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A short-range part of the Coulomb interaction causes splitting and shift of excitons due to exchange interaction and mixing between different valleys in semiconducting carbon nanotubes. In the absence of a magnetic flux only a single exciton is optically active (bright) and all others are inactive (dark). Two bright excitons appear in the presence of an Aharonov- Bohm magnetic flux.

Aharonov-Bohm effects on bright and dark excitons in carbon nanotubes