Geometrodynamics of electrons in a crystal under position and time-dependent deformation

The semiclassical dynamics of Bloch electrons in a crystal under slowly varying deformation is developed in the geometric language of a lattice bundle. Berry curvatures and gradients of energy are introduced in terms of lattice covariant derivatives, with the corresponding connections given by the gradient and rate of strain. A number of physical effects are discussed: an effective post-Newtonian gravity at band bottom, polarization induced by spatial gradient of strain, orbital magnetization induced by strain rate, and the electron energy-stress tensor.