499 research outputs found
T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
We present exact calculations of the zero-temperature partition function for
the -state Potts antiferromagnet (equivalently, the chromatic polynomial)
for families of arbitrarily long strip graphs of the square and triangular
lattices with width 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, and
determine the continuous locus 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 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
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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