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Quantum Hall systems : braid groups, composite fermions, and fractional charge

Author: Lucjan Jacak
Publisher: Oxford, U.K. ; New York : Oxford University Press, c 2003.
Series: Oxford science publications.; International series of monographs on physics (Oxford, England), 119.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
This text presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It covers the fundamental problems of two-dimensional quantum statistics in terms of topology and the Chern-Simons theory.
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Lucjan Jacak
ISBN: 0198528701 9780198528708
OCLC Number: 52991100
Description: xi, 145 pages : illustrations ; 24 cm.
Contents: 2 Topological methods for description of quantum many-body systems 13 --
2.1 Configuration spaces of quantum many-body systems of various dimensions 13 --
2.1.1 Configuration space of a many-particle system in Euclidean space 14 --
2.2 Quantization of many-body systems 16 --
2.3 The first homotopy group for the many-particle configuration space --
braid groups 18 --
2.4 Braid groups for specific manifolds 20 --
2.4.1 Full braid group for the Euclidean space R[superscript 2] 20 --
2.4.2 Full braid group for the sphere S[superscript 2] 24 --
2.4.3 Full braid group for the torus T 25 --
2.4.4 The full braid group for the three-dimensional Euclidean space R[superscript 3] 28 --
2.4.5 Braid groups for the line R[superscript 1] and the circle S[superscript 1] 28 --
3 Quantization of many-particle systems and quantum statistics in lower dimensions 30 --
3.1 Topological limitations of quantum-mechanical description of many-particle systems 30 --
3.2 Quantum statistics and irreducible unitary representations of braid groups for selected manifolds 31 --
3.2.1 Scalar quantum statistics of particles on the plane R[superscript 2] 31 --
3.2.2 Scalar statistics of particles on the sphere S[superscript 2] 32 --
3.2.3 Scalar statistics of particles on the torus T 32 --
3.2.4 Scalar quantum statistics in three-dimensional Euclidean space R[superscript 3] 33 --
3.2.5 Relation between spin and statistics 34 --
3.2.6 Aharonov-Bohm effect 35 --
3.2.7 Fractional statistics in one-dimensional system 37 --
3.3 Non-Abelian statistics 38 --
3.3.1 Projective permutation statistics 39 --
4 Topological approach to composite particles in two dimensions 41 --
4.1 Mathematical model of composite particles 41 --
4.1.1 Factor groups B[subscript N]/[Omega superscript n subscript N] 43 --
4.1.2 Group [Omega superscript n subscript N] 44 --
4.1.3 One-dimensional unitary representations of the group [Omega superscript n subscript N] 45 --
4.2 Configuration space for the system of composite particles 47 --
4.2.1 Configuration space for two looped particles 47 --
4.2.2 Configuration space of the system of three looped particles of the third order 49 --
5 Many-body methods for Chern-Simons systems 54 --
5.1 Random phase approximation for an anyon gas 54 --
5.2 Correlation energy of an anyon gas 59 --
5.3 Hartree-Fock approximation for Chern-Simons systems 62 --
5.4 Diagram analysis for a gas of anyons 65 --
5.4.1 Self-consistent Hartree approximation for a gas of anyons 65 --
5.4.2 Self-consistent Hartree-Fock approximation for gas of anyons 69 --
6 Anyon superconductivity 71 --
6.1 Meissner effect in an anyon gas at T = 0 71 --
6.2 Gas of anyons at finite temperatures 75 --
6.3 Higgs mechanism in an anyon superconductor 77 --
6.4 Ground state of an anyon superconductor in the Hartree-Fock approximation 81 --
7 The fractional quantum Hall effect in composite fermion systems 83 --
7.1 Hall conductivity in a system of composite fermions 83 --
7.2 Ground-state energy of composite fermion systems 88 --
7.3 Metal of composite fermions 91 --
7.4 BCS (Bardeen-Cooper-Schrieffer) paired Hall state 93 --
8 Quantum Hall systems on a sphere 97 --
8.1 Spherical system 97 --
8.2 Composite fermion transformation 101 --
8.3 Hierarchy 105 --
9 Pseudopotential approach to the fractional quantum Hall states 111 --
9.1 Problems with justification of the composite fermion picture 111 --
9.2 Numerical studies on the Haldane sphere 112 --
9.3 Pseudopotential approach 114 --
9.4 Energy spectra of short-range pseudopotentials 116 --
9.5 Definition of short-range pseudopotential 119 --
9.6 Application to various pseudopotentials 120 --
9.7 Multi-component systems 124 --
A Homotopy groups 130 --
B Correlation function for an anyon gas in the self-consistent Hartree approximation 133.
Series Title: Oxford science publications.; International series of monographs on physics (Oxford, England), 119.
Responsibility: Lucjan Jacak [and others].
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Abstract:

This book presents cutting-edge topics in modern theoretical physics - quantum Hall systems - the subject of two Nobel Prizes in 1985 and 1998.  Read more...

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