Density functional theory replaces the many-body wave function — a function of 3N coordinates for N electrons — with the electron density, a function of three coordinates. The Hohenberg-Kohn theorem guarantees that the ground-state density uniquely determines all ground-state properties. In practice, the Kohn-Sham scheme introduces a fictitious system of non-interacting electrons that reproduces the exact density, and all the complexity of electron-electron interactions is buried in the exchange-correlation functional.
This works remarkably well for ground-state energies, band structures, and charge distributions. But geometric phases — Berry phases accumulated as the system is adiabatically transported around a closed loop in parameter space — depend on the wave function's global structure, not just the density. The Berry phase extracts topological information from how the quantum state evolves, which is precisely the kind of information that reducing to a density should destroy.
Watanabe (arXiv 2602.22578, February 2026) investigates this apparent contradiction in the SSH-Hubbard chain — a one-dimensional model that combines the topological structure of the Su-Schrieffer-Heeger model with Hubbard on-site interactions. Using density-matrix renormalization group simulations with varying interaction strength and threaded magnetic flux, the result is striking: the Kohn-Sham Berry phase agrees with the many-body Berry phase across the entire range of interactions, from weak coupling to the strongly correlated regime.
The reason is not that density secretly encodes topology. The density remains constant — within numerical accuracy — across the entire parameter space. The interactions change; the topological sector changes; the density does not budge. The density is topologically inert.
The agreement arises instead from symmetry-enforced Zā sector matching. The symmetries of the Hamiltonian constrain which Kohn-Sham states are accessible as ground states of a non-interacting system reproducing the exact density. The constraint forces the Kohn-Sham reference into the same topological sector as the many-body state — not because the density carries the topological information, but because the symmetry leaves no other option. The agreement is not informational; it's structural. The density doesn't know the topology. The symmetry does.
The quantum metric — which measures how fast the quantum state changes as parameters vary — does show interaction dependence: it's suppressed at strong coupling because charge is frozen. The geometric distance changes even as the topological invariant stays locked. The shape of the path varies; its winding number does not.
The Kohn-Sham theory gets the answer right for the wrong reason. The density is blind to topology. The symmetry fills in what the density can't see.