Spanning forests and the q-state Potts model in the limit q \to 0

Abstract

We study the q-state Potts model with nearest-neighbor coupling v=e^{βJ}-1 in the limit q,v \to 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 \le L \le 10, as well as the limiting curves 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 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal 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 \downarrow 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 conformal 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.

131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal version

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Cited by in corpus (43)

• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
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• The ABC (in any D) of Logarithmic CFT
• The arboreal gas and the supersphere sigma model
• Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
• Phase diagram of the chromatic polynomial on a torus
• Critical manifold of the kagome-lattice Potts model
• Grassmann Integral Representation for Spanning Hyperforests
• Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices
• Odd q-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases
• Transfer matrix computation of critical polynomials for two-dimensional Potts models
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
• Critical points in coupled Potts models and critical phases in coupled loop models
• Non-scalar operators for the Potts model in arbitrary dimension
• Complex-temperature phase diagram of Potts and RSOS models
• Efficient simulation of the random-cluster model
• A multicritical Landau-Potts field theory
• Devil's Staircase Continuum in the Chiral Clock Spin Glass with Competing Ferromagnetic-Antiferromagnetic and Left-Right Chiral Interactions
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
• Renormalization flow for unrooted forests on a triangular lattice
• Q-colourings of the triangular lattice: Exact exponents and conformal field theory
• High-temperature series expansions for the $q$-state Potts model on a hypercubic lattice and critical properties of percolation
• Classical phase transitions in a one-dimensional short-range spin model induced by entropy depletion or complex fields
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Conductivity of Coniglio-Klein clusters
• The phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet
• Hyperforests on the Complete Hypergraph by Grassmann Integral Representation
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories
• Bridges in the random-cluster model
• Critical Behaviour of Spanning Forests on Random Planar Graphs
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Uniqueness of the infinite tree in low-dimensional random forests
• Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices
• Arnold Tongues and Feigenbaum Exponents of the Rational Mapping for Q-state Potts Model on Recursive Lattice: Q<2
• Percolation transition for random forests in $d\ge 3$
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Dualities between fermionic theories and the Potts model
• Spanning forests on the Sierpinski gasket
• Universality of the complete-graph Potts model with $0<q\le 2$
• Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds
• Sweeny dynamics for the random-cluster model with small $Q$
• Anomalous criticality coexists with giant cluster in the uniform forest model
• Universality and exact finite-size corrections for spanning trees on cobweb and fan networks