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

TL;DR

This paper explores the universal scaling behavior of the entanglement spectrum in one-dimensional quantum spin models, revealing a transition at the factorization point linked to ordered entangled structures and quantum phases.

Contribution

It uncovers universal oscillatory scaling patterns in the entanglement spectrum and introduces a global entanglement order parameter to classify quantum phases.

Findings

Universal oscillatory behavior in entanglement spectrum scaling.

Identification of a quantum transition at the factorization point.

Introduction of a global entanglement order parameter.

Abstract

We investigate the scaling of the entanglement spectrum and of the R\'enyi 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 R\'enyi 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 quasi-dimers, quasi-trimers, and quasi-polymers. 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 R\'enyi 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.