Universal higher-order topology from a five-dimensional Weyl semimetal: Edge topology, edge Hamiltonian, and a nested Wilson loop

Higher-order topological insulators (HOTIs) or multipole insulators, hosting peculiar corner states, were discovered [Benalcazar et al., Science 357, 61 (2017) and Schindler et al., Sci. Adv. 4, eaat0346 (2018)]. It was independently discovered [Hasimoto et al., Phys. Rev. B 95, 165443 (2017)] that continuum five-dimensional (5D) Weyl semimetals generically host the corner states, and so do four-dimensional (4D) class A and three-dimensional (3D) class AIII topological insulators. In this paper we further confirm that the 5D Weyl semimetals, upon dimensional reduction, lead to universal higher-order topology. First we explain a discrete symmetry protecting the 5D Weyl semimetals, and describe dimensional reductions of the 5D Weyl semimetals to the popular HOTIs in the continuum limit. We calculate the topological charge carried by edge states of the 5D Weyl semimetal, for the most generic boundary condition. The topological charge is a Dirac monopole, which can also be seen from that edge Hamiltonians, are always of the form of a 3D Weyl semimetal. This edge topology leads to the edge-of-edge states, or the corner states, generically, suggesting that the 5D Weyl semimetal is thought of as a physical structural origin of corner states in HOTIs. In addition, we explicitly calculate a nested Wilson loop of the 5D Weyl semimetal and find that the topological structure is identical to that of a Wilson loop of a Dirac monopole.