Universal fingerprint of topological defect cores
Topological defects are ubiquitous in physics, arising from condensed matter physics to the early universe. Although there exist many universal scaling laws for correlations between topological defects, such as Porod scaling, the fingerprint of topological defect cores has remained largely unexplored. Here, we discover a universal scaling law in the region $k>1/\xi$, where $\xi$ is the healing length of the topological defects, taking the scaling of the form factor $S_f \propto k^{-(d+p+2)}$, where $d$ is the spatial dimension and $p$ is the defect codimension. We analytically prove that this exponent originates from a universal V-shaped cusp at the defect core and is independent of the underlying system and dynamics. Numerical simulations verify this scaling law in four typical frameworks: the time-dependent Ginzburg-Landau and Gross-Pitaevskii equations in the weak-coupling regime, the gauge/gravity duality model in the strong-coupling regime, and the Klein-Gordon equation in the Friedmann-Robertson-Walker background in cosmology. Our work provides a new probe for studying topological defects in systems ranging from superconductors to cosmological phase transitions.