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

TL;DR

This paper extends the concept of $2^n$-root topological phases to two-dimensional models, revealing new topological insulators and semimetals, and explores their spectral properties, including non-Hermitian to Hermitian mappings.

Contribution

It generalizes $2^n$-root topology to 2D systems, introduces models with higher-order boundary modes, and analyzes spectral properties of non-Hermitian models.

Findings

Generalization of $2^n$-root topology to 2D models.

Identification of models with higher-order boundary modes.

Non-Hermitian models can have real spectra and map to Hermitian counterparts.

Abstract

Recently, we have introduced in [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 $2^n$-root TI ($\sqrt[2^n]{\text{TI}}$). In the process, $n$ non-topological 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 non-topological) 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.