  We investigate deep learning autoencoders for the unsupervised recognition of
phase transitions in physical systems formulated on a lattice. We focus our
investigation on the 2-dimensional ferromagnetic Ising model and then test the
application of the autoencoder on the anti-ferromagnetic Ising model. We use
spin configurations produced for the 2-dimensional ferromagnetic and
anti-ferromagnetic Ising model in zero external magnetic field. For the
ferromagnetic Ising model, we study numerically the relation between one latent
variable extracted from the autoencoder to the critical temperature $T_c$. The
proposed autoencoder reveals the two phases, one for which the spins are
ordered and the other for which spins are disordered, reflecting the
restoration of the $\mathbb{Z}_2$ symmetry as the temperature increases. We
provide a finite volume analysis for a sequence of increasing lattice sizes.
For the largest volume studied, the transition between the two phases occurs
very close to the theoretically extracted critical temperature. We define as a
quasi-order parameter the absolute average latent variable ${\tilde z}$, which
enables us to predict the critical temperature. One can define a latent
susceptibility and use it to quantify the value of the critical temperature
$T_c(L)$ at different lattice sizes and that these values suffer from only
small finite scaling effects. We demonstrate that $T_c(L)$ extrapolates to the
known theoretical value as $L \to \infty$ suggesting that the autoencoder can
also be used to extract the critical temperature of the phase transition to an
adequate precision. Subsequently, we test the application of the autoencoder on
the anti-ferromagnetic Ising model, demonstrating that the proposed network can
detect the phase transition successfully in a similar way.
