Topological Superconductors
ISSP, The University of Tokyo, Masatoshi Sato
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Outline
1. What is topological superconductor
2. Topological superconductors in various systems
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What is topological superconductor ?
Topological superconductors
Bulk:gapped state with non-zero topological #
Boundary:gapless state with Majorana condition
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Bulk: gapped by the formation of Cooper pair
In the ground state, the one-particle states below the fermi energy are fully occupied.
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Topological # can be defined by the occupied wave function
Topological # = “winding number”
Entire momentum space
Hilbert space of occupied state
empty band
occupied band
A change of the topological number = gap closing
A discontinuous jump of the topological number
Vacuum( or ordinary insulator) Topological SC
Gapless edge state 7
Therefore,
gap closing
Bulk-edge correspondence
If bulk topological # of gapped system is non-trivial, there exist gapless states localized on the boundary.
For rigorous proof , see MS et al, Phys. Rev. B83 (2011) 224511 .
different bulk topological # = different gapless boundary state
2+1D time-reversal breaking SC
2+1D time-reversal invariant SC
3+1D time-reversal invariant SC
1st Chern #(TKNN82, Kohmoto85)
Z2 number(Kane-Mele 06, Qi et al (08))
3D winding #(Schnyder et al (08))
1+1D chiral edge mode
1+1D helical edge mode
2+1D helical surface fermion
Sr2RuO4Noncentosymmetric SC
(MS-Fujimto(09))3He B
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The gapless boundary state = Majorana fermion
Majorana Fermion
Dirac fermion with Majorana condition
1. Dirac Hamiltonian
2. Majorana condition
particle = antiparticle
For the gapless boundary states, they naturally described by the Dirac Hamiltonian
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How about the Majorana condition ?
The Majorana condition is imposed by superconductivity
[Wilczek , Nature (09)]
Majorana condition
quasiparticle anti-quasiparticlequasiparticle in Nambu rep.
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Topological superconductors
Bulk:gapped state with non-zero topological #
Boundary:gapless Majorana fermion
Bulk-edge correspondence
A representative example of topological SC: Chiral p-wave SC in 2+1 dimensions
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BdG Hamiltonian
with
chiral p-wave
spinless chiral p-wave SC
[Read-Green (00)]
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Topological number = 1st Chern numberTKNN (82), Kohmoto(85)
MS (09)
Fermi surface
Spectrum
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SC
2 gapless edge modes(left-moving , right moving, on different sides on boundaries)
Edge state
Bulk-edge correspondence
Majorana fermion
• There also exist a Majorana zero mode in a vortex
We need a pair of the zero modes to define creation op.
vortex 1vortex 2
non-Abelian anyon
topological quantum computer16
Ex.) odd-parity color superconductorY. Nishida, Phys. Rev. D81, 074004 (2010)
color-flavor-locked phase
two flavor pairing phase
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For odd-parity pairing, the BdG Hamiltonian is
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(B)
Topological SC
Non-topological SC
• Gapless boundary state• Zero modes in a vortex
(A) With Fermi surface
No Fermi surface
c.f.) MS, Phys. Rev. B79,214526 (2009) MS Phys. Rev. B81,220504(R) (2010) 19
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Phase structure of odd-parity color superconductor
Non-Topological SC Topological SC
There must be topological phase transition.
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Until recently, only spin-triplet SCs (or odd-parity SCs) had been known to be topological.
Is it possible to realize topological SC in s-wave superconducting state?
Yes !
A) MS, Physics Letters B535 ,126 (03), Fu-Kane PRL (08)
B) MS-Takahashi-Fujimoto ,Phys. Rev. Lett. 103, 020401 (09) ;MS-Takahashi-Fujimoto, Phys. Rev. B82, 134521 (10) (Editor’s suggestion),J. Sau et al, PRL (10), J. Alicea PRB (10)
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Majorana fermion in spin-singlet SC
① 2+1 dim Dirac fermion + s-wave Cooper pair
Zero mode in a vortex
With Majorana condition, non-Abelian anyon is realized
[Jackiw-Rossi (81), Callan-Harvey(85)]
[MS (03)]
MS, Physics Letters B535 ,126 (03)
vortex
On the surface of topological insulator [Fu-Kane (08)]
Spin-orbit interaction => topological insulator
Topological insulator
S-wave SC
Dirac fermion + s-wave SC
Bi2Se3
Bi1-xSbx
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Hsieh et al., Nature (2008)
Nishide et al., PRB (2010)
Hsieh et al., Nature (2009)
2nd scheme of Majorana fermion in spin-singlet SC
② s-wave SC with Rashba spin-orbit interaction[MS, Takahashi, Fujimoto PRL(09) PRB(10)]
Rashba SO
p-wave gap is induced by Rashba SO int.
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Gapless edge statesx
y
a single chiral gapless edge state appears like p-wave SC !
Chern number
nonzero Chern number
For
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Majorana fermion
Summary
• Topological SCs are a new state of matter in condensed matter physics.
• Majorana fermions are naturally realized as gapless boundary states.
• Topological SCs are realized in spin-triplet (odd-parity) SCs, but with SO interaction, they can be realized in spin-singlet SC as well.
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