Martín Topological Orders with Spins and Fermions

Quantum Phases and Computation
1. Auflage 2019
ISBN: 978-3-030-23649-6
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 176 Seiten, eBook

Reihe: Springer Theses

This thesis deals with topological orders from two different perspectives: from a condensed matter point of view, where topological orders are considered as breakthrough phases of matter; and from the emerging realm of quantum computation, where topological quantum codes are considered the most appealing platform against decoherence.  The thesis reports remarkable studies from both sides. It thoroughly investigates a topological order called the double semion model, a counterpart of the Kitaev model but exhibiting richer quasiparticles as excitations. A new model for symmetry enriched topological order is constructed, which adds an onsite global symmetry to the double semion model. Using this topological phase, a new example of topological code is developed, the semion code, which is non-CSS, additive, non-Pauli and within the stabiliser formalism.
Furthermore, the thesis analyses the Rashba spin-orbit coupling within topological insulators, turning the helical edge states into generic edges modes with potential application in spinstronics. New types of topological superconductors are proposed and the novel properties of the correspondingly created Majorana fermions are investigated. These Majorana fermions have inherent properties enabling braiding and the performance of logical gates as fundamental blocks for a universsal quantum computator.

Zielgruppe

Research

Autoren/Hrsg.

Martín, Laura Ortiz

Inhaltsverzeichnis

1;Supervisor’s Foreword;6
2;Abstract;9
3;Outline of the Thesis;12
4;Contents;14
5;Introduction;17
6;1 Topology in Condensed Matter;18
6.1;1.1 SET Phases: Symmetry Fractionalisation;20
6.2;1.2 SPT Phases: Edge States;21
6.3;References;22
7;2 Topology in Quantum Information;26
7.1;2.1 Topological Quantum Computation with Ground State;27
7.2;2.2 Topological Quantum Computation with Excitations;28
7.3;References;29
8;Spin Systems;31
9;3 The Bilayer Double Semion Model;32
9.1;3.1 Introduction;32
9.2;3.2 Summary of Main Results;34
9.3;3.3 The Double Semion Model;34
9.4;3.4 The String-Flux Mechanism;38
9.5;3.5 The Bilayer DS Model;42
9.5.1;3.5.1 Invariance Under Global Symmetry;48
9.6;3.6 New Paramagnetic Phase;50
9.6.1;3.6.1 The Ground State;54
9.6.2;3.6.2 Edge States;57
9.7;3.7 Outlook;59
9.8;References;61
10;4 Double Semion Model as a Quantum Memory;62
10.1;4.1 Introduction;62
10.2;4.2 Summary of Main Results;63
10.3;4.3 Off-Shell Double Semion: Microscopic Model;64
10.3.1;4.3.1 Double Semion Model on the Vertex-Free Subspace;64
10.3.2;4.3.2 Exactly Solvable Model in the Whole Hilbert Space;66
10.4;4.4 Constructing Open-String Operators;68
10.4.1;4.4.1 Algorithm to Generate Strings;69
10.4.2;4.4.2 Example of an Open-String Operator;73
10.4.3;4.4.3 Completeness of String Operators;74
10.5;4.5 Constructing Closed Strings: Logical Operators;76
10.6;4.6 Error Correction with the Semion Code;79
10.7;4.7 Outlook;80
10.8;References;80
11;Fermionic Systems;82
12;5 Topological Insulators;83
12.1;5.1 Introduction;83
12.2;5.2 Summary of Main Results;85
12.3;5.3 Generic Helical Edge States;86
12.3.1;5.3.1 Phenomenology of the Generic Helical Edge States;86
12.3.2;5.3.2 Realistic GHES: The BHZ Model with Rashba SOC;88
12.4;5.4 Outlook;102
12.5;References;102
13;6 Topological Superconductors;105
13.1;6.1 Introduction;105
13.2;6.2 Summary of Main Results;108
13.3;6.3 Microscopic Characterisation;109
13.3.1;6.3.1 Disorder Analysis;113
13.3.2;6.3.2 Topological Phase Diagrams;115
13.4;6.4 Topological Edge States;117
13.4.1;6.4.1 Edge States from d-wave Superconductor;118
13.4.2;6.4.2 Edge States from Nodeless Pairing;123
13.5;6.5 Outlook;126
13.6;References;127
14;Conclusions;131
15;7 Summary of Results;132
16;8 Outlook;135
16.1;References;137
17;A Technical Details on the Bilayer DS Model;138
17.1;A.1 Remarks on the DS Model;138
17.2;A.2 Proofs of Commutativity of the Bilayer Double Semion;142
18;B Technical Details on the Semion Code;146
18.1;B.1 Commutation of the Plaquette Operators on the Vertex-Free Subspace;146
18.2;B.2 Properties of the Generalised Plaquette Operators;147
18.2.1;B.2.1 Hermiticity and Unitarity of the Plaquette Operators;147
18.2.2;B.2.2 Commutation of the Plaquette Operators;148
18.2.3;B.2.3 The Product of All Plaquette Operators Gives the Identity;152
18.3;B.3 Proofs Regarding Algorithm 1;155
18.3.1;B.3.1 Useful Technical Lemmas;156
18.3.2;B.3.2 Proof of Theorem 1;161
18.4;B.4 Properties of Open Crossing Strings;161
19;C Technical Details on How to Calculate GHESs;165
20;D Technical Details on Derivation of Majorana States;175
20.1;References;176

Über Autor(innen)

Until now, Laura Ortiz Martin's career has been a purely scentific, although she has addressed topics which could be very useful for entrepreneurial environment nowadays, focusing in Quantum Technologies. She obtained her first degree and M.Sc in Madrid, at the Universidad Complutense, then continued her research at CSIC, the National Research Institute, before returning to the Universidad Complutense to for her Ph.D. During her doctoral studies , she attended many international conferences and visited numerous research groups around the world. Currently, she is a Postdoctoral Fellow at the Hebrew University of Jerusalem.