499 research outputs found

    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

    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
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