{"title":"准自由费米子的\\({\\mathbb {Z}}_{2}\\) -拓扑指数","authors":"N. J. B. Aza, A. F. Reyes-Lega, L. A. M. Sequera","doi":"10.1007/s11040-022-09421-w","DOIUrl":null,"url":null,"abstract":"<div><p>We use infinite dimensional self-dual <span>\\(\\mathrm {CAR}\\)</span> <span>\\(C^{*}\\)</span>-algebras to study a <span>\\({\\mathbb {Z}}_{2}\\)</span>-index, which classifies free-fermion systems embedded on <span>\\({\\mathbb {Z}}^{d}\\)</span> disordered lattices. Combes–Thomas estimates are pivotal to show that the <span>\\({\\mathbb {Z}}_{2}\\)</span>-index is uniform with respect to the size of the system. We additionally deal with the set of ground states to completely describe the mathematical structure of the underlying system. Furthermore, the weak<span>\\(^{*}\\)</span>-topology of the set of linear functionals is used to analyze paths connecting different sets of ground states.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A \\\\({\\\\mathbb {Z}}_{2}\\\\)-Topological Index for Quasi-Free Fermions\",\"authors\":\"N. J. B. Aza, A. F. Reyes-Lega, L. A. M. Sequera\",\"doi\":\"10.1007/s11040-022-09421-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We use infinite dimensional self-dual <span>\\\\(\\\\mathrm {CAR}\\\\)</span> <span>\\\\(C^{*}\\\\)</span>-algebras to study a <span>\\\\({\\\\mathbb {Z}}_{2}\\\\)</span>-index, which classifies free-fermion systems embedded on <span>\\\\({\\\\mathbb {Z}}^{d}\\\\)</span> disordered lattices. Combes–Thomas estimates are pivotal to show that the <span>\\\\({\\\\mathbb {Z}}_{2}\\\\)</span>-index is uniform with respect to the size of the system. We additionally deal with the set of ground states to completely describe the mathematical structure of the underlying system. Furthermore, the weak<span>\\\\(^{*}\\\\)</span>-topology of the set of linear functionals is used to analyze paths connecting different sets of ground states.</p></div>\",\"PeriodicalId\":694,\"journal\":{\"name\":\"Mathematical Physics, Analysis and Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-03-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Physics, Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11040-022-09421-w\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-022-09421-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A \({\mathbb {Z}}_{2}\)-Topological Index for Quasi-Free Fermions
We use infinite dimensional self-dual \(\mathrm {CAR}\)\(C^{*}\)-algebras to study a \({\mathbb {Z}}_{2}\)-index, which classifies free-fermion systems embedded on \({\mathbb {Z}}^{d}\) disordered lattices. Combes–Thomas estimates are pivotal to show that the \({\mathbb {Z}}_{2}\)-index is uniform with respect to the size of the system. We additionally deal with the set of ground states to completely describe the mathematical structure of the underlying system. Furthermore, the weak\(^{*}\)-topology of the set of linear functionals is used to analyze paths connecting different sets of ground states.
期刊介绍：
MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas.
The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process.
The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed.
The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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