Critical properties of the
q-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for
qc, the largest value of
q for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.
A modification of the Migdal-Kadanoff (MK) real-space renormalization technique is studied and applied to the q-state Potts model on the square and the simple cubic lattices. A parameter x which describes the boundary condition is introduced to the cluster-decimation (CD) approximation. When x=2, the present method is the same as the CD approximation, and in the limit x\ensuremath{\rightarrow}\ensuremath{\infty} this method reduces to the MK technique. Critical temperatures and exponents of the Potts model are calculated for 0<q\ensuremath{\le}4 by using various boundary conditions. Very good estimates of critical properties are obtained. The choice of the boundary condition is discussed.
Critical properties of the
Q-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of
Q that annihilate as
Q approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For
Q>Qc, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients
a and
b cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation,
a=6.52 and
b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model,
Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
Abstract The critical behavior of the ( n + 1 ) -states Potts model in d -dimensions is studied with functional renormalization group techniques. We devise a general method to derive β -functions for continuous values of d and n and we write the flow equation for the effective potential (LPA’) when instead n is fixed. We calculate several critical exponents, which are found to be in good agreement with Monte Carlo simulations and ϵ-expansion results available in the literature. In particular, we focus on Percolation ( n → 0 ) and Spanning Forest ( n → − 1 ) which are the only non-trivial universality classes in d = 4,5 and where our methods converge faster.
The randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold
p∗ at which the critical temperature goes to zero depends upon q.