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arXiv:cond-mat/0501708 · Physical Review B

Quantum compass model on the square lattice

Julien Dorier, Federico Becca, Frédéric Mila

Published Jul 22, 2005• 129 citations
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Abstract

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.

DOI
10.1103/physrevb.72.024448

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The quantum compass model consists of a two-dimensional square spin lattice where the orientation of the spin–spin interactions depends on the spatial direction of the bonds. It has remarkable symmetry properties and the ground state shows topological degeneracy. The implementation of the quantum compass model in quantum simulation setups like ultracold atoms and trapped ions is far from trivial, since spin interactions in those systems typically are independent of the spatial direction. Ising spin interactions, on the contrary, can be induced and controlled in atomic setups with state-of-the art experimental techniques. In this work, we show how the quantum compass model on a rectangular lattice can be simulated by the use of the photon-assisted tunneling induced by periodic drivings on a quantum Ising spin model. We describe a procedure to adiabatically prepare one of the doubly degenerate ground states of this model by adiabatically ramping down a transverse magnetic field, with surprising differences depending on the parity of the lattice size. Exact diagonalizations confirm the validity of this approach for small lattices. Specific implementations of this scheme are presented with ultracold atoms in optical lattices in the Mott insulator regime, as well as with Rydberg atoms.
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We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a XXXXXX- to ZZZZZZ-ordered phase in the 3d3d3d-Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a ``superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a ppp-wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature (TTT)-dependent dimensional crossover from a high-TTT quasi-1d1d1d insulator-like state, via an incoherent bad-metal-like state at intermediate TTT, to a 2d2d2d symmetry-broken insulator at low TTT, well below the ``orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.