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thermoelectric effects have generated an immense interest for quite some time already because they offer the possibility to convert heat from the environment into electrical work @xcite . this form of energy harvesting is potentially useful for electric circuits on modern computer chips that produce large amounts of heat and currently need to be cooled actively in order not to overheat . one can also use the harvested energy to run auxiliary circuits such as autonomous sensors or recycle the lost energy to charge a battery . unfortunately , even after decades of material research current thermoelectric materials still have a very low efficiency in converting heat into electrical work and deliver only moderate powers . for this reason , thermoelectric energy harvesting so far is restricted to certain niche applications such as in interplanetary spaceships where the fact that a thermoelectric generator does not require any moving parts and therefore no maintenance turns out to be useful . we give a brief summary of how conventional thermoelectric devices work to orient the reader . the building blocks of thermoelectricity are the peltier and seebeck effect . the seebeck effect is the flow of electrical current in response to an applied temperature difference , while the peltier effect is the reverse : the creation of temperature difference in response to an applied electrical voltage . this is intuitively understood by the fact that the hotter charge carriers diffuse faster than the cold ones , creating a flow of thermal energy from hot to cold , and consequently the imbalanced charge builds up an electrical voltage across the material . one challenge in making good thermoelectric devices is that materials that are good electrical conductors tend also to be good thermal conductors . this fact makes it difficult to maintain the necessary temperature difference needed to produce electrical power from the seebeck effect . the need for ways to create systems with high electrical conductance , while maintaining low thermal conductance is an outstanding challenge in this field that the interface - based devices we shall describe can help to solve . commercial devices use doped semiconductors such as bismuth telluride . in an n - type semiconductor , heat flow is in the same direction of the electron ( current ) flow , however , in a p - type semiconductor , heat and current flows are in opposite directions . this is useful because a thermoelectric architecture can be built of alternating p - type and n - type semiconductor elements that are connected electrically in series ( via metallic contacts ) and thermally in parallel . this permits the small electrical voltage ( or power ) produced by a single element to be increased by the number of elements in the device , making a practically useful system . these are placed between ceramics plates which conduct heat , but not electricity , so one side can be heated and the other cooled to produce the power . alternatively , power can be applied to cool the cold side , and heat the hot one , acting as a refrigerator . mesoscopic solid - state physics can help to overcome the limitations of current thermoelectric materials by providing powerful and highly efficient heat engines operating at the nanoscale . soon after initial studies on the thermopower of basic mesoscopic structures like quantum point contacts @xcite and quantum dots @xcite , there were first proposals pointing out that structures of reduced dimension can give rise to an increased thermoelectric figure of merit @xmath0 as compared to bulk structures made from the same materials @xcite . similar ideas where brought forward by mahan and sofo @xcite who showed that sharp spectral features give rise to high thermoelectric performance as characterized by a high value of @xmath0 . nanoscale conductors such as quantum dots naturally provide these sharp spectral features . hence , they are promising candidates for thermoelectric energy harvesters . initial experiments on quantum dots defined in a two - dimensional electron gas showed saw - toothlike oscillations of the thermopower as a function of gate voltage in agreement with theory @xcite . further studies have been carried out on open quantum dots @xcite and carbon nanotubes @xcite . more recently , the thermopower of quantum dots defined in nanowires has been studied @xcite . interestingly , these experiments could observe nonlinear thermoelectric effects @xcite . in another series of experiments the thermopower due to sequential tunneling , cotunneling and the kondo effect has been investigated in gate - defined quantum dots @xcite . the thermopower of double dots was observed in ref . @xcite . two - terminal geometries using mesoscopic conductors have been considered , notably using quantum dots @xcite . in the two - terminal geometry , both temperature and voltage bias are applied to the sample and the thermoelectric response is investigated . it has the advantage of being the analogue of the traditional thermocouple which has wide applications , mainly in its role as a thermometer . however , for the purpose of energy harvesting , it suffers from the fact that different parts of the same electrical circuit must be at different temperatures which makes thermal isolation difficult . sketch of a generic thermoelectric energy harvester : a mesoscopic system situated at the interface between a cold and a hot bath converts a heat current @xmath1 into a charge current @xmath2 that is able to power a load or charge a battery . ] in contrast to the above mentioned works based on two - terminal configurations , the problem of a device that powers a circuit by harvesting energy from the outside environment demands a three - terminal geometry , cf . figure [ fig : harvest ] . two terminals at the same temperature define the conductor that supports a charge current @xmath2 . the third terminal represents the coupling to the heat source from which energy but not charge is absorbed . the harvester converts the heat current @xmath1 into useful power , @xmath3 , that runs a load resistance , represented here by a voltage drop @xmath4 opposed to the generated current . the properties of the mesoscopic region to which the three terminals are connected define the characteristics of the heat engine , in particular its efficiency , @xmath5 . only recently has the investigation of thermoelectric effects in quantum - dot structures in three - terminal geometries generated a lot of attention @xcite . this was motivated by a number of reasons . first of all , such three - terminal setups share a number of features with mesoscopic coulomb - drag setups that have been studied both theoretically @xcite as well as experimentally @xcite in the last years . such coulomb - drag setups consist of two nearby mesoscopic conductors such as quantum dots . the first conductor is subject to a bias voltage that drives a charge current through it . nonequilibrium charge fluctuations ( noise ) due to this current will induce charge fluctuations in the second , unbiased conductor . if the second conductor is intrinsically nonlinear it will rectify these induced charge fluctuations and thus exhibit a charge current without an applied bias voltage . similarly , in a three - terminal heat engine , thermal fluctuations from the hot source can get rectified and drive a directed charge current . here , the nonequilibrium situation is introduced by the temperature imbalance between the hot source and the conductor . in addition , three - terminal heat engines offer the advantage of spatially separating the hot and cold reservoirs , cf . . this becomes most pronounced in a recently proposed microwave - cavity heat engine where two mesoscopic conductors such as double quantum dots are connected via a superconducting microwave cavity over a typical length of a few centimeters @xcite . the separation of hot and cold baths helps to reduce leakage heat currents that are detrimental to achieving a high efficiency of heat to work conversion . finally , three - terminal heat engines also exhibit a crossed flow of heat and charge currents . this is useful for energy harvesting applications because it allows the hot source to be kept electrically separated from the actual energy harvester . thermoelectric configurations including thermometer terminals have also been investigated @xcite . in contrast to bulk thermoelectric materials , nanoscale heat engines often operate in the nonlinear regime . linear response theory is valid if the temperature difference on the scale of the inelastic scattering length is small compared to the average temperature @xcite . for bulk materials , the inelastic scattering length is typically much smaller than the system size , such that even for a sizable temperature bias , linear response theory is a good approximation . for nanoscale setups , system size and inelastic scattering length become comparable . hence , nonlinear effects become important @xcite . in order to properly describe nonlinear thermoelectric effects in mesoscopic structures , a nonlinear scattering matrix theory of thermoelectric transport has been developed recently @xcite . it has been applied to study heat transport through quantum dots where , e.g. , a deviation from the wiedeman - franz law was found @xcite . it was also used to investigate the magnetic - field asymmetry of nonlinear transport coefficients @xcite . experimentally , the nonlocal thermoelectric response of a ballistic four - terminal structure has been investigated @xcite . further studies theoretically analyzed nonlinear thermoelectric transport through molecular junctions @xcite . an important feature of nonlinear thermoelectrics is that the figure of merit @xmath0 is no longer sufficient to characterize the thermoelectric performance @xcite . instead , one has to rely on quantities such as the maximal efficiency , the efficiency at maximum power @xcite or the maximal efficiency at a given output power @xcite . here , we review recent theoretical work on three - terminal thermoelectrics with quantum dots . the review is organized as follows . in sec . [ sec : coulomb ] we discuss systems of coulomb - coupled conductors ranging from quantum dots in the coulomb - blockade regime over chaotic cavities to resonant tunneling quantum dots . the second part , sec . [ sec : boson ] , focusses on quantum dot heat engines that are driven by bosonic degrees of freedom such as phonons , magnons and microwave photons . we finish with conclusions and an outlook in sec . [ sec : concl ] . in the following , we discuss different types of heat engines based on quantum dots that are coupled to a hot electronic reservoir . as a first example , we will analyze a setup based on capacitively coupled quantum dots in the coulomb - blockade regime . we then continue the discussion with a similar system based on chaotic cavities coupled to reservoirs via quantum point contacts with a large number of open transport channels . finally , we demonstrate that heat engines based on resonant tunneling through either quantum dots or quantum wells can yield high power in combination with good efficiency . here , we analyze the thermoelectric properties of two capacitively coupled quantum dots in the coulomb - blockade regime in a three - terminal geometry @xcite . this model emphasizes the contribution of charge fluctuations , with heat being absorbed merely by means of the coulomb interaction between electrons in the two different conductors . we demonstrate how a temperature bias across the system can drive a charge current which in turn can be used to generate electrical power . furthermore , we demonstrate that the device can act as an ideal heat to current converter that can reach carnot efficiency . the bipartite nature of this setup can be exploited for other purposes such as feedback control @xcite , the detection of dynamical rates @xcite or the investigation of information flows @xcite . sketch of the coulomb - blockade heat engine . two single - level quantum dots ( yellow ) are capacitively coupled to each other . the lower gate quantum dot is connected to a single hot reservoir ( red ) . the upper conductor dot is coupled to two cold electronic reservoirs ( blue ) . ] we consider the setup shown schematically in . it consists of two quantum dots @xmath6 in the coulomb - blockade regime . each dot hosts a single level with energy @xmath7 . assuming strong onsite coulomb interaction , the dots can host either @xmath8 or @xmath9 excess electron . in the following , we neglect the electron spin as it will only lead to a renormalization of the tunnel couplings introduced below . the two quantum dots are capacitively coupled to each other such that they can exchange energy but no particles . the strength of the coupling is characterized by the coulomb energy @xmath10 that is needed to occupy both quantum dots at the same time and will parametrize the energy exchange between the two dots . the conductor dot , @xmath11 , is tunnel coupled to two electronic reservoirs @xmath12 . the reservoirs are in local thermal equilibrium and characterized by chemical potentials @xmath13 and temperature @xmath14 . the gate dot , @xmath15 , is coupled to a single reservoir @xmath16 with chemical potential @xmath17 and temperature @xmath18 . in order to obtain a finite thermoelectric response through the conductor dot to a temperature difference @xmath19 , we need to have energy - dependent tunnel coupling strengths . here , we model this energy dependence by choosing the tunnel couplings @xmath20 between the dot and its respective reservoir @xmath21 to depend on the number of electrons , @xmath22 , on the _ other _ quantum dot . there are different ways how to realize this energy dependence in an actual experiment . first of all , the transmission through a tunnel barrier generically depends on energy of the tunneling particle . this effect is most pronounced for energies close to the barrier height . second , quantum dots with an excited state that couples asymmetrically to the two leads and decays quickly into the ground state can mimic energy - dependent tunneling rates as well . finally , additional quantum dots can be added between the conductor dot and its two reservoirs . these additional dots serve as energy filters that only transmit electrons at a given energy and thereby allow to achieve an effectively energy - dependent tunnel coupling . in order to describe the system , we trace out the noninteracting electronic reservoirs . the remaining strongly interacting quantum dot degrees of freedom are then characterized by a reduced density matrix for the double quantum dot . the time evolution of the system is determined by a master equation @xmath23 for the reduced density matrix elements @xmath24 describing the probability to find the double dot empty @xmath25 , occupied with one electron on the conductor @xmath26 or gate dot @xmath27 and doubly occupied @xmath28 . in the sequential tunneling regime @xmath29 the transition rates @xmath30 follow from fermi s golden rule and are given by @xmath31 the transition rates @xmath32 describe tunneling of electrons onto or off the dot through barrier @xmath33 when the other dot has @xmath22 electrons . specifically , they are written as @xmath34 where @xmath35 + 1\}^{-1}$ ] denotes the fermi function with temperature @xmath36 and bias voltage @xmath37 . in order to calculate the charge and heat currents as well as their fluctuations and correlations , we use a full - counting statistics approach . to this end , we multiply the transition rates with a factor @xmath38 where @xmath39 denotes the charge transferred through barrier @xmath33 in the associated tunneling event . the cumulant generating function @xmath40 is then given by the eigenvalue of the resulting matrix @xmath41 that goes to zero as @xmath42 . the charge currents are simply given by the derivative of the cumulant generating function with respect to the counting fields , @xmath43 . similarly , for the energy currents , we introduce counting factors @xmath44 where @xmath45 denotes the energy transferred through barrier @xmath33 in the corresponding tunneling event . higher - order derivatives give the correlations that will be discussed in section [ sec : cbfluct ] . finally , heat currents follow from charge and energy currents as @xmath46 . we remark that while charge and energy currents are conserved , @xmath47 , heat currents in general are not conserved due to joule heating . current at zero bias voltage in units of @xmath48 as a function of the two level positions @xmath49 and @xmath50 . parameters are @xmath51 , @xmath52 , @xmath53 and @xmath54 . ] in the following , we want to drive a charge current through the conductor dot simply by applying a temperature difference between the reservoirs of conductor and gate dot , i.e. without applying a bias voltage across the conductor dot . in order to achieve this goal , two requirements have to be met . first of all , the left - right symmetry of the system has to be broken . this can be achieved by choosing the tunnel couplings of the conductor dot to the left and right reservoirs different . in addition , we also have to break the particle - hole symmetry underlying the system . this is achieved by having energy - dependent tunneling rates for the conductor dot . in the absence of a bias voltage , the charge current @xmath2 through the conductor dot can be simply related to the heat current @xmath1 flowing out of the gate dot , @xcite @xmath55 the ratio of heat and charge currents is determined by the ratio of electron charge and charging energy . furthermore , it depends on the ratio of the different tunnel couplings that enter the problem . the direction of the charge current can be controlled via the asymmetry of tunnel couplings as well as by the direction of the heat current , i.e. by the sign of the temperature bias @xmath19 . the largest current , @xmath56 , can be achieved in an optimal configuration when the conductor dot couples only to , say , the left lead when the gate dot is empty and only to the right lead , when the gate dot is occupied . the heat - driven current as a function of the level positions of the two dots is shown in . as a function of the gate dot level , @xmath50 , the current exhibits a peak at @xmath57 with a width given by the temperature of the hot bath , @xmath18 . as a function of the conductor dot level , @xmath49 , the current has a plateau between @xmath58 and @xmath59 . the borders of the plateau are smeared by the temperature of the cold bath , @xmath18 . the mechanism giving rise to the heat - driven charge current is the following . initially , the double dot is completely empty . in a first step , an electron tunnels onto the conductor dot from , say , the left reservoir . next , an electron tunnels onto the gate dot . this requires the charging energy @xmath10 which is extracted from the hot reservoir . afterwards , the electron from the conductor dot tunnels into the right lead , thereby releasing the charging energy into the cold reservoir . in a final step , the gate dot is emptied such that the system returns to its initial state . in one such transport cycle , one quantum of energy , i.e. the charging energy , is transferred from the hot to the cold , giving rise to correlations of the charge and heat currents . we remark that , in the optimal configuration , one electron is transmitted from the left to the right reservoir for _ every _ absorbed quantum of heat . this is known as the tight - coupling limit . ( a ) power @xmath60 ( solid lines ) and efficiency @xmath61 ( dashed lines ) as a function of the applied bias voltage for different temperatures of the cold reservoir . ( b ) efficiency at maximum power as a function of the carnot efficiency . parameters are @xmath51 , @xmath52 , @xmath62 , @xmath63.,title="fig : " ] ( a ) power @xmath60 ( solid lines ) and efficiency @xmath61 ( dashed lines ) as a function of the applied bias voltage for different temperatures of the cold reservoir . ( b ) efficiency at maximum power as a function of the carnot efficiency . parameters are @xmath51 , @xmath52 , @xmath62 , @xmath63.,title="fig : " ] so far , we demonstrated that our quantum - dot heat engine can convert a heat current into a directed charge current . in a next step , we want to generate a finite output power by adding a load to the system . hence , we apply a bias voltage @xmath4 across the conductor dot against which the heat - driven current can perform work . the output power is then simply given as @xmath3 . it is shown as a function of the applied bias in a ) . for zero bias , the output power obviously vanishes . similarly , it vanishes at the so called stopping voltage @xmath64 where the heat- and bias - driven currents compensate each other such that @xmath65 . in between , the system reaches its maximal output power . for situations close to equilibrium , the maximum power is reached at half the stopping voltage . as the system is taken far away from equilibrium by lowering @xmath14 , the point of maximum power is shifted towards larger bias voltages . another important quantity to characterize the thermoelectric performance of a heat engine is its efficiency of heat to work conversion . it is defined as the ratio between the output power and the input heat which in our case is given by the heat current flowing out of the hot reservoir , i.e. , we have @xmath5 . in the tight - coupling limit , the efficiency grows linearly with the applied bias voltage and we simply have @xmath66 , cf . hence , at the stopping voltage @xmath67 the device reaches carnot efficiency @xmath68 indicating that it operates as an optimal heat to current converter . however , at this point , the heat engine operates reversibly and , therefore , does not produce any output power . a more relevant quantity to consider is the efficiency at maximum power . it is shown as a function of the temperature bias in b ) . for a small temperature bias , it increases linearly as @xmath69 in agreement with a general thermodynamic bound for systems with time - reversal symmetry @xcite . in the nonlinear regime , it grows faster than @xmath69 and even reaches @xmath70 for @xmath71 . we remark , however , that at this point our master equation approach is no longer valid since higher order tunneling contributions that give rise to a broadening of the dot level become important . up to now , we only discussed the average transport of heat and charge through the quantum - dot heat engine . however , at the nanoscale , fluctuations of average quantities turn out to be important @xcite . indeed , we discussed above how the generation of current depends on the correlation of charge fluctuations in the two dots . a way to characterize these fluctuations is to look at the full counting statistics @xmath72 which addresses the probability that @xmath73 electrons have passed through the system in a given time @xmath74 @xcite . in a quantum dot system , this quantity can actually be measured by coupling a charge sensor such as a quantum point contact or a single - electron transistor to the quantum dot @xcite . as the number of electrons on the dot changes , the electrical potential felt by the charge sensor changes , thus leading to a change in the current through it . this allows to measure the charge on the quantum dot in real time . interestingly , in the quantum - dot heat engine we can not only access the full counting statistics of charge but also of heat @xcite . measuring the charge of both , the conductor and the gate dot , e.g. , via a quantum point contact that couples asymmetrically to the two dots , allows us to reconstruct the counting statistics of heat from the counting statistics of charge due to the intimate relationship between heat and charge transfer in the system . this way , non - equilibrium fluctuation relations can be measured that relate the charge and heat currents @xcite in terms of electron counting . this is important because in the presence of temperature gradients , fluctuation relations for charge currents @xcite become configuration - dependent @xcite unless one also considers energy currents . notably , the charge fluctuation theorem becomes universal ( only depending on the thermodynamic forces ) in the tight - coupling limit @xcite . additional insight of the thermoelectric performance of the quantum - dot heat engine can be accessed by investigating the charge and heat current - current correlations @xcite . the interest on electronic heat noise has appeared only very recently @xcite . already in the linear regime the charge - heat cross - correlations are related to seebeck - like and peltier - like coefficients by means of an extended three - terminal fluctuation - dissipation theorem . far from equilibrium , dimensionless quantities for the auto- and cross - correlations ( charge and heat fano factors , and cross - correlation coefficient ) can be defined . divergences of the charge fano factor can be used to measure the non - linear thermovoltage . most interestingly , the charge - heat cross - correlations are maximal in the tight - coupling limit @xcite , leading to carnot - efficient configurations @xcite . as we have just discussed , capacitively coupled quantum dots in the coulomb - blockade regime can operate as optimal heat to charge current converters that can reach carnot efficiency . yet , they are of limited practical use for energy harvesting applications as they provide only small currents and output power . this is a consequence of the fact that transport in these systems occurs via the tunneling of single electrons . in the following we elucidate the thermoelectric performance of a device based on chaotic cavities coupled to electronic reservoirs via quantum point contacts with a large number of open transport channels @xcite . our main aim is to discuss how current , power and efficiency behave as the number of open channels is changed . schematic sketch of the heat engine based on chaotic cavities . the two cavities are capacitively coupled to each other . in addition , the rectifying cavity is connected to two cold electronic reservoirs at temperature @xmath75 via quantum point contacts . similarly , the second cavity is connected to a single hot reservoir at temperature @xmath76 . ( reprinted with permission from @xcite . copyright 2012 american physical society . ) ] we consider two open quantum dots @xmath77 coupled via a mutual capacitance @xmath78 . each cavity is connected to an electronic reservoir @xmath12 via a quantum point contact , cf . . the reservoirs are in local thermal equilibrium and described by a fermi function @xmath79 + 1\}^{-1}$ ] with chemical potential @xmath13 and temperature @xmath36 . interaction effects are captured by capacitive couplings @xmath80 between cavity @xmath81 and reservoir @xmath82 that screen potential fluctuations . we focus on the semiclassical regime where the number of open transport channels @xmath83 of the quantum point contacts is large . we assume that dephasing destroys phase information but preserves energy . in this situation , the cavities are characterized by distribution functions @xmath84 that depend only on energy . for later convenience , we write these distributions as @xmath85 the first term corresponds to the average of the reservoir distribution functions weighted with the transmission @xmath86 of the corresponding qpc . the second term describes fluctuations of the distribution function that have to be determined in the following . in addition , the cavities are characterized by their potential @xmath87 and associated fluctuations @xmath88 . in order to obtain a finite thermoelectric response , we need to have energy - dependent transmissions @xmath89 . here , we model this energy dependence as @xmath90 . while the first term , @xmath91 describes the energy - independent transmissions that depend linearly on the number @xmath83 of open transport channels , the second term captures changes in the transmission due to fluctuations of the cavity potential . we remark that the energy - dependent term @xmath92 is independent of the number of open transport channels . the distribution functions of the cavities obey a kinetic equation of the form ( cf . , e.g. , ref . @xcite ) @xmath93 where @xmath94 denotes the density of states at the fermi energy in cavity @xmath81 . the kinetic equation describes how the charge in a given energy interval changes due to changes of the cavity potential @xmath87 , in- and outgoing electron currents through the quantum point contacts as well as due to fluctuations of these currents @xmath95 . here , the index @xmath96 indicates that we have to sum over all contacts @xmath82 of a given cavity @xmath81 . in a next step , we obtain a relation between the fluctuations of the cavity distributions @xmath97 and potentials @xmath88 by expressing the charge inside each cavity in terms of an integral over the distribution functions as well as in terms of the various capacitances and potentials . using this relation between @xmath97 and @xmath88 , we can transform the kinetic equation into a langevin equation for the potential fluctuations @xmath88 . due to the nonlinearity introduced into the problem by the energy - dependent transmission , the langevin equation has a multiplicative noise term that leads to the it - stratonovich problem in the interpretation of the stochastic integral when converting the langevin equation into a fokker - planck equation @xcite . for the problem at hands , it turns out that only the kinetic prescription by klimontovich @xcite provides a meaningful solution that exhibits vanishing heat and charge currents in global equilibrium . from the fokker - planck equation , we can obtain the expectation values of the potential fluctuations , @xmath98 and @xmath99 , which subsequently allow us to evaluate the charge current between cavity 1 and its contact @xmath82 via the standard scattering matrix expression @xmath100 we first focus on the situation where a finite temperature bias @xmath101 is applied across the system while there is no bias voltage applied across cavity 1 , i.e. @xmath102 . for the charge current through cavity 1 we find to lowest order in the energy - dependent transmission the compact expression @xmath103 the charge current depends on the asymmetry parameter @xmath104 with @xmath105 , @xmath106 and @xmath107 . it characterizes both the breaking of left - right symmetry as well as the breaking of particle - hole symmetry due to energy - dependent transmissions . from eq . , we conclude that in order to have a finite charge current driven by the temperature bias , we need to break both symmetries simultaneously . we remark that a similar antisymmetric combination of energy - dependent transmissions was found in the expression for the charge current through coulomb - blockaded dots , , as there we also have to break left - right and particle - hole symmetry at the same time . in addition , the current also depends on the @xmath108 time of the cavity , @xmath109 . here , @xmath110 denotes the effective conductance of the double cavity while @xmath111 is an effective capacitance that characterizes the coupling between the two cavities . for weakly coupled cavities , it grows as @xmath112 . we now turn to the discussion of how the current depends on the number of open transport channels . as the energy - dependent part of the transmission does not scale with @xmath83 , we find that the current is independent of the number of open channels . for realistic parameters @xcite with @xmath113{ff}$ ] , @xmath114{(mv)}^{-1}$ ] and a temperature bias of , we obtain @xmath115{na}$ ] . hence , the current for the cavity heat engine is about two orders of magnitude larger than for the heat engine operating in the coulomb - blockade regime . the reason for this enhanced current is simply the difference between transport through a tunnel barrier and a fully open quantum channel . in order to generate a finite output power , we have to apply a bias voltage @xmath116 against which the heat - driven charge current through cavity 1 can perform work . the output power is then simply given by @xmath117 . it vanishes when no bias voltage is applied . similarly , it also vanishes at the stopping voltage @xmath118 ( here , @xmath119 denotes the conductance of cavity 1 ) where heat- and bias - driven charge currents exactly compensate each other such that @xmath120 . the maximal power is generated at half the stopping voltage and is given by @xmath121 ^ 2.\ ] ] surprisingly , the output power drops inversely as the number of open transport channels . the reason for this lies in the fact that the energy - dependent transmission is independent of the number of open transport channels . while the heat - driven current thus is independent of the channel number as well , the bias driven current linearly scales with the channel number . hence , the more transport channels are open , the smaller the stopping voltage and , hence , the smaller the output power that can be achieved . we now turn to the discussion of the efficiency @xmath61 given by the ratio of output power to input heat . the input heat is given by the heat current flowing between the cavities . to leading order in the nonlinearity , it is given by @xmath122 i.e. there is a finite heat current even in the absence of energy - dependent transmissions . as the heat current is independent of the applied bias voltage , we find that the maximal efficiency and the efficiency at maximum power coincide and are given by @xmath123 analyzing the scaling behaviour of the efficiency with the number of transport channels , we find that it drops inversely to the square of the number of open channels . this faster decrease as compared to the output power is due to the proportionality of the heat current to the number of transport channels . for realistic parameters , we find that for a few open channels an output power of a few fw can be generated . at the same time , the efficiency reaches at most a few percent of the carnot efficiency for a device operating a liquid helium temperatures . schematic dependence of the output power @xmath60 of a heat engine on its conductance @xmath124 . in the coulomb - blockade regime , power grows linearly with conductance until the coulomb - blockade regime is left . for large conductances , power decays as the inverse conductance . in between , the maximal output power is reached for a heat engine that operates with a single open transport channel . ] so far , we discussed heat engines based on quantum dots in the coulomb - blockade regime and open quantum dots . we found that both deliver a rather small power . in the coulomb - blockade regime , power grows linearly with the conductance but is limited by the fact that transport occurs via the tunneling of single electrons . for chaotic cavities , the power drops inversely with the conductance because the relative importance of the energy - dependent transmission goes down as more and more transport channels open up , cf . the schematic sketch in . the maximal output power can therefore be expected for a heat engine based on transport through a single quantum channel . a paradigmatic realization of such single channel transport is given by resonant tunneling through quantum dots that we analyze below @xcite . similar setups have been considered in their dual role as an electronic refrigerator @xcite and successfully been used to cool a micrometer - sized island from to @xcite . heat engine based on resonant tunneling through quantum dots . a central cavity ( red ) at temperature @xmath125 is coupled via quantum dots to electronic reservoirs ( blue ) at temperature @xmath126 . light areas in the cavity and reservoirs indicate the thermal broadening of the fermi distribution . ( reprinted with permission from @xcite . copyright 2013 american physical society . ) ] we consider a setup consisting of a central cavity connected to two electrodes via quantum dots , cf . . the electrodes are assumed to be in local thermal equilibrium characterized by a fermi distribution with temperature @xmath126 and chemical potential @xmath127 . each quantum dot has a single resonant level relevant for transport . the levels are characterized by their width @xmath128 which we assume to be the same for both dots in the following and their level position @xmath129 . the difference of level positions @xmath130 characterizes the energy an electron can gain in passing through the cavity . it is different from the level spacing within each dot and can be tuned by applying a gate voltage . the central cavity is in thermal contact with a hot bath . this coupling is treated as a third terminal that injects a heat current @xmath1 but no charge in the conductor , as in . for our purposes here , the nature of the hot source needs not to be further specified . in contrast to the previous model , where the cavity was assumed out of equilibrium , here we make the simple assumption that fast relaxation processes via electron - electron and electron - phonon scattering give rise to a fermi distribution of electrons inside the cavity with temperature @xmath125 and chemical potential @xmath131 . the cavity temperature and chemical potential are determined by the conservation of charge and energy flowing into the cavity @xmath132 and @xmath133 where @xmath134\ ] ] denotes the charge current flowing between the cavity and reservoir @xmath135 while @xmath136\ ] ] is the energy current flowing between the cavity and reservoir @xmath137 . in the above expressions for the currents @xmath138 denotes the transmission function of the quantum dot levels @xcite . it is a lorentzian with width @xmath128 centered around the level position @xmath129 . the charge conservation condition can be satisfied by placing the cavity potential symmetrically between the resonant levels of the quantum dots @xmath139 and putting the chemical potentials of the reservoirs symmetrically with respect to @xmath131 , @xmath140 . we first discuss the regime @xmath141 in which an analytical solution of the problem can be obtained that provides us with an intuitive understanding of the underlying physics . afterwards , we will discuss the regime @xmath142 which we numerically find to yield the largest output power . with narrow resonances , an electron that enters the cavity from the left lead with energy @xmath143 has to gain the precise amount of energy @xmath144 in order to be able to leave the cavity into the right lead . in the steady state , rectified charge current @xmath145 and heat current @xmath1 are therefore proportional to each other with the proportionality constant being given by the ratio between electron charge and difference of level positions , @xmath146 again defining a tight - coupling limit . we now assume that no bias voltage is applied between the two electrodes . we inject a heat current @xmath1 in such a way as to keep the cavity at a given temperature @xmath125 . as a result , we find that the charge current @xmath147,\ ] ] flows in response to the temperature difference between the hot cavity and the cold electrodes . the above expression is valid in the limit where @xmath148 . from we conclude that the current and , hence , the output power grow both with the level width as well as with the difference of level positions until these quantities exceed the temperature . the output power that can be generated against an externally applied bias voltage @xmath149 is given by @xmath150 for temperatures larger than @xmath151 and @xmath144 . here , @xmath152 denotes the stopping voltage at which heat- and bias - driven current compensate each other . at the stopping voltage , the device operates reversibly and reaches carnot efficiency . at half the stopping voltage , the output power becomes maximal with @xmath153 at this point , the efficiency of heat to work conversion is given by @xmath69 in agreement with general thermodynamic bounds for time - reversal symmetric systems @xcite . ( a ) maximal power as a function of the energy splitting @xmath144 for @xmath154 and level position and width optimized for maximum power . ( b ) maximum power and ( c ) efficiency at maximum power as a function of @xmath155 . the corresponding optimal values of @xmath156 , @xmath128 and @xmath157 are shown in ( d ) . ( reprinted with permission from @xcite . copyright 2013 american physical society . ) ] we now turn to the situation of arbitrary level width and position . in this case , the integrals in and have to be evaluated numerically . the main results of our analysis are summarized in where we defined the average temperature @xmath158 and the temperature bias @xmath159 . ( a ) shows the output power as a function of the energy difference @xmath144 . for small @xmath144 the power grows quadratically with @xmath144 in agreement with the analytical result . after reaching a maximum at @xmath160 it decays exponentially for large @xmath144 due to the small number of electrons available at very high energies . ( b ) shows the maximal output power we can obtain when optimizing bias voltage , level positions and level width at the same time . the corresponding optimal parameters are depicted in ( d ) . while the optimal bias voltage grows linearly with the applied temperature bias , the optimal level position and width are nearly independent of it and given by @xmath160 and @xmath161 . the maximal power is given by @xmath162 which amounts to about for temperature bias of . at the same time , the efficiency at maximum power is nearly independent of the temperature bias and given by about @xmath163 . compared to the coulomb - blockade regime , we therefore loose a factor of two in efficiency . this is , however , more than compensated by the two orders of magnitude gain in output power . the reason for this dramatic increase in power is the combination of a strong energy dependence of the transmission functions in combination with a large number of electrons that can pass through a fully open quantum channel . we estimate that an area of @xmath164{cm^2}$ ] covered with nanoengines that each have an area of @xmath165{nm^2}$ ] can produce an output power of for a temperature bias of . swiss - cheese sandwich heat engine . a large central cavity ( red ) is connected via layers of self - assembled quantum dots ( yellow ) embedded into an insulating matrix ( transparent ) to two cold electronic reservoirs ( blue ) . ] such a high packing density can be achieved by a strongly parallelized setup shown in . here , the electrodes are connected to a large central cavity via layers of self - assembled quantum dots embedded into an insulating matrix . electrons tunnel through the quantum dots like through holes in a slice of swiss cheese driven by the thermal bias . importantly , the positions of the dots in the two layers do not have to match . apparently , the swiss - cheese sandwich heat engine outperforms a heat engine based on chaotic cavities discussed above . the reason for this lies in the fact that here we put many optimized channels in parallel while for open quantum dots there are many channels in parallel out of which only a single one is relevant for thermoelectric purposes . we remark that the heat engine based on self - assembled quantum dots offers the additional advantage that the irregular nature of the quantum dots layers can help to increase scattering of phonons at the interface and thereby reduce unwanted leakage heat currents between the hot cavity and the cold reservoirs . output power as a function of the width @xmath166 of the gaussian distribution of level positions . in ( a ) the influence of level width that differ from the optimal value are shown . similarly , in ( b ) the influence of deviations from the optimal bias voltage are shown . ( reprinted with permission from @xcite . copyright 2013 american physical society . ) ] so far , we considered the ideal situation where all dots have the same , optimal properties . in a real sample , there will be fluctuations of level positions from dot to dot . in order to investigate in how far these imperfections deteriorate the performance of the heat engine , we assume that the level positions are distributed according to a gaussian with width @xmath166 centered around the average level position @xmath129 . the output power as a function of the distribution width is shown in . as expected the power drops down as the width of the distribution increases . importantly , even for a spread of 10% the power decreases only to 90% of its optimal value , i.e. the proposal can tolerate a certain degree of imperfections . interestingly , for a given degree of fluctuations , choosing nonoptimal values of the level width or bias voltage can even increase the output power . in the previous section we analysed a heat engine based on resonant tunneling through quantum dots that could be scaled up using self - assembled quantum dots in order to deliver a macroscopic output power . in the following , we discuss a related proposal in which the quantum dot layers are replaced by quantum wells @xcite . the possibility to create high thermoelectric figures of merit by using ridged quantum wells has been pointed out @xcite . furthermore , it was shown that quantum well structures can be used for refrigeration @xcite . multilayered thermionic devices which have been proposed for refrigeration purposes are similar in design but differ in the role of the resonance @xcite . compared to heat engines based on resonant tunneling through quantum dots , heat engines based on quantum wells offer a number of potential advantages . first of all , electrons inside a quantum well have transverse degrees of freedom . this gives rise to a larger phase space for tunneling electrons . hence , quantum wells potentially allow for larger currents and output powers . second , fabricating a homogenous quantum well can be easier than growing a layer of quantum dots with identical properties ( even though we have seen above that certain fluctuations in the dot properties can be tolerated ) . finally , narrow quantum wells exhibit subband spacings of the order of several hundreds of mev @xcite which makes them ideal candidates for room temperature applications including spin based thermoelectrics @xcite . another important difference between quantum dots and wells is that the latter transmit any electron with an energy larger than the subband threshold . therefore , quantum wells are much less efficient energy filters which can potentially degrade their thermoelectric properties . schematic of the quantum - well heat engine . a cavity at temperature @xmath18 ( red ) is coupled to electronic reservoirs at temperature @xmath14 ( blue ) . the coupling is established via quantum wells with threshold energies @xmath129 . the light shading inside the quantum wells indicates that electrons with any energy larger than the threshold energy can pass through the quantum well . ( adapted with permission from @xcite . copyright 2013 iop publishing . ) ] we consider a setup similar to the one discussed in section [ ssec : resonant ] . it consists of a central cavity in local thermal equilibrium with temperature @xmath18 and chemical potential @xmath131 . it is coupled to cold electronic reservoirs with temperature @xmath14 and chemical potentials @xmath13 via quantum wells , cf . . we assume that the quantum wells are noninteracting such that charging effects can be neglected . it is an interesting topic of future research to investigate the influence of interactions on the thermoelectric properties of quantum - well heat engines . the cavity potential and temperature are determined by the conservation of charge and energy , @xmath167 and @xmath168 , where @xmath169 and @xmath170 denote the charge and energy current flowing between the cavity and reservoir @xmath82 , respectively . in addition , we have the heat current @xmath1 that is injected into the cavity from a hot thermal bath and later on serves as a drive for the heat engine . the charge and energy currents can be derived within a scattering matrix approach as @xcite , @xmath171,\ ] ] and @xmath172.\ ] ] in these expressions , @xmath173 denotes the density of states of the two - dimensional electron gas inside the quantum well with an effective electron mass @xmath174 . @xmath175 is the surface area of the quantum well . the energies associated with the electron motion in the quantum well plane and perpendicular to it are @xmath176 and @xmath177 , respectively . the transmission of the quantum wells is given by @xcite @xmath178 ^ 2/4}.\ ] ] with the ( energy - dependent ) coupling strengths @xmath179 and @xmath180 between the quantum well and reservoir @xmath82 or the cavity , respectively . the energies @xmath181 denote the subband thresholds at which a new transport channel through the quantum well opens up . for the following discussion , we consider the limit that the quantum wells are only weakly coupled to the reservoirs and the cavity , @xmath182 . in addition , we restrict ourselves to the case where only the lowest subband is relevant for transport . due to the large level spacing of narrow quantum wells , this is a reasonable approximation . a discussion of thermoelectric transport through a single quantum well that takes into account higher subbands as well can be found in ref . @xcite . under the above assumptions , the transmission function of the quantum wells simplifies to a delta peak , @xmath183 . in this limit , the integrals in and can be solved analytically and we obtain for the charge and energy currents @xmath184,\ ] ] and @xmath185,\ ] ] respectively . for simplicity , we denoted the energy of the single relevant subband in each quantum well as @xmath186 . furthermore , we introduced the integrals @xmath187 and @xmath188 with the dilogarithm @xmath189 . from we find that the energy current consists of two different contributions . the first one is proportional to the charge current while the second term breaks this proportionality . it arises due to the transverse degrees of freedom and is absent in the case of quantum dots with sharp levels , cf . [ ssec : qdresult ] . ( a ) maximal output power in units of @xmath190 as a function of the threshold energies of the two quantum wells within linear response for a symmetric configuration @xmath191 . ( b ) efficiency at maximum power in units of the carnot efficiency as a function of the threshold energies of the two quantum wells within linear response for a symmetric setup . ( c ) and ( d ) show the same as ( a ) and ( b ) but for an asymmetric device with @xmath192 . ( reprinted with permission from @xcite . copyright 2013 iop publishing.),title="fig : " ] ( a ) maximal output power in units of @xmath190 as a function of the threshold energies of the two quantum wells within linear response for a symmetric configuration @xmath191 . ( b ) efficiency at maximum power in units of the carnot efficiency as a function of the threshold energies of the two quantum wells within linear response for a symmetric setup . ( c ) and ( d ) show the same as ( a ) and ( b ) but for an asymmetric device with @xmath192 . ( reprinted with permission from @xcite . copyright 2013 iop publishing.),title="fig : " ] ( a ) maximal output power in units of @xmath190 as a function of the threshold energies of the two quantum wells within linear response for a symmetric configuration @xmath191 . ( b ) efficiency at maximum power in units of the carnot efficiency as a function of the threshold energies of the two quantum wells within linear response for a symmetric setup . ( c ) and ( d ) show the same as ( a ) and ( b ) but for an asymmetric device with @xmath192 . ( reprinted with permission from @xcite . copyright 2013 iop publishing.),title="fig : " ] ( a ) maximal output power in units of @xmath190 as a function of the threshold energies of the two quantum wells within linear response for a symmetric configuration @xmath191 . ( b ) efficiency at maximum power in units of the carnot efficiency as a function of the threshold energies of the two quantum wells within linear response for a symmetric setup . ( c ) and ( d ) show the same as ( a ) and ( b ) but for an asymmetric device with @xmath192 . ( reprinted with permission from @xcite . copyright 2013 iop publishing.),title="fig : " ] in the following , we analyze the thermoelectric properties of the quantum - well heat engine in the linear response regime . for simplicity , we assume that both quantum wells are intrinsically symmetric , i.e. we have @xmath193 . we parametrize the tunnel couplings as @xmath194 and @xmath195 where @xmath196 denotes the total coupling strength while @xmath197 describes the asymmetry between the couplings of the left and right quantum well . we , furthermore , introduce the average temperature @xmath198 and the temperature difference @xmath199 . the charge current through the heat engine to linear order in the applied bias voltage @xmath200 and in the applied temperature difference @xmath201 is given by @xmath202,\ ] ] where we introduced the auxiliary functions @xmath203 and @xmath204 from we directly infer that a finite temperature bias @xmath201 can drive a charge current in the absence of an applied bias voltage @xmath151 . the direction of the charge current can be tuned by adjusting the threshold energies @xmath205 and @xmath206 . by applying a bias voltage @xmath151 against the heat - driven current , the heat engine can perform work . the resulting output power @xmath207 becomes maximal at half the stopping voltage where it takes the value @xmath208 in order to evaluate the efficiency of heat to work conversion , again given by the ratio between output power and input heat , we need to calculate the heat current injected from the hot bath . at half the stopping voltage , it takes the form @xmath209 where the function @xmath210 satisfies @xmath211 . its complete analytical expression can be found in ref . hence , the efficiency at maximum power is given by @xmath212 in the following , we discuss the output power and efficiency in more detail , starting with a symmetric setup @xmath191 . in this case , both power and efficiency are symmetric with respect to an interchange of @xmath205 and @xmath206 , cf . . the output power takes its maximal value @xmath213 when one level is deep below the equilibrium chemical potential while the other is located at approximately @xmath214 above it . an explanation for this behaviour will be given below . the efficiency at maximum power takes its maximal value @xmath215 when both levels are above the equilibrium chemical potential and satisfy @xmath216 . however , for these parameters there is only an exponentially suppressed number of electrons that can contribute to transport such that the output power in this regime becomes vanishingly small . for level positions that optimize the output power , we find an efficiency at maximum power of about @xmath217 . hence , the quantum - well heat engine is not as efficient as a quantum - dot based setup in the limit of narrow resonances @xmath218 . the reason for this lies in the different energy - filtering properties of quantum dots and wells . quantum dots with narrow resonances transmit energies only at a single energy . hence , they reach the tight - coupling limit where heat and charge current are proportional to each other . in this situation , the efficiency at maximum power is then given by @xmath219 . quantum wells on the other hand transmit any electron with an energy larger than the threshold voltage as in this case the energy @xmath220 can be decomposed into a part associated with the motion in the plane of the well and perpendicular to it , @xmath221 . hence , high - energy electrons can be transmitted through the well if most of their energy is in the perpendicular degrees of freedom such that @xmath176 matches the resonance condition . as a result , quantum wells act as much less efficient energy filters . given the rather weak energy filtering properties of quantum wells , it is quite surprising that the efficiency at maximum power is only a factor of three smaller than for quantum dots with level widths of the order of @xmath222 the configuration that gives the largest output power , as discussed in sec . [ ssec : qdresult ] . to understand this feature , we consider the situation depicted in . the right quantum well has a threshold energy slightly above the equilibrium chemical potential . as the number of electrons with energies much larger than @xmath206 is exponentially small , it acts as a good energy filter . for the left well , the energy filtering relies on a different mechanism . electrons with energy @xmath220 can enter the cavity only if the associated state in the left reservoir is occupied , @xmath223 and , at the same time , the corresponding state in the cavity is empty , @xmath224 . this defines an energy window of about @xmath222 in which electrons are transmitted through the well . this explains why both quantum - dot and quantum - well based heat engines have similar efficiencies . we now turn to the asymmetric case @xmath225 . in this situation , the output power and efficiency are no longer symmetric with respect to an interchange of @xmath205 and @xmath206 . in fact , as can be seen in c ) and d ) , we find that power and efficiency are strongly suppressed if @xmath226 and @xmath227 for an asymmetry @xmath228 ( the roles of @xmath205 and @xmath206 are interchanged for @xmath229 ) . however , for @xmath230 and @xmath231 we find that the power can be enhanced by up to @xmath232 for an asymmetry of about @xmath233 while the efficiency at maximum power is even nearly doubled compared to the symmetric case . we finally estimate what output power can be expected for a realistic device . for a gaas based structure with an effective electron mass @xmath234 , level width of @xmath235 and asymmetry @xmath192 operating at room - temperature @xmath236{k}$ ] , we obtain a maximal output power of @xmath237{w cm^{-2}}$ ] for a temperature bias of @xmath238{k}$ ] . hence , the quantum - well heat engine is about a factor of two more powerful than the previously discussed quantum - dot heat engine . we remark that materials with higher effective mass can yield even larger output powers . similarly to the quantum - dot case , we also find that a quantum - well heat engine is robust with respect to fluctuations of the threshold energies . as can be seen in c ) the output power is hardly affected at all by fluctuations of the right threshold energy as long as @xmath239 is fulfilled . fluctuations of the left threshold energy are more crucial , but even here we find that variations of as much as @xmath222 reduce the output power by only 20% . ( a ) maximized output power ( red ) and efficiency at maximum power ( blue ) as a function the temperature bias @xmath201 . ( b ) parameters that maximize the output power . ( reprinted with permission from @xcite . copyright 2013 iop publishing.),title="fig : " ] ( a ) maximized output power ( red ) and efficiency at maximum power ( blue ) as a function the temperature bias @xmath201 . ( b ) parameters that maximize the output power . ( reprinted with permission from @xcite . copyright 2013 iop publishing.),title="fig : " ] we now turn to a discussion of the thermoelectric performance of our heat engine in the nonlinear regime . similarly , to the quantum dot heat engine discussed in sec . [ ssec : qdresult ] , we optimize the applied bias voltage , the threshold energies of the wells as well as the asymmetry @xmath240 for a given temperature bias @xmath201 . the corresponding results are shown in b ) . the optimal bias voltage grows linearly with the temperature bias . the optimal asymmetry @xmath241 is independent of @xmath201 . the optimal threshold energy for the right quantum well decreases only slightly with the temperature bias while the threshold energy of the left well should be chosen as @xmath242 independent of @xmath201 . the optimized power plotted in a ) grows quadratically with the applied temperature difference and is approximately given by @xmath243 . due to the quadratic dependence on @xmath201 we obtain the same output power for a given value of @xmath201 in the linear and nonlinear regime however , as the efficiency at maximum power grows linearly with the applied temperature bias , the device should be operated as much in the nonlinear regime as possible . in the extreme case of @xmath244 , the quantum - well heat engine reaches an efficiency at maximum power of @xmath245 , i.e. it is about as efficient as the quantum - dot heat engine but delivers twice the power . so far , we discussed three - terminal heat engines based on electronic degrees of freedom only . in the following , we turn to a different class of energy harvesters where heat is injected from a third _ bosonic _ reservoir . first , we will discuss several heat engines that are powered by phonons @xcite . we will then analyze a magnon - driven heat engine that offers an additional spin degree of freedom , thereby establishing the connection to spin caloritronics @xcite . finally , we will discuss heat engines driven by photons . these photons can either be absorbed from the electromagnetic environment @xcite or be specifically emitted by the hot source into a resonant superconducting microwave cavity that serves as a quantum bus for heat exchanged between spatially separated mesoscopic conductors @xcite . ( a ) a quantum dot ( brown ) is coupled to two electronic reservoirs ( pink ) with temperatures @xmath246 and @xmath247 as well as to a phonon bath ( green ) with temperature @xmath248 . ( reprinted with permission from @xcite . copyright 2010 american physical society . ) ( b ) aharonov - bohm ring with a quantum dot embedded in one of the arms . ( reprinted with permission from @xcite . copyright 2012 american physical society.),title="fig : " ] ( a ) a quantum dot ( brown ) is coupled to two electronic reservoirs ( pink ) with temperatures @xmath246 and @xmath247 as well as to a phonon bath ( green ) with temperature @xmath248 . ( reprinted with permission from @xcite . copyright 2010 american physical society . ) ( b ) aharonov - bohm ring with a quantum dot embedded in one of the arms . ( reprinted with permission from @xcite . copyright 2012 american physical society.),title="fig : " ] as a first example of a boson - driven heat engine , we consider a single - level quantum dot coupled to two electronic reservoirs as well as a to a local phonon mode , cf . a ) @xcite . experimentally , such a system can be realized in a single - molecule junction where generically the electron localized on the molecule couples strongly to the molecular vibrations . this electron - phonon coupling gives rise to a number of interesting phenomena such as franck - condon blockade of transport at small bias voltages @xcite , nonequilibrium phonon distributions @xcite , lasing of optical phonons @xcite , two - terminal thermoelectric effects @xcite and allows the study of fluctuation theorems in the presence of fermionic and bosonic degrees of freedom @xcite . here , we demonstrate that such a setup can also serve as a heat engine driven by the temperature bias between the electronic and phononic reservoirs @xcite . the hamiltonian of the system can be written as the sum of three parts , @xmath249 the first term describes the left , @xmath250 , and right , @xmath251 , electronic reservoir in terms of noninteracting , spinless electrons ( we neglect the spin degree of freedom in the following as it only leads to a trivial renormalization of the tunnel couplings defined below ) @xmath252 the two electronic reservoirs are at temperature @xmath36 and chemical potential @xmath13 , respectively . for later convenience , we parametrize the temperatures and chemical potentials as @xmath253 and @xmath254 . the molecule inside the junction is described by the hamiltonian @xmath255 the molecule hosts a single level relevant for transport with energy @xmath256 . it is coupled to a single dispersionless vibrational mode with energy @xmath257 via coupling strength @xmath128 . while in principle the phonon distribution could be a nonequilibrium one , in the following we consider the situation where it is thermal with temperature @xmath258 due to the coupling to the environment . finally , tunneling between the dot and the electrodes is given by @xmath259 where the tunnel matrix elements @xmath260 are related to the tunnel coupling strength as @xmath261 . for the system at hands , the charge current @xmath2 through the molecule as well as the electronic and phononic heat currents , @xmath262 and @xmath263 , can be evaluated using a nonequilibrium green s function approach that takes into account the tunneling between the dot and the leads exactly and performs a perturbative expansion in the electron - phonon coupling @xmath128 up to second order . in linear response , the currents are related to the corresponding thermodynamic forces @xmath264 , @xmath155 and @xmath265 via the onsager matrix @xmath266 the full analytic expressions for the onsager coefficients can be found in ref . @xcite . while the coefficients @xmath124 , @xmath267 and @xmath268 contain contributions that are both due to elastic and inelastic transport through the molecule , the other onsager coefficients result from inelastic processes only . the coefficient @xmath269 describes a charge current response to a temperature difference between phonons and electrons . in order to have a finite @xmath269 , both the left - right as well as the particle hole symmetry need to be broken , i.e. @xmath270 must be energy - dependent and fulfill @xmath271 . similarly , the electronic heat current due to a temperature difference between electrons and phonons , characterized by the onsager coefficient @xmath272 , requires a breaking of left - right and particle - hole symmetry . an extension of the setup we just discussed consists of an aharonov - bohm geometry with a molecular quantum dot embedded in one of the arms @xcite , cf . b ) . transport through the reference arm of the interferometer is described by the new contribution to the hamiltonian @xmath273 where @xmath274 denotes the aharonov - bohm flux through the ring . due to the aharonov - bohm flux , the onsager matrix becomes @xmath274-dependent , @xmath275 , such that the onsager relations @xcite are satisfied . the analytic expressions for the onsager coefficients that are given in ref . @xcite exhibit three different types of flux dependence . first , there are contributions proportional to @xmath276 that arise from interference contributions . second , there are terms proportional to @xmath277 due to contributions from time - reversed paths . third , there are @xmath278 contributions that emerge from the coupling to the phonons . while the former two contributions are even in the flux , the latter one is odd . the odd contributions in the off - diagonal onsager coefficients can potentially lead to a large efficiency at maximum power @xcite that overcomes the linear response limit @xmath219 that exists for time - reversal symmetric systems @xcite . however , for the system at hands , these efficiency bounds have not yet been investigated . the idea of phonon - driven thermoelectric transport in multi - terminal devices was further discussed in the context of phonon - assisted hopping @xcite . furthermore , a p - i - n - diode structure that drives a charge current by harvesting energy from a hot phonon source was discussed in ref . it was theoretically estimated that such a device based on a bi@xmath279te@xmath280/si superlattice can have a figure of merit larger than 1 when operated at room temperature , thus making it a promising candidate for energy harvesting applications . we now turn to a quantum - dot heat engine that is driven by spin waves from a ferromagnetic insulator @xcite , cf . . compared to the phonon - driven setups that we discussed in the previous section , the magnon - driven heat engine offers a number of advantages . first of all , magnons are potentially easier to control than phonons . as phonons exist in any material , it is hard to avoid leakage heat currents from the hot phonon bath to the cold electronic reservoirs . magnons , in contrast , exist only in magnetic materials and couple via short - range exchange interactions which facilitates coupling them only to the quantum dot degrees of freedom . another advantage of the magnon - heat engine is that it does not rely on energy - dependent tunnel couplings . instead , here the necessary asymmetry between electrons and holes is introduced into the system by the spin - dependence of the tunnel barriers . finally , the magnon - driven heat engine provides an example of a spin caloritronic heat engine @xcite that allows to drive spin - polarized charge currents as well as pure spin currents by thermal gradients . alternative spintronic heat engines based on nanowires with domain walls have been proposed in ref . @xcite . a related setup has been discussed in ref . @xcite where a system consisting of a quantum dot coupled to ferromagnetic electrodes with spin waves has been taken into account . keeping the magnons inside the ferromagnets at a different temperature than the electrons , e.g. , by microwave excitation of spin precession , gives rise to spin - polarized charge currents . while this effect was termed magnon - assisted transport in ref . @xcite in analogy to phonon - assisted tunneling , it can also be viewed as a three - terminal thermoelectric device . schematic sketch of a magnon - driven quantum - dot heat engine . a single - level quantum dot ( blue ) is tunnel - coupled to two ferromagnetic metals ( yellow ) . in addition , it is also exchange - coupled to a ferromagnetic insulator ( green ) serving as a source of spin waves . ( reprinted with permission from @xcite . copyright 2012 iop publishing . ) ] we consider a single - level quantum dot in the coulomb - blockade regime tunnel coupled to two ferromagnetic electrodes @xmath12 at temperature @xmath281 as well as exchange coupled to a ferromagnetic insulator that serves as a source of magnons with temperature @xmath282 . the ferromagnetic electrodes are modeled in the spirit of a stoner model as a noninteracting electron gas with a constant but spin - dependent density of states @xmath283 . it is related to the spin polarization @xmath284 that varies between @xmath285 for a normal metal and @xmath286 for a half - metallic ferromagnet . in the following , we assume identical polarizations for the two leads , @xmath287 . the ferromagnetic insulator is modeled as a heisenberg chain of exchange - coupled spins . using a holstein - primakoff transformation , it can be described in terms of bosonic operators that create and annihilate magnons . at low temperatures where the average magnon number is small , the ferromagnetic insulator behaves as a noninteracting magnon gas with dispersion relation @xmath288 . the quantum dot has a single spin - split level with energy @xmath289 . the zeeman splitting @xmath290 due to an externally applied magnetic field determines the energy of the magnons that the heat engine can harvest . the coulomb energy @xmath10 that is required to occupy the dot with two electrons at the same time is assumed to be infinite . we remark that taking into account a finite value of @xmath10 does not give rise to qualitatively different results . tunneling between the dot and the ferromagnetic electrodes is characterized by the spin - dependent tunnel couplings @xmath291 . for later convenience we also introduce the total tunnel coupling strength @xmath292 . the coupling between the dot and the ferromagnetic insulator is due to an exchange interaction with spectral weight @xmath293 that flips the spin of the dot and emits or absorbs a magnon in the insulator . as in the following we only need the spectral weight evaluated at the zeeman splitting of the dot , we will omit its energy dependence and write @xmath294 . transport through the system is described via a standard master equation approach that integrates out the noninteracting fermionic and bosonic degrees of freedom in the reservoirs and characterizes the quantum dot by its reduced density matrix . within this approach , the spin - resolved electron and magnon currents can then be evaluated in a straightforward way . we count particle currents as positive when they flow from the reservoir into the dot . ( a ) spin - resolved electron and magnon currents as a function of the magnon temperature for parallel and antiparallel geometry . parameters are @xmath295 , @xmath296 , @xmath297 and @xmath298 . ( b ) power and efficiency as a function of applied bias voltage for different polarizations and other parameters as in the left panel . ( reprinted with permission from @xcite . copyright 2012 iop publishing.),title="fig : " ] ( a ) spin - resolved electron and magnon currents as a function of the magnon temperature for parallel and antiparallel geometry . parameters are @xmath295 , @xmath296 , @xmath297 and @xmath298 . ( b ) power and efficiency as a function of applied bias voltage for different polarizations and other parameters as in the left panel . ( reprinted with permission from @xcite . copyright 2012 iop publishing.),title="fig : " ] we now turn to a discussion of the thermoelectric performance of the magnon - driven heat engine . shows the spin - resolved electron and magnon currents that flow in response to a temperature difference between electrons and magnons . for parallely magnetized electrodes , we find that as soon as @xmath299 , we have finite electron and magnon currents flowing through the dot . for any temperature bias , we observe that the currents of spin up and spin down electrons have equal magnitude but opposite sign , @xmath300 . hence , the total charge current @xmath301 vanishes while the spin current @xmath302 is finite , i.e. a pure spin current is generated . the physical picture behind this is that spin up electrons tunnel into the dot , absorb a magnon to flip their spin and then tunnel out again . on average , for each spin up electron entering the dot through a given tunnel barrier , there is a spin down electron leaving the dot through the same barrier such that @xmath303 and @xmath304 . for antiparallel magnetizations of the ferromagnetic electrodes , the magnitudes of @xmath305 and @xmath306 are no longer equal to each other such that now a finite , spin - polarized charge current flows through the dot . it arises as spin up electrons preferably tunnel in from the left electrode whereas spin down electrons preferably tunnel out to the right electrode due to the spin - dependent tunnel couplings . in the following , we focus on the antiparallel case and discuss the output power @xmath60 and efficiency @xmath61 of heat to work conversion when an external bias voltage @xmath4 is applied against the thermally driven current . shows the output power as a function of @xmath4 for different polarizations . it grows from @xmath307 at vanishing bias to a maximal value and then drops down again to zero at the stopping voltage . furthermore , it grows as the polarization is increased because the energy filtering properties of the heat engine become more pronounced , reaching the tight - coupling limit for @xmath308 . the efficiency @xmath61 shown in exhibits qualitatively different behaviour for @xmath309 and @xmath308 . in the former case , it grows from @xmath310 at @xmath311 to a finite value and goes down to zero at the stopping voltage . we remark that the maximal efficiency in general occurs for a different bias voltage than the maximal power . for @xmath308 , the efficiency just grows linearly with the applied voltage and reaches the carnot efficiency at the stopping voltage as expected in the tight - coupling limit . the corresponding efficiency at maximum power @xmath312 is given by @xmath69 in the linear response regime while it satisfies @xmath313 in the nonlinear regime . apart from phonons and magnons , one can also make use of photons to drive mesoscopic heat engines . heat transfer due to photons in nanoscale circuits has been studied theoretically @xcite as well as been experimentally observed @xcite . here , we present two different examples of photon harvesting . in the first case , a quantum - dot based setup is used to harvest microwave photons from the electromagnetic environment @xcite . a similar proposal used to harvest energy from visible light was analyzed in ref . @xcite . in the second example , we consider a system of two double quantum dots connected via a superconducting microwave cavity . the latter serves as a quantum bus for heat flow between the quantum dots . both types of setup offer the advantage of spatially separating the hot and the cold part of the heat engine . this potentially reduces leakage heat currents due to substrate phonons and can therefore help to achieve highly efficient heat engines . ( a ) heat engine that harvests energy from the electromagnetic environment . a quantum dot structure is coupled to two electronic reservoirs at temperature @xmath314 . the circuit connecting the two leads contains an impedance @xmath315 at the environment temperature @xmath316 . realizations of the heat engine with a single dot and a double dot are shown in ( b ) and ( c ) , respectively . ( reprinted with permission from @xcite . copyright 2012 american physical society . ) ] we consider a mesoscopic heat engine as discussed in ref . @xcite that consists of one or two quantum dots coupled to electronic reservoirs at temperature @xmath314 . the electronic reservoirs are connected to each other via an external circuit with impedance @xmath315 that is kept at temperature @xmath316 . we assume that the tunnel coupling between the dots and the leads is weak , such that a lowest order description of tunneling is valid . we , furthermore , assume that the relaxation time of the electromagnetic environment is much shorter than the average time between subsequent tunneling events . in this situation , transport through the system can be described by @xmath317 theory @xcite . the rate for an electron transition from system @xmath81 to @xmath137 is then given by @xmath318 here , @xmath74 denotes the tunnel matrix element for a transition from @xmath81 to @xmath137 . the density of states for electrons @xmath319 is given by @xmath320 for a quantum dot with discrete levels and by @xmath321 for a metallic island or an electrode where @xmath322 denotes the density of states at the fermi energy . similarly , the density of states for holes @xmath323 is given by @xmath320 for quantum dots and @xmath324 $ ] for metallic systems . finally , @xmath325 denotes the probability density that the tunneling electron exchanges the energy @xmath326 with the environment . for a simple tunnel junction without any embedded quantum dot system , there is no directed charge current in the absence of an applied bias voltage as the rectification of thermal fluctuations requires the presence of a nonlinearity in the heat engine . the simplest way to achieve such a nonlinearity is to add a single quantum dot or metallic island into the junction , cf . b ) . in the limit of strong coulomb blockade , the quantum dot or metallic island is either empty or occupied with a single excess electron with energy @xmath220 . the corresponding occupation probabilities @xmath327 and @xmath328 in the stationary state follow from a simple rate equation @xmath329 with @xmath330 . here , @xmath331 denotes the transition rate for an electron tunneling left ( right ) through barrier @xmath82 evaluated according to . in the following , we assume that the right tunnel barrier has a much larger transition rate than the left barrier . we , furthermore , assume that its capacitance is larger than that of the left junction such that it effectively decouples from the environment . under these conditions , we obtain a directed charge current that is simply given by @xmath332 and thus finite even in the absence of an applied bias voltage . we remark that the above expression also describes the current through a junction between two quantum dots with energy levels @xmath205 and @xmath206 such that @xmath333 if the appropriate definitions for the electron and hole density of states are inserted in and the dot is preferably singly occupied . by applying a bias voltage @xmath4 against the thermally driven current , we can generate a finite output power . as for any coulomb - blockade heat engine , the output power is limited by the operation temperature . for a device operating at a temperature of about , one obtains a power in the femtowatt range . the efficiency of the heat engine depends strongly on the junction type . for a junction between two quantum dots with discrete energy levels , the tight - coupling limit is reached as every electron that is transferred between the dots has to absorb a photon with energy @xmath220 . consequently , the heat engine can reach carnot efficiency and achieve a large efficiency at maximum power . for junctions between a discrete level and a metal or between two metals , the situation is different . now photons of different energies can be absorbed , leading to a significantly lower efficiency . interestingly , the efficiency depends on whether the environment is hotter or colder than the electronic system . for a cold electron system , the sharp fermi functions help to achieve unidirectional transport against the bias voltage such that decent efficiencies can be achieved . in contrast , for a hot electron system the thermally smeared fermi function give rise to a current flow with the bias voltage such that only small efficiencies are possible . as a second example of a photon - driven heat engine we now consider a system where a superconducting microwave cavity connects two mesoscopic conductors , cf . . such hybrid structures that allow the investigation the interplay of light and matter at the nansocale have recently gained a lot of interest both theoretically @xcite as well as experimentally @xcite in the context of circuit quantum electrodynamics . similar to the previously discussed energy harvester , the hybrid microwave cavity heat engine @xcite also allows to separate the hot and the cold part of the engine by a macroscopic distance of the order of a centimeter . this suppresses leakage heat currents far more efficiently than vacuum nanogaps that are only a few nanometer wide @xcite . in addition , the heat engine has the advantage that the cavity helps in efficiently transferring photons from the hot to the cold side whereas otherwise photons just get randomly emitted into all directions and can be lost for the energy harvesting purpose . ( a ) sketch of the microwave cavity heat engine . double quantum dots ( yellow ) are coupled to hot ( red ) or cold ( blue ) electronic reservoirs . the level positions of the dots are tunable via voltages applied to gates ( green ) . the two double dots are connected via a superconducting microwave cavity ( gray ) . ( b ) operation scheme of the heat engine . electrons enter the excited state of the hot dqd and leave the ground state after emitting a photon into the cavity . at the cold dot , electrons enter the ground state preferrably from the left and leave the excited state to the right after absorbing a photon from the cavity . as a result of this asymmetry , a directed charge current flows through the cold dot . ( reprinted with permission from @xcite . copyright 2014 american physical society.),title="fig : " ] ( a ) sketch of the microwave cavity heat engine . double quantum dots ( yellow ) are coupled to hot ( red ) or cold ( blue ) electronic reservoirs . the level positions of the dots are tunable via voltages applied to gates ( green ) . the two double dots are connected via a superconducting microwave cavity ( gray ) . ( b ) operation scheme of the heat engine . electrons enter the excited state of the hot dqd and leave the ground state after emitting a photon into the cavity . at the cold dot , electrons enter the ground state preferrably from the left and leave the excited state to the right after absorbing a photon from the cavity . as a result of this asymmetry , a directed charge current flows through the cold dot . ( reprinted with permission from @xcite . copyright 2014 american physical society.),title="fig : " ] the heat engine consists of two double quantum dots @xmath77 , connected to each other via a microwave cavity . due to strong coulomb interactions , the double dots can be either empty or occupied with a single electron . in the following , we neglect the electron spin as it only renormalizes the tunnel couplings in a trivial way . the eigenstates of the singly - occupied dot can be expressed as linear combinations of the left @xmath334 and right @xmath335 double quantum dot states as @xmath336 the mixing angle @xmath337 that characterizes the hybridization of the dot levels depends on the level positions and the interdot tunnel coupling . it can be controlled by gate voltages applied to the double quantum dot . the energy difference between the eigenstates @xmath338 and @xmath339 can be tuned independently of @xmath337 such that it matches the resonance frequency @xmath340 of the microwave cavity . the coupling between the double dots and the cavity with coupling strength @xmath341 then induces transitions between the ground and excited state of the dot accompanied by the absorption or emission of a microwave photon in the cavity . each quantum dot is furthermore tunnel coupled to two electronic reservoirs @xmath342 at temperature @xmath343 and chemical potential @xmath344 with tunnel coupling strength @xmath345 . in the following , we assume for simplicity symmetric tunnel couplings for each double quantum dot , @xmath346 . we remark that the actual transition rates of the system depend not only on the tunnel coupling @xmath347 but also on the mixing angle @xmath337 that can effectively break the left - right symmetry within a double quantum dot . we describe transport through the system again within a generalized master equation approach that integrates out the noninteracting electronic reservoirs and characterizes the remaining quantum system consisting of the two double quantum dots and the microwave cavity by its reduced density matrix . in the limit of strong electron - photon coupling , @xmath348 , coherences between different eigenstates of the quantum system can be neglected to lowest order in the tunnel coupling such that a simple rate equation description holds . it is this limit that we will consider first before taking into account the effects of finite electron - photon coupling as well as of relaxation and dephasing within the double quantum dots . before discussing the thermoelectric performance of the heat engine , we elucidate the basic operation principle that drives a charge current through the cold dot in the absence of any applied bias voltage . to this end , we choose a situation where the mixing angle of the hot dot is @xmath349 . in this case , the eigenstates are the bonding and antibonding states . as they do not break left - right symmetry , there is no directed charge current through the double dot . for the cold double dot , we choose the mixing angle @xmath350 such that the ground state couples more strongly to one electrode while the excited states couples more strongly to the other one . electrons then tunnel into the ground states of the cold double dot from one side , absorb a photon from the cavity to get into the excited state and then tunnel out to the other side . as there is a net flow of heat ( and hence microwave photons ) from the hot to the cold double dot , we thus obtain a finite directed charge current through the cold double dot . we remark that the direction of the charge current depends on the mixing angle as it determines the asymmetry of eigenstates . unlike the energy dependence of tunnel couplings that is required for the heat engines discussed in sec . [ sec : cbfluct ] , the asymmetry can therefore be changed in a controlled way by manipulating @xmath351 via gate voltages . thermoelectric performance of the cavity heat engine . ( a ) heat - driven charge current in units of @xmath352 as a function of the temperature bias for different values of the mixing angle @xmath351 . the analytic results ( dashed lines ) compare well with the full numerics ( solid lines ) . parameters are @xmath353 and @xmath354 . ( b ) output power in units of @xmath355 and ( c ) efficiency over carnot efficiency as a function of the applied bias voltage . the analytical results for the power ( dashed lines ) agree well with the numerics ( solid lines ) . parameters are @xmath356 and @xmath357 . ( d ) efficiency at maximum power and ( e ) maximum power as a function of the temperature bias . the optimal values of the frequency @xmath257 and voltage @xmath358 are shown in ( f ) . ( reprinted with permission from @xcite . copyright 2014 american physical society . ) ] the thermoelectric performance of the microwave cavity heat engine is summarized in . in panel a ) we show that charge current @xmath359 that flows through double quantum dot 2 without an applied bias voltage . it grows linearly with the applied temperature bias and saturates for @xmath360 . for @xmath361 , the tight - coupling limit is reached where one electron is transferred through the cold double dot for each photon that is transferred through the cavity . as a consequence , we find that the charge current @xmath359 is proportional to the heat current @xmath362 flowing out of the hot reservoirs , @xmath363 , where the ratio of these two currents is determined only by the electron charge and the photon energy . in the regime where the cavity is mostly empty , i.e. , @xmath364 the charge current is approximately given by the analytic expression @xmath365 that agrees well with the full numerical result , cf . the dashed lines in a ) . as can be inferred from , the current becomes largest when @xmath366 . we will therefore focus on this limit in the following discussion . the output power against an externally applied bias voltage @xmath358 is depicted in b ) . it shows a typical behaviour by growing from @xmath307 at @xmath367 to a maximal value and dropping to zero at the stopping voltage . for @xmath368 it is well approximated by @xmath369.}\ ] ] the efficiency in general shows behaviour qualitatively similar to the output power . only in the tight - coupling limit it instead grows linearly with the applied bias voltage and reaches carnot efficiency at the stopping voltage . the results for the maximum power @xmath370 and the associated efficiency at maximum power @xmath312 together with the optimizing values of @xmath257 and @xmath358 are shown in d)-f ) . in the tight - coupling limit , @xmath312 grows as @xmath69 in the linear response regime . in the nonlinear regime , it grows more quickly and reaches @xmath70 for @xmath371 . for @xmath372 , the efficiency at maximum power is slightly reduced but exhibits a qualitatively similar behaviour . influence of electronic relaxation and dephasing rates on the heat - driven current . in the optimal case , the current takes the value @xmath373 . ( reprinted with permission from @xcite . copyright 2014 american physical society . ) ] in current experiments on circuit quantum electrodynamics relaxation and dephasing within the double quantum dots are the major obstacles in reaching the strong coupling limit . when taking into account a finite value of the electron - photon coupling @xmath341 together with relaxation and dephasing processes with rates @xmath374 and @xmath375 , respectively , we find that the charge current through the system is reduced to @xmath376 the corresponding current suppression is shown in as a function of @xmath374 and @xmath375 . even when taking these imperfections into account , we estimate that for realistic system parameters one can achieve charge currents of the order of @xmath377{pa}$ ] clearly within the reach of current experiments . in this review we discussed different types of thermoelectric energy harvesters based on multi - terminal quantum dot and well setups . we presented nano heat engines based on coulomb - coupled conductors . in particular , we looked at a setup based on coulomb - blockade quantum dots that turned out to be ideally efficient but only gives small currents and powers . a similar system based on chaotic cavities was shown to yield much larger currents but had a significantly reduced efficiency . the optimal heat engine that has both a large output power in addition to a good efficiency was then found to be based on resonant tunneling through quantum dots or quantum wells . in the second part of this review , we discussed different types of heat engines that are powered by absorbing bosons from the environment . we analyzed heat engines based on molecular junctions where electrons couple to phononic degrees of freedom from which energy is harvested . we then turned to a related setup that is driven by magnons from a ferromagnetic insulator . this type of setup allows one to drive pure spin currents as well as spin - polarized charge currents and therefore makes connection to the emerging field of spin caloritronics . finally , we discussed setups that harvest microwave photons from the electromagnetic environment or from a superconducting microwave cavity thus providing a bridge between energy harvesting and circuit quantum electrodynamics . the field of energy harvesting with mesoscopic conductors of course still faces a number of open questions . while most of the setups presented in this review were described in terms of minimal models , it would be interesting to have more realistic descriptions that include , e.g. , charging effects in quantum wells or provide a microscopic model of heat injection into the central region of the resonant tunneling heat engines . furthermore , the influence of phonons has to be better understood as they not only degrade the efficiency but also make it hard to maintain a given temperature bias across the device . in particular , it would be desirable to investigate systems where an analytic treatment of their influence can be made and to invent setups where they can be controlled in a systematic way . another interesting question addresses the thermoelectric performance of multi - terminal heat engines with broken time - reversal symmetry . theoretically , these devices allow to get arbitrarily close to carnot efficiency at finite output power . but how close can any physical realization actually get ? finally , the field of mesoscopic energy harvesting will greatly benefit from experiments that demonstrate that the theoretical ideas presented in this review can also be put into practice . above all , we are deeply indebted to the support and inspiration of the late markus bttiker during many years of collaborations that led to our works presented in this review . we furthermore acknowledge numerous fruitful discussions with c. bergenfeldt , h. buhmann , c. flindt , f. hartmann , p. jacquod , a. m. lunde , r. lpez , i. martin , l. molenkamp , m. moskalets , p. samuelsson , d. snchez , j. splettstoesser , h. thierschmann , r. whitney and l. worschech . we also acknowledge financial support from the european strep project nanopower , the swiss nsf via the nccr qsit , the spanish micinn juan de la cierva program and mat2011 - 24331 , the cost action mp1209 and the nsf grant dmr-0844899 . 100 url # 1#1urlprefix[2][]#2 shakouri a 2011 _ annu . rev . mater . res . _ * 41 * 399431 http://dx.doi.org/10.1146/annurev-matsci-062910-100445 staring a a m , molenkamp l w , alphenaar b w , van houten h , buyk o j a , mabesoone m a a , beenakker c w j and foxon c t 1993 _ europhysics letters ( epl ) _ * 22 * 5762 issn 0295 - 5075 , 1286 - 4854 http://iopscience.iop.org/0295-5075/22/1/011 svensson s f , hoffmann e a , nakpathomkun n , wu p m , xu h q , nilsson h a , snchez d , kashcheyevs v and linke h 2013 _ new j. phys . _ * 15 * 105011 issn 1367 - 2630 http://iopscience.iop.org/1367-2630/15/10/105011 koch j , von oppen f , oreg y and sela e 2004 _ phys . b _ * 70 * 195107 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/abstract/prb/v70/e195107 kubala b , knig j and pekola j 2008 _ phys . lett . _ * 100 * 066801 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/abstract/prl/v100/e066801 wirkowicz r , wierzbicki m and barna j 2009 _ phys . b _ * 80 * 195409 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/physrevb.80.195409 snchez r , lpez r , snchez d and bttiker m 2010 _ phys . _ * 104 * 076801 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/physrevlett.104.076801 zhang y , dicarlo l , mcclure d t , yamamoto m , tarucha s , marcus c m , hanson m p and gossard a c 2007 _ phys . lett . _ * 99 * 036603 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/physrevlett.99.036603 braggio a , knig j and fazio r 2006 _ phys . _ * 96 * 026805 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/abstract/prl/v96/e026805 flindt c , novotn t , braggio a , sassetti m and jauho a p 2008 _ phys . lett . _ * 100 * 150601 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/abstract/prl/v100/e150601 esposito m , harbola u and mukamel s 2009 _ rev . mod . phys . _ * 81 * 1665 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/revmodphys.81.1665 snchez r , platero g and brandes t 2008 _ phys . rev . b _ * 78 * 125308 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/physrevb.78.125308 van wees b j , kouwenhoven l p , willems e m m , harmans c j p m , mooij j e , van houten h , beenakker c w j , williamson j g and foxon c t 1991 _ phys . b _ * 43 * 1243112453 http://link.aps.org/doi/10.1103/physrevb.43.12431 koch j , von oppen f and andreev a v 2006 _ phys . rev . b _ * 74 * 205438 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/abstract/prb/v74/e205438 koch j , semmelhack m , von oppen f and nitzan a 2006 _ phys b _ * 73 * 155306 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/abstract/prb/v73/e155306 ojanen t and jauho a p 2008 _ phys . lett . _ * 100 * 155902 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/physrevlett.100.155902 ruokola t , ojanen t and jauho a p 2009 _ phys . b _ * 79 * 144306 copyright ( c ) 2010 the american physical society ; please report any problems to [email protected] http://link.aps.org/doi/10.1103/physrevb.79.144306 ingold g l and nazarov y v 1992 charge tunneling rates in ultrasmall junctions _ single charge tunneling _ ( _ nato asi series b _ vol 294 ) ed grabert h and devoret m h ( new york : plenum press ) pp pp | we review recent theoretical work on thermoelectric energy harvesting in multi - terminal quantum - dot setups .
we first discuss several examples of nanoscale heat engines based on coulomb - coupled conductors .
in particular , we focus on quantum dots in the coulomb - blockade regime , chaotic cavities and resonant tunneling through quantum dots and wells .
we then turn towards quantum - dot heat engines that are driven by bosonic degrees of freedom such as phonons , magnons and microwave photons .
these systems provide interesting connections to spin caloritronics and circuit quantum electrodynamics . |
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the first detection of the transit of an exoplanet in front of its parent star ( @xcite ; @xcite ) opened a new avenue to determine the characteristics of these exotic worlds . for all but the most eccentric cases , approximately half - an - orbit after their transits these planets pass behind their star along our line of sight allowing their thermal flux to be measured in the infrared . the first detections of an exoplanet s thermal emission ( @xcite ; @xcite ) came from observations in space with spitzer using the infrared array camera ( irac ; @xcite ) . since then the vast majority of such measurements have been made using spitzer at wavelengths longer than 3 @xmath10 , and thus longwards of the blackbody peak of these `` hot '' exoplanets . recent observations have extended secondary eclipse detections into the near - infrared ; the first detection was from space with nicmos on the hubble space telescope ( @xcite at @xmath112 @xmath10 ) . more recently , near - infrared detections have been achieved from the ground ; the first of these detections include a @xmath116@xmath1 detection in k - band of tres-3b using the william herschel telescope @xcite , a @xmath114@xmath1 detection in z-band of ogle - tr-56b using magellan and the very large telescope ( vlt ; @xcite ) , and a @xmath115@xmath1 detection at @xmath112.1 @xmath10 of corot-1b also with the vlt @xcite . thermal emission measurements in the near - infrared are crucial to our understanding of these planets atmospheres , as they allow us to constrain hot jupiters thermal emission near their blackbody peaks . the combination of spitzer / irac and near - infrared thermal emission measurements allows us to constrain the temperature - pressure profiles of these planets atmospheres over a range of pressures @xcite , better estimate the bolometric luminosity of these planets dayside emission , and thus contributes to a more complete understanding of how these planets transport heat from the day to nightside at a variety of depths and pressures in their atmospheres @xcite . the transiting hot jupiter tres-2b orbits a g0 v star with a period of @xmath112.47 @xmath12 @xcite . according to the @xcite theory this places tres-2b marginally in the hottest , mostly highly irradiated class ( the pm - class ) of hot jupiters and close to the dividing line between this hottest class and the merely warm class of hot jupiters ( the pl - class ) . thus tres-2b could be a key object to refine the dividing line between these two classes , and indicate the physical cause of this demarcation , or reveal whether this divide even exists . recently @xcite used spitzer / irac to measure the depth of the secondary eclipse of tres-2b in the four irac bands . their best - fit eclipses are consistent with a circular orbit , and collectively they are able to place a 3@xmath1 limit on the eccentricity , @xmath2 , and argument of periastron , @xmath3 , of @xmath4@xmath2cos@xmath3@xmath4 @xmath6 0.0036 . their best - fit eclipses at 3.6 , 5.8 and 8.0 @xmath10 are well - fit by a blackbody . at 4.5 @xmath10 they detect excess emission , in agreement with the theory of several researchers ( @xcite ) that predicts such excess due to water emission , rather than absorption , at this wavelength due to a temperature inversion in the atmosphere . one - dimensional radiative - equilibrium models for hot jupiter planets generally show that the atmospheric opacity is dominated by water vapor , which is especially high in the mid - infrared , but has prominent windows ( the jhk bands ) in the near infrared @xcite . one can probe more deeply , to gas at higher pressure , in these opacity windows . models without temperature inversions feature strong emission in the jhk bands , since one sees down to the hotter gas . models with temperature inversions , since they feature a relatively hotter upper atmosphere and relatively cooler lower atmosphere , yield weaker emission in the near - ir ( jhk ) , but stronger emission in the mid - infrared @xcite . near - infrared thermal emission measurements should thus be useful to determine whether tres-2b does or does not harbour a temperature inversion . owing to its high irradiation , with an incident flux of @xmath11@xmath13@xmath14@xmath15 @xmath16@xmath17@xmath18 , and favourable planet - to - star radius ratio ( @[email protected] ) , we included tres-2b in our program observing the secondary eclipses of some of the hottest of the hot jupiters from the ground . here we present ks - band observations bracketing tres-2b s secondary eclipse using the wide - field infrared camera ( wircam ) on the canada - france - hawaii telescope ( cfht ) . we report a 5@xmath1 detection of its thermal emission . we observed tres-2 ( @xmath9=9.846 ) with wircam @xcite on cfht on 2009 june 10 under photometric conditions . the observations lasted for @xmath113.5 hours evenly bracketing the predicted secondary eclipse of this hot jupiter assuming it has a circular orbit . numerous reference stars were also observed in the 21x21 arcmin field of view of wircam . to minimize the impact of flat field errors , intrapixel variations and to keep the flux of the target star well below detector saturation , we defocused the telescope to 1.5 mm , such that the flux of our target was spread over a ring @xmath1120 pixels in diameter ( 6 ) on our array . we observed tres-2 in `` stare '' mode on cfht where the target star is observed continuously without dithering . 5-second exposures were used to avoid saturation . to increase the observing efficiency we acquired a series of data - cubes each containing twelve 5-second exposures . the twelve exposure data - cube is the maximum number of exposures allowed in a guide - cube in queue mode at cfht . to counteract drifts in the position of the stars positions on the wircam chips , which we had noticed in earlier wircam observations of secondary eclipses @xcite , we initiated a corrective guiding `` bump '' before every image cube to recenter the stellar point - spread - function as near as possible to the original pixels at the start of the observation . the effective duty cycle after accounting for readout and for saving exposures was 43% . the images were preprocessed with the iiwi pipeline . this pipeline includes the following steps : applying a non - linearity flux correction , removing bad and saturated pixels , dark subtraction , flat - fielding , sky subtraction , zero - point calibration and a rough astrometry determination . we sky subtract our data by constructing a normalized sky frame built by taking the median of a stack of source - masked and background - normalized on - sky images . our on - sky images consist of 15 dithered in - focus images observed before and after the on - target sequence . for each on - target image the normalized sky frame is scaled to the target median background level and then subtracted . we performed aperture photometry on our target star and all unsaturated , reasonably bright reference stars on the wircam array . we used a circular aperture with a radius of 12.5 pixels . we tested larger and smaller apertures in increments of 0.5 pixels , and confirmed that this size of aperture returned optimal photometry . the residual background was estimated using an annulus with an inner radius of 21 pixels , and an outer radius of 30 pixels ; a few different sizes of sky annuli were tested , and it was found that the accuracy of the resulting photometry was not particularly sensitive to the size of the sky aperture . as tres-2 has a nearby reference star ( 0.17 separation ) that falls in our sky aperture , we exclude a slice of the annulus that falls near this reference star to avoid any bias in background determination to -45@xmath20 degrees as measured from due north towards the east are excluded from our annulus . ] . during our observations , despite the aforementioned corrective `` bump '' to keep the centroid of our stellar point - spread - function ( psf ) as steady as possible , our target star and the rest of the stars on our array displayed high frequency shifts in position ( figure [ figshifts ] ) . to ensure that the apertures for our photometry were centered in the middle of the stellar psfs , we used a center - of - mass calculation , with pixel flux substituted for mass , to determine the x and y center of our defocused stellar rings for each one of our target and reference star apertures . the light curves for our target and reference stars following our aperture photometry displayed significant , systematic variations in intensity ( see the top panel of figure [ figtres2brefstars ] ) , possibly due to changes in atmospheric transmission , seeing and airmass , guiding errors and/or other effects . the target light curve was then corrected for these systematic variations by normalizing its flux to the 11 reference stars that show the smallest deviation from the target star outside of the expected occultation . reference stars that showed significant deviations in - eclipse from that of the target star and other reference stars , as indicated by a much larger root - mean - square in - eclipse than out - of - eclipse due to intermittent systematic effects for instance , were also excluded . for the reference stars that were chosen for the comparison to our target star , the flux of each one of these star was divided by its median - value , and then an average reference star light curve was produced by taking the mean of the lightcurves of these median - corrected reference stars . the target flux was then normalized by this mean reference star light curve . ccc 1 & j19072977 + 4918354 & 10.294 + 2 & j19065501 + 4916195 & 10.737 + 3 & j19071365 + 4912041 & 11.270 + 4 & j19065809 + 4916315 & 9.875 + 5 & j19070093 + 4917323 & 11.337 + 6 & j19074435 + 4915418 & 10.766 + 7 & j19071824 + 4916526 & 11.239 + 8 & j19073380 + 4916035 & 10.712 + 9 & j19071955 + 4911176 & 11.514 + 10 & j19075629 + 4923281 & 9.671 + 11 & j19065548 + 4925404 & 11.454 + [ tablestars ] figure [ figtres2bfullframearray ] marks the 11 reference stars used to correct the flux of our target ; the 2mass identifiers of the reference stars are given in table [ tablestars ] . note that the majority of the reference stars with the smallest out of occultation residuals to our target star are on the same chip as our target , despite the fact that there are other reference stars on other chips closer in magnitude to our target . we believe this is due to the differential electronic response of the different wircam chips , and have noticed this same effect with other wircam observations of other hot jupiter secondary eclipses @xcite . following this correction we noticed that the flux of our target and reference stars displayed near - linear correlations with the x or y position of the centroid of the stellar psf on the chip . given the aforementioned high frequecy of these shifts ( fig . [ figshifts ] ) this suggests that any leftover trend with position and the flux of the star was instrumental in origin . thus these near - linear trends ( figure [ figshiftscorrection ] top panels ) were removed from the data for both the target and reference stars by performing a fit to the x and y position of the centroid of the psf and the normalized flux for the out - of - eclipse photometry . we fit the out - of - eclipse photometric flux to the x and y position of the centroid of the psf with a function of the following form : @xmath21 where @xmath22 , @xmath23 , and @xmath24 are constants . we then apply this correction to both the in - eclipse and out - of - eclipse photometry . the out - of - eclipse photometric data prior to and following this correction are displayed in figure [ figshiftscorrection ] ( bottom panels ) . no other trends that were correlated with instrumental parameters were found . by correcting the flux of our target with these 11reference stars and by removing the above correlation with the x / y position on the chip the point - to - point scatter of our data outside occultation improves from a root - mean - square ( rms ) of 13.7@xmath1410@xmath25 to 0.71@xmath1410@xmath25 per every 58 @xmath26 ( or 5 images ) . the photometry following the aforementioned analysis is largely free of systematics , as evidenced by the fact that the out - of - eclipse photometric precision lies near the gaussian noise expectation for binning the data of one over the square - root of the bin - size ( figure [ figbinbyn ] ) . our observations in ks - band , though , are still well above the predicted photon noise rms limit of 2.3@xmath1410@xmath27 per 58 seconds . for the following analysis we set the uncertainty on our individual measurements as 0.95 times the rms of the out of eclipse photometry after the removal of a linear - trend with time ; we found simply using 1.0 times the rms of the out - of - eclipse photometry resulted in a reduced @xmath28 below one , and thus resulted in a slight over - estimate of our errors . similarly to nearly all our near - infrared photometric data - sets taken with cfht / wircam ( e.g. @xcite ) , our ks - band photometry following the reduction exhibited an obvious background trend , @xmath29 . this background term displayed a near - linear slope , and thus we fit the background with a linear - function of the form : @xmath30 where dt is the time interval from the beginning of the observations , and @xmath31 and @xmath32 are the fit parameters . given that most of our other data - sets display these background trends , it is unlikely , but not impossible , that this slope is intrinsic to tres-2 . we fit for the best - fit secondary eclipse and linear fit simultaneously using markov chain monte carlo ( mcmc ) methods ( @xcite ; @xcite ; described for our purposes in @xcite ) . we use a 5@xmath1410@xmath33 step mcmc chain . we fit for @xmath31 , @xmath32 , the depth of the secondary eclipse , @xmath34 , and the offset that the eclipse occurs later than the expected eclipse center , @xmath35 . we also quote the best - fit phase , @xmath36 , as well as the best - fit mid - eclipse heliocentric julian - date , @xmath37 . we use the @xcite algorithm without limb darkening to generate our best - fit secondary eclipse model . we obtain our stellar and planetary parameters for tres-2 from @xcite , including the planetary period and ephemeris . the results from these fits are presented in table [ tableparams ] . the phase dependence of the best - fit secondary eclipse is presented in figure [ figtres2bcontour ] . the best - fit secondary eclipse is presented in figure [ figtres2b ] . ; solid - line ) , 95.5% ( 2@xmath1 ; dashed - line ) and 99.7% ( 3@xmath1 ; short dashed - line ) credible regions from our mcmc analysis on the secondary eclipse depth , @xmath34 , and phase , @xmath36 . the `` x '' in the middle of the plot denotes the best - fit point from our mcmc analysis . ] , while the bottom panel shows the binned residuals from the best - fit model . in each one of the panels the best - fit best - fit secondary eclipse and background , @xmath29 , is shown with the red line . the expected mid - secondary eclipse is if tres-2b has zero eccentricity . ] ccccc reduced @xmath28 & 1.089@xmath38 & 1.086@xmath39 + @xmath34 & 0.062@xmath0% & 0.064@xmath40% + @xmath41 ( @xmath42 ) & 4.5@xmath43 & 3.8@xmath44 + @xmath37 ( hjd-2440000 ) & 14994.0605@xmath45 & 14994.0600@xmath46 + @xmath31 & 0.00061@xmath47 & 0.00061@xmath48 + @xmath32 ( @xmath49 ) & -0.005@xmath50 & -0.005@xmath51 + @xmath36 & 0.5014@xmath52 & 0.5012@xmath53 + @xmath54 ( @xmath9 ) & 1636@xmath8 & 1646@xmath55 + @xmath56 & 0.0020@xmath57 & 0.0017@xmath58 + @xmath59 & 0.358@xmath60 & 0.367@xmath61 + [ tableparams ] to determine the effect of any excess systematic noise on our photometry and the resulting fits we employ the `` residual - permutation '' method as discussed in @xcite . in this method the best - fit model is subtracted from the data , the residuals are shifted between 1 and the total number of data points ( @xmath62=1056 in our case ) , and then the best - fit model is added back to the residuals . we then refit the adjusted lightcurve with a 5000-step mcmc chain and record the parameters of the lowest @xmath63 point reached . by inverting the residuals we are able to perform @xmath64 total iterations . the best - fit parameters and uncertainties obtained with this method are similar to those found for our mcmc method and are listed in table [ tableparams ] . as the two methods produce similar results we employ the mcmc errors for the rest of this paper . we also test for autocorrelation among the residuals to our best - fit model using the durbin - watson test @xcite ; for the durbin - watson test a test - statistic greater than 1.0 and less than 3.0 ( ideally near 2.0 ) indiciates a lack of autocorrelation . our residuals pass this test with a test - statistic of 1.97 . the depth of our best - fit secondary eclipse is 0.062@xmath0% . the reduced @xmath28 is 1.089 . our best - fit secondary eclipse is consistent with a circular orbit ; the offset from the expected eclipse center is : @xmath41 = 4.5@xmath43 minutes ( or at a phase of @xmath36=0.5014@xmath52 ) . this corresponds to a limit on the eccentricity and argument of periastron of @xmath65 = 0.0020@xmath57 , or a 3@xmath1 limit of @xmath4@xmath2@xmath5@xmath3@xmath4 @xmath6 0.0090 ) . our result is fully consistent with the more sensitive @xmath2cos@xmath3 limits reported from the secondary eclipse detections at the four spitzer / irac wavelengths @xcite . thus our result bolsters the conclusion of @xcite that tidal damping of the orbital eccentricity is unlikely to be responsible for `` puffing up '' the radius of this exoplanet . a secondary eclipse depth of 0.062@xmath0% corresponds to a brightness temperature of @xmath54 = 1636@xmath8 @xmath9 in the ks - band assuming the planet radiates as a blackbody , and adopting a stellar effective temperature of @xmath66 = 5850 @xmath67 50 @xcite . this compares to the equilibrium temperature of tres-2b of @xmath68@xmath111472 @xmath9 assuming isotropic reradiation , and a zero bond albedo . hot jupiter thermal emission measurements allow joint constraints on the bond albedo , @xmath69 , and the efficiency of day to nightside redistribution of heat on these presumably tidally locked planets . the bond albedo , @xmath69 is the fraction of the bolometric , incident stellar irradiation that is reflected by the planet s atmosphere . we parameterize the redistribution of dayside stellar radiation absorbed by the planet s atmosphere to the nightside by the reradiation factor , @xmath70 , following the @xcite definition . if we assume a bond albedo near zero , consistent with observations of other hot jupiters @xcite and with model predictions @xcite , we find a reradiation factor of @xmath59 = 0.358@xmath60 from our ks - band eclipse photometry only , indicative of relatively efficient advection of heat from the day - to - nightside at this wavelength . in comparison , the reradiation factor for an atmosphere that reradiates isotropically is @xmath70=@xmath71 , while @xmath70=@xmath72 denotes redistribution and reradiation over the dayside face only . ; solid - line ) , 95.5% ( 2@xmath1 ; dashed - line ) and 99.7% ( 3@xmath1 ; short dashed - line ) @xmath63 confidence regions on the reradiation factor , @xmath73 , and bond albedo from the combination of our ks - band point and the spitzer / irac measurements @xcite . ] our secondary eclipse depth , when combined with the secondary eclipse depths at the spitzer / irac wavelengths from @xcite , is consistent with a range of bond albedos , @xmath69 , and efficiencies of the day to nightside redistribution on this presumably tidally locked planet ( figure [ figbondreradiation ] ) . the best - fit total reradiation factor , @xmath73 , that results from a @xmath28 analysis of all the eclipse depths for tres-2b assuming a zero bond albedo is @xmath73 = 0.346@xmath74 . thus our ks - band brightness temperature ( @xmath54 = 1636@xmath8 @xmath9 ) and reradiation factor @xmath59=0.358@xmath60 , reveal an atmospheric layer that is similar to , and perhaps slightly hotter , than the atmospheric layers probed by longer wavelength spitzer observations ( @xmath7@xmath111500 @xmath9 from spitzer / irac observations of tres-2b [ @xcite ] ) . the ks - band is expected to be at a minimum in the water opacity @xcite , and thus our ks - band observations are expected to be able to see deep into the atmosphere of tres-2b . our observations suggest that the deep , high pressure atmosphere of tres-2b displays a similar temperature perhaps a slightly warmer temperature to lower pressure regions . another way of parameterizing the level of day - to - nightside heat redistribution is calculating the percentage of the bolometric luminosity emitted by the planet s dayside , @xmath75 , compared to the nightside emission , @xmath76 . measurements of the thermal emission of a hot jupiter at its blackbody peak provide a valuable constraint on the bolometric luminosity of the planet s dayside emission , and by inference its nightside emission @xcite . from simple thermal equilibrium arguments if tres-2b has a zero bond albedo and it is in thermal equilibrium with its surroundings it should have a total bolometric luminosity of @xmath77 = 7.7@xmath1410@xmath78@xmath79 . by integrating the luminosity per unit frequency of our best - fit blackbody model across a wide wavelength range we are able to calculate the percentage of the total luminosity reradiated by the dayside as @xmath1169% ( @xmath75 = 5.3@xmath1410@xmath78@xmath79 ) . the remainder , presumably , is advected via winds to the nightside . 2.15 @xmath10 ) is our own , while the spitzer / irac red points are from @xcite . blackbody curves for isotropic reradiation ( @xmath70=@xmath71 ; @xmath68@xmath111496 @xmath9 ; blue dashed - line ) and for our best - fit reradiation factor ( @xmath70=0.346 ; @xmath68@xmath111622 @xmath9 ; grey dotted - line ) are also plotted . we also plot one - dimensional , radiative transfer spectral models @xcite for various reradiation factors and with and without tio / vo . the models with tio / vo include @xmath70=@xmath71 ( purple dotted line ) , @xmath70=0.31 ( green dashed line ) , and @xmath70 = @xmath72 ( orange dotted line ) ; only the last of the models has a temperature inversion . the model without tio / vo features emission from the dayside only ( @xmath70=@xmath72 ; cyan dot - dashed line ) . the models on the top panel are divided by a stellar atmosphere model @xcite of tres-2 using the parameters from @xcite ( @xmath80=0.98 @xmath81 , @xmath82=1.00 @xmath83 , @xmath66=5850 @xmath9 , and log @xmath84= 4.43 ) . we plot the ks - band wircam transmission curve ( dotted black lines ) and spitzer / irac curves ( solid red lines ) inverted at arbitrary scale at the top of both panels . , title="fig : " ] 2.15 @xmath10 ) is our own , while the spitzer / irac red points are from @xcite . blackbody curves for isotropic reradiation ( @xmath70=@xmath71 ; @xmath68@xmath111496 @xmath9 ; blue dashed - line ) and for our best - fit reradiation factor ( @xmath70=0.346 ; @xmath68@xmath111622 @xmath9 ; grey dotted - line ) are also plotted . we also plot one - dimensional , radiative transfer spectral models @xcite for various reradiation factors and with and without tio / vo . the models with tio / vo include @xmath70=@xmath71 ( purple dotted line ) , @xmath70=0.31 ( green dashed line ) , and @xmath70 = @xmath72 ( orange dotted line ) ; only the last of the models has a temperature inversion . the model without tio / vo features emission from the dayside only ( @xmath70=@xmath72 ; cyan dot - dashed line ) . the models on the top panel are divided by a stellar atmosphere model @xcite of tres-2 using the parameters from @xcite ( @xmath80=0.98 @xmath81 , @xmath82=1.00 @xmath83 , @xmath66=5850 @xmath9 , and log @xmath84= 4.43 ) . we plot the ks - band wircam transmission curve ( dotted black lines ) and spitzer / irac curves ( solid red lines ) inverted at arbitrary scale at the top of both panels . , title="fig : " ] we compare the depth of our ks - band eclipse and the spitzer / irac eclipses @xcite to a series of planetary atmosphere models in figure [ figmodel ] . this comparison is made quantitatively as well as qualitatively by integrating the models over the wircam ks band - pass as well as the spitzer / irac channels , and calculating the @xmath28 of the thermal emission data compared to the models . we first plot blackbody models with an isotropic reradiation factor ( @xmath70=@xmath71 ; blue dotted - line ) and that of our best - fit value ( @xmath70=0.346 ; grey dotted - line ) these models have dayside temperatures of @xmath85@xmath111496@xmath9 and @xmath85@xmath111622@xmath9 , respectively . both blackbody models provide reasonable fits to the data , although the latter model ( @xmath70=0.346 ; @xmath28=4.7 ) provides a definitively better fit than the former isothermal model ( @xmath70=@xmath71 ; @xmath63=9.1 ) as it better predicts our ks - band emission and the spitzer / irac 8.0 @xmath10 emission . this suggests that overall tres-2b has a near - isothermal dayside temperature - pressure profile and is well - fit by a blackbody . we thus also compare the data to a number of one - dimensional , radiative transfer , spectral models @xcite with different reradiation factors that specifically include or exclude gaseous tio and vo into the chemical equilibrium and opacity calculations . in these models when tio and vo are present they act as absorbers at high altitude and lead to a hot stratosphere and a temperature inversion @xcite . however , if the temperature becomes too cool ( tio and vo start to condense at 1670 @xmath9 at 1 mbar [ @xcite ] ) , tio and vo condense out and the models with and without tio / vo are very similar . in the case of tres-2b , for all the models we calculate , except our model that features dayside emission only ( @xmath70=@xmath72 ) , they do not harbour temperature inversions because the atmospheres are slightly too cool and tio / vo has condensed out of their stratospheres . we plot models with tio / vo and reradiation factors of @xmath70=@xmath71 ( purple dotted line ) , @xmath70=0.31 ( green dashed line ) , and @xmath70=@xmath72 ( orange dotted line ) , and without tio / vo with a reradiation factor of @xmath70=@xmath72 ( cyan dot - dashed line ) . @xcite argued that tres-2b experienced a temperature inversion due to the high 4.5 @xmath10 emission compared to the low 3.6 @xmath10 emission , which was predicted to be a sign of water and co in emission , rather than absorption , in tres-2b s presumably inverted atmosphere . we also find that our models without a temperature inversion have difficultly matching the spitzer / irac 5.6 and 8.0 @xmath10 thermal emission ( @xmath28=25.4 for @xmath70=@xmath71 with tio / vo , @xmath28=15.3for @xmath70=0.31 with tio / vo , and @xmath28=5.5 for @xmath70=@xmath72 without tio / vo ) . if the temperature inversion is due to tio / vo , by the time the atmosphere becomes hot enough that tio / vo remains in gaseous form , the thermal emission is too bright to fit the 3.6 , and 5.8 @xmath10 thermal emission ( @xmath28=10.6for @xmath70=@xmath72 with tio / vo ) . the combination of our blackbody and radiative transfer models with our own eclipse depth and those from the spitzer / irac instrument @xcite thus suggest that the atmosphere of tres-2b likely features modest redistribution of heat from the day to the nightside . it is unclear whether the atmosphere of tres-2b requires a temperature inversion . a simple blackbody model ( @xmath70=0.346 and @xmath68@xmath111622 k ) provides an exemplary fit to the data ; this may indicate that tres-2b has a fairly isothermal dayside temperature structure , perhaps similar to hat - p-1b @xcite . an important caveat , on the above result is that our @xmath70=@xmath72 model without tio / vo ( @xmath28=5.5 ) and thus without a temperature inversion returns nearly as good of fit as our best - fit blackbody model ( @xmath70=0.346 ; @xmath28=4.7 ) ; thermal emission measurements at other wavelengths , and repeated measurements at the above wavelengths , are thus necessary to differentiate a blackbody - like spectrum , from significant departures from blackbody - like behaviour , and to confirm that tres-2b efficiently redistributes heat to the nightside of the planet . specifically , the variations between the models displayed in figure [ figmodel ] are largest in the near - infrared j & h - bands and thus further near - infrared constraints if they are able to achieve sufficient accuracy to measure the small thermal emission signal of tres-2b in the near - infrared should prove eminently useful to constrain the atmospheric characteristics of this planet . if the excess emission at 4.5 @xmath10 is due to water emission , rather than absorption , due to a temperature inversion in the atmosphere of tres-2b then the inversion is unlikely to be due to tio / vo . this is because the atmosphere of tres-2b appears too cool to allow tio / vo to remain in gaseous form in its upper atmosphere . if there is a temperature inversion then the high altitude optical absorber is likely to be due to another chemical species than tio / vo . for instance , @xcite have investigated the photochemistry of sulphur - bearing species as another alternative . tres-2b is a promising target for the characterization of its thermal emission across a wide wavelength range . in addition to orbiting a relatively bright star , and having a favourable planet - to - star radius ratio , tres-2 lies within the kepler field . the combination of secondary eclipse measurements already published using spitzer / irac , upcoming measurements with kepler ( @xmath11430 - 900 @xmath86 ; @xcite ) , and j , h and k - band near - infrared measurements that could be obtained from the ground , will allow us to fully constrain tres-2b s energy budget . at the shorter end of this wavelength range it should also be possible to constrain the combination of reflected light and thermal emission . our results predict that even if the geometric albedo of tres-2b is as low as 5% in the kepler bandpass , if kepler is able to detect the secondary eclipse of this planet then it will be detecting a significant fraction of reflected light in addition to thermal emission . this will largely break the degeneracy on the bond albedo and the reradiation factor for this planet , facilitating a more complete understanding of its energy budget . the natural sciences and engineering research council of canada supports the research of b.c . and r.j . the authors would like to thank marten van kerkwijk for helping to optimize these observations , and norman murray for useful discussions . the authors especially appreciate the hard - work and diligence of the cfht staff in helping us pioneer this `` stare '' method on wircam . we thank the anonymous referee for a thorough review . | we present near - infrared ks - band photometry bracketing the secondary eclipse of the hot jupiter tres-2b using the wide - field infrared camera on the canada - france - hawaii telescope .
we detect its thermal emission with an eclipse depth of 0.062@xmath0% ( 5@xmath1 ) .
our best - fit secondary eclipse is consistent with a circular orbit ( a 3@xmath1 upper limit on the eccentricity , @xmath2 , and argument or periastron , @xmath3 , of @xmath4@xmath2@xmath5@xmath3@xmath4 @xmath6 0.0090 ) , in agreement with mid - infrared detections of the secondary eclipse of this planet .
a secondary eclipse of this depth corresponds to a day - side ks - band brightness temperature of @xmath7 = 1636@xmath8 @xmath9 .
our thermal emission measurement when combined with the thermal emission measurements using spitzer / irac from odonovan and collaborators suggest that this planet exhibits relatively efficient day to night - side redistribution of heat and a near isothermal dayside atmospheric temperature structure , with a spectrum that is well approximated by a blackbody . it is unclear if the atmosphere of tres-2b requires a temperature inversion
; if it does it is likely due to chemical species other than tio / vo as the atmosphere of tres-2b is too cool to allow tio / vo to remain in gaseous form .
our secondary eclipse has the smallest depth of any detected from the ground at around 2 @xmath10 to date . |
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the tev source j2032 + 4130 ( aharonian et al . 2002 ) was the first unidentified very high energy ( vhe ) @xmath0-ray source , and also the first discovered extended tev source , likely to be galactic . intensive observational campaigns at different wavelengths have been carried out on tev j2032 + 4130 . butt et al . ( 2003 ) presented an analysis of the co , hi , and infrared emissions , together with first observations by _ chandra _ ( 5 ksec ) and a reanalysis of vla data . these observations showed that the tev source region is positionally coincident with an outlying group of stars ( from the cygnus ob2 core ) , although they failed to identify a counterpart . mukherjee et al . ( 2003 ) analyzed the same _ chandra _ data and provided optical follow - up observations of several of the brightest x - ray sources , confirming that most were either o stars or foreground late - type stars . a deeper _ chandra _ observation ( 50 ksec , butt et al . 2006 ) , found hundreds of star - like sources and yet no diffuse x - ray counterpart emission . a deep ( @xmath12 50 ksec ) _ xmm - newton _ exposure has also been obtained ( horns et al . 2007 ) . after the subtraction of the contribution of known sources from the data , an extended x - ray emission region with a fwhm size of @xmath12 12 arcmin was reported . the centroid of the emission is co - located with the position of tev j2032 + 4130 and was proposed as the counterpart of the tev source . the question whether the result reported by horns et al . can be interpreted as a truly diffuse background , or it could be a result of unresolved x - ray sources , remains disputable . paredes et al . ( 2007 ) and mart et al . ( 2007 ) have provided deep radio observations covering the tev j2032 + 4130 vicinity using the giant metrewave radio telescope and discovered a population of radio sources , some in coincidence with x - ray detections by butt et al . ( 2006 ) and with optical / ir counterparts . at least three of these sources are non - thermal , and one has a hard x - ray energy spectrum . they found extended non - thermal diffuse emission in the radio band apparently connecting with one or two radio sources . it is yet to be determined if one or more of these sources is similar to some of the known @xmath0-ray binaries ( e.g. , aharonian et al . 2006 , albert et al . 2006a ) . several theoretical explanations for the tev emission from j2032 + 4130 have been given . among them , those related with extragalactic counterparts , e.g. , a radiogalaxy ( butt et al . 2006 ) or a proton blazar ( mukherjee et al . 2003 ) , face the difficulty of explaining the extended appearance of the source . gamma - ray production in hypothetical jet termination lobes of cyg x-3 was explored ( aharonian et al . 2002 ) , but the putative northern lobe of cyg x-3 ( now considered a mere thermal hii region , mart et al . 2006 ) is far from the location of the tev source . a yet unknown pulsar wind nebula ( pwn ) was proposed by bednarek ( 2003 ) , although no clear pwn signal was observed . a distant microquasar was proposed by paredes et al . ( 2007 ) , perhaps related with one of the x - ray / radio sources they discovered . if such an association is accepted , the extension of the source could be explained by the diffusion of accelerated particles into a hypothetical nearby molecular enhancement ( see bosch - ramon et al . torres et al . ( 2004 ) and domingo - santamara & torres ( 2006 ) studied the relationship between the tev emission and the known massive stars in the area , through the interaction of relativistic protons with wind ions . the distribution of stars in the neighborhood favors this interpretation ( butt et al . an explanation involving the excitation of giant dipole resonances of relativistic heavy nuclei in radiation dominated environments has also been suggested ( anchordoqui et al . we start by making a brief summary of what has been claimed by other experiments observing at the highest energies . the hegra iact using four years of data ( from 1999 to 2002 ) found a source to the north of cygnus x-3 , steady in flux over the years , extended , with radius [email protected]@xmath100.9@xmath11 arcmin , and exhibiting a hard energy spectrum with a photon index of @xmath9=-1.9@xmath2 [email protected]@xmath4 ( aharonian et al . 2005 ) . its integral flux above 1 tev amounts to @xmath125% of the crab nebula , assuming a gaussian profile for the intrinsic source morphology . the center of the source position was determined quite accurately at @xmath13=20@xmath14 31@xmath15 57@xmath16 @xmath2 6@xmath17 @xmath2 13@xmath18 and @xmath19=41@xmath2029@xmath2156@xmath228@xmath23 1@xmath24 @xmath2 1@xmath25 . the whipple collaboration reported an excess at the position of the hegra unidentified source ( 3.3@xmath1 ) in their archival data of 1989 and 1990 ( lang et al . 2004 ) , with a flux level of @xmath12 12% of the crab nebula for e@xmath26600 gev . the detected flux is in conflict with the hegra flux level and steady nature of the source , assuming they all have the same origin . this large difference between the detected flux levels , if physical , might suggest episodic emission ( with low duty cycle ) or variability over timescale measured in years . nevertheless , the existence of @xmath0-ray variability is difficult to reconcile with the extended appearance of the source . also the large difference might be in part due to unspecified systematic errors on the flux determination . recently , the whipple collaboration reported new observations of this field done with their 10-m telescope for 65.5 hours during 2003 and 2005 ( konopelko et al . their data is consistent with either a point - like or an extended source with less than 6@xmath27 angular size . regarding the position , the hegra and the latest whipple data are barely in agreement : their centers of gravity are @xmath129@xmath27 apart , and only agree when adding up the spatial uncertainties in both data sets in opposite directions . konopelko et al . do not provide a energy spectrum for this source , but give a 8% crab - level flux ( although with no energy threshold specified ) under the assumption of a steep ( crab - like ) energy spectrum . the cygnus region shows an excess in the milagro data ( abdo et al . the flux at 20 tev in a 3x3 square degree region centered at the hegra position is ( [email protected]@xmath102.7@xmath4 ) @xmath510@xmath28 tev@xmath8 @xmath7 s@xmath8 assuming a differential energy spectrum e@xmath29 . this flux is three times the hegra flux extrapolated at 20 tev . the tibet air shower detector recently reported evidence for an excess also in their vhe @xmath0-ray candidate set from this region ( amenomori et al . 2006 ) . in this rich observational and theoretical context we report here on magic telescope observations of tev j2032 + 4130 . the magic single dish imaging air cherenkov telescope ( see e.g. , cortina et al . 2005 for a detailed description ) is located on the canary island of la palma . its angular ( energy ) resolution is approximately 0.09@xmath30 ( 20% ) , and the trigger ( analysis ) threshold is 55 ( 60 ) gev at zenith in dark conditions ( see albert et al . one of the unique characteristics of magic is its capability of observing under moderate moon light illumination ( albert et al . 2007b ) albeit with a slightly elevated threshold . the field of view of tev j2032 + 4130 was observed with magic for more than 100 hours distributed in 2005 , 2006 and 2007 , see table [ tab1 ] . during the first period in summer 2005 , the observation was carried out in on / off mode , that is , the source was observed on - axis while observations from an empty , nearby field of view were used to estimate the background . in summer 2006 and 2007 , the data were taken in wobble mode , using five positions around the hegra position instead of the usual two symmetrical position in order to monitor a wider field of view . quality cuts based on the trigger and after - cleaning rates were applied in order to remove bad weather runs and data spoiled by car or satellite light flashes . after these quality cuts the total observation time is 93.7 h. the energy range for which we report these results is significantly above the aforementioned trigger and analysis threshold energies due to the fact that the observations were scheduled during moonlight and at relatively high zenith angles ( up to 44@xmath31 ) . the data analysis was carried out using the standard magic analysis and reconstruction software ( bretz & wagner 2003 ) . it follows the general stream explained in albert et al . ( 2006b , c , d ) . after calibration and two levels of image cleaning tail cuts ( for image core and boundary pixels , see fegan 1997 ) , the camera images are parameterized by the so - called image parameters ( hillas 1985 ) . the random forest method was applied for the @xmath0/hadron separation ( albert et al . using this method a parameter , dubbed hadronness ( h ) , can be calculated for every event and which is a measure of the probability that the event is not @xmath0 like . the @xmath0 like sample is selected for images with a h below a specified value , which is optimized using a sample of crab nebula data processed with the same analysis stream . an independent sample of monte carlo @xmath0-showers was used to determine the cut efficiency . since part of our observations was recorded during partial moon - shine , we have corrected the efficiency loss due to the increase of ambient light following the procedure outlined in albert et al . ( 2007b ) . the @xmath32-distribution was calculated , being @xmath33 the angular distance between the source direction and the reconstructed arrival direction of the showers . the reconstruction of individual @xmath0-ray arrival directions makes use of the disp method ( domingo - santamaria et al . the expected number of background events is calculated using five regions symmetrically placed for each wobble position with respect to the center of the camera and refered to as anti - sources . figure [ fig1 ] shows the distribution of the @xmath32 parameter for the excess observed from the direction of the source , for a size cut of 800 photoelectrons ( pe ) . this relatively high size cut was selected in order to optimize the sensitivity for a source with such a hard energy spectrum observed during moonlight . therefore , the total number of @xmath0-like excesses after hillas cuts and applying a cut in @xmath340.05 , is @xmath35 , for which a total significance of @xmath36 is obtained . table [ tab2 ] shows the number of excesses above background for the different observing periods . the excess is fitted to a gaussian function folded with the telescope psf , as obtained from monte carlo simulations and validated with crab nebula observations . the source is extended with respect to the magic psf . its intrinsic size assuming a gaussian profile is @xmath37 = [email protected]@xmath380.6@xmath4 arcmin . the exact shape of the source , even if similar to the kev diffuse emission reported by horns et al . 2007 , can not be completely trusted due to limited statistics and telescope pointing systematics . figure [ fig2 ] shows the gaussian - smoothed ( @xmath1=4 ) map ( [email protected]@xmath40 ) of @xmath0-ray ( background subtracted ) around tev j2032 + 4130 for energies e @xmath26 500 gev . the position of a few previously observed @xmath0-ray source candidates are also shown , namely cyg x-3 , the egret source 3eg j2033 + 4118 ( with its confidence contour at 95@xmath41 ) , the wolf rayet star wr 146 , and the whipple and hegra experimental positions . the regions around cyg x-3 , wr 146 and 3eg j2033 + 4118 have been further investigated by us and no detection is obtained for a steady emission . the upper limit fluxes ( rolke et al . 2005 ) for 95@xmath41 confidence level , above 500 gev for a point - like source at these positions are given in table [ tab3 ] . to determine the best position of the magic detection the excess map was fitted to a 2d bell - shaped function . the result is shown in the skymap with a black cross as well as by a circle indicating its size . the best - fit coordinates are ra@xmath42=20@xmath14 32@xmath15 20@xmath43 @xmath2 11@xmath44 @xmath4511@xmath46 and dec@xmath42=41@xmath20 30@xmath21 36@xmath220 @xmath2 1@xmath47 @xmath2 1@xmath48 ( for more details on the systematic uncertainties in the source position determination , see bretz et al . the position found is compatible within errors with the one determined by hegra , and barely compatible with the claims by whipple mentioned above ( in konopelko et al . 2007 ) . the tev j2032 + 4130 energy spectrum was obtained using the tikhonov unfolding technique ( tikhonov @xmath49 arsenin 1979 ) . it can be fitted ( @xmath50 ) by a power law function . the differential flux ( tev@xmath8@xmath7s@xmath8 ) is : @xmath51 the errors quoted are only statistical . the systematic error is estimated to be 35@xmath41 in the flux level and 0.2 in the photon index ( see albert et al . the differential energy spectrum is shown in figure [ fig3 ] . the hegra tev j2032 + 4130 and magic crab nebula measured spectra ( in albert et al . 2007a ) are shown in blue solid line and black dotted line , respectively . the magic energy spectrum is compatible both in flux level and photon index with the one measured by hegra . crab nebula data from the same periods and zenith angle distributions were studied with the same analysis chain to check for any systematic deviation due to the long observation period . no indication of time variability was observed : the source integral flux is constant within errors , at 3@xmath41 of the crab nebula flux . the relative systematic uncertainty in the ratio of both fluxes was estimated to be less than 10@xmath41 . this uncertainty comes mainly from the slightly different atmospheric transmission conditions and differences in the detector parameters during data taking of the source and the crab nebula . for illustrative purposes , the dotted lines in figure [ fig3 ] represent one - zone hadronic and leptonic models of the high energy emission , both consistent with observations at lower energies in the region . under the hadronic scenario , the @xmath52 are obtained from a proton parent population described by a power law ( index @xmath53 ) with exponential cutoff at 100 tev . the cutoff value was adopted to be consistent with the upper limit at the highest energies coming from the hegra spectrum . the inverse compton spectrum is obtained from an electron population with equal index and a 40 tev exponential cutoff scattering off the cmb photons . as in aharonian et al . 2005 , we do not consider here the conditions under which particles are accelerated or how they lose energy . our leptonic fits ( see also the quoted paper for an sed representation ) can only cope with the data if we are actually looking at a compton peak around the energy range of detection , which is not fully discarded within errors . both models are compatible with the high energy emission . confirming the reality of the diffuse emission detected at lower energies is crucial to distinguish between these and more complex models . magic observations confirm the location of tev j2032 + 4130 found by hegra . the magic observation shows an extended source with a significance of 5.6@xmath1 . we find a steady flux with no significant variability within the three year span of the observations ( with the flux being at a similar level of the hegra data of the period 2002 - 2005 ) . we also present the source energy spectrum obtained with the lowest energy threshold to date , which , within errors , is compatible with a single power law . we thank the iac for the excellent working conditions at the orm . the support of the german bmbf and mpg , the italian infn , the spanish cicyt , the eth research grant th 34/04 3 , and the polish mnii grant 1p03d01028 is gratefully acknowledged . albert j. et al . ( magic collab . ) 2007b , `` very high energy gamma - ray observations during moonlight and twilight with the magic telescope '' submitted to astroparticle physics , astro - ph/0702475 . albert j. et al . ( magic collab . ) 2007c , `` implementation of the random forest method for the imaging atmospheric cherenkov telescope magic '' submitted to astroparticle physics , arxiv:0709.3719 . amenomori m. et al . 2006 , science , 314 , 439 . -parameter for events coming from the direction of tev j2032 + 4130 ( size@xmath26800 pe ) , the background distribution subtracted ( black points ) . a convolved radial gaussian fit f = a@xmath5exp(-0.5@xmath32/(@xmath54 ) ) is indicated by the solid black line with @[email protected] arcmin . the @xmath56 was measured from mc simulation and validated with crab nebula observations to be @[email protected] arcmin ( dashed black line ) . ] = 4 ) map of @xmath0-ray excess events ( background - subtracted ) for energies above 500 gev . the magic position is shown with a black cross . the surrounding black circle corresponds to the measured 1@xmath1 width . the last position reported by whipple is marked with a white cross while the hegra position is shown with a blue cross in the center of the field of view . the error bars , in all cases , correspond to the linear sum of the statistical and systematic errors . the green crosses correspond to the positions of cyg x-3 , wr 146 and the egret source 3eg j2033 + 4118 . the ellipse around the egret source marks the 95@xmath41 confidence contour . ] error in the fitted energy spectrum . the flux observed by whipple in 2005 and in the milagro scan are marked with colored squares ( blue and pink , respectively ) . the grey dotted line represents the crab nebula energy spectrum measured by magic . the blue line shows the hegra energy spectrum . theoretical one - zone model predictions are depicted with dashed lines . ] | we observed the first known very high energy ( vhe ) @xmath0-ray emitting unidentified source , tev j2032 + 4130 , for 94 hours with the magic telescope .
the source was detected with a significance of 5.6@xmath1 .
the flux , position , and angular extension are compatible with the previous ones measured by the hegra telescope system five years ago
. the integral flux amounts to ( [email protected]@xmath30.35@xmath4)@xmath510@xmath6 ph @xmath7 s@xmath8 above 1 tev .
the source energy spectrum , obtained with the lowest energy threshold to date , is compatible with a single power law with a hard photon index of @[email protected]@xmath100.2@xmath11 . |
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graphene has been long studied as a theoretical toy model not only to understand it s appealing physical properties,@xcite but also as a basic building block of various carbon allotropes like graphite,@xcite and more recently fullerenes and nanotubes.@xcite while graphite is the three dimensional allotrope of carbon and could be formed by the bernal stacking of graphene sheets , fullerene and nanotubes are the zero and one dimensional allotropes , formed by introducing pentagonal impurities and rolling the graphene sheets , respectively . after its experimental isolation in 2004,@xcite there has been a renewed interest in studying various properties of graphene sheet , both theoretically and experimentally , as well as due to potential technological applications.@xcite graphene consists of a single sheet of carbon atoms arranged on a honeycomb lattice . basic properties of graphene are well described by a tight - binding model for the @xmath3-orbitals which are perpendicular to the graphene plane at each carbon atom . the effective low - energy theory states that the charge carriers in graphene are massless dirac fermions , characterized by a linear dispersion relation and a linear energy dependence of the density of states which vanishes at the fermi level implying a semi - metallic behaviour for graphene.@xcite graphene has attracted a lot of attention recently not only due to its potential technological applications but also for understanding of physics in 2d systems@xcite . its low energy description mimics ( 2 + 1)-dimensional quantum electrodynamics and hence graphene could act as a testing ground for various relativistic phenomena.@xcite early experiments on graphene have revealed that the conductivity at low temperatures is directly proportional to the carrier concentration ( or gate voltage ) except for very low carrier concentration . for zero gate voltage , the conductivity approaches a robust minimum universal value proportional to @xmath4.@xcite this could not be explained by the born approximation which predicts a conductivity independent of carrier concentration.@xcite other interesting properties include anomalous integral quantum hall effect and suppression of weak localization.@xcite recent experiments , however , show that the dependence of conductivity on carrier concentration could vary from sub linear to superlinear for different carrier concentrations.@xcite it has been argued that presence of impurities in graphene is the main contributor towards its electronic properties.@xcite the importance of disorder in graphene could most easily be emphasized by observing that the universal conductivity suggested by the theoretical studies on defectless graphene sheet is 2 - 20 times smaller than the observed conductivity close to the dirac points.@xcite the boltzmann conductivity for graphene is given by @xmath5 . the observed conductivity rises linearly with carrier concentration in graphene and @xmath6 , where @xmath7 is the density of states at the fermi energy and @xmath8 is the carrier density . this implies that the scattering rate , @xmath9 . on the other hand , for weak local scatterers , born approximation predicts @xmath10 where @xmath11 is the impurity concentration.@xcite in view of this discrepancy , various investigations , both theoretical and numerical , have been carried out in order to understand the behavior of graphene under various types of disorder,@xcite such as vacancies,@xcite charged carriers,@xcite on - site disorder,@xcite long range on - site disorder,@xcite off - diagonal disorder,@xcite off - diagonal disorder with sign change probability in the hopping term.@xcite vacancies have been proposed to induce localized states , extended over many lattice sites , which are sensitive to the electron - hole symmetry breaking@xcite detailed studies in the presence of both compensated and uncompensated defects reveal that they could modify the low energy spectrum in graphene drastically like there could be quasi - localized zero modes and introduction of gap in the dos.@xcite for charged scatterers , nomura _ @xcite have argued on the basis of boltzmann transport theory that the linear dependence of conductivity on carrier concentration could be explained . they find that states close to the dirac point are delocalized leading to @xmath12 . also , one could observe antilocalization if the inter - valley scattering is weak . on the other hand , if inter - vally scattering is large , all states could be localized due to accumulation of berry phases . conductivity in the presence of random charged impurity is also studied by hwang _ _ et al.__@xcite they find linear dependence of conductivity on carrier concentration for high carrier density . however , for low carrier density , they argue , that system develops some inhomogenities ( random electron - hole puddles ) which implies that this domain is dominated by localization physics . they also conclude that change of bias voltage may change the average distance between graphene sheet and the impurity in the substrate which could lead to sub- and super - linear conductivity dependence on carrier concentraion . _ @xcite have argued that there could be a `` critical coupling '' distinguishing strong and weak coupling regimes in the presence of unscreened coulomb charges . they also find bound states and strong renormalization of van hove singularities in the dos . _ @xcitehave argued that the intrinsic conductivity of graphene ( ambipolar system ) is dominated by strong electron - hole scattering . it has a universal value independent of temperature . it is shown that conductivity could be proportional to v or @xmath13 depending on the other scattering mechanisms present like those on phonons by charged defects . in the unipolar system , it is argued that electron - hole scattering is not important and conductivity is proportional to @xmath14 . _ @xcite have reported an analytical calculation for boltzmann conductivity with screened coulomb scatterers and both electron - hole coherent and incoherent solutions . they find that the experimentally observed dependence of conductivity on @xmath8 could be explained by the electron - hole coherent solution . for diagonal disorder , it is found that beyond a certain threshold disorder strength , bound states appear beyond the band continuum and resonant states could appear at low energies . in the infinite disorder strength limit , results match with that of vacancies . in the off - diagonal disorder case , strong low - energy resonances appear . there are , however , no bound states.@xcite localization studies on graphene with on - site disorder carried out by xiong _ _ et al.__@xcite reveal that all states are localized in the case of random diagonal disorder which is consistent with anderson localization . lherbier _ _ et al.__@xcite have also studied the energy dependent elastic mean free path , charge mobilities and semi - classical conductivity in the presence of anderson - type disorder , using real space order n kubo formalism , for both two - dimensional graphene and graphene nano - ribbons ( gnrs ) . it was found that the systems undergo a conventioanl anderson transition in the zero temperature limit . lewenkopf _ et al . _ @xcite have studied long range diagonal disorder ( gaussian scatteres ) at finite concentration . they find that conductivity increases as disorder strength is decreased . it is shown that conductivity depends only on disorder strength and ratio of the system size to disorder correlation length . this dependence could vary between sublinear to superlinear depending on disorder strength . for fixed disorder strength , conductivity increases with doping concentration . in the presence of strong long - range impurities , zhang _ _ et al.__@xcite have shown that states close to the dirac points are localized for sufficiently strong disorder strength and kosterlitz - thouless transition between localized and delocalized states is proposed which is seen in terms of the current flow vector . in the case of random off - diagonal disorder ( hopping disorder ) , localization studies by xiong _ _ et al.__@xcite reveal that states close to the dirac point are delocalized due to chiral symmetry . they find that the off - diagonal disorder leads to a shape - dependent conductivity depending on the length to width ratio . however , if a sign change probability is introduced , they find that the conductivity becomes shape independent _ et al.__@xcite have shown that for disorder strength less than the hopping strength i.e. @xmath15 , there is no localization . disorder @xmath1 slows down the dirac quasi particles but preserves their nature . for @xmath16 , localization sets in for states close to the fermi energy , gap at energy close to the dirac point and for @xmath17 existence of mobility edge is proposed which starts at fermi energy and moves towards the edges . states close to the fermi energy are extended . they also propose the existence of disorder induced gap defined as the distance between the upper and lower mobility edges around the fermi point . it should be noted that in all these works , the inter - valley scattering is assumed to be very small and hence not contributing towards conductivity . however , klos _ _ et al.__@xcite have done a comparative study of conductivity using the tight - binding(tb ) landauer approach and on the basis of the boltzmann theory and find a discrepancy between that results obtained by tb calculation and boltzmann approach . despite all the efforts , the issue of localization in graphene is currently highly debated from a theoretical standpoint . the obsered minimal conductivity @xmath18 over a range of mobilities remains to be fully understod . in view of a recent experiment by ponomarenko _ _ et al.__@xcite and katoch _ _ et al.__@xcite , which explores the dominant scatterers in graphene , and horng__et al.__@xcite , which measures the high - frequency conductivity in graphene , unlike believed so far , the primary reason for the linear rise of conductivity with carrier concentration is also debatable . in the present work , we will investigate the finite - frequency electrical conductivity and localization properties of graphene in the presence of diagonal ( on - site ) disorder for various disorder strengths . our exact - eigenstates approach implicitly takes into account the inter - valley as well as the intra - valley scatterings . this paper is organized into eight sections . in section ii and iii , we introduce the two sub - lattice basis and evaluate the exact single - particle green s function within the t - matrix approach for a single impurity on either sublattice . in section iv , we consider disordered graphene . disorder is introduced via random fluctuation of the on - site energies of the @xmath3-orbitals . in section v , we use the kubo - greenwood formula to calculate the frequency - dependent conductivity for different disorder strengths and for different system sizes . the system - size dependence is employed to perform a renormalization group analysis , and for moderate disorder strength contact is made with the weak localization result of abraham s _ et al._. later on , focus on the weak disorder strength . we study the frequency dependence of the averaged current matrix element squared for different locations in the band ( fermi energies ) and hence calculate conductivity and mobility for low - disordered graphene samples . we also study the dependence of energy resolved current matrix elements squared on system size and disorder strength . in section vi , we study the average current matrix elements squared over a range of disorder strength and for different system sizes in order to gain a better insight into disorder induced localization in graphene . in section vii , we make a comparision between graphene and square lattice . we compare the normalized conductivity between the two . we also show the intensity plots for both the lattices for different values of disorder strength . finally , our conclusions are presented in section viii . as already mentioned above , graphene is a single layer of carbon atoms on a honeycomb lattice . a honeycomb lattice , which is not a bravais lattice , could be viewed as a composed of two kinds of sub - lattices . nearest neighbor hopping takes electron from one sub - lattice to another . we start with the tight - binding(tb ) hamiltonian : @xmath19 \label{tbham}\ ] ] where the sum is over the nearest neightbours . this could be written in the two sub - lattice basis as : @xmath20 \label{ham1}\ ] ] where , the hopping term in k - space is : @xmath21\ ] ] it should be noted that this hamiltonian is ( @xmath22 ) and mixes the dirac points . the energy eigenvalues are given by @xmath23 in this basis , the low energy spectrum provides six dirac points in the extended brilluoin zone . these points are located at @xmath24 and @xmath25 . expanding around the dirac points , we find the linear dispersion for carriers in graphene . the free particle green s function is given by the expression : @xmath26 \frac{1}{\omega^2\ ! -\ ! e_{\mathbf k}^2 } % \end{displaymath } } \ ] ] where @xmath27 is the energy eigenvalue derived earlier . the density of single particle states would then be given by @xmath28 & \mbox{if } e > e_f \\ -\frac{1}{\pi}{\sum_{\mathbf k } } im[g^0(\mathbf k,\omega ) ] & \mbox{if } e < e_f \end{array}\right.\ ] ] fig.([fig : dos , gii ] ) shows the density of states ( dos ) for pure graphene ( fig.([fig : dos]))and also the real and imaginary part of the green s function ( fig.([fig : gii ] ) ) in the presence of impurities , the free particle green s function gets modified . we shall , at this stage , look at the effect of a single impurity on the green s function . we shall employ the t - matrix approach which gives exact result in the case of single impurity . as from the scattering theory , the modified green s function based on the diagrams in fig.([fig : t - matrix ] ) is be given by : @xmath29 where the t - matrix is given as @xmath30 $ ] , v is the strength of the impurity . the index @xmath31 and @xmath32 denotes the impurity site and the lattice sites respectively . -matrix approximations for the self energy . the `` x '' , dotted line and line with arrow signify the impurity , impurity potential and the green s function respectively . ] information regarding additional poles is contained in the t - matrix . this implies that there are possibilities of existence of new states if there is an intersection of @xmath33 $ ] and 1/v . the imaginary part is small but finite . if the strength of impurity is small or even moderate , there shall be no intersection with the @xmath34 line . it is only when the strength of the impurity is large , some intersection is expected and this might lead to a new state . therefore , for some value of @xmath35 , a resonance state occurs at the impurity site . this new state would be localized at the impurity site . the full gren s function is given by : @xmath36}{i - v[g^{0}_{ii}(\omega)]}\ ] ] around either @xmath37 or @xmath38 points ( dirac points ) , the @xmath39 has the following form . @xmath40\end{aligned}\ ] ] where @xmath27 is the energy , @xmath41 is the momentum cutoff , @xmath42 is the fermi velocity and @xmath43 is the impurity site . one should note that the free green s function contributes equally for an impurity sitting on either sub - lattice a or b. also , as , @xmath44 , the @xmath45 term in front provides a cut - off to the log divergence . imaginary part gives the density of states as already mentioned above . real part of the above expression is small . if we introduce a magnetic impurity at the some site @xmath43 , there could again be appearance of new states for sufficiently large impurity strength . however , the intersection could now be either above or below the @xmath45 axis . however , since the nature of intersection is same as for the single non - magnetic impurity , the physics is expected to remain same . we will now include diagonal ( on - site ) disorder and consider the following tight - binding hamiltonian : @xmath46 \label{ham2}\ ] ] on a honeycomb lattice . here the random on - site energies @xmath47 are chosen from a uniform distribution on @xmath48 . the second term is the hopping term with summation over nearest - neighbour pairs of sites . from the matrix realization of the above hamiltonian on a finite lattice with periodic boundary condition in both directions , we numerically obtain the eigenfunctions @xmath49 and the eigenvalues @xmath50 of the hamiltonian in eq.([ham2 ] ) by exact diagonalization of the hamiltonian matrix . one should note that the spin index simply runs through the calculation and shall be considered while making comparisions with the experiments . in the following , we have set the hopping term @xmath51 as the unit of energy . from the exact eigenvalues we obtain the local density of states ( ldos ) using : @xmath52 where @xmath53 is the total number of lattice points in the system . fig.([fig : dos_dis ] ) shows the ldos for various disorder strengths . clearly , the van hove singularities in the pure graphene dos are softened with increasing disorder strength . for very small disorder strength , the dos is still linear . however , for @xmath54 , the density of states becomes independent of @xmath45 . this could be understood on the basis of intersection of @xmath55 $ ] and 1/v , as argued in the previous section . we are interested in calculating finite - frequency conductivity for graphene . we use the kubo - greenwood formula@xcite for conductivity which uses the eigenvalues @xmath50 and the eigenfunctions @xmath49 calculated earlier . the kubo - greenwood formula is given by : @xmath56 where , the current matrix element @xmath57 is given by : @xmath58 where @xmath59 and @xmath60 are the eigenstates with @xmath61 and @xmath62 is the angle between the @xmath63-axis and the two other nearest neighbors at each site as shown in fig.([fig : theta ] ) . summation , @xmath64 is over nearest neighbors . here @xmath65 and @xmath66 are the number of unit cells and area of each unit cell , respectively . ( 2.3,-1.00)(4.3,3.05 ) ( 1.00,1.00)(6.00,1.00 ) ( 3.00,-1.00)(3.00,3.00 ) ( 5.00,1.00)(3.00,1.00 ) ( 3.00,1.00)(2.00,2.732 ) ( 3.00,1.00)(2.00,-0.732 ) ( 5.00,1.00 ) ( 3.00,1.00 ) ( 2.00,2.732 ) ( 2.00,-0.732 ) ( 3.05,1.0)0.520.0120.0 ( 3.57,1.73)@xmath62 ( 3.25,0.75)i ( 5.05,0.75)j ( 1.75,2.732)j ( 1.75,-0.732)j ( 6.25,1.00)x ( 3.00,3.20)y the summation over @xmath67 in eq.([kg ] ) is carried out by considering each pair of states @xmath59 and @xmath60 such that the energy difference , @xmath68 , remains fixed . we shall first consider the case for @xmath1 = 5 . when the density of states is independent of @xmath45 ( refer fig . [ fig : dos_dis ] , w=5 ) , the energy levels are nearly equally spaced . for a given random distribution of on - site impurity potentials , the contribution to the matrix elements from each energy pair of eigenvalues with fixed energy difference , @xmath68 could simply carried out by keeping @xmath69 fixed . we average the total contribution by the number of such paired states considered and the number of random configurations , typically 5000 samples of the matrix elements in all . the expression for the finite - frequency conductivity then reduces to : @xmath70 where @xmath71 is the averaged matrix element squared and @xmath72 is the averaged density of states per site at the dirac points . we now define the normalized conductivity @xmath73 which is convenient for plotting the data . ( [ fig : normcond1 ] ) is the data for a ( 20,24 ) lattice for various values of @xmath1 . at high frequency the normalized conductivity is approximately one . for higher @xmath1 , the conductivity falls off at low frequency . this reduction of the low - frequency conductivity can be ascribed to the effects of localization . we show in fig.([fig : normcond2 ] ) the normalized conductivity versus frequency for different system sizes at fixed @xmath1(@xmath74 ) . we observe that the conductivity decreses with increasing system size which exhibits the renormalization group at work . we also note that , despite the fact that the lattice is not symmetric about @xmath63 and @xmath75 axes , the conductivity behaviour turns out to be similar in both directions ( not shown ) . this is due to the fact that the spectrum of graphene is conical . versus @xmath45 for various impurity strengths for a ( 20,24 ) system . in all the cases , the dos is flat . the curves are drawn as a guide to the eye . some data points are omitted for clarity . ] versus@xmath45 for various system sizes . in all the cases , the dos is flat ( w=5 ) . the curves are drawn as a guide to the eye . some data points are omitted for clarity . ] we now turn to the macroscopic renormalization - group ( mrg ) calculation . we start with a certain set of hamiltonian parameters ( @xmath76 and @xmath1 ) and calculate the appropriate macroscopic physical quantites as a function of lattice parameter . the hamiltonian parameters are then renormalized to preserve the physical quantites as the lattice parameter is varied . we consider specifically two systems with different lattice spacing ( @xmath77 and @xmath78 ) but same physical size . for these two systems to represent the same physical problem with different microscopic length scales , we demand that the hamiltonian parameters be so related that the physical properties are preserved . the appropriate physical quantites to be preserved for the localization problem are the one - electron dos at the fermi energy , @xmath72 and the low - frequency electrical conductivity we define the dimensionless conductance which depends only on @xmath1 . @xmath79 then , the one - parameter rg recursion relation is : @xmath80 the other mrg relation : @xmath81 fixes the absolute magnitude of @xmath76 to preserve the dos . for weak disorder , the weak localization scaling theory result in 2d is@xcite @xmath82 where , l is the length scale . for @xmath74 , we compare the ( 20,24 ) lattice with ( 12,14 ) lattice and find that @xmath83 which matches well with the above value : @xmath84 . thus , for @xmath74 , the scaling theory of weak localization is completely obeyed . we now turn our attention to the case when the density of states is not flat ( e.g. for w=2 , refer fig . [ fig : dos_dis ] ) . the earlier method of obtaining the average current matrix elements squared does not work here since the ldos is linear . we , therefore , average the current matrix elements squared , @xmath85 over all @xmath86 and @xmath87 states such that the energy difference @xmath88 lies in a given range ( binning ) . we have checked that for @xmath1=5 the @xmath89 obtained by this method matches with the earlier one . for weak disorder strength , we study the d.c . limit of average current matrix elements squared for different locations in the band . we consider a narrow band of states centred at different energy eigenvalues and calculate their contribution to @xmath90 in the limit @xmath44 . it could be directly seen from the fig.([fig : energyres](a ) ) that in the limit @xmath44 for @xmath1=5 , contribution from all the energy states to conductivity is nearly equal . this implies that all the states are of the similar nature . however , in the same limit for @xmath1=2 , the states close to the fermi energy have a larger contribution and the contribution decreases as we move away from the band centre as seen in fig . ( [ fig : energyres](b ) ) . this implies that , close to band centre , the states are more extended ( compared to the states away from the band centre ) . we shall return to the issue of localization in the next section . + v / s @xmath50 for a ( 16,20 ) graphene lattice and w=2 . the curve is a @xmath91 fit with a= 0.0010 and b = 0.002 ] we may now ask : what is dependence of @xmath92 for @xmath1=2 on energy eigenvalues as we move along the band ? we polynomial fit the data and find ( fig.([fig : omega ] ) ) that the behaviour is inversly proportional to the square of the energy eigenvalue . the dependence is of the form @xmath93 . for w = 2 and system size ( 16,20 ) , we find that @xmath94 , and @xmath95 . we study the scaling behaviour of the constants @xmath96 and @xmath97 with increasing system size . [ fig : scaling ] ) . we have plotted the behaviour of @xmath96 and @xmath97 versus @xmath98 . it could be seen that there is not a significant change in the value of @xmath96 and @xmath97 as we increase the system size and the dependence of these quantities on @xmath98 is almost linear . as @xmath99 , the values of @xmath96 and @xmath97 are 1.06157 and 3.8294 respectively . the first term ( @xmath100 ) is what is expected from the boltzmann theory . the other term , independent of @xmath50 is the non - boltzmann contribution . a particle with the hamiltonian described by eq.([tbham ] ) , which supports both positive and negative energy states(positive and negative bands ) , could be in a linear combination of the positive and negative energy eigenstates . the boltzmann term arises out of transition between term in the same band . off - diagonal terms in the velocity operator correspond to the transition between the same @xmath101 but belonging to different bands and is responsible for the non - boltzmann contributions to the conductivity . the origin of this term could also be understood from the fact that the velocity operator is non - diagonal in the helicity basis . an analytical calculation for boltzmann conductivity with screened coulomb scatterers and incorporating the off - diagonal terms in the velocity operator was carried out by trushin _ _ et al.__@xcite in order to study d.c . conductivity , we go back to eq.([kg ] ) whereby d.c . conductivity could be written as the product of the current matrix elements squared in the limit @xmath44 and dos squared . in order to do this , we polynomial fit the dos squared and the average current matrix elements squared in the d.c . limit . for @xmath1 = 2 , we have seen that dos @xmath102 . carrier concentration , @xmath8 is defined as @xmath103 @xmath104 . putting in the values of the parameters , @xmath76 = 2.8ev and @xmath77 = 1.42 @xmath105 , together with the the bahavior of @xmath106 versus @xmath50 ( obtained in the previous subsection ) and @xmath8 versus @xmath50 , we obtain conductivity as a function of the carrier concentration . here the factor of two for spin multiplicity has already been taken into account . for large carrier concentration , we notice that as @xmath50 increases , the constant term ( `` @xmath107 '' ) in the fit dominates the current matrix element contribution . this together with the dos shows a linear dependence on the carrier concentration . we also note that for n @xmath108 zero , the conductivity does not go to zero . it turns out that for small @xmath1 , both the @xmath85 and the dos together conspire to give a non - zero d.c . conductivity ( @xmath12 ) , independent of @xmath109 , close to the dirac point . comparision of different system size suggests that as the system size is increased the lesser number of eigensatates contribute to @xmath12 for very small doping concentration . however , the @xmath100 dependence is still retained . this translates to the fact that as we increase the system size , the @xmath2 shifts towards the fermi energy as seen in fig.([fig : energyres_w2 ] ) . the value of this minimal conductivity , @xmath12 @xmath110 @xmath111 in confirmity with the previous results obtained with scba@xcite . we also compare this behaviour with respect to the system size and find that for larger system size , the number of such extended states decreases as shown in fig . ( [ fig : energyres_w2 ] ) but still retaining its @xmath100 dependence . thus , away from the zero doping concentration , one obtains a linear dependence of conductivity on the doping concentration in confirmity with the experimental findings . however , there is a quantitative difference between the two . the slope of the @xmath112 v / s @xmath8 ( fig.([fig : sigma_n ] ) ) is very small compared to the experimental results . the dependence if a.c . conductivity on @xmath45 can be obtained from the energy resolved matrix elements squared . for a fixed energy , the dependence of @xmath113 . this together with the _ mobility _ is defined as : @xmath114 . since , @xmath112 goes linearly as @xmath8 , @xmath115 should nearly be a constant for energies away from the dirac points . since @xmath116 , from eq.([sigma ] ) , we find that @xmath117 . @xmath118 in fig.([fig : mobility ] ) , we plot @xmath115 versus gate voltage , @xmath119 where @xmath120 . we find that the mobility rises sharply close to the dirac point . this sharp rise around the dirac points is in a very good qualitative agreement with the experiments . however , the values of mobility is around 1000 times lower than the experimental results . we would like to add that , in the adopted formalism , it is impossible to distinguish the effect of inter - valley scattering or to determine it s contribution towards conductivity . however , since we are dealing with very short range scatterers , it would be impossible to avoid large momentum interactions and hence inter - valley scattering . in order to study localization , we looked at ipr versus energy eigenvaluei(not shown ) and found that ipr is not sensitive enough to capture signatures of localization for small @xmath1 . therefore , we , again , turn towards the energy resolved studies of averaged current matrix elements squared , ( fig . [ fig : energyres ] ) . as mentioned earlier , for a few states close to the band centre have a very large averaged current matrix element squared value . this implies that these state are extended . we notice that this is the case independent of the system size . also , the experimental finding of a universal zero - bias minimum conductivity , @xmath12 suggests that the states close to the band centre should be extended , as we find which is in conformity with the findings of amini _ _ et al.__@xcite . this , however , is in contradiction with the findings of xiong _ _ et al.__@xcite . one might , also , question the @xmath121 dependence on the energy eigenvalues , @xmath50 since it could very well be a finite - size effect . for low disorder strength , the localization length , @xmath122 varies exponentially with the inverse of disorder strength i.e. @xmath123 and in order to see the effects of localization , one needs to change the system size exponentially.we note that the @xmath124 behaviour is present at all system sizes starting from system containing nearly 300 lattice points((8,10 ) ) to nearly 3000 lattice points((24,30 ) ) . also , that for @xmath1=3 , the @xmath125 is independent of @xmath50 ( not shown ) suggesting a disorder induced crossover from localized to delocalized phase for @xmath126 . we check this by studying the averaged current matrix element squared over a range of @xmath1 for different system sizes , at a fixed energy . we have fixed the energy at @xmath127 and have considered ten states around this energy . we study the contribution of these ten states towards the @xmath125 . the results are shown in fig.([fig : m_w ] ) . the inset of fig.([fig : m_w ] ) shows the plot between averaged current matrix element squared over a range of @xmath1 for three different system sizes ( 16,20),(20,24 ) and ( 24,30 ) . the cross - over is highlighted in the main panel which shows the logarithm of averaged current matrix elements squared versus @xmath1 . we indeed find a crossover around @xmath1=3 . however , this crossover is not between localized and delocalized phases but from localized to a very weakly localized behaviour . this is in agreement with the scaling theory of localization which predicts that all states in 2d are localized . we , now , compare the normalized conductivity , @xmath128 between square lattice and graphene for @xmath1 = 5 . this comparision has been shown in fig.([fig : comparsq ] ) . we observe that the conductivity matches well at large frequencies for both the lattices . however , at low frequencies , the grahene has a lower conductivity suggesting that the states are more localized . this could mean that the states in graphene are more susceptible to localized when subject to disorder , as also suggested by xiong _ _ et al.__@xcite we also compare the intensity plots for graphene and square lattice for different values of disorder strengths and fixed system size . the results are shown in fig([fig : comparsq2 ] ) . we find that there is no signature fo puddle formation for square lattice . the reason for the formation of puddle for low carrier concentration in graphene is yet to be understood . in conclusion , we have studied the finite - frequency electrical conductivity in graphene under diagonal ( on - site ) disorder of various strength and in the presence of inter - valley scattering using kubo - greenwood foumula . for moderate disorder strength , we find that for differnt system sizes , fixed @xmath1 , scaling theory is at work . we made contact with the weak localization result of abrahams _ _ et al.__@xcite and found a very good agreement which means that for @xmath74 , logarithmic scaling is obeyed . we compare normalized conductivity of graphene with that of a square lattice and find that graphene is more susceptible to localization when subject to disorder . we have established that , for low disorder strength , @xmath129 , the states away from the band centre are more localized comared to the ones close to the band centre . for low disorder strength , we have calculated the conductivity for low - disordered graphene and have found the results are in disagreement with that of boltzmann conductivity . also , the conductivity and mobility are in qualitative agreement with experiments . also , for weak disorder strength , it is the competition between @xmath130 and dos which gives rise to a universal conductivity minimum . the linear rise of conductivity with carrier concentration is due to a term unaccounted for in the boltzmann expression for conductivity . we have also established that , for low disorder strength , @xmath129 , the states close to the band centre are extended and that there exists a crossover at @xmath1 @xmath110 3 . comparative study of intensity plots for states close to the band centre for graphene with square lattice shows clear signatures of puddle formation in graphene . although we have studied a simple disorder model in graphene , some of the feature studied could be generic . j. w. klos , and i. v. zozoulenko phys . b * 82 * , 081414(r ) ( 2010 ) . l. a. ponomarenko , r. yang , t. m. mohiuddin , s. m. morozov , a. a. zhukov , f. schedin , e. w. hill , k. s. novoselov , m. i. katsnelson , and a. k. geim phys . rev . lett . * 102 * , 206603 ( 2009 ) . jyoti katoch , j - h . chen , ryuichi tsuchikawa , c. w. smith , e. r. mucciolo , and masa ishigami phys . rev . b * 82 * , 081417(r ) ( 2010 ) . jason horng , chi - fan chen , baisong geng , caglar girit , yuanbo zhang , zhao hao , hans a. bechtel , michael martin , alex zettl , michael f. crommie , y. ron shen , feng wang , arxiv:1007.4623 | we employ the exact eigenstate basis formalism to study electrical conductivity in graphene , in the presence of short - range diagonal disorder and inter - valley scattering .
we find that for disorder strength , @xmath0 5 , the density of states is flat .
we , then , make connection , using the mrg approach , with the work of abrahams _ et al . _ and find a very good agreement for disorder strength , @xmath1 = 5 . for low disorder strength , @xmath1 = 2 , we plot the energy - resolved current matrix elements squared for different locations of the fermi energy from the band centre .
we find that the states close to the band centre are more extended and falls of nearly as @xmath2 as we move away from the band centre .
further studies of current matrix elements versus disorder strength suggests a cross - over from weakly localized to a very weakly localized system .
we calculate conductivity using kubo greenwood formula and show that , for low disorder strength , conductivity is in a good qualitative agreement with the experiments , even for the on - site disorder .
the intensity plots of the eigenstates also reveal clear signatures of puddle formation for very small carrier concentration .
we also make comparison with square lattice and find that graphene is more easily localized when subject to disorder . |
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the recent atmospheric , solar , reactor and accelerator neutrino experiments have convinced us that neutrinos are massive particles . however , the problem of absolute values of their masses is still waiting for a solution . neutrino oscillations depend on the differences of neutrino masses , not on their absolute values . apparently three kinds of neutrino experiments have a chance to determine the light neutrino masses : i ) cosmological measurements , ii ) the tritium and rhenium single @xmath0-decay experiments , iii ) neutrinoless double @xmath0-decay experiments . the measurement of the electron spectrum in @xmath0-decays provides a robust direct determination of the values of neutrino masses . in practice , the most sensitive experiments use tritium @xmath0-decay , because it is a super - allowed transition with a low @xmath5-value . the effect of neutrino masses @xmath6 ( @xmath7=1,2,3 ) can be observed near the end point of the electron spectrum , where @xmath8 . @xmath9 is the electron kinetic energy . a low @xmath5-value is important , because the relative number of events occurring in an interval of energy @xmath10 near the endpoint is proportional to @xmath11 . the current best upper bound on the effective neutrino mass @xmath12 given by , @xmath13 has been obtained in the mainz and troitsk experiments : @xmath14 @xcite . @xmath15 is the element of neutrino mixing matrix . in the near future , the karslruhe tritium neutrino mass ( katrin ) experiment will reach a sensitivity of about @xmath16 @xcite . in this experiment the @xmath0-decay of tritium will be investigated with a spectrometer taking advantage of magnetic adiabatic collimation combined with an electrostatic filter . calorimetric measurements of the @xmath0-decay of rhenium where all electron energy released in the decay is recorded , appear complementary to those carried out with spectrometers . the unique first forbidden @xmath0-decay , @xmath17 is particularly promising due to its low transition energy of @xmath18 and the large isotopic abundance of @xmath19 ( @xmath20% ) , which allows the use of absorbers made with natural rhenium . measurements of the spectra of @xmath19 have been reported by the genova and the milano / como groups ( mibeta and manu experiments ) . the achieved sensitivity of @xmath21 was limited by statistics @xcite . the success of rhenium experiments has encouraged the micro - calorimeter community to proceed with a competitive precision search for a neutrino mass . the ambitious project , called the `` microcalorimeter arrays for a rhenium experiment '' ( mare ) , is planned in two steps , mare i and mare ii . mare i might reach a statistical sensitivity of 4 ev after 3 years of data taking @xcite . mare ii is to challenge the katrin goal of 0.2 ev @xcite . the aim of this paper is to derive the form of the endpoint spectrum of emitted electrons for the @xmath0-decay of @xmath19 , which is needed to extract the effective neutrino mass @xmath12 or to place a limit on this quantity from future mare i and ii experiments . the ground - state spin - parity is @xmath22 for @xmath19 and @xmath23 for the daughter nucleus @xmath24 , and the decay is associated with @xmath25 ( @xmath26 , @xmath27 ) of the nucleus , _ i.e. _ , classified as first unique forbidden @xmath0-decay . the emitted electron and neutrino are expected to be , respectively , in @xmath28 and @xmath29 states ( see appendix a ) or vice versa . the emission of higher partial waves is strongly suppressed due to a small energy release in this nuclear transition . or @xmath30 ) for emission of @xmath29 and @xmath28 electrons vs. electron energy @xmath31 for @xmath0-decay of @xmath19 . ] the differential decay rate is a sum of two contributions associated with emission of the @xmath29 and the @xmath28 state electrons . by considering the finite nuclear size effect the theoretical spectral shape of the @xmath0-decay of @xmath32 is @xmath33 \nonumber\\ & & ~~\times \sqrt{(e_0 - e_e)^2-m^2_{k}}~~ \theta(e_0-e_e - m_k ) \label{eq.sp}\end{aligned}\ ] ] with @xmath34 @xmath35 is the fermi constant and @xmath36 is the element of the cabbibo - kobayashi - maskawa ( ckm ) matrix . @xmath37 , @xmath31 and @xmath38 are the momentum , energy , and maximal endpoint energy ( in the case of zero neutrino mass ) of the electron , respectively . @xmath39 and @xmath40 in eq . ( [ eq.sp ] ) are relativistic fermi functions and @xmath41 is a theta ( step ) function . @xmath42 denotes an axial - vector coupling constant . @xmath43 is a coordinate of the @xmath44-th nucleon and @xmath45 is a nuclear radius . the value of nuclear matrix element @xmath46 in eq . ( [ nme ] ) can be determined from the measured half - life of the @xmath0-decay of @xmath19 . -decay for various values of the effective neutrino mass : @xmath47 and @xmath48 ev . ] the @xmath0-decay rate of rhenium is a sum of decay rates for emission of @xmath28 and @xmath29 electrons [ see eq . ( [ eq.sp ] ) ] . in fig . [ fig:1 ] we show the single electron differential decay rate normalized to the particular decay rate as a function of electron energy @xmath31 . the two possibilities offer a different energy distribution of outgoing electrons . close to the end point , there is a more flat distribution for @xmath29 electrons due to the dependence on squared neutrino momentum @xmath49 [ see eq . ( [ eq.sp ] ) ] . because of this factor the two particular decay rates depend on the neutrino mass in a different way . experimentally , it was found that @xmath28-state electrons are predominantly emitted in the @xmath0-decay of @xmath50 @xcite . by performing numerical analysis of partial decay rates associated with emission of the @xmath28 and the @xmath29 electrons ( terms with @xmath51 and @xmath52 in eq . ( [ eq.sp ] ) , respectively ) we conclude that about @xmath53 times more @xmath28-state electrons are emitted when compared with emission of @xmath29-state electrons . the reasons for it are as follows : i ) @xmath54 for @xmath55 ( see appendix b ) , ii ) the maximal momentum of electron ( @xmath56 kev ) is much larger than the maximal momentum of the neutrino ( @xmath57 kev ) . henceforth , we shall neglect a small contribution to the differential decay rate given by an emission of the @xmath29-state electrons . for a normal hierarchy ( nh ) of neutrino masses with @xmath58 the kurie function of the @xmath0-decay of @xmath50 is given by @xmath59 ^{1/2 } , \label{kpre1}\end{aligned}\ ] ] with @xmath60 and @xmath61 . the ratio @xmath62 depends only weakly on the electron momentum in the case of the @xmath0-decay of rhenium . with a good accuracy the factor @xmath63 can be considered to be a constant . the current upper limit on neutrino mass from tritium and rhenium @xmath0-decay experiments holds in the degenerate neutrino mass region ( @xmath64 with @xmath65 ) for which @xmath66 . so far the future rhenium @xmath0-decay experiments will not see any effect due to a smallness of neutrino masses , it is possible to approximate @xmath67 ( k=1,2,3 ) and obtain @xmath68 where @xmath69 . in fig . [ fig:2 ] we show the kurie plot for @xmath0-decay of @xmath19 versus @xmath70 near the endpoint . we see that the kurie plot is linear near the endpoint for @xmath71 . however , the linearity of the kurie plot is lost if @xmath12 is not equal to zero . we note that there is also a possibility of bound - state decay of @xmath19 , where an electron is placed into a bound state above occupied electron shells of the final atom . in ref . @xcite it was shown that the ratio of decay rates of bound - state and continuum @xmath0-decays of @xmath19 is less than @xmath72 . for the first unique forbidden @xmath0-decay of @xmath19 to ground state of @xmath24 the theoretical spectral shape is presented . the decay rate of the process is a sum of particular decay rates associated with emission of @xmath29 and @xmath28 electrons , which depend in a different way on the neutrino mass . the p - wave emission is dominant over the s - wave . so , the kurie function , defined by eq . ( [ bre ] ) , is almost linear in the endpoint region . an observed deviation from the linearity indicates effects of the finite neutrino mass . the analysis of the the kurie function of the first forbidden @xmath0-decay of @xmath19 show that close to the endpoint it coincides up to a factor to the kurie function of superallowed @xmath0-decay of tritium @xcite . these findings are important for the planned mare experiment , which will be sensitive to neutrino mass in the sub ev region . this work was supported in part by the dfg project 436 slk 17/298 and by the vega grant agency under the contract no . and @xmath73 vs. electron energy @xmath31 for the @xmath0-decay of @xmath19 . ] we adopt the relativistic electron wave function in a uniform charge distribution in a nucleus , which is expanded in terms of spherical waves : @xmath74 where of particular interest are @xmath29 and @xmath28 states @xcite : @xmath75 \chi_s \\ { \tilde f}_{+2}(r ) [ ( \mathbf{\hat{r}\cdot\hat{p } } ) ( \mathbf{\sigma\cdot\hat{p } } ) - ( \mathbf{\sigma\cdot\hat{r } } ) ] \chi_s \end{array}\right).\nonumber\\\end{aligned}\ ] ] here , @xmath76 is a position vector , @xmath77 , @xmath78 and @xmath79 . by keeping the lowest power in expansion of @xmath80 the radial wave functions take the form : @xmath81 in approximation up to @xmath82 terms we have @xmath83 relativistic fermi function @xmath84 takes into account the distortion of electron wave function due to electromagnetic interaction of the emitted electron with the atomic nucleus . it takes the form @xcite : @xmath85 where @xmath86 and @xmath87 following ref . @xcite it can be written as @xmath88 the constant @xmath89 is given by @xmath90 ^ 2 . \nonumber\\\end{aligned}\ ] ] the function @xmath91 takes the form @xmath92 we note that @xmath93 the fermi functions @xmath94 and @xmath73 are related to the emission of @xmath29 and @xmath28 electrons , respectively . in fig . [ fig:3 ] , they are plotted as functions of the kinetic energy of electrons emitted in @xmath0-decay of @xmath95 . we note that values of @xmath73 are significantly larger than those of @xmath94 . a quantity of interest is the ratio @xmath96.\end{aligned}\ ] ] as the @xmath5-value of rhenium @xmath0-decay is only @xmath97 kev the above ratio can be to a good accuracy considered to be a constant with the value @xmath98 . 99 e.w . otten , c. weinheimer , rept . prog . phys . * 71 * , 086201 ( 2008 ) . katrin collaboration , a. osipowicz _ et al . _ , arxiv : 0109033 [ hep - ex ] ; l. bornschein _ et al . _ , a * 752 * , 14 ( 2005 ) ; c. weinheimer , nucl . . suppl . * 168 * , 5 ( 2007 ) . mare collaboration , e. andreotti _ et al . _ , nucl . instrum . meth . a * 572 * , 208 ( 2007 ) ; a. nucciotti , arxive : 1012.2290 [ hep - ex ] . c. arnaboldi _ et al . _ , lett . * 96 * , 042503 ( 2006 ) . williams , w.a . fowler , and s.e . koonin , apj * 281 * , 363 ( 1984 ) . f. imkovic , r. dvornick and a. faessler , phys . c * 77 * , 055502 ( 2008 ) . m. doi , t. kotani , and e. takasugi , prog . ( supp . ) * 83 * , 1 ( 1985 ) . m. doi , t. kotani , prog . phys . * 87 * , 1207 ( 1992 ) . | the planned rhenium @xmath0-decay experiment , called the `` microcalorimeter arrays for a rhenium experiment '' ( mare ) , might probe the absolute mass scale of neutrinos with the same sensitivity as the karlsruhe tritium neutrino mass ( katrin ) experiment , which will take commissioning data in 2011 and will proceed for 5 years . we present the energy distribution of emitted electrons for the first unique forbidden @xmath0-decay of @xmath1 .
it is found that the @xmath2-wave emission of electron dominates over the @xmath3-wave . by assuming mixing of three neutrinos the kurie function for the rhenium @xmath0-decay
is derived .
it is shown that the kurie plot near the endpoint is within a good accuracy linear in the limit of massless neutrinos like the kurie plot of the superallowed @xmath0-decay of @xmath4 . |
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in recent years there has been growing interst in graph polynomials , functions from graphs to polynomial rings which are invariant under isomorphism . graph polynomials encode information about the graphs in a compact way in their evaulations , coeffcients , degree and roots . therefore , efficient computation of graph polynomials has received considerable attention in the literature . since most graph polynomials which naturally arise are @xmath6-hard to compute ( see e.g. @xcite ) , a natural perspective under which to study the complexity of graph polynomials is that of _ parameterized complexity_. parameterized complexity is a successful approach to tackling @xmath7-hard problems @xcite , by measuring complexity with respect to an additional _ parameter _ of the input ; we will be interested in the parameters tree - width and clique - width . a computational problem is _ * fixed - parameter tractable * _ ( @xmath1 ) with respect to a parameter @xmath5 if it can be solved in time @xmath8 , where @xmath9 is a computable function of @xmath5 , @xmath4 is the size of the input , and @xmath10 is a polynomial in @xmath4 . many @xmath7-hard problems are fixed parameter tractable for an appropriate choice of parameter , see @xcite for many examples . every problem in the infinite class of decision problems definable in _ monadic second order logic _ ( @xmath0 ) is fixed - parameter tractable with respect to tree - width by courcelle s theorem @xcite ( though the result originally was not phrased in terms of parameterized complexity ) . the computation problem we consider for a graph polynomial @xmath11 is the following : @xmath12 @xmath13 : a graph @xmath14 @xmath15 compute the coefficients @xmath16 of the monomials @xmath17 . for graph polynomials , a parameterized complexity theory with respect to tree - width has been developed . here , the goal is to compute , given an input graph , the table of coefficients of the graph polynomial . the tutte polynomial has been shown to be fixed - parameter tractable @xcite . @xcite used a logical method to study the parameterized complexity of an infinite class of graph polynomials , including the tutte polynomial , the matching polynomial , the independence polynomial and the ising polynomial . @xcite showed that the class of graph polynomials definable in @xmath0 in the vocabulary of hypergraphs in the vocabulary of hypergraphs is denoted @xmath18 , while @xmath0 in the vocabulary of graphs is denoted @xmath19 . ] is fixed - parameter tractable . this class contains the vast majority of graph polynomials which are of interest in the literature . going beyond tree - width to clique - width the situation becomes more complicated . @xcite studied the class of graph polynomials definable in @xmath0 _ in the vocabulary of graphs_. they proved that every graph polynomial in this class is fixed - parameter tractable with respect to clique - width . however , this class of graph polynomials does not contain important examples such as the chromatic polynomial , the tutte polynomial and the matching polynomial . in fact , @xcite proved that the chromatic polynomial and the tutte polynomial are not fixed - parameter tractable with respect to clique - width ( under the widely believed complexity - theoretic assumption that @xmath20 $ ] ) . @xcite proved that the chromatic polynomial and the matching polynomial are _ * fixed - parameter polynomial time * _ computable with respect to clique - width , meaning that they can be computed in time @xmath21 , where @xmath4 is the size of the graph , @xmath22 is the clique - width of the graph and @xmath9 is a computable function . @xcite proved an analogous result for the ising polynomial . the main result of this paper is a meta - theorem generalizing the fixed - parameter polynomial time computability of the chromatic polynomial , the matching polynomial and the ising polynomial to an infinite family of graph polynomials definable in @xmath0 analogous to @xcite . let @xmath23 be an @xmath0-ising polynomial . @xmath23 is fixed - parameter polynomial time computable with respect to clique - width . the class of @xmath0-ising polynomials is defined in section [ se : msolising ] . let @xmath24=\left\ { 1,\ldots , k\right\}$ ] . let @xmath25 be the vocabulary of graphs @xmath26 consisting of a single binary relation symbol @xmath27 . a @xmath5-graph is a structure @xmath28 which consists of a simple graph @xmath29 together with a partition @xmath30 of @xmath31 . let @xmath32 denote the vocabulary of @xmath5-graphs @xmath33 extending @xmath25 with unary relation symbols @xmath34 . the class @xmath35 of @xmath5-graphs of clique - width at most @xmath5 is defined inductively . singletons belong to @xmath35 , and @xmath35 is closed under disjoint union @xmath36 and two other operations , @xmath37 and @xmath38 , to be defined next . for any @xmath39 $ ] , @xmath40 is obtained by relabeling any vertex with label @xmath41 to label @xmath42 . for any @xmath39 $ ] , @xmath43 is obtained by adding all possible edges @xmath44 between members of @xmath41 and members of @xmath42 . the clique - width of a graph @xmath14 is the minimal @xmath5 such that there exists a labeling @xmath45 for which @xmath46 belongs to @xmath35 . we denote the clique - width of @xmath14 by @xmath22 . the clique - width operations @xmath37 and @xmath38 are well - defined for @xmath5-graphs . the definitions of these operations extend naturally to structures @xmath47 which expand @xmath5-graphs with @xmath48 . a @xmath5-expression is a term @xmath49 which consists of singletons , disjoint unions @xmath36 , relabeling @xmath37 and edge creations @xmath38 , which witnesses that the graph @xmath50 obtained by performing the operations on the singletons is of clique - width at most @xmath5 . every graph of tree - width at most @xmath5 is of clique - width at most @xmath51 , cf . @xcite . while computing the clique - width of a graph is @xmath7-hard , s. oum and p. seymour showed that given a graph of clique - width @xmath5 , finding a @xmath52-expression is fixed parameter tractable with clique - width as parameter , cf . @xcite . for a formula @xmath53 , let @xmath54 denote the quantifier rank of @xmath53 . for every @xmath55 and vocabulary @xmath56 , we denote by @xmath57 the set of @xmath0-formulas on the vocabulary @xmath56 which have quantifier rank at most @xmath58 . for two @xmath56-structures @xmath59 and @xmath60 , we write @xmath61 to denote that @xmath59 and @xmath60 agree on all the sentences of quantifier rank @xmath58 . an @xmath62-ary operation @xmath63 on @xmath56-structures is called _ smooth _ if for all @xmath64 , whenever @xmath65 for all @xmath66 , we have @xmath67 smoothness of the clique - width operations is an important technical tool for us : [ smoothness , cf . @xcite ] [ th : smooth ] 1 . for every vocabulary @xmath56 , the disjoint union @xmath36 of two @xmath56-structures is smooth . 2 . for every @xmath68 , @xmath37 and @xmath38 are smooth . it is convenient to reformulate theorem [ th : smooth ] in terms of _ hintikka sentences _ ( see @xcite ) : [ hintikka sentences ] [ prop : hin ] let @xmath56 be a vocabulary . for every @xmath69 there is a finite set @xmath70 of @xmath57-sentences such that 1 . every @xmath71 has a finite model . 2 . the conjunction @xmath72 of any two distinct @xmath73 is unsatisfiable . every @xmath57-sentence @xmath74 is equivalent to exactly one finite disjunction of sentences in @xmath75 . 4 . every @xmath56-structure @xmath59 satisfies a unique member @xmath76 of @xmath75 . in order to simplify notation we omit the subscript @xmath56 in @xmath77 when @xmath56 is clear from the context . let @xmath78 the be the vocabulary consisting of the binary relation symbol @xmath79 and the unary relation symbol @xmath80 . let @xmath81 extend @xmath78 with the unary relation symbols @xmath82 . from theorem [ th : smooth ] and proposition [ prop : hin ] we get : for every @xmath83 : 1 . there is @xmath84 such that , for every @xmath85 and @xmath86 , @xmath87 . 2 . for every unary operation @xmath88\}$ ] , there is @xmath89 such that , for every @xmath90 , @xmath91 . for every @xmath92 , let @xmath93 , where @xmath94 are new unary relation symbols . [ @xmath0-ising polynomials ] [ def : msolising ] for every @xmath95 , @xmath96 and @xmath29 we define @xmath97 as follows : @xmath98 @xmath99 is the sum over partitions @xmath100 of @xmath31 such that @xmath101 satisfies @xmath74 of the monomials obtained as the product of @xmath102 for all @xmath103 and @xmath104 for all @xmath105 . [ ising polynomial ] the trivariate ising polynomial @xmath106 is a partition function of the ising model from statistical mechanics used to study phase transitions in physical systems in the case of constant energies and external field . @xmath106 is given by @xmath107 where @xmath108 denotes the set of edges between @xmath109 and @xmath110 , and @xmath111 denotes the set of edges inside @xmath109 . @xmath106 was the focus of study in terms of hardness of approximation in @xcite and in terms of hardness of computation under the exponential time hypothesis was studied in @xcite . @xcite also showed that @xmath106 is fixed - parameter polynomial time computable . @xmath112 generalizes a bivariate ising polynomial , which was studied for its combinatorial properties in @xcite . @xcite showed that @xmath106 contains the matching polynomial , the van der waerden polynomial , the cut polynomial , and , on regular graphs , the independence polynomial and clique polynomial . the evaluation of @xmath113 at @xmath114 , @xmath115 , @xmath116 , @xmath117 and @xmath118 gives @xmath106 and therefore @xmath106 is an @xmath0-ising polynomial . [ independence - ising polynomial ] the independence - ising polynomial @xmath119 is given by @xmath120 @xmath119 contains the independence polynomial as the evaluation @xmath121 . see the survey @xcite for a bibliography on the independence polynomial . the evaluation @xmath122 is @xmath123 , where @xmath124 is the number of isolated vertices in @xmath14 . @xmath119 is an evaluation of an @xmath0-ising polynomial : @xmath125 where @xmath126 . [ dominating - ising polynomial ] the dominating - ising polynomial is given by @xmath127 @xmath128 where @xmath108 denotes the set of edges between @xmath109 and @xmath110 . @xmath127 contains the domination polynomial @xmath129 . @xmath129 is the generating function of its dominating sets and we have @xmath130 . the domination polynomial first studied in @xcite and it and its variations have received considerable attention in the literature in the last few years , see e.g. @xcite . previous research focused on combinatorial properties such as recurrence relations and location of roots . hardness of computation was addressed in @xcite . @xmath127 encodes the degrees of the vertices of @xmath14 : the number of vertices with degree @xmath131 is the coefficient of @xmath132 in @xmath127 . @xmath127 is an @xmath0-ising polynomial given by @xmath133 , where @xmath134 two classes of graph polynomials which have received attention in the literature are : 1 . @xmath0-polynomials on the vocabulary of graphs , and 2 . @xmath0-polynomials on the vocabulary of hypergraphs . see e.g. @xcite for the exact definitions . the former class contains graph polynomials such as the independence polynomial and the domination polynomial . the latter class contains graph polynomials such as the tutte polynomial and the matching polynomial . every graph polynomial which is @xmath0-definable on the vocabulary of graphs is also @xmath0-definable on the vocabulary of hypergraphs . the class of @xmath0-ising polynomials strictly contains the @xmath0-polynomials on graphs , see fig . the containment is by definition . for the strictness , we use the fact that by definition the maximal degree of any indeterminate in an @xmath0-polynomial on graphs grows at most linearly in the number of vertices , while the maximal degree of @xmath135 in the ising polynomial @xmath136 of the complete bipartite graph @xmath137 equals @xmath138 . every @xmath0-ising polynomial @xmath99 is an @xmath0-polynomial on the vocabulary of hypergraphs , given e.g. by @xmath139 where the summation over @xmath140 is exactly as in definition [ def : msolising ] , and the summation over @xmath141 is over tuples @xmath142 of subsets of the edge set of @xmath14 satisfying @xmath143 , where @xmath144 we use the fact that @xmath145 is a partition of the set of vertices is definable in @xmath0 . + containments of classes of graph polynomials definable in @xmath0 . , title="fig : " ] we are now ready to state the main theorem and prove a representative case of it . [ main theorem][th : main ] for every @xmath0-ising polynomial @xmath99 there is a function @xmath146 such that @xmath147 is computable on graphs @xmath14 of size @xmath4 and of clique - width at most @xmath5 in running time @xmath148 . we prove the theorem for graph polynomials of the form @xmath149 for every @xmath150 . the summation in @xmath151 is over subsets @xmath109 of the vertex set of @xmath14 . the graph polynomials @xmath151 are a notational variation of @xmath99 with @xmath152 , @xmath115 and @xmath153 : for every @xmath154 , @xmath155 , where @xmath156 is obtained from @xmath74 by substituting @xmath157 with @xmath158 and @xmath159 with @xmath160 . the proof for the general case is in similar spirit . for every @xmath55 there is a finite set @xmath161 of @xmath162-ising polynomials such that , for every formula @xmath163 , @xmath151 is a sum of members of @xmath161 ( see below ) . the algorithm computes the values of the members of @xmath161 on @xmath14 by dynamic programming over the parse term of @xmath14 , and using those values , the value of @xmath164 on @xmath14 . more precisely , for every @xmath165 , let @xmath166 and let @xmath167 every @xmath168 also belongs to @xmath169 , and hence there exists by proposition [ prop : hin ] a set @xmath170 such that @xmath171 hence , @xmath172 setting @xmath173 and @xmath174 for all @xmath175 . for tuples @xmath176),(b_{c_{1},c_{2}}:c_{1},c_{2}\in[k])\right)\in[n]^{k}\times[n]^{k^2}$ ] , let @xmath177 be the coefficient of @xmath178 in @xmath179 . given a @xmath5-graph @xmath14 , the algorithm first computes a parse tree @xmath49 as in @xcite . the algorithm then computes @xmath179 for all @xmath165 by induction over @xmath49 : 1 . if @xmath14 is a graph of size @xmath180 , then @xmath181 is computed directly . 2 . let @xmath14 be the disjoint union of @xmath182 and @xmath183 . we compute @xmath184 for every @xmath185 and @xmath186^{k}\times[n]^{k^2}$ ] as follows : @xmath187 3 . let @xmath188 . we compute @xmath184 for every @xmath165 and @xmath186^{k}\times[n]^{k^2}$ ] as follows : @xmath189 where the inner summation is over @xmath190 such that @xmath191 and @xmath192 4 . let @xmath193 with @xmath194 . let @xmath195 be the number of vertices in @xmath14 . we compute @xmath184 for every @xmath165 and @xmath186^{k}\times[n]^{k^2}$ ] as follows : @xmath196 where the summation is over @xmath190 such that @xmath197 and @xmath198 finally , the algorithm computes @xmath151 as the sum from eq . ( [ eq : sumqa ] ) . the main observations for the runtime analysis are : * the size of the set @xmath199 of hintikka sentences is a function of @xmath5 but does not depend on @xmath4 . let @xmath200 . * by definition of @xmath201 , for a monomial @xmath202 to have a non - zero coefficient , it must hold that @xmath203 and @xmath204 , since @xmath205 and @xmath206 are sizes of sets of vertices and sets of edges , respectively . * the coefficient of any monomial of @xmath201 is at most @xmath207 . * the parse tree guaranteed in @xcite is of size @xmath208 for suitable @xmath209 and @xmath210 . the algorithm performs a single operation for every node of the parse tree . * singletons * : the coefficients of every @xmath211 for a singleton @xmath5-graph can be computed in time @xmath212 , which can be bounded by @xmath213 . * disjoint union * , * recoloring * * and * * edge additions * : the algorithm sums over ( 1 ) @xmath214 or pairs @xmath215 and ( 2 ) over @xmath216^{k}\times[n]^{k^2}$ ] or pairs @xmath217^{k}\times[n]^{k^2}\right)^{2}$ ] , then ( 3 ) performs a fixed number of arithmetic operations on numbers which can be written in @xmath218 space . each node in the parse tree requires time at most @xmath219^{k}\times[n]^{k^2}\right)^{2}\right)$ ] . since the size of the parse tree is @xmath208 , the algorithm runs in fixed - parameter polynomial time . we have defined a new class of graph polynomials , the @xmath0-ising polynomials , extending the @xmath0-polynomials on the vocabulary of graphs and have shown that every @xmath0-ising polynomial can be computed in fixed - parameter polynomial time . this result raises the question of which graph polynomials are @xmath0-ising polynomials . in previous work @xcite we have developed a method based on connection matrices to show that graph polynomials are not definable in @xmath0 over either the vocabulary of graphs or hypergraphs . the tutte polynomial does not seem to be an @xmath0-ising polynomial . @xcite proved that the tutte polynomial can be computed in subexponential time for graphs of bounded clique - width . more precisely , the time bound in @xcite is of the form @xmath220 , where @xmath221 for all @xmath222 . is there a natural infinite class of graph polynomials definable in @xmath0 which includes the tutte polynomial such that membership in this class implies _ fixed parameter subexponential time _ computability with respect to clique - width ( i.e. , that the graph polynomial is computable in @xmath223 time for some function @xmath224 satisfiying @xmath225 for all @xmath222 ) ? fedor v. fomin , petr a. golovach , daniel lokshtanov , and saket saurabh . algorithmic lower bounds for problems parameterized with clique - width . in _ proceedings of the twenty - first annual acm - siam symposium on discrete algorithms , soda 2010 , austin , texas , usa , january 17 - 19 , 2010 _ , pages 493502 , 2010 . benny godlin , tomer kotek , and johann a. makowsky . evaluations of graph polynomials . in _ graph - theoretic concepts in computer science , 34th international workshop , wg 2008 , durham , uk , june 30 - july 2 , 2008 . revised papers _ , pages 183194 , 2008 . johann a. makowsky . connection matrices for msol - definable structural invariants . in _ logic and its applications , third indian conference , icla 2009 , chennai , india , january 7 - 11 , 2009 . proceedings _ , pages 5164 , 2009 . johann a. makowsky , udi rotics , ilya averbouch , and benny godlin . computing graph polynomials on graphs of bounded clique - width . in _ graph - theoretic concepts in computer science _ , pages 191204 . springer , 2006 . | graph polynomials which are definable in monadic second order logic ( @xmath0 ) on the vocabulary of graphs are fixed - parameter tractable ( @xmath1 ) with respect to clique - width .
in contrast , graph polynomials which are definable in @xmath0 on the vocabulary of hypergraphs are fixed - parameter tractable with respect to tree - width , but not necessarily with respect to clique - width .
no algorithmic meta - theorem is known for the computation of graph polynomials definable in @xmath0 on the vocabulary of hypergraphs with respect to clique - width .
we define an infinite class of such graph polynomials extending the class of graph polynomials definable in @xmath0 on the vocabulary of graphs and prove that they are fixed - parameter polynomial time ( @xmath2 ) computable , i.e. that they can be computed in time @xmath3 , where @xmath4 is the number of vertices and @xmath5 is the clique - width . |
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in this paper we continue our investigation of @xmath0 scattering at high energies ( see ref.@xcite for our previous attempts to study this process in the dglap dynamics ) . we concentrate our efforts here on the case of two photons with large but almost equal virtualities . it has been argued @xcite that this process is the perfect tool to recover the bfkl dynamics @xcite which is the key problem in our understanding of the low @xmath7 ( high energy ) asymptotic behavior in qcd . it is well known that the correct degrees of freedom at high energy are not quarks or gluon but colour dipoles @xcite which have transverse sizes @xmath8 and the fraction of energy @xmath9 . therefore , two photon interactions occur in two successive steps . first , each virtual photon decays into a colour dipole ( quark - antiquark pair ) with size @xmath8 . at large value of photon virtualities the probability of such a decay can be calculated in pqcd . the second stage is the interaction of colour dipoles with each other . the simple formula ( see for example ref . @xcite ) that describes the process of interaction of two photons with virtualities @xmath10 and @xmath11 ( @xmath12 ) is ( see ) [ pps ] ( q_1 , q_2,w)= d^2 b_t ^n_f_a , b@xmath13 where the indexes @xmath14 and @xmath15 specify the flavors of interacting quarks , @xmath16 and @xmath17 indicate the polarization of the interacting photons where @xmath18 denote the transverse separation between quark and antiquark in the dipole ( dipole size ) and @xmath19 are the energy fractions of the quark in the fluctuation of photon @xmath20 into quark - antiquark pair . @xmath21 is the imaginary part of the dipole - dipole amplitude at @xmath7 given by [ x ] x = for massless quarks ( @xmath22 is the energy of colliding photons in c.m.f . ) . @xmath1 is the impact parameter for dipole - dipole interaction and it is equal the transverse distance between the dipole centers of mass . the wave functions for virtual photon are known @xcite and they are given by ( for massless quarks ) @xmath23 with @xmath24 where @xmath25 denote the faction of quark charge of flavor @xmath14 . since the main contribution in is concentrated at @xmath26 and @xmath27 where @xmath28 is the soft mass scale , we can safely use pqcd for calculation of the dipole - dipole amplitude @xmath29 in . in this paper we study this process in the region of high energy and large but more - less equal photon virtualities ( @xmath30 ) in the framework of the bfkl dynamics . in the region of very small @xmath7 ( high energies ) the saturation of the gluon density is expected @xcite . we will deal with this phenomenon using glauber - mueller formula @xcite which is the simplest one that reflects all qualitative features of a more general approach based on non - linear evolution @xcite . for @xmath0 scattering with large but equal photon virtualities , the glauber - mueller approach is the only one on the market since the non - linear equation is justified only for the case when one of the photon has larger virtuality than the other . in the next section we discuss the dipole - dipole interaction in the bfkl approach of pqcd . the solution to the bfkl equation , that describes the dipole - dipole interaction in our kinematic region , has been found @xcite and our main concern in this section is to find the large impact parameter ( @xmath1 ) behavior of the solution . as was discussed in ref . @xcite , we have to introduce non - perturbative corrections in the region of @xmath1 larger than @xmath31 where @xmath32 is the pion mass . we argue in this section that it is sufficient to introduce the non - perturbative behavior into the born approximation to obtain a reasonable solution at large @xmath1 . section 3 is devoted to glauber - mueller formula in the case of the bfkl emission @xcite . here , we use the advantage of photon - photon scattering with large photon virtualities , since we can calculate the gluon density without uncertainties related to non - perturbative initial distributions in hadronic target . we consider the low @xmath7 behavior of the dipole - dipole cross section and show that the large impact parameter behavior , introduced in the born cross section , fulfills the unitarity restrictions ( unitarity bound @xcite ) . therefore , we confirm that the large @xmath1 behavior can be concentrated in the initial condition ( see refs . @xcite without changing the kernel of the non - linear equation that governs evolution in the saturation region as it is advocated in ref.@xcite . in the last section we summarize our results . in this section we discuss the one parton shower interaction in the bfkl dynamics ( see ) . we start with the born approximation which is the exchange of two gluons ( see ) or the diagrams of without emission of a gluon . these diagrams have been calculated in ref . @xcite using the approach of ref . @xcite and they lead to the following expression for the dipole - dipole amplitude : @xmath33 where @xmath19 is the fraction of the energy of the dipole carried by quarks ; @xmath34 and @xmath35 . @xmath36 is the coordinate of quark @xmath20 ( see ) . all vectors are two dimensional in . each diagrams in is easy to calculate @xcite and the first diagram is equal to [ ba2 ] _ s^2 ^2_1,1^2_2,2. summing all diagrams we obtain . we are interested mostly in the limit of large @xmath37 where the dipole - dipole amplitude can be reduced to a simple form . [ balb ] n^ba ( r_1 , t , r_2 , t ; b_t)_s^2 , after integration over azimuthal angles . therefore , we have a power - like decrease of the dipole - dipole amplitude at large @xmath1 , namely @xmath38 . such behavior can not be correct since it contradicts the general postulates of analyticity and crossing symmetry of the scattering amplitude @xcite . since the spectrum of hadrons has no particles with mass zero , the scattering amplitude should decrease as @xmath39 @xcite . in ref . @xcite we suggested a procedure of how to cure this problem which is based on the results of qcd sum rules @xcite.following this procedure we rewrite the dipole - dipole amplitude as the integral over the mass of two gluons in @xmath40-channel ; and we assume , as in qcd sum rules , that this integral describes all hadronic states on average . restricting the integral over mass by the minimal mass of hadronic states ( 2 @xmath32 ) we obtain the model which provides the exponential fall at large @xmath41 and does not change the power like behavior for small @xmath42 . we choose for the born amplitude the following formula [ bacorr ] n^ba(r_1 , t , r_2 , t ; b_t)=_s^2 k_4(2m_b_t ) . one can easily see that reproduces and leads to [ bacorlb ] n^ba ( r_1 , t , r_2 , t ; b_t)_s^2 e^ - 2m_b_t . at large @xmath43 . the emission of a gluon is described by the bfkl equation @xcite which was solved in ref.@xcite for fixed @xmath1 ( see ref.@xcite for many useful discussion of the different aspects of the solution ) . the solution can be presented in factorized form ( see ) . [ bfkl1 ] n(x , r_1 , t , r_2 , t ; b_t)= @xmath44 with [ omega ] ( ) = ( 2 ( 1 ) -(- i)- ( + i ) ) ; where @xmath45,@xmath46 is euler gamma function and where [ v ] v(r_i , t , r_i;)= ( ) ^ -i using the following notations : @xmath47;@xmath48 is the size of the colour dipole @xmath49 and @xmath50 is the position of the center of mass of this dipole . in function @xmath51 should be found from the initial condition which determines the dipole amplitude at fixed @xmath52 , namely , @xmath53 . it should be stressed that the bfkl equation is a linear equation in which the kernel does not depend on @xmath1 ( see ref.@xcite ) . therefore , @xmath54 could be an arbitrary function on @xmath1 . in we can take the integral over @xmath55 which leads to [ v2 ] v(r_2,t , r_2;-)= ( ) ^ + i. we are interested in the large @xmath1 behavior , namely , @xmath56 . it is instructive to consider two cases : * @xmath57 . + this is so called double log approximation of pqcd ( dla ) in which we consider @xmath58 and @xmath59 while @xmath60 as well as @xmath61 and @xmath62 . we have considered this case in ref.@xcite and found that the emission of gluons does not induce any additional dependence on @xmath1 which is concentrated only in the born amplitude . indeed , we can see this property directly from the solution of . + integrating over @xmath63 we find that the integrand of this integral falls down rapidly for @xmath64 due to @xmath63 dependence of the vertex @xmath65 ( see ) providing a good convergence for the integral . for @xmath66 we can neglect @xmath63 dependence of the vertex @xmath65 and consider it as @xmath67 . the integral over @xmath63 of @xmath68 for @xmath66 gives @xmath69 . + therefore , @xmath70 . finally , taking @xmath71 , the dipole amplitude has a form [ dla1 ] n^dla(x , r_1 , t , r_2 , t ; b_t ) = n^ba(x , r_1 , t , r_2 , t ; b_t)e^ ( ) y + ( -i)(r^2_1,t / r^2_2,t ) considering @xmath72 and taking into account that @xmath73 at @xmath74 one can take the integral in explicitly . the answer is well known ( see ref.@xcite for example ) , namely , at low @xmath7 [ dla2 ] n^dla(x , r_1 , t , r_2 , t ; b_t)=n^ba(x , r_1 , t , r_2 , t ; b_t)i_0 ( 2 ) for fixed coupling constant . ] . * @xmath75 . + for such small values of @xmath76 the integral over @xmath63 is convergent for @xmath77 ( see ref.@xcite ) and , therefore , we neglect the @xmath63 dependence in @xmath78 . introducing a new variable @xmath79 we see that [ df1 ] d^2 r_1 v(r_1,t , r_1 ; ) = ( r^2_1,t)^+ i d^2 ( + ) ^2 ( - ) ^2)^+ i where @xmath80 is a unit vector in the direction of @xmath81 . the integral is a function of @xmath76 only and can be absorbed in @xmath82 in . for @xmath83 at @xmath84 we have [ vdf ] v(r_2,t , b_t,-)=()^+ i therefore , the dipole amplitude is [ ndf ] n^df(x , r_1 , t , r_2 , t ; b_t)= _ in ( ; b_t ) e^()y ( ) ^+ i. we choose @xmath82 to be of in the form [ incon ] _ in ( ; b_t)= _ s(m_b_t)^4k_4 ( 2m_b_t ) . at small values of @xmath76 we can expand @xmath85 [ kerndf ] ( ) = _ l - d^2 with [ omsn ] _ l=42 ; d=14(3 ) ; finally , we can evaluate the integral over @xmath76 in using the method of steepest decent and obtain the following expression for dipole amplitude : [ andf ] n^df(x , r_1 , t , r_2 , t ; b_t)= @xmath86 + at @xmath87 [ andfb1 ] n^df(x , r_1 , t , r_2 , t ; b_t)@xmath88 + while at @xmath89 [ andfb2 ] n^df(x , r_1 , t , r_2 , t ; b_t ) @xmath90 the glauber - mueller approach @xcite takes into account the interaction of many parton showers with the target as is shown in . in our case of more or less equal but large virtualities of both photons this approach gives a unique opportunity to study the high energy asymptotic behavior of the dipole amplitude since other methods based on non - linear evolution equation @xcite do not work in the case of two dipoles with more - less equal sizes . the main idea of this approach is that the colour dipoles are the correct degrees of freedom for high energy scattering ( this idea was formulated by a.h . mueller in ref . @xcite ) . indeed , the change of the value of the dipole size @xmath8 ( @xmath91 ) during the passage of the colour dipole through the target is proportional to the number of rescatterings ( or the size of the target @xmath92 ) multiplied by the angle @xmath93 where @xmath94 is the energy of the dipole and @xmath95 is the transverse momentum of the @xmath40-channel gluon which is emitted by the fast dipole . [ dof1 ] r_t r. since @xmath95 and @xmath8 are conjugate variables and due to the uncertainty principle @xmath96 therefore , [ dof2 ] r_trr_t rr^2_tex . since the colour dipoles are correct degrees of freedom , they diagonalize the interaction matrix at high energy as well as the unitarity constraints , which have the form [ un ] 2n ( x , r_1,t , r_2,t;b_t)=|a_el ( x , r_1,t , r_2,t;b_t)|^2+g_in(x , r_1,t , r_2,t;b_t ) , where @xmath97 is the elastic amplitude of the dipole - dipole interaction and @xmath29 is the imaginary part of @xmath97 ( @xmath98 ) . assuming that the amplitude is pure imaginary at high energy , one can find a simple solution to , namely @xmath99 where @xmath100 is the arbitrary real function . in glauber - mueller approach the opacity @xmath100 is chosen as @xmath101 where @xmath102 is the dipole - dipole amplitude for one parton shower interaction that has been found in the previous section ( see ) . one can see that if we substitute the explicit solution to the bfkl equation of at any fixed @xmath1 the opacity @xmath103 increases at @xmath104 . therefore , the dipole - dipole amplitude given by glauber - mueller formula of tends to unity in the region of low @xmath7 . this statement is called saturation @xcite since the physical interpretation of @xmath29 is the density of colour dipoles at least when @xmath29 is not very large . in this discussion the saturation appears to be the consequence of unitarity for fixed @xmath1 . however , we have learned several examples where the dipole density could reach a maximum value without having any effect on the elastic dipole - dipole amplitude at fixed @xmath1 ( see ref . @xcite and paper of kovchegov and mueller in ref.@xcite ) . however , for @xmath0 scattering of two small dipoles the initial condition is given by born amplitude of which is small . therefore , we have no reason to expect that the dipole density will be high due to the final state interaction . to obtain the unitarity bound for the dipole - dipole cross section we have to integrate over @xmath1 , namely [ ddxs ] ( dipole - dipole)=2d^2b_tn^gm(x , r_1,t , r_2,t;b_t ) = 2d^2b_t(1-e^- n^df(x , r_1,t , r_2,t;b_t ) ) following froissart @xcite , we divide the region of integration over @xmath1 in in two parts [ ddun1 ] ( dipole - dipole)=2 ^b^2_0(x)_0 db^2_tn^gm( ... ;b_t)+^_b^2_0(x ) d b^2_t n^gm( ... ;b_t ) where @xmath105 is defined from the equation [ b0 ] n^df(x , r_1,t , r_2,t;b_0(x))=1 it is easy to see that for @xmath106 @xmath107 since @xmath108 , while for @xmath109 and for @xmath110 @xmath111 therefore , we have the following unitarity bound [ ddun2 ] ( dipole - dipole)2(b^2_0(x)+^_b_0(x)d b^2_t n^gm( ... ;b_t ) ) let us consider two possibilities . the first one that @xmath112 . in this case the solution to follows directly from for the amplitude @xmath113 and for @xmath114 [ b0s ] ( ) = -2y or [ b0s1 ] b^2_0(x)r_1,tr_2,te^y substituting into we can obtain [ un3 ] ( dipole - dipole)2b^2_0(x)\{1 + 2 } e^y where the second term is calculated by integrating first over @xmath1 and after that using saddle point approach . since @xmath115 turns out to be small at low @xmath7 and we neglect it . therefore , in this kinematic region we face a power -like increase of the dipole - dipole cross section as was pointed out in refs . @xcite . however , this power - like increase will stop for @xmath116 . indeed , for such large values of @xmath1 we should use for the dipole amplitude @xmath113 . for such large values of @xmath105 has a solution which at low @xmath7 is [ sol1 ] b_0(x ) = y + o(ln y ) which leads to [ un4 ] ( dipole - dipole)2b^2_0(x)=^2(x_0/x ) which comes from the first term in . it is easy to understand that the second term in this equation gives a term which does not increase with @xmath117 . is the unitarity bound which has the same energy dependence as for hadron - hadron collisions @xcite but in our approach we are able to calculate the coefficient in front of @xmath118 . the bound of is the same as was derived in @xcite . it should be stressed that the diffusion approximation that we used was derived only at small values of saddle point in @xmath76 integration in which is equal to [ daddle ] | _ saddle| = 1 at @xmath119 from . does not have a solution at any values of @xmath120 and @xmath121 ( formally , we obtain a negative values of @xmath105 ) . the same equation at @xmath122 , namely [ satsc ] n(x , r_sat , r_2,t;b_t = 0 ) = 1 , determines the saturation scale . at @xmath123 the opacity @xmath100 in glauber - mueller formula is larger than unity ( @xmath124 ) , has a solution and we are in the saturation region with for the unitarity bound . if @xmath125 , opacity @xmath126 at any value of @xmath1 . this is a domain of perturbative qcd in virtual photon scattering . @xmath127 we can find from integrating over @xmath63 , namely [ bfklb0 ] n(x , r_1 , t , r_2 , t ; b_t=0)= _ in ( ; b_t = 0 ) d^2 r_1 e^ ( ) y v(r_1,t , r_1;)v(r_2,t , r_1;- ) . since @xmath128 from is much smaller than @xmath121 we need to find only for @xmath129 . this observation simplifies the calculations . indeed , the main contribution in the integral over @xmath63 stems from @xmath130 . therefore , we can neglect the @xmath63 - dependence of vertex @xmath131 which has the form [ v2s ] v(r_2,t , r_1;- ) = ( ) ^+ i. to perform the integration over @xmath63 we use the following formula ( see equation 3.198 of ref.@xcite ) [ fp ] b(- i , - i ) ( ) ^ -i= @xmath132 where @xmath133 is the euler beta - function . integrating over @xmath63 using we obtain that [ bfklb01 ] n(x , r_1 , t , r_2 , t ; b_t=0)= _ in ( ; b_t ) e^ ( ) y ( ) ^+ inu the born approximation at @xmath134 and at @xmath135 can be reduced to @xcite [ bab0 ] n^ba(r_1 , t , r_2 , t ; b_t=0)_s^2 . it is easy to choose @xmath136 in such a way that the final answer for @xmath137 is : [ finb0 ] n(x , r_1 , t , r_2 , t ; b_t=0)= _ s^2 , e^ ( ) y+ ( + i ) ( r^2_1,t / r^2_2,t ) . we can find the solution to in the saddle point approximation for the integral over @xmath76 in @xcite . introducing a new variable @xmath138 we have the following equation for the saddle point value of @xmath139 : [ sadga ] |_= _ s y + ln(r^2_1,t / r^2_2,t)=0 substituting in we obtain [ satsc1 ] n(x , r_1 , t , r_2 , t ; b_t=0)e^ ( _ s)y- _ s ( r^2_2,t / r^1_2,t)=e^ y \{(_s)- _ s|_= _ s } using we can solve in semiclassical approximation ( see ref . @xcite in which we can not calculate the numerical factor in front of . indeed , @xmath140 is constant on the line [ crl ] |_= _ 0 y + ln(r^2_sat(x)/r^2_2,t)=0 with @xmath141 is the solution to the equation @xcite was called @xmath142 in ref.@xcite and @xmath143 in ref.@xcite . the numerical solution of leads to @xmath144 . ] [ crg ] = |_= _ 0 leads to a power - like increase of the saturation momentum ( @xmath145 ) at high energies ( low @xmath7 ) . namely , [ satsc2 ] q^2_sat(x ) ( ) ^q^2_2 ( ) ^ actually , the pre - exponential factors in the steepest decent method of taking integral over @xmath146 could change the @xmath7-dependence of the saturation scale adding some log(1/x ) dependence in ( see ref . @xcite for an analysis of such corrections ) . to obtain the unitarity bounds for @xmath0 scattering we need to substitute the unitarity bound for dipole - dipole cross section ( see ) into and to perform integrations over @xmath148 and @xmath19 . @xmath149 is convergent while @xmath150 has a logarithmic divergence that we need to deal with . holds only for @xmath151 since if @xmath152 dipole - dipole cross section is small and proportional to @xmath153 . as has been mentioned we consider @xmath154 . on the other hand @xmath155 at @xmath156 . finally , one can see [ int ] ^1/|q_1_r_sat d^2r_t ^1_0dz_1 |_t(q , z , r_t)|^2= c_q ( q^2_sat(x)r^2_2,t ) , where @xmath157 . we recall that @xmath158 does not depend on @xmath121 . the integration over @xmath121 is concentrated at the limits @xmath159 and leads to @xmath160 finally , for @xmath0 cross sections we have @xmath161 since the saturation scale increases as a power of @xmath162 one can see that the energy behavior of the unitarity constraints is @xmath163 note only @xmath164 has the same energy dependence as hadron - hadron collisions @xcite but even this cross section has a different coefficient in front . @xmath165 as well the as numerical factor @xmath166 come from the photon wave function while @xmath167 reflects the bfkl dynamics making and quite different from the unitarity bound for hadronic reactions . in this paper we use @xmath168 scattering as the laboratory for studying the large impact parameter behavior of the amplitude in the saturation region . at first sight , this processes occurs at short distances for both photons with large virtualities and could be calculated in perturbative qcd . we demonstrated that the non - perturbative qcd corrections have to be introduced for large @xmath1 even for this process . the main result of this paper is the statement that it is enough to include the non - perturbative qcd corrections in the born approximation and neglect them in the kernel of the bfkl equation . this result confirms the mechanism suggested in refs . @xcite but it contradicts the arguments of ref . @xcite . this result does not mean that the bfkl kernel correctly describes the large @xmath1 behavior . the uncertainties in the large @xmath1 tail of the kernel will not affect the high energy asymptotic behavior of the dipole amplitude . let us assume that kernel of the bfkl equation can be written as @xmath169 where @xmath170 is normal bfkl kernel in pqcd and @xmath171 includes the non - perturbative contribution . we know that @xmath172 from general properties of the strong interaction @xcite . let us treat @xmath171 as a small correction and calculate the first digram of the order of @xmath171 ( see ) . the sum of all diagrams in leads to a contribution [ dltk ] k ( 1-n^gm(x , r_1,t , r_2,t , b_t ) ) = k e^ - n^ops(x , r_1,t , r_2,t , b_t ) since for @xmath106 @xmath173 is very close to unity , the above corrections are suppressed . only for @xmath174 we can expect a considerable contribution . however , this contribution is proportional to @xmath175 . therefore they turn out to be very small . this simple discussion shows why the strategy to include the non - perturbative corrections in the born amplitude , works actually , the main result of this paper , namely , is based on a simple physics ( see ref . @xcite ) . we have demonstrated here that the multi rescattering processes embraced by the glauber - mueller formula lead to a different resulting parton cascade than is given by the bfkl approach . the principle difference is the fact that the multi parton shower interaction creates a new scale or mean parton transverse momentum ( saturation scale ) given by . @xmath176 denotes the parton density , consequently the fact that @xmath177 can be understood as the fact that the partons reach a maximal density at low @xmath7 . this phenomenon is called saturation @xcite . therefore , at low @xmath7 we have the parton distribution in the transverse plane presented in : the uniform distribution of partons ( dipoles ) with sizes of the order of @xmath178 in the disc of radius @xmath179 . if one of the dipole inside of the disc will emit one extra parton this emitted parton will interact with others partons and as a result of this interaction its transverse momentum will be of the order of @xmath180 . it means that this emitted gluon will not change its position in impact parameter space since due to uncertainty principle [ unpr ] b_t p_t1 its @xmath181 . however , for the parton at the edge of the disc the situation is different since the emitted parton in the direction outside of the disc can move freely without any interaction . this parton changes the size of the disc by its displacement in @xmath1 , namely @xmath182 . in this estimate we consider the non - perturbative emission with @xmath183 because , as have been discussed , a non - perturbative emission is needed to provide the unitarization of our process . since the emission that leads to a growth of the disc occurs in one direction ( the exterior of the disc ) it leads to @xmath184 where @xmath185 is the number of emission at given @xmath7 . since the emission takes place at the edge of the disc where the parton density is rather small , @xmath186 is determined by the bfkl dynamics only @xcite . in the bfkl approach @xcite@xmath187 since @xmath188 . therefore , we obtain , namely , @xmath189 . we have discussed in this paper the structure of dipole - dipole interaction in the glauber - mueller approach which is the only one on the market for the interaction of two dipoles of the same sizes . however , for two dipole with small but different sizes the non - linear evolution equation @xcite should be solved to which the bfkl emission is only an approximation in the region of small partonic densities . comparison of the result of this paper with the dipole - dipole interaction in , so called , double log approximation @xcite shows that the bfkl dynamics does not change physics at large @xmath1 . the non - linear evolution equation at fixed @xmath1 was solved @xcite in the case when the bfkl kernel was replaced by the double log one . the solution leads to the answer in the saturation region with geometrical scale@xcite [ gescale ] n(x , r_1,t , r_2,t;b_t)= f(=r^2_1,tq^2_sat(x)e^-4m_b_t ) . therefore , we believe that for the bfkl dynamics will hold . this belief is based on the similarity between double log and bfkl approximation for @xmath0 processes . * acknowledgments : * we wish to thank jochen bartels , errol gotsman and uri maor for very fruitful discussions on the subject . one of us ( e.l . ) would like to thank the desy theory group for their hospitality and creative atmosphere during several stages of this work . he is indebted to the alexander - von - humboldt foundation for the award that gave him a possibility to work on low @xmath7 physics during the last year . this research was supported in part by the gif grant @xmath190 i-620 - 22.14/1999 and by israeli science foundation , founded by the israeli academy of science and humanities . the integration over @xmath63 in can be taken explicitly @xcite and can be reduced to [ a1 ] d^2r_1v(r_1,t , r_1 ; ) v(r_1,t,|_1 -_t| ; -)= @xmath191 where @xmath192 is the hypergeometrical function ( see ref . @xcite ) ; @xmath7 is the complex anharmonic ratio : [ a2 ] x= and @xmath193 . @xmath194 gives [ a5 ] xx^*= one sees that is invariant with respect to rotation in the plane . however , one can see that does not reproduce the born term of at @xmath198 . to understand why it is so we should consider the vertex @xmath199 in momentum representation ( see ref . @xcite ) , namely , [ a6 ] v(r_1,t , q ; ) = d^2r_1e^ i v(r_1,t , r_1 ; ) . it turns out @xcite that @xmath200 at small @xmath201 leads to the following behavior of vertex @xmath202 : [ a8 ] v(r-1,t , q ; ) | _ q^2 0()^-i ( 1-(q^2)^2 i()^i 2 ) as have been discussed the matching with the born approximation occurs at @xmath203 . in this limit [ a9 ] v(r_1,t , q ; ) ( 1 - q^2 ) , which has correct analytical behavior actually this behavior dictates the choice of the coefficients @xmath195 and @xmath196 in and . however , at @xmath204 the low @xmath201 behavior has a singularity @xmath205 . therefore the symmetry of with respect to sign of @xmath76 is broken . mueller and tang @xcite pointed out that this problem can be cured by adding to the expression of @xmath206 of , namely , @xmath207 as was found@xcite such terms can be added due to gauge invariance of qcd . in momentum representation ( see ) @xmath208 can be written as a sum of three terms as it is shown in . the mueller - tang vertex leads to the born approximation amplitude in the form of . however , as was discussed in refs . @xcite , it has not been proven that this vertex will satisfy the bfkl equation . the solution in the form of has a different form of the born amplitude , namely , [ a10 ] n^ba ( ) ( ) . however , these two expressions for the born amplitude are equivalent due to gauge invariance of qcd @xcite . using we can calculate the dipole - 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it is shown that the non - perturbative contribution is essential to fulfill the unitarity constraints in the region of @xmath2 , where @xmath3 is pion mass .
the saturation and the unitarization of the dipole - dipole amplitude is considered in the framework of the glauber - mueller approach .
the main result is that we can satisfy the unitarity constraints introducing the non - perturbative corrections only in initial conditions ( born amplitude ) .
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j. phys_. * g#1*#2#3 + * @xmath4-@xmath4 scattering : saturation and * + * unitarization in the bfkl approach .
* _ @xmath5 ) hep department _ + _ school of physics and astronomy _ + _ raymond and beverly sackler faculty of exact science _
+ _ tel aviv university , tel aviv , 69978 , israel _ + 0.3 cm _
@xmath6 desy theory group _ + _ 22603 , hamburg , germany _ |
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in the last few years band offset engineering " ( _ i.e. _ the possibility of tuning the electronic and transport properties of semiconductor heterojunctions through modifications of their valence band offset ( vbo ) induced by strain , growth process , number of layers etc . ) has attracted great attention for both scientific and technological reasons @xcite . only recently , however , have lattice mismatched heterostructures begun to realize their potential , due to new developments in preparation techniques which finally allowed pseudomorphic crystal growth without misfit dislocations @xcite . in a parallel way , some theoretical works @xcite focused on strained heterojunctions , but still much effort is needed to understand what mainly affects the band line - up at the interface . in this work , we consider homopolar isovalent heterostructures . in particular , we examine iii - v superlattices ( sls ) , namely the common - anion system gasb / insb ( lattice mismatch of 5.7 @xmath0 ) and the common - cation system inas / insb ( lattice mismatch of 6.4 @xmath0 ) , concentrating on the effect of ordering direction and strain conditions determined by sl pseudomorphic growth on a given substrate . _ ab - initio _ self - consistent full potential linearized augmented plane wave flapw @xcite local density calculations were performed for ( gasb)@xmath1/(insb)@xmath1 and ( inas)@xmath1/(insb)@xmath1 sls , grown along the [ 001 ] ( tetragonal symmetry ) and [ 111 ] ( trigonal symmetry ) directions , in which the interface was represented by three alternating layers of each binary constituent ( 12 atoms in each unit cell ) . in what follows , it is important to keep in mind that if strain is not taken into account , the valence bands in unstrained gasb and insb are predicted to line - up ( therefore giving zero vbo ) , whereas the unstrained insb topmost valence band is expected to be 0.51 ev _ higher _ in energy than that in unstrained inas ( see ref.@xcite and references therein ) . as pointed out by mailhot and smith @xcite , [ 111 ] ordered strained - layer sls show large polarization fields oriented along the growth direction , which lead to a shift in the electronic energy levels . however , this effect is negligible for ultrathin sls , such as those examined in the present work . moreover , the effect completely vanishes for [ 001 ] ordered sls , due to the symmetry properties of the strain tensor . finally , we neglect interdiffusion processes which lead to interfacial composition changes ( i.e. we consider an atomically abrupt geometry ) and relaxations at the interface of the anion - cation distance ( bulk bond length away from the interface are considered equal to those immediately next to it ) : this , in fact , is expected @xcite to introduce little modification of the charge rearrangement at the junction and hence of the vbo . most of the computational parameters are common to those used previously for [ 001 ] and [ 111 ] ordered ( 1x1 ) sls @xcite , except for the wave function cut - off ( @xmath2 = 2.7 a.u.).[17 ] tests performed by increasing the @xmath2 up to 3.1 showed a change in the vbo of less than 0.01 ev . there is , however , an important difference with respect to the calculations performed for ultrathin ( @xmath3 = 1 ) sls regarding the cation @xmath4-shell : in the present work , the ga 3@xmath4 and in 4@xmath4 electrons are treated as part of the core and not as valence electrons ( previous works @xcite demonstrated a slight dependence ( about 0.03 ev ) of the vbo in common - cation systems on the @xmath4-shell treatment ) - which results in a strong reduction of the computational effort . furthermore , the core charge spilling out of the ga and in muffin tin spheres was treated using an exact overlapping charge method , thus minimizing the error introduced by the treatment of semicore states . the structural parameters ( reported elsewhere @xcite ) are determined according to the macroscopic theory of elasticity ( mte ) , whose validity in predicting the correct structure for the determination of the vbo is established @xcite . our choice is also justified by the results obtained in the case of ultrathin sls @xcite , which were found to be in good agreement with those obtained from total energy minimization . in order to study the dependence of the vbo on the strain , we examined different strain conditions for ( ac)@xmath1/(bc)@xmath1-type [ 001 ] ordered sls ( the common cation case ( ab)@xmath1/(ac)@xmath1 is treated in an analogous way ) : ( i ) pseudomorphic growth of a bc epilayer on an ac substrate ; ( ii ) free standing mode " , equivalent to a system grown on an a@xmath5b@xmath5c substrate ( denoted in the following as av . subs . " ) ; and ( iii ) pseudomorphic growth of an ac epilayer on a bc substrate . on the other hand , the dependence of the vbo on the ordering direction is studied through a comparison of the [ 001 ] and [ 111 ] ordered sls grown on a fixed substrate with average lattice constant , but different crystallographic orientations . this choice of the substrate and the consequent lattice relaxation , leads to a small difference ( about 1.4 @xmath0 ) between the lattice constants of the binary constituents along the [ 001 ] and [ 111 ] growth direction . in analogy with the common experimental approach followed in photoemission measurements , we have evaluated the vbo using core electron binding energies as reference levels @xcite . we have chosen the @xmath6-levels of the common atom c ( _ i.e. _ sb in the common - anion system and in in the common - cation system ) . note that other choices of core levels for different atoms ( _ i.e. _ ga and in in the common - anion system , as and sb in the common - cation system ) would produce a vbo value differing from those reported here by at most 0.06 ev , which has thus to be considered as our numerical uncertainty . the calculation of the vbo , @xmath7 , is done according to the following expression : @xmath8 where the interface term @xmath9 indicates the relative core level alignment of the two c atoms at opposite sides of the interface ( one belonging to the ac side and the other to the bc side ) , while @xmath10 indicates the binding energy difference ( relative to the valence band maximum ( vbm ) ) of the same core levels evaluated in the binary constituents , opportunely strained to reproduce the elastic conditions of the sl . first of all , we focus our attention on the elastically relaxed av . subs . " [ 001 ] and [ 111 ] ordered sls . we should now notice that there are two inequivalent interfaces along the [ 111 ] growth axis @xcite ; for example , in the common - anion sl , we can have the ordering direction parallel either to the insb interface bond or to the gasb interface bond . our results indicate that the @xmath9 term is essentially the same ( within 0.02 ev ) in the two different situations , suggesting that the effect due to the particular geometry at the interface is very small . table [ vbo1 ] lists the contributions due to the interface ( @xmath9 ) and to the strained bulks ( @xmath10 ) and the resulting values of the vbos ( @xmath7 ) as a function of the ordering direction ; the superscripts @xmath11 and @xmath12 indicate respectively the non - relativistic and relativistic ( _ i.e. _ spin - orbit coupling treated in a perturbative approach ) calculations . we note that the interface term has a positive sign , indicating that the sb - core levels are deeper at the gasb ( inas ) side of the common - anion ( common - cation ) interface , compared to the corresponding levels at the insb side . further , the first contribution ( @xmath13 is seen to be sensitive to the crystallographic ordering ( the two values for [ 001 ] and [ 111 ] growth axis differ by 0.1 ev both in the common - anion and the common - cation systems ) , while the second contribution to the vbo ( @xmath10 ) is almost uninfluenced by the ordering direction . on the whole , we do not observe a marked dependence of the vbo on the crystallographic ordering at the interface . since it is well - known for lattice matched structures @xcite that the band line - up is independent of interface orientation , the vbo change we find in going from the [ 001 ] to the [ 111 ] ordered sls has to be related only to the appreciable mismatch which causes a different relaxation of the interface bond - lengths . let us now look at the role of the mismatch in determining the band line - up . our results for [ 001 ] systems with different pseudomorphic growth conditions are shown in table [ vbo2 ] , where the notation is analogous to that of table [ vbo1 ] . note that the interface term @xmath9 is very similar in the same sl grown on the three different substrates , implying that the charge readjustment at the interface is almost independent of the strain conditions . this is confirmed by the results obtained with a small change ( by as much as 0.6 @xmath0 ) of the bond length at the interface : the calculated @xmath9 is consistent with the one obtained for mte structures within 0.03 ev . on the other hand , the @xmath10 term ( _ i.e. _ the bulk contribution to the vbo ) varies dramatically , showing that the core level binding energies in the strained binary suffers an appreciable change when growing the sl on different substrates . in fact , the energy of the topmost valence level ( and hence the binding energy @xmath14 ) is determined by the interplay of the spin - orbit coupling and the non - cubic crystal field " @xcite . in particular , the second of these two effects is critically dependent on strain conditions , and is thus the origin of the large difference between the @xmath15 in table [ vbo2 ] . furthermore , what is remarkable about table [ vbo2 ] is the clear trend shown by the vbo as a function of the substrate lattice constant : the smaller the @xmath16 , the more the insb topmost valence level is raised with respect to the vbm of the other sl constituent . in order to understand more fully the action of the strain on the vbo , we have also estimated the band line - up with respect to unstrained binaries for the [ 001 ] interfaces . thus , we substituted in eq.([equazione ] ) the @xmath9 value obtained for [ 001 ] interfaces ( which was found to be almost independent of strain effects ) and the @xmath10 , evaluated starting from the zincblende bulk unstrained constituents ( i.e. disregarding the effect of strain on the binary s vbm ) . taking into account the spin - orbit coupling , we obtain @xmath17 = 0.03 ev and @xmath17 = 0.49 ev for the gasb / insb and inas / insb heterojunctions , respectively . these results match perfectly those reported in ref . @xcite and the above mentioned valence bands perfect alignment for the common anion system and @xmath17 = 0.51 ev for the common cation case , showing that , if strain is not correctly taken into account , completely different results are found . it is important to notice that the strain acting on the energy of the topmost valence band level is also responsible for the spatial localization of this state . we find , in fact , that in all of the common anion ( common cation ) structures , the hole carriers are mainly localized on the insb side of the heterojunction , while in the gasb / insb system grown on an insb substrate we find a complementary situation . this is clear from the decreasing trend of the vbo as the lattice constant is increased and from the sign change in the insb substrate case ( showing that the vmb in gasb is higher in energy than in insb ) . figures [ fig1 ] and [ fig2 ] , for common - anion and common - cation interfaces respectively , illustrate the linear dependence ( see the solid line in the figures ) of the vbo on the lattice parameter which determines the sl pseudomorphic growth . thus , the two figures show that the gasb / insb and inas / insb sls provide a good opportunity for tuning their vbo : a range of about 0.5 ev for common - anion and of 0.7 ev for common - cation systems is covered by varying the strain conditions determined by the substrate . let us now compare our results with other theoretical predictions , obtained from model @xcite , semi - empirical @xcite and _ ab - initio _ @xcite calculations , as illustrated in figures [ fig1 ] and [ fig2 ] . note that all the predicted values agree with those of the present work ( except those of ref . @xcite ) , within their uncertainty of a few hundredths of an ev @xcite and our error bars , respectively . incidentally , we observe that a similar disagreement between _ ab initio _ results and those obtained by cardona and christensen @xcite was also found in other iii - v isovalent heterojunctions , such as gap / inp @xcite and gaas / inas @xcite . furthermore , the linear trend of the band offset as a function of the strain found in the present work is in excellent agreement with the predictions of other theoretical work @xcite and is reasonably expected to reproduce the real situation . so far , we have completely omitted a discussion of the conduction band offset ( @xmath18 ) , due to well known failures of lda in predicting the correct band gap energies . however , we should now point out that , using an approximate estimate of empirical band gaps ( see ref . @xcite for details ) , we can obtain information on the different kinds of band line - ups as growth conditions are changed . in fact , we find a type i alignment for all the [ 001 ] common anion interfaces , while [ 111 ] gasb / insb shows a type ii staggered alignment ( partial overlap of the band gaps ) . on the other hand , for [ 001 ] common cation grown both on an inas and on an average substrate , we find a type ii broken gap line - up , with the inas conduction band minimum lower in energy than the insb vbm . finally , the [ 001 ] inas / insb grown on insb and the [ 111 ] inas / insb heterojunctions contain a semimetallic compound ( inas ) , thus leading to a type iii alignment . starting from the results obtained from _ ab initio _ calculations @xcite for different [ 001 ] oriented strained layer interfaces ( with similar lattice mismatch and grown on a substrate having lattice constant averaged over those of the constituents ) , it is interesting to discuss the vbo trend as a function of the atomic species involved . we compare the value obtained for the common anion gasb / insb ( lattice mismatch of 5.7 @xmath0 and @xmath7 = 0.07 ev , in good agreement with @xmath7 = 0.04 ev reported in ref . @xcite ) , with those obtained for gap / inp @xcite ( lattice mismatch of 7.4 @xmath0 and @xmath7 = 0.01 ev ) and for gaas / inas @xcite ( lattice mismatch of 5.7 @xmath0 and @xmath7 = 0.00 ev ) . taking into account that these results are obtained by different computational methods and are therefore affected by different error bars , it appears that , under similar strain conditions , the vbo is almost uninfluenced by the change of both anions in the common anion systems . therefore , since the @xmath9 term is expected to be constant ( due to similar ionicity difference of the constituents ) , these results seem to suggest that also the @xmath10 term has to be similar in all the gax / inx ( x = p , as , sb ) structures . a similar @xmath7 is also found , if we compare our results for inas / insb ( lattice mismatch of 6.4 @xmath0 and @xmath7 = 0.54 ev ) with gaas / gasb ( lattice mismatch of 7.2 @xmath0 ) , where both cations are changed ( @xmath7 = 0.65 @xmath19 0.1 ev @xcite ) . this observation can be explained , considering the anionic character ( see ref . @xcite , ref . @xcite and references therein ) of the topmost valence level in the binary constituents , which determines the binding energy contribution ( @xmath10 ) to the valence band offset : for example , provided that the state of strain is similar in the constituent materials grown on an average substrate , we do nt expect a strong difference in the band line - up if we change the cation at both sides of the common cation interface . from the experimental point of view , the high mismatch ( about 6 @xmath0 ) between the lattice constants of the two sl constituents results in great difficulty to grow the sl systems , without misfits and dislocations . only recently , an insb quantum - well has been realized in gasb @xcite ; starting from photoluminescence peak emission energy data and from calculations based on a standard finite square - well model @xcite ( taking into account strain @xcite ) , a vbo of 0.16 ev was obtained that is quite different from the one , 0.34 ev , reported in table [ vbo2 ] . unfortunately , a similar disagreement between theoretical and experimental data , obtained using different techniques , was found also for other homopolar isovalent iii - v interfaces , such as the gaas / inas ( see ref . @xcite and references therein ) and gap / inp ( @xmath7 = 0.01 ev , the theoretical result @xcite against @xmath7 = 0.60 ev , the experimental results from photoluminescence @xcite ) systems , that are reasonably close to the common - anion one studied here . in particular , in a recent work focused on gaas / inas sls , ohler _ et al . _ @xcite obtained ( from ultraviolet photoelectron spectroscopy measurements of the cation @xmath4-core levels ) a @xmath7 value in disagreement ( by as much as 0.3 ev ) with theoretical predictions @xcite . notwithstanding some differences between the experimental and theoretical vbo values , many important observations discussed above are confirmed by ohler _ et al . _ experimental work @xcite . for example , the linear trend found @xcite for @xmath20 ( in gaas / inas sls ) as a function of @xmath16 agrees with theoretical predictions @xcite and with our results ; further , the independence of the @xmath21 term on the strain conditions discussed above , is experimentally confirmed @xcite by the trend of the in 4@xmath22 and ga 3@xmath23 core - level binding energy difference , which is almost unaffected by the different substrate used in growing the heterostructure . as a last comment , we think that it worthwhile to remark that the linear trend found in ref . @xcite for the gaas / inas vbo s as a function of the growth conditions leads to a slope that is almost equal to that of fig . [ fig1 ] for gasb / insb systems . starting from the band offset transitivity rule - which is well established ( to within 0.02 ev ) @xcite for [ 001 ] common - atom superlattices - we can derive the vbo for the inas / gasb system , an almost lattice matched interface ( the two lattice constants differ only by 0.6 @xmath0 ) . this system is attracting more and more attention recently for its unusual type ii broken - gap band line - up . through a linear interpolation of the common - anion band offsets as a function of the substrate lattice constant , we have calculated the vbo for the gasb / insb system as if grown on an inas substrate . as evidenced above ( see fig . [ fig1 ] ) , the linear approximation is expected to be reasonably valid ; furthermore , in this case the extrapolation is obtained for a lattice constant ( @xmath24 ) which differs by only 0.7 @xmath0 from one of our self - consistent results ( @xmath25 ) . we thus report in fig . [ fig3 ] our calculated vbo for the common - cation sl on an inas substrate and the extrapolated value for the ideal common - anion sl grown on an inas substrate . using the transitivity rule , we obtain : @xmath26 so that @xmath27 = 0.88 - 0.40 = 0.48 ev . this result is in good agreement with the available experimental values ( 0.46 ev @xcite , 0.51 ev @xcite ) . in summary , we have studied the valence band offsets in [ 001 ] and [ 111 ] gasb / insb and inas / insb interfaces by means of _ ab - initio _ flapw calculations , focusing our attention on its dependence on ordering direction and strain conditions . our results indicate that , under the same strain conditions , the former has quite a small effect on the band line - up mainly due to the different structural relaxation of the interface atoms at the [ 001 ] and [ 111 ] heterojunctions . on the other hand , a much more important effect is due to pseudomorphic growth on different substrates : the high tunability of the vbo ( about 0.5 ev and 0.7 ev for common - anion and for common - cation sls , respectively ) is evidenced by its linear decreasing trend as the substrate lattice constant is increased , mainly due to the bulk contribution to the band line - up . finally , the transitivity rule was used to determine the inas / gasb valence band offset and good agreement between theory and experiment was obtained . we thank b. w. wessels and m. peressi for stimulating discussions and a careful reading of the manuscript . useful discussions with s. massidda are also acknowledged . work at northwestern university supported by the mrl program of the national science foundation , at the materials research center of northwestern university , under award no . dmr-9120521 , and by a grant of computer time at the nsf supported pittsburgh supercomputing center . partial support by a supercomputing grant at cineca ( bologna , italy ) through the consiglio nazionale delle ricerche ( cnr ) is also acknowledged . .interface term ( @xmath9 ) , strained bulk term ( @xmath10 ) and valence band offset ( @xmath7 ) for elastically relaxed ( gasb)@xmath1/(insb)@xmath1 and ( inas)@xmath1/(insb)@xmath1 -average substrate- superlattices as a function of the ordering direction ( @xmath28 and @xmath29 ) and including ( @xmath30 and @xmath31 ) spin - orbit effects ) . energy differences ( in ev ) are considered positive if the level relative to the insb layer is higher in energy with respect to the gasb ( inas ) layer in the common - anion ( common - cation ) system . [ cols="^,<,>,^,^,^,^",options="header " , ] s. baroni , m. peressi , r. resta , a. baldereschi , _ theory of band offsets at semiconductor heterojunctions _ , in _ proceedings of the 21@xmath32 international conference on the physics of semiconductors _ , edited by ping jiang and hou - zhi zheng ( world scientific , singapore , 1993 ) , p. 689 . | first - principles full potential linearized augmented plane wave ( flapw ) calculations have been performed for lattice - mismatched common - atom iii - v interfaces . in particular , we have examined the effects of epitaxial strain and ordering direction on the valence band offset in [ 001 ] and [ 111 ] gasb / insb and inas / insb superlattices , and found that the valence band maximum is always _ higher _ at the insb side of the heterojunction , except for the common - anion system grown on an insb substrate .
the comparison between equivalent structures having the same substrate lattice constant , but different growth axis , shows that for comparable strain conditions , the ordering direction slightly influences the band line - up , due to small differences of the charge readjustment at the [ 001 ] and [ 111 ] interfaces . on the other hand
, strain is shown to strongly affect the vbo ; in particular , as the pseudomorphic growth conditions are varied , the bulk contribution to the band line - up changes markedly , whereas the interface term is almost constant . on the whole ,
our calculations yield a band line - up that decreases linearly as the substrate lattice constant is increased , showing its high tunability as a function of different pseudomorphic growth conditions .
finally , the band line - up at the lattice matched inas / gasb interface determined using the transitivity rule gave perfect agreement between predicted and experimental results . |
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the heisenberg ferromagnet is defined by the partition function @xmath0 = \int \prod [ d \omega(x ) \exp(-s)\label{eq:1}\ ] ] where @xmath1 ^ 2 \qquad \vec n^2(x ) = 1\label{eq:2}\ ] ] and @xmath2 is the element of solid angle for the orientation of @xmath3 in colour space . the model presents a 2nd order phase transition at @xmath4@xcite . for @xmath5 there is an ordered phase , with order parameter the magnetization @xmath6 ; for @xmath7 , @xmath8 ( disordered phase ) . we shall describe the system from a dual point of view , and show that in the disordered phase vortices condense . the model will be viewed as a @xmath9 dimensional euclidean field theory . vortices will be labelled by the integer valued topological charge of the 2 dimensional spacial configurations , which is a conserved quantity , and defines a @xmath10 symmetry of the system . a disorder parameter @xmath11 will be constructed , which detects spontaneous breaking of this @xmath10 symmetry , or condensation of vortices . the situation is analogous to the condensation of monopoles in the confining phase of gauge theories@xcite or to the condensation of abelian vortices in @xmath12@xcite . to check our construction we shall extract from the numerical determination of @xmath11 at the phase transition the known critical indices . usually the colour frame to which the direction of @xmath3 is referred is a fixed frame , independent of @xmath13 @xmath14 a body fixed frame can be defined@xcite by three unit vectors @xmath15 @xmath16 and @xmath17 . the frame is defined up to an arbitrary rotation around @xmath18 . since @xmath19 @xmath20 or @xmath21 @xmath22 are the generators of the @xmath23 symmetry group . eq.([eq:3 ] ) is nothing but the definition of parallel transport . from eq.([eq:3 ] ) it follows @xmath24\vec\xi_i = 0 $ ] or , by completeness of @xmath25 @xmath26 = \vec t\cdot\vec f_{\mu\nu}(\omega ) = 0 \label{eq:4}\ ] ] @xmath27 @xmath28 is a pure gauge , apart from singularities . the general solution of eq.([eq:3 ] ) is then , for @xmath18 @xmath29 where @xmath30 is the value of @xmath31 at infinity , and the dependence on the path @xmath32 is trivial , because @xmath33 is a pure gauge , eq.([eq:4 ] ) . this is true apart from singularities . we will show that such singularities exist . the current @xmath34 is identically conserved @xmath35 if we look at the theory as the euclidean version of a field theory , with euclidean time on the 3 axis , the corresponding conserved quantity is @xmath36 which is nothing but the topological charge of the 2 dimensional configurations of the theory . @xmath37 can assume positive and negative integer values . by use of eq.([eq:3 ] ) and eq.([eq:4 ] ) it is easy to show that @xmath38 where the path @xmath32 is the contour of the region in the 2 dimensional space ( @xmath39 = const . ) where eq.([eq:4 ] ) holds . since @xmath40 eq.([eq:5 ] ) shows that @xmath41 is not always a pure gauge . this can be explicitely checked on a configuration corresponding to a static 2 dimensional instanton propagating in time ( vortex ) . the conserved current @xmath42 identifies a @xmath10 symmetry . we will show that this symmetry is wigner in the ordered phase @xmath5 , and is spontaneously broken in the disordered phase . let @xmath43 be a @xmath44 dependent singular rotation creating a vortex of charge @xmath45 at the site @xmath46 in a 2 dimensional configuration . @xmath47 the creation operator of a vortex @xmath48 at site @xmath46 , time @xmath49 , will be defined as @xmath50\bigr\ } \label{eq:7}\end{aligned}\ ] ] we measure the correlator @xmath51 by cluster property @xmath52 @xmath53 signals spontaneous breaking of the @xmath10 symmetry ( [ eq:4a ] ) , or condensation of vortices . by use of the definition ( [ eq:7 ] ) it is easy to see that @xmath54}{z[s]}\ ] ] where @xmath55 is obtained from @xmath56 , eq.([eq:2 ] ) by the replacement @xmath57 ^ 2 \hskip-5pt&\to&\hskip-5pt \left[r_q^{-1}(\vec x,0)\vec n(\vec x,1 ) - \vec n(\vec x,0)\right]^2 \\ \left[\delta_0\vec n(\vec x , x_0)\right]^2 \hskip-5pt&\to&\hskip-5pt \left[r_q^{-1}(\vec x,0)\vec n(\vec x , x_0 + 1 ) - \vec n(\vec x , x_0)\right]^2\end{aligned}\ ] ] and that this really amounts to have a vortex propagating from @xmath58 to @xmath59@xcite . instead of @xmath60 itself it proves convenient@xcite to study the quantity @xmath61 . as @xmath62 from eq.([eq:8 ] ) we have for @xmath63 , @xmath64 . since @xmath65 , @xmath66 $ ] . @xmath67 is easier to measure and contains all the information on the transition . the behaviour of @xmath67 is shown in fig.1 . for @xmath68 , @xmath69 finite limit consistent with zero , or @xmath70 , which means condensation of vortices . for @xmath71 @xmath67 can be evaluated in perturbation theory and behaves as @xmath72 @xmath73 is the lattice size . eq.([eq:9 ] ) implies that @xmath74 for @xmath71 in the thermodynamical limit @xmath75 . around @xmath76 a finite size scaling analysis can be performed @xmath77 0.1 in * fig.1 * 0.1 in and since @xmath78 , we get the scaling law@xcite @xmath79 the scaling law is verified , fig.2 , and allows to extract @xmath76 and @xmath80 @xmath81\\ \nu & = & 0.70 \pm 0.02\ ; [ 0.698]\end{aligned}\ ] ] they agree with the values determined from @xmath82 , which are indicated in parentheses@xcite . -10pt 0.1 in * fig.2 * the phase transition to disorder in 3d heisenberg model is produced by condensation of topological solitons . a disorder parameter can be defined and out of it the critical indices can be determined . this work has partially supported by murst . a. di giacomo is grateful to ec , tmr project , erbfmx - ct97 - 0122 for financing the partecipation to the conference . 9 e. brezin , j. zinn - justin , _ nucl . phys . _ * b257 * , 867 , ( 1985 ) . a. di giacomo , g. paffuti , _ phys . rev . _ * d56 * , 6816 , ( 1997 ) . l.del debbio , a.di giacomo , g.paffuti and p.pieri , _ phys . lett._*b 355 * ( 1995 ) 255 . g. di cecio , a. di giacomo , g. paffuti , m. trigiante , _ nucl . phys . _ * b489 * , 739 , ( 1997 ) . a. di giacomo , m. mathur , _ phys . _ * b400 * , 129 , ( 1997 ) . p. peczak , a.l . ferrenberg , d.p . landau , _ phys . rev . _ * 43 * , 6087 , ( 1991 ) . | the 3d heisenberg model is studied from a dual point of view in terms of 2d solitons ( vortices ) .
it is shown that the disordered phase corresponds to condensation of vortices in the vacuum , and the critical indices are computed from the corresponding disorder parameter . |
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it is widely believed that the most natural and appealing explanation of the recent neutrino oscillation results is provided by the seesaw mechanism @xcite incorporated into extensions of the standard model that include a local @xmath0 symmetry . the simplest models with local @xmath0 symmetry are the left - right symmetric models @xcite based on the gauge group @xmath6 . these models have the additional virtue that they explain the origin of parity violation in weak interactions as a consequence of spontaneous symmetry breaking in very much the same way as one explains the strength of the weak interaction in the standard model . stability of the higgs sector under radiative corrections calls for weak scale supersymmetry as in the minimal supersymmetric standard model ( mssm ) . it has recently been shown that if the mssm is embedded into a left right symmetric framework at a high scale @xmath7 gev , as suggested by neutrino oscillation data and by gauge coupling unification , it helps solve some important problems faced by the mssm , viz . , the susy cp problem @xcite , the strong cp problem @xcite and the @xmath8 problem . supersymmetric models with such a high scale embedding are therefore attractive candidates for physics beyond the standard model . it was noted many years ago @xcite that the electric charge formula of the left right symmetric models , @xmath9 , allows one to conclude from pure group theoretic arguments that parity symmetry breaking implies a breakdown of @xmath0 symmetry as well with the constraint that @xmath10 . this simple relation is profoundly revealing . it says that the neutrinos must be majorana particles since the lepton number breaking terms in the theory must obey @xmath11 selection rule . this conclusion follows directly if higgs triplets are used to break @xmath12 symmetry since @xmath13 for triplets , it also holds when higgs doublets are used for this purpose , since gauge invariance requires the presence of two such doublets in the mass term for the neutrinos . secondly , for purely hadronic baryon number violating processes , baryon number must change by at least two units , @xmath14 . this means that models based on left right symmetric gauge structure can lead to the process where a neutron transforms itself into an antineutron ( @xmath2 oscillation@xcite ) , while they may forbid the decay of the proton , which is a @xmath15 process . while the above group theory argument predicts the existence of @xmath2 oscillation in left right symmetric models , its strength will depend on the details of the model . using simple dimensional analysis it is easy to find that the lowest dimensional operators that contributes to @xmath2 oscillation are six quark operators , a typical one being @xmath16 . this operator has dimension 9 and therefore the coupling strength scales as @xmath17 , where @xmath18 is the scale of new physics . it is natural to identify @xmath18 with the scale of @xmath0 ( or parity ) breaking . the current lower limit on @xmath2 oscillation time , @xmath19 sec @xcite , sec @xcite . ] implies an upper limit @xmath20 gev@xmath21 . for @xmath2 oscillations to be observable then , the scale @xmath18 should be rather low , @xmath22 gev . one class of models where @xmath23 transition manifests itself through higgs boson exchange has been discussed in ref . there it was shown that if the @xmath6 model is embedded into the @xmath24 gauge group , then @xmath2 oscillations can arise at an observable level if the @xmath25 breaking scale is in the 100 tev range . in these models , @xmath2 oscillation amplitude is intimately tied to an understanding of small neutrino masses via the seesaw mechanism as well as the breaking of quark lepton degeneracy implied by @xmath25 symmetry . the same higgs field that breaks @xmath25 and generates heavy majorana masses for the right handed neutrinos also mediate @xmath2 oscillations here . with the scale of @xmath25 breaking in the 100 tev range , these models would appear to be incompatible with gauge coupling unification . furthermore , such a low scale of parity breaking would not yield naturally neutrino masses in the range suggested by current experiments . if we raise the scale of parity/@xmath25 breaking to values above @xmath26 gev , so that small neutrino masses in the right range are generated naturally , then @xmath2 transition amplitude becomes unobservably small in these models . oscillation was noted in the context of a susy @xmath27 model in ref . these models possess accidental symmetries that lead to light ( @xmath28 gev ) diquark higgs bosons even though the scale of parity violation is high . as a result , the @xmath2 oscillation operator can have observable strength . unification of gauge couplings is however difficult to achieve in these models . ] does the above arguments mean that @xmath2 oscillations are beyond experimental reach based on current neutrino oscillation phenomenology ? in this letter we will show that this is not the case in a class of attractive seesaw models with local @xmath0 symmetry . we will see that in these models a new class of @xmath29 operators is induced as a consequence of parity breaking . these operators lead to observable @xmath2 oscillation despite the scale @xmath30 of parity breaking being close to the conventional gut scale of @xmath31 gev . in fact , @xmath32 increases with @xmath30 and therefore one has the inverse phenomenon that increasing @xmath30 leads to stronger @xmath2 oscillation amplitude . interestingly , the scale @xmath30 implied by neutrino masses is such that @xmath2 oscillation should be accessible experimentally with a modest improvement in the current limit . we obtain an _ upper _ limit of @xmath33 sec in this class of models . this prediction becomes sharper in a concrete model where flavor symmetries reduce considerably the uncertainties in the estimate of @xmath4 . we emphasize that our upper limit is derived in the context of conventional seesaw models of neutrino mass without using any special ingredients to enhance @xmath2 oscillation amplitude . this should provide new impetus for an improved experimental search for @xmath2 oscillations . the basic framework of our model involves the embedding of the mssm into a minimal susy left right gauge structure at a scale @xmath30 close to the gut scale . the electroweak gauge group of the model , as already mentioned , is @xmath34 with the standard assignment of quarks and leptons left handed quarks and leptons ( @xmath35 ) transform as doublets of @xmath36 , while the right handed conjugate ones ( @xmath37 ) are doublets of @xmath12 . the quarks @xmath38 transform under the gauge group as @xmath39 and @xmath40 as @xmath41 , while the lepton fields @xmath42 and @xmath43 transform as @xmath44 and @xmath45 respectively . the dirac masses of fermions arise through their yukawa couplings to two higgs bidoublet @xmath46 , @xmath47 . the @xmath48 symmetry is broken down to @xmath49 in the supersymmetric limit by @xmath50 doublet scalar fields , the right handed doublet denoted by @xmath51 accompanied by its left handed partner @xmath52 . anomaly cancellation requires the presence of their charge conjugate fields as well , denoted as @xmath53 and @xmath54 . the vacuum expectation values ( vevs ) @xmath55 break the left right symmetry group down to the mssm gauge symmetry . a singlet @xmath56 is also used to facilitate symmetry breaking in the susy limit . it has recently been shown that if there exists a @xmath57 r symmetry , the minimal model just described will solve the strong cp problem and the susy phase problem based on parity symmetry @xcite . furthermore , the @xmath8 term will have a natural origin . one possible @xmath57 assignment was given in ref . @xcite . here we present a slight variant , which yields the same superpotential at the renormalizable level as in ref . @xcite and thus preserves all its success . under this @xmath57 , the superpotential @xmath58 changes sign , as do @xmath59 and @xmath60 . the quark fields @xmath61 are even , while @xmath62 transform as @xmath63 . the fields @xmath64 are all odd under @xmath57 . the gauge invariant superpotential consistent with this @xmath57 r symmetry at the renormalizable level is @xmath65 this superpotential breaks the gauge symmetry to that of the standard model in the susy limit without leaving any unwanted goldstone bosons and induces realistic quark masses and mxings . the baryon number violating processes as well as neutrino masses arise in this model from higher dimensional operators induced by planck scale physics . they will be the main focus of the rest of the paper . we shall pay special attention to the relation between the neutrino mass and the @xmath2 oscillation time . the relevant dimension four operators in the superpotential which are scaled by @xmath66 and are allowed by the @xmath57 symmetry are : @xmath67~,\nonumber\\ { \cal o}_2 & = & f'\left [ q^cq^cq^c\bar{\chi^c } + qqq\bar{\chi } \right]~.\end{aligned}\ ] ] operator @xmath68 gives rise to majorana masses for @xmath69 of order @xmath70 . combining this with the dirac neutrino masses arising from eq . ( 1 ) , small neutrino masses will be generated by the seesaw mechanism . for @xmath71 gev , the magnitude of the light neutrino masses are in the right range to explain the atmospheric and the solar neutrino oscillation data . operator @xmath72 , which is also invariant under the @xmath57 , leads to baryon number violation . while @xmath73 could have their origin in quantum gravity , they may also be induced by integrating out vector states that have @xmath57inviariant masses of order the planck scale . note that operators such as @xmath74 are not allowed by the @xmath57 symmetry . if they were present along with @xmath72 , they would lead to rapid proton decay . note also that the well known proton decay operator @xmath75 is not allowed by the @xmath57 symmetry . in any case its presence would not have been a problem since it is scaled by the planck mass and therefore can lead to a proton lifetime consistent with the present lower limit . to see the connection between neutrino masses and the @xmath2 oscillation time @xmath76 qualitatively , first we note that the operator @xmath68 leads to the majorana mass for the right handed neutrino @xmath77 . the seesaw formula then leads to the relation @xmath78 . on the other hand , the operator @xmath72 leads to a @xmath29 operator with strength @xmath79 . leaving aside the details of the flavor structure of @xmath72 and how actually @xmath80 arises , it is clear that we have a simple linear relation between the neutrino masses and the @xmath2 oscillation time : @xmath81 where @xmath82 is a dimensional constant which depends only on the details of weak scale physics and does not involve the high scale @xmath30 . we will evaluate @xmath82 in the next section . this simple relation makes it clear that our present knowledge of the neutrino masses allows a direct prediction of the @xmath2 oscillation time in the context of the supersymmetric left - right models broken by doublet higgs fields . let us now proceed to examine the expected @xmath2 oscillation time resulting from the @xmath29 operator @xmath72 of eq . an important point to note here is that since @xmath72 is a superpotential term with antisymmetric color contraction it must have antisymmetric flavor contraction as well . the flavor structure of this operator is then of the type @xmath83 , @xmath84 or @xmath85 in terms of the superfields . we must then use flavor mixings to obtain the fermionic operator of the type @xmath86 and then the six quark @xmath2 operator . the dominant contribution to this process comes from the feynman diagram shown in fig . 1 which proceeds through the exchange of a gluino and squarks @xcite and involves two @xmath87 mixings . the strength of the @xmath23 operator resulting from fig . 1 can be estimated to be @xmath88 ^ 2f'^2 \over { m_{\tilde{g } } m_{\tilde{q}}^4}}~,\end{aligned}\ ] ] where @xmath89 is the gluino mass , @xmath90 is the squark mass and @xmath91 is the @xmath87 mixing angle . the effective baryon number violating @xmath84 yukawa coupling in the superpotential is parametrized here as @xmath92 ( see eq . ( 2 ) ) . ( fig1nnbar.epsf width 8 cm ) let us first discuss the origin of the flavor mixing that changes a @xmath84 operator to the required @xmath86 operator . the dominant source for this in the present context turns out to be the mixing of @xmath93 with @xmath94 . such mixings occur in the left right supersymmetric model since the right handed quark mixings are physical above the scale @xmath30 . the renormalization group evolution of the soft susy breaking mass parameters between @xmath95 and @xmath30 will then induce mixings in the right handed down squark sector proportional to the top quark yukawa coupling and the right handed ckm mixings . this is analogous to the rge evolution in the mssm inducing squark mixing in the left handed down squark sector proportional to the left handed ckm angles and the top quark yukawa coupling . we estimate this right handed @xmath87 mixing to be @xmath96 this estimate is obtained by integrating out the rge between @xmath95 and @xmath30 assuming universality of masses at @xmath95 @xcite . since above @xmath30 , both @xmath97 and @xmath98 are part of the same @xmath12 multiplet , unlike in the mssm , @xmath98 yukawa coupling is of order one . in this momentum range , quark yukawa coupling reduces the mass of @xmath93 . in going to the physical basis of the quarks , this effect will induce the squark mixing quoted in eq . ( 5 ) . for the numerical estimate we took @xmath99 and @xmath100 and @xmath101 gev for illustration . there is a second source of flavor violation that induces @xmath102 mixing in general susy models . that is the baryon number violating yukawa couplings themselves . if we write in standard notation , the effective @xmath103violating superpotential arising from eq . ( 2 ) as @xmath104 , the rge evolution from planck scale to the weak scale will induce @xmath87 mixing proportional to @xmath105 . for example , if we keep only the couplings involving the @xmath106 quark , viz . , @xmath107 and @xmath108 , we can estimate the induced @xmath91 by integrating the relevant rge @xcite to be @xmath109 recalling that @xmath110 , we see that while this source of flavor mixing may not be negligible , it would be typically smaller than the ones from the right handed quark mixings of eq . a third source of flavor violation relevant for @xmath2 oscillations has been identified in ref . @xcite involving the exchange of the wino . such diagrams will have an electroweak loop suppression and a chirality suppression necessary to convert the left handed squark to the right handed one . we find that this contribution to @xmath80 has a suppression factor given approximately by @xmath111 ^ 2 \sim 1 \times 10^{-9}$ ] ( valid for small @xmath112 ) which is about two orders of magnitude smaller in this class of models compared to the gluino exchange diagram of fig . 1 . one has to calculate the hadronic matrix element of the six quark operator in order to obtain the @xmath4 . this has been discussed in several places in the literature@xcite . the calculations of this `` conversion '' factor can be done using crude physical arguments , according to which one has to multiply the @xmath32 by @xmath113 to obtain @xmath80 where @xmath114 is the baryonic wave function for three quarks inside a nucleon . on dimensional grounds , one can deduce that @xmath115 , which implies that @xmath116 gev . more detailed bag model calculations have been carried out . rao and shrock in ref . @xcite quote this conversion factor to be @xmath117 . we shall use this number for our numerical illustrations . combining this matrix element with eq . ( 4)-(5 ) we obtain @xmath118 we can rewrite eq . ( 7 ) in a form that makes the connection with the neutrino mass more transparent . the mass of @xmath119 can be expressed through the seesaw formula from eq . ( 2 ) as @xmath120 where @xmath121 denotes the dirac mass of @xmath119 . eliminating the high scale @xmath30 from this , we have from eq . ( 7 ) , @xmath122 since the value of @xmath123 can be determined from the atmospheric neutrino data under certain assumptions , we conclude that within the seesaw framework , measurement of @xmath2 oscillation will be a measure of the dirac mass of the tau neutrino . this can then be used as a way to discriminate between models of neutrino masses . to see the specific prediction for @xmath2 oscillations within the context of the class of models under consideration , we need to know the @xmath119 dirac mass . we can estimate it from the following relations for the dirac masses of the third generation quarks and leptons in the susy left right model : @xmath124 here @xmath125 are the higgs mixing parameters obtained from eq . ( 1 ) ( eg : @xmath126 ) and @xmath127 are the vevs of the mssm doublets . from eq . ( 9 ) it follows that in the limit of @xmath128 and @xmath129 , we get @xmath130 . since at such high scales @xmath131 , this predicts @xmath132 . in fact , we find that unless the two terms in eq . ( 9 ) for @xmath121 are precisely canceled , the dirac mass of @xmath119 will be approximately equal to @xmath133 . using @xmath134 and @xmath135 , @xmath136 , we get a value for the @xmath2 oscillation time which is tantalizingly close to the present experimental lower limit @xcite . for values of @xmath137 10 times smaller than @xmath133 , and taking the supersymmetric particle masses as large as 1 tev , we see that @xmath4 is less than @xmath138 seconds , case should be about 2 , which would reduce the estimate of @xmath4 by a factor of 2 . ] which is in the range accessible to a recently proposed experiment @xcite . it would thus appear that a search for neutron - antineutron oscillation will provide an enormously useful window into neutrino mass models and as such a powerful constraint on the nature of new physics beyond the standard model . the prediction for @xmath2 oscillations can be sharpened if we make use of flavor symmetries to determine the coefficients @xmath139 and @xmath140 in eq . we illustrate this with a specific choice of flavor symmetry @xcite taken to be @xmath141 . the first two families of fermions form doublets of @xmath142 and have a @xmath143 charge of @xmath144 while the third family fermions are singlets under both groups . this flavor symmetry is broken by a pair of doublets @xmath145 and singlets @xmath146 . allowing for effective operators suppressed by a scale @xmath18 larger than the vevs of these fields provides a natural explanation of the fermion mass and mixing angle hierarchy . if we choose @xmath147 and @xmath148 , a resonable fit to all quark and lepton masses is obtained , including neutrinos , for @xmath149 and @xmath150 and all dimensionless couplings being order one @xcite . in this model , we can estimate the couplings @xmath139 and @xmath140 from the horizontal quantum numbers . they are @xmath151 and @xmath152 , so that could have been of order one but in the horizontal model or ref . @xcite , due to large @xmath153 mixing , it is the @xmath154 flavor entry that dominates the atmospheric neutrino mass difference and hence the horizontal suppression factor @xmath155 . the @xmath156 factor is due the fact that the operator must be invariant under @xmath143 . ] @xmath157 this estimate leads to @xmath158 sec from eq . ( 8) . allowing for uncertainties of order 1 in this estimate , we expect that @xmath4 not to exceed about @xmath159 sec . before we conclude a few comments are in order : 1 . the model becomes unacceptable as soon as @xmath160 is embedded into a higher symmetry such as @xmath25 or so(10 ) group because in that case , the @xmath29 operator described in eq . ( 2 ) is accompanied by other r - parity violating operators coming from the same higher dimensional operator @xmath73 due to the higher symmetry . together , they would lead to unacceptable proton decay rate . thus observation of @xmath2 oscillation would be a signal of an explicit @xmath160 symmetry all the way upto the planck ( or string ) scale . 2 . baryogenesis has to proceed through a weak scale scenario since the @xmath29 interactions in the model are in equilibrium down to the tev scale and will wash out any primordial baryon or lepton asymmetry . we note that the baryon number violating interactions contained in @xmath72 themselves can potentially be the source of weak scale baryogenesis @xcite . the lightest neutralino in this model is unstable and will decay via @xmath161 modes due to the presence of the effective @xmath162 operator @xmath72 . this prediction is directly testable at colliders . an alternative candidate for dark matter must be sought . in conclusion , we have found that in a large class of seesaw models for neutrino masses , despite the high scale of sessaw dictated by the current neutrino oscillation data , neutron antineutron oscillation is in the observable range . in fact , unless the dirac masses of neutrinos are far below those deduced under simple and reasonable assumptions , we predict an upper bound on the neutron - antineutron oscillation time in the range of @xmath163 sec . this is very close to the present experimental lower limit on @xmath2 oscillations . in the most conservative theoretical scenario , the measurement of @xmath2 oscillation time would be a measure of the dirac mass for the tau neutrino , given the values of squark masses . this in itself would be an extremely interesting result , since it would discriminate among theoretical models of neutrino masses . this is apart from the fundamental importance that any observation of baryon number violation will carry . we therefore strongly urge a new experimental search for neutron antineutron oscillation . the work of ksb has been supported in part by doe grant # de - fg03 - 98er-41076 , a grant from the research corporation , doe grant # de - fg02 - 01er4864 and by the osu environmental institute . rnm is supported by the national science foundation grant no . ksb is grateful to the elementary particle theory group at the university of maryland for the warm hospitality extended to him during a visit when this work was completed . one of the authors ( r. n. m. ) would like to thank y. kamyshkov and r. shrock for discussions . m. gell - mann , p. ramond and r. slansky , in _ supergravity _ , p. van niewenhuizen and d.z . freedman ( north holland 1979 ) ; t. yanagida , in proceedings of _ workshop on unified theory and baryon number in the universe _ , eds . o. sawada and a. sugamoto ( kek 1979 ) ; r. n. mohapatra and g. senjanovi , phys . lett . * 44 * , 912 ( 1980 ) . imb collaboration , t.w . al . , phys . lett . * 52 * , 720 ( 1984 ) ; kamiokande collaboration , m. takita et . al . , phys . rev . * d34 * , 902 ( 1986 ) ; frejus collaboration , ch . berger et . al . , phys . lett . * b240 * , 237 ( 1990 ) . s. rao and r. shrock , phys 116b * , 238 ( 1982 ) ; j. pasupathy , phys . lett . * 114b * 172 ( 1982 ) ; riazzuddin , phys . rev . * d25 * , 885 ( 1982 ) ; s. misra and u. sarkar , phys . rev . * d28 * , 249 ( 1983 ) ; s. fajfer and r. j. oakes , phys . lett . * b132 * , 432 ( 1983 ) . | we show that in a large class of supersymmetric models with spontaneously broken @xmath0 symmetry , neutron
antineutron oscillations occur at an observable level even though the scale of @xmath0 breaking is very high , @xmath1 gev , as suggested by gauge coupling unification and neutrino masses .
we illustrate this phenomenon in the context of a recently proposed class of seesaw models that solves the strong cp problem and the susy phase problem using parity symmetry .
we obtain an _ upper _ limit on @xmath2 oscillation time in these models , @xmath3 sec .
this suggests that a modest improvement in the current limit on @xmath4 of @xmath5 sec will either lead to the discovery of @xmath2 oscillations , or will considerably restrict the allowed parameter space of an interesting class of neutrino mass models .
epsf.tex ( # 1 width # 2)=#2 |
You are an expert at summarizing long articles. Proceed to summarize the following text:
laser cooled trapped ions offer a very high level of control , both of their motional and internal quantum states . at the same time , the large charge - to - mass ratio of ions makes their motion very sensitive to electric fields , both static and oscillatory . thus , trapped ions recently emerged as a tool in small - force sensing@xcite . more common applications of trapped ions are in quantum information science @xcite and frequency metrology @xcite . all these applications can benefit from scalable ion - trap architectures based on microfabricated ion traps . in particular , a promising route to achieve scalable quantum information processing uses complex electrode structures@xcite . considerable effort is made in developing microfabricated trap architectures on which all trap electrodes lie within one plane @xcite . these so - called planar traps facilitate creation of complex electrode structures and are , in principle , scalable to large numbers of electrodes . moreover , this approach makes use of mature microfabrication technologies and is ideally suited to approaches involving hybrid ion - trap or solid state systems @xcite . despite the advantages of planar trap architectures , a number of issues remain unsolved . to achieve reasonably large trap frequencies , planar traps require shorter ion - electrode distances than conventional three - dimensional traps @xcite . this results in high motional heating rates for the ions @xcite and causes charge buildup via stray light hitting the trap electrodes@xcite . in addition , the proximity of the charges increases the effect of charge buildup as compared to macroscopic three dimensional traps . finally , planar traps do not shield stray electrostatic fields from the environment surrounding the trap as well as the three dimensional trap geometries tend to do . combined , these effects make the operation of planar traps much more sensitive to uncontrolled charging effects . to harness the full advantages of segmented ion traps , ion - string splitting and ion shuttling operations are required@xcite . for the reliable performance of these operations , control of the electrostatic environment over the full trapping region is necessary . typically one employs numerical electrostatic solvers to determine the potential experienced by the ions and generates electrode voltage sequences that will perform the desired ion shuttling @xcite . stray electrostatic fields , however , displace the ions from the rf - null of the trap and thus introduce so - called micromotion@xcite sometimes to the point where trapping is no longer feasible . thus , precise characterization and compensation of stray electric fields in the trapping region is required . conventional methods to sense and compensate the electric stray fields can not easily be extended to planar traps because typically the stray fields are quantified via the the doppler shift induced by the micromotion . it is undesirable to scatter uv light from the trap electrodes , and , thus , for planar traps , the detection laser typically does not have a sizable projection on the motion perpendicular to the plane of the trap . we address these issues by applying a new method to compensate for stray fields well suited for planar trap geometries @xcite . based on the voltages required to compensate the stray fields , we realize a single - ion electric field sensor characterizing the electric stray fields along the trap axis . we observe a strong buildup of stray charges around the loading region on the trap . we also find that the profile of the stray field remains constant over a time span of a few months . the strength of the electric stray fields and its position on the trap is correlated with the high heating rates observed close to the loading region @xcite . we use a planar trap with gold electrodes deposited on a sapphire substrate to trap single @xmath0ca@xmath1ions at a height of 240 @xmath2 m above the trap plane , see fig.[fig : trap ] . ions are created via two step photoionization from a neutral calcium beam using 250 mw/@xmath3 of laser light at 422 nm and 750 mw/@xmath3 of laser light at 375 nm . both the laser beams are focused to a waist size of 50 @xmath2 m . great care has been taken to minimize exposure of the trap surface to the neutral calcium beam . schematic of the trap used for the measurements@xcite . the dc electrodes are drawn in blue , the rf electrode in orange , and the ground plane in gray . details of the bonding pads to the dc electrodes are not shown for simplicity . the axes indicate the origin of the coordinate system . the green line along the z axis on the central dc electrode indicates the range of axial positions in which the stray electric fields shown in fig.[fig : el - field ] were measured . the circular mark on this line indicates the location used as a loading region , around which the highest increase in stray electric fields was observed.,scaledwidth=40.0% ] the rf electrode is driven at a frequency @xmath4 15 mhz , amplified to @xmath5mw and stepped up via a helical resonator in a quarter wave configuration to a voltage of approximately 100 v amplitude . a 2:1 asymmetry in the width of the rf electrode results in a tilt of the radio frequency quadrupole by approximately @xmath6 in the @xmath7 plane . the dc electrodes are used to move the ion along the axial direction and to compensate the stray fields . the dc voltages used for trapping and compensation are between -10 v and 15 v. typical secular frequencies in this work were @xmath8 ( 1.2 , 1.4 , 0.4 ) mhz where the primes refer to the frame of reference rotated by @xmath6 . for doppler cooling and detecting the ions , we use a diode laser at 794 nm , which is frequency doubled using a ring cavity to produce a wavelength of 397 nm . a second diode laser at 866 nm is used as a repump . both lasers are frequency locked to cavities using the pound - drever - hall method , and their frequencies can be varied by changing the cavity lengths with piezoelectric elements . the intensity of the detection laser at 397 nm is adjusted to about 40@xmath9 and the intensity of the repump laser at 866 nm is adjusted to approximately 120@xmath9 . the doppler cooling and repump lasers are overlapped and sent to the trap using a photonic crystal fiber . the laser beam is aligned almost parallel to the surface of the trap and forms an angle of approximately @xmath10 with the @xmath11 and @xmath12 axes . ion fluorescence is collected perpendicular to the trap plane using a lens system of na = 0.29 and split between a pmt and ccd camera on a 90:10 beam splitter . in an ideal paul trap the ion is confined to a position at which the electric field due to the oscillating drive voltage on the rf electrodes is zero . stray dc electric fields , however , push the ion off the rf node and the ion undergoes so - called micromotion driven by the oscillating rf field @xcite . this motion causes broadening of the electronic transitions of the ion and , among other things , leads to a higher temperature limit for doppler cooling @xcite . in addition , micromotion can lead to the heating of trapped ions due to the noise present at the secular sidebands of the micromotion drive @xcite . in order to position the ion on the rf node , the dc potential is carefully adjusted to minimize micromotion . crucial to all minimization schemes is the efficient detection of micromotion in all three spatial directions . different techniques exist for fine - tuning the compensation of micromotion . the photon correlation method@xcite relies on correlating the ion fluorescence to the phase of the rf field . in the resolved sideband method@xcite , the sidebands of a narrow atomic transition are compared with the carrier transition to estimate the modulation index . both methods are widely used to suppress micromotion in 3d traps . however , neither method can be easily extended to surface paul traps because they directly use the doppler shift induced by the ion motion . in the case of surface traps , the geometry typically limits the laser alignment to be in the plane parallel to the trap surface resulting in no doppler shift associated with the oscillations perpendicular to the trap surface unless one directs laser light onto the trap electrodes . however , it has been documented that uv light hitting the trap surface can lead to dramatic charge buildup even to the point where the trap becomes inoperable for days@xcite . to circumvent this obstacle , the infrared repump light in @xmath0ca@xmath1has been used to detect the doppler shift perpendicular to the trapping plane for micromotion compensation @xcite . however , many ion species such as mg@xmath13 , al@xmath13 , hg@xmath13 , cd@xmath13 , be@xmath13 do not have such transitions in the infrared and other methods need to be employed . here we compensate the micromotion perpendicular to the trap plane with the following method . when the ion is displaced from the rf node , any voltage applied on the rf electrode creates an electric field at the ion position . if this voltage contains a frequency component which is in resonance with one of the ion secular frequencies the ion can get excited in the direction of the secular mode provided that the driving field from the rf electrodes has some projection @xcite . experimentally we find that large oscillation amplitudes of each of the three secular motions can be detected as a drop in ion fluorescence . the dynamics of ion fluorescence in the presence of the cooling laser and a resonant excitation are complex and go beyond the scope of this study @xcite . minimizing micromotion is achieved by shifting the ion position via dc potentials until the ion is in the rf minimum and can not be excited by driving the rf electrode at any of the secular frequencies . this method is also being used by nist @xcite and osaska @xcite ion trap groups . in order to implement this method , we first position the ion within 1 @xmath2 m along the @xmath11 direction from the rf null using the ccd camera and varying the rf amplitude . further compensation in the @xmath11 direction is achieved by reducing the linewidth of the s@xmath14-p@xmath14 transition . for this , the detection laser intensity is adjusted close to saturation and red - detuned from the transition so that the fluorescence drops to half that of the value at the resonance . then compensation voltages are adjusted to minimize the fluorescence . both methods detect only micromotion along the _ x _ direction , i.e. the direction which is parallel to the trap surface . for a very coarse compensation along the _ y _ direction ( perpendicular to the trap surface ) , we keep the frequency of 397 nm and 866 nm laser on resonance and maximize the ion fluorescence by adjusting the compensation voltages . once a coarse compensation is achieved using the above methods , we proceed with the method as outlined in the beginning of this section . instead of exciting the secular frequency @xmath15 directly , we excite at a frequency @xmath16 @xcite . to achieve this , the excitation signal from a function generator is mixed with the trap drive @xmath17 before it is amplified and stepped up with a helical resonator , and scanned around @xmath18 . when the frequency of the excitation becomes resonant with @xmath18 , the ion heats up resulting in a decrease in fluorescence ( see fig.[fig : mmdip ] ) , @xcite . a crucial requirement is that the step - up circuit which produces the high - voltage trap drive has a large enough bandwidth . the bandwidth of the helical resonator in our experiment is 270khz and allows compensation with excitation frequencies up to order @xmath192 mhz . the frequency of the doppler cooling laser was detuned between 1 and 5 mhz below the @xmath20 transition to maximize sensitivity of the ion fluorescence to the ion kinetic energy . + for compensation of micromotion in the _ x _ direction , the excitation signal is scanned repeatedly around @xmath21 while adjusting the voltages on the dc electrodes such that between successive scans the static electric field minimum moves predominantly in the _ x _ direction . the compensated position is reached when resonance of the excitation does not result in a decrease of fluorescence . the same process is repeated for compensation in the _ y _ direction . results are shown in fig.[fig : mmcomp ] . fig.[fig : mmx2d ] and fig.[fig : mmy2d ] show the change in the dip depth when the excitation frequency is @xmath21 and @xmath22 , respectively , as a function of ion position along the _ x _ and the _ y _ direction . the gray area in fig.[fig : mmx2d ] is the region were no data is acquired since the excitation would drive the ion out of the trap . fig.[fig : mmx1d ] and fig.[fig : mmy1d ] show cross sections of the 2d plots along the equilibrium positions of the ion when the dc saddle point moves along the _ x _ and _ y _ directions , respectively . the energy gain rate of the ion @xmath23 is expected to be @xmath24 , where @xmath25 is the strength of the exciting electric field . we find that the data can be well fitted with a parabola suggesting that the dip depth is linear in the ion energy for our experiments while the electric field @xmath25 can be described well as a quadrupolar field . the accuracy of our measurements is estimated by calibrating the ion displacement as a function of change in the dc voltages using the ccd camera in the x direction and the detection laser in the y direction . these values were verified by modeling the displacement of the dc minimum with variation of the compensation voltages . by translating the applied voltage into actual displacement , we determine an accuracy of about 50 nm in the x direction and 300 nm in y direction in positioning the ion at the rf minimum . this corresponds to excess micromotion amplitudes of 6 nm and 40 nm in these respective directions . the accuracies could be further improved by increasing the excitation voltage and decreasing the frequency detuning of the detection laser from resonance . one concern with this method is that in practical situations the dc electrodes pick rf voltage , the phase of which might be shifted , or which might depend on the rf excitation frequency . the reason for this is that the dc electrodes may capacitively couple to the rf electrode , with a frequency dependent coupling determined by the filtering circuits connected to the dc electrodes . in our setup , we estimate the rf pickup of the excitation at @xmath18 on the dc electrodes to differ by less than 20 @xmath2v from the pick up at @xmath17 which would shift the rf - null by about 15 pm , and thus not limit the accuracy . it is instructive to extract the size and direction of the stray fields from the compensation voltages along various trapping positions . we derive the stray field from the applied compensation voltages via an accurate electrostatic model of the trapping fields . thus determining the stray fields over an extended region yields important information for shuttling experiments or investigating the mechanism for charge buildup . we model the trap potentials using the boundary element method solver cpo @xcite . since the radial trap axes are tilted , we determine the radial secular frequencies from a 2-dimensional polynomial fit in the @xmath7-plane . the agreement between experiment and simulation is better than 4% , corresponding to a disagreement of less than 20khz for axial secular frequency of 500khz , and 30khz for radial frequencies of between 1 and 2mhz . we attribute the disagreement to details of the trap electrode geometry which were not included in the electrostatic simulation , as well as to the unknown spatial inhomogeneity of the electrostatic stray fields . only the spatially inhomogeneous part of the stray field will contribute to the discrepancy between simulation and experiment , since the electrostatic stray fields are compensated and included in the simulations , as we now describe . the essential part of the stray field determination is to perform micromotion compensation as described above . after the compensation parameters have been determined , we measure the ion position along the z direction using the ccd imaging system . the values of the dc voltages at the compensated configuration and the ion position are then input to a minimization algorithm . the algorithm finds the stray field which results in the observed compensation parameters and position . the accuracy of our measurement scheme is limited by an estimated uncertainty of @xmath262.5 @xmath2 m in measuring the absolute ion position along the trap axis . this position uncertainty arises from the imprecision of the alignment of the microscope objective used for imaging with respect to the trap plane and from the size of trap features used to determine the position along the trap axis . this results in a systematic error of the electric field curves , mainly an offset of the entire curve . we find typical inaccuracies of @xmath27v / m . the precision in determining the electric field at each ion position is on the order of a few v / m , and is limited by the precision to which the compensation parameters and relative ion position are determined . the errors from imperfect compensation lead to stray field imprecisions of @xmath260.4 v / m in the horizontal direction and @xmath262.5 v / m in the vertical direction , where we assume that the precision is limited by the ion displacement steps in fig.[fig : mmcomp ] . the accuracy in determining the axial ion position at any given setting with respect to its positions in neighboring settings is limited by aberrations in the imaging optics and is estimated to be @xmath260.1@xmath2 m . this limitation leads to an imprecision of @xmath28v / m in the electric fields . the above limitations of the measurement leads to a total error of @xmath29v / m , however , the limitations are of technical nature and can be further improved . stray electrostatic field along the trap axis . the three components @xmath30 ( @xmath31 ) , @xmath32 ( @xmath33 ) , @xmath34 ( @xmath35 ) and the total electric field @xmath25 ( @xmath36 ) are shown . the solid curves on the two sides of each curve correspond to the possible systematic offset arising from the uncertainty in the absolute axial position , as described in the text.,scaledwidth=50.0% ] [ sec : electric - stray - fields ] we performed this type of analysis over an extensive part of the trap . measurements were performed during several months . the results obtained in the first weeks of trap installation in vacuum and trap operation were significantly different than those obtained later on . initially , the magnitude of the stray electric fields in the radial directions varied on a day - to - day basis with a mean of 47 v / m and a standard deviation of 52 v / m . measurements during that time were carried out only for one isolated axial ion position at 2240 @xmath2 m . after two months of trap operation , there was an abrupt change in stray electric fields to considerably higher values . this change coincided with a pressure increase in the chamber to approximately @xmath37 mbar because of a temporary ion pump failure . subsequently , the stray field was mapped along the trap axis over a length of approximately 2 mm . the results are shown in fig.[fig : el - field ] . we measured stray fields as large as 1300v / m . intuitively it is conceivable that two localized charge sources located near the z axis are responsible for the observed electric field pattern . however , such inverse problems can not be solved without further assumptions about the charge distribution @xcite . attempts to reconstruct the charge distribution mathematically are in progress and will be reported elsewhere the position of the postulated charges close to the most frequently used trapping positions suggests that exposure of the trap material to laser light caused the long term charging . finally , the high stray fields also appear to be spatially and temporally correlated with the observed high heating rate@xcite . our measurement scheme allowed us to monitor charging of the trap during operation . we found that the stray fields after the abrupt change in trap behavior are semi - permanent , showing a slow temporal drift on the order tens v / m per week . the most striking occurrence of this slow drift is the slight discontinuity in the data of fig.[fig : el - field ] around an axial position of 2450@xmath38 . this drift by 70v / m , corresponding to a fractional change in the stray field by @xmath396% , occurred over a period of 9 days . besides the slow drift of the stray field , short - term charging is observed in the course of a day . this is typically on the order of 100v / m . when the laser light is turned off , discharging occurs over the course of several hours . in this article , we demonstrate a simple , yet efficient method of measuring stray electric fields in planar ion traps . this method permits us to sense electric fields over an extended region , thus providing insight into the undesired charging of ion traps . this ability to characterize electric fields in the trapping region will be a valuable tool for evaluating planar ion traps , for developing ion loading approaches that minimize stray charging , and for compensating stray fields . we expect that this technique will be useful not only for scalable quantum information processing , but also for precision frequency metrology applications of trapped ions . the experiments were supported by the austrian ministry of sciences with a start grant and by the director , office of science , office of basic energy sciences , materials sciences and engineering division , of the u.s . department of energy under contract no . de - ac02 - 05ch11231 . n. daniilidis was supported by the european union with a marie curie fellowship . f. schmidt - kaler acknowledges support from the german - israel foundation and the eu network aqute . 10 t rosenband , d b hume , p o schmidt , c w chou , a a , l lorini , w h oskay , r e drullinger , t m fortier , j e stalnaker , s a a , w c swann , n r newbury , w m itano , d j wineland , and j c bergquist . , 319(5871):18081812 , 2008 . s seidelin , j chiaverini , r reichle , j j bollinger , d leibfried , j britton , j h wesenberg , r b blakestad , r j epstein , d b hume , w m itano , j d jost , c langer , r ozeri , n shiga , and d j wineland . , 96:253003 , 2006 . j britton , d leibfried , j beall , r b blakestad , j j bollinger , j chiaverini , r j epstein , j d jost , d kielpinski , c langer , r ozeri , r reichle , s seidelin , n shiga , j h wesenberg , and d j wineland . . , 2006 d r leibrandt , j labaziewicz , r j clark , i l chuang , r epstein , c ospelkaus , j wesenberg , j bollinger , d leibfried , d wineland , d stick , j sterk , c monroe , c .- s . pai , y low , r frahm , and r e slusher . . , 9(11):0901 , 2009 . | we use a single ion as an movable electric field sensor with accuracies on the order of a few v / m . for this , we compensate undesired static electric fields in a planar rf trap and characterize the static fields over an extended region along the trap axis .
we observe a strong buildup of stray charges around the loading region on the trap resulting in an electric field of up to 1.3 kv / m at the ion position .
we also find that the profile of the stray field remains constant over a time span of a few months . |
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the inflationary stage of the very early universe explains the dynamical origin of the observed isotropic and homogeneous frw geometry . the patch of the frw geometry covers the cosmological horizon and beyond if inflation lasted @xmath3 e - folds or longer . here @xmath4 is the potential energy of the inflation , and @xmath5 is a correction from the ( p)reheating stage after inflation , which is not essential for our discussion . chaotic inflationary models , associated with a large energy ( @xmath6 gut scale ) of @xmath7gev , predict a very large number of inflationary e - folds , @xmath8 . long - lasting inflation erases all classical anisotropies and inhomogeneities of the pre - inflationary stage . however , scalar and tensor vacuum fluctuations during inflation lead to almost scale free post - inflationary scalar and tensor metric inhomogeneities around our smooth observable frw patch . in particular , the amplitude of the gravitational waves generated from the vacuum fluctuations during inflation is proportional to @xmath9 , @xmath10 ( where @xmath11 is the reduced planck mass ) . there are significant efforts to measure the @xmath12-mode of @xmath13 polarizations , since this will provide a direct probe of the scale of inflation . the current @xmath14 c.l . limits on @xmath15 ( ratio of the tensor to scalar amplitudes of cosmological fluctuations ) @xmath16 ( wmap - only ) and @xmath17 ( wmap plus acoustic baryon oscillation , plus supernovae ) @xcite shall be improved to @xmath18 by the planck mission @xcite , to @xmath19 by the @xmath20over @xcite , ebex @xcite , and spider @xcite experiments ( see @xcite for the study of a mission that can improve over these limits ) . while these limits imply a detection in the case of high energy inflation , a number of other inflationary models , including many of the string theory constructions have lower energy , and therefore lead to gw of much smaller amplitude , which are virtually unobservable through @xmath12 mode polarization . up to the level @xmath21 with bbo @xcite or ultimate decigo @xcite direct detection experiments . ] in anticipation of the null signal observation of the primordial gw from inflation , it is worth thinking about other implementations of this result for the theory of inflation , besides putting limits on the energy scale @xmath22 . there are models of inflation ( including many string theory inflationary models ) where the total number of e - folds , @xmath23 , does not exceed the minimum ( [ efold ] ) by a large number . if the extra number of e - folds @xmath24 beyond ( [ efold ] ) is relatively small then pre - inflationary inhomogeneities of the geometry are not erased completely , and their residuals can be subject to observational constraints . in the context of this idea , in this paper we suggest an additional mechanism to have observable gravitational waves associated with inflation . these gravitational waves are very different from the gw generated from the vacuum fluctuations during inflation . firstly , they are the residual tensor inhomogeneities from the pre - inflationary stage . secondly , they can be of a classical , rather than quantum , origin . thirdly , while their initial amplitude and spectrum are given by the initial conditions , they are significantly affected by the number of `` extra '' e - folds @xmath24 . therefore , observational limits on gravity waves result in constraints on a combination of @xmath24 and of the initial amplitude . the choice of the initial geometry of the universe before inflation is wide open . in principle , one may assume an arbitrary geometry with significant tensor inhomogeneities component , and much smaller scalar inhomogeneities . this choice is , however , very artificial . a much more comfortable choice of the pre - inflationary stage will be a generic anisotropic kasner - like geometry with small inhomogeneities around it . the origin of the anisotropic universe with the scalar field can be treated with quantum cosmology , or can be embedded in the modern context of the tunneling in the string theory landscape . in fact , a kasner - like ( bianchi i ) space was a rather typical choice in previous papers on pre - inflationary geometry , see e.g. @xcite . most of the works on an anisotropic pre - inflationary stage aimed to investigate how the initial anisotropy is diluted by the dynamics of the scalar field towards inflation @xcite . the formalism of linear fluctuations about an anisotropic geometry driven by a scalar field toward inflation was constructed only recently @xcite . besides the technical aspects of calculations of cosmological fluctuations , there is a substantial conceptual difference between computations in the standard inflationary setting and in the anisotropic case . for an isotropic space undergoing inflationary expansion , all the modes have an oscillator - like time - dependence at sufficiently early times , when their frequency coincides with their momentum . one can therefore use quantum initial conditions for these modes . this is no longer the case for an expansion starting from an initial kasner singularity . in this case , a range of modes , which can potentially be observed today ( if @xmath24 is not too large ) , are not oscillating initially and therefore can not be quantized on the initial time hyper - surface ; as a consequence , there is an issue in providing the initial conditions for such modes . for this reason we will adopt another perspective , namely , we will consider generic small classical inhomogeneities around the homogeneous background , as an approximation to the more generic anisotropic and inhomogeneous cosmological solution . equipped with this philosophy , we consider an anisotropic expanding universe filled up by the scalar field with a potential @xmath25 which is typical for the string theory inflation . we add generic linear metric fluctuations about this geometry . the evolution of these fluctuations is by itself an interesting academic subject . however , it acquires a special significance in the context of the gw signals from inflation , because of a new effect that we report here of amplification of long - wavelength gw modes during the kasner expansion . this growth terminates when a mode enters the `` average '' hubble radius ( the average of that for all the three spatial directions ) , or , for larger wavelength modes , when the background geometry changes from anisotropic kasner to isotropic inflationary expansion . we perform explicit computations in the case of an isotropy of two spatial directions . in this case the computation becomes much more transparent and explicitly @xmath26 dependent . fluctuations for arbitrary @xmath27 were considered in the formalism of @xcite , where the @xmath26 dependence is not explicit . we verified that our results agree with @xcite in the axisymmetric @xmath28 limit . we find that only one of the two gw polarizations undergoes significant amplification . therefore , even if we assume for simplicity equi - partition of the amplitudes of the three inhomogeneous physical modes of the system ( the scalar and the two gw polarization ) at the initial time , the final spectra that will be frozen at large scales in the inflationary regime will be very different from each other , in strong contrast to what is obtained in the standard inflationary computations . this result can have different consequences , that we explore in the present work . suppose that the growing gw mode is still linear ( but significantly exceeds other modes ) when the space becomes isotropic . then , we can have significant yet linear classical gw fluctuations at the beginning of inflation , say of amplitude @xmath29 . if the modes which correspond to the largest scales that we can presently observe left the horizon after the first @xmath24 e - folds of inflation , their amplitude decreased by the factor @xmath30 in this period . if @xmath24 is relatively small , say @xmath31 the freeze out amplitude of these gw modes would be @xmath32 . the angular spectrum of these gw will rapidly decrease as the multipole number @xmath0 grows , since smaller angular scales are affected by modes which spend more time inside the horizon during the inflationary stage . suppose instead that the growing gw mode becomes non - linear before the onset of inflation . in this case the background geometry departs from the original onset . besides the phenomenological signatures , it is interesting to study the origin of the amplification of the gw mode . it turns out that the effect of gw amplification is related to the anisotropic kasner stage of expansion . therefore we will separately study gw in the pure expanding kasner cosmology . for completeness , we also include the study of gw in a contracting kasner universe , which is especially interesting due to the universality of anisotropic kasner approach to singularity . the plan of the paper is the following . in section [ sec : background ] we discuss the evolution of the anisotropic universe driven by the scalar field towards inflation . in section [ sec - linear ] we briefly review the formalism of the linear fluctuations in the case of a scalar field in an anisotropic geometry , paying particular attention to the gw modes . in section [ sec : decoupled ] we compute the amplification of one of the two gw modes that takes place at large scales in the anisotropic era . in section [ sec : pair ] we discuss instead the evolution of the other two physical modes of the system . in section [ sec : kasner ] we study the evolution of the perturbations in a pure kasner expanding or contracting universe . in section [ sec : obs ] we return to the cosmological set - up , and we compute the contribution of the gw polarization amplified during the anisotropic stage to the cmb temperature anisotropies . in particular , by requiring that the power in the quadrupole does not exceed the observed one , we set some limits on the initial amplitude of the perturbations vs. the duration of the inflationary stage . in section [ sec : sum ] we summarize the results and list some open questions following from the present study , which we plan to address in a future work . the anisotropic bianchi - i geometry is described by @xmath33 where @xmath34 are the scale factors for each of the three spatial directions . we consider a scalar field @xmath35 in this geometry . many string theory inflationary models ( for examples see @xcite ) have a very flat inflationary potential which changes abruptly around its minima . therefore , to mimic this situation , we will use a simple inflaton potential @xmath36 which has quadratic form @xmath37 around the minimum , and is almost flat @xmath38 away from it . to obtain the correct amplitude of scalar metric perturbations from inflation , we set @xmath39 the background dynamics is governed by the einstein equations for the scale factors in the presence of the effective cosmological constant @xmath40 , plus a possible contribution from the kinetic energy of the scalar field . quantum cosmology or tunneling models of the initial expansion favor a small scalar field velocity . therefore we select the small velocity initial conditions @xmath41 . in this case the generic solutions of the einstein equations with cosmological constant for @xmath42 are known analytically ( see e.g. @xcite ) and can be cast as @xmath43^{1/3 } \ , \left[\tanh \left(\frac{3}{2}h_0 t \right ) \right]^{p_i-1/3 } \ , \ ] ] where @xmath44 are the kasner indices , @xmath45 , @xmath46 , and @xmath47 is the characteristic time - scale of isotropization by the cosmological constant , @xmath48 , while @xmath49 are the normalizations of the three scale factors for earlier times @xmath50 the anisotropic regime is described by the vacuum kasner solution @xmath51 which corresponds to an overall expansion of the universe ( the average scale factor @xmath52 is increasing ) , although only two directions are expanding ( two positive @xmath44-s ) while the third one is contracting ( the remaining @xmath44 is negative ) . for later time @xmath53 the universe is isotropic and expanding exponentially @xmath54 where the constant normalizations @xmath55 are typically chosen to be equal . it is instructive to follow the evolution of the curvature in the model . the ricci tensor is ( almost ) constant throughout the evolution up to the end of inflation @xmath56 at earlier times the weyl tensor i.e. the anisotropic component of the curvature tensor gives @xmath57 and , for @xmath50 , @xmath58 is much bigger than the isotropic components ( [ ricchi ] ) . this is why initially the contribution from the effective cosmological constant is negligible , and the vacuum kasner solution ( [ early ] ) is a good approximation . in contrast , at later times @xmath53 @xmath59 and the anisotropic part of the curvature becomes exponentially subdominant relative to its isotropic part driven by the cosmological constant . this is an illustration of the isotropization of the cosmological expansion produced by the scalar field potential . the timescale for the isotropization is @xmath60 . in the following sections we will study the equations for the linear fluctuations around the background ( [ bianchi1 ] ) , ( [ factor ] ) . these equations become significantly simpler and more transparent for the particular choice of an axi - symmetric geometry e.g. when @xmath61 , and the metric is @xmath62 while the effect we will discuss is generic , for simplicity we will adopt the simpler geometry ( [ special ] ) rather than the general bianchi - i space ( [ bianchi1 ] ) . in this case , the early time solution is a kasner background with indices @xmath63 .. the solution is a very special one , since it is the only bianchi - i model with cosmological constant that is regular at @xmath64 ( as can be easily checked by computing the curvature invariants ; e.g. @xmath65 at @xmath66 ) . for @xmath67 , this space is actually minkowski space - time in an accelerated frame . due to its special nature , we disregard this solution in the present study . ] also , it will be useful to define an `` average '' hubble parameter @xmath68 and difference @xmath69 between the expansion rates in @xmath70 and @xmath71 ( or @xmath72 ) directions as @xmath73 at earlier times @xmath74 , while , at late times , @xmath75 . the equation for the homogeneous scalar field is @xmath76 since the value of @xmath68 is very large initially , @xmath77 , the hubble friction keeps the field ( practically ) frozen at @xmath78 during the anisotropic stage . for @xmath79 , the universe becomes isotropic , and it enters a stage of slow roll inflation until @xmath35 rolls to the minimum of its potential . we also will use another form of the metric ( [ special ] ) , with the conformal time @xmath80 . there is ambiguity in the choice of @xmath80 , related to possible different choices of the scale factors in its relation with the physical time . we will use the average scale factor @xmath81 and define @xmath80 through @xmath82 which , at early times , gives @xmath83 . in this variable , the line element ( [ special ] ) reads @xmath84\ .\ ] ] in the following , dot denotes derivative wrt . physical time @xmath85 , and prime denotes derivative wrt conformal time . moreover , we always denote by @xmath86 and @xmath87 the hubble parameters with respect to physical time . in the frw universe with a scalar field there are three physical modes of linear fluctuations . two of them are related to the two polarizations @xmath88 and @xmath89 of the gravitational waves , and one to the scalar curvature fluctuations @xmath90 induced by the fluctuations of the scalar field @xmath91 . all three modes in the isotropic case are decoupled from each other . the formalism for the linear fluctuations on a frw background has been extended to the bianchi - i anisotropic geometry in @xcite . again , there are three physical modes ; however , in the general case of arbitrary @xmath92 the modes are mixed , i.e. their effective frequencies in the bi - linear action @xmath93 are not diagonal , as it is the case in the isotropic limit . in a special case of the axi - symmetric bianchi i geometry ( [ special ] ) one of the three linear modes of fluctuations , namely , one of the gravity waves modes , is decoupled from the other two . this makes the analysis of fluctuations much more transparent than the general @xmath94 case . while the effects we will discuss , we believe , is common for arbitrary @xmath92 , we will consider linear fluctuations around the geometry ( [ special ] ) . the computation follows the formalism of @xcite , where the reader is referred for more details . the most general metric perturbations around ( [ special ] ) can be written as @xmath95 \end{array } \right)\ , . \label{metric}\ ] ] where the indices @xmath96 span the @xmath97 coordinates of the isotropic @xmath98d subspace . the above modes are divided into @xmath98d scalars ( @xmath99 ) and @xmath98d vectors ( @xmath100 , subject to the transversality conditions @xmath101 ) d vectors have one degree of freedom ; contrary to the @xmath102d case , there are no transverse and traceless @xmath98d tensors . ] according to how these modes transform under rotations in the isotropic subspace . the two sets of modes are decoupled from each other at the linearized level . in addition , there is the perturbation of the inflaton field @xmath103 , which is also a @xmath98d scalar . the gauge choice @xmath104 corresponding to @xmath105 , completely fixes the freedom of coordinate reparametrizations . it is convenient to work with the fourier decomposition of the linearized perturbations . we can therefore fix a comoving momentum @xmath106 , and study the evolutions of the modes having that momentum . since modes with different momenta are not coupled at the linearized level , this computation is exhaustive as long as we can solve the problem for any arbitrary value of @xmath106 . more precisely , we denote by @xmath107 the component of the momentum along the anisotropic @xmath70 direction , and by @xmath108 the component in the orthogonal @xmath109 plane . we denote by @xmath110 and by @xmath111 the corresponding components of the physical momentum . finally , we denote by @xmath26 and @xmath2 the magnitudes of the comoving and physical momenta , respectively . therefore , we have @xmath112 to identify the physical modes , one has to compute the action of the system up to the second order in these linear perturbations . one finds that the modes @xmath113 , and @xmath114 , corresponding to the @xmath115 metric perturbations , are nondynamical , and can be integrated out of the action . this amounts in expressing the nondynamical fields ( through their equations of motion ) in terms of the dynamical ones , and in inserting these expressions back into the quadratic action . for instance , for the nondynamical @xmath98d vector mode one finds @xmath116 the analogous expressions for the @xmath98d nondynamical scalar modes can be found in @xcite . in this way , one obtains an action in terms of the three remaining dynamical modes @xmath117 . once canonically normalized , these modes correspond to the three physical perturbations of the system . the canonical variables are @xmath118 \;\;,\;\ ; h_+ \equiv \frac{\sqrt{2 } \ , a_{\rm av } \ , m_p \ , p_t^2 \ , h_b}{h_a \ , p_t^2 + h_b \left ( 2 p_l^2 + p_t^2 \right ) } \psi \label{sca2d}\ ] ] and @xmath119 where @xmath120 is anti - symmetric and @xmath121 ( we stress that @xmath122 encodes only one degree of freedom , since , due to the transversality condition of the @xmath98d vector modes , @xmath123 ) . the dynamical equations for the modes @xmath124 and @xmath4 are coupled to each other , while that of the @xmath122 mode is decoupled @xmath125 the explicit expressions for the frequency matrix @xmath126 are rather tedious and given in @xcite . in the limit of isotropic background , @xmath127 , also the @xmath98d scalars decouple , and the frequencies become @xmath128 therefore , the mode @xmath4 becomes the standard scalar mode variable @xcite , associated to the curvature perturbation , while the modes @xmath124 and @xmath122 are associated to the two polarizations of the gravitational waves . also notice that without the scalar field there are two physical modes which , due to the residual @xmath98d isotropy , are decoupled . since there are no vector sources , the @xmath98d vector system describes a polarization of a gravitational wave obeying the `` free field equations '' @xmath129 ( which reproduce the equation ( [ b3 ] ) and the first of ( [ evol ] ) ) . as we now show , this mode undergoes an amplification , which does not occur for the other two modes . this can be understood from considering the frequency of this mode @xmath130 in the isotropic case , for which @xmath131 is given by ( [ omegiso ] ) , each mode is deeply inside the horizon , @xmath132 , at asymptotically early times . this is due to the fact that @xmath68 is nearly constant , while @xmath133 is exponentially large at early times . as a consequence , @xmath134 in the asymptotic past , and the mode oscillates with constant amplitude . in the present case instead @xmath135 at early times ( @xmath136 ) . namely , as we go backwards in time towards the initial singularity , the anisotropic direction becomes large , and the corresponding component of the momentum of the mode is redshifted to negligible values . on the contrary , the two isotropic directions become small , and the corresponding component of the momentum is blueshifted . however , the magnitude of the two hubble parameters increases even faster . therefore , provided we can go sufficiently close to the singularity , the early behavior of each mode is controlled by the negative term proportional to the hubble parameters in eq . ( [ vec2dminus ] ) . to be more precise , if we denote by @xmath137 and @xmath138 the values of the two scale factors at some reference time @xmath139 close to the singularity , we have @xmath140 \label{omegaapproxearly}\ ] ] where the first term in the expansion comes from the terms proportional to @xmath141 in eq . ( [ vec2dminus ] ) , while the second term from the component of the momentum in the isotropic plane ( cf . the early time dependences with those given in eq . ( [ earlyph ] ) ) . we see that the frequency squared is negative close to the singularity , so that the mode @xmath122 experiences a growth . in a pure kasner geometry , the relations ( [ earlyph ] ) hold at all times . therefore , one would find @xmath142 at asymptotically late times . for brevity , we will loosely say that the mode `` enters the two horizons @xmath143 '' at late times ; the meaning of this is simply that the frequency is controlled by the momentum in this regime , @xmath144 , and the mode @xmath122 enters in an oscillatory regime . this simple description is affected by two considerations : firstly , we do not set the initial conditions for the modes arbitrarily close to the singularity , but at some fixed initial time @xmath145 ; secondly , the geometry changes from ( nearly ) kasner to ( nearly ) de sitter due to the inflaton potential energy . consequently , there are three types of modes of cosmological size . i : modes with large momenta start inside the two horizons at @xmath145 . they oscillate ( @xmath146 ) all throughout the anisotropic regime , and they exit the horizon during the inflationary stage . ii : modes with intermediate momenta , for which ( [ omegaapproxearly ] ) is a good approximation at @xmath139 . these modes enter the horizons , and start oscillating , at some time @xmath147 during the kasner era ; they exit the horizon later during inflation . iii : modes with small momenta , that are always outside the horizons , and never oscillate during the kasner and inflationary regimes . these considerations are crucial for the quantization of these modes . we can perform the quantization only as long as @xmath146 , and the mode is in the oscillatory regime . as we mentioned , during inflation , this is always the case in the past . moreover , the frequency is adiabatically evolving ( @xmath148 ) , and one can set quantum initial conditions for the mode in the adiabatic vacuum . this procedure is at the base of the theory of cosmological perturbations , and results in a nearly scale invariant spectrum at late times , once the modes have exited the horizon and become classical . in the case at hand , we can not perform this procedure for modes of small momenta / large wavelength ( modes iii above ) . if inflation lasts sufficiently long , such modes are inflated to scales beyond the ones we can presently observe , so that the inability of providing initial quantum conditions is irrelevant for phenomenology . however , if inflation had a minimal duration , this problem potentially concerns the modes at the largest observable scales . irrespectively of the value of the frequency , it is natural to expect that the modes possess some `` classical '' initial value at @xmath145 . in the following we discuss the evolution of the perturbations starting with these initial conditions . although we do not have a predictive way to set these initial values , we can at least attempt to constrain them from observations . in addition , we should worry whether the growth of @xmath122 can result in a departure from the kasner regime beyond the perturbative level , in which case the background solution described in the previous section may become invalid ( we discuss this in section [ sec : kasner ] ) . as long as the frequency is accurately approximated by ( [ omegaapproxearly ] ) in the kasner regime , the evolution eq . for the mode @xmath122 is approximately solved by @xmath149 + c_2 \ : \sqrt{\eta } \ , y_3 \left[2\,k_t \left(\frac{a_0}{b_0}\right)^{1/3 } \sqrt{\eta_0\,\eta}\right ] \label{solveccl}\ ] ] where @xmath150 are two integration constants , while @xmath151 and @xmath152 are the bessel and neumann functions . the first mode increases at early times ( small argument in the bessel function ) @xmath153 , while the second one decreases as @xmath154 . we disregard the decreasing mode in the following computations , @xmath155 ( moreover , this mode diverges at the singularity ) . rather than the time evolution of @xmath122 , we show in figure [ evoltcl ] that of the corresponding power . we note the very different behavior obtained for large ( iii ) , intermediate ( ii ) , and small ( i ) scale modes . we choose to define the power spectrum as @xmath156 this definition coincides with the standard one ( see for instance @xcite ) as the universe becomes isotropic . in particular , the power spectrum is frozen at large scales in the isotropic inflationary regime . clearly , there is an ambiguity in this definition at early times , when the two scale factors differ ( there is no a - priori reason for the choice of the average scale factor in this definition ) . this arbitrariness affects the time behavior shown in the figure . however , it does not affect the relative behavior of the large vs. intermediate vs. small scale modes . moreover , if we analogously define the power spectra for the @xmath98d scalar modes , the relative behavior of these two types of mode ( figure [ evoltcl ] vs. figure [ evolpvcl ] ) is also unaffected by this arbitrary normalization . the reality of this instability is demonstrated in section [ sec : kasner ] , where we compute the squared weyl invariant due to these fluctuations ( more precisely , we do so in an exact kasner background , which , as we have remarked , coincides with the cosmological background at asymptotically early times ) . in figure [ powertcl ] we show the power spectrum ( normalized to the initial value for each mode ) for the same background evolution as in the previous figure , at some late time during inflation , when all the modes shown are frozen outside the horizon . as the approximate solution ( [ solveccl ] ) indicates , the growth of @xmath122 occurs as long as the transverse momentum @xmath111 is smaller than the hubble rates @xmath157 . therefore , modes with smaller @xmath108 experience a larger growth . we denote by @xmath158 the angle between the comoving momentum , and the anisotropic direction , @xmath159 therefore , in general , we expect a greater growth at smaller values of @xmath26 ( for any fixed @xmath158 ) and at smaller values of @xmath158 ( for any fixed @xmath26 ) . and @xmath160 . ] this behavior is manifest in figure [ powertcl ] . modes with @xmath161 experience the same growth during the kasner era ( since the leading expression for @xmath131 is independent of the momentum in this regime ) . then , the modes shown in the figure exhibit a very strong @xmath162 dependence for @xmath163 . we also see an increase of the power as @xmath158 decreases . we stress that figure [ powertcl ] shows the contribution of each mode to the power spectrum normalized to the value that that contribution had at the initial time @xmath145 . therefore , if the original spectrum has a scale , or an angular dependence , this will modify the final spectrum ( for comparison , for the isotropic computation of modes with adiabatic quantum initial condition , @xmath164 at early times ) . the large growth at small @xmath158 is more manifest in figure [ powerxi ] . the smallest angles shown in the figure correspond to @xmath165 , but to @xmath166 at the initial time ( this is due to the different behavior of the two scale factors in the anisotropic era ) . in this region , the spectrum exhibits a milder @xmath167 dependence than for intermediate values of @xmath158 . finally , one may also consider smaller angles than those shown in the figure , for which @xmath168 initially . we have found that final spectrum becomes @xmath169independent in this region . we conclude this section by discussing how the growth scales with the initial time . as long as @xmath170 , the power ( [ powtimes ] ) of a mode grows as @xmath171 ( as can be easily seen by combining the time dependences @xmath172 ) . we use the initial value of @xmath69 ( the difference between the two expansion rates , defined in eq . ( [ hubbles ] ) ) as a measure on the initial time , since , contrary to the conformal time , this quantity is not affected by the normalization of the scale factors . starting with a greater value of @xmath173 corresponds to starting closer to the initial singularity , and , therefore , to a longer phase in which @xmath174 grows . since @xmath175 in the kasner regime , the ratio @xmath176 in the region in which the growth takes place . although we do not show this here , we have verified that this scaling is very well reproduced by the numerical results . as discussed in section [ sec - linear ] the two other physical modes of the system are coupled to each other in the anisotropic era . the evolution equations for the coupled system are formally given in ( [ evol ] ) . at early times , we find @xmath177 \nonumber\\ \omega_{12}^2 = \omega_{21}^2 & = & a_{\rm av}^2 \left ( \eta \right ) { \mathcal o } \left ( \eta^0 \right ) \label{omegaapproxearly+}\end{aligned}\ ] ] therefore the coupling between the two modes can be neglected also at asymptotically early times . the main difference with the analogous expression for the decoupled tensor polarization , eq . ( [ omegaapproxearly ] ) , is that the squared eigenfrequencies of the two modes are now positive close to the singularity ; therefore the two modes @xmath178 do not experience the same growth as @xmath179 . indeed , as long as the expressions ( [ omegaapproxearly+ ] ) are good approximations , we find the solution @xmath180 + c_2^{h_+ } \ : \sqrt{\eta } \ , y_0 \left[2\,k_t \left(\frac{a_0}{b_0}\right)^{1/3 } \sqrt{\eta_0\,\eta}\right ] \label{solscacl}\ ] ] and an identical one for @xmath4 with the replacement @xmath181 of the integration constants . close to the singularity , the two modes grow as @xmath182 and @xmath183 , respectively . analogously to what we did for the @xmath122 polarization , we disregard the mode which grows less at early times , @xmath184 . we define the power spectra for the tensor polarization , and for the comoving curvature perturbation @xmath185 with the same prefactor as @xmath174 , cf . ( [ powtimes ] ) , @xmath186 we see that , contrary to what happened for @xmath179 , the coupled perturbations , and the corresponding power spectra , do not grow while outside the horizons during the anisotropic era . this effect is also manifest in figure [ powerpscl ] , where we show the spectra of the tensor mode @xmath124 and the comoving curvature for the same range of momenta , and for the same angles @xmath158 , as those of @xmath174 shown in figure [ powertcl ] . the main result of the previous section was a significant amplification of the mode @xmath122 , compared to the milder amplification of the mode @xmath124 , in the anisotropic background which is undergoing isotropization due to the effect of a scalar field . this growth can be ascribed to the instability of the kasner geometry , either contracting or expanding , against gravitational waves which we are going to report in this section . therefore in this section we consider linearized gravity waves around an expanding and a contracting kasner solution , without the presence of the scalar field , nor its fluctuation . in this case there are only the two decoupled modes @xmath122 and @xmath187 . the claim of instability of the kasner solution against the growth of the gw sounds at first glance heretic , at least for the contracting branch , in the light of the universality of the belinskii - khalatnikov - lifshitz ( bkl ) oscillatory regime of the kasner epochs approaching the singularity . in fact , it is not , and , on the contrary , it is compatible with the bkl analysis . moreover , our finding of the gw instability suggests a new interpretation of the phenomena connected to the instability , discussed by bkl and others for the contracting kasner geometry in very different formalism and language @xcite . in this section we first perform linearized calculations for the classical gravitational waves around expanding and contracting kasner solutions , and demonstrate their instabilities in terms of the evolution of the weyl tensor invariant @xmath188 , which is independent of the gauge choice . we then connect the result with the bkl analysis . the background line element is given by equation ( [ special2 ] ) , with the scale factors @xmath189 this compact notation describes two disconnected geometries , at negative and positive conformal times , respectively . the algebraic expressions below simplify if we introduce the time @xmath190 in which the normalization of the two scale factors coincide . therefore , rather than ( [ abeta ] ) , we can also use @xmath191 the two hubble rates are @xmath192 while the average scale factor is @xmath193 . the physical and conformal times are related by @xmath194 for future use , we also define @xmath195 to be the physical time corresponding to @xmath196 . the solution with negative conformal time describes an overall contracting geometry , @xmath197 , which crunches into the singularity at @xmath198 . the solution with positive conformal time describes instead an overall expanding space , @xmath199 , originating at the singularity at @xmath200 . as in the previous sections , we restrict the computation to the simpler case of a residual @xmath98d isotropy between two spatial directions . we expect that the instability occurs for general kasner indices @xmath201 . now we turn to the linearized perturbations satisfying the vacuum equations @xmath129 . there are two gravitational waves polarization perturbations . we consider a single mode with a given comoving momentum with components @xmath107 and @xmath108 . we denote by @xmath202 the vector in the @xmath109 plane along the direction of @xmath203 , @xmath204 the two gw polarizations obey the equations @xmath205 where the effective frequencies can be written in compact form @xmath206 \ , \nonumber\\ \omega_+^2 & = & a_{\rm av}^2 \left [ p^2 + h_a^2 \ , \frac{16 p_l^4 + 296 p_l^2 \ , p_t^2 + p_t^4}{\left ( 4 p_l^2 + p_t^2 \right)^2 } \right ] \ , \label{omega}\end{aligned}\ ] ] both for the contracting and the expanding backgrounds . we recall that the physical momenta are related to the comoving one by the relations ( [ mom ] ) . after solving the two equations ( [ eqomtomp ] ) , we can compute the metric perturbations ( [ metric ] ) and the weyl tensor of the background plus perturbations . the square of the weyl tensor , once expanded perturbatively , has the following schematic structure @xmath207 where @xmath208 is the weyl invariant for the non - perturbed background solution , @xmath209 is the term linear in @xmath210 and @xmath124 , and @xmath211 is the term quadratic in the perturbations . we do the computation for the two polarizations separately . for instance , for the @xmath98d vector modes , we compute the weyl tensor in terms of the background and of the metric perturbations @xmath212 and @xmath213 . we then relate the two perturbations to @xmath122 through eqs . ( [ b3 ] ) and ( [ canon ] ) , expressing spatial derivatives in terms of the comoving momenta , see eq . ( [ onemode ] ) . in this way , we can write the expression ( [ pweyl ] ) in terms of @xmath214 , and their time derivatives . finally , we insert in this expression the solutions of eq . ( [ eqomtomp ] ) . the procedure for the mode @xmath124 is analogous . we compare the second and third term in ( [ pweyl ] ) with the background term . a growth of the ratios @xmath215 ( denoted as @xmath216 ) , or @xmath217 , signals an instability of the kasner geometry . the background ( zeroth order ) weyl invariant is @xmath218 the first order term @xmath209 vanishes identically for the @xmath122 polarization . it is nonzero for @xmath124 , and it oscillates in space as @xmath219 \,$ ] . the second order term @xmath211 for the fourier modes @xmath122 or @xmath124 has a part which is constant in space , plus two parts that oscillate in space as @xmath220 \,$ ] . in the following , we disregard the oscillatory parts in @xmath221 . the solutions @xmath122 and @xmath124 are either monotonically evolving in time , or oscillating , with an envelope that is monotonically evolving in time . the oscillatory regime takes place when the mode has a wavelength shorter than the hubble radii , @xmath222 , and are absent in the opposite regime . eqs . ( [ earlyph ] ) show the time dependence of the momentum and of the hubble rates for the expanding kasner solution . we see that , if we consider a complete background evolution , any mode starts in the large wavelength regime , and then goes in the short wavelength regime . therefore , we expect that a mode is in the non - oscillatory regime sufficiently close to the singularity , and in the oscillatory regime sufficiently far from it ( this behavior is manifest in the time evolutions shown ) . the same is true for the contracting background solution ( with the obvious difference that early and late times interchange ) . the time dependence of the weyl invariant ( or of its amplitude , when the mode is oscillating ) is summarized in table 1 . more specifically , we show the ratio between the terms proportional to the perturbations and the background one , both for the expanding and the contracting kasner . in the expanding case , a mode evolves from the large scale to the short scale regime . the opposite happens in the contracting case . notice the the short scale behaviors in the expanding and contracting cases coincide . the same is not true for the two large scales behaviors . the reason is that , in the expanding case , we set to zero one of the two solutions of eq . ( [ eqomtomp ] ) that is decreasing at early times , and that would diverge at the initial singularity . the different behaviors are discussed in details in the next subsection and in appendix [ appa ] . [ cols="^,^,^,^,^",options="header " , ] in this subsection , we compute the contribution of the @xmath122 to the square of the weyl tensor , both for an expanding and a contracting kasner geometry . the analogous computation for the mode @xmath124 can instead be found in appendix [ appa ] . plugging ( [ abeta2 ] ) into ( [ omega ] ) , the frequency has the large scales ( early times ) and short scales ( late times ) expansions @xmath223 consequently , we have the asymptotic solutions @xmath224 where @xmath225 are three integration constants . in the early time solution we have disregarded a decaying mode that would diverge at @xmath226 , and where the expression in the second line is the large argument asymptotic expansion of the parabolic cylinder functions @xmath227\ ] ] which are solutions of the evolution equation with the short scales expanded frequency ( [ esp - omx - exp ] ) . an extended computation of the square of the weyl tensor ( performed as outlined after eq . ( [ pweyl ] ) ) leads to @xmath228 ; for the non - oscillatory part of the quadratic term in the fluctuations we find instead @xmath229 since the background square weyl @xmath230 , this computation indicates that the ratio @xmath231 increases as @xmath232 in the large scales regime , while it oscillates in the short scales regime , with an amplitude that increases as @xmath233 . in the left panel of figure [ weyl - exp ] we present the full numerical evolution of @xmath217 for three modes with different momenta . the behavior of the modes that we find here ( pure kasner geometry ) should be compared with that discussed in the previous sections , where the initial kasner evolution was followed by an isotropic inflationary stage . in the evolutions shown in that case ( for instance , [ evoltcl ] and [ evolpvcl ] ) we had isotropization at the time @xmath234 , and we started with an initial time of about @xmath235 . in the present case , the geometry does not undergo isotropization . however , the two scale factors are normalized in such a way that they are equal to each other at the time @xmath195 , cf . eqs . ( [ abeta2 ] ) . therefore , we also choose @xmath236 in the present case . also in analogy to the modes shown in figs . [ evoltcl ] and [ evolpvcl ] , we define @xmath237 to be the comoving momentum of the modes which have parametrically the same size as the average horizon at the time @xmath195 , namely @xmath238 ( cf . the expressions ( [ hubconf ] ) ) . moreover , we choose @xmath239 as in those two figures . figure [ weyl - exp ] confirms that each mode evolves from the non oscillatory large scales regime to the oscillatory short scales regime ( the transition occurs at later times for modes of smaller momenta / larger scales ) . the time dependence of @xmath217 shown in the figure agrees with the one obtained analytically , and summarized in table [ tab1 ] . for comparison we also plot in the figure [ weyl - exp ] the evolutions of the @xmath124 mode during expansion considered in the appendix a(a ) . the results shown in the figure confirm the instability of the background kasner solution against the gw polarization @xmath179 . the growth in the large scales regime ( early times ) agrees with the amplification of the power spectrum shown in figure [ evoltcl ] . however , contrary to what one would deduce from figure [ evoltcl ] , we see that the growth continues also in the short scales ( late times ) regime . we recall that the definition of the power spectrum ( [ powtimes ] ) contains an arbitrariness in the overall time dependence ( since one may have used a different combination of the two scale factors as overall normalization ) . we nonetheless adopted it to show the strong scale dependence of the evolution of the power spectrum ( which is not affected by the overall normalization ) , and the very different evolution experienced by the two gw polarizations ( which is also independent of the arbitrary normalization , since @xmath174 and @xmath240 are normalized in the same way ) . to properly study the instability , one must study invariant and unambiguous quantities , such as the ( scalar ) square of the weyl tensor which is investigated in this section . we consider now the contribution to the square of the weyl tensor from the mode @xmath122 in a contracting kasner geometry . plugging ( [ abeta2 ] ) into ( [ omega ] ) , the frequency of the @xmath122 mode on the contracting background has the short and large scales expansions @xmath241 once expressed in terms of absolute values of the time , the short and late time asymptotic expressions coincide with those of the expanding case , cf . eqs . ( [ esp - omx - exp ] ) . as in the expanding case , the short scales regime occurs asymptotically far from the singularity , while the large scales regime occurs asymptotically close to the singularity ( notice , however , that a mode evolves from the large scales to the short scales regime in the expanding kasner , while from the short scales to the large scales regime in the contracting kasner background ) . consequently , also the short and large scales asymptotical solutions are identical , once expressed in terms of @xmath80 and @xmath196 . @xmath242 however , in contrast to the expanding case , we now keep both solutions for @xmath122 in the large scales regime ) , since it is a decreasing mode in that case . in the contracting case instead this mode dominates at late times . ] . for the non - oscillatory part of the weyl tensor we find @xmath243 notice that the short scales asymptotic evolution agrees with the corresponding one in the expanding case . however , in the large scale regime @xmath211 is now controlled by the mode that was disregarded in the expanding background . the asymptotic behaviors ( [ w - hx - con ] ) are confirmed by the fully numerical evolutions shown in the left panel of figure [ weyl - con ] . for comparison we also plot the evolutions of the @xmath124 mode considered in the appendix a(b ) . how does the instability of gravitational waves which we demonstrated above fit with the classical picture of the universality of the rule of alternation of the kasner epochs during contraction towards a singularity ? let us briefly recall this picture . one of the points of the original paper @xcite was to extend the anisotropic homogeneous contracting kasner solution to a class of generalized kasner solutions , describing more general inhomogeneous anisotropic geometries . it was implicitly assumed that there are large - scale , super - horizon inhomogeneities at the scales exceeding the ( average ) hubble radius . generic anisotropic solution shall contain eight physically different arbitrary functions of the spatial coordinates . however , it was identified by @xcite that the homogeneous kasner solution is unstable against a particular inhomogeneous mode for which @xmath244 ( here @xmath245 is the kasner axes corresponding to the direction that is expanding ) . in other words , this mode is unstable and is growing with time . therefore the monotonic ( stable ) inhomogeneous solution can have only seven arbitrary functions of the spatial coordinates . homogeneous kasner contraction ( in bianchi models with spatial curvature ) occurs in the stochastic regime of alternation of kasner exponents @xcite . this influenced the philosophy of inhomogeneous generalization of the kasner contraction . following @xcite , now one can allow all eight arbitrary functions to describe generalized kasner ( bianchi i ) geometry . according to @xcite , the backreaction of the growing mode of the spatial instability alters the kasner exponents of the patch of the contracting universe in the same manner as they were altered in the homogeneous kasner - oscillating universe . again , in this picture the inhomogeneity scale is larger than the `` hubble '' patch , so that locally , along the time geodesics , the contraction is asymptotically homogeneous @xcite . the instability induced by the spatial curvature associated with the inhomogeneous growing mode , and the resulting rotation of the kasner axes were rigorously studied with the mathematical tools of the theory of dynamical systems in @xcite . in the previous section we found instabilities of both expanding and contracting kasner geometries against gravitational waves . here for comparison with the bkl analysis we focus on the gw instability in the contracting universe . in this background , both polarizations @xmath122 and @xmath124 are unstable in the large wavelength limit . all physical wavelengths of the gw corresponding to the momenta @xmath246 in different directions ( associated with different kasner exponents ) sooner or later will leave the hubble radius @xmath247 , independently of whether they are red- or blue - shifted . the physical frequency of the two modes has the structure @xmath248 , where the function @xmath249 is of order of @xmath141 . for small @xmath85 this function dominates over the momenta terms ; this turns the time dependence of the gw amplitude from the oscillating to the non - oscillating regime . the amplitude of the non - oscillating gw increases with time in this long - wavelength regime , signaling the instability which we described in the previous section . it turns out ( see appendix [ appb ] ) that these unstable gw modes in the long - wavelength limit exactly coincide with the unstable solution @xmath244 found in @xcite . phrased in another way , the unstable solution of @xcite which destroys the monotonicity of the homogeneous kasner contraction is nothing but the gw polarizations that are evolving with time into the long - wavelength regime . this provides us with a new insight into the origin of the generalized inhomogeneous contracting kasner solution : small short - wavelength gw will eventually be stretched to become the long - wavelength inhomogeneous modes , which are unstable . equipped with the bkl conjecture , one may think that for the contracting universe the growth of the unstable gw modes results in an alternation of the kasner exponents . while this conjecture is supported for a contracting universe @xcite , it is not clear how the gw instability evolves for the expanding kasner background . indeed , for contracting kasner universe gw leave the hubble radius and become long - wavelength inhomogeneities , and their backreaction can be described by the impact of the spatial curvature at the `` local '' time evolution of @xmath34 . for an expanding universe , initially long - wavelength gw enter the hubble radius . their backreaction can be treated with the pseudo - tensor @xmath250 for the high - frequency gw . another interesting issue is how our unstable modes correspond to the perturbatively small variations of the kasner exponents ( giving contribution to the diagonal metric fluctuations ) as well as small variations of the kasner axes ( contribution to off - diagonal metric fluctuations ) . finally , one of the most interesting application of the effect is related to zero vacuum fluctuations of gravitons in contracting kasner geometry , which are unstable . they grow and become large scale classical gw inhomogeneities described by the random gaussian field . one may think that the kasner axes will be altered and rotated differently in different spatial domains of that random field . we will expand these considerations in a forthcoming investigation . the growth of the decoupled tensor polarization at large scales can leave an imprint in the amplitude and in the polarization of the cmb anisotropies . a characteristic signature is a non - diagonal correlation between different multipoles in the expansion of the anisotropies , due to initial background anisotropy . such extended phenomenological study is beyond the goals of the present work . however , we want to obtain a crude estimate on the limits that such a study would impose on the model , namely on the physical wavelength beyond which the statistics of the modes is anisotropic , and on the initial amplitude of the gw signal . for this reason , we compute the contribution of the gw mode to the quadrupole of the temperature anisotropies , and we impose that it does not exceed the wmap value @xmath251 @xcite . as we show in appendix [ appc ] , the @xmath252 coefficients of the temperature anisotropies are related to the primordial spectrum of the gw by @xmath253 \ , , \label{c - ell}\end{aligned}\ ] ] where @xmath254 is a `` window function '' , which for a matter dominated universe s , and the period of late accelerated expansion ; this is consistent with the present approximate computation . ] is given in eq . ( [ iell ] ) . the analytic expression for @xmath255 can be found in @xcite . in this case , the function peaks at @xmath256 , while it goes to zero as @xmath257 at small argument , and as @xmath258 at large argument . the quantity @xmath139 is the present conformal time . for a matter dominated universe @xmath259 where here @xmath137 denotes the present value of the scale factor . the comoving momentum @xmath26 of a mode is related to its present wavelength @xmath260 by @xmath261 . therefore the quantity @xmath262 gives the present ratio between the size of the ( particle ) horizon and of the mode with comoving momentum @xmath263 . as shown in figure [ powertcl ] , the power spectrum of the decoupled tensor polarization experiences a growth only for @xmath264 . if the anisotropic stage is followed by a prolonged inflationary stage , these large scale modes are inflated to scales much larger than the present horizon size , @xmath265 , and the window function @xmath254 suppresses the contributions of these modes to the @xmath252 coefficients . we now relate @xmath266 to the duration of inflation through the number of e - folds @xmath267 between the moment in which the mode with comoving momentum @xmath26 leaves the horizon during inflation , and the end of inflation . this quantity is @xcite @xmath268 for a matter dominated universe @xmath269 , cf . ( [ md ] ) ; @xmath40 is the energy density in the universe at the moment in which that mode left the horizon ( for the potential we are discussing , cf . ( [ poten ] ) , this quantity is nearly independent of k ) ; finally , the quantity @xmath5 is sensitive to the details of reheating ( see @xcite for a discussion ) . imposing that all the modes within the horizon today were sub - horizon at the beginning of inflation gives the minimal number of e - folds given in eq . ( [ efold ] ) . we define @xmath270 since @xmath271 is the comoving momentum of the modes leaving the horizon when the universe becomes isotropic , @xmath24 is the difference in e - folds between the duration of the isotropic inflationary stage and the minimal duration ( [ efold ] ) . as we discussed , a large @xmath24 results in a suppressed effect of the amplified gw modes on the observed cmb anisotropies . we compute only the contribution of @xmath122 to the @xmath272 coefficient in eq . ( [ c - ell ] ) . figures [ powertcl ] and [ powerxi ] give the power spectrum of this mode at the end of inflation normalized by the initial power spectrum @xmath273 . this quantity is related to the initial condition at the start of the anisotropic stage , and it was left unspecified . for definiteness , we parametrize it with an overall amplitude and with a power law dependence on the scale @xmath274 where @xmath275 is the initial power at a scale which parametrically corresponds to the present horizon size . we numerically perform the angular integral in ( [ c - ell ] ) , to find @xmath276 where the proportionality to @xmath173 is related to the growth of the power spectrum during the anisotropic era , as explained at the end of section [ sec : decoupled ] . the dependence on @xmath26 is due to the fact that the angular integral is dominated by modes of small angle ( @xmath277 ) , cf . figure [ powerxi ] , where the growth of the spectrum is flat for @xmath278 , and scales as @xmath279 at larger @xmath263 . the function @xmath280 is shown in figure [ barp ] , normalized by @xmath173 and by the initial power spectrum @xmath281 . the dependences on @xmath173 and on @xmath26 given in eq . ( [ eqpbar ] ) are manifest in the figure . ) . we show the result normalized by the initial power spectrum , starting from different values of @xmath173 ( given in units of @xmath282 ; larger values of this quantity corresponds to a longer anisotropic stage ) , and rescaling it by @xmath173 , to explicitly show the linear dependence of the power spectrum on this quantity.,scaledwidth=60.0% ] we can now perform the final integral over momenta in ( [ c - ell ] ) . taking into account the large and small argument dependence of @xmath283 and that of @xmath280 given in eq . ( [ eqpbar ] ) , we see that the integrand behaves as @xmath284 at small @xmath26 , and as @xmath285 at large @xmath26 . therefore , the integral converges for a wide range for the initial slope defined in eq . ( [ defan ] ) , namely @xmath286 . after performing the integral , we impose that the resulting @xmath272 does not exceed the wmap value @xmath251 @xcite . this results in an upper limit on @xmath287 ( the initial overall amplitude , times the growth during the anisotropic phase ) for any value of @xmath271 and @xmath288 . we show this in figure [ final - lim ] . as explained after eq . ( [ ketalambda ] ) , the limit weakens at small values of @xmath266 , corresponding to a longer duration of the inflationary stage . , defined in eq . ( [ dn ] ) , is the number of e - folds of isotropic inflation minus the minimum usually required to homogenize the universe , cf . eq . ( [ efold ] ) . for a longer duration of the inflationary stage , the modes experiencing the growth are inflated to larger scales than the present horizon , and the resulting effect is suppressed ( weaker limit ) . @xmath275 and @xmath289 , are , respectively , the amplitude and the slope of the initial power spectrum , see eq . ( [ defan ] ) , while @xmath173 is proportional to the duration of the anisotropic stage.,scaledwidth=60.0% ] finally , we show in figure [ cl ] the first few @xmath252 s , normalized by the initial amplitude of the gw and the length of the anisotropic era ( controlled by @xmath173 ) , for a few values of the slope ( [ defan ] ) . as expected , for moderate slopes , the spectrum of @xmath252 decreases with @xmath290 . this is due to the fact that larger angular scales ( small @xmath0 ) are affected by the modes with lower momenta , which grow more during the anisotropic stage . the three spectra shown in the figure can be fitted by a single power - law ( with an accuracy up to about @xmath291 ) : @xmath292 coefficients generated by the gw , normalized by the initial amplitude and the duration of the anisotropic era ( controlled by @xmath173 ) . the tree curves corresponds to three different slopes of the initial power in the gw , eq . ( [ defan]).,scaledwidth=60.0% ] we see that the angular spectrum roughly decreases with @xmath0 as @xmath293 , where the exponent @xmath2 depends on the power - spectrum of the classical gw at the initial hypersurface . we found a new effect of instability of the gravitational waves in an expanding and contracting kasner geometries . we demonstrated the effect for a particular choice of the kasner exponents @xmath294 , but we expect it is rather generic . this particular choice allows to simplify the description of gw in such a way that their wave equations is manifestly depending on the momenta . for the contracting kasner geometry , we found that our unstable gw mode is identical to the unstable large - scale inhomogeneous mode first identified in @xcite-@xcite ( for arbitrary kasner exponents ( @xmath295 ) ) . backreaction of this unstable mode is conjectured to alter the kasner exponents , differently in different spatial patches , depending on the spatial profile of the initial gw . the kasner geometry is a rather universal asymptotic solution for many interesting situations : it describes approach towards black holes or cosmological singularities ( including higher dimensional and supersymmetric cases @xcite ) , and it describes generic anisotropic expansion from singularity prior to inflation . all of these situations are inevitably accompanied by quantum fluctuations of the gravitons ) , and , possibly , also by classical gravitational waves . there is a long list of questions arising in connection with a new effect of the gravity waves instability in anisotropic geometry . we have to investigate how the instability growth depends on the kasner exponents @xmath295 . it will be interesting to understand if the instability of the classical gw also results in the instability of the graviton zero vacuum fluctuations . it will also be worth to understand the impact of the gw instability on the structure of the singularity inside the black hole and cosmological singularity . it is also interesting to investigate the backreaction of the gw instability on the contracting and expanding anisotropic geometries depending on the initial gw profile . in this paper we considered the effect of gw instability in the context of the anisotropic pre - inflationary stage . the transition from kasner expansion to inflation terminates the effect of gw instability but leaves classical gw signal as the initial conditions for inflation . if inflation does not last very long , then the residual gw can contribute to the cmb temperature anisotropies . since gw polarizations and the scalar mode of cosmological fluctuations behave differently during anisotropic pre - inflation , we can consider only the leading contribution , namely , @xmath122 mode of the gw polarization . in this paper we calculated its contribution to the total @xmath13 anisotropy angular power spectrum . the angular power spectrum of the signal decreases with @xmath0 in a power - law manner and depends on the initial spectrum of the classical gw fluctuations , @xmath293 , where the exponent @xmath2 depends on the power - spectrum of the classical gw at the initial hypersurface . it is interesting to note that this result is qualitatively similar to the result of @xcite , where the impact of gw from the our - universe - bubble nucleation was considered . the signal from @xmath122 is rather anisotropic , and there is an interesting question about the anisotropy of its multipole structure @xmath296 , which is intriguing in connection with an apparent alignment of the low multipoles of @xmath13 . while such an analysis is beyond the aims of the present work , we have estimated the initial conditions ( initial amplitude of the gw , versus the duration of inflation ) which can lead to potentially observable effects . we leave a more extended analysis to future investigation . * acknowledgements * we thank j.r . bond , c.r . contaldi , t. damour , i. khalatnikov , a. linde , c. pitrou , m. sasaki , a. starobinsky , j.p . uzan and j. weinwright for useful discussions and correspondence . the work of a.e.g . and m.p . was partially supported by the doe grant de - fg02 - 94er-40823 . lk was supported by nserc and cifar . in this appendix we compute the contribution of the @xmath124 mode to the square of the weyl tensor for a pure kasner background . we treat the expanding and contracting cases separately . this presentation follows the analogous one for the @xmath122 mode placed in subsection [ subhx ] of the main text . substituting ( [ abeta2 ] ) into ( [ omega ] ) we have the large and short scale expansions @xmath297 consequently , we have the asymptotic solutions @xmath298 where again in the early time solution we have disregarded a decaying mode that would diverge at the initial singularity . notice that the late time expression is the same as for the @xmath122 mode ( since the late time expansions of @xmath131 and @xmath299 coincide ) . for the linear part of the weyl tensor we find @xmath300 the non - oscillatory part of the @xmath98nd order term in the weyl square is @xmath301 this behavior is confirmed by the fully numerical solutions shown in the right panel of figure [ weyl - exp ] . it is interesting to compare the contribution of the @xmath122 mode and that of the @xmath124 mode . in the large scale regime only the contribution of the @xmath122 mode grows with respect to the background . this is consistent with the amplification of this polarization in the kasner era that we have found in section [ sec : decoupled ] . in the small scales regime , the contribution of both modes is instead growing with respect to the background . although the two modes evolve identically in this regime ( since they satisfy the same asymptotic equation ) , the time evolution of the corresponding @xmath211 term is different , since the two modes enter differently in the weyl tensor . plugging ( [ abeta2 ] ) into ( [ omega ] ) we have the early / late time expansions @xmath302 the similarities and differences between the @xmath124 mode in the expanding and contracting cases are identical to those discussed in the previous subsection for the @xmath122 mode . we have the asymptotic solutions @xmath303 this leads to the following asymptotic evolution for the linear term in the square of the weyl tensor @xmath304 \\ \quad\quad\quad\quad \times \frac { 12 \ , \sqrt{2 } \left ( - \eta _ * \right)^{5/2}}{\pi \ , m_p \ , a_*^5 \ , \left ( - \eta \right)^6 } & \,,{\rm ~large~scales}\\ \end{array}\right\ } + { \rm h.c . } \label{wlin - hp - con}\ ] ] and for the non - oscillatory part in the quadratic term of the square of the weyl tensor @xmath305 this behavior is confirmed by the fully numerical evolutions shown in the right panel of figure [ weyl - con ] . our analysis of the squared weyl invariant shows that the kasner background is unstable due to the amplification of the gw perturbations @xmath124 and @xmath122 . this is manifest by the growth of @xmath306 and @xmath231 summarized in table [ tab1 ] . for the contracting background we found that the contribution to @xmath231 of the polarizations @xmath124 and @xmath122 grow , respectively , as @xmath307 and as @xmath308 asymptotically close to the singularity . conversely , in the expanding case , the weyl invariant is regular near the singularity . the reason for this discrepancy is that , in the expanding case , we disregarded the neumann term @xmath309 ( for @xmath124 ) and @xmath310 ( for @xmath122 ) in the solutions of the evolution equations ( [ eqomtomp ] ) for the two modes . however these two modes are generally excited as the contracting kasner background solution approaches the singularity . the purpose of this appendix is to show explicitly that this instability is not inconsistent with the lore of alternation of kasner exponents @xcite . on the contrary , the instability in the contracting case was pointed out already in @xcite , where it was shown that the kasner background is stable provided one physical condition is imposed . as we now show , the physical condition imposed by @xcite precisely eliminates the unstable @xmath309 and @xmath310 solutions . we start by quoting the results of @xcite where the authors considered a perturbed vacuum kasner solution in synchronous gauge and did a stability analysis near the singularity . in the main text , we describe the contracting kasner background using negative time . in @xcite , instead , positive time was used , and the approach to the singularity was studied in the @xmath311 limit . in this appendix , we adopt the time convention of @xcite . moreover , we specify the analysis of @xcite to the axisymmetric @xmath312 , @xmath313 case . for this choice , the solution obtained in @xcite for @xmath314 reads @xmath315+t^{4/3}\,c_{21 } \ , , \nonumber\\ h_{13 } & = & t^{-2/3 } \left[c_{13}- \frac{9}{32}\,k_2\,\left(k_3\,c_{12 } - k_2\,c_{13}\right ) t^{2/3 } \right]+t^{4/3}\,c_{31 } \ , , \nonumber\\ h_{23 } & = & t^{4/3 } \left(c_{23 } + c_{32}\right)\ , , \label{blsol}\end{aligned}\ ] ] with constants @xmath316,@xmath114 , @xmath317 satisfying the constraints @xmath318 as a result of the synchronous gauge , the solutions ( [ blsol ] ) still have freedom due to gauge artifacts ; the corresponding gauge contributions to the perturbations are given in eqs ( f3 ) of @xcite . using this freedom , the authors of @xcite set @xmath319 . in the general case . however , the time dependences of both terms in @xmath320 are the equal to each other for the axisymmetric @xmath321 case ; therefore , in this case , the removal of the gauge artifact corresponds to @xmath322 . ] after this gauge fixing , one sees that the mode @xmath323 diverges at the singularity . for this reason , ref . @xcite concluded that the kasner background is stable only provided the condition @xmath324 is imposed . we stress that setting @xmath325 is not a gauge choice , but rather the suppression of a physical degree of freedom . this physical constraint , along with the relations ( [ constraints ] ) , implies that @xmath326 . we now turn back to our analysis . for @xmath311 , eqs . ( [ eqomtomp ] ) admit the solutions @xmath327\,,\nonumber\\ h_{\times } & = & t^{1/3}\,\left[e_1\,j_3 ( 3\,k_t\,t^{1/3 } ) + e_2\,y_3 ( 3\,k_t\,t^{1/3 } ) \right]\ , . \label{earlysols}\end{aligned}\ ] ] the modes @xmath328 and @xmath329 are regular at the singularity , whereas @xmath309 and @xmath310 diverge , thus resulting in the instability . we now show that the conditions @xmath330 in @xcite correspond to having @xmath331 in our case . to do such a comparison , we need to change from the gauge chosen in the main text to the synchronous gauge adopted in @xcite . we do so through a general infinitesimal transformation @xmath332 , with @xmath333 where @xmath334 is a transverse 2d vector . from the transformed metric @xmath335 , we find that the parameters @xmath336 which take our gauge to the synchronous coordinates are @xmath337 where @xmath338 are integration constants and @xmath339 . the perturbations @xmath340 , @xmath341 , @xmath12 and @xmath114 are the modes in the decomposition given in the main text . using their relations to the canonical modes @xmath124 and @xmath210 , along with the early time solutions ( [ earlysols ] ) , we calculate the metric perturbations in synchronous gauge . the relevant components for the present discussion are @xmath342 \nonumber\\ & & + \frac{2\,\sqrt{2}}{27\,\pi\,k_t^5\,m_p}\left[-\,32\,i\,k_1\,k_2\,\frac{k_3}{k_t } e_2 -9\,d_2\,k_t^3 \left(k_3 ^ 2-k_2 ^ 2\right ) \right]\ln t \nonumber\\ & & -2\,f_0\left(3\,k_2 ^ 2\,t^{-1/3 } + \frac{2}{3\,t}\right ) + 2\,k_2\left(i\,f_2 + k_2\,f\right ) + \mathcal{o}(t^{2/3})\ , . \label{gkpsol}\end{aligned}\ ] ] in eqs ( [ blsol ] ) , setting @xmath343 ( gauge choice ) and @xmath324 ( physical choice ) eliminates the terms up to @xmath344 in @xmath345 and @xmath323 . on the other hand , from our solution ( [ gkpsol ] ) , we see that these terms in @xmath346 vanish only if @xmath347 . furthermore , the choice @xmath348 implies that @xmath326 , now eliminating the log term in @xmath349 , which corresponds to @xmath350 in our solutions . in other words , the stability conditions derived in @xcite correspond to removing the neumann functions in the early time solutions ( [ earlysols ] ) of both @xmath124 and @xmath122 . here , we prove the relation ( [ c - ell ] ) of the main text . this expression gives the @xmath252 coefficients of the temperature anisotropies in terms of the value of the power spectrum of gw at the end of inflation . therefore , this computation is performed in an isotropic universe . the effects of the anisotropic stage are encoded in the power spectra that , on large scales , depend also on the orientation of the modes . we start by decomposing the temperature perturbations in direction @xmath352 measured by an observer at @xmath353 and at conformal time @xmath139 , into spherical harmonics @xmath354 assuming that the perturbations are gaussian , their statistics are completely specified by the second order correlations , which can be written by inverting ( [ definedt ] ) @xmath355 assuming instantaneous recombination , we can use the well - known solution to the first order boltzmann equation for gravity waves : @xmath356 where @xmath357 . we expand the gravitational perturbations in terms of plane waves as @xmath358 where @xmath359 designates the two polarizations @xmath360 and @xmath361 , @xmath362 is the normalized polarization tensor satisfying @xmath363 , and @xmath364 are gaussian random variables with unit dispersion , ie . @xmath365 . using the solution ( [ sachswolfe ] ) along with the decomposition ( [ decomposeh ] ) in the expression ( [ correl1 ] ) , we get @xmath366 \nonumber\\ & & \quad\quad\quad\quad\times\left [ \int d\tilde{\eta } \left ( \frac{\partial}{\partial \tilde{\eta } } h^{(\lambda ) } ( { \bf k},\tilde{\eta})\right ) { \rm e}^{i\,{\bf k } \cdot { \bf \hat{p}'}(\tilde{\eta}-\eta_0 ) } \epsilon ^ { ( \lambda)}_{ab } ( { \bf k } ) \hat{p}^{\prime\,a } \hat{p}^{\prime\,b } \right]^\star \,,\end{aligned}\ ] ] where we noted that @xmath367 describes the geodesics of the photons with direction @xmath352 , observed at @xmath368 . the contraction @xmath369 can be calculated in terms of the angles of @xmath352 defined with respect to a coordinate system with a @xmath72axis coinciding with @xmath370 , as @xcite @xmath371 \sin^2 \theta_{\hat{p}_{\hat{k}}}\,.\ ] ] to take the angular integral over @xmath372 @xmath373 , we first rotate the spherical harmonics to the angular basis @xmath374 ( @xmath375 ) with @xmath376 where @xmath377 are the wigner coefficients satisfying @xmath378 . finally , the angular integrals can be evaluated as @xcite @xmath379}{\left[k\left(\eta_0-\eta\right)\right]^{5/2 } } \nonumber\\ & & \times\left[\left(\delta_{(+ ) } ^{(\lambda ) } -i\,\delta_{(\times ) } ^{(\lambda)}\right ) \delta_{m,2 } + \left(\delta_{(+ ) } ^{(\lambda ) } + i\,\delta_{(\times ) } ^{(\lambda)}\right ) \delta_{m,-2}\right]\,.\nonumber\\ \label{integwhite}\end{aligned}\ ] ] ignoring the effect on the ( small scale ) modes which enter the horizon during the radiation dominated era , the solutions for the gravity waves during matter domination can be written in terms of the primordial one as @xcite , @xcite @xmath380 using the integral ( [ integwhite ] ) along with the above solution yields @xmath381 \nonumber\\ & & \times\int_0 ^\infty \frac{dk}{k}\,\ , i_\ell^2(k\,\eta_0)\int^{+1}_{-1 } \frac{d\xi}{2}\,\int_0^{2\,\pi}\,\frac{d\phi_k}{2\,\pi}\,d^\ell_{m m}({\bf \hat{k}})\,d^{\ell'\,\star}_{m ' m'}({\bf \hat{k } } ) \,p_{h_\lambda } ( { \bf k})\,,\end{aligned}\ ] ] where @xmath382 , the power spectrum is defined through @xmath383 which coincides with the definitions given in the main text , cf . ( [ powtimes ] ) and ( [ powepv ] ) once the universe has isotropized . finally , the time integral is @xmath384 to be able to compare with observations , we average the diagonal ( @xmath385 , @xmath386 ) part of the correlator over all @xmath387 and obtain @xmath388\ , , \label{c - ell - app}\end{aligned}\ ] ] the final result ( [ c - ell - app ] ) is valid for any model of pre - inflation which results in an anisotropic primordial tensor power spectrum . g. hinshaw _ et al . _ [ wmap collaboration ] , arxiv:0803.0732 [ astro - ph ] . [ planck collaboration ] , arxiv : astro - ph/0604069 . c. e. north _ et al . _ , arxiv:0805.3690 [ astro - ph ] . p. oxley _ et al . _ , proc . . soc . eng . * 5543 * , 320 ( 2004 ) [ arxiv : astro - ph/0501111 ] . c. j. mactavish _ et al . _ , arxiv:0710.0375 [ astro - ph ] . j. bock _ et al . _ , arxiv:0805.4207 [ astro - ph ] . e. s. phinney _ et al . _ ( the big bang observer ) , nasa mission concept study ( 2003 ) ; http://universe.nasa.gov/program/vision.html a. e. gumrukcuoglu , c. r. contaldi and m. peloso , proceedings of the eleventh marcel grossmann meeting on general relativity , ed . h. kleinert , r.t . jantzen & r. ruffini , world scientific , 2007 , arxiv astrophysics e - prints , arxiv : astro - ph/0608405 . t. s. pereira , c. pitrou and j. p. uzan , jcap * 0709 * , 006 ( 2007 ) [ arxiv:0707.0736 [ astro - ph ] ] . a. e. gumrukcuoglu , c. r. contaldi and m. peloso , jcap * 0711 * , 005 ( 2007 ) [ arxiv:0707.4179 [ astro - ph ] ] . d. i. podolsky , g. n. felder , l. kofman and m. peloso , phys . rev . d * 73 * , 023501 ( 2006 ) [ arxiv : hep - ph/0507096 ] . g. hinshaw _ et al . _ [ wmap collaboration ] , astrophys . j. suppl . * 170 * , 288 ( 2007 ) [ arxiv : astro - ph/0603451 ] . m. j. white , phys . d * 46 * , 4198 ( 1992 ) [ arxiv : hep - ph/9207239 ] . m. s. turner , m. j. white and j. e. lidsey , phys . d * 48 * , 4613 ( 1993 ) [ arxiv : astro - ph/9306029 ] . e. m. lifshitz and i. m. khalatnikov , adv . * 12 * , 185 ( 1963 ) . v. a. belinsky , i. m. khalatnikov and e. m. lifshitz , adv . * 19 * , 525 ( 1970 ) . | we show that expanding or contracting kasner universes are unstable due to the amplification of gravitational waves ( gw ) . as an application of this general relativity effect
, we consider a pre - inflationary anisotropic geometry characterized by a kasner - like expansion , which is driven dynamically towards inflation by a scalar field .
we investigate the evolution of linear metric fluctuations around this background , and calculate the amplification of the long - wavelength gw of a certain polarization during the anisotropic expansion ( this effect is absent for another gw polarization , and for scalar fluctuations ) .
these gw are superimposed to the usual tensor modes of quantum origin from inflation , and are potentially observable if the total number of inflationary e - folds exceeds the minimum required to homogenize the observable universe only by a small margin .
their contribution to the temperature anisotropy angular power spectrum decreases with the multipole @xmath0 as @xmath1 , where @xmath2 depends on the slope of the initial gw power - spectrum .
constraints on the long - wavelength gw can be translated into limits on the total duration of inflation and the initial gw amplitude .
the instability of classical gw ( and zero - vacuum fluctuations of gravitons ) during kasner - like expansion ( or contraction ) may have other interesting applications .
in particular , if gw become non - linear , they can significantly alter the geometry before the onset of inflation . |
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it is believed that a neutron star begins its life as a proto - neutron star ( pns ) in the aftermath of a supernova explosion . the evolution of the pns depends upon the star s mass , composition , and equation of state ( eos ) , as well as the opacity of neutrinos in dense matter . previous studies @xcite have shown that the pns may become unstable as it emits neutrinos and deleptonizes , so that it collapses into a black hole . the instability occurs if the maximum mass that the equation of state ( eos ) of lepton - rich , hot matter can support is greater than that of cold , deleptonized matter , and if the pns mass lies in between these two values . the condition for metastability is satisfied if `` exotic '' matter , manifested in the form of a bose condensate ( of negatively charged pions or kaons ) or negatively charged particles with strangeness content ( hyperons or quarks ) , appears during the evolution of the pns . even if collapse to a black hole does not occur , the appearance of exotic matter might lead to a distinguishable feature in the pns s neutrino signature ( _ i.e. _ , its neutrino light curve and neutrino energy spectrum ) that is observable from current and planned terrestrial detectors . this was investigated recently by pons _ @xcite who studied the evolution of a pns in the case where hyperons appeared in the star during the latter stages of deleptonization . although the possibility of black hole formation was first discovered in the context of kaon condensation in neutron star matter @xcite , a full dynamical calculation of a pns evolution with consistent eos and neutrino opacities in kaon condensed matter has not been performed so far . one of the objectives of this paper is to investigate @xmath1 condensation in finite temperature matter , including the situation of trapped neutrinos in more detail . an impetus for this study is the recent suggestion that a mixed phase of kaon - condensed and normal matter might exist which could greatly affect the structure @xcite and its neutrino opacity @xcite . another objective of our study is to identify differences in thermodynamic quantities such as the pressure , entropy or specific heat that might produce discriminating features in the star s neutrino emission . in separate works , we will examine neutrino interactions in kaon - condensed matter and neutrino signals from pns evolution calculations in a consistent fashion . since we wish to isolate the aforementioned effects due to kaons in this paper , we deliberately exclude consideration of hyperons . hyperons and kaons were considered together in refs . @xcite and @xcite . hyperons tend to delay the appearence of kaons in matter , especially if the @xmath2 appears first . however , the @xmath2 couplings are not as well determined as those of the @xmath3 and even in this case the data are restricted to nuclear or subnuclear densities . relatively small variations in the coupling constants can lead to a situation where the threshold density for the appearance of @xmath2 particles is larger than that for kaons . these uncertainties remain unresolved ; further hyper - nuclear experiments are needed to pin down their couplings . the original investigations of kaon condensation in neutron star matter ( _ e.g. _ refs . @xcite and its astrophysical conseqences @xcite ) employed a chiral @xmath4 model in which the kaon - nucleon interaction occurs directly via four point vertices . however , one can also employ an indirect , finite - range interaction which arises from the exchange of mesons . several studies have been performed along these lines @xcite . @xcite found that the chiral and meson exchange approaches give similar results provided that the kaon - nucleon couplings are chosen to yield similar optical potentials in nuclear matter . allowing kaons to interact via the exchange of mesons has the advantage that it is more consistent with the walecka - type effective field - theoretical models usually used to describe nuclear matter @xcite . in most studies of kaon condensation it has been found that the transition to a phase in which kaons condense is second order for modest values of the kaon optical potential , @xmath5 , of order -100 mev . for magnitudes of @xmath5 well in excess of 100 mev , however , the phase transition becomes first order in character . even when the transition is first order , it is not always possible to satisfy gibbs criteria for thermal , chemical and mechanical equilibrium , so a maxwell construction , which satisfies only thermal and mechanical equilibrium , was sometimes employed to construct the pressure - density relation . recently , glendenning and schaffner - bielich ( gs ) @xcite modified the meson exchange lagrangian in such a way that the gibbs criteria for thermal , chemical and mechanical equilibrium in a first order phase transition was possible . the extended mixed phase of kaon - condensed and normal matter which results produces a qualitative difference for the structure of a neutron star , since the eos is softened over a wider region than in the case in which there is no mixed phase . this has implications for the mass - radius relation and the maximum mass , among other properties of the star . in this paper , we investigate the phase transition involving kaon - condensed matter and its influence upon the equation of state . we find that the precise form assumed for the scalar interactions ( particularly , their density dependence ) , both for baryon - baryon and kaon - baryon interactions , determines whether or not the transition is first or second order , and , in the case of a first order phase transition , establishes whether or not a gibbs construction is possible . since the form of the scalar interactions is not experimentally well constrained at present , we have explored several different models in this study of the effects of kaon condensation on the eos and the structure of a pns . for each model , we have performed a detailed study of the thermal properties which are summarized in terms of phase diagrams in the density - lepton content and density - temperature planes . in sec . ii we present the various lagrangians and derive exressions for the thermodynamic properties of each . we also develop the theoretical formalism necessary to describe baryons and kaon condensed matter in both the pure and mixed phases . this is followed by a discussion of the determination of the various coupling constants . section iii contains a comparison of the results for the eos and for the structure of neutron stars for typical values of entropy and lepton content in a proto - neutron star as it evolves . our conclusions and outlook for evolution of a proto - neutron simulation are presented in sec . iv . in appendix a , the extent of the correspondence between a meson exchange formalism and a chiral model to describe kaon condensation in matter is examined . the role of higher order kaon self - interactions in determining the order of the phase transition to a kaon condensed state is studied in appendix b. we begin with the well - known relativistic field theory model of walecka @xcite supplemented by nonlinear scalar self - interactions @xcite . here nucleons ( @xmath6 ) interact via the exchange of @xmath7- , @xmath8- , and @xmath9-mesons . explicitly , the lagrangian is @xmath10 where @xmath11 is the nucleon field , the @xmath9-meson field is denoted by @xmath12 and the quantity @xmath13 is the isospin operator which acts on the nucleons . scalar self - couplings @xcite , which improve the descripton of nuclear matter at the equilibrium density , are included in the potential @xmath14 , with @xmath15 denoting the vacuum nucleon mass . the field strength tensors for the vector mesons are given by the expressions @xmath16 and @xmath17 . in the standard walecka model the nucleon effective mass m^*_gm = m - g_(the label gm refers to the glendenning - moszkowski parameters @xcite that we will use with this expression ) . we shall also study an alternative form due to zimanyi and moszkowski ( labelled by zm ) @xcite : @xmath18 by redefining the nucleon field , @xmath19 , the lagrangian can be written exactly in the form eq . ( [ hyp1 ] ) , but the nucleon effective mass becomes m^*_zm = m(1+g_/m)^-1 . for small values of @xmath7 this is equivalent to the walecka form . however the zm effective mass has the property that , in the limit of large @xmath7 , @xmath20 remains positive whereas @xmath21 can become negative @xcite , which is unphysical . in the mean field approximation the thermodynamic potential per unit volume for both lagrangians is @xmath22 here the inverse temperature is denoted by @xmath23 , @xmath24 and the subscripts @xmath25 or @xmath26 have been suppressed . the chemical potentials are given by @xmath27 note that in a rotationally invariant system only the time components of the vector fields contribute to eq . ( [ hyp2 ] ) and for the isovector field only the @xmath28 component contributes . the contribution of antinucleons is not significant for the thermodynamics of interest for a pns and is ignored . using @xmath29 , the thermodynamic quantities can be obtained in the standard way . the nucleon pressure is @xmath30 , and the number density @xmath31 and the energy density @xmath32 are given by @xmath33 where the fermi distribution function @xmath34 . the entropy density is then given by @xmath35 . the contribution from the leptons and antileptons is adequately given by its non - interacting form , since their interactions give negligible contributions @xcite . thus the thermodynamic potential per unit volume of the leptons and antileptons is : @xmath36\ , , \label{zlept}\end{aligned}\ ] ] where @xmath37 and @xmath38 denote the degeneracy and the chemical potential , respectively , of leptons of species @xmath39 . the degeneracy @xmath37 is 2 for electrons and muons and it is 1 for neutrinos of a given species . since the star is in chemical equilibrium with respect to the weak processes @xmath40 , where the lepton @xmath39 is either an electron or a muon , the chemical potentials obey @xmath41 . if there are no neutrinos trapped in the star the neutrino chemical potentials @xmath42 are zero or , equivalently , the total neutrino concentration @xmath43 , where we define the concentration for particle @xmath44 to be @xmath45 . the pressure , density and energy density of the leptons are obtained from eq . ( [ zlept ] ) in standard fashion . the two kaon lagrangians of the meson - exchange type which have been previously suggested ( in refs . @xcite and @xcite , respectively ) , can both be written in the form @xmath46 where @xmath47 denote the charged kaon fields and we have defined the combined vector field @xmath48 with @xmath49 and @xmath50 denoting @xmath8 and the @xmath28 fields , respectively ; @xmath51 and @xmath52 are coupling constants . since only the time components of the vector fields survive , in practice only @xmath53 is non - zero . the two lagrangians differ in the forms chosen for the quantity @xmath54 . both have the standard vacuum mass term , but the interaction terms differ . specifically , knorren , prakash and ellis ( kpe ) @xcite take @xmath55 with @xmath56 denoting the vacuum kaon mass , while glendenning and schaffner - bielich ( gs ) @xcite choose @xmath57 note that the coupling constant @xmath58 is defined here to be twice that defined in gs . a similar remark applies to the @xmath59 coupling constant @xmath60 . it is remarkable , as pointed out in appendix a , that to leading order in the kaon condensate intensity , the equations obtained with the chiral kaplan - nelson @xcite lagrangian at zero temperature agree precisely with those from the kpe lagrangian . to see the significance of the term involving the vector fields in eq . ( [ algs ] ) , consider the invariance of the lagrangian under the transformation @xmath61 . this allows the conserved kaon current density to be identified as @xmath62 now for the combined gs lagrangian , @xmath63 , the equation of motion for the omega field is @xmath64 since the nucleon current @xmath65 is conserved , as is the kaon current @xmath66 , taking the divergence of eq . ( [ veceom ] ) immediately yields @xmath67 ( and similarly for the @xmath9 field ) . this is the required condition for a vector field @xcite so as to reduce the number of components from four to three . on the other hand , @xmath68 does not contain an @xmath69 term , so that the kaon current does not appear on the right hand side of eq . ( [ veceom ] ) . at the mean field level , however , where the vector fields are constants , any derivative is necessarily zero so that the divergence condition is automatically satisfied . for the coupling of the kaon fields to the scalar @xmath7 field , kpe use a linear coupling , whereas gs have an additional quadratic term there is little guidance on the form that should be used to generate the kaon effective mass so the choice is somewhat arbitrary , although , as we shall see , it can significantly affect the thermodynamics . both the kpe and gs choices lead to problems for sufficiently large values of the @xmath7 field ; in one case the effective mass becomes imaginary , in the other it becomes negative . we are therefore led to consider a third form in the spirit of the zm model for nucleons . for specificity we start with the gs lagrangian which can be written @xmath70 in terms of a covariant derivative @xmath71 , and replace it by @xmath72 we observe that the form of @xmath73 above is one of many possibilities . making the transformation @xmath74 and noting that @xmath7 is a constant mean field , the kaon lagrangian can be put in the form of eq . ( [ kaonlag ] ) with @xmath75 the label @xmath76 denotes this work " . while eqs . ( [ alkpe ] ) , ( [ algs ] ) and ( [ altw ] ) all give @xmath77 for small @xmath7 , they differ at order @xmath78 and beyond , _ i.e. _ , for large values of @xmath7 . the kaon partition function at finite temperature can be obtained for a lagrangian of the form ( [ kaonlag ] ) by generalizing the procedure outlined in kapusta @xcite . first , we transform to real fields @xmath79 and @xmath80 , @xmath81 and determine the conjugate momenta @xmath82 the hamiltonian density is @xmath83 , and the partition function of the grand canonical ensemble can then be written as the functional integral @xmath84[d\pi_2]\int_{periodic}[d\phi_1][d\phi_2 ] \exp\left\{\int\limits_0^{\beta}d\tau\int d^3x\left ( i\pi_1\frac{\partial\phi_1}{\partial\tau}+ i\pi_2\frac{\partial\phi_2}{\partial\tau}-{\cal h}_k(\pi_i,\phi_i ) + \mu j_0(\pi_i,\phi_i)\right)\right\}\;.\label{zint}\ ] ] here the fields obey periodic boundary conditions in the imaginary time @xmath85 , namely @xmath86 , where @xmath87 . the chemical potential associated with the conserved kaon charge density is denoted by @xmath88 and chemical equilibrium in the reaction @xmath89 requires that @xmath90 . the gaussian integral over momenta in eq . ( [ zint ] ) is easily performed . next the fields are fourier decomposed according to @xmath91 where the first term describes the condensate , so that in the second term @xmath92 . the pion decay constant @xmath93 has been inserted so that the condensate angle @xmath94 is dimensionless . the matsubara frequency @xmath95 . the partition function can then be written @xmath96[d\phi_{2,n}({\mbox{\bfp } } ) ] e^s\;,\qquad{\rm where}\nonumber\\ s&=&\thalf \beta v(f\theta)^2(\mu^2 + 2\mu x_0-\alpha ) -\thalf\sum_{n,{\mbox{\scriptsize{\bfp}}}}\bigl(\phi_{1,-n}(-{\mbox{\bfp}}),\phi_{2,-n}(-{\mbox{\bfp}})\bigr ) { \mbox{\bfd}}\left(\matrix{\phi_{1,n}({\mbox{\bfp}})\cr\phi_{2,n}({\mbox{\bfp}})\cr}\right ) \;,\nonumber\\ { \mbox{\bfd}}&=&\beta^2\left(\matrix{\omega_n^2+p^2+\alpha-2\mu x_0-\mu^2 & 2(\mu+x_0)\omega_n\cr-2(\mu+x_0)\omega_n & \omega_n^2+p^2+\alpha-2\mu x_0-\mu^2\cr}\right).\end{aligned}\ ] ] @xmath11 is a normalization constant . we define the @xmath47 energies according to @xmath97 so that the three approaches give @xmath98 using the definition ( [ defom ] ) and suppressing the explicit dependence of @xmath99 on @xmath100 , the determinant of @xmath101 is @xmath102 \left[\omega_n^2+(\omega^++\mu)^2\right]\;,\ ] ] giving @xmath103 where the normalization constant @xmath11 has been dropped since it is irrelevant to the thermodynamics . performing the sum over @xmath104 and neglecting the zero - point contribution , which contributes only beyond the mean field approach and in any case is small @xcite , we obtain the grand potential for the kaon sector : @xmath105\;.\label{zkexch}\ ] ] the kaon pressure , @xmath106 , and the kaon number density is easily found to be @xmath107\;,\ ] ] and the bose occupation probability @xmath108 . the kaon energy density is @xmath109\;,\ ] ] and the kaon entropy density is @xmath110 . it is useful first to define the quantity @xmath111\;.\ ] ] then the mean @xmath8 , @xmath9 and @xmath7 fields , as well as the condensate amplitude @xmath94 , determined by extremizing the total grand potential @xmath112 , can be written @xmath113 \frac{\partial\alpha}{\partial x_0}\right\}\nonumber\\ & & m_{\rho}^2b_0 = \thalf g_{\rho}(n_p - n_n)-g_{\rho k } \left\{\mu(f\theta)^2+n_k^{th}-x_0a_k^{th}-\thalf[(f\theta)^2+a_k^{th } ] \frac{\partial\alpha}{\partial x_0}\right\}\nonumber\\ & & m_{\sigma}^2\sigma = -\frac{du(\sigma)}{d\sigma } -2\frac{\partial m^*}{\partial\sigma}\sum_{n , p } \int\frac{d^3k}{(2\pi)^3 } \frac{m^*}{e^*}f_f(e^*-\nu_{n , p } ) -\thalf\left[(f\theta)^2+a_k^{th}\right]\frac{\partial\alpha } { \partial\sigma}\nonumber\\ & & \theta(\mu^2 + 2\mu x_0-\alpha)= \theta[\mu-\omega^-(0)][\mu+\omega^+(0 ) ] = 0\;. \label{hhyp5}\end{aligned}\ ] ] the derivative @xmath114 is zero for the kpe case and @xmath115 for the gs and tw cases . the derivatives with respect to the @xmath7 field are @xmath116 note that the last of eqs . ( [ hhyp5 ] ) yields either @xmath117 ( no condensate ) or the condition for a condensate to exist . since @xmath88 is positive here , we only have the possibility of a @xmath1 condensate with @xmath118 . note also that the contribution of the condensate to the kaon pressure @xmath119 vanishes , as it should . the remaining condition to be imposed is that neutron star matter must be charge neutral . for a single phase this implies @xmath120 where @xmath121 and @xmath122 are the net negative lepton number densities . the mixed phase thermodynamics is discussed below . the sum of the nucleon and kaon energy densities can be simplified somewhat by using the equations of motion . this gives @xmath123\;.\end{aligned}\ ] ] in the theory discussed above there are two independent chemical potentials , which we can take to be @xmath124 and @xmath88 , each connected with a conserved charge , the baryon number and charge of the system , respectively . glendenning @xcite pointed out that in the presence of a first order phase transition , conservation laws must be globally , not locally , imposed , if possible , in the mixed phase region . a maxwell construction would have been appropriate had there been just a single conserved charge . however , relaxing the condition of local charge neutrality does not guarantee that the model lagrangian , solved in the mean field approximation , will provide a description of the mixed phase , which is only possible if the gibbs criteria can be satisifed . a simple , yet general , procedure to check if the gibbs criteria can be fulfilled by a specific model is discussed in appendix b. denoting the phase containing a condensate with a subscript @xmath94 , and the phase without a condensate with @xmath117 , the total pressures in the two phases must be equal @xmath125 each of the chemical potentials is the same in the two phases . if the volume fraction of the non - condensed phase is @xmath126 , then global conservation of charge requires @xmath127_{\theta=0}+(1-\chi)[n_p - n_k - n_e - n_\mu]_{\theta}=0 \ , . \label{g2}\ ] ] the densities of the individual species in the mixed phase are evident here . the total energy density is the weighted sum of the two phases @xmath128 the total entropy density is obtained through a similar equation . in the effective lagrangian approach adopted here , knowledge of two distinct sets of coupling constants , one parametrizing the nucleon - nucleon interactions , and one parametrizing the kaon - nucleon interactions , are required for numerical computations . these are associated with the exchange of @xmath129 and @xmath9 mesons . we consider each of these in turn . the nucleon - meson coupling constants are determined by adjusting them to reproduce the properties of equilibrium nuclear matter at @xmath130 . the properties used are the saturation density , @xmath131 , the binding energy / particle , @xmath132 , the symmetry energy coefficient , @xmath133 , the compression modulus , @xmath134 , and the dirac effective mass at saturation , @xmath135 . not all of these quantities are precisely known and the values we choose are listed in table i. for completeness , we list the equations needed to obtain the coupling constants , assuming that the scalar self coupling has the form @xmath136 , where @xmath137 . from the equation of motion for the @xmath138 field and the fact that the pressure is zero at saturation density in nuclear matter , the value of @xmath139 is given by @xmath140 where @xmath141 and @xmath142 . the @xmath9 meson coupling constant can be determined for a given symmetry energy through the relation @xmath143 an expression involving the compression modulus can be deduced by differentiating the @xmath7 equation of motion : @xmath144 ^ 2 \left\ { \frac{9n_0m^{*2 } } { e_f^{*2}[k+9(e_a+e_f^*-m)]-3k_f^2e_f^ * } -3 \left ( \frac{n_0}{e_f^*}-\frac{n_s}{m^ * } \right ) \right\ } + n_s \frac{df}{d\phi_0 } - \frac{d^2u(\phi_0)}{d\phi_0 ^ 2 } \;. \label{comp}\ ] ] here @xmath145 , the value of @xmath146 at saturation density , is obtained directly from the dirac effective mass . the function @xmath147 depends on the particular expression used for the effective mass . the scalar density @xmath148 \}$ ] . the @xmath7 equation of motion at saturation can be written in the form @xmath149\;,\ ] ] which together with eq . ( [ comp ] ) allows the @xmath7 coupling to be obtained . finally the constants appearing in the scalar self - coupling @xmath150 are determined from : @xmath151 \nonumber \\ b & = & \frac{1}{2 m \phi_0 } \left[\frac{d^2u(\phi_0)}{d\phi_0 ^ 2 } - 3 c \phi_0 ^ 2 \right ] \,.\end{aligned}\ ] ] the constants determined in this way are given in table i. note that in principle the potential should be bounded from below for large values of the @xmath7 field requiring @xmath152 to be positive ; this is the case for the zm model . in order to investigate the effect of a kaon condensate on the eos in high - density matter , the kaon - meson coupling constants have to be specified . empirically known quantities can be used to determine these constants , but it should be borne in mind that laboratory experiments give information only about kaon - nucleon interaction in free space or in nearly isospin symmetric nuclear matter . on the other hand , the physical setting in this work is matter in the dense interiors of neutron stars which has a different composition and spans a wide range of densities ( up to @xmath153 ) . therefore , kaon - meson couplings as determined from experiments might not be appropriate to describe the kaon - nucleon interaction in neutron star matter , and the particular choices of coupling constants should be regarded as parameters that have a range of uncertainty . with the above caveats in mind , we now examine the relationship between the optical potential of a single kaon in infinite nuclear matter and the kaon - meson couplings in our lagrangian . lagrange s equation for an @xmath154-wave @xmath1 with a time dependence @xmath155 , where @xmath156 is the asymptotic energy , defines the optical potential @xcite for our lagrangian ( [ kaonlag ] ) according to @xmath157~k^-({\bf x } ) & = & [ -2x_0e + \alpha - m_k^2]~k^-({\bf x } ) \nonumber\\ & \equiv & 2~m_k~u_k~k^-({\bf x})\;. \end{aligned}\ ] ] in nuclear matter , @xmath158 , so for a kaon with zero momentum ( @xmath159 ) the optical potential is @xmath160 utilizing the functional forms for @xmath54 in eqs . ( [ alkpe ] ) , ( [ algs ] ) , and ( [ altw ] ) , the optical potentials for the kpe , gs and tw models are easily obtained . for the kpe case this may be written exactly as @xmath161 whereas for the gs and tw cases there are higher order corrections in addition to the terms linear in the fields . we choose @xmath51 to be @xmath162 and @xmath52 to be @xmath163 on the basis of simple quark and isospin counting arguments . note that this value for @xmath51 is also suggested by comparison to the chiral approach ( see ref . @xcite and appendix a ) and it leads to a @xmath164 mev contribution to the optical potential . the total optical potential is shown in table ii for various choices of the @xmath7 coupling . the linear form eq . ( [ linear ] ) , exact for kpe , is an accurate fit to the the gs and tw cases for moderate values of the optical potential . for orientation , chiral models suggest that the magnitude of the optical potential is at most 120 mev @xcite , while fits to kaonic atom data have been reported with values in the range 50200 mev @xcite . we note that glendenning and schaffner - bielich @xcite label their results according to values of the optical potential obtained in the linear approximation ( henceforth , @xmath165 ) . in order to make an apposite comparison with their results , we will parametrize the kaon coupling for each model simply by specifying the value of @xmath166 . the effects of kaon condensation on the eos are more pronounced at zero temperature than at finite temperature , since the fraction of thermally excited kaons increases with temperature relative to the fraction of kaons residing in the condensate . we therefore begin by examining results for the zero temperature case . we have considered two different nucleon lagrangians , gm and zm , and three different kaon lagrangians , kpe , gs and tw . below densities of about @xmath167 , matter is composed of neutron - rich nuclei immersed in a neutron sea . for this regime , we use the potential model results of negele and vautherin @xcite in the range @xmath168 and those of baym , bethe , and pethick @xcite for @xmath169 . for cold stars , the eos in this regime has little effect on maximum masses or stellar radii . furthermore , since the entropy in the stellar mantle @xmath170 is quickly radiated away in neutrinos , the eos in this regime does not substantially affect the results of this paper . in fig . [ fig1 ] , we compare the pressures for the different nucleon and kaon lagrangians as a function of baryon density , @xmath171 . the solid lines show results for both the pure nucleon and kaon condensed phases with no attempt to enforce the gibbs conditions of chemical and mechanical equilibrium . in all cases , a first order phase transition is found to occur , as long as the magnitude of the optical potential @xmath172 is in excess of 100 mev . where possible , the pressure in the mixed phase obtained by imposing gibbs criteria for mechanical and chemical equilibrium is shown as a dashed line . for the gm+kpe , zm+kpe and zm+gs models it was not possible to satisfy gibb s criteria , despite the occurrence of a first order phase transition for large enough @xmath173 . the reason for this is connected with the form of the kaon lagrangian , as discussed below . we also point out in appendix b that non - linear kaon self - interactions lead to a second order , rather than a first order , transition . the qualitative similarity of the results shown in fig . [ fig1 ] for the different nuclear lagrangians enables us to simplify our analysis by allowing us to focus on three , rather than six , possible lagrangian combinations . for a given kaon lagrangian , fairly similar results can be obtained with different nuclear lagrangians by making relatively small shifts in the kaon optical potential @xmath165 . the following discussion will therefore focus on the three cases gm+kpe , gm+gs and zm+tw . the case gm+kpe is chosen to compare with the results of kpe , the case gm+gs is chosen to compare with the results of gs , and the case zm+tw demonstrates the usefulness of lagrangians in which anomalous values of effective masses are implicitly eliminated . the results for the model gm+gs shown here and elsewehere in this paper are identical to those found by gs for the same interactions . note that in all models considered , the phase transition is second order in nature for moderately low values of the optical potential . in figs . [ fig2 ] and [ fig3 ] , the density dependence of the scalar , vector , and iso - vector fields , the electron chemical potential @xmath174 , the condensate amplitude @xmath94 , and the nucleon and kaon effective masses are displayed in the pure nucleon and kaon condensed phases , ignoring any possible mixed phase for the present . for the optical potential chosen , @xmath175 mev , a first order phase transition occurs in all three cases . after the onset of condensation a rapid change in the behavior of the electron chemical potential and some of the field strengths is seen to occur . the differences in the variation of the scalar ( @xmath176 ) and isovector ( @xmath177 fields between the models are particularly illuminating . for gm+kpe , the scalar field exhibits a relatively rapid increase with density after the onset of condensation . this in turn causes both the nucleon and kaon effective masses to drop rapidly with density . in fact , for sufficiently large density , the gm+kpe kaon effective mass vanishes ( see fig . [ fig3 ] ) . the variations of the effective masses in the models gm+gs and zm+tw are more moderate . the variation of the isovector field with density , which in large part controls the variation of the electron chemical potential @xmath88 and hence the electron concentration , is also more dramatic in the case of gm+kpe than in the gm+gs and zm+tw models . notice that in the kpe model it goes to zero for asymptotic densities ( this follows from eq . ( [ hhyp5 ] ) ) , so that the proton and neutron abundances become equal . this does not occur for the other two cases considered here . finally , it is worth noting that in all three models the condensate amplitude rises rapidly once the threshold density is reached . we turn now to a discussion of the results obtained by imposing gibbs criteria for mechanical and chemical equilibrium at zero temperature . in fig . [ fig3a ] , we show the chemical potentials associated with the two conserved charges , charge and baryon number , as functions of each other , for the model gm+gs for a kaon optical potential of @xmath178 mev . quantities associated with the pure nucleon phase , phase i , are shown as solid lines here and in subsequent figures . phase ii refers to the high - density phase in which nucleons and the kaon condensate are in equilibrium , and quantities associated with it are shown as dashed lines . both phases , i and ii , coexist in the mixed - phase region which is displayed as a dotted line . this figure illustrates the way a mixed phase is built from the two pure phases . for electron chemical potentials below the solid curve , matter is positively charged in phase i. a similar interpretation of positive or negative charge for @xmath88 below or above the dashed curve is not possible , since two different types of particles , kaons and leptons , can furnish charge . in other words , a decrease in @xmath88 , or , equivalently , the number of electrons , does not necessarily lead to a positive net charge in phase ii . for @xmath179 mev , only phase i with nucleons and leptons are present . for @xmath180 mev , a mixed phase of positively charged phase i and negatively charged phase ii obeying the gibbs conditions ( [ g1 ] ) is favored . qualitatively , a similar situation is encountered in the construction of the mixed phase for the zm+tw model , but the mixed phase region is quite small . as noted earlier , however , it was not possible to satisfy gibbs criteria for models with the kaon lagrangian kpe . in fig . [ fig3b ] we show the individual charge densities of phase i and ii in the mixed phase , as a function of baryon density . the dotted curve in this figure shows the volume fraction of phase i. the results are for the gm+gs model with @xmath181 mev ( upper panel ) and for the zm+tw model with @xmath182 mev ( lower panel ) . near the lower threshold , matter in phase i is very slightly positively charged and occupies most of the volume . as the density increases , the volume fraction of phase i , @xmath126 , decreases and its charge density increases . note that the negative charge density of matter in phase ii at the lower transition point , @xmath183 @xmath184 , and the positive charge density of matter in phase i at the higher transition point , @xmath185 @xmath184 , are rather large in the case of gm+gs compared to the case zm+tw . this is due to the stronger density dependence of the scalar and isovector densities in the former case . note also that a first order transition allows for the existence of a very dense and nearly isospin symmetric matter in the mixed phase . in figs . [ fig4 ] and [ fig5 ] , we show the magnitudes of the various fields , the electron chemical potential , the nucleon and kaon effective masses , and the condensate amplitude for the gm+gs and zm+tw models , respectively . both models show the same qualitative behavior . at the lower phase boundary , in which phase ii just begins to appear , the scalar field in phase ii is much larger than in phase i and the condensate amplitude @xmath94 in phase ii takes a large value which decreases with increasing @xmath186 through the mixed - phase region . thus , the effective masses of both kaons and nucleons in phase ii are much smaller than in phase i. the densities demarking the mixed phase region and its overall extent are dependent upon the interaction models , and upon the assumed values of the kaon optical potentials , here taken to be @xmath187 mev in the case of gm+gs and @xmath188 mev in the case of zm+tw . the region in density over which the mixed phase extends is much smaller in the latter case , chiefly due to the more moderate behavior of the scalar interaction with density variations in this case . it is instructive to compare the behavior of the two models at the threshold of the mixed phase region . phase i will have a net small positive charge and a volume proportion @xmath126 close to 1 ( see fig . [ fig3b ] ) . this has to be counterbalanced by a large net negative charge in phase ii since it is weighted by the small proportion @xmath189 . focusing on phase ii , the condensate condition for the models gm+gs and zm+tw from the last of eqs . ( [ hhyp5 ] ) is @xmath190 and the kaon number density , which has to be large , is @xmath191 in order to ensure that @xmath192 , the quantity @xmath193 , and hence @xmath194 , has to be positive definite . in the zm+tw model the kaon effective mass is relatively large so that @xmath53 is positive and therefore @xmath94 is relatively small . on the other hand in the gm+gs model @xmath194 is quite small so that @xmath53 is negative and @xmath94 has to be large . the negative value of @xmath195 implies a large negative value of @xmath196 which is clearly sensitive to the value of @xmath52 . in fact , if this coupling is reduced by more than about 15% from our chosen value it is no longer possible to satisfy the gibbs criteria . by comparing the pure phase results in figs . [ fig2 ] and [ fig3 ] with the mixed phase results of figs . [ fig4 ] and [ fig5 ] , it is clear that substantial modifications of the various fields are required to satisfy gibbs criteria . we examine now the kpe model for which it was not possible to satifsfy the gibbs criteria . in this case , eq . ( [ alkpe ] ) and the last of eqs . ( [ hhyp5 ] ) leads to the condensate condition @xmath197 whereas the functional form of the number density of kaons is identical to that in eq . ( [ kden ] ) . ( [ culprit ] ) differs in important ways from eq . ( [ lucky ] ) . for the kpe model , even if @xmath198 is positive , @xmath88 has the proclivity to turn negative for large @xmath124 ( or equivalently , for large baryon densities ) , leading to @xmath199 or imaginary values of the kaon effective mass @xmath194 . this may be seen in fig . [ fig5a ] where we show the electron chemical potential @xmath88 as a function of the ( negative ) charge density in pure phase ii for a typical value of the neutron chemical potential @xmath200 mev . it is now possible to understand qualitatively why a mixed phase can not occur in the case of the kaon lagrangian kpe . in comparison with the gm+gs and zm+tw models , a distinctive feature of the kpe model is that @xmath88 decreases rapidly with the ( negative ) charge density . in constructing a mixed phase , we are attempting to balance the positive charge in phase i with the negative charge in the dense phase ii in which the electron chemical potential , and hence the charge content in leptons , is rapidly decreasing towards zero . the balance never occurs , hence the failure to meet the gibbs criteria . in terms of compositions , the gs or tw lagrangians introduce negative charges in matter by increasing the number density of kaons , while keeping the electron density nearly constant or even slightly increasing with the charge density . the kpe lagrangian , however , rapidly substitutes electrons by kaons , which is detrimental to meeting the gibbs criteria . for these reasons , we will concentrate on results with the other two kaon lagrangians in the remainder of this paper . the influence of the condensate on neutron star structure ( at zero temperature ) is shown in fig . [ fig6 ] in which the gravitational mass is displayed as a function of the star s central baryon number density ( left panel ) and its radius ( right panel ) . for the models shown , the transition is first order and gibbs equations for mechanical and chemical equilibrium are utilized . for all cases shown the central densities of the maximum mass stars lie in the mixed phase . the effects of the condensate are more evident in the case of the gm+gs model in which the mixed phase occurs over a wider region of density than in the zm+tw model . when the effects of the softening induced by the occurrence of the condensate are large , the limiting mass and the radius at the limiting mass are reduced significantly from their values when the condensate is absent . note , however , that the softening effects are limited by the constraint that the maximum mass must exceed that of the binary pulsar psr 1913 + 16 , 1.442 m@xmath201 . in the case of gm+gs , this constraint limits @xmath202 to be smaller than about 125 mev . in such a case , the minimum radius achieved is not as small as in the case @xmath203 mev , as shown in fig . the radii of stars with masses less than 1.2 m@xmath201 are not affected by the choice of the kaon lagrangian or the kaon optical potential , since the condensation threshold is not reached in these cases . the density dependence of @xmath204 , @xmath205 and @xmath206 have been investigated in other works @xcite , but for the most part either in isospin symmetric nuclear matter or pure neutron matter . in general , our results for @xmath204 with @xmath207 mev are consistent with those of refs . @xcite ( for an appropriate comparison , our results are to be compared with results obtained without in - medium pion contributions in ref . @xcite ) and those of ref . @xcite for nuclear matter at both @xmath208 and 3 . there is a relatively small change produced in going from nuclear matter to beta - equilibrated neutron star matter to pure neutron matter for the quantities @xmath204 and @xmath205 . note that a direct comparison of the real parts of the optical potentials between different calculations must also account for the fact that in obtaining fits to data , the imaginary parts are often found to be as large as the real parts , which indicates fragmentation of strength in the quasi - particle spectral function . relatively larger variations are found in the kaon energies in matter with varying amounts of isospin as can be seen from fig . [ newfig ] . in this figure , the top panel provides a comparison of results for beta - equilibrated neutron - star matter for the gm+kpe , gm+gs , and zm+tw models , respectively , for values of @xmath209 at the extreme ends considered here , namely , 80 and 120 mev . the bottom panel shows results for the zm+tw model for @xmath210 mev , for pure neutron matter , neutron - star matter , and isospin symmetric nuclear matter , respectively . at nuclear density where the models are calibrated , @xmath8 decreases by about a few mev in going from pure neutron matter to neutron star matter and by about a few tens of mev in going from neutron star matter to nuclear matter . with increasing density , these differences become progressively larger . this trend is chiefly due to the behavior of the vector fields in matter with different amounts of isospin . at this time , our results for the density dependence of @xmath8 can be compared with those of the potential models in refs . @xcite . for values of @xmath209 near the lower end of the range we explored , in the neighborhood of 80 mev , the behavior of @xmath8 , for example , is quite similar to the potential model results . as the authors in refs . @xcite indicated , kaon condensation may be unlikely in this case . however , the relevant comparision must also include the electron chemical potential @xmath211 , since the density where @xmath212 determines the onset of kaon condensation . as demonstrated in ref . @xcite , the behavior of @xmath211 for neutron star matter at high densities is determined by the density dependence of the nuclear symmetry energy ( see also a similar discussion in ref . potential model calculations ( see , for example ref . @xcite ) tend to have a relatively weak density dependence of the symmetry energy , which generally results in an onset of kaon condensation that is at a rather large density . in field - theoretical and dirac - brueckner - hartree - fock @xcite models , however , the symmetry energy varies relatively rapidly with density . these lead to smaller densities where kaon condensation occurs , for a given behavior of the kaon energy @xmath8 . furthermore , the calculations of ref . @xcite have been performed only for pure neutron matter which further enhaces the values of @xmath8 and discourages kaon condensation . in addition , as @xmath213 is increased in magnitude in field - theoretical models , the role of kaons increases and @xmath8 becomes progressively smaller as a function of density . nevertheless , the lack of effective constraints at high density preclude choosing any model over another at this time . in summary , choosing values of @xmath209 near the lower end of the range we explored either lead to a second order phase transition or no transition at all in a neutron star , in which case the gross properties of the star are relatively unaffected from the case without kaons . on the other hand , values near the higher end of this range lead to a first order phase transition at a relatively low density , depending on the form of the interaction chosen , and a more pronounced effect on the star . our aim has been to provide benchmark calculations in which both possibilities are entertained in order to consider their impact on thermodynamics and their astrophysical implications . we now investigate results at finite temperature and values of the lepton content characteristic of those likely to be encountered in the evolution of a pns . we choose three representative sets of pns conditions which correspond to : the initial conditions within a pns ( entropy / baryon @xmath214 , trapped neutrinos with a lepton fraction @xmath215 ) , a time after several seconds when the interior is maximally heated ( @xmath216 , no trapped neutrinos so @xmath43 ) , and a very late time when the pns has cooled ( @xmath217 identical to the zero temperature case discussed above ) . for a detailed explanation of the evolution of a cooling pns see pons _ _ @xcite . the contribution of the nucleons to the entropy per baryon @xmath218 , with @xmath171 denoting the total nucleon density , in degenerate situations ( @xmath219 ) can be written @xmath220 where @xmath135 and @xmath221 are the effective mass and the fermi momentum of species @xmath44 , respectively . for the temperatures of interest here , and particularly with increasing density , the above relation provides an accurate representation of the exact results for entropies per baryon even up to @xmath222 . the behavior with density of both the fermi momenta and the effective mass controls the temperatures for a fixed @xmath223 . for kaons it is straightforward to show that the contribution to the entropy from @xmath224 mesons can be ignored since it is exponentially suppressed in comparison to the @xmath1 contribution . for the latter , keeping the leading temperature dependence of the simplest approximation scheme for bosons given in ref . @xcite , the kaon entropy per baryon is @xmath225 \frac{n_k^{th}}{n_b}\quad{\rm where } \quad \psi t=\mu+x_0-\sqrt{\alpha+x_0 ^ 2}\;,\ ] ] and @xmath226 is determined from @xmath227 by solving the equation @xmath228 below the kaon condensation threshold as the temperature becomes very small @xmath229 so @xmath230 . above the kaon condensation threshold the last of eqs . ( [ hhyp5 ] ) implies that @xmath231 in which case @xmath232 . this simple approximation provides quite an accurate account of the kaon entropy per baryon which is fairly small for the scenarios examined here since it involves just the thermal contribution and the condensate plays no role . the total entropy per baryon @xmath233 also includes the lepton contributions ; @xmath234 is dominated , however , by @xmath223 . in figs . [ fig7 ] and [ fig8 ] , the relative concentrations of various particles are displayed versus baryon number density for our three pns conditions in the cases gm+gs and zm+tw , respectively . the cases shown allow the gibbs equations to be solved , and the boundaries of the mixed phase regions are indicated by vertical lines . the effect of finite temperature is to allow the existence of @xmath235 and @xmath1 particles at all densities , although kaons become relatively abundant only within the mixed phase region . in the third set of diagrams , trapped neutrinos are present at all densities and the appearance and abundances of the negatively charged particles @xmath235 and @xmath1 are suppressed . furthermore , the critical density for kaon condensation is shifted to higher density . in fig . [ fig9 ] the pressure is displayed as a function of baryon number density for these two lagrangians and the three pns conditions . two choices for the kaon optical potential are shown to highlight differences between cases in which kaons condense in second or first order phase transitions . the reduction of the pressure when kaons condense is obvious . for conditions in which the phase transition is first order , the result of applying the gibbs conditions and the result of assuming pure phases ( thin line ) are both shown . the application of the gibbs conditions leads to further softening of the pressure over a wider density range . in the case of model zm+tw , a first order phase transition occurs only for very low temperatures and low neutrino concentrations . in fig . [ fig10 ] we show the matter temperature as a function of the baryon density for these two lagrangians for the two pns conditions with @xmath236 ( the kaon optical potentials are as in the previous figure ) . the appearance of the kaon condensate generally leads to a reduction in specific heat which is indicated by the abrupt temperature increase which persists to high densities . in the case of first order transitions , applying the gibbs conditions leads to a further enhancement of the temperature in the mixed phase regime . this behavior is in marked contrast to the case in which additional fermionic degrees of freedom , such as hyperons or quarks , are excited @xcite causing the temperature to drop and the specific heat of the matter to be increased . the latter follows from eq . ( [ nucentropy ] ) where , in the absence of any variation of @xmath135 , a system with more components at a given baryon density has a smaller temperature than a system with fewer components ( recall that @xmath237 ) . in the present case the dropping of the effective mass is the dominant effect and this leads to larger temperatures . figure [ fig11 ] shows the phase diagram of kaon condensed matter , for the case gm+gs with @xmath178 mev . the left panel displays results for zero temperature in the density lepton concentration plane . the dashed lines show the minimum lepton concentration allowed at zero temperature ( with @xmath43 ) for each density . note that the minimum lepton concentration increases with density until the phase transition begins ; above this density , the minimum lepton concentration decreases with increasing density . also note that the phase transition to a kaon - condensed phase is pushed to higher densities when trapped neutrinos are present . this implies that in the initial pns core material , in which @xmath238 and the central density is less than 3.5 times the nuclear saturation density , a kaon condensate phase likely does not exist . however , as neutrinos leak from the star the transition density decreases and a kaon condensate eventually forms . the right panel displays results in the density versus temperature plane , assuming no trapped neutrinos ( @xmath43 ) . the phase diagram for kaon condensed matter for the case zm+tw with @xmath182 mev is shown in fig . [ fig12 ] ; the results are qualitatively similar to the gm+gs case in which @xmath239 mev in fig . [ fig11 ] . this is understandable from the perspective that the actual optical potential for these two models are nearly the same . the boundary between phases i and the mixed - phase region are nearly the same for the two cases . the major difference is the much smaller extent of the mixed - phase region for the case zm+tw . note that for both cases the density at which the phase transition begins is relatively independent of temperature , so that the heating which initially occurs in the pns has little effect on the eventual appearance of a kaon condensate . also note that the density range of the mixed phase decreases with increasing temperature , and the mixed phase persists to high temperatures . it appears that the mixed phase exists up to temperatures exceeding 60 mev , for the case gm+gs and @xmath178 mev , or 30 mev for the case zm+tw with @xmath182 mev . it becomes increasingly difficult to determine the properties of a mixed phase near the temperature at which it disappears . in fig . [ fig13 ] the gravitational mass is plotted as a function of central baryon number density for these models . results are shown for our three pns conditions which correspond to snapshots of the pns evolution . the initial configuration ( dotted curves ) has the largest maximum mass . the progression to the dashed and solid curves indicates the evolution with time and we see that the maximum masses decrease . the effect of temperature upon the structure of the pns is significant . thermal kaons play a significant role here , since the net negative charge they contribute to the system partially inhibits the appearance of the condensate which allows hot neutrino free stars to reach higher masses than cold stars . the net decrease in maximum mass during the evolution for either case is seen to be of order 0.20.3 m@xmath201 . thus there is an appreciable range of masses for the pns which will result in metastability with the star ultimately collapsing to a black hole . the central density of the maximum mass , zero temperature star is smaller for the gm+gs case than for the zm+tw case . this is in spite of the apparently softer " gm+gs equation of state in which the kaon condensed mixed - phase region extends over a wider density range . ultimately , the smaller maximum mass of the gm+gs eos leads to a smaller central density at the maximum mass . in this work , we have studied the equation of state of matter , incorporating the possible presence of a kaon condensate , and including the effects of trapped neutrinos and finite temperatures . the calculation of the neutrino spectra of different flavors emitted from a proto - neutron star as it evolves from a hot , lepton - rich state to a cold , neutrino - poor state requires the knowledge of the equation of state of matter at temperatures up to about 5060 mev and lepton fraction up to about 0.4 . since the nucleon - nucleon and kaon - nucleon interactions at high density are relatively poorly understood , we explored several possible field - theoretical models in both sectors . these models are distinguished by the form of the assumed scalar ( and in some cases vector ) interactions which chiefly determine the density dependences of the nucleon and kaon effective masses . these models produce significantly different high density behavior of the eos , even though the kaon - meson couplings in these models are calibrated to give the same the kaon - nucleus optical potential in nuclear matter . the principal findings of our studies at zero temperature were : 1 . the order of the phase transition between pure nucleonic matter and a phase containing a kaon condensate depends sensitively on the choice of the kaon - nucleon interaction . 2 . in one case we studied ( kpe ) , although a first - order phase transition resulted , it was not possible to satisfy gibbs rules for phase equilibrium which would have produced a mixed phase . we performed a detailed analysis of this situation and found that scalar , and to a lesser extent the isovector , interactions that vary rapidly with density were chiefly responsible for this failure . this was confirmed by developing a new kaon - nucleon interaction ( tw ) with more moderate variations in the scalar density and the kaon effective mass in which the gibbs criteria in a first order phase transition would be satisfied . the extent of the mixed phase region was thereby reduced . the significance of the new kaon - nucleon interaction ( tw ) we developed is that it avoids the anomalous behavior for the kaon effective mass that occurs in previous models ( kpe , gs ) at very high density . near the low - density boundary of a mixed phase region , the kaon condensed phase appears with large density , too large for the kpe interaction to produce physically acceptable effective masses . we also made detailed comparisons with earlier work which used the gs form for the scalar interactions . 3 . in all models considered ( kpe , gs and tw ) , a first - order phase transition occurs only for large values of the kaon - nucleus optical potential ; moderate values generally produce a second order phase transition . in the meson exchange models studied here , only linear kaon self - interactions were considered . in the case of a first order phase transition , the condensate amplitude was found to be rather large at the low - density boundary of the mixed phase . we therefore explored the effect of non - linear kaon self - interactions guided by the chiral model in appendix b. we found that introducing higher order interactions , using the lowest order chiral lagrangian , results in a second order , rather than a first order , phase transition . whether this behavior persists when more general higher order operators in the chiral expansion are considered remains an open question . at finite temperatures , we find the effects of condensation , in general , are less pronounced than at zero temperature . for moderate values of the optical potential , when the phase transition is first order at zero temperature , kaon condensation eventually becomes a second order phase transition at high enough temperatures , whether or not neutrino trapping is considered . the temperature at which this occurs is in the range of 3060 mev , depending upon interactions . for the cases at finite temperatures in which the transition is first order , its thermodynamics ( such as the pressure - density relation ) becomes effectively similar to that of a second order phase transition this is because of the existence of thermal kaons and because of nucleonic thermal effects . the condensate is suppressed , and moved to higher densities , both by the existence of trapped neutrinos and by finite temperatures . compared to earlier works , the new aspects of our work are : 1 . the delineation of the phase boundaries in the baryon density versus lepton number and baryon density versus temperature planes . this is helpful to anticipating the possible outcome in a full pns simulation . in particular , the critical temperatures above which the mixed phase disappears are above 30 and 60 mev , depending upon the interaction . this has implications for the temperature dependence of the surface energies , and for the melting temperatures , of the droplets in the mixed phase . the finding that thermal effects on the maximum gravitational mass of neutron stars are comparable to the effects induced by the trapped neutrino content . this is in stark contrast to previously studied cases in which nucleons - only matter , or matter containing hyperons , were considered . furthermore , compared to equations of state previously studied that allow metastable protoneutron stars , those containing hyperons or quark matter , the maximum mass does not significantly decrease during the deleptonization of the protoneutron star because of these thermal effects . only after the temperature in the protoneutron star significantly decreases does the maximum mass appreciably fall . this implies that the possible collapse of a metastable protoneutron star to a black hole occurs during the late stages of cooling , after several tens of seconds , rather than during the late stages of deleptonization , which is somewhat earlier . the support of the u.s . department of energy under contract numbers doe / de - fg02 - 87er-40317 ( jap and jml ) , doe / de - fg06 - 90er40561 ( sr ) , doe / de - fg02 - 88er-40328 ( pje ) , and doe / de - fg02 - 88er-40388 ( mp ) is acknowledged . j. pons also gratefully acknowledges research support from the spanish dgcyt grant pb97 - 1432 , and thanks j.a . miralles for useful discussions . in this appendix , we examine the conditions under which there exists a close correspondence between a meson exchange model and the chiral @xmath240 approach of kaplan and nelson @xcite . such a correspondence is most easily established for the zero temperature case by setting the scalar self - coupling terms , _ i.e. , _ @xmath241 . specializing to the case where the only baryons are nucleons and using the walecka lagrangian for the nucleons , it was shown in ref . @xcite that the chiral thermodynamic potential per unit volume can be written @xmath242 where the primes on the meson fields distinguish them from those used previously and @xmath243 is the heaviside step function . the nucleon effective masses are @xmath244 we employ the values suggested by politzer and weise @xcite , namely @xmath245 mev ( @xmath246 is the strange quark mass ) and @xmath247 mev . @xmath248 is usually taken to lie in the range @xmath249 to @xmath250 mev . if we ignore the fairly small effect of @xmath251 here and in the kaon - nucleon sigma term , @xmath252 , we can write @xmath253 as well as redefining the scalar field , we can redefine the chiral vector fields entering the chemical potentials : @xmath254 substituting in eq . ( [ omkap ] ) we find @xmath255 if we expand this in powers of @xmath94 and retain only the lowest order @xmath256 term , the last term in eq . ( [ omchmes ] ) does not contribute and our thermodynamic potential is exactly of the form given by eqs . ( [ hyp2 ] ) and ( [ zkexch ] ) for the meson exchange model provided that the @xmath68 expression is used . in order for the correspondence to be exact , the parameters for the @xmath7 and @xmath8 meson need to obey @xmath257 these are precisely the conditions found in ref . @xcite for the optical potentials of the chiral and meson exchange models to be the same in nuclear matter . the relation involving the @xmath8 meson couplings is quite well obeyed with our parameters . in addition , for the @xmath9 meson , @xmath258 this indicates that @xmath259 , a condition which is not well obeyed by the parameters used here or in other works . given that the chiral and meson exchange thermodynamic potentials can be put into precise correspondence to lowest order in @xmath256 , it follows that the equations of motion and the thermodynamics will be identical to this order . if scalar self - coupling terms are included , @xmath260 , then the transition from the chiral to the meson exchange approach will couple higher powers of the @xmath7 field to the kaon condensate ( in the braces in eq . ( [ omchmes ] ) ) . it will also introduce higher order terms . these additional contributions may not be negligible so the correspondence between the two approaches becomes less precise . our findings in appendix a naturally raise the question of whether it is sufficient to work at order @xmath256 , involving only linear kaon self - interactions , in the meson exchange models . it clearly will be sufficient at the low - density onset of a second order phase transition where @xmath94 is small . on the other hand , for a first order phase transition , the value of @xmath94 is large at the low - density onset of the mixed phase , particularly for the gs model . it is therefore interesting to explore the effect of non - linear kaon self - interactions guided by the chiral model . the order of the phase transition ( in the mean field approximation ) is determined by the behavior of the thermodynamic potential , @xmath261 , at fixed chemical potentials . a first order transition , with a mixed phase , is possible only if there exists some value of @xmath124 for which @xmath261 exhibits two degenerate minima . at the critical density corresponding to the low - density onset of the mixed phase , the @xmath117 phase should be a local minimum which is degenerate with a minimum at some finite @xmath262 . in the vicinity of the critical density , the @xmath117 phase is nearly charge neutral ( with an infinitesimal excess of positive charge and a volume fraction close to 1 which balances the negative charge in the kaon phase which has an infinitesimal volume fraction ) . this requirement enables us to determine the electron chemical potential at the critical density by charge neutrality . we focus on the gm+gs model for which the thermodynamic potential of nucleons and kaons was given in eqs . ( [ hyp2 ] ) and ( [ zkexch ] ) ; the contribution due to leptons is ignored since it does not contain any @xmath94 dependence at fixed @xmath88 . at zero temperature with a kaon optical potential @xmath175 mev , this model predicts a first order phase transition in the vicinity of @xmath263 mev , as can be deduced from fig . [ fig14 ] where @xmath264 is plotted as a function of @xmath94 . the thermodynamic potential for the model gm+gs is shown as the solid curve labelled @xmath265 . it clearly shows two minima , one at @xmath117 and the other at @xmath266 . the latter corresponds to a kaon number density @xmath267 @xmath184 which is larger than the baryon density . for such a dense condensate one would suspect that non - linear kaon self - interactions might be important . the order @xmath268 corrections to the thermodynamic potential are easily found from eq . ( [ omchmes ] ) : @xmath269 the result of adding this correction to @xmath265 is shown as the dashed curve in fig . [ fig14 ] . it greatly alters the behavior of @xmath261 for @xmath270 . the exsistence of a second minimum suggests that a first order phase transition is still possible , but at larger @xmath124 . however , we find that this is not the case and a second - order transition occurs at @xmath271 mev . it is possible to incorporate all powers of @xmath94 arising from self - interactions in the chiral model . in this case the correction to the grand potential is @xmath272 the result of including this correction is shown as the dot - dashed curve in fig . [ fig14 ] . in this case no first order phase transition is possible in the vicinity of @xmath273 . instead a second order phase transition occurs once again at @xmath271 mev ; this is because kaon self interactions play no role when @xmath94 is small . despite our findings here , it is not clear if kaon self - interactions will generically disfavor a first order transition . this is because we have ignored higher order operators in the chiral expansion which will become important with increasing @xmath94 . the indication from phenomenological chiral perturbation theory @xcite is that such effects can be significant when @xmath274 . the robust finding here is that the higher order kaon self - interactions predicted by the lowest order chiral lagrangian lead to a second order , rather than a first order , phase transition . 99 bigus m. prakash , i. bombaci , m. prakash , p.j . ellis , j.m . lattimer and r. knorren , phys . rep . * 280 * , 1 ( 1997 ) . tpl v. thorsson , m. prakash and j.m . lattimer , nucl . phys . a * 572 * , 693 ( 1994 ) . kj w. keil and h.t . janka , astron . and astrophys . * 296 * , 145 ( 1994 ) . pons , s. reddy , m. prakash , j.m . lattimer and j.a . miralles , astrophys . j. * 513 * , 780 ( 1999 ) . glendenning and j. schaffner - bielich , phys . c * 60 * , 025803 ( 1999 ) . rbp s. reddy , g. bertsch and m. prakash , phys . b 475 , 1 ( 2000 ) . kpe r. knorren , m. prakash and p.j . ellis , phys . c * 52 * , 3470 ( 1995 ) . schaffner j. schaffner and i.n . mishustin , phys . c * 53 * , 1416 ( 1996 ) . kapnel d. b. kaplan and a. e. nelson , phys . b * 175 * , 57 ( 1986 ) ; * 179 * , 409 ( 1986 ) ( e ) . politzer and m.b . wise , phys . b * 273 * , 156 ( 1991 ) . brown , k. kubodera , m. rho and v. thorsson , phys . b * 291 * , 355 ( 1992 ) . mfmt t. maruyama , h. fujii , t. muto and t. tatsumi , phys . b * 337 * , 19 ( 1994 ) . mti t. muto , t. tatsumi and n. iwamoto , phys . d * 61 * , 083002 ( 2000 ) ; _ ibid_. d * 61 * , 063001 ( 2000 ) . ty t. tatsumi and m. yasuhira , nucl . a * 670 * , 218 ( 2000 ) . sew b. d. serot and j. d. walecka , advances in nuclear physics * 19 * ed . negele and e. vogt ( plenum , ny , 1986 ) ; b. d. serot , rep . phys . * 55 * , 1855 ( 1992 ) . bb j. boguta and a. bodmer , nucl * a292 * , 413 ( 1977 ) . gm n.k . glendenning and s.a moszkowski , phys . lett . * 67 * , 2414 ( 1991 ) . zm j. zimanyi and s.a . moszkowski , phys . c * 42 * , 1416 ( 1990 ) . lrp c .- h . lee , s. reddy and m. prakash , proc . of int . workshop xxvi on gross properties of nuclei and nuclear excitations , ed . m. buballa , w. nrenberg , j. wambach and a. wirzba ( hirschegg , austria , 1998 ) p. 86 . ogievetskij and i.v . polubarinov , ann . 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[ cols="^,^,^,^,^,^ " , ] fig . [ fig1 ] : pressure versus baryon number density for the six choices of the nucleon and kaon lagrangians considered in this paper . the temperature @xmath130 and there are no trapped neutrinos ( @xmath43 ) . selected values for the kaon optical potential @xmath278 are indicated . the solid lines show the pressure in the pure phases i ( nucleons only ) and ii ( the high - density nucleon - kaon condensed phase ) . the dashed lines show the pressures obtained by imposing gibbs criteria for phase equilibrium in a mixed - phase region for the case of first order transitions . for the kpe choice of the kaon lagrangian , gibbs criteria could not be satisfied despite the occurence of first order phase transitions in some cases . fig . [ fig2 ] : the density dependences of the scalar , vector , and iso - vector fields for different choices of the nucleon and kaon lagrangians ( @xmath279 ) . the solid curves show the chemical potential @xmath280 . in this figure , results are shown only for the pure phases i and ii ; the mixed phase produced by satisfying gibbs criteria is ignored . [ fig3a ] : the electron chemical potential @xmath88 versus the neutron chemical potential @xmath124 in pure phases i and ii , and in the mixed phase . the pure phase i ( solid curve ) consists of nucleons and leptons . the pure phase ii ( dashed curve ) is comprised of a kaon condensate coexisting with nucleons and leptons . the mixed phase ( dots ) is constructed by satisfying gibbs rules for phase equilibrium . [ fig3b ] : individual charge densities of pure phases i and ii and the volume fraction @xmath126 of phase i in the mixed phase as a function of baryon density . results are for the gm+gs model with @xmath181 mev and for zm+tw model with @xmath281 mev . fig . [ fig4 ] : the density dependences of the scalar , vector , and iso - vector fields for two choices of the nucleon and kaon lagrangians ( @xmath279 ) . phase i is the pure nucleon phase and phase ii is the high - density nucleon - kaon condensed phase . the vertical lines demark the mixed phase region . [ fig5a ] : the electron chemical potential @xmath88 in phase ii matter versus charge density for different models at a fixed neutron chemical potential of @xmath283 mev . in all cases , the optical potential @xmath178 mev . [ fig6 ] : left panel : the gravitational mass as a function of the central baryon number density for the cases gm+gs and zm+tw ( @xmath279 ) . curves are labelled by the values of @xmath278 and the eos includes a mixed phase region . right panel : the gravitational mass as a function of the stellar radius . fig [ newfig ] : the density dependences of the kaon energy @xmath8 in matter with different isospin content . the top panel compares results of gm+kpe , gm+gs and zm+tw models for beta stable neutron star matter for @xmath284 and -120 mev , respectively . the bottom panel shows results for the zm+tw model with @xmath284 mev in pure neutron matter , beta stable neutron star matter and nuclear matter . fig [ fig7 ] : the relative concentrations of hadrons and leptons as functions of baryon number density for three representative snapshots during the evolution of a pns . the results shown are for the model gm+gs . to the left of the vertical line there is no kaon condensate , to the right a mixed phase is present . fig [ fig9 ] : the pressure versus baryon number density for three representative snapshots during the evolution of a pns . the cases shown in the upper panels produce only second order phase transitions . for the cases in the lower panels the transitions are first order , except for zm+tw with @xmath236 . in the lower panels , heavy curves include a mixed phase region and light curves ignore a mixed phase region . fig [ fig11 ] : the phase diagram of kaon condensed matter for the case gm+gs and @xmath175 mev . the left panel shows results at zero temperature in the density versus lepton concentration plane . the dashed curve shows the minimum lepton concentration for each density , which occurs for trapped neutrino concentration @xmath43 . the right panel shows results in the density versus temperature plane for neutrino free matter ( @xmath43 ) . fig [ fig14 ] . the thermodynamic potential as a function of the condensate order parameter @xmath94 . results are shown for the gm+gs model near the critical density ( @xmath286 mev and @xmath287 mev ) with optical potential @xmath175 mev . | we study the equation of state of kaon - condensed matter including the effects of temperature and trapped neutrinos .
several different field - theoretical models for the nucleon - nucleon and kaon - nucleon interactions are considered .
it is found that the order of the phase transition to a kaon - condensed phase , and whether or not gibbs rules for phase equilibrium can be satisfied in the case of a first order transition , depend sensitively on the choice of the kaon - nucleon interaction . to avoid the anomalous high - density behavior of previous models for the kaon - nucleon interaction , a new functional form is developed .
for all interactions considered , a first order phase transition is possible only for magnitudes of the kaon - nucleus optical potential @xmath0 mev .
the main effect of finite temperature , for any value of the lepton fraction , is to mute the effects of a first order transition , so that the thermodynamics becomes similar to that of a second order transition . above a critical temperature , found to be at least 3060 mev depending upon the interaction , the first order transition disappears .
the phase boundaries in baryon density versus lepton number and baryon density versus temperature planes are delineated , which are useful in understanding the outcomes of protoneutron star simulations .
we find that the thermal effects on the maximum gravitational mass of neutron stars are as important as the effects of trapped neutrinos , in contrast to previously studied cases in which the matter contained only nucleons or in which hyperons and/or quark matter were considered .
kaon - condensed equations of state permit the existence of metastable neutron stars , because the maximum mass of an initially hot , lepton - rich protoneutron star is greater than that of a cold , deleptonized neutron star .
the large thermal effects imply that a metastable protoneutron star s collapse to a black hole could occur much later than in previously studied cases that allow metastable configurations . #
10= -.025em0 - 0 .05em0 - 0 -.025em.0433em0 |
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thermonuclear explosions of carbon - oxygen ( co ) white dwarfs ( wds ) , which lead to type ia supernovae ( sne ia ) , could be triggered by the merger of two wds ( the double - degenerate scenario ( dds ) @xcite ) or by the accretion of matter from a non - degenerate star through roche - lobe overflow ( the single - degenerate scenario ( sds ) @xcite ) . in the sds , the companion to the co - wd could be a main - sequence ( ms ) , red giant ( rg ) , or helium - rich ( he ) star . in the dds , the companion could be another co wd or he wd . based on current studies , both scenarios are not ruled out by observations , but are also not proven by them . if both scenarios lead to sn ia , it is still unclear which channel(s ) in the sds and dds dominate(s ) the sne ia , and by what fraction @xcite . recent multi - dimensional hydrodynamics simulations of sn impact on the non - degenerate binary companions in the sds suggest that the companions should survive the sn impact and could be detectable . therefore , one of the direct methods to distinguish between the sds and dds is to search for the surviving companions ( scs ) in sn ia remnants ( ia snrs ) . * hereafter p10 ) and ( * ? ? ? * hereafter p12a ) examine the effects of a sn impact on the non - degenerate binary companions in the sds for ms , rg , and he star binary companions via multi - dimensional hydrodynamics simulations . these simulations include the symmetry - breaking effects of orbital motion , rotation of the binary companion , and roche - lobe overflow ( rlof ) , all of which allow a better description of sn - driven shock compression , heating , and stripping of scs . examined a similar sn impact on ms and he stars with the sph approach using companion models from more sophisticated one - dimensional binary evolutions . however liu et al . did not study the subsequent post - impact evolution and therefore could not predict the properties of scs in historical ia snrs . on the other hand , @xcite and @xcite examined the evolution of a @xmath2 subgiant and a @xmath2 ms companion with ad hoc prescriptions for energy input and mass stripping without performing detailed hydrodynamical calculations , thereby dramatically overestimating the luminosity of scs . in ( * hereafter p12b ) and ( * ? ? ? * hereafter p13 ) , we mapped our detailed three - dimensional hydrodynamical results into a one - dimensional stellar evolution code to simulate the post - impact evolution of ms- and he - scs and thus provided a more realistic treatment of post - impact evolution . in this paper , we calculate the time evolution of the magnitudes and colors of our models and discuss the possibility of searching for sds - scs in specific nearby ia snrs . we present numerical results in 2 and compare them with searches for scs in galactic ia snrs in 3 , in the magellanic clouds in 4 , and in m31 and m82 in 5 . finally , we discuss the evidence for the sds channel for sne ia and present our conclusions in the last section . in this section , we discuss and predict some possible observables of scs in nearby ia snrs and describe differences between sc candidates and unrelated stars in ia snrs . in particular , we calculate the colors , magnitudes , effective temperatures , and photospheric radii of ms- and he - scs as functions of time . we also predict the magnitude and effective temperature changes as functions of time . the linear and rotational velocities of scs are also discussed . we also suggest upper limits for the ni / fe contamination in scs . in simulations of ms- and he - scs , the bolometric luminosity , effective temperature , and photospheric radius are directly determined using the stellar evolution code mesa ( modules for experiments in stellar astrophysics ; @xcite ) . to facilitate direct comparison with optical observations , the bolometric luminosity is converted to broad band magnitudes . additional observable quantities such as the strength of absorption lines require the treatment of detailed radiation transfer effects in stellar atmosphere models , which are not considered in this paper . given the effective temperature and photospheric radius of a sc , the magnitude of the sc can be estimated under the assumption that the photosphere emits a blackbody radiation spectrum . we have considered several different filters with their corresponding sensitivity functions in this study , including the johnson - cousins - glass ubvir system and the hst / wfc3 system . the absolute magnitudes are calculated in the ab magnitude system . for a given extinction , @xmath3 , the extinction curve can be calculated using the fitting formula in @xcite . the stellar parameters of ms- and he - sc models in our previous work are summarized in table [ tab_models ] . figures [ fig_hr_all ] and [ fig_hr_bvi ] illustrate the hertzsprung - russell ( h - r ) diagram in different representations using the b and v wavebands and hst wavebands respectively . he - scs are sdo - like stars that exhibit stronger emission in the @xmath4 and @xmath5 bands , while ms - scs exhibit greater emission in the @xmath6 and @xmath7 bands ( a - k subgiants ) . the absolute magnitudes of ms - scs ( he - scs ) span the range @xmath8 @xmath9 . the brightest phase of ms - scs corresponds to ia snr ages of @xmath10 yr , which are similar to the ages of most known historic nearby ia snrs ( see table [ tab1 ] ) . therefore , if these snrs originated from normal sne ia via the sds ms or the he star channels , scs should be detectable . in the rg donor channel , almost the entire envelope of a rg should be removed during the sn impact ( @xcite ; p12a ) . therefore , the sc in ia snr will no longer be a giant star , but could be a helium degenerate core star with a shallow hydrogen - rich envelope . it should be noted that in our calculations , we assume normal sne ia with chandrasekhar mass explosions , and the explosions are initially spherically symmetric . asymmetric explosions , sub - chandrasekhar , or super - chandrasekhar mass explosions may reduce or enhance the evolution of post - impact luminosity ( p12b ) . in addition to comparisons of the colors and magnitudes of sc candidates , the long term variation of magnitude and effective temperature changes is an alternative way to identify the sc . in figure [ fig_hr_all ] we see that ms - scs show rapid luminosity changes but maintain similar effective temperatures in the early few hundred years after an explosion . therefore , a slight magnitude drop but without color change for a few years after the first observation is expected for ms - scs . on the other hand , he - scs show both magnitude and color changes in the first ten years . figure [ fig_mag ] and figure [ fig_teff ] show the magnitude and effective temperature variations of our ms- and he - scs as functions of time . brighter sc models ( models a , b , and d ) have a maximum magnitude change rate of 0.3 mag yr@xmath11 in the v - band when their ia snrs have ages less than five hundred years . for models c , e , and f , the ms - scs show a smoother rate of change of magnitude during the first thousand years , but the maximum rate is less than 0.01 mag yr@xmath11 , which is very difficult to detect with current optical telescopes . model g radiates the deposited energy immediately and shows a positive magnitude gradient ( gets dimmer ) . the rate of change of magnitude for model g is less than @xmath12 mag yr@xmath11 . the effective temperatures of all our ms - sc models are roughly constants . the highest rate of change is about @xmath13 k yr@xmath11 for models a , b , and d and @xmath14 k yr@xmath11 for other models . however , the rates of change of magnitude and effective temperature for he - scs could be notable . all our he - scs change by @xmath15 mag yr@xmath11 in the b - band during the first three years and maintain @xmath16 mag yr@xmath11 over the first decade . furthermore , he - scs increase in temperature by @xmath17 k yr@xmath11 during the first @xmath18 years and then decrease by @xmath19 k yr@xmath11 in the first decade . therefore , these magnitude and effective temperature changes could be detected in young and nearby ia snrs if the sc is a he star . a sc will have an abnormal speed due to its original orbital speed plus the sn kick . the momentum transfer between the sn ejecta with the donor star will give a kick velocity perpendicular to its orbital velocity . the kick velocity is about @xmath20 km s@xmath11 for ms and he star donor channels , and @xmath21 km s@xmath11 for the rg donor channel ( p10 ; p12a ; ) . by adding the kick velocity to the orbital velocity , the final linear speed at the end of supernova impact is @xmath22 km s@xmath11 for ms - scs , @xmath23 km s@xmath11 for he - scs ( see table [ tab_models ] ) , and @xmath24 km s@xmath11 for rg - scs ( p12a ) . the contribution from the kick velocity could be as high as @xmath25 in the ms donor channel . assuming the co wd has a similar orbital speed ( but opposite direction ) at the time of explosion and the snr is moving with this orbital speed , the observational error circle has to include a radius of at least @xmath26 , where @xmath27 is the age of the snr , and @xmath28 is the inclination angle . the theoretical estimate of the maximum @xmath29 is @xmath17 km s@xmath11 for the he star donor channel . the surface rotational speed of a sc also could be an important signature for sc searches in nearby ia snrs . at the time of the sn ia explosion , the companion should be synchronized by tidal locking and be rapidly rotating , resulting in a surface rotational speed close to @xmath30 km s@xmath11 for ms donors , @xmath31 km s@xmath11 for he star donors , and @xmath24 km s@xmath11 for rg donors ( p12a ; ) . however , because of the angular momentum loss accompanying mass stripping , the surface rotational speed could drop to @xmath32 km s@xmath11 for ms donors and @xmath33 km s@xmath11 for he star donors after the sn impact ( p12b ; p13 ) . once the sn ejecta escape and the sc reaches hydrostatic equilibrium , the rotational speed could keep decreasing or increasing as a result of post - impact contraction or expansion . if we assume the specific angular momentum is conserved during the post - impact evolution , the surface rotational speed could be in the range from @xmath34 km s@xmath11 to @xmath31 km s@xmath11 , depending on the amount and depth of sn energy deposition and the age of the snr ( see figure 11 in p12b and figure 12 in p13 ) . in p12b and p13 , we have shown that the surface rotation speed will first drop to @xmath35 km s@xmath11 after the sn impact and then keep decreasing to @xmath36 km s@xmath11 during the early expansion over a timescale of @xmath37 yr for ms - scs and @xmath38 yr for he - scs . the stellar expansion is due to the release of energy deposited by the sn ejecta . once the deposited energy has all been released , the envelope of the sc will contract , releasing its gravitational energy . in this phase , the rotational speed may increase up to @xmath38 km s@xmath11 for ms - scs and @xmath31 km s@xmath11 for he - scs . for most of the lifespan of ia snrs with ages less than a thousand years , ms - scs are slowly rotating subgiants but he - scs are rapidly rotating sdo / b stars . @xcite have suggested that the contamination from sn ejecta may provide observable features in iron absorption lines , which can be used to identify the scs in ia snrs . in p12a , we showed that the amount of ni / fe contamination at the surfaces of the scs is @xmath39 for ms star companions , @xmath40 for he star companions , and @xmath41 for rg companions . at the current stage , our simplified post - impact evolution method can not predict the strength of any absorption or emission lines during the evolution , but we can provide an order - of - magnitude estimate by assuming that these contaminated material are uniformly mixed in the stellar envelope . the estimated upper limit for the nickel - to - hydrogen - plus - helium ratio is about @xmath42 for ms - scs , @xmath43 for he - scs , and @xmath44 for rg - scs ( p12a ) . the rate of galactic sne ( including type i and type ii ) is about @xmath45 sne per century , and @xmath46 of them are sne ia @xcite . these rates suggest that there should be more than 2,500 snrs in our galaxy , and 300 of them are ia snrs , if snrs are recognizable for ages less than @xmath47 yrs . however , only @xmath48 galactic snrs have been identified @xcite , and only four of them are known as ( or likely to be ) ia snrs ( @xmath49 ) . therefore , a search for a sc in the central region of a snr could also be an alternative indirect method to identify ia snrs among currently - known snrs . table [ tab1 ] shows a summary of the four possible galactic ia snrs ( and/or ia snr candidates ) : sn 1006 , sn 1572 , sn 1604 , and rcw 86 . the young ( @xmath50 yr ) and nearby ( @xmath51 kpc , @xcite ) snr , tycho s snr , has been identified as a normal ia snr by its scattered - light echo @xcite . the non - thermal x - ray arc within tycho s snr could arise from an interaction between the sn ejecta and mass ejected from the companion star , giving support for the sds as the progenitor @xcite . @xcite found a subgiant star , namely tycho g , characterized by a high radial velocity @xmath52 km s@xmath11 that could be related to the original orbital speed of the white dwarf companion in the sds . * hereafter k13 ) recently updated this radial velocity to @xmath53 km s@xmath11 , which is still anomalous for the region . figure [ fig_snrs ] shows our predictions of post - impact conditions of ms - scs compared with observations by ( * ? ? ? * hereafter gh09 ) and k13 . gh09 reported that tycho g is a subgiant with effective temperature @xmath54 k , surface gravity @xmath55 dex , and [ fe / he ] @xmath56 dex , using keck high resolution optical spectra . the distance of tycho g is also comparable to that of tycho s snr . using the same data but with a slightly different analysis tool , k13 determined hotter and less luminous characteristics for tycho g ( @xmath57 k , @xmath58 dex , and [ fe / h]@xmath59 dex ; see k13 for a more detailed comparison of their results with gh09 ) . our model e ( @xmath60 k , @xmath61 dex , and @xmath62 ; see table 3 in p12b ) has a similar effective temperature to that of tycho g , but is brighter than the luminosity reported by gh09 ( @xmath63 , assuming a mass of @xmath2 ) . we note that our model e is more massive ( @xmath64 ) than the mass ( @xmath2 ) assumed in gh09 and k13 , and the stellar radius and evolution stage are also different with what is suggested in ( * ? ? ? * hereafter b14 ) . the evolution of scs after the sn explosion depends not only on the amount of mass lost and energy deposition , but also on the depth of energy deposition . therefore , a less massive companion , different evolutionary stage , asymmetric explosion , or sub - chandrasekhar mass explosion may better match tycho g. if tycho g is indeed similar to our model e , we predict a magnitude change of @xmath65 mag for every 10 years ( figure [ fig_mag ] ) . b14 determined high - accuracy proper motions for @xmath66 sc candidates , including tycho g , in the central region of tycho s snr . they obtained a tangential velocity for tycho g of @xmath67 km s@xmath11 . together with the observed radial velocity ( 80 km s@xmath11 ) of tycho g and the average radial velocity ( 37 km s@xmath11 ) in the direction of tycho s snr from the sun , the linear velocity with respect to the center of tycho s snr can be determined to be @xmath68 km s@xmath11 , if tycho g were the sc ( b14 ) . this value is half that of the linear velocities in our models in table [ tab_models ] . an inclination angle @xmath69 , orbital separation @xmath70 , and orbital period @xmath71 days are also suggested by b14 . these progenitor system data are consistent with the spectroscopic observation by gh09 indicating that tycho g is a g - type subgiant . however , we note that the linear velocity does not equal the orbital speed in the binary system , since the sn kick may contribute up to one - third of its final linear speed ( p12a ; ) . * hereafter k09 ) found an upper limit for the rotational speed of tycho g of @xmath72 km s@xmath11 , updated to @xmath73 km s@xmath11 in k13 , causing them to question tycho g as a sc candidate . p12a , p12b , and studied the sn impact and post - impact conditions of ms - like binary companions and found that this discrepancy can be resolved due to the loss of angular momentum during the sn impact and post - impact expansion . however , even when applying the inclination angle in b14 , our best model ( model e ) still has a @xmath74 km s@xmath11 , which lies above the upper limit observed by k09 and k13 . similar results have been found in as well . therefore , a less massive or less compact companion model is required to better fit with observations . fortunately , using the inclination angle and orbital period derived in b14 , a low rotational speed of @xmath75 km s@xmath11 can be calculated by assuming tidal locking , explaining the non - detection of rotational speed in k09 and k13 . on the other hand , @xcite comments that the absence of an fe i line feature at 372 nm in tycho g argues against the sc interpretation as there is no evidence for absorption due to the sn ejecta in the stellar spectrum . in contrast , tycho e shows a strong blueshifted fe i absorption line without redshifted lines , which implies that it is within the remnant and that its projected position is close to the center of tycho s snr , possibly qualifying it as another sc candidate . however , k13 suggests that tycho e is far behind the snr and hence the low column density on the receding side of the remnant could explain the lack of redshifted lines . the recent observation of tycho e reveals an effective temperature @xmath76 k and surface gravity @xmath77 dex ( k13 ) , which is also close to our model e ( @xmath60 k and @xmath61 dex ) . however , the low radial velocity ( @xmath78 km s@xmath11 ) and large distance ( @xmath79 kpc , in comparison to @xmath80 kpc ) make it less likely to be the sc , although the distance uncertainty is large . finally , gh09 also have suggested that tycho e could be a double lined binary , which does not provide support for it as a sc . further detailed observations of these sc candidates are necessary to establish whether any of them is connected to tycho s sn . the lack of a compact remnant star , and the amount of iron observed inside the snr @xcite indicate that sn 1006 was a sn ia . measurements of its proper motion indicate that sn 1006 is the closest historical ia snr with a distance @xmath81 kpc @xcite and an age of @xmath82 yr . the geometric center of the remnant is also well determined in x - ray and radio @xcite . furthermore , the foreground extinction is low due to its high galactic latitude ( @xmath83 ) , providing a good environment in which to search for sc candidates . searches for sc candidates in sn 1006 have been done using two independent observations by ( * ? ? ? * hereafter gh12 ) and ( * ? ? ? * hereafter k12 ) . both teams suggest that there is no evidence for the survival of a sc in the central region of sn 1006 . k12 observed all stars to a limit of 0.5 @xmath84 at the distance of sn 1006 within the central 2 arcminutes . they found no stars as bright as predicted in @xcite and @xcite , and no stars show significant rotation . similar conclusions have been drawn by gh12 using observations of the central 4 arcminutes . however , we note that the stars b90474 ( @xmath85 cm s@xmath86 , @xmath87 k ) and b14707 ( @xmath88 cm s@xmath86 , @xmath89 k ) in gh12 have surface gravities similar to our model g ( @xmath90 dex , @xmath91 k ) , although with lower effective temperatures ( see figure [ fig_snrs ] ) . furthermore , the star b90474 has a high radial velocity ( @xmath92 km s@xmath11 ) . however , unlike the case of tycho s sn , sn 1006 lies 500 pc above the galactic plane , and therefore , the radial velocities of surrounding stars do not exhibit a simple trend . in addition , b90474 has a larger distance uncertainty ( @xmath93 kpc ) and b14707 ( @xmath94 kpc ) lies at a much closer distance than the snr . in both gh12 and k12 , the authors use the geometric center of the x - ray and radio observations . the error circles included in gh12 and k12 are @xmath95 and @xmath96 respectively , corresponding to a sc moving with a speed of @xmath17 km s@xmath11 for 2,000 and 1,000 yr at a distance of 2.2 kpc . @xcite have suggested that there is an offset between the geometric center and the center of the iron core . furthermore , @xcite and @xcite have shown that the ejecta distribution in sn 1006 is asymmetric and concentrated in the se quadrant , suggesting a @xmath97 pc offset to the geometric center @xcite . @xcite reported that the expansion velocity varies significantly with azimuth ( @xmath98 km s@xmath11 in the nw and @xmath99 km s@xmath11 in the se ) . therefore , the error circles used in gh12 and k12 may not include the real explosion center . on the other hand , the schweizer - middleditch star is a subdwarf ob ( sdob ) star which is located at the center of sn 1006 and has strong fe absorption lines @xcite . it is relatively bright ( @xmath100 , @xmath101 ) , with low foreground extinction ( @xmath102 ; @xcite ) . it is consistent with the he - sc models ; however , the presence of redshifted absorption lines due to supernova ejecta suggests that it is more likely a background star @xcite . kepler s snr has been identified as a ia snr based on x - ray observations of its o / fe ratio @xcite , but its distance is still uncertain ( @xmath103 kpc ; ) . the interaction of the supernova ejecta with circumstellar material in kepler s snr provides evidence that kepler s snr may have originated from the sds in an evolutionary channel consisting of a wd and an asymptotic giant branch ( agb ) star . however , the circumstellar medium could be also explained by the stellar wind from a massive progenitor in core - collpase supernova . the low iron mass of @xmath104 @xcite and a progenitor mass of @xmath105 @xcite suggest that kepler s sn may not be a sn ia . recently , ( * ? ? ? * hereafter k14 ) have ruled out red giants as scs due to the lack of bright stars in the central snr . the observed radial velocity of sc candidates also shows only a small possibility of being associated with the sn explosion . however , 24 stars with @xmath106 at the center of kepler s snr have been found by k14 , and five of them have @xmath107 , perhaps suggesting a relatively higher probability of being a sc . additional observations are required to test these candidates . as we pointed out in 2 , nearly all the envelope of a rg should be removed during the sn impact in the rg donor channel , and this is likely to be the case for the agb donor channel as well . therefore , the sc in kepler s snr would cease to be a agb star , which would explain the non - detection of a giant star in k14 . follow - up observations and a detailed study of the evolution of the agb(rg ) channel in the sds and its scs will be necessary in order to understand the progenitor system of kepler s snr . rcw 86 is the oldest known galactic sn ia . it was recently identified as a ia snr by @xcite and @xcite , who estimate that the integrated fe - k emission corresponds to a total fe mass of about @xmath2 . its possible association with sn 185 is somewhat dubious since it would imply the age of rcw 86 is 1,829 yr . the large size of its radius at an estimated distance of @xmath108 kpc @xcite ( or @xmath109 kpc ; ) suggests a very high shock speed ( @xmath110 km s@xmath11 ) or a much older age . one explanation is that it originated in a cavity explosion , although cavity explosions are more common in core - collapse sne . hence , the connection between rcw 86 and sn 185 has yet to be established . in addition , the derived ambient density ( @xmath111 @xmath112 ) found by @xcite suggests that an unusually low - density cavity surrounds the snr . this can be understood either as resulting from the existence of a strong stellar wind from the progenitor itself or as an outflow from the nearby ob association discovered by @xcite . if rcw 86 was a member of this group , the age of this group may place some constraints on the delay - time of this sn ia , although the ages of these ob stars are still unknown . the wind - blown bubble scenario by @xcite suggests that rcw 86 originated in the sds . therefore , a search for sc candidates in rcw 86 seems reasonable . however , the two expansion fronts in the southwest and the northeast make it difficult to measure the center of the explosion ( k14 ) . in our calculations , all ms - sc models reach the brightest phase ( @xmath113 ) and highest effective temperature at around @xmath114 yr , depending on the model . therefore the nearby distance and low extinction ( @xmath115 ; @xcite ) make it easier to distinguish sc from unrelated stars . the magellanic clouds ( mcs ) are excellent environments to search for scs for several reasons . ia snrs in the large magellanic cloud ( lmc ) and small magellanic cloud ( smc ) are at known distances , 50 kpc and 60 kpc , that are sufficiently close to detect scs with hst based upon the results in 2 . the mcs are nearly face - on galaxies , minimizing confusion along the line of sight @xcite . the foreground galactic extinction is low , and the internal extinction of the mcs is modest @xcite . finally , the star formation history ( sfh ) and metallicity of the mcs are different than those of the milky way @xcite , allowing the possibility of testing sn ia progenitors in different type of galaxies @xcite . ten ia snrs have been reported to lie in the lmc and four in the smc ( see table [ tab1 ] ) . since only four galactic ia snrs have been identified , these fourteen ia snrs provide a larger sample of ia snrs to search for scs . in addition , the distances of ia snrs in the lmc and smc are known more accurately than for galactic ia snrs , which reduces the uncertainties in comparing observations with the theoretical predictions . recently , the youngest ia snr in the lmc , snr 0509 - 67.5 , has been studied by @xcite . snr 0509 - 67.5 formed from a sn 1991t - like sn ia @xmath116 yrs ago . the observation of the central region of snr 0509 - 67.5 within a radius of @xmath117 , corresponding to a distance of 0.36 pc from the center of snr 0509 - 67.5 , shows no star brighter than @xmath118 ( @xmath119 ) in this region . this result rules out all standard sds channels and suggests that snr 0509 - 67.5 originated in the dds @xcite . @xcite studied the central sources within an error circle of @xmath120 radius about the center the @xmath121 yr old ia snr 0519 - 69.0 using hst . they found 27 ms stars brighter than @xmath122 magnitude . this result requires the progenitor of snr 0519 - 69.0 to arise from either the dds or the sds with a supersoft source . we point out that @xcite found a sub - giant star in snr 0519 - 69.0 ( @xmath123 mag , and @xmath124 ) . although this star is consistent with our model g star ( @xmath125 mag , and @xmath126 ; the black line in figure [ fig_snrs ] ) , it lies close to the edge of the possible error circle , which suggests that it may not be the sc of snr 0519 - 69.0 unless the explosion was asymmetric . besides the lmc and smc , other nearby galaxies can provide samples that probe diverse environments for scs . in particular , the metallicities and sfhs in other nearby galaxies differ from those of the milky way and magellanic clouds , which could affect pre - sn conditions , the sn ia explosion itself , and , potentially , the post - impact evolution . in section [ sec_predictions ] , we have shown that our predicted ms - scs ( he - scs ) span a range of absolute magnitude @xmath8 ( @xmath127 ) in f555w ( f438w ) . by using hst / wfc3 ( with u , b , v , and i filters ) , the limiting magnitude with s / n of 10 for point sources is @xmath128 with a one hour exposure , and @xmath129 for a 10 hour exposure , giving maximum distance moduli of @xmath130 and @xmath131 respectively . this corresponds to a maximum detectable distance of @xmath0 mpc ( @xmath1 mpc ) if there is no extinction . however , as extinction varies from galaxy to galaxy , the maximum detectable distance would be smaller in extreme cases . given that he - scs reach their maximum brightness at @xmath38 yr after sn explosions and then fade within @xmath30 yr ( p13 ) , their @xmath5 magnitudes are likely less than 1 for historic ia snrs . this yields a distance limit of @xmath132 mpc . however , the short timescale ( @xmath38 yrs ) of their luminous phase provides the possibility to detect slow transitions of their brightness at the locations of recent nearby sne ia with @xmath133 mpc . we note that the timescale for the decay of the light curve from the supernova may take longer than the luminous phase of the sc and , hence , that the maximum detectable distance could be smaller . snrs in nearby galaxies , including the andromeda galaxy ( m31 or ngc 224 ; ) , ngc 300 @xcite , m82 @xcite , ngc 4214 , ngc 5204 , ngc 5585 , ngc 6946 , m81 , and m101@xcite , have been studied recently . therefore , ia snrs in these galaxies should be targeted for sc searches . however , it is unclear whether these snrs are ia snrs . more than 26 snrs ( including type i and type ii ) have been identified in the andromeda galaxy at 0.79 mpc by @xcite and . m31 has a distance modulus of 24.5 , which makes it possible to detect all of our ms - scs and most he scs using hst / wfc3 . the panchromatic hubble andromeda treasury ( phat ) multi - cycle program has observed about one third of m31 using hst @xcite . its uvis data reached a magnitude limit of @xmath134 in the f275w and f336w bands ; acs data reach maximum depths of @xmath135 magnitudes in f475w and @xmath128 magnitudes in f814w in the uncrowded outer disk . in these same regions , wfc3/ir data reach maximum depths of @xmath136 and @xmath137 in f110w and f160w , respectively . have listed 26 x - ray snrs and 20 x - ray snr candidates in m31 based on their x - ray , optical , and radio emission , which is the most recent complete list of x - ray snrs in m31 . therefore , using their list together with phat s data , it may be possible to identify sc candidates . sn 1885a ( s andromedae ) is a subluminous sn ia in m31 @xcite . its young age and the small size of its remnant make it easier to search for a sc . however , the surrounding light is dominated by the bulge of m31 , making sc searches difficult unless the sc is overluminous . sn 2014j was discovered by stephen fossey and his students on january 21 , 2014 and reported as a sn ia on january 22 , 2014 @xcite . sn 2014j is the closest sn ia ( @xmath138 mpc ; @xcite ) discovered in the past 42 years . the next close one is sn 1972e ( in ngc 5253 ) with a distance of @xmath139 mpc . sn 2011fe is also a nearby sn ia discovered in the modern astronomical era . however , the distance of sn 2011fe ( in m101 ; @xmath140 mpc , @xcite ) is above our predicted maximum distance for ms - scs . a search for scs in sn 2011fe will be very challenging . furthermore , the emptiness of the surrounding medium @xcite and the lack of pre - explosion optical and x - ray source @xcite suggest a dds progenitor for sn 2011fe . sn 2014j is almost twice as close as sn 2011fe , and the distance ( 3.5 mpc ) is within our predicted detection limit ( 4 mpc ) for ms - scs , though the distance is close to the limit . hst archival observations do not rule out sds progenitors such as recurrent novae or some classical novae @xcite . however , even when the snr becomes transparent , any ms - sc would need a few hundred years to become bright enough to be detected by hst . if the sc is a he - sc , a bright luminous ob - like star is expected to be observed in the next few decades . we have determined the colors and magnitudes of seven ms- and four he - sc models based on our stellar evolutionary calculations ( p12b , p13 ) and predicted their rates of change as functions of time . we have also studied the linear and rotational speeds of scs and the potential ejecta contamination of scs . comparisons of the model predictions for scs with the galactic sne 1572 and 1006 are presented , assuming all the candidate stars have the same distance as their host snrs . in particular , it is found that both tycho e ( @xmath76 k , @xmath77 dex ) and tycho g ( @xmath54 k , @xmath55 dex ) approximately fit our predicted ms - sc model e ( @xmath60 k , @xmath61 dex ) , although there are small discrepancies ( k13 ) . furthermore , two sub - giants in the error circle of sn 1006 are also found to have magnitudes consistent with our ms sc model g ( @xmath141 k , @xmath90 dex ) , but have lower effective temperatures @xcite . however it should be noted that the uncertainties in the distances of these sc candidates are large , and that the derived effective temperatures and surface gravities are not consistent in different papers . in addition to the galactic ia snrs , a sub - giant in the snr 0519 - 69.0 is consistent with the magnitude and color of our model g , but the projected position in the snr is too far from the center @xcite . if such a candidate is a confirmed sc , then the original orbital speed ( plus sn kick ) would be much higher than expected unless the explosion was asymmetric . although higher orbital speeds would be expected for he scs , there is no evidence for such a candidate . based on the current sample of sne ia companion searches , it is more likely that most have originated from the dds or peculiar sds channels . peculiar sds channels such as the m - dwarf channel @xcite or the spin up / down channel @xcite may explain the non - detection of sc candidates in ia snrs . to obtain a better understanding of the progenitor systems of sn ia , we encourage further sc searches in other galactic or nearby extragalactic ia snrs . unlike galactic ia snrs , which have large distance uncertainties , the distances of extragalactic ia snrs are relatively well - known . we predict that the maximum detectable distance of ms - scs ( he - scs ) is @xmath0 mpc ( @xmath1 mpc ) , if the apparent magnitude limit is 27 with no extinction , suggesting that the lmc , smc , and m31 are excellent targets in which to search for scs . furthermore , we also predict he - scs will not only show high luminosity and effective temperature ( luminous ob - like stars ) , but also show a rate of magnitude in b of @xmath142 mag yr@xmath11 and a rate of change of effective temperature of @xmath143 k yr@xmath11 in the first decade since explosion . future observations of he - sc candidates should look into their magnitude and color changes as well . similar analysis can be applied to core - collapse ( cc ) sne as well . the observed high fraction of sn ib / c can not be explained by single stellar evolution , suggesting that a high fraction of ccsne were in binary systems @xcite . however , the ejected - mass - and - energy and companion types and separations are very different from those in sn ia . in addition , the companion does not need to be in roche lobe overflow at the time of explosion . therefore , the evolution of scs in ccsne could be very different and vary from case to case . the impact of ccsne on binary companions and their subsequent evolution are important future work in this area . we thank the anonymous referee for his / her valuable comments and suggestions . kcp thanks you - hua chu for useful discussions about ia snrs in the lmc . this work was supported by the computational science and engineering ( cse ) fellowship at the university of illinois at urbana - 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67.5 & @xmath151 & 50 ( lmc ) & 3.6 + n103b & @xmath152 & 50 ( lmc ) & 3.6 + b0519 - 69.0 & @xmath153 & 50 ( lmc ) & 3.9 + dem l71 & @xmath154 & 50 ( lmc ) & 8.6 + b0548 - 70.4 & @xmath155 & 50 ( lmc ) & 12.5 + dem l316a & ? & 50 ( lmc ) & 15 + b0534 - 69.9 & @xmath156 & 50 ( lmc ) & 16 + dem l238 & @xmath157 & 50 ( lmc ) & 21 + dem l249 & @xmath157 & 50 ( lmc ) & 23 + b0454 - 67.2 & @xmath158 & 50 ( lmc ) & 27 + ikt 4 & ? & 60 ( smc ) & @xmath159 + ikt 5 & ? & 60 ( smc ) & 15 + ikt 25 & ? & 60 ( smc ) & 18 + dem s128 & ? & 60 ( smc ) & 26 + | the nature of the progenitor systems of type ia supernovae is still unclear .
one way to distinguish between the single - degenerate scenario and double - degenerate scenario for their progenitors is to search for the surviving companions . using a technique that couples the results from multi - dimensional hydrodynamics simulations with calculations of the structure and evolution of main - sequence- and helium - rich surviving companions , the color and magnitude of main - sequence- and helium - rich surviving companions
are predicted as functions of time .
the surviving companion candidates in galactic type ia supernova remnants and nearby extragalactic type ia supernova remnants are discussed .
we find that the maximum detectable distance of main - sequence surviving companions ( helium - rich surviving companions ) is @xmath0 mpc ( @xmath1 mpc ) , if the apparent magnitude limit is 27 in the absence of extinction , suggesting that the large and small magellanic clouds and the andromeda galaxy are excellent environments in which to search for surviving companions .
however , only five ia snrs have been searched for surviving companions , showing little support for the standard channels in the singe - degenerate scenario . to better understand the progenitors of type ia supernovae
, we encourage the search for surviving companions in other nearby type ia supernova remnants . |
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phase - field models have been introduced by g.caginalp in @xcite ( see also e.g. @xcite for generalizations ) as a relaxation of stefan problems where the development of the sharp interface between phases should be found exactly . in contrast to that , phase field models operate with phase functions that assume values from -1 ( solid state ) to 1 ( liquid state ) at each spatial point and change sharply but smoothly over the solidification fronts so that the phase interfaces become smoothed . mathematically speaking , such models consist of two coupled parabolic equations describing the temperature and phase fields satisfying initial and boundary conditions . phase field models are frequently used to describe processes of melting , solidification , evaporation , and condensation , which is directly related to applications such as metal casting , design of cooling systems , cryopreservation of living tissues etc . ( see e.g. @xcite ) . phase field models are also appropriate for the description of phase changes when modeling co@xmath2 sequestration . it is assumed that the supercritical carbon dioxide , co@xmath2 pressured to a phase between gas and liquid , is injected into a saline aquifer where it may either dissolve in the brine , react with the dissolved minerals and the surrounding rocks , or become trapped in the pore space of the aquifer . in this relation , phase field models are useful for the description of transitions between supercritical , ordinary , and dissolved co@xmath2 phases . in this paper , we study the case where a material subjected to a phase change is located inside of a container with heat conductive walls that are not subjected to phase changes . for example , this can be an ampoule filled with a liquid containing living cells ( see ( * ? ? ? 2.2 ) ) and subjected to cooling applied to the outer surface but not immediately to the liquid to be frozen . the paper is structured as follows . section [ sec : model ] presents the mathematical model . the definition of weak solutions and formulation of main results are given in section [ sec : main : res ] . the existence of weak solutions is shown in section [ sec : galerkin ] . in section [ sec : energy ] , continuity in time of the phase function and the validity of an energy equality are established . the uniqueness of weak solutions and their continuous dependence on the initial and boundary data are proved in section [ sec : uniq ] . first , the phase field model proposed by g.caginalp in @xcite will be recalled , then generalizations considered in this paper will be introduced . let us outline the model proposed by g.caginalp in @xcite . assume that a material subjected to phase changes , e.g. liquid to solid and back , occupies a region @xmath3 . the evolution of the system is described in terms of the temperature @xmath4 and a phase function @xmath5 satisfying the following system of non - linear parabolic equations : @xmath6 the first equation of expresses the balance of heat energy . notice that this equation is scaled so that the term @xmath7 appears without any multiplier . here , @xmath8 is the scaled heat conductivity coefficient which is assumed to be the same in the solid and liquid states , and @xmath9 is the scaled specific latent heat of the phase change . the second equation of is derived from statistical physics , namely from the landau - ginzburg theory of phase transitions @xcite . in equilibrium , @xmath5 minimizes the following free energy functional : @xmath10,\ ] ] where @xmath11 is a length scale , the thickness of the interfacial region between liquid and solid . in non - equilibrium , @xmath5 does not minimize @xmath12 but satisfies the gradient flow equation @xmath13 , where the symbol @xmath14 denotes the variation of @xmath15 in @xmath5 , which yields the second equation of . it is clear that the constant @xmath16 characterizes the relaxation time . the following boundary conditions are usually imposed ( see e.g. @xcite ) : @xmath17 where @xmath18 denotes the outer normal to @xmath19 , and @xmath20 is the scaled overall heat conductivity . the specific of the case considered in this paper is that the temperature @xmath21 is not directly applied to the boundary of the liquid but to the outer surface of the container ( ampoule ) . to include the container into the model , still denote the inner part occupied by the medium subjected to the phase change by @xmath19 , and the solid walls of the container by @xmath22 . moreover , let @xmath23 be the region occupied by the whole system . in this case , the phase function @xmath5 is defined in @xmath19 , the temperature @xmath4 is defined in @xmath24 , and the boundary function @xmath21 is defined on @xmath25 . thus , the modification of the model and reads as follows : @xmath26 u_t - k_d \ , \delta u = 0 & \qquad \text { in } d\times(0,t ) , \\ -k_d \ , \partial_{{{\boldsymbol{\nu}}}}u = \lambda(u - g ) & \qquad \text { on } { { \partial u}}\times(0,t ) , \\[1ex ] u({{\boldsymbol{x}}},0)=u^0({{\boldsymbol{x } } } ) & \qquad \text { in } u , \\[1ex ] \phi({{\boldsymbol{x}}},0 ) = \phi^0({{\boldsymbol{x } } } ) & \qquad \text { in } \omega . \end{aligned}\ ] ] here the indices @xmath22 and @xmath19 denote restrictions to the domains @xmath22 and @xmath19 , respectively , and the symbol @xmath18 is used to indicate outward normals as well to @xmath19 as to @xmath24 , whenever it is not ambiguous . note that the matching conditions imposed on @xmath4 in mean , in particular , the continuity of the temperature and the heat flux across @xmath27 . the main result of this paper is that the problem admits a unique weak solution @xmath28 , and the mapping @xmath29 ; { l^2(u ) } ) \times \mathcal c([0,t ] ; { h^1(\omega ) } ) \end{split}\ ] ] is continuous provided that @xmath24 and @xmath19 are bounded lipschitz domains in @xmath30 , @xmath31 . a precise formulation of this result is given in the next section , see definition [ def : wsol ] and theorem [ thm:2reg ] . problem will be studied in a weak formulation . denote @xmath32 and introduce the following spaces according to @xcite and @xcite : @xmath33 @xmath34 ; { h^1(\omega ) } ) : = \big\ { & \eta\in l^\infty(0,t , { h^1(\omega ) } ) : ~~ t\rightarrow \left<\eta(t ) ; \xi\right>\\ & \mbox{is continuous on~~ } [ 0,t ] \mbox{~~for each~~ } \xi\in ( { h^1(\omega ) } ) ^\prime \big\ } , \end{split } \label{eq : def : cs}\ ] ] where @xmath35 denotes the dual pairing between @xmath36 and @xmath37 . a pair @xmath28 of functions @xmath38 satisfying the initial conditions @xmath39 is a weak solution of problem , if the identities @xmath40 \end{aligned}\ ] ] hold for all test functions @xmath41 and @xmath42 . [ def : wsol ] [ bemcs ] notice that the initial conditions in definition [ def : wsol ] have a sense because @xmath43 ; { l^2(u ) } ) \ ] ] and @xmath44 ; { h^1(\omega ) } ) , \ ] ] see ( * ? ? ? 25.5 ) for the first assertion and ( * ? ? ? 3 , lemma 8.1 ) for the second one . the next theorem states the main result of this paper . [ thm:2reg ] let @xmath24 and @xmath19 be bounded lipschitz domains in @xmath45 with @xmath46 , and @xmath47 be finite . if @xmath48 @xmath49 and @xmath50 , then 1 . there exists a unique weak solution @xmath28 of problem in the sense of definition [ def : wsol ] . 2 . the phase function @xmath5 has the additional regularity @xmath51 ; { h^1(\omega ) } \right)$ ] . 3 . for all @xmath52 $ ] , the following energy equation holds : @xmath53 \\ & \quad = { { \int}_\omega}\left [ \frac{1}{8}\phi(s)^4 - \frac{1}{4}\phi(s)^2 + \frac{\xi^2}2|\nabla\phi(s)|^2 \right ] + \int_s^t { { \int}_\omega}\phi_t \left [ 2u- \tau\phi_t \right ] . \end{aligned } \label{eq : energy}\ ] ] 4 . problem is well - posed in the sense that the mapping @xmath54 ; { h^1(\omega ) } ) \big ) \end{split}\ ] ] is continuous . the proof of theorem [ thm:2reg ] is given in the following sections . the existence of weak solutions of problem can be proved in the similar way as theorem 3.1.2 in @xcite , and therefore we only discuss changes in the construction of approximate solutions and a priori estimates . for details related to the passage to the limit , we refer to @xcite . the next lemma establishes some a priori estimates and the existence of weak solutions to problem . [ lem : exist ] let @xmath24 and @xmath19 be bounded lipschitz domains in @xmath55 , with @xmath46 , and @xmath47 . if @xmath48 @xmath49 and @xmath50 , then there exists a weak solution @xmath28 to problem in the sense of definition [ def : wsol ] . proof . let @xmath56 be a basis of @xmath57 which is orthonormal in @xmath58 , and @xmath59 a basis of @xmath36 which is orthonormal in @xmath60 . consider galerkin approximations of the form @xmath61 where the functions @xmath62 and @xmath63 are to be determined . let @xmath64@xmath65;l^2({{\partial u}}))$ ] be a sequence such that @xmath66 in @xmath67 as @xmath68 . to determine the functions @xmath69 and @xmath70 , we require that @xmath71 and @xmath72 satisfy the relations @xmath73 \\ & + \int_{\partial u } \lambda \left ( u^m(t)-g^m(t ) \right ) \psi^m , \\ 0 & = { { \int}_\omega}\left [ \tau \ , \phi^m_t(t ) - 2u^m(t ) + \frac{1}{2}\left ( ( \phi^m(t))^3-\phi^m(t)\right ) \right]\eta^m \\ & + { { \int}_\omega}\xi^2\nabla\phi^m(t ) \cdot \nabla\eta^m \end{aligned}\ ] ] for all test functions @xmath74 and @xmath75 . substituting ansatz and test functions @xmath76 and @xmath77 , @xmath78 , into equations yields the following system of ordinary differential equations for determining @xmath79 and @xmath80 : @xmath81 since the set @xmath82 is orthonormal in @xmath83 and the set @xmath84 is orthonormal in @xmath85 , equations can be rewritten as the following system of odes : @xmath86 \left [ \begin{matrix } \dot{{\boldsymbol{a}}}^m(t)\\ \dot{{\boldsymbol{b}}}^m(t ) \end{matrix } \right ] + \left [ \begin{matrix } { \boldsymbol{a}}^m({{\boldsymbol{a}}}^m(t),\,{{\boldsymbol{b}}}^m(t))\\ { \boldsymbol{b}}^m({{\boldsymbol{a}}}^m(t),\,{{\boldsymbol{b}}}^m(t ) ) \end{matrix } \right ] = { \boldsymbol{0}},\ ] ] where @xmath87 , @xmath88 ; @xmath89 denotes the @xmath90 identity - matrix ; and @xmath91 and @xmath92 denote the terms of which do not comprise the time derivatives of the unknown functions . initial conditions for @xmath93 and @xmath94 are specified as follows : @xmath95 where @xmath96 and @xmath97 denote the projectors onto the subspaces @xmath98 and @xmath99 in @xmath58 and @xmath36 , respectively . notice that , for each fixed @xmath100 , the functions @xmath101 and @xmath102 are analytic with respect to all their variables . thus , the theory of ordinary differential equations provides that , for each @xmath100 , there exist a nonempty time interval @xmath103 $ ] on which the initial value problem and admits a unique solution @xmath104 $ ] . to continue these solutions to any time interval , establish some independent on @xmath100 a priori estimates on @xmath71 and @xmath72 . first , substitute @xmath105 into the first equation of , integrate it over @xmath106 for some @xmath107 $ ] , and use young s inequality to obtain the estimate @xmath108 for @xmath109 . next , substitute @xmath110 into the second equation of and proceed as before to obtain @xmath111 \\ & \quad \le \frac \tau 2 { { \int}_\omega}|\phi^0|^2 + \frac{1}{2 } { { \int}_0^t}{{\int}_\omega}|\phi^m|^2 + 2 { { \int}_0^t}{{\int}_u}|u^m|^2 . \end{aligned } \label{eq : estm : phi}\ ] ] now , substitute @xmath112 into the second equation of and proceed as before to obtain @xmath113 \\ & \quad \le { { \int}_\omega}\left [ \frac{1}{8 } \ , |\phi^0|^4 + \frac{\xi^2 } 2 |\nabla\phi^0|^2 \right ] + \frac 1 4 { { \int}_\omega}|\phi^m(t)|^2 + \frac{1}{\epsilon } { { \int}_0^t}{{\int}_u}|u^m|^2 \end{aligned } \label{eq : estm : phit}\ ] ] for @xmath109 . choosing @xmath114 sufficiently small ; multiplying inequalities , , and by suitable constants ; adding the resulting inequalities ; and using the embedding @xmath115 yield the estimate @xmath116 \\ & \quad + { { \int}_0^t}{{\int}_u}|\nabla u^m|^2 + { { \int}_0^t}{{\int}_{{{\partial u}}}}|u^m|^2 + { { \int}_0^t}{{\int}_\omega}|\phi^m_t(t)|^2 \\ & \quad \le c \left [ 1 + { { \int}_0^t}{{\int}_\omega}|\phi^m|^2 + { { \int}_0^t}{{\int}_u}|u^m|^2 \right ] , \end{aligned } \label{eq : est : gron1}\ ] ] where the constant @xmath117 depends on @xmath16 , @xmath11 , @xmath9 , @xmath20 , @xmath118 , @xmath119 , @xmath120 , and @xmath114 but is independent on @xmath100 . by gronwall s inequality , the left - hand side of is bounded independently on @xmath100 , which along with the embedding @xmath121 , @xmath122 , implies the following assertions : @xmath123 where the bounds are independent on @xmath100 . this means that the approximate solutions can be continued to any interval @xmath124 $ ] keeping the above mentioned bounds . now , the proof of the lemma can be completed analogously to that of ( * ? ? ? in this section , two lemmas are proved . lemma [ lem : equal : energy ] states a slightly weaker than energy equality , which nevertheless implies the continuity in time of the phase function : @xmath126 ; { h^1(\omega ) } ) $ ] . lemma [ lem : phi3 t ] completes the proof of the energy equality . these results will be used in section [ sec : uniq ] to show the claimed uniqueness of weak solutions and their continuous dependence on the initial and boundary data . the proof of the next lemma is based on the techniques of ( * ? ? ? * chap 3 , sec 8.4 , lemma 8.3 ) . [ lem : equal : energy ] let @xmath28 be a weak solution considered in lemma [ lem : exist ] . then , for all @xmath127 $ ] , the following energy equality holds : @xmath128 = \frac2{\xi^2 } \int_s^t{{\int}_\omega}\phi_t \left [ 2u - \tau\phi_t - \frac{1}{2 } \left(\phi^3 - \phi \right ) \right ] . \end{aligned } \label{eq : energy : weak}\ ] ] moreover , equality implies that @xmath129;h^1(\omega)\right)$ ] . assume that holds . notice that the assertions of provide the integrability of the integrand in the right - hand side of and therefore the convergence of the integral to zero as @xmath130 , which proves the continuity of the function @xmath131 . this , along with the inclusion @xmath132 ; { l^2(\omega ) } ) $ ] provided by the third and forth assertions of , implies the continuity of the function @xmath133 . let @xmath134 $ ] be fixed , and @xmath135 as @xmath136 . denote @xmath137 and compute @xmath138 . utilizing that @xmath139;h^1(\omega))$ ] ( see and remark [ bemcs ] ) implies the convergence @xmath140 , which means that @xmath141 in @xmath142 . now go on to the proof of equality and introduce some notation . similar to @xcite , denote the inner product in @xmath60 or the dual pairing in @xmath143 by @xmath144 , and denote the inner product in @xmath145 or the dual pairing in @xmath146 by @xmath147 . assume that @xmath148 is defined on @xmath149 and possesses the same properties as @xmath148 on @xmath124 $ ] . this can be achieved trough continuation of @xmath148 by means of reflections . in the following , set @xmath150 and @xmath151 in equation . define a bilinear form @xmath152 and an operator @xmath153 by @xmath154 then the second equation of is equivalent to the relation @xmath155 \eta . \label{eq : weak : a}\ ] ] for @xmath156 $ ] and @xmath157 , define the function @xmath158 $ ] by @xmath159 \\ 0 \quad & \mathrm{for } \,\ , t\notin [ 0,t_0 ] \\ \text{linear on } & [ 0,\delta]\cup[t_0-\delta , t_0]\,\ , \text { and continuous on } \mathbb{r}. \end{cases}\ ] ] let @xmath160 be a sequence of non - negative even regularizing functions with @xmath161 and @xmath162 . denote @xmath163 . moreover , as it was already mentioned ( see remark [ bemcs ] ) , @xmath164 ; { h^1(\omega ) } ) . \label{eq : phi : scalcont}\ ] ] since @xmath165 is a symmetric operator , it holds @xmath166 notice that @xmath167 because of the second equation of and the assertions of . this implies that @xmath168 since @xmath169 and @xmath170 depend only on time , and @xmath165 is time independent . therefore , all duality brackets on the right - hand side of equation present the inner product of @xmath171 . consider the passage to the limit in each term of the right - hand side of as @xmath172 and then as @xmath173 . the objective is to show the following three relations : @xmath174 , \label{term1lim } \\[2ex ] & \left| \left ( \rho_n \ast a ( q_\delta - q_0 ) \phi \ , ; \ , \rho_n \ast q_\delta ' \phi \right ) \right| \longrightarrow 0 , \label{term2lim } \\[2ex ] & 2 \left ( \rho_n \ast ( a \ , q_0 \ , \phi ) \ , ; \ , \rho_n \ast q_\delta ' \phi \right ) \longrightarrow \left < a\ , \phi(0 ) \ , ; \ , \phi(0 ) \right > - \left < a\ , \phi(t_0 ) \ , ; \ , \phi(t_0 ) \right > , \label{term3lim } \end{aligned}\ ] ] as first @xmath172 , and then @xmath173 , which along with equation completes the proof of the lemma . _ proof of _ : consider the limit as @xmath175 . by properties of convolutions , it holds @xmath176 because these functions have compact supports in @xmath177 . moreover , @xmath178 in @xmath171 , and therefore , the first term of the right - hand side of the last equation of satisfies @xmath179 as @xmath172 . relation , the second equation of , and time independence of @xmath165 imply that @xmath180 \end{aligned}\ ] ] as @xmath172 . moreover , note that @xmath181 in @xmath171 as @xmath173 , because @xmath182 . therefore , relations and yield . _ proof of _ : using hlder s inequality yields @xmath183 using again the relation @xmath167 and the properties of the convolution operator , one can show that the first factor on the right - hand side of estimate tends to zero . the second factor on the right - hand side of can be estimated using young s inequality as follows : @xmath184 the right - hand side is bounded , because @xmath185 , @xmath186 , and @xmath187 $ ] . thus , the right - hand side of tends to zero as @xmath188 , and relation is proved . _ proof of _ : notice that @xmath189 ; { l^2(\omega ) } ) $ ] because @xmath190 . thus , the function @xmath191 is continuous since @xmath192 ; { l^2(\omega ) } ) $ ] . accounting for the evenness of @xmath169 , the left - hand side of admits the following passage to the limit as @xmath188 : @xmath193 it remains to show that @xmath194 as @xmath173 . consider the first relation of . by the definition of @xmath170 and @xmath195 , we have @xmath196 and @xmath197 because @xmath195 is even . from the definition of the operator @xmath165 ( see ) , it follows that @xmath198 because the function @xmath199 is continuous , see . thus , lemma [ lem : equal : energy ] is proved . @xmath125 the next lemma proves an integral form of the relation @xmath200 , the chain rule . [ lem : phi3 t ] if @xmath201 ; { h^1(\omega ) } ) $ ] , then @xmath202 = \frac{1}{4 } { { \int}_\omega}\left [ |\phi(t)|^4 - |\phi(s)|^4 \right ] - \frac{1}{2 } { { \int}_\omega}\left [ |\phi(t)|^2 - |\phi(s)|^2 \right ] \label{eq : phi : poly}\ ] ] for all @xmath203 $ ] . [ lem : poly : chain ] proof . consider the case @xmath150 . since @xmath204 , it holds @xmath205 , \label{eq : phi2:aim}\ ] ] see e.g. ( * ? ? ? * th . 1.67 ) . to show the relation @xmath206 , \label{eq : phi4:aim}\ ] ] notice that @xmath207;{l^ { 4 } ( \omega)})$ ] , which follows from ( * ? ? ? 8 , cor 4 ) and the compact embedding @xmath208 for @xmath122 . let @xmath209 be a sequence of smooth functions such that @xmath210 ; { h^1(\omega ) } ) \quad \text{as } \epsilon \to 0 . \label{eq : phi4:approx}\ ] ] the chain rule yields @xmath211 . \label{eq : phi4:eps}\ ] ] the passage to the limit on the both sides of equation , as @xmath212 , has to be done . the left - hand side is being processed as follows : @xmath213 = { { \int}_0^t}{{\int}_\omega}\left [ \left ( \phi_\epsilon^3 - \phi^3 \right)\ , ( \phi_\epsilon)_t + \phi^3 \left ( ( \phi_\epsilon)_t - \phi_t \right ) \right ] . \end{aligned } \label{eq : phi4:diff}\ ] ] the first summand in the integral on the right - hand side can be estimated using hlder s inequality and the formula @xmath214 with @xmath215 . it holds @xmath216^{1/3 } \cdot \left [ { { \int}_\omega}\left| \zeta^3 \right| \right]^{3/2 } \right\ } { { \mathrm{d}}t}\cdot { { \int}_0^t}{{\int}_\omega}\left| ( \phi_\epsilon)_t \right|^2 \\ & \quad \le c { \left\| \phi_\epsilon - \phi \right\| _ { { l^ { \infty } ( 0,t ; { h^1(\omega ) } ) } } } ^2 \cdot { \left\| \zeta \right\| _ { { l^ { 9/2 } ( 0,t ; { l^ { 3 } ( \omega ) } ) } } } ^{9/2 } \cdot \|(\phi_\epsilon)_t\|^2_{l^2(q_t)}. \end{aligned } \label{eq : phi4:term1}\ ] ] it is not hard to prove tat @xmath217 . really , young s inequality and the embedding @xmath218 yield the estimate @xmath219 ^ 3 \\ & \le { { \int}_\omega}\phi_\epsilon^6 + { { \int}_\omega}\phi^6 + c \sum_{k=1}^5 { { \int}_\omega}\left| \phi_\epsilon^k \ , \phi^{6-k } \right| \\ & \le { { \int}_\omega}\phi_\epsilon^6 + { { \int}_\omega}\phi^6 + c \sum_{k=1}^5 \left [ \frac{k}{6 } { { \int}_\omega}\phi_\epsilon^6 + \frac{6-k}{6 } { { \int}_\omega}\phi^{6 } \right ] \\ & \le c \left [ { \left\| \phi \right\| _ { { l^ { \infty } ( 0,t ; { h^1(\omega ) } ) } } } ^{6 } + { \left\| \phi_\epsilon \right\| _ { { l^ { \infty } ( 0,t ; { h^1(\omega ) } ) } } } ^{6 } \right ] . \end{aligned } \label{eq : zeta : l3}\ ] ] thus , estimates and yield @xmath220 consider the second summand in the integral on the right - hand side of equation . processing it similarly to the first term yields @xmath221 now , estimates and along with equation yield @xmath222 \longrightarrow 0 \qquad \text{as } \epsilon\to0 . \label{eq : phi4:diff : lim}\ ] ] using the approximation property , and the embedding @xmath218 yields the estimate @xmath223 = { { \int}_\omega}\left [ \left ( \phi(t ) - \phi_\epsilon(t ) \right ) \zeta(t ) \right ] \\ & \quad \le { \left\| \phi(t ) - \phi_\epsilon(t ) \right\|_{l^ { 2 } ( \omega ) } } \cdot { \left\| \zeta(t ) \right\|_{l^ { 2 } ( \omega ) } } \\ & \quad \longrightarrow 0 \end{aligned } \label{eq : phi4:diff4:lim}\ ] ] for all @xmath224 $ ] as @xmath212 . here we have used the abbreviation @xmath225 ; { l^2(\omega ) } ) $ ] . thus , equation follows from equation and relations and , which proves lemma [ lem : phi3 t ] . @xmath125 finally , the energy equality follows from lemmas [ lem : equal : energy ] and [ lem : poly : chain ] . the next lemma completes the proof of theorem [ thm:2reg ] . let @xmath24 and @xmath19 be bounded lipschitz domains in @xmath45 with @xmath46 , and @xmath47 . let @xmath226 , @xmath227 , and @xmath228 be given initial and boundary data of problem ; @xmath229 weak solutions of problem in the sense of definition [ def : wsol ] . if @xmath230 and @xmath231 , then @xmath232 ; { l^2(u ) } ) } \\ & { \left\| \bar u \right\| _ { { l^ { 2 } ( 0,t ; { h^1(u ) } ) } } } \\ & \|\bar \phi\|_{\mathcal c([0,t ] ; { h^1(\omega ) } ) } \\ & \|\bar \phi\|_{h^1(0,t ; { l^2(\omega ) } ) } \end{aligned } \right\ } \le f\big ( \|\bar u^0\|_{l^2(u ) } , \ , \|\bar\phi^0\| _ { { h^1(\omega ) } } , \ , \|\bar g\|_{l^2({{\partial u}}\times(0,t))}\big ) , \label{eq : cont}\ ] ] where @xmath233 is a continuous function with @xmath234 . [ lem : uniq ] proof . denote @xmath235 and bear in mind that @xmath236 is non - negative . obviously , @xmath237 and @xmath238 satisfy the equations @xmath239 \end{aligned}\ ] ] for all test functions @xmath41 and @xmath240 . in particular , test functions @xmath241 , @xmath242 , and @xmath243 will be considered to obtain three estimates . here , @xmath244 denotes the characteristic function of the interval @xmath106 . notice that the proofs of lemmas [ lem : equal : energy ] and [ lem : poly : chain ] show that @xmath243 is an admissible test function . substituting @xmath245 into the first equation of and using young s inequality yield the estimate @xmath246 \end{aligned}\ ] ] for any @xmath109 . substituting @xmath242 into the second equation of and applying young s inequality yield @xmath247 \\ & \quad \le \frac \tau 2 { { \int}_\omega}|\bar\phi^0|^2 + { { \int}_0^t}{{\int}_\omega}\left [ |\bar u|^2 + \frac 3 2 |\bar\phi|^2 \right ] . \end{aligned}\ ] ] substituting @xmath243 into the second equation of and applying young s inequality yield @xmath248 + \int_0^{t}{{\int}_\omega}\frac{\tau}{4 } \ , |\bar\phi_t|^2 \\ & \quad \le { { \int}_\omega}\left [ \frac{1}{4}|\bar\phi(t)|^2 + \frac{\xi^2}2|\nabla\bar\phi^0|^2 \right ] + { { \int}_0^t}{{\int}_\omega}\left [ \frac 2 \tau \ , \bar u^2 + \frac 1 { 4\tau } \ , \zeta^2\ , \bar\phi^2 \right ] . \end{aligned}\ ] ] using hlder s inequality and the embedding @xmath218 yield @xmath249 \le { { \int}_0^t}\left\ { \left [ { { \int}_\omega}|\zeta|^3 \right]^{2/3 } \cdot \left [ { { \int}_\omega}|\bar\phi|^6 \right]^{3 } \right\ } \\ & \quad \le { \left\| \zeta \right\| _ { { l^ { \infty } ( 0,t ; { l^ { 3 } ( \omega ) } ) } } } ^2 { { \int}_0^t } { \left\| \bar\phi(t ) \right\|_{l^ { 6 } ( \omega ) } } ^2 \\ & \quad \le c { \left\| \zeta \right\| _ { { l^ { \infty } ( 0,t ; { l^ { 3 } ( \omega ) } ) } } } ^2 \cdot { { \int}_0^t}{{\int}_\omega}\left [ |\bar\phi(t)|^2 + |\nabla\bar\phi(t)|^2 \right ] . \end{aligned } \label{eq : zeta : bar}\ ] ] now , choose @xmath250 in and combine estimates , , , and to obtain @xmath251 \\ & \quad + c \left [ { { \int}_0^t}{{\int}_u}|\bar u|^2 + c_\zeta^2 { { \int}_0^t}{{\int}_\omega}\left [ |\bar\phi|^2 + |\nabla\bar\phi|^2 \right ] \right ] , \end{aligned } \label{eq : bar : est1}\ ] ] where @xmath252 , and the constant @xmath117 depends on @xmath253 and @xmath16 only . to eliminate the first term on the right - hand side of , multiply inequality by a constant greater than @xmath254 and add the resulting inequality to ( remember that @xmath236 is non - negative ) . this yields @xmath255 \\ & \quad + { { \int}_0^t}{{\int}_u}|\nabla\bar u|^2 + { { \int}_0^t}{{\int}_\omega}|\bar\phi_t|^2 + { { \int}_0^t}{{\int}_{{{\partial u}}}}|\bar u|^2 \\ & \quad \le c \left\ { c_0 + ( 1+c_\zeta^2 ) { { \int}_0^t}\left ( { { \int}_u}|\bar u|^2 + { { \int}_\omega}\left [ |\bar\phi|^2 + |\nabla\bar\phi|^2 \right ] \right ) \right\ } , \end{aligned } \label{eq : bar : est2}\ ] ] where @xmath117 is sufficiently large , and @xmath256 + { { \int}_0^t}{{\int}_{{{\partial u}}}}|\bar g|^2 . \label{eq : def : c0}\ ] ] applying gronwall s inequality to yields @xmath257 \\ & \quad + { { \int}_0^t}{{\int}_u}|\nabla\bar u|^2 + { { \int}_0^t}{{\int}_\omega}|\bar\phi_t|^2 + { { \int}_0^t}{{\int}_{{{\partial u}}}}|\bar u|^2 \\ & \quad \le c_0 \ , c \left [ 1 + t \ , c ( 1+c_\zeta^2 ) \exp \left ( t \ , c ( 1+c_\zeta^2 ) \right ) \right ] . \end{aligned } \label{eq : bar : est3}\ ] ] due to the a priori estimates derived in section [ sec : galerkin ] ( see the assertions of ) , the constant @xmath258 is bounded in terms of the problem data only . thus , estimate proves the continuous dependence of weak solutions on the problem data . in particular , inequality and the definition of @xmath259 ( see ) imply uniqueness of weak solutions.@xmath125 the following simulation deals with a plastic ampoule used for freezing small tissue samples ( see @xcite ) . the diameter of the ampoule is equal to 1 cm , the height equals 5 cm , and the wall thickness equals 0.1 cm . the ampoule is filled with water , and is being cooled with the rate of 1@xmath260c / s applied to the outer surface of the ampoule . the mass and thermal characteristics of plastic and water are easily available in reference books . the length scale @xmath11 and the relaxation time @xmath16 are equal to @xmath261 and @xmath262 , respectively . the first picture of figure [ fig1 ] shows an axial cut of the whole ampoule ( region @xmath24 ) . the others show the water part ( region @xmath19 ) only . the unfrozen part is shown in light , whereas the frozen one becomes dark . ( 9,5 ) ( 0,0 ) ( 0.2,0 ) in the light region , and @xmath263 in the dark one . there is a small transition zone between the unfrozen and frozen parts.,title="fig:",height=188 ] ( 3.0,0 ) in the light region , and @xmath263 in the dark one . there is a small transition zone between the unfrozen and frozen parts.,title="fig:",height=188 ] ( 5.3,0 ) in the light region , and @xmath263 in the dark one . there is a small transition zone between the unfrozen and frozen parts.,title="fig:",height=188 ] ( 7.9,0 ) in the light region , and @xmath263 in the dark one . there is a small transition zone between the unfrozen and frozen parts.,title="fig:",height=188 ] ( 10.5,0 ) in the light region , and @xmath263 in the dark one . there is a small transition zone between the unfrozen and frozen parts.,title="fig:",height=188 ] ( 0,3)@xmath264 ( 2.8,3)@xmath265 ( 5.2,3)@xmath266 ( 7.6,3)@xmath267 ( 10.3,3)@xmath268 we have considered a phase field model describing phase changes of a medium located in a container with heat conductive walls that are free of phase changes . it is shown that the temperature and the phase variables are continuous functions with values in @xmath83 and @xmath142 , respectively , provided that the initial data are from @xmath83 and @xmath142 , respectively . moreover , solutions depend continuously on the initial and boundary data of the problem . + * acknowledgements . * the work was motivated by the project spp 1253 of the german research society ( dfg ) , and was supported by award no . ksa - c0069/uk - c0020 , made by king abdullah university of science and technology ( kaust ) . a. novotny and i. straskraba . _ introduction to the mathematical theory of compressible flow _ , volume 27 of _ oxford lecture series in mathematics and its applications_. oxford university press , august 2004 . | a phase field model proposed by g.caginalp for the description of phase changes in materials is under consideration .
it is assumed that the medium is located in a container with heat conductive walls that are not subjected to phase changes .
therefore , the temperature variable is defined both in the medium and wall regions , whereas the phase variable is only considered in the medium part . the case of lipschitz domains in two and three dimensions
is studied .
we show that the temperature and phase variables are continuous in time functions with values in @xmath0 and @xmath1 , respectively , provided that the initial values of them are from @xmath0 and @xmath1 , respectively .
moreover , continuous dependence of solutions on the initial data and boundary conditions is proved . |
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after bourbaki [ 1989 11 ] we use _ cogebra , bigebra _ and hopf _ gebra _ instead of coalgebra , bialgebra and hopf algebra . let @xmath5 be an @xmath6-space and @xmath7 be an @xmath6-dual @xmath6-space . if @xmath5 is an @xmath6-cogebra and @xmath8 is an @xmath6-algebra then the @xmath6-space @xmath9 inherits the structure of an @xmath6-algebra with a convolution product : this is a convolution @xmath6-algebra , an @xmath6-convolution for short . a dual @xmath6-space @xmath10 inherits a structure of an @xmath6-cogebra with coconvolution coproduct : this is a coconvolutional @xmath6-cogebra , an @xmath6-coconvolution for short . in particular if an @xmath6-space @xmath11 carries an @xmath6-biconvolution algebra & cogebra structure , then do also the @xmath6-spaces @xmath12 as well as all iterated @xmath6-spaces @xmath13 inherit also @xmath6-biconvolution algebra & cogebra structures . if the @xmath6-space @xmath5 is an @xmath6-cogebra with a coproduct @xmath14 then @xmath7 is an @xmath6-algebra with product @xmath15 if an @xmath6-space @xmath8 is a finite dimensional @xmath6-algebra having a binary product @xmath16 then an @xmath6-dual @xmath6-space @xmath17 ( or @xmath18-graded dual in the case @xmath8 is not a finite dimensional @xmath6-space ) is an @xmath6-cogebra with a binary coproduct @xmath19 however there are several important situations ( free tensor algebra , exterior algebra , clifford algebra , weyl algebra , ... ) where the dual space of an algebra is also an algebra in a natural way by construction . if this is the case , then by the above ( @xmath18-graded ) duality , both mutually dual @xmath6-spaces carry both structures , algebra & cogebra , and therefore we have a dual pair of @xmath6-biconvolutions . an unital and associative convolution possessing an ( unique ) antipode is said to be a hopf gebra or an _ antipodal convolution _ ( definition [ dfn2.2 ] ) . this terminology has been introduced in [ oziewicz 1997 , 2001 ; cruz & oziewicz 2000 ] and is different from sweedler s [ 1969 p. 71 ] . a general theory of the finite - dimensional antipodal and antipode - less biconvolutions ( convolutions and coconvolutions ) has been initiated in [ cruz & oziewicz 2000 ] . nill in 1994 and bhm & szlachnyi in 1996 introduced weak bigebras and weak hopf gebras with antipode @xmath20 defined as the galois connection with respect to the binary convolution @xmath21 which does not necessarily needs to be unital [ nill 1998 , nill et all . 1998 ] , @xmath22 if @xmath23 is invertible then @xmath24 if @xmath25 is a clifford @xmath6-algebra , then a dual @xmath6-space @xmath26 is a clifford @xmath6-cogebra . it was shown in [ oziewicz 1997 , and ff . ] that a clifford convolution @xmath27 is antipode - less . the aim of this paper is to show that a clifford convolution @xmath28 for @xmath29 _ i.e._for @xmath30 does posses an antipode iff @xmath31 ( main theorem 4.1 ) . applications to physics have been proposed in [ fauser 2000b ] . ] ] let the convolution be unital , fig . [ conv_unit ] . this is the case if an @xmath6-algebra @xmath8 is unital with unit @xmath32 and an @xmath6-cogebra @xmath5 is counital with counit @xmath33 [ dfn2.1 ] the convolutive inverse , w.r.t . the convolutive unit @xmath34 , of the identity map on @xmath11 , fig . [ antipode ] , is said to be an _ antipode _ , @xmath35 . if an antipode exists w.r.t . an unital associative convolution it must be unique . an antipodal biconvolution defines a unique crossing as given in fig . [ crossing ] in the last section , see [ oziewicz 1997 , 2001 ; cruz & oziewicz 2000 ] . using the axioms of the antipode , fig . [ antipode ] , and biassociativity , this crossing is equivalent to the algebra homomorphism between algebra and crossed algebra , as well as , to the cogebra homomorphism from crossed cogebra to cogebra . a proof is given in [ cruz & oziewicz 2000 ] . the gramann wedge product , the gramann coproduct and the unique antipode , as given by sweedler [ 1969 , ch . xii ] and extended by woronowicz [ 1989 , 3 ] , see section 3.1 below for details , gives the gramann hopf gebra . this motivates the following definition : [ dfn2.2 ] an unital and associative convolution possessing a ( unique ) antipode is said to be a hopf gebra or an antipodal convolution . a closed structure is given by the evaluation and coevaluation [ kelly & laplaza 1980 ; lyubashenko 1995 ] : @xmath36 following [ lyubashenko 1995 , fig . 3 , p. 250 ] the evaluation is represented by cup and coevaluation by cap . the evaluation intertwines the transposed endomorphisms @xmath37 with @xmath38 , fig . [ dual_op ] . up and down arrows indicate the spaces @xmath39 and @xmath11 . . ] a coconvolution is counital if e.g. there exits a dual counit @xmath40 and a dual unit @xmath41 , fig . [ conv_counit ] . ] ] we have to use the dual coconvolution counit @xmath42 in fig . [ conv_counit ] . the gramann hopf gebra was constructed by sweedler [ 1969 , ch . xii ] factoring the couniversal shuffle tensor hopf gebra sh@xmath11 by the _ switch _ , @xmath43 . this construction was generalized to any braid by woronowicz [ 1989 , 3 , an exterior hopf gebra can be defined in terms of an unique braid dependent homomorphism of universal tensor hopf gebra into couniversal tensor hopf gebra and this implies that an exterior hopf gebra is couniversal and braided [ oziewicz , paal & raski 1995 8 ; raski 1996 ; oziewicz 1997 p. 1272 - 1273 ] . ] the tensors @xmath44 are said to be scalar products on @xmath11 or coscalar products on @xmath39 ( @xmath45 is the transpose of @xmath46 ) . the tensors @xmath47 are said to be scalar products on @xmath39 and coscalar on @xmath11 . in particular @xmath48 is the symmetric part of @xmath49 the scalar products are displayed by decorated ( or labelled ) cups and coscalar products by decorated caps , see fig . [ cliffordization ] . rota and stein [ 1994 ] introduced the clifford product as a deformation of exterior biconvolution . this deformation process was called _ a cliffordization introduces an internal loop in a binary product having two inputs and one output , employing the @xmath50-cup ( or @xmath51-cap for the coproduct ) , fig . [ cliffordization ] . clifford biconvolution was defined in [ oziewicz 2001 ] as the @xmath52-bicliffordization of an exterior biconvolution . in sweedler s notation , @xmath53 if the product coproduct duality of fig . [ duality ] is used with cup as an evaluation , then every product on @xmath54 induces a coproduct on @xmath55 and vice versa . if @xmath50-cup s and @xmath56-cap s are used , one gets a correlation between products on @xmath54 and coproducts on @xmath54 . a clifford @xmath6-algebra together with a clifford @xmath6-cogebra on @xmath57 @xmath58 is said to be a clifford @xmath6-convolution . it was shown in [ oziewicz 1997 ] that a clifford @xmath6-convolution for @xmath59 and for @xmath60 is antipode - less . an antipode - less clifford convolution @xmath61 for an _ invertible _ tensor @xmath46 _ can not _ be a deformation of an exterior gramann hopf gebra . the tensors @xmath46 and @xmath56 are said to be dependent if @xmath62 and @xmath63 exist such that one of the following relations hold , @xmath64 if the tensors @xmath65 are independent then the clifford product and coproduct are defined independently by rota & stein s _ deformation_. cliffordization and cocliffordization of the gramann convolution does not change the convolutive unit @xmath66 however since the @xmath18-grading is changed due to the skewsymmetric parts of @xmath46 and @xmath67 the counit is no longer the projection onto @xmath68 [ fauser 1998 - 2000a ] . [ mnthm ] a clifford biconvolution @xmath69 is a clifford hopf gebra iff @xmath70 then @xmath71.\end{aligned}\ ] ] theorem 4.1 was proved for @xmath72 in [ oziewicz 1997 ] . if @xmath73 then for @xmath74 @xmath75 therefore @xmath76 the clifford antipode @xmath77 is computed from fig . [ antipode ] . we give the proof for @xmath78 only . the general case will be treated elsewhere . let @xmath79 and @xmath80 then we find together with ( i ) of the main theorem : @xmath81 an action of @xmath82 on tensors @xmath83 and @xmath84 is given as follows @xmath85 it would be desirable to study @xmath86-orbits on @xmath87 and present full classification of all orbits in terms of invariants . however , this topic exceeds the scope of this paper and will be presented elsewhere . theorem [ mnthm ] solves and improves conjecture 2.2 posed in [ oziewicz 1997 , p. 1270 the above clifford hopf gebra includes as a particular case for @xmath88 the construction made by urevich [ 1994 ] . in the case @xmath89 one can take @xmath90 . also if @xmath91 one can choose @xmath92 . in order to prevent such possibilities we need to supplement the definition of the clifford hopf gebra with the extra conditions : @xmath93 the antisymmetric tensor @xmath94 can be adjusted in such a way that @xmath95 with invertible symmetric tensor @xmath96 we found a clifford antipode for @xmath97 @xmath98 the following matrices for @xmath99 represent the same tensor from @xmath100 or from @xmath101 with respect to the different bases , these matrices are on the same @xmath102 orbit , @xmath103 an antipode for regular scalar and coscalar tensors can be found also . a clifford biconvolution @xmath69 is antipode - less if @xmath104 . in particular this is the case if @xmath105 it was shown [ oziewicz 1997 ] that @xmath106 . what axioms for clifford biconvolution implies the condition @xmath104 ? in particular , does a braid exists for which such a clifford biconvolution is a braided hopf gebra in the usual sense ? if such a braid exits how much freedom remains for choices ? compare with [ oziewicz 1997 , p. 1272 ] where it was shown that for @xmath72 , @xmath107 exists a 12-parameter family of crossings . let @xmath108 and @xmath109 then the following equations are equivalent : @xmath110 @xmath111 according to lemma 5.2 we have to ask that @xmath112 we present three examples of antipode - less clifford biconvolutions @xmath69 , @xmath104 for @xmath78 , @xmath113 and for @xmath65 given by with signature @xmath114 @xmath115 @xmath116 let @xmath117 be a commutative ring . an exact sequence of homomorphisms of @xmath117-modules , @xmath118 @xmath119 splits if @xmath120 is a direct summand of @xmath121 the following statements are equivalent @xmath122 let @xmath123 split the exact sequence of @xmath6-algebra homomorphisms @xmath124 thus @xmath125 in this case the element of the crossed @xmath6-algebra , viz . @xmath126 , is said to be a _ splitting idempotent _ ( a cleft of @xmath127 ) , @xmath128 . if @xmath129 then @xmath130 a crossing defined in fig . [ crossing ] gives a cogebra map @xmath131 and an algebra homomorphism @xmath132 [ oziewicz 1997 , 2001 , cruz & oziewicz 2000 ] . however an algebra homomorphism @xmath133 in general does not need to split . a clifford convolution @xmath134 is said to be a gramann convolution . if @xmath69 is a gramann hopf gebra , @xmath135 then @xmath133 splits , @xmath136 iff @xmath137 a crossing for an antipodal convolution is defined on fig . [ crossing ] . for antipodal convolution ] the crossing @xmath138 fig . [ crossing ] is equivalent that there is a cogebra map @xmath139 and an algebra homomorphism @xmath140 [ oziewicz 1997 , 2001 , cruz & oziewicz 2000 ] . a crossing for @xmath141 and thus for @xmath142 is the involutive graded switch [ sweedler 1969 , ch.xii ] , @xmath143 in the sequel @xmath70 the degree of the minimal polynomial of the crossing @xmath144 we denote by : degree@xmath145 let @xmath146 then @xmath147 let @xmath18 be the ring of integers , @xmath148 $ ] be the ring of polynomials in @xmath149 with coefficients in @xmath150 and @xmath151 $ ] be the following polynomial , @xmath152 let @xmath69 be a gramann hopf gebra , _ i.e._an antipodal gramann convolution ( definitions 2.3 & 6.4 ) with @xmath153 and @xmath78 . the minimal polynomial of the crossing is of order 30 in @xmath154 if @xmath155 then the minimal polynomial of @xmath138 vanishes , as also no antipode exists in this case . moreover @xmath156,\\ & p(\sqrt{\mu}-1)\neq 0,\ ; \text{then the minimal polynomial of the crossing}\\ & \text{is of degree 3.}\\ ( ii)&\quad\text{if}\;p(\sqrt{\mu}-1)=0,\ ; \text{then the the minimal polynomial of the crossing is}\\ & \text{of degree}\;\leq 2.\end{aligned}\ ] ] clifford , a maple v package by rafa abamowicz [ abamowicz , 1995 - 2000 ] and extensions like bigebra , by rafa abamowicz and bertfried fauser , have been used to check and get some of the results . 99 rafa abamowicz , clifford , an maple v package and bigebra , available at http://math.tntech.edu/rafal/ nicolas bourbaki , _ elements of mathematics , algebra i _ , chapter iii , springer verlag , berlin 1989 henry cartan & samuel eilenberg , _ homological algebra _ , princeton university press 1956 jos de jess de cruz guzman and zbigniew oziewicz , _ unital and antipodal biconvolution and hopf gebra _ , ( 2000 ) this volume micho urevich , _ braided clifford algebras as braided quantum groups _ , advances in applied clifford algebras * 4 * ( 2 ) ( 1994 ) 145156 micho urevich , _ braided clifford algebras as quantum deformations _ , international journal of theoretical physics * 40 * ( 1 ) ( 2001 ) samuel eilenberg , _ extensions of general algebra _ , annales soc . polon . math . ( rocznik polskiego towarzystwa matematycznego ) * 21 * ( 1 ) ( 1948 ) 125134 bertfried fauser , _ on an easy transition from operator dynamics to generating functionals by clifford algebras _ , journal of mathematical physics * 39 * ( 1998 ) 49284947 bertfried fauser and rafa abamowicz , _ on the decomposition of clifford algebras of arbitrary bilinear form _ , in : `` clifford algebras and their applications in mathematical physics '' , rafa abamowicz and bertfried fauser , eds . , birkhuser , boston 2000 p. 341366 bertfried fauser , _ on the hopf algebraic origin of wick normal - ordering _ , 2000a , j. phys . a. : math . and gen . accepted ( hep - th/0007032 ) bertfried fauser , _ quantum clifford hopf gebras for quantum field theory _ , talk at `` quantum group symmposium '' at group23 , dubna , 2000b , ( hep - th/0011026 ) alexander j. hahn , _ quadratic algebras , clifford algebras , and arithmetic witt groups _ , springer verlag , berlin 1994 , chap . 2 gregory maxwell kelly & miguel l. laplaza , _ coherence for compact closed categories _ , journal of pure and applied algebra * 19 * ( 1980 ) 193213 volodimir lyubashenko , _ tangles and hopf algebras in braided categories _ , journal of pure and applied algebra * 98 * ( 1995 ) 245278 florian nill , _ axioms for weak bialgebras _ , q - alg/9805104 florian nill , k. szlachnyi & h .- w . wiesbrock , _ weak hopf algebras and reducible jones inclusion of depth 2 . i : from crossed products to jones towers _ , q - alg/9806130 zbigniew oziewicz , eugen paal and jerzy raski , _ derivations in braided geometry _ , acta physica polonica * 26 * ( 7 ) ( 1995 ) 12531273 zbigniew oziewicz , _ clifford hopf gebra and bi - universal hopf gebra _ , czechoslovak journal of physics * 47 * ( 12 ) ( 1997 ) 12671274 zbigniew oziewicz , _ guest editor s note : clifford algebras and their applications _ , international journal of theoretical physics * 40 * ( 1 ) ( 2001 ) 111 richard s. pierce , _ associative algebras _ , graduate texts in mathematics # 88 , springer - verlag , new york 1982 gian - carlo rota & joel a. stein , _ plethystic hopf algebras _ , proceedings of natl . academy of sciences usa * 91 * ( 1994 ) 1305713061 jerzy raski , _ braided antisymmetrizer as bialgebra homomorphism _ , reports in mathematical physics * 38 * ( 2 ) ( 1996 ) 273277 gnter scheja , uwe storch , _ lehrbuch der algebra _ , teil 1 , b.g . teubner , stuttgart , 1980 moss e. sweedler , _ hopf algebras _ , benjamin , new york 1969 stanisaw lech woronowicz , _ differential calculus on compact matrix pseudogroups ( quantum groups ) _ , communications in mathematical physics * 122 * ( 1989 ) 125170 stanisaw zakrzewski , _ quantum and classical pseudogroups . part i. union pseudogroups and their quantization _ , communications in mathematical physics * 134 * ( 1990 ) 347370 | a clifford algebra @xmath0 jointly with a clifford cogebra @xmath1 is said to be a clifford biconvolution @xmath2 we show that a clifford biconvolution for @xmath3 does possess an antipode iff @xmath4 an antipodal clifford biconvolution is said to be a clifford hopf gebra . + * 2000 mathematics subject classification : * 15a66 clifford algebra , 16w30 coalgebra , bialgebra , hopf algebra + * 2000 pacs : * 02.10.tq associative rings and algebras + * keywords : * cliffordization , clifford algebra , clifford cogebra , antipode , hopf gebra , clifford bigebra , gramann algebra |
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matter which is exposed to strong external magnetic fields changes its basic properties and structure and leads to a variety of new phenomena . as a result strong fields are of importance in different branches of physics like atomic , molecular or solid state physics . for atomic and molecular systems there are two prominent possibilities to encounter the strong field regime : highly excited rydberg states in the laboratory and atoms and molecules in the atmospheres of magnetized white dwarfs ( see refs.@xcite for a compilation of the subject ) . from a theorists point of view particle systems in strong fields pose a hard problem due to several competing interactions ( coulomb attraction and repulsion , para- and diamagnetic interactions ) . of particular interest , but most complicated to investigate , is hereby the so - called intermediate regime which is characterized by comparable magnetic and coulomb binding forces . focusing on the low - lying states of atoms and molecules we envisage this regime for those magnetic white dwarfs which possess field strengths in the regime @xmath2 . each magnetic white dwarf possesses a characteristic regime of field strengths which varies , in case of a dipole , by a factor of two from the pole to the equator . to perform a first identification of observed spectra from the atmospheres of these objects one uses the so - called stationary line spectroscopy : characteristic absorption features can appear only for those wavelengths which correspond to an extremum of the transition wavelength with respect to the field strength . in a second step one then performs simulations of the radiation transport in the atmosphere in order to obtain synthetic spectra . in the eighties the above approaches have been used to identify hydrogen in a number of magnetized white dwarfs @xcite . recently the stationary line argument has been successfully @xcite used to obtain strong evidence for helium in the spectrum of the magnetic white dwarf gd229 whose absorption features have been mysterious ever since its discovery @xmath3 years ago . this was only possible due to the enormous progress achieved with respect to our knowledge of the spectrum and transitions of the helium atom in strong magnetic fields @xcite . however , this should not obscur the fact that there are a number of magnetic white dwarfs whose spectra remain unexplained and furthermore new magnetic objects are discovered ( see , for example , reimers et al @xcite in the course of the hamburg survey of the european southern observatory ( eso ) ) . very recently strong candidates for quasimolecular absorption features have been discovered in magnetized white dwarfs @xcite . this raises the demand for a theoretical investigation of molecular properties , in particular of the hydrogen molecule , in such strong fields . several theoretical investigations were performed for molecular systems in strong magnetic fields . most of them deal with the electronic structure of the @xmath4 ion ( see refs . @xcite and references therein ) . very interesting phenomena can be observed already for this simple diatomic system . for the ground state of the @xmath4 molecule the dissociation energy increases and the equilibrium internuclear distance simultaneously decreases with increasing field strength . furthermore it was shown @xcite that a certain class of excited electronic states , which possess a purely repulsive potential energy surface in the absence of a magnetic field , acquire a well - pronounced potential well in a sufficiently strong magnetic field . moreover the electronic potential energies depend not only on the internuclear distance but also on the angle between the magnetic field and the molecular axis which leads to a very complex topological behavior of the corresponding potential energy surfaces @xcite . in contrast to the @xmath4 ion there exist only a few investigations dealing with the electronic structure of the hydrogen molecule in the presence of a strong magnetic field . highly excited states of @xmath5 were studied for a field strength of @xmath6 in ref . for intermediate field strengths two studies of almost qualitative character investigate the potential energy curve ( pec ) of the lowest @xmath7 state @xcite . a few investigations were performed in the high field limit @xcite , where the magnetic forces dominate over the coulomb forces and therefore several approximations can be used . very recently a first step has been done in order to elucidate the electronic structure of the @xmath5 molecule for the parallel configuration , i.e. for parallel internuclear and magnetic field axes @xcite . hereby refs.@xcite apply an exact , i.e. fully correlated approach , whereas refs.@xcite use hartree - fock calculations and focus exclusively on the identification of the global ground state of the molecule . in refs.@xcite the lowest states of the @xmath1 and @xmath8 manifolds were studied for gerade and ungerade parity as well as singlet and triplet spin symmetry . hereby accurate adiabatic electronic energies were obtained for a broad range of field strengths from field free space up to strong magnetic fields of @xmath9 a variety of interesting effects were revealed . as in the case of the @xmath4 ion , the lowest strongly bound states of @xmath1 symmetry , i.e. the lowest @xmath7 , @xmath10 and @xmath11 state , show a decrease of the bond length and an increase of the dissociation energy for sufficiently strong fields . furthermore a change in the dissociation channel occurs for the lowest @xmath11 state between @xmath12 and @xmath13 due to the existence of strongly bound @xmath14 states in the presence of a magnetic field . the @xmath10 state was shown to exhibit an additional outer minimum for intermediate field strengths which could provide vibrationally bound states . an important result of refs . @xcite is the change of the ground state from the lowest @xmath7 state to the lowest @xmath15 state between @xmath16 and @xmath17 this crossing is of particular relevance for the binding properties of the global ground state of the molecule : the @xmath15 state is an unbound state and possesses only a very shallow van der waals minimum which does not support any vibrational level . therefore , the global ground state of the hydrogen molecule for the parallel configuration is an unbound state for @xmath18 . furthermore it has been shown in ref . @xcite that for very strong fields ( @xmath19 ) the strongly bound @xmath20 state is the global ground state of the hydrogen molecule oriented parallel to the magnetic field . finally the complete scenario for the crossovers of the global ground state of the parallel configuration has been clarified in ref.@xcite which contains the transition field strengths for the crossings among the lowest states of @xmath21 and @xmath20 symmetry . the above considerations show that detailed studies of the electronic properties of the hydrogen molecule in a magnetic field are very desirable . the present investigation deals with the excited @xmath1 states of the hydrogen molecule in the parallel configuration which is distinct by its higher symmetry @xcite . we employ a full configuration interaction ( ci ) approach which is most suitable for obtaining detailed information on the electronic structure . our investigation is divided into two separate studies : the first and present work focuses on singlet states of both gerade and ungerade symmetry whereas a later investigation will focus on the corresponding triplet states again for both parities . due to the spin zeeman splitting the spin singlet and triplet manifolds are increasingly separated with increasing field strength . the spin character provides therefore a natural dividing line of our extensive work which contains a large amount of information and data on the behaviour of the excited states of the molecule . the results of our calculations include accurate adiabatic pecs for the complete range of field strengths @xmath22 . up to seven excited states have been investigated for each symmetry . we present detailed data for the total and dissociation energies at the equilibrium internuclear distances as well as the equilibrium positions themselves for the lowest three excitations for each symmetry . due to the large amount of data the evolution of the higher excited states with increasing field strength is presented only graphically . however , further informations like , for example , the positions of the maxima and the accurate heights of the barriers or the complete data of the pec s as well as numerical data on the higher excited states , more precisely , on the fourth up to the seventh excited state , can be obtained from the authors upon request . in detail the paper is organized as follows . in section ii we describe the theoretical aspects of the present investigation , including a discussion of the hamiltonian , a description of the atomic orbital basis set and some remarks on the ci approach . section iii contains the results and an elaborate discussion of the evolution of the electronic structure in the presence of the magnetic field with increasing strength . our starting point is the total nonrelativistic molecular hamiltonian in cartesian coordinates . the total pseudomomentum is a constant of motion and therefore commutes with the hamiltonian @xcite . for that reason the hamiltonian can be simplified by performing a so - called pseudoseparation of the center of mass motion @xcite which introduces the center of mass coordinate and the conserved pseudomomentum as a pair of canonical conjugated variables . further simplifications can be achieved by a consecutive series of unitary transformations @xcite . in order to separate the electronic and nuclear motion we perform the born - oppenheimer approximation in the presence of a magnetic field @xcite . as a first order approximation we assume infinitely heavy masses for the nuclei . the origin of our coordinate system coincides with the midpoint of the internuclear axis of the hydrogen molecule and the protons are located on the @xmath23 axis . the magnetic field is chosen parallel to the @xmath23 axis of our coordinate system and the symmetric gauge is adopted for the vector potential . the gyromagnetic factor of the electron is chosen to be equal to two . the hamiltonian , therefore , takes on the following appearance : @xmath24 the symbols @xmath25 , @xmath26 and @xmath27 denote the position vectors , their canonical conjugated momenta and the angular momenta of the two electrons , respectively . @xmath28 and @xmath29 are the vectors of the magnetic field and internuclear distance , respectively and @xmath30 denotes the magnitude of @xmath29 . with @xmath31 we denote the vector of the total electronic spin . throughout the paper we will use atomic units . the hamiltonian ( [ form1 ] ) commutes with the following independent operators : the parity operator @xmath32 , the projection @xmath33 of the electronic angular momentum on the internuclear axis , the square @xmath34 of the total electronic spin and the projection @xmath35 of the total electronic spin on the internuclear axis . in field free space we encounter an additional independent symmetry namely the reflections of the electronic coordinates at the @xmath36 ( @xmath37 ) plane . the eigenfunctions possess the corresponding eigenvalues @xmath38 . this symmetry does not hold in the presence of a magnetic field ! therefore , the resulting symmetry groups for the hydrogen molecule are @xmath39 in field free space and @xmath40 in the presence of a magnetic field @xcite . in order to solve the fixed - nuclei electronic schrdinger equation belonging to the hamiltonian ( [ form1 ] ) we expand the electronic eigenfunctions in terms of molecular configurations . in a first step the total electronic eigenfunction @xmath41 of the hamiltonian ( [ form1 ] ) is written as a product of its spatial part @xmath42 and its spin part @xmath43 , i.e. we have @xmath44 . for the spatial part @xmath42 of the wave function we use the lcao - mo - ansatz , i.e. we decompose @xmath42 with respect to molecular orbital configurations @xmath45 of @xmath5 , which respect the corresponding symmetries ( see above ) and the pauli principle : @xmath46 \nonumber \\ & = & \sum\limits_{i , j } c_{ij } \left[\phi_i\left(\bbox{r_1}\right)\phi_j\left(\bbox{r_2}\right ) \pm \phi_i\left(\bbox{r_2}\right)\phi_j\left(\bbox{r_1}\right)\right ] \nonumber\end{aligned}\ ] ] the molecular orbital configurations @xmath47 of @xmath5 are products of the corresponding one - electron @xmath48 molecular orbitals @xmath49 and @xmath50 . the @xmath48 molecular orbitals are built from atomic orbitals centered at each nucleus . a key ingredient of this procedure is a basis set of nonorthogonal optimized nonspherical gaussian atomic orbitals which has been established previously @xcite . for the case of a @xmath51molecule parallel to the magnetic field these basis functions read as follows : @xmath52 the symbols @xmath53 and @xmath23 denote the electronic coordinates . @xmath54 , @xmath55 and @xmath56 are parameters depending on the subspace of the h - atom for which the basis functions have been optimized and @xmath57 and @xmath58 are variational parameters . we remark that the nonlinear optimization of the variational parameters @xmath57 and @xmath58 has to be accomplished for typically of the order of @xmath59 atomic orbitals and is done by reproducing many excited states of the hydrogen atom for each field strength separately . it represents therefore a tedious and time consuming work which has , however , to be done with great care in order to obtain precise results for the following molecular structure calculations . for a more detailed description of the construction of the molecular electronic wave function we refer the reader to ref . @xcite . in order to determine the molecular electronic wave function of @xmath5 we use the variational principle which means that we minimize the variational integral @xmath60 by varying the coefficients @xmath61 . the resulting generalized eigenvalue problem reads as follows : @xmath62 where the hamiltonian matrix @xmath63 is real and symmetric and the overlap matrix is real , symmetric and positive definite . the vector @xmath64 contains the expansion coefficients . the matrix elements of the hamiltonian matrix and the overlap matrix are certain combinations of matrix elements with respect to the optimized nonspherical gaussian atomic orbitals . a description of the techniques necessary for the evaluation of these matrix elements is given in ref . we mention here only that the electron - electron integrals needed a combination of numerical and analytical techniques in order to make its rapid evaluation possible . the latter represents the cpu time dominating factor for the construction of the hamiltonian matrix . for the numerical solution of the eigenvalue problem ( [ form3 ] ) we used the standard nag library the typical dimension of the hamiltonian matrix for each subspace varies between approximately 2000 and 5000 depending on the magnetic field strength . depending on the dimension of the hamiltonian matrix , it takes between 70 and 250 minutes for simultaneously calculating one point of a pec of each subspace on a ibm rs6000 computer . the overall accuracy of our results with respect to the total energy is estimated to be typically of the order of magnitude of @xmath65 and for some cases of the order of magnitude of @xmath66 . it should be noted that this estimate is rather conservative ; in some ranges of the magnetic field strength and internuclear distance , e.g. , close to the separated atom limit , the accuracy is @xmath67 or even better . the positions , i.e. internuclear distances , of the maxima and the minima in the pecs were determined with an accuracy of @xmath68 herefore about 350 points were calculated on an average for each pec . it was not necessary to further improve this accuracy since a change in the internuclear distance about @xmath69 results in a change in the energy which is typically of the order of magnitude of @xmath65 or smaller . the total cpu time needed to complete the present work amounts to several years on the above powerful computer . to understand the influence of the external magnetic field on the electronic structure of the hydrogen molecule we first have to remind ourselves of the properties in the absence of the field . accurate data for hydrogen are of great importance both in astrophysics as well as laboratory physics . it is a paradigm for many molecular phenomena like charge transfer , excitation , ionization or scattering processes . indeed our ci calculations on the basis of an anisotropic gaussian basis set provided also significant progress with respect to the knowledge of the field - free excitations of the molecule : several highly excited states have been calculated for the first time and some of the pecs for the lower lying states have been improved . the corresponding results have been presented to some detail in ref.@xcite and contain elaborate information on the first eight excited singlet and triplet states for both gerade and ungerade parity . in the following we will first summarize the main properties of the excited singlet states in the absence of the field and then investigate the electronic structure in the presence of the magnetic field with increasing field strength . we hereby first deal with the gerade and subsequently with the ungerade states . the investigation of the electronic states and pecs has been done for all internuclear distances considered ( @xmath70 ) with the same atomic orbital basis set . the latter has been optimized to yield precise energies ( accuracy @xmath71 ) of the hydrogen atom for the six lowest states for both parities for vanishing atomic magnetic quantum number . additionally , in order to describe correlation effects , we have included basis functions with atomic magnetic quantum numbers @xmath72 . the approximate number of two - particle configurations resulting from the above basis set is @xmath73 . the accuracy of the electronic energies for the higher excited states @xmath74 ( @xmath75 indicates the degree of excitation ) is , due to the above choice of the optimized basis set , lower than that for less excited states . figure 1 shows the pecs for the states @xmath76 where the dotted lines represent the curves for the higher excited states @xmath77 . there is a large energetical gap ( @xmath78 ) between the ground and the excited states of @xmath79 symmetry . the first five excited states are well - known from the literature @xcite . our calculations @xcite show in most cases an agreement within @xmath80 compared to the literature and in several cases also a variationally lower energy . as already mentioned the results on the higher excited states ( @xmath77 ) have for the first time been reported very recently in ref.@xcite . as can be seen from figure 1 all the pecs of the states @xmath76 possess a deep potential well around a minimum located approximately at @xmath81 . a particular feature occuring for most of the considered @xmath82 states is the existence of a second outer minimum and therefore the corresponding pecs exhibit a double well . vibrational states in these outer wells @xcite attracted recently significant experimental interest @xcite since they allow the experimental observation of long - lived and highly excited valence states of the hydrogen molecule . the two minima of the @xmath83 state arise due to the fact that two different configurations of the same symmetry , namely the @xmath84 and the @xmath85 configurations , are energetically minimized at two significantly different internuclear distances . the deep outer wells of the @xmath86 states arise due to a series of avoided crossings between the heitler - london configurations @xmath87 and the ionic configurations @xmath88 . particularly the @xmath89 state possesses a very broad and deep ( @xmath90 ! ) outer potential well which is separated by a broad barrier from the inner well located at @xmath91 . the outer minimum is located at @xmath92 . a series of avoided crossings at very large internuclear distances @xmath93 leads to the energetically equal dissociation limits @xmath94 of the @xmath95 states . the dissociation channel of the @xmath96 state is the ionic configuration @xmath97 . tables 1 to 3 contain ( among the data in the presence of a magnetic field ) the total and dissociation energies at the equilibrium internuclear distances , the equilibrium internuclear distances and the total energies in the dissociation limit for the first to third excited @xmath98 states in the absence of the magnetic field . the subspace of @xmath7 symmetry contains the electronic ground state of the hydrogen molecule in field - free space and in the presence of a magnetic field in the regime @xmath99 for a detailed discussion of the appearance of this state and the global ground state with increasing field strength in general we refer the reader to refs.@xcite ( see also introduction of the present work ) . in the following we investigate the evolution of the excited @xmath100 states with increasing magnetic field strength for the regime @xmath101 . we will first study the changes of the pecs of individual states with increasing field strength and thereafter we present a global view of the evolution of the spectrum . in order to compare the pecs for the same state for different field strengths we subtract from the total energies the corresponding energies in the dissociation limit ( which is different for different field strengths ) , i.e. we show the quantity @xmath102 . in general the dissociation limit of a certain state of @xmath1 symmetry changes with increasing field strength which is due to the reordering of the energy levels of the atoms ( hydrogen , hydrogen negative ion ) in the external field . for the atomic states we will use in the following the notation @xmath103 where @xmath75 specifies the degree of excitation and @xmath104 the atomic magnetic quantum number and z - parity , respectively . let us begin our investigation of the evolution of individual states with increasing field strength with the @xmath105 state whose pecs are shown in figure 2a . the positions of the two minima and the corresponding maximum decrease with increasing field strength . the depth of the inner potential well decreases for @xmath106 and increases rapidly for @xmath107 . the depth of the outer well is monotonically increasing for the complete regime @xmath101 . for @xmath108 and @xmath109 the inner well is therefore deeper than the outer well and vice versa for @xmath110 ( see figure 2a ) . the dissociative behaviour of the pecs changes significantly with increasing field strength . the origin of these changes is the fact that for @xmath111 the dissociation channel is @xmath112 whereas for @xmath113 we have the asymptotic behaviour @xmath114 ( the index @xmath115 stands for spin singlet ) . the appearance of the ionic configuration as the dissociation channel for the low - lying electronic @xmath105 state can be explained as follows . it is well - known that the hydrogen negative ion possesses infinitely many bound states in the presence of a magnetic field of arbitrary strength assuming an infinite nuclear mass @xcite . certain of these bound states show a monotonically increasing binding energy with increasing field strength . the latter surpass then more and more of the energy levels belonging to two hydrogen atoms one being in the global ground state and the other one in the corresponding excited state . for a sufficiently strong magnetic field we therefore expect the configuration @xmath116 to become the dissociation channel particularly for the first excited state of @xmath7 symmetry . due to the long range forces the onset of the asymptotic ( @xmath117 ) behaviour of the corresponding pecs with the ionic channel ( @xmath118 ) is qualitatively different from the pecs with a neutral dissociation limit ( @xmath119 ) . this explains the different asymptotic behaviour of the pecs shown in figure 2a with increasing field strength . finalizing the discussion of the @xmath105 state we remark that its pecs possesses a second maximum for @xmath120 which however occurs at large internuclear distances ( @xmath121 ) and is only of the order of @xmath122 above the dissociation limit . table 1 contains relevant data of the pecs of the @xmath105 state with increasing field strength . next we turn to the second excited i.e. the @xmath123 state whose pecs are shown in figure 2b . the positions of the two minima and the corresponding maximum already present in field - free space decrease monotonically with increasing field strength . starting from @xmath124 the depth of the inner well decreases with increasing field strength whereas it increases for @xmath125 . besides a very small interval of field strengths the depth of the outer well increases with increasing field strength . for @xmath126 the outer well is deeper than the inner one whereas for @xmath127 the deep inner well dominates the shape of the pec . we remark that the curvature at the ( first ) maximum and the outer minimum increases significantly with increasing field strength . the evolution of these increasingly sharper turns can only be fully understood if one looks at the complete spectrum ( see figure 3 and in particular 3(e ) ) with increasing field strength : they develop due to a number of narrow avoided crossing of the first to third excited states in strong fields . an interesting property of the pec of the @xmath123 state is the existence of an additional outer ( third ) minimum for the interval @xmath128 which is shown in figure 2c . this minimum arises due to the interaction with the ionic configuration @xmath118 . in field - free space the lowest and only bound ionic channel @xmath116 is the dissociation channel of the @xmath129 state . with increasing field strength the hydrogen negative ion becomes increasingly stronger bound ( see discussion above ) and therefore it occurs as the dissociation channel for the sequence of excited states @xmath130 finally becoming the dissocation channel of the @xmath105 state for @xmath113 . the existence of the additional outer minimum becomes now understandable : due to the energetical lowering of the ionic dissociation channel with increasing field strength the higher excited states of @xmath7 symmetry evolve outer minima and corresponding wells for certain regimes of the field strength . for the @xmath123 state this outer well is extremely shallow for @xmath131 and therefore almost invisible in figure 2c . for @xmath132 it becomes however well - pronounced . between @xmath133 and @xmath134 there occurs a change with respect to the dissociation channel of the @xmath123 . for @xmath135 the dissociation channel is @xmath136 and for @xmath137 it is @xmath138 . the similar asymptotic behaviour of the pecs belonging to different field strengths ( see figure 2c for @xmath139 and @xmath140 ) arises due to the fact that they possess all the ionic dissociation channel . in the latter regime the position of the ( third ) outer minimum increases with increasing field strength ( for @xmath141 the outer minimum is located at @xmath142 ) . finally there is a second change of the dissociation channel of the @xmath123 state to @xmath143 and therefore the outer minimum disappears for @xmath113 . table 2 contains the total and dissociation energies at the equilibrium internuclear distances , the equilibrium internuclear distances and the total energies in the dissociation limit for the second excited @xmath144 state in the regime @xmath101 next we focus on the third excited @xmath145 whose pecs with increasing field strength are shown in figure 2d . in field - free space it possesses two minima and associated potential wells located at @xmath146 and @xmath147 , respectively . the position of the inner minimum increases with increasing field strength whereas the corresponding dissociation energy decreases . finally for @xmath148 the associated well disappears but reappears for @xmath149 . with further increasing field strength the position of this inner minimum decreases and the depth of the corresponding well increases monotonically for @xmath107 . independently of this first inner minimum and the outer minimum there appears for @xmath150 an additional third minimum and corresponding well ( see table 3 and figure 2d ) for small internuclear distances @xmath151 . although this new minimum and well are energetically well below the dissociation limit for @xmath152 they are separated from the other inner minimum only by a tiny barrier . these facts will become better understandable in the context of our discussion of the evolution of the whole spectrum with increasing field strength ( see below ) . the properties of the pec of the @xmath145 state at large internuclear distances are somewhat analogous to that of the @xmath123 state . the outer minimum has its origin in the interaction of the neutral @xmath119 and ionic @xmath153 configurations . starting with @xmath154 and increasing the field strength the depth of the outer well increases . the first change of the dissociation channel from @xmath155 to @xmath156 occurs in the regime @xmath157 . in the regime @xmath158 the position of the outer minimum increases with increasing field strength ( for @xmath133 it is already @xmath159 ) and the depth of the outer well decreases . due to the further increasing binding energy of the hydrogen negative ion @xmath160 state with increasing field strength we encounter a second change of the dissociation channel at @xmath161 to @xmath162 which causes the disappearance of the outer minimum and well . table 3 provides the corresponding data for the @xmath145 state . the pecs of the @xmath163 and @xmath164 states are shown in figures 2e and 2f , respectively . for both states the positions of the maxima and minima as well as the corresponding total energies show an irregular behaviour as a function of the field strength for @xmath165 . we therefore focus on the main features of these states . for certain regimes of the field strength we observe double well structures for the pecs . analogously to the @xmath166 states there exist additional outer minima and wells due to the interaction with the ionic configuration for certain field strength regimes . for @xmath167 the position of the first inner minimum decreases rapidly with increasing field strength whereas the corresponding dissociation energy increases . also we observe the existence of minima whose energies lie above the dissociation energy , i.e. the corresponding wells contain if at all metastable states . we remark that some of the above - discussed features , in particular those associated with small energy scales , might not be visible in the corresponding figures 2 but only in a zoom of the relevant regimes of internuclear distances of the considered pecs . we again emphasize that due to the large amount of data we do not present full pecs or data on the higher excited states @xmath168 which can be obtained from the authors upon request . in the present subsection we focus on the evolution of the complete spectrum of the excited @xmath169 states with increasing field strength . this will give us the complementary information to the evolution of individual states presented above . figure 3a - f shows the corresponding pecs for the field strengths @xmath170 , respectively . the pecs of the five energetical lowest excited states @xmath171 are hereby illustrated with full lines indicating their higher accuracy whereas the pecs of the electronic states @xmath172 are less accurate and illustrated with dotted lines . before we discuss the evolution with increasing field strength some general remarks are in order . the energy gap between the ground state @xmath173 and the first excited state @xmath105 is of the order of @xmath174 in field - free space and increases montonically with increasing field strength . at the same time the total energies of all states @xmath175 are shifted in lowest order proportional to @xmath176 with increasing field strength which is due to the raise of the kinetic energy in the presence of a magnetic field . in field - free space many of the dissocation channels of the pecs of excited @xmath7 states are degenerate due to the degeneracies of the field - free hydrogen atom ( see figure 1 and 4 ) . the major difference of the pecs in field - free space compared to those for weak fields is the removal of these degeneracies ( see , for example , figure 3a for @xmath177 ) . with increasing field strength figures 3a - d ( @xmath178 ) demonstrate the systematic lowering of the diabatic energy curve belonging to the ionic configuration @xmath97 . this diabatic curve passes through the spectrum with increasing field strength thereby causing an intriguing evolution of avoided crossings and corresponding potential wells for the individual states . at @xmath179 the fourth excited @xmath163 state acquires the ionic dissociation channel . the @xmath180 state thereby looses its outer potential well which was very well - pronounced in the absence of the external field . in the same course the @xmath123 state shows a number of avoided crossings with the @xmath105 state : it develops an additional outer minimum and well which is rather deep at @xmath181 accompanied by the flattening of the first inner well and the deepening of the second inner well . furthermore we observe for @xmath181 the appearance of a large number of avoided crossing among the higher excited states @xmath168 at @xmath182 . at @xmath183 the third excited @xmath145 state acquires the ionic dissociation channel . subsequently , i.e. with further increasing field strength , the second excited @xmath123 state ( see figure 3d ) and finally the first excited @xmath105 state acquire ionic character for sufficiently large internuclear distances . in the high field situation ( see figure 3e for @xmath184 ) only the energetically lowest excited state possess a well - pronounced double well structure and the overall picture is dominated by the fact that the pecs of the considered states possess a very similar shape and are energetically very close to each other in particular around the inner minimum at small internuclear distances . figure 3f shows for @xmath184 a zoom of the series of avoided crossings occuring for the higher excited states @xmath185 in the regime @xmath186 . the four energetically lowest states of @xmath187 symmetry at @xmath154 have been investigated in detail and with high accuracy in the literature @xcite . our results @xcite show a relative accuracy of @xmath65 for the energies of the @xmath188 state and of @xmath67 for the first two excited states i.e. the @xmath189 states . the energies of the @xmath190 state are significantly lower than the data presented in @xcite . the pecs for the @xmath191 presented in ref.@xcite for the first time are estimated to possess an accuracy of @xmath67 for the @xmath192 states and @xmath65 for the @xmath193 states . figure 4 shows the pecs of the ground as well as eight excited states of @xmath187 symmetry in the range @xmath194 on a logarithmic scale . the pec of the ground state @xmath188 of ungerade symmetry possesses a minimum at @xmath195 and a corresponding deep well . a closer look at the wave function reveals its ionic character for @xmath196 . with further increasing internuclear distance the ionic character of the wave function decreases and the corresponding dissociation channel is @xmath197 . the pec of the first excited @xmath198 state is similar to that of the ground state @xmath188 : its equilibrium internuclear distance is @xmath199 the dissociation channel is identical to that of the @xmath188 state . the depth of its single well is however only one third of the depth of the well of the @xmath188 state . for the higher excited states we observe a similar behaviour as in the case of the excited electronic states of @xmath200 symmetry . the pecs of the @xmath201 states possess a deep well around a minimum located approximately at @xmath202 . furthermore the @xmath203 states exhibit additional deep outer potential wells at large internuclear distances which arise due to the avoided crossings of the heitler - london configurations with the corresponding ionic configuration . the outer minimum of the @xmath204 state is located at @xmath205 and the corresponding well possesses a remarkable depth of @xmath206 : it is expected to contain a large number of long - lived vibrational states . tables 4 to 6 contain ( among the data in the presence of the magnetic field ) the total and dissociation energies at the equilibrium internuclear distances , the equilibrium internuclear distances and the total energies in the dissociation limit for the first to third excited @xmath207 states in the absence of the magnetic field . first of all we remark that the dissociation channels of the @xmath208 states coincide with those of the @xmath209 states for @xmath210 in the complete regime @xmath211 . the qualitative behaviour of the pecs of the @xmath209 states at large internuclear distances is therefore similar to that of the @xmath208 states discussed in the previous section . in particular many of the explanations and remarks provided there hold also for the present case of the @xmath209 states . before discussing the behaviour of the pecs of the individual excited @xmath209 states with increasing field strength some remarks concerning the lowest , i.e. ground state of @xmath11 symmetry are in order ( for its pec with increasing field strength see figure 6 ) . its dissociation energy increases monotonically with increasing field strength whereas its equilibrium internuclear distance increases slightly for weak fields and decreases significantly for increasingly stronger fields . as indicated above the asymptotic @xmath117 behaviour of the pecs of the @xmath212 and @xmath105 states is very similar . for @xmath184 the pec of the @xmath212 state possesses a peculiar shape which is largely determined by the ionic dissociation channel @xmath213 ( see figure 6f ) . for more details on this state we refer the reader to ref.@xcite . the first excited @xmath214 state possesses in field - free space an equilibrium internuclear distance @xmath215 . figure 5a shows the corresponding pec with increasing field strength for @xmath216 whereas figure 5b illustrates particularly the behaviour at large internuclear distances . in the regime @xmath217 the dissociation energy decreases slightly and the bond length increases . with further increasing field strength the dissociation energy increases drastically and the bond length decreases . for @xmath218 there exists a maximum and a corresponding additional outer minimum at large internuclear distances ( see figure 5b ) whose origin is again the emergence of the ionic configuration for the wave function of the @xmath214 state . figure 5b also demonstrates the similarity of the asymptotic @xmath219 behaviour of the pecs of the @xmath214 state in the regime @xmath220 . the corresponding data for the pecs of the first excited @xmath214 state are given in table 4 . turning to the second excited @xmath221 state we observe that the depth of the potential well located for @xmath124 at @xmath222 decreases for weak fields whereas it increases significantly for strong fields @xmath223 ( see figure 5c ) . the existence of an additional outer minimum for this state can be seen in figure 5d . in many respects a similar behaviour to that of the @xmath214 state is observed although , of course , the regimes of field strength for which the individual phenomena take place are different . table 5 contains the corresponding data of the pecs of the @xmath221 state . finally figures 5e and 5f show the pecs of the @xmath224 states with increasing field strength , respectively . they exhibit a number of maxima and minima most of which can however hardly be seen in figures 5e , f or occur at large internuclear distances . the origin of their existence are again the different ( ionic and neutral ) dissociation channels . these maxima and minima are present only for certain individually different regimes of the field strength . some of them are located above and some of them below the dissociative threshold . as can be seen the bond length ( belonging to the inner minimum ) decreases monotonically and the dissociation energy increases significantly above some critical value @xmath225 . the inner minimum and associated well possesses a remarkably large dissociation energy for strong fields . table 6 provides data on the pecs of the @xmath226 state . to finalize our discussion on the @xmath11 subspace we show in figure 6 the evolution of the spectrum with increasing field strength . figures 6a - f show the pecs for the @xmath227 states for the field strengths @xmath228 , respectively . analogously to the case of the @xmath7 subspace we observe for weak fields the removal of the degeneracies due to the field - free hydrogen atom in the dissociation limit . with increasing field strength we see the lowering of the diabatic energy line belonging to the ionic configuration which causes the appearance and disappearance of outer maxima , minima and corresponding outer potential wells until finally ( @xmath184 ) the @xmath212 state possess the ionic dissociation channel @xmath229 which is the origin of the peculiar shape of its pec . a number of further observations made for the manifold of the @xmath230 states above can also be seen for the @xmath227 states in figure 6 like , for example , the similar shape of the potential wells of the excited states in the high field limit . the hydrogen molecule is the most fundamental molecular system and of immediate importance in a variety of different physical circumstances . in spite of the fact that it has been investigated over the past decades in great detail and that our knowledge on this system has grown enormously there are plenty of questions and problems to be addressed even for the molecule in field - free space . as an example we mention certain highly excited rydberg states ( @xmath231 ) which , due to the ionic character of the binding for certain regimes of the internuclear distance , possesses a deep outer well at large distances which contains a considerable number of vibrational states . on the other hand the detailed knowledge of hydrogen ( even of highly excited states ) is of utmost importance for our understanding and interpretation of the astrophysically observed interstellar radiation . much less is known about the behaviour of the hydrogen molecule in strong magnetic fields . with increasing field strength the ground state of the molecule undergoes two transitions which are due to a change of the spin and orbital character , respectively . very recently the global ground state configurations have been identified for the parallel configuration ( there are good reasons which lead to the conjecture that the derived results hold for arbitrary angle of the internuclear and magnetic field axis ) both on the hartree - fock level @xcite and via a fully correlated approach @xcite . for low fields the ground state is of spin singlet @xmath7 symmetry , for intermediate fields the spin triplet @xmath15 state represent the ground state whereas in the high field regime the @xmath20 state is the energetically lowest state . the present work goes for the first time beyond the ground state properties and investigates excited states of the hydrogen molecule in the broad regime @xmath101 . we hereby focus on singlet states of both gerade as well as ungerade symmetry : up to seven excited states have been studied for the parallel configuration with a high accuracy of the obtained pecs . a variety of different phenomena have been observed out of which we mention here only the most important ones . double well structures observed in particular for the field - free @xmath232 states are severly modified in the presence of the field thereby showing a coming and going of new maxima and minima as well as corresponding wells . the overall tendency in the strong field limit is the development of deep inner wells containing a large number of vibrational states . in the course of the increasing field strength a fundamental phenomenon occurs which has a strong impact on the overall shape of the pecs . due to the fact that the hydrogen negative ion becomes increasingly bound with increasing field strength we encounter changes in the dissociation channels of individual states from neutral @xmath233 to ionic @xmath234 character . for a certain regime of field strength @xmath235 a certain excited state possesses therefore the ionic dissociation channel thereby modifying the asymptotic behaviour of its pec to an attractive coulombic tail . for weaker fields @xmath236 higher excited states possess this ionic dissociation channel whereas for stronger fields @xmath237 it belongs to increasingly lower excitations . these facts influence the overall appearance of the spectrum thereby creating features like outer potential wells and/or largely changing avoided crossings . the data on the pecs of the excited singlet states obtained here should serve as part of the material to be accumulated for the investigation of quasimolecular absorption features in magnetic white dwarfs . the investigation of excited triplet states of @xmath1 symmetry or of @xmath8 states , which are of equal importance , are left to future investigations . fruitful discussions with w.becken are gratefully acknowledged . the deutsche forschungsgemeinschaft is gratefully acknowledged for financial support . * figure 2 : * the evolution of the potential energy curves for some excited @xmath7 electronic states of the hydrogen molecule in the presence of a magnetic field @xmath101 . in detail are shown the evolution of the pecs for the ( a ) @xmath105 , ( b ) @xmath123 , ( c ) zoom of @xmath123 , ( d ) @xmath145 , ( e ) @xmath163 and ( f ) @xmath164 states , respectively . shown is the quantity @xmath102 where @xmath239 is the total energy . * figure 3 : * the spectrum of potential energy curves for the excited @xmath100 electronic states of the hydrogen molecule in the presence of a magnetic field @xmath101 with increasing field strength . in detail are shown the pecs for ( a ) @xmath240 ( b ) @xmath241 ( c ) @xmath242 ( d ) @xmath243 ( e ) @xmath244 and ( f ) zoom of @xmath245 , respectively . * figure 5 : * the evolution of the potential energy curves for some excited @xmath11 electronic states of the hydrogen molecule in the presence of a magnetic field @xmath101 . in detail are shown the evolution of the pecs for the ( a ) @xmath105 , ( b ) zoom of @xmath105 , ( c ) @xmath123 , ( d ) zoom of @xmath123 , ( e ) @xmath145 and ( f ) @xmath163 states , respectively . shown is the quantity @xmath102 where @xmath239 is the total energy . * figure 6 : * the spectrum of potential energy curves for the excited @xmath227 electronic states of the hydrogen molecule in the presence of a magnetic field @xmath101 with increasing field strength . in detail are shown the pecs for ( a ) @xmath247 ( b ) @xmath241 ( c ) @xmath242 ( d ) @xmath248 ( e ) @xmath249 and ( f ) @xmath245 , respectively . dddddddd & & & & & & & + 0.0 & 1.91 & 0.093122 & -0.718121 & 4.39 & 0.089241 & -0.714240 & -0.624999 + 0.001 & 1.91 & 0.093120 & -0.718117 & 4.39 & 0.089242 & -0.714239 & -0.624997 + 0.005 & 1.91 & 0.093061 & -0.718017 & 4.39 & 0.089252 & -0.714208 & -0.624956 + 0.01 & 1.91 & 0.092878 & -0.717703 & 4.39 & 0.089289 & -0.714114 & -0.624825 + 0.05 & 1.90 & 0.087831 & -0.708672 & 4.39 & 0.090459 & -0.711300 & -0.620841 + 0.1 & 1.88 & 0.077017 & -0.686953 & 4.38 & 0.093325 & -0.703261 & -0.609936 + 0.2 & 1.88 & 0.057630 & -0.633195 & 4.33 & 0.100709 & -0.676274 & -0.575565 + 0.5 & 1.89 & 0.040502 & -0.462472 & 4.08 & 0.122427 & -0.544397 & -0.421970 + 1.0 & 1.78 & 0.044819 & -0.135993 & 3.76 & 0.150618 & -0.241792 & -0.091174 + 2.0 & 1.54 & 0.067740 & 0.612336 & 3.37 & 0.191218 & 0.488858 & 0.680076 + 5.0 & 1.18 & 0.140991 & 3.130994 & 2.88 & 0.269224 & 3.002761 & 3.271985 + 10.0 & 0.95 & 0.245637 & 7.623918 & 2.55 & 0.351756 & 7.517799 & 7.869555 + 20.0 & 0.76 & 0.359056 & 16.960943 & 2.27 & 0.408351 & 16.911648 & 17.319999 + 50.0 & 0.56 & 0.575262 & 45.784366 & 1.96 & 0.471572 & 45.888056 & 46.359628 + 100.0 & 0.44 & 0.821768 & 94.614199 & 1.76 & 0.521822 & 94.914145 & 95.435967 + ddddddddddd & & & & & & & & & & + 0.0 & 2.03 & 0.035466 & -0.660465 & 3.27 & 0.038014 & -0.663013 & & & & -0.624999 + 0.001 & 2.03 & 0.035462 & -0.660458 & 3.27 & 0.038021 & -0.663017 & & & & -0.624996 + 0.005 & 2.03 & 0.035399 & -0.660305 & 3.27 & 0.038045 & -0.662951 & & & & -0.624906 + 0.01 & 2.03 & 0.035225 & -0.659851 & 3.27 & 0.038101 & -0.662727 & 18.97 & 0.000007 & -0.624633 & -0.624626 + 0.05 & 2.04 & 0.033199 & -0.649592 & 3.20 & 0.040478 & -0.656871 & 10.56 & 0.001510 & -0.617903 & -0.616393 + 0.1 & 2.03 & 0.030981 & -0.626596 & 3.06 & 0.048089 & -0.643704 & 10.02 & 0.006992 & -0.602607 & -0.595615 + 0.2 & 1.94 & 0.023894 & -0.563261 & 2.86 & 0.064882 & -0.604249 & 10.43 & 0.026352 & -0.565719 & -0.539367 + 0.5 & 1.90 & 0.030305 & -0.378302 & 2.73 & 0.079773 & -0.427788 & 10.69 & 0.066892 & -0.414907 & -0.348015 + 1.0 & 1.78 & 0.039842 & -0.041327 & 2.50 & 0.091331 & -0.092816 & 10.96 & 0.085978 & -0.087463 & -0.001485 + 2.0 & 1.54 & 0.042657 & 0.717581 & 2.22 & 0.089173 & 0.671065 & 12.14 & 0.079131 & 0.681107 & 0.760238 + 5.0 & 1.19 & 0.075290 & 3.248431 & 1.86 & 0.091565 & 3.232156 & 19.38 & 0.051644 & 3.272077 & 3.323721 + 10.0 & 0.96 & 0.131036 & 7.749942 & 1.63 & 0.093390 & 7.787588 & 85.58 & 0.011423 & 7.869555 & 7.880978 + 20.0 & 0.76 & 0.275591 & 17.095639 & 1.44 & 0.142076 & 17.229154 & & & & 17.371230 + 50.0 & 0.56 & 0.605249 & 45.931209 & 1.22 & 0.250618 & 46.285840 & & & & 46.536458 + 100.0 & 0.45 & 0.976059 & 94.770524 & 1.09 & 0.357751 & 95.388832 & & & & 95.746583 + ddddddddddd & & & & & & & & & & + 0.0 & 1.97 & 0.099408 & -0.654963 & & & & 11.21 & 0.049597 & -0.605152 & -0.555555 + 0.001 & 1.97 & 0.099397 & -0.654944 & & & & 11.21 & 0.049604 & -0.605151 & -0.555547 + 0.005 & 1.97 & 0.099051 & -0.654469 & & & & 11.21 & 0.049686 & -0.605104 & -0.555418 + 0.01 & 1.97 & 0.098011 & -0.653029 & & & & 11.21 & 0.049930 & -0.604948 & -0.555018 + 0.05 & 1.99 & 0.081038 & -0.625896 & 2.93 & 0.068063 & -0.612921 & 11.39 & 0.055502 & -0.600360 & -0.544858 + 0.1 & 2.05 & 0.075239 & -0.597699 & & & & 12.04 & 0.06419 & -0.586650 & -0.522460 + 0.2 & & & & 2.46 & 0.067543 & -0.546109 & 15.62 & 0.059554 & -0.538120 & -0.478566 + 0.5 & 1.91 & 0.022150 & -0.350029 & 2.35 & 0.035929 & -0.363808 & 50.06 & 0.020138 & -0.348017 & -0.327879 + 1.0 & 1.77 & 0.019181 & -0.010815 & 2.17 & 0.030750 & -0.022384 & & & & 0.008366 + 2.0 & 1.53 & 0.053756 & 0.750088 & 1.95 & 0.058173 & 0.745671 & & & & 0.803844 + 5.0 & 1.19 & 0.142822 & 3.283035 & 1.67 & 0.116673 & 3.309184 & & & & 3.425857 + 10.0 & 0.96 & 0.257211 & 7.786043 & 1.50 & 0.176832 & 7.866422 & & & & 8.043254 + 20.0 & 0.76 & 0.427463 & 17.133303 & 1.34 & 0.249318 & 17.311448 & & & & 17.560766 + 50.0 & 0.56 & 0.768452 & 45.971004 & 1.16 & 0.362134 & 46.377322 & & & & 46.739456 + 100.0 & 0.45 & 1.142090 & 94.811929 & 1.05 & 0.464011 & 95.490008 & & & & 95.954019 + dddddddddd & & & & & & & & & + 0.0 & 2.09 & 0.040772 & -0.665771 & & & & & & -0.624999 + 0.001 & 2.08 & 0.040770 & -0.665766 & & & & & & -0.624996 + 0.005 & 2.09 & 0.040697 & -0.665603 & & & & & & -0.624906 + 0.01 & 2.09 & 0.040477 & -0.665103 & & & & & & -0.624626 + 0.05 & 2.12 & 0.037068 & -0.653461 & & & & & & -0.616393 + 0.1 & 2.13 & 0.036237 & -0.631852 & 9.37 & 0.008169 & -0.603784 & 5.16 & -0.602345 & -0.595615 + 0.2 & 2.12 & 0.043756 & -0.583123 & 10.09 & 0.027637 & -0.567004 & 4.60 & -0.555876 & -0.539367 + 0.5 & 2.03 & 0.065201 & -0.413216 & 10.34 & 0.068386 & -0.416401 & 4.01 & -0.382377 & -0.348015 + 1.0 & 1.85 & 0.079251 & -0.080736 & 10.55 & 0.087513 & -0.088998 & 3.66 & -0.038398 & -0.001485 + 2.0 & 1.59 & 0.085790 & 0.674448 & 11.46 & 0.080259 & 0.679979 & 3.34 & 0.744375 & 0.760238 + 5.0 & 1.22 & 0.121736 & 3.201985 & 10.49 & 0.052138 & 3.271583 & 2.95 & 3.349758 & 3.323721 + 10.0 & 0.98 & 0.178133 & 7.702800 & 9.70 & 0.011729 & 7.869204 & 2.69 & 7.955928 & 7.880933 + 20.0 & 0.78 & 0.321481 & 17.049749 & 9.27 & 0.000271 & 17.370959 & 2.46 & 17.464313 & 17.371230 + 50.0 & 0.57 & 0.646721 & 45.889737 & 9.04 & 0.000168 & 46.536290 & 2.21 & 46.635075 & 46.536458 + 100.0 & 0.45 & 1.012902 & 94.733681 & 9.04 & & 95.746735 & 2.05 & 95.846945 & 95.746583 + dddddddddd & & & & & & & & & + 0.0 & 2.03 & 0.081441 & -0.636996 & 11.12 & 0.049964 & -0.605519 & 5.65 & -0.567698 & -0.555555 + 0.001 & 2.03 & 0.081426 & -0.636973 & 11.12 & 0.049971 & -0.605518 & 5.64 & -0.567692 & -0.555547 + 0.005 & 2.03 & 0.081009 & -0.636427 & 11.12 & 0.050053 & -0.605471 & 5.64 & -0.567528 & -0.555418 + 0.01 & 2.03 & 0.080107 & -0.635125 & 11.18 & 0.050291 & -0.605309 & 5.67 & -0.567116 & -0.555018 + 0.05 & 2.04 & 0.074463 & -0.619321 & 11.32 & 0.055802 & -0.600660 & 5.45 & -0.555308 & -0.544858 + 0.1 & 2.05 & 0.071675 & -0.594135 & 12.01 & 0.064313 & -0.586773 & 5.40 & -0.530941 & -0.522460 + 0.2 & 2.03 & 0.062081 & -0.540641 & 15.62 & 0.059632 & -0.538198 & 5.42 & -0.477396 & -0.478566 + 0.5 & 1.94 & 0.035148 & -0.363027 & 50.06 & 0.020138 & -0.348017 & 5.20 & -0.297160 & -0.327879 + 1.0 & 1.78 & 0.033120 & -0.024754 & & & & 4.92 & 0.058491 & 0.008366 + 2.0 & 1.54 & 0.068597 & 0.735247 & & & & 4.56 & 0.858143 & 0.803844 + 5.0 & 1.20 & 0.158473 & 3.267384 & & & & 4.06 & 3.490593 & 3.425857 + 10.0 & 0.96 & 0.272912 & 7.770342 & & & & 3.70 & 8.118636 & 8.043254 + 20.0 & 0.77 & 0.442635 & 17.118131 & & & & 3.38 & 17.648590 & 17.560766 + 50.0 & 0.56 & 0.782055 & 45.957401 & & & & 3.02 & 46.845692 & 46.739456 + 100.0 & 0.45 & 2.154264 & 94.799755 & & & & 2.81 & 96.074966 & 96.954019 + ddddddddddd & & & & & & & & & & + 0.0 & 2.00 & 0.078543 & -0.634098 & 5.67 & 0.010196 & -0.565751 & & & & -0.555555 + 0.001 & 2.00 & 0.078531 & -0.634077 & 5.67 & 0.010194 & -0.565740 & & & & -0.555546 + 0.005 & 2.00 & 0.078211 & -0.633537 & 5.67 & 0.010161 & -0.565487 & 33.89 & 0.000058 & -0.555384 & -0.555326 + 0.01 & 2.00 & 0.077103 & -0.631765 & 5.67 & 0.010059 & -0.564721 & 34.18 & 0.000338 & -0.555000 & -0.554662 + 0.05 & 2.01 & 0.063148 & -0.603904 & 5.56 & 0.009249 & -0.550005 & 42.09 & 0.004106 & -0.544862 & -0.540756 + 0.1 & 2.02 & 0.060038 & -0.577452 & 5.50 & 0.009923 & -0.527337 & 36.87 & 0.005048 & -0.522462 & -0.517414 + 0.2 & 2.00 & 0.057601 & -0.522906 & 5.79 & 0.002238 & -0.467543 & 33.92 & 0.000006 & -0.465309 & -0.465305 + 0.5 & 1.92 & 0.062864 & -0.343463 & & & & 34.57 & 0.000001 & -0.280600 & -0.280599 + 1.0 & 1.76 & 0.082384 & -0.003776 & & & & & & & 0.078608 + 2.0 & 1.53 & 0.123535 & 0.757397 & & & & & & & 0.880932 + 5.0 & 1.19 & 0.224179 & 3.290662 & & & & & & & 3.514841 + 10.0 & 0.96 & 0.348221 & 7.794138 & & & & & & & 8.142359 + 20.0 & 0.76 & 0.528386 & 17.142170 & 3.52 & & 17.707128 & & & & 17.670556 + 50.0 & 0.56 & 0.882660 & 45.981219 & 3.09 & & 46.897984 & & & & 46.863879 + 100.0 & 0.45 & 1.266401 & 94.823274 & 2.82 & & 96.127449 & & & & 96.089675 + | excited states of the hydrogen molecule subject to a homogeneous magnetic field are investigated for the parallel configuration in the complete regime of field strengths @xmath0 . up
to seven excitations are studied for gerade as well as ungerade spin singlet states of @xmath1 symmetry with a high accuracy .
the evolution of the potential energy curves for the individual states with increasing field strength as well as the overall behaviour of the spectrum are discussed in detail .
a variety of phenomena like for example the sequence of changes for the dissociation channels of excited states and the resulting formation of outer wells are encountered .
possible applications of the obtained data to the analysis of magnetic white dwarfs are outlined . |
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granular media are large conglomerations of discrete , solid particles , such as sand , gravel , or powder , with unusual , interesting dynamics @xcite . a one - dimensional system of spherical beads in a lattice is one of the simplest representation of granular media substrates , wherein each bead represents a grain of material . in this approximation , the position of a particular bead is based on forces resulting from its interaction with its two nearest neighbors @xcite . this context has proved especially fruitful for investigating numerous aspects of the nonlinear dynamic response of such bead chain systems @xcite . a particular focal point of emphasis has been on the study of one - dimensional granular crystals . the availability of a wide variety of materials and bead sizes , as well as the tunability of the response within the weakly or strongly nonlinear regime renders such crystals an ideal playground for the investigation of a variety of fundamental concepts ranging from nonlinear waves and discrete breathers to shock waves , defect modes and bifurcation phenomena among many others . however , this tunability also makes these crystals promising candidates for a wide variety of engineering applications such as shock and energy absorbing materials @xcite , actuating and focusing devices @xcite , and sound scramblers or filters @xcite . one aspect that has not been studied , to the best of our knowledge , in such prototypical granular lattices is that of directed transport via the so - called ratcheting effect . directed ratchet transport ( drt ) , is defined as the directed transmission of an entity despite the lack of a net external force acting upon it @xcite . this phenomenon has been associated with applications in dc current in semiconductors @xcite , the motion of fluxons in josephson junctions @xcite , bose - einstein condensates @xcite , cold atoms in optical lattices @xcite , among many others . furthermore , drt occurring in granular systems has been associated with the study of molecular motors @xcite . as detailed in ref . @xcite , the emergence of drt behavior is associated with the breaking of symmetries , which can be achieved by either a reshaping of the system s potential or by introducing an external forcing @xcite . for instance , drt is present when a granular material is placed in a vertically vibrating sawtooth surface profile @xcite . typically , drt is studied ( for a single particle or a collection of particles ) when the external input acts on the system as a whole @xcite . our aim in this work , on the other hand , is to force a _ single _ particle to achieve global drt in the context of granular crystals . in what follows in section ii , we present the basic ( fermi - pasta - ulam type ) model that is widely accepted as representing the one - dimensional dynamics of a granular crystal @xcite with parameters that are adapted from recent experiments on the field such as refs . @xcite . we then proceed to use the tunability of the system through actuating one bead within the chain by means of suitable biharmonic forcing that will be the source of our drt through its induced breaking of time - reversal and half - period time shift symmetries . in particular , in section iii , we will propose a biharmonic forcing of the system involving the simplest pair of two frequencies @xmath0 relevant for such drt ( i.e. , with @xmath1 coprimes and @xmath2 odd ) , namely @xmath3 and @xmath4 @xcite . in section iv , we will develop diagnostic quantities evaluating the relative magnitude of the clearly discernible in our numerical computations drt . we will analyze the dependence of the induced asymmetry in the crystal response on both the frequency of the drive @xmath5 , as well as on the relative strength of the two terms in the biharmonic forcing , as controlled by the corresponding parameter @xmath6 . the former analysis will separate different regimes in our observation of drt , namely the non - permanent deformation of the crystal that we will refer to as temporal ratcheting and the permanent deformation thereof that we will refer to as spatial ratcheting . the clear distinction between these two regimes is an especially intriguing feature of our current setup . the latter analysis ( over @xmath6 ) will provide a means for optimizing the ensuing transport which can both be theoretically understood and , in principle , experimentally exploited . finally , we consider in section v , the modification of the above features in the more experimentally realistic setup incorporating dissipation . we find there that the relevant phenomenology is modified dramatically , including even a potential reversal of the direction of the current ( for sufficiently low driving frequencies ) . finally , in section vi we summarize our findings and present a number of directions for future study . to ensure that the beads remain in contact , we consider a horizontal lattice that is precompressed on both ends with a force @xmath7 resulting in a static bead displacement @xmath8 ( see fig . [ fig : beadlattice ] ) . the existence of the precompression also serves to ensure that a linear spectrum of excitations exists in the lattice ( see details below ) . with these considerations , based on the hertzian law of spherical point contacts , a system comprised of @xmath9 identical beads can be described by the following newtonian equation @xcite : @xmath10_+^\frac{3}{2 } - a[\delta_0+u_{i}-u_{i+1}]_+^\frac{3}{2}\ ] ] where @xmath11_+ = { \rm max}\left\{0,y\right\}$ ] , @xmath12 is the bead mass , @xmath13 is the displacement of the center of the @xmath14th bead from its equilibrium position , and @xmath15 is the hertzian constant calculated as @xmath16 where @xmath17 , @xmath18 , @xmath19 are , respectively , the bead s radius , young s elastic modulus , and poisson s ratio . the static displacement is @xmath20 . in line with the experiments of refs . @xcite , the parameter values listed in table [ tab : defparam ] were used . .default parameters for bead - lattice system . [ cols="^,^,^",options="header " , ] as we will show later , a critical factor affecting the presence / type of drt is the acoustic phonon band cutoff frequency . plane wave solutions to the system follow the dispersion relation @xmath21 @xcite , where @xmath22 is the wave number and @xmath23 is the temporal frequency . we see that this relationship is periodic ( with period @xmath24 ) and that there is a cutoff frequency @xmath25 , above which plane wave solutions can not propagate . the maximal frequency value occurs at the boundaries of the @xmath26 interval , which correspond to the smallest allowable wavelength . substituting @xmath27 into the dispersion relation yields the cutoff frequency , @xmath28 therefore , @xmath29 defines the range of propagating frequencies , called the acoustic band . frequencies @xmath30 lie within the band gap and can not propagate through the lattice as plane waves . with the parameter values given in table [ tab : defparam ] , we have @xmath31 khz . in terms of angular frequency , @xmath32 40.31 @xmath33 , which is the critical frequency used from this point forward . notice that all the frequencies that will be mentioned hereafter will be measured in @xmath33 . typically , drt behavior is observed in the velocity of a single particle or in that of a coherent structure such as a solitary wave @xcite . in the case of the granular crystal though , drt will be observed ( and examined ) throughout the system as a whole . consider a lattice where each bead begins at its equilibrium position with no initial velocity . to introduce energy into the system , the @xmath34th bead , located at the center of the lattice , is controlled by the following biharmonic , periodic function @xmath35,\ ] ] where @xmath36 is time , @xmath37 is the amplitude , @xmath5 is the frequency , @xmath6 is the biharmonic weight , and @xmath38 is a phase . to maintain uniformity on each side of @xmath34th bead , we assume @xmath9 is odd . the motivation for choosing to control the displacement of the central bead is that we envisage the possibility of doing experimental drt studies in the future where the position of the central bead will be controlled by an actuator . an essential characteristic of this functional form is that it has a zero - integral over one period , indicating that the function is not biased in any direction . in other words , the @xmath34th bead s temporal center of mass , relative its equilibrium position , is zero . consequently , any directed behavior observed must be attributed to drt rather than a preferential direction for the input . it is relevant to note that the prescription of the motion of the @xmath34th bead is tantamount to introducing a force , with the same characteristics , into its nearest neighbors through the equations of motion ( [ eomcomp ] ) . for @xmath39 each bead orbit on one side of the lattice corresponds directly to an orbit on the other side of the lattice traveling in the opposite direction , translated by a half - period delay . these orbits exactly cancel each other out and thus there is no drt . however , when @xmath40 , the symmetry of @xmath41 is broken and drt can occur @xcite . the system is numerically solved using a fourth - order runge - kutta scheme . the conservation of total energy is used to determine an appropriate time step . the final integration time @xmath42 varies based on the frequency @xmath5 , but is always selected so that it is an integer multiple of @xmath43 , the period of @xmath41 . @xmath9 also varies with @xmath5 , but is always sufficiently large so that energy from @xmath34th bead s oscillation never reaches the 1st or @xmath9th bead . we consider values of @xmath5 ranging from 10 to 40 and set @xmath6 equal to @xmath44 . in section [ sec : ip&r ] , we demonstrate that these parameters result in drt towards the right - hand side of the lattice . it is possible to change this direction by adding an additional phase mismatch between the two harmonics of the driver @xmath41 ( results not shown here ) . figure [ fig : varyamp ] illustrates drt magnitude ( quantification is discussed in section [ sec : ip&r ] ) for values of @xmath37 ranging from @xmath45 to @xmath46 . these relatively small values of the forcing amplitude @xmath37 ensure that a ( relatively ) small amount of energy is introduced into the system so the beads always remain in contact with each other . the relationship between @xmath37 and the drt magnitude is clearly nonlinear , as the average slope of the lines in the log - log plot in fig . [ fig : varyamp ] is about 2.9725 , which indicates an essentially cubic ( gain ) relationship . based on these findings , the remaining simulations have @xmath47 in order to exploit most of the nonlinear gain but also avoid losing contact between all beads for all times . in order to quantify the drt displayed by the bead chain , let us define the quantity @xmath48 corresponding to the average of the hertzian forces on either side of the @xmath14th bead integrated over time . this choice of drt measure is inspired by the fact that a piezo embedded inside a bead precisely measures the average of the hertzian forces felt by the adjacent beads . it is important to mention at this stage that drt could be captured using many possible measures . in fact , we also used , instead of @xmath48 , the actual forces acting on each bead and other combinations thereof and the results are qualitatively similar ( results not shown here ) . we should point out that it is essential to consider the entire space of possible phases @xmath38 in @xmath41 . this allows the full spectrum of the function to be sampled without biasing any direction based on the initial phase of the driver . to do this in our numerical experiments , we consider sixteen values of @xmath38 , equally spaced throughout one period of @xmath41 and define @xmath49 as the average of @xmath48 over these phases . to create profiles which will allow drt behavior to be observed , we compare the normalized difference of @xmath49 for pairs of beads equidistant from the center bead , that is @xmath50 where @xmath51 ) . by monitoring @xmath52 over time , a drt profile is constructed ( see fig . [ fig : impprof ] ) . + + + in fig . [ fig : impprof ] , for each value of @xmath53 , a contour plot illustrating the drt profile as a function of @xmath54 and @xmath36 is provided . additionally , the right panels show the asymmetry indicator profile of eq . ( [ impdif ] ) ) at a set of particular times , written in terms of oscillations of the center bead . a non - zero value of @xmath52 indicates the preferential transport of force in one direction , that is , drt . we see that , after transient behavior , all significant values of @xmath52 are negative , indicating the presence of drt towards the right hand side of the lattice ( see eq . ) . figures [ fig : impprof](a),(b ) , illustrate the drt profiles for @xmath55 , where both @xmath5 and @xmath56 are below @xmath57 . after a transient time interval , we observe a drt `` wave '' , or cone , advancing as time progresses ( and leaving no drt behind it ) . for this reason we call this behavior _ temporal drt_. for a given time , let @xmath58 denote the value of @xmath54 at the outer edge of the cone ( that effectively travels at the speed of sound within the medium ) , @xmath59 denote the value of @xmath54 at the inner edge of the cone , and @xmath60 denote the value of @xmath54 for which @xmath61 , and therefore drt , is maximal . for @xmath62 , we have @xmath63 since the energy from the forcing function has not yet reached beads offset this far from the center bead . for the region defined by @xmath64 , there exists a positive , approximate - linear relationship ( with respect to @xmath54 ) describing the magnitude of the drt . similarly , for @xmath59@xmath65 , there exists a negative approximate linear relationship describing drt magnitude . the drt wave has already moved through the region defined by @xmath66 . the characteristic property of this class of behavior , which we call class i , is that for @xmath66 , @xmath67 , indicating drt is _ no longer present _ shortly after the wave has left a region . class ii behavior is observed by considering a forcing frequency of 20 , as shown in figs . [ fig : impprof](c),(d ) . here , @xmath57 is greater than @xmath5 and slightly larger than @xmath56 . after allowing for transient time , the regions defined by @xmath62 , @xmath64 , and @xmath59@xmath65 , display the same qualitative behavior as the previous case . however , what clearly distinguishes this region is that for @xmath68 we see a qualitatively different result , namely , for each @xmath54 , @xmath52 is approximately equal to a nonzero constant . this is indicative of an equilibrium drt state defined by the spatial extent of the region through which the ratcheting wave has already passed . this effect and the `` kink''-like pattern that it leads to ( rather than the pulse like structure of class i ) in the context of the asymmetry indicator @xmath52 is hereafter referred to as _ spatial drt_. this fundamental distinction of regimes of temporal and spatial drt is , arguably , one of the most interesting traits observed herein , and to our knowledge , has not been reported before , although we believe that it should be more general than the particular realization considered herein . in figs . [ fig : impprof](e),(f ) , @xmath5 = 30 and thus @xmath69 and @xmath70 . as a result , a different behavior that will be characterized hereafter as belonging to class iiia is observed . the features are similar to class ii , but now the magnitude of the spatial drt is approximately equal to @xmath71 , the maximal temporal drt magnitude . otherwise said , the tail of the kink associated with the asymmetric deformation of the lattice , rather than having the linear profile of class ii , it is essentially flat . as shown in figs . [ fig : impprof](g),(h ) , where @xmath72 , as @xmath5 approaches @xmath57 , the drt profile remains qualitatively the same , but the slope of the ( approximately ) linear relationship for @xmath64 decreases . in our kink - based visualization of the corresponding @xmath52 s , this regime is associated not with the translation of the structure over the lattice which roughly preserves its shape , as in class iiia . instead , it appears associated predominantly with the widening of the relevant spatial structure in this regime that we will refer to as class iiib . for @xmath73 , neither of the plane waves comprising @xmath41 can propagate . as a result , the spatial and temporal drt behavior breaks down . this can be discerned in figs . [ fig : freq - drt ] , and [ fig : freq - vel ] where the drt magnitude and velocity ( that we will now proceed to define more precisely ) approach zero as @xmath5 approaches @xmath57 . to explore the effects of parameter variation on the magnitude of the drt , we now establish an additional ratcheting metric . for each of the drt profiles under consideration , @xmath74 represents the maximal amount of temporal ratcheting at any given time . we use this value as the temporal drt metric . on the other hand , when spatial ratcheting is present , it is first observed by comparing beads adjacent to the @xmath34th bead , that is at @xmath75 . as time progresses , the behavior spreads out from the center bead and is also observed for larger values of @xmath54 . we observe that @xmath52 is approximately the same for each @xmath54 exhibiting spatial drt behavior ; therefore the magnitude of spatial ratcheting can be quantified by means of @xmath76 . both @xmath74 and @xmath76 vary slightly over time . nevertheless , we have verified that this variation is small and , therefore , choose to measure @xmath76 and @xmath74 at the final integration time @xmath77 . figure [ fig : freq - drt ] illustrates the magnitude of spatial and temporal drt , based on the above diagnostic , for a wide range of representative frequencies . for class i ( @xmath78 ) , the magnitude of temporal drt slowly increases with the frequency . this regime corresponds to both input frequencies of the forcing ( @xmath5 and @xmath56 ) being below the cutoff frequency @xmath79 . as it can be noticed in fig . [ fig : freq - drt ] , the spatial drt starts appearing once the second harmonic ( @xmath56 ) of the driver gets closer to the cutoff frequency ( see left vertical dashed line ) . this seems to be an effect of the nonlinear response of the system that `` widens '' the region of the cutoff frequency . in fact , class ii corresponds to the region of frequencies where the second harmonic of the driver transitions from being transmitted to completely being stopped due to the cutoff frequency . it is interesting that for class ii and iiia , defined by @xmath80 corresponding to @xmath69 but @xmath56 close to ( under or over ) @xmath57 , the temporal drt magnitude is approximately constant . however , as the first harmonic ( @xmath5 ) starts getting close to the cutoff frequency ( see right vertical dashed line ) , naturally , both spatial and temporal drt start to disappear and eventually vanish , as expected , once both , first and second , driver harmonics are inside the forbidden gap . the effect of the first harmonic starting to approach the cutoff frequency begins at , approximately , @xmath81 , corresponding to the onset of class iiib behavior , the magnitude of temporal drt begins to sharply decrease as @xmath5 approaches @xmath57 . this lack of drt for higher frequencies is consistent with the drt breaking down for @xmath73 . on the other hand , spatial drt significantly increases as we move from class i to class ii and subsequently iiia , and it also , in turn , sharply decreases in the case of class iiib . we define the _ outer - cone horizon _ as the location of the onset of temporal drt and the _ inner - cone horizon _ as the location of the onset of spatial drt . the velocities of the outer and inner horizon are calculated numerically for values of @xmath5 ranging from 10 to 40 . for each frequency , the velocity of the outer - cone horizon approximately corresponds to the sound velocity of the system , @xmath82 ( see filled dots in fig . [ fig : freq - vel ] ) . this is a consequence of the nonlinearity of the system that `` mixes '' the frequencies introduced by the forcing and thus excites all modes . in contrast to the behavior observed by the outer - cone velocity , as shown in fig . [ fig : freq - vel ] ( see open dots ) , the inner - cone velocity strongly depends on the forcing frequency . for small values of @xmath5 , where both @xmath5 and @xmath56 are smaller than @xmath57 , there is essentially no spatial drt , as the two cones propagate with essentially the same speed forming the `` pulse '' observed in the asymmetry indicator @xmath52 . as the frequency increases through class i and toward class ii , the velocity of the inner - cone horizon decreases significantly until the threshold @xmath70 is crossed . at that point the inner - cone velocity approaches zero . within class ii , the continuously decreasing velocity of the inner - cone forms the tail of the kink discussed in connection to fig . [ fig : impprof ] . past the point of @xmath83 , the inner - cone velocity abruptly increases to a value similar to that for the lower frequencies . subsequently , in class iiia , the velocity decreases as the case where both @xmath5 and @xmath56 are larger than @xmath57 is approached . class iiib corresponds to a larger rate of decreasing velocity . interestingly , the spatio - temporal wave velocities are independent of the choice of @xmath6 . it is interesting to point out at this stage that the presence of drt in our system is a direct consequence of the symmetry breaking provided by the external forcing when @xmath84 [ see eq . ( [ biharmonic2 ] ) ] . however , it is also important to note that nonlinearity is also a key ingredient for the presence of drt . in fact , if the hertzian forces in eq . ( [ eomcomp ] ) are replaced by linear ( hooke ) springs , drt is no longer present . the absence of drt for the linear force case is a consequence of the fact that the drt corresponding to a phase @xmath38 is the negative of the one for phase @xmath85 , where @xmath43 is the period of the driver , and thus providing a cancellation when drt is averaged through all possible phases of the driver . finally , it is also important to stress that , as discussed earlier and depicted in fig . [ fig : varyamp ] , the magnitude of drt is , approximately , proportional to the cube of the forcing amplitude . therefore , nonlinearity in the system is not only necessary for observing drt , but it also provides a nonlinear enhancement of the drt magnitude with respect to the input amplitude . @xmath86{fig6a.eps } \\ \includegraphics[width = 5.5cm]{fig6b.eps } \end{array}$ ] in ref . @xcite , it was shown that when considering the forcing function @xmath41 , the magnitude of drt behavior is due to two competing effects : the increase in the degree of symmetry breaking and the decrease in the transmitted impulse over a half - period , which is denoted as effective symmetry breaking . it was shown that the drt behavior is optimally enhanced for @xmath87 . we now demonstrate that this result holds for the granular crystal case under consideration . we consider forcing frequencies of @xmath5 = 17.5 and 30 . figure [ fig : eta - drt ] illustrates the spatio - temporal drt magnitude as a function of @xmath6 . as is expected , for @xmath88 , the single harmonic does not induce drt behavior . as @xmath6 increases , the magnitude of drt increases until a maximum value for ratcheting is reached at @xmath89 ( in our case since drt is towards the left , the maximum ratcheting effect corresponds to a minimum for drt ) , with the exception of spatial drt for @xmath5 = 17.5 . the magnitude then decreases until drt is again not present at @xmath90 . to further explore and quantitatively appreciate this result , consider a generic biharmonic forcing function @xmath91 , where @xmath92 are amplitudes , @xmath93 are phases , @xmath5 is the frequency and @xmath94 and @xmath95 are coprimes . it can be shown that the ratchet velocity @xmath96 where @xmath97 is a system - dependent constant @xcite . with the parameters in @xmath41 , we have @xmath98 . observe that this function has a minimum ( for @xmath99 ) at @xmath87 , which is consistent with the numerical simulations . by setting @xmath100 equal to the numerically - calculated drt magnitude at @xmath87 , we can solve for the free parameter @xmath97 . these curves are shown in fig . [ fig : eta - drt ] . there is a striking agreement between the calculated numerical drt magnitudes and theoretical curves . we see less of a correspondence for spatial drt for @xmath101 . in general , for other frequencies considered , the temporal drt matched the theoretical curves more consistently than spatial drt , particularly for @xmath5 near @xmath102 and @xmath57 . as a side note , it is possible to draw physical intuition for the optimal biharmonic weight being at 2/3 if one considers the ideal ratcheting forcing : a sawtooth function . expanding a sawtooth function in fourier series and keeping only the first two harmonics it is straightforward to show that their ratio is 2 . in our case , when @xmath103 , we precisely get a ratio between the two harmonics of @xmath104 . in other words , the optimal ratcheting forcing , i.e. @xmath103 , is the best possible approximation to a sawtooth function when using two harmonics . to more closely represent physical reality , dissipation is introduced into the uniform granular crystal by augmenting eq . : @xmath105_+^\frac{3}{2 } - a[\delta_0+u_{i}-u_{i+1}]_+^\frac{3}{2}-\frac{m}{\tau}\dot{u}_i,\ ] ] where @xmath42 is a dissipation constant set equal to @xmath106 . this value was chosen to match the dissipation constant used in the experiments of ref . @xcite . to investigate the effects of friction , we numerically integrate the four representative frequency cases presented earlier and illustrate the corresponding results in fig . [ fig : impproffric ] . in order to allow sufficient time for transient behavior , @xmath77 , the final integration time , and @xmath9 were considerably larger than for the non - friction simulations . + + + when friction is present , we observe that all cases exhibit qualitatively similar behavior . namely , unlike the frictionless case , the velocity of the temporal drt is not set by the sound velocity of the system , but instead decreases as time progresses . furthermore , in all dissipative cases , we have observed that spatial drt is weakened by the presence of dissipation and that all cases present a similar spatial structure which seems to involve a progressively widening ( i.e. , dispersing ) kink state . it is worth noticing that for @xmath107 , the drt profile is slightly different than for the other cases . the maximal temporal drt value , @xmath74 now occurs at @xmath108 . furthermore , as time increases , @xmath109 for @xmath110 . this is indicative of a small amount of spatial drt . in fact , the behavior is similar to class iiib for the frictionless system . for @xmath55 , the presence of friction radically changes the characteristics of the drt profiles . temporal drt is present , but for @xmath111 periods , all non - zero values of @xmath52 are now positive , indicating that the drt direction has _ reversed _ and now favors left propagation . for @xmath112 , the minimum of @xmath52 is less than zero illustrating the remnants of rightward drt . in each of the drt profiles provided in fig . [ fig : impproffric ] , we observe two regimes of drt propagation . initially , the temporal drt `` wave '' travels at the sound velocity of the system . however , as time progresses , the dissipation tend to slow down the propagation of this wave . similarly , dissipation is also responsible for progressively damping out the drt magnitude . eventually , a steady state solution is reached where the energy being pumped into the system by the forcing is balanced by dissipation . finally , it is worth mentioning that , despite taking the optimal value for the forcing amplitude to exploit maximum drt gain , the drt magnitude is on the order of @xmath113@xmath114 which amounts to a @xmath115@xmath116 of biased transport between left and right propagation . although these values are relatively small , drt should be possible to measure in current experimental setups . in this work , a one - dimensional granular chain ( crystal ) was considered where the position of the center bead was prescribed by a biharmonic forcing function . this functional form is known to induce ratcheting . yet , in our case , a distinguishing characteristic was the system - wide emergence ( i.e. in space - time ) of directed ratchet transport ( drt ) in the force profiles . the regimes where temporal ( transient ) ratcheting and spatial ( i.e. , with a permanent spatial `` imprint '' over the lattice ) ratcheting were identified as a function of the system s frequency . the relationship between the frequencies @xmath5 and @xmath56 of the forcing function and the cutoff frequency @xmath57 of the system determined the characteristics of the observed drt and its separation into different classes . in the class i pertaining to temporal ratcheting , a drt wave " traveled away from the center bead at the sound velocity of the system . once the temporal drt wave moved through a region , a steady - state was induced in this region , wherein all bead pairs exhibited similar drt magnitude . if this value was non - zero , it corresponded to class ii and the so - called spatial drt . the modification of the form of spatial ratcheting past the regime where @xmath83 gave rise to yet another regime that was referred to as class iii . the frequency , @xmath5 , and biharmonic weight , @xmath6 of the forcing function were varied so that the response of the magnitude of drt and the velocity of the drt waves " could be determined . while the wave velocity was independent of the biharmonic weight , @xmath87 maximized the magnitude of spatio - temporal drt , in accordance with the expectations of refs . friction was subsequently introduced into the system , leading to weakening of the ratcheting effect and a rather uniform spatial form of its profile in classes ii and iii . yet , it was class i that was most significantly affected by the inclusion of friction within the system , which resulted in drt switching directionality from right to left . having paved the way for the consideration of ratchet effects in granular crystals , there are numerous directions along which the present study can be extended . it is certainly of interest to attempt to expand the range of considered materials and parameters ( as well as that of heterogeneous systems such as dimers , trimers @xcite ) and of a wider range of forcing frequencies and displacement parameters . a key aspect of such a broader parametric effort is to try to maximize the relevant drt , so as to render it more accessible to potential experiments . another important direction of particular interest is to attempt to expand the present considerations to the realm of higher dimensional granular crystals . recent efforts have made these gradually more accessible to experimental investigations @xcite and hence such ratcheting efforts would be extremely timely and relevant to consider . carlin and y.k . proceedings of the ieee , * 53 * , 11 ( 1965 ) 1788 . f. falo , p.j . martnez , j.j . mazo , and s. cilla , europhys . lett . , * 45 * ( 1999 ) 700 . f. falo , p.j . martnez , j.j . mazo , t.p . orlando , k. segal , and e. tras , appl . a , * 75 * ( 2002 ) 263 . ustinov , c. coqui , a. kemp , y. zolotaryuk , and m. salerno . * 93 * ( 2004 ) 087001 . f. jlicher , a. ajdari , and j. prost . * 69 * ( 1997 ) 1269 . s. flach , o. yevtushenko , and y. zolotaryuk . * 84 * ( 2000 ) 2358 . p. reimann . phys . rep . , * 361 * ( 2002 ) 57 . rapaport comp . 147 * ( 2002 ) 141 . n. boechler , g. theocharis , s. job , p.g . kevrekidis , m.a . porter , and c. daraio . * 104 * ( 2010 ) 244302 . | directed - ratchet transport ( drt ) in a one - dimensional lattice of spherical beads , which serves as a prototype for granular crystals , is investigated .
we consider a system where the trajectory of the central bead is prescribed by a biharmonic forcing function with broken time - reversal symmetry . by comparing the mean integrated force of beads equidistant from the forcing bead , two distinct types of directed transport can be observed _spatial _ and _ temporal _ drt . based on the value of the frequency of the forcing function relative to the cutoff frequency ,
the system can be categorized by the presence and magnitude of each type of drt .
furthermore , we investigate and quantify how varying additional parameters such as the biharmonic weight affects drt velocity and magnitude . finally , friction is introduced into the system and is found to significantly inhibit spatial drt .
in fact , for sufficiently low forcing frequencies , the friction may even induce a switching of the drt direction . |
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in the last few years it has become possible to observe details of absorption by singly ionized helium . the observations combine new information about the history of quasars , intergalactic gas , and structure formation . early observations of the heii ly@xmath1 absorption spectral region included the quasars q0302 - 003 ( z=3.285 , jackobsen et al . 1994 , hs 1700 + 64 ( z=2.72 , davidsen et al . 1996 ) and pks 1935 - 692 ( z=3.18 tytler & jackobsen 1996 ) . higher resolution ( ghrs ) observations of q0302 - 003 hogan , anderson & rugers 1997 and he 2347 - 4342 ( z=2.885 , reimers et al . 1997 ) revealed structure in the absorption which could be reliably correlated with hi absorption . heap et al . have followed up with stis on q0302 - 003 ( 1999a ) and he 2347 - 4342 ( 1999b ) . the second observation should be particularly illuminating since he 2347 - 4342 is relatively bright allowing a high resolution grating to be used . the anderson et al . ( 1998 ) observations of pks 1935 - 692 with stis yield good zero level estimates important for estimating the optical depth @xmath6 . preliminary reductions of longer stis integrations of pks 1935 - 692 with the 0.1 slit by anderson et al . ( 1999 ) confirm the 1998 results . taken together , these data now appear to be showing the cosmic ionization of helium by quasars around redshift 3 . although it is possible that the medium is already ionized to heiii by other sources at a lower level ( miralda - escud et al . 1999 ) . all of the objects show absorption with mean @xmath7 at redshifts lower than the quasar . for the higher redshift qso s q0302 - 033 and pks 1935 - 692 ( shown in fig . 1 ) there is a clear shelf of @xmath8 in a wavelength region of order 20 in observed wavelength blueward of the quasar emission line redshift , dropping to a level consistent with zero flux or @xmath9 beyond that . the sharp edge led anderson et al . to conclude that gas initially containing helium as mostly heii was being double - ionized in a region around the quasars . the lack of a strong emission line for heii ly@xmath1 suggests that ionizing flux is escaping so that the 228 flux may be similar to a simple power - law extension of the observed 304 rest frame flux . hogan et al . ( 1997 ) used the same reasoning to estimate the time required for quasars to create the double ionized helium region to be 20 myr for a 20 shelf ( dependent on the hubble parameter , spectral hardness , cosmology , baryon density and the shelf size ) . the features present in heii ly@xmath1 spectra are reflected in the hi ly@xmath1 forest for these quasars . attempts to model the heii absorption with line systems detected in hi suggest that very low column hi absorbers , difficult to differentiate from noise in hi spectra , provide a substantial contribution to the heii absorption . typically , in the shelf region , the ratio of heii to hi ions is of order 20 or more , rising to at least 100 farther away ( the cross - section for heii ly@xmath1 absorption is 1/4 that of hi ) . both pks 1935 - 692 and he 2347 - 4342 display conspicuous voids in the heii absorption near the apparent edge of the heiii bubble with corresponding voids in the hi spectra . reionization and the origin of the heii ly@xmath1 forest can addressed with detailed theoretical treatments ( eg . zheng & davidsen 1995 , zhang et al . 1998 , fardal et al . 1998 , abel & haehnelt 1999 , gnedin 1999 ) . wadsley , hogan & anderson ( 1999 ) presented numerical models of the onset of full helium reionization around a single quasar . the models integrated one - dimensional radiative transfer along lines of sight taken from cosmological hydrodynamical simulations and used flux levels comparable to pks 1935 - 692 . the results reinforced the basic intepretation of pks 1935 - 692 by hogan et al . ( 1997 ) as the growth of an he iii bubble over time in a medium that was mostly heii . in addition void recoveries in the large ionized bubbles similar to that observed for pks 1935 - 692 occurred often among a random set of simulated lines of sight . the gas temperature in such bubbles is strongly affected by the ionization of he ii to he iii , particularly because the first photons to reach much of the gas will be quite hard since the softer photons are absorbed close to the quasar until the gas is optically thin . in particular the underdense medium reaches temperatures of order 15000k . we make a simple analytical argument for this result in the next section . given our fairly good guess as to the physical conditions near pks 1935 - 692 we are in position to attempt to convert the observed optical depths into a density in baryons . comparing optical depths measured for the small sample of voids in the shelf region of pks 1935 - 692 to distributions of void widths and densities from simulations we can build an estimate of @xmath0 , the total cosmic density in baryons . the are still uncontrollable systematic uncertainties in this estimate but these differ from other techniques and can be addressed with better simulations and a larger sample of quasars . the main focus of this paper is the technique . in he ii ly@xmath1 absorption spectra only the voids are sufficiently low density to allow measurements of the optical depth . the highly ionized void gas is optically thin to the ionizing radiation and cool enough to ignore collisional effects , resulting in a fairly simple relation between optical depth and gas density , where @xmath10 is the optical depth for absorption of he ii ly@xmath1 , @xmath11 is the number density of he ii ions , @xmath12 converts from real space to wavelength ( velocity ) space and @xmath13 is the gas density . the recombination rate @xmath14 contains the only temperature dependence . the photo - ionizations per second @xmath15 is determined from the rest frame heii ionizing flux , @xmath16 can be extrapolated from the rest frame flux at @xmath17 which is estimated ( assuming a cosmological model ) from the observed continuum flux at @xmath18 , where @xmath19 is the redshift of the quasar . the fraction of heii depends on temperature roughly as @xmath20 through the recombination coefficient . the radiative transfer models of wadsley et al . ( 1999 ) gave temperatures around 15000k for the underdense gas near quasars . the optical depth to a he ii ionizing photon travelling @xmath21 ( the size of the pks 1935 - 692 shelf region ) at redshift @xmath2 ( scdm , @xmath22 , @xmath23 ) is approximately @xmath24 . the optical depth falls to 1 for photons with energies , @xmath25 . thus the heii in the outer half of the shelf region will be ionized preferentially by photons in the energy range 100 - 200 ev . injecting 100 - 54.4 ev per helium atom is equivalent to a temperature increase of 13000 k when the energy is distributed among all the particles . the cooling time for the underdense gas is very long , dominated by adiabatic expansion . gas elsewhere will gain @xmath26 4000 k due to heiii reionization . as voids evolve they approach an attractor solution resembling empty universes and their relative growth slows when they reach around @xmath27 times the mean density , making that value fairly representative . we used @xmath28-body simulations of 3 cosmologies ( standard cdm , open cdm and @xmath5cdm ) to get probability distributions for the quantity we label normalized opacity @xmath29 as a function of void width . @xmath30 may be directly related to the void optical depths via ( [ opacity ] ) . the set of voids allows a statistical maximum likelihood comparison of the simulated distribution of @xmath30 to the distribution of void widths and optical depths observed . figure [ taunorm ] shows the probability distribution of @xmath30 in voids ( equivalent to void optical depth ) and void widths for the @xmath5cdm cosmology . the optical depth from the simulation was normalized so that mean density gas expanding with the hubble flow gives an optical depth of 1.0 . the effect of gas pressure on the dynamics of gas in the voids should be negligible and was ignored in our simulations . thus the gas follows the dark matter exactly . the simulations were performed using hydra ( couchman 1991 ) with 64 mpc periodic simulation volumes ( @xmath31 @xmath32 at @xmath2 ) containing @xmath33 particles . a preliminary convergence study indicated that the statistics of @xmath30 are relatively insensitive to resolution however our resolution is still lower than that of the best current lyman-@xmath1 forest studies which also include gas . we were constrained by the need to include larger scales so that the large void portion of our sample was reasonable . higher resolution gives slightly larger estimates for @xmath0 . the ratio of @xmath10 to @xmath30 that has the maximum likelihood gives an estimate for the total baryon density as follows , @xmath34 in this equation all the quantities are given as ratios with typical values estimated from pks 1935 - 692 , with @xmath35 being the local density divided by the cosmological mean density , @xmath36 the local expansion rate along the line - of - sight compared to the mean , @xmath37 local hubble parameter value , @xmath1 the spectral slope of the quasar ( @xmath38 ) and @xmath39 the helium mass fraction . the value of @xmath40 is @xmath41 at @xmath2 for an @xmath42 scdm cosmology . the exact systemic redshift of pks 1935 - 692 is uncertain due to an absence of narrow ir emission lines . we use @xmath43 . we used the 5 large voids observed over the range @xmath44 for pks 1935 - 692 . this range should be far enough from the quasar to avoid `` associated '' absorbers caught in the quasar outflow . the fits are shown in figure [ taufit ] . he ii and h i absorption are both plotted with the models for each overplotted as dashed lines . the he ii spectrum was modeled with the known hi ly@xmath1 absorption lines and 5 parameter values for the optical depth in each void . the ratio of the hi to heii optical depths was a sixth free parameter . the models were convolved with the same spectral point spread function as the observations and include same level of photon shot noise . the best model was determined using maximum likelihood and the fitting errors determined with monte carlo realizations . the emptiest void has a reasonable probability according to the simulations , however since it has been suggested that it could be due to a local ionizing source we calculated the effect of removing it , which was a @xmath45 increase in the estimate for @xmath0 . the ionizing flux was inferred from the observed pks 1935 - 692 continuum level at 304 @xmath46 rest frame and extrapolated to @xmath47 and 228 @xmath46 . assuming the quasar redshift is @xmath43 this gives a flux of @xmath48 ( for standard cdm ) at @xmath49 which is in the middle of the voids . the sample of helium quasars is quite small and though pks 1935 - 692 does not have any known variability all quasars are thought to be intrinsically variable due to the nature of the fueling and emission processes . selection effects favour the idea that pks 1935 - 692 is currently brighter than its average value over the last few thousand years ( the ionization response timescale ) . if pks 1935 - 692 has recently brightened then ( [ ombh2eqn ] ) implies that our @xmath0 estimate is biased upward . if there is additional foreground continuum absorption then our estimates are biased low . the results are tabulated below with 1-@xmath4 fitting errors indicated ( monte carlo ) . aside from the uncertainties mentioned above , there is still uncertainty in the temperature through the @xmath50 dependence . if the gas was pre - ionized to heiii the underdense gas could be colder by a factor of order 2 which would lower the @xmath51 estimates by 20 % . a larger sample of quasars would make the greatest improvement in the robustness of this measurement . | a new method to extract @xmath0 from high redshift intergalactic absorption is described , based on the distribution of heii ly@xmath1 optical depths in the voids in the ionization zone of quasars .
a preliminary estimate from recent hst - stis spectra of pks 1935 - 692 at @xmath2 gives @xmath3 ( 1-@xmath4 statistical errors , for a @xmath5cdm cosmology ) consistent with other estimates . |
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the perceptron which was first analyzed with statistical mechanics techniques in the seminal paper of gardner @xcite is by now a well - known and standard model in theoretical studies and practical applications in connection with learning and generalization @xcite . a number of extensions of the perceptron model have been formulated , including many - state and graded - response perceptrons ( e.g. , @xcite ) . here we present some new extensions allowing for so - called coloured or ashkin - teller type neurons , i.e. , different types of binary neurons at each site possibly having different functions . the idea of looking at such a model is based upon our recent work on ashkin - teller recurrent neural networks @xcite . there we showed that for this model with two types of binary neurons interacting through a four - neuron term and equipped with a hebb learning rule , both the thermodynamic and dynamic properties suggest that such a model can be more efficient than a sum of two hopfield models . for example , the quality of pattern retrieval is enhanced through a larger overlap at higher temperatures and the maximal capacity is increased . for more details and an underlying neurobiological motivation for the introduction of different types of neurons we refer to @xcite . in the light of these results an interesting question is whether such a coloured perceptron can still be more efficient than the standard perceptron . in other words , can it have a larger maximal capacity than the one of a standard perceptron , which is known @xcite to be @xmath0 ( for random uncorrelated patterns ) . it has been suggested that this number is characteristic for all binary networks independent of the multiplicity of the neuron interactions . thereby , the capacity is defined as the thermodynamic limit of the ratio of the total number of bits per ( input ) neuron to be stored and the total number of couplings per ( output ) neuron @xcite . we remark that `` input '' and `` output '' refer specifically to the perceptron case . in the sequel the maximal capacity of coloured perceptron models is studied using the gardner approach @xcite . first - step replica - symmetry - breaking effects are evaluated and the analytic results are compared with extensive numerical simulations using various learning algorithms . the rest of this paper is organized as follows . in sec . ii we introduce two ashkin - teller type perceptron models . section iii contains the replica theory and determines the maximal capacity by calculating the available volume in the space of couplings both in the replica - symmetric ( sec . iiia ) and the first - step replica - symmetry - breaking approximation ( sec . section iv describes the results of numerical simulations with algorithms obtained by generalizing various algorithms for simple perceptrons . in sec . v we present our conclusions . finally , two appendices contain some technical details of the derivations . let us first formulate the coloured perceptron models . we consider @xmath1 input patterns @xmath2 consisting out of two different types of patterns @xmath3 and @xmath4 , and a corresponding set of outputs @xmath5 which are determined by @xmath6 where @xmath7 ( @xmath8 ) are the local fields acting on the patterns @xmath9 , @xmath10 and their product @xmath11 respectively @xmath12 both types of input patterns and their corresponding outputs are supposed to be independent identically distributed random variables ( iidrv ) taking the values @xmath13 or @xmath14 with probability @xmath15 . the set of three equations ( [ e.s1])-([e.s3 ] ) defines a mapping of the inputs @xmath16 onto the corresponding outputs @xmath17 . we call it model i. we remark that the specific form of the equations ( [ e.s1])-([e.s3 ] ) is related to the transition probabilities for a spin - flip in the dynamics @xcite . a second model , denoted by ii , is defined by considering only the two equations ( [ e.s1 ] ) and ( [ e.s2 ] ) . when @xmath18 and @xmath19 then the relations ( [ e.s1])-([e.s2 ] ) are satisfied by two ( out of the four possible ) values of the output @xmath20 , otherwise model ii gives the same output as model i. in other words , due to the presence of the @xmath21 and @xmath22 in the gain functions , model ii contains more freedom and , strictly speaking , it is not a mapping . the sequential dynamics of these two models has been studied in the case of low loading with the hebb rule and shown to lead to the same equilibrium behaviour @xcite . however , this is not guaranteed here since we are concerned with optimal couplings maximizing the loading capacity . at this point we remark that when all @xmath23 are equal to zero we find back two independent standard binary perceptron models . in the sequel we take the couplings to satisfy the spherical constraint @xmath24 . the coloured perceptron is trained to store correctly @xmath25 patterns with @xmath26 the loading capacity . the factor @xmath27 follows naturally from the definition of capacity given in the introduction . a pattern is stored correctly when the so - called aligning field @xcite is bigger than a certain constant @xmath28 whereby the latter indicates the stability . it is a measure for the size of the basin of attraction of that pattern . specifically we require that @xmath29 with @xmath30 denoting the configurations in the space of interactions . for @xmath31 all patterns that satisfy equations ( [ e.s1])-([e.s3 ] ) also satisfy ( [ eq : afieldxi])-([eq : afieldxieta ] ) . we remark that for model ii the last inequality is superfluous . ' '' '' ' '' '' the aim is then to determine the maximal value of the loading @xmath26 for which couplings satisfying ( [ eq : afieldxi])-([eq : afieldxieta ] ) can still be found . in particular , the question whether this model can be more efficient than the existing two - state models is relevant . following refs . @xcite we formulate the problem as an energy minimization in the space of couplings with the formal energy function defined as @xmath32\ , . \label{eq : energyi}\end{aligned}\ ] ] we remark that for model ii the third @xmath33-factor is absent . the quantity above counts the number of weakly embedded patterns , i.e. , the patterns with stability less than @xmath34 . therefore , the minimal energy gives the minimal number of patterns that are stored incorrectly . this number is zero below a maximal storage capacity @xmath35 . the basic quantity to start from is the partition function @xmath36 \rangle_{\{j\}}\\ \langle ... \rangle_{\{j\ } } & = & \int \prod_i dj_i \prod _ r \delta \left(\sum_i ( j_i^{(r)})^2 - n \right) ... ~ , \end{aligned}\ ] ] with @xmath37 the inverse temperature . as usual it is @xmath38 which is assumed to be a self - averaging extensive quantity @xcite . the related free energy per site @xmath39 is equal , in the limit @xmath40 , to @xmath41\rangle_{\{j\ } } } { nz(\beta ) } \ , , \label{eq : flimit}\ ] ] which is the minimal fraction of wrong patterns ( recall eq . ( [ eq : energyi ] ) ) . in order to perform the average over the disorder in the input patterns @xmath42 and the corresponding outputs @xmath43 we employ the replica method . the calculations proceed in a standard way although the technical details are much more complex . introducing the order parameters @xmath44 , with @xmath8 and @xmath45 we write following @xcite @xmath46 \int\prod_{r ' } { \rm d}x^{r'\gamma } \exp\left\{\sum_{r ' } ix^{r'\gamma}\lambda^{r'\gamma } \right.\right.\right . \nonumber\\ & - & \frac{1}{2}\left[\left(x^{1\gamma}+x^{3\gamma}\right)^2 + \left(x^{2\gamma}+x^{3\gamma}\right)^2 + \left(x^{1\gamma}+x^{2\gamma}\right)^2\right ] \nonumber\\ & -&\sum_{\tau>\gamma}\left[\left(x^{1\gamma}+x^{3\gamma}\right ) \left(x^{1\tau}+x^{3\tau}\right)q_{\gamma\tau}^{(1 ) } + \left(x^{2\gamma}+x^{3\gamma}\right ) \left(x^{2\tau}+x^{3\tau}\right)q_{\gamma\tau}^{(2)}\right . + \left.\left.\left.\left.\left(x^{1\gamma}+x^{2\gamma}\right ) \left(x^{1\tau}+x^{2\tau}\right)q_{\gamma\tau}^{(3 ) } \right ] \rule{0cm}{0.5cm}\right\ } \rule{0cm}{0.6cm}\right ) \rule{0cm}{0.7cm}\right\}\nonumber \\ g_1&=&\ln\left\{\int\prod_{r,\gamma}\left({\rm d } j^{(r)\gamma}\right ) \exp\left[i\sum_{r,\gamma}\epsilon_\gamma^r((j^{(r)\gamma})^2 - 1)- i\sum_{r,\gamma,\tau > \gamma}\phi_{\gamma\tau}^r j^{(r)\gamma}j^{(r)\tau } \right]\right\},\nonumber\end{aligned}\ ] ] 2 where @xmath47 denotes the average over the patterns , @xmath48 for model i and @xmath49 for model ii . because of the latter we remark that for model ii the formula for @xmath50 can be simplified : the integrals with respect to @xmath51 and @xmath52 are not present and thus @xmath52 , @xmath53 and @xmath51 have to be set to zero . because of this simplification we only outline explicitly the calculations for model ii in the sequel . the corresponding formulas for model i can be found in appendix b. we continue by making the replica - symmetric ( rs ) anzatz @xmath54 . moreover , for convenience , we set @xmath55 . the latter is justified for model i because of the symmetry present in this model . furthermore , since we are going to take all @xmath56 in the gardner - derrida analysis anyway , we keep this equality also for model ii . taking then the limits @xmath40 , @xmath57 and @xmath58 we arrive , in the case of model ii , at @xmath59 with @xmath60 where @xmath61 , @xmath62 is a modified gaussian measure , @xmath63 and @xmath64 takes those values that minimize @xmath65 , the available ' '' '' ' '' '' volume in the space of couplings . for the corresponding expression in the case of model i we refer to appendix b. taking @xmath66 and supposing that the maximal capacity , @xmath67 , is signaled by the gardner - like criterion @xmath68 we obtain @xmath69 this maximal capacity as a function of @xmath70 is shown for both models in figs . [ res1 ] and [ res2 ] as a full line . for model i we obtain , e.g. , @xmath71 , a value that is smaller than the gardner capacity for the simple perceptron . for model ii however , we get the interesting result that @xmath72 . it is straightforward to show geometrically that learning almost antiparallel patterns , i.e. , patterns satisfying @xmath73 results in a splitting of the space of couplings into disconnected regions . this suggests that rs is broken and , consequently , the results for @xmath74 found in sec . iiia are only upperbounds for the true capacity . therefore , we want to improve the rs results by applying the first step of parisi s replica - symmetry - breaking ( rsb ) scheme ( e.g. , @xcite ) . so , we assume that the @xmath75 in equation ( [ zpn ] ) have the following matrix block structure @xmath76 where @xmath77 is the size of the matrix @xmath75 , @xmath78 is the number of diagonal blocks and int(@xmath79 ) denotes the integer part of @xmath79 . for model ii we take @xmath80 reflecting the symmetry of this model . for model i we repeat that all @xmath81 s can be taken equal . we then consider the limits @xmath82 and @xmath58 in such a way that @xmath83 , with @xmath84 , remains finite . after a tedious calculation we arrive at the following expression for the rsb1 maximal capacity for model ii @xmath85 2 with @xmath86 and d@xmath87 a gaussian measure . the explicit form of the function @xmath88 can be found in appendix a. an analogous form for model i is written down in appendix b. the results are presented in figs . [ res1 ] and [ res2 ] as full lines . as expected they lie below the rs results confirming the breaking of rs , e.g. , @xmath89 for model i and @xmath90 for model ii . we remark that the breaking for model ii is stronger than for model i , the reason being that model ii allows more freedom as explained in the introduction . finally , on the basis of results in the literature for the simple perceptron @xcite , @xcite we expect that the rsb1 results are very close to the exact ones . this is further examined by performing numerical simulations as described in the following section . the idea of these simulations is to train the network with a certain learning algorithm in order to learn as many random patterns as possible . the main technical difficulties are to find an efficient algorithm and prove its convergence . we have tried to generalize various algorithms proposed for simple perceptrons @xcite . the most effective ones appeared to be some particular generalization of the adaptive gardner algorithm @xcite and the adatron algorithm @xcite . in the sequel we only report on the results obtained with these two algorithms . we remark that we have chosen @xmath91 in all simulations . one of the algorithms that has demonstrated its efficiency and for which convergence has been shown in the case of the standard perceptron is given in ref . it is an adaptive version of the original algorithm proposed by gardner @xcite . using heuristic arguments presented in @xcite we have constructed for the coloured perceptron model ii the following analogous learning rule @xmath92 \,.\end{aligned}\ ] ] ' '' '' ' '' '' the form of the algorithm for model i is a bit different and given in appendix b. this algorithm should be carried out sequentially over the patterns and sequentially or parallel over the couplings as long as one of the arguments of the @xmath33 functions is positive . it appears to have the characteristics of the most efficient , non - linear algorithm discussed in @xcite . using this learning rule we have trained networks of sizes @xmath93 sites ( depending on the value of @xmath70 ) in order to store perfectly as many randomly chosen patterns as possible . for each value of @xmath70 we have calculated the maximal capacity for different @xmath94 and extrapolated the results to @xmath95 . results for a given value of @xmath70 and @xmath94 are averages over 1000 samples . as shown in figs . [ res1 ] and [ res2 ] this algorithm performs especially well for small values of @xmath70 for both the models i and ii . the second algorithm we report on is the adatron algorithm @xcite which works in a different way . instead of searching the maximal capacity for a given stability it tries to find the maximal stability for a given capacity . the derivation of this algorithm and a proof of its convergence are based upon the assumption that the problem can be formulated as a quadratic optimization with linear constraints @xcite . such a formulation can not be given for the coloured perceptron model , because the three different types of couplings have to be normalized independently and because the stability conditions ( [ eq : afieldxi])-([eq : afieldeta ] ) are more complex . hence , a straightforward generalization similar to the one for the potts model @xcite is not possible . below we describe a learning rule that tries to incorporate the ideas of the adatron approach . we assume that the couplings can be written in the form ( cfr.,@xcite and references therein ) @xmath96 where @xmath97 ( @xmath8 ) are the so called embedding strengths of pattern @xmath98 . then , in the case of model ii the couplings are updated by modifying @xmath97 with the following increments @xmath99 this is done sequentially over the patterns . we remark that again the algorithm for model i is somewhat different ( see appendix b ) . for each value of the capacity we have considered system sizes @xmath100 and extrapolated the results to @xmath95 . the best results were obtained for a learning rate @xmath101 . results for each size are averages over 1000 samples . for small values of the capacity the algorithm gives better results , both in the case of models i and ii than the first algorithm we have discussed , as shown in figs . [ res1 ] and [ res2 ] . for larger values of the capacity , however , it performs worse . the results for the adatron algorithm are displayed only in the region where they are better than the results for the gardner algorithm . we remark that the numerical simulations with the different algorithms give different results and that we have not shown their convergence analytically such that , in principle , the values for @xmath102 obtained here are lower bounds . looking at figs . [ res1 ] and [ res2 ] in more detail we see that for the whole range of @xmath70 the values of the maximal capacity in model ii are larger than those of a standard binary perceptron . for @xmath103 , e.g. , the simulations give @xmath104 , which is bigger than the maximal capacity of the binary perceptron model @xcite and the binary many - neuron interaction model @xcite , both of which have @xmath0 . for model i the maximal capacity at @xmath103 found by simulations is @xmath105 . in this work we have calculated the maximal capacity per number of couplings for two coloured perceptron models . compared with the standard perceptron these models have two neuronal variables per site and a local field that contains higher order neuron terms . the ' '' '' ' '' '' + + method used is a generalization of the gardner approach and both the rs and rsb1 results have been discussed . we expect that the latter give very close upperbounds for the exact values . extensive numerical simulations have been performed for finite systems and extrapolated to @xmath95 . the adaptive gardner algorithm and the adatron algorithm give the best , but different results . hence , the results of the simulations can be considered only as lower bounds for the exact maximal capacity . additional work looking for improved algorithms would be welcome . comparing both the rsb1 results and the results from numerical simulations we conclude that they are in good agreement . for bigger values of @xmath70 they even completely coincide . for model i we find that at @xmath103 the maximal capacity satisfies @xmath106 . this suggests that it is equal to the maximal capacity of the @xmath107-potts perceptron , i.e. , @xmath108 ( after appropriate rescaling of the latter @xcite ) . this would parallel the situation for hebb learning @xcite . for model ii we have for @xmath103 that @xmath109 , which is larger than the maximal capacity of the standard binary perceptron . this is due to the fact that model ii is not a strict mapping such that it allows for more freedom in the determination of the couplings . the authors would like to thank m. bouten and j. van mourik for critical discussions . the function @xmath88 in formula ( [ alpharsb1 ] ) reads @xmath110 \nonumber\\ & + & \frac{1}{2c_1}e^{\varepsilon_2 } \int_{-\infty}^{\frac{c1}{c}(u_1-\delta_2 ) } { \rm d}s\left[1+{\rm erf}\left(\sqrt{\frac{3r}{2c^2 } } \left(-\frac{x_2}{\sqrt{3r}}+\delta_2 + \frac{c}{c_1}s\right)\right)\right ] \nonumber\\ & + & \frac{1}{2c_2}e^{\phi_2 } \int_{-\infty}^{-\frac{c2}{c}(u_1-\gamma_2 ) } { \rm d}s\left[1+{\rm erf}\left(\sqrt{\frac{3r}{2c^2 } } \left(\frac{x_2}{\sqrt{3r}}-\gamma_2 + \frac{c}{c_2}s\right)\right)\right ] \nonumber\\ & + & \frac{1}{2c_2}e^{\phi_3}\int_{-\infty}^{-\frac{c2}{c}(u_1-\gamma_3 ) } { \rm d}s\left[1+{\rm erf}\left(\sqrt{\frac{3r}{2c^2 } } \left(-\frac{x_3}{\sqrt{3r}}-\gamma_3 + \frac{c}{c_2}s\right)\right)\right ] \nonumber\\ & + & \frac{1}{2c'}e^{d_1}\int_{-\infty}^{-\frac{u_1}{c ' } } { \rm d}s\left[{\rm erf}\left(\sqrt{\frac{3r}{2 } } \left(\frac{x_3}{\sqrt{3r}}-b_1-\frac{1}{c'}s\right)\right ) + { \rm erf}\left(\sqrt{\frac{3r}{2}}\left(-\frac{x_2}{\sqrt{3r}}-b_1- \frac{1}{c'}s\right)\right)\right ] \nonumber\\ & + & \frac{1}{2}\int_{-\infty}^{u_1 } { \rm d}s\left[{\rm erf}\left(\sqrt{\frac{3r}{2 } } \left(\frac{x_2}{\sqrt{3r}}-s\right)\right ) + { \rm erf}\left(-\sqrt{\frac{3r}{2}}\left(\frac{x_3}{\sqrt{3r } } + s\right)\right)\right]\nonumber\end{aligned}\ ] ] 2 with d@xmath111 a gaussian measure and @xmath112 ' '' '' ' '' '' @xmath113 for model i the calculations are very similar . some resulting expressions , however , have a somewhat different structure . for completeness we write down these expressions here . finally , the learning algorithms for model i differ in the way that the couplings @xmath122 and @xmath123 are updated . we have for the adaptive gardner algorithm @xmath124 \nonumber\\ j_i^{(2)}&\rightarrow&j_i^{(2)}+\eta_0^{\mu}\eta_i^{\mu}\frac{1}{2 } \left[\left(\kappa_\eta-\lambda_\eta^\mu\right)\right . \theta\left(\kappa_\eta-\lambda_\eta^\mu\right ) \nonumber\\ & + & \left(\kappa_{\xi\eta}-\lambda_{\xi\eta}^\mu\right)\left . \theta\left(\kappa_{\xi\eta}-\lambda_{\xi\eta}^\mu\right)\right ] \nonumber\end{aligned}\ ] ] instead of ( [ j1per ] ) and ( [ j2per ] ) and for the adatron algorithm we take @xmath125 instead of ( [ x1ada ] ) and ( [ x2ada ] ) . 99 e. gardner j. , phys . a * 21 * , 257 ( 1988 ) . j. hertz , a. krogh and r. g. palmer , _ introduction to the theory of neural computation _ ( addison - 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teller type perceptron models are introduced .
their maximal capacity per number of couplings is calculated within a first - step replica - symmetry - breaking gardner approach .
the results are compared with extensive numerical simulations using several algorithms . 2 |
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in many crystal - growing procedures of interest , a nano - scale faceted surface appears and proceeds to evolve , often exhibiting coarsening and even dynamic scaling , whereby characteristic statistics describing the surface remain constant even as the characteristic lengthscale increases through the vanishing of small facets . for many evolving faceted surfaces , a _ facet velocity law _ can be observed @xcite , assumed @xcite , or derived @xcite which specifies the normal velocity of each facet , often in configurational form which depends on the geometry of the facet . in this way , the dynamics of a continuous , two - dimensional surface can be concisely represented by a discrete collection of such velocities , and overall computational complexity reduced to that of a system of ode s ; the resulting system is known as a _ piecewise - affine dynamic surface _ , or pads . such theoretical simplification , in turn , enables the large - scale numerical simulations necessary for the statistical investigation of coarsening and dynamic scaling . the numerics involved in the direct geometric simulation of an arbitrary pads is straightforward for one - dimensional surfaces , requiring nothing beyond traditional ode techniques except simple geometric translation between facet displacement and edge displacement , and a small surface correction associated with each coarsening event . consequently , such simulations accompany many of the above facet treatments of facet dynamics , and have also been independently repeated elsewhere @xcite . however , in two dimensions , the corrections due to coarsening events are much more involved , and any code must be able to deal with a family of non - coarsening _ topological events _ that alter the neighbor relations between nearby facets . consequently , the fewer simulation attempts use either fast but poentially imprecise envelope methods @xcite , or more robust but slower phase - field @xcite or level - set @xcite methods to avoid explicitly performing topological changes . besides the speed / accuracy trade - off exhibited by these approaches , both methods obscure the natural geometric simplicity of the native surface , complicating the extraction of detailed surface statistics which , after all , motivates large simulations in the first place . additionally , as will be seen , the presence of non - unique topological events requires explicit intervention regardless of topological scheme , which negates much of the advantage of a `` hands - free '' treatment . in the previous chapter , we introduced a direct - simulation method which explicitly performs topological events along the way , thus preserving both simulation speed and topological accuracy . in addition , by representing the surface as a collection of facets , edges , and junctions , plus the neighbor relations between them , the method mirrors the natural geometry of the surface being modeled , which allows easy extraction of geometric statistics . there , however , the restricted case of threefold symmetry was chosen for ease of topological implementation ; under this symmetry , a limited number of topological events were observed , and both vanishing facets and non - vanishing surface rearrangements could be handled explicitly using a priori knowledge of the before and after surface states . while many surfaces exhibit threefold symmetry , making the method useful even in this special case , it could not handle other common crystal symmetries , notably fourfold and sixfold . in this chapter , then , we generalize the previous model to allow the simulation of surfaces with arbitrary symmetry groups . we begin in section [ sec : cellular - structure ] with a brief summary of the basic method , including surface representation , facet kinematics , and the application of a dynamics . next , in section [ sec : topological events ] , we provide a careful enumeration of topological events which may occur on surfaces of arbitrary symmetry ; this includes discussion of the far - field reconnection algorithm , by which network holes left by vanishing facets may be consistently repaired without knowledge of the post - event state . then , we provide in section [ sec : discretization and topology ] a careful consideration of the consequences of using ( necessarily discrete ) timesteps during the simulation of a surface whose evolution equations change qualitatively between steps ( at topological events ) ; the issues that arise are discussed in the context of three sample strategies . the completed method is illustrated from three- , four- , and six - fold symmetric surfaces in section [ sec : demonstration ] ; these exhibit all of the topological events likely to be encountered on a real surface , and demonstrate that the method is robust enough to generically simulate faceted surfaces of any symmetry class for which a facet - velocity law is uniquely specified . finally , in addition to detailing the ffr algorithm , the appendix includes a discussion of kinematically non - unique topological events , where two resolutions are possible , and highlights the need to refer to the dynamics or even first principles to decide how the surface should evolve in those cases . we consider the evolution of a single - valued , fully - faceted surface @xmath1 ; this definition explicitly forbids overhangs and inclusions . we assume that the surface bounds a single crystal which exists on exactly one lattice ; thus , we are not treating surfaces with multiple grains . the surface is piecewise - affine , consisting of facets @xmath2 with fixed normals @xmath3 . these are bounded by and meet at edges @xmath4 which are necessarily straight line segments ; edges in turn meet at triple - junctions @xmath5 . this three - component structure is reminiscent of two - dimensional _ cellular networks _ @xcite and indeed , while we consider three dimensional surfaces , the projection of the edge set onto the plane @xmath6 is a 2d cellular network . this structure and the neighbor relations inherent within it suggest a doubly - linked object - oriented data structure , consisting of : ( 1 ) a set of junctions , each having a location , pointing to three edges and three facets ; ( 2 ) a set of edges , each having a tangent , pointing to two junctions and two facets ; and ( 3 ) a set of facets , each having a normal , pointing to @xmath7 edges and @xmath7 junctions . these objects and the associated neighbor relations are illustrated in figure [ fig : neighbor - relations ] ; this structure is the natural structure of the surface , and uniquely and exactly describes it . we now consider each element in more detail . [ fig : neighbor - relations ] a _ junction _ is a point in space formed where edges ( and hence , facets ) intersect . the * order * @xmath8 of a junction is simply the number of edges which meet there . while junctions of any order @xmath9 are possible , we restrict ourselves here to the case of order @xmath10 junctions or `` triple junctions . '' this greatly simplifies analysis and code , as triple junctions are uniquely positioned by the three facets meeting there . the intrinsic geometric information carried by a junction is its location . junctions are stored in a ` junction ` class , which contains this location , as well as pointers to the three edges and three facets which meet there . an _ edge _ is a line segment formed by the intersection of exactly two facets , and bounded by exactly two junctions . the intrinsic geometrical quantity of an edge is its orientation , which is fixed since facets have fixed normals . edges are stored in the ` edge ` class , which records the tangent , as well as pointers to the two neighboring facets and two bounding junctions . at creation , edges are `` directed '' : one junction is arbitrarily deemed the origin , and the other the terminus , establishing a _ tangent_. this has two important consequences . first , if we imagine walking along the edge in the tangent direction , then one neighboring facet may be labeled `` left '' , and the other `` right . '' this information allows us to distinguish between convex and concave edges , and also to determine the clockwise direction around a given facet , which is necessary for effective navigation of the network , as well as the proper calculation of boundary integrals on facets . second , the tangent allows us to detect when an edge `` flips '' ( see @xcite ) ; this will be discussed in more detail in section [ sec : topology - class-1 ] . a facet is a simply - connected planar polygonal region in space , which is bounded by an equal number of edges and junctions . the intrinsic geometric information carried by a facet is its normal , which is fixed . our surface definition @xmath11 requires that the normal of each facet is constrained to be on the hemispherical shell of unit - length vectors with positive @xmath12 component . the imposition of a particular symmetry on the crystal may further restrict available normals , but no such restriction is here assumed . facets are stored in a ` facet ` class , which contains the normal , as well as a list of bounding edges and junctions , sorted in counter - clockwise order . the intrinsic geometric means of characterizing surface evolution is by specifying the normal velocity of each point on the surface . a piecewise - affine surface is composed of a collection of planar , fixed - normal facets , whose motion is limited to displacement along the normal . therefore , the kinematics @xmath13 of the entire surface may be expressed by a discrete set of individual facet velocities @xmath14 . as edges and junctions are merely intersections between two and three facets , respectively , their motion is uniquely specified by the motion of the facets that neighbor them . in particular , if @xmath15 is the location in space of a triple junction , then the velocity of that a triple junction may be calculated through the expression @xmath16 where the rows of @xmath17 and entries of @xmath18 are the unit normals and normal velocity , respectively , of the three facets intersecting to form @xmath15 . in practice , facet velocities are only used indirectly to calculate junction velocities if junctions are moved correctly , edges ( connections between two junctions ) and thus faces ( collections of edges ) are necessarily moved correctly as well . all that remains now is to select a particular dynamics ; that is , to specify an expression for the normal velocity @xmath14 of each facet . having chosen one , we follow @xcite and refer to the resulting evolving structure as a piecewise - affine dynamic surface ( @xmath19 ) . example dynamics describing many different physical situations were listed in the introduction , and the exact dynamics is not of special concern here ( although we will select one for demonstration later ) . it is worth noting here , however , that most of the dynamics proposed to date are _ configurational _ , depending on properties of the facet such as area , perimeter , number of junctions , or mean height . thus , sudden changes in the geometric properties of a facet can lead to sudden changes in its velocity , an issue which will be explored in more detail in section [ sec : discretization and topology ] . we have just discussed how elements of each class ( facet , edge , vertex ) neighbor members from each of the other classes . taken together , the set of all of these neighbor relations comprises the * topological state * of the surface . it is a complete record of every neighbor relationship on the surface , and is unique for a given surface . as the system evolves , these neighbor - relations may change as facets exchange neighbors , join together , split apart , or vanish . each of these cases is an example of a * topological event * , and represents a change to the topological state of the surface ( topological events are a defining feature of evolving cellular networks again see @xcite ) . to maintain an accurate representation of the surface , a direct geometric method like that described here must manually perform topological events as necessary . because actual surface evolution is fairly trivial , this is the main difficulty of our method . a natural first question to ask at this point is `` how many topological events are possible ? '' to begin answering this question , we point out that on a physical surface , topological events occur automatically , and by geometric necessity . if a detected event signals the need to change neighbor relationships at some location on the surface , we may therefore infer that failing to change them would produce a cellular network with `` wrong '' relationships , that do not correspond to a physical surface . we call such erroneous configurations * geometrically inconsistent * ; examples include primarily edge networks that intersect when viewed from above , since these correspond to overhangs and inclusions , which are prohibited . since topological events serve to avoid possible geometric inconsistencies , we may discover what events are possible by considering how inconsistencies may occur . this is most easily accomplished by considering each surface element in turn . we first consider junctions , which are simply a location in space . a junction can , in the course of surface evolution , leave the periodic domain , in which case it is wrapped to the other side . however , this is only a bookkeeping operation , and does not represent a real topological event . turning to edges , we note that edges possess a directed length . as already hinted in section [ sec : edges ] , this length could become negative if the edge were to `` flip '' @xcite a flipped edge has no geometrical meaning on a single - valued surface , and so we introduce a class of * vanishing edge * events which occur when edges reach zero length . finally , we consider facets . since a facet has fixed orientation , its changing properties are loosely its shape and size . specifically , a facet is a _ simply connected _ planar region with _ positive area_. these two defining properties of facets lead , through consideration of their potential violation , to two additional classes of topological event : * facet constriction * events which prevent the formation of self - intersecting facets , and * vanishing facet * events which remove facets from the network when they reach zero area . an adjacent point - point event occurs when an edge shrinks to zero length , and its junctions meet . to consider what might happen to the faceted surface when this occurs , we first label the immediate surroundings of an edge . each edge is composed of two faces of which it is the intersection , its * composite faces * , and stretches between two faces at which it terminates , its * terminal faces*. in addition , we will also use the term * emanating edges * to refer to those edges immediately neighboring the shrinking edge . now , consider the hemispherical shell of available facet normals ( section [ sec : faces ] ) . the ( necessarily distinct ) normals of the composite faces specify a great circle about this hemisphere , which divides it into two parts . the normals of the terminal faces can not lie on this boundary , and unless they are identical ( a special case ) , they form a second great circle around the hemisphere . while terminal normals may not lie on the composite great circle , the reverse is not true , and this fact effectively divides vanishing edge events into three sub - classes . [ fig : shrinking edge classes ] figure [ fig : shrinking edge classes ] illustrates this idea , and gives an example of each of the three possible cases . if the terminal great circle touches neither composite point , then the well - studied * neighbor switch * occurs . if the terminal great circle touches one composite point , then an * irregular neighbor switch * results . finally , if the terminal normals occupy the same point , then no great circle is defined the terminal facets have he same normal , and when the edge between them shrinks to zero , they join into a single facet : a * facet join*. on a general surface , the most common vanishing edge event is the neighbor switch , which is frequently encountered in other evolving cellular networks . in this event , neither composite normal touches the terminal great circle , so any three of the normals involved form a linearly independent set this property is the defining feature of the neighbor switch . when an edge with this configuration shrinks to zero length , the surrounding facets simply exchange neighbors . figure [ fig : t1-illustration ] gives an example of this event . + [ fig : t1-illustration ] _ resolution . _ the neighbor switch is performed by the ` ns_repairman ` class . to resolve this event , it simply deletes the old edge , and creates a new edge . the composite faces which formed the old edge become terminal faces of the new edge , and cease to neighbor each other . conversely , the terminal faces of the old edge become the composite faces of the new edge , and thus become neighbors . this symmetric exchange in neighbor relations is the cause of the name neighbor switch , which comes from the grain - growth literature the less - descriptive name `` t1 process '' in often used in the soap froth literature . in addition to replacing the vanishing edge , the junctions on either side of this edge are replaced . each new junction is formed by the intersection of the deleted edge s ( formerly non - adjacent ) terminal faces with one of its composite faces . _ comments . _ readers familiar with other cellular - network literature will note that the example neighbor switch in figure [ fig : t1-illustration ] lacks the typical `` x '' shape . this is due to the constrained nature of facet normals , and hence , edge orientations . additionally , we note that the neighbor switch is a reversible event ; in fact it is its own reversal . finally , a certain sub - class of neighbor switches posessing `` saddle '' structure are non - unique , as was observed by thijssen @xcite . for a discussion of this non - uniqueness and its consequences , see appendix [ sec : non - uniqueness ] . when the normal of one of the composite faces lies on the great circle formed by the terminal normals , the neighbor switch can not occur . here , the terminal faces can not form a new junction with the offending composite face because the three normals involved are not independent . instead , when an edge with this configuration shrinks to zero , two closely related events are possible , depending on the configuration of the nearby edges . these events are collectively called irregular neighbor switches , with two varieties called a `` gap opener '' and `` gap closer '' that are exact opposites . these are illustrated in figure [ fig : gap opener ] . [ fig : gap opener ] _ resolution . _ the irregular neighbor switch is performed by the ` ins_repairman ` class . because one composite normal lies on the terminal great circle , exactly two of the emanating edges are parallel in @xmath20 . the gap opener occurs when these edges emanate from the shrinking edge in _ opposite _ directions , while the gap closer occurs when the edges emanate in the _ same _ direction . to resolve the gap opener , we select one of the parallel emanating edges to be split apart ( see below ) . the gap will go here , filled by the terminal face that touches the other parallel edge , and will extend all the way to the far end of the split edge , where a new edge is introduced to link the two edges resulting from the split edge . this is all illustrated in figure [ fig : gap opener ] . to resolve the gap closer , simply reverse the steps . _ comments . _ several comments on this pair of events are in order . first , while the gap closer is uniquely resolved , the gap - opener is an inherently non - unique event , as either of the parallel edges could be the one split ( we will discuss this further in section [ sec : non - uniqueness ] ) . second , both resolution options have the potentially dissatisfying property of being non - local in effect , because the collision of two junctions causes an entire edge to split apart . what is perhaps more likely is the nucleation of a new , tiny facet at the moment the junctions collide ; however , we have excluded that possibility from consideration here . finally , while common experimentally - encountered surfaces usually have either high symmetry ( only a few facet orientations ) or no symmetry ( as many orientations as facets ) , the irregular neighbor switch with its three coplanar orientations requires what may be called `` intermediate symmetry , '' where orientations are limited , but many are available . because it poses resolution difficulties , and because it is not encountered in any surfaces we wish to study , we have not yet actually implemented this event . finally , we consider the special case where the terminal normals are identical . when such an edge shrinks to zero length , the terminal faces meet exactly . having the same orientation , they then join to form a larger face . figure [ fig : face - join - illustration ] depicts a representative facet join event . [ fig : face - join - illustration ] _ resolution . _ facet joins are performed by the ` fjoin_repairman ` class . to perform a facet join , a new face is created to replace the joining faces , and all edges and junctions that neighbored the old faces are re - assigned to this new face . next , the vanishing edge and its two junctions are deleted , leaving the four emanating edges to be considered . these are most logically grouped into the ( necessarily parallel ) pairs of edges bordering , respectively , the left and right composite faces of the vanished edge . in the example event shown in figure [ fig : face - join - illustration ] , these two pairs look different : one pair meets side - to - side , while the other pair meets end - to - end . computationally , however , this makes no difference ; each pair is replaced by a single edge connecting their remaining non - deleted junctions . this behavior is generic for all face joins . _ comments . _ we note that the face join is , strictly speaking , non - reversible ( though see section [ sec : face - split ] ) . the exact opposite of the face join would be a facet which spontaneously `` shatters , '' as described in @xcite ; this behavior is certainly worth studying , but is not currently implemented . second , although this is a `` special case '' in general , for high - symmetry crystal surfaces it may be very common indeed , for the case of a cubic crystal with only three available facet orientations considered in chapter 2 , facet joins are the only vanishing edge event exhibited . finally , we note that this event is the only vanishing edge event which does not conserve the number of facets . it is , in fact , one mechanism by which coarsening may occur , and may be the dominant mechanism for high - symmetry surfaces . the second class of topological event occurs whenever a facet ceases to be simply - connected , and results in that facet being split into two new facets . remembering that the edges of a facet trace out a polygon in the plane , we observe that the non - simply connected polygon , if allowed to continue evolving , would become self - intersecting , which clearly has no geometrical interpretation . so , how may an evolving polygon become self - intersecting ? since the boundary consists of edges and junctions , there are three possible modes : ( a ) two non - adjacent junctions meet , ( b ) a junction meets an edge , or ( c ) two edges meet . each case has a distinct `` signature , '' illustrated in figure [ fig : self - intersection - signatures ] , which can be used to tell them apart . [ fig : self - intersection - signatures ] the junction - junction collision shown in figure [ fig : self - intersection - signatures]a represents the formation of a perfect @xmath21 junction . while theoretically interesting , such events are not considered here ; we hypothesize that , given random initial data , two junctions not connected by an edge will never exactly meet . furthermore , by considering figure [ fig : self - intersection - signatures ] , it can be seen that all junction - junction collisions , if perturbed as we hypothesize , result in either junction - edge or edge - edge collisions , and can therefore be resolved accordingly . junction - edge collisions occur when a facet is pinched into two pieces by three of its neighbors , depicted in figure [ fig : self - intersection - signatures]b . there , two adjacent neighbors of the facet , forming a wedge , meet a third neighbor and pierce it . two separate events are possible in this class . in most cases , the wedge simply splits the central facet into two parts , in an event called a * facet pierce*. however , if the normals of the wedge facets and the normal of the central facet lie on the same great circle , then , as the central facet is split , the opposing facet opens up a gap in the wedge : an * irregular facet pierce*. edge - edge collisions occur when a facet is pinched by four neighbors , shown in figure [ fig : self - intersection - signatures]c . in these events , two non - adjacent , exactly parallel edges meet , which requires that the normals of the impinging facets be coplanar with the normal of the pinched facet . again , two variations are possible . if the impinging faces have different normals , the event is called a * facet pinch*. however , if they have the same normal , they join even as they pinch the facet in question , in a process called a * joining facet pinch*. in addition , each event may occur in either symmetrical or asymmetrical flavors , which are shown in figure [ fig : self - intersection - signatures]c1,c2 respectively . the meeting of two edges requires the involvement of two junctions ; these lie on the same edge for the symmetrical case , and on different edges for the asymmetrical case , as seen in the figure . the first self - intersection we will study is the simplest ; the facet pierce . it is a point - line event as described above ; that is , a facet is split when a triangular wedge formed by two adjacent neighboring facets intersects the edge formed with a third , opposing neighbor . the facet pierce is functionally the opposite of a facet join , and is illustrated in figure [ fig : face - split - illustration ] . [ fig : face - split - illustration ] _ resolution . _ each facet pierce is performed by the ` fjoin_repairman ` class . given the constricted facet , as well as the junction and edge which meet , it can label all of the surrounding facet elements and deterministically reconnect them correctly . first , two new facets are created to replace the constricted facet . the junctions and edges that bordered the old facet can be reassigned to these based on the labels created initially . the colliding junction and edge are deleted , to be replaced by three new junctions and two new edges . the locations of the former and neighbor relationships of each can be determined by considering figure [ fig : face - split - illustration ] and using the labels . _ comments . _ first , technically , at the moment of the event , an @xmath22 junction forms , which as shown in figure [ fig : face - split - illustration ] may proceed to break in one of three ways . this does not , however , constitute a non - uniqueness ; rather , the dynamics governing the surface evolution at the moment of topological change specify which exit pathway is chosen . second , while thijssen @xcite rightly objected to this resolution for the case of separate grains , we find it satisfactory for the case of a single crystal considered here . a special modification of the facet pierce just described occurs when the normal of the opposing facet shares a great circle with the normals of the facets forming the wedge . this event is called an irregular facet pierce . recall that three new junctions were created during the facet pierce . however here , since the two newly created facets have identical normals , and the remaining three have normals which are not independent , those junctions can not be created . instead , as the wedge facets meet the opposing facet , one of two things happen either the center edge of the wedge is split apart by the opposing facet ( a `` wedge split '' ) , or the opposing facet is split apart by the wedge ( a `` wedge extension '' ) . we see an illustration of each possibility in figure [ fig : table - split - illustration ] . [ fig : table - split - illustration ] _ resolution . _ the irregular facet pierce is performed by the ` ifp_repairman ` class , which at instantiation is given the constricted facet , as well as the junction and edge which meet . this event is repaired quite similarly to the regular facet pierce , with modifications . as is done there , two new facets are created to replace the constricted facet , and junctions and edges bordering the old facet are reassigned to the new ones . the resolution differs in how to replace the colliding junction and edge . if the `` wedge split''resolution is chosen , then the middle edge of the wedge and its far junction are also deleted these are replaced by two parallel edges and junctions . finally , an edge is formed which links them and borders the facet on the far side of the deleted edge . if the `` wedge extension '' resolution is chosen , not only is the constricted facet split apart , but so is the one opposite the edge split by the wedge . one must first determine which edge of this second split facet the extended wedge will intersect . having done so , that facet is deleted , to be replaced by two new facets . the extension is formed by adding two edges parallel to the middle edge of the wedge , and the edge it intersects is split in two . two new edges and three junctions must be created to link the extension with the edge it intersects . finally , all edges and junctions bordering the deleted facet , plus those created to form the extension , are re - assigned appropriately to the new facets . figure [ fig : table - split - illustration ] is especially helpful here . _ comments . _ the event clearly recalls the `` gap opener '' described above . it shares with that event three coplanar surface normals , and as a result , two possible resolutions . additionally , while the two options here are qualitatively different compared to the symmetric options of the gap opener , they are additionally both non - local effects due to a local cause . again , perhaps the best resolution is to nucleate a new facet , which we do not yet consider . finally , both events require `` intermediate symmetry , '' and for the same reasons discussed above , we have not implemented this event . we now turn to consider the case of edge - edge events , the first of which is called a face pinch . here the normals of the pinching facets are not identical , and so junctions can be created as needed an illustration of this event is shown in figure [ fig : face - pinch - illustration ] . this event is philosophically similar to the face split described above . in each case , a facet is split into two by non - joining neighbors ; the difference is just whether the procedure is `` sharp '' or `` blunt '' ; i.e. , caused by parallel edges or a junction and an edge . [ fig : face - pinch - illustration ] _ resolution . _ each facet pinch is performed by the ` fpinch_repairman ` class . because of the similarities between the facet pierce and facet pinch , the associated ` repairman ` classes behave similarly as well . here , the ` repairman ` class constructor takes the constricted facet and the two colliding edges . with this information , it can label all of the surrounding facet elements and deterministically achieve the change shown in figure [ fig : face - pinch - illustration ] . as with the facet pierce , two new facets are created to replace the constricted facet , and the junctions and edges that bordered the old facet are reassigned as required . the colliding edges are deleted , as are the associated junctions discussed above . five edges and four junctions are created to complete the reconnection , as shown in figure [ fig : face - pinch - illustration ] . _ comments . _ this event , like the irregular neighbor switch and irregular facet pierce , requires a surface with `` intermediate symmetry . '' while it is uniquely resolved and poses no great difficulty of implementation , we have not yet implemented it for this reason . finally , a special modification of the face - pinch occurs when the impinging facets have identical normals . the constricted is split in exactly the same way as in a face pinch ; however , since the two facets doing the `` pinching '' are identically oriented , they join together to form a larger facet . we see an illustration of this situation in figure [ fig : face - swap - illustration ] . [ fig : face - swap - illustration ] _ resolution . _ each joining facet pinch is performed by the ` jfpinch_repairman ` class , which operates similarly to the ` repairman ` classes associated with the facet pierce and facet pinch . this class is again instantiated with the constricted facet and the two meeting edges , which allows the necessary labeling . again , two new facets are created to replace the constricted facet , but in this case the two facets which meet must join , and so another new facet must be created to replace them necessary junction and edge reassignments are again easily carried out . finally , rather than deleting the edges which meet and the associated junctions involved , the meeting edges are simply re - connected as shown in figure [ fig : face - swap - illustration ] . _ comments . _ note that the final configuration is similar to the original configuration ; in fact , with suitable facet motion , the surface could return to its original configuration via another face swap ; the event is thus self - reversible in a sense . also , since both a facet pinch and a facet join occur simultaneously , the total numbers of each surface element remain unchanged during this event . the final class of topological event occurs when a facet shrinks to zero area and is removed . however , as has been noted numerous times previously in the context of cellular networks , very small facets can result in stiff dynamics that are difficult to numerically simulate accurately . for this reason , we follow previous authors by pre - emptively removing facets with areas below some small threshold ( but see section [ sec : pure - predictive - method ] ) . this process is summarized in figure [ fig : facet - removal - illustration ] . there , we see a single small flat facet vanishing into a pentagonal well ( [ fig : facet - removal - illustration]a ) . being smaller than the allowed threshold , it is removed , leaving a `` hole '' in the network ( [ fig : facet - removal - illustration]b ) . the facets and edges bordering this hole we call the * far field * , and they need to be reconnected correctly to patch the hole . the correct reconnection for this particular well is shown in figure [ fig : facet - removal - illustration]c . [ fig : facet - removal - illustration ] the principal difficulty in this process occurs during the reconnection step ( figure [ fig : facet - removal - illustration]c ) . here , we are assigning new neighbor relationships to the far - field facets , which also involves the creation of new edges and junctions to form boundaries between them . in other cellular - network problems , these neighbor relationships ( and hence the reconnection ) is usually chosen randomly , under the reasoning that any error introduced is small enough to neglect and quickly corrected . however , because the faceted - surface network represents a piecewise - planar geometrical surface , we are not free to choose randomly . since each facet in the far field has a normal and a local height , neighbor relationships determine junction locations and thus edge placement . however , the final reconnection must be geometrically consistent all facets must be simply connected , and thus no edges may intersect . if we were to randomly choose our neighbor relationships , the resulting reconnection would likely fail this test , and would thus represent a non - physical `` surface . '' to guarantee a geometrically consistent reconnection , we must search through all _ virtual _ reconnections until we find one that does not result in any self - intersecting facets . several questions immediately arise : 1 : : how can we effectively characterize a `` reconnection '' ? 2 : : how many virtual reconnections are there to search ? 3 : : how can we efficiently list all these choices ? 4 : : can we be sure a good reconnection exists ? 5 : : is this reconnection unique ? for our method to be effective , all but the last of these questions must be answered satisfactorily . the detailed answers to ( 1 - 3 ) are found in the appendix , but we will summarize them here . the edges and junctions created during an @xmath23 reconnection may be effectively characterized as a binary tree with @xmath24 nodes . the number of @xmath25noded binary trees is given by the _ catalan number _ @xmath26 . finally , these trees may be efficiently listed using a greedy recursive algorithm in @xmath27 time . for the fourth question regarding existence , we argue heuristically that a facet reaching zero area proves the existence of its own reconnection , since a surface with a zero - area facet is functionally the same as the surface with that facet removed . we then assume the existence of that same reconnection for some window of time before the facet reaches zero . a fuller proof would appeal to manifold theory . finally , the fifth question regarding uniqueness is addressed in section [ sec : non - uniqueness ] . having established these facts , we have a robust method for reconnecting an arbitrary far field of facets . before considering some special cases of this method , let us summarize the general process so far : whenever facets smaller than a threshold area are detected , we : a : : remove them , leaving a hole in the mesh . b : : list all virtual reconnections ( vr s ) as n - node binary trees . c : : use associated neighbor relationships to find edge locations . d : : test each vr until one with no intersecting edges is found . we note that this approach represents a comprehensive reconnection method for _ any _ cellular network problem . though it is necessary for the faceted surface problem , it may be useful in any situation where a verifiably optimal reconnection is sought . it is possible for _ groups _ of facets to shrink together , in such a way that they can not be removed sequentially . for an example , consider the configurations in figure [ fig : volume - removal ] . in such a case , it is necessary to identify and collect a contiguous group of small facets for simultaneous deletion we call this a * near field*. any facet neighboring the near field is assigned to the far field , which may be reconnected as described previously after the near field is deleted . [ fig : volume - removal ] to gather the near - field facets , we maintain a second , more liberal threshold . whenever a face shrinks below the first threshold , as described above , its neighbors are recursively examined to collect those smaller than the second threshold . this method is rather simplistic , and , in cases of oddly - shaped pyramids , may not return the entire near field . this , in turn , will result in an incorrect far field , which will most likely be non - reconnectable . however , a group of facets vanishing together eventually all head to zero area , and for some window of time before they would physically vanish , all are small enough to be detected in this way . thus , we allow the code to `` skip over '' small facet combinations that it can not remove successfully , and try again during a future timestep . it is also possible , on high - symmetry crystal surfaces , that the small facet or group of facets forms a `` step '' between two much larger facets of identical orientation , but different height . figure [ fig : step - removal ] illustrates this situation , in which the near field is bounded by exactly four facets , two of which have identical orientations . in such a case , the final fate of the surface is that the small facets vanish _ as the large facets join together_. the method described above contains no provision for joining far - field facets during reconnection , and so there is no way to reconnect the far field produced in this case . [ fig : step - removal ] having identified a near field as forming a step , one solution is to delete the small facets , then move the two large faces to the same height and join them . this results in two pairs of unconnected edges , which are each deleted and replaced with an appropriate single edge . since facet groups forming steps are , in fact , bordered by four facets generically , a separate ` repairman ` class could be written to handle this case . however , the small adjustment to the positions of the large facets can lead to subtle problems , as will be seen in section [ sec : discretization and topology ] . therefore , a more robust if less elegant approach is to simply add one of the large parallel facets to the ( step - forming ) near field ; a good choice is the one with fewer edges . since the far field surrounding this modified near field requires no joins , it can be repaired using the ffr method . again , failures are possible as described in the above section , but resolution is always possible near enough to the time the event would physically occur . we have now discussed the general kinematics of a @xmath19 , and surveyed all topological events which may occur as the surface evolves . before our treatment is complete , however , we must consider with care the application of a time - stepping scheme . the accurate performance of topological events under such a scheme is problematic because , while events on a continuously evolving surface happen at precise times ( @xmath28 at @xmath29 ) , any time - stepping method invariably skips over these times . this has three consequences , concerning detection , consistency , and accuracy . after discussing them briefly , we will present three possible timestepping methods which illustrate them in more detail . * detection . * because timestepping will always skip over moments of topological change , we must abandon hope of simply finding topological events ready to perform . instead , we must either look ahead before each timestep and anticipate when events will occur ( a _ predictive _ method ) ; or step before looking , and then by examining the network infer where events should have occurred ( a _ corrective _ method ) . class a events can be easily be detected either way , while class b events are easier to correct , and class c are easier to predict . * consistency . * once the occurrence of an event has been detected by either means , it must be performed in a way that preserves geometric consistency i.e. , the network always corresponds to a physical surface @xmath30 . for example , two joining facets can only be mechanically fused if they exhibit the same local height . if , in addition to the occurrence of an event , a detection scheme can determine the exact time at which it occurred , then one strategy is to move the network to the precise event time , at which resolution is trivial . however , one may wish to attempt resolutions at other times , and the geometric consequences of doing so must be weighed . * accuracy * finally , we must consider the possibility of error that is produced during topological change . this error is most easily understood if we view the evolving surface in its abstract form as a highly nonlinear system of ode s . the ( usually configurational ) evolution function is moderated by the topological state ; thus , topological events can represent sudden , qualitative changes in the evolution function . a naive time - stepping scheme which steps over these without appropriate measures will produce large localized errors at moments of topological change . assume that , at all times , we accurately predict the time and location of the next topological event . then , a straightforward timestepping strategy which avoids consistency and accuracy concerns is to continually calculate the time of the next topological event ( accurate to the the order of the time - stepping method ) , and then step from event to event . under this approach , time is divided into slices with constant equations of motion , guaranteeing that that the system always evolves under the correct equations , and accurately representing the continuously evolving surface . in addition , high - order single - step methods such as runga - kutta methods may be used to obtain high accuracy . though neatly eliminating consistency and accuracy concerns , this method has a serious disadvantage . the frequency of topological events scales with the system size , and since we can never step farther than the next event , we effectively make the timestep dependent on system size @xmath31 . since moving the system through a single timestep is itself an @xmath32 operation , then advancing the system through any @xmath33 period of time takes @xmath34 time . while acceptable for the detailed study of a small surface , it is obviously undesirable for the statistical study of large surfaces this is chiefly because , consistency concerns aside , it makes little sense to halt the entire surface at every single topological event , when each of these involves only a few facets . thus , our next method has as its chief objective the use of timesteps which are independent of system size . a second strategy is to take fixed timesteps , use a corrective method of topological detection , and attempt to perform topological corrections late . since timestep is independent of system size , many events will now occur per timestep , the size of which is chosen to produce a fixed small _ percentage _ of facets undergoing topological change each step . while this approach theoretically eliminates the @xmath34 contribution to running time , it introduces hurdles to event detection , as well as geometrically consistent and accurate resolution . * detection . * we just stated that , in this corrective detection scheme , more than one event occurs per timestep . whether or not this is a problem depends on the * domain of influence * of each event , defined to be the set of network elements that event affects . if these sets contain no common elements , then the associated events occur too far apart in space to affect each other they are independent . consequently , a detection routine can hand them in arbitrary order to the repair routines , there to be confidently performed in isolation . however , occasionally two or more domains of influence overlap . in this situation , called a * discrete compound event * , the associated topological events are no longer independent , and a detection routine can no longer ensure _ a priori _ their correct , consistent resolution when handed off . even worse , the very signatures used to identify separate events may be obscured in the resulting `` tangle , '' such that the routine does not even recognize what has happened . given the variety of event signatures described in section [ sec : topological events ] , and the many combinations in which they might occur , creating a complete list of all dce s would be prohibitive if not impossible . instead , we reason that , on a random surface , no two events will ever occur at exactly the same moment ( it is possible to artificially construct faceted surfaces such two or more events must occur simultaneously we do not consider this case ) . thus , if we simply refine our timestep when necessary , formerly overlapping events can be sorted out , and detected in sequence . a robust strategy for handling compound events is thus to ( a ) _ retrace _ the problematic timestep , ( b ) _ refine _ it into smaller slices , and ( c ) _ repeat _ steps ( a ) and ( b ) recursively , until only single events are detected . * consistency . * since the surface is allowed to evolve unrepaired past numerous topological events per timestep , surface regions near these events will be geometrically inconsistent after the step . to say the same thing , facets involved in the bypassed events will have incorrect neighbor relationships . however , we have already classified all possible events , so having identified which event occurred , and which facets were involved , we know _ a priori _ what the correct neighbor relationships should be after the event . this knowledge , along with knowledge of the position of each facet involved , allows us to reconstruct the consistent surface that should have emerged during the event . unfortunately , not all events can be consistently corrected at a late time in this way . in particular , facet joins and joining facet pinches involve the joining of two facets that meet each other at a single local height . since this condition exists for only a single instant , such events can not be performed in a geometrically consistent way at any time other than the `` correct '' one . to accommodate this requirement while preserving a topology - independent timestep , we are forced to manually adjust the height of the joining facets before the event is performed . besides the error induced by this strategy ( discussed next ) , this need illustrates a second problem that can arise . in a * repair - induced inconsistency * , the very act of performing one event , because it is done late , triggers a second event that was not detected originally . an example is when the just - described height adjustment required for the delayed repair of a facet join triggers , say , a neighbor - switching event . since this newly - triggered event was not originally detected , the system is left in an inconsistent state after all repairs are made . an ad - hoc strategy to find such rii s is infeasible for the same reason as is a complete listing of all possible discrete compound events ( indeed , an rii may produce a dce , which rules out a simple multiple - rechecking strategy ) . thus , a similar retrace / refine / repeat strategy is required , with the added requirement that all events performed prior to detecting the rii must first be undone . * accuracy . * finally , as alluded above , repairing topological events _ after _ they occur can introduce large isolated errors . this can be due to the `` fudging '' required for the delayed repair of facet joins and swaps , but more generally is caused by facets involved in ( uncorrected ) topological events having been evolved under the wrong equations of motion for part of the relevant timestep . consider an event @xmath35 , with domain of influence @xmath36 . since topological events likely correspond to a change in the surface s evolution equation , the facets in @xmath36 are moved using the wrong equations for the time interval @xmath37 $ ] . since the equations guiding @xmath36 are wrong by as much as @xmath33 for a time of order @xmath38 , facets in @xmath36 may accumulate @xmath39 location errors during the timestep in which the event occurs . since the quantity of topological events does not depend on @xmath40 , the method retains first order accuracy globally . however , this error introduces a barrier to achieving higher - order accuracy later on . the previous method , alas , contains one subtle problem that keeps it from being a true @xmath32 method . this problem is that the frequency of dces and riis , though small , still scales with the system size , and these necessitate timestep refinement . so although the late method does not have to explicitly step according to the @xmath41 time between topological events , yet to accurately detect and resolve those events it is still implicitly driven by the refinement strategy to step along a time associated with dces and riis . while this characteristic time is longer than that between individual topological events , and does not greatly slow the simulation of tens of thousands of facets , it still results in a method that is formally @xmath34 , which becomes prohibitive when considering systems of millions of facets . thus , we now sketch a third method , not yet implemented , which eliminates this effect all together . in addition , the method allows us to perform topological events in a way which confers all the accuracy benefits of the first , predictive method . we first re - state that , on any given timestep , _ most _ facets are not involved in any topological changes . while it was therefore obviously wasteful to move the entire surface from event to event in the first method , it is also conceptually wasteful to perform a global retrace / refine / repeat step to dces and riis in the second method . instead , after every timestep , we should identify for each dce / rii the * topological subdomain * containing all facets involved in the event . the few facets within these subdomains would be retraced / refined / repeated as required , while the rest of the ( unaffected ) facets would be left undisturbed in their post - timestep state . since operating on a given , constant number of facets takes @xmath33 time , and since the number of events per timestep scales only like @xmath32 , we see that a single timestep and all associated corrections including dces and riis can now be performed in @xmath32 time , with a final state that is guaranteed to be consistent . this produces a true @xmath32 method . in addition , this `` localized replay '' strategy has an accuracy benefit . regular , recognized topological events also have easily identifiable topological subdomains . if the facets within these domains are retraced , then the predictive detection mechanism of the first method can be applied within the domain to eliminate consistency and accuracy problems associated with late removal . one difficulty remains , however . facets involved in topological events may , under configurational facet - velocity laws , exhibit abrupt changes in velocity a result of the event . during the remaining segment of timestep , these facets may `` break out '' of the subdomain initially created to contain them , and begin interacting with facets outside of it . thus , we would need a mechanism to detect this , and start over with a larger subdomain if it occurs . finally , if subdomains can change size , then there is the possibility that two nearby subdomains will come to overlap as the algorithm progresses . therefore , we must include the ability to merge them if necessary , start over with the new , larger sub - domain , and repeat adaptively until everything can be sorted . this adaptivity ensures the robustness of the method , as highlighted by the method s formal name of * adaptive localized replay*. the reader may note that the pattern of adaptive repetition is similar to that used to resolve dces and riis above , and worry that another , even smaller @xmath34 effect lurks in the shadows . however , in both of the previous methods , such effects were due to the _ global _ response to a _ local _ problem . since this latter method is designed to be localized , there is no longer any mechanism to generate such effects . we demonstrate our method using the sample dynamics derived in chapter 1 , associated with the directional solidification of a strongly anisotropic dilute binary alloy . when a sample is solidified at a pulling velocity which is greater than some critical value , solute gradients caused by solute rejection at the interface create a solute gradient which opposes and overcomes the thermal gradient , resulting in a negative effective thermal gradient . in this environment , facets move away from the freezing isotherm at a rate proportional to their mean distance from the isotherm , as given by the dynamics @xmath42 in figures [ fig : threefold ] , [ fig : fourfold ] , and [ fig : sixfold ] , this dynamics is applied to surfaces with common three- , four- , and six - fold symmetries to illustrate the flexibility of our topology - handling approach . a series of snapshots from the coarsening surface are presented , in which surface configurations near to many of the topological events described above may be observed . ( however , neither the irregular neighbor switch , irregular facet pierce , nor facet pinch occur because no three facet normals are coplanar in these symmetries ; indeed , these events are not expected to occur on most physical surfaces , and were included for theoretical completeness . ) [ fig : threefold ] about half of computational time is spent looking for topological changes , which is significant but not prohibitive . with appropriate timestep choice , using even the still - inefficient timestepping method 2 above , a surface of @xmath43 facets may be simulated to a @xmath44 percent coarsened state in about an hour on currently available workstations . with the implementation of method 3 above , this time should be cut in half , and since method 3 is truly @xmath32 , a single million - facet simulation should take about a day . looking further ahead , since facet velocity calculations and topology checks require only local information , the method should be easily parallelizable , making possible even larger speed gains . we have presented a complete method for the simulation of fully - faceted interfaces of a single bulk crystal , with arbitary symmetry , where an effective facet velocity law is known . the surface , which is reminiscent of two - dimensional cellular networks , is encoded numerically in a geometric three - component structure consisting of facets , edges , and junctions , and the neighbor relationships between them . consistent surface evolution specified by the facet velocity law is accomplished via a simple relationship between facet motion and junction motion . although requiring the explicit handling of topological events , the method is efficient , using the natural structure of the surface , and accessible , allowing easy extraction of geometrical data . this combination makes it ideal for the statistical study of extremely large surfaces necessary for the investigation of dynamic scaling phenomena . a comprehensive listing of all topological events has been presented . these allow single - crystal surface with arbitrary symmetry ( or no symmetry at all ) to be simulated . events are classified into three categories , representing three ways that surface elements can become geometrically inconsistent . these are : edges which approach zero length , facets which become constricted , and facets which approach zero area . resolution strategies for the former two classes can be determined a priori , while repairing surface `` holes '' left by vanishing facets requires a novel far - field reconnection algorithm , which iteratively searches through all virtual reconnections to find one which produces a consistent surface . finally , intrinsic non - uniqueness of several events is discussed ; since ours is a purely kinematic method , decisions regarding resolution of these events must be made ahead of time through consideration of the dynamics or other physics . in addition , a detailed discussion of the issues associated with a discrete time - stepping scheme has been presented . the core issue is that topological events , which occur at discrete times throughout surface evolution , invariably fall between timesteps , with consequences for the detection of events , as well as their geometrically consistent and numerically accurate resolution . since topological change corresponds ( under configurational facet velocity laws at least ) to qualitative changes in the local evolution function , some way to reach these in - between times must be introduced , while recognizing that only a few facets are involved in topological change during each timestep . a comparison of three approaches showed that the optimal solution is one of localized adaptive replay , where large timesteps are taken to improve speed , but local surface subdomains associated with topological change are reverted , and then replayed in a way that re - visits events with the necessary precision as necessary . while further work remains to implement this approach , the method as presented is capable of comparing million - facet datasets via averaged runs . our method of removing facets and facet groups requires patching a `` hole '' in the network left by the deleted facets . this requires selecting a geometrically consistent reconnection from a list of potential , or virtual reconnections . as outlined in the text , this involves searching through a complete list of virtual reconnections and testing each for geometric consistency . in this appendix , we address in more detail questions ( 1 - 3 ) posed in section [ sec : vanishing - facets ] regarding the details of this method . for convenience , we repeat them here : 1 : : how can we effectively characterize a `` reconnection '' ? 2 : : how many potential reconnections are there to search ? 3 : : how can we efficiently list all potential choices ? we show here that an effective means of answer these questions is to think of reconnections as extended binary trees . this characterization enables us to easily count potential reconnections , distinguish them through naming , and suggests an algorithm for efficiently listing them for testing . an exhaustive illustration of the process is given for the case of an @xmath22 far field in figure [ fig : binary tree listing ] . it will be useful to refer to that diagram during the following discussion . [ fig : binary tree listing ] patching network holes always involves finding unknown neighbor relations between a given number of adjacent facets that is , no facets are ever created , only edges and junctions . these are always connected into a single graph . in fact , the edges and junctions created during reconnection ( the `` reconnection set '' ) form a binary tree . in figure [ fig : binary tree listing ] , the trees associated with each possible reconnection are shown in thick blue lines . the far - field edges touching the reconnection set shown in gray represent the _ completion _ of this tree . that is , they take the interior tree , and add leaves to it so that every node of the interior tree is a triple - node . each virtual reconnection corresponds to a unique interior tree and completed tree in this manner . the counting of binary trees is , fortunately , a solved problem of graph theory . given @xmath8 nodes , they may be arranged in @xmath45 distinct binary trees , where @xmath45 is the @xmath46 _ catalan number _ ; @xmath47 now , re - connecting an @xmath23 far field requires the creation of @xmath48 edges and @xmath24 nodes ; this may be visually confirmed for the case @xmath49 in figure [ fig : binary tree listing ] . this creates an @xmath24 noded binary tree , and so an @xmath23 far field has @xmath50 virtual reconnections to search . if take the completed binary tree and arbitrarily select a _ root node _ , then from each virtual interior tree we can generate a unique sequence of letters which identify it and encode its construction . this can be formed in one of two ways . the first way is to specify the tree by a set of recursive function calls . each branch point has left and right branches , each of which may terminate in either a leaf , or another branch point . the middle column of figure [ fig : binary tree listing ] gives such a function for each tree shown in the left column . the second way is to walk around the tree in a counter - clockwise manner , recording each branch point or leaf as it is encountered . either method produces a series of letters that uniquely identify the tree . since the beginning lb and terminal ll are guaranteed , we may use only an abbreviated version consisting of @xmath48 of each letter . the problem is now reduced to generating all possible letter combinations . we can recursively build these combinations letter by letter using a greedy algorithm which chooses b over l if possible . this approach is subject to three restrictions which must be true of a `` legal '' word . at each step , ( a ) @xmath51 , ( b ) @xmath52 , ( c ) @xmath53 . the function we use is sufficiently short that we simply reproduce it here : .... list_trees(n ) { rec_list_trees(n , 0 , 0 , '' ) ; } rec_list_trees(n , leaves , branches , word ) { max = n-3 ; if ( branches < max ) rec_list_trees(n , leaves , branches+1 , word+'b ' ) ; if ( leaves < max and leaves < = branches ) rec_list_trees(n , leaves+1 , branches , word+'l ' ) ; if ( leaves = = max and branches = = max ) print(word ) ; } .... thus , using this algorithm to generate a complete list of reconnection labels , we perform the reconnection associated with each one , and test it for geometric consistency . this allows us to efficiently find the correct reconnection . we have already mentioned that certain topological events are ambiguous in their resolution . here we review the instances of non - uniqueness , discuss their cause and implications , and suggest a method of treating them in a numerical scheme . we begin our list with an important ambiguity not discussed in the main text . as noted by thijssen @xcite , the lowly neighbor switch can be non - unique if the four facets involved have a `` saddle '' configuration where the edges neighboring the vanishing edge ( the emanating edges ) form an alternating sequence of two valleys and two ridges . this is illustrated in figure [ fig : non - uniqueness]a . there , from either starting position on the left , two topological resolutions are possible . one involves changing neighbor - relations , while the other does not . this latter resolution , which we term a `` neutral pass , '' represents a vanishing edge event which needs no resolution . however , it still results in a ( prohibited ) flipped edge , so if it is to actually occur , a bookkeeping operation must take place to correct this . moving on , we recall the non - unique `` gap opener '' flavor of irregular neighbor switch , which is not reproduced here . however , as a vanishing edge event , it may also come in the saddle variety , and admits a neutral - pass resolution option which is displayed in figure [ fig : non - uniqueness]b . finally , the irregular facet pierce event is also not reproduced . none of these cases of ambiguity of resolution can be resolved at the kinematic level . since the particular evolution pathway leading to such ambiguities is the product of a long chain of mathematical reasoning , we therefore look backward in this chain to resolve them . the kinematics are most directly provided by the dynamics , which is a good first place to start . one option is to see which resolution is more strongly self - reinforcing ; if one choice under the given dynamics immediately works to reverse itself , for example , the other option should probably be chosen . moving farther back the chain of reasoning , one may consider the argument used to derive the given dynamics . for example , the dynamics ( [ eqn : solidification - dynamics ] ) comes from the constant reduction of an undercooling energy by moving away from the maximally - expensive @xmath6 isotherm . therefore , an energy - informed choice is to choose the resolution which minimizes this energy , i.e. , maximizes the total integrated distance from @xmath6 . taking another step back , perhaps such arguments come , as this one does , from a partial differential equation describing surface evolution . such equations can be simulated directly , and the resulting resolution choices studied . ultimately , the original physical model may have to be considered , perhaps at the atomic level . for each dynamics we wish to study , one must apply this chain of reasoning to find the correct ambiguity - resolution strategy . ideally , considerations at all levels should produce identical results . the non - uniqueness of particular topological events reflects a deeper , underlying problem . for each non - unique event , facet heights of competing resolutions are identical ; only neighbor relationships between these facets differ . thus , non - uniqueness in topological events indicates the presence of , and indeed is caused by , multiple possible ffr - style reconnections of a given group of far - field facets . what are the implications for the ffr algorithm of early facet removal ? while it may seem helpful at first to list prohibited resolution configurations and feed these to the ffr algorithm , this approach grows increasingly brittle as the number of prohibited configurations grows . instead , a more robust approach is to cause an ffr yielding multiple valid results to fail . having discovered resolution strategies for each individual event as described above , we simply let these rules apply until the far field is small enough to be unique ( similar to @xcite ) . a theory predicting when far fields will have multiple resolutions would aid in this process . finally , we note that the benefits of phase - field and level - set methods , which automatically capture topological change , are only clear if topological changes are unique . these methods are ultimately only kinematic , and offer no clear criteria for resolving kinematic non - uniqueness . thus , a dynamic selection criteria would seem to be required no matter the method used , and indeed , the implicit handling of topology offered by these methods may actually hinder the dynamic selection of kinematically ambiguous events . | we fully generalize a previously - developed computational geometry tool @xcite to perform large - scale simulations of arbitrary two - dimensional faceted surfaces @xmath0 .
our method uses a three - component facet / edge / junction storage model , which by naturally mirroring the intrinsic surface structure allows both rapid simulation and easy extraction of geometrical statistics .
the bulk of this paper is a comprehensive treatment of topological events , which are detected and performed explicitly .
in addition , we also give a careful analysis of the subtle pitfalls associated with time - stepping schemes for systems with topological changes .
the method is demonstrated using a simple facet dynamics on surfaces with three different symmetries .
appendices detail the reconnection of `` holes '' left by facet removal and a strategy for dealing with the inherent kinematic non - uniqueness displayed by several topological events . |
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the idea of quantum pumping , i.e. of producing a dc current at zero bias voltage by time periodic modulation of two system parameters , dates back to the work of thouless @xcite . if the parameters change slowly as compared to all internal time scales of the system , the pumping is _ adiabatic _ , and the average charge per period does not depend on the detailed time dependence of the parameters . using the concept of emissivity proposed by bttiker et al.@xcite , brouwer@xcite related the charge pumped in a period to the derivatives of the instantaneous scattering matrix of the conductor with respect to the time - varying parameters . since then , a general framework to compute the pumped charge through a conductor has been developed for noninteracting electrons@xcite . the interest in the pumping phenomenon has shifted then to the experimental@xcite investigations of confined nanostructures , as quantum dots , where the realization of the periodic time - dependent potential can be achieved by modulating gate voltages applied to the structure@xcite . in case of interacting electrons the computation of the pumped charge becomes rather involved and few works have addressed this issue for different systems@xcite and in specific regimes . as for the case of interacting quantum dots , the pumped charge in a period was calculated by aono@xcite by exploiting the zero - temperature mapping of the kondo problem . a very general formalism was developed in ref . [ ] where an adiabatic expansion of the self - energy based on the average - time approximation was used to calculate the dot green s function while a linear response scheme was employed in ref . more recently , another interesting study@xcite was performed aiming at generalizing brouwer s formula for interacting systems to include inelastic scattering events . in this work we present a general expression for the adiabatic pumping current in the interacting quantum dot in terms of instantaneous properties of the system at equilibrium , generalizing the scattering approach for noninteracting particles and discuss the limit of its validity . to get a pumped current the two model parameters which are varied in time are the tunneling rates between the noninteracting leads and the quantum dot . in particular , we let them vary both in modulus and phase through the adiabatic and periodic modulation of two external parameters ( e.g. gate voltages or magnetic fields ) and show that a rectification - like term arises in the current due to the time - dependent tunneling phase . the plan of the paper is the following . in sec.[sec : model ] we introduce the model and relevant parameters . we develop the scattering matrix approach together with the green s function formalism to derive the formula of the pumped current through an interacting multilevel quantum dot in the adiabatic regime at very low - temperatures . in sec.[sec : single - level ] we specialize on a single - level quantum dot and give the explicit expression of the pumped current . conclusions are given in sec.[sec : conclusions ] . we consider a multi - level quantum dot ( qd ) coupled to two noninteracting leads , with the external leads being in thermal equilibrium . the hamiltonian of the system is given by : @xmath1 where @xmath2 , with @xmath3 the creation ( annihilation ) operator of an electron with spin @xmath4 in the lead @xmath5 and dispersion @xmath6 . the qd is described by the hamiltonian @xmath7 , where @xmath8 with @xmath9 the creation ( annihilation ) operator of the electron with spin @xmath10 and @xmath11 the dot @xmath12-th energy level . the on - site energy @xmath13 describes the coulomb interaction . the tunneling hamiltonian is given by @xmath14 , with time - dependent tunnel matrix elements @xmath15 . for simplicity we assume that @xmath16 are spin independent , i.e. @xmath17 and that both the modulus and the phase of @xmath18 vary in time with frequency @xmath19 , i.e. @xmath20 . their explicit time dependence is determined by two external parameters ( e.g. two gate voltages applied at the barriers of the dot or a gate voltage and a magnetic field ) which are varied adiabatically and periodically in time or by the presence of parasitic bias voltages . two specific examples will be considered below . in particular , we will specialize on the case in which the tunneling phase can vary harmonically or linearly in time . the instantaneous strength of the coupling to the leads is instead characterized by the parameters @xmath21 , where @xmath22 is the density of states in the leads at the fermi level . by varying in time @xmath23 and @xmath24 and keeping them out of phase , the charge @xmath25 pumped in a period @xmath26 is related to the time dependent current @xmath27 flowing through the left barrier , i.e. @xmath28 . while the exact formula for the current depends on time - dependent green s function out of equilibrium , in the following we consider the adiabatic limit where the current depends only on the instantaneous equilibrium properties of the dot , i.e. on the retarded dot green s function ( gf ) . this situation is realized in the two following cases . first , let us consider that only the modulus of the tunneling matrix elements is varied in time , i.e. @xmath29 . under the adiabatic condition , the tunneling rate varies slowly in time , and the quantum dot can be considered time by time in equilibrium with the external leads . the effect of quantum pumping is well described by an adiabatic expansion of the self - energy based on the average - time approximation as described in ref . [ ] and using the equilibrium relations to write the pumped current in terms of the retarded gf only . let us consider now the situation in which the modulus of the tunneling terms is fixed while their phases are modulated in time , i.e. @xmath30 . this situation is equivalent ( by a gauge transformation ) to having a system biased with an ac external signal . in particular , the ac voltage applied to the leads is proportional to the time derivative of the tunneling phase @xmath31@xcite . since the tunneling terms are assumed to vary in time with frequency @xmath19 , the ac signal forcing the quantum dot is proportional to the pumping frequency @xmath19 and thus can be considered as a small perturbation under the adiabatic condition . in particular , if one consider the case in which the tunneling phases vary linearly in time , @xmath32 , this situation corresponds to an interacting quantum dot biased by a dc voltage @xmath33 . following the work by meir and wingreen@xcite , the current @xmath34 flowing through the interacting multilevel dot biased by a dc voltage @xmath33 can be written as : @xmath35tr\{g^a\gamma^r g^r \gamma^l\mathcal{r}\},\end{aligned}\ ] ] where @xmath36 are the fermi functions , @xmath37 are the dot - leads coupling strengths and @xmath38 is the ratio between the fully interacting self - energy and the noninteracting one , responsible for the deviation from the landauer - bttiker formula ( see ref . [ ] , eq.(10 ) ) . in the zero - temperature limit and for a weak bias , @xmath39 at the fermi level and thus eq . ( [ eq : curr - out - eq ] ) can written as @xmath40 , which corresponds to the usual linear response form , even though the green s function are interacting ones . this argument is extensively discussed in refs . [ ] ( see eq.s ( 37 ) and ( 38 ) ) and [ ] . thus in general we expect that , when both the modulus and the phase of the pumping parameters are varied in time , apart the usual dc pumping current a new term arises ( that we call of _ rectification _ ) which is proportional to the time derivative of the tunneling phase : @xmath41 . since , as explained above , in adiabatic regime the current is determined by the instantaneous properties of the dot ( retarded green s function ) and since in the zero temperature limit the usual linear response formula can be adopted for the calculation of the current , the pumping and rectification currents through the interacting quantum dot can be calculated by the scattering matrix approach as well . in fact , as well known , the retarded green s function is related to scattering matrix by the fisher - lee relation . we thus employ the scattering matrix formalism developed in ref . [ ] where the charge current originated by an adiabatic pump is related to an expansion of the quantity @xmath42s^{\dag}(e,\tau)\}_{\alpha\alpha}$ ] with respect to the time derivative operator @xmath43 ( here @xmath44 is the scattering matrix and @xmath45 is the fermi function ) . the first order of this expansion reproduces the famous brouwer s formula@xcite . let us only stress that the scattering matrix formalism is well defined , not only in the noninteracting case , but also for the interacting problem ( e.g. see ref . the expression of the pumped current @xmath46 in terms of the time - dependent scattering matrix@xcite is : @xmath47 where @xmath45 is the fermi function and @xmath44 is the instantaneous @xmath48-matrix of the qd . it is given by the wigner transform @xmath49 , where @xmath50 here @xmath51 is the full retarded qd green s function , @xmath52 and @xmath53_{mn}=2\pi i\rho v^{\ast}_{\alpha , m}(t)v_{\beta , n}(t')$ ] . in the limit of the pumping frequency @xmath54 , i.e. under the adiabatic condition , the scattering matrix @xmath44 is expressed by the instantaneous green s function of the dot as@xcite : @xmath55 when substituting ( [ eq : inst ] ) into ( [ eq : curr ] ) to compute the current we need the time - derivative of the qd green s function which satisfies the relation : @xmath56 where the dot symbol indicates a time - derivative and the matrix notation for the dot green s function has been used . the final expression obtained for the @xmath46 for a multi - level quantum dot is@xcite : @xmath57,\end{aligned}\ ] ] where @xmath58 this expression has been obtained by considering explicitly the time dependence of the modulus and phase of the tunnel matrix elements , and consequently of the leads - dot coupling function @xmath59 . the symbol @xmath60 stands for the l , r lead in correspondence of @xmath61=r , l . the total dc current through the lead @xmath61 is given by : @xmath62.\ ] ] the expression ( [ eq : main ] ) represents our main result . it is valid for a multi - level qd and for any interaction strength in the zero temperature limit under the adiabatic condition . the first term in eq.([eq : main ] ) represents the pumping current , while the second one , proportional to the time - derivative of the tunneling rate phase , is the effective rectification term we have discussed above . it can also be written as @xmath63 , where @xmath64 is the conductance of the structure , while @xmath65 . the last term in ( [ eq : main ] ) contains information on the time derivative of the retarded self - energy and is zero for a single - level quantum dot within the wide band limit . up to now we have developed a theory of the pumped current valid in the case of a multi - level qd . we now specialize eq . ( [ eq : main ] ) to the case of a single level qd . eliminating the trace in ( [ eq : main ] ) and considering the remaining quantities as c - numbers , the expression for the current simplifies to : @xmath66.\end{aligned}\ ] ] when the time - derivative of the tunneling phase is neglected the above formula is equivalent to the pumped current calculated by the self - energy adiabatic expansion@xcite . in the following we consider the case of a single level qd both in the strongly interacting and non - interacting case and describe the behavior of the charge @xmath67 ( in unit of the electron charge @xmath68 ) pumped per cycle in the zero - temperature limit . the fermi energy @xmath69 is set to zero as reference energy level , while the static linewidth @xmath70 is assumed as energy unit ( typical value for @xmath71 is @xmath72 ) . + in the noninteracting case , i.e. when the qd green s function becomes a scalar , the expression for the instantaneous pumping current is explicitly given by : @xmath73.\end{aligned}\ ] ] when @xmath74 , the charge pumped is zero when the level is resonant ( @xmath75 ) . in the case of a strongly interacting quantum dot , i.e. in the infinite-@xmath13 limit , we take the expression of the qd green s function as in ref . the current is : @xmath76,\end{aligned}\ ] ] where the occupation number @xmath77 on the dot has to be determined self - consistently by the relation @xmath78 , where @xmath79 is the qd lesser green s function . + in order to show the effects of the time - dependent tunneling phase we report below the numerical results of the charge @xmath67 pumped per cycle . the pumping cycle is determined by the periodic time variation of the leads - dot coupling strength , where @xmath80 , while for the tunneling phase two cases can be considered . either it varies harmonically @xmath81 with the same frequency of the two external gate voltages ( this case is shown in fig.[fig : fig1 ] ) or it varies linearly in time @xmath82 , e.g. when a parasitic gate voltage is present , ( this case is shown in fig.[fig : fig2 ] ) . the quantity @xmath83 is the pumping phase that we take different from zero between l and r lead . + in fig.[fig : fig1 ] we plot @xmath67 as a function of the energy level @xmath84 by fixing the other parameters as : @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 . in particular , in the upper panel , the charge induced by the pumping ( triangle ) and the charge due to the rectification term ( box ) is shown for a non - interacting dot ( @xmath92 ) . the total charge ( empty circle ) is significantly modified by the presence of the rectification term which is a non - vanishing quantity at the fermi energy ( @xmath75 ) . the lower panel in fig.[fig : fig1 ] shows the behavior of the rectification current as a function of @xmath84 in the strongly interacting limit ( @xmath93 ) and by choosing the remaining parameters as in the upper panel . while the general aspect of the total charge ( empty circle ) is only marginally modified by the strong correlations in the specified region of parameters , the rectified charge ( box ) shows a pronounced asymmetric behavior with respect to the level of the dot . pumped per pumping cycle as a function of the dot level @xmath84 . the pumping ( triangle ) and the rectification ( box ) contribution to the total charge ( empty circle ) are shown for the non interacting case ( @xmath92 ) in the upper panel and for strongly interacting dot ( @xmath94 ) in the lower panel . both figures are computed for the following choice of parameters : @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 . the pumping cycle is determined by : @xmath80 , @xmath81 , with @xmath95 and @xmath96 , @xmath97,title="fig : " ] + pumped per pumping cycle as a function of the dot level @xmath84 . the pumping ( triangle ) and the rectification ( box ) contribution to the total charge ( empty circle ) are shown for the non interacting case ( @xmath92 ) in the upper panel and for strongly interacting dot ( @xmath94 ) in the lower panel . both figures are computed for the following choice of parameters : @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 . the pumping cycle is determined by : @xmath80 , @xmath81 , with @xmath95 and @xmath96 , @xmath97,title="fig : " ] + in fig.[fig : fig2 ] we focus on the case of time - linear variation of the phase @xmath98 , and take the parameters as in fig.[fig : fig1 ] . in the upper panel , the pumped charge @xmath67 is plotted as a function of the dot level @xmath84 in the non - interacting case ( @xmath92 ) . contrary to the previous case , the rectification contribution ( box ) is dominant over the one induced by the pumping mechanism ( triangle ) and thus the total charge ( empty circle ) is mainly affected by a resonant - like behavior . in the lower panel , the results for @xmath94 are shown . apart from a renormalization of the linewidth of the resonance induced by the factor @xmath99 in the numerator of eq . ( [ eq : single - level - interact ] ) , a behavior similar to the one of the non - interacting system is found . a slave boson treatment with the inclusion of a renormalization of the dot energy level , could in principle modify this picture . let us note that when the tunneling phase varies harmonically both the pumping current and the rectification current follow the same @xmath100 behavior w.r.t . the pumping phase @xmath0 . pumped per pumping cycle as a function of the dot level @xmath84 in the non interacting case ( upper panel ) and strongly interacting case ( lower panel ) . the pumping ( triangle ) and the rectification ( box ) contribution to the total charge ( empty circle ) are shown in the zero temperature - limit by setting the remaining parameters as follows : @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 . differently from fig.[fig : fig1 ] , the pumping cycle is determined by : @xmath80 , @xmath82 , with @xmath95 and @xmath96 , @xmath97.,title="fig : " ] + pumped per pumping cycle as a function of the dot level @xmath84 in the non interacting case ( upper panel ) and strongly interacting case ( lower panel ) . the pumping ( triangle ) and the rectification ( box ) contribution to the total charge ( empty circle ) are shown in the zero temperature - limit by setting the remaining parameters as follows : @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 . differently from fig.[fig : fig1 ] , the pumping cycle is determined by : @xmath80 , @xmath82 , with @xmath95 and @xmath96 , @xmath97.,title="fig : " ] + in fig.[fig : fig3 ] we show the behavior of the charge induced by the pumping term ( upper panel ) and by the rectification ( lower panel ) with respect to the pumping phase @xmath0 and by fixing the dot level to @xmath101 and the remaining parameters as done in fig.[fig : fig2 ] . the full line in both panels represents the result for the @xmath92 case , while the full circles ( @xmath102 ) represent the curves computed for the infinite-@xmath13 case . let us note that while the pumping term follows the conventional @xmath100-behavior as in brouwer theory , the rectification contribution takes the form @xmath103 $ ] , where the coefficients @xmath104 and @xmath105 for the @xmath92 case are explicitly given by : @xmath106}{32[\xi+(\gamma_0/2)^2]^3},\end{aligned}\ ] ] with @xmath107 . the different symmetry of the pumping and rectification current has already been reported in experimental works in quantum dots@xcite and some theoretical explanations have been proposed@xcite . furthermore , the analysis of the coefficient @xmath105 shows that the charge transferred by the rectification effect can be significantly increased by coupling the dot region to the leads in an asymmetric way ( @xmath108 ) , e.g. by using tunnel barriers with very different transparencies . the above results remain almost unchanged in the strongly interacting case ( @xmath109 ) . + is reported for @xmath101 and taking the remaining parameters as in fig.[fig : fig2 ] . notice that the pumping term is proportional to @xmath100 , while the rectification one is proportional to @xmath110 . both in the upper and lower panel , the full line represents the non interacting result , while the interacting case is indicated by full circles ( @xmath102).,title="fig : " ] + is reported for @xmath101 and taking the remaining parameters as in fig.[fig : fig2 ] . notice that the pumping term is proportional to @xmath100 , while the rectification one is proportional to @xmath110 . both in the upper and lower panel , the full line represents the non interacting result , while the interacting case is indicated by full circles ( @xmath102).,title="fig : " ] + in the limiting case in which the phase difference between the tunneling barriers is kept zero , i.e. @xmath111 , the pumping term is exactly zero and the rectification contribution acts as a quantum ratchet@xcite . within the green s function and scattering matrix approach we have analyzed the quantum pumping current through an interacting quantum dot when both the modulus and the phase of the model time - dependent parameters , in our case the leads - dot tunneling rate , is adiabatically varied . in this way it has been possible to derive an expression for the pumped current containing an effective rectification term due to the time - dependent phase . such contribution can be written in a landauer - buttiker - like form , even though for the interacting system , when the zero temperature limit and the adiabatic conditions are met . the numerical analysis also show that when the tunneling phase varies linearly in time the rectification term is even with respect to the pumping phase @xmath0 , i.e. of the form @xmath112 , in contrast to the usual pumping term which is odd . the mentioned contribution could be related to the experimentally observed rectification effects in quantum dots @xcite . + in particular , we have been considering an open system , but concerning closed systems ( e.g. annular devices with a quantum dot ) , the tunneling phase contribution to the pumped charge could find a natural interpretation in a complex phase of geometric nature@xcite . this phase would be a berry phase @xcite . thus the detection of a rectification current in addition to the pumping one could be an indirect probe of a berry phase . + the proposed analysis could be easily generalized up to the second order in the pumping frequency @xmath19 allowing to describe features involved in the moderate non - adiabatic limit . we thank dr . adele naddeo for useful suggestions . we regret to acknowledge the passing away of prof . maria marinaro with whom we have shared enlightening discussions during the completion of this work . 99 d. thouless , phys . b * 27 * , 6083 ( 1983 ) . b. altshuler and l. glazman , science * 283 * , 1864 ( 1999 ) . m. bttiker , h. thomas , and a. prtre , z. phys . b * 94 * , 133 ( 1994 ) . p. w. brouwer , phys . b * 58 * , 10135 ( 1998 ) . f. zhou , b. spivak , and b. altshuler , phys . 82 * , 608 ( 1999 ) ; yu . makhlin and a. d. mirlin , phys . lett . * 87 * , 276803 ( 2001 ) ; o. entin - wohlman , a. aharony , and y. levinson , phys . b * 65 * , 195411 ( 2002 ) ; m. moskalets and m. bttiker , phys . b * 66 * , 035306 ( 2002 ) ; _ ibid._*66 * , 205320 ( 2002 ) . m. switkes , c. m. marcus , k. campman , and a. c. gossard , science * 283 * , 1905 ( 1999 ) ; s. k. watson , r. m. potok , c. m. marcus , and v. umansky , phys . lett . * 91 * , 258301 ( 2003 ) . h. pothier , p. lafarge , c. urbina , d. esteve , and m. h. devoret , europhys . lett . * 17 * , 249 ( 1992 ) ; i. l. aleiner and a.v . andreev , phys . lett . * 81 * , 1286 ( 1998 ) ; r. citro , n. andrei , and q. niu , phys . b * 68 * , 165312 ( 2003 ) ; p.w . brouwer , a. lamacraft , and k. flensberg , phys . rev . b * 72 * , 075316 ( 2005 ) . t. aono , phys . 93 * , 116601 ( 2004 ) . j. splettstoesser , m. governale , j. knig , and r. fazio , phys . lett . * 95 * , 246803 ( 2005 ) . yigal meir and ned s. wingreen , phys . lett . * 68 * , 2512 ( 1992 ) ; see the discussion of pag . 2514 , first column , after eq.(9 ) . b. wang and j. wang , phys . b * 66 * , 201305(r ) ( 2002 ) . d. langreth , phys . rev . * 150 * , 516 ( 1966 ) . f. m. souza , j. c. egues , and a. p. jauho , phys . b * 75 * , 165303 ( 2007 ) ; see appendix . e. sela and y. oreg , phys . lett . * 96 * , 166802 ( 2006 ) . d. fioretto and a. silva , phys . 100 * , 236803 ( 2008 ) ; see also ` arxiv:0707.3338 ` . in formula ( [ eq : inst ] ) the sum on the discretized dot levels has been explicitly written . in deriving the expression for the instantaneous current we have been using the relation @xmath113 for the istantaneous green s function @xmath114 of the qd , while we defined @xmath115 . p. w. brouwer , phys . b * 63 * , 121303 ( 2001 ) . when an external d.c . or a.c . bias voltage is applied to the system , a gauge transformation can be perfomed to eliminate the bias and the tunneling term acquires a time - dependent phase proportional to the bias . in our case the time - dependent phase of the tunneling comes from the representation of a complex quantity . a. l. kuzemsky , int . j. mod b * 10 * , no . 15 , 1895 - 1912 ( 1996 ) . s. k. watson , r. m. potok , c. m. marcus , and v. umansky , phys . 91 * , 258301 ( 2003 ) . f. romeo , r. citro and m. marinaro , phys . b * 78 * , 245309 ( 2008 ) and references therein . antti - pekka jauho , ned s. wingreen and yigal meir , phys . b * 50 * , 5528 ( 1994 ) . huan - qiang zhou , urban lundin and sam young cho , j. phys . : matter * 17 * , 1059 ( 2005 ) . r. s. whitney , y. makhlin , a. shnirman and y. gefen , phys . lett . * 94 * , 070407 ( 2005 ) . h. linke _ science * 286 * , 2314 ( 1999 ) ; see also liliana arrachea , phys . b * 72 * , 121306(r ) ( 2005 ) and phys . rev . b * 72 * , 249904(e ) ( 2005 ) . | we derive a formula describing the adiabatically pumped charge through an interacting quantum dot within the scattering matrix and green s function approach .
we show that when the tunneling rates between the leads and the dot are varied adiabatically in time , both in modulus and phase , the current induced in the dot consists of two terms , the pumping current and a rectification - like term .
the last contribution arises from the time - derivative of the tunneling phase and can have even or odd parity with respect to the pumping phase @xmath0 .
the rectification - like term is also discussed in relation to some recent experiments in quantum - dots . |
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non - perturbative investigation of the qcd dynamics in the low energy region by means of the effective lagrangian approach has made considerable progress recently . in the very low energy region ( @xmath2 ) the chiral perturbation theory ( @xmath3pt ) @xcite based on the spontaneously broken chiral symmetry @xmath4 grew into a very successful model - independent tool for description of the green functions ( gf ) of quark currents and related low - energy phenomenology . @xmath3pt is organized as a rigorously defined simultaneous perturbative expansion in small momenta and light quark masses . recent calculations are performed at the next - to - next - to - leading order @xmath5 @xcite . in the intermediate energy region ( @xmath6 ) , however , the situation is less satisfactory . the set of relevant degrees of freedom includes now the low lying resonances and because there is no mass gap existing in the spectrum , the effective theory in this region can not be constructed as a straightforward extension of the @xmath3pt low energy expansion . on the other hand , the considerations based on the large @xmath7 expansions together with high - energy constraints derived from perturbative qcd and operator product expansion ( ope ) allow to introduce another type of effective lagrangian description , corresponding to the leading order in @xmath8 and reflecting the basic features of qcd in the @xmath9 limit . namely , the spectrum consisting of an infinite tower of free stable mesonic resonances exchanged in each channel requires infinite number of resonance fields in the @xmath10 symmetric lagrangian with interaction vertices suppressed by an appropriate power of @xmath11and ( since the @xmath12 expansion is correlated with semiclassical expansion ) only tree graphs have to be taken into account in the leading order . an approximation to this general picture consisting in limiting the number of resonance field to one in each channel and matching the resulting theory in the high energy region with ope is known as resonance chiral theory ( r@xmath3 t ) @xcite . integrating out the resonance fields from the lagrangian of r@xmath3 t in the low energy region and subsequent matching with @xmath3pt has become a very successful tool for estimates of the resonance contribution to the values of the @xmath13 @xcite and @xmath5 @xcite low energy constants ( lec ) entering the @xmath14pt lagrangian . though the usual chiral power counting fails within r@xmath3 t due to the presence of an additional heavy scale ( the mass of the resonances ) and the usual weinberg formula @xcite can not be generalized here ( because of the lack of a scale playing the role analogous to @xmath15 ) , it seems to be fully legitimate to go beyond the tree level r@xmath3 t and calculate the loops @xcite . being suppressed by one power of @xmath8 , the loops allow to encompass such nlo effects in the @xmath8 expansion as resonance widths and final state interaction and to determine the nlo resonance contribution to lec ( and their running with the renormalization scale ) . however , we have to be ready for both technical and conceptual complications connected with renormalization of the effective theory for which no natural organization of the expansion ( other than the @xmath8 counting ) exists . especially , because there is no natural analog of the weinberg power counting in r@xmath3 t , we can expect mixing of the naive chiral orders in the process of the renormalization ( _ e.g _ the loops renormalize the @xmath16 lec and also counterterms of unusually high chiral orders are needed ) . also , lack of appropriate protective symmetry can bring about appearance of new poles in the gf corresponding to new degrees of freedom which are frozen at the tree level . the latter might be felt as a pathological artefact of the not carefully enough formulated theory , particularly because this extra poles might be negative norm ghost or tachyons @xcite . on the other hand , however , we could also try to take advantage of this feature and adjust the poles in such a way that they correspond to the well established resonance states . in the following we will illustrate these problems in more detail . as an explicit example we use the one - loop renormalization of the propagator corresponding to the antisymmetric tensor field which originally describes the @xmath0 vector resonance ( @xmath17 meson ) at the tree level . we will show that the loop corrections to the propagator could lead to the dynamical generation of various types of @xmath0 and @xmath18 states and that the appropriate adjustment of coupling constants allows us to generate in this way the one which could be identified with the @xmath19 meson . the details of the calculations and further discussion will be provided in @xcite . the @xmath0 resonance part of the r@xmath3 t lagrangian within the antisymmetric tensor formalism reads @xcite @xmath20 where @xmath21 with normalized @xmath22 generators @xmath23 , @xmath24 are antisymmetric tensor fields with appropriate quantum numbers and @xmath25 is the usual chiral covariant derivative . @xmath26 is the interaction lagrangian which will be specified later . the full antisymmetric tensor field propagator @xmath27 has in general the following tensor structure @xmath28 where @xmath29 are longitudinal and transverse projectors ( @xmath30 ) @xmath31 note that , for @xmath32 we can express @xmath33 as the polarization sums @xmath34 where @xmath35 and @xmath36 are the usual spin - one polarization vectors with mass @xmath37 . the possible poles @xmath38 of @xmath39 and @xmath40 correspond therefore both to the spin - one states which couple to the fields @xmath41 and @xmath42 respectively and have therefore the same quantum numbers up to the parity . at lo in the @xmath8 expansion the lagrangian ( [ lagrangian ] ) gives @xmath43 and @xmath44 so that the @xmath0 resonance multiplet appears as a pole in @xmath45 . because there is no pole in @xmath40 at this order , no additional @xmath46 state is propagated . beyond lo we get generally @xmath47 where the self - energies @xmath48 are of the order @xmath8 at least . in the next section we present the results of the calculation of the renormalized self - energies @xmath49 in the chiral limit at nlo for a concrete form of the interaction lagrangian @xmath26 a more systematic treatment will be given in @xcite . in what follows we limit ourselves to the interaction lagrangian @xmath50 with at most two derivatives and up to two resonance fields . writing explicitly only those terms that contribute to the one - loop self - energies we have @xcite , @xcite @xmath51\rangle + 2d_1\langle d_\beta u^\sigma \{\widetilde{r}_{\alpha \sigma } , r^{\alpha \beta } \}\rangle\ ] ] @xmath52 where @xmath53 . in the large @xmath7 limit the couplings are @xmath54 and @xmath55 and apparently the intrinsic parity odd part is of higher order . however , the trilinear vertices contributing to the one - loop self - energies are @xmath56 in both cases due to the appropriate power of @xmath57 accompanying @xmath58 . therefore the operators with two resonance fields can not be eliminated using the large @xmath7 arguments . also nonzero @xmath59 are required in order to satisfy the ope constraints for vvp gf at the lo @xcite . in order to cancel the infinite part of the one - loop self - energies we have to introduce a set of counterterms . because the interaction terms are @xmath60 we would expect ( by the analogy with @xmath3pt power counting ) these counterterms to have four derivatives at most . however , the nontrivial structure of the free resonance propagator ( namely the presence of the @xmath61 @xmath62 part ) results in the failure of this naive expectation . in fact we need counterterms with up to six derivatives , namely @xmath63 the complete list of the counterterms and their infinite parts is postponed to @xcite . let us only note that @xmath64 contains a new type of kinetic term @xmath65 . provided such a term was included in the lo lagrangian from the very beginning , the propagator would have an additional pole in @xmath66 . however , interpretation of such a pole as a @xmath1 state would be problematic . according to the sign of @xmath67 this state would be either a tachyon or a negative norm ghost @xcite . evaluating the one - loop feynman graphs and adding the polynomial contributions from the counterterms ( [ ct ] ) we get the @xmath3pt minimally subtracted self - energies @xmath68 . the equation for the poles of @xmath45 has then an approximative perturbative solution @xmath69 corresponding to the original @xmath0 vector resonance with lo mass @xmath70 , which develops a mass correction and a finite width of the order @xmath71 due to the loops and which we identify with the @xmath72 meson . this allows to re - parameterize perturbatively @xmath73 in terms of @xmath74 and @xmath75 and requiring further @xmath76 we get for @xmath77 @xmath78 @xcite @xmath79 \\ & & -\frac{40}9\left ( \frac{m_\rho } { 4\pi f_\pi } \right ) ^2d_3 ^ 2(x^2 - 1)^2% \widehat{j}(x ) \\ { \sigma } _ t^r(x ) & = & \frac 1\pi \frac{\gamma _ \rho } { m_\rho } % \sum_{i=0}^3b_ix^i+\frac{20}9\left ( \frac{m_\rho } { 4\pi f_\pi } \right ) ^2d_3 ^ 2\end{aligned}\ ] ] @xmath80 here we put for further numerical estimates @xmath81 , @xmath82 ( for @xmath83 we take the value from @xcite ) and we have introduced the re - scaled free parameters @xmath84 and @xmath85 with natural size @xmath86 in the large @xmath7 expansion . these can be expressed in terms of renormalization scale independent combinations of the renormalized counterterms couplings and @xmath87logs . the loop functions @xmath88 and @xmath89 are given on the first sheet as @xmath90 , \label{loop_functions}\end{aligned}\ ] ] where we take the principal branch of the logarithm ( @xmath91 ) with a cut for @xmath92 . on the second sheet we have then @xmath93 and similarly for @xmath94 @xmath95 . the equation for the poles of @xmath40 has only non - perturbative solutions of the order @xmath96 . the @xmath1 states corresponding to them therefore decouple in the @xmath9 limit , however , for physical values of @xmath74 , @xmath97 and @xmath98 ( and for reasonable @xmath86 values of the parameters @xmath85 and @xmath99 ) , the position of poles can lie well within the intermediate energy region we are interested in . the nature of the corresponding states , which is also controlled by the free parameters @xmath85 and @xmath99 , is rich and covers bound states @xmath100 , ( which also might be negative norm ghosts ) , tachyonic poles @xmath101 , resonance poles in the lower complex half - plane on the second sheet or even complex conjugated pair of lee - wick poles on the first sheet . it is not straightforward to formulate general conditions for @xmath102 and @xmath99 under which there is _ no _ pole in @xmath103 at all , on the other hand we can rather easily arrange them to obtain the pole corresponding _ e.g. _ to the @xmath19 meson . this can be achieved in many ways _ e.g. _ for the choice @xmath104 , @xmath105 , @xmath106 and @xmath107 . the plot of the denominator of @xmath108 for this particular choice for the section @xmath109 on the second sheet and the shape of @xmath110 on the first sheet for @xmath111 real are depicted in fig [ figo ] . = 6.5 cm ( -158,18 ) we have illustrated the problems connected with loop calculations within r@xmath112 t using the one loop renormalization of the propagator of antisymmetric tensor field which describes @xmath0 resonance multiplet at the leading order of the large @xmath7 expansion as a concrete example . we have found that new @xmath1 states of various nature ( including pathological ones like negative norm ghosts and tachyons ) can be dynamically generated . for example , for a wide range of parameters @xmath85 , one such a state can be identified with @xmath19 meson . further discussion of possible applications of this feature is postponed to @xcite . | we discuss the renormalization of the @xmath0 vector meson propagator within resonance chiral theory at one loop . using the particular form of the interaction lagrangian
we show that additional poles of the renormalized propagator corresponding to @xmath1 degrees of freedom can be generated .
we give a concrete example of such an effect . |
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the recent detection of blue - shifted fe xxv / xxvi absorption lines in the x - ray spectra of several seyferts and quasars suggests the presence of highly ionized and mildly - relativistic outflows in the center of these agns ( e.g. , chartas et al . 2002 , 2003 ; pounds et al . 2003 ; dadina et al . 2005 ; markowitz et al . 2006 ; braito et al . 2007 ; cappi et al . 2009 ; reeves et al . 2009 ; giustini et al . they are possibly directly connected with accretion disk winds / outflows and the high associated outflow rate and mechanical power suggest they might be able to provide an important contribution to the expected agn cosmological feedback ( e.g. , king 2010 ; tombesi et al . 2012 ) . here we describe the systematic analysis and characterization of these so called ultra - fast outflows ( ufos ) on a large sample of both radio - quiet and radio - loud agns . these results are discussed in detail in tombesi et al . ( 2010a , b ; 2011a , b ) . 143 , best - fit model and background spectrum . _ right panel : _ example of @xmath7 distribution from 1000 monte carlo simulations.,title="fig:",width=207,height=188 ] 143 , best - fit model and background spectrum . _ right panel : _ example of @xmath7 distribution from 1000 monte carlo simulations.,title="fig:",width=207,height=151 ] the radio - quiet sample was defined in tombesi et al . ( 2010a ) selecting all the narrow - line seyfert 1 , seyfert 1 and seyfert 2 ( with neutral absorption column density @xmath8 @xmath6 ) in the rxte all - sky slew survey catalog ( revnivtsev et al . 2004 ) and then cross - correlated with the _ xmm - newton _ catalog . we obtained 42 sources for a total of 101 pointed _ xmm - newton _ observations . the sources are local ( @[email protected] ) and x - ray bright . we applied standard screening procedures and extracted the 410 kev epic pn spectra . we carried out a uniform spectral analysis using a phenomenological baseline model composed of an absorbed power - law continuum and gaussian fe k emission lines ( see left panel of fig . 1 ) . then , we performed a blind rearch for emission / absorption lines adding an additional narrow line to the baseline models with free positive / negative intensity and stepped the line energy between 410 kev , recording each time the associated @xmath7 deviations . the resultant energy - intensity contour plots were inspected to select possible features . initially , only the narrow lines ( @xmath11@xmath10100 ev ) with @xmath1299% f - test detection probability were selected and their velocity shifts were estimated using an identification as fe xxv / xxvi 1s2p/1s3p transitions . as discussed in 4 , we subsequently performed additional tests on the significance of the absorption lines at [email protected] kev using extensive monte carlo simulations and selected only those with probability @xmath1395% ( see right panel of fig . 1 ) . thus , in tombesi et al . ( 2010a ) we detected a total of 36 absorption lines , 14 in the range [email protected] kev and 22 at [email protected] kev . in a subsequent paper , tombesi et al . ( 2011a ) , we then performed a detailed curve of growth analysis of the fe xxv / xxvi absorption lines and a photo - ionization modeling using the _ xstar _ code ( see left panel of fig . 1 ) . we considered an ionizing continuum equivalent to the average sed of the seyfert 1s in the sample , i.e. a @xmath15@xmath162 power - law with cut - off at e@xmath16100 kev . we then calculated _ xstar _ grids assuming standard solar abundances and turbulent velocities of 1000 , 3000 and 5000 km / s . given the limited energy resolution of the epic pn , only four absorption lines were resolved with width @xmath11@xmath165,000 km / s and for the other we could place only upper limits . a blind search for the best - fit _ solution(s ) was performed stepping the absorber redshift between 0.1 and @xmath170.4 , leaving the column density and ionization free , and checking for minima in the resultant @xmath18 distribution . this allows to self - consistently take into account the possible presence of lines / edges from ions of different elements . in all cases a good fit ( p@xmath19@xmath099% ) was reached with a single _ xstar _ component and all the absorption lines were indeed consistent with blue - shifted fe xxv / xxvi transitions . the possible degeneracy of the identification in a few cases was included in the relative larger parameter errors . these studies allowed us to derive the distribituion of the main parameters of the ufos in the radio - quiet sample ( tombesi et al . the detection fraction is @xmath040% , which may suggest a large covering factor of @xmath00.4 . the lines have been found to be variable in both ew and velocity shift on time - scales even as short as @xmath1days , indicating compact absorbers . the outflow velocities are mildly - relativistic , in the range @xmath10.030.3c , with a mean value of @xmath10.14c . their ionization is high , in the interval log@[email protected] erg s@xmath3 cm , and the associated column densities are also large , in the range @xmath20@xmath1@xmath4@xmath5 @xmath6 . the estimated location and energetics of the ufos in the radio - quiet sample is reported in a subsequent paper , tombesi et al . we extented the search for ufos also in radio - loud agns and in particular to blrgs , which are the radio - loud counterparts of seyfert 1s . they show strong relativistic radio jets , but the typical inclination of @xmath21@xmath120@xmath2240@xmath22 allows the direct observation of the inner disk in x - rays . therefore , in tombesi et al . ( 2010b ) we performed a systematic 410 kev spectral analysis of the long _ suzaku _ observations of 3c 111 , 3c 390.3 , 3c 120 , 3c 382 and 3c 445 . we applied the same method as for the radio - quiet sample ( see 2 ) and detected blue - shifted absorption lines at e@xmath127 kev in 3/5 sources . their significance was assessed using both f - test and extensive monte carlo simulations ( see 4 ) . through a photo - ionization modeling with _ xstar _ we find that they are consistent with series of fe xxv / xxvi k - stell transitions blue - shifted with mildly - relativistic velocities in the range @xmath10.040.15c . they are highly ionized , log@xmath2@xmath1646 erg s@xmath3 cm , and the associated column densities are @xmath20@xmath0@xmath4 @xmath6 . their estimated location within @xmath10.010.1 pc from the central super - massive black hole suggests a likely origin related with accretion disk outflows . the mass outflow rate of these ufos is comparable to the accretion rate of @xmath11 @xmath23 yr@xmath3 . their estimated kinetic power is high , in the range @xmath1@xmath24@xmath25 erg s@xmath3 , which is comparable to their typical jet power and corresponds to a significant fraction of the bolometric luminosity . therefore , these ufos can potentially play a significant role in the expected agn feedback and can also possibly be directly linked to the jet activity . the galaxy 3c 111 was then the subject of a follow - up study of the variability of its ufo with _ suzaku _ in september 2010 ( tombesi et al . we obtained three pointings of @xmath160 ks spaced by @xmath17 days and observed a @xmath130% flux increase between the first and the second . a galactic absorbed power - law continuum plus a narrow fe k@xmath26 emission line at [email protected] kev provide a good representation of the 410 kev xis spectra . however , an additional emission line at [email protected] kev is observed in the first observation and an absorption line at [email protected] kev in the second . the detection significance of these features is high , @xmath1299% from both f - test and monte carlo simulations . they are also significantly variable among the observations . the emission feature can be modeled with a highly ionized line coming from reflection off the accretion disk at @xmath120100@xmath27 from the black hole . instead , a photo - ionization modeling of the absorption line suggests its association with a highly ionized , log@[email protected] erg s@xmath3 cm , ufo with outflow velocity @xmath10.1c and column density @xmath20@xmath16@xmath28 @xmath6 . the location of the material is constrained at @xmath29@xmath30 pc from variability . this observation may provide the first direct evidence for an accretion disk - wind connection in an agn and is consistent with a picture in which an increased illumination of the inner disk causes an outflow to be lifted at @xmath1100 @xmath27 . this is then possibly accelerated through radiation pressure to the observed terminal velocity of @xmath10.1c , but additional magnetic thrust can not be excluded ( e.g. , ramrez & tombesi 2012 ) . the significance of the absorption lines detected at [email protected] kev was further investigated through extensive monte carlo simulations ( tombesi et al . 2010a , b ) . for each case we simulated 1000 spectra assuming the baseline model without the absorption lines and calculated the @xmath7 distribution of random generated features in the 7.110 kev band . then , we selected only the observed absorption lines with measured @xmath7 corresponding to a monte carlo detection probability @xmath1395% . we can also exclude any contamination due to the epic pn background and calibration uncertainties and we checked that the possible presence of spectral complexities , such as reflection and warm absorption , are typically weak enough to assure a only marginal model dependency of the results . this systematic analysis on a large sample of sources allows to estimate the global detection probability of the absorption lines of @xmath125@xmath11 and to overcame the possible publication bias claimed by vaughan & uttley ( 2008 ) . moreover , we checked that the results are consistent using also the mos detectors and the global random probability in that case is also low , @xmath29@xmath31 . in the left panel of fig . 2 we show the distribution of the significance @xmath11 ( simply estimated from the ratio @xmath29ew@xmath12/error@xmath29ew@xmath12 ) of the blue - shifted absorption lines at [email protected] kev detected by tombesi et al . ( 2010a ) with respect to the 410 kev counts of the relative observations ( filled circles ) . it has been claimed that the tendency of the distribution to lay close to the 3@xmath11 level and not showing a strong enhancement in the significance following an increase in counts might suggest that all the detections are fake . however , plotting also the same distribution for the absorption lines at [email protected] kev ( open triangles ) and the ionized fe k emission lines at [email protected] kev ( crosses ) , we note that they also follow this general trend and it is not peculiar of the [email protected] kev lines . in particular , we state that this does not necessarily mean that the lines are fake , given that their significance was already estimated using the f - test and extensive monte carlo simulation , but that they are intrinsically weak and variable . we support this statement with a simple test . we simulated 15 absorption lines assuming the same parameters as the real data , i.e. 410 kev counts in the range 3300@xmath32 and variable ew@xmath1420 , 40 , 60 ev and e@xmath146 . 8 , 9 kev . as expected , the same trend is followed also by the simulations ( open circles ) . instead , in the right panel of fig . 2 we show the distribution of the significance of the lines detected simultaneously in the epic pn and mos . in this case the open circles refer to the narrow 6.4 kev fe k@xmath26 emission lines . we can see that the points are systematically indicating that the detection significance in the pn is higher than in the mos . this trend is simply expected from the higher effective area and lower background of the epic pn with respect to the mos . important improvements in the detection and characterization of these lines are expected from the higher effective area and supreme energy resolution in the fe k band offered by the micro - calorimeters on board _ astro - h _ and the proposed esa _ athena _ missions . braito , v. , et al . 2007 , apj , 670 , 978 cappi , m. , et al . 2009 , a&a , 504 , 401 chartas , g. , brandt , w. n. , gallagher , s. c. , & garmire , g. p. 2002 , apj , 579 , 169 chartas , g. , et al . 2003 , apj , 595 , 85 dadina , m. , cappi , m. , malaguti , g. , ponti , g. , & de rosa , a. 2005 , a&a , 442 , 461 giustini , m. , cappi , m. , chartas , g. , et al . 2011 , a&a , 536 , a49 king , a. r. 2010 , mnras , 408 , l95 markowitz , a. , et al . 2006 , apj , 646 , 783 pounds , k. a. , et al . 2003 , mnras , 345 , 705 ramrez , j. m. , tombesi , f. 2012 , mnras , 419 , l64 reeves , j. n. , et al . 2009 , apj , 701 , 493 revnivtsev , m. , sazonov , s. , jahoda , k. , & gilfanov , m. 2004 , a&a , 418 , 927 tombesi , f. , cappi , m. , reeves , j. n. , et al . 2010a , a&a , 521 , a57 tombesi , f. , sambruna , r. m. , reeves , j. n. , et al . 2010b , apj , 719 , 700 tombesi , f. , cappi , m. , reeves , j. n. , et al . 2011a , apj , 742 , 44 tombesi , f. , sambruna , r. m. , reeves , j. n. , et al . 2011b , mnras , 418 , l89 tombesi , f. , cappi , m. , reeves , j. n. , braito , v. , 2012 , mnras submitted vaughan , s. , & uttley , p. 2008 , mnras , 390 , 421 | x - ray evidence for ultra - fast outflows ( ufos ) has been recently reported in a number of local agns through the detection of blue - shifted fe xxv / xxvi absorption lines .
we present the results of a comprehensive spectral analysis of a large sample of 42 local seyferts and 5 broad - line radio galaxies ( blrgs ) observed with _
xmm - newton _ and _ suzaku_. we detect ufos in @xmath040% of the sources .
their outflow velocities are in the range @xmath10.030.3c , with a mean value of @xmath10.14c .
the ionization is high , in the range log@xmath2@xmath136 erg s@xmath3 cm , and also the associated column densities are large , in the interval @xmath1@xmath4@xmath5 @xmath6 .
overall , these results point to the presence of highly ionized and massive outflowing material in the innermost regions of agns .
their variability and location on sub - pc scales favor a direct association with accretion disk winds / outflows .
this also suggests that ufos may potentially play a significant role in the agn cosmological feedback besides jets and their study can provide important clues on the connection between accretion disks , winds and jets . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in many scientific studies the knowledge of derivatives of a given quantity is of particular importance . for example in theoretical physics , especially in thermodynamics , many quantities of interest require the calculation of derivatives of an underlying thermodynamic potential with respect to some external parameters such as temperature , volume , or chemical potentials . in many cases the thermodynamic potentials can only be evaluated numerically and one is forced to employ numerical differentiation techniques which are error - prone as any numerical methods . furthermore , the thermodynamic potential has to be evaluated at the physical point defined by minimizing the thermodynamic potential with respect to some condensates yielding the equations of motion ( eom ) . generally , these equations can be solved only numerically and thus introduce additional implicit dependencies which makes the derivative calculations even more complicated . even in cases where the thermodynamic potential and the implicit dependencies on the external parameters are known analytically , the evaluation of higher - order derivatives becomes very complex and tedious and in the end impedes their explicit calculation . in this work we present a novel numerical technique , based on algorithmic differentiation ( ad ) to evaluate derivatives of arbitrary order of a given quantity at machine precision . compared to other differentiation techniques such as the standard divided differentiation ( dd ) method or symbolic differentiation , the ad produces truncation - error - free derivatives of a function which is coded in a computer program . additionally , ad is fast and reduces the work required for analytical calculations and coding , especially for higher - order derivatives . furthermore , the ad technique is applicable even if the implicit dependencies on the external parameters are known only numerically . in ref . @xcite a comprehensive introduction to ad can be found . first remarks about the computation of derivative of implicitly defined functions were already contained in @xcite . however , a detailed description and analysis is not available yet . additional information about tools and literature on ad are available on the web - page of the ad - community @xcite . this work is organized in the following way : for illustrations we will introduce an effective model , the so - called linear sigma model with quark degrees of freedom in sec . [ sec : model ] . this model is widely used for the description of the low - energy sector of strongly interacting matter . as a starting point the basic thermodynamic grand potential and the eom of this model are calculated in a simple mean - field approximation in order to elucidate the technical problems common in such types of calculations . before we demonstrate the power of the ad method by calculating certain taylor expansion coefficients up to very high orders for the first time in sec . [ sec : taylor ] , the ad method itself and some mathematical details are introduced in sec . [ sec : ad ] . details for the calculation of higher - order derivatives of implicit functions are given in the following sec . [ sec : impfun ] . in sec . [ sec : advsdd ] the results of the ad method are confronted with the ones of the standard divided differences ( dd ) method in order to estimate the truncation and round off errors . finally , we end with a summary and conclusion in sec . [ sec : summary ] . in order to illustrate the key points of the ad method we employ a quantum field theoretical model @xcite . this model can be used to investigate the phase structure of strongly interacting matter described by the underlying theory of quantum chromodynamics ( qcd ) . details concerning this effective linear sigma model ( l@xmath0 m ) in the qcd context can be found in reviews , see e.g. @xcite . the quantity of interest for the exploration of the phase structure is the grand potential of the l@xmath0 m . this thermodynamic potential depends on the temperature @xmath1 and quark chemical potential @xmath2 because the particle number can also vary . it is calculated in mean - field approximation whose derivation for three quark flavors is shown explicitly in @xcite . for the l@xmath0mthe total grand potential @xmath3 consists of two contributions @xmath4 where the first part , @xmath5 , stands for the purely mesonic potential contribution and is a function of two condensates , @xmath6 and @xmath7 . the second part , @xmath8 , is the quark contribution and depends on the two condensates as well as on the external parameters temperature @xmath1 and , for simplicity , only one quark chemical potential @xmath2 . since the quark contribution arises from a momentum - loop integration over the quark fields , it is given by an integral which can not be evaluated in closed form analytically . readers who are unfamiliar with the physical details , may simply regard eq . ( [ eq : grand_pot ] ) as an only numerically known function and continue with the reading above eq . ( [ eq : eom ] ) , which introduces an implicit dependency on the parameters @xmath1 and @xmath2 whose treatment with the ad technique is the major focus of this work . explicitly , in mean - field approximation the quark contribution reads @xmath9 where a summation over three quark flavors @xmath10 is included . the usual fermionic occupation numbers for the quarks are denoted by @xmath11 and for antiquarks by @xmath12 respectively . in this example only two different single - particle energies , @xmath13 , emerge @xmath14 the first index @xmath15 denotes the combination of two mass - degenerate light - quark flavors ( @xmath16 ) and the other index @xmath17 labels the heavier strange quark flavor . the expressions in parentheses in @xmath18 are the corresponding quark masses . in this way , the dependency of the grand potential on the condensates , @xmath19 , @xmath20 enter through the quark masses , which has not been indicated explicitly in eq . ( [ eq : quark_pot ] ) . the mesonic potential does not depend on the quark chemical potential nor on the temperature explicitly . it is just a function of the two condensates and reads @xmath21 wherein all remaining quantities , e.g. @xmath22 are constant parameters . since the physical condensates , @xmath23 and @xmath24 , are determined by the extrema ( minima ) of the total grand potential with respect to the corresponding fields , they fulfill the equations of motion @xmath25 this in turn introduces an implicit @xmath1- and @xmath2-dependence of both condensates , @xmath26 these quantities represent the physical order parameters which , together with the grand potential , are the basis of the exploration of the phase structure of the model . we denote the grand potential evaluated at @xmath27 @xmath20 , as @xmath28 in order to find the temperature and chemical potential behavior of the order parameters the integral in eq . ( [ eq : quark_pot ] ) and simultaneously the eom have to be solved numerically . this already is an example suitable for an ad application , because a derivative of a only numerically solvable , implicit function is needed as input . later we will be interested in higher - order derivatives of the grand potential with respect to , e.g. , the chemical potential . for example , the quark number density at the physical point is defined by @xmath29 in cases without an implicit @xmath1- or @xmath2-dependence in the thermodynamic potential some progress can be made by calculating the corresponding derivatives explicitly and solving the corresponding equations numerically . this might be feasible for lower - order derivatives , in particular , if parts of the derivative calculations can be performed by some computer algebra packages like mathematica or maple . but for higher - order derivatives this procedure is error - prone and time - consuming and not applicable anymore . in the following sections the algorithmic differentiation technique for implicitly defined functions is introduced on a general mathematical level . suppose the function @xmath30 , @xmath31 describing an arbitrary algebraic mapping from @xmath32 to @xmath33 is defined by an evaluation procedure in a high - level computer language like fortran or c. the technique of algorithmic differentiation provides derivative information of arbitrary order for the code segment in the computer that evaluates @xmath34 within working accuracy . for this purpose , the basic differentiation rules such as , e.g. , the product rule are applied to each statement of the given code segment . this local derivative information is then combined by the chain rule to calculate the overall derivatives . hence the code is decomposed into a long sequence of simple evaluations , e.g. , additions , multiplications , and calls to elementary functions such as @xmath35 or @xmath36 , the derivatives of which can be easily calculated . exploiting the chain rule yields the derivatives of the whole sequence of statements with respect to the input variables . as an example , consider the function @xmath37 with @xmath38 that can be evaluated by the pseudo - code given on the left column of tab . [ fig1 ] . on the right - hand side , the resulting statements for the derivative calculation @xmath39 are given . @xmath40 for the vector @xmath41 one obtains the first column of the jacobian @xmath42 the vector @xmath43 , @xmath44 . correspondingly , the other unit vectors in @xmath45 yield the other two remaining columns of the jacobian @xmath46 . table [ fig1 ] illustrates the so - called forward mode of ad , where the derivatives are propagated together with the function evaluation . alternatively , one may propagate the derivative information from the dependents @xmath47 to the independents @xmath48 yielding the so - called reverse mode of ad . over the past decades , extensive research activities led to a thorough understanding and analysis of these two basic modes of ad , where the complexity results with respect to the required runtime are based on the operation count @xmath49 , i.e. , the number of floating point operations required to evaluate @xmath50 , and the degree @xmath51 of the computed derivatives . using the forward mode , one computes the required derivatives together with the function evaluation in one sweep as illustrated above . the forward mode yields one _ column _ of the jacobian @xmath52 at no more than three times @xmath49 @xcite . one _ row _ of @xmath52 , e.g. , the gradient of a scalar - valued component function of @xmath53 , is obtained using the reverse mode in its basic form also at no more than four times @xmath49 @xcite . it is important to note that this bound for the reverse mode is completely independent of the number @xmath54 of input variables . this observation is called _ cheap gradient result_. for the application discussed in the present work , the forward mode has been chosen for the efficient computation of higher - order derivatives which is illustrated in the following paragraphs . to this end , we consider taylor polynomials of the form @xmath55 . the expansion is truncated at the highest derivative degree @xmath51 which is chosen by the user . the vector polynomial @xmath56 describes a path in @xmath57 which is parameterized by @xmath58 . thus , the first two vectors @xmath59 and @xmath60 represent the tangent and the curvature at the base point @xmath61 . assuming that the function @xmath62 is sufficiently smooth , i.e. , @xmath51 times continuously differentiable , one obtains a corresponding value path @xmath63 the coefficient functions @xmath64 are uniquely and smoothly determined by the coefficient vectors @xmath65 with @xmath66 . to compute this higher - order information , first we will examine for a given taylor polynomial @xmath67 the derivative computation based on `` symbolic '' differentiation . let us generalize the previous relation @xmath68 given in eq . ( [ eq : ytay ] ) , and consider now a general smooth function @xmath69 as for example the evaluation of a @xmath70-function . this function @xmath71 represents one of the intermediate values computed during the function evaluation as illustrated in table [ fig1 ] . one obtains for the taylor coefficients @xmath72 the derivative expressions @xmath73 hence , the overall complexity grows rapidly with the degree @xmath51 of the taylor polynomial . to avoid these prohibitively expensive calculations the standard higher - order forward sweep of algorithmic differentiation is based on taylor arithmetic @xcite yielding an effort that grows like @xmath74 times the cost of evaluating @xmath75 . this is quite obvious for arithmetic operations such as multiplications or additions , where one obtains the recursion shown in table [ tab : arith ] , .taylor coefficient propagation for arithmetic operations [ cols="^,<,^,^",options="header " , ] similar formulas can be found for all intrinsic functions . this fact permits the computation of higher - order derivatives for the vector function @xmath34 as composition of elementary components . the ad - tool adol - c @xcite uses the taylor arithmetic as described above to provide an efficient calculation of higher - order derivatives . for the application considered here , higher - order derivatives of a variable @xmath76 are required , where @xmath47 is _ implicitly _ defined as a function of some variable @xmath77 by an algebraic system of equations @xmath78 naturally , the @xmath54 arguments of @xmath79 need not be partitioned in this regular fashion . to provide flexibility for a convenient selection of the @xmath80 _ truly _ independent variables @xmath81 , let @xmath82 be a projection matrix with only @xmath83 or @xmath84 entries that picks out these independent variables . hence , @xmath85 is a column permutation of the matrix @xmath86 \in { { \mathbb r}}^{p\times n}$ ] . then the nonlinear system @xmath87 has a regular jacobian , wherever the implicit function theorem yields @xmath47 as a function of @xmath48 . therefore , we may also write with @xmath88 @xmath89 for the seed matrix @xmath90^\top \in { { \mathbb r}}^{n \times p}$ ] . now , we have rewritten the original implicit functional relation between @xmath48 and @xmath91 as an inverse relation @xmath92 . assuming an @xmath93 that is locally invertible we can evaluate the required derivatives of the implicitly defined @xmath94 with respect to @xmath95 using the computation of higher - order derivatives described above in the following way . starting with a taylor expansion eq . ( [ eq : x ] ) of @xmath48 and a corresponding solution @xmath96 of eq . ( [ eq : f ] ) , one obtains for a sufficiently smooth @xmath97 the representation @xmath98 substituting the taylor expansion of @xmath48 into the previous equation yields @xmath99from the comparison of coefficients , it follows that @xmath100 as a next step , the structure of the taylor coefficients @xmath101 is analyzed . for the first three coefficients , one has @xmath102 due to the definition of @xmath103 , where @xmath104 denotes the derivative of @xmath105 with respect to its argument . for the higher - order coefficients , it is now shown that they have the structure @xmath106 where @xmath107 involves only derivatives of order @xmath108 with respect to @xmath58 and hence @xmath109 for @xmath110 , one obtains @xmath111 now , let the assumption hold for @xmath108 . then , one obtains for @xmath112 the equation @xmath113 \bigg|_{t=0 } \\ & = \frac{1}{j } \left({\bf h_z } ( { \bf z}({\bf x}(t ) ) ) \frac{\partial^j}{\partial t^j } { \bf z}({\bf x}(t))\right ) \bigg|_{t=0 } \\ & \phantom{=. } + \frac{1}{j } \left[\left ( \frac{\partial}{\partial t } { \bf h_z } ( { \bf z}({\bf x}(t ) ) ) \right ) \left ( \frac{\partial^{j-1}}{\partial t^{j-1 } } { \bf z}({\bf x}(t ) ) \right)\right ] \bigg|_{t=0 } \\ & \phantom{=.}+ \frac{1}{j}\left ( \frac{\partial } { \partial t } { \bf \tilde{h}}_{j-1 } ( { \bf z}({\bf x}(t)))\right ) \bigg|_{t=0}.\end{aligned}\ ] ] due to the assumptions , the function @xmath114 involves only derivatives of order @xmath108 with respect to @xmath58 since @xmath115 does only contain derivatives of order @xmath116 with respect to @xmath58 . therefore , ( [ eq:14 ] ) is proven and it follows that @xmath117 due to the definition of @xmath103 . combining ( [ eq:15 ] ) with ( [ eq:2 ] ) , one obtains the equations @xmath118 and therefore @xmath119 where the jacobian @xmath120 and its factorization can be reused as long as the argument @xmath121 is the same . for this purpose , the jacobian @xmath120 can be evaluated exactly by using the forward mode of ad . therefore , it remains to provide the missing contributions @xmath122 to compute the desired taylor coefficients @xmath103 . one starts with the taylor expansion @xmath123 for @xmath124 , one performs the following steps 1 . a forward mode evaluation of degree @xmath112 . since @xmath125 this yields only the contribution @xmath122 . 2 . one system solve @xmath126 to compute @xmath103 . this approach provides the complete set of taylor coefficients of the taylor polynomial @xmath127 that is defined by a given taylor polynomial for @xmath48 . these taylor coefficients of @xmath127 are computed for a considerably small number of taylor polynomials @xmath56 to construct the desired full derivative tensor for the implicitly defined function @xmath91 according to the algorithm proposed in @xcite . consider the following two nonlinear expressions @xmath128 describing the relation between the cartesian coordinates @xmath129 and the polar coordinates @xmath130 in the plane . assume , one is interested in the derivatives of the second cartesian and the second polar coordinate with respect to the first cartesian and the first polar coordinate . then one has @xmath131 , @xmath132 , @xmath133 , @xmath134 , and @xmath135 . the corresponding projection and seed matrix are @xmath136 provided the argument @xmath91 is consistent in that its cartesian and polar components describe the same point in the plane , one has @xmath137 . in this simple case , one can derive for the implicitly defined functions @xmath138 and @xmath139 the desired derivatives explicitly by symbolic manipulation : @xmath140 the derivatives up to order 3 of @xmath141 will be used to verify the results from the differentiation of the implicitly defined functions . these derivatives have the following representation : @xmath142 as shown in @xcite , these derivatives can be computed efficiently and exactly from a considerable small number of univariate taylor expansions like the one in eq . . furthermore , this article also proposes a specific choice of the employed taylor polynomials . for the example considered here , i.e. , @xmath132 and @xmath143 , one obtains the taylor expansions @xmath144 then , the procedure described in the previous section yields the following taylor expansions of @xmath145 with the base point @xmath146 @xmath147 { \bf z}^2 = { \bf z}_0 + \left ( \begin{array}{r } \;2 \\ -1 \\ \;1 \\ \!- \frac{2}{5 } \end{array } \right ) t + \left ( \begin{array}{c } \;\;0 \\ \!- \frac{2}{3 } \\ \;\;0 \\ \!- \frac{2}{75 } \end{array } \right ) t^2 + \left ( \begin{array}{c } \;\;0 \\ \!- \frac{2}{9 } \\ \;\;0 \\ \!- \frac{46}{1125 } \end{array } \right ) t^3 \end{array}\end{aligned}\ ] ] @xmath148 { \bf z}^4 = { \bf z}_0 + \left ( \begin{array}{r } 0\\ 5\\ 3\\ \frac{4}{5 } \end{array } \right ) t + \left ( \begin{array}{c } \;\;0 \\ \!- \frac{8}{3 } \\ \;\;0 \\ \!- \frac{68}{73 } \end{array } \right ) t^2 + \left ( \begin{array}{c } 0 \\ \frac{40}{9 } \\ 0 \\ \frac{1508}{1125 } \end{array } \right ) t^3 . \end{array}\end{aligned}\ ] ] from these numerical values , one can derive the desired derivatives in eq . ( [ eq:17 ] ) as given below @xmath149 { \displaystyle}\frac{\partial^2 y_1}{\partial z^2_1 } & = - { \displaystyle}\frac{25}{27 } = \frac{2}{9 } { \bf z}^1_{22 } = { \displaystyle}\frac{2}{9 } * \left(- \frac{25}{6}\right)\\{\displaystyle}\frac{\partial^2 y_1}{\partial z_1 \partial z_3 } & = { \displaystyle}\frac{20}{27 } = - \frac{5}{36 } { \bf z}^1_{22 } + \frac{1}{4 } { \bf z}^2_{22 } + \frac{1}{4 } { \bf z}^3_{22 } - \frac{5}{36 } { \bf z}^4_{22 } \\[.3cm]{\displaystyle}\frac{\partial^2 y_1}{\partial z^2_3 } & = - \frac{16}{27 } = \frac{2}{9 } { \bf z}^4_{22 } = \frac{2}{9 } * \left(- \frac{8}{3}\right ) \\[.3 cm ] { \displaystyle}\frac{\partial^3 y_1}{\partial z^3_1 } & = - { \displaystyle}\frac{100}{81 } = \frac{2}{9 } { \bf z}^1_{32 } = \frac{2}{9 } * - \frac{50}{9 } \\[.3 cm ] { \displaystyle}\frac{\partial^3 y_1}{\partial z^2_1 \partial z_3 } & = { \displaystyle}\frac{95}{81 } = - \frac{5}{27 } { \bf z}^1_{32 } + \frac{2}{3 } { \bf z}^2_{32 } - \frac{1}{3 } { \bf z}^3_{32 } + \frac{2}{27 } { \bf z}^4_{32 } \\[.3 cm ] { \displaystyle}\frac{\partial^3 y_1}{\partial z_1 \partial z^2_3 } & = - { \displaystyle}\frac{88}{81 } = \frac{2}{27 } { \bf z}^1_{32 } - \frac{1}{3 } { \bf z}^2_{32 } + \frac{2}{3 } { \bf z}^3_{32 } - \frac{5}{27 } { \bf z}^4_{32}\\ { \displaystyle}\frac{\partial^3 y_1}{\partial z^3_3 } & = \frac{80}{81 } = \frac{2}{9 } { \bf z}^4_{32 } = \frac{2}{9 } * \frac{40}{9}\end{aligned}\ ] ] where @xmath150 denotes the @xmath151th component of the @xmath152th taylor coefficient of the taylor expansion @xmath112 . as can be seen , the required 24 entries of the first three derivative tensors can be obtained from four univariate taylor expansions . this computation of tensor entries from the taylor expansions , i.e. the exact coefficients for the taylor coefficients , is derived and analyzed in detail in @xcite . in the procedure described above , the higher - order forward mode of ad is applied for each value of @xmath112 for @xmath124 . employing in addition to the higher - order forward mode the higher - order reverse mode , the number of forward and reverse sweeps can be reduced to @xmath153 . in this case , the values of the required @xmath154 is reconstructed from the information available due to the reverse mode differentiation . the ad - tool adol - cprovides a corresponding efficient implementation of this algorithm and will be used in the numerical tests below , where the @xmath155- behavior of the higher - order derivative calculation can be observed in the measured runtimes . in the following the previous general mathematical description of the ad technique is applied to the model example introduced in the beginning . furthermore , the ad results are then compared to those obtained with the standard divided differentiation ( dd ) method . the first step for the calculation of the @xmath152-th order derivatives of the grand potential with respect to @xmath2 , @xmath156 by means of the ad technique requires a suitable formulation of @xmath157 . this can be accomplished by a taylor expansion of the condensates @xmath158 . the required coefficients , i.e. , the derivatives @xmath159 can be calculated by applying the technique described in the previous section for implicit functions . the next step consists in the calculation of the derivatives of @xmath157 w.r.t . @xmath2 by using the taylor expansions of the condensates . in the following the procedure will be exemplified in detail . in this example only one , i.e. @xmath160 , truly independent variable @xmath161 is considered and the temperature @xmath1 plays the role of a constant parameter . the generalization to mixed derivatives with respect to @xmath1 and @xmath2 can also be realized but is omitted for simplicity . firstly , the taylor coefficients for the condensates @xmath162 are needed . this is done via the inverse taylor expansion capabilities of adol - c . for that purpose the following function @xmath163 is introduced . the @xmath164 dimensional argument @xmath165 splits into @xmath132 implicitly defined functions @xmath166 and @xmath167 truly independent variable @xmath168 , cf . ( [ eq : f ] ) . furthermore , the projection matrix reads @xmath169 . in order to obtain the @xmath2-derivatives of the functions @xmath170 for fixed values of @xmath171 the following steps are required : 1 . the numerical solution of the eom , see eq . ( [ eq : eom ] ) , yields the values of the condensates @xmath172 at the potential minimum . for these values the condition @xmath173 with @xmath174 is obviously valid . 2 . prepare the derivative calculation for @xmath175 usingadol - c . 3 . evaluate the taylor coefficients of @xmath176 at @xmath177 up to the highest derivative degree @xmath51 desired by the user . from now on , the taylor expansions of @xmath178 around @xmath179 are labeled as @xmath180 . these taylor expansions are inserted in the grand potential which leads to the definition @xmath181 the function @xmath182 is exact in the explicit @xmath2-dependence but only exact up to order @xmath51 in the implicit dependence . thus , the @xmath152-order derivatives of @xmath182 correspond to the derivatives of the original @xmath183 if the derivatives are evaluated at the expansion point @xmath179 and @xmath184 , i.e. , we have @xmath185 this equation is valid only at the point @xmath179 . in order to obtain the desired @xmath183 derivatives at another @xmath186 point , the expansion coefficients of @xmath187 have to be recalculated for each @xmath186 point . however , this reduces the problem of calculating the derivatives of @xmath183 with only implicitly known functions @xmath162 to the calculation of the @xmath182 derivatives with explicitly known @xmath188 . finally , the calculation of the derivatives of @xmath182 then requires two steps : 1 . prepare the derivative calculation for @xmath189 with adol - cat @xmath179 which were chosen for the evaluation of the coefficients of @xmath188 . 2 . evaluate the taylor coefficients of @xmath189 . another method to approximate derivatives of a function is based on divided differences which is explained in the following . based on the definition of the derivative of a function @xmath10 at a point @xmath190 @xmath191 the simplest linear approximation for @xmath192 is obtained by calculating the right - hand side of eq . ( [ eq : def_deriv ] ) for a small but finite value of @xmath193 @xmath194 the problem with this approximation is that it involves two types of errors ( cf . e.g. @xcite ) . if @xmath193 is too large , the so - called truncation error induced by the used approximation or algorithm to calculate the derivative becomes significant . on the other side , when @xmath193 becomes too small another error , the rounding error yields cancellations in the enumerator of ( [ eq : dd ] ) and spoils the quality of the approximation . since the two error sources compete with each other , one has to find an optimal value of @xmath193 for which the numerical error of the derivative evaluation is smallest . in general , this optimal @xmath193 varies with @xmath190 , the point at which the derivative is calculated . the truncation error is relatively easy to control . by comparing eq . ( [ eq : dd ] ) with a taylor expansion for @xmath195 around @xmath190 one sees that the truncation error of the linear approximation is of @xmath196 , i.e. , the error is a linear function of @xmath193 . thus , decreasing @xmath193 will also decrease the truncation error . in addition , by increasing the degree of the expansion the truncation error can also be further improved . one such improved extrapolation , the richardson expansion , is based on the @xmath54-th order taylor expansion for @xmath195 and @xmath197 around @xmath190 for which a truncation error of the order @xmath198 can be derived . by repeating the algorithm for the determination of the truncation error , a better approximation for the first derivative @xmath192 can be obtained . similar improvements of the truncation error for higher - order derivatives are also known . as an example the corresponding approximations for the second derivative @xmath199 with three grid points @xmath200 @xmath201 + \mathcal{o}(h^{2})\ ] ] and with five grid points @xmath202 + \mathcal{o}(h^{4 } ) \end{gathered}\ ] ] are itemized . the grid points are given by @xmath203 for completeness the fourth - order derivative is quoted @xmath204 + \mathcal{o}(h^{4})\ .\end{gathered}\ ] ] where at least five grid points are needed for its calculation . the disadvantage of such type of improvements is that the function has to be evaluated at several different grid points @xmath200 which are located in the vicinity of @xmath190 . the other error source , the rounding error , depends on the used format of the floating point number representation in the computer . a single precision ieee floating point number is stored in a 32-bit word , where 8 bits are used for the biased exponent and the fractional part of the normalized mantissa is a 23-bits binary number . one bit in the ieee format is always reserved for the sign of the number . a double precision number occupies 64 bits , with the biased exponent stored in 11 bits and the fractional part is stored on the remaining 52 bits . thus , besides the fact that one can represent only a finite subset of all real numbers , all floating point calculations are furthermore rounded resulting in incorrect values . the smallest positive number @xmath205 , where the floating point approximation for @xmath206 is indeed larger than one is called the machine precision . when one rounds to the nearest representable number the machine precision is roughly @xmath207 where @xmath208 is the number of bits used to store the mantissa s fraction . for a single precision representation one finds @xmath209 and for a double precision number calculation @xmath210 . this means that single precision numbers have at most about 7 accurate digits while double precision numbers have about 16 accurate digits . but in general , due to the error propagation during the application of approximate algorithms the number of accurate digits for a numerical solution decreases . therefore , the rounding error will be several orders of magnitude larger for a more complicated calculation such as the one for the thermodynamic potential . to minimize this source of error in the derivative calculation of the thermodynamic potential , a larger value of @xmath193 is reasonable . in order to estimate these numerical errors and verify the quality of the adol - cevaluations the results of the derivative calculation obtained with ad are confronted with the dd method . in figs . [ fig : advsnum_t0 ] and [ fig : advsnum_cep ] the results of a dd evaluation as a function of @xmath193 in comparison with the ad calculation for the second - order and fourth - order derivative of the thermodynamic potential are shown . [ fig : advsnum_t0 ] shows the @xmath2-derivatives of the potential evaluated at @xmath211 mev which is close to the crossover phase transition in the @xmath186 phase diagram . one can clearly see the competition of the truncation and rounding errors . for the second - order derivative the optimal value is around @xmath212 while for the fourth - order derivative a slightly larger value @xmath213 leads to more stable results . in fig . [ fig : advsnum_cep ] the same derivatives are calculated at the point @xmath214 mev which is near the critical end point in the phase diagram . while in the previous fig . [ fig : advsnum_t0 ] the rounding error dominates , the truncation error is now more important . for the second - order derivative almost no rounding error is visible in the resolution shown . since the truncation error for the five - point expression , eq . ( [ eq:2nd_five ] ) , is of the order @xmath215 and of the order @xmath216 for the corresponding three - point equation , eq . ( [ eq:2nd_three ] ) , the results of the five - point derivative is indistinguishable already for @xmath213 while for the three - point formula a smaller value of @xmath217 is required . for the fourth - order derivative the interval where the derivative does not vary with @xmath193 is very small . only for @xmath218 the dd result is close to the ad result . in summary , one realizes that the dd derivatives require a very careful fine - tuning of the @xmath193 value . the dd result coincides always with the dd results where the @xmath193 variation vanishes . one finds that the ad technique is more efficient than the dd method . the dd calculation always requires the evaluation of the function at several points , e.g. , for the fourth derivative five function evaluations are necessary . in our case this involves the solution of the eom at these five nodes . this is a time - consuming disadvantage of the dd method . with the ad the eom need to be solved only once . despite the fact that it is required to generate an internal function representation of the evaluation of the eom solution and of the thermodynamic potential inside of adol - c , the ad implementation is much faster . corresponding runtime measurements are illustrated in fig . [ fig : mwad_runtime ] . the runtime of the dd approach can be described by the linear function @xmath219 where as the ad runtime performs like @xmath220 . this result fits perfectly to the computational complexity of the ad approach described in sec . [ sub:4.3 ] . as previously illustrated , both error sources for a derivative calculation with the dd method are in general difficult to keep under control , in particular , if higher - order derivatives are involved . however , with the ad method it is possible to obtain higher - order derivatives with very high precision . in the following an explicit example is given within the already introduced linear sigma model . higher derivatives are required if one is interested , e.g. , in the extrapolation of monte carlo lattice simulations of strongly interacting matter ( lattice gauge theory ) to finite quark chemical potential . at finite quark chemical potential such types of monte carlo simulations can not be directly performed @xcite . one possible extrapolation to finite quark chemical potential is based on a taylor expansion around zero chemical potential @xcite . for this purpose , we consider the same kind of expansion in the quark - meson system described by the l@xmath0 m . an example is given by the coefficients in the expansion of the pressure @xmath221 which is related to the thermodynamic potential via @xmath222 at fixed temperature and small values of the quark chemical potential the pressure may be expanded in a taylor series around @xmath223 , @xmath224 where the expansion coefficients are given in terms of derivatives of the pressure @xmath225 the series is even in @xmath226 which reflects the invariance of the partition function under the exchange of particles and antiparticles . in fig . [ fig : adtaylor ] the expansion coefficients @xmath227 to @xmath228 are shown as function of the scaled temperature @xmath229 . here , @xmath230 is the pseudocritical temperature at which the crossover transition occurs for vanishing chemical potential . since the first three expansion coefficients @xmath231 and @xmath232 are already known and well - understood we do not show them again @xcite . in lattice gauge theory one can currently calculate the first five coefficients , @xmath233 @xcite . the higher coefficients @xmath234 with @xmath235 vanish for temperatures basically outside of a five percent window around @xmath230 . thus , all coefficients are only shown in the range @xmath236 . all curves are smooth oscillating functions around zero even up to the @xmath237 derivative order . the amplitude of the oscillation and the number of roots around @xmath230 increases with the order @xmath54 . thus , this oscillating behavior of the coefficients obviously requires a smaller @xmath193 in order to decrease the truncation error but then the rounding error increases . already in this example the error sources are dramatic for such a high degree of derivatives . therefore it is not reasonable and actually not possible to obtain the higher coefficients with standard techniques such as the dd method . a novel numerical technique , which is based on algorithmic differentiation , for the calculation of arbitrarily high - order and high - precision derivatives has been presented . the new feature of the technique is the additional treatment of implicitly defined functions . in addition , the basic concepts of the algorithmic differentiation for explicit dependencies is discussed . as a demonstration of the successful extension to implicitly defined functions the ad technique is applied to a quantum - field theoretical model for strongly - interacting matter . in this model the implicitly defined functions are represented by the underlying equations of motion where the implicitly defined order parameter is known only numerically . two important error sources namely , the rounding and truncation error , for a derivative calculation in general are discussed in detail . furthermore , the results with the improved ad method are confronted to those obtained by standard divided difference ( dd ) methods . in the comparison the rounding and truncation errors can clearly be identified . while for a second - order derivative calculation the error sources are still controllable , they become intractable for higher orders . in the model example higher - order derivative coefficients of a taylor expansion for the pressure are calculated up to @xmath237 order . since these coefficients are calculated for the first time , no comparison with other results can be performed . the obtained curves are very stable and smooth functions which demonstrates the power of the novel ad technique . in a forthcoming publication @xcite this method will be applied to the more realistic polyakov - quark - meson model for three quark flavors @xcite . the presented ad technique augmented by implicitly defined dependencies can be applied to a wide class of problems , where high - order derivatives are involved . standard alternative methods for the derivative calculation such as the dd method fail due to uncontrollably increasing errors . especially , in the case of only numerically known implicit dependencies , an analytic solution is actually not possible . here , the ad method is still applicable and displays its exceptional impact . the work of mw was supported by the alliance program of the helmholtz association ( ha216/ emmi ) and bmbf grants 06da123 and 06da9047i . we thank j. albersmeyer , f. karsch , a. krassnigg , r. roth and j. wambach for useful discussions and comments . gell - mann , m. and levy , m. , nuovo cim . * 16 * ( 1960 ) 705 . meyer - ortmanns , h. , rev . * 68 * ( 1996 ) 473 . schaefer , b .- j . and wambach , j. , phys . part . * 39 * ( 2008 ) 1025 . schaefer , b .- wagner , m. , phys . rev . * d79 * ( 2009 ) 014018 . | scientific studies often require the precise calculation of derivatives . in many cases
an analytical calculation is not feasible and one resorts to evaluating derivatives numerically
. these are error - prone , especially for higher - order derivatives . a technique based on algorithmic differentiation
is presented which allows for a precise calculation of higher - order derivatives .
the method can be widely applied even for the case of only numerically solvable , implicit dependencies which totally hamper a semi - analytical calculation of the derivatives . as a demonstration
the method is applied to a quantum field theoretical physical model .
the results are compared with standard numerical derivative methods .
algorithmic differentiation , numerical differentiation , taylor expansion , quantum chromodynamics 02.60.gf , 02.70.bf , 12.38.aw , 11.30.rd |
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it is well known that horizontal seismometers are sensitive to ground rotation at low frequencies.@xcite indeed , due to the equivalence principle , conventional seismometers and tiltmeters can not distinguish between horizontal acceleration and rotation of the ground . while this is a problem in precision seismology , it is especially problematic for seismic isolation in next - generation gravitational - wave detectors , such as advanced laser interferometer gravitational - wave observatory ( aligo ) , where it is believed that rotation noise may limit improvements in low - frequency isolation.@xcite conventional horizontal seismometers can be idealized as horizontal spring - mass systems whose horizontal displacement is sensed relative to the housing . similarly , conventional tiltmeters or inclinometers can be idealized as vertical pendulums whose horizontal displacement is sensed relative to the housing . they are schematically represented in fig . [ tiltaccel ] . by ` tilt ' , we henceforth refer to the angular deflection of a simple pendulum with respect to its suspension platform or enclosure . from the diagram of the horizontal seismometer , a periodic rotation of @xmath5 at angular frequency @xmath6 will look equivalent to an acceleration of @xmath7 or a displacement of @xmath8 . in other words , the displacement in response to a unit rotation for the horizontal seismometer is @xmath9 comparison between a horizontal seismometer , a tiltmeter , and our rotation sensor under the influence of slow horizontal acceleration and rotation . in the first two instruments , an observer inside the box can not distinguish between the displacement of the bob due to rotation and acceleration . ] due to the @xmath10 in the denominator , response to rotation dominates at low frequencies ( typically @xmath11 mhz ) . similarly for a pendulum tiltmeter , the displacement of the bob due to a rotation is indistinguishable from that due to a horizontal acceleration at frequencies well below the resonance frequency . consequently , the rotation in response to a unit displacement is given by the inverse of the right - hand side of eq . ( [ eq0 ] ) . thus , typically , a tiltmeters output is dominated by acceleration at frequencies greater than @xmath12 mhz . an important limitation of the active - control system for seismic isolation in aligo is this inability of horizontal seismometers to distinguish between ground rotation and horizontal acceleration at low frequencies ( @xmath13 to @xmath14 mhz@xcite ) . slow ground rotation , such as that induced by wind , would be misinterpreted as large ground displacements , which could cause a large correction to be applied at the low - frequency active - isolation stages . this large displacement can produce noise in the gravitational - wave signal band through frequency up - conversion mechanisms , non - linearities , and cross - couplings . this problem can be addressed by measuring the absolute ground rotation and removing it from the seismometer channels before it has a chance to enter the isolation system . our rotation sensor design may also be important to the field of rotational seismology.@xcite large ring - laser gyroscopes are the traditional rotation sensors used in this field.@xcite they are traditionally run horizontally to sense rotation about the vertical axis , but can be mounted vertically to sense rotation about a horizontal axis.@xcite our design offers a simpler and more compact alternative . for comparison , it has roughly an order of magnitude better angle sensitivity than the horizontal - axis ring - laser gyroscope described in belfi _ _ et al.__@xcite between 10 to 100 mhz . in this frequency band , our sensor has comparable sensitivity to the angle sensitivity of c - ii : a vertical - axis meter - scale monolithic laser ring gyro@xcite and its sensitivity is surpassed by roughly an order of magnitude by the horizontal - axis 3.5 meter - square g-0 ring - laser gyro.@xcite our rotation sensor is a low - frequency beam balance whose angle with respect to the platform is measured using an autocollimator . above the resonance frequency , the beam balance stays inertial as the platform rotates around it . thus , the autocollimator measures the platform rotation , as shown in fig . [ schematic ] . to decouple the influence of rotation and translation , the center of mass ( com ) of the balance is located as close to the axis of rotation as possible . the relevant parameters of the balance are listed in table [ paratab ] . schematic showing the principle of the rotation sensor . at frequencies above the resonance of the balance , it stays inertial as the platform rotates . thus the autocollimator measures the platform rotation . ] .[paratab]parameters of the balance [ cols="^,^",options="header " , ] to understand the dynamics of the system , we can write down the equations of motion in an inertial frame aligned with gravitational vertical . let @xmath5 be the angle between the beam and the inertial frame s horizontal plane and let @xmath15 be the platform rotation angle with respect to the inertial frame s horizontal plane . the equation of motion for rotation of the balance about the pivot axis in the presence of external torques ( assuming @xmath16 ) is @xmath17 where @xmath18 is the vertical distance from the com and the pivot ( positive sign if the com is below the pivot ) , @xmath19 is the stiffness of the flexure , @xmath20 is the total suspended mass of the balance , @xmath21 is its moment of inertia , @xmath22 is the loss factor of the flexure material ( @xmath23 ) , and @xmath24 is the horizontal displacement of the platform . the external torque @xmath25 includes torques from all sources other than acceleration or rotation , such as brownian motion ( @xmath26 noise)@xcite , temperature gradients , or magnetic fields . accelerations and rotations in other degrees of freedom are ignored in this simple model as they couple only through higher - order effects . to establish the transfer function for the autocollimator output in response to a platform rotation , external torque and acceleration can be set to zero . substituting @xmath27 and @xmath28 and solving for @xmath29 gives @xmath30 the quantity measured by the autocollimator is @xmath31 and the transfer function is @xmath32 the magnitude of the transfer function can be rearranged into the usual form for an oscillator as @xmath33 where @xmath34 , @xmath35 and @xmath36 . the new quality factor @xmath37 can be either enhanced or reduced depending upon the sign of @xmath18 . the transfer function has a zero at @xmath38 and a pole at @xmath39 . at frequencies much less than both @xmath38 and @xmath39 , @xmath40 fig . [ transfunc ] shows the measured transfer function for a positive and a negative @xmath18 . the low frequency response is given by eq . ( [ eq6 ] ) , which is strongly sensitive to @xmath18 . based on the sign of the gravitational spring stiffness @xmath41 there are three possibilities : 1 . when @xmath41 is much larger than @xmath19 , the balance behaves as a simple pendulum , giving a low frequency response of @xmath42 . when @xmath41 is close to zero , the balance behaves as a spring - mass system , giving a low frequency response much less than @xmath42 . 3 . when @xmath41 is @xmath43 , the low frequency response is enhanced . this principle was exploited by building a second balance ( sec . [ rotdata ] ) with a negative and large @xmath18 , which made it very sensitive to tilt . magnitudes of the transfer function between platform rotation and autocollimator output for two configurations of the balance with different com . the measurement and a fit with free parameter @xmath18 are shown for each configuration . a positive @xmath18 results in a real zero in the transfer function whereas a negative @xmath18 yields an imaginary zero . ] thus , a simple way of minimizing @xmath44 is to minimize the low - frequency rotation response of the balance . this measurement technique provides a method for minimizing @xmath44 far more precisely than would otherwise be possible . similarly , we can calculate the transfer function for the autocollimator output in response to a platform acceleration . setting external torque and platform rotation to zero gives @xmath45 rearranging terms as in eq . ( [ tiltresp2 ] ) yields @xmath46 and for @xmath47 , @xmath48 for our balance this is less than @xmath4 rad@xmath49 m . this ratio is equivalent to that of a 33-km - long simple pendulum . a useful way of understanding the noise sources on the balance is to calculate the net torque acting on the system . the torque measured in the non - inertial lab frame can be written as @xmath50 using eqs . ( [ eq1 ] ) and ( [ autocoleq ] ) we can rewrite this as @xmath51 and converting to the frequency domain yields @xmath52 as mentioned previously , the first term includes torques from all non - acceleration sources . the second term in eq . ( [ eq9 ] ) contains the signal we are interested in measuring . this can also be interpreted as a pseudo - torque . the third term is the torque due to accelerations that couple through the gravitational spring . drawing of the rotation sensor showing the beam balance suspended by a pair of flexures . mirrors are located at the center for readout of the angular position of the balance . ] a pair of beryllium - copper flexures are shown on the left . the right side shows an image of the flexure as viewed through a microscope . ] the balance beam consists of an aluminum tube with brass weights at the ends . the tube has a diameter of 38 mm , wall thickness of 1.6 mm , and a length of 0.76 m. the brass weights are solid cylinders of diameter 34 mm and length of 0.23 m , and are fitted into the aluminum tube bringing the end to end length of the balance to 0.97 m. the balance is suspended by two flexures machined from beryllium - copper alloy , shown in figs . [ model ] and [ flex ] . they are notch type flexures , with a thickness of @xmath53 @xmath3 m , radius of 3.2 mm , and width of 6.3 mm . following standard procedures for achieving optimum tensile strength , the flexures were age - hardened at @xmath54 c for about 2 hours . we built a vacuum chamber for the balance using stainless steel cf vacuum components ( see fig . [ can ] ) . the chamber consists of a 6-way cross with two tubes flanged to it . a turbo vacuum pump backed by a diaphragm pump is connected through a long bellows to the 6-way cross . the side - port contains a mechanical - manipulator bellows to adjust the horizontal com while the balance is under vacuum . a viewport on top allows optical access for an autocollimator to measure the angle of the balance . the instrument is located in a temperature - controlled ( @xmath55 c ) underground experimental hall and is surrounded by an extruded - foam enclosure . the instrument is installed on a @xmath56-m - thick 600-kg aluminum plate bolted to three stainless steel feet . the feet sit on a 0.3 m - thick concrete platform that rests on a deep foundation . we installed 6.3-mm - thick aluminum thermal shielding tubes surrounding each balance - arm to provide thermal isolation and reduce thermal gradients . these aluminum tubes are located inside the vacuum vessel and are thermally decoupled from the vacuum chamber and from the balance , but are connected to each other through thick aluminum heat links . university of washington rotation sensor . thermal insulation was removed for this picture . ] the platform - angle - to - autocollimator - output transfer function was measured by applying a periodic horizontal force to the aluminum platform on which the balance sits . this force deflected the platform rotationally ( @xmath57 @xmath3rad ) and induced horizontal translations ( @xmath57 @xmath3 m ) . the periodic force was generated by a spring attached to a @xmath58-mm - diameter crankshaft driven by a stepper motor . the spring was connected to the platform through a @xmath59 m cable which reduced the variation in the direction of the applied force . the circular motion of the motor was thus converted to a linear force on the platform . due to the small @xmath18 during measurements , the response of the balance to displacements of the platform were many orders of magnitude smaller than the rotation response . hence the platform displacement did not affect the transfer function measurement . each point in fig . [ transfunc ] represents a single measurement which consisted of driving the platform at a single frequency for a few thousand seconds and measuring the autocollimator output . schematic drawing of the four reflections of the light beam on the target mirror . ] we used a multi - slit autocollimator , built by our group , that measures the angular deflection of a light beam reflected off a mirror mounted on the balance.@xcite the autocollimator uses a fiber - coupled led and a multi - slit grating as the light source and a sensitive ccd sensor for readout . sensitivity was enhanced by employing multiple reflections of the light beam . our setup used four reflections ( see fig . [ fourb ] ) giving a dynamic range of 5 mrad . the autocollimator is designed for use with a partially reflecting reference mirror in the beam path . subtracting the reference pattern allows a rejection of common - mode noise such as low - frequency distortion of the vacuum can or the autocollimator body . in practice , we found such common - mode noise to be negligible compared with the seismic background . however , this option allows referencing to a surface other than the platform to which the balance is bolted . this would prove useful if we were to read the rotation of a surface that could not support the weight of the entire balance . the reference pattern readout is also a measure of the intrinsic autocollimator noise . the autocollimator was calibrated using one ordinary mirror and one partially reflecting mirror held at a fixed , known angle.@xcite in addition the instrument was calibrated using the gravitational torque from a pair of lead bricks on a turntable located @xmath60 m from the vacuum vessel . the computed gravitational torque agreed with the measured torque to within the @xmath13 percent systematic uncertainty . the rotation sensor output was recorded for several months . typically , the rotation noise was smallest from 22:00 hrs to 04:00 hrs . the rotation in the 10-to 100-mhz frequency range was dominated by local weather , especially wind . to understand the spectral distribution , the raw time series of the autocollimator output is fourier transformed to an amplitude spectral density . to obtain the platform rotation , we divide the autocollimator output by the measured transfer function between them ( eq . ( [ tiltresp ] ) ) . [ tiltdata ] shows the spectral density of the data from a 4000-s measurement during a quiet period . the spectral density shown is typical for most nights with low wind speeds . typical platform rotation spectral density as measured by the rotation sensor during quiet conditions . also shown are the autocollimator sensitivity and the rotation sensor requirement to potentially improve aligo.@xcite ] fig . [ torque ] shows a plot of the torque spectral density as a function of frequency calculated using eq . ( [ torqeq ] ) for the same data set as above . also shown are the thermal or brownian motion noise torque@xcite and the rotation sensor requirement,@xcite interpreted in torque units for our balance . torque sensitivity of the rotation sensor . also shown are the sensor requirements , autocollimator limit and thermal noise limit@xcite in torque units . ] separating the background rotation signal from the instrument noise requires a second instrument of similar or better sensitivity . since the rotation background at frequencies above 50 mhz was sufficiently small during quiet conditions , we focused on measuring the ground rotation near the 10-mhz limit of the required bandwidth . to build a sensitive tiltmeter at these frequencies we exploited the principle of the amplification of rotation using a negative @xmath18 , described in sec . [ subsec : eqnmotion ] . we constructed the tiltmeter from an aluminum plate and brass weights suspended by two stiff flexures ( fig . [ newtilt ] ) . this balance had its com far above the pivot ( @xmath61 mm ) resulting in a negative and large gravitational stiffness . thus , an autocollimator mounted to the frame measured an amplified rotation signal at frequencies below resonance of the tiltmeter . schematic of the tiltmeter . it is suspended by two stiff flexures . the com is located 78 mm above the pivot . the angle of the beam is read - out using an autocollimator . ] this tiltmeter was placed on the same aluminum platform as the rotation sensor . to check the calibration and sensitivity of this instrument , we temporarily modified the rotation sensor to run in tiltmeter mode by changing the com to be @xmath42 mm below the pivot . the difference between the tilt measurements by the two sensors was smaller than @xmath62 nrad/@xmath63 at 10 mhz and less than @xmath42 nrad/@xmath63 at 30 mhz . [ corr ] shows the angle spectral density from a 4000-s dataset recorded during the day with both instruments . the angle measured by the rotation sensor and the ordinary tiltmeter agree well in the 15-to 60-mhz frequency range . above that , the angle measured by the tiltmeter is dominated by acceleration , which the rotation sensor rejects efficiently . fig . [ msc ] shows the mean - squared coherence of the two data sets as a function of frequency . subtracting the two datasets gives a factor of 2 to 3 suppression of rotation noise in the rotation sensor in the 15-to 60-mhz range . the small @xmath18 of the rotation sensor balance implies that the microseismic acceleration affecting the tiltmeter in fig . [ msc ] near @xmath64 hz would produce an angle noise of less than 10 picorad/@xmath63 . this is confirmed by the coherence plot which shows poor coherence at the microseismic frequencies . however , the residual noise after subtraction is worse compared to the noise from the quiet - condition data . we believe this excess noise arises from high - frequency vibration , discussed in the next section . during quiet conditions at night , when background rotations were much smaller , the two instruments showed reduced correlation , indicating that at least one instrument was limited by intrinsic noise . . there are various environmental influences that can produce noise in the rotation sensor . due to the low - pass filtering of torques by the balance and the nature of background influences ( such as weather ) , the noise requirement at 10 mhz is most challenging . other than the autocollimator noise , all other noise sources are practically insignificant above 50 mhz . comparison of the lowest torque noise measured with and without the rubber shims placed under the feet of the support platform , demonstrating the suppression of the torque noise . ] one of the largest noise sources we had to overcome was what we referred to as vibration noise . the spectral shape of the noise was white ( in torque ) , but it had a daily amplitude variation of a factor of @xmath65 , correlated with ambient vibration levels . there are several models which could explain this kind of noise . one possible explanation is that this noise is a result of irregular torques arising from fluctuations of the parameters of the flexure ( parametric down - conversion ) triggered by excessive high - frequency vibrations . regardless of the exact mechanism for the process , we found that placing small rubber shims under the feet of the large aluminum platform to which the rotation sensor was bolted reduced this vibration noise ( fig . [ daily ] ) . the white - noise floor was lowered by a factor of 10 with the introduction of the rubber shims . still , this may remain the current limiting noise source and might be worse in places with large vibrations . we plan to investigate and reduce this noise source further . an important noise source is the presence of a temperature gradient between the two arms of the balance . this armlength effect was analyzed by speake _ et al . _ @xcite using their formulation and a measured upper limit on the temperature gradient , we estimated the upper limit of the angle noise from this effect to be less than @xmath66 rad@xmath0 at 10 mhz . noise from ambient magnetic field fluctuations exerting a torque on the balance are expected to be small . magnetic materials were avoided in the construction of the balance . the magnetic moment of the balance was measured to be @xmath67 @xmath68 j / t . with ambient field fluctuations expected to be no larger than @xmath69 t@xmath0 , the angle noise from magnetic field fluctuation would be less than @xmath70 rad@xmath0 . another possible source of noise was from ambient gravity gradients . our balance has a gravity gradient sensitivity of better than @xmath71 eotvos@xmath0 or @xmath72 s@xmath73 around @xmath13 mhz . for reference , this would allow the rotation sensor to resolve the gradient signal from a 20-ton bus , with closest approach of 80 m at @xmath74 to the beam axis , in @xmath75 seconds to 2@xmath76 under quiet conditions . however , the expected gradient noise due to environmental sources such as nearby trees , buildings , etc . , was estimated from models to be lower than our current best sensitivity by at least an order of magnitude . sensitivity of the rotation sensor . ] we have constructed a prototype absolute rotation sensor comprising of a @xmath77-mhz beam balance and an autocollimator . we also constructed a tiltmeter or inclinometer to measure background rotation at frequencies below @xmath78 mhz . the two instruments show good correlation in the @xmath79-to @xmath80-mhz band during the day when background rotations are larger . under quiet conditions the correlation is small , indicating that the instruments are limited by intrinsic noise . using the best quiet - condition data , we were able to place upper limits on the sensitivity of the instrument as shown in fig . the sensor can reject horizontal acceleration to better than @xmath4 rad / m at frequencies above resonance . by design , the rotation sensor has optimal sensitivity in a frequency band spanning @xmath81few mhz to hz . the low - frequency range of the sensor is limited by the @xmath82 frequency - dependence in sensitivity below the @xmath77 mhz resonance frequency of the balance . above 0.1 hz , the rotation sensor is limited by the @xmath64 nrad@xmath0 flat angular sensitivity of the autocollimator . at the high - frequency end , the sensor is limited by the 3.3 hz downsampling of the autocollimator signal.if desired , it can be operated at frequencies as high as 1.5 khz , excepting acoustic resonances of the balance . this instrument has the potential to improve seismic isolation in advanced ligo by reducing the rotation noise contribution in horizontal seismometers . it meets the requirements described by lantz _ _ et al.__@xcite above 40 mhz . we continue to improve the sensitivity of the instrument at low frequencies and to develop a more robust and compact version . we would like to thank ron musgrave , larry stark and jim greenwell at the uw physics instrument shop for our flexures . we would like to thank the us taxpayers , nsf ( grants : phy0969488 and phy1306613 ) , and the ligo scientific collaboration for funding and supporting this project . in particular , the authors would like to thank vladimir dergachev , brian lantz , fabrice matichard , and rainer weiss for valuable suggestions and discussions . we are also grateful to our colleagues in the eot - wash group for useful discussions . we thank the center for experimental nuclear physics and astrophysics ( cenpa ) for use of its facilities . s. d. angelis and p. bodin , `` watching the wind : seismic data contamination at long periods due to atmospheric pressure - field - induced tilting '' , bull . of the seismological society of america , * 102 * , 1255 - 1265 ( 2012 ) b. lantz and r. schofield and b. oreilly and d. e. clark and d. debra , `` review : requirements for a ground rotation sensor to improve advanced ligo '' , bull . of the seismological society of america , * 99 * , 980 - 989 ( 2009 ) k. u. schreiber and j. n. hautmann and a. velikoseltsev and j.wassermann and h. igel and j. otero and f. vernon and j .- p . r. wells , `` ring laser measurements of ground rotations for seismology '' , bull . of the seismological society of america , * 99 * , 1190 - 1198 ( 2009 ) j. belfi and n. beverini and f. bosi and g. carelli and a. d. virgilio and e. maccioni and a. ortolan and f. stefani , `` a 1.82 m@xmath83 ring laser gyroscope for nano - rotational motion sensing '' , e - print arxiv:1104.0418 ( 2011 ) t. b. arp and c. a. hagedorn and s. schlamminger and j. h. gundlach , `` a reference - beam autocollimator with nanoradian sensitivity from mhz to khz and dynamic range of @xmath84 '' , rev . instrum . , * 84 * , 095007 ( 2013 ) c. c. speake and g. t. gillies , `` the beam balance as a detector in experimental gravitation '' , proceedings of the royal society of london . series a , mathematical and physical sciences , * 414 * , 315 - 332 ( 1987 ) | we have developed a mechanical absolute - rotation sensor capable of resolving ground rotation angle of less than 1 nrad@xmath0 above @xmath1 mhz and 0.2 nrad@xmath0 above @xmath2 mhz about a single horizontal axis .
the device consists of a meter - scale beam balance , suspended by a pair of flexures , with a resonance frequency of 10.8 mhz .
the center of mass is located 3 @xmath3 m above the pivot , giving an excellent horizontal displacement rejection of better than @xmath4 rad / m .
the angle of the beam is read out optically using a high - sensitivity autocollimator .
we have also built a tiltmeter with better than 1 nrad@xmath0 sensitivity above 30 mhz .
co - located measurements using the two instruments allowed us to distinguish between background rotation signal at low frequencies and intrinsic instrument noise .
the rotation sensor is useful for rotational seismology and for rejecting background rotation signal from seismometers in experiments demanding high levels of seismic isolation , such as advanced ligo . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
envelopes of neutron stars ( nss ) are divided in the inner and outer envelopes . properties of the inner envelopes are outlined in a companion paper @xcite . here we focus on the outer ones , at typical densities @xmath6 and temperatures @xmath7 k. these envelopes are relatively thin , but they affect significantly ns evolution . in particular , they provide thermal insulation of stellar interiors . bulk properties of the ns envelopes ( pressure @xmath8 , internal energy @xmath9 ) are determined mainly by degenerate electron gas . the electrons are relativistic at @xmath10 , where @xmath11 , @xmath12 is the fermi wave number , @xmath13 is the electron mass , @xmath14 is the electron number density , @xmath15 , @xmath16 and @xmath17 are the ion charge and mass numbers , respectively . degeneracy is strong at @xmath18 , where @xmath19 and @xmath20 is the lorentz factor . the ionic component of the plasma can form either strongly coupled coulomb liquid or solid , classical or quantum . the classical ion coupling parameter is @xmath21 , where @xmath22 is the ion - sphere radius ( @xmath23 ) . the ions form a crystal if @xmath24 exceeds some critical value @xmath25 ( see below ) . the quantization of ionic motion becomes important at @xmath26 , where @xmath27 is the ion plasma temperature , and @xmath28 is the ion mass . the ionic contribution determines specific heat @xmath29 , unless @xmath26 . the ions and electrons are coupled together through the electron response ( screening ) . the inverse screening ( thomas fermi ) length is @xmath30 where @xmath31 is the electron chemical potential . if @xmath32 , then @xmath33 and @xmath34 , where @xmath35 is the fine structure constant . note that @xmath36 at @xmath37 ; therefore the screening effects do not vanish even at high densities . thermodynamic properties are altered by the magnetic field @xmath38 in the case where the landau quantization of transverse electron motion is important . the field is called _ strongly quantizing _ when it sets all electrons on the ground landau level . this occurs at @xmath39 and @xmath40 ( see , e.g. , ref . @xcite ) , where @xmath41 and @xmath42 g. _ weakly quantizing _ fields ( @xmath43 ) do not significantly alter @xmath8 and @xmath9 but cause oscillations of @xmath29 , other second - order quantities , and electron transport coefficients with increasing density . characteristic domains in the @xmath44@xmath45 diagram are shown in figure [ fig - domains ] . the short - dashed lines on the left panel indicate the region of partial ionization . the dot - dashed lines correspond to @xmath46 ( upper lines ) and @xmath47 ( liquid / solid phase transition ) . long - dashed lines on the right panel separate three @xmath44@xmath45 regions where the magnetic field is strongly quantizing ( to the left of @xmath48 and considerably below @xmath49 ) , weakly quantizing ( to the right of @xmath48 at @xmath50 ) , or classical ( much above @xmath49 ) . 10.2 cm the dotted lines in fig . [ fig - domains ] show profiles @xmath51 in the envelope of a `` canonical '' ns with mass @xmath52 and radius 10 km , and with an effective surface temperature @xmath53 k or @xmath54 k ( the values of @xmath55 are marked near these curves ) . typical temperatures of isolated nss are believed to lie in the hatched region between these two lines . in a magnetized ns , @xmath51 depends on strength as well as direction of the magnetic field . therefore on the right panel we show _ two _ dotted curves for each value of @xmath56 : the lower curve of each pair corresponds to the heat propagation along the field lines ( @xmath57 , i.e. , near the magnetic poles ) and the upper one to the transverse propagation ( @xmath58 , near the magnetic equator ) . thermodynamic functions of the electron gas at arbitrary degeneracy are expressed through the well known fermi - dirac integrals . for astrophysical use , it is convenient to employ analytic fitting formulae for these functions presented , e.g. , in ref . @xcite . nonideal ( exchange and correlation ) corrections for nonrelativistic electrons at finite temperature have been calculated and parameterized in ref . @xcite . for the relativistic electrons at low @xmath45 , an analytic expansion of the exchange corrections is given , e.g. , in ref . @xcite ( in this case , the correlation corrections are negligible ) . a smooth interpolation between these two cases has been constructed in ref . @xcite . for the ionic component at @xmath59 , the main contribution to the thermodynamic functions comes from the ion correlations . strongly coupled one - component coulomb plasmas ( ocp ) of ions in the _ uniform _ electron background have been studied by many authors . the thermodynamic functions of the classical ocp liquid have been calculated @xcite at @xmath60 and parameterized @xcite for @xmath61 . the latter parameterization ensures accuracy @xmath62 ( per particle ) . for the classical coulomb crystal , accurate numerical results and fitting formulae to the free energy @xmath63 ( with anharmonic corrections taken into account ) have been presented in refs . a comparison of the latter results with the fit @xcite for the liquid yields the liquid solid phase transition at @xmath64 ( curiously , this value could be derived in ref . @xcite , but it was first noticed in ref . @xcite ) . in the solid phase , quantum effects may be important at the considered temperatures . these effects can be most easily taken into account in the approximation of a harmonic coulomb crystal . a convenient analytic approximation is provided by a model @xcite which treats two phonon modes as debye modes and the third one as an einstein mode . numerical calculations @xcite of the harmonic - lattice contributions to @xmath63 , @xmath9 , @xmath29 , and the mean - square ion displacement are reproduced by this model at arbitrary @xmath65 within several percent , which is sufficient for many applications . 8 cm finally , there is a contribution from ion - electron ( @xmath66 ) correlations , which can be treated as polarization of the jellium " of the degenerate electrons . in a coulomb liquid , this contribution has been evaluated in refs . @xcite using a perturbation theory which involves the electron dielectric function and the static structure factor of ions . this approximation is justified at @xmath67 ( which is the case at @xmath68 ) . at smaller densities the @xmath66 contribution in the coulomb liquid has been calculated in ref . @xcite using the hnc technique ( heavy dots in fig . [ fig - ie ] ) . in a coulomb solid , the @xmath66 contribution has been evaluated in ref . @xcite using the perturbation theory @xcite and a model lattice structure factor . these results , though approximate , indicate that the polarization corrections may be rather important . in particular , @xmath25 may be shifted by @xmath69% when the @xmath66 corrections in the liquid and solid phases are taken into account . the numerical results have been fitted @xcite by analytic functions of @xmath70 and @xmath24 . the approximation @xcite for the ocp liquid is shown in fig . [ fig - ie ] by solid lines . since the @xmath66 correction is not dominant , the accuracy of the fitting formulae ( @xmath71% ) is sufficient for most of astrophysical applications . an alternative pad approximation , proposed in ref . @xcite for the electron - ion fluid , is also accurate at @xmath72 for small and large @xmath24 , but it is less accurate at intermediate @xmath24 and inapplicable at @xmath73 ( dashed lines ) . as mentioned above , quantizing magnetic fields significantly affect the equation of state . in the fully ionized dense plasma , the main effects are described by the model of ideal electron gas , studied by many authors ( e.g. , @xcite ) . heat conduction in ns envelopes is provided mainly by the electrons . the thermal conductivity tensor @xmath74 is determined mainly by the electron - ion scattering . calculation of @xmath74 should take into account specific features of the coulomb plasmas in ns envelopes , quite different from the terrestrial liquid and solid metals : ( i ) in the liquid phase , an incipient long - range order emerges at @xmath75 ; ( ii ) in the solid phase , there are usually many brillouin zones within the fermi surface ; ( iii ) in the solid phase , the approximation of one - phonon scattering fails near the melting , which necessitates the use of a more general expression for the structure factor of ions . these features have been taken into account in ref . @xcite , where analytic approximations to the electron transport coefficients have also been derived . magnetic field hampers electron transport across the field lines : transverse thermal and electrical conductivities are reduced by orders of magnitude in typical ns envelopes . strongly quantizing field significantly changes also longitudinal conductivities ; weakly quantizing field causes de haas van alphen oscillations . these effects have been outlined , e.g. , in @xcite . a unified treatment of the electron transport coefficients in the domains of classical , weakly quantizing , and strongly quantizing magnetic field has been developed in @xcite . a fortran code which implements this treatment is available at ` http://www.ioffe.rssi.ru/astro/conduct/ ` . using this code and solving the thermal diffusion equation , we have calculated the @xmath51 profiles shown in fig . [ fig - domains ] . ayp is grateful to the theoretical astrophysics group at ecole normale suprieure de lyon for generous hospitality and financial support . the work of ayp and dgy has been supported in part by intas ( grant 96 - 542 ) and rfbr ( grant 99 - 02 - 18099 ) . ayp and dgy thank the organizing committee for support provided to attend pnp10 . 99 , this volume , in the equation of state in astrophysics , ed . g. chabrier , e. schatzman , cambridge univ . , cambridge , uk ( 1994 ) 214 , phys . e * 58 * ( 1998 ) 4941 , phys . rep . * 149 * ( 1987 ) 91 , astron . , * 314 * ( 1996 ) 1024 , phys . rev . e * * ( 2000 ) accepted , physica b * 228 * ( 1996 ) 21 , j. chem . * 111 * ( 1999 ) 6538 , phys . a * 42 * ( 1990 ) 4972 , phys . rev . e * 47 * , 4330 ( 1993 ) . , astrophys . j. * 414 * ( 1993 ) 695 , ph.d . thesis , ioffe phys .- tech . institute , st . petersburg ( 2000 ) , phys . rev . a * 14 * ( 1976 ) 816 , astrophys . space phys * 7 * ( 1989 ) 311 , astron . ( 2000 ) accepted ; astro - ph/0008399 , j. phys . c * 15 * ( 1982 ) 6233 , astron . * 346 * ( 1999 ) 345 , astron . astrophys . * 351 * ( 1999 ) 787 | outer envelopes of neutron stars consist mostly of fully ionized , strongly coupled coulomb plasmas characterized by typical densities @xmath0@xmath1 and temperatures @xmath2@xmath3 k. many neutron stars possess magnetic fields @xmath4@xmath5 g. recent theoretical advances allow one to calculate thermodynamic functions and electron transport coefficients for such plasmas with an accuracy required for theoretical interpretation of observations . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
low mass x - ray binaries ( lmxbs ) are compact binaries where the primary is a compact object and the secondary a low mass star ( @xmath81@xmath7 ) . the secondary is transferring mass via roche - lobe overflow , forming an accretion disk around the compact object that gives rise to the observed x - rays . by far , most of the persistently bright lmxbs are neutron star systems that can be divided into two groups , the z - sources and atoll sources ( hasinger & van der klis 1989 ) . z - sources are usually the brightest lmxbs in x - rays ( they are thought to have mass accretion rates that reach the eddington limit ) and trace a z - like shape in their x - ray colour - colour diagrams . atoll sources on the other hand have lower accretion rates ( @xmath91 - 2 orders of magnitude lower ) and their colour - colour diagrams usually consists of fragmented island - like regions . apart from the difference in accretion rates , the main physical difference between z - sources and atoll sources are thought to be the strength of the neutron star magnetic field and their evolutionary history ( hasinger & van der klis 1989 ) . lmcx-2 is a persistent lmxb that shows the properties of a z - source ( smale et al . 2003 ) , and it is therefore thought to be a neutron star system that has an accretion rate around the eddington limit . it is one of the most x - ray luminous lmxbs known ( @xmath10@xmath910@xmath11 erg s@xmath3 ) , but due to its extra - galactic nature ( it is located in the large magellanic cloud at a distance of @xmath948 kpc ) , its x - ray flux is rather low . its optical counterpart was identified by pakull ( 1978 ) as a @[email protected] blue star . despite being x - ray luminous and having a known optical counterpart , thus far little is known about the system parameters of lmcx-2 . even the estimates for the orbital period range from 6.4 hrs ( motch et al . 1985 ) or 8.2 hrs ( callanan et al . 1990 , smale & kuulkers 2000 ) up to 12 days by crampton et al . ( 1990 ) . to make things even more complicated , no periodic variability was detected in 6 years of macho data ( alcock et al . 2000 ) . in recent years steeghs & casares ( 2002 ) developed a new technique to detect a signature of the donor star in persistent lmxbs . using phase - resolved spectroscopy they detected narrow emission lines in scox-1 , especially in the bowen blend ( a blend of niii and ciii lines between 4630 - 4650 ) , that were interpreted as coming from the irradiated side of the donor star . this discovery in scox-1 was followed by a survey of other lmxbs that are optically bright enough to also resolve these narrow components . thus far these narrow emission lines have been detected in x1822@xmath14371 ( casares et al . 2003 ) , gx339@xmath144 ( hynes et al . 2003 ) , v801ara and v926sco ( casares et al . 2006 ) , grmus ( barnes et al . 2007 ) , aqlx-1 and gx9@xmath159 ( cornelisse et al 2007a , b ) , leading to constraints on their system parameters . in this paper we apply the technique of bowen fluorescence to lmcx-2 . we will show that it is possible to detect a periodic signal in our spectroscopic dataset that we identify as the orbital period . furthermore , similar to the other x - ray binaries thus far , the bowen region shows the presence of narrow emission lines that we identify as coming from the irradiated side of the companion , giving the first ever constraints on the system parameters of lmcx-2 . on november 21 and 22 2004 we obtained a total of 77 spectra of lmcx-2 with an integration time of 600s each , using the fors2 spectrograph attached to the vlt unit 4 ( yepun telescope ) at paranal observatory ( eso ) . each spectrum was taken with the 1400v volume - phased holographic grism using a slit width of 0.7@xmath16 , giving a wavelength coverage of @xmath1@xmath14514 - 5815 and a resolution of 70 km s@xmath3 ( fwhm ) . the seeing during the first night was between 0.4 and 0.7 arcsec , while on the second night it varied between 0.5 and 2.7 arcs . the slit was orientated at a position angle of 7@xmath17 to include a comparison star in order to correct for slit losses . during daytime he , ne , hg and cd arc lamp exposures were taken for the wavelength calibration scale . we de - biased and flat - fielded all the images and used optimal extraction techniques to maximise the signal - to - noise ratio of the extracted spectra ( horne 1986 ) . we determined the pixel - to - wavelength scale using a 4th order polynomial fit to 20 reference lines giving a dispersion of 0.64 pixel@xmath3 and rms scatter @xmath180.05 . we also corrected for any velocity drifts due to instrumental flexure by cross - correlating the sky spectra . finally , we divided all spectra of lmcx-2 by a corresponding low order spline - fit of the comparison star to get the final fluxed spectra . since we did not observe a spectro - photometric standard star , we were not able to correct for instrumental response , and all spectra are therefore in relative fluxes . we created average spectra for each individual night . since we do not have a flux standard to derive an absolute flux for lmcx-2 , we decided to normalise the continuum flux to one by dividing each average spectrum by a low order spline fit . in fig.[spectrum ] we show the results . both spectra are dominated by the very narrow high excitation heii @xmath14686 emission line , while also bowen emission is present in both spectra . however , compared to other x - ray binaries , such as scox-1 , x1822@xmath14371 , v801ara and v926sco the bowen emission is much weaker compared to heii in lmcx-2 ( steeghs & casares 2002 , casares et al . 2003 , casares et al . this might be due to the much lower metal abundances in the large magellanic cloud ( motch & pakull 1979 ) . the most striking difference between the spectra is the dramatic change of h@xmath19 ( see fig.[spectrum ] ) . during the first night ( 21 november ) h@xmath19 is dominated by a weak emission feature superposed on a broad absorption feature , while in the second night ( 22 november ) the emission feature has become almost as strong as heii @xmath14686 . furthermore , also the hei @xmath14922@xmath205016 lines have become more prominent during the second night ( although they might be present during the first night ) . in order to quantify the change in the most prominent emission lines we estimated the equivalent widths and their line fluxes ( in arbitrary units ) for the two nights , and show them in table[width ] . for h@xmath19 we decided to also include the absorption component , giving negative values for the first night . table[width ] shows that there is no change in equivalent width of heii and the bowen region , and the line fluxes have dropped by @xmath940% during the second night . on the other hand the h@xmath19 and hei lines have all increased significantly in both equivalent width and line flux . unfortunately , due to the faintness of these lines ( or the presence of an absorption feature ) , it is not possible to say if they have all changed by the same amount , but it is likely that the same process is responsible for this change in line intensity . finally , in order to see if these changes could be related to a change in brightness we have also created a lightcurve of the continuum flux for lmcx-2 . we have normalised the flux of the first night around unity , and show the result in fig.[light ] . we note that during the second night the continuum flux was @xmath940% lower compared to the first night , a similar fraction as was observed for the line fluxes of the heii lines and bowen region , keeping their equivalent widths the same and suggesting a common origin . we determined phase - resolved radial velocities by cross - correlating each spectrum with a gaussian of width 150 km s@xmath3 centered on the core of the heii @xmath14686 line . since the conditions during the second night were much worse , together with the fact that the source was @xmath940% fainter , we have binned these spectra together in groups of three , and then determined the radial velocity . in fig.[radial ] we show the results . the first thing to note in fig.[radial ] is that during both nights the radial velocity shows a sine - like variation , which we interpret as orbital motion of a region that is co - rotating with the binary , and perhaps is connected to the dynamical properties of the primary . however , during the second night it appears that either the semi - amplitude of the radial velocity or the off - set compared to the rest wavelength has increased . this last behaviour was noted by crampton et al . ( 1990 ) , who observed a long - term variation in the radial velocity of heii . we searched the radial velocity curve for any periodic signal with a duration between 1 hr and 2 days using the lomb - scargle technique ( scargle 1982 ) . apart from the 24 hr alias due to the separation of our 2 observing nights , only two significant peaks with comparable strength were present in the power spectrum ; one peak is at a period of [email protected] days and another at [email protected] day . interestingly , the 0.32 day period is similar to the photometric period detected by callanan et al . ( 1990 ) , suggesting that this is the orbital period . although we can not exclude the possibility that the 0.45 day period is real ( but see sect.3.3 ) , we tentatively interpret the 0.32 day period as the orbital period and use it in the rest of this paper . fitting a sine curve with a period of 0.32 day to the radial velocity curve gives a phase zero at hjd 2,453,[email protected] , an off - set of 344@xmath011 km s@xmath3 , and the semi - amplitude of our sine fit is 41@xmath04 km s@xmath3 . note that we did not attempt to account for the seemingly variable systemic velocity from night to night , but only did a single sine fit to both nights . .lmc x-2 equivalent widths and spectral line fluxes . [ width ] [ cols="<,^,^,^,^ " , ] we used doppler tomography on the most prominent emission lines in order to probe the structure of the accretion disk ( marsh & horne 1988 ) . in order to create the maps , we used the orbital period of [email protected] hrs as determined by callanan et al . ( 1990 ) and a systemic velocity derived from the radial velocity curve in sect.3.2 . to comply with the standard definition of orbital phase 0 in doppler maps ( when the donor star is at inferior conjunction ) we used the phasing derived from the bowen map ( see below ) and applied a shift of @xmath90.5 orbital phase compared to the value derived in sect.3.2 . since heii @xmath14686 is by far the strongest emission line in lmcx-2 we decided to create doppler maps for each individual night , and we show the result in fig.[hedopp ] . both maps show a ring - like structure , but there are some minor differences . during the second night it appears as if the outer edge of the structure is at higher velocities than during the first night ( 218@xmath010 km s@xmath3 compared to 180@xmath010 km s@xmath3 ) . furthermore , during the first night there is a clear emission feature in the lower part of the map , that appears to have shifted toward the lower - right quadrant during the second night . since we only have 2 nights of observations , it is not clear if these changes are real , but it suggests that there is some change in accretion disk structure from night to night . we also created a doppler map of the bowen region by simultaneous fitting all the major niii ( @xmath14634/4640 ) and ciii ( @xmath14647/4650 ) lines using the relative strengths as given by mcclintock et al . the map is dominated by a bright emission feature , that we used to rotate the map ( by @xmath90.5 orbital phase ) until it was located in the top . fig.[bowdopp ] shows the resulting map . the bright emission feature is at a velocity of @xmath21=351@xmath028 km s@xmath3 , and another ( much fainter ) spot is also present in the map . if we interpret the bright spot as arising on the surface of the donor star ( see sect.4.2 ) , the fainter spot is in a region where we could expect an interaction between the accretion disk and the accretion flow . we do note that the velocity of the bright spot is much higher than the disk velocities in the heii doppler map ( fig.[hedopp ] ) , and we will discuss this in sect.4.2 . furthermore , since the radial velocity curve in sect.3.2 could not exclude a 0.45 days orbital period , we also created a bowen map for this period . although there are several spots present in this map , none is as sharp and significant as those in fig.[bowdopp ] . this is further support that the 0.32 day period is the correct orbital period . we have presented phase - resolved spectroscopy of lmcx-2 , one of the brightest x - ray sources in the large magellanic cloud , and also one of the most luminous lmxbs known . this enables us to derive the first constraints on its system parameters , and in particular give new insights on the previously reported orbital periods that ranged from @xmath96 - 8 hrs by motch et al . ( 1985 ) or callanan et al . ( 1990 ) up to @xmath912 days by crampton et al . ( 1990 ) . callanan et al . ( 1990 ) based their claim of a short orbital period on an extended @xmath92 week photometric campaign . a clear 8.15 hr modulation was present in their data that was interpreted as the orbital period , although there is indication of a long term ( @xmath910 day ) variability that was also observed by crampton et al . ( 1990 ) . on the other hand , crampton et al . did not detect the @xmath98 hr period in a @xmath91 week photometric campaign . however , there was a variation in the heii radial velocity curve over a period of 4 nights in their spectroscopic data that they interpreted as a @xmath912 day orbital period . interestingly , they noticed that the h@xmath19 emission lines also changed in strength over those 4 nights ( even going into absorption ) , with maximum line strength occurring at minimum ( continuum ) light . although crampton et al . ( 1990 ) did speculate that the @xmath912 day period is a precession or beat period , they discarded this due to the absence of shorter periods in their data set . in sect.3.2 we have shown that there is a @xmath98 hr period present in the radial velocity of heii @xmath14686 that is similar to the period detected by callanan et al . ( 1990 ) . since the semi - amplitude of this radial velocity curve is rather small ( @xmath941 km s@xmath3 ) , it might have been difficult to detect with the 4 m class telescope and an instrument with a resolution of @xmath134 used by crampton et al . we , therefore , tentatively identify this period with the orbital period . however , that does leave the question of the long term variation observed by both crampton et al . ( 1990 ) and callanan et al . similarly to crampton et al . ( 1990 ) , we also observe a large change in the emission line strength of h@xmath19 , with a stronger line occurring at lower continuum flux levels . unfortunately we only have two nights of observations , and are therefore not able to check if there is also a long term variation in our heii radial velocity curve . however , fig.[radial ] does suggest that during the second night ( when the continuum flux was lower ) either the amplitude or the mean velocity of the radial velocity has increased compared to the first night ( when the continuum flux was higher ) , similar to what crampton et al . observed . one explanation for the observed properties could be the presence of an inclined precessing , warped , accretion disk in lmcx-2 , as is also observed in herculesx-1 and ss433 ( katz 1973 , margon 1984 ) . we will discuss this suggestion in detail in a forthcoming paper by shih et al . ( in preparation ) , but here we will briefly highlight the spectroscopic evidence . during the first night we could be observing the accretion disk more edge on compared to the second night , and this could explain the much lower h@xmath19 and hei line intensities observed . this is further strengthened by the fact that the doppler maps suggest that heii is extending to higher radial velocities during the second night . if true , this could also explain the change in continuum flux observed in fig.[light ] . callanan et al . ( 1990 ) already suggested that a significant contribution to the optical light is either coming from the heated surface of the secondary or the outer disk bulge . if the fraction of the secondary or disk bulge that is in the shadow of the accretion disk changes as a function of precession period , this would lead to a change in the optical , with maximum light occurring when the accretion disk is most edge on . we can compare the characteristics of lmcx-2 to those of xtej1118@xmath15480 , an x - ray transient that is known for having a precessing ( although not necessarily inclined ) accretion disk ( uemura et al . 2000 ) . zurita et al . ( 2002 ) showed that the nightly average h@xmath22 lines changed in both velocity and width that is consistent with a periodic variation on the precession period ( torres et al . 2004 ) . estimating the average wavelength of heii @xmath14686 for the two nights in lmcx-2 gives [email protected] and [email protected] , respectively , suggesting a slight velocity shift . however , we must be careful with this slight shift , since we do not have a full orbital coverage each night and this could lead to a systematic off - set to the average wavelength . a better way to find out if this slight shift in velocity is real , is by examining the two heii doppler maps . they show a bright spot that appears to have moved over night , suggesting the presence of an irradiated region that shows movement on a much longer time - scale than the orbital period , such as the warped and irradiated part of the accretion disk . unfortunately , we only have two nights of data and can therefore not follow the long - term evolution of this bright spot to unambiguously claim that it moves periodically on a longer timescale . a spectroscopic campaign with a vlt - class telescope would be needed to follow the evolution of the emission lines in lmcx-2 over a full expected precession cycle ( of @xmath91 week ) and show that this bright spot in the heii doppler maps is long - lived and connected to a precessing disk . the radial velocity curve of the heii @xmath14686 emission line shows a periodic variability that we have interpreted as the orbital period . this could suggest that it also traces the primary and that we have an estimate for both orbital phase zero and @xmath23 . however , the doppler maps in fig.[hedopp ] show that the heii emission is dominated by the bright spot due to the warped accretion disk . since this spot does not have a similar phasing as the primary ( and even moves due to precession ) , we can not use the radial velocity curve to determine the orbital phasing of the primary . furthermore , if lmcx-2 harbours an inclined precessing accretion disk it is also not likely that semi - amplitude of the heii radial velocity curve traces @xmath23 . in this case the accretion disk , or at least the irradiated side that produced the bright spot in the heii doppler map , is tilted out of the orbital plane thereby changing its radial velocity . this is clear from fig.[radial ] , where it appears that either the average velocity or the semi - amplitude of the radial velocity curve has changed . only from long - term spectroscopic monitoring of lmcx-2 might it be possible to determine the @xmath23 velocity , but currently we can not constrain this value . this also means that currently we can not be certain that the average velocity that we determined corresponds to the systemic velocity @xmath24 . we have detected narrow emission lines in the bowen region that dominate the bowen doppler map . although these lines are not visible in the individual spectra , they become prominent when we create an average spectrum that is shifted into the rest - frame of these narrow lines . as fig.[average ] shows , all important bowen lines are present , and especially the niii @xmath14640 line is very strong , suggesting that this spot is real and not just a noise feature in the doppler map . these narrow lines have been detected in many other x - ray binaries thus far , such as scox-1 , x1822@xmath14371 , gx339@xmath144 , v801ara , v926sco , aqlx-1 , gx9@xmath159 and grmus ( steeghs & casares 2002 , casares et al . 2003 , hynes et al . 2003 , casares et al . 2006 , cornelisse et al . 2007a , b , barnes et al . since there are few compact regions that could produce such narrow emission lines , it was proposed that they arise on the irradiated surface of the donor star . especially in x1822@xmath14371 , but also in v801ara this connection could unambiguously be made , strengthening the claim in all other sources . furthermore , the width of these emission lines in lmcx-2 suggests that they come from a very compact region in the system , and apart from the donor star surface not many other regions in the binary could produce such narrow lines . therefore , following the other systems and given the narrowness of these emission lines we tentatively identify them as coming from the donor star of lmcx-2 , despite the fact that the absolute phasing of the system is unknown . in lmcx-2 there is another problem with identifying the compact spot in the bowen doppler map with the secondary , namely the fact that all emission in the heii doppler maps is at much lower velocities than the compact bowen spot . this would suggest that all emission in the heii map is at sub - keplerian velocities , and not related to the accretion disk . however , such behaviour is not unique to lmcx-2 . also in the heii @xmath14686 doppler maps of many other lmxbs ( such as scox-1 , x1822@xmath14371 , v801ara and v926sco amongst others ) is most , if not all , emission at sub - keplerian velocities ( steeghs & casares 2002 , casares et al . 2003 , 2006 ) . however , if the compact spot in the bowen map is produced on the donor star , lmcx-2 would be the most extreme system in showing too low disk velocities . we do note that one of the assumptions of the technique of doppler tomography is that all motion occurs in the orbital plane . if the accretion disk in lmcx-2 is inclined , as suggested above , these assumptions are no longer fulfilled . this would make the interpretation of the accretion disk structure using the doppler maps more difficult , and could be a reason why we observe these sub - keplerian velocities . for example , in ss433 the precession angle of the jets ( and most likely also the accretion disk ) is thought to be @xmath920@xmath25 ( margon 1984 ) , and if lmcx-2 has a similar precession angle ( and a relatively low inclination ) this could easily account for observed accretion disk velocities that are a factor 2 too low . again , long term spectroscopic monitoring would be needed to give more insight into the real accretion disk velocities . however , we do note that this should not impact our interpretation of the compact spot arising on the secondary star , and we remain confident that we have detected a signature of the donor star . assuming that we have detected the donor star in lmcx-2 we can use these data to constrain the system parameters . firstly , the narrow lines must arise on the irradiated surface , therefore the determined @xmath2 ( 351@xmath028 km s@xmath3 ) must be a lower limit to the center of mass velocity of the secondary ( @xmath4 ) . however , since @xmath2 must be smaller than @xmath4 this already gives a lower limit to the mass function of @xmath26=@xmath27@xmath28@xmath29/(1+@xmath30)@[email protected] @xmath7 ( at 95% confidence ) , where @xmath30 is the binary mass ratio @xmath32/@xmath27 and @xmath29 the inclination of the system . in order to further constrain the mass function the @xmath33-correction must be determined . unfortunately , this depends on @xmath30 , @xmath29 and the disk flaring angle @xmath22 , all of which are unknown for lmcx-2 ( muoz - darias et al . furthermore , the @xmath33-correction by muoz - darias et al . ( 2005 ) assumes an idealised accretion disk that is located in the orbital plane of the system , both of which are most likely not true for lmcx-2 . therefore , we must be very careful with any constraints we will derive in the rest of this section . we can set a strict lower - limit to @xmath4 by assuming that it is equal to @xmath2 , i.e that all emission is coming from the poles . furthermore , we can use the 4th order polynomials for the @xmath33-correction by muoz - darias et al . ( 2005 ) to determine @xmath4 as a function of @xmath30 in the case of @xmath22=0@xmath25 and @xmath29=40@xmath25 ( since lmcx-2 does not show dips or eclipses its inclination must be lower than @xmath970@xmath25 , and therefore the polynomials for the @xmath33-correction when @xmath29=40@xmath25 are a better approximation than in the case of @xmath29=90@xmath25 ) . we show these limits on @xmath4 in fig.[limits ] . note that these limits are still true even if the accretion disk is severely warped or out of the orbital plane . assuming that the width of the narrow emission lines is mainly due to rotational broadening we can derive a lower - limit to @xmath30 using @xmath34@xmath35@xmath29=0.462@xmath4@xmath36(1+@xmath30)@xmath37 ( wade & horne 1988 ) . from fig.[average ] we derive a _ fwhm _ of the narrow lines of [email protected] km s@xmath3 . since these emission lines are expected to arise from only part of the secondary , this value must be a strict lower - limit and the true _ fwhm _ must be higher . furthermore , our estimate still includes the effect of the intrinsic instrumental resolution of 70 km s@xmath3 . following casares et al . ( 2006 ) we accounted for this effect by broadening a strong line in our arc spectrum using a gray rotational profile without limb darkening ( since the fluorescence lines occur in optically thin conditions ) until we reached the observed _ fwhm _ ( gray 1992 ) . we found that a rotational broadening of @xmath38@xmath35@xmath29@xmath660@xmath018 km s@xmath3 reproduced our results . since this is a strict lower - limit on the true rotational broadening this will also give a lower limit on both @xmath30 and @xmath4 that is shown in fig.[limits ] . casares et al . ( 2006 ) derived an upper - limit to @xmath30 by assuming that the secondary is the largest possible zero - age main sequence star fitting in its roche - lobe . however , lmcx-2 shows the x - ray properties of a z - source ( smale et al . z - sources trace a z - like shape in their x - ray colour - colour diagrams and are thought to have more evolved secondaries ( see e.g. hasinger & van der klis 1989 ) . although the evolved nature of the secondary in z - sources is not unambiguously confirmed , it is still reasonable that the assumption of a zero - age main sequence donor is most likely not true for lmcx-2 . we therefore can not use this assumption to derive an upper - limit on the mass ratio , since the evolved donor star could be more massive than a main sequence star ( schenker & king 2002 ) . however , we can use an alternative way to set an upper - limit to @xmath30 , namely using the fact that lmcx-2 has a precessing accretion disk . one of the main criteria to produce a precessing accretion disk is to have an extreme mass ratio ( e.g. whitehurst 1988 ) . from an overview by patterson et al . ( 2005 ) of compact binaries that have a precessing accretion disk and for which the mass ratio is known , we can use the system with the largest measured @xmath30 thus far and assume that it must be smaller for lmcx-2 . this leads to a conservative upper - limit of @[email protected] , and our final limit for the system parameters shown in fig.[limits ] . all these limits constrain an area in fig.[limits ] that is still quite large , and at the moment it is still not possible to constrain any of the system parameters tightly enough in order to say anything about the component masses or the evolution of lmcx-2 . however , we can make two more assumptions , although more speculative , for lmcx-2 to further constrain its system parameters . given that lmcx-2 shows the x - ray properties of a z - source ( smale et al . 2003 ) , we can speculate that its compact object is a neutron star . in this case the maximum mass for the compact object would be @xmath93.2@xmath7 , and we could use this to set tighter upper - limits to @xmath4 than derived from the assumptions we have made thus far . in fig.[limits ] we have therefore also indicated the maximum limit where the compact object can still be a neutron star , i.e. where @xmath27@xmath35@xmath39=3.2@xmath7 , as a solid dark line . however since we can not completely rule out the possibility that the compact object in lmcx-2 is a low mass black hole , we have decided not to use this limit to constrain @xmath4 . finally , we can speculate that the semi - amplitude that we have derived from the heii @xmath14686 radial velocity curve is @xmath23 . in this case we set an upper - limit to @xmath30=@xmath23/@[email protected] . note that we have used this value for @xmath30 and @xmath4=@xmath2 to draw the roche lobe in fig.[bowdopp ] to illustrate that it encompasses the bright spot . however , this set of system parameters is just one possible solution within the allowed region in fig.[limits ] and we do not imply that it represents the true or even preferred system parameters of lmcx-2 . we note that this value of @xmath30=0.12 is similar to the mass ratios determined from other x - ray binaries that show a precessing accretion disk ( odonoghue & charles 1996 ) , suggesting that it might be close to the true mass ratio of lmcx-2 . also , if we assume that the @xmath912 day variation observed by crampton et al . ( 1990 ) is the precession period , and that the 8.2 hr photometric period is a superhump period , we can estimate the fractional period excess of the superhump over the orbital period @xmath40 . we find that @[email protected] and with the @xmath40(@xmath30)-@xmath30 relation by patterson et al . ( 2005 ) we estimate that @[email protected] . this is again close to the mass ratio derived above . although this would set tight constraints to @xmath30 , and makes a black hole nature for the compact object very unlikely , we have argued in sect.4.2 that we can not make this assumption and have therefore not plotted this in fig.[limits ] . finally , we do want to point out that , despite the large range of system parameters possible , lmcx-2 most likely harbours a neutron star that is more massive than the canonical 1.4@xmath7 . since lmcx-2 does not show eclipses or dips we can estimate an upper - limit on the inclination of 65@xmath25 ( see e.g. paczynski 1974 ) . combined with the lower limit on the mass function already gives @xmath41(@xmath27)/@xmath28@[email protected]@xmath7 . for the compact object to be 1.4@xmath7 , this only leaves @[email protected] ( and @[email protected] ) . furthermore , already for @xmath4=309 km s@xmath3 ( and @xmath29@xmath865@xmath25 ) the mass of the compact object will exceed 1.4m@xmath42 . this only leaves a very small corner in the bottom - left part of our allowed system parameters , and makes lmcx-2 another strong candidate to harbour a massive neutron star . we have detected for the first time a spectroscopic period in lmcx-2 that is close to previously published photometric period of 8.15 hrs by callanan et al . we interpret this as the orbital period , while the 12 day period detected by crampton et al . ( 1990 ) is most likely the super - orbital period due to an inclined precessing accretion disk . the signature of such a precessing accretion disk is also present in our data , mainly in the heii doppler maps , but also in the form of a nightly change in semi - amplitude and/or mean velocity of the radial velocity curve of heii @xmath14686 . in a forthcoming paper by shih et al . ( in prep . ) we will explore the consequences of such a precessing accretion disk in more detail . the main result of our spectroscopic data - set is the detection of narrow emission lines in the bowen region . following previous detections of such lines in other lmxbs , we tentatively identify these as arising from the surface of the secondary . this gives us for the first time the possibility to derive the mass function of lmcx-2 and constrain its system parameters . although they point toward a massive neutron star , the constraints on the system parameters are currently not very tight . however , this could change with a better determination of both the spectroscopic and precession period during a long term spectroscopic campaign to study the emission line kinematics . using the relation between the mass ratio @xmath30 and the fractional period excess of the spectroscopic and the photometric period @xmath40 could give strong constraints on @xmath30 and thereby the other system parameters . in particular this would be an excellent way to identify the nature of the donor star to find out if z - sources really have more evolved secondaries . this work is based on data collected at the european southern observatory paranal , chile ( obs.id . 074.d-0657(a ) ) . we acknowledge the use of the molly and doppler software packages developed by t.r . marsh . rc acknowledges financial support from a european union marie curie intra - european fellowship ( meift - ct-2005 - 024685 ) . jc acknowledges support from the spanish ministry of science and technology through project aya2002 - 03570 . ds acknowledges a smithsonian astrophysical observatory clay fellowship as well as support from nasa through its guest observer program . ds acknowledges a pparc / stfc advanced fellowship . 99 alcock , c. , allsman , r.a . , alves , d.r . , axelrod , t.s . , becker , a.c . , bennett , d.p . , charles , p.a . , cook , k.h . , drake , a.j . , freeman , k.c . , and 16 coauthors 2000 , mnras ; 316 , 729 barnes , a.d . , casares , j. , cornelisse , r. , charles , p.a . , steeghs , d. , hynes , r.i . , obrien , k. 2007 , mnras , accepted for publication [ astro - ph/0707.0624 ] casares j. , steeghs d. , hynes r.i , charles p.a . , obrien k. , 2003 , apj , 590 , 1041 casares , j. , cornelisse , r. , steeghs , d. , charles , p.a . , hynes , r.i . , obrien , k.o . , & strohmayer , t.e . 2006 , mnras , 373 , 1235 cornelisse , r. , casares , j. , steeghs , d . , barnes , a.d . , charles , p.a . , hynes , r.i . , obrien , k. 2007a , mnras , 375 , 1463 cornelisse , r. , steeghs , d. , casares , j. , charles , p.a . , barnes , a.d . , hynes , r.i . , obrien , k. 2007b , mnras , accepted for publication [ astro - ph/0707.0449 ] callanan , p.j . , charles , p.a . , van paradijs , j. , van der klis , m. , pedersen , h. , harlaftis , e.t . 1990 , a&a , 240 , 346 crampton , d. , hutchings , j.b . , cowley , a.p . , schmidtke , p.c . , thompson , i.b . 1990 , apj , 355 , 496 hynes , r.i . , obrien , k. 2007 , mnras , 375 , 1463 hasinger , g. , & van der klis , m. 1989 , a&a , 225 , 79 gray , d.f . , 1992 , `` the observation and analysis of stellar photospheres '' , cup , 20 horne k. , 1986 , pasp , 98 , 609 hynes r.i . , steeghs d. , casares j. , charles p.a . , obrien k. , 2003 , apj , 583 , l95 katz , j.i . 1973 , nature , 246 , 87 margon , b. 1984 , ara&a , 22 , 507 marsh t.r . , horne k. , 1988 , mnras 235 , 269 mcclintock , j.e . , canizares , c.r . , & tarter , c.b . 1975 , apj , 198 , 641 motch , c. , chevalier , c. , ilovaisky , s.a . , pakull , m.w . 1985 , space sci . reviews , vol . 40 , p. 239 motch , c. , pakull , m.w . 1989 , a&a , 214 , l1 muoz - darias , t. , casares , j. , & martinez - pais , i.g . 2005 , apj , 635 , 502 odonoghue , d. , charles , p.a . 1996 , mnras ; 282 , 191 paczynski , b. 1974 , a&a , 34 , 161 pakull , m. 1978 , iau circ . , 3313 patterson , j. , kemp , j. , harvey , d.a . , fried , r.e . , rea , r. , monard , b. , cook , l.m . , skillman , d.r . , vanmunster , t. , bolt , g. , et al . 2005 , pasp ; 117 , 1204 scargle , j.d . 1982 , apj , 263 , 835 schenker , k. , king , a.r . 2002 , asp conf.ser . 261 , `` the physics of cataclysmic variables and related objects '' , 261 , 242 steeghs d. , casares j. , 2002 , apj , 568 , 273 smale , a.p . , kuulkers , e. 2000 , apj , 528 , 702 smale , a.p . , homan , j. , kuulkers , e. 2003 , apj , 590 , 1035 torres , m.a.p . , callanan , p.j . , garcia , m.r . , zhao , p. , laylock , s. , kong , a.k.h . 2004 , apj , 612 , 1026 uemura , m. , kato , t. , matsumoto , k. , honkawa , m. , cook , l. , martin , b. , masi , g. , oksanen , a. , moilanen , m. , novak , r. , sano , y. , ueda , y. 2000 , iauc , 7418 wade r.a . , horne k. , 1988 , apj , 324 , 411 whitehurst , r. 1988 , mnras , 232 , 35 zurita , c. , casares , j. , shahbaz , t. , wagner , r.m . , foltz , c.b . , rodriguez - gil , p. , hynes , r.i . , charles , p.a . , ryan , e. , schwarz , g. , & starrfield , s.g . 2002 , mnras , 333 , 791 | two nights of phase - resolved medium resolution vlt spectroscopy of the extra - galactic low mass x - ray binary lmcx-2 have revealed a [email protected] day spectroscopic period in the radial velocity curve of the heii @xmath14686 emission line that we interpret as the orbital period .
however , similar to previous findings , this radial velocity curve shows a longer term variation that is most likely due to the presence of a precessing accretion disk in lmcx-2 .
this is strengthened by heii @xmath14686 doppler maps that show a bright spot that is moving from night to night .
furthermore , we detect narrow emission lines in the bowen region of lmcx-2,with a velocity of @xmath2=351@xmath028 km s@xmath3 , that we tentatively interpret as coming from the irradiated side of the donor star .
since @xmath2 must be smaller than @xmath4 , this leads to the first upper - limit on the mass function of lmcx-2 of @[email protected]@xmath7 ( 95% confidence ) , and the first constraints on its system parameters .
[ firstpage ] accretion , accretion disks stars : individual ( lmcx-2 ) x - rays : binaries . |
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there are certain scientific problems that provide deep connections between many different scientific fields . the study of the low - energy states of classical interacting many - particle systems is an exemplar of a class of such problems because of its manifest importance in physics , materials science , communication theory , cryptography , mathematics and computer science . such many - particle systems have been used with great success to model liquids , glasses and crystals when quantum effects are negligible @xcite . the total potential energy @xmath3 of @xmath4 identical particles with positions @xmath5 in some large volume in @xmath0-dimensional euclidean space @xmath1 can be resolved into separate one - body , two - body , @xmath6 , @xmath4-body contributions : @xmath7 where @xmath8 represents the intrinsic @xmath9-body interaction in excess to the interaction energy for @xmath10 particles . to make the statistical - mechanical problem more tractable , the exact many - body potential ( [ full ] ) is usually replaced by a mathematically simpler form . for example , in the absence of an external field ( i.e. , @xmath11 ) , often one assumes pairwise additivity , i.e. , @xmath12 pairwise additivity is exact for hard - sphere systems and frequently has served to approximate accurately the interactions in simple liquids , such as the well - known lennard - jones pair potential @xcite , and in more complex systems where the pair potential @xmath13 in ( [ pair ] ) can be regarded to be an _ effective _ pair interaction @xcite . an outstanding problem in classical statistical mechanics is the determination of the _ ground states _ of @xmath3 , which are those configurations that globally minimize @xmath3 and hence are the states that exist at absolute zero temperature . while classical ground states are readily produced by slowly freezing liquids in experiments and computer simulations , our theoretical understanding of them is far from complete @xcite . virtually all theoretical / computational ground - state studies of many - particle systems have been conducted for pairwise additive potentials @xcite . often the ground states of short - range pairwise interactions are crystal structures in low dimensions @xcite , but long - range interactions exist that can suppress any kind of symmetry leading to disordered ground states in low dimensions @xcite . moreover , in sufficiently high dimensions , it has been suggested that even short - ranged pairwise interactions possess disordered ground states @xcite . ground states of purely repulsive pair interactions have profound connections not only to low - temperature states of matter but to problems in pure mathematics , including discrete geometry and number theory @xcite , information theory , and computer science . as will be explained further below , it is known that the sphere packing problem and the number variance problem ( closely related to an optimization problem in number theory ) can be posed as energy minimizations associated with an infinite number of point particles in @xmath0-dimensional euclidean space @xmath1 interacting via certain repulsive pair potentials . both of these problems can be interpreted to be optimization problems involving _ point processes _ , which can then be recast as energy minimizations . a point process in @xmath1 is a distribution of an an infinite number of points in @xmath1 at number density @xmath14 ( number of points per unit volume ) with configuration @xmath15 ; see ref . @xcite for a precise mathematical definition . a packing of congruent nonoverlapping spheres is a special point process in which there is a minimal pair separation distance , equal to the sphere diameter . the sphere packing problem seeks to determine the densest arrangement(s ) of congruent , nonoverlapping @xmath0-dimensional spheres in euclidean space @xmath1 @xcite . although it is simple to state , it is a notoriously difficult problem to solve rigorously . indeed , kepler s four - century - old conjecture , which states that the face - centered - cubic lattice in @xmath16 is maximally dense , was only recently proved @xcite . for @xmath2 , the packing problem remains unsolved @xcite . it is well known that the sphere packing problem can be posed as an energy minimization problem involving pairwise interactions between points in @xmath1 ( e.g. , inverse power - law functions in which the exponent tends to infinity ) ; see ref . @xcite and references therein . problems concerning the properties and quantification of density fluctuations in many - particle systems continue to provide many theoretical challenges . of particular interest are density fluctuations that occur on some local length scale @xcite . it has been shown that the minimal number variance associated with points ( e.g. , centroids of atomic or molecular systems ) contained within some window " can also be formulated as a ground - state problem involving bounded repulsive pair interactions with compact support @xcite . for spherical windows in the large - radius limit , the best known solutions in @xmath1 are usually point configurations that are duals " ( in the sense discussed later in the paper ) to the best known sphere packings in @xmath1 . the focus of this paper is on two other optimization problems involving point processes in @xmath1 : the _ covering _ and _ quantizer _ problems . roughly speaking , the covering problem asks for the point configuration that minimizes the radius of overlapping spheres circumscribed around each of the points required to cover @xmath1 . the covering problem has applications in wireless communication network layouts @xcite , the search of high - dimensional data parameter spaces ( e.g. , search templates for gravitational waves ) @xcite , and stereotactic radiation therapy @xcite . the quantizer problem is concerned with finding the point configuration in @xmath1 that minimizes a distance error " associated with a randomly placed point and the nearest point of the point process . it has applications in computer science ( e.g. , data compression ) @xcite , digital communications @xcite , coding and cryptography @xcite , and optimal meshing of space for numerical applications ( e.g. , quadrature and discretizing partial differential equations ) @xcite . heretofore , the covering and quantizers problems were not known to correspond to any ground - state problems . we reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in @xmath1 that generally involve single - body , two - body , three - body , and higher - body interactions . this is done by linking the covering and quantizer problems to certain optimization problems involving the void " nearest - neighbor functions that arise in the theory of random media and statistical mechanics @xcite . these reformulations , which again exemplifies the deep interplay between geometry and physics , enable one to employ theoretical and numerical optimization techniques to solve these energy minimization problems . we find that disordered _ saturated _ sphere packings ( roughly , packings in which no space exists to add an additional sphere ) provide relatively thin ( i.e. , economical ) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions . in the case of the quantizer problem , we derive improved upper bounds on the quantizer error that utilize sphere - packing solutions . these improved bounds are generally substantially sharper than an existing upper bound in low to moderately large dimensions . we also demonstrate that disordered saturated sphere packings yield relatively good quantizers . our reformulation helps to explain why the known solutions of quantizer and covering problems are identical in the first three space dimensions and why they can be different for @xmath2 . in the first three space dimensions , the best known solutions of the sphere packing and number variance problems are directly related to those of the covering and quantizer problems , but such relationships may or may not exist for @xmath2 , depending on the peculiarities of the dimensions involved . we begin by summarizing basic definitions and concepts in sec . [ def ] . because of the connections between the sphere - packing , number - variance , covering and quantizer problems , in sec . [ problems ] , we formally define each of these problems , summarize key developments , and compare the best known solutions for each of them in selected dimensions . this includes calculations obtained for the best known number - variance solutions for @xmath17 and 24 . we then define in sec . [ near ] the void nearest - neighbor functions and represent them in terms of series involving certain integrals over the @xmath9-particle correlation functions , which statistically characterize an ensemble of interacting points . the special case of a single realization of the point distribution follows from this ensemble formulation , which reveals that quantizer and covering problems can be expressed as ground - state solutions of many - body interactions of the general form ( [ full ] ) . section [ reform ] specifically gives these ground - state reformulations and shows how some known solutions in low dimensions can be explicitly recovered using the void nearest - neighbor functions . in sec . [ results - cover ] , we show that _ disordered _ saturated sphere packings provide relatively thin coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions . in sec . [ results - quant ] , we derive improved upper bounds on the quantizer error that utilize sphere - packing solutions . we also show that _ disordered _ saturated sphere packings yield relatively good quantizers . finally , in sec . [ conc ] , we make concluding remarks and comment on the application of the quantizer problem to the search for gravitational waves . for a statistically homogeneous point process in @xmath1 at number density @xmath14 ( number of points per unit volume ) , the quantity @xmath18 is proportional to the probability density for simultaneously finding @xmath9 sphere centers at locations @xmath19 in @xmath1 @xcite . with this convention , each _ @xmath9-particle correlation function _ @xmath20 approaches unity when all of the points become widely separated from one another . statistical homogeneity implies that @xmath20 is translationally invariant and therefore only depends on the relative displacements of the positions with respect to some arbitrarily chosen origin of the system , _ i.e. _ , @xmath21 where @xmath22 . as we will see , statistically homogeneous point processes include as special cases periodic point distributions . the _ pair correlation _ function @xmath23 is a particularly important quantity . if the point process is also rotationally invariant ( statistically isotropic ) , then @xmath24 depends on the radial distance @xmath25 only , _ i.e. _ , @xmath26 . thus , it follows that the expected number of points @xmath27 found in a sphere of radius @xmath28 from a randomly chosen point of the point process , called the _ cumulative coordination function _ , is given by @xmath29 where @xmath30 is the surface area of a @xmath0-dimensional sphere of radius @xmath31 . a _ lattice _ @xmath32 in @xmath1 is a subgroup consisting of the integer linear combinations of vectors that constitute a basis for @xmath1 and thus represents a special subset of point processes . in a lattice @xmath32 , the space @xmath1 can be geometrically divided into identical regions @xmath33 called _ fundamental cells _ , each of which contains the just one point specified by the _ lattice vector _ @xmath34 where @xmath35 are the basis vectors for the fundamental cell and @xmath36 spans all the integers for @xmath37 . we denote by @xmath38 the volume of the fundamental cell . in the physical sciences , a lattice is equivalent to a bravais lattice . unless otherwise stated , we will use the term lattice . every lattice has a dual ( or reciprocal ) lattice @xmath39 in which the sites of the lattice are specified by the dual ( reciprocal ) lattice vector @xmath40 , where @xmath41 . the dual fundamental cell @xmath42 has volume @xmath43 . this implies that the number density @xmath14 of @xmath32 is related to the number density @xmath44 of the dual lattice @xmath39 via the expression @xmath45 . a _ periodic _ point process is a more general notion than a lattice because it is is obtained by placing a fixed configuration of @xmath4 points ( where @xmath46 ) within one fundamental cell of a lattice @xmath32 , which is then periodically replicated . thus , the point process is still periodic under translations by @xmath32 , but the @xmath4 points can occur anywhere in the chosen fundamental cell . common @xmath0-dimensional lattices include the _ hypercubic _ @xmath47 , _ checkerboard _ @xmath48 and _ root _ @xmath49 lattices , defined , respectively , by @xmath50 @xmath51 @xmath52 where @xmath53 is the set of integers ( @xmath54 ) and @xmath55 denote the components of a lattice vector of either @xmath47 or @xmath48 and @xmath56 denote a lattice vector of @xmath49 . the @xmath0-dimensional lattices @xmath57 , @xmath58 and @xmath59 are the corresponding dual lattices ; see ref . @xcite for definitions . the dual lattice @xmath57 is also a hypercubic lattice ( even if the lattice spacing is @xmath60 times the lattice spacing of @xmath47 ) and hence we say that the hypercubic lattice @xmath47 is equivalent ( similar ) to its dual lattice @xmath61 , i.e. , @xmath62 . following conway and sloane @xcite , we say that two lattice are _ equivalent _ or _ similar _ if one becomes identical to the other by possibly a rotation , reflection and change of scale , for which we use the symbol @xmath63 . in fact , the hypercubic lattice @xmath47 is characterized by the stronger property of _ self - duality_. a _ self - dual _ lattice is one with an _ identical _ dual lattice at density @xmath64 , i.e. , without any rotation , reflection , or change of scale @xcite . the @xmath49 and @xmath48 lattices can be regarded to be @xmath0-dimensional generalizations of the face - centered - cubic ( fcc ) lattice because this three - dimensional lattice is defined by @xmath65 . in one dimension , @xmath66 ( equality meaning self - duality ) are identical to the integer lattice @xmath67 . in two dimensions , @xmath68 defines the triangular lattice . in three dimensions , @xmath69 defines the body - centered - cubic ( bcc ) lattice . the @xmath0-dimensional laminated lattice @xmath70 @xcite is of special interest . in dimensions 8 and 24 , @xmath71 , where @xmath72 is the _ self - dual _ root lattice , and @xmath73 is the self - dual leech lattice , are remarkably symmetric and believed to be the densest sphere packings in those dimensions @xcite . thus , @xmath74 and @xmath75 . the laminated lattice @xmath76 , called the barnes - wall lattice , and the coxeter - todd lattice @xmath77 are thought to be the densest lattice packings in sixteen and twelve dimensions , respectively . note that for a single periodic point configuration at number density @xmath14 , the radial pair correlation function can be written as @xmath78 where @xmath79 is the coordination number at radial distance @xmath80 ( number of points that are exactly at a distance @xmath81 from a point of the point process ) such that @xmath82 and @xmath83 is a radial dirac delta function . for cases in which each point is equivalent to any other , which includes all lattices and some periodic point processes , the coordination numbers @xmath79 are integers . for point processes for which the point are generally inequivalent , @xmath79 should be interpreted as the _ expected _ coordination number and hence will generally be a non - integer . substitution of ( [ period ] ) into ( [ cum ] ) gives the coordination function for such a periodic configuration as @xmath84 where @xmath85 is the smallest integer for which @xmath86 . consider any discrete set of points with position vectors @xmath87 in @xmath1 . associated with each point @xmath88 is its _ voronoi cell _ , @xmath89 , which is defined to be the region of space nearer to the point at @xmath90 than to any other point @xmath91 in the set , i.e. , @xmath92 the voronoi cells are convex polyhedra whose interiors are disjoint , but share common faces , and therefore the union of all of the polyhedra is the whole of @xmath1 . this partition of space is called the _ voronoi _ tessellation . the vertices of the voronoi polyhedra are the points whose distance from the points @xmath93 is a local maximum . while the voronoi polyhedra of a lattice are congruent to one another , the voronoi polyhedra of a non - bravais lattice are not identical to one another . a hole in a lattice is a point in @xmath1 whose distance to the nearest lattice point is a local maximum . a _ deep _ hole is one whose distance to a lattice point is a global maximum . the distance @xmath94 to the deepest hole of a lattice is the _ covering radius _ and is equal to the _ circumradius _ of the associated voronoi cell ( the radius of the smallest circumscribed sphere ) . in the case of the @xmath0-dimensional simple cubic ( hypercubic ) lattice @xmath47 , the voronoi cell is a hypercube and there is only one type of hole with covering radius @xmath95 , assuming unit number density @xmath96 . figure [ reg3d - vor ] shows the voronoi cell in the case @xmath97 as well as the corresponding voronoi cells for the three - dimensional body - centered cubic and face - centered cubic lattices . note that the truncated octahedron is the most spherically symmetric of the three voronoi cells shown in fig . [ reg3d - vor ] @xcite . a sphere packing @xmath98 in @xmath0-dimensional euclidean space @xmath1 is a collection of @xmath0-dimensional nonoverlapping congruent spheres . the _ packing density _ or , simply , density @xmath99 of a sphere packing is the fraction of space @xmath1 covered by the spheres . for spheres of diameter @xmath100 and number density @xmath14 , the density is given by @xmath101 where @xmath102 is the volume of a @xmath0-dimensional sphere of radius @xmath28 . a packing is _ saturated _ if there is no space available to add another sphere without overlapping the existing particles . we denote the packing density of such a packing by @xmath103 . a _ lattice packing _ @xmath104 is one in which the centers of nonoverlapping spheres are located at the points of @xmath32 . thus , the density of a lattice packing @xmath105 consisting of spheres of diameter @xmath100 is given by @xmath106 where @xmath38 is the volume of a fundamental cell . periodic _ packing of congruent spheres is obtained by placing a fixed configuration of @xmath4 sphere centers ( where @xmath46 ) within one fundamental cell of a lattice @xmath32 , which is then periodically replicated without overlaps . the packing density of a periodic packing is given by @xmath107 where @xmath108 is the number density . the sphere packing problem seeks to answer the following question : among all packings of congruent spheres , what is the maximal packing density @xmath109 , i.e. , largest fraction of @xmath1 covered by the spheres , and what are the corresponding arrangements of the spheres @xcite ? more precisely , the maximal density is defined by @xmath110 where the supremum is taken over all packings in @xmath1 . the sphere packing problem is of great fundamental and practical interest , and arises in a variety of contexts , including classical ground states of matter in low dimensions @xcite , the famous kepler conjecture for @xmath97 @xcite , error - correcting codes @xcite and spherical codes @xcite . the optimal solutions are known only for the first three space dimensions @xcite . for @xmath111 , the densest known packings are bravais lattice packings @xcite . for example , the checkerboard " lattice @xmath48 , which is the @xmath0-dimensional generalization of the fcc lattice ( densest packing in @xmath16 ) , is believed to be optimal in @xmath112 and @xmath113 . the remarkably symmetric self - dual @xmath72 and leech lattices in @xmath114 and @xmath115 , respectively , are most likely the densest packings in these dimensions @xcite . table [ pack ] summarizes the densest known packings in selected dimensions . & packing & packing density , @xmath116 + 1 & @xmath117 & 1 + 2 & @xmath118 & @xmath119 + 3 & @xmath120 & @xmath121 + 4 & @xmath122 & @xmath123 + 5 & @xmath124 & @xmath125 + 6 & @xmath126 & @xmath127 + 7 & @xmath128 & @xmath129 + 8 & @xmath130 & @xmath131 + 9 & @xmath132 & @xmath133 + 10 & @xmath134 & @xmath135 + 12 & @xmath136 & @xmath137 + 16 & @xmath76 & @xmath138 + 24 & @xmath75 & @xmath139 + for large @xmath0 , the best that one can do theoretically is to devise upper and lower bounds on @xmath109 @xcite . for example , minkowski @xcite proved that the maximal density @xmath140 among all bravais lattice packings for @xmath141 satisfies the lower bound @xmath142 where @xmath143 is the riemann zeta function . it is seen that for large values of @xmath0 , the asymptotic behavior of the _ nonconstructive _ minkowski lower bound is controlled by @xmath144 . note that the density of a _ saturated _ packing of congruent spheres in @xmath1 for all @xmath0 satisfies @xmath145 which has the same dominant exponential term as ( [ mink ] ) . this is a rather weak lower bound on the density of saturated packings because there exists a disordered but _ unsaturated packing construction _ in @xmath1 , known as the ghost " rsa packing @xcite , that achieves the density @xmath144 in any dimension . we will employ these results in sec . [ improve ] . it is also known that there are saturated packings in @xmath1 with densities that exceed the scaling @xmath144 @xcite , as we will discuss in sec . [ cov ] . in the large - dimensional limit , kabatiansky and levenshtein @xcite showed that the maximal density is bounded from above according to the asymptotic upper bound @xmath146 we will employ sphere - packing solutions to obtain heretofore unattained results for both the covering and quantizer problems . in particular , we obtain coverings and quantizers utilizing disordered saturated packings in secs . [ results - cover ] and [ sat - quant ] , respectively . in sec . [ results - quant ] , we use the densest lattice packings to derive improved upper bounds on the quantizer error . we denote by @xmath147 the variance in the number of points @xmath148 contained within a window @xmath149 . the number variance @xmath147 for a specific choice of @xmath150 is necessarily a positive number and is generally related to the _ total correlation function _ @xmath151 for a translationally invariant point process @xcite , where @xmath24 is the pair correlation function defined in sec . [ def ] . in the special case of a spherical window of radius @xmath28 in @xmath1 , the number variance is explicitly given by @xmath152 , \label{variance}\end{aligned}\ ] ] @xmath153 is the dimensionless volume common to two spherical windows of radius @xmath28 ( in units of the volume of a spherical window of radius @xmath28 , @xmath154 ) whose centers are separated by a distance @xmath31 . we will call @xmath153 the _ scaled intersection volume _ , which will play an important role in this paper . the scaled intersection volume has the support @xmath155 $ ] , the range @xmath156 $ ] , and the following alternative integral representation @xcite : @xmath157 } \sin^d(\theta ) \ , d\theta , \label{alpha}\end{aligned}\ ] ] where @xmath158 is the @xmath0-dimensional constant given by @xmath159}. \label{c}\end{aligned}\ ] ] torquato and stillinger @xcite found the following series representation of the scaled intersection volume @xmath153 for @xmath160 and for any @xmath0 : @xmath161 } x^{2n-1 } , \label{series}\end{aligned}\ ] ] where @xmath162 . for even dimensions , relation ( [ series ] ) is an infinite series because it involves transcendental functions , but for odd dimensions , the series truncates such that @xmath153 is a univariate polynomial of degree @xmath0 . for example , in two and three dimensions , respectively , the scaled intersection volumes are given by @xmath163\theta(2r - r ) \quad ( d=2),\ ] ] @xmath164 \theta(2r - r ) \quad ( d=3),\ ] ] where @xmath165 is the heaviside step function . figure [ intersection ] provides plots of @xmath153 as a function of @xmath31 for the first five space dimensions . for any dimension , @xmath153 is a monotonically decreasing function of @xmath31 . at a fixed value of @xmath31 in the interval @xmath166 , @xmath153 is a monotonically decreasing function of the dimension @xmath0 . for spherical windows of radius @xmath28 as a function of @xmath31 for the first five space dimensions . the uppermost curve is for @xmath167 and lowermost curve is for @xmath168 . , width=336 ] for large @xmath28 , it has been proved that @xmath169 can not grow more slowly than @xmath170 , where @xmath171 is a positive constant @xcite . we note that point processes ( translationally invariant or not ) for which @xmath169 grows more slowly than the window volume ( i.e. , as @xmath172 ) for large @xmath28 are examples of _ hyperuniform _ ( or superhomogeneous ) point patterns @xcite . for hyperuniform point processes in which the number variance grows like the surface area of the window , one has @xmath173 \qquad r \rightarrow \infty,\ ] ] where @xmath174 is a dimensionless constant with @xmath158 defined by ( [ c ] ) , @xmath175 is a dimensionless density , and @xmath176 represents some microscopic " length scale , such as the minimum pair separation distance in a packing or the mean nearest - neighbor distance . this class of hyperuniform point processes includes all periodic point patterns , quasicrystals that possess bragg peaks , and disordered hyperuniform point patterns in which the pair correlation functions decay exponentially fast to unity @xcite . it has been shown that finding the point process that minimizes the number variance @xmath169 is equivalent to finding the ground state of a certain repulsive pair potential with compact support @xcite . specifically , by invoking a volume - average interpretation of the number variance problem valid for a single realization of a point process , torquato and stillinger found @xcite : @xmath177 where @xmath178 . \label{bn}\ ] ] the asymptotic coefficient @xmath179 defined by ( [ b ] ) for a hyperuniform point pattern is then related to @xmath180 by the expression @xmath181 these results imply that the asymptotic coefficient @xmath179 obtained in ( [ b ] ) involves an average over small - scale fluctuations in the number variance with length scale on the order of the mean separation between points @xcite . in the special case of a ( bravais ) lattice @xmath32 , one can express the rescaled surface - area coefficient as follows : @xmath182^{1 - 1/d}}{v_f^{1 + 1/d } } \sum_{\mathbf{q}\neq\mathbf{0 } } \frac{1}{\vert\mathbf{q}\vert^{d+1 } } , \label{num}\end{aligned}\ ] ] where we recall that @xmath38 is the volume of the fundamental cell of the lattice @xmath32 and @xmath183 represents a lattice vector in the dual ( or reciprocal ) lattice @xmath39 . the rescaled coefficient @xmath184 renders the result independent of the length scale in the lattice @xcite . finding the lattice that minimizes @xmath179 is directly related to an outstanding problem in number theory , namely , finding the minima of the epstein zeta function @xmath185 @xcite defined by @xmath186 where @xmath187 is a lattice vector of the lattice @xmath32 . note that the _ dual _ of the lattice that minimizes the epstein zeta function at @xmath188 among all lattices will minimize the scaled asymptotic number - variance coefficient ( [ num ] ) among lattices @xcite . certain duality relations have been derived that establish rigorous upper bounds on the energies of such ground states and help to identify energy - minimizing lattices @xcite . because @xmath185 is globally minimized for @xmath167 by the integer lattice @xcite and is minimized for @xmath189 among all lattices by the triangular lattice @xcite , it has been conjectured that the epstein zeta function for @xmath190 is minimized among lattices by the maximally dense lattice packing @xcite . sarnak and str " ombergsson @xcite have proved that the conjecture can not be generally true , but for @xmath191 , @xmath192 and @xmath193 , the densest lattice packing is a strict local minimum . since as @xmath194 , the minimizer of epstein zeta function is the densest sphere packing in @xmath1 for any @xmath0 , it is likely that in the high - dimensional limit the minimizers of this function are non - lattices , namely , disordered sphere packings @xcite . & structure & scaled coefficient , @xmath184 + 1 & @xmath195 & 0.083333 + 2 & @xmath118 & 0.12709 + 3 & @xmath196 & 0.15560 + 4 & @xmath197 & 0.17488 + 5 & @xmath198 & 0.19069 + 6 & @xmath199 & 0.20221 + 7 & @xmath200 & 0.21037 + 8 & @xmath201 & 0.21746 + 12 & @xmath202 & 0.24344 + 16 & @xmath203 & 0.25629 + 24 & @xmath204 & 0.26775 + in table [ var ] , we tabulate the best known solutions to the asymptotic number variance problem in selected dimensions . values for the first three space dimensions were given in ref . @xcite and those for @xmath205 were provided in ref . the values reported for @xmath206 and 24 were ascertained using efficient algorithms based on alternative number - theoretic representations of the epstein zeta function @xmath185 @xcite for the corresponding densest known lattice packings for @xmath188 and then using the duality relations connecting it to the asymptotic surface - area coefficient for the number variance . appendix [ num - ep ] provides details for these computations . surround each of the points of a point process @xmath207 in @xmath1 by congruent overlapping spheres of radius @xmath28 such that the spheres cover the space . the _ covering density _ @xmath208 is defined as follows : @xmath209 where @xmath154 is given by ( [ v1 ] ) . the _ covering problem _ asks for the arrangement of points with the least density @xmath208 . we define the covering radius @xmath94 for any configuration of points in @xmath1 to be the minimal radius of the overlapping spheres to cover @xmath1 . figure [ sq - tri ] shows two examples of coverings in the plane . the covering density associated with @xmath59 at unit number density @xmath96 is known exactly for any dimension @xmath0 @xcite : @xmath210^{d/2}.\ ] ] for the hypercubic lattice @xmath47 at @xmath96 , @xmath211 thus the ratio of the covering density for @xmath59 to that of @xmath47 is given by @xmath212^{d/2}.\ ] ] for large @xmath0 , this ratio becomes @xmath213 and thus we see that @xmath59 provides exponentially thinner coverings than that of @xmath47 in the large-@xmath0 limit . we note that in this asymptotic limit , @xmath214 until recently , @xmath59 was the best known lattice covering in all dimensions @xmath215 . however , for most dimensions in the range @xmath216 , schrmann and vallentin @xcite have discovered other lattice coverings that are slightly thinner than those for @xmath59 . table [ coverings ] provides the best known solutions to the covering problem in selected dimensions . & covering & covering density , @xmath208 + 1 & @xmath217 & 1 + 2 & @xmath118 & 1.2092 + 3 & @xmath196 & 1.4635 + 4 & @xmath218 & 1.7655 + 5 & @xmath219 & 2.1243 + 6 & @xmath220 & 2.4648 + 7 & @xmath221 & 2.9000 + 8 & @xmath222 & 3.1422 + 9 & @xmath223 & 4.3401 + 10 & @xmath224 & 5.2517 + 12 & @xmath225 & 7.5101 + 16 & @xmath226 & 15.3109 + 17 & @xmath227 & 18.2878 + 18 & @xmath228 & 21.8409 + 24 & @xmath75 & 7.9035 + until the present work , there were no known explicit non - lattice constructions possessing covering densities smaller than those of the best lattice coverings in any dimension @xmath0 @xcite . in sec . [ results - quant ] , we provide evidence that certain disordered point patterns give thinner coverings than the best known lattice coverings beginning at @xmath229 . there is a fundamental difference between coverings associated with point patterns that have identical voronoi cells ( i.e. , lattices and periodic point patterns in which each point is equivalent ) and those point processes whose voronoi cells are generally different ( e.g. , irregular point processes ) . this salient point is illustrated in fig . [ tri - rsa - cov ] and explained in the corresponding caption . rogers showed that ( possibly nonlattice ) coverings exist with @xmath230 for @xmath141 . this is a nonconstructive upper bound . this upper bound provides a substantially thinner covering density than that of the @xmath59 lattice in the large-@xmath0 limit [ cf . ( [ a_n^ * ] ) ] , but it is not known whether this bound becomes sharp in the large-@xmath0 limit . the best lower bound on the covering density is given by @xmath231 to define @xmath232 , let @xmath233 be a regular simplex with edge length equal to two . spheres of radius @xmath234 centered at the vertices of @xmath233 just cover @xmath233 . the quantity @xmath232 is the ratio of the sums of the intersections of these spheres with @xmath233 to the volume of @xmath233 . thus , in the large-@xmath0 limit , @xmath235 . is quantized ( rounded - off " ) to the nearest point @xmath90 . left panel : triangular lattice ( best quantizer in @xmath236 @xcite ) . right panel : irregular point process.,width=211 ] consider a point process @xmath237 in @xmath1 with configuration @xmath238 . a @xmath0-dimensional _ quantizer _ is device that takes as an input a point at position @xmath239 in @xmath1 generated from some probability density function @xmath240 and outputs the nearest point @xmath90 of the point process to @xmath241 . equivalently , if the input @xmath241 belongs to the voronoi cell @xmath89 , the output is @xmath90 ( see fig . [ tri - rsa ] ) for simplicity , we assume that @xmath241 is uniformly distributed over a large ball in @xmath1 containing the @xmath4 points of the point process . one attempts to choose the configuration @xmath238 of the point process to minimize the mean squared error , i.e. , the expected value of @xmath242 . specifically , the quantizer problem is to choose the @xmath4-point configuration so as to minimize the _ scaled dimensionless error _ ( sometimes called the _ distortion _ ) @xcite @xmath243 where @xmath244 is a dimensionless error , @xmath245\ ] ] is the expected volume of a voronoi cell , and @xmath246 is the volume of the @xmath247th voronoi cell . the scaling factor @xmath248 is included to compare the second moments appropriately across dimensions . we will denote the minimal scaled dimensionless error by @xmath249 . note that since there is one point per voronoi cell , the number density @xmath14 is set equal to unity . if all of the voronoi cells are congruent ( as they are in the case of all lattices and some periodic point processes ) , we have the simpler expression @xmath250 where the centroid of the voronoi cell @xmath251 is the origin of the coordinate system . lattice quantizer problem _ is to find the lattice for which @xmath252 , given by ( [ gamma2 ] ) , is minimum . thus , @xmath252 can be interpreted as the scaled , dimensionless _ second moment of inertia _ of the voronoi cell . & quantizer & scaled error , @xmath252 + 1 & @xmath217 & 0.083333 + 2 & @xmath118 & 0.080188 + 3 & @xmath196 & 0.078543 + 4 & @xmath197 & 0.076603 + 5 & @xmath253 & 0.075625 + 6 & @xmath199 & 0.074244 + 7 & @xmath254 & 0.073116 + 8 & @xmath255 & 0.071682 + 9 & @xmath256 & 0.071626 + 10 & @xmath257 & 0.070814 + 12 & @xmath202 & 0.070100 + 16 & @xmath203 & 0.068299 + 24 & @xmath258 & 0.065771 + the best known quantizers in any dimension @xmath0 are usually lattices that are the duals of the densest known packings ( see the discussion in sec . [ sphere ] and ref . @xcite ) , except in dimensions 9 and 10 , where the best solutions are still lattices , but are not duals of the densest lattice packings in those dimensions . although the best known solutions of the quantizer and covering problems are the same in the first three space dimensions , they are generally different for @xmath2 @xcite . zador @xcite has derived upper and lower bounds on @xmath249 . conway and sloane @xcite have obtained conjectural lower bounds on @xmath249 . we defer the discussion of these bounds to sec . [ results - quant ] , where we derive sharper upper bounds on @xmath249 , among other results . table [ quant ] provides the best known solutions to the quantizer problem in selected dimensions . table [ comparison ] lists the best known solutions of the quantizer , covering and number - variance problems and sphere - packing problems in @xmath1 for selected @xmath0 . it is seen that in the first two space dimensions , the best known solutions for each of these four problems are identical to one another . for @xmath97 , the densest sphere packing is the @xmath259 or , equivalently , @xmath260 lattice , which is the dual lattice associated with the best known solutions to the quantizer , covering and number - variance problems , which is the @xmath261 lattice . thus , for the first three space dimensions , the best known solutions for each of the four problems are lattices , and either they are identical to one another or are duals of one another . however , such relationships may or may not exist for @xmath2 , depending on the peculiarities of the dimensions involved . & quantizer & covering & variance & packing + 1 & @xmath195 & @xmath195 & @xmath217 & @xmath217 + 2 & @xmath118 & @xmath118 & @xmath118 & @xmath118 + 3 & @xmath196 & @xmath196 & @xmath196 & @xmath120 + 4 & @xmath197 & @xmath218 & @xmath197 & @xmath197 + 5 & @xmath262 & @xmath219 & @xmath198 & @xmath124 + 6 & @xmath199 & @xmath220 & @xmath199 & @xmath126 + 7 & @xmath263 & @xmath221 & @xmath264 & @xmath128 + 8 & @xmath72 & @xmath222 & @xmath72 & @xmath72 + 9 & @xmath256 & @xmath223 & @xmath265 & @xmath132 + 10 & @xmath257 & @xmath224 & @xmath266 & @xmath134 + 12 & @xmath77 & @xmath225 & @xmath267 & @xmath136 + 16 & @xmath268 & @xmath226 & @xmath268 & @xmath76 + 24 & @xmath73 & @xmath73 & @xmath73 & @xmath73 + there is a fundamental difference between the nature of the interactions for the sphere - packing problem and those for the other three problems . any sphere packing ( optimal or not ) consisting of nonoverlapping spheres of diameter @xmath100 is described by a short - ranged pair potential that is zero whenever the spheres do not overlap ( when the pair separation distance is greater than @xmath100 ) and is infinite whenever the pair separation distance is less than @xmath100 . by contrast , the other three problems are described by soft " bounded interactions . in particular , we have seen that the number variance is specified by a bounded repulsive pair potential with compact support [ cf . ( [ volavginterp ] ) ] . we will see in the subsequent sections that the covering and quantizer problems are described by many - particle bounded interactions but of the more general form ( [ full ] ) , which involves single - body , two - body , three - body , and higher - body interactions . one simple reason why the optimal solutions of the sphere - packing and number - variance problems are related to one another either directly or via their dual solutions in the first three space dimensions is that they both involve short - ranged repulsive pair interactions only . the reader is referred to refs . @xcite and @xcite for a comprehensive explanation . the reasons why the optimal solutions to these two problems are sometimes the optimal solutions for the covering and quantizer problems will become apparent in the subsequent sections . the explanation for why the optimal covering and quantizer solutions are generally different for @xmath2 is discussed in sec . [ reform ] . we note that the leech lattice @xmath73 for @xmath269 is an exceptional case in that it provides the optimal solution to all four different problems . the remarkably high degree of symmetry possessed by this self - dual lattice @xcite accounts for this unique property . the only other dimensions where all four optimal solutions are the same are @xmath167 and @xmath189 . we recall the definition of the void " nearest - neighbor probability density function @xmath270 @xcite : @xmath271 the void " exclusion probability @xmath272 is the _ complementary cumulative distribution function _ associated with @xmath270 : @xmath273 and hence is a monotonically decreasing function of @xmath28 @xcite . thus , @xmath272 has the following probabilistic interpretation : @xmath274 there is another interpretation of @xmath275 that involves circumscribing spheres of radius @xmath28 around each point in a realization of the point process . it immediately follows that @xmath272 is the _ expected _ fraction of space not covered by these circumscribing spheres . differentiating ( [ ev - cum ] ) with respect to @xmath28 gives @xmath276 note that these void quantities are different from the particle " nearest - neighbor functions @xcite in which the sphere of radius @xmath28 is centered at an actual point of the point process ( as opposed to an arbitrary point in the space ) . it is useful to introduce the conditional " nearest - neighbor function @xmath277 @xcite , which is defined in terms of @xmath278 and @xmath279 as follows : @xmath280 where @xmath281 is the surface area of a @xmath0-dimensional sphere of radius @xmath28 [ cf . ( [ area - sph ] ) ] . thus , we have the following interpretation of the conditional function : @xmath282 therefore , it follows from ( [ derv - ev ] ) and ( [ def - hv2 ] ) that the exclusion probability can be expressed in terms of @xmath283 via the relation @xmath284 . \label{ev - gv}\end{aligned}\ ] ] it is clear that the void functions have the following behaviors at the origin for @xmath285 @xcite : @xmath286 moments of the nearest - neighbor function @xmath270 arise in rigorous bounds for transport properties of random media @xcite . the @xmath9th moment of @xmath270 is defined as @xmath287 the void functions can be expressed as infinite series whose terms are integrals over the @xmath9-particle density functions @xcite . for example , the void exclusion probability functions for a translationally invariant point process are respectively given by @xmath288 where @xmath20 is the @xmath9-particle correlation function and @xmath289 is the heaviside step function defined by ( [ heaviside ] ) . the corresponding series for @xmath270 is obtained from the series above using ( [ derv - ev ] ) . note that the series ( [ ev - series ] ) can be rewritten in terms of intersection volumes of spheres : @xmath290 where @xmath291 is the intersection volume of @xmath9 equal spheres of radius @xmath28 centered at positions @xmath292 . observe that @xmath293 , where @xmath154 is the volume of a sphere of radius @xmath28 [ cf . ( [ v1 ] ) ] and @xmath153 is the scaled intersection volume [ cf . ( [ alpha ] ) ] and ( [ series ] ) ] . in the special case of a poisson point distribution , @xmath294 for all @xmath9 , and hence ( [ f - series ] ) immediately yields the well - known exact result for such a spatially uncorrelated point process @xmath295 the use of this relation with definition ( [ derv - ev ] ) gives @xmath296 for a single realization of @xmath4 points within a large volume @xmath297 in @xmath1 , we have @xmath298 this formula assumes that @xmath4 is sufficiently large so that boundary effects can be neglected . the second term in ( [ ev ] ) @xmath299 can be interpreted as a sum over one - body terms , which is independent of the point configuration . clearly , the ( @xmath300)th term in ( [ ev ] ) can be interpreted as a sum over intrinsic @xmath9-body interactions , namely , @xmath301 . thus , except for the trivial constant of unity ( the first term ) , @xmath272 can be regarded to be a many - body potential of the general form ( [ full ] ) , which heretofore was not observed . upper and lower bounds on the so - called _ canonical @xmath9-point correlation function _ @xmath302 ( with @xmath303 and @xmath304 ) for point processes in @xmath1 have been found @xcite . since the void exclusion probability and nearest - neighbor probability density function are just special cases of @xmath305 , then we also have strict bounds on them for such models . let @xmath306 represent either @xmath275 or @xmath307 and @xmath308 represent the @xmath309th term of the series for these functions . furthermore , let @xmath310 be the partial sum . then it follows that for any of the exclusion probabilities or nearest - neighbor probability density functions , we have the bounds @xmath311 application of the aforementioned inequalities yield the first three successive bounds on the nonnegative exclusion probability : @xmath312 where @xmath313 is the intersection volume of two @xmath0-dimensional spheres of radius @xmath28 whose centers are separated by the distance @xmath314 and @xmath315 is the scaled intersection volume given by ( [ alpha ] ) . the corresponding first two nontrivial bounds on the nonnegative pore - size density function @xmath270 are as follows : @xmath316 where @xmath317 is the surface area of the intersection volume @xmath318 . bounds on the conditional function @xmath277 follow by combining the bounds above on @xmath279 and @xmath278 and definition ( [ def - hv2 ] ) . for example , the following bounds have been found @xcite : @xmath319 and @xmath320 which should only be applied for @xmath28 such that @xmath321 remains positive . for congruent sphere packings of diameter @xmath100 at packing density @xmath116 , the infinite series expansion for @xmath272 [ cf . ( [ ev - series ] ) ] will truncate after a finite number of terms for a bounded value of the radius @xmath28 . a spherical region of radius @xmath28 centered at an arbitrary point in the space exterior to the spheres can contain at most @xmath322 sphere centers . therefore , series truncates after @xmath323 terms , i.e. , @xmath324 for a spherical region of radius @xmath325 , @xmath326 and hence we have the exact expression that applies for @xmath327 @xmath328 for any sphere packing for which ( [ period ] ) applies such that @xmath329 , we have upon use of ( [ ev - packing ] ) the exact result @xmath330 \left(\frac{r}{d}\right)^d , & d/2 \le r \ge d/\sqrt{3 } , \end{array } \right . \label{ev - packing2}\ ] ] where we have used the fact that @xmath331 . importantly , this relation applies to the densest known lattice packings of spheres , at least for dimensions in the range @xmath332 . now we can reformulate the covering and quantizers problems in terms of the void exclusion probability . in particular , the covering problem asks for the point process in @xmath1 at unit density ( @xmath96 ) that minimizes the support of the radial function @xmath272 . we define @xmath333 the smallest possible value of the covering radius @xmath94 among all point processes for which @xmath334 , which we call the _ minimal covering radius_. this is indeed a special ground state in which the energy " is identically zero ( i.e. , @xmath335 ) @xcite . depending on the space dimension @xmath0 , this special ground state will involve one - body interactions , the first two terms of ( [ ev ] ) , one- and two - body interactions , the first three terms of ( [ ev ] ) , one- , two- and three - body interactions , the first four terms of ( [ ev ] ) , etc . , and will truncate at some particular level , provided that @xmath272 for the point process has compact support . the minimal covering radius @xmath333 increases with the space dimension @xmath0 and , generally speaking , the highest - order @xmath9-body interaction required to fully characterize the associated @xmath272 increases with @xmath0 note that for a particular point process , twice the covering radius @xmath336 can be viewed as the effective interaction range " between any pair of points , since the intersection volume @xmath337 , which appears in expression ( [ ev ] ) for @xmath272 , is exactly zero for any pair separation @xmath338 ; moreover , for such pair separations @xmath339 can be written purely in terms of the lower - order intersection volume @xmath340 . therefore , because @xmath341 for @xmath342 , the effective interaction range between any @xmath9 points for @xmath342 is still given by @xmath336 . the quantizer problem asks for the point process in @xmath1 at unit density that minimizes the scaled average squared error @xmath343 defined as @xmath344 we will call the minimal error @xmath249 . thus , we seek the ground state of the many - body interactions that are involved upon substitution of ( [ ev ] ) into ( [ g ] ) . again , depending on the dimension , this many - body energy will truncate at some particular level , provided that @xmath272 has compact support . again , as @xmath0 increases , successively higher - order interactions in the expression ( [ ev ] ) must be incorporated to completely characterize @xmath272 . it is instructive to express explicitly the void exclusion probabilities for some common lattices and use these functions to evaluate explicitly their corresponding covering densities and scaled average squared errors in the first three space dimensions . in the simplest case of one dimension , the series expansion for @xmath272 [ cf . ( [ ev ] ) ] for the integer lattice @xmath53 truncates after only one - body terms . at unit number density ( @xmath96 ) , it is trivial to show that @xmath345 where the covering radius @xmath346 , @xmath347 , and the nearest - neighbor distance from a lattice point is unity . using the definitions ( [ covering ] ) and ( [ g ] ) in combination with ( [ integer ] ) yield the covering density and scaled average squared error , respectively , for the optimal integer lattice : @xmath348 @xmath349 let us now determine the void exclusion probabilities for the @xmath350 ( square ) and @xmath351 ( triangular ) and lattices for @xmath28 up to their respective covering radii for which @xmath334 . for these lattices , the series expression ( [ ev ] ) for @xmath279 truncates after two - body terms . for the square and triangular lattices at @xmath96 , @xmath352 where @xmath353 is the nearest - neighbor distance from a lattice point and the covering radius @xmath94 , equal to one half of the next - nearest - neighbor distance , which we will denote by @xmath354 . for the square and triangular lattices at @xmath96 , @xmath355 and @xmath356 , and @xmath357 and @xmath358 respectively . figure [ 2d - lattices ] provides plots of @xmath272 for these two @xmath189 lattices . employing the definitions ( [ covering ] ) and ( [ g ] ) in combination with ( [ square ] ) provide the covering density and scaled average squared error , respectively , for the @xmath350 lattice : @xmath359 @xmath349 similarly , the corresponding equations for the @xmath360 lattice yields the covering density and scaled average squared error , respectively , for this optimal structure : @xmath361 @xmath362 for the @xmath350 ( square ) and @xmath351 ( triangular ) lattice have support up to the covering radii @xmath356 and @xmath358 , respectively , at unit number density ( @xmath96 ) . , width=336 ] it is also useful to express explicitly the void exclusion probabilities for the @xmath363 ( simple cubic ) and @xmath261 ( bcc ) lattices for @xmath28 up to their respective covering radii for which @xmath334 . it turns out that calculating @xmath272 in the case of the simple - cubic lattice is more complicated than that for the bcc lattice because the former involves up through four - body terms , i.e. , @xmath364 . however , the symmetry of the geometry for @xmath363 enables one to express @xmath364 purely in terms of @xmath365 and @xmath366 . the calculation of @xmath272 does not involve four - body terms . for the simple cubic lattice at @xmath96 , @xmath367 where @xmath355 , @xmath368 , @xmath369 and @xmath370 is explicitly given by ( [ v3int ] ) in appendix [ v3-int ] for triangles of side lengths @xmath31 , @xmath371 and @xmath372 . here we have used the fact that for @xmath373 , @xmath374 . using the definitions ( [ covering ] ) and ( [ g ] ) in combination with ( [ sc ] ) yield the covering density and scaled average squared error , respectively , for the @xmath363 lattice : @xmath375 @xmath349 for the bcc lattice at @xmath96 , @xmath376 where @xmath377 , @xmath378 , @xmath379 , and @xmath380 is the circumradius of a triangle of side lengths @xmath353 , @xmath353 and @xmath354 , the general expression of which is given by ( [ circum ] ) in appendix [ v3-int ] . employing the definitions ( [ covering ] ) and ( [ g ] ) in combination with ( [ bcc ] ) provide the covering density and scaled mean squared error , respectively , for the @xmath381 lattice : @xmath382 @xmath383 figure [ 3d - lattices ] provides plots of @xmath272 for the simple cubic and bcc three - dimensional lattices . we see that the best known solutions to the covering and quantizer problems are identical for the first three space dimensions . however , there is no reason to expect that the optimal solutions for these two problems to be the same in higher dimensions , except for @xmath269 for reasons mentioned in sec . although both problems involve ground states associated with the many - body interaction " function @xmath272 , such that it possesses compact support for finite @xmath0 , the precise shape of the function @xmath272 for a particular point configuration is crucial in determining its first moment or quantizer error . by contrast , the best covering seeks to find the point configuration that minimizes the support of @xmath272 without regard to its shape . for the @xmath363 ( simple cubic ) lattice and @xmath261 ( bcc ) lattice have support up to the covering radii @xmath369 and @xmath379 , respectively , at unit number density ( @xmath96 ) . , width=336 ] saturated sphere packings in @xmath1 should provide relatively thin coverings . surrounding every sphere of diameter @xmath100 in any saturated packing of congruent spheres in @xmath1 at packing density @xmath103 by spheres of radius @xmath100 provides a covering of @xmath1 , and thus the associated covering density @xmath384 is given by @xmath385 where @xmath386 and @xmath387 are the number density and packing density , respectively , of the saturated packing . what is the thinnest possible covering associated with a saturated packing ? it immediately follows that the thinnest coverings among saturated congruent sphere packings in @xmath1 are given by the saturated packings that have the minimal packing density @xmath388 in that space dimension and have covering density @xmath389 we can also bound the packing density @xmath103 of any saturated sphere packing in @xmath1 from above using upper bounds on the covering density . * lemma 1 : * _ the density @xmath103 of any saturated sphere packing in @xmath1 is bounded from above according to _ @xmath390 the proof is trivial in light of the upper bound ( [ cover - up ] ) on the covering density and relation ( [ theta - den ] ) . the standard random sequential addition ( rsa ) sphere packing is a time - dependent process produced by randomly , irreversibly , and sequentially placing nonoverlapping spheres into a large region @xmath391 @xcite . initially , this large region is empty of sphere centers and subsequently spheres are added provided each attempted placement of a sphere does not overlap an existing sphere in the packing . if an attempted placement results in an overlap , further attempts are made until the sphere can be added to the packing at that time without violating the impenetrability constraint . for identical @xmath0-dimensional rsa spheres , the filling process terminates at the _ saturation limit _ at infinitely long times in the infinite - volume limit . thus , in this limit , the rsa packing is a saturated packing . the saturation density @xmath103 can only be determined exactly in one dimension , where it is known to be @xmath392 @xcite . in higher dimensions , the saturation density can only be determined from computer simulations . earlier work focused on two and three dimensions , where is was found that @xmath393 @xcite and @xmath394 @xcite , respectively . more recent numerical work has reported rsa saturation densities for the first six space dimensions @xcite . it is important to emphasize that upper bound ( [ sat - up ] ) provides a very poor estimate of the rsa saturation density ; for example , for @xmath97 , it gives an upper bound greater than unity ( 2.32224 , which is about 6.1 times larger than the value obtained from simulations ) and for @xmath395 , it provides the upper bound @xmath396 , which is about 7.3 times larger than the value obtained from simulations @xcite . thus , bound ( [ sat - up ] ) could only be realized by rsa saturated packings , if at all , in high dimensions . therefore , it is useful here to compute the corresponding covering densities for rsa packings . the numerical data for rsa saturation densities for @xmath397 reported in ref . @xcite was well approximated ( with a correlation coefficient of 0.999 ) by the following form : @xmath398 where @xmath399 and @xmath400 . this can be used to estimate the rsa saturation densities for @xmath401 . however , even though the upper bound ( [ sat - up ] ) on the packing density of a saturated sphere packing grossly overestimates the rsa value for the first six space dimensions , we include in the fit function for @xmath103 a @xmath402 correction , which will lead to a more conservative estimate of the covering density , as explained shortly @xcite . including such a term , we find the following fitted function for the saturated rsa packing density : @xmath403 with a correlation coefficient of @xmath404 , where @xmath405 , @xmath406 and @xmath407 , does as well as ( [ linear ] ) for @xmath397 . since ( [ log ] ) predicts slightly higher densities than ( [ linear ] ) for @xmath408 @xcite , we use it to obtain the corresponding estimate of the rsa covering density , namely , @xmath409 which is a slightly more conservative estimate , since ( [ linear ] ) would yield a slightly thinner covering density . in table [ rsa ] , we provide estimates of the rsa covering densities for selected dimensions up through @xmath269 . the covering density for @xmath167 is determined from reyi s exact saturation packing density value @xcite and multiplying it by 2 . the values for @xmath410 is are obtained from the reported saturated density values in ref . @xcite and multiplying each density value by @xmath411 . the values reported for @xmath412 are estimates obtained from the fitting formula ( [ log2 ] ) . comparing table [ rsa ] to table [ coverings ] for the best known coverings , we see that saturated rsa packings not only provide relatively thin coverings but putatively represent the first non - lattices that yield thinner coverings than the best known lattice coverings beginning in about dimension 17 . this suggests that saturated rsa packings may be thinner than the previously best known coverings for @xmath413 and probably for some dimensions greater than 24 . & covering density , @xmath384 & packing density , @xmath103 + 1 & 1.4952 & 0.74759 + 2 & 2.1880 & 0.54700 + 3 & 3.0622 & 0.38278 + 4 & 4.0726 & 0.25454 + 5 & 5.1526 & 0.16102 + 6 & 6.0121 & 0.09394 + 7 & 7.0512 & 0.05508 + 8 & 8.0526 & 0.03145 + 9 & 10.0706 & 0.01769 + 10 & 11.0860 & 0.009834 + 12 & 12.1052 & 0.002955 + 16 & 16.2141 & @xmath414 + 17 & 17.2482 & @xmath415 + 18 & 18.2848 & @xmath416 + 24 & 24.5489 & @xmath417 + using the successive lower and upper bounds on the void exclusion probability function @xmath272 given in the previous section , we can , in principle , derive corresponding bounds on the minimal error @xmath249 . moreover , one can obtain a variety of upper bounds on @xmath249 using our approach by utilizing the exact form of the void exclusion probability , when it is known , for some point process at unit density . since a general point process must have an error @xmath343 that is generally larger than the minimal @xmath249 , it trivially follows that @xmath418 to illustrate how we can obtain bounds on @xmath249 using our approach , we begin by rederiving the following bounds due to zador @xcite : @xmath419 consider the lower bound first . combination of relation ( [ g ] ) and lower bound ( [ ev - bound1 ] ) yields at unit density @xmath420 dr=\frac{1}{(d+2)\pi}\gamma(1+d/2)^{2/d},\ ] ] which is seen to be equal to zador s lower bound . here @xmath421 is the zero of @xmath422 . it is clear that a sphere of radius @xmath423 has the smallest second moment of inertia of any solid @xmath0-dimensional solid , and hence establishes the lower bound . the simplest example of a point process for which @xmath272 is known is the poisson point process [ cf . ( [ poisson ] ) ] . substitution of ( [ poisson ] ) into ( [ upper ] ) at unit density yields @xmath424 which is seen to be equal to zador s upper bound . these derivations of the inequalities stated in ( [ zador ] ) appear to be much simpler than the ones presented by zador . in the large-@xmath0 limit , zador s upper and lower bounds become identical , and hence one obtains the exact asymptotic result @xmath425 the convergence of zador s bounds to the exact asymptotic limit is to be contrasted with the sphere - packing problem in which the best upper and lower bounds on the maximal density become exponentially far apart in the high - dimensional limit . improved upper bounds on @xmath249 can be obtained by considering those point processes corresponding to a sphere packing for which the minimal pair separation is @xmath100 and lower bounds on the conditional function @xmath321 for @xmath426 . in what follows , we present two different upper bounds on @xmath249 based on this idea that improve upon zador s upper bound . for any packing of identical spheres with diameter @xmath100 , the following exact relations on the nearest - neighbor quantities apply for @xmath427 @xcite : @xmath428 observe that @xmath321 is a monotonically increasing function of @xmath28 in the interval @xmath429 $ ] . from these equalities , it immediately follows that @xmath430 consider now the class of sphere packings for which the conditional nearest - neighbor function is bounded from below according to @xmath431 this class of packings includes equilibrium ( gibbs ) ensembles of hard spheres along the disordered fluid branch of the phase diagram @xcite , nonequilibrium disordered sphere packings , such as the ghost " random sequential addition ( rsa ) process @xcite , and a large class of lattice packings of spheres , as will be described below . in the equilibrium cases , it is known that @xmath321 is a monotonically increasing function of @xmath28 for @xmath285 and thus using this property together with the equality @xmath432 [ cf . ( [ equality ] ) ] means that the lower bound ( [ gv - bound ] ) is obeyed . for one - dimensional equilibrium rods , " the bound ( [ gv - bound ] ) is sharp ( exact ) @xmath321 for all realizable @xmath433 $ ] . a bound of the type ( [ gv - bound ] ) was used to bound the related particle " mean nearest - neighbor distance from above for different classes of sphere packings for all @xmath0 @xcite . using definition ( [ ev - gv ] ) and inequality ( [ gv - bound ] ) , the void exclusion probability function obeys the following upper bound for @xmath426 : @xmath434\right\ } , \quad r \ge d/2 . \label{ev-1}\ ] ] since any upper bound on the nonnegative function @xmath272 leads to an upper bound on its first moment , we then have upon use of ( [ upper ] ) , ( [ exact ] ) and ( [ ev-1 ] ) , the upper bound @xmath435^{2/d}}{d\pi}\left[\frac{(d+2(1-\phi))}{4(2+d ) } + \frac{(1-\phi)}{2d } \left(\frac{1-\phi } { \phi}\right)^{2/d}\exp\left(\frac{\phi}{1-\phi}\right ) \gamma\left(\frac{2}{d},\frac{\phi}{1-\phi}\right)\right ] , \label{new - upper}\ ] ] where @xmath436 is the incomplete gamma function . observe that the prefactor multiplying the bracketed expression is @xmath437 , where , in light of ( [ v1 ] ) and ( [ den ] ) , @xmath438^{1/d}/\sqrt{\pi}$ ] , assuming unit number density . note also that the upper bound ( [ new - upper ] ) depends on a single parameter , namely , the packing density @xmath116 . thus , there is an optimal packing density @xmath439 $ ] that yields the best ( smallest ) upper bound for any particular @xmath0 , where @xmath440 is the maximal packing density . since the right side of the inequality is a monotonically decreasing function of @xmath116 for any @xmath0 , then the optimal density @xmath441 is , in principle , given by @xmath440 . it is noteworthy that the upper bound ( [ new - upper ] ) for the optimal choice @xmath442 may still be valid for a packing even if the bound ( [ ev-1 ] ) , upon which it is based , is violated for @xmath28 of the order of @xmath100 because the exponential tail can more than compensate for such a violation such that the error [ first moment of @xmath272 ] is overestimated . observe also that because @xmath444 + { \cal o}(1 ) \quad ( d \rightarrow \infty)\ ] ] the upper bound ( [ new - upper ] ) tends to the exact asymptotic result ( [ asympt ] ) of @xmath445 . before discussing the optimal bounds , it is useful to begin with an application of the upper bound ( [ new - upper ] ) for the _ sub - optimal _ case of a disordered sphere packing , namely , the aforementioned _ ghost _ rsa packing process @xcite , which we now show generally improves on zador s upper bound . this represents the only exactly solvable disordered sphere - packing model for all realizable densities and in any dimension , as we now briefly describe . the ghost rsa packing process involves a ( time - dependent ) sequential addition of spheres in space subject to the nonoverlap condition . not only is an attempted addition of a sphere rejected if it overlaps an existing sphere of the packing , it is also rejected if it overlaps any previously rejected sphere ( called a ghost " sphere ) . unlike the standard rsa packing , the ghost rsa packing does not become a saturated packing in the infinite - time limit . all of the @xmath9-particle correlation functions for this nonequilibrium model have been obtained analytically for any @xmath0 , time @xmath372 , and for all realizable densities . for example , one can show that the maximal density ( achieved at infinite time ) is given by @xmath446 and the associated pair correlation function is @xmath447 where @xmath289 is the unit step function , equal to zero for @xmath448 and unity for @xmath449 . it is straightforward to verify that the upper bound on the exclusion probability @xmath272 for this infinite - time case obtained by using ( [ g2-grsa ] ) in the inequality ( [ ev - bound2 ] ) is always below the upper bound ( [ ev-1 ] ) . therefore , the upper bound ( [ new - upper ] ) is valid at the maximal density , i.e. , at @xmath450 , we have @xmath451^{2/d}}{d\pi}\left[\frac{(d+2(1 - 1/2^d))}{4(2+d ) } + \frac{2(1 - 1/2^d)^{(2+d)/d}}{d } \exp\left(\frac{1}{2^d-1}\right ) \gamma\left(\frac{2}{d},\frac{1}{2^d-1}\right)\right ] . \label{upper - grsa}\ ] ] for @xmath167 , 2 and 3 , this upper bound yields @xmath452 , @xmath453 and @xmath454 , respectively , which is to be compared to zador s upper bound , which gives @xmath455 , @xmath456 and @xmath457 , respectively . we note that in the large-@xmath0 limit , the upper bound ( [ upper - grsa ] ) yields the exact asymptotic result ( [ asympt ] ) , which implies that the upper bound ( [ ev-1 ] ) on @xmath272 becomes exact for ghost rsa packings , tending to the unit step function in this asymptotic limit , i.e. , @xmath458 this asymptotic result implies the following corresponding one for the void nearest - neighbor probability density function : @xmath459 scaled by @xmath460 for the optimal lattice packings for @xmath189 ( @xmath461 ) and @xmath191 ( @xmath462 ) , as obtained from ( [ ev - packing2 ] ) , for @xmath463 ( solid curves ) to the corresponding estimates obtained from ( [ ev-1 ] ) for these cases ( dashed curves ) . it is only for the case @xmath189 that estimate ( [ ev-1 ] ) is not a rigorous pointwise upper bound on the exact void exclusion probability for @xmath332 and , likely , for @xmath464 . the exponential tail associated with ( [ ev-1 ] ) more than compensates for the narrow pointwise violation for the special case @xmath189 , resulting in a strict upper bound on the first moment of @xmath272 , i.e. , ( [ new - upper ] ) remains a strict upper bound for @xmath189 . , width=336 ] we now return to finding the optimal ( smallest ) upper bound ( [ new - upper ] ) for each dimension . for @xmath167 , the optimal density is @xmath465 , which produces the sharp bound @xmath466 this bound is exact in this case because the inequality ( [ gv - bound ] ) is exact for all realizable densities for equilibrium hard rods , " including at @xmath467 , which corresponds to the optimal integer lattice packing . this is to be contrasted with zador s upper bound , which yields 1/2 for @xmath167 and is far from the exact result . the improved upper bound ( [ new - upper ] ) in the higher dimensions reported in table [ tab - eta-3 ] is obtained by evaluating it at the densities of the densest known lattice packings in these respective dimensions @xcite . we note that it is only for optimal triangular lattice packing in @xmath236 that the upper bound ( [ ev-1 ] ) on @xmath272 is violated pointwise for a small range of @xmath28 around @xmath468 [ inequality ( [ ev-1 ] ) is obeyed for @xmath469 and in the vicinity of @xmath470 , but the exponential tail associated with ( [ ev-1 ] ) more than compensates for this narrow pointwise violation , resulting in a strict upper bound on the first moment of @xmath272 , i.e. , ( [ new - upper ] ) remains a strict upper bound for @xmath189 . using relation ( [ ev - packing2 ] ) and lattice coordination properties , it is easily verified that the ( [ ev-1 ] ) is a strict upper bound on the void exclusion probability for the densest known lattice packings for all @xmath426 and @xmath471 as well as @xmath167 , and hence inequality ( [ new - upper ] ) provides a strict upper bound on the scaled error for all of these lattices . for illustration purposes , we compare in fig . [ ev - bounds ] the exact result for @xmath275 obtained from ( [ ev - packing2 ] ) to the estimate ( [ ev-1 ] ) for the cases @xmath189 and @xmath191 for @xmath463 . the upper bound ( [ new - upper ] ) is generally appreciably tighter than zador s upper bound for low to moderately high dimensions . @xmath0 & quantizer & scaled error , @xmath252 & conjectured & improved + & & & lower bound & upper bound + 1 & @xmath217 & 0.083333 & 0.083333 & 0.083333 + 2 & @xmath118 & 0.080188 & 0.080188 & 0.080267 + 3 & @xmath196 & 0.078543 & 0.077875 & 0.079724 + 4 & @xmath197 & 0.076603 & 0.07609 & 0.078823 + 5 & @xmath198 & 0.075625 & 0.07465 & 0.078731 + 6 & @xmath199 & 0.074244 & 0.07347 & 0.077779 + 7 & @xmath264 & 0.073116 & 0.07248 & 0.076858 + 8 & @xmath255 & 0.071682 & 0.07163 & 0.075654 + 9 & @xmath256 & 0.071626 & 0.070902 & 0.075552 + 10 & @xmath257 & 0.070814 & 0.070405 & 0.074856 + 12 & @xmath202 & 0.070100 & 0.06918 & 0.073185 + 16 & @xmath203 & 0.068299 & 0.06759 & 0.070399 + 24 & @xmath204 & 0.065771 & 0.06561 & 0.067209 + [ tab - eta-3 ] for any saturated packing of identical spheres of diameter @xmath100 , @xmath272 by definition is exactly zero for @xmath28 beyond the diameter , i.e. , @xmath472 in the particular case of saturated rsa packings , the void exclusion probability can computed using the same techniques described in ref . @xcite for the first six space dimensions . these results are summarized in fig . [ rsa - all ] . the corresponding quantizer errors for these dimensions are listed in table [ rsa - quant ] . we see that the discrepancies between the saturated rsa quantizer error improves as @xmath0 increases as compared to the best known quantizer error reported in table [ quant ] . for saturated rsa packings of congruent spheres of diameter @xmath100 for the first six space dimensions.,width=288 ] & quantizer error , @xmath473 + 1 & 0.11558 + 2 & 0.09900 + 3 & 0.09232 + 4 & 0.08410 + 5 & 0.07960 + 6 & 0.07799 + * lemma 2 : * _ saturated sphere packings in @xmath1 possess void nearest - neighbor functions that tend to the following high - dimensional asymptotic behaviors : _ @xmath474 for any saturated packing at packing density @xmath103 , it is clear that @xmath272 is bounded from above for @xmath475 as follows : @xmath476 let @xmath473 denote the scaled dimensionless quantizer error for a saturated packing . combination of the expression for @xmath272 in ( [ exact ] ) and ( [ rsa - bound ] ) yields the following upper bound on @xmath473 : @xmath477^{2/d}}{d\pi}\left[\frac{(d+2(1-\phi_s))}{4(2+d ) } + \frac{3(1-\phi_s)}{8 } \right ] , \label{new - upper2}\ ] ] the fact that this upper bound becomes exact in the high - dimensional limit ( that is , it tends to @xmath478 ) , implies that @xmath272 and @xmath270 for a saturated packing tends to the unit step function and radial delta function , as specified by ( [ lemma2 ] ) . not surprisingly , the bound ( [ new - upper2 ] ) is not that tight in relatively low dimensions . we have reformulated the covering and quantizer problems as the determination of the ground states of interacting point particles in @xmath1 that generally involve single - body , two - body , three - body , and higher - body interactions ; see sec . [ reform ] . the @xmath9-body interaction is directly related to a purely geometrical problem , namely , the intersection volume of @xmath9 spheres of radius @xmath28 centered at @xmath9 arbitrary points of the system in @xmath1 . this was done by linking the covering and quantizer problems to certain optimization problems involving the void " nearest - neighbor functions . this reformulation allows one to employ theoretical and numerical optimization techniques to solve these energy minimization problems . a key finding is that disordered saturated sphere packings provide relatively thin coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions . in the case of the quantizer problem , we derived improved upper bounds on the quantizer error that utilize sphere - packing solutions . these improved bounds are generally substantially sharper than an existing upper bound due to zador in low to moderately large dimensions . moreover , we showed that disordered saturated sphere packings yield relatively good quantizers . our reformulation helps to explain why the known solutions of quantizer and covering problems are identical in the first three space dimensions and why they can be different for @xmath2 . in the first three space dimensions , the best known solutions of the sphere packing and number variance problems are directly related to those of the covering and quantizer problems , but such relationships may or may not exist for @xmath2 , depending on the peculiarities of the dimensions involved . it is clear that as @xmath0 becomes large , the quantizer problem becomes the easiest to solve among the four ground - state problems considered in this paper , since , unlike the other three problems , the asymptotic quantizer error tends to the same limit independent of the configuration of the point process . the detection of gravitational waves from various astrophysical sources has and will be searched for in the output of interferometric networks @xcite by correlating the noisy output of each interferometer with a set of theoretical waveform templates @xcite . depending upon the source , the parameter space is generally multidimenisonal and can be as large as @xmath229 or larger for inspiraling binary black holes . the templates must _ cover _ the space and the challenge is to place them in some optimal fashion such that the fewest templates are used to reduce computational cost without reducing the detectability of the signals . optimal template placement for gravitational wave data analysis has proved to be highly nontrivial . one solution proposed for the optimal placement in flat ( euclidean ) space is to simply use the optimal solution to the covering problem @xcite . however , this requires every point in the parameter space to be covered by at least one template , which rapidly becomes inefficient in higher dimensions when optimal lattice covering solutions are employed . another approach to template placement consists in relaxing the strict requirement of complete coverage for a given mismatch , and instead require coverage only with a certain confidence @xcite . such stochastic approaches have involved randomly placing templates in the parameter space , accompanied by a pruning " step in which redundant " templates , which are deemed to lie too close to each other , are removed @xcite . the pruning step may be a complication that can be avoided , as discussed below . another approach is to place spherical templates down according to a poisson point process @xcite , which has been claimed to provide good solutions for @xmath479 . the problem with the latter approach is that there will be numerous multiple overlaps of templates , which only increases as the number density of templates ( intensity of the poisson point process ) is increased in order to cover as much of the space as is computationally feasible . the results of the present study suggest alternative solutions to the optimal template construction problem . first , we remark that if the covering of space by the templates is relaxed , then it is possible that the optimal lattice quantizers could serve as good solutions in relatively low dimensions ( @xmath480 ) because the mean square error is minimized . second , in such relatively low dimensions , we have shown that saturated rsa sphere packings provide both relatively good coverings and quantizers , and hence may be useful template - based constructions for the search of gravitational waves . indeed , we have shown that saturated rsa sphere packings are expected to become better solutions as @xmath0 becomes large . however , for @xmath481 , it will be computationally costly to create truly saturated packings , which by definition provide coverings of space ( see sec . [ results - cover ] ) . however , the existing stochastic approaches do not require complete coverage of space and hence an _ unsaturated _ rsa sphere packing that gets relatively close to the saturation state might still be more computationally efficient than either the random placement / pruning technique @xcite ( because pruning is unnecessary ) or the poisson placement procedure @xcite ( because far fewer spheres need to be added ) . moreover , when the template parameter space is curved , which occurs in practice @xcite , the rsa sphere packing would be computational faster to adapt than lattice solutions . in future work , we will explore whether our reformulations of the covering and quantizer problems as ground - state problems of many - body interactions of the form ( [ full ] ) can facilitate the search for better solutions to these optimization tasks . clearly , a computational challenge in high dimensions will be the determination of the intersection volume @xmath339 of @xmath9 spheres of radius @xmath28 at @xmath9 different locations in @xmath1 for sufficiently large @xmath9 . however , it is possible that that the series representations ( [ ev ] ) for @xmath272 and bounds on this quantity [ cf . sec . [ bounds - e ] ] can be used to devise useful approximations of the monotonic function @xmath272 , which should be zero for @xmath482 , where @xmath94 is the bounded covering radius . such approximation could be employed to evolve an initial guess of the configuration of points within a fundamental cell to useful but sub - optimal solutions , which upon further refinement could suggest novel solutions . in short , the implications of our reformulations to discover better solutions to the covering and quantizer problems in selected dimensions have yet to be fully investigated and deserves future attention . i am very grateful to yang jiao and chase zachary for their assistance in creating many of the figures for this manuscript and for many useful discussions . i thank andreas str " ombergsson and zeev rudnick for their generous help in providing efficient algorithms to compute the epstein zeta function for @xmath206 and 24 . i also thank henry cohn for introducing me to the quantizer problem as well as for many helpful discussions . this work was supported by the office of basic energy sciences , u.s . department of energy , under grant no . de - fg02 - 04-er46108 . here we summarize the steps in computing the asymptotic number - variance coefficient ( [ num ] ) for the lattices @xmath77 , @xmath76 and @xmath73 , which correspond to the densest lattice packings in dimensions @xmath483 and 24 , respectively . importantly , the sum in ( [ num ] ) converges slowly . we noted in sec . [ num - var ] that the _ dual _ of the lattice that minimizes the epstein zeta function @xmath185 [ defined by ( [ epstein ] ) ] at @xmath188 among all lattices will minimize the asymptotic number - variance coefficient ( [ num ] ) among lattices . we will exploit number - theoretic representations of the epstein zeta function that enable its efficient numerical evaluation and thus efficient computation of the asymptotic number - variance coefficient ( [ num ] ) using the aforementioned duality relation . first , let us note that epstein zeta function @xmath185 ( [ epstein ] ) can be rewritten as follows : @xmath484 where @xmath79 is the coordination number at a radial distance @xmath485 from some lattice point in the lattice @xmath32 . [ note that the epstein zeta function defined in this way applies to a general periodic point process provided that @xmath79 is interpreted in the generalized sense discussed in ( [ def ] ) . ] the quantities @xmath79 and @xmath485 for many well - known lattices in @xmath1 can be obtained analytically using the _ theta series _ for a lattice @xmath32 , which is defined by @xmath486 and is directly related to the quadratic form associated with the lattice @xcite . this series expression can usually be generated from the simpler functions @xmath487 @xmath488 , and @xmath489 , which are defined by @xcite : @xmath490 specifically , for the @xmath77 , @xmath76 and @xmath73 lattices , the associated theta series are given by @xcite @xmath491 @xmath492\nonumber\\ & = & 1 + 4320q^4 + 61440q^6 + \cdots \end{aligned}\ ] ] @xmath493 ^ 3 - \frac{45}{16}[\theta_2(q)\theta_3(q)\theta_4(q)]^8\nonumber\\ & = & 1 + 196560q^4 + 16773120q^6 + \cdots \end{aligned}\ ] ] where @xmath494 direct evaluation of ( [ zeta ] ) has the same convergence problems that the direct evaluation of the asymptotic number - variance coefficient ( [ num ] ) . however , we can exploit alternative number - theoretic representations of ( [ zeta ] ) to facilitate its evaluation . in particular , there is an expression for the epstein zeta function that can be derived using poisson summation @xcite : @xmath495 where @xmath496 and @xmath497 is the complementary incomplete gamma function . it is important to note that the volumes of the fundamental cells of the lattice and its dual associated with the first and second sums in ( [ f ] ) , respectively , are both taken to be unity here . using the appropriate theta series given above for the lattices corresponding to the densest lattice packings for @xmath17 and 24 and expression ( [ f ] ) for @xmath188 , one finds the corresponding epstein zeta functions to be @xmath498 now since all of these lattices are self - dual ( i.e. , @xmath499 , @xmath500 , @xmath501 ) , we can directly determine the corresponding asymptotic number - variance from ( [ num ] ) by replacing the sum therein with the appropriate evaluation of the epstein zeta function specified by relations ( [ 12])-([24 ] ) . the asymptotic number - variance values for @xmath17 and 24 reported in table [ var ] were obtained in this fashion . for @xmath97 , the intersection volume @xmath370 of three identical spheres of radius @xmath28 whose centers are separated by the distances @xmath31 , @xmath371 , and @xmath372 for @xmath502 is given by @xcite @xmath503 where @xmath504 , @xmath505^{1/2 } } , \label{circum}\ ] ] is the circumradius of the triangle with side length lengths @xmath31 , @xmath371 , and @xmath372 and @xmath506 . h. reiss , h. l. frisch , and j. l. lebowitz , j. chem . phys . * 31 * , 369 ( 1959 ) . this paper considered void " nearest - neighbor functions for the special case of identical hard spheres in equilibrium ( gibbs ensemble ) . s. torquato , b. lu , and j. rubinstein , phys . a * 41 * , 2059 ( 1990 ) . this paper considered _ two types _ of nearest - neighbor functions ( void " and particle " quantities ) for a general nonequilibrium " case of identical of spheres with arbitrary interactions _ e.g. _ , spheres with variable interpenetrability that interact with repulsive / attractive forces . mathematicians usually define a dual bravais lattice to have a fundamental cell volume @xmath507 ( i.e. , without the factor of @xmath508 ) , in which case self - duality is defined with respect to unit density ; see ref . @xcite . a laminated lattice @xmath70 in @xmath1 is built up of layers of @xmath509-dimensional lattices in @xmath0-dimensional euclidean space . note that a general lattice has been denoted by @xmath32 , which should not be confused with the specific laminated lattice @xmath70 . these are common notations for both objects and therefore we adhere to this convention . s. torquato and y. jiao , nature * 460 * , 876 ( 2009 ) ; s. torquato and y. jiao , phys . e * 80 * , 041104 ( 2009 ) . in these papers , the _ asphericity _ of a nonspherical solid body is defined to be ratio of the circumradius to the inradius of the circumsphere and insphere of the nonspherical particle , respectively , which provides a measure of the spherical asymmetry of a solid body . an asphericity equal unity corresponds to a perfect sphere . s. torquato , j. stat . phys . * 45 * , 843 ( 1986 ) . canonical @xmath9-point correlation function _ @xmath302 is a _ hybrid _ probablistic function in that it contains the character of a correlation function , probability function , and probability density function , depending upon its arguments @xmath512 and the values of @xmath513 , @xmath514 and @xmath515 . from @xmath516 , which has been explicitly represented as a series involving certain integrals over the @xmath9-particle correlation function @xmath20 , one can derive any of the statistical descriptors that have arisen in the theory of random media and statistical mechanics as well as their generalizations . the reader is referred to ref . @xcite for a comprehensive discussion of @xmath516 . there is an uncountably infinite number of periodic point configurations in @xmath1 in which circumscribed overlapping spheres of radius @xmath94 around each of the points just cover the space and renders @xmath517 such that for @xmath518 . the point configuration that minimizes the support of @xmath272 , i.e. , minimizes @xmath94 ( called @xmath333 ) is the unique ground - state configuration or the optimal covering . twice the covering radius @xmath336 for a particular point process determines the effective range " of the interaction associated with @xmath272 ; see the text for further explanation . note that leading order term in either ( [ linear ] ) or ( [ log ] ) involving @xmath519 is a dominant dimensional contribution for rsa saturation densities in relatively low dimensions for good theoretical reasons @xcite . the fact that this term does not appear in the upper bound ( [ sat - up ] ) is another reason why it could only be realizable by rsa saturated packings in high dimensions . if true , then the @xmath520 and @xmath521 terms in ( [ sat - up ] ) are high - dimensional asymptotic corrections . observe that the extrapolation of the conservative fit function ( [ log ] ) to @xmath269 gives a density that is about 3.7 percent larger than that predicted by ( [ linear ] ) . this percentage difference between the two fit functions decreases as @xmath0 decreases to @xmath522 . this includes the operational ground - based laser interferometer gravitational - wave observatory ( ligo ) and the space - based laser interferometer space antenna ( lisa ) , which is expected to be operational in the next six years . see the link `` ' ' astro2010 : the astronomy and astrophysics decadal survey . " | it is known that the sphere packing problem and the number variance problem ( closely related to an optimization problem in number theory ) can be posed as energy minimizations associated with an infinite number of point particles in @xmath0-dimensional euclidean space @xmath1 interacting via certain repulsive pair potentials .
we reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in @xmath1 that generally involve single - body , two - body , three - body , and higher - body interactions .
this is done by linking the covering and quantizer problems to certain optimization problems involving the void " nearest - neighbor functions that arise in the theory of random media and statistical mechanics .
these reformulations , which again exemplifies the deep interplay between geometry and physics , allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems .
the covering and quantizer problems have relevance in numerous applications , including wireless communication network layouts , the search of high - dimensional data parameter spaces , stereotactic radiation therapy , data compression , digital communications , meshing of space for numerical analysis , and coding and cryptography , among other examples . in the first three space dimensions , the best known solutions of the sphere packing and number variance problems ( or their dual " solutions ) are directly related to those of the covering and quantizer problems , but such relationships may or may not exist for @xmath2 , depending on the peculiarities of the dimensions involved .
our reformulation sheds light on the reasons for these similarities and differences .
we also show that disordered saturated sphere packings provide relatively thin ( economical ) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions . in the case of the quantizer problem ,
we derive improved upper bounds on the quantizer error using sphere - packing solutions , which are generally substantially sharper than an existing upper bound in low to moderately large dimensions .
we also demonstrate that _ disordered _ saturated sphere packings yield relatively good quantizers .
finally , we remark on possible applications of our results for the detection of gravitational waves . |
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dark matter remains one of the most important unsolved problems in contemporary physics . astronomical observations indicate that the energy density of dark matter exceeds that of ordinary matter by a factor of five @xcite . extensive laboratory searches for weakly interacting massive particle ( wimp ) dark matter through scattering - off - nuclei experiments have failed to produce a strong positive result to date , see , e.g. , refs . @xcite , which has spurred significant interest of late in searching for alternate well - motivated forms of dark matter , such as ultralight ( sub - ev mass ) spin-0 particles that form either an oscillating classical field or topological defects , see , e.g. , refs . @xcite . the idea that the fundamental constants of nature might vary with time can be traced as far back as the large numbers hypothesis of dirac , who hypothesised that the gravitational constant @xmath1 might be proportional to the reciprocal of the age of the universe @xcite . more contemporary theories , which predict a variation of the fundamental constants on cosmological timescales , typically invoke a ( nearly ) massless underlying dark energy - type field , see , e.g. , the review @xcite and the references therein . most recently , a new model for the cosmological evolution of the fundamental constants of nature has been proposed in ref . @xcite , in which the interaction of an oscillating classical scalar dark matter field with ordinary matter via quadratic interactions produces both ` slow ' linear - in - time drifts and oscillating - in - time variations of the fundamental constants @xcite . topological defects , which are stable , extended - in - space forms of dark matter that consist of light scalar dark matter fields stabilised by a self - interaction potential @xcite and which interact with ordinary matter , produce transient - in - time variations of the fundamental constants @xcite . the oscillating - in - time and transient - in - time variations of the fundamental constants produced by scalar dark matter can be sought for in the laboratory using high - precision measurements , which include atomic clocks @xcite , highly - charged ions @xcite , molecules @xcite and nuclear clocks @xcite , in which two transition frequencies are compared over time . instead of comparing two transition frequencies over time , we may instead compare a photon wavelength with an interferometer arm length , in order to search for variations of the fundamental constants @xcite ( see also @xcite for some other applications ) . in the present work , we outline new laser interferometer measurements to search for variation of the electromagnetic fine - structure constant and particle masses ( including a non - zero photon mass ) . we propose a strontium optical lattice clock silicon single - crystal cavity interferometer as a novel small - scale platform for these new measurements . the small - scale hydrogen maser cryogenic sapphire oscillator system @xcite and large - scale gravitational - wave detectors , such as ligo - virgo @xcite , geo600 @xcite , tama300 @xcite , elisa @xcite or the fermilab holometer @xcite , can also be used as platforms for some of our newly proposed measurements . unless explicitly indicated otherwise , we employ the natural units @xmath2 in the present work . alterations in the electromagnetic fine - structure constant @xmath3 , where @xmath4 is the electron charge , @xmath5 is the reduced planck constant and @xmath6 is the photon speed , or particle masses ( including a non - zero photon mass @xmath7 ) produce alterations in the accumulated phase of the light beam inside an interferometer @xmath8 , since an atomic transition frequency @xmath9 and length of a solid @xmath10 , where @xmath11 is the number of atoms and @xmath12 is the bohr radius ( @xmath13 is the electron mass ) , both depend on the fundamental constants and particle masses . alterations in the accumulated phase can be expressed in terms of the sensitivity coefficients @xmath14 , which are defined by : @xmath15 where the sum runs over all relevant fundamental constants @xmath16 ( except photon mass ) . the sensitivity coefficients depend on the specific measurement that is performed . in order to define the variation of dimensionful parameters , such as @xmath13 , we assume that such variations are due to the interactions of dark matter with ordinary matter , see , e.g. , ref . @xcite . the sensitivity coefficients , which we derive below in sections [ sec:3 ] and [ sec:4 ] , are for single - arm interferometers , but are readily carried over to the case of two - arm michelson - type interferometers , for which the observable quantity is the phase difference @xmath17 between the two arms , as we illustrate with a couple of examples in section [ sec:5 ] . one intuitively expects that multiple reflections should enhance observable effects due to variation of the fundamental constants by the effective mean number of passages @xmath18 . this can be readily verified by the following simple derivation . for multiple reflections of a continuous light source that forms a standing wave ( in the absence of variation of the fundamental constants ) , we sum over all possible number of reflections @xmath19 : @xmath20 = \frac{1}{\exp \left ( \kappa - i \phi \right ) - 1 } , \ ] ] where @xmath21 is the attenuation factor that accounts for the loss of light amplitude after a single to - and - back passage along the length of the arm , and @xmath22 ( @xmath23 is an integer ) is the phase accumulated by the light beam in a single to - and - back passage along the length of the arm . for a large effective mean number of passages , @xmath24 , and for sufficiently small deviations in the accumulated phase , @xmath25 , the sum in eq . ( [ app - a_derivation1 ] ) can be written as : @xmath26 & \simeq n_{\textrm{eff } } \exp \left(i n_{\textrm{eff } } \cdot \delta \phi \right ) , \end{aligned}\ ] ] from which it is evident that the effects of small variations in the accumulated phase are enhanced by the factor @xmath18 . variation of @xmath0 and particle masses alters the accumulated phase through alteration of @xmath9 and @xmath10 . there are four main classes of experimental configurations to consider , depending on whether the frequency of light inside an interferometer is determined by a specific atomic transition ( i.e. , when the high - finesse cavity length is stabilised to an atomic transition ) or by the length of a resonator ( i.e. , when the laser is stabilised to a high - finesse cavity ) , and whether the interferometer arm length is allowed to vary freely ( i.e. , allowed to depend on the length of the solid spacer between the mirrors ) or its fluctuations are deliberately shielded ( i.e. , the arm length is made independent of the length of the solid spacer between the mirrors , e.g. , through the use of a multiple - pendulum mirror system ) . we consider each of these configurations in turn . the simplest case is when the frequency of light inside an interferometer is determined by an optical atomic transition frequency and the interferometer arm length is allowed to vary freely ( i.e. , allowed to depend on the length of the solid spacer between the mirrors ) . a strontium clock silicon cavity interferometer in its standard mode of operation falls into this category . in this case , the atomic transition wavelength and arm length are compared directly : @xmath27 where the optical atomic transition frequency @xmath9 is proportional to the atomic unit of frequency @xmath28 . variation of @xmath0 thus gives rise to the following phase shift : @xmath29 we note that the effect of variation of @xmath0 already appears at the non - relativistic level in eq . ( [ sensitivity_coefficient_1 ] ) , with the corresponding sensitivity coefficient @xmath30 . for systems consisting of light elements , the relativistic corrections to this sensitivity coefficient are small and can be neglected . this is in stark contrast to optical clock comparison experiments , for which @xmath31 in the non - relativistic approximation and the contributions to @xmath32 arise solely from relativistic corrections @xcite . for a strontium clock silicon cavity interferometer , which operates on the @xmath33sr @xmath34 @xmath35 @xmath36 transition ( @xmath37 nm ) and for which the cavity length is @xmath38 m @xcite , the phase shift in eq . ( [ sensitivity_coefficient_1 ] ) for a single to - and - back passage of the light beam is : @xmath39 for comparison , in a large - scale gravitational - wave detector of length @xmath40 km and operating on a typical atomic optical transition frequency , the phase shift for a single to - and - back passage of the light beam is : @xmath41 as noted in section [ sec : theory ] , multiple reflections enhance the coefficients in eqs . ( [ sensitivity_coefficient_1a ] ) and ( [ sensitivity_coefficient_1b ] ) by the effective mean number of passages @xmath18 , which depends on the reflectivity properties of the mirrors used . for large - scale interferometers , this enhancement factor is @xmath42 . for small - scale interferometers with highly - reflective mirrors , this enhancement factor can be considerably larger : @xmath43 . another possible system in this category is the hydrogen maser cryogenic sapphire oscillator system , which operates on the @xmath44h ground state hyperfine transition : @xmath45 \left [ \mu_p \frac{m_e}{m_p } \right ] , \ ] ] where @xmath46 is the relativistic casimir factor and @xmath47 is the dimensionless magnetic dipole moment of the proton in units of the nuclear magneton . in this case , changes in the measured phase have the following dependence on changes in the fundamental constants : @xmath48 where @xmath49 is the averaged light quark mass , and where we have used the calculated values @xmath50 @xcite and @xmath51 @xcite . if one performs two simultaneous interferometry experiments with two different transition lines , using the same set of mirrors , then one may search for variations of the fundamental constants associated with changes in the atomic transition frequencies : @xmath52 in particular , note that shifts in the arm lengths ( due to variation of the fundamental constants or undesired effects , such as seismic noise or tidal effects ) cancel in eq . ( [ seismic_shield ] ) . we also note that atomic clock transition frequencies may also be compared by locking lasers to the atomic transitions and using phase coherent optical mixing and frequency comb techniques to measure the laser frequency difference / ratio . if fluctuations in the arm length are deliberately shielded ( i.e. , the arm length is made independent of the length of the solid spacer between the mirrors , e.g. , through the use of a multiple - pendulum mirror system ) , but @xmath9 is still determined by an atomic transition frequency , then changes in the measured phase @xmath53 have the following dependence on changes in the fundamental constants : @xmath54 when a laser is locked to a resonator mode determined by the length of the resonator , @xmath9 is determined by the length of the resonator , which changes if the fundamental constants change . in the non - relativistic limit , the wavelength and arm length ( as well as the size of earth ) have the same dependence on the bohr radius , and so there are no observable effects if changes of the fundamental constants are slow ( adiabatic ) and if the interferometer arm length is allowed to vary freely ( i.e. , allowed to depend on the length of the solid spacer between the mirrors ) . indeed , this may be viewed as a simple change in the measurement units . transient effects due to the passage of topological defects may still produce effects , since changes in @xmath9 and @xmath55 may occur at different times . the sensitivity of laser interferometry to non - transient effects is determined by relativistic corrections , which we estimate as follows . the size of an atom @xmath56 is determined by the classical turning point of an external atomic electron . assuming that the centrifugal term @xmath57 is small at large distances , we obtain @xmath58 , where @xmath59 is the energy of the external electron and @xmath60 is the net charge of the atomic species ( for a neutral atom , @xmath61 ) . this gives the relation : @xmath62 . the single - particle relativistic correction to the energy in a many - electron atomic species is given by @xcite : @xmath63 where @xmath64 is the energy of the external atomic electron , @xmath65 is its angular momentum , @xmath66 is the nuclear charge , and @xmath67 is the effective principal quantum number . variation of @xmath0 thus gives rise to the following phase shift : @xmath68 \frac{\delta \alpha}{\alpha } .\ ] ] here @xmath69 is the atomic number of the atoms that make up the solid spacer between the mirrors of the resonator , while @xmath70 is the atomic number of the atoms that make up the arm . note that the sensitivity coefficient depends particularly strongly on the factor @xmath71 . @xmath72 for light atoms and may be of the order of unity in heavy atoms . if fluctuations in the arm length are deliberately shielded ( i.e. , the arm length is made independent of the length of the solid spacer between the mirrors ) and @xmath9 is determined by the length of the resonator , then changes in the measured phase @xmath73 have the following dependence on changes in the fundamental constants : @xmath74 a large - scale gravitational - wave detector ( such as ligo , virgo , geo600 or tama300 ) in its standard mode of operation falls into this category . a non - zero photon mass alters the accumulated phase through alteration of @xmath9 , @xmath75 ( where @xmath56 is the atomic radius ) and @xmath6 . in particular , if a non - zero photon mass is generated due to the interaction of photons with slowly moving dark matter ( @xmath76 ) , then the energy and momentum of the photons are approximately conserved and the photon speed changes according to : @xmath77 the effects of a non - zero photon mass in atoms are more subtle . the potential of an atomic electron changes from coulomb to yukawa - type : @xmath78 where the sum runs over all remaining atomic electrons . for @xmath79 , the leading term of the corresponding perturbation reads ( we omit the constant terms , which do not alter the atomic transition frequencies and wavefunctions ) : @xmath80 , \ ] ] which for a neutral atom takes the asymptotic forms : @xmath81 in the semiclassical approximation , it is straightforward to confirm that the dominant contribution to the expectation value of the operator ( [ yukawa_perturbation ] ) comes from large distances , @xmath82 , where the external electron sees an effective charge of @xmath83 . therefore , the shift in an atomic energy level @xmath84 is simply : @xmath85 where @xmath86 is the expectation value of the radius operator for state @xmath84 . typically , @xmath87 . assuming that the perturbation ( [ yukawa_perturbation ] ) is adiabatic and that the the dominant contribution to the matrix elements @xmath88 comes from large distances , application of time - independent perturbation theory gives the following shift in the size of the atomic orbit for state @xmath84 : @xmath89 where @xmath90 is the static dipole polarisability of state @xmath84 . static dipole polarisabilities for the electronic ground states of neutral atoms range from @xmath91 @xmath92 in hydrogen to @xmath93 @xmath92 in caesium @xcite . if @xmath9 is determined by an atomic transition frequency and the interferometer arm length is allowed to vary freely ( i.e. , allowed to depend on the length of the solid spacer between the mirrors ) , then a non - zero photon mass produces the following changes in the measured phase @xmath94 : @xmath95 where @xmath96 is the difference in the orbital size between the final and initial states involved in the radiative atomic transition , and @xmath97 is the static dipole polarisability of the atoms that make up the arm . the three separate contributions in eq . ( [ sensitivity_coefficient_1b ] ) scale roughly in the ratio @xmath98 , respectively , meaning that the contribution from the change in the photon speed dominates . if fluctuations in the arm length are deliberately shielded ( i.e. , the arm length is made independent of the length of the solid spacer between the mirrors ) , but @xmath9 is still determined by an atomic transition frequency , then a non - zero photon mass produces the following changes in the measured phase @xmath99 : @xmath100 where we again note that the contribution from the change in the photon speed dominates . if @xmath9 is determined by the length of the resonator and the interferometer arm length is allowed to vary freely ( i.e. , allowed to depend on the length of the solid spacer between the mirrors ) , then a non - zero photon mass produces the following changes in the measured phase @xmath101 : @xmath102 here @xmath103 is the static dipole polarisability of the atoms that make up the solid spacer between the mirrors of the resonator . the phase shift in eq . ( [ sensitivity_coefficient_2b ] ) is suppressed by the factor @xmath104 in the static limit ( compare with eqs . ( [ sensitivity_coefficient_1b ] ) and ( [ sensitivity_coefficient_4b ] ) above ) . however , for time - dependent effects , the phase shift can be significantly larger ( see the examples in section [ sec:5 ] ) . if fluctuations in the arm length are deliberately shielded ( i.e. , the arm length is made independent of the length of the solid spacer between the mirrors ) and @xmath9 is determined by the length of the resonator , then a non - zero photon mass produces the following changes in the measured phase @xmath105 : @xmath106 similarly to eq . ( [ sensitivity_coefficient_2b ] ) , the phase shift in eq . ( [ sensitivity_coefficient_3b ] ) is also suppressed by the factor @xmath104 in the static limit . however , we again note that the phase shift can be significantly larger for time - dependent effects ( see section [ sec:5 ] ) . oscillating classical dark matter exhibits not only temporal coherence @xcite , but also spatial coherence , with a coherence length given by : @xmath107 , where @xmath108 is the dark matter particle mass , and a virial ( root - mean - square ) speed of @xmath109 is typical in our local galactic neighbourhood . our solar system travels through the milky way ( and hence relative to galactic dark matter ) at a comparable speed @xmath110 . an oscillating scalar dark matter field takes the form : @xmath111 meaning that measurements performed on length scales @xmath112 are sensitive to dark matter - induced effects that arise from differences in the spatial phase term @xmath113 at two or more points . ) . , width=226 ] as a specific example , we consider measurements performed using a large - scale gravitational - wave detector with equal arm lengths that are deliberately shielded from fluctuations , @xmath114 , and with the emitted photon wavelength determined by the length of the resonator . since we are considering slowly moving dark matter ( @xmath76 ) , changes in the wavelength of the travelling photon are related to changes in @xmath6 by : @xmath115 ^ 2 / 2 \omega^2 $ ] , where the interaction between the photon field and @xmath116 may be interpreted as the varying photon mass : @xmath117 ^ 2 = ( m_\gamma)_{\textrm{max}}^2 { \cos^2 \left ( m_\phi t - m_\phi \left<{\boldsymbol{v}}\right > \cdot { \boldsymbol{r } } \right ) } $ ] . for the simplest case when the dark matter is incident directly onto one of the detector arms as shown in fig . [ fig : interferometer_spatial_coherence ] , the shift in the accumulated phase difference between the two arms is given by : @xmath118 dz , \ ] ] and to leading order we find : @xmath119 the shift in the accumulated phase difference between the two arms in eq . ( [ ligo_example_b ] ) is suppressed by the factor @xmath120 . topological defect dark matter is intrinsically coherent , both temporally and spatially . as a specific example , we again consider measurements performed using a large - scale gravitational - wave detector with equal arm lengths that are deliberately shielded from fluctuations and with the emitted photon wavelength determined by the length of the resonator . for the case of a 2d domain wall with a gaussian cross - sectional profile of root - mean - square width @xmath121 and which travels slowly ( @xmath122 ) in the geometry shown in fig . [ fig : interferometer_spatial_coherence ] , the interaction between the photon field and @xmath116 may be interpreted as the varying photon mass : @xmath123 ^ 2 = ( m_\gamma)_{\textrm{max}}^2 { \exp[-(z + vt)^2 / d^2 ] } $ ] . calculating the shift in the accumulated phase difference between the two arms , eq . ( [ ligo_example_a ] ) , we find to leading order : @xmath124 \right . \notag \\ & - \left . \frac{\sqrt{\pi}d}{2l } \left[\textrm{erf } \left ( \frac{l + tv + z_0 } { d } \right ) - \textrm{erf } \left ( \frac{tv + z_0 } { d } \right ) \right ] \right\ } , \end{aligned}\ ] ] where erf is the standard error function , defined as @xmath125 . the shift in the accumulated phase difference between the two arms in eq . ( [ ligo_example_c ] ) is largest for @xmath126 . for @xmath127 , the phase shift in ( [ ligo_example_c ] ) is suppressed by the factor @xmath128 . in the case when @xmath129 , the phase shift in ( [ ligo_example_c ] ) is suppressed by the factor @xmath130 when the topological defect envelops arm 2 but remains far away from arm 1 ; however , at the times when the topological defect envelops arm 1 , there is no such suppression . we have outlined new laser interferometer measurements to search for variation of the electromagnetic fine - structure constant @xmath0 and particle masses ( including a non - zero photon mass ) . we have proposed a strontium optical lattice clock silicon single - crystal cavity interferometer as a novel small - scale platform for these new measurements . our proposed laser interferometer measurements , which may also be performed with large - scale gravitational - wave detectors , such as ligo , virgo , geo600 or tama300 , may be implemented as an extremely precise tool in the direct detection of scalar dark matter . for oscillating classical scalar dark matter , a single interferometer is sufficient in principle , while for topological defects , a global network of interferometers is required . the possible range of frequencies for oscillating classical dark matter is @xmath131 ( corresponding to the dark matter particle mass range @xmath132 ) , while the timescale of passage of topological defects through a global network of detectors is @xmath133 s for a typical defect speed of @xmath134 km / s . the current best sensitivities to length fluctuations are at the fractional level @xmath135 in the frequency range @xmath136 hz for a large - scale gravitational - wave detector @xcite and at the fractional level @xmath137 in the frequency range @xmath138 hz for a silicon - based cavity @xcite . we are very grateful to jun ye and fritz riehle for suggesting the strontium clock silicon cavity interferometer as a suitable small - scale platform for our newly proposed measurements , and for important discussions . we are also grateful to an anonymous referee for suggesting the use of phase coherent optical mixing and frequency comb techniques , further to our proposal centred around eq . ( [ seismic_shield ] ) . we would like to thank bruce allen , dmitry budker , federico ferrini , hartmut grote , sergey klimenko , giovanni losurdo , guenakh mitselmakher and surjeet rajendran for helpful discussions . this work was supported by the australian research council . v. v. f. is grateful to the mainz institute for theoretical physics ( mitp ) for its hospitality and support . b. m. roberts , v. a. dzuba , v. v. flambaum , m. pospelov , y. v. stadnik , dark matter scattering on electrons : accurate calculations of atomic excitations and implications for the dama signal . arxiv:1604.04559 . o. k. baker , m. betz , f. caspers , j. jaeckel , a. lindner , a. ringwald , y. semertzidis , p. sikivie , k. zioutas , prospects for searching axionlike particle dark matter with dipole , toroidal , and wiggler magnets . _ phys . rev . d _ * 85 * , 035018 ( 2012 ) . m. pospelov , s. pustelny , m. p. ledbetter , d. f. j. kimball , w. gawlik , d. budker , detecting domain walls of axionlike models using terrestrial experiments . _ _ phys . lett . _ _ * * 110 * * , 021803 ( 2013 ) . y. v. stadnik , v. v. flambaum , axion - 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structure constant @xmath0 and particle masses ( including a non - zero photon mass ) .
we propose a strontium optical lattice clock
silicon single - crystal cavity interferometer as a novel small - scale platform for these new measurements .
our proposed laser interferometer measurements , which may also be performed with large - scale gravitational - wave detectors , such as ligo , virgo , geo600 or tama300 , may be implemented as an extremely precise tool in the direct detection of scalar dark matter that forms an oscillating classical field or topological defects . |
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the presence of large and rapidly varying electric and magnetic fields in relativistic heavy ion collisions results in charge - dependent effects , visible in a series of observables in the final state of the collision . these effects can be used as a new source of information on the space - time evolution of the non - perturbative process of particle production , and on the space - time properties of the system created in the heavy ion collision . to give one example , in 2007 we demonstrated that the distortion which the electromagnetic repulsion ( attraction ) of positive ( negative ) pions induced on charged pion ( @xmath1 ) ratios brought new information on the space - time scenario of fast pion production @xcite . in recent years , the general problematics of electromagnetically - induced effects in ultrarelativistic heavy ion reactions was subject of an important theoretical and experimental interest @xcite as it was connected to very interesting phenomena like the chiral magnetic effect ( cme @xcite ) . in the present paper we review our earlier studies of the electromagnetic distortion of charged pion spectra in the context of our more recent findings on the influence of spectator - induced @xmath4 and @xmath5 fields on the azimuthal anisotropies of charged pions . special attention is put on tracing the utility of both observables for studying the longitudinal evolution of the expanding matter created in the collision . a phenomenological model analysis is presented , aimed at explaining the space - time features of pion production which we deduced from the observed electromagnetic phenomena . of positively and negatively charged pions produced in peripheral pb+pb collisions at @xmath6 gev . the pion invariant density is drawn as a function of transverse momentum in fixed bins of @xmath7 as marked from top to bottom . the subsequent distributions are consecutively multiplied by 0.2 . the arrows point at the regions where the distortion induced by the spectator em - field is most visible . from @xcite.,title="fig:",scaledwidth=80.0% ] + the relatively moderate collision energy range available to the sps makes corresponding fixed - target experiments suitable for studying the electromagnetic influence of the spectator system on charged particle spectra in a large range of available rapidity . importantly , this includes the region of very low transverse momenta where the corresponding effects are expected to be largest . a detailed double - differential study of @xmath8 and @xmath9 densities as a function of longitudinal and transverse pion momentum is presented in fig . [ fig1a ] . the na49 experimental data cover , in the longitudinal direction expressed in terms of the c.m.s . feynman variable @xmath10 , the whole region from `` mid - rapidity '' ( @xmath11 ) up to @xmath12 which is about one unit above beam rapidity at lowest transverse momenta . the smooth exponential - like shape of the transverse momentum distribution gets visibly distorted in the region of low @xmath13 , where a dramatic decrease of invariant @xmath8 density and an accumulation of @xmath9 density is apparent as indicated by the arrows . this `` deformation '' is caused by the spectator system , which modifies the trajectories of charged pions by means of its space- and time - dependent @xmath4 and @xmath5 fields . the ratio of @xmath8 over @xmath9 density , fig . [ fig1](a ) , appears particularly sensitive to the spectator - induced electromagnetic field in the region of higher rapidity ( @xmath14 ) and lower transverse momenta . here , a deep two - dimensional `` valley '' is apparent with the @xmath1 ratio approaching zero in the region @xmath15 ( @xmath16 at low @xmath13 ) . note that with the pb nucleus composed of 39% protons over 61% neutrons , this implies breaking of isospin symmetry which unequivocally confirms the electromagnetic origin of the observed effect . quantitatively , this is confirmed in fig . [ fig1](b ) , where the observed distortion can be fairly well described by means of a simple two - spectator model with the two spectators assumed as lorentz - contracted homegenously charged spheres , and isospin effects being taken into account @xcite . it is important to underline that the unique free parameter in the model is the distance @xmath2 , in the longitudinal direction , between the pion emission point and the center of the spectator system . the reasonable agreement between data and model demonstrated in figs [ fig1](a),(b ) is obtained for values of @xmath2 in the range of 0.5 - 1 fm @xcite ; different values of @xmath2 lead to different detailed shapes of the distortion of @xmath1 ratios as described in @xcite . gev , ( b ) model simulation of this ratio as described in the text , ( c ) our monte carlo prediction for the ( pure ) electromagnetically - induced directed flow of positive pions , compared to the data from the wa98 experiment @xcite , ( d ) directed flow of charged pions in intermediate centrality au+au collisions @xcite , ( e ) , ( f ) electromagnetic component of @xmath8 and @xmath9 directed flow , extracted from star data @xcite and compared to our simulation made assuming @xmath17 fm . from : @xcite ( panels a , b ) , @xcite ( panel c ) , @xcite ( panels d , e , f).,title="fig:",scaledwidth=90.0% ] + in full analogy to charged pion ratios , the _ directed flow _ of charged pions emitted close to beam rapidity is also strongly affected by spectator - induced em effects . this is shown in fig . [ fig1](c ) where our prediction for a _ purely electromagnetic effect _ on the directed flow @xmath0 of positive pions is shown for three different values of the distance @xmath2 : 0 , 0.5 and 1 fm . as it can be seen in the figure , our monte carlo calculation shows that very large values of directed flow can be induced by the sole effect of electromagnetic repulsion of positive pions by the spectator system . our prediction is compared to the measurements provided by the wa98 collaboration at the same energy , @xmath6 gev @xcite . this comparison indicates that a very sizeable part of positive pion directed flow in the region close to beam / target rapidity can in fact come from the electromagnetic origin . at the same time , the wa98 experimental data apparently constrain the possible values of the distance @xmath2 , yielding the possible range of @xmath2 from 0 up to 1 fm . thus consistently from both observables ( @xmath1 ratios , fig . [ fig1](a ) and directed flow , fig . [ fig1](c ) ) , the longitudinal distance between the actual pion emission site and the center of the spectator system appears quite small , in the range below 1 fm . this small distance is to be viewed with respect to the longitudinal extent of the lorentz - contrated spectator system which is itself of the order of about 1 fm at this collision energy . the situation changes significantly when passing to pions produced close to _ central _ rather than _ beam _ rapidity . here experimental data on intermediate centrality au+au reactions exist from the star experiment at rhic @xcite at different collision energies ( from @xmath18 up to @xmath19 gev ) . the directed flow of positive and negative pions at the lowest available energy is presented in fig . [ fig1](d ) . a _ charge splitting _ is apparent between @xmath8 and @xmath9 . as shown in figs [ fig1](e),(f ) , the latter splitting can again be understood as a spectator - induced em effect , provided that a value of @xmath2 far larger than in the preceding case , @xmath17 fm , is assumed . gev . ( a ) subdivision of the nuclear matter distribution into longitudinal `` strips '' . ( b ) kinematical characteristics of the `` strips '' as a function of their position in the perpendicular plane ; the distance @xmath2 is indicated in the plot . ( c ) invariant mass of the `` strips '' projected in the perpendicular @xmath20 plane , where @xmath21 is the direction of the impact parameter vector . ( d ) longitudinal velocity @xmath22 of the `` strips '' as a function of their position . the `` hot '' participant and `` cold '' spectator regions are indicated in the plots.,title="fig:",scaledwidth=80.0% ] + this apparent sensitivity of the electromagnetic distortion of final state charged pion ratios and directed flow to the distance between the pion formation zone and the spectator system provides , in the opinion of the authors , a completely new and very welcome tool for studying the space - time evolution of charged particle production in the soft sector of ultrarelativistic heavy ion collisions . specifically , the elongation of the distance @xmath2 with decreasing pion rapidity is the reflection of the longitudinal evolution of the system created in the collision . summing up the findings from the precedent section , in our studies we obtained : * @xmath23 fm for pions moving at rapidities comparable to @xmath24 ( from our study based on na49 @xcite and wa98 @xcite data ) ; * @xmath17 fm for pions moving at central rapidities ( @xmath25 , from our study based on star data @xcite ) . while the mere fact that @xmath2 evolves with pion rapidity is simply the confirmation of the expansion of the system in the longitudinal direction , the latter is , especially at high pion rapidities , poorly known to hydrodynamical calculations due to the presence of a sizeable baryochemical potential @xcite , and difficult to access experimentally e.g. in lhc experiments ( in contrast to sps energies where the na49 and na61/shine experiments cover the whole region from @xmath26 to @xmath27 and above in the collision c.m.s . @xcite ) . in the present section we discuss this issue in the context of energy - momentum conservation in the initial state of the collision , in a model proposed by a.s . the spatial nuclear matter distribution in the volume of the two colliding nuclei is considered in a two - dimensional @xmath20 projection perpendicular to the collision axis ; peripheral pb+pb collisions at top sps energy are presented in fig . [ fig2](a ) . the resulting `` strips '' of highly excited nuclear ( or partonic ) matter , fig . [ fig2](b ) , define the kinematical properties of the longitudinal expansion of the system as a function of collision geometry . these are shown in figs [ fig2](c ) and ( d ) in the perpendicular @xmath20 plane . for the peripheral collision considered here , the overall energy available for particle production ( invariant mass of the `` strips '' as defined assuming local energy - momentum conservation ) has a well - defined `` hot '' peak at mid - distance between the centers of the two nuclei , and gradually decreases when approaching each of the two `` cold '' spectator systems . on the other hand , the longitudinal velocity @xmath22 of the `` strips '' depends strongly on their position in the @xmath20 plane . a careful comparison of figs [ fig2](c)-(d ) shows that significantly excited volume elements of the longitudinally expanding system can move at very large longitudinal velocities , comparable to that of the spectator system . assuming a given proper hadronization time of the different volume elements , a natural picture emerges . pions produced at high rapidity ( dominantly from `` strips '' moving at large values of the longitudinal velocity @xmath22 ) will emerge at a small distance from the `` cold '' spectator systems ; these originating from `` hot '' central `` strips '' , at smaller values of @xmath28 , will evidently show up at larger values of the distance @xmath2 . altogether , we conclude that a non - negligible amount of experimental data on charge - dependent effects in particle spectra and anisotropic flow exists , and much more can be obtained from existing fixed - target as well as collider experiments . these data can be used to trace the influence of the electric and magnetic fields in heavy ion collisions , which should be useful in future studies related to the chiral magnetic effect , the electromagnetic properties of the quark - gluon plasma , and others . our own studies demonstrate the sensitivity of the em - induced distortions of charged particle spectra and directed flow to the space - time scenario of particle production in heavy ion collisions , and allow us to trace the longitudinal evolution of the expanding matter created in the course of the collision . + + the authors , and especially a.r . , gratefully thank the organizers of the x workshop on particle correlations and femtoscopy ( wpcf 2014 ) , for their invitation and for the excellent organization of such a fruiful and interesting workshop . u. grsoy , d. kharzeev and k. rajagopal , phys . c * 89 * , 054905 ( 2014 ) [ arxiv:1401.3805 [ hep - ph ] ] . v. voronyuk , v. d. toneev , s. a. voloshin and w. cassing , phys . c * 90 * , no . 6 , 064903 ( 2014 ) [ arxiv:1410.1402 [ nucl - th ] ] . h. schlagheck ( wa98 collaboration ) , nucl . a * 663 * , 725 ( 2000 ) [ nucl - ex/9909005 ] . l. adamczyk _ et al . _ ( star collaboration ) , phys . * 112 * , 162301 ( 2014 ) [ arxiv:1401.3043 [ nucl - ex ] ] . a. rybicki and a. szczurek , phys . c * 87 * , 054909 ( 2013 ) [ arxiv:1303.7354 [ nucl - th ] ] , and references therein . a. rybicki , a. szczurek and m. klusek - gawenda , epj web conf . * 81 * , 05024 ( 2014 ) . | the large and rapidly varying electric and magnetic fields induced by the spectator systems moving at ultrarelativistic velocities induce a charge splitting of directed flow , @xmath0 , of positive and negative pions in the final state of the heavy ion collision . the same effect results in a very sizeable distortion of charged pion spectra as well as ratios of charged pions ( @xmath1 ) emitted at high values of rapidity .
both phenomena are sensitive to the actual distance between the pion emission site and the spectator system .
this distance @xmath2 appears to decrease with increasing rapidity of the pion , and comes below @xmath31 fm for pions emitted close to beam rapidity . in this paper
we discuss how these findings can shed new light on the space - time evolution of pion production as a function of rapidity , and on the longitudinal evolution of the system created in heavy ion collisions . |
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laser spectroscopy of optical transitions in highly charged ions ( hcis ) is a subject of considerable interest as it provides access to relativistic effects in few - electron systems and can be used to test bound - state qed in the extremely strong electric and magnetic fields in the vicinity of the ionic nucleus @xcite . experimentally , such magnetic dipole ( m1 ) transitions in mid-@xmath1 hcis have first been studied in electron - beam ion traps ( ebits ) by laser excitation and fluorescence detection @xcite , yielding a relative accuracy of a few ppm for the determination of the wavelength . direct laser spectroscopy of heavy ( high-@xmath1 ) hcis has so far only been performed at the experimental storage ring esr on hydrogen - like bismuth @xmath2bi@xmath3 @xcite and lead @xmath4pb@xmath5 @xcite . in both cases , the transition between the ground state hyperfine levels was induced by pulsed lasers and resonance fluorescence was recorded . these investigations have been extended to the ground - state hyperfine transition in lithium - like bismuth @xmath2bi@xmath6 , which has recently been observed in the experimental storage ring ( esr ) @xcite . this measurement in combination with the measurement on hydrogen - like bismuth will allow the first determination of the so - called specific difference between the hyperfine splittings @xmath7 as suggested by shabaev and co - workers @xcite . the first observation of the transition in @xmath2bi@xmath6 is an important step , but it will not provide sufficient accuracy for a high - precision determination of the qed effects in the specific difference , since the wavelength determination for both transitions ( h - like and li - like ) is still limited in accuracy due to the large doppler width and the uncertainty of additional doppler shifts caused by the relativistic ion motion in the storage ring . this will be considerably improved once high-@xmath1 highly charged ions are available at rest in a clean environment allowing for high - accuracy laser spectroscopy . to this end , the spectrap experiment has been designed @xcite . it is part of the highly charged heavy ion trap ( hitrap ) project @xcite at the gsi helmholtzzentrum darmstadt , which will provide hcis up to u@xmath8 at low energies suitable for capture into a penning trap . the precision achieved in the laser spectroscopy of trapped ions crucially depends on the width of the optical transition of interest and the mechanisms that lead to additional broadening , e.g. doppler broadening . the study of forbidden transitions with high accuracy requires the elimination of doppler broadening . this can be achieved by first - order doppler - free techniques like two - photon transitions or by the trapping and cooling of atoms or ions . there is a variety of corresponding methods for the cooling of the ion motion , for a detailed overview see e.g. @xcite . the evaporative cooling of hcis in an ebit has been used for the laser spectroscopy of ar@xmath9 @xcite and recently in a penning trap on hcis that were produced in an ebit and then transported and re - trapped in a penning trap @xcite . at spectrap we make use of resistive cooling @xcite and laser cooling @xcite . the former is a very effective cooling mechanism for hcis , while the latter is most effective for ions with a level scheme suitable for laser cooling such as be@xmath10 or mg@xmath10 . laser - cooled ions can then be used for sympathetic cooling @xcite of simultaneously trapped hcis . such experiments have so far been performed with be@xmath10 in a penning trap @xcite and are foreseen in a paul trap @xcite . here , we present first studies with laser - cooled mg@xmath10 ions in the spectrap penning trap . we have performed systematic measurements with externally produced mg ions which have been captured in flight and stored . the observation of laser - induced fluorescence ( lif ) down to the single - ion level allows a determination of the ion storage time , ion number and ion temperature . evidence for the formation of ion crystals has been observed . these measurements represent an initial characterization and optimization of the system as an important step towards the sympathetic cooling and precision laser spectroscopy of highly charged ions . penning traps are well - established tools for capture and confinement of externally produced ions . a static homogeneous magnetic field ensures radial confinement , while the electrode arrangement produces an electrostatic potential well which provides axial confinement of charged particles . ions can thus be localized , which allows laser irradiation and fluorescence detection under well - controlled conditions . stored ions can be motionally cooled to reduce the doppler broadening of transition lines to well below the ghz level . the achievable storage time is fundamentally limited only by the residual gas pressure inside the trap , since collisions with gas particles may lead to ion loss . typical storage times range from seconds to minutes , but also storage times of several months have been achieved @xcite . hence , also slow transitions like magnetic dipole ( m1 ) transitions can be investigated with high resolution and statistics . such traps have been realized in numerous variations especially concerning their geometry , for details see @xcite . for the purposes of laser spectroscopy , trap geometries need to be chosen such that they allow both ions and light to enter and leave the trap suitably , as well as to provide the means for observing the fluorescence . the spectrap experiment employs a five - pole cylindrical penning trap with open endcaps @xcite , with an additional pair of capture electrodes , as described in detail in @xcite . the geometry is chosen such that the trap is orthogonal , i.e. the trapping potential depth is independent from the choice of correction voltages used to make the trapping potential harmonic close to the trap centre . the ion motion in such a trap has been discussed in detail in e.g. @xcite . the open endcaps and capture electrodes yield axial access to the trap from both sides . in our case , the ions enter from the top and the cooling laser from below , as shown in fig . [ fig : traplaser ] . the capture of externally produced ions is achieved by fast switching of trap voltages . the ring electrode is radially split into four segments to allow the use of a rotating wall @xcite for ion cloud compression and shaping . a central hole in each ring segment enables the detection of the stored ion fluorescence on radially positioned detectors outside the magnet vessel . the fluorescence light emerging out of the holes is collimated by plano - convex lenses . the geometrical light collection efficiency of this system is the main limiting factor of the total fluorescence detection efficiency . also , reflection and absorption in the lens and the vacuum windows as well as misalignments of the main optical axis reduce the signal . at the wavelength used for laser cooling of mg@xmath11 , the detection efficiency was measured to be about @xmath12 . the trap is installed in a vertical , cold - bore , superconducting magnet with helmholtz configuration , such that direct optical access to the trap centre is possible through four radial ports in the horizontal plane . before it was consigned to gsi , the magnet was used for a similar experiment ( retrap at lawrence livermore national laboratory ) , with a slightly different penning trap configuration and radially cooled @xmath13 ions @xcite . the magnetic field in the trap centre can be set to any value up to 6 t and provides a relative central homogeneity of @xmath14 over a region of 2.5 cm . a liquid helium cryostat is used for cooling both the superconducting solenoids and the trap with its attached electronics . the residual gas pressure in the vacuum system is monitored in the room temperature region at the bottom of the magnet vessel , and typically amounts to @xmath15 mbar during magnet operation . there is no direct separation between the trap and the insulation vacuum of the cryostat , so additional cryopumping of the volume inside the trap is provided by the cold surfaces . hence , the vacuum conditions inside the trap can be assumed to be much better than indicated by the gauge , as will be discussed below . laser beams are guided into the trap along the central vertical axis , from a laser laboratory located under the superconducting magnet setup , as shown in fig.[fig : trapsetup ] . the fluorescence light is detected by a channel photo multiplier attached to the outside of the magnet vessel . it has a quantum efficiency of 18% and a very low dark count of some 20 hz . it is well suited for detection of uv light between 200 and 400 nm . because of its sensitivity to the stray magnetic field it was mounted in a magnetically shielded housing about 1 metre away from the main magnet chamber , as depicted in fig . [ fig : trapsetup ] . to avoid excess heating of the cryostat and prepare the system for injection of externally produced hci , mg ions are produced by an off - line ion source . it consists of a directly heated tungsten crucible filled with grains of mg metal . mg atoms leaving the crucible are ionized inside a cup - formed grid by electrons emitted from a thoriated tungsten filament located outside the grid . the potential of the grid sets the energy of the produced ions . they are collimated with an einzel lens and enter a 90@xmath16 quadrupole deflector , which guides them into the vertical part of the beamline . the quadrupole geometry is chosen to allow injection of ions from both sides of the beamline and to have free access along the vertical axis for the laser beam . two additional einzel lenses in the vertical beamline prepare the ion bunch for injection into the magnetic field and guide them into the trap . the second arm of the horizontal beamline will be connected to an ebit and later to the hitrap cooling trap in order to trap heavy hci provided by the gsi accelerator facility . mg ions are produced in bunches of 1 - 2 @xmath17s length at a rate of a few hz . they are transported towards the trap with a kinetic energy of 200 ev and dynamically captured into the penning trap . one typical trapping cycle is illustrated in fig . [ fig : trappingcycle ] . initially , only the lower capture ( reflector ) electrode is permanently switched high ( closed ) , while the upper capture electrode is switched between a confining potential and a value just below the ion transport energy , synchronized with the arrival time of the ion bunch . it has been experimentally observed that around 50 ev out of 200 ev axial energy are transferred into the radial motion during the ion injection into the magnetic field . that is sufficient for the accumulation of many ion bunches , with minimal losses of ions already stored during the reopening of the capture electrode . typically 50 - 200 such accumulation cycles are repeated before permanently closing the capture electrode . the voltage on the endcaps and correction electrodes is then slowly ( with respect to the ion motion ) ramped up in order to compresses the ion cloud towards the trap centre . afterwards , laser cooling by scanning the laser wavelength as well as electronic ion excitation and detection are performed . a fast cooling method , such as laser cooling , is needed in order to rapidly decrease the energy of stored ions and reduce losses . as previously stated , direct laser cooling is limited to ions with a favourable level structure . such ions can then be used for sympathetic cooling of other ions of interest which are simultaneously stored . a suitable species for laser cooling is the @xmath18mg@xmath0 ion . it can be easily produced and the undisturbed ion provides a closed , ground - state , two - level @xmath19 - @xmath20 transition , with an excited state natural lifetime of only 4 ns . however , magnetic fields of several tesla in the penning trap lead to splitting of the mg sublevels due to the zeeman effect , as shown in fig . [ fig : mgzeeman ] . the zeeman slope coefficient for each sublevel is provided in table [ zeemanshift ] . ground state transition in @xmath21 . the necessary polarisation for driving the corresponding transition is also indicated.,scaledwidth=45.0% ] ccc * level * & * @xmath22 * & * @xmath23(ghz / t ) * + @xmath24s@xmath25 & @xmath26 & @xmath27 + @xmath24s@xmath25 & @xmath28 & @xmath29 + @xmath24p@xmath30 & @xmath31 & @xmath32 + @xmath24p@xmath30 & @xmath26 & @xmath33 + @xmath24p@xmath30 & @xmath28 & @xmath34 + @xmath24p@xmath30 & @xmath35 & @xmath36 + [ zeemanshift ] in this level scheme only the @xmath37 ( here and in further text the @xmath22 quantum numbers ) zeeman transitions remain closed systems . we have chosen the lowest transition @xmath38 for the cooling process . it has a zeeman shift of @xmath27 ghz / t compared to the unperturbed @xmath24s@xmath39p@xmath30 transition frequency of @xmath40 ghz @xcite . considering the required polarization for the cooling laser and the injection direction of the ions , the laser is polarized @xmath41 and sent along the trap axis . a specific issue of the experiment is the relatively high kinetic energy of the captured ions , required for an efficient transport and a small ion bunch width . frequency detuning of the laser corresponding to a typical transport energy of 200 ev together with a very fast adjustment of this detuning to match the dropping energy caused by the cooling would be a serious technical challenge . however , a much simpler approach can be employed at the expense of the cooling speed : since the axial speed of the injected ions varies between a maximum corresponding to 200 ev at the trap centre and zero at the turning points near the endcaps , the laser can be kept fixed at a frequency corresponding to a cold ion . it can then absorb a photon near each turning point @xcite which is still efficient provided that the laser intensity is sufficiently large . the condition is that the rabi frequency for this transition is much larger than the axial frequency of the ion inside the trap , @xmath42 . it should be noted that only the axial ion motion will be directly cooled this way , since there is no cooling force acting on the ion cloud in the radial direction . an all - solid - state laser system at the required wavelength of 279 nm has been set up as depicted in fig . [ fig : laser ] . it has been described in detail in @xcite and comprises a single - mode fiber laser at 1118 nm as well as two cavities for second harmonic generation to obtain frequency quadrupling . the laser is a koheras boostik fiber laser , specified to deliver 1.66 w maximum output power . experimentally , a maximum of 1.2 w including the amplified spontaneous emission ( ase ) was obtained . : the laser beam from a fiber laser is frequency quadrupled using two non - linear crystals in bow - tie resonators with the length stabilized using the hnsch - couillaud locking scheme . the wavelength is controlled using a high - finesse wavemeter . ( @xmath43 and @xmath44 : quarter- and half - wave plate respectively , app : anamorphic prism pair , m1-m8 cavity mirrors , fc : fiber coupler),scaledwidth=43.0% ] since the doubling efficiency for second harmonic generation ( shg ) inside a non - linear crystal is proportional to the square of the fundamental power , bow - tie optical resonators were constructed to enhance the laser power inside the crystal . the first doubler uses non - critical phase - matching in a lithium triborate ( lbo ) non - linear crystal , which has a phase - matching temperature at 1118 nm of @xmath45 90@xmath46c . the second doubler employs critical phase - matching of a beta barium borate ( bbo ) non - linear crystal , using the round , gaussian output of the first doubler . both resonators were designed using the ray transfer matrix analysis and computer simulations in order to obtain the optimal experimental parameters according to the boyd - kleinmann theory . the length of the doublers is actively stabilized using hnsch - couillaud polarization - analysis locking @xcite . for the first doubler , a maximum overall shg efficiency of 33% was obtained , providing 320 mw of laser power at 559.3 nm from 950 mw of the fundamental 1118.5 nm power . for the second doubler , a maximum of 16.7 mw at 279.6 nm was achieved using 210 mw of green power in front of the resonator , equivalent to an overall shg efficiency of 8% . all power levels of the harmonics were measured after appropriate filtering of the fundamental power leaking from the bow - tie resonators . in spite of several problems with the koheras main fiber laser , which greatly affected the available pump power , a sufficient amount of about 2 mw uv laser power was available for the first trapping tests . the parameters of the frequency quadrupling system are summarized in table [ resonatorsummary ] . ccc * parameter * & * @xmath47 resonator * & * @xmath48 resonator * + crystal type & lbo & bbo + phase matching & ncpm type i & cpm type i + crystal length & 20 mm & 7.4 mm + input wavelength @xmath49 & 1118.54 nm & 559.27 nm + crystal cut @xmath50 & 90@xmath46 & 44.4@xmath46 + crystal cut @xmath51 & 0@xmath46 & 0@xmath46 + crystal surfaces & dual ar & brewster - cut + crystal temperature & 96@xmath46c & 50@xmath46c + total cavity length & 1158 mm & 504 mm + focusing mirrors @xmath52 & 70 mm & 50 mm + focusing arm length & 157.5 mm & 104.8 + full folding angle & 38@xmath46 & 18.8@xmath46 + enhancement factor @xmath53 & @xmath4560 & @xmath4570 + coupling efficiency & @xmath5485% & @xmath5485% + input power ( @xmath55 ) & 950 mw & 210 mw + output power ( @xmath56 ) & 320 mw & 16.7 mw + doubling efficiency & @xmath4533@xmath57 & @xmath458@xmath57 + + the storage time constant has been determined by monitoring the laser induced fluorescence of the trapped ions as a function of time . for this measurement , the laser has been tuned to a frequency @xmath45 200 mhz above the resonance frequency of the @xmath58 transition in order to avoid strong laser cooling and the resulting fluctuation of the fluorescence due to the decreasing doppler width of the transition . on the other hand , although a blue - detuned laser frequency leads to heating of the ion cloud , it is important to note that it is still red - detuned for the other mg isotopes which might be present . additionally , resistive cooling contributes to ion cooling with a time constant of around 100 s for @xmath59 stored in spectrap @xcite . thus , an equilibrium between heating and cooling processes is established , resulting in a fluorescence signal proportional to the number of stored ions . a typical trace resulting from this procedure is shown in fig . [ fig : ionlifetime ] . a storage time constant of about 140 s can be extracted from a single exponential fit to the data . since in this measurement the ions were not cooled to sub - k temperature it can be regarded as a lower limit for the ion lifetime in the trap . stored in spectrap . the @xmath58 transition was continuously excited with a 200 mhz blue - detuned laser . the observed fluorescence rate is plotted as a function of time . at @xmath60 loading was complete and the endcaps were closed . the data was fitted with a single exponential function for points with @xmath61 s and the storage time constant of 137(5 ) s was observed.,scaledwidth=42.0% ] after improving the laser stability and the vacuum conditions , longer storage times of up to an hour have recently been observed for a cold cloud of @xmath21 ions @xcite . further efforts in this direction are ongoing . unlike @xmath21 , the storage time of hci can be significantly smaller and an estimate should be made . ion loss is mainly attributed to charge exchange with residual gas particles . although the cross sections for electron capture in ion - neutral collisions at very low energies are largely unknown , they can be estimated using the semi - empirical mller - salzborn formula @xcite @xmath62 \label{eq : schlachter}\ ] ] where @xmath63 is the charge state of the ion and @xmath64 is the ionization potential of the residual gas particle expressed in ev , which amounts to @xmath65 ev for h@xmath66 @xcite and @xmath67 ev for he . partial pressures of all other typical residual gases are much smaller at the cryogenic temperature around the trap , and can be safely neglected . alternatively , the cross section can also be estimated using the so - called classical barrier model @xcite which brings similar results . the rate @xmath68 of electron capture is then calculated by multiplying the cross - section @xmath69 from eq . by the neutral particle density @xmath70 and the relative velocity @xmath71 of the two colliding particles . the expected ion storage time is given through the reciprocal value of this rate @xmath72 where @xmath73 is the boltzmann constant , @xmath74 the pressure , @xmath75 and @xmath76 are the mass and temperature of the residual gas atoms and the ions , respectively . assuming that the pressure inside the trap volume is not worse than @xmath77 mbar , which corresponds to @xmath78 mbar measured in the 300 k region and scaled down to liquid helium temperature , the lifetime of around 160 s and 18 s is calculated using eq . for charge states @xmath79 and @xmath80 , respectively . these lifetime - estimates were calculated for ion temperatures around 1 k or less and rapidly decrease with increasing ion temperature , pointing towards the need for rapid ion cooling , such as sympathetic cooling with laser cooled @xmath21 . the laser cooling time can be estimated by evaluating the cooling force exerted by the photons . generally , for low energy ions , the scattering of the photons leads to a frictional force @xmath81 which slows the ion down . it can be written as @xcite @xmath82 where @xmath83 is the photon momentum , @xmath84 is the transition linewidth , @xmath64 is the intensity of the laser , @xmath85 the saturation intensity and @xmath86 the detuning of the laser frequency . this force is proportional to the laser intensity below the saturation value and it approaches its maximum value @xmath87 for intensities @xmath88 . here , the kinetic energy of the captured ions is typically 200 ev , and the laser frequency can not be scanned fast enough to maintain the cooling condition . the laser is thus kept fixed at a small red - detuning and the scattering force in eq . ( [ fscatt0 ] ) can then be written as @xmath89 where @xmath90 is a scattering frequency inversely proportional to the ion velocity . this equation is a good approximation for an intensity close to or above the saturation intensity . since under our experimental conditions the laser intensity was about 1/3 of the saturation intensity , the cooling force is reduced by a factor of @xmath91 . the deceleration can then be expressed as @xmath92 where @xmath93 is the wavelength of the cooling laser and @xmath94 the mass of @xmath18mg@xmath0 . the stopping time can be calculated accordingly as @xmath95 where @xmath96 is the initial speed of the ions . if @xmath90 is approximated as the maximal scattering frequency reduced by the ratio between the natural linewidth @xmath84 and the doppler width @xmath97 of the transition , in our case this stopping time approximation results in @xmath98 s. the experimental cooling time was determined using the same measurement procedure as described in the previous section for recording the lif signal shown in fig . [ fig : ionlifetime ] . initially , the ions have a large spatial oscillation amplitude between the endcaps while the fluorescence detection system is focused on a small volume at the centre of the trap . hence , the emitted photons can not be recorded and only the background signal is present . as the ions are cooled , they get localized in the centre of the trap and the fluorescence rate per ion rises as the doppler - shifted transition matches the fixed red - detuned laser frequency for an increasing amount of ions . this results in the sharp rise in fluorescence observed in fig . [ fig : ionlifetime ] , which appears about 10 seconds after raising the endcaps to the trapping potential . the measured value is of the same order of magnitude , but smaller than the predicted 40 s because of uneven ion velocity distribution and a finite probability for photon absorption also outside the natural linewidth of the transition . in contrast to trapping a large ion cloud , with the current experimental setup it was also possible to isolate and observe the fluorescence of a single trapped ion . the spectra shown in fig . [ fig : singleions ] were recorded by scanning the laser from -1 ghz with 100 mhz / s across the resonance , using a 0.9 mw laser beam with a diameter of @xmath99 mm . taken under identical conditions and trapping times , they show quantized changes of the laser induced fluorescence , associated with single trapped ions on top of a constant background signal . in order to verify that the single - ion regime was reached the following procedure was carried out : first an integrated number of detected photons over the resonance was determined for each of the recorded few - ion spectra . the error was treated as the statistical uncertainty with an added offset from the average deviation of the background . the number of photons per ion was then varied between 10 and 500 and compared to the recorded spectra , producing a deviation for each point . these deviations were used to calculate the reduced chi - square for each assumed number of photons per ion . the result of this procedure is plotted in fig . [ fig : chisquare ] . the area around @xmath100 corresponds to the most probable number of integrated photons per ion ( roughly @xmath101 ) which was used to determine the number of ions in the spectra shown in fig . [ fig : singleions ] . represents the most probable value of roughly @xmath101 photons.,scaledwidth=42.0% ] according to it , a single stored @xmath21 ion in full resonance yields a rate of around @xmath102 fluorescence photons per second , on top of around @xmath103 background photons per second . the uncertainty of this value is dictated by the fluctuation of the background and the frequency uncertainty of the scanning laser frequency . having that in mind it can be concluded that under the given conditions a single ion yields a fluorescence signal of @xmath104 photons / s . in spite of the large uncertainty of around 20% , this result shows that even a few trapped ions with a fast optical transition can be detected in spectrap via laser induced fluorescence . if the observed number of fluorescence photons is compared to the maximum expected number of photons from a non - fully saturated @xmath21 ion , this yields a total detection efficiency of @xmath105 . disregarding the quantum efficiency of the detector and considering only geometrical factors the detection efficiency amounts to about @xmath106 . additionally , the expected number of photons per ion can be used to quantify a signal from an ion cloud and estimate the number of ions stored under the same conditions . a maximum of about 2000 ions were trapped and cooled using the current ion source and @xmath99 mw / mm@xmath107 of cooling laser intensity . as an addition to lif detection , spectrap can also perform ft - icr ( fourier transform ion cyclotron resonance ) measurements , a well - established technique for non - destructive mass and charge state spectrometry in ions traps @xcite . by a combination of both , it is possible to gauge the electronic signal height obtained in ft - icr to the number of observed ions as measured by lif . hence , a stored ion cloud can be characterized by the ion number and temperature . for ft - icr , the ion motion is excited by a fix - phase burst such that subsequent signal pickup of the ions oscillatory motions is efficient . a transient of that signal is recorded and its fourier transform represents a spectrum of the mass - to - charge spectrum of ions present in the trap . in the present case , the signal for ion excitation is generated by an agilent 33250a frequency generator and processed by the spectrap rotating wall drive @xcite , which splits the input signal into two with a 180@xmath16 phase difference . these two signals are transmitted to two opposing ring segments , while the remaining two segments are kept at dc potential . the signal induced in the trap electrodes by the excited ions is amplified by a cryogenic amplifier mounted next to the trap and processed by a hp3589a spectrum analyser . the dipole excitation was performed by applying 5000 cycles of a 2.555 mhz signal , where the modified cyclotron resonance @xmath108 was expected for a magnetic field of 4 t. the amplitude was set to @xmath109 . because of the short coherence time the spectrum analyser was triggered by the last excitation cycle and averaged over 10 excitation - detection rounds . the observed resonance signal is depicted in fig . [ fig : cycresonance ] . the width of the resonance is @xmath45 100 hz , while the modified cyclotron frequency can be determined with an accuracy of a few hz . the resulting mass resolving power @xmath110 is of the order of @xmath111 , while the magnetic field can be determined with a relative accuracy of @xmath112 . both of these values exceed the requirements of the experiment and show that electronic and optical ion detection can be performed simultaneously . ; @xmath108 is the modified cyclotron frequency.,scaledwidth=42.0% ] the transfer of energy into the cyclotron motion during excitation pushes the ions into larger orbits , where they either have a smaller overlap with the laser beam , or are even lost from the trap . this was used to perform a measurement of the modified cyclotron frequency via lif . the ions were laser - cooled and their fluorescence recorded while applying the dipole excitation to the ring electrode . the excitation frequency was changed stepwise across the expected cyclotron resonance , while the trap was reloaded under identical conditions for each point . this resulted in a fluorescence dip seen in fig . [ fig : cyclaser ] , fitted well with a gaussian function , with the central frequency marking the resonance . it was noticed that the range of possible excitation amplitudes was rather narrow - excitation with more than 400 mv@xmath113 resulted in a total loss of fluorescence or even ion loss , i.e. the fluorescence did not return after switching off the excitation . conversely , amplitudes smaller than 100 mv@xmath113 had little or no observable influence on the ion fluorescence . .,scaledwidth=42.0% ] it can be seen that the central frequencies from the two measurements , shown in fig . [ fig : cycresonance ] and fig . [ fig : cyclaser ] , differ by 1.33 khz . this exceeds the statistical fitting uncertainty and was found in several repeated measurements . the systematic shift corresponds to a magnetic field difference of 2 mt . the discrepancy is ascribed to the non - ideal magnetic field and the different spatial positions where the two measurements were performed : while the ft - icr induces a signal directly in the trap electrodes , the lif - signal depends on the optical axis of the detector system , which is not necessarily aligned exactly along the trap radial axis . adding a camera to the system will allow us to measure position , shape and radial extent of the cloud , which is of special interest when the rotating wall is applied . nevertheless , it was demonstrated that electronic and laser induced fluorescence ion detection methods can be used simultaneously , with reasonably good agreement . by determining the transition linewidth @xmath114 and assuming the absence of line - broadening mechanisms other than doppler broadening , the upper limit to the ion temperature @xmath115 can be calculated according to @xcite @xmath116 a series of measurements was performed in order to determine the transition linewidth of the laser - cooled @xmath21 . the laser frequency was kept 1 ghz red - detuned during ion accumulation , as well as for another 10 seconds after closing the trap . it was observed that due to the large initial ion energy this pre - cooling time was necessary for efficient laser cooling . after pre - cooling , the laser frequency was scanned over the central transition frequency of mg@xmath11 and the fluorescence recorded . a typical result is shown in fig . [ fig : crystal ] , where the recorded fluorescence rate was plotted against the laser frequency detuning . as a function of frequency detuning . the width of the measured transition can be used to set the upper limit to the achieved ion temperature . a precooling peak and an abrupt drop after crossing the zero is typically associated with crystalline structure of the ion cloud.,scaledwidth=42.0% ] after crossing the resonance frequency , ion cooling turns into heating and the fluorescence drops quickly to zero . it can therefore be safely assumed that the total fwhm of the voigt profile is less than twice the observed width of 33(10 ) mhz indicated in fig . [ fig : crystal ] . this value is of the same order of magnitude as the natural linewidth of the transition ( 42 mhz ) , and a deconvolution of the doppler and the natural linewidth contribution to the line profile needs to be performed @xcite . the deconvoluted value for the doppler width can be inserted into eq . and an upper limit for the ion temperature is obtained @xmath117 by experience from a similar experiment @xcite , the typical volume of such an ion cloud is of the order of 0.5 mm@xmath118 , resulting in an ion number density of around @xmath119 ions / mm@xmath118 for @xmath21 ions stored in spectrap . under such conditions the single - particle description begins to break down and the ion cloud has to be treated as a non - neutral plasma . the plasma coupling parameter , describing the ion coulomb coupling intensity in one - component plasmas , is given by @xcite @xmath120{\frac{3}{4\pi n } } \label{eq : plasmacoupling}\ ] ] where @xmath121 is the wigner - seitz radius . gilbert and co - workers have predicted that for coupling parameters @xmath122 the plasma starts gradually to exhibit liquid - like properties @xcite . according to eq . , for @xmath21 trapped and cooled in spectrap ( @xmath123 ions / mm@xmath118 and @xmath124 ) the plasma coupling parameter amounts to @xmath125 , such that strong ion coupling may be assumed . after the planned introduction of hci into the trap , the temperatures of the two components are expected to roughly equalize , resulting in a much larger @xmath126 for high charge states @xcite . studies have already shown that for sufficiently low temperatures , a trapped ion cloud exhibits a structural change and its spectrum resembles the one of single ions @xcite . a similar behaviour was observed for the trapped @xmath21 in spectrap and is shown in fig . [ fig : crystal ] . a small pre - cooling peak appears at the point where the transition s doppler broadened half - width becomes smaller than the laser detuning ( here at @xmath45 400 mhz ) , after which the fluorescence disappears and can be observed again only close to the natural linewidth of the transition . such structures were observed also in @xcite and mark the transition of the stored ion plasma from a non - correlated to a strongly coupled state . because of strong cooling and simultaneous reduction of the doppler width , the fluorescence close to the resonant frequency is characterized by a sharp asymmetric shape , followed by an abrupt drop to zero after crossing the central frequency . the spectrum shown in fig . [ fig : crystal ] was recorded with 1.1 mw / mm@xmath107 of laser power and a 100 mhz / s frequency sweep . it has been observed that different pre - cooling times cause different positions of the pre - cooling peak with respect to the main one , moving them closer together for shorter pre - cooling times . this structure was , however , not observed for very short pre - cooling times below roughly 8 s , which were also typically followed by much smaller or no detectable fluorescence . however , after allowing sufficient pre - cooling time and observing the crystalline structure , a smaller , sharp fluorescence peak was observed at resonance even when scanning the laser frequency in the opposite direction . we have performed systematic measurements with laser - cooled @xmath127mg@xmath128 ions stored in a penning trap . these ions were externally produced , transported , captured and stored in the trap for subsequent measurements . using both optical and electronic non - destructive detection techniques , the properties of stored ion clouds were determined . combining electronic with optical detection , it is possible to determine stored ion numbers down to the single ion level and to characterize the stored ion cloud with respect to its temperature , storage time and related properties . laser cooling was achieved to temperatures below 0.1 k and evidence of ion crystallization was found . such laser - 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the externally produced ions were captured in flight , stored and laser cooled .
laser - induced fluorescence was observed perpendicular to the cooling laser axis
. optical detection down to the single ion level together with electronic detection of the ion oscillations inside the penning trap have been used to acquire information on the ion storage time , ion number and ion temperature .
evidence for formation of ion crystals has been observed .
these investigations are an important prerequisite for sympathetic cooling of simultaneously stored highly - charged ions and precision laser spectroscopy of forbidden transitions in these . |