ABSTRACT - DRUM - University of Maryland

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Group(RG) method, assuming the bare couplings g p and g bs are weak. RG flow of the coupling constants are governed by the standard Kosterlitz-Thouless equations [161]: where y − = dy p dl dy bs = (2 − 2K −1 − )y p = (2 − 2K − )y bs dl d ln K − = 2K− −1 y 2 dl −, (7.7) gp πv − , y bs = g bs πv − are the dimensionless coupling constants and l = ln a a 0 is the flow parameter. When K − > 1(corresponding to attractive intra-chain interaction), y p is relevant and flows to strong-coupling under RG flow, indicating gap formation in the anti-bonding sector, while y bs is irrelevant so can be neglected when considering long-wavelength, low-energy physics. Semiclassically, the θ − is pinned in the ground state. From now on, we will assume K − > 1 and neglect the irrelevant coupling y bs . 7.2 Majorana zero-energy edge states To clarify the nature of the gapped phase in the anti-bonding sector, we study the model at a special point K − = 2, known as the Luther-Emery point [162], where the sine-Gordon model is equivalent to free massive Dirac fermions. First we rescale the bosonic fields: ˜ϕ − = ϕ − √ K− , ˜θ − = √ K − θ − , (7.8) 132
and define the chiral fields by ˜ϕ r− = 1 2 ( ˜ϕ −+r˜θ − ). Neglecting the irrelevant backscattering term, Ĥ − is refermionized to Ĥ − = −iv − (ˆχ † R ∂ x ˆχ R − ˆχ † L ∂ x ˆχ L ) + im(ˆχ † R ˆχ† L − ˆχ L ˆχ R ), (7.9) where the Dirac fermionic fields ˆχ r are given by ˆχ r = 1 √ 2πa0 ˆξr e ir√ 4π ˜ϕ r− , (7.10) with the fermion mass m = gp πa 0 . ˆξr are again Majorana operators. It is quite clear that effective theory (7.9) also describes the continuum limit of a Majorana chain, which is known to support Majorana edge states [28]. However, caution has to be taken here when dealing with open boundary condition(OBC). We impose open boundary condition at the level of underlying lattice fermionic operators [163]: ĉ ia ≈ √ a 0 [ ˆψRa (x)e ik F x + ˆψ La (x)e −ik F x ] , (7.11) where ĉ ia are annihilation operators of fermions and x = ia 0 . Since the chain terminates at x = 0 and x = L, we demand ĉ 0 = ĉ N+1 = 0 where N = L/a 0 is the number of sites on each chain. Let us focus on the boundary x = 0. Thus the chiral fermionic fields have to satisfy ˆψ Ra (0) = − ˆψ La (0). Using the bosonization identity, we find ϕ a (0) = √ π , from which we can deduce the boundary condition of 2 the anti-bonding field: ϕ − (0) = 0. (7.12) Therefore, we obtain the boundary condition of the Luther-Emery fermionic fields as ˆχ R (0) = ˆχ L (0). The Hamiltonian is quadratic in ˆχ and can be exactly 133
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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
Page 91 and 92: like phonons. However, such process
Page 93 and 94: 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: The standard Abelian bosonization r
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 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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