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 = w 0 , where w 0 = −1/4 (resp. w 0 = −0.1753 ± 0.0002) for the square (resp. triangular) lattice. For w > w 0 we find a non-critical disordered phase that is compatible with the predicted asymptotic freedom as w → +∞. For w < w 0 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 = w 0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w 0 , while the correlation length diverges as w ↓ w 0 (and is infinite at w = w 0 ). The critical behavior at w = w 0 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.

Complex-temperature phase diagram of Potts and RSOS models

“…As a matter of fact, these branches extend to infinity (i.e., they are unbounded 14 ), in sharp contrast with the bounded limiting curves obtained using free longitudinal boundary conditions [39,40]. It is important to remark that this phenomenon also holds in the limit p → ∞, as shown in Figure 8.…”

High-temperature series expansions for theq-state Potts model on a hypercubic lattice and critical properties of percolation

“…The Potts model in two dimensions is exhaustively reviewed in (Jacobsen et al, 2005). We repeat here the principal facts: Baxter has determined the exact free energy on Z 2 along the two curves…”

The ABC (in any D) of logarithmic CFT

“…In that case, logarithms appear in OPEs of higherdimensional operators, for instance in φ 2 ab × φ 2 cd . Another case of interest is the Q → 0 limit of the Potts model, which has a geometrical interpretation in terms of spanning forests [97][98][99][100][101].…”