Cover Page
The handle http://hdl.handle.net/1887/22343 holds various files of this Leiden University dissertation
Author: Fulga, Ion Cosma
Title: Scattering theory of topological phase transitions
Issue Date: 2013-11-21
Scattering theory of
topological phase transitions
Proefschrift
ter verkrijging van
de graad vanDoctor aan de Universiteit Leiden, op gezag vanRector Magnificus
prof. mr. C. J. J. M. Stolker,
volgens besluit van hetCollege voor Promoties te verdedigen op donderdag 21 november 2013
klokke 16.15 uur
door
Ion Cosma Fulga
geboren teBoekarest, Roemenië in 1986
Promotiecommissie
Promotor: Prof. dr. C. W. J. Beenakker
Co-promotor: Dr. A. R. Akhmerov (Delft University of Technology) Overige leden: Prof. dr. E. R. Eliel
Dr. V. Vitelli
Prof. dr. F. Hassler (RWTH Aachen)
Prof. dr. C. M. Marcus (Kopenhagen University)
Casimir PhD Series, Delft-Leiden, 2013-27 ISBN 978-90-8593-167-6
Dit werk maakt deel uit van het onderzoekprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM), die deel uit maakt van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
This work is part of the research programme of the Foundation for Fun- damental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
Cover: Trajectories of zeros and poles of the reflection matrix determinant across a topological phase transition in class DIII. Compare with the top right panel of Fig. 3.5.
To my parents.
Contents
1 Introduction 1
1.1 Preface . . . 1
1.2 The scattering matrix . . . 3
1.3 Topological invariants . . . 5
1.4 Finite size scaling . . . 7
1.5 This thesis . . . 9
1.5.1 Chapter 2 . . . 9
1.5.2 Chapter 3 . . . 9
1.5.3 Chapter 4 . . . 10
1.5.4 Chapter 5 . . . 11
1.5.5 Chapter 6 . . . 11
1.5.6 Chapter 7 . . . 13
2 Scattering formula for the topological quantum number of a disordered multi-mode wire 19 2.1 Introduction . . . 19
2.2 Topological quantum number from reflection matrix . . . 20
2.3 Number of end states from topological quantum number . 23 2.3.1 Superconducting symmetry classes . . . 23
2.3.2 Chiral symmetry classes . . . 26
2.4 Superconducting versus chiral symmetry classes . . . 28
2.5 Application to dimerized polymer chains . . . 31
2.6 Appendix . . . 33
2.6.1 Calculation of the number of end states in class DIII 33 2.6.2 Computing the topological quantum number of a dimerized polymer chain . . . 34
vi CONTENTS
3 Scattering theory of topological insulators and superconductors 39
3.1 Introduction . . . 39
3.1.1 Dimensional reduction in the quantum Hall effect 40 3.1.2 Outline of the paper . . . 41
3.2 Scattering matrix from a Hamiltonian . . . 42
3.3 Dimensional reduction . . . 46
3.4 Results for one–three dimensions . . . 51
3.4.1 Topological invariant in 1D . . . 51
3.4.2 Topological invariant in 2D . . . 51
3.4.3 Topological invariant in 3D . . . 54
3.4.4 Weak invariants . . . 55
3.5 Applications and performance . . . 56
3.5.1 Performance . . . 56
3.5.2 Finite size effects . . . 56
3.5.3 Applications . . . 58
3.6 Conclusion . . . 60
3.7 Appendix . . . 61
3.7.1 Introduction to discrete symmetries . . . 61
3.7.2 Calculation of the number of poles . . . 66
4 Topological quantum number and critical exponent from con- ductance fluctuations at the quantum Hall plateau transition 71 4.1 Introduction . . . 71
4.2 Topological quantum number and conductance resonance 72 4.3 Numerical simulations in a disordered system . . . 75
4.4 Discussion and relation to the critical exponent . . . 77
4.5 Appendix . . . 78
4.5.1 Calculation of the topological quantum number . . 78
4.5.2 Calculation of the critical exponent . . . 81
5 Thermal metal-insulator transition in a helical topological su- perconductor 87 5.1 Introduction . . . 87
5.2 Chiral versus helical topological superconductors . . . 88
5.3 Class DIII network model . . . 91
5.3.1 Construction . . . 91
5.3.2 Vortices . . . 93
5.3.3 Vortex disorder . . . 94 5.4 Topological quantum number and thermal conductance . 94
CONTENTS vii
5.5 Topological phase transitions . . . 96
5.5.1 Phase diagram without disorder . . . 96
5.5.2 Scaling of the critical conductivity . . . 96
5.5.3 Phase diagram with disorder . . . 98
5.5.4 Critical exponent . . . 99
5.6 Conclusion . . . 100
5.7 Appendix . . . 102
5.7.1 Location of the critical point in the network model without disorder . . . 102
5.7.2 Finite-size scaling analysis . . . 103
6 Adaptive tuning of Majorana fermions in a quantum dot chain109 6.1 Introduction . . . 109
6.2 Generalized Kitaev chain . . . 112
6.3 System description and the tuning algorithm . . . 113
6.4 Testing the tuning procedure by numerical simulations . . 117
6.5 Conclusion . . . 121
6.6 Appendix . . . 121
6.6.1 System parameters in numerical simulations . . . . 121
7 Statistical Topological Insulators 129 7.1 Introduction . . . 129
7.2 Construction of an STI topological invariant . . . 131
7.3 STIs with reflection symmetry . . . 134
7.4 Numerical Simulations . . . 134
7.5 Conclusions and discussion . . . 138
7.6 Appendix . . . 138
7.6.1 Tight-binding construction for statistical topologi- cal insulators with reflection symmetry . . . 138
7.6.2 Localization on the Majorana triangular lattice by broken statistical reflection symmetry . . . 141
Samenvatting 145
Summary 147
List of Publications 151
Curriculum Vitæ 153
viii CONTENTS
