Symmetry-enforced agreement of Kohn--Sham and many-body Berry phases in the SSH--Hubbard chain

Abstract

We study when a density-matching Kohn--Sham (KS) description can reproduce a many-body Berry phase in a correlated insulator, despite the fact that geometric phases are functionals of the wave function. Focusing on the one-dimensional SSH--Hubbard chain on a ring as a controlled interacting topological model, we introduce a $U(1)$ twist $\theta$ (flux insertion). The many-body ground state along the full twist cycle is computed by the density-matrix renormalization group (DMRG), while the onsite interaction $U$ is tuned from the noninteracting to the strong-coupling regime. At half filling in the inversion-symmetric gapped regime, our DMRG calculations show that the density remains constant within numerical accuracy over the entire $(\theta,U)$ range studied. Thus, the density has no dependence on either the flux $\theta$ or the interaction strength $U$. Accordingly, the symmetry-preserving density constraint collapses the KS reference to an SSH-type quadratic representative with $U$-independent geometric diagnostics. Nevertheless, the many-body wave function exhibits a nontrivial geometric response: the quantum metric associated with the $\theta$-parametrized ground-state manifold depends on $\theta$ at intermediate $U$ and is strongly suppressed at large $U$, consistent with the charge fluctuation freezing. Intriguingly, the KS and many-body Berry phases coincide throughout the gapped regime as $U$ is tuned from weak to strong coupling. We show that this agreement is best understood as symmetry-enforced $\mathbb{Z}_2$ sector matching, rather than as evidence that the density encodes the many-body Berry connection.