Functional Renormalization Group Study of the One Dimensional Hubbard-Holstein Model

The electron-phonon(e-ph) coupling can give rise to interesting phases in low-dimensional systems like molecular crystals. The Holstein model, where the electrons couple to dispersionless phonons, is known to capture this interplay. Early studies of the 1D Hubbard-Holstein model at half-filling revealed the existence of two insulating phases, Charge Density waves (CDW), and Spin Density waves (SDW). However, recent Quantum Monte Carlo studies suggest that a finite metallic region exists near the phase boundaries. Although the existence has been confirmed from other studies, its mechanism is still rather poorly understood. Unlike other numerical methods, the phonon self energy and the vertex functions are readily accessible in the functional renormalization group (FRG) method and these quantities can provide insight into the mechanism of the metallic phase. We systematically investigate the effects of electron-phonon coupling and phonon frequency in a one-dimensional Hubbard-Holstein model using frequency dependent FRG across the CDW transition to understand the mechanism of metallic phase in one-dimension half-filled systems.