spin- topological superconductivity beyond triplet pairingnon-degenerate fs so coupling inversion...
TRANSCRIPT
Spin-π
πtopological superconductivity
beyond triplet pairing
Congjun Wu
University of California, San Diego
July 6, WHU Summer School
π + ππ π β ππx
yz
Wang Yang (UCSD UBC)
Da Wang (Nanjin Univ. UCSD Nanjing Univ)
Yi Li (UCSD Princeton Johns Hopkins)
Tao Xiang (IOP, Chinese Academy of Sciences)
2
Collaborators:
Supported by NSF, AFOSR
Reference
1. W. Yang, Chao Xu, CW, arXiv:1711.05241.
2. W. Yang, Tao Xiang, and CW, Phys. Rev. B 96, 144514 (2017).
3. W. Yang, Yi Li, CW, Phys. Rev. Lett. 117, 075301(2016).
4. Y Li, D. Wang, CW, New J. Phys. 15 085002 (2013)
5. D Wang, Zhou-Shen Huang, CW, PRB 89, 174510 (2014)
Novel unconventional superconductivity
βBoundary of boundaryβ Majorana fermion without
spin-orbit coupling
Spin-3/2 half-Heusler SC β beyond triplet pairing
Majorana flat-band and
spontaneous TR symm. breaking
septet
A. J. Leggett, Rev. Mod. Phys 47, 331 (1975)
L=1, S=1, J=L+S=0
β’ Topological: DIII class (time-reversal invariant)
π
π π β π = 0 B
)(Λ kd
Ξ
β’ Unconventional but isotropic spin-orbit coupled gap function
β’ Is 3He-B alone?
New opportunities in multi-component
fermion systems!
The distinction of the 3He-B phase
π π = π
β’ Cold atom: alkali/alkaline-earth fermions
4-component fermion systems: beyond triplet
Kim, Hyunsoo, et al., Science Advances Vol. 4, eaao4513 (2018).
β’ Hole-doped semiconductors:
CW, J. P. Hu, and S. C. Zhang. PRL 91 186402 (2003).CW, Mod. Phys. Lett, (2006).CW J. P. Hu, and S. C. Zhang. Int. J. Mod. Phys. B 24 311 (2010)
β’ Spin π
π: Quintet and Septet pairings
beyond singlet and triplet.
septet
Wang Yang, Yi Li, CW, PRL 117, 075301 (2016).W. Yang, Tao Xiang, and CW, PRB 96, 144514 (2017).
β’ Experiment: nodal superconductivity in half-Heusler compound YPtBi.
S-wave quintet pairing β Non-Abeliean statistics
CW, Mod. Phys. Lett. (2006)
CW, J. P. Hu, and S. C. Zhang. Int. J. Mod. Phys. B 24 311 (2010)
β’ Half-quantum vortex (HQV) loop οΌAlice String) β the SO(4) Cheshire charge.
β’ Non-Abeliean phase: particle penetrating HQV loop.
|3/2
| 0
|1
|1/2 |β1/2
|2
| ππ§
Isotropic pairings beyond singlet and triplet
d-vector d-tensor
Spherical harmonics
β’ Isotropic pairings:
s-wave + singlet
p-wave + triplet
d-wave + quintet
f-wave + septet
π½ = πΏ + π = 0
Spin tensors (spin, quadrupole, octupole)
β’ Pairing Hamiltonian.
ΞπΏ,πΌπ½ π = ΞπΏ π=βπΏπΏ β πππΏ,βπ
π ππΏππ
Wang Yang, Yi Li, CW, PRL 117, 075301 (2016).
β’ Odd-parity pairing
states are topo. nontrivial.
Pictorial Rep.β spin structure of the gap function
septet
|30 β Ξ3
2
β Ξ1
2
+ Ξβ
1
2
β Ξβ
3
2
β’ Helical basis: π β π|ππΌβ© = πΌ|ππΌβ©
ΞπΌ π : β¨πΌ+ π πΌ+(βπ) β©β’ Intra-helical FS pairings (different phase patterns):
(πΌ = Β±3
2, Β±
1
2)
Topo. index
# =3-1=2
triplet
|πππ§ = |10β© β Ξ3
2
+ Ξ1
2
β Ξβ
1
2
β Ξβ
3
2
High topo.
index # =3+1=4,
distinct from 3He-B
Boundary Majorana modes (f-wave septet)
Bulk Vacuum
β’ Zero modes (π2π· = 0) as chiral eigenstates.
πͺππ is a symmetry only for zero modes
Chiral operator πͺππ = ππͺππͺπ»;
π = +, β, +, β, for πΌ =3
2,1
2, β
1
2, β
3
2.
β’ k.p theory: linear Majorana-Dirac cones.
032, +
012, β
0β
12, +
0β
32, β
πΆπβ ππΌ2π· = 0πΌ , π = π |0πΌ , π β©
States with opposite chiral indices couple
β’ A linear and a cubic Majorana-Dirac
cones.
p-wave boundary Andreev-Majorana modes
β’
π»ππππ
(π||) =βπ
ππΉ
00
00
ππ+2
π(π+3)
ππ+
ππ+2
ππβ2
βππβ
π(πβ3)
ππβ2
00
00
1st order π β π theory
π2π· = 032, +
012, +
0β
12, β
0β
32, β
β’ Zero modes (π2π· = 0) with chiral indices
β’ Band inversion
π 1/2, ππ/π
Spin-3/2 systems: YPtBi half-Huesler semi-metal
β’ Low carrier density β semimetal
h. h. l. h.
π β 2 Γ 1018ππβ3, ππΉ~1
10
1
π
non-degenerate FS
SO coupling
Inversion symmetry broken
ππ/π
ππ/π
β’ Non-centrosymmetric: ππ symmetry
β’ Linear π-dependence of penetration depth β Nodal lines
Kim, Hyunsoo, et al., Science Advances Vol. 4, eaao4513 (2018).
π»πΏ π = Ξ»1 +5
2Ξ»2 π2 β 2Ξ»2 π β π
2
π΄ π = kxππ₯ + kyππ¦ + kzππ§
Band Hamiltonian of YPtBi
β’ Luttinger-Kohn for the hole band (Ξ8: π3/2)
β’ Non-centrosymmetric ππ invariant
ππ₯ = SySxSy β SzSxSz
ππ¦ = SzSySz β SxSySπ₯
ππ§ = SxSzSx β SySzSy
ππ₯
ππ¦
ππ§
π2 rep. of ππ
Inversion βTime reversal βππ group β
β’ Non-degenerate FS
π»ππππ π = π»πΏ π + π΄ π
β‘ P. M. R. Brydon, L. Wang, W. Weinert, D. F. Agterberg, Phys. Rev. Lett. 116 177001 (2016)
Pairing symmetries in speculations
Nodal rings in gap function for βπ
βπ= 0.3 and 0.7
β’ One possibility: π -wave singlet + π-wave septet
πΌ,π½
πππΌβ [(βπ + βππ¨ π )π ]πΌπ½πβππ½
β
Pairing within the same spin-split Fermi surface
Nodal rings around 001 , etc
β‘ P. M. R. Brydon, L. Wang, W. Weinert, D. F. Agterberg,Phys Rev Lett 116 177001 (2016)
π΄ π = ππ₯ππ₯ + ππ¦ππ¦ + ππ§ππ§
D. Agterberg, P. A. Lee, Liang Fu, Chaoxing Liu, I. Herbut, β¦β¦.
β’ Phase sensitive test?
Previous example (YBCO): zero-energy boundary modes
[11] boundary:π πππ = βπ ππππ
++β
β
[10] boundary:π πππ = π ππππ
C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994)
L. H. Greene, et al, PRL 89, 177001 (2002)
πππ
πππ’π‘
πππ
πππ’π‘
++β
β
β’ Surface Brilliouin zone:
Topo-index distribution in
[111]-surface for βπ
βπ= 0.3
A. P. Schnyder, P. M. R. Brydon, and C. Timm. PRB 85.2 (2012): 024522.
(ππ₯2π· , ππ¦
2π·) inside a loop non-trivial topo index Β±1
β’ Loops: projection of the gap nodal rings.
Topo-index for nodal-ring superconductors
Each (ππ₯2π· , ππ¦
2π·) a 1D superconductor
Majorana flat bands on the 111 -surface
e
e
e
o
o
o
π
πe
e
e
o
o
o
π
π
β’ Chiral index (πΆπβ = ππππ») for Majorana surface modes
a symmetry for zero modes (even, odd)
Non-magnetic impurity: odd under πΆπβ 1,3 β; 0,2 β
Magnetic impurity: even under πΆπβ 1,3 β; 0,2 β
β’ Selection rules:
Bright regions: Majorana zero modes
STM: quasi-particle interference (QPI) pattern
Ξππ π π, π
β’ Joint density of states of impurity scattering
Ξππ π π, πFourier transform
β’ Non-magnetic impurity on (111)-surface:
π
π
π
π
πΉπ(βπππ π = π, πβ₯ ) π°π(βπππ π = π, πβ₯ )
Novel unconventional superconductivity
βBoundary of boundaryβ Majorana fermion without
spin-orbit coupling
Spin-3/2 half-Heusler SC β beyond triplet pairing
Majorana flat-band and
spontaneous TR symm. breaking
π + ππ π β ππ
Majorana modes on surfaces of π Β± ππ SC
βBoundary of boundaryβ method,
Surfaces spontaneously magnetized
β’ Strategy one:
1) Single out one Fermi surface in the normal state by spin-orbit coupling.
2) Majorana fermion appears at boundary, or topo-defect (e.g. vortex core)
β’ New strategy -- two-component Fermi surfaces without spin-orbit coupling
Mixed singlet-triplet pairing π + ππ π β ππ
Spontaneous time-reversal symmetry breaking
β’ Ginzburg-Landau analysis:
β’ Pairing breaking time-reversal symmetry!
C. Wu and J. E. Hirsch, PRB 81, 20508 (2010).
20
πΉ = πΌ βπ‘2 β π½ βπ
2 + πΎ1 βπ‘2 βπ
2 + πΎ2(Ξπ‘β βπ‘
ββπ βπ + π. π. )
πΎ2>0 ππ β ππ‘= Β±π
2
βπ‘ + πβπ (βπ‘ + πβπ )| πβ, βπβ + (βπ‘ β πβπ ) πβ, β πβ
Equal in magnitude, opposite in phase.
Invariant under combined parity-time reversal (PT) transf.
βπΉ = 2πΎ2 Ξπ 2 Ξπ
2cos 2(ππ β ππ‘)
Gapped edge modes of 1D ππ§ Β± π π
π»1π· = (ββ2ππ§
2
2πβπ(π§))Iβ¨ππ§ β
Ξπ
ππΉπ
π
ππ§ππ§(πππ¦)β¨ππ₯ β Ξπ ππ¦β¨ππ₯
β’ π -wave pairing: βπ πΆπβ.
Zero modes Β±Ξπ remain eigenstates
β’ Magnetized edges reduced
degrees of freedom
β’ Opposite edges are magnetized
oppositely related by PT symmetry.
ππππ + ππ
πΆ=-1
Majorana zero mode at the magnetic domain
β’ Chiral operator πΆπβ = βππ§β¨ππ₯
π»2π· = ββ2 ππ¦
2+ππ§2
2πβ π π§ Iβ¨ππ§ β
Ξπ
ππΉπ(πππ¦πΌβ¨ππ¦ β ππ§ππ₯β¨ππ₯) βΞπ π¦ ππ¦β¨ππ₯
πΆπβ, π» = 0
β’ Symmetry: reflection + gauge
π π¦ = πΊππ¦
ππ¦: π¦ β βπ¦, πππ¦β¨π0,
β’ Majorana-mode at the magnetic domain: πΆπβ and π π¦ common
eigenstates. π¦
π§
πΊ: ππ0β¨ππ§
ππππ + ππππ β ππππππ + ππππ + ππ
π β π + π π
Ξ¨β
Ξ¨β = Ξ¨β+
β’ Zero mode: chiral and spin locking: πΆ = ππ¦β¨ππ₯ , ππ§: ππ§ β¨ππ§.
β’ 3π»π-B: TR invariant: gapless Majorana-Dirac cone.
β’ Mass by mixing Ξπ π»π = ππ¦β¨ππ₯ = πΆΞπ
πΆ=1, ππ§=β πΆ=-1, ππ§= β
Ξ¨β =
0
πβππ4
πππ4
0
π’0(π§)Ξ¨β =
πβππ4
00
πππ4
π’0(π§)
Surface states of 3π»π-B phase and π β π + ππ
π»πΒ±ππ =Ξπ‘
ππππ₯ππ¦ β ππ¦ππ₯ Β± Ξπ ππ§
β’ Massive Dirac cone and surface magnetization:
3π»π-B
ππ₯
ππ¦
π2π· = 0
Chiral Majorana modes along the π β π Β± ππ boundary
β’ Mass (surface) changes sign across the domain.
β’ Propagating 1D chiral Majorana mode.
β’ Chiral operator πΆβ²: πΆβ² = πΊπ π₯ππβ β πΆβ², π» = 0,
π π₯ is reflection: πππ₯β¨ππ§, π₯ β βπ₯ ,
G is transformation πβ β ππβ .
Ξ¨(ππ₯ = 0) =
1βπ1π
π’0(π§, π¦) πΆβ² = β1,π π¦ = β1
Ξ¨(ππ₯ = 0) =
π1βπ1
π’0(π§, π¦)πΆβ² = 1,
π π¦ = β1
β’ Symmetry: π π¦
π β π + ππ
m>0m<0
π β π β ππ
π
Drag and control by magnetic field
π β π β ππ π β π + ππ
Novel unconventional superconductivity
βBoundary of boundaryβ Majorana fermion without
spin-orbit coupling
Spin-3/2 half-Heusler SC β beyond triplet pairing
Majorana flat-band and
spontaneous TR symm. breaking
π + ππ π β ππ
x
y
z
Majorana edge modes in quasi-1D Toposuperconductors
Andreev Bound States
β 1D or 2D Majorana fermions lattices.
Dispersionless in kx and ky
Kitaev, 2000; Tewari, et al, 2007;Alicea, et al, 2010;etc ...
Andreev bound states localized at ends z0 with energy zero.
x
y
z
Yi Li, Da Wang, Congjun Wu, New J. Phys
15, 085002(2013)
mJ
Majorana Josephson coupling between chains
)2
sin( 221
iJH mt
J
L>>ΞΎ
mJ
)2
'cos( 2
21
iJH mt
L>>ΞΎ
J
2
L>>ΞΎ
z
[Kitaev, 2000; Yakovenko et al, 2004;Fu and Kane, 2009; Xu and Fu, 2010]
22 '
Yi Li, Da Wang, Congjun Wu, New J. Phys
15, 085002(2013)
Superconducting phase β Majorana fermion coupling
jiji
jim
jijit iJJH
,,
)2
sin()cos(
β’ Possibility (I): Uniform phase, time-reversal symm. maintained.
Majorana edge modes decouple β flat edge-bands.
β’ Possibility (II): Spontaneous time-reversal symm. breaking.
Majorana modes coupled and develop dispersion β lowering energy.
But density of states diverges intrinsic instability!!
Self-consistent calculation β spinless fermion
β’ Current distribution β non-quantized vortex-antivortex
β’ Superfluid phase distribution
z
y
π» = β
π
(π‘π§ππ+ππ+π§ + π‘π¦ππ
+ππ+π¦ + β. π. ) β πππ+ππ β π
π
Ξπ,ππ§β ππ+π§ ππ + β. π.
+π
π
Ξπ,π+π§β βπ,π+π§
Local Density of States (LDOS)
2
1 4
1
Non-interacting result
Interaction treated at self-consistent mean-field level
z
y
Summary
β’ Beyond triplet
Septet topo-SC from spin-3/2 electrons
Application to YPtBi
β’ βBoundary of boundaryβ
Majorana zero/chiral modes without spin-orbit coupling
π β π + ππ
m>0 m<0
π β π β ππ
π
π
β’ Majorana flat-band and spontaneous TR symm. breaking
x
yz
Back up!
Drag and control by magnetic field
π β π β ππ π β π + ππ
Bulk topological index
π20 = π30 =
spin
(p-wave triplet)quadrupole
(d-wave quintet)
octupole
(f-wave septet)
β’ Winding number expressed in helicity basis: π3 ππ 4 = β€
3
2+
1
2β
1
2(β) β
3
2(β)
ππ€ = 4
3
2+
1
2(β) β
1
2(-) β
3
2(+)
ππ€ = 0
3
2+
1
2(β) β
1
2(+) β
3
2(β)
ππ€ = 2
β’ Pairing spin structure for π// π§
Mixed triplet and singlet superconductivity
C. Wu and J. E. Hirsch, PRB 81, 20508 (2010).
β’ Ultra-cold fermionic dipolar molecular
CuxBi2Se3, Sn1βxInxTe
Sasaki, et. al., PRL 107, 217001 (2011); Sasaki, et. al., PRL 109, 217004 (2012).
Solid systems:
k
k
k
kk
kk
k
)}()({2
1);( kkVkkVkkV
dplrdplrtr
set πβ² β π
36
zkk Λ)(
zkk Λ//)(
3
8)(,
3
4)(
22 dkkV
dkkV
dplrdplr
0);( kkVtr
set πβ² β βπ 0);( kkVtr
kktrkkV
coscos~);(
β’ Dominant p-wave component
Magnetoelectric Effect
β’ Spatial variation of π, Ξπ , Ξπ induce magnetization
β’ Ginzburg-Landau free energy:
β’ Surface: sudden change of potential.
ΞπΉ(3) =2
3π·ππΉβ« π3 π β β πΌπ[β π»Ξπ Ξp
β + Ξπ π»Ξπβ ]
ΞπΉ(4) = π·β« π3 π β β πΌπ[ π»π ΞsΞπβ β π π»Ξπ Ξp
β + πΞπ π»Ξπβ ]
(π· = ππΉ
1
ππΉ
7π 3
8π 2
1
ππ2)
ππ = βππΉ
πβπ= π·πΌπ Ξπ Ξπ
β π»π for uniform Ξπ , Ξπ
Topology In Nodal Systems
Deform
β’ Topo num for surface momenta: Trivial: enclosing nodal line even timesNon-trivial: enclosing nodal line odd times
a) b)
Topo num distribution in [111]-surface for a) βπ
βπ= 0.3 b)
βπ
βπ= 0.7
β’ Topo # for a path πΏ in π-space:(TR and particle-hole sym)
π»π
π·π
β
π·π
ππ
β
ππ
BlockOff-diagonal
SVD
ππΏ =1
2ππ πΏ
πππππ[ππ
β πππππ], ππ: unitary.
A. P. Schnyder, P. M. R. Brydon, and C. Timm. Phys Rev B 85.2 (2012): 024522.