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arXiv:1907.06699 · Physical review. B./Physical review. B

Non-Hermitian dynamic strings and anomalous topological degeneracy on a non-Hermitian toric-code model with parity-time symmetry

Cui-Xian Guo, Xiaoran Wang, Can Wang, Su-Peng Kou

Published Apr 29, 2020• 41 citations
arXivHTMLPDF

Abstract

In this paper, with the help of Hermitian/non-Hermitian dynamic strings, a theory of non-Hermitian topological orders is developed based on a non-Hermitian Wen-plaquette model. The effective models for bosonic topological excitations ($e$ particles and $m$ particles) are Hermitian tight-binding lattice models; the effective model for fermionic topological excitations ($f$ particles) becomes a non-Hermitian tight-binding lattice model. In addition, the effective pseudospin model for topologically degenerate ground states is derived by calculating the expectation values of Hermitian/non-Hermitian topologically closed dynamic strings. For the topologically degenerate ground states of the non-Hermitian Wen-plaquette model on even-by-odd, odd-by-even, and odd-by-odd lattices, anomalous topological degeneracy occurs, i.e., the number of topologically protected ground states may be reduced from 2 to 1. Now, the effective pseudospin model turns into the typical $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian with spontaneous $\mathcal{PT}$ symmetry breaking. At exceptional points, the topologically degenerate ground states merge with each other and the topological degeneracy turns into non-Hermitian degeneracy. In the end, the application of the non-Hermitian ${Z}_{2}$ topological order and its possible physics realization are discussed.

DOI
10.1103/physrevb.101.144439

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In this paper, with the help of Hermitian/non-Hermitian dynamic strings, a theory of non-Hermitian topological orders is developed based on a non-Hermitian Wen-plaquette model. The effective models for bosonic topological excitations (eee particles and mmm particles) are Hermitian tight-binding lattice models; the effective model for fermionic topological excitations (fff particles) becomes a non-Hermitian tight-binding lattice model. In addition, the effective pseudospin model for topologically degenerate ground states is derived by calculating the expectation values of Hermitian/non-Hermitian topologically closed dynamic strings. For the topologically degenerate ground states of the non-Hermitian Wen-plaquette model on even-by-odd, odd-by-even, and odd-by-odd lattices, anomalous topological degeneracy occurs, i.e., the number of topologically protected ground states may be reduced from 2 to 1. Now, the effective pseudospin model turns into the typical PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonian with spontaneous PT\mathcal{PT}PT symmetry breaking. At exceptional points, the topologically degenerate ground states merge with each other and the topological degeneracy turns into non-Hermitian degeneracy. In the end, the application of the non-Hermitian Z2{Z}_{2}Z2​ topological order and its possible physics realization are discussed.
In this paper, based on a non-Hermitian toric-code model we found the fragileness of intrinsic topological orders under non-Hermitian perturbations. The effect of non-Hermitian avalanche for a designed toric-code model is uncovered: for the designed toric-code model with special external fields, a tiny non-Hermitian perturbation (local imaginary state-selective dissipation) leads to anomalous topological degeneracy and breaks down the bulk-degeneracy correspondence (a correspondence between bulk quasi-particles and topologically protected degenerate ground states). To support our theoretical predictions, we calculated the ground-state degeneracy and fidelity susceptibility of the ground states based on the non-Hermitian toric-code model on lattice and that on lattice. The numerical results are consistent with theoretical prediction.
Abstract The existence of topological zero modes in nontrivial phase of quantum Ising chain results in not only the Kramers-like degeneracy spectrum, but also dynamic response for non-Hermitian perturbation in the ordered phase (2021 Phys. Rev. Lett. 126 116 401). In this work, we investigate the possible response of the degeneracy spectrum for Hermitian perturbations. We provide a single-particle description of the model in the ordered phase, associating with an internal degree of freedom characterized as a pseudospin. The effective magnetic field, arising from both local and nonlocal perturbations in terms of string operators, acts on the pseudospin. We show that the action of string operator can be realized via a quench under the local perturbations. As an application, any ground states and excited states for the Hamiltonian with perturbation can be selected to identify the quantum phase, by adding the other perturbations to trigger a quench and measuring the Loschmidt echo.
Effective non-Hermitian physics for degenerate ground states of a non-Hermitian Ising model with RT\mathcal{RT}RT symmetry
In this paper, based on a one-dimensional non-Hermitian spin model with -invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high-order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists spontaneous (anti) -symmetry breaking for the topological degenerate ground states with tunable parameters in external fields. We also found that even a very tiny imaginary external field applied will drive phase transition.
Landau's spontaneous symmetry breaking theory is a fundamental theory that describes the collective behaviors in many-body systems. It was well known that for usual spontaneous symmetry breaking in Hermitian systems, the order-disorder phase transition with gap closing and spontaneous symmetry breaking occur at the same critical point. In this paper, we generalized the Landau's spontaneous symmetry breaking theory to the cases in non-Hermitian (NH) many-body systems with biorthogonal Z2 symmetry and tried to discover certain universal features. We were surprised to find that the effect of the NH terms splits the spontaneous biorthogonal Z2 symmetry breaking from a (biorthogonal) order-disorder phase transition with gap closing. The sudden change of similarity for two degenerate ground states indicates a new type of quantum phase transition without gap closing accompanied by spontaneous biorthogonal Z2 symmetry breaking. We will take the NH transverse Ising model as an example to investigate the anomalous spontaneous symmetry breaking. The numerical results were consistent with the theoretical predictions.
We introduce the concept of boundary degeneracy, as the ground state degeneracy of topologically ordered states on a compact orientable spatial manifold with gapped boundaries. We emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the Z2{Z}_{2}Z2​ toric code and Z2{Z}_{2}Z2​ double-semion model [more generally, the Zk{Z}_{k}Zk​ gauge theory and the $U{(1)}_{k}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}k}$ nonchiral fractional quantum Hall state at even integer kkk] can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.
In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2AZ2AZ2A or Z2BZ2BZ2B projective symmetry group. We find that the Z2AZ2AZ2A translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translation-symmetry protected topological orders. We introduced four Z2{Z}_{2}Z2​ topological indices ${\ensuremath{\zeta}}_{\mathbf{k}}=0,1$ at k=(0,0)\mathbf{k}=(0,0)k=(0,0), $(0,\ensuremath{\pi})$, $(\ensuremath{\pi},0)$, and $(\ensuremath{\pi},\ensuremath{\pi})$ to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2AZ2AZ2A topological orders for a system on three-dimensional cubic lattice.