ABSTRACT - DRUM - University of Maryland

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Chapter 8 Conclusion and Outlook We first present a summary of the findings in this dissertation and then discuss open problems that can be pursued in the future. In the first Chapter, we reviewed the theory of non-Abelian topological superconductors, focusing on the topological classification and the non-Abelian statistics of quasiparticle excitations and briefly introduced the topological quantum computation scheme based on non- Abelian superconductors. In Chapter 2 we established a general condition under which topological superconductivity can arise in lattice models of interacting spinless fermions, within the framework of BCS mean-field theory. We showed that due to the particular convexity property of the BCS free energy, superconducting order parameters with a chiral pairing symmetry are naturally selected by energetics when the point symmetry group has only multidimensional irreducible representations. We also studied the phase diagram and topological phase transitions in other lattice models which do not satisfy the condition. In Chapter 3 we reviewed the solutions of Bogoliubov-de Gennes equation and derived the analytical expressions of the zero-energy Majorana bound states in vortices. We also made the connection to the general index theorem and provided a physical argument that the Majorana zero modes in vortices admit a Z 2 classification. In Chapter 4 we considered the 142
effect of quasiparticle tunneling on the topological degeneracy that is fundamental to the realization of topological qubits, and calculated the energy splitting of the degenerate states using a generalized WKB method. We found the energy splitting exhibits an oscillatory behavior with the inter-vortex distance, apart from the well-known exponential suppression. The presence of these oscillations has important implications for topological quantum computation, since the energy splitting determines the fusion channel of two non-Abelian vortices. In Chapter 5 we turn to the question of thermal effects on the topological quantum computation scheme based on Majorana quasiparticles. We distinguished two types of fermionic excitations that can possibly spoil the topological protection of qubits, the localized midgap states and extended states above the gap, and considered their effects on the braiding, read out and the lifetime of the qubits. We exploited a density matrix formulation based on physical observables and found the topological braiding remains intact in the presence of thermal excitations. However, thermally excited midgap states do result in decoherence in the read out of topological qubits based on vortex interferometry and we derived an analytical expression for the deduction in the interference visibility using a simplified but still physical model. In Chapter 6 we consider the effect of non-adiabaticity on vortex braiding in a microscopic model of a spinless p x + ip y superconductor. We developed a time-dependent Bogoliubovde Gennes equation approach to describe time evolution of BCS superconductors. With the help of this formalism, we studied the robustness of the braiding operations when non-adibaticity is taken into account and calculated the corrections to the Ivanov’s rule perturbatively. In Chapter 7, we addressed the question of whether 143
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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
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 and 150: quence. The splitting is then propo
Page 151: where α = 1 2 (K + K−1 − 2). A
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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