ИСТИНА

Consider a graphene tube in the shape of a right circular open-ended cylinder whose height and radius are both much greater than the distance between neighboring carbon atoms. A magnetic field everywhere vanishing on the tube surface (or at least tangent to it) is adiabatically switched on, causing the eigenvalues of the tight-binding Hamiltonian describing the electron $\pi$ states in graphene in the nearest neighbor approximation to move in the process. If the magnetic flux through the tube in the final configuration has an integer number of flux quanta, then the electron energy spectrum eventually returns to the original position. Moreover, the spectral flow of the Hamiltonian, defined as the net number of eigenvalues (counted with multiplicities) which pass through $0$ in the positive direction, proves to be zero. A more detailed analysis reveals that the eigenfunctions corresponding to these eigenvalues can be chosen to be localized near the Dirac points $K$ and $K^\prime$ in the momentum space, and if one separately counts the spectral flow for the eigenfunctions localized near $K$ and near $K^\prime$, then one obtains two ``partial spectral flows, which have opposite signs and the same modulus (equal to the number of magnetic flux quanta through the tube). The physical interpretation is that switching on the magnetic field creates electron--hole pairs (or, more precisely, pairs of ``electron and ``hole energy levels) in graphene, the number of pairs being determined by the magnetic flux. If an electron level is created near $K$, then the corresponding hole level is created near $K^\prime$, and vice versa. Further, the number of electron/hole levels created near $K$ equals the spectral flow of the family of Dirac operators approximating the tight-binding Hamiltonian near $K$ (and the same is true with $K$ replaced by $K^\prime$). We assign a precise mathematical meaning to the notion of partial spectral flow in such a way that all the preceding assertions make rigorous sense. This is joint work with M.I. Katsnelson and J.Brüning.