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arXiv:cond-mat/0007050 · Physical review. B, Condensed matter

Optical phonons in a quarter-filled one-dimensional Hubbard model

Philippe Maurel, Marie-Bernadette Lepetit

Published Oct 15, 2000• 7 citations
arXivHTMLPDF

Abstract

The influence of dispersionless quantum optical phonons on the phase diagram of a quarter-filled Hubbard chain is studied using the density-matrix renormalization group technique. The ground state phase diagram is obtained for frequencies corresponding to the intramolecular vibrations in organic conductors. For high vibrational modes the system is only slightly affected by the electron-phonons coupling. It remains in a L\"uttinger liquid phase as long as the electronic repulsion is larger than the polaronic binding energy. For low vibrational modes the phase diagram is very rich. A noticeable point is the existence of a ${4k}_{F}$ charge-density wave phase for small values of the correlation strength. For realistic values of the electron repulsion and of the electron-phonons coupling constant a large phonons-mediated reduction of the L\"uttinger liquid parameter ${K}_{\ensuremath{\rho}}$ was found compared to the pure electronic model.

DOI
10.1103/physrevb.62.10744

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The influence of dispersionless quantum optical phonons on the phase diagram of a quarter-filled Hubbard chain is studied using the density-matrix renormalization group technique. The ground state phase diagram is obtained for frequencies corresponding to the intramolecular vibrations in organic conductors. For high vibrational modes the system is only slightly affected by the electron-phonons coupling. It remains in a L"uttinger liquid phase as long as the electronic repulsion is larger than the polaronic binding energy. For low vibrational modes the phase diagram is very rich. A noticeable point is the existence of a 4kF{4k}_{F}4kF​ charge-density wave phase for small values of the correlation strength. For realistic values of the electron repulsion and of the electron-phonons coupling constant a large phonons-mediated reduction of the L"uttinger liquid parameter K\ensuremathρ{K}_{\ensuremath{\rho}}K\ensuremathρ​ was found compared to the pure electronic model.
The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane for the phonon-mediated interaction gph{g}_{\mathrm{ph}}gph​ and the Coulomb interaction UUU. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency ${\ensuremath{\omega}}_{0}$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical ${U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}$, followed at larger $U\ensuremath{\sim}{g}_{\mathrm{ph}}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on UUU. For the Peierls-Hubbard model, the region of the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency ${\ensuremath{\omega}}_{D}$ decreases. A power-law dependence ${g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term VVV is added to the Hubbard interaction.
We have studied one dimensional acoustic phonon linearly coupled to the electron density. The phonon spectrum is linear with cutoff at the Debye frequency, it is straightforward modification of three dimensional optical phonons. Phonon field results in substituting the dynamical coupling for the screened coulomb interaction. We have shown that the electron-phonon coupling quantitatively changed the phase diagram of Luttinger liquid with a single impurity. The existence of slow and fast polaron modes with different weights in different regimes resulted in different renormalization group flow for a weak scatterer and a weak tunneling link. The resulting phase diagram depending on the parameters of the problem, regimes corresponding to purely metallic or purely insulating behavior and an intermediate regime with two stable fixed points and one unstable, finite conductance fixed point. we have found that the direction of renormalization group flow change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. The obtained results were found in good agreement with previously obtained results.
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (nnn). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.
We consider a one-dimensional system consisting of electrons with short-ranged repulsive interactions and coupled to small-momentum transfer acoustic phonons. The interacting electrons are bosonized and described in terms of a Luttinger liquid which allows us to calculate exactly the one- and two-electron Green function. For non-interacting electrons, the coupling to phonons alone induces a singularity at the Fermi surface which is analogous to that encountered for electrons with an instantaneous attractive interaction. The exponents which determine the presence of singlet/triplet superconducting pairing fluctuations, and spin/charge density wave fluctuations are strongly affected by the presence of the Wentzel-Bardeen singularity, resulting in the favoring of superconducting fluctuations. For the Hubbard model the equivalent of a phase diagram is established, as a function of: the electron-phonon coupling, the electron filling factor, and the on-site repulsion between electrons. The Wentzel-Bardeen singularity can be reached for arbitrary values of the electron-phonon coupling constant by varying the filling factor. This provides an effective mechanism to push the system from the antiferromagnetic into the metallic phase, and finally into the superconducting phase as the electron filling factor is increased towards half-filling.
We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent K(rho) >1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard U, phonon coupling constant, and phonon frequency.
We derive an effective electronic Hamiltonian for the square lattice Hubbard-Holstein model (HHM) in the strong electron-electron $(e\text{\ensuremath{-}}e)$ and electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling regime and under nonadiabatic conditions $(t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1)$, ttt and ${\ensuremath{\omega}}_{0}$ being the electron hopping and phonon frequency respectively. Using the density matrix renormalization-group method, we simulate this effective electronic model on a four-leg cylinder system at quarter filling and present a phase diagram in the $g\text{\ensuremath{-}}U$ plane where ggg and UUU are the $e\text{\ensuremath{-}}ph$ coupling constant and Hubbard on-site interaction respectively. For larger ggg, we find that a cluster of spins, i.e., phase separation (PS), gives way to a charge density wave (CDW) phase made of nearest-neighbor singlets which abruptly goes to another CDW phase as we increase UUU. But for smaller ggg, we find a metallic phase sandwiched between PS and the singlet CDW phase. This phase is characterized by a vanishing charge gap but a finite spin gap, suggesting a singlet superconducting phase.