 In the lecture yesterday, we were discussing how to achieve these negative temperatures in the moving gas of particles. And so I just remind you again of the sequence that we were doing. And the general trick, of course, that you apply is that you basically want to take your Hamiltonian and eigenstate of that Hamiltonian, then you flip the Hamiltonian into minus h. So an eigenstate of h will, of course, stay in eigenstate of minus h. But what has become the lowest energy state in the system now is the highest energy state in the system. So this trick, we basically apply here in this sequence, which was devised actually by Achim Rosskup and also similar proposals by Alad Mosk in the Netherlands. So where you basically make this mod-insulating state in a trap, as I said, so where u is repulsive, you have repulsive interactions and you have a trapping potential. And then you suddenly switch the interactions in your atomic particles from u to minus u and from v, the trap, here, to an anti-trap to minus v. And then we said, well, this is going to be still an eigenstate of this new Hamiltonian where we have attractive interactions and repulsive potentials actually here. And but now we're in the highest energy state of the system. And if we ramp down the lattice, decrease kind of the lattice depth again and increase kind of the kinetic energy to interaction energy ratio, we again form a super fluid. But now because we're in the highest energy state of the system, we form a super fluid at the highest possible state in our band. And the highest possible state is, of course, a condensate at the upper band edge of the system. So one thing you might worry about and we have to discuss is actually why is this whole thing stable? I mean, if you have an anti-trapping potential, if you have a trap, that's good. You can trap your particles. They will stay in here. But if you have an anti-trap, then clearly this is a very unstable situation and you'd usually, you lose your particles. So why, why at all are you going to be stable in this situation? Do you have a question or do you want to answer this question? The question, yes. As you to minus you, so I didn't talk about this, and I think Jean's also not going to talk about this. We have this nice tool, what we call Feshbach resonances, where with the magnetic field we can adjust the scattering properties between the particles. So at the knob of the magnetic field, I can basically change the scattering length and thereby the interaction parameter u from positive to negative. I can also go to the non-interacting case, which is also nice, so I can always compare results to a non-interacting gas of particles as well. So this is something we didn't talk at all about, these scattering resonances that occur. But at the knob, what you just need to know at the knob of the magnetic field, you can tune these interactions. And that's very, very nice. And also you can do this dynamically, so you can subtly change the interactions. This is what we need to do here. Not adiabatic, but non-adiabaticly, rapidly jump from u to minus u. Yes. In position, nothing's happening. If you position, this is position space. This is momentum. Actually, in position space, the particles just sit there, localized on each lattice side. And I'm in a deep lattice, and then when I jump to the anti-trap configuration, they just stay where they were originally. So in position space, nothing happens when I switch, okay? But what changes, of course, that now this isn't the lowest energy state anymore of the new Hamiltonian. It's actually the highest energy state of the new Hamiltonian, with the new Hamiltonian being governed by this minus u parameter and minus v parameter. Remember, you had this minus j a dagger a term. You had this one half u ni ni minus one term. And you had this, well, I don't know how I defined it with one half or without one half, v i squared ni term, the potential energy. Okay, and by flipping the sign of those terms, I'm actually reversing this Hamiltonian. Okay, so, and now then you fake of condensate, actually not in the lowest energy state, but in the highest energy state. Let's have a look at that if that really works, a little bit closer here. Here we go. So here's what happens if I go from our super fluid, our Bose-Einstein condensate with this nice interference pattern when I release it from the lattice. So these are time-of-flight images, the one we discussed in the beginning of the class. When you make the lattice very deep, you go to this mod-insulating regime where you have unit occupancy and you see you lose the interference pattern. So this is an indication that we're in this strongly interacting regime with repulsive interactions. Then I kind of switch, I just ramp back down again, let's do that first. Let's just ramp back down again and you see you get this interference pattern back. It's a little bit deteriorated in quality, but otherwise you see it comes back nicely, so it shows you get Bose-Einstein condensate back again. And now you do the switch, and this is the case where we do the switching. And now we ramp back down again and you see you also get an interference pattern. But you get the interference pattern, remember the interference pattern tells you where, which block state you are in. And this is the interference pattern for not the lowest energy state, but the highest energy state in the band. If you recall, the central peak vanishes and you get these side peaks coming up, which tells us that we really have been creating this super fluid now in the most excited state of the system. Now one thing you might worry about when you make attractive interactions, this is actually a very famous experiment from Eric Cornell and Carl Weiman's group, is that the system actually becomes unstable. And this is what happens to a normal condensate. When you make attraction suddenly turn them from repulsive to attractive. The whole system just collapses on each other as you would expect. You have a repulsive gas, it's stable. And suddenly you make everything attractive and then everything wants to collapse on to each other. And this is what they call kind of a Bose-Nova that happens. So it's kind of a violent kind of explosion of the implosion of this gas that occurs in the system. And you might wonder, actually, does that happen here also? And actually we see that's not the case. Here's the lifetime determined through the visibility of the interference pattern of our gas with repulsive interactions in blue and attractive interactions in this anti-trapped case in red. And you see that even this negative interaction case is even slightly longer than the repulsive interaction in case that we see before. So actually this gas is completely stable. So it's another thing we have to note and we will understand in a second actually. And if we actually now look at the momentum distribution, I can just determine the temperature of the gas by fitting kind of a Bose-Einstein condensation, a Bose-Einstein distribution formula in momentum space to my momentum distribution. So remember for positive temperatures you have the low momentum states which have large occupation and lower occupation to higher momentum states. And for negative temperatures you just have the inverse. So the high energy states have large occupation and the low energy states have low occupation. And by just fitting this Bose-Einstein distribution function in momentum space of the kinetic energy to our data, we can determine the temperature of the system and it turns out to be something like minus 2.2. So in this case it's about a few minus 0.5 nano Kelvin in the system absolute temperature measured directly through this Bose-Einstein distribution function. So let's discuss a few puzzling things in this negative temperature. First of all, let's think what negative temperature actually also implies. So if the gas is stable, as we've seen, this means that the derivative of entropy versus volume at fixed energy, we're working isolated system, has to be larger than zero, has to be positive, right? Because we know in thermodynamics the long term limit is the case where we have increase of entropy of the system and that would be the case. If that would be negative, that would mean the whole gas collapses onto itself. And we saw that that's not what's happening, the gas is stable. So the SDV has to be positive in our system for the system to be stable. And that's what we also see in the experiment. Now if we take just DE TDS minus PDV, our first theorem from thermodynamics. And look at constant energy. We find that this DSDV is nothing than absolute pressure divided by absolute temperature, right? So if DE is zero, then this DSDV is just P over T. And if temperature is negative, but this whole thing has to be positive, that actually means that absolute pressure of this gas is also negative. So this is kind of a strange gas where usually on a gas when you kind of want to compress it, you have to perform work on it, right? If you take air, you want to compress it, you perform work on it. This is a gas which has, does the opposite. So here's a gas where you actually have to perform work on it when you want to expand it, that's the meaning of this absolute pressure being negative. Then another thing that's a little bit puzzling but not really a problem is that if we look at the efficiencies of a Carnot engine that we could form by connecting a positive negative temperature reservoir to each other and making kind of a machine out of it, it turns out that if this, if you calculate this efficiency and plug in kind of the negative temperature of one of the reservoirs, you actually find that this efficiency is larger than one and that's something for you to think about. So don't worry, we're not violating energy conservation here. So we're not creating a perpetual mobility, otherwise you wouldn't believe me right away and rightfully so. But actually there's good reason why this energy, this Carnot efficiency defined in this way actually would be larger than one in this case and this is something where you can think about. Actually some statements of the second law of thermodynamics become invalid also in this regime but most, most theorems still hold in that regime too. There's another kind of curious thing that can happen in this negative temperature gas which would be nice to test maybe in experiments is that typically when we send a particle through a gas with positive temperature, we have friction and entropy increases as the medium heats up. Okay, so the particle slows down when you send an impurity through a positive temperature gas. So here actually for a kind of negative temperature gas, when you send an impurity through that negative temperature gas of mobile particles, the opposite would happen, the entropy would again increase. But in this case this would mean that the system has to cool down, okay? So actually what would happen is that the particle will actually accelerate in the system taking energy from the reservoir and accelerating in this process. So you have something like anti-friction if you would send an impurity through this kind of negative temperature gas. So one final thing I want to discuss, so these are all things for you to think a little bit about. One final thing I want to resolve is why this whole thing doesn't roll down the hill. So why is this stable in this case where we have an anti-trap? Why don't we have to worry about the case that here this particle clearly on the hill will just fold out and fall out of our traps? Why can this not happen in our system here? Why is the system stable still with such an anti-trap configuration here that we see somebody who's talking? Okay, can you speak a little louder? That's not, no, not really, that, why can't it roll? Why doesn't it go roll down, gain energy? Yeah, why negative pressure? Yeah, maybe we're getting there, we can maybe explain it like that also, yes? Yeah, expansion, let's think about it, where can this energy go? I mean, so if this particle were to roll down this hill, it would have to convert that potential energy in some other energy, no? So it has maximum potential energy here, and if it rolls down the hill, it loses potential energy and that energy has to go somewhere. So where can it go? What other energies do we have where this potential energy could be converted to? You can't, what the problem is that the kinetic energy has a maximum possible value here. You cannot dump more than this kinetic energy into the system. The kinetic energy has maximum value, so you cannot convert arbitrary amounts of potential energy into kinetic energy, like for an unbounded spectrum. Of course, if you have a parabolic spectrum, then that's no problem, you just convert all that potential energy into kinetic energy as large as you want. But here you have a maximum amount of kinetic energy in the system, and that's actually already maxed out. So that is already at the maximum, this kinetic energy. So you cannot create more kinetic energy, so you have to stay at that point of maximum potential energy. And that's of course only true, the important message maybe in lesson to learn from this is that this is only true for an isolated quantum system. Because the system is isolated, there are no other forms where this energy can go. This is why it's stable. If there would be other forms where this energy could go, for example, in a solid, there would be maybe phonon excitations in the solid. Then all this breaks down what I'm saying, and this would be a highly unstable situation. So only in this case where we have kind of the isolated system, does this work? Yes? Sure, that's a little bit of course the quantum description of that. You would be for the many body system. But you're starting here and then you have interactions. And then you could think why, okay, principally of course for the, but still for an inverted spectrum here, I mean, there are no bound states anymore in that case, okay? So if you would have, if you would just have the inverted parabola, here you wouldn't be in eigenstate at the top of the potential anymore. So here really the main point is that you cannot excite any other degrees of freedom, all other energy channels are maxed out in the energy and you cannot dump energy into those other channels. Yeah, yeah, I mean, maybe, maybe, yes. Yeah, what will happen? Exactly, yeah, what would happen? Okay, let's think about it. What's maybe a little bit deceiving that here we're just plotting the kinetic energy in the energy channel. So let's think about what would happen to an electron if I do, if the electrons were bosons, let's think about that. Let's imagine we could all put them in the same state here. But now there would be phonons in the system also, lattice vibrations. And now what would happen through the interactions of the electrons with the phonons, that's what we're missing here, yeah? The electrons with the phonons you would basically relax down into this state. So this energy would be given up to the phonons in the system. You would excite a lot of lattice phonons in the system and you would actually relax from this point down to this point. So that's what we don't have, right? We don't have this other energy channel present in the system. Because there are no phonons for us. There's no lattice vibration. And the lattice, remember, is done statically through the optical lattice potential, okay? So in contrast to a real solid where these atoms can vibrate, you have a situation here where you just create the lattice statically on top of them, okay? And that's basically, these other channels are usually what prevent you in looking really at these isolated quantum system behavior, though. And you have your, let's say probe, condense matter probe connected to a thermostat in a fridge, dilution fridge, then you always have these lattice vibrations of the thing you're connecting to the copper block maybe where you mount your sample on or something that can be excited. And therefore, it's really hard in those situations to create conditions where you have this isolation as we have here. So it's really the isolation which makes this stable, okay? All right, so that's all about negative temperatures. I want to tell you in this small interlude, and I actually want to switch gears and come back in the second part of the lecture to quantum magnetism and build a little bit on what we learned in the last class on the strongly interacting, how about physics that we had these mod insulators that would form for strong repulsive interactions? But now I want to continue and discuss with you what the magnetic properties of such systems are. And so for the first thing I want to explain to you is actually why we even have magnetic ordering in these systems. And this was actually surprised to people in the 50s when they were looking at compounds such as this manganese oxide or copper oxide compounds. Where they actually found even though the manganese and oxygen atoms and manganese manganese had large separations between them. So there was no wave function over that. There was still an anti-ferromagnetic ordering present in these systems. And usually you know from your maybe your atomic physics or molecular physics course that in order to have kind of spin ordering, you need to have some exchange interaction. And you need to have some direct overlap of the wave functions for that to occur. So in a molecule, you favor your singlet states over your triplet states because there's wave function overlap of the electron clouds. If the electron clouds are completely separated, there's no such ordering of those particles. In helium atom, you favor kind of triplet orientation compared to singlet orientation also because there's direct overlap of the electron wave functions. So what people were puzzled is how can there be magnetic ordering in such situations where there's really no wave function overlap? And this was explained in the 50s by Phil Anderson. And this is what I want to discuss with you basically how we can go from a system like this where we have our Habad model. We just have this tunneling term and interaction term here in the case of bosons. And actually get to something like a spin-spin interaction term in the system, okay? So we're considering two bosons, a spin up and a spin down boson. In this kind of strongly interacting regime, strongly repulsive regime. And we want to understand how we get kind of to this kind of Heisenberg Hamiltonian, that is the effective Hamiltonian of the system that emerges for strong interactions in unit filling in the system. So a way, good way again to understand this is to go back to our simple double well case and treat it as a model system for our lattice. So we think of a spin up, red, and a spin down, blue boson on those two lattice sites. And we have in the strong interaction regime, we remember we have one boson per side. So we have the red boson here and the blue boson here. Now what can you do now? Let's treat the kinetic energy operator. We're in a strong interaction regime. So this term is dominating. The repulsive interaction part is dominating. So let's treat kinetic energy as a perturbation operator and try to see what it can do. So in first order, it can't do anything because it would change the configuration from 1, 1 to 0, 2. And we know that that's suppressed because the repulsions are so strong. But in second order perturbation theory, we can have a process where this particle, for example, hops here with an amplitude J, creates an intermediate state in second order perturbation theory of energy U. And then it hops back, for example, to the original site with an amplitude J again, or the blue particle hops back to create this configuration. These are final states that are created by this process are obviously, but the second order hopping processes are obviously allowed because they still are in the subspace of single occupancy. They're still in the energy allowed subspace without avoiding double occupancies in the system. So now this basically is the identity. Here, we've just created the same state as before again. And in this kind of operation, we've actually swapped the configuration of those two particles. We've made an exchange operation between the left and right particle, okay? So this is what happens. So, and in case of the bosons, actually this is a plus sign here, one plus exchange left right. If you would have fermions, this would be a minus sign here and that will be important in a second. So the strength of this process, the second order hopping process in that we can calculate from second order perturbation theory is just J squared over U. Remember, you have the second order perturbation theory. You have the coupling amplitudes J, J and the intermediate state energy, which is U, which is in the denominator. So you have a coupling strength of J squared over U. And since you have two of those processes that can happen, the blue particle can hop first or the red particle can hop first. Then you have twice this energy two that you see here, okay? And you have this one plus exchange term that can happen in the system. So now we're looking for the eigenstates of this Hamiltonian. What are the eigenstates of this new effective Hamiltonian that we generate? And for that, we just have to look for the eigenstates of the exchange operator. And of course, the eigenstates of the exchange operator for spin one-half particles would be just the singlet states and the triplet states in the system, okay? So this exchange operator on the singlet state here gives me minus the singlet state. Exchange operator on the triplet state gives me plus the triplet state. Again, as you can directly see when you exchange the position of those two particles on those two lattice sites, okay? So we can basically now take this. We see now this is now basically what we have here. The eigenstates will be singlet and triplet eigenstates will be the singlet and triplet states in our system. In our case here, the lower energy states will be the triplet states for bosons where we have the plus sign here. The triplet states will be lying below the singlet states here. Remember the exchange operator here for the singlet state gives you minus sign. One minus one gives you zero, so that's a zero energy. Whereas the triplet state exchange operator on this triplet state gives you plus the triplet state gives you minus four times j squared over u. That's the energy of the triplet state, okay? So we can rewrite this effective Hamiltonian that we derived in second order perturbation theory as minus j exchange, which is just this four j squared over u term times the projection operator onto the triplet state. So this gives us one if you're in the triplet state. This gives us zero if you're not in the triplet state and corresponds to the spectrum that we've just derived here. Now how can we write this projection operator into the triplet state? Well, here's a trick from atomic physics that's useful to remember. If we have two spin one-half particles, and we basically write down the total spin of those, S left, the spin of the left particle plus the spin of the right particle. And square that, you see you get this basically SLSR term, minus three-quarters in the context of kind of spin one-half particles. And so what you actually now see if you take these spin one-half particles that we have in our system. The total spin here would be the triplet state. Actually just give you the projector into the triplet state here minus three-quarter. So actually you can rewrite this projector into the triplet state as just this spin spin term, spin interaction term, spin on the left side multiplied by the spin on the right side plus three-quarter with the pre-factor being just the energy scale of that kind of super exchange term minus j exchange in the system. And what's important to see here that we basically started out from this original Hamiltonian, the Hubbard Hamiltonian. We were in the regime of strong interactions and the only thing we made use of was exchange symmetry to derive an effective Hamiltonian in this configuration, which is just given by a spin-spin interaction in the system. And whether we have a minus sign here or plus sign here only depends on whether we have bosons or fermions, okay? Because that gave us the one plus exchange term or one minus exchange term, which changes the sign here. So in the case of bosons, this is j exchange is positive and we have a minus sign here, so we favor ferromagnetic alignment. The triplet state is the low energy state whereas for fermions we have a plus sign here and we favor anti-ferromagnetic alignment of the particles. In that case with the singlet state being lower in energy than the triplet state. Okay, so this is just what the spectrum would look like. So if you put those double well, make a double well and put those spin one halves in that double well, you have the triplet state at zero magnetic gradient in your system here and the singlet state energy above with an energy separation of 4j squared over you. And this just comes from the, again, from the strong repulsive interactions in the system and the exchange symmetry that we have in the system. Okay, so let's make an experiment and try to see this kind of super exchange interaction. How we call that, that's how we call it. Let's try to see this S-I-S-J interaction. And a simple way to do that is to just start with the spin up and a spin down particle on the left and right side. And basically decompose this S-I-S-J term into the following way. This would just give you S-X-S-X plus S-Y-S-Y plus S-Z-S-Z term. And the S-X term and the S-Y term we can just rewrite in terms of spin flip operators, whereas plus is the operator that takes a spin down and flips it into a spin up. And S-minus operator is the operator which takes a spin up particle and flips it into a spin down particle in the system. And in our case, we only have to look at this flip flop term here. This is just a constant energy offset for us. If we start with a spin up down configuration like this, this we know is not an eigenstate, spin up down, because it's not the singlet state, neither the triplet state. So it shows dynamics. So this state will actually evolve into the down-up configuration back in the up-down configuration, down-up configuration, precisely through the action of this flip flop term here, which flips those spins and flips them from the up-down to the down-up configuration. Okay, let's have a look at that if that really works. Let me just show you how we detect that. So one way how we detect that is that we take these spins on left and right sides and bring the left spin into a different energy band than the right spin just for detection purposes. And then during time of flight, we separate the two spin components with a magnetic field gradient that allows us to measure where the spin downs are, are they on the left or right side, and where the spin ups are. This was slightly before we had these nice ways of using kind of our quantum gas microscopes to detect this kind of spin ordering in the lattice. Well, let's just look at the dynamics. So here's the density evolution in the system when you start just with the spin up-down configuration as a function of time. And you see in terms of the density, nothing's happening. So you keep the configuration where you have one particle left, one particle right, nothing's happening there. But when you look at the magnetization, so the difference of spins on left and right sides, you actually see that there's kind of a very strong dynamical evolution in the spin. And you actually see for the case of very strong repulsive interaction, there's just one sinusoidal oscillation, and this is precisely the super exchange term we've been looking at. This is the super exchange swap operation we've been looking at. The reason why you can see these additional oscillations, why you can see this slow oscillation together with a fast low amplitude one here for weaker interactions, is because now, you can still have single particle tunneling processes as well, which we kind of suppressed in the very strong interaction regime. So these are kind of two processes that you're seeing super exchange interaction plus kind of these very fast kind of single particle oscillations which where the amplitude is more and more suppressed as we go to stronger interactions in the system. Okay, so this is where we get the spin interactions from. So now we should keep in mind that if we start with this unity filled system, if we make now extend this argument to a lattice, and we just put spin ups and downs on this lattice, we can rewrite this whole complicated system which in principle was described by the Habat model by this effective Hamiltonian which is just the spin-spin interaction on neighboring sides. Okay, that's the one thing you should remember for now. So this whole kind of Hamiltonian that we had before becomes a very simple Hamiltonian which just describes the spin-spin interaction of those spins on that lattice, okay? And that's what we derive from the single double well model, but it applies equivalently also to the whole lattice configuration that you can see here. Now there's no direct magnetic interactions or something between the particles, all comes from this virtual hopping processes which take particles, move them here and move them back, which mediates the spin-spin interaction energy in the system. And that's actually true for strongly correlated electron systems as well. So the most dominant spin-spin interaction that arises and strongly correlated electron systems arises because of the strong repulsive interactions between the electrons and the exchange symmetry which gives us this Heisenberg Hamiltonian which with positive or negative values for J exchange here and with ferromagnetic and antiferromagnetic alignments in the system. Yes, yes you can also get anisotropy. Typically if you have spin dependent lattices also a different interaction energies between spin up and down particles in the case of bosons. So then you can actually also tune the anisotropy of the model. So for the for the rest of you let me just, what your colleague has been saying. So you see we have here this term S i S j and this S i S j, I can just decompose into S x S x i i plus S y S y. I should write i j i j plus S z S z i j. So in that case you see the strength of interactions is equal for the x x y y and z z part. And what your colleague is asking is can you have cases where for example this parameter here would be different from this parameter here. You can have an anisotropic spin coupling in the system. And you can have that if you have spin dependent lattices and if you have kind of tunable kind of interaction, spin interactions. So for bosons for example u up up, interaction between two ups bosons on the side is not equal to the interaction of up down boson. For example on the lattice side. But for the case that the one that you naturally get for spin independent lattices and for equal interaction strengths in bosons or just one interaction strength of course for fermions because there's just this term here for fermions you naturally get the kind of isotropic case. And that's what we're going to realize in the experiment also. Okay so let's look at a simple example what happens. Let's go back what I told you last in the last class that we can prepare this situation where one of the spins is flipped in the system. And we want to know what is it dynamic of the single spin impurity in the system. And now we'll see actually that's actually a very simple problem. Let's do this in the case of two component bosons. Again where we have this kind of now ferromagnetic-Heisenberg interaction. Ferromagnetic because we have the minus sign here and Jake's change is positive. So we favor in the chain ferromagnetic alignment equal alignment of the spins. And we want to know what happens if a single spin is flipped in the system. The way we did this remember is we could go in with this line of light, shine it onto one of the rows of the atoms and flip the spins on that line. And then we can just release the atom, the spin there on that line and look and take photos of where the spin is at a later point in time, where the spin down spin is the later point in time. So let's understand what's happening here. So we have the spin-spin interaction term as ISJ, which we could just rewrite as this S X, S Y, S Z coupling terms. And the S X and S Y terms also remember we could rewrite as this S plus S minings the spin flip terms in the system. If I write it like that, you actually also see a nice analogy. This term here is very much like the a dagger a term we had in the single particle Hubbard Hamiltonian. This is like the kinetic energy that moves the spin through the lattice. Just as a single particle is propagated through a lattice through this a dagger a term in the tunneling of the particle. And this here, this second term here, this describes the interaction of two spins at a certain distance next to each other, okay? So this is kind of the kinetic energy and this is the interaction term that we have in the system. Now for a single flipped spin, what can we say about this term? Well, this term will be constant wherever that spin down is. Wherever you put that spin down on the chain, it always gives you the same interaction energy, right? Because the configuration of all the other particles is the same. And wherever you put that spin down, this is always just going to give you a constant offset. And a constant energy offset we can just drop in the system. So the only term that remains for us to understand the problem of a single flipped spin is this s plus s minus term. And s plus s minus is simple because that's just kinetic energy, that's like a dagger a, that's just like tunneling of a single particle in a lattice. So basically we expect exactly the same result like what we got. We just put a single particle on the lattice and let it tunnel in empty space. This is the same thing as putting a single flipped impurity in a ferromagnetic chain and letting it propagate in that ferromagnetic chain. And this is what we see. So here's the probability distribution of finding that flipped spin that we put in at one specific location. And then we look at a later time. For example, 80 milliseconds later we look where that spin is. We make many, many measurements. We record the probability distribution. And we get, again, this nice agreement with what we expect just from Schrodinger theory, just from this kinetic energy term. So that's basically nice. It explains, it tells us the super exchange really works as advertised. We have this coherent evolution again over long times. But on the other hand, it's maybe a little bit disappointing also because it's not so much different from just a single particle in an empty lattice, so why would I even want bother to do that? And things get more interesting if you do that with more particles, more flipped spins. Because then you can have new states where, for example, these two flipped spins would like to bind together. And why would they like to do that? Because now, if we write down this generalized Heisenberg model, now allowing for an anisotropy between the kinetic energy term and the interaction energy term. Now you can see if we have ferromagnetic interactions, and we have neighboring spins which are aligned to each other, they want to stick together. They will have a lower energy if they stick together. Then if we break them up, then this kind of interaction energy would be larger. So they really want to stick together and form a new particle. And this is kind of what you can think of as a magnetic soliton, as maybe the most elementary magnetic soliton that you can have, where these new two flipped spins form a new quasi-particle. And in the literature, this quasi-particle is called a bound magnon and was first derived through an exact description to this beta ansatz that Hans-Beta developed precisely for this context in his work in the 30s. Now there can actually be many more of these bound states. You can have even strings of three or four particles in the system which stick together and which behave as one new compound object and propagate as this compound object to the lattice. So let's try to understand where this comes from. And let's try to see actually how we could detect such a bound magnon motion. So let's flip two spins. With our line, we can not only flip one spin. We can just make a line which is a little bit wider, which flips two spins. So we now have two spins on neighboring sites. And we can expect two things to happen. Either they propagate together through the lattice as this new compound object, or they would just break up and just separate and propagate individually. So those are two possibilities that we can have in the system. And we can actually look what happens. We can just, first of all, look at theory what happens. And let's first look at the case where we have no interactions. So this S-Z-Z term is zero. No spin-spin interactions between the particles, just the kinetic energy of each of one of them propagating independently. And what I'm plotting here to show their dynamical evolution is the probability of finding one of the spin ups at position x1 and one of the spin downs at position x1 and the other spin down at position x2. So initially, at time t equals 0, they sit right next to each other. So that's why you see this blue point here. And as time goes on, we see this is what happens in the dynamical evolution of the probability of finding the spins at x1 and x2. And what you actually see if you have a large probability of finding the x2 spin at a positive position, the first spin, the other spin, will actually show up at a large negative position value. So they kind of break up. You find one here and the other one at the opposite end. There's a large probability of finding it at the opposite end. So there's clearly absolutely no binding effect present here. But we don't expect that because the binding was set to zero. So they just break up. So let's look at the case where we now have isotropic interactions in the system. So a delta is equal to 1. And now do the same thing. Again, it's not too hard to calculate that. And this is what you get. And now you see, actually, again, there's a little bit of breaking up. But now mostly they stick together and propagate together through the lattice on this diagonal direction. So now they move together and be found next to each other as they propagate through the lattice collectively as a new quasi-particle in the system. And if you could make the interactions even larger, if you could make this anisotropy even larger of the interaction term, then this would happen. The breaking up would be more and more suppressed, would be even further suppressed. They propagate slower because the kinetic energy part has become slower. They've become a heavier particle. But now they kind of stick even more together and they propagate kind of collectively together as a bound state through the system. Now let's look what experiment says. Can we really see that? And this is experiment. This is theory. Remember, we're in the case where delta is 1, where we have the isotropic Heisenberg model. And we indeed see, if you look at this time evolution, that we see this collective propagation of this new quasi-particle, this bound-magnon state, we can even see the break up of the bound-magnon as well in the system. So you can see both of those features that theory tells us that we should see in the system. And we can even check how fast does this new object propagate? How fast does a single spin propagate? We can just measure how far did a single spin make it in a certain propagation time and from this slope determine its velocity. And then we can see how the bound-magnons, the objects that stick together, propagate through the system collectively. And we see that that actually, in this case, is actually just twice a factor of 2 faster for the free-magnon compared to the bound-magnon. And from beta's theory, from the beta ansatz, we would expect this to be precisely two times the anisotropy parameter. Delta is 1 in our case, so it matches beautifully with this kind of exact ab initio result that we see. So again, maybe the message to take home message from this is that we can really beautifully see phenomena that typically we can only see in spectral features in condensed matter systems. We can now see in real space and the real time evolution of these kind of new collective bound states in these systems. OK, let me move on a little bit. Let me skip this part. I'll keep it in for you for the notes, because I think it's interesting to learn about how we can control and detect these kind of spin correlations. I want to come back now in the final part of the lecture to this fermion case. We started out with the Fermi-Habat model, kinetic energy, interaction energy. We said when we lower the temperatures, we get the mod insulator. We now have learned that if we have a spin mixture of fermions, we have anti-ferromagnetic interactions, where we basically have a plus term here. So we expect our spins to orient anti-ferromagnetically on the lattice. And this is what we would expect. If the temperature is lower than this energy scale, then we would expect this term to dominate the spin ordering of the particles on the lattice. So as we extend now our temperature axis from high to low temperatures, we start out with this kind of metallic state. Then we go into this mod insulating state, where we have incompressible state with unit occupancy, but no spin ordering that yet, because the temperature is too large compared to J exchange. But now if the temperature becomes lower than J exchange, then we should be able to see also the spin ordering of the particles as they are in the lattice. And first experiments, actually already in 2008, we're able to detect these mod insulators here, these fermionic mod insulators. Later experiments were then able to take signs of a anti-ferromagnetic ordering in this crossover regime. But newer experiments that I would like to talk about now with these quantum gas microscopes have given us much better evidence of this anti-ferromagnetic ordering in the system. So what we really want to understand, then, if we have such anti-ferromagnetic ordering, like here in these 2D systems, what we would like to understand in such systems is what happens when you put charged effects in these systems. So the anti-ferromagnetic case is actually simple. We understand everything from this effective Heisenberg Hamiltonian. The thing that we don't understand, and that's where a lot of the controversy arises, what the correct solution to this is when you, again, allow for some defects here, for some charged defects, and how these charged defects have what they're interplay with this anti-ferromagnetic environment is. You can do that in 2D, or you can do that also in 1D, where you basically take out a particle or you put in an excess particle, create a doubly occupied side, a double-on, or an empty side, a whole-on in the system, and you now ask how do these double-ons or whole-ons kind of interact with this anti-ferromagnetic background that you have in the system. And while this is a whole story for itself in 1D, again, life in 1D is completely different than 2D, and people like Thierry Jamaki, who spent their whole life in thinking and working out what these 1D systems actually do. All right, so let's go back to the experiments and discuss a little bit what has been seen. So here are some of these quantum gas microscope experiments on Fermi gases. We've seen the band insulator yesterday in Markus Greiner groups. There are these beautiful pictures from the Mott insulator for fermions. So these are two components, spin component up and down fermions in the lattice. And you can see that they actually form this beautifully uniform density plateau that we already understood from the bosonic Mott insulator case again. But you don't see where the spins are yet in the system. And now for what's really nice now that actually we've been able, our group and group of Markus have been able to see and other groups, and Martin Svieland's group and also I've seen Barker's group in Princeton have been able to see kind of different anti-ferromagnetic correlations on short, medium, and kind of long-range length scales in these systems. Here's the result from Markus' group, which I think holds the record in terms of low temperatures, which is important because lower temperature means you are more in the regime of this anti-ferromagnetic ordering. So you start here in this undoped case where you have no charged effects. You have a nice Mott insulator. You cool down below this exchange temperature and then you get this anti-ferromagnetic ordering. And they kind of were able to see this anti-ferromagnetic ordering by in their case removing the spin up components from the system. So they're only left with the spin down components. When you see a hole then in this case where there's no doping, you are sure that where there was a hole that actually was originally a spin up particle. And then you can calculate spin-spin correlation functions from that which are shown here in this 2D plot where you can actually see that for very low temperatures, they see very nice anti-ferromagnetic spin correlations over the entire cloud, extending over the entire cloud. Whereas if they heat up the system, if the system becomes warmer, those anti-ferromagnetic correlations vanish and you just have this constant Mott plateau but no anti-ferromagnetic ordering in the system anymore. So getting temperatures lower means you can really now see this anti-ferromagnetic ordering also in the system. And this was I think really a breakthrough experiment in this context. Let me explain a little bit what we've been doing and how our experiments work and how we can actually achieve spin resolution in the system. How we can see what's going on in the spins and the charges in the experiments. And so here are some pictures of our Mott insulators in the short, short space lattice. So the lattice period is 1.15 micrometer in the x direction, 1.15 micrometer in the y direction. This is what we call the short, short lattice. And then we have another lattice where we basically lattice configuration where we double the period in the y direction and keep the period in the x direction the same. And I'll show you in a second why we actually wanna do that. In both cases you see nice Mott insulators. You see this unit filling. There are some defects. We discussed those. Those are typically temperature defects that occur in the system. But there's nice uniform filling. But what's a little bit unsatisfactory is you don't really see where are the spins. You just see there's an atom there but you don't see from these pictures where's the spin up and where's the spin down particle actually in the system. So how can we achieve spin resolution in our detection in the experiments? And the way we are doing this is actually a very elementary way of doing it. It's actually just using the standard Stengala effect that you all know. So let's imagine you have a 1D chain of particles where you have, for example, configuration like this on this chain. Spin down, spin up, spin down, spin up a hole, a double on, and so forth. And you want to resolve that. Taking a direct picture of this will not show you where the spins are. It will just give you maybe a picture like this where you don't know where the spins are sitting. You know where the holes are sitting. The doublons, but you don't know where the empty sites and where the spin ups and the spin downs are. So the way to resolve that is to do the following to basically take a single well, single of one of those lattice sites on one side and split it in the transverse direction into a double well. Okay? So we take one site and split it into a double well. And now we do this under the action of having a magnetic gradient field present V prime during the splitting process. And what that will do is we'll actually do something different for a spin up or a spin down particle. For example, a spin up particle will, under the action of this magnetic gradient, be pulled to the left side, whereas the spin down particle, due to its opposite magnetic moment, will be pulled to the right side of this double well when we split them. Okay? So it's really Stern-Gerlach experiment, but in a very controlled way at the level of a double well in the system. So what happens now, for example, to this single chain is the following. You take this chain, you split it under the action of this gradient, and then the red particle goes into the lower chain, the green spin up goes to the upper chain, and that happens for every lattice site on the system. So now a configuration like this will be mapped onto these two chains. And those are the ones where we take an image of now. But for these new chains where we take an image now, we know that the upper chains are the spin up particles and the lower chain are the spin down particles. So now I have everything I want. So let me give you an example. So this is a chain that we've split like this, and I want to reconstruct what was the original occupation of the single chain that image in the splitting process. Well, I can read it off directly. That's an empty site, obviously. Another empty site, then there's a spin down, a spin up, a spin down, a spin up, a spin down, a spin down, a spin up, and a spin up and a spin down, a doubly occupied site. So I can directly tell you where the spins are, where the holes are, where the doubly occupied sites are in this single snapshot of the system. That's like if you are coming from Monte Carlo simulations as a theorist, that's like getting a single snapshot in your Monte Carlo simulation. But this is not a simulation. This is the real thing. These are the real atoms, the real fermionic atoms on the lattice. And like that, we can get these snapshots. But of course, then we can also calculate correlations and so forth. So let me show you how this looks like. So here's a case where we have a 2D mod insulator. Charge is resolved. But obviously, spin, when I talk about charge, sorry, I'm using a little bit of lingo. Remember, we're dealing with neutral atoms. So there is no charged object here. It's just the neutral atom which represents the electron in a real solid. And there, people talk of charge, of course, because the electron has charged and we adopt the same language. So don't get confused when I talk of charge. There is no charge here. When I talk about charge, I actually mean density of the particles. Yes. Yes, it also resolves that because doubly occupied sites are also separated on single sites. But only up to occupation of 2, but we don't have more than occupation of 2 for fermions if we're in the single band model. Because we can only have, what can we have? We can have no particle. We can have one spin-up particle. We can have one spin-down particle. Or we can have a spin-up or spin-down particle. We cannot have up-up or down-down, like for the bosons. So for the fermions, it indeed solves that problem completely. If you're in the lowest band, you have full resolution of all those four configurations. We thought about that, indeed, in the context of a terbium. That should be possible also. But there it would be maybe one split it, but you would move different transition frequencies to try to resolve the different spin states individually, sequentially after each other. But that's tricky, and we don't know whether that will work. Because when you have ever have scattering light around, usually to image the particles, it has a damaging effect. So what you want, if you want to just first image one spin component, you want that it doesn't damage the other spin component while you image. So that's subtle. So there should be possibilities, but we have some ideas. But we haven't tried them yet in experiments. I cannot tell you how good this works. All right, so here's charge resolved, spin unresolved. Now we do the splitting. And then we can see, basically, for each of those lines, we see where the spin ups and where the spin downs are. Now we have complete charge and spin resolution. And now we take many, many images, and then you get all the statistics, all the different snapshots, and you can calculate correlation functions from that. So let me start maybe with a simple correlation function, which is a standard two-point correlator, where you just now correlate, how is the spin on site I correlated with the spin away a distance i plus d away from that original site, a distance d away from that original site, and then you just subtract your uncorrelated background that you, of course, always get from that to reveal kind of the correlations in the system, the quantum correlations in the system. And what you see then is that you need in this 1D configuration, you see indeed this nice kind of anti-firm magnetic ordering in the system. Now, as Jean explained in the colloquium yesterday, there's no long range order in low dimensions in 2D or in 1D here, so we cannot expect this to be a constant like what we would get in 3D, in a 3D system. Instead, in 1D, even for zero temperature, we would have decaying spin correlations which have an algebraic decray here in the system. So even for a 1D system at t equals zero, we would still expect this to be algebraically decaying spin correlations in the system. In our case, temperature, finite temperature still dominates kind of exponential decay of those correlations in the system, but you can see, we can still see these spin correlations up to maybe 10 sites out in the system, okay? You can see these anti-firm magnetic correlations. And they're not so weak, they're actually compared to what we would expect from the ground state Hubbard model. If you look at this next neighbor correlation, the correlation of neighboring kind of spins. In our case, this is like minus 0.45 if we would have the 1D Heisenberg model, we would expect minus 0.6. If we take the 1D Hubbard model for our parameters, it would be a little bit lower, minus 0.56. So that's not so far from the result that we have here. So we are indeed in this lower temperature regime of the Hubbard model in this 1D setting. Let me show you a single picture. Too fast here. A single picture. Here's a single snapshot, which is one of these up-down, up-down, up-down configurations in this building result way. However, this picture is very misleading because it suggests that the state that we've created is really this classical state, up-down, up-down, up-down, up-down. That's, I should caution you, this state is the one you see there. That's not at all the ground state of this Heisenberg model. This would be the ground state if we only have the S-Z-Z interaction term in the system, so far I would have S-I-S-J, just Z-Z interaction. Then obviously this state will minimize the interaction energy of this term here. But the S-X and S-Y-A-S-Y terms create kind of much more interesting quantum correlations in the system. And actually, even though we like to show this picture, this is a very low probability picture that you get even if you're in the ground state of this Heisenberg model. But we still like to think of the antifurma net in this classical picture, but keep in mind it's something much more complicated. Something with a much more entangled structure and this kind of classical state that you see here. But nevertheless, in the following, I want to adopt this picture, at least helps us with our intuition to understand what's happening in this antifurma magnetic systems. So with that said, let's think of this antifurma net as up, down, up, down, up, down, knowing that with all the things, reservations that it has. And now we want to move forward in the step of understanding what do these actually holds, what do these charge impurities do in that system? And how does that depend on the dimensionality of our problem, for example, as well? So what we can do is we can take, for example, one spin and just remove it from the system. Let's take this spin and just remove it. What you see now beyond the two holds, you see now a spin, up, spin, up configuration. And now what can happen, this hole will, of course, now tunnel. We said holes are like particles. They tunnel in the 1D configuration, so it will just move through the system. And it will leave behind, as you can see here, another excitation of the spin configuration. This has caused some magnetic energy costs because this is not an antifurma magnetic alignment of the spins. So it will leave behind a spin excitation, which we call the spin on. And here's the whole excitation, which we call the hold on, and they kind of propagate independently through the system. So that's kind of already an interesting thing. You've kind of removed a particle, an elementary particle, all right? And this kind of has fractionalized into two excitations, a spin-on, quasi-particle, and a hold-on particle in the system, okay? This is kind of a very celebrated example of a spin-charge fractionalization that happens. So you take an elementary particle with kind of spin and density, and you remove it, and it fractionalizes, and it breaks up into two elementary excitations of the many-body system that you can see here. Okay, so in this case, actually we see that the hole moves independently and moves through the system independently, and this leaves a spin-on behind. But we can ask, is this really the ground state of the system? Wouldn't it be better? Actually, I'm missing one slide here. One slide, let me see, find that slide. Sorry for that. I wouldn't get that slide for you. Yes, this one. Get it right away up here, but in the notes also, sorry for that. So now we can ask, and that's what happens, we violently remove one particle, we create a hold-on and a spin-on excitation. But let's now ask, actually what is the ground state? And what's not the excited state? What's the ground state of a chain of these spin-on halves where it just took out one particle, where there's one particle missing? And then we have to basically consider two things. We have the hole, the missing particle, and it will have minimum energy when it's maximally delocalized, right? Any single particle has minimum energy if you delocalize it maximally. And the hole is like a single particle, so in order to create a minimum energy state, you actually want to delocalize the hole as much as you can. And the spins, they want to align antifurma magnetically because of this term here, which gives us this lower energy cost when they are antifurma magnetically aligned. So you have these two competing effects that you would like to minimize at the same time, and you ask yourself, what is the best way how I can fulfill that? And the best way how you can fulfill that is the following way by writing down the many-body state in the following kind of cartoon picture as this antifurma magnetic background with the hole being delocalized over the entire chain, so it's a coherent superposition of being everywhere on the chain. So that minimizes its kinetic energy, but at the same time, having kind of an antifurma magnetic configuration around this hole. And not the ferromagnetic configuration that we would get if we would just violently remove this hole and create these excited states, the spinon and the hole on behind, and you would see here a ferromagnetic alignment around the hole, but rather for the ground state, we would actually expect a antifurma magnetic alignment around those holes. That's the lowest energy state that we can have in the system. So what we basically see from that is that these holes for us act like domain walls in an antifurma magnetic background. So whenever you encounter a hole, you basically want to shift the antifurma magnetic order from plus one to minus one. If we denote the ordering, let's see a down, up, down, up order by plus one and the up, down, up, down order by minus one, whenever we encounter a hole, what you want to do to create the low energy state is kind of shift the parity of the antifurma magnetic order from plus one to minus one. So if you want to think about it, these holes are like domain walls or parity kinks in the antifurma magnetic background. Let's see if we can see that. And this is kind of a first example of something you really cannot measure in a standard condensed matter experiment. This is the three-point correlator that allows us to reveal whether around a hole we have antifurma magnetic alignment. So let me go slowly through this. So the correlator we want to measure is the following. We want to know whether a spin on this side, i, how that's correlated with a spin on side i plus two, so two sides away, conditioned on having a hole in between. It's a three-point correlator because you have the two spins that you're correlating, but you have a conditional probability conditioned on having a hole in between them. Okay? So this is the three-point correlator that you see here. And here's the undoped case. If we would now have no hole, you see you have for the C2 correlator here at distance two, you have ferromagnetic alignment of the spins. But if you can actually remove that hole, if you have this doped situation of a hole there, then actually we find indeed antifurma magnetic correlations across that hole. So if we have a hole, then the spins do not align ferromagnetically around that hole, but they aligned antifurma magnetically around that hole. And this is not only so for just the single environment here's in this more complicated diagram, you can actually see that the whole antifurma magnetic order is shifted by the presence of this hole. So here's the correlator again I'm plotting, the spin-spin correlator conditioned on having a hole between or beyond the two spins. So the spins we're correlating are at side i and i plus d. And the hole sits either inside those two correlators, inside those two spins when s is smaller than d or beyond the two spins when s is larger than d. And depending on whether s is larger than d or s is smaller than d, you can see that this up-down, up-down configuration is flipped into a down-up-down, up-configuration. If you look carefully, the colors are just shifted by one column, indicating that this spin correlator has just flipped in sign whenever there's a hole in between the system. And the whole antifurma magnetic background flips. So now let's just discuss and come back maybe to something also that Jean talked about, topological order in his lecture. What happens if you have more holes? So one hole we understand now, one hole makes a single domain wall flip. But what about many holes that you have in the system? So if you have actually many holes in the system, what happens that each of these holes introduces this domain wall kink that we saw before that flips the antifurma magnetic background parity from plus one to minus one. And if we would kind of look at this system a bit closer and you stare at it longer, you'll see that actually if I could somehow just remove the holes, it would just look like an undoped antifurma magnet in this kind of squeezed space. So this is the full system. Each time there's a hole, there's this parity flip. But if you remove the holes and squeeze the system together, you see actually the antifurma magnet you get is a perfect antifurma magnet again, like in the undoped case, okay? So in this call called squeezed space, this dope system looks like a perfect antifurma magnet that we have in the system. And the reason for that is a very special thing in one dimension and this is so-called the essence of spin charge separation in one dimension that was originally derived in 82 using this beta ansatz from Hans beta that he also used for the bound states, Magnum bound states by Wojnarowicz et al and data by Ogata and Chiba, which says the following thing. It actually is a very powerful statement about the many body wave function of this very complicated systems, of the spins moving on the lattice, interacting with each other. It tells us that this many body wave function actually factorizes into two parts. It factorizes into a part of spinless fermions which just determined the position of the particles on the lattice. And the second part, a magnetic part which determines the magnetic structure on this fictitious lattice with M sites where M is smaller than N where just all the holes have been removed and we have a perfect Heisenberg anti-firm magnet on this fictitious lattice living on lattice sites where all the holes have been removed in this fictitious squeezed space. And the fact that this wave function factorizes in this density part and the spin part is really the essence of what we call spin charge separation in 1D. It's a very special feature of one dimensions where we have this kind of factorization of the wave function. You can actually reveal these kind of correlations that are present in the system also by introducing something that we call a string correlator where you now basically correlate the spin on one side with the spin on another side. But remember to reveal the correlations you always have to take care of the fact that whenever there's a hole, the whole spin order will be flipped by minus one. And this is what this intermediate operator takes care of. Whenever it encounters a hole it introduces a minus sign in the spin correlation. And if you don't have that part here you will not be able to see those spin correlations I'm going to show you actually in a second. So this brings me just to an interlude to put this into context of discussion of order parameters, non-local hidden order parameters. Let's go back to our original definition of an order in a many body system. So if we think of a typical many body system and a typical order parameter in this Landau paradigm of phase transitions we would define it in the following way. We would take the correlation function of an observable A, measure the value at point X, correlate it with a value at point Y and with X and Y going to infinity if that goes to a constant value we would say well there's a long range order in the system which extends over the whole system and we can introduce an order parameter here which will be just kind of this expectation value of A. What are the famous examples of this? Well a famous example would be just a ferromagnet all spins pointing in the same direction that's easy to understand. If you know the spin points up here then you know it also points up over here. As far as you go it will point in the same direction. The same thing is true for a condensate wave function. Whenever you know the phase of the condensate wave function in three dimensions and higher dimensions you know it somewhere, you know it perfectly what it's going to be somewhere else. Or also for a Steinert BCS superconductor they are all famous examples of such kind of order parameters defined via this Landau paradigm. But of course you can ask is this a general classification scheme for all phases of matter? Are all phases of matter characterized like that? And what you should have learned by now and also from John's talk that's not the case. There are phases of matter that show have hidden correlations that we can reveal however often through such kind of hidden correlation functions. Why do we, first of all let me just explain what it is and then why we call this hidden. So you see now it's very similar to the normal one. You correlate observable A at point X with observable A at point Y. This is also here. Observable A at point X correlated with observable A at point Y. But in order for this to show that there's some order or that there's some constant value here you not only need to know what's going on at X and Y you need to know what's going on on all the points in between with some observable B. Okay that's a general, for example, definition of such a string order parameter. People have called this a hidden order, non-local order or Y because basically in order to reveal these correlations you not only need to know what's going on at two points you need to know what's going on everywhere on the system. You need to know what's going on at X and Y on all the points between. And people never thought that something like that could be measured because for an electronic system it would mean you may need to measure spin-spin correlations here but you need to know what all the electrons in between those two spins are doing as well. And that's why people talked about these as hidden order parameters or non-local order parameters because you really need a global view on the wave function. You need to know all that's going on on the wave function not just at two points but at these kind of different positions in between as well. So this is a very powerful concept and applies actually in our case as well where we have these hidden correlations. It's actually also at the bottom of Haldane's famous spin-1 model where there's this hidden order and to reveal it you have to use these spin correlations, these kind of string correlators. I don't want to go into this because of time. I want to come back to our simple example and connect it now to the experiment. So again, what are we having? We have this system with this spin chain, spin-1 half particles. We have a lot of holes in them. If I just measure the standard two-point correlation function of spin z at position i with spin correlated at position i plus d, I will find no correlations in the system for this dope system. Only when the spin sits right next to each other you know they're going to be anti-firm magnetically aligned. But for larger distances you basically have no correlations between the spins. So if you see this data you would tell me, well there's no anti-firm magnet present in the system. But remember, there was something there, the holes introduced this deterministic flip. Here we're just not taking care. We're not taking note of that deterministic flip. We're ignoring it. So having a random number of holes between the two spins of course means that those two spins can flip up and down and therefore their correlation function will vanish. So I'm just thinking there are no correlations because I'm not taking care of what the holes are doing. So now if I do take care of what the holes are doing if I do measure the holes and each time there's a hole I kind of introduce this sign flip in my correlation function and I measure, then I do see this anti-firm magnet that's present in the system. So I do see that the system exhibits anti-firm magnetic order and indeed it can reveal this hidden order parameter in the system. So I think this is very nice because it shows us that we can detect topological order through very complicated non-local kind of order parameters and I think there's hope now in experiment at least in these quantum gas experiments or also maybe ion-tribe experiments that we can indeed directly see this hidden order in the experiments. And that's something you could not do in a standard condensed matter experiment, yes. It is the ground state because we're just doping it. We're not introducing the hole violently so we're not making, we're just starting out with a density that's incommensurate with the lattice, okay? That's an important point. So we're always talking about the ground state physics here, we're not talking about the case of violently removing some holes out of the system, introducing holes like that. I'm talking about a system, imagine the following situation. Imagine you have a 2D lattice of 100 sides. If you have 50 spin up particles and 50 spin down particles, then you have unit filling, you can match every lattice side to a particle in the mod-insulating regime. So you will get a nice anti-firm magnet like what I showed before. Let me go back to this, back to this picture here. Okay, so let's say you have, okay this is a little bit larger, but you have as many atoms, as many electrons as you have lattice sites and half of them are spin up and half of them spin down. So you can match every lattice side to a particle. So but let's, and this is the ground state. This Heisenberg anti-firm magnet is the ground state of that configuration, okay? But now let's imagine a situation where you have for example 95 particles, but 100 lattice sites. And now what I'm asking you, what is the ground state of that system? Okay, and that's what we're discussing. Okay, so I'm not removing those, I'm really trying to find the ground state of a doped system where the number of particles in my case doesn't match the number of lattice sites. You have as much as mine, and you have even more, you have the hopping of the particles because we have to go back actually to the Habad model to solve that problem, yeah? That's what I tried to explain in the beginning, that the classic, so we are, and when I draw like this, it's the ground state of this, of the SESC, the classical uncorrelated system. We just like to think of it like this. We picture it like this. The Heisenberg anti-firm magnet is something much more complicated is the ground state of this. Okay, but still what we're doing in the experiment is the same thing. Think about a situation where you have 100 lattice sites, okay? And you make the ground state of your Habad system, but now you just have 95 particles. So you're doping the system effectively, yeah? And that's what we've created. Come back to your question and say, and that's what we did in the experiment. So we didn't go in the experiment, create a uniform system, and then just kick out some particles. We started out with a system which has a lower number of particles, yeah? So we have, let's say, quantum holes in there, let's call them quantum holes, ground state holes, and we try to look how they arrange in the system. We don't, we just measure them. It's in the measurement process. So what is curly, we don't. I mean, this is the point. I mean, if you go back to the ground state of these quantum holes, let me go back to this slide, which was kind of important, I think, to explain how can we think of the ground state of this system, this doped system, okay? We say that we have the charge, the hole, in order to minimize its energy, it's going to be delocalized over the entire lattice, so we don't know at all its position. In the ground state wave function, it can be everywhere on the system. And the spins want to anti-firm magnetically align. And in this cartoon picture, I just write it in this icing notation, up-down spins, so the configuration, actually the ground state configuration will be this one, of a single hole. And you notice that's different from this one where you just remove that particle. If you start from a standard anti-firm magnet up-down, up-down, up-down, you just remove one particle, this would be the right-hand configuration. That's not the ground state. The ground state is this. And what's different from here to here? The difference from here to here is, for example, that across a hole, here you have anti-firm magnetic alignment, whereas here you would have ferromagnetic alignment if you just abruptly remove the hole. And that's what we saw in Malcolm I data, where I indeed showed this. Okay, yes. Again, I cannot, I would not, you're totally right, and that's not how we're doing it. I'm not removing a hole, I'm not removing a particle, and then trying to recool this. That's not how I'm introducing holes into the system. Oh, what you would expect is precisely this, and this is what we're trying to see now in the lab, actually, when you either remove abruptly a particle or you inject, you put another electron on the lattice side. This electron will now break up into this two degrees of excitation of the system. Hold on, a charge degree of excitation, which just carries no spin, but just charge density. And this spin excitation here, where the spin, the spin one-half of this spin on excitation is left behind. So what we're looking for in the experiment, you can also do that. You start with an anti-firm magnetic chain. You abruptly remove one particle, and what you should see now is a hole propagating in the system from that side, but you should also see that correlated with the spin on being propagating also in the system. Okay, so you are basically putting in, and that's what's the beautiful thing about this fractionalization. You have one elementary particle, the hole that you remove, or the particle that you remove, or another particle that you put in, a spin one-half particle of charge, for example, and it breaks up into these two degrees of freedom of the collective system. And that's what we're trying to see. So this is dynamical spin charge separation. This is seeing spin charge separation dynamically. The spin part of the electron and the charge of the electron breaking up into two collective excitations of the many-body system. And what we are seeing, what I've shown you so far, this here, this is the ground state version of spin charge separation. So typically people, I mean, a standard condensed matter physicist would have probably not thought about measuring spin charge separation in the ground state because you would have to measure these non-local order parameters which people thought you could never measure. So everybody told me and still tells me what you have to do in order to measure kind of a new topological phase or something of the system, you have to measure some excitation branches. You have to measure some dynamical new excitation branches that appear in the system, like the spin and charge excitation branches in the system. So what I want to remind you of that actually connected to this dynamical effect in the collective excitations, there is a ground state kind of order in the system that is connected. And this ground state order is typically hard to reveal or impossible to reveal because it's a hidden order parameter. It's a non-local order parameter. Okay? And I think what's new here is that we can really see these things. We can not only see them dynamically through excitations, we can directly see this order, I think in a much more direct way then see also the underlying correlations of that many body systems and just looking at the collective excitation branches. Yes? Okay, so the cooling part is, I didn't at all touch upon the whole cooling part. It's a very good question. It's a very subtle question. So we basically, we start out with this 50-50 mixture of particles, okay? And then we basically, we just evaporate those particles and we didn't talk about this evaporative cooling and we basically just remove the hot atoms out of the system. This we all do in the continuum, not in the lattice. So there's no lattice even there. We just have a gas and a harmonic trap, spin up and down particles. We remove those. We just start to spin up and down particles in the trap. They're just flying around here. There's no lattice yet. There's just a harmonic trap, okay? But the high energy particles, they typically make it further out in the trap. Okay? If you have a particle with a lot of kinetic energy, it will go higher in potential energy. And if we can kind of open up the trap here, get some point, okay? Let's say we can open up the trap, make somehow an avoided crossing here. Then this particle will just fall out. And by that we can remove the most energetic particles from the system. Then we rely on collisions between the remaining particles to reestablish thermal equilibrium in the system. And that will of course then be at a lower temperature. So like this, we try to achieve the lowest possible temperature of the continuum gas, okay? And then we turn on the lattice. That's the point when we turn on the lattice, okay? And then we load into the lattice. And now to determine what the density, how many holes we have in the lattice on average, that is determined now by what the trap is, how tightly confined we make the trap. So make the trap rather loosely confining. We can make the gas spread out over a larger region in space. So we introduce more holes. If we make it more tightly confined, we can compress the gas and compress it to basically create this unity filling region better in the system. Or even create doublons. We compress more. We can create doubly occupied sites in the center, which is another form of doping which we're actually going to come to in a second. Just by comparing to theory of course. And that's basically what we do, right? I mean, when we do these experiments here, we can take reference states. And this is actually all nicely calculatable. So this 1D situation, you can calculate very well. And then from the spin, actually no, I had it before. For example, from the strength of the spin correlations here. Okay, so these are measured spin correlations as a function of distance. And from these spin correlations, we can compare to Monte Carlo simulations, for example, and derive the temperature of the system. Now that becomes of course increasingly difficult if you don't have numerical methods anymore that can help you. And then we have to find other ways and all other reference points to estimate the temperature of the system. So this is how we derive the temperature in the system. In this case, by just comparing to theory. And actually also I should say something important. Remember what we're doing in the experiments where all these changes, when we cool and then load into the lattice, we're not connected to a thermal reservoir. You remember that. We have an isolated quantum system. What we try to remain is what we, so we're trying to actually create not a minimum temperature state, but a minimum entropy state. So we're trying to minimize the entropy of the system. And when we adiabatically turn on the lattice here, the entropy here in the lattice should equal, let's say this is the continuum situation, should equal the entropy of the continuum system. If we do everything adiabatic, if we turn on the lattice very slow, then the entropy here will be the same entropy as here. And then we can try to achieve the lowest temperatures in the system. But temperature throughout all those processes never stays constant for us. Because when you change lattice steps and manipulate parameters, temperature always changes. But the important thing that remains is that the entropy remains constant in those systems. More questions? Comments? Okay, so let's continue a little bit on this 1D situation and discuss a bit more what you can see, because this is not the end of the story and brings us to rather, rather new experiments where we actually look a little bit more carefully how these holes actually behave when you dope the system. So we said we introduced these holes, they introduced these parity flips, but let's take a closer view a little bit what these holes can do to the anti-firm magnet. And actually there's this very famous Luttinger liquid description of these one-dimensional many-body systems which allows us to calculate that in a very effective way. What that predicts is that basically these holes, they kind of squeeze in the anti-firm magnetic order and lead to basically incommensurate spin correlations with the underlying lattice. So if you now calculate the two-point spin correlation function for a given density, for a fixed density, then you would find that the anti-firm magnetic wave vector is actually just stretched by this end density factor when the density at the same way goes below one, above one, you get the same stretching effect for that kind of introduction of the holes in the system. Or if you introduce additional spins in the system, let's say you don't dope it with charges, you introduce additional spins so you magnetize the system so you have more spin ups and spin downs in the system, then you actually also stretch this anti-firm magnetic wave vector in the system. Now in order to see that, you might ask why don't we, can't we see that directly? Well the difficulty in the experiment is to see that you really need to work at a precisely fixed density. If you have varying densities in the system, density fluctuates, then you're just going to average over those different densities and you're just going to average over those different wave vectors, so you're not going to see that. So you need a way how you can actually measure precisely at a given density or at a given magnetization in the system. How do we do that in the experiment? And that comes maybe a little bit back to your question, how do we actually control density magnetization in the system? We only do it to a certain degree, but post-selection is very powerful here, okay? So if I have a chain, I get like many many images. So these are different shots that we take and we analyze them. We analyze them for example, for the number of atoms we have in the chain, for the number of spin ups and downs that we have in the chain, so the magnetization of the chain, for the number of doubly occupied sites or the holes in the system. So you actually see, we don't have such a great control of how many atoms we have in the chain or what the magnetization is. But now I can go in post-analysis and say, okay, let's just look at the chains where we have absolutely equal number of spin ups and downs, where we have zero magnetization. Or just look at the chains where I have one more upspin than when I have downspins, okay? So I have magnetization of one minus one or so forth. So I can just go into my data set and post-select all those data of a very defined magnetization or very defined density and then analyze the data for that. And that's the powerful approach that we can bring to this to looking really at the defined magnetization and defined densities in the system where we basically select the shots in post-analysis for magnetization and density. And by doing that, we can indeed see now this stretching of the antifermagnetic wave vector. So here's the antifermagnetic correlations plotted again as a function of the density in the system. And you can indeed see, as you introduce these holes in the system or excess particles, you see that the antiferminetic wave vector stretches out precisely like in this lutein gel liquid prediction in this linear fashion that was predicted by theory. We also see that in magnetization. So here, if you introduce this magnetization, extra excess magnetization in the system, we see that the spin density wave vector, so the kind of wave vector of this antifermagnetic ordering stretches out linearly as a function of the magnetization of the system that we put in the particles. So that's a really powerful thing that you can just go and post-selection and work at a very precise magnetization, very precise density in the system and analyze for these effect and for the first time really directly see this incommensurate magnetic behavior of these lutein gel liquids in this 1D configuration. All right, how are we doing in time? Okay, I will just, yeah, let's just then go to the final topic and let's just extend this maybe in looking a little bit in the final part, what happens in 2D. So everything I said was one dimension and as you see, the theory is very well established in one dimension. What's much less clearer is actually what's happening in 2D. So again, we want to understand what is the effect of single charge or density impurities in this anti-ferromagnetic background and we can invoke the same idea notion of what we had before. We have this 2D anti-ferromagnetic system, now we have a density impurity in the system and now we want to find the ground state of the system. Now obviously if the excess particle, this doublon is localized to a single site, it will cost us a lot of kinetic energy but we have a low magnetic energy cost so that's not going to be the ground state. On the other hand, if we kind of move this particle around, you actually see that as we delocalize it, it reduces its kinetic energy but when you move this doublon around, you actually see that it leaves behind a string of particles where there's no anti-ferromagnetic alignment anymore. So just picture this and just think of this red particle moving here so the doublon is moved here and it now leaves behind this up configuration on these bonds. Obviously that's not a good anti-ferromagnetic correlation so that's going to cost you magnetic energy. So as you want to move further, if you want to move further, you move the doublon even one side further so you move the spin up here then you see it leaves behind even more kind of flipped spins in the wrong direction. So this actually already shows you a very fundamental difference of the 1D case to the 2D case. In 1D, the excess charge could move completely freely but separate from the spinon excitation. There was no additional energy cost as far as the particle separated. The further the particle separated, there was no more energy cost as a function of distance. Here in the 2D case, we have something like confinement where this doublon cannot separate, cannot move a long distance without kind of leaving behind these kind of magnetically flipped particles. So here there's really restriction of this particle, this whole of this excess kind of density carrier of moving in the system because there will be an excess magnetic cost in the system. So instead of 1D, where you have the separation of spinon charges, we expect a completely different behavior in 2D. Here in 2D this cannot happen and the question now again is what is the ground state for a single impurity for a single hole in a 2D system? And you might think of two things that could happen. Well either the hole again is a coherent superposition of being everywhere and now dragging around with it a kind of slightly more ferromagnetic background around it, a modified antifurromagnetic background around it, or there's an overall maybe reduced antifurromagnetic background which makes it easier for the hole to move in general through the AFM system with less cost. And the question which of these two pictures is actually true? That turns out that the first one is the more likely situation that we should think of this impurity, this charge impurity in the 2D situation now as a whole with a region surrounding it which has ferromagnetic correlations instead of antifurromagnetic correlations such in this local region that the hole can explore, it's easy for it to move, but it can't go beyond that. So you can think of this hole now as a new object, a polaron, which is formed by these ferromagnetic correlations around this single impurity. And there are several kind of simple pictures that you can evoke to calculate the size of this kind of object, this polaron, this modified ferromagnetic region around this density impurity, and it all relies again on this competition of charge and spin energy costs of the kinetic energy costs and the magnetic energy costs in the system to calculate this radius. And we can directly see that. So let me show you that it's now really nice that we can also see these modifications in a 2D environment. So here we're looking at how the C1 correlators, so these are the next neighbor spin correlators, so this is the spin correlator between this side and this side. We plot as an ellipse of some color here, and we see that this is kind of the basically undoped region, and here there's an excess particle. Here's our doublon in the center. And you can see how in the region of the doublon this anti-ferromagnetic environment is actually reduced. The spin correlations are less strong in the region of this doublon that we have here. If we look at the diagonal correlators, it's even more striking. So now we're plotting the spin correlation functions on the diagonal between this spin and this spin, this spin and this spin, this spin and this spin. This is given by this diagonal bond and the strength of the correlator and the sign of the correlator is given by the color value that you can see here. And you can see actually in this vicinity of the doublon there's a very, very dramatic effect. Instead where you have ferromagnetic diagonal spin correlations in the bulk around this single impurity you now change to anti-ferromagnetic correlations around this impurity, okay? Yes? Can a bit louder? There are symmetries while there's a little bit just statistical, just experimental noise also. This, I should say this includes to measure this is really challenging. It has over 20,000 pictures going into this. So a lot of this is just statistical fluctuations. And since in this picture, at least we have a hard time plotting the error bar, you see a little bit of inhomogeneity in the system. But overall the picture is clear, I think. What happens that you have this modified region around the charge carrier, you have this modified anti-ferromagnetic background. And there's clearly a kind of a range of this anti-ferromagnetic background over this density impurity where you have this modified anti-ferromagnetic region. Also here you'd say, and that's probably just statistical fluctuation that this is negative should be positive as well here, okay? And we can see this even in the C2 correlators and now it gets a bit messy and we didn't find any better way to plot this. So we're plotting now how is this spin correlated with the two sides way spin in all directions? Okay, so it gets a bit messy, things are crossing. But what's actually interesting that you see that across this doubloin, what's interesting here, you have kind of, again, anti-ferromagnetic correlations whereas in the normal Heisenberg anti-ferromagnetic, this would be ferromagnetic correlated in correlations between this particle and this particle. But when there's the presence of this doubloin here, you see again that in this local vicinity, we get this strong change of the local spin correlations. So I think what's beautiful now is that we can basically have a complete map of the spin correlations around single impurities. So we can really measure directly how do single impurities affect the magnetic environment or and what is the resulting, how can we map out those spin correlations? And at least for now, we have very good evidence that the particle here, indeed, we have this polaron formation with a finite radius of the system that is formed around this density impurity. Now, one important check you can do is how does the mobility of this charge impurity affect these spin correlations? And for that, we can resort again to these beautiful tools of our quantum gas microscope. We can take a laser beam and focus it onto this density impurity and just keep it pinned, okay? By just focusing on that single lattice site, a kind of laser beam, we can make the potential deep so that particle cannot move anymore. And now you ask for this pinned doublon, for this pinned density impurity, do you actually also see this? And there we have to say, actually, no, for the pinned case, you see we don't see this modification around the charge impurity. So it's really this mobile nature of the charge impurity that modifies its local anti-firm magnetic environment in this subtle way that we can see here. So this is the mobile case where the hole is mobile or the doublon is mobile. And here you have a doublon that's just immobile. And you see the dramatic differences in the both cases. Here you see the Polaron forming. Here for this pinned case, you just have no Polaron forming in this picture. So this is actually, I think, very nice and brings us to exciting outlook, I think, for really understanding at a fundamental level and really directly getting microscopic pictures of what the interplay between charge impurities, density impurities and magnetic environments are. And this many people believe is the key solution to, for example, understanding the Hubbard model, understanding whether it has these connections to high-Tc superconductivity. And one of the elementary things what could go on to do next would be really to look at these Polarons that are forming around individual charge carriers and charge impurities and then trying to see, is there some interaction between those Polarons? Do they want to kind of link up to form new quasi-particles that can, for example, then condense and form a Bose-Einstein condensate and form a superconductor in the system? We can also look maybe at timescale of these Polaron formations, everything, as you've seen, can we can look at dynamical effects in the system? We can look at the mixed dimensions, see how when we go over from 1D to 2D how these kind of dynamical effects change and how dimensionality plays a big role in the formation of those Polarons. So I think it's really a new area of understanding how magnetic correlations and charge impurity interplay interact with each other, and hopefully it will give us insight into this long-standing problem of understanding the Hubbard model and its connections to high-TC superconductivity. Okay, I think that I'll stop and we can just discuss a few questions if you want on the talks. I just want to give credit to the people who did the work, especially on this last part I spent so much time today on. So this is our lithium quantum gas experiment team led by Christian Gross and Guillaume Salomon as a postdoc. Then there's Yanis and Jaya as postdocs here and Timon has left us for Oxford and we've been joined by Mim and Dominic as new postdocs and students and the theory work and people who have consulted us on this and we've been working on this Polaron problem are Eugene Demler and Fabian Guest from Harvard. And with that, I think I'll leave it here and thank you very much for your attention. Thank you.