 Okay. So, welcome everyone. We are very happy to have today Dominic Elsa for the next seminar in our Quantum Encounter series. So Dominic is a, he's a condensed method theorist. He's been doing many things, including some more exotic things like he's been just this year awarded the New Horizons and Physics Prize for his work on time crystals. So congratulations. But for today's seminar, he chose something more classic, condensed method subject metals. So he's going to talk about the merchant symmetries and and the normalism methods. Please Dominic, thank you. Yeah, thank you very much for having me here to talk in a virtual sense. So I'm going to tell you about this work that we'll mainly describe in this paper that I wrote with Ryan Fongren at Central. Although I will also mention some more recent works at Central and I have done as well. Okay, so if we want to talk about a quantum many-body system, you know, many, many quantum degrees of freedom in the thermodynamic limit. In terms of the low temperature properties, there's two different things that could happen. The system could be gapped, which means you have the ground state and then you have a non-zero gap even in a thermodynamic limit to the excited states. And we know that in a gap system, even though you basically just have a ground state, the ground state itself can have many non-trivial topological properties. So this gives rise to topological phases of matter, such as very intuitive fractional quantum whole effect, topological insulators, and they are all related to basically the quantum entanglement in the ground state. And in recent years, I would say that the general theory of these topological phases has been much better appreciated than it was before. But in this talk, I want to talk more about gapless systems, actually, and a specific kind of gapless system, namely a metal. So for the purpose of this talk, you can say metal is just a system of a global UN symmetry, charge conservation, and a no-charge gap. So this tower of gapless states above the ground state, they carry charge. So you have charged excitations going down to the totalist energies. If there was no charge gap, then of course, if there was a charge gap, of course, the system would be an insulator. If it had been more precise, I would have to distinguish in metals and semi-metals. This definition would apply both to metals and semi-metals. But let me not worry about that for the purpose of this talk. So although, as was mentioned, metals might be a very classic thing to consider in condensed-house physics, in this talk, I want to show that you can use ideas that originated in the study of gap topological phases to actually gain a lot of insight into metals. So let's just briefly recap the theories of metals. The simplest kind of metal would be, of course, a non-interacting electron gas. You have electrons that move in space, but they don't interact except through the Fermi statistics. So then the electrons just occupy single-particle states. Single-particle states, assuming you have momentum conservation, is labeled by momentum. And then in the ground state, you would occupy some set of single-particle states. And then the Fermi surface is just the boundary in the momentum space between the occupied and unoccupied states. And the low-energy excitations, of course, are the electrons and holes that sit near the Fermi surface. So now maybe we want to switch on the interactions again. So that brings us to this famous Fermi liquid theory. So once you have interactions, you cannot label the eigenstates just by the single-particle states, because a single-particle state at some momentum would just scatter from the interaction. But nevertheless, the Fermi liquid theory tells us, at least for some class of systems, it's valid to consider still that there's some kind of Fermi surface, sharp Fermi surface. And then the excitations near the Fermi surface can be described in terms of these quasi-particles, which have a very long lifetime. Actually, the lifetime goes to infinity, right, on the Fermi surface. And they're kind of dressed versions of the original electrons. And so in terms of the low-energy physics, at least, you can understand it in terms of these quasi-particles. And then there's this very important result. The Fermi liquid theory, a lot of this theorem, that says that the volume enclosed by the Fermi surface is the same as it would be even without interactions. So even though the interactions can renormalize in general the shape of a Fermi surface, they cannot actually change the volume enclosed. Dominic, can you remind me, please, how do these quasi-particles decay? Like, what can they decay into? Into some quasi-particles, which are even closer to the surface surface, what's happening here, right? Well, the quasi-particle is very close to the Fermi surface to kind of don't decay. So you have to go a little bit off the Fermi surface. And then it kind of decays into the continuum. I don't think there's a simple way to say it. Into the continuum, okay. Yeah, if you create a quasi-particle and then it decays, it kind of goes into a I mean, the quasi-particle is like a dressed version of a single particle state. But then it just goes into some many particle state that I think you can describe simply. But is there here something related to the fact that momentum is not conserved or even the presence of momentum? The momentum is conserved. We will always assume in this talk that momentum is conserved. So whatever final state you get will have the same momentum that it won't be just made up of a single quasi-particle. It will be some many particle state. Okay, thank you. But in any case, the main properties of Fermi look like they come from the infinite, assuming infinite lifetime. The decay is kind of the leading correction to Fermi laboratory, if you like. Okay, so let's firmly look at Fermi. And this is a very famous theory. It successfully describes many materials at low temperatures, but not all materials. As we now know, there are many materials, including some famous classes of materials like cuprates, which also has high temperature superconductivity that Fermi look at theory is just clearly inadequate description of those materials. And then the question becomes if it's, if you cannot describe these materials by Fermi look at theory, then what is the correct theory? And that's a very hard problem. And in many cases, such as the cuprates, not well understood at all at the moment. But in general, if you go beyond Fermi look at say, you have a metal, but it's not the snapback from liquid theory. That's what we call a non Fermi liquid. And so non Fermi liquids, you do often find both experimental and maybe from some theoretical models that they have some kind of sharp Fermi surface. Nevertheless, the sharp Fermi surface is no longer associated with these well defined quasi particles. You have more strongly interacting quantum super particles, they cannot be described just in a quasi particle language. And so then there are different, I mean, apart from the question of, you know, what exactly is the theoretical description of these things, which is maybe a very hard problem to answer, there are some broader conceptual questions that we can ask as well. Like, what exactly do we mean by a Fermi surface in these systems is if it's not the place where the quasi particles are. You know, the volume enclosed by the Fermi surface, does it still obey some kind of Latin just theorem like constraint or not. So these are questions I would like to address in this talk. But also I would like to pose a more general question, which may be really specific questions will be an instance of. So the question basically is so we know probably what's going to happen in for any given material, we probably know what's happening microscopically, it's just going to be described as a Hamiltonian that involves electrons hopping on some lattice. So that's a microscopic Hamiltonian. Of course, as we know, it's very hard to predict from a microscopic Hamiltonian in a quantum many body system, you know, what the nature of OMG physics could be, you know, that's a quantum many body problem is a hard problem. And there are many exotic behaviors that could occur. And so degrees of freedom at low energies, maybe, you know, can be emergent degrees of freedom that don't have much to do with microscopic degrees of freedom. But nevertheless, the question you can ask the following question, you said, can ask what questions does the I offer? So the theory, of course, here, I just mean the, you know, if you like to configure the renormalization, but it's just whatever theory that describes just for low energy physics of material. What properties does the I offer a need to satisfy, if it's going to emerge from a particular microscopic model, like maybe you can't fully predict from a microscopic model what the I offer is going to be, but at least can we place some constraints on, you know, what properties of an IR theory would be compatible with the microscopic model. And that's what we call the question of sorry, the question of emergibility, like, what kind of IR theories can emerge as the as the low energy physics of some particular microscopic model. And so let me say it more precisely now, or pose a more precise question. So the, so the microscopic cell, we have a microscope Hamiltonian in particular, I'm going to assume of a microscopic cell, we have this particular symmetry group. So we have charge conservation symmetry, and lattice translation symmetry, which is an appropriate symmetry group for metals, for example. So that's a microscopic symmetry. And then from a microscopic symmetry, you can define this number. It's a real number, which we call the feeling is a very important number for my talk. And the feeling just represents the average charge per unit cell. So in general, this can be could be any real number. And then the question is, you know, given that we have this microscopic constraints, microscopic symmetry and the microscopic filling, then what constraints does that lead to in terms of the low energy physics? So any real number shouldn't be a rational number? Well, that's going to be important distinction later in metal, but for metal, it can be any real number, because for metal, basically, the God's sphere, I'm safe to say that the volume enclosed by the Fermi surface is equal to new. So because the Fermi surface, you can deform it continuously, there's no constraints. You can continuously tune this number new in a metal. But where do these electrons come from? Sorry for a late question. I'm not an expert clearly, but I thought that these electrons, they just come from some atoms that share the electrons, or you're going to inject some extra electrons into your lettuce from outside to change this feeling friction continuously. Yeah, well, it's sort of a theoretical feeling. Well, I mean, often, you know, there's some chemical chemistry constraint, right? Each atom contributes some number. But theoretically, you can just add more electrons into the system and see what happens. I mean, in practice, you could do this, for example, in a two-dimensional material by gating the material, applying some electric fields that will basically change the equilibrium charge density. Or you can also imagine doping the material with dopants and then that changes the electron concentration. Well, that's a little subtle because it might break the translation symmetry of disorder. But in certain regimes, you can just take into account the change in density from the dopants and not worry about the disorder. Okay. So maybe I'll ask, so in general, when you have a symmetry group, then the charge is an element of what? I mean, given a symmetry group, just confuse the, like you have zd times u1. And then if you just have a symmetry, let's say absolute group g, then the charge would be an element of workspace. Well, I guess in general, charge is an element of the dual of that group. But here I'm really just assuming that here by charge, I mean the u1 charge. So it's the charge under the u1 symmetry. Yes. But so the dual would be the discrete. I think I have the same question as Lava. Like usually when we have u1 symmetry group, that's the reason why the usual charging quantum field theory is discrete because it lies in the character lattice of u1. Yes. So the total charge of the system has to be an integer. But here I'm talking about the charge per unit. So you divide the total charge by the number of, by the volume, the number of unit cells. So that can be some real number. So I guess it's true for any finite system, it has to be rational, but we're always considering here the thermodynamic limits. So there it will be valid. Is there any real number? Yeah, I'm more comfortable now doing it. Sorry, I don't know about Mercedes, but I'm happy. I also have a question about this filling number. If you, so depending on what kind of state you have, it'll tell you what kind of filling number you have. So in particular, you're looking at not just the system, but all states that admit this filling number as well. Or am I misunderstanding something? Well, it's not always. I mean, if you really know the microscope that I'm turning, then you know the filling number. But my question is more like, suppose you only know the nature of the low energy theory, what is the relationship between that and the filling? That's not obvious a priori. It's sort of a question I want to answer in this talk, you know, it's not a totally trivial question, because it's often pretty hard to see from the nature of low energy theory what it has to do with the microscopic filling, at least without the theory that I'm going to give you. Got it. Okay. Thank you. So first point is that suppose the filling wasn't integer, and it doesn't really impose any constraints on the low energy theory, because we know that you can have an integer filling, and then the theory can just be your fluid atomic insulator. So you just have some atoms, and then you put some number of electrons on each atom, and the electrons just stay very tightly bound to the atoms. So they're not moving around or anything. So obviously, you know, then the fluid has to be an integer, because you know, you choose how many electrons you want to put on each atom. But you know, the low energy theory of an atomic insulator is basically completely trivial. Firstly, it's gapped. And secondly, for grounds that doesn't even have any non-trivial topological order or anything, because it's just literally a product state. So, you know, this, you would say the IR theory is literally, you know, the trivial theory. And yet, you know, you can still be any integer. So the integer filling is compatible with totally trivial low energy theories. So the interesting constraints will come about if new is not an integer. And so an example of a result on this line is this famous result of Rachel Smiddish of Kavar-Histins. Suppose you have a system that has U1 chart conservation symmetry and lattice translation symmetry, and suppose a microscope showing is not an integer. So it has some non-trivial fractional part. Then either one of the theorem states that one of these possibilities must be true, either one of the symmetries is spontaneously broken, or the system must have ground set digits in the torus, which I guess would mean you have some kind of topological order, or the system must be gapless. Okay, so that's already a very strong constraint of the nature of low energy physics, just coming from this feeling of not being an integer. But we would like to go a bit beyond this. I mean, this result of our powerful is still a bit vague. First of all, is, you know, these three different possibilities. Secondly, each of these possibilities is still very general, like for example, gapless. I mean, there's so many different kinds of gaplessness. And so we would like to have, you know, a stronger statement than this. And so in order to do this, so, okay, we'll assume that it's non-zero, mod 1, because if it was zero, mod 1 opening in its journey, so then there's no constraint. And then we want to address what the constraints are of low energy theory. In order to do that, I'm going to make connections of the idea of emergent symmetries and anomalies, which is, as I mentioned, also related to the theory of gap-top particle phases. So I'm going to illustrate this by an example, first of all. So I'm going to consider a Lagrangian liquid. So this is a one-dimensional system. So you can think of a non-interacting limit where it's just a one-dimensional non-interacting electron gas. And then, you know, similar to the picture I drew before, you just fill up the single particle states. And then the Fermi surface here is just a pair of points. It's what I call the Fermi points. So these are where the low energy excitations live. You can also switch back on the interactions and you get a, what's called a Lagrangian liquid. Although that turns out that distinction, although very important for many things, will not be very important for what I'm going to say. So if you like, you can just think about the non-interacting limit, but the statement still holds with interactions as well. So this theory actually has an emergent symmetry, which is interesting property. So microscopically, of course, you have a single U1 symmetry. But in an emergent sense of symmetry, if the low energy theory is actually U1 times U1, because the left moving point and the charge of the right moving point is separately conserved, you know, in the low energy theory. So that's an emergent symmetry. And then this emergent symmetry also has an interesting property, which is it has an anomaly. And what that means here is that if you apply an electric field to the system, then the left and right moving charges are no longer separately conserved. Without the electric field, they will be separately conserved. But if you apply the electric field, they're no longer separately conserved. And so we have this non-trivial continuity equation for the left and right moving currents with a non-to the right hand side proportional to the right to the electric field, which is telling you that these left and right moving charges are not separately conserved. Of course, the total charge is still conserved, because if you add these two equations, the right hand side will cancel. But the left and right moving charges are not separately conserved. And so this is a more general example of something that we call a tuft anomaly. I will define what I mean by tuft anomaly in a minute. But first of all, I want to argue in what way this is related to a topological phases. And in order to do that, I'm going to suppose I, so this is an emergent symmetry, but suppose I wanted to have this kind of same kind of structure, which is anomaly, et cetera, but where you, the left and right moving symmetries become microscopic symmetries. So a way to do that would be suppose that you have a symmetry, a system that microscopic has both charge conservation and also conservation of the z-component of spin. So that could be a plausible microscopic symmetry. And then if you wanted to have the same anomaly structure for this microscopic q1 times u1, basically what we would want to say is that the, you have some kind of gap-wise point where the right moving modes carries spin up and the left moving modes carries spin down. So people, this is what people call helical modes. It's like the direction that the electron is moving is locked to the spin. And so then the, you know, if you think about the conservation of up spins and conservation down spins, then that maps onto this conservation of left moving and right moves, which I described on the previous slide. But the interesting thing about these helical modes is that they can only occur on the boundary of a topological phase in one high dimension. It can actually never occur in a strictly one-dimensional system. So you can have a topological phase. So this is, you know, some gapped phase of matter, but where the ground state has some non-trivial topological properties. And then a manifestation of those non-trivial topological properties on the boundary will be, first of all, the boundary is kind of forced to be gap-wise. And secondly, the boundary theory has this anomaly structure that if you apply an electric field, the up spins and down spins are not separately conserved anymore. And so there is this general correspondence, bulk boundary correspondence between two of the non-ways in these spatial dimensions. And so two of the non-ways just really means it's, well, maybe this is not the most general definition, but it's sufficient for this talk. Tuft anomaly is the non-consolidation of charge in response to a background on gauge field. So for example, you applied an electric field and the left and right, left and right movies were not conserved anymore. So that was an example of a tuft anomaly. And so there's a, there's this bulk boundary correspondence where a non-trivial topological phase like a topological insulated, for example, in D plus one spatial dimensions has this boundary theory that carries this tuft anomaly on the, on the boundary. And so in some sense, the tuft anomaly in the topological phase are just two sides of the same coin there. The clarification of tuft anomalies is in one to one cost once with the classification of topological phases in one high dimension, or specifically symmetry-protected topological phases in one high dimension. So, but now coming back to this laundry liquid, I said that these tuft anomalies can only curve a boundary of a topological phase in one high dimension. But then maybe you find this statement strange because, I mean, a laundry liquid is just a one-dimensional metal. Like you would say why it doesn't have to be on the boundary of anything. Well, the key, the key word here is there's an emergent symmetry. This anomaly is defined with respect to this emergent symmetry. So it is, it is perfectly okay for emergent symmetries to have anomalies, even if you're not on the boundary of a high-dimensional topological phase. But if you wanted to make this emergent symmetry macroscopic, that's when you would have this obstruction. Okay, so this laundry liquid of course can exist in 1D because the symmetry is emergent. But the tuft anomaly, you know, it can still be helpful to think about this high-dimensional topological phase just because we know that the classification of the tuft anomalies is in one to one cost once with the classification of topological phases in one high dimension. Sorry, let me try to understand you. So basically, what you are seeing here is that you want the anomaly to be there in the infrared, but along the RG flow is not there because your symmetry is broken. Yeah, I mean, because the microscopic theory doesn't have the symmetry, no, there's no sense in anomaly. Sorry, saying something that I didn't hear. Maybe I don't know if that was a question. That was just a noise. I see. Yeah, you're right. So you cannot match this anomaly from the UV in the eye out because the symmetry is not present in the UV. So now I want to show you how you can derive lunges theorem from perspective with anomaly. So here essentially, I guess I'm following this seminal paper by Yamanaka Oshikawa, Affleck. I'm not phrasing it seems exactly in the way that they did because they didn't talk about anomalies. It was not in those words, but essentially it's the argument that they gave. So the first thing we need to do is we have a microscopic level. We seem to have some lattice translation symmetry. And then you have to say, well, the lattice translation symmetry somehow needs to be manifested in the low energy theory. And so in particular, it's going to look like this. So NL and NL here are the generators of the left and right moving charges, which so these are generators of the U1 times U1 symmetry, the immersion symmetry. So the lattice translation, they just X like this, like KL and KR, if I'm meant to have the left moving and right moving Fermi points, we have some amount of charge here and some amount of charge here, you can see this is how much momentum you're carrying. Notice of the overall, it's a translation in the microscopic actually maps into an internal symmetry of the eye after which is actually a general feature because it's related to the rescaling as you go from the UV to the IR. So this is how we got this translation somebody asked in the IR theory. So now what we're going to do, so this is a picture in real space here now not momentum space, we can take our one dimensional system, we're going to put it on a ring. And then we're going to slowly switch on a magnetic flux. So as a function of time, you go from zero magnetic flux to a two pi magnetic flux. So we know from Faraday's law that, you know, that generates an electric field. So therefore, you can, that's where the tooth normally comes into play, because you apply an electric field to the system. And in particular, you can show that if you do this flux threading process as you go from zero to two pi left moving charges, we shift exactly by one and then the right moving charges by one. And since we've expressed the translation operator in terms of these left and right moving charges, you can now see how the translation operator is going to shift. And so essentially it's the same because translation operators for momentum, that the momentum gets shifted by this process. Of course, you apply an electric field. So it makes sense that the system is going to feel a force and so the momentum will get shifted. So this is all statements that hold, you know, with respect to the low energy theory. Then we also know what's going to happen microscopically. Microscopically, the, you know, the force, the momentum that the system picks up has to be proportional to the charge density. Or more precisely, with ladder sensation symmetry, and you do this two pi flux threading, the translation operator will get shifted by amount proportional to new. So you remember, new is this number I introduced previously. It's the charge per unit cell. Okay, so these are just two descriptions of the same process from two different perspectives. So you have to equate these two shifts. And so you conclude that kr minus kr, which is here, but basically the volume enclosed by the Fermi surface is equal to new, at least mod 1. And so this is London's theorem and through this one dimensional system. But the important thing to note here is that I have not assumed anything about the low energy physics. You know, I actually didn't need to know anything about London's deliquids really. The only thing I needed to know was firstly that you have this U1 times U1 emergent symmetry. Secondly, but it has this Tuft anomaly. And then you need to say, how the translation operator maps into VIR theory. And that was the only ingredients you needed to try for London's theorem. So the point is that you can do this very generally. Sorry, I, yeah, this, so usually one says that the Slottinger's theorem relates interacting from a system to free-firm and system, but here kind of just prove it generally, you didn't assume that the system was, there was no interaction that you had to turn on. Well, usually one says that this has to be some adiabaticity, you relate things to to the free theory. Yeah, exactly. Is there this aspect here present? Yeah, so you don't know, you know, you don't need to see anything like it here, which is the power of this approach. Okay. I mean, that's the beauty of, I mean, that statement is also true for this paper. That's really the beauty of this paper, but it does not require any perturbative argument. It's really true, non-perturbatively. It's a non-perturbative statement. Right. Okay, thanks. So we don't need to restrict ourselves just to take them to liquids. The point is that this flux-frogging argument is something you can do in any theory, you know, whatever the nature of the lone geophysics is, you can do this flux-frogging argument. The conclusion that you obtain is that you only need to know these particular features for the lone geophysics. You need to know it's an emerging symmetry, which I call GIR here. You have to know how a microscopic symmetry is mapped into GIR. So the microscopic symmetry here is assumed to be a translation and charge conservation, so you have to know how he's mapping to the IR theory. So this mathematically just would be like a group homomorphism from Zd times u1 into GIR. And then you need to know the Tuft-Tinomri of the IR symmetry. And then by doing this flux-frogging argument, basically you obtain a general form with it, as you compute the sewing in any lone geophysics whatsoever. You just need to know these ingredients. So obviously the ultimate generalization of this theorem is this statement about a particular kind of theory like, for example, Fermi liquids or Lange liquids. But this result is a statement about any lone geophysics whatsoever. There's a general prescription to compute the sewing. Sorry, sorry. I want to understand more clearly. So this u1-xl, it comes from the translation, lattice translation. I wouldn't say it comes from. I mean, it's just a property of the lone geophysics theory that it has this imaginary symmetry. But it's true that the microscopic translation will map into that axial symmetry. Yes. So my question is this, would this kind of symmetry always be anomalous in the infrared? Basically, I mean, symmetry is that kind of originated from translation. Well, it has to be anomalous if the sewing is non-zero, because if the Tuft-Tinomri was trivial, then the sewing would just be zero, because then the flux-frogging argument kind of doesn't do anything to the system, because there's no anomaly. So then the sewing would be zero. So if you have non-zero sewing, then you'll always need to have an anomaly. Yeah, sorry, non-zero at mod 1. So everything is mod 1 here. So these arguments can be done in any spatial dimension. Previously, I just presented you the one-dimensional version, but here I'm saying you could have d spatial dimensions. And so I want to maybe not do it in arbitrary spatial dimension, although you can do it, but I want to at least consider the d equals 2 case and show you how that works, because and that's maybe also a bit more novel compared with this famous flux-frogging argument. So I'm going to show you how these arguments worked in two spatial dimensions now. And specifically, I'm going to consider a Fermi liquid in two spatial dimensions, although as before, the arguments will hold more generally. So there's an important property of Fermi liquid theory. It's emphasized perhaps by Haldane. It has a very large emergency symmetry group actually. So the larger group has u on times u1, but Fermi liquid has a much larger emergency symmetry. So this is the famous Hamiltonian Fermi liquid theory. And in particular, this Hamiltonian has a property that we charge at every point of a Fermi surface is separately conserved. So in terms of the emergency symmetry, so this is a low energy properties. So the emergency symmetry group is actually infinite dimensional because the generator is like the charge at each point of a Fermi surface. So let me be a bit more precise about what the emergency symmetry group is. I said that the charge at each point of a Fermi surface is separately conserved. It's not really that you have a u1 symmetry for each point of a Fermi surface. I mean, because there's, you know, the Fermi surface with a set of all points of a Fermi surface is an uncountable set. So u1 to the uncountable infinity is not a very well-defined object. So it's not quite that you have u1 for each one of the Fermi surface. The correct statement is that for each point there's a conserved linear charge density. And by charge density, I mean that it's a thing that you integrate over theta. So theta here is just some coordinate parameterizing over one dimensional Fermi surface. The total charge is the integral of this linear charge density. So it's a charge density in theta space, not in like real space. And so the point of this linear charge density is something, it's really something to integrate against test functions. So a general element of the immersion symmetry group can be written as like this. So you take some smooth function f of theta, you integrate over against the n of theta, and then take exponential. So that gives you the most general element of the immersion symmetry group. And then so quantization of total charge will give you a constraint that you want to identify f theta with f theta plus 2 pi. So f of theta really lives on the circle or on u1, if you like. So the total, in summary, the actual precise definition of immersion symmetry group of a 2D filament liquid, which somehow, I don't think anyone actually stated until we wrote our paper, is that it's the group of smooth functions from the circle into u1. And this is something that mathematicians have called the loop group of u1. Okay, so that's the immersion symmetry group of a 2D filament liquid. Is that clear? Physically, it's just reflecting the fact that you have conservation of charge, but each point of affirming surface separately. Because quasi-particles cannot scatter, at least in the low energy theory. There's no quasi-particle scattering. So if you have a quasi-particle somewhere, it just stays there forever. Okay, hopefully that's clear. And so this loop group symmetry actually has an anomaly. So there are many manifestations that you can talk about. Let me just focus on one particular manifestation of this anomaly. It's basically you insert a 2 pi magnet flux system. So unlike in the 1D case, I'm not thinking of this as a function of time. I just consider some static configuration where you have 2 pi flux sitting somewhere in your two-dimensional system. You imagine that the magnet field is very weak, so we spread this 2 pi flux that were a very large area. And so on this object, it turns out that this loop group symmetry actually acts projectively. So without any magnetic field, the loop group symmetry is just an abelian group of infinite-dimensional. But when acting on this particular object, the loop group actually gets a central extension. So in particular, there's this non-trivial commutator. So n of theta here, remember, with the generators of the loop group, there's non-trivial commutator that n of theta has. If the commutator is proportional to this contact term, which involves the derivatives of the direct order function. So this is the manifestation of the anomaly that you have this projected representation when acting on a magnetic flux. And so this coefficient here is actually quantized for a Fermi liquid, at least a spinous Fermi liquid. The coefficient here is just 1. In general, it's consistent to have any integer sitting here. So that's like the quantized anomaly coefficient for spinous Fermi. Can you give an idea of how one derives this result? Supposedly, just for Fermi liquid, what should I do to compute this kind of data? Well, there's different ways you can think about deriving it. I mean, the way we did in the paper, I don't know if it's the best way or not, is we considered the semi-classical equations. We know that the quasi-particles in a Fermi liquid, especially in the non-tracking case, can be represented by some semi-classical equations of motion. The semi-classical equations of motion come from a Hamiltonian, you know, a canonical Hamiltonian in the classical Hamiltonian language. And also you can define Poisson brackets in that picture. And so you can just switch on the matter of field, derive the Poisson brackets of these density operators, and then you just sort of claim in quantum mechanics Poisson brackets will become a commutator. Yeah, I mean, I guess that's still a little formal, but okay, I don't have a clear physical argument in that. Mathematically, it's a precise statement. Can I ask you a question? Yes. Is this symmetry, loop group symmetry, acting only on the zero energy quasi-particles, or also away from the Fermi surface? Well, on the low energy, it's an emergency for low energy theory. So you would, but so it would be defined in respect to the low energy excitations, but not, it doesn't have to be exactly on the Fermi surface, but close enough to the Fermi surface that you're still in the low energy regime. But as you described before, when you are not exactly on the Fermi surface, the excitation decays. Yeah, but it decays, you know, slowly in the sense that, you know, it's like, as you go a mega away from the Fermi surface, the scattering rate is omega square. So those terms that cause the scattering, you should think about it in the language of renormalization loop irrelevant. So you can consider the RG fixed point theory, whether it's irrelevant terms set to zero. Once you put those terms back in, of course, the emerging symmetry will be broken, which is the usual story of emerging symmetries. Okay. So this is the commutator. So this is the anomaly. And so now I'm going to use this anomaly to drive logist firm. So I don't know, for a question of why is this commutator true, maybe, you know, the fact that it reproduces logist firm, maybe that's, okay, that would be kind of a circle argument, but anyway, it will give some insight. So, okay, we have this object that we want to consider this 2 pi magnetic flux. And then we want to, so let's consider microscopically what's going to happen. So next topic, we have this discrete translation operators now in two dimensions that we have x and y translations. And then we can consider this 2 pi magnetic flux and this object t x, t y, t x inverse t y inverse acting on the 2 pi magnetic flux. And so I claim that acting on the 2 pi flux, you get a non trivial phase factor that's proportional to the microscopic feeling. There's a hat, I mean, you can give a precise argument for this as we did in the paper, but let me just give you the hand waving version. If you have a 2 pi flux and then you act on it with this object, it's kind of like you're moving it around one unit cell, but a unit cell contains by definition charge new. So then you get an arrow of bone phase, you know, each of it to buy a new. So that's a rough argument anyway. So that's what happens microscopically and as before, we want to match it with something that's happening in the IR theory. So first of all, you need to say how does the microscopic translations act in the IR theory, so they will act like this. So we have our filmy surface. You know, parametrized, we parametrized the location of the filmy surface by this parameter theta and kx theta, ky theta is the momentum of the filmy surface as a function of theta. So, so this tx and here I just say alpha equals x and y, x or y. And so tx and ty will map in this way into the law in G theory. So again, it's an internal symmetry. Now we can consider this 2 pi by 8 flux and then we know that we have this commutator of the nth fetus. The conclusion is that because the tx is nty is expressed in terms of the nth fetus, now you can compute this commutator of the tx and ty and you'll see that it's proportional to this integral because it's derivative of direct delta function you integrate by parts basically you get this integral and this integral is just computing the volume encoded by the filmy surface. So now all you have to do is you compare these two equations and you'll see that the volume encoded by the filmy surface is equal to the microscopic fill in mod 1. So again, this is London's theorem derived in a non-perturbative way and it only depends on, as before, the emergent symmetry, in this case for loop group symmetry, the mapping of the translations into the law in G theory. You can actually sort of think of this as defining the filmy surface in the most general context because whatever the translation is as to somehow mapping to some operator, this is the most general form of a symmetry operating in the law in G theory. So that you can give us just the definition of a filmy surface. And then, yeah, and then you need the anomaly of the emergent symmetry, which is this equation. And from these agreements, you re-derive this constraint of London's theorem. Okay, is that clear? So as before, it's the same machine that I promised you before. The only ingredients that you need are the emergent symmetry, the mapping from microscopic symmetries into emergent symmetry and the toothed anomaly. And then there's a general form of driver filling just by doing the same procedure in a general, whatever the most general theory is, you insert the tooth by flux and compute the commutator of the translations. It gives you the filling. So, okay, so what can we do with this? I mean, you can try to apply this to many different theories and compute what they're filling is in terms of only physics, which is pretty useful. But there's also things you can do. Once you have a general formula, you can prove general theorems. And so let me tell you about some of the general theorems that we prove based on this formula. So one of them is the following. So suppose the fulfilling is an irrational number. Then the emergent symmetry group cannot be a compact finite dimensional regroup, especially mentioned degrade and equal to two. So D equals one, London's liquid would be a kind of example. But for degrades and equal to two, we have this theorem and that we can drive based on these general arguments. So what is the significance of this theorem? Well, particularly if you have a compressible state, which means that new is continuously tunable, which so generally you expect metals to be compressible. If you have a compressible state, when new is continuously tunable, then in particular, you can tune new to be irrational. And so the consequences of this theorem will then apply. So, okay, insulated is not generally incompressible. So you wouldn't have this result. But for compressible states, we generally expect metals to be compressible, even for non-firm liquids, you have this conclusion. You conclude that the emergent symmetry group cannot be a compact finite dimensional regroup, and especially mentioned degrade and equal to two. So for example, for me, liquid satisfies this theorem because they have this, I mean, they're compressible, but they have this infinite dimensional regroup symmetry. And so the point is that just from this compressibility assumption, basically you conclude that it must be this very non-trivial emergent symmetry structure. I've heard it's an infinite dimensional regroup group. I guess the theorem doesn't include, doesn't necessarily exclude some alternate infinite dimensional group, although we don't really have any known examples of that. But compressibility does at least give you some very non-trivial emergent symmetry structure as a consequence. Okay, so then we can ask, what is the possible emergent symmetry groups for a compressible state of matter? And say it's focused on spatial dimension equals two. Because it cannot be, it definitely cannot be trivial, and it cannot in fact be any finite dimensional regroup. So what could it be? What is the most general possibility? So okay, in Fermi liquid, we know that we have this infinite dimensional glue group symmetry. You could consider some non-Fermi liquid that nevertheless still has the same emergent symmetry group, and that's what we call an Erzatz Fermi liquid. So if you're non-Fermi liquid, you don't have cosy particles anymore. But that doesn't necessarily roll out the possibility that it could be a conserved charge at each point on the Fermi surface. So there's this possibility that non-Fermi liquids still have the same emergent symmetry structure as a Fermi liquid. And if that's the case, then there are many consequences of that, because for example, you can drive the London's theorem even for non-Fermi liquids, because it just comes from the anomaly of this emergent symmetry. An example of such states where this symmetry is present, is it hypothetical or do such examples exist that this Erzatz Fermi liquid? Well, there are some examples. There is theories of non-Fermi liquids where you have a Fermi surface coupled to a critical boson. And in this theory, you see that the actually of over this kind of scattering, there's very like small wavelength scattering in the low energy limit, there's no long wavelength scattering, sorry, no large case scattering from the coupling to the critical boson. And so the conclusion is that you do still have this conservation law associated with each point of Fermi surface. So this is Fermi surface coupled to critical boson is something that's been very extensively studied in the context of non-Fermi liquids, although often not from very well-controlled methods, but recently has been developed and said certain like large end limits in which it's analytically controlled. But anyway, so to these non-Fermi liquids, to exactly understand them from coupling to the critical boson would have this conservation law. Sorry, what would be the experimental signature for between like quasi-partic picture and non-quasi-partic picture? Well, okay, experimental signature is also, I mean, one thing that you often look at is resistivity because the Fermi liquid theory, the resistivity, which is actually determined by the leading irrelevant operator, generally goes either as temperature squared or some high power of T. So if you see a power of, if you measure resistivity and see the scaling of some power of T just less than two, then you know you have a non-Fermi liquid. You can also look at like in our pairs, you measure the spectral function, the quasi-particle versus non-quasi-particle is related to the width of the peak in the spectral function. So in some sense, I mean, as I mentioned, there are series, particularly controlled examples where we have this conservation law, but we sort of think that this conservation law probably has to be general in any non-Fermi liquid just because of this theorem that if it seems it's compressible, it has to be an infinite international symmetry group. Again, in theory, it could be some different international symmetry group. If that was the case, it would be very interesting, but it's not clear whether that could ever happen. I think the most likely possibility is that all non-Fermi liquids actually have this emerging symmetry structure. Same as a Fermi liquid. So there's one other possibility that is interesting to consider, which is that in a super fluid, so super with a spontaneously break charge concentration simply, however, that does not mean that you cannot apply a general framework. A general framework really doesn't care about spontaneous symmetry breaking, as long as it's spontaneous and not explicit symmetry breaking. You can still apply the general framework. So super fluid spontaneously breaks charge conservation symmetry, and it's compressible as well. So, okay, this theorem should apply to this super fluid as well. I mean, the way it actually doesn't have an infinite international symmetry group, the way it evades the theorem is that it has this emerging one form symmetry. Basically, the vortex winding number, which is something that you measure on like closed curves. So that's a one form symmetry. Maybe if you don't know what a one form symmetry is, you don't need to worry too much because it's not the main point I want to make. But anyway, super fluids have this emerging one form symmetry. I didn't mention it when I stated this theorem, but one form symmetries are also a possibility in the presence of when you have a compressible system. Okay, so these are the two options that we know are to get a compressible state of matter. Either you have this loop group symmetry, or you have this one form symmetry like in a super fluid. So are there any other possibilities? I think that's a very important question to ask. And, you know, as far as we know, I mean, we can think about different kinds of known states that matter, exotic or not, but all of them basically want to fall under one of these two categories. I mean, there are slight variations like the loop group could be extended by some finite one form symmetry, which is the case where a fermi-liquid is co-existent of top article order. So that includes things like the fermi-liquid star phase that people sometimes discuss, whether the bloodstream is violated by a constant shift due to a top article order. Yeah, but it's essentially kind of the compressibility still comes from this loop group. So it's just some slight variant of this. So is there anything fundamentally different that's the question I want to pose? I don't have the answer to that. All known examples seem to fit to one of these two categories. So that's an important question for the future to consider. So there's another theorem that I, general theorem that I can state. Well, it actually just, it's only a theorem in D cos 1. It's a conjecture in D greater than 1, D being the spatial dimension. Although this conjecture is true for anything that's like these examples. So the only way this conjecture could be violated would be if there was be something fundamentally different from these. Anyway, so the theorem such conjecture states that the system is compressible if and only if the Lagrangian theory supports dissipation of statistical current, which is to say the local resistivity is exactly zero. So it is sort of clear a super fluid famously has dissipation of statistical current. So that's compatible with theorem. It's compressible when it has this dissipation with current. Maybe I need to say something a bit more about thermo liquid theory. So in a thermo liquid theory, you can, so you have a thermosuppress, you can consider a non-neurogram state where you've created some non-neurogram charges distribution on the thermosuppress like this. So thermo liquid, you know, is also represented as a current carrying state. And moreover, the state kind of has infinite lifetime, at least with respect to the low energy theory because of the emergent symmetry. Like with charges, each point of the thermosuppress is conserved. So this charge distribution gets frozen in place, basically. And so therefore, the low energy description of thermo liquid theory also has exactly zero resistivity because you have this current carrying state that has infinite lifetime. So the current doesn't degrade at all. And in fact, you can argue even beyond thermo liquid theory that just from the statement of this emergent symmetry and Tuft anomaly that this state actually carries electrical current. So even if you didn't have a thermo liquid, and thermo liquid, it's obvious that this state carries current. You just look at the group velocity on the thermosuppress and you compute the current. It's non-zero. But even beyond thermo liquid theory, assuming you know you have an emergent symmetry, the loop group and the Tuft anomaly, you conclude that this state also carries electrical current. And so therefore, from this loop group and anomaly, you get this dissipation of electrical current in great generality. And then the thing to mention is that this is all statements about the low energy theory. So at non-zero temperature, you also have operators, additional operators that are irrelevant to the RG sense. So these operators cause some scattering processes and that would grade the current. So then you'll get a finite resistivity, but kind of high resistivity at low temperatures because the scattering is very slow. So very low resistivity at low temperatures because the scattering is very slow. So let me make this statement, electrical current and Tuft anomaly is more precise. I consider a grand canonical ensemble state, a generalized grand canonical ensemble state because I have this NF theta that is conserved through each theta, NF theta remember this charge density on the thermosurface. So I can introduce a generalized chemical potential now that actually depends on theta. So this is a generalized grand canonical ensemble state. And this is sort of like a non-equilibrium state, but it's really equilibrium with respect to this generalized ensemble that takes into account all the conservation laws. And then if it's generalized grand canonical ensemble state, I don't have time to give you the derivation that if you have the conservation of NF theta, so the root group symmetry, you can show that there's a general formula that competes with current in this state. And it looks like this. So K of theta here again is just tracing out the thermosurface in the momentum space, new theta of these generalized chemical potentials. M here is the quantized anomaly coefficient. So for a thermo liquid, it's just one. And so the dissipation was current is actually protected by the anomaly and by the conservation law. I'm kind of running out of time. So let me skip over these slides, which were not really essential to my talk. I just included them in case I had time. And so I will come to the conclusion, which is that, okay, the conclusion is that in general, the idea of emergent symmetries and anomalies are powerful ways to think about properties of energy theory. And the anomalies are ideas that we originally thought about in contact with gap topological phases and their boundaries. But for emergent symmetries, it's useful to think about anomalies in their own right. And there are many connections to interesting microscopic properties like fueling, compressibility, I talked about electrical resistivity. And I see also our what was central in which we specifically applied these ideas to strange metals, which are exotic metallic phase seen in in cup rates and also materials in which the resistivity is perfectly tilinear down to zero temperature, which is very mysterious. But we were able to try to drive pretty strong constraints on what could be going on in these strange metals based on the general arguments. Okay, so yeah, I will finish now. Okay, thank you, Domenic. So I have a question. Is there some difficulty in going to the larger larger than two? Could there be nonfermal liquids in the larger than two? Yeah, there can be nonfermal liquids in. Well, I think for dynamical reasons, it's sort of easier to get nonfermal liquids in low dimensions, but I don't think there's any fundamental obstruction to having nonfermal liquids in any dimension. Certainly these general arguments can be applied in any dimension. Can you use your methods to prove what in your theory also for DETO3 where it was originally proven? Yes, these arguments can be applied in any dimension, but the degree you think that one just sure. So there's going to be some interesting anomalies. So is there some, so how is this going to generalize? It's going to be, and which is going to now be a function of two variables in the sphere, but the commutator is going to look like what? Well, it's not going to be about the commutator anymore. The commutator is really in D equals 1. It becomes a little bit tricky to explain in simple language what it means, but what it normally means, it's like a projective representation. It looks like the anomaly that would appear at the boundary of a one-dimensional topological phase, like the Haldane phase, for example, has a projective representation of spin on the boundary. So the statement in 3D would be you would look at a 2 pi flux tube, which is now a one-dimensional object, and it will kind of, the symmetry will act on this one-dimensional object as though it was the boundary of a two-dimensional topological phase. And then it's a little bit more subtle to see how the anomalous action of a symmetry works in terms of two-dimensional, two-dimensional topological phase, but there is a way to say it. Yeah, yes, I don't have a simple physical statement. So what replaces the loop group, the maps from S1 to U1 when you go to higher-dimensional sphere? Right. So it will be maps from, like, assuming the Fermi surface is basically like a sphere in two-dimensions, in three-dimensions, at least topologically, it's the same as the sphere. So it will be the maps from the sphere into U1. Okay. Actually, there is one thing that one can say, which is that in general dimension, you can write an anomaly equation. This is what I skipped over. You can write an anomaly equation in terms of the non-consolation of the current. And so it turns out, you know, in 1 plus 1D, the anomaly equation is related to a boundary of a 3D-turned-simonstone. In 2 plus 1D, it's related to a boundary of a 5D-turned-simonstone. And so in 3 plus 1D, it will be the boundary of a 7D-turned-simonstone. I have a question. Yes. So in the strange metals, there is this pseudo-gap phase, right, where the Fermi surface breaks into Fermi arcs. Yeah, I mean, that's what you see experimentally. I mean, yes. And physically, you cannot really have Fermi arcs. I mean, that's actually clear from these general arguments, because if you try to define this anomaly for a Fermi arc, it has an inconsistency. So it's pretty clear that Fermi arcs can never really happen. So for, okay, then this question becomes, what is happening in the pseudo-gap phase? Probably, you know, there's some kind of pocket, but then you just don't see the one part of a Fermi surface, because it has very low oval aperture, you know, with the probe that you're using. I mean, yeah, the pseudo-gap is still mysterious, but it must have to be something like that. It can't be a real Fermi arc. I'm sorry, what is the Fermi arc? Just, yeah, I'm sure. So Fermi arc would be like a Fermi surface, but instead of being a closed surface, it just kind of is some segment of a surface, and then it just ends. And it doesn't have a second boundary, like it's not really like a crescent, it's really an arc. Yeah, a true Fermi arc, that's what it would be, yeah. And so how, how then does this fit in your description? What is the infrared symmetry group? Well, as I said, the Fermi, it cannot, it should be not valid to have an actual Fermi arc. What must really happen is it's some Fermi arc, Fermi surface, closed Fermi surface, and you just not, not seeing part of a Fermi surface. So then the immersion symmetry group would be the same as for any Fermi surface, it would be this liquid symmetry. Last call for questions. I have a question concerning the, the commutator of the density, and yeah. So, so can you explain more how, how this equation works for, say, interacting theory, because you mentioned the semi-classical approach to, to derive this. So does it only work for free theory? No, this equation has to be true in general, because the, well, first of all, we assume that interacting theory is still going to have this liquid symmetry. I mean, because as I mentioned, it sort of seems required by this compressibility theorem. So, okay, if it has the loop group symmetry, then the loop group symmetry must have an, either anomaly or it must be trivial, but the, the anomaly classification of the loop group is just z, and, and it's just given by choosing the integer coefficient here. So the only thing it could be for this commutator is either zero or, you know, non-zero. If it was zero, then it wouldn't be compressible because then the feeling would be zero. So you kind of, by illumination, you can conclude it must look like this, because this is the only possible form of a commutator if it's topological invariant with respect to reprimandizing theta. Okay. Yeah, it basically has to look like this. Okay. Well, we are now for the questions. Thanks a lot, Dominic, for very interesting talk. Thank you.