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

Jesper Lykke Jacobsen, Jesús Salas, Alan D. Sokal

    Research output: Contribution to journalArticle

    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 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.

    Original languageEnglish (US)
    Pages (from-to)1153-1281
    Number of pages129
    JournalJournal of Statistical Physics
    Volume119
    Issue number5-6
    DOIs
    StatePublished - Jun 2005

    Fingerprint

    Spanning Forest
    Potts Model
    Triangular Lattice
    Transfer Matrix
    Square Lattice
    Charge
    Scaling
    scaling
    Correlation Length
    transition points
    Diverge
    Partition Function
    Critical Exponents
    Phase Diagram
    Strip
    Free Energy
    partitions
    Critical point
    Nearest Neighbor
    strip

    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

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

    In: Journal of Statistical Physics, Vol. 119, No. 5-6, 06.2005, p. 1153-1281.

    Research output: Contribution to journalArticle

    Jacobsen, Jesper Lykke ; Salas, Jesús ; Sokal, Alan D. / Spanning forests and the q-state potts model in the limit q →0. In: Journal of Statistical Physics. 2005 ; Vol. 119, No. 5-6. pp. 1153-1281.
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    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 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.",
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    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.

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    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.

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    KW - q → 0 limit

    KW - Spanning forest

    KW - Square lattice

    KW - Transfer matrix

    KW - Triangular lattice

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