ABSTRACT - DRUM - University of Maryland

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maps onto an effective continuum model but with two fermion species: ∫ Ĥ 2h,eff = (ξ k ĥ † +,kĥ+,k + ξ −k ĥ † −,kĥ−,k) + interactions, (2.18) k where ĥ±,k are fermion operators near the two pockets labeled by a pseudospin index σ = ± and the spectrum is asymptotically given by ξ k = k 2 /2m + α(k 3 x − 3k x k 2 y) − E F , (2.19) with k measured from the corner points of the hexagonal Brillouin zone. Note that under a π or ±π/3 rotation, the spectrum transforms as ξ k → ξ −k and this symmetry is preserved if σ → −σ. This leads to a pairing analogous to the s-wave pairing of spin-1/2 fermions, with the order parameter of the inter-pocket pairing defined as ∆ h = g ∫ 〈ĥ+,kĥ−,−k〉. However, this is an f-wave pairing state, because k under a π/3-rotation, σ → −σ and ∆ h (k) changes sign. Since the low-density limit leads to a topological phase and the high-density limit leads to an f-wave topologically trivial state, there must be a quantum phase transition in between. The entire phase diagram can be derived using Eqs. (2.4) and (2.5) and the real-space construction as follows: On a triangular lattice we can define three order parameters on the nearest neighbor bonds corresponding to the three lattice vectors, e n with azimuth angles 2nπ/3 and n = 0, 1, 2: ∆ n = g〈ĉ r ĉ r+en 〉. Two different types of pairing channels are formed by these three order parameters: An f-wave channel with ∆ n = ∆ and a p-wave channel with ∆ n = ∆e ±2πin/3 . The resulting phase diagram is shown in Fig. 2.2. As expected, a topological p x + ip y -wave SC state with ∆ n = ∆e ±2πin/3 is stabilized at low fillings, while an 44
0.2 T/t 0.1 Normal f p x + ip y µ ∗ 0 6 3 0 3 µ/t Figure 2.2: The phase diagram for spinless fermions on a triangular lattice with nearest-neighbor hoppings and attraction (g/t=1). The bottom of the band is located at µ = −6t and the top is at µ = 3t; µ ∗ = 2t corresponds to a van Hove singularity. Two SC phases, with (p x + ip y )- and f-wave symmetries are present. They are separated by a first-order phase transition at µ cr /t ≈ 1.057. The insets (left to right) are the FSs for µ < µ ∗ , µ µ ∗ , and µ > µ ∗ and the dashed lines indicate the nodal directions of the f-wave SC. 45
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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: with the gap operator being ˆ∆ =
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
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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
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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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