arXiv:cond-mat/9803277 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical Ising lines of the d=2 Ashkin-Teller model

G. Kamieniarz, Piotr Kozłowski, Raf Dekeyser

Published Mar 1, 1997• 19 citations

Abstract

The universal critical point ratio Q is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller model on the square lattice. A leading-order expansion of the ratio Q in the presence of a nonvanishing thermal field is found from finite-size scaling and the corresponding expression is fitted by the accurate perturbative transfer-matrix data calculations for the L\ifmmode\times\else\texttimes\fi{}L square clusters with L\ensuremath{\leqslant}9.

DOI: 10.1103/physreve.55.3724

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arXiv:cond-mat/9803277 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical Ising lines of the d=2 Ashkin-Teller model

G. Kamieniarz, Piotr Kozłowski, Raf Dekeyser

Published Mar 1, 1997• 19 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreve.55.3724

Related papers

The critical point ratio for some d = 2 classical models

Gregory Kamieniarz, Piotr Kozłowski, Raf Dekeyser

The universal critical point ratio of the square of the second moment of the order parameter to its fourth moment, denoted as Q, is exploited to determine the position of the critical Ising transition lines in the phase diagram of the Ashkin-Teller (AT) model on the square lattice. A leading-order expansion of the Q ratio in the presence of a non-vanishing thermal field is found from finite-size scaling and the corresponding expression is fitted to the accurate perturbative transfer-matrix data for the L x L squaxe clusters with L <= 9. The same method is used to determine the critical Ising transition line on the phase diagram of the Blume-Capel (BC) model (L <= 11). We have also calculated the Q ratio for the q=3 Potts model in the critical point. The previously predicted scaling relation for this universality class is confirmed by our results for L <= 12.The universal critical point ratio of the square of the second moment of the order parameter to its fourth moment, denoted as Q, is exploited to determine the position of the critical Ising transition lines in the phase diagram of the Ashkin-Teller (AT) model on the square lattice. A leading-order expansion of the Q ratio in the presence of a non-vanishing thermal field is found from finite-size scaling and the corresponding expression is fitted to the accurate perturbative transfer-matrix data for the L x L squaxe clusters with L <= 9. The same method is used to determine the critical Ising transition line on the phase diagram of the Blume-Capel (BC) model (L <= 11). We have also calculated the Q ratio for the q=3 Potts model in the critical point. The previously predicted scaling relation for this universality class is confirmed by our results for L <= 12.

Jan 1, 19960 citations

Phase diagram of the 2d Ashkin-Teller model: numerical transfer-matrix approach

G. Kamieniarz, Piotr Kozłowski, Raf Dekeyser

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The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.

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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.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:1212.5339Physical Review BApr 9, 2013116 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

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.

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.