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Physical review. B, Condensed matter

Strong-coupling expansions for the attractive Holstein and Hubbard models

James K. Freericks

Published Aug 1, 1993• 50 citations
Publisher

Abstract

A strong-coupling expansion for the Holstein (electron-phonon) model is carried out through fourth order. In the large-frequency limit the expansion agrees with known results for the attractive Hubbard model. Mean-field theory is employed to determine transition temperatures of the effective (pseudospin) Hamiltonian and the results are compared with recent Monte Carlo simulations of these models in infinite dimensions.

DOI
10.1103/physrevb.48.3881

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A strong-coupling expansion for the Holstein (electron-phonon) model is carried out through fourth order. In the large-frequency limit the expansion agrees with known results for the attractive Hubbard model. Mean-field theory is employed to determine transition temperatures of the effective (pseudospin) Hamiltonian and the results are compared with recent Monte Carlo simulations of these models in infinite dimensions.
Monte Carlo simulations are performed to examine superconductivity and charge-density-wave fluctuations in the infinite-dimensional electron-phonon Holstein model. The electron-phonon system interpolates between an attractive, static, Falicov-Kimball model that always exhibits charge-density-wave order and an attractive, instantaneous, Hubbard model that always superconducts as a function of phonon frequency. The maximum charge-density-wave transition temperature at half-filling is an order of magnitude smaller than the effective electronic bandwidth and is virtually independent of the phonon frequency. The maximum superconducting transition temperature depends strongly on phonon frequency and is bounded from above by the maximum charge-density-wave transition temperature. The exact solution is compared to both weak-coupling expansions and strong-coupling expansions. The effective phonon potential becomes anharmonic and develops a double-well structure that deepens as the electron-phonon interaction increases.
We study the Holstein-Hubbard model at half filling to explore ordered phases including superconductivity (SC), antiferromagnetism (AF), and charge order (CO) in situations where the electron-electron and electron-phonon interactions are strong (comparable to the electronic bandwidth). The model is solved in the dynamical mean-field approximation with a continuous-time quantum Monte Carlo impurity solver. We determine the superconducting transition temperature Tc{T}_{c}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{T}_{c}Tc​ dome against the effective electron-electron interaction Ueff{U}_{\text{eff}}Ueff​ given by the Hubbard UUU 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 renormalized bandwidth. In addition, we discuss the superconducting gap $\ensuremath{\Delta}$ and $2\ensuremath{\Delta}/{T}_{c}$ to reveal the effect of the retardation and the Coulomb interaction. 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{U}_{\mathrm{eff}}=0Ueff​=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.
The interplay of electron-phonon coupling and strong electronic correlations is studied in the frame of the two-dimensional Hubbard-Holstein model. Static and dynamic properties are determined by quantum Monte Carlo simulations and by Migdal-Eliashberg theory. The comparison allows us to assess the diagrammatic approach. The competition between the phonon-mediated electron-electron attraction and the local Coulomb repulsion leads to a rich phase diagram, which we study in detail for a wide range of parameters. We address the question, to which extent the systems can be described by an effective negative-U Hubbard model.
A strictly truncated (weak-coupling) perturbation theory is applied to the attractive Holstein and Hubbard models in infinite dimensions. These results are qualified by comparison with essentially exact quantum Monte Carlo results. The second-order iterated perturbation theory is shown to be quite accurate in calculating transition temperatures for retarded interactions, but is not as accurate for the self-energy or the irreducible vertex functions themselves. Iterated perturbation theory is carried out through fourth order for the Hubbard model. The self-energy is quite accurately reproduced by the theory, but the vertex functions are not. Anomalous behavior occurs near half-filling because the iterated perturbation theory is not a conserving approximation.