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postgraduate thesis: Topological theory of gapless and gapped fermionic systems
Title  Topological theory of gapless and gapped fermionic systems 

Authors  
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Zhao, Y. [趙宇心]. (2014). Topological theory of gapless and gapped fermionic systems. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5312335 
Abstract  Recently the discoveries of graphene, Weyl semimetal and Dirac semimetal are drawing more and more people’s attentions back to the topological characteristics of Fermi surfaces, which may be tracked back to the pair of Weyl points observed experimentally in the phase A of Helium three in 80s in the last century. Based on the pioneer works by Volovik and Hořava, we classify all kinds of Fermi surfaces with respect to antiunitary symmetries and codimensions of Fermi surfaces. The first chapter of this thesis is attributed to develop symmetrydependent topological invariants to characterize topological properties of Fermi surfaces, and map out the periodic classification tables of Fermi surfaces. Compared with the existing classification of topological insulators (TIs) and superconductors (TSCs), it is observed that there exists a twostep dimension shift from our classification of Fermi surfaces. Actually the two classifications can both be derived rigorously in the framework of Ktheory, a mathematical algebraic topology theory for stable fiber bundles, where the dimension shift can also be derived rigorously by constructing maps between Fermi surfaces and TIs/TSCs. This unified treatment of the two classifications is of mathematical elegance, even providing us deeper understandings of these topological phenomena, and is the subject of chapter II of this thesis.
In the beginning of chapter III, when applying our theory of topological Fermi surfaces on the boundary of TIs/TSCs, a general index theory is conjectured describing a faithful boundarybulk correspondence of TIs/TSCs, which is motivated by the dimension shift in the two classifications. Then we construct all kinds of TIs/TSCs and Fermi surfaces by Dirac matrices, which is actually a physical interpretation of the AtiyahBottShapiro construction as a mathematical theory, and provides us a rigorous proof of our general index theorem. We also provide applications of our theory and its connections to nonlinear sigma models of disordered systems.
The last chapter of this thesis may be regarded as a collection of applications of the boundarybulk correspondence described by the general index theorem for spatially onedimensional systems. Specifically onedimensional superconductor models in the other three nontrivial cases are constructed as generalizations of the Kitaev’s model that is one of four nontrivial cases, and every model is solved in detail by methods similar to that provided by Kitaev. Then we analyze each model in the framework of the general index theorem, focusing on the topological properties of Majorana zeromodes with codimension zero at the ends of these models under the open boundary condition. The possible applications of these models to universal quantum manipulations are also discussed. 
Degree  Doctor of Philosophy 
Subject  Fermions 
Dept/Program  Physics 
Persistent Identifier  http://hdl.handle.net/10722/206330 
DC Field  Value  Language 

dc.contributor.author  Zhao, Yuxin   
dc.contributor.author  趙宇心   
dc.date.accessioned  20141023T23:14:27Z   
dc.date.available  20141023T23:14:27Z   
dc.date.issued  2014   
dc.identifier.citation  Zhao, Y. [趙宇心]. (2014). Topological theory of gapless and gapped fermionic systems. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5312335   
dc.identifier.uri  http://hdl.handle.net/10722/206330   
dc.description.abstract  Recently the discoveries of graphene, Weyl semimetal and Dirac semimetal are drawing more and more people’s attentions back to the topological characteristics of Fermi surfaces, which may be tracked back to the pair of Weyl points observed experimentally in the phase A of Helium three in 80s in the last century. Based on the pioneer works by Volovik and Hořava, we classify all kinds of Fermi surfaces with respect to antiunitary symmetries and codimensions of Fermi surfaces. The first chapter of this thesis is attributed to develop symmetrydependent topological invariants to characterize topological properties of Fermi surfaces, and map out the periodic classification tables of Fermi surfaces. Compared with the existing classification of topological insulators (TIs) and superconductors (TSCs), it is observed that there exists a twostep dimension shift from our classification of Fermi surfaces. Actually the two classifications can both be derived rigorously in the framework of Ktheory, a mathematical algebraic topology theory for stable fiber bundles, where the dimension shift can also be derived rigorously by constructing maps between Fermi surfaces and TIs/TSCs. This unified treatment of the two classifications is of mathematical elegance, even providing us deeper understandings of these topological phenomena, and is the subject of chapter II of this thesis. In the beginning of chapter III, when applying our theory of topological Fermi surfaces on the boundary of TIs/TSCs, a general index theory is conjectured describing a faithful boundarybulk correspondence of TIs/TSCs, which is motivated by the dimension shift in the two classifications. Then we construct all kinds of TIs/TSCs and Fermi surfaces by Dirac matrices, which is actually a physical interpretation of the AtiyahBottShapiro construction as a mathematical theory, and provides us a rigorous proof of our general index theorem. We also provide applications of our theory and its connections to nonlinear sigma models of disordered systems. The last chapter of this thesis may be regarded as a collection of applications of the boundarybulk correspondence described by the general index theorem for spatially onedimensional systems. Specifically onedimensional superconductor models in the other three nontrivial cases are constructed as generalizations of the Kitaev’s model that is one of four nontrivial cases, and every model is solved in detail by methods similar to that provided by Kitaev. Then we analyze each model in the framework of the general index theorem, focusing on the topological properties of Majorana zeromodes with codimension zero at the ends of these models under the open boundary condition. The possible applications of these models to universal quantum manipulations are also discussed.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.subject.lcsh  Fermions   
dc.title  Topological theory of gapless and gapped fermionic systems   
dc.type  PG_Thesis   
dc.identifier.hkul  b5312335   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Physics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5312335   