I shall discuss the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic periodically kicked interacting quantum many-body systems by analytically calculating spectral form factor (SFF), $K(t)$, up to two leading orders in time, $t$. We explicitly consider the presence or absence of time reversal symmetry ($\mathcal{T}$) to investigate all three Dyson symmetry classes. While the number of diagrams that contribute to the SFF in the first and second order in $t$ for $\mathcal{T}$-invariant systems with $\mathcal{T}^2 = 1$ and in the absence of $\mathcal{T}$-symmetry is finite, the number of such diagrams is exponentially large in $t$ for $\mathcal{T}$-invariant systems with $\mathcal{T}^2 = -1$. For this, we have developed a new diagrammatic technique using reduced diagrams to include contributions from many diagrams of different permutations of basis states. In all three cases, the system-size ($L$) scaling of the Thouless time $t^*$, beyond which the SFF takes the universal RMT form, is determined by eigenvalues of a doubly stochastic matrix $M$. For strongly interacting fermionic chains, $M$ is SU(2) invariant in all three cases, leading to $t^* \propto L^2$ in the presence of U(1) symmetry. In the absence of U(1) symmetry, we find $t^* \propto L^0$, due to gapped non-degenerate second-largest eigenvalue of $M$ or $t^* \propto \ln(L)$ due to gapped second-largest eigenvalue with degeneracy $\propto L^\zeta$.
Reference: Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes, Vijay Kumar, Tomaˇz Prosen, and Dibyendu Roy, arXiv: 2502.04152 (2025)
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