Chapter 13 - Phase Transitions: The Renormalization Group Approach

Abstract

This chapter discusses the use of renormalization groups to tackle the problem of phase transitions. The main point of this formulism is that, as the critical point of a system is approached, its correlation length becomes exceedingly large—with the result that the sensitivity of the system to a length transformation gets exceedingly diminished. At the critical point itself, the correlation length becomes infinitely large and with it the system becomes totally insensitive to such a transformation. It is then conceivable that if one is not too far removed from the critical point, the given system with lattice constant a may bear a close resemblance to a transformed system, renormalized so that all distances in it are measured in terms of the new lattice constant. These considerations lead to a convincing argument based on the role played by correlations among the microscopic constituents of the system, which, in the vicinity of the critical point, are so large-scale that they make all other length scales, including the one that determines the structure of the lattice, essentially irrelevant. The chapter discusses finite-size scaling of lattice size, emphasizing the fact that finite-size effects in a given system are quite sensitive to the choice of the boundary conditions imposed on the system.