Analytic Bethe ansatz for fundamental representations of Yangians (original) (raw)

Analytic Bethe Ansatz and TTT-system in C(1)_2C^{(1)}_2C(1)_2 vertex models

1993

Eigenvalues of the commuting family of transfer matrices are expected to obey the TTT-system, a set of functional relation, proposed recently. Here we obtain the solution to the TTT-system for C(1)_2C^{(1)}_2C(1)_2 vertex models. They are compatible with the analytic Bethe ansatz and Yang-Baxterize the classical characters.

Poles of finite-dimensional representations of Yangians

Cornell University - arXiv, 2020

Let g be a finite-dimensional simple Lie algebra over C, and let Y (g) be the Yangian of g. In this paper, we study the sets of poles of the rational currents defining the action of Y (g) on an arbitrary finite-dimensional vector space V. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of V and the inverse of the q-Cartan matrix of g. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double. Contents 1. Introduction 1 2. Yangians 6 3. Finite-dimensional representations and their poles 9 4. Poles and Baxter polynomials 13 5. Poles of simple modules 21 6. Poles and characters 31 7. Poles, cyclicity conditions and R-matrices 36 8. Representations of DY (g) 46 Appendix A. Tensor products of fundamental modules 53 References 56

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

Nuclear Physics B

An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of 'symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.

Bethe Ansatz for the Universal Weight Function

Annales Henri Poincaré, 2009

We consider universal off-shell Bethe vectors given in terms of Drinfeld realization of the algebra U q ( gl N ) . We investigate ordering properties of the product of the transfer matrix and these vectors. We derive that these vectors are eigenvectors of the transfer matrix if their Bethe parameters satisfy the universal Bethe equations [1].

The Yangian of and its quantum R-matrices

J High Energy Phys, 2011

In this paper we study Yangians of sl(n|m) superalgebras. We derive the universal R-matrix and evaluate it on the fundamental representation obtaining the standard Yang R-matrix with unitary dressing factors. For m = 0, we directly recover up to a CDD factor the well-known S-matrices for relativistic integrable models with su(n) symmetry. Hence, the universal R-matrix found provides an abstract plug-in formula, which leads to results obeying fundamental physical constraints: crossing symmetry, unitrarity and the Yang-Baxter equation. This implies that the Yangian double unifies all desired symmetries into one algebraic structure. In particular, our analysis is valid in the case of sl(n|n), where one has to extend the algebra by an additional generator leading to the algebra gl(n|n). We find two-parameter families of scalar factors in this case and provide a detailed study for gl(1|1).

Algebraic Bethe ansatz for deformed Gaudin model

Journal of Mathematical Physics, 2011

The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl2-invariant Gaudin model. As opposed to the sl2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not hermitian. Finally, the inner products and norms of the Bethe states are studied. * The Gaudin model based on the r-matrix (1.4) was studied as the quasiclassical limit of the corresponding quantum spin chain , which is a deformation of the XXX spin chain where the Jordanian twist is applied to the Yang R-matrix . The algebraic Bethe Ansatz for this Gaudin model was fully implemented in , following ideas in .

Rectangular Yang–Baxter Algebras and Alternating A-Type Integrable Vertex Models

International Journal of Geometric Methods in Modern Physics, 2005

Given a couple of Yang–Baxter operators 𝖱[k] and 𝖱[l] corresponding to integrable anisotropic vertex models of Ak-1 and Al-1 type, respectively, we construct and study a class of related lattice models whose monodromy matrices alternate between the mentioned operators. In order to do that, we use a natural generalization of the idea of coproduct in a bialgebra, that appears in the scenario of non-commutative algebraic geometry, related to the notion of internal homomorphisms of quantum spaces. We build up all eigenstates and eigenvalues of the transfer matrix by means of algebraic Bethe ansatz technics, where not only one vector, but a pseudovacuum subspace is needed for the process of diagonalization.

Heun operator of Lie type and the modified algebraic Bethe ansatz

Journal of Mathematical Physics, 2021

The generic Heun operator of Lie type is identified as a certain BC-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We also show that these Bethe roots are intimately associated with the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of O(3) and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.

Bethe Ansatz and Bethe vectors scalar products

An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra U q ( gl 3 ) is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a "new" realization of the quantum affine algebra U q ( gl 3 ).

From S-matrices to the thermodynamic Bethe ansatz

Nuclear Physics B, 2004

We derive the TBA system of equations from the S-matrix describing integrable massive perturbation of the coset G l × G m /G l+m by the field (1, 1, adj) for all the infinite series of simple Lie algebras G = A n , B n , C n , D n. In the cases A n , C n , where the full S-matrices are known, the derivation is exact, while the B n , D n cases dictate some natural assumption about the form of the crossing-unitarizing prefactor for any two fundamental representations of the algebras. In all the cases the derived systems are transformed to the corresponding functional Y-systems and shown to have the correct high temperature (UV) asymptotic in the ground state, reproducing the correct central charge of the coset. Some specific particular cases of the considered S-matrices are discussed.