  We describe a versatile mechanism that provides tight-binding models with an
enriched, topologically nontrivial bandstructure. The mechanism is algebraic in
nature, and leads to tight-binding models that can be interpreted as a
non-trivial square root of a parent lattice Hamiltonian---in analogy to the
passage from a Klein-Gordon equation to a Dirac equation. In the tight-binding
setting, the square-root operation admits to induce spectral symmetries at the
expense of broken crystal symmetries. As we illustrate in detail for a simple
one-dimensional example, the emergent and inherited spectral symmetries equip
the energy gaps with independent topological quantum numbers that control the
formation of topologically protected states. We also describe an implementation
of this system in silicon photonic structures, outline applications in higher
dimensions, and provide a general argument for the origin and nature of the
emergent symmetries, which are typically nonsymmorphic.
