948 research outputs found

    Ground State Entropy in Potts Antiferromagnets and Chromatic Polynomials

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    We discuss recent results on ground state entropy in Potts antiferromagnets and connections with chromatic polynomials. These include rigorous lower and upper bounds, Monte Carlo measurements, large--qq series, exact solutions, and studies of analytic properties. Some related results on Fisher zeros of Potts models are also mentioned.Comment: LATTICE98(spin) 3 pages, Late

    Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice

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    We present exact solutions for the zero-temperature partition function (chromatic polynomial PP) and the ground state degeneracy per site WW (= exponent of the ground-state entropy) for the qq-state Potts antiferromagnet on strips of the square lattice of width LyL_y vertices and arbitrarily great length LxL_x vertices. The specific solutions are for (a) Ly=4L_y=4, (FBCy,PBCx)(FBC_y,PBC_x) (cyclic); (b) Ly=4L_y=4, (FBCy,TPBCx)(FBC_y,TPBC_x) (M\"obius); (c) Ly=5,6L_y=5,6, (PBCy,FBCx)(PBC_y,FBC_x) (cylindrical); and (d) Ly=5L_y=5, (FBCy,FBCx)(FBC_y,FBC_x) (open), where FBCFBC, PBCPBC, and TPBCTPBC denote free, periodic, and twisted periodic boundary conditions, respectively. In the LxL_x \to \infty limit of each strip we discuss the analytic structure of WW in the complex qq plane. The respective WW functions are evaluated numerically for various values of qq. Several inferences are presented for the chromatic polynomials and analytic structure of WW for lattice strips with arbitrarily great LyL_y. The absence of a nonpathological LxL_x \to \infty limit for real nonintegral qq in the interval 0<q<30 < q < 3 (0<q<40 < q < 4) for strips of the square (triangular) lattice is discussed.Comment: 37 pages, latex, 4 encapsulated postscript figure

    Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips

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    We present transfer matrices for the zero-temperature partition function of the qq-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and M\"obius strips of the square, triangular, and honeycomb lattices of width LyL_y and arbitrarily great length LxL_x. We relate these results to our earlier exact solutions for square-lattice strips with Ly=3,4,5L_y=3,4,5, triangular-lattice strips with Ly=2,3,4L_y=2,3,4, and honeycomb-lattice strips with Ly=2,3L_y=2,3 and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a M\"obius strip of a lattice Λ\Lambda and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width LyL_y. New results are presented for the Ly=5L_y=5 strip of the triangular lattice and the Ly=4L_y=4 and Ly=5L_y=5 strips of the honeycomb lattice. Using these results and taking the infinite-length limit LxL_x \to \infty, we determine the continuous accumulation locus of the zeros of the above partition function in the complex qq plane, including the maximal real point of nonanalyticity of the degeneracy per site, WW as a function of qq.Comment: 62 pages, latex, 6 eps figures, includes additional results, e.g., loci B{\cal B}, requested by refere

    Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice

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    We present exact calculations of the partition function of the qq-state Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the honeycomb (brick) lattice of width Ly=2L_y=2 and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the qq plane for fixed temperature and in the complex temperature plane for fixed qq values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and W(q)W(q), the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) Ly=3L_y=3, cyclic, (v) Ly=3L_y=3, M\"obius, (vi) Ly=4L_y=4, cylindrical, and (vii) Ly=4L_y=4, open. In the infinite-length limit we calculate W(q)W(q) and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the Ly=4L_y=4 strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in 10510^5 for moderate qq values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).Comment: 48 pages, latex, with encapsulated postscript figure

    T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology

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    We present exact calculations of the zero-temperature partition function of the qq-state Potts antiferromagnet (equivalently the chromatic polynomial) for Moebius strips, with width Ly=2L_y=2 or 3, of regular lattices and homeomorphic expansions thereof. These are compared with the corresponding partition functions for strip graphs with (untwisted) periodic longitudinal boundary conditions.Comment: 9 pages, Latex, Phys. Lett. A, in pres

    T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions

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    We present exact calculations of the zero-temperature partition function for the qq-state Potts antiferromagnet (equivalently, the chromatic polynomial) for families of arbitrarily long strip graphs of the square and triangular lattices with width Ly=4L_y=4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation (i.e. of torus or Klein bottle type). These boundary conditions have the advantage of removing edge effects. In the limit of infinite length, we calculate the exponent of the entropy, W(q)W(q) and determine the continuous locus B{\cal B} where it is singular. We also give results for toroidal strips involving ``crossing subgraphs''; these make possible a unified treatment of torus and Klein bottle boundary conditions and enable us to prove that for a given strip, the locus B{\cal B} is the same for these boundary conditions.Comment: 43 pages, latex, 4 postscript figure

    Potts Model Partition Functions for Self-Dual Families of Strip Graphs

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    We consider the qq-state Potts model on families of self-dual strip graphs GDG_D of the square lattice of width LyL_y and arbitrarily great length LxL_x, with periodic longitudinal boundary conditions. The general partition function ZZ and the T=0 antiferromagnetic special case PP (chromatic polynomial) have the respective forms j=1NF,Ly,λcF,Ly,j(λF,Ly,j)Lx\sum_{j=1}^{N_{F,L_y,\lambda}} c_{F,L_y,j} (\lambda_{F,L_y,j})^{L_x}, with F=Z,PF=Z,P. For arbitrary LyL_y, we determine (i) the general coefficient cF,Ly,jc_{F,L_y,j} in terms of Chebyshev polynomials, (ii) the number nF(Ly,d)n_F(L_y,d) of terms with each type of coefficient, and (iii) the total number of terms NF,Ly,λN_{F,L_y,\lambda}. We point out interesting connections between the nZ(Ly,d)n_Z(L_y,d) and Temperley-Lieb algebras, and between the NF,Ly,λN_{F,L_y,\lambda} and enumerations of directed lattice animals. Exact calculations of PP are presented for 2Ly42 \le L_y \le 4. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), W(q)W(q). Generalizing qq from Z+{\mathbb Z}_+ to C{\mathbb C}, we determine the continuous locus B{\cal B} in the complex qq plane where W(q)W(q) is singular. We find the interesting result that for all LyL_y values considered, the maximal point at which B{\cal B} crosses the real qq axis, denoted qcq_c is the same, and is equal to the value for the infinite square lattice, qc=3q_c=3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of qcq_c.Comment: 36 pages, latex, three postscript figure

    Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice

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    We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii) (FBC_y,TPBC_x)= M\"obius, (iii) (PBC_y,PBC_x)= toroidal, and (iv) (PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the L_y=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the {\mathbb C}^2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.Comment: latex, with encapsulated postscript figure

    Exact Potts Model Partition Function on Strips of the Triangular Lattice

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    In this paper we present exact calculations of the partition function ZZ of the qq-state Potts model and its generalization to real qq, for arbitrary temperature on nn-vertex strip graphs, of width Ly=2L_y=2 and arbitrary length, of the triangular lattice with free, cyclic, and M\"obius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. Considering the full generalization to arbitrary complex qq and temperature, we determine the singular locus B{\cal B} in the corresponding C2{\mathbb C}^2 space, arising as the accumulation set of partition function zeros as nn \to \infty. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where B{\cal B} reduces to the accumulation set of chromatic zeros. Comparisons are made with our previous exact calculation of Potts model partition functions for the corresponding strips of the square lattice. Our present calculations yield, as special cases, several quantities of graph-theoretic interest.Comment: 43 pages, latex, 24 postscript figures, Physica A, in pres
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