# How can electrons be “topological”?

*The following text is an excerpt from a draft of an article that I co-wrote for the magazine Physics Today, together with Prof. Art Ramirez at UC Santa Cruz. The article will appear (edited, and with more professional figures) in the magazine in September. The article attempts to give some intuition about the concept of “topological” electron bands, which have become very important in modern condensed matter physics.*

## Dawn of the Topological Age?

Historians often label epochs of human history according to their material technologies. For example, we have the bronze age, the iron age and, more recently, the silicon age. From the physicist’s perspective, the silicon age began with the interplay of theory, experiment, and device prototyping of a new type of material, the semiconductor. Semiconductors had been known since the late 1800s as materials with unusual sensitivity to light, to the method of synthesis, and to the direction of current flow. It was not until the early 1930s, however, that a theoretical understanding of semiconductors approached its modern form [1]. The prevailing view at the time saw metals and insulators as opposite limits of electron itineracy, adiabatically controlled by the probability for an electron to hop between atoms. But the recently-developed idea of accessible electron energies (bands) and inaccessible energies (band gaps) provided a natural category for semiconductors – they were more like insulators, but with smaller band gaps, often controlled by impurities. The next fifteen years witnessed breakthroughs in the purification and control of dopants in the elemental semiconductors silicon and germanium that eventually enabled the discovery of transistor action at Bell Labs in 1947. A surprise came during the transistor patent preparation, however. The basic idea underlying the field effect transistor had already been patented in 1930 by Julius Lillienfeld, an Austrian-Hungarian physicist who had emigrated the United States in 1921.

For semiconductors, the path from materials discovery to device implementation was neither linear nor easily predicted. For a relatively new class of materials, called “topological” materials, one can notice compelling analogies with the development of semiconductors, suggesting the tantalizing possibility that we are at the dawn of the “topological age”. In this article, we will describe what it means for materials to be “topological,” and why topology raises the prospect for revolutionary new devices.

## The “shape” of an electron band

The notion of a topological invariant comes from a branch of mathematics called topology, which studies properties of geometric objects that are conserved under continuous deformations. The most famous such property is the *genus*,* g*, which is an integer that counts the number of holes in a three-dimensional (3D) shape (i.e., *g *= 0 for a sphere or a plate, *g = *1 for a donut or a coffee mug, *g = *3 for a pretzel, etc.). The genus is defined via the Gauss-Bonnet theorem, which states that the integral of Gaussian curvature *K* over the surface *S *of an object is quantized:

Here is an integer that is related to the genus by *. *For example, for a sphere with radius , the curvature is a uniform constant, so the integral over the surface area gives , which is consistent with the genus of an object that has no holes in it. The remarkable implication of the Gauss-Bonnet theorem is that, if one stretches the sphere in any arbitrary way, so that some parts of the surface become more curved and other parts become flatter, the integer *n* remains unchanged. The integrated curvature is *invariant *(see Fig. 1).

Much of the recent excitement surrounding “topological electronics” originates in the prospect of finding physical properties of electronic systems that behave like this integer-valued genus. Such an invariant property is necessarily robust to small perturbations or defects, since integers cannot change continuously. In the remainder of this section we will explain the origin of a commonly-discussed invariant, the *Chern number*, which is what defines a “topological electron band”.

Within a single isolated atom, electrons occupy discrete quantum energy levels, or orbitals. When many atoms are arranged into a crystal, the wave functions from neighboring atoms hybridize with each other, and the orbitals broaden into bands of states having a range of energies. Each of the states within a band describes an electron that is shared among many atoms, and its wave function can be written in terms of the momentum with which the electron hops from one atom to another:

.

The factor in this wave function resembles the usual “plane wave” state that describes free electrons. But this plane wave is multiplied by a function , called the “Bloch function” (after Felix Bloch [3]), that repeats periodically for each identical unit (the *unit cell*) of the crystal and describes the way that the electron is affected by atomic nuclei within the unit cell. As we will see, it is the Bloch function that contains information about the topology of the electron band.

Importantly, the electron momentum can take only certain restricted values. Since describes the process of hopping between neighboring crystal lattice sites, the de Broglie wavelength associated with the wave function cannot be shorter than the distance between neighboring unit cells of the crystal. Thus, in a given direction, the momentum has a maximum possible magnitude. The space of allowable momenta is called the *Brillouin zone*, and its shape depends on the arrangement of atoms in the crystal. In discussing the “topology” of an electron band, the Brillouin zone plays the role of a “geometric space”, and closed surfaces within the Brillouin zone are akin to geometric shapes, which can have an integer-valued index akin to the genus. Importantly, exactly opposite points on the zone boundary are equivalent to each other, since they correspond to the same standing wave state with minimal de Broglie wavelength (effectively, the Brillouin zone has periodic boundary conditions).

Constructing an analogy with the Gauss-Bonnet theorem for electron bands requires an analog of the “curvature” that is integrated over a closed surface. As it turns out, this analog of curvature arises from the properties of the Bloch functions . To see the analogy, consider asking the question: for a given momentum , where within the unit cell is the electron wave function centered? The usual way to answer this question is to calculate the expectation value of the position operator within the unit cell. Written in terms of momentum, the position operator is , and one can write its expectation value as

.

This quantity is called the “Berry connection”, and it can be thought of as the momentum-dependent centroid of the electron wave function in real space (see Fig. 2). One caveat is that is not precisely defined because its definition is not gauge-invariant. The Bloch functions are defined only up to an overall phase (a multiplicative factor ) that can be momentum-dependent. Thus the Berry connection is like the vector potential in problems with a magnetic field, and only its curl has a physical meaning. We will show below that this analogy with magnetic field runs much deeper.

Imagine now the hypothetical process of accelerating and then decelerating an electron in such a way that the electron returns to its initial momentum. During the course of this cyclic process the electron traces out a path in momentum space, and its final state is identical to its initial state except for a possible overall phase factor. This phase is an example of what is called a *Berry phase *and its value is

(This expression for the phase is analogous to how a particle traversing a path in position space experiences a phase shift , equal to the number of wavelengths in the path multiplied by ) Traversing the path in the clockwise and counterclockwise directions leads to opposite signs for the phase , since the direction of momentum change along the path is opposite in the two cases.

The Berry phase becomes particularly instructive if we consider how it behaves for paths on a closed two-dimensional (2D) surface of momenta. For example, the Brillouin zone of a 2D system effectively acts like a closed surface because opposite edges of the zone are equivalent to each other. Consider, in particular, the path shown in Fig. 3, which traverses the Brillouin zone boundary of a 2D system. Traversing this path in the clockwise direction yields some Berry phase , while the counterclockwise direction gives . However, because opposite edges of the zone boundary describe physically equivalent states, the clockwise and counterclockwise paths must produce equivalent changes to the wave function. The only way this requirement can be satisfied is if either , or is an integer multiple of , so that the phase factor multiplying the wave function becomes equal to unity. This quantization of the Berry phase in units of can be turned into a more exact analog of the Gauss-Bonnet theorem by using Stokes’ Theorem to relate the contour integral along the boundary of the Brillouin zone to a surface integral across the entire Brillouin zone. This procedure gives

where is called the *Berry curvature*, and is an integer called the *Chern number*.

One can notice that, even though local symmetries do not define the Chern number, having requires that certain symmetries must be broken. In particular, one can see from Fig. 3 that within the Brillouin zone of a system with nonzero Chern number the momenta and are not equivalent, in the sense that they have different values of the Berry connection . Such an inequivalence requires the system to break the symmetry with respect to inversion (inverting the spatial coordinates of the system) or time reversal (playing electron trajectories backward in time). Thus, the search for topological materials has largely focused on materials that break one of these two symmetries. In fact, as we show below, only systems with broken time reversal symmetry can have a nonzero Chern number, and breaking inversion symmetry alone is insufficient. However, the coupling between electron spin and momentum may allow a given spin species (up or down) to have a nonzero Chern number, so long as the two spin-resolved Chern numbers sum to zero.

Below we discuss this idea and other implications of electron topology, as well as its generalization to 3D systems.

**Implications of Topology**

One can notice from Fig. 3 that a nonzero Chern number implies a “winding” or “self-rotation” in the structure of the electron wave function. This self-rotation is associated with a physical angular momentum for electrons. For example, if one imagines making a wave packet using states from some particular region of momentum space, one sees that there is a relation between the position (within the unit cell) and the momentum of the states that comprise the wave packet. This relation between position and momentum implies an angular momentum for the wave packet that depends on the local Berry curvature. In this sense the Berry curvature is again like a magnetic field, created by a broken symmetry in the material itself rather than by any external source, in that it gives electrons an angular momentum.

The analogy of Berry curvature to magnetic field can be made more precise by considering the effects of an applied electric field , which accelerates the electron momentum at a rate , where is the electron charge. If the local Berry connection has a finite curl, then, as the electron accelerates, its centroid location shifts in the transverse direction. This shifting of the electron in the transverse direction is called the *anomalous velocity*, and is given by

.

In this way, the anomalous velocity is like the E-cross-B drift experienced by an electron in crossed electric and magnetic fields. Applying an electric field in a particular direction causes an electron to drift in a direction that is, in part, transverse to both the field and the direction of the (momentum-dependent) Berry curvature, which acts like a magnetic field.

One of the most striking implications of the magnetic field analogy arises from the motion of electrons at the boundary of a sample. In a conductor with no intrinsic Berry phase that is subjected to a magnetic field, electrons near the boundary perform “skipping orbits”, essentially rolling along the boundary in a direction that is defined by the magnetic field direction. No matter how the boundary is shaped, these skipping orbits persist, providing a single conducting channel for current to flow through. In a similar way, the self-rotation implied by a finite Chern number guarantees the existence of traveling edge states. In 2D electron systems with a magnetic field (and sufficiently high electron mobility), the skipping orbit states give rise to the celebrated Quantum Hall Effect, including a quantized electrical conductance whose value is completely universal. 2D materials with finite Chern number have this same universal conductance, even though no magnetic field is present.

In fact, the existence of a topological invariant for electron systems subjected to a magnetic field was first identified by Thouless, Kohomoto, Nightingale and den Nijs (TKNN) [4]. This topological invariant allows for a remarkable universality of the quantum Hall effect among different samples and materials. In fact, the TKNN invariant allows the universal constant to be measured to more than 12 significant figures of accuracy, and it now forms the basis for the metrological standard of the kilogram [5]. The Chern number defined above can be thought of as a generalization of the TKNN result, such that any given material has a particular integer Chern number defined in the absence of any applied fields. Most of the materials we know of have , and the possibility of 2D materials with nonzero Chern number was one of the seminal insights of the topological age.

There is an important way, however, in which the edge states of a topological material can be different from the edge states of a quantum Hall system. In a quantum Hall system, the magnetic field breaks the time-reversal symmetry of the system, since the magnetic field forces electrons to turn in spiral trajectories with a particular handedness, clockwise or counterclockwise, that is set by the magnetic field direction. These spiral trajectories break time reversal symmetry since playing them backwards in time (without reversing the sign of the external magnetic field) produces motion that is inconsistent with the Lorentz force law.

But it is possible to have a topological electron band, and its associated edge states, even in a material that has time-reversal symmetry. The key idea is to combine broken inversion symmetry in the material with a strong coupling between the electron’s momentum and its spin. For example, in the simplest case of a *quantum spin Hall insulator*, these two ingredients combine to allow the two different electron spin states to have nonzero but opposite Chern number. To see how this possibility can arise, consider that the time reversal operation also reverses an electron’s spin, so that, under time reversal, a left-going-spin-up electron becomes a right-going-spin-down electron. Thus, a topological electron band can retain time-reversal symmetry if the bands for spin-up and spin-down electrons have opposite Chern number (Fig. 4). Opposite Chern number means that the two spin species have opposite-moving edge states. This locking of the direction of the edge state to the spin is called the *quantum spin Hall effect*, and it was discovered experimentally in 2007 following its experimental prediction in 2003.

Ultimately, the locking of spin to momentum at edge states arises from the microscopic spin-orbit, or “”, coupling that is present in atomic orbitals. Spin-orbit coupling arises when a fast-moving electron experiences a magnetic field in its reference frame due to the electrostatic potential of the nucleus, and the strength of this field varies as the fourth power of the atomic number. In topological materials with time reversal symmetry, strong spin orbit coupling combines with broken inversion symmetry to produce a Berry curvature and a finite Chern number for each spin, even in the absence of a magnetic field.

**Topological bands in three dimensions**

So far we have only discussed one example of a topological invariant: the Chern number in a 2D band, which gives rise to edge states very much like those in the quantum Hall effect. But a range of 3D materials have been identified with electrical properties that are protected by a topological invariant. These include the 3D *topological insulators*, which are largely made from narrow band-gap semiconductors with strong spin-orbit coupling [6]. In these materials, an electrically insulating interior coexists with surface states that form 2D metals on every free surface and have a similar locking of the electron’s spin perpendicular to its momentum.

The 2D Chern number discussed above can also be applied to understand another class of 3D materials: the *Weyl semimetals*. These materials have special points in momentum space where their “topological charge” is concentrated. To see why this happens, imagine defining an arbitrary closed surface *S* of momentum states within the 3D Brillouin zone of some material (Fig. 5). One can apply to this surface the same arguments about Berry phase that we used for the 2D case, and arrive at the conclusion that the Chern number associated with the surface must also be quantized. In particular,

where, is an integer that depends on the chosen surface and the integral defines a “flux of Berry curvature” through the surface. Since is an integer and cannot change continuously, slight distortions of the surface *S* cannot produce small changes to the integral. The only way for to change is by a discontinuous jump when is expanded to include a particular, special point in momentum space. Such points are *monopoles of Berry flux*, by analogy to the description of electric charges via Gauss’s law, and they always come in pairs of opposite charge. Materials that possess such monopoles are called *Weyl semimetals*, and the special points in momentum space are *Weyl points*.

From a materials perspective, Weyl points arise when strong spin-orbit coupling causes two bands of states with different angular momentum to coincide in energy. Since the orbital character of the wave function must change abruptly upon crossing from one band to another, Weyl points correspond to the locations in momentum space where the two bands touch. In the usual metals and semiconductors that comprise the majority of our electronic technologies, such touching of bands is unusual. Typically, electronic bands cannot coincide in energy because of the phenomenon of avoided crossing — the quantum phenomenon of hybridization of degenerate quantum states into symmetric and antisymmetric combinations that have different energies. However, it was recognized as early as 1937, in a paper by Conyers Herring that presaged much of modern topological band theory [7], that two electron bands could meet in energy because of “accidental degeneracies” that prevent the two bands from hybridizing. In these cases, a perturbation that removes an accidental degeneracy can destroy the band crossing point and open a gap. In Weyl semimetals, however, the Weyl points are protected by the quantization of the Chern number. Only a sufficiently strong perturbation, which effectively brings two oppositely-charged Weyl points together, can destroy the degeneracy and open a gap between two electron bands. Thus, a Weyl semimetal is a topologically-protected gapless system (a *semimetal*). Like the other topological materials, Weyl semimetals have intriguing surface states. In particular, the surfaces of Weyl semimetals exhibit *Fermi arcs*, or momentum states ranging from one Weyl point momentum to another [9].

Experimentally, the study of topological materials is progressing rapidly, with new compounds, even whole classes of topological materials, being discovered routinely. The initial discovery of 3D topological insulating behavior was made using angle-resolved photoemission spectroscopy (ARPES) in Bi_{1-x}Sb_{x}, a mixture of two non-topological semimetals, each with spin orbit coupled bands, that produces a bulk semiconductor for 0.07 < x < 0.22 [12]. Since that discovery, many other topological insulators, as well as Weyl and nodal-line semimetals, have been identified. While ARPES remains a primary tool for revealing electronic band structures, the transport properties of topological materials are also being intensely studied, and raise the possibility of new device functionalities.

**What technologies will topological materials enable?**

The usefulness of many materials comes from their ability to either pass a current of some kind, or to prevent a current from flowing. For example, copper is useful because it allows electric current to flow freely down the length of a wire, while the polymer encasing the wire is useful because it blocks the current from leaking out. Other examples include materials for passing or blocking heat currents (like heat sinks on computer processors, or the insulating foam on the space shuttle) or filtering light (like the lenses on protective sunglasses, which pass some light frequencies while blocking others). Seen from this perspective, the “silicon age” arose because silicon can act as a kind of switchable valve, an “on/off switch” for electrical current. We know now that pure silicon is a good insulator, but if a “gate” voltage is applied to its surface it becomes an electrical conductor.

Thus, for developing new electronic materials, the main goals are usually *filtering* and *sensitivity*. The material should be able to selectively pass or block a generalized current (like silicon can selectively pass electrical current), or exhibit a strong response to some input (like silicon p-n junctions can turn light into electricity). In these questions of filtering and sensitivity, topological materials offer the promise of truly new technologies.

Topological materials can perform interesting kinds of filtering because of their Berry curvature. Since Berry curvature is a kind of winding or handedness that breaks the symmetry between clockwise and counterclockwise motion, topological materials can act like a doorknob, which “opens” when turned in the correct direction and blocks motion in the wrong direction. One striking application is in spin filtering. As illustrated in Fig. 4, the edge states in a topological insulator carry electrons with opposite spin in opposite directions. Such filtering is an essential ingredient for so-called *spintronics*, which aims to build electronic and computer technology based on currents of spin rather than charge [13]. The Berry curvature also implies that different directions of light polarization (clockwise or counterclockwise) couple differently to the electron material, and this effect can be used to create optical filters or logic circuits [14].

Topological materials are also unusually responsive to many kinds of applied fields, owing to their gapless, topologically protected electron bands. For example, the topological edge states associated with finite Chern number offer the promise of dissipationless current-carrying channels, a potential alternative to superconductors for some applications, perhaps even at room temperature. More generally, the *topological protection* of low-energy states in a topological electron band can be exploited in a number of ways, providing an advantage over conventional materials where the low energy states are often heavily distorted by disorder.

This topological protection of the electron band structure may be the reason why some topological materials exhibit enormous electrical mobility (i.e., a very large contribution to the current from each mobile electron) [15]. The Weyl semimetals can also have an extreme sensitivity to light, which may yield a new generation of photo-detectors and night-vision goggles [16]. Topological semimetals also promise an unprecedented thermoelectric effect, which is the ability to convert waste heat into useful electric power [17]. Finally, topological electrons have an unusually sensitive response to magnetic fields, including a wide spacing between quantum levels of the electron’s magnetic field orbit (*Landau levels*), and a strong reduction of electrical resistance when a magnetic field is applied along the current direction (the *chiral anomaly*) [18].

Whether these effects, or the myriad others that are currently being studied, will revolutionize our current electronic technologies remains to be seen. But what is clear is that ideas from topology have established themselves in materials physics, they have led us to predict and observe new materials and new phenomena, and they are here to stay. Who can tell how we will choose to name our current era in the decades to come?

#### References

[1] M. Riordan, and L. Hoddeson, *Crystal Fire *(W. W. Norton & Co., New York, 1997).

[2] L. D. Landau, Zh. Eksp. Teor. Fiz. **7,** 19 (1937).

[3] F. Bloch, Zeitschrift Fur Physik **52,** 555 (1929).

[4] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Dennijs, Physical Review Letters **49,** 405 (1982).

[5] K. von Klitzing, Physical Review Letters **122,**200001,(2019).

[6] C. Kane, and J. Moore, Physics World **24,** 32 (2011).

[7] C. Herring, Physical Review **52,** 0365 (1937).

[9] X. G. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Physical Review B **83,**205101,(2011).

[10] C. Fang, H. M. Weng, X. Dai, and Z. Fang, Chinese Physics B **25,**117106,(2016).

[11] F. Schindler, A. M. Cook, M. G. Vergniory, Z. J. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Science Advances **4,**eaat0346,(2018).

[12] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature **452,** 970 (2008).

[13] I. Zutic, J. Fabian, and S. Das Sarma, Reviews of Modern Physics **76,** 323 (2004).

[14] K. F. Mak, D. Xiao, and J. Shan, Nature Photonics **12,** 451 (2018).

[15] T. Liang, Q. Gibson, M. N. Ali, M. H. Liu, R. J. Cava, and N. P. Ong, Nature Materials **14,** 280 (2015).

[16] C. K. Chan, N. H. Lindner, G. Refael, and P. A. Lee, Physical Review B **95,**041104,(2017).

[17] B. Skinner, and L. Fu, Science Advances **4,**eaat2621,(2018).

[18] A. A. Burkov, Journal of Physics-Condensed Matter **27,**113201,(2015).

[19] B. Bradlyn, L. Elcore, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Atoyo, and B. B. A. Bernevig, Nature, 547, 298 (2017).

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Thanks for the post. I suspect that most traditional QM/CMP books(Sakurai, Ashcroft&Mermin, etc.) lack content on topology in physics.

Could you please suggest a reference (book) for someone who has had a traditional quantum mechanics education but not something topologically flavored?

My question upon leaving topological materials talks has always been “where is the topology?”. I’ve never received a satisfying answer beyond more jargon and unhelpful drawings of doughnuts. This is the clearest explanation I have yet seen of what is going, so bravo!

Amazing article! I had never read a more succinct and easy to understand introduction as this

In your description of Weyl nodes, you have an expression for C_S, the Chern number associated with the surface bounding a subset of the 3D Brillouin zone

By the divergence theorem, the integral of the Berry curvature over any closed surface = the divergence of the Berry curvature over the volume bounded by that surface.

However, the divergence of the curl of a vector field is necessarily zero (and the Berry curvature is the curl of the Berry connection).

Put another way, by the curl theorem, the integral of the curl of a vector field over any closed surface is zero, since

The integral of the curl of a vector field over a surface = the integral of the vector field along the edge bounding that surface (which for a closed surface is non-existent)

So how is it possible that C_S can have a non-zero value?