edge currents in superfluid 3 he-a films riken, may 21, 2012 surface states and edge currents of...

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)


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


‣ 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


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


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).


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


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:


Triplet P-wave CondensatesTriplet P-wave Condensates

Singlet S-wave Condensates

``Scalar BEC’’

``Chiral P-wave molecular BEC’’⟿

Ground State Angular MomentumAngular Momentum Density


Molecular BEC vs. BCS PairingMolecular BEC vs. BCS Pairing

✤Loosely Bound Cooper Pairs:

ξ ≫ a

✤Overlapping Pairs ⟿ Internal Exchange

✤Cancellation of Orbital Currents?

ξ

⟿


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


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:


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


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


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


⟿

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


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


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


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)


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


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)


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


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


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


‣ 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


‣ 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

edge currents in superfluid 3 he-a films riken, may 21, 2012 surface states and edge currents of...

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