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Chiral and helical Majorana fermions in non-centrosymmetric superconductors on honeycomb lattice Chung-Hou Chung 仲崇厚 Department of Electrophysics, National Chiao-Tung University HsinChu, Taiwan

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Chiral and helical Majorana fermions in non-centrosymmetric superconductors on honeycomb lattice

Chung-Hou Chung 仲崇厚 Department of Electrophysics, National Chiao-Tung University

HsinChu, Taiwan

Workshop on recent developments in chiral matter and topology, NTU, Dec. 6-9, 2018 Der-Hau Lee and CH C, Phys. Status Solidi B 2018, 1800114

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• Introduction to Non-centrosymmetric superconductors

Outline

• Topological superconductor on honeycomb lattice:

-- topological phase diagram of the superconducting Kane-Mele t-J model

--singlet-triplet pairing mixture and coexisting helical-chiral Majorana zero mode

• Summary

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity

Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity

Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity

Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”

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Non-centrosymmetric superconductor on honeycomb lattice

Topological superconductor on honeycomb lattice

with mixed chiral and helical Majorana zero modes

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Hassn, Kane, 2010

3D Topological insulators

Gapless surface states due to Spin-orbit coupling

odd integer

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2D symmetry protected Topological insulator—

Quantum Spin Hall (Helical) Edge States in Kane-Mele model

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2D symmetry protected Topological insulator

--Quantum Spin Hall (Helical) Edge States in KM model

Kane, Mele, PRL 2005

Time-reversal invariant 

Pair of counter-propagating edge states helical edge states

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Topological superconductors : Superconductors with large SO coupling

Majorana fermions:

Gapless quasi-particle edge states with Dirac spectrum Particle-hole symmetry

M.Z. Hasan, C.Kane, RMP 2010, X.L. Qi, S.C. Zhang, RMP 2011

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M. Sato’s talk, Dec. 13, 2013, NCTU, Taiwan

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Prof. M. Sato’s talk, Dec. 13, 2013, NCTU, Taiwan

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Prof. Yuval Oreg’s talk, Dec. 13, 2013, NCTU, Taiwan

s-wave superconductors + 1D wire with strong SO coupling

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Spin triplet, p-wave superconductor in a magnetic field

Time-reversal broken topological superconductors Proposals for Majorana fermions

Can helical Majorana fermions exist?

Chiral Majorana fermions

 New route: Helical Majorana fermions without TRS

Via doping the Mott insulator with SO coupling

1. Spin singlet, d-wave, s-wave + SO coupling via proximity effect

2.

 TRITOPs: Time-Reversal-Invariant-Topological-Superconductors

Helical Majorana fermions exist in TRS breaking chiral d-wave superconductors in doped Kane-Mele t-J model

Oreg 2014, Flensberg 2014, Law 2012

S.C. Zhang 2009

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Helical MFs in Time-Reversal-Topological-Superconductors (TRITOPs)

Y.Oreg’s talk, Dec. 13, 2013, NCTU, Taiwan

2D spin-triplet superconductors X-L Qi et al PRL 2009 K-T Law et al PRB 2013

Y. Oreg et. al PRB 2014

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Pairing symmetry of superconductivity on correlated honeycomb lattice

nearest-neighbour pairing

Resonating Valence Bond

t

correlated regime

t~J J

Coulomb U graphene

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Helical Majorana fermions in chiral-d-wave superconductors in doped KM t-J model

Superconducting pairing order parameter

J.P. Hu et al PRB 2008

N=8 CH Chung et al Scientific Reports 2016

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Renormalized Mean-Field Theory (RMFT)

RMFT:

Gutzwiller projected BCS wave function

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Symmetries of H

k

 Particle-Hole Symmetry C:

C

-1

H

k

C = - H

-k

 Sub-lattice symmetry S:

S: CA(k)  CB(k)

S

-1

H

k

S = H

-k

C: C

k

 C

k

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d+id’-wave superconductivity

kx

-5 -4 -3 -2 -1 0 1 2 3 4 5

ky

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 d

 = 0.1 tso = 0.5 J = 0.2

TRS breaking

Normal state Fermi surface CH Chung et al Scientific Reports 2016

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Time-reversal symmetric (TRS) v.s. Time-reversal broken (TRB) superconductors

T

-1

H

k

T = H

-k T

TRS:

*

(-k) = ∆ (k)

TRB:

in graphene superconductivity

chiral edge state

Black-Schaffer PRL 2013

t

so << chiral edge state

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TKNN number in spin-singlet chiral-d-wave superconductors in graphene (tSO0)

chiral superconductors

2 co-propagating chiral edge modes Black-Schaffer et al. J. Phys. C 2014

d +i d-wave pairing breaks time-reversal symmetry (TRS) and parity (P) symmetry TKNN number:

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Bogoliubov excitation spectrum at large tso

t

so

2-fold degeneracy at each kx 2 Dirac-dispersed lines

CH Chung et al Scientific Reports 2016

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Helical Majorana zero modes in d+id’-wave superconductivity in doped KM t-J ribbon

2 pairs of helical MFs

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Helical Majorana fermions protected by the additional symmetry

k

_

CH Chung et al Scientific Reports 2016

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Pseudo-Spin Chern number Charge Chern number

Topological quantum phase diagram

S M Huang, CY Mou, Wei-Feng-Tsai, CH Chung, PRB, 2016 Chiral edge states

Helical edge states

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Topological phase transition between choral and spin-Chern phases

Bulk band gap closes at the phase transition

NTKNN =2 Nspin-Chern=2 NTKNN =2

CH Chung et al Scientific Reports 2016

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Pseudo-spin Symmetry protects helical Majorana

0 0

CH Chung et al

Scientific Reports 2016

Pseudo-spin Symmetry

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p+ip-wave triplet pairing and Rashba SOC on honeycomb lattice

p+ip-wave triplet pairing pairing amplitude

-J’

Via KM SO, ferromagnetic spin coupling on NNN sites

triplet pairing

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NCS on honeycomb lattice

: d + i d-wave singlet pairing

: p + i p-wave triplet pairing Singlet-triplet mixing superconducting pairing

CH Chung et al pssb 2018

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charge Chern number

Topological phase diagram CH Chung et al pssb 2018

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CH Chung et al pssb 2018

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Edge states and mixed helical/chiral Majorana modes on a ribbon

(A,B), (E,F):

helical Majorana (C,D):

chiral Majorana

CH Chung et al pssb 2018

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K. F. Mak et al., Phys. Rev. Lett. 105, 136805 (2010) T. Cao et al. Nature Communication, 3, 887 (2012)

Direct band gap 1.8eV at Dirac points

Monolayer MoS 2

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Gate-tuned superconductivity in MoS2

Y. Saito et al. Nature

Physics volume12, pages144-149 (2016)

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Nat. Phys. 2018

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Y. Matsuda’s talk, IoP, Academia Sinica, Taipei, Nov. 29, 2018

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Y. Matsuda’s talk, IoP, Academia Sinica, Taipei, Nov. 29, 2018

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Summary

 Doped Kane-Mele t-J model

--TRB d+d’-wave singlet superconductivity in doped Kane-Mele t-J model, protected by pseudospin symmetry due to A-B sublattice

--Topological phase transition chiral helical Majorana fermions with increasing KM SO

 Rashba SO on correlated honeycomb lattice favors p+ip-wave triplet superconductivity

 Doped Kane-Mele + Rashba t-J model on honeycomb lattice

--coexistence between d+i d singlet and p+ip-wave triplet superconductivity --coexistence between helical and choral Majorana zero modes

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