 Isian from ICTP. Hello. Yes, that is me. So let me. Who will speak about analog of many body very essential and for critical systems? Yes, Karan. So the floor is yours. Thank you. Thank you for the presentation and thanks the organizers for the opportunity to present my work, our work. So it is going to be about a extension and analog of a quantum many body very essence theorem and with the aim to extend it to critical systems. So I should just upfront let you know that this is going to be a little bit orthogonal to perhaps most of the talks during this conference because it is not a condensed matter per se. But well, hopefully it will be of some interest to you. So this is a paper that has appeared on archive already. And it is it is work done in collaboration with with people from Denmark, Matthias Jogensen, Gabriel Landy, Alvaro Alambra, Jonathan Brask and Martíper now you'll be from Geneva. So well, the point of the talk is going to be a many body system, a generic many body system. So imagine a chair truly fascinates me that you can say so many general nice things even about a chair. So well, it's Hamiltonian is going to be some short-range interacting Hamiltonian which will consist of some atoms A and K and they are going to interact somehow and each atom is going to have a self-potential. So we're going to have this Hamiltonian and very little can in general be said about this Hamiltonian if we look at it as just an operator. No, I mean it's going to be a complicated Hamiltonian with a very complicated spectrum. But the fact that the spectra of Hamiltonians are complicated, of generic Hamiltonians are complicated is not really related to the fact that these Hamiltonians are complicated. Even if you take something very simple such as the quantum icing model, which is very easily solvable, its spectrum is still very complicated. Now, if we want to nevertheless be able to say some things about the spectra, about the energy distributions of the systems, of pretty much all many body systems, then we need to become interested in other kinds of quantities. So if we imagine that our many body systems is in some state raw, then let's look at its cumulative energy distribution. So it is basically this object, which is basically the sum of the probabilities of all energy levels that are below E. So basically what is the probability that I have an energy that is somewhere from 0 to E, where I say that the ground state has energy 0 because it's arbitrary and can be set arbitrary number and we can choose it to be 0. So if it turns out when we ask questions about this JE, about the cumulative distribution, then things become very easy. Well, not very easy, but nevertheless one can say things. So the non-critical Berry Eson theorem that has been proven in this a bit older work by Brandao and Kramer, regards general lattice systems. So imagine you have a lattice, which has n sites. And the Hamiltonian is just a sum of n terms, the sum of h news. And so there are going to be several assumptions in Hamiltonian. So first assumption is going to be that the Hamiltonian is finite range, which means that if we take the vertex v, then this particle v is going to interact only with neighbors, only not necessarily nearest neighbors. So for example, on this graph I represented next to nearest neighbor interactions basically. And each h nu is going to be either at most next to nearest neighbor interacting. You may have a particle that doesn't interact. So the Hamiltonian doesn't even have to be translation invariant. But all the local terms have to be upper bounded. So yes. Now, so this is the requirements on the Hamiltonian. As you may notice, these are very weak requirements because it basically doesn't even require translation invariance and nothing. You just need to have short range interactions. Now coming back to state, the requirement on the state is the following. So imagine you have two regions, this one and this one. Let's call them arbitrarily A and B regions. Then we want the correlations between these two regions to be exponentially decaying, which is quantified in this following way. So rho is the state of the whole many-body system. And A and B are operators that are localized on regions A and B respectively. So A lives in A, B lives in B. So this is the correlation. This is the correlator between them. And this is normalized with the norms of A and B because basically we keep A and B arbitrary. So the point of this equation is that the correlations between any observables localized to these regions is going to be upper bounded by this exponent, where dA, B is the distance between the regions A and B. And psi is going to be basically the correlation length. So just for future convenience, let me denote this trace with rho as just this averaged brackets. So this is just a notation. And the other and last requirement on assumption about the system is that this quantity mu square that I define as variance, which in turn is defined as the standard deviation of the Hamiltonian, should be proportional to n. Now, if you notice, these two things are essentially statements about the system being non-critical because if it was critical, then the psi would diverge, then the variance would diverge because it gives the heat capacity and this kind of thing. So basically, we have a non-critical arbitrary system. Then it can be shown that the energy distribution. So this is the cumulative energy distribution of the system. And this GE is the cumulative distribution of the Gaussian distribution. So we have a normal distribution, which is just the exponent. And then we take a cumulative distribution and we can show that the maximal distance between these two cumulative distributions goes to 0 as n goes to infinity because this is just a logarithm over n and this always goes to 0 as n goes to infinity. And this d is the spatial dimensionality of the lattice. Maybe I forgot to mention that the lattice can be arbitrarily dimensional. Now, so this is a pretty powerful result that says that pretty much any non-critical system is just a Gaussian. Basically, despite the fact that the spectra are very, very complicated, essentially, if you just look at the average energy, then the whole distribution of energy is going to be just a Gaussian distribution around the average. Now, if we become interested in critical systems, this result was about non-critical systems, then what can be said in this case? Can we say anything general in this situation? So what we're able to find is the following. So if you limit yourself to only thermal states to finite temperature phase transitions and translation invariant lattices, so you don't take arbitrary Hamiltonian, you take a translation invariant Hamiltonian and you ask questions about your phase transition takes place at finite temperature and well, and the state of the system is thermal. Now, to somehow approach this problem, I'm going to give you a bird's eye view on the proof of our result, which is going to be basically a step-by-step introduction to the result itself. So now the relevant quantities start with the spectral density. Now, in analogy with the cumulative energy distribution, we can just ask how many energy levels are there below the value of energy. So this is this gamma E. It is going to be the cumulative spectral density. And of course, we can immediately define the spectral density, the usual spectral density as the derivative of the cumulative spectral density. And of course, it is going to be this quantity that gives you the amount of energy levels in a small window around the value of E. So this delta n is the amount of levels and this is the energy window. Now, if we take a thermal state, so as I said, we're going to limit ourselves to thermal states, then we can define the energy distribution of this system, which is obviously the derivative of the cumulative distribution as this quantity. So this E to the minus beta E z tells you the probability of almost any level in the window, because the window is small. So almost all of these guys have the same probability. So this is the probability. And this omega E gives you the number of energy levels in the window. So this QE is going to be the energy distribution of the system. Now, with these definitions, we invoke the following, again, quite powerful result, which is due to Mueller Adlem Masanes Bieber. It's again from 2015. So now it says the following thing. So if U is the energy density and U n is, of course, then the value of energy, if we take the cumulative energy, sorry, spectral density here, and take the logarithm of the amount of states that are below this value of energy and look at its density, then in the thermodynamic limit, it is going to converge to the canonical entropy. So this sounds a little bit either trivial or completely obscure, but let me just explain what this little formula actually means. So first of all, the regime of validity is that it holds at and above the critical point. So below a critical point, your equilibrium is not going to be a Gibbs state. It's not going to be of this form. It is going to be a KMS state. And below the critical point, you have multiple phases, and none of them is described by thermal equilibrium. So you need to be either exactly at the critical point or above. So this is the range of validity of this formula. Now this U, as I said, is already the energy density. So we just set a threshold here. This SU is the density of canonical entropy corresponding to average energy density U. Now this part is tricky. Although these two U's have the same value, they are different things. In one case for the entropy, this U is the average energy of a canonical ensemble. So it is an average energy that this thermal state would give you. And this is the entropy that would correspond to this average energy density. And this just gives you the threshold below which you count the amount of energy levels. Now the physical meaning of this thing is that this logarithm of the amount of energy levels below a certain energy level is the micro-canonical entropy density at energy density U. And this equation, this equality, expresses the equivalence between canonical and micro-canonical ensembles. Because this is basically the density of micro-canonical entropy. And this is density of canonical entropy. And we see that in thermodynamic limit, they converge. So for translationally invariant short-range interacting systems, the canonical and micro-canonical ensembles are equivalent both at and away criticality or above criticality. So how are we going to make use of this fact for our needs? Now again, where we call the formula for energy density, this is the quantity that we are trying to find out. So it is given by the subject. And using the formula for omega e, because again, so let me remind that the omega is the derivative of gamma. Now we have the gamma, so we can find the omega by taking the derivative. And so we will arrive at the following formula. So you have the energy distribution for energy. This u is the energy density. And this object here is basically the free energy minus this free energy functional. Now here, let me invoke the standard thermodynamic equilibrium condition. So we know that the equilibrium in thermodynamics is achieved for the minimum of the free energy. So if we take the free energy density functional, which is energy minus t times the entropy, and then we minimize over all u's, then we're going to obtain the free energy. Right? Sorry. Now if we look more carefully at this expression, we will find that its expansion around the value of energy that delivers the minimum is going to be, of course, the fn beta is going to be the free energy, which appears here. Plus, I'm sorry, here it has to be a plus, not a minus. Plus, because this is a minimum. Plus, there's this object, which is completely positive. It's just a square of energies. And the cn beta is the specific heat. And this un beta is, again, the canonical energy density. OK, so this is going to be the main formula around which our result is built. And basically, this Taylor expansion, we continue this Taylor expansion till the end and then estimate all the terms and thereby obtain a non-perturbative upper bound on this difference here. So the results are as follows. Now before formulating the results, let me just remind you about the critical exponents that we have in finite temperature phase transitions. So if this is the specific heat, then when approaching the critical point, critical temperature, it's going to diverge with some exponent alpha. And since any antibody system has actually finite amount of constituents, so n is finite, there cannot be a true infinity. So if we look at the scaling of the specific heat at the point of phase transition where it should formally be infinite, it should have a diverging scaling with n. Now it is quite standard in textbooks. It can be found in textbooks. It's standard knowledge, more or less, that when alpha is equal to 0, so formally we see no divergence here, the specific heat actually diverges logarithmically with n. So when alpha is equal to 0, then we have logarithmic divergence when alpha is larger than 0, then we have polynomial divergence or algebraic divergence depending on the term. So this is this exponent determined by alpha. Now the very essence theorem that we find for alpha equal to 0 is the following. So when the energy departs from the average only as square root of the logarithm of the variance of H. So this is the standard deviation of the energy. And then when we depart from the average square root of logarithm n times the variation, then the energy distribution is proportional to a Gaussian. So basically in this range, it is just a Gaussian. You just see from this expression with a 1 over logarithm n correction. Basically, this is this thing. And whenever we're outside of this range, whenever the energy is farther from the average than the square root of logarithm n times the standard deviation, oh, sorry, then the total probability contained in that region, which is formally not necessarily Gaussian, is proportional to 1 over n. So it is a very small probability. Now basically, we have a quite complete specification of the probability distribution. And we see it is essentially Gaussian whenever it is relevant. Now this can be actually tested numerically for an exactly solvable model. And 2D Ising model is a paradigmatic example of the case where alpha is equal to 0. Because it's free energy can be computed exactly. There is a formula for finite n. And of course, taking the derivative of the free energy gives you access to all the moments of the energy. So they all can be calculated. And if kappa n is the nth cumulant of energy, if we take this quantifier of non-Gaussianity, basically, we just take the cumulant. We kind of normalize, bring it to the 0 level, and then compare it with the second cumulant, because the variance is just the second cumulant, brought again to the ground 0 level, which is basically the d. This is kappa 2 to the power of 1 half. So this quantity is proportional to 1 over logarithm, square root of logarithm n. So basically when n goes to infinity, all the non-Gaussianity measures go to 0 for the Ising model. So we see this general result somehow manifest itself. Also, when you look at exact numerics, so I'm not showing any plots, but this is what the numerics show, that indeed for also for this exactly solvable example, we have that the distribution is Gaussian. Right. Now, when alpha is strictly larger than 0, there is unfortunately not much that can be said about Qe, except that it is picked around the average energy and is decaying exponentially at the tails. So this is related to the following fact. So if we look at this formula, which I now realize that that is a bit cumbersome, but so this is the free energy. This is the minimized free energy, if you look at it. And this guy is the non-minimized free energy. This is the free energy functional around the minimum. When U is the canonical value, then this guy turns into this guy. Now, it can be rigorously proven that this object is strictly concave with U, right? Which means that once you go further from the minimum, you are inexorably becoming larger and larger, and the derivative is also growing. This means that whenever this U minus, whenever this free energy functional is further from its minimum by any delta, then we know that the Q is going to be e to the minus beta n times delta. So basically, it's going to be an exponential decay of Q once we go further from the minimum. So basically, this is all the information that you can get from this generic expression from the Qe. And maybe there can be other things said about alpha larger than 0. But basically, this is all we have been able to find out. So basically, in all cases, for any situation, you know that your Qe, your distribution is a unimodal distribution that decays quickly at the tails. And you can prove that for alpha equals to 0 or away from criticality, it is a Gaussian distribution. So basically, that's all I have. And thank you for your attention. And I think I finished quite early. Thank you, Karen, for this talk. I clap for behalf of the audience. Thank you. So at the moment, there are no questions. So let us wait for questions to be stopped. So maybe while people are typing, I will go ahead if that's OK. So thank you very much for your talk. I have a couple of questions if you don't mind. So when you talk about the original version of the Berry-Essen theorem, so it's about the fact that when you look at a system that are at criticality or above criticality, you expect the distribution of energy level to be Gaussian, right? Well, whenever you're away from criticality, when you're at criticality, basically, the standard Berry-Essen theorem says nothing. And that was basically the purpose of our work to explore what happens with criticality. Sorry. So when you are above criticality, so sometimes you can be gapless or the less, but sometimes you can still remain gapped. But in this case, do your considerations still hold or? Oh, yes, absolutely, yes. But basically, the thing is that since here, we deal with finite temperature phase transitions. Basically, the presence of the gap is not so. It doesn't change much. Yeah, OK, of course. Because we are macroscopically above whatever happens at the ground state. Yes, yes. Yes, in a sense, this theorem kind of trivially applies to zero temperature case. Because basically, it says that you are at the ground level. And wow, I mean, that's your distribution, right? It's extremely pictorial. OK. And I have another question. It's about possible Hamiltonians that do not verify both the original theorem and your theorem. Can they exist? Although they have a weight of zero when you consider the whole Hamiltonian ensembles. And so there will be some specific Hamiltonian that weights nothing, but that do not belong to the theorem. In particular, some integrable models, typically. Oh, well, let me think for a second about it. Well, I would say that rather no than yes, because basically, of course, if the Hamiltonian, OK, so I mean, for example, things, none of these theorems applies for long range interacting Hamiltonians. And those Hamiltonians, they are not of measure zero, rather the opposite, right? I mean, the short range interacting Hamiltonians are of measure zero in a full space of Hamiltonians. So in that sense, yeah, basically, short-rangeedness is a very strong requirement. But I mean, if I take something very, very stupid, like a Hamiltonian that is always zero, naturally, this, I mean, I guess you can say that it's a Gaussian, but with a zero. Yeah, yeah, I think these sort of pathological cases can be just, again, described by a pathological Gaussian. Yeah, OK. OK, I see, I see. Thank you. I have one question. Could you, so when you were referring to non-critical Bres and theorem, what's the definition of this J of E? This is not the thing which is associated with the Gaussian. OK, OK, it is dependent on the state. It is not the full distribution of energy levels. But this is dependent on rho, as this equation shows. Oh, yes, sure, sure, sure. So basically, you have your distribution of energy, and this J E is basically the cumulative distribution of it. So basically, it's the probability of having any energy below or equal this value E. So this is the actual cumulative distribution of energy of your system. That's what you have, in fact. And well, basically, this one is the Gaussian distribution. Yes, OK, yes. OK, so is anything known about T equal to infinity case? So rho is identity matrix. I'm asking because I am doing some things with many body localization when I look at generic eigenstates in the middle of the spectrum. And usually, I get Gaussian density of states. So I wonder whether there are some more strict results about that. Well, I would say that all these reasoning is that they apply for arbitrarily high temperatures. So in principle, T equals to infinity should also fall under this theorem. Like, I see no reason that that shouldn't work. So only if I have this condition verified with the exponential decay of the correlations between sub-subjections A and B, then I can expect that Gaussian. Most certainly, yes. So in other ways, if I have systems with long range interactions, then it might not be the case. Oh, yeah, definitely. If the range of interactions is not short, let's say, then really, I mean, this theorem says nothing. So basically, it can be anything. Yeah, it can be anything. Sure, sure, thank you. Thank you. All right, so let us wait a minute for further questions. Oh, I suppose we can have a small coffee break before the main coffee break, right? That's exactly. So I was about to suggest to Piotr as a chair. You have to have the last words. But how about we do an impromptu break of half an hour, and we come back with the presentation of Aritra at 11, our schedule. OK, that sounds great for me. So thank you, Karan, again, for this nice talk. And OK, so we can reconvene at 11.