Single-Particle Quantum States in a Crystal with Topological Defects

The influence of frozen-in topological defects in a crystal on the long-wavelength quantum states of a particle is considered. In the continuum limit of a conveniently defined tight-binding model one is led to a covariant Schr\"odinger equation on a Riemann-Cartan manifold. When the tight-binding transfer energies are assumed to depend on the local lattice deformations caused by the defects, additional noncovariant terms are generated in the Hamiltonian. These terms generate bound states of the particle to edge dislocations and enhance the scattering of particles on screw dislocations.