ABSTRACT - DRUM - University of Maryland

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to measure the fermion parity, we need the “dual” version, the so-called Aharonov- Casher effect [41], that when a superconducting vortex moves around a fermion it also acquires a π Berry phase, which is almost a trivially obvious statement if we put our reference frame on the fermion. Based on Aharonov-Casher effect one can design appropriate interferometers to measure the fermion parity. For a detailed exposition we refer the readers to Chapter 6. However, it has been mathematically proved that the braiding operations can not realize all possible single-qubit rotations. In fact, Bravyi has established [42] that the Ising anyon computation model is an intersection of two classically simulatable models, quantum circuits with Clifford gates and fermionic linear optics. So the computational power of Ising anyon model with braidings and measurements is quite limited and is equivalent a classical computer. Thus, Ising anyons only offer topologically protected quantum memory and a limited set of protected gates. Generally speaking non-topological operations are needed in order to perform universal quantum computations [43]. There are also a number of interesting proposals to implement universal quantum computation using Ising anyons in a fully topologically protected way, e.g. by dynamically changing the topology [43], which we will not go into details in this introduction. 1.5 Physical Realizations of TSC In this section we discuss possible realizations of TSC. Although p x +ip y superconductor is a rather simple model in theory, it turns out to be very challenging to 18
find a realistic material in nature. Most electronic superconductors in metals have s-wave pairing, which can be traced back to the electron-phonon mediated pairing mechanism. To have the required p-wave pairing symmetry one clearly needs unconventional pairing mechanisms. There are a number of candidates, though, including the 3 He film in the superfluid A phase [44] and the oxide compound Sr 2 RuO 4 [45]. Although a lot of experimental efforts have been taken, progress in identifying the topological superconductivity/superfluidity in both systems is quite limited. Among the many obstacles we just mention that in both cases, due to the spin degeneracy, to observe a single Majorana zero mode requires creating a half-quantum vortex in the superfluid [46], in which the phase of the order parameter and the Cooper pair spin vector both wind by π. However this type of vortices are not thermodynamically stable: Its free energy diverges logarithmically with the system size. This apparently hinders the observation of Majorana excitations. In addition, the unconventional p-wave pairing symmetry, believed to be caused by ferromagnetic spin fluctuations, results in very low superconducting transition temperature, making the experimental setup very delicate. Recent theoretical progress has revealed a completely new avenue towards realizing chiral p-wave superconductivity, which becomes by far the most promising direction in the search of non-Abelian superconductivity. The approach is to engineer chiral p-wave superconductor from conventional materials instead of trying one’s luck in nature. In particular, the stringent requirement of the p-wave pairing is removed and all the proposals only involve ordinary s-wave superconductivity. In the following we discuss three independent different proposals for the pratical realiza- 19
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: In the case of Ising anyons, 2N vor
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
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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