  In this work, persistent currents in rings made of band insulators are
analyzed theoretically. We first formulate a recipe which determines the Bloch
states of a one-dimensional (1D) ring from the Bloch states of an infinite 1D
crystal created by the periodic repetition of the ring. Using the recipe, we
derive an expression for the persistent current in a 1D ring made of an
insulator with an arbitrary valence band E(k). To find an exact result for a
specific insulator, we consider a 1D ring represented by a periodic lattice of
N identical sites with a single value of the on-site energy. If the Bloch
states in the ring are expanded over a complete set of N on-site Wannier
functions, the discrete on-site energy splits into the energy band. At full
filling, the band emulates the valence band of the band insulator and the ring
is insulating. It carries a persistent current equal to the product of N and
the derivative of the on-site energy with respect to the magnetic flux. This
current is not zero if one takes into account that the on-site Wannier function
and 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 the
ring over the infinite basis of Wannier functions of the constituting infinite
1D crystal and eventually determine the crystal Wannier functions by a method
of localized atomic orbitals. Finally, we estimate the persistent current at
full filling in rings made of real band insulators (GaAs, Ge, InAs). The
current decays with the ring length exponentially due to the exponential decay
of the Wannier functions. In spite of that, it can be of measurable size.
