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Physical review. B, Solid state

Solution of the one-dimensional electron gas on a lattice

Victor J. Emery, Alan H. Luther, Ingo Peschel

Published Feb 1, 1976• 91 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.13.1272

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Umklapp Scattering in Fermi Liquids: A Renormalisation Group Approach

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arXiv:cond-mat/9704226•Physical review. B, Condensed matter•Nov 1, 1997•1 citation
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.
The energy spectrum of the one-dimensional electron gas, solved by Luther and Emery for a particular attractive interaction strength UUU of the backward scattering across the Fermi surface, has been calculated for small deviations from this particular limit. The energy spectrum shows a gap at the Fermi energy, whose magnitude is very sensitive to the interaction UUU. The density of states shows a gap at zero temperature, which is smeared out by the quasiparticle lifetime for $T\ensuremath{\ne}0$.
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.
Asymptotic properties of nearly half-filled one-dimensional conductors coupled with phonons are studied through a renormalization-group method. Due to spin-charge coupling via an electron-phonon interaction, the spin correlation varies with filling as well as the charge correlation. Depending on the relation between cutoff energy scales of the umklapp process and of the electron-phonon interaction, various phases appear. We find a metallic phase with a spin gap and a dominant charge-density-wave correlation near half filling between a gapless density-wave phase (like in the doped repulsive Hubbard model) and a superconductor phase with a spin gap. The spin gap is produced by phonon-assisted backward scatterings that interfere with the umklapp process constructively or destructively depending on the character of electron-phonon coupling. \textcopyright{} 1996 The American Physical Society.
The role of Umklapp scattering for electrons on a lattice is studied. Using renormalisation group methods it is shown that the effective low energy interaction involves not only forward scattering, but also non-forward Umklapp scattering. We show that the latter can happen to almost all of the electrons even when they have a filling of around one-half. The bahavior of these Umklapp couplings is studied in the loop expansion. It is shown that they remain marginal to all loop orders. The discussion is given in the context of two-dimensional lattices, but generalisation to three-dimensions is mentioned. We also briefly discuss the possible consequences of this on Fermi liquid theory and on physical properties of such electronic systems.
The one-dimensional Kondo lattice model with attractive interaction among the conduction electrons is analyzed in the case of half-filling. It is shown that there are three distinct phases depending on the coupling constants of the model. Two phases have a spin and charge gap. While one shows a clear distinction of the spin and charge excitation spectrum the other phase may be characterized as a band insulator type where both excitations are due to two-particle states. The third phase is gapless in both channels and has quasi-long-range order in the spin- and charge-density-wave correlation. In this phase the spin and charge excitations have a clearly separated spectrum. For the analysis we discuss first two limiting cases. Then a density matrix renormalization group calculation on finite systems is applied to determine the phase diagram and the correlation functions in the gapped and gapless phase for general coupling constants.