  Space out of a topological defect of the Abrikosov-Nielsen-Olesen vortex type
is locally flat but non-Euclidean. If a spinor field is quantized in such a
space, then a variety of quantum effects is induced in the vacuum. Basing on
the continuum model for long-wavelength electronic excitations, originating in
the tight-binding approximation for the nearest neighbor interaction of atoms
in the crystal lattice, we consider quantum ground state effects in monolayer
structures warped into nanocones by a disclination; the nonzero size of the
disclination is taken into account, and a boundary condition at the edge of the
disclination is chosen to ensure self-adjointness of the Dirac-Weyl Hamiltonian
operator. In the case of carbon nanocones, we find circumstances when the
quantum ground state effects are independent of the boundary parameter and the
disclination size.
