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Physical review. B, Condensed matter

Functional-integral approach to an exactly soluble one-dimensional electron-phonon system

Yang Chen, Derek K. K. Lee, Marco U. Luchini

Published Oct 15, 1988• 10 citations
Publisher

Abstract

We consider forward-scattering electron-electron and electron-phonon interactions in an extended Tomonaga-Luttinger model. The method of functional integration over anticommuting fields gives us a bosonized effective action. This enables us to obtain the spectrum and the correlation functions of the system exactly. We consider couplings to both acoustic and optical phonons. In the case of acoustic phonons, we obtain $T\ensuremath{\rightarrow}0$ phase diagram in the plane of coupling constants which indicates the existence of a normal phase between the regions of charge-density-wave and superconducting instabilities. This is qualitatively different from previous results.

DOI
10.1103/physrevb.38.8497

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We consider forward-scattering electron-electron and electron-phonon interactions in an extended Tomonaga-Luttinger model. The method of functional integration over anticommuting fields gives us a bosonized effective action. This enables us to obtain the spectrum and the correlation functions of the system exactly. We consider couplings to both acoustic and optical phonons. In the case of acoustic phonons, we obtain T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0 phase diagram in the plane of coupling constants which indicates the existence of a normal phase between the regions of charge-density-wave and superconducting instabilities. This is qualitatively different from previous results.
On the basis of the extended Hubbard model we consider phonon mediated forward scattering in one-dimensional systems. Exact results for the two-particle spectral functions are derived for the case when Umklapp and backward scattering due to electron–electron interaction are neglected. Due to the electron–phonon interaction the phase diagram of the interacting electron system changes remarkably. In dependence on the phonon frequency and on the electron–phonon coupling constant a new divergence may appear in a two-particle spectral function or an existing one may be removed. The inclusion of Umklapp and backward scattering leads to the appearance of an energy gap and a polaron energy shift. The electron–phonon coupling constant is strongly renormalized and vanishes faster than the charge velocity as half-filling is approached. Beyond half-filling the polaron shift may diverge at certain values of the coupling parameters. This divergency is related to the Wentzel-Bardeen singularity.
A one-dimensional electron gas problem is investigated within the framework of the Luttinger model incorporating both the electron-acoustic-phonon interaction and the electron-optical-phonon interaction together with the electron-electron interaction. It is shown that this model is exactly soluble and the energy spectrum is constituted by bosonic excitations. It is also shown that for certain values of the coupling parameters the system exhibits an instability indicating the possibility of a phase transition.
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.
A new bosonisation procedure based on an exact method of functional integration is applied to the one-dimensional Tomonaga (1950)-Luttinger (1963) model with forward scattering. This method circumvents the conventional procedure where the fermion field operator is represented by (bosonic) charge and spin density fields. Fictitious auxiliary fields are introduced to deal with the interaction term so that the fermionic degrees of freedom can be integrated out exactly. In addition to the usual description of the system in terms of charge and spin fluctuations, an effective action in terms of the auxiliary fields can be used. The evaluation of the Green function is facilitated if the latter route is taken. The authors' results for the spinless and spin-1/2 cases agree with previous work.
We consider strongly correlated electrons coupled to low-energy acoustic phonons in one dimension. Using a Luttinger liquid description we calculate the exponents of various response functions and discuss their remarkable sensitivity to the Wentzel-Bardeen singularity induced by the presence of phonons. For the Hubbard model plus phonons the equivalent of a phase diagram is established. By increasing the filling factor towards half filling the Wentzel-Bardeen singularity is approached. This in turn triggers a simple and efficient mechanism to suppress antiferromagnetic fluctuations and to drive the system via a normal metallic state towards a superconducting phase.