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=e^βJ−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 x_T,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 x_T,1=0, and the critical exponents are ν=1/d=1/2 and α=1.