Time-dependent bond-current functional theory for lattice Hamiltonians: Fundamental theorem and application to electron transport

Graphical abstract: A time-dependent bond-current functional theory for lattice systems is formulated. For a biased impurity in the Coulomb blockade regime, the discontinuity of the exchange- correlation potential can prevent the evolution towards a steady state. Research highlights: ► Time-dependent density functional theory (TDDFT) for lattice Hamiltonians is studied. ► Using Peierl's phases we prove the fundamental theorem of TDDFT on lattices. ► A lattice model for transport is studied with a discontinuous Kohn–Sham potential. ► The discontinuity is crucial in the Coulomb blockade regime. ► It may prevent the biased system from reaching a steady state. - Abstract: 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–Gross 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.