Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2404.02196 · Physical review. B./Physical review. B

Ground state properties of the Heisenberg-compass model on the square lattice

Subhankar Khatua, Griffin C. Howson, Michel J. P. Gingras, Jeffrey G. Rau

Published Sep 23, 2024• 5 citations
arXivHTMLPDF

Abstract

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 ${\mathrm{Ba}}_{2}{\mathrm{IrO}}_{4}$---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.

DOI
10.1103/physrevb.110.104426

Related papers

Quantum compass model on the square lattice

Julien Dorier, Federico Becca, Frédéric Mila

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.

arXiv:cond-mat/0501708•Physical Review B•Jul 22, 2005•129 citations

Phase diagram and Ashkin-Teller universality in the classical square-lattice Heisenberg-compass model

Yuchen Fan

We determine the finite-temperature phase diagram and critical behavior of the classical square-lattice Heisenberg-compass model using large-scale Monte Carlo simulations and finite-size scaling. Six symmetry distinct ordered phases are identified. The four phases that simultaneously break the spin-lattice $C_4$ and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two $z$-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.

arXiv:2603.09469•Physical review. B./Physical review. B•Feb 18, 2026•0 citations

Compass-Heisenberg model on the square lattice —Spin order and elementary excitations

Fabien Trousselet, Andrzej M. Oleś, Peter Horsch

We explore the physics of the anisotropic compass model under the influence of perturbing Heisenberg interactions and present the phase diagram with multiple quantum phase transitions. The macroscopic ground-state degeneracy of the compass model is lifted in the thermodynamic limit already by infinitesimal Heisenberg coupling, which selects different ground states with symmetry depending on the sign and size of the coupling constants —then low-energy excitations are spin waves, while the compass states reflecting columnar order are separated from them by a macroscopic gap. Nevertheless, nanoscale structures relevant for quantum computation purposes may be tuned such that the compass states are the lowest-energy excitations, thereby avoiding decoherence, if a size criterion derived by us is fulfilled.

arXiv:1005.1508•Europhysics Letters (EPL)•Aug 1, 2010•46 citations

Magnetic properties of nanoscale compass-Heisenberg planar clusters

Fabien Trousselet, Andrzej M. Oleś, Peter Horsch

We study a model of spins $1/2$ on a square lattice, generalizing the quantum compass model via the addition of perturbing Heisenberg interactions between nearest neighbors, and investigate its phase diagram and magnetic excitations. This model has motivations both from the field of strongly correlated systems with orbital degeneracy and from that of solid-state based devices proposed for quantum computing. We find that the high degeneracy of ground states of the compass model is fragile and changes into twofold degenerate ground states for any finite amplitude of Heisenberg coupling. By computing the spin structure factors of finite clusters with L\'anczos diagonalization, we evidence a rich variety of phases characterized by ${\mathbb{Z}}_{2}$ symmetry that are either ferromagnetic, $C$-type antiferromagnetic, or of the N\'eel type and analyze the effects of quantum fluctuations on phase boundaries. In the ordered phases, the anisotropy of compass interactions leads to a finite excitation gap to spin waves. We show that for small nanoscale clusters with large anisotropy gap the lowest excitations are column-flip excitations that emerge due to Heisenberg perturbing interactions from the manifold of degenerate ground states of the compass model. We derive an effective one-dimensional $XYZ$ model that faithfully reproduces the exact structure of these excited states and elucidates their microscopic origin. The low-energy column-flip or compass-type excitations are robust against decoherence processes and are therefore well designed for storing information in quantum computing. We also point out that the dipolar interactions between nitrogen-vacancy centers forming a rectangular lattice in a diamond matrix may permit a solid-state realization of the anisotropic compass-Heisenberg model.

arXiv:1207.0102•Physical Review B•Oct 15, 2012•36 citations

Anisotropic frustrated Heisenberg model on the honeycomb lattice

Ansgar Kalz, Marcelo Arlego, D. C. Cabra +2 more

We investigate the ground-state phase diagram of an anisotropic Heisenberg model on the honeycomb lattice with competing interactions. We use quantum Monte Carlo simulations, as well as linear spin-wave and Ising series expansions, to determine the phase boundaries of the ordered magnetic phases. We find a region without any classical order in the vicinity of a highly frustrated point. Higher-order correlation functions in this region give no signal for long-range valence-bond order. The low-energy spectrum is derived via exact diagonalization to check for topological order on small-size periodic lattices.

arXiv:1201.1505•Physical Review B•Mar 12, 2012•12 citations

Phase diagrams of two-dimensional frustrated spin systems

Ansgar Kalz

In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.

•Jan 1, 2012•0 citations
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{\mathrm{Ba}}_{2}{\mathrm{IrO}}_{4}Ba2​IrO4​---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.
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 determine the finite-temperature phase diagram and critical behavior of the classical square-lattice Heisenberg-compass model using large-scale Monte Carlo simulations and finite-size scaling. Six symmetry distinct ordered phases are identified. The four phases that simultaneously break the spin-lattice C4C_4C4​ and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two zzz-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.
We explore the physics of the anisotropic compass model under the influence of perturbing Heisenberg interactions and present the phase diagram with multiple quantum phase transitions. The macroscopic ground-state degeneracy of the compass model is lifted in the thermodynamic limit already by infinitesimal Heisenberg coupling, which selects different ground states with symmetry depending on the sign and size of the coupling constants —then low-energy excitations are spin waves, while the compass states reflecting columnar order are separated from them by a macroscopic gap. Nevertheless, nanoscale structures relevant for quantum computation purposes may be tuned such that the compass states are the lowest-energy excitations, thereby avoiding decoherence, if a size criterion derived by us is fulfilled.
We study a model of spins 1/21/21/2 on a square lattice, generalizing the quantum compass model via the addition of perturbing Heisenberg interactions between nearest neighbors, and investigate its phase diagram and magnetic excitations. This model has motivations both from the field of strongly correlated systems with orbital degeneracy and from that of solid-state based devices proposed for quantum computing. We find that the high degeneracy of ground states of the compass model is fragile and changes into twofold degenerate ground states for any finite amplitude of Heisenberg coupling. By computing the spin structure factors of finite clusters with L\'anczos diagonalization, we evidence a rich variety of phases characterized by Z2{\mathbb{Z}}_{2}Z2​ symmetry that are either ferromagnetic, CCC-type antiferromagnetic, or of the N\'eel type and analyze the effects of quantum fluctuations on phase boundaries. In the ordered phases, the anisotropy of compass interactions leads to a finite excitation gap to spin waves. We show that for small nanoscale clusters with large anisotropy gap the lowest excitations are column-flip excitations that emerge due to Heisenberg perturbing interactions from the manifold of degenerate ground states of the compass model. We derive an effective one-dimensional XYZXYZXYZ model that faithfully reproduces the exact structure of these excited states and elucidates their microscopic origin. The low-energy column-flip or compass-type excitations are robust against decoherence processes and are therefore well designed for storing information in quantum computing. We also point out that the dipolar interactions between nitrogen-vacancy centers forming a rectangular lattice in a diamond matrix may permit a solid-state realization of the anisotropic compass-Heisenberg model.
We investigate the ground-state phase diagram of an anisotropic Heisenberg model on the honeycomb lattice with competing interactions. We use quantum Monte Carlo simulations, as well as linear spin-wave and Ising series expansions, to determine the phase boundaries of the ordered magnetic phases. We find a region without any classical order in the vicinity of a highly frustrated point. Higher-order correlation functions in this region give no signal for long-range valence-bond order. The low-energy spectrum is derived via exact diagonalization to check for topological order on small-size periodic lattices.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.