Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transitions in the q-state clock models

Abstract

We demonstrate that a machine learning technique with a simple feedforward neural network can sensitively detect two successive phase transitions associated with the Berezinskii-Kosterlitz-Thouless (BKT) phase in q-state clock models simultaneously by analyzing the weight matrix components connecting the hidden and output layers. We find that the method requires only a data set of the raw spatial spin configurations for the learning procedure. This data set is generated by Monte-Carlo thermalizations at selected temperatures. Neither prior knowledge of, for example, the transition temperatures, number of phases, and order parameters nor processed data sets of, for example, the vortex configurations, histograms of spin orientations, and correlation functions produced from the original spin-configuration data are needed, in contrast with most of previously proposed machine learning methods based on supervised learning. Our neural network evaluates the transition temperatures as T2/J = 0.921 and T1/J = 0.410 for the paramagnetic-to-BKT transition and BKT-to-ferromagnetic transition in the eight-state clock model on a square lattice. Both critical temperatures agree well with those evaluated in the previous numerical studies.

Machine learning classification of two-dimensional vortex configurations

“…This has led to implementations of machine learning methods, for instance, to identify symmetry-broken phases in the field of classical statistical physics [8][9][10], and in some cases neural networks have even been shown to be able to learn an order parameter or other thermodynamical parameters [8,10]. More recently, the machine learning methodology has found applications in the realm of physics problems such as identifying phase transitions of many-body systems [11][12][13][14][15][16][17][18][19][20][21], topological systems [22][23][24][25][26], and finding quantum enhanced learning algorithms [27][28][29].…”

“…For this reason, the unsupervised machine learning methods, that do not rely on prior knowledge, may offer significant benefits over supervised learning methods [31,32]. Inspired by the application of the supervised learning to classify the KT transition [11,23], it is natural to ask whether unsupervised neural networks would be capable of identifying a variety of vortex phases of matter and quantum turbulent flow states in two-dimensional Bose-Einstein condensates? Few unsupervised learning techniques have been previously applied to the XY model.…”