# Topological tight-binding models from non-trivial square roots

## TL;DR

This paper introduces a novel algebraic method to generate topologically nontrivial tight-binding models by taking non-trivial square roots of parent Hamiltonians, leading to new spectral symmetries and protected states.

## Contribution

It presents a new algebraic mechanism for creating topologically nontrivial bandstructures via square-root operations on lattice Hamiltonians, with practical implementations and theoretical insights.

## Key findings

- Spectral symmetries induce independent topological quantum numbers.
- Emergent nonsymmorphic symmetries protect topological states.
- Implementation demonstrated in silicon photonic structures.

## Abstract

We describe a versatile mechanism that provides tight-binding models with an enriched, topologically nontrivial bandstructure. The mechanism is algebraic in nature, and leads to tight-binding models that can be interpreted as a non-trivial square root of a parent lattice Hamiltonian---in analogy to the passage from a Klein-Gordon equation to a Dirac equation. In the tight-binding setting, the square-root operation admits to induce spectral symmetries at the expense of broken crystal symmetries. As we illustrate in detail for a simple one-dimensional example, the emergent and inherited spectral symmetries equip the energy gaps with independent topological quantum numbers that control the formation of topologically protected states. We also describe an implementation of this system in silicon photonic structures, outline applications in higher dimensions, and provide a general argument for the origin and nature of the emergent symmetries, which are typically nonsymmorphic.