  A topological mechanism is a zero elastic-energy deformation of a mechanical
structure that is robust against smooth changes in system parameters. Here, we
map the nonlinear elasticity of a paradigmatic class of topological mechanisms
onto linear fermionic models using a supersymmetric field theory introduced by
Witten and Olive. Heuristically, this approach consists of taking the square
root of a non-linear Hamiltonian and generalizes the standard procedure of
obtaining two copies of Dirac equation from the square root of the linear Klein
Gordon equation. Our real space formalism goes beyond topological band theory
by incorporating non-linearities and spatial inhomogeneities, such as domain
walls, where topological states are typically localized. By viewing the two
components of the real fermionic field as site and bond displacements
respectively, we determine the relation between the supersymmetry
transformations and the Bogomolny-Prasad-Sommerfield (BPS) bound saturated by
the mechanism. We show that the mechanical constraint, which enforces a BPS
saturated kink into the system, simultaneously precludes an anti-kink. This
mechanism breaks the usual kink-antikink symmetry and can be viewed as a
manifestation of the underlying supersymmetry being half-broken.
