Edge States: Topological Insulators and Superconductors
Abstract: Let M be a manifold with boundary ∂M. Let it be occupied by a superconductor with ( London) mass m > 0 for the electromagnetic field. We show that there are low-lying edge states with energies small compared to the energy gap m in the bulk . The system preserves parity (P- )and time reversal(T-) invariance. An associated set of edge states occurs for the Dirac operator D for mass m. Its self- adjointness requires an Atiyah-Patodi-Singer (APS) boundary condition. When it is appropriately chosen, D also has low-lying P -and T-invariant edge states, the spinor analogues of the electromagnetic edge states. Further there is spin- momentum locking. All this suggests that these edge states may provide a description of the edge states of topological insulators. The edge states above occur in all spatial dimensions.
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