Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time

The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable’s matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.

Figure 1

${\mathrm{\Lambda }}^T$ versus $T/{\tau }_{\mathrm{th}}$ for $\mathcal{A}$ with $q=L/2$ and $L=16$. Results obtained by the typicality approach, averaged over 500 states, agree convincingly with ED data. As a comparison, ${\mathrm{\Lambda }}^T$ obtained from a sign-randomized operator [Eq. (7)] yields the GOE value ${\mathrm{\Lambda }}^T=1/2$. Insets show eigenvalue distributions $P(\lambda )$ of ${\mathcal{A}}^T$ and ${\tilde{\mathcal{A}}}^T$ for different energy windows.

Figure 2

${\mathrm{\Lambda }}^T$ versus $T/{\tau }_{\mathrm{th}}$ for the density-wave operator $\mathcal{A}$ with (a) $q=L/2$ and (b) $q=1$. Data are obtained using the typicality approach up to $L=26$, averaged over $500·2^{16-L}$ random states [96]. The dashed line indicates the GOE value ${\mathrm{\Lambda }}_{\mathrm{GOE}}^T=0.5$. Insets show ${\mathrm{\Lambda }}^T$ versus $T/({\tau }_{\mathrm{th}}L)$ and the rescaled correlation function $\tilde{C}(t)$. The dashed vertical line signals ${\tau }_{\mathrm{th}}$ according to our definition in the text. The data collapse of $\tilde{C}(t)$ and $\tilde{C}(t/L^2)$ indicates $L$ independence of ${\tau }_{\mathrm{th}}$ for $q=L/2$ and diffusive behavior ${\tau }_{\mathrm{th}}\propto L^2$ for $q=1$.

Figure 3

Analogous data as in Fig. 2, but now for $\mathcal{B}$.

Figure 4

$F_T(t)$ [Eq. (9)] and $2{\overline{C}}_T(t)$ [Eq. (10)] for $\mathcal{A}$ with $q=L/2$ and $L=16$, for (a) $T={\tau }_{\mathrm{th}}$ and (b) $T=25{\tau }_{\mathrm{th}}$.