We consider the isotropic or anisotropic XY spin chain in the presence of a transversal random magnetic field, with parameters given by random variables. It is shown that eigenfunction correlator localization of the corresponding effective one-particle Hamiltonian implies a uniform area law bound in expectation for the bipartite entanglement entropy of all eigenstates of the XY chain, i.e., a form of many-body localization at all energies. Here, entanglement with respect to arbitrary connected subchains of the chain can be considered. Applications where the required eigenfunction correlator bounds are known include the isotropic XY chain in random field as well as the anisotropic chain in strong or strongly disordered random field.
@cite_7: We study the entropy of entanglement of the ground state in a wide family of one-dimensional quantum spin chains whose interaction is of finite range and translation invariant. Such systems can be thought of as generalizations of the XY model. The chain is divided in two parts: one containing the first consecutive L spins; the second the remaining ones. In this setting the entropy of entanglement is the von Neumann entropy of either part. At the core of our computation is the explicit evaluation of the leading order term as L → ∞ of the determinant of a block-Toeplitz matrix with symbol @math where g(z) is the square root of a rational function and g(1 z) = g −1(z). The asymptotics of such determinant is computed in terms of multi-dimensional theta-functions associated to a hyperelliptic curve ( L ) of genus g ≥ 1, which enter into the solution of a Riemann-Hilbert problem. Phase transitions for these systems are characterized by the branch points of ( L ) approaching the unit circle. In these circumstances the entropy diverges logarithmically. We also recover, as particular cases, the formulae for the entropy discovered by Jin and Korepin [14] for the XX model and Its, Jin and Korepin [12, 13] for the XY model.
@cite_32: 
@cite_3: Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: The entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such "area laws" for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium we review the current status of area laws in these fields. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations.
@cite_31: Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near and at a quantum critical point. Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories. We explore some of the implications of this connection.
@cite_12: A microscopic calculation of ground state entanglement for the XY and Heisenbergmodels shows the emergence of universal scaling behavior at quantum phase transitions,Entanglement is thus controlled by conformal symmetry, Away from the critical point,entanglement gets saturated by a mass scale, Results borrowed from conformal fieldtheory imply irreversibility of entanglement loss along renormalization group trajecto-ries. Entanglement does not saturate in higher dimensions which appears to limit thesuccess of the density matrix renormalization group technique. A possible connection be-tween majorization and renormalization group irreversibility emerges from our numericalanalysis.
The trace identity ) for Fermion systems seems to originate from @cite_31 , see also @cite_12 for a more detailed presentation and @cite_3 for a survey of subsequent results. In these works ) was used as a tool in the study of the ground state entanglement for the XY chain in constant magnetic field, In particular, the dimension reduction from @math to @math accomplished by ) (or @math to @math for the subsystem) allowed numerical predictions for the limit of large subsystems @math (and in the thermodynamic limit of an infinite chain @math ). These were later proven rigorously, see @cite_7 for most complete results as well as references therein. Generally, it was found that the ground state entanglement for the anisotropic chain remains bounded in the size @math of the subsystem, i.e. satisfies an area law. Phase transitions with divergent entanglement appear for @math and for the isotropic chain @math . In particular, @cite_32 shows that in the latter case the ground state entanglement grows as @math .