On Exponentially Localized Wannier Functions in Non-Periodic Insulators

Exponentially localized Wannier functions (ELWFs) are an orthogonal basis for the low energy states of a material consisting of functions which decay exponentially quickly in space. When a material is insulating and periodic, conditions which guarantee the existence of ELWFs in dimensions one, two, and three are well-known and methods for constructing ELWFs numerically are well-developed. In this dissertation, we consider the case where the material is insulating but not necessarily periodic and develop an algorithm for calculating ELWFs.

In Chapter 3, we propose an optimization-free algorithm for constructing Wannier functions in both periodic and non-periodic insulating systems. In this chapter, we rigorously prove that under the assumption of ``uniform spectral gaps'', a technical assumption we introduce, that our algorithm constructs ELWFs.

While the uniform spectral gaps assumption is not always met in practice, in Chapter 4, we prove that for a wide class of systems (both periodic and non-periodic) it is always possible to modify our algorithm so that the uniform spectral gaps assumption holds. As a consequence of this result, we conclude that for both periodic and non-periodic systems our algorithm can construct ELWFs whenever they exist.

The results in this dissertation open the door for extending the theory of topological insulators, a recently discovered class of materials, to fully non-periodic systems.