 Now you're on. Okay. Yeah. Thank you. I'm going to first review the Landau's Fermi liquid theory and then describe, reformulate Landau's Fermi liquid theory as a dynamic of shapes, using the method of the coaxial in orbit. Based on the idea that the state of a Fermi surface can be described as a point on the coaxial in orbit of a group of canonical transformation and that would lead to a reformulation of Landau's Fermi liquid theory. The work that I will describe to you in this talk is based on a paper written this March by Lucca de la Cretas postdoc in Chicago. Yixian Du human meta to students in Chicago and myself. So let's first remind ourselves that Landau's Fermi liquid theory was first formulated by Landau in the 1950s. And the idea that Landau has is that even if we start our description, if our system consists of fermions which may interact strongly with each other, near the Fermi surface, the notion of the quasi particle becomes more and more well defined. And the notion of the Fermi surface survive even when the interaction between the original fermions may be strong. Landau's Fermi liquid theory has been successfully applied to many systems like helium three electrons in metals, neutrons in neutron star and possibly many other examples. The Landau's Fermi liquid theory is sometimes described in the popular press as a theory of free theory theory of free particles. But as we will see the structure of Landau's theory is very different from the example of free field theory that one usually learned from textbooks like free scalar field or free fermion field theory. It is instructive then to first go through the formulation of Landau's Fermi liquid theory as it was originally described by Landau. So Landau tells us that the state of a Fermi liquid at low energies can be characterized by phase space distribution function, f as a function of t, x and p. From the point of view of quantum mechanics this might be and might see that is contradictory to specify the both the position and the momentum of the particles. But once I have to think about this distribution function and describing particles that can be localized only in a certain volume in space. And when this volume is the uncertainty in the definition of x is large enough then we can describe the phase space distribution function both in terms of x and p. I read in a recent interview by Cillian that Cillian claimed that it was he who solved Landau the idea that a Fermi liquid can be described by a phase space distribution function or at least convince Landau that this notion exists. So now let's consider the case of free fermion. Of course for free fermion if we put that free Fermi gas in external electromagnetic fields then one can write down the new wheels equations or one can think about that also as the Vlasov equation or collision less Boltzmann equations. So the f over dt the time derivative of the distribution function is given by the evolution depends on how fast the particle move in phase space. So this v of p is the velocity of the particle in configuration space that is the epsilon over dp the group velocity this particle and here e cross v cross b e plus v cross b is the Lorentz force acting on the system p dot. So this equation completely determine how the phase space distribution function evolved as a function of time. Given external field we can determine the response of a Fermi liquid to such an external perturbation. Moreover one can notice the following if the distribution function is is the zero or one like it is in the ground state of a Fermi liquid then the property that this function is always zero and one preserved under the new wheels evolution and that's because the new wheel equation preserves the phase space and so if we follow each volume of phase space that it move in phase that it move the distribution function remains either one or zero depending on whether it was one or zero in the beginning. And so the new wheel equation preserved the sharpness of the Fermi surface and then one can think about this time evolution as the evolution of the shape of the Fermi surface. So here what happens so I'm drawing here the original Fermi surface say at t equal minus infinity we have a spherical Fermi surface in a Fermi liquid with rotational symmetry that would be the ground state and then we turn on some electromagnetic field and what happened is that this shape deforms so the figure on the right one have to think about that as the particular shape of the Fermi surface at a particular point in space and time at every point in space and time you will have a different shape and our task is try to formulate the time evolution of the shape in a mathematically convenient way. If there is the connectedness of the Fermi surface can it split in several order? I think mathematically this equation doesn't allow it but if the equation doesn't apply itself then the Fermi surface can split but if one can apply this if this equation applies then I think there is the surface cannot split. Given some properties if the property of the function for example epsilon of p is regular function with some nice properties etc the land of Fermi liquid theory is also predictive in the sense that one can use it to compute physical quantities for example the linear response of a system to external say electric field. So here is an example of the calculation of the density-density correlation function called pi zero zero how the density of the system changes under external scalar potential. One should just solve the equation the Liouville equation in an external electric field which is gradient of the potential a zero and then solve it into linear approximation one find the response function to be this integral. Now this integral turned out to be exactly the same as the one compute one would find using the usual Green's function method in the limit of small q. If q is small then this formula would be valid. So that illustrates the validity of the Landau's theory although it doesn't know when we write down the Landau theory it's written not in terms of the individual electrons in terms of some grossly, grossly defined distribution function but it reproduces the correct response function. Now Landau's contribution is actually one on three we'll learn that he figured out how to include interactions between the electron between the original fermion into the effective Liouville description. He showed that the only thing that one need to do is to include the so-called Landau parameter which is a function of two point p and p prime in momentum space or rather to the extent that one needs it only p will be prime near the Fermi surface. And then the modification in the Liouville equation becomes the modification of the energy of the quasi particle with momentum p and at point x. The energy is the bare energy plus the contribution that comes from the distribution of particles at other values of momentum but at the same point in space. So this integral is a linear correction to the energy coming from the deviation of the distribution function at point x and p at the same point x but at different p I forget to write a prime so delta f at x and p prime. And the kinetic equation then written down in a very similar way one has to also include into the time evolution of coordinate is momentum the deviation of the energy from its bare values so here the true velocity and also there is another contribution to the force that depends on derivative of this function with respect to x and that's all that one needs to be ordered to compute reproduce the linear response of an interacting now Fermi system to external electromagnetic field. Now the Landau's Fermi liquid theory as Landau has written down is not a field theory if you can see it's formulated completely in terms of the some equation of motion not in terms of an action and that has some consequences. First of all we lose access to a large two box of field theory effective field theory we don't lose the Wilsonian point of view although Landau's Fermi liquid theory looks like a effective low energy field theory but it's not something that one can understand from a Wilsonian perspective. There is some advantages if one can reformulate Landau's Fermi liquid theory as a field theory and one of the advantage maybe that one can better solve the problem of non Fermi liquid. So these non Fermi liquids presumably exist in some places in nature. One place is two by two plus one dimensional electrons in the high magnetic fields the quantum hole effect. The quantum hole effect the half field Landau level is supposed to be dual to a theory of composite fermions at finite density coupled to a gauge field and that problem of fermions at finite density with a Fermi surface coupled to a gauge field is one of the place where presumably a non Fermi liquid would appear. So to understand the Fermi liquid non Fermi liquid one needs to first try to at least form my perspective one it's useful to reformulate Landau's Fermi liquid theory as a effective field theory. Previous approaches have been there have been some previous approaches towards this goal. So earlier in the 1990s they would work by Benfato Galavoti and Gendruchin and Schanker trying to reinterpret Landau's theory as an effective theory of the Fermi liquid particle and so one write down some Lagrangian or some action that depends on the fields of the fermions and try to include the Landau's interaction as a four Fermi interaction between this quasi particle fermion. So that Lagrangian you will see is not the most simple Lagrangian so one of the the drawback of this formulation is that it has to be done in momentum space and so in momentum space it's very difficult to couple the theory for example to external gauge field. Also another thing that makes this formalism that say maybe something else perhaps to be done is that in this formalism there is a unnatural cancellation between diagrams so if one compute not two point function for example but the three point function one see that there are three diagrams that almost cancel each other exactly in the infrared and that that non-natural cancellation survive to four point five point function and become more and more severe. And so something else probably is here L1 is a momentum in K1 is momentum okay so here actually I lost yeah I'm not I'm not sure yeah it's this L I think it's from Konchinsky and this L may be P maybe K plus L I forget his notation. I'm not saying that I this is important for the the thought I'm just trying to illustrate that it looks like an original Hamiltonian but one has to to interpret this V as some effective interaction so the idea is pretty simple that there is some effective interaction that can be renormalized as one goes into infrared. Another set of idea comes from one plus one dimensions where one knows that a fermion can be bosonized into a boson and during this bosonization the density and the current becomes a derivative of some bosonic field. But to transfer this idea to higher dimension the situation becomes very complicated that's because now instead of one boson one in one plus one dimensions the Fermi surface are just point but in higher dimension the Fermi surface now has an internal extension like it is a line or a surface and so one would have now one chiral boson per point on the Fermi surface and that makes the theory very complicated and there is no general rule to write down such an action one try to one would try to go from the microscopic theory and derive such an effective action but but doing doing that is it technically complicated and doesn't exhibit the general properties. Recently Elz, Tolkien and Senzel try to think about this problem in a perspective of anomalies and that's one thing that they notice is that there is mixing of mode with different theta and that mixing also is very make this bosonization very non-trivial because suppose suppose we put this electron in a magnetic field then suddenly one fermion from one point can move to a point fermion at another point so somehow different bosonic modes of different theta need to couple to each other and how they couple to each other is a very non-trivial question. Now in this top I want to answer a simple classical problem and the the the question is can one formulate reformulate Landau's equations as an equation that comes out from some action s so action depending on some few phi so that delta s over delta phi equal to zero is identical to Landau's kinetic equation and the answer my answer is to this question is yes one can do that and the method that one need to use is the method of co-watching orbit so let's let I'm going to flash to you just a few mathematical uh trans transparencies and then we go to the more physical uh application of this formalism so consider a group that we call g and corresponding to that is the algebra that I also call g but it's with a different font that small g is the algebra and large capital g is the group and for each elements of the algebra one can exponentiate it into a element of the group now we know from the theory of a group and the algebra that there is something called the adjoint representation so for the algebra the adjoint representation is simply realized by the by the by the commutator that is if I have a an element of the algebra g of f then an action of the element g in the algebra on f is simply the um given by the commutator of g with f so the adjoint representation is realized on the algebra and one can exponentiate that to a representation of the lead group uh so for unitary lead group usually we write that f go to u f u minus one u is the exponent of that g now there's something that usually we do not learn in uh as a physicist in the course of of lead representation of lead group uh it's the co adjoint representation because in a lot of time this co adjoint representation is very similar identical to the adjoint representation so mathematicians like to make this distinction between the space and the dual space so the lead algebra we call little g and the dual space g star are the space of all the linear map from g to to to the number real number so for each uh uh elements of the dual space and uh elements of g each of such pair one can put a scalar product that belong to that is a real number and and this dual space also realize a representation of the lead group and the lead algebra uh we define this adjoint co adjoint representation so that if we have an f and a capital f and apply a co adjoint rotation on capital f and co adjoint rotation on this f that scalar product is is preserved so that is how one can define the co adjoint representation okay and then what is the co adjoint orbit the co adjoint orbit is defined as the orbit of a given point in co in in in in the dual space under uh under the action of all possible uh elements of the lead group so if I take one point and then use this co adjoint representation and then rotate this point and one one can define this co adjoint orbit and this co adjoint orbit actually is a coset of the lead group and the subgroup that that keep this reference point fixed in space okay so all this might look a bit abstract but I'm going to go to a physical uh case of a Fermi a Fermi liquid that's relevant for Fermi liquid theory so the idea is the following the idea is that uh all the uh different shapes of the Fermi surface uh that uh evolve that is the result of evolution of original Fermi surface and the Liouville equation can be thought of as the result of a canonical transformation on the original Fermi surface so all the state with a sharp Fermi surface and connected Fermi surface can be obtained from the original spherical Fermi surface by applying a canonical transformation and this canonical transformation acts in the face space of a single particle the canonical transformations um acts as follows uh so first the the the infinitesimal canonical transformation or the elements of the algebra are all functions of a face space x and p so x go to x prime equal to x minus epsilon gradient over p of f and p go to p prime equal p plus epsilon gradient of x you can see here that this is basically a a Hamiltonian evolution under a time epsilon with the Hamiltonian given by f and one can see that this preserved the Poisson brackets like all Hamiltonian evolution uh the Li algebra uh is given by the Poisson brackets that is the commutator of two of such infinitesimal canonical transformation is another another canonical transformation whose function is the Poisson bracket of these two function f and g so it is a Li algebra and exponentiating this Li algebra one get the Li group that is the group of all finite canonical transformation into the power of f now the dual space what is the dual space f so i'm going to define this dual space as also function of face space x and p and the scalar product between f small f and capital f i'm going to define a simply integral over the face space of the product of small f and big f and this equation has a interesting physical interpretation if if one interpret capital f as an observable for example p square over 2 m one possible function capital f then this small f can be interpreted as the distribution function it's a state and the scalar product is the average value of this observable capital f in the state given by f it's very similar to the trace of the product of rho and f in quantum mechanics instead of taking the trace of the product of the density matrix and the operator here which is take the classical integral over the face space of the product of two functions so the adjoin and co-adjoin action of the group in this case are the same i turn out that action of group element exponent of g on is realized in the same ways in the Li algebra in the in the dual space by the same formulas one can see that if we do this adjoin rotation then the scalar product of these two functions remains the same by some properties of the Poisson bracket one can do integrating integration by part and show that any question okay so now let's let's go back to the problem of the Fermi liquid so let's take the our let's try to construct the coadjoin orbit the coadjoin orbit we take the reference state and try to apply different canonical transformation on that reference state and our reference state is simply the state with spherical Fermi surface and let's try to work in perturbation theory that is for small perturbations so let me take the one element of the Li group of the canonical transformation hq now it's exponents of minus phi which minus is just for convention and phi is a function of face space x and p so now using this this rule that we have before a new distribution function is the action of the coadjoin the coadjoin action on this f zero and it's just f zero minus two leading order in phi minus the commutator of phi with f zero by commutator i mean the Poisson bracket so this this can be calculated and rewritten in term of another theta function you see here that the theta function is not different from the original theta function the position of the Fermi surface has now moved this p the this here p hat is the direction of p in momentum space p hat cross dot with gradient of phi is how much the Fermi surface has moved in the direction of p the gradient because gradient of p naught of what pd phi dx because the Poisson bracket so the Poisson bracket between phi and f zero is like px by the p of f zero yeah f zero has p dependent yeah but but but there is um but um all the turnout that all the uh if the the the the coadjoin orbit does not depend on the choice of the reference point you can choose different reference point get the same uh orbit now there is an ambiguity in the um in the uh parameterization of the point of the coadjoin orbit as the elements of the group two different elements of the group can correspond to the same point on coadjoin orbit and that's because if we i multiply this u by u times v but v is the thing that preserve f zero then acting this u v on f zero give me the same as acting u and f zero because v doesn't do anything and this procedure allows us to do a gauge fixing in our phi in particular one can uh think about phi as not a function of x and p but only a function of x in the and in the direction of p we can assume that phi is in but it is the same at all point along the radial uh direction in p depends only on the angle uh parameter of p and so phi is now a function of only x and theta and that's that's how a scalar field at each point in the Fermi surface appears in this formalism okay so now let me write down the actions that turn out to give rise to Landau's equation so the action is written in the following way so there is a very phased term or a wetsumina written term that i'm written in this way so u is remember u is a point parameterizes point in the coadjoin orbit so here is the first derivative how this point moved in time and then this is the Hamiltonian that depends on the distribution function the first term can also be written alternatively as the integral of a two form which in mathematics is called the Kirilov-Gostan-Suryo-Symplectic two form but it turned out to be also possible with some uh um and with some some ignoring some term that is totally derivative as the just integral over time for three for me on this this this omega i can write it but probably i can write it privately for you this this omega i have a slide i can i can show you this uh actually it might be too cryptic for this slide but this omega can be defined as follow if i take a point and then you find two uh tension vector they'll uh i i i i can this decipher this uh for you a bit later uh but i want to uh go to more physical um um um calculation it is it has the properties that it is uh closed uh two form so it can be used as a as a definition of Poisson bracket on the as a symplectic form on the coadjoin orbit so this um this this this is an interesting um place to um to stop and think about the this first term the first term basically is the bary phase term which tells us how much the bary phase a system acute when the Fermi surface changes its shape so we have a Fermi surface that move uh in shape changes and then we have a bary phase and apparently Landau's theory specify what is this bary phase in a precise way the bary phase does not depends on the Landau parameter or the interaction at all it is properties of the phase space itself how did you choose this action why is this one this one is this this Hamiltonian yeah we can check that the equation of motion are all so if we make this choice and we can verify that all the correlation function from Landau's theory the equation of motion the Liouville's equation appears from this action as i i i try to illustrate later in particular the coefficient in front of this term has to be chosen to be one for example then everything should the this fine derivative time derivative term is pretty unique uh yeah i can't see it can you uh unitary perturbation theory and time i think blue but um if i'm not wrong yeah i i was i wasn't aware of this so yeah thank you okay so let's uh now write this action and expand it quadratic order in phi so that's very simple and we get the quadratic action that basically uh chiral boson at each value in phi and the two-point function which in the original theory of free fermion is a one loop graph in this effective theory becomes a three-level diagram with a propagator of phi and the all the formula the calculation can be done and one can check that the two-point function is reproduced but one can also go to higher point function so here the free fermion theory map to an interacting theory of the scalar field the interaction comes from various places it's because for example in this uh wesomino-witton term there are all terms of higher order in phi and we expand the exponent of the boson bracket uh and also in the energy there is also non non non linear terms and when we write it down the result is complicated but very unique the prescription is given we just follow the calculus the the formula and then we can write down all the interaction terms it interestingly free fermion become interacting uh bosons in this approach and bosons a different value of theta coupled to each other in uh unlike in the linear approximation quadratic approximation this is the Hamiltonian uh that is the minus the integral of dt of the Hamiltonian so that Hamiltonian is just the energy associated with the a Fermi surface that is perturbed integral of p square over 2m for example and this swzw is the first derivative term in phi so here we see um the result is complicated but one can do the following check which is quite non trivial that is one can compute three-point diagram three-point functions row row row correlation function and check that it is exactly equal to the sum of all the three graphs that arises from the interaction between the phi fields so uh loop diagrams in fermion is exactly equal to uh the sum of the three graphs and moreover here we see the cancellation between two different Feynman diagram in the case of fermions uh become natural in the bosonic language that is if individually each of these fermion diagram is one over q and cancel each other to be q to the zero of power each of the diagram in the bosonic theory is already automatically of order q to the zero it's a non trivial check thank you and including the deland out parameter in this theory can be done uh uh by just saying that instead of uh the Hamiltonian being the integral of uh energy over distribution function we also include for example the lambda parameter terms so here i write the classic lambda parameter term but there in principle one can also write down all kind of lambda parameters that depends on say derivative of the distribution function or three-point uh the lambda parameter etc so here in principle one can envision a program similar to effective field theory that is one write down effective action based on certain symmetry and then expand it in derivative okay so i have a few minutes left let me mention some a possible extension of this formalism so one thing that one can notice is that the fermion bilinear in a theory of fermion former closed algebra and in the long wavelength limit that closed algebra is actually the algebra of canonical transformation and that means that we can uh extend the theory to include for example spin full permissurface we need to extend the algebra to include not only psi vector psi but psi vector sigma a psi where sigma is a spin matrix pauli matrix and also we can extend the theory to a bcs to include bcs uh other parameter by extending the algebra of canonical transformation to algebra that contain psi psi operator on psi decar psi decar operator this would still be a closed algebra okay so let me conclude the method of co-echoing orbit provide a natural way to write down an effective theory of a fermi liquid uh the the outcome of this formalism is a non-linear posonized version of landau's fermi liquid theory it reproduces uh non-linear and linear both linear and non-linear um fermi liquid responses and perhaps it's a suitable starting point for the study of non fermi liquid thank you very much maybe this is a very naive question but i'm still having trouble understanding where the advantage here is on what people do all the time in the sense of trying to write down an effective ginsburg landau by decomposing exactly the fermi surface into its various shape fluctuations and then taking a four fermi on term doing harvester tonovitch and integrating out is it is it in control is it in the way that you can extend it where exactly is the advantage of using the co-echoing orbit this this formalism give rise to a very unique construction that is all the interaction for example at the three three point three point function uh interactions are all well defined when people do this let me let me just say that this formalism allows one to not think too much and just follow the mathematics um i actually don't know if there is a way more a way to actually arrive to this uh interaction between the bow zones in a way that people have been trying to do so far in the formulation of the Boltzmann equation there is also an approach that you introduce an inner product that looks just remotely similar of course um so the question is could i uh write down the field theory for the Boltzmann equation and um or reversely write down the hydrodynamic version of a fermi liquid uh just as i can write down the hydrodynamic theory of and that follows from a from a Boltzmann equation by the Boltzmann equation you mean just the equation this this uh with the collision term with the collision term well for example because then you have i mean you can write down the Boltzmann equation also in terms of inner products that look very very similar to the ones that you have written down there but you always stay of course at the level of the equation of motion you try to show a type form and so forth but i could now conceive that you write using this action principle to write down the hydrodynamic field theory that would follow from the Boltzmann equation or vice versa that would apply to fermi liquid theories this yeah the collision term has break time reversal so i'm actually not sure if if uh if the simple action uh would be sufficient it might be an action on a double contour but i haven't thought about that uh very interesting uh so if it is a true quantum theory so you you also need to tell us probably what is a measure of integration what's independent fields so is it some kind of hard measure of of your capital use i think so yeah it would be a hard measure of in capital U invariant under the group action yeah and in terms of your phi field it's just supposedly flat that's i'm not actually sure near presumably yeah presumably yeah there is a question in the chat from Andrei Chubukov does this formalism allow one to study non-analytic corrections to fermi liquid example the t-squared terminal specific heat in two dimensions i would hope so that would comes from a loop correction in the bosonic field theory but we have not analyzed it there is one more question in the chat from Ahmed Salah are all elements in coad joint orbital regular values i'm not sure regular values yeah so we are here concentrating on the path of the coad joint orbit near f zero so that makes everything simple but we have not tried to think about the global properties of the coad joint orbit maybe maybe the last question actually this this is probably an extension of your plus Andrei's question so you showed nicely you can very quickly get the polarization so presumably you can get the zero sound collective modes now in helium three it's also known very simply calculation and experiment that you increase temperature and it goes from zero sound to to regular sound you just get this crossover inside can you do this with this theory do you also get it's just increased temperature the crossover probably not exactly this theory because this theory is is zero temperature because f zero we start the coad joint orbit or have sharp for me surface so to extend that to non-zero temperature we have to extend presumably the coad joint orbit to include non-sharp for me surfaces but you could do it with disorder as well probably so it stays your temperature but have a disorder type lifetime but that's also not included right the lifetime of the zero sound in a zero temperature or yeah i give us give the fermions a lifetime by adding disorder but i guess that's also not in this theory and not too in the in the outcast not in the three-level version of the theory but presumably also if it's included in the in the loop some loop diagram