Geometry of variational methods

A key challenge in the theoretical study of quantum many body systems is to overcome the exponential growth of the Hilbert space with the system size. Many successful approaches are variational, i.e., they are based on choosing suitable families of states that capture key properties of the system. Prominent examples range from Gaussian states to matrix product states and tensor networks. In this talk, I will review the geometric structures of variational manifolds and how we can use them to systematically (a) estimate ground state energies, (b) compute approximate excitation spectra and (c) predict the linear response of these systems. Taking the Bose-Hubbard model as example, I show how this gives rise to a systematic extension of the traditional Bogoliubov theory.[based on work with Tommaso Guaita, Tao Shi, Claudius Hubig, Eugene Demler and Ignacio Cirac]