General properties of fidelity in non-Hermitian quantum systems with PT symmetry

TL;DR

This paper explores the fundamental properties of fidelity in non-Hermitian PT-symmetric quantum systems, revealing how fidelity behaves near exceptional points and applying these insights to specific models like the SSH and XXZ chains.

Contribution

It establishes general properties of fidelity in PT-symmetric non-Hermitian systems and demonstrates their implications through analysis of specific quantum models.

Findings

Fidelity is always real for PT-unbroken states.

Re[{ extstyle ext{X}}_F] corresponds to PT partner states in PT-broken phase.

Re[{ extstyle ext{F}}] equals 1/2 at second-order exceptional points.

Abstract

The fidelity susceptibility is a tool for studying quantum phase transitions in the Hermitian condensed matter systems. Recently, it has been generalized with the biorthogonal basis for the non-Hermitian quantum systems. From the general perturbation description with the constraint of parity-time (PT) symmetry, we show that the fidelity $\mathcal{F}$ is always real for the PT-unbroken states. For the PT-broken states, the real part of the fidelity susceptibility $\mathrm{Re}[\mathcal{X}_F]$ is corresponding to considering both the PT partner states, and the negative infinity is explored by the perturbation theory when the parameter approaches the exceptional point (EP). Moreover, at the second-order EP, we prove that the real part of the fidelity between PT-unbroken and PT-broken states is $\mathrm{Re}\mathcal{F}=\frac{1}{2}$. Based on these general properties, we study the two-legged non-Hermitian Su-Schrieffer-Heeger (SSH) model and the non-Hermitian XXZ spin chain. We find that for both interacting and non-interacting systems, the real part of fidelity susceptibility density goes to negative infinity when the parameter approaches the EP, and verifies it is a second-order EP by $\mathrm{Re}\mathcal{F}=\frac{1}{2}$.