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\begin{document}
\titlepage
\title{ All exactly solvable quantum spin 1 chains from Hecke algebra }
\author{ Francisco C. Alcaraz\thanks{This research was supported in
part by Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado
de S\~ao Paulo (~FAPESP-BRASIL), Conselho Nacional de
Desenvolvimento Cient\'ifico e Tecnol\'ogico(~CNPq-BRASIL~)
and by the
National Science Foundation - USA - Grant~\# PHY89-04035.} \\
Institute for Theoretical Physics \\
University of California at Santa Barbara \\
Santa Barbara, CA 93106, USA
\and Roland K\"{o}berle
\thanks{ Supported in part by CNPq-BRASIL.} \\
Instituto de F\'{\i}sica e Qu\'{\i}mica de S\~ao Carlos \\
Universidade de S\~ao Paulo\\
Caixa Postal 676, S\~ao Carlos 13560\\ BRASIL
\and A. Lima - Santos~$\;\textstyle{^\dagger}$\\
Instituto de F\'isica da Universidade de S\~ao Paulo\\
Caixa Postal 20516, S\~ao Paulo \\BRASIL
}
\maketitle
\begin{abstract}
We obtain all exactly integrable spin $1$ quantum chains, which satisfy
the Hecke algebra and discuss their solution via Bethe Ansatz methods.
We also present various generalizations for arbitrary spin $S$.
\end{abstract}
%
\newpage
\sect{Introduction}{s1}
The study of exactly soluble models in two dimensions has been a profound
inspiration and testing ground for many new ideas in several areas of physics.
Eventually the straightjacket of two dimensions has to be broken through.
In order to achieve this aim, the essential aspects and techniques used
in solving these models, should be brought to light as clearly and
economically as possible.
A royal road to obtain new exactly soluble models consists in solving the
Yang-Baxter \cite{YB} factorization equations.
The existence of an infinite number of conservation laws being now guaranteed,
one proceeds then aiming at an exact solution. We discuss some aspects of
this endeavour, such as obtaining at least part of the spectrum through
Bethe Ansatz (BA) methods.
% s2: Hecke algebra & bxterization
% s3: U(1)xU(1) solutions
% s4: U(1) solutions
% s5: Betha Ansatz
The outline of this paper is as follows.
In section \ref{s2}, we present for the readers benefit, well known
algebraic structures, allowing one to set up Yang-Baxter~(YB) equations.
In sections \ref{s3} and \ref{s4} we discuss the solutions of these equations wi
th $U(1) \otimes U(1)$ and $U(1)$ symmetry respectively.
In section \ref{s5} we exhibit the BA for the models with $U(1) \times U(1)$ sym
metry.
%
%******************************************************************************
%
\sect{ Hecke algebra and its baxterization}{s2}
In this section we briefly discuss, how to obtain exactly
soluble models satisfying YB equations from
hamiltonians obeying certain algebraic constraints,
generating in this way a subset of soluble
spin 1 models.These will be spin 1 generalizations of the spin $1/2$ six
vertex models.
First we require nearest neighbor interactions
and rotational $U(1)$ invariance around the $z$~-axis.
Secondly, our hamiltonians are required to satisfy the
{\em Hecke}\,-algebra. This second requirement guarantees the
availability of YB equations, which in turn provide an
infinite number of conservation laws.
This is not the most general algebraic setting providing exact
solubility, but we restrict ourselves to this case for simplicity
and leave more general situations for a future publication.
The most general nearest neighbor $U(1)$-invariant hamiltonian
can be parametrized as follows :
\beq
g_{i+1/2} = A_1 + A_2 S_z(i) + A_3 S_z(i+1) + A_4 S_z^2(i) +
A_5 S_z^2(i+1)+
\label{eq:par1}
\eeq
\bdm
A_6 S_z(i)S_z(i+1) + A_7 S_z^2(i)S_z(i+1)
+ A_8 S_z(i)S_z^2(i+1) + A_9 S_z^2(i)S_z^2(i+1) +
\edm
\bdm
A_{10} S_-(i)S_+(i+1) + A_{11} S_+(i)S_-(i+1) +
A_{12} S_-^2(i)S_+^2(i+1) + A_{13} S_+^2(i)S_-^2(i+1) +
\edm
\bdm A_{14} S_+(i)S_z(i+1)S_-(i+1) +
A_{15} S_z(i)S_+(i)S_-(i+1) + A_{16} S_-(i)S_z(i+1)S_+(i+1) +
\edm
\bdm
A_{17} S_-(i)S_z(i+1)S_+(i+1) + A_{18} S_z(i)S_-(i)S_+(i+1) +
+ A_{19} S_z(i)S_+(i)S_z(i+1)S_-(i+1),
\edm
where the index on $g$ reminds us, that $g_{i+1/2}$ operates in the
three-dimensional complex vector spaces $\bf C_{i}$, resp. $\bf C_{i+1}$ at posi
tions $i$ and $i+1$.
We use the usual spin 1 matrices
\bdm
S_z = [1,1] - [3,3],S_{-} = [2,1]+[3,2],
S_{+}= [1,2]+[2,3],
\edm
where the symbol $[a,b]$ means the matrix, whose $i,j$ element is :
\beq
\left( [a,b] \right)_{i,j} \equiv \delta _{a,i} \delta _{b,j}.
\label{eq:times}
\eeq
If we take the direct product of these matrices :
\beq
[ab|cd] \equiv [a,b] \otimes [c,d],
\eeq
we may
conveniently express the $9\times9$ matrix $g$ as :
\beq
g = \sum_{a,b,c,d=1}^{3} c_{ab|cd} [ab|cd] ,
\label{eq:par2}
\eeq
where the coefficients $c_{ab|cd}$ are linearly related to the $A_i$ and due
to the $U(1)$-symmetry, only $19$ of them don't vanish.
We now require the matrices $g$ to satisfy the following
Hecke algebra
\bdm
g_{i+1/2}\, g_{i+3/2}\, g_{i+1/2 }
= g_{i+3/2}\, g_{i+1/2}\, g_{i+3/2},
\edm
\beq
g_{i+1/2}^2 + I_9 = x\, g_{1+1/2},
\label{eq:hecke}
\eeq
\bdm
[ g_{i+1/2}, g_{j+1/2}] = 0, \hspace{2em} for \;\; |i-j| \geq 2
\edm
where $I_9
$ is the 9-dimensional unit matrix and $x$ is a
{\em deformation } \, parameter of the algebra, allowing for the presence of a
coupling constant occurring in the hamiltonian. We will also use
the parametrization $ x=q/\imath + \imath/q,\, q=\exp (\imath\gamma)$,
where $\gamma$ is in general a complex number.
It is also convenient to express the operators $g_{i+1/2}$ in terms
of {\em Temperley-Lieb}\, operators~:
\beq
e_{i+1/2}=q- \imath g_{i+1/2},
\eeq
satisfying
\bdm
e_{i+1/2}^2 = \beta e_{i+1/2},
\edm
\beq
e_{i+1/2}\; e_{i+3/2}\; e_{i+1/2 } -e_{i+1/2}
= e_{i+3/2}\; e_{i+1/2}\; e_{i+3/2} -e_{i+3/2},
\label{tl}
\eeq
\bdm
[e_{i+1/2},e_{j+1/2}] = 0, \hspace{3em} for \; |i-j|\geq2,
\edm
where $ \beta = q + 1/q $.
When both sides of equ.(\ref{tl}) vanish, the Hecke algebra degenerates into the
Temperley-Lieb algebra.
We will take our hamiltonian for an $N$-site chain to be :
\beq
H(\beta) = - \sum_{j=1}^{N} e_{j+1/2},
\label{eq:ham}
\eeq
which obviously commutes with $\sum_{i=1}^N S_z(i) $.
This may differ from other authors conventions
by at most a linear transformation. It is interesting to point out here,
that the transformation $ \beta \rightarrow -\beta $ implements the
change from {\em ferromagnetic } to {\em anti-ferromagnetic } behavior.
Once with a solution at hand for $g$, i.e. the parameters $A_a,a=1,\ldots,19$
have been adjusted so that $g$ satisfies the Hecke algebra, we have at
our disposal a standard way to {\em baxterize }\, $g$. This means the
introduction of a spectral parameter $u$, which is necessary to write
down the YB equations.
When $\gamma$ is real, the solutions to the YB equations will be written in t
he {\em trigonometric }\, regime $( \beta < 2 )$ and
when $\gamma$ is pure imaginary $ ( \beta > 2 ) $ , they are in the {\em
hyperbolic}\, regime. Based on all examples we know of, we do expect all
models in the trigonometric regime $( \beta \leq 2 )$ to be critical
( massless ), while for $ \beta > 2 $ they will be massive.
The baxterization recipe is :
define a $9\times 9$ matrix $X(u)$ by
\beq
X_{i+1/2}(u)=\sin ( \gamma + u) - \sin(u) e_{i+1/2}.
\eeq
Due to the Hecke algebra $X(u)$ satisfies
\beq
X_{i+1/2}(u)\, X_{i+3/2}(u+v)\, X_{i+1/2} (v)
=
X_{i+3/2}(v)\, X_{i+1/2}(u+v)\, X_{i+3/2} (u),
\eeq
\bdm
[X_{i+1/2}(u),X_{j+1/2}(v)]=0,\hspace{3em}for\;\;|i-j|\geq 2.
\edm
These equations are equivalent to the YB equations for $S$-matrices
( or Boltzmann weights ${\cal L}$ ), if we define them as
\beq
X_{i+1/2}(u) = \sum_{a,b,c,d=1}^{3} S_{ab}^{cd}(u) I^{(1)} \otimes
I^{(2)} \otimes \ldots \otimes
[b,c]^{(i)} \otimes [a,d]^{(i+1)} \otimes \ldots .
\eeq
Here $S_{ab}^{cd}(u)$ describes the scattering of particles
$ a\, +\, b \, \rightarrow \, c\, +\, d $
with {\em colors } $a,b,c,d$ and relative rapidity $\imath u$.
( Row-to-row ) Monodromy and transfer matrices
are built using the $S$-matrices as Boltzmann weights.
% figure 1.
Define the local transition matrix
$\cal L$, acting on {\em horizontal } indices $a$ and $c$ and
{\em vertical} indices $b$ and $d$ as :
\beq
{\cal L}_{ac|bd } (u)=
S_{cd}^{ab}(u).
\eeq
As usual, we will supress the vertical indices and regard the matrix
elements of $\cal L$ ( in horizontal space ) to be operators in the vertical
space. The monodromy matrix $T$ is :
\beq
T_{a,\{b\} }^{c,\{d\} }(u)=
\sum_{a_i} {\cal L}_{aa_1}(u) \otimes
{\cal L}_{a_1a_2}(u)\otimes\ldots
\otimes {\cal L}_{a_{N-1}c}(u).
\eeq
The transfer matrix $\tau (u)$ is the ( horizontal ) trace
of the monodromy matrix : $\tau_{\{b\}}^{\{d\}} (u) =
T_{a,\{b\} }^{a,\{d\} }(u)$ .
%
%**************************************************************************
%
\sect{$U(1)\bigotimes U(1) $-symmetric solutions}{s3}
The solutions of equs.(\ref{eq:hecke}) may be classified
according to their symmetry.
In this section, we discuss solutions with an extra $U(1)$ symmetry,
requiring the hamiltonian to commute also with
$ \sum_{i=1}^N S_z^2(i) $.
In the case of spin 1, we may associate three colors with the spin
states and the extra
$U(1)$-symmetry refers to color conservation. We thus look for
$S$-matrices, which do not allow processes like
$ 1+2 \rightarrow 1+3 $ etc. Our $S$-matrices are therefore of the
form
\beq
% S_{ab}^{cd}(u) =
% \left\{ \begin{array}{l}
% S_{ab}^{ab}(u) \\
% S_{ab}^{ba}(u)
% \end{array}
% \right. .
S_{ab}^{cd}(u) = \left\{ S_{aa}^{aa}(u),\;S_{ab}^{ab}(u),
\;S_{ab}^{ba}(u) \right\}
\eeq
One immediately sees, that this requires
\beq
c_{12|32}=c_{32|12}=c_{21|23}=c_{23|21}=0 .
\label{eq:u1}
\eeq
To simplify the classification of all possible solutions,
we will first scale our matrices $g$ or $e$ to a standard form.
For this, we note the $9\times 9$ matrix $g$ is a direct product of
$3\times 3$ matrices
and can thus be written in terms of the matrices of equ.(\ref{eq:times}) :
$[a,b] \otimes [c,d]$. Since these satisfy
\beq
[a,b]*[c,d]=\delta_{b,c} [a,d],
\eeq
we may rescale them as
\beq
\overline{[a,b]} = \rho_{a,b}\; [a,b]
\eeq
and provided
\beq
\rho_{a,a}=1,\;\rho_{b,a}=1/ \rho_{a,b},\;\;
\rho_{a,b}\; \rho_{b,c} = \rho_{a,c}.
\eeq
The matrices $\overline{[a,b]}$ satisfy the same algebra as $[a,b]$.
In the present case, when equ.(\ref{eq:u1}) is satisfied,
we are free to rescale
such that $|c_{12|21}|=|c_{21|12}|=|c_{13|31}|=|c_{31|13}|=
|c_{23|32}|=|c_{32|23}|=1$ .
In order to satisfy Hecke, we put
\beq
c_{12|21}=c_{21|12}=c_{13|31}=c_{31|13}=
c_{23|32}=c_{32|23}=\imath.
\eeq
Moreover, if $ c_{12|32} = c_{32|12} = 0$ ( resp. $ c_{21|23}= c_{23|21}=0$ ),
the spectrum of $g$ from equ.(\ref{eq:par2})
will not depend upon the
values of $c_{21|23} $ and $ c_{23|21} $ ( resp. $ c_{12|32} and c_{32|12}$ ).
In this way we generate models that, although aparently violating the
$ U(1) \otimes U(1) $-symmetry, do nevertheless share the same spectrum
with the corresponding $ U(1)\otimes U(1) $-symmetric models with
$c_{12|23}=c_{23|21}=0 ( resp. c_{12|32}=c_{23|21}=0 )$
\footnote{This feature of the spectrum was checked numerically.}.
After this rescaling, we have basically two types
of solutions, from which all the others may be obtained by simple
manipulations ( like changing the color indices etc. ).
We now state $e$-matrices for these two cases. The first family depends
on three free parameters $\nu_a;a=1,2,3$, which may assume the values
$0$ or $1$ :
\beq
e^{(1)}(\beta;\nu_1,\nu_2,\nu_3) = \sum_a \beta \nu_a [aa|aa]
+ \sum_{a**1$. In this case we have $n=2s+1$ colors and
$[a,b];a,b=1,\ldots,n$ are $n$-dimensional matrices. The generalization
of $e^{(1)}$ is simply given by equ.(\ref{eq:f1}) and in the periodic
case is related to the $n$-color Perk-Schultz model by an expression
similar to equ.(\ref{eq:rela}).
For $S>1 ( n\geq 4 )$ there are several generalizations of
equ.(\ref{eq:fam2}), since now we can decouple more than two colors.
Decoupling always in pairs, whose number is denoted by $k$,we have for
the $e^{(2)}$ operator:
\beq
e^{(2)}(\beta;\nu_1,\ldots,\nu_{n-k}) =
\sum_{a,b=1;a\neq b}^n [ab|ba] +
\sum_{a < b}^n \left( [aa|bb]\,q + [bb|aa]/q \right ) -
\eeq
\bdm
\sum_{a=1}^k \left( [2a-1,2a|2a,2a-1] + [2a-1,2a-1|2a,2a]\,q +
[2a,2a|2a-1,2a-1]/q \right) +
\edm
\bdm
\nu_1 \sum_{a=1}^k \left( [2a-1,2a-1] + [2a,2a] \right) \otimes
\left( [2a-1,2a-1] + [2a,2a] \right) +
\sum_{a=2k+1}^n \nu_{a-2k+1}[aa|aa]
\edm
%
% ***************************************************************************
%
\sect{Solutions with $U(1)$-symmetry }{s4}
In this case there is only one solution
\footnote{Solving the Hecke algebra for this case, one finds a solution
depending on two free parameters and four signs. After rescaling,
all have the same spectrum, up to at most a linear transformation.},
given by:
\beq
e^{(3)}(\beta) =[11|33]\,\hat{q}^2 + [33|11]/\hat{q}^2 + [22|22] +
\label{TL}
\eeq
\bdm
\left( [12|32]+[21|23]\right) \hat{q} +
\left( [23|21]+[32|12]\right) /\hat{q},
\edm
where now $\beta =\hat{q}^2+1/\hat{q}^2+1$. This model does in fact obey
the Temperley-Lieb algebra and its hamiltonian commutes with the quantum group
$U_q sl(2) $. This solution has been presented in \cite{BB} and \cite{BMNR}.
The $g$-matrix of this model coincides with the $R(u)$-matrix of the
Izergin-Korepin model \cite{IK}, at a particular value of their
spectral parameter $u$. The models are different though, since the
Izergin-Korepin models is a $19$-vertex model, whereas equ.(\ref{TL}) involves
only $15$ vertices.
In terms of spin variables our model describes the anisotropic version of
the bi-quadratic spin-1 model \cite{BB} and is given by:
\beq
e(\beta) = \left( \vec{S}(i)\cdot \vec{S}(i+1)\right)^2 - 1 +
\frac{\beta-3}{4}\left( S_z^2(i) S_z^2(i+1) -
S_z(i) S_z(i+1)\right) +
\eeq
\bdm
\frac{\hat{q}^2-\hat{q}^{-2} }{4}
\left ( S_z(i)S_z^2(i+1) - S_z^2(i)S_z(i+1) \right) +
\edm
\bdm
(\hat{q}-1)\left(S_z(i)S_+(i)S_z(i+1)S_-(i+1) +
S_-(i)S_z(i)S_+(i+1)S_z(i+1)
\right )+
\edm
\bdm
(\hat{q}^{-1}-1)\left(S_+(i)S_z(i)S_-(i+1)S_z(i+1) +
S_z(i)S_-(i)S_z(i+1)S_+(i+1) \right) ,
\edm
where $ \vec{S}(i)\cdot \vec{S}(i+1) $ denotes the standard
rotationally invariant scalar product.
Similarly to the $U(1)\otimes U(1) $ case, we also find here higher
spin generalizations.
In fact, for $S>1$ we present two families of solutions
$ e^{(m)}(S,\beta),m=1,2$. They all satisfy
the Temperley-Lieb algebra and reduce to $e^{(3)}$, when $ S\rightarrow 1$
and are given by:
\beq
e^{(m)}(S,\beta) = \sum_{k=0}^{2S} \sum_{l=0}^{2S} D_{k,l}^{(m)} \,
[1+k,1+l|2S+1-k,2S+1-l],
\eeq
where the coefficients $D^{(m)}$ for the two families are given by:
\beq
D_{k,l}^{(1)} = - (-\hat{q})^{2S-l-k},\hspace{2em} l,k=0,1,\ldots,2S
\eeq
with $\beta$ being
\bdm
\beta = -\sum_{l=0}^{2S} (\hat{q})^{2S-l}.
\edm
and
\beq
D_{k,l}^{(2)} =
\hat{q}^{(\delta_{k,0}+\delta_{l,0}-\delta_{k,2S}-\delta_{l,2S})/2}
\eeq
and
\bdm
\beta = \hat{q} +1/\hat{q} + 2S-1.
\edm
The first family coincides with the one reported earlier in ref.\cite{DWA},
but the second one, which correponds to a different deformation, was not
reported previously, at least to our knowledge.
%
% ****************************************************************************
%
\sect{ Bethe-Ansatz }{s5}
All the models presented above can be {\em solved } by the BA -~or
some variant thereof~-~,
although for lack of space we will present here only the cases with
$U(1) \otimes U(1)$-symmetry, leaving the rest for a future
publication.
We will apply the BA in it's algebraic version pioneered by
Takhtadjian and Fadeev \cite{TF}. In order to find the eigenvalues and eigenvect
ors of the transfer matrix $\tau(u)$,
one proceeds in two steps.
First find a {\em reference } state,
which is a convenient eigenstate $|\Omega \!>$ of $\tau(u)$, such that
the monodromy matrix $L_{a,b}(u)$ applied to
$|\Omega\!>$ takes a simple form.
That is, applying $L_{a,b}(u)$ to $|\Omega\!>$, we obtain either zero or
a state proportional to $|\Omega\!>$ or a new vector. The matrix elements of $L_
{a,b}(u)$ producing a new vector will be called creation operators and those
giving zero are destruction operators.
One also has to make sure, that the YB equations give usable
commutation relations for these craetion and destruction operators.
Second, since our BA will normally be of the nested
type, one has to decide where to start the BA hierarchy. Since all this depends
on the details of the model, it is an art to be exercised on each model ( or
set of models ) in turn.
Let us take the $U(1) \otimes U(1)$ model of family equ.(\ref{eq:f1}) with
$ \nu_1 = \nu_2 = \nu_3 =1 $.
The vacuum to be chosen is
\beq
|\Omega _{2}\!>= \prod_i^N
\left( \begin{array}{c}
0 \\
1 \\
0
\end{array} \right).
\eeq
The local transition matrix $\cal L$, when applied to this vacuum,
takes the form :
\beq
{\cal L}(u)|\Omega_{2}\!> = \left (
\begin{array}{ccc}
c(u) & 0 & 0 \\
\ast & a_+(u) & \ast \\
0 & 0 & c(u)
\end{array} \right )|\Omega _{2}\!>.
\eeq
Here the stars $\ast$ stand for nonzero entries not proportional to
$|\Omega _{2}\!>$, so that we are actually somewhat abusing the notation.
The functions are :
\beq
a_{\pm} (u) = \sin ( \gamma \pm u),\hspace{2em}
c(u) = \frac{\sin (u)}{\sin ( \gamma +u)}.
\eeq
It follows, that $|\Omega _{2}\!>$ is an eigenstate of the transfer-matrix,since
\bdm
T_1^1 (u) |\Omega _{2}\!>=T_3^3 (u) |\Omega _2\!>=
c^N (u) |\Omega _{2}\!>
\edm
\beq
T_2^2 (u) |\Omega _{2}\!> = a_+^N (u) |\Omega _{2}\!>.
\eeq
Starting from element $T_2^2 (u)$, we can now unleash the nested BA machinery an
d obtain the following set of eigenvalues $\tau (u)$ :
\bdm
\tau(u) = \prod_{l=1}^{m_1} \frac{1}{c(u - \lambda _l)} +
c^N (u) \prod_{l=1}^{m_1} \frac{1}{c(\lambda _l - u )}
\tau _1 (u),
\edm
\beq
\tau _1(u) = \prod_{k=1}^{m_2}
\left (
\frac{1}{c(u - \Lambda _k)} +
\frac{1}{c(\Lambda _k - u)}
\prod_{j=1}^{m_1} c(u - \lambda _j)
\right ),
\eeq
where the $\lambda _l$ and $\Lambda _k$, satisfy the equations :
\bdm
\prod_{l=1}^{m_1} c( \Lambda _k - \Lambda _ l ) =
\prod_{j=1,j \neq l}^{m_1}
\frac{ c( \Lambda _j - \Lambda _ k )}
{ c( \Lambda _k - \Lambda _j)}
\edm
\beq
c^N (\Lambda _l )
\prod_{k=1}^{m_2} \frac{1}{ c( \lambda _l - \Lambda _k )} =
\prod_{j=1,j \neq l}^{m_1}
\frac{ c( \lambda _j - \Lambda _l )}
{ c( \Lambda _l - \Lambda _j)} .
\eeq
The corresponding eigenvectors are given by linear combinations of
vectors, obtained by applying
$B_1(u)$ and $B_2(u)$ to $|\Omega _{2}\!>$,
where $ B_1(u)=T_2^1(u)$ and $B_2 (u) = T_3^1(u)$. In detail :
\beq
|_1 \{ \lambda_l,a_l \}_{m_1} > =
\sum_{a_l=1}^{2}
F_1(a_1,\ldots,a_{m_1} ) \prod_{l=1}^{m_1} B_{a_l}(\lambda_l)
|\Omega_{2} \!>,
\eeq
where $F_1(a_1,\ldots,a_{m_1})$ is an now eigenvector of the six vertex transfer
-matrix $T_1(u)$,
constructed form local transition matrices
\beq
l_{ac|bc} (u) =
c(u) \delta _{ab} \delta _{cd} +
b_c (u) \delta _{ad} \delta _{bc}\;\; ,
\hspace{2em} a \neq b \neq c \neq d,
\eeq
\bdm
l_{11}^{11} (u) = l_{22}^{22} (u) = 1.
\edm
Here
\bea
b_1(u) & = & \sin (\gamma) e^{\imath u} \nonumber \\
b_2(u) & = & \sin (\gamma) e^{- \imath u}.
\eea
The same scheme may now be applied to the corresponding decoupled model of equ.
(\ref{eq:fam2}). In this case the two-step nested BA simplifies to a
single step affair. This happens, because after executing the first step, the re
sulting auxiliary six vertex model has a trivial $l(u)$ matrix, proportional
to the unit matrix.
The eigenvalues are then given by:
\beq
\tau(u,\{ \lambda_i \}) =
\prod_{l=1}^m \left (
\frac{ a^N (u) }{ c(\lambda _l - u) } +
\frac{ c^N (u) }{c(u- \lambda _l)} \right),
\eeq
the $\lambda_l$ satisfying:
\beq
c^N (\lambda_k)=\prod_{j=1,j\neq k}^m
\frac{c(\lambda _k - \lambda _j )}
{c(\lambda _j - \lambda _k )}.
\eeq
Thus the eigenvalues are identical to the ones of the six vertex model and out o
f all eigenvalues the BA yields only this subset.
the question of how to retrieve the remaining eigenvalues and the BA for the $U(
1)$-symmetric case is left for a forthcoming publication.
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