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Journal of Physics C Solid State Physics

Screening in the classical two-dimensional Coulomb gas and the one-dimensional fermion problem

Hanoch Gutfreund, Bernardo A. Huberman

Published May 14, 1977• 3 citations
Publisher

Abstract

The screened interaction V(r) between two opposite charges in the classical two-dimensional Coulomb gas is discussed. This interaction appears in certain response functions of the one-dimensional fermion system, when the latter is mapped onto the two-dimensional Coulomb problem. The effect of the power-law behaviour of V(r) on these response functions, when the large momentum transfer coupling constant (g 1 ) goes to zero from below and above the g 1 =0 line, is discussed.

DOI
10.1088/0022-3719/10/9/001

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The screened interaction V(r) between two opposite charges in the classical two-dimensional Coulomb gas is discussed. This interaction appears in certain response functions of the one-dimensional fermion system, when the latter is mapped onto the two-dimensional Coulomb problem. The effect of the power-law behaviour of V(r) on these response functions, when the large momentum transfer coupling constant (g 1 ) goes to zero from below and above the g 1 =0 line, is discussed.
A one-dimensional Coulomb gas is considered in which the particles represent both charges and dipoles. The values of the charges and dipole moments and the fugacities of that gas have been obtained in the preceding paper by mapping the dynamics of a two-level system (TLS) interacting with a degenerate electron gas onto the present problem. The scaling procedure of Anderson, Yuval, and Hamann is applied to eliminate the close pair which corresponds to the short-time behavior of the TLS. The main contribution arises from pairs of zero total charge, but the role of other pairs is considered as well. The scaling equations show that the electron-assisted transitions of the TLS play a very important role if their value is different from zero for the starting Hamiltonian. In that case localization cannot occur in the case of a spin-(1/2 fermionic system. Localization may occur only if the fermion spin has an artificially large value with spin degeneracy Ns{N}_{s}Ns​>2. That clearly demonstrates that a particle moving in a double potential well may have different dissipative behavior depending upon whether it is coupled to a fermionic or a bosonic heat bath. The screening of the TLS by electrons is described by an arbitrarily large phase shift \ensuremath{\delta} which renormalizes towards the value \ensuremath{\delta}=\ensuremath{\pi}/2Ns{N}_{s}Ns​, but the scaling equations lose their validity as the fugacities corresponding to the TLS transitions become of the order of unity. The method developed is a very general one, and thus it can be applied to other similar one-dimensional gases as well.
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.
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 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.
We continue our previous investigation of the properties of a system of either fermions or bosons interacting in one dimension by a two-body potential V(r)=gr2V(\mathcal{r})=\frac{g}{{\mathcal{r}}^{2}}V(r)=r2g​ with periodic boundary conditions. A method is introduced which provides a physically intuitive description of the excited states. The energies are given explicitly for finite and infinite systems, including all corrections due to quasiparticle interaction. The method is shown to apply to other one-dimensional systems as well. In particular, the results are applied to a classical one-dimensional NNN-body model of Dyson---the Coulomb gas in a Brownian medium.