  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.
