Thank you very much for inviting me. I'm very happy to participate in celebration of anniversary of Professor Shatash Shrilley.

Many of his achievements were mentioned earlier in the conference. Today I want to emphasize the role of Samson in integrability, and specifically in birthday-gauge correspondence. I think this photo was not shown before, this new one. I'm just taking credit. It's old ones. Old ones.

Well, but new for the conference. It's this, your sister-in-law is like Georgia got talent. So let's start with non-linear Schrodinger equation. First of all, the model has many, many different names. It's called Leip-Lininger, non-linear Schrodinger,

Bose-Gaz with delta interaction, and some people call it Tonk-Schirardo-Gaz. So there is also several different ways to write down Hamiltonian. And this is the representation in terms of many-body quantum mechanics. Kinetic energy and delta potential, coupling constant will be positive for this lecture.

Later I will write it in terms of the quantum field, Bose fields, like Bose field kappa, standard Bose commutation relation. Delta is the positive number, which I will interpret later, the step of the lattice. So at first, the model was solved by coordinate,

and stayed that way for a long time. Many interesting physical properties were extracted from coordinate, but later it was reformulated in terms of quantum inverse scattering method. That's our matrix, which is i squared to minus 1, permutation, and this is what lambda and mu we call spectral parameters.

So this is a operator, and somehow in the original paper, it was written approximately in the sense of the step of the lattice. The step of the lattice was supposed to be small, and the elevator was written approximately with this precision, delta squared. But later we were able to restore all the next orders

in the step of the lattice, and this is exact elevator on the lattice, I mean, no corrections. So the step of the lattice under the square root, rho is the square root of each one, coupling constant, side by side, number of particles, in the lattice set number j, rho index,

and that's mainly modification of diagonal elements by the square root, but also some correction on the diagonal, this number of particles. So this is exact elevator. This is lattice non-linear Schrodinger, a operator for lattice non-linear Schrodinger. It has involution. It has involution. So sigma x is off-diagonal matrix

with identities on the off-diagonal, and then this is dug as a conjugation of quantum operators. And lambda, later I will argue that lambda can be kept real for the purposes of this lecture.

Excuse me, what is lambda? Oh, the coupling constant. No, no, no, no, no, no, no, no, no, no, no. This is coupling constant which enter Hamiltonian. Lambda is a spectral parameter which is a new parameter which was not in the Hamiltonian. This is like artificial new parameter. These are designed in order to solve the model. There's something new which was not in the Hamiltonian,

but appear, you know, the trick of solution. Also, the dimension two by two of a operator was not mentioned in the Hamiltonian. This is yet another auxiliary object in order designed to solve the model. Okay, so this is entries of this elevator. This is two of diagonal and diagonal,

and I can calculate the commutation relation. This is SU2 algebra, representation of SU2 algebra. But spin is negative. Spin is minus two, kappa divided by delta, and I'm very happy to present this in the presence of Professor Karczemski who's somewhere here because he likes negative spin,

minus one, so I mentioned minus one specifically. Complex even better. Well, for the purpose of this lecture, it will be negative spin. So once again, kappa is positive. Okay, SU2 representation was negative spin. Then quantum inverse scattering method

works very similar to the continuous case. So I can multiply operators in the auxiliary space. I mean like matrices two by two, row by column, and then the entries, the quantum operators in different lecture sets commute with one another. We call it ultra-locality. So this project I can write as explicit matrix

two by two in the auxiliary space. But A, D, and B, and C are complicated combination of quantum operators, all of these S plus minus in all of these letter sets from one to L. I'm sorry for the degeneration of notation. I mean length of the letters is also L. Trace of monodromy matrix A plus D is important,

and the Hamiltonian will be extracted later from this trace of monodromy matrix by means of trace identities. This involution still works for the monodromy matrix. So dagger means Hermitian conjugation of quantum operators.

Zero means this is explicit matrix in this auxiliary space two by two. And sigma S is off-diagonal matrix two by two and multiplication that auxiliary space. So it's easy to invert this matrix. I kind of can use Cramer's formula, but with some shift of spectral parameters. So here, I mean, I can divide T inverse,

and I mean, can multiply by T inverse, and T inverse will be given by Cramer's formula almost with some shift of spectral parameter. And the determinant also given like A, D minus B, C with some shift of spectral parameters. It's like we call it quantum determinant. It's the center of the algebra.

It commutes with all other operators, actually equal to some function of spectral parameters, with quadratic function. So algebraic beta-ansat works in the form which was designed by Professor Tartarzhan with Cuartos. And then it leads to beta-ansat's equation, which we shall analyze in a moment.

So lambda is those spectral parameters. Later, they will play the role of momentum. Momentum is a simple function of the spectral parameter. And then they're even more convenient than momentum, because if I write two-body scattering matrix as a function of momentum, it will depend on both momentum of two colliding particles.

But if I write this in terms of spectral parameters, then scattering matrix will depend on the difference of spectral parameters. It's like more convenient parametrization from the point of view of study of two-body scattering. So and then next, we have to design Hamiltonian, starting from L.A. There are several different ways

to construct Hamiltonian on the lattice. The most popular was constructed by Tartarzhan and Fazieff. Later, I will mention other Hamiltonians. So this one, TTF, people call it F. Well, TTF. The Hamiltonian is written in terms

of the logarithmic derivative of gamma function. And this J depends on the local spins. Here is this. I mean, this is spin in one lattice site, spin in the neighboring lattice site. And then in this equation, everything commutes in this equation. So this quadratic equation, which

I can solve by standard formula. And then, oops, substitute this into here. So this is famous Tartarzhan-Hamiltonian, which we shall study for a while. And then later, we shall switch to another Hamiltonian. So for spin minus 1, specification

for Professor Karczemski. And then the spin minus 1, this is better this equation and this is expression for the energy. Let's study better these equations for a while. We can prove three different theorems for these better equations. So let's prove the theorems. Yes, do I have time for proof of the theorems?

OK, 15 minutes. Theorem number one, this system of equations, first of all, I have n variables, n lambdas, and also n equations. So for k here, supposed to run through 1 to n, so it's not one equation, it's a system of equations. So the theorem number one, if solutions of the system exist,

all of them have to be real numbers. So how can possibly prove this statement? Well, I have to study these factors in the right hand side. So in the right hand side, I can, simple elementary calculation, show that the modulus of each and every factor in the right hand side is larger than 1 only if imaginary part of this lambda is greater than 0.

If it's smaller than 0, then it's smaller than 1. In the left hand side is the other way. So if imaginary part of this lambda is greater than 0, then smaller than 1, and the other way around. So how I can prove it?

First of all, let's, I mean, consider this set of lambdas. Maybe they're imaginary. Let's pick the lambda with a maximal value for imaginary part. If there are several of them, I pick one of them. And then I plug this in the right hand side. So in the right hand side, I will have this difference. This lambda has a largest imaginary part. So the difference actually will have a positive imaginary part.

So the modulus of the right hand side is larger than 1 as well as the left hand side. So left hand side, if imaginary part, I mean, it's larger than 1, then this means that imaginary part is smaller than 0. So all of this show that maximal lambda

with the maximal imaginary part, it's imaginary part is smaller than 0. So all other lambdas also have imaginary part smaller than 0. So all lambdas has imaginary part that's smaller than 0. Then I can go through a similar consideration, can pick up lambda with a minimal imaginary part. And then I can prove that all of the imaginary parts

are greater than 0. So this means that they're equal to 0. So that's like real, right? OK, so I proved is that George Poirier was using same arguments in 1918 in his paper on Riemann-Siedler function. Poirier was using similar arguments. Poirier, 19, 19, 19.

I will have reference for 1856 letters, but for the different subject. Not about this, some other stuff. OK, so we proved that it's a solution. If they exist, they're real. Do they exist? So take logarithm.

Take a logarithm of the birth equation, so l is the length of the letters. Theta is this, logarithm of what was left hand side. And theta is of the difference is this logarithm of the former right hand side. So birth equations, famous. So how I can possibly prove this solution? So I want to prove next theorem,

solution exist and unique. So after I choose integer numbers, so first I pick set of integer numbers, which is the subject of the third theorem. Well, the second, let's pick set of integer numbers, this n, and then after we do that, solution exist and unique. So how can possibly prove this?

Actually, this equation, birth equation, has young action, has action. So this action was suggested by Young and Young. So theta one is integral of the theta of the previous transference, integral. And then this is integer numbers, and this is theta integral of those scattering phase.

So if I consider extremum of this action, that derivative of s with respect to lambda, I will come back to this equation, to this equation. So this equation has action, which was used in many papers by Professor Shadashvili.

So let's prove that this is, it's convex. So let's prove that it's convex, then there is unique minimum, then solution exist and unique. So let's calculate second derivative. Second derivative of this is in diagonal element, k. It's positive, right? Because k is small, coupling content is positive. And this is off-diagonal element, which is also with a minus sign.

So let's calculate quadratic form. So v is a sum vector, a real vector, with real components. So after some massaging, it's equal to L, this k multiplied by the square, and this off-diagonal will be combined to this square of the difference, square of the difference. So anyway, it's positive for if v is real.

So the young action is convex. So unique minimum, nice. So it's theorem number two, solution exist and unique, given by the set of integer numbers. By the way, the determinant of this will pop up again later, when we shall construct eigenvectors of this Hamiltonian,

which was mentioned in the beginning, then the square of the norm of the better wave function is actually given by the determinant of this matrix. Determinant, it was conjectured here first by Michel Godin from Saclay, and then I proved it 10 years later. So now the third theorem, which I promised,

Pauli principle, Pauli principle. Let me come back to this equation. So Pauli principle, the third theorem about this equation. So third theorems is the following. I can prove that if all n's are different, then all lambda's are different, which solutions exist. But if two n's coincide, then corresponding lambda's,

corresponding means with the same index, also coincide. This is not good. If two lambda's coincide, that construction of algebraic better answers will have this b square. And then all these calculations of algebraic better answers will lead to one additional extra equation. So the system of better answer equation will be overfilled, will be too much equations.

And this is the third equation, looks like positive number equal to zero. So there is no solution. So if two n coincide, then there's no corresponding eigenvectors, does not exist. So it means that all n's has to be different. So that's the end of the theorems. And the implication for physics means that the ground state of the Hamiltonian will look like a Fermi sphere. It will look like interacting electrons.

So all n's will, at zero temperature, will fill some interval, and this will minimize the energy. So I started with bosonic theory, but in the momentum space, it looks like Fermi's because of the Pauli principle. So let's move further, maybe something interesting. So this is thermodynamic limit. I have this length of the box, total length of the lattice,

which I assume go to infinity right now. And also, number of particles also go to infinity. So number of particles go to infinity, so length of the box go to infinity, the ratio is fixed, density. And then it can be these lambdas, solution of that equations.

They distribute it along some curve, curve described by Roar. And Roar is a solution of integral equation. Q is the value of spectral parameter on the Fermi sphere, which people sometimes call it Fermi sphere. And as long as Q is finite, then one

can prove that this linear integral operator, I mean, I can move this in the left-hand side, is not degenerated. So there's a unique solution of this equation. It's somewhat similar to a Leiblinger equation. For a Leiblinger equation, I have similar equation. But instead of this one, I have 1 divided by 2 pi. So this is kind of Q deformation of Leiblinger

because I moved to the lattice. So in the continuous, I had this equation, but this k was 1 divided by 2 pi. And now on the lattice, I have this k. So these equations, I mean, we cannot solve it analytically. We don't have an analytical expression. There is very simple decomposition when coupling constant kappa go to infinity,

the integral becomes small. But at small kappa, this is very, very singular. And I mean, Leiblinger studied right now for small values of coupling constant and the coefficient given in terms of our own Riemann z-dot odd arguments, which further similar enough like so much. This is even more singular at small kappa,

which we didn't study yet. So in thermodynamic limit, there's no bound state because all lambdas are real. No bound state. And any energy level can be interpreted at the scattering state of several elementary excitations. So elementary excitation can be described,

well, it has some energy and momentum. So the energy of elementary excitation is this is original energy. This is chemical potential. And this is integral equation, which we already saw later. I mean, as I said that lambda is not really momentum. This is formal spectral parameter.

So momentum as a function of lambda, I will write later, a little bit later. So this is the picture of this elementary excitation. This is my Fermi sphere. So meaning that all the states here are filled in. So I cannot put any more particles into here because of Pauli principle. But I can make a hole. So here I have excitation, which is a hole.

And this is like energy. And out of this interval, there is no particle. So I can put particle into here. And then this is my dispersion curve, dependence of energy on spectral parameter of this elementary excitation. Momentum will come momentarily. So it's gapless. The energy will vanish.

And then later, we shall see that the velocity of sound, sound velocity, is given by the slope of this curve. Well, I have to differentiate with respect to momenta, but we'll do this later. This is formula for momentum, like physical momentum.

This P0 is a log. This is log of the left-hand side of Mybert equations. Theta is a logarithm of the right-hand side. And the rho is something which we saw before. So this is momentum of the particle. This is momentum of the hole. But all of this together describe elementary excitation. It gives the dependence of energy on the momentum of this elementary excitation.

This is scattering metric, these two elementary excitation. And the scattering is elastic, the result in transition. And phase shift is given by the similar integral equation. In the right-hand side, I put this theta. Theta actually has a physical meaning of the phase shift in the bare vacuum, zero density.

And then when I have non-zero density, I have to dress up this phase by this integral equation. So phase shift. V is a derivative of energy with respect to momentum of that elementary excitation on the Fermi age. And it describes, well, by now it's called quench velocity.

Quench velocity means many different things, among other things. If I make some local quantum mechanical measurement in one letter side, then this will cause the entropy wave. And entropy wave will spread with this velocity, so quench velocity. In a moment, I will describe thermodynamics,

young and young thermodynamics. Thermodynamics has an entropy, thermal entropy. But before that, at zero temperature, I have another entropy, which is entanglement entropy, which should not be confused. So first I talk about entanglement entropy, and then we close up zero temperature and move to the positive temperature. Entanglement entropy behaves standardly.

So ground state is unique. So entropy of the whole ground state is zero. But there is some entropy in the block of spins. This is quantum mechanical phenomena, right? Because if I can't see the classical random variable, if the total entropy of the classical random variable is zero, then there is no entropy in any subsystem. One can throw the theorem.

Quantum is not so. Total entropy can be equal to zero, but there is entropy in subsystem. Quantum fluctuations, entanglement entropy. And then, of course, entropy is this complicated function of the size of this block of spins. But for large size of the spin, it's logarithmic dependence with a coefficient 1 third. And it agrees with safety.

It agrees with conformal field theory, because central charge is equal to 1. Rainy entropy. So rho is a density metric of block of that spin. So alpha is some fractional number from zero to 1. And then I take this density matrix, raise it in the power alpha, take trace, log.

This is a rainy entropy. Rainy entropy also depends logarithmically on the size of the block by coefficient in front of the log depends on that alpha. So this is entanglement entropy. Well, Professor Verlinden mentioned this in his lecture, so it might be appropriate. So now let's consider thermodynamics.

Thermodynamics, in principle, for continuous nonlinear Schrodinger, thermodynamic was constructed by Young, C.M. Young, and C.P. Young. In here, in the latest version, it's similar. Construction is similar. Also, equation is different from the continuous case. Difference in here, in homogeneous.

In the continuous case, it was square of lambda square minus chemical potential. In here, I have this, my bare energy. That's the one which I saw when I wrote that equation. This integral looks really, really similar to Young equation. So this is famous Young equations. There's some notation. So epsilon is the ratio of the density of the holes

to the particles. I mean, at zero temperature, I have a concentration of particles for a small moment of minus q. There was no holes on the particle. But at positive temperature, everything mixed up, particles and holes, so the ratio is epsilon. Epsilon also has other meaning because this model is integrable. There is infinitely many conservation laws.

This leads to consequences that even in positive temperature, there is stable excitations which does not decay. And then the energy, this epsilon is actually energy of this stable excitation, which exists for positive temperature and does not decay.

So thermodynamics, free energy. Free energy can be described by, well, this answer in terms of this epsilon. I wrote equation for the epsilon. This equation, the previous Young-Young equation was analyzed. I mean, the only one, mathematical theorem, which was proved that if I start iterating this,

like in the first approximation, epsilon is given by this expression. And then I put this into here and keep iterating. So this iteration converts, so one of the solutions exists. But it's probably unique, but it's not proven. There is no theorem that it's unique. But one solution exists. So this is free energy. This is pressure.

And this is entropy. But this is thermal entropy, right? It's thermal entropy of the whole bulk. So it's not entanglement. I mean, thermal entropy also exists for classical thermodynamics. It doesn't describe quantum mechanics. While entanglement entropy is the difference between classical and quantum, so I should not confuse.

So all of this can be done for other values of negative spin. So here, I just had to deviate from Professor Karczemski and consider not necessarily minus 1, but maybe some other number. Everything goes through. This is the equations. This is thermodynamics, entropy, everything works.

So this was, so far, analysis of the Hamiltonian of Tach-Tajian, Fadev, and Tara. So there is other Hamiltonians of the latest non-linear Schrodinger. I keep our matrix. My armatics is the same like it was in continuous case.

And for the latest, I never change it. It was discovered by Young, actually. So I keep the Young theorem. A operator is the one which was in the beginning. But by now, I want to change it a little bit. I want to make it different in the odd and even latest sites. Well, the purpose of doing this to get some other Hamiltonian.

So this is the shift of the specs of the parameter, which is a little bit different in the odd and even latest sites. And I think it was J before, so I'm sorry for changing notation. So it was J. Now it's N. And it's evidently different. All diagonal elements looks the same, but this inhomogeneity also appears under the square root.

And kappa now becomes size. I just copied from the other paper. And then later, we shall see delta is still length step of the lattice. And this is number of the total number of particles in the whole lattice set. So with this modification of a operator, we can design new Hamiltonian.

This is expression in terms of trace identities. So tau is a trace like a plus z of the product of all of this a operator. And this is a special value of spectral parameter where a operator become one-dimensional projector. The quantum determinants vanish at this point. So that's the reason I can write this Hamiltonian.

It looks even more complicated than fadif, tarasif, and taktajan. So this t, we express in terms of alpha. And alpha is a relatively simple function of the local Bose field. So this liberator actually describe interaction of eight lattice sites.

So people might argue that fadif, tarasif, taktajan is better. OK, so now I'm actually moving to form factors. Before that, well, as I mentioned, the square of the norm is given by this Gordon formula, by the determinant.

Form factors behave similar in the continuous and the lattice case. So I will kind of explain it. First, I will remind what happens with form factors in the continuous case. I mean, they don't exist. The answer is negative. But the behavior is similar on the lattice

and the continuous case. So this is one of form factors. So this j is the psi dagger psi for the continuous case. This is local density of particles in the x and t space time points. This is standard canonical Bose operator. And this q is number of particles on the interval 0x.

So I can take this q and take a matrix element between two birth states. This is like one, and this is another, and normalize it. It actually was a long story. First, some determinant representation was written for this. And determinant representation, I mentioned,

determinant of the matrix of a large size, the size of the matrix equal to n capital number of particles, which goes to infinity. And later, I will mention the similar determinant representation also exists for the lattice non-linear Schrodinger, not identical, but similar. And then this determinant representation was studied,

and the answer was negative. Because when the lengths of the lattice go to infinity, this form factor decays, goes to zero as some fractional power. So f is some solution of integral equation. Actually, this is a phase, scattering phase

of those two elementary excitations divided by 2 pi. This is definition in terms of the lambdas. We call this shift function, but it also coincides with a phase shift of two scattering matrices. So the form factors vanish in the continuous. This is just other form factors also analyzed. They also don't exist.

This is the determinant representation, but maybe it's published somewhere. Maybe it's not interesting to the people right now. So this last couple of transparencies, I was explaining problems with the form factor of the continuous non-linear Schrodinger. Similar problems occur for lattice.

And Takeshi Ota from Japan, he wrote the determinant representation for some operator, or for lattice non-linear Schrodinger. The operator somehow has correct continuous limit. And maybe I'll just come back. And then Karl Kozlowski.

Karl Kozlowski, he analyzed the thermodynamic limit of this determinant of Takeshi Ota, and then it also goes to zero as a fractional power of the lengths of lattice. This is actually the same power. So this is more or less the end of what I was going to say about non-linear Schrodinger on the lattice.

Maybe the last transparency that similar calculation can be done for sine Gordon. This is continuous sine Gordon. And then continuous sine Gordon was sold by algebraic Vidanzas and quantum inverse scattering method by Professor Tachterjian and Fadev. But we have our own lattice version,

which is interesting by itself. But maybe it's a good time for me to stop my lecture and wish happy birthday to Professor Shatashvili again. And here, he was my best student. So that's the end of my lecture.

Any questions or comments? Maybe just some references. The main reference here, the main reference is this. And this was our inspiration. This was Professor Shatashvili's related two-dimensional topological gauge theories to continuous non-linear Schrodinger.

This was the inspiration for our work. So if you look at sine Gordon on the lattice, do you see something of the Kosterlitz-Wallis transition? Is your algebraic methods?

I didn't see that phase transition. And I mean, our contribution was kind of a boring mathematics. I mean, that answers for sine Gordon. I mean, this is a relativistic model. There is one Feynman diagram in divergence. For better answers, we have to learn how to make ultraviolet renormalization.

And here, for sine Gordon, for specific, like for rational values of coupling constant, there is anisotropy parameter, which is coupling constant. For the rational values, we can compactify the local Hilbert space. We don't have to have this infinite dimensional Hilbert space, which we had for lattice non-linear Schrodinger. But we have Feynman dimensional.

I mean, dimension is equal to the denominator of this fraction. So we just use it more like for mathematical justification, everything is rigorous. But the short answer is no to your question. There is another, I learned some times ago,

lattice version of this data function, Schrodinger operator. Oblovitz logic. There's two people in the United States. They live in the Midwest. One is Oblovitz. That's the name of a person. And logic is another person. There is author of that paper, I was told,

is some Dutch person living in Brazil. They changed, they changed. I don't know. Well, I don't know that paper. But Oblovitz logic, you know Oblovitz logic. So Oblovitz logic has another discretization of non-linear Schrodinger. I mean, we kind of insist on ours, because in their case, R matrix is different.

So R matrix depends on the lattice step. And in our case, going from continuous to lattice is like changing the representation of the same algebra. The algebra is given by R matrix, so we just keep R matrix, change the representation. So we think this is an intelligent way to discretize. But I mean, Oblovitz logic, they change R matrix.

Sorry, I knew it was meaning. Your question was? That one is actually this second-order differential operator with delta function potential is replaced by some difference equation, which in the limit becomes that. Well, I mean, if. The wave function there is what's called how little polynomial. Yeah, I don't know.

If I take nonlinear Schrodinger and replace the second derivative by the difference, it won't be integrable. So one should be careful with this. I mean, it's just straightforward discretization. And then there will be a problem, because the phase transition will appear when I send a lattice step to 0. Then it's not integrable and not integrable. And this is kind of problematic.

So it's not regularization. It's like falsification. Right, but I think that this is personal. I don't know that paper, so maybe you show me later. Well, maybe I should also mention that on the lattice nonlinear Schrodinger, the model is equivalent to xxx with negative spin, which has multiple application to a lattice gauge. You know, Professor Karczemski and Lipatov and Fadev.

But also, Kazakov in his fishnet has this xxx with negative spin. So the lattice version has direct application of four-dimensional Yang-Mills. We'll talk to him. Also, I have a question just about you

were saying that when lattice size going to infinity, this land is becoming uniformly distributed with a certain density. Everyone. Did Yang and Yang or you, can it be proved rigorously? I'm just, oh, actually, yes. Yes, yes, yes.

Shodan says yes. Because remember, well, maybe I can answer your question if you permit. OK, so remember, when I wrote that and those equations, I had three theorems. The third, I kind of crumbled. You know, I didn't say clearly. But the third one was like this. I have logarithmic form of that equation. I have this n. So I can subtract this equation,

and I can estimate the difference between two neighboring lambdas in terms of the difference of n's from above and from below. So when l, the total length of this, goes to infinity, I can prove that the difference between lambda will go like 1 divided by l with some finite coefficient. So we can prove that the difference 1 divided by l, and then this is the basis of the proof that rho exists.

So the answer is yes. It's not from Yang. Yang is in my book. OK, Yang is in your book. He didn't prove this. I did. It's in the textbook. Thank you. Thank you.