Reconfigurable topological valley-Hall interfaces: Asymptotics of arrays of Dirichlet and Neumann inclusions for multiple scattering in metamaterials

Abstract

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.

Figures (9)

• Figure 1: Two-dimensional square (a) and hexagonal (c) Bravais lattices in physical space, generated by $\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$. Panels (b) and (d) show the corresponding reciprocal lattices, generated by $\boldsymbol{\beta}_{1}$ and $\boldsymbol{\beta}_{2}$. In all panels, Bravais lattice points are indicated by $\circ$.
• Figure 2: Hexagonal primitive cell with six Dirichlet inclusions (Dirichlet inclusions are shown as black filled inclusions). Parameters are $\boldsymbol{\alpha}_{1} = \cos(\frac{\pi}{6}) \textbf{e}_{x} + \sin(\frac{\pi}{6})\textbf{e}_{y}$, $\boldsymbol{\alpha}_{2} = -\cos(\frac{\pi}{6}) \textbf{e}_{x} + \sin(\frac{\pi}{6})\textbf{e}_{y}$, $\textbf{X}_{1J} = 0.2( \textbf{e}_{x} \cos \frac{(5-J)\pi}{3} + \textbf{e}_{y} \sin \frac{(5-J)\pi}{3})$, and $\eta_{1J} = 0.03$ for $J = 1, \ldots ,6$. Panel (a) shows the Floquet--Bloch dispersion along $\Gamma$--$M$--$K$--$\Gamma$, the black lines are eigenvalues from \ref{['HelmholtzHardScheme_1']} and the black squares are FEM computed results; the four marked points - $\color{myRED}\boldsymbol{\square}$ correspond to (b) & (e) and $\color{myBLUE}\boldsymbol{\bigcirc}$ to (c) & (d) - lie on the two branches adjacent to the degeneracy at $K$. Panels (b),(c) correspond to the two marked points on the $M$--$K$ segment and panels (d),(e) to the two marked points on the $K$--$\Gamma$ segment; they (b)-(e) show the normalised real part of the corresponding Bloch eigenmodes. The dotted line indicates the reflection symmetry line used for parity classification. The unperturbed configuration retains three $\sigma_{v}$ reflection symmetries. The code required to compute all panels of this figure is given in the ancillary files, refer to section \ref{['CodeAv']}.
• Figure 3: Perturbing the structure in Figure \ref{['fig:HexDirac']} by making every other inclusion Neumann (depicted as white or blue filled inclusions) creates the mirror-related chiral unit cells in (a) and (b). Panel (c) shows the Floquet--Bloch dispersion relation for cell (a), with the opened bulk band gap shaded in grey- again the black lines are eigenvalues from \ref{['HelmholtzHardScheme_1']} and the black squares are FEM computed results. Panels (d) and (e) show the Berry curvature over the Brillouin zone for the bands immediately below and above the gap, respectively, exhibiting valley localisation of opposite sign near $K$ and $K'$. The code required to compute panel (c) of this figure is given in the ancillary files, refer to section \ref{['CodeAv']}.
• Figure 4: Ribbon/interface states for the hexagonal chiral pair. Panel (a) shows the ribbon Floquet--Bloch dispersion, derived from the eigenvalues of \ref{['HelmholtzHardScheme_1']}; the shaded region is the projected bulk band gap from Figure \ref{['fig:HexValley']} and the coloured curves are interfacial ZLM branches. Panels (b) and (c) show the normalised magnitude of the corresponding interface-localised eigenmodes (absolute value) at the marked $\kappa$ values at $\boldsymbol{\square}$ and $\boldsymbol{\diamond}$ respectively.
• Figure 5: Plane-wave multiple-scattering simulation (the absolute value of the normalised total field is plotted, solving \ref{['scatteringSoln']} to determine the unknown coefficients in \ref{['mscSOLNtotal']}) for a finite collection of $2031$ hexagonal cells from Figure \ref{['fig:HexValley']}(a),(b), with $\Omega = 4.8304$, $\theta_{\mathrm{inc}} = \frac{7 \pi}{6}$, $A_{\mathrm{inc}} = 1$ and $\textbf{X}_{\mathrm{inc}}$ the centre of the collection. The schematic configurations (a),(c),(e) differ only by the Dirichlet/Neumann assignments on inclusions (geometry fixed), thereby relocating the internal interface; the corresponding field magnitudes (b),(d),(f) show localisation along the moved interface. The frequency is chosen within the topologically non-trivial bulk band gap from fig. \ref{['fig:HexValley']}. The code required to compute panel (d) of this figure is given in the ancillary files, refer to section \ref{['CodeAv']}.