Many-body quantum chaos and time reversal symmetry

We investigate universal signatures of quantum chaos in the presence of time reversal symmetry (TRS) in generic many body quantum chaotic systems (gMBQCs). We study three classes of minimal models of gMBQCs with TRS, realized through random quantum circuits with (i) local TRS, (ii) global TRS, and (iii) TRS combined with discrete time-translation symmetry. In large local Hilbert space dimension $q$, we derive the emergence of random matrix theory (RMT) universality in the spectral form factor (SFF) at times larger than the Thouless time $t_{\mathrm{Th}}$, which diverges with system sizes in gMBQCs. At times before $t_{\mathrm{Th}}$, we identify universal behaviour beyond RMT by deriving explicit scaling functions of SFF in the thermodynamic limit. In particular, in the simplest non-trivial setting - preserving global TRS while breaking time translation symmetry - we show that the SFF is mapped to the partition function of an emergent classical ferromagnetic Ising model, where the Ising spins correspond to the time-parallel and time-reversed pairings of Feynman paths, and external magnetic fields are induced by TRS-breaking mechanisms. Without relying on the large-$q$ limit, we develop a second independent derivation of the Ising scaling behaviour of SFF using space-time duality and parity symmetric non-Hermitian Ginibre ensembles. Moreover, we show that many body effects originating from time-reversed pairings of Feynman paths manifest in the two-point autocorrelation function (2PAF), the out-of-time-ordered correlator (OTOC), and the partial spectral form factor - quantities sensitive to both eigenvalue and eigenstate correlations. We establish that the fluctuations of 2PAF are governed by an emergent three-state Potts model, leading to an exponential scaling with the operator support size, at a rate set by the three-state Potts model. [See full abstract in the paper]