Eigenstate Thermalization Hypothesis (ETH) for off-diagonal matrix elements in integrable spin chains

Abstract

We investigate off-diagonal matrix elements of local operators in integrable spin chains, focusing on the isotropic spin-$1/2$ Heisenberg chain ($XXX$ chain). We employ state-of-the-art Algebraic Bethe Ansatz results, which allow us to efficiently compute matrix elements of operators with support up to two sites between generic energy eigenstates. We consider both matrix elements between eigenstates that are in the same thermodynamic macrostate, as well as eigenstates that belong to different macrostates. In the former case, focusing on thermal states we numerically show that matrix elements are compatible with the exponential decay as $\exp(-L |{M}^{\scriptscriptstyle{\mathcal{O}}}_{ij}|)$. The probability distribution functions of ${M}_{ij}^{\scriptscriptstyle{\mathcal{O}}}$ depend on the observable and on the macrostate, and are well described by Gumbel distributions. On the other hand, matrix elements between eigenstates in different macrostates decay faster as $\exp(-|{M'}_{ij}^{\scriptscriptstyle{\mathcal{O}}}|L^2)$, with ${M'}_{ij}^{\scriptscriptstyle \mathcal{O}}$, again, compatible with a Gumbel distribution.

Figures (9)

• Figure 1: Logarithm of the matrix elements for the one-spin operator $E^{(11)}_j$ with $L = 56,\, M = 14$ as a function of the energy densities $E_i/L$ of the two eigenstates (diagonal elements not reported).
• Figure 2: Scaling of the logarithm $M_{ij}$ (cf. \ref{['eq:Mdef']}) of the matrix elements of the one-spin operator $E_i^{(11)}$ as a function of $E_i-E_j$, with $E_i,E_j$ the energy of the two eigenstates. (a) Scatter plot of the logarithm $M_{ij}$ rescaled by $L$ as a function of $(E_i-E_j)\sqrt{L}$ for $L = 56, 80, 120, 160,$ and $240$ and fixed $M = L/4$. (b) Two-dimensional histogram of the rescaled logarithm $M_{ij}/L$ as a function of $(E_i - E_j)/\sqrt{L}$.
• Figure 3: Distribution of the logarithm $\left\vert M_{ij}\right\vert$ (cf. \ref{['eq:Mdef']}) of the matrix elements of the one-spin operator $E_i^{(11)}$ (cf. \ref{['eq:elementary']}) for eigenstates of the $XXX$ chain with particle number $M = 14, 20, 30, 40,$ and $60$ and chain size $L=4M$. Eigenstates are extracted from the infinite-temperature thermal macrostate. (a) Histogram of $\left\vert M_{ij}\right\vert$. The continuous lines are fits to the Gumbel distribution \ref{['eq:Gumbel']}, with $\beta,\mu$ fitting parameters. In the inset we report the histogram of $\left\vert M_{ij}\right\vert/L$. (b) Histograms of the shifted $M_{ij}/L-c\ln(L)$, with $c$ a parameter obtained by fitting the average $\overline{\left\vert M_{ij}\right\vert}/L$ to the behavior $c\ln(L)+c_0$, with $c_0$ another fitting parameter. The fit is reported as gray-dashed line in the inset. In the inset $\mu_G$ is the mean of the Gumbel distribution, i.e., $\mu_G=\mu+\beta \gamma$, with $\beta,\mu$ as in \ref{['eq:Gumbel']} and $\gamma$ the Euler-Mascheroni constant. To compute $\mu_G$ we used the fitted values of $\beta,\mu$ in panel (a). (c) Double logarithm of the cumulative distribution function (CDF) $F(\left\vert M_{ij}\right\vert/L - c\ln L)$ as extracted from the numerical data.
• Figure 4: Same analysis as in Fig. \ref{['fig:histogram_1spin_rat4_same_macro']} for eigenstates of the $XXX$ chain in the infinite-temperature macrostate with particle density $M/L=1/2$.
• Figure 5: Distribution of $M_{ij}$\ref{['eq:Mdef']} of the one-spin operator $E_i^{(11)}$ in \ref{['eq:elementary']} for eigenstates of the $XXX$ chain in the finite temperature macrostate at $\beta = 0.5$ with particle density $M/L=1/4$ and particle numbers $M = 14, 20, 30$ and $40$. (a) Histogram of $\left\vert M_{ij}\right\vert$. The continuous lines are fits to the Gumbel distribution \ref{['eq:Gumbel']}. In the inset we report the histograms of $\left\vert M_{ij}\right\vert/L$. (b) Histograms of the shifted $\left\vert M_{ij}\right\vert/L - \overline{\left\vert M_{ij}\right\vert}/L$, where $\overline{\left\vert M_{ij}\right\vert}$ is the average of $\left\vert M_{ij}\right\vert$. In the inset we report the averages $\overline{\left\vert M_{ij}\right\vert}$ and we compare them with the mean $\mu_G$ of the Gumbel distribution \ref{['eq:Gumbel']} obtained from the fits in panel (a). (c) Double logarithm of the CDF of $\left\vert M_{ij}\right\vert/L - \overline{\left\vert M_{ij}\right\vert}/L$.