The cluster expansion for potentials with exponential fall-off

Continuing the work of a previous paper, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by potentials that fall off exponentially at large distance. The HamiltonianH ₀+V need be stable in the extended sense thatH ₀+4V+BN0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. A natural system included in the present treatment is that of matter with ther ^–1 potential replaced bye ^–ar/r. The Hamiltonian is stable, but the system would collapse in the absence of the exclusion principle—the potential is unstable. Therefore this system cannot be handled by the classic work of Ginibre, which requires stable potentials.This work was supported in part by NSF Grant MPS 75-10751Michigan Junior Fellow     

The mass gap for Higgs models on a unit lattice

An isomorphism is established between eertain compact and noncompact formulations of Abelian gauge theory on a lattice. For weak coupling, the mass gap predicted by the Higgs mechanism is then established.     

Preparation of ion-selective membranes from crosslinked copolymers of styrene and p-vinylbenzenesulfonic acid

Ion-selective membranes similar to those described by Graydon and Stewart [J. Phys. Chem. 59 , 86 (1955)] were prepared by improved methods involving copolymerization of styrene, n-propyl-p-vinylbenzenesulfonate, and divinylbenzene (DVB) with subsequent hydrolysis of the ester. Both the highly reactive ester and DVB were purified by column chromatography and were well characterized by their infrared and proton magnetic resonance spectra. For preparation of membranes about 25% by weight of N,N-dimethylformamide (DMF) was included with the vinyl monomers in the polymerization mixture. Thermal polymerization was initiated by benzoyl peroxide, and membranes were formed in special molds of two types. In one type a free surface of the mixture was exposed, while in the other the mixture filled the mold. The DMF in the fresh membranes was replaced with aqueous KOH by diffusion, and hydrolysis was then carried out. By this method hydrolysis times were decreased from several days to a few hours. Capacity, water content, and thickness were determined for all membranes, and density, swelling in salt solutions, and Young's modulus were measured for selected membranes. The Vermaas-Hermans-Flory theory of polyelectrolyte gels was found to account satisfactorily for the swelling properties and water content of these membranes, the solvent interaction parameter χ1 increasing with increasing volume fraction of polymer from 0.7 to 1.0 at zero external salt concentration and up to 1.8 for membranes deswollen in 1M KCl. No differences in average water contents for given densities of fixed charge and crosslinking were observed with (1) different molds, (2) commercial or pure DVB, and (3) a swelling agent (DMF) or no swelling agent. The results in general indicate that the state of the membrane at a given temperature and pressure can be specified by any two of the parameters: degree of ionization per unit volume of an average structural unit, concentration of crosslinking agent, and degree of swelling. Young's modulus decreased markedly with increasing water content, but data were not sufficiently extensive or precise to permit a quantitative treatment.     

Erratum: Mayer Expansions and the Hamiltonian–Jacobi Equation. II. Fermions,Dimensional Reduction Formulas

The authors apologize to readers for an egregious error in the proof of Theorem 2.3: contrary to our claim, (2.12) does not follow from Proposition 2.2 and Theorem 2.3 is almost certainly false. We are indebted to Manfred Salmhofer and Christian Wieczerkowski for discovering the error. They are preparing a paper with a corrected theorem proven by different methods. The work in Section 3 is independent of Section 2 and Theorem 3.1 is unaffected.     

Scaling Limits and Critical Behaviour of the 4-Dimensional n-Component |\varphi |^4 Spin Model

We consider the \(n\) -component \(|\varphi |^4\) spin model on \({\mathbb {Z}}^4\) , for all \(n \ge 1\) , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent \(\frac{n+2}{n+8}\) for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for \(n =1,2,3\) ; double logarithmic scaling for \(n=4\) ; and is bounded when \(n>4\) . In addition, for the model defined on the \(4\) -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the \(|\varphi |^4\) model.     

Convergence of Mayer expansions

The tree graph bound of Battle and Federbush is extended and used to provide a simple criterion for the convergence of (iterated) Mayer expansions. As an application estimates on the radius of convergence of the Mayer expansion for the two-dimensional Yukawa gas (nonstable interaction) are obtained.     

Critical (Φ4)3,ε

The Euclidean (⁴)_3, model in R³ corresponds to a perturbation by a ⁴ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter in the range 01. For =1 one recovers the covariance of a massless scalar field in R³. For =0, ⁴ is a marginal interaction. For 0<1 the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for >0, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point. Partially supported by NSERC of CanadaLaboratoire Associé au CNRS. UMR 5825Partially supported by CNR, G.N.F.M. and MURST