transverse force on a magnetic vortex lara thompson phd student of p.c.e. stamp university of...
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Transverse force on a magnetic vortex
Lara ThompsonLara ThompsonPhD student of P.C.E. StampPhD student of P.C.E. Stamp
University of British ColumbiaUniversity of British Columbia
July 31, 2006July 31, 2006
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!Topic of this talk!
Equations of motion controversy
Superfluid/superconductor vortices: vortex effective mass
Estimates range from ~rv2 to order of Ev /v0
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…
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”
Spin System: magnons & vortices
use spherical coordinates (S,, )
with conjugate variables and S cos
System Hamiltonian
MAGNON SPECTRUM
VORTEX PROFILE
Berry’s phase:
Particle description of a vortex
vortex → charged particle in a magnetic field
vorticity q ~ chargepolarization 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→
FM
→FM
→
FC
→
BC’s…
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 ~ Xk
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)
.
Quantum Brownian motion
Feynman & Vernon, Ann. Phys. 24, 118 (1963); Caldeira & Leggett, Physica A 121, 587 (1983)
quantum Ohmic dissipationclassical Ohmic dissipation
damping coeff fluctuating force
Specify quantum system by the density matrix (x,y) as a path integral.Average over the fluctuating force (assuming a Gaussian distribution):
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 linear coupling:
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 (~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!
.
. .
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
Many-vortex equations of motion
Extremize the action in terms ofSetting i = 0 (a valid solution), then xi(t) satisfies:
xi(t)
xi(s) || damping force
refl damping force||
refl
xj(s)
Special case: circular motion
Independent of precise details, for vortex velocity coupling via the Berry’s phase:
Fdamping(t) = ds ║(s) + refl(s)
X(t)
X(t)
X(s1)
X(s2)
.
.
.
refl(s1)
refl(s2)
Fdamping
Damping forces conspire to lie exactly opposing current motion
No transverse damping force!
Results/conclusions/yet to come…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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