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 $3\ensuremath{\le}d\ensuremath{\le}6$.
Based upon the known results of the critical properties of the q state Potts ferromagnets obtained by different methods, we proposed an empirical formula of the model. It gives the critical points and the critical exponents for general Potts spin value q in any d dimensions which are in good agreement with all the available results. It improves the results given by Migdal–Kadanoff renormalization group method. It gives more accurate results than Hajdukovic conjecture to the problem of interest. This empirical formula may serve as the primary reference to the problem of interest.
The critical behavior of the
s-state Ashkin-Teller-Potts model on a square lattice is studied for
s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the
s=3 and
s=4 models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For
s=5, however, the transformation fails to detect the first-order transition that is known to occur.
A generalization of the Niemeijer-van Leeuwen renormalization-group transformation treating disordered cells of spins as vacancies is introduced and applied to the two-dimensional
q-state Potts model. Finite-lattice approximations yield the changeover from continuous to first-order phase transitions predicted by Baxter but not observed in previous renormalization-group calculations. Exact results are conjectured for the tricritical exponents of dilute Potts models.
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.