Unraveling the Hyperfine Structure of Entanglement with the Decomposition of R\'enyi Contour

Entanglement contour and R\'{e}nyi contour reflect the real-space distribution of entanglement entropy, serving as the fine structure of entanglement. In this work, we unravel the hyperfine structure by rigorously decomposing R\'{e}nyi contour into the contributions from particle-number cumulants. We show that the hyperfine structure, introduced as a quantum-information concept, has several properties, such as additivity, normalization, symmetry, and unitary invariance. To extract the underlying physics of the hyperfine structure, we numerically study lattice fermion models with mass gap, critical point, and Fermi surface, and observe that different behaviors appear in the contributions from higher-order particle-number cumulants. We also identify exotic scaling behaviors in the case of mass gap with nontrivial topology, signaling the existence of topological edge states. In conformal field theory (CFT), we derive the dominant hyperfine structure of both R\'{e}nyi entropy and refined R\'{e}nyi entropy. By employing the AdS$_3$/CFT$_2$ correspondence, we find that the refined R\'{e}nyi contour can be holographically obtained by slicing the bulk extremal surfaces. The extremal surfaces extend outside the entanglement wedge of the corresponding extremal surface for entanglement entropy, which provides an exotic tool to probe the hyperfine structure of the subregion-subregion duality in the entanglement wedge reconstruction. This paper is concluded with an experimental protocol and interdisciplinary research directions for future study.