Exact and asymptotic dissipative spectral form factor for elliptic Ginibre unitary ensemble

The dissipative spectral form factor (DSFF) has recently emerged as a diagnostic tool for assessing non-integrability or chaos in non-Hermitian systems. It extends the concept of the spectral form factor (SFF), a widely used measure for similar investigations in Hermitian systems. In this study, we concentrate on elliptic Ginibre unitary ensemble (eGinUE), which interpolates between non-Hermitian limit of Ginibre unitary ensemble (GinUE) and Hermitian limit of Gaussian unitary ensemble (GUE) by varying a symmetry parameter. We derive exact finite-dimension expression and large-dimension approximation for DSFF of eGinUE. A key result is a ''scaling relationship'' connecting the DSFF of eGinUE with that of GUE and GinUE, revealing a previously unobserved link between the DSFF of GinUE and the SFF of GUE. The DSFF for eGinUE displays a dip-ramp-plateau structure in GinUE and GUE limits, as well as in the crossover region, with differences in time scales that are well-explained by the aforementioned scaling relationship. We also estimate Thouless and Heisenberg times for various symmetry regimes, which correspond to dip-ramp and ramp-plateau transitions, respectively. Analytical results align well with Monte Carlo simulations of eGinUE random matrix model. In conclusion, we explore spectral statistics of the crossover SYK model, finding close agreement with our results.