The Quantum Compass Model on the Square Lattice

Using exact diagonalizations, Green's function Monte Carlo simulations and high-order perturbation theory, we study the low-energy properties of the two-dimensional spin-1/2 compass model on the square lattice defined by the Hamiltonian $H = - \sum_{\bm{r}} (J_x \sigma_{\bm{r}}^x \sigma_{\bm{r} + \bm{e}_x}^x + J_z \sigma_{\bm{r}}^z \sigma_{\bm{r} + \bm{e}_z}^z)$. When $J_x\ne J_z$, we show that, on clusters of dimension $L\times L$, the low-energy spectrum consists of $2^L$ states which collapse onto each other exponentially fast with $L$, a conclusion that remains true arbitrarily close to $J_x=J_z$. At that point, we show that an even larger number of states collapse exponentially fast with $L$ onto the ground state, and we present numerical evidence that this number is precisely $2\times 2^L$. We also extend the symmetry analysis of the model to arbitrary spins and show that the two-fold degeneracy of all eigenstates remains true for arbitrary half-integer spins but does not apply to integer spins, in which cases eigenstates are generically non degenerate, a result confirmed by exact diagonalizations in the spin-1 case. Implications for Mott insulators and Josephson junction arrays are briefly discussed.