High-order series expansion of non-Hermitian quantum spin models

TL;DR

This paper develops high-order series expansions to analyze the low-energy properties and phase transitions of non-Hermitian quantum spin models, including the 1D Ising chain and 2D toric code, revealing complex phase behavior and robustness of topological order.

Contribution

It introduces a novel high-order series expansion method for non-Hermitian quantum spin models, providing insights into their phase diagrams and topological robustness.

Findings

Exact solvability of 1D non-Hermitian Ising chain and its phase transitions.

Extension of the toric code phase diagram into non-Hermitian parameter space.

Identification of potential intermediate phases in high-field regimes.

Abstract

We investigate the low-energy physics of non-Hermitian quantum spin models with $PT$-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.