Lattice Nonlinear Schroedinger Equation

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Lattice Nonlinear Schroedinger Equation
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[Music] thank you very much for inviting me I'm very happy to participate in celebration of anniversary of Professor Shah - really many of his achievement mentioned earlier in the conference today I want to emphasize the role of Samson in integral agenda specifically in birth engage correspondence
I think this photo was not shown before this is new one I'm just taking well but new for the conference it's this your sister none it's like Georgia Got Talent
so let's start with nonlinear Schrodinger equation there is um it's first of all the model has is like many many different names it's called leap linear nonlinear Schrodinger bazookas with dealt interaction and some people called tongues here are the gas so there is also several different ways to write down on him Etonian this is the representation terms 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 both the fields like body field copper standard both the commutation relation Delta the positive number which I will interpret later the step of the latest so at first the model was sold by coordinate the transits and stay that way long time many interesting physical properties were extracted from coordinate Betances but later it was reformulated in terms of quantum inverse scattering method um that Sarah matrix which is I squared minus one permutation and this is lambda and mu called spectral parameters um so this is a liberator and somehow in the original paper it was written approximately in the sense of the step of the latest step of the latest was supposed to be small and the elevator was written approximately with the this precision Delta Square but later we was able to restore all the next orders in the step of the ladies and this is exact an operator on the ledges I meant no Corrections so this step of the laces and in under the square root Row is a square to this one coupling constant side agus a number of particles in the legislate number J although index and that's mainly modification of of diagonal elements by the square but also some correction on the diagonal this number of particles so this is exact a liberator this is latest nonlinear Schrodinger a liberator fallacious nonlinear showing it has involution it has involution so Sigma axis of diagonal matrix with identity with identities on the off diagonal and then this is Doug is a conjugation of quantum operators and lambda later will argue that lambda can be kept real for the purposes of this this is coupling question which enter him into onion and lambda the spectral parameter which is a new parameter which was not in the Hamiltonian is like artificial new parameter designed in order to solve the model this is something new which was not in the Hamiltonian but appear you know the trick of solution also the dimension 2x2 of a liberator was not to mention the Hamiltonian this is yet another exhibit object-- you know the designed to solve the model okay so this is entries for this elevators is to of diagonal and diagonal and I can calculate their commutation relation this is su to algebra representation Phi C to algebra but spin is negative spin is minus 2 Kappa divided by Delta and I am very happy to present this in the presence of professor Carr Chomsky who's somewhere here because he likes negative spin minus 1 so I mentioned minus one specifically complex even better well for the purpose of this lecture it will be a negative spin so once again copper is positive okay su to representation is negative spin Oh then quantum inverse scattering method works very similar to the continuous case so I can take multiply L appear it is in the auxiliary space I mean like mitosis two by two row by column and then the entries the quantum operators in different letter size commute with one another we call the ultra locality so this project I can write as explicit matrix 2 by 2 in the auxiliary space but agian being CR complicated combination of quantum operators all of this s plus minus in all of this letter size from 1 to L I'm sorry for the generation of notation I mean length of the lattice is also L trace of the matrix a plus G is important and Hamiltonian will be extracted later from this trace of monetary matrix by means of three identities Oh this involution still works for the model driven matrix so dagger means hermitian conjugate of quantum operators zero muses is an explicit Mattox in this auxiliary space of 2 by 2 and sigma is of diagonal matrix 2 by 2 in 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 G inverse and I mean can multiply by G inverse and G inverse will be given by crimeless / formula almost with some shift of spectral parameter and the determinant also given like a G minus BC with some shift of spectra paramus it's like we call it quantum determinant it's the center of the algebra commutes with all other operators actually equal to some function of spectral parameters this quadratic function so algebraic the times that works in the form which was designed by a professor dr. John with quarters and then it leads to a bit under the question which we shall allies in a moment or the equation so lambda is those spectral parameters later they will play the role of momentum momentum is a simple function of this spectral parameter and then they're even more convenient than momentum because if I read two body scatter expand X as a function of momentum it will depend on both momentum of two colliding particles but if I write this in these terms of spectral parameter than scattering melody would depend on the difference aspects of parameters like more convenient parameter ization from the point of view of study of two bodies scattering so and then next we have to design Kabuto nian of starting from elevated there are several different ways to construct Hamiltonian on the ledges the most popular was constructed by terra's of takhtajan and Rajeev later I will mention other humility audience so this one TGIF people called if well TGIF the human toe nyan is written in terms of the logarithmic derivative of gamma function and this J depend on the local spins here is here this I mean this is spin and one night side spin in a neighboring letter side and then in this equation everything commutes in this equation so I can this quadratic which which I can solve by you know standard formula and then substitute this into here so this is famous dr. john mccain which we shall study for a while and then later we shall switch to another Hamiltonian so for spin -1 specification for professor car Chomsky and then the spin -1 this is better this equation and this is expression for the energy let's study better than this equations for a while we can prove three different theorems for this both equation so let's prove the serums yes don't have time for proof
of the theorems ok 15 minutes Oh serum number 1 this system of equations first of all I have n variables and lambdas and also an equation so for K here supposed to run through to answer it's not one equation system equations so a serum 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 calculations show that the models of each and every factor in the right hand side is larger than one only if imaginary part of this lambda is greater than zero if it's smaller than zero then it's smoother than one in the left hand side in the other way so if I'm actually part of this lambda of this lambda is greater than zero then small N one and the other way around so how I can prove it first of all let's I mean the consider this set of lambdas maybe they imagine let's pick the lambda with a maximal value of 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 dimension apart so the difference actually will have a positive imaginary part this one so the models of the right hand side is larger than one as well as the left hand side so left hand side if imaginary part I mean it's larger than one then this means that imaginary part is smaller than zero so all of this shows that maximum lambda is the maximal imaginary part it's imagined about the smaller than zero so all others lambdas also have a major part smaller than zero so all lambdas has imaginary part is smaller than zero then I can go through similar consideration can consider can pick up lambda with the minimal imaginary part and then I can prove that all of the imaginary part greater than zero so this means that they equal to zero so that's like real right okay so I proved this George for here was using the same arguments in they 19 I will have reference for 1856 later so we'll but for that for the different subject not about this some other stuff okay so we prove that it's same solution if they exist darriel do they exist so take logarithm
take a logarithm of the birth equation so L the length of the latest seed is this logarithm of the what was left hand side and theta is of the difference is this logarithm of the former right hand side so built equations famous so how I can possibly prove this solution so I want to prove next it in the solution exists 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 in unique so how can possibly prove this actually this young this equations but equation has young action has action so this action was suggested by young and young so theta one is integral of the seat of the previous transparency integral and then this is integer numbers and this is theta integral of those scattering phase so if I consider extremum of this action the derivative of s with respect to lambda I will come
back to this equations to this equation so this bit equation has action which was used in many papers by professor - really so let's prove that this is its convex so let's projected convex then address unique minimum of then solution existent unique so let's calculate second derivative second derivative this is in diagonal element case this it's positive right because case small coupling constant is positive and this is over diagonal element which is also good luck with the - set so let's calculate quadratic form so V is some vector a real vector with real components so after some massaging is equal to L this K multiplied by L Square and this of diagonal will be combined it is square of the difference square of the difference so anyway it's positive for if the Israel so the infection is convex so unique minimum nice so step number two solution exists and unique given by the set of integer numbers by the way the determinant of this play will pop up again later when we shall construct eigenvectors of this Hamiltonian which was mentioned in the beginning than the square of the norm of the better wave function is actually given by determinant of this matrix determinant it was conjectured here first by Michelle Gordon from Saqlain then I proved the ten years later so now the third theorem which I promised Peloponnese for Powell principle let me come back to this equation so Pauli principle theorem about this equation so third theorems is the following I can prove that if all ends are different or that all lambdas are different which solutions exist but if two ends coincide and corresponding lambdas correspond means with the same index also coincide this is not good if to alarm the score inside that construction algebraic but the answers will have this B Square and then all these calculations of algebraic Betances will lead to one additional extra equation so this system of buttons as equation will be over the field will be too much equations and this the third equation looks like positive number equal to zero so there is no solution so if to end coincide that there's no corresponding eigenvector this does not exist so I mean that all else has to be different so that the end of the theorems and the implication for physics mean that the ground state of the cemetery will look at thermosphere it will look like interacting electrons so all ends will at zero temperature will feel some interval and this will minimize the energy so I started with buzzer Onyx city but in the momentum space it looks like thermals because of the power principle so let's move so
this is thermodynamic limit I have this length of the box total length of the ledges 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 the book is going to engineer a short sticks density and then it can be this this lambdas solution both equations they distributed along some curve curve described by Rho and Rho is a solution of integral equation q is the value of spectral parameter on the thermosphere which people sometimes call it thermosphere and as long as Q is finite then one can prove that this the this linear integral operator I mean I can move this in the left hand side is not the generated so there is solution of this equation it's somewhat similar to a linear equation for loop linear equation I have similar equation but instead of this one I have learned by by 2pi so this is kind of two deformation or flip linear because I moved to the legends so in the continuous I had this equation but this disc a was the replacement was 1/2 PI and now in the latest I have this ok so these equations I mean we cannot solve it analytically we don't have analytical expression there is very simple decomposition when coupling constant Kappa go to infinity the integral becomes small but at small copper this is very very singular and I mean lip line graph studies right now for small values of coupling constant and the coefficient given in terminal in terms of our own element Z that other arguments which furthers in turn off like so much this is even more singular at at small copper which we didn't sell it yet okay so in thermodynamic limit there is no bound stage right because all lambdas are you know bound state and any energy level can be interpreted at the scattering state of several elementary excitations so Elementary excitation can be described well has some energy and momentum so the enemies of elementary excitation is this is a 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 specter parameter so momentum as a function of lambda I will write later a little bit later um so this is the picture of this elementary excitation this is my thermosphere so meaning that all the states here I filled in so I cannot put any more particles into here because of poly principle but I can make a hole so here I have excitation which is the hole and this is like energy and in in out out of this interval there is no particle so I can put particle into here and then this is my dispersion curve depends dependence of energy on spectral parameter of this elementary excitation momentum will come momentarily so it's gapless the energy 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 okay this is formula for momentum like physical momentum this P Zero is a log this is log of the left-hand side of my bat equations Ct is a logarithm of the right hand side and the ROI is something which we saw before so this is momentum of the particle this momentum of the whole but all of this together describe elementary excitation gives the dependence of energy of all the momentum of this elementary excitation this is scattering terrific this to elementary excitation they scatter and the scattering is elastic there is own transition and phase shift is given by the similar integral equation in the right hand side I put this Cetus it actually has a physical meaning of the phase shift in the bare wacom zero density and then when I have nonzero density I have to dress up this phase by this integral equation so phase shift V is the derivative of energy with respect to momentum of that elementary excitation on the thermal age and it describes Willits by now it's called quenching loss in sequential loss it's a means many different things among other things if I make some local quantum mechanical measurement and one letter side then it will cause the entropy wave and entropy wave will spread with this velocity sequentialized in the moment are goto will describe thermodynamics your can yank thermodynamics thermic has an entropy theory and Ruby but before that at zero temperature I have another entropy which is entangled entropy which should not be confused so first I talked 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 in classical if I can say the classical random variable if the total entropy of the classical random variable is zero then there is no entropy name subsystem one can through the theorem quantum is not so total entropy can be equal to zero but there is entropy in subsystem quantum fluctuations you know
entanglement entropy and then of course entropy is a complicated function of the size of this block of spins but for large size of the spin is logarithmic dependence with the coefficient one third and it agrees with safety it agrees with quantum field conformal field theory because central charge they put one rainy entropy so Rho is a density symmetric of block of that spin so alpha is some fractional number from 0 to 1 and then I take this density matrix rates in the power alpha at x-rays log this is Iranian turbulent B also it depends logarithmically on the size of the block but coefficient in front of the log depends on that alpha so this isn't a common entropy well professor Linda mention this in his lecture so much might be appropriate so now let's consider thermodynamics thermodynamics in principle for continuous nonlinear dynamic was constructed by Yangtze in yarn SCP yarn in here in the latest version it's similar construction is similar also equation is not it's different from the continuous case difference in here in the 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 a young young equation so this is famous young equations there's some notation so epsilon is a ratio of the density of the holes to the particles I mean zero temperature I have a concentration of particles for small momentum minus Q there was no holes on the particle but that positive tempted every sink mixed-up particles in holes so the ratio is epsilon epsilon also has added meaning it because this model is integrable there is infinitely many conservation laws this leads to a consequences that even in positive temperature there's stable excitations which does not decay and then 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 what the sense in terms of this epsilon I wrote equation for the epsilon this equation the previous yungang equation was analyzed I mean the only one mathematical setting which was proved that if I start it erasing 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 converge so one of the solution exist but it's probably unique but it's not proven there is no serum that is unique but one solution exists so this is free energy this is pressure and this is entropy so but is it thermal entropy right its term entropy of the whole bulk so it's not entanglement I mean it's thermal integrals exist for classical thermodynamics it doesn't describe quantum mechanics so while entanglement is the difference between classical and quantum should not confuse so all of this can be done for other values of negative spin so here I just say took deviates of professor car Chomsky and cancels not necessarily minus one but maybe some other number everything goes through this is the equations or this is thermodynamics entropy everything works okay so this so far analysis of the Hamiltonian of dr. John Rajeev and Tara's of there is other Hamiltonians all the latest nonlinear Schrodinger so I keep aromatics are amazing my aromatics is the same like it was in continuous case and for the ladies I never change it it was discussed by young texture so I keep the young for me a liberator 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 old and even latest sites well the purpose of doing this to get some other Hamiltonian so this is the shift of the specs at a parameter which is little bit different than the Orden in let us say so and I think it was J before so I'm sorry for changing notation so towards J now it's m and it's evidently different all diagonal elements looks the same but this in homogeneity also appears under the square root and Kappa now becomes size so I just copy from the other paper so and then later we shall see Delta still likes step of the ledges and this is number of the total total number of particles in the whole data set so with this modification of a liberator 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 le Pretre and this is a special value of spectral parameter where LEP later become one dimensional projector the quantum determinant vanish at this point so that series and I can write this Hamiltonian loops even more complicated than fuzzy if there are sufficient oxygen so this T we express in terms of alpha and alpha is a relatively simple function of the local baza field so this elaborate
reaction describe interaction of eight later sites so people might argue that Friday stars of detergents better okay so now I'm actually moving to form factors before that well as I measure the square of the norm is given by the wooden formal about the determinant form factors in the behaved similar in the continuous in the latest case so I will kind of explaining first I will remind what happens with foreign factories in the continuous case I mean like they don't exist the answers like negative but I mean the behavior is similar on the letters in the continuous case so this is someone of form factors so this J is the side I guess I for the continuous case this is local density of particles in the X and T space ten-point is a standard canonical operator and if Q is number of particles on the interval of 0 X so I can take this Q and taken matrix element between two better states this is like 1 and this is neither and normalize it it's actually was long story first some determined representation was written for this and determinate representation I mentioned determinant of the matrix of the large size the number the size of the matrix equal to m capital number of particles which goes to infinity and later I will mention the similar determinate representation also exists for the latest nonlinear surely not identical but similar then this determinant representation was studied and the answer was negative because when the lengths of the letters go to infinity this for a fox that decays goes to 0 as some fractional power so f is some solution of integral equation actually this is a face scattering phase of those two elementary excitation divided by 2 pi this this is definition in terms of the lambdas this we call the shift function but it also coincide with a phase shift of to schedule many so the four boxes vanish in the continuous this
is just R the photo factors also was analyzed they also don't exist
this is the tournament representation but maybe it's published somewhere maybe it's not interesting to the people right now okay so this last a couple of transparencies I was explaining problems with the form factor the continuous nonlinear shading similar problems occur for leches and Takeshi water from Japan he wrote the tournament representation for some operator of all edges nonlinear injure the operator somehow has correct continuous limit and maybe I just come back so and then Karl kosnovski Karl kozlovskiy he analyzed thermodynamic limit of this determinant of Takeshi Obata and then it also goes to zero as a fractional power of the length of this is actually the same power so this is more or less and of what I was going to say about nonlinear Schrodinger on the ledges will be the last transparency that similar calculation can be done for sine garden this is continuous and garden and then continue same garden was sold by algebraic Betances and quantum annealer scattering methods by Professor Tata Jen and Fridays but we have our own latest version which is interesting by itself but maybe maybe it's a good time for me to stop my lecture and wish happy birthday the professor shot actually again and he'll hear you was my best student so that's the end of my lecture [Applause] maybe maybe just some references or the main reference yeah the main reference is this and this was our inspiration this was a professor should actually related two-dimensional topological gauge theories to nonlinear to continuous nonlinear change this was inspiration for our work so if you look at signed Walton the on the like this do you see something of the coastal a trellis transition is your I didn't see that phase transition I mean our contribution was kind of a boring mathematics I mean that answers for San Goran I mean this relativistic model is one Feynman diagram in divergent 4-bit answers we have to learn how to make ultraviolet renormalization and here for saying world for specific like for rational values of coupling question there is that anisotropy parameter which is coupling constants for the rational value so we can compactify the local Hilbert space we don't have to have this infinite dimensional hilbert space which we had for leches non linear showing it but we have a finite dimension or I mean dimension is equal to the denominator of this fraction so we just use it's more like for mathematical justification everything is rigorous but sure dance is now to your question it's logic there's two people in the United States that they live in the Midwest one is up limits that's the name of a person and logic is another person I don't know well I don't know that KP but I believe it's logic you know a beloved static so our beloved slightly has another discretization of nonlinear and yet I mean we kind of insist on always because in their case aromatics is different so our metrics depends on the latest step and in our case in our case like going from continuous to leches is like changing the representation of the same algebra right the algebra is given by our metrics so which keep our metrics change the presentation so we think like this is intelligent way to criticize to discretize but I mean a blob is logically change our matrix sorry and you was meaning that one is actually by some difference equation which in the limit becomes a well I mean it functions if I take nonlinear shading Gary replace its second derivative by the difference it won't be integrable so one should be careful with this I mean it's just straightforward discretization but just keep and then then from the problem because there will the phase transition will appear when I sent a later step to zero then it's not integrable the internal integrable and this is kind of problematic so it's not regularization it's like falsification I don't know that paper so maybe you show me later well maybe I should also mention that on the latest the nonlinear Schrodinger the model is equivalent to x equals negative spin which has multiple application to latest gauge you know professor Chomsky and le part of and but also to him is yes because remember well maybe I can ask you a question if you permit okay so remember when I wrote that equations I had three syrian so they 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 end so I can subtract this equation and I can estimate the difference between two neighboring lambdas in terms of difference of ends from above and from below so when L total length of s go to infinity I can prove that the difference between lambda 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 Roy exists so the answer is yes it's not from Yankee Yankees in my book they I did it is it's in the textbook [Applause] [Music]