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Journal of Physics A Mathematical and General

Logarithmic corrections for the percolative properties of the four-state Potts model

Carlo Vanderzande, John F. Marko

Published Dec 21, 1993• 1 citation
Publisher

Abstract

Clusters in the Potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state. Droplets can be obtained by adding bonds with probability p=1-exp(-K) between the sites in a cluster (where K is the Potts model's inverse temperature). When q to 4, renormalization group (RG) fixed points describing clusters and droplets will coalesce, leading to logarithmic corrections. We calculate the precise form of these corrections used in a differential RG method. Our predictions are then tested using extensive Monte Carlo calculations. Theory and the simulations are found to be in excellent agreement.

DOI
10.1088/0305-4470/26/24/014

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Clusters in the Potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state. Droplets can be obtained by adding bonds with probability p=1-exp(-K) between the sites in a cluster (where K is the Potts model's inverse temperature). When q to 4, renormalization group (RG) fixed points describing clusters and droplets will coalesce, leading to logarithmic corrections. We calculate the precise form of these corrections used in a differential RG method. Our predictions are then tested using extensive Monte Carlo calculations. Theory and the simulations are found to be in excellent agreement.
The critical behaviour of several spin models can be simply described as percolation of some suitably defined clusters or droplets: the onset of the geometrical transition coincides with the critical point, and the percolation exponents are equal to the thermal exponents. It is still unknown whether, given a model, one can define at all the droplets. In the cases where this is possible, the droplet definition depends in general on the specific model at study and can be quite involved. We propose here a simple general definition for the droplets: they are clusters obtained by joining nearest-neighbour spins of the same sign with some bond probability p CK , which is the minimal probability that still allows the existence of a percolating cluster at the critical temperature T c . This definition has been recently introduced and indeed satisfies the conditions required for the droplets, for many classical spin models, discrete and continuous, in two dimensions. By means of lattice Monte Carlo simulations we show here that the definition works as well in three dimensions. In particular, our prescription allows us to describe exactly the confinement–deconfinement transition of SU ( N ) gauge theories as Polyakov loop percolation.
The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at T c in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.
The fractal dimensions of the largest clusters of like spins connected by nearest-neighbor bonds in the critical two-dimensional Q-state Potts model have been determined via Monte Carlo simulations to be df{\mathit{d}}_{\mathit{f}}df​=1.946(3), 1.922(5), and 1.909(8) for Q=2, 3, and 4, respectively. These dimensions are close to the magnetic tricritical exponents of the (Q-1)-state dilute Potts model (d-x=187/96=1.947..., 77/40=1.925, and 40/21=1.904... for Q-1=1, 2, and 3, respectively). The critical Q-state model and the tricritical (Q-1)-state model are related to a two-component Potts lattice gas; this model generates percolative susceptibilities of critical Potts clusters, and suggests a renormalization-group argument for df{\mathit{d}}_{\mathit{f}}df​=d-x.
Based on the connection between the q-state Potts model (QPM) and the q-state bond-correlated percolation model (QBCPM), we have proposed percolation renormalization-group (PRG) methods to calculate the free energy, the critical point, and critical exponents for the QPM. Our methods are free from inconsistency in Larsson's method. We have carried out the ${\ensuremath{\lambda}}_{1}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{1}$ to ${\ensuremath{\lambda}}_{2}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{2}$ renormalization-group (RG) transformations in our methods for several values of original cell sizes ${\ensuremath{\lambda}}_{1}$ and final cell sizes ${\ensuremath{\lambda}}_{2}$ with various boundary conditions. We find that such RG transformations with large ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ and periodic boundary condition usually give accurate physical quantities for the QPM. The advantages and generalization of our approach are also discussed.
We introduce a procedure which considerably simplifies the calculation of clusters like those commonly appearing in real-space renormalization-group treatments of bond-percolation and pure and random Ising and Potts problems. The method is illustrated through two applications for the qqq-state Potts ferromagnet.
The qqq-state Potts models in two dimensions are studied with the Monte Carlo renormalization-group method. The results are in agreement with known properties. Combining our data with the qualitative picture of Nienhuis, Berker, Riedel, and Schick, we obtain a consistent description of the behavior of these models, including nearly singular behavior for q>4q>4q>4 (as proposed by Kim and Joseph) and suggesting logarithmic corrections to scaling for q=4q=4q=4.