Renormalisation group flow and geodesics in the O( N) model for large N

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 three 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 ultraviolet, are not geodesics. The geodesic flow is interpreted in terms of a maximisation principle for the relative entropy.