 So, now the way it is done is to recognize you can go over that you recognize that you can decompose the partition function just like we did you can decompose the partition function in terms of a matrices and these matrices are very simple matrix these are the matrices that it comes so these are the diagonal terms and you can essentially use this matrix to propagate the nearest number of correlations to all order and you do very similar thing that you have done here to the bar n and then but the you cannot decompose completely like that I have done here because of the b b introduce an asymmetry in the problem the symmetry that I have in this problem that is broken because of this term because here now here two down comes with the same term with two up but here that is reflected here two up and two down are not the same because see b is the magnetic field so when you are going down then magnetic field if the spin is up then both the two spins down will be unfavorable so that will come as a plus sign that otherwise it will be both spins up that will come with a negative sign so this symmetry of this ising Hamiltonian in the absence of the magnetic field is broken as it should in this in the presence of the magnetic by this term and then one can solve this let me tell you the by transform matrix method with the diagonalization the reason of my not doing it it uses some theorems of again value analysis which will take a long time to do that I am not going to do but I will tell you the and there is some examples done you will see that particularly what happened when you have these kind of spins which are kind of things that I have a little better than than that how one configurations gets energy and how you can calculate the eigen values of such things but let me tell you the result okay and there is at least this interesting but not almost the same kind of thing that we have so this is the partition function now is that is what I was saying it includes the calculation of the eigen values which you find most of the books and assume the eigen values of this matrix this matrix you have to find the eigen values of that matrix and then you notice that one eigen value is larger than the other eigen value and then you can like here these two eigen values and it is done here and one is larger than the other one so in the end going to infinity one of them remains but let me tell you the basic result the final result is this the partition function like ui pricing in the presence of the magnetic field is a considerably more complicated but still analytically doable you can go through this these have to be done analysis then you have to do certain it is not too difficult very similar thing that we did in absence of the magnetic field same factorization and then the sum or all these things these are the theorem I was saying I am not comfortable doing that that quantity is the trace of m to the power n and then of course you have to have certain kind of unitary this is a symmetric matrix the unitary transformation of this symmetric matrix and then partition function come as this diagonal matrix diagonal to the power n and then that is this to the power n this is the diagonal matrix to the power n is exactly for that so I do not want to go through all this but these are mathematics you can do and they are not difficult mathematics but you know you have to do them all right so this is the but I this is all given there so the partition function then becomes this look at that how much more complication we have when the electric field magnetic field is there and then you go to if I would be equal to zero then it it will go over to the old old form then we can calculate the free energy cavity l and q I think using that I think and then you can gauge the as I said one of the again values one of the again values survive because other ones is more so n go to infinity limit this is very visible with respect to that one these are the asymptotic analysis and then you get this free energy and this is very interesting once you have free energy then we can do some work with that we couldn't do something over there in the earlier one that I have worked out here because within the magnetic field I could not do entropy and I could have done specifically but I couldn't do the most important thing that people want which is the magnetization but if I have the free energy I can do the magnetization now can you tell me how to get the magnetization from this free energy you know I am dumping some equations on you but then not really very difficult beyond beyond what I have done here was the tricky little thing after that it is same no you have to do the unitary matrix transformation you have to do the eigen value analysis all these things more than more than what you need to do how do I get when I have the free energy how do I get the magnetization lighter lighter yeah so I have to all I have to do d i d b and I get the I get the magnetization okay that is done here then magnetization is with this magnetic field and h is same as b magnetic moment taken out okay so this is then this quantity then one can calculate going back to same as if this lambda plus and all these things and okay and then you get call that or combine this and why and then we have taken with this magnetic field and that that's why you have to take care of the magnetic moment we come here and there change the variable after I absorb it here because this is the quantity that I need as my a magnetization I want to go back the variable I have in my work which is which is which is is y and y is this quantity beta b and then I can take the derivative these are the ugly things that one doesn't want to do in the class but you can do it fortunately it is done here so I can do the derivative of this l n so these idiots sorry it is that comes in the derivative and you take the derivative you get two terms okay taking derivative of course and all these things is no sudden you not the most pleasant thing in the world so because there are two terms there but you can do it but essentially I think everything is clear here right nothing should be complicated doing the board is complicated this is where the sign I am mortally afraid of science okay now we get this beautiful expression however at the end of the day you can simplify that that remains these remains and there is this gets simplified and I think we have to define redefine variables as x not x may p and q small p and small q and again do that and this should become the same as this I think we have to think of where to go back to your x and e to the power minus x or e to the power y and e to the power minus y and square all the same then we have to combine terms I think here also you have to combine term these and this is the same and that is different from this one but whatever this and this is the same so I think there is sudden manipulation one needs to do that I have not done that for a very long time but you can do that but at the end of the day you get the magnetic moment which is experimented observable quite easily in these magnetic systems you get a completely analytical form that is the main thing that is a cinch and cinch is as analytic as cauch hyperbolic cauch so and now I plot that against one over temperature now if there are phase transition what would I expect I am plotting is one over temperature so this is a high temperature limit and this is a low temperature limit right if there is a magnetic transition to magnetic transition then what will happen now tell me what is a ferromagnetic state tell me what is a ferromagnetic state see one of the reasons that physics course they learned it very quickly very rigorously in your past year you learned it but not in terms of spins in your in your schools they should have taught in terms of spins they need not solve the ising model but they should have told you about the ising model but that's not done not in indian context but ising is a very beautiful thing and this is the little thing sorry the little thing i did the little thing that i did here could have been done in high school also but doesn't matter what the solution forget about that what has not been done not done now tell me what is a ferromagnetic state see we are doing phase transition and now the most important phase transition is a magnetic transition then metal insulator transition which is huge amount of industry runs from that right then semiconductor superconductivity did you did anybody teach you magnetism magnetism has not been taught at all but in in in the indicate phd it has not been taught yet they should have taught one magnetism course electrostatics electromagnetism because i we had that a little bit in our honors course we had it one small 16 a half of a course in calcine university magnetism and then we i took lot of courses there from physics department but i tell you that it's really rewarding those of you going to material science and physical chemistry i hope some of you are going there because that's the most fascinating idea now and let me tell you alternatives is not that fascinating when it's going on and on and on the same kind of synthesis you can get some job but it's very boring so most exciting things are happening at the world at the world of material science and world of biology materials is particularly exciting now as you can see your tv getting changed your mobile getting changed every technology is getting changed in every two three years huge amount of what's going on you know this you know company library and some soon you should do the more of this now the artificial intelligence is going integrated with the okay ferromagnetic is that when all these spins are parallel each other that the ferromagnetic state and parametric state is that when the spins are random so ferromagnetic state is the state with an ordered state and parametric is the disorder state so what you are talking of an order disorder transition is very important is like order disorder transition is an example of order disorder transition which is same in melting which is same when you have a beta brass which i believe is zinc and copper right and that undergoes a fabulous transition which is a huge change of material properties so the and we use that material properties many times you see when you use the valve safety valve that we put an alloy and a little change of temperature change the properties of the alloy that is the basic of this many of the valve in these complex systems so these are important thing to know you know because chemists make these things and then we just give it to for theories is to do everything that if we understand we could have make much better material probably okay so parametric the order disorder transition then what would happen in a real system you see that there would be a hysteresis there would be sudden jump of magnetization this is one over temperature so it will be other way around if i put temperature it will be like that otherwise it will be like that go like that so that's a and this system so the hysteresis and as i told you a system that told shows hysteresis is the first order phase transitions however the beauty you see in a in a in a two dimensional system that even in the absence of the magnetic field there is an order disorder transition where the specific heat compress the magnetic susceptibility so that that diverges and that the characteristics of an order disorder or a critical phenomena so the one of the thing the rising model can use us and that i am going to use to take you to a critical phenomena okay so before we go so this is the as for one-dimensionalizing model goes i have derived a part of it the basic basic basic thing and i have it is given here we'll put it in the a if we want to make a little changes i'll probably make a little changes of this chapter and send it to you but you can see there's even a transformation going to diagonal then is lambda one lambda n and here you have to work it out find out the eigenvalues and here the same thing i have done exactly same thing does here that you go to a factorization the same factorization comes here that allowed me to write it in varen same thing empty to varen i go here okay but then you have to go by eigenvalue analysis i could sum it up here because i did have the symmetry which is not here so now uh these are a hugely important thing an extremely difficult thing at the same time the two-dimensionalizing model i will just tell you the result now and i'll do something else which i want to teach you something else which is the other part of rising model many other techniques which are very useful so the but let me continue a little bit with two-dimensionalizing model i'll come back to the two-dimensionalizing model in a normal course of event i would not have two-dimensionalizing model at this stage i will have a little later and that's what the way i did so two-dimensionalizing model when i have a square lattice and i put these pins the beauty of this that this model exists a phase transition and this played a very important role as i told you in the whole history of almost history of science that can be solved exactly and that is the was done by the evolution of the partition function is extremely highly non-trivial the feat was achieved by olsager still considered the most brutal and difficult calculation done in the entire physics and that has a very complicated function which we are not going to and but most important thing that has this is the model shows and this is what you see in experiment and it has this fall has certain exponent which is called critical exponent and that is the things critical phenomena that would do a little later study of critical now i want to do not the solution this is a highly non-trivial solution but i want to do this is what i want to do now and i want to start is there are certain ways okay there is a very important thing in my imagination which is the kuri point now can you tell me what is the kuri point i i i really strongly recommend that you guys i'll send you some material but you read up the magnetism it is really not fair not to know about magnetism that is not fair even if you are doing i i at least indicate basis students are after all doing ph right and you would need even you are doing organic chemistry the magnetic property you are studying and many of the things the magnetic body is of organic materials organic that you don't know why magnetic is so important magnetic properties are studied the reason is that sometimes you have a don't pair or some characterization in this so you need to know the basics of magnetism okay so these are the we'll make a beginning this so the way i'll go now let me write down and i'll come back and forth few things you should certainly do go through the derivation this will take you time about an hour or maybe a little bit more but this is what you're doing so the flow that that i'm going to do is we have done one-dimensionalizing model we'll not do and we done the bandageation and everything and so there is no phase transition no phase transition there was a little bit hint of that why it's the phase transition that we'll come back later essentially because the new phase cannot be stabilized with only two nearest neighbor and next what we'll do is they call mean field theories a huge list here in this context of lattice is the Bragg Williams and the same as van der Waals much simpler and more elegant way is Landau Landau theory i did not do Landau theory yet right so we'll do the Landau theory after we finish these things so in the mean field theories is you have done one example of mean field theory that's what i'm asking kuri temperature of the molecular field what is the kuri temperature and also called kuri wise theory or equation what is that magnetization equal to one minus t by tc remember that the magnetization goes to zero but magnetization goes to zero with exponent one if i go t minus tc then t minus t minus one and that is uh not correct we found that we saw the today i think it's a very different experiment so the kuri the theory the way it was developed we'll see that was a mean field theory mean field theory is the following concept that we have a is a very important concept but fortunately they are understandable in a very simple physical way and that's what we'll do and you will like the Bragg Williams i once i do i'm not dragging but i once had the good fortune to teach at harvard a really formal course a course like that stat mech and there's a great hall called Pfizer hall where many great people have presented their results and the next to it was great bobbled words laboratory when you are that legend lives come and all the legends are there but when you teach at that hall the beauty of that is that when i go to teach has to teach from 11 to 12 three times a week everything is pakka that means there there nine nine uh one screen and screen can be taken out and nine boards time board in the following sense three row three row three column and uh you press you start from one side do this do that then three goes up you press and then one but more important that's not but what i'm saying when you enter the secretary i told them secretary is in the morning i want my coffee for me and my students well they are rich so there is a coffee there and everything was clean all the chalks were ready you know that's what the secretary is are supposed to do here god knows what the secretary is do much of it and they don't do anything and we are not assigned job to them so coming back so this now i have got something nice i complain a lot but that's it's improved by complaining only okay so the main field theory means i have so let's stay nearest neighbor so i have spins here spins here spins here spins here and these are fast nearest neighbor right and second nearest neighbor at these four okay now only everywhere there is a spin and spin sir so what is the main effect of the of the of the other spins on my central spin think a little because this will be continued within next class so let us consider a very high dimensional solid higher than three dimensional then stop at three but you know in your mind you can consider their number of nearest neighbor increasing okay here i have four 3d is simple 3d could be six like that now the i can have a magnetic field or not have a magnetic field these spins are what are they doing on this they're interacting how they're interacting they're interacting their moments now for a moment if i say i freeze this one then i'll add the effects of all these things what my spin will see then my spin will see an average effect so that is average and mean is the same right so i if i then i'll have a my spin here is seeing a mean field so if all of them are up then my spin here will also want to be up so now can i take a description little bit coarse grain means opaque restoration little bit make it little not completely microscopic little bit less mic we call it coarse graining where i would be able to talk of the i write the hamiltonian now in terms of the mean field now what i do these guys also acting on that guy i can construct another mean field and i can make themselves consistent okay that is one probably our upper theory so this is the way one can get beautiful solutions they show the phase transitions we don't have to do as much what and i told you anyway three-dimensionalizing model we cannot solve anything nobody in the world hasn't able to solve but you will see this is just like vanduva's equation so we'll do these beautiful mean field theories they are the very favorite of people because this much easier to do the calculations and much easier to apply otherwise you have to do the volcanic in three dimension and these mean field theories as i said very favorite of us and we will continue with that