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[AUDIO LOGO] SPEAKER 1: Thank you all for coming. It's my great pleasure to introduce Charlie Kane for his second seminar of the series this week. He is the Christopher H. Brown distinguished Professor of physics at the University of Pennsylvania. And he got his PhD from MIT in 1989. And after a post-doc at IBM, joined the faculty at UPenn in 1991.
Now, this is the last AEP seminar. So this is more of a specialized audience. So by now, I think many of us are familiar with Charlie's work. But I'll still go through it briefly, that he worked on many electronic states of matter, including quantum Hall systems, legendary liquids, carbon nanotubes, and in particular, topological insulators. And his work in that field and its generalizations have-- at least I can say for me, they've definitely impacted me very deeply professionally, since my entire PhD was based on 2D and 3D topological insulators and their experiments.
So he's won a number of prizes. And Chao-Ming introduced those yesterday. I understand his tradition not necessarily to read out everything on all three lectures. So without further ado, Charlie Kane.
CHARLES KANE: OK, thank you.
[APPLAUSE]
All right. So again, I'm really happy to be here. And so I want to tell you a story about topology. Of course-- yeah, so topology for the last several decades has been a big story in physics. And most of the stories of topological insulators, turn insulators, quantum Hall effect, what it's about, it's about topology and Hilbert space.
And the remarkable thing is that topology gives us a tool for distinguishing phases of matter-- there we are-- distinguishing phases of matter that have the same symmetry. And one of the hallmarks of this is the quantized Hall effect, where the different topological phases are characterized by quantized responses, this remarkable quantized Hall resistance, which reflects the Chern number, which is a topological invariant that characterizes the sort of winding of the quantum state as a function of momentum as it goes around the Brillouin zone.
And so this idea has spawned many, many new things, topological insulators, topological superconductors, topological crystalline insulators. There are many, many versions of this type of topology.
And it's not just insulators. It also applies to metallic systems. So for example, a while semimetal has a Chern number associated with the wild points. And so this topological protection associated with the winding in of the wave function in momentum space has a lot of consequences.
Now, so this is not what I'm going to talk about today. I'm going to talk about something completely different. And so what I want to talk about today-- so over the course of the pandemic, I spent a lot of time sitting at home by myself and thinking about things, trying to think of something interesting. And one of the things that popped into my mind was this picture of the Fermi surface of copper.
And so of course, this is a picture which is familiar to anybody who has studied Ashcroft and Mermin, which in my youth, I studied Ashcroft and Mermin and I learned about this picture of the Fermi surface of copper. And of course, this is a well-known picture.
But when I thought about it when I was sitting by myself, it occurred to me that, it's kind of a donut. One could view this Fermi surface as a topological object. And in fact, it's like a donut that has four handles on it. And so there's a number, four, associated with the Fermi surface of copper.
And in fact, any metal is going to have some topological number associated with its Fermi surface. And so the question I asked myself is, does that have any significance? Does that show up in anything? Is there any kind of quantized response or anything else that knows about this number that every metal has? So that's the question that I posed myself.
And so I want to tell you the story of some of the things that I came up with about this. So in fact, I think the topology of a metal does show up in some things. It shows up in a kind of quantized, nonlinear response that I will describe to you. That's the first idea I had.
So it also shows up in a quantized response, and what I'll tell you is sort of landauer-type response in a Josephson pi junction. So I'll explain to you what this is.
So this is actually very recent. We just posted the paper about this last night. So this is hot off the press. So this is the first time I've talked about this number two here. But it sort of belongs in the middle.
And then, I'm going to talk about another thing that we realized, is that this topology of the Fermi surface shows up in the structure of the quantum entanglement that is present in a Fermi liquid. And so I'll try to get to all three of these. Hopefully I will have time.
So again, I started off sitting in sitting in my room by myself thinking about this. So that's the first paper.
And then for the entanglement, I was joined by my PhD student, Pok Man Tam, who's really a phenomenal PhD student who unfortunately for me is going to be graduating soon. But lucky for somebody is going to get him as a post-doc. So that will be a lucky person. And also my colleague, Martin Klaassen, who knows more about entanglement than I do. And so he helped on that as well. And then the recent work is with Pok and me, OK? And so this is what we're going to try to do.
OK, so I was sitting by myself. And of course, my first thought was, look, this is physics of the 1960s and '70s. How could there be any unknown territory there? So my first guess is that there must not be anything interesting about the topology of the Fermi surface.
But then when I thought about it more, I realized that I already knew an example where the topology of the Fermi surface is important and plays a role similar to what I'm looking at. And this is something that we've all known about for a long time, which is what happens in one dimension.
So in one dimension, if you measure the conductance of a one-dimensional wire, and if it's sufficiently ballistic, you measure a quantized conductance. So this is called the Landauer formula.
So for those of you who haven't been through this, let me remind you how the landauer formula works. So if I just have a one-dimensional electron gas P squared over 2m, then Landauer's idea is that the electronic states get populated according to the chemical potential of the reservoir that they came out of. So the states with a positive group velocity which are moving from left to right, they're in equilibrium with the left reservoir. And the left-moving states are in equilibrium with the right reservoir that they came out of.
So if you put a voltage difference across them, then there's a mismatch between the chemical potential of the right movers and the left movers. And so to compute the current, all you have to do is add up the current that's carried by every state. So that's just going to be sum over K of the charge times the velocity.
And now the velocity, of course, is the group velocity DEDK. And so you can see that there's a magic happens. The DKs cancel, and so you just get an integral over energy, which gives you then e squared over h times the voltage. And so this is the Landauer formula for the conductance of a ballistic conductor, where there's no backscatter.
So actually, this argument is very similar to the same argument that you'd give to explain the quantization of the Hall conductance and the quantum Hall effect due to edge states. It's the same story. And there in the quantum Hall effect, that number that you measure, which is the number of edge states, is a topological property of the interior. It tells you about the Chern number that characterizes the interior of the two-dimensional Chern insulator.
But you see, this is different because we don't have an interior of the one-dimensional metal. So there's no Chern number. But I want to argue that it really is still topological because it doesn't depend on the-- I can change this band structure. I can make it like that and it stays exactly the same. I can keep changing it until I split the Fermi surface into two pieces, and then it goes from one to two.
And so it's a topological property, but it's not a topological property of a two-dimensional bulk. It's a topological property of the one-dimensional Fermi sea. And that's what I was after. So the Landauer formula does exactly what I wanted.
Now, one thing about this though-- so basically, what the Landauer conductance measures is, it measures the number of components of the one-dimensional Fermi sea. That's a topological property of the one-dimensional-- it's kind of a trivial topological property, but nonetheless, it's a topological property.
Now of course, the quantized Hall effect is great, one part in a billion. Now, this is not as good as that. Because in order for this to be quantized, it requires that you have ballistic transport, that there's no scattering. Electrons can can't turn around. Now in the quantum Hall effect, you get that for free because the right movers and the left movers are spatially separated from each other.
But here, they're on top of each other. So in order for this to be quantized, we have to declare that the transport is ballistic. But let's be willing to do that. And then we have some quantized response.
So I said, it's not as good as the quantum Hall effect. But that doesn't mean that you can't see it. And so in fact, this quantized Landauer conductance, this is an old story going back to the 1980s. It's been observed in quantum point contacts, in sort of lithography really patterned quantum wires, carbon nanotubes, semiconductor nanowires. So this quantized Landauer conductance can be observed, provided you have a system which is good enough. It doesn't have to be perfect in order to be able to see it.
Now of course, you're not seeing one part in a billion here. But it's still good enough, OK.
So what I'd like to do is, I'd like to try to generalize this one-dimensional topology of the Fermi sea to higher dimensions. And so in order to do that, I want to think about, how do I topologically characterize the Fermi sea? And so I'd like to introduce the Euler characteristic as a topological number that characterizes.
And so the Euler characteristic has a mathematical definition, so one can write it as a sum of something what are called the betti numbers. I don't want to dwell on this mathematical definition too much. But let's talk specifically about one, two, and three dimensions. And there, it's easy to understand what the Euler character is.
So in one dimension, it's precisely just the number of components of the Fermi sea. So it's exactly the thing that the Landauer formula measures.
In two dimensions, what it is-- so in two dimensions, every Fermi surface is just a loop, a circle. But it either encloses electrons or it has electrons on the outside, so it encloses holes. So every Fermi surface is either electron-like or hole-like. And what this Euler characteristic is, is it's the difference between the number of electron-like Fermi surfaces and the number of hole-like Fermi surfaces.
In three dimensions, every Fermi surface is a two-dimensional surface. It could be a sphere. It could be a donut. So every Fermi surface has a genus. And so the Euler characteristic is just related to the genus of every component of the Fermi surface, like that.
Now actually, there's a fine distinction that I want to make. You could talk either about the Fermi sea or you could talk about the Fermi surface, which is the boundary of the Fermi sea. So the Fermi sea is what's on the inside. That's where the electrons are. And the Fermi surface is the boundary.
And the two are related to each other. So for instance, in one dimension, obviously every component of the Fermi sea has two boundaries. So the Euler characteristic of the Fermi surface is twice that. The same thing as true in three dimensions. Perhaps you know the Gauss-Bonnet theorem, which tells you that the Euler characteristic of a surface is twice 1 minus the genus. And the Euler characteristic of the interior is 1 times 1 minus the genus. So the same factor of 2.
But in even dimension, in two dimensions, every Fermi surface is a circle, whether it's electron-like or hole-like. It's just a circle. And the circle doesn't know which is the inside and the outside.
So every Fermi surface is trivial. So it turns out that in even dimensions, the Euler characteristic of the Fermi surface is not the same as the Euler characteristic of the Fermi sea. It has less information. So there's a sense in which-- that's why I want to focus this on the Euler characteristic of the Fermi sea. In even dimensions, that has a little bit more information.
Now, there's another way of thinking about this topology, which is actually kind of useful, which in mathematics is called Morse theory. And it has to do with, you can understand the topology by thinking about the critical points of what's called a Morse function. And for us, this Morse function is just the energy as a function of K, the dispersion. And so one can characterize the topology of the Fermi sea by thinking about the points that are either a minimum or maximum or a saddle point in the dispersion energy of K and count those that are inside the Fermi sea.
And you count them up with a plus or minus sign, which depends on how many downward directions you have. So a saddle point, you have a minus 1. And so it's a theorem that says that if you add those up, that that's equal to this Euler number.
And it makes sense that something like this has to be the case, because you can ask the question, how can the Euler characteristic change? How can the topology of the Fermi surface change? And so what has to happen is, one of these critical points has to pass through the Fermi surface. And so that would be a topological transition in the Fermi surface, which actually has-- this is something that's been known for a long time. These are called Lifshitz transitions.
And for example, imagine that you have a energy dispersion that has a saddle in it. And so if the Fermi energy is a little bit below the saddle point, then the Fermi sea is sort of split into two pieces. But if you raise the Fermi energy above the saddle point, then those two pieces get connected to each other. And so that's a topological transition.
And so it makes sense that topological transitions occur when the critical points pass through the Fermi surface. And so there's a connection between the Fermi surface topology and the critical points. And that's actually going to be useful for the way we think about this.
OK, so let me tell you what the quantized response that I have in mind is. I want to try to generalize the Landauer formula to higher dimensions. And so let me just cast the one-dimensional version in a slightly different form.
So what the one-dimensional Landauer says is, I apply a voltage, and then I get a current which is e squared over h times the voltage. And so let me imagine that I have an infinite one-dimensional wire, and I'm going to put a voltage on half of it and 0 voltage on the other. And then I'm going to measure-- I could either measure the current that flows, maybe as a function of frequency, or I could measure the charge in the right hand side as a function of frequency. And they're just related to each other by a time derivative. So in the frequency domain, they're just going to be a factor of a power of omega different.
And so that defines if I define the charge in the right hand side as a function of the voltage on the left hand side. Then this response function is just going to be the Landauer conductance divided by i omega.
And so in order to generalize this to higher dimensions, what I'm going to do is, instead of dividing the line into two pieces, I'm going to divide the plane, the two-dimensional plane into three pieces. And then I am going to apply voltages to two of these pieces, for the first one at a frequency omega 1, the second one at a frequency of omega 2. And then I want to measure the current that flows in the third region at the sum frequency, which since I'm doing it at the sum frequency, it has to be due to both of these.
And so this will define a second order nonlinear response. And so my claim that I want to argue to you is that this second order nonlinear response has a universal term in it, which involves two powers of omega in the denominator, fundamental constants e cubed divided by h squared, and then this Euler characteristic of the two-dimensional Fermi sea. So this is what I want to try to convince you of. So there's a sort of universal quantized response here.
So in order to do that, I'm going to try to-- I'm going to present to you a kind of simple argument. And this argument is inspired by the argument that Laughlin introduced to explain the quantization and the quantized Hall effect. So that was my inspiration for coming up with this argument.
And so let me first give you the version of this argument that explains the one-dimensional Landauer formula. So again, what I want to do is, so I have my infinitely long one-dimensional wire. And so what I'm going to do is instead of applying a constant voltage, I'm going to apply a voltage pulse. And I'm going to apply a voltage pulse in such a way that the time integral of the voltage pulse is h divided by e, which is Planck's constant divided by the electric charge.
Now, the reason I want to do this is because this is analogous to what Laughlin-- so what Laughlin did is, he imagined he had a ring. So if I imagine having this one-dimensional thing wrap back onto itself so it forms a circle, then this voltage pulse is like threading exactly one quantum of magnetic flux through the ring. By Faraday's law, there will be an EMF that gets generated, which gives precisely this voltage pulse.
And so then I want to ask, what happens in this case? And so if you think about Laughlin's construction, so what happens due to this voltage pulse is, the electrons get an impulse, they get a kick. And that kick transfers a momentum of exactly 2kf to the electron gas. So it's as if exactly one electron gets moved from the left side of the Fermi surface to the right side of the Fermi surface.
Another way you can think about it, which again, is inspired by Laughlin is if I put it in a ring and I idiomatically thread the quantum of magnetic flux, of course, in the ring, then all of the states are quantized. So you have an integer number of wavelengths that fits in the ring. And when you thread the flux, every state moves over by one space. And so at the end of the day, there's one empty state left here and there's one extra electron here.
So this is what happens when we do this h over e voltage pulse. And then what's going to happen is that this extra right moving electron is going to go off into the right lead, and the extra left moving hole is going to go off into the left lead. So the end result is exactly one electron gets transferred to the right lead.
Now so this is actually telling us something kind of deep. Because what this is related to is the fact that if I just think about the right movers by themselves, then it's as if the charge was not conserved in the right movers. And so that reflects an anomaly. It reflects the chiral anomaly.
And so why does the chiral anomaly happen? Well, the chiral anomaly happens because the left moving electrons and the right moving electrons are actually not completely independent of each other. They're connected to each other at the critical points where the velocity is 0.
So one way we can think about what's happening here when an electron gets transferred from here to here is that at every critical point, when I thread the flux, exactly one electron changes its mind about what direction it's going in. And actually so when I Drew it here, one goes from being a left over to a right over here and also here. But here, one goes from being a right mover to a left mover. So the net result is the difference between the number of maxima and the number of minima, which is exactly the Morse theory way of writing down the Euler characteristic. So you see this connection.
All right. So now, this is what I want to generalize to two dimensions. And so now, this next argument is going to be my main point.
So what I want to do is, I want to consider my two-dimensional plane that I've divided into three regions now. And just for convenience, I'm going to assume that the first region, the boundary of the first region is a straight line. So the first region is the half plane and the second two regions are the other two quadrants.
This region, I'll relax this assumption a little bit later. So it's not essential. But it helps me because now what I'm going to do is, I'm going to apply-- in real time, I'm going to first apply h over e voltage pulse to region 1. And after that, I'm going to apply an h over e voltage pulse to region two. And then what I want to ask is, I want to ask how much charge goes into region three?
But now of course, after the first pulse, there's going to be a whole bunch of charge that goes into region three, because this boundary is long. And likewise after the second pulse, there's going to be a bunch of charge. So what I have to do is, I have to subtract off the charge that would go into region three if I applied the pulses independently. So I only want to keep the connected piece, the piece that depends on both pulses. But that's a well-defined thing. I can subtract off the independent pulses.
And so that's what I want to do. And so in order to think about this, the reason having this straight line is useful is because after the first pulse, I still have translation symmetry in the y direction. So I can talk about each value of ky one at a time, separately.
And for each value of ky, it's just like the one-dimensional problem that I talked about before. I apply this voltage pulse. And for each value of ky after the pulse, there is going to be one extra electron propagating to the right, and then there will be one extra hole propagating to the left. But if I'm on the right hand side, which is what I care about, then that's not relevant. I just have the extra electrons that are propagating to the right.
So now, I apply the second pulse. And when I subtract off the independent pulses, the only thing I need to worry about is the effect of the second pulse on these extra electrons. And so now, what's going to happen? What's the effect of the pulse on the extra electron? Well there's the extra electron that's up at the top here. It's basically going up. And it's going to go into this lead whether I apply the pulse or not.
Likewise, the pulse down here or here, these are definitely going to go-- this is definitely not going to go into region three. The only ones that are going to be affected by the second pulse are the ones that change their mind because of the second pulse. And the second pulse is going to move everything up by a space. So it's the electrons that are at these critical points that are where the velocity is, basically in the X direction. Then the second pulse is going to cause these electrons to change their mind about which lead they go into.
And so the upshot of this is that we can count the number of excess electrons that goes into region three by counting the critical points on the Fermi surface, where basically the velocity is in the X direction. But there are two kinds, because you see, this one, where the Fermi surface is sort of concave, it's sort of like an electron-like Fermi surface, then this one, when I move up, the dispersion is going to be like a upward-going parabola as a function of ky here. So this is going to go from being a down mover to an up mover.
Whereas on this side, on this point, which is a convex Fermi surface which looks like a hole-like Fermi surface, it's a downward moving parabola as a function of ky. So it's going to go in the opposite direction.
So what I need to do is, I need to take the difference between the number of concave critical points and the number of convex critical points. And so in this one, you can see, there are two concave ones and there are three convex ones. And so I'm going to get 2 minus 3, which is minus 1.
So I want to argue to you that this minus 1 is exactly the Euler characteristic of this Fermi surface. So in fact, it is because the Euler characteristic is the difference between the number of electron Fermi surfaces and hole Fermi surfaces, which is, there's one electron Fermi surface and there are two hole Fermi surfaces. So the Euler characteristic is indeed minus 1.
And one thing that it's not too hard to convince yourself of is that you can change this Fermi surface. So for instance, I could pull this point out so that I only have one critical point on the side, which would be a concave critical point. I sort of get rid of these two. But that doesn't change the difference. So this really is a topological invariant. And so the upshot is that this pulse argument really does reflect the topology of the Fermi sea.
So now, this is a little bit of a heuristic argument. So I can back this up with some calculations. Let me just tell you what kind of calculations we did. So one calculation is sort of a semi-classical analysis where we solve the Boltzmann equation. And the collisionless Boltzmann equation. And if we have ballistic transfer, that kind of is the right calculation to do. And then we can solve it order by order in the electric field. And this reproduces this pulse argument. We can exactly reproduce that by solving the Boltzmann equation.
Another class of calculations you can do is, you can do the analog of quantum linear response theory, but now it's second order, so nonlinear response theory. So this is sort of a second order version of the Kubo formula. And that amounts to evaluating some Feynman diagram that has three legs on it. So we can do that too. And in a smoothly varying limit, this quantum calculation exactly reproduces the semi classical calculation. So these are really the same calculations. Both of them reproduce the pulse argument. And then we can then generalize those calculations.
So for example, I can relax my argument that one of the boundaries had to be straight. So I can do it for arbitrary angles. And moreover, I can do the calculation in the frequency domain. So rather than doing these real time pulses, I can have one of the voltages be an omega 1, the other one be an omega 2, and then measure the sum frequency.
And so what I do is, I can verify then that I have this universal intrinsic term. There's also another term, which is not universal. It depends on the details, but it has a distinct frequency dependence. So one can separate these two. And so this is the one that I'm interested in.
So one can ask the question, is it conceivable that one could actually measure this in an experiment? And the honest answer I have to that is, I don't know.
So let me tell you what some of the issues are. So one issue is that it is essential that if you're going to do a measurement, you have to think about what the role of the context is, or the fact that it has a finite size. So my calculation is exact if I have, well, two assumptions. One is that I'm doing the calculation-- I did the calculation for non interacting electrons. And the other assumption was that I had an infinite plane.
Now, so if the plane is finite-- of course, it has to be in a real situation, then one thing that you can see, you know is that you know there's got to be a problem if I try to go to a very low frequency. Because if you think in the time domain, imagine I do I do my first pulse on Monday and then I come back and I do the second pulse on Tuesday. So what's going to happen is that on Monday, all of the effects of the first [INAUDIBLE] are going to go out in the leads, and they're going to be gone by Tuesday. So there will be no correlation left if I have a very long time separation between the pulses.
So what I need to do is, I need to do the two pulses within a time which is the amount of time that the electrons stay inside the two-dimensional electron gas of your sample.
And so that then puts a bound on what the frequencies-- it gives you a lower bound for the frequency. And in order to have the intrinsic term dominate over the extrinsic term that depends on the details, I need to have one of the frequencies be much bigger than the other. And then, everything breaks down if the frequencies are over to the Fermi energy. And so this is a pretty restrictive frequency criterion that one has to have. So I don't know. This frequency is not necessarily so small. So this could be an issue.
Now, there's another issue, which is the role of electron-electron interactions. And so those are something which one could imagine putting back into the calculation. One could imagine dressing this bubble with some sort of corrections. And those corrections are not zero.
And actually, there's a history to this question in one dimension. So it turns out if you think about the finite frequency conductance of a one-dimensional interacting electron gas, a one-dimensional Luttinger liquid, then at finite frequency with the length to infinity, the conductance of a Luttinger liquid is affected by the electron-electron interactions. And in fact, it's the Luttinger parameter k times e squared over h. So it's not e squared over h, just k times e squared over h.
Now of course, at 0 frequency in a finite one-dimensional system with leads, it goes back to being exactly equal to e squared over h. But we don't have the luxury of taking the frequency much less than vf over l. We have to take frequency bigger than vf over l. So we're in this regime where the interactions affect it. And there's a similar interaction effect that happens in the higher dimensional calculation as well, OK? So in order to do this, one has to be in a regime-- you have to find a system where you have weak interactions in order for this to work.
So my best attempt at this is to think about graphene. So graphene is nice because graphene has the Dirac point. And at the Dirac point, the density of states is very small. So the interactions are going to be small close to the Dirac point. So maybe it can be sufficiently weakly interacting.
And moreover, if you compare being just above the Dirac point to being just below, it's actually different. You have an electron-like Fermi surface here and you have a hole-like Fermi surface here. So they're different. So you might expect that as a function of the filling, you'd get a jump in this nonlinear conductance. So maybe it's worth trying to think about this.
So I don't want to be too naive. I expect that there are serious issues that experimentalists would need to confront in order to do this. But I don't know that it's impossible. Yes, question.
SPEAKER 2: So is this number for two-dimensional electron gas connected to the quantized Hall response?
CHARLES KANE: No, no, no, it's not. I mean first of all, there's no magnetic field. So the quantized Hall response depends on how many Landau levels are filled, and that depends on the magnetic field.
SPEAKER 2: You're talking about this [INAUDIBLE] that goes across the Dirac point [INAUDIBLE]. You have change before.
CHARLES KANE: Well--
SPEAKER 2: [INAUDIBLE].
CHARLES KANE: Look, I mean, the fact that you have a 0 Landau level in graphene is related to the fact that you have a Dirac point. And this is related to the fact that you have a Dirac point. So in that sense, yes, they're both related to the fact that you have a Dirac.
And it is-- you're right, it does depend on the number of Dirac points that you have. The fact that it's four is because you have two valleys and two spins. And there's a similar-- that factor of 4-- actually for the Hall conductance, you have to divide it by 2, right? But it's 4 divided by 2 for the Hall conductance. So OK. Yeah, yeah. So look, so graphene has a Dirac point. And that is featuring in this. OK.
So this is one possibility. Let me float another possibility, and I'm not an expert on cold atomic gases. But I'd like to learn more.
So there have been a couple of papers suggesting the idea that this might not be so crazy to implement this real time pulse construction in a gas of fermionic atoms in an atom trap.
And so the idea is, you have your atoms in a trap and then you can apply these pulses. And then you measure the number in one of these quadrants and you take the difference. And it's really implementing precisely this real time pulse construction.
And so the virtue of this is, in these cold atom systems, you don't have to worry about impurities. And the interactions in these atoms can be tuned so that they're very weakly interacting. So it's not crazy to think about a very weakly interacting system. There's no context to worry about.
And one of the things that these papers did is, they generalized this pulse construction to account for the fact that you have a trap. And they seem to think that this is actually something which is feasible with current technology. So that's one possible alternative venue for this physics.
I want to tell you another idea. This is the idea we recently have, the one we just posted last night. And this is a rather different kind of experiment. But it makes it more closely related to the one-dimensional Landauer transport.
So what I want to suggest is the following experimental setup which is, I want to have my two-dimensional electron gas. And my pointer is dying, unfortunately. But I think it will survive.
So I have my two-dimensional electron gas. And I want to deposit superconductors on top of it in such a way that the two-dimensional electron gas is proximitized by the superconductors. And in particular, I want to consider the case where there's a one-dimensional channel between these superconductors and the superconductors have a phase difference of pi.
And so the reason why this is an interest-- and then the experiment that I want to propose is to-- so now of course, the superconductors sort of end here. So now I have two-dimensional electron gas on this side, and I also have a two-dimensional electron gas on this side that are connected by this one-dimensional channel. And so I want to apply a voltage on this side, and I want to measure the current that flows into this side.
Now, if the phase difference isn't 0, then basically, the superconductors sort of get glued together and nothing I do over here is going to affect anything over here as long as I'm putting voltage below the superconducting gap. But when the phase difference is pi, then there will be a 0 energy Andrea bound states that get bound to the junction.
And the way to think about this is that, let's think about an infinitely long, one-dimensional channel. Then I should be able to solve the electronic modes as a function of K in the X direction if I have an infinitely long channel.
And each one, I have to solve the problem in the y direction. I have to solve the motion in the y direction. And in the y direction, what I have is, I have a superconducting pair term for positive y and a superconducting pair term for negative y. Those open up an energy gap. But since the phase difference is pi, they have opposite signs.
And so it's just like the [INAUDIBLE] domain wall problem, the [INAUDIBLE] domain wall problem. There will be a 0 mode that gets bound to the domain wall.
And so at least as long as kx is less than kf, then there's going to be a 0 mode. So if I think about the spectrum of this infinite wire as a function of K, there's going to be 0 modes. So there's actually going to be a pair of 0 modes. And as a function of the phase difference, if I go away from pi, then what's going to happen is, those 0 modes exactly cross each other at pi.
But then, if I think about K outside the Fermi surface, if my K is much bigger than kf, then this energy is basically just going to go up to be K squared over 2m. It's going to be very big and it's not going to know anything about superconductivity. There's going to be no 0 modes.
So what has to happen is that these 0 modes have to go up and merge with the gap at kx. And these 0 modes are really associated with the critical points on the Fermi surface where the electrons are moving parallel to the junction. And so now, one more piece of information. So what I want to be doing is, I want to be doing Landauer transport in these 0 modes.
And so the question you have to ask-- so this 0 mode here, it's dispersing. It has a positive slope, so it's propagating from right to left. So this quasiparticle that's in this state is going to keep propagating. It's going to end up on the right hand side, and it came from the left hand side.
So the question you have to ask is, what is it going to become when it goes into the lead, and what was it when it came from the other lead. And in this case-- so there are two ways I can think about it. I can think about having the superconducting conductor get wider and wider, or I can imagine that I just make the energy gap get smaller and smaller. I can imagine idiomatically turning off the superconductivity in the lead. And then eventually it just turns into a normal metal.
But what's going to happen is if I turn down the superconducting gap, then the energy of this quasiparticle is going to go from being below the gap to being above the gap. And when it's above the gap, it just turns into an electron.
So for this electron-like Fermi surface, these quasiparticle states turn into an electron on the outside. And of course, the ones on the bottom here, these are just a phantom copy of the ones on the top. That's what happens when you do the BCS theory of superconductivity, is you have to make two copies of your original band structure, one with plus and one with minus [INAUDIBLE].
So the upshot of this is that if I apply a positive voltage to this left lead, then I will populate these quasiparticle states, and they will end up as electrons flowing into the lead. And so I will get a current which is 2 e squared, 2 is for the spin, 2 e squared over h times the voltage.
Whereas if I apply a negative voltage, then I'm not going to be populating these states, because these are the states which were electrons. And so nothing happens for negative voltage. So we get a kind of interesting-looking IV character, so it looks kind of a diode. And so this is what happens.
And I want to emphasize that this is associated with what happens at this critical point. So now, you can imagine what happens if you have a more complicated Fermi surface. So let's imagine I have-- this one looks like the thing that had Euler characteristic minus 1.
So this has a bunch of critical points going to the right. It has this one, this one, this one. But this one is electron-like and this one is hole-like. And so what happens is that the quasiparticle in this one turns into a electron, but the quasiparticle associated with this one turns into a hole. And so the upshot is that the current that flows when I go across this junction, it's going to be my two e squared over h times-- for positive voltage, it's going to be the number of electron-like Fermi points, which are the red ones, the convex ones. And for negative voltage, it's going to be the number of hole-like Fermi points, which are the blue ones. So you're going to have two different slopes. And the difference of those two different slopes is precisely the Euler characteristics.
Remember, this is the same counting that I did when I did the pulse argument, counting the difference between the number of concave and convex critical points on the Fermi surface. So it's exactly the same topological invariant.
So actually, one way you could get at this is as with a diode. A diode gives you rectification. So I apply an AC voltage and then you get a DC current. And the DC current is going to depend on the difference of these, so this rectified current actually pulls out the Euler characteristic. So this is-- yeah.
SPEAKER 3: Is it important that the superconductor be Noblis?
CHARLES KANE: Be what?
SPEAKER 3: Noblis?
CHARLES KANE: Yeah, because I think if it's not Noblis, then the Andreev bound states can leak out into the superconductor. So that I don't want. So I want to have a situation where in the superconductor, I really have a gap so that I just have this one channel that connects the source and the drain. Yeah, yeah. So I think it is important to have a gap, have a superconductor with a gap.
OK, so I have a few more minutes. So I want to tell you one more story. And so this story has to do with the structure of entanglement. And there's some history to thinking about-- there's a deep connection between the topology of a state and the structure of quantum entanglement. And so there are two important examples of this.
One of them is if you have a two-dimensional topological phase, like the fractional quantum Hall effect or a spin liquid, something like that, then Kitaev and Preskill and also Levin and Wen showed that there's a deep connection between the total quantum dimension, which I can think of that as a topological index that characterizes the topological phase of matter. There's a connection between that and what they call the topological entanglement entropy, which occurs if you consider your system and you consider three subsystems, and you form a combination of the bipartite entanglement entropies of those regions that looks like this.
And this combination actually has important physics and an important name. It's called the mutual information between these three regions. And so somehow what this mutual information characterizes is, it characterizes the sense in which the three regions are entangled among each other. And so this characterizes what I call the tripartite entanglement of those three regions.
So this is one version of topology and entanglement. There's another version that arises in one dimension. If you have a one-dimensional gapless system that's described by a conformal field theory, then John Cardy and Calabrese showed that the bipartite entanglement entropy, if I cut it in half and ask how is region A entangled with region B, that's characterized by an entanglement entropy.
And what they showed is that the bipartite entanglement entropy has a logarithmic divergence as a function of the system size, and the coefficient of that log has topological information. In particular, it reflects the central charge C that characterizes the conformal field theory.
And so these are two things. And so the question that one can ask is-- oh, and one more point I will make is that for a one-dimensional Fermi gas-- that's the simplest free Fermi nonconformal field theory. That has C equals 1. And it has this log. So noninteracting electrons have this log in it, in their bipartite entanglement entropy.
And so I want to ask if this is something that can be generalized to a higher dimensional Fermi liquid. And so there's actually some history to this question. So Israel Klich and his collaborator ask the question, if I have a twe-dimensional or three-dimensional Fermi gas, what's the bipartite entanglement entropy?
And the answer to that is actually very simple to think about if I just imagine putting my two-dimensional Fermi gas on a cylinder, so I have a infinitely long cylinder but it's finite in the second direction. So then I'm going to quantize all my modes around. And every mode, every one-dimensional sub band is just going to be a one-dimensional system that's going to have Cardi's log in it.
So the upshot of that is, there's going to be a log divergence but the coefficient is going to be the number of sub bands, which I can think of as kf times the circumference. And so you get an area law entanglement, but the area law has a log divergence.
So that's that. But that's not what I'm after because that depends on kf. It's not topological. And the coefficient of the log depends on the system size. So that's not what I want.
You see, the problem is that in one dimension, of course, two regions meet at a point. So the boundary doesn't have any dimension to it. Whereas in two dimensions, two regions meet on a line, and so the entanglement is going to be proportional to the length of that line. In three dimensions, they meet on an area so you get an arial line.
And so what I'd like to do is, I'd like something that in higher dimensions, depends only on a point. And so the observation we made is that in two dimensions, generically three regions meet at a point, and in three dimensions, generically four regions meet at a point. So this is like a peace sign. And this picture is, of course, I realize it's a little bit hard to parse. But what this picture is supposed to represent is sort of like a three-dimensional peace sign, if you can picture that, so where these four regions are meeting at this one point.
And so what we have considered is, we've considered the multipartite entanglement between these regions and showed that is topological. So the specific thing that we have done is-- so let me just talk about the three-dimensional case, just like copper, the Fermi surface of copper.
So what we did is, we introduced a fancy version of this mutual information that characterizes the mutual information among four regions. And so this is designed to subtract off all of the area law contributions and all of the line law contributions. So all of those get subtracted off. So the only thing that's left is the entanglement that involves all four regions.
And so what we've been able-- so we've computed this, we figured out how to do it for free fermions. And what we've shown is that it also has a universal log divergence. And it's actually log cubed. The coefficient of this log cubed is precisely the Euler characteristic of the Fermi sea. So in a sense, this is a kind of generalization of the Cardy Calabrese formula.
And one thing that we have a check-- so we did the calculation for free fermions. It turns out, we can add interactions performatively to that calculation. And we've been able to show that actually, this mutual information is not perturbed by interactions. So we believe that this is really a characterization of the interacting Fermi liquid phase.
So unlike my nonlinear response which only worked for non interacting electrons, this should be true as long, I think, as you have as you have a Fermi liquid. And so therefore, this topological index which characterizes the genus of your Fermi surface, it distinguishes distinct phases, topologically distinct phases which have a topologically distinct pattern of entanglement.
And one other thing that I want to say, the way we did this calculation-- so it turns out for free fermions, the way that one can calculate this mutual information is to relate it to a correlation function among the number of particles in the different regions. So I have a charge in region A, charge in region B, charge in region C, charge in region D. And what this mutual information for free fermions is related to is the equal time connected correlation function of those four charges.
And actually, of course, entanglement is hard to measure. But in principle, this is something which in principle could be measured. And so it's worth thinking about this.
And in the process of doing this, we discovered a remarkable fact about a free Fermi gas. And so this will be the last thing I say.
And so I still don't completely appreciate why this is true. But it is a fact that a gas of free fermions has topological correlations in its density.
And so in particular, let me tell you what happens in one dimension, because that case is easy. So if I consider f of q, which is the equal time density-density correlation function in one dimension. So this is actually-- f of q is what you measure with X-ray scattering.
And so you can calculate that for a Fermi gas. It's a simple calculation. You just do ck dagger ck plus q, calculate the expectation value, you use Wick's theorem. And so that ends up being a sum over the Fermi occupation k, and then 1 minus the Fermi occupation at k plus q.
So what this basically counts is, it counts the volume in momentum space that's inside the Fermi surface but outside the Fermi surface shifted by q. And so it's just this red volume here, which is obviously linearly proportional to q.
And in fact, it's exactly q over 2 pi times the number of Fermi surfaces. So it has the Euler characteristic in it. The thing that we discovered that is remarkable is that this formula has a generalization to higher dimensions. And the generalization in two dimensions, it's to a third order equal time density correlation function. And so if I do it as a function of q, then the q's have to add up to 0. So I have row of q1, row of q2, and then row of minus q1, minus q2. And so what that is equal to is, it's basically q1 cross q1, the two-dimensional cross-product times the Euler characteristic of the Fermi surface.
In three dimensions, we do a fourth order equal time density correlation. So this is the thing I use to do the mutual information in three dimensions. And so it's a function of three q's, and it's going to be the Euler characteristic times the triple product of the three q's.
I feel like there must be something deep about this, which I'm not quite sure I appreciate yet. But it points to-- I told you, entanglement's hard to measure, but density correlations maybe are possible. And so this is something where I think an ingenious experimentalist maybe could figure out how to measure the equal time third order density correlations in a weakly interacting two-dimensional electron gas.
And so one thought I had is, let's suppose you have an atomic gas of weakly interacting fermionic atoms. Can you take a picture of it and see where all of the electrons are at any time? That should have information about these correlations.
Then you have to take that picture over and over again and get an average. But those pictures have correlations which know about the topology of the Fermi surface. And that would be a remarkable thing to demonstrate experimentally.
OK. So I think that's it for now. So yeah, I don't want to-- I'm going to skip this. So main lesson, which is, maybe it's obvious once you say it, is that the Fermi surface is a topological object. But maybe it has some consequences. So it's reflected in this nonlinear conductance, which I have to put caveats on, non-interacting and there's this issue about the frequencies.
Maybe this Andreev transport is a more robust signature. I actually think this should survive with interactions. And so the question is, can one do experiments on that?
There's this more theoretical aspect of the being reflected in the multipartite entanglement. And maybe this is a fundamental characterization of a Fermi liquid, the entanglement present in a Fermi liquid. Maybe there are other probes of the topology of the Fermi surface. I think it's worth thinking about. So thank you very much.
[APPLAUSE]
Yeah, [INAUDIBLE]. Oh.
SPEAKER 4: I was just wondering, [INAUDIBLE] how. Does that affect the [INAUDIBLE]?
CHARLES KANE: The entanglement?
SPEAKER 4. [INAUDIBLE].
CHARLES KANE: Yeah, I think that's probably right. I think probably if you have disorder, then things are going to get scrambled on the scale of the main [INAUDIBLE] path. That's my first guess, yeah, yeah, yeah. I think that's right.
SPEAKER 4: [INAUDIBLE] there's discrimination between the [INAUDIBLE] version and [INAUDIBLE]. Is it just a [INAUDIBLE], or is there something [INAUDIBLE]?
CHARLES KANE: It's the divergent part. Yeah, well so, yeah. So what I can say is that in general, the mutual information can be written as a cumulative expansion. And where I expand, I have higher and higher order. So the cumulative expansion is a sum over higher and higher order correlation functions.
And what I can say is that the higher order ones are not divergent. The log divergence is only in the first term. So if I'm talking about the log, then I'm justified in ignoring all the other terms. Yeah?
SPEAKER 5: I have [INAUDIBLE] question [INAUDIBLE]. Especially with regards to some of the transport based ideas, [INAUDIBLE] extending this to the case where you have a [INAUDIBLE] that has an [INAUDIBLE] Fermi surface, which can be deduced in some situations, like when you have [INAUDIBLE], or things like this?
CHARLES KANE: A superconductor where I'm proximitizing the entire two DAG with that or--?
SPEAKER 5: No, no, an actual superconductor that still maintains a [INAUDIBLE] Fermi surface, because of some sort of-- you have to add something. But [INAUDIBLE] has these ideas about--
CHARLES KANE: I see, I see.
SPEAKER 5: [INAUDIBLE] Fermi surface.
CHARLES KANE: Right, I see, I see. So you want-- OK. So you want to view the superconductor as the thing that has a Fermi surface whose topology you want to interrogate. No, I haven't thought about-- I haven't thought about that.
Yeah, there's no reason one couldn't think about that, yeah, yeah.
SPEAKER 1: Thank you. Any other questions?
SPEAKER 7: How do you stabilize [INAUDIBLE]?
CHARLES KANE: How do I what?
SPEAKER 7: Stabilize [INAUDIBLE]?
CHARLES KANE: Oh, they have to be connected to each other. They have to be connected to each other and then you have to have some magnetic flux that gives you the pi-- yeah. Yeah.
SPEAKER 7: So if there is a tiny invasion from pi, [INAUDIBLE]
CHARLES KANE: Well so, if there's a tiny deviation from pi, then what will happen is, there will be splitting in these 0 modes. And if that splitting is small, then it'll affect what happens at very small voltage. But at higher voltage bigger that gap, then it'll be OK.
So as long as the deviation is small, then I think it's OK. But what's important is that the size of the mini gap that you open up be small compared to the gap. Yeah.
SPEAKER 1: All right. [INAUDIBLE] again.
CHARLES KANE: All right. Thank you.
[APPLAUSE]
The Fermi sea in a metal is a topological object characterized by an integer topological invariant called the Euler characteristic, ¿F. In this talk we will argue that for a 2D Fermi gas ¿F is reflected in a quantized frequency dependent non-linear 3 terminal conductance that generalizes the Landauer conductance in D=1. We will critically address the roles of electrical contacts and Fermi liquid interactions, and we will propose experiments on 2D Dirac materials, such as graphene, using a triple point contact geometry. We will go on to show that for a D dimensional Fermi gas, ¿F is also reflected in the multipartite entanglement characterizing D+1 regions that meet at a point. This generalizes a well-known result that relates the bipartite entanglement entropy of a 1+1D conformal field theory to its central charge c. We will argue that for an interacting 3D Fermi liquid, ¿F distinguishes distinct topological Fermi liquid phases.
