Systematic generation of Hamiltonian families with dualities

Dualities are hidden symmetries that map seemingly unrelated physical systems onto each other. The goal of this work is to systematically construct families Hamiltonians endowed with a given duality and provide universal description Hamiltonian near self-dual points. We focus on tight-binding models (also known as coupled-mode theories), which an effective composed coupled harmonic oscillators across domains. start by considering the general case in group-theoretical arguments suffice dualities combining irreducible representations operation parameter space operator space. When additional constraints due system-specific features present, purely group-theoretic approach no longer sufficient. To overcome complication, we reformulate existence root-finding problem, amenable standard optimization numerical continuation algorithms. illustrate generality our method designing concrete toy photonic, mechanical, thermal metamaterials dualities.