Continuous-Mixture Autoregressive Networks Learning the Kosterlitz–Thouless Transition

We develop deep autoregressive networks with multi channels to compute many-body systems with continuous spin degrees of freedom directly. As a concrete example, we demonstrate the two-dimensional XY model with the continuous-mixture networks and rediscover the Kosterlitz–Thouless (KT) phase transition on a periodic square lattice. Vortices characterizing the quasi-long range order are accurately detected by the generative model. By learning the microscopic probability distributions from the macroscopic thermal distribution, the networks are trained as an efficient physical sampler which can approximate the free energy and estimate thermodynamic observables unbiasedly with importance sampling. As a more precise evaluation, we compute the helicity modulus to determine the KT transition temperature. Although the training process becomes more time-consuming with larger lattice sizes, the training time remains unchanged around the KT transition temperature. The continuous-mixture autoregressive networks we developed thus can be potentially used to study other many-body systems with continuous degrees of freedom.