Time-dependent density functional theory on a lattice

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 $v$ 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 locally in time $v$ 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.

Figure 1

Each normalized state in the Hilbert space $\mathcal{H}$ maps to a point on the Bloch sphere. The north $|1\rangle$ and the south $|2\rangle$ poles correspond to the particle on sites 1 and 2. The line $k_{12}=0$ divides the sphere into two (left and right) hemispheres corresponding to two disconnected parts of the $v$-representability subset $\Omega$.