On the metric of a non-Hermitian model

Ebru Ergün

Xin Wang, Haodi Liu, Libin Fu

Marcos J. Hernández, Bogar Díaz, J. David Vergara

Jun-Feng Ren, Jing Li, Hai-Tao Ding +1 more

The geometric properties of quantum states are fully encoded by the quantum geometric tensor. The real and imaginary parts of the quantum geometric tensor are the quantum metric and Berry curvature, which characterize the distance and phase difference between two nearby quantum states in Hilbert space, respectively. For conventional Hermitian quantum systems, the quantum metric corresponds to the fidelity susceptibility and has already been used to specify quantum phase transitions from the geometric perspective. In this paper, we extend this wisdom to the non-Hermitian systems for revealing non-Hermitian critical points. To be concrete, by employing numerical exact diagonalization and analytical methods, we calculate the quantum metric and corresponding order parameters in various non-Hermitian models, which include two non-Hermitian generalized Aubry-Andr\'e models and non-Hermitian cluster and mixed-field Ising models. We demonstrate that the quantum metric of eigenstates in these non-Hermitian models exactly identifies the localization transitions, mobility edges, and many-body quantum phase transitions with gap closings, respectively. We further show that this strategy is robust against the finite-size effect and different boundary conditions.

Kangyi Hu, Menghua Deng, Fuxiang Li

Abstract Due to the nonunitary time evolution and possibly complex energy eigenvalues in non-Hermitian systems, it is still under debate how to properly deal with the dynamics of time-dependent non-Hermitian Hamiltonian. Recently a quantum metric framework has been proposed to study the dynamics of generated defects of a non-Hermitian system under linear quench. Here, we provide an explicit expression for the endowed Hamiltonian under quantum metric for a general two-level non-Hermitian system. Then we propose two exactly solvable models for the study of nonadiabatic dynamics of non-Hermitian systems, and analyze the defect production using the metric method. We find that, in contrast to the direct normalization method, the metric method can reproduce the symmetry of generated defects. The power-law scaling of generated defects with respect to quench time is also obtained.

Ken-Ichiro Imura, Kohei Kawabata

The non-Hermitian skin effect is nonreciprocity-induced localization phenomena in which a macroscopic number of eigenstates accumulate anomalously at the boundary, accompanied by the extreme sensitivity to boundary conditions. Here, we develop a geometric characterization of the non-Hermitian skin effect. We demonstrate that the localization length scale associated with the skin effect is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. Moreover, we show that the quantum metrics can exhibit the power-law divergences at gapless points that depend on the different boundary conditions. We also reveal that cusps of the generalized Brillouin zone in non-Bloch band theory are signaled by discontinuities in the quantum metrics. We illustrate these behavior using prototypical non-Hermitian models, such as the Hatano-Nelson model and the non-Hermitian, nonreciprocal Su-Schrieffer-Heeger model.