Competing interactions and symmetry breaking in the Hubbard-Holstein model

Competing interactions are often responsible for intriguing phase diagrams in correlated electron systems. Here we analyze the competition of instantaneous short range Coulomb interaction $U$ with the retarded electron-electron interaction induced by an electron-phonon coupling $g$ as described by the Hubbard-Holstein model. The ground state phase diagram of this model in the limit of large dimensions at half filling is established. The study is based on dynamical mean field theory combined with the numerical renormalization group. Depending on $U$, $g$, and the phonon frequency $\omega_0$, the ground state is antiferromagnetically (AFM) or charge ordered (CO). We find quantum phase transitions from the AFM to CO state to occur when $U-\lambda\simeq 0$, where $\lambda$ characterizes the phonon induced effective attraction. The transition is continuous for small couplings and large phonon frequencies $\omega_0$ and becomes discontinuous for large couplings and small values of $\omega_0$. We comment on the possible relevance of this work for Ba${}_{1-x}$K${}_x$BiO${}_3$.