Spectral statistics in spatially extended chaotic quantum many-body systems

TL;DR

The spectral form factor K(t) is computed analytically and numerically and it is shown that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, t_{Th}.

Abstract

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, $t_{\rm Th}$. We obtain a striking dependence of $t_{\rm Th}$ on the spatial dimension $d$ and size of the system. For $d>1$, $t_{\rm Th}$ is finite in the thermodynamic limit and set by the inter-site coupling strength. By contrast, in one dimension $t_{\rm Th}$ diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form.

Figures (7)

• 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: $\langle r \rangle$ vs $\varepsilon$ for $q=3$ with system sizes $L=4, 5, 6, 7$. Upper and lower horizontal lines indicate the values expected in an ergodic Atas and an MBL phase OganesyanHuse, respectively.
• Figure S1: $K(t)$ at small $t$, showing deviations from RMT form that grow with $L$. Data for $L=4,6, \dots, 14$ (from bottom to top).