Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos

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.

Figure 1

$K(\tau ,{\tau }^{*})$ vs $|\tau |$ for the GinUE and a Poissonian random spectrum [defined above (5)] with matrix size $N=256$. Writing $\tau =|\tau |e^{i\theta }$, numerical simulations of the GinUE (Poisson) for fixed $\theta =0$ to $\theta =\pi /2$ in steps of $\pi /4$ are plotted in blue colors (red colors) in multiple shades. The analytical solutions of $K_{\text{GinUE}}$ [Eq. (3)] and $K_{\mathrm{Poi}}$ [Eq. (5)] are plotted as the purple and green lines on top of the numerical data. The connected part of $K_{\text{GinUE}}$ [first and third terms in Eq. (3)] is plotted as the orange line. Inset: two-dimensional histogram of $\{z_{mn}\equiv z_m-z_n\}$ of the GinUE for $N=1024$, where increasing values are plotted with deeper blue colors. Note that the bin at the origin is occupied by $N$ (diagonal) contributions of $\{z_{nn}=0\}$, and there is a dip around the origin due to level repulsion. The computation of the DSFF for fixed $\theta$ as a function of $|\tau |$ is equivalent to the computation of the SFF as a function of $|\tau |$ of $\{z_m\}$ projected onto the axis defined by $\theta$. The sample sizes are 5000 and 2000 for Poisson and the GinUE, respectively.

Figure 2

DSFF of QKT with dissipation for $j=35$ and Gaussianly distributed $p\in \mathcal{N}(2,2/3)$ and $k_0\in \mathcal{N}(10,3)$. The blue (red) lines in different shades are for the DSFF of QKT with the kick with $k_1=8$ (without the kick with $k_1=0$) for fixed $\theta \in [\pi /16,7\pi /16]$ in steps of $\pi /16$. The two cases fit the connected part of GinUE (orange line) and Poisson (green line) predictions as expected. Inset: DSFF of QKT and GinOE for angles $\theta =0$, $\pi /2$ [69]. The sample sizes for QKT with and without the kick are 2500 and 4300, respectively.

Figure 3

DSFF of the random classical stochastic ensemble induced by CUE for $N=1024$ with sample size 5000. The blue colors are for fixed $\theta \in [\pi /16,7\pi /16]$ in steps of $\pi /16$. The data can be fitted with the connected part of the GinUE (orange line). Inset: DSFF for CS and GinOE at angles $\theta =0$, $\pi /2$ [69].