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 $H=-{\sum }_r(J_x{\sigma }_{\mathit{r}}^x{\sigma }_{\mathit{r}+e_x}^x+J_z{\sigma }_{\mathit{r}}^z{\sigma }_{\mathit{r}+e_z}^z)$. 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 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.

Figure 1

Ground-state energy of the ferromagnetic states defined by ${\mathit{S}}_{\mathit{r}}=\mathit{S}\left({\theta }_0\right)=S\cos {\theta }_0{\mathit{e}}_x+S\sin {\theta }_0{\mathit{e}}_z$, including zero-point energy as a function of the angle ${\theta }_0$. $E_{cl}=-JN{S}^2$ is the classical ground-state energy and $N$ is the number of sites.

Figure 2

Energy of the low-lying states vs anisotropy parameter $\theta$ for $4\times 4$ and $5\times 5$ lattices obtained by exact diagonalization $(J_x=J\cos \theta ;J_z=J\sin \theta )$.

Figure 3

Gap ${\Delta }_2$ (see text) as a function of the number of sites $N$.

Figure 4

Log-linear plot of $LP\left(L\right)$ vs the linear lattice size $L$. The squares are the results obtained numerically, and the dotted line is an exponential fit.

Figure 5

Ground-state energy $E_0$ vs the number of sites $N$ for various values of the asymmetry parameter $\theta$.

Figure 6

Gap $\Delta$ vs linear lattice size $L$ for various values of the asymmetry parameter $\theta$. For $L=2$, 3, 4, 5 the results were obtained from exact diagonalization.

Figure 7

$\alpha$ vs $J_z∕J_x$, where $\alpha$ is the gap-scaling constant $\Delta \sim {\alpha }^L$. The symbols are the results obtained with Green’s function Monte Carlo, and the dotted line is fitted to these values.

Figure 8

Long-range correlations in the ground state along $z$ direction $C^z\left(\infty \right)$ (see text) as a function of the linear lattice size $L$ for various values of the anisotropy parameter $\theta$.

Figure 9

Spectrum vs asymmetry parameter $\theta$ for linear lattice sizes $L=2$ and $L=3$ obtained by exact diagonalization ($J_x=J\cos \theta$ and $J_z=J\sin \theta$).