[Paper Review] Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos

Jiachen Li, Tomaž Prosen+1 more|arXiv (Cornell University)|Mar 8, 2021

Quantum chaos and dynamical systemsPhysics and Astronomy112 references100 citations

TL;DR

This paper introduces the dissipative spectral form factor (DSFF), a complex-time Fourier transform of the density-density correlation of complex eigenvalues, to diagnose dissipative quantum chaos in non-Hermitian matrices. It analytically solves DSFF for the complex Ginibre ensemble (GinUE) and Poissonian spectra, showing a dip-ramp-plateau structure with a quadratic ramp in |τ|, contrasting the linear ramp in Hermitian systems, and establishes universality in quantum kicked top and classical stochastic matrices.

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 $ au$. Analogous to the spectral form factor (SFF) behaviour for GUE, DSFF for GinUE shows a ``dip-ramp-plateau'' behavior in $| au|$: 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 $| au|$, 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.

Motivation & Objective

• To develop a spectral diagnostic for non-Hermitian and non-Unitary matrices that captures correlations across arbitrary energy and time scales.
• To characterize dissipative quantum chaos via a complex-time correlation function of eigenvalues.
• To establish the dissipative spectral form factor (DSFF) as a universal probe for chaotic and integrable dissipative systems.
• To demonstrate universality of DSFF behavior across quantum kicked top models and classical stochastic matrices.

Proposed method

• Proposes the DSFF as the α-th power of the 2D Fourier transform of the two-level correlation function of complex eigenvalues, defined as Kα(t,s) = ⟨|∑_{m,n} e^{i(xn−xm)t + i(yn−ym)s}|^α⟩.
• Expresses DSFF in terms of complex time τ = |τ|e^{iθ}, with K(τ,τ*) depending on the projection of eigenvalue differences onto the θ-axis in the complex plane.
• Analytically computes DSFF for the GinUE (chaotic) and Poissonian (integrable) ensembles as minimal models.
• Uses numerical simulations to verify analytical results and study dependence on matrix size and eigenvalue density.
• Applies DSFF to the quantum kicked top with dissipation and to random classical stochastic matrices, showing universality with GinUE.
• Defines a critical angle θ* to quantify the range of θ where DSFF behavior aligns with GinUE universality, using error functions and curvature analysis.

Experimental results

Research questions

• RQ1 Does the DSFF exhibit universal behavior in dissipative quantum chaotic systems, and how does it differ from the standard SFF in Hermitian systems?
• RQ2 What is the functional form of the DSFF for the GinUE, and does it show a dip-ramp-plateau structure with a quadratic ramp in |τ|?
• RQ3 Can the DSFF distinguish between dissipative chaotic and integrable systems, such as in the quantum kicked top model?
• RQ4 Is the DSFF behavior of classical stochastic matrices universal and equivalent to that of the GinUE?
• RQ5 How does the critical angle θ* scale with matrix size N, and what does this imply about the range of universal behavior?

Key findings

• The DSFF for the GinUE exhibits a dip-ramp-plateau structure as a function of |τ|, with the ramp increasing quadratically in |τ|, in contrast to the linear ramp in Hermitian systems.
• For large matrix size, the DSFF of real and quaternion real Ginibre ensembles coincides with GinUE behavior except on a set of complex time of measure zero.
• The DSFF for Poissonian spectra remains constant except in a region dependent on eigenvalue density, indicating lack of correlations.
• Numerical results confirm that the quantum kicked top with dissipation falls into the GinUE universality class when the kick is on, and into the Poisson class when off.
• The critical angle θ* scales as θ* ∝ N^{-1/2}, indicating that the universal regime shrinks with increasing matrix size.
• Classical stochastic matrices and random matrix ensembles derived from CUE or GinUE also fall into the GinUE universality class, confirming broad applicability of DSFF.