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.