PPT-Topological Inference

Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course London May 2014 Many thanks to Justin Chumbley Tom Nichols and Gareth Barnes for slides. Syed. Ali . Raza. Supervisor: Dr. . Pervez. . Hoodbhoy. What are Topological insulators?. Fairly recently discovered electronic phases of matter.. Theoretically predicted in 2005 and 2007 by Zhang, . and Beyond. Kai. . Sun. University . of Maryland, College Park. Outline. Topological state of matter. Topological nontrivial structure and topological index. Anomalous quantum Hall state and the . Chern. in a collaboration between the PFC at JQI . and . CalTech. have . shown that it may be possible to take a conventional semiconductor and endow it with topological properties without subjecting the material to extreme environmental conditions or fundamentally changing its solid state structure. . “Topologically . Robust Transport of Photons in a Synthetic Gauge Field. ," S. Mittal, J. Fan, S. . Faez. , A. . Migdall. , J.M. Taylor, M. . Hafezi. , Phys. Rev. . Lett. ., 113, 087403 (2014. ). Topological transport of light is the photonic analog of topological electron flow in certain semiconductors. In the electron case, the current flows around the edge of the material but not through the bulk. It is “topological” in that even if electrons encounter impurities in the material the electrons will continue to flow without losing energy. Not continuously deformable. Topological. Invariant. Topology & Topological Invariant. Number of Holes. Manifold . of wave functions in the . Hilbert space . r. xy. r. xx. Quantum Hall system:. D. Hilbert. Weisong. . Tu. Department of Physics and Astronomy. University of Tennessee. Instructor: Dr. . George . Siopsis. Introduction. Quantum Hall Effect. The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. In the quantum hall effect, and the conductivity can be represented as. Polarization. : . an E. ntanglement Measure . of . Topological Spin and Chiral Central Charge. Xiao-Liang Qi. Stanford University. Banff, 02/06/2013. Reference: . Hong-. Hao. . Tu. , Yi Zhang, Xiao-Liang Qi, . Dimitrie Culcer. D. Culcer, PRB 84, 235411 (2011). . D. Culcer, . Physica. E 44, 860 (2012) – review on TI transport . Outline. Introduction to topological insulators. . and. . Topological. . phases. . of. matter. b. y. Reichmann Alexander. Overview. Phase . transitions. Topology. Quantum Hall . effect. Superconductivity. Applications. Phase . transition. Different . Hafezi. , S. Mittal, J. Fan, A. . Migdall. , J.M. Taylor, Nature Photonics, . (. 2013. ) . doi:10.1038/nphoton.2013.274. . Topology . -- the understanding of how things are connected -- remains abstract, even with the popular example of doughnuts and coffee cups. This concept, esoteric as it appears, is also neat because it is the basis for creating . . Cumrun Vafa. . Oct. 31, 2017. . 20 . Years . Later: The Many Faces of . AdS. /CFT. Princeton University. ECE 580. 12. Graph Theory, Topological Analysis - Terms. Topological Analysis: General, systematic, suited for CAD. Graph. : Nodes and directed branches, describes the topology of the circuit, ref. direction. Helps visualize CAD. Graph Traversals. Spring 2015. Yanling He. Graphs. A Graph G = (V, E). Represents relationships among items. Can be directed or undirected. Complexity is O(|E|+|V|) is O(|V|^2). Graph Data Structure. . - Insulating State, Topology and Band Theory. . II. Band Topology in One Dimension. . - Berry phase and electric polarization. - Su Schrieffer . Heeger. model : . domain wall states and .