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

Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher