We show 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 the spectral form factor (SFF), $K(t)$, up to two leading orders in time, $t$. We explicitly consider the presence or absence of time-reversal ($\mathcal{T}$) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for $\mathcal{T}$-invariant systems with ${\mathcal{T}}^2=1$, the circular unitary ensemble (CUE) in the absence of $\mathcal{T}$ symmetry, and the circular symplectic ensemble (CSE) for $\mathcal{T}$-invariant systems with ${\mathcal{T}}^2=-1$. Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in $t$ for the COE and CUE classes is finite, the number of such diagrams is exponentially large in $t$ for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time $t^{*}$, the SFF takes the form $K(t)\simeq 2t-2t^2/\mathcal{N}$ up to second order in time, where $\mathcal{N}$ is the Hilbert space dimension. For the CUE class, we show that $K(t)\simeq t$ beyond $t^{*}$, and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that $K(t)\simeq 2t+2t^2/\mathcal{N}$ up to two orders in time beyond $t^{*}$. The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of $t^{*}$ with system size $L$ is determined by the eigenvalues of a doubly stochastic matrix $\mathcal{M}$. For strongly interacting fermionic chains, $\mathcal{M}$ 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 the gapped, nondegenerate, second-largest eigenvalue of $\mathcal{M}$, or $t^{*}\propto \ln (L)$, due to the gapped second-largest eigenvalue with degeneracy proportional to $L^{\zeta }$, $\zeta >0$. Our calculation of the SFF is plausible in higher space dimensions as well, where similar system-size scalings of $t^{*}$ can be obtained.

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• Quantum chaos
• Dynamical systems
• Floquet systems
• Chaos & nonlinear dynamics
• Many-body techniques
• Random matrix theory