Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model

Abstract

Spectral rigidity in Hermitian quantum chaotic systems signals the presence of dynamical universal features at timescales that can be much shorter than the Heisenberg time. We study the analog of this timescale in many-body non-Hermitian quantum chaos by a detailed analysis of long-range spectral correlators. For that purpose, we investigate the number variance and the spectral form factor of a non-Hermitian $q$-body Sachdev-Ye-Kitaev (nHSYK) model, which describes $N$ fermions in zero spatial dimensions. After an analytical and numerical analysis of these spectral observables for non-Hermitian random matrices, and a careful unfolding, we find good agreement with the nHSYK model for $q>2$ starting at a timescale that decreases sharply with $q$. The source of deviation from universality, identified analytically, is ensemble fluctuations not related to the quantum dynamics. For fixed $q$ and large enough $N$, these fluctuations become dominant up until after the Heisenberg time, so that the spectral form factor is no longer useful for the study of quantum chaos. In all cases, our results point to a weakened or vanishing spectral rigidity that effectively delays the observation of full quantum ergodicity. We also show that the number variance displays nonstationary spectral correlations for both the nHSYK model and random matrices. This nonstationarity, also not related to the quantum dynamics, points to intrinsic limitations of these observables to describe the quantum chaotic motion. On the other hand, we introduce the local spectral form factor, which is shown to be stationary and not affected by collective fluctuations, and propose it as an effective diagnostic of non-Hermitian quantum chaos. For $q=2$, we find saturation to Poisson statistics at a timescale of $\log D$, compared to a scale of $\sqrt{D}$ for $q>2$, with $D$ the total number of states.

47 pages, 19 figures. v2: minor corrections, version as published

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• Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis
• Robustness of quantum chaos and anomalous relaxation in open quantum circuits
• Entanglement Transition and Replica Wormhole in the Dissipative Sachdev-Ye-Kitaev Model
• Probing quantum chaos through singular-value correlations in sparse non-Hermitian SYK model
• Breakdown of the Quantum Distinction of Regular and Chaotic Classical Dynamics in Dissipative Systems
• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
• Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking
• Theory of Irreversibility in Quantum Many-Body Systems
• Universal hard-edge statistics of non-Hermitian random matrices
• Correspondence principle, dissipation, and Ginibre ensemble
• Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking
• Surmise for random matrices' level spacing distributions beyond nearest-neighbors