ABSTRACT - DRUM - University of Maryland

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their braiding against thermal effects and non-adiabaticity, unavoidable in any realistic systems. We apply the formalism of density matrix and master equation and characterize the topological qubits in terms of physical observables. Based on this formulation, we show that the topological qubits are robust against both localized and extended fermionic excitations even when gapless bosonic modes are present. Finally, we explore the non-perturbative effect of strong fluctuations of superconducting order parameter, when the mean-field description in terms of Bogoliubov quasiparticles is invalidated. We consider a model of two-leg ladder of interacting fermions with only quasi-long-range superconducting order and derive the low-energy effective field theory using bosonization techniques. We find that although the whole spectrum is gapless, one can identify degeneracies of low-energy states resulting from Majorana edge modes. In the presence of certain impurity scatterings, we show that the splitting of the degeneracy has a power-law decay with the size of the system.
MAJORANA QUBITS IN NON-ABELIAN TOPOLOGICAL SUPERCONDUCTORS by Meng Cheng Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2013 Advisory Committee: Professor Sankar Das Sarma, Chair/Advisor Professor Victor Yakovenko Assistant Professor Michael Levin Professor Charles Clarke Professor Christopher Jarzynski
Page 1: ABSTRACT Title of dissertation: MAJ
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 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
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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
Page 69 and 70:
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
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
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eaks down in this regime and one sh
Page 85 and 86:
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
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like phonons. However, such process
Page 93 and 94:
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
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
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states: ν(ε) = 2ν 0 ε √ ε2
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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 ˜
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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
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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