Identifying new phases of matter that have unusual properties is a key goal of condensed-matter physics. A famous recent example is the theoretical prediction of crystalline materials known as topological insulators (TIs), several of which have now been identified in the laboratory . TIs are electronic insulators in their d-dimensional interior (bulk) but allow metallic conduction on their (d−1)-dimensional boundaries. This is because in their bulk these materials have an energy gap between the ground and first excited states of electrons, but at their boundaries electrons can move, and hence conduct charge, without paying an energy penalty. Such “gapless” boundary states are unusually robust to the detrimental effects of impurities, and they are responsible for exotic properties that emerge when TIs are coupled to magnets or superconductors. For instance, they endow superconducting vortices with “non-Abelian” quantum statistics that could make the vortices a robust platform for quantum computing.
Now, four teams of researchers [2–6] have identified a new class of TIs in dimensions d>1. These “higher-order”  TIs have (d−1)-dimensional boundaries that, unlike those of conventional TIs, do not conduct via gapless states but instead are themselves TIs. An nth order TI instead has gapless states that live on (d−n\)-dimensional subsystems. For instance, in three dimensions, a second-order TI has gapless states located on 1D "hinges" between distinct surfaces, whereas a third-order TI has gapless states on its 0D corners (Fig. 1). Similarly, a second-order TI in two dimensions also has gapless corner states. Such higher-order systems constitute a distinctive new family of topological phases of matter.