  The LaAlO$_3$/SrTiO$_3$ interface hosts a plethora of gate-tunable electronic
phases. Gating of LaAlO$_3$/SrTiO$_3$ interfaces are usually assumed to occur
electrostatically. However, increasing evidence suggests that non-local
interactions can influence and, in some cases, dominate the coupling between
applied gate voltages and electronic properties. Here, we sketch quasi-1D
ballistic electron waveguides at the LaAlO$_3$/SrTiO$_3$ interface as a probe
to understand how gate tunability varies as a function of spatial separation.
Gate tunability measurements reveal the scaling law to be at odds with the pure
electrostatic coupling observed in traditional semiconductor systems. The
non-Coulombic gating at the interface is attributed to the existence of a
long-range nanoelectromechanical coupling between the gate and electron
waveguide, mediated by the ferroelastic domains in SrTiO$_3$. The long-range
interactions at the LaAlO$_3$/SrTiO$_3$ interface add unexpected richness and
complexity to this correlated electron system.
