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

S. M. Giampaolo, S. Montangero, F. Dell’Anno, S. De Siena, and F. Illuminati
Phys. Rev. B 88, 125142 – Published 30 September 2013

Abstract

We investigate the scaling of the entanglement spectrum and of the Rényi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the Rényi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing value of the factorizing field for this particular model. In general, the transition occurring at the factorizing field between two different scaling regimes of the entanglement spectrum corresponds to a quantum transition to the formation of finite-range, ordered structures of quasidimers, quasitrimers, and quasipolymers. This entanglement-driven transition is superimposed to and independent of the long-range magnetic order in the broken symmetry phase. Therefore, it conforms to recent generalizations that identify and classify the quantum phases of matter according to the structure of ground-state entanglement patterns. We characterize this form of quantum order by a global order parameter of entanglement defined as the integral, over blocks of all lengths, of the Rényi entropy of infinite order. Equivalently, it can be defined as the integral of the bipartite single-copy or geometric entanglement. The global entanglement order parameter remains always finite at fields below the factorization point and vanishes identically above it.

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  • Received 29 July 2013

DOI:https://doi.org/10.1103/PhysRevB.88.125142

©2013 American Physical Society

Authors & Affiliations

S. M. Giampaolo1, S. Montangero2, F. Dell’Anno3, S. De Siena4,5, and F. Illuminati4,5,*

  • 1University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria
  • 2Institut für Quanteninformationsverarbeitung, Universität Ulm, D-89069 Ulm, Germany
  • 3Liceo Statale P. E. Imbriani, via Pescatori 155, I-83100 Avellino, Italy
  • 4Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II, I-84084 Fisciano (SA), Italy
  • 5CNISM - Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Unità di Salerno, I-84084 Fisciano (SA), Italy

  • *Corresponding author: illuminati@sa.infn.it

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Issue

Vol. 88, Iss. 12 — 15 September 2013

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