 Felly dyna'n dweud ymchwil yn y cyfnodol. Mae'r cyfan yn gweithio'r cyfnodol. Yn ymchwil yn ymgyrchu'r cyffredigau o'r cyffredigau, mae'n mynd i'r cyfrifau. Yn ymgyrch ar hyn, yma'n cyfrifau, ond ymgyrch yn mynd i ddailwt, oherwydd y byddwch yn cyfrifau'n cyfrifau, ac mae'n mynd i'r cyfrifau yn ymwil. Mae'n ymwil yn cyfrifau'i cyfrifau ar hynny, ond maen nhw'n ddynamiadau yn fglwyd. Yndw i'n gwybod, rwy'n unrhyw arall unrhyw gyda wrathau wahanol. Mae'n gobeithas, felly mae'n amser cyfrwaretodd yn ddynamiadau bydd yn amlwgol arall. Mweld, mae'n gyflawn cyfeonol i'r ddynamiadau. I gael ddynamiadau ei lei'r ddynamiadau, ych yn ddeunam hwnny o ddynamiadau? A fynd i gyda unrhyw mewn ddynamiadau? Fyd ychydig i'r holl ddynamiadau, One Universal aspect of Dynamics is the Singularities it turns out that closed singularities as we'll see, what catastrophe theory tells you is that you can only have certain types of singularities and in a way singularities are the friend very in the neighbourhood of the singularity everything behaves in a universal and predictable way. In fact if I don't make it at the end of my talk Or autobiograff wedi'urol y gwyl, yw'r yswynt gwybod yng Ngheraru'i. Felly, mae oedd yn hawl 3 uchwch argynod, sy'n gydigodiaeth gyda'r symwy, fel aethau sydd wedi gyrraff agnod yn cael ei fod yn ymwneud. Nid yn rhan o un, rhan o un, rhan o un, rhan o un. Stoedd y syniadau yn eistedd, i fynd, o'r nu. So, mae'n gwybod ymddiadau deisgu, a mae'n gwybod yw ddatblygu, fel yw'r gymreithio agorolol hefyd, ond yn ddweud o'r olygu. If you zoom in a bit more, the black line is the geometric caustic, but you get interference patterns. If this is a phenomena of light, the light is propagating from, let's say, above and is bouncing off the back wall of the mug and is forming this kind of focus, except it's not a perfect focus. If it was a perfect focus, it would be a spot. If the back wall was a parabolic mirror, it would focus a spot. But the point is that these catastrophes are generic things. If you break the symmetry away from being a perfect parabola, that focus point is unstable and explodes into this CUSP structure. And if we go one, so we get waves, and if we go one further and we ask, well, in the quantum world, if you have a quantum field, then you can look for these things in Fox-space action, then you find that they're discretised. And these are the quanta of the excitations. So we'll come to all of this. Oh, so, before I get started, so Ryan, where are you, Ryan? Hope you're here. There he is. So he has a poster on this, and it's kind of on the back wall there. It's kind of over there. OK, so I'm going to give you a quick run-through in about catastrophe theory, and then we're going to apply it to the dynamics of what I've learnt here is the model, one of the models of the Hamiltonian mean field model but applied to cold atoms. And then we'll also see catastrophes in the transverse field ising model. So here's a sort of zoo of catastrophes. So this simple one, this coffee cup caustic, but also the bright lines on the bottom of the swimming pool. This is focusing of sunlight by the waves on the surface. A rainbow that's focusing in angle space by water droplets, of course, they're not perfect lenses. So again, you don't get a perfect focus. Gravitational lensing is another example in nature. In fact, there's five different images of the same galaxy. If you line up A with A and B with B, I can't really see them from here, but maybe you can. Then you get five different images of the same galaxy, which is behind some mass distribution and it gets refocused in various ways. And there's various other examples like rogue waves at sea and shipwaves and so on. And then there's other fancier things like Hawking radiation. We saw yesterday about the Anderson transition, so it turns out that the wave packet shapes at the Anderson transition are catastrophes, in fact. So once you have the eyes to see these things, you in fact see them everywhere. So here's the geometric picture, actually Leonardo da Vinci, since we're in Italy. He had to go and understand this, and I guess Italians like coffee, so maybe they're into your coffee cup. So we imagine these light rays, just half of them are shown here, are coming and then they bounce off the back wall of the coffee cup and then the caustic is the envelope. And the point is that these are structurally stable, so we've already perturbed away from being a perfect parabola, say, which would give a focus point, but now imagine this cup was plastic. If I pressed into it, this structure would be shifted and stretched and so on, but it fundamentally wouldn't change its shape. It would be sheared and all sorts of things, but it would still have that fundamental shape. So in that sense, they're structurally stable and hence generic. And the other thing is that within the geometric theory of optics, then, if you work out the intensity along this caustic line, you'll find out that it's infinite. So that's interesting, because that's a place where one theory breaks down and you have to go to a better theory. Of course, we know what the better theory is for light, it's waves. More pertinent to the subject of this conference, as I said, when you look for these things, you see them everywhere. So some years ago, there was this conference, which I was at and I had a chapter on a completely different subject in this book and I was looking at the book when it arrived in the post some years after the conference and the front cover intrigued me. And so these are in fact examples from the Hamiltonian mean field model. So the one example you can think of is particles distributed around a ring. And so the density of particles is plotted along the top here as a function of angle. This is initial time up here. And okay, if you start with them perfectly symmetrically around the ring, then the forces in each direction are equal and nothing happens. But if you perturb the particles slightly from their equilibrium positions and let it go, it spontaneously forms these by clusters. So these density spontaneously focuses as we heard actually on the first day in Julian's talk. And so this is described then by these long range cosine interactions. I put in this slide, I must say I'm not, I don't, this is probably all I know about these gravity driven examples, but I don't know what happened to this theory after it came out. If anyone from the gravity community has heard of it, but the claim is, so Arnold was one of the inventors of catastrophe theory, but working with Zeldovic, they predicted that certain structures that are gonna form under gravitation would have, that you start off with a smooth distribution and you can develop singular structures. So maybe afterwards I'd be happy to know if anything ever came of that work. Okay, so this is the kind of the picture behind catastrophe theory. So in every dimension of space, there are only certain allowed types of singularity. And what I mean by this, these are singularities of gradient maps and gradient maps include things like classical mechanics and quantum mechanics. In fact, any theory that relies on a variational principle is a gradient map, and that's a very large chunk of physics. If you have a theory relying on a variational principle, then the idea is that these are the structurally stable singularities. That's not to say you can't get other singularities, but those other singularities generically will vanish if you move away from some special symmetry or something like that. So on a line, in one dimension, the singularity can be a point. In two dimensions, it's this cusp. In three dimensions, then you have these three different classes. They're distinct. One can't be mapped onto the other by any transformation. So there's a swallowtail, elliptic, umbilic, and hyperbolic. Umbilic, and the higher ones contain the lower ones. So if you look on the swallowtail where you've got cusps up the top here, so you have these singular sheets and lines and ultimately points. And so this is the mathematical description. So each of these things for every dimension of space, these are the allowed ones. And they're generated by a generating function. So I don't know, some things in life are so beautiful, they make you want to cry. And in my case, it's one of these. I don't know if anyone, Andrea, are you crying? No, I don't see you in tears. This is remarkable. I mean, what they're saying is, so this is the theory due to René Tom originally, a French mathematician and Arnold, a Russian mathematician, is that the structurally stable singularities of these gradient maps can be described by these generating functions. And you can think of these as actions. Okay, they are linear in the control parameters. Those are the space, the parameters of space, x, y and z and so on. And then they're non-linear, they're polynomials in the state variable s and that's something that sort of labels the classical path, if you like. I'll show you some examples. Well, there's an example here. There's these different state variables. Okay, so for example, yeah, this is for the cusp, this is how it works. So the cusp, you have this, this quartic action and so we know, well, we know how to get classical mechanics. Okay, by Fermat's principle, we look for the stationary, the stationary path where the action is stationary. So we take the first derivative with respect to s and we get a cubic and then to get a core stick, it's one more. It's stationary to one higher order. We take another derivative, we get this equation and then between these two equations, we can eliminate s and we left purely with an equation for c1 and c2 and actually this is the equation for a cusp. But I said at a core stick that the geometric theory, the classical theory breaks down and you need to go one step further. Well, how do you do that? Well, you just basically do a path integral. If we know the action, we can come up with a path integral. And if we, in the case of the cusp, we have, so we have this quartic action again and so to get the path integral, what we're doing is we're summing over all the label of the paths and s labels all the paths. We get this function. It's a function now of the spatial coordinates c1 and c2. C1 goes across and c2 goes up. This is a complex number in general, so it has an amplitude and a phase. Actually, if you look at the phase, these are vortex, anti-vortex pairs. They're dislocations in other words that proliferate as you go up in time. There's also a line of vortices up either side. So this is a special function in fact. It's not, many people might not realise that, but if you look in this, in the new addition of what I would call Abramovitz and Stagon, the new NIST, the Handbook of Mathematical Functions, you can find it in the last chapter. They're called Diffraction Integrals there, a chapter by Michael Berry. If you look at what happens at the cusp then to this action, so as you change c1 and c2, as you change these two parameters, you change the form of this function. So it goes from being this double well thing with three classical solutions in the middle. When you go across one of these, these are called fold lines and they meet at a cusp point, two of the solutions annihilate. You just have one solution outside on either side and all three of them annihilate at once. So this is the brightest part of the caustic, the most singular part. Okay, so applying this to the Hamiltonian mean field then. So we've kind of heard, let me remind you though that you can think of the Hamiltonian mean field model as being particles on a ring interacting with this cosine potential, which just depends on the difference in their two angles. And in that sense, it's not really a mean field theory in my book, okay? Then this can be perfectly exact. Whereas if you think of it as the X, Y model, then normally you would think of these rotors spread out over a long, in space, then it becomes mean field when you have long range interactions and they all see each other identically. Then you get this sum, it doesn't die off in any way. So you can imagine all those rotors then sitting on top of each other on, so sitting on top of each other on this circle here. So how can we realize this with cold atoms? Okay, so there was an experiment in 2012 by Tillman Esslinger's group in Zurich and he realized these laser induced interactions between atoms. So the idea is you have this cloud of atoms, a BC actually in a cavity and you illuminate it from the side, not through the end mirrors, but from the side. Then what happens is a photon comes in and it can then get scattered to one of the neighboring atoms before it then rejoins the laser beam. And so after it's exited, your atoms are both back in their ground state and you're left with, if you integrate out the photons, a long range potential. But it's not quite what we want. So if you look up here, so there's was in two dimensions, we'll only need this in one dimension so you can confine the atoms so that they're kind of in a wave guide if you like. So if you look at say, imagine that the long axis of the cavities is the z direction, then you have z and z prime, so it's a product of two cosines, but that's not cosine of z minus z prime, of course. So one way to get that is rather than to consider a fabric per o cavity with this fixed where the boundary conditions really fix your mode, your mode structure, you consider a ring cavity. And in a ring cavity, the modes are running waves and you're much closer to the free space situation. So this is the real part of the potential in the laser induced dipole-dipole interaction. You have two atoms, this is supposed to represent a sort of cloud of atoms, but you have two atoms somewhere in here and you illuminate it with a laser. Here r is the distance between the two particles. You get this long range interaction. The r cubed here is going to cancel with this r squared. You get a 1 upon r term, in fact, outside there. But when you put it in a cavity, it doesn't die away at all. You get this, so the cavity gives you this infinite range because you're interacting purely through this cavity mode that extends forever, but you do then get this, because the ring cavity, what turns out is you get the difference between the two positions. In that way, I believe, you can realise, there's one way to realise the Hamiltonian mean field model rather directly. Okay, so imagine we can put a condensate and the experimentalists can do amazing things. They have made condensates inside these optical cavities. So then the equation that describes a Bose-Eye-Sein condensate is called the Gross-Batewski equation, so this is the equation for the condensate wave function. So it's a mean field equation, actually, because we've reduced a many-body problem to a function of a single position. If this was the full many-body wave function, it'd be positions of all the particles, but in the mean field theory, it's just a function of one position. And then we add in, so this is basically a Schrodinger equation, but now we add in this non-linear term, which is just the mean field potential so at any particular angle is due to that propagated to it from all the, by cause, from all the other points. Now something, so this system has been worked on by Chaveny and looks at various instabilities and so on, one thing that actually my student Ryan proved is that this potential is always a cosine potential. Okay, but it has, in general, a time varying amplitude, which one has to work out, and also its position around the ring can move with time, but it still always has the form of a cosine potential, which is kind of neat, and that means you can solve this problem using Mathieu equations, because then this equation becomes the Mathieu equation. You can solve it with Mathieu functions. So what do you get? So on the left is a sort of brute force, numerical solution of the Gross-Petefsky equation from a particular initial state, so we had to really whack it. I mean this isn't a perturbation really, so this would be uniform around the ring and then we've really put in a large density modulation around the ring. The reason we had to put a large density modulation is that the thing damps, that the potential damps away very quickly. I suspect that this is due to land out damping but that's something we're still looking at, but learn behold what do you get. So this isn't at all, this is for this sort of condensate wave function. Nevertheless, catastrophe theory would tell you, well if you're going to get any singularities in the system, they're going to have to be cusps and it's a wavy system, it's a quantum wave in this case, so you're going to get piercing functions and indeed that's what you see and they carry on going although they do begin to die out. If you put, if you solve, you go back and you say, oh what was the mean field potential at each moment in time and then you solve for the trajectories in for a sort of point particle trajectory in that this is what you get and indeed you see these classical trajectories and what's lying behind all of this is of course this cosine potential. So if this was a perfectly harmonic potential and was static you would focus all of these if the property of a harmonic potential just like a lens is it's isochronous and all the trajectories no matter how large their amplitude would focus perfectly to focus points but we don't get that, we get these caustics here because of the non-linearity of this potential and you get these repeating caustics. Another example where you can get these where I found these caustics is in this, in the Ising model you look at now for long range interactions so you have, so every spin is interacting with every other. And, but if we, so if we're in this with these long range interactions we can replace these, we can replace the spin operators for the individual particles by a total spin operator, total spin Z and total spin X and then our Hamiltonian takes this form and this is a famous Hamiltonian that came up in nuclear physics in the 1960s, Lipkin-Meshkoff Glick model and so what you have is this kind of giant spin, this macroscopic spin. It's still a quantum object. Well, depending on how long you make it but if it's not too long it's still going to be a principal quantum object. If you look at the classical limit of it so you associate say the spin, the Z direction with this variable N here and then SX, that's a component along the X direction looks like this then you get an equation that looks like the equation for a pendulum but it's a pendulum whose length depends on if we treat N as its angular momentum on the angular momentum so it's length changes and this is a studied system and in particular you get various types of dynamics you can think of this giant spin living on the surface of a sort of block sphere or many particle block sphere and you get two types where you get ellipses below a certain excitation energy but then you can also get more interesting hyperbolic fixed points and things like that. In particular you can realise this model with basically a bosonic Josephson junction some of the original theory was done by Augusto Smirzi and they've been experiments by Marcus Oberthaler's group and Gershawn here and Steinhauer, Jeff Steinhauer at the Technion and so the idea is you have two condensates in a double well trapping that N represents the number difference between the two difference in the number of atoms and phi represents the phase difference between the two sides and so you can realise this Hamiltonian or a quantum version of it even. Another example would be two rotational states of a BC on a ring or a superfluid on a ring you also get a sort of pendulum Hamiltonian in that situation. So if you solve again so this time I've got rid of this square root one minus N squared that can be important but to see if your number difference is rather small you can set that equal to one so you just look at the low energy excitations then you really just have the pendulum equation and so again we have this cosine potential and if you look at say so each of these lines is a solution of the Josephson equation or in other words the Gross-Batewski equation you can either look for the phase as a function of time or the number as a function of time now this is for a particular so I've drawn a bunch of trajectories here to get these caustic curves so this is the problem, a well-known problem in condensed matter what happens if you put two superfluids that have never seen each other before you put them together? Okay well if they initially start separate they have a well-defined number difference but due to quantum mechanics number difference and phase difference are conjugate variables there so you have no idea of the phase so before they're put in contact they have no phase relationship between them and they have to build it up and so this is the initial condition we take here that you start off in a well-defined number difference I've taken it to be zero I don't have to because of the structural stability of caustics in fact but this is the closest the classical the mean field theory because each of these is a solution of these mean field equations can get to the quantum is by superimposing all these different trajectories say with all possible initial phases and again you see we get these caustics if you look at the quantum version well then the number difference between the two connotates atoms can't be smoothly can't be smoothly they come in little packets of one and so this number difference which is actually a fox-base coordinate is quantised it's normalized here to one but so then you find that that you get this discretised pattern and what's interesting about this is so if you do the mean field theory with all these trajectories and you sum up all these trajectories you get the caustics which is a bit like the classical theory and they are singular here so this is the probability density if you did a measurement at some time this is 3.3 pi it's right over here this is the probability density you would find this number difference you know you would redo the experiment many many times to build up this pattern but it's singular so what you have to do in this theory to get non-singularity to remove the singularity to make things well-behaved is to actually quantise it second quantise it and that is what regularises these singularities these mean-field singularities and so this is an example of a quantum catastrophe at each place where a theory fails you have to go up to the next theory so if you didn't know that atoms existed you believed all fluids were just like water continuum fluids you would have discovered atoms I suppose here okay so in summary I hope you convinced you or at least picked your interest that these catastrophe things are generic objects in dynamics there's at least three levels of structure there's geometric, classical classical wave quantum you can also get polarisation singularity I didn't discuss that so we've looked at the very simple case of just two-mode fields basically then the cusp is the singularity you're going to get there I didn't emphasise it but the cusp has a lot of because it's a well-studied mathematical function it has lots of scaling properties and things like that so once you know you have one of these catastrophes you know a lot you know a lot about your many-body wave function around that region they are regularised by various different things either by going to a wave theory or by going to a quantum field theory where you need really discrete excitations and so the role of the long-range interactions in all of this was to provide this this overall long-range potential this collective potential this focusing potential that generated these structures so an obvious question is well okay this is a two-mode quantum field what do we do let's say you can if you want to go to a multi-mode field then these things okay so the catastrophes are known in higher and higher dimensions but they get increasingly complex and there's too much information at least for me so then I think you'd want to go over into a statistical theory which does exist actually for catastrophes where you look at how moments diverge and so okay thank you very much for your attention