The first of these workshops will take place April 11-14, 2016 at Wuppertal University.

This workshop will focus on the lattice realization of quantum field theories with non-compact degrees of freedom and on functional equations for the calculation of spectra and correlation functions of integrable lattice models.

P. Dorey, University of Durham

S. Eggert, TU Kaiserslautern

The emergence of new properties from low-dimensional building blocks is a universal theme in different areas in physics. The investigation of transitions between isolated and coupled low-dimensional systems promises to reveal new phenomena and exotic phases. Interacting 1D bosons, which are coupled in a two-dimensional array, are maybe the most fundamental example of a system which illustrates the concept of a dimensional phase transition. However, recent experiments using ultracold gases have shown a surprising discrepancy between theory and experiment. We discuss how the off-diagonal correlations in isolated chains can be used to predict the nature of the phase transition to a Bose-Einstein condensate. Comparison with large scale quantum Monte Carlo simulations show surprising discrepancies at low filling, which demonstrate that a careful analysis of different non-commuting limiting cases is necessary.

Y. Ikhlef, Université Pierre et Marie Curie

Non-Linear Integral Equations (NLIEs) were introduced in the 1990s as a powerful tool to study the scaling limit of integrable lattice models. Generically, the scaling theory describing the critical point of a spin chain is a Conformal Field Theory (CFT) with a discrete spectrum of scaling dimensions, which can be extracted from the NLIEs by a well-controlled procedure. In this talk, I will present a simple integrable spin-chain model whose Bethe Ansatz equations are governed by a singular kernel: I will explain how to derive the NLIEs in this situation, and how to treat the singularity to obtain the CFT spectrum. It turns out that the corresponding CFT is the SL(2, R)/U (1) “black hole” WZW model, a toy model of CFT with non-compact target space.

J.L. Jacobsen, ENS Paris

We describe how non-compact continuum limits arise from a_N-1⁽²⁾ spin chains based on the second baxterisation of the so(N) Birman-Murakami-Wenzl algebra. These chains can be physically realised, for N=3, as the Izergin-Korepin 19-vertex model or the dilute O(n) loop model, and for N=4 as two coupled antiferromagnetic Potts models. The non-compact physics arises in the so-called regime III, which for N=3 contains a theta point of polymers (n → 0 limit) with self-attraction.

The continuum limit of these models is identified by a series of arguments, including level-rank duality, RSOS restrictions, and numerical resolution of the Bethe Ansatz equations. For N=3, the corresponding conformal field theory turns out to be the Euclidean black hole sigma model. In particular, we show that the discrete states in the black hole model emerge from the non-compact continuum upon changing the twist of the spin chains. Some implications of non-compactness for the logarithmic behaviour of observables in the polymer problem are given.

C. Matsui, Tokyo University

There are few far-from-equilibrium systems which are analytically solvable. One of those examples is the asymmetric simple exclusion process (ASEP). The ASEP is an integrable two-state stochastic process in one dimension. The integrability of the model lies in the U_q(sl₂)-invariance of the bulk part. We consider the multi-state extension of the ASEP based on the fact that the Markov matrix of this process satisfies the Temperley–Lieb algebra. Besides the construction of steady states, we derive the exact expressions of particle-density profiles and currents on the steady states under the closed boundary condition. Although strong restrictions are imposed on hopping rates to keep integrability, we show that they are simplified in the limit q to 0.

Kh. Nirov, University of Wuppertal and INR Moscow

Modern approaches to quantum integrable systems are based on the notion of quantum groups. Here, the choice of a representation in the auxiliary space defines an integrability object, and by a representation in the quantum space one fixes a model subject to consideration. The functional relations between integrability objects follow from the characteristics of the representations of the quantum group. We discuss various representations of quantum loop algebras giving rise to different integrability objects and functional relations. Specifically, we give a comparative analysis of q-oscillator and prefundamental representations of the corresponding Borel subalgebras.

B. Pozsgay, Hungarian Academy of Sciences

In this talk we will discuss non-equilibrium situations of the XXZ chain, in particular time evolution from simple product states such as the N'eel state or the dimerized state. The focus will be on calculating the long-time limit of local observables, which can be performed using the Quench Action method. Two very important ingredients for this method are the overlaps with the initial state and the calculation of short range correlators in arbitrary excited states. We will discuss the latter topic in detail, and present a conjectured formula which calculates the correlation functions in arbitrary excited states of the finite XXZ chain. Our result builds on the theory of factorization of correlation functions, and it calculates the physical part of the construction using a finite set of Bethe roots. In the thermodynamic limit the formula leads to TBA-like sets of linear equations which can be solved effectively for arbitrary Bethe root distributions.

V. Schomerus, DESY Theory

The conformal bootstrap programme relies on the expansion of 4-point functions into kinematically determined conformal blocks. I will explain that conformal blocks of scalar 4-point functions in a d-dimensional conformal field theory can mapped to eigenfunctions of a 2-particle hyperbolic Calogero-Sutherland Hamiltonian. The link makes considerable mathematical developments in integrability and the modern theory of special functions available for conformal field theory.

F. Smirnov, Université Pierre et Marie Curie

In this talk I shall discuss the CFT limit of the fermionic basis and its connection with the reflection relations. Emphasis will be done on including the descendants created by the local integrals of motion.

J. Suzuki, Shizuoka University

The recent advance on the form factor expansion approach to correlation functions of the spin 1/2 XXZ model in massive regime will be reviewed. We put emphasis on the advantage in starting from the finite temperature problem with a non-vanishing magnetic field. This yields relatively simple expressions, without complicate multiple contour integrals. The comparison with other numerical approaches will be briefly commented. The talk is based on a collaboration with M. Dugave, F. Göhmann and K. K. Kozlowski.

V. Terras, Université Paris Sud

During the last decades, important progresses have been made concerning the computa- tion of form factors and correlation functions of simple models solvable by algebraic Bethe Ansatz (ABA) such as the XXZ spin-1/2 chain or 1D Bose gas with periodic boundary conditions. However, the generalization of these results to more complicated models or different types of integrable boundary conditions is for the moment limited by the range of applicability of ABA or by some difficulties of the method. In this talk, we discuss the solution of Heisenberg spin chains (XXX, XXZ or XYZ) in the framework of a complementary approach, Sklyanin’s quantum Separation of Variables approach. This enables us notably to consider for these models various types of boundary conditions (quasi-periodic, open…) not directly solvable by Bethe ansatz. More precisely, we discuss in this framework some new results and open problems concerning the description of the spectrum by means of solutions of a functional T- Q equation (or equivalently in terms of Bethe-type equations). We also discuss the problem of the computation of the eigenstates scalar products and of the form factors of local operators.

X. Zotos, University of Heraklion

I will discuss recent results on the evaluation of spin and thermal Drude weight in the spin-1/2 easy-axis Heisenberg chain and an application in far-out of equilibrium transport.

If you are interested in participating in this workshop, please send an e-mail message to any of the PIs.

PDF of abstracts: for_2316_workshop_2016_abstracts.pdf

The second workshop will take place September 05-08, 2017 at the Institute of Theoretical Physics, Leibniz Universität Hannover.

The second workshop on Correlations in Integrable Quantum Many-Body Systems will focus on applications of exactly solvable models to

• cold gases
• anyons and topological matter
• non-equilibrium, dynamics and transport
• form factors on higher rank (super-)spin chains

Stockholm University

In this talk, I will give an introduction to anyonic chains, starting from their mathematical formulation, using so-called anyon models, or more precisely, modular tensor categories. The main ingredients are the F-symbols, which can be obtained using quantum groups. The physics of anyonic chains will be discussed, focussing on the rich phase diagrams, and in particular, the integrability at special points.

SISSA

We present the theory of quantum quenches in near-critical one-dimensional systems [1,2]. Aspects that are discussed include role of interaction, role of integrability, appearance of timescales, analytic determination of one-point functions and their long time behavior.

References: [1] G. Delfino, Quantum quenches with integrable pre-quench dynamics, J. Phys. A 47 (2014) 402001 [2] G. Delfino and J. Viti, On the theory of quantum quenches in near-critical systems, J. Phys. A 50 (2017) 084004

University of Oxford

The spin-1/2 Heisenberg chain exhibits a quantum critical regime characterized by quasi long-range magnetic order at zero temperature. The strength of quantum fluctuations in the ground state can be quantified by determining the probability distributions of the components of the (staggered) subsystem magnetization. Some of these are shown to exhibit scaling and the corresponding universal scaling functions are determined by free fermion methods and by exploiting a relation with the boundary sine-Gordon model.

Université Grenoble Alpes

A universal decay power-law of the large-momentum tails of the momentum distribution, fixed by Tan’s contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. We derive the Tan’s contact of one-dimensional harmonically trapped gases, both in the case of a multicomponent Fermi gas and of a Bose gas. In the case of a multicomponent mixture, we find a direct correspondence between the value of the Tan’s contact and the symmetry of the state. We show that a local density approximation (LDA) on the Bethe-Ansatz equation of state for the homogeneous gas is in excellent agreement with the results for the harmonically confined gas and predicts a scaling behavior of the Tan’s contact. This provides useful analytical expressions for the dependence on the number of particles, number of components and on interaction strength. Based on work with J. Decamp, J. Jünemann, G. Lang, M. Albert, M. Rizzi, and P. Vignolo

Bergische Universität Wuppertal and Institute for Nuclear Research / Moscow

Universal integrability objects are determined by the choice of representations of the quantum group in the auxiliary space. We consider various representations of Borel subalgebras of quantum loop algebras to define such objects. We are especially interested in the highest ℓ-weight representations. Here we compare the q-oscillator and prefundamental representations and argue that the latter can be obtained by tensoring the former. We also discuss how relations between the corresponding highest ℓ-weights reproduce functional relations between integrability objects.

Institute for Space Science / Bucharest

For the most general case of an one-dimensional multi-component system (bosons, fermions or an arbitrary mixture) interacting through a delta function potential and subjected to an external potential we show that the large momentum distribution of these systems behaves like nσ∼Cσ/k4 with Cσ the contact of species σ which can be computed from the thermodynamic properties of the system (derivatives of appropriate thermodynamic potentials with respect to scattering lengths). We obtain short distance expansions for the Green's function and pair distribution function and show that the coefficients of these expansions can be expressed in terms of the density, kinetic energy and contact. In addition we derive universal thermodynamic identities for homogeneous and inhomogeneous systems connecting the pressure, total energy, trapping energy and contact. Based on work with A. Klümper.

Laboratoire de Physique Théorique / Annecy

We apply the nested algebraic Bethe ansatz to integrable models. 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 of two types: the twisted scalar product and the zero modes method.

Tokyo University of Science

In this talk, I will present our recent results on spin transport properties in the spin-1/2 XXZ spin chain at finite temperatures. Based on the functional relations among the row-to-row transfer matrices (T-systems) and their certain combinations (Y-system), the spin Drude weight and its size dependence are evaluated. This talk is based on joint work with Andreas Klümper.

Utrecht University

We study the effect of interactions on Kitaev's toy model for Majorana wires. We demonstrate that even though strong repulsive interaction eventually drive the system into a Mott insulating state the competition between the (trivial) band-insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions. We show that the exact ground states can be obtained analytically even in the presence of interactions when the chemical potential is tuned to a particular function of the other parameters. The ground states obtained are two-fold degenerate and differ in fermion parity, as is the case with the Kitaev/Majorana chain in a topological phase. We prove that the ground state is unique in each fermion parity sector and that there exists an energy gap. Furthermore, we investigate the effect of disorder in the chemical potential. We find that, like the non-interacting system, moderate disorder supports the topological phase, while at large disorder strengths the system becomes trivial. Based on work with Niklas Gergs, Fabian Hassler, Hosho Katsura, Masahiro Takahashi

Steklov Mathematical Institute

We apply the nested algebraic Bethe ansatz to the models with gl(2|1) symmetry. We obtain explicit representations for the Bethe vectors scalar products. In some particular cases we find determinant formulas for the scalar products. Starting from these formulas and using the zero modes method we obtain compact determinant representations for the form factors of the monodromy matrix entries. The latter, in their turn, lead us to determinant formulas for form factors of local operators.

Université Pierre et Marie Curie

It has been shown that using the fermionic basis the correlation functions for XXZ model on a cylinder are expressed in universal form for arbitrary Matsubara data. We use this arbitrariness in order to fix the coefficients in the expansion of operators in the fermionic basis. For XXX case this allows to compute the correlation functions up to 11 sites. The dependence of coefficients on the lattice spacing exhibits remarkable regularity.

Shizuoka University

We discuss a quantitative analysis on the correlation functions of spin 1/2 XXZ model based on the quantum transfer matrix. The static and the dynamical aspects of the correlations will be discussed within the framework of form factor expansions. The content of the talk is based on collaborations with M. Dugave, F Göhmann, A. Klümper and K. K. Kozlowski.

Brookhaven National Lab

We have constructed a 2D lattice model of spins 1/2 interacting with nearest neighbor 2- and 3-spin interactions which have gapped bulk and gapless chiral excitations on the edges. The bulk excitations are 2D Majorana fermions and 1D solitons. The 2D propagating Majorana's are bound states of fractionalized 1D solitons. The model has two phases – Abelian and non-Abelian topological ones.

SISSA

While much progress has been made over the last years in understanding the relaxation mechanisms taking place in the non-equilibrium dynamics of quantum many-body systems, very few analytical results exist concerning the full time evolution of physical observables, however urgently called for by the ongoing advances in cold-atomic experiments as well as by the recently emerged subject of dynamical phase transitions. A reason for this fact is that despite the existence of prototypical integrable models, the time dynamics involves contributions of arbitrarily excited eigenstates of the Hamiltonian which render exact calculations prohibitively difficult. In this seminar I will present a method to tackle the exact-time dynamics of quantum observables in a prototypical interacting integrable quantum many-body system, the Heisenberg XXZ spin chain, starting with a simple observable that is the Loschmidt echo (or quantum fidelity). The latter measures the overlap between the system's state at a given time and its initial state, and has attracted a renewed interest recently in the context of dynamical phase transitions, which it signals through its non-analyticities as a function of the time. Using a reformulation of the problem in terms of an auxiliary boundary quantum transfer matrix, the Loschmidt echo is written as the solution of a set of Non Linear Integral Equations, which allows for its exact determination at arbitrarily large time. This method overcomes the time limitations experienced by numerical approaches, and allows to for an analytic approach to dynamical transitions. I will further discuss perspectives concerning the classification of “integrable initial states” allowing for such exact computations, as well as the extension of this method to the study of more general physical observables. This is based on L Piroli, B Pozsgay, E Vernier, J. Stat. Mech., 023106 (2017), as well as some ongoing work.

The third workshop will take place September 03-07, 2018 at Wuppertal University

The third workshop on Correlations in Integrable Quantum Many-Body Systems will focus on lattice realizations of integrable quantum field theory and on applications of integrable models to cold atomic gases and condensed matter physics.

Australian National University

In this talk we revisit some unsolved problems of Quantum Field Theory, in particular, the canonical quantization of two-dimensional non-linear sigma models (NLSM) in two dimensions. We review the ways how the long-standing “non-ultralocality” problem can be resolved. In particular, we consider the O(3) NLSM and its one-parameter deformation — the sausage model. Our consideration is based on the continuous version of the the Quantum Inverse Scattering Method enhanced by a powerful ODE/IQFT correspondence, which connects stationary states of Integrable QFT models with special solutions of classical integrable equations. This approach leads to new efficient methods for computation of vacuum eigenvalues of the continuous analogs of quantum transfer-matrices in QFT.

Institut Optique Palaiseau

Losses are always present in a cold atom gases and they are dominated either by one-body, two-body or three-body loss processes. Although they strongly impact the state that can be prepared in cold atoms experiments, their effect has not been investigated in detail. As a dissipative process, they are usually considered as detrimental for cold gases. However, it was recently shown that a simple loss process, characterized by a one-body loss rate, independent of the atoms energy, could lead to cooling [B. Rauer et al., Phys. Rev. Lett. 116, 030402 (2016)]. We generalize those studies and investigate theoretically the effect of loss process on Bose Einstein condensates or quasicondensates in any dimension, for any j-body loss process, and for homogeneous gases as well as cloud confined in a smoothly varying trapping potential. We focus on low energy collective modes, the phonon modes. On the one hand losses produce a reduction of the energy in each phonon mode since a smaller amplitude of density modulations lowers the interaction energy of each phonon mode. On the other hand, the shot noise due to the discrete nature of losses is responsible for an increase of the density fluctuations in the gas, and thus increases the energy in each mode. The competition between these two processes leads to a decay of the temperature such that the ratio between $k_B T$ and $mc^2$, where $m$ is the atomic mass and $c$ the speed of sound, becomes asymptotically a constant, of the order of unity. On the experimental side, we demonstrate cooling by 3-body losses in a 1D Bose gas in the quasi-condensate regime and we identify the asymptotic value of the ratio $k_BT/(mc^2)$.

University of Amsterdam

Recent years have demonstrated that integrability can be used to compute many physical properties of experimentally-accessible magnetic or cold atomic systems. Besides their rich equilibrium dynamics, such systems also host relaxation and equilibration behaviour which cannot be simply described by traditional textbook methods, and can lead to long-lived non-thermal equilibrium states. This talk will provide an overview of recent work in this area, and introduce some new methods to treat quenched and driven systems in various experimentally-relevant contexts.

University of Durham

ENS de Paris

We show how the recently developed generalized hydrodynamic theory - an exact hydrodynamic description of the homogeneous non-equilibrium time evolution of integrable models - has been very efficient to

a) determine the non-equilibrium steady states emerging from joining two different chains in two different equilibrium states,

b) provide an exact value of Drude weights of integrable chains, including XXZ and Fermi-Hubbard, for any finite temperature state and also give a rigorous and group-theory related interpretation of ballistic transport at finite temperature,

c) construct a theory for diffusive transport in integrable models, where two-body scatterings among the quasiparticles completely provide the diffusion terms, namely the Navier-Stokes corrections to the Euler-scale hydrodynamic equations.

King's College London

I will discuss propositions for exact asymptotic of certain dynamical correlation functions in generalised Gibbs ensembles (GGEs) of integrable models, at large space-time scales. These propositions are based on the theory of generalised hydrodynamics (GHD), which, in its current form, combines principles of hydrodynamics with the thermodynamic Bethe ansatz formulation of GGEs. I will overview the main elements of GHD. I will then explain how correlation functions at large space-time distances — the so-called Euler scale — are studied within the statistical theory of fluids, and apply this to GHD. The results conjecturally hold for a wide family of integrable models, including quantum and classical field theory and chains, and for many (but not all) observables of interest. I will present numerical confirmations in the classical sinh-Gordon model. If time permits, I will also discuss correlation functions in inhomogeneous, non-stationary states that can be locally described by local GGEs. This is based on various works on GHD, including in particular works with Herbert Spohn and with Alvise Bastianello, Gerard Watts and Takato Yoshimura.

University of Florence

Ultracold gases of neutral atoms are a powerful resource for engineering synthetic many-body quantum systems. In a “quantum simulation” perspective, it is possible to control the atomic state to provide direct experimental realizations of fundamental theoretical models, and to achieve “extreme“ states of matter with no counterpart in conventional materials.

I will report on recent experiments performed at University of Florence with degenerate gases of ultracold 173Yb fermions. These two-electron atoms exhibit a rich internal structure, with distinct degrees of freedom – nuclear spin and electronic state – that can be both manipulated in a quantum coherent way. In particular, the control of the nuclear spin allowed us to study the many-body physics of multicomponent fermions with SU(N) interaction symmetry, with intriguing properties emerging when the atoms are trapped in low dimensions [1]. Furthermore, by coupling different internal states we have demonstrated the possibility of engineering “synthetic dimensions”, in which effective lattice dynamics are encoded in the internal Hilbert space of single atoms. By using this approach, we have demonstrated new techniques for the production of synthetic gauge fields for neutral atoms and observed the emergence of edge currents in fermionic ladders with tunable flux [2,3].

[1] G. Pagano et al., Nature Phys. 10, 198 (2014).

[2] M. Mancini et al., Science 349, 1510 (2015).

[3] L. F. Livi et al., Phys. Rev. Lett. 117, 220401 (2016).

UFRGS Porto Alegre

In this work we investigate some integrable models in the context of ultracold atoms. Our main focus will be a general construction of integrable models for boson tunneling in multi-well systems, with a particular emphasis on the triple-well Hamiltonian and on a four-well ring model for bosons, where the tunneling couplings between nearest-neighbour wells are not restricted to be equal. Algebraic aspects of this construction and the effects of breaking the integrability in some cases will be presented. As an application we discuss how to engineer an atom-tronic switching device by breaking the integrability of the triple-well system.

University of Aberdeen

In the talk we survey the recent work of Nekrasov, Shatashvili, Okounkov and others on the connection between objects in the title.

Wigner Institute, Budapest

The 6-vertex model with appropriately chosen inhomogeneities gives the so-called light-cone lattice regularization of the Massive Thirring model. This regularization, through the quantum inverse scattering method offers an appropriate framework for computing the finite volume form-factors of local operators of the theory. This approach works efficiently, when the diagonal matrix elements of local operators are considered.

In this talk I will discuss:

- the computation of pure solitonic diagonal matrix elements of the U(1) current and the trace of the stress energy tensor,

- the large volume behavior of these finite volume diagonal form-factors,

- and the continuum limit of the norm of Bethe-eigenstates, as well.

ENS de Lyon

I will discuss the progress achieved recently in the description and thorough characterisation of dynamical correlation functions of the XXZ spin-1/2 chain, this for the zero- and the finite-temperature cases.

In particular, I will present the large-distance and long-time asymptotic expansion of the zero temperature two-point dynamical correlation functions in the massless regime of the model. I will also outline the singular behaviour in the momentum k and energy ω plane of zero temperature dynamic response function, viz. space and time Fourier transforms of dynamical two-point functions. Finally, I will discuss some of the features of the series of multiple-integral representations for the finite temperature dynamical two- point functions.

Universität zu Köln

Low-dimensional quantum spin systems offer an ideal playground to study the generic behavior close to magnetic-field induced quantum phase transitions. Of particular interest is the XXZ spin-1/2 chain model with the anisotropy parameter ∆=Jz/Jxy between the exchange couplings acting either on the z- or the x(y)-spin components. For ∆=0, 1 and ∞, respectively, the exactly solvable XY, Heisenberg and Ising chain models are covered, but real materials are typically located in between these models and there are additional intra- and/or inter-chain couplings. In this contribution, I will present our recent experimental studies of model materials with field-induced quantum phase transitions. Cu(C$_4$H$_4$N$_2$)(NO$_3$)$_2$ is an almost idealrealization of the Heisenberg chain with weak intra-chain coupling, such that we could study its quantum critical behavior in great detail and we find nearly perfect agreement with Bethe-Ansatz calculations [1]. The Co$_2$+ ions of BaCo$_2$V$_2$O$_8$ represent effective spin-1/2 chains with pronounced Ising anisotropy. Here, sizeable inter-chain couplings cause long-range antiferromagnetic order with complex magnetic-field temperature phase diagrams [2]. For a particular transverse-field direction, however, we find that the field-induced suppression of the three-dimensional magnetic order is well separated from the critical field of the one-dimensional transverse-field Ising quantum phase transition [3].

[1] O. Breunig, T.L. et al., Science Advances 3, eaao3773 (2017).

[2] S. Niesen, T.L. et al., Phys. Rev. B 87, 224413 (2013).

[3] Zhe Wang, T.L. et al., Phys. Rev. Lett. 120, 207205 (2018).

Rutgers University

A common approach to the quantization of integrable models 
starts with the formal substitution of the Yang-Baxter Poisson algebra with 
its quantum version.
 However it is
 difficult to discern 
 the presence of such an algebra
 for the so-called non-ultralocal models. 
 The latter includes the class of
 non-linear sigma models which are most interesting
 from the point of view of applications.
 In this talk I discuss the emergence of the Yang-Baxter Poisson algebra
 in a non-ultralocal system 
which is related to integrable deformations of the Principal Chiral Field.

ENS de Lyon

I will present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. I will show how this general scheme applies concretely to fundamental models associated to the $Y(gl_2)$ and $Y(gl_3)$ $R$-matrices leading to the full characterization of their spectrum. For $Y(gl_2)$ and its trigonometric deformation a particular case of the method reproduces Sklyanin's construction of separate variables. For $Y(gl_3)$ it gives new results, in particular through the proper identification of the shifts acting on the separate basis. This method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including in particular trigonometric and elliptic spin chains, open chains with general integrable boundaries, and higher rank cases like $Y(gl_n)$. This is a common work with G. Niccoli that appeared recently as arXiv:1807.11572.

University of Tokyo

We discuss two representative problems of the XXZ spin chain in non-equilibrium state.

1. The relaxation state. Recently, the eigenstate thermalization hypothesis (ETH) has been proposed as the mechanism for isolated quantum systems to thermalize. While the ETH holds for all energy eigenstates in non-integrable cases, not all energy eigenstates obey the ETH in integrable cases. Instead, it is believed that the generalized Gibbs ensemble (GGE), which is the generalization of the Gibbs ensemble including many conserved charges, describes the relaxation state. We discuss which set of conserved charges are enough to describe the relaxation state.

2. Ballistic transport of spin current. There is a long history about the discussion whether current in one-dimensional integrable system remains finite or not at finite temperature. It has been shown that the overlap between the current operator and conserved charges leads to a ballistic channel of current. By focusing on the spin current of the XXZ model, we discuss the locality of conserved charges and how locality works in ballistic transport of spin current.

Institute for Space Science Bucharest

We investigate the universal thermodynamics of the two-component one-dimensional Bose/Fermi gas with contact interactions in the vicinity of the quantum critical point separating the vacuum and the ferromagnetic liquid regime/Tomonaga-Luttinger liquid. We find that the quantum critical region belongs to the universality class of the spin-degenerate impenetrable particle gas which, surprisingly, is very different from the single-component case and identify its boundaries with the peaks of the specific heat. In the Tonks-Girardeau regime the Tan contact develops a pronounced minimum, reflected in a counter-intuitive narrowing of the momentum distribution as we increase the temperature. This momentum reconstruction, also present at low and intermediate momenta, signals the transition from the Tomonaga-Luttinger/ferromagnetic regime to the spin-incoherent Luttinger liquid phase and can be detected in current experiments with ultracold atomic gases in optical lattices. Based on joint work with A. Klümper and A. Foerster.

Budapest University of Technology and Economics

We review recent progress in exact methods for quantum quenches, focusing on a sub-class of problems called integrable quenches. In these problems the initial state for the time dynamics shows a number of special features related to integrability, for example the existence of factorized overlap formulas that display the so-called pair structure for the Bethe roots. The integrable states can be embedded into the framework of boundary integrability by relating them to integrable boundary conditions in the so-called mirror (or rotated) channel, in close analogy with the classic results of Ghoshal and Zamolodchikov. This leads to a fusion hierarchy, and generalized TBA equations for the Loschmidt amplitude. Furthermore it provides a way to rigorously derive the extensive part of the overlap formulas, even in cases that were unaccessible to other methods. The construction can be generalized to certain Matrix Product States that were obtained recently in the AdS/CFT setting: they correspond to operator valued solutions to the Boundary Yang-Baxter relation. We review a number of cases in SU(N) and SO(N) symmetric models for general N, with more detailed computations for N=2 and N=3.

Institute for High Energy Physics Protvino

We extend the graphical approach to investigation of integrable quantum vertex statistical models and the corresponding quantum spin chains. The most general form of the commutativity conditions for the transfer operators of lattices with boundary are derived by the graphical method.

Universität Duisburg-Essen

The confinement phenomenon occurs when the constituents of a compound particle cannot be separated from each other and therefore cannot be observed directly. A prominent example in high-energy physics is the confinement of quarks in hadrons. It is remarkable, that confinement can also be realized in such condensed matter systems, as quantum quasi-one-dimensional ferro- and anti-ferromagnets. After a brief review of recent progress in the experimental and theoretical study of this phenomenon, I describe analytical perturbative calculations of the energy spectra of the two-kink bound states in the antiferro- magnetic massive XXZ spin-chain model in the presence of a weak staggered longitudinal magnetic field. The two-kink bound-state spectra in this model have recently been studied numerically by A. K. Bera et al. [Phys. Rev. B 96, 054423 (2017)] in order to interpret the magnetic excitation energy spectra in the quasi-one-dimensional compound SrCo$_2$V$_2$O$_8$ investigated by neutron scattering. The obtained analytical results for the two-kink energy spectra are in a nice agreement with the available numerical results by A. K. Bera et al.

Shizuoka University

The recent development on the equilibrium dynamics of quantum spin chains based on quantum transfer matrix formalism will be reviewed. Some explicit numerical evidences showing the efficiency of the method will be presented. The content of the talk is based on the collaboration with K. K. Kozlowski and F. Göhmann.

Penn State University

I will describe experiments that observe the evolution of the momentum distributions of expanding 1D Bose gases. The equilibrium momentum distribution is fairly peaked (Bose-like), and during expansion it rapidly evolves into a flattened distribution (Thomas-Fermi like). The final distribution directly maps onto the distribution of rapidities, which are the underlying conserved quantities in these very nearly integrable, quantum degenerate, many-body systems. We can do this experiment in both the intermediate and strong coupling limits

The fourth workshop will take place April 11-13, 2019 at Wuppertal University

The fourth workshop on Correlations in Integrable Quantum Many-Body Systems will focus on subjects related to the work of our research group member and dear friend Holger Frahm. We will take the opportunity to congratulate him on the occasion of his 60th birthday.

University of Oxford

I will discuss applications of quantum integrability to open quantum systems described by Lindblad equations. One such example is the imaginary-U Hubbard model.

Université de Bourgogne, Dijon

We propose a method based on the algebraic Bethe ansatz leading to explicit results for the form factors of quantum spin chains in the thermodynamic limit. Starting from the determinant representations we retrieve in particular the formula for the two-spinon form factors for the isotropic XXX Heisenberg chain obtained initially in the framework of the q-vertex operator approach.

Bergische Universität Wuppertal

We evaluate the energy level curvatures of typical states of thermal ensembles by combining the thermodynamic Bethe ansatz with the functional relations of row-to-row transfer matrices of $Y$-system type. As a result, the level curvatures taken at zero twist angle converge to the results obtained by Zotos (1999), however with very slow convergence upon increase of the system size. This strong size dependence may explain that extrapolations from various numerical approaches yield conflicting results.

In a second part we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators which prove conjectures within the generalized hydrodynamics formalism. We numerically evaluate the Drude weight for anisotropies $\Delta=\cos(\gamma)$ with $\gamma = n\pi/m$, $n\leq m$ integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general $\gamma=\pi n/m$ agree with the bound which has been obtained by the construction of quasi-local charges. We observe fractal behaviour of $D(T,\Delta)$ as function of $\Delta$ for any $T>0$.

ENS de Lyon

In this talk, I will explain how, starting from the recently obtained thermodynamic limit of a form factor series expansion in the massless regime of the XXZ chain, one can extract the long-time and large-distance asymptotic behaviour of two-point correlation functions. The analysis unravels two structurally different regimes of the asymptotics. Furthermore, it connects the asymptotic properties of the correlators to the singularity structure of the model’s form factors.

Universität zu Köln

Universality is a well-established central concept of equilibrium physics. However, in systems far away from equilibrium, a deeper understanding of its underlying principles is still lacking. Up to now, a few classes have been identified. Besides the diffusive universality class with dynamical exponent z=2, another prominent example is the superdiffusive Kardar-Parisi-Zhang (KPZ) class with z=3/2. The latter is closely related to the asymmetric simple exclusion process which has a time evolution generated by a nonsymmetric Heisenberg quantum spin chain. We will summarize recent work which has shown that both universality classes are only part of an infinite discrete family of nonequilibrium universality classes that have dynamical exponents which are given by ratios of neighboring Fibonacci numbers.

Steklov Institute, Moscow

In the last decade, there has been a significant progress in calculating scalar products of Bethe vectors in the nested algebraic Bethe ansatz solvable models. Determinant representations were obtained for many particular cases of scalar products. We describe a new generating function for these determinants. This generating function allows one to obtain both already known determinant formulas and new representations.

Université Pierre et Marie Curie

Using the fermionic basis we conjecture exact expressions for the diagonal finite volume matrix elements of the exponential operators and their descendants in the sinh-Gordon theory. Our expressions sum up the LeClair Mussardo type infinite series generalized by Pozsgay and can be used to generate symmetric and connected form factors efficiently. We checked our formulas against the Liouville three-point functions for small, while against Pozsgay’s expansion for large volume.

Joint work with Z. Bajnok

Shizuoka University

Recently we proposed a novel framework in the quantitative study of the equilibrium dynamics of quantum spin chains based on the quantum transfer matrix method. As a prototype, here we consider the transverse correlation function of the spin 1/2 XX chain as a concrete example. A Fredholm determinant representation, which seems different from the one discussed by Colomo, Izergin, Korepin and Tognetti, efficiently produces high-precision data for a wide range of temperature, magnetic field, space and time. The content of the talk is done in collaboration with F. Göhmann, K. K. Kozlowski and J. Sirker.

Brookhaven National Laboratory

I demonstrate that the exchange interactions between itinerant and localized electrons on quasi-1D Kondo lattice may give rise to an exotic spin liquid state combining staggered odd-frequency superconducting order with quasiparticles. This state has zero Hall response and becomes metallic when superconductivity is destroyed by magnetic field. I argue that this explains recent experiments in the striped LBCO.

The fifth workshop will take place March 30 - April 1, 2020 at the Technical University of Kaiserslautern

The fifth workshop on Correlations in Integrable Quantum Many-Body Systems is the opening workshop of the second funding period of our research unit. One purpose is to discuss the perspectives of our projects with our guests.

Université de Lorraine, Nancy

After a brief introduction to “Generalized Hydrodynamics” [1,2], I will present the experimental results obtained recently by I. Bouchoule and M. Schemmer (Institut d’Optique, Palaiseau) which confirm that Generalized Hydrodynamics is the right framework to describe the dynamics of the Lieb-Liniger gas [3]. If time permits, I will also discuss a first attempt at quantizing Generalized Hydrodynamics, which results in a generalized Luttinger liquid [4].

[1] O. Castro-Alvared, 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] P. Ruggiero, P. Calabrese, B. Doyon, J. Dubail, “Quantum Generalized Hydrodynamics”, arXiv:1910.00570

Technische Universität Kaiserslautern

The recent development of Floquet engineering has made periodic driving a versatile tool for achieving new phases not accessible in static equilibrium systems. We now study the exact Floquet steady states of the periodically driven Tomonaga-Luttinger liquid without resorting to any high frequency approximations. We show that the timedependent Schrödinger equation can be solved analytically for a large class of driven interacting 1D systems, which give the resulting nonequilibrium steady states. Remarkably, we observe regions of instabilities as a function of total momentum where the solution is not of Floquet form, which implies a loss of time translational invariance and therefore heating of excitations. For small driving amplitudes the instabilities are close to the naively expected resonance condition nω = 2vq, but for stronger driving the heating regions separate a rich structure of bands of steady state solutions. Physical consequences are discussed.

Eberhard Karls Universität Tübingen

Synthetic quantum many-body systems made of ultracold atoms in optical lattices open new possibilities to study fundamental many-body effect in a extremely clean and well controlled environment. Here we discuss recent experiments on Fermi-Hubbard chains, where we studied correlations (or their absence) between spins and charges microscopically. The capability to detect the spin state of each atom in the chain provides access to the full counting statistics and allows to measure non-local multipoint correlations. Their analysis reveals signatures of spin-charge separation in equilibrium systems, an effect which we also probe by following the real time dynamics after a local quench.

Bergische Universität Wuppertal

We study short-distance correlation functions of the homogeneous XXZ chain by means of the hidden fermionic structure discovered by Boos et.al. Short-distance correlation functions appear in various physical applications as e.g. in the calculation of moments of ESR spectral lines. Using this framework we consider two possible approaches. The first one makes use of the so called fermionic basis and the JMS-Theorem and the second one uses the construction known as the exponential form of the density matrix. Both approaches are used to explicitly calculate correlation functions on the computer. We will show results for two-point functions at distance $n=5$ for a wide range of temperatures and magnetic fields which have been previously unknown.

ENS de Lyon

Within the approach of the bootstrap program, the physically pertinent observables in a massive integrable quantum field theory in 1+1 dimensions are expressed by means of the so-called form factor series expansion. This corresponds to a series of multiple integrals in which the $n$th summand is given by a $n$-fold integral. While being formally effective for various physical applications, so far, the question of convergence of such form factor series expansions was essentially left open. Still, convergence results are necessary so as to reach the mathematical well-definiteness of such construction and appear as necessary ingredients for the justification of numerous handlings that are carried out on such series.

In this talk, I will describe the technique I recently developed that allows one to prove the convergence of the form factor series that arise in the context of the simplest massive integrable quantum field theory in 1+1 dimensions: the Sinh-Gordon model. The proof amounts to obtaining a sufficiently sharp estimate on the leading large-$n$ behaviour of the $n$-fold integral arising in this context. This appeared possible by refining some of the techniques that were fruitful in the analysis of the large-$n$ behaviour of integrals over the spectrum of $n \times n$ random Hermitian matrices.

Helmholtz Zentrum Berlin and TU Berlin

Complex bound states of magnetic excitations, known as Bethe string, were predicted almost a century ago to exist in one-dimensional quantum magnets. The dispersions of the string states have so far remained the subject of intensive theoretical studies. By performing neutron scattering experiments on the one-dimensional Heisenberg–Ising antiferromagnet SrCo2V2O8 in high longitudinal magnetic fields, we reveal in detail the dispersion relations of the string states over the full Brillouin zone, as well as their magnetic field dependences. Furthermore the characteristic energy, the scattering intensity and linewidth of the observed string states exhibit excellent agreement with precise Bethe Ansatz calculations. Our results establish the important role of string states in the quantum spin dynamics of one-dimensional systems, and will invoke studies of their dynamical properties in more general many-body systems.

Work jointly with Anup Kumar Bera, Jianda Wu, Wang Yang, Robert Bewley, Martin Boehm, Jianhui Xu, Maciej Bartkowiak, Oleksandr Prokhnenko, Bastian Klemke, A. T. M. Nazmul Islam, Joseph Mathew Law, Zhe Wang

Université catholique de Louvain

The connection between statistical mechanics and the combinatorics of alternating sign matrices is known since the work of Razumov and Stroganov on the spin-1/2 XXZ chain. One important example of this combinatorial relation occurs in the study of the emptiness formation probability $EFP_{N,m}$. This observable is defined as the sum of the squares of the ground state components for the chain of length $N$, restricted to components where $m$ consecutive spins are aligned. At the combinatorial point $\Delta = - 1/2$, it takes the form of a simple product of integers. This was proven in 2012 by L. Cantini, who also found a combinatorial interpretation for these probabilities in terms of plane partitions.

In joint work with C. Hagendorf and L. Cantini, we define a new family of overlaps $C_{N,m}$ for the spin-1/2 XXZ chain. It is equal to the linear sum of the groundstate components that have $m$ consecutive aligned spins. For reasons that will be discussed, we refer to the ratio $C_{N,m}/C_{N,0}$ as the boundary emptiness formation probability. We compute $C_{N,m}$ at the combinatorial point as a simple product of integers.

Institute for Space Science Bucharest

We report results on the asymptotics of the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions and fractional statistics. Using the Fredholm determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry.

Bergische Universität Wuppertal

Bethe Ansatz equations for rapidities (or Bethe roots) of integrable systems sometimes allow for special singular solutions: namely, Bethe roots giving zero contribution to the energy. We study circumstances under which such “fantom” singular Bethe roots appear, which can be mixed with usual nonsingular Bethe roots.

In open spin chain, singular Bethe roots appear for a subset of root of unity anisotropies. In open XXZ spin chain, singular Bethe roots appear for fine tuned off-diagonal boundary fields. In both closed and open cases, singular Bethe roots signalize appearance of special chiral eigenstates of the respective integrable Hamiltonians, completely factorized spin-helix state being an example. We describe the structure and hierarchy of these chiral states. The number of chiral eigenstates in the spectrum can be exponentially large, while the states themselves, remarkably, can be generated in the XXZ spin chain with boundary dissipation in the Zeno regime.

Our results clarify and add to previous studies [Wang, Y., Yang, W.-L., Cao, J. and Shi, K.,Off-Diagonal Bethe Ansatz for Exactly Solvable Models, Springer 2016]

University of Ljubljana

Recently, clear observations of Kardar-Parisi-Zhang dynamical scaling of spin-spin correlations have been made in isotropic Heisenberg (XXX) spin 1/2 chain. Similar observations have later been made in classical integrable counterparts of SU(2) chains, the so-called lattice Landau-Lifshitz models.

In my talk, I will define a general class of classical symplectic many-body dynamics defined on a discrete space-time lattice with general nonabelian compact Lie group symmetry, say SU(N) or USp(2N) for arbitrary N, defined in terms of explicit matrix rational functions. After demonstrating integrability by explicitly constructing conserved transfer matrices generating local or quasi-local conservation laws, I will review numerical results which clearly suggest KPZ superuniversality with dynamical exponent 3/2 independent of the non-abelian symmetry group of the model.

Université Pierre et Marie Curie

Computing the ratios of the one-point functions of the primary field in the sine-Gordon model one arrives at certain consistency relations for the function $\omega$ which is the building block of the entire procedure. In this talk I shall explain the problem and prove these consistency relations.

Shizuoka University

A new framework has been proposed in the evaluation of the dynamical correlation function based on QTM [1]. The efficiency of the method has been exemplified for the XX model [2]. Here we are applying the scheme to a truly interacting spin system. Preliminary results will be presented on the ground state correlation function of the antiferromagnetic XXZ spin chain. The content of the talk is done in collaboration with F. Göhmann, K. K. Kozlowski and J.Sirker.

[1] F. Göhmann, M. Karbach, A. Klümper, K. K. Kozlowski, J. Suzuki, JSTAT (2017) 113106.

[2] F. Göhmann, K. K. Kozlowski, J. Sirker, J. Suzuki, Phys. Rev. B 100, 155428 (2019), J. Math. Phys. 61, 013301 (2020).

University of Manitoba

Diffusive and ballistic transport channels are known to simultaneously coexist in the XXZ model. This is demonstrable at low-temperatures from a combined bosonization/ self-energy/ memory matrix approach. At zero temperature the conductivity is entirely contained in the Drude peak (ballistic channel). Whereas for finite temperatures the Drude peak melts through Umklapp scattering and spreads into a Lorentzian in frequency space. This Lorentzian picture indicates that the diffusion constant should depend on the sub-leading corrections to the low-temperature Drude weight.

To obtain the low-temperature corrections to the Drude weight involves analyzing Zotos' formula, which is found to depend very strongly on the small rapidity region. This structure simplifies the auxiliary function formalism and consequently result in the low-temperature correction. Qualitatively, the resulting diffusion constants agree with those from generalized hydrodynamics.

The sixth workshop will take place August 24-26, 2021 at the University of Wuppertal.

The sixth workshop on Correlations in Integrable Quantum Many-Body Systems is our first attempt to resume something like normal scientific communication after long months of being affected by the prevailing Covid 19 pandemic.

• Z. Bajnok, Wigner Research Center for Physics Budapest
• S. Belliard, LMPT, Université de Tours
• A. Morin-Duchesne, MPI für Mathematik Bonn
• E. Granet, University of Oxford
• G. Kotoousov, DESY Hamburg
• V. Pasquier, CEA Saclay
• D. Serban, CEA Saclay
• G. Takacs, Budapest University of Technology and Economics

In this talk I will review the recent developments of the finite size effects of form factors in integrable relativistic quantum field theories. I will focus only on diagonally scattering theories, but cover both diagonal and non-diagonal form factors.

I will discuss a way to calculate spectrum and states for models without U(1) symmetries such as the open XXZ spin chain. The modified algebraic Bethe ansatz will be described in that case, and off-shell action of the related transfer matrix will be established. It will allows to calculate Slavnov’s formula for scalar product of the associated states.

The calculation of out-of-equilibrium dynamics in integrable models has been the object of increasing interest in the past decade. I will explain how to compute out-of-equilibrium expectation values and correlation functions from form factor expansions in non-interacting models and perturbations around them, by outlining the main difficulties and their solutions. Specifically, I will take the examples of the XY model in arbitrary varying magnetic fields and anisotropy and of the Lieb-Liniger model at strong coupling. While the Quench Action remains the preferred approach to interacting models, I will show that in the XY model the use of coherent states is more powerful and natural.

The inhomogeneous six-vertex model is a multi-parametric integrable 2D statistical system. With the anisotropy parameter |q|=1, the model is critical and covers a variety of interesting universality classes. For instance, in the homogeneous case, the lattice system is related to the XXZ spin 1/2 chain, whose critical behaviour is governed by a free boson. The so-called staggered six-vertex model has attracted a lot of attention in the past 15 years in connection with the 2D Euclidean and Lorentzian black hole sigma models. The goal of the talk is to describe a conjecture from ref. arXiv:2106.01238. It predicts the scaling limit of the general inhomogeneous six-vertex model and its spin J = 1, 3/2, 2, … generalizations in a certain regime of the anisotropy |q|.

In the dense loop model, the scaling properties of critical random curves are encoded in the correlators of its connectivity operators. Each such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.

Some years ago, Fabricius and McCoy observed extra degeneracies in the finite size spectrum of the XXZ spin chain Hamiltonian at roots of unity. They attributed it to a affine sl₂ symmetry and the existence of maximal length exact strings. They also characterized the representations appearing using the so called Drinfeld polynomial but could not relate its roots to the Bethe roots. I shall construct a Q-matrix having the roots of the Drinfeld polynomial among its roots and point out possible applications.

Work done in collaboration with Jules Lamers and Yuan Miao.

Some large N gauge theories in higher dimensions, known to be dual to string theories, were shown to be integrable in the planar limit. A detailed mapping to 2d integrable field theories was devised. While a lot is known about the spectrum of these theories, the computation of physically significant correlation functions is still under way. A big step forward was done by formulating the problem in terms of form factors for non-local operators, but some conceptual and practical steps to extract useful information in particular regimes of parameters are still out of reach. The talk will give an introduction to the subject and present a few particular examples of objects of interest.

First I discuss the deep connection recently found between two classes of deformation of integrable models: TTbar and related irrelevant deformations of integrable quantum field theories on the one hand, and long range integrable deformations of quantum spin chains on the other. These two classes were developed entirely independently, but they turn out to have the same algebraic origin. As a consequence, their detailed structures are formally identical and therefore they share many features. These considerations lead to a novel concept of weak integrability breaking, which defines a novel class of perturbations of integrable models. I then demonstrate that there is a profound difference between ordinary (strong) and weak integrability breaking perturbations, which can be made manifest using the level spacing distribution.

The seventh workshop will take place July 24-28, 2023 at the University of Wuppertal. The conference venue is the seminar room F13.11.

The seventh workshop on Correlations in Integrable Quantum Many-Body Systems is most probably the last one in this series. We will take the opportunity to sum up the activities of our DFG-funded research group and to discuss new directions with our guests.

• Fabian Essler, University of Oxford
• Rob Klabbers, Humboldt Universität Berlin
• Christian Korff, University of Glasgow
• Gleb Kotousov, Leibniz Universität Hannover
• Karol K. Kozlowski, ENS de Lyon
• Fedor Levkovich-Maslyuk, IPhT Saclay
• Oleg Lisovyi, LMTP, Université de Tours
• Sergei Lukyanov, Rutgers University
• Masaki Oshikawa, ISSP, University of Tokyo
• Ovidiu Patu, Institute for Space Sciences Bucharest
• Giuliano Pavan Ribeiro, U Federal de São Carlos
• Vladislav Popkov, University of Ljubljana and BUW
• Balázs Pozsgay, Eötvös Loránd University Budapest
• Tomohiro Sasamoto, Tokyo Institute of Technology
• Fedor Smirnov, Sorbonne Université - UPMC
• Véronique Terras, Université Paris-Saclay
• Andrew Urichuk, Université Cergy-Pontoise
• Éric Vernier, Sorbonne Université - LPTMS

Markov-chain Monte Carlo (MCMC), the field of stochastic algorithms built on the concept of sampling, has countless applications in science and technology. The overwhelming majority of MCMC algorithms are time-reversible and satisfy the detailed-balance condition, just like physical systems in thermal equilibrium. The underlying Markov chains typically display diffusive dynamics, which leads to a slow exploration of sample space. Significant speed-ups can be achieved by non-reversible MCMC algorithms exhibiting non-equilibrium dynamics, whose steady states exactly reproduce the target equilibrium states of reversible Markov chains. Such algorithms have had successes in applications but are generally difficult to analyze, resulting in a scarcity of exact results. Here, we introduce the “lifted” TASEP (totally asymmetric simple exclusion process) as a paradigm for lifted non-reversible Markov chains. Our model can be viewed as a second-generation lifting of the reversible Metropolis algorithm on a one-dimensional lattice and is exactly solvable by an unusual kind of coordinate Bethe ansatz. We establish the integrability of the model and present strong evidence that the lifting leads to faster relaxation than in the KPZ universality class.

The ubiquitous Heisenberg XXX spin chain admits many deformations, for example the XXZ and XYZ spin chains, which have deformed spin interactions compared to XXX. The Inozemtsev spin chain is another deformation of XXX: one can interpret it as a deformation of the potential controlling the strength of interactions, causing it to have interactions between all pairs of spins. Its deformation parameter controls the effective interaction distance, going from the nearest-neighbour XXX chain to the truly long-range and integrable Haldane-Shastry spin chain.

I will present the construction of a deformation of the Inozemtsev spin chain that interpolates between a Heisenberg XXZ spin chain and the long-range q-deformed Haldane-Shastry model, presenting an answer to a long-standing question. This deformation is integrable by virtue of its connection via freezing to a novel elliptic dynamical spin-Ruijsenaars model. I will discuss its properties and show how it encompasses several interesting new models, including a dynamical XXZ chain, reachable by various limits. The talk is based on recent work with Jules Lamers.

The ring of symmetric functions plays a central role in representation theory. It connects with exactly solvable lattice models by observing that many Bethe wave functions are symmetric in the Bethe roots. For periodic boundary conditions so-called cylindric symmetric functions emerge whose product and co-product expansions lead to 2D topological quantum field theories. For infinite lattices and with suitable boundary conditions at infinity, one can use the transfer matrices of exactly solvable lattice models to obtain combinatorial formulae for vertex operators of symmetric functions. This links the area of lattice models (via the boson-fermion correspondence) with integrable hierarchies of PDEs such as the Kadomtsev-Petiashvili equation where it is known that particular solutions, tau-functions, are given by symmetric functions. The more recent insight is that these functions can also be re-interpreted as partition functions of lattice models.

The subject matter of the talk is an integrable spin chain associated with the staggered six-vertex model that possesses an additional ${\cal Z}_2$ symmetry. In the case of periodic boundary conditions it was originally studied by (Ikhlef), Jacobsen and Saleur in a particular regime of the parameters, where the model is critical. They determined that the spectrum of scaling dimensions is continuous and conjectured that its scaling limit is related to the so-called 2D black hole sigma models. In this talk we focus on the spin chain with U_q(sl(2)) invariant open boundary conditions imposed. We show how to fully characterize the low energy spectrum of the model. Its relation to the Euclidean 2D black hole sigma model, where the end points of the fields are restricted to line along a brane, will also be briefly discussed. The talk is based on work in progress with Prof. Holger Frahm and Sascha Gehrmann.

The quantum transfer matrix is an auxiliary tool allowing one to significantly simplify the problem of effectively calculating the the per site free energy as well as the correlation functions of a one dimensional quantum spin chain model at finite temperature. It is conjectured that certain universal features arising in the long-distance asymptotic behaviour of multi-point functions of critical one-dimensional quantum spin chains directly at zero temperature also manifest themselves on the level of the low-temperature behaviour of various quantities related with the associated quantum transfer matrix. In particular, if a given conformal field theory captures the long distance behaviour in the model at zero temperature, than the spectrum of this conformal field theory should arise in the low- temperature behaviour of the spectrum of the quantum transfer matrix.

In the case of the XXZ chain spin-1/2 chain, the quantum transfer matrix may be chosen to be integrable, what allows one, in principle, to study the mentioned universality properties of its spectrum by means of the Bethe Ansatz. It is well know that the latter can be reduced to finding solutions to certain non-linear integral equations. In this talk, I will describe how the relevant non-linear problem can be put on rigorous grounds, especially in view of taking the infinite Trotter number limit and subsequently the low-temperature limit. I will also provide a classification of its solutions. Finally, I will explain how those results then allow one to access to the universal features of the spectrum of the quantum transfer matrix by showing that a subset thereof explicitly contains, in the low-temperature limit, the spectrum of the c = 1 free Boson conformal field theory. This is a joint work with S. Faulmann and F. Göhmann.

Recent years have seen great progress in developing and applying separation of variables (SoV) in quantum integrable models. I will describe the main results achieved in this program based on a series of papers with my collaborators. In particular, I will present the SoV construction for gl(N) integrable spin chains. I will also show how to resolve the longstanding problem of computing the SoV measure, and how it leads to new highly compact determinant results for a large class of correlators and wavefunction overlaps. I will also demonstrate the power of SoV in 4d integrable CFTs such as the fishnet theory, and present related results for Yangian symmetry of a new large class of Feynman graphs. Lastly I will outline highly promising applications in computation of exact correlators in N=4 super Yang-Mills theory.

I will discuss a semiclassical approach to form factors in the sinh-Gordon field theory in the background of a radial classical solution that describes a heavy exponential operator placed at the origin. I will introduce and study new special functions which generalize the Bessel functions and have a nice interpretation in the Tracy–Widom theory of the Fredholm determinant solutions of the classical sinh-Gordon model.

The inhomogeneous six-vertex model is a 2D multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, remain unexplored. The main subject of the talk is a certain critical quantum 
spin 1/2 chain 
associated with the inhomogeneous 
six-vertex model with an additional $Z_r$ symmetry. 

The CFT underlying its critical behaviour possesses remarkable features. 
Among them is an infinite 
degeneracy of 
the conformal 
primary states and the presence of a continuous component in the spectrum 
in the case of even $r$.

Nonlinear conductivities are of increasing experimental and theoretical interest recently, but the subject is largely open. I will discuss our recent approach to the fundamentals of the nonlinear optical conductivities. An application of a spatially uniform electric field can be formulated as an insertion of Aharonov-Bohm flux. By considering the energy gain of the system in the two opposite limits of flux insertion — adiabatic and sudden, we can derive, in a unified manner, the two renowned formulae on linear conductivities: Kohn formula for Drude weights and the f-sum rule. Furthermore, they can be naturally generalized to all orders of nonlinear conductivities. The nonlinear Kohn formula can be evaluated exactly for the S=1/2 XXZ chain, utilizing the integrability of the model. However, the nonlinear Drude weights thus obtained often exhibit pathological behaviors such as divergence in the thermodynamic limit. To clarify the issue, we propose to distinguish the “Kohn Drude weight” (obtained from the Kohn formula) from “bulk Drude weight” which is defined by taking the thermodynamic limit before the zero frequency limit. We demonstrate that, in a simple model of one-dimensional electrons in a single impurity potential, the pathological behaviors of the Kohn Drude weights are indeed removed in the bulk Drude weight.

We report on the derivation of determinant representations for the space-, time- and temperature-dependent correlation functions of the trapped Gaudin-Yang model with infinite repulsion and arbitrary statistics. Our representations are valid in equilibrium and in nonequilibrium scenarios initiated by a quench of the trapping potential. In the case of a sudden change in the frequency of the trapping potential, we predict a many-body bounce effect that causes a nontrivial periodic narrowing of the momentum distribution, not present in the evolution of the density profile, with an amplitude that depends on statistics. We also present results for the dynamics of the Gaudin-Yang model in the quantum Newton's cradle setup. In contrast with Generalized Hydrodynamics or other analytical methods, our exact representation allows for the investigation of hydrodynamization which was recently observed in the Lieb-Liniger model [Le et al., Nature (2023)].

We are going to speak about the partition function per site of the integrable Sp(2n) vertex model on the square lattice. We establish a set of transfer matrix fusion relations for this model. The solution of these functional relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental Sp(2n) representation of the vertex model.

We construct the nonequilibrium steady state (NESS) of a XYZ spin- chain in contact with a dissipative bath forcing the edge spins to maintain arbitrary spin polarizations. The NESS is given by a matrix product ansatz with Lax operators being infinite-dimensional bidiagonal site-dependent matrices. As an application, we show how an interplay between a coherent and a dissipative dynamics allows to generate a family of chiral quasi-separable states carrying large current of magnetization, very different from the ground state or thermal states.

We review a number of recent results about factorized correlation functions in selected quantum spin chains. We treat the Heisenberg XXZ and XYZ models, the higher SU(N)-related models, and also the spin chains given by representations of the Temperley-Lieb algebra. We explain a fundamental result about the mean values of generalized current operators, which seems to hold rather generally. These mean values in question can be seen as building blocks for certain correlation functions. In technical terms, these operators are those for which the mean values are all linear in the so-called omega function. This appears to hold in all models that we consider.

Propagation of correlation fronts in quantum many-body systems have attracted much attention in recent years. In this talk we discuss such propagating correlation fronts for a free fermionic system, starting from the alternating initial state. We find that all the moments of dynamical fluctuations around the correlation fronts are described by the universal GOE and GSE random matrix correlations in the long time limit. This may be comparable to the previous work of Eisler and Racz that all particle number correlations are described by the universal GUE random matrix correlations.

The talk is based on a joint work with Kazuya Fujimoto.

There are two kinds of formulas for form factors: the ones obtained by bootstrap and the ones following from the fermionic description.The equivalence of the two is not obvious, we shall explain that it follows from a Grassmannian hidden in the off-shell Bethe Ansatz.

We review some recent results concerning the computations of correlation functions in open XXZ spin 1/2 chains with boundary fields. In the case of longitudinal boundary fields, correlation functions at zero temperature can be computed within the algebraic Bethe Ansatz framework in the form of multiple integrals in the half-infinite chain limit. We discuss the extension of this result to two different cases: (1) the temperature case; (2) the case of more general boundary fields at zero-temperature. In the temperature case (1), form factor series for correlation functions can be written within the quantum transfer matrix framework. The case of non-longitudinal boundary fields (2) can be considered within the Separation of Variables approach. If the two boundary fields are related by one constraint, we can compute the matrix elements of a set of local operators as multiple integrals, similarly as in the longitudinal case.

An infinite number of evolution equations dictate the dynamics of the Lieb-Liniger model according to generalized hydrodynamics (GHD). At the Euler scale with T=0 the infinite GHD equations truncate and are known to reduce to conventional Euler hydrodynamics, which is also known as zero entropy GHD. By carrying out a low-temperature expansion for the diffusive kernel the diffusive GHD equations are likewise found to truncate. This yields an identification between viscous GHD, written in a universal fashion by use of Luttinger liquid parameters, and conventional viscous hydrodynamics.

The understanding of integrable models as it emerged from the 80s is intimately related to algebraic structures called quantum groups, which offer a systematic construction of solutions of the Yang-Baxter equation. In this talk we will see how certain key integrable models (the six-vertex model and its higher spin generalization) allow for a description in some sense “more elementary” than in terms of quantum groups. We will start by reviewing a recent construction by Bazhanov and Sergeev, where such models are derived from an underlying Ising-like geometry and the Yang-Baxter equation results from multiple applications of the Star-Triangle relation. We will then specialize to the “root of unity” points, where this construction can be extended and allows to illuminate remarkable new features of such models, in particular, their symmetry under the Onsager algebra, an object which first appeared in the solution of the two-dimensional Ising model.