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In this work, persistent currents in rings made of band insulators areanalyzed theoretically. We first formulate a recipe which determines the Blochstates of a one-dimensional (1D) ring from the Bloch states of an infinite 1Dcrystal created by the periodic repetition of the ring. Using the recipe, wederive an expression for the persistent current in a 1D ring made of aninsulator with an arbitrary valence band E(k). To find an exact result for aspecific insulator, we consider a 1D ring represented by a periodic lattice ofN identical sites with a single value of the on-site energy. If the Blochstates in the ring are expanded over a complete set of N on-site Wannierfunctions, the discrete on-site energy splits into the energy band. At fullfilling, the band emulates the valence band of the band insulator and the ringis insulating. It carries a persistent current equal to the product of N andthe derivative of the on-site energy with respect to the magnetic flux. Thiscurrent is not zero if one takes into account that the on-site Wannier functionand consequently the on-site energy of each ring site depend on magnetic flux.To derive the current analytically, we expand all N Wannier functions of thering over the infinite basis of Wannier functions of the constituting infinite1D crystal and eventually determine the crystal Wannier functions by a methodof localized atomic orbitals. Finally, we estimate the persistent current atfull filling in rings made of real band insulators (GaAs, Ge, InAs). Thecurrent decays with the ring length exponentially due to the exponential decayof the Wannier functions. In spite of that, it can be of measurable size.

Bloch-Wannier theory of persistent current in a ring made of the band  insulator: Exact result for one-dimensional lattice and estimates for  realistic lattices