 elbow, da couldn't have much for lunch so you won't be sleeping now. So the next lecture is by Marcus亡r in introduction to the nAKE plans transiton . How many lectures are you.. Four hours until four hours. Four hours. Very good. So this is two hours. Yes. Okay, very good, alright. Thank you. V bistvu benzina nekaj je poslednja, da bom še iznamenja, da so da samo da moždem izložimo za taj koncept termitev za dinamizovane zelo, vzbeni takimi, ki bilo takimi ili termitev za nekalibriju režimu. Zato ki ozbiti pravaj sveta, V pravdu lektu smo prišli v kvantom vrte biti, in v pravdu stavili nekolibriom vrte dinamizu kvantom nekolibrijev. If you are interested in knowing more or beyond those things I will tell you in this lecture, you can always have a look at this rather recent review summarizing the recent developments in this context. Let me also quickly emphasize that many of the things I will tell you would not have been possible with the help of many others that you can see listed over here. Ok, so now of course we are all familiar with phase transitions in our everyday life. One of the examples which are best known to all of us is probably boiling water. So take your liquid water and upon heating it up we see that we can transform obviously liquid water into gas upon increasing temperature and experiment every one of us has been doing at some point. If you look at this process from a thermodynamic point of view, for example by studying the entropy of our system we would observe that upon increasing the temperature that because this is a first order phase transition, the entropy will exhibit some kind of jump at the boiling temperature. So increasing suddenly and drastically at the boiling temperature. And this is already one incarnation. Here you can see already one incarnation of a phase transition, which is that thermodynamic quantities become non-analytic as you tune the control parameter of your transition, which is temperature here in this context. But now let's take a different point of view and do the experiment actually in real time. So what will happen now? Again we will start with the liquid initially at a temperature below the boiling temperature and now we start to heat it up. But monitor now the properties of the system not as a function of the control parameter, but really as a function of time as we see our water starting boiling. For example here let's measure for example the temperature of our system as a function of time as we heat it up. Now the situation is a bit different than before. Initially we will see of course that the temperature of the liquid water will rise up. But then as soon as it hits now at a boiling temperature, the temperature will actually not increase anymore, which they constant over some time, and then eventually it will start to rise again. First of all, and that is most important for what I will discuss in the following, if you studied this experiment or your phase transition as a function of real time, actually the properties you study are actually smooth, but non-analytic as we have seen before when you study your phase transition as a function of a control parameter. So the temperature is very smoothly reaching the boiling temperature here and then rising again. The reason why we have this intermediate plateau for this first order kind of phase transition is just that we have a latent heat here. We have to provide the latent heat and since we only can provide a finite power with our heating machine, this takes some time until we have provided the latent heat to bring water into its gas state. So everything is now smooth when monitored as a function of real time and the main point now of the lecture that I would like to show you in the following is that when you go to quantum systems things can be actually different. What you see here, and I will discuss this in more detail later on, you see here some quantity which is called lambda, I will define it later, as a function of time and as you can see this function can become non-analytic and not as a function of a control parameter but as a function of time as you let your system evolve. And this is now what has been termed a dynamical quantum phase transition and I will discuss it now in quite some length in the following. And let me emphasize that this here is actually not theory data, that's an experiment and there are many more experiments that have been appeared rather recently which all show signatures of this non-analytic real time behavior. So it's not only a theoretical concept but also has been subject to experimental verification in quite many cases. Ok, and with that let me give, tell you a bit about the outline of this lecture. Initially since this is the first lecture on quantum systems and non-equilibrium real time dynamics, I will actually would have to put actually a point zero, initially I will try to motivate in some detail why we are interested in studying systems like this and then I will give you a rather extended introduction into the basics of these dynamical quantum phase transitions and in the remaining parts of the lecture I will try to address some more specific properties that these dynamical transitions can have and which are also important when you want to connect to the concept of equilibrium phase transitions which is about scaling and universality, then essentially about one of, in more detail, about one of these experiments that I've been showing you before which is about dynamic transitions in systems with symmetry breaking in equilibrium and in the last two parts let's see what I make it there about dynamical transitions in systems with topological properties and finally how you could use this concept to dynamically characterize quantum walks. So before going on let me also emphasize that I would be happy if you interrupt me at any point if there is something unclear or when you want to know more. Ok, so now let me start with a rather general introduction into non-equilibrium dynamics of quantum many body systems and in particular what I will consider in the following is the purely unitary dynamics, unitary real-time evolution of closed quantum many body systems so of a quantum system which is decoupled from any environment such that its evolution is governed only by Schrodinger equation which you all will know. Yeah, let me say that in the following I will always use units where we set h bar equal to one. If we have a time independent Hamiltonian that's the case I will consider mostly in the following we can solve formally this equation rather easily in the following form. So in general one can imagine variety of different aspects that one could study in the context of such non-equilibrium systems so we always have initially a system prepared in some well-defined initial states here some particles localized on a lattice then we act on our system with some external force which induces some real-time dynamics in intermediate time scales here particles start to hop around the lattice and finally if we wait long enough if we let our system evolve we will see some relaxation to some long-time state which here is depicted as these particles maybe delocalized over the full lattice. About this long-time steady states there are many important and fundamental questions associated to it I will not discuss them in very much detail you will see some of them probably in later lectures so there are important questions of thermalization about the foundations of statistical mechanics or about some metastable precursors of thermalization called prethermalization or about some robust absence of thermalization in systems with strong disorder which goes under the name of many body localization what I will talk about is not associated to this long-time dynamics but rather something which concerns the dynamics on short-time or intermediate time scales these dynamically quantum phase transitions are a phenomenon which deal with this intermediate stages of time evolution so why should we currently care actually about such a scenario why should we care about such non-equilibrium quantum real-time dynamics and the reason is that these kind of scenarios can nowadays be realized experimentally studied and probed experimentally in various kind of systems which go under the name of quantum simulators here you can see a list which is not fully complete like systems made up of ultra-cold atoms optical lattices or trapped ions Rydberg systems superconducting qubits but also more so all of them have the special property that they are well isolated from the environment so they represent to a very high degree of accuracy closed quantum systems for which you can induce some non-equilibrium real-time dynamics and in the following I would like to flash a few few phenomena that purely dynamical phenomena that in the meantime have been already observed experimentally and which should are supposed to highlight why that this is not a field which is at its infancy but rather has managed to study rather intricate quantum-anybody problems so on the one hand I mentioned already this phenomenon of many body localization which is some non-agodic phase of interacting quantum matter in the presence of strong disorder and for example here you can see two experiments one done in ultra-cold atoms and another one done in trapped ions which managed to observe this phenomenon another maybe even more prominent one which has attracted a lot of attention recently is that of time crystals which even made it to star tricks someone told me which in the experiments that have been performed there is something like a period doubling in a phenomenon in periodically driven quantum systems which realize some time translational sumoji breaking and there are mostly two pioneering I would like to mention the two pioneering experiments one realized in a system of nature nitrogen vacancy centers another one realized in trapped ions and what you see there is some kind of a signal at half the driving frequency here and here which corresponds precisely to this period doubling phenomenon frequency halving phenomenon that realizes time translational symmetry breaking but also to others that I would like to mention the other one is so-called pre-thermalization which is a long-lived metastable state in weekly interacting systems or another generic or inherent dynamical phenomenon of particle-antiparticle production in gauge theories which has been realized in trapped ion systems so all of these examples are supposed to show you that from the experimenter side there is a lot of things that you can nowadays do and which motivate for us to study non-equilibrium quantum real-time dynamics but now what is now the main challenge in studying theoretically such kind of systems and that somehow almost the definition the states that are generated by this close non-equilibrium dynamics are quantum states for which you cannot write down a free energy there is no description in terms of a free energy that you could use to describe these states in other words there is no thermodynamic description so we cannot use the tools of thermodynamics to understand principles of quantum real-time dynamics but maybe that's not only bad one can also take a look at it from a different point of view it also means that we are not subject to some equilibrium we are not subject to some equilibrium constraints such as the principle of equal a priori probability that all states at a given energy have to be equally populated in equilibrium you always have that but for these quantum states you principle have a way to generate states that do not have to satisfy those rules which gives you the possibility to generate new quantum states which are hopefully interesting examples of that you have seen on previous slides like this many body localized phase or discrete time crystals they necessarily require precisely to break these equilibrium constraints these are phases of matter that you can only generate when you don't have a description in terms of a thermodynamic ensemble so now having a free energy however immediately leads to some major questions and that is since we cannot use a thermodynamic description how can we then describe such non-equilibrium states in some at least partially unified manner or in different terms can we identify some general principles in unitary dynamics that require thermodynamic description and these are now questions I would like not to address in full glory but parts of it at least now I would like to discuss the definition of these dynamical quantum phase transitions and to discuss some basic principles ok so for the following this specific non-equilibrium scenario to illustrate everything however let me also emphasize that what I will tell you in the following does not rely on this particular non-equilibrium scenario of a quantum quench and quantum quench is conceptually very simple what we do is we prepare initially our system described by some quantum Hamiltonian in the ground state of some what we call initial Hamiltonian H naught and this ground state we denote in the following as psi naught and then at time t equal to zero we suddenly switch a parameter in our Hamiltonian such that when we have switched the parameter that our system is then described by a new Hamiltonian H and because of that when we solve Schrodinger's equation the time evolution of our state will now be of the time evolved state at some time t would then be of this form ok so now having that let me introduce now the central object that will appear throughout this lecture many times and I will call it in the following lo-Schmidt amplitude and that is nothing than the which is denoted by this curly G here it's nothing but the overlap of the time evolved state with the initial state itself and for this quantum quench protocol I was showing you on the previous slide we can write this of course in the following form this object depending on the context you are this quantity appears in various context and depending on the context it also has different names it also goes under the name of return amplitude for instance amplitude fidelity you can choose the name you wish in the following I will use the notion of a lo-Schmidt amplitude and from time to time it will also be useful to study the corresponding probability I will call lo-Schmidt eco which is nothing but the modulus squared of this amplitude so why is this quantity now interesting I know let me say one thing before so this quantity has some very important property that will appear in many places in the following in that it will have a particular dependence on the number of degrees of freedom and you will see later on why it has to have that it has to this lo-Schmidt amplitude exhibits large deviation scaling meaning that it depends exponentially on system size or number of spins in general the number of degrees of freedom you have in your system that I will denote by n and the small function g will call in the following a rate function in other words this quantity in curly g does not have a well defined limit a thermodynamic limit it's only the log of this curly g essentially the small g which has a well defined thermodynamic limit that's why we will study this one in the following in two, three slides and you will also see why this is the case and the same thing also is true of course then for the corresponding probability this curly l of t it also has this large deviation scaling which allows us to introduce this rate function lambda and because this is the modular square of the curly g this lambda is nothing but two times the real part of this other rate function small g of t ok, now what is now a dynamical quantum phase transition you have seen it already on one of the previous slides it's a phase transition which is not happening as a function of some external control parameter so some parameter that you can control from the exterior like temperature pressure or magnetic field it's a phase transition which occurs as a function of time so occurring only due to internal changes that occur in your system not imposed by the outside but only occurring due to internal changes and more concretely defined as a non-analytic behavior in such a rate function like the small g or small lambda as a function of time as you have seen in one of the previous plots so here you see a measurement of this small lambda of t which is minus one over n and number of degrees of freedom log of the and as a function of some dimension rescale time you see that there are kinks appearing and those points where these non-analyticities appear I will call dynamical phase transitions in the following no, not really let me I think on the next slide you will see maybe some way of how you might think about it it's not a simple I would not consider it a simple correlation function I will actually discuss this particular case in quite some detail later on for that particular experiment or that particular experiment was realizing some kind of long range transverse field icing model but you will see later on the details about this experiment yes, not thermodynamic limit you will see later on that the thermodynamic limit is essential and why n, like a system of 10 spins is sufficient here to say make statements about phase transitions you will also see later on ok, yes no, this transition does not have to do anything in principle with a transition of this form let me take the chance here maybe to make the following statements that there are various other notions of dynamical transitions in the literature, for example for other generally non-equilibrium transitions like the many body localization transition but these are transitions of different kind than those that I will discuss here in the following because those are transitions which occur at some finite time and the ones are typically associated with some long time limit although some of these other dynamical transitions connect also to those ones but that is not a general statement ok, so now that's the definition this non-analytic behavior in this rate function and the remaining part or the remaining time of the lecture I want mainly to address two different questions here the first one is why can this be why can this function be non-analytic as a function of time how is this at all possible and the second one which is the more non-trivial one is what does it mean to have a non-analytic behavior in this quantity as a function of time so let me first this will now require a few slides let me first try to argue that why this quantity can become non-analytic and that in particular that this is nothing accidental but something which is as generic as or like that these dynamical transitions can happen as generic as there are also equilibrium phase transitions and the main observation here that leads to this conclusion is that these amplitudes that I was introducing before actually formally resemble quite a lot equilibrium phase transitions up here you see the canonical partition function of a system described by a Hamiltonian at some inverse temperature beta already here you might recognize some similarity to this Lo-Schmidt amplitude in that it is a some kind of an average of an evolution operator but there is an even more like formally even more closer connection to a certain class of partition functions and these are called boundary partition functions you can see them written down here which appear when you have systems subject to boundary conditions consider for example a Kazimir effect where you have two metal plates in between some in closing some medium and when you want to describe the equilibrium physics of such a system then you can show that you can write down or can describe the properties in terms of such boundary partition functions where these two so-called boundary states encode somehow the boundary condition R denotes the distance of these two boundaries and H is the Hamiltonian of the bulk in between now having such a boundary partition function this is almost exactly the same structure than for this Lo-Schmidt amplitude I was showing you before except that we would replace this R by some IT by a complex number so you can think of this Lo-Schmidt amplitude as a partition function but a partition function at complex parameters or in a different way you can think of this Lo-Schmidt amplitude as a boundary partition function where the boundary conditions are not imposed in real space but in real time so it's your initial condition your time evolved in your initial state and you project at some point T later in time back on your initial condition which realizes the two boundary conditions in time so from this analogy formal analogy we see that these Lo-Schmidt amplitudes are partition functions at complex parameters already from there you might believe me that this quantity can become non-analytic but I would actually in the following few slides would like to show you that one can make this much more rigorous and in particular to connect to other equilibrium principles R is the distance between the two boundaries the spatial distance of course here in this notation I'm using certain units for space you can define it for generic for example if you would like to describe the Kazmi effect you could do it by solving this quantity but here I will only use it to draw some formal analogies so it's important that also to emphasize although there is this formal similarity there are also differences that I will point out at various points later on that there is a formal equivalence but there are also differences on the physics side so now it's actually an interesting point that already in the 60s people started to realize that on a formal level it's useful to study partition functions in complex parameter planes at that point it was more like it was a purely formal mathematical way of studying phase transitions as I will show you in the following so completely abstract concept here it becomes physically relevant and what has been done in this context for studying complex partition functions is that you take some parameter here like inverse temperature but you could also take a magnetic field or a coupling and extend it to the complex plane artificially so give it a complex number replace it by a complex number z when you do that so why should you do that you do that because you can then say two things the first one is the following like when you do it for inverse temperature when you replace inverse temperature by some complex number z and study its partition function written down here the partition function is only a sum of exponential functions which means it's an analytic function as long as your Hilbert space is finite so when you consider for example a system of fermions, or spins on a finite lattice when your Hilbert space is finite then your partition function is an analytic function and this analytic property you can then use to understand the structure of partition functions in bit more detail particular due to the following rather mathematical theorem but which will turn out to be rather important so if you have an analytic function in the complex plane then there is the Weierstrass factorization theorem which tells you that those analytic functions actually behave almost like a polynomial in that you can write this function in terms of a smooth part apart which always remains smooth this mu of z and as some kind of polynomial which includes all the zeros of this function so like you would do for a polynomial for a polynomial you would not have this smooth pre-factor only this product of zero this factorized product involving the zeros on the right-hand side which means in the end that if you are interested in some non-analytic structures of your partition function or later on because we now know that partition functions are formally low Schmidt amplitudes if you are interested in the non-analytic properties of that quantities this information is only contained in those zeros because this function mu of z is by definition always a smooth function it will never lead to some non-analytic behavior so now that we know that we can write partition functions in this form we can also write down this low Schmidt amplitude in this form involving only the zeros and some smooth part which we will not care so much about because we are interested in non-analytic behavior so that's fine we now know that the non-analytic part is contained in these zeros what can we how can we use this now in the folic statement is only because this mu of z is always a smooth function so so if there is something non-analytic in this z in this function z since it cannot come from this part it can only come from this zeros zeros part yes you yes yes yes I am a bit sloppy here but you will see precisely this point of taking the free energy will be in the end be non-analytic and that's what I will discuss on the next 2-3 slides so first of all it has to be analytic in the whole complex plane if it is that then I think that's already sufficient from a practical point of view I would say all partition functions in the complex plane which are described some more realistic systems will do that so the number of zeros increases as you increase the number of degrees of freedom so there is a connection but I would not be able to tell you more details can be more concrete but it increases when you increase the number of degrees of freedom so in order to understand what now what these zeros mean and how they can lead to non-analytic behavior it's actually turns out to be useful to study not the amplitude but rather the associated probability if you have non-analytic behavior in one or the other quantity or like it doesn't matter if you have non-analytic in this L of Z in G and vice versa but it will be in terms of to get some physical intuition it will be more useful to study this L of Z so it's the absolute square of this amplitude so you will have there's actually a two missing here you would take two times the real part of the smooth function and the absolute square of all these expressions involving factors involving the zeros but as you all know actually the relevant quantities to study in the thermodynamic limit also in terms of thermodynamics you don't study partition functions but you study free energies which means taking the logarithm of partition function and here and studying the intensive contribution like the corresponding density meaning that you divide by the number of degrees of freedom so concretely what I previously called the rate function lambda due to the connection to partition functions also think about dynamical analog of a free energy density again this is a formal identification there are many this is not a thermodynamic property so defined as like the logarithm of this L of Z and then dividing by the number of degrees of freedom now we will ignore completely the smooth part here because we are only interested in non-analytic contributions so we only consider the product involving the zeros so taking the log of a product we can rewrite this as a sum of logs I think I hope this is somewhat obvious when we are interested in the singular part here you will see can you say it again so these zeros depend not only on your final Hamiltonian which is doing the dynamics it only depends on your initial condition so they will change so like the precise location of your zeros will change both upon changing your initial condition and upon changing your Hamiltonian however this does not mean that like they change in an arbitrary way they change in a certain way that you can still get some meaningful information out of studying the zeros for a concrete problem so now let me so why I am doing that because I would like to get some physical intuition about give you some physical intuition about these zeros let me define a density of zeros row of z down here which is minus 2 over n sum over n delta function of the delta z n minus z if I am doing this I can rewrite the singular part of my free energy now in terms of an integral involving this row of z namely in the following form that the singular part of the dynamically analog of the free energy is like integral over the complex plane over this density of complex partition function zeros and then this log which we had from before and now this quantity that you see here has a very nice analogy to some problem you already know from electro dynamics lectures namely you can think of this lambda s as an electric field generated an electric field in two dimensions generated by some charge distribution row of z and why? because this log of z bar minus z is nothing but the green's function in two dimensions so using this analogy you can immediately see when this lambda, your electric field can become non analytic if your charge distribution is analytic your electric field is also analytic is also a smooth function the only possibility to get ah, can you speak a bit louder? ah, you're right yeah, sorry so what kind of charge distributions can you have so you can have either point charges you can have that your charges form lines or you can have that these charges form areas in the complex plane here the complex plane is made up of some real time, one axis imaginary time, the other axis so now whenever as I said your charge distribution is smooth also your lambda will be smooth so meaning in those regions where there are no charge distributions where there is no charge of course the charge distribution is smooth and therefore also your potential lambda has to be smooth however for example you know that at surfaces charge distributions necessarily have to be non smooth in order to like this can only happen in some non analytic manner so you know that at those surfaces or lines where some area of charge distribution becomes zero you have to have a zero like you have to have a non analytic behavior of this lambda in this way you can see how the the zeros completely determine whether your dynamical free energy is an analytic or non analytic function at a particular point and also you see that it can happen so there is nothing accidental in all that I showed what I was using is that the formal analogy to a partition function and then using this general analysis of complex partition functions to show you that the zeros of the partition function or the corresponding distribution of zeros determines where you will find non analytic behavior and where not so this I already told you so like can only be non analytic at a point charge upon crossing such a line or on the surface of an area so that these are the points where your lambda is non analytic ok so now a quick wrap up so we have I already said that essentially so we have seen that this amplitude has this formal structure of a partition function however at complex parameters the singular contribution to the this dynamical free energy you can think of an electrostatic problem and from that electrostatic problem understand where you can get non analyticities and where you cannot and therefore it tells us that these functions can become non analytic there is nothing mysterious to that and the non analytic structure is only given by this distribution of the zeros of course what you have to have is that so let me go maybe back to the definition of this density of zeros so that's the definition of course what you what I've assumed somehow in the following that this function has a well defined thermodynamic limit but I can assure you in the generic case it will have that so you need this 1 over n suppression to get a well defined limit for that quantity yes no it turns out actually that the typical scenario is that somewhere in the complex plane that you will find so in one dimension you typically find lines in two dimensions and higher you generically find areas that's the generic behavior from examples I know both from equilibrium and now this non equilibrium context so somehow these zeros have the tendency to form structures so they are not randomly placed somewhere in the complex plane but somehow they tend to form structures like lines or areas why this is the case I don't know honestly but from all the examples I know this is typically what happens ok so now I this was more like a formal part because these things can be non analytic so why these dynamical transitions can appear but now comes the more important part and that is of course equilibrium phase transitions are much more than just non analytic behavior of a free energy for example there are the powerful concepts of scaling and universality there is the notion of an order parameter we know how to macroscopically describe phase transitions in terms of landau or more generally field theories there is a certain kind of robustness of phase transitions to perturbations and many more one could list more much more of these points in what I will tell you is not that we already have a full understanding how to or to which extent all of these important aspects apply also to these dynamical transitions but you can see already here on the right hand side some references where all of these points have been addressed for a particular for particular models and particular problems we don't have full understanding yet but at least already at this point I would like to point out that many of these important properties of equilibrium phase transitions also take over and many I would like to discuss this in the following but also let me emphasize that there are many similarities but there are also differences so it's not like just replacing temperature by time and then you get all the same physics again, there are differences but also many things which are behaving the same way and with this actually I'm very happy right now so I would suggest now maybe to take a 10 minutes break before we continue with this first lecture, thank you