Vortices at the A-B phase boundary in superfluid 3 He

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1 R. Hänninen Low Temperature Laboratory Helsinki University of Technology Finland Theory: E.V. Thuneberg G.E. Volovik Experiments: R. Blaauwgeers V.B. Eltsov A.P. Finne M. Krusius

2 Outline:. Introduction 2. Experimental setup 3. Hydrostatic theory in 3 He 4. A-B phase boundary 5. Results 6. Summary R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 2

Introduction: two main phases: A and B phase several textures, especially in the A phase new experiments with the phase boundary what is the effect of the A-B phase

3 Introduction: two main phases: A and B phase several textures, especially in the A phase new experiments with the phase boundary what is the effect of the A-B phase boundary? we have calculated the effect on the A phase vortices two different textures obtained R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 3

4 Experimental setup: R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 4

5 Hydrostatic theory in 3 He: p-wave spin triplet = order parameter A µi is a 3 3 matrix A phase order parameter: A µj = A ˆdµ ( ˆm j + iˆn j ), where ˆm ˆn vector ˆd defines the axis along which the spin of the Cooper pair vanishes. vector ˆl = ˆm ˆn gives the orbital angular momentum axis superfluid velocity: v sa = h 2m 3 i ˆm i ˆn i R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 5

6 Different vortex textures in 3 He-A: R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 6

7 Hydrostatic free energy: F = d 3 r(f d + f h + f g ) dipole term: magnetic anisotropy term: f d = 2 λ d(ˆd ˆl) 2, f h = 2 λ h(ˆd H) 2, kinetic terms + gradient energy density (v n = ): 2f g = ρ vsa 2 + (ρ ρ )(ˆl v sa ) 2 + 2Cv sa ˆl 2C (ˆl v sa )(ˆl ˆl) + K s ( ˆl) 2 + K t (ˆl ˆl) 2 + K b ˆl ( ˆl) 2 + K 5 (ˆl )ˆd 2 + K 6 [(ˆl ) iˆdj ] 2. Characteristic scales: dipole length: ξ d = ( h/2m 3 ) ρ /λ d µm i,j dipole field: H d = λ d /λ h 2 mt dipole velocity: v d = λ d /ρ mm/s R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 7

(ˆl ˆl) 2 + K b ˆl ( ˆl) 2 + K 5 (ˆl )ˆd 2 + K 6 [(ˆl ) iˆdj ] 2.

8 B phase order parameter: A µj = B R µj (ˆn, θ)e iφ, R µj rotation matrix around ˆn with angle θ superfluid velocity: Hydrostatic free energy: kinetic term (v n = ): v sb = h 2m 3 φ f K = 2 ρ sv 2 sb, dipole term: f D = λ D (R ii R jj + R ij R ji ) = θ = 4 gradient term: plus some other terms f G = λ G i R αi j R αj + λ G2 i R αj i R αj = R µi = const R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 8

D = λ D (R ii R jj + R ij R ji ) = θ = 4 gradient term: plus some other terms f G = λ G i R αi j R αj + λ

9 Vortices in the B phase: µi A µi 2 singular (in the scale of ξ d ) larger critical velocity (v cb v ca ) carry one quantum of circulation R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 9

10 A-B phase boundary: requirements at the A-B boundary (normal ŝ = ẑ Ω): ˆd = R ŝ ( ˆm + iˆn) ŝ = e iφ ˆl ŝ = coordinate system where vn = R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko

11 Simplified model: R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko

12 Results: calculations done for v sa = deep in the A phase assumed GL-region (T T c ) minimization using conjugate gradient method two different textures obtained texture depends on the rotation velocity (density of the vortices at the boundary) both textures have half-quantum vortex cores at the phase boundary ˆd ˆx everywhere independent of the initial ansatz R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 2

the vortices at the boundary) both textures have half-quantum vortex cores at the phase boundary ˆd ˆx

13 Low density texture (low rotation velocity): 4 A phase (z > ) 3 z/ξ d B phase (z < ) y/ξ d R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 3

14 Low density texture (low rotation velocity): n m z/ξ d l x/ξ d 3 l y 2 3 l x y/ξ d 7 8 R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 4

15 v s for the low density texture: 3 Vortices at the A-B phase boundary in superfluid 3 He z/ξ d y/ξ d R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 5

16 Superfluid current for the low density texture: z/ξ d 2 A phase B phase y/ξ d R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 6

17 High density texture (high rotation velocity): z/ξ d A phase (z > ) B phase (z < ) y/ξ d R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 7

18 High density texture (high rotation velocity): n m z/ξ d l 2 x/ξ d l y 2 l x 3 4 y/ξ d 2 5 R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 8

19 v s for the high density texture: 2 Vortices at the A-B phase boundary in superfluid 3 He z/ξ d y/ξ d R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 9

20 Superfluid current for the high density texture:.5 z/ξ d A phase.5.5 B phase y/ξ d R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 2

21 Summary: calculated the vortex structure at the A-B boundary two different textures obtained (low density & high density) half-quantum vortex cores To be done: generalize to other pressures possible other textures?? calculate the NMR spectrum difficult to measure R. Hänninen International Workshop on Superfluidity under Rotation 23, Nikko 2

Contents. Goldstone Bosons in 3He-A Soft Modes Dynamics and Lie Algebra of Group G:

Contents. Goldstone Bosons in 3He-A Soft Modes Dynamics and Lie Algebra of Group G: ... Vlll Contents 3. Textures and Supercurrents in Superfluid Phases of 3He 3.1. Textures, Gradient Energy and Rigidity 3.2. Why Superfuids are Superfluid 3.3. Superfluidity and Response to a Transverse