 Right, so today we will go on with our study of phase transitions and we will have a general introduction to what is called the Landau theory. Now I am not going to do this, we do not have time to do this in great detail or at any length but what we are going to do is to take off from a familiar model namely the Ising model which we have all been looked at and then I will simply make a statement that a very large class of phase transitions will fall under the so called Ising universality class. So whatever happens here is generic to whatever happens in general and I will use the same symbols that we use for the Ising problem. In other words the order parameter is a magnetization, I use a little m for that, we will continue to do that but the physical significance of this m could change from problem to problem and you will see that the structure of the theory is more or less generic, it is quite general and then we will go on to some introduced fluctuations and go on to what is called the Ginsburg-Landau theory and hopefully finally we will talk about dynamic critical phenomena. So we take this in several steps, let us start with some familiar material and it is as follows. I will draw a number of pictures today to show you what the phase transitions, what various diagrams, phase diagrams look like. So wherever the algebra gets a little unnecessarily complicated we will simply draw a picture and go ahead with things and there will be a lot of hand waving arguments which can be made rigorous but I want to communicate the essential physics of what we are talking about. So we start again with the Ising model for which if you recall we had an equation of state. So we had an equation of state for the Ising model and we are dealing now with the critical region, we are looking at the region near the critical point. If you recall the critical point is characterized by a temperature Tc, the Curie temperature at which the system goes from paramagnetic to ferromagnetic. The critical value of the field which is equal to 0 in this case because it is a 0 field that you cross over from paramagnet to a ferromagnet, spontaneous magnetization is in the absence of the field and of course the critical value of the magnetization is also 0 because it takes off from 0. Now just to recall to you what the diagrams were in this case, I plot an h versus m then the critical isotherm did this, critical isotherm. So it c was 0, m sub c was 0 and then we also had a figure, let me draw this again, just to recall to you we had m versus t and here is Tc in this case and the magnetization in the absence of a field m not was 0 beyond t above Tc and then it went down like this or the ferro down was a branch like this and this branch became unstable. So this was m not, this spontaneous magnetization and again by critical region I mean this region, just as I mean this region, critical region and finally there was the h versus t graph and the h versus t graph was a line like this and then at Tc. So again this is the critical region, so hc was 0 and Tc is at this point here. So it is this circle region that we are dealing with, we are talking about and what happened in this circle region was that in the Ising model in main field theory we ended up getting an equation of state and that equation of state read like this, we had tan hyperbolic h over kT but essentially in the critical region h is near 0, so it was h over kT was equal to m minus tan hyperbolic mTc over T divided by 1 minus m tan hyperbolic mTc over T. Now we are going to focus on the critical region namely T near the critical temperature, so let us introduce the reduced temperature T equal to T minus Tc over Tc. Then I can write this equation here in terms of critical quantities very close to the critical region in terms of little t and little m, m minus mc is the same as m because mc is 0, h minus hc is again just h because hc is 0. So this equation can be written in the form on this side mT that is the leading term and then the next term is of order m cube and then plus order Tm cubed, so we will drop this it is a higher order term. Spontaneous magnetization is found by putting h equal to 0 in this and then the equation is m cube plus mT is equal to 0, so you have m cubed is minus mT, m equal to 0 is always a root but for T less than Tc that is unstable and for T negative namely T less than Tc out here this region, m squared is equal to T mod T was the solution and therefore you got plus or minus square root of little t out here, so that was the mechanism. Now we want to make this systematic and one way to do this is to argue and this is Landau's argument it proceeds in a large number of steps, so let me say what it was start with one way to look at it is to say there is probably there is it is possible to introduce a potential after all this equation of state has essentially a reason by saying some thermodynamic potential is at a minimum so that you are in thermal equilibrium. In this case essentially the equivalent of the Gibbs free energy because we have an h here the analog of the pressure and we have the analog of the volume out here and the temperature that is the equation of state, so at a given value of the magnetic field h 0 or otherwise it is equivalent to saying at a given value of the pressure and temperature so equal to the Gibbs free energy. We are not going to write that down let us go back and do a bit of phenomenology and say essentially this is the equation that I have so I understand from this equation how you get this square root here the same equation gave us this cubic here the same equation gave us the susceptibility also remember that we plotted the susceptibility as a function of t I plotted the isothermal susceptibility and it diverged at the point tc like 1 over little t all these things came out of this equation of state here. So let us do the following let us introduce some potential on differentiating that potential with respect to m and setting the result equal to 0 I get this equation of state you could call it the Landau functional or something like that. So this potential it is not exactly the Gibbs free energy it is not exactly the Hemolds free energy because those things have to satisfy certain convexity properties these potentials do not this is just an empirical this is an equation of state at the critical region. So let us define a phi which is a function of m t and h equal to some constant phi naught which is a function of t and h does not involve m and I want to produce this equation of state so minus m h. Let me just for convenience you could write this t as tc plus a small correction the correction is a higher order correction so it is essentially tc here and we can subsume it in h so it is not an essential factor here I have already done that see you see I have already got a tc here this is essentially tc because if I multiply through it is going to the correction t minus tc is going to give me higher order terms no it will be of order t squared it will be of order t squared because if I write this as tc plus little t essentially and I take that little t across there it goes away only the tc part contributes the next term is of order little t squared which I have neglected yeah you need that yes okay you have okay good question you have to check and I am not going to do this here you have to check that is consistent to subsume this t here and call it just tc yes good exercise check that like I have neglected this term for instance okay exactly exactly yeah exactly so this is the consistent this term is not relevant I call it h okay so if you do this minus mh plus t m squared it is this this term here plus m4 and then you compute delta phi over delta m and you set it equal to 0 you get precisely this put a 2 here put a 4 here if you like these are constants which I am not paying attention to so do you agree that if I take this phi and I differentiate with respect to m and set it equal to 0 if you like yeah if I set delta phi over delta m equal to 0 I get this equation so my claim is that this potential whatever it is this phi has a certain geometric shape and its minimum will give me the equilibrium state we can plot that we can plot that fellow in order I am going to generalize that this is the original random functional because the point is eventually what I am going to argue is that the factors here the signs are very important there could be numerical factors which I have not bothered about so I am going to have a term which is proportional to the product of the order parameter and the field then a term proportional to the square of the order parameter with a little t in the coefficient as a leading coefficient and then a term which is essentially a constant coefficient times the 4th power of the order parameter that will generically give me this second or this continuous phase transition and it will generate for me all these magnets so that is the idea and then I am going to argue that these coefficients could in general be temperature dependent but we have looked at the leading temperature dependence near the critical point okay. So let us draw some pictures and see what happens notice also that you could do the same thing with fixed field suppose you plotted the magnetization versus t for small positive field then the curve would be this is the paramagnetic region and then this would be the furrow region eventually it would saturate similarly on the negative side a small negative field if you do not switch it off will behave in this fashion so what you are doing is drawing these isotherms for non-zero value these various isotherms and looking at what happens as you change keep h fixed and you decrease the temperature from above tc to below tc so as you go up here or as you go down here in the case of the magnetization versus temperature it means if you keep the field fixed at some positive value you are here at this point and as you change the temperature you come down for a fixed value of the field you are going to jump this make a transition here as you change the field from positive to negative values keeping the temperature constant you are going to jump here that is equivalent to saying if I go from here to here keeping the temperature constant I cut across this phase transition line that is like the liquid gas coexistence line and there is a discontinuous change in the magnetization from this point to this point so this is a line of discontinuous transitions ending in the critical point where the transition becomes continuous in other words the discontinuity in the magnetization keeps decreasing till it vanishes at the critical point which is why it is called a continuous transition okay well that is going to be encapsulated in the figures that we can draw we can now draw ask what is what is this potential look like in these cases so let us draw these potential this potential so let us draw let us have t greater than tc t equal to tc and then t less than tc let us draw these 3 cases separately and let us draw them for h equal to 0 0 feet so this fellow is gone and I am going to draw phi minus phi naught so this is just a constant as far as m is concerned I move it aside and plot this graph here so I have tm squared plus m4 and that is now easy to see what is going to happen for t greater than tc so I plot phi minus phi naught always and what does the potential look like as a function of m always the order parameter m then at h equal to 0 this is gone t greater than tc this is positive this coefficient starts with a parabolic behavior and then takes off like a fourth power so the minimum is a simple minimum it is not quite a parabola but it is like this 40 greater than tc at t equal to tc the same thing as a function of n little t is 0 now and the field is 0 so you have a pure quartic term therefore this looks like a very flat and t less than tc you have an inverted parabola the field is 0 and then you have the m4 which takes you up so let me plot with a cross or dot the equilibrium value which is the state of thermal equilibrium and give you the order parameter at thermal equilibrium it is obviously 0 here it is still 0 here but now it is either this or that these are the two values of the spontaneous magnetization that you saw in the m versus t m naught versus t graph either up or down up or down this is for h equal to 0 so let us plot the same things take this up a bit t greater than tc t equal to tc t less than tc let us plot the same thing for h small positive on this side now h is positive this fellow is positive so near the origin you have a linear behavior with a negative slope so this curve for t greater than tc looks like something like this because near the origin you have this thing then you can find the minimum by working that out by taking its minimum there t equal to tc what happens at t equal to tc in this case but h is positive t equal to tc this fellow goes away and you still have a curve which looks like this it changes shape here a little bit so this guy gets a little flatter and then he does this and for t less than tc what does it do it will still be like that but it will be it will bias it towards this so near the origin if h is less than 0 there is a linear term so essentially there is a thing like this and there is something like that bias towards the positive side so the absolute minimum is still here on the positive side on the other hand for h less than 0 for h less than 0 for t greater than tc you have a positive slope here and t greater than tc that fellow is still a square and so on it will be simply a reflection something like that that is the equal thing and it will do the same thing you come down and then broaden out and let me do this again it is on the negative side but now for h less than 0 and t smaller than tc we will have this slope but we will have a deeper deeper minimum and we will have a negative thing so at t equal to tc we have seen what happened already positive slope I mean h positive you have a minimum at a positive value h negative you have a minimum at a negative value at tc you have this at 0 right so this corresponds to what do these 3 graphs correspond to they correspond in this h versus m graph on the critical isotherm you are describing this point this point and this point that is the critical isotherm so on this thing case you are describing this point here corresponds to this minimum this value it passes through corresponds to this flat minimum and this value here corresponds to this minimum on the other hand when you go to t less than tc here you are going through a first order phase transition because now you are looking at this here is t here is tc and you had the spontaneous magnetization curve like this and then you had the other curves in the presence of small positive and negative fields like this but now t is less than tc so you are to the left of this curve you are going like this as you are going from positive to negative values of the field so you are really crossing this is m if you do this in the t versus h graph you had this graph with tc you are crossing this line you are going through a first order phase transition from a positive magnetization to a negative magnetization so the way the free energy graph or the energy function or the landoff functional changes shape is first you have a graph with two minima but this is the global minimum and therefore the equilibrium state is a positive value of magnetization at the critical value at tc the two minima are equal and below for negative fields the left hand side minimum becomes more pronounced in the top so this is exchange of stability that has happened in some sense so from this minimum you went to case were equally probable and then this became lower so you went from here to here discontinuously crossing this line here so you can see that this generic form already encapsulates in it all that is happening here both the first order transition below the critical point as well as the critical region itself within main field in the simplest of approximations so the way to generalize this is now deep very deep this is such a generic thing that one starts by saying that now let us look at this picture in general we still not introduced any fluctuations remember that the crucial point the assumption was that we had a term in the Hamiltonian which look like ij jij si sj so the Hamiltonian was this minus h time summation over i si and I wrote this as equal to minus summation hi si over i where hi was equal to summation j equal to nearest neighbor of i jij expectation sj out here plus h applied field so essentially I replace the expectation of this term by si by this term by si times expectation sj in other words I neglected the fluctuation in sj so essentially I wrote sj equal to sj plus sj minus sj and neglected this in mean field that is really what has happened I neglected the fluctuations so we have still we have not gone we have not taken that into account really so we are within mean field theory this in this model mean field theory simply means this I think more than that and we got an m which is spatially independent of where you are in the lattice it is completely homogenous etc etc we will subsequently put in spatial dependence a little bit later because we want to include fluctuations in some sense but the original Landau theory itself starts by saying that for a given problem with an order parameter I introduce a functional phi which as a power series in the order parameter is something like summation n equal to 0 to infinity in principle some coefficients C n which are functions of the set of all the exchange constants the field and all the other parameters so it is functions of all these quantities multiplied by temperature etc multiplied by m to the power n this is the order parameter so I construct such a functional out here and now I ask what are the possible values of n so if I were to write this out this would be of the form C naught plus C 1 times m plus C 2 times m square plus C 3 times m cube plus C 4 times m 4 plus dot dot dot now as far as the critical region is concerned we have seen that all that you need is to keep up to the 4th order term in it do not need anything else so we are trying to describe that sort of a scenario here so first step this is replaced by 4 so we do not have to go beyond that second point this is a constant and we can drop it it is not doing anything when I differentiate etc etc so this is a harmless constant but this term here cannot exist because if it existed it would say that there is a solution m not equal to 0 above T c because if I differentiate at this fellow here and end up with the constant term here so in the potential you cannot have a term which is linear in the order parameter look at what happened there because if you did you can easily check for yourself that if you had such a term in the potential if I differentiate delta phi over delta m and put it equal to 0 I get a constant C 1 independent of m which would not go away even above T c so it would end up with a solution for m not equal to 0 above T c which we do not want because we started by defining our phase transition as such that the order parameters 0 above the critical point and non-zero below the critical point so this term goes away these terms exist but now the argument is wait now this is with H not equal to where with for for H set equal to 0 no C 1 if H equal to 0 yeah I should say that if H is not 0 then of course there is a linear term that is very much there that is what is giving you the susceptibility and so on yes we are going to do that in a second the reason I am singling this out is because you want this if H is equal to 0 this is definitely got to be 0 because by the very statement that the order parameters 0 above the critical point then the next point is that in the absence of the field I want to have symmetry between m and minus m you saw all these figures all that happened was m and m minus m got exchanged as you made the first order transition since you want to retain that symmetry this term is 0 because you want this thing to be invariant under m goes to minus m you can cause this in fancier language you can say that the probability of a given configuration is e to the minus beta times phi and phi of m must be equal to phi of minus m otherwise the probabilities will change in the absence of a symmetry breaking field. So then you have this and you have this and now we have seen that these coefficients could be temperature dependent themselves so this term here would be some C 2 2 0 plus C 2 1 t plus C 2 2 t squared etc. Yes we have assumed that it is analytic in the coefficients and all these exchange constants in the field and then the order parameters. I am not saying this is a convergent power series you are going to truncate it is a finite time finite point and 4 in fact etc. So this term has to be 0 because the only way in which you can get this phase transition critical point is for this term to change sign as you cross little t equal to 0 as you cross t c which means the leading term must be 0 and it must start with the term proportional to t as indeed it is the case here it is because this term change sign that you had solutions for m which are not equal to 0 plus or minus square root of minus mod t to the half of mod t to the half you got that because there was a linear term there. So this term is 0 and we retain only this term. Then similarly C 4 will also have a C 4 0 plus C 4 1 times t plus C 4 2 times t squared etc. Now the only role of that 4 power term is that you want stable minima. So when m becomes very large in the positive or negative side you want the well to point up the concave upwards then downwards. So for stability thermodynamic stability you want this to have a positive coefficient. The temperature dependence of it is not very significant. So you may as well retain only this term and throw out all these terms and you end up with precisely this. So now let us write that in kind of fancy notation. So we are going to take phi equal to some constant phi naught minus m h plus then you standard notation a t plus m h plus m h plus m h square plus half d m 4. This is the land of and all the information I want comes out of this potential. Now the question is what is the guarantee that this mean field is valid. So when is it valid? We need to know when this is valid. Well let us ask that question separately. Let us ask. We are going to put in all the information that we already have in this business. I have already said that all these critical exponents come from just the correlation function, essentially the behavior of the correlation function. That is the correct way to look at this. So I am going to use that information without actually proving it. We have said certain things about this correlation function. I am going to use that without actually using without actually proving it in some sense. So I would say that it is valid as long as you can neglect fluctuations. Now when can you neglect fluctuations? Because what is really happening is that the system is correlated. You cannot say all the spins are independent within some correlation length xi and we also I mentioned that the xi diverges as you hit the critical point. So if you recall we had a G which I called ri minus rj and I call this SiSj minus SiSj this fellow here. So product of expectations and the non-zero nature of this G probes fluctuations because correlations because if this is completely uncorrelated then this thing would be 0 identically. So good measure of this and we also saw several properties of this. I related this to the susceptibility, static susceptibility and so on. Now a measure of this would be to say that if within a correlation length you compare this quantity to the magnetization itself namely to the factored form below. So divide this by SiSj and compare how big this is compared to this guy. That will give you some idea of how accurate this mean field theory is. So if this ratio is much much less than 1 I would say this is the relative error in this quantity and that is a good measure of the fluctuations. So that would be one way of doing this. Let us sum over this i and j in any case. So finally this whole thing reduces to the following. An integral in d dimension so let me be working d dimensions, spatial dimensions d dr and then this is G of r which is mod Si minus Xj etc. We computed this quantity divided by integral d dr m squared of r. So I have course grained in a volume of size, linear size the correlation length and I am comparing the two. So I got to integrate over a volume, over a volume V where V is of orders right to the power d. We are going to sum over i and j on top and then below and find out what is the relative error in this whole business. So that is how I get this. And the volume of integration is not the whole sample. I do not need to do that. I find out within one correlation length because outside it the correlation is 0 anyway essentially. So I need to compute this quantity. But notice that we computed this quantity and showed that it was essentially the susceptibility. So this thing goes to chi t or susceptibility on top. That is essentially what it is divided by the magnetization could be taken to be constant inside a correlation length and it is this value m whatever this is. And we know that in the critical region it goes like t to the beta. That is the magnetization index. So let us write that out effectively. This is t to the minus gamma mod t. Let us put modulus everywhere mod t to the minus gamma divided by this factor is t to the 2 beta because the magnetization exponent was beta the square root of half in mean field theory and this gives me t to the 2 beta. And then a volume d dimensions of linear size xi. So this is xi to the d. So apart from constants of order 1 this thing is mod t to the minus gamma divided by we know that xi itself goes like t to the minus mu. It diverges. So mod t to the 2 beta minus mu d. This must be much much less than 1 as t goes to 0. Then you are safe in neglecting fluctuations. So where does that get us? It says 1 is much much greater than 1 over mod t to the power or rather let us write it like this. Mod t to the power 2 beta plus gamma minus mod t to the 2 beta plus gamma minus mu d must be much much greater than 1 as t tends to 0. That means this exponent must be negative mu d or d greater than 2 beta plus gamma over mu. So only if the spatial dimensionality is high enough is this correct. This must be high enough in any given problem. If it is equal to or less then fluctuations are very important and the smaller it is the fluctuations get more and more important. This quantity this it is called the upper critical dimensionality. Not yet. Yes. So it says that in any of these critical phenomena whenever you have a critical point then depending on what kind of universality class you have there exists an upper critical dimension above which you may as well work in mean field theory because the fluctuations are relatively unimportant and vanish in the thermodynamic limit. They do not affect the exponents or anything like that. Now let us check what it is for the Ising problem because we have an answer in mean field theory. In mean field theory in our problem we had beta equal to half, gamma equal to 1 because magnetization went like plus or minus square root of T C minus T in the critical region. The susceptibility went like 1 over T minus T C. This is the wise Curie-Weiss law. So gamma was 1 and mu was equal to a half in this case. So we have 1 plus 1 divided by half which is 4. So that is the reason you find this statement made in standard textbooks in critical phenomena that in D greater than equal, greater than 4 dimensions the fluctuations are unimportant. Obviously we live in 3 and we have 2 dimensional magnets. So it gets more and more significant. The deviation from the mean field exponents will become more and more significant as D becomes further and further away from the upper critical dimensionality. That is however a lower critical dimensionality below which the critical point itself vanishes. I mean you do not have a critical point. You do not have a phase transition at all because the disorder can never be overcome by the interaction. The effect of entropy is too strong. In the Ising model the lower critical dimensionality is 2. So in less than 2 dimensions you do not have a phase transition. In greater than 4 dimensions, spatial dimensions, mean field theory is good enough but the interesting physics lies in 2 and 3. It is very, very non-trivial in 2 and 3. And that seems to be more or less the case always. So somewhere we should see that we can attach our mean field. Right, exactly. So you have to... So the question, the point is when we computed mean field and we found the phase transition, it said that it predicted a phase transition in every dimensionality. It did not say anything at all. I mean it said independent of D. D did not appear at all. So that was wrong. Mean field theory overlooks that. We know the critical exponents are very dependent on the dimensionality whereas the mean field theory completely ignores that altogether. So experimental evidence rules out the possibility that mean field theory is correct for the Ising model. We know that it is wrong. We can compute in 2 dimensions. We can compute. Yeah. Yeah. Yeah. If you do the renormalization group, you will see the thing at 4 and above. At greater than 4, mean field exponents are exact exponents. Oh yeah, yeah, yeah. Exactly. Now it is a question of whether you know what sort of accuracy you want. We have perturbation methods. We have this D minus, 4 minus epsilon expansions and so on. We need to keep the sufficient number of terms to get that reasonable results. So the computation of the critical exponents for D less than upper critical dimensionality is hard. You could ask what happens at the critical dimensionality itself. Typically of all these cases, very typically, there will be logarithmic corrections. So the corrections are not power law corrections but log corrections. Similarly, you could ask what is the critical region? How small should T be? I have said T is going to 0 but how small should it be? That is called the Ginsburg criterion and I will talk about it next time. We will see how small it should be in order for this to happen. By the way, this whole thing is, the whole renormalization group approach to critical phenomena started off with a lot of empirical observations on scaling exponents. Now let me just write those down since we are not going to use them anywhere. I need to write them down so that you have that at the back of your mind. For the Ising model, for the Ising universality class, yes. For other universality classes, all the mean field itself is different. The Ising universality class includes the Van der Waals kind of model because those are identical to this. They are in the same universality class. So they all have to do with an order parameter which is a scalar. If you looked at the Heisenberg ferromagnet, the order parameter is a 3 dimensional vector and then you are finished. It is a different class. If you looked at the XY model, the order parameter is a 2 vector moving in a plane with 2 components and that is different. So I said that it depends on the spatial dimensionality. It depends on the dimensionality of the order parameter, number of components and it depends on the range of the interaction. Basically these are the only things that govern which universality class a given Hamiltonian belongs to. Now what has been found is that in the critical region and this is how it started in the 1960s, the whole theory of scaling, a lot of these functions of several variables were found to be scale invariant in the sense that they were generalized homogeneous functions. They were functions of combinations of the independent variables. And they led to a lot of empirical scaling relations which today we can justify by somewhat more rigorous methods. Let me just write them down so that you have some information. So let me call it scaling laws between critical exponents. First of all, I said that everything is dependent on the correlation function. So we really need to see what the correlation function does so that we can have this quantity g of r basically s i, s j, minus etc, etc. This quantity went like e to the minus r over psi and then I mentioned what it did at the critical point. This was psi to the power d minus r to the power d minus 1 over 2 psi to the power d minus 3 over 2. This is for r much, much greater than psi. And at the critical point, it goes like 1 over r to the d minus 2 plus 8. And remember we defined the critical exponent for this guy. We also found that psi goes like mod t to the minus mu. So two exponents are introduced in the correlation function. One is the way it diverges at the critical point, the critical region. And the second is what happens to the correlation which changes from an exponentially damped function to a pure algebraic function, a power law, inverse power. This eta is called the non-classical exponent because if you did mean field theory, eta is 0. So let me also write down mean field values in this problem, MFT values, mu equal to half, eta equal to 0. Now what the scaling relations tell you are the following. It is a set of 4 relations so that all the exponents are written in terms of eta and mu. So the first is called the Rushbrook equality in this case. Alpha plus 2 beta plus gamma is equal to 2. This is the specific heat exponent. Specific heat goes like 1 over mod t to the power alpha. This is the magnetization of the order parameter exponent. This guy here is the susceptibility exponent. The second relation says 2 minus alpha equal to mu d. This is called hyper scaling for a reason which I will just mention in a second. So if you give me the dimensionality of space and you give of the system that you are working with and nu then I give you alpha, the specific heat exponent. And then beta plus gamma equal to beta delta. Delta is the critical isotherm exponent. And then the next one is gamma equal to nu times 2 minus eta. So with the help of these relations you can express everything in terms of mu and eta. And what do the mean field values do? MFT values. It says nu equal to half, eta equal to 0. It says alpha equal to 0. It is a finite discontinuity. Mean field predicts a finite discontinuity and jump discontinuity in the specific heat. Beta equal to half, gamma equal to 1, delta equal to 3. What happened here? You notice that all these relations are valid with these values. Alpha plus 2 beta plus gamma is 0 plus 1 plus 1 is 2. Beta plus gamma, half plus 1 is 3 halves. This is 3. This is half. This guy is 1. This is 0. This is half. This is 2. But this, this gets violated by these numbers. Notice that this is the only law, only scaling relation which is dependent on the dimensionality. All the others are independent of the dimensionality. But this depends on the dimensionality. Now this mean field values are valid only above the critical dimensions, upper critical dimensionality. So unless you put d equal to upper critical dimensionality, this relation will not be valid with those values, right? And then indeed it is valid because if you put this equal to half and that equal to 4, then and this equal to 0, that is perfectly consistent. Yeah, but the, well it will work, okay. Two statements here. One is that this relation here for it to be valid with mean field values, you have to put the upper critical dimensionality here. The second point is, it is a separate problem to show, separate thing to show that the effect of fluctuations is negligible above the upper critical dimensionality. That is a separate story. So this is not on the same footing as the rest of it. That is what I am trying to point out. So what we need to do is to pay more attention to fluctuations. This means that our original Landau functional where we just took M to be independent of R is not good. To include fluctuations, you have to allow the fact that the order appears in patches of size Xi typically, which gets larger and larger as you approach criticality. So we need to create a functional that will be the Ginsberg Landau functional in which we have to put in the spatial dependence of the order parameter in some coarse grain fashion, not at each lattice site. But when you coarse grain it into blocks of say size equal to the correlation line, then we should be able to get an effective free energy functional or Landau functional which includes spatial variations. And we will see what happens when we do that. Again we use very general principles. We will use the symmetry arguments. We will use the fact that we want a scalar, etc, etc. Yeah, there will be dependence on the correlation length in the indirect kind of way. Yeah, sure. It will certainly appear. So that will be the next step. Before we do time dependence, we will do that. Then the kinetics of this transition is something we have not talked about at all. That will be the last thing we do where critical slowing down will appear. Just as the lengths, the correlation length diverges, the correlation time also diverges. So things slow down at the critical point and it will create its own exponent. We need to get, we need to deal with that as well. Yeah, that will be the time dependence program. And then we have to include fluctuations in it which we will with a random force and we will be back to our larger model. But this time for not a single particle but for the order parameter itself. So the basic ideas are very straightforward, very simple but the implementation and then the renormalization group is something I have not talked about here at all. We do not have time for that but that is a separate story altogether. All right. So let me stop here.