Speaker: Jiyuan Zhang (University of Melbourne) (slides)
Title: Sums and products of invariant ensembles
Abstract: An invariant ensemble is a random matrix that is unchanged under an adjoint action of a group of invariance. In this talk we will focus on sums/products of such ensembles, where general formulae for those eigenvalue probability density functions are provided, connecting eigenvalue PDFs to additive/multiplicative weights. There, a matrix addition/multiplication corresponds to convolution of their weights. Such connections can be shown for the additive spaces of the Hermitian, Hermitian anti-symmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. This is a joint work with Mario Kieburg (arXiv:2007.15259 [math-ph]).
Speaker: Alexandre Lazarescu (UC Louvain-la-Neuve) (slides)
Title: Einstein's fluctuation relation and Gibbs states far from equilibrium
Abstract: I will present a class of one-dimensional non-equilibrium interacting particle models characterized by a so-called "gradient condition" which generalizes detailed balance and guarantees the existence of Gibbs-type local homogeneous stationary states.
I will show how, defining appropriate boundary conditions, this leads to a special symmetry of the models under time and space reversal which, rephrased in terms of the large deviations function of stationary currents of conserved quantities, yields a novel fluctuation relation under reservoir exchange, unrelated to the standard Gallavotti--Cohen symmetry.
I will then show that this relation can be interpreted as a non-equilibrium and nonlinear generalization Einstein's relation, which points to the existence of a Langevin-type hydrodynamic equation for the macroscopic behavior of those models.
Speaker: Nicolas Crampé (Université de Tours) (slides)
Title: Free-fermion entanglement and Leonard pairs
Abstract: I study the entanglement entropy for free-fermion models. I recall how it is related to the computation of the chopped correlation matrix. Then I present the construction of the algebraic Heun operator which commutes with this correlation matrix and simplifies the computation of its eigenvalues. I show why the concept of Leonard pairs is important in this context.
Speaker: Eric Vernier (Univerité de Paris) (slides)
Title: Boundary integrability, “integrable states” and “integrable quenches”
Abstract: It is well known how to construct integrable boundary conditions for integrable 2D lattice models or quantum spin chains : these are encoded in "reflection matrices", which obey the so-called "reflection equation", or "boundary Yang-Baxter equation", in addition to the usual Yang-Baxter equation. From there the usual construction of commuting transfer matrices, conserved charges and Bethe ansatz follows. I will show here that integrable boundary conditions play a fundamental role in a seemingly unrelated problem, namely the out-of-equilibrium evolution of 1D quantum systems following a "quench", that is when these systems are prepared in some state at t=0 and let evolve from there. More precisely, integrable boundary conditions are in correspondence which "integrable initial states", from which the dynamics can be computed exactly. This will motivate the search for integrable boundary conditions of a more general type, in particular of the "matrix product state" form, which I will then present. This is based on collaboration with Lorenzo Piroli (Munich) and Balazs Pozsgay (Budapest), see in particular : arXiv:1709.04796 and arXiv:1812.11094.
Speaker: Christophe Charlier (KTH Stockholm) (slides)
Title: Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface
Abstract: I will talk about matrix polynomials that satisfy a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the non-Hermitian matrix orthogonality is equivalent to a scalar orthogonality in a Riemann surface. If this Riemann surface has genus 0, it can be mapped to the plane and the matrix orthogonality is equivalent to a scalar orthogonality in the plane. I will also describe some applications of the results to the theory of tiling models.
Speaker: Sandrine Brasseur (UC Louvain-la-Neuve) (slides)
Title: Sum rules for the supersymmetric eight-vertex model
Abstract: In this talk, we are interested in the eight-vertex (8V) model on the square lattice with an odd number of columns and periodic boundary conditions. We consider a particular sub-family of parameters of the system, the "supersymmetric" or “combinatorial line”, dubbed so due to its surprising links to combinatorics. More precisely, we are investigating a certain eigenvector of the transfer matrix of the model whose associated eigenvalue has been shown to have a remarkably simple expression. Incidentally, this vector also corresponds to the ground state of the XYZ spin chain Hamiltonian.
Our goal is to evaluate specific scalar products (sum rules) involving this vector. These will ultimately allow us to obtain exact expressions for conjectured XYZ ground-state components, as well as for 8V partition functions associated to different boundary conditions.
This talk is based on joint work with Christian Hagendorf (arXiv:2009.14077 [math-ph]).
Speaker: Hadewijch De Clercq (U Gent) (slides)
Title: Quantum symmetric pairs and the universal K-matrix
Abstract: Quantum symmetric pairs encode the symmetries in an integrable quantum system with reflecting boundary conditions. They consist of a quantum Kac--Moody algebra U_q(g) and a coideal subalgebra determined by a fixed g-involution. In this talk I will give a gentle introduction to this topic and an overview of some recent results. I will describe the construction of the coideals and cover some elements of their representation theory.
A quantum symmetric pair is naturally endowed with a universal K-matrix, a boundary analog of the universal R-matrix. Consequently, it carries a canonical solution to the reflection equation or boundary type Yang-Baxter equation. I will outline the theory of universal K-matrices as developed by Balagovic and Kolb. Special attention will be drawn to the so-called split case, where the quantum symmetric pair coideals become (generalized) q-Onsager algebras. I will conclude with a presentation of the coideals by generators and relations, based on my recent work arXiv:1912.05368, and some directions for future research.
Speaker: Alan Groot (KU Leuven) (slides)
Title: From matrix orthogonality to scalar orthogonality in the 2-periodic hexagon: a case study on the asymptotic behavior of zeros
Abstract: In this talk, we will consider matrix-valued orthogonal polynomials related to the 2-periodic hexagon tiling model and express these explicitly in terms of scalar orthogonal polynomials. We will show how this connection allows us to find the limit distribution of the zeros of the determinant of the matrix-valued orthogonal polynomials. Moreover, we investigate the zero distribution of one of the entries of the matrix-valued orthogonal polynomial. Under a certain assumption, we are able to find the zero distribution explicitly. This assumption is supported by numerical evidence. This talk is based on joint work with Arno Kuijlaars.
Speaker: Anna Maltsev (Queen Mary University) (slides)
Title: Covariance formulas for fluctuations of linear spectral statistics for Wigner matrices with few moments
Abstract: I will discuss our recent work on central limit theorems for the fluctuation of the linear spectral statistics for Wigner matrices with entries that have a finite variance but no finite 4th moment. This fluctuation tends to a Gaussian process and we obtain a closed form expression for its covariance. We also extract its integral kernel, and give it a heuristic interpretation. This is joint work with Asad Lodhia.
Speaker: Milivoje Lukic (Rice University) (slides)
Title: Stahl--Totik regularity for continuum Schroedinger operators
Abstract: This talk describes joint work with Benjamin Eichinger: a theory of regularity for one-dimensional continuum Schroedinger operators, based on the Martin compactification of the complement of the essential spectrum. For a half-line Schroedinger operator $-\partial_x^2+V$ with a bounded potential $V$, it was previously known that the spectrum can have zero Lebesgue measure and even zero Hausdorff dimension; however, we obtain universal thickness statements in the language of potential theory. Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer--Levin condition, and moreover, the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \frac{a}{2\sqrt{-z}} +o(\frac 1{\sqrt{-z}})$ as $z \to -\infty$. The constant $a$ in its asymptotic expansion plays the role of a renormalized Robin constant and enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t) dt$. This leads to a notion of regularity, with connections to the exponential growth rate of Dirichlet solutions and limiting eigenvalue distributions for finite restrictions of the operator. We also present applications to decaying and ergodic potentials.
Speaker: Harriet Walsh (ENS Lyon) (slides)
Title: Multicritical random partitions with higher-order Tracy--Widom edge statistics
Abstract: I will present new multicritical measures on integer partitions, defined as Schur measures with tuned Newton powersums. These integrable models correspond to one-dimensional free fermion theories, and I will show that the asymptotic laws of the first parts of random partitions distributed by our measures are the higher order Tracy--Widom GUE distributions found to govern fermion momenta at the edge of flat traps by Le Doussal, Majumdar and Schehr. The distributions of the first parts are also exactly equivalent to certain unitary matrix model expectation values previously studied by Periwal and Shevitz; thus I will give generalizations of two theorems of Baik, Deift and Johansson to our measures. Finally I will show the limit shapes of Young diagrams for two explicit multicritical measures (asymmetric and symmetric cases) at various orders of multicriticality. This talk is based on joint work with Dan Betea and Jérémie Bouttier.
Speaker: Jerôme Dubail (Université de Lorraine) (slides)
Title: Generalized hydrodynamics in the one-dimensional Bose gas
Abstract: I will give a brief introduction to “Generalized Hydrodynamics”, a hydrodynamic description of one-dimensional integrable systems discovered in 2016 [1, 2]. I will describe the theory in the context of the one-dimensional Bose gas, where it is particularly simple. I will briefly review how “Generalized Hydrodynamics” is successfully used to describe modern cold atoms experiments [3, 4]. One experimental effect that is not taken into account in the 2016 theory is the atom losses: I will discuss perspectives on how to include these [5, 6].
[1] O. Castro-Alvaredo, B. Doyon, T. Yoshimura, “Emergent hydrodynamics in integrable quantum systems out of equilibrium", Phys. Rev. X 6, 041065 (2016)
[2] B. Bertini, M. Collura, J. de Nardis, M. Fagotti, “Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents", Phys. Rev. Lett. 117, 207201 (2016)
[3] M. Schemmer, I. Bouchoule, B. Doyon, J. Dubail, “Generalized Hydrodynamics on an Atom Chip”, Phys. Rev. Lett. 122, 090601 (2019)
[4] N. Malvania, Y. Zhang, Y. Le, J. Dubail, M. Rigol, D. Weiss, “Generalized Hydrodynamics in strongly interacting 1D Bose gases", arXiv:2009.06651
[5] I. Bouchoule, B. Doyon, J. Dubail, “The effect of atom losses on the distribution of rapidities in the one-dimensional Bose gas”, SciPost Phys. 9, 044 (2020)
[6] A. Hutsalyuk, B. Pozsgay, “Integrability breaking in the one dimensional Bose gas: Atomic losses and energy loss”, arXiv:2012.15640
Speaker: Jean-Marie Stéphan (Université Lyon-1) (slides)
Title: Fermionic limit shapes
Abstract: I study a 1d translational invariant free fermions model in imaginary time, with nearest neighbor and next-nearest neighbor hopping terms, for various types boundary conditions. This type of model typically arises when trying to solve a 2d classical models such as dimer models, or Ising models, through the transfer matrix formalism. In the absence of the next-nearest neighbor perturbation our fermion model is known to give rise to phases separation and so-called arctic curves, which have been studied extensively. The perturbation considered turns out to not be always positive, that is, the corresponding statistical mechanical model does not always have positive Boltzmann weights. I discuss the conditions under which this is the case, and what sense to make of it when not.
Speaker: Gilles Parez (UC Louvain-la-Neuve) (slides)
Title: Free fermions and asymptotics in entanglement-related problems
Abstract: The bipartite fidelity is an entanglement measure introduced in 2011 by J. Dubail and J.-M. Stéphan. For a quantum system, it is defined as the overlap between the ground state of the whole system and the tensor product of the ground states of two complementary subsystems. In this talk, I will describe how we used free-fermion techniques and asymptotic methods to obtain exact results for the bipartite fidelity and how they match universal predictions of conformal field theory. I will also discuss the evolution of entanglement after a so-called quantum quench. The use of free fermions allow one to gain strong insights into the many-body dynamics for more generic models. I shall put a special effort to give physical motivations to the problems I consider, while inserting some chosen mathematical technicalities, in a pedagogical fashion. This talk is based on the papers arXiv:1902.02246, arXiv: 2008.08952 and arXiv:2010.09794.
Speaker: Jhih-Huang Li (University of Warwick) (slides)
Title: (1+1)-dimensional quantum Ising model
Abstract: This is an ongoing joint work with Rémy Mahfouf. In this talk, I will introduce the planar graphical representation of the (1+1)-dimensional quantum Ising model. I will explain how to make sense of the Kadanoff-Ceva order-disorder operators in this context and describe how to use them in order to understand the scaling limit of the correlation functions. I will also say a few words on how orthogonal polynomials arise, which allow us to compute the magnetization.
Speaker: Filippo Colomo (INFN Firenze) (slides)
Title: Correlation functions of the domain wall six-vertex model
Abstract: The six-vertex model is an exactly solvable two-dimensional lattice model of statistical mechanics; in a suitable scaling limit, and under suitable choices of boundary conditions, it develops limit shapes and Arctic curves. In particular, the six-vertex model with domain wall boundary conditions can be regarded as an `interacting' generalization of the famous `free-fermion' problem of domino tilings of the Aztec Diamond.
After reviewing the state of the the art in the determination of the limit shape of the model, we shall become more technical, and present recent progresses in the calculation of multiple integral representations for the correlation functions of the mode. Here a crucial role is played by some antisymmetrization identity, generalizing a relation obtained by Tracy and Widom in the context of the asymmetric simple exclusion process.
Based on works with L.Cantini, G. Di Giulio, A.Pronko, A.Sportiello.
Speaker: Hugo Duminil-Copin (IHES) (slides)
Title: Emergent symmetries in 2D percolation
Abstract: A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact progressed tremendously in two dimensions with the introduction of the Schramm--Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent proof of rotational invariance for a large class of such models. This represents an important progress in the direction of proving full conformal invariance.
Speaker: Sofia Tarricone (Université d'Angers & Concordia University) (slides)
Title: Higher order finite temperature Airy kernels and an integro-differential Painlevé II hierarchy.
Abstract: In this talk we will study Fredholm determinants of a finite temperature version of the higher order Airy kernels that recently appeared in statistical mechanics literature. The main result is an expression of these Fredholm determinants in terms of distinguished solutions of an integro-differential Painlevé II hierarchy. Our result generalizes the case n = 1, already studied by Amir, Corwin and Quastel some years ago for a special choice of the weight function. This latter can be seen as a generalization of the well known formula connecting the Tracy--Widom distribution for GUE and the Hastings--McLeod solution of the Painlevé II equation. The proof of our result, for generic n, strongly relies on the study of some operator-valued Riemann--Hilbert problem that builds up the bridge between the description of the Fredholm determinants and the derivation of a Lax pair for some integro-differential hierarchy. This talk is based on a joint work with Thomas Bothner and Mattia Cafasso, available at https://arxiv.org/pdf/2101.03557.pdf.
Speaker: Iván Parra (KU Leuven) (slides)
Title: Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
Abstract: Universality in the eigenvalue correlations of random matrices has attracted both mathematicians and physicists in the last few decades. In the early sixties Ginibre started the study of Gaussian random matrices without symmetry constraint, whose entries are complex, quaternion or real random variables. In the former ensemble much more is known about the local eigenvalues fluctuations (given in terms of a determinantal point process) — including general potentials in the plane, such as the normal matrix model — and it has been found that the complex eigenvalue statistics provided by the complex Ginibre ensembles are universal in the bulk and at the edge of the spectrum. The rich literature available on orthogonal polynomials (OP) and related kernel functions have a significant contribution in our understanding of such ensembles.
In this talk we focus on Pfaffian point processes in the complex plane — naturally linked to non-Hermitian random matrices with symplectic symmetry, as the symplectic Ginibre ensemble (GinSE) — these point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials (SOP). Its existence is always guaranteed by the partition function in the GinSE, which automatically gives rise to a representation of the SOP in terms of Pfaffian of Gram matrices. However this is not very useful for the actual computation of the SOP since it involves the evaluation of Pfaffians (or determinants in the OP case). Here, we exploit the particularly simple structure of the skew-product in the GinSE. It allows to relate the skew-product to the multiplication acting on the standard Hermitian inner product. New examples of exactly soluble symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are presented, from which the conjectured universality of the elliptic GinSE and its chiral partner follow in the limit of strong non-Hermiticity at the origin.
Join work with Gernot Akemann and Markus Ebke (Bielefeld University).
Speaker: Emma Bailey (University of Bristol) (slides)
Title: Branching random walks, characteristic polynomials, and zeta: log-correlation, combinatorics, moments, and extrema
Abstract: In this talk I will introduce three log-correlated processes and present results on their moments (and moments of moments), and how these relate to their extremes. This study features connections with integrable systems (in particular Toeplitz and Hankel determinants), RH problems, the Fyodorov--Hiary--Keating conjectures, Painlevé equations, Young diagrams and Gelfand--Tsetlin patterns, large deviations and more. This talk will include work joint with Louis-Pierre Arguin, Theo Assiotis, and Jon Keating.
Speaker: Michael Wheeler (University of Melbourne) (slides)
Title: Fermionic vertex models and LLT polynomials
Abstract: This talk is based on recent work with Amol Aggarwal and Alexei Borodin (https://arxiv.org/abs/2101.01605). I will outline the distinction between bosonic and fermionic vertex models, and how each arises from fusion of the fundamental R matrix associated to a quantized affine Lie (super) algebra. From there, I will say something about the different types of symmetric polynomials that can be constructed as partition functions within these models. In the fermionic case, the well-known LLT polynomials appear. If time permits, I will talk about various applications of the fermionic vertex models to studying combinatorial properties of the LLT polynomials (including Cauchy identities and expansions over the modified Hall--Littlewood basis).
Speaker: Myrthe D’Haen (KU Leuven)
Title: Asymptotics for the multiple Laguerre polynomials
Abstract: Master's seminar
Speaker: Patrik Ferrari (University of Bonn) (slides)
Title: Stationary half-space last passage percolation
Abstract: We present our result on study stationary last passage percolation (LPP) in half-space geometry. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry.
Speaker: Thomas Wolfs (KU Leuven)
Title: The irrationality measure of pi and Hermite--Padé approximation
Abstract: Master's seminar
Speaker: Hjalmar Rosengren (Chalmers University and University of Gothenburg) (slides)
Title: Deformed Ruijsenaars operators
Abstract: The Ruijsenaars operators are a commuting family of difference operators with elliptic coefficients, which define an integrable system of relativistic quantum particles. In the trigonometric limit, they have Macdonald polynomials as joint eigenfunctions. Through the work of Chalykh, Feigin, Silantyev, Veselov and others, it has become apparent that even more general "deformed" or "super" operators exist. We will describe how to obtain the main properties of such operators in a direct way, which works also in the elliptic setting. In particular, we can prove that the deformed elliptic Ruijsenaars model is integrable, in the sense of constructing a sufficiently large family of algebraically independent operators that commute with the Hamiltonian.
The talk is based on joint work in progress with Martin Hallnäs, Edwin Langmann and Masatoshi Noumi.
Speaker: Robert Weston (Heriot--Watt University)
Title: Q operators for open spin chains: part II*
Abstract: Baxter's Q-operator plays a central role in the description of integrable quantum spin chains, and its algebraic construction within the QISM/quantum groups picture of closed spin chains is fairly well understood. It has been a goal of the speaker and his collaborators B. Vlaar and A. Cooper to extend this algebraic picture to open chains. Part I of this talk has been delivered elsewhere, and concerned the construction of a Q operator for the open XXZ model in terms of infinite dimensional auxiliary spaces, and the derivation of TQ relations using fusion - a.k.a. short exact sequences of tensor products of infinite and finite dimensional Borel-subalgabra representations. The main technical task was to understand how the projections and embeddings of the short exact sequences moved through the boundaries.
This talk, that is part II, takes the alternative approach involving an isomorphism of products of infinite dimenionsal Borel subalgebra representations, and the relation of this tensor product to the Verma module V. Here one arrives at a relation of the form T_V = Q \bar{Q}. Again this approach is reasonably well understood for closed chains. In this talk I shall describe how this method extends to open chains and how the isomorphism moves through the boundary. The picture is conceptually clear although algebraically more complicated due to the presence of five algebras: the quantum group, its Borel subalgebras and the codieal subalgebras associated with the two boundaries.
* It is not required to have attended part I. In fact I shall mostly just draw lots of pictures in this talk.
Speaker: Romain Vasseur (UMass Amherst) (slides)
Title: Spin hydrodynamics and anomalous transport in integrable spin chains
Abstract: In this talk, I will review recent progress in the understanding of finite-temperature transport in the XXZ quantum spin chain. Spin transport is ballistic in the easy-plane regime, diffusive in the easy-axis regime, and superdiffusive at the isotropic Heisenberg point. Energy transport, by contrast, is ballistic in all cases. I will explain how the framework of generalized hydrodynamics offers an elementary explanation of these phenomena in terms of the quasiparticle content of the XXZ model. I will also argue that superdiffusion is generic for integrable models with nonabelian symmetries, and stems from the presence of stable solitons made up of Goldstone modes.
Speaker: Lenart Zadnik (Université Paris-Saclay) (slides)
Title: Hydrodynamics in the folded XXZ model
Abstract: The recent decade has witnessed several breakthroughs in the description of relaxation of local observables in quantum many-body systems prepared far from equilibrium. In contrast, the time evolution on time scales that precede relaxation towards the stationary state has remained much less explored. Strong coupling expansions offer a good starting point for investigation of such problems, since they provide a natural time scale on which prerelaxation can occur. I will discuss the microscopic and ballistic-scale mesoscopic dynamics, described by the effective Hamiltonian that arises in the large-coupling limit of the anisotropic Heisenberg spin-1/2 chain. Starting with the particle content that forms the basis of the Bethe Ansatz solution, I will move towards the ballistic scale hydrodynamics emerging after a sudden junction of two thermal or chemical reservoirs. In its framework, the nonselective emergence of discontinuities in the profiles of local observables suggests inherent nonballistic behaviour. If time permits, I will conclude with a brief discussion of local perturbations in the jammed sector of the model.
Speaker: Eric Ragoucy (Université Savoie Mont Blanc) (slides)
Title: Integrable models: their Bethe vectors, scalar products and form factors
Abstract: We apply the nested algebraic Bethe ansatz to integrable models based on algebras with rank higher than 1. We present some explicit representations for the Bethe vectors and their scalar products, in the framework of periodic generalized models, that encompass all integrable spin chain models with (twisted) periodic boundary conditions. We review what has been (or can be) done, depending on the algebra which underlies the model (Yangian, super-Yangian or quantum group). Starting from these formulas we present some general methods that allow to deduce the form factors of the models. They are essentially of three types: the twisted scalar product, the zero modes method and the coproduct property.
Speaker: Alexandre Krajenbrink (SISSA) (slides)
Title: A journey from classical integrability to the large deviations of the Kardar-Parisi-Zhang equation
Abstract: In this talk, I will revisit the problem of the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing a novel approach which combines field theoretical, probabilistic and integrable techniques. My goal will be to expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and to unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. I will show that this approach paves the path to understand the large deviations for general initial geometry.
Speaker: Véronique Terras (Université Paris-Saclay) (slides)
Title: Correlation functions by separation of variables: the Heisenberg spin chain
Abstract: The quantum version of the Separation of Variables approach (SoV) has recently been quite systematically developed to solve a wide variety of quantum integrable models, i.e. to characterize the spectrum and eigenstates of the corresponding Hamiltonian. However, the computation in this framework of other physical quantities such as correlation functions is a difficult problem which still remains widely open. In this talk, we review recent progresses that have been made in this context on the particular and simple example of the Heisenberg spin 1/2 chain.