Algebraic localization-delocalization phase transition in moving potential wells on a lattice

The localization and scattering properties of potential wells or barriers uniformly moving on a lattice are strongly dependent on the drift velocity owing to violation of the Galilean invariance of the discrete Schr\"odinger equation. Here a type of localization-delocalization phase transition of algebraic type is unravelled, which does not require any kind of disorder and arises when a power-law potential well drifts fast on a lattice. While for an algebraic exponent $\alpha$ lower than the critical value $\alpha_c=1$ dynamical delocalization is observed, for $\alpha> \alpha_c$ asymptotic localization, corresponding to an asymptotic frozen dynamics, is instead realized. At the critical phase transition point $\alpha_=\alpha_c=1$ an oscillatory dynamics is found, corresponding to Bloch oscillations. An experimentally-accessible photonic platform for the observation of the predicted algebraic phase transition, based on light dynamics in synthetic mesh lattices, is suggested.