Machine learning of phase transitions in the percolation and XY models

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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 $\nu$.
• 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.