Spectral statistics of non-Hermitian matrices and dissipative quantum chaos

Jiachen Li (Author), Tomaž Prosen (Author), Amos Chan (Author)

Abstract

We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and nonunitary) 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 scale (and timescale). Specifically, we provide the exact solution of DSFF for the complex Ginibre ensemble (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 the DSFF exhibits an exact rotational symmetry in its complex time argument $\tau$. Analogous to the spectral form factor (SFF) behavior for Gaussian unitary ensemble, the DSFF for GinUE shows a “dip-ramp-plateau” behavior in |$\tau$|: the DSFF initially decreases, increases at intermediate timescales, and saturates after a generalized Heisenberg time, which scales as the inverse mean level spacing. Remarkably, for large matrix size, the “ramp” of the DSFF for GinUE increases quadratically in |$\tau$|, in contrast to the linear ramp in the SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that the DSFF takes a constant value, except for a region in complex time whose size and behavior depend on the eigenvalue density. Numerically, we verify the above claims and additionally show that the DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behavior, except for a region in the 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 or Markov chains and show that these models again fall under the Ginibre universality class.

Keywords

kvantna mehanika;kvantni kaos;statistična fizika;quantum mechanics;quantum chaos;statistical physics;

Data

Language:	English
Year of publishing:	2021
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	536.93
COBISS:	81850115
ISSN:	0031-9007
Views:	224
Downloads:	174
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary keywords:	kvantna mehanika;kvantni kaos;statistična fizika;
Pages:	str. 170602-1-170602-7
Volume:	ǂVol. ǂ127
Issue:	ǂiss. ǂ17
Chronology:	2021
DOI:	10.1103/PhysRevLett.127.170602
ID:	13729711