  In this thesis, I study Interface Conformal Field Theories (ICFT) and their
holographic dual, which is composed of two asymptotically Anti-de-Sitter (AdS)
spaces glued through a thin gravitating membrane. I restrict the study to
simple minimal models, which allow for analytic control while providing
universally applicable results. The analysis is set in 2D ICFT/3D gravity, but
I expect much of the results to be generalizable to higher dimensions. I first
consider this system at equilibrium and at finite temperature. By solving the
equations of motion in the bulk, I find the allowable solution landscape.
Classifying the rich set of solutions among 3 thermodynamical phases, I draw
the phase diagram outlining the nature of the various phase transitions. I then
examine a simple out-of-equilibrium situation arising from connecting at an
interface two spatially infinite CFTs at different temperatures. Then a
"Non-Equilibrium Steady State" (NESS) describes the growing region where the
interaction has settled into a stationary phase. I determine the holographic
dual of this region, composed of two spinning planar black holes glued at the
membrane. I find an expression for the deformed out-of-equilibrium event
horizon. This geometry suggests that the field theory interface acts as a
perfect scrambler, a property that until now seemed unique to black hole
horizons. Finally, I study the entanglement structure of the aforementioned
geometries by means of the Ryu-Takayanagi prescription. After reviewing a
construction in the vacuum state, I present partial results for more general
geometries at finite temperature. For this purpose, I need to introduce
numerical methods. I outline the main difficulties in their application and
conclude by mentioning the Quantum Null Energy Condition (QNEC), an inequality
linking entanglement entropy and energy, that can be used to test the
consistency of the models.
