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Journal of Physics A Mathematical and Theoretical

The low-energy states and directional long-range order in the two-dimensional quantum compass model

Wen‐Long You, Guang-Shan Tian, Hai-Qing Lin

Published Jun 11, 2010• 15 citations
Publisher

Abstract

In the present paper, we study the properties of the two-dimensional quantum compass model by a mathematically rigorous approach. Based on the Perron?Fr?benius theorem and the reflection positivity method, we show that the low-energy spectrum of this model is of one-dimensional type. Furthermore, its global ground states have a directional long-range order. Our results confirm several previous conclusions obtained by either numerical or perturbation calculations

DOI
10.1088/1751-8113/43/27/275001

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In the present paper, we study the properties of the two-dimensional quantum compass model by a mathematically rigorous approach. Based on the Perron?Fr?benius theorem and the reflection positivity method, we show that the low-energy spectrum of this model is of one-dimensional type. Furthermore, its global ground states have a directional long-range order. Our results confirm several previous conclusions obtained by either numerical or perturbation calculations
In the present paper, we investigate quantum phase transition in the one-dimensional compass model introduced by Brzezicki et al. [Phys. Rev. B 75, 134415 (2007)]. Unlike the previous approach based on the Jordan-Wigner transformation, we use directly the standard pseudospin representation in our investigation. Therefore, our method can be potentially applied to study rigorously the properties of the same model in higher dimensions. By applying the reflection positivity technique, we are able to determine degeneracy of the global ground state and the quantum phase-transition point of this model. We find also that the transition is of the first order. These results are consistent with the previous conclusions. However, some differences are also uncovered.
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∕21∕21∕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{2}^{L}2L states which collapse onto each other exponentially fast with LLL, a conclusion that remains true arbitrarily close to Jx=Jz{J}_{x}={J}_{z}Jx​=Jz​. At that point, we show that an even larger number of states collapse exponentially fast with LLL 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=6L=6L=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=6L=6L=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.
I show the symmetries of the compass model and spin transformations making the Hamiltonian block‐diagonal. I present the new form of the Hamiltonian and explain how the diagonal blocks are related by translational symmetry and by the isotropy of interactions. I reveal the hidden symmetry of the lowest‐energy block and resulting identities in four‐point dimer‐dimer correlators. Using exact diagonalization I show that the ground state has classical order with local quantum fluctuation vanishing in a long range and that the energy spectrum consists of discrete and continuous part.
In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters β = 1/8 and the cental charges c = 1/2 at the critical points show that the topological phase transitions between the phases belong to an Ising type of universality classes. In addition to the string order parameters, the singularities of the second derivative of the ground state energies per site, the continuous and singular behaviors of the Von Neumann entropy and the pinch points of the fidelity per lattice site manifest that the phase transitions between the phases are of the second-order, in contrast to the first-order transition suggested in previous studies.