ABSTRACT - DRUM - University of Maryland

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Majorana zero modes can survive under strong quantum fluctuations, especially in one dimension. We first considered a continuum field theory of spinless fermions on a two-leg ladder with pair tunneling, in the presence of quasi-long-range superconducting order. Using bosonization technique we analyze non-perturbatively the strong-coupling phase and found interesting degeneracies of low-energy states that can be interpreted as Majorana zero-energy edge states. We discussed the stability of these degeneracies under various perturbations. Then we proposed a possible lattice realization of this field theory. We now discuss possible future research directions. It is interesting to explore the possible vortex lattice phase in a topological superconductor where low-energy physics can be described by Majorana fermions hopping on the lattice with hopping amplitudes determined by the energy splitting calculated in Chapter 3. More work needs to be done to fully understand the robustness of topological qubits, including the effect of disorder and possible low-energy impurity bound states, and how they affect the braiding and the read out schemes. The effect of quantum fluctuations on Majorana zero modes in higher dimensions remains a open problem although the one-dimensional case has been rather well understood. It would be very interesting to generalize the bosonization approach to higher dimensions. 144
Appendix A Derivation of the Pfaffian Formula for the Chern Parity In this appendix we derive the Pfaffian formula for the parity of the Chern number in a class D topological superconductor, defined by BdG Hamiltonian H(k) satisfying Ξ −1 H(k)Ξ = −H ∗ (−k). (A.1) Assume that H(k) is a 2N × 2N matrix. After diagonalizing we get 2N bands ε m (k). Due to particle-hole symmetry energy eigenvalues come in pairs so we label the bands as ε −m (−k) = ε m (k). We choose a gauge such that the eigenvectors u m (k) satisfy u −m (−k) = Ξu ∗ m(k). The topological invariant for class D superconductors is the Chern number: C = ∑ m
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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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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: effect of quasiparticle tunneling o
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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