ABSTRACT - DRUM - University of Maryland

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etween these Majorana fermions, which is equivalent to the process of tunneling of a Majorana quasiparticle from one place to another, split the topological degeneracy. The energy splitting determines the fusion channel of two non-Abelian vortices. However, such tunneling process has to overcome the bulk gap, very similar to tunneling of a quantum-mechanical particle through a potential barrier that is higher than its kinetic energy. So the splitting is exponentially suppressed in the topological phase as e −R/ξ where R is the separation between anyons and ξ is the correlation length (the coherence length in a superconductor). Another possibility is thermal excitations of non-Majorana fermionic modes. The process (1.33) has non-vanishing probability to occur at finite temperature and as such, represents a thermal decoherence of Majorana qubits. This issue is particularly pronounced when there are low-energy (but not zero) bound states present together with the Majorana zero-energy states which is the case in superconducting vortices. At a more fundamental level, the BCS theory which all our discussions are based on, is a mean-field theory neglecting all quantum and thermal superconducting fluctuations. In three-dimensional electronic superconductors the fluctuations are gapped due to the famous Anderson-Higgs mechanism [69] and the relevant energy scale is the plasmon frequency. However, since non-Abelian topological superconductors all exist in dimensions smaller than three, the fluctuation effect needs to be reconsidered. For example, in quasi-two-dimensional superconductors the London penetration length is inversely proportional to the thickness of the system. In the limit of vanishing thickness, the superconducting fluctuations are essentially 28
gapless and should not be neglected for low-energy physics. 1.7 Outline of the Thesis In this thesis, we present a systematic study of non-Abelian topological superconductors, focusing on the interplay between non-topological aspects and the topological degrees of freedom, motivated by the question of how these non-universal effects affect topological quantum computation. The model system that is primarily concerned is the chiral p-wave superconductor. In Chapter 2 we address the question of how chiral p-wave superconductivity arises from microscopic lattice models with four-fermion interaction. By analyzing the BCS energetics of a minimal “Hubbard” model of spinless fermions with nearest-neighbor interaction on a two-dimensional lattice, we show that the pairing symmetry selected by the energetics strongly depends on the spatial symmetry. We prove a general theorem specifying a set of necessary conditions that guarantees the existence of chiral p-wave superconductivity. In Chapter 3, we review the Majorana bound states as zero-energy solutions of BdG equations when a superconducting vortex is present in the order parameter. These analytical expressions will be used frequently in the rest of the thesis. From Chapter 4 we turn to the study of non-topological effects. In Chapter 4 we calculate the splitting of topological degeneracy due to quasiparticle tunneling when there are multiple non-Abelian vortices. We devise a WKB-like method to calculate the energy splitting for p x +ip y superconductors and also TI/Superconductor 29
Page 1 and 2: ABSTRACT Title of dissertation: MAJ
Page 3 and 4: MAJORANA QUBITS IN NON-ABELIAN TOPO
Page 5 and 6: Acknowledgments First and foremost,
Page 7 and 8: Table of Contents List of Figures L
Page 9 and 10: List of Figures 1.1 Braiding of Maj
Page 11 and 12: Chapter 1 Introduction The subject
Page 13 and 14: closed throughout the path. If one
Page 15 and 16: its original configuration, we requ
Page 17 and 18: mathematics. It is easy to check th
Page 19 and 20: Here τ x is the Pauli matrix actin
Page 21 and 22: y constructing the ground state pro
Page 23 and 24: here and leaves its proof to Append
Page 25 and 26: the zero mode (see Chapter 3 for de
Page 27 and 28: In the case of Ising anyons, 2N vor
Page 29 and 30: find a realistic material in nature
Page 31 and 32: states near the Fermi circle. We fi
Page 33 and 34: pairing: ( ) p H = ψ † 2 2m −
Page 35 and 36: dimensional non-chiral Majorana mod
Page 37: is formed out of two fermions, does
Page 41 and 42: zero modes. We study a one-dimensio
Page 43 and 44: of possible paired states which is
Page 45 and 46: where d ≡ (d 1 , d 2 , d 3 ) = (R
Page 47 and 48: If the order parameter is proportio
Page 49 and 50: 0.015 T/t 0.01 Normal p x + ip y 0.
Page 51 and 52: trivial p x - and p y -states lead
Page 53 and 54: with the gap operator being ˆ∆ =
Page 55 and 56: 0.2 T/t 0.1 Normal f p x + ip y µ
Page 57 and 58: 2.3 Discussion and Conclusions In c
Page 59 and 60: Chapter 3 Majorana Bound States in
Page 61 and 62: satisfied for m = 0. The singlevalu
Page 63 and 64: We summarize this section by provid
Page 65 and 66: Here γ a and Γ a are 4 × 4 Dirac
Page 67 and 68: with λ = µξ/v. It has the follow
Page 69 and 70: where n = (R∆, −I∆) field des
Page 71 and 72: Chapter 4 Topological Degeneracy Sp
Page 73 and 74: ound states are usually splitted an
Page 75 and 76: sponding quasiparticle operator can
Page 77 and 78: Of particular importance is the sig
Page 79 and 80: [61]. Although analytical expressio
Page 81 and 82: eyond perturbation theory and the r
Page 83 and 84: eaks down in this regime and one sh
Page 85 and 86: 4.3 Conclusions In this chapter, we
Page 87 and 88: the fermion bath, and consequently
Page 89 and 90:
tion for the reduced density matrix
Page 91 and 92:
like phonons. However, such process
Page 93 and 94:
We also notice that in this formula
Page 95 and 96:
etween two topological p x + ip y s
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neglect the AC phase. The Hamiltoni
Page 99 and 100:
which in principle lead to a strong
Page 101 and 102:
We conclude that the geometric phas
Page 103 and 104:
The first-order term vanishes due t
Page 105 and 106:
apply to any realistic system where
Page 107 and 108:
ound states are typically close to
Page 109 and 110:
Chapter 6 Non-adiabaticity in the B
Page 111 and 112:
minor modifications in the details,
Page 113 and 114:
{α(T )} to {α(0)}: |α(T )〉 =
Page 115 and 116:
educed to 2 N−1 by fermion parity
Page 117 and 118:
Berry connection of this state then
Page 119 and 120:
wavefunction is u n (r), v n (r). W
Page 121 and 122:
asis. The most general form of BdG
Page 123 and 124:
6.2.1 Effect of Tunneling Splitting
Page 125 and 126:
Because E + ∼ ∆ 0 e −R/ξ , |
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tunneling amplitudes between them a
Page 129 and 130:
ˆΓ i = iˆγ i ˆξiˆη i , i =
Page 131 and 132:
considerations, we should have β 1
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states: ν(ε) = 2ν 0 ε √ ε2
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necessary as a matter of principle
Page 137 and 138:
Chapter 7 Majorana Zero Modes Beyon
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We then demonstrate that in a finit
Page 141 and 142:
The standard Abelian bosonization r
Page 143 and 144:
and define the chiral fields by ˜
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of (7.2) has a product of the Klein
Page 147 and 148:
7.3 Stability of the Degeneracy We
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quence. The splitting is then propo
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where α = 1 2 (K + K−1 − 2). A
Page 153 and 154:
effect of quasiparticle tunneling o
Page 155 and 156:
Appendix A Derivation of the Pfaffi
Page 157 and 158:
which allows further simplification
Page 159 and 160:
List of Publications Publications t
Page 161 and 162:
Bibliography [1] K. von Klitzing, G
Page 163 and 164:
[46] S. Das Sarma, C. Nayak, and S.
Page 165 and 166:
[92] A. W. W. Ludwig, D. Poilblanc,
Page 167 and 168:
[133] M. V. Berry, Proc. R. Soc. Lo
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