arXiv:1005.1508 · Europhysics Letters (EPL)

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

Fabien Trousselet, Andrzej M. Oleś, Peter Horsch

Published Aug 1, 2010• 46 citations

Abstract

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.

DOI: 10.1209/0295-5075/91/40005

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1 The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins

Johannes Richter, J. Schulenburg

Jan 1, 200991 citations

arXiv:1005.1508 · Europhysics Letters (EPL)

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

Fabien Trousselet, Andrzej M. Oleś, Peter Horsch

Published Aug 1, 2010• 46 citations

arXiv HTML PDF

Abstract

DOI: 10.1209/0295-5075/91/40005

Related papers

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.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:1207.0102Physical Review BOct 15, 201236 citations

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

Subhankar Khatua, Griffin C. Howson, Michel J. P. Gingras +1 more

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.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.

arXiv:2404.02196Physical review. B./Physical review. BSep 23, 20245 citations

Compass models: Theory and physical motivations

Zohar Nussinov, Jeroen van den Brink

Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. A simple illustrative example is furnished by the 90\ifmmode^\circ\else\textdegree\fi{} compass model on a square lattice in which only couplings of the form ${\ensuremath{\tau}}_{i}^{x}{\ensuremath{\tau}}_{j}^{x}$ (where ${{\ensuremath{\tau}}_{i}^{a}{}}_{a}$ denote Pauli operators at site $i$ ) are associated with nearest-neighbor sites $i$ and $j$ separated along the $x$ axis of the lattice while ${\ensuremath{\tau}}_{i}^{y}{\ensuremath{\tau}}_{j}^{y}$ couplings appear for sites separated by a lattice constant along the $y$ axis. Similar compass-type interactions can appear in diverse physical systems. For instance, compass models describe Mott insulators with orbital degrees of freedom where interactions sensitively depend on the spatial orientation of the orbitals involved as well as the low-energy effective theories of frustrated quantum magnets, and a host of other systems such as vacancy centers, and cold atomic gases. The fundamental interdependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors, including the frustration of (semi-)classical ordered states on nonfrustrated lattices, and to enhanced quantum effects, prompting, in certain cases, the appearance of zero-temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. In this article, compass models are reviewed in a unified manner, paying close attention to exact consequences of these symmetries and to thermal and quantum fluctuations that stabilize orders via order-out-of-disorder effects. This is complemented by a survey of numerical results. In addition to reviewing past works, a number of other models are introduced and new results established. In particular, a general link between flat bands and symmetries is detailed.Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. A simple illustrative example is furnished by the 90\ifmmode^\circ\else\textdegree\fi{} compass model on a square lattice in which only couplings of the form ${\ensuremath{\tau}}_{i}^{x}{\ensuremath{\tau}}_{j}^{x}$ (where ${{\ensuremath{\tau}}_{i}^{a}{}}_{a}$ denote Pauli operators at site $i$) are associated with nearest-neighbor sites $i$ and $j$ separated along the $x$ axis of the lattice while ${\ensuremath{\tau}}_{i}^{y}{\ensuremath{\tau}}_{j}^{y}$ couplings appear for sites separated by a lattice constant along the $y$ axis. Similar compass-type interactions can appear in diverse physical systems. For instance, compass models describe Mott insulators with orbital degrees of freedom where interactions sensitively depend on the spatial orientation of the orbitals involved as well as the low-energy effective theories of frustrated quantum magnets, and a host of other systems such as vacancy centers, and cold atomic gases. The fundamental interdependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors, including the frustration of (semi-)classical ordered states on nonfrustrated lattices, and to enhanced quantum effects, prompting, in certain cases, the appearance of zero-temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. In this article, compass models are reviewed in a unified manner, paying close attention to exact consequences of these symmetries and to thermal and quantum fluctuations that stabilize orders via order-out-of-disorder effects. This is complemented by a survey of numerical results. In addition to reviewing past works, a number of other models are introduced and new results established. In particular, a general link between flat bands and symmetries is detailed.

Reviews of Modern PhysicsJan 12, 2015312 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.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:2603.09469Physical review. B./Physical review. BFeb 18, 20260 citations

Exact ground states and phase diagram of the quantum compass model under an in-plane field

A. D. S. Richards, Erik Schwartz Sørensen

We consider the square lattice $S=1/2$ quantum compass model (QCM) parametrized by ${J}_{x},{J}_{z}$ , under a field, $\mathbf{h}$ , in the $x\text{\ensuremath{-}}z$ plane. At the special field value $({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})$ , we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at $({h}_{x}^{★},{h}_{z}^{★})$ and states obtained for fields within the vicinity of $({h}_{x}^{★},{h}_{z}^{★})$ . This gapped phase occupies most of the in-plane field phase diagram. For some values of ${h}_{x}>{h}_{z}$ and ${h}_{z}>{h}_{x}$ at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.We consider the square lattice $S=1/2$ quantum compass model (QCM) parametrized by ${J}_{x},{J}_{z}$, under a field, $\mathbf{h}$, in the $x\text{\ensuremath{-}}z$ plane. At the special field value $({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})$, we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at $({h}_{x}^{★},{h}_{z}^{★})$ and states obtained for fields within the vicinity of $({h}_{x}^{★},{h}_{z}^{★})$. This gapped phase occupies most of the in-plane field phase diagram. For some values of ${h}_{x}>{h}_{z}$ and ${h}_{z}>{h}_{x}$ at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.

arXiv:2310.01384Physical review. B./Physical review. BJun 26, 20243 citations

1 The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins

Johannes Richter, J. Schulenburg

We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1

Jan 1, 200991 citations

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.

{\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.

i

) are associated with nearest-neighbor sites

i

and

j

separated along the

x

axis of the lattice while ${\ensuremath{\tau}}_{i}^{y}{\ensuremath{\tau}}_{j}^{y}$ couplings appear for sites separated by a lattice constant along the

y

axis. Similar compass-type interactions can appear in diverse physical systems. For instance, compass models describe Mott insulators with orbital degrees of freedom where interactions sensitively depend on the spatial orientation of the orbitals involved as well as the low-energy effective theories of frustrated quantum magnets, and a host of other systems such as vacancy centers, and cold atomic gases. The fundamental interdependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors, including the frustration of (semi-)classical ordered states on nonfrustrated lattices, and to enhanced quantum effects, prompting, in certain cases, the appearance of zero-temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. In this article, compass models are reviewed in a unified manner, paying close attention to exact consequences of these symmetries and to thermal and quantum fluctuations that stabilize orders via order-out-of-disorder effects. This is complemented by a survey of numerical results. In addition to reviewing past works, a number of other models are introduced and new results established. In particular, a general link between flat bands and symmetries is detailed.

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.

We consider the square lattice

S=1/2

quantum compass model (QCM) parametrized by

{J}_{x},{J}_{z}

, under a field,

\mathbf{h}

, in the $x\text{\ensuremath{-}}z$ plane. At the special field value

({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})

, we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at

({h}_{x}^{★},{h}_{z}^{★})

and states obtained for fields within the vicinity of

({h}_{x}^{★},{h}_{z}^{★})

. This gapped phase occupies most of the in-plane field phase diagram. For some values of

{h}_{x}>{h}_{z}

and

{h}_{z}>{h}_{x}

at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.