Piezoelectricity: Quantized Charge Transport Driven by Adiabatic Deformations

We study the (zero temperature) quantum piezoelectric response of Harper-like models with broken inversion symmetry. The charge transport in these models is related to topological invariants (Chern numbers). We show that there are arbitrarily small periodic modulations of the atomic positions that lead to nonzero charge transport for the electrons. The Harper model can be interpreted as a tight-binding quantum Hamiltonian describing the dynamics of non-interacting electrons on a two dimensional lattice in the presence of magnetic fields. It is known to have interesting Hall transport properties. Here we study the electric response of Harper-like models to adiabatic changes in the hopping amplitudes. Changes in the hopping amplitudes have a natural interpretation as elastic deformation of the underlying lattice. As we shall show, such deformations can drive electron transport. We shall refer to this kind of response as piezoelectricity. Like the Hall conductance in the integer Hall effect [1, 2], and in quasi-one dimensional systems [3], the Thouless pump [4, 5], the Magnus force [6], adiabatic charge transport in networks [7], adiabatic spin transport [8], and adiabatic viscosity [9], it is a transport phenomenon related to the adiabatic curvature and Berry’s phases [10]. Let us first summarize the central findings: 1. Harper-like models with broken time reversal and broken inversion symmetry have, in general, nontrivial piezoelectric response. 2. Appropriate periodic modulations of the atomic positions lead to integral charge transport given by appropriate Chern integers.