Topological insulators: a beautiful revolution

Niccolò Traverso Ziani, Luca Vannucci, Maura Sassetti


1 Introduction

A topological insulator is a material having an insulating bulk, which cannot carry current at low temperature and small applied voltage, opposed to a special metallic surface. Electrons on the surface, due to the exotic (topological) properties of the bulk, can move freely even in a sample with impurities and defects. Such a material could hence realize a long standing dream of the Silicon Valley: as S.-C. Zhang from Stanford University often tells, topological insulators are highways for electrons. To enrich the story, the electrons on the surface have another very special property: their spin is locked to the direction of motion. For example, when the surface is one-dimensional, all electrons moving right have a well-defined spin polarization (let us say, up) while electrons moving left are oppositely polarized (down). This behavior is extremely interesting for spintronics, a research field in which the electron spin is used to store and transport information instead of the charge, for speed up and low-consumption purposes.

Topological insulators, however, are not only interesting as standalone devices. Rather, some of their most thrilling applications (up to now) occur when they are coupled to other materials, such as superconductors. When a conventional superconductor is placed in contact to the topological surface, very peculiar states can be engineered, which behave much like the elusive and sought-after Majorana fermions. Such particles, first discussed in 1937 by E. Majorana while looking for a symmetric solution of the Dirac equation, have the unique property of being their own antiparticles. The issue is interesting per se: while the fate of the physicist himself is still mysterious, his real fermions can now be closely inspected. However, the main source of interest lies in the fact that Majorana fermions could represent the building blocks of the quantum computer, which would allow to efficiently tackle computationally expensive tasks, such as the factorization of big numbers, which is crucial for the encryption schemes commonly used when surfing the Internet.

The path that led to the development of such interesting materials is extremely winding and rich of ideas, which, throughout fifty years, have changed most of the basic paradigms of condensed matter physics. Here, we provide a possible storyline through the concepts that made topological insulators one of the most exciting fields in physics. For the sake of simplicity, we mainly focus on the milestones towards modern topological devices based on free electronic systems. Interesting and related phenomena such as the fractional quantum Hall effect, topological phases in spin and bosonic systems and the role of interactions are hence not discussed. We refer to the exhaustive literature in the field for a more detailed analysis.

2 The early stages: Beyond Landau’s classification and unavoidable boundary states

Symmetry is a powerful concept in physics. The correspondence between symmetries and conservation laws usually provides the quickest and most effective method to tackle any physical problem. Thus, it is somewhat disappointing to learn that Nature often breaks symmetries by its own, or, as people say, spontaneously. Fortunately enough there is still a silver lining, as even broken symmetries are so interesting that states of matter and transitions among them can actually be characterized by inspecting which symmetries are broken and which are not. The examples are countless. When a solid forms, continuous translational symmetry is broken and atoms arrange in a lattice. Similarly, in ferromagnets, a basic symmetry in the spin space is violated, and the average magnetization is not zero even if the system is invariant under rotations in spin space. L. Landau and V. Ginzburg first understood that symmetries can indeed characterize almost all phases and their transitions, in the so called Ginzburg-Landau theory.

However, even the most powerful ideas have exceptions, sometimes of fundamental importance. In the early ‘70s, V. Berezinskii, J. Kosterlitz and D. Thouless (the last two were Nobel prize recipients in 2016 for these discoveries) astonished the community by showing a phase transition, related to topological defects, without any symmetry breaking, demonstrating that symmetries are not the only principle allowing to classify phases of matter. The first brick toward topological insulators, that are not characterized by symmetry breaking, was set. The tag “topological” in this context is however related to something happening in real space, while topological insulators have a topologically nontrivial structure in the reciprocal space. It is thus time to move on with the historical development.

During the same decade as the Berezinskii-Kosterlitz-Thouless theory, R. Jackiw and C. Rebbi discovered something which is of fundamental importance for understanding the surfaces of topological insulators. In a simplified version of their argument, imagine relativistic electrons moving in one dimension along the x axis subjected to a special kind of potential (a mass term m (x)) which is positive for x > 0, and negative for x < 0 as shown in fig. 1.

If the potential is ±M at ±∞ respectively, physical intuition tells us that only states with energy, in absolute value, larger than M should be allowed. The striking result is that this is not the whole story: there is also a single fermionic state (blue curve in fig. 1) located close to x=0! Very importantly, the energy of this state is zero, so in the middle of the gap opened by the mass term m (x). The only requirement for the existence of such a state is that the potential at infinity and the potential at minus infinity must have opposite sign. This zero energy mode is the first example of topological boundary state. Topology is related to the fact that the existence of the zero-energy state only depends on global properties of the potential (its sign at plus and minus infinity), while the “boundary” nature emerges from the special location of this state, which lives around the abrupt change in the mass term.

This is the first signature of a link between the topological order and the boundary properties of a system, a recurrent feature in the field of topological states of matter.

By the end of the ‘70s, the two seeds for the development of topological insulators were hence already on the stage: not all phases are characterized by symmetries and boundary states with zero energy can exist at the interface between two different gapped bulks. However the time was not ripe for the complete appreciation of this physics, since the experimental relevance was not deep enough to drive intense investigation.

3 The quantum Hall effect and the era of topology

The experimental discovery in 1980 of the quantum Hall effect and of its precise conductance quantization gave the decisive impulse to the development of ideas related to topology in condensed matter physics. In a very simplified picture, the physics of the quantum Hall effect is the following: a gas of electrons is forced to move in a twodimensional plane and is immersed in a strong, perpendicular magnetic field. When such a system is prepared, a very peculiar situation takes place: the transverse (Hall) resistance can only take values which are integer fractions of a resistance quantum (see fig. 2). Such values depend on two fundamental constants only, namely the electron charge e and Planck’s constant h:

(1) $R_{H} = \frac{h}{ne^{2}}$ , $n \in \mathbb{N}$.

Jumps among the different plateaus happen abruptly as the magnetic field is varied, but when the resistance is flat, it is really very flat. So flat that it has been used in metrology. Important questions hence emerge: why is the resistance so stable? Which is the nature of the phase transition that takes place when the system jumps from one plateau to another? Furthermore, energy levels for a charged particle in a perpendicular magnetic field (called Landau levels) are highly degenerate and separated by finite steps in energy. Then, who is carrying the current, since the bulk of the system appears to be gapped?

The answers to those questions opened the way to the development of topological insulators. The first two answers have been given by D. Thouless, M. Kohmoto, P. Nightingale and M. den Nijs by means of one of the more elegant and remarkable formulas of condensed matter physics: the one defining the TKNN invariant. In their original work, the authors considered a twodimensional system of electrons in the presence of a periodic crystalline potential and a strong perpendicular magnetic field, and they computed the conductance by using the Kubo formula. The presence of the periodic potential is crucial because it generates a Brillouin zone, which has its precise topological properties. In the case of a square lattice with periodic boundary conditions, the Brillouin zone is, topologically speaking, a torus. What they were able to show is that the Hall conductance GH(n) can be expressed as the integral, over the Brillouin zone, of a quantity that does not depend on the energy levels, but on all the occupied eigenstates. The TKNN result is

See eq. (2) in PDF

where kx and ky are the two components of the crystalline momentum, $|u^{(n)}_{k_{x},k_{y}} \rangle$ are the Bloch states of the n-th band, and the sum is extended over all filled bands. Equation (2) immediately sounds odd: we usually learn that the current, in linear response, is mainly a property of the states close to the Fermi energy, something very different from the result of Thouless and coworkers. Moreover, and most importantly, the value of GH(n) can only be an integer multiple of the conductance quantum e2/h. There is a profound reason for this precise quantization. Indeed, both the integration space (the Brillouin zone) and the integrand function in eq. (2) have well-defined topological properties. The latter (which is related to the so-called Berry phase) is a curvature defined over a closed surface, and its integral must be an integer multiple of 2π, as we know from the branch of mathematics called “topology”. The word “topological” used here is ultimately due to this concept.

The phase transition among different states (i.e. different Hall plateaus) is hence not associated with a symmetry breaking and the behavior of a local order parameter. It is instead related to an integral, hence a global property, changing value as soon as the conductance jumps from one plateau to the next one (see Box 1 for further details).

Still, who carries the current in this gapped system? B. Halperin, inspired by a beautiful Gedanken experiment by Nobel laureate R. Laughlin, demonstrated that the current is carried by metallic states at the edges of the sample. To understand why, consider an intuitive semiclassical picture of electrons constrained in a finite region of the plane by hard-wall boundary conditions. In the presence of a strong perpendicular magnetic field the trajectory is circular, with a radius that decreases with increasing magnetic field as 1/B. For sufficiently high magnetic fields, particles describe very small circular orbits whose radius is much smaller than the linear dimensions of the sample, and no conduction can take place through the bulk. But what happens at the edges? Here, electrons hit the walls before completing a full cycle and, as a result of multiple collisions, they move along the edge describing the chiral skipping orbits shown in fig. 3.

The fact that the current flows at the edge brings several important implications. First, the motion is chiral and electrons on the same edge move along the same direction, dictated by the external magnetic field. They would reverse their direction of motion only by tunneling to the opposite edge, where they can find a backward-propagating channel. But this is impossible (or, at least, exponentially suppressed) since the bulk is insulating! The chiral edge states are thus topologically protected from backscattering and realize a perfect, dissipationless conductor. A vanishing longitudinal resistance is indeed always measured in correspondence of transverse resistance plateaus.

Second, it is now clear that all details about the host sample (shape, presence of impurities, material) are irrelevant. What really matters is the boundary between the electron gas and the vacuum, were current-carrying edge states develop.

Finally, if the Hall conductance in the presence of a single edge state is G0=e2/h, n boundary states generate a conductance GH=nG0, accounting for the step-like behavior RH=1/GH. Jumping from one resistance plateau to the next one corresponds to closing or opening a new channel at the edge (see fig. 2, bottom panel).

With the quantum Hall effect and its topological interpretation, the various aspects started to get together: a plethora of states of matter which are defined by their topology rather than by their symmetries have been discovered and understood. Moreover, when forced to have an interface with something topologically different (most often, the vacuum), those states necessarily show robust metallic edges.

2016 Nobel laureate F. D. M. Haldane gave a seminal contribution along those lines in 1988, when he realized that it was not necessary to have a net magnetic flux in order to observe quantum Hall physics. He considered a two-dimensional system on the honeycomb lattice shown in fig. 4, with unconventional hopping terms between second neighbor sites breaking time-reversal symmetry. He showed that such a system can have a gapped bulk and metallic edge states of topological nature even in the absence of a magnetic field. The new hopping term introduced by Haldane is, in other words, able to open a gap of topological nature. The result is striking for many reasons. There is no net magnetic flux, and the original discrete translational symmetry is thus preserved, but the Hall conductivity can be non-zero. More specifically it can be ±e2/h in the topological sectors of the model. The sign has a very direct physical interpretation: when G=+e2/h electrons circulate along the edge, say, clockwise, while for negative sign they circulate anticlockwise. Along with this first surprising point comes a crucial second point: it is not the magnetic field itself, but the breaking of time-reversal symmetry that makes Hall conductivity possible. Indeed, when time-reversal symmetry is preserved the Hall conductance is zero.

At the end of the ‘80s the theoretical background was hence quite well established, but the lack of experimental evidence of non-interacting topological phases beyond the quantum Hall effect slowed down the field of topological states of matter.

4 Interlude

New impulse came from different fields. In 1990 S. Datta and B. Das demonstrated that spin-orbit coupling could be employed to build a spin transistor without the need of external magnetic fields, a dream for spintronics. Qualitatively speaking, spin-orbit coupling plays the role of an effective magnetic field, whose magnitude depends on the velocity of the electrons. The basic mechanism is the following: crystal structures with particular symmetries offer a non-homogeneous background for the propagation of electrons. When electrons are fast (so preferably in heavy materials) the crystalline electric field $\vec{\nabla} V$ is perceived as a magnetic field in their own reference frame, and hence it couples to the electron spin $\vec{\sigma}$, generating a term proportional to $\vec{\sigma} \cdot (\vec{k} \times \vec{\nabla} V)$ in the Hamiltonian. Still, spin-orbit coupling does not break time-reversal symmetry, as both $\vec{\sigma}$ and the crystalline momentum $\vec{k}$ change sign under time inversion. The scenario opened by spin-orbit coupling is huge. Without breaking time-reversal symmetry (that is, in this contest, without applying external magnetic fields), the electron spin becomes something more than just a multiplication by a factor two. In particular, a lot of the physics of electrons in magnetic fields can be recovered, but in a very special way. The system does not become spin-polarized, it becomes equivalent to two electron gases, with equal number of electrons, experiencing opposite effective magnetic fields due to spin-orbit coupling. The quantum Hall effect, in some form, or the Haldane model could in principle become feasible even without time-reversal symmetry breaking, thanks to spin-orbit coupling.

In 2004 a new star begun to shine in the condensed matter physics sky: graphene. Graphene is an atomically thin layer of carbon atoms, arranged in a honeycomb lattice. In other words, it is a single layer of graphite. Most remarkably, the low-energy properties of electrons in graphene can be described by a Dirac-like equation, just like the one considered in the Jackiw-Rebbi model and in the Haldane model.

One year after the discovery of graphene, C. Kane and E. Mele put graphene, spin-orbit coupling and topology together: the era of the experimental realization of symmetryprotected topological insulators was ready to start.

5 From theory to experiments

The Haldane model represents a proof of concept rather than a proposal for an experiment. It deals with spinless fermions which hop on a lattice in a rather strange way. In 2005 Kane and Mele brought the scenario of the Haldane model closer to material science. They reintroduced spin in the game, and they realized that spin-orbit coupling naturally provides the exotic hopping term thought by Haldane. But in a very particular way. Electrons with spin up are equivalent to a Haldane model with positive Hall conductance, while electrons with spin down have negative Hall conductance. So, graphene with spin-orbit coupling is a material with a bulk gap and two counter-propagating edges. Moreover, electrons propagating clockwise have a spin projection that is opposite to the one describing electrons propagating anticlockwise. This fact has a crucial consequence: an electron, in order to be backscattered, must flip its spin. But, in the absence of interactions, this is forbidden unless time-reversal symmetry gets broken, for example by magnetic fields or by magnetic impurities. The counter-propagating edges cannot mix, and hence cannot get gapped.

A qualitative change had taken place, and it is worth to summarize it. The Haldane model breaks time-reversal symmetry, and is characterized by a topological chiral edge state and a finite Hall conductance. The Kane-Mele model preserves time-reversal symmetry (the Hall conductance is thus zero), and exhibits two gapless counter-propagating edge modes with opposite spin, as shown in fig. 5. These modes, called helical edge states, cannot be gapped out unless time-reversal symmetry is broken: as people often say, they are symmetry-protected topological edge states. The Kane-Mele model hence provides the first example of experimentally relevant symmetry-protected topological insulator.

Unfortunately, spin-orbit coupling in graphene is too small to observe the intriguing phenomena described, but the edge states of the Kane-Mele model immediately appeared very promising to the spintronics community: they can carry spin even when no charge current flows. A more feasible proposal for their implementation was hence needed.

Not so surprisingly, the answer came from one of the most explored fields in condensed matter: the physics of semiconductor heterostructures. In a very influential paper published in Science in 2006, B. A. Bernevig, T. Hughes and S.-C. Zhang (BHZ) theoretically conjectured that a structure constituted by a thin layer of Mercury Telluride (HgTe) grown between two bulk Cadmium Telluride (CdTe) crystals could accommodate one-dimensional helical edge states protected by time-reversal symmetry, exactly as in the case of the Kane-Mele model. Soon after, in Würzburg, the BHZ proposal was experimentally realized by the group lead by L. Molenkamp, bringing symmetry-protected topological insulators to reality. Since the physics of the BHZ model is reasonably easy to appreciate, contains a nice summary of the ideas discussed so far, and gives a hint of the kind of experiments which are performed on two-dimensional topological insulators, we give some details about the BHZ model and experiments that followed. The reader uninterested in the details can skip the next two sections and continue the storyline from sect. 8.

6 BHZ model

The BHZ model is based on a two-dimensional semiconducting quantum well composed of two thick layers of CdTe, separated by a thin layer of HgTe as illustrated in fig. 6. Since the aim is to build a two-dimensional structure, it is important to explicitly state what is two-dimensional here. The band structure of the whole system is of course three-dimensional. However, some states are mainly confined in the (still three-dimensional) HgTe bulk. Those states, which are given by a mixture of the states in all the parts of the heterostructure, have quasi-momenta kx, ky and kz. The reference frame is fixed so that the short length dimension of the HgTe part is along the z direction, and periodic boundary conditions along the x and y axes are set. The lengths in those two directions are large enough to consider kx and ky as continuous variables. The quasi-momentum kz, on the other hand, can only take discrete values, and the energy difference between wave functions with the same kx and ky but different kz becomes large. Close to the Fermi energy, only one particular value of kz is relevant, and hence we are left with an effectively two-dimensional structure. The aim of the BHZ model is then to engineer a heterostructure whose two-dimensional bands are similar to the bands characterizing the Kane-Mele model. The heterostructure that was proposed is called Type III quantum well. The barrier material (CdTe) is a semiconductor with the usual ordering of electronic bands: the so-called $\Gamma_{6}$ band, which has s-wave character, is above the $\Gamma_{8}$ band, which has p-wave character. The well material (HgTe), on the other hand, is a zero-gap semiconductor with inverted band structure. This means that the $\Gamma_{8}$ band is placed above, in energy, the $\Gamma_{6}$ band. In both cases the gap is minimum at the $\Gamma$ point (namely, the center of the Brillouin zone). The effective Hamiltonian describing low-energy physics of the quantum well around the $\Gamma$ point is then built in the ordered basis {$|E,+\rangle$, $|H,+\rangle$, $|E,-\rangle$, $|H,-\rangle$} where $E$ and $H$ are the first sub-bands derived from $\Gamma_{6}$ and $\Gamma_{8}$in the presence of confinement, and the sign ± refers to spin up/down. They are both degenerate because of time-reversal symmetry. The presence of axial, inversion and time-reversal symmetries forces the Hamiltonian to be block-diagonal, with the two blocks related by time-reversal. Moreover, the fact that $E$ and $H$ states have opposite parity (since they originate from s-type and p-type bands, respectively) gives strong constraints on the coupling terms (even or odd in k). These considerations lead to the Hamiltonian matrix

See eq. (8) in PDF

with $\epsilon (\vec{k})$ and $d_{i}(\vec{k})$ given by

(9) $ d_{1}(\vec{k}) + id_{2}(\vec{k}) = A (k_{x} + i k_{y}) $, $ d_{3}(\vec{k}) = M - B ( k_{x}^{2} + k_{y}^{2} ) $, $\epsilon (\vec{k}) = C- D (k_{x}^{2} + k_{y}^{2} ) $.

Here $\sigma_{i}$ are the Pauli matrices and A, B, C, and M are parameters that depend on the materials and on the geometry of the sample. The most important parameter is M. For M>0 the model is equivalent to the Kane-Mele model in its topologically trivial sector (gapped bulk and no edge states), while for M<0 the model is topologically non-trivial and hence exhibits helical edge states circulating around the gapped bulk, as already shown in fig. 5 for the Kane-Mele model. When M=0 the two-dimensional bulk becomes gapless as expected, since we are in the presence of a (topological) quantum phase transition. Physically speaking, M represents the energy difference (with its sign) between the more electron-like band E and the more hole-like band H at the $\Gamma$ point. When the HgTe layer is very thin, the low-energy states of the quantum well are dominated by CdTe, and M is positive. When HgTe is thick enough to dictate the band ordering in the quantum well, M is negative. The critical value of the thickness, dc=6.4 nm, marks the topological quantum phase transition (see fig. 6, right panel).

As already discussed, topological edge states can only emerge in the presence of a boundary between the topological material and the vacuum. Imagine, for instance, that electrons live in the right half of the xy plane (x>0). The resulting dispersion relation is

(10) $\epsilon_{\pm} ( k_{y}) = \pm \hbar v k_{y}$,

showing the characteristic linear spectrum of a massless Dirac equation, with propagation velocity v of the order of 5·105 m/s. Thus, in the inverted regime (M<0) a pair of topological edge states propagate along the boundary, with opposite spin polarization corresponding to opposite direction of motion (see fig. 7).

The key point of the scheme proposed in the BHZ model is that it relies on well-studied semiconductor quantum wells, with strong enough spin-orbit coupling and a sufficient bulk energy gap (of the order of 20 meV). Thus, the experimental realization of a symmetry-protected topological state of matter was finally within reach. The wait was not long, as results from L. Molenkamp’s lab at University of Würzburg were published just one year after the theoretical proposal.

7 Spotting the edge

The BHZ model predicts that the CdTe/HgTe/CdTe heterostructure is a two-dimensional topological insulator when the HgTe layer exceeds the critical thickness dc=6.4 nm. But what are the experimental signatures of such a particular phase of matter? Of course, the existence of time-reversal protected edge states plays a big role in the transport properties. In particular, conductance measurements can clearly distinguish between trivial and topological phases.

It is instructive to first focus on a single pair of helical edge states and compute their conductance with a simple (and apparently naive) calculation. Consider a system of length $L$ connected to two electronic reservoirs. Linear bands of the Dirac-like Hamiltonian are given in eq. (10) and populations of the two bands follow the Fermi distributions of the reservoirs

(11) $f_{R/L} = \frac{1}{e^{[\epsilon_{\pm} (k) - \mu_{R/L}]/(k_{B}T)} + 1}$.

Consider, for instance, a symmetric voltage drop between the two reservoirs, namely $\mu_{R/L}=\mu_{0}\pm eV/2$. Each band provides a contribution to the current given by

(12) $I_{R/L} = \frac{e}{L} \sum_{k} v_{R/L} (k) f_{R/L} (k) = \frac{e}{L} \sum_{k} \frac{1}{\hbar} \frac{\partial \epsilon_{\pm} (k)}{\partial k} f_{R/L} (k)$.

Switching to continuous momenta and transforming the sum into an integral, the total current $I=I_{R}+I_{L}$ reads

(13) $I = \frac{e}{2\pi}\int_{-\infty}^{+\infty} \mathrm{d} kv \left[ \frac{1}{e^{(\hbar vk - \mu_{0} -eV/2)/(k_{B}T)} +1} - \frac{1}{e^{(\hbar vk - \mu_{0} + eV/2)/(k_{B}T)} +1}\right] = \frac{e^{2}}{h} V $.

Thus, the conductance quantum G0=e2/h emerges once again. What makes helical edge states so interesting is that their conductance is protected by the particular spin polarization of the modes. Backscattering events, which would invalidate the present result, are only allowed in the presence of a spin-flipping mechanism, which is however prohibited by time-reversal symmetry. So, this simple calculation turns out to describe extremely well the transport properties of helical edge states for symmetry reasons.

However, experimentalists never measure the conductance of a single helical channel between two reservoirs and always resort to multiterminal setups (as shown in fig. 2 for the case of a quantum Hall measurement). The motivation is two-fold. First, a multiterminal setup is more flexible and allows for different measurements on the same sample. Second, it probes the nonlocal nature of transport for a two-dimensional topological insulator, which is a unique characteristic of these systems.

Figure 8 reports measurements performed in Würzburg using a six-terminal geometry. The plotted quantity is Rij,mn, i.e. the resistance obtained by passing current between terminals i and j and measuring the voltage drop between m and n. All curves show a common behavior of the resistance while sweeping the gate voltage (not to be confused with the source-drain voltage responsible for current injection!), that is, varying the Fermi level. As long as the Fermi level crosses the valence or conduction bands, the system behaves as a conventional conductor, with relatively low resistance. This accounts for the behavior of Rij,mn when the gate voltage is tuned away from V*=0 in the top panel of fig. 8. However, when the Fermi level is in the bulk gap (V*≈0), the current is carried by the one-dimensional edges and the multiterminal structure can be approximated with a resistor network, with each component contributing with R0=h/e2. Values of Rij,mn obtained from this analogy are all consistent with measurements at V*=0. It is worth noting that R0 corresponds to a nicely detectable value, R0=25.8 kΩ. In contrast, a conventional insulator would yield a much higher resistance of the order of 10 MΩ. Results in fig. 8 give a clear indication that we are not dealing with a conventional trivial insulator, and are consistent with a picture in which current flows at the edges, circulating around the whole sample.

There is actually another, much more intuitive way to visualize the edge. Despite flowing in a very particular way, currents in the 2D topological insulator do exactly what every other current does: they generate a magnetic field. If one were able to measure the local magnetic field at the vicinity of the edge, he would access the magnitude and the spatial distribution of the current in a remarkably direct fashion. This is basically what K. Moler and collaborators did in a beautiful experiment performed at Stanford. They made use of a SQUID (an extremely sensitive superconducting device relying on the Josephson effect to measure magnetic fluxes) to scan the top surface of a 2D topological insulator. Their imaging of the current flow across the 6-terminal device (see fig. 9) represents a direct evidence for the existence of edge states in the topological regime.

8 Three-dimensional topological insulators

Two-dimensional topological insulators, by means of their one-dimensional metallic edge states, allow to envision a topology-based technology. A three-dimensional version of the same concepts, however, would be even more appealing. A gapped three-dimensional bulk with Dirac-like gapless electrons at the surface would add topological protection and spin-momentum locking to the already tantalizing properties of graphene.

A first theoretical analysis of three-dimensional topological insulators came from three independent groups in 2006. The topological aspects of such materials are more cumbersome than in the two-dimensional case, but the physics is similar: a peculiar gapped bulk that supports protected gapless surface states at its terminations.

Following a proposal formulated in 2007 by L. Fu and C. Kane, the experimental demonstration of a these ideas was carried out on Bismuth Antimonide (BiSb) alloys. These alloys were already well known for their thermoelectric properties. The group lead by M. Z. Hasan, in Princeton, demonstrated the topological character of the structure by means of angle-resolved photoemission spectroscopy. Despite its importance, there are several down-sides in this topological insulator. First, the structure is not stoichiometric: it is an alloy with random substitutional disorder, so that the bands are only defined within the mean-field approximation. Moreover, impurities tend to generate in-gap states which can mask the protected surface transport. Finally, the number of surface states is odd, but it is at least five, so that no simple model is possible. A new playground was soon found, again by S.-C. Zhang and co-workers. They demonstrated that the stoichiometric crystals Bismuth Selenide (BiSe) and Bismuth Telluride (BiTe) are indeed strong three-dimensional topological insulators with a single surface state. This properties were soon demonstrated, in 2009, by experimental groups in Princeton and Stanford (see fig. 10). Moreover, the bulk gap in BiSe is around 0.3 eV, allowing to envision room temperature topological electronics. Room temperature two-dimensional topological insulators are starting to be available as well, in Bismuthene, as demonstrated by the group lead by R. Claessen in Würzburg. The time has come for full technological application of topological insulators.

9 Latest developments and devices

Research is now moving in different directions. On one side, once the surfaces and edges are created, the electrons have to be manipulated. This means that different kinds of contacts should be connected to topological insulators. Normal contacts are relatively easy to implement: the very same material, with the chemical potential above the bulk gap, is enough for most purposes. This scheme allowed to test the nonlocal conductance in HgTe quantum wells described previously. On the other hand, superconducting and magnetic leads are more attractive but problematic as well. The implementation of magnetic leads, desirable for spintronics, is still cumbersome and mainly restricted to three-dimensional topological insulators. Superconducting leads, on the other hand, have been implemented successfully on both three-dimensional (2011) and two-dimensional (2014) topological insulators. The main interest in inducing superconductivity by proximity effect is that the single-particle Hamiltonian characterizing the d-dimensional boundary states of topological insulators is very special, in the sense that it cannot be realized by a d-dimensional lattice model (according to a theorem due to H. Nielsen and M. Ninomiya) and hence does not exist elsewhere. In particular, the boundary states of topological insulators can be thought to be half of ordinary Dirac-like metals. While a usual quantum wire has four channels (two directions of motion times two spin polarizations), spin-momentum locking imposes that electrons circulating with opposite momentum must have opposite spin. Similarly, the surface of a conventional three-dimensional topological insulator can be thought of as being a quarter of graphene. As a consequence, the pairing mechanism of the Cooper pairs injected on the boundaries of topological insulators are bounded to be particular. Fu and Kane realized that a very special superconducting state can be induced in three-dimensional topological insulators: it supports zero-energy modes at vortexes or at boundaries with regions of the surface that are gapped by ferromagnetic insulators. These zero-energy states are remarkable because they are Majorana fermions, that is, they are quasiparticles that are equal to their antiparticles. Particles of that kind were proposed, and never experimentally detected, by E. Majorana in the context of particle physics and are now available as quasiparticles in condensed matter systems. If the search for zero-energy Majorana fermions is nowadays one of the most active research areas in the whole condensed matter community, it is due to their peculiar statistics. They have, in fact, non-Abelian statistics. This means that if two Majorana fermions are exchanged, the quantum state that is obtained is not just the original state modulo a phase, but it is a genuinely different state. This property can be exploited for quantum computation, with the advantage that the errors that usually harm quantum states, related for example to decoherence, are way more unlikely in this context. As shown in fig. 11, information is in fact encoded in the way Majorana fermions are braided around each other. In usual proposals for quantum computers, on the other hand, the information lies in the quantum state of some object, which is usually a way less robust host. Pictorially speaking, this mechanism is similar to what the Incas used to do in their quipus: instead of using letters on paper, messages were encoded in collections of properly knotted ropes, and were hence way more difficult to damage.

At present, different systems known as spin-orbit coupled quantum wires are the most promising candidates for the creation and the detection of Majorana fermions. Nevertheless topological insulators remain a solid alternative. Moreover, superconducting hybrid structures involving topological insulators can be useful in superconducting spintronics, a discipline aiming to merge the advantages of spintronics and the peculiar properties of the superconducting state.

The search for topological materials has recently extended to more complex realizations which could be more suitable from the mechanical point of view. A considerable effort is devoted to characterize all possible band structures in terms of their topological properties and to link such topological properties to the chemistry of the inter-atomic bonds. This emergent field is called topological quantum chemistry. Most notably, the symmetry that protects the boundary states in topological insulators is, for the cases discussed so far, time-reversal. However, other symmetries, related to those of the lattice, can do the same job. When such a thing happens, the material is called “topological crystalline insulator”. The role of spin-orbit coupling, so crucial in usual topological insulators, is taken, in crystalline topological insulators, by orbital degrees of freedom. The theoretical proposal is due to L. Fu and dates back to 2011. A very remarkable application was found in 2016 in Würzburg in M. Bode’s experimental group. The main idea is the following. The topological character is controlled by crystal symmetries, and surfaces can break such symmetries. Thus, one may wonder if making steps in the surface of a crystalline topological insulator can lead to the emergence of topologically protected one-dimensional edge states, located at the steps in the surface. This behavior was indeed experimentally observed and is shown in fig. 12. These one-dimensional channels are robust with respect to temperature, impurities and magnetic fields, and this hierarchy of topological boundary states can open the path to unpredictable developments.

10 Conclusions

A path which has been lasting for around fifty years enlarged the classification of phases from a picture based on symmetries and excitations that live in the ordinary space-time, to the inclusion of emergent states where, as Hasan and Moore brilliantly wrote, "the notion of a metric disappears entirely, and macroscopic physics is controlled by properties that are insensitive to spacetime distance and described by the branch of mathematics known as topology". The main consequence is that the classification into conductors, semiconductors and insulators breaks down. States, called topological insulators, which have a gapped bulk but extremely stable conducting boundaries emerge. The properties of such boundaries are very exotic: spin and direction of motion are locked, a behavior that is promising for both spintronics and quantum computation. Moreover, the materials in which such peculiar properties are realized are more and simpler than originally expected, so that a huge technological impact of the beautiful ideas behind topological insulators can indeed be expected.

Before concluding, it is worth noticing that the revolutionary concepts that topology brought into physics have far reaching consequences across different fields. Similarities with the physics of topological insulator were recently discovered in mechanical systems, governed by the laws of classical physics. It is now possible to use geometrical and topological tools to design very peculiar materials with unconventional response designed ad hoc, called topological mechanical metamaterials.

The latest gift of topology (but, likely, not the last) has been unveiled in the dynamics of geophysical flows. Equatorial waves that are present in both the tropical atmosphere and oceans have indeed a topological origin and can be described in terms of topological invariants. We expect the powerful ideas that led to the birth of topological insulators to play a major role in many new discoveries in years to come.

Acknowledgements

The authors acknowledge enlightening discussions with B. Trauzettel. N.T.Z. is supported by the DFG (Grants No. SPP1666 and No. SFB1170 “ToCoTronics”) and the Helmholtz Foundation (VITI). L.V. and M.S. acknowledge support from CNR-SPIN through Seed project “Electron quantum optics with quantized energy packets” and from University of Genova.