Phase Transitions and Generalized Biorthogonal Polarization in Non-Hermitian Systems

TL;DR

This paper advances the understanding of non-Hermitian systems by generalizing biorthogonal polarization as a real-space topological invariant and analyzing phase transitions with exact solutions.

Contribution

It extends biorthogonal polarization to systems with multiple boundary modes and proposes a method to find all bulk states in non-Hermitian models.

Findings

Biorthogonal polarization is invariant under basis and local unitary transformations.

The generalized method accurately finds bulk states in non-Hermitian systems.

Analysis reveals unique non-Hermitian features at phase transitions.

Abstract

Non-Hermitian (NH) Hamiltonians can be used to describe dissipative systems, and are currently intensively studied in the context of topology. A salient difference between Hermitian and NH models is the breakdown of the conventional bulk-boundary correspondence invalidating the use of topological invariants computed from the Bloch bands to characterize boundary modes in generic NH systems. One way to overcome this difficulty is to use the framework of biorthogonal quantum mechanics to define a biorthogonal polarization, which functions as a real-space invariant signaling the presence of boundary states. Here, we generalize the concept of the biorthogonal polarization beyond the previous results to systems with any number of boundary modes, and show that it is invariant under basis transformations as well as local unitary transformations. Additionally, we propose a generalization of a perviously-developed method with which to find all the bulk states of system with open boundaries to NH models. Using the exact solutions in combination with variational states, we elucidate genuinely NH aspects of the interplay between bulk and boundary at the phase transitions.