Q-state Potts model by Wilson's exact renormalization-group equation

Critical properties of the $Q$-state Potts model for dimensions $3\le d\le 6$ 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 $d=4-\epsilon$, lines of critical and tricritical Potts fixed points are found as functions of $Q$ that annihilate as $Q$ approaches a critical value $Q_c=\frac{2+{\epsilon }^2}{a+O\left({\epsilon }^3\right)}$. For $Q>\mathrm{}{Q}_c$, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\nu }=2-\frac{1}{6}[\epsilon \pm {({\epsilon }^2-a\delta )}^{\frac{1}{2}}]$, $\frac{\phi }{\nu }=\mp {({\epsilon }^2-a\delta )}^{\frac{1}{2}}$, and $\eta =\frac{{[\epsilon \pm {({\epsilon }^2-a\delta )}^{\frac{1}{2}}]}^2}{216}+b\delta$, where $\epsilon =4-d$, $\delta =Q-2\mathrm{}$, and $\delta \le {\delta }_c=\frac{{\epsilon }^2}{a}+O\left({\epsilon }^3\right)$. While the form of the $\epsilon$ and $\delta$ dependences is exact, the coefficients $a$ and $b$ cannot be obtained systematically by $\epsilon$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ne 2$. In our truncation, $a=6.52\mathrm{}$ and $b=0.065\mathrm{}$. The results have been extended to dimensions $3.4\le d\lesssim 4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, $Q=1\mathrm{}$, is also investigated and the critical exponents ${\nu }^P,{\phi }^P$, and ${\eta }^P$ determined as functions of dimension for $3\le d\le 6$.