8,630 research outputs found
Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
We present transfer matrices for the zero-temperature partition function of
the -state Potts antiferromagnet (equivalently, the chromatic polynomial) on
cyclic and M\"obius strips of the square, triangular, and honeycomb lattices of
width and arbitrarily great length . We relate these results to our
earlier exact solutions for square-lattice strips with ,
triangular-lattice strips with , and honeycomb-lattice strips with
and periodic or twisted periodic boundary conditions. We give a
general expression for the chromatic polynomial of a M\"obius strip of a
lattice and exact results for a subset of honeycomb-lattice transfer
matrices, both of which are valid for arbitrary strip width . New results
are presented for the strip of the triangular lattice and the
and strips of the honeycomb lattice. Using these results and taking the
infinite-length limit , we determine the continuous
accumulation locus of the zeros of the above partition function in the complex
plane, including the maximal real point of nonanalyticity of the degeneracy
per site, as a function of .Comment: 62 pages, latex, 6 eps figures, includes additional results, e.g.,
loci , requested by refere
Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs
We present exact calculations of chromatic polynomials for families of cyclic
graphs consisting of linked polygons, where the polygons may be adjacent or
separated by a given number of bonds. From these we calculate the (exponential
of the) ground state entropy, , for the q-state Potts model on these graphs
in the limit of infinitely many vertices. A number of properties are proved
concerning the continuous locus, , of nonanalyticities in . Our
results provide further evidence for a general rule concerning the maximal
region in the complex q plane to which one can analytically continue from the
physical interval where .Comment: 27 pages, Latex, 17 figs. J. Phys. A, in pres
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
We study the chromatic polynomial P_G(q) for m \times n square- and
triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary
conditions. This polynomial gives the zero-temperature limit of the partition
function for the antiferromagnetic q-state Potts model defined on the lattice
G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn
representation for such lattices and obtain the accumulation sets of chromatic
zeros in the complex q-plane in the limit n\to\infty. We find that the
different phases that appear in this model can be characterized by a
topological parameter. We also compute the bulk and surface free energies and
the central charge.Comment: 55 pages (LaTeX2e). Includes tex file, three sty files, and 22
Postscript figures. Also included are Mathematica files transfer4_sq.m and
transfer4_tri.m. Journal versio
Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
We present exact solutions for the zero-temperature partition function
(chromatic polynomial ) and the ground state degeneracy per site (=
exponent of the ground-state entropy) for the -state Potts antiferromagnet
on strips of the square lattice of width vertices and arbitrarily great
length vertices. The specific solutions are for (a) ,
(cyclic); (b) , (M\"obius); (c)
, (cylindrical); and (d) ,
(open), where , , and denote free, periodic, and twisted
periodic boundary conditions, respectively. In the limit of
each strip we discuss the analytic structure of in the complex plane.
The respective functions are evaluated numerically for various values of
. Several inferences are presented for the chromatic polynomials and
analytic structure of for lattice strips with arbitrarily great . The
absence of a nonpathological limit for real nonintegral in
the interval () for strips of the square (triangular)
lattice is discussed.Comment: 37 pages, latex, 4 encapsulated postscript figure
Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
We present exact calculations of the partition function of the -state
Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the
honeycomb (brick) lattice of width 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
plane for fixed temperature and in the complex temperature plane for fixed
values. We also give exact calculations of the zero-temperature partition
function (chromatic polynomial) and , the exponent of the ground-state
entropy, for the Potts antiferromagnet for honeycomb strips of type (iv)
, cyclic, (v) , M\"obius, (vi) , cylindrical, and (vii)
, open. In the infinite-length limit we calculate and determine
the continuous locus of points where it is nonanalytic. We show that our exact
calculation of the entropy for the strip with cylindrical boundary
conditions provides an extremely accurate approximation, to a few parts in
for moderate 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
We present exact calculations of the zero-temperature partition function of
the -state Potts antiferromagnet (equivalently the chromatic polynomial) for
Moebius strips, with width 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
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