The effect of an on-site electron-phonon (e-ph) interaction is studied within the framework of an extended Hubbard model in one dimension. We have eliminated the local phonon coordinates using a modified Lang-Firsov transformation and diagonalized the effective electronic Hamiltonian by a real space renormalization group technique to determine the transition from a charge density wave (CDW) state to a spin density wave (SDW) state. It turns out that the CDW phase is favored by the e-ph coupling with an appreciable band-narrowing effect. In the SDW phase the distortions produced at nearest neighbor sites due to the e-ph interaction become comparable to the on-site distortions while this retardation effect is negligible in the CDW phase. \textcopyright{} 1996 The American Physical Society.
The effect of an on-site electron-phonon (e-ph) interaction is studied within the framework of an extended Hubbard model in one dimension. We have eliminated the local phonon coordinates using a modified Lang-Firsov transformation and diagonalized the effective electronic Hamiltonian by a real space renormalization group technique to determine the transition from a charge density wave (CDW) state to a spin density wave (SDW) state. It turns out that the CDW phase is favored by the e-ph coupling with an appreciable band-narrowing effect. In the SDW phase the distortions produced at nearest neighbor sites due to the e-ph interaction become comparable to the on-site distortions while this retardation effect is negligible in the CDW phase. \textcopyright{} 1996 The American Physical Society.
The ground-state phase diagram of the half filled one-dimensional Holstein-Hubbard model contains a charge-density-wave (CDW) phase, driven by the electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling, and a spin-density-wave (SDW) phase, driven by the on-site electron-electron $(e\text{\ensuremath{-}}e)$ repulsion. Recently, the existence of a third phase, which is metallic and lies in a finite region of parameter space between these two gapped phases, has been claimed. We study this claim using a renormalization-group method for interacting electrons that has been extended to include also $e\text{\ensuremath{-}}ph$ couplings. Our method [Tsai et al., Phys. Rev. B 72, 054531 (2005); Philos. Mag. B 86, 2631 (2006)] treats $e\text{\ensuremath{-}}e$ and $e\text{\ensuremath{-}}ph$ interactions on an equal footing and takes retardation effects fully into account. We find a direct transition between the SDW and CDW states. We study the effects of retardation, which are particularly important near the transition, and find that umklapp processes at finite frequencies drive the CDW instability close to the transition.
We study the effects of a bond-charge interaction (or correlated hopping) on the properties of the extended [i.e., with both on-site (U) and nearest-neighbor (V) repulsions] Hubbard model in one dimension at half-filling. Energy gaps and correlation functions are calculated by Lanczos diagonalization on finite systems. We find that, irrespective of the sign of the bond-charge interaction, X, the charge-density-wave (CDW) state is more robust than the spin-density-wave (SDW) state. A small bond-charge interaction term is enough to make the differences between the CDW and SDW correlation functions much less dramatic than when X=0. For X=t and fixed V2t (t is the uncorrelated hopping integral), there is an intermediate phase between a charge ordered phase and a phase corresponding to singly occupied sites, the nature of which we clarify: it is characterized by a succession of critical points, each of which corresponds to a different density of doubly occupied sites. We also find an unusual slowly decaying staggered spin-density correlation function, which is suggestive of some degree of ordering. No enhancement of pairing correlations was found for any X in the range examined. \textcopyright{} 1996 The American Physical Society.
The Holstein–Hubbard model is considered to study the competition between the electron-phonon interaction and the electron-electron interaction in a one-dimensional correlated Fermi systemat half-filling. First, a series of canonical transformations that incorporate phonon coherence and correlation effectsis performed and then the transformed Hamiltonian is averaged with respect to a zero-phonon state. The resulting effective electronic Hamiltonian is finally solved exactly by the Bethe-ansatz technique. It is shown that the present method suggests the existence of a wider intermediate metallic phase at the SDW-CDW crossover region than the one predicted by the simple variational method of Takada and Chatterjee. The presence of this metallic phase is indeed a favorable situation from the point of view of high temperature superconductivity.
We study extended Hubbard and Peierls-Hubbard models with up to N=10 sites and half-filled bands with modified boundary conditions. For the extended Hubbard model, the finite-size dependence of the transition from the charge-density-wave regime (CDW) to the spin-density-wave (SDW) regime and of the condensation transition is critically examined, clarifying some previous results. For the Peierls-Hubbard model and N\ensuremath{\le}10 with modified periodic boundary conditions, we always see a decrease of dimerization with U. Extrapolating these results to infinite N, we find that dimerization is enhanced by U for small electron-phonon coupling \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}0.75, whereas it decreases with U for 0.75\ensuremath{\le}1. By extending this model by a nearest-neighbor repulsion V, the amplitude of dimerization reaches a maximum at the same value U\ensuremath{\le}2V, where the CDW-to-SDW transition occurs in the extended Hubbard model.
We extend previous calculations of the 2kF and 4kF (${k}_{F}\ensuremath{\equiv}\mathrm{Fermi}\mathrm{wave}\mathrm{vector}$) spin- and charge-density autocorrelation functions of the one-dimensional Hubbard model by including the nearest-neighbor Coulomb interaction V. The results are computed for finite chains containing up to eight sites using a computer renormalization-group technique previously developed by us. We find, inter alia, that the real part of the 4kF charge-density wave (CDW), which is comparable to the real part of the 2kF CDW when only large onsite Coulomb repulsion is considered, becomes greatly enhanced relative to the 2kF CDW when V is positive. This finding implies that the V interaction is important in explaining the 4kF instability of some quasi-one-dimensional materials.
Using a multistep renormalization group method, we study the low-temperature phases of the interacting one-dimensional (1D) electron gas coupled to phonons. We obtain analytic expressions for the weak-coupling quantum phase boundaries of the 1D extended Holstein-Hubbard model and the 1D extended Peierls-Hubbard model for general band-filling and phonon frequency. Away from half-filling, the phase diagrams are characterized by a delicate competition between spin density wave, charge density wave, and superconducting orders. We study the dependence of the ground state on the electron-phonon (el-ph) and electron-electron (el-el) coupling strengths, the screening length, electron bandwidth, phonon frequency, doping, and type of phonon. Unlike the case in Fermi liquids, in 1D the el-ph coupling is strongly renormalized, often to stronger values. Even when the bare phonon-induced attraction is weak compared to the bare el-el repulsion, a small amount of retardation can cause the renormalized el-ph interaction to dominate the problem. We find cases in which a repulsive el-el interaction enhances the superconducting susceptibility in the presence of a retarded el-ph interaction. The spin gap and superconducting susceptibility are found to be strongly dependent on the deviation from half-filling (doping). In some cases, the superconducting susceptibility varies nonmonotonically with doping and exhibits a maximum at a particular doping. For a quasi-1D array of weakly coupled, fluctuating 1D chains, the superconducting transition temperature Tc also exhibits a maximum as a function of doping. The effect of changing the ion mass (isotope effect) on Tc is found to be largest near half-filling and to decrease rapidly with doping.