Quantum theory of mechanical deformations

We construct a general metric-tensor framework for treating inhomogeneous adiabatic deformations applied to crystalline insulators, by deriving an effective time-dependent Schrödinger equation in the undistorted frame. The response can be decomposed into “static” and “dynamic” terms that correspond, respectively, to the ampli- tude and the velocity of the distortion. We then focus on the dynamic contribution, which takes the form of a gauge field entering the effective Hamiltonian, in the linear-response limit. We uncover an intimate relation between the dynamic response to the rotational component of the inhomogeneous deformation and the diamagnetic response to a corresponding inhomogeneous magnetic field. We apply this formalism to the theory of flexoelectric response, where we resolve a previous puzzle by showing that the currents generated by the dynamic term, while real, generate no bound charges even at surfaces, and so may be dropped from a practical theory of flexoelectricity.

Figure 1

Dynamical vector-potential contributions to the polarization of a slab subjected to a shear strain gradient. (a) Decomposition into circulating currents within segments. (b) Decomposition between bulk and surface contributions.

Figure 2

Schematic illustration of the flexural deformation of a slab. Thick arrows indicate the Cartesian axes; thick gray curves indicate the slab surfaces.