 אני אעשה לך על קרן הז braking ומאוקרדה על כפות מ��טלס וזה הוורכה בבקשה עкой יובל ברמת תזהו ומקרה של יווכת ביחס theresberg ובאת הלב קר contribution פרמצר 1 אלולכיים מרונות היווכן ושפט lawsuit כלל ‫כדי שזה, לדבר, ‫זה הגבוה, נראה, ‫אם אני כבר זכיר, ‫אז אני רחילה לצאת מהפכה ‫בסבירי, אז לא נתקדת, ‫אז זכירה לא תקדת, ‫אם לא נתקדת. ‫אז אני גלדה להיות פה. ‫אוקיי, והשעה היא, ‫הוא, קצת כדי להסתכל ‫מה זה problem ‫שאנחנו בואים, וזה בעצם 2 ספירים לספרמנטליה. אני sẽ תראה את עמירות שהייתה של ספרמנטליה ובאודי ספרמנטליה שייתה לספרמנטליה. ואני שאולי תמיד לתת לך מה ספרמנטליה שילה מה פרמנטליה שהיה ולפתרות מה האתרון מגנטיקה. ופעולה ככה, אני אעשה לך איך נשאר את העמירות ‫למרות מופקדים ‫בפשרות על המתלרות של הקבילים, ‫פרטיקולות כבילים ‫בשרות על מה שאנחנו זוכרים. ‫אז אחרתיים סטאבים דברים. ‫תשעות קצתים שאין סלאב ‫בסלחת סמימנטל. ‫ואן הכפי כנת ‫שאחרתיים מביאים פה, ‫התשעה של סלאב ‫היא לל, ‫שעות כנת ‫הפתוחה על סלאב ‫שחרר תשעות קצת פתות richer much smaller than L. We drive a car from one contact to the other and measure the voltage on the other edge at a similar distance, basically it emit a mirror image of the same points to contact measure the voltage. Now, the question is what is the voltage on the lower to contact as a function of the voltage on the upperline? For the standard metal, of course we solve Kirchoff equations and Ohm's law in just For the reason which I cannot produce now, we use the notation capital sigma for the conductivity rather than small sigma, but other than that, it's really trivial, and what you find is, since this distance is much louder than the distance between the two contacts, very little of the current gets to the other side and therefore the voltage between these two contacts will be small, and whatever you have here will be parallel to the voltage between the upper two contacts. Now for a vile semimetal in a magnetic field, magnetic field oriented in the z direction, what we find is that there will be a significant voltage on the lower surface, and it will be anti-parallel to the voltage here. A little bit, some similarity but I think coincidental with what Leonid was talking about yesterday in the context of hydrodynamics. So this is the first experimental setup. Second experimental setup, same story, same slab, but now instead of putting a DC voltage and contacts and so on, we send in radiation from above, electromagnetic radiation, more or less at the microwave range, and ask how much of it is transmitted to the other side. Now again, this distance is L, and for conventional metal, of course, this is a textbook problem, and you solve Maxwell equation, you have Ohm's law, and you find that the transmission goes down exponentially with the thickness, with the characteristic length being the skin depth. What I'm going to argue is that for vile semimetals under magnetic field in the z direction, there will be transmission resonances in which most of the absorption in the metal turns into transmission. So there will be frequencies at which the vile semimetal becomes transparent. This has some similarity to an old effect found many years ago by Mark Asbel, by Karna and by Gantt Marker, still working in the Soviet Union, and in the effect they were talking about, you take a slab of a metal, you radiate, and you get a transmission resonance under conditions at which the frequency of the radiation corresponds to the cyclotone frequency of the electrons under a magnetic field, and the thickness of the slab corresponds to the cyclotone radius. In that case, the magnetic field you should apply perpendicular, so the radiation goes this way, the magnetic field should be perpendicular to the radiation. In the case I'm talking about, as I said, the magnetic field is in the z direction, and so is also the propagation of the radiation. So it has some similarity, but it's a different effect. Now the source of both of these effects is one. It is the conductivity of the vile semimetal, and in particular the fact that the conductivity is non-local in space. Of course, as we know, Omslo tells us that the current at the point R and T is linear in the electric field and point R prime and T prime, so we have a conductivity, and sigma tilde means that we're talking about real space, real time conductivity, a conductivity that's a function of the point where the electric field is R prime, the point where the current is measured R, and the time difference T minus T prime. For a standard metal, this conductivity is pretty much a delta function in space, not quite a delta function, but close to it. For a vile semimetal, as I will explain, this is not the case. Instead what happens is, remember the slab geometry, you put an electric field in one surface, you get a current flowing in the other surface, this is when there is a magnetic field in the z direction. It all depends on that magnetic field. So that's more or less the introduction to what I wanted to say. Now let me step back a few steps and review what vile semimetals are and how is this magnetic field, which I said is crucial to everything, how does it affect them? So we're talking about the three-dimensional lattice, so think about the three-dimensional band structure that has energy that's a function of kx, ky and kz, and wave functions which are functions of kx, ky and kz. Now fix ky. So if you fix ky, you get a two-dimensional system because ky is fixed, so everything is a function only of kx and kz. So for every ky, this two-dimensional system has a chair number c of ky. Now just for concreteness, it's not crucial, but for concreteness, let's think about a system that's globally termed versus symmetric before we put in the magnetic field. Then the chair number as a function of ky must be 0, both for ky equals 0 and for ky equals pi. So here's the chair number as a function of ky, and you see it's 0 here, and of course it's 0 here because it's periodic, and it's 0 here also. Now it may be 0 everywhere. And then there's nothing, it's not a wild semi-metal, but it may be that it's not. It may be that for parts of the ky axis, it will be non-zero. Of course there's time versus symmetry, so if it's positive here, it will be negative here, but in any case, what's important is that it's non-zero. When that happens, this is a wild semi-metal. And this was discovered in a series of works, and I have a more extensive list later. So this is a wild semi-metal. Now here's the longer list. This was studied in the last few years by many people, both theoretically and experimentally, and experimentally people saw mostly with ARPES, the Ben Structure that's characteristic of this, and maybe I should say, if as a function of ky, there are regions where the chair number is non-zero, then there must be transition points to each the chair number changes, and therefore the gap closes. So this cannot be gapped everywhere, and this is why it becomes a semi-metal. Each of these points becomes a diracone or a wild cone, depending who you want to give the credit to. And those cones were seen in ARPES, and also in some transport measurements, and there are various materials where that takes place. Now, I'd like to get to this non-local conductivity. So let's keep on fixing ky as we did before. So now we have these regions of ky where the system has a chair number. So as a function of kx and kz, as a function of x and z, the system is a quantum whole state. And if it's a quantum whole state, it means that it has f states on the surfaces. So for values of ky, for which the chair number is non-zero, there will be surface states, or edge states on the z surfaces at which the gap closes and at which there is chiral motion, like any quantum whole state. However, this will be limited only for those regions of ky for which the chair number is non-zero. So the surface which has a spectrum, the z surface which has a spectrum of ky will have what's been called Fermi arcs. We'll have lines of states in momentum space which start and end. Now they don't really end. What happened is that they penetrate into the bulk, into the third dimension which is not down here. Now, with no magnetic field, they solve the block states and there are constants of motion. The crystal momentum is a constant of motion. But once you put in a magnetic field, the wave vector changes as a function of time according to Newton's second law. So ky will, so the electron will flow along lines of constant energy in momentum space which means if it is on the surface, it will flow along the Fermi arc and it gets to its end. When it gets to its end, it must penetrate into the bulk because there are no more states on the surface. So that gives rise to very interesting cyclotron trajectory. So to see that, let's now think about the bulk. So what happens in the bulk if we have a three-dimensional metal in a magnetic field? In the standard metal, you just separate the x-y direction from the z-direction. You have Landau levels in the x-y plane and plane waves in the z-direction in which motion is not affected by the magnetic field. This cannot happen for a vile semi-metal because as we said, as we put the magnetic field, the electron flows along the Fermi arc and when it gets to the end of the arc, there is no choice but to penetrate into the bulk. So it must be able to penetrate into the bulk and not come back to that point. So this point cannot... must have some... there must be a way for the electron to get into the bulk and not come back. And indeed if you solve the spectrum of the... of a three-dimensional diracone in a magnetic field in the z-direction, you get the following type of spectrum. Each one of these points is a Landau level and this is now the spectrum as a function of kz of the momentum in the third dimension. So each of these curves is a Landau level which disperses as a function of kz. Now the zero energy Landau level, as you see, becomes a chiral state with a motion that's directed in one direction in the z-direction. Of course, this is one end of the Fermi arc. The other end looks just the mirror image and has zero energy Landau level which flows in the opposite direction. So now to understand what happens as a function or to understand how the cycloton orbits look like as we turn on a magnetic field in such a situation, let's think first about this energy. So in this energy, the bulk has only one Landau level you can occupy and as opposed, of course, to what happens here which we will discuss a little bit later. So the Landau level has... the bulk has only one Landau level and it has an upwards-moving state in one end of the Fermi arc and a downwards-moving state excuse me, in the other end of the Fermi arc. So that gives rise to a cycloton orbit that looks the following way. Remember, there's this surface, that surface, and the bulk. If the electron starts on the upper surface, it moves in momentum space along the arc which in real space, like always, means motion in a direction perpendicular to the motion in momentum space but anyway, it's a motion along the surface. By the time the electron gets to the end of the arc it gets to the bulk and flows all the way to the other surface. When it gets to the other surface it moves backwards on the other surface and then back. So the cycloton orbit instead of looking the familiar planar circle that we have in a conventional metal it will have sort of a half a circle on one surface then in the bulk to the other surface and then the other half of a circle. We need another blackboard but you get the idea, I hope. So this is the crucial point. The cycloton orbit has four parts, if you want. One in the upper surface, one in the bulk, one in the lower surface and back in the bulk and then it goes on and on. This is the source of the non-local responses that we now see. Now there's a period to this motion and as you can guess by this structure I outlined the period will have two parts which are in the bulk and two parts which are on the surfaces. In the bulk the time it takes to go from one surface to the other of course depends on thickness if the thickness divided by the velocity of the cone and the time to go on the arc depends on how long the arc is divided by the speed at which divided by k dot the speed at which you go in momentum space which is determined by the low end force. So this was worked out by this a group in Berkeley. So now let's get back to the problem I started with. We have the slab and we'd like either to look at the DC problem or to look at the radiation. In both cases what we need is the conductivity. What we need is the non-local conductivity that comes out in a wireless semi-metalline magnetic field. So to analyze this let's look at this real space, real time conductivity sigma tilde for some time t and for coordinates which are one surface and the other surface and to see that to calculate that basically what it says is we need to apply a pulse or a pulse of an electric field on the upper surface at t equals zero and ask how much current we get at the lower surface at the time t. So to apply the electric field at time zero we apply a vector potential AX which is applied at a thin layer close to the upper surface at time t equals zero should actually be a delta function of t sorry the electric field goes like delta a delta function of t and the application of the vector potential sorry, this is a mistake it's a delta function the application of the vector potential shifts the Fermi surface at the upper surface at time t equals zero and initiates a cycloton motion at the upper surface now as I said before it takes some time but this motion gets to the lower surface so the current at the lower surface after a while it starts at zero and then goes up when the electrons from the upper surface get to the lower one and then it goes down when they start coming back and then it repeats itself in a periodic fashion so this is the conductivity this is the real time conductivity for the non-local one having the electric field in one surface and the current calculated on the other surface and we can substitute that into the ohm's law and substitute ohm's law into Kirchhoff's laws or into Maxwell's equations for the two problems as it's discussed before and when we do that we find that this is the voltage as a function of the z direction in the absence of for the DC experiment in the absence of magnetic field as I said earlier there is no voltage developing on the opposite surface at all in the presence of magnetic field we get a voltage that's anti-parallel and is appreciable compared to the voltage in the upper surface and as I said before there are transmission resonances and the physics of these transmission resonances is there is a cyclotone frequency associated with this motion between the two surfaces if this period commensuates with the period of the radiation it's a little bit like the Aspelkana effect the electron absorbs a photon when it gets to the upper surface and emits it when it gets to the lower surface so this is the the story this is the story for a clean case now what should we worry about that can change the picture so there are a few things I'd like to mention in particular two of them because they are interesting first is impurity scattering so this story was all block equations and block bands and with the application of electric and magnetic field what happens if there is scattering so I'd like to distinguish between two types of scattering scattering within the same diracone or wild cone and scattering between different wild cones let me start with the same wild cone because it's more interesting so imagine that before we talked about the chemical potential being here and then if the chemical potential is here there's nowhere to scatter to within the same diracone now let's imagine that the chemical potential is here so now if you move along this chiral mode that has only a direction in the zid only motion in the up direction only motion upwards in the zid direction now you can scatter to higher lander levels which have motion both up and down now it turns out that there's very little effect to this and I'd like to explain that and the analog which I find illuminating to think about is a chiral quantum whole edge coupled to many chiral wires so let's imagine the following situation imagine that we have a new equals one you have a new equals one quantum whole state here and now this is only new equals one and this is there's an edge mode that flows chirally to the right and now there are you coupled this mode to a set of wires here and to a set of wires here these wires are not chiral the mapping is basically the geometry of the cycloton orbit that I explained unfolded into a line this is motion along the upper surface, this is the fermi arc this is what happens when you enter to the bulk you have now lander levels to scatter to this is the motion in the lower surface where you move along the fermi arc and this is the motion in the bulk on the way back again you have lander levels to scatter to you cannot scatter so far until we talk until we get later you cannot scatter from one well cone to the other but you can scatter to this set of lander levels now you see the reason for the relative insensitivity to this type of disorder think about the quantum whole analog send the current from here it will diffuse its way into these wires back and forth but eventually it must get to here there is no way it can get back because this is chiral mode and these are finite wires so all the current that terminates from here must eventually make it to the right which is to say all the current that starts on the upper surface must eventually get to the lower surface and on so what will happen what can these wires do well they can slow things down they can make the motion slower and we'll talk about this in a second and so the distance, the time distance the time separation between the recurring repetitions of the cyclotron motion get longer and longer and these peaks can be boredom because what the electrons will do they will diffuse in between these wires or in between these Landau levels going part of the way down part of the way up diffusing but eventually going down if this is the direction of this state so these peaks will get boredom now when we think about DC we integrate over DC conductivity implies integrating sigma tilde over all times which means bordering or slowing down make no difference what that okay so the DC effect is not affected because we integrate over this as to the AC effect the slowing down changes the period and a cool estimate tells us that there's a slowing down factor of 2n plus 1 where n is the number of Landau levels and that there's a bordering that limits the number of harmonics of resonance that we can observe but it does not destroy the effect altogether there's no exponential there's nothing that depends exponentially on the thickness divided by the mean free path so this is this type of disorder other troubles to worry about second is issues of phase coherence and the like and those are completely irrelevant this is not a quantum effect this is all semi classical transport this is not troubnik of Daas it's basically cyclotone resonance so there is no exponential suppression with temperature and there is no as I said exponential suppression with mean free path third issue to worry about is scattering between different wild nodes this is trouble just like various effects in graphene and so on once you introduce coupling between different diracons the story is over so there is no protection in the sense of the quantum effect the protection is only to the extent that the sample is thinner than the mean free path for inter node scattering and the last thing I'd like to mention in a minute or two is there's something called chiral anomaly should we worry about that thinking about anomalies I notice that normality is in the eye of the beholder so what is normal or anomalous may depend on your view in this particular case this is the equation known as chiral anomaly and it is the fact that the number of electrons in one node minus the number of that electrons flow from one node to another as you put an electric field that's parallel to a magnetic field it looks very mysterious it's completely trivial if you ask me if you just write it in terms of the current in the z direction as a function of the electric field in the z direction the statement of chiral anomaly is the statement that the current in the z direction goes like one over i omega times the electric field in the z direction forget all the other factors what is important is that the current goes like one over i omega so this is the statement that there is ballistic transport along the z direction which is exactly the statement I was talking about before that as the current enters the bulk as long as you don't have scattering between the two diracons it must get all the way to the other surface in this particular case we don't have we send radiation in the z direction so the electric field is only in the xy plane so the electric field is perpendicular to the magnetic field it has no effect on the ac on the azbel-kanat type phenomenon and the effect on the dc I don't have time to explain is not crucial okay the regime we think of is we're talking about micro frequencies of course we don't want interbent transitions nothing optical and so on and it turns out the thickness you want for the slab is a few microns a few microns ten microns something like that so to summarize what I told you is that there are unique cyclotone orbits that go between the two surfaces of a slab sample of וילסמימטל and they give rise to two interesting transport phenomena one is the fact that you push a current between two contacts here and get opposite voltage or opposite current between these two contacts down here and the other is that you get electromagnetic resonant transmission at frequencies that commensuate with the frequency of the cycloneura orbits and the the other thing I told you about which I'd like to emphasize in the summary is that the intranode scattering is limited in the way it can affect this phenomenon thank you very much