486 research outputs found
Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice
We present exact solutions for the zero-temperature partition function of the
-state Potts antiferromagnet (equivalently, the chromatic polynomial ) on
tube sections of the simple cubic lattice of fixed transverse size and arbitrarily great length , for sizes and and boundary conditions (a) and (b)
, where () denote free (periodic) boundary
conditions. In the limit of infinite-length, , we calculate the
resultant ground state degeneracy per site (= exponent of the ground-state
entropy). Generalizing from to , we determine
the analytic structure of and the related singular locus which
is the continuous accumulation set of zeros of the chromatic polynomial. For
the limit of a given family of lattice sections, is
analytic for real down to a value . We determine the values of
for the lattice sections considered and address the question of the value of
for a -dimensional Cartesian lattice. Analogous results are presented
for a tube of arbitrarily great length whose transverse cross section is formed
from the complete bipartite graph .Comment: 28 pages, latex, six postscript figures, two Mathematica file
Families of Graphs With Chromatic Zeros Lying on Circles
We define an infinite set of families of graphs, which we call -wheels and
denote , that generalize the wheel () and biwheel ()
graphs. The chromatic polynomial for is calculated, and
remarkably simple properties of the chromatic zeros are found: (i) the real
zeros occur at for even and for odd;
and (ii) the complex zeros all lie, equally spaced, on the unit circle
in the complex plane. In the limit, the zeros
on this circle merge to form a boundary curve separating two regions where the
limiting function is analytic, viz., the exterior and
interior of the above circle. Connections with statistical mechanics are noted.Comment: 8 pages, Late
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.
On the Unification of Gauge Symmetries in Theories with Dynamical Symmetry Breaking
We analyze approaches to the partial or complete unification of gauge
symmetries in theories with dynamical symmetry breaking. Several types of
models are considered, including those that (i) involve sufficient unification
to quantize electric charge, (ii) attempt to unify the three standard-model
gauge interactions in a simple Lie group that forms a direct product with an
extended technicolor group, and, most ambitiously, (iii) attempt to unify the
standard-model gauge interactions with (extended) technicolor in a simple Lie
group.Comment: 24 pages, ReVTe
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
Exact Results for Average Cluster Numbers in Bond Percolation on Lattice Strips
We present exact calculations of the average number of connected clusters per
site, , as a function of bond occupation probability $p$, for the bond
percolation problem on infinite-length strips of finite width $L_y$, of the
square, triangular, honeycomb, and kagom\'e lattices $\Lambda$ with various
boundary conditions. These are used to study the approach of , for a given
and , to its value on the two-dimensional lattice as the strip
width increases. We investigate the singularities of in the complex $p$
plane and their influence on the radii of convergence of the Taylor series
expansions of about and .Comment: 16 pages, revtex, 7 eps figure
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.
Implications of Dynamical Generation of Standard-Model Fermion Masses
We point out that if quark and lepton masses arise dynamically, then in a
wide class of theories the corresponding running masses exhibit
the power-law decay for Euclidean momenta
, where is a fermion of generation , and
is the maximal scale relevant for the origin of . We estimate
resultant changes in precision electroweak quantities and compare with current
data. It is found that this data allows the presence of such corrections. We
also note that this power-law decay renders primitively divergent fermion mass
corrections finite.Comment: 4 pages, late
The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities
We present an analysis of the structure and properties of chromatic
polynomials of one-parameter and multi-parameter families
of planar triangulation graphs , where is a vector of integer parameters. We use these to study the
ratio of to the Tutte upper bound ,
where and is the number of vertices in . In particular, we calculate limiting values of this ratio as for various families of planar triangulations. We also use our
calculations to study zeros of these chromatic polynomials. We study a large
class of families with and and show that these have
a structure of the form for , where
, , and , and for . We derive properties of the
coefficients and show that has a
real chromatic zero that approaches as one or more of
the . The generalization to is given. Further, we
present a one-parameter family of planar triangulations with real zeros that
approach 3 from below as . Implications for the ground-state
entropy of the Potts antiferromagnet are discussed.Comment: 57 pages, latex, 15 figure
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
- …