Flat Bands and Transition into Localization in Interacting Systems

We present a theory of interaction-induced band-flattening in strongly correlated electron systems. We begin by illustrating an inherent connection between flat bands and index theorems, and presenting a generic prescription for constructing flat bands by periodically repeating local Hamiltonians with topological zero modes. Specifically, we demonstrate that a Dirac particle in an external, spatially periodic magnetic field can be cast in this form. We derive a condition on the field which leads to flat bands. Furthermore, we explore an interacting model of Dirac fermions in a spatially inhomogeneous field. We show that certain Hubbard-Stratonovich configurations exist that ``rectify'' the field configuration, inducing band flattening. We present an explicit model where this localization scenario is energetically favorable -- specifically in Dirac systems with nearly flat bands, where the energy cost of rectifying textures is quadratic in the order parameter, whereas the energy gain from flattening is linear. In conclusion, we discuss alternative symmetry-breaking channels, especially superconductivity, and propose that these interaction-induced band-flattening scenarios represent a generic non-perturbative mechanism for spontaneous symmetry breaking, pertinent to many strongly-correlated electron systems.