Bulk-edge correspondence

Topological phase is a new concept, that is independent of the standard phase classification by symmetry breaking, and is characterized by quantized quantities (topological numbers) of the gapped bulk which may not be always experimentally observable. Physical quantities to be measured are gapless excitations (edge states) near the boundaries of the systems. These low energy modes are stable against local perturbations protected by the bulk gap. This is the topological stability. The relation between the topological number of the bulk and the edge states is the bulk-edge correspondence. It has been first established for quantum Hall states. Today not only quantum but also many classical phenomena are recognized as under the control of the bulk-edge correspondence. Universality of the bulk-edge correspondence is extending rapidly over different areas. We also mentioned on the unique feature of the bulk-edge correspondence in topological pump and the extension for symmetry protected topological phases.