Chinese Physics B

Two-dimensional quantum compass model in a staggered field: some rigorous results

Pei-Song He, Wen‐Long You, Guang-Shan Tian

Published Jan 1, 2011• 4 citations

Abstract

We study the properties of the two-dimensional quantum compass model in a staggered field. Using the Perron—Fröbenius theorem and the reflection positivity method, we rigorously determine the low energy spectrum of this model and its global ground state ψ 0 . Furthermore, we show that ψ 0 has a directional long-range order.

DOI: 10.1088/1674-1056/20/1/017503

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Chinese Physics B

Two-dimensional quantum compass model in a staggered field: some rigorous results

Pei-Song He, Wen‐Long You, Guang-Shan Tian

Published Jan 1, 2011• 4 citations

Publisher

Abstract

DOI: 10.1088/1674-1056/20/1/017503

Related papers

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Journal of Physics A Mathematical and TheoreticalJun 11, 201015 citations

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arXiv:1306.0791Physical Review BJun 21, 201328 citations

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

{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.

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.