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arXiv:cond-mat/0503610 · Physical Review E

Level spectroscopy of the square-lattice three-state Potts model with a ferromagnetic next-nearest-neighbor coupling

Hiromi Otsuka, Koutaro Mori, Yutaka Okabe, K. Nomura

Published Oct 5, 2005• 13 citations
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Abstract

We study the square-lattice three-state Potts model with the ferromagnetic next-nearest-neighbor coupling at finite temperature. Using the level-spectroscopy method, we numerically analyze the excitation spectrum of the transfer matrices and precisely determine the global phase diagram. Then we find that, contrary to a previous result based on the finite-size scaling, the massless region continues up to the decoupling point with Z3 x Z3 criticality in the antiferromagnetic region. We also check the universal relations among excitation levels to provide the reliability of our result.

DOI
10.1103/physreve.72.046103

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We study the square-lattice three-state Potts model with the ferromagnetic next-nearest-neighbor coupling at finite temperature. Using the level-spectroscopy method, we numerically analyze the excitation spectrum of the transfer matrices and precisely determine the global phase diagram. Then we find that, contrary to a previous result based on the finite-size scaling, the massless region continues up to the decoupling point with Z3 x Z3 criticality in the antiferromagnetic region. We also check the universal relations among excitation levels to provide the reliability of our result.
We investigate the triangular-lattice antiferromagnetic Ising model with a spatially anisotropic next-nearest-neighbor ferromagnetic coupling, which was first discussed by Kitatani and Oguchi. By employing the effective geometric factor, we analyze the scaling dimensions of the operators around the Berezinskii-Kosterlitz-Thouless (BKT) transition lines, and determine the global phase diagram. Our numerical data exhibit that two types of BKT-transition lines separate the intermediate critical region from the ordered and disordered phases, and they do not merge into a single curve in the antiferromagnetic region. We also estimate the central charge and perform some consistency checks among scaling dimensions in order to provide the evidence of the six-state clock universality. Further, we provide an analysis of the shapes of boundaries based on the crossover argument.
We study a square-lattice three-state Potts antiferromagnet with a staggered polarization field at finite temperature. Numerically treating the transfer matrices, we determine two phase boundaries separating the model-parameter space into three parts. We confirm that one of them belongs to the ferromagnetic three-state Potts criticality, which is in accord with a recent prediction, and another to the Ising-type; these are both corresponding to the massless renormalization-group flows stemming from the Gaussian fixed points. We also discuss a field theory to describe the latter Ising transition.
The three-state Potts model on a square lattice with general nearest-neighbor interaction and ferromagnetic second-neighbor interaction is studied. At zero temperature the model with antiferromagnetic nearest-neighbor interactions is mapped to the FFF model. By comparing the excitations generated at nonzero temperature to those that lead to the eight-vertex model we obtain an explicit expression for the critical index describing such excitations and demonstrate the existence of a critical fan for ferromagnetic second-neighbor interactions. The model with purely nearest-neighbor interactions is critical only at zero temperature. Explicit expressions for the scaling indices of the color-color correlation function in the critical phase are also obtained. Phenomenological renormalization-group methods are applied to determine the general boundaries of the critical fan and to verify our expressions for critical indices. A physical system which might be expected to undergo a transition in the same universality class as that of the above model and to exhibit a critical phase is proposed: an equal mixture of krypton and xenon adsorbed on graphite.
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.
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.