PCE STAMP

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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
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Q. VORTICES ARE EVERYWHERE
RIGHT Vortices vortex rings in He-4 BELOW
pulsar, structure of its vortex lattice
ABOVE vortices penetrating a superconductor RIGH
T Vortices in He-3 A
Conjectured structure of cosmic string, of a
cosmic tangle of These in early universe
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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

Arises from Berry phase
Transverse force from quasiparticles scattering
off vortex
Iordanskii force
Longitudinal force from quasiparticles
scattering off vortex
Drag force
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Topological Solitons in MAGNETS
SOLID 3He
There are many of these. Here are 2 examples
SOLITONS in 2D MAGNETS
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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
(Berry phase)
where
VORTEX PROFILE
MAGNON SPECTRUM
Core Radius
Spin Wave velocity
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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
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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
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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
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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 equation.
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 complicated 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
• 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.

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RESULTS for VORTEX DYNAMICS
The top inset shows the necessary ? of Ohmic
damping to fit full simulated motion. Note the
strong upturn at low speeds!
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)
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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
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VORTICES IN A BOSE SUPERFLUID
Lets 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
Magnus term
with
and also
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Fourier transform
where
and
Equation of motion
Influence functional
Defining
We get the equation of motion for the com
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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.
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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
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