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arXiv:cond-mat/9708171 · Journal of Physics A Mathematical and General

A constrained Potts antiferromagnet model with an interface representation

J K Burton, Christopher L. Henley

Published Dec 21, 1997• 62 citations
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Abstract

We define a 4-state Potts model ensemble on the square lattice, with the constraints that neighbouring spins must have different values, and that no plaquette may contain all four states. The spin configurations may be mapped into those of a two-dimensional interface in a 2 + 5 dimensional space. If this interface is in a Gaussian rough phase (as is the case for most other models with such a mapping) then the spin correlations are critical and their exponents can be related to the stiffness governing the interface fluctuations. Results of our Monte Carlo simulations show height fluctuations with an anomalous dependence on the wavevector, intermediate between the behaviours expected in a rough and in a smooth phase; we argue that the smooth phase (which would imply long-range spin order) is the best interpretation.

DOI
10.1088/0305-4470/30/24/007

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We define a 4-state Potts model ensemble on the square lattice, with the constraints that neighbouring spins must have different values, and that no plaquette may contain all four states. The spin configurations may be mapped into those of a two-dimensional interface in a 2 + 5 dimensional space. If this interface is in a Gaussian rough phase (as is the case for most other models with such a mapping) then the spin correlations are critical and their exponents can be related to the stiffness governing the interface fluctuations. Results of our Monte Carlo simulations show height fluctuations with an anomalous dependence on the wavevector, intermediate between the behaviours expected in a rough and in a smooth phase; we argue that the smooth phase (which would imply long-range spin order) is the best interpretation.
We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a 2+2-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagome lattice and to bond-coloring models on the triangular and hexagonal lattices. We also calculate critical exponents for the ground state ensemble of the model. We find that the exponents governing the spin-spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang-Swendsen-Kotecky cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices with free boundary conditions, or more generally on lattices possessing no non-contractible loops of infinite order. To guard against biases introduced by lack of ergodicity, we therefore perform our simulations on both the torus and the projective plane.
We develop an interface approach to bulk phase transitions and ordered states by using Monte Garlo simulations, and apply it to antiferromagnetic q -state Potts models with q =3∼6 on the simple cubic lattice. A stiffness exponent a defined by Δ F ∼ L a , where Δ F is the interface free energy for a system of size L , is introduced as a measure of the stiffness of the ordered phase against an external stress. Applying finite-size scaling to Δ F and to the squared order parameter enable one to determine each of T c , ν, β and γ in order where T c is the critical temperature and the others are critical exponents. This approach also provides a means to study properties of the ordered state through a and interface profiles. In the q =3 and 4 models the estimated exponents ν, β, γ and a indicate new universality classes. The q =5 model shows a phase transition, while the q =6 model does not. The q =3∼5 models have non-integer values of a at low temperatures. The phase transitions in these models are confirmed to be due to entropy gains.
We study critical properties of the four-coloring model, which is given by the equal-weighted ensemble of all possible edge colorings of the square lattice with four different colors. We map the four-coloring model onto an interface model for which we propose an effective Gaussian field theory, which allows us to calculate correlation functions of operators in the coloring model. The critical exponents are given by the stiffness of the interface, which we calculate exactly using recent results on the statistical topography of rough interfaces. Our numerical exponents, from Monte Carlo simulations of the four-coloring model, are in excellent agreement with the analytical calculations. These results support the conjecture that the scaling limit of the four-coloring model is given by the Su(4)k=1{)}_{\mathit{k}=1})k=1​ Wess-Zumino-Witten model. Moreover, we show that our effective field theory is the free-field representation of the Su(4)k=1{)}_{\mathit{k}=1})k=1​ Wess-Zumino-Witten model. Finally, we discuss connections to loop models, and some predictions of finite temperature properties of a particular Potts model for which the four-coloring model is the ground state.
The q-state Potts model with long-range correlated disorder is studied by means of large-scale Monte Carlo simulations for q=2, 4, 8, and 16. Evidence is given of the existence of a Griffiths phase, where the thermodynamic quantities display an algebraic finite-size scaling, in a finite range of temperatures. The critical exponents are shown to depend on both the temperature and the exponent of the algebraic decay of disorder correlations, but not on the number of states of the Potts model. The mechanism leading to the violation of hyperscaling relations is observed in the entire Griffiths phase.
We map the ground-state ensemble of antiferromagnetic Ising model of spin-S on a triangular lattice to an interface model whose entropic fluctuations are proposed to be described by an effective Gaussian free energy, which enables us to calculate the critical exponents of various operators in terms of the stiffness constant of the interface. Monte Carlo simulations for the ground-state ensemble utilizing this interfacial representation are performed to study both the dynamical and the static properties of the model. This method yields more accurate numerical results for the critical exponents. By varying the spin magnitude in the model, we find that the model exhibits three phases with a Kosterlitz-Thouless phase transition at ${\mathrm{S}}_{\mathrm{KT}}$2 and a locking phase transition at SL{\mathrm{S}}_{\mathrm{L}}SL​\ensuremath{\leqslant}3. The phase diagram at finite temperatures is also discussed.