16747983
Non-Abelian quantum Hall states and their quasiparticles: From the pattern of zeros to vertex algebra
In the pattern-of-zeros approach to quantum Hall states, a set of data (n;m;S{sub a}|a=1,...,n;n,m,S{sub a} is n element of N) (called the pattern of zeros) is introduced to characterize a quantum Hall wave function. In this paper we find sufficient conditions on the pattern of zeros so that the data correspond to a valid wave function. Some times, a set of data (n;m;S{sub a}) corresponds to a unique quantum Hall state, while other times, a set of data corresponds to several different quantum Hall states. So in the latter cases, the pattern of zeros alone does not completely characterize the quantum Hall states. In this paper, we find that the following expanded set of data (n;m;S{sub a};c|a=1,...,n;n,m,S{sub a} is an element of N;c is an element of R) provides a more complete characterization of quantum Hall states. Each expanded set of data completely characterizes a unique quantum Hall state, at least for the examples discussed in this paper. The result is obtained by combining the pattern of zeros and Z{sub n} simple-current vertex algebra which describes a large class of Abelian and non-Abelian quantum Hall states PHI{sub Z{sub n}{sup sc}}. The more complete characterization in terms of (n;m;S{sub a};c) allowsmore » us to obtain more topological properties of those states, which include the central charge c of edge states, the scaling dimensions and the statistics of quasiparticle excitations.« less
54896614
Galois conjugates of topological phases
Galois conjugation relates unitary conformal field theories and topological quantum field theories (TQFTs) to their nonunitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois-conjugated Hamiltonians are typically non-Hermitian, we find that their ground-state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground-state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such nonunitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this, we rigorously prove that no local change of basis can transform the ground states of the Galois-conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasilocal Hamiltonians. A similar statement holds for many other nonunitary TQFTs. One consequence is that these nonunitary TQFTs do not describe physical realizations of topological phases. In particular, this implies that the ``Gaffnian'' wave function can not be the ground state of a gapped fractional quantum Hall state.
54954168
From String Nets to Nonabelions
We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.
11394439
Identifying topological order by entanglement entropy
Topological entanglement entropy provides a robust measure for detecting the long-range entanglement that characterizes quantum ground states displaying topological order. A new method for calculating this entropy isolates minimally entangled states from the ground states of a topological phase—offering a reliable test for identifying topological spin liquids.
18058928
Density of the SO(3) TQFT Representation of Mapping Class Groups
We show that for an odd prime r>3 and an integer g>1, in the projective representation given by the SO(3) Witten-Reshitikhin-Turaev theory at an rth root of unity, the image of the mapping class group of a surface of genus g is dense.
19017470
Topological properties of Abelian and non-Abelian quantum Hall states classified using patterns of zeros
It has been shown that different Abelian and non-Abelian fractional quantum Hall states can be characterized by patterns of zeros described by sequences of integers ${{S}_{a}}$. In this paper, we will show how to use the data ${{S}_{a}}$ to calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types and their quantum numbers, as well as the actions of the quasiparticle tunneling and modular transformations on the degenerate ground states on torus.
54712303
Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model
We study a quartic matrix model with partition function $$Z=\int d\ M\exp \mathrm{Tr}\ (-\Delta M^2-\frac{\lambda }{4}M^4)$$Z=∫dMexpTr(-ΔM2-λ4M4). The integral is over the space of Hermitian $$(\varLambda +1)\times (\varLambda +1)$$(Λ+1)×(Λ+1) matrices, the matrix $$\Delta $$Δ, which is not a multiple of the identity matrix, encodes the dynamics and $$\lambda >0$$λ>0 is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series and hence is a well-defined analytic function of the coupling constant in certain analytic domain of $$\lambda $$λ, by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual $$\phi ^4$$ϕ4 theory on the 2-dimensional Moyal space also called the 2-dimensional Grosse–Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable.
99565521
Study of the psi(2S) decay to p anti-p pi+ pi- (pi0)
16091296
Dielectron cross-section measurements in nucleus-nucleus reactions at 1-A/GeV
We present measured dielectron production cross sections for Ca+Ca, C+C, He+Ca, and d+Ca reactions at 1.0 A GeV . Statistical uncertainties and systematic effects are smaller than in previous dilepton spectrometer (DLS) nucleus-nucleus data. For pair mass M{le}0.35 GeV/c{sup 2} we obtain (1) the Ca+Ca cross section is larger than the previous DLS measurement and current model results, (2) the mass spectra suggest large contributions from {pi}{sup 0} and {eta} Dalitz decays, and (3) d{sigma}/dM{proportional_to}A{sub P}A{sub T}. For M{gt}0.5 GeV/c{sup 2} the Ca+Ca to C+C cross section ratio is significantly larger than the ratio of A{sub P}A{sub T} values. {copyright} {ital 1997} {ital The American Physical Society}
124095561
THE QUANTUM R-MATRIX VIA CANONICAL QUANTIZATION IN THE LIOUVILLE THEORY
Starting from the classical Liouville theory, we study its quantum theory through canonical quantization. We find that if the Poisson bracket relations between two vectors are dominated by the classical r-matrix in the classical case, their quantum analogue is replaced by the exchange relations dominated by the quantum R-matrix. The quantum group structure in the quantum Liouville theory is studied and the central charge of the quantum Liouville theory is also obtained.
55540815
Abelian and non-Abelian anyons in integer quantum anomalous Hall effect and topological phase transitions via superconducting proximity effect
We study the quantum anomalous Hall effect described by a class of two-component Haldane models on square lattices. We show that the latter can be transformed into a pseudospin triplet p + ip-wave paired superfluid. In the long wavelength limit, the ground-statewave function is described by Halperin's ( 1,1,-1) state of neutral fermions analogous to the double-layer quantum Hall effect. The vortex excitations are charge e/2 Abelian anyons which carry a neutral Dirac fermion zero mode. The superconducting proximity effect induces "tunneling" between "layers" which leads to topological phase transitions whereby the Dirac fermion zero mode fractionalizes and Majorana fermions emerge in the edge states. The charge e/2 vortex excitation carrying a Majorana zero mode is a non-Abelian anyon. The proximity effect can also drive a conventional insulator into a quantum anomalous Hall effect state with a Majorana edge mode and the non-Abelian vortex excitations.