Information Geometry and the Renormalization Group

TL;DR

This paper demonstrates how renormalization group flow equations can be used to construct and analyze the information metric near critical points in classical and quantum systems, revealing universal scaling properties.

Contribution

It introduces a universal method to derive the information metric from RG flow equations near criticality for both classical and quantum systems.

Findings

Establishment of the scaling properties of the information metric.

Identification of scaling exponents related to critical phenomena.

Interpretation of scalar curvature and geodesic distance in the context of RG.

Abstract

Information theoretic geometry near critical points in classical and quantum systems is well understood for exactly solvable systems. Here we show that renormalization group flow equations can be used to construct the information metric and its associated quantities near criticality, for both classical and quantum systems, in an universal manner. We study this metric in various cases and establish its scaling properties in several generic examples. Scaling relations on the parameter manifold involving scalar quantities are studied, and scaling exponents are identified. The meaning of the scalar curvature and the invariant geodesic distance in information geometry is established and substantiated from a renormalization group perspective.