Gaudin Model Research Articles

On Bethe vectors in the slN+1 Gaudin model

On Bethe vectors in the slN+1 Gaudin model

Bäcklund transformations for the rational Lagrange chain

We consider a long–range homogeneous chain where the local variables are the generators of the direct sum of N interacting Lagrange tops. We call this classical integrable model rational “Lagrange chain” showing how one can obtain it starting from rational Gaudin models. Moreover we construct one- and two–point integrable maps (Bäcklund transformations).

A rigid body dynamics derived from a class of extended Gaudin models: An integrable discretization RID A-7283-2010

A rigid body dynamics derived from a class of extended Gaudin models: An integrable discretization RID A-7283-2010

TheA(2)2Gaudin model and its associated Knizhnik–Zamolodchikov equation

The semiclassical limit of the algebraic Bethe Ansatz for the Izergin-Korepin 19-vertex model is used to solve the theory of Gaudin models associated with the twisted $A_{2}^{(2)}$ R-matrix. We find the spectra and eigenvectors of the $N-1$ independents Gaudin Hamiltonians. We also use the off-shell Bethe Ansatz method to show how the off-shell Gaudin equation solves the associated trigonometric system of Knizhnik-Zamolodchikov equations.

Exactly-solvable models derived from a generalized Gaudin algebra

We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen–Cooper–Schrieffer, Suhl–Matthias–Walker, Lipkin–Meshkov–Glick, the generalized Dicke and atom–molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.

Formula omitted] Gaudin magnet and its associated Knizhnik–Zamolodchikov equation

The semiclassical limit of the algebraic Bethe Ansatz method is used to solve the theory of Gaudin models for the s l ( 2 | 1 ) ( 2 ) R-matrix. We find the spectra and eigenvectors of the N − 1 independent Gaudin Hamiltonians. We also use the off-shell Bethe Ansatz method to show how the off-shell Gaudin equation solves the associated trigonometric system of Knizhnik–Zamolodchikov equations.

Znelliptic Gaudin model with open boundaries

The n elliptic Gaudin model with integrable boundaries specified by generic non-diagonal K-matrices with n+1 free boundary parameters is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained.

Exact solution of the XXZ Gaudin model with generic open boundaries

The XXZ Gaudin model with generic integrable boundaries specified by generic non-diagonal K-matrices is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained.

Self–self-dual spaces of polynomials

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self–self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self–self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians. We show that the self–self-dual spaces correspond to G 2-populations related to the Bethe Ansatz of the Gaudin model of type G 2 and prove that a G 2-population is isomorphic to the G 2 flag variety.

Elliptic Linear Problem for the Calogero–Inozemtsev Model and Painlevé VI Equation

We introduce $3N\times 3N$ Lax pair with spectral parameter for Calogero-Inozemtsev model. The one degree of freedom case appears to have $2\times 2$ Lax representation. We derive it from the elliptic Gaudin model via some reduction procedure and prove algebraic integrability. This Lax pair provides elliptic linear problem for the Painlev{\'e} VI equation in elliptic form.

Opers on the Projective Line, Flag Manifolds and Bethe Ansatz

We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a oper on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold. The Bethe Ansatz equations can be written for an arbitrary Kac-Moody algebra, and we prove an analogue of the last result in this more general setting. For the Lie algebras of types A,B,C similar results were obtained by other methods by I.Scherbak and A.Varchenko and by E.Mukhin and A.Varchenko.

Gaudin models and bending flows: a geometrical point of view

In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the 'bending flows' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.

Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models

A unifying scheme based on an ancestor model is proposed for generating a wide range of integrable discrete and continuum as well as inhomogeneous and hybrid models. They include in particular discrete versions of sine-Gordon, Landau–Lifshitz, nonlinear Schrödinger (NLS), derivative NLS equations, Liouville model, (non-)relativistic Toda chain, Ablowitz–Ladik model, etc. Our scheme introduces the possibility of building a novel class of integrable hybrid systems including multicomponent models like massive Thirring, discrete self-trapping, two-mode derivative NLS by combining different descendant models. We also construct inhomogeneous systems like Gaudin model including new ones like variable mass sine-Gordon, variable coefficient NLS, Ablowitz–Ladik, Toda chains, etc. keeping their flows isospectral, as opposed to the standard approach. All our models are generated from the same ancestor Lax operator (or its q→1 limit) and satisfy the classical Yang–Baxter equation sharing the same r-matrix. This reveals an inherent universality in these diverse systems, which become explicit at their action-angle level.

Elementary excitations of the BCS model in the canonical ensemble

We have found the elementary excitations of the exactly solvable BCS model for a fixed number of particles. These turn out to have a peculiar dispersion relation, some of them with no counterpart in the Bogoliubov picture, and unusual counting properties related to an old conjecture made by Gaudin. We give an algorithm to count the number of excitations for each excited state and a graphical interpretation in terms of paths and Young diagrams. For large systems the excitations are described by an effective Gaudin model, which accounts for the finite size corrections to BCS.

Trigonometric osp(1|2) Gaudin model

The problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(1|2) classical r-matrix. The eigenvectors of the trigonometric osp(1|2) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik–Zamolodchikov system is discussed both in the rational and in the trigonometric case.

Critical Points of Functions, sl2 Representations, and Fuchsian Differential Equations with only Univalued Solutions

Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z_1, ..., z_n with exponents (a_1,b_1), ..., (a_n,b_n). Let the exponents at infinity be (A,B). Then for fixed generic z_1,...,z_n, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl_2 representation of dimension |A-B| in the tensor product of irreducible sl_2 representations of dimensions |a_1-b_1|, >..., |a_n-b_n|. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl_2 KZ equation and of the Bethe vectors in the sl_2 Gaudin model. As a byproduct of this study we conclude that the Bethe vectors form a basis in the space of states for the sl_2 inhomogeneous Gaudin model.

Relationships among three kinds of finite dimensional integrable Hamiltonian systems

A new type of finite dimensional integrable Hamiltonian systems containing many restricted soliton flows are considered. Their relations with the generalized classical Gaudin models and the generalized classical Gaudin models with boundary are established. As an application, the relationships among three different restricted mKdV flows are exhibited.

Supersymmetrict JGaudin models and KZ equations

Supersymmetric t-J Gaudin models with both periodic and open boundary conditions are constructed and diagonalized by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the Gaudin systems are derived, and used to construct and solve the KZ equations associated with $sl(2|1)^{(1)}$ superalgebra.

Bethe ansatz for the Gaudin model and its relation with Knizhnik–Zamolodchikov equations

We recall the construction of the common eigenvectors of Gaudin Hamiltonians based on the Bethe ansatz. In the case of an arbitrary Lie algebra, this construction can be done either recursively or explicitly and we prove the equivalence of the two methods. We also prove that Bethe vectors are singular only if the Bethe equations are satisfied. In each eigenspace of the spin operator we construct additional common eigenvectors, having the same eigenvalue as the vacuum vector and which can not be obtained by the Bethe ansatz. These eigenvectors are not singular. We also recall the connection between Bethe vectors and integral solutions of the KZ equation. In an analogous way, the additional vectors lead to solutions of KZ equation which are not singular vectors and do not have an integral representation.

The dynamical elliptic quantum Gaudin models and their solutions

In this paper, we construct the Hamiltonians of both periodic and open elliptic quantum Gaudin models and show their relations with the elliptic quantum group, and the boundary elliptic quantum group, respectively. We define the eigenstates of these two models to be the Bethe vectors with η=0 of the elliptic quantum group and the boundary elliptic quantum group, respectively. Then, the Hamiltonians are exactly diagonalized by using the algebraic Bethe ansatz method.