# Detection of Berezinskii--Kosterlitz--Thouless transitions for the two-dimensional $q$-state clock models with neural networks

## TL;DR

This paper demonstrates that a simple supervised neural network can detect and accurately determine the transition temperatures of BKT phase transitions in 2D clock models without prior model information.

## Contribution

It introduces a minimal neural network architecture capable of identifying topological phase transitions and estimating critical points in 2D clock models.

## Key findings

- Successfully detects BKT transitions in 6- and 8-state clock models
- Accurately estimates transition temperatures
- Shows potential for universal neural network applications in phase transition studies

## Abstract

Using the technique of supervised neural networks (NN), we study the phase transitions of two-dimensional (2D) 6- and 8-state clock models on the square lattice. The employed NN has only one input layer, one hidden layer of 2 neurons, and one output layer. In addition, the NN is trained without any prior information about the considered models. Interestingly, despite its simple architecture, the built supervised NN not only detects both the two Berezinskii--Kosterlitz--Thouless (BKT) transitions but also determines the transition temperatures with reasonable high accuracy. It is remarkable that a NN, which has an extremely simple structure and is trained without any input from the studied models, can be employed to study topological phase transitions. The outcomes shown here as well as those previously demonstrated in the literature suggest the feasibility of constructing a universal NN that is applicable to investigate the phase transitions of many systems.