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
• 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
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity
Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity
Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity
Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
M. Sigrist’s Talk “Key symmetries of superconductivity Inversion and time reversal symmetry”
Non-centrosymmetric superconductor on honeycomb lattice
Topological superconductor on honeycomb lattice
with mixed chiral and helical Majorana zero modes
Hassn, Kane, 2010
3D Topological insulators
Gapless surface states due to Spin-orbit coupling
odd integer
2D symmetry protected Topological insulator—
Quantum Spin Hall (Helical) Edge States in Kane-Mele model
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
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
M. Sato’s talk, Dec. 13, 2013, NCTU, Taiwan
Prof. M. Sato’s talk, Dec. 13, 2013, NCTU, Taiwan
Prof. Yuval Oreg’s talk, Dec. 13, 2013, NCTU, Taiwan
s-wave superconductors + 1D wire with strong SO coupling
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
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
Pairing symmetry of superconductivity on correlated honeycomb lattice
nearest-neighbour pairing
Resonating Valence Bond
t
correlated regime
t~J J
Coulomb U graphene
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
Renormalized Mean-Field Theory (RMFT)
RMFT:
Gutzwiller projected BCS wave function
Symmetries of H
k Particle-Hole Symmetry C:
C
-1H
kC = - H
-k Sub-lattice symmetry S:
S: CA(k) CB(k)
S
-1H
kS = H
-kC: C
k C
kd+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
Time-reversal symmetric (TRS) v.s. Time-reversal broken (TRB) superconductors
T
-1H
kT = H
-k TTRS:
∆
*(-k) = ∆ (k)
TRB:
in graphene superconductivity
chiral edge state
Black-Schaffer PRL 2013
t
so << chiral edge stateTKNN number in spin-singlet chiral-d-wave superconductors in graphene (tSO0)
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:
Bogoliubov excitation spectrum at large tso
t
so2-fold degeneracy at each kx 2 Dirac-dispersed lines
CH Chung et al Scientific Reports 2016
Helical Majorana zero modes in d+id’-wave superconductivity in doped KM t-J ribbon
2 pairs of helical MFs
Helical Majorana fermions protected by the additional symmetry
k
_
CH Chung et al Scientific Reports 2016
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
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
Pseudo-spin Symmetry protects helical Majorana
0 0
CH Chung et al
Scientific Reports 2016
Pseudo-spin Symmetry
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
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
charge Chern number
Topological phase diagram CH Chung et al pssb 2018
CH Chung et al pssb 2018
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
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
Gate-tuned superconductivity in MoS2
Y. Saito et al. Nature
Physics volume12, pages144-149 (2016)
Nat. Phys. 2018
Y. Matsuda’s talk, IoP, Academia Sinica, Taipei, Nov. 29, 2018
Y. Matsuda’s talk, IoP, Academia Sinica, Taipei, Nov. 29, 2018
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