Natalie Wolchover wrote an interesting Quanta Magazine article
This seemingly boring crystal has been known to behave as a topological insulator at low temperatures. The crystal structure is simple: create a cubic lattice out of samarium atoms. And in each cube (which may be associated with one samarium atom at the left lower front corner), place an octahedron with six boron atoms at the vertices.
Experiments in the USSR and the Bell Labs since the 1960s have shown that this crystal behaves in unusual ways that makes it a "hybrid" of a conductor and an insulator. At low temperature, it was believed to behave as a topological insulator. The experimental discovery of "topological insulators" is usually quoted to have taken place in 2007. It immediately became a hot subject – not only it's mathematically cool, in the way that high-energy physicists may understand very well, but also because topological insulators are one of the promising paths to construct quantum computers.
What does a "topological insulator" mean? It's a particular hybrid of metals and insulators. How can you hybridize these two opposite types of materials? Well, be patient.
Metals
A metal has some "free electrons" that are being shared by the whole crystal. These electrons may exist in many states. They choose to minimize the energy – at least at low temperatures. So these electrons sit in the states with momenta \(|\vec p|\lt p_{\rm fermi}\), if I oversimplify the shape of the Fermi surface and imagine it's a sphere and the occupied electron states exist in a ball.
When you attach an electric current, it becomes energetically favored for the electrons to sit around a nonzero value of the momentum. So they basically fill the ball \(|\vec p-K\cdot \vec J|\lt p_{\rm fermi}\) – which is just the previous ball displaced by a vector that increases with the current. \(K\) is some constant. It's possible to continuously move the ball of "occupied electron states" because there was no gap around the original ball.
This ability to continuously change the set of electron states that are occupied is what gives the metal its "flexibility" and the ability to conduct electricity. The average velocity of the electrons becomes nonzero once you attach the voltage – and that's another way of saying that there is an electric current going through the material.
Insulators
On the other hand, the insulators don't allow that because they have a gap. The spectrum of the states of the "shared electrons" in the whole material looks like a ball – perhaps with some concentric shells – but there are no available states at \(|\vec p| = p_{\rm fermi} +\varepsilon\), if you get my point. So even when you attach an electric current, the occupied electron states stay the same. Their average velocity doesn't change which is why the material doesn't conduct electricity.
Semiconductors would deserve a special discussion here. The gap may be small and the electrons may tunnel through it and other things may happen. The conductivity may be small but nonzero. But here we're not interested in semiconductors at all.
Topological insulators
A topological insulator is a different kind of a compromise between metals and insulators. This compromise preserves the properties of both metals and insulators in their full glory.
It is a material that behaves like an insulator in the 3-dimensional bulk – the occupied electron states are rigid and can't be moved. However, every material in the real world has a finite volume and has to have a surface. One has to solve some special equations for the surface phenomena. In topological insulators, he finds out that there exist additional electron states linked to the surface and those can be shifted. The topological insulator can therefore conduct electricity through the electrons near the surface.
Why is this possibility called "topological"? Topology is a property of a shape or object that doesn't change when you continuously deform the object in any way – and we especially mean the deformations of the metric tensor that defines the shape of the manifold. The doughnuts have a different topology than the spherical M&M's. You surely know what topology is, otherwise you wouldn't be able to open this blog in your Internet browser. (The non-readers of this blog will find the logic of the previous sentence too difficult so let me translate that sentence to their simpler language: you are idiots!)
But you may have even heard of "topological field theories". Those are theories whose defining action,\[
S = \int d^D x\,{\mathcal L} [\phi_i(x^\mu)]
\] is a topological invariant. So if you change the fields \(\phi_i(x^\mu)\) in the interior (bulk) of the material in any way, the action won't change at all. If the spacetime manifold is closed, the action is invariant under all continuous deformations of everything. Consequently, it may only depend on the "non-continuous", abrupt changes – like the change of an M&M to a doughnut. In other words, the action may still depend on the topological classes.
Well, what I wrote in the previous paragraph isn't quite right. A more accurate definition of topological field theories is that the action doesn't depend on the metric tensor at all – it is therefore invariant under the changes of the metric tensor. For example, the Chern-Simons action \((k/4\pi)\int [A\wedge dA+(2/3) A\wedge A \wedge A] \) only uses the product of three "lower indices" from \(A_\mu\) and \(\partial_\mu\) which are contracted against the three "upper indices" from the integration measure – and the metric tensor isn't needed at any point. (The Christoffel symbol terms, if you would include them, cancel due to the antisymmetrization.)
If the action has this property, what are the implications for the waves and other phenomena that take place in the material? Well, the consequences are striking. Waves don't know where they should be moving because all deformations of the spacetime – including time-dependent ones – are equally good. Because of this invariance under the deformations of the background manifold, all correlation functions must be independent of the locations (of their nonsingular continuous changes). No waves moving through the bulk can really be physical.
In other words, topological field theories are those that don't allow any waves to exist in the "bulk" at all.
You may see that it's the same feature that we attributed to the topological insulators, too. Because the conduction electrons might produce "waves" if they existed and if you changed the external voltage, topological insulators are examples of a material whose interior is governed by laws that may be classified as "topological field theory"!
You may check e.g. this 2013 paper to be sure that for a few years, \({\rm SmB}_6\) was considered to be a "candidate topological Kondo insulator". It was a candidate because the condensed matter physicists weren't quite sure whether the classification was right.
They were named "Kondo" insulators after Aeppli and Fisk ;-) who found two examples of such materials in 1992. They just chose that name because they were modest enough. The word is after Jun Kondo – a theorist who was the first one to perform a third-order perturbative calculation in order to successfully describe the Kondo effect. For the experimenters, the Kondo effect is a characteristic dependence of electrical resistivity on temperature. Theorists explain this by some scattering of conduction electrons that was made much more quantitative by Kondo. This scattering is another face of the correlations between the electrons. Quite generally, "Kondo things" are things that qualitatively depend on strong correlations between the electrons.
Fine. This is almost certainly important for \({\rm SmB}_6\) so this crystal was considered a probable "topological Kondo insulator". It conducts on the surface but not in the bulk and the gap in the bulk probably depends on some strong interactions between the shared electrons in the bulk.
Sebastian and 15 co-authors have accidentally observed some quantum oscillations. Their dependence on the orientation may be exploited to reconstruct the shape of the Fermi surface – the boundary of the electron states in the momentum space that are occupied. They didn't expect any oscillations but there were oscillations. They looked like those you expected in the best metals but a seemingly universal formula for the metals – the Lifshitz-Kosevich formula – seems to be brutally violated.
From these comments, one quickly gets to the cutting edge at which many questions seem open. Why do the electrons conspire in this way and produce a behavior that is stranger than the already pretty strange behavior of a topological Kondo insulators?
String theory has something to say
String theory's AdS/CFT correspondence – whose application to these "highly applied" disciplines of science is sometimes referred to AdS/CMT and similar acronyms – gives us a possible way to think about the materials. Already in 2009, Hartnoll and Hofman studied similar materials using the AdS/CFT methods – which always means that a piece of this material is studied as if it were a black hole in the 5-dimensional anti de Sitter space (whose boundary at infinity is identified with the 4D spacetime we know). And they actually predicted a violation of the Lifshitz-Kosevich formula.
Many predictions seem be perfectly OK with these stringy tools but the AdS/CMT physicists still don't seem to understand why the material behaves as an insulator, at the end. The electrons in the bulk are apparently capable of moving by macroscopic distances and they're as good at it as the electrons in metals. However, at the end, something tells them that they have to stop and reverse their motion in time Рthey are just not allowed to exploit their freedom of motion and establish a logistics or electric utility business. Some people are talking about Schr̦dinger-cat-like superpositions of insulators and metal states and similar things; although all superpositions are allowed, of course, it doesn't seem to be a viable candidate for the right final answer to me.
I would bet that these questions will get at least qualitatively resolved within a few years.
Paradoxical Crystal Baffles Physicistsabout a research direction in condensed matter physics that is important for many reasons – and the apparent links to AdS/CFT are among them. Suchitra Sebastian, a female Indian physicist, and 15 co-authors (Cambridge UK, Florida, New Mexico) just published some experimental findings in Science. She claims that the crystal of samarium hexaboride \({\rm SmB}_6\) – named after the six boring Greek Samaritans who discovered it ;-) – behaves in even stranger ways than previously believed.
This seemingly boring crystal has been known to behave as a topological insulator at low temperatures. The crystal structure is simple: create a cubic lattice out of samarium atoms. And in each cube (which may be associated with one samarium atom at the left lower front corner), place an octahedron with six boron atoms at the vertices.
Experiments in the USSR and the Bell Labs since the 1960s have shown that this crystal behaves in unusual ways that makes it a "hybrid" of a conductor and an insulator. At low temperature, it was believed to behave as a topological insulator. The experimental discovery of "topological insulators" is usually quoted to have taken place in 2007. It immediately became a hot subject – not only it's mathematically cool, in the way that high-energy physicists may understand very well, but also because topological insulators are one of the promising paths to construct quantum computers.
What does a "topological insulator" mean? It's a particular hybrid of metals and insulators. How can you hybridize these two opposite types of materials? Well, be patient.
Metals
A metal has some "free electrons" that are being shared by the whole crystal. These electrons may exist in many states. They choose to minimize the energy – at least at low temperatures. So these electrons sit in the states with momenta \(|\vec p|\lt p_{\rm fermi}\), if I oversimplify the shape of the Fermi surface and imagine it's a sphere and the occupied electron states exist in a ball.
When you attach an electric current, it becomes energetically favored for the electrons to sit around a nonzero value of the momentum. So they basically fill the ball \(|\vec p-K\cdot \vec J|\lt p_{\rm fermi}\) – which is just the previous ball displaced by a vector that increases with the current. \(K\) is some constant. It's possible to continuously move the ball of "occupied electron states" because there was no gap around the original ball.
This ability to continuously change the set of electron states that are occupied is what gives the metal its "flexibility" and the ability to conduct electricity. The average velocity of the electrons becomes nonzero once you attach the voltage – and that's another way of saying that there is an electric current going through the material.
Insulators
On the other hand, the insulators don't allow that because they have a gap. The spectrum of the states of the "shared electrons" in the whole material looks like a ball – perhaps with some concentric shells – but there are no available states at \(|\vec p| = p_{\rm fermi} +\varepsilon\), if you get my point. So even when you attach an electric current, the occupied electron states stay the same. Their average velocity doesn't change which is why the material doesn't conduct electricity.
Semiconductors would deserve a special discussion here. The gap may be small and the electrons may tunnel through it and other things may happen. The conductivity may be small but nonzero. But here we're not interested in semiconductors at all.
Topological insulators
A topological insulator is a different kind of a compromise between metals and insulators. This compromise preserves the properties of both metals and insulators in their full glory.
It is a material that behaves like an insulator in the 3-dimensional bulk – the occupied electron states are rigid and can't be moved. However, every material in the real world has a finite volume and has to have a surface. One has to solve some special equations for the surface phenomena. In topological insulators, he finds out that there exist additional electron states linked to the surface and those can be shifted. The topological insulator can therefore conduct electricity through the electrons near the surface.
Why is this possibility called "topological"? Topology is a property of a shape or object that doesn't change when you continuously deform the object in any way – and we especially mean the deformations of the metric tensor that defines the shape of the manifold. The doughnuts have a different topology than the spherical M&M's. You surely know what topology is, otherwise you wouldn't be able to open this blog in your Internet browser. (The non-readers of this blog will find the logic of the previous sentence too difficult so let me translate that sentence to their simpler language: you are idiots!)
But you may have even heard of "topological field theories". Those are theories whose defining action,\[
S = \int d^D x\,{\mathcal L} [\phi_i(x^\mu)]
\] is a topological invariant. So if you change the fields \(\phi_i(x^\mu)\) in the interior (bulk) of the material in any way, the action won't change at all. If the spacetime manifold is closed, the action is invariant under all continuous deformations of everything. Consequently, it may only depend on the "non-continuous", abrupt changes – like the change of an M&M to a doughnut. In other words, the action may still depend on the topological classes.
Well, what I wrote in the previous paragraph isn't quite right. A more accurate definition of topological field theories is that the action doesn't depend on the metric tensor at all – it is therefore invariant under the changes of the metric tensor. For example, the Chern-Simons action \((k/4\pi)\int [A\wedge dA+(2/3) A\wedge A \wedge A] \) only uses the product of three "lower indices" from \(A_\mu\) and \(\partial_\mu\) which are contracted against the three "upper indices" from the integration measure – and the metric tensor isn't needed at any point. (The Christoffel symbol terms, if you would include them, cancel due to the antisymmetrization.)
If the action has this property, what are the implications for the waves and other phenomena that take place in the material? Well, the consequences are striking. Waves don't know where they should be moving because all deformations of the spacetime – including time-dependent ones – are equally good. Because of this invariance under the deformations of the background manifold, all correlation functions must be independent of the locations (of their nonsingular continuous changes). No waves moving through the bulk can really be physical.
In other words, topological field theories are those that don't allow any waves to exist in the "bulk" at all.
You may see that it's the same feature that we attributed to the topological insulators, too. Because the conduction electrons might produce "waves" if they existed and if you changed the external voltage, topological insulators are examples of a material whose interior is governed by laws that may be classified as "topological field theory"!
You may check e.g. this 2013 paper to be sure that for a few years, \({\rm SmB}_6\) was considered to be a "candidate topological Kondo insulator". It was a candidate because the condensed matter physicists weren't quite sure whether the classification was right.
They were named "Kondo" insulators after Aeppli and Fisk ;-) who found two examples of such materials in 1992. They just chose that name because they were modest enough. The word is after Jun Kondo – a theorist who was the first one to perform a third-order perturbative calculation in order to successfully describe the Kondo effect. For the experimenters, the Kondo effect is a characteristic dependence of electrical resistivity on temperature. Theorists explain this by some scattering of conduction electrons that was made much more quantitative by Kondo. This scattering is another face of the correlations between the electrons. Quite generally, "Kondo things" are things that qualitatively depend on strong correlations between the electrons.
Fine. This is almost certainly important for \({\rm SmB}_6\) so this crystal was considered a probable "topological Kondo insulator". It conducts on the surface but not in the bulk and the gap in the bulk probably depends on some strong interactions between the shared electrons in the bulk.
Sebastian and 15 co-authors have accidentally observed some quantum oscillations. Their dependence on the orientation may be exploited to reconstruct the shape of the Fermi surface – the boundary of the electron states in the momentum space that are occupied. They didn't expect any oscillations but there were oscillations. They looked like those you expected in the best metals but a seemingly universal formula for the metals – the Lifshitz-Kosevich formula – seems to be brutally violated.
From these comments, one quickly gets to the cutting edge at which many questions seem open. Why do the electrons conspire in this way and produce a behavior that is stranger than the already pretty strange behavior of a topological Kondo insulators?
String theory has something to say
String theory's AdS/CFT correspondence – whose application to these "highly applied" disciplines of science is sometimes referred to AdS/CMT and similar acronyms – gives us a possible way to think about the materials. Already in 2009, Hartnoll and Hofman studied similar materials using the AdS/CFT methods – which always means that a piece of this material is studied as if it were a black hole in the 5-dimensional anti de Sitter space (whose boundary at infinity is identified with the 4D spacetime we know). And they actually predicted a violation of the Lifshitz-Kosevich formula.
Many predictions seem be perfectly OK with these stringy tools but the AdS/CMT physicists still don't seem to understand why the material behaves as an insulator, at the end. The electrons in the bulk are apparently capable of moving by macroscopic distances and they're as good at it as the electrons in metals. However, at the end, something tells them that they have to stop and reverse their motion in time Рthey are just not allowed to exploit their freedom of motion and establish a logistics or electric utility business. Some people are talking about Schr̦dinger-cat-like superpositions of insulators and metal states and similar things; although all superpositions are allowed, of course, it doesn't seem to be a viable candidate for the right final answer to me.
I would bet that these questions will get at least qualitatively resolved within a few years.
\({\rm SmB}_6\) seems more stringy than a plain topological insulator
![\({\rm SmB}_6\) seems more stringy than a plain topological insulator]()
Reviewed by DAL
on
July 04, 2015
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