  Disorder or interactions can turn metals into insulators. One of the simplest
settings to study this physics is given by the Falikov-Kimball model, which
describes itinerant fermions interacting with a classical Ising background
field. Despite the translational invariance of the model, inhomogenous
configurations of the background field give rise to effective disorder physics
which lead to a rich phase diagram in two (or more) dimensions with finite
temperature charge density wave (CDW) transitions and interaction-tuned
Anderson versus Mott localized phases. Here, we propose a generalised
Falikov-Kimball model in one dimension with long-range interactions which shows
a similarly rich phase diagram. We use an exact Markov Chain Monte Carlo method
to map the phase diagram and compute the energy resolved localisation
properties of the fermions. We compare the behaviour of this transitionally
invariant model to an Anderson model of uncorrelated binary disorder about a
background CDW field which confirms that the fermionic sector only fully
localizes for very large system sizes.
