Modified algebraic Bethe ansatz for XXZ chain on the segment – III – Proof

Avan, Jean; Belliard, S.; Grosjean, Nicolas; Pimenta, Rodrigo A.

doi:10.1016/j.nuclphysb.2015.08.006

Cited by 66 publications

(85 citation statements)

References 57 publications

(175 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The equations (33,34) are the central relations in the MABA. 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: 97%

“…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%

“…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%

“…The main object is the R-matrix, solution of the Yang-Baxter equation, R ab (u − v)R ac (u − w)R bc (v − w) = R bc (v − w)R ac (u − w)R ab (u − v), (1) defined in the tensor product of complex vector spaces V a ⊗ V b ⊗ V c with V i = C 2 . In the case of the XXX spin chain, the associated R-matrix is called the rational R-matrix and it is given by, R(u) = u + P , (2) where P = σ…”

mentioning

confidence: 99%

“…for any Q in GL (2). For quantum integrable models on the segment additional objects must be considered, namely, the K-matrix, solution of the reflection equation,…”

mentioning

confidence: 99%

See 4 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.

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: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…for any Q in GL (2). For quantum integrable models on the segment additional objects must be considered, namely, the K-matrix, solution of the reflection equation,…”

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.

View full text Add to dashboard Cite

show abstract

Exact solution of the quantum integrable model associated with the twisted $$ {\mathrm{D}}_3^{(2)} $$ algebra

Hao

et al. 2022

J. High Energ. Phys.

View full text Add to dashboard Cite

We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted $$ {D}_3^{(2)} $$ D 3 2 algebra (or the $$ {D}_3^{(2)} $$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the $$ {D}_{n+1}^{(2)} $$ D n + 1 2 model and other high rank integrable models.

show abstract

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

Belliard

Slavnov

2018

J. High Energ. Phys.

View full text Add to dashboard Cite

We consider $$ \mathfrak{g}{\mathfrak{l}}_2 $$ g 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.

show abstract

Modified algebraic Bethe ansatz for XXZ chain on the segment – III – Proof

Cited by 66 publications

References 57 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

Exact solution of the quantum integrable model associated with the twisted $$ {\mathrm{D}}_3^{(2)} $$ algebra

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

Contact Info

Product

Resources

About