 Thank you very much. And I would like to start also by thanking the organizers for inviting me here. So what I would like to tell you about in this first talk of these mini symposium on topology is our work on topology of density matrices and their detection. Now, topological states of meta, of course, interesting or very interesting, and I probably don't have to tell this, basically for two reasons. The first reason is that topological systems can have very exciting, exotic quantum states, like anionic excitations on top of a fraction quantum Hall system, which can be used also in connection with quantum information. And likewise, since topological systems are characterized by integer quantum numbers, whenever you have a spatial border between the two systems having different quantum numbers, something has to happen. And what's happened there is that there are edge states which are protected, so you can have protected edge states and edge transport. So all of these features are very nice, but there are a couple of questions open. And the first question I want to address is simply what is left from topology if you go to a system which has a finite temperature? So most of the things which we know about topological systems are actually defined as properties of the ground state wave function of a single body or a many body problem. And maybe even more interesting is the question can we extend the notion of topology to a driven systems out of equilibrium? Why could that be interesting? Well, if we have, say, the steady state of a driven dissipative system, which is a unique attractor of the dynamics, then it basically means that if you perturb the system slightly that the system tries to bring itself back to that state, so there's a built-in protection of these kind of systems. And so the idea is, can we actually connect topological protection, which protects you, for example, the edge current of a quantum whole system is insensitive to this order, but it's not insensitive to losses. So the question is, can we combine this and maybe even make it insensitive to losses? OK, so in fact, I will not try to talk about topology in the very general sense of open systems, but I will restrict myself to non-interacting fermion models. And the reason is that even for closed systems, this is a field which is mostly understood because we know that just by characterizing the property of a single particle Hamiltonian with respect to time reversal, charge conjugation, and the combination of the two, we can get a complete classification of all sorts of topological insulators in free fermion models. So in essence, I would like to concentrate on something similar to free fermion models. And the analog of free fermion systems for mixed states are Gaussian systems, Gaussian fermionic systems. Gaussian systems are characterized by a density matrix which is completely determined by single particle correlations. And in fact, you can write the density matrix in this way where this 2 by 2 matrix can just be written as any sort of combination of single particle correlations. And Gaussian systems are systems where the Hamiltonian is a big quadratic in fermion operators, and maybe there's a coupling to reservoirs which are described by Lindblad operators, which are linear in the fermions. So this is more or less the setting which I would like to discuss now. OK, so the first question to ask is if we go from a closed system to open systems, what are potential topological invariants which we can use to classify these systems? And in fact, I would like to point out that many of these known topological invariants actually fail if you go to mixed states. However, there is one which can be used, and this is the polarization. And the polarization is actually directly connected to topological charge pumps. So after first introducing what I suggest as a topological invariant for open systems, I will discuss topological pumps at finer temperature, and also this external drive and dissipation. And out of this, I will define a generalization of a geometric phase, what we call the ensemble geometric phase, which allows us to classify topology in these open systems. And at the end, I will tell a little bit about how to actually detect polarization and how to realize what we call an effective Hamiltonian which will come up in that context. Okay, so let me start and tell you a little bit about things which you may know already very well, namely about topological invariants. For the systems I'm interested in, they are basically described by geometric phases, like the berry phase or in the case of a lattice Hamiltonian and the Zuck phase. And if you have a one-dimensional system and you have a Hamiltonian depending on a certain parameter and you change this parameter in a closed loop, then this phase here can have some, has to return to itself, multiples of two pi, and this winding number which shows up there is actually a topological invariant. Now, in 2D systems, you don't even need to have an external parameter because then typically you have two internal parameters like a momentum in X and Y direction, and that defines a churn number, which again is an integer invariant characterizing the topological insulator. Now, both of them are actually tightly connected to a measurable quantity, and this has been pointed out by Tauels, Komodo, Nightingale, and Denise in the early 80s who showed that in a topological insulator if the Zuck phase has a non-trivial winding, this is related to a transport of charge or transport of particles in the bulk of the system. Now, just to illustrate this a little bit, let's have a look at a very simple model, maybe the simplest one which you can think of in one dimension, this is the Zuschriefer-Hegger model. These are basically fermions hopping on a bipartite lattice with alternating hopping amplitudes T1 and T2 at half filling, and this model has chiral symmetry, and if you calculate for this model the Zuck phase, you find that it can take on only two values, namely either zero or pi, and in fact you can go from here to here by exchanging the two tunneling rates T1 and T2. Now, if you look at the bulk of the system, you immediately recognize that these two phases don't, they're not different, they're actually identical. However, if you make a cut somewhere, then you realize that the two cases differ by the properties of the edge here, so here there's a single edge state, here there's none. This is a two-band model, so we have of course, the spectrum is just consists of two bands, and it looks identical in the case that T1 is less than T2 or T1 is larger than T2. Now, if you want to go from here to here directly by just changing the two hopping constants, then we see that it's not possible directly without closing the energy gap. So in other words, you cannot connect this one phase with the other without going through a phase transition. There's a way around that, however, and this way around has been looked at by Reiss and Meiler also in the 80s. They just said, well, let's add a staggered potential, so let's move the energy of one letter side down and the other one and the next one up by the same amount. This then actually breaks the chiral symmetry, and now the situation is slightly different because now you have a two-dimensional parameter space, you have the difference of the two hopping rates and you have this staggered potential. The SSH model lives on this green line here and the center point is the one where there is a phase transition where the two bands touch. Now it is however possible to connect the points on the two sides if you just go around this singularity and this is what is called a charge pump. So let's see what's gonna happen here. So let's start at this place over here and let's introduce a cut which has an edge state and let's assume this edge state is empty in the beginning and now let's just go around in the parameter space. The system always stays in the gap phase, so there's no gap closing because we avoid this point and if you go one cycle around, you realize that now there is one particle sitting here at the edge. So this is exactly this relation put forward by towels is that the quantization of the change of the Zuck phase is directly related to topological pumps. Okay, so this is all well known and now the question is what is gonna change if you go to system with a finer temperature or even systems which are out of equilibrium, then the system is described by a density matrix and so in general there is no easy way of defining a geometric phase. So the second question is what about these pumps, these charge pumps and what one realizes is actually that as soon as you add for example temperature to the system, the transport of charge is no longer quantized. This is a paper by Matthias Trojas' group and here you see the quantized charge as a function of the temperature in units of the gap and as you can see, as soon as the temperature gets comparable to the energy gap the transport is no longer quantized and this is what you would expect because you partially occupy the excited band which has a transport in the opposite direction so these two counteract and eventually cancel each other completely. So this means that also topological pumps or the charge transport is no longer a good quantity to look at topological properties at a finer temperature. However, there is a third quantity which in the for ground state systems is directly related to the other two and this is the polarization and it has been shown by King Smith and Thunderbolt in the also in the early 80s that also the change of a polarization in units of the cell length of the unit cell is identical to the change of the Zuck phase. Now what the polarization is is in essence just the center of mass of the electron wave function within the unit cell. It turns out however that this definition of polarization is not so useful so what we are looking at is in the generalization of this put forward by Resta which is applicable also to systems which have periodic boundary conditions. So if you would for a moment just pull the logarithm inside here you would immediately see that these two here just agree because as X is nothing else in the center of mass on this complete ring. One thing which you notice however is what the polarization is actually measuring it measures the phase of the expectation value of a unitary and since it's a phase it changes have to be quantized by construction if you make a closed loop and parameter space because the phase has to go back to itself or to multiple or multiples of two pi if you make a closed loop and parameter space. The good thing however is that polarization you can evaluate for all states. It does not have to be ground states does not have to be non-interacting systems can be an interacting system and it can in particular also be a mixed state. So we can evaluate this with a density matrix. Okay so let's do that. Let's have a look what polarization is actually doing if we look at topological pumps. So the first model I want to look at is again this Reismiller model is a very simple model but now at final temperature. Now if we look at zero temperature and plot the polarization as a function of the two parameters the difference of the two hoppings and the staggered potential then you can see that if we go around the parameter space around this degeneracy point then the polarization winds by one unit and that's what we would expect because that's related to the change of the Zuck phase that's related to the particle transport. Now the question is what happens if we add temperature to the game and let's add a temperature which is much higher than the energy gap in the system. So this is here T equal to 100 in units of the energy gap at a certain point in parameter space and now what you see that the polarization qualitatively changes but the important thing is what does not change is the winding. So if you go around the singularity you still pick up a winding of exactly one unit in the system and the winding is not dying out. And in fact what we just see even more interesting is that this property remains if you crank up the temperature even more and it only changes if you hit T equal to infinity since this is a two band model we can in principle go above T infinity and go to negative temperatures and as soon as we do this if you go from positive temperatures to negative temperatures then all what's gonna happen is that now the sign of the polarization winding changes. So it looks like that the polarization in some sense picks up the topology of the most populated band for positive temperatures. It's a lower one and a two band model and for the negative temperatures is the upper one and that is what seems to happen here. Now let me introduce the second model which doesn't even have a Hamiltonian and there is no temperature which is a system which is entirely driven by external reservoirs. So again this is a one dimensional model and a one dimensional model of spins and as you can see from the color code I'm going to set it up in a way that it has a unit cell of two sides and the way I do this is I couple the system to reservoirs only. So there's no Hamiltonian one could add one but it's not really needed. So it's essential that we have only a Lindbladian drive and I assume that there are two types of Lindblad operators in order to have something non-trivial they should talk to at least two neighboring sides. So I consider some reservoirs LA which always couple the sides within one unit cell and then I consider a second type of Lindbladians which couple the sides of neighboring unit cells and most importantly these Lindblad operators are linear and they're far beyond creation and annihilations so there is a steady state and the steady state is Gaussian and therefore we can calculate everything and determine all the properties almost analytically. Now let's see what these Lindblad generators what they are doing. So there are two parameters there's this parameter epsilon there's this parameter lambda so let's first look at this parameter lambda if we put lambda equal to plus one then only these terms here remain and what the in essence are doing is that they take out excitation from the green sides and put this them into the blue sides and likewise the other one takes out excitation from the green side and puts it in the blue sides and if we take lambda equal to minus one then only this term survives and then it's exactly the other way around that excitation is taken from the blue side into the green sides and also here and then there's a second parameter which is this epsilon which basically controls the coupling strengths to the upper reservoirs and the coupling strength to the lower reservoir so we have a two parameter space Lindbladian and therefore the steady state of the system is determined by these two parameters now if we calculate for this model also the polarization as a function of these two parameters then we find that the polarization actually also shows a winding so therefore we can argue that there's a topological feature here namely that if you go on parameter space around this critical point here then you see a winding of one unit of the polarization now this is an open driven system and I started off with the motivation that these open driven systems could show some enhanced robustness against perturbations and so let me look at this a little bit more in detail so if we add to the model Hamiltonian disorder so what I plotted here is the polarization when you go around this loop proportional to the angle phi and as you can see Hamiltonian disorder does change the polarization slightly at some points but does not affect the winding and by the way does also not affect the symmetry protected points at pi and zero which are exactly the same now you could ask the question what's gonna happen if I add losses so if I add dissipation to the system is it still surviving and so this is a result here these are homogeneous local losses so independent losses on all the sides and as you can see again there is almost no effect the winding stays the same and the symmetry protected points have exactly the same values as before so this is an example where in a driven system you can have protection topological protection against losses okay now the question is maybe by now I have convinced you that polarization is a good measure to detect topological properties for mixed states now the question is can we understand this a bit better can we actually construct something like a geometric phase for the ensemble and for this let me go back and show you this numerical calculation for the final temperature rise mailer model this is done for a small system of eight sides this again is the angle and this is for different temperatures for low temperatures or t equal to zero essentially is the line in the center and then for high temperatures are these curves which wind which have these steps now if you increase the system size so if you go to 64 sites then the picture looks like this if you go to 256 side the picture looks like this and if you go to 512 sites the picture looks like this so what you can see here is that if we increase the system size the polarization at final temperature seems to approach the value at zero temperature and in fact this is what we could show analytically you can show that the polarization of the steady state of any unique steady state of a Gaussian system it does not have to be a thermal system it can be any Gaussian system it's actually given by the polarization of a pure state plus some correction which scales away with the inverse of the system size and this state here turns out to be the ground state of some Hamiltonian and this Hamiltonian is this effective Hamiltonian which is where the single particle Hamiltonian matrix is in essence the single particle correlation function of the fermionic system at final temperature or in the steady state now and now this of course and then there comes something on top of it if you look at the winding of the polarization I told you this has to be an integer so the winding of this has to be an integer the winding of this has to be an integer so therefore this has to be an integer as well but it also has to scale away with the system size so the only integer which is compatible to this is zero and as a consequence the winding of the polarization in any Gaussian steady state is actually identical to the winding of a pure state polarization of in a pure state and this pure state is the ground state of this effective Hamiltonian now for finite temperature models one can actually show that this matrix is directly related to the Hamiltonian so that's why we saw in the in the Reiss-Meyler model that we pick up essentially what we get at T equal to zero so now since we have this now we can define a geometric phase and this is nothing else than the Zuck phase of this pure state and this can be applied not only to systems at a finite temperature but also at steady states of open systems where there is even not a Hamiltonian but there is a single particle correlation matrix okay so what this shows is that one can classify the topology of Gaussian fermionic models just in terms of the effective of this effective Hamiltonian so in terms of the single particle correlators and so this was actually already conjectured in a paper by Charles Barlin and Sebastian some time ago and is now more or less now proven by this is actually the case and there's two interesting conclusions from that is namely we can also understand when a topological phase transition can happen well it can happen under two conditions and the first condition is that the steady state is no longer unique so if a damping gap closes and there's a second solution which is also infinitely long-lived then there can be a topological phase transition this is the analog in essence to the closing of an Hamiltonian gap and the second possibility is that the gap of this effective Hamiltonian here closes and there may actually be interesting extension to interacting systems but I don't have time to talk about this now unless you ask me in a question session and now we can understand why for example in the finite temperature rise Miele model the phase transition happened at infinite temperature because for a thermal state the correlation matrix is just given by the Hamiltonian itself multiplied by the inverse temperature so topological transition can happen either when the energy gap of the Hamiltonian underlying Hamiltonian closes or when the temperature goes to infinity and that's exactly what we observed here we can also understand why in this reservoir induced topological pump at this point here at the center is a topological singularity because if you calculate for this model the damping spectrum first of all you don't see anything interesting happening here but if you calculate the spectrum of the single particle correlator you find that it has actually a touching point exactly at the center and that is the reason for the topological phase transition okay so now I told you that polarization or I hope I convinced you that polarization and the related ensemble geometric phase are good candidates for classifying topology and mixed systems but now of course we know there is no quantized transport so how do we actually detect these properties this quantity and so here's a proposal based on an interferometric detection scheme so this is a simple machsina interferometer in one arm there's a fixed phase shift and in the other arm we put our one-dimensional fermion lattice and we assume that the fermions interact off-resonantly with a light mode which propagates through here and in essence what this gives is just a kernel linearity proportional to the density of fermions and now if we choose the mode function of the light mode going through here to have a gradient and along the axis of the lattice then in essence the effective Hamiltonian is proportional to the center of mass of the fermions so this is the center of mass operator and as a consequence the photons which goes through the interferometer will pick up a unitary if you choose the parameter in the right way will pick up a unitary which exactly corresponds to the expectation to the exponent of two pi i over l times x which is the operator which determines the many-body polarization if the photon runs through this arm and if it runs through the upper arm it just picks up a reference phase and therefore detecting the different signal of the two will pick up the argument of the expectation value of the t operator and this is exactly the polarization okay so in the last two minutes let me also tell you one other idea which we have and how we can actually realize this effective Hamiltonian by topology transfer so the idea here is let's consider our final temperature or driven model and let's couple it dispersively to a one-dimensional fermion lattice which we can keep at low temperature so close at t equal to zero say and for the sake of discussion let's assume that we have a density-density type coupling in momentum space if you want to know a bit more details of how to realize it, ask me later but if we have something like this where alpha and alpha prime are the indices for the bands because we have here maybe a multi-band problem if this coupling here is sufficiently weak then the dynamics of the auxiliary system can be treated in mean field approximation so we basically can just replace the operators here of the system to be probed by the average value and once we do this you immediately realize that the Hamiltonian, the actual Hamiltonian of the auxiliary system has a single particle Hamiltonian matrix which is just given by the correlation matrix of the system to be probed so as a consequence the winding of the polarization in the open system is just the winding of the polarization in the auxiliary system and since this is operating at t equal to zero this leads directly to a quantized transport so we can detect the quantized transport in the auxiliary system okay just to show you that this is not just it does not only work in mean field these are numerical simulations the winding of the polarization of the open system and this is a transport in the auxiliary one that it makes us jumpy it's just because we calculate this on a cylinder so you see the winding is picked up by the transport in the auxiliary system okay so let me summarize I hope I have convinced you that Gaussian fermionic systems at final temperature or a non-equilibrium can be classified by an effective Hamiltonian whose Hamiltonian matrix is a single particle correlator we can introduce a topological invariant which is related to the Zuck phase or the polarization of the ground state of this Hamiltonian and I have shown you that how to detect the polarization or how to realize this effective Hamiltonian via topology transfer and I would like to thank the people involved there are people in my group Dominic Lindsner, Lukas and Bill and we had a collaboration with Charles Sebastian and Alex and I should also thank for previous contributions to Dr. Gost and with this I thank you for your attention