Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the $\mathfrak{gl}(m|n)$-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 $\mathfrak{gl}(m)$ case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.
We consider $\mathfrak{gl}_2$-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 onto twisted off-shell Bethe vectors.
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.
Supersymmetric composite generalized quantum integrable models solvable by the algebraic Bethe ansatz are studied. Using a coproduct in the bialgebra of monodromy matrix elements and their action on Bethe vectors, formulas for Bethe vectors in the composite models with supersymmetry based on the super-Yangians $Y[\mathfrak{gl}(2|1)]$ and $Y[\mathfrak{gl}(1|2)]$ are derived.
We study Bethe vectors of integrable models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$. Starting from the super-trace formula, we exhibit recursion relations for these vectors in the case of $Y(\mathfrak{gl}(2|1))$ and $Y(\mathfrak{gl}(1|2))$. These recursion relations allow to get explicit expressions for the Bethe vectors. Using an antimorphism of the super-Yangian $Y(\mathfrak{gl}(m|n))$, we also construct a super-trace formula for dual Bethe vectors, and, for $Y(\mathfrak{gl}(2|1))$ and $Y(\mathfrak{gl}(1|2))$ super-Yangians, show recursion relations for them. Again, the latter all…
We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}_3$-invariant $R$-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.
We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$-invariant $R$-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products…
We study the linear problem associated with modified affine Toda field equation for the Langlands dual $\hat{\mathfrak{g}}^\vee$, where $\hat{\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the $Q$-functions for $\mathfrak{g}$-type quantum integrable models. The $ψ$-system for the solutions associated with the fundamental representations of $\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\hat{\mathfrak{g}}$. We also study the $A^{(2)}_{2r}$ affine …
We study integrable models with $\mathfrak{gl}(2|1)$ symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.
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 m…
We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(2|1)$ symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors i…
We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary quantum spaces are irreducible $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian $Y^\pm(\mathfrak{gl}_{2n})$. We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-symmetric open spin chain to that of a $\mathfrak{gl}_{n}$-symmetric periodic spin chain. We explicitly derive th…
We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by $\mathfrak{gl}(m|n)$ superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.
We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor R matrices of low rank orthogonal algebras and the corresponding RTT algebras. Coincidences with fundamental R matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.
We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current generators of the Yangian double $DY(\mathfrak{gl}(m|n))$. More specifically, we use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian doub…
We consider quantum integrable models associated with $\mathfrak{so}_3$ algebra. We describe Bethe vectors of these models in terms of the current generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this approach we use isomorphism between $R$-matrix and Drinfeld current realizations of the Yangians and their doubles for classical types $B$, $C$, and $D$ series algebras. Using these results we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We show that these action formulas lead to recursions for off-shell Bethe vectors and Bethe equations f…
We apply the nested algebraic Bethe ansatz to a model of one-dimensional two-component Bose gas with delta-function repulsive interaction. Using a lattice approximation of the L-operator we find Bethe vectors of the model in the continuous limit. We also obtain a series representation for the monodromy matrix of the model in terms of Bose fields. This representation allows us to study an asymptotic expansion of the monodromy matrix over the spectral parameter.
We find the explicit form of the basic monodromy operators for the case of the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$. Expressions for the quantum Casimir elements of the quantum groups $\mathrm U_q(\mathfrak{gl}_3)$ and $\mathrm U_q(\mathfrak{sl}_3)$ are obtained as a by-product.
We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients. In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin…
We study integrable models solvable by the nested algebraic Bethe ansatz and described by $\mathfrak{gl}(2|1)$ or $\mathfrak{gl}(1|2)$ 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.
We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(m|n)$-invariant $R$-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system of Bethe equations obeys the same properties. In this way we prove a generalized Gaudin hypothesis for the norm of the Hamiltonian eigenstates.
We study the ODE/IM correspondence for all states of the quantum $\widehat{\mathfrak{g}}$-KdV model, where $\widehat{\mathfrak{g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak{g}$. We construct quantum $\widehat{\mathfrak{g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\widehat{\mathfrak{g}}$-KdV opers s…
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $\mathrm Y(\mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $\mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $\mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
II. Institut für Theoretische Physik, Universität Hamburg（汉堡大学第二理论物理研究所）；Zentrum für Mathematische Physik, Universität Hamburg（汉堡大学数学物理中心）；A.I. Alikhanyan National Science Laboratory, Yerevan Physics Institute（阿·伊·阿利哈尼扬国家科学实验室，叶里凡物理研究所）；Beijing Institute of Mathematical Sciences and Applications（北京数学与交叉科学研究院）
We construct exact eigenvectors and eigenvalues for $U_q(\mathfrak{sp}_{2n})$- and $U_q(\mathfrak{so}_{2n})$-symmetric closed spin chains by means of a nested algebraic Bethe ansatz method. We use a fusion procedure to construct higher-dimensional Lax operators. Our approach generalises and extends the results obtained by Reshetikhin and De Vega-Karowski. We also present a generalisation of Tarasov-Varchenko trace formula for nested Bethe vectors.
We describe a reproduction procedure which, given a solution of the $\mathfrak{gl}_{M|N}$ Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family $P$ of other solutions called the population. To a population we associate a rational pseudodifferential operator $R$ and a superspace $W$ of rational functions. We show that if at least one module is typical then the population $P$ is canonically identified with the set of minimal factorizations of $R$ and with the space of full superflags in $W$. We conjecture that the singular eigenvectors (up to rescal…
We consider closed XXX spin chains with broken total spin $U(1)$ symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on the modified Bethe vectors. The obtained formulas of the multiple actions allow us to calculate the scalar products of the modified Bethe vectors. We find an analog of Izergin-Korepin formula for the scalar products. This formula involves modified Izergin determinants and can be expressed as sums over partitions of the Bethe parameters.
We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $Ψ$-system among subdominant solutions. We then use the $Ψ$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explici…
We present a supermatrix realisation of q-deformed spinor-spinor and spinor-vector R-matrices. These R-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.
We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $Ψ$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.
The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras form a natural family of commutative subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundament…
We reformulate nested relations between off-shell $U_q(\widehat{\mathfrak{gl}}_N)$ Bethe vectors as a certain equation on generating series of strings of the composed $U_q(\widehat{\mathfrak{gl}}_N)$ currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal …
An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ 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 $U_q(\hat{\mathfrak{gl}}_3)$.
We study quantum integrable models with GL(3) trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ onto intersections of different types of Borel subalgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromy matrix.
We consider a composite generalized quantum integrable model solvable by the nested algebraic Bethe ansatz. Using explicit formulas of the action of the monodromy matrix elements onto Bethe vectors in the GL(3)-based quantum integrable models we prove a formula for the Bethe vectors of composite model. We show that this representation is a particular case of general coproduct property of the weight functions (Bethe vectors) found in the theory of the deformed Knizhnik-Zamolodchikov equation.
We give combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for tensor products of irreducible evaluation modules over the Yangian $Y({\mathfrak{gl}}_N)$ and the quantum affine algebra $U_q(\widetilde{\mathfrak{gl}_N})$. The results of the paper were obtained in 1998 and were used in math.QA/9905137, math.QA/0302148, math.QA/0610517.
We study integrable models solvable by the nested algebraic Bethe ansatz and possessing $GL(3)$-invariant $R$-matrix. Assuming that the monodromy matrix of the model can be expanded into series with respect to the inverse spectral parameter, we define zero modes of the monodromy matrix entries as the first nontrivial coefficients of this series. Using these zero modes we establish new relations between form factors of the elements of the monodromy matrix. We prove that all of them can be obtained from the form factor of a diagonal matrix element in special limits of Bethe parameters. As a resu…
The zero modes method is applied in order to get action of the monodromy matrix entries onto off-shell Bethe vectors in quantum integrable models associated with $U_q(\mathfrak{gl}_N)$-invariant $R$-matrices. The action formulas allow to get recurrence relations for off-shell Bethe vectors and for highest coefficients of the Bethe vectors scalar product.
In this paper we compare two constructions of weight functions (off-shell Bethe vectors) for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$. The first construction comes from the algebraic nested Bethe ansatz. The second one is defined in terms of certain projections of products of Drinfeld currents. We show that two constructions give the same result in tensor products of vector representations of $U_q(\hat{\mathfrak{gl}}_N)$.
We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a $\mathfrak{gl}_3$-invariant $R$-matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.
We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin - 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- …
A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with M{\" o}bius like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring is discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological na…
We consider quantum integrable models with $\mathfrak{gl}(2|1)$ symmetry. We derive a set of multiple commutation relations between the monodromy matrix entries. These multiple commutation relations allow us to obtain different representations for Bethe vectors.
Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis
Japan Women’s University（日本女子大学）；Faculty of Marine Technology, Tokyo University of Marine Science and Technology（东京海洋大学海事技术学部）；Department of Physics, Tokyo University of Science（东京理科大学物理学系）
We compute an universal weight function (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in arXiv:math/0610398.
