Geodesic Renormalisation Group Flow

TL;DR

This paper interprets the renormalisation group flow as a dynamical system akin to viscous fluid flow, revealing geodesic behavior in specific models and limits, thus providing a geometric perspective on RG flows.

Contribution

It introduces a geometric interpretation of RG flow as a geodesic or viscous fluid flow in coupling space, applicable to scalar theories and the Ising model.

Findings

RG flow in free scalar theory is geodesic in 2D

Flow is approximately geodesic in certain limits for D ≠ 2

RG flow in 1D Ising model is geodesic with zero external field

Abstract

It is shown that the renormalisation group flow in coupling constant space can be interpreted in terms of a dynamical equation for the couplings analogous to viscous fluid flow under the action of a potential. For free scalar field theory the flow is geodesic in two dimensions, while for $D \neq 2$ it is only geodesic in certain limits, e.g. for vanishing external source. For the 1-D Ising model the renormalisation flow is geodesic when the external magnetic field vanishes.