On the Symmetry of Real-Space Renormalisation

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.

16 pages, RevTeX, 3 postscript figures

References in corpus (3)

• Exact evolution equation for the effective potential
• On Truncations of the Exact Renormalization Group
• Geodesic Renormalisation Group Flow

Cited by in corpus (17)

• Geometric Quantum Mechanics
• Quantum information and physics: some future directions
• Information geometry in vapour-liquid equilibrium
• The Information Geometry of the Spherical Model
• Information geometric approach to the renormalisation group
• The Information Geometry of the One-Dimensional Potts Model
• The Information Geometry of the Ising Model on Planar Random Graphs
• Thermodynamics and Thermodynamic geometry of Park black hole
• The renormalisation group via statistical inference
• Geometrothermodynamics of the Kehagias-Sfetsos Black Hole
• Para-Sasakian geometry in thermodynamic fluctuation theory
• Geometrothermodynamics in Horava-Lifshitz gravity
• A unified thermodynamic picture of Hořava-Lifshitz black hole in arbitrary space time
• Renormalization of composite operators
• Thermodynamic Geometry of the Born-Infeld-anti-de Sitter black holes
• The geometry of RG flows in theory space
• Stochastic formulation of the renormalization group: supersymmetric structure and topology of the space of couplings