One-dimensional $2^n$-root topological insulators and superconductors

TL;DR

This paper generalizes the concept of square-root topology to $2^n$-root topological insulators and superconductors in one dimension, establishing a systematic construction method and connecting these models to known topological phases.

Contribution

It introduces a systematic framework for constructing $2^n$-root topological models and links them to known topological insulators/superconductors using graph theory concepts.

Findings

Systematic construction rules for $2^n$-root topological models.

Introduction of arborescence concept connecting models through squaring operations.

Extension potential to higher-dimensional topological systems.

Abstract

Square-root topology is a recently emerged subfield describing a class of insulators and superconductors whose topological nature is only revealed upon squaring their Hamiltonians, i.e., the finite energy edge states of the starting square-root model inherit their topological features from the zero-energy edge states of a known topological insulator/superconductor present in the squared model. Focusing on one-dimensional models, we show how this concept can be generalized to $2^n$-root topological insulators and superconductors, with $n$ any positive integer, whose rules of construction are systematized here. Borrowing from graph theory, we introduce the concept of arborescence of $2^n$-root topological insulators/superconductors which connects the Hamiltonian of the starting model for any $n$, through a series of squaring operations followed by constant energy shifts, to the Hamiltonian of the known topological insulator/superconductor, identified as the source of its topological features. Our work paves the way for an extension of $2^n$-root topology to higher-dimensional systems.