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**Topological Superconductors**

ISSP, The University of Tokyo, Masatoshi Sato

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**What is topological superconductor **

Outline What is topological superconductor 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**

empty band occupied band Entire momentum space Hilbert space of occupied state Topological # = “winding number”

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**( or ordinary insulator)**

A change of the topological number = gap closing gap closing A discontinuous jump of the topological number Therefore, Vacuum ( or ordinary insulator) Topological SC Gapless edge state

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**Bulk-edge correspondence**

If bulk topological # of gapped system is non-trivial, there exist gapless states localized on the boundary. 厳密な証明もできる [MS et al (11)] For rigorous proof , see MS et al, Phys. Rev. B83 (2011)

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**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 Sr2RuO4 Noncentosymmetric SC (MS-Fujimto(09)) 3He B

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**The gapless boundary state = Majorana fermion**

Dirac fermion with Majorana condition Dirac Hamiltonian Majorana condition Boundary state は Majorana fermionになることを書く 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 quasiparticle in Nambu rep. quasiparticle anti-quasiparticle Majorana condition [Wilczek , Nature (09)]

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**Topological superconductors**

Bulk-edge correspondence Bulk: gapped state with non-zero topological # Boundary: gapless Majorana fermion

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**A representative example of topological SC: **

Chiral p-wave SC in 2+1 dimensions [Read-Green (00)] BdG Hamiltonian spinless chiral p-wave SC with chiral p-wave

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**Topological number = 1st Chern number**

TKNN (82), Kohmoto(85) MS (09)

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**Bulk-edge correspondence**

Edge state SC Fermi surface 2 gapless edge modes (left-moving , right moving, on different sides on boundaries) Spectrum Boundary conditionを示す図がほしい Majorana fermion Bulk-edge correspondence

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**There also exist a Majorana zero mode in a vortex**

We need a pair of the zero modes to define creation op. vortex 2 vortex 1 non-Abelian anyon topological quantum computer

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**Ex.) odd-parity color superconductor**

Y. Nishida, Phys. Rev. D81, (2010) color-flavor-locked phase two flavor pairing phase

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**For odd-parity pairing, the BdG Hamiltonian is**

自分の論文の引用

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**(A) With Fermi surface Topological SC (B) No Fermi surface**

Gapless boundary state Zero modes in a vortex (B) No Fermi surface \muによる相転移を指摘 Non-topological SC c.f.) MS, Phys. Rev. B79, (2009) MS Phys. Rev. B81,220504(R) (2010)

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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 ! Odd-parity SC 以外ではtopological SC は可能か MS, Physics Letters B535 ,126 (03), Fu-Kane PRL (08) MS-Takahashi-Fujimoto ,Phys. Rev. Lett. 103, (09) ; MS-Takahashi-Fujimoto, Phys. Rev. B82, (10) (Editor’s suggestion), J. Sau et al, PRL (10), J. Alicea PRB (10)

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**Majorana fermion in spin-singlet SC**

MS, Physics Letters B535 ,126 (03) 2+1 dim Dirac fermion + s-wave Cooper pair vortex Zero mode in a vortex [Jackiw-Rossi (81), Callan-Harvey(85)] With Majorana condition, non-Abelian anyon is realized [MS (03)]

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**Topological insulator**

On the surface of topological insulator [Fu-Kane (08)] Bi1-xSbx Hsieh et al., Nature (2008) Dirac fermion + s-wave SC S-wave SC Topological insulator Nishide et al., PRB (2010) Bi2Se3 Hsieh et al., Nature (2009) まずは、spin-singlet superconductorであっても、マヨラナフェルミオンが実現可能であることを説明したいと思います。 Spin-orbit interaction => topological insulator

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**2nd scheme of Majorana fermion in spin-singlet SC**

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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**a single chiral gapless edge state appears like p-wave SC !**

Gapless edge states x y Majorana fermion For vortexの話も書く a single chiral gapless edge state appears like p-wave SC ! Chern number nonzero Chern number

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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. 文献を書く

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