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Physical review. B, Solid state

Importance of cutoffs in the one-dimensional electron gas

Gary S. Grest

Published Dec 1, 1976• 11 citations
Publisher

Abstract

Several cutoff procedures adopted for the one-dimensional electron gas are investigated. The ultraviolet cutoff introduced by Luther and Emery for their exactly soluble model is compared to physical cutoffs which are introduced for models which have short- or long-ranged interactions. Lowest-order scaling equations are derived for a model which includes long-range forces.

DOI
10.1103/physrevb.14.5114

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Journal of Physics C Solid State Physics•Nov 28, 1979•53 citations
Several cutoff procedures adopted for the one-dimensional electron gas are investigated. The ultraviolet cutoff introduced by Luther and Emery for their exactly soluble model is compared to physical cutoffs which are introduced for models which have short- or long-ranged interactions. Lowest-order scaling equations are derived for a model which includes long-range forces.
We discuss a variety of renormalization-group methods for treating the one-dimensional interacting electron gas when the cutoff for the energy transfer is a phonon frequency ${\ensuremath{\omega}}_{D}$, while the electron bandwidth is of order EF{E}_{F}EF​, with ${\ensuremath{\omega}}_{D}<{E}_{F}$. Our results, consistent only in the region of weak coupling, give the scaling trajectories for the coupling constants and cutoffs. The relationship to the conventional one-cutoff renormalization-group approach is discussed.
The model one-dimensional electron gas, defined by a particular ultraviolet cut-off procedure, is shown to decompose into two parts, a Tomonaga-Luttinger model and a model equivalent to the two-dimensional Coulomb gas. For a particular value of one of its coupling constants, this latter model is shown to be equivalent to a model of free fermions. These results, familiar features of the boson representation of the one-dimensional electron gas, are here derived entirely within the context of fermions. The cut-off procedure used differs from those normally considered in the literature. The significance of these differences 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 $T\ensuremath{\ne}0$.
We study the one-dimensional electron gas with parabolic dispersion and a repulsive delta-function interaction potential. Using the ladder theory we obtain analytical results for the ground-state energy and the compressibility which are in agreement with exact results for weak and strong coupling. We calculate the short- and the long-distance behaviour of the pair-correlation function g(z) and derive critical exponents from . We evaluate the connection between the local-field correction and critical exponents.
A slightly corrected version of the 'boson representation' of fermions in the Luttinger model (that avoids the alpha to 0 limit) is used to obtain explicitly the lowest-order interactions between charge- and spin-density waves in the generalised model that includes backscattering. This is in contrast to the decoupling of these degrees of freedom reported by Luther and Emery. Instead, the results obtained by these authors are identified as applying to a model with long-range coupling between spin densities at different points.