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Journal of Physics C Solid State Physics

Coupling between charge and spin degrees of freedom in the one-dimensional Fermi gas with backscattering

F. D. M. Haldane

Published Nov 28, 1979• 53 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/12/22/020

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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.
A recent solution by Luther and Emery of a one-dimensional interacting Fermi-gas model contains a flaw in its final step which when corrected yields qualitatively different behavior. In particular spin-density wave and triplet superconducting pairing excitations are not divergent. The result is consistent with a gap in the uniform spin susceptibility and the gap can now be given a simple physical interpretation. The bulk of the Luther-Emery solution including exponents for the charge-density wave and singlet pairing remains correct.
The Jordan boson representation and point-splitting regularisation are applied to the one-dimensional two-fermion model (TFM) with momentum-transfer cut-off and zero-mode terms included. It is shown that the correlation functions of the Tomonago-Luttinger model obtained in this way coincide with those given by diagram summation. The energy gaps entering the well known Luther-Emery-Peschel back-scattering and Umklapp scattering solutions acquire a physical dependence on the momentum-transfer cut-off. First-order perturbation calculations of the charge-density function give different results for the spin-parallel 'back-scattering' contribution to the bosonised TFM and corresponding Fermi gas model, thus indicating that these models are not equivalent.
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.
The composite-spin representation of the spin-1 Heisenberg chain is used to transform it through the Jordan-Wigner transformation to the one-dimensional fermion gas. To properly include the xy couplings between spins, we also consider the bosonized version of the fermion model. Phase diagrams deduced from the two versions of the fermion model are compared against numerical results for finite Heisenberg chains. One of the symmetries of the spin model is lost in the fermionization, and this leads to a topologically incorrect phase diagram in at least one part of the parameter space. There are clear indications of significant coupling of spin and charge degrees of freedom in the fermion model and of strong renormalization of the coupling constants in the continuum limit. The existence of the Haldane gap is discussed in connection with these results.
The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.