PCE STAMPPhysics & AstronomyUBCVancouverPacific Institute for Theoretical PhysicsWhat is the EQUATION of MOTION of a QUANTUM VORTEX?

Q. VORTICES ARE EVERYWHERERIGHT: Vortices & vortex rings in He-4BELOW: pulsar, & structure of its vortex latticeABOVE: vortices penetrating a superconductorRIGHT: Vortices in He-3 AConjectured structure of cosmic string, & of a cosmic tangle of These in early universe

FORCES on a QUANTUM VORTEXFor the last 40 years there has been a very strenuous debate going on about the form of the equation of motion for a quantum vortex, focusing in particular on (i) what are the dissipative forces acting on it (ii) what is its effective mass

Quite incredibly, the fundamental question of quantum vortex dynamics is still highly controversialThe discussion is typically framed in terms of the forces acting on a vortex; the following terms are discussed:Magnus force:=Iordanskii force: Drag force: Arises from Berry phaseTransverse force from quasiparticles scattering off vortexLongitudinal force from quasiparticles scattering off vortex

Topological Solitons in MAGNETSSOLITONS in 2D MAGNETSSOLID 3HeThere are many of these. Here are 2 examples:

Part (b) Quantum Vortex in 2D Easy-plane FerromagnetLattice HamiltonianContinuum LimitThe action is:(Berry phase)VORTEX PROFILECore RadiusMAGNON SPECTRUMSpin Wave velocityL. Thompson, PCE Stamp, to be publishedL Thompson, MSc thesis (UBC)where

MAGNETIC VORTEX DYNAMICSSUMMARY of RESULTSLEFT: Profile of a moving vortexRIGHT: Difference between moving & stationary vortexLEFT: Magnus forces on a vortex this is a Berry phase effectBELOW: Forces on a moving vortexBELOW: remarkable circular dynamics of a magnetic vortexHowever, the forces on a vortex are actually very complicated the main question is to know what they are

Dynamics OF THE MAGNETIC QUANTUM VORTEXFrom the Berry phase one immediately recovers the gyrotropic Magnus force:Density matrix propagator(1) MAGNUS FORCE TERM(2) PATH INTEGRAL FORMULATION VELOCITY EXPANSIONRecall that we can always formulate the dynamics for the reduced density matrix aswhere However we are NOT now going to do the usual Caldeira-Leggett trick of assuming a coupling between vortex and magnons which is linear in the magnon variables. As mentioned above, this is not even true for a soliton coupled to its environment. What we need is another expansion parameter, and there is one if the vortex moves slowly we can expand the coupling in powers of the VORTEX VELOCITY.If we do this we get a result for the effective Lagrangian of the system, given byLagrangian for Moving VortexLagrangian for magnons coupled to static solitonLinear velocity coupling between magnons and vortexwhich we can now use to derive an influence functional

Effective couplingINFLUENCE FUNCTIONALEffective bath propagator:From the Lagrangian one findsan influence functional of form:We begin with the phase term then we can derive equations of motion for the 2 coupled paths, which are best written in the variablesThen, in addition to the Magnus force, we find another force acting on the vortex, given bywherewith frictional termsThe definition of the reflection direction is shown we reflect the velocity vector at time t about the vector connecting the present position with the earlier position. Thus the force contains a memory of the previous path traced by the vortexNow we can always write the influence functional in the formPHASE TERM

Thus the real dynamics of a vortex, magnetic or otherwise, has both reactive and dissipative terms that are more complex than those that have been discussed so far. There is definitely a transverse dissipative force having the symmetry of the Iordanskii term, but it is now part of a more complicated time-varying term with memory whose size and form depends on the previous path of the vortex We also have an imaginary term in the influence functional which can be thought of as supplying a quantum noise term in the coupled dynamics of the 2 paths. CONCLUSIONS for Dynamics of a SINGLE MAGNETIC VORTEXThere is no reason whatsoever to exclude transverse dissipative forces. In fact they are much more complex than previously understood(2) The equations of motion for an assembly of vortices involve all sorts of forces (non-local in time and space) that have not previously been studied. This gives a quantum noise term on the right hand-side of a Quantum Langevin equation. However the noise is not only non-Markovian (highly coloured in fact) but also non-local.DECOHERENCE FUNCTIONAL

If we set the vortex into motion with a -kick, we find decaying spiral motion dependent on the initial vortex speed (shown in fractions of v0 = c/rv)The top inset shows the necessary of Ohmic damping to fit full simulated motion. Note the strong upturn at low speeds!RESULTS for VORTEX DYNAMICS

EXPERIMENTS on VORTICES in MAGNETS?The experimental techniques exist already to test these predictions.

It should be very interesting to check them out at low T

VORTICES IN A BOSE SUPERFLUIDLets assume a somewhat simplified model Bose superfluid, with the actionWhere we define a small fluctuation field by and a vortex field byWhere the bare vortex field isWe now separate off the vortex from the fluctuations; defineThen we havewithand alsoMagnus term

Fourier transformwhereandEquation of motionInfluence functionalDefiningWe get the equation of motion for the com:

Total PhaseSo that, eg., the longitudinal Phase terms are, etc.DYNAMICS OF VORTEX ASSEMBLYWe can generalise all this theory to the assembly and find the dynamics. The phase and damping/decoherence terms are more complex, but manageable. For example:whereandetc.Using these equations one can solve for the dynamics of an assembly of vortices, finding the spectrum of collective modes, etc. This takes us too far from this course.

Multi-vortex damping/noise term:with propagator:Assembly of Vortices:Which implies a NON-LOCAL MASS:(2) Mixed memory term:(3) Transverse Damping term: whereConsider now an assembly of magnetic vortices, so that

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