 So, I think we can start. So, today I will like to discuss the mathematical mechanism of phase transitions and the way Lee and Young theory of phase transitions in terms of zeros of partition function in the complex plane, complex activity plane and then I will give some example of how to calculate these zeros you know positions of these zeros. So, that is all that we plan to do. Second is this Lee Young mechanism phase transitions and then we just give some examples explicit calculation lines of zeros in Lee Young theory and well there is might be some time left. So, it will be the general behavior edge singularity. So, let us start with one. So, while now it may appear rather obvious to lot of you that you know there is some Hamiltonian and you can use it to calculate partition functions and then you can get phase transitions. It was not very obvious to lot of people even around 1900 or 1920. The point being that you know the partition function is written as sum over e to the power minus beta e i sum over states. So, it is a continuous and smooth function of beta fairly structure less and you sum over lot of these nice exponentials and how is it that you can hope to get some known analytic function of beta as a result of this because sum of analytic functions is also an analytic function ok. So, this seem to be sort of a mathematical issue, but there was also a psychological point attached to it you know there was this thing called the theory of solids. In the theory of solids people say that ok you have atoms they interact with each other there is a minimum energy configuration which is a periodic arrangement and then there are vibrations about these and may be non-linearities and all kinds of stuff ok, but once you heat it up enough then the stuff melts and then the description in terms of the periodic solid with perturbations is no longer valid. So, how can you have a theory in which you can have the starting point of the theory which was this periodic solid is no longer there. So, the theory is not there you cannot use the same theory which describes the solid the liquids. So, they used to be you know in the past they used to be a theory of solids and there is a theory of liquids and the usual idea was that you can calculate something like this free energy there will be something like this this will be solids, theory of solids and there will be a theory of liquids and at some point the liquid state has a lower free energy than the solid state and that is where the thermodynamic change occurs. But the idea that you can have a same theory which describes both the solid and liquid phase is not automatic and obvious and it took a fair amount of work well yeah took several years no 20 30 years for people to realize that no no no you can actually start with a single Hamiltonian like the Lennard Jones interacting molecules and it is able to give both the solid and the liquid phases. So, this was first realized in terms. So, there is a mathematical mechanism of phase transitions which is the following they said that oh well if you have the finite summation for finite number of particles then it is true everything is a smooth and analytic function of beta. But if you take the large n limit then you can develop singularities in the free energy per particle. So, the non analytic function comes because of the thermodynamic limit of large n responsible for phase transitions so this is of course true, but it is easy to see in a simple example. So, let me write a partition function is sort of very schematic, but it is a mathematical example of how large n limit can give you discontinuities in the non analyticities in the partition function. So, I write you know it is sort of a very contrived example, but let me still go ahead with it. This is the partition function for system on n sides. Then I define g of z is equal to limit n goes to infinity log omega n of z divided by n and then I want to look at that. So, omega n is a nice and analytic function of z for any n. However, if I take the limit of large n and look at the behavior of this function g of z then you will recognize let us keep z to be positive real number just now. So, this will be one this is some other power this is some other power. So, the largest power dominates. So, this will be the log of the maximum of these three terms one root to z because either this term or this term or this term will dominate. And then you can see that g of z is a smooth function of z except the point where the jump occurs from one value to another. So, this is a rather straightforward mathematical mechanism of generating singularities in functions from nice exponentials polynomials. So, I think I should plot this one g of z versus log z. So, for very small values of z this term dominates and g of z is 0. Then when z is less than 1 by 2 then this term dominates, but if z is bigger than 1 by 2 this term will be bigger than this. So, log z is minus log 2 then this term becomes non zero and this term becomes bigger. If z is bigger than 2 then this is root to z this is z this is bigger. So, this becomes there is a term at log 2 and above that it is g of z is proportional to z like that not very well drawn. So, you have two discontinuities in this function. So, that is a mathematical mechanism, but you know where do you get functions like these these are not very reasonable looking partition functions. So, it was actually Cramer-Sainte-Vaniel 1930 who realized that if you write a partition functions in terms of transfer matrices they developed the general formalism of transfer matrices say forizing model. So, this is 1 b it is u work with transfer matrices this is very well known. So, I will only sketch the argument you take some lattice there are ising spins at each side there is some coupling with nearest neighbors I write a transfer matrix T n C C prime is equal to some matrix which gives you the interaction between one column C and one column C prime and you know then you would keep on transferring. This matrix is a 2 to the power n by 2 to the power n matrix the elements of this are just e to the some e to the power minus something or the other. So, it is a matrix of positive numbers in this case they are all positive sometime they may be some 0 elements, but they are non-negative numbers and then there is this famous theorem Perron Frobenius theorem which says that the largest eigenvalue will be non-degenerate and it will be a smooth function of these coupling parameters. So, then and the free energy is equal to the largest eigenvalue. So, the largest eigenvalue is a smooth function of the parameter beta. So, let me make for finite n log lambda max which is the free energy per site is an analytic function of beta on the real axis. So, that does not give you any possibility that if you change beta amongst positive values of beta then you will get a phase transition which is of course also known we understand that you know if for finite system these are not finite this infinite in one direction, but that is not enough it will not give you phase transitions. So, Cramer's and Vanier already realized that yeah what matters is. So, this partition function z I should write big you know z is equal to lambda's max to the power n m plus lambda 1 to the power m plus lambda 2 to the power m and if these two values are no longer non-degenerate sorry the largest value is no longer non-degenerate then the argument does not work. Then you have degenerate eigenvalues then the eigenvector is not you will have different choices they depend on boundary conditions all kinds of stuff. So, the mechanism as explained by Cramer's in one year was that delta lambda is equal to the gap in the eigenvalues of lambda in the eigenvalues of the transfer matrix tends to 0 as n tends to infinity well not always if a phase transition slightly bad language, but let me write it just now like that. So, what they said was that you look at this gap delta lambda and now for a fixed beta you vary n and you see what happens to the gap sometimes the gap will remain finite even when n goes to infinity in that case there is a finite correlation length and you are in some phase in which you have a finite correlation length for other values of t it may happen that as n goes to infinity then for a range of t. So, this is the only change I had to put if this delta lambda goes to 0 then you have a phase transition and this is what is seen in the Ising model ok, but again trying to see this gap closing at some value of lambda etcetera is not so easy to see and there is a very nice paper written by Lee and Yang in 1950 I have the reference yes by definition no I will give the definition in one phase the correlation length is finite in the other phase the correlation length is infinite. So, I can call them distinct phases and hence there is a phase transition it depends on the definition of what is the phase and we are going to define these as qualitatively distinct phases and so there is a phase transition. It does not immediately follow that it leads to singularities of other variables, but actually it does follow because then the detailed properties of observables depend on the boundary condition if the Eigen value is largest Eigen value is degenerate then the value of an observable you will see will depend on what boundary conditions you have put in. So, if the answer you get depends on the details of the boundary condition that is called spontaneous symmetry breaking that is the word and so this is a technical definition of spontaneous symmetry breaking slightly different from the one given in some textbooks and so the then it one phase will be symmetry broken phase and one phase will be non-symmetry broken phase okay. So, I giving the reference to the Lee Young paper Young. So, it is a really very nice paper to read and I sort of request all of you to go and read that paper it is fairly short it is quite easy to read and it is very insightful and they actually explain at great depth how they got around to this problem by starting with the Ising model and so on and so forth. So, I will actually now try to summarize the Lee and Young argument and idea. So, Lee and Young start by saying let us consider a lattice gas model you have a lattice it does not have to be a square lattice just a lattice and the sides can be occupied or unoccupied. So, at each side there is a variable called Ni which is 0 or 1 at most okay. So, that is called the hard core constraint and which is the reason why you know in that word occurred in my title. Now, there is a Hamiltonian which is just the potential energy term submission phi ij and i nj summation over nearest neighbors and so this phi ij can be extended hard core in the sense it came maybe it is infinite even when the distance between sites is 1 or 2. But at least at same site you cannot put two particles that is the beginning point of their discussion. Actually some of the details of the argument will not work very well if the phi ij is infinite at nearest neighbors at least in the original paper. But you know you can fix it later and you can write more careful presentation in which phi ij are not necessarily bounded. So, they say that phi ij should be greater than minus infinity all the time but it may be plus infinity sometime. But if you make it plus infinity sometime some of the later equations will require suitable taking of limits you know they do not work immediately but they can work within some limiting sense okay. So, we will not quibble about that just now let us take some reasonable range interaction and then they said that this thing can be thought of as an Ising model with some coupling constants. That is rather obvious now but that was actually the first time this was observed. So, ni is equal to si plus 1 by 2. So, si take values plus minus 1 and ni take value 0 and 1 and so this Hamiltonian then in terms of si becomes summation si sj jij minus constant time summation si plus constant. I will not write down the values of constant you just put ni equal to si plus 1 by 2 here you will get ss term which will be the coefficient that will be called jij then there will be a term which are linear in s that will be of course independent of the si because j will be some doger and so will be a term like summation si and then there will be some remaining constant and so this is called Ising model in a field okay and so you have to solve for the phase transitions in this Ising model in a field and they will correspond to phase transitions in this lattice gas model and if the interaction is taken to be something like the Lena Jones model then this describes the phase transition in real liquid solids. Yes n is the number of sites is right now doesn't have to be fixed you can fix it but it is that is a question of ensemble. So, details you know like the Ising model can be studied with all spins taking all possible values or when the total number of up spins is held constant and these are different ensembles but they give you the same thermo dynamical behavior. The constraint the total number of particles is fixed or not fixed is a matter of you know whether you work in the canonical ensemble or in the grand canonical ensemble and we have not discussed the equivalence of ensembles but you know take it as a result which has been established somewhere else that the equivalence of ensembles holds and what physical quantities derived using the canonical ensemble matched with the ones derived using the grand canonical ensemble at least in some set of conditions. Sometimes it doesn't work but some most of the time it works. There is a condition under which the equivalence of ensemble breaks down but we are not going to worry about them here. I mean I am not saying that it always holds but I am saying that let us not worry about that now. That is a technical question we can discuss it even more detail privately but let me go on here. So, now comes the very interesting point. So, this is the Lee Young theory. So, they said it is nice to work in the language of lattice case. So, I define omega n of z omega v is equal to partition function of lattice case and lattice case e partition function grand partition function lattice is n. So, this is equal to summation z to the power n q n of beta where q n is this configurational integral we have defined before it is just the sum over all possible configurations and z n z is e to the power beta mu is equal to the active it is called the activity and it is equal to the weight per particle in the grand sum grand partition sum. So, the if you write a configuration with some particles the weight is z to the power n e to the power minus beta e and there you sum over this overall configuration. So, that is z thank you 0 to n and so they said that this omega is actually a polynomial in z because it has an upper bound you cannot puts more than so many particles in this volume v. So, it has an upper bound on the number and this number increases of course, if v becomes bigger. So, the polynomial has a degree which is proportional to v, but for any given v this function can be expressed as a product 1 minus z over z i where z i is at the roots of the partition function grand partition function. So, this is a polynomial if you put z equal to 0 then only the term n equal to 0 comes into picture and then the answer is 1 by definition and so this form is good there is no multiplicative constant outside at z equal to z i it has to vanish and if there are multiple 0 then I will put 2 z i is equal. So, this and this is good then log omega of v divided by v equal to summation over log 1 minus z over z i. So, then they said that we can think of this as a question please yes please z i is at the 0's of the at the roots of the grand partition 0's of this polynomial. This is a polynomial it has n 0's and these are the 0's of the polynomial. So, they said that this can be thought of as a electrostatic potential is set charges equal charges kept at z i. So, of course, the 0's will be complex they you know even if you write a polynomial with real coefficient there is no guarantee that the roots will be real. So, you have to work in the complex plane you cannot work on the line and so then I work in the complex plane complex plane complex z plane and the 0's are like that they are I guess they are complex because the z is a real polynomial they will come in pairs complex conjugate pairs, but other than that I do not know too much you know there will be some scattered over, but then log omega is equal to summation I am writing it again because it is convenient for me 1 minus z over z i. So, this is the potential due to a charge at z equal to z i yes sir and it is the highest power of this the total number of particles maximum number of particles you can put in that volume. So, in our case by definition it is n, but it may be that you know maybe there is a nearest neighbor exclusion as well. So, you can put only half as many particles in which case that degree will be a little bit lower and you can work with you can you know all the arguments will go through and you can check. So, that the number of 0's is proportional to n, but does not have to be equal to n. So, we wrote 1 by n there is a summation over i ok. So, this is the potential due to these point charges at the 0's. So, now I can think of this log omega as the potential, but it is convenient now we have already introduced complex z. So, I just think of this equation of course, it is defined when z is complex right here I wrote this definition I guess I was thinking of z as a real number so far, but it can be thought of as a complex number by this called analytic continuation. Function is defined on the real line, but you just continue it in the complex plane. So, log omega is also defined in the complex plane by this equation of course, I realize that this log is also multivalued function. Yeah, we work with multivalued I mean I know log z for complex z it does not really cause me much problem it is multivalued, but I am ok with it. The real part is of course, unique and the real part corresponds to the real part of this and on the real axis all the complex part will vanish and you know you will get back to the standard stuff otherwise there is some 2 pi i n kind of arbitrary factor you can add to log omega ok we do not worry about that just now ok. So, now so this is the answer, but suppose now you change the volume you change take a bigger volume then all the zeros will change there will be more zeros and the zeros will have new positions and so you can take a bigger volume there will be more zeros there will be newer position. So, what happens in the limit of large n that seems like a difficult question and this is where you know at that time when there were very few computers it requires somebody with the insight of young to only to realize that you know actually the zeros will lie on some lines they cannot lie everywhere. So, they guessed large n zeros lines ok. So, they said that what will happen is like this that let us see there is a line like this all the zeros lie on this line no zero lies outside it. So, if you have n zeros and there are n zeros on this line if you have 2 n zeros then there are 2 n zeros on this line and 3 n 3 n zeros on this line. So, I can ask what fraction of zeros lie within a given range given an interval on this line I have not decided what is the position of the line, but suppose there is a line such that the zeros lie on that line. Then and the limit exists such that as n goes to infinity a finite fraction lies within a finite interval then you can define a density of zeros and interval 2 of s you know. So, interval let us measure the distance like this. So, this is s and this is d s and rho s d s is equal to fractional number of zeros in the interval centered at s of which g s let us write s s plus d s it is very clear the definition is clear whether it happens or not is not yet clear to me if it happens like this then I can write in the limit of large n limit of large ah large letters also limit large n rho omega divided by n tends to integral rho s d s rho s minus z s ok is this ah transition clear there is a additive. So, I had a term here log z 1 minus z by z i, but that is the same as log z i minus z by z i ok. So, up to additive constants the partition function is z i z minus z i integral over z i which is what we have written here ok ah this is being hidden by something no this this part of the text I will rewrite it because it is important yes just once again no ah the zeros can lie on things like this in which case I just integrate over a all the density of zeros yeah this s is perhaps not a good symbol, but let this is given it is a notation yes sir very good. So, that is a good question we should figure out what is the structure of the line. So, that comes next ah yeah. So, I would I was writing log omega n z by z limit n goes to infinity by n equal to integral d s rho z minus z s rho s this is the density of zeros and this is the integral and that is the partition function. Now, before we get into the structure of the line what follows immediately this is the potential phi is the potential due to this line charge point charges or line charge in the limit thermodynamic limit becomes a line charge line charge is some density of charges ok. So, how will the potential vary as a function of z given a particular line charge density? So, the answer is that wherever there is no charge the potential is an analytic function of z because you know this is a electrostatic theory when you come here and you vary z a little bit everything will be well behaved as a function of z the singularities of omega z will only appear wherever there is some non zero density of zeros singularities in omega log omega appear this says that you take the limit appear only where only where rho z is non zero ok. So, so there is a constraint the constraint is that if this is a polynomial and it is a polynomial with positive coefficients. So, if you put any real number z then it will always give you a positive answer. So, there are no zeros of the polynomial on the positive real line. However, you can have zeros which are just so this is my the zeros may be like this they may be very close to the real line, but not on the real line like that and as n tends to infinity these zeros may come close and may touch the real axis pinching on the real axis ok. So, before we go further I guess I should just say that then the very interesting result which Li and Yang found showed that if phi i j is less than 0 this potential of interaction is always attractive. Suppose you consider the case very on-site repulsion, but otherwise it is always an attractive potential and then all the zeros lie on the circle mod z equal to 1 they cannot be any zero outside ok. That is a very nice theorem it is actually quite short the proof is only one page in the published paper. However, it is a little bit of not very transparent proof like if you ask me now you know can I reproduce the proof it will be hard for me I will do it, but you know it will be hard for me. So, I do not want to do it on the blackboard in any case you are welcome to go and read the proof. It is a little bit technical of course, because it is working in sort of general state, but in the end it boils down to verifying a few lines of algebra and it works out, but I am not able to give you a very good feel for the stuff, but let me state the thing. So, this let me phi ij does not have to be even translationally invariant you can take any graph in 2D, 3D whatever you like put any interaction phi ij whatever you like and construct this omega and all the zeros will lie on the unit circle. So, that is very remarkable of course you know the proof has to be very general also clearly, because it is such a general result. So, what then Lee and Young argued like this they said that suppose you consider the Ising model with nearest neighbor coupling we know that it undergoes some phase transition. So, how will that happen how that happens is like this. So, this is the mechanism of phase transition which we will see it says that on one at one value of beta there is a region of the circle which is occupied and there is a region which is empty. Now, I change beta to a lower value then the zeros move all the zeros change their position, but in the end the density of zeros changes to become non-zero in a given bigger interval. Then I lower the temperature further then the zeros move in further to the real axis and at one particular temperature they will meet they will touch the real axis like this and for t less than t c there is a finite density of zeros along this circle everywhere it is not uniform, but if there is a finite density everywhere ok. So, what happens then is that if you look at the real part of the electrostatic potential across a charged line charge density there is a electric field has a discontinuity given by the Gauss-Ritz law you know you take some tiny bit here and construct the e out here e out here that is equal to rho inside even minus e 2 equal to rho there is some electrostatic condition which is the solution of the Laplace equation, but the derivative of phi with z is the density d log omega by d z d log z is the density. So, when there is a discontinuity in the derivative there is a discontinuity the density as complex density as a function of z. So, all along these lines the derivative of log omega has a discontinuity even here the derivative of omega will have a discontinuity. So, that means that when you change the z which is in our case it corresponds to the magnetic field if you change the field slowly then there will be a jump in the magnetization which is the density of lattice gas. So, that is the mechanism of phase transitions. So, to summarize there is a complex plane there are zeros in the complex plane which lie on lines as you change the temperature these zeros move at some temperature they may come and pinch in on the real axis. If they pinch in on the real axis then as you go across the real axis even then you will get a some non-analyticity because of the potential function becoming non-analytic there and that will be the singularity in the omega z and that is the phase transition. So, what happens is that the temperature is an implicit variable inside here because this q n which are the coefficients of the polynomial are functions of temperature. So, as I change the temperature this coefficient change and the zeros change no no no no this is the z plane ah sorry I understand your point I am sorry I did not say it very well. So, the variable here is the z which is the variable which is coupled to the occupancy number. So, it is analogous to the magnetic field ok there is log z is the magnetic field H because the if the particle is up the weight is e to the power beta H if the spin is if the if the particle if the particle is present then the weight is z in the Ising language it is it you know z or 1 or e to the power beta H or e to the power minus beta H this is no particle present ok. So, the equation is z equal to e to the power 2 beta H ok. Now, if I study the free energy of the Ising model as a function of the field then at high temperatures free energy as a function of field has a smooth behavior it looks like this, but at low temperatures. So, the derivative of F with H which is the magnetization is smooth and continuous. So, I will plot M which is the d F by d H minus is a beta and it goes like this or it goes like above T c it is a continuous function below T c is a discontinuous function as I vary H. So, here I am varying H when I go along this line, but sometimes I get a smooth behavior and sometimes I do not get a smooth behavior depending on temperature. So, that is the mechanism of phase transitions it is very nice and it is very general and you can adapt it to other cases where you know the control parameter is not some kind of chemical potential or it is not a function of let us leave it let forget that ignore that comment ok. So, now, so let us say for a magnet I construct M H graph in equilibrium in equilibrium and the statement is that it goes like this something like this for T greater than T c M as a function of H is a smooth function continuous function of H for all H T greater than T c M H has a discontinuity at H equal to 0 for T less than T c and this discontinuity is so called as spontaneous magnetization. So, that is my definition magnetization as a function of H is continuous for all H for T bigger than T c and it has a discontinuity at H equal to 0 for T less than T c. So, in this case in our language this will become that omega of z is continuous and actually d by dz is continuous for T bigger than T c and is discontinuous discontinuous at z equal to 1 for T less than T c is this point clear ok. So, now, now I can actually go and vary I can cross z from here to here and see how the omega z varies from here to here and that will give me some discontinuity also, but now the density has become a complex number and this complex number varies from some value to some value here and that is a little bit trickier to understand, but again and also we remember that log omega was a multivalued function. So, that was pretty bad and then I took 1 by n and I summed over a lot of stuff. So, the imaginary part can be almost anything, but if I look at this picture now, let us say this one and I define the potential of this stuff. It has an imaginary part which is somewhat non trivial, but the real part is pretty unique and well defined. So, I define the function to be real along this real line from here to here and by analytical continuation I define it everywhere else and then the imaginary part will also show discontinuity when I go from here to there. It just requires you to think of the partition function as a potential due to some line charge and then what that will correspond to some discontinuities and that is what this gives and then we should break at some stage. There is a very nice picture in Lien Yang which looks like this. So, Eising model only has one phase transition, but you know in the real liquid sense you have more than one phase transition as a function of various parameters. So, here you will find that as you suppose the charge density does this, then you will find one phase transition here and one there at two different values of z, at two different densities there will be discontinuity in density. So, I guess this part of the curve will be called gas phase and this will be called the liquid phase and this will be called the solid phase and so on. So, that is the general theory of phase transitions. Then in the next part we will discuss the we will actually calculate the density of zeros on some model or the other to get a better feel of this and we will discuss the general properties of the Lien Yang H singularity. Thank you.