Physical review. B, Condensed matter

Z(4) model on the triangular lattice

Yoel Stavans, Eytan Domany

Published Mar 1, 1983• 7 citations

Abstract

Monte Carlo studies of the $Z(4)$ (or Ashkin-Teller) model on the triangular lattice were carried out. In particular, we investigated whether the massless phase, which was shown to exist on the boundary of the physical parameter space, may extend into a region of finite width. The existence of such a phase would imply that ${N}_{c}$, the critical value above which $Z(N)$ models have intermediate phases, is nonuniversal, i.e., lattice and interaction dependent. Although, on the basis of our Monte Carlo results we cannot completely rule out this possibility, most aspects of our findings indicate consistency with existing theories of the $Z(4)$ model which exclude the possibility of such an intermediate massless phase.

DOI: 10.1103/physrevb.27.3043

Related papers

On the Z(4) spin model

Vera Lúcia Vieira Baltar, Maria Elena Pol

The authors study the Z(4) spin model in two dimensions in the region of parameters where a critical phase may occur. The authors results, obtained by Monte Carlo simulations, Monte Carlo renormalisation group technique and a study of the elementary excitations near the SOS model, indicate that there is no critical phase in the Z(4) model.

Journal of Physics A Mathematical and GeneralApr 1, 19860 citations

N -color Ashkin-Teller model

Gary S. Grest, Michael Widom

Physical review. B, Condensed matterDec 1, 198175 citations

Emergent critical phases of the Ashkin-Teller model on the Union-Jack Lattice

Changzhi Zhao, Wanzhou Zhang, Yuan Huang +2 more

arXiv:2605.09453May 10, 20260 citations

Z(N) generalisation of the Baxter-Wu model

Francisco Castilho Alcaraz, L Jacobs

A two-dimensional model of the dynamics of Z(N) spins on a triangular lattice with three-body interactions is analysed. The properties of the model under duality are shown to be identical to those of a wide class of models in two, three and four dimensions. Monte Carlo simulations reveal the existence of three phases for N>4, with an intermediate phase which is most likely soft.

Journal of Physics A Mathematical and GeneralJul 1, 19828 citations

Phase diagram of the Z(4) model

Vera Lúcia Vieira Baltar, Gilson Carneiro, Maria Elena Pol +1 more

Systems with Z(4) global symmetry defined on a square lattice are studied, in the whole parameter space, both by Monte Carlo simulations and by the Migdal-Kadanoff renormalisation group scheme. A phase diagram is obtained with quantitative predictions as to the location of the boundaries between the different phases. Four distinct phases are observed.

Journal of Physics A Mathematical and GeneralJul 11, 19849 citations

Monte Carlo simulations of Ising-like phase transitions in the three-dimensional Ashkin-Teller model

Grzegorz Musiał, L. Dębski, G. Kamieniarz

Physical review. B, Condensed matterJun 28, 200227 citations

Physical review. B, Condensed matter

Z(4) model on the triangular lattice

Yoel Stavans, Eytan Domany

Published Mar 1, 1983• 7 citations

Publisher

Abstract

Monte Carlo studies of the

Z(4)

(or Ashkin-Teller) model on the triangular lattice were carried out. In particular, we investigated whether the massless phase, which was shown to exist on the boundary of the physical parameter space, may extend into a region of finite width. The existence of such a phase would imply that

{N}_{c}

, the critical value above which

Z(N)

models have intermediate phases, is nonuniversal, i.e., lattice and interaction dependent. Although, on the basis of our Monte Carlo results we cannot completely rule out this possibility, most aspects of our findings indicate consistency with existing theories of the

Z(4)

model which exclude the possibility of such an intermediate massless phase.

DOI: 10.1103/physrevb.27.3043

Related papers

On the Z(4) spin model

Vera Lúcia Vieira Baltar, Maria Elena Pol

Journal of Physics A Mathematical and GeneralApr 1, 19860 citations

N -color Ashkin-Teller model

Gary S. Grest, Michael Widom

We study a model Hamiltonian consisting of $N$ Ising models coupled pairwise through a four-spin interaction ${K}_{4}$ . When $N=2$ this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of ${K}_{4}=0$ . To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for $N=3$ . The results show that $N=2$ is special because the coupling between Ising models is marginal over a range of values of ${K}_{4}$ . We discuss the effect of the coupling ${K}_{4}$ for $N\ensuremath{\ne}2$.We study a model Hamiltonian consisting of $N$ Ising models coupled pairwise through a four-spin interaction ${K}_{4}$. When $N=2$ this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of ${K}_{4}=0$. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for $N=3$. The results show that $N=2$ is special because the coupling between Ising models is marginal over a range of values of ${K}_{4}$. We discuss the effect of the coupling ${K}_{4}$ for $N\ensuremath{\ne}2$.

Physical review. B, Condensed matterDec 1, 198175 citations

Emergent critical phases of the Ashkin-Teller model on the Union-Jack Lattice

Changzhi Zhao, Wanzhou Zhang, Yuan Huang +2 more

The Ashkin-Teller (AT) model is a classic spin model in statistical mechanics. For traditional homogeneous lattices like triangular and kagome lattices, even when frustration exists, the model only has one ferromagnetic-paramagnetic critical line in the $J>0$ and $K<0$ region. However, in this paper, for the Union Jack lattice, where the lattice coordination numbers are 4, 8, and 8 and which also contains a large number of small triangular units, using Metropolis Monte Carlo method, we find that, the critical line of the AT model splits into two Berezinskii-Kosterlitz-Thouless(BKT) boundaries, and a critical phase emerges in the intermediate region. This phenomenon is the combined result of frustration, lattice inhomogeneity and the two coupled spin degrees of freedom inherent to the AT model. In detail, the novel critical phase characterized by a power-law decay of magnetization with system size, where the correlation length ratio $ξ/L$ remains finite even in the thermodynamic limit. We also introduce the susceptibility $\widetildeχ = \text{d}\langle m \rangle /\text{d}J$ as a key probe, and through this probe, pseudo-critical points $J_c(L)$ are observed to scale proportionally to $(\ln L)^{-2}$ , a behavior consistent with BKT criticality. Since superfluids, superconductors, and supersolids all possess quasi-long-range order and fall into the category of critical phases, our results could also inspire the exploration of such quantum phases.The Ashkin-Teller (AT) model is a classic spin model in statistical mechanics. For traditional homogeneous lattices like triangular and kagome lattices, even when frustration exists, the model only has one ferromagnetic-paramagnetic critical line in the $J>0$ and $K<0$ region. However, in this paper, for the Union Jack lattice, where the lattice coordination numbers are 4, 8, and 8 and which also contains a large number of small triangular units, using Metropolis Monte Carlo method, we find that, the critical line of the AT model splits into two Berezinskii-Kosterlitz-Thouless(BKT) boundaries, and a critical phase emerges in the intermediate region. This phenomenon is the combined result of frustration, lattice inhomogeneity and the two coupled spin degrees of freedom inherent to the AT model. In detail, the novel critical phase characterized by a power-law decay of magnetization with system size, where the correlation length ratio $ξ/L$ remains finite even in the thermodynamic limit. We also introduce the susceptibility $\widetildeχ = \text{d}\langle m \rangle /\text{d}J$ as a key probe, and through this probe, pseudo-critical points $J_c(L)$ are observed to scale proportionally to $(\ln L)^{-2}$, a behavior consistent with BKT criticality. Since superfluids, superconductors, and supersolids all possess quasi-long-range order and fall into the category of critical phases, our results could also inspire the exploration of such quantum phases.

arXiv:2605.09453May 10, 20260 citations

Z(N) generalisation of the Baxter-Wu model

Francisco Castilho Alcaraz, L Jacobs

Journal of Physics A Mathematical and GeneralJul 1, 19828 citations

Phase diagram of the Z(4) model

Vera Lúcia Vieira Baltar, Gilson Carneiro, Maria Elena Pol +1 more

Journal of Physics A Mathematical and GeneralJul 11, 19849 citations

Monte Carlo simulations of Ising-like phase transitions in the three-dimensional Ashkin-Teller model

Grzegorz Musiał, L. Dębski, G. Kamieniarz

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.

Physical review. B, Condensed matterJun 28, 200227 citations

Monte Carlo studies of the

Z(4)

{N}_{c}

, the critical value above which

Z(N)

Z(4)

model which exclude the possibility of such an intermediate massless phase.

We study a model Hamiltonian consisting of

N

Ising models coupled pairwise through a four-spin interaction

{K}_{4}

. When

N=2

this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of

{K}_{4}=0

. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for

N=3

. The results show that

N=2

is special because the coupling between Ising models is marginal over a range of values of

{K}_{4}

. We discuss the effect of the coupling

{K}_{4}

for $N\ensuremath{\ne}2$.

J>0

and

K<0

region. However, in this paper, for the Union Jack lattice, where the lattice coordination numbers are 4, 8, and 8 and which also contains a large number of small triangular units, using Metropolis Monte Carlo method, we find that, the critical line of the AT model splits into two Berezinskii-Kosterlitz-Thouless(BKT) boundaries, and a critical phase emerges in the intermediate region. This phenomenon is the combined result of frustration, lattice inhomogeneity and the two coupled spin degrees of freedom inherent to the AT model. In detail, the novel critical phase characterized by a power-law decay of magnetization with system size, where the correlation length ratio

ξ/L

remains finite even in the thermodynamic limit. We also introduce the susceptibility

\widetildeχ = \text{d}\langle m \rangle /\text{d}J

as a key probe, and through this probe, pseudo-critical points

J_c(L)

are observed to scale proportionally to

(\ln L)^{-2}

, a behavior consistent with BKT criticality. Since superfluids, superconductors, and supersolids all possess quasi-long-range order and fall into the category of critical phases, our results could also inspire the exploration of such quantum phases.