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 $θ$ (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 $(θ,U)$ range studied. Thus, the density has no dependence on either the flux $θ$ 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 $θ$-parametrized ground-state manifold depends on $θ$ 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.

Figures (5)

• Figure 1: Minimum excitation gap along the twist cycle. For each $U$, we evaluate $\Delta_{\mathrm{MB}}^{\min}(U)\equiv \min_{\theta_j}\Delta_{\mathrm{MB}}(\theta_j,U)$ on the discretized twist grid and plot it as a function of $U$.
• Figure 2: Quantum metric $L^2\langle g_{\theta\theta}\rangle(U)$ as function of $U$. Here $\langle\cdots\rangle_{\rm unwrap}$ denotes the mean over the discretized twist links excluding the wrap-around link $\theta_{N_\theta-1}\to\theta_0$.
• Figure 3: Heatmap of the geometric response $g_{\theta\theta}(\theta,U)$, normalized by the unwrap average $\langle g_{\theta\theta}\rangle_{\rm unwrap}(U)$ at each fixed $U$. The normalization is introduced solely for visualization purposes, to improve the visibility of the $\theta$ dependence. We restrict to the weak-coupling regime $U\le 5$, since at larger $U$ the $\theta$ variation becomes too small to be visually resolved on the same scale. Additional $U$ points are included to sharpen the $U$ dependence of the pattern: $U=0.6,\,0.9,\,1.2,\,1.5,\,1.8,\,2.1,\,2.4,\,2.5,\,2.7,\,3.5,\,4.5,\,5.0$.
• Figure 4: Heatmap of the KS geometric response $g^{\mathrm{KS}}_{\theta\theta}(\theta,U)$, normalized by the unwrap average $\langle g^{\mathrm{KS}}_{\theta\theta}\rangle_{\rm unwrap}(U)$ at each fixed $U$. The normalization is introduced solely for visualization purposes, to improve the visibility of the $\theta$ dependence. We restrict to the weak-coupling regime $U\le 5$, since at larger $U$ the $\theta$ variation becomes too small to be visually resolved on the same scale.
• Figure 5: $\cos(\gamma)$ as a function of $U$ for the interacting many-body state (MB, DMRG) and the Kohn--Sham (KS) reference. Filled circles denote MB results, while open squares denote KS results. The horizontal line at $y=1$ serves as a visual guide.