Higher Order Topological Phases: A General Principle of Construction

In this talk, we discuss a general principle for constructing higher-order topological (HOT) phases [1]. We argue that if a D-dimensional first-order or regular topological phase involves m Hermitian matrices that anti-commute with additional p-1 mutually anti-commuting 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 co-dimension 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 representation. The former system permits an unprecedented realization of a 4^th-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.

[1] D. Călugăru, V. Juričić, and B. Roy, arXiv:1808.08965.