Symmetry properties and spectra of the two-dimensional quantum compass model

We use exact symmetry properties of the two-dimensional quantum compass model to derive nonequivalent invariant subspaces in the energy spectra of $L \times L$ clusters up to L=6. The symmetry allows one to reduce the original $L \times L $ compass cluster to the $\left ( L - 1 \right ) \times \left ( L - 1 \right )$ one with modified interactions. This step is crucial and enables (i) exact diagonalization of the 6 \times 6 quantum compass cluster and (ii) finding the specific heat for clusters up to L = 6, with two characteristic energy scales. We investigate the properties of the ground state and the first excited states and present extrapolation of the excitation energy with increasing system size. Our analysis provides physical insights into the nature of nematic order realized in the quantum compass model at finite temperature. We suggest that the quantum phase transition at the isotropic interaction point is second order with some admixture of the discontinuous transition, as indicated by the entropy, the overlap between two types of nematic order (on horizontal and vertical bonds), and the existence of the critical exponent. Extrapolation of the specific heat to the L\rightarrow \infty limit suggests the classical nature of the quantum compass model and high degeneracy of the ground state with nematic order.