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Physical Review Letters

Equivalence of a One-Dimensional Fermion Model and the Two-Dimensional Coulomb Gas

S. T. Chui, Patrick A. Lee

Published Aug 4, 1975• 137 citations
Publisher

Abstract

We show that the one-dimensional Luttinger model generalized to include spin and backward scattering is equivalent to a two-dimensional Coulomb gas. Scaling equations are derived and correlation functions are given simple physical interpretation in terms of the Coulomb gas; e.g., existence of an energy gap can be understood in terms of Debye screening. We conclude that an energy gap exists for ${U}_{\ensuremath{\parallel}}<|{U}_{\ensuremath{\perp}}|$ so that triplet excitations are nondivergent, and we provide physical arguments to support the exponents proposed by Luther and Emery for singlet excitations for general coupling constant.

DOI
10.1103/physrevlett.35.315

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We show that the one-dimensional Luttinger model generalized to include spin and backward scattering is equivalent to a two-dimensional Coulomb gas. Scaling equations are derived and correlation functions are given simple physical interpretation in terms of the Coulomb gas; e.g., existence of an energy gap can be understood in terms of Debye screening. We conclude that an energy gap exists for U\ensuremath∥<∣U\ensuremath⊥∣{U}_{\ensuremath{\parallel}}<|{U}_{\ensuremath{\perp}}|U\ensuremath∥​<∣U\ensuremath⊥​∣ so that triplet excitations are nondivergent, and we provide physical arguments to support the exponents proposed by Luther and Emery for singlet excitations for general coupling constant.
The equivalence of the one-dimensional two-fermion model, on one hand, with the two-dimensional Coulomb gas and sine-Gordon models on the other is worked out by employing the Jordan boson representation and the point-splitting regularization. The distinction is made throughout between the bandwidth cut-off and momentum-transfer cut-off. It is shown that the coupling constants, both in the weak-coupling regime and near the Luther-Emery exact solution, scale with the momentum-transfer cut-off. The relationship between the bosonized form of the two-fermion model and the corresponding Fermi gas model is briefly discussed.
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.
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.
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.
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 Fermi gas and the one-particle Green's functions are calculated within the one-body (soliton) approximation by perturbation expansion from the particular limit solved by Luther and Emery. Fermion properties, in particular the Ward identity, of the Luttinger model are explicitly used to calculate the backward scattering amplitude. The gap in the excitation spectrum is in agreement with the results of other authors. At nonzero temperature the quasiparticles have a lifetime. As a consequence, for instance, the specific heat at low temperatures is linear in TTT and not exponentially activated.