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Topological Orders with Spins and Fermions Quantum Phases and Computation

:	Laura Ortiz Martín
:	Topological Orders with Spins and Fermions Quantum Phases and Computation
:	Springer-Verlag
:	9783030236496
:	1
:	CHF 85.50
:

:	Atomphysik, Kernphysik
:	English

:	176
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

Th s 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.

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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.

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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.

	Supervisor’s Foreword	6
	Abstract	9
	Outline of the Thesis	12
	Contents	14
	Introduction	17
	1 Topology in Condensed Matter	18
	1.1 SET Phases: Symmetry Fractionalisation	20
	1.2 SPT Phases: Edge States	21
	References	22
	2 Topology in Quantum Information	26
	2.1 Topological Quantum Computation with Ground State	27
	2.2 Topological Quantum Computation with Excitations	28
	References	29
	Spin Systems	31
	3 The Bilayer Double Semion Model	32
	3.1 Introduction	32
	3.2 Summary of Main Results	34
	3.3 The Double Semion Model	34
	3.4 The String-Flux Mechanism	38
	3.5 The Bilayer DS Model	42
	3.5.1 Invariance Under Global Symmetry	48
	3.6 New Paramagnetic Phase	50
	3.6.1 The Ground State	54
	3.6.2 Edge States	57
	3.7 Outlook	59
	References	61
	4 Double Semion Model as a Quantum Memory	62
	4.1 Introduction	62
	4.2 Summary of Main Results	63
	4.3 Off-Shell Double Semion: Microscopic Model	64
	4.3.1 Double Semion Model on the Vertex-Free Subspace	64
	4.3.2 Exactly Solvable Model in the Whole Hilbert Space	66
	4.4 Constructing Open-String Operators	68
	4.4.1 Algorithm to Generate Strings	69
	4.4.2 Example of an Open-String Operator	73
	4.4.3 Completeness of String Operators	74
	4.5 Constructing Closed Strings: Logical Operators	76
	4.6 Error Correction with the Semion Code	79
	4.7 Outlook	80
	References	80
	Fermionic Systems	82
	5 Topological Insulators	83
	5.1 Introduction	83
	5.2 Summary of Main Results	85
	5.3 Generic Helical Edge States	86
	5.3.1 Phenomenology of the Generic Helical Edge States	86
	5.3.2 Realistic GHES: The BHZ Model with Rashba SOC	88
	5.4 Outlook	102
	References	102
	6 Topological Superconductors	105
	6.1 Introduction	105
	6.2 Summary of Main Results	108
	6.3 Microscopic Characterisation	109
	6.3.1 Disorder Analysis	113
	6.3.2 Topological Phase Diagrams	115
	6.4 Topological Edge States	117
	6.4.1 Edge States from d-wave Superconductor	118
	6.4.2 Edge States from Nodeless Pairing	123
	6.5 Outlook	126
	References	127
	Conclusions	131
	7 Summary of Results	132
	8 Outlook	135
	References	137
	A Technical Details on the Bilayer DS Model	138
	A.1 Remarks on the DS Model	138
	A.2 Proofs of Commutativity of the Bilayer Double Semion	142
	B Technical Details on the Semion Code	146
	B.1 Commutation of the Plaquette Operators on the Vertex-Free Subspace	146
	B.2 Properties of the Generalised Plaquette Operators	147
	B.2.1 Hermiticity and Unitarity of the Plaquette Operators	147
	B.2.2 Commutation of the Plaquette Operators	148
	B.2.3 The Product of All Plaquette Operators Gives the Identity	152
	B.3 Proofs Regarding Algorithm 1	155
	B.3.1 Useful Technical Lemmas	156
	B.3.2 Proof of Theorem 1	161
	B.4 Properties of Open Crossing Strings	161
	C Technical Details on How to Calculate GHESs	165
	D Technical Details on Derivation of Majorana States	175
	References	176