ENVIRONMENTALLY FRIENDLY RENORMALIZATION

We analyze the renormalization of systems whose effective degrees of f reedom are described in terms of fluctuations which are ''environment' '-dependent. Relevant environmental parameters considered are: tempera ture, 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 renor malization group which is necessarily environment-dependent. Using suc h a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a rele vant 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, d(eff) = 4 - gamma(lambda), which rep resents the effects of the leading irrelevant operator. We analyze the crossover of an O(N) model on a d-dimensional layered geometry with p eriodic, antiperiodic and Dirichlet boundary conditions. Explicit resu lts to two loops for effective exponents are obtained using a [2/1] Pa de-resummed coupling, for: the ''Gaussian model'' (N = -2), spherical model (N = infinity), Ising model (N = 1), polymers (N = 0), XY model (N = 2) and Heisenberg (N = 3) models in four dimensions. We also give two-loop Pade resummed results for a three-dimensional Ising ferromag net in & 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 exponen ts are in excellent agreement with known results.