Modified Algebraic Bethe Ansatz: Twisted XXX Case Modified Algebraic Bethe
Ansatz: Twisted XXX Case

Samuel BELLIARD †‡, Nikita A. SLAVNOV § and Benoit VALLET ‡

S. Belliard, N.A. Slavnov and B. Vallet

† Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes
Energies,
† LPTHE, F-75005, Paris, France<EMAIL_ADDRESS>
‡ Institut de Physique Théorique, DSM, CEA, URA2306 CNRS Saclay,
‡ F-91191, Gif-sur-Yvette, France<EMAIL_ADDRESS>
§ Steklov Mathematical Institute of Russian Academy of Sciences, Moscow,
Russia<EMAIL_ADDRESS>
Received April 09, 2018, in final form May 28, 2018; Published online June 07,
2018

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.

integrable spin chain; algebraic Bethe ansatz; Baxter T-Q equation; quantum
Wronskian equation

82R53; 81R12

##  1 Introduction

The question of the spectral problem of the lattice quantum integrable models
without ${\rm U}(1)$ symmetry, as the XXZ Heisenberg spin chain on the segment
with the most general integrable boundary condition [27], has led to the
development of new techniques to perform the Bethe ansatz. Among the proposed
methods such as the off diagonal Bethe ansatz [29] or the separation of the
variables [13, 28], the modified algebraic Bethe ansatz (MABA) proposes to
understand this new method from the algebraic Bethe ansatz point of view. In
this paper we study this method and determine what the changes are with
respect to the case with ${\rm U}(1)$ symmetry and what has to be kept.

The new features in the Bethe ansatz for models without ${\rm U}(1)$ symmetry
are as follows. Firstly, the Baxter T-Q equation has a new term [9, 10].
Secondly, the Bethe vectors are linear combinations of all Bethe vectors of
the associated model where the ${\rm U}(1)$ symmetry is restored, and these
vectors are factorized in terms of a modified creation operators [2, 3, 4],
that was proved in [30, 31] by means of the Baxter T-Q equation111Contrary to
this method, the MABA provides a constructive way to build Bethe vectors and
clarifies the algebraic origin of the new term in the Baxter T-Q equation..
Thirdly, the number of Bethe roots is fixed and depends on the model under
consideration [10, 23].

The MABA also can be applied to the models with quasi-periodic boundary
conditions such as the XXX spin Heisenberg chain with an arbitrary twist. In
particular, one can use this approach for studying XXX spin-$\tfrac{1}{2}$
chain

$\displaystyle
H=\sum_{k=1}^{N}\big{(}\sigma^{x}_{k}\otimes\sigma^{x}_{k+1}+\sigma^{y}_{k}\otimes\sigma^{y}_{k+1}+\sigma^{z}_{k}\otimes\sigma^{z}_{k+1}\big{)},$
(1.1)

subject to the following non-diagonal boundary conditions:

$\displaystyle\gamma\sigma^{x}_{N+1}=\frac{\tilde{\kappa}^{2}+\kappa^{2}-\kappa_{+}^{2}-\kappa_{-}^{2}}{2}\sigma^{x}_{1}+i\frac{\kappa^{2}-\tilde{\kappa}^{2}-\kappa_{+}^{2}+\kappa_{-}^{2}}{2}\sigma^{y}_{1}+(\kappa\kappa_{-}-\tilde{\kappa}\kappa_{+})\sigma^{z}_{1},$
$\displaystyle\gamma\sigma^{y}_{N+1}=i\frac{\tilde{\kappa}^{2}-\kappa^{2}-\kappa_{+}^{2}+\kappa_{-}^{2}}{2}\sigma^{x}_{1}+\frac{\tilde{\kappa}^{2}+\kappa^{2}+\kappa_{+}^{2}+\kappa_{-}^{2}}{2}\sigma^{y}_{1}-i(\tilde{\kappa}\kappa_{+}+\kappa\kappa_{-})\sigma^{z}_{1},$
$\displaystyle\gamma\sigma^{z}_{N+1}=(\kappa\kappa_{+}-\tilde{\kappa}\kappa_{-})\sigma^{x}_{1}+i(\tilde{\kappa}\kappa_{-}+\kappa\kappa_{+})\sigma^{y}_{1}+(\tilde{\kappa}\kappa+\kappa_{+}\kappa_{-})\sigma^{z}_{1}.$

The twist parameters
$\\{\kappa,\tilde{\kappa},\kappa_{+},\kappa_{-}\\}\in\mathbb{C}^{4}$ are
generic and $\gamma=\tilde{\kappa}\kappa-\kappa_{+}\kappa_{-}$. The Pauli
matrices222$\sigma^{z}=\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)$, $\sigma^{+}=\left(\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right)$, $\sigma^{-}=\left(\begin{smallmatrix}0&0\\\
1&0\end{smallmatrix}\right)$, $\sigma^{x}=\sigma^{+}+\sigma^{-}$,
$\sigma^{y}=i(\sigma^{-}-\sigma^{+})$. $\sigma^{\alpha}_{k}$ with
$\alpha=x,y,z$ act non-trivially on the $k$th component of the quantum space
${\mathcal{H}}=\otimes_{k=1}^{N}V_{k}$ with $V_{k}=\mathbb{C}^{2}$. This model
provides a simple example where the new properties of the method can be
studied. In particular, in the case of the Hamiltonian (1.1), several
conjectures about the MABA were formulated in [7]. There, a special
transformation of the twist matrix was proposed. One more conjecture of [7],
formulated on the base of direct calculations for the chains of small length,
concerns a special off-shell action of the modified creation operator, which
generates a new term of the Baxter T-Q equation.

In this paper we consider the most general case of the closed XXX Heisenberg
spin chain with an arbitrary twist and arbitrary positive (half)-integer spins
transfer matrix333Explicit Hamiltonian for this case is not known. It can be
formulated in the more restrictive case of $L_{0}$-regular spin chains [5],
where $L_{0}$ arbitrary positive (half)-integer spins are periodically
repeated. The case $L_{0}=1$ with spin $\frac{1}{2}$ corresponds to the
Hamiltonian (1.1).. We prove the conjecture about the multiple action of the
modified creation operator on the pseudo-vacuum state given in [7]. In fact,
we prove this property independently of the action on the pseudo-vacuum state
and go beyond the proofs done for other models [1, 11, 12]. Moreover, we
consider two different bases for solving the spectral problem of this family
of models and we relate the two solutions by a modified quantum Wronskian
equation.

This paper is organized as follows. In Section 2 we introduce a relevant
notation and recall the main tools to implement the ABA: the Yangian of
$\mathfrak{gl}_{2}$ and the associated representation to define arbitrary
positive (half)-integer spins transfer matrix. In Section 3 we recall, for the
case with diagonal twist ($\kappa^{+}=\kappa^{-}=0$), the ABA scheme and the
associated quantum Wronskian equation. Section 4 is devoted to the case with
non-diagonal twist ($\kappa^{+}\neq 0$ and $\kappa^{-}\neq 0$) and to the
study of the MABA scheme. Here we give two basis for the Bethe vectors as well
as their corresponding spectrum, which allows us to obtain the modified
quantum Wronskian equation. Concluding remarks are given in Section 5.

##  2 The Yangian of $\boldsymbol{\mathfrak{gl}_{2}}$ and Bethe subalgebra

Let us recall the simplest rational $R$-matrix, which is an element of
$\operatorname{End}\big{(}\mathbb{C}^{2}\otimes\mathbb{C}^{2}\big{)}$:

$\displaystyle R(u)=\frac{u}{c}I+P.$

Here $c$ is a constant, $I=\sum\limits_{i,j=1}^{2}E_{ii}\otimes E_{jj}$ is the
identity operator on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$,
$P=\sum\limits_{i,j=1}^{2}E_{ij}\otimes E_{ji}$ is the permutation operator on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, and
$(E_{ij})_{kl}=\delta_{ik}\delta_{jl}$. This $R$-matrix is a solution of the
Yang–Baxter equation

$\displaystyle
R_{ab}(u-v)R_{ac}(u-w)R_{bc}(v-w)=R_{bc}(v-w)R_{ac}(u-w)R_{ab}(u-v).$

Here the $R$-matrix is understood as an element of
$\operatorname{End}(V_{a}\otimes V_{b}\otimes V_{c})$ with $V=\mathbb{C}^{2}$.
Namely, $R_{ab}(u)=R(u)\otimes\mathbb{I}$, $R_{bc}(u)=\mathbb{I}\otimes R(u)$,
$R_{ac}(u)=P_{bc}R_{ab}(u)P_{bc}$, and $\mathbb{I}$ is the identity operator
in $\mathbb{C}^{2}$.

The $R$-matrix (2) is $\mathfrak{gl}_{2}$-invariant, and thus,

$\displaystyle[R_{ab}(u),K_{a}K_{b}]=0,$

for any matrix $K\in\operatorname{End}\big{(}\mathbb{C}^{2}\big{)}$.

Starting from this $R$-matrix, we can define a quantum group algebra, so
called Yangian of $\mathfrak{gl}_{2}$. For this, we introduce a monodromy
matrix

$\displaystyle T(u)=\left(\begin{matrix}t_{11}(u)&t_{12}(u)\\\
t_{21}(u)&t_{22}(u)\end{matrix}\right).$

This matrix satisfies the RTT relation

$\displaystyle R_{ab}(u-v)T_{a}(u)T_{b}(v)=T_{b}(v)T_{a}(u)R_{ab}(u-v).$ (2.1)

Equation (2.1) encodes the commutation relations of the entries $t_{ij}(u)$
(see Appendix A), which generate the Yangian algebra.

The transfer matrix

$\displaystyle t(u)=\mathop{\rm Tr}_{a}\big{(}K_{a}T_{a}(u)\big{)}$

generates a commutative subalgebra of the Yangian of $\mathfrak{gl}_{2}$,
i.e., $[t(u),t(v)]=0$. Its expansion as a series in $u$ leads to the conserved
charges of the model that depend on the specific representation. To perform
the MABA we need to consider a finite dimensional representation. However,
this is not necessarily needed for the usual ABA.

Let us define a highest weight representation by
$\mathcal{V}(\lambda_{1}(u),\lambda_{2}(u))$, where $\lambda_{i}(u)$ are some
complex valued functions, and the highest weight vector $|0\rangle$ is defined
by

$\displaystyle t_{ii}(u)|0\rangle=\lambda_{i}(u)|0\rangle,\qquad
t_{21}(u)|0\rangle=0.$ (2.2)

On the other hand, we can define a lowest weight representation
$\mathcal{V}(\lambda_{2}(u),\lambda_{1}(u))$, and the lowest weight vector
$|\hat{0}\rangle$ is defined by

$\displaystyle t_{ii}(u)|\hat{0}\rangle=\lambda_{3-i}(u)|\hat{0}\rangle,\qquad
t_{12}(u)|\hat{0}\rangle=0.$ (2.3)

This provides enough tools to consider the usual ABA but not to accomplish the
MABA. In the latter case we need to explicitly fix the representation to be
finite dimensional. All of them are given by the highest (lowest) weight
representation [14]. More precisely, we consider finite dimensional
representations with an arbitrary positive (half)-integer spin $s_{i}$ for
each site of the chain. In this case the weight vectors and the functions
$\lambda_{i}(u)$ can be constructed from the $\mathfrak{gl}_{2}$ Lie algebra’s
spin $s$ finite dimensional representation and the evaluation representation
of the Yangian $\mathfrak{gl}_{2}$.

The $\mathfrak{gl}_{2}$ Lie algebra is generated by
$\\{S^{z},S^{-},S^{+},I\\}$ that satisfy the commutation relations

$\displaystyle[S^{+},S^{-}]=2S^{z},\qquad[S^{z},S^{\pm}]=\pm
S^{\pm},\qquad[S^{i},I]=0.$ (2.4)

The spin $s$ finite dimensional representation of the $\mathfrak{gl}_{2}$ Lie
algebra is given by the highest (lowest) weight representation. Let us define
the highest weight vector of spin $s$ representation by $|s\rangle$. The
action of the $\mathfrak{gl}_{2}$ generators are given by

$\displaystyle S^{z}|s\rangle=s|s\rangle,\qquad
S^{+}|s\rangle=0,\qquad(S^{-})^{2s+1}|s\rangle=0,\qquad I|s\rangle=|s\rangle.$

The action of the $\mathfrak{gl}_{2}$ generators on the lowest weight vector
$|{-}s\rangle$ of spin $s$ representation is

$\displaystyle S^{z}|{-}s\rangle=-s|{-}s\rangle,\qquad
S^{-}|{-}s\rangle=0,\qquad(S^{+})^{2s+1}|{-}s\rangle=0,\qquad
I|{-}s\rangle=|{-}s\rangle.$

The lowest and the highest vector are related by the formula

$\displaystyle|{-}s\rangle=\frac{(S^{-})^{2s}}{\prod\limits_{j=1}^{2s}\sqrt{j(2s-(j-1))}}|s\rangle.$

To construct the representation of the Yangian of $\mathfrak{gl}_{2}$ we take
the highest (lowest) weight vector as a tensor product of $\mathfrak{gl}_{2}$
highest (lowest) weight vectors with different spins. They are given by

$\displaystyle|0\rangle=\bigotimes_{i=1}^{N}|s_{i}\rangle,\qquad|\hat{0}\rangle=\bigotimes_{i=1}^{N}|{-}s_{i}\rangle.$

The associated weight functions are

$\displaystyle\lambda_{1}(u)=\prod_{i=1}^{N}\frac{u-\theta_{i}+c\big{(}s_{i}+\frac{1}{2}\big{)}}{c},\qquad\lambda_{2}(u)=\prod_{i=1}^{N}\frac{u-\theta_{i}-c\big{(}s_{i}-\frac{1}{2}\big{)}}{c}.$

The evaluation representation is given by the Lax operator

$\displaystyle\mathcal{L}_{ai}(u)=\left(\begin{matrix}\dfrac{u}{c}+\dfrac{1}{2}+S_{i}^{3}&S_{i}^{-}\vspace{1mm}\\\
S_{i}^{+}&\dfrac{u}{c}+\dfrac{1}{2}-S_{i}^{3}\end{matrix}\right)_{a},$

in each site of the spin chain (see, e.g., [28]). Due to the commutation
relations (2.4), the Lax operator satisfies the RTT relation (2.1) and the
$\mathfrak{gl}_{2}$ invariance [28]

$\displaystyle[\mathcal{L}_{ai}(u),K_{a}{\mathcal{K}}_{i}]=0.$

Here

$\displaystyle{\mathcal{K}}_{i}=k_{0}\exp{\big{(}k^{+}S_{i}^{+}+k^{-}S_{i}^{-}+2k_{3}S_{i}^{3}\big{)}},$
$\displaystyle
K=k_{0}\exp{\big{(}k^{+}\sigma^{+}+k^{-}\sigma^{-}+k_{3}\sigma^{3}\big{)}}=\left(\begin{matrix}\tilde{\kappa}&\kappa^{+}\\\
\kappa^{-}&\kappa\end{matrix}\right),$

and $\sigma^{\pm}$, $\sigma^{z}$ are Pauli matrices.

The monodromy matrix for $N$ arbitrary positive (half)-integer spins
$\bar{s}=\\{s_{1},\dots,s_{N}\\}$ is then given by

$\displaystyle
T_{a}(u)=\mathcal{L}_{a1}(u-\theta_{1})\cdots\mathcal{L}_{aN}(u-\theta_{N}),$
(2.5)

where $\\{\theta_{1},\dots,\theta_{N}\\}$ are inhomogeneity parameters. The
monodromy matrix contains the $\mathfrak{gl}_{2}$ Lie algebra generators
$\\{S^{+},S^{-},S^{z},I\\}$ realized as
$S^{\alpha}=\sum\limits_{i=1}^{N}S^{\alpha}_{i}$,

$\displaystyle
T(u)=\left(\frac{u}{c}\right)^{N}+\left(\left(\begin{matrix}\dfrac{I+S^{z}}{2}&S^{-}\\\
S^{+}&\dfrac{I-S^{z}}{2}\end{matrix}\right)-\sum_{i=1}^{N}\theta_{i}\right)\left(\frac{u}{c}\right)^{N-1}+\cdots.$

Using the RTT relation we can extract the relation between the
$\mathfrak{gl}_{2}$ Lie algebra generators and the $t_{ij}(u)$, in particular
we have:

$\displaystyle[S^{z},t_{ii}(u)]=0,\qquad[S^{z},t_{12}(u)]=t_{12}(u),\qquad[S^{z},t_{21}(u)]=-t_{21}(u).$

We say that the transfer matrix has a ${\rm U}(1)$ symmetry if
$[S^{z},t(u)]=0$, that is, in the case of a diagonal twist $K$. In fact, if
$\mathop{\rm Det}(K)\neq 0$, then due to the global $\mathfrak{gl}_{2}$
invariance we can always restore the ${\rm U}(1)$ symmetry, by diagonalizing
the twist matrix and doing a proper similarity transformation [26], to perform
usual ABA444In the case of the XXZ models one should consider a more
sophisticated face-vertex transformation of the twist (see, e.g., [3, 30] and
references therein).. Here, as we want to study the MABA, we do not use this
possibility.

Starting form the highest or the lowest weight vector we can construct two
possible basis, that we call Bethe vectors, to span the Hilbert space of the
model $\bigotimes_{i=1}^{N}\mathbb{C}^{2s_{i}+1}$. Let $\bar{u}$ denote a set
of arbitrary complex parameters $\\{u_{1},\dots,u_{M}\\}$ with cardinality
$\\#\bar{u}=M$, and let $S=\sum\limits_{i=1}^{N}2s_{i}$. The representation
space of the Yangian can be spanned by

$\displaystyle\mathbb{B}^{M}(\bar{u})=\prod_{i=1}^{M}t_{12}(u_{i})|0\rangle$
(2.6)

with $\\#\bar{u}=M$ and $M=0,1,\dots,S$. Or equivalently, by

$\displaystyle\hat{\mathbb{B}}^{\hat{M}}(\bar{v})=\prod_{i=1}^{\hat{M}}t_{21}(v_{i})|\hat{0}\rangle$
(2.7)

with $\\#\bar{v}=\hat{M}$ and $\hat{M}=0,1,\dots,S$.

These two basis are related by a morphism of the Yangian. Let us consider the
mappings

$\displaystyle\psi_{1}(T(u))=T(-u),\qquad\psi_{2}(T(u))=T^{t}(u),$

where $t$ is the usual transposition in the auxiliary space,
$A_{ij}^{t}=A_{ji}$. They define anti-morphism of the RTT relation (see, e.g.,
[22]). Taking the composition of these mappings we obtain an automorphism of
the Yangian

$\displaystyle\phi(T(u))=\psi_{1}\circ\psi_{2}(T(u))=T^{t}(-u).$ (2.8)

This mapping relates the two different bases of the representation space of
the Yangian. Indeed we have

$\displaystyle\phi\big{(}\mathbb{B}^{M}(\bar{u})\big{)}=\hat{\mathbb{B}}^{M}(-\bar{u}),$

where we have to apply the rule $\phi(\lambda_{i}(u))=\lambda_{3-i}(-u)$ and
$\phi(|0\rangle)=|\hat{0}\rangle$ to preserve the structure of the action.
Thus, $\phi$ relates the highest weight and the lowest weight representations.
We remark also that $\phi(t(u))=t(-u)$, and therefore the action of the
transfer matrix on one of the basis defines the action on the other.

##  3 Algebraic Bethe ansatz and quantum Wronskian equation

Let us recall the algebraic Bethe ansatz [6, 15, 16] for generalized
$\mathfrak{gl}_{2}$ invariant integrable models with ${\rm U}(1)$ symmetry.
For this we introduce new notation. Let us define rational functions

$\displaystyle g(u,v)=\frac{c}{u-v},\qquad f(u,v)=1+g(u,v)=\frac{u-v+c}{u-v},$

and a shorthand notation for the products of the $f$ functions and the $g$
functions

$\displaystyle g(z,\bar{u})=\prod_{i=1}^{M}g(z,u_{i}),\qquad
f(\bar{u},z)=\prod_{i=1}^{M}f(u_{i},z),\qquad
f(u_{i},\bar{u}_{i})=\prod_{j=1,j\neq i}^{M}f(u_{i},u_{j}),$ (3.1)

where $\bar{u}_{i}=\\{\bar{u}\setminus u_{i}\\}$ is the set complementary to
the element $u_{i}$.

The ABA allows to solve the spectral problem for the transfer matrix

$\displaystyle t_{d}(z)=\mathop{\rm
Tr}_{a}\big{(}K_{a}^{(d)}T_{a}(z)\big{)}=\tilde{\kappa}t_{11}(z)+\kappa
t_{22}(z),$

with the diagonal twist matrix

$\displaystyle K^{(d)}=\left(\begin{matrix}\tilde{\kappa}&0\\\
0&\kappa\end{matrix}\right).$ (3.2)

In this case we have the ${\rm U}(1)$ symmetry regarding to the operator
$S^{z}$, i.e., $[S^{z},t_{d}(z)]=0$ and simple eigenstates are given by the
highest and lowest weight vectors $\\{|0\rangle,|\hat{0}\rangle\\}$

$\displaystyle
t_{d}(z)|0\rangle=(\tilde{\kappa}\lambda_{1}(z)+\kappa\lambda_{2}(z))|0\rangle,\qquad
t_{d}(z)|\hat{0}\rangle=(\kappa\lambda_{1}(z)+\tilde{\kappa}\lambda_{2}(z))|\hat{0}\rangle.$
(3.3)

We remark that for $K^{(d)}\propto I$, i.e., $\kappa=\tilde{\kappa}$, that
corresponds to the periodic boundary condition, both eigenvalues are the same
and the model has additional symmetries (full $\mathfrak{gl}_{2}$ invariance).
We should recall that the method applies for any highest weight representation
(2.2) without specifying the functions $\lambda_{i}(u)$. In particular it also
applies for an infinite dimensional representation as in the case of the Bose
gas model.

We have seen in the previous section that the space of states can be spanned
by Bethe vectors (2.6) for a given set of generic variables
$\bar{u}=\\{u_{1},\dots,u_{M}\\}$ of the cardinality $\\#\bar{u}=M$. Due to
the ${\rm U}(1)$ symmetry the spectrum is decomposed regarding to subspaces of
the total spin $S^{z}$. The Bethe vectors (2.6) form a basis for the total
spin

$\displaystyle
S^{z}\mathbb{B}^{M}(\bar{u})=\left(\frac{S}{2}-M\right)\mathbb{B}^{M}(\bar{u}).$

Then $M$ is a given quantum number and we can diagonalize the transfer matrix
for each invariant subspace labeled by $M$.

######  Theorem 3.1 ([15]).

The action of the transfer matrix onto the Bethe vector (2.6) is given by

$\displaystyle
t_{d}(z)\mathbb{B}^{M}(\bar{u})=\Lambda_{d}^{M}(z,\bar{u})\mathbb{B}^{M}(\bar{u})+\sum_{i=1}^{M}g(u_{i},z)E_{d}^{M}(u_{i},\bar{u}_{i})t_{12}(z)\mathbb{B}^{M-1}(\bar{u}_{i}),$

where

$\displaystyle\Lambda_{d}^{M}(z,\bar{u})=\tilde{\kappa}\lambda_{1}(z)f(\bar{u},z)+\kappa\lambda_{2}(z)f(z,\bar{u}),$
$\displaystyle
E_{d}^{M}(u_{i},\bar{u}_{i})=-\tilde{\kappa}\lambda_{1}(u_{i})f(\bar{u}_{i},u_{i})+\kappa\lambda_{2}(u_{i})f(u_{i},\bar{u}_{i}).$

If the set $\bar{u}$ satisfies the Bethe equations

$\displaystyle\frac{\tilde{\kappa}\lambda_{1}(u_{i})}{\kappa\lambda_{2}(u_{i})}=\frac{f(u_{i},\bar{u}_{i})}{f(\bar{u}_{i},u_{i})}=-\prod_{j=1}^{M}\frac{u_{i}-u_{j}+c}{u_{i}-u_{j}-c},$

for $i=0,1,\dots,M$, then the Bethe vectors (2.6) are eigenvectors of the
transfer matrix and are called on-shell Bethe vectors.

###### Proof.

Multiplication of the original monodromy matrix by the diagonal twist reduces
to the redefinition of the functions $\lambda_{i}(u)$. Since within the
framework of ABA these functions are not fixed, the proof then follows from
the standard considerations of the ABA [15]. It is based on the commutation
relations of the operators $t_{ij}(u)$ (see Appendix A) and the action on the
highest weight vector (2.2). ∎

Using the mapping $\phi$ (2.8) we can find the transfer matrix action on the
Bethe vectors constructed from the lowest weight vector. This gives another
parametrization for the spectrum of the transfer matrix.

######  Corollary 3.2.

The action of the transfer matrix onto the Bethe vector
$\hat{\mathbb{B}}^{\hat{M}}(\bar{v})$ is given by

$\displaystyle
t_{d}(z)\hat{\mathbb{B}}^{\hat{M}}(\bar{v})=\hat{\Lambda}_{d}^{M}(z,\bar{v})\hat{\mathbb{B}}^{\hat{M}}(\bar{v})+\sum_{i=1}^{\hat{M}}g(z,v_{i})\hat{E}_{d}^{\hat{M}}(v_{i},\bar{v}_{i})t_{21}(z)\hat{\mathbb{B}}^{\hat{M}-1}(\bar{v}_{i}),$

where

$\displaystyle\hat{\Lambda}_{d}^{\hat{M}}(z,\bar{v})=\kappa\lambda_{1}(z)f(\bar{v},z)+\tilde{\kappa}\lambda_{2}(z)f(z,\bar{v}),$
$\displaystyle\hat{E}_{d}^{\hat{M}}(v_{i},\bar{v}_{i})=-\kappa\lambda_{1}(v_{i})f(\bar{v}_{i},v_{i})+\tilde{\kappa}\lambda_{2}(v_{i})f(v_{i},\bar{v}_{i}).$

If the set $\bar{v}$ satisfies the second family of Bethe equations

$\displaystyle\frac{\kappa\lambda_{1}(v_{i})}{\tilde{\kappa}\lambda_{2}(v_{i})}=\frac{f(v_{i},\bar{v}_{i})}{f(\bar{v}_{i},v_{i})}=-\prod_{j=1}^{\hat{M}}\frac{v_{i}-v_{j}+c}{v_{i}-v_{j}-c}$

for $i=0,1,\dots,\hat{M}$, then the on-shell Bethe vectors (2.7) are
eigenvectors of the transfer matrix.

We can rewrite the two results of the previous section in terms of Baxter
Q-polynomials. Let us define the polynomials

$\displaystyle
q_{+}^{M}(z)=\prod_{i=1}^{M}\frac{z-u_{i}}{c}=\big{(}g(z,\bar{u})\big{)}^{-1}$

and

$\displaystyle
q_{-}^{\hat{M}}(z)=\prod_{i=1}^{\hat{M}}\frac{z-v_{i}}{c}=\big{(}g(z,\bar{v})\big{)}^{-1}.$

This allows us to rewrite the spectral problem in terms of Baxter T-Q
equations

$\displaystyle\Lambda_{d}(z)q_{+}^{M}(z)=\tilde{\kappa}\lambda_{1}(z)q_{+}^{M}(z-c)+\kappa\lambda_{2}(z)q_{+}^{M}(z+c),$
(3.4)

and

$\displaystyle\Lambda_{d}(z)q_{-}^{\hat{M}}(z)=\kappa\lambda_{1}(z)q_{-}^{\hat{M}}(z-c)+\tilde{\kappa}\lambda_{2}(z)q_{-}^{\hat{M}}(z+c).$
(3.5)

The two characterizations of the spectrum by Baxter T-Q equation satisfy a
quantum Wronskian condition.

######  Theorem 3.3.

The two Baxter Q-polynomials $q_{+}^{M}(z)$ and $q_{-}^{\hat{M}}(z)$ are
related by the quantum Wronskian equation [25]

$\displaystyle W_{d}(z)-W_{d}(z+c)=0,$ (3.6)

with

$\displaystyle
W_{d}(z)=\frac{\tilde{\kappa}q_{+}^{\hat{M}}(z-c)q_{-}^{M}(z)-\kappa
q_{+}^{\hat{M}}(z)q_{-}^{M}(z-c)}{\lambda(z)},$

and

$\displaystyle\frac{\lambda_{1}(z)}{\lambda_{2}(z)}=\frac{\lambda(z+c)}{\lambda(z)},\qquad\lambda(z)=\prod_{i=1}^{N}\prod_{k=1}^{2s_{i}}\frac{z-\theta_{i}+c\big{(}s_{i}-k+\frac{1}{2}\big{)}}{c}.$

This implies $\hat{M}+M=S=\sum\limits_{i=1}^{N}2s_{i}$ and
$W_{d}(z)=(\tilde{\kappa}-\kappa)$.

###### Proof.

For a given eigenvalue $\Lambda_{d}(z)$ of the transfer matrix, the quantum
Wronskian equation follows from the difference between (3.4) times
$q_{-}^{\hat{M}}(z)$ and (3.5) times $q_{+}^{M}(z)$, and from the fact that
$\frac{\lambda_{1}(z)}{\lambda_{2}(z)}=\frac{\lambda(z+c)}{\lambda(z)}$. Then,
due to (3.6) we conclude that $W_{d}(z)$ is a constant. Taking the limit
$z\to\infty$ we find that $\hat{M}+M=S$ and
$W_{d}(z)=(\tilde{\kappa}-\kappa)$. ∎

##  4 Modified algebraic Bethe ansatz
and quantum Wronskian equation

For a non-diagonal twist, the transfer matrix

$\displaystyle t(z)=\mathop{\rm
Tr}_{a}\big{(}K_{a}T_{a}(z)\big{)}=\tilde{\kappa}t_{11}(z)+\kappa
t_{22}(z)+\kappa^{+}t_{21}(z)+\kappa^{-}t_{12}(z)$ (4.1)

with

$\displaystyle K=\left(\begin{matrix}\tilde{\kappa}&\kappa^{+}\\\
\kappa^{-}&\kappa\end{matrix}\right),$

does not commute with the operator $S^{z}$. We have
$[S^{z},t(z)]=\kappa^{+}t_{21}(z)-\kappa^{-}t_{12}(z)$. As a consequence, the
highest and lowest weight vectors are not anymore eigenvectors of the transfer
matrix. The action of the twisted transfer matrix on the highest weight vector
(2.2), which is not anymore eigenstate unless $\kappa^{-}=0$, is given by

$\displaystyle
t(z)|0\rangle=(\tilde{\kappa}\lambda_{1}(z)+\kappa\lambda_{2}(z))|0\rangle+\kappa^{-}t_{12}(z)|0\rangle.$

Similarly the lowest weight vector (2.3) is not an eigenstate either (unless
$\kappa^{+}=0$), and the action of the twisted transfer matrix on it is

$\displaystyle
t(z)|\hat{0}\rangle=(\tilde{\kappa}\lambda_{2}(z)+\kappa\lambda_{1}(z))|\hat{0}\rangle+\kappa^{+}t_{21}(z)|\hat{0}\rangle.$

The MABA allows us to construct the modified Bethe vector keeping the highest
or lowest weight vectors as a starting point. The idea of the modified
algebraic Bethe ansatz [1, 2, 3, 4, 7] relies in the construction of modified
operators that preserve the operator algebra structure555For XXZ spin chain
the underling operator algebra structure is not preserved but mapped in a more
complex one (see [1, 2, 3] and references therein).. The transformation of the
monodromy matrix

$\displaystyle\bar{T}(z)=AT(z)B=\left(\begin{matrix}\nu_{11}(z)&\nu_{12}(z)\\\
\nu_{21}(z)&\nu_{22}(z)\end{matrix}\right),$

where $A$ and $B$ are two arbitrary two by two matrices, is an automorphism of
the Yangian of $\mathfrak{gl}_{2}$, i.e., new operators satisfy the same
commutation relations as the operators $t_{ij}(z)$ (see Appendix A). We remark
that the modified operators can be seen as a transfer matrix for a model with
a twist that has null determinant:

$\displaystyle\nu_{ij}(z)=\mathop{\rm Tr}_{a}\big{(}(V_{ji})_{a}T_{a}(z))$

with

$\displaystyle V_{ji}=BE_{ji}A,$

that satisfies $\mathop{\rm Det}(V_{ji})=0$. We call such operators null
twisted transfer matrices. The Yangian generators $t_{ij}(u)$, the modified
operators [4, 7], as well as ‘good’ operators [18] are such objects.

We look for a deformation of the action of the transfer matrix on the highest
weight vector in terms of the modified creation operator $\nu_{12}(u)$ of the
following form:

$\displaystyle
t(z)|0\rangle=\big{(}(\tilde{\kappa}-\rho_{1})\lambda_{1}(z)+(\kappa-\rho_{2})\lambda_{2}(z)\big{)}|0\rangle+\eta\nu_{12}(z)|0\rangle.$
(4.2)

Here $\rho_{i}$ and $\eta$ are some scalars that can be seen as deformation
parameters that go to zero when we restore the ${\rm U}(1)$ symmetry and
reduce (4.2) to (3.3). The matrix $V_{21}$ is uniquely determined in terms of
$\\{\kappa,\tilde{\kappa},\kappa^{-},\kappa^{+},\rho_{1},\rho_{2},\eta\\}$.

We do the same for the action on the lowest weight vector and look for the
modified creation operator $\nu_{21}(z)$ such that

$\displaystyle
t(z)|\hat{0}\rangle=\big{(}(\tilde{\kappa}-\rho_{1})\lambda_{2}(z)+(\kappa-\rho_{2})\lambda_{1}(z)\big{)}|\hat{0}\rangle+\hat{\eta}\nu_{21}(z)|\hat{0}\rangle.$

The matrix $V_{12}$ is uniquely determined in terms of
$\\{\kappa,\tilde{\kappa},\kappa^{-},\kappa^{+},\rho_{1},\rho_{2},\hat{\eta}\\}$.
We fix also the factorisation of the twist

$\displaystyle K=BDA,$ (4.3)

with

$\displaystyle D=\left(\begin{matrix}\tilde{\kappa}-\rho_{1}&0\\\
0&\kappa-\rho_{2}\end{matrix}\right).$

Thus, we have the diagonal modified transfer matrix

$\displaystyle t(z)=\mathop{\rm
Tr}(D\bar{T}(z))=(\tilde{\kappa}-\rho_{1})\nu_{11}(z)+(\kappa-\rho_{2})\nu_{22}(z).$

Then, it is easy to see that $A$ and $B$ are given by666We have some freedom
that follows from the transformation $A\to C^{-1}A$ and $B\to BC$ with
$[C,D]=0$ that leaves (4.3) invariant.

$\displaystyle
A=\sqrt{\mu}\left(\begin{matrix}1&\dfrac{\rho_{2}}{\kappa^{-}}\\\
\dfrac{\rho_{1}}{\kappa^{+}}&1\end{matrix}\right),\qquad
B=\sqrt{\mu}\left(\begin{matrix}1&\dfrac{\rho_{1}}{\kappa^{-}}\\\
\dfrac{\rho_{2}}{\kappa^{+}}&1\end{matrix}\right),\qquad\mu=\frac{1}{1-\frac{\rho_{1}\rho_{2}}{\kappa^{+}\kappa^{-}}},$

provided $\rho_{1}$ and $\rho_{2}$ enjoy the following relation:

$\displaystyle\kappa^{+}\kappa^{-}-(\rho_{1}\kappa+\rho_{2}\tilde{\kappa})+\rho_{1}\rho_{2}=0.$

We can now express the modified operators as linear combination of the
original operators $t_{ij}(z)$

$\displaystyle\nu_{11}(z)=\mu\left(t_{11}(z)+\frac{\rho_{2}}{\kappa^{+}}t_{12}(z)+\frac{\rho_{2}}{\kappa^{-}}t_{21}(z)+\frac{\rho_{2}^{2}}{\kappa^{-}\kappa^{+}}t_{22}(z)\right),$
(4.4)
$\displaystyle\nu_{22}(z)=\mu\left(t_{22}(z)+\frac{\rho_{1}}{\kappa^{+}}t_{12}(z)+\frac{\rho_{1}}{\kappa^{-}}t_{21}(z)+\frac{\rho_{1}^{2}}{\kappa^{-}\kappa^{+}}t_{11}(z)\right),$
(4.5)
$\displaystyle\nu_{12}(z)=\mu\left(t_{12}(z)+\frac{\rho_{1}}{\kappa^{-}}t_{11}(z)+\frac{\rho_{2}}{\kappa^{-}}t_{22}(z)+\frac{\rho_{1}\rho_{2}}{(\kappa^{-})^{2}}t_{21}(z)\right),$
(4.6)
$\displaystyle\nu_{21}(z)=\mu\left(t_{21}(z)+\frac{\rho_{1}}{\kappa^{+}}t_{11}(z)+\frac{\rho_{2}}{\kappa^{+}}t_{22}(z)+\frac{\rho_{1}\rho_{2}}{(\kappa^{+})^{2}}t_{12}(z)\right),$
(4.7)

with

$\displaystyle\mu=\frac{1}{1-\frac{\rho_{1}\rho_{2}}{\kappa^{-}\kappa^{+}}}.$

######  Remark 4.1.

In the case $\rho_{1}=\rho_{2}=\rho$ we recover the decomposition of the twist
matrix given in [7] with $A=B=L$.

######  Remark 4.2.

With the constraint $\kappa^{+}=\kappa^{-}=0$ we recover the diagonal twist
(3.2) and the action formulas (4.9)–(4.14) take the usual ABA form. From the
point of view of the decomposition of the twist matrix $K=BDA$ we can recover
the diagonal twist in two ways. On the one hand, taking $\rho_{i}=0$ and then
$\kappa^{+}=\kappa^{-}=0$ we send the modified operators $\nu_{ij}(z)$ to the
initial operators $t_{ij}(z)$. On the other hand, taking first
$\kappa^{+}=\kappa^{-}$, redefining $\rho_{i}=\kappa^{+}\bar{\rho}_{i}$ with
$\bar{\rho}_{i}\neq 0$ and then specifying $\kappa^{+}=0$ we keep the modified
operators $\nu_{ij}(z)$ provided
$\bar{\rho}_{1}\kappa+\bar{\rho}_{2}\tilde{\kappa}=0$. The $B^{\rm good}$
operator in $\mathfrak{gl}_{2}$ case proposed in [18] belongs to this special
case of the modified operators.

Using (4.4), (4.5) we can calculate the actions of the modified operators on
the highest weight and the lowest weight vectors.

######  Proposition 4.3.

If we consider $\nu_{12}(z)$ as a modified creation operator, then the actions
of the modified operators $\\{\nu_{11}(z),\nu_{22}(z),\nu_{21}(z)\\}$ on the
highest weight vector (2.2) are given by

$\displaystyle\nu_{11}(z)|0\rangle=\lambda_{1}(z)|0\rangle+\frac{\rho_{2}}{\kappa^{+}}\nu_{12}(z)|0\rangle,$
$\displaystyle\nu_{22}(z)|0\rangle=\lambda_{2}(z)|0\rangle+\frac{\rho_{1}}{\kappa^{+}}\nu_{12}(z)|0\rangle,$
(4.8)
$\displaystyle\nu_{21}(z)|0\rangle=\left(\frac{\rho_{1}}{\kappa^{+}}\lambda_{1}(z)+\frac{\rho_{2}}{\kappa^{+}}\lambda_{2}(z)\right)|0\rangle+\frac{\rho_{1}\rho_{2}}{(\kappa^{+})^{2}}\nu_{12}(z)|0\rangle.$

On the other hand, if we choose $\nu_{21}(z)$ as a modified creation operator,
then the actions of the modified operators
$\\{\nu_{11}(z),\nu_{22}(z),\nu_{12}(z)\\}$ on the lowest weight vector (2.3)
are given by

$\displaystyle\nu_{11}(z)|\hat{0}\rangle=\lambda_{2}(z)|\hat{0}\rangle+\frac{\rho_{2}}{\kappa^{-}}\nu_{21}(z)|\hat{0}\rangle,$
$\displaystyle\nu_{22}(z)|\hat{0}\rangle=\lambda_{1}(z)|\hat{0}\rangle+\frac{\rho_{1}}{\kappa^{-}}\nu_{21}(z)|\hat{0}\rangle,$
$\displaystyle\nu_{12}(z)|\hat{0}\rangle=\left(\frac{\rho_{2}}{\kappa^{-}}\lambda_{1}(z)+\frac{\rho_{1}}{\kappa^{-}}\lambda_{2}(z)\right)|\hat{0}\rangle+\frac{\rho_{1}\rho_{2}}{(\kappa^{-})^{2}}\nu_{21}(z)|\hat{0}\rangle.$

###### Proof.

The proposition follows from the explicit form of the modified operators in
terms of the original operators $t_{kl}(z)$ given by (4.4)–(4.7) and the
action on the weight vectors (2.2), (2.3). ∎

######  Remark 4.4.

For a given weight vector $|0\rangle$ or $|\hat{0}\rangle$ the role of
$\nu_{12}(z)$ and $\nu_{21}(z)$ can be inverted. Indeed, as example, from the
last equation of (4.8) we can express $\nu_{12}(z)|0\rangle$ in term of
$\nu_{21}(z)|0\rangle$ and $|0\rangle$ and then consider $\nu_{21}(z)$ as a
creation operator on $|0\rangle$.

It follows from the action of the modified operators that the actions of the
transfer matrix on the weight vectors are given by

$\displaystyle
t(z)|0\rangle=\big{(}(\tilde{\kappa}-\rho_{1})\lambda_{1}(z)+(\kappa-\rho_{2})\lambda_{2}(z)\big{)}|0\rangle+\frac{\kappa^{-}}{\mu}\nu_{12}(z)|0\rangle,$

and

$\displaystyle
t(z)|\hat{0}\rangle=\big{(}(\tilde{\kappa}-\rho_{1})\lambda_{2}(z)+(\kappa-\rho_{2})\lambda_{1}(z)\big{)}|\hat{0}\rangle+\frac{\kappa^{+}}{\mu}\nu_{21}(z)|\hat{0}\rangle.$

Let us extend the shorthand notation (3.1) to the product of commuting
operators, for example,

$\displaystyle\nu_{12}(\bar{u})=\prod_{i=1}^{M}\nu_{12}(u_{i}),\qquad\nu_{21}(\bar{v})=\prod_{i=1}^{\hat{M}}\nu_{21}(v_{i}).$

Then the action of $t(z)$ on the modified Bethe vector
$\nu_{12}(\bar{u})|0\rangle$ with $\\#\bar{u}=M$ has been derived in [7] and
is given by

$\displaystyle
t(z)\nu_{12}(\bar{u})|0\rangle=\frac{\kappa^{-}}{\mu}\nu_{12}(z)\nu_{12}(\bar{u})|0\rangle+\Lambda_{1}^{M}(z,\bar{u})\nu_{12}(\bar{u})|0\rangle$
$\displaystyle\hphantom{t(z)\nu_{12}(\bar{u})|0\rangle=}{}+\sum_{i=1}^{M}g(u_{i},z)E_{1}^{M}(u_{i},\bar{u}_{i})\nu_{12}(z)\nu_{12}(\bar{u}_{i})|0\rangle,$
(4.9)

where

$\displaystyle\Lambda_{1}^{M}(z,\bar{u})=(\tilde{\kappa}-\rho_{1})\lambda_{1}(z)f(\bar{u},z)+(\kappa-\rho_{2})\lambda_{2}(z)f(z,\bar{u}),$
(4.10) $\displaystyle
E_{1}^{M}(u_{i},\bar{u}_{i})=(\kappa-\rho_{2})\lambda_{2}(u_{i})f(u_{i},\bar{u}_{i})-(\tilde{\kappa}-\rho_{1})\lambda_{1}(u_{i})f(\bar{u}_{i},u_{i}).$
(4.11)

On the other hand, the action of $t(z)$ on the second family of Bethe vectors
given by $\nu_{21}(\bar{v})|\hat{0}\rangle$ with $\\#\bar{v}=\hat{M}$ reads

$\displaystyle
t(z)\nu_{21}(\bar{v})|\hat{0}\rangle=\frac{\kappa^{+}}{\mu}\nu_{21}(z)\nu_{21}(\bar{v})|\hat{0}\rangle+\hat{\Lambda}_{1}^{\hat{M}}(z,\bar{v})\nu_{12}(\bar{v})|\hat{0}\rangle$
$\displaystyle\hphantom{t(z)\nu_{21}(\bar{v})|\hat{0}\rangle=}{}+\sum_{i=1}^{\hat{M}}g(v_{i},z)\hat{E}_{1}^{\hat{M}}(v_{i},\bar{v}_{i})\nu_{21}(z)\nu_{21}(\bar{v}_{i})|\hat{0}\rangle,$
(4.12)

where

$\displaystyle\hat{\Lambda}_{1}^{\hat{M}}(z,\bar{v})=(\kappa-\rho_{2})\lambda_{1}(z)f(\bar{v},z)+(\tilde{\kappa}-\rho_{1})\lambda_{2}(z)f(z,\bar{v}),$
(4.13)
$\displaystyle\hat{E}_{1}^{\hat{M}}(v_{i},\bar{v}_{i})=(\tilde{\kappa}-\rho_{1})\lambda_{2}(v_{i})f(v_{i},\bar{v}_{i})-(\kappa-\rho_{2})\lambda_{1}(v_{i})f(\bar{v}_{i},v_{i}).$
(4.14)

For the finite dimensional representation given by the monodromy matrix (2.5),
the multiple product of a null transfer matrix $\nu(z)=\mathop{\rm Tr}(VT(z))$
with $\mathop{\rm Det}(V)=0$ (such as the modified creation and annihilation
operators) satisfies the following theorem.

######  Theorem 4.5.

Let $\bar{u}$ be a set of arbitrary parameters of cardinality
$\\#\bar{u}=\sum\limits_{i=1}^{N}2s_{i}=S$. For a finite dimensional
representation of the monodromy matrix given by (2.5), the following operator
identity holds

$\displaystyle\nu(z)\nu(\bar{u})=\mathop{\rm
Tr}(V)\left(F(z)g(z,\bar{u})\nu(\bar{u})+\sum_{i=1}^{S}g(u_{i},z)F(u_{i})g(u_{i},\bar{u}_{i})\nu(z)\nu(\bar{u}_{i})\right),$
(4.15)

where $\nu(\bar{u}_{i})=\prod\limits^{S}_{j=1,\,j\neq i}\nu(u_{j})$ and

$\displaystyle
F(z)=\prod_{i=1}^{N}\prod_{k=0}^{2s_{i}}\frac{z-\theta_{i}+c\big{(}s_{i}-k+\frac{1}{2}\big{)}}{c}.$
(4.16)

The l.h.s. of (4.15) involves the product of $S+1$ operators $\nu$ and each
term of the r.h.s. involves the product of $S$ operators $\nu$.

###### Proof.

It is proved in the Appendix B that for $\mathop{\rm Tr}(V)\neq 0$ and for
arbitrary inhomogeneity parameters
$\bar{\theta}=\\{\theta_{1},\dots,\theta_{N}\\}$, the operator $\nu(z)$ has
simple spectrum. Moreover, its inverse multiplied by the function $F(z)$ has
polynomial eigenvalues of degree $S=\sum\limits_{i=1}^{N}2s_{i}$ with the
leading term given by

$\displaystyle F(z)\nu^{-1}_{12}(z)=\mathop{\rm
Tr}(V)^{-1}\left(\frac{z}{c}\right)^{S}+\cdots,\qquad z\to\infty.$

Let $\\#\bar{u}=S+1$. Consider a product of operators

$\displaystyle F(z)\nu^{-1}_{12}(z)\nu(\bar{u})g(z,\bar{u}).$

It has simples poles at the points $u_{i}$ and behaves as $z^{-1}$ at
infinity. Then, taking the sum of all residues we find that

$\displaystyle\nu(\bar{u})=\mathop{\rm
Tr}(V)\sum_{i=1}^{S+1}F(u_{i})g(u_{i},\bar{u}_{i})\nu(\bar{u}_{i}).$

Setting $u_{S+1}=z$ we complete the proof. ∎

######  Corollary 4.6.

We can specify Theorem 4.5 by taking $V=BE_{21}A$ to find

$\displaystyle\frac{\kappa^{-}}{\mu}\nu_{12}(z)\nu_{12}(\bar{u})=(\rho_{1}+\rho_{2})\left(\\!F(z)g(z,\bar{u})\nu_{12}(\bar{u})+\sum_{i=1}^{S}g(u_{i},z)F(u_{i})g(u_{i},\bar{u}_{i})\nu_{12}(z)\nu_{12}(\bar{u}_{i})\\!\right),$

and by taking $V=BE_{12}A$ to find

$\displaystyle\frac{\kappa^{+}}{\mu}\nu_{21}(z)\nu_{21}(\bar{v})=(\rho_{1}+\rho_{2})\left(F(z)g(z,\bar{v})\nu_{21}(\bar{v})+\sum_{i=1}^{S}g(v_{i},z)F(v_{i})g(v_{i},\bar{v}_{i})\nu_{21}(z)\nu_{21}(\bar{v}_{i})\right).$

These examples show the algebraic origin of the inhomogeneous term of the
modified Baxter T-Q equation [9, 10, 29]. It was conjectured and then proved
in [1, 2, 3, 7, 11] for models on a segment. Here we go beyond and prove this
property independently of the action on the highest weight vector in the case
the twisted XXX spin chain with arbitrary positive (half)-integer spins.

######  Remark 4.7.

For the fundamental representation, $s_{i}=\frac{1}{2}$, we have
$F(z)=\lambda_{1}(z)\lambda_{2}(z)$. This proves the conjecture of [7].

######  Remark 4.8.

The entries of the monodromy matrix can be treated as null transfer matrices
$t_{ij}(z)=\mathop{\rm Tr}(E_{ji}T(z))$. Then we have from Theorem 4.5

$\displaystyle
t_{ii}(z)t_{ii}(\bar{u})=F(z)g(z,\bar{u})t_{ii}(z)t_{ii}(\bar{u})+\sum_{i=1}^{S}F(u_{i})g(u_{i},z)g(u_{i},\bar{u}_{i})t_{ii}(z)t_{ii}(\bar{u}_{i}),$
$\displaystyle t_{ij}(z)t_{ij}(\bar{u})=0,$

for $i\neq j$, $\\#\bar{u}=S$, and the function $F(z)$ given by (4.16).

######  Theorem 4.9.

For finite dimensional representation of the monodromy matrix given by (2.5),
the action of the transfer matrix on the Bethe vector
$\nu_{12}(\bar{u})|0\rangle$ with $\\#\bar{u}=\sum\limits_{i=1}^{N}2s_{i}=S$
is given by

$\displaystyle
t(z)\nu_{12}(\bar{u})|0\rangle=\Lambda(z,\bar{u})\nu_{12}(\bar{u})|0\rangle+\sum_{i=1}^{S}g(u_{i},z)E(u_{i},\bar{u}_{i})\nu_{12}(z)\nu_{12}(\bar{u}_{i})|0\rangle,$

where

$\displaystyle\Lambda(z,\bar{u})=(\tilde{\kappa}-\rho_{1})\lambda_{1}(z)f(\bar{u},z)+(\kappa-\rho_{2})\lambda_{2}(z)f(z,\bar{u})+(\rho_{1}+\rho_{2})F(z)g(z,\bar{u}),$
$\displaystyle
E(u_{i},\bar{u}_{i})=(\kappa-\rho_{2})\lambda_{2}(u_{i})f(u_{i},\bar{u}_{i})-(\tilde{\kappa}-\rho_{1})\lambda_{1}(u_{i})f(\bar{u}_{i},u_{i})+(\rho_{1}+\rho_{2})F(u_{i})g(u_{i},\bar{u}_{i}).$

The action of the transfer matrix on the Bethe vector
$\nu_{21}(\bar{v})|\hat{0}\rangle$ with
$\bar{v}=\sum\limits_{i=1}^{N}2s_{i}=S$ is given by

$\displaystyle
t(z)\nu_{21}(\bar{v})|\hat{0}\rangle=\hat{\Lambda}(z,\bar{v})\nu_{21}(\bar{v})|\hat{0}\rangle+\sum_{i=1}^{S}g(v_{i},z)\hat{E}(v_{i},\bar{v}_{i})\nu_{21}(z)\nu_{21}(\bar{v}_{i})|\hat{0}\rangle,$

where

$\displaystyle\hat{\Lambda}(z,\bar{v})=(\kappa-\rho_{2})\lambda_{1}(z)f(\bar{v},z)+(\tilde{\kappa}-\rho_{1})\lambda_{2}(z)f(z,\bar{v})+(\rho_{1}+\rho_{2})F(z)g(z,\bar{v}),$
$\displaystyle\hat{E}(v_{i},\bar{v}_{i})=(\tilde{\kappa}-\rho_{1})\lambda_{2}(v_{i})f(v_{i},\bar{v}_{i})-(\kappa-\rho_{2})\lambda_{1}(v_{i})f(\bar{v}_{i},v_{i})+(\rho_{1}+\rho_{2})F(v_{i})g(v_{i},\bar{v}_{i}).$

Thus, when the inhomogeneous Bethe equations are satisfied, i.e.,
$E(u_{i},\bar{u}_{i})=0$ and $\hat{E}(v_{i},\bar{v}_{i})=0$ for $i=1,\dots,S$,
the vectors $\nu_{12}(\bar{u})|0\rangle$ and
$\nu_{21}(\bar{v})|\hat{0}\rangle$ are eigenvectors of the transfer matrix.

###### Proof.

The theorem follows from the actions (4.9)–(4.12) and Corollary 4.6. ∎

Then we can rewrite the two eigenvalues by two inhomogeneous Baxter T-Q
functional equation. We define the functional Q-operators
$Q^{+}(z)=(g(z,\bar{u}))^{-1}$ and $Q^{-}(z)=(g(z,\bar{v}))^{-1}$, that are
two polynomials in $z$ of degree $S$, to find that

$\displaystyle\Lambda(z)Q^{+}(z)=(\tilde{\kappa}-\rho_{1})\lambda_{1}(z)Q^{+}(z-c)+(\kappa-\rho_{2})\lambda_{2}(z)Q^{+}(z+c)+(\rho_{1}+\rho_{2})F(z)$
(4.17)

and

$\displaystyle\Lambda(z)Q^{-}(z)=(\kappa-\rho_{2})\lambda_{1}(z)Q^{-}(z-c)+(\tilde{\kappa}-\rho_{1})\lambda_{2}(z)Q^{-}(z+c)+(\rho_{1}+\rho_{2})F(z).$
(4.18)

Then we can construct quantum Wronskian equations [25].

######  Theorem 4.10.

The two functional Baxter Q-operators $Q^{\pm}(z)$ are related by the modified
quantum Wronskian equation given as

$\displaystyle W(z)-W(z+c)=(\rho_{1}+\rho_{2})(Q^{+}(z)-Q^{-}(z)),$
$\displaystyle
W(z)=\frac{(\tilde{\kappa}-\rho_{1})Q^{+}(z-c)Q^{-}(z)-(\kappa-\rho_{2})Q^{+}(z)Q^{-}(z-c)}{\lambda(z)}.$

###### Proof.

For a given eigenvalue $\Lambda(z)$ of the transfer matrix, one should
multiply (4.17) and (4.18) respectively by $Q^{-}(z)$ and $Q^{+}(z)$, and
consider the difference of the resulting expressions. Then the use of
$\frac{\lambda_{1}(z)}{\lambda_{2}(z)}=\frac{\lambda(z+c)}{\lambda(z)}$ and
$\lambda_{1}(z)\lambda(z)=F(z)$ directly leads to the modified quantum
Wronskian equation. ∎

######  Remark 4.11.

For invertible twist, the transfer matrix (4.1) can also be characterized from
the usual Baxter T-Q equations [26]. Defining $\alpha^{\pm}$ to be eigenvalues
of the twist matrix $K$ that satisfy
$\alpha^{+}+\alpha^{-}=\tilde{\kappa}+\kappa$ and
$\alpha^{+}\alpha^{-}=\mathop{\rm Det}K$. Then we have

$\displaystyle\Lambda(z)q^{M}(z)=\alpha^{+}\lambda_{1}(z)q^{M}(z-c)+\alpha^{-}\lambda_{2}(z)q^{M}(z+c).$

We can construct Wronskian type equation between this parametrization and the
inhomogeneous one (4.17)

$\displaystyle
W(z)-W(z+c)=\left(\frac{\alpha^{-}}{\tilde{\kappa}-\rho_{1}}\right)^{z/c}(\rho_{1}+\rho_{2})q(z),$

with

$\displaystyle
W(z)=\left(\frac{\alpha^{-}}{\tilde{\kappa}-\rho_{1}}\right)^{z/c}\frac{\alpha^{+}q(z-c)Q^{+}(z)-(\tilde{\kappa}-\rho_{1})q(z)Q^{+}(z-c)}{\lambda(z)}.$

Here we used the identity
$\alpha^{+}\alpha^{-}=(\tilde{\kappa}-\rho_{1})(\kappa-\rho_{2})$.

##  5 Conclusion

In this paper we presented several proofs of the new steps needed to perform
the algebraic Bethe ansatz for the models without ${\rm U}(1)$ symmetry. We
showed that the appearance of the new term in the Baxter T-Q equation follows
from the analysis of the product of the modified creation operators and that
it is a general property of the null twisted transfer matrix. It is not
necessary to consider the action on a weight vector.

Then the Bethe ansatz characterization of the spectral problem of the XXX
Heisenberg spin chain with an arbitrary twist and arbitrary positive
(half)-integer spin at each site of the chain is fully understood by means of
the MABA. We also derived, for arbitrary positive (half)-integer spins, a
modified quantum Wronskian equation that relates two different
characterizations of the spectral problem. It should be of interest to relate
the modified quantum Wronskian equation with Hirota equation [17]. Moreover
finding the numerical solutions of the Bethe equations remains a challenging
open problem (see recent development in [19, 20, 21]), and modified quantum
Wronskian equation can be used to address it.

The new actions of the modified operators on the weight vectors deserve to be
studied in details. They also appear in the context of the separation of
variable by introduction of the $B^{\rm good}$ operators [18] (see also recent
work of two of the authors [8] that considers the $\mathfrak{gl}_{2}$ case
from the point of view of the ABA). These actions provide formulas for the
development of the modified Bethe vector in terms of the original $t_{12}(u)$
creation operator. The multiple action of the modified operators should lead
to generalization of the results of [6]. These multiple action formulas are
very useful for the calculation of Bethe vector’s scalar products, form
factors, and correlation functions and will be given in a forthcoming
publication. In the framework of the MABA for the twisted XXX models, these
multiple actions lead to new compact formulas for the scalar products of the
modified Bethe vectors and should open a path to prove the modified
determinant formulas conjectured in [7].

An important step should be to achieve the MABA for the twisted XXX
$\mathfrak{sl}_{3}$ spin chain and to other types of boundary conditions or
other higher rank algebras. For a lower block twist, there is no new
difficulties, as the first step of the nested Bethe ansatz preserves the ${\rm
U}(1)$ symmetry. However, for the upper block twist some new tools have to be
found.

##  Appendix A Commutation relations of the $\boldsymbol{t_{ij}(u)}$ and
multiple actions
on the Bethe vector

The RTT relation (2.1) yields the following commutation relations:

$\displaystyle t_{ij}(v)t_{ij}(u)=t_{ij}(u)t_{ij}(v),$ $\displaystyle
t_{ij}(v)t_{ik}(u)=f(u,v)t_{ik}(u)t_{ij}(v)+g(v,u)t_{ik}(v)t_{ij}(u),$
$\displaystyle
t_{ij}(v)t_{kj}(u)=f(v,u)t_{kj}(u)t_{ij}(v)+g(u,v)t_{kj}(v)t_{ij}(u).$

They imply the following actions on the products of $M$ operators:

$\displaystyle
t_{ij}(v)t_{ik}(\bar{u})=f(\bar{u},v)t_{ik}(\bar{u})t_{ij}(v)+\sum_{i=1}^{M}g(v,u_{i})f(\bar{u}_{i},u_{i})t_{ik}(v)t_{ik}(\bar{u}_{i})t_{ij}(u_{i}),$
$\displaystyle
t_{ij}(v)t_{kj}(\bar{u})=f(v,\bar{u})t_{kj}(\bar{u})t_{ij}(v)+\sum_{i=1}^{M}g({u_{i},v})f(u_{i},\bar{u}_{i})t_{{kj}}(v)t_{{kj}}(\bar{u}_{i})t_{{ij}}(u_{i}).$
(A.1)

The same commutation relations are valid for the modified operators
$\nu_{ij}(u)$.

##  Appendix B SoV basis for the modified creation operator
$\boldsymbol{\nu_{12}(z)}$

Let us construct the Separation of variables basis [24] of the modified
creation operator $\nu_{12}(z)$. We assume that the inhomogeneity parameters
$\bar{\theta}$ are generic complex numbers and that the product $AB$ has non
zero entries.

Let us introduce a vector

$\displaystyle|y\rangle={\mathcal{A}}^{-1}_{1}{\mathcal{A}}^{-1}_{2}\cdots{\mathcal{A}}^{-1}_{N}|\hat{0}\rangle,$

where

$\displaystyle{\mathcal{A}}_{i}=a_{0}\exp{\big{(}a^{+}S_{i}^{+}+a^{-}S_{i}^{-}+2a_{3}S_{i}^{3}\big{)}},$

and the parameters $\\{a_{0},a^{+},a^{-},a_{3}\\}$ are fixed by the equality

$\displaystyle
A=\sqrt{\mu}\left(\begin{matrix}1&\dfrac{\rho_{2}}{\kappa^{-}}\\\
\dfrac{\rho_{1}}{\kappa^{+}}&1\end{matrix}\right)=a_{0}\exp{\big{(}a^{+}\sigma^{+}+a^{-}\sigma^{-}+a_{3}\sigma^{3}\big{)}}.$

Using $\mathfrak{gl}_{2}$ invariance we can show that

$\displaystyle\nu_{12}(z)|y\rangle=\frac{\mu}{\kappa^{-}}(\rho_{1}+\rho_{2})\lambda_{2}(z)|y\rangle.$

Indeed, we have

$\displaystyle\nu_{12}(z)|y\rangle=\mathop{\rm
Tr}(E_{21}AT(z)B){\mathcal{A}}^{-1}_{1}{\mathcal{A}}^{-1}_{2}\cdots{\mathcal{A}}^{-1}_{N}|\hat{0}\rangle={\mathcal{A}}^{-1}_{1}{\mathcal{A}}^{-1}_{2}\cdots{\mathcal{A}}^{-1}_{N}\mathop{\rm
Tr}(E_{21}T(z)AB)|\hat{0}\rangle$
$\displaystyle\hphantom{\nu_{12}(z)|y\rangle}{}=(AB)_{12}\lambda_{2}(z)|y\rangle.$

Consider a set of vectors

$\displaystyle|Y(\bar{k})\rangle=\prod_{i=1}^{N}\prod_{j=1}^{k_{i}}\nu_{22}\left(\theta_{i}+c\left(s_{i}+\frac{1}{2}-j\right)\right)|y\rangle,$

where $\bar{k}$ is a set of $N$ integers such that
$k_{i}\in\\{0,1,\dots,2s_{i}\\}$ for $i=1,\dots,N$. Then it follows from the
commutation relation (A.1) that

$\displaystyle\nu_{12}(z)\prod_{j=1}^{n}\nu_{22}(u-cj)=\left(\prod_{j=1}^{n}f(z,u-cj)\right)\left(\prod_{j=1}^{n}\nu_{22}(u-cj)\right)\nu_{12}(z)$
$\displaystyle\qquad{}+g(u-c,z)\left(\prod_{j=2}^{n}f(u-c,u-cj)\right)\nu_{22}(z)\left(\prod_{j=2}^{n}\nu_{22}(u-cj)\right)\nu_{12}(u-c).$

Here we have used $f(x-c,x)=0$ and
$\lambda_{2}\big{(}\theta_{i}+c\big{(}s_{i}-\frac{1}{2}\big{)}\big{)}=0$.
Hence, the action of $\nu_{12}(z)$ on $|Y(\bar{k})\rangle$ reads

$\displaystyle\nu_{12}(z)|Y(\bar{k})\rangle=\frac{\mu}{\kappa^{-}}(\rho_{1}+\rho_{2})\frac{F(z)}{\Lambda_{12}(z,\bar{k})}|Y(\bar{k})\rangle,$

where $F(z)$ given by (4.16) and

$\displaystyle\Lambda_{12}\big{(}z,\bar{k}\big{)}=\prod_{i=1}^{N}\left(\prod_{j=1}^{k_{i}}h(z-c,\theta_{i}+c\left(s_{i}+\frac{1}{2}-j\right)\right)\left(\prod_{j=k_{i}+1}^{2s_{i}}h(z,\theta_{i}+c\left(s_{i}+\frac{1}{2}-j\right)\right),$

with $h(u,v)=f(u,v)/g(u,v)$.

######  Remark B.1.

Using $\mathfrak{gl}_{2}$ invariance we can show that

$\displaystyle\big{\langle}\hat{0}|{\mathcal{A}}^{-1}_{1}{\mathcal{A}}^{-1}_{2}\cdots{\mathcal{A}}^{-1}_{N}|Y(\bar{k})\big{\rangle}=\prod_{i=1}^{N}\prod_{j=1}^{k_{i}}(AB)_{22}\lambda_{1}\left(\theta_{i}+c\left(s_{i}+\frac{1}{2}-j\right)\right)\neq
0.$

Thus, all the vectors under consideration are non zero.

Thus, we have constructed $\prod\limits_{i=1}^{N}(2s_{i}+1)$ different
eigenvectors of the modified creation operator with different eigenvalues. As
the vectors are independent and the representation is finite (being of
dimension $\prod\limits_{i=1}^{N}(2s_{i}+1)$), this implies that the modified
creation operator has simple spectrum and it is invertible. Thus, we have

$\displaystyle
F(z)(\nu_{12}(z))^{-1}|Y(\bar{k})\rangle=\frac{\kappa^{-}}{\mu(\rho_{1}+\rho_{2})}\Lambda_{12}(z,\bar{k})|Y(\bar{k})\rangle.$

Therefore, $F(z)(\nu_{12}(z))^{-1}$ is a polynomial in $z$ of degree $z^{S}$.

Observe that the modified creation operator

$\displaystyle\nu_{12}(z)=\mathop{\rm Tr}(V_{12}T(z)),\qquad
V_{12}=\mu\begin{pmatrix}\dfrac{\rho_{1}}{\kappa^{-}}&\dfrac{\rho_{1}\rho_{2}}{(\kappa^{-})^{2}}\vspace{1mm}\\\
1&\dfrac{\rho_{2}}{\kappa^{-}}\end{pmatrix}$

is a null twisted transfer matrix, because $\mathop{\rm Det}(V_{12})=0$. Since
we can consider $\rho_{i}$, $\mu$, and $\kappa^{-}$ as free parameters in our
construction, the result applies to any null twisted transfer matrix.

### Acknowledgements

S.B. thanks R. Pimenta, N. Crampé, P. Baseilhac, V. Pasquier, D. Serban, A.
Faribault for discussions. N.S. would like to thank the hospitality of the
Institute de Physique Théorique at the CEA de Saclay where a part of this work
was done. S.B. and B.V. would like to thank the hospitality of the LMPT of
Tours where a part of this work was done. S.B. was supported by a public grant
as part of the Investissement d’avenir project, reference
ANR-11-LABX-0056-LMH, LabEx LMH. N.S. was supported by the Russian Foundation
RFBR-18-01-00273a.

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