CRITICAL FINITE-RANGE SCALING IN SCALAR-FIELD THEORIES AND ISING-MODELS

We develop a critical finite-force-range scaling theory for D-dimensio nal scalar phi(n) field theories that is based on a scaling ansatz equ ivalent to a Ginzburg criterion. To investigate its relationship to ot her scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak l ong-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension D(c), we arrive at a criterion of critical eq uivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a phi4 Ginzburg-Landau Ham iltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N X infinity quasi-one-dimensional Ising (Q1 DI) model, in which each spin interacts with O(N) others. Near the Gau ssian mean-field critical point the Ginzburg-Landau Hamiltonians for a ll three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D = 1 quartic field theory, and for the Q1DI model the y are those of a cylindrical Ising system above D(c). Results of speci alized numerical scaling techniques applied to transfer-matrix calcula tions for the Q1DI model with N less-than-or-equal-to 1024 strongly su pport our theoretically obtained scaling relations.