Eigenstate thermalization hypothesis in two-dimensional $XXZ$ model with or without SU(2) symmetry

We investigate the eigenstate thermalization properties of the spin-1/2 $XXZ$ model in two-dimensional rectangular lattices of size $L_1\times L_2$ under periodic boundary conditions. Exploiting the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up to system size $4\times 7$ and of the energy eigenstates up to $4\times 6$. Numerical analysis of the Hamiltonian eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis supports that the two-dimensional $XXZ$ model follows the eigenstate thermalization hypothesis. When the spin interaction is isotropic, the $XXZ$ model Hamiltonian conserves the total spin and has SU(2) symmetry. We show that the eigenstate thermalization hypothesis is still valid within each subspace where the total spin is a good quantum number.

Figure 1

(a) Rectangular lattice of size $L_1\times L_2$ under periodic boundary conditions in the horizontal (${\mathit{e}}_1$) and vertical (${\mathit{e}}_2$) directions. (b) Commutation relations among the $XXZ$ Hamiltonian and symmetry operators. Mutually commuting operators are connected with a solid line. A dashed line connect operators which are commuting only within the subspace with specific quantum numbers of the symmetry operator. The Hamiltonian and ${\mathit{S}}^2$, connected by a dashed-dotted line, commute only when $\mathrm{\Delta }=1$.

Figure 2

Distributions of the ratio of consecutive energy gaps of the $XXZ$ model with $\lambda =1$ on the rectangular lattice of size $4\times 7$. These data are obtained from the half of the energy eigenvalues in the middle of the entire spectrum. They are compared with the corresponding distribution from the Poisson-distributed energy spectrum, $P_{\mathrm{Poisson}}\left(r\right)$, and the random matrix spectrum in the Gaussian orthogonal ensemble, $P_{\mathrm{GOE}}\left(r\right)$. The peculiar shape of the distribution at $\mathrm{\Delta }=1$ is ascribed to the SU(2) symmetry, which will be analyzed in detail in Sec. 4. The dotted and dashed lines are from a mixture of replicated spectra, which will be also explained in Sec. 4.

Figure 3

Matrix elements $O_{mn}^Z$ in (a) and $O_{mn}^P$ in (b) in the Hamiltonian eigenstate basis with $m,n=0,\cdots ,1391$. The lattice size is $4\times 5$ and model parameters are $\mathrm{\Delta }=2$ and $\lambda =1$.

Figure 4

Diagonal matrix elements $O_{nn}=\langle E_n\left|O\right|E_n\rangle$ versus energy density $e_n=E_n/N$ at three different lattice sizes with $\mathrm{\Delta }=2$ and $\lambda =1/2$. Rectangular boxes represent energy windows $W(e_{c}N,\delta E)$ of width $\delta E=0.5,1.0,2.0$ for $O^Z$ and $\delta E=2.0$ for the other observables for the system of size $4\times 6$.

Figure 5

Distribution of detrended diagonal matrix elements $d={\tilde{O}}_{nn}$ within the energy window $W(E_c=e_{c}N,\delta E)$ depicted with the rectangular boxes in Fig. 4. Model parameters are $\mathrm{\Delta }=2$ and $\lambda =1/2$. (a) We compare the distributions for the operator $O^Z$ with the choice of three different values $\delta E$ when the lattice is of size $4\times 6$. The solid curves represent the Gaussian distribution of the same mean and variance as the histogram data. The dotted line is the probability distribution of the bare diagonal elements, after being subtracted by their mean value, with $\delta E=1.0$. (b)–(d) We compare the distributions obtained from the lattices of size $4\times 5$ and $4\times 6$. All the distributions are consistent with the Gaussian distributions (solid curves). Numerical values of ${\sigma }_d^2{N}^{2\theta }D$ (see main text) are annotated in (b)–(d).

Figure 6

Distribution of off-diagonal matrix elements $O_{mn}$ with $m\ne n$ among energy eigenstates within the energy window $W(E_c=e_{c}N,\delta E)$ of width $\delta E=0.5$ (filled symbols and solid lines) and $\delta E=1.0$ (open symbols and dashed lines) centered at the energy density $e_c=0.0$ or $-0.2$. The curves represent the Gaussian distribution with the same mean and variance as the histogram data. The model parameters are $\mathrm{\Delta }=2$ and $\lambda =1/2$.

Figure 7

Variance ratio $q_n={\sigma }_o^2/{\sigma }_d^d$ for the model with $\mathrm{\Delta }=2$ and $\lambda =0.5$ and $L_1\times L_2=4\times 5$ (dotted line) and $4\times 6$ (solid line). The mean value and the standard deviation of the ratios $\left\{q_n\right\}$ within the energy interval $-0.3<e_n<0.3$ are presented in each panel (broken line) for $L_1\times L_2=4\times 6$.

Figure 8

Diagonal elements of the operators $O^Z$ and $O^P$ in the energy eigenstate basis when $\mathrm{\Delta }=1, \lambda =1/2$, and $L_1\times L_2=4\times 6$. Note that the diagonal elements of $O^P$ are quantized to the values $s(s+1)/N$ with $N=24$ and $s=0,2,\cdots ,12$ as indicated by arrows.

Figure 9

Diagonal elements of the operators (a) $O^Z$ and (b) $O^J$ when $\mathrm{\Delta }=1, \lambda =1/2$, and $L_1\times L_2=4\times 6$. Diagonal elements are plotted with different symbols depending on their total spin quantum number $s$. The inset in (b) shows $D_s$, the number of eigenstates in the SU(2) subsector of total spin quantum number $s$.

Figure 10

(a) Distribution of the ratio of consecutive energy gaps. (b) Ratio of the variance of off-diagonal elements to the variance of diagonal elements of the operator $O^J$ in the SU(2) subsector of $s=2$. $\langle q\rangle$ denotes the average of $q_n$ within the energy interval $-0.2\le E_n/N\le 0.3$. Both data are obtained from the system with $\mathrm{\Delta }=1, \lambda =1/2$, and $L_1\times L_2=4\times 6$.