Directional Ballistic Transport for Partially Periodic Schr\"odinger Operators on $\mathbb{Z}^2$

We study the transport properties of Schr\"{o}dinger operators on $\mathbb{Z}^2$ with potentials that are periodic in one direction and compactly supported in the other. Such systems are known to produce surface states that are weakly confined near the support of the potential. We demonstrate that surface states exhibit what we describe as directional ballistic transport, characterized by a strong form of ballistic transport along the periodic direction and its absence in the compactly supported one. By showing that the scattering states exhibit ballistic transport, we obtain ballistic transport for a dense subset of all of $\ell^2(\mathbb {Z} ^2)$.