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Physica Scripta

Scaling equations for the one-dimensional two-fermion model

C. Mantea, M Aposto, Victor Bârsan, A. Corciovei, Rosario Fazio, G Giaguinta

Published Mar 1, 1989• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/39/3/002

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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 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.
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.
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.
Recently, critical exponents in the continuum Baxter model have been calculated using an equivalence of this model to the Luttinger model of a one-dimensional spinless fermion gas. We discuss here the extension of this method to the two-dimensional planar model, using the Stoeckly-Scalapino representation of the transfer matrix. This representation is shown to be equivalent to the spin-1/2 fermion gas in one dimension, which further separates into a Luttinger model and the quantum-mechanical sine-Gordon equation. We emphasize the role that soliton bound states play in determining the critical properties. We find that a critical temperature Tc{T}_{c}Tc​ exists below which the susceptibility is infinite but without long-range order. At Tc{T}_{c}Tc​, the correlation-function exponent $\ensuremath{\eta}$ takes the values 181\sqrt{8}18​, and decreases as the temperature is lowered, consistent with low-temperature results. As in the Baxter-model calculations, we argue that these results are exact for the continuum limit of the transfer matrix, and therefore provide a solution for the asymptotic properties near Tc{T}_{c}Tc​. We discuss the correlation length and susceptibility, and sugget that nonuniversality is responsible for the disagreement between different numerical calculations.