The exact result for the free energy per particle in systems described by a hamiltonian of the type NP(VN), where NVN denotes a set of short-range operators, is reformulated in terms of a convex-envelope construction. A comparison is given with results obtained for classical systems with...

In this paper we give an exact evaluation of the free energy per particle for systems with separable many-particle interactions described by a hamiltonian of the type H = ∑kT(k) + NP (N-1 ∑k V(k)), where P is an arbitrary polynomial. In the proof use is made of a fundamental theorem due to...

The critical behaviour of a reference system with short-range interactions under small perturbations is investigated in detail for the special case of two (relevant) thermodynamic fields h1 and h2. As reference system we choose Schofield's linear model and we consider perturbations of the type...

The treatment of a previous paper on systems with many-particle interactions is generalized to hamiltonians containing an analytic function of a number of short-range interaction operators V, which are normalized. An exact expression for the free energy per particle in the thermodynamic limit...

The critical behaviour of systems with short-range interactions in which the free energy has divergent second derivatives is shown to be unstable under small perturbations. The perturbations can arise from additional terms in the hamiltonian with a long-range nature, but also from hidden...

A particular class of periodic Ising models with diluted frustration is studied using a simple recursion relation for spinor correlation functions. A modulated phase is found at T = 0 which extends into a modulated paramagnetic phase for finite temperatures. The disorder line at which this...

The variational principle is presented for a continuous renormalization transformation. It is shown that a strict application of the variational method leads to a singularity in the variational parameter which is related to that in the free energy. The equations governing the behaviour of the...

A real space renormalization is applied to the Ising model for d ⩾ 3, allowing for the generation of vacancies. A fixed point structure is obtained that reflects the cross-over to classical behaviour as known from ϵ-expansion. The exponents found are an improvement over those obtained by the...

In this note it is shown that gradients of bounded operators do not contribute to singularities in the bulk free energy. The formal scaling index of these operators describes at best, depending on the boundary conditions, singularities in the surface free energy. This observation is used in the...

The fixed point structure resulting from the approximate renormalization group equations obtained by shifting bonds on the square Ising lattice is considered as a function of a free parameter h appearing in the definition of these equations. Next to the fixed point S considered by Kadanoff which...