Realization of the Topological Superconductor Phases Protected by Chiral Symmetry

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Realization of the Topological Superconductor Phases Protected by Chiral Symmetry

Ching-Kai Chiu (邱靖凱)

Kavli Institute for Theoretical Sciences University of Chinese Academy of Sciences

Chiral Matter and Topology Workshop at National Taiwan University

December 6, 2018

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The ten-fold classification of topological phases

D DIII

AII CII C CI AI A AIII

Z

2

0 Z 0 0 0 Z 0

Z

2

Z

2

0 Z 0 0 0 Z

Z Z

2

Z

2

0 Z 0 0 0

0 Z Z

2

Z

2

0 Z 0 Z

0 0 Z Z

2

Z

2

0 Z

Z 0 Z 0 Z

0 0d 1d 2d 3d 4d

+1 +1 0 -1 -1 -1 0 0 0 C

0 -1 -1 -1 0 +1 +1 0 0 T

0 1 0 1 0 1 0

BDI +1 +1 1 Z

2

Z 0 0 0

0 1 S

Time-Reversal Symmetry

Particle-hole symmetry

Chiral

Symmetry Integer Quantum Hall Effect, Quantum Anomalous Hall Effect TKNN number

Z2 Time Reversal Symmetric Topological Insulators

k

2d

3d

XL Qi, TL Hughes, Shou-Cheng Zhang, Physical Review B 78 (19), 195424 (2008)

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The ten-fold classification of topological phases

D DIII

AII CII C CI AI A AIII

Z

2

0 Z 0 0 0 Z 0

Z

2

Z

2

0 Z 0 0 0 Z

Z Z

2

Z

2

0 Z 0 0 0

0 Z Z

2

Z

2

0 Z 0 Z

0 0 Z Z

2

Z

2

0 Z

Z 0 Z 0 Z

0 0d 1d 2d 3d 4d

+1 +1 0 -1 -1 -1 0 0 0 C

0 -1 -1 -1 0 +1 +1 0 0 T

0 1 0 1 0 1 0

BDI +1 +1 1 Z

2

Z 0 0 0

0 1

S

^Z² Topological Superconductors hosting

Majorana zero modes

Topological Superconductors

Z Topological Superconductors hosting Majorana chiral edge modes

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Majorana Workshop at the Kavli Institute for Theoretical Sciences

Z2 Topological Superconductors hosting Majorana zero modes

Z Topological Superconductors hosting Majorana chiral edge modes

In Beijing, Jan 8th to 11th, 2019

Invited Speakers

Hong Ding, IOP CAS

Hao Zhang, Tsinghua University Dong Liu, Tsinghua University Jun-Yi Ge, Shanghai University Tetsuo Hanaguri, Riken

Tadashi Machida, Riken

Roland Wiesendanger, University of Hamburg Jinfeng Jia, Shanghai Jiao Tong University Hao Zheng, Shanghai Jiao Tong University Hai-Hu Wen, Nanjing University

Donglai Feng, Fudan University Jun He, Riken

Organizers

Fuchun Zhang, Ching-Kai Chiu, KITS

Class D

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Time-reversal symmetry

D DIII

AII CII C CI AI A AIII

Z

2

0 Z 0 0 0 Z 0

Z

2

Z

2

0 Z 0 0 0 Z

Z Z

2

Z

2

0 Z 0 0 0

0 Z Z

2

Z

2

0 Z 0 Z

0 0 Z Z

2

Z

2

0 Z

Z 0 Z 0 Z

0 0d 1d 2d 3d 4d

+1 +1 0 -1 -1 -1 0 0 0 C

0 -1 -1 -1 0 +1 +1 0 0 T

0 1 0 1 0 1 0

BDI +1 +1 1 Z

2

Z 0 0 0

0 1

S

Z Time-reversal symmetric topological

Superconductors

Topological Superconductors

Z2 Topological Superconductors hosting Majorana Helical edge modes

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Topological Superconductor phases with chiral symmetry

DIII 0 Z

2

Z

2

Z 0

0d 1d 2d 3d 4d

+1 C

-1 T

1 BDI +1 +1 1 Z

2

Z 0 0 0

S

Z Time-reversal symmetric topological

Superconductors

Topological Superconductors

Z2 Topological Superconductors hosting Majorana Helical edge modes

Time-reversal symmetry

Particle-hole symmetry

Chiral symmetry

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Topological Superconductors

1d 2d 3d C

T S

BDI +1 +1 1 Z 0 0

Spinless time-reversal symmetric topological superconductors 1D: Multiple Majorana bound states

Majorana operator

Time reversal symmetry

Two Majorana coupling

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Interaction of four Majorana fermions emerges

The interaction preserves all of the symmetries

This is a well-studied Majorana interacting system 1. Majorana strip order

(Y Kamiya, A Furusaki et al, PRB 98 (16), 161409 (2018))

2. Enlarge topological region

(HH Hung, CKC et al, Scientific Reports (2017), Jian-Jian Miao, Fuchun Zhang, et al, Scientific Reports (2018))

3. Majorana fermion surface code for universal quantum computation (

^S

Vijay, TH Hsieh, L Fu, PRX 2015)

4. Emergent Supersymmetry

(A Rahmani et al, PRL 2015)

The realization of the class BDI platform

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Setup: Fu-Kane model

Abrikosov lattice of vortices in the s-wave superconducting (SC) surface of a strong topological insulator (STI).

A Majorana ZERO mode in a vortex

L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).

CK Chiu, DI Pikulin, M Franz, Physical Review B 91 (16), 165402 (2015)

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Hamiltonian of the superconducting surface of a STI

Particle-hole symmetry is automatically preserved.

Class D implies multiple Majorana zero modes are not stable due to Z

2

invariant.

J.C.Y. Teo and C.L. Kane, PRB 82, 115120 (2010)

When chemical potential vanish, chiral symmetry is preserved.

Time reversal symmetry operator T=SC.

M. Cheng et al, PRB 82, 094504 (2010)

Class BDI

Break time-reversal symmetry

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Estimate interaction strength

in real superconducting topological insulators

The heterostructure of NbSe

2

and Bi

2

Te

3

Jin-Feng Jia et al, Phys. Rev. Lett. 112, 217001 (2014)

Microscopic origin of the interaction terms

Where

CK Chiu, DI Pikulin, M Franz, Physical Review B 91 (16), 165402 (2015) HH Hung, CKC et al, Scientific Reports (2017)

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2D Topological Superconductors

1d 2d 3d C

T

D 0 +1 0 Z

2

Z 0

S

Time-Reversal symmetry breaking topological superconductors 1D: Majorana Chains (Majorana bound state FeTe

1-x

Se

x

)

2D: p+ip topological superconductors (Chiral Majorana edge mode)

DIII -1 +1 1 Z

2

Z

2

Z

Spinfull Time-Reversal Topological Superconductors 1D: Majorana kramers’ pair

2D: Helical Majorana edge mode

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Start from 2D Topological Chern Insulator

edge bulk

bulk

Quantum Anomalous Hall Insulator (QAHI, Chern Insulator)

= Integer Quantum Hall Insulator without magnetic field

Two Majoranas = One electron

QAHI

Superconducting Nambu basis (particle-hole)

edge bulk

bulk

D Z

2d +1

C 0

T

0 S

Chang, C.-Z., et al., Science 340, 167 (2013)

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Single Chiral Majorana Edge Mode

An s-weve superconductor on the top of Quantum Anomalous Hall Insulator (QAHI)

Two Majoranas = One electron

QAHI

QAHI QAHI

(TSC)

TSC

Edge and bulk spectrum

X.-L. Qi, T. L. Hughes, S.-C. Zhang, Phys. Rev. B 82, 184516 (2010)

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QAHI

QAHI QAHI

TSC

Single Chiral Majorana Edge Mode

(TSC)

Green Region Topological Phase Transition

One electron mode = Two Majorana modes

Bulk Gap closing Single Chiral Majorana mode

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Two-terminal Conductance

QAHI

QAHI QAHI

TSC

Longitudinal Conductance

Lead 1 Lead 2 Lead 1 Lead 2

Majorana is a half electron

Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Mu- rata, X. Kou, Z.

Chen, T. Nie, Q. Shao, Y. Fan, S.-C. Zhang, K. Liu, J. Xia, and K. L. Wang, Science 357, 294 (2017) J. Wang, Q. Zhou, B. Lian, S.-C. Zhang, Phys. Rev. B 92, 064520 (2015).

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2D Topological Superconductors

1d 2d 3d C

T

D 0 +1 0 Z

2

Z 0

S

Time-Reversal symmetry breaking topological superconductors 1D: Majorana Chains (Majorana bound state FeTe

1-x

Se

x

)

2D: p+ip topological superconductors (Chiral Majorana edge mode)

DIII -1 +1 1 Z

2

Z

2

Z

Spinfull Time-Reversal Topological Superconductors 1D: Majorana kramers’ pair

2D: Helical Majorana edge mode

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Time Reversal Symmetry

Integer Quantum Hall Effect in 1980 Chiral Electron Mode

Quantum Spin Hall Effect in 2005 ~2007

Helical Electron Mode protected by time reversal symmetry

Time Reversal Symmetry

AII A

Z

2

Z 2d

0 0 C

-1 0 T

0 0 S Quantum Spin Hall Insulator

edge

bulk bulk

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Time Reversal Symmetry

Chiral Topological Superconductor in 2016 Chiral Majorana Mode

D DIII

Z Z

2

2d +1

+1 C 0

-1 T

0 1 S Time Reversal Symmetry

k

Helical Topological Superconductor

Helical Majorana Mode

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Proposals to realize Helical Majorana edge mode

M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).

S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. Lett. 108, 147003 (2012).

J. Wang, Y. Xu, and S.-C. Zhang, Phys. Rev. B 90, 054503 (2014).

J. Wang, Phys. Rev. B 94, 214502 (2016) C.-X. Liu and B. Trauzettel, Phys. Rev. B 83, 220510 (2011).

J. C. Y. Teo and C. L. Kane, Phys. Rev. B 82, 115120 (2010).

Exotic Superconducting Pairing Phase difference

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Idea

QAHI QAHI

S-wave superconductor

QSHI QSHI+ QSHI

s-wave SC

QSHI s-wave SC QSHI QSHI

Quantum Anomalous Hall Insulator: QAHI

Quantum Spin Hall Insulator: QSHI

Chiral Majorana edge mode

Helical Majorana edge mode?

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Quantum Spin Hall Insulator with S-wave SC

Yingyi Huang, CKC, Physical Review B 98 (8), 081412 (R)(2018) Quantum Spin Hall Insulator (QSHI)

S-wave superconductivity destroys the helical electron mode

QSHI QSHI+ QSHI

s-wave SC

s-wave pairing potential

QSHI s-wave SC QSHI QSHI

Electron, not Majorana

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Antiferromagnetic order — Different spins located at different atoms

A B

Antiferromagnetic Quantum Spin Hall Insulator

A

B

A helical electron mode

A B

B A Spins flip

Effective time reversal symmetry (Use crystalline symmetry)

Z. Wang, H. Zhang, D. Liu, C. Liu, C. Tang, C. Song, Y. Zhong, J. Peng, F. Li, C. Nie, et al., Nature Materials 15, 968 (2016) R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev. B 81, 245209 (2010)

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Model of the Antiferromagnetic Quantum Spin Hall Insulator

A B B A Spins flip

spin orbital

Effective time reversal symmetry operator

Hamiltonian

Time-reversal symmetry

Yingyi Huang, CKC, Physical Review B 98 (8), 081412 (R)(2018)

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Non-Trivial

Helical Electron Edge Mode

A

B

Two Helical Majorana modes

DIII Z

2

2d +1

C -1

T

1 S

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Introduce s-wave superconductivity

Helical Electron Edge Mode

One helical Majorana Edge Mode Topological Phase Transition

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How to observe the Helical Majorana Edge Mode

Yingyi Huang, CKC, Physical Review B 98 (8), 081412 (R)(2018)

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Disorders breaks the effective time-reversal symmetry

Potential difference disorder Disorder destroys helical electron mode

The helical Majorana edge mode can be present in weak disorders

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Conclusion

A Majorana zero mode is trapped in a vortex on the

superconducting Dirac cone surface. When the chemical potential is at the neutral point of the Dirac cone, the chiral symmetry emerges; the Majorana interactions dominate.