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.

/pdf/entanglement-in-many-body-systems-sqpatuxt14.pdf

Entanglement in many-body systems

It is demonstrated that an isolated generic quantum many-body system does relax to a state well described by the standard statistical mechanical prescription The thermalization happens at the level of individual eigenstates, allowing the computation of thermal averages from knowledge of any eigenstate in the microcanonical energy window An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive Recently, meaningful experimental studies1,2 of the problem have become possible, stimulating theoretical interest3,4,5,6,7 In generic isolated systems, non-equilibrium dynamics is expected8,9 to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible10 For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete11 Some recent studies4,5 even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch12 and Srednicki13 A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result

Thermalization and its mechanism for generic isolated quantum systems

We review recent developments in the physics of ultracold atomic and molecular gases in optical lattices. Such systems are nearly perfect realisations of various kinds of Hubbard models, and as such may very well serve to mimic condensed matter phenomena. We show how these systems may be employed as quantum simulators to answer some challenging open questions of condensed matter, and even high energy physics. After a short presentation of the models and the methods of treatment of such systems, we discuss in detail, which challenges of condensed matter physics can be addressed with (i) disordered ultracold lattice gases, (ii) frustrated ultracold gases, (iii) spinor lattice gases, (iv) lattice gases in “artificial” magnetic fields, and, last but not least, (v) quantum information processing in lattice gases. For completeness, also some recent progress related to the above topics with trapped cold gases will be discussed. Motto: There are more things in heaven and earth, Horatio, Than are dreamt of in your...

/pdf/ultracold-atomic-gases-in-optical-lattices-mimicking-tfesylj4h7.pdf

Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

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

We calculate, within the pseudopotential approximation, a one-dimensional scattering amplitude and effective one-dimensional interaction potential for atoms confined transversally by an atom waveguide or highly elongated ``cigar''-shaped atomic trap. We show that, in the low-energy scattering regime, the scattering process degenerates to a total reflection, suggesting an experimental realization of a famous model in theoretical physics---a one-dimensional gas of impenetrable bosons (``Tonks'' gas). We give an estimate for suitable experimental parameters for alkali atoms confined in waveguides.

https://arxiv.org/pdf/cond-mat/9804130.pdf

Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons

The book contain detailed explanation of Bethe Ansatz, Quantum Inverse Scattering Method and Algebraic Bether Ansatz as well. Main Models are Nonlinear Schrodinger equation (one dimensional Bose gas), Sine-Gordon and Thiring models. Heisenberg Antiferromagnet and Hubbard models. It is explained in detail, how to calculate correlation functions.

Introduction The quantum inverse scattering method (QSIM) appears as the quantized form of the classical inverse scattering method. It allows us to reproduce the results of the Bethe Ansatz and to move ahead. QISM is now a well developed branch of mathematical physics. In this chapter the fundamentals of QISM are given and illustrated by concrete examples. In section 1 the general scheme of QISM, which allows the calculation of commutation relations between elements of the transfer matrix (necessary to construct eigenfunctions of the Hamiltonian in Chapter VII) is presented, and the quantum R-matrix is introduced. As in the classical case, the existence of an R-matrix and trace identities ensures that a Lax representation for the model exists. Thus, there are infinitely many conservation laws. The Yang-Baxter equation, which is satisfied by the R-matrix, is discussed in section 2. Some important features of the R-matrix are also mentioned. The trace identities for the quantum nonlinear Schrodinger equation are proved in section 3. The general scheme of QISM is applied to the quantum sine-Gordon and Zhiber-Shabat-Mikhailov models in section 4. Spin models of quantum statistical physics are discussed in section 5. It is shown that a fundamental spin model can be constructed with the help of any given R-matrix. The connection between classical statistical models on a two-dimensional lattice and QISM is established in section 6. QISM is the generalization of the classical inverse scattering method.

N. M. Bogoliubov

Papers

Quantum Inverse Scattering Method and Correlation Functions

Quantum Inverse Scattering Method and Correlation Functions

Quantum Inverse Scattering Method and Correlation Functions

The Quantum Inverse Scattering Method

Critical exponents for integrable models