 So the talk is given by Jan Kunisch now from the Technical University of Vienna, please. Okay, good morning everyone. I would like to start by thank you to the organizers for inviting me to this workshop. And also thank the chairman and the previous speaker for introducing the method. So my talk will be about calculation of dynamical susceptibilities with dynamical mean field theory. Our main motivation was to actually test the method. But usually you want to use it on some physically interesting problem. So I will use a dynamical mean field theory and a calculation of the susceptibilities within this method to study the excitonic magnetism. And I will explain a little bit what excitonic magnetism is. Let me start by acknowledging the people that contributed to this work. So most of the results I'm going to show you were obtained by my postdoc Dominic Zewkoa and the other people that contributed are Atsushi Ariki, also a postdoc in my group and PhD students in groups of Carsten Held and George Sanjewany, Josef Kaufman, Patrick Gunaker and Andreas Haussuel. So my talk will be mostly about models, but let me start by motivating the work by looking at materials. And the first material I want to point out is lantern cobaltate. What is excitonic magnetism about in lantern cobaltate? Well, excitonic magnetism is a magnetism of materials where the atomic ground states are non-magnetic. That sounds very boring, but what makes it interesting is that the atomic excited states carry a magnetic moment, and if these atomic excited states are not very high in energy, it may lead to interesting effects. And so in lantern cobaltate, the ground state is a non-magnetic low spin state of cobalt3+, but the excited states are called either intermediate spin state, carrying spin1 or high spin state carrying spin2, and these are usually treated as just atomic excitation, just crystal field excitations of atoms. What we've suggested and what our experimental colleagues observed recently is that these atomic excitations are not actually atomic, they are mobile. They have rather large dispersion, and you can see here the results of Rick's LX measurement, these experimental points on top of the contour plot, which is theoretical prediction, and you see this is the dispersion of our exciton. It's quite sizable, it's of about half an electron wall. Well, another material which is of great interest of us is another cobaltate compound, or family of compounds, prosidium cobaltates. This family of compounds exhibit a phase transition between 60 and 130 Kelvin, and there are lots of experimental data that suggests that this phase transition can be understood in terms of condensation of the excitons. I don't want to go through all of these, I just want to point out that there are a number of experiments which are consistent with this picture. However, we don't have a real smoking gun, an experiment that could specifically probe the condensate or tell us this is an excitonic condensate, it cannot be explained by any other physical theory. So one of the questions I want to ask in my talk and provide an answer to is how can we detect an excitonic magnet with today's experimental techniques, and the other question is how does mean field theory perform in phases with long-range order, with spontaneously broken symmetry, questions like do we observe Goldstone modes, what about Higgs modes, and some other fancy names that one can sell papers with. Okay, so what is our model? The model we want to study is the simplification of the real material. Instead of T2G and EG bands of real cobaltates, we will have just two bands, which we will call A and B, so we will study a two-orbital hubbub model on a square lattice. We can think about it as let's say X, Y, and X squared minus Y squared orbitals. We will consider nearest neighbor hopping, which is primarily diagonal between the orbitals of the same type, but we will also include the effect of cross-hopping between orbitals of different type. And we will include a local Coulomb interaction with the Hund's coupling, which is for magnetic, which is an important part of our problem or model that we want to study. The method we are going to use is dynamical mean field theory, and you have already heard about it. So I just want to make, well, dynamical mean field theory can be understood from many different perspectives. The perspective I want to point out that we can understand it as an effective method. Dynamical mean field theory provides us an auxiliary problem, which allows us to calculate the irreducible vertices of many body theory. So the common use is to calculate the one-particular reducible vertex, which is a self-energy, but we can also calculate the two-particular reducible vertex, which then gives us access to a two-particle response to susceptibilities, which are typically more experimentally relevant quantities or more commonly studied experimentally. To calculate these two-particle response functions, what we do technically is to solve two beta-salpeter equations. We start with the impurity problem, where we obtain the two-particle correlation function and the one-particle propagators, and we solve this beta-salpeter equation to get the two-particle reducible vertex, and then we take it and plug it into the lattice problem, and we solve this lattice beta-salpeter equation to get the physical two-particle susceptibility. So let me make just a few technical points. So in this approach, we have the full frequency structure, so we have the bosonic frequency and two fermionic frequencies. What is simplified is the case-space structure, or it's better to say it in the real space, our vertex is purely local, so it's sitting on one atom. Our approach is multi-orbital, and we impose no symmetric constraints, which allows us to go to phases with broken symmetry. Technically, the implementation is performed in a Matsubara representation, and so we have to perform analytic continuation at the end to get real physical quantities in real physical frequency. So let me make a few remarks about the model. So this is the phase diagram, kind of the general phase diagram of the two-band Hubbard model, studied by many people, where we have a crystal field splitting on one axis and the interaction strengths on the other axis. In the right, the low corner, when the crystal field dominates, this is all at half-filling. So at half-filling, when the crystal field dominates, we get a band insulator, when the interaction dominates, we get a mod insulator, which would be described by S1 Heisenberg model, and in the wedge where the bandwidth or hopping is comparable with the other two parameters, we get a metallic phase. But this model is actually much more interesting. When you allow for long-range ordering, of course, the high-spin mod insulator would form a Heisenberg anti-ferromagnet. The anti-ferromagnetic phase could extend into the metallic phase, but we can also get phases like a spin state order, which would be a lattice decorated by high-spin, low-spin, high-spin, low-spin. Or we can get an S1 excitonic condensate, which is the phase I'm actually going to talk about, and so I will tell you more about it in a moment. And the results I'm going to show you fall into this region. So we will use the crystal field splitting as a parameter. We will be close to the spin state crossover where low-spin and high-spin are approximately balanced in energy, and we will introduce another direction which will be the temperature, which you can think about as perpendicular direction here. So this is the actual phase diagram calculated by changing a crystal field. So what we have here is the normal phase, the low-spin, and then we have the excitonic condensate. So let me try to briefly explain what the excitonic condensate is. And I think the most intuitive picture is to use the strong coupling picture where we think in terms of the atomic states, the low-spin state and the high-spin state. You can think about the low-spin state as a vacuum, so a lattice full of low-spin states can be viewed as vacuum, and the high-spin state, which comes in three species because it's S1 state, can be viewed as an excitation. And because of processes like this one, these excitations are not locked on the atom, but they can move through the lattice, and that's why they can condense, essentially both Einstein condense, and then they form a state which can be approximately described by a wave function, which is just a product wave function where we have the combination of low-spin and some high-spin on each atomic side. So this would be a state which we would call a polar condensate. The condensate can come in many different symmetries because we still have the degree of freedom of the high-spin states. What is actually realized in our model is so-called polar condensate, where the S equal minus 1 states of equal weights, so this state has no ordered moment, but it still breaks the spin-rotational symmetry. Well, we are not doing a Bosonic theory, we are doing actually numerics for the Hubbard model, which is fermionic theory, so we have to look at fermionic observables. The broken symmetry can be detected, for example, by looking at this spin-triplet combination of orbital of diagonal expectation value, which is if we have the original symmetry of the Hamiltonian, this should be zero by symmetry. If we go to the ordered phase, this order parameter acquires a finite expectation value. Before I go to the results, let me make one more remark about the symmetry of our Hamiltonian. The Coulomb interaction that we use is actually the so-called density-density form of the Coulomb interaction. This Coulomb interaction has only... The spin symmetry of this interaction is a uniaxial symmetry. We have u1 spin rotation. This is the symmetry we have always in our model. Then when we put this cross-hopping to zero, we have one additional symmetry, which is a gauge invariance with respect to this gauge transformation. Basically, it reflects the fact that the A charge and B charge are conserved separately, so we cannot convert an A electron to a B electron if this term is missing. Why am I pointing this out? Well, the broken symmetries and both of these symmetries, this u1 times u1 in this case and the u1 in this case, are broken at the excitonic transition. As you know, the number of broken symmetries or the type of broken symmetries determines the nature of the Goldstone modes in your ordered phase, so one of the questions, basically the main reason we did this study was to see do we get these corresponding Goldstone modes with dynamical mean-field theory or not. So these are the results and let me just quickly flesh the one-particle spectra. Basically, the one-particle spectra are pretty boring, so what we have here is, again, the cartoon of the phase diagram and if we move along this line, if we drive the transition by reducing the crystal field, we go from this bend structure to this bend structure. We have always gaps, so on one particle level there is not much interesting happening and the gap is actually pretty large, it's about 0.3 in these units, but if you look at the two-particle level, it's much more interesting. So what is shown here? Each of the rows is one point in this phase along this line, so going from top to the bottom, we are reducing the crystal field. Here is the phase transition, so the upper two rows correspond to the normal phase, the lower rows correspond to the ordered phase. The first column is just a spin-spin correlation function, so this is what you would call a dynamical spin structure factor. The other four columns are excitonic susceptibilities, so I don't want to go into details, but you can imagine we can take these expectation values and calculate them in different spin directions, so this is the x and y that corresponds to spin direction and we can make them in a symmetric or anti-symmetric combination or you can also call it a density-like or current-like combination or we started to call it real and imaginary, so we call it r and i. So these are some kind of excitonic susceptibilities, so these are matrix elements of our susceptibility matrix. And what is shown here is the dispersion, so this is a cut through a two-dimensional Brillouin zone from gamma to x to m to gamma. So what you see in the upper line is the dispersion of our exciton. Indeed, in our thermionic calculation, when we calculate the susceptibility, we see the dispersion of an exciton. When we reduce the crystal field, the excitonic gap goes down and at the transition the gap is closed. And you see in the normal phase all these pictures are the same. These modes are degenerate. As we proceed down to the ordered phase, we see the development of the two Goldstone modes. You see one Goldstone mode here and this one is associated with breaking of the spin rotation symmetry and there is another Goldstone mode here which is associated with breaking of this gauge symmetry. And you can extract the sound velocities and show that these sound velocities are indeed different and also the linear regime of these different Goldstone modes is different. So the first question we ask, do we get the Goldstone modes and does it correspond to the general expectations from the general theory based on symmetry breaking? Yes, we observe these Goldstone modes even with our numerical implementation. Now let us remove one of the symmetries. So let us switch on this cross-hopping and remove the gauge symmetry. What happens now? So again, going in the same direction. We see the Goldstone mode associated with the spin rotation is still there but the other mode that was associated with the phase rotation is now gapped. And if you can extract the value of this gap for different values of this hopping parameter and you see in the normal phase the gap closes. It depends on the crystal field. It does not really depend on V but in the ordered phase this is how the gap opens. So we get something what could be called a Higgs gap. It is a gap that is closed exactly at the phase transition and then opens again when we go into the ordered phase. But there is something else that is quite interesting about this mode, namely where the spectral weight lives. And let me show you the same picture when zooming around the gamma point. So again, here we have the normal phase. Here we go into the ordered phase and we see that if we are close to the phase transition the main spectral weight lives in this column. The order parameter here is a vector in the space of spin direction and also in the space of this phase variable. So the order parameter is imaginary and pointing along the y-axis and the response here is exactly, or this element describes the response to the field pointing in the same direction. So this response is amplitude response. We apply a field which is pointing along the order parameter and we ask how does the order parameter respond and we see that this can be called an amplitude mode or amplitude fluctuation. But as we proceed into the ordered phase this amplitude mode loses its spectral weight and it moves to what was originally the Goldstone mode. So if we were zero, the Goldstone mode which went all the way to zero frequency was sitting in this direction or in this matrix element. So what do we see here is what we can call a gap Goldstone mode. So what is happening? Well this situation, by the way, is very common if you take a magnetic material where you have some magnon dispersion and you introduce some magnetic crystalline and isotropy due to spin orbit coupling you open a spin gap. This would be precisely that kind of gap. So what is controlling where the spectral weight lives? Well it is the relative strength of the vice field so we have two symmetry breakings here. We have a symmetry break in which we placed explicitly into the Hamiltonian and a symmetry break which appears spontaneously. If the spontaneous symmetry break in represented by the vice field dominates over the explicit symmetry break in well we get this gap Goldstone situation. But if we have, if the explicit symmetry break in dominates over the vice field well we get a real amplitude, real amplitude response something what you would normally call a Higgs mode. So in the rest of my time let me go back to the question how do we, how do we detect, how can we detect the excitonic condensate and go back to what I already show you and focus now on the first column. So this is the, this is a spin structure factor. This is something that you can routinely measure with inelastic neutron scattering and you can observe something interesting. In the normal phase there is absolutely no response. Well that's not surprising because what you essentially have is a band insulator which is a gap much larger than the temperature scale so there is nothing really that can be magnetically active. But as you proceed into the ordered phase where you get rather sharp, sharp kind of dispersion with increasing weight and if you look carefully you perhaps notice that there is a similarity between this spectra and this spectra in this column and this, this is not, this is not a coincidence. This has actually a very simple explanation of the strong coupling, strong coupling picture. So if we go back to our bosonic picture the magnetic moment along the z direction has just this simple representation, right? There's a number of excitons with s equals one minus the number of excitons with s equals minus one but the natural representation that we use in polar condensate is to use the xy representation. So if you go to the Cartesian representation the spin moment has this form. Now in the condensate one of these operators acquires a finite expectation value so you can replace it with a number and in this case it is an imaginary, it's pointing in the y direction and it is imaginary. So if you put imaginary number here you find that in the ordered phase the sz operator is proportional to this drx operator and that's why their susceptibilities are proportional to each other. The proportionality constant is the order parameter so that's why there is no coupling in the normal phase but in the ordered phase there is a clear coupling and we can see the fingerprint of the condensed excitons in the spin structure factor. So before I run out of time let me show you that we can observe a similar behavior when we drive the transition thermally so now we are moving along this line. In the normal phase now we have some magnetic response. Why? Because at high temperature we excite some of the high spin states but they do not form condensate so they basically respond like isolated weakly interacting high spins so you see a featureless in the reciprocal space response low class at very low frequencies but as we cool down and go into the ordered phase this picture changes qualitatively and eventually we get this sharp transition or sharp dispersion with a large cap away from the gamma point. This is something which was actually recently observed in an elastic neutron scattering experiment in one of these prasidium cobaltate materials. This was a pretty difficult experiment to interpret because there is a big effect from the prasidium ions which has to be subtracted but nevertheless they claim and they could base on their data conclude that indeed they see that away from the gamma point when one cools down through the transition temperature a gap opens in the response in the spin response. So this takes me to my conclusion so the answers to the questions I posted at the beginning how to detect an excitonic magnet well we show that there is a qualitative change in the dynamical spin structure factor that can be used as a fingerprint of this excitonic condensate I should stress that this is not a magnon there are no ordered moment there is no moment pointing in any direction in this condensate nevertheless we see almost magnon-like dispersion which however vanishes its spectral weight vanishes very close to the gamma point there is of course still a possibility to observe the excitons directly with methods like ricks but the resolution of ricks is probably not enough and also the ricks at allergies of transition metals cannot probe the full Brillouin zone because of momentum constraints the second question how does a dynamical mean field theory performs in phases with long range order so I showed you that the Goldstone theorem is fulfilled and that actually to rather high accuracy we can detect actually a small symmetry breaking already leads to opening of a detectable gap and we started as a by-product of this investigation the crossover from a gap Goldstone mode to a Higgs amplitude mode as a function of the relative strength of the symmetry breaking term a nice field that creates this spontaneous symmetry breaking in our system thank you for your attention