14 - Phase transitions: the renormalization group approach

Abstract

The renormalization group approach is a powerful calculational tool for studying phase transitions and critical phenomena. The approach uses a partial trace over the interacting degrees of freedom which leads to a length transformation or change of scale that reduces all length scales in the system, including the correlation length. The rescaled system can be described by a similar system but with renormalized couplings. If the system is at a critical point with divergent correlation length, then renormalized couplings are unchanged by the rescaling, resulting in a fixed point of the transformation. Linear expansions of the renormalized couplings about the fixed point allows the critical exponents and scaling functions to be calculated. The renormalization group provides a theoretical framework for understanding universality classes of critical points. Applying the renormalization group as an expansion in the dimensionality about four dimensions in the ε-expansion allows the calculation of critical exponents as functions of spatial and spin dimensionality.