Exact diagonalization and quantum Monte Carlo study of the spin-$\frac{1}{2}$ $\mathrm{XXZ}$ model on the square lattice

The spin-$\frac{1}{2}$ $\mathrm{XXZ}$ model on an infinite square lattice at zero temperature has been studied and is presented here. Our methods using finite square lattices have been the exact diagonalization and quantum Monte Carlo methods. The physical properties estimated per lattice vertex include ground-state and first-excited-state energy, staggered magnetization, and susceptibility in the parallel and perpendicular directions, spin stiffness, and spin-wave velocity. In the $\mathrm{XXZ}$ model range from the Ising model to the Heisenberg model our estimates of physical properties compare very well with estimates by several other methods published in several other papers. In the $\mathrm{XXZ}$ range between the Heisenberg and $\mathrm{XY}$ models there is very little in the literature to compare with our finite lattice estimates of physical properties.