Environmentally Friendly Renormalization

TL;DR

This paper develops a environment-dependent renormalization group framework to analyze crossover phenomena and effective critical exponents in various models, incorporating boundary conditions and external parameters.

Contribution

It introduces a formal two-loop renormalization approach for environment-dependent systems, providing explicit results for effective exponents across different models and boundary conditions.

Findings

Effective exponents obey scaling laws across crossovers.

Explicit two-loop results for multiple models and boundary conditions.

Asymptotic results agree with known critical exponents.

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 \lq\lq 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, $d\ef=4-\gl$, 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] Pad\'e resummed coupling, for: the ``Gaussian model'' ($N=-2$), spherical model ($N=\infty$), Ising Model ($N=1$), polymers ($N=0$), XY-model ($N=2$) and Heisenberg ($N=3$) models in four dimensions. We also give two loop Pad\'e 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 Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.