  Strong zero modes provide a paradigm for quantum many-body systems to encode
local degrees of freedom that remain coherent far from the ground state.
Example systems include $\mathbb{Z}_n$ chiral quantum clock models with strong
zero modes related to $\mathbb{Z}_n$ parafermions. Here we show how these
models and their zero modes arise from geometric chirality in fermionic Mott
insulators, focusing on $n=3$ where the Mott insulators are three-leg ladders.
We link such ladders to $\mathbb{Z}_3$ chiral clock models by combining
bosonization with general symmetry considerations. We also introduce a concrete
lattice model which we show to map to the $\mathbb{Z}_3$ chiral clock model,
perturbed by the Uimin-Lai-Sutherland Hamiltonian arising via superexchange. We
demonstrate the presence of strong zero modes in this perturbed model by
showing that correlators of clock operators at the edge remain close to their
initial value for times exponentially long in the system size, even at infinite
temperature.
