Characterizing quantum morphology through topological data analysis

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Hamilton, Gregory

Contributors dc:contributor

• Clark, Bryan K
• Stone, Michael
• Leditzky, Felix
• Eckstein, James

• Topological Data Analysis
• Many-body Localization
• Quantum Information

Statement dc:rights

• Copyright 2023 Gregory Hamilton

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120325

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120325