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

Non-Euclidean geometry, nontrivial topology and quantum vacuum effects