arXiv subjects

Robert Shrock

Publications and source records attributed to Robert Shrock.

At least 181 records · Page 10Linked to original sources

arXiv2000

T=0 Partition Functions for Potts Antiferromagnets on Square Lattice Strips with (Twisted) Periodic Boundary Conditions

We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and (i) periodic (ii) twisted periodic boundary conditions in the longitudinal direction, so that the strip graphs are embedded on a (i) torus (ii) Klein bottle. In the limit of infinite length, we calculate the exponent of the entropy, W(q), show it to be the same for (i) and (ii), and determine its analytic structure.

cond-mat.stat-mech↗

arXiv2000

Exact Potts Model Partition Functions on Ladder Graphs

We present exact calculations of the partition function $Z$ of the $q$-state Potts model and its generalization to real $q$, the random cluster model, for arbitrary temperature on $n$-vertex ladder graphs with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed.

cond-mat.stat-mech↗

arXiv1999

Matter Effects on Long Baseline Neutrino Oscillation Experiments

We calculate matter effects on neutrino oscillations relevant for long baseline neutrino oscillation experiments. In particular, we compare the results obtained with simplifying approximations for the density profile in the Earth versus results obtained with actual density profiles. We study the dependence of the oscillation signals on both $E/Δm^2_{atm.}$ and on the angles in the leptonic mixing matrix. The results show quantitatively how matter effects can cause significant changes in the oscillation signals, relative to vacuum oscillations and can be useful in amplifying these signals and helping one to obtain measurements of mixing parameters and the magnitude and sign of $Δm^2$.

hep-ph↗

arXiv1999

Ground-State Degeneracy of Potts Antiferromagnets on Two-Dimensional Lattices: Approach Using Infinite Cyclic Strip Graphs

The q-state Potts antiferromagnet on a lattice $Λ$ exhibits nonzero ground state entropy $S_0=k_B \ln W$ for sufficiently large q and hence is an exception to the third law of thermodynamics. An outstanding challenge has been the calculation of W(sq,q) on the square (sq) lattice. We present here an exact calculation of W on an infinite-length cyclic strip of the square lattice which embodies the expected analytic properties of W(sq,q). Similar results are given for the kagomé lattice.

cond-mat.stat-mech↗

arXiv1999

T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology

We present exact calculations of the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently the chromatic polynomial) for Moebius strips, with width $L_y=2$ or 3, of regular lattices and homeomorphic expansions thereof. These are compared with the corresponding partition functions for strip graphs with (untwisted) periodic longitudinal boundary conditions.

cond-mat.stat-mech↗

arXiv1999

Exact Partition Functions for Potts Antiferromagnets on Cyclic Lattice Strips

We present exact calculations of the zero-temperature partition function of the $q$-state Potts antiferromagnet on arbitrarily long strips of the square, triangular, and kagomé lattices with width $L_y=2$ or 3 vertices and with periodic longitudinal boundary conditions. From these, in the limit of infinite length, we obtain the exact ground-state entropy $S_0=k_B \ln W$. These results are of interest since this model exhibits nonzero ground state entropy $S_0 > 0$ for sufficiently large $q$ and hence is an exception to the third law of thermodynamics. We also include results for homeomorphic expansions of the square lattice strip. The analytic properties of $W(q)$ are determined and related to zeros of the chromatic polynomial for long finite strips.

cond-mat.stat-mech↗

arXiv1999

Collider Signatures from the Brane World

We discuss some collider signatures of the brane world. In addition to the usual bulk (closed string) fields and brane (open string) fields in the Type I string picture, there are closed string fields, namely, twisted modes, which are not confined on the branes but nonetheless are localized in the extra compactified dimensions. While the coupling of the Standard Model (brane) fields with a bulk mode (such as the graviton) is suppressed by powers of the Planck mass, their coupling to a twisted mode is suppressed only by powers of the string scale M_s, which can be as low as a few TeV. This means these localized twisted fields can have important observable effects in the TeV range, including resonances in dijet invariant mass distributions in \bar p p, pp \to jets + X. Given the current lower bound on the fundamental higher-dimensional Planck scale, the experimental effects of these twisted fields may turn out to be larger than the effects of virtual and real KK gravity modes. The collider signatures of anomalous U(1) gauge symmetries as well as other phenomenological implications of the brane world are also discussed.

hep-ph↗

arXiv1999

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 $S_0 > 0$.

cond-mat.stat-mech↗

arXiv1999

New Constraints on Chiral Gauge Theories

Recently, a new constraint on the structure of a wide class of strongly coupled field theories has been proposed. It takes the form of an inequality limiting the number of degrees of freedom in the infrared description of a theory to be no larger than the number of underlying, ultraviolet degrees of freedom. Here we apply this inequality to chiral gauge theories. For some models we find that it is always satisfied, while for others we find that the assumption of the validity of the inequality implies a strong additional restriction on the spectrum of massless composite particles.

hep-th↗

arXiv1999

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 (Möbius) 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.

cond-mat.stat-mech↗

arXiv1998

Ground State Entropy of Potts Antiferromagnets: Homeomorphic Classes with Noncompact W Boundaries

We present exact calculations of the zero-temperature partition function $Z(G,q,T=0)$ and ground-state degeneracy $W(\{G\},q)$ for the $q$-state Potts antiferromagnet on a number of families of graphs $G$ for which (generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$) the boundary ${\cal B}$ of regions of analyticity of $W$ in the complex $q$ plane is noncompact, passing through $z=1/q=0$. For these types of graphs, since the reduced function $W_{red.}=q^{-1}W$ is nonanalytic at $z=0$, there is no large--$q$ Taylor series expansion of $W_{red.}$. The study of these graphs thus gives insight into the conditions for the validity of the large--$q$ expansions. It is shown how such (families of) graphs can be generated from known families by homeomorphic expansion.

cond-mat.stat-mech↗

arXiv1998

Some Remarks on Theories with Large Compact Dimensions and TeV-Scale Quantum Gravity

We comment on some implications of theories with large compactification radii and TeV-scale quantum gravity. These include the behavior of high-energy gravitational scattering cross sections and consequences for ultra-high-energy cosmic rays and neutrino scattering, the question of how to generate naturally light neutrino masses, the issue of quark-lepton unification, the equivalence principle, and the cosmological constant.

hep-ph↗

arXiv1998

Ground State Entropy of Potts Antiferromagnets: Cases with Noncompact W Boundaries Having Multiple Points at 1/q = 0

We present exact calculations of the zero-temperature partition function, $Z(G,q,T=0)$, and ground-state degeneracy (per site), $W({G},q)$, for the $q$-state Potts antiferromagnet on a number of families of graphs ${G}$ for which the boundary ${\cal B}$ of regions of analyticity of $W$ in the complex $q$ plane is noncompact and has the properties that (i) in the $z=1/q$ plane, the point $z=0$ is a multiple point on ${\cal B}$ and (ii) ${\cal B}$ includes support for $Re(q) < 0$. These families are generated by the method of homeomorphic expansion. Our results give further insight into the conditions for the validity of large--$q$ series expansions for the reduced function $W_{red.}=q^{-1}W$.

cond-mat.stat-mech↗

arXiv1998

Chromatic Polynomials for $J(\prod H)I$ Strip Graphs and their Asymptotic Limits

We calculate the chromatic polynomials $P$ for $n$-vertex strip graphs of the form $J(\prod_{\ell=1}^m H)I$, where $J$ and $I$ are various subgraphs on the left and right ends of the strip, whose bulk is comprised of $m$-fold repetitions of a subgraph $H$. The strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction. This extends our earlier calculations for strip graphs of the form $(\prod_{\ell=1}^m H)I$. We use a generating function method. From these results we compute the asymptotic limiting function $W=\lim_{n \to \infty}P^{1/n}$; for $q \in {\mathbb Z}_+$ this has physical significance as the ground state degeneracy per site (exponent of the ground state entropy) of the $q$-state Potts antiferromagnet on the given strip. In the complex $q$ plane, $W$ is an analytic function except on a certain continuous locus ${\cal B}$. In contrast to the $(\prod_{\ell=1}^m H)I$ strip graphs, where ${\cal B}$ (i) is independent of $I$, and (ii) consists of arcs and possible line segments that do not enclose any regions in the $q$ plane, we find that for some $J(\prod_{\ell=1}^m H)I$ strip graphs, ${\cal B}$ (i) does depend on $I$ and $J$, and (ii) can enclose regions in the $q$ plane. Our study elucidates the effects of different end subgraphs $I$ and $J$ and of boundary conditions on the infinite-length limit of the strip graphs.

cond-mat.stat-mech↗

arXiv1998

Ground State Entropy of Potts Antiferromagnets on Homeomorphic Families of Strip Graphs

We present exact calculations of the zero-temperature partition function, and ground-state degeneracy (per site), $W$, for the $q$-state Potts antiferromagnet on a variety of homeomorphic families of planar strip graphs $G = (Ch)_{k_1,k_2,Σ,k,m}$, where $k_1$, $k_2$, $Σ$, and $k$ describe the homeomorphic structure, and $m$ denotes the length of the strip. Several different ways of taking the total number of vertices to infinity, by sending (i) $m \to \infty$ with $k_1$, $k_2$, and $k$ fixed; (ii) $k_1$ and/or $k_2 \to \infty$ with $m$, and $k$ fixed; and (iii) $k \to \infty$ with $m$ and $p=k_1+k_2$ fixed are studied and the respective loci of points ${\cal B}$ where $W$ is nonanalytic in the complex $q$ plane are determined. The ${\cal B}$'s for limit (i) are comprised of arcs which do not enclose regions in the $q$ plane and, for many values of $p$ and $k$, include support for $Re(q) < 0$. The ${\cal B}$ for limits (ii) and (iii) is the unit circle $|q-1|=1$.

cond-mat.stat-mech↗

arXiv1997

Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits

We calculate the chromatic polynomials $P((G_s)_m,q)$ and, from these, the asymptotic limiting functions $W(\{G_s\},q)=\lim_{n \to \infty}P(G_s,q)^{1/n}$ for families of $n$-vertex graphs $(G_s)_m$ comprised of $m$ repeated subgraphs $H$ adjoined to an initial graph $I$. These calculations of $W(\{G_s\},q)$ for infinitely long strips of varying widths yield important insights into properties of $W(Λ,q)$ for two-dimensional lattices $Λ$. In turn, these results connect with statistical mechanics, since $W(Λ,q)$ is the ground state degeneracy of the $q$-state Potts model on the lattice $Λ$. For our calculations, we develop and use a generating function method, which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant $W(\{G_s\},q)$ function in the limit $n \to \infty$. From this, we obtain the exact continuous locus of points ${\cal B}$ where $W(\{G_s\},q)$ is nonanalytic in the complex $q$ plane. This locus is shown to consist of arcs which do not separate the $q$ plane into disconnected regions. Zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities ${\cal B}$. We find that as the width of the infinitely long strips is increased, the arcs comprising ${\cal B}$ elongate and move toward each other, which enables one to understand the origin of closed regions that result for the (infinite) 2D lattice.

cond-mat.stat-mech↗

arXiv1997

Complex-Temperature Partition Function Zeros of the Potts Model on the Honeycomb and Kagomé Lattices

We calculate complex-temperature (CT) zeros of the partition function for the $q$-state Potts model on the honeycomb and kagomé lattices for several values of $q$. These give information on the CT phase diagrams. A comparison of results obtained for different boundary conditions and a discussion of some CT singularities are given. Among other results, our findings show that the Potts antiferromagnet with $q=4$ and $q=5$ on the kagomé lattice has no phase transition at either finite or zero temperature.

cond-mat.stat-mech↗

arXiv1997

A Mapping Relating Complex and Physical Temperatures in the 2D $q$-state Potts Model and Some Applications

We show an exact equivalence of the free energy of the $q$-state Potts antiferromagnet on a lattice $Λ$ for the full temperature interval $0 \le T \le \infty$ and the free energy of the $q$-state Potts model on the dual lattice for a semi-infinite interval of complex temperatures (CT). This implies the existence of two quite different types of CT singularities: the generic kind, which does not obey universality or various scaling relations, and a special kind which does obey such properties and encodes information of direct physical relevance. We apply this observation to characterize CT properties of the Potts model on several lattices, to rule out two existing conjectures, and to determine the critical value of $q$ above which the Potts antiferromagnet on the diced lattice has no phase transition.

cond-mat.stat-mech↗