$Q$-Laguerre spectral density and quantum chaos in the Wishart-Sachdev-Ye-Kitaev model

Abstract

We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\hat{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number $N$ of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara $Q$-Laguerre polynomials. For $\hat{q}=3,4$, the $Q$-Laguerre prediction, with $Q=Q(\hat{q},N)$ also computed analytically, agrees well with exact diagonalization results for $30<N\le 34$ while we observe some deviations for $\hat{q}=2$. The most salient feature of the spectral density is that, for odd $\hat{q}$, low-energy excitations grow as a stretched exponential, with a functional form different from that of the supersymmetric SYK model. For $\hat{q}=4$, a detailed analysis of level statistics reveals quantum chaotic dynamics even for time scales substantially shorter than the Heisenberg time. More specifically, the spacing ratios in the bulk of the spectrum and the microscopic spectral density and the number variance close to the hard edge are very well approximated by that of an ensemble of random matrices that, depending on $N$, belong to the chiral or superconducting universality classes. In particular, we report the first realization of level statistics belonging to the chGUE universality class, which completes the tenfold-way classification in the SYK model.

37 pages, 10 figures. v2: typos corrected. v3: scope of results clarified, comparison with numerics for fixed $\hat{q}$ expanded into new Sec. 6, references added. v4: minor modifications, added clarifications. v5: as published

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• Lindbladian dynamics of the Sachdev-Ye-Kitaev model
• Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model
• Probing quantum chaos through singular-value correlations in sparse non-Hermitian SYK model
• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
• Emergent Topology in Many-Body Dissipative Quantum Matter
• Universal scaling of higher-order spacing ratios in Gaussian random matrices
• Sixfold Way of Traversable Wormholes in the Sachdev-Ye-Kitaev Model
• Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions
• A bipartite Sachdev-Ye-Kitaev model: Conformal limit and level statistics