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Journal of Physics C Solid State Physics

On the instabilities of the one-dimensional interacting electron gas

D. Baeriswyl, J. J. Forney

Published Jun 20, 1980• 9 citations
Publisher

Abstract

The model Hamiltonian of the one-dimensional electron gas with different types of interaction (forward and backward scattering, Umklapp processes) is studied by assuming a sharp bandwidth cut-off. Exact equalities among different response functions are obtained for specific values of coupling constants from simple symmetry considerations. The ground-state energy derived in mean-field approximation and the asymptotic behaviour of response functions as calculated by various authors are discussed with respect to these exact relations. The agreement of phase diagrams as obtained with different methods is traced back to the underlying symmetry. Giving up this symmetry, e.g. by assuming different cut-offs above and below the Fermi energy, changes the phase diagram. Furthermore this allows one to construct a mean-field ground state where superconductivity and density waves coexist in a finite region of coupling space. If the electron-phonon interaction is also taken into account it follows that the charge-density wave instability (together with a periodic lattice distortion) is favoured.

DOI
10.1088/0022-3719/13/17/012

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•Jan 1, 2010•0 citations
The model Hamiltonian of the one-dimensional electron gas with different types of interaction (forward and backward scattering, Umklapp processes) is studied by assuming a sharp bandwidth cut-off. Exact equalities among different response functions are obtained for specific values of coupling constants from simple symmetry considerations. The ground-state energy derived in mean-field approximation and the asymptotic behaviour of response functions as calculated by various authors are discussed with respect to these exact relations. The agreement of phase diagrams as obtained with different methods is traced back to the underlying symmetry. Giving up this symmetry, e.g. by assuming different cut-offs above and below the Fermi energy, changes the phase diagram. Furthermore this allows one to construct a mean-field ground state where superconductivity and density waves coexist in a finite region of coupling space. If the electron-phonon interaction is also taken into account it follows that the charge-density wave instability (together with a periodic lattice distortion) is favoured.
An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.
Several exact results are proved for two models of one-dimensional conductors, a model with transfer cutoff and spin- and charge-carrying interactions and the usual backscattering model with a bandwidth cutoff. A set of Ward identities is derived which connects the three-point charge vertex parts with single-particle Green's functions. These are combined with a modified renormalization-group scheme to prove the scale invariance of the couplings between charge densities in the absence of umklapp processes. The Ward identities are also used for the computation of the correlation functions in the charge sector of the system. Then some canonical transformations are introduced which allow one to relate various response functions among themselves. From these relations the phase boundaries of the ground state can be established. It is shown that the transfer cutoff and the backscattering models have the same phase diagram.
The effects of Umklapp scattering on electronic states are studied in one spatial dimension at absolute zero. The model is basically the Hubbard model, where parameters characterizing the normal (UUU) and Umklapp (VVV) scattering are treated independently. The density of states is calculated in the t-matrix approximation by taking only the forward and Umklapp scattering into account. It is found that the Umklapp scattering causes the global splitting of the density of states. In the presence of sufficiently strong Umklapp scattering, a pole in the t-matrix appears in the upper half plane, signalling an instability towards the 'G/2−G/2-G/2−pairing' ordered state (GGG is the reciprocal lattice vector), whose consequences are studied in the mean field approximation. It turns out that this ordered state coexists with spin-density-wave state and also brings about Cooper-pairs. A phase diagram is determined in the plane of VVV and electron filling nnn.
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.
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.
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.