An Exact Conjugation Identity for the Many-Body Wilson Loop Beyond Quantization
Constraints on unquantized many-body holonomies, such as Wilson loops (or their Berry phases), are less explored than those on their quantized counterparts. Here, we realize an unquantized regime by tuning the bond dimerization $δ$ and the staggered potential $Δ$ in a half-filled dimerized staggered Hubbard ring. For the tuned parameter sets, a finite excitation gap persists along the $U(1)$ twist cycle $θ\in[0,2π]$, so that the ground state $|ψ_δ(θ)\rangle$ remains separated from the excited states. The many-body Wilson loop is therefore well defined from the ground-state family $\{|ψ_δ(θ)\rangle;\,θ\in[0,2π]\}$. In this setup, we show an exact many-body Wilson loop conjugation identity, $W(-δ)=W(δ)^*$, accumulated along a cycle parametrized by $θ$. Importantly, the identity persists in regimes where the Berry phase $γ\equiv-\arg W$ 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 $θ$-cycle is mapped to the reversed cycle. 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 $δ$ and $-δ$ and averaging $W(δ)$ with $W(-δ)^{*}$.