Quantum compass model on the square lattice
Julien Dorier, Federico Becca, Frédéric Mila · Physical Review B · 2005
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 $H=\ensuremath{-}{\ensuremath{\sum}}_{r}({J}_{x}{\ensuremath{\sigma}}_{\mathbit{r}}^{x}{\ensuremath{\sigma}}_{\mathbit{r}+{e}_{x}}^{x}+{J}_{z}{\ensuremath{\sigma}}_{\mathbit{r}}^{z}{\ensuremath{\sigma}}_{\mathbit{r}+{e}_{z}}^{z})$. When ${J}_{x}\ensuremath{ e}{J}_{z}$, we show that, on clusters of dimension $L\ifmmode\times\else\texttimes\fi{}L$, 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 $2\ifmmode\times\else\texttimes\fi{}{2}^{L}$. 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.