V-Representability Beyond Singular Times in Time-Dependent Density Functional Theory: The Hubbard Dimer Case

Abstract

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.

The paper addresses whether an interacting density trajectory can be reproduced by non-interacting Kohn–Sham dynamics after density-to-potential inversion becomes singular. It treats the two-electron Hubbard dimer restricted to the spin-singlet sector, for conventional TDDFT with a real potential (VP) and a geometric formulation with an imaginary effective potential (GP), including a version with a prescribed real potential (GPc). For every finite interval and external control $u\in L^1$, global representability is proved: VP requires matching the initial density and current, whereas GP/GPc requires only matching the initial density. A compatible VP initial Slater state always exists. Complete necessary-and-sufficient conditions for general density trajectories and classifications of all reproducing trajectories and potentials are also given. For interacting densities, singular times are finite. VP continuations have binary branch choices; geometric continuations have angular parameters. The potentials satisfy $|v|\le\sqrt{2}|u|+\tfrac12\sqrt{U_1^2+U_2^2}$ and $|w|\sqrt{\rho_1\rho_2}\le2\tau$. The proofs reconstruct trajectories on the Bloch sphere and use a density–current discriminant bound to control the apparent VP singularity. A separate analysis shows that geometric potentials generically diverge as $1/|t-t_z|$ when a site density vanishes, even for smooth controls; global geometric representability therefore uses measurable potentials with weighted integrability. Exceptional branches permit integrable potentials under an additional Dini-type condition. Bang-bang-control examples illustrate the results. The theorems are restricted to this dimer sector and do not establish global representability for larger lattices or continuum systems.