Electrostatics in Periodic Slab Geometries I

We propose a new method to sum up electrostatic interactions in 2D slab geometries. It consists of a combination of two recently proposed methods, the 3D Ewald variant of Yeh and Berkowitz, J. Chem. Phys. 111 (1999) 3155, and the purely 2D method MMM2D by Arnold and Holm, to appear in Chem. Phys. Lett. 2002. The basic idea involves two steps. First we use a three dimensional summation method whose summation order is changed to sum up the interactions in a slab-wise fashion. Second we subtract the unwanted interactions with the replicated layers analytically. The resulting method has full control over the introduced errors. The time to evaluate the layer correction term scales linearly with the number of charges, so that the full method scales like an ordinary 3D Ewald method, with an almost linear scaling in a mesh based implementation. In this paper we will introduce the basic ideas, derive the layer correction term and numerically verify our analytical results.Comment: 10 pages, 7 figure

Electrostatics in Periodic Slab Geometries II

In a previous paper a method was developed to subtract the interactions due to periodically replicated charges (or other long-range entities) in one spatial dimension. The method constitutes a generalized "electrostatic layer correction" (ELC) which adapts any standard 3D summation method to slab-like conditions. Here the implementation of the layer correction is considered in detail for the standard Ewald (EW3DLC) and the PPPM mesh Ewald (PPPMLC) methods. In particular this method offers a strong control on the accuracy and an improved computational complexity of O(N log N) for mesh-based implementations. We derive anisotropic Ewald error formulas and give some fundamental guidelines for optimization. A demonstration of the accuracy, error formulas and computation times for typical systems is also presented.Comment: 14 pages, 7 figure

Bipartite induced density in triangle-free graphs

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.Comment: 20 pages; in v2 added note of concurrent work and one reference; in v3 added more notes of ensuing work and a result towards one of the conjectures (for list colouring

Gluon mass generation in the massless bound-state formalism

We present a detailed, all-order study of gluon mass generation within the massless bound-state formalism, which constitutes the general framework for the systematic implementation of the Schwinger mechanism in non-Abelian gauge theories. The main ingredient of this formalism is the dynamical formation of bound states with vanishing mass, which give rise to effective vertices containing massless poles; these latter vertices, in turn, trigger the Schwinger mechanism, and allow for the gauge-invariant generation of an effective gluon mass. This particular approach has the conceptual advantage of relating the gluon mass directly to quantities that are intrinsic to the bound-state formation itself, such as the 'transition amplitude'' and the corresponding 'bound-state wave function.'' As a result, the dynamical evolution of the gluon mass is largely determined by a Bethe-Salpeter equation that controls the dynamics of the relevant wave function, rather than the Schwinger-Dyson equation of the gluon propagator, as happens in the standard treatment. The precise structure and field-theoretic properties of the transition amplitude are scrutinized in a variety of independent ways. In particular, a parallel study within the linear-covariant (Landau) gauge and the background-field method reveals that a powerful identity, known to be valid at the level of conventional Green's functions, also relates the background and quantum transition amplitudes. Despite the differences in the ingredients and terminology employed, the massless bound-state formalism is absolutely equivalent to the standard approach based on Schwinger-Dyson equations. In fact, a set of powerful relations allows one to demonstrate the exact coincidence of the integral equations governing the momentum evolution of the gluon mass in both frameworks

An improved procedure for colouring graphs of bounded local density

We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The leading term in this bound is best possible as $\sigma\to0$. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erd\H{o}s-Ne\v{s}et\v{r}il conjecture and Reed's conjecture. We prove that the strong chromatic index is at most $1.772\Delta^2$ for any graph $G$ with sufficiently large maximum degree $\Delta$. We prove that the chromatic number is at most $\lceil 0.801(\Delta+1)+0.199\omega\rceil$ for any graph $G$ with clique number $\omega$ and sufficiently large maximum degree $\Delta$.Comment: 21 page

QCD effective charge from the three-gluon vertex of the background-field method

In this article we study in detail the prospects of determining the infrared finite QCD effective charge from a special kinematic limit of the vertex function corresponding to three background gluons. This particular Green's function satisfies a QED-like Ward identity, relating it to the gluon propagator, with no reference to the ghost sector. Consequently, its longitudinal form factors may be expressed entirely in terms of the corresponding gluon wave function, whose inverse is proportional to the effective charge. After reviewing certain important theoretical properties, we consider a typical lattice quantity involving this vertex, and derive its exact dependence on the various form factors, for arbitrary momenta. We then focus on the particular momentum configuration that eliminates any dependence on the (unknown) transverse form factors, projecting out only the desired quantity. A preliminary numerical analysis indicates that the effective charge is relatively insensitive to the numerical uncertainties that may afflict future simulations of the aforementioned lattice quantity. The numerical difficulties associated with a parallel determination of the dynamical gluon mass are briefly discussed