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arXiv:2108.05730 · Physical review. D/Physical review. D.

Galois conjugates of string-net model

Chao‐Yi Chen, Bing-Xin Lao, Xin-Yang Yu, Ling-Yan Hung

Published Mar 21, 2022• 6 citations
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Abstract

We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate that these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate biorthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the biorthogonal basis the topological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates.

DOI
10.1103/physrevd.105.065009

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We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate that these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate biorthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the biorthogonal basis the topological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates.
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.
We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the ((5±5)/2)((5\pm\sqrt{5})/2)((5±5​)/2)-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.
In quantum mechanics, string-net condensed states---a family of prototypical states exhibiting nontrivial topological order---can be classified via their long-range entanglement properties, in particular, topological corrections to the prevalent area law of the entanglement entropy. Here we consider classical analogs of such string-net models whose partition function is given by an equal-weight superposition of classical string-net configurations. Our analysis of the Shannon and Renyi entropies for a bipartition of a given system reveals that the prevalent volume law for these classical entropies is augmented by subleading topological corrections that are intimately linked to the anyonic theories underlying the construction of the classical models. We determine the universal values of these topological corrections for a number of underlying anyonic theories including SU(2)k{}_{k}k​, SU(N)1{(N)}_{1}(N)1​, and SU(N)2{(N)}_{2}(N)2​ theories.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of projected entangled pair states, topological order is characterized by an entanglement symmetry of the local tensor that describes the model. This symmetry can take the form of a tensor product of group representations [for quantum double models D(G)D(G)D(G) of a group G]G]G], or in the more general case a correlated symmetry action in the form of a matrix product operator (MPO), which encompasses all string-net models, including those that are not quantum double models. Among other things, these entanglement symmetries allow for the succinct description of ground states and topological excitations (anyons). Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and we use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net model and its Galois conjugate, namely the non-Hermitian double Yang-Lee (DYL) string-net model. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and we show that these can be mapped to certain restricted solid on solid (RSOS) models, which are equivalent to the $[(5\ifmmode\pm\else\textpm\fi{}\sqrt{5})/2]$-state Potts model, respectively. Moreover, the order parameter for condensation maps to the RSOS order parameter. The known exact solutions of the statistical models subsequently allow us to locate the critical points of the models, and to predict the critical exponents for the order parameters from conformal field theory. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the behavior of order parameters for condensation and deconfinement.
We ask which topological phases can and can not be realized by exactly soluble string-net models. We answer this question for the simplest class of topological phases, namely, those with Abelian braiding statistics. Specifically, we find that an Abelian topological phase can be realized by a string-net model if and only if (i) it has a vanishing thermal Hall conductance and (ii) it has at least one Lagrangian subgroup, a subset of quasiparticles with particular topological properties. Equivalently, we find that an Abelian topological phase is realizable if and only if it supports a gapped edge. We conjecture that the latter criterion generalizes to the non-Abelian case. We establish these results by systematically constructing all possible Abelian string-net models and analyzing the quasiparticle braiding statistics in each model. We show that the low-energy effective field theories for these models are multicomponent U(1)\text{U}(1)U(1) Chern-Simons theories, and we derive the KKK-matrix description of each model. An additional feature of this work is that the models we construct are more general than the original string-net models, due to several new ingredients. First, we introduce two new objects $\ensuremath{\gamma},\ensuremath{\alpha}$ into the construction which are related to Z2{\mathbb{Z}}_{2}Z2​ and Z3{\mathbb{Z}}_{3}Z3​ Frobenius-Schur indicators. Second, we do not assume parity invariance. As a result, we can realize topological phases that were not accessible to the original construction, including phases that break time-reversal and parity symmetry.