Spectral Statistics in Spatially Extended Chaotic Quantum Many-Body Systems

We study spectral statistics in spatially extended chaotic quantum many-body systems, using simple lattice Floquet models without time-reversal symmetry. Computing the spectral form factor $K(t)$ analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, $t_{\text{Th}}$. We obtain a striking dependence of $t_{\text{Th}}$ on the spatial dimension $d$ and size of the system. For $d>1$, $t_{\text{Th}}$ is finite in the thermodynamic limit and set by the intersite coupling strength. By contrast, in one dimension $t_{\text{Th}}$ diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form. Lastly, our Floquet model exhibits a many-body localization transition, and we discuss the behavior of the spectral form factor in the localized phase.

Figure 1

Diagrammatic representation of the Floquet operator, where $W_1$ contains unitary 1-gates, and $W_2$ contains diagonal 2-gates with random phases.

Figure 2

$K(t)$ at small $t$, showing deviations from RMT form that grow with $L$. Data for $L=4$, 6, 8, and 10 (from bottom to top). Inset: $K(t)$ vs $t$ for $L=6$.

Figure 3

${\xi }_L(t)$ vs $t$. Main panel: $L=4$, 6, 8, and 10 (from bottom to top). Inset: $L=10$ (red) and $L=100$ (purple) at short times.

Figure 4

Identification of MBL and ergodic phases: $⟨r⟩$ vs $ϵ$ for $q=3$ with system sizes $L=4$, 5, 6, 7. Upper and lower horizontal lines indicate the values expected in an ergodic [22] and an MBL phase [21], respectively.