Transverse force on a magnetic vortex

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Transverse force on a magnetic vortex

• Lara Thompson
• PhD student of P.C.E. Stamp
• University of British Columbia
• July 31, 2006

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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

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Topic of this talk!
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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. (Berrys 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
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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

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Spin System magnons vortices
System Hamiltonian

• use spherical coordinates (S,??, ?)
• with conjugate variables ? and S cos?

Berrys phase
MAGNON SPECTRUM
VORTEX PROFILE
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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,
BCs
(in SI units)
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Vortex-magnon interactions

• Add fluctuations about vortex configuration
• Introduce fourier decomposition of magnons
• Integrate out spatial dependence Magnus force,
  inter-vortex force, perturbed magnon eoms,
  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 arent here

.
Stamp, Phys. Rev. Lett. 66, 2802 (1991) Dubé
Stamp, J. Low Temp. Phys. 110, 779 (1998)
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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)
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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 lt(x-x0)2gt, x0 X
Such damping/fluctuating force correlator result
from coupling particle x with an Ohmic bath of
SHOs with linear coupling
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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)
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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
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Many-vortex equations of motion

• Extremize the action in terms of
• Setting ??i 0 (a valid solution), then xi(t)
  satisfies

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Special case circular motion

• Independent of precise details, for vortex
  velocity coupling via the Berrys phase
• Fdamping(t) ?ds ?(s) ?refl(s)

Damping forces conspire to lie exactly opposing
current motion
No transverse damping force!
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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 Berrys phase
  (d?/dt)2 ?