Long-Range Falikov Kimball Model in One Dimension: Phase Transition, Disorder-Free Localisation and Delocalisation

Both disorder or interactions may turn metals into insulators. One of the simplest settings to study this physics is given by the Falikov-Kimball model describing itinerant fermions interacting with a classical background field. Despite the translational invariance of the model, inhomogenous configurations of the background field give rise to effective disorder physics leading to a rich phase diagram in two (or higher dimensions) with finite temperature charge density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalised \ac{FK} model in one dimension with long-range interactions which shows a similarly rich phenomenology. In the high temperature phase, the fermions propagate against a disordered background field which leads to exponential localisation of all eigenstates. Below the CDW transition enabled by the power-law interactions, the thermal fluctuations of the order parameter induce correlated disorder which leads to the coexistence of localised and delocalised states at separate energies. Using a combination of exact diagonalisation and \acl{MCMC} we map out the phase diagram and compute the energy resolved localisation properties of the fermions.