Hunting for the non-Hermitian exceptional points with fidelity susceptibility

TL;DR

This paper extends the concept of fidelity susceptibility to non-Hermitian quantum systems, revealing that exceptional points can be identified by a divergence to negative infinity, unlike the Hermitian case.

Contribution

It introduces a geometric approach to fidelity susceptibility in non-Hermitian systems and demonstrates how exceptional points are characterized by a divergence to negative infinity.

Findings

Exceptional points occur when fidelity susceptibility approaches -∞.

Fidelity susceptibility can be computed algebraically or numerically away from EP.

Application to PT-symmetric and SSH models confirms the theoretical predictions.

Abstract

The fidelity susceptibility has been used to detect quantum phase transitions in the Hermitian quantum many-body systems over a decade, where the fidelity susceptibility density approaches $+\infty$ in the thermodynamic limits. Here the fidelity susceptibility $\chi$ is generalized to non-Hermitian quantum systems by taking the geometric structure of the Hilbert space into consideration. Instead of solving the metric equation of motion from scratch, we chose a gauge where the fidelities are composed of biorthogonal eigenstates and can be worked out algebraically or numerically when not on the exceptional point (EP). Due to the properties of the Hilbert space geometry at EP, we found that EP can be found when $\chi$ approaches $-\infty$. As examples, we investigate the simplest $\mathcal{PT}$ symmetric $2\times2$ Hamiltonian with a single tuning parameter and the non-Hermitian Su-Schriffer-Heeger model.