Invariant Integrable Models Research Articles

Algebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$-invariant integrable models

A class of $\mathfrak{o}_{2n+1}$-invariant quantum integrable models is investigated in the framework of algebraic Bethe ansatz method. A construction of the $\mathfrak{o}_{2n+1}$-invariant Bethe vector is proposed in terms of the Drinfeld currents for the double of Yangian $\mathcal{D}Y(\mathfrak{o}_{2n + 1})$. Action of the monodromy matrix entries onto off-shell Bethe vectors for these models is calculated. Recursion relations for these vectors were obtained. The action formulas can be used to investigate structure of the scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant models.

Actions of the monodromy matrix elements onto -invariant Bethe vectors

Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the -invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.

A note on mathfrak{g}{mathfrak{l}}_2 -invariant Bethe vectors

We consider mathfrak{g}{mathfrak{l}}_2 -invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries on the twisted off-shell Bethe vectors.

Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models

We study $\mathfrak{gl}(2|1)$ symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors.

Form factors of the monodromy matrix entries in [formula omitted]-invariant integrable models

We study integrable models solvable by the nested algebraic Bethe ansatz and described by gl(2|1) or gl(1|2) superalgebras. We obtain explicit determinant representations for form factors of the monodromy matrix entries. We show that all form factors are related to each other at special limits of the Bethe parameters. Our results allow one to obtain determinant formulas for form factors of local operators in the supersymmetric t–J model.

Baxter Q-operator and functional relations

We obtain the Baxter Q-operators in the Uq(slˆ2) invariant integrable models as a special limits of the quantum transfer matrices corresponding to different spins in the auxiliary space. We derive the Baxter equation from the well-known fusion relations for the transfer matrices. Our method is valid for an arbitrary integrable model corresponding to the quantum group Uq(slˆ2), for example for the XXZ-spin chain.

Form factors in SU(3)-invariant integrable models

We study SU(3)-invariant integrable models solvable by a nested algebraic Bethe ansatz. We obtain determinant representations for form factors of diagonal entries of the monodromy matrix. This representation can be used for the calculation of form factors and correlation functions of the XXX SU(3)-invariant Heisenberg chain.

Bethe vectors of GL(3)-invariant integrable models

We study GL(3)-invariant integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the Bethe vectors and the actions of the generators of the Yangian \U0001d4b4(gl3) on the Bethe vectors are considered. These actions are relevant for the calculation of correlation functions and form factors of local operators of the underlying quantum models.

The algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models

We study SU(3)-invariant integrable models solvable by a nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of the XXX SU(3)-invariant Heisenberg chain.

Highest coefficient of scalar products in SU(3)-invariant integrable models

We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of a bilinear combination of their highest coefficients. We obtain various different representations for the highest coefficient in terms of sums over partitions. We also obtain multiple integral representations for the highest coefficient.

Pullback of the volume form, integrable models in higher dimensions and exotic textures

A procedure allowing for the construction of Lorentz invariant integrable models living in d+1 dimensional space time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized zero curvature formulation and infinitely many conserved quantities. A close relation between the Lagrange density of the integrable models and the pullback of the pertinent volume form on target space is established. Moreover, we show that the conserved currents are Noether currents generated by the volume-preserving diffeomorphisms. Further, we show how such models may emerge via Abelian projection of some gauge theories. Then we apply this framework to the construction of integrable models with exotic textures. Particularly, we consider integrable models providing exact suspended Hopf maps, i.e., solitons with a nontrivial topological charge of π4(S3)≅Z2. Finally, some families of integrable models with solitons of πn(Sn) type are constructed. Infinitely many exact solutions with arbitrary value of the topological index are found. In addition, we demonstrate that they saturate a Bogomolny bound.

Highest-weight Uq(sl(n)) modules and invariant integrable n-state models with periodic boundary conditions

Weights are computed for the Bethe vectors of an RSOS-type model with periodic boundary conditions obeying Uq(sl(n)) (q=exp(i pi /r)) invariance. They are shown to be highest-weight vectors. The q-dimensions of the corresponding irreducible representations are obtained.

Quantum group invariant integrable n-state models with periodic boundary conditions

Using the methods of topological quantum field theory we construct aU q [sl(n)] invariant integrable transfer matrix for the case ofq being a root of unity. It corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. its interior. By means of the nested Bethe ansatz method we analyse conformai properties and discuss the representational content of the Bethe ansatz solutions.