ABSTRACT - DRUM - University of Maryland

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elevant to Majorana fermions in 2D p x + ip y superconductor since bound states in vortices often dominate at low energy. However, for Majorana fermions in onedimensional systems, zero-energy state is the only subgap state and coupling between zero-energy state and continuum of excited states may become important. Again we write time-dependent BdG equation in the instantaneous eigenbasis iċ 0,1 = − ∑ λ (β 1λ c λ − β ∗ 1λc λ ) iċ 0,2 = − ∑ (β 2λ c λ − β2λc ∗ λ ) λ . (6.44) ic˙ λ = (ε λ − α λ )c λ − β1λc ∗ 0,1 − β2λc ∗ 0,2 ic˙ λ = −(ε λ − α λ )c λ + β 1λ c 0,1 + β 2λ c 0,2 Here we still use λ to label the excited states. As in the case of bound states, we ignore the coupling between excited states since these only contribute higher order terms to the dynamics of zero-energy states. In another word, we treat each excited state individually and in the end their contributions are summed up. To proceed we need to determine matrix elements β 1λ and β 2λ . We will consider the spinless one-dimensional p-wave superconductor as an example. The zeroenergy states localize at two ends of the 1D system which lie in the interval [0, L]. We also assume reflection symmetry with respect to x = L/2. Without worrying about the tunneling splitting, we can consider the two ends near x = 0 and x = L independently. Then we make use of the fact that BdG equations near x = 0 and x = L are related by a combined coordinate and gauge transformation x → L − x, ∆(x) → −∆(L − x). Therefore, the bound state and the local part of the extended states near x = 0 and x = L are related by gauge transformations. Based on these 120
considerations, we should have β 1λ = β 2λ ≡ β λ , up to exponentially small corrections. Similar argument also applies to vortices in 2D p x + ip y superconductors. As mentioned above, we will make the approximation that each excited state can be treated independently. So we consider the effect of one of the excited states first and omit the label λ temporarily. Again we write ˆd = 1 √ 2 (ˆξ + iˆη). Without loss of generality we also assume that β is real. We find from the solution of timedependent BdG equation that ) ˆγ 1 → 4β2 Et sin2 E2 2 ˆγ 1 + (1 − 4β2 Et sin2 ˆγ E2 2 + 2√ √ 2βε sin 2 Et 2 2β sin Et ˆξ + ˆη 2 E 2 E ) ˆγ 2 → − (1 − 4β2 Et sin2 ˆγ E2 1 − 4β2 Et sin2 2 E2 2 ˆγ 2 + 2√ √ 2βε sin 2 Et 2 2β sin Et ˆξ + ˆη E 2 E ˆξ → 2√ ( ) √ . 2βε sin 2 Et 2 (−ˆγ E 2 1 + ˆγ 2 ) + cos Et + 8β2 sin 2 Et 2 2ε sin Et ˆξ − ˆη E 2 E ˆη → 2√ 2β sin 2 Et 2 (−ˆγ 1 + ˆγ 2 ) + E Here again E = √ ε 2 + 4β 2 . ε sin Et E ˆξ + cos Et ˆη (6.45) At first glance the physics here is very similar to what has been discussed for local bound states: non-adiabatic transitions cause changes of fermion parity in the ground state subspace. The crucial difference between local bound states and a continuum of extended states is that, in the former case, local fermion parity is still conserved as long as we count fermion occupation in the excited states while in the latter, it is impossible to keep track of the number of fermions leaking into the continuum so the notion of local fermion parity breaks down. These non-adiabatic effects may pose additional constraints on manufacturing of topological qubits. Let’s consider to what extent the braiding statistics is affected. A useful quantity to look 121
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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
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Here τ x is the Pauli matrix actin
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y constructing the ground state pro
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here and leaves its proof to Append
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the zero mode (see Chapter 3 for de
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In the case of Ising anyons, 2N vor
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find a realistic material in nature
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states near the Fermi circle. We fi
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pairing: ( ) p H = ψ † 2 2m −
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dimensional non-chiral Majorana mod
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is formed out of two fermions, does
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gapless and should not be neglected
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zero modes. We study a one-dimensio
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of possible paired states which is
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where d ≡ (d 1 , d 2 , d 3 ) = (R
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If the order parameter is proportio
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0.015 T/t 0.01 Normal p x + ip y 0.
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trivial p x - and p y -states lead
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with the gap operator being ˆ∆ =
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0.2 T/t 0.1 Normal f p x + ip y µ
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2.3 Discussion and Conclusions In c
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Chapter 3 Majorana Bound States in
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satisfied for m = 0. The singlevalu
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We summarize this section by provid
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Here γ a and Γ a are 4 × 4 Dirac
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with λ = µξ/v. It has the follow
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where n = (R∆, −I∆) field des
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Chapter 4 Topological Degeneracy Sp
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ound states are usually splitted an
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sponding quasiparticle operator can
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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
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 , |
Page 127 and 128: tunneling amplitudes between them a
Page 129: ˆΓ i = iˆγ i ˆξiˆη i , i =
Page 133 and 134: states: ν(ε) = 2ν 0 ε √ ε2
Page 135 and 136: necessary as a matter of principle
Page 137 and 138: Chapter 7 Majorana Zero Modes Beyon
Page 139 and 140: 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 ˜
Page 145 and 146: of (7.2) has a product of the Klein
Page 147 and 148: 7.3 Stability of the Degeneracy We
Page 149 and 150: quence. The splitting is then propo
Page 151 and 152: 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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ABSTRACT - DRUM - University of Maryland

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