  In this article we investigate the possibility of generating piezoelectric
orbital polarization in graphene-like systems which are deformed periodically.
We start with discrete two-level 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 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.
