Localization-delocalization transition in a two-dimensional Holstein-Hubbard model

► Self-trapping (ST) transition is studied in a 2D Holstein-Hubbard (HH) model. ► Series of canonical transformations are performed to eliminate the phonons. ► Resultant effective Hubbard model is solved by a mean-field method. ► ST transition turns out to be continuous in the anti-adiabatic regime. ► In the adiabatic case, ST transition is found to be discontinuous.