ABSTRACT - DRUM - University of Maryland

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can only be regarded as an approximation, although a very good one. Understanding quantitatively the non-topological aspects of topological qubits is of fundamental importance to the practical implementation of topological quantum computation. A large part of this thesis will be devoted to the study of problems related to this topic. We briefly review the subject here in the context of Majorana qubits. We first consider the stability of Majorana zero modes. Due to the particlehole symmetry, the only way a single Majorana fermion can decohere is to couple to another Majorana fermion since it has no internal degrees of freedom. Given a Majorana fermion γ, one can write a mass term: H mass ∝ iγ(uψ + u ∗ ψ † ). (1.33) Here u is a complex number and ψ is a fermion. However, since there is a superconducting gap everywhere in the bulk except at the defect where the Majorana fermion resides, any other single-particle excitation must have a gap. The above Hamiltonian (1.33) is thus irrelevant at energy below the bulk gap. We therefore conclude that a single Majorana fermion in a gapped superconductor is stable. On contrary, two or more Majorana fermions are generically not stable, since one can pairwisely gap them out. They can be stabilized if there is a certain symmetry in the system that prohibits the mass term. The existence of a bulk gap for single-particle excitations plays a crucial role for the stability of Majorana fermions. In fact, it implies an important conservation law in fermionic systems. In a superconductor, due to the condensation of Cooper pairs the total fermion number is not conserved: adding or removing a Cooper pair, which 26
is formed out of two fermions, does not cost any energy. However, the parity of the fermion number is conserved at low energy since exciting a single particle/hole costs a lot of energy. The conservation of the discrete physical observable, the fermion parity, plays an important role not only in protecting the Majorana fermions from developing a gap but also the measurement of the topological qubits. We give here a concise derivation of the non-Abelian statistics of Majorana fermions based on fermion parity conservation. Let us consider adiabatically exchanging two Majorana fermions γ 1 and γ 2 , the net effect of which is a unitary transformation U. Since before and after the exchange the configuration is exactly the same, we expect the Majorana nature of the excitations remains intact. Therefore, Uγ 1 U † = s 1 γ 2 , Uγ 2 U † = s 2 γ 1 . (1.34) Here s 2 1 = s 2 2 = 1, required by the Majorana condition γ 2 1,2 = 1. It follows that Uiγ 1 γ 2 U † = −s 1 s 2 · iγ 1 γ 2 . (1.35) Since iγ 1 γ 2 = 1 − 2c † c measures the fermion parity, it should be invariant through the entire process of the adiabatic braiding. Therefore we must have s 1 s 2 = −1, i.e. s 1 and s 2 must have opposite signs. We therefore reproduce the Ivanov’s rule derived previously for Majorana fermions in superconducting vortices. The implication is that the non-Abelianess of the braiding is closely related, or even a consequence of, the conservation of fermion parity and the adiabaticity of the braiding process. There are a number of ways that the fermion parity protection can be spoiled. We have mentioned that in any realistic superconductors Majorana fermions must come in pairs, but they can be very well separated from each other. The couplings 27
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 and 24: here and leaves its proof to Append
Page 25 and 26: the zero mode (see Chapter 3 for de
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: dimensional non-chiral Majorana mod
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 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 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
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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
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
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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 ˜
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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