  A variety of topological Hamiltonians have been demonstrated in photonic
platforms, leading to fundamental discoveries and enhanced robustness in
applications such as lasing, sensing, and quantum technologies. To date, each
topological photonic platform implements a specific type of Hamiltonian with
inexistent or limited reconfigurability. Here, we propose and demonstrate
different topological models by using the same reprogrammable integrated
photonics platform, consisting of a hexagonal mesh of silicon Mach-Zehnder
interferometers with phase-shifters. We specifically demonstrate a
one-dimensional Su-Schrieffer-Heeger Hamiltonian supporting a localized
topological edge mode and a higher-order topological insulator based on a
two-dimensional breathing Kagome Hamiltonian with three corner states. These
results highlight a nearly universal platform for topological models that may
fast-track research progress toward applications of topological photonics and
other coupled systems.
