On exotic sphere fibrations, topological phases, and edge states in physical systems

Abstract

We suggest that exotic sphere fibrations can be mapped to band topologies in condensed matter systems. These fibrations can correspond to geometric phases of two double bands or state vector bases with second Chern numbers m+n and -n respectively. They can be related to topological insulators, magneto-electric effects, and photonic crystals with special edge states. We also consider time-reversal symmetry breaking perturbations of topological insulator, and heterostructures of topological insulators with normal insulators and with superconductors. We consider periodic TI/NI/TI/NI' heterostuctures, and periodic TI/SC/TI/SC' heterostuctures. They also give rise to models of Weyl semimetals which have thermal and electrical transports.

23 pages; journal version

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• Weyl semimetals with short-range interactions
• Finite-size-effect-induced topological phase transition in a topological crystalline insulator
• Differential topology of semimetals
• Weyl Semimetal in the Strong Coulomb Interaction Limit
• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII