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

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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 ?