\section{Introduction}
A diluted magnetic semiconductor (DMS), in which a host semiconductor is doped with a low concentrations of magnetic ions, is a material of interest for spintronics research. In particular, ferromagnetic DMSs have attracted considerable attention because the itinerant carriers are considered to mediate the magnetic interaction between doped ions \cite{NatPhys_10_Sawicki, Science_11_Yamada}.
This type of magnetism is called ``carrier-induced ferromagnetism'' and enables the possibility of manipulating both electronic charge and spin degrees of freedom.
III-V based DMS Ga$_{1-x}$Mn$_x$As is an archetypical ferromagnetic DMS and has been studied from both fundamentally and in applications as a ferromagnetic DMS \cite{RMP_06_Jungwirth}. Spintronic devices using Ga$_{1-x}$Mn$_x$As have been fabricated and their effectiveness has been demonstrated \cite{Nature_00_Ohno, Nature_08_Chiba}, although the Curie temperature ($T_{\mathrm{C}}$) of Ga$_{1-x}$Mn$_x$As is below room temperature ($T_{\mathrm{C}} < 200$ K).

Understanding the mechanism of ferromagnetism in Ga$_{1-x}$Mn$_x$As is strongly desirable to develop DMS for application in spintronic devices.
Several physical models of the ferromagnetism have been proposed, e.g., the Zener $p$-$d$ exchange model \cite{Science_00_Dietl, PRB_07_Jungwirth, NatPhys_10_Sawicki}, the Mn $3d$ impurity band model \cite{PRL_01_Berciu, PRL_06_Burch, NatPhys_11_Ohya}, and the magnetic polaron model \cite{PRL_02_Kaminski, PRB_02_Mayr}. These models depend on the localization of hole carriers around the Mn ions, the energy position of the Mn $3d$ states, and the strength of hybridization between the Mn $3d$ orbital and the ligand band.
To obtain a fundamental understanding of ferromagnetism, it is essential to know the Mn $3d$ electronic configurations in the ground state. If the valence state of Mn in the ground state is divalent (Mn$^{2+}$), the Mn ion acts as an acceptor supplying a hole to the host GaAs, supporting the Zener $p$-$d$ exchange model.
Thus, the electronic structure of Mn $3d$ ions in Ga$_{1-x}$Mn$_x$As has been intensively investigated experimentally \cite{NatPhys_11_Ohya, PRB_98_Okabayashi, PRB_01_Okabayashi, PRL_11_Edmonds}. However, the presence of divalent (Mn$^{2+}$) or trivalent (Mn$^{3+}$) states has not been conclusively determined, and the degree of the localization of the Mn $3d$ states remains a matter of dispute.

In this Letter, to address the electronic structure of the doped Mn ions, we report the results of Mn $L_3$ X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) measurements of Ga$_{1-x}$Mn$_x$As ($x=0.04$).
RIXS is a powerful tool to investigate electronic excitations in element- and symmetry-specific ways including $d$-$d$ and charge-transfer (CT) excitations for open shell $3d$ orbitals \cite{PRB_06_Ghiringhelli}, and magnetic excitations for spin/charge-ordered systems \cite{NatPhys_11_LeTacon}.
These excitations are sensitive to electron correlation, crystalline symmetry, and the strength of hybridization with ligand band.
Moreover, RIXS is less sensitive to the surface structure because of its long probing depth ($>100$ nm).
The RIXS spectra obtained for Ga$_{1-x}$Mn$_x$As are compared with configuration-interaction (CI) cluster-model calculations \cite{JSPS_94_Tanaka}, and the electronic structure parameters are estimated.
It is found that the bulk electronic structures of Mn ions in Ga$_{1-x}$Mn$_x$As consist predominantly of charge-transferred states.
Based on the experimental findings, we shall discuss the Mn $3d$ electronic state in Ga$_{1-x}$Mn$_x$As.

Ga$_{1-x}$Mn$_x$As ($x=0.04$) thin films with a thickness of 20 nm were grown on a GaAs(001) substrate at 270 $^{\circ}$C under ultra-high vacuum by a molecular beam epitaxy method. To avoid surface oxidation, the samples were covered by GaAs (2 nm) and an As capping layer (1 nm) after the deposition of Ga$_{1-x}$Mn$_x$As layer to produce a structure of As/GaAs/Ga$_{0.96}$Mn$_{0.04}$As/GaAs(buffer)/GaAs(001).
The thin film sample after the deposition was cut into two pieces and one half of the sample was annealed at 270 $^{\circ}$C for 2 hours.
$T_\mathrm{C}$ values of the as-grown and annealed samples were $\sim 65$ K and $\sim 100$ K, respectively, as determined by the Arrott plot of the magnetic circular dichroism.
The RIXS and XAS experiments were performed using the high-resolution soft X-ray emission station HORNET \cite{RSI_12_Harada} at the long undulator beamline BL07LSU of SPring-8, Japan \cite{NIMPRESA_11_Senba}. The total energy resolution for the RIXS experiments was about 170 meV at the Mn $L_3$ edge ($E/\Delta E > 3750$). The RIXS spectra were measured with linear horizontal polarization at room temperature under a vacuum of $1.0 \times 10^{-5}$ Pa.
The XAS signals were collected in the total-fluorescence-yield (TFY) mode. The XAS spectra of MnO were measured in the total-electron-yield (TEY) mode as a reference.

Figure~\ref{GMA_RIXS}(a) shows the Mn $L_3$ XAS spectra of the Ga$_{1-x}$Mn$_x$As thin films.
The XAS spectra show a broad profile compared with the reference MnO, similar to the XAS specta previously reported for chemically etched Ga$_{1-x}$Mn$_x$As thin films \cite{APL_04_Edmonds, PRB_05_Edmonds}.
The XAS spectra exhibit the same features as those of the ``intrinsic Mn component'' [see Fig.~\ref{GMA_CIcalc}(a)], which consists mainly of Mn ions substituting Ga sites and partial filling of some interstitial site \cite{PRL_08_Takeda}.
In Ga$_{1-x}$Mn$_x$As, there are ferromagnetic and paramagnetic Mn components, which come from the intrinsic Mn and extrinsic oxidized Mn ions, respectively.
Details for the assignment of Mn species are described in Ref.~\cite{PRL_08_Takeda}.
It has been reported that low-temperature annealing of Ga$_{1-x}$Mn$_x$As induces interstitial Mn defects to diffuse toward the surface \cite{PRL_04_Edmonds, NatMater_12_Dobrowolska} creating Mn oxides at the surface \cite{PRB_05_Ishiwata}.
However, because of bulk sensitivity of the fluorescence yield mode, the XAS spectra do not show the oxide components and reflects the bulk electronic structure.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=8.8cm]{FIG1_GaMnAs_XAS-RIXS.eps}
\caption{Mn $L_{3}$ resonant inelastic X-ray scattering spectra of the as-grown and annealed Ga$_{1-x}$Mn$_x$As ($x = 0.04$) thin films.
(a) Mn $L_3$ XAS spectra taken in the TFY mode.
The vertical lines denote the excitation photon energy for the RIXS measurements.
(b) Mn $L_{3}$ RIXS spectra. The excitation energies are 639.6, 640.0, and 640.6 eV.
The spectra of MnO are also shown as a reference.
}
\label{GMA_RIXS}
\end{center}
\end{figure}

Figure~\ref{GMA_RIXS}(b) shows the Mn $L_3$ RIXS spectra of the Ga$_{1-x}$Mn$_x$As thin films taken with various excitation (incident photon) energies $h\nu$ across the $L_3$ edge in the XAS spectra. The spectra are normalized by area and plotted as a function of energy loss ($E_L$).
The line shape of the RIXS spectra strongly depends on the excitation energy. As a reference, Mn $L_3$ RIXS spectra of MnO for the corresponding excitation energy are also shown in Fig.~\ref{GMA_RIXS}(b).
The RIXS spectra of MnO demonstrate sharp $d$-$d$ excitation peaks from $E_L = 2 - 6$ eV and broad features taht aoriginate from CT excitation above $E_L = 6$ eV \cite{PRB_06_Ghiringhelli}. Notably, the energy resolution is high enough to resolve individual $d$-$d$ excitation peaks in MnO.
In contrast, the RIXS spectra of Ga$_{1-x}$Mn$_x$As show broad line shapes even at the same high energy resolution, which are similar to those previously reported at lower energy resolution \cite{PRB_05_Ishiwata}.
Therefore, the broad features of the Mn $L_3$ RIXS spectra are expected to accurately reflect the electronic structure of the Mn ions in Ga$_{1-x}$Mn$_x$As.
Since the CT excitation features from MnO are absent, the CT excitation energy is likely to be reduced to an energy level comparable to that of the $d$-$d$ excitation ($E_L = 2 - 6$ eV) and contribute to the broadening of the $d$-$d$ transition peaks \cite{PRB_06_Taguchi}.
There are differences in the RIXS spectra taken at $h\nu = 640$ and 640.6 eV between the as-grown and annealed samples.
These changes in the RIXS spectra taken at $h\nu = 640$ and 640.6 eV by annealing may be attributed to out-diffusion of interstitial Mn atoms during annealing.

The intrinsic Mn $L_{2,3}$ XAS and X-ray magnetic circular dichroism (XMCD) spectra are compared with CI cluster-model calculations, as shown in Fig.~\ref{GMA_CIcalc}(a) and \ref{GMA_CIcalc}(b), respectively. The calculations are performed for Mn ions in divalent (Mn$^{2+}$) or trivalent (Mn$^{3+}$) states in a crystal field of tetrahedral ($T_d$) symmetry.
Both the calculated spectra for the Mn$^{2+}$ and Mn$^{3+}$ states reproduce the experimental spectra. Common electronic structure parameters are used to fit bot the XAS and XMCD: the ligand-to-$3d$ CT energy $ \Delta = 1.5$ eV, the $d$-$d$ Coulomb interaction energy $U_{dd} = 3.5$ eV, the Slater-Koster parameter $(pd\sigma) = -1.0$ eV, and the crystal-field splitting $10Dq = 0$ eV.
Here, the CT energy $\Delta$ is defined as the energy difference between the $d^5$ and $d^6\underline{L}$ states for Mn$^{2+}$, and between $d^5\underline{L}$ and $d^6\underline{L}^2$ states for Mn$^{3+}$, where $\underline{L}$ denotes a hole in the ligand bands.
The values of $\Delta$, $U_{dd}$, and $(pd\sigma)$ are the same as those estimated from the photoemission experiments \cite{PRB_99_Okabayashi}.
We note that spectra calculated using $10Dq$ larger than 0.3 eV could not reproduce the experimental spectra.
The small crystal-field splitting is similar to that of the iron pnictide BaFe$_2$As$_2$ \cite{FPC_10_Ma} and the chalcopyrite CuFeS$_2$ \cite{pssa_09_Sato}, as expected from the small electronegativity of the heavy chalcogenide anions compared with that of oxygen \cite{FN_CF}.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=8.8cm]{FIG2_GaMnAs_ClusterCalc.eps}
\caption{RIXS spectra compared with the configuration-interaction (CI) cluster-model calculations.
(a), (b) Comparison of the intrinsic Mn $L_{2,3}$ XAS and XMCD spectra with the CI calculations, respectively. The experimental XAS and XMCD spectra are extracted from the magnetic-field dependence of XMCD \cite{PRL_08_Takeda}.
(c) Comparison of experimental RIXS spectra with CI calculations for Mn$^{2+}$ and Mn$^{3+}$ states. The CI calculations for RIXS were performed using the parameters common to those of XAS (XMCD). The calculated spectra are broadened by the experimental energy resolution of 170 meV.
}
\label{GMA_CIcalc}
\end{center}
\end{figure}

The CI calculations for the Mn$^{2+}$ and Mn$^{3+}$ states were also performed for the RIXS spectra using the same electronic structure parameters as those for XAS and XMCD.
Figure~\ref{GMA_CIcalc}(b) shows the Mn $L_3$ RIXS spectra of the Ga$_{1-x}$Mn$_x$As thin film compared with the CI calculations broadened by a Gaussian of the experimental energy resolution. The calculated spectra for the ``Mn$^{3+}$'' state reproduce the energy distribution of the experimental spectra except for the linewidths of each $d$-$d$ excitation peak, while the calculated spectra for the Mn$^{2+}$ state are different from the experimental RIXS spectra.
It should be noted that the profiles of the RIXS spectrum for the annealed film are also close to the calculated spectrum for the Mn$^{3+}$ state. Since annealing drives the interstitial Mn atoms towards the surface \cite{PRB_05_Edmonds} and increases the number of Mn in substitutional sites relative to the interstitial site \cite{NatMater_12_Dobrowolska}, the spectral change induced by the annealing may be attributed to the relative increase of Mn atoms in substituted sites.
Interstitial Mn defects are expected to act as double donors and have similar electronic structure to the CI calculations for Mn$^{2+}$. The relative decrease of Mn atoms in interstitial sites induced by annealing may also cause the peak at $E_L = 2$ eV to be more pronounced.
The results suggest that the doped Mn atoms in bulk Ga$_{1-x}$Mn$_x$As are predominantly in the formally trivalent Mn$^{3+}$ state. In addition, the electronic configurations in the Mn$^{3+}$ ground state are calculated as $3d^4: 3d^5\underline{L}: 3d^6\underline{L}^2: 3d^7\underline{L}^3 = 9\%: 69\% : 21\% : 1\%$.
Here, it is clearly shown that the delocalized charge-transferred states ($3d^5\underline{L}$ and $3d^6\underline{L}^2$) are dominant in the electronic configurations of the Mn$^{3+}$ ground state, and are inherently different from the ``Mn$^{2+}$'' ground state with a localized $d^5$ state hybridized with $d^6\underline{L}$ state.
Thus the Mn $L_3$ RIXS spectra are very sensitive to the valence state of Mn and our results conclusively show ``Mn$^{3+}$'' in the ground state of Ga$_{1-x}$Mn$_x$As.

Based on our experimental findings, we discuss the detail of the Mn $3d$ electronic states in Ga$_{1-x}$Mn$_x$As.
The transport properties of Ga$_{1-x}$Mn$_x$As have been reported to show a metal-insulator transition at $T_\mathrm{C}$ with metallic behavior below $T_\mathrm{C}$ \cite{PRB_98_Matsukura}, implying that the hole carriers are not itinerant above $T_\mathrm{C}$.
This transport property has been qualitatively well explained by bound magnetic polarons, in which Mn ions weakly bind holes \cite{PRB_03_Kaminski}.
Because the occupied Mn $3d$ levels are located below the As $4p$ ligand band \cite{PRB_99_Okabayashi} and the present results suggest that the hybridization between the $3d$ orbital and the As $4p$ ligand is strong enough, the Anderson impurity model predicts the formation of a split-off state above the valence-band maximum \cite{PRB_01_Okabayashi}.
The split-off state acts as an acceptor level and leads to the formation of the impurity band \cite{PRB_01_Okabayashi}, in which the Mn ion weakly binds a hole, which may be the origin of the bound magnetic polarons.
Our RIXS results show that the dominant charge transferred states are the trivalent Mn$^{3+}$ ground state and indicates the presence of a ligand hole bound to the Mn configuration.
This arrangement will significantly reduce ionized impurity scattering and support the presence of quantum states in Ga$_{1-x}$Mn$_x$As quantum wells observed by resonant tunneling spectroscopy \cite{NatPhys_11_Ohya, PRL_10_Ohya}.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=6cm]{FIG3_GaMnAs_LorBroadening.eps}
\caption{Lorentzian broadening of the RIXS spectra calculated by the CI cluster model.
The experimental RIXS spectra of the annealed sample are compared with the calculations for the Mn$^{3+}$ configuration convolved by Lorentzian lineshapes. The full-width-half-maximum values of the Lorentzian the broadening are 0.3, 0.6, and 0.9 eV.
}
\label{GMA_LB}
\end{center}
\end{figure}

The RIXS spectra are now well reproduced by the CI cluster-model calculation using a common parameter set with XAS and XMCD. However, the experimental RIXS profile is still broader than the calculated profile with 0.17 eV broadening because of the experimental resolution.
To evaluate the additional broadening of the Mn $d$-$d$ excitation, the CI calculations are further broadened.
The calculated RIXS spectra with additional broadening are shown in Fig.~\ref{GMA_LB} and broadened spectra with a full-width-half-maximum of 0.6 eV reproduce the experimental RIXS spectra relatively well.
According to Ref.~\cite{PRB_06_Taguchi}, the RIXS spectra can be broadened because of stronger hybridization between $d$-$d$ excitations and particle-hole pairs in the host valence and conduction bands itself.
However, this effect should be accompanied by distortion of the overall RIXS profile \cite{PRB_06_Taguchi} and may fail to explain either the experimental RIXS or XAS/XMCD spectra.
Another reason could be a double exchange interaction between the $d$ orbitals \cite{PR_51b_Zener} but this hardly contributes to the broadening of the Mn $3d$ states because hole carriers have $p$-type ligand character as discussed above.
Alternatively, we can attribute the additional broadening to a lifetime broadening in the RIXS final state where fast decay of the $d$-$d$ excitation into an electron-hole pair in the host valence and conduction bands occurs through the hybridization between the Mn $3d$ state and the As $4p$ ligand band.
To account for this, we have applied Lorentzian, rather than the Gaussian, broadening to the RIXS spectra as shown in Fig.~\ref{GMA_LB}.
The additional broadening implies two steps for the RIXS decay process; First, the resonantly excited core-hole state decays into a $d$-$d$ excitation involving a localized bound hole state (split-off state) because of the strong hybridization between the Mn $3d$ and As $4p$ orbitals. Then, through the interaction with the As $4p$ band, the $d$-$d$ excitation decays into the electron-hole pair in the host valence and conduction bands.
With increasing energy resolution, RIXS may provide information about the time evolution of the decay process, and expand the applications of RIXS in the near future.

In conclusion, we have conducted Mn $L_3$ RIXS and XAS measurements on Ga$_{1-x}$Mn$_x$As ($x=0.04$) thin films to investigate the electronic structure of the doped Mn ions.
The XAS spectrum taken in the TFY mode includes few contributions from extrinsic surface Mn oxides, suggesting that the TFY-XAS spectra largely reflect the bulk electronic structure of Ga$_{1-x}$Mn$_x$As.
The RIXS spectra, which also reflect bulk sensitive information, show a broad profile even though at energy resolutions high enough to distinguish individual $d$-$d$ excitation peaks of MnO.
Analysis by the CI calculations indicates that the Mn ground states mainly consists of the Mn$^{3+}$ electronic configuration composed of the charge-transferred states ($d^5\underline{L}$ and $d^6\underline{L}^2$), in which the ligand hole is weakly bound to the Mn $3d^5$ state, rather than the pure Mn$^{2+}$ state.
The width of the Mn $d$-$d$ excitation has been estimated to be $\sim 0.6$ eV by the Lorentzian broadening of the calculated spectra.
Therefore, the broad $d$-$d$ excitation peaks in the RIXS spectra can be attributed to the lifetime broadening in the final state of the RIXS process where fast decay of the $d$-$d$ excitations to an electron-hole pair in the host valence and conduction bands occurs because of the hybridization between the Mn $3d$ orbital and the ligand band.

This work was carried out by a joint research in the Synchrotron Radiation Research Organization, The University of Tokyo (Proposal No. 2011A7403 and 2009BS03).
M.K. acknowledges support from the Japan Society for the Promotion of Science.


\section{#1}}

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   \large \bf
Junpeng Cao${}^{a}$,~Wen-Li Yang${}^b\footnote{Corresponding author:
<EMAIL>}$,
 ~ Kangjie Shi${}^b$ and~Yupeng Wang${}^a\footnote{Corresponding author: <EMAIL>}$
 \end{center}}
\newcommand{\Address}{\begin{center}

     ${}^a$Beijing National Laboratory for Condensed Matter
           Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
           100190, China\\
     ${}^b$Institute of Modern Physics, Northwest University,
     Xian 710069, China
   \end{center}}
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   {\large \sf #1}\\ \vspace{1mm}{\small \sf Accepted for Publication}
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\preprint
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\bigskip\bigskip\bigskip

\Title{Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions}
\Author

\Address
\vspace{1cm}

\begin{abstract}
Employing the off-diagonal Bethe ansatz method  proposed recently by
the present authors, we exactly diagonalize the $XXX$ spin chain
with arbitrary boundary fields. By constructing a functional
relation between the eigenvalues of the transfer matrix and the
quantum determinant, the associated $T-Q$ relation and  the Bethe
ansatz equations are derived.

\vspace{1truecm} \noindent {\it PACS:} 75.10.Pq, 03.65.Vf, 71.10.Pm

\noindent {\it Keywords}: Spin chain; reflection equation; Bethe
Ansatz; $T-Q$ relation
\end{abstract}
\newpage
\section{Introduction}
\label{intro} \setcounter{equation}{0}

Our understanding of quantum phase transitions and critical
phenomena has been greatly enhanced by the study of exactly solvable
models (or quantum integrable systems) \cite{Bax82}. Such exact
results have provided valuable insight into the important
universality classes of quantum physical systems ranging from modern
condensed matter physics \cite{Kas98} to string and super-symmetric
Yang-Mills theories \cite{Dol03}. Since Yang and Baxter's pioneering
works \cite{yang2,bax1,Bax82}, the quantum Yang-Baxter equation
(QYBE), which define the underlying algebraic structure,  has become
a cornerstone for constructing and solving the integrable models.
There are several well-known methods for deriving the Bethe ansatz
(BA) solution of integrable models: the coordinate BA
\cite{Bet31,Bax82,Alc87,Bel13,Cra12}, the T-Q approach \cite{Bax82,Yan06}, the
algebraic BA \cite{Skl78,Tak79,Kor93}, the analytic BA \cite{Res83},
the functional BA \cite{Skl92} and others
\cite{And84,Baz89,Nep04,Cao03,Yan04,Gie05,Mel05,Doi06,Bas07,Gal08,Fra11,Nie09,Gra10,Nic12}.

Generally speaking, there are two classes of integrable models. One
possesses $U(1)$ symmetry and the other does not. Three well known
examples without $U(1)$ symmetry are the $XYZ$ spin chain
\cite{bax1,Tak79}, the $XXZ$ spin chain with antiperiodic boundary
condition \cite{Yun95,Bat95,Gal08,Fra11,Nie09,Nic12} and the ones
with unparallel boundary fields
\cite{Nep04,Cao03,Yan04,Gie05,Bas07,Gal08,Fra11,Nie09,Gra10,Nic12}.
It has been proven that most of the conventional Bethe ansatz
methods can successfully diagonalize the integrable models with
$U(1)$ symmetry. However, for those without $U(1)$ symmetry, only
some very special cases such as the $XYZ$ spin chain with even site
number \cite{bax1,Tak79} and the $XXZ$ spin chain with constrained
unparallel boundary fields \cite{Cao03,Yan04,Yan07,Gie05-1} can be
dealt with due to the existence of a proper ``local vacuum state" in
these special cases. The main obstacle applying the algebraic Bethe
ansatz and Baxter's method to general integrable models without
$U(1)$ symmetry lies in the absence of such a ``local vacuum". A
promising method for approaching such kind of problems is Sklyanin's
separation of variables method \cite{Skl92} which has been recently
applied to some integrable models \cite{Fra11,Nie09,Gra10,Nic12}.
However, before the very recent work\cite{Cao13}, a systematic
method was still absent to derive the Bethe ansatz equations for
integrable models without $U(1)$ symmetry, which are crucial for
studying the physical properties in the thermodynamic limit.

As for integrable models without $U(1)$ symmetry, some off-diagonal
elements of monodromy matrix enter into expression of the transfer
matrix. This breaks down the usual $U(1)$ symmetry. Very recently,
we have proposed a  method \cite{Cao13} for dealing with the
integrable models without $U(1)$ symmetry. The central idea of the
method is to construct the functional relations between eigenvalues
$\Lambda(\lambda)$ of the transfer matrix (the trace of the
monodromy matrix) and its quantum determinant $\Delta_q(\lambda)$,
i.e.,
$\Lambda(\theta_j)\Lambda(\theta_j-\eta)\sim\Delta_q(\theta_j)$ (see
below (\ref{Main})) based on the zero points of the product of
off-diagonal elements of monodromy matrix $B(u)B(u-\eta)=0$. Since
the trace and the determinant are two basic quantities of a matrix
which are independent of the representation basis, this method could
overcome the obstacle of absence of a reference state which is
crucial in the conventional Bethe ansatz methods.

Our primary motivation for this work comes from the long standing
problem of solving the open spin-$\frac{1}{2}$ XXX spin chain with
unparallel boundary fields, defined by the Hamiltonian
\cite{Skl88,Veg93}
\begin{eqnarray}
H=\sum_{j=1}^{N-1}{\vec \sigma}_{j} {\vec \sigma}_{j+1}+h_N
\sigma_{N}^{z} + h_1^x \sigma_{1}^{x}+h_1^z \sigma_{1}^{z}.
\label{ohami}
\end{eqnarray}
$N$ is the site number of the system and $\sigma_{j}^{\alpha}$
$(\alpha=x, y, z)$ is the Pauli matrix on the site $j$ along the
$\alpha$ direction. The parameters $h_N$, $h^x_1$ and $h^z_1$ are related to
boundary fields. Solving this problem for generic values of these three parameters
is a crucial step in formulating the thermodynamics of the spin chain, due to the
fact that this problem has important applications in condensed matter physics and statistical
mechanics. In this paper, we shall use the method developed in \cite{Cao13} to solve the eigenvalue problem of the
above Hamiltonian with generic $h_N$, $h^x_1$ and $h^z_1$.

The paper is organized as follows.  Section 2 serves as an
introduction our notation and some basic ingredients. We briefly
describe the inhomogeneous open XXX chain with non-diagonal boundary
terms.  In Section 3, we derive the exchange relations among the
matrix entries of the monodromy matrix algebras (or the Yang-Baxter
algebras). In section 4 after obtaining some properties of the
eigenvalue as a function of spectrum parameter $u$, we derive the
very relation between the eigenvalue and the quantum determinant of
the double-row monodromy matrix. This allows us to construct a
generalized $T-Q$ relation type solution of eigenvalue. In section
5, we consider the homogeneous limit of the results of the previous
section and give the energy spectrum of the Hamiltonian of the open
spin-$\frac{1}{2}$ XXX spin chain with unparallel boundary fields.
In section 6, we summarize our results and give some discussions.
Some detailed technical proof is given in Appendix A.

\section{ Transfer matrix}
\label{XXZ} \setcounter{equation}{0}

Throughout, ${\rm\bf V}$ denotes a two-dimensional linear space. The well-known rational
six-vertex model R-matrix $R(u)\in {\rm End}({\rm\bf V}\otimes {\rm\bf V})$
is given by \bea
R(u)=\lt(\begin{array}{llll}u+1&&&\\&u&1&\\
&1&u&\\&&&u+1\end{array}\rt).
\label{r-matrix}\eea Here $u$ is the spectral parameter. Without losing the generality we have
set the so-called bulk anisotropy parameter (or crossing parameter) $\eta=1$. The R-matrix satisfies
the quantum Yang-Baxter equation (QYBE)
\bea
R_{12}(u_1-u_2)R_{13}(u_1-u_3)R_{23}(u_2-u_3)
=R_{23}(u_2-u_3)R_{13}(u_1-u_3)R_{12}(u_1-u_2),\label{QYB}\eea
and the properties,
\bea &&\hspace{-1.5cm}\mbox{ Initial
condition}:\,R_{12}(0)= P_{12},\label{Int-R}\\
&&\hspace{-1.5cm}\mbox{ Unitarity
relation}:\,R_{12}(u)R_{21}(-u)= -\xi(u)\,{\rm id},
\quad \xi(u)=(u+1)(u-1),\label{Unitarity}\\
&&\hspace{-1.5cm}\mbox{ Crossing
relation}:\,R_{12}(u)=V_1R_{12}^{t_2}(-u-1)V_1,\quad
V=-i\s^y,
\label{crosing-unitarity}\\
&&\hspace{-1.5cm}\mbox{ PT-symmetry}:\,R_{12}(u)=R_{21}(u)=R^{t_1\,t_2}_{12}(u),\label{PT}\\
&&\hspace{-1.5cm}\mbox{ Antisymmetry}:\,R_{12}(-1)=-(1-P)=-2P^{(-)}.\label{Ant}
\eea
Here $R_{21}(u)=P_{12}R_{12}(u)P_{12}$ with $P_{12}$ being
the usual permutation operator and $t_i$ denotes transposition
in the $i$-th space. Here and below we adopt the standard
notations: for any matrix $A\in {\rm End}({\rm\bf V})$, $A_j$ is an
embedding operator in the tensor space ${\rm\bf V}\otimes
{\rm\bf V}\otimes\cdots$, which acts as $A$ on the $j$-th space and as
identity on the other factor spaces; $R_{ij}(u)$ is an embedding
operator of R-matrix in the tensor space, which acts as identity
on the factor spaces except for the $i$-th and $j$-th ones.

One introduces the ``row-to-row"  (or one-row ) monodromy matrices
$T(u)$ and $\hat{T}(u)$, which is an $2\times 2$ matrix with elements being
operators acting on ${\rm\bf V}^{\otimes N}$,
\bea
T_0(u)&=&R_{0N}(u-\theta_N)R_{0\,N-1}(u-\theta_{N-1})\cdots
R_{01}(u-\theta_1),\label{Mon-V-1}\\
\hat{T}_0(u)&=&R_{01}(u+\theta_1)R_{02}(u+\theta_{2})\cdots
R_{0N}(u+\theta_N).\label{Mon-V-2}
\eea Here $\{\theta_j|j=1,\cdots,N\}$ are
arbitrary free complex parameters which are usually called
inhomogeneous parameters.

Integrable open chain can be constructed as follows \cite{Skl88}.
Let us introduce a pair of K-matrices $K^-(u)$ and $K^+(u)$. The
former satisfies the reflection equation (RE)
 \bea &&R_{12}(u_1-u_2)K^-_1(u_1)R_{21}(u_1+u_2)K^-_2(u_2)\no\\
 &&~~~~~~=
K^-_2(u_2)R_{12}(u_1+u_2)K^-_1(u_1)R_{21}(u_1-u_2),\label{RE-V}\eea
and the latter  satisfies the dual RE \bea
&&R_{12}(u_2-u_1)K^+_1(u_1)R_{21}(-u_1-u_2-2)K^+_2(u_2)\no\\
&&~~~~~~= K^+_2(u_2)R_{12}(-u_1-u_2-2)K^+_1(u_1)R_{21}(u_2-u_1).
\label{DRE-V}\eea For open spin-chains, other than the standard
``row-to-row" monodromy matrix $T(u)$ (\ref{Mon-V-1}), one needs to
consider  the
 double-row monodromy matrix $\mathbb{T}(u)$
\bea
  \mathbb{T}(u)=T(u)K^-(u)\hat{T}(u).
  \label{Mon-V-0}
\eea Then the double-row transfer matrix of the XXX chain with open
boundary (or the open XXX chain) is given by \bea
\t(u)=tr(K^+(u)\mathbb{T}(u)).\label{trans}\eea The QYBE (\ref{QYB})
and (dual) REs (\ref{RE-V}) and (\ref{DRE-V}) lead to the fact that
the transfer matrices with different spectral parameters commute
with each other \cite{Skl88}: $[\t(u),\t(v)]=0$. Then $\tau(u)$
serves as the generating functional of the conserved quantities of
the corresponding system, which ensures the integrability of the
open XXX chain.

In this paper in order to study the system described by the Hamiltonian (\ref{ohami}),
we consider the K-matrix $K^{-}(u)$ which is a diagonal
solution to the RE (\ref{RE-V}) associated the six-vertex
model R-matrix  \cite{Veg93,Gho94}
\begin{eqnarray}
K^-(u)=\left(\begin{array}{cc}
p+u & 0\\
0 & p-u
\end{array}\right). \label{K-}
\end{eqnarray}
At the same time, we introduce  the corresponding {\it dual\/}
K-matrix $K^+(u)$ which is a generic solution to the dual
reflection equation (\ref{DRE-V})
\begin{eqnarray}
K^+(u)=\left(\begin{array}{cc}
q+u+1 & \xi(u+1)\\
\xi(u+1) & q-u-1
\end{array}\right). \label{K+}
\end{eqnarray}
The first order derivative of logarithm of the
transfer matrix $\tau(u)$ (\ref{trans}) with the K-matrices $K^{\pm}$ given
by (\ref{K-}) and (\ref{K+}) yields   the Hamiltonian (\ref{ohami})
\begin{eqnarray}
&&H=\frac{\partial \ln \tau(\lambda)}{\partial
\lambda}|_{\lambda=0,\theta_j=0}-N\nonumber \\
&&\quad =2\sum_{j=1}^{N-1}P_{j,j+1}
+\frac{{K_{N}^-}^\prime(0)}{K^-_N(0)}+2\frac{{K^+_1}^\prime(0)}{K^+_1(0)}-N \nonumber\\
&&\quad =\sum_{j=1}^{N-1}{\vec \sigma}_{j} {\vec
\sigma}_{j+1}+\frac{1}{p} \sigma_{N}^{z} +\frac{1}{q}(
\sigma_{1}^{z}+\xi \sigma_{1}^{x}). \label{oh}
\end{eqnarray}
Therefore, $h_N=1/p$, $h_1^x=\xi/q$ and $h_1^z=1/q$.

\section{ Exchange relations of the monodromy matrice}
\label{T-QR} \setcounter{equation}{0}

The Yang-Baxter algebra is a corner stone of the QISM and it has
been successfully explored in order to construct integrable systems
and to get their exact solutions.

From the quantum Yang-Baxter equation (\ref{QYB}), one may derive the following ``RLL" relations
among the one-row monodromy matrices $T(u)$ and $\hat{T}(u)$
\bea
&&R_{12}(u-v)T_1(u)T_{2}(v)=T_{2}(v)T_1(u)R_{12}(u-v),\label{RLL-1}\\
&&R_{12}(v-u)\hat{T}_2(v)\hat{T}_{1}(u)=\hat{T}_{1}(u)\hat{T}_2(v)R_{12}(v-u),\label{RLL-2}\\
&&\hat{T}_2(v)R_{12}(u+v)T_1(u)=T_1(u)R_{12}(u+v)\hat{T}(v).\label{RLL-3}
\eea
The crossing relation of the R-matrix (\ref{crosing-unitarity}) allows us to derive the following relation
between $T(u)$ and $\hat{T}(u)$
\bea
 \hat{T}_0(u)=(-1)^{N-1}V_0\,T^{t_0}(-u-1)\,V_0.\label{Crossing-TT}
\eea Let us decompose the monodromy matrices $T(u)$ and $\hat{T}(u)$
in terms of its components \bea
T(u)=\left(\begin{array}{cc} \alpha(u) & \beta(u) \\
\gamma(u)& \delta(u)
  \end{array}\right),\quad\quad
\hat{T}(u)=(-1)^{N-1}\left(\begin{array}{cc}
\delta(-u-1)&-\beta(-u-1) \\
-\gamma(-u-1)&\alpha(-u-1)
\end{array}\right). \label{Decomp}
\eea One may  find the commutation relations among $\a(u)$, $\b(u)$,
$\g(u)$ and $\d(u)$. Here we give the relevant ones for our
purpose,
\bea
&&\beta(\lambda)\beta(\mu)=\beta(\mu)\beta(\lambda),\label{Exch-1}\\
&&\beta(\lambda)\gamma(\mu)=\gamma(\mu)\beta(\lambda)+\frac{1}{\lambda-\mu}[\delta(\mu)\alpha(\lambda)-\delta(\lambda)\alpha(\mu)],
\label{Exch-2}\\
&&
\alpha(\lambda)\beta(\mu)=\frac{\lambda-\mu-1}{\lambda-\mu}\beta(\mu)\alpha(\lambda)+\frac{1}{\lambda-\mu}\beta(\lambda)\alpha(\mu),
\label{Exch-3}\\
&&
\delta(\lambda)\beta(\mu)=\frac{\lambda-\mu+1}{\lambda-\mu}\beta(\mu)\delta(\lambda)-\frac{1}{\lambda-\mu}\beta(\lambda)\delta(\mu).
\label{Exch-4}
\eea
Let us decompose the double-row monodromy matrix $\mathbb{T}(u)$ in terms of its components which can be expressed in terms
of the components of $T(u)$
\begin{eqnarray}
\mathbb{T}(\lambda)&=&\left(\begin{array}{cc} A(\lambda) & B(\lambda) \\
C(\lambda)& D(\lambda)
  \end{array}\right)\no\\
&=& \left(\begin{array}{cc}
\alpha(\lambda)& \beta{\lambda}\\
\gamma(\lambda) & \delta(\lambda)
\end{array}\right)
\left(\begin{array}{cc}
p+\lambda & 0\\
0 & p-\lambda
\end{array}\right)
\left(\begin{array}{cc}
\delta(-\lambda-1)&-\beta(-\lambda-1) \\
-\gamma(-\lambda-1)&\alpha(-\lambda-1)
\end{array}\right).\no
\end{eqnarray} Namely,
\begin{eqnarray}
&&A(\lambda)=(p+\lambda)\alpha(\lambda)\delta(-\lambda-1)-(p-\lambda)\beta(\lambda)\gamma(-\lambda-1),\nonumber  \\
&&B(\lambda)=-(p+\lambda)\alpha(\lambda)\beta(-\lambda-1)+(p-\lambda)\beta(\lambda)\alpha(-\lambda-1),\label{B-operator} \\
&&C(\lambda)=(p+\lambda)\gamma(\lambda)\delta(-\lambda-1)-(p-\lambda)\delta(\lambda)\gamma(-\lambda-1),\nonumber  \\
&&D(\lambda)=-(p+\lambda)\gamma(\lambda)\beta(-\lambda-1)+(p-\lambda)\delta(\lambda)\alpha(-\lambda-1).\no
\end{eqnarray}
Moreover, the RE (\ref{RE-V}) and the ``RLL" relations (\ref{RLL-1})-(\ref{RLL-3}) enable us to derive exchange relation
of the double-row monodromy matrix $\mathbb{T}(u)$
\bea &&R_{12}(u_1-u_2)\mathbb{T}_1(u_1)R_{21}(u_1+u_2)\mathbb{T}_2(u_2)\no\\
 &&~~~~~~=
\mathbb{T}_2(u_2)R_{12}(u_1+u_2)\mathbb{T}_1(u_1)R_{21}(u_1-u_2),\label{RE-Operator}
\eea which allows one to obtain  the commutation relations among $A(u)$, $B(u)$,
$C(u)$ and $D(u)$.
Here we give the relevant ones given by \cite{Skl88} for our
purpose,
\bea
C(\lambda)B(\mu)&=&B(\mu)C(\lambda) +\frac{\lambda+\mu}{(\lambda+\mu+1)(\lambda-\mu)(2\lambda+1)}A(\mu){\bar D}(\lambda)\nonumber\\
&&+\frac{(\lambda-\mu+1)(2\lambda)}{(\lambda+\mu+1)(\lambda-\mu)(2\lambda+1)}A(\mu)A(\lambda) -\frac{2\lambda[A(\lambda){\bar D}(\mu)+A(\lambda)A(\mu)]}{(\lambda-\mu)(2\lambda+1)(2\mu+1)}\no\\
&&-\frac{[{\bar D}(\lambda){\bar D}(\mu)+{\bar D}(\lambda)A(\mu)]}{(\lambda+\mu+1)(2\lambda+1)(2\mu+1)},\label{BC}\\
A(\lambda)B(\mu)&=&\frac{(\lambda+\mu)(\lambda-\mu-1)}{(\lambda-\mu)(\lambda+\mu+1)}B(\mu)A(\lambda)
-\frac{1}{(\lambda+\mu+1)(2\mu+1)}B(\lambda){\bar D}(\mu)\nonumber\\
&&+\frac{2\mu}{(\lambda-\mu)(2\mu+1)}B(\lambda)A(\mu),\label{AB}\\
{\bar
D}(\lambda)B(\mu)&=&\frac{(\lambda-\mu+1)(\lambda+\mu+2)}{(\lambda-\mu)(\lambda+\mu+1)}B(\mu){\bar
D} (\lambda)-\frac{2(\lambda+1)}{(\lambda-\mu)(2\mu+1)}B(\lambda){\bar D}(\mu)\nonumber\\
&&+\frac{4(\lambda+1)\mu}{(2\mu+1)(\lambda+\mu+1)}B(\lambda)A(\mu).\label{DB}
\eea
Here we have introduced  the operator $\bar D(\lambda)=(2\lambda+1)D(\lambda)-A(\lambda)$. At this stage, we have provided
most of the ingredients required to obtain a fundamental functional relation the eigenvalue of the
transfer matrix (\ref{trans}) with K-matrices given by (\ref{K-}) and (\ref{K+}).

\section{ Functional relations and the T-Q relation}
\label{BAE} \setcounter{equation}{0}

Following the method in \cite{Yan08,Beh96} and using the crossing relation of the
R-matrix (\ref{crosing-unitarity}) and the explicit expressions of the K-matrices
(\ref{K-}) and (\ref{K+}), one can show that the corresponding transfer matrix
$\tau(u)$ has the following
properties,
\bea
&&\hspace{-1.5cm}\mbox{Crossing
symmetry}:\,\quad\tau(-u-1)=\tau(u),\label{T-Cro}\\
&&\hspace{-1.5cm}\mbox{Initial
condition}:\,\tau(0)=2p\,q\prod_{j=1}^N(1-\theta_j)(1+\theta_j)\times\,{\rm id},\label{T-In}\\
&&\hspace{-1.5cm}\mbox{Asymptotic behavior}:\,
\tau(u)\sim 2u^{2N+2}\times \,{\rm id}+\ldots,\quad {\rm for}\, u\rightarrow
\pm\infty.\label{T-Asy} \eea

The commutativity of the transfer matrix $\tau(u)$ implies that one
can find the common eigenstate of $\tau(u)$, which indeed does not
depend upon $u$. Suppose $\langle\Psi|$ is an eigenstate of
$\tau(u)$ with an eigenvalue $\Lambda(u)$, namely, \bea
\langle\Psi|\tau(u) =\Lambda(u)\langle\Psi|.\no \eea The properties
of the transfer matrix $\tau(u)$ given by
(\ref{T-Cro})-(\ref{T-Asy}) imply that the corresponding eigenvalue
$\Lambda(u)$ satisfies the following relations: \bea
&&\hspace{-1.5cm}\mbox{Crossing
symmetry}:\,\quad\L(-u-1)=\L(u),\label{Eig-Cro}\\
&&\hspace{-1.5cm}\mbox{Initial
condition}:\,\L(0)=2p\,q\prod_{j=1}^N(1-\theta_j)(1+\theta_j)=\L(-1),\label{Eig-In}\\
&&\hspace{-1.5cm}\mbox{Asymptotic behavior}:\,
\L(u)\sim 2u^{2N+2}+\ldots,\quad {\rm for}\, u\rightarrow
\pm\infty.\label{Eig-Asy} \eea
The analyticity of the R-matrix and K-matrices and independence on $u$ of the eigenstate
lead to that the eigenvalue $\Lambda(u)$ further obeys  the property
\bea
\hspace{-0.2cm}\mbox{Analyticity}:\,\quad \L(u) \mbox{, as a function of $u$, is a polynomial of degree $2N+2$}.\label{Eigen-Anal}
\eea
Therefore the values of $\L(u)$ at generic $2N+3$ points suffice to determine  the function uniquely. However, we have already
obtained the corresponding values of  $\L(u)$ at points $u=0,\,-1,\,\infty$. For this purpose, we shall find the associated
equations to determine the values of  $\L(u)$ at the other $2N$ points, for example at  $u=\theta_j,-\theta_j-1$.

Let us define the state $|0\rangle=\otimes |\uparrow\rangle_j$ which
now has nothing to do with the eigenstate of the transfer matrix
$\tau(u)$ as it does in the conventional algebraic Bethe ansatz
\cite{Kor93}. On the other hand,  the components of the double-row
monodromy matrix $\mathbb{T}(u)$ act on the states, giving rise to:
\bea
&& C(u)|0\rangle=0, \label{C-1} \\
&&
A(u)|0\rangle=(p+u)\prod_{j=1}^{N}(u-\theta_j+1)(u+\theta_j+1)|0\rangle
=a(u)|0\rangle, \label{a(u)} \\
&& {\bar
D}(u)|0\rangle=2u(p-u-1)\prod_{j=1}^{N}(u-\theta_j)(u+\theta_j)|0\rangle=d(u)|0\rangle.
\label{d(u)} \eea From the explicit expression (\ref{K+}) of the
K-matrix $K^+(u)$, one can express the transfer matrix in terms of
the components of the monodromy matrix $\mathbb{T}(u)$, \bea
\tau(u)&=&(q+u+1)A(u)+\xi(u+1)[C(u)+B(u)]+(q-u-1)D(u)\nonumber\\
&=&
\frac{2(u+q)(u+1)}{2u+1}A(u)+\xi(u+1)[B(u)+C(u)]+\frac{q-u-1}{2u+1}{\bar D}(u).
\eea
Now let us compute the action of $\tau(\theta_j)\tau(\theta_j-1)$ on the state $|0\rangle$.
Keeping in mind  the fact that the function $a(u)$ (or $d(u)$) vanishes at $u=\theta_j-1$ (or $u=\theta_j$), namely,
\bea
a(\theta_j-1)=0=d(\theta_j), \quad\quad j=1,...N,\label{Identity}
\eea
and using the relations (\ref{C-1})-(\ref{d(u)}), we have
\bea
\tau(\theta_j)\tau(\theta_j-1)|0\rangle&=&\frac{2(\theta_j+q)(\theta_j+1)(q-\theta_j)}{(2\theta_j+1)(2\theta_j-1)}
 a(\theta_j)d(\theta_j-1)|0\rangle+\xi^2\theta_j(\theta_j+1)C(\theta_j)B(\theta_j-1)|0\rangle\no\\
&&\hspace{-2.2truecm} +\frac{\xi(q-\theta_j)(\theta_j+1)}{2\theta_j-1}d(\theta_j-1)B(\theta_j)|0\rangle
+\frac{2\xi\theta_j(\theta_j+q)(\theta_j+1)}{2\theta_j+1}A(\theta_j)B(\theta_j-1)|0\rangle\no\\
&&\hspace{-2.2truecm}+\frac{\xi\theta_j(q-\theta_j-1)}{2\theta_j+1}\bar{D}(\theta_j)B(\theta_j-1)|0\rangle
+\xi^2\theta_j(\theta_j+1)B(\theta_j)B(\theta_j-1)|0\rangle\no\\
&&\hspace{-2.6truecm}=\frac{2(\theta_j+1)(q^2-(1+\xi^2)\theta^2_j)}{(2\theta_j-1)(2\theta_j+1)}a(\theta_j)d(\theta_j-1)|0\rangle
+\xi^2\theta_j(\theta_j+1)B(\theta_j)B(\theta_j-1)|0\rangle.\label{BB-T}
\eea In the derivation of the second equality of the above equation, we have used the exchange relation (\ref{BC})-(\ref{DB})
and (\ref{Identity}). One can compute the last terms of (\ref{BB-T}) by expanding the operator $B(u)$ in terms of the components
of one-row monodromy matrix $T(u)$ via (\ref{B-operator}). Direct calculation shows that
\bea
B(\theta_j)B(\theta_j-1)=0.\label{BB-Main}
\eea
The proof of the above very relation is relegated to Appendix A. Finally, we obtain
\bea
\tau(\theta_j)\tau(\theta_j-1)|0\rangle=\frac{2(\theta_j+1)(q^2-(1+\xi^2)\theta^2_j)}{(2\theta_j-1)(2\theta_j+1)}a(\theta_j)d(\theta_j-1)
|0\rangle.
\eea
Multiplying the eigenstate $\langle\Psi|$ from the left on the both sides of the above equation, we have
\bea
 \langle\Psi|\tau(\theta_j)\tau(\theta_j-1)|0\rangle=
 \frac{2(\theta_j+1)(q^2-(1+\xi^2)\theta^2_j)}{(2\theta_j-1)(2\theta_j+1)}a(\theta_j)d(\theta_j-1)
 \langle\Psi|0\rangle,
\eea which leads to the following equations\footnote{Here we assume that the scalar product $\langle\Psi|0\rangle$ is nonzero. The relations (\ref{BB-Main}) and (\ref{Main})
were also obtained previously by the Sklyanin's separation of variables method for the boundary parameters with some constraint \cite{Fra08}.}
\bea
\Lambda(\theta_j)\Lambda(\theta_j-1)&=&\frac{2(\theta_j+1)(q^2-(1+\xi^2)\theta^2_j)}{(2\theta_j-1)(2\theta_j+1)}a(\theta_j)d(\theta_j-1)\no\\
&=& \frac{\Delta_q(\theta_j)}{(1-2\theta_j)(1+2\theta_j)},
\quad\quad j=1,\ldots,N.\label{Main}
\eea The expression of the quantum determinant $\Delta_q(u)$ is given by \cite{Mez90,Zho96}
\bea
\Delta_q(u)={\rm Det}\{T(u)\}\, {\rm Det}\{\hat{T}(u)\}
\,{\rm Det}\{K^-(u)\}\,{\rm Det}\{K^+(u)\},
\eea and the corresponding determinants can be directly calculated
 \bea
\rm{Det}\lt\{T(u)\rt\}\,{\rm
id}&=&tr_{12}\lt(P^{(-)}_{12}T_1(u-1)T_2(u)P^{(-)}_{12}\rt)
=\prod_{j=1}^N(u-\theta_j+1)(u-\theta_j-1)\,{\rm
id},\no\\
\rm{Det}\lt\{\hat{T}(u)\rt\}\,{\rm
id}&=&tr_{12}\lt(P^{(-)}_{12}\hat{T}_1(u-1)\hat{T}_2(u)P^{(-)}_{12}\rt)
=\prod_{j=1}^N(u+\theta_j+1)(u+\theta_j-1)\,{\rm
id},\no\\
\rm{Det}\lt\{K^-(u)\rt\}&=&tr_{12}
\lt(P^{(-)}_{12}K^-_1(u-1)R_{12}(2u-1)K^-_2(u)\rt)=2(u-1)(p^2-u^2),\no\\
\rm{Det}\lt\{K^+(u)\rt\}&=&tr_{12}
\lt(P^{(-)}_{12}K^+_2(u)R_{12}(-2u-1)K^+_1(u-1)\rt)=2(u+1)((1+\xi^2)u^2-q^2).\no
\eea

The equations (\ref{Main}) play an important role in our method,
which gives rise to the functional relations between eigenvalues
$\Lambda(u)$ of the transfer matrix (the trace of the monodromy
matrix algebra) and its quantum determinant $\Delta_q(u)$, i.e.,
$\Lambda(\theta_j)\Lambda(\theta_j-1)\sim\Delta_q(\theta_j)$ when
the spectrum $u$ is taken at some particular points $\{\theta_j\}$
which are the zero points of $B(u)B(u-1)|0\rangle=0$. Since the
trace and the determinant are two basic quantities of a matrix which
are independent of the representation basis, this method can
overcome the obstacle of absence of a reference state which is
crucial in the conventional Bethe ansatz methods. Moreover, the
equations (\ref{Eig-Cro})-(\ref{Eigen-Anal}) and (\ref{Main}) enable
one to determine the functions $\Lambda(u)$. In the following part
of the section, we shall express the solutions of these equations in
terms of a generalized  $T-Q$ relation formulism.

Let us introduce some functions $\bar{a}^{(\pm)}(u)$ and $\bar{d}^{(\pm)}(u)$
\bea
&&\bar{a}^{(\pm)}(u)=\frac{2u+2}{2u+1}(u\pm p)(\sqrt{1+\xi^2}\,u\pm q)\prod_{j=1}^{N}(u+\theta_j+1)(u-\theta_j+1),\label{a-function}\\
&&\bar{d}^{(\pm)}(u)=\frac{2u}{2u+1}(u\mp p+1)(\sqrt{1+\xi^2}\,(u+1)\mp q)
\prod_{j=1}^{N}(u+\theta_j)(u-\theta_j)\no\\
&&\quad\quad\quad =\bar{a}^{(\pm)}(-u-1).\label{d-function}
\eea We can construct the solutions of (\ref{Eig-Cro})-(\ref{Eigen-Anal}) by the following ansatz
\bea
\Lambda^{(\pm)}(u)&=& \bar{a}^{(\pm)}(u)\frac{Q^{(\pm)}(u-1)Q_1^{(\pm)}(u-1)}{Q^{(\pm)}(u)Q_2^{(\pm)}(u)}+
\bar{d}^{(\pm)}(u)\frac{Q^{(\pm)}(u+1)Q_2^{(\pm)}(u+1)}{Q^{(\pm)}(u)Q_1^{(\pm)}(u)}\no\\
&&+2(1\hspace{-0.12truecm}-\hspace{-0.12truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
u(u\hspace{-0.12truecm}+\hspace{-0.12truecm}1)\frac{\prod_{j=1}^N
(u\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_j)(u\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_j)
(u\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(u\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)}
{Q^{(\pm)}(u)Q_1^{(\pm)}(u)Q_2^{(\pm)}(u)}.
\label{T-Q}
\eea The functions $Q^{(\pm)}(u)$, $Q_1^{(\pm)}(u)$ and $Q_2^{(\pm)}(u)$ are parameterized by $N$  Bethe roots
$\{\lambda^{(\pm)}_j|j=1,\ldots N-2M\}$, $\{\mu^{(\pm)}_j|j=1,\ldots,M\}$ and $\{\nu^{(\pm)}_j|j=1,\ldots,M\}$ with
$M=0,\ldots,\lt[\frac{N}{2}\rt]$ as follows,
\bea
 Q^{(\pm)}(u)&=&\prod_{j=1}^{N-2M}(u-\lambda^{(\pm)}_j)(u+\lambda^{(\pm)}_j+1),\label{Q-1}\\
 Q_1^{(\pm)}(u)&=&\prod_{j=1}^{M}(u-\mu^{(\pm)}_j)(u+\nu^{(\pm)}_j+1),\label{Q-2}\\
 Q_2^{(\pm)}(u)&=&\prod_{j=1}^{M}(u-\nu^{(\pm)}_j)(u+\mu^{(\pm)}_j+1).\label{Q-3}
\eea These $N$ parameters are different from each other and  satisfy  the following Bethe ansatz equations
\bea
&&-\frac{\bar{a}^{(\pm)}(\lambda^{(\pm)}_j)} {\bar{d}^{(\pm)}(\lambda^{(\pm)}_j)}
-\frac{Q^{(\pm)}(\lambda^{(\pm)}_j+1)Q_2^{(\pm)}(\lambda^{(\pm)}_j)Q_2^{(\pm)}(\lambda^{(\pm)}_j+1)}
{Q^{(\pm)}(\lambda^{(\pm)}_j-1)Q_1^{(\pm)}(\lambda^{(\pm)}_j)Q_1^{(\pm)}(\lambda^{(\pm)}_j-1)}\no\\
 &&\quad\quad=\frac{(1\hspace{-0.08truecm}-\hspace{-0.08truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
 (\lambda^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)(2\lambda^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
 \prod_{l=1}(\lambda^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_l\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(\lambda^{(\pm)}_j\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_l\hspace{-0.08truecm}+\hspace{-0.08truecm}1)}
{(\lambda^{(\pm)}_j\mp p+1)(\sqrt{1+\xi^2}(\lambda^{(\pm)}_j+1)\mp q)
Q^{(\pm)}(\lambda^{(\pm)}_j-1)Q_1^{(\pm)}(\lambda^{(\pm)}_j)Q_1^{(\pm)}(\lambda^{(\pm)}_j-1)},\no\\
&&\quad\quad j=1,\ldots,N-2M,\label{BAE-M-1}\\
&&\frac{(1\hspace{-0.08truecm}-\hspace{-0.08truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
 (\mu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)(2\mu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
 \prod_{l=1}(\mu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_l\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(\mu^{(\pm)}_j\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_l\hspace{-0.08truecm}+\hspace{-0.08truecm}1)}
{(\mu^{(\pm)}_j\mp p+1)(\sqrt{1+\xi^2}(\mu^{(\pm)}_j+1)\mp q)
Q^{(\pm)}(\mu^{(\pm)}_j+1)Q_2^{(\pm)}(\mu^{(\pm)}_j)Q_2^{(\pm)}(\mu^{(\pm)}_j+1)}=-1,\no\\
&&\quad\quad j=1,\ldots,M,\label{BAE-M-2}\\
&&\frac{(1\hspace{-0.08truecm}-\hspace{-0.08truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
 \nu^{(\pm)}_j(2\nu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
 \prod_{l=1}(\nu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_l)
(\nu^{(\pm)}_j\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_l)}
{(\nu^{(\pm)}_j\pm p)(\sqrt{1+\xi^2}\nu^{(\pm)}_j\pm q)
Q^{(\pm)}(\nu^{(\pm)}_j-1)Q_1^{(\pm)}(\nu^{(\pm)}_j)Q_1^{(\pm)}(\nu^{(\pm)}_j-1)}=-1,\no\\
&&\quad\quad j=1,\ldots,M.\label{BAE-M-3}
\eea

Some remarks are in order.  The completeness \cite{Nep03,Yan06} of
the two sets of  eigenvalues in the XXZ spin chain with constrained
unparallel boundary fields case suggests that in our case, the
eigenvalues $\Lambda^{(-)}(u)$ and $\Lambda^{(+)}(u)$ {\it together}
constitute the complete set of eigenvalues of the transfer matrix
$t(u)$ of the open XXX spin chain. The last term of our generalized
$T-Q$ relation (\ref{T-Q}) (c.f. the conventional type \cite{Bax82}
) is crucial, which has encoded the contribution from the
off-diagonal element of the K-matrix and vanishes when two boundary
fields are parallel. Taking the limit: $\xi\rightarrow 0$, the set
of parameter $\{\mu_j\}$ coincides with that of $\{\nu_j\}$, the
corresponding $T-Q$ relation (\ref{T-Q}) becomes the usual type \bea
\Lambda^{(\pm)}(u)&=&
\bar{a}^{(\pm)}(u)\frac{\bar{Q}^{(\pm)}(u-1)}{\bar{Q}^{(\pm)}(u)}+
\bar{d}^{(\pm)}(u)\frac{\bar{Q}^{(\pm)}(u+1)}{\bar{Q}^{(\pm)}(u)},\label{T-Q-Ordinary}
\eea where $\bar{Q}^{(\pm)}(u)=Q^{(\pm)}(u)Q_1^{(\pm)}(u)$. Then the
resulting solutions are reduced to those in \cite{Skl88}.

\section{Homogeneous limit}
\label{Hom} \setcounter{equation}{0}

The results of the previous sections focus on  the inhomogeneous spin-$\frac{1}{2}$ XXX open chain.
In this section we consider the homogeneous limit,  i.e. $\{\theta_j\}\longrightarrow 0$, of the above results.
The corresponding
eigenvalue $\Lambda(u)$ of the transfer matrix of the homogeneous model satisfies the following relations:
\bea
&&\hspace{-1.5cm}\mbox{Crossing
symmetry}:\,\quad\L(-u-1)=\L(u),\label{H-Eig-Cro}\\
&&\hspace{-1.5cm}\mbox{Initial
condition}:\,\L(0)=2p\,q=\L(-1),\label{H-Eig-In}\\
&&\hspace{-1.5cm}\mbox{Asymptotic behavior}:\,
\L(u)\sim 2u^{2N+2}+\ldots,\quad {\rm for}\, u\rightarrow
\pm\infty,\label{H-Eig-Asy}\\
&&\hspace{-1.5cm}\mbox{Analyticity}:\,\quad \L(u) \mbox{, as a
function of $u$, is a polynomial of degree
$2N+2$}.\label{H-Eigen-Anal} \eea The corresponding relations
(\ref{Main}) now read \bea \frac{\partial^{l}}{\partial
u^l}\lt\{\Lambda(u)\Lambda(u-1)\rt\}|_{u=0}=
\frac{\partial^{l}}{\partial
u^l}\lt\{\frac{\bar{\Delta}_q(u)}{(1+2u)(1-2u)}\rt\}|_{u=0},\,\,l=0,1,\ldots,2N-1.\label{Main-H}
\eea Here the quantum determinant $\bar{\Delta}_q(u)$ becomes \bea
\bar{\Delta}_q(u)=4(u^2-1)(p^2-u^2)((1+\xi^2)^2u^2-q^2)(u+1)^{2N}(u-1)^{2N}.
\eea Actually the equations (\ref{H-Eig-Cro})-(\ref{Main-H}) enable
one to determine the functions $\Lambda(u)$. We shall express the
solutions of these equations in terms of a generalized  $T-Q$
relation formulism. For this purpose, let us introduce the following
functions $H^{(\pm)}(u)$ and $\Lambda^{(\pm)}(u)$ \bea
H^{(\pm)}(u)&=&\frac{2u+2}{2u+1}(u\pm p)(\sqrt{1+\xi^2}\,u\pm q)(u+1)^{2N},\label{H-function}\\
\Lambda^{(\pm)}(u)&=& H^{(\pm)}(u)\frac{Q^{(\pm)}(u-1)Q_1^{(\pm)}(u-1)}{Q^{(\pm)}(u)Q_2^{(\pm)}(u)}+
H^{(\pm)}(-u-1)\frac{Q^{(\pm)}(u+1)Q_2^{(\pm)}(u+1)}{Q^{(\pm)}(u)Q_1^{(\pm)}(u)}\no\\
&&\quad\quad+\frac{2(1-\sqrt{1+\xi^2})
u^{2N+1}(u+1)^{2N+1}}{Q^{(\pm)}(u)Q_1^{(\pm)}(u)Q_2^{(\pm)}(u)},
\label{H-T-Q}
\eea where the functions $Q^{(\pm)}(u)$, $Q_1^{(\pm)}(u)$ and $Q_2^{(\pm)}(u)$ are also given by (\ref{Q-1})-(\ref{Q-3}).
Then the functions $\Lambda^{(\pm)}(u)$ become the solutions of
the equations (\ref{H-Eig-Cro})-(\ref{Main-H}) provided that the parameters
$\{\lambda^{(\pm)}_{j}\}$, $\{\mu^{(\pm)}_{j}\}$  and $\{\nu^{(\pm)}_{j}\}$ satisfy the following Bethe ansatz equations respectively
\bea
&&-\frac{H^{(\pm)}(\lambda^{(\pm)}_j)} {H^{(\pm)}(-\lambda^{(\pm)}_j-1)}-\frac{Q^{(\pm)}(\lambda^{(\pm)}_j+1)Q^{(\pm)}_2(\lambda^{(\pm)}_j)Q^{(\pm)}_2(\lambda^{(\pm)}_j+1)}
{Q^{(\pm)}(\lambda^{(\pm)}_j-1)Q^{(\pm)}_1(\lambda^{(\pm)}_j)Q^{(\pm)}_1(\lambda^{(\pm)}_j-1)}\no\\
 &&\quad\quad=\frac{(1\hspace{-0.08truecm}-\hspace{-0.08truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
 (\lambda^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)(2\lambda^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(\lambda^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)^{2N}}
{(\lambda^{(\pm)}_j\mp p+1)(\sqrt{1+\xi^2}(\lambda^{(\pm)}_j+1)\mp q)Q^{(\pm)}(\lambda^{(\pm)}_j\hspace{-0.08truecm}-\hspace{-0.08truecm}1)
Q^{(\pm)}_1(\lambda^{(\pm)}_j)Q^{(\pm)}_1(\lambda^{(\pm)}_j\hspace{-0.08truecm}-\hspace{-0.08truecm}1)},\no\\
&&\quad\quad\quad\quad\quad\quad j=1,\ldots,N-2M,\label{BAE-M-H-1}\\
&&\frac{(1\hspace{-0.08truecm}-\hspace{-0.08truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
 (\mu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)(2\mu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(\mu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)^{2N}}
{(\mu^{(\pm)}_j\mp p+1)(\sqrt{1+\xi^2}(\mu^{(\pm)}_j+1)\mp q)
Q^{(\pm)}(\mu^{(\pm)}_j+1)Q_2^{(\pm)}(\mu^{(\pm)}_j)Q_2^{(\pm)}(\mu^{(\pm)}_j+1)}=-1,\no\\
&&\quad\quad\quad\quad\quad\quad j=1,\ldots,M,\label{BAE-M-H-2}\\
&&\frac{(1\hspace{-0.08truecm}-\hspace{-0.08truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
 \nu^{(\pm)}_j(2\nu^{(\pm)}_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
 {\nu^{(\pm)}_j}^{2N}}
{(\nu^{(\pm)}_j\pm p)(\sqrt{1+\xi^2}\nu^{(\pm)}_j\pm q)
Q^{(\pm)}(\nu^{(\pm)}_j-1)Q_1^{(\pm)}(\nu^{(\pm)}_j)Q_1^{(\pm)}(\nu^{(\pm)}_j-1)}=-1,\no\\
&&\quad\quad\quad\quad\quad\quad j=1,\ldots,M.\label{BAE-M-H-3}
\eea

With the help of the relation (\ref{oh}) between the Hamiltonian
(\ref{ohami}) and the transfer matrix $\tau(u)$, the energy value
$E$ of the Hamiltonian is given as \bea
&&E^{(\pm)}=2\sum_{j=1}^{N-2M}\frac{1}{\lambda^{(\pm)}_j(\lambda^{(\pm)}_j+1)}
+2\sum_{j=1}^{M}\lt(\frac{1}{\nu^{(\pm)}_j}-\frac{1}{\mu^{(\pm)}_j+1}\rt)
\no \\
&&\qquad \quad
+N-1+\frac1p+\frac{\sqrt{1+\xi^2}}{q},\label{Spectrum} \eea where
the parameters $\{\lambda^{(\pm)}_{j}\}$ and $\{\mu^{(\pm)}_{j}\}$
need to satisfy the associated  Bethe ansatz equations
(\ref{BAE-M-H-1})-(\ref{BAE-M-H-3}) respectively. It would be
believed that the eigenvalues $E^{(-)}(u)$ and $E^{(+)}(u)$ {\it
together} constitute the complete set of energy spectrum of the
Hamiltonian (\ref{ohami}).

\section{Discussions}
\label{Con} \setcounter{equation}{0}

In this paper, the open spin-$\frac{1}{2}$ XXX spin chain with
unparallel boundary fields defined by the Hamiltonian (\ref{ohami})
is exactly diagonalized by the off-diagonal Bethe ansatz method
proposed in \cite{Cao13}. There are two sets of eigenvalues
$E^{(\pm)}$ given by (\ref{Spectrum}), which are expressed in terms
of the roots of the associated Bethe ansatz equations
(\ref{BAE-M-H-1})-(\ref{BAE-M-H-3}).

As for integrable models without $U(1)$ symmetry, some off-diagonal
elements of monodromy matrix enter into expression of the transfer
matrix. This breaks down the usual $U(1)$ symmetry. However the
central idea of our method is to construct the functional relations
such as (\ref{Main}) between eigenvalues $\Lambda(\lambda)$ of the
transfer matrix (the trace of the monodromy matrix) and its quantum
determinant $\Delta_q(\lambda)$ based on the zero points of the
product of off-diagonal elements of monodromy matrix
$B(u)B(u-\eta)=0$. In fact, with some operator product identities,
we can demonstrate \footnote{We have directly proven such an operator identity in \cite{Cao13-1}.}
\begin{eqnarray}
\tau(\theta_j)\tau(\theta_j-1)=
\frac{\Delta_q(\theta_j)}{(1-2\theta_j)(1+2\theta_j)}, \quad\quad
j=1,\ldots,N,\label{Main2}
\end{eqnarray}
which is completely independent of the representation basis and thus
avoids the obstacle of absence of a reference state which is crucial
in the conventional Bethe ansatz methods. Although the functional
relations between eigenvalues $\Lambda(\lambda)$  and the quantum
determinant $\Delta_q(\lambda)$ at some particular points can be
obtained by different ways \cite{Cao13,Nic12,Fra11}, the very
relations would play an important role in the method. In our case,
based on the relation (\ref{Main}) and some properties
(\ref{Eig-Cro})-(\ref{Eigen-Anal}) of $\Lambda(\lambda)$ we can give
a generalized $T-Q$ relation type solution (\ref{T-Q}), which
modifies the usual $T-Q$ relation by adding an extra term. Such an
extra term encodes the contribution of the off-diagonal element of
the associated K-matrix.

The numerical results\cite{Nep03} strongly suggest that a fixed $M$
might give a complete set of eigenvalues of the transfer matrix. In such a sense,
different $M$ might only give different parametrization of the eigenvalues but not different
states. This allows us to simplify further the generalized $T-Q$ relation as follows.
For the case of $N$ being even, let $M=\frac{N}{2}$ and the corresponding
eigenvalue $\Lambda(u)$ can be parameterized by
\bea
\Lambda^{(\pm)}(u)&=& \bar{a}^{(\pm)}(u)\frac{Q_1^{(\pm)}(u-1)}{Q_2^{(\pm)}(u)}+
\bar{d}^{(\pm)}(u)\frac{Q_2^{(\pm)}(u+1)}{Q_1^{(\pm)}(u)}\no\\
&&+2(1\hspace{-0.12truecm}-\hspace{-0.12truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
u(u\hspace{-0.12truecm}+\hspace{-0.12truecm}1)\frac{\prod_{j=1}^N
(u\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_j)(u\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_j)
(u\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(u\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)}
{Q_1^{(\pm)}(u)Q_2^{(\pm)}(u)}. \label{T-Q-1} \eea For the case of
$N$ being odd, let $M=\frac{N+1}{2}$ and the corresponding
eigenvalue $\Lambda(u)$ can be parameterized by \bea
\Lambda^{(\pm)}(u)&=&
\bar{a}^{(\pm)}(u)\frac{Q_1^{(\pm)}(u-1)}{Q_2^{(\pm)}(u)}+
\bar{d}^{(\pm)}(u)\frac{Q_2^{(\pm)}(u+1)}{Q_1^{(\pm)}(u)}\no\\
&&+2(1\hspace{-0.12truecm}-\hspace{-0.12truecm}\sqrt{1\hspace{-0.08truecm}+\hspace{-0.08truecm}\xi^2})
u^2(u\hspace{-0.12truecm}+\hspace{-0.12truecm}1)^2\frac{\prod_{j=1}^N
(u\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_j)(u\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_j)
(u\hspace{-0.08truecm}+\hspace{-0.08truecm}\theta_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)
(u\hspace{-0.08truecm}-\hspace{-0.08truecm}\theta_j\hspace{-0.08truecm}+\hspace{-0.08truecm}1)}
{Q_1^{(\pm)}(u)Q_2^{(\pm)}(u)}. \label{T-Q-2} \eea The above $T-Q$
relations lead to the Bethe ansatz equations which allow one to
investigate the distribution of roots of these equations and compute
the physical properties in the thermodynamic limit by the usual
method \cite{Kor93}.

\section*{Acknowledgments}

The financial support from  the National Natural Science Foundation
of China (Grant Nos. 11174335, 11075126 and 11031005), the National
Program for Basic Research of MOST (973 project under grant
No.2011CB921700) and the State Education Ministry of China (Grant
No. 20116101110017 and SRF for ROCS) are gratefully acknowledged.
Two of the authors (W. Yang and K. Shi) would like to thank IoP/CAS
for the hospitality and they enjoyed during their visit there.

\section*{Appendix A: Proof of (\ref{BB-Main}) }
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}

In this appendix, we prove the relation (\ref{BB-Main}). From the definition (\ref{Mon-V-1}) of the
one-row monodromy matrix $T(u)$  and the exchange relations (\ref{Exch-1})-(\ref{Exch-4})
of its components, we have
\begin{eqnarray}
&& \alpha(\theta_j)\beta(\theta_j-1)|0\rangle=0=\delta(\theta_j)\alpha(\theta_j-1)|0\rangle, \label{usr1} \\
&& \alpha(\theta_j)\alpha(\theta_j-1)|0\rangle=0=\delta(\theta_j)\delta(\theta_j-1)|0\rangle,\label{usr2}\\
&& \gamma(\theta_j)\beta(\theta_j-1)|0\rangle=-\tilde a(\theta_j)\tilde d(\theta_j-1)|0\rangle, \label{usr3}\\
&& \delta(\theta_j)\beta(\theta_j-1)|0\rangle=-\tilde d(\theta_j-1)\b(\theta_j)|0\rangle, \label{usr4} \\
&& \alpha(\theta_j)\delta(\theta_j-1)|0\rangle=\tilde a(\theta_j)\tilde d(\theta_j-1)|0\rangle, \label{usr5} \\
&& \beta(\theta_j)\beta(\theta_j-1)|0\rangle=0,  \label{usr6}
\end{eqnarray}
where
\begin{eqnarray}
\tilde a(\lambda)=\prod_{j=1}^N(\lambda-\theta_j+1), \quad \tilde
d(\lambda)=\prod_{j=1}^N(\lambda-\theta_j).
\end{eqnarray}
One can check the last equation (\ref{usr6}) by induction. Expanding the operator $B(u)$ in terms of the components of the
one-row monodromy matrix (\ref{B-operator}) and using the relations (\ref{usr1})-(\ref{usr6}), we have
\begin{eqnarray}
B(\theta_j)B(\theta_j-1)|0\rangle&=&\lt\{-(p+\theta_j)\alpha(\theta_j)\beta(-\theta_j-1)
+(p-\theta_j)\beta(\theta_j)\alpha(-\theta_j-1)\rt\}\nonumber \\
&&\times\lt\{-(p+\theta_j-1)\alpha(\theta_j-1)\beta(-\theta_j)+(p-\theta_j+1)\beta(\theta_j-1)\alpha(-\theta_j)\rt\}|0\rangle\nonumber \\
&=&\frac{(2\theta_j+1)(\theta_j-p+1)(2p+(\theta_j-p-1)2\theta_j)}{(\theta_j+1)(2\theta_j-1)}\nonumber \\
&& \times \tilde a(-\theta_j)
\tilde a(-\theta_j-1)\beta(\theta_j)\beta(\theta_j-1)|0\rangle\nonumber \\
&=&0.
\end{eqnarray}
Due to the fact that for a generic value of  $\{u_j\}$ the set of vectors $\lt\{\prod_{j=1}^M B(u_j)|0\rangle|,M=0,\ldots,N\rt\}$ spans the whole
vector space ${\rm\bf V}^{\otimes N}$ and the fact that the commutativity between the operators $B$ with different spectrum, one can conclude that
$B(\theta_j)B(\theta_j-1)=0$. This completes the proof of (\ref{BB-Main})\footnote{One may also check that the operator $B(u)B(u-1)$,
when $u=\theta_j$, becomes zero by using the very properties (\ref{Int-R}) and (\ref{Ant}) of the R-matrix and QYBE and RE relations.}.