Unsupervised Machine Learning Method for the Phase Behavior of the Constant Magnetization Ising Model in Two and Three Dimensions

Machine learning methods have been important in the study of phase transitions. Unsupervised methods are particularly attractive because they do not require prior knowledge of the existence of a phase transition. In this work we focus on the constant magnetization Ising model in two (2D) and three (3D) dimensions. While there have been many studies using machine learning for the critical behavior of these systems, we are not aware of any studies for the phase diagram at off-critical magnetizations below the critical temperature. Previous work has used the raw spins as the input feature. We show that a more robust input feature is the local affinity, where the value of the feature at each site is determined by the spin and its neighbors. When coupled with a variational autoencoder, the method is able to predict the phase behavior of the 2D and 3D Ising models (including the critical exponent β) in quantitative agreement with conventional simulations. The choice of activation functions in the autoencoder is crucial, and this requires physical insight into the nature of the phase transition. The method is general and can be applied to any lattice or off-lattice system.