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Synopsis

The notion of electronic bands is fundamental to condensed matter physics. It is a consequence of Bloch's theorem which provides the basis for approaches such as tight binding, ab initio methods, etc. Recent discoveries of 'mesoscopic' crystals with large unit cells are pushing the envelope in this regard. Mesoscopic lattices also occur in ordered phases with spatial textures, e.g., in skyrmion crystals in magnets. Can we develop an analogue of Bloch’s theorem and its associated machinery for such systems?

Motivated by this question, we study a conventional type-II superconductor. The relevant excitations here are not electrons, but rather ‘collective modes’ that are composed of order parameter fluctuations. We first describe an isolated vortex where the low energy excitations are ‘gyrotropic’ and ‘breathing’ modes. Moving to a vortex lattice, the collective modes obey a suitable form of Bloch’s theorem. We develop a tight binding approach with localized excitations that 'hop' from a vortex to its neighbours. We discuss possible signatures, e.g., in spectroscopy with circularly polarized light.