Proposal of a machine learning method for detection of the Berezinskii-Kosterlitz-Thouless transitions in the q-state clock models

Yusuke Miyajima^1*, Masahito Mochizuki¹

¹Department of Applied Physics, Waseda University, Shinjuku-ku, Tokyo, Japan

* Presenter:Yusuke Miyajima, email:y.m-ichikawa@akane.waseda.jp

We develop a machine learning technique for detecting successive Berezinskii-Kosterlitz-Thouless (BKT) transitions in the q-state clock models using a simple feedforward neural network [1]. The BKT transition is a topological transition, which is characterized neither by trivial order parameter nor by spontaneous symmetry breaking, and thus is difficult to detect by machine learning techniques. In fact, a few techniques have succeeded in detecting the BKT transitions so far, but most of them require prior knowledge of the phase transitions (e.g., critical temperatures and the number of phases) and feature engineering by preprocessing the raw spin configuration data into, e.g., vortex configurations and histogram of spin orientations. In this sense, previous techniques can never be exploited for exploration of new physics in the BKT transitions.
Under these circumstances, we have developed a technique previously proposed for Ising models in Ref. [2] by introducing a new correlation function for the analysis of a weight matrix in the trained neural network and have demonstrated that this method can determine the BKT transition temperatures with high precision. Importantly, this method requires only the spatial spin configurations generated by Monte Carlo thermalization to train the neural network. Because neither prior knowledge of the model nor feature engineering in advance are required, our technique has significant advantages over those previously proposed and is anticipated to be a powerful and versatile tool to study the BKT transitions in general spin models.

[1] Y. Miyajima, Y. Murata, Y. Tanaka and M. Mochizuki, Phys. Rev. B 104, 075114 (2021).
[2] A. Tanaka and A. Tomiya, J. Phys. Soc. Jpn. 86, 063001 (2017).

Keywords: Machine learning , Berezinskii-Kosterlitz-Thouless transition, Phase diagram