ABSTRACT - DRUM - University of Maryland

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Its bosonic representation is O T = 2 πa 0 cos √ 2πϕ − cos √ 2πθ − . (7.17) First let us consider the case when the operator is taken in the bulk of the chain away from any of the boundaries. Because θ − is pinned in the ground states, ϕ − gets totally disordered and therefore 〈O T 〉 ∝ 〈cos √ 2πϕ − 〉 = 0, which is just equivalent to the fact that the Luther-Emery fermions are gapped. However, this is no longer true as one approaches the ends of the chains, since there exists zero-energy edge states. Let us focus on the left boundary x = 0. The boundary condition of the anti-bonding boson field ϕ − has been derived: ϕ − (0) = 0. With the boundary condition, we proceed with Luther-Emery solution at K − = 2 and find O T (0) ∼ ˆχ(0)+ ˆχ † (0). Thus O T (0) has non-vanishing matrix element between the two ground states, independent of the system size. As a result, the two-fold degeneracy is splitted. We now turn to the inter-chain backscattering O B = ˆψ † 2R ˆψ 1L + ˆψ † 2L ˆψ 1R + h.c. = 2 πa 0 cos √ 2πϕ + cos √ 2πθ − . (7.18) An analysis similar to the single-particle tunneling leads to the conclusion that backscattering at the ends also splits the degeneracy. However, even if the backscattering occurs in the middle of the chain, it still causes a splitting of the ground states decaying as a power law in system size L. To see this, let us consider a single impurity near the middle of the chain, modeled by O B (x) where x ≈ L/2. We assume that the backscattering potential is irrelevant under RG flow and study its conse- 138
quence. The splitting is then proportional to 〈cos √ 2πϕ + (x)〉 since cos √ 2πθ − has different expectation values on the two ground states. Because ϕ + is pinned at x = 0, 〈cos √ 2πϕ + (x)〉 ∼ 1/x K + . Therefore the splitting of the ground states due to a single impurity in the middle of the system scales as 1/L K + . We thereby conclude that the ground state degeneracy is spoiled by the singleparticle inter-chain tunneling near the boundaries and the backscattering processes in the bulk. To avoid the unwanted tunneling processes near the ends, one can put strong tunneling barriers between the two chains near the ends, or the chains can be bended outwards so that the two ends are kept far apart, as depicted in Fig. 7.1. 7.4 Lattice Model We now show that the field theory (7.6) can be realized in lattice models of fermions. We consider the model of two weakly coupled chains of spinless fermions [169, 170, 159, 171]. The Hamiltonian reads Ĥ = −t ∑ (ĉ † i+1,aĉia + h.c.) + ∑ ∑ V (r)ˆn iaˆn i+r,a − t ⊥ (ĉ † i2ĉi1 + h.c.). (7.19) i,a i,a,r i Here a = 1, 2 labels the two chains. We assume the filling is incommensurate to avoid complications from Umklapp scatterings. V (r) is an intra-chain shortrange attractive interaction between two fermions at a distance r (in units of lattice spacing). Thus without inter-chain coupling, each chain admits a Luttinger liquid description with two control parameters: charge velocity v and Luttinger parameter K (we assume V is not strong enough to drive the chain to phase separation). We bosonize the full Hamiltonian and write the theory in the bonding and anti- 139
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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
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 and 146: of (7.2) has a product of the Klein
Page 147: 7.3 Stability of the Degeneracy We
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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