DIMENSIONAL CROSSOVER AND FINITE-SIZE-SCALING BELOW TC

Using the formalism developed in earlier work, dimensional crossover on a d-dimensional layered Ising-type system satisfying periodic boundary conditions and of size L is considered below T(c)(L), T(c)(L) being the critical temperature of the finite-size system. Effective critical exponents delta(eff) and beta(eff) are shown explicitly to crossover between their d- and (d-1)-dimensional values for xi(L) --> infinity in the limits L/xi(L) --> infinity and L/xi(L) --> 0, respectively, xi(L) being the correlation length in the layers. Using an L-dependent renormalization group, the effective exponents are shown to satisfy natural generalizations of the standard scaling laws. In addition, L-dependent global scaling fields which span the entire crossover ate defined and a scaling form of the equation of state in terms of them derived. All the above assertions are verified explicitly to one loop in perturbation theory, in particular effective exponents and a universal crossover equation of state are obtained and shown in the above asymptotic limits to be in good agreement with known results.