Majorana Fermions in Superconducting Chains
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1 16 th December 2015 Majorana Fermions in Superconducting Chains Matilda Peruzzo
2
3 Fermions (I) Quantum many-body theory: Fermions Bosons
4 Fermions (II) Properties Pauli exclusion principle
5 Fermions (II) Properties Pauli exclusion principle
6 Fermions (II) Properties Pauli exclusion principle Anticommutation relations
7 Fermions (II) Properties Pauli exclusion principle Anticommutation relations
8 Fermions (II) Properties Pauli exclusion principle Anticommutation relations
9 Fermions (II) Properties Pauli exclusion principle Anticommutation relations
10 Majorana fermions Properties Hermitian creation operators Chargeless linear combination of electron and hole
11 Majorana fermions Where to look for Majorana fermions? Superconductivity h + Cooper pair e - +
12 Kitaev wire Finite superconducting chain A.Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi, 2001
13 Kitaev wire site occupation
14 Kitaev wire hopping between neighboring sites
15 Kitaev wire addition of a cooper pair in neighboring sites
16 Majorana operators Hermitian operators Majorana from different sites satisfy fermion commutation relations Two Majorana operators correspond to one fermion
17 Majorana operators Hermitian operators Majorana from different sites satisfy fermion commutation relations Two Majorana operators correspond to one fermion
18 Finite superconducting chain
19 Kitaev hamiltonian Analytically solvable cases: Trivial phase Topological phase
20 Kitaev hamiltonian Edge state
21 Numerical calculation Topological phase Trivial phase
22 Numerical calculation 0 3t Chain site Edge state wavefunction
23 Infinite superconducting chain
24 Closing the chain Periodic boundary conditions Fourier transform into momentum space
25 Momentum space Transformation Hamiltonian becomes decomposable
26 Momentum space Problem becomes 2-dimentional
27 Energy bands - 3t 3t E Gap closing at μ = -2t and μ = 2t
28 Topological invariant Quantity that can identify the topological phase of the system
29 Topological invariant Quantity that can identify the topological phase of the system
30 Topological invariant Trivial phase Topological phase
31 Topological invariant Invariant quantity In the trivial phase In the topological phase More general Hamiltonian
32 Topological invariant Trivial phase Q = 1 Topological phase Q = -1
33 Experimental realization and results
34 s & p superconductors S-pairing P-pairing
35 System requirements Spinless p-wave superconductor Band gap and cooper pairing
36 System requirements Spinless p-wave superconductor Allows for momentum dependent band gap
37 System requirements Spinless p-wave superconductor
38 System requirements Spinless p-wave superconductor
39 System requirements Spinless p-wave superconductor
40 System requirements Superconductivity Magnetic field Spin orbit coupling
41 System requirements Electron chain + superconductivity
42 System requirements Electron chain + superconductivity + spin orbit coupling
43 System requirements Electron chain + superconductivity + spin orbit coupling + magnetic field
44 System requirements Electron chain + superconductivity + spin orbit coupling + magnetic field Condition:
45 Andreev reflection metal superconductor barrier
46 Andreev reflection metal superconductor barrier
47 Andreev reflection metal superconductor barrier Current 2e
48 Conductance
49 Experimental results V. Mourik et al., Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices, Science, 2012
50 Extensions Wire circuits Higher dimensions Exchange operations
51 Thank you for listening
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