ABSTRACT - DRUM - University of Maryland

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f-wave state with ∆ n = ∆ is favored at high densities. These phases are separated by a first-order transition. As shown in Fig. 2.2, the van Hove singularity µ = µ ∗ gives rise to a maximal T c and is located inside the f-wave superconducting dome. This point represents another type of a quantum transition that separates two qualitatively different topologically trivial paired states: (1) For µ < µ ∗ , there is just one electron-type Fermi pocket that is cut by the nodes of the f-wave gap in the directions, θ (m) node = mπ/3 + π/6. This gives rise to gapless quasiparticles. (2) For µ > µ ∗ , no FS can be cut and the nodal quasiparticles disappear. The phase becomes fully gapped and eventually crosses over to the two-specie continuum model (2.18). Experimentally, the two types of f-wave phases can be distinguished by different T -dependence of the heat capacity. We also present the Bogoliubov-de Gennes equations for the f-wave pairing state in high-density limit µ → 3t. Their derivation goes along the same lines as that given in Section 2.2.1 for p x + ip y pairing SC. However, in the f-wave case, the momentum space order parameter is given by ∆ k = ∆(sin k·e 1 +sin k·e 2 +sin k·e 3 ). Therefore, the order parameter reads: ( ) r + r ′ ∆ rr ′ = ∆ cos(3θ r 2 ′ −r). Since e 1 +e 2 +e 3 = 0, the leading term in the expansion is ∼ a 3 . With some algebra one can show that the gap operator is ˆ∆ = a3 24 2∑ {∂ n , {∂ n , {∂ n , ∆(r)}}} (2.20) n=0 with ∂ n ≡ ∇ · e n and the BdG equation takes the form of Eq. (2.17). 46
2.3 Discussion and Conclusions In conclusion, we discover that topological superconducting phases breaking time -reversal symmetry emerge naturally within a large class of spinless fermion models. The technique we apply here has a close relation to BCS mean field theory of a spin-triplet superfluid 3 He [78, 44], which concluded that the B-phase with isotropic gap is stabilized compared to anisotropic A-phase (The A-phase can be stabilized under high pressure. In this case, due to strong spin-fluctuations in liquid 3 He, the ground state energy departs from the BCS theory, which is not a contradiction to our conclusion). However, we have shown that similar conclusion can be generalized to any band structures, filling factors, and interactions, as long as the system satisfies proper (discrete) rotational group symmetries. More importantly, our proof is insensitive to the existence of the “nodes”. In continuum, it has been argued that a p x state is unstable against the p x + ip y pairing state, because the former has nodes thus having smaller condensation energy. However, the stability of a nodeless p x state, which could exists in lattice models, was unclear before this our work. We should also emphasize that although the discussions above focus on spinless fermions, all the conclusions can be generalized to the triplet pairing channels of spin- 1/2 fermions, because these pairing channels also correspond to the space-inversion odd representations of the symmetry group. In addition, we note that any pairing state that spontaneously breaks a lattice rotational symmetry must have at least one degenerate state for both spinless and spin-1/2 fermions. Our theorem indicates that 47
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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 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: 0.2 T/t 0.1 Normal f p x + ip y µ
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
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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.
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[92] A. W. W. Ludwig, D. Poilblanc,
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[133] M. V. Berry, Proc. R. Soc. Lo
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ABSTRACT - DRUM - University of Maryland

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