An Exact Conjugation Identity for the Many-Body Wilson-Loop Beyond Quantization

Constraints on the unquantized many-body holonomy are less explored than their quantized counterparts. Here we realize an unquantized regime by tuning the bond dimerization $\delta$ and the staggered potential $\Delta$ in a dimerized staggered Hubbard ring at half filling. For the tuned parameter sets, a finite excitation gap persists along the $U(1)$ twist cycle $\theta\in[0,2\pi]$, so that the ground state $|\psi_{\delta}(\theta)\rangle$ is separated from the excited states. The many-body Wilson loop is therefore well defined from the ground-state family $\{|\psi_{\delta}(\theta)\rangle;\,\theta\in[0,2\pi]\}$. In this setup, we show an exact many-body Wilson loop conjugation identity, $W(-\delta)=W(\delta)^*$, accumulated along a cycle parametrized by $\theta$. Importantly, the identity persists in regimes where the Berry phase $\gamma\equiv-\arg W$ varies continuously. We demonstrate the identity numerically using the density-matrix renormalization group (DMRG) method. The identity extends to other models where the flux-threaded ground-state family along the closed $\theta$-cycle is mapped to the reversed cycle. More generally, the identity can be viewed as a Wilson-loop-level constraint that contains the Berry phase pinning as a fixed-point corollary. Beyond its conceptual content, the identity provides a symmetry-based consistency check for numerical evaluations of Berry phases in interacting systems. It also justifies the signal-to-noise ratio improvement in Monte Carlo simulations by performing simulations at both $\delta$ and $-\delta$ and averaging $W(\delta)$ with $W(-\delta)^{*}$.