Table of contents 1. Introduction 2. Non-Fermi-liquid features of Fermi liquids: 1D physics in higher dimensions 3. Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model 4. Renormalization group for interacting fermions 5. Single impurity in a 1D system: scattering theory for interacting electrons 6. Bosonization solution 7. Transport in quantum wires 7.1 Conductivity and conductance 7.2 Dissipation in a contactless measurement 7.3 Conductance of a wire attached to reservoirs 7.4 Spin component of the conductance 7.5 Thermal conductance: Fabry-Perrot resonances of plasmons 8. Appendices
Table of contents 1. Introduction 2. Non-Fermi-liquid features of Fermi liquids: 1D physics in higher dimensions 3. Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model 4. Renormalization group for interacting fermions 5. Single impurity in a 1D system: scattering theory for interacting electrons 6. Bosonization solution 7. Transport in quantum wires 7.1 Conductivity and conductance 7.2 Dissipation in a contactless measurement 7.3 Conductance of a wire attached to reservoirs 7.4 Spin component of the conductance 7.5 Thermal conductance: Fabry-Perrot resonances of plasmons 8. Appendices
This chapter discusses the technique of bosonization for studying systems of interacting fermions in one dimension. It reviews the low-energy properties of Fermi and Luttinger liquids and presents some of the relations between bosonic and fermionic operators in one dimension. The chapter explores these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. It explains the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension and transport in clean quantum wires and in the presence of isolated impurities. The basic idea of bosonization is that there are certain objects which can be calculated either in a fermionic theory or in a bosonic theory, and the two calculations give the same answer. The chapter also discusses another application of bosonization, which is to study transport, in particular the DC conductivity in one-dimensional wires of interacting fermions.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension and transport in clean quantum wires and in the presence of isolated impurities.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension, and transport in clean quantum wires and in the presence of isolated impurities.
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.
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.
The electronic systems in one dimension have features quite distinct from those in higher dimensions. The onedimensional Fermi surface consists of two discrete points, while in higher dimensions it is continuous. The low-energy particle-hole pairs’ spectrum, which arises from momentum and energy conservations for wave vector between zero and twice of Fermi wave vector does not exist in one dimension. In higher dimensions, this region of missing states is filled in. The Fourier transform of one-dimensional Coulomb potential diverges. This led to wide spread belief that one-dimensional interacting electronic systems should exhibit non-Fermi liquid (known as Luttinger liquid) behaviour. However, there is still no unambiguous evidence both for the real existence or non-existence of Luttinger liquid behaviour in a realistic quantum wire. This talk briefly reviews key aspects of Fermi liquid approach and Tomonaga-Luttinger liquid approach for a onedimensional system. In this paper, structure factor, pair distribution function and electron-electron scattering, which exhibit similarity between two approaches in low energy and momentum regime, is reviewed.