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

Ground States in Relatively Bounded Quantum Perturbations of Classical Lattice Systems

D.A. Yarotsky

arXiv:math-ph/0412040Communications in Mathematical PhysicsNov 4, 200545 citations

Local-Channel Stability of Classical $\mathbb Z_2$ Lattice Gauge ModelsLocal-Channel Stability of Classical $\mathbb Z_2$ Lattice Gauge Models

Guanxi Xun, Weiguang Cao, Haruki Watanabe

We establish sufficient conditions for two-way local quantum-channel stability in classical $\mathbb Z_2$ lattice gauge models with spins on $m$ -dimensional cells of a $d$ -dimensional lattice. Each interaction multiplies the spins on the boundary of an $(m+1)$ -cell. This family, defined on the full spin Hilbert space without a Gauss-law constraint, includes the Ising model at $m=0$ and the magnetic sector of the toric code at $m=1$ . The low-temperature proof combines two ingredients: locally indistinguishable flat sectors and extended magnetic defects of dimension $d-m-1$ . For $1\le m\le d-2$ , sufficiently weak local classical perturbations at sufficiently low positive temperature preserve equivalence to the canonical flat ensemble. No internal or gauge symmetry is required of the perturbation. For weak noncommuting perturbations, we prove the same conclusion in a smaller sufficient window by reducing the diagonal probabilities to a weak classical interaction and recovering the coherences with a local channel. The conversion channels use product ancillary inputs, have polylogarithmic causal radius, and achieve any prescribed inverse-power whole-system trace-norm error. Sufficiently high-temperature states are instead two-way equivalent to a product state. Independent finite-dimensional onsite systems can be included with weak local couplings. An optimized covariance between two distant neighborhoods of parallel $m$ -tori is nonincreasing under one-way local conversion and invariant under two-way conversion. Its values $\mathcal F_m=1$ and $0$ distinguish the proved low- and high-temperature domains without a protecting symmetry. These sufficient domains do not determine the full phase diagram.We establish sufficient conditions for two-way local quantum-channel stability in classical $\mathbb Z_2$ lattice gauge models with spins on $m$-dimensional cells of a $d$-dimensional lattice. Each interaction multiplies the spins on the boundary of an $(m+1)$-cell. This family, defined on the full spin Hilbert space without a Gauss-law constraint, includes the Ising model at $m=0$ and the magnetic sector of the toric code at $m=1$. The low-temperature proof combines two ingredients: locally indistinguishable flat sectors and extended magnetic defects of dimension $d-m-1$. For $1\le m\le d-2$, sufficiently weak local classical perturbations at sufficiently low positive temperature preserve equivalence to the canonical flat ensemble. No internal or gauge symmetry is required of the perturbation. For weak noncommuting perturbations, we prove the same conclusion in a smaller sufficient window by reducing the diagonal probabilities to a weak classical interaction and recovering the coherences with a local channel. The conversion channels use product ancillary inputs, have polylogarithmic causal radius, and achieve any prescribed inverse-power whole-system trace-norm error. Sufficiently high-temperature states are instead two-way equivalent to a product state. Independent finite-dimensional onsite systems can be included with weak local couplings. An optimized covariance between two distant neighborhoods of parallel $m$-tori is nonincreasing under one-way local conversion and invariant under two-way conversion. Its values $\mathcal F_m=1$ and $0$ distinguish the proved low- and high-temperature domains without a protecting symmetry. These sufficient domains do not determine the full phase diagram.

quant-pharXiv:2610.27939Oct 21, 20260 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.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, 20120 citations

Existence of phase transitions for quantum lattice systems

J. Ginibre

Communications in Mathematical PhysicsSep 1, 196973 citations

{\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.

Local-Channel Stability of Classical

\mathbb Z_2

Lattice Gauge Models

We establish sufficient conditions for two-way local quantum-channel stability in classical

\mathbb Z_2

lattice gauge models with spins on

m

-dimensional cells of a

d

-dimensional lattice. Each interaction multiplies the spins on the boundary of an

(m+1)

-cell. This family, defined on the full spin Hilbert space without a Gauss-law constraint, includes the Ising model at

m=0

and the magnetic sector of the toric code at

m=1

. The low-temperature proof combines two ingredients: locally indistinguishable flat sectors and extended magnetic defects of dimension

d-m-1

. For

1\le m\le d-2

, sufficiently weak local classical perturbations at sufficiently low positive temperature preserve equivalence to the canonical flat ensemble. No internal or gauge symmetry is required of the perturbation. For weak noncommuting perturbations, we prove the same conclusion in a smaller sufficient window by reducing the diagonal probabilities to a weak classical interaction and recovering the coherences with a local channel. The conversion channels use product ancillary inputs, have polylogarithmic causal radius, and achieve any prescribed inverse-power whole-system trace-norm error. Sufficiently high-temperature states are instead two-way equivalent to a product state. Independent finite-dimensional onsite systems can be included with weak local couplings. An optimized covariance between two distant neighborhoods of parallel

m

-tori is nonincreasing under one-way local conversion and invariant under two-way conversion. Its values

\mathcal F_m=1

and

0

distinguish the proved low- and high-temperature domains without a protecting symmetry. These sufficient domains do not determine the full phase diagram.