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

Abstract

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 $δ$ and the staggered potential $Δ$ 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 $θ\in[0,2π]$, so that the ground state $|ψ_δ(θ)\rangle$ is 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. 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 $δ$ and $-δ$ and averaging $W(δ)$ with $W(-δ)^{*}$.