{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120325"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120325","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Characterizing quantum morphology through topological data analysis","abstract":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","abstract_html":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","abstract_has_math":false,"creators":["Hamilton, Gregory"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Physics","degree_department":null,"school":null,"contributors":["Clark, Bryan K","Stone, Michael","Leditzky, Felix","Eckstein, James"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:57Z","subjects":["Topological Data Analysis","Many-body Localization","Quantum Information"],"languages":["en","eng"],"rights":["Copyright 2023 Gregory Hamilton"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120325","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Clark, Bryan K","Stone, Michael","Leditzky, Felix","Eckstein, James"]},{"key":"dc:creator","label":"Author","values":["Hamilton, Gregory"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2022-12-13"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Topological Data Analysis","Many-body Localization","Quantum Information"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Gregory Hamilton"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120325"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","The student, Gregory Hamilton, accepted the attached license on 2022-12-10 at 08:22.","The student, Gregory Hamilton, submitted this Dissertation for approval on 2022-12-10 at 08:38.","This Dissertation was approved for publication on 2022-12-13 at 09:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18798 on 2023-09-01 at 17:12:15","The onslaught of machine learning approaches to many-body problems has demonstrated the power of data-analytic tools in describing phenomena where standard techniques fail. One niche in this new regime of scientific study is topological data analysis (TDA), a field firmly rooted in algebraic topology and only recently explored in the context of quantum information and many-body physics. TDA offers a topological lens to quantum information amenable to direct computation and inputtable to machine learning pipelines. Given its relative lack of application in condensed matter physics, TDA poses a tantalizing route towards observables and insights invisible to traditional approaches. In this thesis we explore applications of this burgeoning field to identify and describe the “shape” of quantum information. Three broad topics comprise our testbed to evaluate TDA’s efficacy: many-body localization, relativistic heavy-ion collisions, and multipartite entanglement. While diverse and largely orthogonal, each of these areas of research can be probed by tools from TDA, illustrating its flexibility and versatility. The outline of this thesis is as follows. In Ch. 1, we give an overview of the primary tool of topological data analysis, persistent homology. Through numerous examples we plot out how TDA can be leveraged to identify large and small-scale topological structure in high-dimensional datasets relevant to quantum information and quantum many-body physics. We also introduce two main topics of study in this work, many-body localization and entanglement. In Ch. 2 we outline algorithms for simplicial and persistent homology as well as other hierarchical clustering techniques. We give an overview of exact diagonalization and shift-invert methods for determining eigenpairs before briefly describing the Wegner-Wilson flow, a central topic of Ch. 4. We also define the Delaunay triangulation field estimation which features heavily in Ch. 6. In Ch. 3 we explore Fock space localization landscapes through a topological lens, by identifying how observables derived from persistent homology indicate morphological shifts in eigenstructure across the MBL phase transition. We give phenomenological arguments regarding the scaling of several persistent observables connected to percolation theory and introduce a novel persistent fractal dimension. Keeping with MBL, in Ch. 4 we investigate a fixture of MBL, the quasilocal integrals of motion, and one methodology for generating them, the Wegner-Wilson Flow (WWF). We identify the metrical structure inherent in upper bounds on entanglement growth and relate the WWF to arc lengths and geodesics in the projective Hilbert space. We demonstrate that a key quantum information quantity, the information fluctuation complexity, dictates the correspondence between unitary flow in the projective Hilbert space and entanglement generation. In Ch. 5 we formalize a protocol to generate persistence modules from multipartite entanglement. We give numerous examples of persistence filtrations and show that many familiar entropic quantities appear as topological summaries of persistence diagrams. Interestingly, we show that an entanglement monotone called the n-tangle equates to an integrated Euler characteristic. We explore correspondences between persistent homology and magnitude homology as a tantalizing direction for future study of this perhaps surprising result. We also conjecture different operational interpretations for our persistent homology construction as well as examine the statistics of integrated Betti numbers for a specific entanglement functional over the class of Haar random states. In Ch. 6 we leverage persistent homology to understand signals of hydrodynamical flow in relativistic heavy-ion collisions. Using the Delaunay triangulation field estimation we propose novel observables to discriminate collisions from background noise and offer interesting new measures for multi-particle correlations and anisotropic flow. In Ch. 7 we pivot from topological data analysis into studying topological superconductivity, via an investigation into the topological classification and behavior of bulk-doped Weyl semimetals. Finally, in Ch. 8 we summarize our work."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Characterizing quantum morphology through topological data analysis"]}]}],"canonical_facts":{"dc:contributor":["Clark, Bryan K","Stone, Michael","Leditzky, Felix","Eckstein, James"],"dc:creator":["Hamilton, Gregory"],"dc:date":["2023-05","2022-12-13"],"dc:description":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","The student, Gregory Hamilton, accepted the attached license on 2022-12-10 at 08:22.","The student, Gregory Hamilton, submitted this Dissertation for approval on 2022-12-10 at 08:38.","This Dissertation was approved for publication on 2022-12-13 at 09:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18798 on 2023-09-01 at 17:12:15","The onslaught of machine learning approaches to many-body problems has demonstrated the power of data-analytic tools in describing phenomena where standard techniques fail. One niche in this new regime of scientific study is topological data analysis (TDA), a field firmly rooted in algebraic topology and only recently explored in the context of quantum information and many-body physics. TDA offers a topological lens to quantum information amenable to direct computation and inputtable to machine learning pipelines. Given its relative lack of application in condensed matter physics, TDA poses a tantalizing route towards observables and insights invisible to traditional approaches. In this thesis we explore applications of this burgeoning field to identify and describe the “shape” of quantum information. Three broad topics comprise our testbed to evaluate TDA’s efficacy: many-body localization, relativistic heavy-ion collisions, and multipartite entanglement. While diverse and largely orthogonal, each of these areas of research can be probed by tools from TDA, illustrating its flexibility and versatility. The outline of this thesis is as follows. In Ch. 1, we give an overview of the primary tool of topological data analysis, persistent homology. Through numerous examples we plot out how TDA can be leveraged to identify large and small-scale topological structure in high-dimensional datasets relevant to quantum information and quantum many-body physics. We also introduce two main topics of study in this work, many-body localization and entanglement. In Ch. 2 we outline algorithms for simplicial and persistent homology as well as other hierarchical clustering techniques. We give an overview of exact diagonalization and shift-invert methods for determining eigenpairs before briefly describing the Wegner-Wilson flow, a central topic of Ch. 4. We also define the Delaunay triangulation field estimation which features heavily in Ch. 6. In Ch. 3 we explore Fock space localization landscapes through a topological lens, by identifying how observables derived from persistent homology indicate morphological shifts in eigenstructure across the MBL phase transition. We give phenomenological arguments regarding the scaling of several persistent observables connected to percolation theory and introduce a novel persistent fractal dimension. Keeping with MBL, in Ch. 4 we investigate a fixture of MBL, the quasilocal integrals of motion, and one methodology for generating them, the Wegner-Wilson Flow (WWF). We identify the metrical structure inherent in upper bounds on entanglement growth and relate the WWF to arc lengths and geodesics in the projective Hilbert space. We demonstrate that a key quantum information quantity, the information fluctuation complexity, dictates the correspondence between unitary flow in the projective Hilbert space and entanglement generation. In Ch. 5 we formalize a protocol to generate persistence modules from multipartite entanglement. We give numerous examples of persistence filtrations and show that many familiar entropic quantities appear as topological summaries of persistence diagrams. Interestingly, we show that an entanglement monotone called the n-tangle equates to an integrated Euler characteristic. We explore correspondences between persistent homology and magnitude homology as a tantalizing direction for future study of this perhaps surprising result. We also conjecture different operational interpretations for our persistent homology construction as well as examine the statistics of integrated Betti numbers for a specific entanglement functional over the class of Haar random states. In Ch. 6 we leverage persistent homology to understand signals of hydrodynamical flow in relativistic heavy-ion collisions. Using the Delaunay triangulation field estimation we propose novel observables to discriminate collisions from background noise and offer interesting new measures for multi-particle correlations and anisotropic flow. In Ch. 7 we pivot from topological data analysis into studying topological superconductivity, via an investigation into the topological classification and behavior of bulk-doped Weyl semimetals. Finally, in Ch. 8 we summarize our work."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120325"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Gregory Hamilton"],"dc:subject":["Topological Data Analysis","Many-body Localization","Quantum Information"],"dc:title":["Characterizing quantum morphology through topological data analysis"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Physics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:57Z"}