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arXiv:2003.10099 · Europhysics Letters (EPL)

Effective non-Hermitian physics for degenerate ground states of a non-Hermitian Ising model with $\mathcal{RT}$ symmetry

Can Wang, Meng-Lei Yang, Cui-Xian Guo, Xiao-Ming Zhao, Su-Peng Kou

Published Jan 20, 2020• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/128/41001

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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.
The ground-state degeneracy of the quantum spin system is a characteristic of nontrivial topology when it is gapped and robust against disordered perturbation. The corresponding quantum phase transition (QPT) is usually driven by a real parameter. We study a non-Hermitian Ising chain with two transverse fields, one real and another imaginary, based on the exact solution and numerical simulation. We show that topological degeneracy still exists and can be obtained by an imaginary transverse field from a topologically trivial phase of a Hermitian system. The topological degeneracy is robust against the random imaginary field and therefore expected to be immune to disordered dissipation from the spontaneous decay in experiment. The underlying mechanism is the nonlocal symmetry, which emerges only in thermodynamic limit and relates two categories of QPTs in the quantum spin system, rooted from topological order and symmetry breaking, respectively.
We investigate the low-energy physics of non-Hermitian quantum spin models with PTPTPT symmetry. To this end, we consider the one-dimensional Ising chain and the two-dimensional toric code in a non-Hermitian staggered field. For both systems, dual descriptions in terms of non-Hermitian staggered Ising interactions in a conventional transverse field exist. We perform high-order series expansions about the high- and low-field limit for both systems to determine the ground-state energy per site and the one-particle gap. The one-dimensional non-Hermitian Ising chain is known to be exactly solvable. Its ground-state phase diagram consists of second-order quantum phase transitions, which can be characterized by logarithmic singularities of the second derivative of the ground-state energy and, in the symmetry-broken phase, the gap closing of the low-field gap. In contrast, the gap closing from the high-field phase is not accessible perturbatively due to the complex energy and the occurrence of exceptional lines in the high-field gap expression. For the two-dimensional toric code in a non-Hermitian staggered field, we study the quantum robustness of the topologically ordered phase by the gap closing of the low-field gap. We find that the well-known second-order quantum phase transition of the toric code in a uniform field extends into a large portion of the non-Hermitian parameter space. However, the series expansions become unreliable for a dominant anti-Hermitian field. Interestingly, the analysis of the high-field gap reveals the potential presence of an intermediate region.
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 In this manuscript, we explore the properties of a non-Hermitian spin-1/2 XY model subjected to alternating imaginary and transverse magnetic fields. Focusing on a two-spin system, we systematically construct the ground state phase diagram and provide an exact calculation of the ground state entanglement via the Negativity measure. Our findings reveal that, within eigenstates influenced by the anisotropy parameter, real magnetic fields, and imaginary magnetic fields, the anisotropy parameter significantly enhances entanglement, whereas the real magnetic field tends to diminish or even annihilate it. Notably, in the ΡΤ-symmetry broken phase, quantum entanglement demonstrates increased resilience to variations in the real magnetic field with the strengthening of the imaginary magnetic field. For eigenstates governed purely by the imaginary magnetic field, we observe that the two-spin entanglement remains maximal (i.e., value of 1) within the ΡΤ-symmetry region, while it gradually declines in the ΡΤ-symmetry broken region as the parameter η0 decreases. An intriguing observation is that the first derivative of the Negativity shows non-analytic behavior at the critical points, highlighting the role of Negativity as a reliable and effective indicator of phase transitions in this non-Hermitian system.
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.