Quantum phase transition between antiferromagnetic and charge order in the Hubbard-Holstein model

We explore the quantum phase transitions between two ordered states in the infinite dimensional Hubbard-Holstein model at half filling. Our study is based on the dynamical mean field theory (DMFT) combined with the numerical renormalization group (NRG), which allows us to handle both strong electron-electron and strong electron-phonon interactions. The transition line is characterized by an effective electron-electron interaction. Depending on this effective interaction and the phonon frequency $\omega_0$ one finds either a continuous transition or discontinuous transition. Here, the analysis focuses on the behavior of the system when the electron-electron repulsion $U$ and the phonon-mediated attraction $\lambda$ are equal. We first discuss the adiabatic and antiadiabatic limiting cases. For finite $\omega_0$ we study the differences between the antiferromagnetic (AFM) and charge order, and find that when present the AFM state has a lower energy on the line.