Abstract

The key idea behind the renormalization group (RG) transformation is that properties of physical systems with very different microscopic makeups can be characterized by a few universal parameters. However, finding the optimal RG transformation remains difficult due to the many possible choices of the weight factors in the RG procedure. Here we show, by identifying the conditional distribution in the restricted Boltzmann machine (RBM) and the weight factor distribution in the RG procedure, an optimal real-space RG transformation can be learned without prior knowledge of the physical system. This neural Monte Carlo RG algorithm allows for direct computation of the RG flow and critical exponents. This scheme naturally generates a transformation that maximizes the real-space mutual information between the coarse-grained region and the environment. Our results establish a solid connection between the RG transformation in physics and the deep architecture in machine learning, paving the way to further interdisciplinary research.

Highlights

  • The renormalization group (RG) [1] formalism provides a systematic method for quantitative analysis of critical phenomena

  • Among all the RG schemes, the real-space renormalization group (RSRG), first proposed by Kadanoff [2], is the most intuitive and natural way to perform RG transformations on lattice models [3]. These methods allow for a straightforward construction of the critical surface and calculation of the critical exponents using numerical methods such as the Monte Carlo renormalization group (MCRG) [4,5,6]

  • We address a different question: How can we train a deep neural networks (DNNs) to obtain a good RSRG transformation? This issue is partially addressed from an information-theoretic perspective [25,26], where an optimal RG transformation is obtained by maximizing the real-space mutual information (RSMI)

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Summary

INTRODUCTION

The renormalization group (RG) [1] formalism provides a systematic method for quantitative analysis of critical phenomena. Among all the RG schemes, the real-space renormalization group (RSRG), first proposed by Kadanoff [2], is the most intuitive and natural way to perform RG transformations on lattice models [3]. These methods allow for a straightforward construction of the critical surface and calculation of the critical exponents using numerical methods such as the Monte Carlo renormalization group (MCRG) [4,5,6]. We demonstrate the accuracy of this approach for the two- and three-dimensional classical Ising models

PARAMETRIZATION OF THE REAL-SPACE RENORMALIZATION GROUP
STOCHASTIC OPTIMIZATION FOR THE OPTIMAL CRITERION
TWO-DIMENSIONAL ISING MODEL
THREE-DIMENSIONAL ISING MODEL
REAL-SPACE MUTUAL INFORMATION
CONCLUSION
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