Event № 881
Graphene is an allotrope of carbon consisting of a single layer of carbon atoms that are densely packed in a honeycomb crystal lattice. Suppose that one take a graphene sheet with holes and start to change the magnetic flux through the holes. When the changes of the magnetic fluxes become integer, the energetic spectrum as a whole should return to its initial state. However, the individual eigenenergies are not necessarily periodic; they can cross the boundary between hole and electron states. The number of such crossings (counted with sign) is called the spectral flow. The situation of non-zero spectral flow is important for physicists and is called the Aharonov-Bohm effect.
Single-layer graphene is described by the Dirac operator acting on a two-or four-component spinor. There is also a bilayer form of graphene, which is described by self-adjoint differential operators of non-Dirac type. In the talk I will show how the spectral flow can be computed, using topological methods, for both single-layer and bilayer graphene (and, more generally, for a one-parameter family of arbitrary first order self-adjoint elliptic operators over a compact surface with classical boundary conditions). The talk is based on my papers arXiv:1108.0806 and arXiv:1703.06105.