On the Symmetry of Real-Space Renormalisation

TL;DR

This paper investigates the geometric and symmetry properties of real-space renormalisation group flow, revealing a gradient structure with no global geodesic symmetry, exemplified by the 1D Ising model in an external field.

Contribution

It introduces a geometric framework for RG flow using a Hilbert space embedding and analyzes symmetry properties, including explicit beta function expressions for the 1D Ising model.

Findings

RG flow has a gradient structure with a projective automorphism.

No global geodesic symmetry exists in the RG flow on parameter space.

Approximate conformal symmetry appears near the critical point.

Abstract

A natural geometry, arising from the embedding into a Hilbert space of the parametrised probability measure for a given lattice model, is used to study the symmetry properties of real-space renormalisation group (RG) flow. In the projective state space this flow is shown to have two contributions: a gradient term, which generates a projective automorphism of the state space for each given length scale; and an explicit correction. We then argue that this structure implies the absence of any symmetry of a geodesic type for the RG flow when restricted to the parameter space submanifold of the state space. This is demonstrated explicitly via a study of the one dimensional Ising model in an external field. In this example we construct exact expressions for the beta functions associated with the flow induced by infinitesimal rescaling. These constitute a generating vector field for RG diffeomorphisms on the parameter space manifold, and we analyse the symmetry properties of this transformation. The results indicate an approximate conformal Killing symmetry near the critical point, but no generic symmetry of the RG flow globally on the parameter space.