ABSTRACT - DRUM - University of Maryland

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We now turn to the limit where µ is very close to Dirac point, i.e. µ → 0, k F ξ ≪ 1. We evaluate the integral for ξ ≪ R ≪ k −1 F . E + ≈ −√ 2µ ( ) 3/2 ( R exp − R ) , (4.11) π ξ ξ where we have made use of asymptote of N 3 in the limit k F ξ ≪ 1. Eq. (4.11) implies that for fixed R the splitting vanishes as µ approaches Dirac point. Actually this fact can be easily seen from (4.8) without calculating the integral. Because at µ = 0 either χ ↑ or χ ↓ vanishes, the splitting which is proportional to the product of χ ↓ and χ ↑ is zero. The same result for splitting at µ = 0 has also been obtained in Ref. [96]. We now show that vanishing of the splitting at µ = 0 is a direct consequence of chiral symmetry. At µ = 0 zero modes carry chirality which labels the eigenvalues of γ 5 . More specifically, wave function is an eigenstate of γ 5 : γ 5 Ψ i = λΨ i . Consider an arbitrary perturbation represented by O to the ground state manifold expanded by these local zero modes. To leading order in perturbation theory its effect is determined by matrix element O ij = 〈Ψ i |O|Ψ j 〉. Now assume that Ψ i and Ψ j have the same chirality(which means that vortices i and j have identical vorticity). If the perturbation O preserves chiral symmetry, i.e. {γ 5 , O} = 0, then 〈Ψ i |{γ 5 , O}|Ψ j 〉 = 2λ〈Ψ i |O|Ψ j 〉 = 0. (4.12) Therefore matrix element 〈Ψ i |O|Ψ j 〉 vanishes identically. Tunneling obviously preserves chiral symmetry so there is no splitting between two vortices with the same vorticity from this line of reasoning. As discussed below this fact actually holds 70
eyond perturbation theory and the robustness of zero modes in the presence of chiral symmetry is ensured by an index theorem. Now we can fit our splitting calculation into the general picture set by index theorem. As being argued above, Majorana zero modes in spinless p x +ip y superconductor is classified by Z 2 . When there are two vortices in the bulk, the topological index of order parameter is 2 thus there is no zero mode and we find the splitting as expected. The same applies to two vortices in TI/SC heterostructure with µ ≠ 0. However, as we have seen in the calculation the splitting vanishes for µ = 0. This should not be surprising since according to index theorem, there should be at least two zero modes associated with total vorticity 2 which is the case for two vortices. 4.1.3 Comparison with the splitting calculations in other systems. Recently numerical calculations of the degeneracy splitting have been performed for other systems supporting non-Abelian Ising anyons [87, 88, 90]. In all these calculations it was found that the splitting has qualitatively similar behavior - there is an exponential decay with the oscillating prefactor which stems from the spatial oscillations of Majorana bound states. In the case of Moore-Read quantum Hall state [88], the splitting between two quasiholes exponentially decays and oscillates with the magnetic length l c = eB since there only one length scale in the problem. The oscillatory behavior is also predicted for pair of vortex excitations in the B-phase of Kitaev’s honeycomb lattice model in an external magnetic field [90]. 71
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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
Page 29 and 30: find a realistic material in nature
Page 31 and 32: states near the Fermi circle. We fi
Page 33 and 34: pairing: ( ) p H = ψ † 2 2m −
Page 35 and 36: dimensional non-chiral Majorana mod
Page 37 and 38: is formed out of two fermions, does
Page 39 and 40: gapless and should not be neglected
Page 41 and 42: zero modes. We study a one-dimensio
Page 43 and 44: of possible paired states which is
Page 45 and 46: where d ≡ (d 1 , d 2 , d 3 ) = (R
Page 47 and 48: If the order parameter is proportio
Page 49 and 50: 0.015 T/t 0.01 Normal p x + ip y 0.
Page 51 and 52: trivial p x - and p y -states lead
Page 53 and 54: with the gap operator being ˆ∆ =
Page 55 and 56: 0.2 T/t 0.1 Normal f p x + ip y µ
Page 57 and 58: 2.3 Discussion and Conclusions In c
Page 59 and 60: Chapter 3 Majorana Bound States in
Page 61 and 62: satisfied for m = 0. The singlevalu
Page 63 and 64: We summarize this section by provid
Page 65 and 66: Here γ a and Γ a are 4 × 4 Dirac
Page 67 and 68: with λ = µξ/v. It has the follow
Page 69 and 70: where n = (R∆, −I∆) field des
Page 71 and 72: Chapter 4 Topological Degeneracy Sp
Page 73 and 74: ound states are usually splitted an
Page 75 and 76: sponding quasiparticle operator can
Page 77 and 78: Of particular importance is the sig
Page 79: [61]. Although analytical expressio
Page 83 and 84: eaks down in this regime and one sh
Page 85 and 86: 4.3 Conclusions In this chapter, we
Page 87 and 88: the fermion bath, and consequently
Page 89 and 90: 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 =
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considerations, we should have β 1
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states: ν(ε) = 2ν 0 ε √ ε2
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necessary as a matter of principle
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Chapter 7 Majorana Zero Modes Beyon
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We then demonstrate that in a finit
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The standard Abelian bosonization r
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and define the chiral fields by ˜
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of (7.2) has a product of the Klein
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7.3 Stability of the Degeneracy We
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quence. The splitting is then propo
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where α = 1 2 (K + K−1 − 2). A
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effect of quasiparticle tunneling o
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Appendix A Derivation of the Pfaffi
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which allows further simplification
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List of Publications Publications t
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Bibliography [1] K. von Klitzing, G
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[46] S. Das Sarma, C. Nayak, and S.
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[92] A. W. W. Ludwig, D. Poilblanc,
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[133] M. V. Berry, Proc. R. Soc. Lo
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