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

Critical properties of the Q-state Potts model for dimensions 3≤d≤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-ε, lines of critical and tricritical Potts fixed points are found as functions of Q that annihilate as Q approaches a critical value Qc=2+ε²a+O(ε³). 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: 1ν=2-1/6[ε±(ε²-aδ)1/2], φν=∓(ε²-aδ)1/2, and η=[ε±(ε²-aδ)1/2]²216+bδ, where ε=4-d, δ=Q-2, and δ≤δc=ε²a+O(ε³). While the form of the ε and δ dependences is exact, the coefficients a and b cannot be obtained systematically by ε expansion, since the upper critical dimensionality of the Potts model is six when Q≠2. In our truncation, $a=6.52$ and $b=0.065$. The results have been extended to dimensions 3.4≤d4 by solving the renormalization-group equations numerically. The percolation limit of the Potts model, $Q=1$, is also investigated and the critical exponents νP,φP, and ηP determined as functions of dimension for 3≤d≤6.