  The cornerstone of time-dependent (TD) density functional theory (DFT), the
Runge-Gross theorem, proves a one-to-one correspondence between TD potentials
and TD densities of continuum Hamiltonians. In all practical implementations,
however, the basis set is discrete and the system is effectively described by a
lattice Hamiltonian. We point out the difficulties of generalizing the
Runge-Groos proof to the discrete case and thereby endorse the recently
proposed TD bond-current functional theory (BCFT) as a viable alternative.
TDBCFT is based on a one-to-one correspondence between TD Peierl's phases and
TD bond-currents of lattice systems. We apply the TDBCFT formalism to
electronic transport through a simple interacting device weakly coupled to two
biased non-interacting leads. We employ Kohn-Sham Peierl's phases which are
discontinuous functions of the density, a crucial property to describe Coulomb
blockade. As shown by explicit time propagations, the discontinuity may prevent
the biased system from ever reaching a steady state.
