  This paper presents 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 to produce perfectly flat bands
and provide an exact analytical solution for the flat band wave functions.
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.
