Gaudin Model Research Articles

Elliptic Gaudin models and elliptic KZ equations

The Gaudin models based on the face-type elliptic quantum groups and the XYZ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding face-type Knizhnik–Zamolodchikov equations and their solutions are given.

Exact correlation functions of the BCS model in the canonical ensemble.

We evaluate correlation functions of the BCS model for a finite number of particles. The integrability of the Hamiltonian relates it with the Gaudin algebra G[sl(2)]. Therefore, a theorem that Sklyanin proved for the Gaudin model, can be applied. Several diagonal and off-diagonal correlators are calculated. The finite-size scaling behavior of the pairing correlation function is studied.

Integrable models for confined fermions: applications to metallic grains

We study integrable models for electrons in metals when the single particle spectrum is discrete. The electron–electron interactions are BCS-like pairing, Coulomb repulsion, and spin-exchange coupling. These couplings are, in general, nonuniform in the sense that they depend on the levels occupied by the interacting electrons. By using the realization of spin-1/2 operators in terms of electrons the models describe spin-1/2 models with nonuniform long range interactions and external magnetic field. The integrability and the exact solution arise since the model Hamiltonians can be constructed in terms of Gaudin models. Uniform pairing and the resulting orthodox model correspond to an isotropic limit of the Gaudin Hamiltonians. We discuss possible applications of this model to a single grain and to a system of few interacting grains.

Creation operators and Bethe vectors of the osp(1|2) Gaudin model

A Gaudin model based on the orthosymplectic Lie superalgebra osp(1|2) is studied. The eigenvectors of the osp(1|2) invariant Gaudin Hamiltonians are constructed by algebraic Bethe ansatz. Corresponding creation operators are defined by a recurrence relation. Furthermore, explicit solution to this recurrence relation is found. The action of the creation operators on the lowest spin vector yields Bethe vectors of the model. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation of the corresponding super-conformal field theory is established.

Trigonometric SU(N) Gaudin model

In this paper, we obtain the eigenstates and the eigenvalues of the Hamiltonians of the trigonometric SU(N) Gaudin model based on the quasi-classical limit of the trigonometric SU(N) chain with the periodic boundary condition. By using the quantum inverse scattering method, we also obtain the eigenvalues of the generating function of the trigonometric SU(N) Gaudin model.

The Open Boundary Dynamical Elliptic Quantum Gaudin Model and Its Solution**The project supported by National Natural Science Foundation of China

We construct the Hamiltonians of open elliptic quantum Gaudin model and show its relation with the open boundary elliptic quantum group. We define eigenstates of the model to be Bethe vectors with of the boundary elliptic quantum group. Then, the Hamiltonian is exactly diagonalized by using the algebraic Bethe ansatz method.

Rational SU(3) Gaudin Model**The project supported by National Natural Science Foundation of China

The Hamiltonians of the SU(3) Gaudin model are constructed based on the nonrelativistic limit of the SU(3) chain. After the quantum determinant being well defined, the eigenvectors and eigenvalues of the Hamiltonians of the SU(3) Gaudin model are given. These results can be generalized to any number of constituting spins (SU(N)).

Singular and nonsingular eigenvectors for the Gaudin model

We present a method to construct a basis of singular and nonsingular common eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie algebra SL(2). The subset of singular vectors is completely described by analogy with covariant differential operators. The relation between singular eigenvectors and the Bethe Ansatz is discussed. In each weight subspace the set of singular eigenvectors is completed to a basis, by a family of nonsingular eigenvectors. We discuss also the generalization of this method to the case of an arbitrary Lie algebra.

Rational SU(N) Gaudin Model

We propose the eigenstates and eigenvalues ofHamiltonians of the rational SU(N) Gaudin model basedon the quasi-classicallimit of the SU(N) chain under the periodic boundary condition.Using the quantum inverse scattering method,we also obtain the eigenvalues of thegeneration function of the rational SU(N) Gaudin model.

GeneralizedSUq(1|2) Gaudin model

In this paper we propose the Hamiltonians of the generalized SUq(1|2) Gaudin model corresponding to the periodic generalized t-J model. With the help of the well defined graded quantum determinant, we obtain the eigenstate and eigenvalues of the generating function and the Hamiltonians of the Gaudin model in the fermionic background in the framework of the graded quantum inverse scattering method. The Bethe ansatz equations are also obtained.

Off-shell Bethe ansatz equation for osp(2∣1) Gaudin magnets

The semi-classical limit of the algebraic Bethe Ansatz method is used to solve the theory of Gaudin models. Via the off-shell method we find the spectra and eigenvectors of the N-1 independent Gaudin Hamiltonians with symmetry osp(2|1). We also show how the off-shell Gaudin equation solves the trigonometric Knizhnik- Zamolodchikov equation.

Supersymmetric SU (1|2) Gaudin model

In this paper, we propose a supersymmetric SU(1|2) Gaudin model and have derived its eigenvalues. We also present the well-defined eigenstates through the quasi-classical limit of the eigenstates in the supersymmetric t-J model.

On the relationship between classical Gaudin models and BC-type Gaudin models

Two invertible transformations of parameters and variables which connect classical sl(2,) Gaudin models with BC-type Gaudin models are proposed. The explicit corresponding formulae of the Lax matrices and the r-matrix relations are exhibited. As applications, we establish a relation of two kinds of restricted Kaup-Newell flows, present the BC-type counterpart of the well known restricted AKNS flow and obtain the Gaudin model corresponding to the Hikami system.

Integrable Many-Body Systems via the Inosemtsev Limit

The Inozemtsev limit (IL), or the scaling limit, is known as a procedure applied to the elliptic Calogero-Moser model. It is a combination of the trigonometric limit, infinite shifts of particle coordinates, and coupling-constant rescalings. This results in an interaction of the exponential type. We show that the IL applied to the sl(N,ℂ) elliptic Euler-Calogero-Moser model and to the elliptic Gaudin model produces new Toda-like systems of N interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit of sl(n,ℂ). The limits corresponding to the complete degeneration of the orbital degrees of freedom lead to recovering only the known periodic and nonperiodic Toda systems. We classify the systems appearing in the IL in the sl(3,ℂ) case. This classification is represented on a two-dimensional plane of parameters describing infinite shifts of particle coordinates. This space is subdivided into symmetric domains. In this picture, a mixture of the Toda and trigonometric Calogero-Sutherland potentials emerges on lower-dimensional domain walls. Because of obvious symmetries, this classification can be generalized to an arbitrary number of particles. We also apply the IL to the sl(2,ℂ) elliptic Gaudin model on a two-punctured elliptic curve and discuss the main properties of its possible limits. The limits of Lax matrices are also considered.

Solvable gaudin models for higher rank symplectic algebras

We study generalisations of theXXX-model to higher rank underlying symplectic algebras in the semi-classical approximation. With the help of the metaplectic representation ofsp(2n,ℝ) we establish general results that are useful in the analysis of these algebras, and hence solve the algebraic Bethe ansatz explicitly in this representation. Thesu(n) multiplet structure of the Bethe eigenstates is revealed.

Separation of variables for quantum integrable systems on elliptic curves

We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ by difference operators. One model of statistical mechanics to which this method applies is the IRF model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher order theta functions.

Osp (1|2) off-shell Bethe ansatz equations

The semiclassical limit of the algebraic quantum inverse scattering method is used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz equations of an integrable representation of the graded osp(1|2) vertex model we find the spectrum of the N - 1 independents Hamiltonians of Gaudin. Integral representations of the N-point correlators are presented as solutions of the Knizhnik-Zamolodchikov equation. These results are extended for highest representations of the osp (1|2) Gaudin algebra.

Separation of Variables in the Elliptic Gaudin Model

For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest one-degree-of-freedom case is studied in detail.

QUASI-EXACTLY SOLVABLE DEFORMATIONS OF GAUDIN MODELS AND "QUASI-GAUDIN ALGEBRAS"

A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra.

Gaudin Model, KZ Equation and an Isomonodromic Problem on the Tours

This Letter presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe within the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural r-matrix structure. These Hamiltonians are used to build a nonautonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.