The recent discovery of three-dimensional（3D） topological insulators（TIs） has provided a fertile ground for obtaining further insights into electron localization in condensed matter systems.In the past few years,a...The recent discovery of three-dimensional（3D） topological insulators（TIs） has provided a fertile ground for obtaining further insights into electron localization in condensed matter systems.In the past few years,a tremendous amount of research effort has been devoted to investigate electron transport properties of 3D TIs and their low dimensional structures in a wide range of disorder strength,covering transport regimes from weak antilocalization to strong localization.The knowledge gained from these studies not only offers sensitive means to probe the surface states of 3D TIs but also forms a basis for exploring novel topological phases.In this article,we briefly review the main experimental progress in the study of the localization in 3D TIs,with a focus on the latest results on ultrathin TI films.Some new transport data will also be presented in order to complement those reported previously in the literature.展开更多
Sufficient conditions were given to assert that between any two Banach spaces over K, Fredholm mappings share at least one .value in a specific open ball. The proof of the result is constructive and based upon continu...Sufficient conditions were given to assert that between any two Banach spaces over K, Fredholm mappings share at least one .value in a specific open ball. The proof of the result is constructive and based upon continuation methods.展开更多
This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a du...This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.展开更多
基金Project supported by the National Basic Research Program of China(Grant Nos.2012CB921703 and 2015CB921102)the National Natural Science Foundation of China(Grant Nos.61425015,11374337,and 91121003)the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant No.XDB070202)
文摘The recent discovery of three-dimensional（3D） topological insulators（TIs） has provided a fertile ground for obtaining further insights into electron localization in condensed matter systems.In the past few years,a tremendous amount of research effort has been devoted to investigate electron transport properties of 3D TIs and their low dimensional structures in a wide range of disorder strength,covering transport regimes from weak antilocalization to strong localization.The knowledge gained from these studies not only offers sensitive means to probe the surface states of 3D TIs but also forms a basis for exploring novel topological phases.In this article,we briefly review the main experimental progress in the study of the localization in 3D TIs,with a focus on the latest results on ultrathin TI films.Some new transport data will also be presented in order to complement those reported previously in the literature.
基金Project supported by D.G.E.S. Pb 96-1338-CO 2-01 and the Junta de Andalucia
文摘Sufficient conditions were given to assert that between any two Banach spaces over K, Fredholm mappings share at least one .value in a specific open ball. The proof of the result is constructive and based upon continuation methods.
基金supported by the National Natural Science Foundation of China(Nos.11371093,11261062,11471167)
文摘This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.
