Reconfigurable topological valley-Hall interfaces: Asymptotics of arrays of Dirichlet and Neumann inclusions for multiple scattering in metamaterials
We study two-dimensional periodic metamaterials in which idealised cylindrical inclusions are modelled by boundary conditions. In the scalar time-harmonic setting, the background field satisfies the Helmholtz equation, and high-contrast inclusion limits reduce to Dirichlet or Neumann conditions, with direct analogues in dielectric and acoustic media. By switching the condition assigned to selected inclusions, we break point-group symmetries of the primitive cell and thereby lift symmetry-induced degeneracies in the Floquet--Bloch spectrum of hexagonal and square lattices, opening valley-type band gaps with Berry curvature localised near opposite valleys. To analyse infinite and finite structures within a unified framework, we derive matched-asymptotic point-scatterer approximations for mixed Dirichlet--Neumann arrays. For doubly periodic systems, this yields a finite-dimensional generalised eigenvalue problem for the Floquet--Bloch spectrum; for finite arrays, it yields a generalised Foldy multiple-scattering system. In both hexagonal and square lattices, geometrically identical crystals can realise distinct valley-Hall phases solely through boundary-condition assignment while retaining an overlapping bulk gap. Spatially varying this assignment therefore creates and relocates internal interfaces without altering the underlying geometry, enabling the associated valley-Hall interfacial modes to be repositioned within the same crystal.