Eigenstate thermalization hypothesis and integrability in quantum spin chains

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}-\frac{1}{2}$ isotropic Heisenberg $\left(XXX\right)$ chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as $L^{-1/2}$, with $L$ the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the $XXX$ chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).

Figure 1

Finite-size scaling of the ETH for the two-spins reduced matrix ${\rho }_2$ in the $XXX$ chain. (a)–(d) Matrix elements ${\rho }_2[1,1],{\rho }_2[3,3]$, and ${\rho }_2[2,3]$ plotted versus the eigenstates energy density $E/L$. Panels (c) and (d) plot the real and imaginary parts of ${\rho }_2[2,3]$, respectively. Circles, squares, and crosses denote chain sizes $L=20,40,160$. The data correspond to $\sim {10}^4$ eigenstates obtained from real solutions of the Bethe equations at fixed particle density $x\equiv M/L=1/4$, with $M$ the number of down spins (particles). In all panels the rhombi denote the result obtained from the ERS. All the matrix elements are well described by the ERS in the limit $L\to \infty$. (e) Eigenstate average $\langle {\rho }_2\rangle$ plotted versus $L^{-1/2}$. Different symbols correspond to different matrix elements [cf. (g)]. The dashed-dotted lines are the results obtained from the ERS. (f) Eigenstate-to-eigenstate fluctuations of ${\rho }_2$. Histograms of ${\rho }_2[3,3]$ obtained from different eigenstates and $L=40,160$. The histogram for $L=160$ is shifted vertically for visibility. The dashed line is a Gaussian fit. (g) Fluctuations of ${\rho }_2$: Standard deviation $\sigma \left({\rho }_2\right)$ plotted versus $L^{-1/2}$. The dashed-dotted lines are linear fits. $\sigma \left({\rho }_2\right)$ is vanishing in the limit $L\to \infty$.

Figure 2

Finite-size scaling of the eigenstate-to-eigenstate fluctuations of the three-spins reduced density matrix ${\rho }_3$ in the $XXX$ chain. (a) Eigenstate average $\langle \left|{\rho }_3\right|\rangle$ of ${\rho }_3$ plotted against $L^{-1/2}$ for chains with $L=20,40,80,160$. The average is over $\sim {10}^4$ eigenstates in the sector with fixed particle density $x\equiv M/L=1/4$, $M$ being the number of down spins (particles). Full and empty symbols denote the diagonal and off-diagonal matrix elements of ${\rho }_3$, respectively. The dash-dotted lines denote ${\rho }_3^{\mathrm{ERS}}$. (b) Fluctuations of $|{\rho }_3|$: Standard deviation $\sigma \left(\right|{\rho }_3\left|\right)$ plotted versus $L^{-1/2}$. The dash-dotted lines are linear fits. The data suggest that $\sigma \left(\right|{\rho }_3\left|\right)\to 0$ in the thermodynamic limit.

Figure 3

EEVs of the conserved charges $I_n$ ($n=2,3,4$) of the $XXX$ chain. (a)–(c) Histograms of $I_n/L$ for several chain lengths $L=20,40,80,160$. Here $I_2$ and $I_3$ are the energy $E$ and the energy current $J_E$, respectively. For each $L$ the data are obtained from $\sim {10}^4$ eigenstates with fixed particle density $x\equiv M/L=1/4$, $M$ being the number of down spins. The histograms for the larger chains are shifted vertically for visibility. The dashed lines are Gaussian fits. In the limit $L\to \infty$, one has $I_2/L\approx -0.3$, whereas $I_3/L\approx I_4/L\approx 0$. (d) Rescaled fluctuations $\sigma \left(I_n\right)L^{-1/2}$ plotted versus $L^{-1/2}$. The dotted lines are linear fits. Notice the finite extrapolations in the thermodynamic limit.

Figure 4

Entanglement entropy of the eigenstates of the spin-$\frac{1}{2}XXX$ chain. (a) Two-spin von Neumann entropy $S_2$ plotted versus the eigenstates energy density $E/L$. Different symbols correspond to chain lengths $L=20,40,80,160$. The rhombus denotes $S_2^{\mathrm{ERS}}$. (b) $S_{\ell }^{\mathrm{ERS}}$ as a function of $\ell =1,\cdots ,5$, and for $L=80,160$. Circles and triangles correspond to particle densities $x\equiv M/L=1/4,1/10$, respectively. Here $M$ is the number of down spins (particles). The dashed-dotted lines are linear fits. For both $x=1/4,1/10$ the volume law $S_{\ell }^{\mathrm{ERS}}\propto \ell$ is clearly visible.