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We investigate the scaling of the R\'enyi $\alpha$-entropies inone-dimensional gapped quantum spin models. We show that the block entropieswith $\alpha > 2$ violate the area law monotonicity and exhibit dampedoscillations. Depending on the existence of a factorized ground state, theoscillatory behavior occurs either below factorization or it extendsindefinitely. The anomalous scaling corresponds to an entanglement-driven orderthat is independent of ground-state degeneracy and is revealed by a nonlocalorder parameter defined as the sum of the single-copy entanglement over allblocks.

Scaling of the R\'enyi entropies in gapped quantum spin systems: Entanglement-driven order beyond symmetry breaking