  We investigate the application of deep learning techniques employing the
conditional variational autoencoders for semi-supervised learning of latent
parameters to describe phase transition in the two-dimensional (2D)
ferromagnetic Ising model and the two-dimensional XY model. For both models, we
utilize spin configurations generated using the Wolff algorithms below and
above the critical temperatures. For the 2D Ising model we find the latent
parameter of conditional variational autoencoders is correlated to the known
order parameter of magnetization more efficiently than their correspondence in
variational autoencoders used previously. It can also clearly identify the
restoration of the $\mathbb{Z}_2$ symmetry beyond the critical point. The
critical temperature extracted from the latent parameter at larger lattices are
found to be approaching its correct value. Similarly, for the 2D XY model, we
find our chosen network with the latent representation of conditional
variational autoencoders is equally capable of separating the two phases
between the high and low temperatures, again at the correct critical
temperature with reasonable accuracy. Together these results show that the
latent representation of conditional variational autoencoders can be employed
efficiently to identify the phases of condensed matter systems, without their
prior knowledge.
