Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Introduction
- 2 The Hubbard model
- 3 The magnetic instability of the Fermi system
- 4 The renormalization group and scaling
- 5 One-dimensional quantum antiferromagnets
- 6 The Luttinger liquid
- 7 Sigma models and topological terms
- 8 Spin-liquid states
- 9 Gauge theory, dimer models, and topological phases
- 10 Chiral spin states and anyons
- 11 Anyon superconductivity
- 12 Topology and the quantum Hall effect
- 13 The fractional quantum Hall effect
- 14 Topological fluids
- 15 Physics at the edge
- 16 Topological insulators
- 17 Quantum entanglement
- References
- Index
6 - The Luttinger liquid
Published online by Cambridge University Press: 05 March 2013
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- 1 Introduction
- 2 The Hubbard model
- 3 The magnetic instability of the Fermi system
- 4 The renormalization group and scaling
- 5 One-dimensional quantum antiferromagnets
- 6 The Luttinger liquid
- 7 Sigma models and topological terms
- 8 Spin-liquid states
- 9 Gauge theory, dimer models, and topological phases
- 10 Chiral spin states and anyons
- 11 Anyon superconductivity
- 12 Topology and the quantum Hall effect
- 13 The fractional quantum Hall effect
- 14 Topological fluids
- 15 Physics at the edge
- 16 Topological insulators
- 17 Quantum entanglement
- References
- Index
Summary
One-dimensional Fermi systems
We will now consider the case of one-dimensional (1D) Fermi systems for which the Landau theory fails. The way it fails is quite instructive since it reveals that in one dimension these systems are generally at a (quantum) critical point, and it will also teach us valuable lessons on quantum criticality. It will also turn out that the problem of 1D Fermi systems is closely related to the problem of quantum spin chains. This is a problem that has been discussed extensively by many authors, and there are several excellent reviews on the subject (Emery 1979; Haldane, 1981). Here I follow in some detail the discussion and notation of Carlson and coworkers (Carlson et al., 2004).
One-dimensional (and quasi-1D) systems of fermions occur in several experimentally accessible systems. The simplest one to visualize is a quantum wire. A quantum wire is a system of electrons in a semiconductor, typically a GaAs–AlAs heterostructure built by molecular-beam epitaxy (MBE), in which the electronic motion is laterally confined along two directions, but not along the third. An example of such a channel of length L and width d (here shown as a two-dimensional (2D) system) is seen in Fig. 6.1. Systems of this type can be made with a high degree of purity with very long (elastic) mean free paths, often tens of micrometers or even longer. The resulting electronic system is a 1D electron gas (1DEG). In addition to quantum wires, 1DEGs also arise naturally in carbon nanotubes, where they are typically multi-component (with the number of components being determined by the diameter of the nanotube).
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- Field Theories of Condensed Matter Physics , pp. 145 - 188Publisher: Cambridge University PressPrint publication year: 2013
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