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 $3\ensuremath{\le}d\ensuremath{\le}6$.
Wilson's renormalization-group method is used for investigating the critical properties and the role of quantum fluctuations for a d-dimensional quantum sine-Gordon model. In the classical regime a phase transition is found in d=2-\ensuremath{\epsilon} (\ensuremath{\epsilon}\ensuremath{\ge}0) dimensions and the static critical exponents are evaluated to leading order in \ensuremath{\epsilon}>0. Within the same formalism, the dynamic critical exponent z is also obtained as a characteristic of the intrinsic dynamics of the model. The quantum regime is briefly discussed, and no phase transition is found for d>1. We argue that a quantum-classical crossover occurs for 1<d\ensuremath{\le}2 in the low-temperature limit as a consequence of a dimensional crossover d\ensuremath{\rightarrow}d+1 which shifts the upper critical dimensionality from
dcu=2 to
dcu=1.
The phase transition of the classical two-dimensional Potts model, in particular the order of the transition as the number of components
q increases, is studied by constructing renormalization-group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical
qc separating the small-
q and large-
q regions with different critical behaviors. The physically accessible fixed point for
q>qc is a discontinuity fixed point where the specific-heat exponent $\ensuremath{\alpha}=1$, and therefore the transition is of first order.
The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.
The effect of quadratic symmetry breaking (QSB) on two representations for the Potts vectors in a continuum-field model are studied to two-loop order in renormalized perturbation theory in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, in extension of an earlier group-theoretical analysis by Wallace and Young. The explicit dependence of the crossover exponent $\ensuremath{\varphi}$ that corresponds to QSB that destroys the equivalence between pairs of Potts vectors is obtained as a function of
d and
n for the
p-state model with
p=n+1. It is shown that this exponent follows from the calculation of vertex functions in a representation due to Wallace and Young, whereas a second crossover exponent $\overline{\ensuremath{\varphi}}$, that can be identified with the critical exponent $\ensuremath{\beta}$, and which corresponds to QSB that favors a single Potts vector against the others, follows from a calculation using the representation of the Potts vectors due to Priest and Lubensky.
A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for
p>2. For
2<p<310 the transition is first order in spite of the fact that there is an accessible stable fixed point for all
p<310. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature
Tc, due to instantons, which is of the same form as that of the Ising model at
H=0.