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