Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet

We study the four-state antiferromagnetic Potts model on the triangular lat tice. We show that the model has six types of defects which diffuse and ann ihilate 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 he nce show that the model is equivalent to the three-state Potts antiferromag net on the Kagome: lattice and to bond-coloring models on the triangular an d honeycomb lattices. We also calculate critical exponents for the ground-s tale ensemble of the model. We find that the exponents governing the spin-s pin correlation function and spin fluctuations violate the Fisher scaling l aw because of constraints on path length which increase the effective wavel ength of the spin operator on the height lattice. We confirm our prediction s by extensive Monte Carlo simulations of the model using the Wang-Swendsen -Kotecky cluster algorithm. Although this algorithm is not ergodic on latti ces with toroidal boundary conditions, we prove that it is ergodic on latti ces whose topology has no noncontractible loops of infinite order, such as the projective plane. To guard against biases introduced by lack of ergodic ity, we perform our simulations on both the torus and the projective plane.