  We develop deep autoregressive networks with multi channels to compute
many-body systems with \emph{continuous} spin degrees of freedom directly. As a
concrete example, we embed the two-dimensional XY model into 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 autoregressive neural networks. By
learning the microscopic probability distributions from the macroscopic thermal
distribution, the neural networks compute the free energy directly and find
that free vortices and anti-vortices emerge in the high-temperature regime. 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.
