ABSTRACT - DRUM - University of Maryland

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Notice that we choose the first Brillouin zone k x ∈ [−π, π], k y ∈ [−π, π]. We now start simplifying the expression for C. The particle-hole symmetry implies A m (k) = A −m (−k), F m (k) = F −m (−k). Therefore ∑ m
which allows further simplification of C: C = 1 ∑ (∫ π dk x a m (k x , π) − π m 0 ∫ π 0 ) dk x a m (k x , 0) . We can relate the gauge fields a m to the eigenstates of the Hamiltonian. Define the unitary matrix U(k) as U † (k)H(k)U(k) = D(k), where D(k) is the diagonal matrix of eigenvalues ordered in descending order of value. It is easy to show that ∑ A m (k) = −i∇ k ln det U(k). m Therefore As a result, ∑ a m (k, θ) = −i∂ k ln det U(k, θ). m C = 1 [∫ π dk ∂ k ln det U(k, π) − πi 0 ∫ π It can be written in a more natural form: 0 ] dk ∂ k ln det U(k, 0) = 1 det U(π, π) det U(0, 0) ln πi det U(0, π) det U(π, 0) . e iπC = det U(π, π) det U(0, 0) det U(0, π) det U(π, 0) . (A.3) which is the desired result. Although in the derivation we choose a certain gauge, the result is certainly gauge-invariant. We now further simplify this results. (A.3) basically means that the parity of Chern number of a 2D particle-hole symmetric insulator is determined by the 147
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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
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which in principle lead to a strong
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We conclude that the geometric phas
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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 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: Appendix A Derivation of the Pfaffi
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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