Geometric tangent plane methods and the Bose-Hubbard model

The paradigmatic Bose-Hubbard model describes interacting bosons which can be experimentally realized with ultracold atoms in optical lattices. In this talk, I will focus on the superfluid phase and introduce geometric methods to find approximate ground states and quasi particle excitations. For the variational class of Gaussian states, we can systematically extend Bogoliubov theory and compute quasi particle excitations, life times and spectral functions. To our surprise, we find a bound state below the continuum. Depending on the interaction strength, the dispersion relation of the bound state enters the continuum for large momentum. I will conclude with discussing observable effects and how we can test if this finding is physical.