Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes

TL;DR

This paper analytically calculates the spectral form factor in chaotic quantum many-body systems, revealing universal behaviors across all Dyson symmetry classes and their dependence on system size and symmetries.

Contribution

It provides the first analytical derivation of spectral form factor beyond leading order for all Dyson classes in chaotic quantum systems, including symmetry-dependent second-order corrections.

Findings

Spectral form factor matches RMT predictions for all symmetry classes.

System size scaling of Thouless time varies with symmetry and presence of U(1) symmetry.

Beyond leading order, spectral correlations exhibit universal forms consistent with RMT ensembles.

Abstract

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 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's symmetry classes. Our derivation only assumes random phase approximation to enable ensemble average. For $\mathcal{T}$-invariant systems with $\mathcal{T}^2=1$, we show that beyond the 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. This is identical to the result from circular orthogonal ensemble of RMT. In the absence of $\mathcal{T}$-symmetry, we show that $K(t)\simeq t$ beyond $t^*$, and there is no universal term in the second order, unlike the $\mathcal{T}^2=1$ case, in agreement with the result of circular unitary ensemble. For $\mathcal{T}$-invariant systems with $\mathcal{T}^2=-1$, we show that $K(t)\simeq 2t+2t^2/\mathcal{N}$ up to two orders in time beyond $t^*$, in agreement with the result of circular symplectic ensemble. In all three cases, the system-size, $L$, scaling of $t^*$ is determined by 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 gapped non-degenerate second-largest eigenvalue of $\mathcal{M}$ or $t^*\propto \ln(L)$ due to gapped second-largest eigenvalue with degeneracy $\propto L^\zeta$. Our calculation of SFF is plausible in higher space dimensions as well, where similar system-size scalings of $t^*$ can be obtained.