 Fantastic, right first of all I'd like to thank the organizers for inviting me to this absolutely spectacular and diverse event. I hope I'll be able to contribute to the diversity of this meeting. So my talk here is on the subjects which is in a sense the exact opposite of what the previous speaker was talking about. I'm going to discuss adiabatic protocols as opposed to the quench protocols, very slow protocols in driven many body systems and I'll present a rather general results relating to those systems. It can be applied to strongly-related systems. It can be applied to weakly-related systems. Initially our research on this subject was motivated by a strongly-related example, but I'm not going to consider it today because it is very special, probably not very exciting for many people. I will focus on something a lot more simple to demonstrate how this technique works in the many body systems. So what I'm going to do today is I briefly discuss motivation, what adiabatic protocols are, the relationship to the adiabatic theorem and what extensions test. I'll state the main question or the main deficiency of the adiabatic theorem and how people are working to mend this today and then I'll state our main results. I'll try to keep it simple. I'll try to write as few equations as I can and these are going to be again as opposed to the previous speaker. These are going to be short equations. So hopefully I'll be able to explain in a very simple terms our results. Now I'll discuss an example. I have to apologize to the organizers. There's not going to be a strongly-related example. It is going to be a weakly-related system or in fact an uncorrelated system. However, if you are interested in interacting many body systems, applications to those systems of our results, you can look on the archive for two recent papers published by this waters. Okay. So adiabatic protocols. The idea of an adiabatic protocol is shown in this picture. So let me explain what is shown here. Suppose I have a Hamiltonian which depends on a number of parameters and this yellow thing here is the parameter space for this Hamiltonian. Let's imagine that at some point the Hamiltonian was placed at this point of the parameter space. To each point of the parameter space you can associate a point in the projective Hilbert space of the system which is the ground state of this Hamiltonian. This denotes denoted phi in this picture. So phi is the ground state of the Hamiltonian taken at a particular point of the parameter space. Now let's imagine that we prepared our system in the ground state initially and then we slowly change the parameters of our Hamiltonian so it describes some trajectory in this parameter space. Then the initial state will evolve along a certain curve under Schrodinger's evolution. That's the trajectory shown here. The physical state which results from this evolution is called psi. There is another trajectory which I can draw on the same projective Hilbert space which is the trajectory described by the ground states of instantaneous ground states of the Hamiltonian which are denoted phi here. These two trajectories generally do not have to be the same of course. However an important result was proven in 1928 which says that if the driving is sufficiently slow I can keep these two trajectories as close to each other as possible. So this is the statement by Born and Faulk. This theorem can be extended to the general ground state. I'm not going to dig into that. For us this formulation, this statement will be sufficient. This theorem has plenty of applications in physics and what I'm going to mention here are practical applications which are almost engineering. One example is the Tautilis pump. More genuinely adiabatic transport in mesoscopic systems. This is a very schematic representation of this device. We have two reservoirs which are connected by a region of a gapped material. There are lots of particles inside this region and it has a gap inside. This is the region whose parameters I can control experimentally. This is the parameter space and let's imagine that I perform some cycle, some closed curve in the parameter space as a function of time. Also keeping inside this cycle a region where this segment of my system is gapless. I perform a cycle around this singularity in the parameter space. Then, as was pointed out by Tautilis, if the driving is adiabatic, so if adiabatic conditions are fulfilled, that is the system is not excited from its current state as it moves around. Exactly one particle will be transferred from left reservoir to the right reservoir. The current to the system will be quantized. That's a fantastic example and later on I'll show you recent experimental realizations of this protocol. It was realized actually only two years ago. Another example of application, possible application of adiabaticity or adiabatic protocols is non-abilian braiding for the purposes of quantum computation. These are the ideas which were originally merged in late 80s, early 90s, late 80s, early 90s, but they became popular in the beginning of this century because due to the ketifs bugs. The idea is certain strongly related systems, or actually there are weakly related examples as well, can have degenerate ground states, topologically protected degenerate ground states if I punch holes in the space where these quantum systems reside. One example is of course the archetypal example is the new equal five-half quantum pole system. The holes are vortex-like excitations in the system. Now if I move adiabatically slowly, move these vortices around each other, if I braid them, then that is equivalence to performing a unitary operation on the degenerate ground state manifold. This operation does not care about the exact way I move the particles around. The only thing it cares is what element of the braid group is represented by this braid, not part of the details of the trajectories. This is another example and it is being explored very intensively in different communities. People are trying to implement this in topological superconductors looking for ketive latices, fermions with P-wave pairing. Just another example that I'd like to mention, it is important. It is an example of adiabatic quantum computations. There is an excellent review which appeared earlier this year which covers this subject extensively. I recommend you to have a look at this review if you are interested in this subject. The idea of the adiabatic quantum computation is that you prepare a Hamiltonian system in a way that you understand and control its ground state. Then you gradually deform your Hamiltonian into a messy one for which you do not know the ground state. Adiabaticity ensures you that the state that you obtain as a result of this evolution is the ground state of this messy Hamiltonian. This is the ground state you are interested in. There are multiple applications of this idea in quantum computation. Let me briefly summarize what we've seen. In all these examples, the protocols are performed on large, many body systems. Some of them can be strongly correlated. Some do not have to be. The correct operation of the protocol requires adiabaticity. The actual state of the system does not deviate from the instantaneous ground state. The adiabaticity theorem tells us that indeed it is possible to achieve adiabaticity in the system, but it has nothing to say about the conditions for the adiabaticity. How slowly I need to drive the system to achieve adiabatic conditions. From the practical point of view, this is an extremely important question. We need to know if our system is adiabatic or not. The question is how slowly do we need to drive a quantum system to achieve adiabatic conditions. Before discussing the answer to this question, I'll introduce slightly more formal notation. We have this parameter space and there is a trajectory in this parameter space, which I parameterized by a single parameter lambda. H as a function of x of lambda will be called h sub lambda. Without the loss of generalities, because that depends on my parameterization, without the loss of generalities, I can say that lambda as a function of t is a linear function of t multiplied by gamma. Gamma is called the driving rate or the ramp rate in the system. The instantaneous ground states are parameterized by lambda and are given by this equation. Schrodinger's equation can be recast in this form as an equation for the wave function as a function of lambda and gamma appears as a parameter on the left-hand side of the equation. This is pretty straightforward. The initial condition for the equation is that the initial physical state coincides with the instantaneous ground state and what we are looking at is the adiabatic fidelity. The adiabatic fidelity is essentially the overlap between the instantaneous ground state and the physical state squared. If adiabatic fidelity is close to one, we are happy. If it is very, very small, we are unhappy. That's the idea which can, of course, be reformulated in rigorous mathematical terms. The adiabatic theorem has a very, very short statement. The adiabatic fidelity goes to one if the driving rate goes to zero. This convergence is, of course, a non-uniform and stuff, but I'm not going to discuss this in detail. That's the rigorous statement of this theorem. I'll give you a very brief overview of known results. The archetypal model, which is an exactly solvable model where you can derive the adiabaticity condition, is the Landau-Zehner model, again 1932, very early in the days of quantum mechanics. The Hamiltonian there is a two-level Hamiltonian. This one, there is a parameter delta, a gap parameter in the system. There is a polymatrix sigma z here. This is the initial Hamiltonian, and this is the perturbation, which contains the polymatrix matrix sigma x. These are the levels of this model as a function of time. We want the system to follow the instantaneous ground state, which corresponds to this level here. The exact solution of this model shows that the adiabatic fidelity is given by this expression here. It is 1 minus e to the power minus pi delta over gamma. If the driving is very, very slow, this exponential function goes to 0. If it is large, the fidelity is 0, which is not what we want to achieve. The conclusion is the fidelity is close to 1 as long as gamma is much less than delta. This exact, exactly solvable model is very nice. However, it does not easily admit generalizations to systems with large Hilbert spaces, let alone many body systems. Many body systems have huge Hilbert spaces. There are lots and lots of dense set of levels above the ground state. Actually, the speaker after me is going to discuss many level generalizations of this problem, which are exactly solvable. This is an interesting direction of research. However, generally, if you are dealing with the many body systems, you have exactly zero chance of solving the time evolution of a time-dependent Hamiltonian. It is a very, very complex problem, a lot more complex than finding the ground state of the system, for example. What do we have instead? The only hope then is you can try and find some reasonable bounds on the speed of the diabetic process. There has been a generalization of the Landau-Zehner bound, which is still used by people in different communities, just to say that if I take an overlap between the time-dependent ground state and the derivative of an excited state, if I take the maximum of this expression, it has to be less than the excitation gap. If this is true, this, by the way, is the criterion, exactly the criterion that you have in the Landau-Zehner problem, but it does not work generally. There has been some discussion of this in different literature. This is actually a very, very bad criterion for large systems. This is just an example of this way of thinking about conditions for the diabeticity still encountered in literature. There have been efforts to overcome these difficulties using rigorous mathematics, using the operator theory. There is a whole school of thought which was initiated by Katza in 1950, which tries to generalize the Landau-Zehner criterion to arbitrary Hamiltonians in arbitrary Hilbert spaces using norm estimates and the known spectral properties of the Hamiltonians. This is a typical result from this community. This is not Katza's result. This is the result from subsequent work. I'm showing it to you here just to demonstrate that the main idea in this calculation is you take a second or third derivative of your Hamiltonian with respect to the parameter. You calculate the spectral norm of this operator divided by the gap to some power, and then you demand that your evolution time is greater than this expression. These are rigorous, sufficient conditions for diabeticity. It turns out they are not extremely good in many body systems. The main reason, of course, is that on the right hand side of this expression I have a norm of an operator. The norm of an operator in a huge Hilbert space, well, in an actual Hilbert space, it tends to be unbounded. The expression on the right hand side is just meaningless unless you impose some cutoffs on the Hilbert space of your system, and then your answers depend on these cutoffs, and the whole thing gets very, very complicated. Even if you do that, you end up with an expression which is very, very large on the right hand side because of nature of having a big Hilbert space. That makes your sufficient conditions very stringent. This approach also doesn't give you a necessary condition for adiabaticity. Now we are gradually moving to our main observation. There are some interesting insights into adiabaticity conditions from the many-body physics side, or from the strongly related side. There were two publications in 2008 where two groups analyzed in different contexts adiabaticity conditions in many-body systems. Actually the conclusions were similar, and I just boxed the abstract of this publication here, and it is important. So essentially they say that the structure of the theory suggests that the... sorry, they reformulate and approximately solve a specific many-body generalization of the Landau-Zener problem. Unlike the single particle Landau-Zener problem, our system does not abide in adiabatic ground state, even at very slow driving rates. Our solution can be used to understand the example, behavior of two-level systems, coupled to electromagnetic fields, and blah, blah, blah, but the main observation is that if the system is large, if the system is a many-body system, then for some reason the driving has to be extremely slow to ensure adiabaticity in the system, and they speculate in their paper that this could be some general effect pertaining to all many-body systems. Many-body character of the system is an interesting aspect of this problem. I want to make another argument, which is quite general. If I want to do adiabaticity conditions from the point of view of quantum mechanics, pure quantum mechanics, I have to solve some nasty Schrodinger's equation for the amplitude of the ground state, or the system in the instantaneous ground state. It takes this form, and the energies on the right-hand side and the states on the right-hand sides are all the states in the Hilbert space of my system, which are exponentially large. If I have a big many-body system, it's an enormous number of terms on the right-hand side. Now this equation, if I write it in this form, makes no assumption about the Hamiltonian, no assumption about the eigenvalue problem, doesn't know anything apart from the linear structure of Schrodinger's equation. So it is essentially an equation from linear algebra. However, many-body systems that we know of have extra structure embedded in them. They have sense of space, they have sense of locality, dimensionality, correlations, and stuff like that. They are a lot more structure-rich than this equation. So we need to, we thought we need to exploit this structure to move forward to find some better understanding of adiabaticity. And we hit upon the most important property of the many-body system, which is the most relevant property of the many-body system, which is generalized octagonality catastrophe. It turns out that if a physical system exhibits generalized octagonality catastrophe, then we can find a new bound on adiabaticity and the new adiabaticity condition, which happens to be a necessary adiabaticity condition. So let's imagine that we have a many-body system and we are looking at the overlap between the ground state at lambda equals zero and a small lambda. So this corresponds to finding an overlap of the ground state of a Hamiltonian and a slightly deformed Hamiltonian. Then, as many of you know, there is a broad range of examples where the logarithm of this quantity behaves as minus c and lambda squared plus the correction term and this c and this coefficient, which I'll call susceptibility. The susceptibility increases with growing n, which means that this overlap function decays as a function of the system size. So the system size in this context is an important asymptotic parameter. This phenomenon was first discussed by Anderson, I think, in the context of a spin impurity embedded in the metal and in that case lambda is just the coupling constant between the impurity and the metal. I'm sorry. There are different formulations of the same. Another formulation is that I just take a metal and I insert a non-magnetic impurity, just a simple impurity inside and I calculate the overlap between the ground state without an impurity and with an impurity and that will give me the same result. I'll have cn, which is equal to log n. So local impurity inside the metal gives rise to orthogonality catastrophe with susceptibility logarithmically increasing in the system size. There is another requirement, an additional requirement on this parameter here, which is technically fulfilled in all systems that I know of. If you find the system where it is not, that would be interesting. And I'm discussing this in complete generality and I'm making this the defining property of the class of the systems that I'm going to look at. This works, yeah, this works for gap systems as I'll show you in the subsequent examples. It depends on the type of driving. So the sort of deformation of the Hamiltonian. So we can deform the Hamiltonian of a gap system in a natural way that gives rise to orthogonality catastrophe. I'll show an example later on. So this is a defining property. Again, I'm not going to prove that this is a general case. There are exceptions from this and this is not a theorem. This is a defining property of the system that I'm going to look at. Okay, so our main results we have established new rigorous bounds on the fidelity of adiabatic process in general quantum theory for systems exhibiting orthogonality catastrophe and our estimates are given in terms of the parametric susceptibility C and only. And these results give useful necessary condition for adiabaticity of quantum protocols. I'll give you the main idea. So the main idea is we exploit the metric structure of the projective Hilbert space. Projective Hilbert space is a set of vectors from the point of view of linear algebra, but it is also can be thought of as a metric space where the metric, the distance between two points is defined this way, for example. There are more than one metric actually. So it is one minus the overlap between the two states squared. Another important ingredient is the quantum speed limits. Again, you can look at the PIFAS review about quantum spin limits. The idea is if you have an initial state, phi naught, and you begin to deform the Hamiltonian of the system, then the physical state cannot travel arbitrarily far away from the initial point. The distance it can travel in time is limited by the quantum spin limit written here, where delta H is the average, the time average of the uncertainty of the Hamiltonian in the initial ground state. So if the Hamiltonian's uncertainty is zero, the state of course cannot travel from the initial point. Otherwise it travels. For generic driving, which is smooth, this delta H here is just proportional to lambda multiplied by some number V. Okay. Now here comes the main trick that we use. So this is the projective Hilbert space, and this is the initial point. Now if I deform the Hamiltonian, the orthogonality catastrophe tells me that the state phi lambda, rather quickly, if the system is large, rather quickly is going to travel very, very far away from the initial point, whilst the actual physical state can't do this because of the quantum spin limit. Moreover, I can give a rigorous estimate using the triangle inequality. So if I consider this triangle here, A is A plus B is greater than C, and this gives me A is greater than C minus B. This directly translates into the statement that the fidelity, the adiabatic fidelity of my system minus the exponential of the, minus the overlap between these two states is less than lambda square at the overgum. So this, for small lambda, for sufficiently small lambda, these two quantities are very close. Now this quantity is very easy to calculate. This is simple because this is just an overlap of two ground states. This quantity is very, very hard to calculate, requires the knowledge of the complete evolution of the Hamiltonian in the hover space. But with, for sufficiently small lambda, they are close to each other and one can exploit this. So let's, let's see how this works. So this, this is a particular system and this, this is the fidelity phi lambda. This is the overlap squared of two functions and this is, these are the bounds. This is a system where I only have 10 particles. As you can see, the bounds are not very useful here. They, they do not tell me that these two quantities are close enough. However, if I go to n equal to 100, the bounds become quite strong. The two susceptibilities must be close to each other within this range. And as you can see, before the bounds become useful, useless. So the bounds become useless somewhere in this region because the bounds increases as number squared. However, before it becomes useless, the fidelity drops by factor of e. If I increase the system size further, I'll get better and better bounds on, on c, on, on f of lambda, which makes it very close to c of lambda. So that's, that's the idea. And this is written in algebraic terms. f of lambda is e to the power minus lambda squared cn. This we know from orthogonality catastrophe, plastic erection due to the bound. If n is large, the system completely loses adiabaticity after, after having traveled this distance. Now the star is equal one over square root cn provided that I can neglect this term. And I can neglect this term as if we over gamma cn tends to zero and, and goes to zero. And that gives me the necessary condition for adiabaticity in the system, which is written here. Gamma has to be less than v over over cn. Okay. I'll just show two slides, two or three slides actually to, to illustrate how this works in the experimental system called the Tali's pump. So this, the Tali's pump that I mentioned earlier was realized in, in these two recent experiments. So this is the one-dimensional optical lattice. There are many particles sitting there and the cycle is performed in the parameters of the lattice here. And this is the Reiss-Meyer model, which describes this and it has two parameters, staggering potential and staggering hopping. And the cycle is performed in the space of these two parameters. Now, we calculated from, from our relationships, we calculated the, the adiabaticity conditions and what we can see that despite the system being gapped, despite the system being gapped, the necessary condition for adiabaticity is gamma less than v over cn. Scales is one over square root n because cn scales is n and v scales is square root n. So the adiabaticity in the system collapses with increasing system size if the driving rate is fixed. And, okay. I'll, I'll skip this one. I'll just show you what happens to the pumped charge if this, if this happens. So this is the adiabaticity per cycle, the fidelity per cycle is good here. The pumped charge is one. This is the pumped charge. Now, when fidelity drops, the pumped charge goes berserk. It's, it's, it's, the quantization is completely lost. Again, incomplete agreement with our criterion. Okay. So this is, these are the conclusions of my talk. And thank you for listening.