Yesterday's post about how nobody cares about condensed matter physics produced a surprising number of comments of the form "I was really hoping you would post about topological insulators," which surprised me a bit. Anyway, since people asked for it, I'll give it a shot. The important caveats here are that 1) this isn't my field, and 2) I have not read a great deal of the primary literature on this, so my understanding is not that deep.

We'll do this in Q&A format, as that's been working well for ResearchBlogging posts lately.

**So, what's a "topological insulator," anyway? You make a material in the right shape, and it won't conduct electricity?** The name is a little confusing. It doesn't have anything to do with the shape, and the interesting feature is not the insulating. A topological insulator is a material which will not conduct current through the bulk of the material, but will carry current along the surface. The current that it carries has some special properties, too, that arise from the quantum nature of the material.

**Wait, it won't conduct in the center of the material, but does along the edges? How do you make that happen?** The main ways to do this are illustrated in a figure from this Science article (paywalled, alas):

The left column of this shows a simple picture of what's going on in different types of insulating states, and that's what you want to focus on here. The key idea is to recognize that an insulator is a material in which there are no electrons that aren't bound to specific places inside the material. That's the picture shown at the top (the electrons aren't really tracing out little elliptical orbits around specific atoms, but that gets the basic idea). This happens in specific types of materials, and depends on things like the number of electrons per atom, and how those atoms are arranged in a solid.

What you want to make a topological insulator is a system where the electrons in the center of the material are stuck in place, but the ones on the edges are free to move.

**So, wait, you want to take something that conducts, and make it only conduct on the edges? How do you do that?** The easiest way is by applying a magnetic field, through something called the "Quantum Hall Effect," which is represented in the second picture down in the left-hand column.

The "Quantum Hall Effect," as you might guess, is a quantum version of the ordinary Hall effect, which occurs when you apply a magnetic field to a piece of material with electrons moving inside it. When you put an electron in a magnetic field and start it moving, it feels a force at right angles to the direction of motion and the magnetic field. This causes the electron to move in a circular path, with the radius of the circle depending on the applied magnetic field.

If the moving electrons are inside a piece of material-- a semiconductor, say-- this effect will divert some of the electrons from the straight-through path, and push them down toward one side. Some of those electrons will end up hitting the edge, and collect there. So, if you take a rectangular piece of semiconductor, send a current through it (running north-south, say), and put it in a magnetic field that's perpendicular to the current (vertical), you'll find a voltage developing sideways (east-west) across the material, and that voltage will depend on the magnetic field.

**That's pertty cool.** Yeah. It's useful, too-- that's how we make probes for magnetic fields. If you measure the voltage, and know the properties of the material you're using, that tells you the strength of the field.

**It's not quantum, though. How do you make it quantum?** The normal Hall effect uses a classical picture of electrons as little particles moving through the material in well-defined paths. That's not really what happens, though-- the electrons are waves, and just like the electrons in an atom can only exist in certain orbits due to their wave properties, when you look at the Hall effect in a quantum sense, the electrons can only be moving in circles whose radius satisfies the same sorts of properties as the allowed orbits in an atom.

Most of the time, this doesn't matter, because if you're dealing with ordinary material at room temperature, there are a lot of effects that mess up the quantum behavior-- the atoms in the sample are vibrating, and collide with moving electrons, knocking them off course, and there are impurities that do the same thing.

If you make a high-quality sample, though, and get the whole thing to low temperature, and use a big magnetic field, you can see quantum effects in the Hall effect. Basically, the electrons can only occupy orbits of certain radii, and as you increase the magnetic field, you change the radius of the orbit the electrons want to be in, but the electrons can't move to a new orbit until you hit the next special value of the field. This shows up as a series of flat steps in the conductance of the material as you increase the magnetic field. The conductance will take a certain value, and stay at that value as you increase the field until the next resonance, when it will make a rapid jump to a different value, and so on.

**OK, but what does that have to do with making an insulator?** Well, if you look at what's going on in the middle of the material, it looks a lot like what's going on in the insulator. If you think in terms of electrons moving through the material, each electron is sort of stuck in place making little circular orbits. You don't get much current flowing through the material as a result.

**Right, but what about the topological whatsis?** The quantum Hall effect gives you something insulator-like in the middle of a big chunk of material, but the edges screw things up. Sticking with the individual electron picture, if you look near the edges, the electron orbits get interrupted by the surface, and the electron will tend to get bounced back into the material, reversing its velocity. Once that happens, though, it will start making another circular orbit, until it hits the edge again. the process repeats over and over, and the electron ends up following a path like the series of semicircles at the bottom of the second picture in the left-hand column.

**So, current can flow to the right along the bottom edge?** Exactly. The electrons that are in the middle of the material just revolve in place, but the ones close to the bottom edge find themselves moving to the right. So you have a surface current. The same sort of picture along the top edge gives you a current flowing to the left.

The net effect of this is to create a very funny sort of material, where the bulk of the material doesn't conduct current effectively, but you have these surface states that carry current in one direction or another. More importantly, these surface currents are protected against impurities-- even if you put something on the surface that should interrupt the flow of current, the current will go right by it.

**The what will who now? How does that work?** Well, if you think about the way the surface states work, you can see that there's no way for the current to be turned back. Even if you put an impurity on the surface that an electron should bounce off, reversing its direction, there's no way for the electron to head back where it came from. The magnetic field will continue to bend the trajectories along circular arcs, and that motion has a very definite direction, either clockwise or counter-clockwise. The direction of current is completely determined by the magnetic field.

In the full quantum treatment, of course, you can't think of the electrons as following particular semi-circular paths, but the end result works out to be the same. Electrons on the top surface can *only*move to the left, and electrons on the bottom surface can *only* move to the right, and that means that the current will flow right around any surface impurities.

**Hey, that's pretty cool. So, this is a topological insulator?** Sort of. This quantum Hall state has the properties we're looking for, but it can only exist in the presence of a magnetic field. What we really want is a material that does this sort of thing *without* a magnetic field.

**Yeah, but can't you find some clever type of material that does the same thing?** No, because of fundamental physics. We expect real physical systems to have time-reversal symmetry-- that is, for everything to look the same if the flow of time was reversed. A material where electrons on the top move left and electrons on the bottom move right would violate time reversal-- when you change the direction of time, the top electrons now move right, and the bottom electrons now move left.

(The magnetic field provides an extra element that makes the full situation symmetric under time reversal. That is, if you reverse the direction of time and also reverse the direction of the magnetic field, everything looks the same. The magnetic field reversal makes sense if you think of how you make the field. If the field is due to current flowing in an electromagnet, when you reverse time, you automatically reverse the magnetic field, because the current is now flowing in the opposite direction.)

**So, it's hopeless?** No, because there's another element that we haven't accounted for in this, namely the electron spin. Electrons are not literally spinning balls of charge, but they do have intrinsic angular momentum, called "spin," that gives you an extra factor that you can change.

What you can do, then, is to construct a material where the spin is correlated with the direction of motion, shown in the third row of the figure above. Electrons with their spin up can move to the right along the bottom surface, while electrons with their spin down can move to the left. On the top surface, the directions are reversed-- spin-up electron move left, and spin-down electrons move right.

**How does this help anything?** It gets rid of the time-reversal symmetry problem. When you reverse the direction of time, you also flip the spin-- the electron isn't literally spinning, but behaves as if it were, and thus the spin reverses direction. So electrons that were originally spin-up and moving to the right are now spin-down and moving to the left. But that's exactly the behavior we had for spin-down electrons originally, so everything stays the same.

**Yeah, but doesn't this mean that your currents aren't protected any more? I mean, if you put an impurity on the surface that both reflects the electron and flips its spin, then the current can be diverted after all, right?** You might think that, but if you look closely at the situation, quantum mechanics bails you out.

You can think of the spin as being represented by a little arrow pointing either up or down. In order to flip the spin from spin-up to spin-down, you can either rotate the up arrow clockwise by 180 degrees, or rotate it counter-clockwise by 180 degrees. As there's no way to detect which of those actually happens, quantum mechanics says that they *both* happen-- that is, you need to include a term for each of those processes in the calculation.

But when you go through the math, it turns out that the two processes cancel each other out. That is, the mathematical term for the clockwise rotation is exactly the same size as the term for the counter-clockwise rotation, only with a negative sign. So when you add the two together, they add up to zero, meaning that it's not actually possible for an impurity to both reflect an electron and flip its spin. So, just like in the quantum Hall effect case, the electron flows right past a surface impurity, as if it wasn't there.

**So, quantum mechanics is magic?** In this limited sense, yes. For reasons that I can't really explain, electron spins have the property that when you rotate them by 360 degrees, you get back to where you started, only with a negative sign in front of the wavefunction. That's what leads to the difference between clockwise and counter-clockwise rotations, which leads to the cancellation of these terms, and prevents scattering by impurities.

**So, if you can make a material where spin-up moves left and spin-down moves right on one side, and the reverse on the other side, you can get this effect even without a magnetic field?** Right. It turns out that, with the right choice of material, you can create exactly this sort of state. Mercury telluride (HgTe) turns out to be a material that exhibits the right properties, if you prepare it in the right way.

**How does that work?**. Ummmm..... mumble, garble rhubarb spin-orbit coupling mumble narf. I don't understand it well enough to put it in terms suitable for this blog post. The key is the interaction between the spin angular momentum of the electron and the orbital angular momentum of the electron in the atoms making up the material, which pushes the energy levels around in a way that lets you make a spin-dependent surface state. The spin-orbit effect gets stronger for heavier atoms, which is why mercury is involved, but I don't have a good picture of how it works.

The full explanation involves details of the band structure of the relevant materials, but I doubt you want to hear that.

**That's ok. You kind of lost me back at time-reversal symmetry, to be honest.** I thought that might be the case. Sorry.

Anyway, the take-home message from all of this is that when you include spin in the calculation, you can make a material that has the properties of a topological insulator: it does not allow electrons to flow through the middle of a solid chunk of material, but will allow electrons to flow along the surface of the material. Moreover, the flow of electrons along the surface will be unaffected by defects and impurities on the surface (within reason, anyway-- if you put enough impurities in, you effectively make it a surface of a different material, and then everything changes).

**What's topological about this, again? You haven't said anything about coffee cups and donuts yet, so I don't see where topology comes into it.** It's a little difficult to explain in a picture involving real particles moving around. When you move to the more abstract "band structure" picture, it turns out that the band structure of a topological insulator has features that cannot be changed by allowable distortions of the band structure, in much the same way that you can turn a doughnut into a coffee cup with simple stretching operations, but can't make either one into a sphere. The mathematical structure of these things is identical to some of the math you find in topology, so they're "topological insulators."

**Isn't that an awfully confusing name?** Look, it's not my field. Nobody consulted me.

**OK, so why should people care about these things?** Well, for one thing, the fact that we can make these sorts of materials is a little bit surprising, and deepens our understanding of the physics of materials. Since there's an awful lot of stuff we don't understand about the way materials work (*cough*high-T_c Superconductors*cough*), particularly where surfaces are involvedevery little bit helps.

These states are also amenable to experimental exploration in a lot of ways that don't work so well for bulk materials. You can study these states in amazing detail with scanning probe microscopes, which makes them a great test bed for exploring the implications of materials physics.

Finally, they might have some practical applications. Insensitivity to surface impurities is an awfully nice feature, given how hard it is to make and maintain really clean surfaces. It also might be possible to put these sorts of materials together in ways that would be useful for quantum computation and fun things like that. There's also some connection to things that happen in graphene, which is everybody's favorite magic material of the future (at the moment, anyway), which bumps the interest up a little more.

It's still a young field, so the ultimate applicability of these states is not yet known, but there are a lot of ideas floating around about fun things you could maybe do with them, some of which might be useful.

**So, if I want to learn more about this stuff, what should I do?** The above explanation was put together by looking at this Science article, this Physics article, and this piece from Nature Physics (warning: PDF), as well as listening to this lecture on YouTube. If you don't have the patience to listen to an hour-long physics lecture on YouTube, this Physics Today article co-written by the speaker appears to be a print version of the same explanation, or at least contain the same graphics (I haven't read it closely, as I only spotted it in my RSS feed this morning). There's also this background article from Nature going through the history of the field and why it's exciting.

If you don't have access to some or all of those, I can't help you.

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This is kind of a quantum mechanics 101 question, and not exactly about topological insulators, but it's something I feel I

should knowand never did learn. Why is the electron's intrinsic angular momentumnotappropriately described as the electron being a spinning ball of charge (whose position is uncertain, of course)?If you try to explain the electron's angular momentum (or, rather, the magnetic moment that's associated with it) as being the result of a spinning ball of charge in a classical sense, it can't work. You find that the charge on the surface would need to be moving at some large-ish multiple of the speed of light.

It's also got this strange property that a 360 degree rotation of the spin direction picks up a factor of -1 in the wavefunction, and you need to rotate it by 720 degrees to get back to the initial state. You can't really reproduce that behavior with ordinary rotating objects.

See, now I am distracted from painfully boorish classwork and stuck contemplating the fun things a person could make of this in a good scifi story. Or conversely, how this might translate practicably to building materials (yes, I have an obsession of questionable health for both) and why one might want it to. Given my obsession (of equally questionable health) with combining the aforementioned obsessions, into an obsession with stratosphere piercing archologies (that also function as chemical propulsion free launch platforms), my mind is agog.

Unfortunately, materials just don't enough play in scifi.

If only I could figure out how to apply this - or at least myself, to my world security essay responses, I would be all set.

I just put your rss feed back in a prominent position on my aggregator... I'm not sure why it fell off in the first place. This post made me take notice though. It was interesting and provided a good introduction at my level. What a wonderful nugget of knowledge to discover today, Thanks!

BTW I would usually just lurk but it sounds like you need some encouragement for these kinds of posts. Keep it up :)

Thanks for the really interesting post, I've been hoping to read a basic-introduction to topological insulators for quite a while and this post certainly did the job! The suppression of scattering of surface states certainly suggests interesting electronic and possible quantum computing applications (much like graphene, as you suggest). There's certainly substantial overlap with graphene, with the problem of substrate removed, as the conducting layer is sat upon the same material already with no lattice mismatch and (presumably) no doping.

Thank you! :)

Thanks a lot! (This inspires me to try explaining things I don't really know much about -- I would surely learn much myself, even if I could not find as many interested readers as you do.)

Thanks for explaining it. Now that I know what it is I have to agree that it's quite difficult to make it interesting to general public. The main problem is the lack of impressive applications which could pique interest.

Good post on a subject I generally find to be very, very dry (although that's mainly do to the fact that I think the name was chosen a little too sensationally, as topological insulators really have very little to do with actual topology).

Ok, so the "features" of this band structure... Does "topological" arise in terms of a homotopy between the edge state curves you have in the picture?

Ok, so the "features" of this band structure... Does "topological" arise in terms of a homotopy between the edge state curves you have in the picture?The coffee cup analogy pretty much taps out my knowledge of topology, so it's hard for me to say.

A feature that gets mentioned a lot in these systems is that the edge states for a topological insulator cross the Fermi surface (basically, the curve giving the energy of the most energetic electron inside the material as a function of the momentum components in the different directions of motion) at an odd number of places, while a quantum Hall state or other non-topological insulating state has edge states that cross at an even number of places. With some sort of variable transform, with even number of crossings can be made into zero crossings, making the system an insulator everywhere, while an odd number of crossings can only ever be reduced to one, which leaves you with a conducting edge state no matter what you do.

I think the connection is that these Fermi surface crossings are analogous to holes in topology in that the fact that you can never get rid of an odd number of them allows you to classify the surfaces into two different groups. But I'm way out of my depth, here, and could be misunderstanding the whole point of the crossing business.

...as topological insulators really have very little to do with actual topology).Sure they do, it's just that Z2 invariants are less sexy than the Chern number that's relevant in the quantum Hall effect. But there's an index theorem and everything.

Somewhat dumb question I suspect but: You have current moving only in one direction on one surface (the top, say), and only in the other at the opposite surface (bottom). What, exactly, happens on the two remaining surfaces? Or if the material is a cylinder, how does behaviour evolve as we move from the top side to the bottom?

Is it simply a weakening of the effect until it's zero at right angles? And if so, does that mean it acts as a normal conductor at that point, or as an insulator? My guess is the latter?

Thanks for writing this. Even mentioning the odd-even thing as a difference between the quantum Hall effect and topological insulators was useful. Sometimes waving one's hands is all one needs. I took a probability course from Giancarlo Rota who would now and then start a proof and show the direction it would take; then he would wave his hands and announce "Italian proof".

Thanks for writing this. It's always good to see condensed matter physics get some attention!

Einstein was right about the shortcomings of Quantum Mechanics and so therefore String Theory is also the incorrect approach. As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the

value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity and is located at: The website called Super Relativity. This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.

"I have to agree that it's quite difficult to make it interesting to general public."

I'm a non-science lurker here. To the degree that I understood it, I thought it was REALLY interesting. Yes, the 'time reversal' thing was a good place to say ??Hm??.

I have looked for CMP books for the non-scientist - I don't think they exist. I don't want to BECOME a scientist - I just want to know more about it.

Uncertain Principles is one of my favorite blogs - even if I don't always understand what is going on.

Janne, in the quantum Hall effect and spin Hall effect examples, the sample is a two-dimensional object (like a piece of paper). At the top and bottom *edges*, current is flowing, but not in the middle. In practice, of course, the sample isn't exactly two-dimensional, but it is constructed as close as possible. Good samples are *very* carefully fabricated so that the edge is just a single atomic crystal layer, or as close as possible.

The big advantages of topological insulators over traditional quantum Hall materials are that they work without a magnetic field, but also that they are easier to fabricate (as 3D samples), able to handle more impurities and surface imperfections.

Chad,

I am of course out of my depth here... so I'll ramble and see if something catches....

Whenever I think "topological," I think of two different types of mappings in particular:

1) Homotopy - The continuous deformation of curves/functions into other functions. For example, a real homotopy would be a parametric function F over [0,1] and {some set of real functions} that continuously deforms a real function f into a real function g.

2) Homeomorphism - A bijective, open, continuous map f:S-> T between two topological spaces S and T. (Equivalently, f is continuous and the inverse map of f is continuous.) When a homeomorphism exists between S and T, S and T are called topologically equivalent. Topological equivalences preserve all sorts of topological invariants for obvious reasons.

So, when you say this: "A feature that gets mentioned a lot in these systems is that the edge states for a topological insulator cross the Fermi surface (basically, the curve giving the energy of the most energetic electron inside the material as a function of the momentum components in the different directions of motion) at an odd number of places, while a quantum Hall state or other non-topological insulating state has edge states that cross at an even number of places. With some sort of variable transform, with even number of crossings can be made into zero crossings, making the system an insulator everywhere, while an odd number of crossings can only ever be reduced to one, which leaves you with a conducting edge state no matter what you do."

What immediately comes to mind is a listing of all possible edge states which would likely be types of discrete spaces since we're dealing with quanta. So here's what I'm thinking, and let me know if I'm anywhere near the mark:

For a topological insulator, the possible edge states E1, E2,... admit a topological equivalence between them. As operation "topologically equivalent to" is an equivalence relation, all of the edge states form an equivalence class under this topological equivalence. A physical translation of this would be that the possible states of the material never (or is very unlikely to) "open holes," effectively blocking current across the insulator. Now, this would be the case in the center of the material. For the edge of the material, current could still flow around the edge as a parametrized surface would have an orientation, so the topology of the surface would be nontrivial.

So, this would be different from my first intuitive idea of the importance of some homotopy between the edge state curves. How close/far am I off of the mark?

Have you ever blogged about those qualities of sub-atomic particles some of which you mentioned in this two-parter: spin, charge, color? If so could you post a reference? If not, would you consider blogging something about that to explain it to us poor plebians?

I tried to read Zhang's Physics Today article on topological insulators. But I got stuck where he said "for the QSH state to be robust, the edge states must consist of an odd number of forward movers and an odd number of backward movers".

I don't get this AT ALL.

He says that if you have a spin-up channel moving in one direction and another spin-up channel moving in the opposite direction, then you can get scattering on impurities.

But I don't know where the part about the odd number of channels comes in.

He says this is the basic reason why it's "topological", because what you care about is the number of channels in a given direction, mod 2.