WEBVTT

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Rm 330: Justin Wilson.

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Rm 330: So it's a pleasure to have Jeffrey Wilson here today.

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Rm 330: Justin was a graduate student from the University of Maryland with Victor Galitsky. Then he moved to Caltech.

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Rm 330: Where he worked with Gail Raphael, and then he was a postdoc with our own Ed Pickley here, and now he's a faculty member at LSU.

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Rm 330: We're thrilled to have him back.

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Rm 330: Justin has a broad range of interests.

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Rm 330: going from quantum materials to quantum information to analog gravity. And today, he's going to tell us about rare regions, instantons, and the fate of the wild node. So take it away. Thank you.

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Rm 330: Well, thank you, Pragny. And thanks… thanks for having me. This, should… should, I hope, be a fun, fun talk. So, I'll be talking about.

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Rm 330: As that's pretty said, rare regions, instantons, and the fate of the vowel nodal disorder. Is there a pointer, by chance, or…

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Rm 330: Wildly gesture. That's fine. Sorry, I… I don't care. No, that's okay.

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Rm 330: Oh, thank you. I'm sorry, I didn't realize I had to bring it with you. No, it's fine, I thought maybe it would be a stick. I don't need anything fancy.

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Rm 330: I think Sean is gonna have the first book. Oh.

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Rm 330: Fantastic. Yes.

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Rm 330: Thank you. Appreciate it. So.

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Rm 330: So yeah, this is a problem that actually has a fair bit of a history to it, so there's 40 years of, of.

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Rm 330: the question I'm going to ask, which is, is the NCS space, at the node for Val Femel, is it an order parameter for a transition? And there's, there's work saying yes, no, maybe, and I hope by the end of this.

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Rm 330: Seminar will convince you that the answer is pretty definitively no.

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Rm 330: So this is work that I put recently on the archive, but I'll give you the whole broad history for this problem. My work will be just in the last third of the talk. Just to make it clear, in a pure disorder-free case, the tensor state at the node is zero. Yes. Right? Yes. Okay, exactly. And then what does disorder do, is the question. Short-range disorder, I should say.

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Rm 330: So, first I want to be a little bit Goshen, like, mention a couple of things that are happening at LSU. So we're… we just got approved a couple of weeks ago for six quantum faculty lines, two in physics, one in computational materials. These job ads will be out

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Rm 330: Within a couple of weeks. And, this is joint with Physics and Engineering, but it'll be

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Rm 330: most likely tenure home in physics. We don't know that for sure, but I'm pretty confident in it. And, two experimental hires, and devices and materials. So if you know anyone, please… So, junior? Yeah, junior, so, assistant professor is at the assistant professor, level here.

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Rm 330: So we have the appropriate startup and everything to that.

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Rm 330: Promise to us.

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Rm 330: So, great, there's also some in engineering and chemistry as well. Fantastic, good for you. So, if you know anyone, they can reach out to me. The job ads aren't on there, but I helped craft this, so I know what we want.

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Rm 330: Great, so the other thing I want to say is that we have an invitation, so we have this new institute, which Piers knows well about now, because I've been bothering him a fair bit about this. It's the top floor of a new building at LSU, and we're going to have kind of a workshop there every so often, different subjects in math and physics, and potentially also biology and chemistry, you know, those other subjects.

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Rm 330: But, fall 2027 will be a quantum materials program, and spring will be a quantum computing information and dynamics, program. So.

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Rm 330: keep an eye out for that, or if you're interested, please, please, please let me know. We have a whole bunch of office space up here for people to come for, you know, if you want to come for a couple of weeks and interface with people.

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Rm 330: We are hoping that we can build something kind of like that.

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Rm 330: I'm there.

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Rm 330: So anyway, let me get into the actual, physics.

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Rm 330: So the first thing that I want to mention, of course, is, well, I'll be talking about Dirac and Weyl fermions, and these, as we know now, these… since 2007, we've known pretty well that these, exist in the actual lab. So, graphene, of course.

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Rm 330: has, has 2D direct cones at the K and K prime points, and we also know now that there are 3D versions of these, so, Tensil Mars and I, for instance, we can… we get a direct point that can split into a pair of vial nodes.

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Rm 330: You can get this by breaking time reversal symmetry or breaking inversion symmetry. And… but also, these can be built in the, lab with, say, rubidium-87 atoms, where you have some artificial 3D spin-orbit coupling that's.

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Rm 330: that you can tune in just the right way to actually get vial nodes there. This is particularly nice because you could, for example, pattern a disorder that you wanted in the system, as opposed to just relying on the natural disorder and the materials.

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Rm 330: So every one of these is or can be dirty, and so what does that actually mean for the vial or DRAC system?

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Rm 330: So, let me just set up the problem as I'm going to be, approaching it. This is just the simple, vile Hamiltonian. We have this kinetic term, where these are the real poly matrices, all three of them, dotted into Nabla, and…

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Rm 330: We're going to add a potential term, some V of X. This V of X will be random, and it'll average to zero, so this over bars is the average of it. And, if we look at, how V of X is correlated over space.

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Rm 330: It'll be with some strength W squared, and some correlation which, for most of the analytic stuff I'm going to be talking about.

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Rm 330: The functional form of this C bar won't matter.

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Rm 330: But whenever I show you analytics, I will be using this, this sort of, Gaussian and space correlated. So this is a short-ranged, disorder, very, very crucially. Just to be clear, X and Y are each three-dimensional vectors? Yes. Yeah, X and Y are not one-dimensional, they're three-dimensional vectors in space, that's right.

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Rm 330: Umm.

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Rm 330: So, with all of this, you, you can introduce a dimensionless disorder parameter in terms of this correlation length.

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Rm 330: And the actual disorder strength.

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Rm 330: Very… one thing that I… one question that I kind of get here sometimes is, what happens if this C bar is a delta function? And, for this problem, you can't really have delta function correlated disorder, because

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Rm 330: the vial Hamiltonian has no link scale associated with it. You need to introduce a link scale somewhere. Delta function doesn't really do that. As a result, you'll get a whole bunch of infinities when you start to deal with that. So you really need this, correlation link scale to cut things off. Yeah?

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Rm 330: Yes, that's true. Does that help, or… That, you know, that absolutely helps. That'll take the place of this CAI.

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Rm 330: So, this guy is a… will act as a momentum cutoff in the end, the same as a lattice spacing would do it. Are you imagining correlation across unit cells, or within the same unit cell? So how deeply are… So for this problem, I'm 100% in the continuum, and so everything I'm going to be talking about is going to be in this long wavelength approximation.

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Rm 330: Here, so really it's across a lot of unit cells, right? So that I'm actually in the long wavelength. So I won't…

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Rm 330: Aside from references to other materials, most things I'm going to be talking about are really close to the bottom.

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Rm 330: Question? Yeah. The real world has chemical potentials and sometimes a mass to the biophermion, and so, which would be also, I guess, a magnetic field in that context.

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Rm 330: Right. Are those all relevant perturbations to what you're going to say now? They are relevant perturbations. So, one thing that… so, a couple of things, to… to… to note there. Of course, the maps are going to just, like, open up again, right? So, that's… that's going to be… How do you get that?

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Rm 330: What? How do you get out a three-dimensional… this is a three-dimensional… Yeah, so, this three-dimensional cone.

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Rm 330: What? There's an M. Yeah, I guess… Yeah, here… here, the single vial node, you can't really gap it out. You can think of this as the surface of a four-dimensional topological insulator.

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Rm 330: So, it's hard to gap out on its own, but on the lattice, you absolutely can. You can have scattering between nodes to gap it out.

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Rm 330: Just to be clear, now you are neglecting this scattering. Yes. I'm neglecting all… the only scattering I'm looking is within a single vial node.

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Rm 330: the intra… notes gathering.

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Rm 330: So basically, you're assuming that Cai sufficiently

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Rm 330: correlated so that you don't scatter between the two. Yes, yes, yeah. No, that's… that's right. It turns out when we do lattice simulations, that doesn't really matter. Then all this physics will, like, survives, whenever you do have scattering even between nodes.

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Rm 330: Yeah, but that's just kind of an empirical observation from our simulations. And this Hamiltonian without the disorder part, does it have two… the minimum is just two wild nodes, or just… this is just a single… Just one single wild node. This is a single wild node. Yeah, yeah, exactly, exactly.

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Rm 330: So, so we all know that the vowel node will look like this in K space, you know, this little cartoon of K and E. If I look at the density of states, the density of states is parabolic.

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Rm 330: And, the, the key thing about this is at zero energy, the density of states vanishes.

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Rm 330: So we say it's a semi-metal, because if I have a little bit of a chemical potential, I will get a finite density of states immediately. But there's not enough states here to really have too much conduction.

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Rm 330: Just because you're sitting there right at the vowel node.

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Rm 330: Any any questions about the starting in 3 dimensions.

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Rm 330: First. Oh, I'll go… I'll relax that in a bit, so I'll have arbitrary dimension, because it's useful to know when the disorder is relevant or relevant.

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Rm 330: things like that, and putting in a dimension will be very helpful. Also, the reason you call this wild node instead of giraffe node is because, some… because of spin parties. Yeah, this is… these are only the 2x2 matrices, so, you know, I'm neglecting all anomalies and stuff, so this is, like, kind of a… this chiral-type model, right?

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Rm 330: Yes, last question. Yeah. So in a lattice realization, does it matter if you have — remember, you break 10 reverse of inversion and you have inversion, I think. And you have the two binodes at different energies, and therefore you have a finite.

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Rm 330: Right. Every surface in other part of the momentum doesn't matter, or as long as there is no scattering, it doesn't… they are independent? Yeah, so, like, I guess one thing that I didn't even answer with, with, with Pierce, because it dropped my mind… from my mind, is when you have the chemical potential up in a vial node, you're up here, there will be effects of what I'm going to talk about. There's going to be something, that's going to be kind of a transition that could occur in the model.

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Rm 330: And that will affect finite energy states. You should still see that physics, but for what I'm talking about here, that won't be relevant, because I'm specifically going to be looking at the density of states at the bio node itself. So if the other bio node is there polluting it with a whole bunch of states, that will obviously mask everything I'm talking about.

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Rm 330: Right.

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Rm 330: All right. Anything else?

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Rm 330: Alright, so let's…

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Rm 330: So, the question I'm going to ask here is, is there a critical disorder strength below which this density of states at zero is exactly zero? And, in other words, is the semi-metal a stable point when I put in disorder?

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Rm 330: All right?

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Rm 330: So, there are two ways of thinking about this, and we'll go through, kind of, both of them. And so we're gonna have quenched disorder, which is to say V of X is frozen in space.

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Rm 330: And the first way of thinking about this is perturbatively. We're going to be looking at waves that have a wavelength of 1 over K, because I'm getting close to the vial node, which is when K goes to 0, this wavelength gets very broad.

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Rm 330: When I do that, then I have to ask, what is the effective disorder seen by this long wavelength object? And here we're going to get the central limit theorem, or an Emory Ma argument, which is going to tell us that disorder will be irrelevant for three dimensions. This is what this argument will give.

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Rm 330: However, this is complicated by the fact that there are rare and exponentially unlikely events that can bind exactly a zero energy state, right, where that little whale occurs. And I'll talk about that in a lot more… a lot more detail.

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Rm 330: Okay?

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Rm 330: So…

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Rm 330: These… these wells, though, they are going to be both, the… they are very… they're going to be quite… quite rare, and they get rarer the smaller the… the… the disorder, but they can dominate the average quantity of the density of states.

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Rm 330: Especially because it's zero plus a small non-zero number.

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Rm 330: So, first, let's talk about what this disorder usually does. I think this is kind of a useful thing, because we know that disorder can lead to something called Anderson localization.

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Rm 330: So, in this kind of argument, a very cartoon level, we'll have extended states that conduct. They can think about these as plane wave states, and then there's going to be states that localize in space, and these localized states will not contribute to conduction. They'll be exponentially localized in space.

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Rm 330: And so, very importantly, localization, these type of states, they will cause transport to actually change.

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Rm 330: But the density of states does not actually see this. You can… as you tune through an Anderson transition, the density of states will look the same, but the nature of the states changes drastically.

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Rm 330: So that is going to be this part of this phase diagram here, where if you have a diffusive metal, you tune up the disorder, you localize all your states, and you get an Anderson insulator. But this will be when we have finite density of states.

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Rm 330: The question that we're going to be asking today is what happens at the smaller disorder, where potentially the density of states remains zero, we have a good semi-metal, and

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Rm 330: And at some point, there will be a transition that the density of states will see. That's the claim, at least. And when that occurs, we'll become a diffusive metal.

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Rm 330: So, that is kind of what we want to ask, is rho of zero, you know, is zero to non-zero? That's an order parameter. So, is it an order parameter for a different type of transition than our usual Anderson localization transition?

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Rm 330: Yeah. Justin, when you talk about density of states, do you mean the average density of states? The average density of states, not the typical density of states, because the typical density of states will see this transition here.

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Rm 330: Yeah.

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Rm 330: Thank you. Yeah.

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Rm 330: Okay, so this is what we're going to be focusing on.

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Rm 330: Why would you expect Rho of 0 to be greater than 0 if you have particles all symmetric.

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Rm 330: system.

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Rm 330: Couldn't you imagine a situation which zero would remain?

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Rm 330: Zero with a mod-E behavior over a finite range of disorder. That's certainly possible, but even in the particle-hole symmetric systems that we've looked at, it becomes finite.

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Rm 330: That's yeah.

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Rm 330: So, for example, if I had quadratic band touch.

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Rm 330: Oh, the same case. The quadratic band touching will, that should already have density of states, right? There's enough states there. But it depends on the dimension, no? So you're saying 3D, I think, right? Am I remembering that? No, no, I am. I mean, couldn't this just be that as you gradually vary the chemical potential, you dunk it down, you push it up?

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Rm 330: And therefore you average over the average of MODI.

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Rm 330: over some variation in chemical potentials will be finite? Yeah, so… Such a dumb picture. So the thing is, I don't know if this picture, really, will hold, because that's kind of a perturbative argument that you… sometimes a little bit up, you're a little bit down, but that picture kind of works for the rare regions.

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Rm 330: Except for that those are very special regions, so I feel like it doesn't quite work, but… And there's no length scale in this problem, so… Yes. No matter how much you make psi, you can't really talk about smooth or whatever. Right.

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Rm 330: But, the link fill is very important, even for the, even for the… On a real lattice, there would be a length scale. Yeah, right. So you could imagine smooth variations in the chemical potential, though nevertheless random. You could choose Xi to be much bigger than the

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Rm 330: the size of the lattice is your psi much bigger than the size of the lattice. Yeah. Oh, but then we can talk about smooth.

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Rm 330: Yeah. Variations, and then we can talk about dunking the, Yeah, but I think that that ends up being an irrelevant perturbation, that those dunkings down and up just aren't big enough to overcome the, the averaging.

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Rm 330: I mean, a counter example is the case of putting a magnetic field on a D-wave superconductor, where you…

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Rm 330: Where you do… Oh, but there is… You do velocity, you go up and down and push them up, and then you get a finite density of states. That's two dimensions, isn't it?

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Rm 330: Oh, 3? Oh, okay. So, okay, well, I might be off here a little bit. Okay, yeah, sorry.

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Rm 330: So, anyway, let me give the, where this ends up in the end of the day. Here, I'm doing 1 over W squared, and less disorder is then all the way over here at minus infinity, or at plus infinity. More disorder is over here, and

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Rm 330: the… I'm showing two different things here. The answer is going to be that rho of 0 is going to be 0… greater than 0, no matter what.

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Rm 330: So this is a log linear plot. And the key result here, these blue dots are dots that are data points that I took while actually here at Rutgers.

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Rm 330: In 2020, with Jed and David Hughes and Shankar Dasarma. And, so, we computed these to argue then that it would be finite.

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Rm 330: The new result is that we have a no-fit curve here. This black line is the result of an instanton calculation that I've done, which matches

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Rm 330: the slope, and pretty well with the other data. The log curve obscures us a little bit. They do actually differ by a factor of 3 quarters, so, like, they look a little bit closer together than they actually are, but, but nonetheless, this is, quite a, I think, quite a good fit, without any fitting parameters.

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Rm 330: parameters for your previous one? The previous one, there was no fitting parameters, we just plotted it like this, and saw that this looked approximately linear. Okay.

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Rm 330: Now, you see that there's a sneaky factor of W to the fourth here? That actually makes it more linear, which is just very satisfying. Yeah. Because maybe this is so maybe willing to ask, curious, you know, but there's just the total number of states.

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Rm 330: Should be constant, isn't it?

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Rm 330: So the sum should be conserved. Yeah, yeah. So if the density — They definitely come from like — Where do they come from? It comes from the higher energy state. From the higher — not nearby? Not nearby. So we need the momentum cutoff. And in fact, the momentum cutoff plays a very crucial role in this physics even showing up to begin with. So you should really think when you start to get to — on the order of the bandwidth — Oh, I get it.

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Rm 330: that zero energy, then actual, really, the overall, the states really broke. Yeah. It comes from far outside, I see. Yeah, and we'll see how the DCSS looks when I, when I do this, too. This is the fit you did?

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Rm 330: So, there's no fit.

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Rm 330: A e to the minus something over W. Oh, yeah, e to the… if there's a constant, e to the minus S star, this will be the instance on action in the end of the day, but there's a constant that's… so this will go, like, minus 1 over W squared. E to the minus 1 over W squared times a constant.

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Rm 330: this will become a lot more clear as we go through. I just wanted to, you know, not bury the lead. This is where things are ending up.

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Rm 330: So, let's go to part one. So, part one is… oh man, is the density of states at the nodal point an order parameter?

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Rm 330: So, this is the argument for it being a good order parameter. So we can take a long wavelength state and see how it's going to average out the disorder. We have a correlation link for our disorder. This could also be the lattice spacing.

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Rm 330: Cai, and you could think that within the Cai, there's going to be a box that'll be some just

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Rm 330: disorder value, right? It can be positive, negative, whatever you, whatever you want.

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Rm 330: And our state, our state is going to have a wavelength of 1 over K.

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Rm 330: As a result, we can ask, what is the effective disorder strength that this 1 over K wave sees?

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Rm 330: And this will be, like, kind of a central limit theorem of, like, of the disorder string W divided by 1 over square root of number of boxes that are inside of here. That's what this N is, the number of these chi boxes.

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Rm 330: And you can… so you can just count them, right? It's just, this K times chi to the D over 2. Because of the square root, you get a 1 over, 2.

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Rm 330: As a result, the, like, this is the effective size of the disorder that's seen by the wave, and we can compare that with the kinetic energy.

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Rm 330: So now this is asking, does the kinetic energy win, or does the… does the potential win? And when you do that, the K is going to cancel one of these Ks, and you get this D over 2 minus 1.

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Rm 330: And this is basically the entire argument for when disorder is relevant or irrelevant. Now we can see that when D is less than 2, this effective disorder strength compared to the kinetic energy will vanish.

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Rm 330: And so their disorder will always, will always win, or it becomes… sorry, it goes… it goes… doesn't vanish, it goes up and up and up. So their disorder will always win.

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Rm 330: At D equals 2, this is the case of graphene, then this is marginal, so you get things like logs and whatnot, but disorder will still destroy your semi-metal pretty quickly.

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Rm 330: For D greater than 2, this will shrink.

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Rm 330: And so a weak disorder, we say, is irrelevant at the node by this Emory Ma type argument.

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Rm 330: All right.

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Rm 330: So, this ends up being kind of what was seen by, first, Fradkin in 1986. So the, the argument is that because of this.

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Rm 330: irrelevance, you get a… you get a phase transition where this row of zero on one side is flowing to zero, that's… that we call the semi-metal.

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Rm 330: And then after a certain critical value, it goes to a diffusive metal. And so he has a field theory argument, some large-end perturbative argument that actually sees this transition point.

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Rm 330: On the other hand, if we go to 2D… In this scenario, is the density of states becomes zero as soon as you pass through WC? It goes from zero to a non-zero value.

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Rm 330: When you go through WC. Is it zero along the left half of the… Yes, it's, identically zero in this, in this region. Okay.

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Rm 330: Yeah, and so, for the 2D, case, of course, this is marginally relevant, so we put a little bit of disorder, and it'll flow to the diffusive metal right out of the way. Why do you compare Dirac versus Weil? The difference… It makes no difference, actually, yeah. This could be Weil, this could be Dirac. It'll be the… it'll be the same exact picture.

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Rm 330: Yeah.

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Rm 330: Questions? So, on the semi-metal side, do you really mean that rho of V is proportional to modi, or… I mean, rho of 0 could mean a gap.

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Rm 330: Row… so over here, row of E will go, like, E squared up until this critical value, and then it'll have a critical, it'll change. I'll actually show this in just, I think, maybe the next slide. Okay.

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Rm 330: So you don't just mean rho of 0 equals 0, you mean rho of E equals E squared. Yeah, yeah, and in fact, this is kind of the picture you should have in mind. So you could do a perturbative RG analysis.

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Rm 330: Like, I have not done the perturbative RG analysis myself, but it's been done in a fair number of publications, where you get a flow equation for your disorder variance, so you can… this G hat is the square of the disorder.

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Rm 330: And you get this equation here where at d equals 3, you have an unstable fixed point. And you can see very clearly that d equals 2 is marginal.

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Rm 330: And if you do this analysis, you can actually get that there is a transition, and the transition has critical exponents. It has a correlation length at one loop, it's one, a dynamic exponent of three halves, and at the transition, the density of states goes like D over Z minus one.

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Rm 330: So what that means is, at the transition, you actually go from E squared to, a linear, dispersion.

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Rm 330: And the way this shows up, if we're a little bit below the critical point, you would expect something like this for the density of states.

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Rm 330: When you go to a large enough value, proportional to how far away from the transition you are, you start to pick up the criticality. So now it's no longer a parabola, it's a straight line. But if you go close enough, where we're still not at the transition, there will be a point where you see this E squared still.

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Rm 330: So that's, that's sort of how all of the criticality is encoded into the density of states.

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Rm 330: All right.

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Rm 330: And… Crucially, this analysis tells you that rho of 0 should be 0, exactly.

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Rm 330: Any questions? And what's the experimental verdict on what this is? Oh, experimental. Yeah, this is a very, hard experiment. I mean, there's been some experiments on seeing, like, the Fermi arcs melt into the bulk and things like that.

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Rm 330: As you'd expect from a picture like this, but, like, a true blue, like, what short-range disorder does to the density of states, I think there's not a definitive experiment on. It's a bit of a… it's kind of an academic question, to be completely honest.

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Rm 330: Because there's too much difficulty in preparing… Well, you have to get… you have to make sure your chemical potential is right at the right point, there's no other… yeah, there's no thing flying around, and also, there's no, there's no charge disorder, there's no Coulomb disorder. And you have to control the disorder, there would be… Yeah.

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Rm 330: Yeah, yeah, yeah. The… and the Coulomb disorder, all of this picture goes out the window. The SCSH just becomes finite with Coulomb disorder. You could imagine in a condo vial semi-metal, there might be something that locks

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Rm 330: Oh, really? Locks it to neutrality. Oh, that's interesting.

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Rm 330: Yeah, that makes sense. It worked, and they claim that's the case. But I don't know if it's true. But yeah.

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Rm 330: And would, charges be… well, anyway, we should… yeah, chat a bit more. I have other questions. But anyway, so…

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Rm 330: This is the perturbative RG analysis, and the numerics agreed with this, with this, analysis. So this is some of the early, numerics, here by your own, jet with, disarmament, Paul Goswami, where they, saw that they got a nice semi-metal.

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Rm 330: quantum critical at some finite temperature, and a diffusive metal over here. So, the critical point I was telling you about was at zero temperature, down here. You could kind of think of the higher energy states, or for the density of states, as telling you about

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Rm 330: finite temperature, where you start to see this quantum critical fan. This linear density of states starts to help govern a lot of the physics, and you can kind of pull out stuff there.

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Rm 330: But there's a… it looks like there's a nice transition, and you can characterize these by, by the specific heat of T cubed, T squared, and T for semi-metal, quantum critical, and diffusive metal.

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Rm 330: And in fact, the, the critical exponents seem to agree quite well, too. Z is 1.5 to quite good accuracy. In fact, if you do these transport calculations that were done by Bjorn Sabirski and Pete Brower and company, you can pull out this 3 halves with a pretty… with a pretty small error bars.

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Rm 330: And this is how the conductivity scales with chemical potential. So it goes as 1 over… chemical potential to the 1 over Z.

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Rm 330: you can pull out that. New is somewhere between 1.1 point… between 1 and 1.5, it's still argued.

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Rm 330: And it looks like rho zero is a good order parameter. So maybe, maybe this isn't just asking about this particular work, but in general, this is a non-interacting system, right? Yes, this is a temperature error. What temperature does here, exactly? Where is the… Temperature helps you sample the states around zero energy. Because, like, you're saying smearing, so there's… Yeah, yeah, right, right. And then the usual way of a step function for your Fermi function, you have to…

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Rm 330: You know, cut, or whatever.

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Rm 330: Okay, so, great. It looks like there's a complete picture here. There's a… this is a good order parameter, we have nice criticality, we're pulling out exponents.

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Rm 330: Well, there's a complication, and that's that somewhere there is a very, a very deep well.

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Rm 330: Even though this is poetic, this isn't quite true. Somewhere there's an optimal well, would be the more provided way of saying it. But anyway, the thing is, every dirty band has

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Rm 330: Lifshitz tails associated with it. And so, this is my cartoon for Lifshitz tails. We're here to make it abundantly clear how it works. I've done… I've put… I've made up an artificial band that doesn't really exist, where the density of states vanishes, like E squared, and then it stops.

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Rm 330: If I add…

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Rm 330: disorder into this, then I can get states… I'll in fact get a whole bunch of states below this, but these states will be localized, exponentially localized.

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Rm 330: And there will be a mobility edge where these states even though they're there they don't contribute to conduction. Up here they do contribute to to conduction. And as you tune energy this is actually an Anderson transition and the energy itself between localized and delocalized.

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Rm 330: So, because of this, you might expect that, in fact, the density of states should be finite. This is a very simple argument of just, like, you make random states below your band when you add disorder.

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Rm 330: Why is that not true here in the middle? Why were we saying this has to vanish exactly? What's special about this point that would stop this from happening? And does it, in fact, look like

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Rm 330: Look like this, instead.

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Rm 330: So, this is kind of, like, very rough intuition, which actually ends up holding. This will lift off of zero. Here, it's exaggerated, but, yeah.

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Rm 330: So

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Rm 330: And the end result is going to be that this density of states is going to go, like, 1 over W squared. So whenever I'm teaching, like, quantum mechanics or something, I very much like to throw this, throw this up and ask them, like, what is the Taylor expansion of this E to the minus C over W squared?

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Rm 330: If you do it, of course, you know, 0 plus 0 plus 0 plus 0, it's just a whole bunch of zeros. Every term in perturbation theory is technically zero, because it has, you know, the rates of convergence.

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Rm 330: 0.

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Rm 330: So if you do any perturbation theory, you cannot pick this up. You have to do something else to actually see it. And in fact, that's what we have to do here.

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Rm 330: So, what actually occurs is, so in 2014, and let me actually put the reference up now.

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Rm 330: Nakashora Hughes and Sandi actually, gave this argument that, in fact, if you look, exactly at zero energy, you will occasionally get a rare deep well and a rare high barrier. These are related to each other just by parity, and, you get, you'll actually bind a state

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Rm 330: you'll occasionally bind the state exactly to these rare wells and rare barriers. And these states, if you look at them, they're so close to a band that they won't be exponentially localized. They will, in fact, be power law localized, like 1 over R squared.

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Rm 330: This is a log scale here, so any curvature is a power law.

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Rm 330: And, there… so they did this argument all the way back in 2014, and argued that rho of 0 has… there's some constant factor which they did not compute, and they argued that this has to go, like, 1 over e to the minus here, S star is just a constant, times 1 over W squared.

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Rm 330: Okay, so… Why is it not exponential? What's the argument behind long range?

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Rm 330: So… Right, right, right. So, the argument, I think, is actually quite, quite, quite, quite simple. So, it's, it's like, whenever you make a, state in a band gap.

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Rm 330: then the, how exponentially localized it is is related to how far away from the band gap you are, right? And as you get closer to the band gap, that localization length gets broader and broader.

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Rm 330: This exists exactly at a band gap, kind of. This is in the middle. This, of course, only… it is a while or drop node, basically. Yeah, yeah, it has to be exactly this node. There's extended states so close that it can only go like a power law. There's no, way for it to get, exponentially localized.

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Rm 330: So… Great. Oh, yeah, so, now…

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Rm 330: I'm going to describe now how this actually, how we can actually understand this… how this comes about, and what the typical well size is that, that will bind us to state at zero energy. This is going to be…

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Rm 330: I had instantons in my slide. The next slide is going to show you kind of how I visualized instantons coming about in the problem. So, what we do, we have some disorder, we come along, we find a state, we pull it down, we find another one.

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Rm 330: We pull it down. This is the disorder that does it. And we keep bringing it down and averaging it to what we had before. So we're averaging over all of the states that could possibly give us a zero energy state, barriers and wells. Keep every time we find one, we pull it down, average.

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Rm 330: And as this goes along, this will converge to some profile. This profile will be what we'll call the instanton profile, and this is kind of the optimal well that will bind the state at zero energy. And it really is just averaging a whole bunch of the configurations that could do it together.

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Rm 330: Okay?

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Rm 330: So this shape will… will come back. This will be… this is actually a solution to a nonlinear, Weyl equation, a nonlinear integral Weyl equation.

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Rm 330: I'm… So.

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Rm 330: when people went back to the numerics, and in particular, when Jed went back to the numerics with David Hughes and Shankar Basarma, they actually saw that, in fact.

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Rm 330: they were able to explain how they saw the criticality and how the density of states lifts off of zero. And so, in particular, they can still see this quantum critical fan at finite energy or temperature.

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Rm 330: And you see something that looks kind of semi-metallic away from E equals zero, and they see something kind of like a diffusive metal over here. However, if you look exactly at zero energy.

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Rm 330: they're, the diffusive metal creeps all the way down to zero, and in fact, this sigma kind of tunes how rare your rare states are. You can never get rid of them entirely, but you can tune it down quite a fair bit.

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Rm 330: And so, being very careful with the numerics, you can actually see

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Rm 330: It's just a parameter that, that, is tuning… it's tuning, between, like, it's kind of a Gaussian

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Rm 330: disorder to something that's more like binary disorder, where it's, rare states are just much harder. You should think that sigma zero is harder to make rare states, not impossible, just harder. It's much easier for sigma of 1. It's just, it's just a numerical parameter in the problem.

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Rm 330: So, the end result was that this was probably, an avoided quantum critical point. So, as a result, we expect that this is actually diffusive metal all the way down here, but, just, suppressed.

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Rm 330: By this 1 over e to the minus 1 over W squared, and, we can only access critical properties by looking away from it.

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Rm 330: All right.

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Rm 330: So, this was, of course, seen, in the density of states as well, so if you looked at the log of the density of states over… with 1 over W squared, this is giving you that.

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Rm 330: exponential, that I was showing, showing before, and, with both KPM and Langshus, Jed was able to show that this is just a straight line. So that confirms, kind of, the instanton picture numerically very, very well over orders of magnitude.

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Rm 330: And then we, in 2020, came back to the… this was on a lattice, and now this is in the continuum, where we saw very similar features, where this, this is 1 over W squared, a straight line indicates that exponential, and it seems to fit, quite, quite well.

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Rm 330: You might notice that there's a little bit of curvature to this red line. It's not exactly straight. That is because of an extra factor in the fluctuations of the instanton, which we didn't know at that time.

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Rm 330: Umm.

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Rm 330: All right.

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Rm 330: Why do you call it instant one?

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Rm 330: It's…

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Rm 330: It's kind of a historical reason, actually, but it's a saddle point to some, when we average out the disorder, we get an effective action, and the saddle point to that we call the instanton.

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Rm 330: So, then, 2018 came around, and, then these, these authors, Buko Diehl and, and Altland, came and said that this picture was actually incorrect.

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Rm 330: That, in fact, potentially the latitude mirrors that we did had band curvature, node coupling, finite resolution, and the slope that we were seeing, was actually not, not the physics.

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Rm 330: If we were to consider a perfectly linear while node with disorder, they claim that it should be, if not exactly zero, it should be super exponentially suppressed.

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Rm 330: So, it's either a straight… instead of a straight line, maybe it dives off, like, some super exponential.

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Rm 330: And they had some argument, which depended on some, supersymmetric field theory to, to, to argue this, but

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Rm 330: Today, I'm going to… describe a term that I believe that they missed.

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Rm 330: I have to say that I'm not just coming out of the blue to say that they're wrong about this. I have discussed with them privately about this, and I don't want to speak for them, but I think that we're in more or less agreement at this point.

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Rm 330: About how this picture is actually not… not correct.

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Rm 330: So you're going back to the blue one again? Yeah, yeah, yeah, so they were arguing for the orange, and I'm saying that, in fact, the blue is the… The blue is actually correct.

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Rm 330: So, what is the argument, though? So, the argument actually is entirely in a prefactor that should multiply this exponential that I was telling you about.

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Rm 330: And this prefactor, here is the fluctuations. This is the fluctuations about the saddle point itself. So, in fact, no one disagrees about the saddle point. They, in fact, in their paper, they compute the saddle point as well, and they get this factor. They then argue that this P in front is 0.

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Rm 330: And so now I'm going to argue that, in fact, it's not… it's not zero.

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Rm 330: And if we were to look at the numerics, this would be… this factor, according to their super exponential argument, would be, like, e to the minus 70. It would be so, so, so very, very small.

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Rm 330: So, let's now look at the fluctuations.

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Rm 330: So, the end result's going to be that, not sure if Hughes and Sande were right, the rare regions do survive their own fluctuations, and here is the actual calculation. So…

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Rm 330: I'll be using a supersymmetric field theory. I won't go into all the nitty-gritty details. I'll try to give you a very high-level view of how the calculation goes, but just to…

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Rm 330: set it up. What we do is, take, a… what we call a superfield, where we break things up into bosons and fermions.

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Rm 330: This is only a trick. There aren't really bosons and fermions in this system. This is not the electron, either. It's just some random fermion that I make up.

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Rm 330: And the nice thing about this is we create a partition function in terms of some action that has our potential within it, and this partition function will be exactly equal to 1.

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Rm 330: This is very, very convenient for disorder calculations, because we often have to look at quantities that have 1 over Z, and 1 over Z with a potential in it is really, really hard to take averages of. So if Z is equal to 1, that's really easy to average, right? It's just 1.

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Rm 330: So, even though we have to look at things with these bosons and fermions, that is a complication, but this simplification.

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Rm 330: kind of balances it out, makes it a lot easier to deal with. So, but the important point about this is that it's supersymmetric. Every boson mode has a fermion partner associated with it. And if you neglect one of these sectors, either the boson sector or the fermion sector, you're likely to mess up your problem and get wrong results.

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Rm 330: Okay. How do you impose the condition that the partition function is exactly 1?

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Rm 330: So this… these are, free bosons, free fermions. The differential operator that applies to both is the same. The, the boson gives you a 1 over determinant factor, the fermion gives you a determinant factor. They just cancel.

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Rm 330: It's, so you're just using this, like, kind of neat trick about Grassmannian integrals versus complex integrals.

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Rm 330: So…

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Rm 330: The first thing that you can do is actually ask, what is the saddle point? And this is where that picture I was showing you with the animation, how it became some average well.

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Rm 330: That is what this calculation gives you precisely. So the saddle point is actually this nonlinear Weyl equation, where we have this Weyl kinetic term.

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Rm 330: And this is the wave function that we're trying to solve for. But now, there's a self-consistent potential. So this wave function is sourcing itself.

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Rm 330: as a potential term, where the disorder is entering, here. So you see the correlation length is entering there, smearing it out a little bit.

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Rm 330: And it turns out you can numerically compute this, what the solution to this is. It decomposes into angular momentum, and we can concentrate on the J equals 1 half sector. Is that a question? So… Yeah.

259
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Rm 330: There's no… I thought this was non-interacting. Is this an interaction? This is now kind of interacting, yeah. So when you integrate out… so the, the original problem is non-interacting fermions, and that is very important, that it's non-interacting fermions.

260
00:45:35.040 --> 00:45:47.499
Rm 330: But then when we integrate out the disorder, disorder couples quadratically to the fermions. But when you integrate it out, you get something that's fourth order in your fermions.

261
00:45:47.580 --> 00:45:52.920
Rm 330: So you generate interacting terms by integrating out the disorder term.

262
00:45:53.750 --> 00:45:55.290
Rm 330: Leading to something like this.

263
00:45:55.790 --> 00:46:07.510
Rm 330: So, Justin, just to make sure I understand, supersymmetry here is in place of doing a replica trick, right? Yes. Another way of… related to Bonson's question, another way of doing.

264
00:46:07.690 --> 00:46:11.899
Rm 330: disorder is to do replica symmetry breaking, which also generates

265
00:46:12.010 --> 00:46:15.419
Rm 330: interactions. Yes, yeah, you… you can do it with

266
00:46:15.760 --> 00:46:32.050
Rm 330: replicas, comes through Keldysh. There's a couple different ways of doing it, but all of it will… Has this problem been approached from that standpoint? I think there are problems with the… with the… with the replica version, but I am interested in actually… I haven't actually looked at doing it those… those ways. I think it's… there's interesting…

267
00:46:32.050 --> 00:46:45.000
Rm 330: This is kind of an academic question. The other question I asked is, is it obvious that there's just one subtle point, one inciton? I mean, in principle, there could be many. Yeah, and in fact, there is more than one, because this is…

268
00:46:45.150 --> 00:46:53.580
Rm 330: This is a J equals one half, and there's obviously M one half and M minus one half. So why are you only talking about one?

269
00:46:53.730 --> 00:47:12.479
Rm 330: So, because, well, that is a solution I know I can get, I guess. I guess I can't prove that there's only one… one solution to… to this. The, I can say that I… I… I really tried to break this. Yeah, there should be more than one. Plenty of ways to solution. Well, at E equals zero, it's… it's, you know.

270
00:47:13.490 --> 00:47:19.660
Rm 330: There's, yeah, there's obviously a plane wave solution. That will not contribute to the density of states, though. Sure.

271
00:47:20.020 --> 00:47:28.759
Rm 330: All right. Yeah, so, if I make you a constant, just to be… but it, of course, violates your average being zero.

272
00:47:28.850 --> 00:47:45.240
Rm 330: Right? Then clearly the density of states would now become finite. Yes, I agree. Right? So there's gotta be a limit in which slow variations of U with a finite amplitude just creates a, creates a finite density of states. So I think this is a fundamental flaw in your instanton argument.

273
00:47:45.430 --> 00:47:48.609
Rm 330: But I don't think it's a… You're not forgetting about the lattice.

274
00:47:48.790 --> 00:47:52.179
Rm 330: Yeah, I'm definitely forgetting about the labs on purpose.

275
00:47:52.840 --> 00:47:57.610
Rm 330: But the real world lives on an atlas. That's the regularization. Of course.

276
00:47:58.480 --> 00:48:00.060
Rm 330: Yeah.

277
00:48:00.380 --> 00:48:10.610
Rm 330: Should I just go on, or do you want me to… Well, just… is this just an artifact of not working on an artist? But maybe, Pierce, you can discuss this later, okay? Because you're going to talk to him later.

278
00:48:11.210 --> 00:48:14.290
Rm 330: Okay. But what about our tradition of having

279
00:48:14.720 --> 00:48:28.080
Rm 330: I'm trying to say, like, off of the last year, because that's where they started, but the…

280
00:48:28.080 --> 00:48:36.940
Rm 330: But nonetheless, the — In the real world. In the real world. Seminars go on for hours, and I'm not sure if you're — There's a lot of scattering in this.

281
00:48:37.010 --> 00:48:52.890
Rm 330: Yeah. There is internode scattering, but certainly there are optimal, there are optimal wells that look like exactly what you're talking about, too, that do bind states at zero energy. But, yeah, I don't…

282
00:48:53.220 --> 00:49:01.959
Rm 330: I'm just I don't know if it's I don't know if it's entirely the dumb argument, which is yeah, it just dunks it in and out, and you're gonna find a density of six.

283
00:49:01.990 --> 00:49:18.380
Rm 330: Why do I need to go to instantons and supersymmetry when an in-your-face obvious answer works perfectly well? Yeah, so… The Hugh Sandy, Mantkashore was a… Yeah. But not only that, but I…

284
00:49:18.420 --> 00:49:33.580
Rm 330: I think it actually might be due to the fact there's just not a lot of states nearby, so even if you do change it, it is E squared, and so that should be accessing this perturbative regime where you just have, you know, it's quadratically suppressed

285
00:49:33.580 --> 00:49:38.990
Rm 330: There, and you wouldn't be able to pick up enough states by moving the chemical potential up and down.

286
00:49:39.190 --> 00:49:44.799
Rm 330: Let's see… Okay, I'm gonna keep quiet now, but… Yes, please do.

287
00:49:45.110 --> 00:50:06.739
Rm 330: It's unsatisfying. I think it ends up being the same as the Emory Ma argument that I was giving about why it's irrelevant. There must be a simple argument why this doesn't work. I'm happy to engage further. You used up your token.

288
00:50:09.130 --> 00:50:26.220
Rm 330: You have to pay the API cost now. Anyway, so anyway, the point is that I can actually find a self-consistent solution that binds a bound state, and very crucially, it dies off, like, 1 over R squared, far, far away from the middle.

289
00:50:26.220 --> 00:50:35.190
Rm 330: And if I do this, I can compute this constant that I showed you before. You can just, compute it exactly, and then you get about 6.412.

290
00:50:35.230 --> 00:50:43.690
Rm 330: And you get, you also have to modify this coupling to put it at zero energy. That also has a value that you can compute.

291
00:50:43.880 --> 00:50:52.400
Rm 330: And here, this solution, this J equals 1 half, is a hedgehog solution, so I have a Sonic the Hedgehog for that.

292
00:50:54.120 --> 00:51:03.320
Rm 330: That'll come back. Anyway, so this, this disorder strength, I think this is kind of a nice calculation, because the…

293
00:51:03.470 --> 00:51:10.870
Rm 330: The disorder strength itself actually enters, exactly as Planck's constant does in a semi-classical analysis.

294
00:51:10.960 --> 00:51:17.910
Rm 330: So if you've done semi-classics with a path integral, ever, you see that you get, there's action divided by H bar.

295
00:51:17.930 --> 00:51:36.550
Rm 330: And you want to take H-bar to be very small, and you can do a saddle point analysis about your classical action. You can do this even with just regular quantum mechanics. It turns out that you can re-parameterize the problem that I just told you, so that you get W squared showing up exactly where that H-bar was.

296
00:51:36.550 --> 00:51:50.810
Rm 330: And so this W squared being small actually controls this calculation. It is the control parameter for the instanton, just the same as H bar being small controls the semi-classical argument.

297
00:51:53.380 --> 00:51:54.310
Rm 330: So…

298
00:51:54.420 --> 00:52:08.669
Rm 330: So now, three things about the, the fluctuations. This will be three slides. One thing will be that the bosons are going to cancel the fermions one for one, and this will give us a nice finite number in the end of the day.

299
00:52:08.670 --> 00:52:22.899
Rm 330: We also will need to go back and count the zero modes. So, very important when you're doing, any kind of, mean field, calculation, you need to know what your goldstone modes are. We need to… so we find our goldstone modes, both in the bosonic and the fermionic sectors.

300
00:52:23.060 --> 00:52:29.949
Rm 330: And then we look at what is left to argue that it's impossible to suppress this, fluctuation

301
00:52:30.220 --> 00:52:32.059
Rm 330: Determinants in any other way.

302
00:52:33.240 --> 00:52:34.100
Rm 330: So…

303
00:52:34.370 --> 00:52:50.359
Rm 330: Let's look at the, the fluctuation determinant by itself to see what happens. So, when we look at these fermions and bosons, they will give us two determinants, and here the prime is I've removed the zero modes. We'll come back to those.

304
00:52:50.360 --> 00:52:57.620
Rm 330: I've removed the zero modes by knowing what they are beforehand. I'm not artificially throwing out zeros.

305
00:52:57.680 --> 00:53:19.110
Rm 330: And, you don't need to worry about this, direct sum. I break a U1 symmetry, and I get, Nambu, I get a Nambu space as a result. It's a kind of technical detail. You should think about this as just the, vial Hamiltonian, here, the free, free vial, K, oh, sorry, K is the free vial plus the self-consistent, potential.

306
00:53:19.140 --> 00:53:23.919
Rm 330: And H3 is just the free vial Hamiltonian.

307
00:53:23.960 --> 00:53:28.309
Rm 330: And if you look at the fluctuation determinant for fermions.

308
00:53:28.600 --> 00:53:34.250
Rm 330: you get this dashed, orange line here at the bottom. So this is…

309
00:53:34.300 --> 00:53:44.059
Rm 330: the log of the fluctuation determinant, you can see that it is actually quite small. So the fluctuation determinant for fermions, we can compute it.

310
00:53:44.080 --> 00:54:00.760
Rm 330: It is quite a small value. This, RC, I've actually taken this fluctuation determinant, and I've filtered it in space, based on how much of the wave function is there. So we can see that in the core of the instanton, you get most of the… most of the spectral weight.

311
00:54:00.800 --> 00:54:08.910
Rm 330: Until, until eventually you start looking at more and more space. Here is the full fluctuation determinant,

312
00:54:09.300 --> 00:54:14.369
Rm 330: For a box of size 60, and it kind of, saturates.

313
00:54:14.760 --> 00:54:15.950
Rm 330: At some value.

314
00:54:16.060 --> 00:54:19.789
Rm 330: But then, if you look at the bosonic.

315
00:54:19.980 --> 00:54:22.829
Rm 330: Determinant, you also get that it is very small.

316
00:54:23.110 --> 00:54:35.099
Rm 330: And in fact, they're both not only very small, but they're, they cancel each other almost exactly up to a very small minus .02.

317
00:54:35.230 --> 00:54:42.510
Rm 330: And when we take the exponential of that, that'll be some order 1 number as a… as a result.

318
00:54:42.680 --> 00:55:01.930
Rm 330: So this is kind of what I argue was done incorrectly in the previous paper. They were looking at the fluctuator determinant of fermions and arguing that it's really small, but in fact, if you look at the bosonic sector, which they did not look at at all, that actually cancels the fermionic part, almost exactly.

319
00:55:02.040 --> 00:55:12.860
Rm 330: Giving you an order one number for this… this DREL is the fluctuations. This is basically almost all of what goes in front, that constant P.

320
00:55:13.750 --> 00:55:31.839
Rm 330: Okay, you talk about bosons and fermions, but why aren't we talking about electrons? What do you mean by bosons? These are completely artificial. These bosons… both bosons and the fermions are not real bosons or real fermions. They're there only to assist us in the calculation.

321
00:55:33.500 --> 00:55:37.859
Rm 330: Does this mean that they're thinking in the language of, I mean.

322
00:55:38.070 --> 00:55:42.470
Rm 330: Because it doesn't mean that they've not properly done the average over the disorder.

323
00:55:42.830 --> 00:56:00.790
Rm 330: Well, the main thing… They've actually, broken supersymmetry. Yeah, but take away the supersymmetry, and just talk about the fact they're only taking one part of the thing. Yeah. Does that mean that they've just not done the proper disorder average?

324
00:56:00.990 --> 00:56:20.600
Rm 330: it… yeah, it does mean that, and maybe a better way of thinking about it is, is, is, so, you know, when you compute a T matrix, right, you, you get, like, like, 1 over, like, 1 minus G times your potential, right? What they did is effectively, like, they computed a self-consistent Greens function.

325
00:56:20.600 --> 00:56:23.369
Rm 330: And they didn't do the 1 over 1 minus

326
00:56:23.650 --> 00:56:31.980
Rm 330: you know, G that they actually needed to do. They did the top half of the T matrix and not the bottom half of the T matrix.

327
00:56:32.050 --> 00:56:46.169
Rm 330: So they didn't consider the full scattering, they just considered what happens with the… So it's a perturbative approach. It's not exactly perturbative, because it breaks in a number of ways. But we'll actually see something that'll look very much like a T matrix in a little bit.

328
00:56:47.380 --> 00:57:05.810
Rm 330: I mean, they just made it worse, they just dropped the bosonic sector completely. Yeah, yeah, they… You're doing supersymmetry. You can't do that. So… So anyway, if we go, and we actually count the zero modes, so, BDA's picture is that there should be a whole band of soft fermionic zeros.

329
00:57:05.810 --> 00:57:13.949
Rm 330: But we can now just, look at them exactly. And if we look in the bosonic.

330
00:57:13.950 --> 00:57:24.450
Rm 330: the sector, there is going to be 6 bosonic zero modes, because we've broken 6 symmetries. 3 translations, 2 rotations, and 1 phase.

331
00:57:24.650 --> 00:57:40.289
Rm 330: And so, of course, I had to put here, you can see Sonic, who is a hedgehog solution. He obviously breaks translation, because you can move him around. And when you spin him, he also looks different, so he breaks rotational symmetry, too.

332
00:57:40.650 --> 00:57:45.260
Rm 330: Sorry, I'm way too proud of that.

333
00:57:46.720 --> 00:57:55.039
Rm 330: On the other side, if we look at the fermionic modes, this operator K is just the vial Hamiltonian plus the self-consistent potential.

334
00:57:55.270 --> 00:58:07.770
Rm 330: And by definition, this has zero modes. Those zero modes are the instanton solutions, and there's only two of them, plus one half and minus one half. So I put, of course.

335
00:58:08.010 --> 00:58:13.839
Rm 330: Shadow and Sonic for the two solutions to the Instaton equation here.

336
00:58:14.140 --> 00:58:33.199
Rm 330: So, so we've accounted for all these zero modes. There are no other zero modes in the problem. You can actually prove, prove that actually there's a simple kind of numerically assisted proof that there are no… that there are no, other zero modes by some proofer bound, if you're familiar, but it's not super important.

337
00:58:33.430 --> 00:58:40.480
Rm 330: And… So…

338
00:58:40.770 --> 00:58:59.200
Rm 330: If we account for all those zero modes, we can come back to our problem and just remove them and look at this object that I showed before, this, which is the… everything that's left, all the massive modes that are left. This looks like… so this is where you get something that looks kind of like the determinant of a T matrix.

339
00:58:59.200 --> 00:59:03.010
Rm 330: This T is not the T matrix, the 1 plus T is.

340
00:59:03.240 --> 00:59:18.759
Rm 330: But this, with the zero modes removed, this, this is constrained to be something of order 1, because this T actually, it traces to a finite number, so it can't be anything other than a number of order 1.

341
00:59:19.070 --> 00:59:29.009
Rm 330: Once you remove all the zero modes. And in order for, the BDA argument to be true, there would have to be an exponential amount of, of

342
00:59:29.130 --> 00:59:34.689
Rm 330: of zero modes, this object, this determinant would have to grow exponentially.

343
00:59:34.800 --> 00:59:39.150
Rm 330: Which it, clearly, does not.

344
00:59:41.410 --> 00:59:46.210
Rm 330: Okay? And there's a whole bunch of technical details, which I've obviously swept under the rug here.

345
00:59:46.760 --> 00:59:58.889
Rm 330: But the end result is that we get this, non-cultural Hughes-Sandi form with the exponential, and now a completely computed.

346
00:59:58.900 --> 01:00:15.689
Rm 330: constant out front. The only input to this was solving numerically for the instanton, and then using it to compute everything else. You can compute the amplitude, and you can compute this S star, and because of the zero modes, you get this 1 over W to the fourth out front.

347
01:00:17.350 --> 01:00:35.430
Rm 330: It's actually, if you were to do this argument by Sydney Coleman about, like, what the power should be, you would… might naively end up with a W to the 6th, because of every bosonic… you break 6 bosonic zero modes, and that argument tells you that you should get a W to the 6 out front.

348
01:00:35.680 --> 01:00:40.590
Rm 330: Turns out, supersymmetry cancels one of those, and you go down to W to the fourth.

349
01:00:41.160 --> 01:00:42.050
Rm 330: Umm.

350
01:00:42.500 --> 01:00:56.939
Rm 330: But that's just another technical detail. Are you winding up? Yeah, I'm winding up. So… so, going back to what we have, now with this W to the fourth, this came from the zero modes, and that, curve now becomes even straighter.

351
01:00:57.010 --> 01:01:04.789
Rm 330: And it now fits quite well, modulo this three quarters that I mentioned.

352
01:01:05.050 --> 01:01:17.529
Rm 330: And, I think I also understand the difference between these curves. One is that there's higher… higher order effects which could easily account for this difference, but also, if you look back at the numerical paper where we found these blue dots.

353
01:01:17.530 --> 01:01:28.749
Rm 330: We looked at smaller correlation links, and those were actually a little bit below, and a little bit below. So it looks like it's actually converging to the correct result if I go to larger correlation links in the numerics.

354
01:01:28.850 --> 01:01:31.759
Rm 330: Which is, I think, plausible.

355
01:01:31.880 --> 01:01:40.150
Rm 330: Or maybe I got super lucky, and this just happened to line up with my curve very well. That's also, I guess, hasn't been ruled out.

356
01:01:40.680 --> 01:01:55.589
Rm 330: So, let me just wrap up with what this does not settle. This is the mean density of states. It's finite and smooth through the potentially avoided transition. This analysis does not touch typical observables. Those

357
01:01:55.750 --> 01:01:57.820
Rm 330: Could, in principle, still be critical.

358
01:01:57.930 --> 01:02:10.550
Rm 330: However, numerically, it looks like they are not critical. It looks like those also track the avoided criticality, like, like, we've seen the connectivity goes as rho of 0 squared.

359
01:02:10.550 --> 01:02:17.950
Rm 330: And this could be due to a strong hybridization between these power law states, which can then lead to conduction through the entire band.

360
01:02:18.920 --> 01:02:34.810
Rm 330: But we don't technically… there's no proof of that, and we don't even know exactly why it's a square. So these are kind of open questions. How do the typical observables look like? Could they potentially be multifactorial up to some actual critical point? We don't know for sure.

361
01:02:34.890 --> 01:02:50.499
Rm 330: And is there a full theory where you have both the perturbative RG and the instantons side by side? We don't have that either. And could you have a controlled theory for a dilute gas of instantons? Everything I've had was one instanton. I just averaged it over all of space.

362
01:02:50.560 --> 01:02:55.199
Rm 330: But you could have a theory with multiple instantons that could interact with each other.

363
01:02:55.800 --> 01:02:59.560
Rm 330: With that, I think I will leave it there, so thank you.

364
01:03:03.280 --> 01:03:09.079
Rm 330: So it seems like the term instant turn existed for quite a long time.

365
01:03:09.080 --> 01:03:24.050
Rm 330: Yeah. Yeah. And is instanton ever, let's say, discovered or anything similar to this ever experimented or something? Is it completely new or conceptual? So the classic example of instanton is tunneling in the double well.

366
01:03:24.050 --> 01:03:38.209
Rm 330: So tunneling events in a double well is an instanton. That's sort of the quintessential example of an instanton. So high energy people might have another way of looking at instantons, but at least that's my basic understanding.

367
01:03:38.210 --> 01:03:53.109
Rm 330: of them. Those are, saddle points of, of the inverted double well, because that's how you get… that's how you get the tunneling. This is also a saddle point, sort of the analysis is very, very similar, but the word…

368
01:03:53.570 --> 01:03:55.479
Rm 330: Slightly a misnomer, in my opinion.

369
01:03:57.730 --> 01:04:10.209
Rm 330: Yeah, you have looked at the destiny states of one node, but there is also, I guess, a possible topological or quantum geometric transition. You have some anomaly, some chirality of the by node now. Yeah. If you have two inner lattice.

370
01:04:10.220 --> 01:04:18.039
Rm 330: Like, there should be a transition from having this chirality, even if they are… even if you have some chiral diffusive metal.

371
01:04:18.220 --> 01:04:38.779
Rm 330: So that is controlled by the inter-node scattering. Do you have anything to say about that? Yeah, yeah, so you do actually… so Jed has some very nice work where they actually looked at the chiral anomaly for this case, and I think the result is, like, right, it's still there, right?

372
01:04:39.100 --> 01:04:50.780
Rm 330: If you have two nodes, then you have internet scattering. Yeah, you definitely get internet. It's part of the chiral nominal magneto resistance that, you know, there's a regime where you get it, and eventually you'll have internet scattering.

373
01:04:50.990 --> 01:05:09.219
Rm 330: cut it off. Yeah, but did you guys have a potential where you had it to where it didn't actually affect it? Like, only in one node? Oh, that is… Yeah, if it's in one node, you still get a nice chiral anomaly, but then you, yeah, make it bigger and source the scatter? Oh, anomaly itself is robust, but when you have

374
01:05:09.220 --> 01:05:11.690
Rm 330: In terms of scattering, there's always a scale.

375
01:05:11.870 --> 01:05:19.269
Rm 330: In any material. Any material. Any disorder is trying to gain, like, breath, like, renormal. Eventually, it'll break it.

376
01:05:19.570 --> 01:05:20.610
Rm 330: Because you can't.

377
01:05:20.850 --> 01:05:22.509
Rm 330: Truly get rid of internet scatter.

378
01:05:24.290 --> 01:05:25.690
Rm 330: throw my my

379
01:05:27.110 --> 01:05:37.400
Rm 330: my, credit came back, so I wanted to… what I was saying earlier. So… so, if I understand right, this is all continuum discussion.

380
01:05:37.500 --> 01:05:42.830
Rm 330: here. Everything is continuous. Yes, everything is continuous. And you argue that anytime I put on

381
01:05:42.830 --> 01:06:02.699
Rm 330: In a lattice, I'm going to introduce the other nodes, and it's a different problem. But you did allude to… you did allude to one possibility which you didn't talk about, which is, supposing I took these to be the surface states of a four-dimensional churn insulator, right? Then I could have a lattice, right?

382
01:06:03.010 --> 01:06:17.870
Rm 330: So I can still ask my question. If I did a lattice in that context, and I now allowed for the sloshing up and down of my vial, no, would I not just get a finite density of states from just taking the average of the density of states over the sloshing?

383
01:06:17.870 --> 01:06:28.310
Rm 330: Yeah, so I — So isn't this even wrong in that example? So I believe in that example that the average of the sloshing, there's not —

384
01:06:28.310 --> 01:06:47.839
Rm 330: the kinetic energy is sort of too strong for that sloshing to actually help contribute. There's just not enough states nearby that sloshing… It's like a little spacing to the average to sort of strengthen it. It'd just be stupid. The density of states goes like mg squared. If the sloshing average is mu, or delta mu, it'll go like delta mu squared.

385
01:06:48.550 --> 01:07:02.280
Rm 330: A direct contradiction of your result. Direct. What's wrong with that? Yeah. So I think there is something wrong with that. What's protecting your result against my claim that there should be a delta mu squared effect?

386
01:07:03.290 --> 01:07:16.959
Rm 330: in the example of the surf. There's no inter node scattering, because I've very carefully given myself a churn insulator in 4 dimensions. This created a beautiful single Dirac node, and I go ahead.

387
01:07:17.560 --> 01:07:19.480
Rm 330: and just use my dumb argument.

388
01:07:19.730 --> 01:07:21.599
Rm 330: What's wrong with the dumb argument?

389
01:07:21.750 --> 01:07:33.389
Rm 330: Yeah, so… What's wrong with the… with… with the dumb argument? The… So, I guess, like, I…

390
01:07:33.510 --> 01:07:47.140
Rm 330: there's going to still be a scale for that sloshing, and the vial node itself is at an even longer scale, nonetheless. The wavelength will still be over an even longer scale, because it's 1 over K.

391
01:07:47.210 --> 01:08:07.190
Rm 330: near… near that point. And since it's sampling things that are even broader than any sloshing you can possibly imagine, that'll still, by the, Emory Ma argument, be an effective disorder size. That'll vanish. I think the wavelength of my mode to be larger than a… much larger than a lattice.

392
01:08:08.190 --> 01:08:17.749
Rm 330: And I can definitely move it up and down, right? But the way I think of the mode is it's going to be… it's going to be larger than whatever sloshing that you actually have in your potential.

393
01:08:18.960 --> 01:08:25.060
Rm 330: Because you're… Would you agree that if I made the wavelength into that, I definitely can move the chemical potential up and down?

394
01:08:25.250 --> 01:08:26.060
Rm 330: Right?

395
01:08:26.370 --> 01:08:35.739
Rm 330: So definitely infinite wavelength. I can change change the density of states. Yeah. So now I just need to go in between infinite and the lattice spacing.

396
01:08:36.100 --> 01:08:44.880
Rm 330: Right? Yeah, I just don't know how you get… how you… you… you get around the fact that these waves need to, like, they, they're going to…

397
01:08:45.700 --> 01:08:58.940
Rm 330: sample, like, their effective disorder strength, even if it is a slow disorder, is going to, like, tune to zero. Just because the wavelength has to be very, very long at the node itself.

398
01:08:59.180 --> 01:09:03.079
Rm 330: It's not that the average is zero, but the root mean square is finite.

399
01:09:03.960 --> 01:09:15.399
Rm 330: But it gets bigger and bigger the closer you get to the node, or the effective size that you see, which should scale. I just didn't hear anything that got rid of my dumb argument.

400
01:09:15.420 --> 01:09:25.430
Rm 330: Okay, well, maybe, maybe you can argue about this later. This is an American argument. Maybe you can argue about this later. Are there any other questions?

401
01:09:26.399 --> 01:09:28.170
Rm 330: All right, thank you, Jeff.

402
01:09:31.000 --> 01:09:52.309
Rm 330: I mean, like, when it curves off, you'll run into maybe the… What is the next order? Okay, good. Correction here, I just…

403
01:09:52.310 --> 01:09:56.570
Rm 330: But don't you think it's some sort of they're very typical, they're not rare. Don't get me.

404
01:09:56.630 --> 01:10:00.319
Rm 330: Right. If I'm not clear on that brief thing, Helpful.

405
01:10:00.320 --> 01:10:23.400
Rm 330: Oh, you're very correct. I mean, that is right, it's just the argument of the meaning of the path and the level spacing. Yeah, which is memory mar. I know, I know. I don't know what the issue is with that. Did you write on the board form and compare it to level spacing? I think that didn't pan out, again, because…

406
01:10:23.980 --> 01:10:48.689
Rm 330: Yeah, I showed that argument on the board. Thank you. No, I can look them up. Yeah, exactly right. No, it applies to everything.

407
01:10:48.690 --> 01:10:58.450
Rm 330: Is there any relation because there was like four fermionic things?