Phase transitions and dimensional reduction

Using field theoretic methods a formalism is presented within which the critical behaviour of a system undergoing a dimensional reduction may be investigated. As a paradigm we study an Ising-like system on S1 x R3-ε. If the size of the system is L, and the correlation length ξ, then as L/ξ varies it is possible to get critical behaviour associated with two different fixed points. By exploiting a set of renormalization schemes which lead to manifest dimensional reduction in the loop expansion, and utilizing the renormalization group and an expansion about the fixed point of the finite system, we quantitatively investigate such crossover behaviour in its entirety. In particular, effective susceptibility and correlation length exponents are defined and computed. These exponents interpolate between those associated with a (4-ε)-dimensional and a (3-ε)-dimensional Ising model. (orig.)