 Thank you. So as I said in the following I would like to study different class of models, topological models, and discuss a few aspects. So let me directly jump in. So these are the type of models we will consider in the following. Essentially non-interacting fermions, translation variants such that we can decompose the Hamiltonian into a different momenta or quasi momenta when you have an underlying lattice Hamiltonian. So here overall it means the Hamiltonian decomposes into different components for each momentum k and the Hamiltonian for a given momentum k can be written in this quadratic form where this ck is a spinner and depending on what is what are the microscopic details this spinner can be like of this form where this c's are fermionic annihilation operators that might correspond to different sublattices in real space A and B or you could have a spinner of this form which you might well know when you are familiar with ketife models. But actually so the details are not so important just keep in mind that these are models of non-interacting fermions with translation and variance. There's an addition symmetry imposed here which is some particle whole symmetry which somehow will be important later on. Now due to this decoupling into different momentum sectors the actual information about the physical properties of the given model is solely contained in this matrix hk which is typically in this Bogolubov-Degen formulation expressed in terms of this vector d and this vector of Pauli matrices tau. Essentially this matrix hk is a 2 by 2 matrix which you can express in terms of Pauli matrices in this form. This is something you might well know from your quantum mechanics course and yes all the information is contained in hk or in this equivalently in this vector dk. This is just to set the stage yeah and one example if you are familiar with the one-dimensional ketife chain for example then this vector dk would for example have this form where delta is the the pairing amplitude mu is the chemical potential of the fermions and J is some nearest neighbor hopping. But actually the only thing that I will use throughout the talk here is that non-intacting fermions and translation invariance that everything decomposes into different momentum sectors. The Hamiltonian can then be diagonalized for each momentum separately so which means you diagonalize a 2 by 2 matrix which you can do essentially analytically. Despite of course this makes these problems analytically is rather easily tractable there is further important property I will use and that is because the different momentum sectors are decoupled in the Hamiltonian it also implies that the ground state of or also excited states actually of the system factorize also in different momentum sectors as illustrated here. When you do now time evolution that's what we are interested in in the following mainly this does also not change so as long as the Hamiltonian which is doing the time evolution is also of non-interacting form and has translational invariance so that we can essentially study the problem for each momentum separately. So if we have such an expression for our time evolved state again due to translation invariance also this Lo-Schmidt amplitude in the end factorizes everything in these models so we can write this Lo-Schmidt amplitude as a product of different momentum where this Lo-Schmidt amplitude for a given K is nothing but the overlap of the time evolved wave function single particle wave function at the given momentum K and its overlap with the initial condition. Okay so now let us use one of the insights I was presenting you in the yesterday which is that dynamic phase transitions are associated with zeros of the partition function. At that point it was mostly a way to argue on general grounds that these quantities can be non-analytic here we can actually also make use of it because to determine conditions for when we can have a dynamical phase transition in this kind of models since we have to require that at a dynamical transition the partition function or here this Lo-Schmidt amplitude at this critical time has to vanish exactly has to be zero but since this is a product it also means that it's equivalent of saying that there is at least one what is called critical momentum K star which at this time G star or this amplitude becomes zero so in order to to study whether this given protocol supports a dynamical transition you would just have to solve for for this condition which is rather easy to analyze for a concrete model and now for now that's where topology now comes in when you have these two bands to bogeyub of the gen models with particle host symmetry it turns out actually that they're rather general statements about when these dynamical phase transitions can occur and it turns out that they are rather strongly related to the underlying equilibrium ground state properties of the model although we are studying a system far away from equilibrium at energy densities which are far beyond of the ground far beyond the ground state for example we are there are proofs which show that in one dimension we always have to get dynamical phase transitions in this in these models when our initial and final Hamiltonian are topologically different so I have ground states which are of the different topological nature in two dimensions the situations situation is slightly different and it's not sufficient in general to have just a different topologically different ground states but actually the absolute value of the chair number has to differ and then we are guaranteed to have these dynamical phase transitions okay and this you can deduce just from in the end it's not very difficult from from checking under which conditions conditions this equation can be satisfied so in the end there is now a rather general understanding when whether these for two-band models for high bands it's actually much much more non-chevia and for two-band models and under which conditions these dynamic transitions can appear yes yes so that the ground states of the two respective Hamiltonians would have different chair number say or different absolute value of the journal and actually this can be especially in one dimension for example this strong connection between the equilibrium and dynamical transitions can be used to detect for example equilibrium ground state phase boundaries by just studying dynamical phase transitions so there's a one-to-one correspondence there in two dimensions it's not yet settled yes so half filling you have yes yes it can happen but you're not guaranteed that it happens so in some sense they're not like when you can satisfy one of these two conditions here your dynamical transition to some extent is topologically protected so you have to have to get it in other cases you might also get it but then it might depend on fine-tuning okay and now I promised you this already quite a lot of times for these classes of models we can now construct some order parameters and the basic underlying quantity we need for that is something which has also different names I will use the notion of a pancheratnam geometric phase it's a generalization of very phase two non adiabatic and non-cyclic evolution so it measures gives you a geometric content of geometric a geometric content of two non orthogonal states and that this goes as follows so since so we know already that the overall low Schmidt amplitude factorizes so we can study it at each individual a momentum separately and this is what I've been writing down here let's do a polar decomposition of that object so in terms of our absolute value and a phase so this phase now is nothing but the phase acquired during non equilibrium evolution starting from this initial condition yeah so RK measures the overall probability or R square me measures the probability of returning to your initial condition and 5k measures the phase acquired during your non equilibrium protocol yes that's what I've been writing on here now it's important as you're maybe familiar from from various phase this phase acquired although you're not doing a cyclic path in your in in your parameter space or Hilbert space this phase contains a purely geometric part and goes under the same along the same lines as for for various phase when you subtract a dynamical contribution which in our case is nothing but the what you expect like linearly increasing in time phase where the pre-factors given by essentially the the corresponding energy of the mold and this is now this celebrated pancher geometric phase it's the geometric content of a non adiabatic phase acquired by the system for a given moment and now comes a crucial point and that's actually the only point where topological properties start to be important now let me consider in the following just a one-dimensional case first and now comes the important point when you have particle host symmetry this pancher atom geometric phase you can show is pinned for two different momenta namely for men for momentum k equals zero and for momentum k equal to pi I said implicitly let the spacing equal to one here and it's pinned exactly to zero so what what does that mean it means that as a function of momentum this phase describes a closed loop on on a circle and for closed loops on a circle we can define a winding number and since they are closed loops and that's ensured by this condition the winding number of course has to be an integer okay and as it turns out this integer is a order parameter for these dynamic transitions so you can then show that whenever a dynamic phase transition occurs this winding number has to change its quantized its integer value and the way maybe to see that why this phase can change at at a dynamic phase transition let me quickly maybe go back and forgot to put this on the slide I told you that on a even previous slide that the condition for having a dynamic transition is that there is one momentum for which at the given time this Loschmitt amplitude becomes exactly zero and when this Loschmitt amplitude becomes exactly zero of course the phase becomes undefined and can change its value and that is then reflected in the change of this of this winding number let me show some data here on the left hand side you can see this is a quantum quenching a key type chain a one-dimension key type chain and here you can see as a function of time momentum resolved the pancher atom geometric phase on the one hand you can see this pinning at k equals zero and at k equal pi that's the pancher atom phase is always zero and you can see that your depending on which time you're looking at the your winding number starts to increase so here's definitely the phase is almost zero everywhere so there's no winding number associated to that but for example here we we start to cross once through the full circle we cover once the full circle so we get a winding number of one and indeed here on the right hand side you can see this is the lambda that I was the rate function of the Loschmitt equal and the corresponding winding number and whenever you have a dynamical transition here here and so on you see that this winding number jumps here by a value of one now this construction is very general for these two-band models whenever you have a two-band model with particle host symmetry you can define this winding number and whenever you have a dynamical transition you you can show you have to have a change of this value so it can operate as a winding number so the more interesting case maybe is even the two-dimensional case you can define also a pancher at them geometric phase for two dimensions of course then your brain on zone is not a one-dimensional object as before but rather two-dimensional that's why you see here two-dimensional color plots so this are the two this is kx and ky and as a color scale you see here some phase it's actually not the pancher at them phase but something which is equivalent and in two dimensions you of course don't have directly a winding number but the you also see the emergence of some topological like topological defects namely that whenever you have a dynamical phase transition you see the appearance or disappearance of vortices in this phase profile so here this is actually also this is the experimental data of obtained in a system of ultra cold atoms in the group of is the Weitembeck and clausenck stock you see this at the muta phase but it's what you see is equivalent to the pancher at them phase we solved over the full beyond zone for different times here and what you observe at specific points in time that here enclosed by these red circles pairs of vortices in this phase profile up here they walk around and at some point they recombine and vanish completely again so when you measure for example the the number of dynamically generated vortices you would find zero initially then one pair is generated to and then at some point they recombine and will disappear again so in one dimension summarizing one dimension for these models we have we can define rather generally these winding number whereas in two dimensions you can define this number of dynamically generated vortices that have to appear whenever you have a dynamical transition in such two bent models yes no that's momentum space so that's a vortex in momentum space this is ky and this is kx yeah please so these red circles enclose a vortex so a structure when you measure this phase in the circle around this this core you would find that you get the one non-zero winding so you mean what what sets the location for example that is fixed by by this critical momentum that I mentioned on the previous slide so there's for a given a model and given initial condition final Hamiltonian they are fixed to appear at certain positions and also like the the move they take along certain lines that's all determined by initial and final initial state and final Hamiltonian yeah so like good question so like in the end actually there should be some symmetry in these plots but the experiments were actually we're not realizing the symmetry required here that so they were not good enough in realizing the Hamiltonians that you would observe the same vortices appearing also at some other other other places of the beyond zone so that was in the ideal case there would be more of these vortices but in the experiment they were not able to realize it as perfect as they would have please say it again I saw I think I didn't get your point so you mean this this plot here and there should be you mean there should be some symmetry involved here yeah yeah it should be but it's not realized here yeah yeah yes that the point is in principle yeah in principle there should be some relation this one here they realized is probably not one of these topologically protected ones where you can make strong statements but a dynamical transition which requires more fine-tuning and there we cannot make as general statements as for the other ones but otherwise it's related to as you say to I think to the difference in absolute value of chair number something like that I'm not 100% sure honestly so they should I'm not 100% sure I'm sorry at least within some range it does not depend on that okay so now I want like I would like to come now to a to my last part which I which is a very very recent experiment two three weeks ago was put on the archive which I an application which I like actually quite a lot and that is about quantum walks and how to use the the the insights that I've already presented you before can be used to characterize such processes so let me first say what quantum walks are and why I think that this is an interesting application so overall a quantum walk is a very like rather simple quantum process it's the quantum analog of a classical random walk it goes as follows you have some lattice or like I will only consider now a discrete quantum walks there are also continuous ones which are not constrained to let the sides but rather live in the continuum so in a quantum walk you have a lattice as indicated here in real space and you have some internal space which is typically called a coin space it just means that at each lattice sides that there are two degrees internal degrees of freedom which we label by up and down and initially before we start the process we put one particle on one particular lattice side with which we typically choose as you so one excitation here in some superposition of this in in this internal coin state up and down and then according to some protocol I will show you in a next slide we will we just let the system evolve but it's a single particle problem let me also emphasize this here okay I will consider a specific class of quantum walks which are called split step quantum walks and they have the following show the order dynamics is governed by the following sequence of individual gates so first of all you do some rotation are in the internal coins coin space so you somehow redistribute your amplitudes between up and down on let the on all your lattice sides then you do a conditional shift by this T up operator what it does it moves your particle your walker one let the site to the right but only if under the condition that the particle is in state up locally okay and when it the system when for those when the when the particles isn't in state down it does nothing then there comes another rotation with some angle theta 2 in this coin space and then we do a conditional shift again but now precisely opposite we move the walker to the left when the walker is in state down only then if it's in up we don't do anything and now what we do is this is one step of your quantum walk and then we repeatedly apply the same sequence and essentially study how the single particle this starts to distribute among your lettuce which is initially particle which is initially localized so now you can think of this problem in a different way namely that it's an effectively periodic time-dependent problem so you have some sequence of four operations and these four operations you up you repeat on and on it on in a periodic fashion in the end and for such sequences or periodic sequences you can equivalently think of a Hamiltonian formulation so before the operations for the evolution I was showing you were not I didn't show you any Hamiltonian these were just operations on a particle itself but you can think of defining a Hamiltonian a flockey Hamiltonian hf which generates the same dynamics or Hamiltonian which generates the same dynamics as the full unitary you here and now of course you could ask yourself how does what properties does this Hamiltonian have this is somehow shown here there's a phase diagram for this Hamiltonian depending on these two angles theta 1 and theta 2 which describe the internal rotation in the coin space and you find some phase boundaries here between topologically non-trivial like so you can find a phase diagram and for for the following situation so like this hf actually mimics models that we had I have discussed on the previous part like momentum is a good quantum number for this perfect lattice so also this hf decomposes in different momentum sectors and it's a non-interacting single particle problem this hf therefore looks like as a model that would be realized by non-interacting particles in solids for an insulator say or for a topological insulator and for that we can write down a phase diagram or we can determine a phase diagram and it shows also topologically non-trivial phases that's why the split step quantum August has attracted quite some attention and there are also topologically trivial phases but you have to keep in mind this is a single particle problem so you're not dealing with ground states of a half-filled half-filled quantum anybody system it's a single particle which is moving on your lattice so it does not make sense to think about ground state properties or chair numbers a priori for this problem okay so now that leads us to the challenge now what it's an inherent dynamical process a quantum walk you prepare your particle locally and then it starts to spread there is no equilibrium so state associated to a quantum walk the particle will just move somewhere in contrast to classical random walks there's not even a limiting distribution so there's not a lot there's not a there's no steady state for that problem there's no distribution asymptotically this particle will attain so the there's nothing which one can think of a steady state or some equilibrium state you can associate to that problem so how can we then characterize such an inherent dynamical process for which we don't have cannot apply any thermodynamics or something else and of course you will probably I guess that so there will be somehow connection to the maker phase transitions not directly because also these dynamical transitions only occur in a thermodynamic limit here we have a single particle but we can still do the following so now let me discuss a bit the state of your of this quantum walker so that's the wave function psi after t time steps you can write it as a superposition in the following way where x denotes a spatial point on your lettuce and mu the internal coin space so which can be up or down so that's now the space at the basis in which I expand and the system has a respective amplitude to be at a given time step t at this position x and having this particular in value mu for the internal coin space we could also think so that's now a real space representation of that momentum here's a good quantum number or quasi momentum crystal momentum we can also equivalently equivalently think about the momentum space representation of that so we can just fully transform these amplitudes which gives us this psi tk of mu and now in momentum basis and now let us define performing the sum or the internal coin space to get a the wave function of the quantum walker at a given momentum okay involving both of the internal possible orientations in the in the coin space so that our final wave function has this form so it's just a sum over all these momentum components so as opposed to the previous one like when I was discussing in the previous section the state was a product over different momentum sectors the quantum walker is now a super position over the different momentum sectors but formally that's actually the only difference so it's superposition in terms of instead of a product so now we could equivalently think of the low Schmidt amplitude at a given momentum it does not make sense to think about the many body wave function but still we can think about a low Schmidt amplitude at a given momentum so we can again define our phase acquired at a different at a given momentum and also can think about a pancher atnam geometric phase and we can think about a this dynamical topological order parameter which does not require to have a many body wave function it does only need to have a phase which depends on momentum and which performs closed loops on a circle but this we can define equivalent equivalently for a quantum walk for that we don't need a many body wave function so it's still a single particle problem but of course there's an equivalent partner many body system where you replace the superposition by a product okay but in this way we can now transfer everything I was telling you before now to this to this quantum walk problem and now what you can see here on the left hand side is an experiment where these and dynamical topological order parameter here denoted by WD was measured for a quantum walk and you see here that this dynamical topological order parameter can behave differently depending on which parameters you choose in your split step quantum walk here you see again the the phase diagram hypothetical phase diagram of the model it would have an equilibrium if it would be a quantum many body system and if it would acquire a steady state so that these like these these are the different parameter sets are here indicated by different symbols and you find them also in the respective respective plots now what they were what they managed to do in the experiment or like this is something something very interesting also concern when you want to connect to these dynamical transitions is that there are certain lines and parameter space here where the your Hamiltonian this effective flockey Hamiltonian would have flat bands so that eigenstates are are localized objects in real space this you can only have for flat for flat bands so if you choose your initial state properly you can think about that it corresponds to to prepare your system in some eigenstate of a Hamiltonian which is single particle eigenstate of the Hamiltonian in this in the space diagram and now you do dynamics a different values of a different parameter sets here and here and here and as you can see so this is the phase boundary of the hypothetical equilibrium problem you see that for the star and the square this topological order parameter always this stays zero throughout the dynamical process and as soon as you cross the hypothetical equilibrium phase boundary the triangle and the circle you see that now the this dynamical topological order parameter starts to show jumps and to show an increase so below here these red red data is the rate function of the equivalent many bodies system but I think so let me emphasize I think it's not correct here to think about a many body system it's a single particle problem for which you can nevertheless define a dynamical topological order parameter which is this omega d and with this you can now characterize different dynamical properties your quantum walk and half so qualitatively different properties so either your your the omega d can stay constant or it can increase they even more complicated processes that can appear but for that for these parameter sets you can only have these two options and for example it allows you to get even information about the hypothetical phase diagram of this effective flockey Hamiltonian although you never are in any of far away from that in terms of energy density or you're seeing a studying a single particle problem there are also actually many other experiments on similar problem similar scenarios like quantum walks or a single particle qubit experiments which see similar properties okay so what let me just summarize this part so the for me this is like a prime example of of of quantum process the quantum walk which in no way has any equilibrium equilibrium meaning so there's not there's not even a limiting distribution in some long-time limit even furthermore it's a single particle process but nevertheless the knowledge we had from these dynamical transitions we have been able to dynamically characterize the the time evolution okay and with that I'm actually too early at the end of the lecture and so let me just very briefly close like I with a summary and some open questions so I give you gave you a rather detailed introduction into this concept of dynamical phase transitions I showed you some central properties and also today started to give you some hints towards interpreting this phenomenon I discussed also quite a lot of x experiments that have been done in this direction so it's not it is also a an experimentally observable phenomenon but there are also many many open questions and some of the big ones I've been listing here so that the first one is the most one of the most pressing ones and that is why is the equilibrium also like the equilibrium theory first phase transitions so successful mainly because we have effective descriptions in terms of land or effective field theories but we have some first attempts for particular models but we are very far away from having something equivalent so to see whether to which extent we can find analogs of land or field theories effective macroscopic descriptions for such phenomena that would be a very important second very important aspect is higher dimensions so many of the things that have been showing you were essentially one-dimensional problems maybe two dimensions for exactly solvable two dimensional problems but we also know in equilibrium that many of the interesting phases or phase transition phenomena occur not in one dimension but rather in interacting systems in two and three dimensions and for that we just don't know at the moment how to compute the dynamics so that's more like a computational problem and how to compute this loch mid-amplitude which has the complexity of a full partition function but maybe even worse because you have to deal with complex numbers so Monte Carlo will not help you probably in general because you immediately run into sign problems and another thing is order parameters so for the for these topological systems we have somehow an understanding but away from that we are lacking a lot of insight in this direction so like there are many dots many points I could add therefore these dots and with this I'd like to thank you very much for joining me quite a long time here and hope you will enjoy a future lectures