Physical review. B, Condensed matter

Z(3) criticality in three dimensions: Study of extended Potts models

Rajiv V. Gavai, F. Karsch

Published Jul 1, 1992• 7 citations

Abstract

We investigate numerically the three-dimensional three-state Potts models with nearest-neighbor (NN) ferromagnetic coupling and next-nearest-neighbor (NNN) antiferromagnetic coupling of relative strength \ensuremath{\gamma} on ${\mathit{L}}^{3}$ lattices with L=12, 16, 20, 32, and 40. For all the \ensuremath{\gamma} values that we studied, 0.0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}0.8, we find indications of a first-order phase transition between the ordered and disordered phases. In the neighborhood of \ensuremath{\gamma}=0.25, the latent heat becomes rather small, making it necessary to use still larger lattices to rule out a higher-order phase transition at \ensuremath{\gamma}=0.25. We also studied the boundary regimes of the three different ordered phases and find no criticality along them, thus suggesting a lack of criticality in these extended Potts models.

DOI: 10.1103/physrevb.46.944

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Physical review. B, Condensed matter

Z(3) criticality in three dimensions: Study of extended Potts models

Rajiv V. Gavai, F. Karsch

Published Jul 1, 1992• 7 citations

Publisher

Abstract

{\mathit{L}}^{3}

lattices with L=12, 16, 20, 32, and 40. For all the \ensuremath{\gamma} values that we studied, 0.0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}0.8, we find indications of a first-order phase transition between the ordered and disordered phases. In the neighborhood of \ensuremath{\gamma}=0.25, the latent heat becomes rather small, making it necessary to use still larger lattices to rule out a higher-order phase transition at \ensuremath{\gamma}=0.25. We also studied the boundary regimes of the three different ordered phases and find no criticality along them, thus suggesting a lack of criticality in these extended Potts models.

DOI: 10.1103/physrevb.46.944

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Gary S. Grest, Jayanth R. Banavar

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Kai-Wei Huang, Chien-Ting Chen, Shang-Wei Li +1 more

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{\mathit{L}}^{3}

Results of Monte Carlo studies of the

q

-state Potts model on a square lattice with antiferromagnetic nearest-neighbor coupling and ferromagnetic next-nearest-neighbor coupling are presented for

q>~3

. The three-state Potts model shows a variety of unusual transitions whereas for

q>~4

, we find two first-order phase transitions separated by a broken sublattice symmetry state.

Phase transitions of the antiferromagnetic

J_1

J_2

J_3

Ising model on the cubic lattice

Motivated by the conflicting findings in prior studies regarding the nature of the phase transition for the three-dimensional antiferromagnetic

J_1

J_2

J_3

Ising model on the cubic lattice with

J_1 = J_2 = J_3 = 1.0

, in this study we conduct a detailed investigation of this model using both Monte Carlo simulations and neural networks (NN). Here

J_1

and

J_2

are the intralayer nearest-neighbor and next-nearest-neighbor couplings, respectively, and

J_3

is the nearest-neighbor interlayer interaction. By investigating the data obtained from much larger lattices than those used in previous studies, we find that the phase transition for

J_1 = J_2 = J_3 = 1.0

is second order without any ambiguity. Hence we bring a resolution to this debate. Apart from

J_1 = J_2 = J_3 = 1.0

, we study the critical phenomenon associated with

J_2/J_1 = 4/7

and

J_3 = J_1 = 1.0

as well. In addition, by employing a dedicated training and testing strategy, our NN computation not only is much more efficient than those utilizing the standard procedures commonly used in the literature, but also requires only a per-mille fraction of the storage capacity needed by a conventional NN calculation when a large linear system size

L

is considered. The method employed here provides an extremely resource-efficient route for applying NN techniques to study the critical behaviors of higher-dimensional systems.

The three-state Potts model is numerically investigated on three-dimensional simple cubic lattices of up to

128^3

volume, concentrating on the neighborhood of the first-order phase transition separating the ordered and disordered phases. In both phases clusters of like spins are observed with irregular boundaries. In the ordered phase the two different non-favored spins are found to attract each other with a long but finite range. As a result, the neighborhoods of the non-favored spins are interpreted as domains of the disordered phase. This explains why the first-order phase transitions associated with the global

Z_3

symmetry, including the SU(3) pure-gauge one, are so weak.

Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling

{J}_{\mathrm{NN}}

between nearest neighbors and additional interactions

{J}_{\mathrm{NNN}}

between next-nearest neighbors and

{J}_{3\mathrm{N}\mathrm{N}}

between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=

{J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}

and R'=

{J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}

. In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.