Abstract

The problem of quantum particles in periodic potentials is one of the pillars of modern condensed matter physics. For some years now, atoms can be prepared in optical lattices, and even light in photonic lattices can emulate quantum physics in periodic media under certain conditions. In this case, the single particle problem in the infinite size limit is solved by Bloch’s theorem as a consequence of discrete, rather than continuous translation symmetry. Even for finite and semi-infinite lattices, Bloch’s theorem is easily adapted for bulk states, while its extension to complex quasi-momenta may be utilised to extract topological edge modes when the system has non-trivial topology—and these states have very unique properties. The situation drastically changes if the particles move in a superposition of periodic potentials whose periods are incommensurate with each other, i.e. when their ratio is an irrational number: (i) Since periodicity is lost, Bloch’s theorem does not apply; (ii) energy bands cannot be defined, so one cannot in principle decide whether states with energies lying within spectral gaps (in the semi-infinite limit) are topological edge modes; (iii) the system is inhomogeneous and therefore it is unclear whether particle-particle or even potential scattering can happen at all; (iv) there are generally localised and delocalised phases and regions of the spectrum depending. In this talk, I will consider and give a solution to points (i) to (iii) above in the tight-binding approximation and in the delocalised phase. The results I will present are fully general, but will use as an illustration Harper’s or Hofstadter’s model.