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We investigate the scaling of the entanglement spectrum and of the R\'enyiblock entropies and determine its universal aspects in the ground state ofcritical and noncritical one-dimensional quantum spin models. In all cases, thescaling exhibits an oscillatory behavior that terminates at the factorizationpoint and whose frequency is universal. Parity effects in the scaling of theR\'enyi entropies for gapless models at zero field are thus shown to be aparticular case of such universal behavior. Likewise, the absence ofoscillations for the Ising chain in transverse field is due to the vanishingvalue of the factorizing field for this particular model. In general, thetransition occurring at the factorizing field between two different scalingregimes of the entanglement spectrum corresponds to a quantum transition to theformation of finite-range, ordered structures of quasi-dimers, quasi-trimers,and quasi-polymers. This entanglement-driven transition is superimposed to andindependent of the long-range magnetic order in the broken symmetry phase.Therefore, it conforms to recent generalizations that identify and classify thequantum phases of matter according to the structure of ground-stateentanglement patterns. We characterize this form of quantum order by a globalorder parameter of entanglement defined as the integral, over blocks of alllengths, of the R\'enyi entropy of infinite order. Equivalently, it can bedefined as the integral of the bipartite single-copy or geometric entanglement.The global entanglement order parameter remains always finite at fields belowthe factorization point and vanishes identically above it.

physical review. b, condensed matter and materials physics

Universal aspects in the behavior of the entanglement spectrum in one  dimension: scaling transition at the factorization point and ordered  entangled structures