Hunting for the non-Hermitian exceptional points with fidelity susceptibility

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 +infinity 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 the EP, we found that the EP can be found when chi approaches -infinity. As examples, we investigate the simplest PT symmetric 2 x 2 Hamiltonian with a single tuning parameter and the non-Hermitian Su-Schriffer-Heeger model.