Two-spin entanglement distribution near factorized states

We study the two-spin entanglement distribution along the infinite $S=1/2$ chain described by the XY model in a transverse field; closed analytical expressions are derived for the one-tangle and the concurrences $C_r$, $r$ being the distance between the two possibly entangled spins, for values of the Hamiltonian parameters close to those corresponding to factorized ground states. The total amount of entanglement, the fraction of such entanglement which is stored in pairwise entanglement, and the way such fraction distributes along the chain is discussed, with attention focused on the dependence on the anisotropy of the exchange interaction. Near factorization a characteristic length-scale naturally emerges in the system, which is specifically related with entanglement properties and diverges at the critical point of the fully isotropic model. In general, we find that anisotropy rule a complex behavior of the entanglement properties, which results in the fact that more isotropic models, despite being characterized by a larger amount of total entanglement, present a smaller fraction of pairwise entanglement: the latter, in turn, is more evenly distributed along the chain, to the extent that, in the fully isotropic model at the critical field, the concurrences do not depend on $r$.Comment: 14 pages, 6 figures. Final versio

Renyi Entropy of the XY Spin Chain

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power $\alpha$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $\lambda$ - function. We study the limiting entropy as a function of its parameter $\alpha$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map $\alpha \to \alpha^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $\alpha$ is a integer power of 2, this includes the purity $tr \rho^2$. We also analyze the behavior of the entropy as $\alpha \to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model].Comment: 28 Pages, 1 Figur

Entanglement and Density Matrix of a Block of Spins in AKLT Model

We study a 1-dimensional AKLT spin chain, consisting of spins $S$ in the bulk and $S/2$ at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension $(S+1)^{2}$. This subspace is described by non-zero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to $\ln(S+1)^{2}$.Comment: Revised version, typos corrected, references added, 31 page

A short review on entanglement in quantum spin systems

We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.Comment: 20 pages and 6 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