Unsupervised Machine Learning Method for the Phase\nBehavior of the Constant Magnetization Ising Model in Two and Three\nDimensions

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