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.
","abstract_html":"<p>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.</p><p>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.</p><p>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.</p><p>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.</p>","abstract_has_math":false,"creators":["Stubbs, Kevin"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Lu, Jianfeng"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021","date_published":"2021","updated_at":"2026-07-24T02:06:59Z","subjects":["Mathematics","Physics","Applied physics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/22997","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Lu, Jianfeng"]},{"key":"dc:creator","label":"Author","values":["Stubbs, Kevin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2021-05-19T18:07:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2021-05-19T18:07:56Z"]},{"key":"dc:date.issued","label":"Date","values":["2021"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physics","Applied physics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/22997"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["On Exponentially Localized Wannier Functions in Non-Periodic Insulators"]}]}],"canonical_facts":{"dc:contributor.advisor":["Lu, Jianfeng"],"dc:creator":["Stubbs, Kevin"],"dc:date.accessioned":["2021-05-19T18:07:56Z"],"dc:date.available":["2021-05-19T18:07:56Z"],"dc:date.issued":["2021"],"dc:description.abstract":["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.
"],"dc:identifier.uri":["https://hdl.handle.net/10161/22997"],"dc:subject":["Mathematics","Physics","Applied physics"],"dc:title":["On Exponentially Localized Wannier Functions in Non-Periodic Insulators"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:06:59Z"}