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Generating configurations of increasing lattice size with machine learning and the inverse renormalization group

Dimitrios Bachtis

TL;DR

The paper addresses the challenge of generating equilibrated configurations for large lattice volumes without encountering critical slowing down. It surveys the inverse renormalization group (IRG), emphasizing CNN- and wavelet-based inverse transformations that iteratively enlarge lattice size while preserving equilibrium properties, with demonstrations in phi^4 lattice field theory and the Edwards-Anderson spin glass. The results show IRG can locate critical fixed points via observable intersections and extract critical exponents consistent with known universality classes, potentially reaching lattice volumes beyond current supercomputers. The work suggests future directions including incorporating numerical exactness and applying IRG to phase transitions encountered during machine-learning training, broadening the method's applicability in statistical mechanics and disordered systems.

Abstract

We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations. Inverse renormalization group methods enable the iterative generation of configurations for increasing lattice size without the critical slowing down effect. We discuss the construction of inverse renormalization group transformations with the use of convolutional neural networks and present applications in models of statistical mechanics, lattice field theory, and disordered systems. We highlight the case of the three-dimensional Edwards-Anderson spin glass, where the inverse renormalization group can be employed to construct configurations for lattice volumes that have not yet been accessed by dedicated supercomputers.

Generating configurations of increasing lattice size with machine learning and the inverse renormalization group

TL;DR

The paper addresses the challenge of generating equilibrated configurations for large lattice volumes without encountering critical slowing down. It surveys the inverse renormalization group (IRG), emphasizing CNN- and wavelet-based inverse transformations that iteratively enlarge lattice size while preserving equilibrium properties, with demonstrations in phi^4 lattice field theory and the Edwards-Anderson spin glass. The results show IRG can locate critical fixed points via observable intersections and extract critical exponents consistent with known universality classes, potentially reaching lattice volumes beyond current supercomputers. The work suggests future directions including incorporating numerical exactness and applying IRG to phase transitions encountered during machine-learning training, broadening the method's applicability in statistical mechanics and disordered systems.

Abstract

We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations. Inverse renormalization group methods enable the iterative generation of configurations for increasing lattice size without the critical slowing down effect. We discuss the construction of inverse renormalization group transformations with the use of convolutional neural networks and present applications in models of statistical mechanics, lattice field theory, and disordered systems. We highlight the case of the three-dimensional Edwards-Anderson spin glass, where the inverse renormalization group can be employed to construct configurations for lattice volumes that have not yet been accessed by dedicated supercomputers.
Paper Structure (5 sections, 4 equations, 5 figures)

This paper contains 5 sections, 4 equations, 5 figures.

Table of Contents

  1. Inverse Monte Carlo renormalization group
  2. Inverse renormalization group of quantum field theories
  3. Inverse renormalization group of disordered systems
  4. Conclusions
  5. Acknowledgements

Figures (5)

  • Figure 1: Illustration of the application of a standard renormalization group transformation (RG) with the majority rule and its approximate inversion (IRG) with the implementation of machine learning algorithms.
  • Figure 2: Absolute value of the magnetization versus the squared mass for an original $L$ and a renormalized $L'$ system of identical lattice size. The renormalized system has been obtained via the application of a standard renormalization group transformation. Figure from Ref. PhysRevLett.128.081603.
  • Figure 3: Absolute value of the magnetization versus the squared mass for an original $L$ and a renormalized $L'$ system of identical lattice size. The renormalized systems have been obtained via the application of an inverse renormalization group transformation. Figure from Ref. PhysRevLett.128.081603.
  • Figure 4: The transition to the overlap configurations for the case of the three-dimensional Edwards-Anderson model and the implementation of a standard renormalization group transformation on the effective spin glass which comprises overlap degrees of freedom. Figure from Ref bachtis2023inverse.
  • Figure 5: The iterative application of inverse renormalization group transformations for the case of the three-dimensional Edwards-Anderson. Starting from a lattice size of $L=16$ we apply the inverse transformations until we construct $L'=128$. Figure from Ref bachtis2023inverse.