Recent Advances in Quantum Integrable Systems
RAQIS'12: Talks, posters and PDFs

AUFGEBAUER Britta     Finite temperature correlation functions from discrete functional equations
We present an approach to the static finite temperature correlation functions of the Heisenberg chain based on functional equations. An inhomogeneous generalization of the n-site density operator is considered. The lattice path integral formulation with a finite but arbitrary Trotter number allows to derive a set of discrete functional equations with respect to the spectral parameters. We show that these equations yield a unique characterisation of the density operator. Our functional equations are a discrete version of the reduced q-Knizhnik-Zamolodchikov equations which played a central role in the study of the zero temperature case.
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AVAN Jean     Dynamical reflection algebras: examples from Calogero-Moser models
The presence of boundaries to a quantum integrable system imposes to complement the Yang-Baxter algebra RTT = TTR by the boundary equation RKRK = KRKR. In parallel consistent deformations of the YB algebra have been defined, leading to so-called dynamical Yang Baxter equations. The association of both structures yields so-called dynamical boundary algebras. Two such structures were known until last year when we constructed a third one. We will describe this new algebraic structure and unravel its connection with the higher Poisson structures of the Calogero-Moser model, focusing on the rational potential case v(r) = 1/r².
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BALOG Janos     Hybrid-NLIE for the AdS/CFT spectral problem
Hybrid-NLIE equations, an alternative finite NLIE description for the spectral problem of the super sigma model of AdS/CFT and its gamma-deformations are derived by replacing the semi-infinite SU(2) and SU(4) parts of the AdS/CFT TBA equations by a few appropriately chosen complex NLIE variables, which are coupled among themselves and to the Y-functions associated to the remaining central nodes of the TBA diagram. Our equations differ substantially from the recently published finite FiNLIE formulation of the spectral problem.
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BELAVIN Alexander     Instantons and bases in CFT
We use AGT correspondence for the construction of the bases of OPE in Conformal field theory
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BELLIARD Samuel     SU(3) Bethe vectors
Different formulations of SU(3) Bethe vectors will be presented. The action of the generators of the Yangian on these vectors will be given. These results are important for the calculation of form factors and correlation functions from the quantum inverse scattering problem and scalar products of Bethe vectors. This is a joint work with S. Pakuliak, E. Ragoucy and N. Slavnov.
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BYTSKO Andrei     Tetrahedral Y-systems
Using a solution to the Zamolodchikov tetrahedron equation, we construct an identity involving quantum dilogarithms. Suitable reductions of this identity yield quantum dilogarithm identities related to Y-systems of the A-type.
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CAUDRELIER Vincent     Electronic transport on quantum graphs
Combining the formalism of integrable quantum field theory with results from the theory of quantum graphs, we will present a general framework to describe electronic transport on arbitrary geometries that can be modeled by quantum graphs. In particular, the formalism allows for the computation of equilibrium and out of equilibrium correlation functions at finite temperature and with an external magnetic field. This is interesting in view of applications to networks of quantum wires. As an illustration, we will present results for the conductance and the noise in a quantum ring pierced by a magnetic flux and connected to three (or more) thermal reservoirs.

CHENG Zhang     Yang-Baxter and reflection maps from factorization of vector soliton interactions on the half-line
We consider the Vector Nonlinear Schrödinger equation on the half-line. First via a Bäcklund transformation method, we derive two classes of integrable boundary conditions, a mirror image technique is applied to compute exact soliton solution. Then using dressing transformations, factorization of soliton-soliton and soliton-boundary interactions is showed. Soliton interactions correspond to so-called Yang-Baxter map which are solutions of the set-theoretical Yang-Baxter equation. We introduce a new mathematical object, which we call reflection map corresponding to soliton-boundary interaction, that satisfies the set-theoretical reflection equation, which we also introduce. Basic aspects of reflection map are discussed.

DE LEEUW Marius     Quantum deformations in AdS/CFT
In this talk I will discuss the quantum deformed su(2|2) algebra. This superalgebra describes a quantum deformed version of the Hubbard model and the corresponding S-matrix is closely related to the S-matrix of the AdS5xS5 superstring. I will introduce the algebra, highlight some of its features and discuss some applications in the context of the AdS/CFT correspondence.
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DOBREV Vladimir     Group-Theoretical Classification of BPS States in D=4 Conformal Supersymmetry
We give explicitly the reduction of supersymmetries of the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). Using this we give the classification of BPS and possibly protected states.
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DOYON Benjamin     Quantum steady states out of equilibrium
Quantum steady states out of equilibrium are states admitting a nonzero constant local current density. In many-body integrable systems, there are infinitely-many local conserved charges, hence current densities. I will explain how to describe the associated steady states in integrable systems. These steady states emerge after connecting two baths with different chemical potentials associated to the conserved charges. The description is valid both in spin chains and in quantum field theory (i.e. near their critical points). Specialising to the energy flow, I will describe the consequences in conformal field theory (non-equilibrium CFT), and in the Ising quantum field theory (non-equilibrium form factors). Various parts of this work are done in collaboration with Denis Bernard and my students Yixiong Chen and Marianne Hoogeveen.
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DUGAVE Maxime     Form factors for the XXZ chain and non-linear integral equations
We have worked out exact expressions for the thermal form factors of the XXZ chain. This poster is based on joint work with Frank Goehmann and is related to his talk.
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FADDEEV Ludwig D     A survey on modular quantum dilogarithm
TBA

FEIGIN Misha     Generalized Macdonald-Mehta integrals and Baker-Akhiezer functions
We consider Baker-Akhiezer functions associated with the special arrangements of hyperplanes with multiplicities; these are particular eigenfunctions for the generalized Calogero-Moser operators. We establish integral identity for the Baker-Akhiezer functions that may be viewed as a generalization of the self-duality property of the usual Gaussian with respect to the Fourier transformation. We derive that the value of the Baker-Akhiezer function at the origin is given by the integral of Macdonald-Mehta type. In contrast to the standard Macdonald-Mehta integrals arising in random matrix theory our integrand has singularities on the arrangement so a regularisation is needed. We explicitly compute these regularised integrals for known Coxeter and non-Coxeter arrangements admitting the Baker-Akhiezer functions. In the Coxeter cases we use analytic continuation of the usual Macdonald-Mehta integral. In the two-dimensional examples we use the explicit form of the Baker-Akhiezer functions found by Berest, Cramer and Eshmatov. In the higher-dimensional non-Coxeter examples we evaluate these integrals by taking a special limit of the Dotsenko-Fateev integral arising in conformal field theory. This is a joint work with M.A. Hallnas and A.P.Veselov.
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FINCH Peter     Excitations of the integrable D(D3) anyon chain
In this talk I will consider integrable quantum chains constructed from the quasi-triangular Hopf algebra DD3. The conformal field theories along with the excitation spectra are presented for the periodic chain in the usual spin basis as well as in a fusion path (anyon) basis.

FODA Omar     Slavnov scalar products in Yang-Mills theories
I would like to review how Slavnov scalar products are used to compute tree-level and 1-loop 3-point functions in su(2) integrable sectors in planar Yang-Mills theories with various supersymmetries including QCD.
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FOERSTER Angela     Exactly solvable models and ultracold atoms
We investigate some Bethe ansatz integrable models in the context of ultracold atom systems. First we briefly discuss the integrability and mathematical construction of some simple models. Then exactly solvable models of ultracold Fermi gases are examined through the thermodynamic Bethe Ansatz. and the phase diagrams of two- and three-component one-dimensional attractive fermions with population imbalance are obtained. The results for the strong coupling regime provide a description of the quantum phases which are applicable to experiments with cold fermionic atoms confined to one-dimensional tubes [1,2]. In addition, other integrable models relevant in this ultracold scenario, such as spin-1 bosons in a 1D harmonic trap are also investigated and the phase diagrams and density profiles are obtained [3].
[1] Phase Diagrams of Three-Component Attractive Ultracold Fermions in One-Dimension, C.C. N. Kuhn and A. Foerster New J. Phys. 14 (2012) 013008 [2] Exactly solvable models and ultracold Fermi gases, C.C. N. Kuhn, X. W. Guan, A. Foerster and M. Batchelor J. Stat. Mech. (2010) P12014 [3] Quantum criticality of spin-1 bosons in a 1D harmonic trap, C.C. N. Kuhn, X. W. Guan, A. Foerster and M. Batchelor arXiv:1111.2375.
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FONSECA Tiago     Higher Spin Generalization of the 6-Vertex Model and Macdonald Polynomials
It is known that the 6-Vertex model is a quantum integrable model, therefore we know, at least in theory, everything about it. For example, in the case of Domain Wall Boundary Conditions, the partition function is a relatively simple determinant (Izergin, 1987) and it is related to a Schur polynomial. In a more recent work, Caradoc, Foda and Kitanine (2006) tell us how to generalize this result for higher spins. Based in their work, one can prove that the new partition function is related to a Macdonald polynomial (dF and Balogh, to appear). In this talk, I will describe the 6-vertex model, explain how to create the higher spin model from the original model. And finally, I will sketch how one can prove that this is indeed a Macdonald polynomial.
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GAINUTDINOV Azat     A limit of affine Temperley-Lieb algebras at a root of unity and bulk logarithmic CFT
We study spin-chains based on affine Temperley-Lieb (TL) algebras and their centralizer constructions at a root of unity case. An analogue of the Howe dulaity in such systems allowed to take the direct limit of these spin-chains when the number of tensorands (sites) goes to infinity. The limit of the representation spaces turns out to be a logarithmic conformal field theory in the bulk, while the affine TL algebra in the limit is expressed by a representation of the affine Lie algebra of infinite rank sp_∞ and contains left and right Virasoro algebras. The description of this limit in field-theoretic terms led us to a notion of the interchiral algebra generalizing chiral algebras in the non-chiral (bulk) case.
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GEPNER Doron     Nonstandard parafermions and string compactification
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GOMES Jose Francisco     The higher grading structure of the WKI hierarchy and the two-component short pulse equation.
A higher grading affine algebraic construction of integrable hierarchies, containing the Wadati-Konno-Ichikawa (WKI) hierarchy as a particular case, is proposed. We show that a two-component generalization of the Schäfer-Wayne short pulse equation arises quite naturally from the first negative flow of the WKI hierarchy. The conserved charges, both local and nonlocal, are obtained from the Riccati form of the spectral problem. The loop-soliton solutions of the WKI hierarchy are systematically constructed through gauge followed by reciprocal Bäcklund transformation, establishing the precise connection between the whole WKI and AKNS hierarchies. The connection between the short pulse equation with the sine-Gordon model is extended to a correspondence between the two-component short pulse equation and the Lund-Regge model.
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GOEHMANN Frank     Form factors of the quantum transfer matrix and asymptotics of temperature correlators of the XXZ chain
We have derived expressions for the form factors of the quantum transfer matrix of the spin-1/2 XXZ chain which allow us to take the limit of an infinite Trotter number. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite temperature correlation functions of the model. We consider the longitudinal as well as the transversal two-point correlation functions and also indicate how our formulae have to be modified to cover the case of ground state correlation functions of finite length chains. In the temperature case we show how known results for the high-temperature asymptotics are recovered from the form factor expansion. In the zero-temperature limit we show by example how to obtain the `critical behaviour' of the form factors from our formulae. This talk is based on joint work with Maxime Dugave.
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JIMBO Michio     Representations of quantum toroidal algebras: Recent topics
Quantum toroidal algebras were first introduced in 1995 as further affinization of the quantum affine algebras. In the type A case, basic structure theory has been established by Miki. However their representation theory remains largely open. Here we explain an elementary construction of a new family of representations, in which bases are parametrized by combinatorial objects (ordinary or plane partitions) and the matrix coefficients for the generators are given in factorized form. This talk is based on joint works with Feigin, Miwa and Mukhin.
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KAZAKOV Vladimir     Classical integrability for quantum spin chains and sigma models
I will review recent advances in applications of classical Hirota dynamics for the analysis and solution of quantum integrable systems. In quantum (super)spin chains, it allows to interpret the (eigenvalues of) transfer matrices as special type of mKP tau-functions and give a new construction of Baxter's Q-functions. In sigma-models, such as principal chiral field or the superstring model related to AdS/CFT duality, it allows to efficiently reduce the spectral Y-system related to TBA, to a finite non-linear system of integral equations (FiNLIE).

KLUEMPER Andreas     Correlation functions for integrable higher spin su(2) quantum chains
We use the unique characterisation of density operators of inte- grable su(2) quantum chains by suitable functional equations. The lattice path integral formulation with a finite but arbitrary Trotter number allows to derive a set of discrete functional equations and an- alytical properties with respect to the spectral parameters. The n-site density operator is considered for integrable higher spin-S quantum chains at arbitrary temperature T . Explicit results are given for the case of S = 1 and n = 2 and 3 in terms of non-linear and associated lin- ear integral equations avoiding the multiple integral expressions previ- ously derived in the literature. The correlation functions on arbitrarily many n sites factorize similarly to the by now well understood spin-1/2 case.
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KONNO Hitoshi     Elliptic Quantum Groups U_q,p and E_τ,η;
Up to now, we know three formulations of the face type elliptic quantum group, B_q,λ(g⁽¹⁾), U_q,p(g⁽¹⁾) and E_τ,η(g). These are distinguished by their generators, Chevalley type (B_q,λ), Drinfeld type (U_q,p) and L-operator (E_τ,η), as well as by their co-algebra structures, quasi-Hopf (B_q,λ) and Hopf algebroid (U_q,p and E_τ,η). In this talk, we discuss some relations among them. In particular, we propose a FRST formulation of E_τ,η associated with affine Lie algebra g⁽¹⁾ (a central extension of Felder's E_τ,η(g)) and discuss an isomorphism between U_q,p(g⁽¹⁾) and E_τ,η(g⁽¹⁾) for g⁽¹⁾=sl_N⁽¹⁾.
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KORFF Christian     The quantum Ablowitz-Ladik model: completeness of the Bethe ansatz
We discuss a discrete version of the quantum nonlinear Schr"odinger model, the quantum Ablowitz-Ladik model, and give a completeness proof of the algebraic Bethe ansatz. Using this result we establish an algebra isomorphism between the conserved charges of the model and an affine version of the spherical Hecke algebra. In the crystal (strong coupling) limit this algebra becomes the Verlinde or fusion algebra of the su(n) WZW model.
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KOZLOWSKI Karol     Surface Free energy of the open XXZ spin-1/2 chain.
In this talk, I will report on the possibility to represent the finite Trotter number approximant of the surface free energy of the open XXZ spin-1/2 chain subject to diagonal boundary fields in terms of Tsuchiya determinants. I will explain how to take the infinite Trotter limit of such expressions. This provides an expression for the surface free energy of the open chain in terms of series of multiple integrals. This integral representation allows one to extract the low-temperature asymptotic behavior of the boundary magnetization at finite external magnetic field. This is a joint work with B. Pozsgay.
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KULISH Petr     Integrable spin systems and representation theory
Quantum integrable spin chains are related to classical and quantum Lie (super-) algebras and their finite and infinite dimentional representations. It is pointed out that with appropriate boundary condition (open spin chains) corresponding algebra can be identified with the symmetry algebra of integrals of motion (transfer matrix). According to a generalized Schur - Weyl duality this yields a multiplet structure of the spectrum and invariant subspaces. This approach includes spin chains related to non-quasiclassical quantum algebras as well.
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LISOVYY Oleg     Conformal field theory of Painlevé VI
Generic Painlevé VI tau function τ(t) may be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c=1. Full and completely explicit expansion of τ(t) near the singular points can thus be obtained using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants..
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MALLICK Kirone     Current fluctuations in the exclusion process
The asymmetric simple exclusion process is a model used as a template to study various aspects of non-equilibrium statistical physics. It appears as a building block in more realistic descriptions for low-dimensional transport with constraints. In the steady state, a non-vanishing current is carried through the system. The statistical properties of this current are archetypal observables for non-equilibrium behaviour. It this talk, we explain how to derive the full statistics of the current in the ASEP. We present exact combinatorial formulas valid for all system sizes and all values of the system parameters. Our results are obtained using integrability techniques borrowed from the theory of quantum integrable systems such as the Bethe Ansatz and the Matrix Product Representation.
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MATSUI Chihiro     The ground state of SUSY sine-Gordon model with the Dirichlet boundary conditions via lattice regularization
We discuss the ground state of SUSY sine-Gordon model with the Dirichlet boundary conditions via the lattice regularization. The Bethe roots which characterize the ground state are specified from the nonlinear integral equations, which are found to have different forms depending on the value of boundary parameters. The boundary energy is calculated based on the analyticity of the transfer matrices.

MIRAMONTES J Luis     Aspects of the q-deformed AdS(5)x S(5) superstring S-matrix
I will review recent work aimed to the construction of an S-matrix theory that interpolates between the usual non-relativistic AdS(5)xS(5) superstring S-matrix and the relativistic S-matrix corresponding to its Pohlmeyer reduction. The talk will be based on arXiv:1206.0010, arXiv:1112.4485 and arXiv:1107.0628.
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MOTEGI Kohei     Exact relaxation dynamics in the totally asymmetric simple exclusion process
We study the exact relaxation dynamics of the totally asymmetric simple exclusion process on a ring. The full relaxation dynamics of the local densities and currents are examined by the algebraic Bethe ansatz method. Moreover, we find the scaling exponents of the asymptotic amplitudes. We also find that the relaxation times starting from the step and alternating initial conditions are governed by different eigenvalues of the Markov matrix. This is based on joint work with K. Sakai and J. Sato.
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NIEMI Antti     Discrete Nonlinear Schrödinger Equation and Polygonal Solitons with Applications to Collapsed Proteins
We introduce a novel generalization of the integrable discrete nonlinear Schrodinger equation. It supports dark solitons that we utilize to model proteins in the biologically active collapsed phase. As an example we consider the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. We use its backbone as a template to explicitely construct a two soliton configuration. Each of the two solitons describe well over 7.000 supersecondary structures of folded proteins in the Protein Data Bank with sub-Angstrom precision. Our results suggest that these new solitons are abundant in nature.
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OSHIMA Kazuyuki     The Elliptic Algebra U_q,p(B_N⁽¹⁾) and Vertex Operators
We introduce an elliptic quantum algebra of type B_N⁽¹⁾. By using the elliptic analogue of the Drinfeld currents, we construct the L-operator, which satisfies the dynamical RLL relations characterizing the elliptic quantum algebra U_q,p(B_N⁽¹⁾). We then derive the two types of vertex operators of arbitrary level U_q,p(B_N⁽¹⁾)-modules. As an example, we give a level one free field representation. This is a joint work with Hitoshi Konno.

PALMAI Tamas     Sine-Gordon form factors in finite volume
We verify exact form factors coming from bootstrap in the sine-Gordon model, a prototype theory with nondiagonal scattering. This is achieved by finding convincing agreement between finite volume form factors calculated from the truncated conformal space approach and from bootstrap via results of the finite volume form factor program. Evaluation of the nontrivial multi-soliton form factors is made possible by a newly developed regularization scheme.

ROLLET Geneviève     Classification of Non-Affine Non-Hecke Dynamical R-Matrices
A complete classification of non-affine dynamical quantum R-matrices obeying the Gl_n(C)-Gervais-Neveu-Felder equation is given without assuming either Hecke or weak Hecke conditions. More general dynamical dependences are observed. It is shown that any solution is built upon elementary blocks, which individually satisfy the weak Hecke condition. Each solution is in particular characterized by an arbitrary partition of the set of indices {1..n} into classes and an arbitrary family of signs on this partition. The weak Hecke-type R-matrices exhibit an analytical behaviour with general trigonometric or rational dynamical dependencies (arXiv:1204.2746v2).
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SAKAI Kazumitsu     Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
We provide multiple Schramm-Loewner evolutions (SLE) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models are described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct a multiple SLE with the additional Lie algebra symmetry. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE interfaces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. By analyzing the correlation functions, we also discuss the topology of configurations of the SLE curves.
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SKRYPNYK Taras     Quantum integrable systems and classical non-skew -symmetric r-matrices
We will make a review of a recent advances in the theory of quantum integrable systems associated with non-skew-symmetric classical r-matrices. Such the systems, in general, are not connected with the quantum qroups or related structure and constitute a separate family of integrable models. We will illustrate our approach on the examples of integrable spin systems of the generalized Gaudin type, integrable boson and spin-boson systems. We will present the physical applications of the obtained integrable models, in particular, to BCS-type models (theory of superconductivity), Jaynes-Cummings-Dicke-type models and Bose-Hubbard-type dimers (quantum optics). We will consider in details the case of classical elliptic r-matrices and the corresponding quantum integrable models.
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SLAVNOV Nikita     Algebraic Bethe ansatz for scalar products in SU(3)-invariant models
We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU(3)-invariant Heisenberg chain.
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TERRAS Véronique     ABA approach to correlation functions of the cyclic SOS model
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VICEDO Benoit     Alleviating the non-ultralocality of coset sigma models through a generalized Faddeev-Reshetikhin procedure
The property of non-ultralocality in certain classically integrable field theories is known to pose a serious obstacle to quantization. A way around this problem was proposed in '86 by Faddeev and Reshetikhin in the case of the principal chiral model on SU(2). Their procedure begins by replacing the problematic non-ultralocal Poisson brackets by ultralocal ones and modifying the corresponding Hamiltonian to reproduce the same dynamics. I will show how these first steps maybe be generalized to symmetric space sigma-models. The modified theory in this case is classically equivalent to the Pohlmeyer reduction of the original sigma-model and is described by a gauged Wess-Zumino-Witten action with an integrable potential. I will discuss the first steps towards constructing an integrable lattice discretization of these latter models. (based on arXiv:1204.0766, 1204.2531, 1206.6050)
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VOLKOV Alexander     Tetrahedral Y-system
TBA

WHEELER Michael     Factorization formula for SU(3) scalar products
The calculation of scalar products between Bethe vectors is an important topic of quantum integrable models. In the case when both vectors are on-shell, the scalar product is the norm-squared of the Bethe eigenstate. More generally, off-shell scalar products (in which one vector is not an eigenvector of the transfer matrix) are important to the study of correlation functions. In the case of models based on the SU(2)-invariant R-matrix, it is known from the work of N. Slavnov that certain off-shell scalar products have a determinant expression. A difficult unsolved problem is the generalization of Slavnov's formula to models based on SU(n). This talk will describe recent work on the SU(3) problem, where it was shown that by sending half of the Bethe variables in the scalar product to infinity, it factorizes into a product of two determinants.
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YAMADA Yuji     Classification of solutions to the reflection equation
We classify and list up all the meromorphic solutions K(z) to the reflection equation associated to the critical Z_N-symmetric vertex model under two assumptions that none of the diagonal elements is constantly zero and that there is at least a pair of elements K^a_b(z)K^b_a(z)≠ 0 We make explicit the matrix elements of K(z), parameters they have and the relations among parameters.
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YOUNG Charles     Quantum affine algebras and loop-weight algebras
Quantum affine algebras have been the topic of much research over the last 25 or so years. However, even in the category of their finite-dimensional representations, many problems remain open. I will briefly discuss some of these: classifying the prime objects, understanding possible extensions, and even elementary questions like computing dimensions of irreducibles. Then I want to suggest a new approach, which builds on the notion, due to Frenkel and Reshetikhin, of q-characters. The key idea is that, in analogy with the weight lattice of simple Lie algebras, quantum affine algebras have a notion of "loop-weights" and "loop-roots". The analogy with usual weight theory, however, breaks down after a certain point. I shall argue that it is possible to overcome this problem, by obtaining representations as pullbacks of representations of a new algebra, to be defined. Based in part on recent joint work with Evgeny Mukhin, Vyjayanthi Chari and Adriano Moura: arXiv:1206.6657, arxiv:1204.2769 and arxiv:1112.6376
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