ABSTRACT - DRUM - University of Maryland

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onding basis. Hamiltonian in the bonding sector is just a theory of free bosons. In the anti-bonding sector, it reads Ĥ= v 2 [K(∂ x θ) 2 + 1 K (∂ xϕ) 2 ] + 2t ⊥ πa 0 cos √ 2πϕ cos √ 2πθ. (7.20) The bosonic fields ϕ and θ are in the anti-bonding basis. The perturbation (t ⊥ ) term has nonzero conformal spin which implies that two-particle processes are automatically generated by RG flow even when they are absent in the bare Hamiltonian. Therefore, one has to include two-particle perturbations in the RG flow Ĥ 2 = g 1 (πa 0 ) 2 cos √ 8πϕ + g 2 (πa 0 ) 2 cos √ 8πθ. (7.21) The RG flow equations for weak couplings have been derived by Yakovenko [169] and Nersesyan et al. [170]. Here we cite their results [159]: dz ( dl = 2 − K + ) K−1 z 2 dy 1 = (2 − 2K)y 1 + (K − K −1 )z 2 dl , (7.22) dy 2 = (2 − 2K −1 )y 2 + (K −1 − K)z 2 dl dK = 1 dl 2 (y2 2 − y1K 2 2 ) where the dimensionless couplings are defined as z = t ⊥a 2πv and y 1,2 = g 1,2 πv . Since we are interested in the phase where the pair tunneling dominates at low energy, we assume K > 1 so y 1 is irrelevant and can be put to 0. Also we neglect renormalization of K. Integrating the RG flow equations with initial conditions z(0) = z 0 ≪ 1, y 2 (0) = 0 we obtain y 2 (l) = z 2 0 K −1 − K [ e 2(1−α)l − e ] 2(1−K−1 )l , (7.23) 2α 140
where α = 1 2 (K + K−1 − 2). Assume K −1 < α, then the large-l behavior of y 2 is dominated by e 2(1−K−1 )l . y 2 becomes of order of 1 at l ∗ ≈ − ln z 0 /(1 − K −1 ), where the flow of z yields z(l ∗ ) ≈ z (α−K−1 )/(1−K −1 ) 0 ≪ 1 given z 0 ≪ 1. This means that if K > √ 2 + 1 (so K −1 < α), then y 2 reaches strong-coupling first. Thus the strong-coupling field theory is given by (7.6). 141
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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
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[61]. Although analytical expressio
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eyond perturbation theory and the r
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eaks down in this regime and one sh
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4.3 Conclusions In this chapter, we
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the fermion bath, and consequently
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tion for the reduced density matrix
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like phonons. However, such process
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We also notice that in this formula
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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 , |
Page 127 and 128: 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
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: quence. The splitting is then propo
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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