Learning thermodynamics and topological order of the 2D XY model with generative real-valued restricted Boltzmann machines

TL;DR

This paper introduces a new generative real-valued restricted Boltzmann machine that effectively models the 2D XY model, capturing thermodynamics and topological order, and enables detection of the Kosterlitz-Thouless transition without prior physics knowledge.

Contribution

The paper develops a novel RBM with nonlinear cos/sin activation that learns the 2D XY model's distribution and detects phase transitions from weight matrices.

Findings

RBM-CosSin accurately models XY configurations

It captures thermodynamics and topological order

Phase transition info can be extracted from weights

Abstract

Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical 2D XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued RBM (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generate authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.