thank you good morning I will need so this this will be the second lecture and just I will start with a very very very brief reminder what what was discussed last time yesterday um so yesterday we started with the classical transport in normal metals and there are two basic equations one is the diffusion equation Z this is the diffusion equation that you can apply not just to metals to to any of that propes fusion and equation we said that object the diff which is the propagator ofation [Music] and is dimensionality and the second relation was theind side relation that that to the new States and this is as I explained this is a very general relation that appes to any material and in a particular case when you have met which is mod by Spectrum from here can which is and from this question you immediately see that it's very specific because it contains must must very good so um just one other comment this um the Spectrum surface energy all here and this equation it contains which is C of the spectum and it contains the totality which means the total number of states whereas this equation contains two parameters D which only know about what happens near theg this is the den of States at the fir level so Most states you have on the fir surface and the diffusion constant knows only about fir velocity which is the velocity here also so if you not parab Spectrum but some more complicated but still surface can app this equation but you can okay so this is this is this is what happened on the first lecture and then uh we started to discuss the quantum description of of the metal and Quantum description again this this is the repetition of yesterday this qu description is usually done in terms of the green function GRE function is definition can Beed and just the inverse operator energy plusus i0e function was we discussed as a part the depends on momentum and corresponds to the dispersion relation like that plus potential and that corresponds to this is hold dis poal and for this potential we have C distribution so a is zero and solution so it is fully determined by the second moment the second moment is just function power so the stronger the smaller is power the B from here whenever we talking about the metal we always assume that the energy time is large or same momentum so the first goal was to average function respect to distribution of the disorder average function was in the Dy way is is the function whichs some self energy so it equals um 1/ epon -- Sig and sigma is a set of diagrams starting with the this simp just this function then next one is plus and we diam initial before we okay we had in terms of strength of then comb the diagrams right and then we can place like Sigma up to some order and then line G using this Sigma right but at that moment when we combine G we like do it to much biger order because we like take into account a lot of diagrams yes yes yes yes so takes into account with much higher order ination then we actually like yeah I I got question actually this is a very typical situation it happens almost always when you do a DI expansion and the reason for this is that a single function without sign has a singularity at the surface eps S zero it is diver so in the diam which contains two functions three functions all the same momentum because it's the exal these simularities multiply and become stronger and stronger that's why you have to sum up this whole ser and the result is that you get Sigma in the denominator which actually removes the singularity when you at the fir surface and these two things are zero your GRE function is no longer diverent I mean but like speak we take into account lot of expansion terms and we just take some random of them not R we pick on the most diverent ah okay okay okay so like all those like non Ruble diagrams they are like the most and and so all [Music] so right so actually the peration theory as just mentioned is the peration theory for Forma has the first of the terms second the terms third of the terms and what we also did last diam which approximation approximation and and we found that that it is equal to some real part which is forver but has no physical meaning it just the position of the chical potential and in the imaginary part was plus I 2 to depending on whe we consider function and this to comes from so the next thing we are going to do yeah very Bas thetion so the actual parameter this is large the inverse so next thing we are going to do if there are no further questions we will consider these two and so there next but as but but there is one additional interesting thing about this diagrs mainly let me just SE so this is the first one and we have some external momentum see that ENT here and same from there each dash line is a Delta function which means it's just one momentum presentation so it's just a number but it also means that any momentum can go along this l so inste we have some other momentum P1 in this L some other momentum P2 this then here and this object computer have to multiply three functions this integrate B1 and2 now if I draw second [Music] which here i1 this line I this yeah excuse me you WR out what this I I want just to finish this thing that momentum in the in this mdle so to answer your question I can means 2 and then is interal over P1 and2 and three new functions 1 2 3us the expression so we have functions of two and this Das line I can say that we have a momentum here and momentum P1 here so there is a momentum pus P1 [Music] isim and then there is another line momentum here is p1- P2 then this momentum comes back p1- P2 becomes then this momentum comes eus 1 actually you can do this I I don't want to really spend time more on this topic but uh you can do it like this you have functions and you have you can write realation function points points then function you have to transform to moment representation then you then you see that integration oros of this point gives you the concentation of momentum so momentum need for each momentum integration yeah the only thing I want to why I draw this two things here is to explain what what possible vales ofum to you can think of it in a different ways this is your firm surface so this is momentum space and you have all the states all field and up to this so here and P is the momentum of the exal somewhere at then on the first point some position and the momentum changes and then so real space one point s point then function return Then first it's not cross real space these are the process thats this this process here here [Music] questions to this I I just I will not Cate I just wanted to say that U as I already mentioned question by um the green function is very singular it diverges and both and from this first diagram we have inside two functions at the same momentum1 which means that when this P1 this P1 is z P1 surface sing s and independently we can also have this function while this diagram we can make the first function s put P the last function if on but this middle one appears somewhere it's not necessar on this combination 1 2 p can be anywhere therefore first diagram actually is much larger second just because the integr moment space we can pick up the larger intergral here and this much larger if you analyze it further in it's much larger in this so our is large me and this is actually a very general situation you can do the same calculation for diagrams with more lines and you will see that whenever the two dash lines cross each other the diagram becomes smaller compared to the same diagram so there is a very important uh so again using our small parameter is very important observation that we can effectively function Crossing di di diose each other give the rain [Music] contribution turns out quite easy to Sol all [Music] such yeah we will get to this later sorry another question I just want have some kind of classification of small parameters we use so first all we made an expansion and the formal parameter is was like or well in the sense if you like we we take the whole series so it's an expansion but it's not necessarily small okayy yes this diagram is larger than this because just has an extra line extra from top and these two diagrams compared to each other it's also but you you will so actually our our ultimate goal is to Su all the diagrams completely everywhere we will just do it in in steps so I me is just a little bit weird then that we consider like second order and it turned out that in the second order there is some like third order I mean we second right we St that this Crossing di smaller by yes so it like should be somewhere well first of all it's not exactly from I believe from the so it should be compared to the fourth order but as I said eventually we will sum all the so it will not be a problem we will just do it step by step selecting first some smaller set of diagrams and then looking what happens we add ex so the the idea of this this Di as a name called self consistent approximation so it's also approximation but so in the B approximation we had which is given just by a single di function self consistent approximation means that now we will insert here inside this self energy not thean function but function that takes yes so is a approximation and the way to Su up all it's just so if we say that s contains inside the average function what does it mean very simple if this function is also averaged it means that this di also account line here is the part of the aage and if you say that Sigma is like this and then average G is itself G inverse inverse so this Ser function here you will so so these two equations completely determine both the average G and so these two equations the two unnown things G these are the self consistent approximations and G should be so we found find Sigma find the sigma then we reinert thisg back to compute itself so if you write out all the diagrams you will see that this Sigma contains simple form diagram plus plus diagram this G inside plus instance so all and it's actually quite easy to do because let's let's write this equation Sigma is 1 2 integral and then function have here so this is theist equation is on the left side also inside the interal so if we can Findation willut [Music] and from the B approximation we know that se is just some imaginary shift some number it's independent of moment actually this Sigma will also of momentum this it does not depend on moment is just the number anation we should find and actually turns out that it is it simp just because this part is is the residual hole and the only thing changed is that we shifted this pole from was very close to the real is shift by five but the res so four di the third so supp that I have one uh which will be also like of the third and we did not include it seems like something correct I actually I already answered this question before we do not compute all the diagrams we compute only a subset of diagrs at this point which is called self consistent this subset of di are the diagrams process it doesn't mean that the diags we thr away are smaller than everywhere okay but but this subset it is it contains all theion terms any and sub terms might be not fully correct but we will consider them later as I said we are going to all the what so is actually the full all the but then I do some approximation I canate Sigma approxim okay and there are different approximations oim just a single di then there is a self consist approximation thats to the single Di diam and then sometime later it will be even more okay this isim we of all the diams so this is like an approximation okay so the question is this isim what you can guess we okay okay look that's that's actually not so easy to see because yes each dash line has one top so looks like each dash line should another also have the integration of a momentum which can be diver so it's not true that this di is not actually smaller than this one it's the same reason why we don't take just this have to add up Sigma many times because time we have newma we have a new Singularity as well ah okay but andity is like so it's not ex we want to [Music] and yes just to put another Point here we have Orin approximation which produces par I to it corresponds to the first now we add up all these diagrams is the same to it's the same to so in in many cases people wouldn't bother just approximation get the correct result actually it's not always the case sometimes self consistent approximation gives you a different result from simple for approximation and this happens I can tell you when this happens when the density of states is not con so uh in in our metal that we consider we have big F surface we replace St St replace this byal over side the same so this is Conant then you always get the same number but if new is dependent on momentum so if you have to if you take interal the curv Spectrum then there are not local then will depend so this still can now this becomes a function function again it doesn't have any Simple Solution but for any particular Spectrum any particular form of okay so just to summarize this this part um we have we found that diag of cross lines have some extra smallness and we can neglect them and if we neglect them then we can some other diags if you like levation or cross approximation some approximation and the result is sequence given here just and function is just so with this thing we can proceed further and so so what what we compute next is the defone propagator that we had from the previous lecture the propagator of the diffusion equation we will compute it using green functions and the basic thing Start Formula actually it's quite tricky topic which I don't want to go into how this formul is part of mechanics which is describ in many text books very thing I will just give you without ination form for is the following thing one average two functions function function en sorry energy plus once function of integrated over all moment so this this is an equation which I will not Rive but actually it's very general whatever you measure can be always expressed in terms of new functions no not enery that's why don't this this is no this isor ofus well I I have removed some extra things here so originally there was integration of energy differently but for our purposes this simplified ver will be enough no en anything but it will drop out from the result so once youate momentum result the result it's the trick okay so this this thing can be represented then that goes back and we have frequency [Music] right so this function the momentum lar so the first thing to do uh how do we average product of two functions first thing we do we just multiply two average this object I will denote by B so is it's my defin this is theal over all of the prodct average average should be and as before instead of over momentum I will write as the and myage function meus - s of p + q+ I is the function and the second one is so we we should integrate over p and p enters both here and here and as I already mentioned previously when when we integrate for example s function it has a singularity surface so we replace integration by integration which is valid near now we have two denominators shifted by the moment Q which means that one function becomes as maximum surface and the second one is the maximum of surface which is shifted by this are C and I will not do this expression exactly I will assume addition I will make addition ass is small is small compared to so this shift is small compar R oh well actually yeah okay this is true it's more consider low energ and since Q is much smaller we here P small fraction so we can expand this function side in Q and this expansion means that I can write pus B3 * Q scale these two things in now since I have single out of P here here now I can say that instead OFA I will this and this is a simple integral again you just complete the integration T take take a res and result is here eals 2 Ed by 1 plus IUS I this thing should be averaged over the whole F over the direction and already here you see that this does not depend on ESS of course of course small and new and the only thing we do we we take residual side so we have one pole in the upper half plan the other one in the lower half plane you can complet control any way you like result the same and EPS cancels because it's just a common part well so this is still not not the complete result uh because now I will add another another approximation which is more this I will additionally assume that Q is much smaller than 1 K and omeg is much smaller than one to so I so small human that these two terms are smaller than one sorry I still have some what I mean from begin we [Music] we now where like we about what what what we before Sy one is the average function this thick function just means average now there is something which is called Kuba formula Kuba formula relates the diffusion propagator to the grill functions for any system it's very general thing without any average and what so D is the product of these two functions in general now I want to average this of and then I have to app to the that that is what this is exactly the point that I wanted to avoid Fusion the propagator Den is slightly different it also involves energy don't want to do the energy that's why diffusion and this way how I def here really any other [Music] questions a small scale is this approximation just something we want or is this actually very good question actually diffusion only has meaning on this scale because M path is just the distance between scatter to establish diffusion you should scatter many times which means you should go to a distance which is much larger than which means that momentum smaller than one and actually the is intermediate scale momentum is smaller than momentum but larger than this called bis propagation a ballistic propagator versus diffus is there any way to see that the diff diffusion well the diffus is a quickly fluctuating quantity that is can be averaged and that's enough or all in well this equation already implies that we aage D itself over disorder and as we know like perally true that a physical quantity is just average of disorder of some some physical quantity in in disordered particular realization depends of the quantity is the way to see that D to see this you should go back to previous question by consider the propag of that that is well I know this I just want to see if there's any arum which allows easily to see that this particular quantity is based on it let's say pression for like that well in a sense I will answer this question the much later okay so let's proceed so these two things here I now assume that they both these two things are small to one front so I can expanders things and what I get is 2 * 1 plus andus I so this this thing I should exp to the [Music] second dire so iand to the second what I gu what I get isus uh B * Q also squar and a over the the aage of Direction the and this Bec and what we recover here is exactly the diffusion so again this is 2 1- d and it doesn't look like a diffusion propor and the reason why it does not look like diffusion propagator is exactly because the average of the prodct of two functions is not the same as the two AAG so this here should be the following it's um product of two average fun like I just plus [Music] same and also for the cross lines but as I explained before Crossing D lines always gives some smallness so we now do nonoss approximation but in this cross approximations we should in all and first term here is just my function and the second term here is b² one power functions second and single thir 2 so this this is just a geometric series you can sum sum it up and 2 function insert [Music] the any any number of any number then you say average means that you should connect to process with a dash line in all possible ways and one so this will become part function this I because it's not part of so if you do all possible things here without lines they will all each function also and the same go a separately and on top of this I add this one this is one two I can put moment so and it's over1 and then theine can pass any moment so the next2 so each line is a Del function space can provide extensions trans any momentum so all moment in this section so I have this for example car yeah so this is what and now you see why I want to to do this expansion here so I have one plus some small dire one small dire 2 2 so I I [Music] this one so if you substitute B here hereat get the num you can just okay so this is a demonstration how very simple diff equation with a very simple propagator can be derived from Green functions so it's just a um very complicated way to d a how question question what time should 25 more minutes great then um if this thing yes question that part F and we will now drive one possible result options now be better than before so the next thing is hold activity is also very nice quantity that can be measured computed and H it can be [Music] describe it's not ver now very similarity is 1/ 2 average of the two GRE functions just like and it has two operators here and another here and if you apply electric field in X Direction and if you me Curr on the X direction will be C Vector which is electric charge of remove just some something here which represents the that interested in here this is what is called um DC uniform Q is exactly zero is ex again this is not the most General is another version which also and more one this okay so this culation is EAS proor actually go like this we have one 2 and we have single interal of now I will [Music] just two green functions Ander I have a product of two velocity each one is like this so I have e s and I should divide by D I take only one component among de component okay this is now straightforward integral you just compl T side doesn't matter and you get thing e s s here and all the rest here all the this here we immedately ride [Music] this that much straight forward um yeah that's very simple um consider di with just a single FL line the the new think cas we have operators they are vector so we have if we do it we have one moment here here and the integration of P1 will keep zero because it integrates over all dire on the surface so average X component will if you like VOR integ will notations this first thing and other the next thing um you still can do this calculation and if you have no computers you have to here and what you get in the very end you will replace appears here with the Trans in a system where scattering depends on the El you have distinction between Quantum scattering time and trans time so the momentum relaxation time and phe relaxation and the whole transer qu trans this distinction between these two is exactly exactly [Music] okay so just to summarize what we had so far uh we discussed classical transport and uh we had the Einstein expression for postivity in terms class diffusion Conant and we have a diffusion propagator which which is also propagator of the class equation hereat and then I tried to show you how this same result can be mechanics in terms of the green functions average and uh as far as we our and this I will answer the question right approximation equals yes yes question and if you want to Crossing diagrams you will get some Quantum effects that are Beyond question that our [Music] our yeah so why like over because we are considering situations when Quantum effects are weak corrections to CL results in the sameet our are actually telling us that they are control controls everything that's that's not so easy to tell because yeah okay well H bar is hidden in many equations here I'm not sure if you can trace really to the single parameter but but [Music] maybe yeah in a sense when we don't cross mes we actually I I can draw many diagrams and without Crossing you can always see that each green function connecting two impurities is always accompanied by a second green function so it's always targ Advan pair that con purities which means that we consider not the quantum electron propagator but like class density step so so this this is the reason how this oim reproduces the result okay I I don't have much time left so I will start very but and actually be start a new topic once we have understood what the classical transport means and how it is describe um so the next topic is it's already something about which is which no and it's very so the the idea is the foll imagine you have a piece of it's a cube and it has a size a different if it's metal you can attach to contact and measure its resistance or what people usually discuss this conductance conductance is inverse resistance so for a piece of metal state is a condu is resistance and there is a conductivity Sigma which is a material parameter conductivity per unit length Pro so conductance equals conductivity times the cross-section The Wider is simple the better conduct crosssection is a power d d Dimension object in One Direction crosssection D diens and the longer is sample the larg resistance the smaller conductance so is also divided by or if you like this is how conductance of depends on size this [Music] is and TOS now consider you have the same piece of material but it's not a metal it's insent does notu you apply voltage there is no current but actually there is current so using Al ternative expression very approximate the conductance of the p ofor is exponentially small it's exponent of minus the size sample just because it's finite object so some Cur can just J here T exp more current FL and typical L scale is loation L but doesn't matter for now so this this law is for an insul so two alternative formulas Oneal the other one insulator and then is a very deeper step which is called the St hypothesis this is the some very smart people for long time and decided that Dera of thei condu with respect to logarith the size this this function should be the function of conductance itself not s this if this is true if this derivative can be expressed as the function G if this is correct it means that the system has this scaling so the scaling hpo is the HP this identity exist and uh there are some arguments why it should exist but I will we can be way we can be closes me that g can be any well if G look um if you have just a single function G of L you can always do it like this but now mention many different samples of many different sizes so we have many functions G1 ofl G2 ofl G3 of and if all of them oby the same equation this means yeah to some extent and these two limits of G in the first Cas just for it's and we see that depends on the dimension and nothing else it's very Universal so you can have any metal you can have copper aluminum iron gold they all have and for the insulator for this exponential function if you this derivative it will be just a itself so of g isus a the derivative is just a and you can read also Universal function it doesn't contain any material par side for example drops out from here right so we originally assumed that the system which is a good metal has large conductivity G is large and insor exponentially small conductance G is small so in the sense if this hypothesis is correct there should be some function of G which behaves like this large Val of G this so should be does move as a function of here which is just straight line one straight line second third this is for Dimension this and for the small G here this function should be just and the if the SC is correct there should be one universal function and again just is correct whenever we have Dimension this is not metal I take something mect say that I here so it's thre dimensional object my cond then I say that here is positive which means that if I start to increase the size of my sample some material size conduct grow so I move this direction and I end up somewhere far this and if I have a sample and it's conductance somewhere here where the derivative is negative I started I so point this function cross zero is the phe transition point between what we call metal State what an insulat State this is the this this is in a this is a cond so yes I mean uh you may have some strangely known isotropic materials which have different conductance in different directions but then you should this is not what ask oh no no no no no fact is beyond this will discuss later so for now this is just for usual us so once again there are two so to say states of not states of metal but two types of materials metals and insulations metals are not the material that has zero resistance has some finite resistance but the point is that when you increase the size it reaches some finite conductivity per unit length and insulator something increase theze the resistance goes to infinity and there is a single point which separates these two things so you can have many materials you can always classify them these are metals these are insul B are on this diagram and what this thing also suggests is when you consider two dimensional one dimension materials this function is negative so presumably there are no metals to one dimension in the sense that even a very clean system two dimensional when you start to make it larger larger eventually you will become this is all based on the scaling hypothesis I gave you zero grounds why this hypothesis should be correct so now you don't need to believe it but actually later later Lees I will kind of prove it so this this is actually the thing and this is how nature works and with this today's lecture after answer last question def What is INS materal that becomes more and more resist with if you have some small piece piece can measure the resistance it's fine you don't know if it's to distinguish you have to see how changes any other questions okay if not then thank you