  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 Hamiltonians 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 minimization problem which is amenable to standard
optimization and numerical continuation algorithms. Combined with existing
procedures to physically implement coupled-resonator Hamiltonians, our approach
enables on demand design of photonic, mechanical, thermal, or electronic
metamaterials with dualities.
