Recent Advances in Quantum Integrable Systems

Titles and abstracts of talks


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Many thanks to Samuel Belliard who scanned all the transparencies


J. Avan:    From quadratic exchange to dynamical reflection

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J. Balog:    Structure functions and form factor clustering in the 2-dimensional O(n) non-linear sigma-model

We study (the 1+1 dimensional analog of) DIS and structure functions in the 2-dimensional (asymptotically free) non-linear O(n) sigma-models using the non-perturbative S-matrix bootstrap program. The exact small (Bjorken) x behaviour is derived.

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P. Baseilhac:    New exact results in XXZ open spin chain

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A. Belavin:    Higher Equations of Motion in N = 1 SUSY Liouville Field Theory.

Similarly to the ordinary bosonic Liouville field theory, in its N=1 supersymmetric version an infinite set of operator valued relations, the ``higher equations of motions'', hold. Equations are in one to one correspondence with the singular representations of the super Virasoro algebra and enumerated by a couple of natural numbers (m,n). We demonstrate explicitly these equations in the classical case, where the equations of type (1,n) survive and can be interpreted directly as relations for classical fields. The general form of higher equations of motion is established in the quantum case, both for the Neveu-Schwarz and Ramond series.

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V. Belavin:    Bootstrap in Supersymmetric Liouville Field Theory.

A four point function of basic Neveu-Schwarz exponential fields is constructed in the N=1 supersymmetric Liouville field theory. Although the basic NS structure constants were known previously, we present a new derivation, based on a singular vector decoupling in the NS sector. This allows to stay completely inside the NS sector of the space of states, without referencing to the Ramond fields. The four-point construction involves also the NS blocks, for which we suggest a new recursion representation, the so-called elliptic one. The bootstrap conditions for this four point correlation function are verified numerically for different values of the parameters.

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E. Buffenoir:    Universal Vertex-IRF Transformations

We construct a universal Vertex-IRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical Yang-Baxter equation, in the finite dimensional and affine case. In the fundamental evaluation representation of Uq(A1(1)), this object is equal to standard Vertex-IRF transformation introduced by Baxter in his study of 8-vertex model. This universal Vertex-IRF transformation satisfies the Quantum Dynamical coBoundary equation. This solution has a simple Gauss decomposition which appears to be intimately related to the structure of quantum version of Whittaker vectors. We will present the consequences of our algebraic structures in the study of Whittaker Functions.

J.S. Caux:    The dynamics of integrable spin chains and Bose gases

Heisenberg quantum spin chains and interacting Bose gases in one dimension are two fundamental models of strongly-correlated physics with direct experimental relevance. Recent progress in the theory of integrable models have opened the door to extremely accurate computations of their dynamical correlation functions, and even to some analytical results for specific cases. We review these developments, and their application in the description of experimental data on representative systems.

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T. Deguchi:    The sl(2) loop algebra of the twisted XXZ spin chain and the Onsager algebra

We discuss the sl(2) loop algebra symmetry of the twisted XXZ spin chain at roots of unity. Here the chain is under the twisted boundary conditions. We then apply the result to the Onsager algebra symmetry of the superintegrable chiral Potts model. The connection of the Onsager algebra should be nontrivial. (The latter part of the talk is in collaboration with Akinori Nishino.)

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V.K. Dobrev:    Parabolic Subalgebras and Invariant Differential Operators

Our approach to the construction of invariant differential operators requires the explicit description of parabolic subalgebras. This is easily generalised to the supersymmetric and quantum group settings and is applicable to string theory and integrable models.

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A. Doikou:    Boundary integrals of motion and modified Lax pairs

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P. Dorey:    Aspects of ODE/IM

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L. Feher:    Calogero-Sutherland type models from Hamiltonian reduction

We first survey general results on classical and quantum Hamiltonian reductions of the free geodesic motion on complete Riemannian manifolds under polar actions of compact symmetry groups, i.e., isometric actions that admit regularly embedded, closed, connected submanifolds meeting all orbits orthogonally in the configuration space. We then explain that if the original configuration space is a Lie group, or a symmetric space, and the orthogonal `section' of the orbits can be realized as a suitable Abelian subgroup, then the Hamiltonian reductions of the free particle typically yield spin Calogero-Sutherland type integrable models. We present several examples that fit in this framework, including the standard (spinless) BC(n) Sutherland models with three independent coupling constants.

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G. Feverati:    Generalized integrable Hubbard models

We construct the XX and Hubbard-like models based on unitary superalgebras gl(N|M) generalizing Shastry's and Maassarani's approach. We introduce the R-matrix of the gl(N|M) XX-type model; the one of the Hubbard-like model is defined by "coupling" two independent XX models. In both cases, we show that the R-matrices satisfy the Yang-Baxter equation. We derive the corresponding local Hamiltonian in the transfer matrix formalism and we determine its symmetries. We give a description of the two-particle scattering. A perturbative calculation "a la Klein and Seitz" is performed. Some explicit examples are worked out.

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A. Foerster:    Integrable models for Bose-Einstein condensates

We will discuss some integrable models for different types of Bose-Einstein condensates. The Bethe ansatz solution for these models will be presented. We will also analyse the classical and quantum dynamics for some of these systems.

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A. Hegedus:    Finite size effects and 2-string deviations in the spin-1 XXZ chains

We present and study the nonlinear integral equations (NLIE) governing the finite size effects of the spin-1 XXZ chain in the regime 0

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F. Göhmann:    Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

I report on the status of our work on spatial correlation function of the XXZ spin chain. We study spatial correlations by means of the density matrix of a finite segment of the chain. In two cases we succeeded to derive a multiple integral representation for this density matrix: 1. for the infinite chain exposed to a heat bath and to an external longitudinal magnetic field, and 2. for the ground state of the finite chain with possibly twisted periodic boundary conditions. We are now studying the mutiple integrals in order to obtain explicit results that can be compared with experiments and in order to understand the general structure of all static correlation function in the spin chain. Using the insight gained for the ground state by Boos, Jimbo, Miwa, Smirnov and Takeyama and combining it with explicit claculations on the multiple integrals and with the high temperature expansion technique we were able to conjecture an explicit exponential formula for the density matrix in case 1. Since the magnetic field is included it yields new formula even in the zero temperature limit. Our conjecture implies that all static correlation functions of the XXZ chain should be polynomials in only two transcendental functions and their derivatives with coefficients of purely algebraic origin.

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M. Jimbo:    Fermionic structure in the XXZ chain: toward creation operators

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V. Kazakov:    Quantum integrability of super-spin chains from discrete Hirota dynamics

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H. Konno:    Elliptic Quantum Group, Hopf Algebroid and Elliptic 6j-symbols

Elliptic algebra Uq,p(A1(1)) is an elliptic deformation of the Drinfeld currents of Uq(A1(1)). It is known to give a realization of the face type elliptic quantum group in the quasi-Hopf formulation. In this talk, we introduce an H-Hopf algebroid structure in the sense of Etingof and Varchenko to Uq,p(A1(1)) and formulate it as an elliptic quantum group. Giving finite dimensional representations, we investigate a sub-module structure of the tensor product of two evaluation representations of Uq,p(A1(1) ). We then show that the elliptic analogue of the very well poised balanced hypergeometric series, 12V11, introduced by Frenkel-Turaev as an elliptic analogue of 6j-symbol appears as a coefficients connecting certain weight vectors in the sub-module with vectors in the tensor product space.

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C. Korff:    PT symmetry and the quantum group invariant XXZ spin-chain

This talk is based on joint work with Robert Weston: JPA 40 (2007) 8845-8872. We revisit the quantum group invariant XXZ spin-chain and discuss the Hermiticity of the Hamiltonian when the deformation parameter q lies on the unit circle. We connect the quantum group reduction technique introduced by Reshetikhin and Smirnov with the recent discussion of non-Hermitian, PT invariant quantum Hamiltonians by giving two explicit, algebraic constructions for Bender's C-operator. Both are related by the quantum analogue of Schur-Weyl duality. Also some new results (not contained in the above paper) will be presented: a GNS construction leading to a graphical calculus analogous to the one of Khovanov and Frenkel.

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K. Kozlowski:    Correlation functions of the boundary XXZ model.

There is a large interest in computing correlation functions of lattice integrable models. There are two approaches to derive the correlators of the bulk XXZ spin 1/2 chain. One is based on the Vertex Operator (VO) formalism and yields the correlators of the infinite chain. The other method, also applicable to finite chains in an overall longitudinal magnetic background, is based on the Algebraic Bethe Ansatz (ABA). Yet, most of the physically relevant spin chains have other types of boundary conditions. Although the VO formalism gave results concerning the correlators of the half-infinite XXZ chain with a longitudinal magnetic field acting on its boundary, there are no results based on the ABA. Sklyanin developed an ABA suited for the diagonalization of boundary hamiltonians. In this talk, after a short introduction to the ABA, I will present the general method we developed to obtain the exact expressions for the correlation functions of the boundary XXZ spin-1/2 finite chain subject to longitudinal magnetic fields at the boundaries. We were able to recast, in the half-infinite size limit, the so-called elementary blocks of m adjacent operators as m-fold integrals. These results are similar to the ones obtained in the bulk model. In the end of this talk, I will present some cases where the integrals can be separated thus leading to results in a closed form. This is a joint work with N. Kitanine, J-M. Maillet, G. Niccoli, N. Slavnov and V. Terras.

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P. Kulish:    Algebraic features of integrable spin chains

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O. Lisovyy:    Painleve VI transcendents related to the Dirac operator on the hyperbolic disk

Dirac hamiltonian on the Poincare disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for the Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branching points on the Poincare disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a class of solutions of Painleve VI equation.

B. McCoy:    New results in the Ising and the 8 vertex models

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M. Mintchev:    Quantum fields and scale invariance on graphs

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R. Nepomechie:    The QCD spin chain S matrix

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A. Nishino:    The sl2 loop algebra symmetry of the XXZ-type spin chain associated with the superintegrable chiral Potts model

The sl2 loop algebra L(sl2) symmetry is found in a sector of the XXZ-type spin chain at a root of unity whose transfer matrix commutes with that of the superintegrable chiral Potts model. The regular Bethe state which is an eigenstate of the spin chain is shown to be a highest weight vector of L(sl2). The highest weight representation space generated by the Bethe state gives a L(sl2)-degenerate eigenspace of the spin chain. The Drinfeld polynomial which characterizes the L(sl2)-degenerate eigenspace is calculated. The Drinfeld polynomial is equivalent to the superintegrable chiral Potts polynomial which characterizes a subspace with the Ising-like spectrum of the superintegrable chiral Potts model. (Joint work with Tetsuo Deguchi)

S. Pakuliak:    Universal Bethe vectors and universal Bethe equations

In case of the quantum affine algebra Uq(\widehat{glN) we proved that projections of the Drinfeld currents onto intersection of the different type Borel subalgebras can be treated as universal Bethe vectors provided their parameters satisfy the universal Bethe equations.

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P. Pearce:    Polymers, percolation and fusion

Dense polymers and critical percolation are the first two members of the Yang-Baxter integrable series of logarithmic minimal models. Integrable boundary conditions for these models will be considered on the strip. These give rise, in the continuum scaling limit, to representations of the Virasoro algebra. The various (irreducible, reducible, indecomposable) representations will be described along with their fusion rules found empirically from the lattice approach.

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Y.H. Quano:    Vertex operator approach for correlation functions of Belavin's (Z/nZ)-symmetric model

Belavin's (Z/nZ)-symmetric model is considered on the basis of bosonization of vertex operators in the An-1(1) model and vertex-face transformation. CTM Hamiltonian of (Z/nZ)-symmetric model and tail operators are expressed in terms of bosonized vertex operators in the An-1(1) model. Correlation functions of (Z/nZ)-symmetric model can be obtained by using these objects, in principle. In particular, we calculate spontaneous polarization, which reproduces the result by myself in 1993.

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O. Ragnisco:    Backlund transformations for trigonometric Gaudin models

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F. Ravanini:    On the analytic continuation of NLIE from Sine-Gordon to Sinh-Gordon

The lagrangians of Sine-Gordon and Sinh-Gordon are related one another by the analytic continuation of the parameter β to i β. It is well known that the S-matrix of Sine-Gordon lightest breather and Sinh-Gordon particle are related accordingly. We here investigate the behaviour of the analyitic continuation of the NLIE governing the finite size effects of Sine-Gordon. It turns out that the first determination of the NLIE disappears into an essential singularity, while the second determination can be analytically continued to negative values of β2 giving exactly the Sinh-Gordon NLIE/TBA derived by other means. Possible applications of this result to non-compact integrable models of interest in various areas of Physics are briefly commented.

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M. Rossi:    On the commuting charges for the highest dimension SU(2) operator in planar N=4 SYM

We consider the highest anomalous dimension operator in the SU(2) sector of planar N=4 SYM at all-loop, though neglecting wrapping contributions. In any case, the latters enter the loop expansion only after a precise length-depending order onwards. In the thermodynamic limit we write both a linear integral equation for the Bethe root density and a linear system obeyed by the commuting charges. Consequently, we determine the leading strong coupling contribution to the density and from this an approximation to the charges: the charge Qr shows a λ1/4-r/2 scaling which generalises the Gubser-Klebanov-Polyakov energy (∝ λQ2) law. In the end, we briefly extend these considerations to finite lenghts by using the idea of a non-linear integral equation.

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G. Satta:    Global treatment of gl(m|n) spin chains through analytical Bethe Ansatz

We compute the Bethe Ansatz equations for spin chains based on the gl(m|n) superalgebra, for any representation and Dynkin diagram. The analytical Bethe Ansatz approach allows to treat open chains with general boundary conditions.

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K. Shigechi:    Generalized O(n) loop model

After a brief review of recent developments around the Razumov-Stroganov conjectures, I present a new graphical way of constructing generalized link patterns for Uq(slN) spin chain by using the rhombus tiling. This tiling method may be applied to spin chains with boundaries. Finally, the Razumov-Stroganov conjecture for O(1) loop model is generalized to the higher-rank case as the positivity conjecture for the solution of Uq(slN) quantum Knizhnik-Zamolodchikov equation.

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J. Shiraishi:    Kernel function for Koornwinder's operator and their applications

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J. Suzuki:    The ODE/IM correspondence in excited states will be discussed

C. Zambon:    Purely transmitting defects in affine Toda field theory

The possibility to have purely transmitting defects in affine Toda field theory is discussed. Making use, as an example, of the affine Toda model based on the a2 Lie algebra, the quantum transmission matrix describing the interaction between a defect and a soliton is proposed. This matrix represents an infinite dimensional solution of the triangular equations investigated, and its choice will be fully justified. Unlike the sine-Gordon model, the a2 model displays intriguing disparities between the classical and the quantum pictures such as the presence, in the quantum context, of an unstable bound state, which is missing in the classical framework.

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