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arXiv:cond-mat/9605148 · Physical review. B, Condensed matter

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

Mitsuhiro Itakura

Published Jan 1, 1997• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.55.48

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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.
Results of Monte Carlo studies of the qqq-state Potts model on a square lattice with antiferromagnetic nearest-neighbor coupling and ferromagnetic next-nearest-neighbor coupling are presented for q> 3q>~3q> 3. The three-state Potts model shows a variety of unusual transitions whereas for q> 4q>~4q> 4, we find two first-order phase transitions separated by a broken sublattice symmetry state.
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 L3{\mathit{L}}^{3}L3 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.
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We study the phase diagram of the three-state Potts model on a triangular lattice with general interactions (ferro- and/or antiferromagnetic) between nearest-neighbor spins. When the interactions along two lattice-vector directions are antiferromagnetic and infinitely strong, this model becomes equivalent to a six-vertex model and exhibits a first-order potassium dihydrogen phosphate (KDP) transition from an ordered phase into a critical phase. Comparing the excitations that occurred by relaxing the restriction of infinite-strength interactions and those in the eight-vertex model, we analytically obtain the critical index for those excitations and demonstrate the existence of a critical phase for the case of finite antiferromagnetic interactions in two directions and ferromagnetic interactions in the other direction. When the interactions are antiferromagnetic in all three directions, Monte Carlo simulations show that a first-order line emerges from the KDP point and separates completely an ordered phase and a disordered phase. Along the special line where all three antiferromagnetic interactions have the same strength, the cell-spin analysis reveals that the symmetry of the ground states is dual to the symmetry of the n=3 ferromagnetic cubic model which is known to exhibit a first-order phase transition.
The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.
3×3\sqrt{3}\times\sqrt{3}3​×3​ Structure on the Triangular Lattice
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