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

Yi-Ting Tu1, Iksu Jang2, Po-Yao Chang2, and Yu-Chin Tzeng3,4

1Department of Physics, University of Maryland, College Park, MD, USA
2Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan
3Physics Division, National Center for Theoretical Sciences, Taipei 106319, Taiwan
4Center for Theoretical and Computational Physics, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan

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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 $\mathrm{Re}\mathcal{F}=\frac{1}{2}$.

There are various definitions of fidelity in non-Hermitian systems, leading to confusion among researchers on which definition to use and the potential for different results. The present study proposes metricized fidelity, which has many desirable general properties. In PT-symmetric non-Hermitian systems, the PT-unbroken state is characterized by a real fidelity, while for PT-broken states, the real part of the fidelity susceptibility approaches negative infinity as the parameter approaches the exceptional point. Furthermore, the study proves that the real part of the fidelity between PT-unbroken and PT-broken states is always 1/2 at the second-order exceptional point. This definition provides clarity and consistency in the study of non-Hermitian systems, potentially enabling more accurate and comprehensive investigations of non-Hermitian systems in the future.

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