Eigenstate thermalization hypothesis through the lens of autocorrelation functions

Matrix elements of observables in eigenstates of generic Hamiltonians are described by the Srednicki ansatz within the eigenstate thermalization hypothesis (ETH). We study a quantum chaotic spin-fermion model in a one-dimensional lattice, which consists of a spin-1/2 XX chain coupled to a single itinerant fermion. In our study, we focus on translationally invariant observables including the charge and energy current, thereby also connecting the ETH with transport properties. Considering observables with a Hilbert-Schmidt norm of one, we first perform a comprehensive analysis of ETH in the model taking into account latest developments. A particular emphasis is on the analysis of the structure of the offdiagonal matrix elements $|\langle \alpha |\widehat{O}{|\beta \rangle |}^2$ in the limit of small eigenstate energy differences $\omega =E_{\beta }-E_{\alpha }$. Removing the dominant exponential suppression of $|\langle \alpha |\widehat{O}{|\beta \rangle |}^2$, we find that (1) the current matrix elements exhibit a system-size dependence that is different from other observables under investigation and (2) matrix elements of several other observables exhibit a Drude-like structure with a Lorentzian frequency dependence. We then show how this information can be extracted from the autocorrelation functions as well. Finally, our study is complemented by a numerical analysis of the fluctuation-dissipation relation for eigenstates in the bulk of the spectrum. We identify the regime of $\omega$ in which the well-known fluctuation-dissipation relation is valid with high accuracy for finite systems.

Figure 1

Diagonal matrix elements $O_{\alpha \alpha }$ of the eight observables defined in Sec. 3, plotted as a function of $E_{\alpha }/\left(L{t}_0\right)$. Symbols from the back to the front represent results for $L=14$ (red), $L=15$ (dark blue), and $L=16$ (light blue), respectively.

Figure 2

Diagonal matrix elements $T_{1,\alpha \alpha }$ of the spin correlator ${\widehat{T}}_1$ defined in Eq. (14). Results are plotted as a function of $E_{\alpha }/\left(L{t}_0\right)$ for three system sizes $L=14$ (red), $L=15$ (dark blue), and $L=16$ (light blue). The spin-exchange coupling is set to (a) $J=0$ and (b) $J/t_0=0.2$.

Figure 3

Scaled offdiagonal matrix elements $\overline{|O_{\alpha \beta }{|}^2}\mathcal{D}$ for (a) the charge current ${\widehat{J}}_{\mathrm{c}}$ and (b) the energy current ${\widehat{J}}_{\mathrm{e}}$, shown as a function of $\omega /t_0$ in a logarithmic scale (main panels) and in a linear scale (insets). Overlapping solid lines are results for different system sizes from $L=12$ to $L=16$ at the target energy ${\overline{E}}_{\mathrm{tar}}=0$ as explained in the context of Eq. (21). Dashed-dotted lines are results for $L=16$, averaged over all energies [$\mathrm{\Delta }\to \infty$ in Eq. (21)]. Horizontal lines are fits of a constant to the data at $\omega /t_0\ll 1$.

Figure 4

Offdiagonal matrix elements $\overline{|O_{\alpha \beta }{|}^2}$ as a function of $\omega /t_0$ for the observables (a) ${\widehat{T}}_1$, (b) ${\widehat{S}}^z$, (c) ${\widehat{V}}_1$, and (d) ${\widehat{H}}_{\mathrm{kin}}$. Results are shown for different lattice sizes from $L=12$ to $L=16$. Dashed horizontal lines are fits of the data at $\omega /t_0\ll 1$ to a constant. Dashed-dotted lines denote the scaling $\propto {(\omega /t_0)}^{-2}$. The arrow in (d) marks the peak at $\omega /t_0\approx 4$. Insets show the scaled matrix elements $\overline{|O_{\alpha \beta }{|}^2}{\omega }^2\mathcal{D}L$. In the insets, we choose $\delta \omega$ in Eq. (21) as $\delta \omega /t_0=0.08$.

Figure 5

Scaled offdiagonal matrix elements $\overline{|O_{\alpha \beta }{|}^2}\mathcal{D}$ at $\omega \approx 0$ as a function of $L$ for (a) the observables ${\widehat{T}}_1$, ${\widehat{V}}_1$, ${\widehat{S}}^z$, and ${\widehat{H}}_{\mathrm{kin}}$ and (b) for the currents ${\widehat{J}}_{\mathrm{c}}$ and ${\widehat{J}}_{\mathrm{e}}$. The values of $\overline{|O_{\alpha \beta }{|}^2}\mathcal{D}$ at $\omega \approx 0$ correspond to the smallest nonzero $\omega$ according to Eq. (21), i.e., they are averaged over 700, 1000, 2000, 3000, and 4000 matrix elements for $L=12,13,14,15,16$, respectively. Dashed lines are fits to the results for $13\le L\le 16$. In (a) we fit the function $\propto L^{\gamma }$, where $\gamma =1.3,1.4,1.2,2.0$ for ${\widehat{T}}_1$, ${\widehat{V}}_1$, ${\widehat{S}}^z$, and ${\widehat{H}}_{\mathrm{kin}}$, respectively. In (b) we fit a constant (dashed lines) to the data.

Figure 6

(a–c) Scaled offdiagonal matrix elements $\overline{|O_{\alpha \beta }{|}^2}\mathcal{D}/L$ for the operators ${\widehat{T}}_1$, ${\widehat{S}}^z$, and ${\widehat{V}}_1$, respectively, and (d) $\overline{|O_{\alpha \beta }{|}^2}\mathcal{D}/L^2$ for the fermion kinetic energy ${\widehat{H}}_{\mathrm{kin}}$. Results are shown as a function of $(\omega /t_0)L$ for different lattice sizes from $L=12$ to $L=16$.

Figure 7

Scaled offdiagonal matrix elements $\overline{|O_{\alpha \beta }{|}^2}\mathcal{D}$ as a function of ${(\omega /t_0)}^2$ for four observables and different system sizes $L=12,...,16$. Dashed lines are fits of a Gaussian function $\propto e^{-{\zeta }_O{(\omega /t_0)}^2}$ to the results for $L=16$ and ${(\omega /t_0)}^2>30$, with the values of ${\zeta }_O$ given in the legends.

Figure 8

Variances of the offdiagonal matrix elements $\langle |O_{\alpha \beta }{{|}^2\rangle }_{\mathcal{D}}$, see Eq. (26), for the nine observables defined in Sec. 3. Symbols are numerical results for systems with $L=10,\cdots ,16$, while the lines are fits of the function $a_0{\mathcal{D}}^{-\gamma }$ to the data for $L\ge 13$. The resulting values of $\gamma$ are shown in the figure legend for each observable.

Figure 9

Scaled variances of the diagonal matrix elements ${\langle O_{\alpha \alpha }^2\rangle }_{\mu }\mathcal{D}$ for (a) the charge current ${\widehat{J}}_{\mathrm{c}}$ and (b) the energy current ${\widehat{J}}_{\mathrm{e}}$, as a function of the lattice size $L$. Lines are fits of the data for $L\ge 13$ to a constant. We choose $\mu$ in Eq. (28) such that we include results for 70%, 50%, and 20% eigenstates around the center of the spectrum (see the legend).

Figure 10

Scaled variances of the diagonal matrix elements for the spin correlator ${\widehat{V}}_1$ as a function of the lattice size $L$. Variances are scaled as (a) ${\langle V_{1,\alpha \alpha }^2\rangle }_{\mu }\mathcal{D}$ and (b) ${\langle V_{1,\alpha \alpha }^2\rangle }_{\mu }\mathcal{D}/L$. The line in (b) is a fit to a constant for $L\ge 13$. We choose $\mu$ in Eq. (28) such that we include results for 70%, 50%, and 20% eigenstates around the center of the spectrum (see legend).

Figure 11

Ratio of variances of diagonal versus offdiagonal matrix elements ${\mathrm{\Sigma }}_{\alpha ,\mu =100}^2\left(O\right)$ [see Eq. (29)] at $L=16$ as a function of the eigenstate index $\alpha$. Each panel includes $2^{16}-99$ values of ${\mathrm{\Sigma }}_{\alpha ,\mu =100}^2\left(O\right)$. In the inset, the moving average (MA) is performed over the 2000 nearest values. Results are shown for the operators ${\widehat{J}}_{\mathrm{c}}$, ${\widehat{J}}_{\mathrm{e}}$, and ${\widehat{V}}_1$ in (a)–(c), respectively. Horizontal dashed lines represent the averages $\overline{{\mathrm{\Sigma }}_{\mu =100}^2}\left(O\right)$, which are obtained by averaging over those sets of ratios of variances for which the mean energy $\overline{E_{\alpha }}={\sum }_{\rho =\alpha }^{\alpha +99}{E}_{\rho }/100$ lies in the window $\overline{E_{\alpha }}/\left(L{t}_0\right)\in [-1/6,1/6]$ (such average includes $83.6\%$ of all values in the main panel). We get $\overline{{\mathrm{\Sigma }}_{\mu =100}^2}\left(J_{\mathrm{c}}\right)=1.994$, $\overline{{\mathrm{\Sigma }}_{\mu =100}^2}\left(J_{\mathrm{e}}\right)=1.995$, and $\overline{{\mathrm{\Sigma }}_{\mu =100}^2}\left(V_1\right)=1.984$.

Figure 12

Numerical verification of the Kubo-Martin-Schwinger relation in Eq. (46) for the means of ratios of the eigenstate spectral densities $S_O^{\left(\alpha \right)}\left(\omega \right)$ and $S_O^{\left(\alpha \right)}(-\omega )$. Results are shown for the observables (a) ${\widehat{J}}_{\mathrm{c}}$, (b) ${\widehat{J}}_{\mathrm{e}}$, (c) ${\widehat{T}}_1$, and (d) ${\widehat{V}}_1$. Horizontal lines are analytical predictions at $\beta =0$, and symbols are numerical results for $L=12,14,16$. Full symbols represent the typical values $R_{O,\mu }^{\left(\mathrm{typ}\right)}\left(\omega \right)$ from Eq. (49), and open symbols represent the average values $R_{O,\mu }^{\left(\mathrm{avr}\right)}\left(\omega \right)$ from Eq. (50). In Eqs. (49) and (50), we choose $\mu$ such that the means include 10% of all eigenstates w.r.t. the center of the spectrum. We discretize the $\omega /t_0$ axis by choosing 30 (15) points for $R_{O,\mu }^{\left(\mathrm{typ}\right)}\left(\omega \right)\left[R_{O,\mu }^{\left(\mathrm{avr}\right)}\left(\omega \right)\right]$ that are equidistant on a log scale from ${10}^{-3}$ to ${10}^1$.

Figure 13

Ratios of averages of the eigenstate spectral densities ${\langle S_O^{\left(\alpha \right)}\left(\omega \right)\rangle }_{\mu }$ and ${\langle S_O^{\left(\alpha \right)}(-\omega )\rangle }_{\mu }$, defined in Eq. (51). Horizontal lines are results for the Kubo-Martin-Schwinger relation in Eq. (46) at $\beta =0$. Filled symbols are numerical results for $L=12,14,16$. We choose $\mu$ in Eq. (51) such that the average includes 10% of all eigenstates w.r.t. the center of the spectrum. We discretize the $\omega /t_0$ axis by choosing 30 points that are equidistant on a log scale from ${10}^{-3}$ to ${10}^1$. Open symbols are numerical results for the finite-size correction $exp\left\{\omega \frac{\partial ln|f_O(\overline{E},\omega ){|}^2}{\partial \overline{E}}\right\}$ [see Eq. (55)] at $L=16$. We calculate the discrete derivative over $\overline{E}$ by first calculating $|f_O(\overline{E},\omega ){|}^2$ at $\overline{E}=0$ and $\overline{E}=-\mathrm{\Delta }$, where $\mathrm{\Delta }/L=0.0025$. For each target $\overline{E}$, the results are averaged over all matrix elements for which the mean energy is in the interval $[\overline{E}-\mathrm{\Delta }/2,\overline{E}+\mathrm{\Delta }/2]$.

Figure 14

Time integration of the autocorrelation functions. Solid lines: Integrated autocorrelation functions $D_O^{\left(\alpha \right)}\left(t\right)$ [see the first row in Eq. (56)], averaged over (a, b) $10\%$ of eigenstates $|\alpha \rangle$ in the middle of the spectrum and (c, d) over all eigenstates. Observables are the charge current ${\widehat{J}}_{\mathrm{c}}$ (a, c) and the energy current ${\widehat{J}}_{\mathrm{e}}$ (b, d). The horizontal dashed lines represent the $\omega /t_0\to 0$ limits of the spectral densities, evaluated for the corresponding matrix elements of the operators and the density of states as given by the r.h.s. of Eq. (57).

Figure 15

Time evolution (a–c) of the autocorrelation functions $C_O\left(t\right)={\langle C_O^{\left(\alpha \right)}\left(t\right)\rangle }_{\mu }$ for the observables ${\widehat{T}}_1$, ${\widehat{S}}^z$, and ${\widehat{V}}_1$, respectively, and (d) of the scaled autocorrelation function $C_O\left(t\right)/L$ for the kinetic energy ${\widehat{H}}_{\mathrm{kin}}$. The latter scaling is consistent with the additional division by $L$ of the offdiagonal matrix elements of ${\widehat{H}}_{\mathrm{kin}}$ in Fig. 6. We choose $\mu$ such that $C_O^{\left(\alpha \right)}\left(t\right)$ [see Eq. (4)] is averaged over all eigenstates $|\alpha \rangle$. Results are shown for different system sizes $L$ (see legend) versus the scaled time $t{t}_0/L$. The dashed line is a fit to the $L=16$ results for $t{t}_0/L>1$.