“ENVIRONMENTALLY FRIENDLY” RENORMALIZATION

Abstract

We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are “environment”-dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions and external fields. The points in the space of “coupling constants” at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment-dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale g. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, deff=4−γλ, which represents the effects of the leading irrelevant operator. We analyze the crossover of an O(N) model on a d-dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2/1] Padé-resummed coupling, for: the “Gaussian model” (N=−2), spherical model (N=∞), Ising model (N=1), polymers (N=0), XY model (N=2) and Heisenberg (N=3) models in four dimensions. We also give two-loop Padé resummed results for a three-dimensional Ising ferromagnet in a transverse magnetic field and corresponding one-loop results for the two-dimensional model. One-loop results are also presented for a three-dimensional layered Ising model with periodic, Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.