Physical Review Letters

Monte Carlo Study of the Antiferromagnetic Potts Model in Two Dimensions

Gary S. Grest, Jayanth R. Banavar

Published Jun 1, 1981• 52 citations

Abstract

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.

DOI: 10.1103/physrevlett.46.1458

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Physical Review Letters

Monte Carlo Study of the Antiferromagnetic Potts Model in Two Dimensions

Gary S. Grest, Jayanth R. Banavar

Published Jun 1, 1981• 52 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.46.1458

Related papers

A Monte Carlo study of the three-states antiferromagnetic Potts model in two dimensions

Francesco Fucito

Journal of Physics A Mathematical and GeneralOct 1, 19838 citations

Mean-field analysis of antiferromagnetic three-state Potts model with next-nearest-neighbor interaction

Mitsuhiro Itakura

The three-state Potts model with antiferromagnetic nearest-neighbor (NN) and ferromagnetic next-nearest-neighbor (NNN) interaction is investigated within a mean-field theory. We find that the phase diagram contains two kinds of ordered phases, the so-called broken-sublattice-symmetry and permutationally-symmetric-sublattice phases, separated by a discontinuous phase transition line. The order-disorder transition is continuous for weak NNN interactions and becomes a discontinuous transition, via a tricritical point, when the strength of the NNN interaction is increased.The three-state Potts model with antiferromagnetic nearest-neighbor (NN) and ferromagnetic next-nearest-neighbor (NNN) interaction is investigated within a mean-field theory. We find that the phase diagram contains two kinds of ordered phases, the so-called broken-sublattice-symmetry and permutationally-symmetric-sublattice phases, separated by a discontinuous phase transition line. The order-disorder transition is continuous for weak NNN interactions and becomes a discontinuous transition, via a tricritical point, when the strength of the NNN interaction is increased.

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Abstract The Monte Carlo method has been used to study phase transitions and critical properties of the antiferromagnetic 3-state Potts model on a body-centered cubic lattice with interactions of the first (J1) and second (J2) nearest neighbors. The competition of exchange interactions in the range –2/3 < J2 < 0 was determined to result in a phase separation. In the low-temperature region, an ordered phase, lying inside the phase with broken sublattice symmetry, was found. The magnetic structures of the ground state corresponding to different regions of the phase diagram have been obtained. The analysis of the phase transition order has been carried out using the Binder cumulant method and the histogram method. It has been revealed that the interaction of the second nearest neighbors leads to a change in the type of phase transition. Static critical exponents have been calculated based on scaling relations. The critical exponents correspond to the universality class of the three-dimensional XY model.Abstract The Monte Carlo method has been used to study phase transitions and critical properties of the antiferromagnetic 3-state Potts model on a body-centered cubic lattice with interactions of the first (J1) and second (J2) nearest neighbors. The competition of exchange interactions in the range –2/3 < J2 < 0 was determined to result in a phase separation. In the low-temperature region, an ordered phase, lying inside the phase with broken sublattice symmetry, was found. The magnetic structures of the ground state corresponding to different regions of the phase diagram have been obtained. The analysis of the phase transition order has been carried out using the Binder cumulant method and the histogram method. It has been revealed that the interaction of the second nearest neighbors leads to a change in the type of phase transition. Static critical exponents have been calculated based on scaling relations. The critical exponents correspond to the universality class of the three-dimensional XY model.

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D. P. Landau, Kurt Binder

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.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.

Physical review. B, Condensed matterMay 1, 1985202 citations

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.

{\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.

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.