Renormalization-Group Theory of the Heisenberg Model in d Dimensions

TL;DR

This paper applies renormalization-group theory to analyze the classical Heisenberg model across different spatial dimensions, calculating phase transition properties and revealing the nature of ordered phases.

Contribution

It introduces a Fourier-Legendre expansion approach to solve the Heisenberg model in arbitrary dimensions, recovering known results and exploring phase behavior.

Findings

Exact solution in d=1 matches Fisher's result

No ordered phase in d=2, consistent with theory

Ordered phase appears for d>2

Abstract

The classical Heisenberg model has been solved in spatial d dimensins, exactly in d=1 and by the Migdal-Kadanoff approximation in d>1, by using a Fourier-Legendre expansion. The phase transition temperatures, the energy densities, and the specific heats are calculated in arbitrary dimension d. Fisher's exact result is recovered in d=1. The absence of an ordered phase, conventional or algebraic (in contrast to the XY model yielding an algebraically ordered phase), is recovered in d=2. A conventionally ordered phase occurs at d>2. This method opens the way to complex-system calculations with Heisenberg local degrees of freedom.