Machine learning vortices at the Kosterlitz-Thouless transition

TL;DR

This paper explores whether neural networks can classify phases in the 2D XY model by learning topological vortices, highlighting the challenges and potential of deep learning for topological phase detection.

Contribution

The study demonstrates the necessity of feature engineering for vortex classification and introduces a deep network capable of learning vortices without manual features.

Findings

Single-layer networks fail to classify phases accurately.

Convolutional networks can classify based on magnetization.

Deep networks can learn vortices but with limited accuracy for small systems.

Abstract

Efficient and automated classification of phases from minimally processed data is one goal of machine learning in condensed matter and statistical physics. Supervised algorithms trained on raw samples of microstates can successfully detect conventional phase transitions via learning a bulk feature such as an order parameter. In this paper, we investigate whether neural networks can learn to classify phases based on topological defects. We address this question on the two-dimensional classical XY model which exhibits a Kosterlitz-Thouless transition. We find significant feature engineering of the raw spin states is required to convincingly claim that features of the vortex configurations are responsible for learning the transition temperature. We further show a single-layer network does not correctly classify the phases of the XY model, while a convolutional network easily performs classification by learning the global magnetization. Finally, we design a deep network capable of learning vortices without feature engineering. We demonstrate the detection of vortices does not necessarily result in the best classification accuracy, especially for lattices of less than approximately 1000 spins. For larger systems, it remains a difficult task to learn vortices.