  We study the emergence of Berezinskii-Kosterlitz-Thouless (BKT) phases in
(PbTiO$_3$)$_3$/(SrTiO$_3$)$_3$ superlattices by means of second-principles
simulations. Beyond a threshold tensile epitaxial strain of $\epsilon = 0.25
\%$ the local dipole moments within the superlattices are confined to the
film-plane, and thus the polarization can be effectively considered as
two-dimensional. The analysis of the decay of the dipole-dipole correlation
with the distance, together with the study of the density of defects and its
distribution as function of temperature, supports the existence of a BKT phase
in a range of temperature mediating the ordered ferroelectric (stable at low
$T$), and the disordered paraelectric phase that appears beyond a critical
temperature $T_{\rm BKT}$. This BKT phase is characterized by quasi-long-range
order (whose signature is a power-law decay of the correlations with the
distance), and the emergence of tightly bounded vortex-antivortex pairs whose
density is determined by a thermal activation process. The proposed
PbTiO$_{3}$/SrTiO$_{3}$ superlattice model and the imposed mechanical boundary
conditions are both experimentally feasible, opening the door for the first
experimental observation of these new topological phases in ferroelectric
materials.
