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

TL;DR

This work presents a unified, analytic framework to derive the spectral form factor (SFF) for generic chaotic, periodically driven many-body quantum systems across all Dyson circular ensembles (COE, CUE, CSE) using random phase averaging (RPA). By decomposing SFF contributions into X, Y, and Z-type diagrams and introducing reduced diagrams, the authors isolate universal (Type I) terms that reproduce RMT predictions beyond the Thouless time t*, while nonuniversal (Type II/III) terms either cancel or decay, respectively. They show t* is set by the second-largest eigenvalue of a doubly stochastic matrix M that encodes phase-space returns, with distinct system-size scalings depending on U(1) symmetry. The results extend to higher dimensions and provide explicit leading- and second-order SFF corrections, unveiling a mechanism for the universal emergence of RMT statistics in strongly interacting quantum chaos. The findings have potential experimental relevance for measuring SFF and spectral correlations in quantum simulators and cold-atom platforms.

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 $\propto L^ζ$. Our calculation of SFF is plausible in higher space dimensions as well, where similar system-size scalings of $t^*$ can be obtained.

Figures (25)

• Figure 1: (a) A diagrammatic representation of a transposition for systems with $\mathcal{T}^2=1$ and without $\mathcal{T}$-symmetry, where states $\underline{n}_{\tau_1}$ and $\underline{n}_{\tau_2}$ are interchanged. The dashed blue circle represents initial configuration of states, $\{\underline{n}_1,...,\underline{n}_{\tau_1},...,\underline{n}_{\tau_2},...,\underline{n}_{t}\}$. The subscripts $(1,...,t)$ are in increasing order along the counterclockwise direction on the blue circle. The red curve represents the configuration of states after transposition $\{\underline{n}_1,...,\underline{n}_{\tau_2},...,\underline{n}_{\tau_1},...,\underline{n}_{t}\}$. (b) A diagrammatic representation of a transposition for $\mathcal{T}^2=-1$ case, where states $\underline{n}_{\tau_1}$ and $\underline{n}_{\tau_2}$ are interchanged and $\sigma_{\tau_1}=\sigma_{\tau_2}=1,\sigma_{\tau}=0$ for $\tau=1,\dots,t$ excluding $\tau_1,\tau_2$. The inner black dashed circle represent time reversed version of the states on the outer circle.
• Figure 2: A reduced diagram obtained by inserting a factor $1/\mathcal{N}$ for a red arc in a diagram representing a transposition.
• Figure 3: Reduced diagrams of (a) and (b) are identical. The green circles in diagram (b) indicate that the states at those time steps are identical.
• Figure 4: Reduced diagrams of (a) and (c) are identical. Reduced diagrams of (b) and (d) are identical. The green circles in (c) indicate that the states at those time steps are identical. The green and white circle in (d) indicate that the states at those time steps are related by time reversal.
• Figure 5: A diagram representing a sub-sequence reversal which reverses the order of states from $\tau_1$ to $\tau_2$. States at time steps $\tau_1$ and $\tau_2$ are denoted by the labels $b$ and $c$, $b\equiv\underline{n}_{\tau_1},c\equiv\underline{n}_{\tau_2}$. The other labels $a$ and $d$ denote states at the corresponding time steps, $a\equiv\underline{n}_{\tau_1-1},d\equiv\underline{n}_{\tau_2+1}$.