Features of edge states and domain walls in chiral ...nqs2017.ws/Slides/5th/Sigrist.pdfChiral superconductors Chiral superconductors - candidates UPt 3 SrPtAs hexagonal crystal structure

Features of edge states and domain walls in chiral superconductors

Manfred Sigrist

kk

E

NQS2017,YITPKyoto

Chiralsuperconductors

Chiralsuperconductors-candidatesSr2RuO4 URu2Si2

tetragonalcrystalstructure

µSRpolarKerreffect polarKerreffect

odd-parity

chiralp-wave

chirald-wave

even-parity

�~k = �0(kx ± iky)

�~k = �0(kx ± iky)kz

Lz = ±1

Chiralsuperconductors

Chiralsuperconductors-candidatesUPt3 SrPtAshexagonal

crystalstructureUPt odd-parity

µSRpolarKerreffect

µSR

even-paritychiralf-wave

chirald-wave

�~k = �0(kx ± iky)2kz

�~k = �0(kx ± iky)2

Lz = ±2

Content

edgestatesandedgecurrentsinachiralp-waveSC

manyothercollaborators:Y.Imai,K.Wakabayashi,A.Furusaki,M.Matsumoto,C.Honerkamp,M.Fischer,T.M.Rice,J.Goryo,W.Huang,...

chiraldomains

SarahEtter AdrienBouhon

formerdoctorstudentsatETHZurich

focusonSr2RuO4asachiralp-waveSC

Maenoetal1994

Sr2RuO4

layeredcrystalstructure

quasi-2Dmetal

possibleodd-parityspin-tripletstates

(~k) =

✓0 kx ± iky

kx ± iky 0

◆

(~k) =

✓�kx + iky 0

0 kx + iky

◆

A-phase chiralphase

B-phase helicalphase

=

✓ "" "# #" ##

◆pairwavefunction

Maenoetal1994

Sr2RuO4-chiralp-wavesuperconductor


quasi-2Dmetal

(~k) =

✓0 kx ± iky

kx ± iky 0

◆

(~k) =

✓�kx + iky 0

0 kx + iky

◆

A-phase chiralphase

B-phase helicalphase

identifaction

•  intrinsicmagnetism•  inplanespinpolarizable•  multi-component•  polarKerreffect•  phase-sensitiveSQUID

Maenoetal1994

Sr2RuO4-chiralp-wavesuperconductor


quasi-2Dmetal A-phase chiralphase

identifaction

•  intrinsicmagnetism•  inplanespinpolarizable•  multi-component•  polarKerreffect•  phase-sensitiveSQUID

(~k) =

✓0 ke±i✓k

ke±i✓k 0

◆

phasewindingaroundtheFS

�k = |�0|e+i✓k

�k = |�0|e�i✓k

nodelessgap

2-folddegenerate

chiraldomains

gapfunction

chiral-brokentimereversalsymmetry

intrinsic magnetism in Sr2RuO4 ?

randomlocalmagnetism

Luke et al (1998)

µSR - zero-field relaxtion

''edgecurrents''aroundinhomogeneities&defects

muon-spindepolarization

intrinsicmagnetism

edgestatecurrentsscanningprobesatmesoscopicdiscs

T>Tc T<Tc

N SC

SC-N

disc

Mackenziegroup(2014)

R ⇡ 5µm

scanningHallprobe

Edgecurrents

edgestatesforthechiralp-wavestate

Sr2RuO4-edgestatespectrum

scatteringofquasiparticlesatthesurfacesolutionofBogolyubov-deGennesequations

subgapboundstates(closeorbitsinparticle-holespace)“Andreevreflection”

surface

electron

hole

θ

specularscattering

kk

phaseshiftsatturningpoints

k1 k2

for �k1 + �k2 = ⇡ + ✓k2 � ✓k1

1

~

Ip · ds+ �k1 + �k2 = 2⇡n

|p| ⇡ E

vF✓k2 � ✓k1 = ⇡ E = 0

�k = |�0|ei✓k |E| ⌧ |�0|

Bohr-Sommerfeldquantization


Sr2RuO4-edgestatespectrum


subgapboundstates(closeorbitsinparticle-holespace)


2✓ = ⇡ � (✓~k2� ✓~k1

)

phaseshifts

“Andreevreflection”

E = Ekk = �0 sin ✓ = �0kkkFsu

rface

electron

hole

θ

specularscattering

kkk1 k2

�k = |�0|ei✓k


Sr2RuO4-bulkandedgespectrum


subgapboundstates(closeorbitsinparticle-holespace)


2✓ = ⇡ � (✓~k2� ✓~k1

)

phaseshifts

“Andreevreflection”

E = Ekk = �0 sin ✓ = �0kkkFkk+kF

�kF

E

+�0

��0

continuum

continuum

note:✓~k2� ✓~k1

= ±⇡ E = 0resultoftopologicalproperty

Sr2RuO4-bulkandedgespectrum

kk+kF

�kF

E

+�0

��0

continuum

continuum

occupied

surface

electron

holecharge

spontaneouschargecurrent

orderparameterdeformation

x

driving currents

Bz

�screeningcurrents

fields

Bz ⇠ 10 G

�~k = ⌘xkx + ⌘yky

|⌘y|

|⌘x||�0|

Areedgecurrentsauniquetopologicalproperty?latticeversionofchiralp-wavesuperconductor(tight-binding):

⇠~k = �2t(cos kxa+ cos kya) + ....� ✏F

zerosof�~k

+1-winding

-1-winding

Chernnumberskx

ky

1stBrillouinzone

N = +1

�~k = �0(sin kxa+ i sin kya)

Topologyandedgecurrents


Areedgecurrentsauniquetopologicalproperty?latticeversionofchiralp-wavesuperconductor(tight-binding):

⇠~k = �2t(cos kxa+ cos kya) + ....� ✏F

zerosof�~k

+1-winding

-1-winding

Chernnumbers


kx

ky

1stBrillouinzone

N = +1� 4⇥ 1

2

= �1⇥ 1 = �1

kx

ky

kx

ky

kk

E


kk

E

N = +1 N = �1

kk

E

kk

E

kx

ky

kx

ky


N = +1 N = �1

vq =1

~dEkk

dkk

quasiparticlevelocity

vq > 0 vq < 0

kx

ky

kx

ky

kk

E


kk

E

N = +1 N = �1

surface

electron

holecharge

chargecurrent

kk

oppositechirality,butidenticalcurrentdirection

backwardbackward

Topology-thermalHalleffect

“Spontaneous”Righi-Leduceffect

chiralQPedgestates T1

heatcurrent


tempe

rature

gradient

xy =⇡k

2BT

12~ N +O(e��

kBT )

Read&Green;Qin,Niu&Shi;Sumiyoshi&Fujimoto;…

Chernnumber

12~ xy

⇡k2BT

µ

N = +1

N = �1

Lishitz-transition(topologicaltransition)

propertyofthe(subgap)quasiparticles

T ⌧ Tc

Topology-thermalHalleffect

“Spontaneous”Righi-Leduceffect



tempe

rature

gradient

xy =⇡k

2BT

12~ N +O(e��

kBT )

Read&Green;Qin,Niu&Shi;Sumiyoshi&Fujimoto;…

Chernnumber

12~ xy

⇡k2BT

µ

N = +1

N = �1

analogtoν=1QuantumHall

effect

Lishitz-transition(topologicaltransition)

heatcurrent

propertyofthe(subgap)quasiparticles

T ⌧ Tc

vQP > 0 vQP < 0

Topologyandedgecurrents-afurthertwist


particle-holesymmetry

�~k+~Q = ��~k ~Q =⇡

a(1, 1)

kx

kykk

Ex

y ~n = (1, 0)

~k1~k2

kk

E




�~k+~Q = ��~k ~Q =⇡

a(1, 1)

kx

kykk

E

x

y

~n = (1, 0)

~k1

~k2

x

y

~n = (1, 1)

kk

E




�~k+~Q = ��~k ~Q =⇡

a(1, 1)

kx

kykk

E

x

y

~n = (1, 0)

~k1

~k2

x

y

~n = (1, 1)

kk

E




�~k+~Q = ��~k ~Q =⇡

a(1, 1)

kx

kykk

E

x

y

~n = (1, 0)

~k1

~k2

x

y

~n = (1, 1)

~Q

✓~k2� ✓~k1

= ±⇡ E = 0

newzero-energymomenta

nochangeinChernnumber


kk

E~n = (1, 1)

kk

E

kx

ky

~k1

~k2~Qkx

ky

~k1

~k2

currentreversal


kk

E~n = (1, 1)

kk

E

currentreversal

x

y

x

y

circularsupercurrent non-circularsupercurrentA.Bouhon

Ginzburg-Landauapproach-chiralp-wave

F =

ZdV

�a|⌘|2 + b1|⌘|4 + b2(⌘

⇤2x ⌘2y + ⌘2x⌘

⇤2y ) + b3|⌘x|2|⌘y|2

+K1(|⇧x⌘x|2 + |⇧y⌘y|2) +K2(|⇧x⌘y|2 + |⇧y⌘x|2)

+ [K3(⇧x⌘x)⇤(⇧y⌘y) +K4(⇧x⌘y)

⇤(⇧y⌘x) + c.c.]

+K5|⇧z⌘|2 + (r⇥A)/8⇡

orderparameter: d(k) = z⌘ · k = z(⌘xkx + ⌘yky)

⇧ =~ir+

2e

cAand


F =

ZdV

�a|⌘|2 + b1|⌘|4 + b2(⌘

⇤2x ⌘2y + ⌘2x⌘

⇤2y ) + b3|⌘x|2|⌘y|2

+K1(|⇧x⌘x|2 + |⇧y⌘y|2) +K2(|⇧x⌘y|2 + |⇧y⌘x|2)

+ [K3(⇧x⌘x)⇤(⇧y⌘y) +K4(⇧x⌘y)

⇤(⇧y⌘x) + c.c.]

+K5|⇧z⌘|2 + (r⇥A)/8⇡


lengthscalesforamplitudemodulationsforthetwoorderparametercomponents

⇧ =~ir+

2e

cAand


F =

ZdV

�a|⌘|2 + b1|⌘|4 + b2(⌘

⇤2x ⌘2y + ⌘2x⌘

⇤2y ) + b3|⌘x|2|⌘y|2

+K1(|⇧x⌘x|2 + |⇧y⌘y|2) +K2(|⇧x⌘y|2 + |⇧y⌘x|2)

+ [K3(⇧x⌘x)⇤(⇧y⌘y) +K4(⇧x⌘y)

⇤(⇧y⌘x) + c.c.]

+K5|⇧z⌘|2 + (r⇥A)/8⇡


edgecurrentsforchiralphase

⇧ =~ir+

2e

cAand


F =

ZdV

�a|⌘|2 + b1|⌘|4 + b2(⌘

⇤2x ⌘2y + ⌘2x⌘

⇤2y ) + b3|⌘x|2|⌘y|2

+K1(|⇧x⌘x|2 + |⇧y⌘y|2) +K2(|⇧x⌘y|2 + |⇧y⌘x|2)

+ [K3(⇧x⌘x)⇤(⇧y⌘y) +K4(⇧x⌘y)

⇤(⇧y⌘x) + c.c.]

+K5|⇧z⌘|2 + (r⇥A)/8⇡


cylindricallysymmetricbands1

3K1 = K2 = K3 = K4 � K5

⇧ =~ir+

2e

cAand

Ginzburg-Landauapproach-edgecurrents

x

y

K1|⇧x⌘x|2 +K2|⇧x⌘y|2

lengthscalesoforderparametercomponents

⇠2x/⇠2y = K1/K2

currentdensityalongedge:

jy = 16⇡e~✓K3|⌘x|

@

@x|⌘y|�K4|⌘y|

@

@x|⌘x|

◆+

c��2

4⇡Ay

drivingcurrent screeningcurrent

note:lengthscalesimportant

sampleed

ge


K1(|⇧x⌘x|2 + |⇧y⌘y|2) +K2(|⇧x⌘y|2 + |⇧y⌘x|2)

kx

ky

~k1

~k2~Qkx

ky

~k1

~k2

K1 > K2K1 < K2

x

y

x

ydetermines

currentpatter

(specularscattering)


K1(|⇧x⌘x|2 + |⇧y⌘y|2) +K2(|⇧x⌘y|2 + |⇧y⌘x|2)

kx

ky

~k1

~k2~Q

K1 < K2

x

y

γ

-bandismostlikelydominantandcrossesUmklappdiamond�

Diskshapedsample


diskshapedsample S.Etter

currentandfluxpatternforallsurfaceorientations

~n

��

�

specularscattering

⌘?

⌘k

componentperpendiculartosurface

R = 40⇠0

isotropic

componentparalleltosurface

Bz(r, ✓)

✓r

Re[ � ]

Re[�]

��

�

r /ξ0

η/η 0

��

��

��

r /ξ0

Bz(γξ02 )

��

�� -��

��

��

��

r /ξ0

J θ(4πγ

ξ 03/c)


differentboundaryconditions

��

�

��

�

��

�

specularscattering totallypairbreaking diffusescattering

⌘?

⌘k⌘?

⌘k⌘?

⌘k

rotationsymmetric

Re[ � ]

Re[�]

��

�

r /ξ0

η/η 0

��

��

��

r /ξ0

Bz(γξ02 )

��

�� -��

��

��

��

r /ξ0

J θ(4πγ

ξ 03/c)

K1

K2= 3

Bz(r, ✓)


boundaryconditionsspecular pairbreaking diffuse

Aniso

trop

y

isotropic

scanninganisotropyandboundaryconditions

Bz(r, ✓)



Aniso

trop

y

isotropic


Bz(r, ✓)



Aniso

trop

y

isotropic


maximalfluxesFurusaki,Matsumoto&MS

Bz(r, ✓)


boundaryconditionsspecular

Aniso

trop

y

isotropic


Bz(r, ✓)


boundaryconditionsspecular

Aniso

trop

y

isotropic


netfluxindiskvanishes

Chiraldomaindomainwalls

Chiraldomainsanddomainwalls-chiralp-wave

chiralp-wavestatediscretedegeneracy2

2typesofdomains

in-planedomainwalls

x

y

x

⌘x

�i⌘y

x

⌘x

�i⌘yType1 Type2

phaseswitch~0 phaseswitch~π

+ -

�+k = z�0(kx + iky)

��k = z�0(kx � iky)

�k = ⌘xkx + ⌘yky

+ -


“parabolicbandrotationsymmetric”

γ-bandlikeFermisurfaceinBZ

kx

ky

~k1

~k2~Q

kx

ky

~k1

~k2

γ

K1 > K2 K1 < K2

x

y

+ -s-waveSC



π - loop


x

y

Jy / Im(⌘⇤s (n⇥ ⌘)z) = |⌘s||⌘x| sin(�s � �x)

Josephsoncoupling

conservationoftotalangularmomentumcouplingwiththex-component �x ⇡changesby

throughthedomainwall

intrinsicphasetwist

+ -s-waveSC


γ-bandlikeFermisurfaceinBZnarrows-p-junctions

Sr2RuO4-Cu-PbIn

Bahr&vanHarlingen

PbIn

500 nmSr2RuO4

π - loop

interferencepattern


+ -

+ -s-waveSC



π-loop


+ -s-waveSC

extendedinterfacemanydomains

vanHarlingengroup

+ -

+ -s-waveSC



π-loop


+ -s-waveSC

extendedinterfacemanydomains

vanHarlingengroup

irregular–historydependentinterference

I(H) = max�

Zdx jc sin

✓�� x(x)�

2⇡Hd

�0x

◆�Interferencepatterninamagneticfield

Conclusions

edgecurrents“invisible”•  supercurrentsarenotuniversal•  vanishingwouldbehoweveraccidential

•  note:quantumthermalHalleffectuniversal

domainwalls•  defectsofcondensatehistorydependence•  visiblethroughinterferenceeffects

seealsoC.Kallingroup

Documents

Features of edge states and domain walls in chiral ...nqs2017.ws/Slides/5th/Sigrist.pdfChiral superconductors Chiral superconductors - candidates UPt 3 SrPtAs hexagonal crystal structure