ABSTRACT - DRUM - University of Maryland

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ameter correlations. It is therefore an important question to understand the fluctuation effects on the topological aspects, particularly the Majorana zero modes in TSC, since they do live in one or two dimensions. From a more general perspective, the interplay between interaction (since fluctuations are essentially caused by interactions) and topological classification of non-interacting systems is a fundamental problem which we only began to understand quite recently. A remarkable progress is that the topological classification of one-dimensional non-interacting fermionic systems with timereversal symmetry is dramatically changed by interactions [152, 153, 154]. Several theoretical studies on the effects of interactions on Majorana fermions in proximityinduced TSC have been performed recently [155, 156, 157, 158], confirming the stability of Majorana fermions against weak and moderate interactions. In this chapter we present an attempt to understand the fate of Majorana zero modes when quantum fluctuations are strong enough that only quasi-longrange superconducting order can exist. We consider a generic theoretical model of spinless fermions on two-chain ladders. The model generalizes the simplest onedimensional TSC, namely spinless fermions with p-wave pairing (also known as Majorana chain) [28], to interacting two-chain systems. Instead of introducing pairing by proximity effect, the effective field theory includes inter-chain pair tunneling with inter-chain single-particle tunneling being suppressed. Therefore the fermion parity on each chain is conserved. When the pair-tunneling interaction drives the system to strong coupling, localized Majorana zero-energy states are found on the boundaries, which represents a nontrivial many-body collective state of the underlying fermions. 128
We then demonstrate that in a finite-size system the Majorana edge states lead to (nearly) degenerate ground states with different fermion parity on each chain, thus revealing its analogy with the Majorana edge states in non-interacting TSC. The degeneracy is shown to be robust to any weak intra-chain perturbations, but interchain single-particle tunneling and backscattering can possibly lift the degeneracy. We also discuss a lattice model where such field theory is realized at low energy. 7.1 Field-Theoretical Model We start from an effective field-theoretical description of the model for the purpose of elucidating the nature of the Majorana edge states. We label the two chains by a = 1, 2. The low-energy sector of spinless fermions on each chain is well captured by two chiral Dirac fermions ˆψ L/R,a (x). The non-interacting part of the Hamiltonian is simply given by Ĥ0 = ∫ dx Ĥ0(x) where Ĥ 0 = −iv F ∑ a ( ˆψ† Ra ∂ ˆψ x Ra − ˆψ † La ∂ ˆψ ) x La . (7.1) Four-fermion interactions can be categorized as intra-chain and inter-chain interactions. Intra-chain scattering processes (e.g., forward and backward scattering) are incorporated into the Luttinger liquid description of spinless fermions and their effects on the low-energy physics are completely parameterized by the renormalized velocities v a and the Luttinger parameters K a . We assume that the filling of the system is incommensurate so Umklapp scattering can be neglected. For simplicity we assume the two chains are identical so v 1 = v 2 = v, K 1 = K 2 = K. We now turn to inter-chain interactions. Those that can be expressed in terms 129
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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
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
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: Chapter 7 Majorana Zero Modes Beyon
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
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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