 Right, so we go resume where we left off, we left off at the magnetic equation of state for blazing model in mean field theory and if you recall the magnetization per particle in suitable units M, this was given by the magnetic field if you recall H over K Boltzmann T, we solved for it and we discovered there was a solution of the form M minus tan hyperbolic M times T C over T divided by 1 minus the product of these 2. Oh yeah, tan hyperbolic is transcendental of this was M T C over T and T C was if I recall T C was equal to it certainly was 2 D and then there was a J which is the exchange constant, the coupling constant divided by K Boltzmann I think that was it, that was wrong. So there is a dimensionality dependence here in this thing in a trivial sense because it just includes increases the number of nearest neighbors, right. So now we are going to look at it in the critical region and from this we can extract all the critical exponents which we already saw in various ways. So critical region remember that in this problem the critical value analogous to P C, V C and T C the critical values are M critical equal to 0 because it takes off from 0 from the paramagnetic phase to the paramagnetic. H C is also 0 because remember that H equal to 0 corresponded to this flat line ending in a critical point in the H versus T plane and of course T C is non-zero it is some number given by this. So the critical region says when M H are small very near 0 and T is very close to T C that is the critical region. So let us see what this looks like to lean to leading order this equation of state becomes H over K Boltzmann T is equal to M minus tan hyperbolic X here tan hyperbolic X goes like X minus X cubed over 3 the leading behavior. So it is M minus M cubed T C cubed over T cubed in this case plus etcetera times now this guy here already has an M out here and it is going to be multiplied by another M oh yeah M M T C over T that is important otherwise I am not going to get the exponents you see they assume I made no mistake so they got rid of this I have to prove them wrong. So M minus M T C over T plus M cubed T C cubed over T cubed dot dot dot times the denominator you have 1 minus because tan hyperbolic this has got to be X minus X cubed because it comes down and saturates so it has got to be negative and that becomes this divided by 1 minus M squared T C over T and then the next term is T M cubed here and there is an M here so it becomes M 4 we will throw that out. So it is 1 minus M squared T C over T inverse which is 1 plus in this fashion so if you write this out this is equal to M times this so this is 1 minus T C over T that is this portion and then plus M cubed times the first term is T C over T and then there is minus T C squared over T squared from this fellow and then this time here plus T C cubed over T cubed oh yeah this is a 3 factorial thanks so this is M cubed over 3 so we are ready to read off various results let us give this some name let us let us give this T C over T some name use little t I am going to use little t let us fix it this is equal to T minus T C divided by T C so we could in principle write it in terms of little t this whole thing but what are the things we want to see immediately well the first thing we want to do first is to find the susceptibility remember that chi T equal to delta M over delta H delta H at constant T and H equal to 0 that was our definition so all I have to do is to differentiate both sides out here and this is M cubed already so it is clear that the leading linear behaviour comes from here and it immediately says 1 over K Boltzmann T is equal to if I differentiate both sides with respect to H this is equal to chi T into T minus T C over T or chi T in these units is 1 over K Boltzmann T minus T C. So it diverges at the critical temperature like 1 over T minus T C now this is for T greater than T C so that this is positive I want to show that the same result obtains for T less than T C so show that chi T is proportional to 1 over T C minus T for T less than of the order of T C just below T C the constant of proportionality will not be this it will be some other number some number possibly so it is the same critical exponent both are to power 1 I have used the fact that M is yeah so I have used the fact that this quantity except ferromagnet it is not a paramagnet so it orders in the direction of the field and the susceptibility has to be positive so implicitly I assume that T is greater than T C you have to now solve for T less than T C and find the right branch in M as a function of H this happens to be the right branch for T greater than T C because this is the paramagnetic branch yeah you have to solve the cubic because the susceptibility for T less than T C is the slope at a point where M is not 0 but has finite intercept and you have to find that root and then get back to this either of the roots will do because it was the graph the graph looked like this it was like this and like this and we are discussing this slope or this loop and that is going to go like this so that is a good point I implicitly by using this alone I said we are near M equal to 0 but that is only above the critical point below the critical point M is away from the stable roots are M not equal to 0 in the absence of a field they are the spontaneous magnetization okay that is the first result then the second one is we can also see what happens at T equal to T C so we need this result to at T equal to T C we want to get hold of this exponent so this is M versus H we want to get this exponent here how does it behave what is the power law we would like to find that so you set T equal to T C and of course you immediately see that this cancels this cancels this cancels you are left with this so it immediately shows M is proportional to H cube sorry H is proportional to there is a cubic point on the critical isotherm that follows immediately from this because this term goes away this to cancel and you are left with the leading behavior H proportional to M cube at the critical isotherm in other words this is a cubic curve then let us ask what happens to the spontaneous magnetization in the absence of the field so we need to find out 3 at H equal to 0 now we are trying to find out here is T here is M M not and remember that it goes like this is what we had said this is T C and we are trying to find the behavior here having set H equal to 0 so this is equal to 0 and you have to solve this equation for M not with this set equal to 0 the root M equal to 0 is always a root as we saw but it turned out to be an unstable root we want the nonzero roots we want these roots not the zero root here so you can cancel out M for that and then I leave it to you to show this fairly simple you cancel this out you get M squared and then you move it to this side and remember now T is less than T C so T C over T minus 1 will be a positive quantity you move it to this side and that will be equal to M squared times something or the other here so you get 2 real roots show that M goes like plus or minus square root of 3 times T C minus T over T C to the this whole thing to the power of half of course it is immediately obvious yeah that is right yeah you have to be a little careful because there is similar term sitting here exactly exactly exactly you have to check that that is not 0 identically that is straight forward to the several ways in which you can show from the exact equation that it is going to be a square root singularity so the critical exponent is a half now these are exactly the same exponents which you get for the van der Waals equation also in the critical region however they are not the experimental exponents experimentally what happens is these exponents are somewhat different yeah I am going to do that that is exactly what we are going to do experimentally you have a whole set of critical exponents rigorous way of defining it but in a heuristic way there are various power laws of what happens in the critical region near a critical point now these critical exponents have values which are universal for different universality classes so for the so-called Ising model universality class which is what we are dealing with here started with the Ising model you have one set of exponents for the Heisenberg universality class you have another set etc etc now the simplest instance and then you have what are called mean field exponents these are mean field exponents here so this specific heat is supposed to behave C is supposed to behave like I am not specifying C V or C P I am not going to do that for a minute this diverges like 1 over mod T minus T C but let me write everything in terms of this little this measures how far away you are from the critical temperature so this goes like 1 over mod T to the power alpha that defines the exponent alpha then the order parameter either the magnetization the one that distinguishes the 2 phases and we are going to take typically one of them to be the one order parameter in the high temperature phase to be 0 and non-zero below just in analogy with the magnetization this order parameter goes in the critical region goes like mod T to the power beta in fact it goes like T minus T C so it goes like T C minus T so minus T for T less than T C while it is 0 for T greater than T C yeah T C so the order parameter is 0 and then it branches off in this fashion and we are talking about what happens in that region to the stable root the order parameter there is many ways of define there is no uniqueness about an order parameter you could take other things as well I mean in the magnetization example you could ask why did they take magnetization to the power 3 or 7 as order parameter no reason why not but the most convenient and simplest one now there is no reason why it should be a scalar for example in not the Ising class but the Heisenberg class the magnetization is a vector then there are situations where it is a planar vector no matter how many physical dimensions you are in for the lattice the magnetization itself magnetic moments can only move in a plane say that is the so called XY model in the Ising model they move only in one direction up or down so it is a scalar otherwise a 2 dimensional vector otherwise a 3 dimensional vector may be an n component object in very complicated systems like in nothing like liquid crystal and nematic liquid crystal for example it is an axis liquid crystal and nematic liquid crystal consists of rod like molecules which are arranged every which way in the disordered phase and it is low the temperature these guys get ordered so on the average they all point in this fashion of course there will be small fluctuations about it but on the average they point along this way but there is no distinction between the head and tail so it is not an arrow unlike the magnetic moment and so very profound consequences follow because it is not an arrow but only a line and that implies that the order parameter is not a vector but an axis here so in this case it turns out to be a tensor of rank 2 because that will specify an axis headless vector it is also called a director I did not name it I mean it produces a line field and it is got defects because it is got a line field and so on if you look at your thumb for instance I should not get digressed but if you look at your thumb you got the words like this the thumb print and then at one point there is something like this this is a defect a topological defect on the surface here you should really look at it as something a point a line defect really but in two dimensions it is a point defect here and the directors are supposed to be like this such a thing cannot happen if you had arrows because then it means that you are going like this then what happens on this line it is a point defect but what happens on this line it is indeterminate completely so a two dimensional ferromagnet cannot have a point singularity of that kind but a liquid crystal can it is called a 180 degree discrimination and it is got a physical effects and so on very real thing so we are all carrying topological defects on our thumbs so this is a complicated order parameter if you go to more complicated substances like liquid helium super fluid liquid helium in the helium 4 then the order parameter is a wave function of what is called the condensate the super fluid condensate that is now as you know a wave function is a complex number so it is a modulus and a phase that is the order parameter a complex number if you look at helium 3 we have a rare isotope of helium consists of fermions that too can become a super fluid and it is got all sorts of magnetic properties and so on that order parameter is pretty complicated it is SO 3 cross SO 3 cross O 2 so 9 by 9 or whatever it is some 18 18 dimensional object so it is got a lot of physical information buried in it but the order parameter can be very complex in many cases you could ask what is the order parameter in a liquid gas as we said it could be the difference in densities between gas and the liquid but what is it in a crystal as opposed to a liquid because the liquid and crystal have practically the same densities most substances when they freeze they do not become very much more dense nice it actually expands the other way but they are equal to each other to within 10 percent so what would be a good order parameter in a crystal something that reflects the nature of the order namely that atoms only sit at regular intervals and so on so it would be if you take the density of the crystal the local density wherever there is an atom there is a big spike and then there is nothing etc and you do its Fourier transform then it will have components of all the wave vectors corresponding to the reciprocal lattice that would that set of amplitudes would be your order parameter so it is not a trivial job finding the order parameter in many cases in some cases but we know it when we see it and the order parameter exponent is called beta in mean field so whatever we have done is called mean field theory this alpha is 0 it turns out that in mean field theory specific heat is predicted to merely be discontinuous the finite jump and not divergent this is its infinite as t tends to 0 when alpha is positive this becomes infinite on the other hand you have a discontinuity the order parameter beta is a half in this problem then the so called susceptibility order parameter time this goes like 1 over t to the power gamma t greater than tc and goes like 1 over t to the gamma prime for t less than tc 1 over minus t to the power gamma prime in mean field theory I just showed that gamma is 1 and gamma prime I asserted was also equal to 1 so the susceptibility exponent is 1 then you ask on the critical isotherm what does the critical isotherms curvature look like so critical isotherm this is delta because we found h is proportional to m cube p is proportional to minus v cube and so on in mean field field in general this is some delta and in mean field theory this delta so alpha equal to this beta equal to half gamma equal to 1 equal to gamma prime delta equal to 3 there are 2 more actually a few more there are 2 more which I will introduce very shortly so we sort of extracted whatever we want from this thing here as much as we can but I must now we must now go back and ask where is all this coming from what about the corrections to it etc but first some experimental facts in real life if you look at magnets like the 3 dimensionalizing model or the real liquids for instance then these exponents are very different for example alpha is for liquids very close to 0 some small number 0.1 or something like that or less in the case of super fluid helium sorry in the case of the 2 dimensional easing model alpha is 0 but the specific heat diverges logarithmically so there is a log divergence there now the specific heat itself depends on what is kept constant it is either Cp or Cv or C with the constant field etc depends on the system that we are looking at so that is why I did not specify which one it is for instance for the van der Waals model for fluids Cv continues to be that of an ideal gas in the critical region but Cp diverges and it is related to the divergence of the susceptibility of the susceptibility in that case the compressibility so various possibilities of this kind can happen in 3 dimensional liquids beta is not of the order of half but nearer 0.321 325 something like that this exponent gamma is of the order of 1.25 it is larger this delta is of the order of 4 to 5 4.75 something like that in the 2 dimensional easing model it is a very special model again exactly solvable all the exponents can be shown to be rational fractions exactly for instance this turns out to be a log discontinuity 2d ising alpha equal to 0 but it is a log divergence log mod t minus t c then beta is equal to 1 8 gamma is 7 4th delta is what eclipse of the mind I mean I am not I do not remember what is delta and I am frantically trying to find the relations I mean there are relations between these exponents I am trying to see given this can I find it for instance alpha plus 2 beta plus gamma is equal to 2 and that does not help us here but it works here as you can see alpha plus 2 beta plus gamma is equal to 2 it will work it works here too but I am trying to think of what is the simplest it will come back but that is because it is dimension dependent 2d ising model the 3d ones are much closer to real life now I want to get straight to the point that what is underlying this whole business and you will see in a minute where it comes from is the divergence as I said of something called a correlation length so we have to define a correlation length and that will give us a big handle on what to do next okay so let me define that what is it that I mean by this correlation length you see in let us go back to the ising problem the magnetization or the moment at each site lattice site is your is measuring the order parameter the average value of this moment okay but you could ask in a thermodynamically homogenous medium in equilibrium at every local site you do not have something which points on the average it is fluctuating all the time so there are fluctuations and you could ask given the average what is the next the mean square what is the deviation look like from the average some kind of generalized covariance you asked for and what would it be in the case of the spin problem you see if you recall we said that the effective field at every point and if you recall we wrote the Hamiltonian again I go back to this as minus j summation over ij si sj minus h times summation over i si I started by writing this and then I said look this could be written as minus this field plus j times summation over j nearest neighbor of i sj summation over i si so the field that this guy is seeing this moment at the site is this this is exact in the ising model in the mean field case we replace this fellow by its expectation value in other words we wrote this field as equal to minus h plus j summation j nearest neighbor of i sj summation over i si but it is not very elegant notation so let us summation over i si so this is the effective field seen by the id spin times si I added this instead of this so I got to put that back right so plus summation over i summation over j j sj minus sj you could put a j ij here just in case it is in homogenous so this term cancels out and I get back this oh this is also in si the whole thing acting on si now this guy here represents the fluctuation about the average value of these variables and mean field theory drops this fluctuation that is all it is done it is just dropped it but we would like to know how important this is that is the fluctuation at every point so now let us define the autocorrelation let us define let us call it g minus rj it is a function of the difference in positions of the ith and jth moments is equal to expectation value of si minus average si sj minus average sj it is clearly by translation variance in thermodynamic limit clearly a function of i minus j this is like delta si delta sj and it will be a function of i minus j of course you can also write this as equal to si sj minus si you can also write it like that by trivial piece of algebra right it is a generalization of the mean square deviation at some point but it is now spatially dependent on 2 indices i and j okay now let us go back and ask what are these expectation values because the next target is to relate this to the susceptibility that is going to give us the static susceptibility formula and you will immediately recognize linear response theory in it so the derivation I start with this again the density matrix are you yeah expectations already there is no averaging of an average it is already average right so trace so z the partition function is trace e to the minus beta trace over the fact that each si can be either plus 1 or minus 1 that gives you all the 2 to the power n for n of them and then you take the thermodynamic limit okay so this is equal to trace e to the beta h summation i si plus beta j summation i j nearest neighbours si the trace of this whole thing now what is si itself equal to this is equal to trace si e to the minus beta h over trace e to the minus beta which is equal to by the usual trick I want to pull out an si out here right so what should I do I take a derivative with respect to h right that gives me the summation over i this guy so let us do that equal to and there is an extra beta which comes out right so I have to divide by this so 1 over delta z that summation over i si because there is summation over i right by the way you already know this formula it is trivial you already know this from thermodynamics because you see let us connect it up that is useful exercise because remember that the magnetization m will appear in thermodynamics through m dh like v dp right and it will be equal to minus delta f over delta h keeping time temperature constant so m little m this is capital M equal to minus 1 over n delta f over delta h at constant t that is equal to minus 1 over n delta over delta h of minus k Boltzmann t log with canonical partition function that is the formula for the free energy minus k t log z so this is equal to 1 over n beta delta so this is equal to 1 over n beta z delta z over delta h that is what I got here the same formula remember by translation variance this expectation value is independent of i and the n of these fellows so each of them is 1 over n times this that is your m little m so it matches this thermodynamics so all I have done is to write that show how that arises directly by doing this but now comes the interesting part what is this equal to what is summation over i j s i s j equal to I do two derivatives with respect to h first time it will pull down s i second time it will pull down s j it is not this term because this fellow the summation over j is restricted to nearest neighbors of i for each i but this what I am calculating here is over all i and j and that comes by taking this down and differentiating twice and that becomes equal to 1 over each time I pull down a beta so it should be 1 over beta square and then a z d 2 z over d h 2 so now let us calculate our green function or our correlation function so the correlation function was g of r i minus r j this guy here summed over i j sum over it this is equal to summation over i j s i s j minus minus what minus summation over i j s i s j this square of this sum I call sum over i j s i s j this factors that is equal to what let us put this stuff in it is equal to 1 over beta square z d 2 z over delta h 2 minus square of this 1 over beta square z square delta z over delta h whole square that is this guy this correlation or if you like s i minus expectation s i s j minus expectation what happens if I differentiate m with respect to h I should get chi so it is clear that chi equal to delta m over delta h which is equal to 1 over n beta the derivative of this guy the field appears here in z out here so the first term is 1 over z d 2 z over delta h 2 minus 1 over z square delta z over delta whole square but there is just this guy here is it not this is an extra beta so it says you tell me where the beta goes so this is equal to if I multiply beta so it is equal to 1 over k t summation over i j g of r i minus r j this is also an n this is also an n somewhere this is an n sitting where 1 over n but this is a function of r i minus r j so I can fix the j and some over fix the i sum over j s and then fix the next i sum over j s you are going to get the same sum so I can remove one of the summations and call this coordinate some relative coordinate the distance between i and j and I can therefore write this as equal to 1 over k Boltzmann t let us not forget the k Boltzmann t summation over i g of x i where x i means you are centered at i and you are now calculating all the distances to all the other lattice sites in the thermodynamic limit you could actually convert this to an integral you may put a lattice spacing convert it to an integral and let us do that in d them by the way this is called the static susceptibility formula the fact that this guy here is equal to this does not that remind you see this measures what it does in an external field and this is now telling you what the autocorrelation is so it is exactly like linear response formula precisely so now let us see what this does so again we are on chi t is now approximately in the continuum limit 1 over k Boltzmann times t now we have a lattice with lattice constant l in d dimensions I think this is beta squared on top so that gives you it is okay I mean I go back to linear response theory it is beta times a dot of 0 b is 1 over k t so chi t is 1 over k t let us put a lattice constant l and it is in d dimensions and then you have an integral d d of r g of r now we need a formula for this guy we need something for this correlation which requires hard work which requires little bit of work but let me state the result and if time permits we will try to derive it we expect this is going to die down as r increases the correlation between the spin and spins very far away is going to die down now it turns out that this fellow here for r much much greater than some correlation length xi g of r goes like e to the minus r over xi and this can be shown so I am going to assert the result and then we will see the consequence and then I will prove it later on divided by r to the power d minus 1 over 2 xi to the power d minus 3 over 2 away from the critical region t not equal to t c and that is chi so now look at what is happening if I put in what I already know for chi goes like t minus t c inverse so it says t mod t minus 1 goes like on that side this crazy integral this does not do any harm we just replace this by t c but you have to do this integral so you have to do an integral it says only the radial coordinate matters so it is 0 only what happens near infinity matters really r to the power d r r to the power d minus 1 e to the minus r over xi over r to the power d minus 1 over 2 xi to the power d minus 3 over 2 and this integral is 0 to infinity some cut off to infinity we do not have to and we do not care it is not 0 and then this blows up so now there is a paradox you have this fellow by assertion I said that g looks like this exponentially damped there is some power of r floating around by the way this becomes d minus 1 over 2 this cancels against this this is the phase space factor r to the d minus 1 in d dimensions now this diverges as t goes to t c but it is equal to an integral which has got this very strong converging factor here what does it mean well xi certainly cannot go to 0 because if it goes to 0 this will kill it faster than any power here you do not care xi blows up xi has to blow up so xi must tend to infinity as t tends to t c this correlation between spins I already said the fluctuation effect is going to become extremely strong at the critical region and so much so that any correlation length just diverges in an infinite system it goes to infinity itself the question is how that is easily seen from here because all you have to do is to scale out by this to change variables that is change variables so this integral goes like integral d r is xi d u so there is a d u then there is a xi to the power d minus 1 over 2 e to the minus u and then divided by xi there is a xi d u so there is that guy and then d minus 3 by 2 from 0 to infinity which is xi to the power d over 2 minus half that is this part plus 1 that is this part minus d over 2 that is this part plus 3 half times the number so this goes like xi this cancels 3 half minus half is 1 plus 1 is t so we find that in mean field theory in mean field theory xi goes like mod t to the minus half because the square goes like 1 over t so xi goes like 1 over square root of t in other words the correlation length diverges like 1 over t c minus t to the power half that is there is in general xi goes like 1 over mod t to the power nu and the mean field exponent nu is a half in the framework of mean field theory pardon me there is no effect of d as far as this is concerned but we will see in a minute what really happens one could ask what happens at the critical point what happens to the correlation function at the critical point what does it behave like I said this formula is true away from the critical point so you could ask exactly at the critical point what happens this fellow becomes infinite and then what happens is this formula still true this is the question may have come this is e to r over xi is r over infinity which is 1 this goes away so there is a power law so the question you are asking now is is there a power law which is does it blow up how does it blow up it turns out that you can show independently that at t c exactly at t c g of r goes like 1 over r to the power d minus 2 plus an exponent eta you have to introduce one more exponent eta is not as bad as it sounds because everything this thing is in terms of the correlation function so the idea is that away from the critical point the correlation 2 point correlation dies down exponentially with the correlation line at as you approach the critical point that correlation line diverges like with temperature in this power law fashion and at the critical point it becomes an algebraic function of power law dk which is d minus 2 plus another exponent eta and all these other exponents can be written in terms of eta and nu and I will write those relations now in mean field theory nu equal to half and eta equal to 0 so if you put those 2 pieces of information in all the other mean field exponents that we got will jump out automatically so everything is now hinging upon the correlation 2 point correlation and characterized by these 2 exponents eta and nu how this happens how this happens and how this happens requires more careful analysis we will try to do that that is the starting point of the modern theory of equilibrium phase transitions this is the starting point the fact the recognition that what happens at that point the reason thermodynamics fails is because the correlation length becomes you cannot neglect fluctuations whereas thermodynamics neglect fluctuations it also says something more than that even mean field theory does not work it gives the wrong exponents and now you could ask what is the region how close to Tc should I be in order to see the new exponents this is given by a rule of thumb called the Ginzburg criterion which I will try to mention but it is not a rigorous statement take this case by case as you can expect because what is exact is the universality class in each case so it will give you values of exponents but to tell you how good that value is or when the mean field starts becoming a bad approximation depends on the system so it is not universal in that sense but there are criteria which I will tell you for instance if there are long range forces then mean field theory is very good so if you ask what about this normal to superconducting transition in mirrors right mirrors become superconductors and the suitable conditions then it turns out that the temperature range in which the renormalization in which you need to correct the mean field exponents is of the order of 10 to the minus 8 degrees which is negligible unless you hit exactly on the critical point it does not matter so you can get away with mean field theory because there is a long range Coulomb force involved in the problem but in other places spin models and so on it can the corrections can start appearing much more significantly so that is not a precise question but there is such a criteria for this so we will take it from this point and the next time I will introduce the relations between the exponents scaling generalized homogeneous