Machine learning of phase transitions in the percolation andXYmodels

Abstract

In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two dimensional lattices, whose transitions involve non-local or topological properties, including site and bond percolations, the XY model and the generalized XY model. We find that using just one hidden layer in a fully-connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects-vortices and anti-vortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter ∆ values, where ∆ is the relative weight of pure XY coupling. Besides using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L × L can be used to the phase diagram for other sizes (L ′ × L ′ , where L ′ = L) without any further training.

Machine-Learning Studies on Spin Models

“…One is a higher BKT transition, T 2 , between the disordered phase and the BKT phase of QLRO, and the other is a lower transition, T 1 , between the BKT phase and the ordered phase. The recent numerical estimates of T 1 and T 2 for the 6-state clock model are 0.701 (5) and 0.898(5), respectively [18]. The output layer averaged over a test set as a function of T for the 2D 6state clock model is shown in Fig.…”

Direct detection of plasticity onset through total-strain profile evolution

“…The application of modern statistical methods to identify transitions between phases with well-defined order parameters in materials has become quite common [5]. The recent theme in the physics and engineering communities is the application of machine learning methods so that the identification of a precise characterization parameter is avoided and automatic detection is achieved [6][7][8][9][10][11][12][13]. However, the absence of detailed knowledge may lead to overfitting artifacts and unsuccessful machine learning training.…”

Markov Chain Monte Carlo Methods in Quantum Field Theories

“…Recent developments in the implementation of Artificial Intelligence (AI) for physical systems, particularly those that can be formulated on a lattice, show promising evidence in identifying the underlying phase structures [20,21,22,23,24,25,26,27,28,29]. The methods such as the Principal Component Analysis (PCA) [21,22,26,30], Supervised Machine Learning [23,29,31] (ML) and auto-encoders [26,25] are shown to be able to identify different phases of classical statistical systems, such as the two-dimensional Ising model. These techniques have also been applied on quantum statistical systems, such as the Hubbard model [24], which describes the transition between conducting and insulating systems.…”