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arXiv:2310.12631 · Physical review. B./Physical review. B

Inverse renormalization group of spin glasses

Dimitrios Bachtis

Published Oct 29, 2024• 4 citations
arXivHTMLPDF

Abstract

We propose inverse renormalization group transformations to construct approximate configurations for lattice volumes that have not yet been accessed by supercomputers or large-scale simulations in the study of spin glasses. Specifically, starting from lattices of volume $V={8}^{3}$ in the case of the three-dimensional Edwards-Anderson model we employ machine learning algorithms to construct rescaled lattices up to ${V}^{\ensuremath{'}}={128}^{3}$, which we utilize to extract two critical exponents. We conclude by discussing how to incorporate numerical exactness within inverse renormalization group methods of disordered systems, thus opening up the opportunity to explore a sustainable and energy-efficient generation of exact configurations for increasing lattice volumes without the use of dedicated supercomputers.

DOI
10.1103/physrevb.110.l140202

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We propose inverse renormalization group transformations to construct approximate configurations for lattice volumes that have not yet been accessed by supercomputers or large-scale simulations in the study of spin glasses. Specifically, starting from lattices of volume V=83V={8}^{3}V=83 in the case of the three-dimensional Edwards-Anderson model we employ machine learning algorithms to construct rescaled lattices up to V\ensuremath′=1283{V}^{\ensuremath{'}}={128}^{3}V\ensuremath′=1283, which we utilize to extract two critical exponents. We conclude by discussing how to incorporate numerical exactness within inverse renormalization group methods of disordered systems, thus opening up the opportunity to explore a sustainable and energy-efficient generation of exact configurations for increasing lattice volumes without the use of dedicated supercomputers.
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.
We propose inverse renormalization group transformations within the context of quantum field theory that produce the appropriate critical fixed point structure, give rise to inverse flows in parameter space, and evade the critical slowing down effect in calculations pertinent to criticality. Given configurations of the two-dimensional ϕ^{4} scalar field theory on sizes as small as V=8^{2}, we apply the inverse transformations to produce rescaled systems of size up to V^{'}=512^{2} which we utilize to extract two critical exponents. We conclude by discussing how the approach is generally applicable to any method that successfully produces configurations from a statistical ensemble and how it can give novel insights into the structure of the renormalization group.
The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S= 1 / 2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, d c , for spin glass order is slightly greater than 2, although possibly d c =2 exactly. The specific heat is calculated and found to exhibit a broad maximum above T c , with no discernable singularity at the transition since the exponent alpha is large and negative.
The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin- 1 / 2 Ising model. The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0. For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature. A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition. It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random J ij matrix whereas the latter has at least some localized eigenvectors.
A renormalization group transformation suitable for spin glasses, and more generally for disordered systems, is presented. The approach is non-standard for two reasons. 1. The coarse graining transformation is performed on the distribution of site overlaps, which clearly is not a Boltzmann-Gibbs distribution. 2. The additional interactions are realized introducing a gauge-field disorder distribution, while the Hamiltonian is kept fixed. Universality classes are thus naturally defined on a large set of models, going from ℤ2 and Gaussian spin glasses to Ising and fully frustrated models, and others. The role of frustration and of the deconfinement transition can also be analysed in terms of fixed points. The proposed analysis is tested numerically on the ℤ2 Edwards-Anderson model in d=4. Good estimates of the critical index ν and of Tc are obtained, and an RG flow diagram is sketched for the first time.
We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.