ABSTRACT - DRUM - University of Maryland

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Hamiltonian: ⎛ H k = ⎜ ⎝ h k ∆ † k ∆ k −h T −k ⎞ ⎟ ⎠ (1.11) h k is the single-particle Hamiltonian and ∆ k represents BCS pairing. Notice that h k must be a Hermitian matrix h k = h † k and the order parameter matrix ∆ k always satisfies ∆ −k = −∆ T k . Let τ x be the Pauli matrix in Nambu (particle-hole) space, we have ⎛ ⎞ h T −k −∆ † k τ x H k τ x = − ⎜ ⎟ ⎝ ⎠ = −H∗ −k. (1.12) −∆ k h k The last equality is derived from h † k = h k, ∆ ∗ −k = −∆† k . This defines the particle-hole symmetry represented by τ x . A non-interacting fermionic Hamiltonian with such a symmetry is said to be in the class D [23]. The Hamiltonian can be diagonalized by Bogoliubov transformation, as we did for the spinless p x + ip y superconductor. The ground state is defined to be the state with no positive-energy Bogoliubov excitations. At the level of topological classification, it is useful to consider the superconducting ground state as a state in which all “hole”-like states(i.e. with negative energy eigenvalue) are filled. It is in many ways like an insulator with the Hamiltonian matrix the same as the BdG Hamiltonian and the Fermi energy set to zero. Assume H k is a 2N × 2N matrix and for each k the negative energy eigenstates (the “occupied” states) are denoted by |u nk 〉 where n = 1, . . . N. Mathematically, the ground state wavefunction defines a U(1) vector bundle on the Brillouin zone, which is a two-dimensional torus T 2 [24, 25, 26]. This vector bundle can be specified 10
y constructing the ground state projection operator P k = ∑ n |u nk〉〈u nk |. Topologically, we can characterize such a vector bundle by its first Chern number C [27], which is the integral of the first Chern character: ch 1 (F) = Tr ( ) iF , (1.13) 2π and ∫ C = ch 1 (F) = 1 ∫ F k . (1.14) T 2π 2 T 2 Here F k is the curvature form derived from the connection one-form: A k = i ∑ n 〈u nk |∇|u nk 〉. (1.15) There are a couple of equivalent representations of the Chern number. It can expressed solely in terms of the ground state projector [24] C = 1 ∫ 2πi Tr(P k dP k ∧ dP k ) = 1 ∫ 2πi [ ( ∂Pk ∂P k dk Tr P k − ∂P )] k ∂P k . (1.16) ∂k x ∂k y ∂k y ∂k x The first Chern number has very intuitive physical meaning. Any superconducting system with a non-zero Chern number support chiral Majorana edge modes, the number of which is equal to the Chern number [19]. The Majorana edge mode carries energy, leading to quantized thermal Hall effect [19]. It also determines the existence of unpaired Majorana zero modes in topological defects, which will be discussed later. We now apply the formula to the spinless p x + ip y superconductors with only one band, so the BdG Hamiltonian is a 2 × 2 matrix. We write the Hamiltonian in terms of Pauli matrices in Nambu space: H k = d k · τ . (1.17) 11
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 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: Here τ x is the Pauli matrix actin
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
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
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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
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 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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