Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions

The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems by conjecturing statistical properties of matrix elements of typical operators in the (quasi)energy eigenbasis. Here we study the distribution of matrix elements for a class of operators in dual-unitary quantum circuits in dependence of the frequency associated with the corresponding eigenstates. We provide an exact asymptotic expression for the spectral function, i.e., the second moment of this frequency resolved distribution. The latter is obtained from the decay of dynamical correlations between local operators which can be computed exactly from the elementary building blocks of the dual-unitary circuits. Comparing the asymptotic expression with results obtained by exact diagonalization we find excellent agreement. Small fluctuations at finite system size are explicitly related to dynamical correlations at intermediate times and the deviations from their asymptotical dynamics. Moreover, we confirm the expected Gaussian distribution of the matrix elements by computing higher moments numerically.

Figure 1

Diagrammatic representation of $\mathcal{U}={\mathcal{U}}_2{\mathcal{U}}_1$ with ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ given by Eq. (2) for $2L=10$.

Figure 2

Spectral function of the operator $A$, Eq. (18), for five representative realizations of the dual-unitary circuit $\mathcal{U}$ with $2L=16$ labeled by A–E. The corresponding asymptotic expressions (26) are depicted as dashed black lines. The lower panel (b) is a magnification of panel (a).

Figure 3

Autocorrelation functions of the operator $A$ versus time for systems A–E with $2L=16$ on a semilogarithmic scale (colored symbols connected by lines). The asymptotic autocorrelation functions $\alpha {\lambda }_{\nu }^t$ ($t\ge 1$) are depicted as black dashed lines.

Figure 4

System F at system size $2L=16$. Panel (a) depicts the numerically obtained (thin brown line) and the asymptotic (black dashed line) spectral function. Panel (b) shows the corresponding autocorrelation function (brown triangles connected by a thin line) and the asymptotic autocorrelation function $\alpha {\lambda }_{+}^t$ ($t\ge 1$) (black dashed line). The dotted black line indicates the asymptotically slowest possible decay proportional to ${\lambda }_{-}^t$.

Figure 5

Difference ${\mathrm{\Delta }}_{{\left|f\right|}^2}$ of the spectral function and its asymptotic value versus system size $L$. The lines connecting the symbols serve as a guide to the eye.

Figure 6

Distribution of real parts of matrix elements for $2L=16$. In panel (a) the distribution for system A at various frequencies (see legend) is shown while (b) depicts the distribution at all frequencies in system A, C, and D. The black dashed lines show a Gaussian distributions corresponding to the variance determined by (a) ${\left|f\left(\omega \right)\right|}^2$ at $\omega \in \{0,0.15\pi ,0.31\pi ,0.95\pi \}$ and (b) ${\left|f\left(\omega \right)\right|}^2=1$.

Figure 7

Rescaled fourth (thick colored line) and sixth (thin gray line) moment of the distribution of matrix elements for systems A–F with $2L=16$. The dashed black line corresponds to a Gaussian distribution.