We study the topological magnetoelectric effect on a conical topological insulator when a point charge $q$ is near the cone apex. The Hall current induced on the cone surface and the image charge configuration are determined. We also study a kind of
gravitational Aharonov-Bohm effect in this geometry and realize a phase diference betwen the components of the wavefunctions (spinors) upon closed parallel transport around the (singular) cone tip. Concretely, a net current flowing towards cone apex (or botton) shows up, yielding electric polarization of the conical topological insulator. Such an effect may be detected, for instance, by means of the net accumulated Hall charge near the apex. Once it depends only on the geometry of the material (essetially, the cone apperture angle) this may be faced as a microscopic scale realization of (2+1)-dimensional Einstein gravity.
Magnetization ground states are studied in toroidal nanomagnets. The energetics associated to the ferromagnetic, vortex and onion-like configurations are explicitly computed. The analysis reveals that the vortex appears to be the most prominent of su
ch states, minimizing total energy in every torus with internal radius $rgtrsim10,{rm nm}$ (for Permalloy). For $rlesssim10,{rm nm}$ the vortex remains the most favorable pattern whenever $R/ell_{ex}gtrsim1.5$ ($R$ is the torus external radius and $ell_{ex}$ is the exchange length), being substituted by the ferromagnetic state whenever $R/ell_{ex}lesssim1.5$.
