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Features of edge states and domain walls in chiral superconductors
Manfred Sigrist
kk
E
NQS2017,YITPKyoto
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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
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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
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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
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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
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Maenoetal1994
Sr2RuO4-chiralp-wavesuperconductor
layeredcrystalstructure
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
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Maenoetal1994
Sr2RuO4-chiralp-wavesuperconductor
layeredcrystalstructure
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
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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
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Edgecurrents
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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
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edgestatesforthechiralp-wavestate
Sr2RuO4-edgestatespectrum
scatteringofquasiparticlesatthesurfacesolutionofBogolyubov-deGennesequations
subgapboundstates(closeorbitsinparticle-holespace)
Bohr-Sommerfeldquantization
2✓ = ⇡ � (✓~k2� ✓~k1
)
phaseshifts
“Andreevreflection”
E = Ekk = �0 sin ✓ = �0kkkFsu
rface
electron
hole
θ
specularscattering
kkk1 k2
�k = |�0|ei✓k
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edgestatesforthechiralp-wavestate
Sr2RuO4-bulkandedgespectrum
scatteringofquasiparticlesatthesurfacesolutionofBogolyubov-deGennesequations
subgapboundstates(closeorbitsinparticle-holespace)
Bohr-Sommerfeldquantization
2✓ = ⇡ � (✓~k2� ✓~k1
)
phaseshifts
“Andreevreflection”
E = Ekk = �0 sin ✓ = �0kkkFkk+kF
�kF
E
+�0
��0
continuum
continuum
note:✓~k2� ✓~k1
= ±⇡ E = 0resultoftopologicalproperty
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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|
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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
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Topologyandedgecurrents
Areedgecurrentsauniquetopologicalproperty?latticeversionofchiralp-wavesuperconductor(tight-binding):
⇠~k = �2t(cos kxa+ cos kya) + ....� ✏F
zerosof�~k
+1-winding
-1-winding
Chernnumbers
�~k = �0(sin kxa+ i sin kya)
kx
ky
1stBrillouinzone
N = +1� 4⇥ 1
2
= �1⇥ 1 = �1
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kx
ky
kx
ky
kk
E
Topologyandedgecurrents
kk
E
N = +1 N = �1
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kk
E
kk
E
kx
ky
kx
ky
Topologyandedgecurrents
N = +1 N = �1
vq =1
~dEkk
dkk
quasiparticlevelocity
vq > 0 vq < 0
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kx
ky
kx
ky
kk
E
Topologyandedgecurrents
kk
E
N = +1 N = �1
surface
electron
holecharge
chargecurrent
kk
oppositechirality,butidenticalcurrentdirection
backwardbackward
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Topology-thermalHalleffect
“Spontaneous”Righi-Leduceffect
chiralQPedgestates T1
heatcurrent
chiralQPedgestates T2
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
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Topology-thermalHalleffect
“Spontaneous”Righi-Leduceffect
chiralQPedgestates T1
chiralQPedgestates T2
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
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Topologyandedgecurrents-afurthertwist
�~k = �0(sin kxa+ i sin kya)
particle-holesymmetry
�~k+~Q = ��~k ~Q =⇡
a(1, 1)
kx
kykk
Ex
y ~n = (1, 0)
~k1~k2
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kk
E
Topologyandedgecurrents-afurthertwist
�~k = �0(sin kxa+ i sin kya)
particle-holesymmetry
�~k+~Q = ��~k ~Q =⇡
a(1, 1)
kx
kykk
E
x
y
~n = (1, 0)
~k1
~k2
x
y
~n = (1, 1)
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kk
E
Topologyandedgecurrents-afurthertwist
�~k = �0(sin kxa+ i sin kya)
particle-holesymmetry
�~k+~Q = ��~k ~Q =⇡
a(1, 1)
kx
kykk
E
x
y
~n = (1, 0)
~k1
~k2
x
y
~n = (1, 1)
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kk
E
Topologyandedgecurrents-afurthertwist
�~k = �0(sin kxa+ i sin kya)
particle-holesymmetry
�~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
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Topologyandedgecurrents-afurthertwist
kk
E~n = (1, 1)
kk
E
kx
ky
~k1
~k2~Qkx
ky
~k1
~k2
currentreversal
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Topologyandedgecurrents-afurthertwist
kk
E~n = (1, 1)
kk
E
currentreversal
x
y
x
y
circularsupercurrent non-circularsupercurrentA.Bouhon
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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
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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)
lengthscalesforamplitudemodulationsforthetwoorderparametercomponents
⇧ =~ir+
2e
cAand
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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)
edgecurrentsforchiralphase
⇧ =~ir+
2e
cAand
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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)
cylindricallysymmetricbands1
3K1 = K2 = K3 = K4 � K5
⇧ =~ir+
2e
cAand
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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
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Ginzburg-Landauapproach-edgecurrents
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)
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Ginzburg-Landauapproach-edgecurrents
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�
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Diskshapedsample
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Ginzburg-Landauapproach-edgecurrents
diskshapedsample S.Etter
currentandfluxpatternforallsurfaceorientations
~n
�� �� �� ��
�
specularscattering
⌘?
⌘k
componentperpendiculartosurface
R = 40⇠0
isotropic
componentparalleltosurface
Bz(r, ✓)
✓r
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Re[ � ]
Re[�]
�� �� ���
�
r /ξ0
η/η 0
�� �� �� ��
��
����
r /ξ0
Bz(γξ02 )
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�� �� ��-����
��
����
����
r /ξ0
J θ(4πγ
ξ 03/c)
Ginzburg-Landauapproach-edgecurrents
differentboundaryconditions
�� �� �� ��
�
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�
�� �� �� ��
�
specularscattering totallypairbreaking diffusescattering
⌘?
⌘k⌘?
⌘k⌘?
⌘k
rotationsymmetric
Re[ � ]
Re[�]
�� �� ���
�
r /ξ0
η/η 0
�� �� �� ��
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r /ξ0
Bz(γξ02 )
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r /ξ0
J θ(4πγ
ξ 03/c)
K1
K2= 3
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Bz(r, ✓)
Ginzburg-Landauapproach-edgecurrents
boundaryconditionsspecular pairbreaking diffuse
Aniso
trop
y
isotropic
scanninganisotropyandboundaryconditions
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Bz(r, ✓)
Ginzburg-Landauapproach-edgecurrents
boundaryconditionsspecular pairbreaking diffuse
Aniso
trop
y
isotropic
scanninganisotropyandboundaryconditions
![Page 39: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/39.jpg)
Bz(r, ✓)
Ginzburg-Landauapproach-edgecurrents
boundaryconditionsspecular pairbreaking diffuse
Aniso
trop
y
isotropic
scanninganisotropyandboundaryconditions
maximalfluxesFurusaki,Matsumoto&MS
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Bz(r, ✓)
Ginzburg-Landauapproach-edgecurrents
boundaryconditionsspecular
Aniso
trop
y
isotropic
scanninganisotropyandboundaryconditions
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Bz(r, ✓)
Ginzburg-Landauapproach-edgecurrents
boundaryconditionsspecular
Aniso
trop
y
isotropic
scanninganisotropyandboundaryconditions
netfluxindiskvanishes
![Page 42: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/42.jpg)
Chiraldomaindomainwalls
![Page 43: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/43.jpg)
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
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+ -
Chiraldomainsanddomainwalls-chiralp-wave
“parabolicbandrotationsymmetric”
γ-bandlikeFermisurfaceinBZ
kx
ky
~k1
~k2~Q
kx
ky
~k1
~k2
γ
K1 > K2 K1 < K2
x
y
![Page 45: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/45.jpg)
+ -s-waveSC
“parabolicbandrotationsymmetric”
γ-bandlikeFermisurfaceinBZ
π - loop
Chiraldomainsanddomainwalls-chiralp-wave
x
y
Jy / Im(⌘⇤s (n⇥ ⌘)z) = |⌘s||⌘x| sin(�s � �x)
Josephsoncoupling
conservationoftotalangularmomentumcouplingwiththex-component �x ⇡changesby
throughthedomainwall
intrinsicphasetwist
![Page 46: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/46.jpg)
+ -s-waveSC
“parabolicbandrotationsymmetric”
γ-bandlikeFermisurfaceinBZnarrows-p-junctions
Sr2RuO4-Cu-PbIn
Bahr&vanHarlingen
PbIn
500 nmSr2RuO4
π - loop
interferencepattern
Chiraldomainsanddomainwalls-chiralp-wave
![Page 47: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/47.jpg)
+ -
+ -s-waveSC
“parabolicbandrotationsymmetric”
γ-bandlikeFermisurfaceinBZ
π-loop
Chiraldomainsanddomainwalls-chiralp-wave
+ -s-waveSC
extendedinterfacemanydomains
vanHarlingengroup
![Page 48: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/48.jpg)
+ -
+ -s-waveSC
“parabolicbandrotationsymmetric”
γ-bandlikeFermisurfaceinBZ
π-loop
Chiraldomainsanddomainwalls-chiralp-wave
+ -s-waveSC
extendedinterfacemanydomains
vanHarlingengroup
irregular–historydependentinterference
I(H) = max�
Zdx jc sin
✓�� �x(x)�
2⇡Hd
�0x
◆�Interferencepatterninamagneticfield
![Page 49: 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](https://reader036.vdocuments.us/reader036/viewer/2022071417/611477c0acd3ea79746e7ad4/html5/thumbnails/49.jpg)
Conclusions
edgecurrents“invisible”• supercurrentsarenotuniversal• vanishingwouldbehoweveraccidential
• note:quantumthermalHalleffectuniversal
domainwalls• defectsofcondensatehistorydependence• visiblethroughinterferenceeffects
seealsoC.Kallingroup