Chaotic many-body quantum dynamics, spectral correlations, and energy diffusion

TL;DR

The authors present a tractable, time-independent, spatially structured chaotic quantum model with local interactions, showing that energy diffusion is governed by a classical master equation in the large-N, weak-coupling limit and that spectral correlations (via the spectral form factor) can be computed from this diffusion dynamics. They establish a precise link between energy-transfer processes and SFF by decomposing the many-body transition probability into subsystem partitions, yielding early-time multiplet contributions and a diffusion-driven ramp at late times. A detailed two-site solution validates the approach and numerical simulations support diffusive energy transport and diffusion-controlled SFF behavior in many-site systems, including an N=2 spin-1/2 test showing similar qualitative features. Overall, the work connects microscopic random-matrix-based constructions with hydrodynamic diffusion, clarifying how energy diffusion shapes universal chaotic spectral statistics in extended quantum systems.

Abstract

We study chaotic many-body quantum dynamics in a minimal model with spatial structure and local interactions. It has a time-independent Hamiltonian, in contrast to quantum circuits and Brownian models, and is simple at the single-site level, in contrast to Sachdev-Ye-Kitaev chains. It is analytically tractable for large local Hilbert space dimension and weak intersite coupling. In this limit we show that energy dynamics is described by a classical master equation and is diffusive. We also show that the spectral form factor can be expressed exactly in terms of the solution to this master equation. For a two-site system we obtain closed-form expressions for both the two-point correlator of energy density and the spectral form factor, in essentially perfect agreement with numerical simulations. For an $L$-site system we show at late times how a linear ramp emerges in the spectral form factor, as universally expected from level repulsion in chaotic quantum systems. Conversely, at earlier times we identify two distinct mechanisms for an increase of the spectral form factor above its ramp value. One of these is associated with energy diffusion and is effective until the Thouless time, which varies as $L^2$. The other involves contributions like those that would appear if the system were composed of many uncoupled subsystems: they generate a large enhancement of the spectral form factor, and are suppressed on a timescale varying as $(\ln L)^2$. Besides being exact for the limit considered, we believe our approach provides the natural approximation even for small local Hilbert space dimension and strong intersite coupling. We present a numerical study of a spin-half chain, finding an early-time enhancement of the spectral form factor which is qualitatively similar to that in our solvable model.

Figures (12)

• Figure 1: Schematic behaviour of the SFF in a system of $L$ weakly coupled sites, shown as $\ln K(t)$ vs $\ln t$ (black). Dashed lines indicate the behaviours $K(t) \propto t^L$ (green) and $K(t) \propto t$ (blue). See text for discussion of indicated timescales.
• Figure 2: Comparison between numerical and analytical results for the spectral form factor $K(t)$ and the two-point correlation function of energy density $C_{mn}(t)$ with $N=130$ and $L=2$. Intersite coupling strengths are $\lambda=0.1$ for panels (a) and (b), and $\lambda = 0.05$ for panels (c) and (d); panels (e) and (f) are independent of $\lambda$ provided it is small. Panels (a) and (c) show the intermediate-time regime for $K(t)$: numerical data (black), analytical prediction (blue), and the short-time form $K_1(t)^2$ (red). The analytical results are obtained by adding the contributions $K^{(1)}(t)$ and $K^{(2)}(t)$, Eqns. \ref{['eq:K^(1)']} and \ref{['eq:K^(2)']} respectively. Panels (b) and (d) show the long-time regime for $K(t)$: numerical data (black), a linear ramp with gradient fixed by the late-time form of Eq. \ref{['eq:K^(1)']} (blue), and the analytical prediction taking into account the shape of the density of states [Eq. \ref{['eq:K^(1)generalised']}] (red). (e) $K(t)$ at short times: numerical results (black) and comparison with analytics (blue). (f) Comparison between numerical results (black) and analytical predictions [Eq. \ref{['eq:predC11']}] (blue) for $C_{11}(t)$ (solid) and $C_{12}(t)$ (dashed). All numerical data are averaged over 5000 random realizations.
• Figure 3: A contribution to $P(\{\varepsilon_k \},\{ \varepsilon^\prime_k \};t)$ at order $\lambda^4$ for a system of three sites. The three horizontal lines directed to the right represent timelines of sites contributing to $e^{-i{\cal H}t}$ and the three lines directed to the left are the equivalent for $e^{i{\cal H}t}$. The green lines represent Wick-paired factors of $T_{n,n+1}$ after the ensemble-average. The Wick pair shown here with time labels $t_1$ and $\tilde{t}_1$ is a vertex contribution, and the one with time labels $t_2$ and $t_3$ is a self-energy contribution.
• Figure 4: Illustration of the Bethe-Salpeter equation satisfied by the transition probability $P(\{\varepsilon_n \},\{ \varepsilon_n^\prime \};t)$ in the case of a two-site system. The boxes labeled $P(t)$ and $P(t+\Delta)$ represent the transition probability evaluated at times $t$ and $t+\Delta$.
• Figure 5: A vertex insertion, labeled as in Eq. \ref{['eq:energyconservation']}.