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Progress of Theoretical Physics Supplement

Tomonaga's Theory for Collective Motion of Fermions: Basic Concept of One-Dimensional Correlated Electrons

Norio Kawakami

Published Jan 1, 2007• 2 citations
Publisher

Abstract

In 1950, Tomonaga published a pioneering paper on collective motion of interacting fermions in one spatial dimension, which now establishes the basic concept of one-dimensional quantum critical phenomena, the so-called Tomonaga-Luttinger liquid. We briefly discuss fundamental properties of the Tomonaga-Luttinger liquid in comparison with the Fermi liquid. We then explain how (1+1) dimensional conformal field theory can be used to describe universal properties of the Luttinger liquid. Some examples of recent experimental observations are also addressed.

DOI
10.1143/ptps.170.185

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In 1950, Tomonaga published a pioneering paper on collective motion of interacting fermions in one spatial dimension, which now establishes the basic concept of one-dimensional quantum critical phenomena, the so-called Tomonaga-Luttinger liquid. We briefly discuss fundamental properties of the Tomonaga-Luttinger liquid in comparison with the Fermi liquid. We then explain how (1+1) dimensional conformal field theory can be used to describe universal properties of the Luttinger liquid. Some examples of recent experimental observations are also addressed.
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.
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.
It is well known that, generically, one-dimensional interacting fermions cannot be described in terms of a Fermi liquid. Instead, they present a different phenomenology, that of a Tomonaga-Luttinger liquid: the Landau quasiparticles are ill defined, and the fermion occupation number is continuous at the Fermi energy. We demonstrate that suitable fine tuning of the interaction between fermions can stabilize a peculiar state of one-dimensional matter, which is dissimilar to both Tomonaga-Luttinger and Fermi liquids. We propose to call this state a quasi-Fermi liquid. Technically speaking, such a liquid exists only when the fermion interaction is irrelevant (in the renormalization group sense). The quasi-Fermi liquid exhibits the properties of both a Tomonaga-Luttinger liquid and a Fermi liquid. Similar to a Tomonaga-Luttinger liquid, no finite-momentum quasiparticles are supported by the quasi-Fermi liquid; on the other hand, its fermion occupation number demonstrates a finite discontinuity at the Fermi energy, which is a hallmark feature of a Fermi liquid. A possible realization of the quasi-Fermi liquid with the help of cold atoms in an optical trap is discussed.
The properties of fermions in one dimension are markedly different from those in higher dimensions. In two or three dimensions, the many-body behaviour of fermions is described by the Fermi liquid model, whose most prominent result is that, instead of studying the behaviour of individual particles, it is better to describe the emergent phenomena of the whole system, known as quasiparticles. These are collective modes of excitations born out of particle-particle interactions, which resemble the constituent single particles, albeit with renormalised mass. In one dimension, however, the formation of such a correlated state is unfavourable and the quasiparticles are completely absent. Instead, 1D fermions are described by a model known as the Tomonaga-Luttinger liquid. Central to this model is the process of bosonisation, a mathematical technique which maps the excitation of fermions to bosonic modes of spin-density and charge-density waves. As a prerequisite of the bosonisation technique, the fermionic single-particle dispersion is approximated as a linear function. Consequently, the Luttinger liquid model is limited in its scope of applicability. Specifically, it is valid only in the vicinity of the Fermi points, where the excitation energy is low. Recent advancement in the theoretical field has expanded our understanding of 1D interacting fermions beyond the limit of linearity. The mobile-impurity model, proposed by L. Glazman and his co-authors, is a generalised model of 1D fermions which extends the scope of applicability of the Tomanaga-Luttinger model, accounting for the curvature of single-particle dispersion relations. It extends the region of validity of the tradition TLL theory from the neighbourhood of the Fermi points to a band which follows the single-particle dispersion. This thesis presents the results of an experimental work which provides direct evidence of non-linear behaviour predicted by the new model. It revolves around the measurement of the spectral function of electrons in surface-gate-defined quantum wires on a double-quantum-well, GaAs/AlGaAs device, and the comparison and fitting of the result to the non-linear theory. The spectral function is a theoretical concept frequently used in discussions on many-body problems. Its physical interpretation makes it closely related to experimental observables, such as the tunnelling current between the double quantum wells in this work. This thesis consists of three parts. Part one concerns the theories of the various many-body models and the experimental technique: Chapter 1 briefly introduces the concept of Fermi liquid, before focusing on a detailed account of the Luttinger-liquid model. The mobile-impurity model will be introduced in the framework of the Luttinger liquid model afterwards, followed by a review of experimental works reported in the literature on the study of TLL physics. In Chapter 2, the principles of the experimental tunnelling spectroscopy technique are described. The second part of the thesis details the structure of the test device used in the experiment and its fabrication. In particular, we will discuss the intricacies of the three-dimensional EBL fabrication process developed specifically for this project. The setup and running of the measurement circuit will also be discussed in this part of the thesis, in the context of a prerequisite sample characterisation measurement. In the final part of the work, data from the main experiment will be presented and analysed, in search for the electron-electron interaction effects which are predicted by the theoretical discussions in the first chapter.
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.