 Moj način, svoj način. So, after this long introduction, I hope to have convinced you. So, after this long introduction, I hope to have convinced you that is interesting to try to understand the physics of system having many local minima in the free energy or energy landscape. This is due to frustration and to disorder that induced frustration. And so, the first thing that we are going to do now is to restrict our attention to mean field model, in particular to fully connected models, and try to define properly what is a state. So, let me say, one minimum of this typical drawing I do. So, I want to understand how can I define these points, which are, let me say, the minima of the free energy in some space of, say, magnetizations or parameters, mean field parameters that describe the system. And I want to understand how can I define and how can I count these points. And in turn, then I will try to understand what is the physics of system of complex system, by complex system in a system having many states. There is already one case that you know very well, because Matteo spent the first two lectures on this model and we already know one system having two minima, which is the Curivize model for temporal smaller than the critical temperature, where you do have, if you plot the free energy as a function of the global magnetization, you do have something like this. So, you have two local minima. So, the one could say, okay, since we know how to define this point in the Curivize model, can we generalize this tool to a situation where I have a model with many, many minima in the free energy function. And the problem is that in a Curivize model we know how to select one of these two states by just applying an external field. So, essentially we put an external field and we choose one of the states. So, the first problem we have that in general in a complex model like this, we don't know what is the direction of this local minima, so we don't know the field I should put in order to select that direction. So, in the Curivize model everything is very simple, here is much more complicated. And moreover, even in the Curivize model you have to be careful, and it depends how you do the computation, because if you just apply an external magnetic field, you should have seen in a material lecture that actually if you apply a magnetic field and you compute the free energy in a magnetic field and then you do the Legendransform to obtain the free energy as a function of the magnetization, you don't get this. You get the convex envelope of this free energy. Actually, you get this. Why? Just to remind you, if you compute the free energy in presence of an external field that I call B just to keep the same notation of Francesco Zamponi notes, so you can then look there the computation. So, you compute the free energy as the sum over all the spins of exponential minus beta h of the spins plus this term beta sum over i si bi. So, you see what I did. I apply an external field. So, this is the original Hamiltonian. I put on top of the Hamiltonian, which eventually already contains a field. I don't care about what is in the Hamiltonian. But I put on top of this a new field, I call B, bi, in order to select one state. In the case of the Curivize model, I can just put a uniform field because the direction of the minima is along all spins aligned so the fields can be the same. And by doing this summation, I'm computing a free energy in presence of a field and I make this dependence explicit. Obviously, the free energy also depends on the inverse temperature of beta, but I don't put the explicit dependence on beta. And from this expression, it is easy to obtain that the mean value of the spin in presence of this external field. So, this is the mean value of a psi when the average is done on this measure here. You can obtain just by taking the derivative of this free energy as a function of bi. You see, if you take the, there is a factor, maybe I shouldn't put the beta there. Ah, no, yes, there is a beta here. OK, everything is fine. So, you see that if you want to fix a given magnetizacional spin s, you have to choose the right external field b such that the derivative of f in respect to bi is exactly the magnetizacional that you want to enforce. And this f of b is concave. So, concave is means downward curvature because if you take the second derivative, the second derivative of f with respect to bi and with respect to bj, you see, if you take the second derivative, you are computing bmi over dbj and this is the susceptibility. The susceptibility is the derivative of the magnetizacional with respect to the external field. And in the canonical ensemble, I hope you have seen this, this equivalent to minus beta, the connected correlation between spin si and spin sj. If you have not seen it, it is an exercise for you. If you have not seen it, you have an exercise which is in the canonical ensemble showed the derivative of the mean value of si in respect to bj is equal to minus beta si minus mi sj minus mj. Now, this is a connected correlation which is strictly positive, this matrix is positive defined and so the second derivative of f because of this sign is negative defined and so f of b is a concave function in the space of the n-dimensional space of fields. This means that now if you take the Legendre transform, if you take the Legendre transform in order to get back an expression of the free energy in terms of magnetization because we are looking for this free energy as a function of the magnetization in order to locate all the minimum of the free energy as a function of the magnetization. So you do the Legendre transform which amounts a computing keeping the notation of Zamponi lecture notes. This is defined as minus beta the maximum over b of f of b plus the sum over i bi mi. You see f of b is concave. So this is concave. You add a linear term, a concave function plus a linear term is still concave. So you take the maximum, so the maximum is unique and you compute this free energy gamma as a function of m. However, because of the properties of Legendre transformation, if you Legendre transform a concave function, so a function which has a second derivative which is always of the same sign, in this way you get a convex function. So a function which always has the second derivative of the same sign. So this function is convex and it's the one I plot here by the dash curve. So this means that even in the Curivize model if we try to select states that we know they exist because by doing this micro canonical computation at fixed magnetization I got a function with two minima. So the minima are there. But if I do a canonical computation in presence of a field, then I remove the field in order to keep the magnetization because obviously now these fields are used to fix the magnetization here. What I get is a convex function. So I can fix any magnetization but the function will be convex. So I don't get the function with two minima but just the convex envelope which is this one. Now in the Curivize model I can still understand something because I can look for the first point where essentially this becomes flat and so I can argue that I can still identify the, let me say, m0 and minus m0 the position. But this because it's the Curivize model and everything is very simple. If you try to do in this landscape essentially the gamma function is providing you no information. So we have to look for a different root in order to end up with this point. And this is what I call the top states. I could leave it because... OK, so let me... Sorry. So let me explain to you why we should be able to do some kind of computation of those minima. And so let me discuss a little bit more what I mean by states. OK? Because at the moment states are the minimum of free energy function that we are still unable to compute. So it's not a good definition, the minimum of function that we are not able to compute. So let's try to have a better definition of states. So a general results from mathematics say that any Gibbs measure in finite dimension, in infinite dimension is more difficult than in pure states. What I mean by this you take a probability measure over n variables, our spins, but it can be very kind. You can always write as the sum this is approximated but in the thermodynamic limit is really equal to a sum over states with a given weight and this probability distribution p alpha is restricted to a state. Just to be concrete to no, let me give you a more precise definition then we do an example. So what is the property of this probability distribution within a state? Here is a key property that correlation decays. This is called the clustering property in physics and in mathematics you say that this probability distribution is almost factorized. In fully connected models is almost factorizable. So in some sense is it to approximate. The clustering property essentially says that all correlations decay very fast. So if you take the average over this distribution so let me call this sum over spins over this distribution within one state alpha. On this measure on this probability distribution if you compute any connected correlation function it decays very fast when the spins are far apart. This infinite dimension very well defined pure states. So essentially this means that if you compute the connected correlation function within one state this goes to zero when the distance between i and j is very large. This is the clustering property. Why this helps you because essentially it says that within one state the probability distribution if you don't have a very strong correlation this joint probability of n variables can be well approximated. This is a general rule if you have a joint probability of n variables and these are strongly correlated there is no way you approximate it. But if they are weakly correlated eventually if they are independent and so the probability distribution is a product probability distribution this much much better. And so this clustering property says variables are weakly correlated many are correlated up to a given distance but no more. Now if you take this property in a fully connected model there is no notion of distance all variables are at a distance one and indeed in fully connected models this property amounts at say that the probability distribution of n variables essentially is the factorization of all the sites of single site probability distributions you have no notion of distance so if the correlation must decay they must decay immediately at distance one they already must have decay and so only if the product only if the measure is a product measure so it is a factorized measure this connected correlation is zero ok so in this product measure any s i s j is equal to s i s j they are factorized now since we are using easy variables you can also write this single site marginals in a simple form so I can rewrite this as the product over i always from 1 to n of 1 plus m i alpha s i divided by 2 ok so you see that essentially what we are saying is that in general we can they compose the joint probability distribution of our variables in this way and in particular in fully connected models on this factorized probability distribution which depends only on the magnetizations so in practice what we are saying is that under this hypothesis our joint probability distribution of spins can be well described by a set of numbers which are the weights that enters here and the magnetization that enters in each state ok so this is our definition of states ok so these are our states so essentially a state is one set of magnetization that tells us in which direction the probability measure is strongly concentrated in Q device you just have two spots but in complex system you have many spots and you have to identify all these directions so in particular this description exam Q device this description so the probability distribution of Q device can be well described by two numbers which is one half if you are a zero external field otherwise one of the two minima will dominate and the magnetization so this is the in the case of Q device where obviously M naught solves the self consistency equation that you derive for Q device ok so now our aim is to try to identify this magnetization and in a full analogy of what you did in Q device when you identify this magnetization by solving a self consistency equation we want to derive self consistency equations for this more complicated variables because now we are in the presence of disorder so the disorder will induce states in in the action which are uncorrelated and and more complicated to identify so let's try to write the self consistency equation of this kind but for the general case and let me use this so I'm using this model which is the usual one but I just want to rewrite before starting the computation and you can also add an external field if you like there is no problem ok this is the Hamiltonian we are using now let's try to write the self consistency equation for this magnetization so let's do the following I consider a a spin I and all the spins he's attached to they can be all some in fully connected model they will be all ok and so to which spin J is connected to a coupling J at J and then eventually there is also an external field here so the same naiv mean field approximation that leads to this equation can be written in the following form you you just consider a factorized measure for for these spins so you say if all these spins have a factorized measure so the probability of all these spins is a product of m J s J ok then when you write down the equation for m I what you have m I which is the average value of s I which is the average value so you are considering this subsystem and obviously there is all the the rest of the system which produce the correlation between these variables and so you have when you compute the magnetization of this variable you have to consider the effect of all its neighbors and the explicit computation provides you hyperbolic tangent of beta the sum over J J a J s J plus beta h I this is an exact equation because I am just rewriting the spin variable in terms of its neighbors now is that these still these are eventually correlated but if you take the naivni in field approximation which means I take all the neighbors completely uncorrelated you can rewrite this average value now you just have to replace s with m if you don't believe that this is the right thing to do you take this you expand in power teror series you take the average of each term and then you see that you substitute s with m and you re-sum the power series teror series and you get this expression with m instead of s and so this just the hyperbolic tangent of beta sum over J J a J m J plus beta h I ok this is exactly the naivni in field equation that you derive for the coolivized model but now magnetization can be different in any point so this equation is fine for the coolivized model and is fine also for ferromagnetic models where the couplings are different so you see now the couplings are not all the same also the external fields so this equation which we call naiv in field equation are fine for ferromagnetic models non-frustrated models unfortunately for spring glasses are not and now I will try to convince you that you need to add one more term here which is called the on-sugger reaction term in order to get a sensible result but first let me try to convince you of where is the problem the problem is that these spins are not perfectly uncorrelated ok so this factor is not really exact ok why well first of all because they are connected to this spin ok so this is the main source of correlation among these spins it's true that they are also connected to other spins but I will try to convince you that the correlation induced by the presence of this spin when you propagate on this are zero essentially if you give a value to this spin they induce some correlation here and when you take this correlation you propagate through some couplings which are uncorrelated from this they are random and different the net effect is zero but is not zero for the correlation induced by these spins so this spin induce some weak, very weak correlation in these neighbors and when they use these neighbors to obtain back the field in this way on this spin these correlation are relevant so we have to do the computation more carefully and so less split less compute the magnetization of this spin but splitting the local field that they felt in two terms one term arriving from i and one term arriving from the rest of the world the rest of the spin including these other guys ok so we can let me write so more explicitly this system in this way so this i these spins are I call it j this j a j and now this field here I call it let me call it hj cavity is defined as the local field felt by j by sigma j sj when i is not present ok this is why I call it cavity because you do a hole ok so essentially hj cavity is the field felt by sj when all this part of the system is absent ok and now we want to solve this very simple system the problem is that in principle we don't have access to h cavity we have to compute it we don't have access to the magnetization of j in the full system so the first thing to do is to estimate h cavity and how to do it well the full field felt by sj is the sum of h cavity plus the term coming from here so the magnetization on spin j is equal to the hyperbolic tangent of beta and here you have two terms j cavity which is what you get from the rest of the system plus j i j s i ok so what you get from here and actually since let's put mi ok notice that this term is very small this term is order 1 while j is order 1 over square root of n ok so this is a tiny correction and what we did when we derived the naivni field equation essentially is to to this regard about this term and say the field this cavity field is essentially the same in the full system or without this spin and if you are in a customized model where j is 1 over n approximation is fine but here j is 1 over square root of n so it is going to 0 much much slower ok so from this you can expand and so this becomes essentially hyperbolic tangent of beta hj cavity plus you do the first derivative and you get beta 1 minus hyperbolic tangent square of beta hj cavity times jajj mi and this part here this term you can rewrite it also as since we are already in the correction you can substitute to hyperbolic tangent of beta hj cavity which is this term you can substitute mj because these are equal at leading order so in the correction you can put the leading order 1 minus mj square and you recognize that this is nothing but the diagonal element of the susceptibility matrix ok so this beta sj sj minus sj sj this one and this mj square so this is the spin j when you apply a field on j is the local response so if you perturb a little bit that spin this how much it responds ok and you see you are perturbing that spin with mi times j which is a small quantity so this is a small perturbation and so you can understand that beta hj cavity is equal to mj which is the magnetization in the full system minus this term here so minus beta 1 minus mj square jj mi so what we understood is that if we keep the first correction the cavity magnetization which is this object is different from the full magnetization because there is a term coming from the from the height spin and now we can solve this system which is very easy to solve because we have few variables we have to some of the old variables so I leave you this as an exercise and if you solve this system and you compute the magnetization of these variables of this height variable what you get is mi is equal to hyperbolic tangent of beta hi plus beta the sum of the neighbors j jajj times the cavity magnetization because you see the cavity magnetization then you propagate and i is summing all the contribution coming from the neighbors but you have to take the cavity magnetization so here you have to put this cavity magnetization because you are solving really this system which is linear now and so you have to put this expression here mj minus beta 1 minus mj mj ok ok, so what we got while this were the Naimi in field equation now these are called the TAP equation TAP stands for Paulus under Anderson Palmer that derived this equation in 1977 ok you see the difference between these because here you have one more term so if you forget about this term which is what is called the onsager reaction term if you forget about the onsager reaction term you have exactly the Naimi in field equation now we are we are adding this one more term and I will try to convince you that these relevant for spin glasses useless for Q device so let's always here indeed let me rewrite it by just put in evidence the first and the second term so these TAP equation are hyperbolic tangent of beta h i last beta sum over j j i j mj which is the Naimi in field minus beta square sum over j j a j square 1 minus mj square times mi that you can put outside or whatever you want ok I just rewrite the expression so we notice two things that first of all this looks like a second actually term in a expansion in beta j you see this 0 order term beta j, first order term beta square j square moreover the importance of this term obviously depends on the variance of the j if you are in the Q device model where j are of order 1 over n and j square of order 1 over n square since the sum is over all your neighbors in a fully connected model all your neighbors are n this term is 1 over n so this is relevant compare to this term which is order 1 because it's n times 1 over n so this term is always order 1 because it's the local field it must be order 1 if we scale correctly the coupling but this not why in a spin glass case ok, I forget to tell you that the fully connected version of the spin glass model is called Sherrington-Kirpatrick model is what I know that Sherrington-Kirpatrick solve it first so in this Sherrington-Kirpatrick model in the fully connected spin glass model j square which are of order 1 over n and so this term which is n times 1 over n is of order 1 so you cannot disregard it ok so in this way we see that we have to strongly modify the new field equation in order to correctly describe the system and this is because j square so what is happening is what I was saying to you before i is inducing some correlation among these spins via the coupling j and that correlation is propagated back via the coupling j and so what you get at the end is j square sum over the neighbor so if the couplings are very weak correlation and giving it back is irrelevant if the couplings are 1 over square root of n you propagate a very weak correlation among all your neighbors then you can let it back and you feel yourself ok and so this is a very important point to understand that even a system a mean field system where correlation goes to zero because correlation goes to zero like 1 over square root of n and so you have to be careful and this is why you can build in this fully connected models very non-trivial correlation leading to many states so this equation will have much more solutions than this equations because of this on-sugger reaction term now the problem is that ok we are very happy we derive self-consistency equation for our magnetization we can find this what I will call from now on TAP tap states because are the states defined by the magnetization that solves the TAP equation maybe I am not the right one because in principle you can solve this equation even in a model that doesn't satisfy the hypothesis in which you derive it so you can take a three-dimensional model that try to solve this equation it will be useless you find maybe not very interesting thing but you can do and you are looking for the TAP states which are maybe not the right pure states of that model this is an approximation the problem is that ok, I would like to solve this equation but suppose that I do it I don't have any idea about the free energy these states corresponds to because I just wrote a self-consistency equation for magnetization but there is no free energy written on the blackboard and the weights of my states I expect it to be proportional to exponential of minus beta and the free energy of the states so if I am not able so let me say this is the computed on on the magnetization corresponding to that state so I need to compute a free energy which is still not on the blackboard so now what we will do next is to compute the free energy computing the exact free energy is obviously very difficult so what we are going to do is to compute a high temperature expansion to the free energy which has also the nice property to have many minima and each minima corresponds to this, the magnetization solved in this equation so we are going to build a high temperature expansion such that the second order of that high temperature expansion for the free energy do corresponds to a function which have the minima exactly on the magnetization solved in this equation this is not the true canonical free energy because the true canonical free energy is convex this is a high temperature expansion to the free energy which is somehow better for counting metastable states because the canonical free energy is always dominated by the low line free energy states but we are interested in counting them all even the metastable states of a higher free energy so just to make you the usual Q revise example if you take the Q revise in a field the free energy that you want to compute is this one because we are interested in the two minima but if you do canonical computation what you get is this one with just one minima you lose the other metastable state but you are interested in that because if the metastable state becomes many many you cannot lose them because if you lose them then you don't understand what the dynamics starting from here do because if you let the system relax from here it reaches the metastable states and to leave the metastable states it takes a very large time if you are interested in understanding what happened on human time scales because we want to see what happened in hours not in the life of the universe you need to count these states and so doing the standard canonical computation you will lose it and you get only this so it's much better to have an approximated free energy which has new minima hopefully this new minima we hope they are meaningful and solving the TAP equation is already a good point to start being meaningful at least in the fully connected model and even if is still before noon I think we can stop now we do 5 minutes stop and then I will try to derive the high temporal free energy sponge ok, so an announcement Erika just told me that we are not going to do the picture after the lecture so we are going to do it next week because Matteo is not here now and since he will organize it will be not very nice to have the the picture without the organizer so you are free after the lecture ok, so let's concentrate for the 4 hour in a row but now I will I will mainly follow Francesco Zamponi lecture notes from page 12 so if you miss some computation you will find it written in his lecture notes so so our aim now is always for the usual Hamiltonian I have already written several times now I want to find a high temperature expansion to the free energy high temperature or weak coupling expansion is the same, essentially you want to have beta j very small ok and so this holds both for mean field models where j is smaller you have to just keep some terms it holds a high temperature where beta is smaller and so either high temperature or weak coupling with expansion so in the notes the notation is that of paper by George Giedidia in 91 but actually this expansion was proposed by Plefka in 82 but the notation that we are going to use is that because Plefka essentially did the expansion up to the second order and so in order to go on George Giedidia did to the fourth order if I remember correctly they introduce a better notation that we are going to use the notation that we use is in the George Giedidia paper in 91 ok so we start from the free energy in presence of an external field so again we do the same trick as before we take the free energy we put an external field in order to force to have exactly the magnetization that we want on each spin and then we try to do the we try to understand how much is the free energy fixed in the magnetization so the weight the Gibbs weight among configuration that have that specified magnetization in order to simplify the rotation we introduce this function a which depends on the temperature but mainly on the magnetization and this equal to minus beta f of the magnetization so we are trying to expand this f at fixed magnetization so it is a high temperature expansion magnetization and then we introduce the lambda i which depends also on beta we write it explicitly which is defined as the inverse temple times the external field that we add in order to fix the magnetization so this with this new formalism this is the log of the sum of all the spins exponential of minus beta the Hamiltonian which depends on the spins plus the sum of i lambda i beta s i minus m i this is just you can take the definition of the of gamma because essentially we are expanding gamma but we want to obtain a result where gamma is not the convex function but thanks to the high temperature expansion we obtain a new expression which is non convex and here we have to fix this external field such that the magnetization is equal to m so such that the average over this measure s i minus m i is equal to zero so the average of s i is equal to m i I write this way because when in this way you realize that essentially lambda is related to the derivative of a dysfunction with respect to m i so from this you can realize that lambda i is equal to m i indeed if you take the derivative with respect to m i you get essentially this term down and so you you get lambda i beta so once you know the function a you can understand also what are the parameters of the magnetization that you want in order to do the computation faster you have to introduce an observable u which depends on the spins which is defined this way is the Hamiltonian which depends on the spins minus the average value overall on this measure so the average all the average values are carried on the measure in presence of the external fields minus the sum over i the derivative of lambda i of beta with respect to beta times s i minus m i ok, so this define why we introduce this observable because it has nice feature essentially is related to the derivative of any observable to beta which is what we need we want to span this in beta so obviously we have to take derivatives of beta and we want to make the computation as simple as possible this function u has the nice property which is the average value of u is 0 this is pretty obvious because the average value of h cancel the average value of h and the average value of s i minus m i is equal to 0 for any i and so the average value of u is 0 but what is more important for our purpose is that if you take the derivative of any average value or the average value of any observable with respect to beta what you get let me write here the expectation so is simple to show it so the expectation that we are using is this so sum over sigma exponential of minus beta h plus the sum of i lambda i s i minus m i divided by the sum of s so this is the the measure that we are using so when you take the derivative with respect to beta what happen when you take the derivative of the numerator minus h down so the first term is minus is minus o h and you also get this term the derivative of lambda in respect to beta so you have one more term which is plus the average value of o times the derivative of lambda are the derivative of the numerator and then when you take the derivative of the denominator what you get is the average value no and the derivative of the numerator is zero because what you get are the connected are the connected parts but no sorry you get the connected parts so you get the plus o h and the connected part of this is zero because the average value of s i minus m i is zero and you realize that all this expression and finally you have one more term which is the derivative of the observer with respect to beta so you have one more term which is this now all this you realize is minus o times u because you see o times h average of o times average of h and average of o times this term sorry, here there is one summation of a i i miss so what we understand is that the derivative of any observer with respect to beta is the average of the derivative of the observable minus the observable time u and this is why we introduce this observable u because it is the operator that you need to compute in order to take derivative with respect to beta as an example which will be useful if you compute the derivative of a fixed number which is the magnetization with respect to beta because the magnetization is a fixed number we are taking a high temporal expansion of fixed magnetization but the magnetization is also the average value of sigma i and so if you use the formula we just derived we can compute this with capital O being s i and so this turns out to be equal to minus s i minus m i times u ok sorry no no no so you are perfectly right I jump one so I first need to write s i times u now you realize that the average value of u is zero so I can add m i times the average value of u because this is zero so finally you get minus the average value of u times s i minus m i but since you started from zero you just realize that this particular average value so u when you multiply by s i minus m i is zero these are useful relation now that we do the expansion we are going to use to make the computation easier ok so now we can really start doing derivatives so we can for example take the first derivative of a with respect to beta and if you take I just erase this but ok this is essentially taking the derivative of the numerator so it is like the average value of minus h and the derivative with respect to beta you get minus h plus the sum over i of the derivative of lambda i of beta with respect to beta s i minus m i and thanks to the fact that the average value of s i minus m i is zero this is nothing but minus the average value of h here I am not at the end I will be interested in a Taylor expansion in beta equal zero but for the moment I am just taking derivative at any value of beta so this expression is valid at any value of beta then eventually I will compute each term in beta equal zero because I am interested in a high temporal expansion and if you compute the second derivative well this is nothing but the first derivative of the first derivative so you want to do the first derivative and for the first derivative of h you again use this expression and using this expression what you get is that the Hamiltonian does not depend explicitly on the temperature so this is zero you just get this and so you get the average value of h times u and the average value of h times u you realize that is equal to the following let me add zero zeros I can add one term which is the average value of h times the average value of u because the average value of u we know is zero minus I want to reconstruct this and then I add more zeros minus the sum of the i times the average value si minus mi which again is zero so I am adding zeros and what I get finally is that this can be written as u square because you take h, average value of h and on this terms you reconstruct u ok and so the second derivative of a let me write here is u square this is why they introduce this u operator which is very comfortable yes so you mean ok, because the now I am going that the average value of h because this average is made at any temperature and the average value of the energy does depend on the temperature no, sorry ok so when I take the derivative of the average value of h with respect to beta here you have the derivative of h with respect to beta which is zero because the observable h doesn't have beta inside so you just have this term yes so it's true the average value of h depends on beta by this expression but h itself doesn't depend on beta so the first term is zero and you have only the second term and then you have zero and you reconstruct u square and you can go on for example I can write for you the the third order term and ok I don't want to spend too much time in doing computation because I'm interested more in reaching the final result but if you do a similar computation you realize that the third order term is minus the average value of u to the cube ok so you see that all these derivatives are pretty simple expression and the only other things that you need so you remember that in u we also have derivatives of lambda so as long as you take the average of the lowest moment of u you don't need it you just need the derivative of lambda with respect to beta in beta equals zero but in principle if you want to go on with this expansion you will need to compute even higher order derivatives of lambda which is the external field in order to enforce the local magnetization with respect to beta square beta cube so in general starting from this expression that I didn't cancel in purpose you can do any n derivative of lambda beta with respect to beta n times but just taking minus the derivative with respect to mi of the derivative n derivative of a with respect to beta n times so while you compute all these terms by taking these terms and then in the river and arriving with respect to mi you get the derivatives of lambda but for our purpose essentially we just need the first one so the first one corresponds to take changing sign the derivative of h and we are interested in computing these derivatives in beta equals zero and so for example if you are interested in the first one so the lambda i beta in beta in beta equals zero is just the derivative with respect to mi of the Hamiltonian the average value of the Hamiltonian by zero I mean computed with beta equals zero and so this you will see is pretty simple because for beta equals zero you can compute easily the average value of the Hamiltonian because the measure is the one with beta equals zero so completely uncoupled spins I will show you and so this average value are simple so you can take the derivative very very very simply and you can also compute the second derivative by taking the derivative of u squared computed with beta equals zero but we just need the first one so these are essentially the terms of this expansion and now let's make a concrete example for this Hamiltonian and let's compute these terms just to see what are the first terms of this high temperature expansion so so let's compute for example the zero order term the zero order term so a zero of the magnetization this is the free energy of a non-interacting spins because beta enters in front of the Hamiltonian and so you have non-interacting spins you can compute in different way because this still depends on the derivative of the of this parameter that enters in the definition of it depends on this parameter that enters the definition of a naught so let's write it explicitly this is the log of the sum of all the spins of exponential of the sum of a i lambda i computed a beta equal zero si minus mi ok and you see that even if there is no interaction there is no Hamiltonian anymore the the magnetization are non-zero because we are forcing them to be non-zero so you have a set of non-interacting spins each one having its own magnetization because you are putting the right external field in two ways you can write the explicit expression for the external field that would make an isolated spin have magnetization mi which is very simple and is satisfied if you take the derivative you can compute for example si minus mi equal to zero taking the derivative and you realize that the magnetization is given in a verbolic tangent of the field which is exactly what you will expect or even more simply you can think of these as the the log of the partition function of n non-interacting spins each one having its own magnetization and you should already know the answer because this is the you only have the entropy term there is no interaction it is isolated and so what you get is the usual entropy term that you have already seen in the Q-revise model but where each spin has a different magnetization and so the final result is just the entropy term 1 plus mi half log 1 plus mi half plus 1 minus mi half log 1 minus mi half so you can do it in two ways so either you take this expression for lambda i you plug it there and you take really the sum or you much easier you recognize that what you are computing is the partition function of n non-interacting spins so is the product of the partition function and the partition function of a spin taking the value plus with probability 1 plus mi half and value minus with probability 1 minus mi half is this one you are right but yes because we want to reconstruct this so here you have to put an average value of u no no sorry let's put it here because you want to reconstruct so this term u times si minus mi such that you reconstruct exactly u square yes yes you take the derivative with respect to mi of this function beta so this says that you need if you want to enforce in a spin which is isolated you need a non-zero field these are isolated spins so essentially the the magnetization of each spin is the hyperbolic tangent of the field you are putting on no this is why we introduce this function because if I call this beta h and I am expanding around beta equal to 0 I need formally an infinite field but since the true field that enters the measure is beta times h is much better to call lambda beta times h because they will always enter coupled and I need a finite value of lambda which formally will correspond to an infinite value of h but makes no sense because you are taking the beta equal to zero limit so essentially in this expansion we are happy because the zero order term is providing the entropy so non-interactive spins when you compute the partition function you just get the entropy and now we compute the first order term and the first order term of the derivative is minus the average value of h and so the first order derivative computed in zero so db computed in beta equal to zero you get minus the average value of h on zero remember that for beta equal to zero the measure is really factorized you see now we are really allowed to do the computation on a factorized measure because for beta equal to zero spins are independent and so when you take the Hamiltonian which is this one and you take the average over factorized spins as before you just have to replace s with m and so what you get is the sum over i and j j i j m i m j plus the sum over i h i m i ok, so just taking the Hamiltonian and substituting and from this we get for example the first derivative of lambda that we need for the next order term so for example if you want to compute this object you need to take the derivative of this expression with respect to m i and so you get an explicit expression for the derivative of lambda with respect to beta which is what enters in the average value of u so in computing u square you will need this ok, so now let me yes so let me skip some details in the computation and writing the final result otherwise let me just derive the mean value of u at beta equals 0 because that is the quantity that we will find also in the next so when you compute the derivative of lambda i with respect to beta once at beta equals 0 you have to take the derivative of this expression with respect to m i changing sign yes, because there is a minus here and so what you get is minus the sum over j j a j m j minus h i ok and so now with this expression and this expression you can compute what we call which is the average value of u at beta equals 0 and let me just skip the algebra we have everything because we have the amiltonian, the average value of the amiltonian at at beta equals 0 we have it then we have the amiltonian which is written here and we have the first derivative of lambda with respect to beta which is written here you put everything together and what you get is a nice expression which is let's do it without the one half sum over i smaller than j j a j s i minus m i s j minus m j ok and it is very easy to realize that the average value of this observable is 0 because these are connected correlation function ok, you go on and you compute terms and the final result what that you get is the following so let me go to the final result that you can find written here up to the third order term in the paper by georgia dida up to the fourth order term if I remember correctly is the following minus beta f of m is the first term is this entropic part so sum over i of 1 plus m i half log 1 plus m i half plus 1 minus m i half log 1 minus m i half obviously this term changes if you change the nature of the spins this is for easy spins if you take spherical spins you will have a different entropic term but you always have the zero the entropic part of uncorrelated variables very easy to compute ok then the first term is plus beta because it's a an expansion in beta so it's beta times the first derivative and the first derivative we have it here is this so it's beta times the sum over i smaller than j j i j m j plus beta sum over i of h i m i so you see the first order term is the energy in the mean field approximation is the energy where you substitute the spin variables by their mean values ok so entropic part, energetic part and this would be the naive mean field approximation I will show you that taking the derivative of this you get the naive mean field approximation and then you have the second order term that actually provides you the onsager reaction term and if you remember it's not written here but the second derivative of a was the average value of u square so if you now compute this at beta equals zero you just have to take that expression you just have to take this expression this you have to square it and take the average by putting all sigma equal to m if you do this and you manipulate a little bit the expression what you get is something like beta square half the sum over i smaller than j j a j square this is a second order term 1 minus m i square I leave you as an exercise to do this so take this expression take the square and then take the average and you get this ok and then there are higher order terms I don't want to write them but just to say that there is a term where you have j a j cube to the term beta cube you have something like this and terms where you have also j a j j j j k and j k i you have these two types of terms at the cubic term you write this expression in the lecture notes ok so look at this expression so first of all we are very happy because we have an expansion where we are able to give a meaning to each term essentially and especially the first the zero order and also the second order term and I will show you now that taking the derivatives of this expression we get back either Naim field or TAP equation and of this also we are also very happy because we are exactly looking for a free energy eventually an approximated free energy which had the local minima exactly on the magnetization TAP equation and so hopefully using this free energy for also waiting those minima can be meaningful because we are looking for a weight to assign to solution to TAP equation so we have the magnetization when we are looking for the free energy to assign a weight this is a good candidate because at least there are points that solve the TAP equation so let's show that the derivative of this expression provides back the TAP equation and the Naim field equation depending on how many terms you take so let's do the derivative of this equation so let's do the derivative of in respect to Mi so the derivative of the first term is you can do it explicitly but I will write it for you is just the minus the inverse of the hyperbolic tangent the arc hyperbolic tangent of Mi this is minus one half log of one plus Mi one minus Mi you can do the derivatives every time you have this expression when you do the derivative with respect to the argument of the log they cancel because you get one and minus one so the only important derivative are those with respect to the M which are outside so you get logarithm of one plus and now the interesting terms you take the derivative of this guy and you have a beta sum over j j i j m j plus beta h i and you realize that if you set this to zero you get the Naimian field equation because M is hyperbolic tangent of this which is exactly the argument of the Naimian field equation now let's do also the derivative in the second order term the derivative of the second order term you get plus you see if you derive with respect to Mi essentially Mi where essentially you have a two that cancel these two and since we are taking the sum over the ordered pairs you just have to count one so if you have i j you never have j i is enough to take one derivative out so you get beta square sum over j j i j square now this term is the one you derive and so you get Mi and this term is the one you don't derive and you get one minus m j square and you realize that this is the unsuggest reaction term that we wrote in the there is a minus because it's minus Mi square so you get a minus and so this is the unsuggest reaction term so we realize that taking the second order expansion we get back the TAP equation if you want you can take also the third order expansion but the third order term it will be sub leading why because even in the spin glass case j of order root of n j square will be one over square root n to the three halves and the sum in front is always over j over the neighbors this case will be different because you have a double sum so you have to work a little bit in order to realize that is sub dominating but at the end you can convince yourself that essentially in this expansion if you are interested in solving the quidivized model is enough to keep the two the lowest order terms for spin glass SK model you need to keep also this term and the other terms when they are interesting in all the other cases because you can run this expansion on any model even in a finite dimensional model you see these are very generic Hamiltonian j such that they are mostly zero and on zero only among first neighbors and the high-temper expansion is perfectly fine so you can use this expansion either for finite dimensional models this means that if a friend of you arrives and say oh you are very good in safety mechanics can you tell me something about this Hamiltonian and provide you some very nasty j's you say ok I don't want to run a Monte Carlo code that would be the right way of sampling the Gibbs measure but let me start with something simple I take this expansion I plug into the couplings and they are and the fields that my friends brought me and I look for the shape for the minimum of this free energy maybe it's more difficult than running the Monte Carlo but in principle you can do it so these are high-temper expansion that you can use in any situation and I do also one more interesting comment about this ah, but let me add something about this I studied this approximation a few years ago because obviously you said that approximation is meaningful if using more and more terms you get better and better results unfortunately this is not always the case if you use this approximation for example at a very low temperature this is a high-temper expansion and nobody is telling you that at low temperature it will be good you find that actually at the more terms you do worse and this typical thing of mean field approximation you can prove they are good only in the high-temperature regime you can use them wherever but keep in mind that using it in a region where you are not expected to use it maybe it is better to keep few terms rather than increasing the number of terms so increase the number of terms either the approximation is very good or if the series is not converging adding more times is worse because in a non-converging series adding more times you make them even larger indeed in this respect is worse looking back to the original Plevka paper of 1982 and what he did after deriving this expansion actually he derived up to this term because he was interested in showing that TAP equation can be obtained by high-temperature expansion he obviously been a serious expansion he asked what is the radius of convergence what are the temperature and magnetizations where I can expect that if I fix the magnetization to some values and then I make the temperature lower and lower what is the radius of convergence where should I stop and so he tried to compute the radius of convergence of the series which is non-trivial but then you can realize that essentially the radius of convergence of this series is related to to the radius of convergence of the series giving you the first derivative of H if you remember is the Hamiltonian so he managed to write a simple simple series and realize this Plevka condition for convergence convergence of the series which is the following the temperature must be larger than sum over I so you fix the magnetization and you try to do this high temperature expansion of the free energy at fixed magnetization this Plevka condition is telling you you are allowed to use that expansion only for temperature high enough this for for the SK model ok so when the couplings are 0 mean so average is equal to 0 is 1 over n ok what does it mean if you look at the TAP equation for example you always have one solution which is easy to understand what is the solution all magnetization is equal to 0 that is always a solution look at this 0 equal to 0 everything is 0 which is the paramanetic state is always a solution that if you put M equal to 0 here you get 1 and this say that temperature must be larger than 1 and if you solve the SK model we are not going to do it but it's known that the critical temperature of the model is 1 so what the Plevka condition says you cannot use this expansion for computing the free energy of the paramanetic phase below the critical temperature if you do it you get a wrong result ok and indeed this is very important because if you start looking for the solution of this TAP equation unfortunately you find that the paramanetic solution is always very attractive and you can any algorithm for trying to find the solution of this solution because now you have a free energy so you are happy say ok now I solve this equation and every time I find a solution I compute the free energy so I know the weight and so I can compute everything unfortunately you run any algorithm for solving a question at your will and you find that the paramanetic solution is very attractive and so when you try to compute states in the SK model below the critical temperature because above the critical temperature the only solution is m equal to 0 you want to do it below critical temperature you get very much attracted by this solution but this solution is wrong doesn't satisfy the Plevka condition so you are not allowed to use this free energy expansion on that solution and so the number you get this is wrong ok so one more comment before leaving for today and if you take this expansion you see already at the third order terms that there are terms involving always the same link between i and j and terms involving links that form loops ok and this will be true also at high order so if you go to the fourth order you will find a term which is jij to the fourth and terms with loops one nice thing of this expansion is that if you keep all the terms involving just one link so you forget loops you can run this expansion on any graph suppose that I run it on a random graph on a tree no loops so each term involving a loop is zero because at least one of the j is zero so I cancel all the terms with loops I keep all the terms with just jij to any power I resum and what I get is the better free energy that we are going to study later in these lectures so the better free energy can be seen as the resumation of this series this is very good because we know that the better free energy is exact on random graphs or on trees and having a power series that on loopless graph has an infinite number of terms that if you resum you get the right result is very good for this approximation so on loopless graph this is nothing but the high temporal expansion of the exact free energy and so this also very good for this kind of approximation just a last comment about the application of this TAP free energy so from now on I will call this the TAP free energy and those are the TAP equation so the TAP equation are just the derivative of the TAP free energy from and if you didn't read the Plefka paper and maybe you derived by yourself you can say well I want to check that I'm not so good as Plefka in understanding what is the radius of convergence of this series so I just keep the first three terms and but I understood that this solution must be minima of the TAP free energy so instead of I can try to propose a way of discriminating good solution for non-good solution looking whether the solution is a real minimum of the free energy it may be a subtle point it may be a maximum so suppose that I I don't know the Plefka condition and I do this in order to in order to to compute the Hessian so the Hessian is the element ij is defined as the second derivative is the second derivative of f with respect to mi mj so if the point I just found is a good minimum this matrix Hessian must be positive defined and suppose that I do this computation and I do the computation on a very simple point which is the magnetic fixed point all magnetization equal to zero so we take the second derivative you set m equal to zero and this expression for m equal to zero this is simple it looks like t plus one over t or beta plus one over beta delta ij minus jaj so look what happened you have a matrix compute the spectrum of this matrix is not hard because these are random gaussian matrix so his spectrum is a bigger semi-circular law so the spectrum of this matrix is between 2 and minus 2 now these are diagonal matrix so the eigenvalues are t plus one minus t so the smallest eigenvalue the first that may become negative because you want to have all the eigenvalue positive is taking minus 2 for the random matrix and t plus t so the smallest eigenvalue is minus 2 plus t plus one over t you realize that this quantity as a function of t is always positive touches zero in one and is always positive so what is telling you is that integral one which is the critical point you have a zero mode in this free energy so is marginal indeed is not really a minimum of the free energy but what you don't like is that below the critical temperature the paramagnetic minimum so the minimum in m equal to zero is again a minimum sorry the point m equal to zero in the TAP free energy is again a minimum while the prefica condition is telling us that the free energy m equal to zero this free energy shouldn't be used so what's happening so in this you have to be careful you are using an approximation that approximation looks like it have a nice minimum in m equal to zero below the critical temperature but you don't have to use that point and so the prefica condition is telling you this and indeed is very simple that you cannot use the paramagnetic fixed point below the critical temperature because if you compute the entropy and it is a model of discrete variables the entropy must be positive you realize that the entropy of the paramagnetic fixed point because negative slightly below the critical temperature so clearly the entropy of the paramagnetic solution becomes negative which is unphysical so clearly below t equal one for the sk model you don't have to take the m equal to zero solution unfortunately it is very attractive so then finding the TAP solution for sk model is an open problem so this is a new open problem find the TAP states for sk below the critical temperature essentially we still don't have a good algorithm there are just a couple of works where people minimize this function but at a very low temperature it is like minimizing the energy so you are looking for the minimum of the energy the problem is that you would like to compute really states in all the temporal range between zero and one so this is still an open problem it is very simple in stating because it solved this equation but it is very hard and if you are interested I can tell you why it is hard for today it is enough