Speak last.
Night that I made a bet.
Of fourteen years.
If you're fortunate enough to us to
perform karaoke and you're a terrible
singer then the best you can hope for is
that the person before you is even worse
than you are but unfortunately a falling
rain became me and that's impossible.
So you have to take inspiration
from him so I did that
looks at the at the same no he
changed the color of his shirt but.
Randy is always moving forward these
are the things I'm going to have today
he's a singular sensation remember
he had those singularities
those A defects that Steven Strogatz
is a great name for top logical D.
flicks the fix and he's not flat
he also likes to ask a lot of
questions so Randi who is that you know.
Now.
I actually do not.
Know.
German.
In the other guesses that's Mobius OK so.
This is well done with search.
Very good.
They give you half a point.
Yes this work down with my students
actually is in the works with both
Christina market and myself and what
there's lots of things as well there's
the support we've had for this work.
There are basically ingredients in
the work I'm going to talk about today at
least the first part I want to get at
the end if I can to some brand new
work an act of the medics that came
out on the archive last night.
But the two ingredients are dynamical
symmetry breaking a dynamic overjoyed and
of spontaneous symmetry breaking might
be one of the greatest ideas of the last
century of physics and spatial curvature
that is basically all I need and
associative phenomena to give you.
A logical sound so some really kind
of some pole but interesting element
example the top logical made of
material a softer version of
the soul of town coupled to affirming or
atop a logical insulator OK so.
So will be playing around with things
Victorian things OK things are the head
spins flying spins they're moving
forward so I'd be looking a lot
what's various kinds of
Victor fields lately so
here's a koan you can get of the earth so
these are winds on the surface of the.
So there are all kinds of the defects that
Randy was talking about there's I'm a mere
exist so these avoiders cs is
a plus ones minus ones and
so on Cyclons empty Cyclons both
plus ones as Randy was describing.
I've also been sailing since and
send a virus of these a lot
than try to avoid these.
So beautiful map examples you can
find everywhere a victim fields.
There are also there are so
many call examples from morphogenesis
in the biological field so
these are the Thiel cells from developing
fruit fly from Sebastian strike and
group at U.C.S.B.
work of Barsh Raman these
the surface of the developing
fruit fly in embryo annual So
you're tracking these cells and
you can look at the velocity field and
their positional order and maybe even
a matter color maybe Smith order and
these systems defects develop here and
effects having no singular points Randy
was talking about a great sites for
biological functional as ation of some
disorder something's going to happen they
think there's a natural place for
it to happen misery loves company.
So here we go we're going to look at.
A model of flying spends this is more
than twenty years old now developed by
John Palmer and you're into going
to look at these flying spins.
Victor fields anything with a here and.
Then on the curved surface and
in particular I'm going to discuss here
the simplest kind of thing which is
a uniform positively the surface
of a ball a two sphere.
We've also looked at the case of negative
coverage of things the curvature
doesn't really matter too much
as long as you have curvature
alist take constant positive curvature so
there's going to be a day into the field
these things are going to be compressed
of all and there's going to be
an order parameter in this case and
magnetize ation order parameter.
Or pole color to polarize ation and
that is going to be ideal for these things
are moving going to spontaneously move
that's a dynamical cemetery breaking they
can move in the they have some consuming
some energy they can locally they can move
anywhere but they spontaneously
align when they're moving
forward at some lossy that's a dynamical
cemetery breaking it's not put
in by hand it emerges somebody because
the thing is active consuming energy so
the key thing here is that the order
parameter is identified was
the Valar Citi Field these two things
are the same and that will be crucial so
you can you have the conservation
law standard conserved current and
you have an equation of motion.
For this polarization field and
the second and on the right hand side
here this is actually all our need
is your five fourth like
potential it's going to spawn.
Tamia sleeve developing minima
it non-zero values of the order parameter
there's going to be a lend our vision like
phase transition that drives
the dynamical symmetry breaking now.
So here's the dynamical symmetry
breaking here this potential here
the stone can these kind of opposite
side driving the symmetry breaking.
Here is the key thing
about spatial curvature.
You have to have any activity you're
going to have an air victor of term
here because things are moving so
things can change because they change and
things can change because you move to
a different place on the system so
you're required to have this
Ed Victor derivative term and
they're big Whenever you have derivatives
of surfaces there's a basically a third
ingredient here those derivatives have to
be covariant derivative because you have
to parallel transport from one place
to another on a curved surface.
So this drives the pole up or
down or vector like or
that this is the activity with
the ED victor of the river turn and
then you have you noticed you have
made the choose the things and
those have to be calculated with
speak to a curved metric and
you have derivatives as I said as I
said they have to be calculated for
curved metric in general you will have
various kinds of dissipate viscosity
we won't need that here notice that this
viscous term also has a contribution from
the curvature and the extra term when you
drive now the A starts on a curved surface
because you have to commute covariant
derivative and they don't commute
they give the gas in curvature terms so
it's a kind of interesting term.
But I see some implications of this but
that's nonzero viscosity wrongly that we
won't discuss it here and
the system is compressible so
there's this compressibility modulus
the one OK so you can easily.
Right there on the mic truck
on the on the two sphere and
notice that it has the sign
squared theta term later is the.
Latitude like coordinate going this
way and fires and there's a mode for
coordinate and there's a non-trivial
factor here in the metric sine squared.
Gassing curvature of the constant and
I'm going to take this as
a multiple cemetery case OK so
you automatically you have a conserved
current in that case because of the sum of
trees there is only one symmetry direction
one so-called calling that because you
have the symmetry moving around in
the as a move for direction and so the.
The connection call for
shows that you need to compute
on the curve service that modify
the derivative term there particularly
some pole because nothing is changing
in the fire direction so you only have tow
nonzero ones and they depend on fate or.
The.
Third and the of cotangent
pay the just the river toes
of this term of the spec to
favor that is all you need.
OK.
Where's the potential there for
the right sign above some critical
density the the quadratic term here
changes side these two things her this
term and that term which is the quadratic
in the quarter term in the free energy
there's a linear term and a cubic
term in the equation of motion they will
have opposite sign to the density above
the critical didn't study that's
going to be the condition for
the thing to stop these flights flying
spins to start locking together.
So we're going to take this
case above the critical D.N.C.
You have non-zero flocking
at these minima in this.
As you go deeper and
deeper into the flocking ordered phase
this minimal get deeper and
it will move out.
Now.
Much like Randy was saying.
The system.
Has various kinds of defects and
then immediately if you take flocking
on the skirt surface it cannot be uniform
immediately once you have direct the it's
like a band of flocking once you
met these flocking States on to.
It's going to have problems when
it gets near the poles cannot.
Cannot uniformly flock near the poles.
So it's automatically
spatially in homogeneous
rather than uniform somebody by
wrecking it on the sphere so
that's a nice ingredient these are the
kind of defects you have except we have
particle number conservation so
we can have these sink like the Hicks.
You can only have these
ones which are vortices So
we have no sinks or monopoles sinks or
saucers we only have
these kinds of de fix this an example
of one defect on the sphere.
A single plus two D.
fit on this here which is a dipole.
That satisfies the top logical
constraints but more typically for
energetic reasons you get past ones and
we'll only have these ones as
a particle number conservation.
So that's nice consequence of having
to deal with spatial and homogeneous.
Immediately because of the change so
let's look at this polar
clock on the sphere so
the first thing that we have to deal with
is actually finding a steady state
solution because that's no longer trivial
as I said because of the space of a moment
of unity so that was our first challenge.
And indeed we found that there is
an analytic steady state solution.
And the key thing is that it has
to it's depending on face charge
this is a kind of picture
of this flocking state and
you see it's concentrated Here's
the profile this is the equator
here it's concentrated on the equator
unmodulated itself to go to zero.
At the poles here and
as you go this is weakly ordered
this is an increasing order and this is
increasing order here as you increase
the order it focuses down on
a band around the equator this is
the equator here is a special point
because it's a geodesic on the surface so
what your record on the two sphere
is going to be a geodesic path.
Minimal path and that will form
your equator just spontaneously
chosen I'll call that the equator and then
the system will be all flop concentrated.
Around the equator by the way this
is another way physical systems
in chemical and
mathematical can deal with singularities
instead of having an actual singularity
they can do geometry they can excise
the region around a defect the method
Titian's do the surgery so
you can have the feel that you're
talking about have no support on
the region around the defect just cut it
up so you kind of see that here the street
is like cut out and instead of having the
hole to sphere you have an angular band.
So that's kind of call The shows
up on this profile and
if this were morphogenesis like say
a thing or any kind of setting where you
have this the set up you have a natural
place here to attach things or to do stuff
the system is either isotropic
there because of the defect or
there's nothing there there
are actually two holes there.
Spontaneously in the sense here changes
it's topology if it does that it says.
Anyway this dependence on the angular this
power coordinate theta is interesting for
me it's a power of theta with
an experiment that's governed by
the compressibility modulus and
the depth of the Mexican
potential so it's dependent on the degree
of some a tree breaking as I was
telling you here and it has
the factor that to compare it appears
in front of the addict of derivative
because the system is not Galileo and so
this lambda doesn't have to be one or one
of the density it won't be important here
as a non-trivial exponent here
that comes out of this solution.
OK so.
This system's also been studied
prior to this numerically
by rest of us can it make and still Hank
us by looking at Vector like particles.
To say yeah here's the simulation
you see the alignment and
you see it for and nothing to the T.V.
you see these bands forming and
here they really have form holes.
This band is some noise there's some
rotation of the North is the fusion
it wanders around a little bit twists and
turns which is also interesting.
It's very dynamic call
object all driven as
I said by the dynamical symmetry breaking
the flying's And by the curvature.
Now.
What I want to look at in the system
is not just this ordering but
what happens when you perturb that.
Particular it's compressible so
we can look at the density fluctuation
where they are at this point
are there any questions yes.
Yeah the symmetry breaking is driven
by activity which could be noise or
any local consumption of energy
doesn't matter how you break it
you have to dynamically break it.
You.
Know so
here the external field has to be local It
has to be level of the individual units
it's not coming from outside otherwise
you don't have local spontaneous motion.
Or the noise has to be distributed
across the whole system.
So that's a key point here these systems
are not driven by some external field
Yeah but question in the others
OK So
let's look at sound modes in the system.
You have a day into the fluctuation
which is primarily almost
as a Mosel direction because that's
where the thing is concentrated and
you have some extra braking so
you have a gold star mode.
You will have two kinds of in general two
kinds of sound modes that in this long
which are not as the move for
flocking direction around here or.
Look here in this direction of this
band and then you'll have because of
the symmetry breaking lever transverse
Goldstone sound mode there's
another key feature of the system degrees
of freedom that are acting transversely.
The.
Along with the fast agree a freedom
rapidly decays has some finite.
Frequency and you can integrate
it out we're going to be looking
at the long wavelength softer
modes that dominate the system so
we integrate out the longer sound mode and
we have a density fluctuation in the
transverse sound mode first two degrees of
freedom we're going to look at so
right this says a vector I mean there's a.
Two column vector with these two modes and
then right it's equation of
motion that I gave before as.
In the form of the.
Schroedinger Equation put in I and here.
So on the right hand side
we have this matrix of Q.
that comes from that equation
of motion at the on the So
there's a two by two matrix dynamical
matrix governing the evolution
of the two components system so
this is what it looks like.
There's a parameter which is
just the sine multiplying along.
A momentum way Victor Q.
of X.
is the X.
is a lonely as the most all directions
just the relabeling wires along the theta
direction to make it look tatty Asian
even though it's not to have Q.
why you have complex numbers
in here because you have
derivatives give you an I and
the other terms Darrent So
this gives you a cube plus i
times M M tells you where you
are which led it to you are in the text
of the cotangent from those derivatives
of connection cooperation cotangent
of the latitude angle favors zero.
Notice that this M.
vanishes pile of the two which is the
equator and it's negative in the northern
hemisphere and positive in
the southern hemisphere what's going
to matters place just the sign of this
call for him and it's vanishing at.
Equator new is just another parameter
controlling the symmetry breaking and
the alignment and given by the lead which
has the compressibility module as three
scaled and this is zero here so
you can simply work out the spec
dispersion relation of the system
it has a linear term purist and
it has this term here that
involves if you exclude.
Because it's different has
this differently from Q.
why and then you have a term
that's non-zero when M.
is non-zero and there's vanishes for M.
equals zero These these things and
here so this has a different
form in the northern hemisphere in the
southern hemisphere it is because zero.
Has no image you get is
zero frequency mode but
in non-zero develops a get two
components system that has two bands
it looks like this has the case
because zero sitting on the equator.
Now.
This system is you don't put
in any spontaneous rotation.
But if you move in the rest
frame of the flocking system
then the substrate will
appear to rotate backwards so
these equations of motion have
an analog of a Coriolis term in
the Coriolis term as you know drives you
in towards the equator like there's a wind
current vector field I was showing
you before so the system focuses in
on the special geodesic which is all
you need focus isn't on the equator
you have that density fluctuation
in the transverse sound mode.
So round the equator where the thing
is concentrating you have a zero mm of.
The two bands you have no band get.
That shown here this is up of them and
love them and two bands and.
No band get prim because they are as
soon as you go away in the northern
hemisphere and the southern
hemisphere you develop a band get.
Here So you have a gap.
You have broken time reversal
symmetry here because the thing
is flocking once it moves in the
spontaneously chosen direction like that
cannot go backwards that's
different the dynamical some tree
breaking automatically breaks
time reversal some of.
These are the tone greedy and so you need
for having a top or logically protector
sound mode in the system or
what's usually called in the language of.
Top a logical insulators was
usually called an edge mode.
Atop a logical insulators
insulating in the boke and
conducting on the surface which is a and
there Judge Judy and you can in three D.
and two D.
It really is an image so there's no
age here there seems to be no it here.
Because we're on a closed source but
there is an image.
There's a northern
hemisphere of the band get.
And there's a southern
hemisphere with the band gap and
they have opposite signs of this.
Critical them.
So we have to take kind of like putting
charts on when you build a two sphere from
from sets we have to go through
the northern hemisphere the band gap and
the southern hemisphere with the band get
along the special geodesic the equator.
So the edge is formed by this equation.
And to cross from northern hemisphere to
southern hemisphere we have to go throw
the Sequoia or edge and
get vanishes there so
I have to change the sign of
this in that controls the get.
The gap.
Is given by the square root
of the gassing of a child so
you can see here this is so that's just
one of the radius of the system so
here you have the spatial
curvature drives the bandgap and
it depends on the single fater So
it depends where you are on the sphere.
So.
This means that there is the simplest
analog which is before top of logical
insulators is a one dimensional for
me on coupled to a solid ton think
like a wall feel the main wall down
like this this is studied by Jackie
even ready in the seventies and
there's a particular mode zero
at the kink of the wall
that is the edge Mug This
is most here is just like that all
this the closest analogy is thing so
you can solve the equations of motion for
this particular zero moments as this
linear dispersion relation corresponding
to this direct like count here
that develops at the equator I'll
show you pictures in a minute.
So automatically you can take over
all the machinery of berry lot it's
very connections and Berry coverture
from just from the structure of
the dynamical make it secure that governs
the time evolution of the system so
there's a connection call
fission given by gradients and.
Wave it to space acting on
the eigenvectors of that system and
there's a phlox just the curl of the earth
and there's an integrated curvature
which here it has a kind of interesting
value here it's plus or minus.
Essentially we can take
this as one is plus or
minus a half that is a basic units
there's about called the churn number.
But there's nothing within churn none
of the churn number how it's integer.
But the value in the northern hemisphere
in the southern hemisphere is not
in there and
what's in them then is the difference
between the two because you've got
to glue them the difference is one.
Of the REAL well defined
churn number is one for
the system there are only two bands and
the usual counting is that
if you have in bands there in minus one
top logically protective modes so we
have two bands and one mode the simplest
kind of example that you can have.
One protected age modes so
you can actually solve the GO BACK
solve the differential equations and
see that there is this age
mud that was in the band gap.
And it's just linear in Q of it and
here are the bulk mouths.
At some value of the symmetry breaking and
this edge modes runs that connects
this bottom band linearly up to
the top band just in the main
manner of quantum all effect up
a logical insulators it situates it.
This is a picture of density perturbations
in the system worth the worthless
edge protective mode
hair is a simulation of propagation
of this through an obstacle.
So what's well known in these kinds
of systems that you can test here
is that when you have this if things
propagate through an obstacle.
Time reversal symmetry breaking means
they cannot go back scatter directly
because they cannot go back and they
cannot propagate into the interior because
of the top a logical protection they have
the changing atop a logical invariant so
they should go around
obstacles that's the idea so
here's one of these edge modes
propagating around first this
big obstacle here it's generated here
this density fluctuation does around and
you can see it just go
straight around this obstacle
this is developing in time this
is going around this as the most
important five and
this is the the other schools
on the equator here's a case
of a bigger obstacle.
You know the density fluctuation here and
right around the Solve school doesn't
go that doesn't go into the interior of
the now lots of non examples
of people here have worked on
these kinds of things of these kinds of
top of logically protective modes in these
calm classical but
active kind of systems with North or
some analog of a temperature
you know quantum fluctuations
can be replaced by thermal fluctuations so
the system the equally rich and
it's equally rich in terms of typology
fate these classical examples predate.
The quantum examples.
There's another call example
started recently by Brad mast and
then two French collaborators from
Leon which has to do with atmospheric
waves on the Earth Atmospheric ocean waves
on the earth this is Connect the worth.
For there are now
an atmospheric ocean waves
that propagate opposite direction to
the trade winds for they probably go east.
From west to east across
the Pacific Ocean around the equator.
They come up here against
South America here Sheila and
they are probably getting this way
trade winds are going this way but
when programs get weak
they propagate this way.
And these are well known long
lived modes of oscillations.
And you can look at the band gets in the
system this doesn't come out very clearly
but there are three bands in
the system and there are so
these are just shallow water waves
three bands in the system and so to
topple logically protected age modes and
they have names Calvin ways and you and
I way this is a well known to the partial
differential equations community or
atmospheric physics
are physics community but
it was not known why this so long lived.
And so in the Science paper doll class.
And collaborator showed that this is also
top a logic protect the mother's
kind not also just very recently.
The These bring these waves which
is this just height fluctuations or
surface temperature fluctuations because
they're related these mouths coming and
heat the surface water so
they keep cold water
they prevent cold water because
of buoyancy upwelling and so
they reduce nutrients that come to fish
in the fish population drops off and
that's an El Nino effect that's
of great concern around here.
So that's another call example of course
as we know now now in the in the in
the last few years many examples of top
a logical metamaterials with the kinds of
edge protective modes they can make
various kinds of active systems
these groups here they're arranged in this
case they're arranged in kind of a lattice
has a special emphasis that will
give you a protective modes and
there are caustic examples there photonic
examples there are mechanical examples and
the one I just gave you in
our case it's continuum model
it's uniform I mean it's a more
suspect around there's no let us
there's no turning of geometry
required here apart from the spatial.
QUESTION Yes.
Yeah it's.
Known
and you can't
continuously destroy the Stop
the logical invariant and
in the case I describe going to dissipate
which will dampened out that some rate so
working in the rate with
the active drive is.
More important than any dissipates.
Yes yes the less dissipation you
have the longer lived it will be but
still topple logically protected so
it will still not backscatter
because it's blocking it will still
not go into the interior because of
top logical protection by will
dissipate and now it's really
really clean example you don't need even
any active stresses you don't need in.
Any other fix which is why I
like it's really minimalistic
all the other terms also very
complicated to complicated.
I think there's good question that.
You would have to really break.
The band structure.
To not get this so.
This is the summary of already summarized
all of these questions lots of open
questions about where it might
appear in physics or chemistry or
mathematics and then the last one to have.
Excellent OK so.
In the last few minutes I would like
to pass to a different topic which is
changing the order from pole
order to nametag order.
So the best to so
this came out as I said on the.
At But it just might turn Ramaswamy.
So what end is now some of the medical
board again locally consuming energy so
they're active but
lots of beautiful examples which the the
movies got switched here so this is well
known example forms of one of the logic
Scroope we have assemblies of microtubules
form bundles they magically order
even the microtubules of polar and
they slow in the this back flow and
this flow of the DE thicks which become
self propelled particles which are like my
flying objects but they're flying in
the Matins not flying them it's so
you can see Randi's one
half to fix here and
you can see minus one half defect here.
Too deep.
So those are continuously connected.
By the way little Randy didn't
have time to say it but
liquid crystals are absolutely
brilliant examples of
wealth of top alogical defects because
they have three of the four non-basic
homotopic defect they don't have the main
walls but they have line defects so
the discriminations Randy discussed
they have monopole like defects aligned
if it's arise from non contract the wall
loops there monopole defects that arise
from non-controlled double two spheres and
they have so-called textured D.
fix that involve mapping of in
three dimensions mapping of.
Three D.
space onto to do.
So is three on two S.
to their three of the four that's called
screaming on the particle physics three of
the four basic known top logical
defects simple ones in the same system.
Some of the richest system
that are not in that sense so
here we should this is
some relations from.
From earlier and from our group that
look at the self-propelled particles so.
The core thing it's the core thing
is it's going to hear you have
distinct annihilation and
you have defect peer creation.
And once you have peer creation you have
automatically have a many body system
just like going from quantum mechanics to
a field theory you have peer creation when
she appeared creation cannot have one body
system and an infinite number of particles
so you need a kind of mini body theory
of these active defects to understand
the system you know the source of the
beautiful coupling between the bat flower
So here you see the magical
order here lets you see.
The defect or the medical order and you
see the velocity Theo's of the backflow
this is a fluid it's a liquid crystal and
here is a created we have
a counter circulating water sees
that match with the add up because
the counter-rotating here you have
a lot of shear a lot of vorticity and
that creates defects in the order
field in the numeric field so defects
in the fluid backflow drive formation
of defects in the order field and
vice versa they couple So there's just
the effects everywhere just the term and
this whole system the thing driven so
to understand you see that these defects
are moving all over the place so
I understand the system people have been
trying to understand the full dynamics
of this act of defects so that is what
we've done we solve the Four Hundred them
equations or the act of peers of these
active defects when she appears you
can have many body system where
your peers of defects and.
There's the big question here is why
you have any made it order at all
once you have swirling the effects whose
core is disordered that disorder can
just disorder the whole system is like a
flux line in the superconductor high T.C.
superconductor that's wandering around
all over the place and it moves so
drives the system normal everywhere you do
nothing and just completely disorders so
why is there an active pneumatic at all so
what we showed.
Is that this motion of these defects
is not persistent there's rotational
noise but that means that they
don't move out the fuselage and
the sort of the system they turn and
bin and come back so
the net effect is that you just soften
a defect unbinding like costal
it's the Alice transition and the.
You have this log respected.
Between plus one half and
say minus one half that some ple weakened
by the interactions active
talks of the defects and
the stable configurations are always if
you take a plus one half moving apart
from a minus one up the plus one half as
a non-zero self-propelled velocity in
the minus house zero because of symmetry
reasons the stable systems under active
talks are always a plus one moving
away from a minus one house.
And.
It doesn't depend on whether the system is
contract or extensible and then if you
have two plus one half the stable defects.
Then moving the show on
here in the opposite.
That show them moving together.
And then that case the vortex pattern
the defect pattern is different
contract Bell Systems generate
vortex like configurations and
extensible systems generate these asked
like configurations of large distances
just two different kinds of plus ones
the equation of motion has the active
drive and has a logarithmic attraction.
You just solve the system you get in
model you get a potential like this.
And you're working down here and up here
the system and binds and the strands.
Is a cost let's fellas like
transition which takes us back to
apology top a logical and for those
discussed about cells transition and so
on are only that the effect of
happening here is softened by the.
Dr So it's just very normalised
postulates to Alice Like transition OK
thanks thanks.