Exact Results in Quantum Theory

Experiments with ultra-cold atoms in an optical lattice provide a highly-tunable quantum simulator of the Hubbard model. However, theoretical treatment of this model poses a formidable challenge. Analytic solutions are possible only in few, special cases and in general the problem needs to be simplified. In the presentation an approximation scheme of dynamical mean-field theory (DMFT) is discussed. The concept of this method is explained and it is argued why this approach is superior to the standard mean-field one. The bosonic counterpart of the DMFT method is introduced and several results for the Bose-Hubbard model are presented along with the comparison to the mean-field results.