Many-body quantum chaos and emergence of Ginibre ensemble

We establish a connection between many-body quantum chaotic (MBQC) systems and the Ginibre random matrix ensemble (GinUE) - namely that non-Hermitian GinUE behaviors emerge in spatially-extended MBQC systems in the space direction, just as Hermitian random matrix behaviours emerge in MBQC systems in the time direction. We demonstrate the emergence of GinUE firstly in translational invariant (TI) MBQC systems, which can be associated with dual transfer matrices with complex-valued spectra, before generalizing to spatially-random systems. We argue and demonstrate that the dual spectra of TI MBQC systems necessarily have non-trivial and universal correlations due to the existence of a linear ramp in the spectral form factor (SFF). We show that, firstly, the spectral statistics of the dual spectra, probed by the dissipative spectral form factor and spacing distribution, falls under the universality class of the GinUE, and in particular displays level repulsion in the complex plane. Secondly, we obtain an exact analytical expression of SFF for the GinUE, and show that it universally describes the SFF of TI MBQC systems in the scaling limit where $t$ and $L$ are large, while the ratio between $L$ and $L_{\mathrm{Th}}$, the many body Thouless length is fixed. Thirdly, we propose variations of Ginibre models accounting for time periodicity, such that the SFF of Floquet MBQC (with or without TI) can be analytically described, thereby removing the necessity of TI for the emergence of GinUE. Lastly, we remark that locality and the many-body nature of MBQC systems are required for the emergence of the GinUE in large system sizes