1. Transverse force on a magnetic vortex Lara Thompson PhD student of P.C.E. Stamp University of British Columbia July 31, 2006

2. Vortices in many systems Classical fluids  Magnus force, inter-vortex force Superfluids, superconductors  Inter-vortex force  Magnus force, inertial mass, damping forces Spin systems  Magnus → gyrotropic force  Inter-vortex force  Inertial mass, damping forces ?    Topic of this talk!

3. Equations of motion controversy Superfluid/superconductor vortices: vortex effective mass  Estimates range from ~  r v 2 to order of E v /v 0 2 effective Magnus force = bare Magnus force + Iordanskii force?  magnitude of Iordanskii force?  existence of Iordanskii force?!? …denied by Thouless et. al. (Berry’s phase argument); affirmed by Sonin Ao & Thouless, PRL 70, 2158 (1993); Thouless, Ao & Niu, PRL 76, 3758 (1996) Sonin, PRB 55, 485 (1997) and many many more…

4. Vortices in a spin system Similarities same forces present: “Magnus” force, inter- vortex force, inertial force, damping… Differences 2 topological indices: vorticity q + polarization p Magnus → gyrotropic force  p, can vanish! no “superfluid flow”

5. Spin System: magnons & vortices use spherical coordinates (S, ,  ) with conjugate variables  and S cos  System Hamiltonian MAGNON SPECTRUMVORTEX PROFILE Berry’s phase:

6. Particle description of a vortex vortex → charged particle in a magnetic field vorticity q ~ charge polarization p ~ perpendicular magnetic field inter-vortex force → 2D Coulomb force: fixes particle charge = gyrotropic force → Lorentz force: fixes magnetic field, Promote vortex center X to dynamical variable → effective equations of motion → (in SI units) B → FMFM → FMFM → FCFC → BC’s…

7. Vortex-magnon interactions Add fluctuations about vortex configuration Introduce fourier decomposition of magnons: Integrate out spatial dependence: Magnus force, inter- vortex force, perturbed magnon eom’s, vortex-magnon coupling  first order velocity coupling ~  X  k  second (+ higher) order magnon couplings (no first order!) Gapped vs ungapped systems: velocity coupling is ineffective for gapped systems (conservation of energy) → higher order couplings must be considered – aren’t here Stamp, Phys. Rev. Lett. 66, 2802 (1991); Dubé & Stamp, J. Low Temp. Phys. 110, 779 (1998).

8. Quantum Brownian motion Feynman & Vernon, Ann. Phys. 24, 118 (1963); Caldeira & Leggett, Physica A 121, 587 (1983) quantum Ohmic dissipation classical Ohmic dissipation damping coefffluctuating force Specify quantum system by the density matrix  (x,y) as a path integral. Average over the fluctuating force (assuming a Gaussian distribution):

9. Consider terms in the effective action coupling forward and backward paths in the path integral expression for  (x,y): Then, defining new variables: Introduces damping forces, opposing X and along .. → normal damping for classical motion along X → spread in particle “width”, x 0 ~ X Such damping/fluctuating force correlator result from coupling particle x with an Ohmic bath of SHO’s with linear coupling:

10. Brownian motion of a vortex vortex and magnons arise from the same spin system → no first order X  coupling can have a first order V  coupling Rajaraman, Solitons and Instantons: An intro to solitons and instantons in QFT (1982); Castro Neto & Caldeira, Phys. Rev. B56, 4037 (1993) Path integration of magnons result in modified quantum Brownian motion: instead of a frequency shift (~  x 2 ), introduce inertial energy → defines vortex mass! ½ M v X 2 must integrate by parts to get XY – YX damping terms: changes the spectral function (frequency weighting of damping/force correlator not Ohmic → history dependent damping!...

11. Vortex influence functional Extended profile of vortices makes motion non-diagonal in vortex positions, eg. vortex mass tensor: History dependant damping tensors: In the limit of a very slowly moving vortex, mismatch between cos and Bessel arguments:   loses history dependence

12. Many-vortex equations of motion Extremize the action in terms of Setting  i = 0 (a valid solution), then x i (t) satisfies: xi(t)xi(t) xi(s)xi(s) || damping force refl damping force || refl xj(s)xj(s)

13. Special case: circular motion Independent of precise details, for vortex velocity coupling via the Berry’s phase: F damping (t) =  ds  ║ (s) +  refl (s) X(t) X(s 1 ) X(s 2 )...  refl (s 1 )  refl (s 2 ) F damping Damping forces conspire to lie exactly opposing current motion No transverse damping force!

14. Results/conclusions/yet to come… damping forces are temperature independent: hard to extract from observed vortex motion What about higher order couplings?  May introduce temperature dependence  May have more dominant contributions! vortex lattice “phonon” modes… Changes for systems in which Berry’s phase ~ (d  /dt) 2 ?