RECENT TOPICS IN THE THEORY OF SUPERFLUID 3 He

Transcription

1 RECENT TOPICS IN THE THEORY OF SUPERFLUID 3 He Lecture at ISSP Tokyo ErkkiThuneberg Department of physical sciences, University of Oulu Janne Viljas and Risto Hänninen Low temperature laboratory, Helsinki university of technology

2 Content Introduction to superfluid 3 He I. Josephson effect II. Vortex sheet in 3 He-A III. 3 He in aerogel These lecture notes will be placed at

3 Phase diagram of superfluid 3 He 3 He is the less common isotope of helium. It is a fermion. Pressure (MPa) superfluid phases B A solid normal Fermi liquid Temperature (K) classical liquid gas

4 Length scales 1 m 1 mm hydrodynamic scale 1 µm density, viscosity,... 1 nm atom atomic scale 1 pm nucleus nuclear scale charges masses,...

5 Length scales in superfluid 3 He 1 m 1 mm hydrodynamic theory 1 µm Ginzburg-Landau theory quasiclassical theory α', β i, K, γ density, superfluid density, stiffness parameters, viscosity parameters 1 nm atom quantum theory density, Fermi velocity, interaction parameters,t c,... 1 pm nucleus spin = 1/2 charge = 2 e mass = kg gyromagnetic ratio = /Ts

6 Cooper pairs in superfluid 3 He z Orbital wave functions (L=1) Spin wave functions (S=1) x y S =0: x S =0: y S =0: z (- + ) i( + ) ( + ) A xx A xy A xz A yx A yy A yz A zx A zy A zz order parameter

7 Part I. JOSEPHSON EFFECT superfluid L weak link superfluid or superconductor superfluid R Ψ L = A L exp(iφ L ) Ψ R = A R exp(iφ R ) current 0 J J 0 sin( φ) π 2π 3π 4π phase difference φ = φ R φ L energy F F 0 cos( φ) 0 π 2π 3π 4π φ Experiments: - superconductors: 1960 s - 3 He: Avenel and Varoquaux 1988

8 New experiments in 3 He-B - S. Backhaus, S. Pereverzev, R. Simmonds, A. Loshak, J. Davis, R. Packard, A. Marchenkov (1998, 1999) - O. Avenel, Yu. Mukharsky, E. Varoquaux (1999) current 0 π 2π 3π 4π φ energy π state 0 π 2π 3π 4π φ

9 Superfluid B phase z Orbital wave functions (L=1) Spin wave functions (S=1) x y S =0: x (- + ) A xx A xy A xz R xx R xy R xz S =0: y i( + ) A yx A yy A = A 0 e iφ yz R yx R yy R yz S =0: z ( + ) A zx A zy A zz order parameter R zx R zy R zz amplitude (real) phase factor (complex) rotation matrix (real) angle θ = 104 axis n

10 Pinhole size ξ 0 10 nm ξ 0 current J ( φ, R L µj,rr µj ) =2m 3 Av F N(0)πT ɛn d 2ˆk 4π (ˆk ẑ )g(ˆk, R hole,ɛ n ) path of a quasiparticle

11 Pinhole current-phase relation n n current T/T c = 0.1 T/T c = phase π state π π phase 2π π state occurs only at very low temperatures not consistent with experiments

12 Aperture array array of pinholes Current 0 π 2π φ ˆn Energy spin-orbit rotation axis ˆn 0 π 2π φ Anisotextural π state

13 Results - asymmetry of the junction - one fitting parameter η Current [ng/s] η ˆn 0.9T c 0.3T c 0 π 2π φ ˆn 0 π 2π φ Experiment: S. Backhaus, S. Pereverzev, R. Simmonds, A. Loshak, J. Davis, R. Packard, A. Marchenkov (1998, 1999) Theoretically strong dependence on magnetic field and surroundings - to be tested experimentally

14 Large aperture y Order parameter = A = A xx A xy A xz A yx A yy A yz A zx A zy A zz Boundary condition = A = exp(± i 2 φ) when z ± W x Ginzburg-Landau theory - valid only for T near T c - partial differential equation: z j j A µi +(γ 1) i j A µj =[ = A + β 1 = A Tr( = A = A T ) R c +β 2 = A Tr( = A = A T )+β 3 = A = A T = A + β 4 = A = A T = A + β 5 = A = A T = A] µi D - at surfaces = A = current π 2π φ 0 π 2π 0 π 2π

15 Part II. ROTAT ING SUPERFLUID An uncharged superfluid cannot rotate homogeneously Rotation takes place via vortex lines

16 Vortex cores in 3 He-B Numerical minimization of Ginzburg-Landau functional in 2 D j j A µi + (γ 1) i j A µj =[ A+β 1 A Tr(AA T )+β 2 ATr(AA T ) +β 3 AA T A + β 4 AA T A+β 5 A A T A] µi. (1) broken symmetry in vortex cores y y x pressure (MPa) solid B-superfluid 1 normal 0 fluid temperature (mk) A

17 Virtual half-quantum vortices half-quantum (π) vortex y half-quantum (π) vortex z x

18 The A phase The order parameter A µj = ˆd µ (ˆm j + iˆn j ) ( +i )( + ) m n orbital wave function d spin wave function A phase factor e iχ corresponds to rotation of ˆm and ˆn around ˆl: e iχ (ˆm + iˆn) = (cos χ + i sin χ)( ˆm + iˆn) = (ˆm cos χ ˆn sin χ)+i(ˆm sin χ +ˆn cos χ). Superfluid velocity v s = 2m χ = 2m j ˆm j ˆn j. (2)

19 Vortices in the A phase Consider the structure field: Here ˆl sweeps once trough all orientations (once a unit sphere). ˆm and ˆn circle twice around ˆl when one goes around this object. This is a two-quantum vortex. It is called continuous, because (the amplitude of the order parameter) vanishes nowhere.

20 Hydrostatic theory of 3 He-A Assume the order parameter ( ˆm, ˆn, ˆl, ˆd) changes slowly in space. Then we can make gradient expansion of the free energy F = d 3 r [ 1 2 λ D(ˆd ˆl) λ H(ˆd H) ρ v (ρ ρ )(ˆl v) 2 + Cv ˆl C 0 (ˆl v)(ˆl ˆl) K s( ˆl) K t ˆl ˆl K b ˆl ( ˆl) K 5 (ˆl )ˆd K 6[(ˆl ) iˆd j )] 2 ]. (3)

21 Vortex phase diagram in 3 He-A Vortex sheet (VS) field: d constant Continuous unlocked vortex (CUV) field: rotation velocity (rad/s) Singular vortex (SV) field: hard core soft core d constant magnetic field (mt) d constant Locked vortex 1 (LV1) = d field: Locked vortex 3 (LV3) = d field: z y x

22 Vortex sheet Vortex sheets are possible in 3 He-A Dipole-dipole interaction (3) f D = 1 2 λ D(ˆd ˆl) 2 d or d d d domain wall soliton Vortex sheet = soliton wall to which the vortices are bound. Sheets were first suggested to exist in 4 He, but they were found to be unstable. field: d constant Why stable in 3 He-A?

23 Shape of the vortex sheet The equilibrium configuration of the sheet is determined by the minimum of F = d 3 r 1 2 ρ s(v n v s ) 2 + σa. Here A is the area of the sheet and σ its surface tension. The equilibrium distance b between two sheets is velocity v n = Ω r v s b = ( 3σ ρ s Ω 2 ) 1/3. (4) This gives 0.36 mm at Ω = 1 rad/s. The area of the sheet A 1 b Ω2/3, (5) as compared to N vortex line Ω. b radius Connection lines with the side wall allow the vortex sheet to grow and shrink when angular velocity Ω changes.

24 NMR spectra of vortices Locked vortex NMR absorption Vortex sheet Singular vortex Soliton Larmor frequency Frequency shift (khz) potential energy for spin waves Ψ bound state vortex sheet

25 Part III. IMPURE SUPERFLUID 3 He 3 He is a naturally pure substance Impurity can be introduced by porous aerogel - strands of SiO 2 - typically 98% empty - small angle x-ray scattering homogeneous on scale above 100 nm

26 Experimental phase diagram 4 Solid Pressure (MPa) Superfluid Normal fluid Temperature (mk) Porto et al (1995), Sprague et al (1995), Matsumoto et al (1997)

27 Models Main effect of aerogel is to scatter quasiparticles Better models: Simplest model: homogeneous scattering model Random voids - not simple enough to calculate Periodic voids - not simple enough to calculate Isotropic inhomogeneous scattering - spherical unit-cell approximation - (quasi)periodic boundary condition only qualitative agreement with experiments trajectory of a quasiparticle

28 Isotropic inhomogeneous scattering model (IISM) Isotropic on large scale Hydrodynamics the same as in bulk but renormalized parameter values Better fit with experiment than in HSM realistic parameter values 1.0 T c T c0 0.5 HSM R = l, gentle R = l, steep Slab model R = 2l, steep ξ 0 /L

29 Conclusion Superfluid 3 He forms a very rich system because of the several different length scales. 1 m 1 mm 1 µm hydrodynamic theory Ginzburg-Landau theory quasiclassical theory Josephson effect vortices superfluid in aerogel 1 nm 1 pm atom nucleus quantum theory