ABSTRACT - DRUM - University of Maryland

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product of det U(Γ) where Γ is a particle-hole symmetric momentum (Γ = −Γ + G where G is reciprocal lattice vector). We can relate the det U to the Hamiltonian. Define a matrix W (Γ) = H(Γ)Ξ. First we can show that it is anti-symmetric: W T (Γ) = Ξ T H T (Γ) = −Ξ T Ξ −1 H(−Γ)Ξ = −H(Γ)Ξ = W (Γ). On the other hand, we have W (Γ) = H(Γ)Ξ = U(Γ)D(Γ)U † (Γ)Ξ. Notice that ΞU(k) = U ∗ (k)Λ, so U(Γ)D(Γ)U † (Γ)Ξ = U(Γ)D(Γ)ΛU T (Γ). One can show that D(Γ)Λ is anti-symmetric. Therefore Pf W (Γ) = Pf [ U(Γ)D(Γ)ΛU T (Γ) ] = det U(Γ)Pf [D(Γ)Λ] . Combining, we finally obtain e iπC = ∏ Γ Pf W (Γ). 148
List of Publications Publications this thesis is based on: 1. Meng Cheng, Roman Lutchyn and S. Das Sarma, Topological Protection of Majorana Qubits, Phys. Rev. B. 85, 165124 (2012) 2. Meng Cheng and Hong-Hao Tu, Majorana Edge States in Interacting Twochain Ladder of Fermions, Phys. Rev. B. 84, 094503 (2011) 3. Meng Cheng, Victor Galitski and S. Das Sarma, Non-Adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors, Phys. Rev. B 84, 104529(2011) 4. Meng Cheng, Roman Lutchyn, Victor Galitski, and S. Das Sarma, Tunneling of anyonic Majorana excitations in topological superconductors, Phys. Rev. B. 82, 094504 (2010) 5. Meng Cheng, Kai Sun, Victor Galitski and S. Das Sarma, Stable topological superconductivity in a family of two-dimensional fermion models, Phys. Rev. B. 81, 024504 (2009) 6. Meng Cheng, Roman Lutchyn, Victor Galitski, and S. Das Sarma, Splitting of Majorana-fermion modes due to intervortex tunneling in a p x + ip y superconductor, Phys. Rev. Lett. 103, 107001 (2009) 149
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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
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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 and 156: Appendix A Derivation of the Pfaffi
Page 157: which allows further simplification
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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