A new approach to the critical behaviour of systems exhibiting a dimensional crossover

Abstract

The critical behaviour of systems which exhibit a dimensional crossover is studied. The key new ingredient being an explicitly L dependent renormalization group, L being the characteristic “size” of the system, and a set of renormalization schemes which lead to manifest dimensional reduction in the loop expansion. As a paradigm we study an Ising-like system on S 1×R 3−ϵ, (0≤ϵ≤1). t = T−T c(L) is determined to be the relevant variable rather than T−T c(∞), and a perturbative expansion about the L dependent reduced fixed point performed as opposed to the bulk fixed point. A running coupling constant, effective susceptibility and correlation length exponents are defined and computed to lowest order in perturbation theory. These interpolate between the behaviours expected of an Ising model in 4−ϵ and 3−ϵ dimensions as tL 2→∞( t→0), and tL 2→0, respectively. A scaling law γ eff = ν eff (2− η eff ) is conjectured and shown to hold to lowest order in perturbation theory. The formalism generalizes to higher orders.