# Integrable Models

## Introduction

Classical and quantum integrable models constitute a privileged class of models, ranging from mechanics, through hydrodynamics and up to statistical mechanics and field theory, that can be solved explicitly. This means that one is able to compute exactly, in closed explicit form, and deep in the non-perturbative regime, observables that are of interest to the physics of such models. These can range from the exact trajectory in classical mechanics or the per-site free energies in the statistical mechanics or one dimensional quantum mechanics settings at finite temperature, to more advanced quantities that bare a direct connection with experiments, like scattering matrices, form factors, correlation functions and associated dynamical structure factors. This is in outstanding contrast with the case of generic models where, at most, one might expect to access to similar observables solely in a perturbative regime and only for a few orders in the perturbative expansion.

The field of integrable models underwent several important breakthroughs over the last forty years, notably the emergence of new algebraic structures, related to classical and quantum R-matrices and associated Yang-Baxter algebras, that are at the root of integrability both in field theory and statistical mechanics in two dimensions. It led to the introduction of powerful algebraic techniques allowing one to construct and solve even larger classes of models. It also gave access to closed expressions for the correlation functions of several prominent quantum integrable models like the XXZ spin-1/2 chain, the system of Bosons subject to delta function interactions in one-dimension or the two-dimensional six-vertex model to name a few.

In particular, the Lyon group developed a new method based on the resolution of the quantum inverse scattering problem within the Yang-Baxter algebras and associated algebraic Bethe Ansatz frameworks, for computing exact form factors and correlation functions for integrable lattice models. Hence it allowed one to study, from first principle based calculations -viz. free from ad hoc arguments or approximations- various regimes of interest to the physics of these models, in particular in the range of couplings which gives rise to a critical model. There the main problem was to access directly to the universal behavior of the correlation functions, as predicted from various heuristics. For that purpose, the Lyon group devised new analytical methods which allowed one to study the universal critical regime and thus access directly, for the models of interest, to the critical exponents and also to the corresponding non-universal amplitudes that are model dependent. This was possible not only in the case of static correlation functions but as well for the dynamic ones, which are much less understood, even on heuristic grounds. The recent works of the Lyon group allowed to test the validity and range of application of the non-linear Luttinger liquid approach which aims at predicting the universal of dynamical correlation function in one dimensional gapless models. Moreover, the exact analysis pointed out several universal features which were missed by the heuristic approach. Also, one should note that the exact formulae for the form factors allowed the Lyon group to carry out high precision numerics and plot graphs of the dynamical structure factors -directly measured in experiments-, way beyond what can be achieved by more conventional tools (exact diagonalization, DMRG,...).

Another important activity of the Lyon group concerns integrable sigma models in 1+1 dimensions. Sigma models are of great interest both for high energy theories like string theories but also for condensed matter. One of the most studied cases related to string theories is the one describing the super-string evolution on AdS5 x S5. This theory plays a central role in the AdS/CFT correspondence; its integrable structure at classical level has been studied in great details in our group. Also, the development of a systematic method allowing for deformations of integrable sigma models while keeping their integrability properties is of great interest to the group. It enabled us to obtain new integrable sigma models and to study deformation families of such models to get further insights in their integrability structures and also their renormalization properties.

Recent and ongoing research works

Form factors and correlation functions in integrable spin chains

Building on earlier results, the correlation functions of Heisenberg spin chains in various regime (massless and massive) have continued to be investigated including the dynamical and the non-zero temperature cases. This lead to a detailed analysis of the dynamical response functions for the XXZ spin-1/2 chain in the massless regime and a close formula for the spin conductivity in the massive regime. Using the quantum transfer matrix framework, a thermal form factors expansion has been devised to obtain important exact results for the XXZ dynamical correlation functions at non-zero temperature, including the open spin chains case, and with very precise study of the XX limit.

##### Universality in quantum integrable models

The recent research of K.K. Kozlowski evolved around developing a better understanding of the emergence of a universal behavior for the correlation functions of an integrable model at criticality. A substantial progress was achieved in the understanding of the behavior of dynamical correlation functions in the XXZ chain be it at zero temperature or in the low-temperature regime. Furthermore, for various integrable models related to the six-vertex model (the Fortuin-Kasteleyn percolation, the Potts model) it was established rigorously that various non-trivial correlation functions exhibit rotational invariance in the scaling regime where universality is supposed to hold.

Quantum Separation of Variables

J.M. Maillet and G. Niccoli have focused their recent research activity on developing a new method allowing for the construction of separation of variable (SoV) bases for generic quantum integrable lattice models. It uses the very integrable structure of such models, i.e., a complete set of commuting conserved charges, and their associative (and commutative) algebra structure constants under quantum operator product. Being not an Ansatz, it provides the full description of the complete spectrum, and of the scalar products of states, and hence has also the potential to determine the dynamics of these models (form factors and correlation functions).

On the spectrum, the main results are for quantum integrable lattice models associated to different representations of higher rank Yang-Baxter (super-) algebras, up to the inhomogeneous Hubbard model, and under the most general integrable boundary conditions. It led to the complete characterization of the spectrum in terms of finite difference equations, known as quantum spectral curves, for higher rank generalization of Baxter’s equations, enhancing the central role played by the higher rank Q-operators for both fundamental and non-fundamental representations. On the dynamics, the main results are the exact computation of scalar products of Bethe states and of correlation functions for the class of rank-1 XXX and XXZ spin 1/2 quantum chains with general open integrable boundary conditions not accessible in the algebraic Bethe Ansatz framework. While for the higher rank case first results are about the scalar products of separate states which take a workable rank-1 form under special choices of the set of commuting conserved charges.

Integrable non-linear sigma-models

During the last years, F. Delduc and M. Magro have continued their investigations of non-linear integrable sigma models in 1+1 dimensions and their integrable deformations, including their renormalization properties, and their dualities, which are of interest for the corresponding string theories. These works raised the question of the possibility of going beyond the AdS/CFT correspondence and in particular if there exists a theory dual to the integrable deformation of the string theory on AdS5 x S5 that we constructed. On the quantum level, an important issue is to overcome obstructions to the quantization of sigma models using the standard Yang-Baxter structures that has been noticed thirty years ago. One possible way is to concentrate on the algebra of classical local conserved charges and the relations with Gaudin type models.

## Selected Publications

- N. Kitanine, J.M. Maillet, V. Terras, "Form factors of the XXZ Heisenberg spin-1/2 finite chain", Nucl. Phys. B 554 (1999) 647-678.
- N. Kitanine, J.M. Maillet, V. Terras, "Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field", Nucl. Phys. B 567 (2000) 554-582.
- N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, "Master equation for spin-spin correlation functions of the XXZ chain", Nucl.Phys. B712 (2005) 600-622.
- J.S. Caux and J.M. Maillet, "Computation of dynamical correlation functions of Heisenberg chains in a field", Phys. Rev. Lett. 95 (2005) 077201.
- N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov and V. Terras, "Algebraic Bethe Ansatz approach to the asymptotic behavior of correlation functions", J. Stat. Mech.: Th. and Exp. P04003 (2009).
- N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "Form factor approach to dynamical correlation functions in critical models", J. Stat. Mech.: Th. and Exp. P09001 (2012).
- F. Delduc, M. Magro, B. Vicedo, "On classical q-deformations of integrable sigma-models", JHEP (2013) 192.
- F. Delduc, M. Magro, B. Vicedo, "Integrable deformation of the AdS5 x S5 superstring action", Phys. Rev. Lett. 112 (2014) 051601.
- F. Delduc, M. Magro, B. Vicedo, "Derivation of the action and symmetries of the q-deformed AdS(5) x S (5) superstring", JHEP (2014) 132.
- K.K. Kozlowski and J.M. Maillet, "Microscopic approach to a class of 1D quantum critical models", J. Phys. A: Math and Theor. Baxter anniversary special issue, 48 (2015) 484004.
- F. Göhmann, M. Karbach, A. Klümper, K.K. Kozlowski and J. Suzuki, "Thermal form-factor approach to dynamical correlation functions of integrable lattice models.", J. Stat. Mech.: Th. and Exp. (2017) 113106.
- S. Lacroix, M. Magro, B. Vicedo, "Local charges in involution and hierarchies in integrable sigma-models", JHEP (2017) 117.
- J.M. Maillet and G. Niccoli, "On quantum separation of variables", J. Math. Phys. 59 (2018) 091417.
- F. Delduc, S. Lacroix, M. Magro, B. Vicedo, "Integrable Coupled Sigma-Models", Phys. Rev. Lett. 122 (2019) 041601.
- J.M. Maillet, G. Niccoli, L. Vignoli, "On Scalar Products in Higher Rank Quantum Separation of Variables", SciPost Phys. 9 (2020) 086.
- H. Duminil-Copin, K.K. Kozlowski, D. Krachun, I. Manolescu and M. Oulamara, "Rotational invariance of the planar random-cluster model.", math.pr:2012.11672.
- K.K. Kozlowski, "On convergence of form factor expansions in the infinite volume quantum Sinh-Gordon model in 1+1 dimensions.", math-ph:200701740
- C. Babenko, F. Göhmann, K.K. Kozlowski and J. Suzuki, "A thermal form factor series for the longitudinal two-point function of the Heisenberg-Ising chain in the antiferromagnetic massive regime.", J. Math. Phys. 62 (2021) 041901.
- K.K. Kozlowski, "On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain", J. Math. Phys. 62 (2021) 063507.
- G. Niccoli and V. Terras, "Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields", J. Phys. A: Math. Theor. 55 (2022) 405203.