We calculate the partition function of the $q$-state Potts model on arbitrary-length cyclic ladder graphs of the square and triangular lattices, with a generalized external magnetic field that favors or disfavors a subset of spin values ${1,...,s}$ with $s le q$. For the case of antiferromagnet spin-spin coupling, these provide exactly solved models that exhibit an onset of frustration and competing interactions in the context of a novel type of tensor-product $S_s otimes S_{q-s}$ global symmetry, where $S_s$ is the permutation group on $s$ objects.
We present generalized methods for calculating lower bounds on the ground-state entropy per site, $S_0$, or equivalently, the ground-state degeneracy per site, $W=e^{S_0/k_B}$, of the antiferromagnetic Potts model. We use these methods to derive improved lower bounds on $W$ for several lattices.
In technicolor theories using an SU($N_{TC}$) gauge group, the value of $N_{TC}$ is not, {it a priori}, determined and is typically chosen by phenomenological criteria. Here we present a novel way to determine $N_{TC}$ from the embedding of a one-family technicolor model, with fermions in the fundamental represention of SU($N_{TC}$), in an extended technicolor theory, and use it to deduce that $N_{TC}=4$ in this framework.
We calculate zeros of the $q$-state Potts model partition function on $m$th-iterate Sierpinski graphs, $S_m$, in the variable $q$ and in a temperature-like variable, $y$. We infer some asymptotic properties of the loci of zeros in the limit $m to infty$ and relate these to thermodynamic properties of the $q$-state Potts ferromagnet and antiferromagnet on the Sierpinski gasket fractal, $S_infty$.
We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $Lambda$ by $ell$ bonds connecting the same adjacent vertices, thereby yielding the lattice $Lambda_ell$. This relation is used to calculate the bond percolation threshold on $Lambda_ell$. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality $d ge 2$ but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the $N to infty$ limits of several families of $N$-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as $N to infty$.
We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.
We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, Mobius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(Lambda,L_y times L_x,q,v,w)=sum_{d=0}^{L_y} tilde c^{(d)} Tr[(T_{Z,Lambda,L_y,d})^m]$, where $Lambda$ denotes the lattice type, $tilde c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,Lambda,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $Lambda=sq, tri (hc)$, respectively. An analogous formula is given for Mobius strips, while only $T_{Z,Lambda,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,Lambda,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,Lambda,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,Lambda,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.
We reconsider the two different facets of $pi$ and $K$ mesons as $q bar q$ bound states and approximate Nambu-Goldstone bosons. We address several topics, including masses, mass splittings between $pi$ and $rho$ and between $K$ and $K^*$, meson wavefunctions, charge radii, and the $K-pi$ wavefunction overlap.
The successful description of current data provided by the Standard Model includes fundamental fermions that are color-singlets and electroweak-nonsinglets, but no fermions that are electroweak-singlets and color-nonsinglets. In an effort to understand the absence of such fermions, we construct and study {it gedanken} models that do contain electroweak-singlet chiral quark fields. These models exhibit several distinctive properties, including the absence of any neutral lepton and the fact that both the $(uud)$ and $(ddu)$ nucleons are electrically charged. We also explore how such models could arise as low-energy limits of grand unified theories and, in this more restrictive context, we show that they exhibit further exotic properties.
We present a new perspective on the nature of quark and gluon condensates in quantum chromodynamics. We suggest that the spatial support of QCD condensates is restricted to the interiors of hadrons, since these condensates arise due to the interactions of confined quarks and gluons. An analogy is drawn with order parameters like the Cooper pair condensate and spontaneous magnetization experimentally measured in finite samples in condensed matter physics. Our picture explains the results of recent studies which find no significant signal for the vacuum gluon condensate. We also give a general discussion of condensates in asymptotically free vectorial and chiral gauge theories.
