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Continuous topological defects in 3 He-A in a slab

Andrei Golov:. Trapping of vortices by a network of topological defects in superfluid 3 He-A. Continuous topological defects in 3 He-A in a slab Models for the critical velocity and pinning (critical states). Vortex nucleation and pinning (intrinsic and extrinsic):

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Continuous topological defects in 3 He-A in a slab

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  1. Andrei Golov: Trapping of vortices by a network of topological defects in superfluid 3He-A Continuous topological defects in 3He-A in a slab Models for the critical velocity and pinning (critical states). Vortex nucleation and pinning (intrinsic and extrinsic): - Uniform texture: intrinsic nucleation and weak extrinsic pinning - Texture with domain walls: intrinsic nucleation and strong universal pinning Speculations about the networks of domain walls P.M.Walmsley, D.J.Cousins, A.I.Golov Phys. Rev. Lett. 91, 225301 (2003) Critical velocity of continuous vortex nucleation in a slab of superfluid 3He-A P.M.Walmsley, I.J.White, A.I.Golov Phys. Rev. Lett. 93, 195301 (2004) Intrinsic pinning of vorticity by domain walls of l-texture in superfluid 3He-A

  2. 3He-A: order parameter l γ d vs l vs β n m α p-wave, spin triplet Cooper pairs Two anisotropy axes: l - direction of orbital momentum d-spin quantization axis (s.d)=0 Order parameter: 6 d.o.f.: Aμj=∆(T)(mj+inj)dµ Velocity of flow depends on 3 d.o.f.: vs = -ħ(2m3)-1(∇γ+cosβ∇α) Continuous vorticity: large length scale Discrete degeneracy: domain walls

  3. =0 vs=0 vs >0 >0 >0 Groundstates, vortices, domain walls: (slab geometry, small H and vs)

  4. Topological defects (textures) Domain walls (lz, dz)= Rcore~ 0.2D Vortex and wall can be either dipole-locked or unlocked Two-quantum vortex Azimuthal component of superflow

  5. l-wall ATC-vortex (l) dl-wall ATC-vortex (dl) Vortices in bulk 3He-A(Equilibrium phase diagram, Helsinki data) LV2 similar to CUV except d = l (narrow range of small )

  6. Models for vc (intrinsic processes) vc H vd~1 mm/s ħ ~ v vc∼vd ħ c 2m Rcore ~ v 1 mm/s = 3 c x 2 m vc∝H 3 D vc∝D-1 HF=2-4 G Hd≈25 G When l is free to rotate: Hydrodynamic instability at Soft core radius Rcore vs. D and H : ♦H= 0 :Rcore ∼D→vc∝ D-1 ♦2-4 G < H< 25 G:Rcore ∼ξH ∝ H-1→vc ∝H ♦H> 25 G :Rcore ∼ξd = 10 μm →vc∼1 mm/s (Feynman 1955, et al…) or When l is aligned with v (Bhattacharyya, Ho, Mermin 1977): Instability of v-aligned l-texture: at

  7. lz=+1 dz=+1 lz=-1 dz=+1 lz=+1 dz=+1 lz=+1 dz=-1 lz=-1 dz=-1 lz=-1 dz=+1 lz=+1 dz=-1 lz=-1 dz=-1 dl-wall l-wall d-wall Groundstate (choice of four) or Multidomain texture (metastable) (obtained by cooling at H=0 while rotating) (obtained by cooling while stationary)

  8. lz=+1 dz=+1 lz=+1 dz=+1 lz=+1 dz=-1 lz=-1 dz=-1 Also possible: dl-walls only d-walls only (obtained by cooling at H=0 while rotating) (obtained by cooling while stationary)

  9. Fredericksz transition (flow driven 2nd order textural transition) 2 2 æ ö æ ö v H ç ÷ ç ÷ + = 1 ç ÷ ç ÷ 2 walls vF HF è ø è ø Orienting forces: - Boundaries favourl perpendicular to walls (“uniform texture”, UT) - Magnetic field Hfavours l (via d) in plane with walls (“planar”, PT) - Superflow favoursl tends to be parallel to vs (“azimuthal”, AT) vF =FR vF ~D-1HF ~D-1 Theory (Fetter 1977):

  10. uniform uniform rotation rotation H azimuthal domain walls vortices planar Ways of preparing textures Uniform l-texture: cooling through Tc while rotating: Initial preparation NtoA (moderate density of domain walls): cooling through Tc at  = 0 BtoA (high density of domain walls): warming from B-phase at  = 0 Applying rotation,  > F,H = 0: makes azimuthal textures Applying H > HF at  0: makes planar texture, then  > F: twodl-walls on demand Rotating at  > vcR introduces vortices Value of vc and type of vortices depend on texture (with or without domain walls)

  11. Rotating torsional oscillator H  vn=r vs = 0 v vn=r r 0 v vs = 0 r 0 Disk-shaped cavity, D = 0.26 mm or 0.44 mm, R=5.0 mm The shifts in resonant frequency vR ~ 650 Hz and bandwidth vB ~ 10 mHz tell about texture Because s < swe can distinguish: Normal Texture Azimuthal Texture Textures with defects Rotation produces continuous counterflowv= vn - vs Vs Vs Vs

  12. Superfluid circulation Nκ : vs(R) = Nκ(2πR)-1 N vortices  Rotating normal component : vn(R) = R Rotation Principles of vortex detection If counterflow | vn - vs | exceeds vF , texture tips azimuthally TO detection of counterflow

  13. WF Wc Main observables 2.Hysteresis due to pinning 1.Hysteresis due to vc > 0 vs or vs

  14. ? strong, vp> vc vs trap weak, vp< vc c Horizontal scale set by c = vc/R Vertical scale set by trap = vp/R no pinning max c 2c Strong pinning: trap = c Because trapcan’t exceedc (otherwise antivortex nucleates) Hysteresis due to pinning

  15. Uniform texture, positive rotation (H = 0) WF Wc Four fitting parameters: WF Wc R-Rc Dn D = 0.26 mm: R - Rc = 0.30 ± 0.10 mm D = 0.44 mm: R - Rc = 0.35 ± 0.10 mm Vortices nucleate at ~ D from edge vc=cR vc = 4vF ~ D-1, in agreement withvc∼ħ(2m3ac)-1

  16. Critical velocity vs. core radius Adapted from U. Parts et al., Europhys. Lett. 31, 449 (1995)

  17. Uniform texture, weak pinning

  18. Uniform texture, weak pinning

  19. Handful of pinned vortices D=0.44mm

  20. When no pinned vortices leftCan tell the orientation of l-texture One MH vortex with one quantum of circulation

  21. Negative rotation: strange behaviour (only for D = 0.44mm) c Vc2 Vc1 No hysteresis! Vc F D (mm) V+c V-c V-c1 V-c2 Vc(walls) (mm/s) 0.26 0.5 0.3 -- -- 0.2 0.44 0.3 0.2 0.2 0.5 0.2

  22. Bulk dl-wall (theory: Kopu et al. Phys. Rev. B (2000)) What difference will two dl-walls make? Critical velocity:

  23. Just two dl-walls: pinning in field Three times as much vorticity pinned on a domain wall at H=25 G than in uniform texture at H=0. Other possible factors: - Pinning in field might be stronger (vortex core shrinks with field). - Different types of vortices in weak and strong fields. Vortices AT UT PT D=0.26mm

  24. D = 0.44 mm Theory: bulk dl-wall (Kopu et al, PRB 2000) D = 0.26 mm Theory: bulk l-wall NtoA after rotation in field H >Hd: l–walls With many walls in magnetic field: vc

  25. Trapped vorticity vs vs(R) = Nκ0(2πR)-1, trap = vs/R In textures with domain walls: total circulation of ~ 50 0 of both directions can be trapped after stopping rotation

  26. Pinning by networks of walls Strong pinning: single parameter vc : c = vc/R trap = vc/R

  27. ++ (lz=+1, dz=+1) +- (lz=+1, dz=-1) -+ (lz=-1, dz=+1) -- (lz=-1, dz=-1) dl-wall l-wall d-wall can carry vorticity Web of domain walls 3-wall junctions might play a role of pinning centres Trapping of vorticity by defects of order parameter is intrinsic pinning vs.pinning due to extrinsic inhomogeneities (grain boundaries or roughness of container walls) Intrinsic pinning in chiral superconductors In chiral superconductors, such as Sr2RuO4, UPt3 orPrOs4Sb12,vortices can be trapped by domain walls between differently oriented ground states [Sigrist, Agterberg 1999, Matsunaga et al. 2004] Anomalously slow creep and strong pining of vortices are observed as well as history dependent density of domain walls (zero-field vs field-cooled) [Dumont, Mota 2002]

  28. D=0.26mm D=0.44mm Energy of domain walls

  29. ++ (lz=+1, dz=+1) +- (lz=+1, dz=-1) -+ (lz=-1, dz=+1) -- (lz=-1, dz=-1) dl-wall l-wall d-wall can carry vorticity Web of domain walls l dl Edl= El = Ed Edl<< El» Ed (expected for D >> ξd = 10 μm) d l dl d

  30. Then vortices could be trapped too What if only dl-walls? ++ (lz=+1, dz=+1) -- (lz=-1, dz=-1) dl-wall To be metastable, need pinning on surface roughness E.g. the backbone of vortex sheet in Helsinki experiments No metastability in long cylinder

  31. Summary In 3He-A, we studied dynamics of continuous vortices in different l-textures. Critical velocity for nucleation of different vortices observed and explained as intrinsic processes (hydrodynamic instability). Strong pinning of vorticity by multidomain textures is observed. The amount of trapped vorticity is fairly universal. General features of vortex nucleation and pinning are understood. However, some mysteries remain. The 2-dimensional 4-state mosaic looks like a rich and tractable system. We have some experimental insight into it. Theoretical input is in demand.

  32. v > vc v > vM v FM Unpinning mechanisms to remove an existing vortex (vM) or to create an antivortex (vc)? Pinning potential is quantified by “Magnus velocity” vM= Fp /s0D (such that Magnus force on a vortex FM = sD0vequalspinning force Fp) Weak pinning, vM < vc Strong pinning, vM > vc Annihilation with antivortex Unpinning by Magnus force In experiment, vp = min (vM, vc) (i.e. the critical velocity is capped by vc)

  33. Model of strong pinning All vortices are pinned forever Maximum pers is limited to c due to the creation of antivortices

  34. strong, vp> vc trap weak, vp< vc c no pinning max c 2c Two models of critical state • Pinning force on a vortex Fp equals Magnus force FM= (sD0) v • Counterflow velocity v equals vc (nucleation of antivortices) two critical parameters: vc and vp (because Magnus force ~ vs): (anti)vortices can nucleate anywhere when |vn-vs| > vc existing vortices can move when |vn-vs| > vp If vc< vp (strong pinning), |v| = vc If vc > vp (weak pinning), |v| = vp vp=Fp/ s0D In superconductors, vp (Bean-Levingston barrier) is small but flux lines can not nucleate in volume, hence superconductors are normally in the pinning-limited regime |v| = vp even though vc< vp .

  35. Trapping by different textures

  36. uniform rotation rotation rotation azimuthal domain walls planar planar domain walls

  37. Trapped vorticity vs In textures with domain walls: total circulation of ~ 50 0 of both directions can be trapped after stopping rotation vs(R) = Nκ0(2πR)-1 trap = vs/R

  38. vc vc~1 mm/s vc∝H vc∝D-1 H Hd≈25 G HF=2-4 G ħ ħ = = v 1 mm/s ~ v c x c 2 m 2m D 3 D 3 Theory for vc (intrinsic nucleation) Hydrodynamic instability at vc∼ħ(2m3ac)-1 (Feynman) (when l is free to rotate) Soft core radius ac can be manipulated by varying either: slab thickness D ♦H= 0 :ac ∼D→vc∝D-1 or magnetic field H ♦2-4 G < H< 25 G:ac ∼ξH ∝H-1→vc ∝H ♦H> 25 G :ac ∼ξd = 10 μm →vc∼1 mm/s Alternative theory

  39. vc ~ D-1 : Why? Not quite aligned texture! (numerical simulations for v = 3 vF)

  40. lz=+1 dz=+1 lz=+1 dz=-1 lz=+1 dz=+1 lz=-1 dz=-1 lz=-1 dz=+1 lz=-1 dz=+1 dl-wall l-wall d-wall However, these are also possible: or dl-walls only unlocked walls present

  41. Models of critical state ? vs Horizontal scale set by c = vc/R Vertical scale set by trap = vp/R Strong pinning (vM>vc): Single parameter, vc : c = vc/R trap = vc/R Weak pinning (vp<vc): Two parameters, vc and vM: c = vc/R trap = vp/R

  42. Hysteretic “remnant magnetization” (p.t.o.) ? vs Horizontal scale set by c = vc/R Vertical scale set by trap = vp/R What sets the critical state of trapped vortices?

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