Comment on “Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis”

Figure 1

(a) Diagonal and (b) off-diagonal (for $|E_{\alpha }+E_{\beta }|/N\le 0.1$) matrix elements of the nearest-neighbor spin correlations ${\widehat{\sigma }}_{\mathbf{i}}^z{\widehat{\sigma }}_{{\mathrm{nn}}_{\mathbf{i}}}^z$ for the ferromagnetic transverse-field Ising model ($g=J$) in two dimensions [14, 15]. Continuous lines in (b) depict running averages. The inset in (a) shows that the fluctuations of the diagonal matrix elements are different in the two sectors shown. Inset in (b): ratio $\mathcal{R}$ between the running averages of the off-diagonal matrix elements. The dashed line shows that, as expected from the ETH [6, 19], the ratio $\mathcal{R}$ is very close to the square root of the inverse ratio between the Hilbert space dimension $\mathcal{D}$ of the sectors. ${\lambda }_{{\widehat{Z}}_2}$, ${\lambda }_{{\widehat{S}}_x}$, ${\lambda }_{{\widehat{S}}_y}$, ${\lambda }_{{\widehat{S}}_{xy}}$ are the eigenvalues of the spin-flip, mirror-$x$, mirror-$y$, and mirror along the $x=y$ line symmetries, respectively. The results shown are for the zero momentum sector of a lattice with $N=5\times 5$ sites (see Ref. [15] for further details).