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

Abstract

We establish an exact Wilson-loop conjugation identity, $W(-δ)=W(δ)^*$, for the many-body overlap Wilson-loop $W(δ)$ accumulated along a $U(1)$ flux-threading (twist) cycle parametrized by $θ\in[0,2π]$, where $δ$ denotes the bond-dimerization parameter. A minimal sufficient condition is the existence of a composite antiunitary mapping acting on the flux-threaded Hamiltonian family that implements $(δ,θ)\mapsto(-δ,-θ)$. As a concrete demonstration, we construct such a mapping microscopically in a dimerized staggered Hubbard ring at half filling. We then verify the conjugation identity using the density-matrix renormalization group (DMRG) for gapped, nondegenerate ground states along the twist cycle. Importantly, the identity persists in depinned gapped regimes where the Berry-phase $γ\equiv-\arg W$ is not symmetry-quantized; as a corollary, $γ(-δ)=-γ(δ)$ (mod $2π$). More generally, the same conjugation relation applies to any lattice model whose flux-threaded Hamiltonian family is closed under an orientation reversal of the bond pattern (a suitable permutation of link-hopping parameters) combined with a reversal of the flux orientation.

Figures (2)

• Figure 1: Complex-plane plots of the many-body Wilson-loop $W(\delta)$ and its normalized version $\widetilde{W}(\delta)\equiv W(\delta)/|W(\delta)|$ for representative $\theta$-grid $N_\theta=48$. Panels (a,b) [top row]: $U=6.0$; panels (c,d) [bottom row]: $U=7.0$. Markers indicate $\delta\in\{-0.01,-0.005,0,0.005,0.01\}$. Marker shape and color encode $|\delta|$ (shared by $\pm\delta$), and filled (open) markers indicate $\delta>0$ ($\delta<0$). Dashed circles are shown as guides to the eye.
• Figure 2: Berry-phase $\gamma(\delta)\equiv-\arg W(\delta)$ plotted versus $1/N_\theta$. Top (bottom) panel: $U=6.0$ ($U=7.0$). The data points are for $\delta\in\{-0.01,-0.005,0,0.005,0.01\}$ and $N_\theta\in\{8,12,16,20,24,32,40,48\}$. Dashed curves indicate quadratic fits in $1/N_\theta$ to each data set. Marker shape and color encode $|\delta|$ (shared by $\pm\delta$), and filled (open) markers indicat e $\delta>0$ ($\delta<0$).