Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos

TL;DR

This paper introduces the dissipative spectral form factor (DSFF), a new measure to analyze spectral statistics of non-Hermitian matrices, revealing signatures of dissipative quantum chaos and integrability with exact solutions and numerical verification.

Contribution

The paper defines DSFF for non-Hermitian matrices, provides exact solutions for GinUE and Poisson spectra, and demonstrates its effectiveness in diagnosing dissipative quantum chaos and integrability.

Findings

DSFF exhibits a dip-ramp-plateau behavior similar to SFF in Hermitian ensembles.

For GinUE, the ramp increases quadratically with time, contrasting with linear SFF ramps.

Numerical results confirm DSFF's universality across different ensembles and physical models.

Abstract

We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and non-Unitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos, and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy (and time) scale. Specifically, we provide the exact solution of DSFF for the GinUE and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems respectively. For dissipative quantum chaotic systems, we show that DSFF exhibits an exact rotational symmetry in its complex time argument $\tau$. Analogous to the spectral form factor (SFF) behaviour for GUE, DSFF for GinUE shows a ``dip-ramp-plateau'' behavior in $|\tau|$: DSFF initially decreases, increases at intermediate time scales, and saturates after a generalized Heisenberg time which scales as the inverse mean level spacing. Remarkably, for large matrix size, the ``ramp'' of DSFF for GinUE increases quadratically in $|\tau|$, in contrast to the linear ramp in SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that DSFF takes a constant value except for a region in complex time whose size and behavior depends on the eigenvalue density. Numerically, we verify the above claims and show that DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behaviour except for a region in complex time plane of measure zero in the limit of large matrix size. As a physical example, we consider the quantum kicked top model with dissipation, and show that it falls under the Ginibre universality class and Poisson as the `kick' is switched on or off. Lastly, we study spectral statistics of ensembles of random classical stochastic matrices, and show that these models fall under the Ginibre universality class.