Crystals and other condensed matter systems described by density waves often exhibit dislocations. Here we show, by considering the topology of the ground state manifolds (GSMs) of such systems, that dislocations in the density phase field always split into disclinations, and that the disclinations themselves are constrained to sit at particular phase values in the GSM. This constraint on the location of the disclinations results in an energy barrier to dislocation glide. Consequently, the topology of the GSM alone gives rise to a Peierls–Nabarro barrier.

A continuum theory of linearized Helmholtz-Kirchoff point vortex dynamics about a steadily rotating lattice state is developed by two separate methods: firstly by a direct procedure, secondly by taking the long-wavelength limit of Tkachenko’s exact solution for a triangular vortex lattice. Solutions to the continuum theory are found, described by arbitrary anti-holomorphic functions, and give power-law localized edge modes. Numerical results for finite lattices show excellent agreement to the theory.