Spanning Forests and the q-State Potts Model in the Limit q→0

Abstract

We study the q-state Potts model with nearest-neighbor coupling v=eβJ−1 in the limit q,v → 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2≤ L ≤ 10, as well as the limiting curves B of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w0, where w0 =−1/4 (resp. w0=−0.1753 ± 0.0002) for the square (resp. triangular) lattice. For w>w0 we find a non-critical disordered phase that is compatible with the predicted asymptotic freedom as w → +∞. For w<w0 our results are compatible with a massless Berker–Kadanoff phase with central charge c=−2 and leading thermal scaling dimension xT,1 = 2 (marginally irrelevant operator). At w=w0 we find a “first-order critical point”: the first derivative of the free energy is discontinuous at w0, while the correlation length diverges as w↓ w0 (and is infinite at w=w0). The critical ehavior at w=w0 seems to be the same for both lattices and it differs from that of the Berker–Kadanoff phase: our results suggest that the central charge is c=−1, the leading thermal scaling dimension is xT,1=0, and the critical exponents are ν=1/d=1/2 and α=1.