Maxwell plates and phonon fractionalization

TL;DR

This paper explores topological states in continuum elastic media near mechanical instability, introducing Maxwell plates with holographic floppy modes and thin plates with fractional excitations, expanding topological mechanics beyond discrete lattices.

Contribution

It extends the Maxwell-Calladine index theorem to nonlinear continua, classifies elastic media by stress release capability, and identifies two novel topological elastic phases.

Findings

Maxwell plates exhibit holographic floppy modes.

Thin plates with negative Gaussian curvature show fractional excitations.

Topological states in continuum elasticity are classified and characterized.

Abstract

In the past a few years, topologically protected mechanical phenomena have been extensively studied in discrete lattices and networks, leading to a rich set of discoveries such as topological boundary/interface floppy modes and states of self stress, as well as one-way edge acoustic waves. In contrast, topological states in continuum elasticity without repeating unit cells remain largely unexplored, but offer wonderful opportunities for both new theories and broad applications in technologies, due to their great convenience of fabrication. In this paper we examine continuous elastic media on the verge of mechanical instability, extend Maxwell-Calladine index theorem to continua in the nonlinear regime, classify elastic media based on whether stress can be fully released, and identify two types of elastic media with topological states. The first type, which we name ``Maxwell plates'', are in strong analogy with Maxwell lattices, and exhibit a sub-extensive number of holographic floppy modes. The second type, which arise in thin plates with a small bending stiffness and a negative Gaussian curvature, exhibit fractional excitations and topological degeneracy, in strong analogy to $Z_2$ spin liquids and dimerized spin chains.