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.
Highlights
The overlap of two states, or fidelity Fh(|ψ, |φ ) = ψ|φ φ|ψ to be more specific, is used in the quantum information sciences as an estimation of the similarity of two quantum states
The fidelity Fh losses its meaning in the non-Hermitian quantum systems because the inner product in conventional quantum mechanics leads to all kinds of weird behaviors, e.g., faster-than-light communication [19], and entanglement increasing under local operations and classical communications (LOCC) [20]
There are few non-Hermitian generalizations of fidelity suggested in the literature very recently [39,40], we propose a more natural generalization by taking the aforementioned Hilbert space geometry into account
Summary
The fidelity Fh losses its meaning in the non-Hermitian quantum systems because the inner product in conventional quantum mechanics leads to all kinds of weird behaviors, e.g., faster-than-light communication [19], and entanglement increasing under local operations and classical communications (LOCC) [20]. To generalize the fidelity for non-Hermitian systems, it is necessary to use a proper inner product. The exceptional point (EP) [25,26,27,28] of the non-Hermitian systems is a special point in the parameter space where both eigenvalues and eigenstates merge into only one value and state. The eigenstates of a non-Hermitian Hamiltonian are not orthogonal with each other using the conventional inner product. There are few non-Hermitian generalizations of fidelity suggested in the literature very recently [39,40], we propose a more natural generalization by taking the aforementioned Hilbert space geometry into account
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