Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

It is a fundamental assumption of statistical mechanics that a closed system with many degrees of freedom ergodically samples all equal energy points in phase space. To understand the limits of this assumption, it is important to find and study systems that are not ergodic, and thus do not reach thermal equilibrium. A few complex systems have been proposed that are expected not to thermalize because their dynamics are integrable. Some nearly integrable systems of many particles have been studied numerically, and shown not to ergodically sample phase space. However, there has been no experimental demonstration of such a system with many degrees of freedom that does not approach thermal equilibrium. Here we report the preparation of out-of-equilibrium arrays of trapped one-dimensional (1D) Bose gases, each containing from 40 to 250 87Rb atoms, which do not noticeably equilibrate even after thousands of collisions. Our results are probably explainable by the well-known fact that a homogeneous 1D Bose gas with point-like collisional interactions is integrable. Until now, however, the time evolution of out-of-equilibrium 1D Bose gases has been a theoretically unsettled issue, as practical factors such as harmonic trapping and imperfectly point-like interactions may compromise integrability. The absence of damping in 1D Bose gases may lead to potential applications in force sensing and atom interferometry.

http://absimage.aps.org/image/DAMOP06/MWS_DAMOP06-2006-000603.pdf

A quantum Newton's cradle

The physics of one-dimensional interacting bosonic systems is reviewed. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. The effects of various perturbations on the Tomonaga-Luttinger liquid state are discussed as well as extensions to multicomponent and out of equilibrium situations. Finally, the experimental systems that can be described in terms of models of interacting bosons in one dimension are discussed.

/pdf/one-dimensional-bosons-from-condensed-matter-systems-to-4vp39jxn82.pdf

One dimensional bosons: From condensed matter systems to ultracold gases

Submitted for the MAR07 Meeting of The American Physical Society Dynamical spin structure factor for the anisotropic spin-1/2 chain RODRIGO G. PEREIRA, JESKO SIRKER, JEAN-SEBASTIEN CAUX, ROB HAGEMANS, JEAN MICHEL MAILLET, STEVE WHITE, IAN AFFLECK — The longitudinal spin structure factor for the XXZ-chain at small wave-vector q is obtained using Bethe Ansatz, field theory methods and the Density Matrix Renormalization Group. It consists of a peak with peculiar, non-Lorentzian shape and a high-frequency tail. We show that the width of the peak is proportional to q2 for finite magnetic field compared to q3 for zero field. For the tail we derive an analytic formula without any adjustable parameters and demonstrate that the integrability of the model directly affects the lineshape. Rodrigo Pereira University of British Columbia Date submitted: 22 Nov 2006 Electronic form version 1.4

http://absimage.aps.org/image/MAR07/MWS_MAR07-2006-007712.pdf

Dynamical spin structure factor for the anisotropic spin-1/2 chain

Determinant representations of form factors are used to represent the spontaneous magnetization of the Heisenberg XXZ chain (Delta >1) on the finite lattice as the ratio of two determinants. In the thermodynamic limit (the lattice of infinite length), the Baxter formula is reproduced in the framework of Algebraic Bethe Ansatz. It is shown that the finite size corrections to the Baxter formula are exponentially small.

http://arxiv.org/pdf/solv-int/9812021.pdf

Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain

We describe a method for calculating dynamical spin-spin correlation functions in the finite isotropic and anisotropic antiferromagnetic Heisenberg models. Our method is able to produce results with high accuracy over the full parameter space.

How to calculate correlation functions of Heisenberg chains

We compute all dynamical spin-spin correlation functions for the spin-1/2 $XXZ$ anisotropic Heisenberg model in the gapless antiferromagnetic regime, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier transform our results back to real space, allowing us in particular to make a comparison with known exact formulas for equal-time correlation functions for small separations in zero field, and with predictions for the zero-field asymptotics from conformal field theory.

J. M. Maillet

Papers

Dynamical spin structure factor for the anisotropic spin-1/2 chain

Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain

How to calculate correlation functions of Heisenberg chains

Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime