ABSTRACT - DRUM - University of Maryland

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Figure 7.1: Schematic view of the two chains coupled by pair tunneling(denoted by dashed lines). The chains are bended near the two ends to avoid the single-particle tunneling. move away from the Luther-Emery point, the ˆχ fermions start to interact with each other. Since the Majorana edge states are protected by the bulk gap as well as the single-chain fermion parity [155, 156, 157, 158], we expect the qualitative features hold for the whole regime K − > 1 based on adiabatic continuity. Notice that the bonding sector remains gapless. In our field-theoretical model, the bonding and anti-bonding degrees of freedom are completely decoupled so the gaplessness of the bonding boson does not affect the degeneracy in the anti-bonding sector. 136
7.3 Stability of the Degeneracy We now examine whether the ground state degeneracy we have found has a topological nature. Here we define a topological degeneracy of the ground states by the following criteria: the two degenerate ground states are not distinguishable by any local order parameters(i.e. the difference of the expectation values of any local order parameters in the two ground state must be exponentially small in system size ). By local, we mean local operators in the original fermionic operators , otherwise we can easily find such an operator in the bosonic representation. For example, in the model (7.6) the operator O(x) = cos √ 2πθ − (x) can distinguish the two degenerate ground states. But the operator itself is highly non-local in terms of the original fermionic operators. First of all, by analogy with Majorana chain it is quite obvious that any local operators that involve even numbers of fermion operators on each chain are not able to distinguish the two ground states because such operators always commute with single-chain fermion parity operator. Therefore we only have to consider operators that consist of odd number of single-chain fermion operators. They change the single-chain fermion parity and thus presumably connect the two degenerate ground states. Since all such operators can be decomposed into products of single-particle inter-chain tunneling and backscattering operators, it is sufficient to consider these single-particle operators. Let us start with single-particle inter-chain tunneling O T = ∑ ( ˆψ † ˆψ 2r 1r + h.c.). (7.16) r=R,L 137
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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
Page 95 and 96: etween two topological p x + ip y s
Page 97 and 98: 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: of (7.2) has a product of the Klein
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 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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