Universal Continuum Theory for Topological Edge Soft Modes

Topological edge zero modes and states of self stress have been intensively studied in discrete lattices at the Maxwell point, offering robust properties concerning surface and interface stiffness and stress focusing. In this paper we present a continuum topological elasticity theory and generalize the Maxwell-Calladine index theorem to continuous elastic media. This theory not only serves as a macroscopic description for topological Maxwell lattices, but also generalizes this physics to continuous media. We define a \emph{gauge-invariant} bulk topological index, independent of microscopic details such as choice of the unit cell and lattice connectivity. This index directly predicts the number of zero modes on edges, and it naturally extends to media that deviate from the Maxwell point, depicting how topological zero modes turn into topological soft modes.