Fig. 3.

(A) Phonon spectrum for the undistorted kagome lattice. Dashed lines depict frequencies at k^′ = 0 and full lines at k^′ > 0. The inset shows the Brillouin zone with symmetry points Γ, M, and K. Note the line of zero modes along ΓM when k^′ = 0, all of which develop nonzero frequencies for wavenumber q > 0 when k^′ > 0 reaching on a plateau beginning at defining a length scale l^∗ = 1/q^∗. (B) Phonon spectrum for α > 0 and k^′ = 0. The plateau along ΓM defines and its onset at q_α ∼ ω_α defines a length l_α ∼ 1/| sin α|.