 our goal is remember that we want to see how this Ginsburg-Landau equation or the Ginsburg-Landau energy functional which is a macroscopic functional arises from some microscopic model which really takes the individual electrons into account in some scaling limit and what I discussed last time was a spatially homogeneous variant of this model, right? So let me write down the functional from last time. I want to say a few more words about this and then we move on to the real problem. So this is the kinetic energy of the particles. Then we have a term which involves the alpha alone and then there's a coupling term in between which is given by the entropy and by this I mean simply you form this capital gamma matrix which has gamma as the left upper entry, alpha as the right upper entry and then you continue it and then you compute the entropy of this 2 by 2 matrix for every momentum psi and then afterwards you integrate it over r, okay? There was I heard some comments after lectures they should perhaps explain a little bit more where the physics of this comes from. I'm happy to do this however I would like to do this a little bit later when we really go to the full model, okay? So the thing that I want to say now only I mean to repeat only is this alpha thing this represents superconductivity. If we have an alpha nonzero alpha then we are in the in a superconducting state otherwise it's a normal state, okay? Now the gamma here represents the particles the ordinary particles. Now it depends a little bit on what you are modeling here if this is really a superconductivity then the gamma models the electrons in the solid. If you have this in a superfluidity context then rather it's the particles they're really neutral atoms and in some in some ultra cold gas and then the gamma stands for for the atoms. So whatever the gamma models some fermionic particles ordinary particles this would be without the alpha term this would be the stand-up model that has been used forever in this context, okay? So the new thing here is to allow a second parameter couple them to the to the gamma in such a way that if the alpha is zero we get the old model back but we have now the possibility of allowing as an additional alpha which means that the system goes into this superconducting state. The the interpretation of gamma hat is gamma hat that's the momentum distribution of these fermionic particles and in particular the well the total integral of this distribution right which I don't know there's a 2 pi to the d over 2 and then it's gamma of zero that's the the total particle number and you might also say that see that I did when I minimized I did not fix the particle number however bigger okay however I did not fix the particle number however instead I introduced this parameter mu and you see from this formula this is really mu times the particle number okay so that is just a trivial mathematical step I'm just taking a Le Chantre transform so if I can compute this energy for every mu then I take a Le Chantre transform with respect to mu and I get the energy at every fixed particle number okay conversely if I know it for every fixed particle number I do the Le Chantre transform in the other way and I get the energy at each for each chemical potential it's just a mathematical trick as I understand that physically it's perhaps more intuitive to think that the particle number around the density I should say because we're in a gas the density is somehow fixed but mathematically it's much easier if you if you have just add this to your Hamiltonian and and do this minimization problem but physically it's the same thing okay now there was a question I heard no that no okay anyway so that was one physical comment and then one more thing I wanted to say let me draw the picture again and I'm drawing here oops I forgot T that's of course the most important thing that's a temperature because we're interested in a phase transition when the temperature changes and so let me draw again this free energy as a function of the temperature it's concave decreasing function and then there was a TC and for temperatures larger than the TC the energy is really that's the free energy of the normal state right which is the one that we computed last time by alpha is equal to zero and over here we have a bifurcation and that's really the ground state energy and then what I told you last time is we computed here at this point we computed the left derivative we saw that the derivative was continuous but the second derivative is discontinuous and it was a quadratic behavior and the thing that I interpreted as the baby model of Ginsberg-Landau is that the coefficient which is in front here the E can be obtained by minimizing something and the something is exactly of the form minus quadratic plus quartic okay and which is exactly what we want to derive later in the spatially non-homogeneous case now the one more remark which is perhaps interesting to some of you the the expression for E was particularly nice if this effective operator so there was which came as the the what is the Hessian right the Hessian in the in the alpha direction of this functional this when you took the second derivative with respect to alpha at the normal state we arrived at this operator and the assumption that under which I could give you a nice formula for the E was that the kernel of this thing is non degenerate right so if we have this then the expression was really just a minus lambda 2 psi squared plus lambda 3 psi 4 that was the E was equal to the inf of this website is a complex number okay now the surprising thing I should say which is just a side remark is that this assumption is not generic I mean it's not ungeneric either but there is a right the parameters in the game or the mu and the V and now you ask yourself somehow in this parameter space is it generic that this has dimension one or not and actually turns out there's an open interval where this degeneracy can be something else and actually it grows as the mu I think goes to zero this degeneracy grows in in open intervals and it becomes unbounded okay so this is perhaps surprising because for Schroding operators we know that the ground state is unique by some maximum principle for elliptic equations or some positivity of heat kernel or something but this operator has the the large frequency behavior as the Schrodinger case but still the symbol is different and therefore these uniqueness results do not hold and and therefore we have this this non degeneracy which one can study in which gives rise to different Ginsberg-Landau functionals which you see here you have a U1 invariance right which is a trivial way of complicated we're saying that you can multiply psi by phase and nothing changes but if you have these degeneracies you can have very big symmetry groups under which you invariant here all right so that was a bit of a review from last time now I want to do one computation here which I hope clarifies a little bit what's going on and how how to think about this and this is I want to compute the Euler equation for minimizer okay now I told you that we really we don't care about solutions of the equation we're really more interested in the ground states but I think computing the Euler equation still gives us some some way of seeing the problem and also motivates how we choose the trial state when we do a variational computation okay and so the so Euler equation for so it's not necessarily minimizer but let's say a stationary well okay minimizer minimizer gamma alpha and the the good way to do this is that you you take some some convex combinations so gamma is the minimizer and now you take a convex combination which means that you have your t times some gamma tilde minus gamma and then some alpha plus t times alpha tilde minus alpha right and now we differentiate this thing twice at t equal to zero and then this thing should be equal to zero and now let me write this so this term that's obvious right or D so this is Xi square minus mu times this perturbation here gamma tilde minus gamma hat right and then we get for the entropy term you have to well I mean either you do it for the eigenvalues I can write down the eigenvalues last time or you use some general formula how to differentiate traces either way you'll find true a C2 trace the logarithm of gamma hat of Xi over 1 minus gamma hat of Xi multiplied by gamma tilde hat of Xi gamma hat right so that's this Xi and then of course this term that's the the term that's easy to differentiate that's plus twice the real part of integral v alpha bar alpha tilde minus alpha dx and now the point what I want to do is I want to include this term here into that term that sounds first a bit strange but I want to get to this matrix I want to I want to say that the important thing is here this capital gamma matrix so in other words I want to apply here plus Schubert's identity okay and then this thing is just the the one-two entry of this matrix so I can write this as there's a half appearing or D that's a big trace of Xi square minus mu and now there is a delta hat of Xi delta hat of Xi bar I apologize Delta has nothing to do with the Laplacian but it's the standard notation in the physics books so I prefer to keep it I hope it doesn't create any any confusion okay so plus this term okay so this was so you see this was just a well just plunge or identity I just wrote this here is the entry but now this is very convenient because right this has to be zero for every choice of gamma tilde and so therefore I conclude that Xi square minus mu oops and I didn't tell you what with delta is equal to V times alpha so delta of X is equal to V of X alpha of X right that's this guy here now this is as usual right when you do an Euler Lagrange equation this has to be zero for every choice of this thing so therefore this trace has to be equal to zero so we get an operator identity that Xi squared minus mu delta hat of Xi delta hat of Xi bar minus Xi squared plus mu this thing plus t over 2 the logarithm of gamma hat of Xi 1 minus gamma hat of Xi is equal to zero for almost every Xi okay and now I can solve this equation there this is a 2 by 2 matrix we know these matrices commute so it makes sense right if I divide these matrices and take logarithms of those right so now I solve this equation for gamma hat and what I find is after little computation 1 plus e to the 1 over t times this matrix okay so the optimizer this is a convenient form of writing the gamma and the alpha of the optimizer it's again given by a Fermi Dirac distribution just like for our normal state except that something has appeared on the off diagonal okay and so this what appeared on the off diagonal obviously creates some off diagonal for the gamma so this thing this is often called the the effective BCS Hamiltonian okay it's and it's this operator that we have to study and later on in the real full model for which we have to do semi-classics and and do all the thing but that's really the equation and you see the non- linearity you see appears I mean this is self-consistent equation because we know that the gamma is equal to the alpha right and alpha is this entry that's that's the Euler equation in the translation variant model okay and now this gives us a hint how we should construct a trial state right if we talk about the upper bound we want to say here look at this picture we want to say that the energy is smaller by quadratic amount quadratic in t c minus t by quadratic amount smaller than the the free energy of the free energy of the normal state so well what's a natural guess well we just take this a state gamma so construct trial state of the form gamma hat of xi right this whenever you do this this automatically is between 0 and 1 so it's admissible in the sense that we have you have to work a little bit that this if this guy is nice then the the one to entry actually is in in h1 so there's some regularity that you have to do but that's not too difficult and anyway so that's our trial state 1 over t xi square minus mu and I put here h to the a hat of xi and here the same thing h to v a hat of xi bar minus xi square plus mu okay right I told you that we have to guess something and that's a very good guess the parameter that we still haven't decided is this a now but we know somehow that in the optimal case the a will be given somewhere in this nonlinear way by the alpha but now let's forget that constraint just take any old a plug this in compute the energy for this right and I put an h I didn't really tell you why this really is of order h but you can take another small parameter and then see that this actually wants to be of the size h plug it into the equation and then compute okay and what you find is so if you take this for an arbitrary a then compute to find compute to find that's a little bit wrong what I'm writing but I know how wrong let me see alpha is equal okay so there is this of course because h is small 0 but then I have phi kt plus v tc if you want phi a which has an h squared in front plus oh it's a little okay and this to first approximation think of this just as a okay it's an order one thing but you see somehow there's an h squared term remember h is square tc minus t over tc so this thing is linear in tc minus t we want to prove something quadratic so we want to kill that term okay so therefore we want to have the phi a be an eigenvalue of this ktc plus v and then because phi a is closely related to a this translates so into having a an eigenvector of this want this equal to 0 so right because we want to make this as small as possible so what this equal to 0 which gives a in this kernel I'm saying the no I mean right yes exactly right so I mean right I have to come up with a gas for the gamma and the alpha which is a complicated thing to do so I motivated my guess by saying well the true thing is of that form where whether alpha here is coupled to the this off diagonal delta now I just ignore that non-linear coupling okay so I just take alpha whatever plug it in so it has a good form I think right whatever a then I compute and I see that the best thing I can do so I mean think really there's an a it's really everything else it just makes it look more complicated I want to make this come as small as possible so therefore I choose a to be in that kernel so if I understand what you are saying that if you have a minimizer with alpha close to zero most likely should look like that exactly right exactly right that's somehow that's what's done in the lower bound right in the upper bound I guess something I try to make it that but then lower bound I have to say that really everybody as you say really every out for the for every minimizer the alpha really looks like that exactly okay thank you and then so now we have found out so this thing the vanishing of the H squared term has told us what the A should be okay and now we just compute further and compute the H to the four-term now use the well okay now compute well okay now compute H4 and you use that it's in this kernel to to cancel a few a few terms okay so that's that now of course as he said the difficult thing would now be to show that every minimizer is approximately of this form this you get somehow from a h1 gap in equality which I perhaps I explain a little bit in the last lecture but that I want to skip at this point but I rather want to tell you perhaps it's I'm so this was really the proof of the theorem but I want to tell you something else again related to this perhaps this for some is more attractive or more intuitive okay I show you a different way of get arriving at the same result and it's a way that is perhaps even more important if you do a time dependent analysis where you really work with an equation more than with a with an energy okay so the and this well so the point is right I mean this is a two by two matrix and we know how to take the exponential of a two by two matrix I mean we have to to compute but it's nothing impossible and we also know how to take inverses of two by two matrices so if we really sit down and do this computation then what we find just for the for the one two entry is that well I guess I should put it up here this is that the one two entry that's alpha head of Xi on this side is equal to and now let me do the find this thing here so that's minus delta hat of Xi over two times tanh Xi square minus mu squared plus delta hat of Xi divided by 2t everything divided by Xi square minus mu squared plus delta hat of Xi squared okay so that's the Euler equation I mean or at least part of the Euler equation rewritten okay now except for a moment that the the delta is small right that's what we want to show we want to show that somehow if the temperature is close to the critical then the the superconductivity disappears so alpha is small so delta is also small so to leading order we can ignore here the the the delta term and what do we get we get exactly this tanh of Xi square minus mu divided by Xi square minus mu which is exactly our old kt operator 1 over 8 okay so to leading order this equation to leading order on the left side there's nothing to say about leading order but on the right side I have minus delta let me write this v alpha hat of Xi there's the two's the cancel each other and then there is a well tanh or 2tc divided by Xi square minus mu and that's it right but now if you look at this equation that's exactly alpha is equal so minus so what do I have I have v alpha I have 1 over ktc yes this the kt was in Fourier space to multiply by 1 over this so once again in this way I arrived that from this solution I arrived at this conclusion that a is in this kernel a denotes the leading order of alpha but now let's go on if this works okay let's expand this to next order well to next order there are two contributions the first contribution comes now I cannot neglect the delta hat squared in there okay so that gives me something so continue go on so I have zero on the left side and there I have I don't well perhaps I should not write everything here but there's something there's certainly a delta a delta hat from here and then I get a delta hat squared right times something which is some function of Xi plus but what is the other change that I must not forget that here I have temperature T whereas I'm interested in temperature TC so there's a change in temperature so therefore is plus so I differentiate this thing with respect to T and therefore I get a TC minus T divided by TC and this is multiplied by single delta hat and there again there's some function of of Xi so here this you see this is this cubic Schroding equation that you want to get of course when the spatially homogeneous case so there's no gradient yet but you see that this is so the delta is proportional to alpha alpha has this coefficient psi and that this is exactly the psi that that solves this equation you also see from this equation afterwards justified the delta hat squared has that order of magnitude which is h squared so therefore delta hat is equal to h order of magnitude in case you were wondering why did I put an h here why was this the right choice okay so that's on the equation level and if you want to do something time-dependent how you should think of this and how the the non-linearity arises so that let me say the non-linearity the power psi to power 3 comes from here from doing the expansion this is expanding this Fermi Dirac distribution and then the linear term in the equation that the psi term comes from the temperature difference and has the the smallness of the temperature in front good so that was what I wanted to say about this let me see are there any questions so far because now I think I want to move on and I want to really talk about the the real problem okay good so again there will be lots of definitions at the beginning and I hope that but they hope that the things are now more familiar to you after having seen it in this special case so this derivation of Ginsberg-Landau theory from BCS theory and I will follow here two papers together with Heinzel, Siringer and Soloway the first one appeared in Journal of the American Math Society in 2012 and the second one in Communications in Math Physics this year okay and you I hope you can find the details in there okay good so essentially the setup is similar we have a functional free energy functional that we want to minimize this functional sometimes will have an alpha which is zero and sometimes it's non-zero and we want to compute now a t for which this transition happens between superconductivity in normal state and then we want to say just like we did in this picture we want to zoom in at this point where t is close to tc and there we want to get an effective equation for how this alpha looks like okay so we've seen here that the upshot of this discussion was to leading order the the alpha looks like a solution of this equation right and now but now this was the in Fourier space and now we also want to take spatial variations into account and that's what this paper does so I have to so let's talk about the admissible states what's more complicated here now is that the the states of which we are optimizing our operator operators gamma capital gamma and these are operators on L2 Rd plus L2 Rd okay so you can think of it as a two by two matrix where there are four operators such that well what do we want well they should be self-adjoint and not only that they should lie between zero and one in the sense of operators right if you test against some element some normalized element then you should get a number which lies between zero and one now the next assumption is that which looks a little bit complicated so this is u gamma u star is equal to one minus gamma bar okay now I have to explain you what these things are where you zero one minus one zero so that's some of flips in the matrix around a bit and gamma bar is equal gamma surrounded on the left and on the right by complex conjugation now I know this a little bit strange in this theory but we need complex conjugation the reason if you want to really understand that the basics from this is usually we identify Hilbert space with its dual right that's his research theorem however sometimes gets lost is that this is actually anti-unitary right so what really happens is this operator does not in L act in L2 plus L2 but really it acts in L2 plus the dual of L2 okay and so this dual of L2 that gives you somehow complex conjugation now I just it's just confusing if we think of this elements as conjugate complex conjugates of L2 elements so therefore I move over the conjugation to the operators just ignore that point this is really not important it's a formalism that Bogolubov came up with this it nice it's nice it works perfectly but there is no no complicated mathematical content in it so let's let's just ignore this if you really want to read more there's this review of Heinz-Lenz Saringham and some older paper of Soloway and for practical purposes what this means is that we have a gamma up there there's an alpha and there is alpha okay let me say a bar and there's a gamma bar okay where alpha bar is equal to alpha star and yeah that's what we get so there are a couple more assumptions the next assumption is again related to something we cannot do but would like to do so we assume gamma is cd periodic okay which means that if torque a denotes translation then this gamma come commutes with these translations what this really means in practice is that we cannot deal with the boundary that's unfortunate because the boundary is really important in Ginsberg-Lander theory and there are some effects especially on the on at the onset of superconductivity where boundary conditions matter and it is somehow understood by the physicists that what boundary conditions you should get and particularly interesting is if the electrons here satisfied directly boundary conditions then the Ginsberg-Lander theory should have Neumann boundary conditions okay so that's rather you can already I mean from this you can already see that there's some complicated mathematics going on that we do not know at the moment so we ignore this issue by just saying well let's just talk about periodic operators everything takes place on the torus if you want and then there's no boundary and that's I mean that tells us what the bulk superconductivity looks like so and then the other thing is just technical condition and technical I mean just because we do this variational problems you want to have a well-defined energy so we want to have that this thing has finite kinetic energy it's an H1 condition the thing that you should ask me at this point well this is a periodic operator this is a periodic operator how can a periodic operator be trace class well it can't so this thing here really is the trace per unit volume it's just like I mean you cannot compute an L2 norm of a periodic function so what you do is you just compute it over period there's some some things going on here especially when you want to prove bounds but let's let's ignore that for the moment okay so this is the setup these are the operators now I keep saying that last time we talked about the the spatially homogeneous case what does this now mean so that's this important example so let I give them till this be as last time so I also defined admissible states last time yesterday okay and so these are functions now in on R3 there are some because of the these funny conditions conjugation conditions I want to assume that they are reflection symmetric then the important thing is and I define this operator capital gamma in terms of a kernel and it's just convolution operator right of x minus y and I insert another parameter h in there whose importance will be clear later alpha tilde x minus y over h alpha tilde say y minus x over h bar and then here I want to have one so this is kernel delta I guess I have to take this out don't worry too much about these formulas minus gamma tilde x minus y over h bar okay I just define it as a con the old operator as a convolution kernel okay which I can in addition scale then this is admissible in this sense in the sense in the new sense so now we need a functional so we have the same microscopic data as before the same assumption on the micro data as before microscopic data there was the chemical potential mu the temperature t and the interaction potential V and there were some LP conditions and now there are some new things some macroscopic data okay and now these are a that's a magnetic vector potential it's assumed to be periodic and there's an electric potential again periodic and we need slight regularity slight regularity conditions namely that the Fourier coefficients are summable okay so that's the function will be in particular continuous and there's a a hat and there has to be even some some control on the derivatives so they will be c1 it's I think I want to say that these are still rather weak assumptions and I'll explain that in in a minute but I want to distinguish these two this is what we've seen this is what the electrons see this is what you do in your lab right you have your lab here's your superconducting sample and now you turn on a magnetic field which really lives on your laboratory scale on electric potential so they these two things live on different scales and then the energy functional becomes the following so there is a I put in front I put an H to the D fact in front just to get rid of lots of H to the minus D's afterwards as a trace minus I H nabla plus H a squared plus H squared w minus mu gamma okay that's similarly as before that's a term that involves gamma alone then there's a coupling term which is just a trace of gamma log gamma and then finally there's a term which involves the alpha alone it's an integral over our D times t3 TD and it's scaled x minus y over H alpha of x comma y squared dx dy okay and now before we look at this in more detail let me prep say that this thing reduces to the old thing if you do this thing for the states and if you don't have any external fields that's a computation which somehow tells you more or less what this trace per unit volume is so compare if so if gamma arises from gamma tilde alpha tilde as above and if a and w are identically 0 then f t of gamma is equal to there's a unimportant factor of 2 pi to the D in front and now this is f t I should emphasize this is the old functional of gamma tilde alpha tilde yes that's a computation and and I should say that I mean this is the old functional but let me stress that it's independent of H so it's really just psi square minus mu there is no H in front and there is just a V and not a V of x divided by H and now of course I have to tell you why but let me yeah alpha is the integral sorry I should say it's the integral kernel yes that is okay alpha is a periodic operator on RD wherever did I write it here okay so therefore if you think how does an periodic operator on RD look like well you can I mean what happens here if I shift the idiot it means it means that alpha of x plus k y plus k is equal to alpha of x comma k if you move both at the same time so therefore if I would integrate here with the rest I would it would become infinity but this piece I have to integrate yes I mean what this really stands for is I mean it there is a limit taken but since it's a periodic function you restrict yourself to some kind of a fundamental cell where where this thing is is not true I mean just I really think it's a good perhaps I should even do it I mean if we do it in this case right if we plug in a convolution operator if there's a x minus y see what's the good thing about this we can change variables instead of integrating dx dy we can integrate dx plus y perhaps over 2 and dx minus y so if we do the dx minus y integral we get exactly what we had before and the dx plus y over 2 that just lives between zero and one okay I think that's perhaps the better way of thinking about this okay and right you you're right I mean what I'm always doing here is if I have trace class operators I just write the operator of x comma y to denote its integral kernel okay so this of x comma y that's the integral kernel of the operation these traces are in this unit volume sense so in particular I mean this what you might this is here again this in the purposes is not so trivial this what I said if you have the trace of of a convolution operator okay then this means that you integrate the Fourier transform okay that's somehow that you see somehow if you do right if there's no w is no a so this is everything is is diagonal in Fourier space and so this trace is just the integral that we had here on the on the board up there okay any other questions I know it's it's a lot of stuff a lot of material some are needs it's not so easy to digest at the beginning I want what I want to stress is somehow here see the things that break the translation variants are the a and the w and they are small they have h's and h squares in front so therefore to leading order we are still in this old regime and therefore we need to first understand the translation variant case which we hopefully have done at least to some extent and then we see now how do these spatial variations that are introduced change the results that we have so there are two parameters now one parameter is this H and the other parameter is the how far we are away from the critical temperature and eventually we will couple them and look at the joint limit and that's how we will get kinsburg-lander theory let me explain you how the H is why the H is arise there okay so there are two points of use that you can take can either view it from the point of view of a particle those ordinary particles that you want to model or from the point of view of the lab so this is from the point of view of the particle that's what the particle sees well the particle has a kinetic energy minor which is given by minus a laplacian okay and there's an interaction potential how does it interact with its neighbors v of x right so here's the particle and somehow this v has some LP condition so it goes to zero at infinity so I just see some finite range which has size one okay but now I have my lab and if the particle has size one then the lab has a huge size and so I call the size of the lab H to the minus one which I just give you the letter right H is from your small letter so I call one over H is a large letter okay so that that's a big thing and on this big scale that's where the lab is there I apply my external magnetic field a and my my w okay so these are so the external fields I still have to tell you how strong they are so let me denote the strength of the fields by g let's now let's just call talk about the scales where they live so this is H of x right it varies only on the lab scale and the other thing that we have is the magnetic field that's the curl of a that also varies on the labs scale now what you also can do and what's perhaps a little bit more natural when you really want to think from your lab perspective well this is your lab your lab has size one right and now there's something tiny tiny little thing going on with the electrons now you say the scale where the electrons live that has size H which is the same thing I'm just denoting scales differently but I have to be careful what the what the how these quantities scale so the kinetic energy now has become right I'm rescaling space so the kinetic energy has become minus h squared laplacian and the interaction has become v of x over h this is exactly what we see over here right if you ignore the external a then there's a h squared in front of the laplacian and there's a one over h inside the interaction potential now the how about the external fields well they are just g w of x and g b of x now one thing that we should remember is right what I want to address now is both here but g in front the same strength whereas over here it seems like the a has an h and the w has an h squared that comes from the fact that what is the magnetic field it's the the derivative of the I mean the curl of the the vector potential but now this curl here is computed on these big scales so therefore I lose an h so this g over h curl of a of x okay if I compute the curl here in my macroscopic coordinates so I hope this motivates at least I mean I haven't told you why g is equal to h squared why choose these particular strengths but at least I hope it it I told you where this h suddenly comes from it's not a blunt constant it's if you want it's an effective blunt constant but it's really this ratio between the lap size and the the the atom or the electronic size okay and eventually this h will be a small parameter okay I also explained where this comes from so now I should tell you a little bit how the why I want to have it h squared this by the way is stuff that that is not in the physics literature I mean it's even very far from being there this all that that it took us years to to figure this out that these are the correct scales now I mean it looks obvious but just just saying okay so what we've seen here the energy change so change in free energy that was tc minus t squared right that's here from this picture that's how if we are a little bit below the critical temperature by how much the the energy changes now on the other hand we have to edit these external fields okay they over there we we said they they have strength g now the external fields they couple to alpha they I mean they also couple to gamma of course that's how it looks like here but the the fact that they couple to gamma that's subtracted because I only compute relative to the normal state so really how they affect the system is how they couple to alpha and so therefore what I get is so the external fields there so I called the influence g or let me okay let me write it immediately as h but then they couple to alpha and really they couple to alpha squared because that's the if you look at this a gamma is like alpha squared so they have a tc minus t over tc okay that's their strength that's the size with which they're multiplied now in order to get something non-trivial so non-trivial regime is when tc minus t over tc is equal to h squared if you did not follow that derivation or did not believe me that's okay I mean you can just take this as a definition I'm just trying to motivate why we're balancing two effects and that's there is a certain regime the temperature has to be close and the the the strength of the external fields and now well I still have time so let me continue here let's look at this let's study this thing a little bit and let's do the same thing as what we did last time right the first step that we have to do is we have to know our enemy we have to find what is the normal state right now that's what we have to to compute so we take alpha equal to zero here this becomes a diagonal matrix you see and so then we can compute and minimize explicitly and it's more or less the same lemma as what we had last time so what's the normal state what we have to compute is the minimum now just over gamma of the trace minus ih nabla plus h a squared plus h squared w minus mu gamma plus t times the trace trace of the log gamma log gamma plus one minus gamma log one minus gamma okay I can since I can move the complex conjugation out of the the trace because it's anti-unitary okay and so the analog of this lemma that you believed me yesterday and so I hope you believe me today again is let h be a lower semi-bounded operator and let's fix some temperature t or what we want to do is last time yesterday remember we minimized over numbers rho now we minimize our operators okay same thing so we're computing a Legendre transform and what we have to prove is that in fact the rho actually commutes with the h and therefore the whole problem becomes commutative and you get the same answer as yesterday except that you have to put the the trace I mean sum over the individual eigenvalues if you understand the proof for for matrices then you also understand it for operators there's no problem with infinite dimensions of stuff so and the inf is attained if and only if rho is this Fermi Dirac distribution which is one plus e to the minus plus h over t inverse okay so one has to do something but once one has this one concludes so here for us in this minimization problem gamma is this thing with this one body operator and therefore the capital gamma we know if we know what the little gamma is we also know what the capital gamma is and I call it gamma zero so that is one plus e to the h over t inverse one plus e to the minus h bar over t inverse there are zeros I'll tell you in a second what h bar is and this is the same as one plus e to the minus I'm saying plus one over t h minus h bar inverse where h script so script h is this minus i h nabla plus h a squared plus h squared w minus mu okay it's I'm just applying this here this this h is the script h and so I computed this entry and then because of this formula I know that if I have this entry then I get the entry down there by taking one minus the complex conjugate and therefore I from this I get that entry and then I do a little computation and see that it's actually that I can also write it like this okay and perhaps you already smell what will be coming namely when we have the external fields there will be some off diagonal entries okay and the off diagonal entries will be small but they will no longer just I mean this is no longer translation variant operator so therefore these off diagonal entries will no longer be translation variant right so but the the the the translation variance is broken only by very small amount so some are up to commutators we are this and if we compute the commutators to high enough order then we will get the gradient psi against burglar theory let me make was there a question this is script h sorry oh this is complex conjugation just in this sense in this sense that if I write exactly so what this means is the magnetic field changes sign okay right so there is in this Hamiltonian we let me put here plus and no sorry let me put the i over here and then I have to make a plus here I think that's right then you see this is a real operator this doesn't do anything but this when you apply to a complex conjugated wave function then it changes sign and similar since w is real nothing happens there okay so this is this because we're really working in the the dual of l2 and not in in l2 itself okay what I want to notice however is that this operator is not bounded from below this whole operator right I mean this operator is bounded from below but in the lower entry of an operator which is bounded above but not below it's a little bit like a dirac operator but it's one has to be later on when we do semi-classics one has to be rather careful about the structure and what what one does there and one more thing is so if I I can also compute the free energy now of the normal state right here it is minus t trace of the logarithm of 1 plus e to the minus this operator and you see somehow to leading order when I ignore these things then it's equal to 0 and I put remember I put the h to the d there so this is equal to that's sometimes called by law which roughly says that you can replace the the trace by the integral over over phase space and so you have a logarithm 1 plus e to the minus xi square minus mu over t d xi over 2 pi to the d okay plus little of 1 that's called by asymptotics we'll talk more about those long now what people can do is they can expand this further okay in particular there is since the the sub-principle symbol here has some some cancellation property there is no term of order h and then this term really would be of order h squared if the a and the w were nice enough so that I could do semi-classics however these assumptions are weaker than that so the way I will prove my theorem is not by computing the energy the minimizer to some large precision and then computing the energy of the normal state to very large precision and then subtract it from each other but I rather will compute the difference of both okay this allows me to to work under these more modest regularity conditions and it's also very important somehow when you deal with I mean we really as I said we go to fifth order and you really have to to try to minimize your your the amount you have to work let me draw a picture to finish this lecture this thing that's exactly the energy that we had last time it's exactly the same thing we also see seed from this formula on the third line up there okay we turned on our a and the w what happened was that this thing somehow got smeared out I mean not smeared out it's there's definitely change of order h so this thing here this width here this is order h squared this is this amount by which the the free energy changes as h changes and this was our old critical temperature and the preview of what I will prove is that this thing here will also change to order h squared okay and somewhere so there's a region where we a small region where we don't really know what to do and then this thing the the energy will bifurcate here I'm superconducting here I'm normal and in there this is where I will find kinsburg-landau okay once again it's kind of a slope as it goes down there but everything is somehow has these yes too high order in h so that's the idea I think I mean I now how should we do this I can explain some things about the quantum mechanics but perhaps I do this just for people who are interested and I let everybody else go to go to lunch and afterwards we we talk how this really come that comes about is that fine it's I mean I can say a few words now let's do it like this I say a few words five minutes okay right pips even less and then I explain more if people want to hear more okay so there are several layers I guess to to quantum mechanics are to these effective models the first layer is Schrodinger theory okay this is a linear theory where you're the states of which you have to optimize a wave function of the number of variables that you have okay so these are huge objects but they are linear the second level is that of density matrices okay that's what we have here the objects over which you optimize they are operators but the good thing is that these operators are just operators on R3 or something okay so you got rid of lots of variables but what you lost is some of this commutativity I mean you you have well or I mean it's operators now that you have to deal with and then there are the simple things like in Spraglander theory where you just have functions on R3 okay and so there are step these three blocks and we would like to go from the lowest to the highest from the most complicated to the simplest what I'm doing here I'm going from the second thing to the third thing what I don't understand as far as I understand nobody really understands it's how to go from the first thing to the second thing okay now that being said I should say nobody understands how to do this rigorously the way I mean what PCS did is they did an upper bound okay so they derived this PCS functional really by by let me explain you what they did so instead of multi minimizing of all wave functions psi of I don't know n variables where n is a huge number you mean restrict yourself to a certain small class and you just minimize over those so that obviously gives you a higher energy and what you would like is that in some asymptotic regime actually they they are the same right but that's what nobody can prove so they came up PCS came up with a certain class of states over which when I optimize over those then I get this functional so these are states that are characterized by this gamma and this alpha okay these are so-called a quasi-free states now their idea is to explain the idea one has to go to fox space which sounds complicated but is really really not so complicated fox space just means that you take block matrices okay so you have the one body theory in one block and you have the two body theory in a second block the three body theory in a third block and so on you have this big you put the matrices together and now you would say well what's the the lowest point in spectrum of block matrix well it's the lowest point in every entry of this every block right so however thinking of this as a block matrix allows you to to take more functions right even I mean you would when you compute the lowest eigenvalue of block matrix you would just take a trial function which lives in one block however sometimes purely computationally it's simpler to take something which is lives in all the blocks a thing that you might have heard of are these coherent states so coherent states that's kind of a random distribution of all these blocks but they are somehow strongly localized at one certain block so that's what people for instance did in the derivation of these cross beta-yevsky non-linear shading equation they studied the evolution on on coherent states but then they showed that really somehow if you start so you have first a real state which lives in a fixed block then you extend it somehow to to all the all the blocks but then it stays kind of at the block where you really wanted to stay okay but somehow because you look at the evolution in this much bigger picture it's it's nice so you have better computational tools now the idea what in coherent states it's still true so you you want to do something in one block but you occupy all the other blocks but still you only do it in a diagonal way the idea of budding cooper and trefer was to not preserve the particle number okay so the trial state somehow mix the the different block i mean well not mix but really go go away from the diagonal okay which sounds i mean think i mean right think just of a block matrix a two by two block matrix try to minimize this so you would take a state that also occupies the other things the other entries okay it's there is no way i mean it sounds not clear to me why it works but somehow you made such a big step when at the very beginning you said you optimize only over a certain class of states that perhaps you lost so much there but therefore you still have wiggle room somehow to make another mistake in the other direction such that such that you at the end the result is correct so at the moment i think this is really just a computational tool it's one that works i mean they got the Nobel prize and everything for this i mean that's they no doubt that it works it's just we don't know any rigorous regime where it can be can be derived and where the true answer comes from this so the upshot they take quasi-free states quasi-free states depend on two matrices alpha and a gamma gamma is the ordinary thing the the particle number preserving thing that people in Hartree-Fock did for ages i mean since the the 20s 1920s the new idea is somehow to have this non-particle number preserving part that's what's encoded in the alpha and you can occupy this or you cannot occupy this okay and well if it's occupied then they call it cooper where cooper pair wave function it's a pairing mechanism and well and it works i guess that's what i want to say i explain it to in more details if somebody asks thanks that's good so next time you are going to to make us understand this small piece here yes exactly so i mean the idea coming back to what i just said having these levels mathematics these theories on different levels this always boils down to semi-classical analysis essentially where where there's an effective blank constant and one has to do something there and one can understand this Hawkins-Burglander theorizes very easily in terms of a wild calculus okay now the problem that we will have and that i will address next time is that in order for the wild calculus you need a lot of smoothness so if you compose two operators wild calculus tells you how they are the composition what the symbol of the decomposition is but that's very hard i mean you need more derivatives than you actually have to compute this and we do a variational theory for us the psi is only in h1 so we have to do a wild calculus really at the level of regularity that we have we don't have more we cannot spend more than we have and so therefore we have to do a lot ourselves and then re-proof these things and do some kind of pseudo differential calculus use this specific structure that's the preview okay thanks