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CRC Press eBooks

Polarons in One Dimension: Fröhlich–Toyozawa–Luttinger Model

Ashok Chatterjee, Soma Mukhopadhyay

Published Oct 9, 2018• 0 citations
Publisher

Abstract

This chapter presents the energy spectrum and the momentum distribution of the Frohlich–Toyozawa–Luttinger liquid. The one-dimensional many-electron problem has generated renewed interest after the discovery of quasi-one dimensional solids such as carbon nanotubes, quantum wires, and other one-dimensional systems. Tomonaga was the first to solve the one-dimensional interacting fermion problem exactly using bosonization and certain approximations regarding the commutator of the density operators. Marino considered a one-dimensional acoustic many-polaron Hamiltonian within the framework of the Luttinger model and showed that for a certain range of the coupling constants the system would exhibit a metallic behavior because the spectrum contains gapless collective excitations of polaronic plasmons. Rao et al. have exactly solved the one-dimensional interacting electron problem within the framework of Luttinger model incorporating both the electron-optical-phonon interaction and the electron-acoustic-phonon interaction along with the electron-electron interaction to obtain the energy spectrum.

DOI
10.1201/9781315118635-17

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This chapter presents the energy spectrum and the momentum distribution of the Frohlich–Toyozawa–Luttinger liquid. The one-dimensional many-electron problem has generated renewed interest after the discovery of quasi-one dimensional solids such as carbon nanotubes, quantum wires, and other one-dimensional systems. Tomonaga was the first to solve the one-dimensional interacting fermion problem exactly using bosonization and certain approximations regarding the commutator of the density operators. Marino considered a one-dimensional acoustic many-polaron Hamiltonian within the framework of the Luttinger model and showed that for a certain range of the coupling constants the system would exhibit a metallic behavior because the spectrum contains gapless collective excitations of polaronic plasmons. Rao et al. have exactly solved the one-dimensional interacting electron problem within the framework of Luttinger model incorporating both the electron-optical-phonon interaction and the electron-acoustic-phonon interaction along with the electron-electron interaction to obtain the energy spectrum.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
A one-dimensional electron gas problem is investigated within the framework of the Luttinger model incorporating both the electron-acoustic-phonon interaction and the electron-optical-phonon interaction together with the electron-electron interaction. It is shown that this model is exactly soluble and the energy spectrum is constituted by bosonic excitations. It is also shown that for certain values of the coupling parameters the system exhibits an instability indicating the possibility of a phase transition.
On the basis of the extended Hubbard model we consider phonon mediated forward scattering in one-dimensional systems. Exact results for the two-particle spectral functions are derived for the case when Umklapp and backward scattering due to electron–electron interaction are neglected. Due to the electron–phonon interaction the phase diagram of the interacting electron system changes remarkably. In dependence on the phonon frequency and on the electron–phonon coupling constant a new divergence may appear in a two-particle spectral function or an existing one may be removed. The inclusion of Umklapp and backward scattering leads to the appearance of an energy gap and a polaron energy shift. The electron–phonon coupling constant is strongly renormalized and vanishes faster than the charge velocity as half-filling is approached. Beyond half-filling the polaron shift may diverge at certain values of the coupling parameters. This divergency is related to the Wentzel-Bardeen singularity.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.
We have studied one dimensional acoustic phonon linearly coupled to the electron density. The phonon spectrum is linear with cutoff at the Debye frequency, it is straightforward modification of three dimensional optical phonons. Phonon field results in substituting the dynamical coupling for the screened coulomb interaction. We have shown that the electron-phonon coupling quantitatively changed the phase diagram of Luttinger liquid with a single impurity. The existence of slow and fast polaron modes with different weights in different regimes resulted in different renormalization group flow for a weak scatterer and a weak tunneling link. The resulting phase diagram depending on the parameters of the problem, regimes corresponding to purely metallic or purely insulating behavior and an intermediate regime with two stable fixed points and one unstable, finite conductance fixed point. we have found that the direction of renormalization group flow change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. The obtained results were found in good agreement with previously obtained results.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low-energy spectrum of noninteracting fermions is identical to the one for a harmonic chain. This allows us to describe interacting fermions as a system of coupled oscillators. The exact solution of the Tomonaga–Luttinger model is discussed and results for the momentum distribution and spectral functions are presented. The general Luttinger liquid phenomenology is briefly discussed and various attempts at the experimental verification of the theoretical predictions are examined. Extended defects in semiconductors might also be candidate systems.