Spin-𝟑

𝟐topological superconductivity

beyond triplet pairing

Congjun Wu

University of California, San Diego

July 6, WHU Summer School

𝒑 + 𝒊𝒔 𝒑 − 𝒊𝒔x

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

“Boundary of boundary” Majorana fermion without

spin-orbit coupling

Spin-3/2 half-Heusler SC – 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

• Topological: DIII class (time-reversal invariant)

𝑑

𝑆 𝑑 ∙ 𝑆 = 0 B

)(ˆ kd

Δ

• Unconventional but isotropic spin-orbit coupled gap function

• Is 3He-B alone?

New opportunities in multi-component

fermion systems!

The distinction of the 3He-B phase

𝑑 𝑘 = 𝑘

• Cold atom: alkali/alkaline-earth fermions

4-component fermion systems: beyond triplet

Kim, Hyunsoo, et al., Science Advances Vol. 4, eaao4513 (2018).

• 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)

• Spin 𝟑

𝟐: 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).

• Experiment: nodal superconductivity in half-Heusler compound YPtBi.

S-wave quintet pairing – Non-Abeliean statistics

CW, Mod. Phys. Lett. (2006)

CW, J. P. Hu, and S. C. Zhang. Int. J. Mod. Phys. B 24 311 (2010)

• Half-quantum vortex (HQV) loop （Alice String) – the SO(4) Cheshire charge.

• Non-Abeliean phase: particle penetrating HQV loop.

|3/2

| 0

|1

|1/2 |−1/2

|2

| 𝑆𝑧

Isotropic pairings beyond singlet and triplet

d-vector d-tensor

Spherical harmonics

• Isotropic pairings:

s-wave + singlet

p-wave + triplet

d-wave + quintet

f-wave + septet

𝐽 = 𝐿 + 𝑆 = 0

Spin tensors (spin, quadrupole, octupole)

• Pairing Hamiltonian.

Δ𝐿,𝛼𝛽 𝑘 = Δ𝐿 𝜈=−𝐿𝐿 − 𝜈𝑌𝐿,−𝜈

𝑘 𝑆𝐿𝜈𝑅

Wang Yang, Yi Li, CW, PRL 117, 075301 (2016).

• Odd-parity pairing

states are topo. nontrivial.

Pictorial Rep.– spin structure of the gap function

septet

|30 → Δ3

2

− Δ1

2

+ Δ−

1

2

− Δ−

3

2

• Helical basis: 𝜎 ⋅ 𝑘|𝑘𝛼⟩ = 𝛼|𝑘𝛼⟩

Δ𝛼 𝑘 : ⟨𝛼+ 𝑘 𝛼+(−𝑘) ⟩• Intra-helical FS pairings (different phase patterns):

(𝛼 = ±3

2, ±

1

2)

Topo. index

# =3-1=2

triplet

|𝑆𝑆𝑧 = |10⟩ → Δ3

2

+ Δ1

2

− Δ−

1

2

− Δ−

3

2

High topo.

index # =3+1=4,

distinct from 3He-B

Boundary Majorana modes (f-wave septet)

Bulk Vacuum

• Zero modes (𝑘2𝐷 = 0) as chiral eigenstates.

𝑪𝒄𝒉 is a symmetry only for zero modes

Chiral operator 𝑪𝒄𝒉 = 𝒊𝑪𝒑𝑪𝑻;

𝜈 = +, −, +, −, for 𝛼 =3

2,1

2, −

1

2, −

3

2.

• k.p theory: linear Majorana-Dirac cones.

032, +

012, −

0−

12, +

0−

32, −

𝐶𝑐ℎ 𝑘𝛼2𝐷 = 0𝛼 , 𝜈 = 𝜈 |0𝛼 , 𝜈 ⟩

States with opposite chiral indices couple

• A linear and a cubic Majorana-Dirac

cones.

p-wave boundary Andreev-Majorana modes

•

𝐻𝑚𝑖𝑑𝑝

(𝑘||) =∆𝑝

𝑘𝐹

00

00

𝑐𝑘+2

𝑂(𝑘+3)

𝑖𝑘+

𝑐𝑘+2

𝑐𝑘−2

−𝑖𝑘−

𝑂(𝑘−3)

𝑐𝑘−2

00

00

1st order 𝑘 ⋅ 𝑝 theory

𝑘2𝐷 = 032, +

012, +

0−

12, −

0−

32, −

• Zero modes (𝑘2𝐷 = 0) with chiral indices

• Band inversion

𝑠1/2, 𝒑𝟑/𝟐

Spin-3/2 systems: YPtBi half-Huesler semi-metal

• Low carrier density → semimetal

h. h. l. h.

𝑛 ≈ 2 × 1018𝑐𝑚−3, 𝑘𝐹~1

10

1

𝑎

non-degenerate FS

SO coupling

Inversion symmetry broken

𝒑𝟑/𝟐

𝒔𝟏/𝟐

• Non-centrosymmetric: 𝑇𝑑 symmetry

• Linear 𝑇-dependence of penetration depth → Nodal lines

Kim, Hyunsoo, et al., Science Advances Vol. 4, eaao4513 (2018).

𝐻𝐿 𝑘 = λ1 +5

2λ2 𝑘2 − 2λ2 𝑘 ∙ 𝑆

2

𝐴 𝑘 = kx𝑇𝑥 + ky𝑇𝑦 + kz𝑇𝑧

Band Hamiltonian of YPtBi

• Luttinger-Kohn for the hole band (Γ8: 𝑝3/2)

• Non-centrosymmetric 𝑇𝑑 invariant

𝑇𝑥 = SySxSy − SzSxSz

𝑇𝑦 = SzSySz − SxSyS𝑥

𝑇𝑧 = SxSzSx − SySzSy

𝑘𝑥

𝑘𝑦

𝑘𝑧

𝑇2 rep. of 𝑇𝑑

Inversion ✖Time reversal ✔𝑇𝑑 group ✔

• Non-degenerate FS

𝐻𝑏𝑎𝑛𝑑 𝑘 = 𝐻𝐿 𝑘 + 𝐴 𝑘

‡ 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 ∆𝑠

∆𝑝= 0.3 and 0.7

• One possibility: 𝑠-wave singlet + 𝑝-wave septet

𝛼,𝛽

𝑐𝑘𝛼† [(∆𝒔 + ∆𝒑𝑨 𝒌 )𝑅]𝛼𝛽𝑐−𝑘𝛽

†

Pairing within the same spin-split Fermi surface

Nodal rings around 001 , etc

‡ P. M. R. Brydon, L. Wang, W. Weinert, D. F. Agterberg,Phys Rev Lett 116 177001 (2016)

𝐴 𝑘 = 𝑘𝑥𝑇𝑥 + 𝑘𝑦𝑇𝑦 + 𝑘𝑧𝑇𝑧

D. Agterberg, P. A. Lee, Liang Fu, Chaoxing Liu, I. Herbut, …….

• Phase sensitive test?

Previous example (YBCO): zero-energy boundary modes

[11] boundary:𝜟 𝒌𝒊𝒏 = −𝜟 𝒌𝒐𝒖𝒕

++−

−

[10] boundary:𝜟 𝒌𝒊𝒏 = 𝜟 𝒌𝒐𝒖𝒕

C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994)

L. H. Greene, et al, PRL 89, 177001 (2002)

𝑘𝑖𝑛

𝑘𝑜𝑢𝑡

𝑘𝑖𝑛

𝑘𝑜𝑢𝑡

++−

−

• Surface Brilliouin zone:

Topo-index distribution in

[111]-surface for ∆𝑠

∆𝑝= 0.3

A. P. Schnyder, P. M. R. Brydon, and C. Timm. PRB 85.2 (2012): 024522.

(𝑘𝑥2𝐷 , 𝑘𝑦

2𝐷) inside a loop non-trivial topo index ±1

• Loops: projection of the gap nodal rings.

Topo-index for nodal-ring superconductors

Each (𝑘𝑥2𝐷 , 𝑘𝑦

2𝐷) a 1D superconductor

Majorana flat bands on the 111 -surface

e

e

e

o

o

o

𝟎

𝟐e

e

e

o

o

o

𝟏

𝟑

• Chiral index (𝐶𝑐ℎ = 𝑖𝑇𝑃𝐻) for Majorana surface modes

a symmetry for zero modes (even, odd)

Non-magnetic impurity: odd under 𝐶𝑐ℎ 1,3 ✔; 0,2 ✖

Magnetic impurity: even under 𝐶𝑐ℎ 1,3 ✖; 0,2 ✔

• Selection rules:

Bright regions: Majorana zero modes

STM: quasi-particle interference (QPI) pattern

Δ𝜌𝑠𝑓 𝜔, 𝑟

• Joint density of states of impurity scattering

Δ𝜌𝑠𝑓 𝜔, 𝑞Fourier transform

• Non-magnetic impurity on (111)-surface:

𝟏

𝟑

𝟏

𝟑

𝑹𝒆(∆𝝆𝒔𝒇 𝝎 = 𝟎, 𝒒∥ ) 𝑰𝒎(∆𝝆𝒔𝒇 𝝎 = 𝟎, 𝒒∥ )

Novel unconventional superconductivity

“Boundary of boundary” Majorana fermion without

spin-orbit coupling

Spin-3/2 half-Heusler SC – beyond triplet pairing

Majorana flat-band and

spontaneous TR symm. breaking

𝒑 + 𝒊𝒔 𝒑 − 𝒊𝒔

Majorana modes on surfaces of 𝒑 ± 𝒊𝒔 SC

“Boundary of boundary” method,

Surfaces spontaneously magnetized

• 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)

• New strategy -- two-component Fermi surfaces without spin-orbit coupling

Mixed singlet-triplet pairing 𝒑 + 𝒊𝒔 𝒑 − 𝒊𝒔

Spontaneous time-reversal symmetry breaking

• Ginzburg-Landau analysis:

• Pairing breaking time-reversal symmetry!

C. Wu and J. E. Hirsch, PRB 81, 20508 (2010).

20

𝐹 = 𝛼 ∆𝑡2 − 𝛽 ∆𝑠

2 + 𝛾1 ∆𝑡2 ∆𝑠

2 + 𝛾2(Δ𝑡∗ ∆𝑡

∗∆𝑠∆𝑠 + 𝑐. 𝑐. )

𝛾2>0 𝜑𝑠 − 𝜑𝑡= ±𝜋

2

∆𝑡 + 𝑖∆𝑠 (∆𝑡 + 𝑖∆𝑠)| 𝑘↑, −𝑘↓ + (∆𝑡 − 𝑖∆𝑠) 𝑘↓, − 𝑘↑

Equal in magnitude, opposite in phase.

Invariant under combined parity-time reversal (PT) transf.

∆𝐹 = 2𝛾2 Δ𝑠2 Δ𝑝

2cos 2(𝜑𝑠 − 𝜑𝑡)

Gapped edge modes of 1D 𝑝𝑧 ± 𝑖 𝑠

𝐻1𝐷 = (−ℏ2𝜕𝑧

2

2𝑚−𝜇(𝑧))I⨂𝜏𝑧 −

Δ𝑝

𝑘𝐹𝑖

𝑑

𝑑𝑧𝜎𝑧(𝑖𝜎𝑦)⨂𝜏𝑥 − Δ𝑠𝜎𝑦⨂𝜏𝑥

• 𝑠-wave pairing: ∆𝑠𝐶𝑐ℎ.

Zero modes ±Δ𝑠 remain eigenstates

• Magnetized edges reduced

degrees of freedom

• Opposite edges are magnetized

oppositely related by PT symmetry.

𝒑𝒛𝝈𝒛 + 𝒊𝒔

𝐶=-1

Majorana zero mode at the magnetic domain

• Chiral operator 𝐶𝑐ℎ = −𝜎𝑧⨂𝜏𝑥

𝐻2𝐷 = −ℏ2 𝜕𝑦

2+𝜕𝑧2

2𝑚− 𝜇 𝑧 I⨂𝜏𝑧 −

Δ𝑝

𝑘𝐹𝑖(𝑖𝜕𝑦𝐼⨂𝜏𝑦 − 𝜕𝑧𝜎𝑥⨂𝜏𝑥) −Δ𝑠 𝑦 𝜎𝑦⨂𝜏𝑥

𝐶𝑐ℎ, 𝐻 = 0

• Symmetry: reflection + gauge

𝑅𝑦 = 𝐺𝑀𝑦

𝑀𝑦: 𝑦 → −𝑦, 𝑖𝜎𝑦⨂𝜏0,

• Majorana-mode at the magnetic domain: 𝐶𝑐ℎ and 𝑅𝑦 common

eigenstates. 𝑦

𝑧

𝐺: 𝑖𝜎0⨂𝜏𝑧

𝒑𝒚𝝈𝒚 + 𝒑𝒛𝝈𝒛 − 𝒊𝒔𝒑𝒚𝝈𝒚 + 𝒑𝒛𝝈𝒛 + 𝒊𝒔

𝒑 ⋅ 𝝈 + 𝒊 𝒔

Ψ↓

Ψ↑ = Ψ↓+

• Zero mode: chiral and spin locking: 𝐶 = 𝜎𝑦⨂𝜏𝑥 , 𝑆𝑧: 𝜎𝑧 ⨂𝜏𝑧.

• 3𝐻𝑒-B: TR invariant: gapless Majorana-Dirac cone.

• Mass by mixing Δ𝑠 𝐻𝑠 = 𝜎𝑦⨂𝜏𝑥 = 𝐶Δ𝑠

𝐶=1, 𝑆𝑧=↑ 𝐶=-1, 𝑆𝑧= ↓

Ψ↓ =

0

𝑒−𝑖𝜋4

𝑒𝑖𝜋4

0

𝑢0(𝑧)Ψ↑ =

𝑒−𝑖𝜋4

00

𝑒𝑖𝜋4

𝑢0(𝑧)

Surface states of 3𝐻𝑒-B phase and 𝑝 ⋅ 𝜎 + 𝑖𝑠

𝐻𝑝±𝑖𝑠 =Δ𝑡

𝑘𝑓𝑘𝑥𝜎𝑦 − 𝑘𝑦𝜎𝑥 ± Δ𝑠𝜎𝑧

• Massive Dirac cone and surface magnetization:

3𝐻𝑒-B

𝑘𝑥

𝑘𝑦

𝑘2𝐷 = 0

Chiral Majorana modes along the 𝑝 ⋅ 𝜎 ± 𝑖𝑠 boundary

• Mass (surface) changes sign across the domain.

• Propagating 1D chiral Majorana mode.

• Chiral operator 𝐶′: 𝐶′ = 𝐺𝑅𝑥𝑇𝑃ℎ ⇒ 𝐶′, 𝐻 = 0,

𝑅𝑥 is reflection: 𝑖𝜎𝑥⨂𝜏𝑧, 𝑥 → −𝑥 ,

G is transformation 𝑐† → 𝑖𝑐†.

Ψ(𝑘𝑥 = 0) =

1−𝑖1𝑖

𝑢0(𝑧, 𝑦) 𝐶′ = −1,𝑅𝑦 = −1

Ψ(𝑘𝑥 = 0) =

𝑖1−𝑖1

𝑢0(𝑧, 𝑦)𝐶′ = 1,

𝑅𝑦 = −1

• Symmetry: 𝑅𝑦

𝒑 ∙ 𝝈 + 𝒊𝒔

m>0m<0

𝒑 ∙ 𝝈 − 𝒊𝒔

𝜎

Drag and control by magnetic field

𝒑 ∙ 𝝈 − 𝒊𝒔 𝒑 ∙ 𝝈 + 𝒊𝒔

Novel unconventional superconductivity

“Boundary of boundary” Majorana fermion without

spin-orbit coupling

Spin-3/2 half-Heusler SC – beyond triplet pairing

Majorana flat-band and

spontaneous TR symm. breaking

𝒑 + 𝒊𝒔 𝒑 − 𝒊𝒔

x

y

z

Majorana edge modes in quasi-1D Toposuperconductors

Andreev Bound States

→ 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>>ξ

mJ

)2

'cos( 2

21

iJH mt

L>>ξ

J

2

L>>ξ

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 – Majorana fermion coupling

jiji

jim

jijit iJJH

,,

)2

sin()cos(

• Possibility (I): Uniform phase, time-reversal symm. maintained.

Majorana edge modes decouple – flat edge-bands.

• Possibility (II): Spontaneous time-reversal symm. breaking.

Majorana modes coupled and develop dispersion – lowering energy.

But density of states diverges intrinsic instability!!

Self-consistent calculation – spinless fermion

• Current distribution – non-quantized vortex-antivortex

• Superfluid phase distribution

z

y

𝐻 = −

𝑖

(𝑡𝑧𝑐𝑖+𝑐𝑖+𝑧 + 𝑡𝑦𝑐𝑖

+𝑐𝑖+𝑦 + ℎ. 𝑐. ) − 𝜇𝑐𝑖+𝑐𝑖 − 𝑉

𝑖

Δ𝑖,𝑖𝑧∗ 𝑐𝑖+𝑧 𝑐𝑖 + ℎ. 𝑐.

+𝑉

𝑖

Δ𝑖,𝑖+𝑧∗ ∆𝑖,𝑖+𝑧

Local Density of States (LDOS)

2

1 4

1

Non-interacting result

Interaction treated at self-consistent mean-field level

z

y

Summary

• Beyond triplet

Septet topo-SC from spin-3/2 electrons

Application to YPtBi

• “Boundary of boundary”

Majorana zero/chiral modes without spin-orbit coupling

𝒑 ∙ 𝝈 + 𝒊𝒔

m>0 m<0

𝒑 ∙ 𝝈 − 𝒊𝒔

𝟏

𝟑

• Majorana flat-band and spontaneous TR symm. breaking

x

yz

Back up!

Drag and control by magnetic field

𝒑 ∙ 𝝈 − 𝒊𝒔 𝒑 ∙ 𝝈 + 𝒊𝒔

Bulk topological index

𝑆20 = 𝑆30 =

spin

(p-wave triplet)quadrupole

(d-wave quintet)

octupole

(f-wave septet)

• Winding number expressed in helicity basis: 𝜋3 𝑆𝑈 4 = ℤ

3

2+

1

2−

1

2(−) −

3

2(−)

𝑁𝑤 = 4

3

2+

1

2(−) −

1

2(-) −

3

2(+)

𝑁𝑤 = 0

3

2+

1

2(−) −

1

2(+) −

3

2(−)

𝑁𝑤 = 2

• Pairing spin structure for 𝑘// 𝑧

Mixed triplet and singlet superconductivity

C. Wu and J. E. Hirsch, PRB 81, 20508 (2010).

• Ultra-cold fermionic dipolar molecular

CuxBi2Se3, Sn1−xInxTe

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 𝑘′ → 𝑘

36

zkk ˆ)(

zkk ˆ//)(

3

8)(,

3

4)(

22 dkkV

dkkV

dplrdplr

0);( kkVtr

set 𝑘′ → −𝑘 0);( kkVtr

kktrkkV

coscos~);(

• Dominant p-wave component

Magnetoelectric Effect

• Spatial variation of 𝑉, Δ𝑠, Δ𝑝 induce magnetization

• Ginzburg-Landau free energy:

• Surface: sudden change of potential.

Δ𝐹(3) =2

3𝐷𝜖𝐹∫ 𝑑3 𝑟 ℎ ⋅ 𝐼𝑚[− 𝛻Δ𝑠 Δp

∗ + Δ𝑠𝛻Δ𝑝∗ ]

Δ𝐹(4) = 𝐷∫ 𝑑3 𝑟 ℎ ⋅ 𝐼𝑚[ 𝛻𝑉 ΔsΔ𝑝∗ − 𝑉 𝛻Δ𝑠 Δp

∗ + 𝑉Δ𝑠𝛻Δ𝑝∗ ]

(𝐷 = 𝑁𝐹

1

𝑘𝐹

7𝜁 3

8𝜋 2

1

𝑇𝑐2)

𝑀𝜇 = −𝜕𝐹

𝜕ℎ𝜇= 𝐷𝐼𝑚 Δ𝑠Δ𝑝

∗ 𝛻𝑉 for uniform Δ𝑠, Δ𝑝

Topology In Nodal Systems

Deform

• 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) ∆𝑠

∆𝑝= 0.3 b)

∆𝑠

∆𝑝= 0.7

• Topo # for a path 𝐿 in 𝑘-space:(TR and particle-hole sym)

𝐻𝑘

𝐷𝑘

†

𝐷𝑘

𝑄𝑘

†

𝑄𝑘

BlockOff-diagonal

SVD

𝑁𝐿 =1

2𝜋𝑖 𝐿

𝑑𝑘𝑙𝑇𝑟[𝑄𝑘

†𝜕𝑘𝑙𝑄𝑘], 𝑄𝑘: unitary.

A. P. Schnyder, P. M. R. Brydon, and C. Timm. Phys Rev B 85.2 (2012): 024522.