We address and quantify the role of nonadiabaticity (``memory effects'') in the exchange-correlation (xc) functional of time-dependent density functional theory (TDDFT) for describing nonlinear dynamics of many-body systems. Time-dependent resonant processes are particularly challenging for available TDDFT approximations, due to their strong nonlinear and nonadiabatic character. None of the known approximate density functionals are able to cope with this class of problems in a satisfactory manner. In this work we look at the prototypical example of the resonant processes by considering Rabi oscillations within the exactly soluble two-site Hubbard model. We construct the exact adiabatic xc functional and show that (i) it does not reproduce correctly resonant Rabi dynamics, and (ii) there is a sizable nonadiabatic contribution to the exact xc potential, which turns out to be small only at the beginning and at the end of the Rabi cycle when the ground-state population is dominant. We then propose a ``two-level'' approximation for the time-dependent xc potential which can capture Rabi dynamics in the two-site problem. It works well both for resonant and for detuned Rabi oscillations and becomes essentially exact in the linear response regime.
We study the global
V-representability problem in time-dependent density functional theory (TDDFT) for the two-particle Hubbard dimer model in the spin-singlet sector. In particular, we investigate whether the time-dependent density of the interacting system obtained for a given one-body external potential
u (the tunable control variable) can be reproduced by a non-interacting Kohn--Sham system beyond the singular times at which the usual inversion formulas break down. We deal with both the conventional formulation of TDDFT based on some time-dependent variational principle and the recently introduced geometric formulation of TDDFT (Phys. Rev. Lett. 136 (2026) 256401, Phys. Rev. A 113 (2026) 062222). For both principles, we prove global-in-time
V-representability, show that singular times induce non-uniqueness of the time-dependent effective potential, and classify the set of solutions.