Sublattice-selective acoustic attenuation via a band-index-conserved flat channel in a Lieb-patterned ferromagnetic nematic liquid-crystal array.

A Lieb lattice of ferromagnetic nematic liquid-crystal elements yields a sublattice-programmable acoustic response, switched on or off by the drive geometry rather than enhanced attenuation. The central result is a point-group-protected drive-selection rule: at the Brillouin-zone center, the B/C-antisymmetric drive projects entirely onto the B1 irrep of the C4v little co-group while the uniform and A-only drives contain no B1 component, so the corresponding drive overlaps ηflat = 1, 0, 0 are exact symmetry consequences, independent of the range of the intersite coupling. Two independent features of the Lieb sublattice structure enable the wider construction. First, the bipartite chiral symmetry pins a zero-eigenvalue mode whose eigenvector has vanishing amplitude on the corner sublattice, making the associated collective channel insensitive to uniform corner-sublattice detuning. Second, when the on-site frequencies and couplings are sublattice-symmetric, the Lieb band index is conserved and the flat magnetic channel hybridizes only with the flat acoustic channel, so the acoustic loss feature at the single-site frequency ωm(H) is sourced by the flat-channel overlap of the driving field. An interdigitated transducer with oppositely biased combs on the B- and C-edge sublattices maximizes this overlap at Γ and gives the largest Brillouin-zone-averaged overlap among the drive geometries considered; a uniform drive suppresses it. We quantify the selection by an integrated contrast sel ≈ 10. We further show that a direct B-C hopping generated by dipolar coupling lifts the flat band and reduces the global band-index-conservation selection rule away from Γ, but does not compromise the C4v-protected drive gating at Γ; realizing the stronger full-Brillouin-zone flat-channel calibration would require engineered short-range interactions that reduce the effective B-C coupling, and an elastic/anchoring contribution is one possible, but presently unverified, route. The Lieb geometry selects a symmetry-filtered acoustic function distinct from the broad, integrated response of frustrated hexagonal flat-band lattices.