ABSTRACT - DRUM - University of Maryland

More documents

Recommendations

Info

equation (6.40), we find ˆγ → (1 − 4β2 Et sin2 E2 2 ˆξ → 2√ 2εβ sin 2 Et 2 ˆη → E 2 √ 2β sin Et ˆγ − E ˆγ+ )ˆγ+ 2√ 2βε sin 2 Et 2 E 2 ( cos Et + 4β2 sin 2 Et 2 E 2 ε sin Et E ˆξ + cos Et ˆη √ 2β sin Et ˆξ− ˆη E ε sin Et )ˆξ+ E ˆη (6.42) which should be followed up by the basis transformation ˆB. Here we have defined E = √ ε 2 + 4β 2 . By using (6.24) we can work out explicitly how the ground state wavefunctions transform. Physically, the non-adiabatic process causes transitions of quasiparticles residing on the zero-energy level to the excited levels. Superficially these transitions to excited states significantly affect the non-Abelian statistics, since the parity of fermion occupation in the ground state subspaces is changed as well as the quantum entanglement between various ground states [20, 146]. This can also be directly seen from (6.42) since starting from |g〉 the final state is a superposition of |g〉 and ˆd † ˆd † 0 λ |g〉. So we might suspect that errors are introduced to the gate operations. However, noticing that the excited states are still localized, they are always transported together with the zero-energy Majorana states. Threfore the parity of the total fermion occupation in the ground state subspace and local excited states are well conserved. This observation allows for a redefinition of the Majorana operators to properly account for the fermion occupation in local excited states, as being done in [111]. We therefore have to represent the fermion parity in the following way: ∏ ˆP 12 = −iˆγ 1 ˆξ1ˆη 1ˆγ 2 ˆξ2ˆη 2 = iˆγ 1ˆγ 2 (1 − 2 ˆd † ˆd i i ) (6.43) shared by defects 1 and 2. Accordingly, we define the generalized Majorana operators 118 i=1,2
ˆΓ i = iˆγ i ˆξiˆη i , i = 1, 2. Since the couplings between the two vortices are exponentially small, we treat their dynamics independently. ˆΓi is invariant under local unitary evolution, which can be checked explicitly using (6.42). An even general proof proceeds as following: according to (6.23), the three Majorana operators ˆγ i , ˆξ i , ˆη i must transform by a SO(3) matrix V. Then it is straightforward algebra to show that ˆΓ i → det V·ˆΓ i . Since V ∈ SO(3), det V = 1 which means that ˆΓ i is unchanged. Thus the effect of the braiding comes only through the basis transformation matrix ˆB. As a result, the fermion parities, being the expectation values of (6.43), transform exactly according to the Ivanov’s rule under the braiding. This result can be easily generalized to the case where many bound states exist in the vortex core. From the perspective of measurement, to probe the status of a topological qubit it is necessary and sufficient to measure the fermion parity as defined in (6.43). It is practically impossible(and unnecessary) to distinguish between the fermion occupations in ground state subspace and excited states as long as they are both localized and can be considered as a composite qubit. To conclude, non-Adiabatic population of fermions onto the low-lying excited bound states has no effect on the non-Abelian statistics due to the fact that the fermion parity shared by a pair of vortices is not affected by such population. 6.2.3 Effects of Excited Extended States We next consider the effect of excited states that are extended in space. Usually such states form a continuum. The scenario considered here may not be very 119
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 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: tunneling amplitudes between them a
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

Create successful ePaper yourself

Delete template?

Save as template?