2n-root weak, Chern, and higher-order topological insulators, and 2n-root topological semimetals

Recently, we have introduced [A. M. Marques et al., Phys. Rev. B 103, 235425 (2021)] the concept of ${2}^{n}$-root topology and applied it to one-dimensional systems. These models require $n$ squaring operations to their Hamiltonians, intercalated with different constant energy downshifts at each level, in order to arrive at a decoupled block corresponding to a known topological insulator (TI) that acts as the source of the topological features of the starting ${2}^{n}\text{\ensuremath{-}}\mathrm{root}$ TI $(\sqrt[{2}^{n}]{\text{TI}})$. In the process, $n$ nontopological residual models with degenerate spectra and in-gap impurity states appear, which dilute the topologically protected component of the starting edge states. Here, we generalize this method to several two-dimensional models, by finding the 4-root version of lattices hosting weak and higher-order boundary modes (both topological and nontopological) of a Chern insulator and of a topological semimetal. We further show that a starting model with a non-Hermitian region in parameter space and a complex energy spectrum can nevertheless display a purely real spectrum for all its successive squared versions, allowing for an exact mapping between certain non-Hermitian models and their Hermitian lower root-degree counterparts. A comment is made on the possible realization of these models in artificial lattices.