Higher-order topological phases: A general principle of construction

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a D -dimensional first-order or regular topological phase involves m Hermitian matrices that anticommute with additional p -1 mutually anticommuting matrices, it is conceivable to realize an n th -order HOT phase, where n =1 ,...,p , with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An n th -order HOT phase accommodates zero modes on a surface with codimension n . We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo)spin-1/2 or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.