Physical review. E

Phase transitions in two-dimensional Potts models in an external field

Shan-Ho Tsai, L. S. Ferreira, J. A. Plascak, D. P. Landau

Published Dec 5, 2025• 2 citations

Abstract

The q-state Potts model on a square lattice and in the presence of an external magnetic field that couples to only one state is studied by using extensive Monte Carlo simulations. A hybrid algorithm that combines the single spin-flip Metropolis method with Wolff cluster updates is employed. Single-histogram reweighting techniques and the field-mixing approach are also used to obtain the corresponding universal probability distributions. By analyzing the finite-size scaling behavior of the order parameters and universal probability distributions, the phase diagram in the temperature-versus-external field plane is obtained in the thermodynamic limit for some values of state q≥5. For positive fields and higher values of q, there is a clear first-order transition line that starts at the zero-field transition point and ends at an isolated critical point, which is in the two-dimensional Ising universality class. As the value of state q decreases, the corresponding critical field also decreases. The present results strongly suggest that the critical field vanishes when q→5^{+}. However, for negative fields, another first-order transition line occurs without any critical or multicritical behavior for q≥6. The special case with q=5 undergoes a weak first-order transition at zero field and a line of second-order transitions for negative fields. This second-order transition line is in the same universality class as the q=4 Potts model.

DOI: 10.1103/s9x8-h61n

Related papers

Two-dimensional dilute Baxter-Wu model: Transition order and universality

A. R. S. Macêdo, Alexandros Vasilopoulos, M. Akritidis +3 more

arXiv:2304.12379Physical review. EAug 25, 20236 citations

Crossover scaling in the two-dimensional three-state Potts model

Nagai, Okamoto, Janke

arXiv:1307.3865Condensed Matter PhysicsJan 1, 20138 citations

Two-dimensional Potts model with invisible states on a square lattice

Jae Hwan Lee, Wanki Park, J. M. Kim

Physical review. EMay 22, 20250 citations

The critical behaviour of Potts models with symmetry- breaking fields

Francisco Castilho Alcaraz, J. C. Xavier

arXiv:cond-mat/9510124Journal of Physics A Mathematical and GeneralJul 7, 19960 citations

Phase diagram and critical behavior of the antiferromagnetic Ising model in an external field

Bruno Jeferson Lourenço, Ronald Dickman

Journal of Statistical Mechanics Theory and ExperimentMar 1, 20168 citations

Phase transitions in the frustrated Ising model on the square lattice

Songbo Jin, Arnab Sen, Wenan Guo +1 more

arXiv:1212.5339Physical Review BApr 9, 2013116 citations

Physical review. E

Phase transitions in two-dimensional Potts models in an external field

Shan-Ho Tsai, L. S. Ferreira, J. A. Plascak, D. P. Landau

Published Dec 5, 2025• 2 citations

Publisher

Abstract

DOI: 10.1103/s9x8-h61n

Related papers

Two-dimensional dilute Baxter-Wu model: Transition order and universality

A. R. S. Macêdo, Alexandros Vasilopoulos, M. Akritidis +3 more

We investigate the critical behavior of the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal-field coupling Δ with the goal of determining the universality class of transitions along the second-order part of the transition line as one approaches the putative location of the multicritical point. We employ extensive Monte Carlo simulations using two different methodologies: (i) a study of the zeros of the energy probability distribution, closely related to the Fisher zeros of the partition function, and (ii) the well-established multicanonical approach employed to study the probability distribution of the crystal-field energy. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram supports previous claims that the transition belongs to the universality class of the four-state Potts model. For positive values of Δ, we observe the presence of strong finite-size effects, indicative of crossover effects due to the proximity of the first-order part of the transition line. Finally, we demonstrate how a combination of cluster and heat-bath updates allows one to equilibrate larger systems, and we demonstrate the potential of this approach for resolving the ambiguities observed in the regime of Δ≳0.We investigate the critical behavior of the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal-field coupling Δ with the goal of determining the universality class of transitions along the second-order part of the transition line as one approaches the putative location of the multicritical point. We employ extensive Monte Carlo simulations using two different methodologies: (i) a study of the zeros of the energy probability distribution, closely related to the Fisher zeros of the partition function, and (ii) the well-established multicanonical approach employed to study the probability distribution of the crystal-field energy. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram supports previous claims that the transition belongs to the universality class of the four-state Potts model. For positive values of Δ, we observe the presence of strong finite-size effects, indicative of crossover effects due to the proximity of the first-order part of the transition line. Finally, we demonstrate how a combination of cluster and heat-bath updates allows one to equilibrate larger systems, and we demonstrate the potential of this approach for resolving the ambiguities observed in the regime of Δ≳0.

arXiv:2304.12379Physical review. EAug 25, 20236 citations

Crossover scaling in the two-dimensional three-state Potts model

Nagai, Okamoto, Janke

We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional threestate Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields.Our data set has been generated by STM simulations of several square lattices with sizes up to 160×160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points.We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size.We apply simulated tempering and magnetizing (STM) Monte Carlo simulations to the two-dimensional threestate Potts model in an external magnetic field in order to investigate the crossover scaling behaviour in the temperature-field plane at the Potts critical point and towards the Ising universality class for negative magnetic fields.Our data set has been generated by STM simulations of several square lattices with sizes up to 160×160 spins, supplemented by conventional canonical simulations of larger lattices at selected simulation points.We present careful scaling and finite-size scaling analyses of the crossover behaviour with respect to temperature, magnetic field and lattice size.

arXiv:1307.3865Condensed Matter PhysicsJan 1, 20138 citations

Two-dimensional Potts model with invisible states on a square lattice

Jae Hwan Lee, Wanki Park, J. M. Kim

We study the two-dimensional (q+r)-state ferromagnetic Potts model with q interacting visible states and r noninteracting invisible states on a square lattice. We concentrate on the case q=2 for various r values. The density of states is obtained by the Wang-Landau Monte Carlo simulation method, by which the specific heat, the partition function zeros, and the probability distribution of the internal energy are calculated. There exists a second-order phase transition for r<r_{c}, while a first-order phase transition appears for r>r_{c}, where r_{c} is the critical value. From all analyses, we conclude that r_{c}≈29 for q=2. Calculating the density of visible spins using the Metropolis algorithm, we find that the first-order phase transition is due to the abrupt density change of the visible spins. The critical temperatures are measured to compare with a conjecture.We study the two-dimensional (q+r)-state ferromagnetic Potts model with q interacting visible states and r noninteracting invisible states on a square lattice. We concentrate on the case q=2 for various r values. The density of states is obtained by the Wang-Landau Monte Carlo simulation method, by which the specific heat, the partition function zeros, and the probability distribution of the internal energy are calculated. There exists a second-order phase transition for r<r_{c}, while a first-order phase transition appears for r>r_{c}, where r_{c} is the critical value. From all analyses, we conclude that r_{c}≈29 for q=2. Calculating the density of visible spins using the Metropolis algorithm, we find that the first-order phase transition is due to the abrupt density change of the visible spins. The critical temperatures are measured to compare with a conjecture.

Physical review. EMay 22, 20250 citations

The critical behaviour of Potts models with symmetry- breaking fields

Francisco Castilho Alcaraz, J. C. Xavier

The Q-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general special choices of these magnetic fields produce effective models with smaller symmetry . The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behaviour, for finite values of the symmetry-breaking fields, in the cases where Q > 4 is also analysed. Our results indicate that for effective models with symmetry the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.The Q-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general special choices of these magnetic fields produce effective models with smaller symmetry . The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behaviour, for finite values of the symmetry-breaking fields, in the cases where Q > 4 is also analysed. Our results indicate that for effective models with symmetry the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.

arXiv:cond-mat/9510124Journal of Physics A Mathematical and GeneralJul 7, 19960 citations

Phase diagram and critical behavior of the antiferromagnetic Ising model in an external field

Bruno Jeferson Lourenço, Ronald Dickman

Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.

Journal of Statistical Mechanics Theory and ExperimentMar 1, 20168 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.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

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.