Topological Boundary Modes in Isostatic Lattices

TL;DR

This paper links topological boundary modes in isostatic lattices to topological band theory, predicting new phases with boundary modes and demonstrating this in one- and two-dimensional models.

Contribution

It establishes a connection between topological mechanical modes and electronic topological band theory, introducing models that exemplify this topological phenomenon.

Findings

Boundary modes are topologically protected and insensitive to local perturbations.

New topological bulk phases with distinct boundary modes are predicted.

Models in 1D and 2D demonstrate the existence of these topological boundary modes.

Abstract

Frames, or lattices consisting of mass points connected by rigid bonds or central force springs, are important model constructs that have applications in such diverse fields as structural engineering, architecture, and materials science. The difference between the number of bonds and the number of degrees of freedom of these lattices determines the number of their zero-frequency "floppy modes". When these are balanced, the system is on the verge of mechanical instability and is termed isostatic. It has recently been shown that certain extended isostatic lattices exhibit floppy modes localized at their boundary. These boundary modes are insensitive to local perturbations, and appear to have a topological origin, reminiscent of the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators. In this paper we establish the connection between the topological mechanical modes and the topological band theory of electronic systems, and we predict the existence of new topological bulk mechanical phases with distinct boundary modes. We introduce model systems in one and two dimensions that exemplify this phenomenon.