Bethe states of the XXZ spin-12chain with arbitrary boundary fields

Abstract

Based on the inhomogeneous T − Q relation constructed via the off-diagonal Bethe Ansatz, the Bethetype eigenstates of the XXZ spin-1 2 chain with arbitrary boundary fields are constructed. It is found that by employing two sets of gauge transformations, proper generators and reference state for constructing Bethe vectors can be obtained respectively. Given an inhomogeneous T − Q relation for an eigenvalue, it is proven that the resulting Bethe state is an eigenstate of the transfer matrix, provided that the parameters of the generators satisfy the associated Bethe Ansatz equations.

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

“…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.…”

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

“…In this section, we adopt the method developed in [37,38] (see also [30]) to construct eigenstates of the su(3) spin torus based on the inhomogeneous T − Q relations given by (2.18) [59] and the basis introduced in the previous section. For the su(3) case, the monodromy matrix is expressed in terms of the operators (3.1)-(3.2) as…”

“…Due to the fact that the left states { θ p 1 , · · · , θ pm 2 ; θ p m 2 +1 , · · · , θ pm |, m 2 = 0, · · · , m; m = 0, · · · , N } given by (3.9) form a basis of the dual Hilbert space, the eigenstate |Ψ is completely determined (up to an overall scalar factor) by the following scalar products [29,37,38]…”

“…Following [37,38], let us consider the quantities θ p 1 , · · · , θ pm 2 ; θ p m 2 +1 , · · · , θ pm |t(θ p m+1 )|Ψ . Acting t(θ p m+1 ) to the right gives rise to the relation…”

“…Several long-standing models [29,[31][32][33][34][35][36] have since been solved. It should be noted that besides ODBA [37,38] some other methods such as the q-Onsager algebra method [39][40][41][42], the separation of variables (SoV) method [43][44][45][46][47][48][49] and the modified algebraic Bethe ansatz method [50][51][52][53] were also used to obtain the eigenstates of the XXZ spin chains with generic boundary conditions. Remarkably, ODBA allows us to obtain eigenvalues of the U(1)-broken models associated with higher-rank algebras such as the su(n) spin chain with generic integrable boundary The paper is organized as follows.…”