Application Of Braid Groups In 2d Hall System Physics: Composite Fermion Structure

by Janusz Jacak, Ryszard Gonczarek, Lucjan Jacak, Ireneusz Jozwiak

Overview

In the present treatise progress in topological approach to Hall system physics is reported, including recent achievements in graphene. The homotopy methods of braid groups turn out to be of particular convenience in order to grasp peculiarity of 2D charged systems upon magnetic field resulting in Laughlin correlations. The real progress in understanding of structure and role of composite fermions in Hall system is provided. The crucial significance of carrier mobility apart from interaction in creation of the fractional quantum Hall effect (FQHE) is described and supported by recent graphene experiments. Recent progress in FQHE field including topological insulators and optical lattices was reviewed and commented in terms of braid group approach. The braid group methods are presented from more general point of view including proposition of pure braid group application.

Product Details

ISBN-13:	9789814412025
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/07/2012
Pages:	160
Product dimensions:	6.00(w) x 9.00(h) x 0.40(d)

Table of Contents

Preface v

Acknowledgments vii

1 Introduction 1

2 Elements of Hall system physics in 2D spaces 5

2.1 Laughlin function 6

2.2 Composite fermions 7

2.2.1 Composite fermions in Jain's model 7

2.2.2 Composite fermions in Read's model 9

2.2.3 Local gauge transformations corresponding to Jain's flux tubes and Read's vortices in the structure of composite fermions 12

3 Topological methods for the description of many particle systems at various manifolds 15

3.1 Braid groups 15

3.1.1 Full braid groups for R³, R², sphere S² and torus T 16

3.1.2 Pure braid group 17

3.2 Feynman integrals over trajectories and the relation with the one-dimensional unitary representations of the full braid group 19

3.3 Bosons, fermions, anyons and composite particles 20

3.3.1 Anyons on the plane, sphere and torus 20

3.3.2 Quantum statistics and braid groups 21

3.4 Multidimensional unitary irreducible representations of braid groups 23

4 Cyclotron braids for multi-particle-charged 2D systems in a strong magnetic field 25

4.1 Insufficient length of cyclotron radii in 2D systems in a strong magnetic field 25

4.2 Definition of the cyclotron braid subgroup and its unitary representations 27

4.3 Multi-loop trajectories-the response of the system to cyclotron trajectories that are too short 30

4.4 Cyclotron structure of composite fermions 34

4.5 The role of the Coulomb interaction 38

4.6 Composite fermions in terms of cyclotron groups 41

4.7 Hall metal in the description of cyclotron groups 44

4.8 Comments on restrictions for the multi-loop structure of cyclotron braids 45

4.8.1 Periodic character of wave packets' dynamics 47

4.8.2 Quasi-classical character of quantization of the magnetic field flux 49

4.9 Cyclotron groups in the case of graphene 52

5 Recent progress in FQHE field 59

5.1 The role of carrier mobility in triggering fractional quantum Hall effect in graphene 59

5.2 Development of Hall-type experiment in conventional semiconductor materials 70

5.3 Topological insulators-new state of condensed matter 73

5.3.1 Chern topological insulators 76

5.3.2 Spin-Hall topological insulators 79

5.4 Topological states in optical lattices 84

6 Summary 89

7 Comments and supplements 93

7.1 The wave function for a completely filled lowest Landau level 93

7.2 Paired Pfaffian states 97

7.2.1 Fermi sea instability toward the creation of Cooper pairs in the presence of particle attraction 99

7.3 Basic definitions in group theory 101

7.4 Homotopy groups 105

7.4.1 Definition of homotopy 106

7.4.2 Homotopic transformations 106

7.4.3 Properties of homotopy 106

7.4.4 Loop homotopy 107

7.5 Configuration space 112

7.5.1 First homotopy group of configuration space for many particle systems 114

7.5.2 Covering space 116

7.6 Braid groups for the chosen manifolds 117

7.6.1 Braid group for a two-dimensional Euclidean space R² 117

7.6.2 Braid group for a sphere S² 122

7.6.3 Braid group for a torus T 123

7.6.4 The braid group for the three-dimensional Euclidean space R³ 127

7.6.5 Braid group for a line R¹ and a circle S¹ 128

7.7 Exact sequences for braids groups 129

7.8 The use of pure braid groups in classic information processing 130

Bibliography 135

Index 145