Topological polarization in graphene-like systems

In this paper we investigate the possibility of generating piezoelectric orbital polarization in graphene-like systems which are deformed periodically. We start with discrete two-band models which depend on control parameters; in this setting, time-dependent model Hamiltonians are described by loops in parameter space. Then, the gap structure at a given Fermi energy generates a non-trivial topology on parameter space which then leads to possibly non-trivial polarizations. More precisely, we show the polarization, as given by the King-Smith-Vanderbilt formula, depends only on the homotopy class of the loop; hence, a necessary condition for non-trivial piezo effects is that the fundamental group of the gapped parameter space must not be trivial. The use of the framework of non-commutative geometry implies that our results extend to systems with weak disorder. We then apply this analysis to the uniaxial strain model for graphene which includes nearest-neighbor hopping and a stagger potential, and show that it supports non-trivial piezo effects; this is in agreement with recent physics literature.