ABSTRACT - DRUM - University of Maryland

More documents

Recommendations

Info

out for the model of spinless p x + ip y superconductors [91]. Generally energy splitting due to intervortex tunneling is determined by the wave function overlap of localized Majorana bound states. In this chapter we calculate the splitting for both spinless p x + ip y superconductor and a model of Dirac fermions interacting with the scalar superconducting pairing potential realized in a TI/SC heterostructure. In both cases, besides the expected exponential decay behavior, it is found that the prefactor exhibits an oscillatory behavior with the intervortex distance which originates from the interference of different bound state wave functions oscillating with the Fermi wave length. This is generic situation for weak coupling superconductors where the Fermi energy E F is much larger than the superconducting gap ∆. In this chapter, we also consider several cases where the Fermi wavelength is much larger than the coherence length. This scenario is relevant, for example, for TI/SC heterostructure as well as some other systems involving the proximity-induced superconductivity. When chemical potential is tuned to the Dirac point (µ → 0), we find indeed that the splitting in TI/SC heterostructure vanishes. This fact can be related to the Atiyah-Singer index theorem an additional symmetry possessed by the system at µ = 0 - the chiral symmetry. 4.1 Degeneracy splitting due to intervortex tunneling The ground state degeneracy, which is crucial for topological quantum computation with non-Abelian anyons, heavily relies on the assumption that intervortex tunneling is negligible. When tunneling effects are taken into account zero energy 62
ound states are usually splitted and the ground state degeneracy is lifted. Besides, the sign of energy splitting is important for understanding many-body collective states [92]. We now discuss a general formalism to calculate the energy splitting. We focus on the case of two classical vortices each with vorticity l = 1 located at certain fixed positions R 1 and R 2 . To develop a physical intuition, it is useful to view a vortex as a potential well, which may host bound states including zero-energy states, while the regions where superconducting gap is finite play the role of a potential barrier. Therefore, the two-vortex problem resembles the double-well potential problem in single-particle quantum mechanics (sometimes referred to as the Lifshitz problem in the literature [93]). The solution to this simple problem in one-dimensional quantum mechanics is readily obtained [93] by considering symmetric and antisymmetric combinations of single-well wave-functions (which can be taken within the quasiclassical approximation for high barriers) and the overlap of these wave-functions always selects the symmetric state as the ground state in accordance with the elementary oscillation theorem (i.e., the ground state has no nodes). We note that both quasiclassical approximation and the Lifshitz method are not specific to the Schrödinger equation, but actually represent general mathematical methods of solving differential equations of certain types. Moreover, these methods can be applied to rather generic matrix differential operators, and such a generalization has been carried out by one of the authors in a completely different context of magnetohydrodynamics, [94] where interestingly the relevant differential operator appears to be mathematically similar to the BdG Hamiltonian. These considerations suggest 63
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 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
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: Chapter 4 Topological Degeneracy Sp
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
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
show all

ABSTRACT - DRUM - University of Maryland

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?