Elasticity and Response in Nearly Isostatic Periodic Lattices

The square and kagome lattices with nearest-neighbor springs of spring constant k are isostatic with a number of zero-frequency modes that scale with the perimeter. We analytically study the approach to this isostatic limit as the spring constant k' for the next-nearest-neighbor bond vanishes. We identify a characteristic frequency ω* ~ √(k') and length ω* ~ √(k/k' ) for both lattices. The shear modules C44 = k' of the square lattice vanishes with k', but that for the kagome lattice does not.