Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2002-05-20
J. Stat. Phys. 112, 815-879 (2003)
Physics
Condensed Matter
Statistical Mechanics
66 pages, latex, 13 figures; dedicated to F. Y. Wu on his 70'th birthday
Scientific paper
We present exact calculations of flow polynomials $F(G,q)$ for lattice strips of various fixed widths $L_y$ and arbitrarily great lengths $L_x$, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face $fl$ in the infinite-length limit. We study the zeros of $F(G,q)$ in the complex $q$ plane and determine exactly the asymptotic accumulation sets of these zeros ${\cal B}$ in the infinite-length limit for the various families of strips. The function $fl$ is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case $L_y=1$, the maximal point, $q_{cf}$, where ${\cal B}$ crosses the real axis, is universal on cyclic and M\"obius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value $q_{cf}=3$ for the infinite square lattice.
Chang Shu-Chiuan
Shrock Robert
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