Andrei Golov Trapping of vortices by a network

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): - 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

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

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

Topological defects (textures) Two-quantum vortex Domain walls (lz, dz)= Rcore~ 0. 2 D Azimuthal component of superflow Vortex and wall can be either dipole-locked or unlocked

Vortices in bulk 3 He-A (Equilibrium phase diagram, Helsinki data) l-wall ATC-vortex (l) dl-wall LV 2 similar to CUV except d = l (narrow range of small ) ATC-vortex (dl)

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

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

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

Fredericksz transition (flow driven 2 nd order textural transition) 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 vs (“azimuthal”, AT) 2 walls v. F = FR Theory (Fetter 1977): æ v çç è v. F 2 ö ÷÷ + ø 2 æ H ö çç ÷÷ = 1 è HF ø v. F ~ D-1 HF ~ D-1

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

Rotating torsional oscillator Disk-shaped cavity, D = 0. 26 mm or 0. 44 mm, R=5. 0 mm H The shifts in resonant frequency v. R ~ 650 Hz and bandwidth v. B ~ 10 m. Hz tell about texture Because s < s we can distinguish: Normal Texture Azimuthal Texture vs = 0 Textures with defects 0 vs = 0 vn= r v Rotation produces continuous counterflow v = vn - vs Vs Vs Vs r

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

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

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

Uniform texture, positive rotation (H = 0) WF 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 Wc vc = c. R vc = 4 v. F ~ D-1, in agreement with vc∼ħ(2 m 3 ac)-1

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

Uniform texture, weak pinning

Uniform texture, weak pinning

Handful of pinned vortices D=0. 44 mm

When no pinned vortices left Can tell the orientation of l-texture One MH vortex with one quantum of circulation

Negative rotation: strange behaviour (only for D = 0. 44 mm) c Vc 2 Vc 1 No hysteresis! Vc F D (mm) 0. 26 0. 44 V+c 0. 5 0. 3 V-c 0. 3 0. 2 V-c 1 -0. 2 V-c 2 -0. 5 Vc(walls) (mm/s) 0. 2

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

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: Vortices - Different types of vortices in weak and strong fields. AT UT - Pinning in field might be stronger (vortex core shrinks with field). PT D=0. 26 mm

With many walls in magnetic field: vc Theory: bulk dl-wall (Kopu et al, PRB 2000) D = 0. 44 mm D = 0. 26 mm Theory: bulk l-wall Nto. A after rotation in field H >Hd: l–walls

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

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

Web of domain walls ++ (lz=+1, dz=+1) +- (lz=+1, dz=-1) -+ (lz=-1, dz=+1) -- (lz=-1, dz=-1) dl-wall can carry vorticity l-wall d-wall 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 Sr 2 Ru. O 4, UPt 3 or Pr. Os 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]

Energy of domain walls D=0. 26 mm D=0. 44 mm

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

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

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.

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

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

Two models of critical state 1. Pinning force on a vortex Fp equals Magnus force FM= ( s. D 0) v 2. 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 trap strong, vp> vc c existing vortices can move when |vn-vs| > vp If vc< vp (strong pinning), |v| = vc weak, vp< vc no pinning c 2 c max If vc > vp (weak pinning), |v| = vp vp=Fp/ s 0 D 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.

Trapping by different textures

rotation planar domain walls azimuthal rotation planar rotation domain walls uniform

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

Theory for vc (intrinsic nucleation) Hydrodynamic instability at vc∼ħ(2 m 3 ac)-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 vc vc~1 mm/s vc ∝D-1 HF=2 -4 G vc∝H H Hd≈25 G

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

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

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

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