Applications and connections between twisted equivariant K-theory, quantum mechanics and condensed matter
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Sheinbaum Frank, Daniel
issued: 2020-02-26T21:52:43Z
issued: 2019
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: The present thesis consists of 2 parts. Chapter 1 is about applications of twisted equivariant K-theory to condensed matter. We consider non-interacting electrons on a half-crystal (a crystal with a boundary), with a gapped bulk condition, under quasi-adiabatic evolution. In A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 we found that Fermi surfaces for these systems under quasi-adiabatic evolution are classified by the K⁻¹-group of the surface Brillouin zone Td⁻¹. Systems with time-reversal and particle-hole symmetry were also considered and we obtained different KR-groups for the different cases. In Chapter 1 I rewrite A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 in a more function-analytic language and further solve technical issues to extend it to include crystallographic symmetries on the directions parallel to the boundary. In Chapter 2 I reproduce the relevant parts of my joint work with C. Okay (C. Okay and D. Sheinbaum arXiv:1905.07723). There we explored a connection between twisted equivariant K-theory to contextuality in quantum mechanics. We also reformulated the sheaf-theoretic framework of S. Abramsky and A. Brandenburger New Journal of Physics, 2011 for contextuality and connect it to another one employing a group cohomology approach of C. Okay, S. Roberts, S.D Bartlett, and R. Raussendorf Quantum Information and Computation, 2017. This leads to the construction of a classifying space for contextuality, from which Wigner functions are classes in its twisted K-theory.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/73611
note: Applications and Connections betweenTwisted Equivariant K-theory,Quantum Mechanics and CondensedMatterbyDaniel Sheinbaum FrankB.Sc., Universidad Nacional Auto´noma de Me´xico, 2013M.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Posdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2019c© Daniel Sheinbaum Frank 2019The following individuals certify that they have read, and recommendto the Faculty of Graduate and Postdoctoral Studies for acceptance, thedissertation entitled:Applications and Connections between Twisted Equivariant K-theory,Quantum Mechanics and Condensed Mattersubmitted by Daniel Sheinbaum Frank in partial fulfilment of the re-quirements for the degree of Doctor of Philosophy in Mathematics.Examining Committee:Alejandro Adem, MathematicsSupervisorCihan Okay, PhysicsCo-supervisorLior Silberman, MathematicsSupervisory Committee MemberGordon Walter Semenoff, PhysicsSupervisory Committee MemberSven Bachmann, MathematicsSupervisory Committee MemberKalle Karu, MathematicsUniversity ExaminerFei Zhou, PhysicsUniversity ExamineriiAbstractThe present thesis consists of 2 parts. Chapter 1 is about applicationsof twisted equivariant K -theory to condensed matter. We consider non-interacting electrons on a half-crystal (a crystal with a boundary), with agapped bulk condition, under quasi-adiabatic evolution. In A. Adem, O. An-tol´ın, G. Semenoff and D. Sheinbaum JHEP, 2016 we found that Fermi sur-faces for these systems under quasi-adiabatic evolution are classified by theK−1-group of the surface Brillouin zone Td−1. Systems with time-reversaland particle-hole symmetry were also considered and we obtained differentKR-groups for the different cases. In Chapter 1 I rewrite A. Adem, O.Antol´ın, G. Semenoff and D. Sheinbaum JHEP, 2016 in a more function-analytic language and further solve technical issues to exte…
has type: Thesis/Dissertation
date issued: 2020-05
is shown at: 10.14288/1.0388728
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/73611