Inverse renormalization group of spin glasses

TL;DR

This work proposes 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, and discusses how to incorporate numerical exactness within inverse renormalization group methods of disordered systems.

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 $V'=128^{3}$, 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.

Figures (5)

• Figure 1: The approximate inversion of a standard renormalization group transformation with the use of machine learning. Two replicas $\sigma$, $\tau$ are mapped to an overlap configuration $\rho$ which is renormalized (RG) via the majority rule to obtain $\rho'$. The standard renormalization group is then inverted (IRG) with the use of convolutions to reproduce $\rho$ from $\rho'$.
• Figure 2: Application of the inverse renormalization group. Two replicas $\sigma$, $\tau$ are mapped to an overlap configuration $\rho$. Inverse renormalization group transformations are applied iteratively to construct equilibrated configurations $\rho^{(1)}$, $\rho^{(2)}$ and $\rho^{(3)}$ up to lattice volumes $V=128^{3}$ without requiring additional Monte Carlo simulations on the larger lattices.
• Figure 3: Observables of the original system of $L=16$ and two inversely renormalized systems $L'=8 \rightarrow L=16$ and $L"=4 \rightarrow L'=8 \rightarrow L=16$ versus the inverse temperature $\beta$. $q$ corresponds to the overlap order parameter, $1nn$ is the interaction of a given spin with the first nearest-neighbor, $nnn$ is a three-spin interaction with the first and second nearest-neighbors, and $nnnn$ is a four-spin interaction with the first, second, and third nearest-neighbors. The machine learning algorithm has not been explicitly trained to reproduce the $nnn$ and $nnnn$ observables.
• Figure 4: Calculation of the critical exponent $y_{h}$ versus the inverse temperature $\beta$. The results are obtained based on the inversely renormalized configurations, starting from $L=16$.
• Figure 5: Numerical fits which approximate the linear region in the vicinity of the fixed point using as an observable the overlap order parameter. The fixed point has been obtained by two inversely renormalized systems of identical lattice size $L'=128$, one obtained starting from $L=8$ and another from $L=16$.