Engineering topology in waveguide arrays

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 $\epsilon =\pi$, even in one dimension. The protecting symmetry – $\mathit{\text{shifted}}$-particle-hole symmetry – applies equally to higher-dimensional Floquet waveguides.