Many body quantum chaos and time reversal symmetry

We investigate universal signatures of quantum chaos with time-reversal symmetry (TRS) in generic many-body quantum chaotic systems using three minimal random-circuit classes: (i) local TRS, (ii) global TRS, and (iii) TRS with discrete time-translation symmetry. For large local Hilbert dimension qqq, the spectral form factor (SFF) shows random-matrix (RMT) universality for times beyond the Thouless time​, which grows with system size. For times before Thouless time​, we derive explicit thermodynamic-limit scaling functions revealing universal, non-RMT behavior. In the simplest nontrivial case—global TRS with broken time-translation symmetry and broken local TRS—we map the SFF to a classical ferromagnetic Ising model: spins encode time-parallel vs time-reversed Feynman-path pairings, while TRS breaking acts as an external field. Independently of the large-qqq limit, we recover the same Ising scaling via space-time duality and parity-symmetric non-Hermitian Ginibre ensembles. Many-body effects from time-reversed pairings appear in the two-point autocorrelation function (2PAF), the out-of-time-order correlator (OTOC), and the partial SFF—observables sensitive to eigenvalue and eigenstate correlations. With TRS, 2PAF generically favors time-reversed over time-parallel pairings, yielding negative or suppressed values for antisymmetric operators; summing 2PAF over a complete operator basis gives a third route to the Ising scaling of SFF. We further show that 2PAF fluctuations are governed by an emergent three-state Potts model, leading to exponential scaling with operator support size at a Potts-set rate—a distinctive TRS chaos signature. Finally, leading-order OTOC is TRS-insensitive for spatially separated operators but becomes TRS-sensitive when their supports initially overlap. Numerical simulations of two one-dimensional circuit models corroborate these results.