surface states and edge currents of superfluid 3 he in confined geometries
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Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Surface States and Edge Currents ofSurface States and Edge Currents ofSuperfluid Superfluid 33He in Confined GeometriesHe in Confined GeometriesSurface States and Edge Currents ofSurface States and Edge Currents of
Superfluid Superfluid 33He in Confined GeometriesHe in Confined Geometries
James A. SaulsPhysics
Lake Michigan
DMR-0805277
✓ Bose Condensation of Molecules vs. Cooper Pairs✓ Chiral Edge States in Superfluid 3He-A Films✓ Ground-State Angular Momentum of 3He-A✓ Temperature Dependence of Lz (T)✓ Sensitivity to Boundary Scattering and Topology
M. Stone and R. Roy, Phys. Rev. B 69, 184511 (2004).T. Kita, J. Phys. Soc. Jpn. 67, 216 (1998).G. E. Volovik, JETP Lett 55, 368 (1992).J. A. Sauls, Phys. Rev. B 84, 214509 (2011)
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Bulk Phase Diagram of Superfluid Bulk Phase Diagram of Superfluid 33HeHeBulk Phase Diagram of Superfluid Bulk Phase Diagram of Superfluid 33HeHe
B - phase (``isotropic’’)Balian-Werthamer
A - phase (``axial’’)Anderson-MorelNodal QuasiparticlesChiral Axis: Lz = ℏ
Fully Gapped
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
‣ Balian & Werthamer Balian & Werthamer (1963)(1963)
Weak Nuclear Dipole Weak Nuclear Dipole EnergyEnergy
violation::violation::A. Legge
tt
A. Legge
tt
Nuclear Spin Nuclear Spin DynamicsDynamics
Nuclear Spin Nuclear Spin DynamicsDynamics
Superfluid Superfluid 33He-B He-B
Approximate Approximate particle-holeparticle-hole symmetrysymmetry
violation::violation::
Possible SuperSolid Phase
Possible SuperSolid Phase
A.Vorontsov & JAS
A.Vorontsov & JAS
FS
Translational InvarianceTranslational Invariance
Fully Gapped, TRI Superfluid with Spontaneously generated Spin-Orbit Coupling
GeneratorGenerator
Broken Broken relativerelative spin-orbit spin-orbit symmetrysymmetry
Broken Broken relativerelative spin-orbit spin-orbit symmetrysymmetry
Transverse SoundTransverse Sound
Acoustic Faraday Acoustic Faraday EffectEffect
Transverse SoundTransverse Sound
Acoustic Faraday Acoustic Faraday EffectEffect
G. Moor
es
& JAS
G. Moor
es
& JAS
Y. Lee et al.
Nature 1999
Y. Lee et al.
Nature 1999
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Superfluid Superfluid 33He-A (``Axial phase’’) He-A (``Axial phase’’)
Broken Broken time-reversal time-reversal symmetrysymmetry
Ground state Orbital Angular Ground state Orbital Angular MomentumMomentum
Broken relative spin-orbit Broken relative spin-orbit symmetrysymmetry
Broken relative spin-orbit Broken relative spin-orbit symmetrysymmetry
Broken relative gauge-orbit symmetry
Broken relative gauge-orbit symmetry
‣Anderson & Morel Anderson & Morel (1962)(1962)‣Anderson & Morel Anderson & Morel (1962)(1962)
Ans: Chiral Edge States and Edge Currents
Ans: Chiral Edge States and Edge Currents
Lz =(N/2)ℏ
(Δ/Ef)p
p = 0,1,2 ?
Lz =(N/2)ℏ
(Δ/Ef)p
p = 0,1,2 ?
Chirality: Lz = ℏBroken 2D ParityBroken T-Symmetry
Chirality: Lz = ℏBroken 2D ParityBroken T-Symmetry
Broken Broken 2D 2D parityparity
Spin-Mass Vortices
Spin-Mass Vortices
Chiral FermionsChiral Fermions
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Chiral Superfluids
Spin AFM Orbital FM
✤Chiral Spin-Triplet Superconductivity
UPt3Sr2RuO4
‣A-phase of A-phase of 33HeHe
‣A-phase of A-phase of 33HeHe
Anderson & Morel (PR,1962)Anderson & Morel (PR,1962)
strong spin-orbit coupling
tetragonal
?
hexagonal
DDDD
Y.Nagato and K.Nagai, Physica B (2000).
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Bose-Einsten CondensationBose-Einsten CondensationMacroscopically Occupied Single Particle
State
One-Particle Density Matrix
Long-Range Order
Penrose & Onsager Phys. Rev. 1956
Order Parameter ≝ Macroscopically Occupied State
Thermodynamic State FunctionThermodynamic State Function Superfluidity & Quantum InterferenceSuperfluidity & Quantum Interference
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Molecular BECMolecular BEC
❖Cold Fermions with attractive interactions - e.g. 6Li, 40K ...
✤Molecular Wave Functionξ
✤Tightly Bound Bose Molecules: ξ≪ a
✤Internal Spin & Orbital Degrees of Freedom, e.g. s1=s2=½
➡Odd Parity, Spin Triplet (S=1):
➡Even Parity, Spin Singlet (S=0):
Macroscopically Occupied Two-Particle Wave Function
Even Orbital Angular Momentum:
Odd Orbital Angular Momentum:
Order Parameter:
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Triplet P-wave CondensatesTriplet P-wave Condensates
Singlet S-wave Condensates
``Scalar BEC’’
``Chiral P-wave molecular BEC’’⟿
Ground State Angular MomentumAngular Momentum Density
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Molecular BEC vs. BCS PairingMolecular BEC vs. BCS Pairing
✤Loosely Bound Cooper Pairs:
ξ ≫ a
✤Overlapping Pairs ⟿ Internal Exchange
✤Cancellation of Orbital Currents?
ξ
⟿
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Molecular BEC
FermiSea
✤Momentum Space: Pair Correlations on the Fermi Shell
✤Angular Momentum Density in the BCS limit
# of pair-correlated Fermions# of pair-correlated Fermions
vs BCS Condensation
A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. Mineev JETP 1976Mineev JETP 1976
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Angular Momentum Paradox
✤Real Space Formulation in Cylindrical Geometries
✤Integrated Angular Momentum Density in the BCS
~10-6
... vs ...BEC limits
A. Leggett RMP 1975A. Leggett RMP 1975
M. Ishikawa M. Ishikawa (1977)(1977)
z
independent of (a /ξ)!
✤McClure-Takagi Theorem:
M. McClure, S. Takagi, PRL (1979)M. McClure, S. Takagi, PRL (1979)For any cylindrically symmetric chiral texture defined by and pair wave function that vanishes on the
boundary: ✤Uniform State: ✤Mermin-Ho
Texture:
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
N. D. Mermin P. Muzikar PRB N. D. Mermin P. Muzikar PRB (1980) (1980)
M. Ishikawa M. Ishikawa (1977)(1977)
McClure-Takagi gives the correct answer for Lz , but ...
where are the currents?
✤Gradient Expansion for
BEC or BCS
TwistTwist currentcurrent
z
Sheet Current
AmpereanAmperean currentcurrent
Bulk Bulk SupercurrentSupercurrent
Uniform Uniform TextureTexture
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
FermiSea
✤Angular Momentum Paradox
✤Theory of Inhomogeneous BCS States
Loosely Bound Cooper Pairs: ξ ≫ a
BCS Pairing & the Quasiclassical Scale
Inhomogeneous Edge: a ≪ ξ ≪ L
2D A-phase/ 3D A-phase Film
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Coupled Fermions & PairsNambu spinors
Quasiparticle Spectral functionQuasiparticle Spectral function
Order parameter - pair spectrumOrder parameter - pair spectrum
Quasiclassical propagators
Gorkov Equations à la Eilenberger
Gorkov’s Propagator
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
⟿
2D Chiral A-phase with
Bulk SolutionBulk Solution
Bulk spectrumBulk spectrum Bound State PoleBound State Pole
Propagators for States Near an Edge
Propagators for States Near an Edge
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Surface Surface Confinement ...Confinement ...
Surface Surface Confinement ...Confinement ...
Edge StatesEdge States
occupied
unoccupied
Pair of Time-Reversed Edge States
➡ ➡
Chiral Edge StatesChiral Edge StatesChiral Edge StatesChiral Edge States
Edge CurrentEdge CurrentEdge CurrentEdge Current
≪ L a ≪
Weyl Fermion G. E. VolovikWeyl Fermion G. E. Volovik
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Local Spectral Local Spectral DensityDensity
in
out
Pair Time-reversed Trajectories ⟿ Pair Time-reversed Trajectories ⟿ Spectral Current Spectral Current DensityDensity
Pair Time-reversed Trajectories ⟿ Pair Time-reversed Trajectories ⟿ Spectral Current Spectral Current DensityDensity
in
out
α
p’
_p’
x = 0.5 ξΔx = 0.5 ξΔ
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Bound-State Current & Angular Bound-State Current & Angular Momentum Momentum
z
Rx
✤Number of Fermions:
✤Galilean Invariance:
r
Mass CurrentMass Current
⨉ 2 Too Big vs. MT ⨉ 2 Too Big vs. MT
Continuum States determine
Edge CurrentsM. Stone & R. Roy PRB 2006JAS, PR B 84, 214509 (2011)
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
confined?
confined?
ξ
CR
Resonance Resonance EffectEffectResonance Resonance EffectEffect
T = 0T = 0
+iΔ
-iΔ
C1
C2
Continuum Spectral Continuum Spectral CurrentCurrent
Continuum Continuum ResponseResponse to the Edge ⟿ McClure- to the Edge ⟿ McClure-Takagi ResultTakagi ResultContinuum Continuum ResponseResponse to the Edge ⟿ McClure- to the Edge ⟿ McClure-Takagi ResultTakagi Result
Exactly Exactly Cancels Cancels Bound State Bound State LLzz
Exactly Exactly Cancels Cancels Bound State Bound State LLzz
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
Generalized Yosida Function Generalized Yosida Function for Lfor Lzz
Generalized Yosida Function Generalized Yosida Function for Lfor Lzz
Finite Finite TemperatureTemperature
ξ
CR
C1
+iΔ
C2
Matsubara Matsubara RepresentationRepresentation
Matsubara Matsubara RepresentationRepresentation
Takafumi Kita’s ``conjecture’’Takafumi Kita’s ``conjecture’’J. Phys. Soc. Jpn. 67 (1998) pp. 216-J. Phys. Soc. Jpn. 67 (1998) pp. 216-
224224
Takafumi Kita’s ``conjecture’’Takafumi Kita’s ``conjecture’’J. Phys. Soc. Jpn. 67 (1998) pp. 216-J. Phys. Soc. Jpn. 67 (1998) pp. 216-
224224
3D Mesoscale (R≃ 23D Mesoscale (R≃ 2ξξ))Numerical BdG Numerical BdG
3D Mesoscale (R≃ 23D Mesoscale (R≃ 2ξξ))Numerical BdG Numerical BdG
??
LLzz(T) is ``soft’’ (2D or 3D) due to thermal excitation of (T) is ``soft’’ (2D or 3D) due to thermal excitation of Excited Edge StatesExcited Edge StatesLLzz(T) is ``soft’’ (2D or 3D) due to thermal excitation of (T) is ``soft’’ (2D or 3D) due to thermal excitation of Excited Edge StatesExcited Edge Statesρρs|| s|| (T) is ``soft’’ (3D) due to thermal excitation (T) is ``soft’’ (3D) due to thermal excitation of of Nodal QPsNodal QPsρρs|| s|| (T) is ``soft’’ (3D) due to thermal excitation (T) is ``soft’’ (3D) due to thermal excitation of of Nodal QPsNodal QPs
LLzz(T)(T)
ρρs|| s|| (T)(T)
ρρss⊥⊥ (T)(T)
YYzz(T) ≈ 1- (T) ≈ 1- c c TT22
T T ≠≠ 0 0
⨯⨯⨯⨯
JAS, PR B 84, 214509 (2011)
Tsutsumi & Machida,PR B 85, 100506(R) (2012)
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
RR1,1, R R2,2, (R (R11 - R - R22) ) ⋙⋙ ξΔΔRR1,1, R R2,2, (R (R11 - R - R22) ) ⋙⋙ ξΔΔ
Edge Currents in a Toroidal Edge Currents in a Toroidal GeometryGeometry
Specular EdgeSpecular EdgeSpecular EdgeSpecular Edge
Angular MomentumAngular MomentumAngular MomentumAngular Momentum
Sheet Current
x
Counter-Propagating CurrentsCounter-Propagating CurrentsCounter-Propagating CurrentsCounter-Propagating Currents
MT ResultMT ResultMT ResultMT Result
J1
VolumeVolumeVolumeVolume
J2
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
NoNo Chiral Currents Chiral CurrentsNoNo Chiral Currents Chiral Currents
Robustness of the Chiral Edge Robustness of the Chiral Edge StatesStates
Chiral Edge StatesChiral Edge StatesChiral Edge StatesChiral Edge States
Specular Specular ReflectionReflectionSpecular Specular
ReflectionReflection
in
out
➡
in
out
pp_
TinyTiny Angular Angular MomentumMomentumTinyTiny Angular Angular MomentumMomentum
Facetted Facetted SurfaceSurfaceFacetted Facetted SurfaceSurface
Chirality Invisible!Chirality Invisible!Chirality Invisible!Chirality Invisible!
Retro ReflectionRetro ReflectionRetro ReflectionRetro Reflection
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
RR1,1, R R2,2, (R (R11 - R - R22) ) ⋙⋙ ξΔΔRR1,1, R R2,2, (R (R11 - R - R22) ) ⋙⋙ ξΔΔ
Non-Extensive Scaling of Non-Extensive Scaling of LLzz
Non-Specular Non-Specular ScatteringScattering
Non-Specular Non-Specular ScatteringScattering
Fraction of Forward Scattering Fraction of Forward Scattering TrajectoriesTrajectories
Fraction of Forward Scattering Fraction of Forward Scattering TrajectoriesTrajectories
Sheet Current - Non-Specular Edge
Incomplete Screening of Counter-Propagating Incomplete Screening of Counter-Propagating CurrentsCurrents
Incomplete Screening of Counter-Propagating Incomplete Screening of Counter-Propagating CurrentsCurrents
LLzz ≉ ≉ VVLLzz ≉ ≉ VV
J1
J2
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
‣ Thermal Excitation of Edge Thermal Excitation of Edge States: States:
≈ ≈ 1- 1- c Tc T22
‣Toroidal GeometryToroidal Geometry & & Non-Non-specular Surfacesspecular Surfaces⇓⇓LLzz is Non-Extensive: is Non-Extensive:
LLzz > (N/2) > (N/2)ℏ ℏ or Lor Lzz < - (N/2) < - (N/2)ℏ ℏ ⇓⇓Direct Evidence of Edge CurrentsDirect Evidence of Edge Currents
‣ Thermal Excitation of Edge Thermal Excitation of Edge States: States:
≈ ≈ 1- 1- c Tc T22
‣Toroidal GeometryToroidal Geometry & & Non-Non-specular Surfacesspecular Surfaces⇓⇓LLzz is Non-Extensive: is Non-Extensive:
LLzz > (N/2) > (N/2)ℏ ℏ or Lor Lzz < - (N/2) < - (N/2)ℏ ℏ ⇓⇓Direct Evidence of Edge CurrentsDirect Evidence of Edge Currents
Detecting Chiral Edge Currents
‣Gyroscopic Dynamics of Toroidal Disks of 3He-A‣Gyroscopic Dynamics of Toroidal Disks of 3He-A
J. Clow and J. Reppy, Phys. Rev. A 5, 424–438 (1972).
DissipationleDissipationlessss
Chiral Edge Chiral Edge CurrentsCurrents
DissipationleDissipationlessss
Chiral Edge Chiral Edge CurrentsCurrents
EquilibriumEquilibriumAngular MomentumAngular Momentum
EquilibriumEquilibriumAngular MomentumAngular Momentum
Non-Specular EdgeNon-Specular Edge
Specular EdgeSpecular Edge
‣Engineered surfaces - differential Edge scattering <-> Edge Currents ‣Engineered surfaces - differential Edge scattering <-> Edge Currents
Edge Currents in Superfluid 3He-A Films RIKEN, May 21, 2012
‣ Ground-State Currents Confined to Edge on Scale ~ a ≪ ξ ≪ L‣ Edge Current Originates from Contiuum disturbed by the Surface Bound State‣ Lz = (N/2)ℏ originates from Edge currents on Specular Boundaries‣Kita Conjecture: Lz (T) ≅ (N/2)ℏ (ρs||(T)/ρ) is a accidental
‣ Ground-State Currents Confined to Edge on Scale ~ a ≪ ξ ≪ L‣ Edge Current Originates from Contiuum disturbed by the Surface Bound State‣ Lz = (N/2)ℏ originates from Edge currents on Specular Boundaries‣Kita Conjecture: Lz (T) ≅ (N/2)ℏ (ρs||(T)/ρ) is a accidental
Resumé
‣ Soft Temperature Dependence of Lz (T) due thermally excited Weyl Fermions ‣ Soft Temperature Dependence of Lz (T) due thermally excited Weyl Fermions
⟿ Direct Evidence of Edge Currents
‣Topology and Non-specular Scattering ⟿ Lz is Non-Extensive: Lz >> (N/2)ℏ or Lz << - (N/2)ℏ
⟿ Direct Evidence of Edge Currents
‣Edge Currents are Not Robust to Surface Scattering: Lz < (N/2)ℏ
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