Influence of the ferroelectric quantum critical point on SrTiO${}_3$ interfaces

We study a model SrTiO${}_3$ interface in which conduction $t_{2g}$ electrons couple to the ferroelectric (FE) phonon mode. We treat the FE mode within a self-consistent phonon theory that captures its quantum critical behavior, and show that proximity to the quantum critical point leads to universal tails in the electron density of the form $n(z)\sim (\lambda +z{)}^{-2}$, where $\lambda \sim T^{2-d/\mathfrak{z}}$,with$d=3$thedimensionalityand$\mathfrak{z}=1$ the dynamical critical exponent. Implications for the metal-insulator transition at low electron density are discussed.