Engineering topology in waveguide arrays

TL;DR

This paper explores how the topological classification of Floquet photonic waveguide arrays relates to their structural symmetries, revealing new ways to realize topologically protected states even without conventional symmetries.

Contribution

It establishes a systematic link between lattice symmetries and Altland-Zirnbauer classes in 1D waveguide arrays and shows non-bipartite networks can support topological states without traditional symmetries.

Findings

Bipartite structure and z-reflection symmetry determine AZ class.

Non-bipartite networks can host topological boundary states at quasienergy π.

Shifted-particle-hole symmetry protects topological states in higher dimensions.

Abstract

The topological classification of a system depends on the discrete symmetries of its Hamiltonian. In Floquet photonic waveguide arrays, the abstract symmetries of the Altland--Zirnbauer (AZ) scheme -- chiral, particle-hole, and time-reversal (for photonics, $z$-reversal) -- arise from structural properties of the lattice, yet a systematic correspondence has not been established. Here, we illustrate this correspondence for a simpler system of one-dimensional waveguide arrays with real coupling coefficients, showing how bipartite structure and $z$-reflection symmetry alone determine the whole AZ class. We further demonstrate that non-bipartite networks -- lacking conventional particle-hole symmetry, chiral symmetry, and $z$-reversal symmetry -- can nonetheless support topologically protected boundary states at quasienergy $\varepsilon = \pi$, even in one dimension. The protecting symmetry -- \textit{shifted}-particle-hole symmetry -- applies equally to higher-dimensional Floquet waveguides.