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VORTICES in
SUPERFLUIDS & SUPERCONDUCTORS
CIFAR Q MATERIALS SUMMER SCHOOL
(May 14-16, 2012)
LECTURE 2
VORTICES
Quantum Vortices in Superfluids
Suppose we look at a vortex in a superfluid- ie., fluid circulating around a
core. From what we know about atoms (ie., from de Broglie) this tells us
we have probability waves circulating round the core with wavelength
λ = h/p = h/mv where v is the velocity of the particles circulating around the
core. But then, as noted by Onsager in 1950, as in atoms, only certain
L Onsager
(1903-1976)
velocities are allowed, if we are to fit the waves around the core. Hence we
find that the total circulation is quantized- we have ‘quantized vortices’.
In this simple picture the core is like a string, in which the superfluid
density goes to zero at the origin - the core has a finite diameter, because
the singularity in the phase cannot persist in bulk superfluid. In He-4 this
diameter is very small (only about 1 Angstrom!), but in other superfluids
like He-3 it is much larger (~150 Angstroms), & so the core is itself very
RP Feynman
complex.
(1920-1987)
The vortices themselves are quantum excitations- so they also have a
probability density! They have fascinating properties, many of which were first discussed
by Feynman in the early 1950’s, as fully quantum-mechanical objects. We now know that
most of the flow properties of He superfluid are governed
by the vortices in them, which can form very complex
patterns. They can form closed ‘vortex rings’, which are
also quantum objects, and which can tunnel and form
state superpositions. The macroscopic properties of the
superfluid are typically determined by vast ‘vortex
A quantized vortex ring
tangles’ of intertwined vortex loops.
VORTICES: The
EXPERIMENTAL PROBLEM
In recent years experiments have been
done on a variety of rotating systems,
where the number of vortices can be
controlled (in superconductors, the
same is done using an external flux)
SUPERFLUIDS: ions,
or H atoms (the tools
used to see the
vortices) completely change their
dynamics.
SUPERCONDUCTORS: Vortices interact
with each other (to form
lattices), and also very
strongly with defects
(which pin them). This
also completely changes
their dynamics
PROBLEM:
how to do experiments
on individual vortices
without changing their
dynamics
COLD BEC GASES: No problem with
defects or ions. But so far very hard
to actually manipulate individual
vortices.
OBSERVING SUPERFLUID VORTICES
The first indirect observation of a quantum vortex
was in 1956, in a famous experiment by Vinen – a
vortex attached to a vibrating wire altered its
dynamics (see RIGHT).
Later experiments imaged
vortices in rotating
cylinders by allowing
electrons to attach to
them (a hydrodynamic
effect) and then sucking
them off with an electric field. However the electrons form a
large ‘bubble’ around themselves in the superfluid, which has a
very large hydrodynamic effective mass, and this radically
alters the vortex dynamics.
More recent imaging
experiments attach H2
molecules to the vortices,
and the H trackers are
then observed by tuning
a laser to one of the H
spectral lines
PROBING VORTEX DYNAMICS
Vortex nucleation (He-4 & He-3)
The tunneling rate &
dynamics of vortex rings &
other vortex configurations
are influenced by the
quasiparticles in interesting ways.
PC Hendry et al., PRL 60, 604 (1988)
Turbulence
This is a much more complicated problem, since it can in principle involve very fast
motions of the vortex lines, and it is intrinsically 3-dimensional and multi-vortex in
nature.
P Walmsley et al., PRL 99, 265302 (2007)
P Walmsley, A Golov, PRL 100, 245301 (2008)
VORTICES IN ROTATING COLD GASES
It is actually much easier to image vortices in cold gases
without disturbing the vortex dynamics. Some remarkable
experiments have looked not only at the dynamics and
different phases of vortex lattices in these systems, but
also at the nucleation and dynamics of single vortices in
both 3d and even 2d BECs.
Z Hadzibabic et al.,
Nature 441, 1118 (2006)
DESCRIPTION of a BOSE SUPERFLUID VORTEX
For a stationary vortex we assume:
Then, since
we get quantized circulation:
Velocity field falls off like
classical vortex. Density
profile depends on form of
energy functional
The vortex inertial mass is
given in terms of the energy by:
The vortex energy, obtained by integrating
energy functional over entire volume,
depends on sample shape (cf. method of
images at right)
Eg., for cylinder
VORTEX in the GRAVITATIONAL ANALOGY
If we describe a vortex in the GR language, we get an
interesting ‘cosmic string’ situation: spacetime has to be ‘cut’,
then repasted together, with a time jump across the cut.
for string with
where
Time jump:
angular momentum J
Outside the vortex core,
the metric is then flat:
constant phase ‘eikonal’ plot
M Stone, Phys Rev B61, 11780 (2000)
There is actually a lot more interesting physics to be gained from this analogy.The
usual ‘gravitational bending of light’ (ie., of phonons) doesn’t happen here. Instead,
phonons are deflected in the same sense, no matter
which side of the vortex core they pass.
One can also ask about the precession of a ‘gyroscope’
attached to a phonon. We expect 2 contributions – a
‘de Sitter’ geodetic contribution, and a ‘Lense-Thirring’
frame dragging contribution.
Both should be present
here; the AdS term
comes from the
acceleration of the
phonon, and the ALT term
from the interaction with
the vortex angular momentum
2-FLUID PHENOMENOLOGY INCLUDING VORTICES
STANDARD “HVI” PHENOMENOLOGY:
add the Magnus force:
+ quasiparticle force:
let
This gives MUTUAL FRICTION:
where
These equations are phenomenological – as we shall see they have been very
controversial. If one accepts the general form they can be applied equally to both
superfluids and superconductors (one simply has to find the coefficients in each case).
REFS:
GENERALIZED 2-FLUID EQUATIONS
By balancing forces between the normal fluid quasiparticles, the superfluid, and the
vortices (taken to have some coarse-grained distribution) we then get the following
force equations:
VORTICES in SUPERFLUID 3He
The diversity of vortices in the various phases is
so large that whole books have been written about
them. Experiments use rotating cryostats, and the
vortices can be observed in various ways
(notably NMR). The core size is ~ 15 nm at T=0. A
2-fluid phenomenology can also be developed.
Some Examples (of a very complicated topic)
Quantum Vortices in
Superconductors
Superconductivity is a condensation
of pairs of electrons, all into a single
state. If we try to disturb this quantum
state by applying an external magnetic
field, the supercurrents in the system,
AA Abrikosov
flowing without resistance, simply
(1928- )
adjust to block the field from entering
the superconductor (the ‘Meissner effect’). However, as
shown by Abrikosov in1957, in some materials the field can
get in via vortices, like those in superfluids- again, the
circulating current is quantised.
If we have a loop of
superconducting material
we can trap magnetic flux
inside it- this is kept out
of the superconductor by
currents in it, as before.
Again, the circulating
current is quantised, and
thus so is the flux- in units
Magnetic Field through
of a flux quantum h/e
superconducting ring
PCES 5.55
TOP: magnetic field lines around
a superconductor
MIDDLE: vortices penetrate
BOTTOM: close-up of vortex in
superconductor
OBSERVING SUPERCONDUCTING VORTICES
The analogue of the Hess-Fairbank in a superconductor is the Meissner effect – an
equilibrium effect. One can also of course have metastable supercurrents set up –
this is the basis of superconducting technology – which also relies on the pinning of
vortices in a finite supercurrent.
Meissner effect in type-I superconductor
Meissner effect in type-II superconductor
Vortex Imaging using Aharonov-Bohm
effect with electrons
SUPERCONDUCTING VORTEX DYNAMICS
We can set up a set of equations analogous to the HVI equations for a neutral
superfluid, by looking at the forces on the vortices in a classical analysis. These
forces are then as follows:
Force from superfluid
Force from normal fluid
(Lorentz force)
(quasiparticle drag)
Now we can always write the current in the form
The two are related by:
Longitudinal
‘Ohmic’ current
Transverse
‘Hall’ current
But this is exactly the same as the HVI ‘mutual friction’ equations. To see this
just rewrite the above in the form:
in which
NEUTRON STAR DYNAMICS
This is observed in the famous ‘glitches’ that exist in the
rotation rate. These are caused by vortex depinning.
Vortices are pinned in the
solid crust – but the slow
spin-down of the star
causes the vortex rotation
to go out of equilibrium
with the internal
superfluid rotation.
Q. VORTICES in MAGNETS
A vortex-like topological excitation can
exist in a 2-d ordered spin system, or in
a thin film.
These vortices can be imaged in a
variety of ways (magneto-optical,
or using MFM techniques.
In this way one can also watch the
dynamics of individual magnetic
vortices under external influences.
TYPICAL EXPERIMENTAL SYSTEM:
PERMALLOY
A well-controlled system where many experiments
on vortices have been done is permalloy, Ni80Fe20,
which has very small (quartic) magnetoelastic
coupling. The vortex dynamics is very important
for hard drive technology, and is described using
the phenomenological eqtn:
We will be interested in a disc
geometry (see above for others). Key
parameters:
Magnon velocity:
Exchange length:
where:
so that:
Quantum Vortex in 2D Easy-plane Ferromagnet
Hamiltonia
+
Continuum
The action is:
where
The vortex is a ‘skyrmion’, with profile:
Vortex core radius
MAGNON
with Spin Wave velocity:
with
(Berry