  Two-dimensional materials exhibit a variety of mechanical instabilities
accompanied by spontaneous symmetry breaking. Here we develop a continuum
description of the buckling instability of antimonene sheets. Regions of
oppositely directed buckling constitute domains separated by domain walls that
are solitons in our model. Perturbations about equilibrium propagate as waves
with a gapped dispersion in the bulk but there is a gapless mode with linear
dispersion that propagates along the domain walls in a manner reminiscent of
the electronic modes of topological insulators. We establish that monolayer
antimonene is a mechanical topological insulator by demonstrating a mapping
between our continuum model and an underlying Dirac equation of the symmetry
class BDI which is known to be a topological insulator in one dimension and a
weak topological insulator in two dimensions. Monolayer antimony can be
produced by exfoliation as well as epitaxy and the effects predicted in this
paper should be accessible to standard experimental tools such as scanning
probe microscopy and Raman spectroscopy. We surmise that the effects studied
here (namely low scale symmetry breaking, strain solitons and gapless edge
modes) are not limited to antimonene but are common features of two dimensional
materials.
