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=-ₑ (Jₗₑ^xₑ+₄ₗ^x+Jₙₑ^zₑ+₄ₙ^z). When JₗJₙ, we show that, on clusters of dimension 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 Jₗ=Jₙ. 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 22^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.