Algebraic Bethe ansatz for the six vertex model with upper triangular<i>K</i>-matrices

Pimenta, Rodrigo A.; Lima-Santos, A.

doi:10.1088/1751-8113/46/45/455002

Cited by 27 publications

(43 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The new terms in (29,30), i.e., the proportional terms to B(u)|Ψ M (ū) and Ψ M (ū)|C (u), are characteristic of MABA approach. If the number of creation operators equals the length of the chain, i.e., M = N , these terms are given by,…”

Section: ω|C (U) (28)mentioning

confidence: 99%

“…Their proof follows from the rational limit of the proof given in [2] for the XXZ case. We arrive, combining (29) and (33), to the final off-shell equation satisfied by the left and right Bethe vectors,…”

Section: ω|C (U) (28)mentioning

confidence: 99%

“…In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].Recently, the ABA has been generalized to include models with general boundary couplings [3,5,11,6,2]. The modified algebraic Bethe ansatz (MABA) has a distinct feature: the creation operator used to construct the eigenstates has an off-shell structure which leads to an inhomogeneous term in the eigenvalues and in the Bethe equations of the model.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Belliard¹,

Pimenta²

2016

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Abstract. We consider the isotropic spin− 1 2Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin-Korepin formula, i.e., the square of the norm, is also obtained.Introduction. The algebraic Bethe ansatz (ABA) [31,30] is a powerful technique to study the spectral problem of quantum integrable models, as well as to construct their correlation functions and compute physical quantities [23]. The possibility of expressing scalar products in a compact form [17,18,22,32] is a crucial aspect of the method. Nevertheless, the application of the usual ABA to obtain the spectrum and the eigenvectors of quantum integrable models becomes problematic for some important cases, in particular in the presence of the non-diagonal boundaries. In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].Recently, the ABA has been generalized to include models with general boundary couplings [3,5,11,6,2]. The modified algebraic Bethe ansatz (MABA) has a distinct feature: the creation operator used to construct the eigenstates has an off-shell structure which leads to an inhomogeneous term in the eigenvalues and in the Bethe equations of the model. We remember that such inhomogeneous term was firstly proposed in the context of the off-diagonal Bethe ansatz (ODBA) method (see [33] for a review) and recovered in the separation of variables (SoV) framework [24]. Let us also remark that the SoV basis [13] has been used to prove the off-shell equation for the creation operator in the MABA context [2] as well as to obtain the on-shell Bethe vector in the ODBA method [36,37].Once the spectral level is understood, the next natural step is to consider the evaluation of scalar products between Bethe vectors obtained in the MABA framework. The calculation of scalar products within this context is a primordial step to access the physical behavior of systems with general integrable open boundaries, since it paves the way to consider form factors and correlation functions. This task has been recently initiated in the case of the twisted XXX spin chain [7], a prototype model that can be described by the MABA. Modified Slavnov and Gaudin-Korepin formulas, i.e., compact expressions for the scalar product between an on-shell and an off-shell Bethe state and the square of the norm, have been conjectured. We propose in this note similar formulas for the XXX spin chain on the segment. They are given in t...

show abstract

Section: ω|C (U) (28)mentioning

confidence: 99%

Section: ω|C (U) (28)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Belliard¹,

Pimenta²

2016

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…There is a renewed interest in applying the algebraic Bethe ansatz to the open XXX chain with non-periodic boundary conditions compatible with the integrability of the systems [9][10][11][12]. Other approaches include the ABA based on the functional relation between the eigenvalues of the transfer matrix and the quantum determinant and the associated T-Q relation [13], functional relations for the eigenvalues of the transfer matrix based on fusion hierarchy [14] and the Vertex-IRF correspondence [15].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model

2014

View full text Add to dashboard Cite

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 relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.

show abstract

Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Baseilhac

Kojima

2014

Nuclear Physics B

View full text Add to dashboard Cite

The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime.Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values σ a 1 with a = z, ± are obtained. Exploiting the spin-reversal property, relations between n-fold integrals of elliptic theta functions are extracted.Beyond the Ising model, a large class of solvable lattice models have been discovered. In the context of quantum integrable systems on the lattice, spin chains are among the most studied examples with applications which range from condensed matter to high energy physics. Given a Hamiltonian, finding analytical expressions for the exact spectrum, identifying the structure of the space of the eigenvectors and deriving explicit expressions for correlation functions are essential steps in the non-perturbative characterization of the system's behavior which can be compared with experimental data.Among the simplest examples considered in the literature, the XXZ spin chain with different boundary conditions has received particular attention. Over the years, different approaches have been proposed in order to understand the Hamiltonian's spectral problem and derive the correlation functions. For models with periodic boundary conditions, the spectral problem can be handled by methods such as the Bethe ansatz (BA) [1], or the corner transfer matrix method (CTM) in the thermodynamic limit [2]. The computation of the correlation functions, however, is a much more difficult problem in general.Apart from the simplest example -namely the XXZ spin chain with periodic boundary condition -for which correlation functions have been proposed by the quantum inverse scattering method (QISM) [4,5] arising from the BA, the generalization of this result to models with higher symmetries requires a better understanding of mathematical structures, for instance, of determinant formulae of scalar products that involve the Bethe vectors [7] (see some recent progress in [8]). However, in the thermodynamic limit, this problem can be alternatively tackled using the q-vertex operator approach (VOA) [3] arising from the CTM. The space of states is identified with the irreducible highest weight representation of U q ( sl 2 ) or higher rank quantum algebras. Correlation functions can be obtained using bosonizations of the q-vertex operators for U q ( sl 2 ) or higher rank quantum algebras [3,9,10,11,12,13]. Either within the QISM or the VOA, correlation functions are obtained in the form of integrals of meromorphic functions in the thermodynamic limit. The situation for integrable spin chains with open boundaries is more difficult. On one hand, for the finite XXZ open chain with diagonal boundaries [14], related non-diagonal boundaries [15, 16] or q a root of unity [17], the BA makes it possible to deriv...

show abstract

Algebraic Bethe ansatz for the six vertex model with upper triangularK-matrices

Cited by 27 publications

References 32 publications

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model

Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Contact Info

Product

Resources

About