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arXiv:1011.2715 · Physical Review B

Time-dependent current density functional theory on a lattice

I. V. Tokatly

Published Jan 31, 2011• 51 citations
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Abstract

A rigorous formulation of time-dependent current density functional theory (TDCDFT) on a lattice is presented. The density-to-potential mapping and the $\mathcal{V}$-representability problems are reduced to a solution of a certain nonlinear lattice Schr\"odinger equation, to which the standard existence and uniqueness results for nonlinear differential equations are applicable. For two versions of the lattice TDCDFT, we prove that any continuous-in-time current density is locally $\mathcal{V}$-representable (both interacting and noninteracting), provided that in the initial state the local kinetic energy is nonzero everywhere. In most cases of physical interest, the $\mathcal{V}$-representability should also hold globally in time. These results put the application of TDCDFT to any lattice model on a firm foundation, and pave the way for studying exact properties of exchange-correlation potentials.

DOI
10.1103/physrevb.83.035127

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A rigorous formulation of time-dependent current density functional theory (TDCDFT) on a lattice is presented. The density-to-potential mapping and the V\mathcal{V}V-representability problems are reduced to a solution of a certain nonlinear lattice Schr"odinger equation, to which the standard existence and uniqueness results for nonlinear differential equations are applicable. For two versions of the lattice TDCDFT, we prove that any continuous-in-time current density is locally V\mathcal{V}V-representable (both interacting and noninteracting), provided that in the initial state the local kinetic energy is nonzero everywhere. In most cases of physical interest, the V\mathcal{V}V-representability should also hold globally in time. These results put the application of TDCDFT to any lattice model on a firm foundation, and pave the way for studying exact properties of exchange-correlation potentials.
Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is vvv representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time vvv representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.
This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.
This chapter discusses time-dependent current-density-functional theory (TDCDFT). It first asks under what circumstances one should choose the current density as basic variable, instead of the particle density. There are several reasons: a current-based formulation is more suitable for constructing functionals beyond the adiabatic approximation. Furthermore, the current is the natural variable to describe coupling to electromagnetic fields and vector potentials. The basic existence theorems of TDCDFT are proved, and several exact properties and the linear-response regime are discussed. The second half of the chapter deals with the so-called Vignale-Kohn functional, which is a local and frequency-dependent approximation to the exchange-correlation vector potential. The Vignale-Kohn functional is expressed in terms of viscoelastic stresses in the electron liquid. Several applications in the linear-response regime and in real time are presented. The Vignale-Kohn functional works well for describing dissipative effects in collective, plasmon-like electron dynamics.
We study the mapping between time-dependent densities and potentials for noninteracting electronic systems on lattices. As discovered recently by Baer [J. Chem. Phys. 128, 044103 (2008)], there exist well-behaved time-dependent density functions on lattices which cannot be associated with any real time-dependent potential. This breakdown of time-dependent V-representability can be tracked down to problems with the continuity equation which arise from discretization of the kinetic-energy operator. Examples are given for lattices with two points and with N points, and implications for practical numerical applications of time-dependent density-functional theory are discussed. In the continuum limit, time-dependent noninteracting V-representability is restored.
The formally exact expression of the exchange-correlation potential in the local-scaling density functional theory (LSDFT) was derived recently by the author, and it is equivalent to that of Holas and March. The basic equation of the LS-DFT was also derived, and it is equivalent to the differential virial theorem (DVT). The LS-DFT has been extended to the local-scaling current density functional theory (LS-CDFT). This has been compared with the formulation of Holas and March. In this note, the LS-CDFT is extended to the timedependent system; i.e., the local-scaling time-dependent current density functional theory (LS-TDCDFT) is formulated. As a straightforward extension of ref. 3, the density and the paramagnetic current density jp are adopted as the basic variables, which is the same procedure used by Colwell et al. It is explicitly shown that the time-dependent exchange-correlation scalar potential vxc and the vector one Axc are same functionals of firstand second-order density matrices as those of the time-independent case, while the total effective potentials are not so. The equation of motion of jp provides the DVT for the time-dependent system in the magnetic field. The time-dependent term of the DVT is equal to that of the nonmagnetic time-dependent system. Throughout the entire formulation, the notation of the tensor product is suppressed. Here, an N-electron system in an external scalar potential vextðr; tÞ, an external vector potential Aextðr; tÞ, and an interaction potential uðr; r0Þ 1⁄4 =jr r0j is considered, where is the coupling constant treated as a variable parameter. In a time-independent theory, the quantity to be minimized is the energy. But in the time-dependent case, the action S plays the role: