 As usual starts with an announcement. Just to remind you that we'll have a reception today at 4 on the terrace of the main building. This is this building here, and in case we don't have enough prisons to celebrate there is one more. Yesterday was was the birthday of Dirac and as per tradition the winners of this year are Dirac Medal are announced and here they are Super Sajidiv Dam Tsong and Shao-Gen Wen. Can you send me? All of the three are well known to this audience. There's personally and by there are contributions and the citation for the medal is for novel for for their contributions toward understanding of novel phases, I guess in strongly interacting many body systems introducing original cross-disciplinary techniques. So this is very much other subject of this conference So it's very nice that we're all here once it happened. We probably need to book the time for the next workshop now. Okay, and please join me in a remote and relations to Subir, Dam Tsong and Shao-Gen Wen. So once again they used to be empty boards on the left and on the right of this auditorium. I think they are gone by now but there will be movable boards in the lobby. So you have ample time to mount your poster before 2.30 today. And with that I'm leaving the podium to our local organizer Professor Fazio from ICTP who will be sharing this session. So it's a pleasure to welcome the first speaker of the section, Nathan Andre, and the talk will be about quench dynamics in the sign Gordon modern. Thank you very much. It started five minutes later. Okay. Okay. It's a great pleasure to be here. It's a beautiful place to celebrate Pierce in particular. I will talk about a work that was done in a direction that has not been discussed very much thus far, but which I think is one of the main intellectual challenges of future condensed matter, namely non-equilibrium dynamics. So I will talk about quenches and the work that I will describe was done with my student who is about to finish calling Rylans. So non-equilibrium is of course a very old topic going back at least to Boltzmann, but what has brought it to the forefront more in the last 10, 20 years is the ability to carry out very precise experiments where one looks at isolated systems where non-equilibrium effects are not washed out by coupling to environment. Such systems as in particular cold atom, system various nano devices, systems that are isolated and for which one has very fine control and one can pose questions in a very precise way. So one of the simplest protocols of carrying out non-equilibrium is called quench. What one does then, one starts with a system that is totally isolated prepared in a particular state and then evolves it in time and asks various questions. How does the system evolve? How do its properties change? How do what is the entropy that is produced in the process and so on and so forth. So one can do it in various ways. One way is to do with the time-dependent Hamiltonian. One prepares the system in a state A, evolves it, it reaches the state B, and one begins to ask questions. This quench that I'm going to talk about is called a sudden quench where one prepares the system in a particular state and turns on the Hamiltonian, a time-independent Hamiltonian, and allows the systems to evolve. And as I said, there are many experiments and even more questions. How does thermalization work? How does one reach thermal state if one does? There are more non-thermal state like non-equilibrium steady state, many states that are not equilibrium and one has to ask the question what happens in the long-time limit? How does one classify currents, entropy production, spreading of entanglement? So there are many, many questions. Here is an experiment, one of the earliest ones, which will be relevant to the work I'm talking about, where one prepares a system in a mod insulator and allows, so here is the mod, I'm reversing the time order. So here it is in a superfluid, suddenly one raises the barriers, one, the Hamiltonian is a mod insulator, the initial state is a superfluid. How does the superfluid involve under a mod like Hamiltonian? And one sees it evolves back and forth. That's because it's a finite time. The work that I'm going to describe is where one starts with a mod, initial state, and evolves it under a superfluid Hamiltonian. Another famous experiment is the Newton's cradle, where one allows a gas of boson to bounce back and forth, interact, and one asks whether it's going to thermalize. Here's an experiment that I hope is going to be done because we are calculating what, how does the condo effect evolve in time? You have a lead, you have a dot, which is isolated, a time t equals to zero, one lowers the barrier, the electrons begin to hop back and forth on the dot, and the condo peak begins to evolve in time. How does the system reach its ground state? Okay, so one has many, many questions. One, I will focus on a particular question, what is the work done when you quench the system? So as you quench the system, you pump in a lot of energy into it in the form of work. So work is typically E final minus E initial. Remember that the system is isolated, so there is no heat flowing. But what is work in a quantum system, it's, in fact, it's a, as was pointed out by Hengie et al, it's a random variable. It evolves to measurements, an initial and final energy. In other words, I prepared the system initially in a state, phi i, phi initial, with a given energy. So the probability then is p i is one, and then I evolve the system, and I do another measurement. I measure energy E n with probability of the overlap phi i psi n, absolutely square, and it has energy E n. So the final energy fluctuates. I could of course start from also an initial distribution, let's say finite temperature, but for simplicity I'll start with a well-defined initial state. So for such a quench, the work distribution is given by p w, namely the probability to measure a certain work is given by a initial energy, final energy minus initial energy being equal to w times all the probabilities that would lead to this result. So if you look at that, then you see that this distribution of work has the form of a spectral function, and it has many properties that follow just from its form. The work has to be larger than delta E, which is the difference between the final ground state of the final Hamiltonian minus the energy of the initial state. And there is going to be a delta function when the work is at this point, it's weighted by the fidelity, which is the overlap square between an initial and ground state. And if the Hamiltonian is gapped, then there would be a continuum of excited states into which phi i can transition, and we have to add to delta E the minimum amount of 2m, and then from there on we'll have a continuum of states of two particle spectrum. So there will be power-like behavior that one expects with some critical exponent alpha. Similarly, when we have four particles, we'll have another threshold with another critical exponent. If we have bound states, we'll be able to transition to poles that correspond to these bound states and they will appear again in the distribution function. So a lot of work was done, particularly here at CISA, started by Silva, Gambassi, Palma, Sotiriani, Smusado, Calabrese, Gould. A lot of people worked and understood much of this. And there are many, many other questions. If the question is, if one carries out the processes in a reversible, irreversible way, is there how much entropy is produced, how the entanglement is spreading. There are beautiful fluctuation theorems that relate various equilibrium and non-equilibrium relations, how the system moves forward or backwards in time, questions that I'm not going to go into. Okay, so let me now begin by introducing the main actor, which is the Loschmitt Echo. Loschmitt, of course, and his debates with Boltzmann have a long history. Here I'll simply define the Loschmitt Echo as, I don't know why it's twice, the overlap probability between the final state, which means Phi I, which has been evolved by E minus IHT, and the initial state. And one can show easily that the work probability is just a Fourier transform of the Loschmitt distribution. And what makes it non-equilibrium interesting and hard is that unlike thermodynamics, it doesn't probe only ground state and low line state, it probes the full Hilbert space. So the Loschmitt amplitude, as you can see from this expression, where I've just put in here a complete set of eigenstates of the Hamiltonian, wanders all around Hilbert space, and it's characterized by probabilities which tell us from what state we have started and what part of the Hilbert space we are going to probe. And such a time evolution may exhibit, for example, dynamical phase transitions. Namely, suppose I start with a system, let's take a simple model like a quantum spin model, a quantum ising model where I prepared the initial state as being up, and then I propagate it with a Hamiltonian where the spin is down. So in one region it will be mainly the overlaps will place the state up and then there is another region in Hilbert space where the spins are down. So as it goes from one region to another region, there may be phase transition in time. But whether such phase transitions occur or not is not only the property of the Hamiltonian as it would have been in statistical mechanics, it's also a dynamic property that is characterized by the initial state. It has to tell us where we start. And there are experiments that probe precisely the system that I was talking about, and you can see that in the case of the quantum ising, there are quantum phase transitions, namely singularities in time. I will talk about another model, more complicated one, which is the Saint-Gordon model. So I have a quantum field theory with interacting quantum field theory with a cosine interaction. It's quite a ubiquitous theory. It describes many, many low energy. It's the low energy dynamics of many other systems, spin chains, interacting boson, the Bose-Herbert model. Even quantum impurities could be related to Saint-Gordon. Of course, costally stylus could be analyzed in this language. Turns out, of course, this model should be known as quantum integrable. Brothers Zamologichov have already analyzed it in 78, but classically it was known earlier. It is equivalent to another model through bosonization to a model which is called the massive Turing model, where you have right and left movers, plus minus, you have a mass term, and you have a density-density interaction. And there is a very precise way to relate the parameters of the theory here, which are m and beta, to the parameters here, which are m0 and g. Now, this relationship is not universal. Both models are renormalizable field theories, and they need to be renormalized, and the way you renormalize affects the precise relations, but once you have decided what you are doing, it's well-defined. So, again, the Turing model is an integrable model, was sold by Berknoff and Thacker in 78. We know what the spectrum is. It consists of solitons and anti-solitons of mass m. M is not the same as the parameter m0 of the Hamiltonian. It's renormalized. This is the actual pole in the Green Functions. And for repulsive interaction, when beta is in the range, beta square is between 4 pi and 8 pi. That's the spectrum, and we have, however, very complex scattering at very interesting S matrix. If we have attractive interaction, there are also bound states between solitons and anti-solitons, which are typically called breathing modes. And there is a very interesting limit. Obvious in the language of the Turing model, but in the language of sign Gordon, if we choose beta square to be equal to 4 pi, this model is actually a free model. Namely, it becomes a free model where solitons and anti-solitons go through each other freely. Okay, so that's the model, and here is the quench I'm interested in. So this is the effective low energy of a Bose Hubbard model. And as I said, the experiment has been done. Imagine that I start it here from a mod state and quench it. And then I'm going over to a superfluid as time evolves, and I want to understand the dynamics of this. So the quench I'm going to do, of course, this is done in two dimension. I'll do my quench in one dimension. So imagine that my initial state is described here. The bosons are in the lattice. The potential is very high, infinitely high. And at time t equals to zero, I lower the barrier. Interactions become active. The bosons begin to flow and interact and scatter. So here is the quench. It's a sudden quench. I start with an M square, which is the amplitude of the potential, and lower it from infinity to a finite value. So I'm going from here to here suddenly. Another initial state would be very interesting is where all the bosons are on top of the hill. It's a maximal energy eigenstate of the original Hamiltonian, or it's the ground state of the Hamiltonian, where M square is minus infinity, where I reversed the spin. So the quench that I'm going to do is from M square being either infinite or minus infinite to a finite value. And this is one realization. Many other realizations exist. For example, Gritzavetal have proposed to do it in condensates of interacting atoms, and there are many systems. So that's the question, and now we are beginning to work. I will introduce a representation of the massive Turing model due to this tree and the Vega, where they define the model on a space-time lattice while maintaining the integrability of the model. So here you have lines in space-time of right movers, lines left movers. You have to think about it as spin degrees of freedom emanating from each vertex, which are the spin degrees of freedom interact via microscopic transition amplitude that I've written here. And what they have shown is that if we take the continuum limit, namely the spacing delta, which is L over N, we take it fixed first, taking N and the size of the system L to infinity, holding the mass parameter fixed, then this model goes over via Jordan-Wigner to the massive Turing model. So this is a very beautiful way of putting the Hamiltonian on a lattice in a way that you can handle it elegantly. So in this language now, you see time evolution, which means evolution in this direction consists of two movements, one to the right and one to the left. So I'm going, theta is a cut-off parameter, I should have mentioned it, which I sent to infinity at the end. So e to the ih delta, namely I move only one direction delta forward, consists of moving this way and this way by a matrix theta, which moves me this way, the whole line, and theta inverse, which moves me in the opposite direction forward, again product of these arm matrices, which describes the local interaction in space-time of the left and right movers. And then if I want to evolve it to a time t, which is m times the amount of delta, then all I have to take is to this expression to the power m and I get the evolution Hamiltonian, e minus ht, this is a normalization, tau inverse, tau to the power m. So what I have here is some initial state, let's say some spin configuration that I put initially at time t equals to zero and then evolve in time successively by this operator, which is the time evolution operator until I reach a time t, at which point I take the overlap with the final state. So here is the description, in the spatial direction I will take periodic boundary condition, some initial state, evolve it in time, and then take the overlap with the final state. So here is my Loschmidt amplitude, I start with phi i, evolve it in time with this trans matrix of evolutions and take the overlap with phi i in the end. I can simplify it with some identities and I rewrite it in a nicer way. And you see now this problem has become a classical two dimensional lattice problem where you have, you are moving forward on a space time lattice, which allows you to rewrite it in a more elegant way. So here is the original problem, I want to move phi i forward and end up with phi i. Let me do now a rotation in space time and look at the problem in this direction and think about this is now the time evolution in this direction, then the states phi i initial and final become boundary condition on the time evolution of the state on a finite sequence. So now what was spatial direction, namely here, that was here, now it becomes periodic in time while what was initial and final now become boundary condition on this time evolution. And therefore the Loschmidt echo becomes, I have to take now a trace, what used to be a trace in space because of periodic boundary condition now becomes a trace in time of operators that evolve in this direction so we can think about it as an open spin chain but with the boundaries prescribed. I can have a finite size spin chain with boundaries prescribed in the beginning and the end and the system evolves in time. So people sometimes call this a quantum transfer matrix. Okay now what initial conditions can we have? So you see this is the initial boundary condition and it could be very complicated, it could be changed on any direction. If we want to be able to make progress we have to choose some initial state that we can handle. So we are going to think about states that can be written as products of two lines. So I'll have a state V on each double line that is a linear combination of a spin up, spin down on these links here and this is going to be my initial boundary condition. So that translates when you go back precisely to the states that I was talking about. If I choose the relation between C1 and C2 which relates spin up, spin down and down up in a particular way I will have in the bosonized language the ground state. If I choose them in the opposite phase relationship I will get maximally excited states. A point to make about comparison to experiment for example the superfluid motor transition is the fact we are talking about an effective low energy like the sign Gordon model but a sign Gordon model is a good description let's say of a Bose Hubbard model for low energy but I just explained that when you do a quench you wander over the full space of the Hamiltonian and then you have to worry whether the post quench Hamiltonian is still a good description of the original model like the Bose Hubbard model but I will not discuss this issue now. Okay, so here is what I have to do then to compute the Loschmidt echo in the limit where N which is the size of the system goes to infinity. Remember this is the direction which now is periodic. Since N goes to infinity then we know from particular work of Baxter that if I diagonalize this Hamiltonian I get product of all eigenvalues to the power N and if N goes to infinity only lambda max will survive. So the question is then how to calculate the maximal eigenvalue of this transform matrix in time and of course if we ask when would the lambda max coincide with the next eigenvalue that is precisely the point where we have the dynamically quantum phase transition. Okay, so here comes a big technical jump and I may lose people here but I get lost when somebody shows me quantum Monte Carlo doesn't explain how it does it, it shows that's what I get. So here I have a machinery that's what I get. Okay, however this machinery is beautiful and I recommend everybody to learn it, learn the beauty of beta ansatz. So here is I was telling you I want to know what is the maximal eigenvalue. So using this beautiful formalism which has been developed over the years it is given by this complicated expression no reason to go into detail. The point is that it is characterized by certain parameters lambda 1 to lambda m, remember m is the time it's t over delta and these beta parameters must satisfy these very complicated beta ansatz equations. So this is a set of transcendental algebraic complicated equations which you have to solve for lambdas. You end up solving for densities of lambdas once you have those densities you can calculate this maximal eigenvalue and then you get the Loschmidt echo. So again technology instead of solving directly this beta ansatz equation one introduces a technique due to Destri and the Vega one introduces an auxiliary function a which I've written down here and if you look at it and if you compare to this equation I have written down a function which is such that the beta ansatz equation is given by a I have to look at values u that are equal to minus 1. All I did is bring this to the other side I say that this product has to be equal to minus 1 and then the lambdas have to be a solution. Of course this equation depends on its own eigenvalues which means that I have I can translate it into a nonlinear integral equations once I solve it I can calculate what I want. I want to a so the two states that I'm going to characterize are characterized are given by phase i pi over 2 or 0 if you remember this is the combination of up down down up at the edge states so the initial state the quench from the ground state is characterized by xi i pi over 2 and the maximal excited state is xi equals to 0. So once I have found what is the this auxiliary function then in a long appendix we can prove that the log of the Loschmitt echo is very nicely expressed in terms of this integral so it's given by minus i e not t log f so e not t is the ground state energy of the post quench Hamiltonian f remember is the overlap and here it's given exactly for these states and integrals over the auxiliary function and its inverse so you see the auxiliary function plays in a sense the role of density of states out of equilibrium and you can see in a very simple way the analog so remember that g was namely the Loschmitt echo was given by some over n to e minus e n t c n square where c n's where the overlaps if I take the log log of this I get an expression minus i e not t plus log c n square so this is this expression here is log of the fidelity and this complicated sum which probes the full Hilbert space weighted by the initial state is now encoded in the auxiliary function oops thank you okay so one writes a nonlinear integral equation for the auxiliary function and there is different physics if the system is attractive or non attractive the boundaries that you have to integrate over in one case do not include poles in the attractive case they are going to include poles which tell us that we have bound states and we can solve those equations but to test them we'll first do the case of free fermions it's just a sanity check we'll choose beta square equals to 4 pi in which case the sine Gordon model becomes a free model we can solve explicitly the auxiliary function plug it in the expression for the Loschmitt echo and get a very nice expression for the echo and for the fidelity and of course since these are free field theories we can obtain the same by doing simple Bogolubov rotation and Alessandro Silva did it already in 08 where he introduced many of the questions that I'm discussing here we can find the work distribution for the free case when we start from a ground state we do the Fourier transform of this expression we find as expected a delta function for the multiplied by the fidelity and here we find a critical exponent of half characterizing the threshold work distribution for a quench if I start from the maximal energy initial state would be similar but with new thresholds and the critical exponent is going now to be minus three half okay so let me now turn on interactions very quickly now I have a new auxiliary function plug it in to compute the Loschmitt echo carry out a Fourier transform to obtain the momentum distribution and we find the following results there are turns out again they are characterized by critical exponent half for quench from a ground state and minus three half for quench from the maximal energy excitation and here it's what it looks like around the threshold we have this kind of behavior the black line means no interaction turning on interaction you see the effect is to reduce the power here you see the excitation the distribution around the other initial state where you start from maximal energy if we have attractive interaction then poles appear in the expression for the auxiliary function and we have new expressions which for the work distribution which reflect now that we can go into the bound states here is the initial threshold critical exponents and one can obtain an exact expression to any region let me summarize what I showed is how to calculate the Loschmitt echo and word statistics for some quenches I didn't discuss it but for this particular initial states we showed that there is no dynamical phase transition we didn't have time to show that either new non-equilibrium dualities occur between strong and weak interactions and one can realize this quench in experiment many things to do, connect, obtain other non-equilibrium thermodynamics go into deeper description of the system calculate the entropy production fluctuation theorem show that this say non-linear integral equation describes also how observables evolve in time as I said that this auxiliary function describes non-equilibrium density of states which will describe how observables evolve it's interesting to do it for small and large system in small system fluctuations are enhanced and one can connect it with fluctuation theorems it's interesting to do quench across critical points study defect production, the kibble Zurich dynamics, scaling universality other type of quenches either slow drives or flow k periodic so non-equilibrium is just at its infancy and I hope you all contribute thank you very much