Duncan Haldane observed, pointedly, that “just knowing the correct laws of quantum mechanics does not mean that we understand all the strange phenomena that it allows.” So he spoke in his banquet speech, just having been awarded the 2016 Nobel Prize in Physics, together with David Thouless and Michael Kosterlitz. The work of the trio, and others, on topological phase transitions and topological phases has flung open the gate to the discovery of countless mind-boggling properties and behaviours of matter — “‘really cool things’ that had never been guessed at before”, as Haldane put it.

In the two years since Haldane’s speech, we have seen many more ‘really cool’ developments: a case in point being higher-order topological insulators, which feature prominently in the current issue of Nature Physics.

Topological insulators (TIs) are, in a nutshell, systems with an insulating bulk and topologically protected (and therefore robust) excitations at their boundary. ‘Conventional’ two-dimensional TIs exhibit such fractionalization on their edges, three-dimensional ones at their surface. In contrast, higher-order topological insulators — or, of course, HOTIs — are topologically protected at the boundary of the boundaries, further reducing the dimensionality. Following the first proposals1 for such exotic states, experimental realizations of two-dimensional electric quadrupole insulators with protected corner states were soon reported. Intriguingly, these demonstrations came on a variety of classical platforms: photonic crystals2, mechanical pendula3, microwave resonators4 and, as Stefan Imhof and colleagues5 report in this issue, electrical circuits.

Whereas the ability to create topologically non-trivial phases in such metamaterials is in itself no longer a surprise, the fact that electric multipole insulators have first been realized in analogue systems, rather than in ‘natural’ electronic systems, is noteworthy. There could even be an opportunity for science outreach, especially in the context of the work of Imhof et al., who use (as a few other groups do) RLC circuits to emulate the behaviour of topological matter. As Ling Lu6 writes in his accompanying News & Views, the ability to explore one of the most active areas of physics using a set of off-the-shelf electronics components should be a blessing for students and teachers alike. No more explaining TIs with coffee mugs morphing into doughnuts.

But the story does not end with these analogue experiments. The concept of electric multipole insulators led to a broader theory of HOTIs, generalizing the topological bulk–boundary correspondence7. In this issue, Frank Schindler and colleagues8 report that bismuth — a material that has already presented us with so many surprises9 — is a realization of a HOTI. And bismuth is unlikely to remain alone. In a series of very recent computational studies10, thousands of materials have been predicted to possess topological electronic properties, rather than the few hundred that are currently known, among them a substantial number of HOTIs. We’re only just scratching the surface, with further ‘strange phenomena’ undoubtedly waiting underneath.