ABSTRACT - DRUM - University of Maryland

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The projector is then given by P k = 1 2 ( 1 + d ) k · τ . (1.18) |d k | Denote ˆd k = d k /|d k | as the normalized d vector, we find C = 1 ∫ 4π d 2 k ˆd k · ( ) ∂ˆd k × ∂ˆd k . (1.19) ∂k x ∂k y This formula has a simple geometrical interpretation. The normalized vector d defines a mapping from the Brillouin zone T 2 to the two-dimensional unit sphere S 2 and the integral is nothing but the area of the image of T 2 . Since the total area of the unit sphere is 4π, C actually counts how many times the image of the torus is “wrapped” around the sphere. It is also known as the degree of the mapping in mathematics. For p x + ip y superconductors, the d vector is given by d k = (∆k x , −∆k y , k2 2m − µ). Direct evaluation of the integral yields ⎧ ⎪⎨ 1 µ > 0 C = . (1.20) ⎪⎩ 0 µ < 0 Therefore p x + ip y superconductor is topological with Chern number 1 if the Fermi energy is above the band bottom. More generally, if the pairing order parameter ∆(k) ∝ (k x + ik y ) n and µ > 0, the Chern number is n. Evaluating the Chern number analytically is a cumbersome task if the superconductor has multiple bands. Fortunately, there is a great simplification if we only want to know the parity of the Chern number which determines whether the superconductors have non-Abelian excitations or not [28, 29]. We present the formula 12
here and leaves its proof to Appendix A: (−1) C = ∏ K Pf[H K Ξ]. (1.21) Here the product is taken over all symmetric points K satisfying K ≡ −K in the first Brillouin zone. 1.4 Majorana Zero Modes and TQC As we have reviewed in the previous section, superconductors have particlehole symmetry as a result of the redundant representation. The energy spectrum is symmetric with respect to zero and γ −E = γ † E . Thus the zero-energy states are very special because γ E=0 = γ † E=0 . Such a quasiparticle is self-conjugate, being identical to its “antiparticle”. Given two self-conjugate quasiparticles at E = 0 denoted by γ 1 and γ 2 , it is straightforward to check that they still obey fermionic commutation relation: {γ i , γ j } = δ ij . It immediately follows that γ 2 = 1 . Such self-conjugate 2 fermionic quasiparticles are called Majorana fermions(MF). Loosely speaking it can be regarded as half of an ordinary fermion, since an ordinary fermion can always be represented as two degenerate MFs: given a fermion annihilation operator c satisfying {c, c † } = 1, we can form two MFs γ 1 = c+c† √ 2 , γ 2 = c−c† √ 2i and c = γ 1 + iγ 2 . One may then wonder what is special about MF here in a topological superconductor since they are just the usual fermionic operators in disguise. It is important to clarify that what we are interested are unpaired (non-degenerate), localized Majorana fermionic excitations, which can not be obtained by naively rewriting a usual fermionic operator as two MF operators. The particle-hole symmetry ensures that 13
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 and 20: Here τ x is the Pauli matrix actin
Page 21: y constructing the ground state pro
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
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 and 138:
Chapter 7 Majorana Zero Modes Beyon
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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
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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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