Quelques applications de l'Ansatz de Bethe (Some applications of the Bethe Ansatz)

TL;DR

This paper discusses the Bethe Ansatz method's general framework and its applications in quantum integrable models, including finite size corrections and thermodynamic properties of specific models, highlighting its connection with quantum groups.

Contribution

It provides a comprehensive overview of the Bethe Ansatz method, its connection to quantum groups, and applies it to finite size corrections and thermodynamic analysis of integrable models.

Findings

Finite size corrections calculated using Non-Linear Integral Equations.

Thermodynamic properties derived via Thermodynamic Bethe Ansatz Equations.

Spectrum of low energy excitations characterized for the multi-channel Kondo model.

Abstract

The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.