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PCE STAMP
What is the EQUATION of MOTION
of a QUANTUM VORTEX?
Physics & Astronomy
UBC
Vancouver
Pacific Institute
for
Theoretical Physics
Q. VORTICES ARE EVERYWHERE
RIGHT: Vortices & vortex rings in He-4
BELOW: pulsar, & structure of its
vortex lattice
ABOVE: vortices penetrating
a superconductor
RIGHT: Vortices in He-3 A
Conjectured structure of cosmic
string, & of a ‘cosmic tangle’ of
These in early universe
FORCES on a QUANTUM VORTEX
For 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 controversial
The 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 phase
Transverse force from quasiparticles
scattering off vortex
Longitudinal force from quasiparticles
scattering off vortex
Topological Solitons in MAGNETS
There are many of these. Here are 2 examples:
SOLITONS in 2D MAGNETS
SOLID
3He
Part (b) Quantum Vortex in 2D Easy-plane Ferromagnet
L. Thompson, PCE Stamp, to be published
L Thompson, MSc thesis (UBC)
Lattice Hamiltonian
Continuum Limit
The action is:
where
(Berry phase)
VORTEX PROFILE
MAGNON SPECTRUM
Core Radius
Spin Wave velocity
MAGNETIC VORTEX DYNAMICS
SUMMARY of RESULTS
LEFT: Profile of a moving
vortex
RIGHT: Difference between
moving & stationary vortex
LEFT: Magnus forces on a
vortex – this is a Berry phase
effect
BELOW: remarkable circular
dynamics of a magnetic vortex
However, the forces on a vortex are actually very
complicated – the main question is to know what
they are
BELOW: Forces on a moving vortex
Dynamics OF THE MAGNETIC QUANTUM VORTEX
(1) MAGNUS FORCE TERM
From the Berry phase one immediately recovers the gyrotropic “Magnus” force:
(2) PATH INTEGRAL FORMULATION – VELOCITY EXPANSION
Recall that we can always formulate the dynamics for the reduced density matrix as
Density matrix propagator
where
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 by
which we can now
use to derive an
influence functional
Lagrangian for
Moving Vortex
Lagrangian for magnons
coupled to static soliton
Linear velocity coupling
between magnons and vortex
INFLUENCE FUNCTIONAL
From the Lagrangian one finds
an influence functional of form:
Effective coupling
Effective bath propagator:
PHASE TERM
Now we can always write the influence functional in the 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 variables
Then, in addition to the Magnus force, we find another force acting on the vortex,
given by
where
with frictional terms
The 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 vortex
DECOHERENCE FUNCTIONAL
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.
This gives a “quantum noise” term on the right hand-side of a Quantum Langevin equati
However the noise is not only non-Markovian (highly coloured in fact) but also non-local.
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 complicate
time-varying term with memory whose size and form depends on the previous path of the vortex
CONCLUSIONS for Dynamics of a SINGLE MAGNETIC VORTEX
(1) There 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.
RESULTS for VORTEX DYNAMICS
simulated vortex motion
motion with Ohmic damping
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!
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 SUPERFLUID
Let’s assume a somewhat simplified model Bose superfluid, with the action
Where we define a small fluctuation field by
and a vortex field by
Where the ‘bare’ vortex field is
We now separate off the vortex from the fluctuations; define
Then we have
with
and also
‘Magnus’ term
Fourier transform
where
Equation of motion
Influence functional
Defining
We get the equation of
motion for the com:
and
DYNAMICS OF VORTEX ASSEMBLY
We 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:
Total Phase
So that, eg., the longitudinal Phase terms are
where
, etc.
and
etc.
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.
Assembly of Vortices:
Consider now an
assembly of
magnetic vortices,
so that
Which implies a NON-LOCAL MASS:
where
(2) ‘Mixed’ memory term:
(3) Transverse Damping term:
Multi-vortex damping/noise term:
with propagator:
see
http://pitp.physics.ubc.ca/