ABSTRACT - DRUM - University of Maryland

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The adiabatic exchange of any two particles can be mathematically implemented by adiabatically changing the positions of two particles, say R i and R j , in such a way that in the end they are interchanged. This means that R i (T ) = R j (0), R j (T ) = R i (0). (6.1) According to the adiabatic theorem [139], it results in a unitary transformation within the subspace: if the system is initially in state |ψ(0)〉, then |ψ(T )〉 = Û|ψ(0)〉. To determine Û, we first consider initial states |ψ(0)〉 = |α({R i(0)})〉. Under this evolution the final state can be written as |ψ α (T )〉 = Û0|α(T )〉. (6.2) Here the matrix Û0 is the non-Abelian Berry phase:[133, 140, 141] ( ∫ T ) Û 0 = P exp i dt ˆM(t) , (6.3) 0 where P denotes path ordering and matrix element of the Berry’s connection ˆM is given by ˆM αβ (t) = i〈α(t)| ˙β(t)〉. (6.4) Although the exchange defines a cyclic trajectory in the parameter space of Hamiltonian, the final basis states can be different from the initial ones (e.g, |α({R i })〉 can be multivalued functions of R i , which is allowed if we are considering quasiparticles being collective excitations of many-body systems). The only requirement we impose is that the instantaneous eigenstates {α(t)} are continuous in t. Therefore, we have another matrix ˆB defined as ˆB αβ ≡ 〈α(0)|β(T )〉, relating 102
{α(T )} to {α(0)}: |α(T )〉 = ˆB αβ |β(0)〉. Combining with (6.2), we now have the expression for Û: |ψ(T )〉 = Û0 ˆB|ψ(0)〉. (6.5) Therefore ( ∫ T ) Û = Û0 ˆB = P exp i dt ˆM(t) 0 ˆB. (6.6) In fact, the factorization of Û into Û0 and ˆB is somewhat arbitrary and gaugedependent. However, their combination Û is gauge-independent provided that the time-evolution is cyclic in parameter space(positions of particles). The unitary transformation Û defines the statistics of quasiparticles. 6.1.2 Non-Abelian Majorana Fermions We now specialize to the non-Abelian statistics of Majorana fermions in topological superconductors, carefully treating the effect of Berry phases. We mainly use spinless superconducting fermions as examples of topological superconductors in both 1D and 2D since all known topological superconducting systems supporting non-Abelian excitations essentially stem from spinless chiral p-wave superconductors.[19, 61, 47] In the BCS mean-field description of superconductors, the Hamiltonian is particle-hole symmetric due to U(1) symmetry breaking. In terms of Nambu spinor ˆΨ(r) = ( ˆψ(r), ˆψ ∫ † (r)) T , the BCS Hamiltonian is expressed as ĤBCS = 1 2 d 2 r ˆΨ † (r)H BdG ˆΨ(r). 103
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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
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 and 80: [61]. Although analytical expressio
Page 81 and 82: eyond perturbation theory and the r
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: minor modifications in the details,
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 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
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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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