Unconventional Superconductivity Topology, Symmetry, and … · 2018. 10. 21. · Unconventional...

Unconventional Superconductivity – Topology, Symmetry, and Strong Correlation

Congjun WuUniversity of California, San Diego

Sept 10, 2018, IOP, CAS.

𝒑 + 𝒊𝒔 𝒑 − 𝒊𝒔

Wang Yang (UCSD UBC)

Yi Li (UCSD Princeton Johns Hopkins)

Da Wang (UCSD Nanjing Univ.)

Tianxing Ma (Visiting scholar from Beijing Normal Univ. )

Tao Xiang (IOP, Chinese Academy of Sciences)

2

Collaborators:

Supported by NSF, AFOSR

Reference

1. Tianxing Ma, Da Wang, Congjun Wu, arXiv:1806.03652.

2. Wang Yang, Chao Xu, Congjun Wu, arXiv:1711.05241.

3. Wang Yang, Tao Xiang, and Congjun Wu, Phys. Rev. B 96, 144514 (2017).

4. Wang Yang, Yi Li, Congjun Wu, Phys. Rev. Lett. 117, 075301(2016).

Novel unconventional superconductivity

“Boundary of boundary” Majorana fermion without

spin-orbit coupling

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

Extended s-wave SC by doping AFM (Mott)

insulators


Unconventional superconductivity

k

k

• Conventional: s-wave pairing symmetry.

)(k

s-wave

• Unconventional: high partial wave symmetries.

22 yxd

++

-

-

k

Δ(𝑘)

d-wave (singlet) : high Tc cupratesp-wave (triplet ) : superfluid 3He

𝐻𝑃 𝑘 = ∆ 𝑘 𝑃+ 𝑘 + ∆∗ 𝑘 𝑃 𝑘

𝑃+ 𝑘 = 𝑐↑+ 𝑘 𝑐↓

+ −𝑘

Hg, Pb, MgB2 …

Majorana boundary zero modes (spinless p-wave)

• Single-component fermion p-wave pairing – Kitaev 2001.

𝜟 𝒌𝒊𝒏 = −𝜟 𝒌𝒐𝒖𝒕𝛾 = 𝑑𝑥 𝑢0 𝑥 𝜓(𝑥) + 𝑣0 𝑥 𝜓+(𝑥)

𝑢0 𝑥 = 𝑣0∗ 𝑥

• Spin-orbit coupled supercond. wire under magnetic field.

L. P. Kouwenhouven, et al, Science 336 1003

(2012);

C. M. Marcus et al, PRB 87, 241401 (2013).

P. A. Lee, arxiv 0907.2681

Sau, Lutchyn, Das Sarma PRL 2010.

Liu, Potter, Law, Lee, PRL 109, 267002

(2011);

He, Ng, Lee, Law, PRL (2014).

X. J. Liu, A. Lobs, PRB (2013).

Topo-index (BDI): 1𝐷 spinless 𝑝-wave SC

• Anderson pseudo spin (Nambu Rep):

• Winding #: 𝑆1 → 𝑆1

𝑘+∞𝑘𝐹0−𝑘𝐹−∞

±∞

−𝑘𝐹 𝑘𝐹

0

ℎ 𝑘

ℎ

𝜓 𝑘 =𝑐𝑘

𝑐−𝑘†

𝐻 𝑘 = ℎ 𝑘 ⋅ 𝜏 =𝜉𝑘 Δ(𝑘)

Δ∗(𝑘) −𝜉(−𝑘)

ℎ 𝑘 = (Δ 𝑘 , 0, 𝜉 𝑘 )

𝜉𝑘 =ℏ2𝑘2

2𝑚− 𝜖𝐹 ,

Δ 𝑘 =Δ𝑝

𝑘𝐹𝑘

2D spinless 𝑝𝑥 + 𝑖𝑝𝑦 superconductivity

ℎ 𝑘 = (Δ𝑝

𝑘𝑥

𝑘𝐹, Δ𝑝

𝑘𝑦

𝑘𝐹, 𝜉𝑘 )Δ 𝑘𝑥, 𝑘𝑦 =

Δ𝑝

𝑘𝐹(𝑘𝑥+𝑖𝑘𝑦)

𝑘𝑓

𝑘𝑦

𝑘𝑥

ℎ 𝑘𝑥, 𝑘𝑦 winding #: 𝑆2 → 𝑆2

• chiral Majorana edge modes• vortex core zero Majoranamode

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

Time-reversal invariant topo-SC (DIII class)

• Topo-pairing (𝑇2 = −1, 𝐶2 = 1).

𝑁𝑤 =1

4𝜋 𝑆

𝑑𝑘2 𝑑 ∙ 𝜕𝜇 𝑑 × 𝜕𝜈

𝑑 = ±1

• Topo index from the d-vector

• Quaternionic pseudospin 3D

skyrmion (SU(2))

(𝜖 𝑘 − 𝜇) + Δ(𝑖𝑑𝑥(𝑘) + 𝑗𝑑𝑦(𝑘) + 𝑘𝑑𝑧 𝑘 )

• Surface modes: Majorana-Dirac cone

B

Δ

𝑑(𝑘)

winding #: 𝑆3 → 𝑆3

winding #: 𝑆2 → 𝑆2

10

High 𝑇𝑐 superconductivity

• Intrinsically strong coupling problem -- complicated structures.

• Many competing phases: d-wave superconductivity by doping the parent AFM Mott insulating state

Novel topologicalsuperconductivity


insulators

“boundary of boundary” Majorana fermion without

spin-orbit coupling

Spin-3/2 half-HeuslerSC – beyond triplet

pairing

e

e

e

o

o

o

𝟏

𝟑

• 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:

C. Wu, J. P. Hu, and S. C. Zhang. PRL 91 186402 (2003).C. Wu, Mod. Phys. Lett, (2006).C. Wu, J. P. Hu, and S. C. Zhang. Int. J. Mod. Phys. B 24 311 (2010)Wang Yang, Yi Li, C. Wu, PRL 117, 075301 (2016).

• Spin 𝟑

𝟐: Quintet and Septet pairings

beyond singlet and triplet.

septet

Wang Yang, Yi Li, C. Wu, PRL 117, 075301 (2016).W. Yang, Tao Xiang, and C. Wu, PRB 96, 144514 (2017).

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

S-wave quintet pairing – Non-Abeliean statistics

C. Wu, Mod. Phys. Lett. (2006)

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

• Half-quantum vortex (HQV) loop carrying spin – 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, C. Wu, PRL 117, 075301 (2016).

• Odd-parity pairing

states are topo. nontrivial.

Pictorial Rep.– spin structure of the gap function

tripletseptet

|𝑆𝑆𝑧 = |30 → Δ3

2

− Δ1

2

+ Δ−

1

2

− Δ−

3

2

• Helical basis: 𝜎 ⋅ 𝑘|𝑘𝛼⟩ = 𝛼|𝑘𝛼⟩

Δ𝛼 𝑘 : ⟨𝛼+ 𝑘 𝛼+(−𝑘) ⟩

|10 → Δ3

2

+ Δ1

2

− Δ−

1

2

− Δ−

3

2

• Intra-helical FS pairings (different phase patterns):

• Applicable to solid state spin-orbit coupled systems.

(𝛼 = ±3

2, ±

1

2)

Topo. index

# =3-1=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∥ )



spin-orbit coupling


pairing



insulators

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).

27

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+𝜕𝑧

2

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



spin-orbit coupling


pairing


insulators

QuantumMonte Carlo

“Poll”Importance Sampling

Numerical “exactness”

Beautiful mathematical structure

QMC: Tame the large Hilbert space stochastically

Dimension of Hilbert Space

~ 𝒆#𝑵

• QMC: scalable, sufficient accuracy at 𝐷 ≥ 2.

35

Hubbard-Stratonovich(HS)

path integral over space-time HS fields

𝑍 = Tr𝑒−𝛽𝐻 = lim𝑀⟶∞

𝑃

𝜌𝑃

𝜌𝑃 = Tr

𝑘=1

𝑀

𝑒−Δ𝜏𝐻0 𝑒−Δ𝜏𝐻𝐼(𝜏𝑘) = det(I +

𝑘=1

𝑀

𝑒−Δ𝜏ℎ0𝑒−Δ𝜏ℎ𝐼(𝜏𝑘))

J. E. Hirsch, PRB 28, 4059 (1983)

Auxiliary Field QMC – Path IntegralBlankenbecler, Scalapino, and Sugar. PRD 24, 2278 (1981)

Fermions Grassmann variables probability

• Integrate out fermions and the resulting fermion functional determinants work as statistical weights.

• Sign problem: 𝜌𝑃 is not positive definite – statistical errors grow exponentially with sample size.

Kramers positivity for Dirac fermion

• Theorem 1: For any HS field config., if there exists an anti-unitary T,

)()(,,1 10

10

2 II hTThhTThT

𝐵 =

𝑘=1

𝑀

𝑒−Δ𝜏ℎ0𝑒−Δ𝜏ℎ𝐼(𝜏𝑘)then 𝜌𝑃 = det(I + 𝐵) ≥ 0, where

• Eigenvalues complex-conjugate pairwised (l, l*).

• Real l double degeneracy.

0)())(()det( **

22

*

11 nnBI llllll

• T needs not be the physical time reversal (TR)-operator.

• I+B may not be diagonalizable.

C. Wu and S. C. Zhang, PRB 2005; S. Hands, et al, Eur. Phys. J. C 17, 285 (2000).

Proof:

37

The bi-layer Scalapino-Zhang-Hanke ModelD. Scalapino, S. C. Zhang, and W. Hanke, PRB 58, 443 (1998)

)1()1()())((

)(}.{}.{

,,2

1

,,2

1

,,

//

di

i

cicii

ciij

idic

i

ji

i

jij

ij

i

nnVdcnnUSSJ

inchdctchddcctH

//t

t V J

U c

d

• Intra-rung interactions: U, V, J.

38.

T-invariant decoupling (Time-reversal*flip two layers)

JVUgJ

VUgJVUg

ingingining

inintchddcctH

c

i i

cAF

i

curtbond

ii

bondjij

ij

iSZH

4

334,

44,

4

34

)2)(()()}()({

)()(}.{

2222

//

• When g, g’, gc>0, T-invariant H-S decoupling absence of

the sign problem.

• T=Time reversal × layer flip

𝑛𝐴𝐹𝑀(𝑖) =𝑖

2(𝑐𝑖

+ 𝜎𝑐𝑖 − 𝑑𝑖+ 𝜎𝑑𝑖 )

𝑛𝑏𝑑(𝑖) =1

2(𝑐𝑖,𝜎

+ 𝑑𝑖,𝜎 + 𝑑𝑖,𝜎+ 𝑐𝑖,𝜎 )

𝑐

𝑑

T-even operators

𝑛𝑐𝑢𝑟(𝑖) =𝑖

2(𝑐𝑖,𝜎

+ 𝑑𝑖,𝜎 − 𝑑𝑖,𝜎+ 𝑐𝑖,𝜎)

S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).

AFM and SC

𝑉 = 𝑡⊥ = 0, 𝑈 =5

8, 𝐽⊥ = 1/4, 𝐽𝑧 = 2.

• AFM ordering with Isinganisotropy appear at half-filling, and is weakened by hole-doping.

• Intra-rung singlet pairing appear as doping.

• Extended-s-wave: sign switching in the bonding and anti-bonding band bases.

Δ𝑠 = 𝑐𝑖↑+𝑑𝑖↓

+ − 𝑐𝑖↓+𝑑𝑖↑

+

Superconductivity by doping Mott insulators

• Coexistence between SC and AFM at 0 < 𝑥 < 𝑥𝑐 = 0.11

• Triplet pair density wave appears in the coexistence region.

• The 3D Ising class of the magnetic transition.

𝜂 ≈ 0.036, 𝑧 = 1

Δ𝑧 𝑄 = − 𝑖(𝑐𝑖↑+𝑑𝑖↓

+ + 𝑐𝑖↓+𝑑𝑖↑

+)

Staggered inter-layer current phase

• High Tc, heavy fermion……

i) 𝑡⊥ = 0.5 ii) 𝑈 = 𝑉 = 0.3, 𝐽 = 1.6

iii) 1/8-doping

Long-range staggered current order:

𝑡⊥ = 0.1, 𝑈 = 0, 𝑉 = 0.5, 𝐽 = 2.suppression of order

S. Capponi, C. Wu and S. C. Zhang, PRB 70, 220505 (R) (2004).

Summary

• Doping AFM Mott insulator

extended s-wave superconductivity

• Beyond triplet

Septet topo-SC from multi-component electrons (half-heusler)

• “Boundary of boundary”

Majorana zero/chiral modes without spin-orbit coupling

𝒑 ∙ 𝝈 + 𝒊𝒔

m>0 m<0

𝒑 ∙ 𝝈 − 𝒊𝒔

Unconventional Superconductivity Topology, Symmetry, and … · 2018. 10. 21. · Unconventional...

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