ABSTRACT - DRUM - University of Maryland

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Other publications and preprints that I contributed to: 1. Xiong-Jun Liu, Zheng-Xin Liu and Meng Cheng, Manipulating Topological Edge Spins in One-Dimensional Optical Lattice, arXiv:1209.2990 2. Meng Cheng, Superconducting Proximity Effect on the Edge of Fractional Topological Insulators, arXiv:1204.6085 3. Meng Cheng and Roman Lutchyn, Josephson Current through a Semiconductor Nanowire: effect of strong spin-orbit coupling and Zeeman splitting, Phys. Rev. B 86, 134522 (2012) 150
Bibliography [1] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). [2] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). [3] X. G. Wen, Phys. Rev. B 40, 7387 (1989). [4] P. W. Anderson, Basic Notions of Condensed Matter Physics (Westview Press, 1997). [5] L. D. Landau, Sov. Phys. JETP 3, 920. [6] L. D. Landau, Sov. Phys. JETP 8, 70 (1958). [7] X.-G. Wen, Quantum Field Theory of Many-body Systems (Oxford University Press, 2004). [8] C. Nayak et al, Rev. Mod. Phys. 80, 1083 (2008). [9] X.-G. Wen, Phys. Rev. B 41, 12838 (1990). [10] X.-G. Wen, Adv. Phys. 44, 405 (1995). [11] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). [12] D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984). [13] J. M. Leinaas and J. Myrheim, Il Nuovo Cimento B 37B, 1 (1977). [14] R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). [15] M. G. G. Laidlaw and C. M. DeWitt, Phys. Rev. D 3, 1375 (1971). [16] E. Fadell and L. Neuwirth, Math. Scand. 10, 111 (1962). [17] K. Druhl, R. Haag, and J. E. Roberts, Commun. Math. Phys. 18, 204 (1970). [18] F. Wilczek, Phys. Rev. Lett. 49, 957 (1982). [19] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000). [20] A. Stern, F. von Oppen, and E. Mariani, Phys. Rev. B 70, 205338 (2004). [21] N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947). [22] P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966). 151
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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
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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
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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
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sponding quasiparticle operator can
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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
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4.3 Conclusions In this chapter, we
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the fermion bath, and consequently
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tion for the reduced density matrix
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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
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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,
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 =
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: List of Publications Publications t
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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