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 Letter, we present a topological elasticity theory for general continuous media where a gauge-invariant bulk topological index independent of microscopic details is defined. This index directly predicts the number of zero modes on edges at long length scales, and it naturally extends to media that deviate from the Maxwell point, depicting how topological zero modes turn into topological soft modes.