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Surface States and Edge Currents of Superfluid 3 He in Confined Geometries

Physics. Lake Michigan. Surface States and Edge Currents of Superfluid 3 He in Confined Geometries. James A. Sauls. DMR-0805277. Bose Condensation of Molecules vs. Cooper Pairs Chiral Edge States in Superfluid 3He-A Films Ground-State Angular Momentum of 3He-A

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Surface States and Edge Currents of Superfluid 3 He in Confined Geometries

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  1. Physics Lake Michigan Surface States and Edge Currents ofSuperfluid 3He in Confined Geometries • James A. Sauls 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)

  2. Bulk Phase Diagram of Superfluid 3He A - phase (``axial’’) Anderson-Morel Nodal Quasiparticles Chiral Axis: Lz = ℏ B - phase (``isotropic’’) Balian-Werthamer Fully Gapped

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

  4. Broken 2D parity Superfluid 3He-A (``Axial phase’’) • Anderson & Morel (1962) Chirality: Lz = ℏ Broken 2D Parity Broken T-Symmetry Broken time-reversal symmetry Ground state Orbital Angular Momentum Spin-Mass Vortices Lz=(N/2)ℏ (Δ/Ef)p p = 0,1,2 ? Chiral Fermions Broken relative gauge-orbit symmetry Broken relative spin-orbit symmetry Ans: Chiral Edge States and Edge Currents

  5. Y.Nagato and K.Nagai, Physica B (2000). ? D Chiral Superfluids • A-phase of 3He Anderson & Morel (PR,1962) Orbital FM Spin AFM • Chiral Spin-Triplet Superconductivity UPt3 Sr2RuO4 hexagonal tetragonal strong spin-orbit coupling 6

  6. Bose-Einsten Condensation Macroscopically Occupied Single Particle State One-Particle Density Matrix Long-Range Order Order Parameter ≝ Macroscopically Occupied State Superfluidity & Quantum Interference Thermodynamic State Function Penrose & Onsager Phys. Rev. 1956

  7. Odd Parity, Spin Triplet (S=1): • Even Parity, Spin Singlet (S=0): Even Orbital Angular Momentum: Odd Orbital Angular Momentum: Order Parameter: ξ ξ≪ a • Tightly Bound Bose Molecules: Molecular BEC Macroscopically Occupied Two-Particle Wave Function • Cold Fermions with attractive interactions - e.g. 6Li, 40K ... • Molecular Wave Function • Internal Spin & Orbital Degrees of Freedom, e.g. s1=s2=½ Two-Particle Density Matrix

  8. Triplet P-wave Condensates Singlet S-wave Condensates ``Scalar BEC’’ ``Chiral P-wave molecular BEC’’ ⟿ Angular Momentum Density Ground State Angular Momentum

  9. ξ ξ ≫ a • Loosely Bound Cooper Pairs: Molecular BEC vs. BCS Pairing • Overlapping Pairs ⟿ Internal Exchange • Cancellation of Orbital Currents? ⟿

  10. Fermi Sea Molecular BEC vs BCS Condensation • Momentum Space: Pair Correlations on the Fermi Shell # of pair-correlated Fermions • Angular Momentum Density in the BCS limit A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. Mineev JETP 1976 11

  11. For any cylindrically symmetric chiral texture defined by and pair wave function that vanishes on the boundary: • Uniform State: z Angular Momentum Paradox • Integrated Angular Momentum Density in the BCS ... vs ...BEC limits ~10-6 A. Leggett RMP 1975 • Real Space Formulation in Cylindrical Geometries M. Ishikawa (1977) independent of (a /ξ)! • McClure-Takagi Theorem: M. McClure, S. Takagi, PRL (1979) • Mermin-Ho Texture: 12

  12. McClure-Takagi gives the correct answer for Lz , but ... • Gradient Expansion for z BEC or BCS where are the currents? M. Ishikawa (1977) N. D. Mermin P. Muzikar PRB (1980) Amperean current Twist current Bulk Supercurrent Sheet Current Uniform Texture 13

  13. BCS Pairing & the Quasiclassical Scale 2D A-phase/ 3D A-phase Film Fermi Sea • Angular Momentum Paradox • Theory of Inhomogeneous BCS States Loosely Bound Cooper Pairs: ξ ≫ a Inhomogeneous Edge: a ≪ ξ ≪ L

  14. Coupled Fermions & Pairs Nambu spinors Gorkov’s Propagator Quasiclassical propagators Gorkov Equations à la Eilenberger Quasiparticle Spectral function Order parameter - pair spectrum

  15. ⟿ 2D Chiral A-phase with Bulk Solution Propagators for States Near an Edge Bound State Pole Bulk spectrum

  16. Edge States Surface Confinement ... unoccupied a ≪ ≪ L Chiral Edge States occupied Weyl Fermion G. E. Volovik Pair of Time-Reversed Edge States Edge Current

  17. in p’ p’ out _ α out in Local Spectral Density Pair Time-reversed Trajectories ⟿ Spectral Current Density x = 0.5 ξΔ

  18. Number of Fermions: r z • Galilean Invariance: R x Bound-State Current & Angular Momentum Mass Current Continuum States determine Edge Currents M. Stone & R. Roy PRB 2006 JAS, PR B 84, 214509 (2011) ⨉ 2 Too Big vs. MT

  19. C1 CR ξ C2 +iΔ -iΔ Continuum Spectral Current confined? T = 0 Resonance Effect Exactly Cancels Bound State Lz M-T !! Continuum Response to the Edge ⟿ McClure-Takagi Result

  20. Finite Temperature C1 ⨯ CR ξ ⨯ C2 ⨯ ⨯ +iΔ T ≠ 0 Matsubara Representation Generalized Yosida Function for Lz Takafumi Kita’s ``conjecture’’ J. Phys. Soc. Jpn. 67 (1998) pp. 216-224 3D Mesoscale (R≃ 2ξ) Numerical BdG ? Yz(T) ≈ 1- c T2 ρs|| (T) ρs⊥(T) Lz(T) Lz(T) is ``soft’’ (2D or 3D) due to thermal excitation of Excited Edge States ρs|| (T) is ``soft’’ (3D) due to thermal excitation of Nodal QPs JAS, PR B 84, 214509 (2011) Tsutsumi & Machida,PR B 85, 100506(R) (2012)

  21. Edge Currents in a Toroidal Geometry R1, R2, (R1 - R2) ⋙ξΔ x Sheet Current J2 J1 Volume Specular Edge Angular Momentum Counter-Propagating Currents !! MT Result

  22. Robustness of the Chiral Edge States Specular Reflection out in Facetted Surface Chiral Edge States No Chiral Currents out p _ p in Retro Reflection Chirality Invisible! Tiny Angular Momentum !!

  23. !! J2 J1 Non-Extensive Scaling of Lz Sheet Current - Non-Specular Edge Non-Specular Scattering R1, R2, (R1 - R2) ⋙ξΔ Fraction of Forward Scattering Trajectories Incomplete Screening of Counter-Propagating Currents Lz≉ V

  24. Detecting Chiral Edge Currents • Gyroscopic Dynamics of Toroidal Disks of 3He-A • Engineered surfaces - differential Edge scattering <-> Edge Currents • Thermal Excitation of Edge States: • ≈ 1- c T2 • Toroidal Geometry & Non-specular Surfaces • ⇓ • Lz is Non-Extensive: • Lz > (N/2)ℏ or Lz < - (N/2)ℏ • ⇓ • Direct Evidence of Edge Currents Dissipationless Chiral Edge Currents Specular Edge Non-Specular Edge Equilibrium Angular Momentum J. Clow and J. Reppy, Phys. Rev. A 5, 424–438 (1972).

  25. ⟿ Direct Evidence of Edge Currents ⟿ Direct Evidence of Edge Currents Resumé • 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)ℏoriginates from Edge currents on Specular Boundaries • Kita Conjecture: Lz (T) ≅ (N/2)ℏ (ρs||(T)/ρ) is a accidental • Soft Temperature Dependence of Lz (T) due thermally excited Weyl Fermions • Edge Currents are Not Robust to Surface Scattering: Lz < (N/2)ℏ • Topology and Non-specular Scattering ⟿ Lz is Non-Extensive: • Lz >> (N/2)ℏ or Lz << - (N/2)ℏ

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