Knowledge of different phases and phase transitions in low-dimensional systems is essential for the understanding and control of nanoscale devices. Theoretical models aided by first-principles calculations have proven invaluable in this respect. However, observed behaviour often differs from what is predicted theoretically. One common reason for this is that in many systems several different phase transitions are possible, and the phases typically compete, or even coexist.

The aim of this thesis is to provide a theoretical and computational study of competing phases and phase transitions in several interesting nanoscale systems. The first is polar-nonpolar interfaces such as LaAlO$3$/SrTiO$3$, where an insulator--metal transition occurs at the interface via the formation of a two-dimensional electron gas (2DEG) which screens the polar discontinuity there. In spite of the interest, tunability, and very special properties of the 2DEG, important uncertainties remain about the character of the metal--insulator transition induced by film thickness and/or perpendicular electric field. We show that a rich variety of scenarios is possible for the apearance of free carriers in such systems due to the coupling of the dielectric response of the film with common structural distortions of the perovskite systems employed, namely the rotations of oxygen octahedra (tilts). We show that continuous or discontinuous, and simultaneous or separate appearances of tilts and carriers can occur, depending very sensitively on the energetics of the tilts, carriers, the biquadratic coupling between them, and the correlation length of the tilts in the polar film.

\noindent The second is ferroelectric thin films, which tend to fall victim to depolarization effects, limiting their use in practical applications. It is well-known that depolarization can be mitigated by the formation of complex domain structures, but recently it has been proposed theoretically and signalled experimentally that the formation of a 2DEG at the surfaces of the films may make it possible to sustain a polarization without domains. Previous theoretical studies have suggested that the effects compete, and that either domains or a 2DEG will form in the thin and thick limits, respectively. However, recent experimental observations suggest they can coexist. We propose a model of ferroelectric interfaces in which the polar discontinuities can be screened both by domains and a 2DEG. We find that the polydomain and monodomain phases are separated by a region of coexistence in which both phenomena are simultaneously observed.

\noindent The final example comes from twistronics, an emerging field in which the properties of layered systems are tuned by introducing a relative twist or lattice mismatch between the layers (moir'e superlattice). Recently, ferroelectricity was observed in a typically non-polar system, facilitated by twisting, although the phenomenon has not yet been understood theoretically. It is well-known that stacking domains form in moir'e superlattices due to the competition between the interlayer van der Waals forces and intralayer elastic forces, which can be recognized as polar domains due to the local spontaneous polarization in bilayers without centrosymmetry. We propose a theoretical model which captures the effect of an applied electric field on the domain structure. The coupling between the spontaneous polarization and field leads to uneven relaxation of the domains, and a net polarization in the superlattice at nonzero fields, which is sensitive to the moir'e period. We show that the dielectric response to the field reduces the stacking energy and leads to softer domains in all bilayers.