Bound on Eigenstate Thermalization from Transport

TL;DR

This work shows that macroscopic thermalization and transport impose nontrivial correlations on ETH matrix elements, challenging the idea that ETH reduces to Random Matrix Theory below the Thouless energy. It derives a bound showing the RMT onset scale ΔE_RMT must be parametrically smaller than the inverse slowest-thermalization time, tying the onset of RMT behavior to the breakdown of hydrodynamic transport. The authors connect an inequality involving time-domain observables to spectral-band fluctuations, and substantiate their claims with numerical simulations of a 1D diffusive Ising chain. Overall, the paper clarifies the relationship between ETH structure, transport, and the emergence of universal spectral statistics at late times.

Abstract

We show that macroscopic thermalization and transport impose constraints on matrix elements entering the Eigenstate Thermalization Hypothesis (ETH) ansatz and require them to be correlated. It is often assumed that the ETH reduces to Random Matrix Theory (RMT) below the Thouless energy scale. We show this conventional picture is not self-consistent. We prove that energy scale at which the RMT behavior emerges has to be parametrically smaller than the inverse timescale of the slowest thermalization mode coupled to the operator of interest. We argue that the timescale marking the onset of the RMT behavior is the same timescale at which hydrodynamic description of transport breaks down.

Figures (7)

• Figure 1: Visualization of the band matrix $(\delta A_T)_{ij}$\ref{['dAT']}.
• Figure 2: Plots of the LHS and the RHS of \ref{['inequality']} in logarithmic scale: $\ln |\langle \Psi|\delta A_T|\Psi\rangle|^2$ (blue) and $\ln x^2(T)$ (orange). Also shown in brown $\ln \delta A(t,\Psi)$. Its approximately linear form (before saturation) confirms exponential decay \ref{['decay']}. Inset: plot of autocorrelation function. All plots are for non-integrable Ising spin chain with $L=24$ spins with open b.c., see SM for details.
• Figure 3: Schematic visualization of the split \ref{['H0']}. Circles correspond to lattice sites (spins). Interaction term $H_{\rm int}$ is visualized by the blue bond in the middle, gray bonds correspond to $H_L$ and $H_R$. Distance between $A$ and $H_{\rm int}$ is denoted by $z$.
• Figure 4: Plot of $\langle A(t) A(0)\rangle_0^{\rm full}$\ref{['infT']} for different system sizes superimposed with $C/\sqrt{t}$ fit, where $C$ is a constant. Inset: asymptotic value of \ref{['infT']}$A_L^2$ vs $L$ plotted in log-log units, together with a linear fit. The fitted slope $1.04$ is in a reasonable agreement with the asymptotic behavior \ref{['tauL']} with $d=1$.
• Figure 5: Plot of $\langle A(t) A(0)\rangle_0^{\rm full}$\ref{['infT']} for $L=24$ together with the $(t_D/t)^\alpha$ fit plotted in log-log units. The fitted value of $\alpha=0.48$ is in good agreement with the diffusion value $\alpha=1/2$. Orange highlight marks the region used for the fit. It ends at the point taken as the Thouless time $\tau\approx 40$. Brown line is the linear fit (it extends beyond the orange region for visualization purposes).