Factorization and Criticality in the Anisotropic XY Chain via Correlations

In this review, we discuss the zero and finite temperature behavior of various bipartite quantum and total correlation measures, the skew information-based quantum coherence, and the local quantum uncertainty in the thermal ground state of the one-dimensional anisotropic XY model in transverse magnetic field. We compare the ability of considered measures to correctly detect or estimate the quantum critical point and the non-trivial factorization point possessed by the spin chain.Comment: 29 pages, 8 figures. A review paper accepted for publication in the special issue Entanglement Entropy in the journal Entrop

Global entanglement and quantum criticality in spin chains

Entanglement in quantum XY spin chains of arbitrary length is investigated via a recently-developed global measure suitable for generic quantum many-body systems. The entanglement surface is determined over the phase diagram, and found to exhibit structure richer than expected. Near the critical line, the entanglement is peaked (albeit analytically), consistent with the notion that entanglement--the non-factorization of wave functions--reflects quantum correlations. Singularity does, however, accompany the critical line, as revealed by the divergence of the field-derivative of the entanglement along the line. The form of this singularity is dictated by the universality class controlling the quantum phase transition.Comment: 4 pages, 2 figure

Quantum discord and information deficit in spin chains

We examine the behavior of quantum correlations of spin pairs in a finite anisotropic $XY$ spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one way-information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.Comment: 14 pages, 5 figure

Single-copy entanglement in critical spin chains

We introduce the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results -- which refine previous results on the divergence of block entropy as the rate at which EPR pairs can be distilled from many identically prepared chains, and which apply to single systems as encountered in actual experimental situations -- are made quantitative for general isotropic translationally invariant spin chains that can be mapped onto a quasi-free fermionic system, and for the anisotropic XY model. For the XX model, we provide the asymptotic scaling of ~(1/6) log_2(L), and contrast it with the block entropy. The role of superselection rules on single-copy entanglement in systems consisting of indistinguishable particles is emphasized.Comment: 5 pages, RevTeX, final versio

Effective mapping of spin-1 chains onto integrable fermionic models. A study of string and Neel correlation functions

We derive the dominant contribution to the large-distance decay of correlation functions for a spin chain model that exhibits both Haldane and Neel phases in its ground state phase diagram. The analytic results are obtained by means of an approximate mapping between a spin-1 anisotropic Hamiltonian onto a fermionic model of noninteracting Bogolioubov quasiparticles related in turn to the XY spin-1/2 chain in a transverse field. This approach allows us to express the spin-1 string operators in terms of fermionic operators so that the dominant contribution to the string correlators at large distances can be computed using the technique of Toeplitz determinants. As expected, we find long-range string order both in the longitudinal and in the transverse channel in the Haldane phase, while in the Neel phase only the longitudinal order survives. In this way, the long-range string order can be explicitly related to the components of the magnetization of the XY model. Moreover, apart from the critical line, where the decay is algebraic, we find that in the gapped phases the decay is governed by an exponential tail multiplied by algebraic factors. As regards the usual two points correlation functions, we show that the longitudinal one behaves in a 'dual' fashion with respect to the transverse string correlator, namely both the asymptotic values and the decay laws exchange when the transition line is crossed. For the transverse spin-spin correlator, we find a finite characteristic length which is an unexpected feature at the critical point. We also comment briefly the entanglement features of the original system versus those of the effective model. The goodness of the approximation and the analytical predictions are checked versus density-matrix renormalization group calculations.Comment: 28 pages, plain LaTeX, 2 EPS figure

Entanglement and magnetic order

In recent years quantum statistical mechanics have benefited of cultural interchanges with quantum information science. There is a bulk of evidence that quantifying the entanglement allows a fine analysis of many relevant properties of many-body quantum systems. Here we review the relation between entanglement and the various type of magnetic order occurring in interacting spin systems.Comment: 29 pages, 10 eps figures. Review article for the special issue "Entanglement entropy in extended systems" in J. Phys. A, edited by P. Calabrese, J. Cardy and B. Doyo