# Universal Continuum Theory for Topological Edge Soft Modes

## TL;DR

This paper develops a continuum topological elasticity theory that generalizes the Maxwell-Calladine index theorem, enabling prediction of edge zero modes in continuous media and extending topological mechanics beyond discrete lattices.

## Contribution

It introduces a gauge-invariant bulk topological index for continuous elastic media, bridging topological lattice theories and continuum elasticity.

## Key findings

- Defines a gauge-invariant topological index for continuous media.
- Predicts the number of edge zero modes from bulk properties.
- Extends topological zero modes to media away from the Maxwell point.

## Abstract

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.