Waves cornered

The experimental discovery of materials known as higher-order topological insulators corroborates theoretical predictions and expands the toolbox for integrated optics and mechanical devices.
Michel Fruchart is at the Lorentz Institute for Theoretical Physics, University of Leiden, 2333 Leiden, the Netherlands.

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Vincenzo Vitelli is at the James Frank Institute and in the Department of Physics, University of Chicago, Chicago, Illinois 60637, USA.

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When waves encounter an obstacle, they are typically scattered in all directions. But at the edges of materials called topological insulators, waves are topologically protected, which means that they can propagate in spite of structural imperfections. Two papers in Nature, by Serra-Garcia et al.1 and Peterson et al.2, and a third published on the arXiv preprint server by Imhof et al.3, now report experimental evidence for a new type of topological insulator that supports protected waves at its corners, rather than at its edges. Such materials could find applications in the design of waveguides (structures that restrict wave propagation) and in integrated optics and mechanics. More importantly, they are the first confirmation of a theoretical description that could unify observations previously thought to be unrelated in condensed-matter physics4.

A good chocolate is hard on the outside, but soft on the inside. Topological insulators are the opposite. The d-dimensional interior (bulk) of a topological insulator is ‘hard’ in the sense that it will not react to external stimuli at certain frequencies: there is a range of frequencies, known as a gap, at which waves cannot propagate. By contrast, the (d − 1)-dimensional boundaries not only allow wave propagation, but also guarantee the existence of topologically protected oscillations (modes) at the gap frequencies. Such oscillations are localized in dimension d = 1, for which the boundaries are points, and propagate along the boundaries in d > 1 (Fig. 1a).

Figure 1 | Types of topological insulator. Materials known as topological insulators consist of a d-dimensional interior (grey) whose boundaries can host oscillations called topologically protected modes (red and blue; the two colours correspond to opposite electric charges). a, In first-order topological insulators, modes are localized in dimension d = 1 (dipole topological insulators), travel along one-dimensional channels in d = 2 and exist on surfaces in d = 3. b, Three papers13 report evidence for second-order topological insulators in d = 2 (quadrupole systems), for which modes are localized to corners. Second-order topological insulators do not exist in d = 1, and modes are supported along one-dimensional hinges in d = 3. c, At third order, the minimum dimension is d = 3, for which modes exist on corners (octupole topological insulators).

Crucially, the existence of these protected edge modes can be traced to the physics of the bulk material. One can summarize the mathematical description of wave propagation inside a topological insulator as an intricate knot and that outside it as a simple loop. The knot must be cut at the edges to match the ‘untwisted’ wave propagation outside. Cutting the knot allows modes that have otherwise forbidden frequencies to be present.

In 2017, the theory of topological insulators was extended5,6 to include higher-order examples79, such that ordinary topological insulators appear at the first order. A higher-order insulator can be thought of as having a nested topological structure. For example, in second-order topological insulators, the properties of the bulk cause the (d − 1)-dimensional boundaries to have frequency gaps. However, the boundaries themselves are topological insulators — protected modes are supported on (d − 2)-dimensional corners or hinges (Fig. 1b). In third-order insulators, the boundaries of the boundaries are topological insulators, and protected modes exist on (d − 3)-dimensional corners (Fig. 1c). The result of this hierarchical process is more subtle than merely adding together topologically protected edges without the bulk of a higher-order insulator. For instance, in a quadrupole topological insulator (a second-order insulator in dimension d = 2), there is only one mode in each corner, yet each mode is shared between two edges.

The three current papers report experimental evidence for higher-order topological insulators. More specifically, they identify the topologically protected corner modes associated with quadrupole insulators. The authors achieved this feat using artificial structures called metamaterials, which are engineered to have properties not found in nature10.

Serra-Garcia et al. obtained the required topological structure by tuning the vibrational excitations of connected vibrating plates. Peterson et al. used coupled light-trapping devices known as microwave resonators. Finally, Imhof et al. used a network of electrical components (capacitors and inductors) that were linked to one other. All three teams showed that the corner modes of their topological insulators exist at frequencies not permitted in the bulk — a clear indication that such modes originate from the bulk’s topology. Peterson and colleagues went a step further by explicitly demonstrating the robustness of the corner modes to deformation of the edges.

The theoretical prediction of higher-order systems rests on a generalization of electric dipole moments to multipole moments that are quantized (having only specific discrete values)5,6. Whereas conventional topological insulators are related to dipoles, higher-order insulators are related to quadrupoles, octupoles, and so on. This theory has been corroborated by the authors’ experimental realizations of quadrupole systems. However, the experiments did not directly measure the responses of the topological insulators to electromagnetic fields, which would prove whether or not a quantized quadrupole moment is present. Such higher-order-bulk responses could be measured in electronic systems, in which higher-order insulators were demonstrated earlier this year4. Future work could also extend the theoretical formalism to general external fields, rather than solely electromagnetic fields.

In terms of potential applications, it is not yet clear whether higher-order topological modes localized to corners or hinges have practical advantages over their conventional counterparts. For instance, higher-order topological insulators rely on the existence of crystal symmetries that typically limit the robustness of the edge modes. Moreover, it has been shown that protected modes can also be localized to points or lines of dimensionality lower than (d – 1) in ordinary topological insulators that have material defects1114.

Finally, one can speculate about such systems beyond third order — in other words, beyond the octupole moment. However, these are difficult to realize because of the unfortunate lack of spatial dimensions in our everyday world. Possible ways of overcoming this difficulty include resorting to ‘synthetic’ dimensions provided by internal degrees of freedom (such as the oscillation modes of a resonator), or artificially enhancing the connectivity of crystal lattices using long-range links15.

The authors’ experimental evidence for higher-order topological insulators illustrates the rapid transition from theoretical proposals to experimental realizations in current research on topological materials. We expect the next few years will be the time for such materials to prove their engineering worth.

Nature 555, 318-319 (2018)

doi: 10.1038/d41586-018-02868-4


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