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Transverse force on a magnetic vortex

Transverse force on a magnetic vortex. Lara Thompson PhD student of P.C.E. Stamp University of British Columbia July 31, 2006. Vortices in many systems. Classical fluids Magnus force, inter-vortex force Superfluids, superconductors Inter-vortex force

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Transverse force on a magnetic vortex

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  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 ~rv2 to order of Ev /v02 • 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 Berry’s phase: MAGNON SPECTRUM VORTEX PROFILE

  6. → • Promote vortex center X to dynamical variable • → effective equations of motion → B → → → FM FM FC 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, BC’s… (in SI units)

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

  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-x0)2>, x0 ~ X Such damping/fluctuating force correlator result from coupling particle x with an Ohmic bath of SHO’s with linearcoupling:

  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 Path integration of magnons result in modified quantum Brownian motion: • instead of a frequency shift (~x2), introduce inertial energy → defines vortex mass! ½ MvX2 • 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! . . . Rajaraman, Solitons and Instantons: An intro to solitons and instantons in QFT (1982); Castro Neto & Caldeira, Phys. Rev. B56, 4037 (1993)

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

  13. . X(s1) refl(s1) . X(t) refl(s2) X(t) . X(s2) Fdamping Special case: circular motion Independent of precise details, for vortex velocity coupling via the Berry’s phase: Fdamping(t) = ds ║(s)+ refl(s) 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 ?

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