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SOME COMMENTS ON THE ONE-DIMENSIONAL MANY-BODY PROBLEM.

Élliott H. Lieb

Published Jan 1, 1965• 0 citations
OpenAlex

Abstract

In the first half, the implication and significance of one dimensional models for many-body physics is discussed. In the second part, a brief historical survey is given of existing models and results obtained to date. (Author)

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Quantum Inverse Scattering Method and Correlation Functions

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.

Cambridge University Press eBooks•Aug 19, 1993•2,422 citations
Considerable progress has taken place in recent years in modeling the large-scale dynam- ics of one-dimensional quantum many-body systems that are integrable or nearly inte- grable, thanks to the advent of Generalized Hydrodynamics. In particular, Generalized Hydrodynamics provides a computationally efficient tool for simulating experiments on one-dimensional Bose gases. In this habilitation thesis I review the main results I ob- tained with my co-authors, both on theory developments and on the application to cold atom experiments. I also discuss our recent attempts at describing the effects of quantum fluctuations of out-of-equilibrium quantum gases within that framework.I also present a few results on other topics in low-dimensional quantum many-body physics. These include basic results on ‘operator entanglement’, an indicator of the complexity of quantum operators and of their approximability by Matrix Product Operators in one- dimensional quantum systems, as well as results on two-dimensional chiral topological phases (e.g. quantum Hall states, Chern bands, or p+ip superfluids) from the point of view of their entanglement properties.
The universal character of non-relativistic large three-body bosonic systems is addressed, with focus on highly interesting situations that occur in low-energy nuclear and atomic physics. Investigations on the trajectory of the first excited Efimov state, in a renormalized zero-range three-body model for a system with two bound and one virtual two-body subsystems, are reported. The approach is applied to n-n-18C, where the n - n virtual energy and the three-body ground state are kept fixed. It is shown that the real part of the elastic s-wave phase-shift ([Formula: see text]) presents a zero, or a pole in [Formula: see text], when the system has an Efimov excited or virtual state. A brief discussion is given on the relevance of the approach for ultracold atom physics with tunable scattering lengths.
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.