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.
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 .
@cite_7 @cite_32 @cite_3 @cite_31 @cite_12
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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. 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. We investigate the creation of highly entangled ground states in a system of three exchange-coupled qubits arranged in a ring geometry. Suitable magnetic field configurations yielding approximate Greenberger-Horne-Zeilinger and exact W ground states are identified. The entanglement in the system is studied at finite temperature in terms of the mixed-state tangle tau. By generalizing a conjugate gradient optimization algorithm originally developed to evaluate the entanglement of formation, we demonstrate that tau can be calculated efficiently and with high precision. We identify the parameter regime for which the equilibrium entanglement of the tripartite system reaches its maximum. We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 .