Renormalization Group as a Koopman Operator

2019/12/30 by William T. Redman, William T Redman · 1 voice · 5 citations
Mathematics · Physics and Astronomy · #Connection (principal bundle) #Critical dimension #Critical exponent #Critical phenomena #Critical point (mathematics) #Fixed point #Functional renormalization group #Group (periodic table) #Infrared fixed point #Mathematical analysis #Mathematical physics #Mathematics #Model Reduction and Neural Networks #Observable #Operator (biology) #Phase transition #Physics #Quantum many-body systems #Quantum mechanics #Quantum, superfluid, helium dynamics #Renormalization #Renormalization group #Universality (dynamical systems) #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.101.060104

published in Physical review. E 101(6), 060104 (American Physical Society) · 13 pages, 1 figure

arxiv published 2019/12/30 · arxiv created 2020/06/09 · openalex publication_date 2020/06/19 · arxiv updated 2020/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Koopman operator theory is shown to be directly related to the renormalization group. This observation allows us, with no assumption of translational invariance, to compute the critical exponents η and δ, as well as ratios of critical exponents, of classical spin systems from single observables alone. This broadens the types of problems that the renormalization group framework can be applied to and establish universality classes of. In addition, this connection may allow for a new, data-driven way in which to find the renormalization group fixed point(s), and their relevant and irrelevant directions.

Citations

• Lectures on Phase Transitions and the Renormalization Group
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• An exact mapping between the Variational Renormalization Group and Deep\n Learning
• Monte Carlo Methods
• The renormalization group: Critical phenomena and the Kondo problem
• Dynamical Systems of Continuous Spectra
• Spectral Properties of Dynamical Systems, Model Reduction and Decompositions
• Spectral analysis of nonlinear flows
• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory
• Applied Koopmanism

Discussions

• Related? arxiv.org/abs/1912.13010 [bsky, 1 points, 0 comments]

Related

• The renormalization group: Critical phenomena and the Kondo problem
• The Renormalization Group
• The Renormalization-Group peculiarities of Griffiths and Pearce: What have we learned?
• The density matrix renormalisation group and critical phenomena
• Universal Ratios of the Renormalization Group