Systematic construction of square-root topological insulators and superconductors

TL;DR

This paper introduces a graph-theoretic method to construct square-root Hamiltonians of topological systems, revealing in-gap edge states at non-zero energies and extending to non-Hermitian models.

Contribution

It presents a novel scheme to systematically build square-root topological insulators and superconductors using subdivided graphs, expanding the understanding of topological edge states.

Findings

Square-root Hamiltonians exhibit in-gap edge states at non-zero energies.

The scheme applies to Hermitian and non-Hermitian topological models.

Examples include square roots of SSH, Kitaev, and Haldane models.

Abstract

We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$ describing a square root of an original Hamiltonian $H_{\text{original}}$ based on the graph theory. The square-root Hamiltonian is defined on the subdivided graph of the original graph of $H_{\text{original}}$, where the subdivided graph is obtained by putting one vertex on each link in the original graph. When $H_{\text{original}}$ describes a topological system, there emerge in-gap edge states at non-zero energy in the spectrum of $H_{\text{root}}$, which are the inherence of the topological edge states at zero energy in $H_{\text{original}}$. In this case, $H_{\text{root}}$ describes a square-root topological insulator or superconductor. Typical examples are square roots of the Su-Schrieffer-Heeger (SSH) model, the Kitaev topological superconductor model and the Haldane model. Our scheme is also applicable to non-Hermitian topological systems, where we study an example of a nonreciprocal non-Hermitian SSH model.