ABSTRACT - DRUM - University of Maryland

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This conclusion can be generalized to the case where multiple vortices are present. In that case the right hand side is basically the sum of vorticities of all vortices. The index theorem (3.37) requires chiral symmetry which is broken by presence of a finite chemical potential µ ≠ 0. Now we argue that when chiral symmetry is broken the Majorana zero modes admit a Z 2 classification corresponding to evenodd number of zero energy solutions. Generally speaking, a small chiral symmetry breaking term cause coupling between zero modes and split them away from zero energy. However, due to particle-hole symmetry, the number of zero modes that are split by chiral symmetry breaking term must be even. So the parity of the topological index is preserved in the generic case. This is consistent with an explicit solutions of zero mode in TI/SC heterostruture with finite chemical potential. Thus, we conclude that without chiral symmetry the Majorana zero modes bound to vortices are classified by Z 2 corresponding to even or odd number of zero modes. 60
Chapter 4 Topological Degeneracy Splitting of Majorana Zero Modes In the topological quantum computation scheme based on Majorana non- Abelian vortices in topological superconductors, the quantum information is encoded in the degenerate ground states when there are multiple non-Abelian vortices present. The degeneracy is crucial for the topological protection of the qubits as well as the braiding operations on them. Understanding the fate of ground-state degeneracy of many-anyon system in realistic solid-state structures is a difficult problem of fundamental importance and of relevance to practical realization of topological quantum computing. In this chapter we address one mechanism that may lift the ground state degeneracy associated with the tunneling processes between spatially separated vortices. The presence of the bulk gap protects ground state degeneracy from thermal fluctuations at low temperature leaving out only processes of Majorana fermion quantum tunneling between vortices. Generic features of tunneling of topological charges have been explored recently [86]. The lifting of ground state degeneracy due to intervortex tunneling for a pair of vortices have been studied numerically for ν = 5/2 quantum Hall state [87, 88], p x + ip y superconductor [89] and Kitaev’s honeycomb lattice model [90]. Analytical calculation has been carried 61
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ABSTRACT Title of dissertation: MAJ
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MAJORANA QUBITS IN NON-ABELIAN TOPO
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Acknowledgments First and foremost,
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Table of Contents List of Figures L
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List of Figures 1.1 Braiding of Maj
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Chapter 1 Introduction The subject
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closed throughout the path. If one
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its original configuration, we requ
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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 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: where n = (R∆, −I∆) field des
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
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
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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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