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 $H={-}{{∑}}ᵣ({J}ₓ{{σ}}_{{r}}ˣ{{σ}}_{{r}+{e}ₓ}ˣ+{J}z{{σ}}_{{r}}ᶻ{{σ}}_{{r}+{e}z}ᶻ)$. When ${J}ₓ{≠}{J}z$, we show that, on clusters of dimension $L×{}L$, 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 $2×{}{2}L$. We also extend the symmetry analysis of the model to arbitrary spins and show that the twofold degeneracy of all eigenstates remains true for arbitrary half-integer spins but does not apply to integer spins, in which cases the eigenstates are generically nondegenerate, a result confirmed by exact diagonalizations in the spin-1 case. Implications for Mott insulators and Josephson junction arrays are briefly discussed.