -1σ +1σ Best Fit -1σ +1σ α3−8µm 1σ 850µm 1 B5 15400 3.1 1.0 8.7 3.230 2.778 3.787 -2.638 0.102 0.005 2 A2 8970 3.2 1.1 8.5 2.067 1.774 2.656 -1.396 0.127 0.009 3 A0 9520 3.9 3.0 4.7 2.073 1.708 2.329 -2.794 0.110 0.000 4 F0 7200 2.3 1.0 7.3 1.614 1.435 2.168 -2.786 0.091 0.004 5 G8 5520 7.7 5.5 9.9 1.306 1.045 1.556 -1.389 0.160 0.014 6 G3 5830 3.5 1.9 7.4 1.209 1.010 1.645 -1.972 0.079 0.011 7 A0 9520 1.7 0.0 8.1 1.642 1.235 2.383 -2.788 0.071 0.009 8 A2 8970 1.6 0.0 8.0 1.509 1.272 2.234 -2.532 0.101 0.008 9 G8 5520 5.3 3.7 7.7 1.032 0.839 1.304 -2.894 0.134 0.012 10 F2 6890 2.1 0.8 7.0 1.153 0.976 1.705 -2.827 0.054 0.057 a The best fit AV from χ 2 fits and the lower and upper 1σ limits from SED fitting. b The derived log luminosity at J band derived at the best fit AV and the lower and upper 1 sigma AV values; for sources without spectral types, AV and L/L� estimates were derived assuming a K7 spectral type. c SED parameters: sources without 1σ fit uncertainties in α3−8µmwere detected in less than 3 Spitzer IRAC bands; in all cases these values correspond to 95% upperlimits at 850µm in a 20′′ beam. See Table 5 for SCUBA detections of protostars. class III members not identified using our disk based crite- ria. We first use the ratio of class II to class III sources in the Luhman et al. (2003b) completeness region (70 / 186 = 0.38) to extrapolate from our class II census. We find 90 class II members within the 10.′33 radius of the Muench et al. (2003) survey and, thus, we estimate there should be a population of 227 class III sources or a total population estimate of 327 members in a r ∼ 1pc region. Although this is consistent with but slightly larger than the 303 ± 28 members estimated by Muench et al. using a 2µm luminosity function analysis, it suggests that about 30 more class III members remain uncon- firmed in this r ∼ 1pc portion of the IC 348 nebula. Probably the most efficient way to find the 30 predicted class III sources would be to employ deep X-ray surveys; unfortunately, existing X-ray surveys are much smaller than our Spitzer survey, and only roughly cover the 20′ M03 re- gion. They also miss most of the protostellar ridge. A second means to identify young stars is by monitoring for periodic (variable) stars over large cluster areas. Combining archived X-ray data and recent literature results (e.g. Cieza & Baliber 2006) for these two techniques (see Appendix C), we cata- loged 27 class III candidates of which 17 fall within the M03 Spitzer census of IC 348 21 + Xray portion of our Spitzer survey region. Considering that both of these techniques have their own (different) complete- ness limits (less than 1/3rd of the confirmed IC 348 members are periodic while 2/3rds are detected in X-rays) these 17 class III candidates confirm our prediction of 30 missing class III members as accurate. Wider field X-ray surveys are clearly warranted, especially to elucidate the radial MF variation we observe for the class II members. On the larger 2.5pc spatial scale of our Spitzer survey, we conclude that the 118 known class II members suggest a to- tal population size perhaps as large as ∼ 420 IC 348 sources. This assumes that the ratio of class II/III members does not vary much over the survey area. Thus, we predict approxi- mately 60 class III sources remain either unidentified or lack- ing spectroscopic follow up within the immediate vicinity of IC 348. In total our findings (confirmed or extrapolated) rep- resent a substantial (30%) increase to the traditional popula- tion estimate of∼ 300 sources for IC 348 (Lada & Lada 1995; Muench et al. 2003; Lada & Lada 2003). It is, however, un- clear where the boundaries of this cluster are and thus where we should stop looking for missing members. The 2MASS surface density excess identified by Cambrésy et al. (2006) extends beyond the borders of our Spitzer survey (but appears to underestimate its membership); from a cursory analysis of archival Spitzer data21 it is quite clear that not far from IC 348 there are small aggregates of young stars, including those around LkHα 330 30′ to the NE and around MSX6C G160.2784-18.4216 (Kraemer et al. 2003; Cambrésy et al. 2006) 30′ to the SE, which may or may not be associated with the star formation we observe within the nebula. Even if we were to include all these groups and account for subsequent generations of stars yet to form in the protostellar ridge it is clear that in a physically similar volume of space, the IC 348 nebula will produce about an order of magnitude fewer stars than the Orion nebula. 4.2. Physical structure of the IC 348 star cluster Our Spitzer census of the IC 348 nebula has revealed a couple of new facts about the structure of the associated em- bedded star cluster, which we discuss briefly in this section. First, our analysis of the composite spectral energy distribu- tions of probable cluster members confirms that a population of embedded sources along the nebula’s southern edge are in- fact class 0/I protostars, as suggested by previous observa- tions of jets and outflows (Tafalla et al. 2006; Walawender et al. 2006). The protostars follow a ridge of molecular ma- terial, are characterized by low spatial surface densities, and are anti-correlated spatially with the cluster’s much more cen- trally condensed class II and class III population. On the other hand these protostars are correlated spatially with a popula- tion of millimeter cores, which we find however to be mostly starless using our Spitzer data. Our analysis of protostars in Trapezium cluster using ground based 3µm data found somewhat similar results: the youngest stars are distributed into an elongated ridge fol- 21 Wider field Spitzer data of the IC 348 nebula was obtained by an Spitzer Legacy Science project entitled “Cores to Disks,” (Evans et al. 2003); the IRAC data was analyzed in Jørgensen et al. (2006). We downloaded the fluxes from their third incremental release which were posted on a web site (http://data.spitzer.caltech.edu/popular/c2d/ 20051220enhancedv1/). We then applied our α3−8µm selection crite- ria to find additional candidates. Unfortunately, the boundaries of the c2d IRAC survey are irregular and one could not simply expand our study to larger cluster radii. lowing the densest molecular gas. The youngest Trapez- ium members appear segregated in subclusters with radii of . 0.1 pc, populations of 10-20+ members (Lada et al. 2004; Grosso et al. 2005). We find something rather different in IC 348 where the protostars are less clustered and have sur- face densities at least one order of magnitude lower than in Orion (peak ∼ 200 stars · pc−2 in IC 348 as opposed to the ∼ 2000 − 3000 stars · pc−2 we found behind the Trapez- ium (Lada et al. 2004) ). Moreover, the IC 348 nebula is sufficiently nearby that we can resolve individual protostel- lar cluster members in our Spitzer data and possibly identify the smallest fragmentation scale, which nonetheless appears unresolved (< 6000 AU). This is in contrast, for example, to the Spokes embedded cluster in NGC 2264, where the bright MIPS sources are spaced by 0.1pc yet appear mostly singular in the Spitzer data (Teixeira et al. 2006). It is pos- sible that higher resolution data will find that the singular Spokes sources will break up into multiples or even small clusters (Young et al. 2006), but it will be interesting to learn whether their protostellar object densities will approach those we find in Orion or are more similar to those we discuss here in IC 348. We also found that the cluster in the IC 348 nebula is more simply structured than previously thought. Using the ex- panded cluster boundaries provided by our Spitzer census, we found that the spatial surface density of confirmed members is fairly smooth and that most of the substructure previously re- ported in IC 348 is not apparently significant. Only those sub- structures which appear correlated with molecular gas appear to be clusterings of actual members. To further examine the structure of the IC 348 cluster, we calculated the (spherically symmetric) radial surface density profile of confirmed IC 348 members. The cluster’s surface density drops off smoothly as r−1 out to a radius of 1pc, which means the space density of stars goes as r−2. This means that the apparent flattening of the radial profile for r > 4′ seen by Herbig (1998) and M03 was the result of variable background contamination, which was also the likely cause for the insignificant sub-clusterings found by LL95. 4.3. History of star formation in the IC 348 nebula Our wide field Spitzer census permits us to reconstruct a more complete history of star formation in the IC 348 neb- ula. Foremost, we found spatially correlated and nearly equal sized populations of class 0/I protostars and starless MM sources in a filamentary ridge that is 1pc from the central B star and lying behind the nebula’s apparent edge. This finding clearly indicates that star formation in the nebula is not fin- ished but is infact ongoing. As pointed out by Hatchell et al. (2005) the large concentration (relative to the entire Perseus cloud) of MM cores near IC 348 is infact consistent with a present day star formation rate equivalent to that which built the older central cluster, assuming that each core will eventu- ally produce 1-3 stars. That our Spitzer data indicate that the majority of these cores appear starless suggests star formation can continue at this rate into the near future. Figure 21a plots the histogram of inferred ages from the HR diagram, and at first glance the SFH for the cluster ensemble is quite broad and suggests a peak at around 2.5 My with per- haps a decline to the present. The interpretation of such a “peak” depends upon the accurate counting of the population of embedded protostars, which could not be included into pre- vious studies of cluster age spreads (e.g. Palla & Stahler 2000) http://data.spitzer.caltech.edu/popular/c2d/20051220 enhanced v1/ http://data.spitzer.caltech.edu/popular/c2d/20051220 enhanced v1/ 22 Muench et al. as they lacked the deep mid-IR data provided by Spitzer 22. Even if the protostars in the SW ridge were long lived (as postulated in White et al. 2006, τ ∼ 1 My for class I), the star formation rate in the southern molecular ridge is increas- ing and approaching ∼ 50 stars/My, which already exceeds the average star formation rate in the cluster halo (Figure 21). The sum of the SFH for the cluster ensemble and the proto- stellar ridge confirms essentially a constant star formation rate of ∼ 50 stars/My over the past 5 My. Attempts to further quantify of the duration of star forma- tion in IC 348 are very difficult. Besides intrinsic differ- ences in birth times, the observed luminosity spread is inflated by the propagated uncertainties in the derivation of L/L� (Kenyon & Hartmann 1990; Hartmann 2001). On one hand we have the fact that extremely “old” members on the HR diagram could be the result of gross underestimates of the intervening extinction caused perhaps by the sources being seen edge-on. On the other hand, the existence of a real lu- minosity spread on the HR diagram (or color-magnitude di- agram) is fairly clear evidence for the stars having a range of radii and thus a range of contraction ages independent of systematic uncertainties in the theoretical tracks used to inter- pret them. From the V − IC vs V color-magnitude diagram Herbig (1998) argued that star formation in the IC 348 neb- ula was not coeval. Instead Herbig found the spread of star formation ages in IC 348 was of order 5 My, which is larger than the members’ median age. Using members drawn from a much larger survey covering the entire nebula we come the same conclusion: if we conservatively ignore the tails of the observed SFH and use the half power points in our derived age distribution functions (Figure 21; cluster ensemble) as the age spread we find that non-negligible star formation began at least 4 My ago. Put another way, if we ignore structure in the SFH and infer a constant star formation rate to the present, the derivation of a median age of 2.5 My, implies a star for- mation duration of ∼ 5 My. Again, the presence of primar- ily starless mm cores suggests this duration will continue to lengthen. Note, if we were to assume that the IC 348 nebula were closer (250 pc; Scholz et al. 1999; Belikov et al. 2002) then the inferred median age and duration of star formation (using the half power points of the SFH) would increase by roughly 0.5 My and 2 My, respectively. 4.4. The origin & evolution of the IC 348 star cluster Using the structure and star forming history derived from our Spitzer census, we can address a few questions about the origin and evolution of the IC 348 star cluster. Foremost, we observe a difference between the structure of the more pop- ulous, centrally condensed and somewhat older cluster and the filamentary ridge of likely younger protostars. As already discussed, there is evidence that the youngest stars in other regions, such as Orion (Lada et al. 2004) and the Spokes clus- ter in NGC 2264 (Teixeira et al. 2006), are also arranged in small subclusters along a filamentary structure. Scally & Clarke (2002) used numerical simulations of the cluster in the Orion Nebula to show that despite the youth of that cluster (τ < 1 My) its current structure could be explained by the merger and evaporation of many (NS ∼ 100) very small sub- clusters similar perhaps to the protostellar ridge in IC 348. 22 Lada et al. (2000) used the statistics of protostellar candidates detected at 3.8µm in the Trapezium core of the Orion Nebula Cluster to draw a similar conclusion about the quite vigorous present day star formation rate in that nebula. FIG. 21.— History of star formation in the IC 348 nebula. As in Figure 20 the star formation rate (stars per Myr) is plotted in bins of roughly equal width of logarithmic age and normalized by the bin width (in age). The plot compares the SFH of the IC 348 core and the halo as defined in text; they appear to peak at around 2.5 Myr ago. The star forming history of the pro- tostellar ridge is a combination of the ages of members seen in projection toward that ridge and placed on the HR diagram (Figure 19b) and the ages of the protostars, assuming that the protostars have formation ages in the past 1 Myr. It appears to be increasing with time; regardless, the ensemble star forming history of IC 348 is consistent with roughly constant star formation over the past 4 Myr. Thus, it is possible that the centrally condensed, older cluster looks different from the protostellar ridge because of signifi- cant dynamical evolution due to stellar interactions. We can examine such a hypothesis by deriving the relevant timescales for dynamical evolution to act upon the stars in the IC 348 nebula. Consider the central cluster of members in the IC 348 nebula: within a roughly 1pc radius region there is a total stellar mass of 165 ( N?330 ) × ( 0.5 ) M�. Were this clus- ter virialized (by its own stellar mass excluding the natal cloud) it would have a 3 dimensional velocity dispersion (σ3d) of 0.86 km/sec. Assuming a star forming efficiency (SFE) less than unity increases this value; for example, a SFE of 0.3, would increase the isotropic virial σ3d by a factor of 2. Rewriting the cluster crossing time, τc = R/v (Binney & Tremaine 1987), as τc ∼= 1.2 · SF E · )3 My, we find the central cluster has maximum τc ∼ 1.2 My, assum- ing SF E = 1. The relatively simple radial profile we find for central cluster members and the lack of substructure outside of the molecular cloud are consistent with the conclusion that the IC 348 cluster is at least one crossing time old (Tan et al. 2006). Indeed, our somewhat conservative estimate for the duration of star formation in the nebula (3-5 My) suggests that the central cluster is at minimum 3-5 crossing times old. For systems older than one crossing time, stellar interactions are important and, subsequently after one relaxation time, 0.1 · N · τc, they will undergo a change in their velocity of order their ve- locity; this is also the equipartition time for a system (Bin- ney & Tremaine 1987). For the central cluster in the IC 348 nebula the relaxation time corresponds to about five crossing times, which is of order the duration of star formation in the Spitzer census of IC 348 23 nebula. We note (again) that reasonable changes to any of these assumptions, e.g. the cluster were initially smaller or had an SFE < 1, would only increase the dynamical age of IC 348 as expressed in crossing (or relaxation) times. Thus, we safely conclude the stars in the nebula have had enough time to undergo an initial relaxation. We believe that the mass segregation we observe is thus the product of the equiparti- tion of energy during these dynamical encounters and is not primordial. Put another way which is independent of whether or not the cluster is relaxed, if there were primordial mass segregation then its precise functional form has likely been erased since the cluster is more than a few crossing times old. Given that sufficient time has passed for the central cluster to undergo dynamical evolution we find it difficult to differen- tiate between two viable models for this cluster’s origin. The current cluster configuration (centrally condensed, smooth ra- dial profile, lack of subclusters) could be the byproduct of the infall and dissolution of stars or small subclusters that formed in filamentary cloud structures, similar to the proto- cluster ridge. The fact that protostellar populations are often observed to be aligned in filamentary structures, including, for examples, the Spokes cluster in NGC 2264 and the embedded subclusters behind the Trapezium in Orion, lends support to this hypothesis. Yet to build up the IC 348 cluster in 3-5 My requires an (constant) infall rate (in stars) of about 30M� per My; there is infact evidence for infall of gas onto the central cluster (see below). In an alternative model the cluster forms from in single, massive (> 200M�) core and the protostel- lar ridge is a subsequent but separate star forming event. In this latter case, for example, we could be observing a process of sequential star formation in which the nebula’s expansion, induced by the presence of the newly formed cluster, swept up the ridge and triggered a second generation or new burst of star formation within it. A more detailed comparison of the radial velocities of the stars and gas could provide some clarity. Stellar radial ve- locities are, unfortunately, known for only 10% of the cluster members (Nordhagen et al. 2006, very recently published v sin i and heliocentric radial velocities for 27 stars). These measurements, which have a typical uncertainty of 3 km/sec, yield a median heliocentric radial velocity of 16.5 km/sec for the stars. This converts to 10 km/sec in the local standard of rest, with a range from 8 to 12 km/sec. No radial velocities are known for the protostars but the southern molecular fil- ament that appears to surround the protostars is blue-shifted relative to the cluster stars (vr,? ∼ −1.5 km/sec). On the one hand the blue velocity shift of the southern filament relative to the stars is consistent with it being swept up (and pushed outward) by the nebula. Since these relative radial velocities are of order the escape speed at the distance of the protostel- lar ridge ( (2)σ3d or ∼ 1.2 km/sec assuming the star clus- ter’s potential can be treated as that of a uniform sphere of mass 165M�) the protostars may escape. On the other hand the relative radial velocities of the stars and the gas provide evidence for continued global infall of gas onto the cluster stars: the central filament, which lies behind the cluster stars, also has a radial velocity of -2 km/sec in the rest frame of the stars and this gas is therefore colliding with or falling in toward the cluster. Perhaps a future study combining addi- tional, higher precision stellar radial velocities and a more de- tailed map of the gravitational potential well created by the star cluster and the molecular gas will provide an origin and fate for the youngest members of the nebula. There are a few additional conclusions we can draw about the IC 348 nebula and its members. First, class II and III sources have the same median “age” (τ ∼ 2.5My) and the same luminosity spread on the HR diagram. This means that external to the protostellar ridge, disked and diskless stars are in general co-spatial and “coeval;” there is abso- lutely no evidence that the halo represents generations of stars which formed before (e.g. Herbig 1998) or more recently (e.g. Tafalla et al. 2006) than the cluster core. Put another way, we have no information from the spatial distribution of disked and non-disked sources (outside the ridge) to indicate when or where they were created. Interestingly, a uniform spatial and temporal distribution of class II and III sources suggests that there is a wide dispersion in the timescale for (inner) disk evo- lution, regardless of the stars’ initial configurations. We can state this same point another way and suggest that since the age spread in IC 348 is of order the disk dissipation timescale as derived from young clusters with a range of median ages (Haisch et al. 2001), the dispersion observed in such a corre- lation is probably real instead of a byproduct of uncertainties in age or disk excess measurement. This uniform spatial and temporal mixing of class II and III members also affirms the notion that accretion does not significantly alter the locations of the stars on the HR diagram. 5. CONCLUSIONS Using sensitive Spitzer mid-IR observations we have per- formed a census of disk-bearing members of the IC 348 young cluster in Perseus, including class II T-Tauri stars and embed- ded class 0/I protostars. Using spectral indices indicative of excess mid-infrared emission, we identified and scrutinized roughly 200 candidate YSOs about which we can draw the following conclusions: 1) There are a total of 118 class II members within a 2.5 pc region in and around the IC 348 nebula. Using exten- sive existing and new spectroscopy we determine that 118 of 136 candidate class II sources are actual mem- bers, indicating that the spectral diagnostic, α3−8µm, is fairly robust for identifying class II stars. 2) We catalog a population of 31 protostars, of which ∼ 20 are high quality candidates (confirmed via other source characteristics such as spectra). Three appear likely to be in the youngest class 0 phase. The catalog of protostars includes 11 faint class I candidates though this faint sample still appears contaminated by back- ground sources which are unassociated with the molec- ular gas cloud. Some of these ∼ 30 protostars have been previously associated with Herbig-Haro jets and molecular outflows, but lacked an SED analysis appro- priate to their classification. Using SED diagnostics to identify class I members was much less effective than for finding class II YSOs; more than half of the ini- tial sample of SED selected class I candidates were eliminated as non-member background contaminants with strong PAH emission features suggestive of extra- galactic sources. Reconnaissance spectroscopy of very faint class I candidates reveal only interlopers which are probably all background galaxies. 3) The size of the class II population suggests a total cluster size of approximately 420 members, which includes a prediction of 60 new class III members that were not 24 Muench et al. uncovered by our Spitzer survey. This estimate is re- inforced by a search of archival X-ray data that cov- ers a much smaller area than our Spitzer data but that nonetheless allow us to identify candidates correspond- ing to about half of these predicted members (also, Ap- pendix C). Comparing various techniques for finding young stars in IC 348, we find that disk excess surveys were successful at identifying approximately 1/3rd of the population, which is similar to the fraction of mem- bers that are periodic photometric variables. On the other hand, 60-80% of the known population are de- tected in X-rays. We further analyzed the properties of the YSOs we identi- fied in the IC 348 nebula, including plotting their spatial dis- tributions, deriving their clustering properties and estimating their physical properties by placing them on the HR diagram. From this analysis we draw the following conclusions about star formation in the IC 348 nebula: 1) Protostars and class II/III YSOs are spatially anti- correlated, with protostars restricted to a narrow fila- mentary ridge 1pc SW of the exposed cluster’s core. The existence of this protostellar ridge illustrates the need for Spitzer surveys to identify securely a cluster’s protostellar population before conclusions are drawn about that young cluster’s structure or star forming his- tory. 2) The stars forming in this protostellar ridge are charac- terized by a lower spatial surface density than either the central cluster core or those protostellar subclusters found in Orion; they also display no preferred resolved spacings which could trace the fragmentation scale of the dense molecular gas in the region. A few small pairs or triples trace the highest order of multiplicity in the region but most protostars appear essentially soli- tary (down to 400 AU). 3) The structure of the central cluster is much simpler than previously supposed. Using confirmed cluster members we found that we do not recover most of the small sub- clusterings previously reported in the halo of the central cluster. Instead the central cluster displays a smooth r−1 radial surface density profile out to a radius of 1 pc. That the exposed cluster shows little substructure indi- cates that nebula is more than a crossing time old (Tan et al. 2006). 3) The star forming history of the IC 348 nebula is con- sistent with essentially constant star formation (∼ 50 stars per Myr) over the past 2.5- 5 Myr. The star for- mation rate in the southern molecular ridge is roughly the same as that spatially averaged rate which formed the foreground cluster, and an ensemble of ∼ 15 star- less mm cores mixed with the protostars indicates star formation will continue at a similar rate in the SW ridge into the near future (Hatchell et al. 2005). Star formation in the vicinity of the IC 348 nebula has been relatively long lived, corresponding to at least a few cluster crossing times. The cluster is also relaxed as evidenced by the segregation of low mass members to the cluster halo, which was reported previously by M03 but is confirmed here using the HR diagram. On the one hand this relatively long dura- tion of star formation means that we cannot determine a pre- cise origin for the central cluster based simply on its struc- ture; such information about its primordial structure appears to have been erased. On the other hand, because the youngest protostars in IC 348 have a filamentary distribution and this distribution matches what is observed in other embedded clus- ters, e.g. the Orion and Spokes clusters, we tend to favor a model where the central cluster was built from members that formed in filaments or perhaps small subclusters and that have since fallen into the central cluster’s potential well. The rela- tive radial velocities of the stars and gas in IC 348 are infact consistent with global infall of molecular gas onto the cluster. In summary, we believe that what we have observed in the protostellar ridge 1pc SW of the central IC 348 cluster rep- resents the primordial building blocks for young embedded clusters. Finally, the argument that star formation is “fast,” i.e., beginning rapidly after parts of an initially turbulent cloud passes some critical gravitational threshold, should not pre- clude the idea that star formation may also be long lived. Un- til either the natal gas reservoir is depleted, resulting in a rel- atively high star formation efficiency, or the infall of gas and new stars is disrupted by an ionizing member, star formation continues. Clearly, neither circumstance has yet been reached for the IC 348 nebula. As star formation in the IC 348 neb- ula does not appear destined to soon cease, a fairly long pe- riod of star formation (> 2.5 Myr) in a fairly small volume (R ∼ 1 pc) of space should be considered when examin- ing numerical renditions of cloud collapse or the dynamics of young stars. We thank Alyssa Goodman for discussions regarding the molecular gas in IC 348 and James Di Francesco for the SCUBA 850µm image, which was provided in advance of publication. We are grateful for comments and questions pro- vided by an anonymous referee. K. L. was supported by grant NAG5-11627 from the NASA Long-Term Space Astro- physics program. This work is based [in part] on observations made with the Spitzer Space Telescope, which is operated by the Jet Propul- sion Laboratory, California Institute of Technology under a contract with NASA. Some of the data presented herein were obtained at Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative Agreement with the National Aeronautics and Space Administration and at the W.M. Keck Observatory, which is operated as a scien- tific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. This Observatory was made possible by the generous financial support of the W.M. Keck Foun- dation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Based [in part] on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. Facilities: Spitzer (IRAC, MIPS), IRTF (SpeX), Keck (NIRC), MMT (Blue Channel), Magellan (IMACS), Chandra (ACIS), XMM-Newton Spitzer census of IC 348 25 FIG. 22.— Effects of reddening on Spitzer spectral energy distributions of T-Tauri stars. Panels (A) and (B) show the typical median spectral energy distribution for K0 diskless members and K6 class II star+disk members of IC 348 (Paper I, Table 3) reddened by extinctions up to AV = 200. Panels (C) and (D) plot the change in three spectral indices with increasing extinction; error bars at each point are the 1σ fit quality and thus chart the departure of the chosen spectral index from a power-law. The slope of the relationship between these spectral indices and AV are given in Table 9. APPENDIX A. EXTINCTION EFFECTS ON α3−8µM We explored the influence of dust extinction on our preferred spectral index, α3−8µm. Using a diskless K0 IC 348 member from Paper I (Table 3) as a template, we reddened the observed photosphere by extinctions as large as AV = 200 using the reddening law from Indebetouw et al. (2005). Ten of these reddened SEDs are shown for illustration in Figure 22a with passbands from K to MIPS 24 micron included. While the IRAC slope of the SED requires AV > 100 to inflect to a positive slope, the K − 3.6µm slope of the SED is inflected by AV > 40. Note that even for AV ∼ 200 the 5.8− 24µm SED slope of background stars remains negative. Since such large column densities within typical molecular clouds occur only in regions very close to embedded YSOs, this SED slope proves that background sources with normal Raleigh-jeans SEDs cannot mimic class I sources except if they were seen through the protostellar envelope of a class I source. Figure 22b plots the explicit dependence of α on AV , from which we can derive the reddening law for these spectral indices. We calculate the α index for the IRAC, K+IRAC and 5.8 − 24µm portions of the SED, plotting them versus fit quality to demonstrate the degree of departure from a true power-law as a function of AV . These yield the relationships given in Table 9 and used in Figure 1. We repeated this experiment with empirical SEDs for thick disk classical T-Tauri stars in IC 348 (Figure 22ab). Reddening the median observed SED of K6-M0 IC 348 member (Paper I, Table 3), we find that AV > 40 cause both the Spitzer based SED indices to inflect. Actually, for AV > 100 the α3−8µm index becomes steeper than the 5.8 − 24µm slope, a result that would rarely occur for background field stars. In principle, background stars could be differentiated from cluster members by a rising α3−8µm slope coupled with a negative 5.8−24µm slope. Again, indices using K , which include those indices calculated by Jørgensen et al. (2006), are very sensitive to extinction causing source classifications including that band to become degenerate for AV > 20. A fixed value of α3−8µm = −1, for example, could correspond either to a typical class II YSO or to an extremely heavily reddened (AV ∼ 75) diskless star. However, the IRAC SEDs of heavily reddened diskless stars are distinct from those of typical Class II star+disk sources: the power-law IRAC SEDs of class II objects are intrinsically shallow but heavily reddened diskless 26 Muench et al. TABLE 9 Aα,SE D SED Range Aα,SE D K − 8.0 0.0514 3.6− 8.0 0.0226 5.8− 24 0.0134 FIG. 23.— SpeX near-IR spectra of 34 candidate class II and protostellar members of IC 348 and a sample of optically-classified members of IC 348 and Taurus. The candidates are labeled with the types derived from a comparison to the IR spectra of the optically-classified objects (“ir”). The spectra are ordered according to the spectral features in these data. They have a resolution of R = 100, are normalized at 1.68µm, and are dereddened (§B.2). Object names containing five digits or less apply to IC 348, while all other names refer to Taurus members. Protostar #904 (M3.5 ir; K ∼ 14.3) is shown here. stars are bent downward at 2-5 micron; their poor power-law SED fits should distinguish them as being diskless. Finally, we see that characteristic dip at 8.0µm, often seen in the SEDs of embedded YSOs (see Figure 7) can be produced by large (AV > 100) reddenings of normal cTTs. Such a dip does not necessitate particular envelope geometries, although the observation of such large column densities may only be possible through an envelope (Myers et al. 1987). Spitzer census of IC 348 27 FIG. 24.— Dereddened SpeX near-IR spectra of candidate IC 348 YSOs. Same as in Figure 23. B. SPECTROSCOPY OF NEW MEMBERS B.1. Infrared Spectra We selected for spectroscopy objects in the IRAC images that display IRAC SEDs indicative of disk excess, are sufficiently bright for the spectrometer employed (K . 15), and have not been previously classified as field stars or cluster members. A resulting sample of 39 candidate cluster members with spectroscopic confirmation is provided in Table 1. We also observed a sample of 36 known late-type members of IC 348 and Taurus (Briceño et al. 1998, 2002; Luhman et al. 1998a, 2003a,b; Luhman 1999, 2004), which are listed in Figures 23-26. These optically-classified objects will be used as the standards during the classification of the candidates in §B.2. These data were collected with the spectrometer SpeX (Rayner et al. 2003) at the NASA Infrared Telescope Facility (IRTF) on the nights of 2004 November 11-13 and 2005 December 12-14. The instrument was operated in the prism mode with a 0.′′8 slit, producing a wavelength coverage of 0.8-2.5 µm and a resolution of R ∼ 100. The spectra were reduced with the Spextool package (Cushing et al. 2004), which included a correction for telluric absorption (Vacca et al. 2003). B.2. Classification To measure spectral types for the candidate members of IC 348 that we observed spectroscopically in §B.1, we used the absorption bands of VO and TiO (λ < 1.3 µm) and H2O (λ > 1 µm). These bands are the primary spectral classification 28 Muench et al. FIG. 25.— Dereddened SpeX near-IR spectra of candidate IC 348 YSOs. Same as in Figure 23. Note the H2 emission of protostellar candidate #234. Another protostellar candidate, #30003, is faint (K ∼ 15.2) and embedded in a scattered light cavity. diagnostics for late-type objects (Kirkpatrick et al. 1991; Leggett et al. 2001; Reid et al. 2001) and are broad enough to be easily detected at the low resolution of our data. Because near-IR H2O absorption bands are stronger in young objects than in field dwarfs at a given optical spectral type (Luhman & Rieke 1999; Lucas et al. 2001; McGovern et al. 2004), spectral types of young objects derived from H2O with dwarf standards will be systematically too late. Instead, to arrive at accurate spectral types, optically-classified young objects rather than dwarfs should be used when measuring spectral types of young sources from steam (Luhman & Rieke 1999; Luhman et al. 2003b), which is the approach we adopted in our classification of the candidates in IC 348. To facilitate the comparison of the band depths between the candidates and the optically-classified known members, we have dereddened the spectra to the same slope as measured by the ratios of fluxes at 1.32 and 1.68 µm. These dereddened spectra are not meant to be precise estimates of the intrinsic, unreddened appearance of these stars since the slopes likely vary with spectral type. As shown in Figures 23-26, we first arranged the dereddened spectra of the previously known, optically-classified members of IC 348 and Taurus in order of the strengths of their molecular absorption features. With a few minor exceptions, the IR features change monotonically with optical type. We then measured a spectral type for each candidate by visually comparing the absorption features in its spectrum to those in the data of the optically-classified objects. Through this analysis, we found that 34 of the 39 candidates in our sample exhibit M types. We have inserted these 34 sources in the sequence of optically- classified objects in Figures 23-26 and have labeled them with the types derived from these IR spectra, which have uncertainties of ± 0.5 subclass unless noted otherwise. The detection of late-type, stellar photospheric features now demonstrates that these objects indeed are young stars, and thus Spitzer census of IC 348 29 FIG. 26.— Dereddened SpeX near-IR spectra of candidate IC 348 YSOs. Same as in Figure 23. members of IC 348. Other available evidence of youth and membership for these objects is listed in Table 1, which is based on the diagnostics described by Luhman et al. (2003b) and Luhman et al. (2005b). Of the 35 sources with new spectral types, 32 are classified as class II T-Tauri stars, while 3 have flat or rising mid-IR SEDs and are classified as protostellar. The composite SEDs of these 32 class II sources are shown in Figure 28. B.3. Optical Spectra We obtained optical spectra of 20 Spitzer selected sources in IC 348 using the Blue Channel spectrograph on the MMT during the nights of 2004 December 10 and 11 and with the Inamori Magellan Areal Camera and Spectrograph (IMACS) on the Magellan I telescope at Las Campanas Observatory during the night of 2005 January 4. The resulting spectra have a wavelength coverage of 6300-8900 Å and a resolution of 3 Å. The procedures for the collection and reduction of these data were similar to those described by Luhman (2004). The sources were classified in the same manner as data for Taurus taken on the same nights (Luhman 2006). Of the 20 targets, 17 were members and 3 were determined to be non-members (the infrared excess of which were very weak; Table 3). Of the members, 13 sources had both optical and SpeX IR spectra and the spectral types derived from them in general agreed very well, except for two class II members whose infrared spectra were indeterminate (1905, 1933). Four sources have only optical spectral types, including two new class II members (1890 and 10120) and two class III members (Appendix C). The reduced, dereddened, optical spectra of these 17 members are displayed in Figure 27. In addition to spectral types, we measured particularly useful optical spectral features (e.g. Hα) for these members (10). C. CLASS III MEMBERSHIP Our Spitzer census cannot uniquely identify diskless cluster members and we did not attempt an exhaustive search for anemic disk candidates. In this appendix we describe how we used archival X-ray and recent optical monitoring results to tabulate candidate members lacking strong disk signatures (class III), which we used to justify our extrapolated population estimate given in Section 4.1. Matching the 220 Chandra X-ray sources identified in the uniformly processed ANCHORS data23 to our source catalog provides the following statistics: 15% (31) of these sources have no match to our Spitzer catalog or are lacking near-IR photometry – these are all likely from background galaxies; 12 X-ray sources with α3−8µm > −0.5, consisting of 4 flat spectrum protostars, 4 low luminosity candidate class I and 4 rejected low luminosity class I sources; 2 known foreground stars and 162 known members. We inspected the composite SEDs of the remaining 13 sources; on the I − J vs I color-magnitude diagram three of them fall below the main sequence at the distance of IC 348 and were rejected; the remaining 10 fall into the locus of X- 23 ANCHORS: an Archive of Chandra Observations of Regions of Star Formation; Chandra Archival Proposal 06200277; S. Wolk, PI. See http:// hea-www.harvard.edu/˜swolk/ANCHORS/. Data (53ksec ACIS; Chandra ObsId 606) originally observed (2000-09-21) and published by Preibisch & Zinnecker (2001). http://hea-www.harvard.edu/~swolk/ANCHORS/ http://hea-www.harvard.edu/~swolk/ANCHORS/ 30 Muench et al. FIG. 27.— Optical spectra of 17 new IC 348 members identified in this work. Spectra were obtained with the IMACS instrument on the Magellan I telescope and the Blue Channel spectrograph on the MMT. The IMACS spectra were obtained in multi-slit mode and some of the spectra fell across two CCDs, resulting in gaps in the spectra. The spectra have been corrected for extinction, which is quantified in parentheses by the magnitude difference of the reddening between 0.6 and 0.9 µm (E(0.6− 0.9)). The data are displayed at a resolution of 8 Å and are normalized at 7500 Å. ray detected known members. Similarly, a cross match of our master catalog to the 71 unique X-ray sources in wider-field XMM data (Preibisch & Zinnecker 2004)24 yielded a further 11 candidate X-ray members along with 39 probable extragalactic sources, 17 confirmed members and 3 known non-members. Lastly, we searched a recent catalog of IC 348 periodic sources (Cieza & Baliber 2006) and found 5 periodic unconfirmed members that fell inside our Spitzer survey region but outside of pre- existing X-ray surveys. Of these five, one falls below the main sequence and we did not consider it a member (Cieza & Baliber, source #140); thus, within the boundaries of our Spitzer census the total number of candidate IC 348 member identified by these two techniques is 25. Only two of these 25 candidate members (#104 and 185) were detected at 24µm; both are anemic disk members with α3−8µm = −2.39 & − 2.48, respectively. The SEDs of the remaining 23 candidate members are consistent with stellar photospheres. 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http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2006AJ....132..467W&db_key=AST http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2005AJ....129.2308W&db_key=AST http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2004ApJS..154....1W&db_key=AST http://arxiv.org/abs/astro-ph/0406223 http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2006astro.ph..4081W&db_key=PRE http://arxiv.org/abs/astro-ph/0604081 http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2004ApJ...617.1177W&db_key=AST http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2006ApJ...642..972Y&db_key=AST http://arxiv.org/abs/astro-ph/0601300 Spitzer census of IC 348 33 1 Introduction 2 Spitzer census 2.1 SED selected young stellar objects 2.2 Class II census results 2.2.1 Membership 2.2.2 Completeness 2.3 Protostellar census 2.3.1 Low luminosity protostellar candidates 2.3.2 MIPS survey of dark cloud cores near IC 348 3 Analysis 3.1 Spatial distribution of members 3.2 Comparison of gas, mid-IR dust emission and young stars 3.3 The protostars of the southern filament 3.3.1 Spitzer & SCUBA correlations 3.3.2 Clustering 3.3.3 Near-Infrared Images 3.4 Inferred cluster properties 4 Discussion 4.1 Total young star population of the IC 348 nebula 4.2 Physical structure of the IC 348 star cluster 4.3 History of star formation in the IC 348 nebula 4.4 The origin & evolution of the IC 348 star cluster 5 Conclusions A Extinction effects on 3-8m B Spectroscopy of new members B.1 Infrared Spectra B.2 Classification B.3 Optical Spectra C Class III membership

0704.0204

Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots

7 Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots Marco G. Pala1, Michele Governale2, Jürgen König2 1 IMEP-MINATEC (UMR CNRS/INPG/UJF 5130), 38016 Grenoble, France 2 Institut für Theoretische Physik III, Ruhr-Universität Bochum, 44780 Bochum, Germany E-mail: [email protected], [email protected], [email protected] Abstract. We present a theory of transport through interacting quantum dots coupled to normal and superconducting leads in the limit of weak tunnel coupling. A Josephson current between two superconducting leads, carried by first-order tunnel processes, can be established by non-equilibrium proximity effect. Both Andreev and Josephson current is suppressed for bias voltages below a threshold set by the Coulomb charging energy. A π-transition of the supercurrent can be driven by tuning gate or bias voltages. PACS numbers: 74.45.+c,73.23.Hk,73.63.Kv,73.21.La Submitted to: New J. Phys. http://arxiv.org/abs/0704.0204v2 Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 2 1. Introduction Non-equilibrium transport through superconducting systems attracted much interest since the demonstration of a Superconductor-Normal-Superconductor (SNS) transistor [1]. In such a device, supercurrent suppression and its sign reversal (π-transition) are achieved by driving the quasi-particle distribution out of equilibrium by means of applied voltages [2, 3, 4, 5]. Another interesting issue in mesoscopic physics is transport through quantum dots attached to superconducting leads. For DC transport through quantum dots coupled to a normal and a superconducting lead, subgap transport is due to Andreev reflection [6, 7, 8, 9, 10, 11]. Also transport between two superconductors through a quantum dot has been studied extensively. The limit of a non-interacting dot has been investigated in [12]. Several authors considered the regime of weak tunnel coupling where the electrons forming a Cooper pair tunnel one by one via virtual states [13, 14, 15]. The Kondo regime was also addressed [13, 16, 17, 18, 19]. Multiple Andreev reflection through localized levels was investigated in [20, 21]. Numerical approaches based on the non-crossing approximation [22], the numerical renormalization group [23] and Monte Carlo [24] have also been used. The authors of [25] compare different approximation schemes, such as mean field and second-order perturbation in the Coulomb interaction. In double-dot systems the Josephson current has been shown to depend on the spin state of the double dot [26]. Experimentally, the supercurrent through a quantum dot has been measured through dots realized in carbon nanotubes [27] and in indium arsenide nanowires [28]. In this Letter we study the transport properties of a system composed of an interacting single-level quantum dot between two equilibrium superconductors where a third, normal lead is used to drive the dot out of equilibrium. A Josephson coupling in SNS heterostructures can be mediated by proximity-induced superconducting correlations in the normal region. In case of a single-level quantum dot, superconducting correlations are indicated by the correlator 〈d↓(0)d↑(t)〉, where dσ is the annihilation operator of the dot level with spin σ. To obtain a large pair amplitude, i.e. the equal- time correlator 〈d↓d↑〉, at least two conditions need to be fulfilled: (i) the states of an empty and a doubly-occupied dot should be nearly energetically degenerate and (ii) the overall probability of occupying the dot with an even number of electrons should be finite. For a non-interacting quantum dot, i.e. vanishing charging energy U for double occupancy, this can be achieved by tuning the level position ǫ in resonance with the Fermi energy of the leads, ǫ = 0 [12]. In this case, the Josephson current can be viewed as transfers of Cooper pairs between dot and leads and the expression of the current starts in first order in the tunnel coupling strength Γ. The presence of a large charging energy U ≫ kBT,Γ destroys this mechanism since the degeneracy condition 2ǫ+U ≈ 0 is incompatible with a finite equilibrium probability to occupy the dot with an even number of electrons. Nevertheless, a Josephson current can be established by higher-order tunnelling processes (see, for example, [13, 14, 15]), associated with a finite superconducting correlator 〈d↓(0)d↑(t)〉 at different times. The Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 3 amplitude of the Josephson coupling is, however, reduced by a factor Γ/∆, i.e., the current starts only in second order in Γ, and the virtual generation of quasiparticles in the leads suppresses the Josephson current for large superconducting gaps ∆. In particular, it vanishes for ∆ → ∞. The main purpose of the present paper is to propose a new mechanism that circumvents the above-stated hindrance to achieve a finite pair amplitude in an interacting quantum dot, and, thus, restores a Josephson current carried by first-order tunnel processes that survives in the limit ∆ → ∞. For this aim, we attach a third, normal, lead to the dot that drives the latter out of equilibrium by applying a bias voltage, so that condition of occupying the dot with an even number of electrons is fulfilled even for 2ǫ+ U ≈ 0. We relate the current flowing into the superconductors to the nonequilibrium Green’s functions of the dot. In the limit of a large superconducting gap, ∆ → ∞, the current is only related to the pair amplitude. The latter is calculated by means of a kinetic equation derived from a systematic perturbation expansion within real- time diagrammatic technique that is suitable for dealing with both strong Coulomb interaction and nonequilibrium at the same time. 2. Model We consider a single-level quantum dot connected to two superconducting and one normal lead via tunnel junctions, see figure 1. The total Hamiltonian is given by H = ΓΓS S Figure 1. Setup: a single-level quantum dot is connected by tunnel junctions to one normal and two superconducting leads with tunnelling rates ΓN and ΓSL,R , respectively. η=N,SL,SR (Hη +Htunn,η). The quantum dot is described by the Anderson model σdσ +Un↑n↓, where nσ = d σdσ is the number operator for spin σ =↑, ↓, ǫ is the energy level, and U is the charging energy for double occupation. The leads, labeled by η = N, SL, SR, are modeled by Hη = kσ ǫkc ηkσcηkσ− η−k↓ +H.c. , where ∆η is the superconducting order parameter (∆N = 0). The tunnelling Hamiltonians are Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 4 Htunn,η = Vη ηkσdσ +H.c. . Here, Vη are the spin- and wavevector-independent tunnel matrix elements, and cηkσ(c ηkσ) and dσ(d σ) represent the annihilation (creation) operators for the leads and dot, respectively. The tunnel-coupling strengths are characterized by Γη = 2π|Vη| k δ(ω − ǫk). 3. Current formula We start with deriving a general formula for the charge current in lead η by using the approach of Ref. [29] generalized to superconducting leads. Similar formulae that relate the charge current to the Green’s function of the dot in the presence of superconducting leads have been derived in previous works, in particular for equilibrium situations, see e.g. Refs. [16, 22]. The formula derived below is quite general, as it allows for arbitrary bias and gate voltages, temperatures, and superconducting order parameters for a quantum dot coupled to an arbitrary number of normal and superconducting leads. For this, it is convenient to use the operators ψηk = (cηk↑, c η−k↓) and φ = (d↑, d T in Nambu formalism. The current from lead η is expressed as Jη = e 〈dNη/dt〉 = i(e/h̄)〈[H,Nη]〉 = i(e/h̄)〈[Htunn,η, Nη]〉 ‡, with Nη = ηkτ3ψηk, where τ1, τ2, τ3 indicate the Pauli matrices in Nambu space and e > 0 the electron charge. Evaluating the commutator leads to Jη = − Re {Tr [τ3VηG D,ηk(ω)]} , (1) with Vη = Diag(Vη,−V η ) and the lead–dot lesser Green’s functions (G D,ηk(ω))m,n that are the Fourier transforms of i〈ψ ηkn(0)φm(t)〉. In the following, we assume the tunnelling matrix elements Vη to be real (any phase of Vη can be gauged away by substituting ∆η → ∆η exp(−2i arg Vη)). The Green’s function G D,ηk is related to the full dot Green’s functions and the lead Green’s functions by a Dyson equation in Keldysh formalism: G<D,ηk(ω) = G R(ω)V†ηg ηk(ω)+G <(ω)V†ηg ηk(ω), where G R(<)(ω) is the retarded (lesser) dot Green’s function, and and g ηk (ω) the lead advanced (lesser) Green’s function. Using this relation and assuming energy-independent tunnel rates Γη, we obtain for the current Jη = J1η + J2η with J1η = ΓηDη(ω)Im ω − µη 2GR(ω)fη(ω) +G , (2) J2η = ΓηSη(ω)Re , (3) where ∆η = ∆∗η 0 , and fη(ω) = [1 + exp(ω − µη)/(kBT )] −1 is the Fermi function, with T being the temperature and kB the Boltzmann constant. The dot Green’s functions (G<D(ω))m,n and GRD(ω) are defined as the Fourier transforms ‡ Note that [Hη, Nη] 6= 0 but 〈[Hη, Nη]〉 = 0 for η = SL,R. Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 5 of i〈φ†n(0)φm(t)〉 and −iθ(t)〈{φm(t), φ n(0)}〉, respectively. The two weighting functions Dη(ω) and Sη(ω) are given by Dη(ω) = |ω − µη| (ω − µη)2 − |∆η|2 θ(|ω − µη| − |∆η|) Sη(ω) = |∆η|2 − (ω − µη)2 θ(|∆η| − |ω − µη|). The terms J1η and J2η involve excitation energies ω above and below the superconducting gap, respectively. For η = N, only the part of J1η that involves normal (diagonal) components of the Green’s functions contributes, and the current reduces to the result presented in [29]. For superconducting leads, this part describes quasiparticle transport that is independent of the superconducting phase difference. The other part of J1η involves anomalous (off-diagonal) components of the Green’s functions and is, in general, phase dependent. The contribution to the Josephson current stemming from this term is the dominant one in the regime considered in [13, 14, 15]. The excitation energies above the gap are only accessible either for transport voltages exceeding the gap or by including higher-order tunnelling, involving virtual states with quasiparticles in the leads, and, therefore, J1η vanishes for large |∆η|. In this case J2η, that involves only anomalous Green’s functions with excitation energies below the gap, dominates transport. It is, in general, phase dependent, and describes both Josephson as well as Andreev tunnelling. In the following we consider the limit |∆η| → ∞, where the current simplifies to Γη|〈d↓d↑〉| sin(Ψ− Φη) , (4) with Φη being the phase of ∆η and 〈d↓d↑〉 = |〈d↓d↑〉| exp(iΨ) the pair amplitude of the dot that has to be determined in the presence of Coulomb interaction, coupling to all (normal and superconducting) leads and in non-equilibrium due to finite bias voltage. We now consider a symmetric three-terminal setup with ΓSL = ΓSR = ΓS, ∆SL = |∆| exp(iΦ/2) and ∆SR = |∆| exp(−iΦ/2), and µSL = µSR = 0. The quantities of interest are the the current that flows between the two superconductors (Josephson current) Jjos = (JSL − JSR)/2 and the current in the normal lead (Andreev current) Jand = JN = −(JSL + JSR). Furthermore, we focus on the limit of weak tunnel coupling, ΓS < kBT . In this regime, an Josephson current through the dot in equilibrium would be suppressed even in the absence of Coulomb interaction, U = 0, since the influence of the superconductors on the quantum-dot spectrum could not be resolved for the resonance condition ǫ ≈ 0. This can, e.g., be seen in the exactly-solvable limit of U = 0 together with ΓN = 0, where the Josephson current is Jjos = (e/2h̄)Γ S sin(Φ) [f(−ǫA(Φ))− f(ǫA(Φ))] /ǫA(Φ) with ǫA(Φ) = ǫ2 + Γ2S cos 2(Φ/2). This provides an additional motivation to look for a non-equilibrium mechanism to proximize the quantum dot. Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 6 4. Kinetic equations for quantum-dot degrees of freedom The Hilbert space of the dot is four dimensional: the dot can be empty, singly occupied with spin up or down, or doubly occupied, denoted by |χ〉 ∈ {|0〉, | ↑〉, | ↓〉, |D〉 ≡ d ↓|0〉}, with energies E0, E↑ = E↓, ED. For convenience we define the detuning as δ = ED −E0 = 2ǫ+ U . The dot dynamics is fully described by its reduced density matrix ρD, with matrix elements P ≡ (ρD)χ2χ1 . The dot pair amplitude 〈d↓d↑〉 is given by the off-diagonal matrix element P 0D. The time evolution of the reduced density matrix is described by the kinetic equations P χ1χ2 (t) + (Eχ1 −Eχ2)P (t) = (t, t′)P (t′). (5) We define the generalized transition rates byW −∞ dt (t, t′), which are the only quantities to be evaluated in the stationary limit. Together with the normalization condition χ Pχ = 1, (5) determines the matrix elements of ρD. Furthermore, in (5) we retain only linear terms in the tunnel strengths Γη and the detuning δ. Hence, we calculate the rates W to the lowest (first) order in Γη for δ = 0. This is justified in the transport regime ΓS,ΓN, δ < kBT . The rates are evaluated by means of a real-time diagrammatic technique [30], that we generalize to include superconducting leads. This technique provides a convenient tool to perform a systematic perturbation expansion of the transport properties in powers of the tunnel-coupling strength. In the following, we concentrate on transport processes to first order in tunnelling (a generalization to higher orders is straightforward). This includes the transfer of charges through the tunnelling barriers as well as energy-renormalization terms that give rise to nontrivial dynamics of the quantum-dot degrees of freedom. We find for the (first-order) diagonal rates Wχ1χ2 ≡ W the expressions Wσ0 = ΓNfN(−U/2);W0σ = ΓN[1 − fN(−U/2)];WDσ = ΓNfN(U/2);WσD = ΓN[1 − fN(U/2)]. The N lead also contributes to the rates WDD00 = (W ∗ = −ΓN[1 + fN(−U/2) − fN(U/2) + iB] where B = U/2−µN 2πkBT −U/2−µN 2πkBT , with µN being the chemical potential of the normal lead and ψ(z) the Digamma function. Notice that B vanishes when µN = 0 or U = 0. The superconducting leads do not enter here due to the gap in the quasi-particle density of states. These leads, though, contribute to the off-diagonal rates W 000D = W W 0D00 W 0DDD = −WD000 = −W − (W 00D0) WDDD0 = −iΓS cos(Φ/2). For an intuitive representation of the system dynamics we define, in analogy to [31], a dot isospin by PD0 + P ; Iy = i PD0 − P ; Iz = PD − P0 . (6) Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 7 From (5), we find that in the stationary limit the isospin dynamics can be separated into three parts, 0 = dI/dt = (dI/dt)acc + (dI/dt)rel + (dI/dt)rot, with [1− fN(−U/2)− fN(U/2)]êz (7) = − ΓN[1 + fN(−U/2)− fN(U/2)]I (8) = I×Beff (9) where êz is the z-direction and Beff = {2ΓS cos(Φ/2), 0,−ΓNB − 2ǫ− U} is an effective magnetic field in the isospin space. The accumulation term (7) builds up a finite isospin, while the relaxation term (8) decreases it. Finally, (9) describes a rotation of the isospin direction. 5. Non-equilibrium Josephson current In the isospin language the current in the superconducting leads is JSL,R = ΓS [Iy cos(Φ/2)± Ix sin(Φ/2)] , (10) where the upper(lower) sign refers to the left(right) superconducting lead. The Iy component contributes to the Andreev current, while Ix is responsible for the Josephson current. To obtain subgap transport, we first need to build up a finite isospin component along the z-direction, i.e. we need a population imbalance between the empty and doubly occupied dot [(this is generated by the accumulation term in (7)]; second, we need a finite Beff which rotates the isospin so that it acquires an inplane component. In order to have a finite Josephson current (Ix 6= 0), we need the z-component, −ΓNB − 2ǫ−U , of the effective magnetic field producing the rotation to be non zero. The Josephson current and the Andreev current read Jjos = − [2ǫ+ U + ΓNB]ΓS sin(Φ) |Beff |2 + Γ N[1 + fN(−U/2)− fN(U/2)] 1− fN(−U/2)− fN(U/2) 1 + fN(−U/2)− fN(U/2) Jand = 2ΓNΓS[1 + cos(Φ)] |Beff |2 + Γ N[1 + fN(−U/2)− fN(U/2)] × [1− fN(−U/2)− fN(U/2)]. (12) These results take into account only first-order tunnel processes, i.e. the rates W are computed to first order in Γη. The factor [1 − fN(−U/2) − fN(U/2)] ensures that no finite dot-pair amplitude can be established if the chemical potential of the normal lead, µN, is inside the interval [−U/2, U/2] by at least kBT . In this situation both the Josephson and the Andreev currents vanish. On the other hand, this factor takes the value −1 if µN > U/2 and the value +1 if µN < −U/2. Hence, the sign of the Josephson current can be reversed by the applied voltage (voltage driven π-transition). Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 8 The considerations above establish the importance of the non-equilibrium voltage to induce and control proximity effect in the interacting quantum dot. In figure 2 we show in a density plot (a) Jjos and (b) Jand for Φ = π/2 as a function of the voltage µN and the level position ǫ. Both the control of proximity effect by the chemical potential µN and the voltage driven π-transition are clearly visible. If the detuning is too large, |δ + ΓNB| > Γ2N + 4Γ S cos 2(Φ/2), it becomes difficult to build a superposition of the states |0〉 and |D〉, which is necessary to establish proximity. As a consequence, the Josephson and the Andreev current are algebraically suppressed by δ−1 and δ−2, respectively. Figure 3 shows the Josephson current as a function of δ = 2ǫ + U . The fact that the Josephson current is non zero for δ = 0 is due to the term ΓNB, i.e. of the interaction induced contribution to the z-component of the effective field Beff acting on the isospin. The term |B| has a maximum at µN = U/2, which causes this effect to be more pronounced at the onset of transport. The fact that the value of the Josephson current varies on a scale smaller than temperature indicates its nonequilibrium nature. A π-transition of the Josephson current can also be achieved by changing the sign of δ+ΓNB, as shown in figure 4 where Jjos is plotted as a function of the phase difference Φ for different values of the level position. Notice that the current for δ = 0 (ǫ = −U/2) is different from zero only due to the presence of the term ΓNB acting on the isospin. 6. Conclusion In conclusion, we have studied non-equilibrium proximity effect in an interacting single-level quantum dot weakly coupled to two superconducting and one normal lead. We propose a new mechanism for a Josephson coupling between the leads that is qualitatively different from earlier proposals based on higher-order tunnelling processes via virtual states. Our proposal relies on generating a finite non-equilibrium pair amplitude on the dot by applying a bias voltage between normal and superconducting leads. The charging energy of the quantum dot defines a threshold bias voltage above which the non-equilibrium proximity effect allows for a Josephson current carried by first-order tunnelling processes, that is not suppressed in the limit of a large superconducting gap. Both the magnitude and the sign of the Josephson current are sensitive to the energy difference between empty and doubly-occupied dot. A π- transition can be driven by either bias or gate voltage. In addition to defining a threshold bias voltage, the charging energy induces many-body correlations that affect the dot’s pair amplitude, visible in a bias-voltage-dependent shift of the π-transition as a function of the gate voltage. Acknowledgments We would like to thank W. Belzig, R. Fazio, A. Shnirman, and A. Volkov for useful discussions. M.G. and J.K. acknowledge the hospitality of Massey University, Palmerston North, and of the CAS Oslo, respectively. Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 9 Figure 2. Density plot of the a) Josephson and b) Andreev current, for fixed superconducting-phase difference Φ = π/2, as a function of the dot-level position ǫ and of the chemical potential of the normal lead µN. The symbols ± refer to the sign of the current. The other parameters are ΓS = ΓN = 0.01U , and kBT = 0.05U . Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots10 -0.05 -0.025 0 0.025 0.05 /U=0.5 Figure 3. Josephson current, for fixed superconducting-phase difference Φ = π/2, as a function of the detuning δ = ED − E0 = 2ǫ+ U for different values of the chemical potential. The other parameters are ΓS = ΓN = 0.01U and kBT = 0.05U . 0 π/2 π 3π/2 2π δ/U=0.01 δ/U=0 δ/U=-0.01 Figure 4. Josephson current as a function of the superconducting-phase difference Φ for different values of the detuning. The other parameters are ΓS = ΓN = 0.01U , µN = U , and kBT = 0.05U . 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Lett. 93 047002 [25] Vecino E, Mart́ın-Rodero A and Levy Yeyati A 2003 Phys. Rev. B 68 035105 [26] Choi M-S, Bruder C and Loss D 2000 Phys. Rev. B 62 13569 [27] Buitelaar M R, Nussbaumer T and Schönenberger C 2002 Phys. Rev. Lett. 89 256801; Cleuziou J-P, Wernsdorfer W, Bouchiat V, Ondarçuhu T, and Monthioux M 2006 Nature Nanotechnology 1 53; Jarillo-Herrero P, van Dam J A and Kouwenhoven L P 2006 Nature 439 953; Jørgensen H I, Grove-Rasmussen K, Novotný T, Flensberg K and Lindelof P E 2006 Phys. Rev. Lett. 96 207003 [28] van Dam J A, Nazarov Y V, Bakkers E P A M, De Franceschi S and Kouwenhoven L P 2006 Nature 442 667; Sand-Jespersen T, Paaske J, Andersen B M, Grove-Rasmussen K, Jørgensen H I, Aagesen M, Sørensen C, Lindelof P E, Flensberg K and Nyg̊ard J Preprint cond-mat/0703264 [29] Meir Y and Wingreen N S 1992 Phys. Rev. Lett. 68 2512 [30] König J, Schoeller H and Schön G 1996 Phys. Rev. Lett. 76 1715; König J, Schmid J, Schoeller H and Schön G 1996 Phys. Rev. B 54 16820 [31] Braun M, König J and Martinek J 2004 Phys. Rev. B 70 195345 http://arxiv.org/abs/cond-mat/0703264 Introduction Model Current formula Kinetic equations for quantum-dot degrees of freedom Non-equilibrium Josephson current Conclusion

0704.0205

Discovery of X-ray emission from the young radio pulsar PSR J1357-6429

Astronomy & Astrophysics manuscript no. 7480 October 24, 2018 (DOI: will be inserted by hand later) Discovery of X-ray emission from the young radio pulsar PSR J1357−6429 P. Esposito1,2, A. Tiengo2, A. De Luca2, and F. Mattana2,3 1 Università degli Studi di Pavia, Dipartimento di Fisica Nucleare e Teorica and INFN-Pavia, via Bassi 6, I-27100 Pavia, Italy e-mail: [email protected] 2 INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica Milano, via Bassini 15, I-20133 Milano, Italy 3 Università degli Studi di Milano - Bicocca, Dipartimento di Fisica G. Occhialini, p.za della Scienza 3, I-20126 Milano, Italy Received / Accepted Abstract We present the first X-ray detection of the very young pulsar PSR J1357−6429 (characteristic age of 7.3 kyr) using data from the XMM-Newton and Chandra satellites. We find that the spectrum is well described by a power-law plus black- body model, with photon index Γ = 1.4 and blackbody temperature kBT = 160 eV. For the estimated distance of 2.5 kpc, this corresponds to a 2–10 keV luminosity of ∼1.2 × 1032 erg s−1, thus the fraction of the spin-down energy channeled by PSR J1357−6429 into X-ray emission is one of the lowest observed. The Chandra data confirm the positional coincidence with the radio pulsar and allow to set an upper limit of 3× 1031 erg s−1 on the 2–10 keV luminosity of a compact pulsar wind nebula. We do not detect any pulsed emission from the source and determine an upper limit of 30% for the modulation amplitude of the X-ray emission at the radio frequency of the pulsar. Key words. stars: individual (PSR J1357−6429) – stars: neutron – X-rays: stars 1. Introduction X-ray observations of radio pulsars provide a powerful diag- nostic of the energetics and emission mechanisms of rotation- powered neutron stars. Due to the magnetic dipole braking, a pulsar loses rotational kinetic energy at a rate Ė = 4π2IṖP−3, where I is the moment of inertia of the neutron star, assumed to be 1045 g cm2, and P is the rotation period. Though pulsars have traditionally been mostly studied at radio wavelengths, only a small fraction (10−7 to 10−5, e.g., Taylor et al. 1993) of the “spin-down luminosity” Ė emerges as radio pulsations. Rotation power can manifest itself in the X / γ-ray energy range as pulsed emission, or as nebular radiation produced by a rela- tivistic wind of particles emitted by the neutron star. Residual heat of formation is also observed as soft X-ray emission from young neutron stars. Such thermal radiation, however, can also be produced as a result of reheating from internal or exter- nal sources. The growing list of observable X-ray emitting rotation-powered pulsars allows the study of the properties of the population as a whole. Young pulsars constitute a particu- larly interesting subset to investigate owing to their large spin- down luminosities (&1036 erg s−1). The discovery of PSR J1357−6429 during the Parkes multi- beam survey of the Galactic plane (see Lorimer et al. 2006 and references therein) is reported in Camilo et al. (2004). The pulsar is located near the supernova remnant candidate G309.8−2.6 (Duncan et al. 1997) for which no distance or age information is available. With a spin period of 166 ms, a pe- riod derivative of 3.6 × 10−13 s s−1, and a characteristic age τc = P/2Ṗ ≃ 7300 yr, this pulsar stands out as one of the ten youngest Galactic radio pulsars known (see the ATNF Pulsar Catalogue1, Manchester et al. 2005). The other main properties of this source derived from the radio observations are the spin- down luminosity of 3.1 × 1036 erg s−1 and the surface magnetic field strength of 7.8 × 1012 G, inferred under the assumption of pure magnetic dipole braking. Based on a dispersion measure of ∼127 cm−3 pc (Camilo et al. 2004), a distance of ∼2.4 kpc is estimated, according to the Cordes-Lazio NE2001 Galactic Free Electron Density Model2. Here we report the first detection of PSR J1357−6429 in the X-ray range using the XMM-Newton observatory and we present its spectral properties in the 0.5–10 keV energy band. We also made use of two short Chandra observations to con- firm the identification and to probe possible spatial extended emissions, taking advantage of the superb angular resolution of the Chandra telescope. 2. XMM-Newton observation and data analysis In this section we present the results obtained with the EPIC in- strument on board the XMM-Newton X-ray observatory. EPIC consists of two MOS (Turner et al. 2001) and one pn CCD 1 See http://www.atnf.csiro.au/research/pulsar/psrcat . 2 See http://rsd-www.nrl.navy.mil/7213/lazio/ne model and references therein. http://arxiv.org/abs/0704.0205v2 2 P. Esposito et al.: X-ray observations of PSR J1357−6429 detectors (Strüder et al. 2001) sensitive to photons with ener- gies between 0.1 and 15 keV. All the data reduction was per- formed using the XMM-Newton Science Analysis Software3 (SAS version 7.0). The raw observation data files were pro- cessed using standard pipeline tasks (epproc for pn, emproc for MOS data). Response matrices and effective area files were generated with the SAS tasks rmfgen and arfgen. The observation was carried out on 2005 August 17 and had a duration of 15 ks, yielding net exposure times of 11.7 ks in the pn camera and 14.5 ks in the two MOSs. The pn and the MOSs were operated in Full Frame mode (time res- olution: 73.4 ms and 2.6 s, respectively) and mounted the medium thickness filter. PSR J1357−6429 is clearly detected in the pn and MOS images (see Figure 1) at the radio pul- sar position (Right ascension = 13h 57m 02.4s, Declination = −64◦ 29′ 30.2′′ (epoch J2000.0); Camilo et al. 2004). The 13:57:12.0 13:57:00.0 13:56:48.0 -64:27:35.8 -64:28:47.8 -64:29:59.8 -64:31:11.8 Right ascension Figure 1. Field of PSR J1357−6429 as seen by the EPIC cam- eras in the 0.5–10 keV energy range. The radio pulsar posi- tion (Camilo et al. 2004) is marked with the white diamond sign. The angular separation of the centroid of the X-ray source (computed using the SAS task emldetect) from the radio pul- sar position is (3.5 ± 0.6)′′ (1σ statistical error). Considering the XMM-Newton absolute astrometric accuracy of 2′′ (r.m.s.), the X-ray and radio positions are consistent. source spectra were extracted from circular regions centered at the position of PSR J1357−6429. The whole observation was affected by a high particle background that led to the selec- tion of a 20′′ radius circle in order to increase the signal-to- noise ratio in the pn detector, particularly sensitive to particle background, and a 40′′ radius for both the MOS cameras. The background spectra were extracted from annular regions with radii of 140′′ and 220′′ for the MOSs, and from two rectan- gular regions with total area of ∼104 arcsec2 located on the sides of the source for the pn. We carefully checked that the choice of different background extraction regions does not af- fect the spectral results. We selected events with pattern 0–4 3 See http://xmm.vilspa.esa.es/ . Table 1. Summary of the XMM-Newton spectral results. Errors are at the 90% confidence level for a single interesting param- eter. Parameter Value PL PL +BB NH (10 22 cm−2) 0.14+0.07 −0.06 0.4 Γ 1.8+0.3 −0.2 1.4 ± 0.5 kBT (keV) – 0.16 +0.09 −0.04 a (km) – 1.4+2.9 Fluxb (10−13erg cm−2 s−1) 2.3 3.6 Blackbody fluxb (10−13erg cm−2 s−1) – 1.3 χ2r / d.o.f. 1.00 / 72 0.85 / 70 a Radius at infinity assuming a distance of 2.5 kpc. b Unabsorbed flux in the 0.5–10 keV energy range. and pattern 0–12 for the pn and the MOS, respectively. The re- sulting background subtracted count rates in the 0.5–10 keV energy range were (4.2 ± 0.3) × 10−2 cts s−1 in the pn and (1.9 ± 0.2) × 10−2 cts s−1 in the two MOS cameras, while the background rate expected in the source extraction regions is about 50% of these values. The spectra were rebinned to have at least 20 counts in each energy bin. Spectral fits were per- formed using the XSPEC version 12.3 software4. The spectra from the three cameras were fitted together in the 0.5–10 keV energy range with a power law and with a power-law plus blackbody model (see Table 1). The latter model provides a slightly better fit, with less structured resid- uals (see Figure 2). Furthermore, considering the distance of 2.5 kpc, the interstellar absorption along the line of sight de- rived with the power-law fit is too low if compared to the typical column density of neutral absorbing gas in that direc- tion of approximately 1022 cm−2 (Dickey & Lockman 1990). The resulting best-fit parameters for the power-law plus black- body model are photon index Γ = 1.4, blackbody temperature kBT = 0.16 keV, and absorption NH = 4× 10 21 cm−2 with a re- duced χ2 of 0.85 for 70 degrees of freedom. The corresponding luminosity in the 0.5–10 keV band is 2.7 × 1032 erg s−1. Young pulsars are often associated with pulsar wind neb- ulae: complex structures that arise from the interaction be- tween the particle wind powered by the pulsar and the supernova ejecta or surrounding interstellar medium (see Gaensler & Slane 2006 for a review). Inspecting the EPIC im- ages in various energy bands, we find only a marginal (≈3σ) evidence of diffuse emission, in the 2–4 keV energy band con- sisting of a faint elongated (∼20 arcsec to the north-east, see Figure 1) structure starting from PSR J1357−6429. We took that excess as an upper limit for a diffuse emission: assuming the same spectrum as the point source, it corresponds to a 2–10 keV luminosity of ≈6 × 1031 erg s−1. For the timing analysis we applied the solar system barycenter correction to the photon arrival times with the SAS task barycen. We searched the data for pulsations around the spin frequency at the epoch of the XMM-Newton observations, predicted assuming the pulse period and the spin-down rate measured with the Parkes radio telescope (Camilo et al. 2004). 4 See http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/ . P. Esposito et al.: X-ray observations of PSR J1357−6429 3 1 2 5 Energy (keV) Figure 2. EPIC pn spectrum of PSR J1357−6429. Top: Data and best-fit power-law (dashed line) plus blackbody (dot- dashed line) model. Middle: Residuals from the power-law best-fit model in units of standard deviation. Bottom: Residuals from the power-law plus blackbody best-fit model in units of standard deviation. As glitches and / or deviations from a linear spin-down may al- ter the period evolution, we searched over a wide period range centered at the value of ∼166 ms. We searched for significant periodicities using two methods: a standard folding technique and the Rayleigh statistic. No pulsation were detected near to the predicted frequency with either method but, since the pn timing resolution (73 ms) allows to only poorly sample the 166 ms pulsar period, a reliable upper limit on the pulsed frac- tion cannot be set. 3. Chandra observations and data analysis PSR J1357−6429 was observed by means of the Chandra X-ray Observatory during two exposures of ∼17 ks duration each on 2005 November 18 and 19. The observations were car- ried out with the Spectroscopic array of the High Resolution Camera (HRC-S; Murray et al. 2000) used without transmis- sion gratings. The HRC is a multichannel plate detector sen- sitive to X-ray over the 0.08–10 keV energy range, although essentially no energy information on the detected photons is available. The HRC-S time resolution is 16 µs. We started from “level 1” event data calibrated and made available through the Chandra X-ray Center5. The level 1 event files contain all HRC triggers with the position information corrected for instrumental (degap) and aspect (dither) effects. After standard data processing with the Chandra Interactive Analysis of Observations (CIAO ver. 3.3), a point-like source has been clearly detected in both the observations at a position consistent with that of PSR J1357−6429 (see Figure 3). For the timing analysis we corrected the data to the solar system barycenter with the CIAO task axbary and then we fol- 5 See http://cxc.harvard.edu/. 13:57:02.9 13:57:02.4 13:57:02.0 -64:29:27.4 -64:29:31.0 -64:29:34.6 Right ascension Figure 3. Chandra 0.08–10 keV HRC-S image centered on the radio pulsar position, marked with a diamond sign (Camilo et al. 2004). The CIAO celldetect routine yields a best-fit position for the X-ray source at an angular distance of (0.9 ± 0.2)′′ (1σ statistical error) from the radio pulsar. This value consistent with the Chandra pointing accuracy of 0.8′′ (99% confidence level). lowed the same procedure described in Section 2, but we again did not detect the source pulsation. By folding the light curve of PSR J1357−6429 on the radio frequency and fitting it with a sinusoid, we determine a 90% confidence level upper limit of ∼30% on the amplitude of a sinusoidal modulation. We stress that this upper limit depends sensitively on data time binning and on the assumed pulse shape. We used the CIAO task merge all to generate a combined image of the source. Our main purpose was to search for diffuse structures on scales smaller than the XMM-Newton angular res- olution. We compared the radial profile of the pulsar emission with the Chandra High-Resolution Mirror Assembly point- spread function at 1 keV generated using Chandra Ray Tracer (ChaRT) and Model AXAF Response to X-rays (MARX). We found that the emission we detect from PSR J1357−6429 (∼100 counts concentrated within a ∼0.5′′ radius circle) is con- sistent with that from a point source. We used the Chandra data and the PIMMS software6 to determine an upper limit on the luminosity of a possible spatial extended emission. The 3σ upper limit on a pulsar wind nebula brightness (in counts s−1) has been estimated as 3(bA)1/2τ−1, where b is the background surface brightness in counts arcsec−2, A is the pulsar wind nebula area, and τ is the exposure duration. Assuming the interstellar absorption value from the XMM-Newton best-fit model (NH = 0.4 × 10 22 cm−2, see Section 2) and typical parameters for a pulsar wind nebula (radius of ∼2× 1017 cm, that corresponds to ∼5′′ for a distance of 2.5 kpc, and power-law spectrum with photon index Γ = 1.6, see, e.g., Gotthelf (2003)), this upper limit corresponds to a 2–10 keV luminosity of ≈3 × 1031 erg s−1 for a uniform dif- fuse nebula. No significant diffuse excess was found even at 6 See http://heasarc.gsfc.nasa.gov/docs/tools.html . 4 P. Esposito et al.: X-ray observations of PSR J1357−6429 larger angular scale, but the corresponding upper limit for dif- fuse emission is less constraining than that derived using the XMM-Newton data. 4. Discussion We have presented the results of the first X-ray observations of PSR J1357−6429 by means of the XMM-Newton and Chandra observatories. The source has been positively detected in all the instruments although, probably due to the low statistics, we could not detect the source pulsation. The high angular resolu- tion Chandra observations favor the picture in which most of the counts belong to a point source. We found that the spectrum is well represented by either a power-law with photon index Γ = 1.8+0.3 −0.2 or by a power-law plus blackbody model. In the latter case the best-fit parameters are for the power-law com- ponent a photon index Γ = 1.4 ± 0.5 and, for the blackbody component, radius7 of ∼1.4+2.9 −0.2d2.5 km and temperature corre- sponding to kBT = 0.16 +0.09 −0.04 keV. It is generally believed that a combination of emission mechanisms are responsible for the detected X-ray flux from rotation-powered pulsars (see, e.g., Kaspi et al. 2006 for a re- view). The acceleration of particles in the neutron star magne- tosphere generates non thermal radiation by synchrotron and curvature radiation and / or inverse Compton processes, while soft thermal radiation could result by cooling of the surface of the neutron star. A harder thermal component can arise from polar-cap reheating, due to return currents from the outer gap or from close to the polar-cap. The dominant emission mechanism is likely related to the age of the pulsar. In pulsar younger than ≈104 yr the strong magnetospheric emission generally prevails over the thermal radiation, making difficult to detect it. As discussed in Section 2, we tend to prefer the power-law plus blackbody spectral model for PSR J1357−6429. The re- sulting blackbody size of ∼1.5d2.5 km may suggest that the soft emission (.2 keV) is coming from hot spots on the surface due to backflowing particles, rather than from the entire surface. However this hint should be considered with caution, as the surface temperature distribution of a neutron star is most likely non uniform (since the heath conductivity of the crust is higher along the magnetic field lines) and the small and hot blackbody could result from a more complicated distribution of tempera- ture. Moreover, currently we lack of reliable models of cooling neutron star thermal emission and thus we cannot exclude that the soft component is emitted from surface layers of the whole neutron star. To date, thermal emission has been detected in only a few young radio pulsars. Among these, the properties of PSR J1357−6429 are similar to those of the young pul- sars Vela (PSR B0833−45; τc = 11 kyr, P = 89 ms, Ė = 6.9 × 1036 erg s−1, and distance d ≃ 0.2 kpc; Pavlov et al. 2001) and PSR B1706–44 ( τc = 17.5 kyr, P = 102 ms, Ė = 3.4 × 1036 erg s−1, and d ≃ 2.5 kpc; Gotthelf et al. 2002). Notably, the efficiency in the conversion of the spin-down energy loss into X-ray luminosity for PSR J1357−6429 is L0.5−10 keV/Ė ≃ 8d 2.5×10 −5, significantly lower than the typical 7 We indicate with dN the distance in units of N kpc. value of ≈10−3 (Becker & Truemper 1997), and similar to that of PSR B1706–44 (∼10−4) and Vela (∼10−5). Although a pulsar wind nebula would not came as a surprise for this young and energetic source, we did not find clear evidence of diffuse X-ray emission associated with PSR J1357−6429. However, some known examples of wind nebulae (see Gaensler & Slane 2006), rescaled to the distance of PSR J1357−6429, would hide below the upper limits derived from the XMM-Newton and Chandra data. New deeper exposures using XMM-Newton or Chandra would help determine if a thermal component is present in the emission of PSR J1357−6429 as our spectral analysis suggests, and possibly detect a pulsed emission. High sensitivity obser- vations would also serve to address the issue of the presence of a pulsar wind nebula. Although there is not any EGRET γ-ray source coincident with PSR J1357−6429 (Hartman et al. 1999), young neutron stars and their nebulae are often bright γ-ray sources and PSR J1357−6429 in particular, given its high “spin-down flux” Ė/d2 and similarity with Vela and PSR B1706–44, is likely to be a good target for the upcoming AGILE and GLAST satellites and the ground based Cherenkov air showers telescopes. Acknowledgements. This work is based on data from observations with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA member states and NASA. We also used data from the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory Center on be- half of NASA. The authors thank the anonymous referee for helpful comments and acknowledge the support of the Italian Space Agency and the Italian Ministry for University and Research. References Becker, W. & Truemper, J. 1997, A&A, 326, 682 Camilo, F., Manchester, R. N., Lyne, A. G., et al. 2004, ApJ, 611, L25 Dickey, J. M. & Lockman, F. J. 1990, ARA&A, 28, 215 Duncan, A. R., Stewart, R. T., Haynes, R. F., & Jones, K. L. 1997, MNRAS, 287, 722 Gaensler, B. M. & Slane, P. O. 2006, ARA&A, 44, 17 Gotthelf, E. V. 2003, ApJ, 591, 361 Gotthelf, E. V., Halpern, J. P., & Dodson, R. 2002, ApJ, 567, Hartman, R. C., Bertsch, D. L., Bloom, S. D., et al. 1999, ApJS, 123, 79 Kaspi, V. M., Roberts, M. S. E., & Harding, A. K. 2006, in Compact stellar X-ray sources, ed. W. H. G. Levin and M. van der Klis (Cambridge: Cambridge University Press), 279 Lorimer, D. R., Faulkner, A. J., Lyne, A. G., et al. 2006, MNRAS, 372, 777 Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, 1993 Murray, S. S., Austin, G. K., Chappell, J. H., et al. 2000, in Proc. SPIE Vol. 4012, X-Ray Optics, Instruments, and Missions III, ed. J. E. Truemper & B. Aschenbach, 68 Pavlov, G. G., Zavlin, V. E., Sanwal, D., Burwitz, V., & Garmire, G. P. 2001, ApJ, 552, L129 Strüder, L., Briel, U., Dennerl, K., et al. 2001, A&A, 365, L18 P. Esposito et al.: X-ray observations of PSR J1357−6429 5 Taylor, J. H., Manchester, R. N., & Lyne, A. G. 1993, ApJS, 88, 529 Turner, M. J. L., Abbey, A., Arnaud, M., et al. 2001, A&A, 365, L27 Introduction XMM-Newton observation and data analysis Chandra observations and data analysis Discussion

0704.0206

Resonant activation in bistable semiconductor lasers

Resonant activation in bistable semiconductor lasers Stefano Lepri1, ∗ and Giovanni Giacomelli1 Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy (Dated: November 4, 2018) We theoretically investigate the possibility of observing resonant activation in the hopping dynam- ics of two-mode semiconductor lasers. We present a series of simulations of a rate-equations model under random and periodic modulation of the bias current. In both cases, for an optimal choice of the modulation time-scale, the hopping times between the stable lasing modes attain a minimum. The simulation data are understood by means of an effective one-dimensional Langevin equation with multiplicative fluctuations. Our conclusions apply to both Edge Emitting and Vertical Cavity Lasers, thus opening the way to several experimental tests in such optical systems. PACS numbers: 42.55.Px, 05.40.-a, 42.65.Sf I. INTRODUCTION It is currently established that stochastic fluctuations may have a constructive role in enhancing the response of nonlinear systems to an external coherent stimulus. Relevant examples are the enhancement of the decay time from a metastable state (noise–enhanced stability) [1, 2], the synchronization with a weak periodic input sig- nal (stochastic resonance) [3] or the regularizaton of the response at an optimal noise intensity (coherence reso- nance) [4]. Another instance is the phenomenon of resonant acti- vation that was discovered by Doering and Gadoua [5]. They showed that the escape of an overdamped Brown- ian particle over a fluctuating barrier can be enhanced by suitably choosing the correlation time of barrier fluc- tuations themselves. In other words, the escape time from the potential well attains a minimum for an optimal choice of such correlation time. Since its discovery, the phenomenon received a considerable attention from the- orists (see e.g. Refs. [6, 7, 8, 9, 10]). Detailed studies by means of analog simulations have also been reported for both Gaussian and dichotomous fluctuations [11]. More recently, the phenomenon has been shown to occur also for the case in which the barrier oscillates periodically [12, 13]. To our knowledge, experimental evidences of resonant activation were only given for a bistable electronic cir- cuit [14] and, very recently, for a colloidal particle sub- ject to a periodically–modulated optical potential [15]. It is therefore important to look for other setups where the effect could be studied in detail. As a matter of fact, multimode laser systems are good candidates to investigate noise–activated dynamics like the switching among modes induced by quantum fluctuations (sponta- neous emission) [16]. In particular, semiconductor lasers proved to be particularly versatile for detailed experi- ∗Electronic address: [email protected] mental investigations of modulation and noise-induced phenomena like stochastic resonance [17, 18] and noise– induced phase synchronization [19]. In those previous studies, the resonance regimes are attained by a suit- able random modulation of the bias current which can be tuned in a well-controlled way. It is thus natural to argue about the possibility of observing resonant activa- tion with the same type of experimental setup. In this paper, we theoretically demonstrate the phe- nomenon of resonant activation in a generic rate– equations model for a two-mode semiconductor laser un- der modulation of the bias current. The basic ingredients that act in the theoretical descriptions are a fluctuating potential barrier and some activating noise. In the laser system, the latter is basically provided by spontaneous emission while current fluctuations, that appear addi- tively into the rate equations, effectively act multiplica- tively if a suitable separation of time scales holds [20]. In a previous paper [21], we have explicitely demonstrated such multiplicative–noise effects on the mode–hopping dynamics. This was shown by a reduction to a bistable one–dimensional potential system with both multiplica- tive and additive stochastic forces. Several predictions drawn from such a simplified model are in good agree- ment with the experimental observations carried out for a bulk, Edge-Emitting Laser (EEL) [21]. In the present context, we will show that this reduced description is of great help in the interpretation of simulation data. The outline of the paper is the following. In Sec. II we recall the model for a two-mode semiconductor laser. In Sec. III we present the numerical simulation for two physically distinct cases displaying resonant activation. These results are discussed and interpreted by compar- ing with the reduced one–dimensional Langevin model mentioned above (Sec. IV). We draw our conclusions in Sec. V. II. RATE EQUATIONS Our starting point is a stochastic rate-equation model for a semiconductor laser that may operate in two longi- http://arxiv.org/abs/0704.0206v1 mailto:[email protected] tudinal modes whose complex amplitudes are denoted by E±. Both of them interact with a single carrier density N that provides the necessary amplification. The two modes have very similar linear gains, provided that their wavelengths are almost equal and they are close to the gain peak. Let J(t) denote the bias (injection) current, the model can be written as [21] Ė+ = (1 + iα)g+ − 1 2DspN ξ+ (1a) Ė− = (1 + iα)g− − 1 2DspN ξ− (1b) Ṅ = γ J(t)−N − g+|E+|2 − g−|E−|2 where γ is carrier density relaxation rate, α is the linewidth enhancement factor [22]. The modal gains read N ± ε(N −Nc) 1 + s|E±|2 + c|E∓|2 , (2) where ε determines the difference in differential gain among the two modes while Nc defines the carrier density where the unsaturated modal gains are equal. The parameters s and c are respectively the self- and cross-saturation coefficients. The ξ± are two indepen- dent, complex white noise processes with zero mean [〈ξ±(t)〉 = 0] and unit variance [〈ξi(t)ξ∗j (t′)〉 = δijδ(t−t′)] that model spontaneous emission. The noise terms in Eqs. (1a) and (1b) are gauged by the spontaneous emis- sion coefficient Dsp. All quantities are expressed in suitable dimensionless units. In particular, time is normalized to the photons’ lifetime, which for semiconductor laser is typically of the order of a few picoseconds or less (see e.g. [22, 23, 24]) A detailed analysis of the stationary solutions of Eqs. (1) is reported in Ref. [25]. For a constant bias cur- rent J(t) = J0 and Dsp = 0, Eqs. (1) admit four different steady state solutions: the trivial one E± = 0, two single- mode solutions — E+ 6= 0, E− = 0 and viceversa — and a solution where both modes are lasing, E± 6= 0. For Nc > 1, and c > s, there exist a finite interval of J0 val- ues for which the two single–mode solutions coexist and are stable while the E± 6= 0 is unstable (bistable region). Here, for Dsp > 0 the laser performs stochastic mode- hopping, with the total emitted intensity remains almost constant while each mode switches on and off alternately at random times. We point out that the emission in each mode is nonvanishing even in the “off” state, as the av- erage power spontaneously emitted in each mode at any time is given by 4DspN [recall that Eqs. (1) are usually interpreted in Itô sense [23]]. Observation of this be- haviour has been reported in several experimental works on EELs [26, 27, 28]. We remark that while Eqs. (1) aim at modeling EELs, the results presented henceforth would apply also to po- larization switching in Vertical Cavity Surface Emitting Lasers (VCSELs). Indeed, experimental data [29] show strong similarities between this phenomenon and the lon- gitudinal mode dynamics. On the theoretical side, this analogy is supported by the fact that the polarization dy- namics in VCSELs is described by models that are math- ematically similar to the one discussed here [30, 31, 32]. In the following, we will focus on the effect of the ex- ternally imposed fluctuation/modulation of the injected current. This situation is modeled by letting J(t) = J0 + δJ(t) . (3) The DC value J0 sets the working point and will be al- ways chosen to be in the bistability region. We focus on the case in which δJ is a Ornstein-Uhlenbeck process with zero average 〈δJ(t)〉 = 0 and correlation time τ : ˙δJ = − ξJ (4) that means 〈δJ(t)δJ(0)〉 = DJ exp(−|t|/τ) . (5) This choice is suitable to model a finite-bandwith noise generator. Notice that τ and the variance of fluctuations DJ = 〈δJ2〉 can be fixed independentely. Another case of experimental interest that we will con- sider is using the current modulation δJ = A sinΩt (6) To assess the nature of the stochastic process at hand, it is important to introduce the relevant time scales. We define first of all the switching or relaxation time TR as the typical time for the emission to change from one mode to the other. The main quantities we are interested in are the Kramers or residence times T± defined as the average times for which the emission occurs in each mode. In semiconductor lasers T± are generally much larger than TR. Typically, TR ∼ 1− 10ns while residence times may range between 0.1 and 100 µs [29, 33]. The third time- scale is of course given by the characteristic time of the external driving, namely, τ and 2π/Ω respectively. In the following, we will study how the hopping dy- namics changes upon varying these latter parameters as well as the strength of the perturbation. III. NUMERICAL SIMULATIONS In this Section we present the outcomes of a series of numerical simulation of Eqs. (1). In Ref. [21] it was observed that the sensitivity of each of the T± on the imposed current fluctuations may be notably different depending on the parameters’ choice. This is a typi- cal signature of the multiplicative nature of the stochas- tic process. In particular, one can argue [21] that such “simmetry-breaking” effects mostly depend on the ratio εσ/δ where , δ = . (7) The parameter σ represents the gain saturation induced by the total power in the laser, while δ describes the reduction in gain saturation due to partitioning of the power between the two modes. The possibility of obtaining qualitatively different re- sponses depending on the actual parameters corresponds to the different experimental observations reported for both EELs [21, 28, 33] and VCSELs [17, 29]. Those two classes of lasers were indeed found to display markedly different simmetry-breaking effects under current modu- lation. To account for those features, we consider two different sets of phenomenological parameters. For defi- niteness, in both cases we fix ε = 0.1, s = 1.0, Nc = 1.1, γ = 0.01 and change the values of c and Dsp (see Table I). The first set (δ = 0.05) corresponds to the case in which added modulation changes the hopping time scale in an almost symmetric way. On the contrary, in the second case (δ = 0.15) the asymmetry effect of the noise is stronger [21]. We can thus consider the two as rep- resentative of the VCSELs and EELs case respectively. The value of J0 has been empirically adjusted to yield T+ ≃ T− ≡ Ts and an almost symmetric distribution of intensities in absence of modulation. The actual values are about 10% above the laser threshold. The sponta- neous emission coefficient Dsp has been chosen to yield a value of the residence times of the same order of magni- tude of the experimental ones. In the following, we decide to set α = 0 which is ap- propriate for our EEL model where the phase dynamics is not relevant [21]. This choice may however not be fully justified for the VCSEL case. In this respect, the sim- ulations presented below are representative of the VC- SEL dynamics only in a qualitative sense. Nonetheless, it should be pointed out that a 1D Langevin model in- dependent of α describes also the VCSEL case [30, 31]. Since resonant activation is mainly due to the multiplica- tive noise effect described by such equations [see Eq. (9) below] we consider this as an indirect proof that phe- nomenology we will report below should be observable also in the VCSEL case. The largest part of the simulations were performed with Euler method with time steps 0.01-0.05 for times in the range 107 − 108 time units depending on the val- ues of τ and Ω. For comparison, some checks with Heun method [34] have also been carried on. Within the sta- tistical accuracy, the results are found to be insensitive the the choice of the algorithm. A. Stochastic modulation Let us start illustrating the results in the case of stochastic current modulation (Eq. (4)). In Fig. 1 and TABLE I: The parameter values used in the two series of simulations of Eqs. (1), the other values are given in the text. c Dsp J0 δ σ 1.1 0.7 × 10−5 1.197 0.05 1.05 1.3 1.5 × 10−5 1.194 0.15 1.15 2 we report the measured dependence of the residence times T± on the correlation time τ for the two parameter sets given in Table I and different values of the noise vari- ance DJ . In all cases, the curves display well-pronounced minima at an optimal value of τ . This is the typical sig- nature of resonant activation. The minima are almost located between the relaxation time TR and the hopping time Ts (marked by the vertical dashed lines). The val- ues of TR reported in the figures have been estimated from the reduced model discussed in the next Section, see Eq. (14) below. The effect manifest in a different way for the second parameter set. In the case of Fig. 1 both times attain a minimum, albeit with different values. On the contrary the data of Fig. 2 show that one of the two times is hardly affected from the external perturbation regardless of the value of τ . In other terms, we can tune the current corre- lation in such a way that emission along only one of the two modes is strongly reduced (about a factor 10 in the simulation discussed here). B. Periodic modulation Let us now turn to the case of sinusoidal current mod- ulation (Eq. 6). In Fig. 3 and 4 we report the measured dependence of the residence times T± on the frequency Ω for the two parameter sets given in Table I and different values of the amplitude A. For comparison with the pre- vious case we choose A such that the RMS value of (6) is roughly equal to the variance of (4), i.e. A ≃ 2DJ . As in the previous case, the curves display resonant activation at an optimal value of Ω. For the second set of parameters, one of the two hopping times is more re- duced than the other (compare Fig. 4 with Fig. 2). It should be also noticed that the data in Fig. 2 display some statistical fluctuations while the curves for the pe- riodic modulation are smoother. IV. INSIGHTS FROM A REDUCED MODEL In order to better understand the activation phe- nomenon it is useful to reduce the five–dimensional dy- namical system (1) to an effective one-dimensional sys- tem. This has been accomplished in Ref. [21]. For com- pleteness, we only recall here some basic steps of the derivation. In the first place, we introduce the change of Correlation time τ =4 10 =4 10 =4 10 FIG. 1: (Color online) Simulations of the rate equations with Ornstein-Uhlenbeck current fluctuations, parameter set with c = 1.1 (see text and Table I): residence times T+ (squares) and T− (circles) for increasing values of the current variance DJ . The values of the relaxation time TR and the hopping time Ts (in absence of modulation) are marked by the vertical dashed lines. coordinates E+ = r cosφ exp iψ+, E− = r sinφ exp iψ− . (8) In these new variables, r2 is the total power emitted by the laser, and φ determines how this power is partitioned among the two modes. The values φ = 0, π/2 correspond to pure emission in mode + and − respectively. The phases ψ± do not influence the evolution of the modal amplitudes and carrier density and can be ignored. In order to simplify the analysis, we assume that (i) The difference between modal gains is very small, i.e., Nc >∼ 1, ε≪ 1, c >∼ s; (ii) the laser operates close enough to threshold, so that r2 ≪ 1 and the saturation term is small: in this limit, r and N decouple to leading order from φ; (iii) r and N can be adiabatically eliminated and (iv) only their fluctuations around the equilibrium val- ues due to J are retained. This last assumption holds for weak spontaneous noise and amounts to say that r and Correlation time τ =1 10 =5 10 =1 10 FIG. 2: (Color online) Simulations of the rate equations with Ornstein-Uhlenbeck current fluctuations, parameter set with c = 1.3 (see text and Table I): residence times T+ (squares) and T− (circles) for increasing values of the current variance N are stochastic processes given by nonlinear transfor- mations of J (see Eqs. (16) in Ref. [21]). This requires that J does not change too fast. For example, in the case of the Orstein–Uhlenbeck process, Eq. (4), τ should be larger than the relaxation time of the total intensity. The validity of the above reduction has been carefully checked against simulations of the complete model [21]. For the scope of the present work, we performed a fur- ther check by comparing the spectrum of fluctuations of r2 with the imposed one, Eq. (4). Indeed, the behaviour is the same for τ > TR while for shorter τ some differ- ences are detected. This means that the reduced descrip- tion discussed below becomes less and less accurate. On the other hand, in this regime spontaneous fluctuation should dominate and this limitation become less relevant for our purposes. Altogether, the hopping dynamics is effectively one- dimensional and is described by the slow variable φ. Its Modulation period 2π/Ω A=0.020 A=0.090 A=0.009 FIG. 3: (Color online) Simulations of the rate equations with sinusoidal modulation of the current, parameter set with c = 1.1 (see text and Table I): residence times T+ (squares) and T− (circles) for increasing values of modulation amplitude A. evolution is ruled by the effective Langevin equation φ̇ = −1 a cos 2φ+ b sin 2φ + tan 2φ 2Dφ ξφ (9) where, together with (7) we have defined the new set of parameters (1 + σ)Nc − 1 1 + σ (J − 1) (11) 1 + σ (J − Js) (12) (1 + σJ)2 (1 + σ)(J − 1) Dsp . (13) We remind in passing that the same equation (9) has been derived by Willemsen et al. [30, 31] to describe po- larization switches in VCSELs (see also Ref. [35] for a similar reduction). The starting point of their derivation Modulation period 2π/Ω 6 A=0.014 A=0.030 A=0.055 FIG. 4: (Color online) Simulations of the rate equations with sinusoidal modulation of the current, parameter set with c = 1.3 (see text and Table I): residence times T+ (squares) and T− (circles) for increasing values of modulation amplitude A. is the San Miguel-Feng-Moloney model [36]. The physi- cal meaning of the variable φ is different from here as it represents the polarization angle of emitted light. This supports the above claim that, upon a suitable reinterpre- tation of variables and parameters, many of the results presented henceforth may apply also to the dynamics of VCSELs. In absence of modulation (δJ = 0), Eq. (9) is bistable in an interval of current values where it admits two stable stationary solutions φ± and an unstable one φ0 (double- well). This regime correspond to the bistability region of model (1). Notice that for J0 = Js, b = 0 the hopping between the two modes occurs at the same rate. The above definitions allows an estimate of relaxation time TR defined above. This is is the inverse of the curvature of the potential in φ0. For J0 = Js this is straightforwardly evaluated to be (1 + σ) δ(Js − 1) For the two parameter sets given in Table I one finds TR = 210 TR = 77.0, respectively. These are the values emploied to draw the leftmost vertical lines in Figs. 1-4. The effect of a time-dependent current is to make the coefficients a, b and Dφ fluctuating. It can be shown [21] that the effect onDφ can be recasted as a renormalization of the intensity of the spontaneous-emission noise. How- ever, for the parameters employed in the present work it turns out that this correction is pretty small and will be neglected henceforth by simply considering Dφ as con- stant [38]. For simplicity, we also disregard the depen- dence of Dφ on δJ in the drift term of Eq. (9). Under those further simplifications the Langevin equation can be rewritten as φ̇ = −U ′(φ)− V ′(φ) δJ + 2Dφ ξφ (15) where we have express the force term as derivatives of the “potentials” U(φ) = − δ(J0 − 1) 16(1 + σ) cos 4φ− εσ(J0 − Js) 4(1 + σ) cos 2φ −Dφ ln sin 2φ (16) V (φ) = − δ 16(1 + σ) cos 4φ− εσ 4(1 + σ) cos 2φ. (17) Langevin equations of the form (15) with (4) have been thoroughly studied in the literature (see e.g. [7, 8, 9, 10, 11] and references therein) as prototypical examples of the phenomenon of activated escape over a fluctuating barrier. In view of their non-Markovian nature, their full analytical solution for arbitrary τ is not generally feasible. Several approximate results can be provided in some limits. For an arbitrary choice of the parameters, V has a different symmetry with respect to U meaning that the effective amplitude of multiplicative noise is different within the two potential wells. If this difference is large enough, current fluctuation will remove the degeneracy between the two stationary solutions. This is best seen by computing the istantaneous potential barriers ∆U±(t) close to the symmetry point J0 = Js . For weak noise and δJ ≪ (Js − 1), they are given to first-order in δJ(t) ∆U±(t) ≃ 8(1 + σ) (Js − 1) + δ ± 2εσ 8(1 + σ) δJ(t) . (18) Obviously, this last expression makes sense only when the fluctuating term is sub-threshold i.e. whenever the system is bistable. In the case of periodic modulation, formula (18) allows estimating the range of amplitude values for a sub-threshold driving δ(Js − 1) δ ± 2εσ . (19) Using this condition, along with the parameter values at hand, we deduce that the cases displayed in lower panels of Figs. 3 and 4 correspond to superthreshold driving. However, while the minima are much more pronounced than in the other panels, there is no qualitative difference in the system response. In the case of stochastic modu- lation, the same remark applies in a probabilistic sense for the last panels of Figs. 1 and 2. Altogether, the mode switching can be seen as an acti- vated escape over fluctuating barriers given by Eq. (18). The statistical properties of the latter process is con- trolled by the current fluctuations. We now discuss the properties of various regimes. For simplicity, we refer to the case of stochastic modulations. Most of the re- marks and formulas reported in the following Subsection should apply also to the periodic case by replacing τ and DJ with 2π/Ω and A 2/2 whenever appropriate. A. Fast barrier fluctuations: τ < TR ≪ T± As we already pointed out, in this regime the reduc- tion to Eq. (15) is not justified. We may thus only expect some qualitative insight on the behaviour of the rate- equations. From a mathematical point of view, some analytical approximations for equations like (15) are fea- sible in this limit (see e.g. Ref. [8] for the stochastic case). For our purposes, it is sufficient to note that in this regime the effect of δJ is hardly detected for both types of driving (see again Figs. 1-4). Note also that working at DJ fixed means that for τ → 0 the fluctua- tion become negligible. B. Resonant activation: TR < τ ≪ T± If TR < τ we are in the colored noise case. The prob- lem is amenable of a kinetic description which amounts to neglect intrawell motion and reduce to a rate model describing the statistical transitions in terms of transition rates. If we consider τ as a time scale of the external driv- ing we can follow the terminology of Ref. [39] and refer to this situation as the “semiadiabatic” limit of Eq. (15). In this regime, the residence time is basically the short- est escape time, which in turn correspond to the lowest value of the barrier (the noise is approximatively con- stant in the current range considered henceforth). For the case of interest, δ < 2εσ we can use (18) to infer that the minimal values of ∆U± should be attained for δJ ∝ ∓ DJ respectively. This yields T± ≃ Ts exp −K 2εσ ± δ 1 + σ where K is a suitable numerical constant. Notice that δ controls the asymmetry level: if δ ≪ 2εσ the two resi- dence times decrease at approximatively the same rate. This prediction is verified in the simulations and also in the experiment [21]. As a further argument in support of the above rea- soning, we also evaluated the probability distributions of the residence times obtained from the simulation of the rate equations. In Fig. 5, we show two representa- tive cumulative distributions. The data are well fitted by a Poissonian P (T ) = 1 − exp(−T/T±) for both the stochastic and periodic modulation cases. This confirms that hopping occours preferentially when a given (mini- mal) barrier occurs. 0 0.5 1 1.5 0 1 2 3 4 FIG. 5: (Color online) Cumulative distributions of the resi- dence times in the resonant activation region, parameter set with c = 1.3 (see text and Table I). Left panel: stochastic modulation with DJ = 5×10 −4, τ = 1.638×103 . Right panel: periodic modulation with A = 0.03 and period 1.286 × 104. We report only the histograms for the times whose averages are denoted by T+ in the text. Solid line is the cumulative Poissonian distribution with the same average. C. Slow barrier, frequent hops: TR ≪ T± ≪ τ This corresponds to the adiabatic limit in which the time scale of the external driving is slower than the intrin- sic dynamics of the system [39]. To a first approximation we can here treat current fluctuations in a parametric way. Correction terms may be evaluated by means of a suitable perturbation expansion in the small parame- ter 1/τ [10]. If δJ is small enough for the expression (18) to make sense, the escape time can be estimated as the average of escape times over the distribution of bar- rier fluctuations, i.e. 〈T±〉δJ . For the case of Eq. (4), the variable δJ is Gaussian and we can use the identity 〈expβz〉 = exp(β2〈z2〉/2) to obtain [11] T± ≃ Ts exp [2(δ ± 2εσ)2 (1 + σ)2D2 . (21) This reasoning implies that for large τ the residence times should approach two different constant values. A closer inspection of the graphs (in linear scale) reveals that this is not fully compatible with the data of Fig.1 even for the smallest value of DJ . In several cases, T± continue to increase with τ and no convincing evidence of saturation is observed. We note that the same type of behaviour was already observed in the analog simulations data of Ref. [11]. There, an increase of hopping times duration at large τ was found. The Authors of Ref. [11] explained this as an effect of a too large value of the noise fluctuation forcing the system to jump roughly every τ . We argue that the same explanation holds for our case. This is also consistent with the fact that the exponential factors in Eq. (21) evaluated with the simulation parameters turn out to be much larger than unity. V. CONCLUSIONS In this paper, we have explored numerically and ana- lytically the effects of external current fluctuations on the mode-hopping dynamics in a model of a bistable semicon- ductor laser. To the best of our knowledge, this setup provides the first theoretical evidence of resonant activa- tion in a laser system. As the phenomenon has hardly received any experimental confirmation in optics, we be- lieve that our study may open the way to future research in this subfield. The model we investigated is based on a rate-equation description, where the bias current enters parametrically into the evolution of the modal amplitudes. We consid- ered, two kinds of current flutuations, namely, a stochas- tic process ruled by an Orstein-Uhlenbeck statistics, and a coherent, sinusoidal modulation. These choices are mo- tivated by the aim of proposing a suitable setup for an experimental verification of our results. Upon varying the characteristic time-scale of the imposed fluctuations, we have shown that the residence times attain a minimum for a well-defined value, which is the typical signature of resonant activation. The magnitude of the effect can be different depending on the parameters of the model. Moreover, the response of the system appears very much similar for both periodic and random modulations. The reduction of the rate equations to a one- dimensional Langevin equation allowed us to recast the problem as an activated escape over a fluctuating bar- rier. To first approximation, the fluctuating barrier (mul- tiplicative term) is mainly controlled by current modu- lations while the spontaneous noise act as an additive source. This simplified description has allowed us to draw some predictions (e.g. the dependence of residence times on noise strength) and to better understand the role of the physical parameters. Given the generality of the description, our results should apply to a broad class of multimode lasers, including both Edge Emitting and Vertical Cavity Lasers. From an experimental point of view, driving the laser in a orders-of-magnitude wide range of time-scales is more feasible in the case of a sinusoidal modulation than for a colored, high frequency noise. 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0704.0207

Quark matter and the astrophysics of neutron stars

Quark matter and the astrophysics of neutron stars M Prakash Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA E-mail: [email protected] Abstract. Some of the means through which the possible presence of nearly deconfined quarks in neutron stars can be detected by astrophysical observations of neutron stars from their birth to old age are highlighted. 1. Introduction Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial question is whether observations of neutron stars from their birth to death through neutrino, photon and gravity-wave emissions can unequivocably reveal the presence of nearly deconfined quarks instead of other possibilities such as only nucleons or other exotica such as strangeness-bearing hyperons or Bose (pion and kaon) condensates. 2. Neutrino signals during the birth of a neutron star The birth of a neutron star is heralded by the arrival of neutrinos on earth as confirmed by IMB and Kamiokande neutrino detectors in the case of supernova SN 1987A. Nearly all of the gravitational binding energy (of order 300 bethes, where 1 bethe ≡ 1051 erg) released in the progenitor star’s white dwarf-like core is carried off by neutrinos and antineutrinos of all flavors in roughly equal proportions. The remarkable fact that the weakly interacting neutrinos are trapped in matter prior to their release as a burst is due to their short mean free paths in matter, λ ≈ (σn)−1 ≈ 10 cm, (here σ ≈ 10−40 cm2 is the neutrino-matter cross section and n ≈ 2 to 3 ns, where ns ≃ 0.16 fm−3 is the reference nuclear equilibrium density), which is much less than the proto-neutron star radius, which exceeds 20 km. Should a core-collapse supernova occur in their lifetimes, current neutrino detectors, such as SK, SNO, LVD’s, AMANDA, etc., offer a great opportunity for understanding a proto-neutron star’s birth and propagation of neutrinos in dense matter insofar as they can detect tens of thousands of neutrinos in contrast to the tens of neutrinos detected by IMB and Kamiokande. The appearence of quarks inside a neutron star leads to a decrease in the maximum mass that matter can support, implying metastability of the star. This would occur if the proto-neutron star’s mass, which must be less than the maximum mass of the http://arxiv.org/abs/0704.0207v1 Quark matter and the astrophysics of neutron stars 2 Figure 1. Evolutions of the central baryon density nB, ν concentration Yν , quark volume fraction χ and temperature T for different baryon masses MB. Solid lines show stable stars whereas dashed lines showing stars with larger masses are metastable. Diamonds indicate when quarks appear at the star’s center, and asterisks denote when metastable stars become gravitationally unstable. Figure after Ref. [2] hot, lepton-rich matter is greater than the maximum mass of hot, lepton-poor matter. For matter with nucleons only, such a metastability is denied (see, e.g., [3]). Figure 1 shows the evolution of some thermodynamic quantities at the center of stars of various fixed baryonic masses. With the equation of state used (see [2] for details), stars with MB ∼< 1.1 M⊙ do not contain quarks and those with MB ∼ 1.7 M⊙ are metastable. The subsequent collapse to a black hole could be observed as a cessation in the neutrino signals well above the sensitivity limits of the current detectors (Figure 2). 3. Photon signals during the thermal evolution of a neutron star Multiwavelength photon observations of neutron stars, the bread and butter affair of astronomy, has yielded estimates of the surface tempeartures and ages of several neutron stars (Fig. 3). As neutron stars cool principally through neutrino emission from their cores, the possibility exists that the interior composition can be determined. The star continuosly emits photons, dominantly in x-rays, with an effective temperature Teff that tracks the interior temperature but that is smaller by a factor ∼ 100. The dominant neutrino cooling reactions are of a general type, known as Urca processes [4], in which thermally excited particles undergo beta and inverse-beta decays. Each Quark matter and the astrophysics of neutron stars 3 Figure 2. The evolution of the total neutrino luminosity for stars of indicated baryon masses. Shaded bands illustrate the limiting luninosities corresponding to a count rate of 0.2 Hz in all detectors assuming 50 kpc for IMB and and Kamioka, 8.5 kpc for SNO, SuperK, and UNO. Shaded regions represent uncertanities in the average neutrino energy from the use of a diffusion scheme for neutrino transport in matter. Figure after Ref. [2]. reaction produces a neutrino or anti-neutrino, and thermal energy is thus continuously lost. Depending upon the proton-fraction of matter, which in turn depends on the nature of strong interactions at high density, direct Urca processes involving nucleons, hyperons or quarks lead to enhanced cooling compared to modified Urca processes in which an additional particle is required to conserve momentum. However, effects of superfluidity abates cooling as sufficient thermal energy is required to break paired fermions. In addition, the poorly known envelope composition also plays a role in the inferred surface temperature (Fig. 3). The multitude of high density phases, cooling mechanisms, effects of superfluidity, and unknown envelope composition have thus far prevented definitive conclusions to be drawn (see, e.g., [5]). 4. Mesured masses and their implications Several recent observations of neutron stars have direct bearing on the determination of the maximum mass. The most accurately measured masses are from timing observations of the radio binary pulsars. As shown in Fig. 4, which is compilation of the measured neutron star masses as of November 2006, observations include pulsars orbiting another Quark matter and the astrophysics of neutron stars 4 Figure 3. Observational estimates of neutron star temperatures and ages together with theoretical cooling simulations forM = 1.4 M⊙. Models and data are described in [6]. Orange error boxes (see online) indicate sources from which both X-ray and optical emissions have been observed. Simulations are for models with Fe or H envelopes, with and without the effects of superfluidity, and allowing or forbidding direct Urca processes. Models forbidding direct Urca processes are relatively independent of M and superfluid properties. Trajectories for models with enhanced cooling (direct Urca processes) and superfluidity lie within the yellow region, the exact location depending upon M as well as superfluid and Urca properties. Figure adapted from Ref. [7]. neutron star, a white dwarf or a main-sequence star. One significant development concerns mass determinations in binaries with white dwarf companions, which show a broader range of neutron star masses than binary neutron star pulsars. Perhaps a rather narrow set of evolutionary circumstances conspire to form double neutron star binaries, leading to a restricted range of neutron star masses [9]. This restriction is likely relaxed for other neutron star binaries. A few of the white dwarf binaries may contain neutron stars larger than the canonical 1.4 M⊙ value, including the intriguing case [10] of PSR J0751+1807 in which the estimated mass with 1σ error bars is 2.1 ± 0.2 M⊙. In addition, to 95% confidence, one of the two pulsars Ter 5 I and J has a reported mass larger than 1.68 M⊙ [11]. Whereas the observed simple mean mass of neutron stars with white dwarf companions exceeds those with neutron star companions by 0.25 M⊙, the weighted means of the two groups are virtually the same. The 2.1 M⊙ neutron star, PSR J0751+1807, is about 4σ from the canonical value of 1.4 M⊙. It is furthermore the case that the 2σ errors of all but two systems extend into the range below 1.45 M⊙, so caution should be exercised before concluding that firm evidence of large neutron star masses exists. Continued observations, which will reduce the observational errors, are necessary to clarify this situation. Masses can also be estimated for another handful of binaries which contain an Quark matter and the astrophysics of neutron stars 5 Figure 4. Measured and estimated masses of neutron stars in binarie pulsars (gold, silver and blue regions online) and in x-ray accreting binaries (green). For each region, simple averages are shown as dotted lines; error weighted averages are shown as dashed lines. For labels and other details, consult Ref. [8]. accreting neutron star emitting x-rays. Some of these systems are characterized by relatively large masses, but the estimated errors are also large. The system of Vela X-1 is noteworthy because its lower mass limit (1.6 to 1.7M⊙) is at least mildly constrained by geometry [12]. Raising the limit for the neutron star maximum mass could eliminate entire families of EOS’s, especially those in which substantial softening begins around 2 to 3ns. This could be extremely significant, since exotica (hyperons, Bose condensates, or quarks) generally reduce the maximum mass appreciably. Ultimate energy density of observable cold baryonic matter Measurements of neutron star masses can set an upper limit to the maximum possible energy density in any compact object. It has been found [13] that no causal EOS has a central density, for a given mass, greater than that for the Tolman VII [14] analytic solution. This solution corresponds to a quadratic mass-energy density ρ dependence Quark matter and the astrophysics of neutron stars 6 Figure 5. Model predictions are compared with results from the Tolman IV and VII analytic solutions of general relativistic stucture equations. NR refers to non- relativistic potential models, R are field- theoretical models, and Exotica refers to NR or R models in which strong softening occurs, due to hyperons, a Bose condensate, or quark matter as well as self-bound strange quark matter. Constraints from a possible redshift measurement of z = 0.35 is also shown. The dashed lines for 1.44 and 2.2 M⊙ serve to guide the eye. Figure taken from Ref. [13]. on r, ρ = ρc[1− (r/R)2], where the central density is ρc. For this solution, ρc,T V II = 2.5ρc,Inc ≃ 1.5× 1016 g cm−3 . (1) A measured mass of 2.2 M⊙ would imply ρmax < 3.1× 1015 g cm−3, or about 8ns. Figure 5 displays maximum masses and accompanying central densities for a wide wariety of neutron star EOS’s, including models containing significant softening due to “exotica”, such as strange quark matter. The upper limit to the density could be lowered if the causal constraint is not approached in practice. For example, at high densities in which quark asymptotic freedom is realized, the sound speed is limited to 3. Using this as a strict limit at all densities, the Rhoades & Ruffini [15] mass limit is reduced by approximately 1/ 3 and the compactness limit GM/Rc2 = 1/2.94 is reduced by a factor 3−1/4 to 1/3.8 [16]. In this extreme case, the maximum density would be reduced by a factor of 3−1/4 from that of Eq. (1). A 2.2 M⊙ measured mass would imply a maximum density of about 4.2ns. 5. Gravitational wave signals during mergers of binary stars Mergers of compact objects in binary systems, such as a pair of neutron stars (NS- NS), a neutron star and a black hole (NS-BH), or two black holes (BH-BH), are expected to be prominent sources of gravitational radiation [17]. The gravitational- wave signature of such systems is primarily determined by the chirp mass Mchirp = Quark matter and the astrophysics of neutron stars 7 Figure 6. Physical and observational variables in mergers between low-mass black holes and neutron stars or self-bound quark stars. The total system mass is 6 M⊙ and the initial mass ratio is q = 1/3 in both cases. The initial radii of the neutron star and quark star were assumed to be equal. The time scales have arbitrary zero points. Upper panel displays semi-major axis a (thick lines) and component mass MNS,MQS (thin lines) evolution. Lower panel displays orbital frequency ν (thick lines) and strain amplitude |h+r| evolution. Solid curves refer to the neutron star simulation and dashed curves to the quark star simulations. Figure taken from Ref. [8]. (M1M2) 3/5(M1 +M2) −1/5, where M1 and M2 are the masses of the coalescing objects. The radiation of gravitational waves removes energy which causes the mutual orbits to decay. For example, the binary pulsar PSR B1913+16 has a merger timescale of about 250 million years, and the pulsar binary PSR J0737-3039 has a merger timescale of about 85 million years [18], so there is ample reason to expect that many such decaying compact binaries exist in the Galaxy. Besides emitting copious amounts of gravitational radiation, binary mergers have been proposed as a source of the r-process elements [19] and the origin of the shorter-duration gamma ray bursters [20]. Observations of gravity waves from merger events can simultaneously measure masses and radii of neutron stars, and could set firm limits on the neutron star maximum mass [21, 22]. Binary mergers for the two cases of a black hole and a normal neutron star and a black hole and a self-bound strange quark matter star (Fig. 6) illustrate the unique opportunity afforded by gravitational wave detectors due to begin operation over the next decade, including LIGO, VIRGO, GEO600, and TAMA. A careful analysis of the gravitational waveform during inspiral yields values for not only the chirp mass Mchirp, but for also the reduced mass MBHMNS/M , so that both MBH and MNS can be found [23]. The onset of mass transfer can be determined by the peak in ω, and the value of ω there gives a. A general relativistic analysis of mass transfer conditions then allows the determination of the star’s radius [22]. Thus a point on the mass-radius diagram can be estimated [24]. The combination h+ω depends only on a function of q, so the ratio of that combination and knowledge of qi Quark matter and the astrophysics of neutron stars 8 should allow determination of qf . From the Roche condition and knowledge of af from ωf , another mass-radius combination can be found. The sharp contrast between the evolutions during stable mass transfer of a normal neutron star and a strange quark star should make these cases distinguishable. For strange quark matter stars, the differences in the height of the frequency peak and the plateau in the frequency values at later times are related to the differences in radii of the stars at these two epochs. It could be an indirect indicator of the maximum mass of the star: the closer is the star’s mass before mass transfer to the maximum mass, the greater is the difference between these frequency values, because the radius change will be larger. Together with radius information, the value of the maximum mass remains the most important unknown that could reveal the true equation of state at high densities. Acknowledgments This work was supported in part by the U.S. Department of Energy under the grant DOE/DE-FG02-93ER40756. References [1] Collins J C and Perry M J 1975 Phys. Rev. Lett. 30, 1353 [2] Pons J A, Steiner A W, Prakash M and Lattimer J A 2001 Phys. Rev. Lett. 86, 5223 [3] Ellis P J, Lattimer J M and Prakash M 1996 Comments in Nuclear and Particle Physics 22, 63 [4] Lattimer J M, Pethick C J, Prakash M and Haensel P, 1991, Phys. Rev. Lett. 66, 2701 [5] Page D, Prakash M, Lattimer J M, and Steiner A W, 2000 Phys. Rev. Lett. 85, 2048 [6] Page D, Lattimer J M, Prakash M and Steiner A W 2004 Astrophys. Jl 155, 623 [7] Lattimer J M and Prakash M 2004 Science 304,536 [8] Lattimer J M and Prakash M 2006, astro-ph/0612440 [9] Bethe H A and Brown G E 1998 Astrophys. Jl 506, 780 [10] Nice D J et al. 2005 Astrophys. Jl. 634, 1242 [11] Ransom S M 2005 Science 307, 892 [12] Quaintrell et al. 2003 Astron. Astrophys. 401, 303 [13] Lattimer J M and Prakash M Phys. Rev. Lett. 94, 111101 [14] Tolman R C 1939 Phys. Rev. 55, 364 [15] Rhoades C E and Ruffini R 1974 Phys. Rev. Lett. 32, 324 [16] Lattimer J M, Prakash M, Masak D, and Yahil A 1990 Astrophys. Jl. 355, 241 [17] Thorne K S, 1973 Three Hundred Years of Gravitation, ed. S. W. Hawking and W. Israel, Cambridge Univ. Press, Cambridge, Ch. 9 [18] Lyne A. G. et al., 2004 Science 303, 1153 [19] Lattimer J M and Schramm D, 1976 Astrophys. Jl. 210, 549 [20] Eichler D et al., 1989 Science 340, 126 [21] Prakash M and Lattimer J M 2003, J. Phys. G. Nucl. Part. Phys. 30, S451 [22] Ratkovic S, Prakash M and Lattimer J M 2005, astro-ph/0512136 [23] Cutler C and Flanagen E E 1994, Phys. Rev. D49, 2658 [24] Faber J A et al., 2002 Phys. Rev. Lett. 89, 231102 http://arxiv.org/abs/astro-ph/0612440 http://arxiv.org/abs/astro-ph/0512136 Introduction Neutrino signals during the birth of a neutron star Photon signals during the thermal evolution of a neutron star Mesured masses and their implications Gravitational wave signals during mergers of binary stars

0704.0208

Some non-braided fusion categories of rank 3

SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 TOBIAS J. HAGGE AND SEUNG-MOON HONG Abstract. We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful varia- tion of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity. 1. Introduction Let k be an algebraically closed field. A fusion category C over k is a k-linear semi-simple rigid monoidal category with finitely many (isomorphism classes of) simple objects, finite dimensional morphism spaces, and End(1) ∼= k. See [7] or [4] for definitions, and [2] for many of the known results about fusion categories. The rank r of C is the number of isomorphism classes of simple objects in C. Let {xi}1≤i≤r be a set of simple object representatives. The fusion rules for C are a set of r × r N-valued matrices N = {Ni}1≤i≤r, with (Ni)j,k denoted Nkij or, when convenient, Nxkxixj , such that xi⊗xj ∼= 1≤k≤r N ijxk. In the sequel, assume k = C. Fusion categories appear in representation theory, operator algebras, conformal field theory, and in constructions of invariants of links, braids, and higher dimen- sional manifolds. There is currently no general classification of them. Classifications of fusion categories for various families of fusion rules have been given in work by Kerler ([3]), Tambara and Yamagami ([12]), Kazhdan and Wenzl ([5]), and Wenzl and Tuba ([13]). For a given set of fusion rules, there are only finitely many monoidal natural equivalence classes of fusion categories. This property is called Ocneanu rigidity (see [2]). It is not known whether or not the number of fusion categories of a given rank is finite. If one assumes a modular structure, the possibilities up to rank four have been classified by Belinschi, Rowell, Stong and Wang in [11]. Ostrik has classified fusion categories up to rank two in [10], and constructed a finite list of realizable fusion rules for braided categories up to rank three in [9], in which the number of categories for each set of fusion rules is known. The rank two classification relies in an essential way on the theory of modular tensor categories; Ostrik shows that the quantum double of a rank two category must be modular, and uses the theory of modular tensor categories to eliminate most of the possibilities. The classification of modular tensor categories is of independent interest; in many contexts one must assume modularity. We consider the only set of rank three fusion rules which is known to be realizable as a fusion category but which has no braided realizations. Ostrik conjectured in http://arxiv.org/abs/0704.0208v2 2 TOBIAS J. HAGGE AND SEUNG-MOON HONG [9] that a classification for this rule set completes the classification of rank three fusion categories. The axioms for fusion categories over C reduce to a system of polynomial equa- tions over C. In this context, Ocneanu rigidity, roughly translated, says that nor- malization of some of the variables in the equations gives a finite solution set. In this case, one can compute a Gröbner basis for the system and obtain the solutions (see [1]). However, normalization becomes complicated when there are i, j, k such that Nkij > 1. The fusion rules we consider are the smallest realizable set with this property. 2. Main theorem and outline Theorem 1 (Main Theorem). Consider the set of fusion rules with three simple object types, x, y and 1. Let 1 be the trivial object, and let x⊗ x ∼= x⊕ x⊕ y⊕ 1, x⊗ y ∼= y ⊗ x ∼= x and y ⊗ y ∼= 1. Then the following hold: (1) Up to monoidal natural equivalence, there are four semisimple tensor cat- egories with the above fusion rules. A set of associativity matrices for one of these categories is given in Appendix A. Applying a nontrivial Galois automorphism to all of the coefficients gives a set of matrices for any one of the other three categories. (2) The categories in part 1 are fusion categories. (3) The categories in part 1 do not admit braidings. (4) The categories in part 1 are spherical. The structure of the remainder of the paper is as follows: Section 3 describes the notation and categorical preliminaries used in later parts of the paper. It constructs a canonical representative for each monoidal natural equivalence class of fusion categories. This construction is really just two well known constructions, skeletization and strictification, applied in sequence. These constructions, taken together, form a bridge between the category theoretic lan- guage in the statements of the theorems and the algebra appearing in the proofs. For some of the calculations in this paper the translation between the category the- ory and the algebra is already widely known, but there are some subtleties when discussing pivotal structure that justify the treatment. Section 3 concludes by de- scribing the algebraic equations corresponding to the axioms for a fusion category, using the language of strictified skeletons. Section 4 proves part 1 of Theorem 1. The proof amounts to solving the variety of polynomial equations defined in the previous section, performing normalizations along the way in order to simplify calculations. The section ends by arguing that the nature of the normalizations guarantees that the solutions obtained really are monoidally inequivalent. Section 5 proves part 2 of Theorem 1 by explicitly com- puting rigidity structures. Part 3 of Theorem 1 follows from Ostrik’s classification of rank three braided fusion categories in [9]. Section 6 gives a direct proof by showing that there are no solutions to the hexagon equations. Section 7 defines pivotal and spherical structures and discusses their properties. The focus is on the question of whether every fusion category is pivotal and spher- ical. A novel and elementary proof that the quadruple dual functor is naturally isomorphic to the identity functor is given. This proof makes use of the strictified skeleton construction developed in Section 3. The section concludes by describing SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 3 what a pivotal category which does not admit a spherical structure would look like. In particular, it must have at least five simple objects. Section 8 proves part 4 of Theorem 1 by computing explicit pivotal structures for the four categories given in Section 4, and invoking a lemma from Section 7 for sphericity. 3. Preliminaries and notational conventions This paper uses the “composition of morphisms” convention for functions as well as morphisms, and left to right matrix multiplication. For calculations of the fusion rules, our treatment is similar to [12], but the notation differs superficially for typographic reasons. The notation captures algebraic data sufficient to classify a fusion category up to monoidal natural equivalence, and is reviewed later in this section. We assume that the reader is familiar with the notions of a monoidal category, a monoidal functor and a monoidal equivalence; for precise definitions, see [7]. Recall that a monoidal category is equipped with an associative bifunctor ⊗ and a distinguished object 1. Reassociation of tensor factors in a monoidal category is described by a natural isomorphism of trifunctors α : (−⊗−)⊗− → −⊗ (−⊗−). Tensor products with 1 have natural isomorphisms ρ : −⊗1 → − and λ : 1⊗− → −. These isomorphisms are subject to a coherency condition, namely that for any pair of multifunctors there is at most one natural isomorphism between them which may be constructed from λ, ρ, α and their inverses, along with Id and ⊗. This coherency condition is well known to be equivalent to the statement that the category satisfies the pentagon and triangle axioms (see [7] for a proof). The triangle equations are the equations ρx⊗y = αx,1,y ◦ (x⊗λy) for all ordered pairs (x, y) of objects. Here, and in the sequel when the context is unambiguous, the name of an object is used as a shorthand for the identity morphism on that object. The pentagon equations, defined for all tuples of objects (w, x, y, z), are as follows (see also Figure 1): (αx,y,z ⊗ w) ◦ αx,y⊗z,w ◦ (x⊗ αy,z,w) = αx⊗y,z,w ◦ αx,y,z⊗w, When studying fusion categories up to monoidal equivalence, one may choose categories within an equivalence class which have desirable attributes. Since the categorical properties considered in this paper (fusion rule structure, monoidality, pivotality, sphericity, presence of braidings) are all well known to be preserved un- der monoidal equivalence, the desirable attributes may be assumed without loss. In particular, one may construct, given an arbitrary fusion category, a canonical rep- resentative for that category’s equivalence class in which one may replace instances of the words “is isomorphic to” with “equals”. 3.1. Skeletization. The skeleton CSKEL of an arbitrary category C is any full subcategory of C containing exactly one object from each isomorphism class in C. If C is semi-simple, every object in C is isomorphic to a direct sum of simple objects in CSKEL. One may then assume without loss that the objects of CSKEL consist of simple object representatives and direct sums of such. It is a well known fact that CSKEL may be given a monoidal structure such that CSKEL and C are monoidally equivalent. The proof is a straightforward but tedious extension of Maclane’s proof of the natural equivalence an ordinary category 4 TOBIAS J. HAGGE AND SEUNG-MOON HONG and its skeleton (see section IV.4 in [7]). In that proof, one defines a family of isomorphisms ix from objects x to their isomorphic representatives in CSKEL and uses it to construct a pair of functors F and G which give a natural equivalence. For the extension, the ix are used to define the tensor product functor on CSKEL, as well as α, λ, ρ and the monoidal structures for F and G. One then writes out all of the relevant commutative diagrams and removes any compositions ix ◦ i−1x . The result in each case is a commutative diagram in C. 3.2. Strictification. Given a monoidal category C, one may construct a strict monoidal category CST R equivalent to C. In a strict monoidal category, α, λ and ρ are the identity. It is common practice to assume that a monoidal category is strict without explicit reference to the construction. However, by using the construction explicitly we will be able to pick a canonical representative for an equivalence class of monoidal categories and provide a natural interpretation of the graphical calculus. Strictification of a monoidal category is analogous to the construction of a tensor algebra; it gives an equivalent strict category CSTR by replacing the tensor product with a strictly associative formal tensor product. The objects of CSTR are finite sequences of objects in C. Morphism spaces of the form Mor((a1, a2, . . . , am−1, am), (b1, b2, . . . , bn−1, bn)) are given by Mor(a1 ⊗ (a2 ⊗ . . . (am−1 ⊗ am) . . . ), b1 ⊗ (b2 ⊗ . . . (bn−1 ⊗ bn) . . . )) in C. The tensor product on objects is just concatenation of sequences, for mor- phisms it is the tensor product in C pre and post-composed with appropriate asso- ciativity morphisms. Monoidal equivalence of C with CST R is proven in section XI.3 of [7]. It is not usually possible to make a fusion category strict and skeletal at the same time. However, the category (CSKEL)ST R, while not a skeleton, is still unique up to strict natural equivalence. Also, it is a categorical realization of a graphical calculus, as will shortly become clear. The next subsection describes what strictified skeleta of fusion categories look like, up to strict equivalence. 3.3. Strictified skeletal fusion categories. A strictified skeletal fusion category C is as follows: Let N be a set of fusion rules for a set of objects S. Then the objects in C are multisets of finite sequences of elements of S. C has a tensor product ⊗, which is defined on objects by pairwise concatenation of sequences, distributed over elements of multisets. Direct sum of objects is given by multiset disjoint union. A strand is an object which is a sequence of length one. Strands correspond to simple object types. If x, y and z are strands, define Mor(x ⊗ y, z) to be a k vector space isomorphic to kN xy . For brevity, V yx will denote Mor(x, y), and tensor products will be omitted when the context is clear. A morphism is (n,m)-stranded if its source and target are sequences of length n and m, respectively. A morphism is (n)-stranded if it is (m,n−m)-stranded for some 0 ≤ m ≤ n. Semi-simplicity of C means that for all objects w, x, y and z there are vector space isomorphisms v∈S V xy ⊗ V zwv ∼= V zwxy ∼= v∈S V wx ⊗ V zvy . The first isomorphism is given by f ⊗ g → (Idw ⊗ f) ◦ g, the inverse of the second by h⊗ l → (h⊗ Idy) ◦ l. The composition of the two isomorphisms is denoted αzw,x,y. Additionally, each morphism space V zxy has an algebraically dual space V z , in the sense that there are bases {vi}i ⊂ V zxy and {wi}i ⊂ V xyz such that wi ◦ vj = δijIdz. SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 5 In a strictified skeleton, the trivial object 1 is the zero length sequence. There is a strand which is isomorphic to the trivial object, but not equal. This strand shall be taken to be 1 in the sequel. This choice makes the category non-strict, since λ and ρ are no longer the identity, but it is convenient for graphical calculus purposes. (Right) rigidity in a strictified skeleton means that there is a set involution ∗ on the strands, and each strand x has morphisms bx : 1 → x⊗ x∗ and dx : x∗ ⊗ x → 1 such that (Idx∗ ⊗ bx) ◦ (dx ⊗ Idx∗) = Idx∗ and (bx ⊗ Idx) ◦ (Idx ⊗ dx) = Idx. This implies that Nzxy = N z∗x. Left rigidity is similar, and in the sequel, the right rigidity morphism for x∗ will be defined to be the left rigidity morphism for x. Define ∗, b, and d on concatenations of strands such that dx⊗y = (Idy∗ ⊗ dx ⊗ Idy) ◦ dy and extend to direct sums. Then there is a contravariant (right) dual functor ∗ which sends f ∈ V yx to f∗ = (Idy∗ ⊗ bx)◦ (Idy∗ ⊗f ⊗ Idx∗)◦ (dy ⊗ Idx∗) ∈ y∗ . The definition of a left dual functor is similar, and the two duals are inverse functors by rigidity. Monoidality for a strictified skeletal fusion category implies that the α are iden- tity morphisms. For this to be true, it is necessary and sufficient that the following equation holds for all objects x, y, z, w, and u. Each instance will be referred to as Pux,y,z,w in the sequel. αty,z,wV V syzα x,s,w ◦ αtx,y,zV V szwV V syzV V syzV V sxyV is equal to V szwα x,y,s ◦ τ ◦ V sxyα s,z,w : V szwV V szwV V sxyV V sxyV Here ⊗ for vector spaces and morphisms are omitted, and τ is the isomorphism interchanging the first and the second factors of vector space tensor products (see Figure 1). 3.4. Remarks. (1) Every fusion category is monoidally naturally equivalent to a strictified skeleton. Also, two naturally equivalent strictified skeleta have an invert- ible equivalence functor that takes strands to strands. This implies that equivalences are given by permutations of strands along with changes of basis on the (2, 0) and (2, 1)-stranded morphism spaces. (2) The functor ∗∗ fixes objects. The isomorphisms Jx,y : x∗∗⊗y∗∗ → (x⊗y)∗∗ associated with ∗∗ in the definition of a monoidal functor (see [7]) may be taken to be trivial. There is an invertible scalar worth of freedom in the choice of each bx, dx pair. (3) Semi-simplicity allows every morphism to be built up from (3)-stranded morphisms. Choosing bases for the (3)-stranded morphisms allows mor- phisms in C to be characterized as undirected trivalent graphs with labeled 6 TOBIAS J. HAGGE AND SEUNG-MOON HONG ((x⊗ y)⊗ z)⊗ w αx,y,zw uukkk αxy,z,w (x⊗ (y ⊗ z))⊗ w αx,yz,w (a) (x ⊗ y)⊗ (z ⊗ w) αx,y,zw x⊗ ((y ⊗ z)⊗ w) xαy,z,w // x⊗ (y ⊗ (z ⊗ w)) s,t V s,t V V sxyα s,z,woo s,t V αtx,y,zV s,t V s,t V V syzα x,s,w s,t V y,z,wV V szwα x,y,s Figure 1. (a) Pentagon equality and (b) corresponding equality edges and vertices, subject to associativity relations given by the penta- gon equations. The labels for the edges are isomorphism types of simple objects; the labels for the vertices are basis vectors for the corresponding morphism spaces. This gives a categorically precise interpretation of an arrowless graphical calculus for C. (4) If C is pivotal (see Section 7 for the definition), a well known construction allows one to add a second copy of each object and get a strict pivotal category. This construction gives a graphical calculus with arrows on the strands. (5) Strictified skeleta give any categorical structure preserved under natural equivalence (and any functorial property preserved under natural isomor- phism) a purely algebraic description. 4. Proof of Theorem 1 part 1:possible tensor category structures In this section we classify, up to monoidal equivalence, all C-linear semisimple tensor categories with fusion rules given in Theorem 1. This amounts to solving the matrix equations described in the previous section. The simplest equations (those involving 1× 1 matrices) are solved first, and normalizations are performed as necessary in order to simplify the equations. 4.1. Setting up the pentagon equations. The fusion rules are given by x⊗x ∼= 1⊕y⊕x⊕x, x⊗y ∼= y⊗x ∼= x, and y⊗y ∼= 1. The non-trivial vector spaces are V 111, V x1x ,V x1, V 1y , V y1, V xy, V yx, V yy, V xx, V xx,and V xx, and they are all 1-dimensional except the last space which is 2-dimensional. Let’s choose basis vectors in each space. If we fix any non-zero vector v111 ∈ V 111, then there are unique vectors vx1x ∈ V x1x, vxx1 ∈ V xx1,v 1y ∈ V 1y, and v y1 ∈ V y1 such that the triangle equality holds. For the other spaces, choose any non-zero vectors SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 7 in each space and denote them by vxxy ∈ V xxy, vxyx ∈ V xyx, v1yy ∈ V 1yy, v1xx ∈ V 1xx, vyxx ∈ V yxx, v1 and v2 ∈ V xxx where the two vectors v1 and v2 are linearly independent. There are 30 associativities. It is a well known fact that if at least one of the bottom objects is 1 then the associativity is trivial. That is, with the above basis choices the matrix for αzu,v,w is trivial if at least one of the u, v and w is 1. Now we have ten non-trivial 1-dimensional associativities, αyy,y,y,α x,y,y,α y,y,x,α x,y,x, αyx,y,x,α y,x,y,α x,x,y,α x,x,y, α y,x,x, and α y,x,x, five non-trivial 2-dimensional ones, αxx,y,x,α x,x,y,α y,x,x,α x,x,x, and α x,x,x, and one 6-dimensional one, α x,x,x. 4.2. Normalizations. With the above basis choices we obtain a basis for each tensor product of vector spaces in a canonical way and can parameterize each associativity and pentagon equation. However, at this point our basis elements have not been uniquely specified, and we should expect to obtain solutions with free parameters. As the calculation progresses it will be convenient to simplify the pentagon equations by requiring certain coefficients of certain associativity matrices to be 1 or 0. These normalizations should be thought of as restrictions on the basis choices made above. Normalizations simplify the equations and have an additional advantage: once the set of possible bases is sufficiently restricted, Ocneanu rigidity [2] guarantees a finite set of possibilities for the associativity matrices of fusion categories, which can be found algorithmically by computing a Gröbner basis. 4.3. Associativity matrices. The following are the 1-dimensional associativities: αyy,y,y : v y1 7→ ayy,y,yv1yyv αxx,y,y : v x1 7→ axx,y,yvxxyvxxy αxy,y,x : v yx 7→ axy,y,xv1yyvx1x α1x,y,x : v xx 7→ a1x,y,xvxxyv1xx αyx,y,x : v xx 7→ ayx,y,xvxxyvyxx αxy,x,y : v yx 7→ axy,x,yvxyxvxxy α1x,x,y : v xx 7→ a1x,x,yvyxxv1yy αyx,x,y : v xx 7→ ayx,x,yv1xxv α1y,x,x : v yy 7→ a1y,x,xvxyxv1xx αyy,x,x : v y1 7→ ayy,x,xvxyxvyxx where associativity coefficients are all non-zero. For 2-dimensional and 6-dimensional associativities we need to fix the ordering of basis elements in each Hom vector space. The orderings are as follows: {vxyxv1, vxyxv2} for V xx(yx), {vxxyv1, vxxyv2} for V x(xy)x, {vxxyv1, vxxyv2} for V xx(xy), {v1v xy, v2v xy} for V x(xx)y, {v1vxyx, v2vxyx} for V xy(xx), {v yxv1, v yxv2} for V x(yx)x, {v1v1xx, v2v1xx} for V 1x(xx), {v1v xx, v2v xx} for V 1(xx)x, {v1vyxx, v2vyxx} for V x(xx) , {v1vyxx, v2vyxx} for V (xx)x {v1xxvxx1, vyxxvxxy, v1v1, v1v2, v2v1, v2v2} for V xx(xx), and {v1xxvx1x, vyxxvxyx, v1v1, v1v2, v2v1, v2v2} for V x(xx)x. With these ordered bases, each associativity has a matrix form (recall that we are using the right multiplication convention). That is, αxx,y,x is given by the invertible 2 × 2 matrix axx,y,x, and αxx,x,y is given by the invertible 2 × 2 matrix axx,x,y,etc., and finally αxx,x,x is given by the invertible 6× 6 matrix axx,x,x. 8 TOBIAS J. HAGGE AND SEUNG-MOON HONG 4.4. Pentagon equations with 1× 1 matrices. Considering only nontrivial as- sociativities, there are 17 1-dimensional pentagon equations, 14 2-dimensional pen- tagon equations, 6 6-dimensional ones, and 1 16-dimensional one. Without redun- dancy, the following are the 1-dimensional equations: P xx,y,y,y : a y,y,y a x,y,y = a x,y,y. P 1x,x,y,y : a x,y,ya x,x,y a x,x,y = 1 , P 1x,y,x,y : a y,x,y a x,y,x = a x,y,x , P yx,y,x,y : :a y,x,y a x,y,x = a x,y,x , P 1x,y,y,x : a y,y,x a x,y,y = (a x,y,x) P yx,y,y,x : a y,y,x a x,y,y = (a x,y,x) P xy,x,y,y :(a y,x,y) 2 = 1 , P 1y,y,x,x : a y,x,x a y,x,x a y,y,x = 1 , P 1y,x,x,y : a x,x,y a y,x,x = a y,x,x a x,x,y If we normalize the basis we may assume axy,y,x, a x,y,x and a x,x,y to be 1 (for normalization see [12] or [6]), and we can solve the above 1-dimensional equations. Here is the solution: ayy,y,y = a x,y,y = a x,x,y = 1, a y,x,y = a x,y,x = ±1, a1y,x,x = ayy,x,x = ±1. Let’s say g := axy,x,y = a x,y,x and h := a y,x,x = a y,x,x in the sequel. Also let A := axx,y,x, B := a x,x,y, D := a x,x,x, E := a x,x,x, F := a y,x,x and Φ := a x,x,x for brevity. 4.5. Pentagon equations with 2× 2 or 6× 6 matrices. Now, the following are the 2-dimensional pentagon equations using the above 1-dimensional solutions: P xy,y,x,x : F 2 = Id2 P xy,x,y,x : gAF = FA P xx,y,y,x : A 2 = Id2 P xy,x,x,y : gBF = FB P xx,y,x,y : gBA = AB P xx,x,y,y : B 2 = Id2 P 1y,x,x,x : EF = D P yy,x,x,x : DF = E P 1x,y,x,x : FDA = D P yx,y,x,x : FEA = gE P 1x,x,y,x : ADB = D P yx,x,y,x : AEB = gE P 1x,x,x,y : BE = D P yx,x,x,y : BD = E It should be noted that for this particular category the large number of one dimensional morphism spaces gives us q-commutativity relations and matrices with ±1 eigenvalues, which are of great help when simplifying the pentagon equations by hand. To analyze 2-dimensional and 6-dimensional pentagon equations, at first let’s look at the isomorphism τ interchanging the first and the second factors of tensor products. This change of basis is necessary for 6-dimensional pentagon equations because the image basis of the matrix for αux,y,zw and the domain basis of the matrix for αuxy,z,w may not be the same. For P x,y,x,x, τ is an isomorphism from the space V 1xxV x1⊕V yxxV xxyV xxy⊕V xxxV xxyV xxx to V xxyV 1xxV xx1⊕V xxyV yxxV xxy⊕V xxyV xxxV xxx, both of SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 9 which correspond to Hom((x⊗ y)⊗ (x⊗x), x). With the canonically ordered basis {v1xxvxxyvxx1, vyxxvxxyvxxy, vivxxyvj} and {vxxyv1xxvxx1, vxxyvyxxvxxy, vxxyvivj}, respectively, τ turns out to be I6. For P y,x,x,x, P x,x,y,x and P x,x,x,y, τ is also I6. But for P x,x,x,x, it is τ1, and for P x,x,x,x, it is τ2, defined as follows: τ1 := 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 , τ2 := 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 Here are the 6 6-dimensional pentagon equations: P xy,x,x,x ; Φ(I2 ⊕ I2 ⊗ F ) ⊕ F ⊗ I2 ⊕ I2 ⊗ F P xx,y,x,x ; ⊕ F ⊗ I2 ⊕A⊗ I2 = (I2 ⊕ I2 ⊗A)Φ P xx,x,y,x ; ⊕A⊗ I2 ⊕B ⊗ I2 = Φ(I2 ⊕ I2 ⊗A) P xx,x,x,y ; ⊕B ⊗ I2 (I2 ⊕ I2 ⊗ B)Φ = Φ ⊕ I2 ⊗B P 1x,x,x,x ; (I2 ⊕ I2 ⊗D)Φ = (I2 ⊕ I2 ⊗D)τ1 ⊕ I2 ⊗D P yx,x,x,x ; Φ ⊕ I2 ⊗ E Φ = (I2 ⊕ I2 ⊗ E)τ2 ⊕ I2 ⊗ E If we normalize the basis {v1, v2} of V xxx, we may assume A is of the form and then get g = −1 and h = 1 using the above equations. Following is the computation for this: At first we may assume that matrix A is of the Jordan canonical form, then A = ±I2 or from P xx,y,y,x. We eliminate the possibility A = ±I2 from P xy,x,y,x, P x,y,x,x and P y,x,x,x which imply respectively that g = 1, F = ±I2 and then det(Φ) = 0, since the first two columns of Φ are scalar multiples of each other. So we conclude A = . Now we eliminate the possibility g = 1 using P xy,x,y,x, P xy,y,x,x and P y,x,x,x, which imply F is a diagonal matrix, with entries ±1 and then det(Φ) = 0, respectively. For the case A = and g = −1, F is of the form [ 0 f 1/f 0 from P xy,x,y,x and P y,y,x,x, and B is of the form 1/b 0 from P xx,y,x,y and P xx,x,y,y. If h = −1, the first column of Φ has to be zero by comparing the first and the second columns of P xy,x,x,x, P x,y,x,x, P x,x,y,x, and P x,x,x,y. At this point we have fixed all 1-dimensional associativity matrices. From the above equations, we get [ 0 f 1/f 0 for F and 1/b 0 for B with the relation f2 + b2 = 0 from P xy,x,x,y. We note that the diagonalization of A defines each basis element v1 and v2 only up to choice of a nonzero scalar. By using up one of these degrees of freedom, we may assume f = 1. Then from the above 6-dimensional equations, we get the following: D = d E = d φ φ −wb w w −wb φ −φ −wb w −w wb x x −yb z y −zb x x −zb y z −yb x −x −yb z −y zb −x x zb −y z −yb 10 TOBIAS J. HAGGE AND SEUNG-MOON HONG 4.6. The pentagon equation with 16 × 16 matrices. Now we analyze the 16- dimensional pentagon equation P xx,x,x,x. It is convenient to express each Hom vector space in two different ways and put basis permutation matrices into the pentagon equation. The following are two expressions with ordered direct sum. Hom(x(x(xx)), x) : V xxxV x1 ⊕ V xxxV yxxV xxy ⊕ V 1xxV xx1V xxx ⊕ V yxxV xxyV xxx ⊕ V xxxV xxxV xxx, and V 1xxV xx ⊕ V yxxV xxyV xxx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx Hom(x((xx)x)), x) : V xxxV x1 ⊕ V xxxV yxxV xxy ⊕ V 1xxV x1xV xxx ⊕ V yxxV xyxV xxx ⊕ V xxxV xxxV xxx, and V 1xxV xx ⊕ V yxxV xyxV xxx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx Hom((x(xx))x), x) : V 1xxV xx ⊕ V yxxV xxyV xxx ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx, and V xxxV 1x ⊕ V xxxV yxxV xyx ⊕ V 1xxV xx1V xxx ⊕ V yxxV xxyV xxx ⊕ V xxxV xxxV xxx Hom((((xx)x)x), x) : V xxxV 1x ⊕ V xxxV yxxV xyx ⊕ V 1xxV x1xV xxx ⊕ V yxxV xyxV xxx ⊕ V xxxV xxxV xxx, and V 1xxV xx ⊕ V yxxV xyxV xxx ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx Hom((xx)(xx), x) : V 1xxV x1 ⊕ V yxxV xxxV xxy ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx, and V 1xxV 1x ⊕ V yxxV xxxV xyx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx where each direct summand space has canonical ordered basis. For exam- ple V xxxV xx has basis {vivjvk} where (i, j, k) range from 1 to 2 in the order (1, 1, 1), (1, 1, 2), (1, 2, 1), etc., and V xxxV 1x has {v1v1xxvx1x, v2v1xxvx1x}. τ3 := ⊕ I8 and τ4 := 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 Then the pentagon equation P xx,x,x,x is of the form: (D ⊕ E ⊕ Φ⊗ I2)τ3(I2 ⊕A⊕ Φ̃)τ3(D ⊕ E ⊕ Φ⊗ I2)τ3 = τ3(I2 ⊕B ⊕ Φ̃)τ4(I2 ⊕ F ⊕ Φ̃) where φ 0 φ 0 −wb w w −wb 0 0 0 0 0 φ 0 φ 0 0 0 0 −wb w w −wb φ 0 −φ 0 −wb w −w wb 0 0 0 0 0 φ 0 −φ 0 0 0 0 −wb w −w wb x 0 x 0 −yb z y −zb 0 0 0 0 x 0 x 0 −zb y z −yb 0 0 0 0 x 0 −x 0 −yb z −y zb 0 0 0 0 −x 0 x 0 zb −y z −yb 0 0 0 0 0 x 0 x 0 0 0 0 −yb z y −zb 0 x 0 x 0 0 0 0 −zb y z −yb 0 x 0 −x 0 0 0 0 −yb z −y zb 0 −x 0 x 0 0 0 0 zb −y z −yb SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 11 4.7. Solutions. We may assume x = 1 once we normalize basis vector vxxy. Then from the equations, we get four explicit solution sets for the parameters b, φ, d, w, y, and z. We list one solution here. All of its values lie in the field Q( 3, i); the other solutions are obtained by applying Galois automorphisms. The full set of associativity matrices for this solution is given in Appendix A. b = i, φ = , d = e7πi/12, w = e2πi/3, (e−πi/3 + i), z = e5πi/6. 4.8. Inequivalence of the solutions. To see that these solutions are monoidally inequivalent, recall from the previous section that for strictified skeletons a natural equivalence between two solutions to the pentagon equations is limited to change of basis on the (2, 0) and (2, 1)-stranded morphism spaces, along with permutation of the strands. In our case permutation of strands does not preserve the fusion rules. Therefore, we must show that it is not possible to replicate the effect of a nontrivial Galois automorphism by change of basis choices for the (2, 0) and (2, 1)-stranded morphism spaces. The Galois automorphism that fixes 3 and sends i to −i changes the eigenvalues of the matrix a1x,x,x. However, a x,x,x is determined by the basis choices v1 and v2, and its rows and columns are indexed by v1 and v2. Thus changes to v1 and v2 conjugate a1x,x,x by a change of basis matrix, which doesn’t affect its eigenvalues. Therefore this automorphism does not correspond to a change of basis. The other two Galois automorphisms send 3 to − 3 and thus change the value of the (1, 1) entry of axx,x,x. But this entry is invariant under change of basis. Therefore no Galois automorphism corresponds to a change in basis, and the four solutions given above are mutually monoidally inequivalent. 5. Proof of Theorem 1 part 2: rigidity structures This section explicitly computes rigidity structures for the categories given in the previous section. Rigidity implies that these categories are fusion categories. Given v1xx ∈ V 1xx, choose a vector vxx1 ∈ V xx1 such that vxx1 ◦v1x,x = id1(see Figure 2). Now we define right death and birth, dx := v xx : x ⊗ x → 1, bx := 1φv 1 : 1 → x⊗ x (see Figure 3). With these definitions, right rigidity is an easy consequence by direct computa- tion. The following is a graphical version of it: = = = idx, = = = idx where the first and the third equalities are from the definitions above, and the second equalities are the associativity αxxxx and (α −1, respectively. 12 TOBIAS J. HAGGE AND SEUNG-MOON HONG , = 1 Figure 2. Graphical notation of v1xx and v 1 and property := =: , := 1 = = 1 Figure 3. Definitions of bx and dx, and elementary properties (yx)z αy,x,z // y(xz) ycx,z (xy)z cx,yz ;;wwwwwwwww αx,y,z y(zx) x(yz) cx,yz // (yz)x αy,z,x ;;wwwwwwwww (yx)z αy,x,z // y(xz) yc−1z,x (xy)z c−1y,xz ;;wwwwwwwww αx,y,z y(zx) x(yz) c−1yz,x // (yz)x αy,z,x ;;wwwwwwwww Figure 4. Hexagon equalities The same morphisms give a left rigidity structure when treated as left birth and left death. Treat the objects y and 1 analogously by replacing φ with 1. 6. Proof of Theorem 1 part 3: the absence of braidings The categories under consideration are known not to be braided (see [9]). How- ever, once associativity matrices are known it is in principle not difficult to classify braidings by direct computation. In this section we perform this computation and show that no braidings are possible. A braiding consists of a natural family of isomorphisms {cx,y : x ⊗ y → y ⊗ x} such that two hexagon equalities hold: (cx,y ⊗ z) ◦ αy,x,z ◦ (y ⊗ cx,z) = αx,y,z ◦ cx,yz ◦ αy,z,x and ((cy,x) −1 ⊗ z) ◦ αy,x,z ◦ (y ⊗ (cz,x)−1) = αx,y,z ◦ (cyz,x)−1 ◦ αy,z,x. SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 13 = rzx,y = r̄zx,y Figure 5. Isomorphisms Rzx,y and R̄ We define isomorphisms Rzx,y : V yx → V zxy by f 7→ cx,y ◦ f and R̄zx,y : V zyx → V zxy by f 7→ (cx,y)−1 ◦ f for any f ∈ V zyx. Figure 5 shows the 1-dimensional case where rzx,y is nonzero and r̄ x,y = (r −1. For higher dimensional spaces it can be expressed as an invertible matrix, also denoted rzx,y on the canonically ordered basis as before. With this linear isomorphism, the hexagon equations are equivalent to the equa- tions ⊕sRsx,zV tys ◦ αty,x,z ◦ ⊕sRsx,yV tsz = αty,z,x ◦ ⊕sV syzRtx,s ◦ αtx,y,z, and ⊕sR̄sx,zV tys ◦ αty,x,z ◦ ⊕sR̄sx,yV tsz = αty,z,x ◦ ⊕sV syzR̄tx,s ◦ αtx,y,z, which we still call hexagon equations, referred to as Htx,y,z and H̄ x,y,z, respectively. These are illus- trated graphically in Figure 6). We show the absence of a braiding by assuming the existence and deriving a contradiction. We need five 2-dimensional hexagon equations as follows: Hxy,x,x : R y,x ⊗ I2 ◦ αxx,y,x ◦Rxy,x ⊗ I2 = αxx,x,y ◦ I2 ⊗Rxy,x ◦ αxy,x,x H̄xy,x,x : R̄ y,x ⊗ I2 ◦ αxx,y,x ◦ R̄xy,x ⊗ I2 = αxx,x,y ◦ I2 ⊗ R̄xy,x ◦ αxy,x,x Hxx,y,x : R x,x ⊗ 1 ◦ αxy,x,x ◦Rxx,y ⊗ I2 = αxy,x,x ◦ 1⊗Rxx,x ◦ αxx,y,x H̄xx,y,x : R̄ x,x ⊗ 1 ◦ αxy,x,x ◦ R̄xx,y ⊗ I2 = αxy,x,x ◦ 1⊗ R̄xx,x ◦ αxx,y,x H1x,x,x : R x,x ⊗ 1 ◦ α1x,x,x ◦Rxx,x ⊗ 1 = α1x,x,x ◦ I2 ⊗R1x,x ◦ α1x,x,x These are of the following forms, respectively: (rxy,x) = rxy,x 1/b 0 (rxx,y) = (rxx,y) 1/b 0 rxx,y (rxy,x) ]−1 [ ]−1 [ = d2r1x,x where represents the matrix rxx,x. From the first four equations, we get rxy,x = b, r x,y = 1/b, −n = rxx,yk, m = rxx,yl, −n = rxy,xk, which imply k = n = 0 since rxx,y 6= rxy,x as above. Now from the final one we get l2 = dr1x,x(b+1) and −blm = dr1x,x(1+ b), and the later equality means l2 = −dr1x,x(1 + b) by substituting m = rxx,yl. We get easily a contradiction for either case b = ±i. 14 TOBIAS J. HAGGE AND SEUNG-MOON HONG x y z ⊕sRsx,yV x y z αty,x,zoo x y z Htx,y,z x y z ⊕sRsx,zV ]];;;;;;;; αty,z,x���� x y z αtx,y,z ]];;;;;;;; x y z ⊕sV syzR x,soo x y z ⊕sR̄sx,yV � x y z αty,x,zoo x y z H̄tx,y,z x y z ⊕sR̄sx,zV αty,z,x���� x y z αtx,y,z ]];;;;;;;;α x,y,z ]];;;;;;;; x y z ⊕sV syzR̄ x,soo Figure 6. Equivalent hexagon equalities 7. Pivotal structures and sphericity Let C be a rigid monoidal category. A pivotal structure for C is a monoidal natural isomorphism π from ∗∗ to Id. A strict pivotal structure is a pivotal structure which is the identity. In a pivotal monoidal category, the right trace trr of an endomorphism f : x → x is given by trr(f) = bx ◦ (f ⊗ Idx∗) ◦ (π−1x ⊗ Idx∗) ◦ dx∗ ∈ End(1) ∼= C. The left trace trl is given by trl(f) = bx∗ ◦ (f∗ ⊗ Idx∗∗) ◦ ((πx)∗ ⊗ Idx∗∗) ◦ dx∗∗ . A pivotal monoidal category is spherical if trr = trl. Pivotal structures may not be unique. For example, in a fusion category with object types given by a finite group G, group multiplication as tensor product and trivial associativity matrices, any group homomorphism G → C induces a pivotal structure. Furthermore, pivotal structures depend on choices of rigidity. However, if one chooses a new rigidity structure with b′x = cbx and d x = c −1dx, then π x = c gives a new pivotal structure π′ inducing the same traces as π. For a strictified skeletal fusion category, we shall assume the rigidity structures described in Section 3.3. Then ∗∗ is an object fixing monoidal endofunctor. The isomorphisms Ja,b : a ∗∗ ⊗ b∗∗ → (a ⊗ b)∗∗ associated with ∗∗ considered as a SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 15 monoidal functor may be taken to be the identity on a⊗ b. If such a category has a pivotal structure π, it must take the following form: for each strand (x), there is a scalar tx such that πx = txIdx, and π(x1,...,xn) = tx1 . . . txnId(x1,...xn). Then for all sequences (x1, . . . , xm) and (y1, . . . , yn), and all f : (x1, . . . , xm) → (y1, . . . , yn), f∗∗ = tx1 . . . txmt . . . t−1yn f . This implies that, in particular, t1 = 1. Writing out the diagrams for b∗∗x and d x and applying rigidity gives that b x = bx and d∗∗x = dx. Thus one must have txtx∗ = 1, and for self-dual strands, tx = ±1; in this case tx is called the Frobenius-Schur indicator for x. Furthermore, in a strictified skeleton the left and right trace on a strand may be rewritten as follows: trr(f) = t x bx ◦ (f ⊗ Idx∗) ◦ dx∗ , trl(f) = txbx∗ ◦ (Idx∗ ⊗ f) ◦ dx. Lemma 2. Every pivotal fusion category with self-dual simple objects is spherical. Proof. The result holds since if x is a self dual strand then bx = bx∗ , dx = dx∗ and tx = t x . Thus for each f : x → x we have trr(f) = trl(f). � Kitaev has shown in [6] that every unitary category admits a spherical structure. A more general property called pseudo-unitarity is shown in [2] to guarantee a spherical structure. However, it is not known if every fusion category admits a pivotal or spherical structure. For arbitrary fusion categories, one has that ∗ ∗ ∗∗ ∼= Id. This was shown in [2], using an analog of Radford’s formula for S4 for representation categories of weak Hopf algebras, which was developed in [8]. The following theorem shows that, in a strictified skeletal fusion category, a convenient choice of rigidity makes ∗ ∗ ∗∗ the identity on the nose. Extending the result to general fusion categories via natural equivalences gives an elementary proof that ∗ ∗ ∗∗ ∼= Id. Theorem 3. In a strictified skeletal fusion category, there is a choice of rigidity structures such that ∗ ∗ ∗∗ = Id. Proof. The functor ∗ ∗ ∗∗ is the identity on (2)-stranded morphisms by rigidity; it suffices to prove the result for (2, 1) stranded morphisms. Let V = V zxy be a (2, 1) stranded morphism space with a basis {vi}, and let {wi} be an algebraically dual basis for the space W = V xyz , in the sense that wi ◦ vj = δijIdz . For any simple object z, define the right pseudo-trace ptrr of an endomorphism f : z → z by ptrr(f) = bz ◦ (f ⊗ Idz∗) ◦ dz∗ , and the left pseudo-trace ptrl by ptrl(f) = bz∗ ◦ (Idz∗ ⊗f)◦dz. This definition is possible because ∗∗ is the identity on objects. Scale rigidity morphisms if necessary so that for any strand z, ptrr(Idz) = ptrl(Idz). Because dz and bz∗ are nonzero elements of one dimensional algebraically dual morphism spaces, ptrr(Idz) 6= 0. One may now exchange left pseudo-traces for right pseudo-traces, just like with traces in a graphical calculus for a spherical category. Figure 7 gives the proof. On the left side, bending arms and pseudo-sphericity implies that the algebraic dual basis of the basis {w∗∗i } is {∗∗vi}. However, on the right side the functoriality of the double dual implies that the algebraic dual basis of {w∗∗i } is {v∗∗i }. Since the left and right double dual are inverse functors, ∗ ∗ ∗∗ is the identity. � 16 TOBIAS J. HAGGE AND SEUNG-MOON HONG Figure 7. In a strictified skeletal fusion category, with the right choice of rigidity structures the quadruple dual is the identity. Even if a category admits a pivotal structure it is not known whether it admits a spherical pivotal structure. Pictorial considerations do not readily provide an answer. It is possible, however, to partially describe what a pivotal strictified skeleton which did not admit a spherical structure would look like. Let C be a pivotal strictified skeletal fusion category which does not admit a spherical structure. Choose rigidity morphisms which give a pseudo-spherical struc- ture as above, and a matching pivotal structure. Then for any object x, one has the following: trl(Idx) trr(Idx) txptrl(x) t−1x ptrr(x) = t2x. Therefore, C is spherical iff there exists a pivotal structure such that all of the tx are ±1. Thus there must be some strand x such that tx 6= ±1. For strands u and v, u ⊗ v has a nontrivial morphism to some object w, and tutv(tw) −1 = ±1, since ∗ ∗ ∗∗ = Id. Thus the set of scalars t and their additive inverses forms a finite subgroup G of C. Note that we can apply any group ho- momorphism that preserves ±1 to the set of scalars t and get a new set of scalars t′ which also gives a pivotal structure. At least one product tutv(tw) −1 must be equal to −1, or else we could apply the trivial homomorphism to the set of scalars t to get a new pivotal structure with t′u = 1 for all strands u, which would make C spherical. Every finite subgroup of C is a cyclic group of roots of unity. We have |G| = 2k for some k, and since C is not spherical, |G| ≥ 4. Using a homomorphism which preserves −1 we may switch to a new pivotal structure which gives |G| = 2k for some k, where k ≥ 2 to contradict sphericity. Pick an object v with t2v = −1. Then v is not self dual, and for a simple summand w in v⊗ v, one has w 6= 1 and t2w = 1. Therefore, C has at least four objects, v, v∗, w and 1. The set of objects u such that t2u = 1 generates a spherical subcategory C′ with at least two simple objects, and missing at least two. SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 17 Lemma 4. Any fusion category which is pivotal but admits no spherical structure contains at least five simple object types. Proof. Assume that C has four simple object types, 1, w, v and v′ as above. Then C′ has two simple object types, and by the classification of fusion categories with two simple object types in [10], its fusion rules are given by w⊗w ∼= nw⊕1, where n ∈ {0, 1}. The pivotal structure places limitations on the fusion rules, for example v⊗w ∼= av⊕bv′ for some a and b in N. An easy calculation shows that C admits only one associative fusion ring, in which objects and tensor products are given by the group Z4. Any such category is pseudo-unitary and therefore spherical, as described in [2], which contradicts the assumption. Therefore, a pivotal fusion category which can’t be made spherical must have at least five simple object types. � 8. Proof of Theorem 1 part 4: spherical structure calculations In this section we explicitly compute pivotal structures for the categories found in Section 4. Since these categories have self dual simple objects, Lemma 2 implies that they are spherical. It is not hard to determine whether or not a fusion category is pivotal once a set of associativity matrices is known. One way is to perform the calculations directly using the associativity matrices, but there is an easier calculation. In order to explain this calculation, it is convenient to extend the definition of composition of morphisms over extra-categorical direct sums of morphism spaces. Suppose f ∈ Mor(a, b) and g ∈ Mor(c, d). Define f◦g as usual if b = c, and f◦g = 0 ∈ Mor(a, d) otherwise. Extend this definition over direct sums of morphism spaces, distributing composition over direct sum. Given a strictified skeletal fusion category C and a set of associativity matrices, choose bases for the (2, 0) and (2, 1)-stranded morphism spaces compatible with the associativity matrices and choose rigidity so that for each strand x, the basis element for V 1x∗x is dx. Define morphisms b = ⊕xbx, d = ⊕xdx, and I = ⊕xIdx, taking sums over the strands. Then B acts on x,y,z V xy as follows: B(f) = (I ⊗ I ⊗ b) ◦ (I ⊗ f ⊗ I) ◦ (d⊗ I) For a single (2, 1)-stranded morphism space, this action amounts to “bending arms”. The cube of B is the double dual. The action of B on a morphism f ∈ V zxy is given by the associativity matrix a1z∗,x,y, since (Idz∗ ⊗ f) ◦ dz = (g⊗ idy) ◦ dy for some g ∈ V y z∗x implies that B(f) = (Idz∗⊗Idx⊗by)◦(Idz∗⊗f⊗Idy∗)◦(dz⊗Idy∗) = (Idz∗ ⊗ Idx ⊗ by) ⊗ (g ⊗ Idy ⊗ Idy∗) ◦ (dy ⊗ Idy∗) = g by rigidity. For the fusion rules at hand, the matrix for B is as follows: v1 v2 v v1 (a xxx)1,1 (a xxx)1,2 0 0 0 v2 (a xxx)2,1 (a xxx)2,2 0 0 0 vyxx 0 0 0 a yxx 0 vxyx 0 0 0 0 a vxxy 0 0 a xxy 0 0 For all of the solutions given in Section 4, B3 is the identity matrix, so the corre- sponding strictified categories have a strict pivotal structure. Non-strict pivotality 18 TOBIAS J. HAGGE AND SEUNG-MOON HONG would mean that B3 is a diagonal matrix with eigenvalues determined by a family of invertible scalars t, coherent as described in Section 7. Appendix A. Associativity matrices In this section, we give explicit associativity matrices for the categorical realiza- tion given in Section 4. ayy,y,y = a x,y,y = a y,y,x = a x,y,x = a x,x,y = a x,x,y = a y,x,x = a y,x,x = 1, ayx,y,x = a y,x,y = −1, axx,y,x = axx,x,y = axy,x,x = a1x,x,x = e7πi/12 ayx,x,x = e7πi/12 axx,x,x = eπi/6 1− e2πi/3 1− e2πi/3 1− eπi/6 eπi/6 1− e2πi/3 − 1− e2πi/3 − 1− eπi/6 1 1 − 1 (eπi/6−1) 1 e5πi/6 1 (e−πi/3+i) 1 eπi/3 1 1 1 eπi/3 1 (e−πi/3+i) 1 e5πi/6 − 1 (eπi/6−1) 1 −1 − 1 (eπi/6−1) 1 e5πi/6 − 1 (e−πi/3+i) − 1 eπi/3 −1 1 − 1 eπi/3 − 1 (e−πi/3+i) 1 e5πi/6 − 1 (eπi/6−1) References [1] Bruno Buchberger. A theoretical basis for the reduction of polynomials to canonical forms. SIGSAM Bull., 10(3):19–29, 1976. [2] Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. On fusion categories. Ann. of Math., 162(2):581–642, 2005. [3] J. Frohlich and T. Kerler. Quantum groups, quantum categories, and quantum field theory, chapter 4. Number 1542 in Lecture Notes in Mathematics. 1993. [4] Christian Kassel. Quantum Groups. Springer-Verlag, 1995. [5] Kazhdan and Hans Wenzl. Reconstructing monoidal categories. Adv. Soviet Math., 16:111– 136, 1993. [6] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2– 111, 2006. [7] Saunders Mac Lane. Categories for the Working Mathematician, Second Edition. Springer- Verlag, 1978. [8] Dmitri Nikshych. On the structure of weak hopf algebras. And. Math., 170:257–286, 2002. [9] Victor Ostrik. Pre-modular categories of rank 3. math.CT/0507349. [10] Victor Ostrik. Fusion categories of rank 2. Math. Res. Lett., 10(2-3):177–183, 2003. [11] Eric Rowell, Richard Stong, and Zhenghan Wang. in preparation. [12] Daisuke Tambara and Shigeru Yamagami. Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra, 209:692–707, 1998. [13] Hans Wenzyl and Imre Tuba. On braided tensor categories of type bcd. J. Reine. Angew. Math., 581:31–69, 2005. Department of Mathematics, Indiana University, Bloomington, Indana E-mail address: [email protected],[email protected] 1. Introduction 2. Main theorem and outline 3. Preliminaries and notational conventions 3.1. Skeletization 3.2. Strictification 3.3. Strictified skeletal fusion categories 3.4. Remarks 4. Proof of Theorem ?? part ??:possible tensor category structures 4.1. Setting up the pentagon equations 4.2. Normalizations 4.3. Associativity matrices 4.4. Pentagon equations with 1 1 matrices 4.5. Pentagon equations with 2 2 or 6 6 matrices 4.6. The pentagon equation with 16 16 matrices 4.7. Solutions 4.8. Inequivalence of the solutions 5. Proof of Theorem ?? part ??: rigidity structures 6. Proof of Theorem ?? part ??: the absence of braidings 7. Pivotal structures and sphericity 8. Proof of Theorem ?? part ??: spherical structure calculations Appendix A. Associativity matrices References

0704.0209

Chandra Observations of Supernova 1987A

Chandra Observations of Supernova 1987A Sangwook Park∗, David N. Burrows∗, Gordon P. Garmire∗, Richard McCray†, Judith L. Racusin∗ and Svetozar A. Zhekov∗∗ ∗Department of Astronomy and Astrophysics, 525 Davey Lab., Pennsylvania State University, University Park, PA 16802, USA †JILA, University of Colorado, Box 440, Boulder, CO 80309, USA ∗∗Space Research Institute, Moskovska Strasse 6, Sofia 1000, Bulgaria Abstract. We have been monitoring Supernova (SN) 1987A with Chandra X-Ray Observatory since 1999. We present a review of previous results from our Chandra observations, and some pre- liminary results from new Chandra data obtained in 2006 and 2007. High resolution imaging and spectroscopic studies of SN 1987A with Chandra reveal that X-ray emission of SN 1987A origi- nates from the hot gas heated by interaction of the blast wave with the ring-like dense circumstellar medium (CSM) that was produced by the massive progenitor’s equatorial stellar winds before the SN explosion. The blast wave is now sweeping through dense CSM all around the inner ring, and thus SN 1987A is rapidly brightening in soft X-rays. At the age of 20 yr (as of 2007 January), X-ray luminosity of SN 1987A is LX ∼ 2.4 × 10 36 ergs s−1 in the 0.5−10 keV band. X-ray emission is described by two-component plane shock model with electron temperatures of kT ∼ 0.3 and 2 keV. As the shock front interacts with dense CSM all around the inner ring, the X-ray remnant is now expanding at a much slower rate of v ∼ 1400 km s−1 than it was until 2004 (v ∼ 6000 km s−1). Keywords: supernova remnants; supernovae; SN 1987A; X-rays PACS: 98.38.Mz 1. INTRODUCTION Supernova (SN) 1987A, the nearest SN in four centuries, occurred in the Large Margel- lanic Cloud (LMC). The identification of a Type II SN from a blue supergiant progenitor and the detection of neutrino bursts associated with the SN indicate a core-collapse ex- plosion of a massive star [1]. SN 1987A, providing these fundamental parameters and being located at a near distance (d = 50 kpc), is a unique opportunity for the study of a massive star’s death and the subsequent birth of a supernova remnant (SNR) in unprece- dented detail. About 10 yr after the SN explosion, the blast wave started to interact with the “inner ring” of dense circumstellar medium (CSM) [5], which is believed to be produced by the equatorial stellar winds of the massive progenitor star. This shock-CSM interaction resulted in a dramatic brightening of SN 1987A in soft X-rays, which provides an excellent laboratory for the X-ray study of the evolution of an optically thin thermal plasma in nonequilibrium ionization (NEI) as the shock propagates through a complex density gradient of the dense CSM. As the rapidly brightening X-rays begin to illuminate the interior of the SN, metal-rich ejecta expelled from the massive star’s core will begin to glow optically, allowing us to study SN nucleosynthesis yields. Then, a few decades from now, when these newly-formed elements begin to cross the reverse shock surface, we will be able to measure the distribution of these elements in more detail through their http://arxiv.org/abs/0704.0209v1 TABLE 1. Chandra Observations of SNR 1987A Observation ID (Age)∗ Instrument (Subarray) (ks) Counts 124+1387† 1999-10-6 (4609) ACIS-S+HETG 116.1 690∗∗ 122 2000-1-17 (4711) ACIS-S3 (None) 8.6 607 1967 2000-12-07 (5038) ACIS-S3 (None) 98.8 9030 1044 2001-4-25 (5176) ACIS-S3 (None) 17.8 1800 2831 2001-12-12 (5407) ACIS-S3 (None) 49.4 6226 2832 2002-5-15 (5561) ACIS-S3 (None) 44.3 6427 3829 2002-12-31 (5791) ACIS-S3 (None) 49.0 9277 3830 2003-7-8 (5980) ACIS-S3 (None) 45.3 9668 4614 2004-1-2 (6157) ACIS-S3 (None) 46.5 11856 4615 2004-7-22 (6359) ACIS-S3 (1/2) 48.8 17979 4640+4641+5362+5363+6099† 2004-8-26∼9-5 (∼6400) ACIS-S+LETG 289.0 16557∗∗ 5579+6178† 2005-1-12 (6533) ACIS-S3 (1/8) 48.3 24939 5580+6345† 2005-7-14 (6716) ACIS-S3 (1/8) 44.1 27048 6668 2006-1-28 (6914) ACIS-S3 (1/8) 42.3 30940 6669 2006-7-28 (7095) ACIS-S3 (1/8) 36.4 30870 7636 2007-1-19 (7271) ACIS-S3 (1/8) 33.5 32798 ∗ Days since SN † These observations were splitted by multiple sequences which were combined for the analysis. ∗∗ Photon statistics are from the zeroth-order data. X-ray emission. Neutrino bursts were a strong support for a core-collapse explosion, and thus for the creation of a neutron star which should become bright in X-rays. High resolution imaging and spectroscopic studies of SN 1987A with Chandra X- Ray Observatory are an ideal tool for the X-ray study of SN 1987A. We have thus been observing SN 1987A with Chandra since its launch in 1999, roughly twice a year, in order to monitor the earliest stages of the evolution of the X-ray remnant of SN 1987A. We here review previous results from our Chandra observations of SNR 1987A [2, 6, 7, 8, 9, 10, 12, 13], and present some preliminary results from the latest Chandra observations. 2. OBSERVATIONS Our Chandra observations of SNR 1987A are listed in Table 1. As of 2007 January, we have performed a total of sixteen Chandra observations of SNR 1987A, including two deep gratings observations. Data reduction and analysis process have been described in the literatures [2, 6, 8, 12]. 3. X-RAY IMAGES Broadband Chandra ACIS images of SNR 1987A are presented in Fig. 1. We applied a subpixel resolution method [11], deconvolved images with the detector point spread function (PSF), and then smoothed. The ring-like overall morphology of the X-ray rem- 2007−1−19 (7271)2006−7−28 (7095) 2006−1−28 (6914)2005−7−14 (6716)2005−1−12 (6533)2004−7−22 (6359) 2004−1−2 (6157)2003−7−8 (5980)2002−5−15 (5561) 2002−12−31 (5791) 2001−12−12 (5407)2001−4−25 (5176)2000−12−7 (5038)2000−1−17 (4711) 1 arcsecond FIGURE 1. Chandra ACIS (in the 0.3−8 keV band) false-color images of SNR 1987A. In each panel, the observation date and age (days since the SN, in parentheses) are presented. nant is evident. SNR 1987A has been brightening and expanding for the last 7 yr. Ini- tially, the eastern side was brighter, but then the western side began brightening in early 2004 (day ∼6200). SNR 1987A is now bright all around the ring. Early images showed that the soft X-ray band images (E < 1.2 keV) were correlated with the optical images while the hard band (E > 1.2 keV) image matched the radio images [7]. These differ- ential X-ray morphologies supported our interpretation that soft X-rays are produced by the decelerated shock entering dense protrusions of the inner ring and that hard X-rays originate from the fast shock propagating through less dense regions between protru- sions. Recent data show that the X-ray morphology is now nearly identical between the hard and soft bands, which is perhaps expected as an increasing fraction of the blast FIGURE 2. Radial expansion of SNR 1987A (taken from Racusin et al. in preparation). Data taken with the gratings are excluded. Day 4711 has also been excluded because of the low photon statistics. wave shock front is reaching dense CSM all around the inner ring [8]. The 0.3−8 keV band count rate is now ∼0.98 c s−1, which is ∼14 times brighter than it was in 2000. Assuming the apparent X-ray morphology of SNR 1987A (i.e., an elliptical torus superposed with 3−4 bright lobes), we model X-ray images to derive the best-fit radius at each epoch. The details of our image modeling are presented in the literature (Racusin et al. in preparation). Measured radii indicate that the X-ray remnant is expanding with an overall expansion rate of v ∼ 3900 km s−1 (Fig. 2), which is consistent with our previous estimates [8]. It is, however, intriguing to note that the expansion rate is significantly reduced to v ∼ 1400 km s−1 since day ∼ 6200 (Fig. 2). Deceleration of the expansion rate is in fact in good agreement with our interpretation of the shock reaching dense CSM all around the inner ring on days ∼6000−6200 [9]. The putative neutron star has not yet become visible [2, 7, 8]. If the extinction for the SNR’s center were similar to that for the entire SNR, an upper limit of LX(2−10 keV) ∼ 1.5 × 1034 ergs s−1 has been estimated for an embedded point source [8]. 4. X-RAY SPECTRUM The X-ray spectrum of SNR 1987A is line-dominated, indicating a thermal origin (Fig. 3). As the shock interacts with increasing amount of dense CSM, multiple com- ponents of hot optically thin plasma are required to adequately fit the observed X-ray 5 10 15 20 25 Wavelength (Anstrom) 10.5 2 51 Energy (keV) (a) (b) FIGURE 3. (a) The ACIS spectrum of SNR 1987A as of 2007-1-19. The best-fit two-component plane shock model is overlaid. (b) The LETG spectrum of SNR 1987A as of 2004-8 (taken from [12]). TABLE 2. Best-Fit Parameters from the Two-Shock Model Fit of SNR 1987A (days) kT(soft) (keV) kT(hard) (keV) net(hard) (1011 cm−3 s) EM(soft) (1058 cm−3) EM(hard) (1058 cm−3) χ2/ν 6914 0.31+0.04 −0.02 2.21 +0.16 −0.07 2.24 +0.48 −0.40 29.28 +5.86 −6.75 3.54 +0.27 −0.21 178.3/142 7095 0.29+0.01 −0.01 2.03 +0.13 −0.12 2.63 +0.62 −0.44 37.89 +1.53 −0.87 4.65 +0.30 −0.30 240.5/141 7271 0.31+0.07 −0.01 1.96 +0.09 −0.07 3.63 +1.05 −0.78 40.80 +3.00 −13.80 5.61 +0.44 −0.33 183.6/142 ∗ Days since SN spectrum [8, 10, 13]. In fact, a two-temperature NEI plane shock model fits the observed ACIS spectrum of SNR 1987A (Fig. 3a). The soft and hard components characteristi- cally represent the decelerated shock (by dense protrusions of the inner ring) and the fast shock propagating into less-dense medium, respectively. Results from two-component plane shock model fits of the ACIS spectrum for the latest three epochs, which have not been published, are presented in Table 2. The foreground column is fixed at NH = 2.35 × 1021 cm−2 [10]. Metal abundances are fixed at values measured by Zhekov et al. [13], which are generally consistent with the LMC abundances. Ionization timescales for the soft component (kT ∼ 0.3 keV) are high (net > 10 12 cm−3 s), indicating the hot gas is in collisional ionization equilibrium due to the shock interaction with dense CSM. The high resolution dispersed spectrum obtained by the deep LETG observation re- vealed detailed X-ray emission lines from various elemental species (Fig. 3b, [12]). The high-quality LETG spectrum showed that the continuous distribution of the shock tem- perature is represented by two dominant components (kT ∼ 0.5 and 2.5 keV) [13]. The LETG spectrum indicated LMC-like metal abundances with a moderate enhancement in N [13]. X-ray line broadening measurements using the deep LETG observation in- dicated shock velocities of v ∼ 300−1700 km s−1 [12] which are significantly lower than that deduced from the HETG observation performed ∼5 yr earlier (v ∼ 3400 km s−1, [6]). These results are consistent with the ACIS spectral analysis, supporting the interpretation of the blast wave recently interacting with the entire inner ring. TABLE 3. Chandra Flux and Luminosity of SNR 1987A Age∗ fX(0.5−2 keV)† fX(3−10 keV)† LX(0.5−10 keV)∗∗ 4711 1.61±0.66 0.84±0.57 1.54 5038 2.40±0.22 0.92±0.21 2.22 5176 2.71±0.54 1.22±0.41 2.59 5407 3.55±0.43 1.20±0.44 3.24 5561 4.19±0.46 1.49±0.64 3.79 5791 5.62±0.45 1.82±0.46 5.05 5980 6.44±0.52 1.95±0.62 5.71 6157 7.73±0.62 2.38±0.57 6.82 6359 11.48±0.69 2.40±0.60 9.54 6533 16.29±0.65 2.80±0.73 13.58 6716 19.41±0.97 3.26±0.68 16.06 6914 21.96±1.10 3.45±0.69 17.99 7095 25.56±1.28 3.84±0.77 20.58 7271 29.62±1.48 4.41±0.88 23.54 ∗ Days since SN † Observed flux in units of 10−13 ergs cm−2 s−1 ∗∗ In units of 1035 ergs s−1, after corrected for NH = 2.35 × 10 21 cm−2. 1000 2000 3000 4000 5000 6000 7000 8000 Square: ROSAT (0.5-2 keV) Triangle: Chandra ACIS 3-10 keV Circle: Chandra ACIS 0.5-2 keV Dot: ATCA 3, 6, 13, 20 cm Days since SN . FIGURE 4. X-ray and radio light curves of SNR 1987A. Radio fluxes are arbitrarily scaled. The solid line is the best-fit model by [9] 5. X-RAY LIGHT CURVES We present the soft (0.5−2 keV) and hard (3−10 keV) band X-ray light curves in Table 3 and Fig. 4. We also present the ROSAT [4] and radio1 light curves (Fig. 4). The soft X- ray light curve has been increasing nearly exponentially for the last several yr, with apparent “upturns” on days ∼3500−4000 and days ∼6000−6200. These features were interpreted as the time when the blast wave first made contact with the dense protrusions, and the time when the shock reached the main body of the inner ring [9]. The latest data points (days > 6700) suggest that the soft X-ray flux is still rapidly increasing, but probably less steeply than it was for the previous ∼2 yr (Fig. 4). This latest behavior of the soft X-ray light curve might have implications for the details of the density structure of the inner ring. Periodic monitoring of the soft X-ray flux is important to study the details of the density and abundance structures of the inner ring. The hard X-ray light curve is increasing at a lower rate than the soft X-ray light curve (Fig. 4). This slow increase rate appears to be roughly consistent with the radio light curve (Fig. 4). Hard X-rays in SNR 1987A might thus originate from the same synchrotron radiation as radio emission does. However, the morphology of hard X-ray images is no longer distinguishable from that of soft X-ray images [9]. The origin of hard X-ray emission from SNR 1987A is thus uncertain. Periodic monitoring of hard X-ray and radio light curves and searching for X-ray lines in the hard band (e.g., Fe K lines) will be important to reveal the origin of hard X-ray emission. 6. THE ACIS PHOTON PILE-UP Based on their XMM-Newton data analysis, Haberl et al. [3] argued that our Chandra soft X-ray light curve [9] was significantly contaminated by the ACIS photon pileup. They re-estimated the 0.5−2 keV band ACIS fluxes of SNR 1987A using archival Chandra data, and calculated pileup correction factors for the measured ACIS fluxes. Their 0.5−2 keV band flux correction factors were up to ∼25%, especially for recent epochs of days ∼6533 and 6716. Thus, they claimed that the reported upturn of the soft X-ray light curve on days ∼6000−6200 [9] was an artifact caused by the photon pileup. Haberl et al., however, misunderstood our Chandra instrument setup for three epochs: we used the HETG on day 4609 and a 1/8 subarray of the ACIS on days 6533 and 6716, while they assumed the bare ACIS on day 4609 and a 1/2 subarray of the ACIS on days 6533 and 6716. Their ACIS flux corrections for these epochs were thus incorrect. We note that the ACIS photon pileup is not the sole contamination, and Haberl et al. did not consider other issues such as the charge transfer inefficiency and the time-dependent quantum efficiency degradation of the ACIS data. A moderate discrepancy ( <10%) is also known between XMM-Newton and Chandra due to the imperfect cross-calibration between them. Furthermore, SNR 1987A is an extended source as observed with the Chandra ACIS, whereas a pointlike source was apparently assumed by Haberl et al. 1 Radio data obtained with Australian Telescope Compact Array (ATCA) have been provided by L. Staveley-Smith. Considering these technical issues, we have re-analyzed the possible effects of ACIS photon pile-up on our Chandra observations using three independent methods: PIMMS/XSPEC simulations, ACIS event grade distribution analysis, and the (modified) standard ACIS photon pileup model. Our results from these three analyses agree that flux correction factors due to the ACIS photon pileup are roughly several % or less, with the exception of day 6157 where ∼15% of the soft X-ray flux appeared to be lost due to photon pileup (see Park et al. [in preparation] for the detailed results). Based on these results, we confirm that the scientific conclusions by Park et al. [9] were not affected by the ACIS photon pileup. ACKNOWLEDGMENTS This work was supported in part by Smithsonian Astrophysical Observatory under Chandra grant GO6-7047X. REFERENCES 1. W. D. Arnett, J. N. Bahcall, R. P. Kirshner, and S. E. Woosley, Annual Review of Astronomy and Astrophysics 27, 629–700 (1989). 2. D. N. Burrows, E. Michael, U. Hwang, R. McCray, R. A. Chevalier, R. Petre, G. P. Garmire, S. S. Holt, and J. A. Nousek, The Astrophysical Journal 543, L149–L152 (2000). 3. F. Haberl, U. Geppert, B. Aschenbach, and G. Hasinger, Astronomy & Astrophysics 460, 811–819 (2006). 4. G. Hasinger, B. Aschenbach, and J. Trümper, Astronomy & Astrophysics 312, L9–L12 (1996). 5. E. Michael, R. McCray, C. S. J. Pun, P. Garnavich, P. Challis, R. P. Kirshner, J. Raymond, K. J. Borkowski, R. A. Chevalier, A. V. Filippenko, C. Fransson, P. Lundqvist, N. Panagia, M. M. Phillips, G. Sonneborn, N. B. Suntzeff, L. Wang, and C. J. Wheeler The Astrophysical Journal 542, L53–L56 (2000). 6. E. Michael, S. A. Zhekov, R. McCray, U. Hwang, D. N. Burrows, S. Park, G. P. Garmire, S. S. Holt, and G. Hasinger The Astrophysical Journal 574, 166–178 (2002). 7. S. Park, D. N. Burrows, G. P. Garmire, J. A. Nousek, R. McCray, E. Michael, and S. A. Zhekov The Astrophysical Journal 567, 314–322 (2002). 8. S. Park, S. A. Zhekov, D. N. Burrows, G. P. Garmire, and R. McCray, The Astrophysical Journal 610, 275–284 (2004). 9. S. Park, S. A. Zhekov, D. N. Burrows, and R. McCray, The Astrophysical Journal 634, L73–L76 (2005). 10. S. Park, S. A. Zhekov, D. N. Burrows, G. P. Garmire, J. L. Racusin, and R. McCray, The Astrophysical Journal 646, 1001–1008 (2006). 11. H. Tsunemi, K. Mori, E. Miyata, C. Baluta, D. N. Burrows, G. P. Garmire, and G. Chartas, The Astrophysical Journal 554, 496–504 (2001) 12. S. A. Zhekov, R. McCray, K. J. Borkowski, D. N. Burrows, and S. Park, The Astrophysical Journal 628, L127–L130 (2005). 13. S. A. Zhekov, R. McCray, K. J. Borkowski, D. N. Burrows, and S. Park, The Astrophysical Journal 645, 293–302 (2006). Introduction Observations X-Ray Images X-Ray Spectrum X-Ray Light curves The ACIS Photon Pile-Up

0704.0210

Classification of superpotentials

CLASSIFICATION OF SUPERPOTENTIALS A. DANCER AND M. WANG Abstract. We extend our previous classification [DW4] of superpotentials of “scalar curvature type” for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in [DW4], i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kähler-Einstein product. 0. Introduction In this paper we continue the study we began in [DW4] of superpotentials for the cohomogeneity one Einstein equations. These equations are the ODE system obtained as a reduction of the Einstein equations by requiring that the Einstein manifold admits an isometric Lie group action whose principal orbits G/K have codimension one [BB], [EW]. As discussed in [DW3], these equations can be viewed as a Hamiltonian system with constraint for a suitable Hamiltonian H, in which the potential term depends on the Einstein constant and the scalar curvature of the principal orbit, and the kinetic term is essentially the Wheeler-deWitt metric, which is of Lorentz signature. For any Hamiltonian system with Hamiltonian H and position variable q, a superpotential is a globally defined function u on configuration space that satisfies the equation (0.1) H(q, duq) = 0. From the classical physics viewpoint, u is a C2 (rather than a viscosity) solution of a time- independent Hamilton-Jacobi equation. The literature for implicitly defined first order partial differential equations then suggests that such solutions are fairly rare. It is therefore not unrea- sonable to expect in our case that one can classify (at least under appropriate conditions) those principal orbits where the associated cohomogeneity one Einstein equations admit a superpotential. The existence of such a superpotential u in our case leads naturally to a subsystem of equations of half the dimension of the full Einstein system. One way to see this is via generalised first integrals which are linear in momenta, described in [DW4]. Schematically, the subsystem may be written as q̇ = J∇u where J is an endomorphism related to the kinetic term of the Einstein Hamiltonian. String theorists have exploited the superpotential idea in their search for explicit metrics of special holonomy (see for example [CGLP1], [CGLP2],[CGLP3], [BGGG] and references in [DW4]). The point here is that the subsystem defined by the superpotential often (though not always) represents the condition that the metric has special holonomy. Also, the subsystem can often be integrated explicitly. In [DW4], §6, we obtained classification results for superpotentials of the cohomogeneity one Ricci-flat equations. Besides assuming that G and K are both compact, connected Lie groups such that the isotropy representation of G/K is multiplicity-free, we also mainly restricted our attention to superpotentials which are of the same form as the scalar curvature function of G/K, i.e., a finite sum with constant coefficients of exponential terms. Almost all the known superpotentials are of this kind. Date: revised October 24, 2018. The second author was partly supported by NSERC grant No. OPG0009421. http://arxiv.org/abs/0704.0210v1 2 A. DANCER AND M. WANG However, the above classification results were further subject to the technical assumption that the extremal weights for the superpotential did not lie in the null cone of the Wheeler-de Witt metric. In [DW4] we gave some examples of superpotentials which do not satisfy this hypothesis. These included several new examples which do not seem to be associated to special holonomy. In this paper, therefore, we attempt to solve the classification problem without the non-null assumption on the extremal weights. As in [DW4], we use techniques of convex geometry to analyse the two polytopes naturally associated to the classification problem. The first is (a rescaled translate of) the convex hull conv(W) of the weight vectors appearing in the scalar curvature function of the principal orbit. The second is the convex hull conv(C) of the weight vectors in the superpotential. In [DW4] we showed that the non-null assumption forces these polytopes to be equal, so we could analyse the existence of superpotentials by looking at the geometry of conv(W). In the current paper, conv(C) may be strictly bigger than conv(W) because of the existence of vertices outside conv(W) but lying on the null cone of the Wheeler-de Witt metric. Our strategy is to consider such a vertex c and project conv(W) onto an affine hyperplane separating c from conv(W). We can now analyse the existence of superpotentials in terms of the projected polytope. The analysis becomes considerably more complicated because, whereas in [DW4] we could analyse the situation by looking at the vertices and edges of conv(W), now, because we have projected onto a subspace of one lower dimension, we have to consider the 2-dimensional faces of conv(W) also. We find that in this situation the only polytopes conv(W) arising from principal orbits with more than three irreducible summands in their isotropy representations are precisely those coming from principal orbits which are circle bundles over a (homogeneous) Fano product. In the latter case, the solutions of the subsystem defined by the superpotential correspond to Calabi-Yau metrics, as discussed in [DW4]. After a review of basic material in §1, we state the main classification theorem of the paper in §2 and give an outline of the strategy of the proof there. 1. Review and notation In this section we fix notation for the problem and review the set-up of [DW4]. Let G be a compact Lie group, K ⊂ G be a closed subgroup, and M be a cohomogeneity one G-manifold of dimension n + 1 with principal orbit type G/K, which is assumed to be connected and almost effective. A G-invariant metric g on M can be written in the form g = εdt2 + gt where t is a coordinate transverse to the principal orbits, ε = ±1, and gt is a 1-parameter family of G-homogeneous Riemannian metrics on G/K. When ε = 1, the metric g is Riemannian, and when ε = −1, the metric g is spatially homogeneous Lorentzian, i.e., the principal orbits are space-like hypersurfaces. We choose an Ad(K)-invariant decomposition g = k ⊕ p where g and k are respectively the Lie algebras of G and K, and p is identified with the isotropy representation of G/K. Let (1.1) p = p1 ⊕ · · · ⊕ pr be a decomposition of p ≈ T(K)(G/K) into irreducible real K-representations. We let di be the real dimension of pi, and n = i=1 di be the dimension of G/K (so dimM = n+ 1). We use d for the vector of dimensions (d1, · · · , dr). We shall assume that the isotropy representation of G/K is multiplicity free, i.e., all the summands pi in (1.1) are distinct as K-representations. In particular, if there is a trivial summand it must be 1-dimensional. We use q = (q1, · · · , qr) to denote exponential coordinates on the space of G-invariant metrics on G/K. The Hamiltonian H for the cohomogeneity one Einstein equations with principal orbit G/K is now given by: H = v−1J + εv ((n− 1)Λ− S) , CLASSIFICATION OF SUPERPOTENTIALS 3 where Λ is the Einstein constant, v = 1 ed·q is the relative volume and (1.2) J(p, p) = which has signature (1, r − 1). The scalar curvature S of G/K above can be written as where Aw are nonzero constants and W is a finite collection of vectors w ∈ Z r ⊂ Rr. The set W depends only on G/K and its elements will be referred to as weight vectors. These are of three types (i) type I: one entry of w is −1, the others are zero, (ii) type II: one entry is 1, two are -1, the rest are zero, (iii) type III: one entry is 1, one is -2, the rest are zero. Notation 1.1. As in [DW4] we use (−1i,−1j , 1k) to denote the type II vector w ∈ W ⊂ Rr with −1 in places i and j, and 1 in place k. Similarly, (−2i, 1j) will denote the type III vector with −2 in place i and 1 in place j, and (−1i) the type I vector with −1 in place i. Remark 1.2. We collect below various useful facts from [DW4] and [WZ1]. Also, we shall use standard terminology from convex geometry, as given, e.g., in [Zi]. In particular, a “face” is not necessarily 2-dimensional. However, a vertex and an edge are respectively zero and one-dimensional. The convex hull of a set X in Rr will be denoted by conv(X). (a) For a type I vector w, the coefficient Aw > 0 while for type II and type III vectors, Aw < 0. (b) The type I vector with −1 in the ith position is absent fromW iff the corresponding summand pi is an abelian subalgebra which satisfies [k, pi] = 0 and [pi, pj] ⊂ pj for all j 6= i. If the isotropy group K is connected, these last conditions imply that pi is 1-dimensional, and the pj , j 6= i, are irreducible representations of the (compact) analytic group whose Lie algebra is k⊕ pi. (c) If (1i,−1j ,−1k) occurs in W then its permutations (−1i, 1j ,−1k) and (−1i,−1j , 1k) do also. (d) If dim pi = 1 then no type III vector with −2 in place i is present in W. If in addition K is connected, then no type II vector with nonzero entry in place i is present. (e) If I is a subset of {1, · · · , r}, then each of the equations i∈I xi = 1 and i∈I xi = −2 defines a face (possibly empty) of conv(W). In particular, all type III vectors in W are vertices and (−1i,−1k, 1j) ∈ W is a vertex unless both (−2i, 1j) and (−2k, 1j) lie in W. (f) For v,w ∈ W (or indeed for any v,w such that vi or wi = −1), we have (1.3) J(v + d,w + d) = 1− For the remainder of the paper, we shall work in the Ricci-flat Riemannian case, that is, we take ε = 1 and Λ = 0. As in [DW4], any argument that does not use the sign of Aw would be valid in the Lorentzian case. We shall also assume that conv(W) is r− 1 dimensional. This is certainly the case if G is semisimple, as W spans Rr (see the proof of Theorem 3.11 in [DW3]). The superpotential equation (0.1) now becomes (1.4) J(∇u,∇u) = ed·q S, where ∇ denotes the Euclidean gradient in Rr. As in [DW4] we shall look for solutions to Eq.(1.4) of the form (1.5) u = Fc̄ e 4 A. DANCER AND M. WANG where C is a finite set in Rr, and the Fc̄ are nonzero constants. Now Eq.(1.4) reduces to, for each ξ ∈ Rr, (1.6) ā+c̄=ξ J(ā, c̄) FāFc̄ = Aw if ξ = d+ w for some w ∈ W 0 if ξ /∈ d+W. We shall assume henceforth that r ≥ 2 since the superpotential equation always has a solution in the r = 1 case, as was noted in [DW4], and J is of Lorentz signature only when r ≥ 2. The following facts were deduced in [DW4] from Eq.(1.6). Proposition 1.3. conv(1 (d+W)) ⊂ conv(C). Proof. If w ∈ W, then Eq.(1.6) implies that d + w = ā + c̄ for some ā, c̄ ∈ C, and hence that (d+ w) = 1 (ā+ c̄) ∈ conv(C). Proposition 1.4. If ā, c̄ ∈ C and ā+ c̄ cannot be written as the sum of two non-orthogonal elements of C distinct from ā, c̄ then either J(ā, c̄) = 0 or ā+ c̄ ∈ d+W. In particular, if c̄ is a vertex of C, then either J(c̄, c̄) = 0, or 2c̄ = d + w for some w ∈ W and J(c̄, c̄) F 2c̄ = Aw. In the latter case, J(d+w, d+w) has the same sign as Aw so is > 0 if w is type I and < 0 if w is type II or III. As mentioned in the Introduction, for the classification in [DW4] we made the assumption that all vertices c̄ of C are non-null. Under this assumption, the second assertion of Prop 1.4 implies that all vertices of C lie in 1 (d+W). Hence conv(C) is contained in conv(1 (d+W)), and by Prop 1.3 they are equal. This meant that in [DW4], subject to the non-null assumption, we could study the existence of a superpotential in terms of the convex geometry of W. The aim of the current paper is to drop this assumption. We still have conv( (d+W)) ⊂ conv(C), but can no longer deduce that these sets are equal. The problem is that a vertex c̄ of conv(C) may lie outside conv(1 (d+W)) if it is null. In fact, it is clear from the above discussion that conv(1 (d+W)) is strictly contained in conv(C) if and only if C has a null vertex. For if c̄ is a null vertex of C and 2c̄ = d + w for some w ∈ W, then Eq.(1.6) fails for ξ = d+ w. We conclude this section by proving an analogue of Proposition 2.5 in [DW4]. The arguments below using Prop 1.4 are ones which will recur throughout this paper. Henceforth when we use the term “orthogonal” we mean orthogonal with respect to J unless otherwise stated. Theorem 1.5. C lies in the hyperplane {x̄ : x̄i = (n−1)} (possibly after subtracting a constant from the superpotential). Proof. We can assume 0 /∈ C by subtracting a constant from the superpotential. We shall also use repeatedly below the fact that as J has signature (1, r− 1) there are no null planes, only null lines. Denote by Hλ the hyperplane x̄i = λ, so (d + W) lies in H 1 (n−1). Suppose there exist elements of C with x̄i > (n − 1). Let λmax denote the greatest value of x̄i over C. If ãc̃ is an edge of conv(C) ∩Hλmax , then Prop 1.4 shows that ã, c̃ are null, and that c̃ is orthogonal to the element of C closest to it on the edge. Hence c̃ is orthogonal to the whole edge. Now J is totally null on Span{ã, c̃}, so since there are no null planes, ã, c̃ are proportional, which is impossible as they are both in Hλmax . So C ∩Hλmax is a single point c̃max, which is null. Next we claim that all elements of C lying in the half-space x̄i > (n − 1) must be multiples of c̃max. If not, let λ∗ be the greatest value such that there is an element of C, not proportional to c̃max, in Hλ∗ . Let ã be a vertex of conv(C) ∩Hλ∗, not proportional to c̃max. Now, by Prop 1.4, J(ã, c̃max) = 0, and so ã is not null. Since λ∗ > (n− 1), we see ã+ ã must be written in another CLASSIFICATION OF SUPERPOTENTIALS 5 way as a sum of two non-orthogonal elements of C. This sum must be of the form µc̃max + f̃ . But c̃max is orthogonal to ã and to itself, hence to f̃ , a contradiction establishing our claim. Similarly, all elements of C lying in x̄i < (n − 1) are multiples of an element c̃min, should they occur. (Note that J is negative definite on H0 and we have assumed 0 /∈ C so λmin 6= 0.) We denote the sets of elements lying in these open half-spaces by C+ and C− respectively. Note that, when non-empty, C+ and C− are orthogonal to all elements of C ∩ H 1 (n−1). (For if ã ∈ C ∩ H 1 (n−1) then ã + c̃max cannot be written in another way as a sum of two non-orthogonal elements of C.) In particular, if c̃max and c̃min are orthogonal, then c̃max is orthogonal to all of conv(C), which is r-dimensional by assumption. So c̃max is zero, a contradiction. The same argument implies that C+ and C− are both non-empty. Let νc̃min and µc̃max be respectively the elements of C− and C+ closest to H 1 (n−1). Suppose that c̃max+νc̃min = c̃ (1)+ c̃(2) with c̃(i) ∈ C and J(c̃(1), c̃(2)) 6= 0. Non-orthogonality means the c̃(i) cannot belong to the same side of H 1 (n−1) and by the choice of ν, they cannot belong to opposite sides of (n−1). Both therefore lie inH 1 (n−1). But by the previous paragraph, J(c̃max+νc̃min, c̃ (1)+c̃(2)) = 0. This means that c̃max + νc̃min is null, which contradicts J(c̃max, c̃min) 6= 0. Hence c̃max + νc̃min lies in d+W ⊂ Hn−1. Applying the same argument to c̃min+µc̃max, we find that in fact µ = ν = 1, i.e., C+ = {c̃max} and C− = {c̃min}. Now C ∩ H 1 (n−1) (and hence its convex hull) is contained in the hyperplanes c̃ max, c̃ min in (n−1). These hyperplanes are distinct as c̃max is orthogonal to itself but not to c̃min. Hence d+W ⊂ (C + C) ∩Hn−1 is contained in the union of the point c̃max + c̃min and the codimension 2 subspace c̃⊥max ∩ c̃ min of Hn−1. So conv(d+W) is contained in a codimension 1 subspace of Hn−1, contradicting our assumption that dimconv(d+W) = r − 1. Remark 1.6. A notational difficulty arises from the fact that, as seen above, points of C are on the same footing as points in 1 (d+W) rather than points of W. Accordingly, we shall use letters c, u, v, ... to denote elements of the hyperplane ui = −1 (such as elements of W), and c̄, ū, v̄, ... to denote the associated elements 1 (d+ c), 1 (d+u), 1 (d+v), · · · of the hyperplane ūi = (n−1) (such as elements of C or of 1 (d+W)). Note that for any convex or indeed affine sum (j) of vectors ξ(j) in Rr, we have λjξ(j) = λj ξ(j). Since we now know that the set C, like 1 (d +W), lies in H 1 (n−1) := {x̄ : x̄i = (n − 1)}, we will adopt the convention, as in the last paragraph, that when we refer to hyperplanes such as c̄⊥ in the rest of the paper, we mean “affine hyperplanes in H 1 (n−1)”. 2. The classification theorem and the strategy of its proof We can now state the main theorem of the paper. Theorem 2.1. Let G be a compact connected Lie group and K a closed connected subgroup such that the isotropy representation of G/K is the direct sum of r pairwise inequivalent R-irreducible summands. Assume that dimconv(W) = r−1, where W is the set of weights of the scalar curvature function of G/K (cf §1). (This holds, for example, if G is semisimple.) If the cohomogeneity one Ricci-flat equations with G/K as principal orbit admit a superpotential of form (1.5) where C contains a J-null vertex, then we are in one of the following situations (up to permutations of the irreducible summands): (i) W = {(−1)i, (11,−2i) : 2 ≤ i ≤ r}, d1 = 1, C = (d + {(−11), (11,−2i) : 2 ≤ i ≤ r}) and r ≥ 2; 6 A. DANCER AND M. WANG (ii) r ≤ 3. Remark 2.2. As mentioned before, the situation where C has no null vertex was analysed in [DW4]. Hence, except for the r ≤ 3 case, Theorem 2.1 completes the classification of superpotentials of scalar curvature type subject to the above assumptions on G and K. Remark 2.3. The first case of the above theorem is realized by certain circle bundles over a product of r−1 Fano (homogeneous) Kähler-Einstein manifolds (cf. Example 8.1 in [DW4], and [BB], [WW], [CGLP3]), and the subsystem of the Ricci-flat equations singled out by the superpotential in these examples corresponds to the Calabi-Yau condition. For more on the r = 2 case, see the concluding remarks in §10. Remark 2.4. Theorem 2.1 remains true if we replace the connectedness of G and K by the connectedness of G/K and the extra condition on the isotropy representation given by the second statement in Remark 1.2(d), i.e., if pi is an irreducible summand of dimension 1 in the isotropy representation of G/K, then [pi, pj] ⊂ k⊕ pj for all j 6= i. This weaker property does hold in practice. For example, the exceptional Aloff-Wallach space N1,1 can be written as (SU(3) × Γ)/(U1,1 · ∆Γ), where U1,1 is the set of diagonal matrices of the form diag(exp(iθ), exp(iθ), exp(−2iθ)) and Γ is the dihedral group with generators 0 1 0 −1 0 0 0 0 1 e2πi/3 0 0 0 e−2πi/3 0 0 0 1 In order to prove Theorem 2.1 we have to analyse the situation when there is a null vertex c̄ ∈ C. As discussed in §1, conv(C) now strictly includes conv(1 (d + W)) as c̄ is not in conv(1 (d +W)). Our strategy is to take an affine hyperplane H separating c̄ from conv(1 (d+W)), and consider the projection ∆c̄ of conv(1 (d+W)) onto H from c̄. Roughly speaking, whereas in [DW4] we could analyse the situation by looking at the vertices and edges of conv(1 (d+W)), now, because we have projected onto a subspace of one lower dimension, we have to consider the 2-dimensional faces of conv(1 (d+W)) also. This is a natural method of dealing with the situation of a point outside a convex polytope. It has some relation to the notion of “lit set” introduced in a quite different context by Ginzburg- Guillemin-Karshon [GGK]. The analysis in the next section will show that the vertices of the projected polytope ∆ can be divided into three types (Theorem 3.8). We label these types (1A), (1B) and (2). Roughly, these correspond to vertices orthogonal to c̄, vertices ξ̄ such that the line through c̄ and ξ̄ meets conv(1 (d+W)) at a vertex, and vertices ξ̄ such that this line meets conv(1 (d+W)) in an edge. In the remainder of the paper we shall gradually narrow down the possibilities for each type. In §3 we begin a classification of type (2) vertices. In §4 we are able to deduce that conv(1 (d +W)) lies in the half space J(c̄, ·) ≥ 0. We are able to deduce an orthogonality result for vectors on edges in conv(1 (d + W)) ∩ c̄⊥. This is analogous to the key result Theorem 3.5 of [DW4] that held (in the more restrictive situation of that paper) for general edges in conv(1 (d+W)). In §5 we exploit this result and some estimates to classify the possible configurations of (1A) vertices (i.e. vertices in c̄⊥), see Theorem 5.18. In §6 we attack the (1B) vertices, exploiting the fact that adjacent (1B) vertices give rise to a 2-dimensional face of conv(1 (d +W)). This is the most laborious part of the paper, as it involves a case-by-case analysis of such faces. We show that adjacent (1B) vertices can arise only in a very small number of situations (Theorem 6.18). In §7 we exploit the listing of 2-dim faces to show that there is at most one type (2) vertex, except in two special situations (Theorem 7.1). In §8 and 9, we eliminate more possibilities for adjacent (1B) and type (2) vertices. We find that if r ≥ 4 then we are either in case (i) of the Theorem or there are no type (2) vertices and no adjacent type (1B) CLASSIFICATION OF SUPERPOTENTIALS 7 vertices. Using the results of §4, in the latter case we find that all vertices are (1A) except for a single (1B). Building on the results of §5 for (1A) vertices, we are able to rule out this situation in §10, see Theorem 10.15 and Corollary 10.16. 3. Projection onto a hyperplane We first present some results about null vectors in H 1 (n−1). Remark 3.1. From Eq.(1.3), the set of null vectors in the hyperplane H 1 (n−1) form an ellipsoid x2i /di = 1}. If c̄ is null, then the hyperplane c̄ ⊥ in H 1 (n−1) is the tangent space to this ellipsoid. So any element x̄ 6= c̄ of c̄⊥ satisfies J(x̄, x̄) < 0. Lemma 3.2. Let x, y satisfy yi = −1. Suppose that J(x̄, x̄) and J(ȳ, ȳ) ≥ 0. Then J(x̄, ȳ) ≥ 0, with equality iff x̄ is null and x̄ = ȳ. In particular, if x̄, ȳ are distinct null vectors then J(x̄, ȳ) > 0. Proof. This follows from Eq.(1.3) and Cauchy-Schwartz. Proposition 3.3. Let H = {x̄ : h(x̄) = λ} be an affine hyperplane, where h is a linear functional such that conv(1 (d + W)) lies in the open half-space {x̄ : h(x̄) < λ}. Then there is at most one element of C in the complementary open half-space {x̄ : h(x̄) > λ}. Such an element is a null vertex of conv(C). Hence any element of C outside conv(1 (d+W)) is a null vertex of C. Proof. Suppose the points of C with h(x̄) > λ are c̄(1), · · · , c̄(m) with m > 1. Our result is stable with respect to sufficiently small perturbations of H, so we can assume that h(c̄(1)) > h(c̄(2)) ≥ h(c̄(3)), · · · , h(c̄(m)). Now c̄(1) + c̄(1) and c̄(1) + c̄(2) cannot be written in any other way as the sum of two elements of C. Hence, by Prop. 1.4, c̄(1) is null and J(c̄(1), c̄(2)) = 0. The only other way c̄(2) + c̄(2) can be written is as c̄(1) + c̄ for some c̄ ∈ C. But then c̄ = 2c̄(2) − c̄(1), so J(c̄(1), c̄) = 0, and such sums will not contribute. Hence J(c̄(2), c̄(2)) = 0, contradicting Lemma 3.2. Corollary 3.4. For distinct elements c̄, ā of C, the line segment c̄ā meets conv(1 (d+W)). This gives us some control over the extent to which conv(C) can be bigger than the set conv(1 Lemma 3.5. Let A ⊂ H 1 (n−1) be an affine subspace such that A ∩ conv( (d +W)) is a face of conv(1 (d+W)). Suppose there exists c̄ ∈ C ∩A with c̄ /∈ conv(1 (d+W)). Let x̄ ∈ A. If x̄ = 1 (ā+ ā′) with ā, ā′ ∈ C, then in fact ā, ā′ ∈ A. Proof. If ā or ā′ equals c̄ this is clear. We know by Cor 3.4 that if ā, ā′ 6= c̄ then the segments āc̄, ā′c̄′ meet conv(1 (d+W)). So there exist 0 < s, t ≤ 1 with tā+ (1− t)c̄ and sā′ + (1− s)c̄ in conv(1 (d+W)). Hence (tā+ (1− t)c̄) + sā′ + (1− s)c̄ ∈ conv( (d+W)). As it is an affine combination of x̄, c̄ this point also lies in A, so it lies in A∩ conv(1 (d+W)). Also, it is a convex linear combination of the points tā+ (1 − t)c̄ and sā′ + (1− s)c̄ of conv(1 (d+W)). Hence, by our face assumption, both these points lie in A, so ā, ā′ lie in A. Remark 3.6. The above lemma will be very useful because it means that in all our later calculations using Prop 1.4 for a face defined by an affine subspace A, we need only consider elements of C lying in A. 8 A. DANCER AND M. WANG Proposition 3.7. Let vw be an edge of conv(W) and suppose v̄, w̄ ∈ C. (i) If there are no points of W in the interior of vw, then J(v̄, w̄) = 0. (ii) If u = 1 (v+w) is the unique point of W in the interior of vw, J(v̄, w̄) > 0, and u is type II or III, then Fv̄, Fw̄ are of opposite signs. Proof. Part(i) is a generalization of Theorem 3.5 in [DW4] and we will be able to apply the proof of that result after the following argument. Let the edge v̄w̄ of conv(1 (d + W)) be defined by equations 〈x̄, u(i)〉 = λi : i ∈ I where 〈x̄, u (i)〉 ≤ λi for i ∈ I and x̄ ∈ conv( (d + W)). (In the above, 〈 , 〉 is the Euclidean inner product in Rr.) Note that Span {u(i) : i ∈ I} is the 〈 , 〉-orthogonal complement of the direction of the edge. Let H be a hyperplane whose intersection with conv(1 (d+W)) is the edge v̄w̄. We can take H to be defined by the equation 〈x̄, i∈I biu (i)〉 = i∈I biλi where bi are arbitrary positive numbers summing to 1. If ā, ā′ are elements of C whose midpoint lies in v̄w̄, then either they are both in H or one of them, ā say, is on the opposite side of H from conv(1 (d+W)). In the latter case ā is null and the only element of C on this side of H so, by Prop 1.4, is J-orthogonal to v̄, w̄. Hence, as 1 (ā + ā′) is an affine combination of v̄, w̄, we see that J(ā, ā′) = 0, and so such sums do not contribute in Eq.(1.6). We may therefore assume that ā, ā′ are in H. But as this is true for all H of the above form, the only sums that will contribute are those where ā, ā′ are collinear with v̄w̄. Now if ā, say, lies outside the line segment v̄w̄, then it is null and J-orthogonal to v̄ or w̄, and hence to the whole line. So the only sums which contribute are those where ā, ā′ lie on the line segment v̄w̄. Now the proof of Theorem 3.5 in [DW4] gives (i). Turning to (ii), note first that the above arguments and Prop 1.4 give (ii) immediately if no interior points of the edge v̄w̄ lie in C. If there are m interior points in C, we again proceed as in the proof of Theorem 3.5 in [DW4] and use the notation there. We may assume that Lemma 3.2 (and hence Cor 3.3 and Lemma 3.4) of [DW4] still holds; for the only issue is the statement for λm+1, but if c (0) + c(λm+1) cannot be written as c(λj) + c(λk) (0 < λj , λk < m + 1) then what we want to prove is already true. Now Lemma 3.4 in [DW4] and our hypothesis J(v̄, w̄) > 0 imply that J00 < 0 and Jλi,λj > 0 except in the three cases listed there. The proof that the elements of C are equi-distributed in v̄w̄ carries over from [DW4] since the midpoint ū is not involved in the arguments. Suppose next that the points in C ∩ v̄w̄ are equi-distributed. In the special case where m = 1, we have J(v̄, ū) = 0 = J(ū, w̄), which imply J(ū, ū) = 0. So the midpoint does not contribute to the equation from c(0) + c(λm+1). If m > 1, we write down the equations arising from c(0) + c(λm+1) and c(λm−1) + c(λm+1). The formula for Fλj in [DW4] still holds for 1 ≤ j ≤ m, and using this and the second equation we obtain the analogous formula for Fλm+1 . Putting all the above information together in the first equation and using Au < 0, we see that Fm+1λ1 /F 0 is positive if m is even and negative if m is odd. In either case it follows immediately that F0Fλm+1 < 0, as required. We shall now set up the basic machinery of the projection of our convex hull onto an affine hyperplane. Let c̄ be a null vector in C and let H be an affine hyperplane separating c̄ from conv(1 (d+W)). Define a map P : conv(1 (d + W)) −→ H by letting P (z̄) be the intersection point of the ray c̄z̄ with H. We denote by ∆ the image of P in H. (P and ∆ of course depend on c̄ and the choice of H. When considering projections from several null vertices, we will use the vertices as superscripts to distinguish the cases, e.g., ∆c̄,∆b̄.) Let us now consider a vertex ξ̄ of ∆. We know that c̄ and ξ̄ are collinear with a subset P−1(ξ̄) of conv(1 (d+W)). As ξ̄ does not lie in the interior of a positive-dimensional subset of ∆, we see that CLASSIFICATION OF SUPERPOTENTIALS 9 no point of P−1(ξ̄) lies in the interior of a subset of conv(1 (d +W)) of dimension > 1. So P−1(ξ̄) is a vertex or an edge of conv(1 (d+W)). If P−1(ξ̄) is a vertex x̄, then 2x̄ ∈ d +W and in Lemma 3.5 we can take the affine subspace A to be the line through c̄, ξ̄, x̄. Using this lemma and also Prop 1.4 and Cor 3.4 we see that either x̄ ∈ C (in which case J(x̄, c̄) = 0), or x̄ /∈ C and x̄ = (ā+ c̄)/2 for some null element ā ∈ C ∩A. We have therefore deduced Theorem 3.8. Let ξ̄ be a vertex of ∆. Then exactly one of the following must hold: (1A) ξ̄ (and hence P−1(ξ̄)) is orthogonal to c̄; (1B) The line through c̄, ξ̄ meets conv(1 (d + W)) in a unique point x̄, and there exists a null ā ∈ C such that (ā+ c̄)/2 = x̄; (2) ξ̄ is not orthogonal to c̄, and c̄ and ξ̄ are collinear with an edge v̄w̄ of conv(1 (d+W)), (and hence c and ξ are collinear with the corresponding edge vw of conv(W)). Remark 3.9. If (1B) occurs, then ā = 2x̄− c̄ being null is equivalent to J(x̄, x̄) = J(x̄, c̄), that is, (3.1) In particular xi and ci are nonzero for some common index i. We will from now on refer to this situation by saying that the vectors x and c overlap. We make a preliminary remark about (1A) vertices. Lemma 3.10. Suppose that u ∈ W and ū ∈ c̄⊥. (a) If u = (−2i, 1j) then ci 6= 0. (b) Suppose that K is connected. If u = (−1i,−1j , 1k), then ci, cj , ck are all nonzero. Proof. After a suitable permutation, we may let 1, · · · , s be the indices a with ca 6= 0. We need In case (a) this is impossible if ci = 0 (that is, i /∈ {1, · · · , s}) as then we need dj = 1 = cj and ca = 0 for a 6= j, contradicting k=1 ck = −1. Next, Cauchy-Schwartz on ( ua√ )sa=1, ( )sa=1 shows ≥ 1. In case (b), if, say, ck = 0, then since 1 ≥ 1 and di, dj ≥ 2 (see Remark 1.2(d)) we must have di = dj = 2. The equations then imply ci = cj = −1 and ca = 0 for a 6= i, j, also giving a contradiction. Similar arguments rule out ci = 0 or cj = 0. In the next two sections we shall get stronger results on (1A) vertices. Let us now consider type (2) vertices. Theorem 3.11. Consider a type (2) vertex ξ̄ of ∆. So c and ξ are collinear with an edge vw of conv(W). Suppose there are no points of W in the interior of vw. Then we have (i) c = 2v − w or (ii) c = (4v − w)/3. In (i) the points of C on the line through c̄, ξ̄ are c̄ and w̄. In (ii) they are c̄, w̄ and c̄(1) = (2v̄ + w̄)/3 = (c̄+ w̄)/2. We need J(c̄(1), w̄) = 0. Proof. This is very similar to the arguments of §3 in [DW4]. We apply Lemma 3.5 to the line through v̄, w̄. (A) We write the elements of C on the line as c̄ = c̄(0), c̄(1), · · · , c̄(m+1) with m ≥ 0. So c̄(m+1) is either null or is w̄. No other c̄(j) can lie beyond w̄, by Cor 3.4. 10 A. DANCER AND M. WANG By assumption c̄ = c̄(0) is not orthogonal to the whole line. As c̄ is null, this means c̄ is not orthogonal to any other point on the line. So c̄(0) + c̄(j) is either 2v̄, 2w̄ or else is a sum of two other c̄(i). In particular, c̄(0)+ c̄(1) = 2v̄. In fact c̄(0)+ c̄(j) is never 2w̄; for the only possibility is for c̄(0) + c̄(m+1) = 2w̄, in which case c̄(m+1) is null, and so c̄(m) + c̄(m+1) = 2w̄, contradicting v̄ 6= w̄. We deduce that for j > 1, we have c̄(0) + c̄(j) = c̄(k) + c̄(p) for some 1 ≤ k, p ≤ j − 1. (B) Let c̄(m+1) be null. Since the segment c̄(0)c̄(m+1) lies in the interior of the null ellipsoid, Lemma 3.2 implies that J(c̄(i), c̄(j)) > 0 unless i = j = 0 or m + 1. Arguments very similar to those in §3 of [DW4] enable us to determine the signs of the Fc̄(j) in (1.5) and show that the contributions from the pairs summing to c̄(1) + c̄(m+1) cannot cancel. So we have a contradiction unless c̄(1) + c̄(m+1) = w̄, which can only happen if m = 1, i.e., c̄(0) + c̄(1) = 2v̄, c̄(1) + c̄(2) = 2w̄ and c̄(0) + c̄(2) = 2c̄(1) (otherwise c̄(0) + c̄(2) cannot cancel). Hence we have c = (3v − w)/2 ; c(1) = (v + w)/2 ; c(2) = (3w − v)/2. Writing Fj for Fc̄(j) , we need 2F0F2J(c̄, c̄ (2)) + F 21 J(c̄ (1), c̄(1)) = 0 so that the contributions from c̄(0) + c̄(2) and c̄(1) + c̄(1) cancel. As J(c̄, c̄(2)) and J(c̄(1), c̄(1)) > 0, we need F0 and F2 to have opposite signs. Now, as J(c̄, c̄(1)), J(c̄(1), c̄(2)) > 0, we see that Av and Aw have opposite signs. So we may let w be type I and v be type II or III, as long as the asymmetry between c̄(0) and c̄(2) is removed. Note that v,w cannot overlap if v is type II, as then Remark 1.2(c) means w is not a vertex. The possibilities are (up to permutation) v w c(0) = 1 (3v − w) c(2) = 1 (3w − v) (1) (−2, 1, 0, · · · ) (−1, 0, · · · ) (−5 , 0, · · · ) (−1 , 0, · · · ) (2) (−2, 1, 0, · · · ) (0,−1, 0, · · · ) (−3, 2, 0, · · · ) (1,−2, 0, · · · ) (3) (−2, 1, 0, · · · ) (0, 0,−1, 0, · · · ) (−3, 3 , · · · ) (1,−1 , 0, · · · ) (4) (1,−1,−1, 0, · · · ) (0, 0, 0,−1, · · · ) (3 , · · · ) (−1 , · · · ) Now, it is clear in (1) and (2) that c̄(0) and c̄(2) can’t both give null vectors. For (3) and (4), we find that the nullity equations for c̄(0) and c̄(2) have no integral solutions in di (in fact d3 (resp. d4) must be 5/2). Therefore in fact c̄(m+1) cannot be null. (C) Now suppose that c̄(m+1) = w and m > 0. Since J(c̄, v̄) 6= 0, v̄ must lie between c̄(0) and c̄(1). So J(c̄(0), ·) and J( · , c̄(m+1)) are affine functions on the line, vanishing at c̄(0) and c̄(m) respectively. Hence J(c̄(0), c̄(i)) (i ≥ 1) and J(c̄(i), c̄(m+1)) (0 ≤ i ≤ m − 1) are the same sign as J(c̄(0), c̄(1)). It follows that J(c̄(i), ·) is an affine function on the line, taking the same sign as J(c̄(0), c̄(1)) at c̄(0), c̄(m+1) (for 1 ≤ i ≤ m − 1). Thus J(c̄(i), c̄(j)) is the same sign as J(c̄(0), c̄(1)) except for the cases J(c̄(0), c̄(0)) = 0 = J(c̄(m), c̄(m+1)) : sign J(c̄(m+1), c̄(m+1)) = −sign J(c̄(0), c̄(1)). It then follows that the sign and non-cancellation arguments of (B) (taken from §3 of [DW4]) still hold, except in the case m = 1. These give the two cases of the Theorem. If m = 0, we have c(1) = w and c(0) = 2v − w as c(0) + c(1) = 2v. If m = 1, then c(2) = w, c(0) + c(1) = 2v and c(0) + c(2) = 2c1 (for cancellation). Hence c(0) = (4v − w)/3, c(1) = (2v + w)/3, as well as J(c̄(1), c̄(2)) = 0. Remark 3.12. If there are points of W in the interior of vw, we can still conclude that c(0)+c(1) = 2v. Hence c = λv + (1 − λ)w for 1 < λ ≤ 2, since if λ > 2 then c̄(1) is beyond w̄. It must then be null, and m = 0, so there is no way of getting 2w̄ as a sum of two elements in C. Lemma 3.13. For case (i) in Theorem 3.11 (i.e., c = 2v −w), either w is type I, or w is type III and vi = −1, wi = −2 for some index i. CLASSIFICATION OF SUPERPOTENTIALS 11 Proof. It follows from above that J(w̄, w̄)F 2w̄ = Aw so J(w̄, w̄) is positive if w is type I and negative if w is type II or III. In the latter case, > 1, but by nullity, c = 2v − w satisfies Hence for some i we have |wi| > |ci| = |2vi − wi|. As vi, wi ∈ {−2,−1, 0, 1}, it follows that vi = −1, wi = −2. We are now able to characterise the case where c is a type I vector. Theorem 3.14. If c is a type I vector, say (−1, 0, · · · ) for definiteness, then W is given by {(−1)i, (11,−2i) : i = 2, · · · , r}. Remark 3.15. Equivalently, W is as in Ex 8.1 of [DW4], where the hypersurface in the Ricci-flat manifold is a circle bundle over a product of Kähler-Einstein Fano manifolds. A superpotential was found for this example in [CGLP3]. Proof. Nullity of c̄ implies d1 = 1, so (−2 1, 1i) /∈ W. Also (−11,−1j , 1k) /∈ W, as then c would be in conv(W). Let us consider the vertices ξ̄ in ∆. ξ̄ cannot be of type (1A); otherwise ξ1 = −1, which implies the existence of a type II vector in W with a nonzero first component, contradicting the above. There can also be no ξ̄ of type (1B) since by Remark 3.9 the vector x̄ satisfies 0 < −x1, which we ruled out above. Hence all vertices of ∆ are of type (2), i.e., correspond to edges vw of conv(W) such that c = λv + (1− λ)w and λ > 1. From this equation it follows that v,w are of the form v = (−1i), w = (11,−2i) for some i > 1. As ∆ (being a (r− 2)-dimensional polytope in an (r− 2)-dimensional affine space) has at least r − 1 vertices, such vectors occur for all i 6= 1. Now no type II vector can be in W, otherwise v would not be a vertex. Also (1i,−2j) with i, j 6= 1 cannot be in W, as then (−1j) would not be a vertex. We have already seen (−21, 1i) is not in W. So W is as claimed. We shall henceforth exclude this case, i.e. case (i) of Theorem 2.1, from our discussion. We conclude this section by giving a preliminary listing of the possibilities for c when we have a type (2) vertex. These are given by cases (i) and (ii) of Theorem 3.11, as well as the possible cases when there is a point of W in the interior of vw. For Theorem 3.11(i) the possible v,w, c are: v w c = 2v − w (1) (−1, 1,−1, · · · ) (−2, 1, · · · ) (0, 1,−2, · · · ) (2) (−1,−1, 1, · · · ) (−2, 1, · · · ) (0,−3, 2, · · · ) (3) (−1, 0,−1, 1, · · · ) (−2, 1, · · · ) (0,−1,−2, 2, · · · ) (4) (−2, 1, · · · ) (−1, 0, · · · ) (−3, 2, · · · ) (5) (−2, 1, · · · ) (0, 0,−1, · · · ) (−4, 2, 1, · · · ) (6) (−1, 0, · · · ) (0,−1, · · · ) (−2, 1, · · · ) (7) (1,−1,−1, · · · ) (0, 0, 0,−1, · · · ) (2,−2,−2, 1, · · · ) Table 1: c = 2v −w cases where · · · denotes zeros as usual. To arrive at this list, recall from Lemma 3.13 that w is either type I or type III with vi = −1, wi = −2 for some i. Note also that if w is type I and v is type II then v cannot overlap with w as w cannot then be a vertex. Furthermore, the other possibility with w type I and v type III is excluded as we are assuming in Theorem 3.11 that there are no points of W in the interior of vw. Finally, the case w = (−2, 1, · · · ), v = (−1, 0, · · · ) can be excluded as this just gives the example in Theorem 3.14. In order to list the possibilities under Theorem 3.11(ii), recall that we need J(c̄(1), w̄) = 0 where c(1) = (2v + w)/3. Equivalently, we need (3.2) 2J(v̄, w̄) + J(w̄, w̄) = 0. 12 A. DANCER AND M. WANG This puts constraints on the possibilities for v,w. For instance, w cannot be type I, as for such vectors J(v̄, w̄) ≥ 0 and J(w̄, w̄) > 0. Also, if w is type II or III, then from the superpotential equation we need J(w̄, w̄) < 0, so J(v̄, w̄) > 0. If w is type III, say (−2, 1, 0, · · · ), then since d1 ≥ 2, we have 4 ≤ 3, and the above equation gives J(v̄, w̄) ≤ 1 with equality iff d1 = 2, d2 = 1. By the above remarks and the nullity of c̄, after a moderate amount of routine computations, we arrive at the following possibilities, up to permutation of entries. In the table we have listed only the minimum number of components for each vector and all unlisted components are zero. Note that the entries (12)-(16) can occur only if K is not connected (cf. Remark 5.9). v w c(1) = (2v +w)/3 c = (4v − w)/3 (1) (0, 0,−2, 1) (−2, 1, 0, 0) (−2 (2) (−2, 0, 1) (−2, 1, 0) (−2, 1 ) (−2,−1 (3) (−1, 0, 0, ) (−2, 1, 0) (−4 , 0) (−2 (4) (0, 0,−1) (−2, 1, 0) (−2 (5) (−1, 0, 1,−1) (−2, 1, 0, 0) (−4 ) (−2 (6) (−1, 1,−1) (−2, 1, 0) (−4 , 1,−2 ) (−2 , 1,−4 (7) (0, 0, 1,−1,−1) (−2, 1, 0, 0, 0) (−2 (8) (0, 1,−1,−1) (−2, 1, 0, 0) (−2 , 1,−2 , 1,−4 (9) (0,−1,−1, 1) (1,−1,−1, 0) (1 ,−1,−1, 2 ) (−1 ,−1,−1, 4 (10) (1,−1, 0,−1) (1,−1,−1, 0) (1,−1,−1 ) (1,−1, 1 (11) (1,−2, 0) (1,−1,−1) (1,−5 ) (1,−7 (12) (1, 0, 0,−2) (1,−1,−1, 0) (1,−1 ) (1, 1 (13) (0, 0, 0,−2, 1) (1,−1,−1, 0, 0) (1 ) (−1 (14) (0,−1, 0, 1,−1) (1,−1,−1, 0, 0) (1 ,−1,−1 ) (−1 ,−1, 1 (15) (1, 0, 0,−1,−1) (1,−1,−1, 0, 0) (1,−1 ) (1, 1 (16) (0, 0, 0, 1,−1,−1) (1,−1,−1, 0, 0, 0) (1 ) (−1 Table 2: c = 1 (4v − w) cases We will also need a listing of those cases for which vw has interior points lying in conv(W). v w c (1) (1,−2, · · · ) (−2, 1, · · · ) (3λ− 2, 1− 3λ, · · · ) (2) (1,−2, · · · ) (−1, 0, · · · ) (2λ − 1, −2λ, · · · ) (3) (−1, 0, · · · ) (1,−2, · · · ) (1− 2λ, 2λ− 2, · · · ) (4) (−2, 1, 0, · · · ) (0, 1,−2, · · · ) (−2λ, 1, 2λ− 2, · · · ) (5) (1,−1,−1, · · · ) (−1, 1,−1, · · · ) (2λ− 1, 1 − 2λ,−1, · · · ) Table 3: Cases with interior points Recall from Remark 3.12 that 1 < λ ≤ 2 and · · · denote zeros. Note that except in (4) all interior points which may lie in W actually do. 4. The sign of J(c̄, w̄) Theorem 4.1. conv(1 (d + W)) lies in the closed half-space J(c̄, ·) ≥ 0, i.e., the same closed half-space in which the null ellipsoid lies. Proof. We know that if ξ̄ is a vertex of ∆c̄ then there are three possibilities, given by (1A), (1B) and (2) of Theorem 3.8. If (1A) occurs, then by definition J(c̄, ξ̄) = 0. If (1B) occurs, let ā be the null vector in Theorem 3.8. Then by Lemma 3.2, J(c̄, ā) > 0, which in turn implies that J(c̄, ξ̄) > 0. It is now enough to show that J(c̄, ξ̄) ≥ 0 if ξ̄ is a type (2) vertex of ∆c̄, since it then follows that ∆c̄, and hence conv(1 (d+W)), lies in the half-space J(c̄, ·) ≥ 0. CLASSIFICATION OF SUPERPOTENTIALS 13 Suppose then that ξ̄ is a type (2) vertex with J(c̄, ξ̄) < 0. By Remark 3.12, c = λv + (1 − λ)w for some v,w ∈ W with 1 < λ ≤ 2, and both J(c̄, v̄), J(c̄, w̄) < 0. In particular, from Remark 3.1 and Lemma 3.2, J(v̄, v̄), J(w̄, w̄) < 0 since v̄, w̄ lie on the side of c̄⊥ opposite to the null ellipsoid. 0 = 4J(c̄, c̄) = J(d+ λv + (1− λ)w, d+ λv + (1− λ)w) = J(λ(d + v) + (1− λ)(d + w), λ(d+ v) + (1− λ)(d+ w)) = λ2J(d+ v, d+ v) + 2λ(1− λ)J(d+ v, d + w) + (1− λ)2J(d+ w, d + w). It follows from the above remarks that J(d+ v, d + w) < 0, that is One then checks that this condition is only satisfied in the following cases (up to permutation of indices and interchange of v and w): (a) v = (−2, 1, 0, · · · ), w = (−2, 0, 1, 0, · · · ) with 1 < d1 < 4; (b) v = (−2, 1, 0, · · · ), w = (−1, 1,−1, 0, · · · ) with d1 = 2, or (d1, d2) = (3, 2), or d2 = 1; (c) v = (1,−1,−1, 0, · · · ), w = (1,−1, 0,−1, 0, · · · ) with d1 = 1 or d2 = 1; (d) v = (1,−1,−1, 0, · · · ), w = (0,−1,−1, 1, 0, · · · ) with d2 = 1 or d3 = 1. In case (a), c = (−2, λ, 1 − λ, 0, · · · ). The condition d1 < 4 is incompatible with the nullity of c̄. Interchanging v and w reverses only the role of λ and 1− λ. A similar argument rules out case (b) with v,w as shown, as here c = (−λ − 1, 1, λ − 1). If we interchange v and w, then c = (λ − 2, 1,−λ, · · · ). Theorem 3.11 tells us λ = 4/3 or 2, so c = (−2/3, 1,−4/3, · · · ) or (0, 1,−2, · · · ). In the former case c(1) := (2v+w)/3 = (−4/3, 1,−2/3, · · · ), so the condition J(w̄, c̄(1)) = 0 gives 8/3d1 + 1/d2 = 1. Thus (d1, d2) = (3, 9) or (4, 3) but in neither case is c̄ null. In the latter case nullity means 1/d2 + 4/d3 = 1, so J(c̄, v̄) = (1− 1/d2 − 2/d3) > 0, a contradiction. In case (c), c = (1,−1,−λ, λ − 1) and if v and w are interchanged, the last two components of c are interchanged. But c̄ cannot be null if d1 = 1 or d2 = 1. A similar argument works for case Corollary 4.2. conv(1 (d+W)) ∩ c̄⊥ is a (possibly empty) face of conv(1 ((d+W)). This enables us to adapt Theorem 3.5 of [DW4] to the elements of c̄⊥. Corollary 4.3. Let vw be an edge of conv(W) and suppose v̄ and w̄ are in c̄⊥. Suppose further that there are no elements of W in the interior of vw. Then J(v̄, w̄) = 0. Proof. This is essentially the same as the proof of Theorem 3.5 of [DW1]. As conv(1 (d+W))∩ c̄⊥ is a face of conv(1 (d + W)), Lemma 3.5 shows that for calculations in c̄⊥ we need only consider elements of C in this hyperplane. Note that by Cor 3.4, no elements of C lie on the opposite side of c̄⊥ to conv(1 (d+W). Any vertex of conv(C) ∩ c̄⊥ outside conv(1 (d +W)) ∩ c̄⊥ is, by Prop 1.4, null, so must be c̄ by Lemma 3.2. Now Cor 3.4 shows that c̄ is the only element of conv(C)∩c̄⊥ outside conv(1 (d+W))∩c̄⊥. But any sum c̄+ ā with ā ∈ c̄⊥ does not contribute, so in fact we are in the situation of Theorem 3.5 of [DW4]. We introduce the following sets: Ŝ1 = {i ∈ {1, · · · , r} : ∃ unique w ∈ W with w̄ ∈ c̄ ⊥ and wi = −2} Ŝ≥2 = {i ∈ {1, · · · , r} : ∃ more than one w ∈ W with w̄ ∈ c̄ ⊥ and wi = −2} These are similar to the sets S1, S≥2 of [DW4], but now we require that the vectors w to lie in c̄ It is immediate from Cor 4.3 that di = 4 if i ∈ Ŝ≥2, (cf Prop 4.2 in [DW4]). 14 A. DANCER AND M. WANG We next prove a useful result about which elements of 1 (d +W) can be orthogonal to c̄. This will give us information about when (1A) vertices can occur. Lemma 4.4. Assume that we are not in the situation of Theorem 3.14 (i.e., c is not of type I ). Let u ∈ W be such that ū ∈ c̄⊥. Then: (a) there exists i with ci 6= 0 and −2 < ci < 1; (b) if c ∈ Zr then there is at most one such u, and hence at most one (1A) vertex (wrt c). Proof. (a) The condition J(ū, c̄) = 0 means = 1, and nullity of c̄ means = 1. As ui ∈ {−2,−1, 0, 1}, if the condition in (a) does not hold, then uici ≤ c i for all i so we must have equality for all i. Now ci = ui for all i with ci nonzero. As ci = −1 and c 6= u (since c /∈ W by definition), this means c is a type I vector and we are in the situation of Theorem 3.14. (b) We see from the previous paragraph that we need uici > c i for some i. If c ∈ Z r this means ci = −1 and ui = −2. The orthogonality condition is now = 1 where uj = 1. As di 6= 1 we see cj ≥ 0. If cj = 0 then di = 2. If cj > 0 then di ≥ 3 so , where the second inequality is due to the nullity requirement 1 ≤ 1. So cj = 1 or 2. Moreover, the latter implies (di, dj) = (3, 6) and c = (−1i, 2j), which contradicts ci = −1. We see that either cj = 1 and (di, dj) = (4, 2) or (3, 3), or cj = 0 and di = 2. Cor 4.3 implies that if there is more than one such u (say (−2i, 1j) and (−2i, 1k)) for a given i, then di = 4, so (di, dj , dk) = (4, 2, 2), and (ci, cj , ck) = (−1, 1, 1), contradicting the nullity of c. It now readily follows that the nullity condition prevents there being more than one u ∈ W with ū ∈ c̄⊥ except when c = (−1,−1, 1, 0, · · · ) with d = (4, 4, 2, · · · ) or (3, 3, 3, · · · ) and u = (−2, 0, 1, 0, · · · ), (0,−2, 1, 0, . . .). But in this case if both u occur then c ∈ conv(W), a contradic- tion. We shall study (1A) vertices for non-integral c in the next section. The following results will be useful. Proposition 4.5. Let v = (−2i, 1j) and w = (−2k, 1l) be elements of W such that v̄, w̄ ∈ c̄⊥. Suppose that i ∈ Ŝ1 and {i, j} ∩ {k, l} = ∅. Then k ∈ Ŝ≥2 and (di, dk, dl) = (2, 4, 2). Proof. By Remark 1.2(e) the affine subspace {x̄ : xi+xk = −2, xj +xl = 1}∩ c̄ ⊥ meets conv(1 W)) in a face, whose possible elements are v,w, u = (−2k, 1j), y = (−1i, 1j ,−1k) and z = (−1i,−1k, 1l) (since i ∈ Ŝ1). As J(v̄, w̄) = 1 , we see from Thm 4.3 that vw is not an edge so z is present in the face. Now Cor 4.3 on vz implies di = 2. Also, u must be present, otherwise y is present and Cor 4.3 on zw and yw gives a contradiction. So k ∈ Ŝ≥2, and Cor 4.3 on uw implies dk = 4. Now considering zw implies dl = 2. Remark 4.6. This is similar to the proof of Prop 4.6 in [DW4]. But we cannot now deduce that dj = 1 as the proof of this in [DW4] relied on the existence of t = (−1 i,−1j , 1k), and although we know this is in W we do not know if t̄ lies in c̄⊥. Proposition 4.7. If i ∈ Ŝ1 and v = (−2 i, 1j) gives an element of c̄⊥ then w = (−1i,−1j , 1k) cannot give an element of c̄⊥. Proof. This is similar to Prop. 4.3 in [DW4]. Since i ∈ Ŝ1, the vectors v̄, w̄ lie on an edge in the face {x̄ : 2xi + xj = −3} ∩ c̄ ⊥ of conv(1 (d+W)), and J(v̄, w̄) = 1 (1− 2 ) 6= 0 since di 6= 1. Corollary 4.8. With v as in Prop 4.7, there are no elements w = (−2j , 1k) with w̄ in c̄⊥. CLASSIFICATION OF SUPERPOTENTIALS 15 Proof. This is similar to Prop 4.4 in [DW4]. If k = i, then the type I vector u := (−1i) = 1 (2v+w) lies in W and ū ∈ c̄⊥. By Lemma 5.1 below, u = c, contradicting c /∈ W. We can therefore take k 6= i. Now v̄, w̄ lie on an edge in the face {x̄ : 3xi +2xj = −4} ∩ c̄ ⊥ (this is a face by Prop 4.7 and the assumption i ∈ Ŝ1). But J(v̄, w̄) = (1 + 2 ) 6= 0. 5. Vectors orthogonal to a null vertex In this section we analyse the possibilities for 1 (d+W)∩ c̄⊥. This will give us an understanding of the vertices of type (1A). We first dispose of the case of type I vectors. Lemma 5.1. If u is a type I vector and ū ∈ c̄⊥ then c = u, so we are in the situation of Theorem 3.14. Proof. Up to a permutation we may let u = (−1, 0, · · · ). The orthogonality condition implies c1 = −d1. But nullity implies c2i /di = 1, so d1 = 1 and ci = 0 for i > 1. (Note that in particular u /∈ W.) We shall therefore assume from now on there are no type I vectors giving points of c̄⊥. Lemma 5.2. (i) Two type II vectors whose nonzero entries lie in the same set of three indices cannot both give elements of c̄⊥. (ii) Two type III vectors (−2i, 1j) and (1i,−2j) cannot both give elements in c̄⊥. (iii) Three type III vectors whose nonzero entries all lie in the same set of three indices cannot all give rise to elements in c̄⊥. Proof. These all follow from Lemma 5.1 by exhibiting an affine combination of the given vectors which is of type I. Let u, v ∈ W be such that ū and v̄ ∈ c̄⊥. It follows that λū + (1 − λ)v̄ ∈ c̄⊥ for all λ. Hence Remark 3.1 shows that for all λ 0 ≥ J(d+ λu+ (1− λ)v, d + λu+ (1− λv)) = J(λ(d+ u) + (1− λ)(d+ v), λ(d + u) + (1− λ)(d+ v)) = λ2(J(d + u, d+ u) + J(d+ v, d+ v)− 2J(d+ u, d+ v)) + 2λ(J(d + u, d+ v)− J(d+ v, d + v)) + J(d+ v, d+ v). Equality occurs if and only if λu+ (1− λ)v = c, as c̄ is the only null vector in c̄⊥. Multiplying by −1, using Eq.(1.3), and recalling that the minimum value of a quadratic αλ2 + βλ+ γ with α > 0 is γ − (β2/4α), we deduce the following result. Lemma 5.3. If u, v ∈ W and ū, v̄ ∈ c⊥ then (5.1) Moreover, equality occurs if and only if c = λu+ (1− λ)v for some λ. Remark 5.4. By definition, c does not lie in conv(W). So in the case of equality in Eq.(5.1) we cannot have 0 ≤ λ ≤ 1. This observation will in many cases show that equality cannot occur. Remark 5.5. The right-hand side of Eq.(5.1) is maximised when = 1 (i.e., J(ū, v̄) = 0). In this case Eq.(5.1) just follows from ≥ 1, which is true for any two vectors in c̄⊥. If J(ū, v̄) 6= 0, we get sharper information. 16 A. DANCER AND M. WANG Corollary 5.6. Suppose that K is connected. If u, v are type II vectors in W with ū, v̄ ∈ c̄⊥ then with equality if and only if c = λu+ (1 − λ)v for some λ, in which case all the di = 2 whenever i is an index such that ui or vi is nonzero. Proof. Writing X = and Y = we see that 1 ≤ X,Y ≤ 3 . The lower bound arises from ū, v̄ being in c̄⊥, while the upper bound follows from Remark 1.2(d) and the assumption that u, v are type II vectors. Now X +Y −XY = 1− (1−X)(1−Y ) is minimised for X,Y in this range if X = Y = 3 , when it takes the value 3 . The inequality Eq.(5.1) now gives the result. When K is connected, it follows that any two such type II vectors must overlap. Moreover, if they have only one common index then we are in the case of equality in Cor 5.6. The nullity of c̄ implies that λ = 1 in this case, contradicting Remark 5.4. Combining this remark with Cor 5.6 and Lemma 5.2 (i) , we deduce the following result. Corollary 5.7. Assume K is connected. If u, v are type II vectors in W with ū, v̄ ∈ c̄⊥, then either u = (−1a,−1b, 1i), v = (−1a,−1b, 1j) or u = (1a,−1b,−1i), v = (1a,−1b,−1j). Hence the collection of all such type II vectors is of the form, for some fixed a, b: (i) (−1a,−1b, 1i) : i ∈ I for some set I; or (ii) (1a,−1b,−1i) : i ∈ I for some set I; or (iii) (1,−1,−1, 0, · · · ), (1,−1, 0,−1, · · · ), (1, 0,−1,−1, · · · ). We now investigate type III vectors. Lemma 5.8. Suppose K is connected. If u is a type II vector and v a type III vector in W with ū, v̄ ∈ c̄⊥, then Proof. With the notation of Cor 5.6 we have 1 ≤ X ≤ 3 and 1 ≤ Y ≤ 3. So X + Y − XY = 1− (1 −X)(1 − Y ) ≥ 0, and Eq.(5.1) gives the desired inequality. Also, the case of equality (i.e., X = 3 , Y = 3) leads to λ = 2 , again contradicting Remark 5.4. Remark 5.9. While Cor 5.6 - Lemma 5.8 are stated under the assumption that K is connected, the actual property we used is that in Remark 2.4. By contrast, the next two results do not require this property. Lemma 5.10. Any two type III vectors u, v giving elements of c̄⊥ must overlap. Proof. Write u = (−2i, 1j) and v = (−2k, 1l). By Cor 4.3, if i, k ∈ Ŝ≥2 then di = dk = 4. Since J(c̄, ū) = 0 we have (by Cauchy-Schwartz) Hence 1 + dj If u and v do not overlap, then the above and the analogous result from considering J(c̄, v̄) = 0, together with the nullity of c̄, imply that dj = 1 = dl and the only nonzero components of c are ci = ck = −1, cj = cl = . But then c is the midpoint of uv, contradicting c /∈ conv(W). So if u and v do not overlap, we can take i ∈ Ŝ1. Proposition 4.5 shows that k ∈ Ŝ≥2 and (di, dk, dl) = (2, 4, 2). Hence 2 < X ≤ 3 and Y = , so X + Y − XY ≥ 0 and Non-overlap means that equality holds. But then λ = 1/3, contradicting Remark 5.4. CLASSIFICATION OF SUPERPOTENTIALS 17 Lemma 5.10, together with Lemmas 5.2 and 4.8, implies the following corollary. Corollary 5.11. The type III vectors associated to elements of 1 (d+W)∩ c̄⊥ are, up to permutation of indices, either of the form (a) (−21, 1i), i ∈ I, (with d1 = 4 if |I| ≥ 2), or (b) (11,−2i), i ∈ I, for some subset I ⊂ {2, · · · , r}. Having found the possible configurations for type III vectors in c̄⊥, we start to analyse the type II vectors for each such configuration. For the rest of this section we will assume that K is connected (cf Remark 5.9). Remark 5.12. Lemma 5.8 now shows that in case (a) of Cor 5.11, if |I| ≥ 2, then every type II vector associated to an element of c̄⊥ must have “-1” in place 1. Similarly, in case (b), if |I| ≥ 3, then every such type II vector has “1” in place 1. (So if a type II is present then d1 6= 1). If |I| = 2, the only possible type II vectors with “0” in place 1 are (01,−12,−13, 1i) where i ≥ 4, and all type II vectors whose first entry is nonzero actually must have first entry equal to 1. Lemma 5.13. In case (a) of Cor. 5.11 with |I| ≥ 2 there are no type II vectors associated to elements of c̄⊥. Proof. Let v = (−21, 1k) and w = (−11, 1i,−1j) give elements of c̄⊥ with k 6= i, j. Consider the face {x̄ : xi + xk = 1, x1 + xj = −2} ∩ c̄ ⊥. Other than v,w the possible elements in this face come from u = (−11, 1k,−1j) and s = (−21, 1i). As d1 = 4, J(v̄, w̄) 6= 0, so vw is not an edge and u must be present. But J(ū, w̄) = 1 (1 − 1 ) 6= 0 since d1 = 4, giving a contradiction. So k = i or j for every such v,w. Hence if such a w exists there are at most two type III vectors. Now if |I| = 2 and the type IIIs are (−2, 1, 0, · · · ), (−2, 0, 1, · · · ), we cannot have w = (−1, 1,−1, · · · ) or (−1,−1, 1, · · · ) as then a suitable affine combination of the above vectors give a type I vector. (cf Lemmas 5.1, 5.2). So in fact no type II vectors give rise to elements of c̄⊥. Lemma 5.14. The vectors v = (−2, 1, 0, · · · ) and w = (0, 1,−1,−1, 0, · · · ) are not both associated to elements of c̄⊥, unless (0, 1,−2, 0, · · · ) or (0, 1, 0,−2, 0, · · · ) is also. Proof. Suppose (0, 1,−2, 0, · · · ), (0, 1, 0,−2, 0, · · · ) are absent. Consider the face {x̄ : x2 = 1, x1 + x3 + x4 = −2} ∩ c̄ ⊥. The other possible elements of this face come from t = (−1, 1,−1, 0, · · · ) and y = (−1, 1, 0,−1, 0, · · · ). Both these must be present, as J(v̄, w̄) 6= 0. Applying Cor 4.3 to wt, vt and wy we obtain (d1, d2, d3, d4) = (4, 2, 2, 2). Now we have equality (for y, t) in Eq.(5.1), as both sides equal 15/16. We find that λ = 1/2, giving a contradiction again to Remark 5.4. Combining this with Lemma 5.8 (and using Lemma 4.7) yields: Corollary 5.15. If there is a unique type III vector u = (−21, 12) with ū in c̄⊥, then the type II vectors associated to elements of c̄⊥ all have “-1” in place 1. Moreover (−11,−12, 1i) cannot be present. Also, if (−11, 12,−1i) is present for some i ≥ 3 then (d1, d2) = (4, 2) or (3, 3) and the index i is unique. For the last assertion, observe that (−11, 12,−1i) and the type III vector are joined by an edge, so Cor 4.3 shows the dimensions are as stated. If we have two such type II for i0 and i1 then Eq.(5.1) implies di0 + di1 ≤ 4. Hence since K is connected, di0 = di1 = 2 and we have equality in Eq.(5.1) with λ = 1 , giving a contradiction. Lemma 5.16. Let the type III vectors be as in Cor 5.11(b), i.e., they are (11,−2a), a ∈ I. Assume that |I| ≥ 2. If we have a type II vector w = (11,−1i,−1j) with w̄ in c̄⊥ then i, j ∈ I 18 A. DANCER AND M. WANG Proof. Suppose for a contradiction that w = (11,−1i,−1j) is present (so d1 6= 1) and (1 1,−2j) absent (i.e. j /∈ I). Since |I| ≥ 2, we can consider v = (11,−2k) where k ∈ I (so k 6= j) and k 6= i. Consider the face {x̄ : x1 = 1, xi + xj + xk = −2} ∩ c̄ ⊥. As well as v,w the possible elements of W in the face giving elements of c̄⊥ are y = (11,−1i,−1k), t = (11,−1j ,−1k) and u = (11,−2i). As d1 6= 1, vw is not an edge so t is present. Now Cor 4.3 applied to vt and tw gives d1 = dj = 2 and dk = 4. Moreover, if i ∈ I then u is present, so the edge wu gives di = 4. Thus we have shown that da = 4 for all a ∈ I. Now considering (11,−2a) and (11,−2b) with a, b ∈ I, we see that we have equality in Eq.(5.1) (both sides equal 3 ). In fact c is the average of these two vectors (i.e., λ = 1 ), so as in Remark 5.4 we have a contradiction. Lemma 5.17. Let the type III vectors be as in Cor 5.11(b), i.e., they are (11,−2a), a ∈ I. Assume that |I| ≥ 3. Then d1 = 1. Proof. Each pair v,w of type III vectors gives an edge, and if d1 6= 1, then we have J(v̄, w̄) > 0. By Theorem 4.3 all the midpoint vectors (11,−1a,−1b) are present for a, b ∈ I. Now Prop 3.7 shows that Fv̄ and Fw̄ have opposite signs, so we have a contradiction if |I| ≥ 3. Putting together our results so far, we obtain a description of the possibilities for c̄⊥∩ 1 (d+W). Theorem 5.18. Assume that r ≥ 3 and K is connected, and that we are not in the situation of Thm 3.14. Up to permutation of the irreducible summands, the following are the possible configurations of vectors in W associated to elements of 1 (d+W) ∩ c̄⊥. (1) {(−21, 1i), 2 ≤ i ≤ m} for fixed m ≥ 2. There are no type II vectors, and d1 = 4 if m ≥ 3. (2) {(11,−2i), 2 ≤ i ≤ m} for fixed m ≥ 3 and d1 = 1. There are no type II vectors. (3)(i) {(11,−22), (11,−23), (−12,−13, 1i), 4 ≤ i ≤ m} with d1 = 1, d2 = d3 = 2. (ii) {(11,−12,−13), (11,−22), (11,−23), (−12,−13, 1i), 4 ≤ i ≤ m}, d1 6= 1, d2 = d3 = 2. (4) {(1,−2, 0, 0, · · · ), (1, 0,−2, 0, · · · ), (1,−1,−1, 0, · · · )} with d1 6= 1. (5) A unique type III (−2, 1, 0, · · · ). Possible type II vectors are (i) (−1, 1,−1, 0, · · · ) with either (d1, d2) = (4, 2) or (3, 3); or (ii) {(−11, 13,−1i), 4 ≤ i ≤ m} for fixed m ≤ r and with d1 = 2; or (iii) {(−11,−13, 1i), 4 ≤ i ≤ m} for fixed m ≤ r and with d1 = 2. (6) No type III vectors. Possible type II vectors are (i) {(−11,−12, 1i), 3 ≤ i ≤ m} for fixed m ≤ r, with d1 = d2 = 2 if m ≥ 4; or (ii) {(11,−12,−1i), 3 ≤ i ≤ m} for fixed m ≤ r, with d1 = d2 = 2 if m ≥ 4; or (iii) {(11,−12,−13), (11,−12,−14), (11,−13,−14)} with d1 = d2 = d3 = d4 = 2. Proof. Cor 5.11 gives the possibilities for the type III vectors in c̄⊥. If there are none then Cor 5.7 gives the possibilities in (6). If there is a unique type III vector, then Cor 5.15 and Cor 5.7 give us the cases listed in (5) (or (1) with m = 2 if there are no type II). If we have two or more type III vectors with −2 in the same place then Lemma 5.13 shows we are in case (1). If we have more than two type III vectors with 1 in the same place a, then da = 1 by Lemma 5.17. Remark 5.12 then implies there are no type II vectors and we are in case (2). If we have exactly two type III vectors with 1 in the same place, e.g., (1,−2, 0, · · · ) and (1, 0,−2, 0, · · · ), then the proof of Lemma 5.17 shows that if the type II vector (1,−1,−1, 0, · · · ) is absent we must have d1 = 1. On the other hand, if d1 = 1 we are, by Remark 5.12 and the connectedness of K, in case (2) or (3)(i). If d1 6= 1, then by the above, Remark 5.12, and Cor 5.7, we are in case (3)(ii) or (4). The statements about values of the di follow from straightforward applications of Cor 4.3 to the obvious edges of conv(1 (d+W)) ∩ c̄⊥. CLASSIFICATION OF SUPERPOTENTIALS 19 Remark 5.19. The possibilities in Theorem 5.18 can be somewhat sharpened. In cases (1), (2), and (3), m cannot be r; in other words the maximum number of vectors is not allowed. This follows easily from looking at the system of equations expressing the nullity of c̄, the orthogonality of the vectors to c̄ and the fact that the entries of c sum up to −1. Similarly, r 6= 3 in (5)(i) and r 6= 4 in (6)(iii). When m ≥ 5 in (5)(ii) or (5)(iii), the segment joining two type II vectors is an edge, so Cor 4.3 gives d3 = 2. 6. Adjacent (1B) vertices We now turn to (1B) vertices. Let ξ̄, ξ̄′ be adjacent (1B) vertices of ∆. Then there exist vertices x̄, x̄′ of conv(1 (d+W)) such that c̄, ξ̄, x̄ are collinear and c̄, ξ̄′, x̄′ are collinear. Moreover, there exist null vectors ā, ā′ such that x̄ = (ā+ c̄)/2 and x̄′ = (ā′+ c̄)/2. By Cor 3.4, there must be an element ȳ of conv(1 (d+W)) on āā′, so P−1(ξ̄ξ̄′) contains the convex hull of x̄, x̄′, ȳ and hence is 2-dimensional. As ξ̄ξ̄′ is by assumption an edge of ∆, P−1(ξ̄ξ̄′) is a 2-dimensional face of conv(1 (d+W)). So we need to analyse the 2-dimensional faces of conv(W) containing vertices x, x′ such that (6.1) x = (a+ c)/2, x′ = (a′ + c)/2, ā, ā′ null, and such that c lies in the 2-dimensional plane defining this face. The lines through x, c (resp. x′, c) only meet conv(W) at x (resp. x′). Most 2-faces of conv(W) are triangular. We list below (up to permutation of components) all the possible non-triangular faces. For further details regarding how this listing is arrived at, see [DW5]. We emphasize that only the full faces are being listed, i.e., configurations formed by all the possible elements of W in a given 2-dimensional plane. As the set of weight vectors for a given principal orbit may be a subset of the full set of possible weight vectors, these full faces may degenerate to subfaces or even lower-dimensional faces (see Remark 6.2). Listing convention: In the interest of economy and clarity, we make the convention that when we list vectors in W belonging to a 2-face we will use the freedom of permuting the summands to place nonzero components of the vectors first and we will only put down the minimum number of components necessary to specify the vectors. Hexagons: There are 3 possibilities. (H1) This is the face in the plane {x1 + x2 + x3 = −1; xa = 0, for a > 3}. Points of W are (−2i, 1j), (−1i, 1j ,−1k), (−1i) where i, j, k ∈ {1, 2, 3}. The type III vectors form the vertices of the hexagon. (H2) The plane here is {x1 + x2 = −1, x3 + x4 = 0, xi = 0 (i > 4)}. Points of W are vertices u = (−2, 1, 0, 0), v = (1,−2, 0, 0), y = (−1, 0, 1,−1), y′ = (0,−1, 1,−1), z = (−1, 0,−1, 1), z′ = (0,−1,−1, 1), and the interior points α = (−1, 0, 0, 0), β = (0,−1, 0, 0). (H3) The plane is {x2 = −1, x1 + x3 + x4 = 0, xi = 0 (i > 4)}. Points of W are the vertices u = (−1,−1, 1, 0), v = (0,−1, 1,−1), w = (1,−1, 0,−1), x = (1,−1,−1, 0), y = (0,−1,−1, 1), z = (−1,−1, 0, 1) and the centre t = (0,−1, 0, 0). Square: (S) with midpoint t = (0,−1, 0, 0, 0) and vertices v = (−1,−1, 1, 0, 0), u = (0,−1, 0, 1,−1), s = (0,−1, 0,−1, 1), w = (1,−1,−1, 0, 0). 20 A. DANCER AND M. WANG Trapezia: We have vertices v, u, s, w, t with 2v − s = 2u − w and t = 1 (s + w), i.e., these are symmetric trapezia. Below we list the possible v, u, s, w. v u s w (T1) (−2, 1, 0, 0) (−2, 0, 1, 0) (0, 0,−2, 1) (0,−2, 0, 1) (T2) (−2, 0, 1, 0) (−2, 1, 0, 0) (0,−1, 1,−1) (0, 1,−1,−1) (T3) (−1,−1, 0, 1) (0,−1,−1, 1) (−2, 1, 0, 0) (0, 1,−2, 0) (T4) (0, 0, 1,−1,−1) (1, 0, 0,−1,−1) (−2, 1, 0, 0, 0) (0, 1,−2, 0, 0) (T5) (−1, 0, 0, 1,−1) (0, 0,−1, 1,−1) (−2, 1, 0, 0, 0) (0, 1,−2, 0, 0) (T6) (1,−1,−1, 0, 0) (1,−1, 0,−1, 0) (0, 0,−1, 1,−1) (0, 0, 1,−1,−1) Table 4: Possible trapezoidal faces Note that the configuration with vertices (−1,−1, 1, 0, 0), (−1,−1, 0, 1, 0), (0, 0, 1,−1,−1), and (0, 0,−1, 1,−1) is equivalent to (T6) under the composition of a permutation and a J-isometric involution. Parallelograms: We have vertices v, u, s, w with v − u = s− w. v u s w (P1) (−2, 1, 0, 0) (−1, 0,−1, 1) (−2, 0, 1, 0) (−1,−1, 0, 1) (P2) (−2, 1, 0, 0, 0) (−2, 0, 1, 0, 0) (0, 1, 0,−1,−1) (0, 0, 1,−1,−1) (P3) (−2, 1, 0, 0, 0) (−2, 0, 1, 0, 0) (0, 0,−1,−1, 1) (0,−1, 0,−1, 1) (P4) (−2, 1, 0, 0) (−1, 0, 1,−1) (−1,−1, 0, 1) (0,−2, 1, 0) (P5) (−2, 1, 0, 0, 0) (−1, 0, 0, 1,−1) (−1, 0, 1,−1, 0) (0,−1, 1, 0,−1) (P6) (−2, 1, 0, 0, 0) (−1, 0, 0,−1, 1) (−1, 0,−1, 1, 0) (0,−1,−1, 0, 1) (P7) (−2, 1, 0, 0, 0) (0, 1,−1,−1, 0) (−1, 0, 0, 1,−1) (1, 0,−1, 0,−1) (P8) (1,−1,−1, 0, 0, 0) (0, 0, 0, 1,−1,−1) (1,−1, 0,−1, 0, 0) (0, 0, 1, 0,−1,−1) (P9) (1,−1,−1, 0, 0, 0) (1,−1, 0, 0,−1, 0) (0, 0,−1, 1, 0,−1) (0, 0, 0, 1,−1,−1) (P10) (1,−1,−1, 0, 0, 0) (1, 0, 0, 0,−1,−1) (0,−1,−1, 1, 0, 0) (0, 0, 0, 1,−1,−1) (P11) (0, 0, 1,−1,−1, 0) (0,−1, 0,−1, 0, 1) (1, 0, 0, 0,−1,−1) (1,−1,−1, 0, 0, 0) (P12) (1, 0,−1, 0,−1) (1,−1,−1, 0, 0) (0, 0,−1, 1,−1) (0,−1,−1, 1, 0) (P13) (−1, 0,−1, 0, 1) (0, 0,−1,−1, 1) (0,−1,−1, 1, 0) (1,−1,−1, 0, 0) (P14) (−1, 0, 1, 0,−1) (−1,−1, 1, 0, 0) (0, 0,−1, 1,−1) (0,−1,−1, 1, 0) (P15) (−1, 0, 1, 0,−1) (−1,−1, 1, 0, 0) (0, 0, 1,−1,−1) (0,−1, 1,−1, 0) (P16) (−2, 1, 0, 0) (0, 1,−2, 0) (−1, 0, 1,−1) (1, 0,−1,−1) (P17) (−2, 1, 0, 0) (0, 1, 0,−2) (−2, 0, 1, 0) (0, 0, 1,−2) Table 5: Possible parallelogram faces Remark 6.1. (P1), (P2), (P3), and (P17) are actually rectangles. (P16) also includes the midpoints y = (u+ v)/2 = (−1, 1,−1, 0) and z = (s+ w)/2 = (0, 0, 0,−1). The rectangle (P17) also includes the midpoints y = (u+ v)/2 = (−1, 1, 0,−1) and z = (s+ w)/2 = (−1, 0, 1,−1). Remark 6.2. We must also consider subshapes of the above. Each symmetric trapezium contains two parallelograms. The two rectangles with midpoints (P17), (P16) will contain asymmetric trapezia. (P17) also contains parallelograms and squares. (For (P16), note that s is present iff w is.) Furthermore, there are numerous subshapes of the hexagons. The regular hexagon (H3) contains rectangles with midpoint (by omitting opposite pairs of vertices). Besides triangles, the hexagon (H2) contains pentagons, rectangles and squares (with midpoints), and kite-shaped quadrilaterals (e.g. y′uz′v). For (H1) see the discussion before Theorem 6.12. Finally, the triangle with midpoints of all sides (where the vertices are the three type III vectors with 1 in the same place) contains a trapezium (by omitting one vertex) and hence parallelograms. Remark 6.3. We also note for future reference that there are examples where we can have four or more coplanar elements of W but the plane cannot be a face. These examples are not of course CLASSIFICATION OF SUPERPOTENTIALS 21 relevant to the case of adjacent (1B) vertices, but some will be relevant when we consider multiple vertices of type (2). The examples which we will need in that context are the following three trapezia v u s w (T ∗1) (0, 1,−1,−1) (1, 0,−1,−1) (−2, 1, 0, 0) (1,−2, 0, 0) (T ∗2) (0,−1, 1,−1) (1,−1, 0,−1) (−2, 1, 0, 0) (0, 1,−2, 0) (T ∗3) (1,−1,−1, 0) (1,−1, 0,−1) (−1, 0,−1, 1) (−1, 0, 1,−1) Table 6: Further trapezia In (T*2),(T*3), as in (T1)-(T7), we have 2v − s = 2u− w. In these examples t = 1 (s+w) may also be present. In (T*1) we have s− w = 3(v − u), and the vectors t = (2s + w)/3 = (−1, 0, 0, 0) and r = (s+ 2w)/3 = (0,−1, 0, 0) will also be present. As an example, we explain why the trapezium (T*2) can never be a face. As u is present in W, so are u′ = (−1, 1, 0,−1) and u′′ = (−1,−1, 0, 1). Now (2u′ + u′′)/3 = (2s + u)/3 = (−1, 1 , 0,−1 ) is in the plane, but u′ is not, so this plane cannot give a face. Similar arguments involving (1, 0,−1,−1), (−1, 0, 0, 0) (resp. (−1, 0,−1, 1), (−1, 0, 1,−1)) show (T*1) (resp. (T*3)) cannot be faces. These arguments also show several parallelograms cannot be faces, but these will not be relevant for our purposes. We now begin to classify the possible 2-faces which arise from adjacent (1B) vertices. We shall repeatedly use Prop 1.4, Cor 3.4, and Lemma 3.5. Let E denote the affine 2-plane determined by the 2-face being studied. Theorem 6.4. Suppose we have adjacent (1B) vertices corresponding to a parallelogram face vusw of conv(W). So we have ū = (ā+c̄)/2 and w̄ = (ā′+c̄)/2 for null ā, ā′. Suppose the vertices v, u, s, w are the only elements of W in the face. Then u,w are adjacent vertices of the parallelogram, and either (i) C ∩ E = {c̄, ā, ā′, ē} where ē is null with v = (a+ e)/2 and s = (a′ + e)/2; or (ii) v̄, s̄ ∈ C and J(ā, v̄) = J(ā′, s̄) = J(s̄, v̄) = 0. Moreover, if none of v, u, s, w is type I, then (i) cannot occur. Proof. We may introduce coordinates in the 2-plane E using the sides sv and sw to define the coordinate axes. In this way we can speak of “left” or “right”, “up” or “down”. If we extend the sides of the parallelogram to infinite lines, these lines divide the part of the plane outside the parallelogram into 8 regions, and c̄ must be in the interior of one such region. We first observe that if c̄ is in one of the four regions which only meet the parallelogram at a vertex, then āā′ does not meet the parallelogram, contradicting Lemma 3.4. (A) Let c̄ then lie in a region which meets the parallelogram in an edge. Without loss of generality we may assume the edge is uw. By Cor 3.4, all elements of C ∩ E lie on or between the rays from c̄ through ā, ā′. Hence, by Lemma 3.2, J(b̄, c̄) > 0 for all b̄ ∈ C \ {c̄}. If b̄ is a rightmost element of (C ∩E) \ {c̄}, then as b̄+ c̄ cannot be written in another way as a sum of two elements of C, we deduce from Prop 1.4 that b̄+ c̄ ∈ d+W. So b̄ is either ā or ā′. All other elements of C ∩ E lie to the left of āā′. Note also that a rightmost element of (C ∩ E) \ {c̄, ā, ā′} satisfies b+ c = a+ a′, 2v or 2s. (B) Next let ē = 2v̄ − ā. Observe that as well as v̄ = (ā + ē)/2, we have s̄ = (ā′ + ē)/2, since 2v̄ − ā = 2(v̄ − ū) + c̄ = 2(s̄ − w̄) + c̄ = 2s̄− ā′. If ē ∈ C, then it must be null, and the same argument as above shows that no elements of (C ∩ E) \ {ē} lie to the left of ā, ā′, so we are in case (i). Now, Lemma 3.2 shows J(h̄, k̄) > 0 for all h̄ 6= k̄ ∈ C ∩E. If v, u, s, w are all type II/III, we see that Fc̄, Fē are of one sign and Fā, Fā′ the other sign. But now the contributions from ā+ ā′ and c̄+ ē in the superpotential equation cannot cancel. 22 A. DANCER AND M. WANG If ē /∈ C then, as in the argument before Theorem 3.8, s̄, v̄ ∈ C and we are in case (ii). Prop 3.7 shows v̄, s̄ are orthogonal. Moreover, note that the remark at the end of (A) shows that v + c or s+ c is left of a+ a′. Lemma 6.5. In case (ii) of Theorem 6.4, we have J(v̄, v̄) = J(s̄, s̄). Proof. As c̄ and ā = 2ū− c̄ are both null, and similarly c̄ and ā′ = 2w̄− c̄ are both null, we deduce (cf Remark 3.9) (6.2) J(ū, ū) = J(ū, c̄) : J(w̄, w̄) = J(w̄, c̄). We also have (6.3) 2J(ū, v̄) = J(c̄, v̄) : 2J(w̄, s̄) = J(c̄, s̄) from the orthogonality conditions on ā, v̄ and ā′, s̄. Now J(s̄, s̄)− J(v̄, v̄) = J(s̄, s̄)− J(w̄− ū− s̄, w̄− ū− s̄), which, on expanding out and using the second relations of Eqs.(6.2),(6.3), becomes J(2ū− c̄, w̄ − s̄)− J(ū, ū). Now J(2ū− c̄, w̄ − s̄)− J(ū, ū) = J(2ū− c̄, ū− v̄)− J(ū, ū) = J(2ū− c̄, ū)− J(ū, ū) = J(ū− c̄, ū) = 0. We have used the first relations of Eqs.(6.3), (6.2) in the second and fourth equalities. Remark 6.6. We must also consider the case when the midpoint of one side or a pair of opposite sides of the parallelogram face is in W. This can happen for (P16) and (P17). Note that v, u, s, w are type II/III in these cases. In fact, the argument of Theorem 6.4 is still valid if one or both of the midpoints of vu, sw is in W and c lies in the region to the right of uw (or the left of vs). Keeping c in the region to the right of uw, we now need to consider the case where one or both of the midpoints of vs, uw is in W. The conclusions (in 6.4(ii)) still hold except that we no longer have J(v̄, s̄) = 0. However, we have to make slight modifications to the arguments as 1 (ā + ā′) may be in C ∩ E. If ē ∈ C, then, as ā + ā′ is not in d + W, the usual sign argument shows that the terms in the superpotential equation summing up to ā + ā′ do not cancel, which is a contradiction. So ē /∈ C and our previous arguments hold except for the use of Prop 3.7. Note that we also have to consider the possibility that a, a′, and e lie on the line through vs. But now the midpoint of uw must be present and C ∩E = {c̄, ā, ā′, 1 (ā+ ā′)}, with v + s = a+ a′. The usual sign argument then forces the midpoints of uw and vs to be present and of type I. Hence this special configuration cannot occur in (P16) or (P17). Lastly, since the proof of Lemma 6.5 makes no mention of midpoints, it remains valid if midpoints are present. The conditions of Theorem 6.4 and Lemma 6.5, together with the nullity of ā, ā′, c̄, put very strong constraints on vusw and the dimensions. In fact, one can check that these constraints cannot be satisfied for any of our parallelograms (including those of Remark 6.2) with one exception. This is the rectangle yy′z′z in (H2) with c = (−2, 1, 0, · · · ) and 1 , which will be dealt with in Lemma 8.5. We now give an example of how to apply the above conditions in a specific case. Example 6.7. Consider parallelogram (P8). The equation of the 2-plane E containing the paral- lelogram is (6.4) x2 = −x1, x5 = x6, x2 + x5 = −1, x1 + · · · + x6 = −1 and xi = 0 for i > 6. As all vertices are type II/III, we must be in case (ii) of Theorem 6.4. (A) Take c to face the side uw. Note that vs and uw have equation x1 = 1, x1 = 0 respectively, so c1 < 0. Also, the remarks at the end of parts (A) and (B) in the proof of Theorem 6.4 shows that c1 > − , as v + c or s+ c is left of a+ a′ so 1 + c1 > −2c1 CLASSIFICATION OF SUPERPOTENTIALS 23 The condition J(v̄, s̄) = 0 implies d1 = d2 = 2 and Lemma 6.5 implies d3 = d4. Eqs.(6.2) and (6.3) give four linear equations in ci. Now d3 = d4 and Eq.(6.3) show c3 = c4, so the equations for the plane give c = (1 − c4, − + c4, c4, c4,− − c4, − − c4). Next d1 = d2 = 2 and Eq.(6.3) show c4 = 3d4/(2d4 + 2) and c1 = (1− 2d4)/(2d4 + 2). But the condition −1 < c1 < 0 now implies d3 = d4 = 1, and it follows that c cannot be null. (B) The argument if c faces vs is very similar. We have d3 = d4 and d5 = d6 = 2, and the orthogonality equations imply c3 = c4. So c has the same form as in the second paragraph of (A) above. We find c4 = −3d4/(2d4+2)) and c1 = (1+4d4)/(2d4 +2). But we now have the inequality 1 < c1 < , so again d3 = d4 = 1, violating nullity. (C) If c faces vu or sw then we need J(s̄, w̄) = 0 (resp. J(v̄, ū) = 0), which is impossible. Example 6.8. The example of the square (S) with midpoint can be treated in essentially the same way as the parallelograms. By symmetry, we may assume that c lies in the region that intersects uw. However, because 1 (a + a′) may now be the midpoint and hence in W, the configuration of Theorem 6.4(i) can occur, even though all vertices are type II. We have C ∩ E = {c̄, ā, ā′, ē} with a = (−1,−1, 1, 1,−1, · · · ), a′ = (1,−1,−1,−1, 1, . . .), c = (1,−1,−1, 1,−1, · · · ), and e = (−1,−1, 1,−1, 1, · · · ) with nullity condition = 1. We will be able to rule this case out in §7. On the other hand, the configuration of Theorem 6.4(ii) cannot occur, as one easily checks. Next assume that adjacent (1B) vertices in ∆c̄ determine a trapezium vusw as shown in the diagram below: II III VVIVII where t is the midpoint of sw and vu is parallel to sw. We assume that v, u, s, w ∈ W but our conclusions hold whether or not t lies in W. We will now derive constraints on the 2-face and E∩C resulting from having c lie in one of the regions shown above. For theoretical considerations, we need only treat the cases where c lies in regions I to VI. In practice, for an asymmetric trapezium, we must consider c lying in the remaining regions as well. In the following we will adopt the convention that ā, ā′ always denote null vectors in C. (I) c in region I: This is impossible because then s̄ = 1 (c̄ + ā) and w̄ = 1 (c̄ + ā′) for some ā, ā′, and so āā′ would not intersect conv(1 (d+W), a contradiction to Cor 3.4. (II) c in region II: Then v̄ = 1 (c̄+ ā), ū = 1 (c̄+ ā′) for some ā, ā′. We get a contradiction to Cor 3.4 if ā, ā′ lie below the line sw. They also cannot lie on the line sw since the argument in (A) in the proof of Theorem 6.4 and Cor 3.4 imply that C ∩ E = {c̄, ā, ā′}, and the terms corresponding to s̄, w̄ in the superpotential equation would be unaccounted for. 24 A. DANCER AND M. WANG Let e = 2s − a, e′ = 2w − a′. These points lie in region VI, and since we have a trapezium, e 6= e′. We may now apply Theorem 3.8 to ā and ā′ to obtain the possibilities: (i) s̄, w̄ ∈ C; J(ā, s) = 0 = J(ā′, w̄), (ii) s̄ ∈ C, J(ā, s̄) = 0; w /∈ C, ē′ ∈ C is null, J(ē′, s̄) = 0, (iii) w̄ ∈ C, J(w̄, ā′) = 0; s̄ /∈ C, ē ∈ C is null, J(ē, w̄) = 0. Note that the last condition in (ii) (resp. (iii)) results from applying Theorem 3.8 to ē′ (resp. ē). (III) c in region III: We have v̄ = 1 (c̄+ ā), w̄ = 1 (c̄+ ā′) for some ā, ā′ lying respectively in regions VIII and VI (in view of Cor 3.4). Applying Theorem 3.8 we obtain the possibilities: (i) s̄ ∈ C, J(ā, s̄) = 0 = J(ā′, s̄), (ii) s̄ /∈ C, 2s̄ = ā+ ā′ (which implies c+ s = v + w). (IV) c in region IV: We have ū = 1 (c̄+ ā), w̄ = 1 (c̄+ ā′) for some ā, ā′ ∈ C ∩E. If a lies in region IX, then Cor 3.4 implies that ā′ lies in region VI. Applying Theorem 3.8 to ā and ā′ we obtain the possibilities: (i) s̄ ∈ C, J(ā, s̄) = 0 = J(ā′, s̄), (ii) 2s = a+ a′, i.e., c+ s = u+ w. If a lies on the line sv, then we may apply Theorem 3.8 to ā′. We cannot have 2s̄ = ā′ + ē′ with ē′ ∈ C and null, otherwise āē′ would not intersect conv(1 (d+W)). So we have (iii) s̄ ∈ C and J(s̄, ā′) = 0. If a lies in region II, then ā′ lies in region VI. Let ē = 2v̄ − ā and ē′ = 2s̄ − ā′. As we have a trapezium, ē 6= ē′. Now e lies in region VII or VIII while e′ lies in region VIII or IX, so by Cor 3.4 ē and ē′ cannot both lie in C and hence be null. Theorem 3.8 now gives the possibilities: (iv) v̄, s̄ ∈ C, J(ā, v̄) = 0 = J(ā′, s̄), (and by Prop 3.7 J(v̄, s̄) = 0), (v) v̄ ∈ C, J(ā, v̄) = 0, ē′ ∈ C is null, and J(ē′, v̄) = 0, (vi) s̄ ∈ C, J(ā′, s̄) = 0, ē ∈ C is null, and J(ē, s̄) = 0. (V) c in region V: We have ū = 1 (c̄+ ā′), s̄ = 1 (c̄+ ā) for some ā, ā′ lying respectively in regions VIII and II (by Cor 3.4). Theorem 3.8 now gives the possibilities: (i) v̄ ∈ C, J(ā, v̄) = 0 = J(ā′, v̄), (ii) v̄ /∈ C, 2v̄ = ā+ ā′ (which implies c+ v = u+ s). (VI) c in region VI: We have s̄ = 1 (c̄ + ā), w̄ = 1 (c̄ + ā′) for some ā, ā′ lying respectively in regions VIII and IV (by Cor 3.4). (To rule out ā, ā′ lying in the line vu, we proceed as in case (II), except that when t ∈ W, we conclude instead that C ∩E = {c̄, ā, ā′, 1 (ā+ ā′)}. One can still check that v̄, ū cannot be both accounted for.) Now let ē = 2v̄ − ā and ē′ = 2ū − ā′. Again, having a trapezium means ē 6= ē′ and Theorem 3.8 now gives the possibilities: (i) ū, v̄ ∈ C, J(ā, v̄) = 0 = J(ā′, ū), (and J(ū, v̄) = 0 by Prop 3.7), (ii) v̄ ∈ C, J(ā, v̄) = 0, ū /∈ C, ē′ ∈ C is null, (iii) ū ∈ C, J(ā′, ū) = 0, v̄ /∈ C, ē ∈ C is null. Remark 6.9. We mention a useful inequality which holds in (II) and (VI) above, as well as in parallelogram faces with the same configuration (cf Example 6.7(A)). Let us consider (II), where we choose in E coordinates such that the first coordinate axis is parallel to s̄w̄ (assumed to be horizontal) and the second coordinate axis is arbitrary, with the second coordinate increasing as we go up. As in (A) in the proof of Theorem 6.4, all points in (C ∩ E) \ {c̄, ā, ā′} must lie below the line āā′. Let b̄ be a point among these with largest second coordinate. Since we have seen above that either s̄ or w̄ lies in C∩E, we have s2 ≤ b2. Furthermore, as b̄ + c̄ cannot lie in d +W it must be balanced by sums of elements in C ∩ E, with the limiting configuration given by ā + ā′. So we have 1 (b2 + c2) ≤ a2 = a 2 = 2v2 − c2. Combining the two inequalities we get 3c2 ≤ 4v2 − s2. Equality in the above holds iff b̄ lies in s̄w̄ and b̄ + c̄ = ā + ā′. In particular, b̄ is unique, so in II(i), the inequality above is strict. CLASSIFICATION OF SUPERPOTENTIALS 25 Note that we only need v̄ū and s̄w̄ to be parallel and the presence or absence of t in W is immaterial. Hence in Theorem 6.4(ii) we also have an analogous strict inequality, which we have already used, e.g., in (B) of Example 6.7. (For a parallelogram, there may be midpoints on the pair of non-horizontal sides lying in 1 (d+W), but 1 (b̄+ c̄) can never equal these midpoints, so we still get the inequality we want.) For the configuration in (VI), we still have an analogous inequality, but since 1 (ā + ā′) ∈ C, we lose uniqueness of b̄ and hence the strict inequality. We will also have occasion to apply the above analysis to appropriate trapezoidal regions in hexagon (H3). The method described above together with Remark 6.9 can now be used to rule out the trapezia (T1)-(T6) as well as those mentioned in Remark 6.2. Example 6.10. For the trapezium (T3), the vectors v, u, s, w are given in Table 4, and lie in the 2-plane {x1 + x2 + x3 + x4 = −1, x2 +2x4 = 1}. vu is given by x4 = 1 while sw is given by x4 = 0. sv is given by x3 = 0 and wu is given by x1 = 0. The vector c that we are looking for has the form (−c3 + c4 − 2, 1 − 2c4, c3, c4). Since the trapezium is symmetric, an explicit symmetry being induced by interchanging x1 and x3, we need only consider c lying in regions II-VI. (A) If c lies in region III, then c1 > 0, c4 > 1. Since a = 2v − c, we obtain a = (c3 − c4,−3 + 2c4,−c3, 2 − c4). Similarly, a ′ = (c3 − c4 + 2, 1 + 2c4,−4 − c3,−c4). If we are in case (ii), then c = v + w − s = (1,−1,−2, 1), which violates c4 > 1. So we must be in case (i). It follows from J(ā, s̄) = 0 = J(ā′, s̄) that d1+3 = −2c3+4c4 and J(w̄, s̄) = J(v̄, s̄). The second equality implies that d1 = d2. Using this together with the first equality and the null condition for ā′ (in the form J(w̄, w̄) = J(w̄, c̄), see Remark 3.9) we get c4 = d1(d1 − 1)/(4d1 + 2d3). Since c4 > 1, we have d1(d1 − 5) > 2d3, so d1 > 5. But by Remark 3.1, J(s̄, s̄) < 0, which gives d1 < 5 (since d1 = d2), a contradiction. (B) Let c lie in region IV, so that c1 > 0, 0 < c4 < 1. We obtain a = (2 + c3 − c4, 2c4 − 3,−2 − c3, 2 − c4) and a ′ = (2 + c3 − c4, 1 + 2c4,−4 − c3,−c4). We claim that a3 > 0, so that a lies in region IX. To see this, we solve for c3, c4 using the null conditions J(ū, ū) = J(c̄, ū) and J(w̄, w̄) = J(c̄, w̄) for ā, ā′ respectively. We obtain c4 = (d2d3 + 2d3d4 − d2d4)/(d3(d2 + 3d4)) and a3 = −2− c3 = (d2d3 + 2d3d4 − d2d4)/(d2(d2 + 3d4)). Since c4 > 0 we obtain our claim. Since a lies in region IX, we first check if (ii) holds. In this case, c = (2,−1,−3, 1) which contradicts c4 < 1. The equations in (i) together imply the contradiction 0 = −4/d2. (C) Suppose c lies in region V, so that c1 > 0, c4 < 0. We obtain a = (c3−c4−2, 1+2c4,−c3,−c4) and a′ = (c3−c4+2, 2c4−3,−2−c3, 2−c4). If (ii) holds then c = (−1, 1,−1, 0) and this contradicts c1 > 0. Hence (i) must hold. By Remark 3.9, the null condition for ā is J(s̄, s̄) = J(s̄, c̄), which is c3 − ( 1 )c4 = 0. The two equations in (i) imply J(ū, v̄) = J(s̄, v̄) and J(v̄, v̄) < 0, which in turn give d1 = 2. Using this, the null condition for ā, and J(ā′, v̄) = 0 we obtain c4 = and c3 = − d2(d2+1) . But c1 = c4 − c3 − 2 > 0, which simplifies to 1 > d2(d2 + 1), a contradiction. (D) Let c lie now in region II. Then c1 < 0, c3 < 0, 1 < c4 ≤ where the last upper bound comes from the inequality in Remark 6.9. We obtain a = (c3 − c4, 2c4 − 3,−c3, 2 − c4) and a′ = (2 + c3 − c4, 2c4 − 3,−2− c3, 2− c4). The null conditions for ā, ā ′ then give c3 = − 2d1d4 + d1d2 − d2d4 (d1 + d3)(d2 + 2d4)− d2d4 , c4 = (d1 + d3)(d2 + 2d4) (d1 + d3)(d2 + 2d4)− d2d4 Suppose we are in case (i). The two equations and the above values of c3, c4 combine to give (d1 − d3)((d1 + d3)(d2 + 2d4) − d2d4) = 0. However, the upper bound c4 ≤ translates into (d1 + d3)(d2 + 2d4) ≥ 4d2d4. So the second factor is positive and we have d1 = d3. Putting this 26 A. DANCER AND M. WANG information into the equation J(ā, s̄) = 0, we get d1d2d4(d2 + 15) = 2d 1d2(d2 + 1)− 6d1d 2 + 2d4(2d 1d2 + 2d 1 + d By Remark 3.1 we also have J(s̄, s̄) < 0, i.e., 1 < 4 , so either d2 = 1 or d1 < 8. Substituting these values into the equation above and using c4 ≤ we obtain in each instance a contradiction. If we are in case (ii), then by adding the equations J(ā, s̄) = 0 and 2J(w̄, s̄) − J(ā′, s̄) = 0 (equivalent to J(ē′, s̄) = 0), we obtain 1 = 2 . Hence (d1, d2) = (4, 2) or (3, 3). One then checks that these values are incompatible with the null condition for ē′, J(ā, s̄) = 0, and the bound c3 < 0. An analogous argument works to eliminate case (iii), where we now need the bound c4 ≤ instead. (E) Lastly suppose c lies in region VI, so c1, c3 < 0 and − ≤ c4 < 0, where the lower bound for c4 results from Remark 6.9. We have a = (c3 − c4 − 2, 1 + 2c4,−c3,−c4) and a ′ = (2 + c3 − c4, 1 + 2c4,−4− c3,−c4). Using the null conditions for ā, ā ′, we obtain c1 = − 2(d2 + d3) d1 + d2 + d3 , c2 = d1 + 5d2 + d3 d1 + d2 + d3 , c3 = −2(d1 + d2) d1 + d2 + d3 , c4 = d1 + d2 + d3 If we are in case (i), J(ū, v̄) = 0 gives d2 = d4 = 2. The other two equations and the above values of c3, c4 then give 3(d1 + d3 + 2)(d1 + d3 − 4) = 4(3d1 + 3d3 − 2). The lower bound − becomes d1 + d2 + d3 ≥ 6d2. Using this inequality in the above Diophantine relation leads to a contradiction. (Alternatively, observe the relation is a quadratic in d1 + d3 with no rational roots). For case (ii), using the two equations and the above values for c3, c4, we arrive at the relation (d1 + d2 + d3)((d1 − 5)d2d4 + d1d4 + d2d3 + 2d3d4) = 2d2(d1d2 + 2d1d4 − d2d4 + d2d3 + 2d3d4). Using the lower bound −1 ≤ c4 in the above relation we see that d1 ≤ 3. By direct substitution, we further obtain d1 6= 3. Finally, if d1 = 2, the null condition for c̄ gives 1 > c21 and so d2 + d3 ≤ 4. The lower bound on c4 now implies d2 = 1. Since c2 > 1, the null condition for c̄ is violated. Case (iii) reduces to case (ii) upon interchanging the first and third summands. Therefore, the trapezium (T3) has been eliminated. We discuss next the hexagons (H1)-(H3). As the three cases are similar, we will focus on (H3) and refer to the following (schematic) diagram: IVIV′ VII′ VII Example 6.11. The hexagon (H3) lies in the 2-plane given by {x2 = −1, x1 + x3 + x4 = 0}. So c has the form (−c3 − c4,−1, c3, c4). The lines vw and zy are given respectively by x1 + x3 = 1 and CLASSIFICATION OF SUPERPOTENTIALS 27 x1 + x3 = −1. Similarly, the lines uv and yx are given by x3 = 1 and x3 = −1 respectively. The lines uz and wx are given by x1 = −1 and x1 = 1 respectively. Interchanging x1 and x3 induces the reflection about the perpendicular bisector of vw, while (x1, x2, x3, x4) 7→ (−x3, x2,−x1,−x4) induces the reflection about ux. These symmetries reduce our consideration to those c lying in regions I-VI. Moreover, (H3) is actually a regular hexagon. The symmetry (x1, x2, x3, x4) 7→ (−x4, x2,−x3,−x1) induces the reflection about zw, which swaps region II with region IV and region I with region VI. Finally, the symmetry (x1, x2, x3, x4) 7→ (−x3, x2,−x4,−x1) induces the rotation in E about t taking x to w, and maps region V to region III. Therefore, we need only consider c lying in regions I, II, and V. In the discussion below we again adopt the convention that ā, ā′ always denote null vectors in C. If c lies in region I, then ū = 1 (c̄+ ā), x̄ = 1 (c̄+ ā′) for some ā, ā′, and we immediately see that āā′ cannot meet conv(1 (d+W)), a contradiction to Cor 3.4. c lying in region II: We have c1, c3, < 1 and c1 + c3 > 1. The assumption of adjacent (1B) vertices means that v̄ = 1 (c̄+ ā) and w̄ = 1 (c̄+ ā′) for some ā, ā′ ∈ E ∩ C. Hence a = (c3 + c4,−1, 2− c3,−2− c4) and a′ = (2 + c3 + c4,−1,−c3,−2− c4). One checks easily that ā ′ lies in region IV and ā lies in region IV′. Moreover, the null conditions for these vectors yield d3 + d4 d1 + d3 + d4 , c3 = d1 + d4 d1 + d3 + d4 , c4 = − d1 + d3 + 2d4 d1 + d3 + d4 Let e := 2u − a and e′ := 2x − a′. These lie respectively in regions VII′ and VII. We can now apply Theorem 3.8 to ā and ā′ to obtain the following possibilities: (i) ū, x̄ ∈ C and J(ā, ū) = 0 = J(ā′, x̄); (ii) ū ∈ C, J(ā, ū) = 0, x̄ /∈ C, ē′ ∈ C is null; (iii) x̄ ∈ C, J(ā′, x̄) = 0, ū /∈ C, ē ∈ C is null; (iv) ū, x̄ /∈ C, ē, ē′ are both null. We can eliminate (i)-(iii) by noting that the two equations in each case together with the values of c3, c4 above imply that 1 = . Using this relation (and the values of c3, c4) in the null condition for c̄ then leads to a contradiction. For case (iv) we can again apply Theorem 3.8 to the null vertices ē and ē′. The conditions J(ē, z̄) = 0 and J(ē′, ȳ) = 0 lead, as above, to 1 = 1 and 1 = 1 respectively. Using this in the null condition for c̄ again leads to a contradiction. Hence z̄, ȳ /∈ C and q̄ := 2z̄− ē and q̄′ := 2ȳ− ē′ are null vectors in E ∩C. In fact we now find that q = q′, so caeqe′a′ is a hexagon circumscribing (H3). Let us consider the pair of null vertices c̄, q̄. We apply the argument in (A) of the proof of Theorem 6.4 to the wedge with vertex c̄ bounded by the rays c̄ā and c̄ā′. All elements of (C ∩ E) \ {ā, ā′, c̄} lie below the line āā′. Let b̄ be a highest (with respect to x1 + x3) element among these. Since ē ∈ C, b1+ b3 > −1 and so c̄+ b̄ cannot equal 2ū, 2t̄, 2x̄. Hence c̄+ b̄ = ā+ ā ′, and we compute that b1 + b3 = d1+d3−2d4 d1+d3+d4 . The analogous argument applied to the wedge bounded by the rays q̄ē and q̄ē′ gives a lowest element b̄′ of (C ∩E) \ {q̄, ē, ē′} satisfying b̄′ + q̄ = ē+ ē′ and b′1 + b 2d4−d1−d3 d1+d3+d4 avoid a contradiction, we must have d1 + d3 ≥ 2d4. We can repeat the above argument with the null vertex pairs {ē, ā′} and {ē′, ā}, obtaining the inequalities d3 + d4 ≥ 2d1 and d1 + d4 ≥ 2d3 respectively. The three inequalities then imply that in fact d1 = d3 = d4 and c = ( ,−1, 2 ). Furthermore, C ∩ E = {ā, ā′, c̄, ē, ē′, t̄, q̄} and the null condition for c̄ gives (d1, d2) = (3, 9) or (4, 3). By looking at the terms in the superpotential equation corresponding to the vertices (all of type II), we find that the coefficients Fc̄, Fē, Fē′ have the same sign, which is opposite to that of Fā, Fā′ , Fq̄. Next we note that the only ways to write d+( ,−1, 1 ) (resp. d+(−1 ,−1, 2 as a sum of element of C are t̄+ c̄ = ā+ ā′ (resp. t̄+ ā = c̄+ ē). The superpotential equation then 28 A. DANCER AND M. WANG gives FāFā′J(ā, ā ′)+Ft̄Fc̄J(t̄, c̄) = 0 and Fc̄FēJ(c̄, ē)+Ft̄FāJ(t̄, ā) = 0. Since J(ā, ā ′), J(c̄, ē), J(t̄, c̄) and J(t̄, ā) are all positive, the above equations and facts imply that Fc̄ and Fā have the same sign, a contradiction. So c cannot lie in region II. c lying in region V: We have c3 < −1 < −c4 < 1 < c1. The adjacent (1B) vertices assumption implies that w̄ = 1 (c̄+ā) and ȳ = 1 (c̄+ā′) for some ā, ā′ ∈ C∩E. It follows that a = (2+c3+c4,−1,−c3,−2−c4) and a′ = (c3 + c4,−1,−2− c3, 2− c4). The null conditions on these vectors give (2d1 + d4)(d3 + d4) d4(d1 + d3 + d4) , c3 = − 2d3 + d4 d1 + d3 + d4 , c4 = d3 − d1 d1 + d3 + d4 Since a3 = −c3 > 1, a lies above the line uv. Also, a 1 = c3 + c4 = −c1 < −1, so a ′ lies below the line uz. We can therefore apply Theorem 3.8 to ā and ā′ to get the following possibilities: (i) ū ∈ C, J(ā, ū) = 0 = J(ā′, ū), (ii) ū /∈ C, e := 2u− a, e′ := 2u− a′ lie in C ∩ E and are null. If (i) occurs, then the two orthogonality conditions imply that d1 = d3, so c4 = 0, c1 = −c3 = 1 + d1 . Substituting these values of ci into J(ā ′, ū) = 0 gives 1 = 2 . But the null condition for c̄ is 1 = 1 (1 + d1 )2 > 1 = 1, which is a contradiction. Hence (ii) must occur. Note that if the above diagram is rotated so that the lines x1 + x3 = κ (for arbitrary constants κ) are horizontal, then the lines x1 − x3 = κ would be vertical. u is the only point in the hexagon lying on x1 − x3 = −2. Observe that a1 − a3 = a 1 − a 3 ≥ −2, otherwise āā′ would not intersect conv(1 (d+W)), which contradicts Cor 3.4. If, however, a1−a3 > −2, then ēē′ would not intersect conv(1 (d+W)). So in fact a = e′, e = a′ and u all lie on x1 − x3 = −2. In other words, the hexagon is circumscribed by the triangle caa′ with intersections at w, u and y. It follows easily from the above that c = (2,−1,−2, 0), d1 = d3 = d4, and the null condition for c̄ is 1 = 8 . Also, we have C ∩E = {c̄, ā, ā′, t̄}. Since w, u, y are type II, by Lemma 3.2, we see that the signs of Fā, Fc̄, and Fā′ in the superpotential equation cannot be chosen compatibly. We have thus shown that the hexagon (H3) cannot occur. The hexagon (H2) is not regular, but has reflection symmetry about uv and the perpendicular bisector of yy′. It can be eliminated by similar arguments, but we now have to consider c lying in regions III and IV as well. The hexagon (H1) can also be eliminated by the above methods. Here the hexagon is invariant under the symmetric group permuting the coordinates x1, x2, x3. Together with Cor 3.4, this fact reduces our consideration to those c lying in three of the regions formed by extending the sides of the hexagon. As mentioned in Remark 6.2, we also need to rule out subshapes of the hexagons. For (H2) and (H3) the methods used above can also be applied to rule out all the sub-parallelograms and trapezia except the rectangle yy′z′z of (H2) (see Lemma 8.5 and the discussion immediately before Ex 6.7). All sub-triangles will be dealt with at the end of this section. (There is a triangle with midpoint in (H2) but that can be dealt with by similar methods.) For (H2) this leaves the pentagon yy′vz′z and the kite y′uz′v, both of which can still be eliminated using the above methods. The possible subshapes of (H1) are rather numerous. However, if r ≥ 4 we will be able to eliminate all of them in Lemma 8.6. Without this assumption, the above methods can be used to eliminate those subshapes which do not contain all three type I vectors. Of course the following discussion will handle the sub-triangles. Lastly, we consider triangular faces. Theorem 6.12. Suppose we have adjacent (1B) vertices in ∆c̄ corresponding to a triangular face x̄x̄′x̄′′ of conv(1 (d +W)). Let E be the affine 2-plane determined by the triangular face. So there are null vectors ā, ā′ in C ∩ E such that x = 1 (a+ c), x′ = 1 (a′ + c). CLASSIFICATION OF SUPERPOTENTIALS 29 Suppose the vertices of the triangle are the only elements of W in the face. Then we are in one of the following two situations: (i) C ∩ E = {c̄, ā, ā′, x̄′′}, with c+ x′′ = a+ a′ and J(x̄′′, ā) = J(x̄′′, ā′) = 0; (ii) C ∩ E = {c̄, ā, ā′} where 1 (a + a′) = x′′, one of x, x′, x′′ is type I, and the others are either both type I or both type II/III. Proof. PPPPPPPPPPPPP (i) ❝✟ ❵❵❵❵❵❵❵❵ (A) We may introduce coordinates in E so that x̄x̄′ is vertical and to the right of c̄. As āā′ must meet conv(1 (d + W)), we see x̄′′ is on or to the right of āā′. Let b̄ be any leftmost point of (C ∩E) \ {c̄}. As in Theorem 6.4, we see that b̄+ c̄ ∈ d+W, so all elements of C ∩E except c̄, ā, ā′ are to the right of āā′. (B) Considering āx̄′′ and ā′x̄′′ we see (using Theorem 3.8 and Cor 3.4) that either (1) x̄′′ ∈ C and J(x̄′′, ā) = 0 = J(x̄′′, ā′), or (2) x̄′′ /∈ C and x′′ = 1 (a+ a′). In case (1), (x̄′′)⊥ ∩E is the line through āā′. By Prop. 3.3 and Cor. 3.4, observe that all elements of (C∩E)\{x̄′′} are left of x̄′′. Let b̄ be a rightmost element of (C∩E)\{x̄′′}. So either J(b̄, x̄′′) = 0 or b̄+ x̄′′ ∈ d+W. Since b̄ is not to the left of āā′, the second alternative cannot hold and so b̄ must lie on āā′. Combining this with our results in (A), we see C ∩ E is as in (i). Also, as J(ā, ā′) > 0 and ā+ ā′ /∈ d+W, we see a+ a′ must equal c+ x′′. In case (2), by Cor 3.4 there are no elements of C ∩E right of āā′. Hence C ∩E is as in (ii). Now J(b̄, ē) > 0 for all b̄ 6= ē in C ∩ E, so the last statement of (ii) follows. Remark 6.13. We must also consider the case when some midpoints of the sides of our trian- gular face lie in W. (This could happen if two vertices were (1,−1,−1, · · · ), (−1, 1,−1, · · · ) or (1,−2, · · · ), (1, 0,−2, · · · ) or (1,−2, · · · ), (−1, 0, · · · ).) Let us denote the midpoints of xx′, xx′′ and x′x′′ respectively by z, y, t. If z is absent, the arguments of (A) in the proof of Theorem 6.12 still hold, so we have the alternatives (1),(2) in (B). If (1) holds then, choosing b̄ as above, if b is right of aa′, we have (b + x′′) ∈ W. This gives a contradiction since 1 (b + x′′) cannot be y or t as b 6= x, x′. Now C ∩ E = {c̄, ā, ā′, x̄′′}, and as c + x′′ /∈ 2W it must equal a+ a′. It follows that the midpoints y, t cannot arise. If instead (2) holds, then C ∩ E = {c̄, ā, ā′} and again no midpoints can be present. Suppose now the midpoint z of xx′ is present. The argument of (A) shows that to account for (ā+ ā′) ∈ C, and all elements of (C ∩E) \ {c̄, ā, ā′, (ā+ ā′)/2} are right of āā′. We still have the alternatives (1) and (2), but (2) immediately gives a contradiction. In (1) we see as before there are no elements of C ∩ E lying to the right of āā′, so C ∩ E = {c̄, x̄′′, ā, ā′, (ā+ ā′)/2}. Note that J(ā, (ā+ ā′)/2) and J(ā′, (ā+ ā′)/2) > 0. If c+x′′ = a+ a′, we find after some algebra that ā+(ā+ ā′)/2 6= 2ȳ and also cannot be written as a different sum of elements of C, giving a contradiction. If c̄ + x̄′′ 6= ā + ā′ then one sees that c̄ + x̄′′ /∈ d +W, and by relabelling x and x′, a and a′ we may assume that c̄ + x̄′′ = ā + 1 (ā + ā′) and also ā + ā′ = 2ȳ and ā′ + 1 (ā + ā′) = 2t̄ = x̄′ + x̄′′. These relations imply a = x′, a contradiction. So no triangle with any midpoints present can arise. 30 A. DANCER AND M. WANG Remark 6.14. There are also triangular faces with two points of W in the interior of an edge. This can only happen if two vertices are (−2, 1, 0, · · · ) and (1,−2, 0, · · · ) (up to permutation). The other sides of the triangle now have no interior points in W unless the triangle is contained in the hexagon (H1). We can again modify the proof of Theorem 6.12 to treat this situation. If the interior points z, w lie on xx′, then (2ā + ā′)/3, (ā + 2ā′)/3 must be in C, and all points of C ∩ E except for these two and c̄, ā, ā′ lie to the right of āā′. By Prop 3.3, alternative (1) must now hold. The usual argument shows x̄′′ is the only element of C ∩ E on the right of āā′. Now again J(ā, 1 (2ā′ + ā)) > 0, J(ā′, 1 (ā+ 2ā′)) > 0, and the sums a+ (2a′ + a)/3 and a′ + (a+ 2a′)/3 cannot give points in 2W. Since they also cannot both be cancelled by c+x′′ in the superpotential equation, we have a contradiction. The other possibility for two interior points is, after relabelling the vertices if necessary, when z = (2x+ x′′)/3 and w = (2x′′ + x)/3. As usual all elements of C ∩ E except for c̄, ā, ā′ are on the right of āā′. Alternative (1) must hold, or else we cannot account for z, w. The usual argument shows either x̄′′ is the only element of C ∩ E right of āā′, or z ∈ C is the rightmost element of (C ∩ E) \ {x̄′′} (so (z + x′′)/2 = w). In the former case we cannot get both z and w, as (c+ x′′)/2 can’t equal both z and w. In the latter, considering āz̄ shows J(ā, z̄) = 0. But as J(ā, x̄′′) = 0, this means ā is orthogonal to x̄ and hence to c̄, a contradiction. So no triangle with points of W in the interior of an edge can arise (except possibly for a subtriangle of (H1)). Nullity of c̄, ā, ā′ and the conditions in Theorem 6.12(i),(ii) again put severe constraints on x, x′, x′ and the dimensions. The possible triangles for case (i) are as follows, where (Tr11)-(Tr22) occur only if K is not connected, and we have also listed the vectors c, a, a′ for future reference. Further details of how the following listing is arrived at can be found in [DW5]. x′′ x x′ (Tr1) (−2, 1, 0, 0, 0) (0, 0,−2, 1, 0) (0, 0,−2, 0, 1) (Tr2) (−2, 1, 0, 0) (0, 1,−2, 0) (0, 1,−1,−1) (Tr3) (0, 0, 0,−2, 1) (−2, 1, 0, 0, 0) (0, 1,−2, 0, 0) (Tr4) (−2, 1, 0, 0, 0, 0) (0, 0,−2, 1, 0, 0) (0, 0, 0, 1,−1,−1) (Tr5) (−2, 1, 0, 0, 0) (0, 1,−1, 0,−1) (0, 1,−1,−1, 0) (Tr6) (−2, 1, 0, 0, 0, 0) (0, 0, 1,−1,−1, 0) (0, 0, 1,−1, 0,−1) (Tr7) (−2, 1, 0, 0, 0, 0) (0, 0, 1,−1,−1, 0) (0, 0,−1,−1, 0, 1) (Tr8) (−2, 1, 0, 0, 0, 0) (0, 0,−1,−1, 1, 0) (0, 0,−1,−1, 0, 1) (Tr9) (−2, 1, 0, 0, 0, 0, 0) (0, 0, 1,−1,−1, 0, 0) (0, 0, 1, 0, 0,−1,−1) (Tr10) (−2, 1, 0, 0, 0, 0, 0) (0, 0,−1, 1,−1, 0, 0) (0, 0,−1, 0, 0, 1,−1) (Tr11) (0, 0, 0, 1,−1,−1) (−2, 1, 0, 0, 0, 0) (−2, 0, 1, 0, 0, 0) (Tr12) (0, 1, 0,−1,−1) (−2, 1, 0, 0, 0) (−1, 1,−1, 0, 0) (Tr13) (0, 0, 0,−1,−1, 1) (−2, 1, 0, 0, 0, 0) (0, 1,−2, 0, 0, 0) (Tr14) (0, 0, 0,−1,−1, 1) (0, 1,−1,−1, 0, 0, 0) (−2, 1, 0, 0, 0, 0, 0) (Tr15) (0, 0, 0, 1,−1,−1, 0) (1,−1,−1, 0, 0, 0, 0) (1,−1, 0, 0, 0, 0,−1) (Tr16) (0, 0, 0, 1,−1,−1, 0) (1,−1,−1, 0, 0, 0, 0) (0,−1, 1, 0, 0, 0, 1) (Tr17) (0, 0, 0, 1,−1,−1, 0) (1,−1,−1, 0, 0, 0, 0) (0,−1,−1, 0, 0, 0, 1) (Tr18) (0, 0, 0, 1,−1,−1, 0) (1,−1,−1, 0, 0, 0, 0, 0) (1, 0, 0, 0, 0, 0,−1,−1) (Tr19) (0, 0, 0, 1,−1,−1, 0) (1,−1,−1, 0, 0, 0, 0, 0) (0,−1, 0, 0, 0, 0,−1, 1) (Tr20) (−1,−1, 1, 0, 0, 0) (0, 0, 1,−1,−1, 0) (0, 0, 1,−1, 0,−1) (Tr21) (−1, 1,−1, 0, 0, 0) (0, 0,−1, 1,−1, 0) (0, 0,−1, 1, 0,−1) (Tr22) (−1, 1,−1, 0, 0, 0) (0, 0,−1,−1, 1, 0) (0, 0,−1,−1, 0, 1) CLASSIFICATION OF SUPERPOTENTIALS 31 3c 3a 3a′ (Tr1) (2,−1,−8, 2, 2) (−2, 1,−4, 4,−2) (−2, 1,−4,−2, 4) (Tr2) (2, 3,−6,−2) (−2, 3,−6, 2) (−2, 3, 0,−4) (Tr3) (−4, 4,−4, 2,−1) (−8, 2, 4,−2, 1) (4, 2,−8,−2, 1) (Tr4) (2,−1,−4, 4,−2,−2) (−2, 1,−8, 2, 2, 2) (−2, 1, 4, 2,−4,−4) (Tr5) (2, 3,−4,−2,−2) (−2, 3,−2, 2,−4) (−2, 3,−2,−4, 2) (Tr6) (2,−1, 4,−4,−2,−2) (−2, 1, 2,−2,−4, 2) (−2, 1, 2,−2, 2,−4) (Tr7) (2,−1, 0,−4,−2, 2) (−2, 1, 6,−2,−4,−2) (−2, 1,−6,−2, 2, 4) (Tr8) (2,−1,−4,−4, 2, 2) (−2, 1,−2,−2, 4,−2) (−2, 1,−2,−2,−2, 4) (Tr9) (2,−1, 4,−2,−2,−2,−2) (−2, 1, 2,−4,−4, 2, 2) (−2, 1, 2, 2, 2,−4,−4) (Tr10) (2,−1,−4, 2,−2, 2,−2) (−2, 1,−2, 4,−4,−2, 2) (−2, 1,−2,−2, 2, 4,−4) (Tr11) (−8, 2, 2,−1, 1, 1) (−4, 4,−2, 1,−1,−1) (−4,−2, 4, 1,−1,−1) (Tr12) (−6, 3,−2, 1, 1) (−6, 3, 2,−1,−1) (0, 3,−4,−1,−1) (Tr13) (−4, 4,−4, 1, 1,−1) (−8, 2, 4,−1,−1, 1) (4, 2,−8,−1,−1, 1) (Tr14) (−4, 4,−2,−2, 1, 1,−1) (4, 2,−4,−4,−1,−1, 1) (−8, 2, 2, 2,−1,−1, 1) (Tr15) (4,−4,−2,−1, 1, 1,−2) (2,−2,−4, 1,−1,−1, 2) (2,−2, 2, 1,−1,−1,−4) (Tr16) (2,−4, 0,−1, 1, 1,−2) (4,−2,−6, 1,−1,−1, 2) (−2,−2, 6, 1,−1,−1,−4) (Tr17) (2,−4,−4,−1, 1, 1, 2) (4,−2,−2, 1,−1,−1,−2) (−2,−2,−2, 1,−1,−1, 4) (Tr18) (4,−2,−2,−1, 1, 1,−2,−2) (2,−4,−4, 1,−1,−1, 2, 2) (2, 2, 2, 1,−1,−1,−4,−4) (Tr19) (2,−4,−2,−1, 1, 1,−2, 2) (4,−2,−4, 1,−1,−1, 2,−2) (−2,−2, 2, 1,−1,−1,−4, 4) (Tr20) (1, 1, 3,−4,−2,−2) (−1,−1, 3,−2,−4, 2) (−1,−1, 3,−2, 2,−4) (Tr21) (1,−1,−3, 4,−2,−2) (−1, 1,−3, 2,−4, 2) (−1, 1,−3, 2, 2,−4) (Tr22) (1,−1,−3,−4, 2, 2) (−1, 1,−3,−2, 4,−2) (−1, 1,−3,−2,−2, 4) Remark 6.15. In making the above table, it is useful to observe from the nullity and orthogonality conditions that x′′ cannot be type I, and that if x′′ is type III, say, (−2i, 1j), then xi = x i iff xj = x The possibilities for Theorem 6.12(ii) are as follows (up to permutation of x, x′, x′′ and the corresponding permutation of c, a, a′): x′′ x x′ (Tr23) (−1, 0, 0, 0, 0) (0,−2, 1, 0, 0) (0, 0, 0,−2, 1) (Tr24) (−1, 0, 0, 0) (0, 1,−2, 0) (0,−1,−1, 1) (Tr25) (−1, 0, 0, 0, 0, 0) (0, 1,−2, 0, 0, 0) (0, 0, 0, 1,−1,−1) (Tr26) (−1, 0, 0, 0, 0) (0, 1,−1,−1, 0) (0,−1,−1, 0, 1) (Tr27) (−1, 0, 0, 0, 0, 0, 0) (0, 1,−1,−1, 0, 0, 0) (0, 0, 0, 0, 1,−1,−1) (Tr28) (−1, 0, 0) (0,−1, 0) (0, 0,−1) c a a′ (Tr23) (1,−2, 1,−2, 1) (−1,−2, 1, 2,−1) (−1, 2,−1,−2, 1) (Tr24) (1, 0,−3, 1) (−1, 2,−1,−1) (−1,−2, 1, 1) (Tr25) (1, 1,−2, 1,−1,−1) (−1, 1,−2,−1, 1, 1) (−1,−1, 2, 1,−1,−1) (Tr26) (1, 0,−2,−1, 1) (−1, 2, 0,−1,−1) (−1,−2, 0, 1, 1) (Tr27) (1, 1,−1,−1, 1,−1,−1) (−1, 1,−1,−1,−1, 1, 1) (−1,−1, 1, 1, 1,−1,−1) (Tr28) (1,−1,−1) (−1,−1, 1) (−1, 1,−1) Remark 6.16. In drawing up the above listing, recall from Theorem 6.12 that one of the vectors, without loss of generality x′′, is of type I. We write x′′ = (−1, 0, 0, · · · ). It now easily follows from nullity and the relations between x, x′, x′′ and c, a, a′ that x1 = x Also, observe that as x′′ is a vertex of W, no type II vector may have a nonzero entry in the first position. 32 A. DANCER AND M. WANG In contrast to the earlier listing of non-triangular faces, the above lists result from examining all triangular faces, including ones which arise from other faces because certain vertices are absent from W. The restrictions on the dimensions of the corresponding summands are as follows: (Tr1) (2, 1, 16, 4, 4, · · · ) (Tr2) (2, 3, 12, 4, · · · ) (Tr3) (16, 4, 16, 2, 1, · · · ) (Tr4) (2, 1, 16, 4, d5 , d6, · · · ), (Tr5) (2, 3, 6, 6, 6, · · · ) (Tr6) (2, 1, d3, d4, 4, 4, · · · ), (Tr7) (2, 1, 12, 3, 12, 12, · · · ) (Tr8) (2, 1, d3, d4, 4, 4, · · · ), (Tr9) (2, 1, 4, d4 , d5, d6, d7, · · · ), (Tr10) (2, 1, 4, d4 , d5, d6, d7, · · · ), (Tr11) (16, 4, 4, 1, 1, 1, · · · ) (Tr12) (12, 3, 4, 1, 1, · · · ) (Tr13) (16, 4, 16, 1, 1, 1, · · · ) (Tr14) (16, 4, d3, d4, 1, 1, 1, · · · ) (Tr15) (d1, d2, 4, 1, 1, 1, 4, · · · ), (Tr16) (12, 3, 12, 1, 1, 1, 12, · · · ) (Tr17) (4, d2, d3, 1, 1, 1, 4, · · · ), (Tr18) (4, d2, d3, 1, 1, 1, d7 , d8, · · · ), (Tr19) (d1, 4, d3, 1, 1, 1, d7 , d8, · · · ), (Tr20-22) (1, 1, 3, 6, 6, 6, · · · ), (1, 2, 2, 8, 8, 8, · · · ), or (2, 1, 2, 8, 8, 8, · · · ) (Tr23) 1 (Tr24) d3 = 2d2 : (Tr25) 1 (Tr26) d2 = d3 : (Tr27) (Tr28) 1 Note that (Tr28) is a subtriangle of (H1), (Tr2) is a subtriangle of a triangle with midpoints of all sides in W, and (Tr12) is a subtriangle of a triangle with the midpoint of one side. Let us now illustrate by an example how one arrives at the above tables. Example 6.17. One possible triangle has vertices V1 = (0, 0, 0,−2, 1), V2 = (−2, 1, 0, 0, 0), V3 = (0, 1,−2, 0, 0) with the midpoint V4 = (−1, 1,−1, 0, 0) of V2V3 in W. The triangle has a symmetry given by interchanging the first and third entries. It therefore suffices to consider V1, V2, V4 as pos- sibilities for x′′. Of course, by Remark 6.13 the full triangle cannot occur. The possible subtriangles xx′x′′ are V2V3V1, V2V4V1, V4V1V2, V3V1V2, and V3V1V4. Now 3c = 2x+2x ′−x′′, 3a = 4x−2x′+x′′, and 3a′ = −2x+ 4x′ + x′′ can be used to compute these vectors in each case. For V2V4V1 one gets 3c = (−6, 4,−2, 2,−1), 3a = (−6, 2, 2,−2, 1) and so c̄ and ā cannot be both null. Similarly, for the last three possibilities, ā′ and c̄ cannot be both null. That leaves the first case, which gives (Tr3). The condition J(ā, x̄′′) = 0 is 3 = 4 , which implies (d4, d5) = (2, 1). Putting this into the null conditions for c̄, ā, ā′ gives the equations CLASSIFICATION OF SUPERPOTENTIALS 33 The last two equations imply that d1 = d3 and the first two equations give d1 = 4d2. These in turn give (d1, d2, d3) = (16, 4, 16), as in the tables above. Putting all the results in this section together we obtain Theorem 6.18. If we have two adjacent (1B) vertices, then the associated 2-face of W is given by a triangle in the list (Tr1) − (Tr27), the square with midpoint (S), a proper subshape of the hexagonal face (H1) containing all three type I vectors, or the sub-rectangle yy′zz′ of (H2). We note for future reference the following properties of the c vector of the non-triangular faces appearing in the above theorem: for (S), all nonzero entries have the same absolute value, and there are only 3 (resp. 2) nonzero entries for the subfaces of (H1) (resp. (H2)). 7. More than one type (2) vertex In this section we shall now show there is at most one type (2) vertex in ∆c̄, except in the situation of Theorem 3.14 and one other possible case. Suppose we have two type (2) vertices of V . Then we have elements v,w, v′, w′ of W with c, v, w collinear and c, v′, w′ collinear. So we have four coplanar elements v,w, v′, w′ of W where vw and v′w′ are edges. Moreover, the edges vw and v′w′ meet at c outside conv(W). Hence vwv′w′ do not form a parallelogram or a triangle. From our listing of polygons in §6 and considering their sub-polygons we see that the possibilities for further analysis are the following: • Trapezia (T1)-(T6): We must have c = 2v− s = 2u−w. Also, we note for future reference that sw is always an edge of conv(W) in (T3) and (T5), regardless of whether or not the whole trapezium is a face, since sw can be cut out by {x2 = 1, x1 + x3 = −2} (cf 1.2(e)). • Hexagons (H1)-(H3) • Rectangle with midpoints (P17): While the rectangle itself cannot occur, we need to con- sider the trapezia obtained by omitting one vertex, so that the edges are a side of the rec- tangle and the segment joining the remaining vertex to the opposite midpoint. As above, note that the longer of the two parallel sides of the trapezium is always an edge of conv(W). • Parallelogram with midpoints (P16): This case is similar to (P17). The sub-polygons to consider are the trapezia obtained by omitting one vertex of the parallelogram. By symmetry, we are reduced to omitting either u or w. But since s occurs, w cannot be omitted. • Triangle with midpoints of all sides: We need to consider the trapezia obtained by omitting a vertex. By symmetry all three trapezia are equivalent. This triangle is always a face of conv(W) as it is cut out by {x2 = 1, x1 + x3 + x4 = −2} (cf 1.2(e)). • Trapezia (T*1),(T*2), (T*3): By Rmk 6.3 these cannot be faces of conv(W), so cannot come from adjacent type (2) vertices of V . For (T*1), besides the full trapezium, we need to consider the two trapezia obtained by omitting either s or w. By symmetry these are equivalent. For (T1),(T2),(T4),(T6),(T*2),(T*3) we must have c = 2v − s = 2u − w, and so Lemma 3.13 applied to vs gives a contradiction. The same argument works for (P16), as up to permutations, c = 2v − s = 2y − w. For (T1*), since Theorem 3.11 rules out c = (3v − s)/2 = (3u − w)/2 (corresponding to the full trapezium), the only other possible c is 2v − s = 2u − r, and again Lemma 3.13 rules this out. For (T3),(T5) and the trapezium coming from the triangle with midpoints, we need more infor- mation from the superpotential equation. Since J(c̄, s̄), J(c̄, w̄) > 0, while Av, Au < 0, Fs̄, Fw̄ must have the same sign, which must be opposite to that of Fc̄. Since stw is always an edge by earlier remarks, the nullity of c̄ implies J(s̄, w̄) > 0, contradicting Prop 3.7(ii). 34 A. DANCER AND M. WANG Essentially the same argument works for (P17), as up to permutations c = 2s − v = 2z − u = (−2,−1, 2, 0). For (H3) most quadruples cannot give pairs of edges. For we observe that u (resp. v,w) is present iff x (resp. y, z) is. Thus, if u is missing, so is x, and v,w, y, z must all be present (otherwise we do not have a 2-dimensional polygon). But we now get a rectangle, which is not admissible. Hence all vertices are present and by symmetry we may assume that one of our edges is uz or uv. From this we quickly find that the two possible c (up to permutations) are (−1,−1,−1, 2) = 2y − x = 2z − u and (1,−1, 1,−2) = 2v − u = 2w − x. Both cases are ruled out by Lemma 3.13. For (H2), observe that y (resp. y′) is present iff z (resp. z′) is. As these four vectors cannot all be absent (otherwise we do not have a 2-dim polygon), by the symmetries of (H2), we can assume y′ is present. If v is present, then all possibilities are eliminated by Theorem 3.11. (Note that although 2α − y′ = 2z − z′ it is impossible for αy′ and zz′ to both be edges.) On the other hand, if v is absent, then y′z′ is an edge. Since the polygon cannot be a parallelogram or a triangle, it follows that u is present and the polygon is a pentagon. In this case, the only possibility compatible with Theorem 3.11 is c = (0,−1, 2,−2) = 2y−u = 3 y′− 1 z′. (This is not a priori ruled out by Theorem 3.11 as y′z′ has an interior point β). To discuss (H1), we write u = (−2, 1, 0), p = (0, 1,−2), v = (1, 0,−2), w = (0,−2, 1), s = (1,−2, 0), q = (−2, 0, 1) for the vertices, x = (−1, 1,−1), y = (−1,−1, 1), z = (1,−1,−1), for midpoints of the longer sides, and α = (−1, 0, 0), β = (0, 0,−1), γ = (0,−1, 0) for the interior points, with the understanding that the rest of the components of the above vectors are zero. As before we consider pairs of vectors which can form edges of an admissible polygon. We then compute the possibilities for c and apply Theorem 3.11. This will eliminate most possibilities. (For many quadruples of points we can see, as in (H3), that they cannot all be vertices.) So up to permutations, the remaining possibilities are as follows. If no type II is present: (1) c = 2α− u = 2β − p = (0,−1, 0, · · · ) (2) c = 2u− v = 2q − s = (−5, 2, 2, · · · ) (3) c = 2u− p = 2q − γ = (−4, 1, 2, · · · ) (4) c = 2q − u = 2α− p = (−2,−1, 2, · · · ) If all type II are present: (5) c = 2u− x = 2q − y = (−3, 1, 1, · · · ) (6) c = 2u− y = 2x− z = (−3, 3,−1, · · · ) (7) c = (3y − z)/2 = 2q − u = (−2,−1, 2, · · · ) (8) c = (3p − u)/2 = (3v − s)/2 = (1, 1,−3, · · · ) Again, we cannot immediately rule out (7) and (8) using 3.11 because of the presence of interior points. However for (8) we easily see using the arguments of 3.11 that the elements of C on the line through v, s are c̄, c̄1 = (v̄+ s̄)/2 = z̄ and c̄2 = (3s̄− v̄)/2. Now as s, v are type III we need Fc̄ and Fc̄2 to have the same sign, which is the opposite sign to Fc̄1 . But the superpotential equation now gives a contradiction to the fact that Az < 0. In (2)-(7) Lemma 3.13 applied respectively to uv, up, qu, ux, uy, qu gives a contradiction. Note that case (4) only occurs when K is not connected, as the vectors β, γ are absent (cf 1.2(b)). We are left with (1), which is precisely the situation of Theorem 3.14. We have therefore proved Theorem 7.1. Apart from the situation of Theorem 3.14, the only other possible case where we can have more than one type (2) vertex is, up to permutation of summands, when two type (2) CLASSIFICATION OF SUPERPOTENTIALS 35 vertices are adjacent and the 2-plane determined by them and c̄ intersects conv(1 (d +W)) in the pentagon with vertices uyy′zz′ contained in the hexagon (H2). We will be able to rule this case out in §8. 8. Adjacent (1B) vertices revisited We now return to our classification of when adjacent (1B) vertices can occur. The idea is as follows: each of the configurations of §6 involves, as well as the null vector c̄, two new null vectors ā, ā′. Hence the arguments of earlier sections also apply to ā, ā′. That is, we may consider the associated polytopes ∆ā and ∆ā The following lemma is useful when applied to ∆c̄,∆ā and ∆ā Lemma 8.1. Suppose we have a (1B) vertex with exactly k adjacent (1B) vertices. Then r ≤ #((1A) vertices) + #((2) vertices) + k + 2. Suppose we have a (1B) vertex with no adjacent (1B) vertices. Then r ≤ #((1A) vertices) + #((2) vertices) + 2. If there are no (1B) vertices then r ≤ #((1A) vertices) + #((2) vertices) + 1. Proof. By our assumption that dim conv(W) = r− 1, it follows that ∆c̄ is a polytope of dimension r − 2. Any vertex in it has at least r − 2 adjacent vertices. So for a (1B) vertex, the first two statements follow immediately. If there are no (1B) vertices the third inequality follows because ∆c̄ has at least r − 1 vertices. Lemma 8.2. Configurations (Tr1) - (Tr22) cannot arise from adjacent (1B) vertices. Proof. The strategy is to count the number of type (1A), (2) and adjacent (1B) vertices in ∆ā or ∆c̄ and apply Lemma 8.1 to get a contradiction. (i) We first observe that for these configurations c and a have at least four nonzero entries (at least five except for (Tr2)), so they cannot be collinear with an edge vw with points of W in the interior of vw (see Table 3 in §3). So if ∆c̄ or ∆ā has a type (2) vertex, by Theorem 3.11, c or a must equal 2v − w or (4v − w)/3. It is easy to check that this is impossible except for c in (Tr3), using the forms of c in Tables 1, 2 in §3. (ii) Next we consider (1A) vertices. For (Tr1)-(Tr10) we have |ai/di| ≤ for all i. For Tr(1), Tr(6),Tr(8) there are three i where equality holds. In these cases one of the associated di equals 1. Moreover for (Tr1) and (Tr8) two of these ai/di equal 1/3 and the third is −1/3, wheras for (Tr6) it is the other way round. For (Tr2)-(Tr5), (Tr7) and (Tr9)-(Tr10) there are only two i where equality holds. Further, for (Tr2) and (Tr5) |ci/di| ≤ for all i, with equality for just two i, and here ci/di = . It follows that for (Tr1), (Tr8) there are at most two (1A) vertices in ∆ā, while for (Tr2)-(Tr5), (Tr7) and (Tr9)-(Tr10) there is at most one. In the case of (Tr6) there are at most three (1A) vertices in general but at most two if K is connected. For (Tr2) and (Tr5) there are no (1A) vertices in ∆c̄. By means of similar considerations, we find that there is one (1A) vertex (corresponding to x̄′′) in ∆ā for (Tr12), (Tr13),(Tr14) (Tr16), (Tr18), (Tr19) and at most two (1A) vertices for (Tr11), (Tr17), (Tr20) and the d = (1, 1, 3, 6, 6, 6, · · · ) case of (Tr21), (Tr22). For (Tr15) there are at most four (1A) vertices in ∆ā, and for the remaining cases of (Tr21) and (Tr22) there are at most r − 4 (1A) vertices (r − 6 of those correspond to (−23, 1j) where j > 6). (iii) Finally, consider the (1B) vertex ξ̄ in ∆ā corresponding to x̄ in each of the triangles. In order for there to be an adjacent (1B) vertex, ā must be (up to permutation) of the form of the null vector c̄ in the 2-faces in Theorem 6.18. Now observe that for examples (Tr1), (Tr3), (Tr4), (Tr6)- (Tr20) 36 A. DANCER AND M. WANG the null vector a does not appear in the list of possible c. Hence ξ̄ has no adjacent (1B) vertices. From above, type (2) vertices cannot occur, so combining the bounds for (1A) vertices in ∆ā with Lemma 8.1 gives an upper bound for r less than the minimum required by each configuration, a contradiction. (iv) Let us now consider (Tr21) and (Tr22). The vector a of (Tr21) has the same form as c in (Tr22) and vice versa. An adjacent 2-face containing aξc of (Tr21) can only be a triangle of type (Tr22) containing 1 (1,−1,−3,−2,−2, 4, 0, · · · ). Thus ξ has at most one adjacent (1B) vertex, and we get the bound r− 2 ≤ 2+1+0 in the d = (1, 1, 3, 6, 6, 6, · · · ) subcase and r− 2 ≤ (r− 4)+1+0 in the other two subcases, a contradiction. (v) For the remaining two triangles (Tr5) and (Tr2) we consider ∆c̄ instead. For (Tr5), observe that c determines the plane xx′x′′ and does not occur as a possible null vector for any other configurations. So we have at most one adjacent pair of (1B) vertices in ∆c̄. From above, there are no type (1A) or (2) vertices. But r ≥ 5, giving a contradiction. For (Tr2), consider the vertex ξ̄′ of ∆c̄ corresponding to x′. If there is a (1B) vertex adjacent to it, we have a 2-dim face including c, x′. By Theorem 6.18 the only one is the face including x, so there is at most one (1B) vertex of ∆c̄ adjacent to ξ̄′. Also, type (1A) and (2) vertices cannot occur, so r ≤ 3, a contradiction. Lemma 8.3. Configurations (Tr23)-(Tr27) cannot arise from adjacent (1B) vertices. Proof. Note first that all entries of c, a, a′ are integers, so Lemma 4.4 shows in each case there is at most one (1A) vertex, and for (Tr23),(Tr24) one checks that there are no (1A) vertices in ∆c̄. Note also that for all these configurations, as x′′ is a vertex, there are no type II vectors with nonzero entry in place 1. Observe as in Lemma 8.2 that there are no type (2) vertices in ∆c̄,∆ā or ∆ā . (For (Tr24), we need to rule out the possibility that c has the form (4) in Table 3 in §3 with λ = 3/2. This follows since the interior point in that case would be a type II vector with nonzero entry in place 1.) So in all cases if we have a (1B) vertex with exactly k (1B) vertices adjacent to it, then by Lemma 8.1 we have r ≤ k + 3. For (Tr23), (Tr24) (using ∆c̄), we have r ≤ k + 2. We will work with ∆c̄ below. First consider (Tr23) and look for (1B) vertices adjacent to ξ̄ where ξ̄ corresponds to the vertex x. We need a 2-face including c, x. By Theorem 6.18 such a face must be of type (Tr23), and having fixed c and a, the only freedom lies in assigning 1 in the third null vector to the first or fifth place. So k ≤ 2, which gives r ≤ 4, a contradiction. Similarly, for (Tr24), since a type (H1) face cannot contain c and x, we need only consider faces of type (Tr24), for which there are again two possibilities. However, as mentioned above, in one of these possibilities the vector “x′” has a 1 in place 1 and hence cannot occur. So there is at most one (1B) vertex adjacent to ξ, and we deduce r ≤ 3, a contradiction. For (Tr25),(Tr27) we similarly deduce that the only 2-face containing x, c is itself because as above we cannot have any type II vectors with nonzero entry in place 1. So r ≤ 4, a contradiction. Finally, for (Tr26) the above argument still works since (Tr24) has been ruled out (the vectors a, a′ of (Tr24) are of the same form as c, a of (Tr26), so a priori (Tr24) could be an adjacent 2-face). Lemma 8.4. Configuration (S) (square with midpoint) cannot arise from adjacent (1B) vertices. Proof. We refer to §6 for the expressions for the vertices vusw of the square. The null vertex c̄ corresponds to (1,−1,−1, 1,−1, 0, · · · ) and the 2-dimensional face is cut out by x2 = −1, x1+x3 = 0 = x4 + x5, and xk = 0, for k > 5. Lemma 4.4(b) shows that there is at most one (1A) vertex in ∆c̄. As r ≥ 5 and all the nonzero entries of c have the same absolute value, it follows that there are no type (2) vertices. Let ξ̄ denote the vertex of ∆c̄ so that ξ is collinear with u and a = 2u − c = (−1,−1, 1, 1,−1, 0, · · · ). A (1B) vertex adjacent to ξ̄ gives a 2-dimensional face CLASSIFICATION OF SUPERPOTENTIALS 37 including c̄, ū. By what we have analysed so far about 2-faces given by adjacent (1B) vertices, this face must again be a face of type (S), and the only possibilities are itself or the face obtained from this by swapping indices 2 and 5. Hence there are at most two (1B) vertices adjacent to ξ̄, and at vertex ξ̄, we have 3 ≤ r − 2 ≤ 1 + 2. Thus r = 5 is the remaining possibility, in which case ξ̄ has exactly two adjacent (1B) vertices and one adjacent (1A) vertex. Let us denote by ξ̄′ the (1B) vertex such that ξ′ is collinear with w and a′ := 2w − c = (1,−1,−1,−1, 1). Let η̄ denote the other (1B) vertex adjacent to ξ̄. Then the 2-face deter- mined by c, ξ, η is cut out by x5 = −1, x1 + x3 = 0 = x2 + x4. The ray cη intersects conv(W) at z = (1, 0,−1, 0,−1) and b := 2z − c = (1, 1,−1,−1,−1) corresponds to a null vertex. Similarly, there is a (1B) vertex η̄′ (besides ξ̄) adjacent to ξ̄′, and the corresponding 2-face (also of type (S)) is cut out by x3 = −1, x1+x2 = 0 = x4+x5. The ray cη ′ intersects conv(W) at z′ = (0, 0,−1, 1,−1). The vector b′ := 2z′ − c = (−1, 1,−1, 1,−1) corresponds to a null vertex. Let us examine the (1A) vertex in ∆c̄ more closely. Let y ∈ W such that J(ȳ, c̄) = 0. As r = 5, the null condition for c̄ implies that di ≥ 2 with at most one equal to 2. Also, for some j ∈ {2, 3, 5} (i.e., j is an index for which the corresponding entry of c is −1) we must have yj = −2, so y is type III. Let i be the index such that yi = 1. Then i ∈ {1, 4} (i.e., i is an index for which the corresponding entry of c is 1), and the orthogonality condition implies (di, dj) = (2, 4) or (3, 3). There are thus six possibilities for y, but only one can actually occur. With the possible exception of the existence of the (1A) vertex, the above arguments apply equally to the projected polytopes ∆ā, ∆b̄, ∆ā , and ∆b̄ as the entries of a, b, a′ and b′ are just permutations of those of c. We claim that whichever possibility for y occurs in ∆c̄, there is another projected polytope with no (1A) vertex. Applying the above arguments to this polytope would result in the contradiction r − 2 ≤ 2 and complete our proof. We can use ∆ā for the contradiction if d1 = 2 or if any of (d1, d2), (d3, d4), (d1, d5) = (3, 3). If d4 = 2 or if (d2, d4) = (3, 3) we can use ∆ ā′ instead. Finally, if (d1, d3) = (3, 3) we can use ∆ b̄′ and if (d4, d5) = (3, 3) we can use ∆ For example, when (d1, d3) = (3, 3) (so y = (1 1,−23)), the null condition for c̄ implies that d2, d4 in particular cannot equal 2 or 3. In order to have a (1A) vertex in ∆b̄ , we must have a type III vector (1i,−2j) with i ∈ {2, 4}. But this requires one of d2, d4 to be 2 or 3. When (d4, d5) = (3, 3), then d1, d2 cannot be 2 or 3. But in ∆ b̄ a (1A) vertex corresponds to (1i,−2j) with i ∈ {1, 2}, which implies that one of d1, d2 is 2 or 3. The remaining cases are handled similarly. Lemma 8.5. The subrectangle yy′zz′ of (H2) cannot arise from adjacent (1B) vertices. Proof. Recall c = (−2, 1, 0, · · · ), so by Lemma 4.4 there are no (1A) vertices of ∆c̄. Moreover, using Tables 1-3 in §3, one may check that there are no type (2) vertices either. (Note that type II vectors other than y, z with a nonzero entry in place 1 cannot occur as then the subrectangle cannot be a face. Similarly the line through c, α, β and (1,−2, 0, · · · ) will not give a type (2) vertex as this line cannot be an edge.) Let η̄ denote the vertex of ∆c̄ collinear with c and y. Any (1B) vertex adjacent to η̄ will give rise to a face containing c and y, which cannot be of type (H1), and must therefore be of type (H2), since we have eliminated all other possibilities. In fact, it must be the face we started with. So there is just one (1B) vertex adjacent to η̄, and from above there are no (1A) or (2) vertices. As r ≥ 4 for (H2), this contradicts Lemma 8.1. Lemma 8.6. If r ≥ 4, configuration (H1) or subshapes cannot arise from adjacent (1B) vertices. Proof. We first note some special properties of W. Since (H1) is a face, there can be no type II vectors in W with nonzero entry in a place ∈ {1, 2, 3} and in a place /∈ {1, 2, 3}. Also, if (−2i, 1k) with i ∈ {1, 2, 3}, k /∈ {1, 2, 3}, then (−1k) must be absent, which has strong implications, as noted in Remark 1.2(b). 38 A. DANCER AND M. WANG Let ξ̄ be a (1B) vertex in the plane. We have at most one (1B) vertex adjacent to ξ̄, as the associated face must again be of type (H1) and is now determined by c̄, ξ̄. It also readily follows that c cannot be collinear with an edge of W not in the face (assuming as usual we are not in the situation of Theorem 3.14). Now the special properties of W in the first paragraph imply that (1A) vertices in ∆c̄ can correspond only to type III vectors in W which overlap with c. A straight-forward check using the null condition for c̄ shows that the possible type IIIs have form (−2i, 1k) with i ∈ {1, 2, 3}, k /∈ {1, 2, 3} and ci/di = −1/2. It follows that di = 2 or 3 and hence, by nullity, the index i is unique. So there are at most r−3 (1A) vertices. By Lemma 8.1, all r−3 (1A) vertices must occur. Applying Cor 4.3 we conclude that r ≤ 4 (as di 6= 4 and r > 4 forces i ∈ Ŝ≥2). We will now improve this estimate to r ≤ 3. Let the vertices of (H1) be as in §7. If r = 4 then a (1A) vertex does exist and we can take it to come from t = (−2, 0, 0, 1) with (−14) absent. It follows that besides t the only other possible members of W lying outside the 2-plane containing (H1) are (0,−2, 0, 1) and (0, 0,−2, 1). As noted just before Theorem 6.12 we may assume the type I vectors α, β, γ are all present. (If K is connected, d4 = 1 and so this last fact follows without having first to eliminate those subshapes not containing one of the type I vectors.) As noted above, d1 = 2 or 3, and c1 = −1 or − respectively. First consider c1 = −1, so d1 = 2. Now c = (−1, c2,−c2, 0), and by swapping the 2, 3 coordinates if necessary, we may take c2 > 0. Observe u, q are absent, as if u is present or if u is absent but q is present, then u (resp. q) gives a (1B) vertex, which contradicts nullity as the associated a would have a1 = −3. Now the type II vectors x, y, z are absent, as if one is present they all are, and we have a type (2) vertex. We deduce α gives a (1B) vertex so a = (−1,−c2, c2, 0). The other (1B) vertex cannot correspond to w since β, γ are present. It also cannot be given by v, s as this violates nullity, so must correspond to p or β. If it is p, we have a′ = (1, 2− c2, c2−4, 0). Now Remark 3.9 implies c2 = (d3+4d2)/(d3 +2d2) so 1 < c2 < 2. But now no entry of a ′ equals −1 or −3 . We can now check that there are no (1A) or (2) vertices with respect to a′, so there is at most one vertex of ∆ā adjacent to p, a contradiction. If it is β then p, v must be absent. Now a′ = (1,−c2, c2 − 2) and Remark 3.9 implies c2 = 1. Hence c = (−1, 1,−1), a = (−1,−1, 1), a′ = (1,−1,−1), and nullity implies 1 . It is easy to check by considering the vertices of ∆ā that w, s must also be absent, so W just contains the three type I vectors, t and possibly one or both of (0,−2, 0, 1), (0, 0,−2, 1). But we can check that, if present, these three latter vectors give respectively vertices with respect to a, a′ which cannot satisfy any of the conditions (1A), (1B) or (2). So in fact we have r = 3. Similar arguments rule out the case c1 = − Lemmas 8.2-8.6 give the following improvement of Theorem 6.18. Theorem 8.7. It is impossible to have adjacent (1B) vertices except possibly when r = 3, in which case conv(1 (d+W)) is a proper subface of (H1) containing all three type I vectors (e.g., the tri-warped example (Tr28)). We are now in a position to strengthen Theorem 7.1 by eliminating the remaining case of the pentagon. Theorem 8.8. Let c̄ be a null vertex of conv(C) such that ∆c̄ contains more than one type (2) vertex. Then we are in the situation of Theorem 3.14. Proof. We just have to eliminate the case of the pentagon uyy′zz′ in Theorem 7.1. Recall r ≥ 4 for this configuration, and c is (0,−1, 2,−2, · · · ). Using the nullity of c̄ we check that the only elements of W which can give an element of c̄⊥ are (−22, 1i) where i > 4 and we have d2 = 2. Note that (1 1,−22) cannot be present as then y′z′ is not an edge. By Cor 4.3, at most one such vector can arise. So there is at most one (1A) vertex, which occurs only if r ≥ 5. CLASSIFICATION OF SUPERPOTENTIALS 39 If we can show there are no (1B) vertices, then we are done because if we look at the adjacent vertices of the type (2) vertex associated to y (in the pentagon), besides one (1A) possibility, the other possibility is the type (2) vertex associated to y′ (by Theorem 7.1). As there must be at least r− 2 adjacent vertices, we deduce r− 2 ≤ 2, so r = 4. But now, from above there is no (1A) vertex so in fact we get r − 2 ≤ 1, a contradiction. We now use Remark 3.9 to make a list of the possible x ∈ W associated to (1B) vertices of ∆c̄. These are (0, 1,−1,−1, 0, · · · ), (0,−2, 1, 0, · · · ), (13,−1i,−1j), (−14, 1i,−1j), (13,−14,−1i), (−13,−14, 1i) and (13,−2i) where i, j 6= 2, 3, 4. Note that type II vectors with nonzero entries in places 2, k,m cannot occur except for y′, z′ as then y′z′ is not an edge. For each x in this list, we consider the projected polytope ∆ā, where a = 2x − c. By looking at the form of a, we see from Theorem 7.1 that there is at most one type (2) vertex in ∆ā. Also, the nonzero components of a are either ≥ 1 or ≤ −2. By Lemma 4.4(a), there are no vertices of type (1A) in ∆ā. Since r ≥ 4, by Theorem 8.7, the type (1B) vertex in ∆ā corresponding to x̄ has no adjacent (1B) vertices. So we have a contradiction to Theorem 8.1. The above result together with Theorem 8.7 and Lemma 8.1 gives us lower bounds on the number of (1A) vertices. Theorem 8.9. Let c̄ be a null vertex of conv(C) and ∆c̄ be the corresponding projected polytope. Suppose further that c is not type I, i.e., we are not in the case of Theorem 3.14. (i) If there are no (1B) vertices in ∆c̄, then there are at least r − 2 type (1A) vertices. (ii) If either there is a type (2) vertex or r ≥ 4, then there are at least r − 3 type (1A) vertices in ∆c̄. Hence there are at least r − 3 elements of 1 (d+W) orthogonal to c̄. 9. Type (2) vertices In this section we consider again type (2) vertices of ∆c̄. In view of Theorem 8.8, it remains to deal with the case of a unique type (2) vertex in ∆c̄. By Theorem 8.7 there are no adjacent (1B) vertices in this situation. Let c be collinear with an edge vw of conv(W). We first consider the situation where there are no interior points of vw lying in W. By Theorem 3.11, we have the two possibilities c = 2v − w and c = (4v − w)/3. Moreover, a preliminary listing of the cases appears in Tables 1 and 2 of §3. Case (i): c = 2v − w We have to analyse cases (1)-(7) in Table 1 of §3. The idea is to determine the number of (1A) and (1B) vertices using respectively Lemma 4.4 and Remark 3.9, and then get a contradiction (sometimes using Theorem 8.9). Note that J(w̄, w̄) < 0 for (1)-(3). In (1), (2) and (4)-(7), Lemma 4.4 shows that there are no elements of 1 (d+W) orthogonal to c̄ (recall c /∈ W), so Theorem 8.9 shows that r ≤ 3. This already gives a contradiction in case (7). (Note that when r = 3 and w is type I, since w is a vertex there are no type II vectors in W.) In (1) the only x ∈ W that could satisfy Eq.(3.1) and give a (1B) vertex with respect to c̄ is (1,−1,−1). But the associated a = 2x− c is (2,−3, 0) and it easily follows that ā, c̄ cannot both be null. For (2), the possible x ∈ W which correspond to (1B) vertices are (1,−2, 0) and (1,−1,−1) respectively. In each case we find the nullity of c̄ and Remark 3.9 imply J(w̄, w̄) > 0, a contradiction to F 2w̄ J(w̄, w̄) = Aw < 0 (as w is type III). In (5) with r = 3, one checks that the only possible x ∈ W corresponding to a (1B) vertex is (0, 1,−2). Let us consider the distribution of points of W in the plane x1 + x2 + x3 = −1. The point (0, 1,−2), if present, would lie on one side of the line vw while the point (−1, 0, 0) lies on the other side. Now (−1, 0, 0) must lie in W as otherwise v cannot be present by Remark 1.2(b). So since vw is an edge by assumption, (0, 1,−2) cannot lie in W, which gives a contradiction to Theorem 8.9(i). Hence in (1),(2),(5) Theorem 8.9 shows r ≤ 2, which is a contradiction. 40 A. DANCER AND M. WANG In case (4) the nullity of c̄ translates into 1 = 9/d1 + 4/d2. Hence d2 6= 1, so if K is connected (0,−1, 0, · · · ) is present and w is not a vertex, which is a contradiction. If r = 3 and K is not connected, by Remark 1.2(b), (1,−2, 0) and (0,−2, 1) must be absent, and, from Remark 3.9, the possibilities for x ∈ W associated to the (1B) vertex are x = (−2, 0, 1) and y = (0, 1,−2). In the first case, conv(W) is the triangle with vertices v,w, x and a = 2x − c = (−1,−2, 2). Now J(ā, w̄) > 0, contradicting the superpotential equation. In the second case, a = (3, 0,−4) with J(ā, ȳ) > 0, and aw intersects conv(W) in an edge. By Theorem 3.11, t = (1, 0,−2) ∈ W and conv(W) is a parallelogram with vertices v, y, w, t. Moreover, Remark 3.12 implies that a and w are the only elements of C in aw. But then the midpoint (0, 0,−1) of wt is unaccounted for in the superpotential equation. For (6) with r = 3, there should be at least two vertices in ∆c̄. But we find there are no (1B) vertices, a contradiction. So r = 2, and we are in the situation of the double warped product Example 8.2 of [DW4]. In case (3), Lemma 4.4 shows 1 (d + W) ∩ c̄⊥ has at most one element. Hence ∆c̄, which has dimension ≥ 2 since r ≥ 4, must contain at least one (1B) vertex. By Theorem 8.7, such a (1B) vertex has at most 2 adjacent vertices. It follows that r = 4 and (1,−2, 0, 0) corresponds to the (1A) vertex; also d2 = 2. Also, since (−1, 0, 0, 0) ∈ W, (0,−1, 0, 0) cannot be a vertex of conv(W). But now routine computations using Eq.(3.1) show there are no (1B) vertices, a contradiction. So the only possible case if K is connected is that giving Example 8.2 of [DW4]. If K is disconnected there is the further possibility of (4) with r = 2, i.e., W = {(−2, 1), (−1, 0)}. This is discussed in the third paragraph of Example 8.3 of [DW4]. An example in the inhomogeneous setting is treated there and in [DW2]. An example where the hypersurface is a homogeneous space G/K is discussed in the concluding remarks at the end of section §10. Case (ii): c = (4v − w)/3 For clarity of exposition let us assume K is connected, using the assumption as indicated in Remark 5.9. We examine the cases (1)-(11) in Table 2 of §3. Some of these cases can be immediately eliminated. In (3), Eq.(3.2) implies (d1, d2) = (3, 3) or (4, 1). In neither case is c̄ null. In (11) Eq.(3.2) and J(v̄, w̄) > 0 imply (d1, d2, d3) = (2, 4, 4), (2, 5, 2) or (3, 3, 3), and again c̄ is not null. In (4) and (6) Eq.(3.2) implies (d1, d2) = (2, 1) and (d1, d2) = (3, 9) or (4, 3) respectively. In neither case does the nullity condition have an integral solution in Further cases can be eliminated by finding the possible (1A) vertices (using Lemmas 3.10 and 4.4) for the given value of c and using Theorem 8.9. In particular, we get a contradiction whenever r ≥ 4 and there are no (1A) vertices. In (1), Eq.(3.2) implies (d1, d2) = (2, 1), and nullity of c̄ implies = 3. But we now find that 1 (d+W) ∩ c̄⊥ is empty, giving a contradiction as r ≥ 4. In (5) Eq.(3.2) and nullity imply (d1, d2) = (3, 3) and {d3, d4} = {3, 8}. One can now check that (d+W) ∩ c̄⊥ is empty, which is a contradiction as r ≥ 4. In (7), Eq.(3.2) implies (d1, d2) = (2, 1) and nullity implies . One can now check that the only possible elements ū orthogonal to c̄ correspond to u = (1, 0, 0,−2, · · · ) if d4 = 4 and (1, 0, 0, 0,−2, · · · ) if d5 = 4. The nullity condition means that at most one of these can occur. This is a contradiction as r ≥ 5. In (8) Eq.(3.2) gives (d1, d2) = (2, 3) and nullity of c̄ gives . Again one can check that 1 (d+W) ∩ c̄⊥ is empty, a contradiction as r ≥ 4. In (9), Eq.(3.2) and the nullity of c̄ give (d1, d4) = (2, 16) and {d2, d3} = {2, 3}. The only u which can give ū ∈ c̄⊥ are (0,−2, 0, 0, 1i , · · · ) if d2 = 2 or (0, 0,−2, 0, 1 i , · · · ) if d3 = 2, where i ≥ 5. In each case, i is unique since d2 (resp. d3) 6= 4. Since r ≥ 4, Theorem 8.9 now implies r = 4. But now these u are not present (as i ≥ 5). Therefore there are actually no (1A) vertices, a contradiction to r = 4. CLASSIFICATION OF SUPERPOTENTIALS 41 In (10) we have (d3, d4) = (2, 16) and {d1, d2} = {2, 3}. The only u which can give an element of c̄⊥ is (0,−2, 0, 0, 1i , · · · ) (for i unique and ≥ 5) if d2 = 2. The final argument in (9) now applies equally here. Finally, we can eliminate (2) by an analysis of both the (1A) and (1B) vertices. First, Eq.(3.2) and nullity of c̄ force (d1, d2, d3) = (6, 1, 8). Next we check that (d+W) ∩ c̄⊥ is empty, so r = 3. Using Remark 3.9 we then find there can be no (1B) vertices, giving a contradiction. So case (ii) cannot occur if K is connected. Remark 9.1. Case (ii) is the only part of this section that relies on the connectedness of K. In fact, the analysis of the cases where w is type III does not use this assumption. If K is not connected, using the same methods and with more computation we obtain the following additional possibilities (all of which are associated to a w of type II). v w c = (4v − w)/3 d r (9∗) (0,−1,−1, 1) (1,−1,−1, 0) (−1 ,−1,−1, 4 ) (1, 2, 6, 8) 4, 5 (1, 6, 2, 8) 4, 5 (10∗) (1,−1, 0,−1) (1,−1,−1, 0) (1,−1, 1 ) (3, 3, 1, 8) 4 (6, 2, 1, 8) 4, 5 (14) (0,−1, 0, 1,−1) (1,−1,−1, 0, 0) (−1 ,−1, 1 ) (1, 3, 1, d4 , d5) 5 In (9*) and (10*) there is always a (1B) vertex in ∆c̄, and r = 4 or 5 according to whether the cardinality of c̄⊥ ∩ 1 (d+W) is 1 or 2. The dimensions d4, d5 in (14) must satisfy (i.e., {d4, d5} = {5, 20}, {6, 12}, {8, 8}) and again there is always a (1B) vertex in ∆ Interior points Finally, we must consider the cases, listed in Table 3, §3, when there may be points of W in the interior of vw. As in the earlier cases, we analyse the possible (1A) and (1B) vertices for these c. For (1) and (2), as 1 < λ ≤ 2, the nonzero entries of c are either < −2 or > 1. Hence by Lemma 4.4 there are no (1A) vertices. By Theorem 8.9 we have r = 2 or 3. In case (3) the nullity of c̄ implies that a vector u ∈ W not collinear with vw and with ū ∈ c̄⊥ must be of the form (−2, 0, 1j). So 2λ− 1 = d1 and c = (−d1 ,−1 + d1 , 0, · · · ). From the range for λ and the nullity of c̄, we have d1 = 3, d2 = 1. But d2 6= 1 since w ∈ W. So again there are no (1A) vertices and by Theorem 8.9 we have r ≤ 3. In case (4), a straight-forward preliminary analysis reduces the possibilities of u ∈ W such that ū ∈ c̄⊥ to the choices u = (−2, 0, 1, · · · ), (0,−2, 1, · · · ) or (−1,−1, 1, · · · ). Note that c1 < −2 and c2 = 1, so by Lemma 4.4(a) we see c3 < 1, i.e. λ < . Now the second vector cannot occur because the orthogonality equation and λ ≤ 2 imply that d3 = 1, contradicting the presence of w. Since the three vectors are collinear, if two satisfy the orthogonality equation then all do. So there is at most one (1A) vertex and so r ≤ 4 by Theorem 8.9. This can be improved to r ≤ 3 as follows. If the third vector (−1,−1, 1, · · · ) occurs then the orthogonality relation, the bound on λ, and nullity imply that d1 = 5, d3 = 2 and λ = 2 + 1 . Now the nullity equation may be written as a quadratic in 2+ 1 with no rational root. If the first vector (−2, 0, 1, · · · ) occurs then orthogonality implies λ = d1(d3+2) 4d3+2d1 , and the bound on λ gives 6 > 1. Nullity implies d1 ≥ 5 and d1 > d We can deduce d3 = 2 and λ = , and one can check that nullity fails. For case (5), again a straight-forward analysis of the orthogonality condition with the help of the nullity of c̄ gives the following u ∈ W as possibilities such that J(c̄, ū) = 0: (a) (−23, 1i), i ≥ 4 and d3 = 2, (b) (1,−2, 0, · · · ), (c) (−22, 1i), i ≥ 4 and d2 = 3, (d) (0,−2, 1, 0, · · · ). 42 A. DANCER AND M. WANG Note (1, 0,−2, · · · ) cannot be in W as then vw is not an edge. It follows from Cor 4.3 that among (a) only one vector can occur and among (b), (c), (d) also only one vector can occur. (The orthogonality conditions of (b) and (d) are incompatible with 1 < λ ≤ 2.) So c̄⊥∩ 1 (d+W) contains at most two elements. If it has two elements, one must then come from (a) and the other from (b)-(d). Together they give an edge of c̄⊥ ∩ conv(1 (d + W)) with no interior points in 1 (d +W). Using Cor 4.3 and the null condition, we find that all these two-element cases cannot occur. Hence r ≤ 4. If r = 4 then the possible u with J(c̄, ū) = 0 are given by (a)-(d) with i = 4. Now, we can show using techniques similar to those of Theorem 3.11 that c∗ = (1− 2λ, 2λ− 1,−1, 0) also gives a null element of C. The possible vectors orthogonal to this element come from (a) and the vectors (b∗), (c∗), (d∗) obtained from (b), (c), (d) by swapping places 1 and 2. If (a) does not give an element in c̄⊥ ∩ 1 (d + W), it is straightforward to show, using the orthogonality and nullity conditions for c̄ and c∗ together, that the (1A) vertices for c̄ and c∗ are given by (b) and (b∗) respectively. Also we must have c = (4 ,−1, 0) and d = (4, 4, 9, d4). We need a (1B) vertex outside x4 = 0. From Remark 3.9, the only possible (1B) vertices for c̄ correspond to (1, 0, 0,−2) and (1,−1, 0,−1). In particular there can be no vertices, and hence no elements of W, with x4 > 0. Hence (−1,−1, 0, 1) and therefore (1,−1, 0,−1) are not in W. So the (1B) vertices for c̄ and c∗ are given by (1, 0, 0,−2) and (0, 1, 0,−2) respectively. Now the line joining the corresponding null vectors a, a∗ misses conv(W), a contradiction. The remaining case is when (a) gives the element in c̄⊥ ∩ 1 (d+W) and in c∗ (d+W). Now for vw to be an edge we need (−14) absent, so by Remark 1.2(b) the only possible members of W lying outside {x4 = 0} are the three type IIIs with x4 = 1. In particular, all vectors in conv(W) have x4 ≥ 0. As (−1,−1, 1, 0) ∈ W, there must be (1B) vertices lying in {x4 = 0} for both c̄ and c∗. We then find that the only possibilities for such a (1B) vertex are given by (b), (b∗) respectively. It follows that d1 = d2, but now nullity is violated. We conclude that there are no (1A) vertices, so r ≤ 3. Theorem 9.2. Let c̄ be a null vertex in C such that ∆c̄ contains a type (2) vertex corresponding to an edge vw of conv(W). Suppose we are not in the situation of Theorem 3.14. (i) If there are no points of W in the interior of vw, then either we are in the situation of Example 8.2 of [DW4] or K is not connected and we are in one of cases in the table of Remark 9.1 or in the situation of the third paragraph of Example 8.3 of [DW4]. (ii) If there are interior points of vw in W then r ≤ 3. For further remarks about the r = 2 case see the concluding remarks at the end of §10. 10. Completing the classification Throughout this section we will assume that K is connected (and we are not in the situation of Theorem 3.14). Theorems 8.7 and 9.2 then tell us that if r ≥ 4 there are no type (2) vertices and no adjacent (1B) vertices in ∆c̄, for any null vector c̄ ∈ C. Since all (1B) vertices lie in the half-space {J(c̄, ·) > 0} bounded by the hyperplane c̄⊥ containing the (1A) vertices, we must therefore have in each ∆c̄ exactly one (1B) vertex, with the remaining vertices all of type (1A). So if r ≥ 4 the only remaining task is to analyse such a situation. As dim∆c̄ = r − 2, we see dim(∆c̄ ∩ c̄⊥) = r − 3. In particular, there must be at least r − 2 elements of W giving elements of c̄⊥ ∩ 1 (d+W). Theorem 5.18 lists the possible configurations of such elements when r ≥ 3. The above discussion, together with Remark 5.19, shows that in cases (1), (2), (3) we can take m = r− 1, in cases (5)(ii), (5)(iii), (6)(i), (6)(ii) we can take m = r, and in (6)(iii) we have r = 5. Finally, since the vectors in (4) are collinear, so that dim(∆ ∩ c̄⊥) = 1, we have r = 3 or 4. If r = 3, it follows that c̄ and the edge in conv(1 (d + W)) determined by the vectors in (4) are CLASSIFICATION OF SUPERPOTENTIALS 43 collinear. This contradicts the orthogonality condition for the configuration of vectors in (4) and we conclude that r = 4. For each configuration we can consider the possible vectors u ∈ W giving the (1B) vertex. Besides the nullity condition on c̄ and the condition that c̄ should be orthogonal to the (1A) vectors, we have a further relation coming from the null condition in Remark 3.9. In most cases, routine (but occasionally tedious) computations show that these relations have no solution. As a result, we obtain the following possibilities for u (up to obvious permutations): case (1A) vectors possible (1B) vector (1) (−21, 1i), 2 ≤ i ≤ r − 1 (−1r), (12,−2r), (−11,−12, 1r) (−11, 12,−13), (−11, 12,−1r) (2) (11,−2i), 2 ≤ i ≤ r − 1 (−12), (−12,−13, 1r) (3) (11,−22); (11,−23); possibly(11,−12,−13); (−12,−13, 1i), 4 ≤ i ≤ r − 1 (14,−2r), (11,−2r), (−1r) (4) (11,−22), (11,−23), (11,−12,−13) (−14), (11,−24), (11,−12,−14), (5)(i) (−21, 12), (−11, 12,−13) (−11), (−14) (5)(ii) (−21, 12); (−11, 13,−1i), 4 ≤ i ≤ r (13,−24), (−11, 12,−14), (13,−14,−15) (5)(iii) (−21, 12); (−11,−13, 1i), 4 ≤ i ≤ r (−22, 14), (−11,−12, 14), (−13, 14,−15) (6)(i) (−11,−12, 1i), 3 ≤ i ≤ r (−11,−13, 14) (6)(ii) (11,−12,−1i), 3 ≤ i ≤ r (11,−13,−14), (−12, 13,−14), (11,−23) (6)(iii) (11,−12,−1i), i = 3, 4; (11,−13,−14) (−15) Table 7: Unique 1B cases Remark 10.1. The possibilities for the (1B) vertex in cases (5)(ii) and (5)(iii) only apply to the r ≥ 5 situation. When r = 4, the two cases become the same if we switch the third and fourth summands and the possibilities are discussed in Lemma 10.14 below. Remark 10.2. Note that u such as (−12), (−13) in (4), (−14) in (5)(ii) or (−1i) with i > 2 in (6)(ii) cannot arise because they will not be vertices, due to the presence of the type II vectors in Remark 10.3. The dimensions must satisfy certain constraints in each case. Some such constraints were stated in Theorem 5.18 and Remark 5.19. We also have constraints coming from the nullity conditions for c̄ and ā. These typically involve the requirement that some expression in the di is a perfect square. The following is a summary of general constraints in each case: case (1): d1 = 4; case (2): d1 = 1; case (3): d2 = d3 = 2; case (4): d2 + d3 ≤ 4d1/(d1 − 1) and d2, d3,≥ 2; case (5)(i): (d1, d2) = (4, 2), (3, 3); case (5)(ii,iii): d1 = 2 and if r ≥ 5 also d3 = 2; case (6)(i, ii): d1 = d2 = 2; case (6)(iii): d1 = d2 = d3 = d4 = 2, d5 = 25. Our strategy now is reminiscent of that in §8. We have a (1B) vertex corresponding to ū and a second null vector ā satisfying a = 2u − c. Now we may apply our arguments to ā, and conclude that the vectors in ā⊥ ∩ 1 (d+W) are also of the form given in the above table, up to permutation. The resulting constraints will allow us to finish our classification. In some cases we can actually show that ā⊥∩ 1 (d+W) is empty and we have a contradiction. A simple example when this happens is case 6(iii), where we now have c = (6 ,−1) and (ai/di) = (− ). Other cases are treated in Lemma 10.4 below. Next we shall show that cases (6)(i)(ii) cannot arise (cf Lemma 10.6), so in all remaining cases there must be at least one type III vector w with w̄ in ā⊥. We now use our explicit formulae for c and a to derive inequalities on the entries of a and find when there can be such a type III vector orthogonal to ā. For each such instance we then check whether ā⊥ ∩ 1 (d + W) forms a configuration equivalent to one of those in Table 4. This turns out to be possible in only two 44 A. DANCER AND M. WANG situations (cf Lemmas 10.7, 10.11 and Lemma 10.9). These have such distinctive features that W can be completely determined and judicious applications of Prop 3.7 lead to contradictions. This yields our main classification theorem. Lemma 10.4. The following cases cannot arise: case (4) with u = (1,−1, 0,−1), case (6)(ii) with u = (1, 0,−2, · · · ), case (4) with u = (0, 0, 0,−1) except for the case c = (4 ,−1) with d = (4, 2, 2, 9). Proof. (α) For case (4) with u = (1,−1, 0,−1) we find that the nullity and orthogonality conditions and Remark 3.9 leave us with the following possibilities: d c (ai/di) (2, 5, 3, 20) (1,−5 , 0) (1 (2, 6, 2, 12) (1,−3 , 0) (1 (3, 4, 2, 12) (1,−4 , 0) (1 (5, 3, 2, 15) (1,−6 , 0) (1 It is easy to see that we can never have = 1 for w ∈ W, so ā⊥ ∩ 1 (d+W) is empty and we have a contradiction. (β) Similarly, nullity, orthogonality and Remark 3.9 give: d c (ai/di) (2, 2, 225, d4 , . . . , dr) ( ,− d4 , . . . ,− dr ) (109 , . . . , 1 (2, 2, 98, d4 , . . . , dr) ( ,− d4 , . . .− dr ) (23 , . . . , 1 (2, 2, 36, d4 , . . . dr)) ( ,−1,−d4 , . . . ,−dr , . . . 1 Moreover n equals 962, 226, 50 respectively. It is easy to see that we can never have for w ∈ W, so we have a contradiction. (γ) For case (4) with u = (0, 0, 0,−1) we find that the nullity and orthogonality conditions and Remark 3.9 leave us with the following possibilities, up to swapping places 2 and 3: d c (ai/di) (2, 2, 4, 25) (6 ,−1) (−3 (2, 3, 3, 25) (6 ,−1) (−3 (3, 2, 3, 121) (15 ,−1) (− 5 (2m− 2, 2, 2,m2) ( 2(m−1) , 1−m , 1−m ,−1) (− 1 , m−1 , m−1 It is now straightforward to see that we cannot have w ∈ W with = 1, except in two cases (both associated to the last entry of the table). One is the case stated in the Lemma. The other occurs if m = 2, so a = (−1, 1 ,−1) which is orthogonal to (−1, 0, 1,−1), (−1, 1, 0,−1). But as (0, 0, 0,−1) is a vertex, neither of these vectors can be in W. So ā⊥∩ 1 (d+W) is still empty, giving the desired contradiction. As discussed above, we now turn to showing that case (6) cannot occur. The following remark is useful in finding when type III vectors can give elements of ā⊥ ∩ 1 (d+W). Lemma 10.5. If w = (−2i, 1j) and w̄ ∈ ā⊥, then ai < 0 (assuming we are not in the situation of Theorem 3.14). Proof. We need (10.1) = 1 + CLASSIFICATION OF SUPERPOTENTIALS 45 so if ai ≥ 0 then ≥ 1. Hence ≥ dj ≥ 1. As ā is null this means a = (−1 j) and we are in the situation of Theorem 3.14. Lemma 10.6. Configurations of type (6) cannot arise. Proof. Recall that we have dealt with (6)(iii) and we have d1 = d2 = 2. Case (6)(i): We have u = (−1, 0,−1, 1, · · · ), and from the nullity and orthogonality conditions we deduce that (ci/di) = 2(m+ 1) 2(m− 1) m2 − 1 , · · · , m2 − 1 where 1 2(m+1) and n− 1 = m2 for some positive integer m. We have, therefore, (ai/di) = 2(m+ 1) 2(m− 1) m2 − 1 m2 − 1 m2 − 1 , · · · , m2 − 1 Let us estimate the size of the entries in (ai/di). First observe that, as d3, d4 > 2(m + 1) from above, we have 4(m+ 1) < d3 + d4 ≤ n− d1 − d2 = n− 4 = m 2 − 3 so we deduce m ≥ 6. Hence we have 3 ≤ |a1 | < 1 ≤ |a2 | < 1 , |a3 | ≤ 6 , |ai | ≤ 1 for i ≥ 5. Also note that a4 . Finally, a4 > 0, else we would have 2n − 4 = 2m2 − 2 ≤ d4, which is impossible. Consider now a type III vector w = (−2i, 1j) with w̄ ∈ ā⊥. By Remark 10.5, we need i = 1, 3 or ≥ 5. If i = 1 then, by Eq.(10.1),we have ∈ (0, 1 ]. So we must have j = 4 and 1 contradicting our above remarks. If i = 3 then , which is impossible. Similarly, if i ≥ 5 , which is impossible. Hence there are no such type III vectors, so we are in case (6) with respect to ā. We cannot be in 6(ii) as then the null vector has exactly one positive entry (see below), but a has two positive entries. For 6(i), the null vector has exactly two negative entries. Now a has r− 2 negative entries, so r = 4. But for 6(i) the negative entries have modulus < 2, while a3 = −2− m2−1 , a contradiction. Case 6(ii): Here there are two possibilities. Subcase (α): u = (0,−1, 1,−1, · · · ). Then, as above, the null condition for c̄ gives (ci/di) = (m− 1)(m+ 2) 2(m+ 1)2 2(m+ 1) (m+ 1)2 , · · · , (m+ 1)2 where 1 2(m+1) and n− 1 = m2. So the vector (ai/di) is given by (m− 1)(m+ 2) 2(m+ 1)2 2(m+ 1) (m+ 1)2 (m+ 1)2 (m+ 1)2 , · · · , (m+ 1)2 As before, we have m ≥ 6. We deduce |a1 | ≤ 1 ≤ |a2 | < 1 , |a3 | ≤ 8 , |a4 | ≤ 1 , |ai | ≤ 1 i ≥ 5. Also a4 < 0, else d4 ≥ 2(m+ 1) 2 > 2n, which is impossible. We look for vectors w = (−2i, 1j) with w̄ ∈ ā⊥. By Lemma 10.5 we have i = 1, 2 or 4. If i = 1 = m+3 (m+1)2 , so j = 3, but now Eq.(10.1) contradicts 2 . A similar argument works if i = 2, while if i = 4, Eq.(10.1) implies , a contradiction. So 6(ii) must hold for ā, as we have already ruled out 6(i). But now we need a to have exactly one positive entry, which has modulus < 2. So r = 4 and this positive entry is a3, but we have a3 > 2, a contradiction. 46 A. DANCER AND M. WANG Subcase (β): u = (1, 0,−1,−1, · · · ). We similarly have (ci/di) = (m− 2)(m+ 1) 2(m− 1)2 2(m− 1) (m− 1)2 , · · · , (m− 1)2 (ai/di) = m2 − 3m+ 4 2(m− 1)2 2(m− 1) (m− 1)2 (m− 1)2 (m− 1)2 , · · · , (m− 1)2 where n− 1 = m2 and 1 = m+1 2(m−1)2 . The last two equations easily imply that m ≥ 6, and so | < 1 for all i. A type III (−2i, 1j) giving an element of ā⊥ must have i = 3 or 4, by Lemma 10.5. In both cases we find from Eq.(10.1) that , which is impossible. So 6(ii) holds for a, which is impossible as a has at least two positive entries. Lemma 10.7. The only possible example in case (4) is when c = , u = (0, 0, 0,−1), and d = (4, 2, 2, 9). ∆ā is then in case (1) with a = (−4 ,−1) and ā⊥ ∩ 1 (d +W) consists of (−2, 1, 0, 0), (−2, 0, 1, 0). Proof. By Lemma 10.4 we just have to eliminate the possibility u = (1, 0, 0,−2). Now (ai/di) = 8− 2d1 + d4 d1(4 + 2d1 + d4) (d1 − 1)(2d1 + d4) 2d1(4 + 2d1 + d4) (d1 − 1)(2d1 + d4) 2d1(4 + 2d1 + d4) 2(d1 − 1) d1(2d1 + d4 + 4) The null condition is −(d1 − 1) d 4 − 4(d 1 − d1 − 1) d 4 + 4d1(3d 1 + d1 + 8)d4 + 16d 1(d1 + 2) 2 = 0. For w = (−2i, 1j) with w̄ ∈ ā⊥ we need, by Lemma 10.5, i = 1 or 4. If i = 1, then for j = 2 or 3, Eq.(10.1) can be rewritten as 2d21 + d1d4 + 2d1 = −5d4 − 32, which is absurd. For j = 4 it can be rewritten as −1 − 2 = 14+2d4−2d1 d1(2d1+d4+4) . So the right hand side is < −1, which on clearing denominators is easily seen to be false. If i = 4, then for j = 1 Eq.(10.1) becomes 4 = 1. So (d1, d4) = (2, 8), (3, 6) or (5, 5), all of which violate the null condition. For j = 2, 3 we obtain from Eq.(10.1) the equation 4d4(d1 − 1) = (2d1 + d4 + 4)(d1d4 + d4 − 8d1), which can only have solutions if d4 ≤ 9. On the other hand, the null condition has no integer solutions if d4 ≤ 9. So no such type III exists, contradicting Lemma 10.6. Lemma 10.8. Configurations of type (5)(i) cannot occur. Proof. It is useful to note that the null condition for c̄ implies that d3 ≤ 4 when (d1, d2) = (4, 2) and d3 ≤ 3 when (d1, d2) = (3, 3). One further finds the following possibilities: u d c (ai/di) (−1, 0, 0, 0) (3, 3, 1, 2) (−1, 1,−1 ) (−1 (3, 3, 2, 1) (−1, 1,−2 ) (−1 (4, 2, 1, 3) (−1, 1,−1 ) (−1 (4, 2, 2, 2) (−1, 1,−1 ) (−1 (4, 2, 3, 1) (−1, 1,−3 ), (−1 (0, 0, 0,−1) (3, 3, 2, 121) (− 9 ,−1) ( 3 (4, 2, 2, 25) (−4 ,−1) (1 One easily checks that ā⊥ ∩ 1 (d +W) is empty in the last two cases, and consists only of type II vectors in the third to fifth cases, giving a contradiction to Lemma 10.6. For the first two cases, note that ā⊥ ∩ 1 (d + W) contains 1 (d + (−1,−1, 1, 0)) since by hypothesis for 5(i) (−1, 1,−1, 0) is in W. Hence (1),(2) cannot hold with respect to ā. Also the vector d of dimensions rules out (3),(4) and (5), so we have a contradiction. CLASSIFICATION OF SUPERPOTENTIALS 47 Lemma 10.9. The only possible example for case (3) is when c = , u = (0, 0, 0, 0,−1), d = (2, 2, 2, 2, 9). Then a = (−2 ,−1) and ∆ā is again in case (3). Proof. (A) Let u = (−1r). The null condition for c̄ gives 4dr = (δ+d1+2) 2, where δ = d4+· · ·+dr−1. In particular, dr is a square. Also, (ai/di) = 1− 1√ 1− 1√ , −1√ , · · · , −1√ If dr = 4, then we find there are no type III vectors in ā ⊥, a contradiction. So dr ≥ 9 and we have |ai | ≤ 1 for i = 1, 4, · · · , r − 1, ≤ 1 for i = r and < 1 for i = 2, 3. Lemma 10.5 shows i 6= 2, 3. From Eq.(10.1) and the above estimates, we first get i 6= r, and for the remaining values of i, we have > 0, so that j = 2 or 3. Also, dr = 9 (and hence d1 + d4 + · · · + dr−1 = 4), and (ai/di) = (− , · · · ,−1 ). Upon applying Theorem 5.18 to ∆ā together with Theorem 8.9 and the above Lemmas, we deduce that we are in case (3)(ii) with r = 5 and d1 = d4 = 2, giving the example in the statement of the Lemma. (B) Next let u = (1, 0, · · · ,−2). Now (ai/di) = ( − α, 1−α , 1−α ,−α, · · · ,−α, (n−2−dr)α−5 where, as a consequence of the null condition for c̄, we have (10.2) α = n− 2−m : m2 = dr(n− 1). Next we get the identity (n− 2)2 −m2 = (n− 1)(d1 + 1+ δ) + 1 = m2(d1+1+δ) +1, where δ is as given in (A) above. We deduce m < n−3(n− 2), and hence α < d1+1+δ n−dr−3 ≤ min( So ai is positive for i ≤ 3 and negative for 3 < i ≤ r−1. Note also that (n−2−dr)α < 2(n−2−dr) n−3−dr = 1 + 1 n−3−dr . In particular, ar < − 5 As usual, we look for (−2i, 1j) giving an element of ā⊥. By Lemma 10.5, i 6= 1, 2, 3. If 4 ≤ i ≤ r−1 then Eq.(10.1) says = 1 − 2α > 0, so j = 1, 2 or 3. If j = 1 we obtain α = 1 − 2 . Comparing this with Eq.(10.2) shows d1 = 3 and α = , but now ā is not null. If j = 2 or 3, we obtain α = 1 we deduce from Eq.(10.2) that d1 ≤ 5, and again one can check that all possibilities violate nullity. So all type III (−2i, 1j) have i = r. Therefore we must be in case (1) or (5) with respect to ā, and dr = 4 or 2 respectively. If j = 1, 2, 3 then > 0. Now Eq.(10.1) combined with the estimate above for ar show that > ar > − , so dr > 5, a contradiction. If 4 ≤ j ≤ r − 1, then in the case dr = 4, we find Eq.(10.1) gives α = n−4 . Combining with Eq.(10.2) we get n = 10, which is incompatible with dr = 4 and r ≥ 5. If dr = 2 we find similarly that α = 4 n−3 and m satisfies 3m 2 − 8m− 4 = 0; but this has no integral roots. So u = (1, 0, · · · − 2) cannot occur. (C) For u = (14,−2r), we have (ai/di) = (−α, , 1−α − α,−α, · · · ,−α, (n−2−dr)α−5) ) and Eq.(10.2) still holds. The arguments of case (B) carry over to this case, on swapping indices 1, 4. Lemma 10.10. Case (2) cannot occur. Proof. (A) Consider u = (−12). Now (ai/di) = ( − 1, −1 , · · · , 1 (2− n+1−dr Nullity implies d2 ≥ 3, so −1 < and |ai | ≤ 1 for 2 ≤ i ≤ r − 1. Also we have dr(n− 1) = m 2, and for this choice of u we have m = n+1− 2d2, so = dr−m which is positive if m < 0 and negative if m > 0. By Lemma 10.5 and the fact that d1 = 1, we only have to consider (−2i, 1j) with i = 2 or r. If i = 2, then Eq.(10.1) says = 1 − 2 > 0 so j ≥ 3. If 3 ≤ j ≤ r − 1 then Eq.(10.1) shows d2 = 3, so m = n− 5 and dr = (n−5)2 n−1 = n− 9 + n−1 . As n ≥ 7 we must then have (dr, n) = (9, 17) or (2, 9). Imposing the nullity condition on ā shows there are only three possibilities, corresponding 48 A. DANCER AND M. WANG to d = (1, 3, 2, 2, 9), (1, 3, 4, 9), (1, 3, 3, 2). In the first two there is only one type III in ā⊥, as d1 = 1 and d2 6= 4, so we are in case (5) with respect to ā, contradicting the fact that d2 6= 2. In the last case we must be in case (4) with respect to ā, but now (0,−1, 1,−1) is present, so u is not a vertex. If j = r then Eq.(10.1) becomes d2 = 3dr−n−1 dr−2 which is less than 3, a contradiction. (We cannot have dr = 2 and 3dr = n+ 1 as n ≥ 7.) Hence all such (−2i, 1j) have i = r and we are in case (1) or (5) with respect to ā. For case (1) we need r− 2 of the (j < r) equal. This can only happen for our a if d2 = 3, which is ruled out as in the previous paragraph. For case (5) we have dr = 2 and = 3 − n−1 . The possibilities on the left-hand side are 2 − 1,− 1 respectively. On using our relations for m,n, dr we find that only the third possibility can occur, and d2 = 3. The argument in the previous paragraph again eliminates this case. (B) Consider u = (0,−1,−1, · · · , 1). Now (ai/di) = (2β−1, β− , β− 2 , β, · · · , β, 4−(n+1−dr)β where again from the null condition of c̄ we have (10.3) β := (1− c1) = n+ 1−m 3 + dr( n+ dr + 1 : dr(n− 1) = m 2 (m > 0). Now 0 < β < 1 (the case β = 1 leads to r = 4, d2 = d3 = 2 and a = (0,−1,−1, 1) which violates nullity). So for w̄ ∈ ā⊥ we just have to consider w = (−2i, 1j) with i = 2, r (as d1 = 1 we can’t have i = 1; also by symmetry the case i = 3 is treated the same way as i = 2). If i = 2 then Eq.(10.1) says = 2β + 1 − 4 . If 4 ≤ j ≤ r − 1 we get β = 4 − 1; the only possibility consistent with our bounds on β is d2 = 3, β = and it is straightforward to check this is incompatible with the null condition for ā. If i = 2 and j = 1 then Eq.(10.1) implies d2 = 2 and again one checks that nullity for c̄ fails. If j = 3 Eq.(10.1) says β = 4 − 1, so as β > 0 either d2 = 2 and β = 1 − or d2 = 3 and β = 1 . In the former case the bound β < 1 shows d3 = 3 and β = , and now nullity for ā fails. In the latter case the bound β > 0 shows d3 > 6. Substituting this into the quadratic which must vanish for nullity of c̄, we see δ = d4 + · · · + dr−1 is < 4. Checking the resulting short list of cases yields no examples where nullity holds. If j = r then Eqs.(10.1) and Eq.(10.3) imply = 1+ 1 and one check that the possible (d2, dr) yield no examples where nullity of c̄ holds. So all such type III have i = r, and case (1) or (5) holds for ā. For case (1), then, as in (A), r−2 of the (j ≤ r− 1) must be equal. So either r = 5 and d2 = d3 with β = 1− or r = 4 and one of the preceding equalities holds. If β = 1 − 2 holds, then the bounds on β show d2 = 3, β = and as usual nullity for ā fails. If d2 = d3 holds, then using our formulae for β and substituting into the null condition for ā gives a quadratic with no integer roots. If case (5) holds, then dr = 2. Now Eq.(10.1) gives = 5 − (n − 1)β. If j = 1, j = 2, or 4 ≤ j ≤ r − 1 we get β = 6 5+(2/d2) respectively. (As usual, the case j = 3 is treated in just the same way as j = 2.) Now using the equations in Eq.(10.3) relating n,m in each case gives a quadratic with no real roots. Lemma 10.11. For case (1) the only possibility is when c = (−4 ,−1) with u = (0, 0, 0,−1) and d = (4, 2, 2, 9). ∆ā is then in case (4). Proof. (A) Consider u = (0, · · · , 0,−1). From the null condition for c̄ we see that dr = k 2, n− 1 = (k + 1)2 for some positive integer k and (ai/di) = ( (1 − 1 ),− 1 , · · · ,− 1 ). Note that since d1 = 4, n > 5 and so k 6= 1. We must consider solutions of Eq.(10.1). By Lemma 10.5, i 6= 1. If i = r we have = 1 − 2 The resulting equation has no solution in integer k > 1 for any choice of j. If 2 ≤ i ≤ r − 1, we = 1 − 2 . We only obtain a solution k > 1 if j = 1; in this case k = 3, so n = 17, dr = 9 CLASSIFICATION OF SUPERPOTENTIALS 49 and we see r = 4 with {d2, d3} = {2, 2} or {3, 1}. The former case is that in the statement of the Lemma. In the latter case we can have just one type III and one type II in ā⊥ (since d2 or d3 is 1, one potential type III is missing), so we must be in case (5) with respect to ā; but no di is 2, a contradiction. (B) Consider u = (0, 1, 0, · · · , 0,−2). Now (ai/di) = ( (1− β), 2 − β,−β, · · · ,−β, (n−dr−2)β−5 where β = dr+6d2 d2(2n−dr−4) . The nullity condition for c̄ implies d2 ≥ 3, d2 > δ and dr > 2d2 + 4, where δ now denotes d3 + · · · + dr−1. We can then deduce that β < 1, 0 < , 0 < a2 particular β < 2 . By Lemma 10.5, we must consider elements of ā⊥ coming from vectors (−2i, 1j) with i ≥ 3. If 3 ≤ i ≤ r−1, Eq.(10.1) says = 1−2β. As β < 1, this immediately rules out 3 ≤ j ≤ r−1. If j = 1 we get β = 1 . Combining this with our formula above for β we get 2d2(d2+δ−7)+(d2−3)dr = 0. The only possibilities are d2 = 3, δ = 4 which violates the null condition, or d = (4, 4, 2, 8) which violates the condition that dr(n − 1) should be a square. If j = 2, we get β = 1 − . Since we saw above that β < 2 we get d2 = 3 and β = , which is ruled out as above. If j = r, Eq.(10.1) implies β = dr+5 2+d2+δ+2dr . Comparing this with the formula for β above leads to a contradiction. The remaining possibility is for i = r. So we are in case (1) or (5) with respect to ā, and dr = 4 or 2 respectively. But dr > 2d2 + 4, so this is impossible. (C) Let u = (−1,−1, 0, · · · , 0, 1). Now (ai/di) = (− β,− 2 −β,−β, · · · ,−β, 1+(n−dr−2)β ) where dr(d2−4) 2d2(2n+dr−4) , so 0 < β < (noting that the nullity condition for c̄ implies d2 ≥ 5). We look for vectors (−2i, 1j) giving elements of ā⊥. Now Lemma 10.5 rules out i = r, while if 3 ≤ i ≤ r − 1 we need = 1− 2β > 2 . So j = r, and Eq.(10.1) yields β = dr−1 n+dr−2 . Equating this to the expression above for β gives an equation which may be rearranged so that it says a sum of positive terms is zero, which is absurd. If i = 1 then Eq.(10.1) says = 1−β. Clearly this can only possibly hold if j = r. The equation then gives β = dr−1 n−2 , and equating this with the earlier expression for β leads, as in the previous paragraph, to a contradiction. So the only possibility is i = 2, and we are therefore in case (1) or (5) with respect to ā. But d2 ≥ 5 so this is impossible. (D) Let u = (−1, 1,−1, 0, · · · ). Now (ai/di) = (− − β,− 2 − β,−β, . . . ,−β, (n−dr−2)β−1 and β = 1 −2( 1 ). An analysis of the nullity condition for c̄ shows that it can only be satisfied , so d2, d3 ≥ 5 and 0 < β < Let us now consider solutions to Eq.(10.1). If i = r, we have = 1− 2 2(n−dr−2)β . If j 6= 2, this equation implies that the positive quantity (1 + 2(n−dr−2) )β (or (1 2(n−dr−2) )β if j = 1) equals a nonpositive quantity (recall dr > 1 as i = r). If j = 2, we get that it equals But d2 ≥ 5 so dr = 2 or 3, and in each case we find the nullity condition for c̄ is violated. If i = 1, Eq.(10.1) says = 1 − β > 9 , so j = 2 or r. But for j = 2 we get d2 = 2, which is impossible as we know d2 ≥ 5, so in fact j = r. If i = 2, Eq.(10.1) is = 1 + 4 − 2β. We cannot then have j = 1, 3 or 4 ≤ j ≤ r − 1 as they lead to β > 2 , > 1, > 1 respectively. So we must have j = r. If i = 3, we see = 1 − 4 − 2β. If j = 1, 2 or 4 ≤ j ≤ r − 1, we see in all cases (using our bounds on d2, d3) that β > , a contradiction. Hence again j = r. If 4 ≤ i ≤ r − 1, then = 1 − 2β > 4 so j = 2 or r. If j = 2 we obtain β = 1 − 2 contradicting our earlier inequality for β; so again we have j = r. We have shown that any (−2i, 1j) giving an element of ā⊥ has j = r, so we are in case (3), (4) or (5) with respect to ā. It cannot be case (3) as we know from Lemma 10.9 that then each di is 50 A. DANCER AND M. WANG 2 or 9, and we have d1 = 4. If we are in case (4), then Lemma 10.7 tells us that d = (4, 2, 2, 9). Moreover, as (−2, 0, 0, 1), (0,−2, 0, 1) are the elements of ā⊥ ∩ 1 (d+W), we must have β = 4 ; but now β > 1 , a contradiction. If it is case (5), then we have di = 2 for some i, which we can take to be 4. Now ā must be orthogonal to vectors associated to (−14, 15,−1k) or (−14,−15, 1k), and either case is incompatible with our expressions for ai/di. (E) Consider u = (−1, 1, 0, · · · , 0,−1). Now (ai/di) = (− − β,−β, · · · ,−β, ((n−dr−2)β−3) and β = 8d2−dr(d2−4) 2d2(2n−dr−4) . It is easy to check that β < . Also, the nullity condition for c̄ implies d2 ≥ 3 and ( − 1)dr + 8 > 0; hence β > 0. The analysis is similar to that in (D). If i = r then Eq.(10.1) implies that a positive quantity times β equals a positive linear combination of reciprocals of di, minus 1. This sum of reciprocals is therefore > 1, which gives us upper bounds on dr. The only case where Eq.(10.1) and the null condition can hold is if j = 2 and d2 = 7, dr = 4, d3 + · · ·+ dr−1 = 11 If i = 1 then Eq.(10.1) says = 1− β > 0, so j = 2 or r. But j = 2 implies d2 = 2, which from above cannot hold, so j = r. Now Lemma 10.5 rules out i = 2. If 3 ≤ i ≤ r − 1, we have = 1 − 2β. If j = 1 then we get β = 2 , which cannot hold. If j = 2 then β = 1 − 2 , and as β < 2 we deduce d2 = 3 and β = which violates the null condition for c̄. If 3 ≤ j ≤ r − 1, then β = 1, which is impossible. So we have j = r. So in all cases we have j = r, except in the exceptional case discussed above where we can have i = r and j = 2. But our list (1)-(6) of possible configurations in ā⊥ shows that if the (i, j) = (r, 2) case occurs then no other type III can be in ā⊥. So we are in case (5), which is impossible as dr = 4 6= 2 for this example. Hence the exceptional case cannot arise. We see therefore that j = r in all cases. So, as in (D), we must be in case (3), (4) or (5) with respect to ā. As before, the fact that d1 = 4 rules out case (3). For case (4) we need the dk to be 4, 2, 2, 9 and dr to be 4 (as the (−2 i, 1j) in ā⊥ have j = r), but this contradicts d1 = 4. So we are in case (5). Now the orthogonality condition for the family of type II vectors leads to β > 2 , which is impossible. Lemma 10.12. Case (5)(ii) cannot occur if r ≥ 5. Proof. (A) Consider u = (0, 0, 1,−2, 0, · · · ). We have (ai/di) = (1− β, 1− 2β, n−2d2−3 n−d2−2 − ( n−6−3d2 n−d2−2 )β, 2(d2+2)β n−d2−2 − n−d2−2 − 2(d2+2)β n−d2−2 − n−d2−2 , · · · ) where all terms from the fifth onwards are equal and where β := 1 + c1 8(n−d2−2)+d4(n+d2) 2d4(n+d2+2) The nullity condition for c̄ implies d4 > 52, so n > 56 and we deduce 0 < β < . Hence > 0. It is also easy to show that ai > 0 for i ≥ 5 and a3 So if (−2i, 1j) gives an element of ā⊥ we need i = 2 or 4. As d4 > 52, Lemmas 10.6 - 10.11 show that case (5) must hold with respect to ā. In particular di = 2, so we cannot have i = 4. Hence i = 2 and d2 = 2. Now Eq.(10.1) implies β = , 2n−5 3n−4 , 4(n−2) + d4(n−2) or 4(n−2) , depending on whether j = 1, 3, 4 or ≥ 5. In all cases this contradicts the bound β < 3 and n > 56. (B) Consider u = (−1, 1, 0,−1, 0, · · · ). We have (ai/di) = (−β, 2 + 1− 2β, − d2+1 n−d2−2 − ( n−6−3d2 n−d2−2 )β, 2(d2+2)β n−d2−2 − n−d2−2 − 2(d2+2)β n−d2−2 − n−d2−2 , · · · ) where all terms from the fifth onwards are equal and β := 1 + c1 = n+d2−2 2(n+d2+2) + n−2 d2(n+d2+2) + n−d2−2 d4(n+d2+2) The nullity condition for c̄ implies d2 ≥ 9 and d4 ≥ 4. It is now easy to check that < β < 31 and that ai > 0 for i ≥ 5. As in (A), case (5) must hold with respect to ā. Now if (−2i, 1j) gives an element of ā⊥ we need di = 2. This, combined with Lemma 10.5, means i = 1 or 3. CLASSIFICATION OF SUPERPOTENTIALS 51 If i = 1, Eq.(10.1) immediately shows j cannot be 2. Moreover, if j = 3 or ≥ 5, Eq.(10.1) yields a value for β that violates the nullity condition for c̄. If j = 4 we obtain β = n−1 + n−d2−2 . As we are in case (5) with respect to ā, we need to consider the elements in ā⊥∩ 1 (d+W) corresponding to type II vectors. Their number and pattern, as stipulated by Theorem 5.18, together with orthogonality to ā, imply further linear relations among the components of (ai/di) and small upper bounds for r (usually of the form r = 5, 6). In all cases these additional constraints can be shown to be incompatible with the above values of β. As an illustration of the above method, note that our type III vector is (−2, 0, 0, 1, 0, · · · ). If ∆ā is in case (5)(iii), Theorem 5.18 says that the possible type IIs must have a −1 in place 1 and a 0 in place 4. Since r ≥ 5, the remaining −1 must be in a place whose corresponding dimension is 2. As d3 = 2 we can have (−1, ∗,−1, 0, ∗, · · · ) where ∗ indicates a possible location of the 1 in the type II. The other possibility is for −1 to be in place k for some k ≥ 5. After a permutation we can assume k = 5, and d5 = 2 must hold. The type II is then of the form (−1, ∗, ∗, 0,−1, ∗, · · · ) where ∗ again indicates possible positions for the 1. In the first case, the orthogonality conditions imply a2 which gives β = n−1 + n−d2−2 . Comparing with the value of β from Eq.(10.1), we get d2 = d4. Using this in the first value of β in (B) gives a contradiction after some manipulation. In the second case, the argument we just gave implies that we can only have r = 5 and the orthogonality condition implies a2 , which gives β = n−1 n+d2+2 2(n−d2−2) d2(n+d2+2) . After a short computation, one sees that the two values of β are again incompatible. If ∆ā is in case (5)(ii), the argument is essentially the same, as we only have to switch the places of the second −1 and the 1 in the type IIs. Let us now take i = 3. If j = 1, Eq.(10.1) implies β = 3d2+4−n 5d2+10−n . If the denominator is negative, then β > 1, which is a contradiction. If it is positive we find that this is incompatible with the inequality β > n+d2−2 2(n+d2+2) which comes from the displayed expression for β above. If j = 2, we get β = d2+1 2(d2+2) + n−d2−2 2d2(d2+2) . As above, we can rule this out by considering the vectors in ā⊥ ∩ 1 (d+W) associated to type II vectors. A similar argument works for j ≥ 5, where we find β = n−2d2−3 2(n−2d2−4) , and for j = 4, where we have β = n−d2−2 d4(n−2d2−4) + n−2d2−3 2(n−2d2−4) . (C) Next let u = (0, 0, 1,−1,−1, 0, · · · ). We have (ai/di) = (1 − β, 1 − 2β, n−2d2−3 n−d2−2 − (n−6−3d2 n−d2−2 )β, 2(d2+2) n−d2−2β − n−d2−2 − 2(d2+2) n−d2−2β − n−d2−2 − 2(d2+2) n−d2−2β − n−d2−2 , · · · ) where all terms from the sixth on are equal and β := 1 + c1 = n+d2 2(n+d2+2) n−d2−2 (n+d2+2) The nullity condition for c̄ implies d4 and d5 ≥ 13. It is now easy to check that 0 < β < > 0. As in (A), we find also that ai > 0 for i ≥ 6, and that a3 As in (A) again, case (5) must hold with respect to ā, so if (−2i, 1j) gives an element of ā⊥ we need di = 2. This, combined with Lemma 10.5, means i = 2. In this situation in all cases Eq.(10.1) gives a value of β incompatible with our bounds on n and β. Lemma 10.13. Case (5)(iii) cannot arise if r ≥ 5. Proof. This is similar to the proof of the previous Lemma so we will be brief. (A) Consider u = (0,−2, 0, 1, 0, · · · ). Now (ai/di) is given by (1− β,− 4 + 1− 2β, (n−3)+(n+d2−2)(β−1) n−d2−2 , 2d2β−(d2+1) n−d2−2 , 2d2β−(d2+1) n−d2−2 , · · · , 2d2β−(d2+1) n−d2−2 ) where β := 1 + c1 (d2+4d4)(n−2−d2−d4)+4d24 2d2d4(2n−d2−4) . The nullity condition for c̄ implies d4 ≥ 8, d2 ≥ 27 and d2 > 2d4, and it readily follows that < β < 1 . In particular, a1 As before, we see that case (5) holds with respect to ā, so for ā⊥ we have to consider type III vectors (−2i, 1j) where di = 2. So we need only consider i = 3 or i ≥ 5. 52 A. DANCER AND M. WANG In either situation, we proceed as in part (B) of the proof of Lemma 10.12, and obtain inconsis- tencies in the equations involving β or contradiction to the bounds on β or the dimensions. (B) Let u = (0, 0,−1, 1,−1, 0, · · · ). The nullity condition on c̄, which has a symmetry in d4 and d5, now implies d4, d5 ≥ 46 and > 28d2. Now (ai/di) is given by (1− β, 1− 2β, (n+d2−2 n−d2−2)β − n−d2−2 , 2d2β−(d2+1) n−d2−2 , 2d2β−(d2+1) n−d2−2 , 2d2β−(d2+1) n−d2−2 , · · · ) where all terms from the sixth on are equal and β := 1 + c1 n+d2−2 + ( )(n−d2−2 n+d2−2). We deduce that 1 < β < 3 and so 1 > 0. ∆ā must be in case (5), and for (−2i, 1j) associated to elements of ā⊥ ∩ 1 (d+W), we have di = 2 and we need only consider i = 2, 3 or ≥ 6. If i = 2, Eq.(10.1) and the upper bound on β imply . As d2 = 2, one checks that this never holds. If i ≥ 6, Eq.(10.1) implies . The bound d4, d5 > 28d2 can be used to show that this never happens. If i = 3, Eq.(10.1) and the expression for β above are seen to be incompatible if we use the bounds on d4, d5 and β. (C) Consider u = (−1,−1, 0, 1, 0, · · · ). The nullity condition on c̄ gives d4 ≥ 4, d2 ≥ 5. The vector (ai/di) is (−β, 1− 2β − 2 (n+d2−2)β−(d2+1) n−d2−2 , 2d2β−(d2+1) n−d2−2 , 2d2β−(d2+1) n−d2−2 , · · · ) where all terms from the fifth on are equal and β := 1 + c1 d24+(d2+d4)(n−d2−d4−2) d2d4(3n−d2−6) . Now as 1 , we see that 1 < β < 1 Again ∆ā is in case (5) and we consider vectors (−2i, 1j) associated to elements of ā⊥∩ 1 (d+W), where we must have di = 2, so i 6= 2, 4. If i = 1, Eq.(10.1) becomes = 1 − 2β > 0. This immediately means j 6= 2, 5, · · · , r. If j = 3, the value of β from Eq.(10.1) and the above expression for β lead to d4 ≤ 10/3. For j = 4, we obtain a contradiction by the method of part (B) in the proof of Lemma 10.12. If i ≥ 5, Eq.(10.1) says 4d2β+(n−3d2−4) n−d2−2 . If i = 3, Eq.(10.1) say 2(d2+1) n−d2−2 + 2(n+d2−2)β n−d2−2 . In both situations, we can again apply the method of part (B) in the proof of Lemma 10.12 to obtain contradictions. The last case to consider is case (5)(ii) with r = 4, which is the same as case (5)(iii) with r = 4 if we interchange the third and fourth summands. Lemma 10.14. No configurations for case (5)(ii) with r = 4 can occur. Proof. When r = 4 we no longer have d3 = 2, but the nullity condition for c̄ implies that Hence either {d3, d4} is one of {3, 3}, {3, 4}, {3, 5}, {3, 6}, {4, 4} or one of d3 or d4 is 2. Using this together with the nullity conditions for ā, c̄ and the orthogonality conditions, we see that we only need to consider u = (0,−2, 1, 0), (0, 0, 1,−2), (−1, 1, 0,−1) and (−1,−1, 1, 0). (A) Let u = (0,−2, 1, 0). From the nullity condition for c̄ we deduce that d4 = 2, d2 ≥ 13, and d3 ≥ 3. Now (ai/di) = (1− β, 2β − 1− 2d2β−(d2+1) (2d2+d3+2)β−(d2+1) n−d2−2 ) where β := 1 + c1 d2+4d3+2d d2d3(d2+2d3+4) . We find that 11 < β < 1 and so a1 The above facts imply that ∆ā is again in case (5)(ii), and for (−2i, 1j) associated to an element of ā⊥ we must have i = 4. If j = 1, 2, Eq.(10.1) leads to a contradiction to the above dimension restrictions. The case j = 3 can be eliminated using the method of part (B) in the proof of Lemma 10.12. (B) Next let u = (0, 0, 1,−2). The nullity condition for c̄ implies that d3 = 2 and d4 > 32d2+14 ≥ 46. Now (ai/di) is given by CLASSIFICATION OF SUPERPOTENTIALS 53 (1− β, 1− 2β, d4−d2+1 + (2d2+2−d4 )β, − 4 2(d2+2)β−(d2+1) where β := 1 + c1 d24+2d2d4−16 2d4(2d2+d4+6) . One easily sees that 1 < β < 1, so that 0 < a1 . Since d4 > 2d2 + 2 we obtain 0 < Therefore, ∆ā is in case (5)(ii) and for (−2i, 1j) associated to an element of ā⊥ we must have (by Lemma 10.5) i = 2 and so d2 = 2. Putting this value of d2 into the nullity condition for c̄ gives a cubic equation in d4 with no integral roots, a contradiction. (C) Consider now u = (−1, 1, 0,−1). From the nullity condition for c̄ we deduce that d2 ≥ 4, d3 = 2, d4 ≥ 3 and 4d2 > 3d4. Also, (ai/di) = (−β, + 1− 2β, (2d2+2−d4)β−(d2+1) , − 2 2(d2+2)β−(d2+1) where β := 1 + c1 2d2+2d4+d d2d4(2d2+d4+6) . It follows that 1 < β < 3 and a2 Now we see that ∆ā is either in case (1) or (4) or (5)(ii). In the first two instances, by Lemmas 10.11, 10.7 d is a permutation of (4, 2, 2, 9). Since 3d4 < 4d2 we have d2 = 9, d4 = 4. But then the null condition for c̄ is violated. So we are in case (5)(ii). For (−2i, 1j) associated to an element of ā⊥, as di = 2, we have i = 1 or 3. If i = 1, then Eq.(10.1) becomes = 1 − 2β < 0, so j = 3, 4. When j = 3 the value of β given above together with Eq.(10.1) imply that d4 = 3 or 4. But then the null condition for c̄ is violated. For j = 4 we may use the argument of part (B) of the proof of Lemma 10.12. If i = 3, using β > 1 in Eq.(10.1), we see that j = 2, 4. In either case, applying our bounds for the dimensions in Eq.(10.1) lead to contradictions. (D) Let u = (−1,−1, 1, 0). The null condition for c̄ implies that d4 = 2, d3 ≥ 3, d2 ≥ 5. With β := 1 + c1 , we have (ai/di) = (−β, 1 − 2β − 2d2β−(d2+1) (2d2+d3+2)β−(d2+1) One computes that β = 1 2d2+d 3+2d3 d2(3d 3+2d2d3+6d3) , and from the dimension bounds one gets 5 ≤ β < . ∆ā cannot be in case (1) or (4), otherwise as d2 ≥ 5, we must have d2 = 9, d3 = 4, and the null condition for c̄ is violated. So ∆ā is in case (5)(ii). For (−2i, 1j) associated to an element of ā⊥, we must then have i = 1, 4. If i = 1, then Eq.(10.1) is = 1 − 2β > 0, so j = 3 or 4. In either situation, we may apply the argument of part (B) of the proof of Lemma 10.12 to get a contradiction. If i = 4, Eq.(10.1) together with the dimension bounds above show first that we can only have j = 3. In that case, a more detailed look at Eq.(10.1) leads to a contradiction. We can summarise our discussions thus far by Theorem 10.15. Let r ≥ 4 and K be connected. Suppose that we are not in the situation of Theorem 3.14. Assume that c̄ ∈ C is a null vector such that ∆c̄ has the property that there is a unique vertex of type (1B) and all other vertices are of type (1A). Then the only possibilities are given by Lemmas 10.7/10.11 and 10.9, up to interchanging ā and c̄ and a permutation of the irreducible summands. We will now sharpen the above Theorem using Proposition 3.7. Corollary 10.16. Let r ≥ 4. Assume that K is connected and we are not in the situation of Theorem 3.14. Then the possibilities given by Lemmas 10.7 and 10.9 (and hence Lemma 10.11) cannot occur. Proof. We will discuss the r = 4 case (i.e. that in Lemmas 10.7, 10.11) in detail and leave the details of the r = 5 case (from Lemma 10.9) to the reader, as the arguments are very similar. In the r = 4 case, first observe that C has exactly two null vectors, c̄ and ā in the notation of Lemma 10.7, as the entries of a, c are determined by the vector d of dimensions. Hence, by Prop. 3.3, these are the only elements of C outside conv(1 (d+W)). 54 A. DANCER AND M. WANG Since (−14) is a vertex of W, all type II vectors in W must be zero in place 4. As (1,−1,−1, 0) is associated to an element of c̄⊥, it, together with (−1, 1,−1, 0), (−1,−1, 1, 0) are the only type II vectors in W. Next we analyse vectors in W and see if they are associated to elements of C, this last property being important for applying Prop 3.7. Recall that (1,−2, 0, 0), (1, 0,−2, 0), (−2, 1, 0, 0), (−2, 0, 1, 0) must be in W. The first two give elements of c̄⊥, the last two give elements of ā⊥. First consider v = (1,−2, 0, 0). Now v̄ is a vertex of conv(1 (d + W)). By the superpotential equation, d+ v = 2v̄ can be written as c̄(α)+ c̄(β), with c̄(α), c̄(β) ∈ C. Since v̄ is a vertex, every such expression must involve ā or c̄, unless it is the trivial expression v̄ + v̄ and v̄ ∈ C. By computing 2v−a, 2v−c we find that these cannot lie in conv(W) (it is enough to exhibit one component < −2 or > 1). Thus v̄ ∈ C. Now an analogous argument shows that if w = (0,−2, 1, 0) is in W then w̄ also lies in C. But vw is an edge of conv(W) with no interior points in W. So Prop 3.7 gives = 4J(v̄, w̄) = 0, a contradiction to d2 = 2. Hence w /∈ W. Similarly we see (0,−2, 0, 1) /∈ W. Next consider z = (−1,−1, 1, 0). By Remark 1.2(e) and the above, z is a vertex of conv(W). As above we can show that z̄ ∈ C. Now v, z are the only elements of the face {x1+2x2 = −3}∩conv(W) (cf. proof of Prop. 4.3 in [DW4]). So applying Prop 3.7 to vz we obtain 0 = 4J(v̄, z̄) = 1 contradiction. To handle the r = 5 case (from Lemma 10.9), first observe that null elements of C must have entries 2 in two of the places 1, · · · , 4 and −2 in the other two places. For a, c as in Lemma 10.9, we can take (1,−2, 0, 0, 0), (1, 0,−2, 0, 0), (0, 1, 0,−2, 0), (−2, 1, 0, 0, 0) to lie in W. The argument above to show that v̄ is in C still works for such type III vectors v. As above, we can use Prop 3.7 to show the other type III vectors (−2i, 1j) (i ≤ 4) do not lie in W; hence the ā, c̄ of Lemma 10.9 are the only null elements of C. Let z = (−1, 1,−1, 0, 0) (it lies in W since (1,−1,−1, 0, 0) is associated to an element of c̄⊥) and v = (1, 0,−2, 0, 0). As above we find z̄ is in C, and the arguments of Prop 4.3 in [DW4] show vz is an edge of conv(W). A contradiction results as above from applying Prop 3.7 to vz. The discussion at the beginning of this section now tells us that if K is connected and r ≥ 4 the only case when we have a superpotential of the kind under discussion is that of Theorem 3.14. The proof of Theorem 2.1 is now complete. Concluding remarks. 1. When r = 2, then c is collinear with the elements of W. In other words, the projected polytope ∆c̄ reduces to a single vertex, which must be of type (2). The possible elements of W are (−2, 1), (−1, 0), (0,−1), (1,−2). If W has just two elements then Theorem 9.2 tells us we are either in the situation of Theorem 3.14 (the Bérard Bergery examples), or in Example 8.2 or the third case of Example 8.3 in [DW4]. In fact one can show that this last possibility can be realised in the class of homogeneous hypersurfaces exactly when (d1, d2) = (8, 18). An example for these dimensions is provided by G = SU(2)9 ⋉ Sym(9) (where Sym(9) acts on SU(2)9 by permutation) and K is the product of the diagonal U(1) in SU(2)9 with Sym(9). The arguments of [DW2] show that this in fact gives an example where the cohomogeneity one Ricci-flat equations are fully integrable. If W has three elements, we may adapt the proof of Theorem 3.11 to derive a contradiction. Here the essential point is that whenever we had to check that a sum of two elements of C does not lie in d+W, such a fact remains true because the interior point of vw is the midpoint. If W contains all four possible elements, then k ⊂ g is a maximal subalgebra (with respect to inclusion). We suspect that this case also does not occur. In any event, it is of less interest because the only way to obtain a complete cohomogeneity one example is by adding a Z/2-quotient of the principal orbit as special orbit. 2. The only parts of this paper which depend on K being connected (or slightly more generally, on the condition in Remark 2.4) are parts of §5, Case (ii) of §9, and all of §10. To remove this CLASSIFICATION OF SUPERPOTENTIALS 55 condition, the main task would be generalizing Theorem 5.18 by getting a better handle on the type II vectors associated to (1A) vertices (cf Lemmas (5.6)-(5.8)). References [BB] L. Bérard Bergery: Sur des nouvelles variétés riemanniennes d’Einstein, Publications de l’Institut Elie Cartan, Nancy (1982). [BGGG] A. Brandhuber, J. Gomis, S. Gubser and S. Gukov: Gauge theory at large N and new G2 holonomy metrics, Nuclear Phys. 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Jesus College, Oxford University, Oxford, OX1 3DW, United Kingdom E-mail address: [email protected] Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada E-mail address: [email protected] http://www.maths.ox.ac.uk/ http://www.math.mcmaster.ca/mckenzie 0. Introduction 1. Review and notation 2. The classification theorem and the strategy of its proof 3. Projection onto a hyperplane 4. The sign of J(, ) 5. Vectors orthogonal to a null vertex 6. Adjacent (1B) vertices 7. More than one type (2) vertex 8. Adjacent (1B) vertices revisited 9. Type (2) vertices 10. Completing the classification References