Cold Atoms on Frustrating Lattices

Ultracold atoms in optical lattices undergo a quantum phase transition from a superfluid to a Mott insulator as the lattice potential depth is increased. We present a theory of the ground state and the elementary excitations of cold atoms in which the potential $\Sigma_i$ which induces coherence between different number states on a given site is elevated from a variational parameter to a quantum degree of freedom. In this approach mean-field theory is equivalent to minimizing the energy with respect to the $\Sigma_i$. The theory is applied to the Boson Hubbard model of optical lattice systems, to frustrated lattice models for rotating atoms, and to inhomogenous systems with a harmonic trapping potential superimposed on the lattice potential.