Critical finite-range scaling in scalar-field theories and Ising models

We develop a critical finite-force-range scaling theory for D-dimensional scalar ${\mathrm{\phi }}^{\mathit{n}}$ field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension ${\mathit{D}}_{\mathit{c}}$, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a ${\mathrm{\phi }}^4$ Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N×∞ quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all 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 they are those of a cylindrical Ising system above ${\mathit{D}}_{\mathit{c}}$. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N≤1024 strongly support our theoretically obtained scaling relations.