### 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 ehavior 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1153-1281 |

Number of pages | 129 |

Journal | Journal of Statistical Physics |

Volume | 119 |

Issue number | 5-6 |

DOIs | |

State | Published - Jun 2005 |

### Fingerprint

### Keywords

- Beraha-Kahane-Weiss theorem
- Berker-Kadanoff phase
- Conformal field theory
- Fortuin-Kasteleyn representation
- Phase transition
- Potts model
- q → 0 limit
- Spanning forest
- Square lattice
- Transfer matrix
- Triangular lattice

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*119*(5-6), 1153-1281. https://doi.org/10.1007/s10955-005-4409-y

**Spanning forests and the q-state potts model in the limit q →0.** / Jacobsen, Jesper Lykke; Salas, Jesús; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 119, no. 5-6, pp. 1153-1281. https://doi.org/10.1007/s10955-005-4409-y

}

TY - JOUR

T1 - Spanning forests and the q-state potts model in the limit q →0

AU - Jacobsen, Jesper Lykke

AU - Salas, Jesús

AU - Sokal, Alan D.

PY - 2005/6

Y1 - 2005/6

N2 - 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 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 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.

AB - 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 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 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.

KW - Beraha-Kahane-Weiss theorem

KW - Berker-Kadanoff phase

KW - Conformal field theory

KW - Fortuin-Kasteleyn representation

KW - Phase transition

KW - Potts model

KW - q → 0 limit

KW - Spanning forest

KW - Square lattice

KW - Transfer matrix

KW - Triangular lattice

UR - http://www.scopus.com/inward/record.url?scp=23644456375&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23644456375&partnerID=8YFLogxK

U2 - 10.1007/s10955-005-4409-y

DO - 10.1007/s10955-005-4409-y

M3 - Article

AN - SCOPUS:23644456375

VL - 119

SP - 1153

EP - 1281

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -