Symmetry breaking from conformality loss on holographic Rényi defects

The Rényi entropies of a ball are computed by a conformal defect, the Rényi defect, dual in holography to a hyperbolic black hole. At an index $n_*$ its charged operator merges with its alternate-quantization partner, and the defect loses conformality. Beyond $n_*$ it flows with no source to a conformal defect that breaks the U(1) symmetry; below $n_*$ that defect needs a symmetry-breaking source, so at large $N$, conformality loss makes the breaking spontaneous. The scalar hair beyond $n_*$, expected to break the symmetries of hyperbolic space, thus preserves them outside a thin layer at the entangling surface, as conformal symmetry requires. The region outside the layer sets the universal term of the Rényi entropy. We demonstrate the mechanism in the probe Abelian Higgs model in AdS$_5$ at the Bogomolny point, where the saddles are in closed form and numerical methods confirm the flows. On a sphere we prove that a positive source-free flow exists exactly when $n>n_*$ and the sphere is large enough; it is then unique and ends at the symmetry-breaking defect. The mechanism needs a scalar mass in a finite window, and for potentials attractive at small amplitude, the symmetry can break before conformality is lost. For operators of low enough dimension, already the second Rényi entropy lies beyond the merger. Classical backreaction does not move the merger at any order; we argue that away from $n_*$ the flows persist at first order.