Ordered phases in the Holstein-Hubbard model: Interplay of strong Coulomb interaction and electron-phonon coupling

TL;DR

This paper investigates the phase diagram of the Holstein-Hubbard model at half filling, revealing how strong electron-electron and electron-phonon interactions influence superconductivity, antiferromagnetism, and charge order using dynamical mean-field theory.

Contribution

It provides a detailed analysis of ordered phases in the Holstein-Hubbard model, including the effects of retardation and Coulomb interactions on superconductivity and phase boundaries.

Findings

Superconducting transition temperature (Tc) is reduced and shifted due to phonon retardation and Coulomb interactions.

Hysteretic regions of antiferromagnetism and charge order appear around effective interaction Ueff=0.

Phase diagram includes paramagnetic metal and insulator phases depending on coupling strength.

Abstract

We study the Holstein-Hubbard model at half filling to explore ordered phases including su- perconductivity (SC), antiferromagnetism (AF), and charge order (CO) in situations where the electron-electron and electron-phonon interactions are strong (~ electronic bandwidth). The model is solved in the dynamical mean-field approximation using a continuous-time quantum Monte Carlo impurity solver. We determine the superconducting transition temperature (Tc) and the SC order parameter and show that the phonon-induced retardation or the strong Coulomb interaction leads to a significant reduction and shift of the Tc dome if one interprets the system as having an effective static interaction Ueff given by the Hubbard U reduced by the phonon-mediated attraction in the static limit. This behavior is analyzed by comparison to an effective static model in the polaron representation with a reduced bandwidth. We also determine the finite-temperature phase diagram including AF and CO. In the moderate-coupling regime, there is a hysteretic region of AF and CO around Ueff = 0, while the two phases are separated by a paramagnetic metal in the weak-coupling regime and a paramagnetic insulator in the strong coupling regime.