Renormalisation Group Flow and Geodesics in the O(N) Model for Large N

TL;DR

This paper introduces a geometric framework for the large N limit of the O(N) model, analyzing the parameter space metric and its curvature, and explores the renormalisation group flow as geodesics, revealing insights into fixed points and flow trajectories.

Contribution

It develops a geometric approach to the renormalisation group flow in the O(N) model, linking geodesics to flow trajectories and entropy maximisation, which is a novel perspective.

Findings

Ricci curvature diverges at the Gaussian fixed point

Curvature tends to a negative constant at the Wilson-Fisher fixed point

The critical crossover line is a geodesic in the parameter space

Abstract

A metric is introduced on the space of parameters (couplings) describing the large N limit of the O(N) model in Euclidean space. The geometry associated with this metric is analysed in the particular case of the infinite volume limit in 3 dimensions and it is shown that the Ricci curvature diverges at the ultra-violet (Gaussian) fixed point but is finite and tends to constant negative curvature at the infra-red (Wilson-Fisher) fixed point. The renormalisation group flow is examined in terms of geodesics of the metric. The critical line of cross-over from the Wilson-Fisher fixed point to the Gaussian fixed point is shown to be a geodesic but all other renormalisation group trajectories, which are repulsed from the Gaussian fixed point in the ultra-violet, are not geodesics. The geodesic flow is interpreted in terms of a maximisation principle for the relative entropy.