TY - JOUR

T1 - Transfer matrices and partition-function zeros for antiferromagnetic potts models. I. General theory and square-lattice chromatic polynomial

AU - Salas, Jesús

AU - Sokal, Alan D.

N1 - Funding Information:
The authors’ research was supported in part by U.S. National Science Foundation grant PHY-9900769 (J.S. and A.D.S.) and CICyT (Spain) grant AEN97-1680 (J.S.). It was completed while the second author was a Visiting Fellow at All Souls College, Oxford, where he was supported in part by Engineering and Physical Sciences Research Council grant GR/M 71626 and aided by the warm hospitality of John Cardy and the Department of Theoretical Physics.

PY - 2001/8

Y1 - 2001/8

N2 - We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) PG(q) for m×n rectangular subsets of the square lattice, with m≤8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n→∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B2, B3, B4, A5 are limiting points of partition-function zeros as n→∞ whenever the strip width m is ≥7 (periodic transverse b.c.) or ≥8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph.

AB - We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) PG(q) for m×n rectangular subsets of the square lattice, with m≤8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n→∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B2, B3, B4, A5 are limiting points of partition-function zeros as n→∞ whenever the strip width m is ≥7 (periodic transverse b.c.) or ≥8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph.

KW - Antiferromagnetic Potts model

KW - Beraha numbers

KW - Beraha-Kahane-Weiss theorem

KW - Chromatic polynomial

KW - Chromatic root

KW - Fortuin-Kasteleyn representation

KW - Square lattice

KW - Temperley-Lieb algebra

KW - Transfer matrix

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U2 - 10.1023/A:1010376605067

DO - 10.1023/A:1010376605067

M3 - Article

AN - SCOPUS:0035430046

VL - 104

SP - 609

EP - 699

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -