Eigenstate thermalization in the two-dimensional transverse field Ising model

We study the onset of eigenstate thermalization in the two-dimensional transverse field Ising model (2D-TFIM) in the square lattice. We consider two nonequivalent Hamiltonians: the ferromagnetic 2D-TFIM and the antiferromagnetic 2D-TFIM in the presence of a uniform longitudinal field. We use full exact diagonalization to examine the behavior of quantum chaos indicators and of the diagonal matrix elements of operators of interest in the eigenstates of the Hamiltonian. An analysis of finite size effects reveals that quantum chaos and eigenstate thermalization occur in those systems whenever the fields are nonvanishing and not too large.

Figure 1

Clusters with periodic boundary conditions used in this work. All clusters, with the exception of the nontilted lattice with 20 sites (bottom right), support the Néel state. Each cluster displays the basis in which translation symmetry operations are implemented.

Figure 2

Distribution of the ratio of consecutive energy gaps in the spectra for (a) the ferromagnetic case with $g=1.0$ ($\varepsilon =0$) and (b) the antiferromagnetic case with $\varepsilon =g=1$. The results were obtained in the cluster 20A (see Fig. 1). Results are reported for $P\left(r\right)$ averaged over all momentum sectors with $\mathbf{k}\ne (0,0)$ and $\mathbf{k}\ne (\pi ,\pi )$, in which $Z_2$ (for the ferromagnetic case) and parity under inversion [for $\mathbf{k}=(0,\pi )$ and $\mathbf{k}=(\pi ,0)$] are the only additional symmetries. and they are resolved. We also show the average $P\left(r\right)$ between the momentum sectors $\mathbf{k}=(0,0)$ and $\mathbf{k}=(\pi ,\pi )$ when divided in the even (${\lambda }_{\widehat{I}}=+1$) and odd (${\lambda }_{\widehat{I}}=+1$) parity sectors under inversion. In those momentum sectors inversion is not the only space symmetry. (Insets) Average value of $r$ as a function of the strength of the fields. The horizontal dashed lines depict the average predicted by $P_{\text{GOE}}\left(r\right)$ (top) and $P_{\text{P}}\left(r\right)$ (bottom). All results were obtained using the the central half of the spectrum in each subspace.

Figure 3

Structural entropy in all symmetry sectors of the ferromagnetic 2D-TFIM ($J=-1$ and $\varepsilon =0$) for different system sizes (see Table 1). The narrowing of the support of the values of the structural entropy with increasing system size, in any given energy window, is an indication of the occurrence of quantum chaos.

Figure 4

Same as Fig. 3 for the antiferromagnetic 2D-TFIM ($J=1$) with a longitudinal field of strength $\varepsilon =g$.

Figure 5

Structural entropy of the ferromagnetic model with $g=1.0$ ($\varepsilon =0$) for $N=12$ and 16 sites. (a) Results after accounting for translational, $Z_2$, and inversion symmetry (when present). (b) No symmetry is used when fully diagonalizing the Hamiltonian.

Figure 6

Energy-eigenstate expectation values of the ferromagnetic structure factor, ${\left(S_{\text{F}}\right)}_{\alpha \alpha }=\langle \alpha |{\widehat{S}}_{\text{F}}|\alpha \rangle$, in the ferromagnetic 2D-TFIM ($\varepsilon =0$). The narrowing of the support of the eigenstate expectation values with increasing system size is an indication of the occurrence of eigenstate thermalization. Vertical dashed lines depict the critical energies $E_c$ [Eq. (9)] below which the system is expected to display long-range order.

Figure 7

Energy-eigenstate expectation values of the antiferromagnetic structure factor, ${\left(S_{\text{AF}}\right)}_{\alpha \alpha }=\langle \alpha |{\widehat{S}}_{\text{AF}}|\alpha \rangle$, in the antiferromagnetic 2D-TFIM with a longitudinal field of strength $\varepsilon =g$. As for the ferromagnetic case, the narrowing of the support of the eigenstate expectation values with increasing system size is an indication of the occurrence of eigenstate thermalization.

Figure 8

Largest, fifth largest, tenth largest, and average value of (a) ${\left(\mathrm{\Delta }{S}_{\text{F}}\right)}_{\alpha }$ for the ferromagnetic 2D-TFIM ($g=1.5$ and $\varepsilon =0$) and (b) ${\left(\mathrm{\Delta }{S}_{\text{AF}}\right)}_{\alpha }$ for the antiferromagnetic 2D-TFIM in the presence of a longitudinal field with $\varepsilon =g=1.5$, plotted as a function of the number of lattice sites in the cluster. All those quantities are computed within two windows of eigenstates characterized by $x_{\mathrm{thr}}=50\%$ (filled symbols) and $x_{\mathrm{thr}}=90\%$ (open symbols). See text for the definition of $x_{\mathrm{thr}}$.

Figure 9

Distribution of: (a) ${\left(\mathrm{\Delta }{S}_{\text{F}}\right)}_{\alpha }$ for the ferromagnetic 2D-TFIM ($g=1.5$ and $\varepsilon =0$) and (b) ${\left(\mathrm{\Delta }{S}_{\text{AF}}\right)}_{\alpha }$ for the antiferromagnetic 2D-TFIM in the presence of a longitudinal field with $\varepsilon =g=1.5$. The distributions were computed for $x_{\text{thr}}=0.5$. (Insets) Density of states in the clusters.

Figure 10

Energy-eigenstate expectation values of the ferromagnetic structure factor, ${\left(S_{\text{F}}\right)}_{\alpha \alpha }=\langle \alpha |{\widehat{S}}_{\text{F}}|\alpha \rangle$, in the ferromagnetic 2D-TFIM ($\varepsilon =0$) for the two clusters with $N=20$ (20A and 20B; see Fig. 1). For all the values of the transverse field depicted, the support of the eigenstate expectation values is narrower in cluster 20A. One can then conclude that this cluster suffers from smaller finite size effects than cluster 20B. All results shown in the main text for $N=20$ are for cluster 20A.