1. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 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): - 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 3 He-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 3 He-A Trapping of vortices by a network of topological defects in superfluid 3 He-A Andrei Golov:

2. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 3 He-A: order parameter vsvs vsvs l d p-wave, spin triplet Cooper pairs Two anisotropy axes: l - direction of orbital momentum d - spin quantization axis (s.d)=0 Continuous vorticity: large length scale Discrete degeneracy: domain walls l n m γ β α Order parameter: 6 d.o.f.: A μj =∆(T)(m j +in j )d µ Velocity of flow depends on 3 d.o.f.: v s = -ħ(2m 3 ) -1 ( ∇ γ+cosβ ∇ α)

3. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Groundstates, vortices, domain walls: (slab geometry, small H and v s )  =0 v s =0  >0 vsvs

4. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Topological defects (textures) Azimuthal component of superflow Two-quantum vortex Vortex and wall can be either dipole-locked or unlocked R core ~ 0.2D (l z, d z )= Domain walls

5. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Vortices in bulk 3 He-A (Equilibrium phase diagram, Helsinki data) LV2 similar to CUV except d = l (narrow range of small  ) dl-wall l-wall ATC-vortex (dl) ATC-vortex (l)

6. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 When l is free to rotate: Hydrodynamic instability at Soft core radius R core vs. D and H : ♦ H = 0 : R core ∼ D → v c ∝ D -1 ♦ 2-4 G < H < 25 G : R core ∼ ξ H ∝ H -1 → v c ∝ H ♦ H > 25 G : R core ∼ ξ d = 10 μm → v c ∼ 1 mm/s H F =2-4 G vcvc H v d ~1 mm/s H d ≈25 G v c ∝ D -1 vc∝Hvc∝H vc∼vdvc∼vd Models for v c (intrinsic processes) When l is aligned with v (Bhattacharyya, Ho, Mermin 1977): Instability of v-aligned l-texture: at or  R core 2m v 3 c ħ mm/s1 2 3 = D m v  ħ  c (Feynman 1955, et al…)

7. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 l z =+1 d z =-1 l z =+1 d z =+1 l z =-1 d z =-1 l z =-1 d z =+1 d-wall dl-wall l-wall l z =+1 d z =-1 l z =+1 d z =+1 l z =-1 d z =+1 l z =-1 d z =-1 orGroundstate (choice of four) Multidomain texture (metastable) (obtained by cooling at H=0 while rotating) (obtained by cooling while stationary)

8. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Also possible: d-walls only dl-walls only (obtained by cooling at H=0 while rotating) (obtained by cooling while stationary) l z =+1 d z =+1 l z =-1 d z =-1 l z =+1 d z =+1 l z =+1 d z =-1

9. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Fredericksz transition (flow driven 2 nd order textural transition) v F =  F R Orienting forces: - Boundaries favour l perpendicular to walls (“uniform texture”, UT) - Magnetic field H  favours l (via d) in plane with walls (“planar”, PT) - Superflow favours l tends to be parallel to v s (“azimuthal”, AT) Theory (Fetter 1977): vFvF 1 22                   HFHF H v v F ~ D -1 H F ~ D -1 2 walls

10. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Ways of preparing textures vortices uniform rotation azimuthaldomain walls rotation planar uniform H Initial preparation Uniform l-texture: cooling through T c while rotating: NtoA (moderate density of domain walls): cooling through T c at  = 0 BtoA (high density of domain walls): warming from B-phase at  = 0 Applying rotation,  >  F, H = 0: makes azimuthal textures Applying H > H F at  0: makes planar texture, then  >  F : two dl-walls on demand Rotating at  > v c R introduces vortices Value of v c and type of vortices depend on texture (with or without domain walls)

11. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Rotating torsional oscillator Disk-shaped cavity, D = 0.26 mm or 0.44 mm, R=5.0 mm The shifts in resonant frequency v R ~ 650 Hz and bandwidth v B ~ 10 mHz tell about texture Rotation produces continuous counterflow v = v n - v s  H VsVs VsVs VsVs Normal Texture Azimuthal Texture Textures with defects Because  s  <  s  we can distinguish: 0 r v s = 0 vn= rvn= r v 0 r vn= rvn= r v

12. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Principles of vortex detection Superfluid circulation Nκ : v s (R) = Nκ(2πR) -1 N vortices Rotating normal component : v n (R) =  R  Rotation If counterflow | v n - v s | exceeds v F, texture tips azimuthally TO detection of counterflow

13. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Main observables FF cc 1. Hysteresis due to v c > 0 2. Hysteresis due to pinning vsvs or vsvs

14. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Hysteresis due to pinning no pinning weak, v p < v c Horizontal scale set by  c = v c /R Vertical scale set by  trap = v p /R  trap cc cc 2c2c  max vsvs strong, v p > v c ? Strong pinning:  trap =  c Because  trap can’t exceed  c (otherwise antivortex nucleates)

15. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Uniform texture, positive rotation (H = 0) Four fitting parameters:  F  c R-R c  D = 0.26 mm: R - R c = 0.30 ± 0.10 mm D = 0.44 mm: R - R c = 0.35 ± 0.10 mm Vortices nucleate at ~ D from edge FF cc vc = cRvc = cR v c = 4v F ~ D -1, in agreement with v c ∼ ħ(2m 3 a c ) -1

16. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Critical velocity vs. core radius Adapted from U. Parts et al., Europhys. Lett. 31, 449 (1995)

17. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Uniform texture, weak pinning

18. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Uniform texture, weak pinning

19. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Handful of pinned vortices D=0.44mm

20. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 When no pinned vortices left Can tell the orientation of l-texture One MH vortex with one quantum of circulation

21. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Negative rotation: strange behaviour (only for D = 0.44mm) VcVc V c1 V c2 D (mm)V +c V -c V -c1 V -c2 V c(walls) (mm/s) 0.26 0.50.3 -- --0.2 0.44 0.30.20.20.50.2 No hysteresis! cc FF

22. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Bulk dl-wall (theory: Kopu et al. Phys. Rev. B (2000)) What difference will two dl-walls make? Critical velocity:

23. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 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. AT UT PT Vortices D=0.26mm

24. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Theory: bulk l-wall Theory: bulk dl-wall (Kopu et al, PRB 2000) D = 0.44 mm D = 0.26 mm With many walls in magnetic field: v c NtoA after rotation in field H >H d : l–walls

25. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Trapped vorticity In textures with domain walls: total circulation of ~ 50  0 of both directions can be trapped after stopping rotation v s (R) = Nκ 0 (2πR) -1,  trap = v s /R vsvs

26. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Pinning by networks of walls Strong pinning: single parameter v c :  c = v c /R  trap = v c /R

27. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Web of domain walls vs. pinning due to extrinsic inhomogeneities (grain boundaries or roughness of container walls) Intrinsic pinning in chiral superconductors In chiral superconductors, such as Sr 2 RuO 4, UPt 3 or PrOs 4 Sb 12,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] Trapping of vorticity by defects of order parameter is intrinsic pinning 3-wall junctions might play a role of pinning centres dl-wall l-wall d-wall can carry vorticity ++ (l z =+1, d z =+1) +- (l z =+1, d z =-1) -+ (l z =-1, d z =+1) -- (l z =-1, d z =-1)

28. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Energy of domain walls D=0.26mm D=0.44mm

29. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Web of domain walls dl-wall l-wall d-wall can carry vorticity ++ (l z =+1, d z =+1) +- (l z =+1, d z =-1) -+ (l z =-1, d z =+1) -- (l z =-1, d z =-1) E dl = E l = E d E dl << E l  E d (expected for D >> ξ d = 10 μm ) dl d l l d

30. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 What if only dl-walls? To be metastable, need pinning on surface roughness dl-wall ++ (l z =+1, d z =+1) -- (l z =-1, d z =-1) E.g. the backbone of vortex sheet in Helsinki experiments No metastability in long cylinder Then vortices could be trapped too

31. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Summary In 3 He-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. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Unpinning by Magnus force Annihilation with antivortex In experiment, v p = min (v M, v c ) (i.e. the critical velocity is capped by v c ) Unpinning mechanisms FMFM v v > v M v > v c to remove an existing vortex (v M ) or to create an antivortex (v c )? Pinning potential is quantified by “Magnus velocity” v M = F p /  s  0 D (such that Magnus force on a vortex F M =  s D  0 v equals pinning force F p ) Weak pinning, v M < v c Strong pinning, v M > v c

33. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Model of strong pinning All vortices are pinned forever Maximum  pers is limited to  c due to the creation of antivortices

34. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004

35. Two models of critical state 1.Pinning force on a vortex F p equals Magnus force F M = (  s D  0 ) v 2.Counterflow velocity v equals v c (nucleation of antivortices) In superconductors, v p (Bean-Levingston barrier) is small but flux lines can not nucleate in volume, hence superconductors are normally in the pinning-limited regime |v| = v p even though v c < v p. If v c < v p (strong pinning), |v| = v c If v c > v p (weak pinning), |v| = v p v p =F p /  s  0 D strong, v p > v c  trap cc cc 2c2c  max no pinningweak, v p < v c two critical parameters: v c and v p (because Magnus force ~ v s ): (anti)vortices can nucleate anywhere when |v n -v s | > v c existing vortices can move when |v n -v s | > v p

36. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Trapping by different textures

37. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 domain walls rotation planar uniform rotation azimuthaldomain walls rotation planar

38. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004

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

40. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004

41. Hydrodynamic instability at v c ∼ ħ(2m 3 a c ) -1 (Feynman) (when l is free to rotate) Soft core radius a c can be manipulated by varying either: slab thickness D ♦ H = 0 : a c ∼ D → v c ∝ D -1 or magnetic field H ♦ 2-4 G < H < 25 G : a c ∼ ξ H ∝ H -1 → v c ∝ H ♦ H > 25 G : a c ∼ ξ d = 10 μm → v c ∼ 1 mm/s H F =2-4 G vcvc H v c ~1 mm/s H d ≈25 G v c ∝ D -1 vc∝Hvc∝H Theory for v c (intrinsic nucleation) Alternative theory mm/s1 2 3 ~ D m v  ħ  c  D2m v 3 c ħ

42. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 v c ~ D -1 : Why? Not quite aligned texture! (numerical simulations for v = 3 v F )

43. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004

44. However, these are also possible: l z =+1 d z =-1 l z =+1 d z =+1 l z =-1 d z =-1 l z =-1 d z =+1 d-wall dl-wall l-wall l z =+1 d z =+1 l z =-1 d z =+1 unlocked walls presentdl-walls only or

45. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Models of critical state Strong pinning (v M > v c ): Single parameter, v c :  c = v c /R  trap = v c /R Weak pinning (v p < v c ): Two parameters, v c and v M :  c = v c /R  trap = v p /R Horizontal scale set by  c = v c /R Vertical scale set by  trap = v p /R vsvs ?

46. Quantum Phenomena at Low Temperatures, Lammi, 10 January 2004 Hysteretic “remnant magnetization” Horizontal scale set by  c = v c /R Vertical scale set by  trap = v p /R vsvs ? (p.t.o.) What sets the critical state of trapped vortices?