381 research outputs found
Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
Denoting as the chromatic polynomial for coloring an -vertex
graph with colors, and considering the limiting function , a fundamental question in graph theory is the
following: is analytic or not at the origin
of the plane? (where the complex generalization of is assumed). This
question is also relevant in statistical mechanics because
, where is the ground state entropy of the
-state Potts antiferromagnet on the lattice graph , and the
analyticity of at is necessary for the large- series
expansions of . Although is analytic at for many
, there are some for which it is not; for these, has no
large- series expansion. It is important to understand the reason for this
nonanalyticity. Here we give a general condition that determines whether or not
a particular is analytic at and explains the
nonanalyticity where it occurs. We also construct infinite families of graphs
with functions that are non-analytic at and investigate the
properties of these functions. Our results are consistent with the conjecture
that a sufficient condition for to be analytic at is
that is a regular lattice graph . (This is known not to be a
necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in
Phys. Rev.
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
Ground State Entropy of Potts Antiferromagnets: Bounds, Series, and Monte Carlo Measurements
We report several results concerning , the
exponent of the ground state entropy of the Potts antiferromagnet on a lattice
. First, we improve our previous rigorous lower bound on for
the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to
the first eleven terms with the large- series for . Second, we
investigate the heteropolygonal Archimedean lattice, derive a
rigorous lower bound, on , and calculate the large- series
for this function to where . Remarkably, these agree
exactly to all thirteen terms calculated. We also report Monte Carlo
measurements, and find that these are very close to our lower bound and series.
Third, we study the effect of non-nearest-neighbor couplings, focusing on the
square lattice with next-nearest-neighbor bonds.Comment: 13 pages, Latex, to appear in Phys. Rev.
Confinement contains condensates
Dynamical chiral symmetry breaking and its connection with the generation of
hadron masses has historically been viewed as a vacuum phenomenon. We argue
that confinement makes such a position untenable. If quark-hadron duality is a
reality in QCD, then condensates, those quantities that were commonly viewed as
constant empirical mass-scales that fill all spacetime, are instead wholly
contained within hadrons; viz., they are a property of hadrons themselves and
expressed, e.g., in their Bethe-Salpeter or light-front wave functions. We
explain that this paradigm is consistent with empirical evidence, and
incidentally expose misconceptions in a recent Comment.Comment: 10 pages, 2 figure
Ground State Entropy of the Potts Antiferromagnet on Cyclic Strip Graphs
We present exact calculations of the zero-temperature partition function
(chromatic polynomial) and the (exponent of the) ground-state entropy for
the -state Potts antiferromagnet on families of cyclic and twisted cyclic
(M\"obius) strip graphs composed of -sided polygons. Our results suggest a
general rule concerning the maximal region in the complex plane to which
one can analytically continue from the physical interval where . The
chromatic zeros and their accumulation set exhibit the rather
unusual property of including support for and provide further
evidence for a relevant conjecture.Comment: 7 pages, Latex, 4 figs., J. Phys. A Lett., in pres
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