PPT - Sauls
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Superfluid 3He in Confined Geometries Broken Symmetry, Excitations and Possible New Phases Physics James A. Sauls Lake Michigan Anton B. Vorontsov Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Superfluid 3He in Confined Geometry ❖ Introduction ‣ Bulk Phases of 3He - Symmetry ‣ Superfluid 3He Near Surfaces - Pair Breaking ❖ Kinks, Domain Walls - Confined Fermions ❖ Chiral Edge States in 3He-A - 2D limit ‣ Edge Currents and Angular Momentum of 3He-A ‣ Robustness: Non-Specular Boundary Conditions ❖ Surface Excitation Spectrum - 3He-B ‣ Majorana Fermions ‣ Andreev Fermions ❖ Phase Diagram for 3He Films Superfluid‣ 3Translationally He in Confined Geometries Invariant Phases AWG - P-wave , RHUL 1. Introduction to Superfluid 3He Unconventional BCS Superfluid: S=1 - Spin Triplet L=1 - Orbital p-wave Cooper Pair Amplitude L=1 S=1 Inhomogeneous States: - relative momentum (p) - Center-of-Mass (R) px py pz 9 complex amplitudes Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Bulk Phase Diagram of Superfluid 3He A - phase (``axial’’) Anderson-Morel Nodal Quasiparticles Chiral Axis: Lz = ℏ Gapped B - phaseFully (``isotropic’’) Balian-Werthamer Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Fully Gapped, TRI Superfluid with Spontaneously generated Spin-Orbit 3 Superfluid He-B Coupling ‣ Balian & Werthamer (1963) FS Translational Invariance Weak Nuclear Dipole Energy violation: Nuclear Spin Dynamics Generator Approximate particle-hole symmetry Broken relative spin-orbit symmetry G. Moores & JAS Transverse Sound Acoustic Faraday Effect A. Leggett Y. Lee et al. Nature 1999 Superfluid 3He in Confined Geometries violation: Possible SuperSolid Phase A.Vorontsov & JAS AWG - P-wave , RHUL Superfluid 3He-P (``Planar phase’’) Non-Chiral Axis Nodal Quasiparticles Degenerate with the Axial State 2D TRI ``B-phase’’ Possible Ground State in Confined 3He Films Superfluid 3He in Confined Geometries Strong-Coupling Fluctuations Un-realized in Bulk 3He AWG - P-wave , RHUL Superfluid 3He-A (``Axial phase’’) ‣Anderson & Morel (1962) Chirality: Lz = ℏ Broken 2D Parity Broken TSymmetry Broken time-reversal symmetry Spin-Mass Vortices Ground state Orbital Angular Momentum Chiral Fermions Broken 2D parity Broken relative gauge-orbit symmetry Broken relative spin-orbit symmetry Superfluid 3He in Confined Geometries Lz =(N/2)ℏ (Δ/Ef)p p = 0,1,2 ? Ans: Chiral Edge States and Edge Currents AWG - P-wave , RHUL Chiral Superfluids ‣A-phase of 3He Spin AFM Anderson & Morel (PR,1962) Orbital FM ✤Chiral Spin-Triplet Superconductivity Sr2RuO4 ? UPt3 tetragonal hexagonal ? strong spinorbit coupling Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Dirac Fermions Mass Degenerate Vacuum States: Domain Wall ❖ Zero Energy Bound State Superfluid 3He in Confined Geometries R. Jackiw and C. Rebbi, Phys. Rev. D 1976 AWG - P-wave , RHUL Quasiparticle States Confined near Boundaries and Interfaces Toplogical defect (kink) - Jackiw,Rebbi (PRD 1976) ⟿ due to interface scattering C.-R. Hu 1994, Buchholtz et al. 1995 |Δ| - Quantum Well Particle-hole Interference A. Andreev 1964 ⇉ ⇉ L Δ ei2 Δ ei1 L grows - more states enter from continuum Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Andreev States and Edge Currents in d-wave SCs Andreev Surface States d-wave near pairbreaking surface ZEBS Walter et al PRL ’98 Burkhart, Rainer & JAS Surface states Paramagnetic Meissner current Anomalous edge currents Doppler Splitting of the ZEBS Large N(0) ⟿ Broken Time-reversal Edge currents Surface d+i s Superconductivity Tunnel Splitting from dI/dV Covington et al PRL ’97 Sub-dominant pairing: d+is Fogelström,Rainer & JAS PRL’97 Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Zero Modes, Sub-Gap States & Bound States in ... Josephson Point Contacts Vortex Core Excitations Δe+iφ Δe+iφ/2 Δe-iφ/2 φ Edge States of dx2-y2 Superconductors Sub-Gap States in Superfluid 3He Films Specular [110] Diffuse Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Chiral Edge States in 3He-A l - parallel to the edge edge currents ‣Chiral Edge State DipoleLocked Specular - no surface bound states - no pairbreaking Diffuse - gapless band - surface pairbreaking - Weyl Fermion G. E. Volovik M. Stone & R. Roy Skew Scatterng ‣Majorana Fermion at Chiral p-wave 2D Chiral p-wave TRI p-wave Superfluid 3He in Confined Geometries N AWG - P-wave , RHUL Molecular BEC Singlet S-wave Condensates ``Scalar BEC’’ Triplet P-wave Condensates ``Chiral P-wave molecular BEC’’ ⟿ Intrinsic Angular Momentum Density Superfluid 3He in Confined Geometries Ground State Angular Momentum AWG - P-wave , RHUL Molecular BEC vs. BCS Pairing ✤Loosely Bound Cooper Pairs:ξ ≫ a ✤Overlapping Pairs ⟿ Internal Exchange ξ ✤Cancellation of Orbital Currents? Superfluid 3He in Confined Geometries ⟿ AWG - P-wave , RHUL Molecular BEC vs BCS Condensation ✤Momentum Space: Pair Correlations on the Fermi Shell Fermi Sea # of pair-correlated Fermions ✤Intrinsic Angular Momentum Density in the BCS limit ... vs ...BEC limit A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. Mineev JETP 1976 Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Angular Momentum Paradox ✤Integrated Angular Momentum Density in the ... vs ...BEC limits BCS ~10-6 z P.W. Anderson and P. Morel 1962 & M. Cross 1975, A. Leggett RMP 1975 ✤Real Space Formulation in Cylindrical Geometries ✤McClure-Takagi Result: M. McClure, S. Takagi, PRL (1979) For any cylindrically symmetric chiral texture and pair wave function that vanishes on the boundary: independent of (a /ξ)! Superfluid 3He in Confined Geometries AWG - P-wave , RHUL McClure-Takagi gives the correct answer: but ... so ... Currents are on the boundary G. E. Volovik V. P. Mineev M. Ishikawa P. Muzikar D. Mermin T. Kita M. Stone A. Garg R. Roy ... M. Stone & R. Roy, PRB 2006 J. A. S., PRB 2011 Superfluid 3He in Confined Geometries AWG - P-wave , RHUL 2D Chiral A-phase with Bulk Solution ⟿ Propagators for States Near an Edge Bulk spectrum Superfluid 3He in Confined Geometries Bound State Pole AWG - P-wave , RHUL Surface States ➡ ➡ Chiral Edge States unoccupied Surface Confinement ... occupied a≪ ≪L Surface Current Pair of Time-Reversed Edge States Superfluid 3He in Confined Geometries Asymmetry in the Occupation AWG - P-wave , RHUL Local Spectral Density Pair Time-reversed Trajectories ⟿ Spectral Current Density in out p’ Asymmetry Exact in Cancellation the Occupation x = 0.5 ξΔ out p _’ α in Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Bound-State Current & Angular Momentum z x r R Mass Current ✤Galilean ✤Number Invariance: of Fermions: ⨉ 2 Too Big vs. MT Superfluid 3He in Confined Geometries Continuum States contribute to the Edge Currents M. Stone & R. Roy PRB 2006 J.A.S. PRB 2011 AWG - P-wave , RHUL Continuum Spectral Current Resonance Effects T=0 ξ C2 +iΔ C1 CR -iΔ Superfluid 3He in Confined Geometries MT !! AWG - P-wave , RHUL Finite Temperature ⨯ ⨯ C2 ⨯ ⨯ Matsubara Representation ξ ✤ +iΔ C1 CR T. Kita ``conjecture’’ J. Phys. Soc. Jpn. 67 (1998) pp. 216-224 3D Mesoscale (R≃ 2ξ) Numerical BdG ? Yz(T) = 1- c T2 ρs|| (T) Lz(T) ρs⊥ (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 Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Robustness of the Chiral Edge States Specular Reflection out ➡ in Facetted Surface Chiral Edge States No Chiral Currents p _p out in Retro Reflection Chirality Invisible! Tiny Angular Momentum Superfluid 3He in Confined Geometries !! AWG - P-wave , RHUL Edge Currents in a Toroidal Geometry R1, R2, (R1 - R2) ⋙ ξΔ x J2 J1 Sheet Current Volume Specular Edge Angular Momentum Counter-Propagating Currents MT Result Superfluid 3He in Confined Geometries AWG - P-wave , RHUL !! Non-Extensive Scaling of Lz Sheet Current - Non-Specular Edge J2 J1 Non-Specular Scattering R1, R2, (R1 - R2) ⋙ ξΔ Fraction of Forward Scattering Trajectories Incomplete Screening of Counter-Propagating Currents Lz ≉ V Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Conclusions ‣Topology and Non-specular Scattering ⟿ Lz is Non-Extensive: Lz >> (N/2)ℏ or Lz << - (N/2)ℏ ⟿ Direct Evidence of Edge Currents ‣Detailed models of surface scattering <-> Edge Currents ‣Gyroscopic Dynamics of Toroidal Disks of 3He-A ‣A.C. rotational dynamics of Edge States and Edge currents Superfluid 3He in Confined Geometries AWG - P-wave , RHUL 2. New States of 3He in Confined Geometry Multi-component Order Parameter with Broken Spin- Orbital and Gauge Symmetries. A & B are Topological Superfluids ↳ Low-Energy Topological Surface/Interface States Chiral p-wave Stripe Phase TRISuperSolid p-wave N ↳ Confinement: Strongly deformed Order Parameter ↳ low energy transport & thermodynamics ↳ phase transitions ➡ D≈10ξ0 Interactions of Surface & Interface States ↳ new superfluid phases in confined geometries S=1 Superfluid 3He in Confined Geometries L=1 AWG - P-wave , RHUL Superfluid 3He-B near a wall Pair Breaking & Pair Enhancement 2D Translationally invariant B-planar state s Specular Trajectories Specular Superfluid 3He in Confined Geometries Diffuse AWG - P-wave , RHUL Surface Fermionic Spectrum of Specular Scattering Non-Specular Scattering 3He-B ϴ ϴ is conserved specular ‣Majorana Fermion at ‣Dispersion: ‣Continuum Edge disperses 3He in Confined Geometries Superfluid is not conserved diffuse ‣Spectral Wt.: N(0)≠0 for all ‣New Andreev Bound States ‣Broad Low-energy Band AWG - P-wave , RHUL 4. Edge Spectrum in the Film ‣ QP spectroscopy: ‣ Ballistic Emmitters ‣ Momentum-resolved Detectors ‣Longitudinal and Transverse Acoustic Impedance Spectroscopy ‣Transverse Acoustic Impedance Spectroscopy R. Nomura et al. PRL 2009 ‣ Wall: Angle-resolved Superfluid 3He in Confined Geometries FS averaged AWG - P-wave , RHUL Translationally invariant phases of 3He films deformed bulk B- P- & A- phases confinement in z - invariant in xy 3He-B-planar A & P phases are degenerate Quantum Critial Point Re-entrance or Inhomogeneous Phase? Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Crystalline Phase of Superfluid ‣ Spontaneously Broken Translation film 3He A. Vorontsov & JAS, PRL 2007 & Review Symmetry in the 2012 x-y plane of the ‣ new OP components small ‣ Dc2 - Single-Q Mode Instability ‣2nd order Superfluid 3He in Confined Geometries ‣ Dc1 - Domain Walls Proliferation ‣Degenerate Vacua AWG - P-wave , RHUL Mechanism at Dc1 = Domain Wall Proliferation Surface - Domain Wall Interaction “perpendicular DW” “parallel DW” Condensaton Energy Condensation Energy loss to surface loss to Andreev states states Net Energy gain from a domain wall Perpendicular domain ‣Different ``healing wall for ⊥ and || costs less enegy/length lengths’’ components than ‣ξ⊥ = √3ξ|| 3 Superfluid He in Confined Geometriesa Parallel domain wall AWG - P-wave , RHUL Spontaneous Currents in d-wave films SC - Normal transition in films A. Vorontsov, PRL ’09 Inhomogeneous State: New structure of the order parameter ! Free Energy gain from paramagnetic edge states undergoing spontaneous TRS breaking Superfluid 3He in Confined Geometries AWG - P-wave , RHUL TRI Superfluid Planar-Stripe transition Order Parameter Equation - Mode-Instability - Dc2 ↳ n=0 - Broken symmetry “Vacuum” - Planar ↳ n=1 - new state (perturbation - linearized) Coupling of Δ with Δ* in the presence of non-trivial “vacuum” azx ~ 0.6 azz Non-trivial “vacuum” TRI Superfluid 3He in Confined Geometries ➡Couples q and -q at the instability AWG - P-wave , RHUL 4. Edge & Domain Wall Fermions in the Film spectral weight transfer to lower energy C3 ⇒ C1 spectral weight transfer to higher energy E3 ⇒ E1 Azz ‣ Wall: Angle-resolved FS averaged 1 2 3 Center Edge Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Angular resolved DOS in Films C1 Andreev States near continuum C3 spectral weight transfer to lower energy Azz E1 1 Edg e Cente 3 r Continuum Tomasch Low Energy States E3 spectral weight transfer to higher energy ‣ QP spectroscopy: Ballistic Emmitters and Momentum-resolved Detectors Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Induced DLRO ~ Density Modulation Superfluid 3He in Confined Geometries AWG - P-wave , RHUL OP and Density Wave in the Stripe Phase OP Domain Wall domain wall variations in the film δn/n ~ 10-5 Azz δn / n0 Axx ‣ Possible Detection: Light Scattering by density fluctuations? Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Superfluid film formation ‣ Competition: gravity vs. Van der Waals attractions of atoms to the substrate or wall ‣ Non-uniform film surface: surface tension & density modulation ‣ Affects surface waves as well (third sound, etc) chemical potential per particle hydrostatic equlibrium Superfluid 3He in Confined Geometries density = (n0 m) ~ 81.5 kg/m3 surface tension σ ~ 0.156 mN/m mass of 3He atom ~ 5 .10 -27 kg vdW constant α ~ 10 .10-9 m4/3 see e.g. Steel, Harrison et al JLTP 95, 1994 “Film flow on a round rim beaker” AWG - P-wave , RHUL Crystalline Phase Film thickness variations D ~ 10 ξ0 = 750 nm ‣ ‣ average density variations ~10-5 energy scales VdW / gravity film height driven by density fluctuations and α(n) dependence ‣ Surface tension / gravity ‣ Dominates at D ~ 10 ξ0 ‣ Overall change in film thickness ζ~0.1 Å Possible Capacitive Detection? A. Schechter et al., Nature 396, 554 (1998). Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Summary Crystalline Superfluid Phases ‣ Confinement ⟷ Surface States ‣ Majorana & Andreev States ‣ Interactions Between Surface States Momentum-Resolved Fermion Spectrum ‣ Broken Translational Symmetry ‣ Density Wave ⟶ ``SuperSolid’’ ‣ Particle-hole asymmetry: ✓ variations of the film height ✓ tension dominates at D ~ 10 ξ0 ‣Possible Detection: ✓ Momentum Resolved QP spectroscopy ✓Capacitance detection of height fluctuations ✓Optical detection of density fluctuations (?) ✓NMR and Mass/Heat Transport Superfluid 3He in Confined Geometries AWG - P-wave , RHUL Spin Triplet Pairing in UPt3 C C. Choi & JAS, PRL (1991) H [kOe] Anisotropic Pauli limiting B. Shivaram et al. PRL (1986) T [mK] pair breaking = No pair breaking ➡ E2u Spin-Triplet, w/ strong Spin-Orbit Coupling Superfluid 3He in Confined Geometries AWG - P-wave , RHUL return C L. Gorkov (1987) Unconventional Pairing in UPt3 C. Choi & JAS, PRL (1991) J. A. Sauls, Adv. Phys. 43, 113 (1994). H-T phase diagram - Tetracritical point - E2u Anisotropic Pauli limiting - S=1 S. Adenwalla et al. PRL (1990) NFL C J. Kycia et al., PRB (1998) B A B. Shivaram et al. PRL (1986) pair breaking No pair breaking Weak Symmetry Breaking - AFM order D. Hess, et al., J. Phys.: Cond. Mat. 1, 8135 (1989). ➡ Spin-Triplet, E2u, w/ strong Spin-Orbit Coupling R. A. Fisher et al., Phys. Rev. Lett. 1989. M. Graf, S.K. Yip & JAS, PRB (1996) Cv/T ✓Heat Capacity Anomalies ✓Anisotropy Transverse Sound ✓Anistropic Thermal Conductivity ➡ E2u orbital symmetry Realignment of the flux-line lattice in UPt3 Andrew Huxley et al. Nature (2000). ➡ E2u orbital symmetry T. Champel & V. Mineev (2001) T/Tc B. Ellman et al.., Phys. Rev. B 54 (1996) Superfluid 3He B. Lussier et al.., Phys. Rev. B 53 (1996) in Confined Geometries AWG - P-wave , RHUL return return
