In this article, we define a non-commutative deformation of the "symplectic invariants" of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergmann kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expected to be related to the enumeration problem of discrete non-orientable surfaces of arbitrary topologies.
The double row transfer matrix of the open O(N) spin chain is diagonalized and the Bethe Ansatz equations are also derived by the algebraic Bethe Ansatz method including the so far missing case when the residual symmetry is O(2M+1)$\times$O(2N-2M-1). In this case the boundary breaks the "rank" of the O(2N) symmetry leading to nonstandard Bethe Ansatz equations in which the number of Bethe roots is less than as it was in the periodic case. Therefore these cases are similar to soliton-nonpreserving reflections.
We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU(3)-invariant Heisenberg chain.
In these notes we study integrable structures of conformal field theory with $\textrm{BCD}$ symmetry. We realise these integrable structures as affine Yangian $\mathfrak{gl}(1)$ "spin chains" with boundaries. We provide three solutions of Sklyanin $\textrm{KRKR}$ equation compatible with affine Yangian $R$-matrix and derive Bethe ansatz equations for the spectrum.
We introduce a generalization of the original Coordinate Bethe Ansatz that allows to treat the case of open spin chains with non-diagonal boundary matrices. We illustrate it on two cases: the XXX and XXZ chains. Short review on a joint work with N. Crampe (L2C) and D. Simon (LPMA), see arXiv:1009.4119, arXiv:1105.4119 and arXiv:1106.3264.
We prove the modified algebraic Bethe Ansatz characterization of the spectral problem for the closed XXX Heisenberg spin chain with an arbitrary twist and arbitrary positive (half)-integer spin at each site of the chain. We provide two basis to characterize the spectral problem and two families of inhomogeneous Baxter T-Q equations. The two families satisfy an inhomogeneous quantum Wronskian equation.
We present the construction of the full set of eigenvectors of the open ASEP and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This "matrix coordinate Bethe Ansatz" can be viewed as a non-commutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in
Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin-1/2 chain as examples, we show that for a given inhomogeneous T-Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.
Using Functional Bethe Ansatz technique, factorizing Drinfel'd Twists for any finite dimensional irreducible representations of the Yangian Y(sl(2)) are constructed.
In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solut…
We present a nested algebraic Bethe ansatz for one-dimensional open so(2n)- and sp(2n)-symmetric spin chains with diagonal boundary conditions and described by the extended twisted Yangian. We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a so(2n)- or sp(2n)-symmetric open spin chain to that of a gl(n)-symmetric open spin chain. We explicitly derive the structure of Bethe vectors and the nested Bethe equations.
In this article we have discovered a close relationship between the (algebraic) Bethe Ansatz equations of the spin $s$ XXZ model of a finite size and the $q$-Sturm-Liouville problem. We have demonstrated that solutions of the Bethe Ansatz equations give rise to the polynomial solutions of a second order $q$-difference equation in terms of Askey-Wilson operator. The more general form of Bethe Ansatz equations and the mathematical problems relevant to the physics of XXZ model are discussed. Furthermore, the similar correspondence between Bethe Ansatz equations of XXX model and the Sturm-Liouvill…
We use the algebraic Bethe ansatz to obtain the eigenvalues and eigenvectors of the spin-1 Temperley-Lieb open quantum chain with "free" boundary conditions. We exploit the associated reflection algebra in order to prove the off-shell equation satisfied by the Bethe vectors.
We study the partition function of the six-vertex model in the rational limit on arbitrary Baxter lattices with reflecting boundary. Every such lattice is interpreted as an invariant of the twisted Yangian. This identification allows us to relate the partition function of the vertex model to the Bethe wave function of an open spin chain. We obtain the partition function in terms of creation operators on a reference state from the algebraic Bethe ansatz and as a sum of permutations and reflections from the coordinate Bethe ansatz.
With the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the $XXX$ spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated $T-Q$ relation and the Bethe ansatz equations are derived.
We show that the scalar products of on-shell and off-shell Bethe vectors in the algebra1ic Bethe ansatz solvable models satisfy a system of linear equations. We find solutions to this system for a wide class of integrable models. We also apply our method to the XXX spin chain with broken $U(1)$ symmetry.
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.
This course of lectures on the algebraic Bethe ansatz was given in the Scientific and Educational Center of Steklov Mathematical Institute in Moscow. The course includes both classical well known results and very recent ones.
We present a unified algebraic Bethe ansatz for open vertex models which are associated with the non-exceptional $A^{(2)}_{2n},A^{(2)}_{2n-1},B^{(1)}_n,C^{(1)}_n,D^{(1)}_{n}$ Lie algebras. By the method, we solve these models with the trivial K matrix and find that our results agree with that obtained by analytical Bethe ansatz. We also solve the $B^{(1)}_n,C^{(1)}_n,D^{(1)}_{n}$ models with some non-trivial diagonal K-matrices (one free parameter case) by the algebraic Bethe ansatz.
We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the ``beyond the equator'', infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.
The generic quantum $τ_2$-model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with periodic boundary condition is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix (solutions of the recursive functional relations in $τ_j$-hierarchy) with generic site-dependent inhomogeneity parameters are given in terms of an inhomogeneous T-Q relation with polynomial Q-functions. The associated Bethe Ansatz equations are obtained. Numerical solutions of the Bethe Ansatz equations for small number of sites indicate that the inhomogeneous T-Q relation…
We provide a conjecture for the following two quantities related with the spin-$\frac{1}{2}$ isotropic Heisenberg model defined over rings of even lengths: (i) the number of the solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie-Wang. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.
The quantum $τ_2$-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T-Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.
We prove that physical solutions to the Heisenberg spin chain Bethe ansatz equations are exactly obtained by imposing two zero-remainder conditions. This bridges the gap between different criteria, yielding an alternative proof of a recently devised algorithm based on $QQ$ relations, and solving its minimality issue.
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.
The Heun-Racah operator is diagonalized with the help of the modified algebraic Bethe ansatz. This operator is the most general bilinear expression in two generators of the Racah algebra. A presentation of this algebra is given in terms of dynamical operators, and allows the construction of Bethe vectors for the Heun-Racah operator. The associated Bethe equations are derived for both the homogeneous and inhomogeneous cases.
The spin-1/2 XYZ model with both periodic and anti-periodic boundary conditions is studied via the off-diagonal Bethe ansatz method. The exact spectra of the Hamiltonians and the Bethe ansatz equations are derived by constructing the inhomogeneous T-Q relations, which allow us to treat both the even N (the number of lattice sites) and odd N cases simultaneously in an unified approach.
A Bethe Ansatz study of a self dual Z_N spin model is undertaken for even spin system. One has to solve a coupled system of Bethe Ansatz Equations (BAE) involving zeroes of two families of transfer matrices. A numerical study on finite size lattices is done for identification of elementary excitations over the Ferromagnetic and Antiferromagnetic ground states. The free energies for both Ferromagnetic and Antiferromagnetic ground states and dispersion relation for elementary excitations are found.
We formulate the Bethe Ansatz equations for the open super spin chain based on the super Yangian of osp(M|2n) and with diagonal boundary conditions. We then study the bulk and boundary scattering of the osp(1|2n) open spin chain.
The theory of classically integrable nonlinear wave equations, and the Bethe Ansatz systems describing massive quantum field theories defined on an infinite cylinder, are related by an important mathematical correspondence that still lacks a satisfactory physical interpretation. In this paper we shall describe this link for the case of the classical and quantum versions of the (Tzitzéica-)Bullough-Dodd model.
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the corresponding Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the Bethe vectors through the so-called quasi-classical limit.
We study the highest weight representations of the RTT--algebras for the R--matrix sp(4) type by the nested algebraic Bethe ansatz. These models were solved by Reshetikhin for sp(2n) but using a very special type of representation. The explicit construction of the Bethe vectors is carried out in the general case and the explicit formulae for the Bethe equations are derived. In conclusion, the direct generalization for sp(2n) is formulated.
This paper continues our recent studies on the algebraic Bethe ansatz for the RTT-algebras of sp($2n$) and o($2n$) types. In these studies, we encountered the RTT-algebras which we called An. The next step in our construction of the Bethe vectors for the RTT-algebras of type sp($2n$) and o($2n$) is to find the Bethe vectors for the RTT-algebras $\mathcal{A}_n$. This paper deals with the construction of the Bethe vectors of the RTT-algebra An using the Bethe vectors of the RTT-algebra $\mathcal{A}_{n-1}$.
We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated.
In this proceeding we present the nested Bethe ansatz for open spin chains of XXX-type, with arbitrary representations (i.e. `spins') on each site of the chain and diagonal boundary matrices $(K^+(u),K^-(u))$. The nested Bethe anstaz applies for a general $K^-(u)$, but a particular form of the $K^+(u)$ matrix. We give the eigenvalues, Bethe equations and the form of the Bethe vectors for the corresponding models. The Bethe vectors are expressed using a trace formula.
We study the highest weight representations of the RTT algebras for the R matrix of sp_q(2n) type by the nested algebraic Bethe ansatz. It is a generalization of our study for R matrix of sp(2n) and so(2n) type
The eigenvectors of the Hamiltonian ${\cal H}_{N}$ of $N$-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang's model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by …
We study solutions of the Bethe ansatz equation for the XXZ-type integrable model associated with the Lie algebra sl_N. We give a correspondence between solutions of the Bethe ansatz equations and collections of quasi-polynomials. This extends the results of E.Mukhin and A.Varchenko for the XXX-type model and the trigonometric Gaudin model.
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 diagonalize Q-operators for rational homogeneous sl(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.
The small polaron, an one-dimensional lattice model of interacting spinless fermions, with generic non-diagonal boundary terms is studied by the off-diagonal Bethe ansatz method. The presence of the Grassmann valued non-diagonal boundary fields gives rise to a typical $U(1)$-symmetry-broken fermionic model. The exact spectra of the Hamiltonian and the associated Bethe ansatz equations are derived by constructing an inhomogeneous $T-Q$ relation.
We establish a correspondence between classical $A_n^{(1)}$ affine Toda field theories and $A_n$ Bethe Ansatz systems. We show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model.
These notes arose from three lectures presented at the Summer School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland, on September 11-18, 2002. We review rigged configurations and the Bethe Ansatz. In the first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged configurations label the solutions of the Bethe equations. This yields the bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov and Reshetikhin. In the second part, we discuss a generalization of t…
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 prove the completeness of the Bethe ansatz eigenfunctions of the Laplacian on a Weyl alcove with repulsive boundary condition at the walls. For the root system of type A this amounts to the result of Dorlas of the completeness of the Bethe ansatz eigenfunctions of the quantum Bose gas on the circle with repulsive delta-function interaction.
The well known string solution to Bethe ansatz equations is shown to be inconsistent in its widely accepted form. A valid refinement demanding higher order corrections in subsequent roots is identified. A new alternative string solution is proposed for finite long chains, consistent with the non-Bethe strings observed earlier.
In this note we construct Q-operators for the spin s open Heisenberg XXX chain with diagonal boundaries in the framework of the quantum inverse scattering method. Following the algebraic Bethe ansatz we diagonalise the introduced Q-operators using the fundamental commutation relations. By acting on Bethe off-shell states and explicitly evaluating the trace in the auxiliary space we compute the eigenvalues of the Q-operators in terms of Bethe roots and show that the unwanted terms vanish if the Bethe equations are satisfied.
We consider open XXX spins chain with two general boundary matrices submitted to one constraint, which is equivalent to the possibility to put the two matrices in a triangular form. We construct Bethe vectors from a generalized algebraic Bethe ansatz. As usual, the method also provides Bethe equations and transfer matrix eigenvalues.
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 $U_q(\hat{\mathfrak{gl}}_3)$.
The spectral problem of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment is investigated within a modified algebraic Bethe ansatz framework. We consider in this work the most general boundaries allowed by integrability. The eigenvalues and the eigenvectors are obtained. They are characterised by a set of Bethe roots with cardinality equal to $N$, the length of the chain, and which satisfies a set of Bethe equations with an additional term.
We derive the coordinate space wave function for the inhomogeneous six-vertex model from the Algebraic Bethe Ansatz. The result is in agreement with the result first obtained long time ago by Yang and Gaudin in the context of the problem of one-dimensional fermions with delta- function interaction.
We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel exp(-u(theta)-u(theta'))/cosh[(1/2)(theta-theta')]
By means of an algebraic Bethe ansatz approach we study the Zamolodchikov-Fateev and Izergin-Korepin vertex models with non-diagonal boundaries, characterized by reflection matrices with an upper triangular form. Generalized Bethe vectors are used to diagonalize the associated transfer matrix. The eigenvalues as well as the Bethe equations are presented.
