Bound on Eigenstate Thermalization from Transport

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 that this conventional picture is not self-consistent. We prove that the 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 the hydrodynamic description of transport breaks down.

Figure 1

Visualization of the band matrix $(\delta A_T{)}_{ij}$ (7).

Figure 2

Plots of the lhs and the rhs of Eq. (11) in logarithmic scale: $\ln |⟨\mathrm{\Psi }|\delta A_T|\mathrm{\Psi }⟩{|}^2$ (blue lines) and $\ln x^2(T)$ (orange lines). Also shown in brown is $\ln \delta A(t,\mathrm{\Psi })$. Its approximately linear form (before saturation) confirms exponential decay (3). Inset: plot of the autocorrelation function. All plots are for a nonintegrable Ising spin chain with $L=24$ spins with open bc; see Supplemental Material [34] for details.