ABSTRACT - DRUM - University of Maryland

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2. What are non-Abelian statistics? 3. How is a non-Abelian superconductor related to quantum computation? 1.1 Topological Phases: An Overview The 1980 discovery of Integer Quantum Hall Effect [1] opened the door to the fascinating world of topological phases in condensed matter systems. The remarkably precise quantization of Hall conductance, insensitive to many microscopic details such as disorder and geometry, is the first example of the “universal”, exact features common in topological phases, that are robust against any small perturbations to the system. The even more striking discovery of fractional quantum Hall effect [2] led to the conceptual formulation of the notion of topological order [3]. To understand its meaning, let us take a grand view of gapped quantum phases. In one sentence, topological order can be regarded as a “periodic table” of all gapped phases. Well-known examples of gapped phases are insulators: band insulators, Mott insulators, etc. All the excitations in these phases are gapped so correlation functions of any local observables decay exponentially in the limit of large space-time separation. However, it does not mean that they are all alike: IQH states, albeit gapped, have quantized Hall conductance while ordinary band insulators do not. Therefore a more refined notion of the equivalence classes between gapped quantum phases is needed, which is provided by the concept of adiabatic continuity [4]. Two gapped quantum phases are said to be adiabatically connected, if there exists a parameter path to connect their Hamiltonians such that the spectral gap is not 2
closed throughout the path. If one adiabatically follows the path of the Hamiltonian, the ground state smoothly evolves from one to the other. It is also clear that under such equivalence relations, there has to be a quantum phase transition if one wants to connect two different gapped phases. Interestingly, although it seems very natural to consider adiabatic continuity between gapped phases, historically the first application of adiabatic continuity in condensed matter physics was Landau’s Fermi liquid theory [5, 6], where a Fermi liquid is in fact defined as a state adiabatically connected to a non-interacting Fermi gas, a gapless phase. Topological phases are then defined as those gapped phases that can not be adiabatically connected to trivial phases. One may wonder what are trivial phases to be compared with. The canonical example of a trivial gapped phase is an atomic insulator, in which all electrons occupy localized atomic orbitals and the many-body wavefunction is just a Slater determinant of all real-space atomic orbitals. This definition of topological phases is quite general since we have not even invoked any physical characterizations. Theoretically, it is an extremely complicated problem to find all topological phases. Still, after thirty years of research, our understanding of topological phases has been greatly enriched [7, 8]. The above definition of topological phases gives no hint on how to characterize topological phases physically. It defines topological phases by what they are not. Conventionally, phases of matter are often associated with broken symmetries, characterized by local order parameters and correlation functions. This conventional approach has been very successful in describing many solid state systems such as magnets, superfluids and superconductors, but completely fails for topologi- 3
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: Chapter 1 Introduction The subject
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
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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
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Chapter 6 Non-adiabaticity in the B
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minor modifications in the details,
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{α(T )} to {α(0)}: |α(T )〉 =
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educed to 2 N−1 by fermion parity
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Berry connection of this state then
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wavefunction is u n (r), v n (r). W
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asis. The most general form of BdG
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6.2.1 Effect of Tunneling Splitting
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Because E + ∼ ∆ 0 e −R/ξ , |
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tunneling amplitudes between them a
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ˆΓ i = iˆγ i ˆξiˆη i , i =
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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
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Chapter 7 Majorana Zero Modes Beyon
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We then demonstrate that in a finit
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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
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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
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effect of quasiparticle tunneling o
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Appendix A Derivation of the Pfaffi
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which allows further simplification
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List of Publications Publications t
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Bibliography [1] K. von Klitzing, G
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[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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