We introduce an exact spin transformation that maps frustrated
Zi,jZi,j+1 and
Xi,jXi+1,j spin interactions along the rows and columns of the quantum compass model (QCM) on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice to $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ quantum spin models with $2(L\ensuremath{-}1)$ classical spins. Using the symmetry properties we unravel the hidden dimer order in the QCM, with equal two-dimer correlations
⟨Xi,iXi+1,iXk,lXk+1,l⟩ and
⟨Xi,iXi+1,iXl,kXl+1,k⟩ in the ground state, which is independent of the actual interactions. This order coexists with Ising-type spin correlations which decay with distance.
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 ${J}_{x}\ensuremath{\ne}{J}_{z}$, we show that, on clusters of dimension $L\ifmmode\times\else\texttimes\fi{}L$, the low-energy spectrum consists of
2L states which collapse onto each other exponentially fast with
L, a conclusion that remains true arbitrarily close to
Jx=Jz. 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.
We use exact symmetry properties of the two-dimensional quantum compass model to derive nonequivalent invariant subspaces in the energy spectra of $L\ifmmode\times\else\texttimes\fi{}L$ clusters up to
L=6. The symmetry allows one to reduce the original $L\ifmmode\times\else\texttimes\fi{}L$ compass cluster to the $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ one with modified interactions. This step is crucial and enables (i) exact diagonalization of the $6\ifmmode\times\else\texttimes\fi{}6$ 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 $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit suggests the classical nature of the quantum compass model and high degeneracy of the ground state with nematic order.
Compass models provide insights into the properties of Mott-insulating materials that host bond-dependent anisotropic interactions between their pseudospin degrees of freedom. In this article, we explore the classical and quantum ground-state properties of one such model relevant to certain layered perovskite materials akin to
Ba2IrO4---namely, the Heisenberg-compass model on the square lattice. We first investigate the ground-state phase diagram of this model using classical Monte Carlo simulations. These reveal that the low-temperature classical phase diagram is divided into six different classes of long-range ordered phases, including four phases that exhibit an order by disorder selection and two phases that are stabilized energetically. This model admits a special duality transformation, known as the Klein duality, conveniently allowing to map one region of coupling parameters onto another and constraining the phase diagram, and which we exploit in our study. From the analysis of the zero-point energy and the free energy of the spin waves, we find that order by quantum disorder at zero temperature and order by thermal disorder select the same orderings as those found from classical Monte Carlo simulations. We further investigate the quantum ground states of this model using numerical exact diagonalization on small clusters by exploiting the translational symmetry of the square lattice. We obtain a ground-state phase diagram bearing close resemblance to that found from the classical analysis.
We study the one-dimensional quantum compass model with two independent parameters by means of an exact mapping to the quantum Ising model. This allows us to uncover hidden features of the quantum phase transition in the ordinary one-parameter quantum compass model, showing that it occurs at a multicritical point where a line of first-order transitions intersects a line of second-order symmetry-breaking transitions of Ising type. We calculate the concurrence and the block entanglement entropy in the four ground-state phases and find that these entanglement measures accurately signal the second-order, but not the first-order, transitions.