  Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of
the Fermi projection of a material consisting of functions which decay
exponentially fast away from their maxima. When the material is insulating and
crystalline, conditions which guarantee existence of ELWFs in dimensions one,
two, and three are well-known, and methods for constructing the ELWFs
numerically are well-developed. We consider the case where the material is
insulating but not necessarily crystalline, where much less is known. In one
spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be
constructed as the eigenfunctions of a self-adjoint operator acting on the
Fermi projection. In this work, we identify an assumption under which we can
generalize the Kivelson-Nenciu-Nenciu result to two dimensions and higher.
Under this assumption, we prove that ELWFs can be constructed as the
eigenfunctions of a sequence of self-adjoint operators acting on the Fermi
projection. We conjecture that the assumption we make is equivalent to
vanishing of topological obstructions to the existence of ELWFs in the special
case where the material is crystalline. We numerically verify that our
construction yields ELWFs in various cases where our assumption holds and
provide numerical evidence for our conjecture.
