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

Physical review. B, Condensed matter

Critical behavior of a frustrated spin-1 Ising model

Yasushi Honda, Kazuyuki Tanaka, Tsuyoshi Horiguchi

Published Jun 1, 1995• 3 citations
Publisher

Abstract

We find that the ordinary universality does not hold for a frustrated spin-1 Ising model with nearest-neighbor bilinear interaction ${\mathit{J}}_{1}$ and next-nearest-neighbor bilinear interaction ${\mathit{J}}_{2}$ on the square lattice. We estimated critical indices \ensuremath{\nu}, \ensuremath{\eta}, and ${\mathit{y}}_{\mathit{s}}$ by applying the Monte Carlo renormalization-group method to the model. The critical index \ensuremath{\nu} varies with ${\mathit{J}}_{1}$/${\mathit{J}}_{2}$ in a region where the ground state is of the super-antiferromagnetic configuration. Although the behavior of \ensuremath{\nu} is analogous to that of the spin-1/2 Ising model as a function of ${\mathit{J}}_{1}$/${\mathit{J}}_{2}$, the value of \ensuremath{\nu} appears to be smaller than that of the corresponding spin-1/2 Ising model. The value of \ensuremath{\eta} agrees with the value predicted by the Onsager universality class. The crossover exponent ${\mathit{y}}_{\mathit{s}}$ agrees with the value estimated for the spin-1/2 Ising model.

DOI
10.1103/physrevb.51.15260

Related papers

Critical properties of the antiferromagnetic layered Ising model on a cubic lattice with competing interactions

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович

The critical properties of the antiferromagnetic layered Ising model on a cubic lattice with regard to the nearest-neighbor and next-nearest-neighbor interactions are investigated by the Monte Carlo method using the replica algorithm. The investigations are carried out for the ratios of exchange nearest-neighbor and next-nearest-neighbor interactions r = J 2/J 1 in the range of 0 ≤ r ≤ 1.0. Using the finite-size scaling theory, the static critical indices of specific heat α, order parameter β, susceptibility γ, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is retained in the range of 0 ≤ r ≤ 0.4. It is established that the change in the next-nearest-neighbor interaction value in this model in the range of r > 0.8 leads to the same universality class as the three-dimensional fully frustrated Ising model on the cubic lattice.

Physics of the Solid State•Sep 1, 2017•6 citations

Analysis of the phase transition for the Ising model on the frustrated square lattice

Ansgar Kalz, A. Honecker, Marion Moliner

We analyze the phase transition of the frustrated ${J}_{1}$-${J}_{2}$ Ising model with antiferromagnetic nearest- and strong next-nearest-neighbor interactions on the square lattice. Using extensive Monte Carlo simulations we show that the nature of the phase transition for $1/2<{J}_{2}/{J}_{1}\ensuremath{\lesssim}1$ is not of the weakly universal type---as commonly believed---but we conclude from the clearly doubly peaked structure of the energy histograms that the transition is of weak first order. Motivated by these results, we analyze the phase transitions via field-theoretic methods; i.e., we calculate the central charge of the underlying field theory via transfer-matrix techniques and present, furthermore, a field-theoretic discussion on the phase-transition behavior of the model. Starting from the conformally invariant fixed point of two decoupled critical Ising models (${J}_{1}=0$), we calculate the effect of the nearest-neighbor coupling term perturbatively using operator product expansions. As an effective action we obtain the Ashkin-Teller model.

arXiv:1105.4836•Physical Review B•Nov 9, 2011•55 citations

Investigation of critical phenomena of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions by the Monte Carlo method

Рамазанов Магомедшейх Курбанович, Akai Kurbanovich Murtazaev

Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.

Phase Transitions•Aug 16, 2017•5 citations

Phase transitions in the frustrated Ising model on the square lattice

Songbo Jin, Arnab Sen, Wenan Guo +1 more

We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions ${J}_{1}<0$ (nearest neighbor, ferromagnetic) and ${J}_{2}>0$ (second neighbor, antiferromagnetic). The striped phase breaks a ${Z}_{4}$ symmetry and is stabilized at low temperatures for $g={J}_{2}/|{J}_{1}|>1/2$. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The ${J}_{1}$-${J}_{2}$ model has a line of very weak first-order phase transitions in the whole region $1/2<g<{g}^{*}$, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from $g={g}^{*}$ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point ${g}^{*}$ in the ${J}_{1}$-${J}_{2}$ model. The continuous transitions near ${g}^{*}$ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of $g>1/2$. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.

arXiv:1212.5339•Physical Review B•Apr 9, 2013•116 citations

Phase transitions and critical characteristics in the layered antiferromagnetic Ising model with next-nearest-neighbor intralayer interactions

Рамазанов Магомедшейх Курбанович, Akai Kurbanovich Murtazaev

Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.

Journal of Experimental and Theoretical Physics Letters•May 1, 2015•25 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
We find that the ordinary universality does not hold for a frustrated spin-1 Ising model with nearest-neighbor bilinear interaction J1{\mathit{J}}_{1}J1​ and next-nearest-neighbor bilinear interaction J2{\mathit{J}}_{2}J2​ on the square lattice. We estimated critical indices \ensuremath{\nu}, \ensuremath{\eta}, and ys{\mathit{y}}_{\mathit{s}}ys​ by applying the Monte Carlo renormalization-group method to the model. The critical index \ensuremath{\nu} varies with J1{\mathit{J}}_{1}J1​/J2{\mathit{J}}_{2}J2​ in a region where the ground state is of the super-antiferromagnetic configuration. Although the behavior of \ensuremath{\nu} is analogous to that of the spin-1/2 Ising model as a function of J1{\mathit{J}}_{1}J1​/J2{\mathit{J}}_{2}J2​, the value of \ensuremath{\nu} appears to be smaller than that of the corresponding spin-1/2 Ising model. The value of \ensuremath{\eta} agrees with the value predicted by the Onsager universality class. The crossover exponent ys{\mathit{y}}_{\mathit{s}}ys​ agrees with the value estimated for the spin-1/2 Ising model.
The critical properties of the antiferromagnetic layered Ising model on a cubic lattice with regard to the nearest-neighbor and next-nearest-neighbor interactions are investigated by the Monte Carlo method using the replica algorithm. The investigations are carried out for the ratios of exchange nearest-neighbor and next-nearest-neighbor interactions r = J 2/J 1 in the range of 0 ≤ r ≤ 1.0. Using the finite-size scaling theory, the static critical indices of specific heat α, order parameter β, susceptibility γ, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is retained in the range of 0 ≤ r ≤ 0.4. It is established that the change in the next-nearest-neighbor interaction value in this model in the range of r > 0.8 leads to the same universality class as the three-dimensional fully frustrated Ising model on the cubic lattice.
We analyze the phase transition of the frustrated J1{J}_{1}J1​-J2{J}_{2}J2​ Ising model with antiferromagnetic nearest- and strong next-nearest-neighbor interactions on the square lattice. Using extensive Monte Carlo simulations we show that the nature of the phase transition for $1/2<{J}_{2}/{J}_{1}\ensuremath{\lesssim}1$ is not of the weakly universal type---as commonly believed---but we conclude from the clearly doubly peaked structure of the energy histograms that the transition is of weak first order. Motivated by these results, we analyze the phase transitions via field-theoretic methods; i.e., we calculate the central charge of the underlying field theory via transfer-matrix techniques and present, furthermore, a field-theoretic discussion on the phase-transition behavior of the model. Starting from the conformally invariant fixed point of two decoupled critical Ising models (J1=0{J}_{1}=0J1​=0), we calculate the effect of the nearest-neighbor coupling term perturbatively using operator product expansions. As an effective action we obtain the Ashkin-Teller model.
Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0{J}_{1}<0J1​<0 (nearest neighbor, ferromagnetic) and J2>0{J}_{2}>0J2​>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4{Z}_{4}Z4​ symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2g={J}_{2}/|{J}_{1}|>1/2g=J2​/∣J1​∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1{J}_{1}J1​-J2{J}_{2}J2​ model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗1/2<g<{g}^{*}1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗g={g}^{*}g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗{g}^{*}g∗ in the J1{J}_{1}J1​-J2{J}_{2}J2​ model. The continuous transitions near g∗{g}^{*}g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2g>1/2g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.
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.