Well, good afternoon and welcome to the colloquium today.
It's a great pleasure to welcome Professor Nicholas Bolton here.
He's going to be giving a colloquium. Nicolette is her first degree in the University of Cambridge.
After that, she moved to do her Ph.D. in Berkeley.
And after that, some periods in Yale and then the University of California, Santa Barbara.
She moved in 2010 to her current position as professor in the DCH.
And Sir Nick has won a large number of prizes.
She's a fellow of the American Physical Society. She has won the Cooper Prize, the James McCarthy Prize at the American Physical Society.
And this year she has been awarded a L'Oreal Mexico women in science laureates in Europe,
which is which is another one of these that's been on this lecture is probably best known for her work in multiple works.
She writes a very beautiful paper explaining why there are so few for electrics.
So she's made enormous contributions to the various condensed matter.
But as you will discover this this afternoon, her interest is not only in condensed matter, but I'm looking at some of the applications more deeply.
You can see from her title today, studying the universe under a microscope,
that there are some applications in cosmology from the work that she's been doing.
Thank you very much. Physics.
So this picture shows a piece of force microscopy image of a furrow electric domain structure of a popular multi heroic material yttrium manganese.
And the black and white regions are opposite orientations of the ferroelectric polarisation measured using piezo force microscopy,
and it's embedded in what you may be recognised as a picture from the Hubble Telescope of the cosmic microwave background.
So my goal for the next 45 minutes or so is to explain to you the connection between these two two properties.
I have to make it. I should make a disclaimer before I start, which is that I'm a condensed matter physicist.
So everything I say about cosmology, I learned from my cosmology friends in the last two or three years.
And so I'm just reporting what, what, how I understood what what they told me.
Are there any cosmologists or astrophysicists in the audience? Okay, so please shut up.
If I say anything that's really flagrant is all or too much of an oversimplification.
So let me start then with how to, as a material scientist, my early universe history to a material scientist.
So we start with the Big Bang, which happened about 14 billion years ago.
And after the Big Bang,
it's believed that there was basically there was nothing in the universe that the universe was a vacuum and that it was a homogeneous vacuum.
And we it's believed that the universe was was expanding very rapidly or to a condensed matter person.
This is equivalent to cooling,
but it remained this homogeneous vacuum that was expanding for quite some time until after about ten to the -30 7 seconds.
The first real event in early universe history after the Big Bang happened and after ten to the -30 7 seconds,
what's proposed is that there was a spontaneous symmetry lowering phase transition called the grand unification transition,
in which the vacuum, which was still a vacuum, lowered its symmetry.
So there was a vacuum to vacuum phase transition from high symmetry, homogeneous background to a lower symmetry vacuum.
And again, this is this is a this is a proposal.
Of course, nobody was around at the time. And this is proposed to be described by a free energy that has a mexican hat type form for the potential.
So that's my understanding of kind of early universe history for the first subtract fraction, tiny fraction of a of a second Z.
Okay. From the cosmology people and you think too glaringly wrong.
Okay, good. So let me for the sake of the condensed matter, people describe kind of the equivalent scenario that we're, I think,
all much more familiar with, which is a spontaneous symmetry lowering phase transition in a ferromagnetic material.
I want to emphasise two aspects of spontaneous symmetry lowering phase transitions that I think is so familiar
to us that we that we kind of we forget about what we didn't have to think about what this term means.
Spontaneous symmetry during phase transition. So here's the,
the scenario that from a condensed matter side we're all very familiar with and we'll come back and look at how this applies in the early universe.
So in a format that at low temperature, these little arrows are supposed to.
Represent magnetic moments on electrons at low temperature.
If the exchange interaction between the the electrons or the atoms is is ferromagnetic, then they all line up.
And this is a low symmetry state because if I sit on this electron and look to the right or the left,
I see something different than if I look too up, up or down.
And then as I heat up, eventually there's enough thermal energy that the spins get disordered.
And this is a high symmetry state because wherever I look around this electron, I see the same.
I see a disordered, disordered spin. Or if I come from the other direction, I start in the high symmetry state.
I cool through some critical temperature and I lower the symmetry.
And if my thorough magnet is uni axial, then there are two equivalent low symmetry states,
one where the magnetic moments are pointing up and one where the magnetic moments are pointing down.
And so if I plot the energy as a function of some order parameter, which in this case is how much these magnetic moments, these spins are lined up,
then I have this characteristic double well potential spins pointing up on this side and spins pointing down on the side.
So I have a potential, potential energy surface describing my symmetry lowering phase transition.
So this is the first aspect of spontaneous symmetry breaking. I want to emphasise one, it happens spontaneously,
so it lowers the energy to to become the lowest symmetry state and there are multiple equivalent grounds to get states.
In this example, there are only two that the system can choose when it goes through this symmetry, low and phase transition.
There's a second aspect of symmetry lowering phase transitions, which is not something that's something historically I ever thought about as much,
which is that there's often defect formation associated with the formation of the low symmetry state.
And so here's the example again of the furrow magnet.
And I'm cooling my paramagnetic disordered material and starting to approach the phase transition.
And one region might adopt the up spin state and another region might spontaneously adopt the down spin state.
And then these two regions will launch large as I approach the phase transition until at the critical temperature,
I transform to all the low symmetry state.
But because of the initial choice of the different regions of the two available ground states, I have a defect where where they meet.
So my spot, my low symmetry state is not perfect here an up spin electron with would like an up spin never has a downstream
neighbour and I have a defect and of course we familiar with this defect as a as a ferromagnetic domain wall.
Okay. So here's the roughly how many students are in the audience section.
Okay. My student my review for the students is for you guys especially.
So this is my review for the students of one of the key factors that we need to remember for spontaneous symmetry lowering phase transitions.
They lower the symmetry. That's kind of obvious.
What the spontaneous refers to is that they lower the energy and that there's a choice of multiple equivalent low symmetry states to in this case.
And then the last point, which is going to become really important in the discussion of early universe behaviour,
is that defects form at the intersections between the equivalent low symmetry states.
Okay. So back to the early universe. Any questions on the.
Okay. So remember off ten to the -30 7 seconds,
we had our symmetry lowering phase transition in the early universe from the homogeneous vacuum to the vacuum of lower symmetry.
And this is analogous then to our paramagnetic magnetic system lowering at symmetry to a fairer magnet in the universe.
The expansion was the was the variable that caused this transition in the in of course, in the paramagnetic it's usually temperature.
And remember we said that the rather than the double well potential we looked at for the case of the
ferromagnetic potential describing this phase transition as this Mexican hat Mexican hat shape potential.
And so as I approach the famous transition from the high symmetry, from the homogeneous vacuum to the lower cemetery vacuum,
then just as in the foreign magnet, different regions of the material could adopt different orientations of the low cemetery state.
Different regions of space can choose low cemetery vacuum with different choice of angle.
So perhaps this region over here will adopt this angle here.
Another region will adopt another angle and so on. And initially, these regions are small, as I as I'm approaching the phase transition.
But as I start to cross the phase transition, these fluctuations of low symmetry vacuum out of the high symmetry vacuum start to grow.
And eventually they meet. And where they meet. Is a defect.
And this is one of the proposals of how mass could evolve from the vacuum in the early universe.
The symmetry lowering phase transition to the lowest symmetry state.
There are defects which of course have higher energy than the surrounding region.
It's not so obvious to see on this picture of plus, this defect looks like a point,
but actually it's a line defect which is set by the symmetry of the of the potential.
So Mexican hat type potential, when one lowers the symmetry with such a potential, one ends up with with line defects in the low symmetry phase.
And this defect is, is what's known as a cosmic string.
So cosmic strings, I should emphasise, are completely different from string theory strings.
These are proposed defects in the, in the structure of the universe that emerged from this symmetry lowering such as symmetry,
law and phase transition in the early universe behaviour. Okay.
So let me tell you a couple of details.
So the next couple of starts are a little bit detailed about the physics of these these cosmic strings,
because there were distinct predictions made about cosmic string formation, which would like to be able to test.
So the first prediction is what determines the distance between the cosmic strings?
How many cosmic strings should there be in the universe?
And so the size of the domains that form which separate the,
the the defects is a competition between the speed at which information propagates through the system.
And this is believed to be set by the speed of light. So it determines whether if one region of the universe has has lowered its symmetry.
How much time does it or how long does it take to tell its neighbours to adopt the same choice of face off the same angle?
So the speed of information propagation, if my information propagation is very fast,
then I would expect to get large domains with the same of the same the same phase angle.
It's also dependent, of course, by how much time you spend at the transmission.
How long do you have to for your your kind of freedoms to grow?
And so if my transition happens more slowly again, my domains become quickly become larger.
So if I expand slowly, then different regions can communicate their choice of phase.
So I get large regions of the same choice and the low density of defects.
Not so many cosmic strings. If I expand quickly, then there's not much time to communicate the choice of phase.
I have many smaller regions with different choices of phase and a high density of defects.
So this is kind of the physics of of such a phase transition.
Sorry. One thing I realise I forgot to mention is that this cosmic string,
because of the nature of the symmetry of this phase transition, is what's called topologically protected.
It's a one dimensional and it's also topologically protected.
And what this means is that the cosmic string can't go away unless it gets pushed out of the outside edges of the system,
or unless two cosmic strings with opposite parity meet each other and annihilate.
So this. This is a consequence of the symmetry.
So there are two factors proposed for cosmic string formation.
One is the type of logical protection, and one is this that the size of the domain separating them.
And the first piece of physics was developed by by Tom Kebble and the second by Voytek Zurek.
So when one combines these two properties together, one can write down a scaling law,
which is called Kibble Direct Scaling for Cosmic String Formation,
which basically just takes the physics that just described the competition between the speed
of information propagation and how much time you have to grow domains and quantifies it.
So I'm not going to derive this. It's not so it's it's rather straightforward to derive this expression here is if you
assume that you're expanding linearly or you're crossing the phase transition linearly.
And so it tells you that the size of the domains, the distance between the topologically protected defects,
the distance between the cosmic strings is given by the correlation length.
It's given by a factor tor q which is the volume at the critical point divided
by the rate of expansion through the transition divided by the relaxation time,
which is just the correlation length, divided by the speed of information transfer.
And it's, it's a scaling law and it's the scaling is determined by some critical exponents which might be fluctuation dominated or,
or might be mean field or could be somewhere in between.
So this is the prediction.
This is the quantifiable prediction for the kind of physics I've just described to you that tells me really for for a particular expansion rate,
for the grand unification transition and a particular speed of information transfer, how many cosmic strings I should have them.
So this is what we would like to be able to like to be able to test.
Okay. Questions? Objections.
Well. Okay.
So so this is all very nice, I think. I think it's extremely creative, extremely conceptual, creative and very pleasant theory.
But of course, one would like to know is, is it true to cosmic strings exist?
And if so, how can how can we measure them? How can we detect them and how can we study them?
And to to study anything in physics directly, then we need to have a probe which has a similar energy scale.
So the thing that we'd like to study, of course, and the energy scale for Cosmic Strings is of the order of ten to the 15 giga electron volts.
So if one had the number that I was told us, if one had a11 kilometre long cosmic strings, we'd have about the mass of the earth.
So if one just kind of wandered in, we wouldn't have much opportunity to detect it before we were obliterated.
I guess ten to the 15 giga electron volts is very energetic.
The highest energy probes we have available at the moment, the largest Hadron Collider.
There are about ten to the ten of the four giga electron volts.
So we're really very far from any direct study of of cosmic string formation.
So what one can do, rather, is observe to look for signatures of of cosmic strings in astronomical data.
And in particular, there are well-defined predictions of signatures that should occur in the cosmic microwave background,
which to date have not been seen. So there is an upper bound on the number of cosmic strings that can occur in the universe.
But there's not based on observations of the cosmic string in.
One can, of course, also do computer simulations. And this picture here is from my colleague Martin Quince at the University of Geneva,
who simulated a symmetry lowering phase transition through that Mexican hat potential that we looked at earlier.
And what he found was the formation of domains and domain intersections, which really form is one dimensional, one dimensional strength.
So these are the two, I'd say, primary roots to studying cosmic string formation in the early universe.
What we would like to do instead is to study the and equivalent to the grand unification transition in our laboratory.
And this started as the Ph.D. thesis of Janet Griffin,
who was a graduate student in my group and who's now a postdoc at the Molecular Foundry in Berkeley.
And so our plan then is first to identify a material, a real condensed matter,
solid with a symmetry lowering phase transition that's described by the same mathematics as that proposed for the grand unification transition.
So what we want is a spontaneous symmetry breaking. That's described by a mexican hat potential.
And then we can do or we will do all the experiments on our material that we would like to do on the early universe,
which we can't do because we can't keep replaying the big bang. So we will look at the cosmic strings exist when we run through this transition.
Do we find line defects in our system? Did they form as we think?
In particular, do they follow this direct type scaling that I introduced to you earlier?
How do they evolve when we wait and watch them? Do they recombine that?
They move around. And what are their properties? Okay.
So this is where I'm headed. Then for the second half of the talk,
I'll show you a material that has a symmetry lowering phase transition that meets these
chemical requirements for the formation of topologically protected one dimensional defects,
and then will use electronic structure calculations to calculate all the parameters in
that cables in the right formula and to predict if cables or a physics is being followed.
How many defects, how many or what the domain size should be and how many domain intersections
should form as a function of of how quickly we move through the transition.
In our case, it's by cooling.
Then we'll go to the lab and we'll measure how many topologically protected defects form as a as a function of cooling rate.
And so we'll be able to answer the question.
Does a system that's described by the same physics and symmetry as the grand unification transition exhibit the predicted Zurek behaviour.
Okay. So any suggestions?
If I'm looking for material that I want to have a spontaneous symmetry lowering phase transition described by a mexican hat type potential?
Where might I look? Anything with a TV or anything with a 2D order parameter?
Yeah. Any suggestions from the students?
Stephen gave you a clue when he started it up. He said that I worked a lot on multiple heroic materials,
so we started looking and videos of the multi multi for rock and the multi heroic material is a material that has multiple Farrokh orders.
And the reason we chose multiple heroic materials is because each of these Farrokh orders can have a has a spontaneous
symmetry lowering phase transition associated with it that has defects associated with that phase transition.
And so they're all candidates for, for such behaviour. So.
So I guess we were not particularly smart, we just thought if we had three times as many possible possibilities in a single material,
we had more chance of finding a suitable order parameter.
So multiple heroic then as a material that combines for electricity and or for magnetism and also elasticity in the same phase.
And so it has a it can have a spontaneous magnetisation that's switchable by a magnetic field,
like the three magnets we looked at earlier, a spontaneous electric polarisation that's switchable by an applied electric field.
It's a for electric or a spontaneous defamation that's switchable by an applied stress that's a foreign elastic.
And multiple rocks are actually not so uncommon or some types of multiple rocks are not a rather common.
So, for example,
thermoelectric electric elastics are very common when one has a structural phase transition that results in an electric dipole moment,
usually associated with this, there's an elastic defamation.
And what this leads to is a cross coupling between the stress and the polarisation of the electric field and the internal strain.
And this gives you electric behaviour. So this is for example, how sonar converts a signal,
a sound wave into an electrical signal through the of electric behaviour and likewise associated with a magnetic phase.
Transition is often an elastic defamation and so thorough magnetic flow elastics are also rather common.
And these are used, for example, for magneto, mechanical actuation.
But many of us have been interested for,
for some years in this side of the triangle materials that are for magnetic and for electric, which are rather uncommon.
That's not what I'm going to focus on today.
But they have the interesting behaviour that with them with an electric field, one can modify the magnetism.
So one has the the potential for electric field control of magnetic phenomena, vice versa.
With a magnetic field one can modify the electric polarisation.
So this is all kept us busy trying to make materials that are for electric and for magnetic and to exploit their properties for many years.
Today, what I want to focus on, rather, is that the defects that are that form from spontaneous,
from spontaneous symmetry, lowering phase transitions in these various types of heroic order.
So we mentioned already the thorium at the domain walls in ferromagnetic between opposite orientations of the magnetisation.
There are similar domain walls in for electric materials between opposite orientations of the electric polarisation and in for elastics the
mechanical twin boundaries one can also think of as a as a domain wall or is spontaneous a defect formed by the spontaneous symmetry line.
So this was our motivation for looking in multiple rubrics for this behaviour.
So the material that we we chose is it is a multi farai material called yttrium manganese yttrium manganese O3 which has a rather unusual structure.
And this was the, the material I showed you showed you right on the first slide I showed you the is a force microscopy image of this material.
So it has a crystal structure which consists of layers of manganese oh five triggered all by pyramids.
So if I look from the top, I have a manganese atom in the middle of a triangle of oxygens which are corner shared and above.
The manganese is an oxygen and also below the manganese. So that's an oxygen on top and an oxygen.
So these are triggered by permits because they have a beautiful triangular lattice of the manganese ions.
And these planes of manganese of five treadmill by pyramids are separated by planes
of yttrium ions and at high temperature the structure has a centre of inversion,
its power electric and there's no electric polarisation.
This is how it looks like I had to show you this this picture, because I'm trained as an electronic structure theorist.
And when I moved to the 8880 build, built an oxide single crystal growth lab.
And so I'm very excited to show you the sample of it through manganese at three because it was actually grown in our laboratory.
This is a beautiful cyber style furnace by Frank Lichtenberg as a member of my group.
So you can see it's this is an insulator it's a through electric so rather good
insulator it's kind of a dirty black colour because the bandgap is rather small.
The bandgap is less than electron volt because of the, the, the electron states that are close to the Fermi energy.
So this is all material. What's important for us is that it undergoes a symmetry lowering phase transition at around 1000
Kelvin and at the cemetery lowering phase transition to two things happened to the structure.
One is that there is a tilting of the trigger by pyramids in threes.
So this one moves in, this one moves forward, and this one moves, moves to the left.
So a traumatisation tilting. So on the view from the top, I've put arrows on the top oxygens to show the way that they tell.
So there's a three foot three fold folding over or tilting of the pyramids.
And at the same time or accompanying that there was a shift of the yttrium ions and there's the,
the net shift of the it reminds us all in one direction. And because the it reminds a positive and the manganese of five layer is overall negative,
this gives a net polarisation and so the low temperature state is far away.
Questions on the crystal structure. Okay.
Let's look a little bit in more detail at these structural distortions,
and then I'll kind of crystallographic you, then I'll show you mathematically how they look at.
So what point about this tilting, this trimer ization is let's say I decide to try MRI's about this point, which I called Alpha.
So I tilt this oxygen outwards and this oxygen outwards in this option outwards around alpha.
If I choose Alpha as my origin for the tilting in the crystal structure,
then you can see straightforwardly that I can choose the Saturn beta as my origin.
Because if I don't, you might tell me my origin. This oxygen would have to tilt up and left.
And I've already, if you like, used it up, tilting up and right. And likewise, I can't choose Gamma as my origin if I've already chosen Alpha.
So I have three possible origins which I can choose for this, this tilting.
So this already gives me three three domains, three symmetry equivalent options for the for the low symmetry state.
You might be thinking, well, I don't have to tilt out. What's your right? I could also tilt inwards.
And so that gives me now six equivalent tilt things like a tilt in or I could tilt out at any of three origins.
And this is rather convenient because if I tilt out then the yttrium iron that
was sitting on top of this hole drops down to fill the space that I've made,
and that will give me one orientation of my ferroelectric polarisation.
Whereas if I tilt inwards, the yttrium iron that was sitting here in the power electric structure has to shuffle up to get out of the way.
So this gives me the opposite origin of my ferroelectric polarisation.
So I have six possible minima in my symmetry load state.
And these are, if you like, a coupled to two opposite polarisations.
And if one looks in detail at the symmetry, one finds that.
Always this will this will always alternate.
So immediately, I can understand this domain structure, this ferroelectric domain structure that I showed you right at the start.
So again, this is the piece of force microscopy image of of yttrium magnet and the black and
white regions correspond to opposite orientations of the for electric polarisation.
What you can't see as a force microscopy is the origin of the tilting.
This is not a parameter that shows up when you measure only the polarisation.
But what you see is that there are always six domains of alternating polarisation that are meeting at a point.
I should emphasise, I should point out also this of course is a two dimensional slice through the through the structure, it's the surface.
And so the fact that these are meeting at a at a point, if one were to look in three dimensions, these are actually line defects.
So these are going to be my strings. I'll come back to that in a second. Okay.
So that's very hand-waving and qualitative, if you like, looking at atoms and crystal structures.
One can also, of course, really make a symmetry analysis and write down the land of free energy as a function of the amplitude,
of the tilting and the angle of the tilting.
So here's my order parameter, which is a combination of the amplitude and the angle of the tilting,
which is how pointed out is a two dimensional order parameter, which allows me then to have the Mexican hat type type potential.
Okay. So this is this is this is what it looks like.
And let me not go through the through the the details of the mathematics, but show you how it looks.
If I actually plot this with these parameters all determined from electronic structure calculations.
And so this is the plot of my energy is a function of as I come out from the middle.
Q So the amplitude of the tilting as I come out from the middle. And as I go around the hat, the angle of the tilting.
And what you can see is that my phase transition is described by a potential which I'd say is rather Mexican hat.
Like it's not a perfect Mexican hat because it's a structural phase transition in a system that has six fold symmetry.
And if the crystal structure has six fold symmetry, then that has to show up in the potential.
And it does show up in the fact that there are six minima. So when I get to to the to the ground state, my potential has six minima.
And so there's, there's a gradual crossover from the Z six symmetry of the ground state to an almost you one like symmetry at higher temperature.
And you can see if you look in the in the expression that the fi dependence is coming it
only at the angle dependence comes in only very high powers of of the tilting amplitude.
So one might think, well, really what we wanted was a mexican hat potential.
So this is really not perfect.
But actually the fact that this is not a perfect Mexican hat is very convenient for us because it allows us to actually be able to rather,
simply, rather trivially measure the defects when they form.
So remember my six minimum I my six ground states have alternating orientations of the four electric polarisation and so.
If I paint my alternating orientations black and white, corresponding to up and down polarisation,
I can image the formation of the domain intersections, the formation of the defects using force microscopy.
So this picture was taken by Martin Lilian Blum, who's a Ph.D. student in the and the group of of Manfred Physics.
We have some related materials that have much closer that have basically a perfect Mexican hat type potential for a similar phase transition.
And they're, of course, aesthetically, it's rather pleasing because it's really a an equivalent to the great unification transition,
but then we can't measure the domain. And so this is this is rather helpful.
So the meeting points to these ferroelectric domains, which look like points on this on this intersection are, in fact, one dimensional strings.
This is a 3D simulation just of the of the potential energy I just just showed you.
And you see that the the straight the the domain intersections are marked in red.
And you can see a remarkable resemblance to the simulation of the early universe that I showed you right at the start,
because essentially it's simulating the same potential with smaller small changes at low energy.
We try very hard to image these strings in the in the real, real material.
We have samples that we if anybody has an idea for how to do this directly, experimentally, we have many samples.
We'll be very happy to send you. If you would like to like to try, it'd be really fun, I think, just to see them really experimentally in real life.
Okay. So hopefully I can convince you then that the structural phase transition in multiple
rocketry of mechanised provides us with an analogue to the grand unification transition.
So in the early universe, the transition from the high symmetry vacuum to the low symmetry vacuum described by a mexican
had potential forming cosmic strings and yttrium manganese at the transition between the High
Symmetry Space Group to the low symmetry space group described by a mexican hat like potential
with these six minima in the ground state that allow us to image our cosmic string analogues.
So we have our we have our our material, our little cosmic string formation simulator, which we now go to the laboratory and to do measurements on.
So what experiment would we like to do on the early universe?
Then we'd like to expand it at different rates and cross the grand unification transition and see how many cosmic streams form in each case.
So this would be it would allow us to test the so-called triple direct scaling.
And since we can't do that, but we now have our our little cosmic stream formation simulator.
Instead, we cool it. We're Mennonite a different race through the structural phase transition and we count how many domain intersections form.
So here are two pictures of cooling yttrium manganese through the structural
phase transition and the subsequent at different rates and the subsequent.
PSA force microscopy images. And I want to emphasise that the scale bar is four microns.
In each case. I didn't just take one of them and blow them up,
although if I had he would not be able to tell the difference because they're they're they're self, self similar.
And we got extremely lucky with yttrium manganese because these three images correspond to.
Cooling through the transition by just switching off the furnace and leaving the sample in there,
cooling through the transition by taking the sample out of the furnace and leaving it on the lab bench and cooling
through the transition by taking the sample out of the furnace and putting it putting it in liquid nitrogen.
So these are, if you like, the three most experimentally accessible cooling rates.
And you can see that they give really distinctly different domain sizes and distinctly different concentrations of cosmic stream analogues.
We were, of course, not the first people to have tried this.
It's a very intriguing game to play, to try to use condensed matter systems to to replicate them,
to replicate behaviours in other kinds of physics, but to get all of the factors to come together, to get the right kind of physics of your,
have your transition, to get the right potential describing your transition,
and then to get all the parameters right so you could actually access cooling rights that give you that gives you different results.
One has to be extraordinarily lucky for those things to come together, I think. Okay.
So then we have a, of course,
a first year undergraduate research student sit and count how many germane intersections one gets in each different coding.
Right. And we can then really go test this predicted Zurich scaling off of the defect information.
Okay. So remember the domain size, the properties that it depends on.
It depends on the zero temperature correlation length, which in a thorough electric is usually equated with this domain wall width.
And this is something which at the time we we calculated my postdoc at the time, Joachim,
a guy who's now a assistant professor in Kyoto University, made a big box of victory.
A man tonight in the computer made to do domains and let the structure relax and measured how wide the intersection between the domains was.
And this is very interesting. The domain, Woolworths was basically vanishing.
It was a very abrupt transition from one ferroelectric domain to the next.
This has since been confirmed with high resolution transmission, electron microscopy.
The zero temperature relaxation time. This is, I'd say, rather obvious what one should use for this.
We we took our zero temperature correlation length and divided by the speed of sound,
which we also calculated from density functional theory we can discuss later whether that's the best measure of the relaxation time,
the critical exponents.
We used in that data I'm going to show you we we we use a ratio of 0.58 which which one sees from Monte Carlo simulations for 3D X Y model.
And this is the number which one obtains for a fluctuation dominated transition.
And again, I'll come back to that to discuss whether that's valid later.
If one uses point five, that would describe a mean field type transition.
And then the last parameter here t c the critical temperature divided by the cooling.
The critical temperature we know from experimentally. And the cooling. Right.
This is a thing we vary. So what we're what we want to calculate is d is a function of of cooling.
Right. Okay.
So then we can do that. We can calculate the vortex density as a function of, of cooling.
Right. And this is what we find. So the red line is our calculations with a, a zero temperature correlation length of 0.06 angstroms.
So this, if you like, was a fit. I told you that our domain wall, which was basically very close to zero.
And so we just basically adjusted this line and that 0.06, it fits rather well with the data, which is the red point.
So these are measured by two different groups by my colleague Mensa thebig and the group of Sang Chong in in Raqqa.
So you can see that there's really a remarkable agreement. So we were very happy for a while.
We thought we were we're still very happy, I guess, but we were even more happy for a while because we really thought we have the
first solid state verification of cables are scaling in a in a real material.
Which is really remarkable if you think about all the junk that exists in a real
material to actually get some real fundamental physics scaling physics out as is,
I think, really, really surprising. Oh, she made me point out here that if we taken a critical exponent of point five rather than .58,
I would say that it gives it more or less equally good fit on the scale.
It's not so clear that that this has to be fluctuation dominated.
Okay. So we were extremely happy for a while and then we got intrigued that things were not as straightforward as we thought.
So when my colleague Manfred Seaburg went and took more data at lower cooling rates and at higher cooling rates,
we found some deviations from the expected people's direct scaling behaviour.
And again, maybe this we should think this makes us more happy because we still have something to do.
So these are very, very slow, slow cooling rates is a 100th of a kelvin per minute.
This is extremely slow. And what Martin, the student of Manfred, has been studying is that we think when one calls this slowly,
that one actually has annihilation of the domain walls,
that we actually have a drop off from Kepler's direct behaviour because the domains are growing,
because the, the, the strings are starting to meet and annihilate each other.
And the reason that we think we have a difference between these two data sets is because these
samples have grown under different conditions in which they have more real physical defects.
And this gives the domain walls much higher mobility.
So, Martin, in so many tests in this region and so we think we very much understand this behaviour and this deviation from people's direct behaviour.
The deviation and fast cooling is much more. Its explanation is proving much more elusive, shall I say?
And I say we still don't know for sure whether this is something intrinsic physics,
whether we've really discovered some antique people Zurich or beyond people's behaviour or it's something just with the.
At these very fast cooling rates we can't expect. We can't expect really to be able to to to study any intrinsic physics at all.
We also find an extremely unexpected dependence on the chemistry.
So the initial data I showed you were for urban manganese, then the initial data set.
These all have exactly the same crystal structure, exactly the same type of phase transition, very similar.
They should have very similar correlation then similar speeds of sound,
nothing that would expect us to have attended the full difference in vortex density for the same cooling rate.
So this is something that's definitely not captured by standard people's direct theory and we really don't understand at all.
So I guess this young this is what we should be happy about because it keeps us busy.
This is not the Ph.D. student of the of the next student to work on this project.
Quintin Meyer So we try to understand what causes this chemistry dependence.
Is it something fundamental or is it just something in the processing of the different materials?
What's the origin of the turn around? Is there really some physics in this beyond people Zurich regime?
And I'd say the answer to that is that we don't know.
We certainly don't know whether it's relevant for early universe behaviour.
But in the last couple of minutes let me just show you our current favourite idea for why this turnover might be happening.
And I think that we have many other ideas, but none of them do we have really any any reason to believe is is correct.
So what I think would be rather charming is if one had a crossover from fluctuation dominated cable,
Zurich, which has an exponent of 0.58 to mean field dominated Zurich.
So this would be scale regions with different slopes. And one could convince oneself that perhaps as the cooling rate gets faster,
one would move from a fluctuation dominated regime to a mean field dominated regime and so on.
And then to get between these two slopes, one would have to drop down in between.
And this might explain our turnover. So now we try to cool it really very fast rates to see if we see a turn around again, which would verify this.
This I wouldn't even call a model. It's a kind of idea.
But I have yet no basis for thinking this is particularly correct.
Good. So then let me summarise. We think that yttrium manganese, it seems to provide the first example of zurek scaling in a condensed matter system.
And that cosmologist got it right. Cosmic strings, if they exist at all form the way they think.
Hopefully I've persuaded you that the use of real materials to explore questions in cosmology is really a lot of fun.
I like this article in The Economist now is a few years ago there was a quote from Cliff Burgess, who's a theoretician at the Perimeter Institute.
He has his doubts that we do anything useful, but he thinks the experiments are worth pursuing, like tap dancing snakes.
He says. The point is not that they do it well, but that they do it at all.
So thanks a lot for listening and I'll be happy to take any questions or comments you have.