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Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories / NICHOLAS SALE

Swansea University Author: NICHOLAS SALE

DOI (Published version): 10.23889/SUthesis.61816

Abstract

This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of t...

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Published: Swansea 2022
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Giansiracusa, Jeffrey ; Lucini, Biagio
URI: https://cronfa.swan.ac.uk/Record/cronfa61816
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first_indexed 2022-11-08T12:05:13Z
last_indexed 2023-01-13T19:22:50Z
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fullrecord <?xml version="1.0"?><rfc1807><datestamp>2022-11-08T12:14:01.2501799</datestamp><bib-version>v2</bib-version><id>61816</id><entry>2022-11-08</entry><title>Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories</title><swanseaauthors><author><sid>38dcae65204c8d1f606b578c99679c1f</sid><firstname>NICHOLAS</firstname><surname>SALE</surname><name>NICHOLAS SALE</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2022-11-08</date><abstract>This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical &#x3B2; and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries.</abstract><type>E-Thesis</type><journal/><volume/><journalNumber/><paginationStart/><paginationEnd/><publisher/><placeOfPublication>Swansea</placeOfPublication><isbnPrint/><isbnElectronic/><issnPrint/><issnElectronic/><keywords>Mathematics, Topological Data Analysis, Lattice Gauge Theory, Statistical Physics</keywords><publishedDay>8</publishedDay><publishedMonth>11</publishedMonth><publishedYear>2022</publishedYear><publishedDate>2022-11-08</publishedDate><doi>10.23889/SUthesis.61816</doi><url/><notes>ORCiD identifier: https://orcid.org/0000-0003-2091-6051</notes><college>COLLEGE NANME</college><CollegeCode>COLLEGE CODE</CollegeCode><institution>Swansea University</institution><supervisor>Giansiracusa, Jeffrey ; Lucini, Biagio</supervisor><degreelevel>Doctoral</degreelevel><degreename>Ph.D</degreename><degreesponsorsfunders>Swansea University Research Excellence Scholarship</degreesponsorsfunders><apcterm/><funders/><projectreference/><lastEdited>2022-11-08T12:14:01.2501799</lastEdited><Created>2022-11-08T12:02:49.4056535</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>NICHOLAS</firstname><surname>SALE</surname><order>1</order></author></authors><documents><document><filename>61816__25692__c1c4269de8604e968451dd53e25c2f91.pdf</filename><originalFilename>Sale_Nicholas_PhD_Thesis_Final_Redacted_Signature.pdf</originalFilename><uploaded>2022-11-08T12:10:28.4090439</uploaded><type>Output</type><contentLength>2720222</contentLength><contentType>application/pdf</contentType><version>E-Thesis &#x2013; open access</version><cronfaStatus>true</cronfaStatus><documentNotes>Copyright: The author, Nicholas Sale, 2022.</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807>
spelling 2022-11-08T12:14:01.2501799 v2 61816 2022-11-08 Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories 38dcae65204c8d1f606b578c99679c1f NICHOLAS SALE NICHOLAS SALE true false 2022-11-08 This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical β and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries. E-Thesis Swansea Mathematics, Topological Data Analysis, Lattice Gauge Theory, Statistical Physics 8 11 2022 2022-11-08 10.23889/SUthesis.61816 ORCiD identifier: https://orcid.org/0000-0003-2091-6051 COLLEGE NANME COLLEGE CODE Swansea University Giansiracusa, Jeffrey ; Lucini, Biagio Doctoral Ph.D Swansea University Research Excellence Scholarship 2022-11-08T12:14:01.2501799 2022-11-08T12:02:49.4056535 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics NICHOLAS SALE 1 61816__25692__c1c4269de8604e968451dd53e25c2f91.pdf Sale_Nicholas_PhD_Thesis_Final_Redacted_Signature.pdf 2022-11-08T12:10:28.4090439 Output 2720222 application/pdf E-Thesis – open access true Copyright: The author, Nicholas Sale, 2022. true eng
title Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
spellingShingle Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
NICHOLAS SALE
title_short Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
title_full Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
title_fullStr Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
title_full_unstemmed Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
title_sort Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories
author_id_str_mv 38dcae65204c8d1f606b578c99679c1f
author_id_fullname_str_mv 38dcae65204c8d1f606b578c99679c1f_***_NICHOLAS SALE
author NICHOLAS SALE
author2 NICHOLAS SALE
format E-Thesis
publishDate 2022
institution Swansea University
doi_str_mv 10.23889/SUthesis.61816
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical β and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries.
published_date 2022-11-08T04:20:56Z
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score 11.030825

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