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 $30 < N \leq 34$ 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.
Highlights
We propose an ansatz for the spectral density of the M > 1 circular WSYK model in Appendix D
We speculate that a Q-deformed random matrix ensemble, like those studied in Refs. [80,81], has the potential to describe the deviations from the random matrix results but we could not find a suitable calculation scheme based on the combinatorial and diagrammatic techniques employed for the calculation of the spectral density
It would be interesting to find out whether a similar classification applies to non-Hermitian SYK models [84] and, in each case, to identify the combinatorial problem, the relevant Touchard-Riordan expression, and the explicit form of the spectral density
Summary
The Sachdev-Ye-Kitaev (SYK) model [1,2,3,4,5,6,7], a model of N Majorana, or Dirac, fermions in zero spatial dimensions with q-body infinite range interactions, is attracting a lot of attention in different research fields because, despite being analytically tractable, it can still reveal intriguing features of both strongly correlated systems and, through the use of holographic dualities, quantum gravity in near-AdS2 backgrounds [8,9] An example of this is Kitaev’s analytical calculation [1] showing that, in the strong-coupling low-temperature limit, the Lyapunov exponent [10] of the SYK model saturates a universal bound on quantum chaos [11]. We note that Iyoda et al [54] studied a related Wishart extension of the complex-fermion Sachdev-YeKitaev model with q 1⁄4 2 and found it to be integrable as it can be mapped to the Richardson-Gaudin model, which is known to be solved by the Bethe ansatz
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