The toroidal dipole operator in nanostructures

The parity violation in nuclear reactions led to the discovery of new class toroidal multipoles. Since then, it was observed that multipoles are present electromagnetic structure systems at all scales, from elementary particles, solid state and metamaterials. dipole ${\bf T}$ (the lowest order multipole) is most common. In quantum systems, this corresponds operator $\hat{\bf T}$, with projections $\hat{T}_i$ ($i=1,2,3$) on coordinate axes. Here we analyze a particle system cylindrical symmetry, which typical moments appear. We find expressions for Hamiltonian, momenta, operators adequate curvilinear coordinates, allow us analytical eigenfunctions momentum operators. While hermitian, not self-adjoint, but set coordinates $\hat{T}_3$ splits into two components, one (only) whereas other self-adjoint. self-adjoint component physically significant represents an observable. Furthermore, numerically diagonalize Hamiltonian their eigenvalues. write partition function calculate thermodynamic quantities ideal particles torus. Besides proving therefore observable (a finding fundamental relevance) such open up possibility making metamaterials exploit quantization properties dipoles.