Generalized Potts-Models and their Relevance for Gauge Theories

Abstract

We study the Polyakov loop dynamics originating from finite-temperature Yang- Mills theory. The effective actions contain center-symmetric terms involving powers of the Polyakov loop, each with its own coupling. For a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and antiferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents and at the continuous transition between symmetric and antiferromagnetic phases are the same as for the 3-state spin Potts model. Symmetry constraints and strong coupling expansion for the effective action describing the Polyakov loop dynamics of gauge theories lead to effective field theories with rich phase struc- tures. The fields are the fundamental characters of the gauge group with the fundamental domain as target space. The center symmetry of pure gauge theory remains a symmetry of the effective models. If one further freezes the Polyakov loop to the center Z of the gauge group one obtains the well known vector Potts spin-models, sometimes called clock models. Hence we call the effective theories for the Polyakov loop dynamics generalized Z-Potts models. We review our recent results on generalized Z3-Potts models (1). These results were obtained with the help of an improved mean field approximation and Monte Carlo simulations. The mean field approximation turns out to be much better than expected. Probably this is due to the existence of tricritical points in the effective theories. There exist four distinct phases and transitions of first and second order. The critical exponents and at the second order transition from the symmetric to antiferromagnetic phase for the generalized Potts model are the same as for the