__Some materials are insulators on the inside, but conductors at the extremities. These surprising properties are due to the topological characteristics of the quantum states of electrons. Initially theoretical, experimental work on these materials is growing rapidly.__

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__Half a century ago, the Nobel Prize for physics winner George Gamow observed that, along with the theory of number, topology was the only field of mathematics that had no application in physics. Today, it has to be said that the considerable growth in new phases of matter, such as topological insulators, leads us instead to wonder which branch of physics will escape the influence of topology.__

__The discovery of topological properties in physics in the 1970s came as something of a surprise. Topology is a branch of mathematics which studies continuous deformations of an object into another object - without the use of glue or scissors. These objects can be of very different natures, such as a space, a function or a surface. For example, the surface of a spoon can be continuously deformed into that of a banana. Similarly, a mug, with its handle, can be continuously deformed into a doughnut. However, the surface of a banana cannot be continuously deformed into that of a doughnut, as the banana would have to be pierced, or its two ends glued together.__

__Through the filter of topology, the spoon and the banana are identical, but both are different from the mug and the doughnut, which also different from spectacle frames. Here it is understood that only the number of holes on the surface counts; zero in the first case, one in the second and two for the glasses. The number of holes is an example of topological number: this is a whole number that can include all the equivalent surfaces from the point of view of their overall shape. With these concepts, physicists have discovered new states of matter that do not fall into the classification between electrical insulators and conductors.__

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__FROM THE ATOM TO THE CRYSTAL__

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__The ability of a material to allow an electrical current to pass through it is governed by the behaviour of the atoms within it. These electrons, originating from the atoms comprising the material, each carry an elementary electrical charge. While some cause atoms to stick together, by forming chemical bonds, others, in excess, are free to circulate. It is the movement of these free electrons that generates an electrical current. This movement however is a far cry from a flow of little electrically-charged marbles. At nanometric scales (10-9 m), the behaviour of elementary particles is dictated by the laws of quantum mechanics, which allow electrons to interfere, as do waves. According to quantum mechanics, the electrons in an atom can only occupy specific energy states. This is known as an atom’s discrete energy spectrum. This feature is quite similar to the possible modes of vibration of an acoustic wave in a flute, which correspond to the various notes of the instrument; while these notes can be played, the sounds at frequencies between these notes cannot.__

__Band theory, developed in the 1930s under the drive of Felix Bloch in Switzerland and Allan Wilson in England, extends the understanding of the operation of atoms to that of a crystal (*), which can contain a disproportionately large number of them. By combining to each other, atoms’ discrete energy bands become continuous, and are called energy bands. These bands are usually separated from each other by gaps of energy, “forbidden” bands, reminiscent of the discrete spectrum of each atom. As a result, the electronic waves that exist in a crystal, called Bloch waves, also have energies that are inaccessible. All the electrons in the crystal occupy electronic states by minimising their energy. Electrons however, belong to the family of quantum particles (fermions) that forbids them from sharing the same quantum state, and in particular to occupy the lowest energy state at the same time. By doing so, they must spread out and fill the bands; first the lowest energy band, then the subsequent bands and so on until each electron has found a place. The highest energy level reached (Fermi energy), separates the last level occupied by the electrons from the first unoccupied level. Either the last occupied band is partially full, and the Fermi level is located in this band. Or all the bands are filled and the Fermi level lies in a gap.__

__The difference between these two scenarios is radical. In the first case, the electrons whose energy is close to the Fermi level can move if they obtain a little energy, for example by applying an electrical voltage to the material. In the second case, the electrons are blocked, and any energy provided by an external voltage is not sufficient to fill the gap and send them into the upper energy band. This is the difference between a metal which is a good conductor, in the first case, and an insulator, in the second.__

__Waveguides__

__Changing the amplitude of the gap or the position of the Fermi level is a game physicists have been playing for decades, with controlling electronic transport as the prize, particularly with semi-conductors that can change an insulator into a conductor and vice-versa. The success of band theory is responsible for the transistor and for photovoltaic cells. Recently discovered materials however, topological insulators, do not follow this simple (!!!) scheme. While they behave as excellent insulators inside, they are excellent conductors at their edges! This means that electronic states, located at the edges of the material, are available in the energy gap, where the Fermi level lies. Thus, the edge states are the only ones that can contribute to carrying the electrical charge. They act like real electronic wave guides. A topological insulator is not really an insulator; the charge is carried, a little like an electronic invisibility cape, that the current cannot penetrate into the heart of the material and has to avoid.__

__Moreover, any electrons passing through these edge wave guides do so all in the same direction and cannot turn around. They are thus insensitive to various disturbances they may encounter on their path - this is known as topological protection.__

__As a result, they provide electrical conductivity without dissipation, i.e. with no Joule-effect energy loss (**). In some classes of topological insulators, this feature is so evident that electrical conductivity (the opposite of resistance) can only have values that are multiples of e 2/h, where h is Planck’s constant used in quantum physics, and e is the elementary electrical charge. This is called the conductance quantum.__

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__CONBING A HAIRY BALL__

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__This phase can be seen in the quantum Hall effect, discovered in Grenoble in 1980, and which won the Nobel Prize for physics for the German Klaus von Klitzing. The effect is obtained in gases of two-dimensional electrons imprisoned in semi-conducting heterostructures (***), such as gallium arsenide, subject to a strong perpendicular magnetic field of several Teslas (or about ten thousand times more intense than the Earth’s magnetic field). The conductance takes on quantized values that are so precise that they can be used for calibration in metrology. The result is all the more spectacular as the sample, it should be remembered, usually contains several defects of varying types. The idea therefore is to deduce universal and fundamental properties from the “dirty” material.__

__The conductance quantum is similar to the energy spectrum of atoms in discrete levels, from which quantum physics takes its name. Their origin is different however, as shown by David Thouless and his colleagues Mahito Kohmoto, Peter Nightingale and Marcel den Nijs, by revealing that the Hall conductance can be expressed as a topological number, which as such, can only have whole number values.__

__We should first insist on the fact that this topological number does not actually represent holes in the material itself, unlike the example of the mug mentioned at the start of the article! While what the number represents is more difficult to imagine, the so-called hairy ball theorem can guide us in the right direction. Imagine a ball covered in hairs over its whole surface. If you try to comb the hairs flat, without them standing on end, you will realise that there is always one point around which the hairs cannot be aligned, as can be seen on the crown of the head of people with short hair. This point is a hairdressing fault which cannot be eliminated, although its position is not fixed. As long as the surface being combed is a ball, a spoon, a banana or any other surface without a hole, there must necessarily be this fault.__

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__IINDIFFERENT TO DETAILS__

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__If a mug or a doughnut were combed, no such fault would exist. This fault can be qualified as topological as its existence only depends on the number of surface holes being combed. In David Thouless’ calculation, the role of the hair is played by a quantum state, and the combing corresponds to all the electronic states of a filled energy band. The question is whether this hair can be combed without a fault. David Thouless’ work showed precisely that this is not the case for the quantum Hall effect, unlike all the other insulators previously known.__

__This result provides many lessons. Firstly, the extreme precision of Hall conductance is easier to envisage: as it is topological, it cannot continuously change value. Of course, various samples may vary between one another, through the impurities they contain. For the electrons in the bands, these differences resemble various “hairdos” for which the number of faults (controlling the value of the electrical conductivity) remains unchanged. Topological numbers in physics convey the crucial idea of indifference to details, and thus the robustness of the phenomenon. Then, David Thouless’ results revealed that there is important, measurable information hidden in the way the quantum states in a band are arranged with respect to each other. This is exactly the point that had escaped physicists for half a century and on which David Thouless put his finger, justifying his winning of the Nobel Prize in 2016. He then understood that the existence of this topological property goes hand in hand with the existence of wave guides at the edges of the system, providing them with robustness qualified as topological.__

__The topological nature of the quantum Hall effect was a curiosity for twenty-five years, until Charles Kane and Eugene Mele from the American University of Pennsylvania, understood that other topological phases could exist in matter. In the new phase they proposed in 2005, the edge states have a whole other aspect. They can only exist in pairs. Each partner in the pair moves in the opposite direction and carries with it an intrinsic magnetic moment (spin) also opposite. This is a very consequential point, as it means not only that the flow of electrons can be manipulated, but also their spin, while guiding them in a controlled manner. This is the challenge of spintronics, which aims at benefitting from this additional information accessible in the quantum world for daily electronic applications.__

__Once the breach was opened by Charles Kane and Eugene Mele, everything accelerated. One year after their pioneering work, the experimental confirmation of the existence of this new topological phase was announced by Laurens Molenkamp and his group from Würzburg university in Germany, in collaboration with the theoretical team of Shoucheng Zhang in Stanford. This confirmation was obtained in heterostructures based on mercury, cadmium and tellurium. Soon thereafter, the first three-dimensional topological phase was predicted and observed in a material based on bismuth and antimony. This time, the electrons circulating on the surface behave as if they had no mass, like ultra-relativistic particles (the electrons in a graphene sheet, a carbon material one atom thick also discovered in 2005, move in the same way). In just a few years, a host of topological insulators was discovered in various alloys often containing heavy elements such as antimony, tellurium, mercury and bismuth.__

__In parallel, it quickly appeared that many other topological phases could be obtained, for example in some types of exotic superconductors, or by combining superconductors with topological insulators. The particles confined to the edges of such devices are said to have the singular property of being... their own antiparticle! The delicate observation and manipulation of these exotic, weightless particles with no magnetic charge, known as Majorana particles, is a fast-growing field. Fundamentally first, as such a particle has never been observed as a fundamental particle in nature. But also due to the unique applications that could be offered by the storage of information and quantum calculations.__

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__BEYOND ELECTRONS__

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__The fundamental concepts of topological robustness and edge waveguides quickly moved beyond electron physics where they saw the light and found a surprising echo in these new fields. Be they optical, acoustic, atomic or mechanical, topological states similar to those mentioned above, are already a reality in laboratories (see opposite). These concepts have even been used in geophysics; some ocean and atmospheric waves at the equator, one of which is a precursor to the El Niño phenomenon, have been formally interpreted as topological edge states stuck between the two hemispheres. The upheaval brought by the discovery of topological phases to the world of physics is not yet over.__

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__(*) A crystal is a solid in which the atoms are arranged in an ordered, regular, symmetrical way. For example, metals are crystals.__

__(**) The Joule effect is the dissipation of the energy in an electrical current as heat when passing through a material that puts up an electrical resistance.__

__(***) A heterostructure is an assembly of several different semi-conducting materials.__

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__UNIQUE POSSIBILITIES IN OPTICS__

__Topology is an ancient notion in optics - optical vortices have been generated using light rays for twenty years. More recently, the concepts presented in this article have become a source of inspiration. Scientists are attempting to guide the propagation of visible light or microwaves by finely structuring materials called photonic crystals. They are now trying to create perfect waveguides for light inspired by the way topological insulators carry electricity without losing their edges. Optics also offers unique possibilities, such as controlling the loss or gain of the material (i.e. increasing or reducing the number of photons locally), allowing lasers with topological properties to be envisaged. This research remains very fundamental, with no industrial applications in the short term.__

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__> AUTHOR__

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__Pierre Delplace__

__physicist__

__Pierre Delplace is a physicist at the ENS physics laboratory in Lyon. Pierre Delplace is a theoretical physicist, working on the topological properties of electrons in solids and their analogues in superficial systems.__