Families of Graphs With Chromatic Zeros Lying on Circles

We define an infinite set of families of graphs, which we call $p$-wheels and denote $(Wh)^{(p)}_n$, that generalize the wheel ($p=1$) and biwheel ($p=2$) graphs. The chromatic polynomial for $(Wh)^{(p)}_n$ is calculated, and remarkably simple properties of the chromatic zeros are found: (i) the real zeros occur at $q=0,1,...p+1$ for $n-p$ even and $q=0,1,...p+2$ for $n-p$ odd; and (ii) the complex zeros all lie, equally spaced, on the unit circle $|q-(p+1)|=1$ in the complex $q$ plane. In the $n \to \infty$ limit, the zeros on this circle merge to form a boundary curve separating two regions where the limiting function $W(\{(Wh)^{(p)}\},q)$ is analytic, viz., the exterior and interior of the above circle. Connections with statistical mechanics are noted.Comment: 8 pages, Late

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

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 $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial $P$) on tube sections of the simple cubic lattice of fixed transverse size $L_x \times L_y$ and arbitrarily great length $L_z$, for sizes $L_x \times L_y = 2 \times 3$ and $2 \times 4$ and boundary conditions (a) $(FBC_x,FBC_y,FBC_z)$ and (b) $(PBC_x,FBC_y,FBC_z)$, where $FBC$ ($PBC$) denote free (periodic) boundary conditions. In the limit of infinite-length, $L_z \to \infty$, we calculate the resultant ground state degeneracy per site $W$ (= exponent of the ground-state entropy). Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the analytic structure of $W$ and the related singular locus ${\cal B}$ which is the continuous accumulation set of zeros of the chromatic polynomial. For the $L_z \to \infty$ limit of a given family of lattice sections, $W$ is analytic for real $q$ down to a value $q_c$. We determine the values of $q_c$ for the lattice sections considered and address the question of the value of $q_c$ for a $d$-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 $K_{m,m}$.Comment: 28 pages, latex, six postscript figures, two Mathematica file

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, $W$, 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, ${\cal B}$, of nonanalyticities in $W$. 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 $S_0 > 0$.Comment: 27 pages, Latex, 17 figs. J. Phys. A, in pres

Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0

Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex graph $G$ with $q$ colors, and considering the limiting function $W(\{G\},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, a fundamental question in graph theory is the following: is $W_r(\{G\},q) = q^{-1}W(\{G\},q)$ analytic or not at the origin of the $1/q$ plane? (where the complex generalization of $q$ is assumed). This question is also relevant in statistical mechanics because $W(\{G\},q)=\exp(S_0/k_B)$, where $S_0$ is the ground state entropy of the $q$-state Potts antiferromagnet on the lattice graph $\{G\}$, and the analyticity of $W_r(\{G\},q)$ at $1/q=0$ is necessary for the large-$q$ series expansions of $W_r(\{G\},q)$. Although $W_r$ is analytic at $1/q=0$ for many $\{G\}$, there are some $\{G\}$ for which it is not; for these, $W_r$ has no large-$q$ 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 $W_r(\{G\},q)$ is analytic at $1/q=0$ and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with $W_r$ functions that are non-analytic at $1/q=0$ and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for $W_r(\{G\},q)$ to be analytic at $1/q=0$ is that $\{G\}$ is a regular lattice graph $\Lambda$. (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: Bounds, Series, and Monte Carlo Measurements

We report several results concerning $W(\Lambda,q)=\exp(S_0/k_B)$, the exponent of the ground state entropy of the Potts antiferromagnet on a lattice $\Lambda$. First, we improve our previous rigorous lower bound on $W(hc,q)$ for the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to the first eleven terms with the large-$q$ series for $W(hc,q)$. Second, we investigate the heteropolygonal Archimedean $4 \cdot 8^2$ lattice, derive a rigorous lower bound, on $W(4 \cdot 8^2,q)$, and calculate the large-$q$ series for this function to $O(y^{12})$ where $y=1/(q-1)$. 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

Extended Technicolor Models with Two ETC Groups

We construct extended technicolor (ETC) models that can produce the large splitting between the masses of the $t$ and $b$ quarks without necessarily excessive contributions to the $\rho$ parameter or to neutral flavor-changing processes. These models make use of two different ETC gauge groups, such that left- and right-handed components of charge $Q=2/3$ quarks transform under the same ETC group, while left- and right-handed components of charge -1/3 quarks and charged leptons transform under different ETC groups. The models thereby suppress the masses $m_b$ and $m_\tau$ relative to $m_t$, and $m_s$ and $m_\mu$ relative to $m_c$ because the masses of the $Q=-1/3$ quarks and charged leptons require mixing between the two ETC groups, while the masses of the $Q=2/3$ quarks do not. A related source of the differences between these mass splittings is the effect of the two hierarchies of breaking scales of the two ETC groups. We analyze a particular model of this type in some detail. Although we find that this model tends to suppress the masses of the first two generations of down-type quarks and charged leptons too much, it gives useful insights into the properties of theories with more than one ETC group.Comment: 14 pages, 4 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 $S_0$ for the $q$-state Potts antiferromagnet on families of cyclic and twisted cyclic (M\"obius) strip graphs composed of $p$-sided polygons. Our results suggest a general rule concerning the maximal region in the complex $q$ plane to which one can analytically continue from the physical interval where $S_0 > 0$. The chromatic zeros and their accumulation set ${\cal B}$ exhibit the rather unusual property of including support for $Re(q) < 0$ and provide further evidence for a relevant conjecture.Comment: 7 pages, Latex, 4 figs., J. Phys. A Lett., 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 $p$ and $\Lambda$, 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 $p=0$ and $p=1$.Comment: 16 pages, revtex, 7 eps figure