Dimensional crossover and finite-size scaling belowT c

Using the formalism developed in earlier work, dimensional crossover on ad-dimensional layered Ising-type system satisfying periodic boundary conditions and of sizeL is considered belowT _c (L), T _c (L) being the critical temperature of the finite-size system. Effective critical exponents _eff and _eff are shown explicitly to crossover between theird- and (d–1)-dimensional values for _L in the limitsL/ _L andL/ _L 0, respectively, _L , being the correlation length in the layers. Using anL-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 are 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.