Ghost-RISB for Correlated Electron-Phonon Systems: Application to the Hubbard-Holstein Model
We develop a generalization of the ghost-rotationally-invariant slave-boson (ghost-RISB) method that incorporates local phonon modes coupled to arbitrary on-site electronic degrees of freedom, enabling a nonperturbative treatment of electron-electron and electron-phonon interactions within an efficient variational framework. The method extends the ghost-orbital construction to capture dynamical self-energy effects and phonon-induced renormalizations beyond static slave-boson approaches. Benchmarking against dynamical mean-field theory (DMFT) results for the Hubbard-Holstein model, we find excellent quantitative agreement for electron quasiparticle weights and phonon properties across a wide range of coupling strengths, and it captures accurately the competition between electron-electron and electron-phonon interactions. We show that the inclusion of the ghost orbitals is crucial to accurately describe the regime of low-frequency, strongly dynamical, phonons. The extended ghost-RISB achieves this accuracy at a fraction of the computational cost of DMFT, due to its self-consistency rooted in static observables instead of dynamical ones, enabling rapid exploration of correlated electron-phonon phase diagrams. We exploit this advantage to characterize the most demanding regime of strong coupling and adiabatic phonons. Our analysis shows a suppression of the superconducting order parameter, which is interpreted as a Franck-Condon-like reduction of the overlap between the phonon wavefunctions associated with empty and doubly-occupied sites in the bipolaronic regime.