Q-State Potts-Model By Wilsons 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−e, lines of critical and tricritical Potts fixed points are found as functions of Q that annihilate as Q approaches a critical value Qc=2+e2a+O(e3). 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=2−16[e±(e2−aδ)12], φν=∓(e2−aδ)12, and η=[e±(e2−aδ)12]2216+bδ, where e=4−d, δ=Q−2, and δ≤δc=e2a+O(e3). While the form of the e and δ dependences is exact, the coefficients a and b cannot be obtained systematically by e 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≤d≲4 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.