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

Abstract

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...

Correlation functions by separation of variables: the XXX spin chain

“…Another advantage with respect to the ABA approach concerns the fact that scalar products of separate states 4 can be generically expressed in the form of determinants, at least for rank one models 5 , [100, For z-oriented boundary magnetic fields, the model is solvable by ordinary Bethe Ansatz or by the q-vertex operator approach, and there exist exact representations for the correlation functions [27,28,63,64]. 2 Here we are referring not only to the Separation of Variables -the subject of the present article -but also to an interesting modification of the Bethe Ansatz (the so-called modified algebraic Bethe Ansatz), introduced in [65][66][67][68] and developed further in [69][70][71] in what concerns the computations of scalar products of Bethe states, a first step towards correlation functions. Let us also mention in this context the so-called off-diagonal Bethe Ansatz which was proposed to describe the spectrum of models without U(1) symmetry [72] (the corresponding eigenstates being anyway constructed through SoV).…”

“…Let us mention here that such inhomogeneous TQ-equations also appear naturally in the context of the modified algebraic Bethe Ansatz[65][66][67][68][69][70][71]133] 7. i.e.…”

Scalar products of Bethe vectors in the 8-vertex model

“…In 1998, Kitanine, Maillet and Terras [36] obtained this result by a different method and showed that the matrix elements of the matrix participating in the determinant representation of scalar products are expressed through derivatives of eigenvalues of the transfer matrix. Later similar results were obtained for models with the 6-vertex R-matrix with non-periodic boundary conditions [44][45][46][47]. In paper [48], a determinant representation was obtained for scalar products in the elliptic solid-on-solid (SOS) model, which is closely related to the 8-vertex model.…”

“…it has N zeros in the fundamental domain. The above properties imply that the dependence on x and y factorizes and the function φ (ν,µ) (v, r, x, y) can be represented in the form 47) where the function φ (ν,µ) 1…”

Correlation functions for open XXX spin 1/2 quantum chains with unparallel boundary magnetic fields

“…See[137] and also[138] for the original idea developed first for periodic chains and see also[20] for a first conjecture on these determinant formulae and[139][140][141] for previous determinant representations under special boundary constraints 7. One should notice that a priori the scalar products analyzed in[136] are between left C-gauged Bethe like states and right B-gauged Bethe like states so a priori different w.r.t.…”