ABSTRACT - DRUM - University of Maryland

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MF has zero excitation energy, implying a vanishing superconducting gap at where the MF emerges. Thus the zero-energy MFs can only exist at “defects” in the superconducting order parameter, such as domain walls or vortices. These defects can be spatially very well separated. Then one can form an ordinary fermion out of two Majorana fermions very far from each other, a highly non-local object. This nonlocal fermionic mode can be occupied or empty, yielding two degenerate many-body states. When there are 2N Majorana fermions, we can group them pairwise and construct the Hilbert space from the N fermionic modes, leading to 2 N degenerate states. If there are no other zero-energy modes in the system, there is a further superselection rule that the total fermion parity must be fixed. States with different global fermion parity belong to different superselection sectors and can not be connected by any physical matrix elements. Thus the actual degeneracy is reduced to 2 N−1 . These degenerate states have no difference if only local measurements are concerned. The only way to distinguish them is to measure the fermion parity stored in pairs of topological defects, which certainly requires non-local measurements. We therefore call such degeneracy “topological degeneracy”. For spinless p x + ip y superconductors, a Majorana zero mode can be found in the core of an Abrikosov vortex [30, 22] around which the phase of the order parameter winds by 2π: ∆ 0 (r) = f(r)e iϕ . (1.22) Here (r, ϕ) is the polar coordinate of the two-dimensional plane. f(r) represents the profile of the order parameter. One can explicitly solve the BdG equation to find 14
the zero mode (see Chapter 3 for details). Besides the zero modes, there are other eigenstates with energy eigenvalues well below the bulk gap, the so-called midgap states [31, 32, 33, 34]. A semiclassical argument, treating the vortex core as a hole of size ξ ∼ v F ∆0 , gives an estimate of the energy of midgap states to be of the order 1 mξ 2 ≈ ∆2 0 E F where E F is the Fermi energy. As long as k F ξ ≫ 1, this energy scale is much smaller than the bulk gap. 2 1 2 1 Figure 1.1: Braiding of Majorana fermions bound to vortices. We now demonstrate the very peculiar non-Abelian braiding statistics of MFs in superconducting vortices, first derived by Ivanov [35]. It is crucial to keep track of the branch cut where the superconducting phase jumps by 2π to uniquely define the superconducting phase everywhere (except at the vortex cores). Pictorially we can attach a “string” to each vortex, which goes all the way to infinity (or the system boundary) to represent the branch cut. As the vortices are transported adiabatically, the branch cuts are also “dragged” along with the vortices and we have to make sure that the vortices do not cross the branch cuts, as depicted in Fig. 1.1. We denote the local phases seen by vortices 1 and 2 by χ 1 and χ 2 respectively. Before the exchange, χ 1 = π + 0 + , χ 2 = 0 + and after the exchange χ ′ 1 = 2π − 0 + , χ ′ 2 = π + 0 + . The gauge transformation then implies γ 1 picks up a phase e i(χ 1−χ ′ 1 )/2 = −1. Consequently, γ 1 is replaced by γ 2 but γ 2 is replaced by −γ 1 . We therefore conclude that the 15
Page 1 and 2: ABSTRACT Title of dissertation: MAJ
Page 3 and 4: MAJORANA QUBITS IN NON-ABELIAN TOPO
Page 5 and 6: Acknowledgments First and foremost,
Page 7 and 8: Table of Contents List of Figures L
Page 9 and 10: List of Figures 1.1 Braiding of Maj
Page 11 and 12: Chapter 1 Introduction The subject
Page 13 and 14: closed throughout the path. If one
Page 15 and 16: its original configuration, we requ
Page 17 and 18: mathematics. It is easy to check th
Page 19 and 20: Here τ x is the Pauli matrix actin
Page 21 and 22: y constructing the ground state pro
Page 23: here and leaves its proof to Append
Page 27 and 28: 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
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[61]. Although analytical expressio
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eyond perturbation theory and the r
Page 83 and 84:
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
Page 89 and 90:
tion for the reduced density matrix
Page 91 and 92:
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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The first-order term vanishes due t
Page 105 and 106:
apply to any realistic system where
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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,
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{α(T )} to {α(0)}: |α(T )〉 =
Page 115 and 116:
educed to 2 N−1 by fermion parity
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Berry connection of this state then
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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
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Because E + ∼ ∆ 0 e −R/ξ , |
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tunneling amplitudes between them a
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ˆΓ i = iˆγ i ˆξiˆη i , i =
Page 131 and 132:
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
Page 141 and 142:
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
Page 147 and 148:
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
Page 153 and 154:
effect of quasiparticle tunneling o
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Appendix A Derivation of the Pfaffi
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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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