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

Abstract

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 $\delta$ and the staggered potential $\Delta$ in a half-filled dimerized staggered Hubbard ring. 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 $|{\psi }_{\delta }(\theta )⟩$ remains separated from the excited states. The many-body Wilson loop is therefore well defined from the ground-state family $\{|{\psi }_{\delta }(\theta )⟩;\theta \in [0,2\pi ]\}$. In this setup, we show an exact many-body Wilson loop conjugation identity, $W(-\delta )=W(\delta {)}^{\ast }$, 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. 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 $W(-\delta {)}^{\ast }$.

arXiv comment4 pages, 2 figures. v2: Improved readability of the manuscript and added Supplemental Material as an ancillary file. v3 : Updated for the SCES 2026 proceedings. The Supplemental Material was also revised to provide a rigorous Slater-determinant treatment of the periodicity of the $\theta$ cycle within the band