1. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Surface States and Edge Currents of Superfluid 3 He in Confined Geometries James A. Sauls Physics Lake Michigan DMR-0805277 ✓ B ose Condensation of Molecules vs. Cooper Pairs ✓ C hiral Edge States in Superfluid 3He-A Films ✓ G round-State Angular Momentum of 3He-A ✓ T emperature Dependence of Lz (T) ✓ S ensitivity to Boundary Scattering and Topology M. Stone and R. Roy, Phys. Rev. B 69, 184511 (2004).Phys. Rev. B 69, 184511 (2004) T. Kita, J. Phys. Soc. Jpn. 67, 216 (1998).98). G. E. Volovik, JETP Lett 55,G. E. Volovik, JETP Lett 55, 368 (1992). J. A. Sauls, Phys. Rev. B 84, 214509 (2011)

2. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Introduction to Superfluid 3 He Unconventional BCS Superfluid: S=1 - Spin Triplet L=1 - Orbital p-wave Cooper Pair Amplitude Inhomogeneous States: - relative momentum (p) - Center-of-Mass (R) 9 complex amplitudes S=1 L=1 pxpx p x pypy p y pzpz p z

3. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Bulk Phase Diagram of Superfluid 3 He B - phase (``isotropic’’) Balian-Werthamer A - phase (``axial’’) Anderson-Morel Nodal Quasiparticles Chiral Axis: L z = ℏ Fully Gapped

4. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 ‣ Balian & Werthamer (1963) Weak Nuclear Dipole Energy : violation: A. Leggett Nuclear Spin Dynamics Superfluid 3 He-B Approximate particle-hole symmetry : violation: Possible SuperSolid Phase A.Vorontsov & JAS FS Translational Invariance Fully Gapped, TRI Superfluid with Spontaneously generated Spin-Orbit Coupling Generator Broken relative spin-orbit symmetry Transverse Sound Acoustic Faraday Effect Transverse Sound Acoustic Faraday Effect G. Moores & JAS G. Moores & JAS Y. Lee et al. Nature 1999 Y. Lee et al. Nature 1999

5. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Superfluid 3 He-A (``Axial phase’’) Broken time-reversal symmetry Ground state Orbital Angular Momentum Broken relative spin-orbit symmetry Broken relative gauge-orbit symmetry ‣ Anderson & Morel (1962) Ans: Chiral Edge States and Edge Currents L z =(N/2) ℏ (Δ/E f ) p p = 0,1,2 ? L z =(N/2) ℏ (Δ/E f ) p p = 0,1,2 ? Chirality: L z = ℏ Broken 2D Parity Broken T- Symmetry Chirality: L z = ℏ Broken 2D Parity Broken T- Symmetry Broken 2D parity Spin-Mass Vortices Chiral Fermions

6. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Chiral Superfluids Spin AFM Orbital FM 6 ✤ Chiral Spin-Triplet Superconductivity UPt 3 Sr 2 RuO 4 ‣ A-phase of 3 He Anderson & Morel (PR,1962) strong spin-orbit coupling tetragonal ? hexagonal DD Y.Nagato and K.Nagai, Physica B (2000).

7. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Bose-Einsten Condensation Macroscopically Occupied Single Particle State One-Particle Density Matrix Long-Range Order Penrose & Onsager Phys. Rev. 1956 Order Parameter ≝ Macroscopically Occupied State Thermodynamic State Function Superfluidity & Quantum Interference

8. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Molecular BEC ❖ Cold Fermions with attractive interactions - e.g. 6 Li, 40 K... Two-Particle Density Matrix ✤ Molecular Wave Function ξ ✤ Tightly Bound Bose Molecules: ξ ≪ a ✤ Internal Spin & Orbital Degrees of Freedom, e.g. s 1 =s 2 =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:

9. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Triplet P-wave Condensates Singlet S-wave Condensates ``Scalar BEC’’ ``Chiral P-wave molecular BEC’’ Ground State Angular Momentum Angular Momentum Density

10. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Molecular BEC vs. BCS Pairing ✤ Loosely Bound Cooper Pairs: ξ ≫ a ✤ Overlapping Pairs Internal Exchange ✤ Cancellation of Orbital Currents? ξ

11. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 11 Molecular BEC Fermi Sea ✤ Momentum Space: Pair Correlations on the Fermi Shell ✤ Angular Momentum Density in the BCS limit # of pair-correlated Fermions vs BCS Condensation A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. Mineev JETP 1976

12. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 12 Angular Momentum Paradox ✤ Real Space Formulation in Cylindrical Geometries ✤ Integrated Angular Momentum Density in the BCS ~10 -6... vs...BEC limits A. Leggett RMP 1975 M. Ishikawa (1977) z independent of (a /ξ)! ✤ McClure-Takagi Theorem: 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:

13. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 13 N. D. Mermin P. Muzikar PRB (1980) M. Ishikawa (1977) McClure-Takagi gives the correct answer for L z, but... where are the currents? ✤ Gradient Expansion for BEC or BCS Twist current z Sheet Current Amperean current Bulk Supercurrent Uniform Texture

14. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 McClure-Takagi gives the correct answer: but... so... Currents are on the boundary z G. E. Volovik V. P. Mineev M. Ishikawa P. Muzikar D. Mermin T. Kita A. Garg M. Stone R. Roy... JAS, PR B 84, 214509 (2011) Tsutsumi & Machida,PR B 85, 100506(R) (2012)

15. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 N TRI p-wave Chiral p-wave Specular - no surface bound states - no pairbreaking Confined 3 He-A Confined 3 He-A l - parallel to the edge ⇒ edge currents Dipole- Locked Edge Reflections ‣ Chiral Edge State - Weyl Fermion G. E. Volovik ‣ Chiral Edge State - Weyl Fermion G. E. Volovik 2D Chiral p-wave Y.Nagato and K.Nagai, Physica B (2000).

16. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Fermi Sea ✤ 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

17. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Coupled Fermions & Pairs Nambu spinors Quasiparticle Spectral function Order parameter - pair spectrum Quasiclassical propagators Gorkov Equations à la Eilenberger Gorkov’s Propagator

18. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 2D Chiral A-phase with Bulk Solution Bulk spectrum Bound State Pole Propagators for States Near an Edge

19. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Surface Confinement... Edge States occupied unoccupied Pair of Time-Reversed Edge States ➡ ➡ Chiral Edge States Edge Current ≪ L≪ L a ≪ Weyl Fermion G. E. Volovik Weyl Fermion G. E. Volovik

20. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Local Spectral Density in out Pair Time-reversed Trajectories Spectral Current Density in out α p’p’ _ p’p’ x = 0.5 ξ Δ

21. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Local Spectral Density in out Pair Time-reversed Trajectories Spectral Current Density in out α p’p’ _ p’p’ Exact Cancellation Asymmetry in the Occupation x = 0.5 ξ Δ Zero Net Current into the Boundary

22. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Bound-State Current & Angular Momentum z R x ✤ Number of Fermions: ✤ Galilean Invariance: r Mass Current 2 Too Big vs. MT Continuum States determine Edge Currents M. Stone & R. Roy PRB 2006 JAS, PR B 84, 214509 (2011)

23. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Continuum Spectral Current +Δ+Δ -Δ-Δ Odd:Odd: & confined?

24. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 confined ? ξ CRCR Resonance Effect T = 0 +iΔ -iΔ C1C1 C2C2 M-T !! Continuum Spectral Current Continuum Response to the Edge McClure-Takagi Result Exactly Cancels Bound State L z Exactly Cancels Bound State L z

25. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 Generalized Yosida Function for L z Finite Temperature ξ CRCR C1C1 +iΔ C2C2 Matsubara Representation Takafumi Kita’s ``conjecture’’ J. Phys. Soc. Jpn. 67 (1998) pp. 216-224 67 Takafumi Kita’s ``conjecture’’ J. Phys. Soc. Jpn. 67 (1998) pp. 216-224 67 3D Mesoscale (R ≃ 2ξ) Numerical BdG 3D Mesoscale (R ≃ 2ξ) Numerical BdG ?? L z (T) is ``soft’’ (2D or 3D) due to thermal excitation of Excited Edge States ρ s|| (T) is ``soft’’ (3D) due to thermal excitation of Nodal QPs L z (T) ρ s|| (T) ρ s ⊥ (T) Y z (T) ≈ 1- c T 2 T ≠ 0 JAS, PR B 84, 214509 (2011)Tsutsumi & Machida,PR B 85, 100506(R) (2012)

26. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 R 1, R 2, (R 1 - R 2 ) ⋙ Δ R 1, R 2, (R 1 - R 2 ) ⋙ ξ Δ Edge Currents in a Toroidal Geometry Specular Edge Angular Momentum Sheet Current x Counter-Propagating Currents MT Result !! J1J1 VolumeVolume J2J2

27. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 No Chiral Currents Robustness of the Chiral Edge States Chiral Edge States Specular Reflection in out ➡ in out p p _ Tiny Angular Momentum !! Facetted Surface Chirality Invisible! Retro Reflection

28. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 R 1, R 2, (R 1 - R 2 ) ⋙ Δ R 1, R 2, (R 1 - R 2 ) ⋙ ξ Δ Non-Extensive Scaling of L z Non-Specular Scattering Fraction of Forward Scattering Trajectories Sheet Current - Non-Specular Edge Incomplete Screening of Counter-Propagating Currents L z ≉ V !! J1J1 J2J2

29. Edge Currents in Superfluid 3 He-A Films RIKEN, May 21, 2012 ‣ Thermal Excitation of Edge States: ≈ 1- c T 2 ≈ 1- c T 2 ‣ Toroidal Geometry & Non-specular Surfaces ⇓ L z is Non-Extensive: L z > (N/2) ℏ or L z (N/2) ℏ or L z < - (N/2) ℏ⇓ Direct Evidence of Edge Currents ‣ Thermal Excitation of Edge States: ≈ 1- c T 2 ≈ 1- c T 2 ‣ Toroidal Geometry & Non-specular Surfaces ⇓ L z is Non-Extensive: L z > (N/2) ℏ or L z (N/2) ℏ or L z < - (N/2) ℏ⇓ Direct Evidence of Edge Currents Detecting Chiral Edge Currents ‣ Gyroscopic Dynamics of Toroidal Disks of 3 He-A J. Clow and J. Reppy, Phys. Rev. A 5, 424–438 (1972). Dissipationless Chiral Edge Currents Dissipationless Equilibrium Angular Momentum Equilibrium Non-Specular Edge Specular Edge ‣ Engineered surfaces - differential Edge scattering Edge Currents

30. Edge Currents in Superfluid 3 He-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 ‣ L z = (N/2) ℏ originates from Edge currents on Specular Boundaries ‣ Kita Conjecture: L z (T) ≅ (N/2) ℏ (ρ s|| (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 ‣ L z = (N/2) ℏ originates from Edge currents on Specular Boundaries ‣ Kita Conjecture: L z (T) ≅ (N/2) ℏ (ρ s|| (T)/ρ) is a accidental Resumé ‣ Soft Temperature Dependence of L z (T) due thermally excited Weyl Fermions Direct Evidence of Edge Currents ‣ Topology and Non-specular Scattering L z is Non-Extensive: L z >> (N/2) ℏ or L z << - (N/2) ℏ Direct Evidence of Edge Currents ‣ Edge Currents are Not Robust to Surface Scattering: L z < (N/2) ℏ