# Quantum Integrable Models

## Introduction

Integrable models in field theory and statistical mechanics are models for which it is possible to compute exactly non-perturbative quantities like the mass spectrum (including bound states) and the S-matrix (for a relativistic quantum field theory) or the partition functions and critical exponents (for a statistical model). Conformal field theories in two dimensions and certain topological models are particular examples. The main motivation for the study of such models is their importance as theoretical laboratories in the construction of non-perturbative or even exact methods and their applications in quantum field theory and statistical mechanics.

In the last thirty years, fundamental concepts such as infinite symmetry algebras and quantum groups have emerged from the study of integrable models. It enables the construction of powerful algebraic structures and their associated resolution methods applying to both field theory and statistical mechanics. This domain of research, soon became an extraordinary arena of interactions between various branches of theoretical physics and mathematics, and led to numerous applications in condensed matter physics (Kondo effect for example), in field theory (like in topological models in low dimension) having in the same time strong links with several domains of mathematics (Knot invariants, topology of low dimensional manifolds, quantum groups and non-commutative geometry).

## Research works

In this domain our research activities have been mainly concentrated on one of the most important problem in the domain of quantum integrable models : computing exactly their correlation functions. It is a fundamental problem in order to enlarge the possibilities of applications of these models in particular in condensed matter physics, but also from the theoretical point of view, to really "solve" them completely.

From 1995, J. M. Maillet started to develop a new method to compute correlation functions of lattice quantum integrable models (like Heisenberg spin chains) in the framework of the algebraic Bethe ansatz. The first results obtained in 1996 (J. M. Maillet, J. Sanchez de Santos) on the construction of factorising Drinfel'd twists shed new lights on the structure of the space of states of the XXZ Heisenberg spin chain. These ideas enabled to considered the computation of correlation functions. The results obtained in 1999-2000 (N. Kitanine, J. M. Maillet, V. Terras) give in particular exact determinant representations of the form factors of local spins and the computation of elementary blocks of correlation functions as multiple integrals for the spin-1/2 XXZ Heisenberg chain in a magnetic field. For zero field, it gives a proof of the results and conjectures obtained previously by the Kyoto group (Jimbo, Miwa, and collaborators) both in the massive and massless regimes. At the root of this method is the resolution of the so-called quantum inverse scattering problem, i.e., the reconstruction of the local spin operators in terms of the monodromy matrix operators entries which satisfy Yang-Baxter quadratic algebra. This is a very general result (it holds for a large class of lattice quantum integrable models) opening the route towards the exact computation of correlation functions.

## Main recent results :

- Multiple integral representations of spin-spin correlation functions of the XXZ Heisenberg chain (2002). (N. Kitanine, J. M. Maillet, N. Slavnov, V. Terras)
- Exact computation of some correlation functions (including spin-spin) of the XXZ spin-1/2 chain at anisotropy Delta = 1/2 (2003 - 2005). (N. Kitanine, J. M. Maillet, N. Slavnov, V. Terras)
- Asymptotic evaluation of the so-called emptiness formation probability of the XXZ chain (massless and massive regime) (2003 - 2005). (N. Kitanine, J. M. Maillet, N. Slavnov, V. Terras)
- Master formula and multiple integral representations of dynamical correlation functions of the XXZ chain (2004 - 2005) (N. Kitanine, J. M. Maillet, N. Slavnov, V. Terras)
- Computation (analytic + numerics) of dynamical structure factors of the XXZ chain in a magnetic field (2005). (J. S. Caux, J. M. Maillet, R. Hagemans)