ABSTRACT - DRUM - University of Maryland

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these type of states must have a complex pairing order parameter to be energetically stable. 48
Chapter 3 Majorana Bound States in Topological Defects In this chapter, we review the analytic solutions of the Bogoliubov-de Gennes equation for Majorana zero modes in a p x + ip y superconductor and at a topological insulator/superconductor interface. From the explicit solutions we deduce the generic Z 2 classification of Majorana zero-energy modes in superconducting vortices, as well as the Z classification for Dirac-type Hamiltonian when an additional chiral symmetry is present. Some of the results are used in the later chapters. 3.1 Bound states in p x + ip y superconductors The BdG equation for p x + ip y superconductor has been derived in: ⎛ ⎞ ⎛ ⎞ u(r) H BdG ⎜ ⎟ ⎝ ⎠ = E u(r) ⎜ ⎟ ⎝ ⎠ , (3.1) v(r) v(r) where the explicit form of the BdG Hamiltonian in real space is given by ⎛ ⎞ − ∇2 2m − µ 1 {∆(r), ∂ x + i∂ y } k ⎟ F H BdG = ⎜ ⎝ − 1 k F {∆ ∗ (r), ∂ x − i∂ y } ∇ 2 2m + µ with anti-commutator being defined as {a, b} = (ab + ba)/2. ⎟ ⎠ (3.2) The particle-hole symmetry of BdG Hamiltonian is represented by Ξ = τ x K with K being complex conjugation operator and τ x being Pauli matrix in Nambu(particle- 49
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ABSTRACT Title of dissertation: MAJ
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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 and 38: is formed out of two fermions, does
Page 39 and 40: gapless and should not be neglected
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: 2.3 Discussion and Conclusions In c
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
Page 97 and 98: 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
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Chapter 6 Non-adiabaticity in the B
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minor modifications in the details,
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{α(T )} to {α(0)}: |α(T )〉 =
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educed to 2 N−1 by fermion parity
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Berry connection of this state then
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wavefunction is u n (r), v n (r). W
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asis. The most general form of BdG
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6.2.1 Effect of Tunneling Splitting
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Because E + ∼ ∆ 0 e −R/ξ , |
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tunneling amplitudes between them a
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ˆΓ i = iˆγ i ˆξiˆη i , i =
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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
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Chapter 7 Majorana Zero Modes Beyon
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We then demonstrate that in a finit
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The standard Abelian bosonization r
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and define the chiral fields by ˜
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of (7.2) has a product of the Klein
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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
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effect of quasiparticle tunneling o
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Appendix A Derivation of the Pfaffi
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which allows further simplification
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List of Publications Publications t
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Bibliography [1] K. von Klitzing, G
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[46] S. Das Sarma, C. Nayak, and S.
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[92] A. W. W. Ludwig, D. Poilblanc,
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[133] M. V. Berry, Proc. R. Soc. Lo
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ABSTRACT - DRUM - University of Maryland

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