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text:
\section{Introduction}
\setcounter{equation}{0}
\renewcommand{\theequation}{1.\arabic{equation}}

There have been much interests in
strongly correlated electron systems.
Among the exactly solvable ones,
the one-dimensional (1D) Hubbard model,
\begin{align}
\mathcal{H} = - \sum_{j=1}^N \sum_{\sigma = \uparrow \downarrow}
       (c_{j+1 \sigma}^{\dagger} c_{j \sigma}
                  + c_{j \sigma}^{\dagger} c_{j+1 \sigma})
       + U \sum_{j=1}^N
    (n_{j \uparrow}-\frac{1}{2})(n_{j \downarrow}-\frac{1}{2}) .
\label{hamiltonian.hubbard}
\end{align}
is the most interesting model.
Here $c^{\dagger}_{j \sigma}$ and $c_{j \sigma}$
are the fermion creation and annihilation operators
and $n_{j \sigma}$ is the fermion number operator,
\begin{align}
n_{j \sigma} = c^{\dagger}_{j \sigma} c_{j \sigma}.
\end{align}
The parameter ${U}$ is the coupling constant
describing the Coulomb interaction between electrons
with opposite spins on the same site.
Lieb and Wu \cite{lieb} diagonalized the Hamiltonian
(\ref{hamiltonian.hubbard}) by means of
the coordinate Bethe ansatz method
under the periodic boundary condition (PBC),
\begin{align}
 c^{\dagger}_{N+1, \sigma} = c^{\dagger}_{1 \sigma},
 \hspace{5mm}
 c_{N+1, \sigma} = c_{1 \sigma},
 \hspace{2cm}               \sigma = \uparrow \downarrow.
\label{pbcfermion}
\end{align}
The bulk properties of the 1D Hubbard model have been investigated
by analyzing the associated Bethe ansatz equations. \cite{korepin}

The integrability of the 1D Hubbard model has been studied
by Shastry \cite{shastry1,shastry2,shastry3}
and Wadati {\it et al.}. \cite{wadati1,wadati2,wadati3}
Shastry introduced the Jordan-Wigner transformation,\cite{jordan}
\begin{align}
   c_{j \uparrow} =& \exp \Bigl\{
         \mbox{i} \pi
            \sum_{k=1}^{j-1} (n_{k \uparrow} - 1) \Bigr\}
            \sigma_j^-,
\nonumber\\
   c_{j \downarrow} =& \exp \Bigl\{
         \mbox{i} \pi
            \sum_{k=1}^{j-1} (n_{k \downarrow} - 1) \Bigr\}
                       \exp \Bigl\{
         \mbox{i} \pi
            \sum_{k=1}^{N} (n_{k \uparrow} - 1) \Bigr\}
            \tau_j^- ,
\label{jwtransf}
\end{align}
to change the fermionic Hamiltonian (\ref{hamiltonian.hubbard})
into an equivalent coupled spin model,
\begin{align}
H = \sum_{j=1}^N (\sigma_{j+1}^+ \sigma_j^- + \sigma_j^+
                  \sigma_{j+1}^-)
    + \sum_{j=1}^N (\tau_{j+1}^+ \tau_j^- + \tau_j^+
                    \tau_{j+1}^-)
    + \dfrac{U}{4} \sum_{j=1}^N \sigma_j^z \tau_j^z,
\label{coupledspin}
\end{align}
where ${\sigma}$  and ${\tau}$ are two species
of the Pauli matrices commuting each other, and
\begin{align}
\sigma_j^{\pm} = \frac{1}{2} (\sigma_j^x \pm {\rm i}
\sigma_j^y),
\hspace{5mm}
\tau_j^{\pm} = \frac{1}{2} (\tau_j^x \pm {\rm i} \tau_j^y).
\end{align}
Shastry constructed the $L$-operator and $R$-matrix
which satisfy the Yang-Baxter relation,
\begin{align}
R_{12}(u_1,u_2) L_{a1}(u_1) L_{a2}(u_2)
 = L_{a2}(u_2) L_{a1}(u_1) R_{12}(u_1,u_2).
\label{spinyangr}
\end{align}
In ref. \cite{shastry3}, the decorated Yang-Baxter equation (DYBE)
for the free-fermion model was used to prove the Yang-Baxter relation
for the coupled spin model, (\ref{spinyangr}). It is a relation similar to,
but not equivalent to the Yang-Baxter equation (YBE),
Recently Shiroishi and Wadati \cite{shiroishi1,shiroishi2}
found that the YBE and the DYBE for the free-fermion model
can be generalized to a larger set of relations among
the $L$-operators called tetrahedral Zamolodchikov algebra (TZA).
\cite{korepanov}
By use of the TZA, the Yang-Baxter equation for the $R$-matrix,
\begin{align}
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)
\end{align}
can be proved. \cite{shiroishi1,shiroishi2,shiroishi3}

The coupled spin model (\ref{coupledspin})
is often referred to as the 1D Hubbard model,
since it is related to the fermionic Hamiltonian
(\ref{hamiltonian.hubbard}) through
the Jordan-Wigner transformation (\ref{jwtransf}).
However, because of the non-locality of the Jordan-Wigner
 transformation (\ref{jwtransf}), the PBC for the fermion model
(\ref{pbcfermion}) does not correspond to the PBC for
the coupled spin model.
Moreover the ${SO(4)}$ symmetry \cite{heilmann,yang1,yang2,pernici,affleck},
which is one of the most important properties
of the Hamiltonian (\ref{hamiltonian.hubbard}),
is not clear in the coupled spin model
(\ref{coupledspin}).

To discuss the fermionic Hamiltonian directly,
we can employ the fermionic formulation of
the Yang-Baxter relation
(graded Yang-Baxter relation). \cite{wadati2}
In this formulation, the transfer matrix
that respects the PBC (\ref{pbcfermion}) can be
constructed by taking the supertrace of the monodromy matrix.
Recently it has been shown that the transfer matrix
naturally enjoys the ${SO(4)}$ symmetry.
\cite{murakami,shiroishi4}
Based on this fermionic formulation,
the algebraic Bethe ansatz method has been developed
by Ramos and Martins. \cite{ramos,martins}

The aim of this paper is to present yet another approach
to discuss the integrability of the 1D Hubbard model
using the fermionic ${R}$-operator.
The notion of the fermionic ${R}$-operator was introduced
in relation to the ${XXZ}$ fermion model.\cite{destri,umeno}
The fermionic ${R}$-operator consists of the fermion operators
and is a solution of the YBE.

In this paper, we first consider the ${XYZ}$ fermion model.
The fermionic ${R}$-operator for the ${XYZ}$ fermion model,
which satisfies the YBE, is explicitly obtained. The YBE ensures
the commutativity of the transfer operators. The logarithmic
derivative of the transfer operator yields the Hamiltonian of
the ${XYZ}$ fermion model under the PBC.

Next we consider the DYBE for the fermionic ${R}$-operator
and find that the fermionic ${R}$-operator related to
the ${XY}$ fermion  model fulfills the DYBE.
Combining the YBE and the DYBE of the fermionic ${R}$-operator
for the ${XX}$ fermion models,
we construct a new fermionic ${R}$-operator and an ${L}$-operator
for the 1D Hubbard model. The transfer operator constructed
from the ${L}$-operator generates the Hamiltonian and
other conserved operators under the PBC.

We also prove the YBE
for the new integrable ${R}$-operator
by considering the fermionic variant of
the tetrahedral Zamolodchikov algebra (TZA).
Then we consider the generalized transfer operator which is
constructed from the fermionic ${R}$-operator.
A new fermionic Hamiltonian is obtained from the transfer
operator.

It is possible to discuss the ${SO(4)}$ symmetry of
the 1D Hubbard model using the fermionic ${R}$-operator.
The (anti) commutation relations between the fermionic ${R}$-operator
and the generators of the ${SO(4)}$ algebra are obtained.
Those relations are extended to the  monodromy operator and
the transfer operator. We can show the ${SO(4)}$ invariance
of the local conserved operators,
though the transfer operator itself is not ${SO(4)}$ invariant.

The outline of this paper is as follows.
In \S 2, we study the Yang-Baxter equation (YBE)
for the ${XYZ}$ fermion model.
Fermionic ${R}$-operator related to the ${XYZ}$ fermion model is
explicitly shown by use of Baxter's parameterization.
In \S 3, the decorated Yang-Baxter equation (DYBE)
for the fermionic ${R}$-operator is introduced.
It is necessary to impose the free-fermion condition
on the Boltzmann weights in order that the DYBE is fulfilled.
Some connections to the ${XY}$ fermion model
with an external field are also discussed.
In \S 4, we discuss the integrability of the 1D Hubbard model.
The fermionic ${R}$-operator for the 1D Hubbard model
is constructed from  the ${R}$-operators
related to the ${XX}$ fermion models with spin up and down.
The YBE and the DYBE are utilized to show the Yang-Baxter relation,
where a new ${L}$-operator is also introduced.
 In \S 5, we prove the YBE
for the fermionic ${R}$-operator of the Hubbard model.
We use the fermionic variant of the TZA. The discussions proceed
just in parallel with the case of the coupled spin model
(\ref{coupledspin}). We also use a generalized transfer
operator which consists of the fermionic ${R}$-operators.
In \S 6, we give a new approach
to the ${SO(4)}$ symmetry of
the 1D Hubbard model. The fermionic ${R}$-operator has a natural
${SO(4)}$ symmetry. We show that all the local fermionic
conserved operators derived from the transfer operator
are ${SO(4)}$ invariant.
The last section is devoted to the concluding remarks.

\section{Fermionic ${R}$-Operator for the ${XYZ}$ Fermion Model}
\setcounter{equation}{0}
\renewcommand{\theequation}{2.\arabic{equation}}
It is well understood that the Yang-Baxter equation (YBE)
\begin{align}
\mathcal{R}_{12}(u-v) \mathcal{R}_{13}(u) \mathcal{R}_{23}(v)
= \mathcal{R}_{23}(v) \mathcal{R}_{13}(u) \mathcal{R}_{12}(u-v),
\label{str}
\end{align}
plays the most important role in the quantum integrable models.
\cite{baxter,kulish,sklyanin,korepin2,gomez}

We look for a solution of the YBE (\ref{str}) in the form
\begin{align}
       \mathcal{R}_{jk}(u) =& a(u) \{- n_j n_k + (1-n_j)(1-n_k)\}
           + b(u) \{n_j (1-n_k) + (1-n_j) n_k\}
\nonumber\\
           +  & c(u) (c_j^{\dagger} c_k + c_k^{\dagger} c_j)
           - d(u) (c_j^{\dagger} c_k^{\dagger} + c_j c_k),
\label{R}
\end{align}
where ${c_{j}^{\dagger}}$ and ${c_{j}}$ are the fermion creation
annihilation operators satisfying the canonical anti-commutation
relations
\begin{align}
\big\{ c_{j}, c_{k} \big\} =
\big\{ c_{j}^{\dagger}, c_{k}^{\dagger} \big\} = 0,\hspace{1cm}
\big\{ c_{j}^{\dagger}, c_{k} \big\} = \delta_{jk},
\label{anticommu}
\end{align}
and ${n_{j} = c_{j}^{\dagger} c_{j}}$.

Substituting (\ref{R}) into (\ref{str}) and using (\ref{anticommu}),
we get the following relations among the functions ${a(u)}$,
${b(u)}$, ${c(u)}$ and
${d(u)}$,
\begin{align}
a(u-v)c(u)a(v) + d(u-v)a(u)d(v) &= b(u-v)c(u)b(v) + c(u-v)a(u)c(v),
\nonumber\\
a(u-v)b(u)c(v) + d(u-v)d(u)b(v) &= b(u-v)a(u)c(v) + c(u-v)c(u)b(v),
\nonumber\\
c(u-v)b(u)a(v) + b(u-v)d(u)d(v) &= c(u-v)a(u)b(v) + b(u-v)c(u)c(v),
\nonumber\\
a(u-v)d(u)b(v) + d(u-v)b(u)c(v) &= b(u-v)d(u)a(v) + c(u-v)b(u)d(v),
\nonumber\\
a(u-v)a(u)d(v) + d(u-v)c(u)a(v) &= b(u-v)b(u)d(v) + c(u-v)d(u)a(v),
\nonumber\\
d(u-v)a(u)a(v) + a(u-v)c(u)d(v) &= d(u-v)b(u)b(v) + a(u-v)d(u)c(v).
\label{baxterr}
\end{align}
These are nothing but the relations among the Boltzmann weights of the 8-vertex model (Baxter model). \cite{baxter} Then, using Baxter's parameterization,
we can solve (\ref{baxterr}) as follows,
\begin{align}
a(u) &= \sn (u+2\eta),
\hspace{5mm}
b(u) = \sn (u),
\hspace{5mm}
c(u) = \sn (2\eta),
\nonumber\\
d(u) &= k a(u) b(u) c(u) = k \sn (2\eta) \sn (u) \sn (u+2\eta),
\label{baxterpara}
\end{align}
where ${\sn u= \sn (u,k)}$ is Jacobi's elliptic function with the modulus ${k}$ and ${\eta}$ is the anisotropy parameter.

Thus we have found a new type of solution for the YBE (\ref{str}).
We call (\ref{R}) the fermionic ${R}$-operator (related
to the ${XYZ}$ fermion model). We remark that Destri and Segalini
\cite{destri} have discussed the special case (${k=0}$) of (\ref{R}).
The fermionic $R$-operator (\ref{R}) can also be obtained from the
${R}$-matrix of the ${XYZ}$ spin model (Appendix A).

It is remarkable that $\mathcal{R}_{jk}(u)$ has a property,
\begin{align}
\mathcal{R}_{jk}(0) = \sn 2 \eta \mathcal{P}_{jk},
\label{regularity}
\end{align}
where ${\mathcal{P}_{jk}}$ is the permutation operator,
\begin{align}
&\mathcal{P}_{jk} = 1 - (c_{j}^{\dagger} - c_{k}^{\dagger})
(c_{j} - c_{k}) = 1 - n_{j} - n_{k} + c_{j}^{\dagger} c_{k}
+ c_{k}^{\dagger} c_{j}, \nonumber \\
&\mathcal{P}_{jk} c_j = c_k \mathcal{P}_{jk}, \ \ \ \
\mathcal{P}_{jk} c_j^{\dagger} = c_k^{\dagger} \mathcal{P}_{jk}.
\label{fermionicpermutation}
\end{align}
This property is useful when we derive a related Hamiltonian
from the transfer operator.

Unlike the ${R}$-matrix of the usual ${XYZ}$ spin model,
the fermionic ${R}$-operator is not symmetric, ${\mathcal{R}_{12}(u)
\ne \mathcal{R}_{21}(u)}$. Instead, the following relation holds
\begin{align}
&\mathcal{R}_{12}(u;k) = \mathcal{R}_{21}(u;-k),
\label{modulusinverse}
\end{align}
where we write the modulus ${k}$ dependence explicitly.
The fermionic ${R}$-operator (\ref{R}) also has the unitarity
relation in the form
\begin{align}
\mathcal{R}_{12}(u) \mathcal{R}_{21}(-u) =
\left( - \sn^{2} u + \sn^{2} 2 \eta \right) {\bf 1},
\end{align}
as well as the property
\begin{align}
\left[ \mathcal{R}_{jk}(u), (1 - 2 n_{j})(1 - 2 n_{k}) \right] = 0.
\label{generalproperty}
\end{align}

Now we introduce the monodromy operator
\begin{align}
\mathcal{T}_{a}(u) = \mathcal{R}_{aN}(u) \ldots \mathcal{R}_{a1}(u),
\label{monodromy}
\end{align}
where the suffix {\it a} means the auxiliary space,
The transfer operator is given by the ``supertrace" of the monodromy
operator
\begin{align}
\tau(u) &= \fstr_{a} \mathcal{T}_{a}(u).
\label{transfer}
\end{align}
Here we define the supertrace  ${\fstr}$ as
\begin{align}
\fstr_{a} X = {}_{a} \langle 0 | X | 0 \rangle_{a}
         - {}_{a} \langle 1 | X | 1 \rangle_{a},
\label{fstr}
\end{align}
with the auxiliary fermion Fock space being
\begin{align}
c_{a} | 0 \rangle_{a} = 0,
\hspace{5mm}
| 1 \rangle_{a} = c_{a}^{\dagger} | 0 \rangle_{a},
\hspace{5mm} {}_a \langle 0 | = ( | 0 \rangle_{a})^{\dagger},
\hspace{5mm} {}_{a} \langle 1 | = {}_{a} \langle 0 | c_{a},
\end{align}
\begin{align}
{}_a \langle  0 | 0 \rangle_a
= {}_a \langle  1 | 1 \rangle_a=1.
\end{align}
When we write the monodromy operator ${\mathcal{T}_{a}(u)}$ as
\begin{align}
\mathcal{T}_{a}(u) = A(u) (1 - n_{a}) + B(u) c_{a} + C(u) c_{a}^{\dagger}
+ D(u) n_{a},
\label{ABCD}
\end{align}
 the transfer operator (\ref{transfer}) is given by
\begin{align}
\tau(u) = A(u) - D(u).
\label{AD}
\end{align}

The YBE for the fermionic ${R}$-operator leads to the global
Yang-Baxter relation for the monodromy operator
(\ref{monodromy}),
\begin{align}
             \mathcal{R}_{12}(u-v)
     \mathcal{T}_1(u) \mathcal{T}_2(v)
           &=      \mathcal{T}_2(v) \mathcal{T}_1(u)
     \mathcal{R}_{12}(u-v).
\label{gyangrxyz}
\end{align}
Rewriting (\ref{gyangrxyz}) as
\begin{align}
             \mathcal{R}_{12}(u-v)
     \mathcal{T}_1(u) \mathcal{T}_2(v) \mathcal{R}_{12}^{-1} (u-v)
           &=      \mathcal{T}_2(v) \mathcal{T}_1(u),
\end{align}
and taking the supertrace (\ref{fstr}) of both sides,
we can show the commutativity
of the transfer operators
\begin{align}
\left[ \tau(u), \tau(v) \right] = 0,
\label{trans}
\end{align}
which ensures the existence of enough number of the fermionic
conserved operators.
In this way we can establish the integrability of the
fermion model by means of the fermionic ${R}$-operator.

Using (\ref{regularity}) in (\ref{transfer}) with (\ref{monodromy}), we have
\begin{align}
\tau(0) &= \left( \sn 2 \eta \right)^{N} \ \fstr_{a} (\mathcal{P}_{aN} \ldots
                      \mathcal{P}_{a2} \mathcal{P}_{a1}) \nonumber \\
        &= \left( \sn 2 \eta \right)^{N} \ \mathcal{P}_{12} \mathcal{P}_{23}
                          \ldots \mathcal{P}_{N-1,N},
\end{align}
where we have used some fundamental properties of the permutation operators
\begin{align}
&\mathcal{P}_{aj} \mathcal{P}_{ak} = \mathcal{P}_{jk} \mathcal{P}_{aj},
\hspace{1cm} \mathcal{P}_{jk} = \mathcal{P}_{kj}, \nonumber \\
&\fstr_{a} \mathcal{P}_{aj} = 1.
\end{align}
Thus we find that ${\tau(0)}$ is proportional to the (left) shift
operator ${\hat{U}}$ for fermions,
\begin{align}
\hat{U} = \mathcal{P}_{12} \mathcal{P}_{23}
                          \ldots \mathcal{P}_{N-1,N} .
\end{align}
It is easy to confirm that the shift operator ${\hat{U}}$
acts on the fermion operators as
\begin{align}
&\hat{U} c_{j} = c_{j+1} \hat{U}, \hspace{3cm} j=1, \ldots, N-1,
\nonumber \\
&\hat{U} c_{N} = c_{1} \hat{U}.
\end{align}

In a similar way as the spin models, the Hamiltonian of the ${XYZ}$
fermion model is derived from the logarithmic derivative of
the transfer operator
\begin{align}
\mathcal{H}^{\rm XYZ} &=  \tau(0)^{-1} \frac{\rm d}{{\rm d} u}
\tau(u) \Big|_{u=0}  \nonumber \\
&= \sum_{j=1}^N  \mathcal{H}_{j,j+1}^{\rm XYZ},
\label{xyz}
\end{align}
where the two-point Hamiltonian density is given by
\begin{align}
\mathcal{H}_{j,j+1}^{\rm XYZ}
     =& \frac{1}{\sn 2 \eta} \Big\{ a'(0) \left( n_{j} n_{j+1}
                                 + (1-n_{j})(1-n_{j+1}) \right)
\nonumber\\
     & \hspace{5mm} + b'(0) ( c_{j}^{\dagger} c_{j+1} +
                              c_{j+1}^{\dagger} c_{j} )
     + d'(0) ( c_{j}^{\dagger} c_{j+1}^{\dagger}
               - c_{j} c_{j+1}) \Big\}.
\label{XYZhamiltonian}
\end{align}
Here and hereafter, ${'}$ means the derivative with respect to the spectral
parameter ${u}$,
\begin{align}
a'(0) = \cn 2 \eta \ \dn 2 \eta,
\hspace{5mm} b'(0) = 1, \hspace{5mm}
d'(0) = k \ \sn^{2} 2 \eta.
\end{align}
In (\ref{xyz}), we have used a fact that $\tau(0)^{-1}$
is proportional to the shift operator.
In terms of an equivalent ${R}$-operator
${\check{\mathcal{R}}_{jk}(u) \equiv \mathcal{P}_{jk} %
\mathcal{R}_{jk}(u)}$ (c.f. Appendix A),
the two-point Hamiltonian density is expressed as
\begin{align}
\mathcal{H}_{jk}^{\rm XYZ} = \dfrac{\rm d}{{\rm d} u}
                             \check{R}_{jk}(u) \Big|_{u=0}.
\end{align}

One may wonder if  our discussion using the fermionic ${R}$-operator
goes fully in parallel with the standard treatment of the integrable
spin models and no new aspect appears. In fact we can proceed the discussion almost in the same way as the spin models. However we like to point out an important difference between
the spin model and the fermion model here.
While the Hamiltonian (\ref{XYZhamiltonian}) is hermitian,
it is not invariant under the space inversion
\begin{align}
j \longrightarrow N-j+1.
\label{inversion}
\end{align}
It is easy to see that the modulus ${k}$
in the Hamiltonian (\ref{XYZhamiltonian})
changes its sign under the transformation (\ref{inversion}).
This point makes a difference between the ${XYZ}$ spin model
and the ${XYZ}$ fermion model. Actually it is a consequence of
the property (\ref{modulusinverse}).

In the case of $k=0$, the fermionic ${R}$-operator (\ref{R})
reduces to the one with
\begin{align}
a(u) &= \sin (u+2\eta)
\hspace{5mm}
b(u) = \sin u
\hspace{5mm}
c(u) = \sin 2\eta
\hspace{5mm}
d(u) = 0,
\end{align}
which we have used in ref. \cite{umeno} to discuss
 the integrability of the ${XXZ}$ fermion model
\begin{align}
\mathcal{H}^{\rm XXZ}
     =& \frac{1}{\sin 2 \eta} \sum_{j=1}^{N}
 \Big\{  c_{j}^{\dagger} c_{j+1} + c_{j+1}^{\dagger} c_{j}
        + \cos 2 \eta \left( n_{j} n_{j+1} + (1-n_{j})(1-n_{j+1})
\right) \Big\}.
\label{XXZHamiltonian}
\end{align}
In this case, the fermionic ${R}$-operator becomes symmetric
and the Hamiltonian
 (\ref{XXZHamiltonian}) is invariant under
the space inversion (\ref{inversion}).

We note that the periodic boundary condition (PBC)
for the fermion operators is satisfied in (\ref{XYZhamiltonian}),
\begin{align}
c_{N+1} = c_{1}, \hspace{1cm} c_{N+1}^{\dagger} = c_{1}^{\dagger}.
\end{align}
It is sometimes necessary to consider the anti-periodic
boundary condition (APBC) for the fermion operators
\begin{align}
c_{N+1} = - c_{1}, \hspace{1cm} c_{N+1}^{\dagger} = - c_{1}^{\dagger}.
\label{apbc}
\end{align}
To explain this, we consider the following twisted monodromy operator
\begin{align}
\mathcal{\tilde{T}}_{a}(u) &= (1 - 2 n_{a}) \mathcal{T}_{a}(u) \nonumber \\
                           &= (1 - 2 n_{a}) \mathcal{R}_{aN}(u)
\ldots \mathcal{R}_{a1}(u).
\label{twistedmonodromy}
\end{align}
Then, thanks to the property (\ref{generalproperty}),
the global Yang-Baxter relation
for the twisted monodromy operator
 holds,
\begin{align}
             \mathcal{R}_{12}(u-v)
     \mathcal{\tilde{T}}_1(u) \mathcal{\tilde{T}}_2(v)
           &= \mathcal{\tilde{T}}_2(v) \mathcal{\tilde{T}}_1(u)
\mathcal{R}_{12}(u-v),
\end{align}
which leads to the commutativity of
the twisted transfer operator ${\tilde{\tau}(u) = \fstr_{a}
\mathcal{\tilde{T}}_a(u)}$,
\begin{align}
\left[ \tilde{\tau}(u), \tilde{\tau}(v) \right] = 0.
\end{align}
We can show that the twisted transfer operator generates
the Hamiltonian (\ref{XYZhamiltonian}) under the APBC
(\ref{apbc}). In fact the logarithmic derivative of
${\tilde{\tau}(u)}$ yields a Hamiltonian
\begin{align}
\mathcal{\tilde{H}}^{\rm XYZ}
= & \tilde{\tau}(0)^{-1} \frac{\rm d}{{\rm d} u}
\tilde{\tau}(u) \Big|_{u=0}  \nonumber \\
= & \sum_{j=1}^{N-1}  \mathcal{H}_{j,j+1}^{\rm XYZ}
    + \frac{1}{\sn 2 \eta} \Big\{ a'(0) \left( n_{N} n_{1}
                                 + (1-n_{N})(1-n_{1}) \right)
\nonumber\\
     & \hspace{3.5cm} - b'(0) ( c_{N}^{\dagger} c_{1} +
                              c_{1}^{\dagger} c_{N} )
     - d'(0) ( c_{N}^{\dagger} c_{1}^{\dagger}
               - c_{N} c_{1}) \Big\}.
\end{align}
Here we have used a relation
\begin{align}
(1 - 2 n_{N}) \fstr_{a} \left\{ (1 - 2 n_{a}) \mathcal{P}_{aN}
\mathcal{H}_{a1}^{XYZ} \right\}
= (1 - 2 n_{N}) \mathcal{H}_{N1}^{XYZ} (1 - 2 n_{N}),
\end{align}
as well as the identities
\begin{align}
( 1 - 2 n_{N}) c_{N} (1 - 2 n_{N}) &= - c_{N}, \hspace{5mm}
( 1 - 2 n_{N}) c_{N}^{\dagger} (1 - 2 n_{N}) = - c_{N}^{\dagger}.
\end{align}
Note that if we express ${\mathcal{T}_a(u)}$ as (\ref{ABCD}),
the twisted transfer operator takes the following form,
\cite{destri}
\begin{align}
\tilde{\tau}(u) = A(u) + D(u).
\end{align}

\section{Decorated Yang-Baxter Equation}
\setcounter{equation}{0}
\renewcommand{\theequation}{3.\arabic{equation}}
The decorated Yang-Baxter equation (DYBE) is an algebraic relation
similar to the YBE (\ref{str}), but not equivalent to the YBE.
Shastry \cite{shastry3} showed that the free-fermion condition on
the Boltzmann weights is necessary
so that the solution of the YBE also satisfies
the DYBE. \cite{shastry3,shiroishi2}
The DYBE is considered to be a hidden algebraic structure of
the free-fermion model (equivalently, ${XY}$ model).

We discuss the DYBE in the context of the fermionic $R$-operator.
The DYBE for the fermionic ${R}$-operator is defined by
\begin{align}
\mathcal{R}_{12}(u+v) (2n_1-1) \mathcal{R}_{13}(u) \mathcal{R}_{23}(v)
= \mathcal{R}_{23}(v) \mathcal{R}_{13}(u) (2n_1-1) \mathcal{R}_{12}(u+v).
\label{dstr}
\end{align}
The DYBE (\ref{dstr}) looks alike the YBE (\ref{str}), but the
operator ${(2n_{1}-1)}$ inserted in both sides
can not be absorbed by the redefinition of ${R_{jk}(u)}$.
It is possible to simplify the DYBE by use of the YBE as follows.

First,
by multiplying ${(2n_1-1)(2n_2-1)}$ to the
YBE (\ref{str})
from the left and $(2n_3-1)$ from the right,
we get a relation
\begin{align}
&\mathcal{R}_{12}(u-v) (2n_1-1) \mathcal{R}_{13}(u)
(2n_2-1) \mathcal{R}_{23}(v) (2n_3-1) \nonumber \\
&= (2n_2-1) \mathcal{R}_{23}(v) (2n_3-1)
\mathcal{R}_{13}(u) (2n_1-1) \mathcal{R}_{12}(u-v),
\label{modSTR}
\end{align}
where we have used a general property (\ref{generalproperty})
and an equivalent relation,
\begin{align}
(2n_j-1) \mathcal{R}_{jk}(u) (2n_k-1) =
(2n_k-1) \mathcal{R}_{jk}(u) (2n_j-1).
\end{align}
Then comparing eq.\ (\ref{modSTR}) with the definition of
the DYBE (\ref{dstr}),
we find a condition
\begin{align}
(2n_j-1) &\mathcal{R}_{jk}(u) (2n_k-1)
    = \mathcal{R}_{jk}(-u).
\label{condition}
\end{align}
Hence we see that the DYBE is equivalent to the condition (\ref{condition})
when we have the YBE.
In terms of the Boltzmann weights,
the condition (\ref{condition}) is rephrased as
\begin{align}
a(u) = a(-u), \hspace{5mm} -b(u) = b(-u), \hspace{5mm}
c(u) = c(-u), \hspace{5mm} -d(u) = d(-u).
\label{condition2}
\end{align}
Using the explicit parameterization (\ref{baxterpara}),
we find that the condition (\ref{condition2}) corresponds to
\begin{align}
2 \eta = K,\ \  3K,
\end{align}
where ${K}$ is the complete elliptic
integral of the first kind.
In this case, the free-fermion condition
\begin{align}
a^2(u) + b^2(u) = c^2(u) + d^2(u),
\label{FFcond}
\end{align}
is fulfilled. Thus as in the case of the spin models,
we have shown that the free-fermion condition is required
for the validity of the DYBE (\ref{dstr}).

Hereafter we restrict our consideration to the case
${2 \eta = K}$.
Then, from (\ref{baxterpara}), the Boltzmann weights
${a(u),b(u),c(u)}$
and ${d(u)}$ are explicitly given as
\begin{align}
&a(u) = \cd u, \hspace{5mm} b(u)= \sn u,
\hspace{5mm} c(u)=1,
\nonumber \\
&d(u) = k \cd u \ \sn u,
\label{XYBoltzmann}
\end{align}
where ${\cd u = \cn u / \dn u}$.
The corresponding Hamiltonian is the ${XY}$ fermion model
\begin{align}
\mathcal{H}^{\rm XY}
= \sum_{j=1}^{N} \Big\{
c_{j}^{\dagger} c_{j+1} + c_{j+1}^{\dagger} c_{j}
+ k \left( c_{j}^{\dagger} c_{j+1}^{\dagger} -
 c_{j} c_{j+1} \right) \Big\}.
\label{XYhamiltonian}
\end{align}
In the special case $k=0$, this Hamiltonian reduces
to the $XX$ fermion model
\begin{align}
\mathcal{H}^{\rm XX}
     = \sum_{j=1}^{N} ( c_{j}^{\dagger} c_{j+1} +
c_{j+1}^{\dagger} c_{j}).
\label{XXhamiltonian}
\end{align}

As an interesting application of the YBE (\ref{str})
and the DYBE (\ref{dstr}),
we shall prove the integrability of the ${XY}$ fermion model
in an external field,
\begin{align}
\mathcal{H}^{\rm XY \mu} = \sum_{j=1}^{N}
  \Big\{ c_{j}^{\dagger} c_{j+1} + c_{j+1}^{\dagger} c_{j}
  + k \left( c_{j}^{\dagger} c_{j+1}^{\dagger} - c_{j} c_{j+1} \right)
+ \mu \left( n_{j} - \frac{1}{2} \right) \Big\}.
\label{XYfield}
\end{align}
The external field ${\mu}$ can be regarded as the chemical
potential for the ${XY}$ fermion model.
Since the term
${\sum_{j=1}^{N} (n_{j} - 1/2)}$ does not commute with
the Hamiltonian (\ref{XYhamiltonian}) unless ${k=0}$,
the addition of the extra term is not trivial at all.
It is well known that the model (\ref{XYfield}) can be
diagonalized
by means of the Fourier transformation through the Bogoliubov
transformation.
 \cite{lieb2} The case ${k=\pm 1}$
corresponds to the transverse Ising model. \cite{pfeuty}

The following argument is in common with the one
explored for the spin models. \cite{shastry1,shiroishi2}
From now on, let  ${\mathcal{R}_{jk}(u)}$ denote
the fermionic ${R}$-operator
for the ${XY}$ fermion model.
Taking a linear combination of the YBE (\ref{str}) and
the DYBE (\ref{dstr}), we have
\begin{align}
& \Big\{ \alpha \mathcal{R}_{12}(u-v)
  + \beta \mathcal{R}_{12}(u+v)(2n_1-1) \Big\}
     \mathcal{R}_{13}(u) \mathcal{R}_{23}(v)
\nonumber\\
&= \mathcal{R}_{23}(v) \mathcal{R}_{13}(u)
     \Big\{ \alpha \mathcal{R}_{12}(u-v)
           + \beta (2n_1-1) \mathcal{R}_{12}(u+v) \Big\},
\label{com}
\end{align}
where ${\alpha}$ and ${\beta}$ are not operators but constants.

Now we look for a solution of the  Yang-Baxter relation
\begin{align}
         \mathcal{R}^{\mu}_{12}(u,v)
         \mathcal{L}_{13} (u)
         \mathcal{L}_{23} (v)
    &=    \mathcal{L}_{23} (v)
         \mathcal{L}_{13} (u)
         \mathcal{R}^{\mu}_{12} (u,v),
\label{yang}
\end{align}
assuming the ${R}$-operator ${\mathcal{R}^{\mu}_{12}(u,v)}$
and the ${L}$-operator ${\mathcal{L}_{jk} (u)}$ in the form,
\begin{align}
&\mathcal{R}^{\mu}_{12}(u,v)  = \alpha \mathcal{R}_{12}(u-v)
  + \beta \mathcal{R}_{12}(u+v)(2n_1-1), \label{Rxy} \\
&\mathcal{L}_{jk}(u) = \mathcal{R}_{jk}(u)
\exp \left\{ h(u) ( 2n_{j}-1)
\right\}.
\label{Lxy}
\end{align}
Here ${\alpha,\beta}$ and ${h(u)}$ are functions
of the spectral parameters to be specified later.

Comparing eq.\ (\ref{yang}) with eq.\ (\ref{com}),
we get a relation
\begin{align}
&\mathcal{I}_1(u) \mathcal{I}_2(v)
 \mathcal{R}^{\mu}_{12}(u,v) \mathcal{I}_1(u)^{-1}
  \mathcal{I}_2(v)^{-1}
 = \alpha \mathcal{R}_{12}(u-v) +
   \beta (2n_1-1)\mathcal{R}_{12}(u+v),
\label{condxy}
\end{align}
where
\begin{align}
\mathcal{I}_j(u) = \exp\{h(u) (2n_j-1)\}.
\end{align}

For $k \neq 0$ (that is, the genuine $XY$ fermion model), the
relation (\ref{condxy}) gives two conditions,
\begin{align}
  \dfrac{\beta}{\alpha} = \tanh (h(u)-h(v))
\label{ratio1}
\\
 \dfrac{\beta d(u+v)}{\alpha d(u-v)} = \tanh (h(v)+h(u)).
\label{ratio2}
\end{align}
Equations (\ref{ratio1}) and (\ref{ratio2}) give the ratio of
$\alpha$ to $\beta$
and constraints on $h(u)$ and ${h(v)}$. Substituting
(\ref{XYBoltzmann}) into (\ref{ratio2}),
we find that the constraints are given by
\begin{align}
\frac{\sinh 2 h(u)}{ \bsc 2 u} =
\frac{\sinh 2 h(v)}{ \bsc 2 v}
= {\rm constant},
\label{sc2}
\end{align}
where ${\bsc u = \sn u / \cn u}$.
In conclusion, we have proved that the fermionic ${R}$-operator
\begin{align}
\mathcal{R}_{12}^{\mu}(u,v) = \mathcal{R}_{12}(u-v) +
\tanh (h(u) - h(v)) \mathcal{R}_{12}(u+v) (2n_1-1),
\end{align}
satisfies the Yang-Baxter relation (\ref{yang}) with
the ${L}$-operators
(\ref{Lxy}) under the constraints (\ref{sc2}) .

From eq.\ (\ref{yang}) we get the global Yang-Baxter relation for
the monodromy operator
\begin{align}
\mathcal{T}_{a}(u) =
\mathcal{L}_{aN}(u) \ldots \mathcal{L}_{a1}(u),
\end{align}
as
\begin{align}
             \mathcal{R}^{\mu}_{12}(u,v)
     \mathcal{T}_1(u) \mathcal{T}_2(v)
           =     \mathcal{T}_2(v) \mathcal{T}_1(u)
     \mathcal{R}^{\mu}_{12}(u,v),
\end{align}
which leads to a one-parameter family of the commuting
transfer operators,
\begin{align}
&  [ \tau(u) , \tau(v)] = 0, \nonumber \\
&\tau(u) = \fstr_a \mathcal{T}_a(u).
\end{align}

If we identify the constant in eq.\ (\ref{sc2}) with ${\mu/2}$,
we recover the Hamiltonian (\ref{XYfield}) by
the logarithmic derivative of the transfer operator
\begin{align}
\mathcal{H} =& \tau(0)^{-1} \frac{{\rm d}}{{\rm d} u} \tau(u)
\Big|_{u=0}
\nonumber\\
            =&  \mathcal{H}^{\rm XY}  +
\mu \sum_{j=1}^{N} \left( n_j-\frac{1}{2}
\right).
\end{align}
Here we have used the relations ${h(0) = 0, \ \ h'(0) = \mu/2}$.

\section{Fermionic $R$-operator for the 1D Hubbard model}
\setcounter{equation}{0}
\renewcommand{\theequation}{4.\arabic{equation}}
The Hamiltonian of the 1D Hubbard model
(\ref{hamiltonian.hubbard}) consists of two $XX$ fermion models
(\ref{XXhamiltonian}) for up and down spins with
an interaction term between them.
The fermionic $R$-operator without an interaction term is
given by
\begin{align}
\bar{\mathcal{R}}_{jk}(u)
= \mathcal{R}^{(\uparrow)}_{jk}(u)
\mathcal{R}^{(\downarrow)}_{jk}(u),
\end{align}
where ${\mathcal{R}_{jk}^{(\sigma)} \ (\sigma = \uparrow \downarrow)}$ denote the fermionic ${R}$-operators for the ${XX}$ fermion model,
i.e.,
\begin{align}
a(u) = \cos u, \hspace{5mm}
b(u) = \sin u, \hspace{5mm}
c(u)=1, \hspace{5mm}
d(u) =0.
\end{align}
Since both ${\mathcal{R}^{(\uparrow)}_{jk}(u)}$ and
${\mathcal{R}^{(\downarrow)}_{jk}(u)}$
satisfy the YBE and the DYBE,
the product ${\bar{\mathcal{R}}_{jk}(u)}$ also satisfies the YBE
\begin{align}
\bar{\mathcal{R}}_{12}(u-v) &\bar{\mathcal{R}}_{13}(u)
 \bar{\mathcal{R}}_{23}(v)
= \bar{\mathcal{R}}_{23}(v)
 \bar{\mathcal{R}}_{13}(u) \bar{\mathcal{R}}_{12}(u-v)
\label{hstr}
\end{align}
and the DYBE
\begin{align}
\bar{\mathcal{R}}_{12}(u+v)
&(2n_{1 \uparrow}-1) (2n_{1 \downarrow}-1)
\bar{\mathcal{R}}_{13}(u) \bar{\mathcal{R}}_{23}(v)
\nonumber\\
&= \bar{\mathcal{R}}_{23}(v) \bar{\mathcal{R}}_{13}(u)
(2n_{1 \uparrow}-1) (2n_{1 \downarrow}-1)
\bar{\mathcal{R}}_{12}(u+v).
\label{hdstr}
\end{align}
A linear combination of (\ref{hstr}) and (\ref{hdstr}) yields
\begin{align}
&\Bigl\{ \alpha \bar{\mathcal{R}}_{12}(u-v)
     + \beta \bar{\mathcal{R}}_{12}(u+v)
        (2n_{1 \uparrow}-1)(2n_{1 \downarrow}-1)
            \Bigr\}
     \bar{\mathcal{R}}_{13}(u) \bar{\mathcal{R}}_{23}(v)
\nonumber\\
&= \bar{\mathcal{R}}_{23}(v) \bar{\mathcal{R}}_{13}(u)
     \Bigl\{ \alpha \bar{\mathcal{R}}_{12}(u-v)
         + \beta (2n_{1 \uparrow}-1)(2n_{1 \downarrow}-1)
             \bar{\mathcal{R}}_{12}(u+v)
                \Bigr\}.
\label{hcom}
\end{align}
For a moment, $\alpha$ and $\beta$ are arbitrary.
As in the case of the ${XY}$ fermion model in an external field,
we look for a solution of the Yang-Baxter relation,
\begin{align}
         \mathcal{R}^{\rm h}_{12}(u,v)
            \mathcal{L}_{13}(u)
            \mathcal{L}_{23}(v)
&=       \mathcal{L}_{23}(v)
            \mathcal{L}_{13}(u)
         \mathcal{R}^{\rm h}_{12}(u,v)
\label{YBRhubbard}
\end{align}
in the form
\begin{align}
\mathcal{R}^{\rm h}_{12}(u,v)  &=
\alpha \bar{\mathcal{R}}_{12}(u-v)
+ \beta \bar{\mathcal{R}}_{12}(u+v)
(2n_{1 \uparrow}-1)(2n_{1 \downarrow}-1), \label{hubbardR} \\
   \mathcal{L}_{jk}(u)
       =& \bar{\mathcal{R}}_{jk}(u)
   \exp{ \{ - h(u) (2n_{j \uparrow}-1)(2n_{j \downarrow}-1) \}}.
\label{hubbardL}
\end{align}

Comparing eq.\ (\ref{hcom}) with the Yang-Baxter relation
(\ref{YBRhubbard}),
we get a relation
\begin{align}
\nonumber\\
&\mathcal{I}_1(u) \mathcal{I}_2(v)
\mathcal{R}^{\rm h}_{12}(u,v)
\mathcal{I}_1(u)^{-1} \mathcal{I}_2(v)^{-1} \nonumber \\
&\hspace{15mm}
= \alpha \bar{\mathcal{R}}_{12}(u-v)
+ \beta (2n_{1 \uparrow}-1)(2n_{1 \downarrow}-1) \bar{\mathcal{R}}_{12}(u+v),
\label{rllcondition}
\end{align}
where
\begin{align}
&\hspace{5mm}
\mathcal{I}_j(u)
= \exp\{ - h(u) (2n_{j \uparrow}-1)(2n_{j \downarrow}-1)\}.
\label{S}
\end{align}
From (\ref{rllcondition}), we obtain two conditions
\begin{align}
  \dfrac{\beta a(u+v)}{\alpha a(u-v)}
       =  - \tanh (h(u)-h(v)) ,
\hspace{5mm}
  \dfrac{\beta b(u+v)}{\alpha b(u-v)}
       = - \tanh (h(u)+h(v)).
\label{hrel}
\end{align}
which give the ratio of $\alpha$ to $\beta$
and constraints on $h(u)$ and ${h(v)}$.
The constraints are given more explicitly by
\begin{align}
\dfrac{\sinh 2h(u)}{\sin 2u} = \dfrac{\sinh 2h(v)}{\sin 2 v} = \frac{U}{4}.
\label{hcond}
\end{align}

To sum up, we have obtained the fermionic ${R}$-operator for the
1D Hubbard model
\begin{align}
\mathcal{R}^{\rm h}_{12}(u,v)  =&
\mathcal{R}^{(\uparrow)}_{12}(u-v)
\mathcal{R}^{(\downarrow)}_{12}(u-v)
-  \dfrac{\cos (u-v)}{\cos (u+v)} \tanh (h(u) - h(v)) \nonumber \\
& \times \mathcal{R}^{(\uparrow)}_{12}(u+v)
\mathcal{R}^{(\downarrow)}_{12}(u+v)
(2n_{1 \uparrow}-1)(2n_{1 \downarrow}-1),
\label{fermionicRhubbard}
\end{align}
which satisfies the Yang-Baxter relation (\ref{YBRhubbard}).
One of the remarkable properties of the fermionic ${R}$-operator
(\ref{fermionicRhubbard}) is that its dependence on
the the spectral parameters is not a difference type ${u-v}$.
We call it the ``non-additive" property.
The non-additive property of the fermionic ${R}$-operator
(\ref{fermionicRhubbard}) allows us to generalize
the Hamiltonian of the 1D Hubbard model (see \S 5).

From (\ref{YBRhubbard}) we get the global Yang-Baxter relation,
\begin{align}
             \mathcal{R}^{\rm h}_{12}(u,v)
     \mathcal{T}_1(u) \mathcal{T}_2(v)
           =&      \mathcal{T}_2(v) \mathcal{T}_1(u)
     \mathcal{R}^{\rm h}_{12}(u,v)
\label{globalHubYBR}
\end{align}
for the monodromy operator
\begin{align}
     \mathcal{T}_a(u) =  \mathcal{L}_{aN}(u) \ldots \mathcal{L}_{a1}(u).
\label{monodromyhubbard}
\end{align}
Now we define the supertrace $\fstr^{\rm h}$ for the 1D Hubbard model as
\begin{align}
     \fstr^{\rm h}_a X = {}_a \langle  0| X | 0 \rangle_a
           - {}_a \langle  \uparrow | X | \uparrow \rangle_a
           - {}_a \langle  \downarrow | X | \downarrow \rangle_a
           + {}_a \langle  \downarrow \uparrow | X | \uparrow \downarrow \rangle_a,
\end{align}
where the auxiliary fermion Fock space is given by
\begin{align}
c_{a \sigma} | 0 \rangle_a = 0, \hspace{5mm} | \uparrow \rangle_a &=  c_{a \uparrow}^{\dagger} | 0 \rangle_a, \hspace{5mm}
| \downarrow \rangle_a =  c_{a \downarrow}^{\dagger} | 0 \rangle_a, \hspace{5mm}| \uparrow \downarrow \rangle_a =  c_{a \uparrow}^{\dagger} c_{a \downarrow}^{\dagger} | 0 \rangle_a, \nonumber \\
{}_{a} \langle 0 | = (| 0 \rangle_a)^{\dagger}, \hspace{5mm}
{}_{a} \langle \uparrow | &= {}_{a} \langle 0 | c_{a\uparrow}, \hspace{5mm}
{}_{a} \langle \downarrow | = {}_{a} \langle 0 | c_{a\downarrow}, \hspace{5mm}
{}_{a} \langle \downarrow \uparrow| = {}_{a} \langle 0 | c_{a\downarrow} c_{a\uparrow}.
\end{align}

Then the transfer operator of the 1D Hubbard model is defined by
\begin{align}
\tau(u) = \fstr^{\rm h}_a \mathcal{T}_{a}(u).
\end{align}
From the global Yang-Baxter relation (\ref{globalHubYBR}), we can show the existence of a commuting family of the transfer operators
\begin{align}
    [ \tau(u) , \tau(v)] = 0,
\end{align}
which proves the integrability of the model.

Using the relations ${h(0) = 0}$ and ${h'(0) = U/4}$,
we can obtain the Hamiltonian of the 1D Hubbard model
as a logarithmic derivative of the transfer operator,
\begin{align}
\mathcal{H} &= - \tau(0)^{-1}
              \dfrac{{\rm d}}{{\rm d} u} \tau(u) \Big|_{u=0}
\nonumber\\
            &=  - \sum_{j=1}^N
               \sum_{\sigma = \uparrow \downarrow}
                  \mathcal{H}^{{\rm XX}(\sigma)}_{j,j+1}
        + \dfrac{U}{4} \sum_{j=1}^{N}
        (2n_{j \uparrow} - 1)(2n_{j \downarrow} -1),
\end{align}
where the PBC for the fermion operators (\ref{pbcfermion}) is satisfied.

In this way we have proved the integrability of
the 1D Hubbard model using the fermionic ${R}$-operator.
It should be emphasized that we have not used the Jordan-Wigner
transformation at all, which changes the boundary condition
and the symmetry of the model. \cite{shiroishi4}
 We shall see in \S 6 that the fermionic ${R}$-operator
(\ref{fermionicRhubbard}) naturally enjoys the ${SO(4)}$ symmetry.

\section{Yang-Baxter Equation for the New Fermionic $R$-Operator}
\setcounter{equation}{0}
\renewcommand{\theequation}{5.\arabic{equation}}
In the previous section we have shown that
the fermionic ${R}$-operator (\ref{fermionicRhubbard})
satisfies the Yang-Baxter relation (\ref{YBRhubbard})
with the ${L}$-operator (\ref{hubbardL}).
It is natural to expect that the fermionic ${R}$-operator itself
fulfills the Yang-Baxter equation (YBE),
\begin{align}
    \mathcal{R}_{12}^{\rm h}(u_1,u_2)
  \mathcal{R}_{13}^{\rm h}(u_1,u_3) \mathcal{R}_{23}^{\rm h} (u_2,u_3)
=   \mathcal{R}_{23}^{\rm h} (u_2,u_3)\mathcal{R}_{13}^{\rm h} (u_1,u_3)
  \mathcal{R}_{12}^{\rm h} (u_1,u_2).
\label{syang}
\end{align}

In the case of the coupled spin model, the YBE of
the ${R}$-matrix was proved in ref.\ \cite{shiroishi1}
using the tetrahedral Zamolodchikov algebra (TZA).
\cite{korepanov,shiroishi3}
In this section, we introduce the TZA for the fermionic ${R}$-operator
and proves the YBE (\ref{syang}).

The TZA is defined by the following set of relations
among operators
${\mathcal{L}_{jk}^{0}}$ and ${\mathcal{L}_{jk}^{1}}$,
\begin{align}
\mathcal{L}_{12}^{a} \mathcal{L}_{13}^{b}
\mathcal{L}_{23}^{c} &= \sum_{d,e,f=0,1} S_{def}^{abc} \mathcal{L}_{23}^{a} \mathcal{L}_{13}^{b}
\mathcal{L}_{12}^{c}, \hspace{2cm} a, \ldots, f =0,1,
\label{TZA}
\end{align}
where ${S_{def}^{abc}}$ are some scalar coefficients.

Let us take ${\mathcal{L}_{jk}^{(0)}}$ and ${\mathcal{L}_{jk}^{(1)}}$ as
\begin{align}
\mathcal{L}_{jk}^{0 (\sigma)} &= \mathcal{R}_{jk}^{(\sigma)}(u_j-u_k), \nonumber \\
\mathcal{L}_{jk}^{1 (\sigma)} &= \mathcal{R}_{jk}^{(\sigma)}(u_j+u_k) (2 n_{j\sigma} - 1), \hspace{2cm} \sigma = \uparrow \downarrow, \label{L0L1}
\end{align}
where ${\mathcal{R}_{jk}^{(\uparrow)}(u)}$ and ${\mathcal{R}_{jk}^{(\downarrow)}(u)}$ are the fermionic ${R}$-operators for the ${XX}$ fermion model as before.
Then we have found the following relations which give the TZA (\ref{TZA}),
\begin{align}
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{23}
&= \mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12},
\hspace{5mm}
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{23}
= \mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12}
\label{strd}
\\
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{23}
&= \mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12},
\hspace{5mm}
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{23}
= \mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12},
\label{dstrd}
\\
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{23}
&=S_{001}^{111}
\mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12}
+S_{010}^{111}
\mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12}
+S_{100}^{111}
\mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12},
\\
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{23}
&=S_{111}^{001}
\mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12}
+S_{100}^{001}
\mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12}
+S_{010}^{001}
\mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12},
\\
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{23}
&=S_{111}^{010}
\mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12}
+S_{100}^{010}
\mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12}
+S_{001}^{010}
\mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12},
\\
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{23}
&=S_{111}^{100}
\mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{12}
+S_{010}^{100}
\mathcal{L}^{0 (\sigma)}_{23} \mathcal{L}^{1 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12}
+S_{001}^{100}
\mathcal{L}^{1 (\sigma)}_{23} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{0 (\sigma)}_{12},
\label{lidel}
\end{align}
where ${\sigma = \uparrow}$ or ${\downarrow}$, and the coefficients
${S_{def}^{abc}}$ are given by
\begin{align}
S_{001}^{111} &=
\dfrac{\sin(u_1+u_2) \cos(u_1+u_3)}{\cos(u_1-u_2) \sin(u_1-u_3)},
\hspace{5mm}
S_{010}^{111} =
- \dfrac{\sin(u_1+u_2) \sin(u_2+u_3)}{\cos(u_1-u_2) \cos(u_2-u_3)},
\nonumber\\
S_{100}^{111} &=
- \dfrac{\sin(u_2+u_3) \cos(u_1+u_3)}{\cos(u_2-u_3) \sin(u_1-u_3)},
\hspace{5mm}
S_{111}^{001} =
\dfrac{\sin(u_1-u_2) \cos(u_1-u_3)}{\cos(u_1+u_2) \sin(u_1+u_3)},
\nonumber\\
S_{100}^{001} &=
\dfrac{\sin(u_1-u_2) \sin(u_2+u_3)}{\cos(u_1+u_2) \cos(u_2-u_3)},
\hspace{5mm}
S_{010}^{001} =
\dfrac{\sin(u_2+u_3) \cos(u_1-u_3)}{\cos(u_2-u_3) \sin(u_1+u_3)},
\nonumber\\
S_{111}^{010} &=
\dfrac{\sin(u_1-u_2) \sin(u_2-u_3)}{\cos(u_1+u_2) \cos(u_2+u_3)},
\hspace{5mm}
S_{100}^{010} =
\dfrac{\sin(u_1-u_2) \cos(u_1+u_3)}{\cos(u_1+u_2) \sin(u_1u_3)},
\nonumber\\
S_{001}^{010} &=
\dfrac{\sin(u_2-u_3) \cos(u_1+u_3)}{\cos(u_2+u_3) \sin(u_1-u_3)},
\hspace{5mm}
S_{111}^{100} =
- \dfrac{\sin(u_2-u_3) \cos(u_1-u_3)}{\cos(u_2+u_3) \sin(u_1+u_3)},
\nonumber\\
S_{010}^{100} &=
\dfrac{\sin(u_1+u_2) \cos(u_1-u_3)}{\cos(u_1-u_2) \sin(u_1+u_3)},
\hspace{5mm}
S_{001}^{100} =
- \dfrac{\sin(u_1+u_2) \sin(u_2-u_3)}{\cos(u_1-u_2) \cos(u_2+u_3)}.
\end{align}
Note that eq.\ (\ref{strd}) and eq.\ (\ref{dstrd}) are equivalent to the YBE (\ref{hstr}) and the DYBE (\ref{hdstr}) respectively. In this sense, the TZA (\ref{TZA}) can be regarded as a generalization of the YBE and the DYBE.

We also remark that the products $\mathcal{L}^{a (\sigma)}_{12} \mathcal{L}^{b (\sigma)}_{13} \mathcal{L}^{c (\sigma)}_{23}$
${(a,b,c= 0,1)}$ are not linearly independent and satisfy the following relations,
\begin{align}
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13} \mathcal{L}^{0 (\sigma)}_{23}
&=x
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13} \mathcal{L}^{1 (\sigma)}_{23}
+y
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13} \mathcal{L}^{1 (\sigma)}_{23}
+z
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13} \mathcal{L}^{0 (\sigma)}_{23},
 \label{linear1} \\
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13} \mathcal{L}^{1 (\sigma)}_{23}
&=x
\mathcal{L}^{1 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13} \mathcal{L}^{0 (\sigma)}_{23}
+y
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{1 (\sigma)}_{13} \mathcal{L}^{0 (\sigma)}_{23}
+z
\mathcal{L}^{0 (\sigma)}_{12} \mathcal{L}^{0 (\sigma)}_{13}
\mathcal{L}^{1 (\sigma)}_{23},
\label{linear2}
\end{align}
where
\begin{align}
x &=
- \dfrac{\sin(u_1-u_3) \cos(u_2-u_3)}{\sin(u_1+u_3) \cos(u_2+u_3)},
\hspace{5mm}
y =
\dfrac{\cos(u_1-u_2) \cos(u_2-u_3)}{\cos(u_1+u_2) \cos(u_2+u_3)},
\nonumber\\
z &=
\dfrac{\cos(u_1-u_2) \cos(u_1-u_3)}{\cos(u_1+u_2) \cos(u_1+u_3)},
\hspace{5mm}
x' =
- \dfrac{\sin(u_1+u_3) \cos(u_2+u_3)}{\sin(u_1-u_3) \cos(u_2-u_3)},
\nonumber\\
y' &=
\dfrac{\cos(u_1+u_2) \cos(u_2+u_3)}{\cos(u_1-u_2) \cos(u_2-u_3)},
\hspace{5mm}
z' =
\dfrac{\sin(u_1+u_3) \cos(u_1+u_2)}{\sin(u_1-u_3) \cos(u_1-u_2)}.
\label{linearcoe}
\end{align}
These relations (\ref{strd}) - (\ref{linearcoe}) have been verified
using mathematica.

Now taking into account the form of the fermionic ${R}$-operator
(\ref{fermionicRhubbard}),
we look for a solution of the YBE (\ref{syang}) in the form
\begin{align}
\mathcal{R}_{jk}^{\rm h} (u_j,u_k)
 = \mathcal{L}^{0 (\uparrow)}_{jk}(u_j,u_k)
   \mathcal{L}^{0 (\downarrow)}_{jk}(u_j,u_k)
 + \alpha_{jk} \mathcal{L}^{1 (\uparrow)}_{jk}(u_j,u_k)
   \mathcal{L}^{1 (\downarrow)}_{jk}(u_j,u_k).
\label{S1}
\end{align}
We substitute eq.\ (\ref{S1}) into the YBE (\ref{syang}).
Using the TZA (\ref{TZA}) and the relations (\ref{linear1}) and (\ref{linear2}), we obtain the conditions
\begin{align}
&\alpha_{13} \sin 2(u_1+u_3)
=\alpha_{12} \sin 2(u_1+u_2) + \alpha_{23} \sin 2(u_2+u_3)
\nonumber\\
&=\dfrac{1}{\alpha_{12}} \sin 2(u_1-u_2)
- \dfrac{1}{\alpha_{13}} \sin 2(u_1-u_3).
\label{scond}
\end{align}
If we assume ${\alpha_{ij}}$ in the form
\begin{align}
\alpha_{jk}= - \dfrac{\cos(u_j-u_k)}{\cos(u_j+u_k)}
                   \tanh(h(u_j)-h(u_k)).
\end{align}
and impose the constraints
\begin{align}
\dfrac{\sinh 2 h(u_j)}{\sin 2 u_{j}} = \frac{U}{4}, \hspace{2cm} j=1,2,3,
\label{constraints}
\end{align}
then one can easily find that the conditions (\ref{scond}) are satisfied.
In conclusion, the fermionic ${R}$-operator for the 1D Hubbard model,
\begin{align}
\mathcal{R}^{\rm h}_{jk}(u_j,u_k)  =&
\mathcal{R}^{(\uparrow)}_{jk}(u_j - u_k)
\mathcal{R}^{(\downarrow)}_{jk}(u_j-u_k)
-  \dfrac{\cos (u_j-u_k)}{\cos (u_j+u_k)} \tanh (h(u_j) - h(u_k)) \nonumber \\
& \times \mathcal{R}^{(\uparrow)}_{jk}(u_j+u_k)
\mathcal{R}^{(\downarrow)}_{jk}(u_j+u_k)
(2n_{j \uparrow}-1)(2n_{j \downarrow}-1),
\label{Rhubbard}
\end{align}
is a solution of the YBE (\ref{syang}) under the constraints
(\ref{constraints}).

Besides the YBE (\ref{syang}), the fermionic ${R}$-operator
(\ref{Rhubbard}) has the following properties
\begin{align}
&\mathcal{R}_{jk}^{\rm h}(u,0) = \dfrac{1}{\cosh h(u)} \mathcal{L}_{jk}(u), \label{fundamental} \\
&\mathcal{R}_{jk}^{\rm h}(u_0,u_0) = \mathcal{P}_{jk}, \label{permutator} \\
&\mathcal{R}_{jk}^{\rm h}(u_j,u_k) \mathcal{R}_{kj}^{\rm h}(u_k,u_j)
= \rho(u_j,u_k) {\bf 1},
\end{align}
where
\begin{align}
\rho(u_j,u_k) = \cos^2 (u_j-u_k) \left\{ \cos^2 (u_j-u_k) -
            \tanh^2 (h(u_j) -h(u_k)) \cos^2 (u_j + u_k) \right\}.
\end{align}
Here the permutation operator is defined by
\begin{align}
\mathcal{P}_{jk} &= \mathcal{P}_{jk}^{(\uparrow)} \mathcal{P}_{jk}^{(\downarrow)} \nonumber \\
\mathcal{P}_{jk}^{(\sigma)} &= 1 - (c^{\dagger}_{j \sigma} - c^{\dagger}_{k \sigma})(c_{j \sigma} - c_{k \sigma}), \hspace{2cm} \sigma = \uparrow \downarrow.
\end{align}
Due to eq.\ (\ref{fundamental}), we can recover the Yang-Baxter relation (\ref{YBRhubbard}) by putting ${u_3=0}$ in the YBE (\ref{syang}).

Using the ${R}$-operator (\ref{Rhubbard}),
we can introduce an inhomogeneous model as
\begin{align}
\mathcal{T}_{a}(u,\{u_j\}) = \mathcal{R}_{aN}^{\rm h} (u,u_N) \ldots \mathcal{R}^{\rm h}_{aj}(u,u_j) \ldots \mathcal{R}_{a1}^{\rm h} (u,u_1),
\end{align}
where ${u_{j} \  (j=1, \ldots,N)}$ are the inhomogeneous parameters
obeying the constraints
\begin{align}
\dfrac{\sinh 2 h(u_j)}{\sin 2 u_{j}} = \frac{U}{4}, \hspace{2cm} j=1,\ldots,N.
\end{align}
Then from the YBE (\ref{syang}), we have the global Yang-Baxter relation
\begin{align}
    \mathcal{R}_{12}^{\rm h} (u,v)
  \mathcal{T}_{1}(u,\{u_j\}) \mathcal{T}_{2}(v,\{u_j\})
=&   \mathcal{T}_{2}(v,\{u_j\})\mathcal{T}_{1}(u,\{u_j\})
  \mathcal{R}_{12}^{\rm h} (u,v),
\label{globalYBE}
\end{align}
which leads to the commutativity
\begin{align}
\left[ \tau(u,\{u_j\}), \tau(v,\{u_j\}) \right] = 0.
\end{align}

Because of the non-additive property of
the ${R}$-operator (\ref{Rhubbard}), the simplest choice
\begin{align}
\mathcal{T}_{a}(u,u_{0}) = \mathcal{R}_{aN}^{\rm h} (u,u_0) \ldots
\mathcal{R}^{\rm h}_{aj}(u,u_0) \ldots
\mathcal{R}^{\rm h}_{a1}(u,u_0)
\label{genetransfer}
\end{align}
still generalizes eq.\ (\ref{monodromyhubbard}).
Actually we can derive a fermionic integrable Hamiltonian
which generalize the 1D Hubbard model.
Note that ${\tau(u_0,u_0)}$ is the shift operator
regardless of the parameter ${u_0}$,
\begin{align}
\tau(u_0,u_0) &= \hat{U} \nonumber \\
              &\equiv \mathcal{P}_{12} \mathcal{P}_{23}
                          \ldots \mathcal{P}_{N-1,N}.
\end{align}
Then the logarithmic derivative of the generalized transfer operator yields a new fermionic Hamiltonian as follows,
\begin{align}
\mathcal{H} =& - \tau(u_0,u_0)^{-1} \dfrac{\rm d}{{\rm d} u} \tau(u,u_0) \Big|_{u=u_0}  \nonumber \\
=& - \sum_{j=1}^N \sum_{\sigma=\uparrow \downarrow}
  \left( c_{j \sigma}^{\dagger} c_{j+1 \sigma}
+ c_{j+1 \sigma}^{\dagger} c_{j \sigma} \right) \nonumber \\
&+ \dfrac{U}{4 \cosh 2h(u_0)} \sum_{j=1}^{N}
  \Big\{
  \cos ^2 u_0 (2n_{j \uparrow}-1) - \sin ^2 u_0 (2n_{j+1 \uparrow}-1) \nonumber \\
 & \hspace{7cm} +\sin 2u_0
     (  c_{j+1 \uparrow}^{\dagger} c_{j \uparrow}
   -  c_{j \uparrow}^{\dagger}  c_{j+1 \uparrow}  )
  \Big\}
\nonumber\\
 & \hspace{3cm} \times \Big\{
 \cos ^2 u_0 (2n_{j \downarrow}-1) - \sin ^2 u_0 (2n_{j+1 \downarrow}-1) \nonumber \\
 & \hspace{7cm} +\sin 2u_0
     (  c_{j+1 \downarrow}^{\dagger}  c_{j \downarrow}
   -  c_{j \downarrow}^{\dagger}  c_{j+1 \downarrow}  )
  \Big\}.
\label{modifiedhub}
\end{align}
If we take ${u_0 = 0}$ or ${\pi/2}$, the Hamiltonian (\ref{modifiedhub}) reduces to
the 1D Hubbard model (\ref{hamiltonian.hubbard}).
The local higher conserved operators can also be obtained by the formula
\begin{align}
I_n = \frac{{\rm d}^n}{{\rm d} u^n} \ln
\left( \tau(u_0,u_0)^{-1} \tau(u,u_0)  \right) \Big|_{u=u_0}, \hspace{2cm}
n=1,2, \ldots,
\label{conserved}
\end{align}
where ${I_{1} = - \mathcal{H}}$. The meaning of the insertion of
the term
${\tau(u_0,u_0)^{-1}}$ in (\ref{conserved}) will be clarified when we discuss the ${SO(4)}$ symmetry in \S 6.

Remark that in terms of
${\mathcal{\check{R}}_{jk}(u,u_0) \equiv \mathcal{P}_{jk} \mathcal{R}_{jk}(u,u_0)}$ we can express the Hamiltonian (\ref{modifiedhub}) as
\begin{align}
\mathcal{H} = - \sum_{j=1}^{N} \dfrac{\rm d}{{\rm d} u} \mathcal{\check{R}}_{jk}(u,u_0) \Big|_{u=u_0}.
\label{hamiltonianRcheck}
\end{align}
The higher conserved operators ${I_n}$ are also expressed
in terms of ${\mathcal{\check{R}}_{jk}(u,u_0)}$,
though it is very difficult to get their explicit forms.

\section{SO(4) Symmetry}
\setcounter{equation}{0}
\renewcommand{\theequation}{6.\arabic{equation}}
The ${SO(4) (= [ SU(2) \times SU(2)]/Z_{2})}$ symmetry is one of the most
important properties of the Hubbard Hamiltonian
(\ref{hamiltonian.hubbard}) \cite{heilmann,yang1,yang2,pernici,affleck}.
The two ${SU(2)}$ come from the spin ${SU(2)}$
\begin{align}
S^{+} &= \sum_{j=1}^{N} c_{j\uparrow}^{\dagger} c_{j\downarrow}, \hspace{5mm}
S^{-}  = \sum_{j=1}^{N} c_{j\downarrow}^{\dagger} c_{j\uparrow}, \hspace{5mm}
S^{z} = \frac{1}{2} \sum_{j=1}^{N} (n_{j\uparrow} - n_{j\downarrow})
\label{spinSU(2)}
\end{align}
and the charge ${SU(2)}$ (${\eta}$-pairing ${SU(2)}$)
\begin{align}
\eta^{+} &= \sum_{j=1}^{N} (-1)^{j} c_{j\uparrow}^{\dagger}
c_{j\downarrow}^{\dagger}, \hspace{5mm}
\eta^{-}  = \sum_{j=1}^{N} (-1)^{j} c_{j\downarrow} c_{j\uparrow},
\hspace{5mm}
\eta^{z} = \frac{1}{2} \sum_{j=1}^{N} (n_{j\uparrow}+n_{j\downarrow}-1).
\label{eq.chargeSU(2)}
\end{align}
We assume that the lattice has an even number of sites
under the PBC (\ref{pbcfermion}). Then the six generators above
commute with the Hamiltonian
(\ref{hamiltonian.hubbard}).

In ref. \cite{murakami,shiroishi4},
the ${SO(4)}$ symmetry of the fermionic
transfer matrix was discussed based on the fermionic formulation by
Olmedilla {\it et al.}. \cite{wadati2} In the following we discuss
the ${SO(4)}$ symmetry of the fermionic ${R}$-operator and the generalized
transfer operator (\ref{genetransfer})
using the similar discussion (c.f. ref. \cite{takhtajan}).
For this purpose,
we introduce the local generators of the ${SO(4)}$ algebra as follows,
\begin{align}
S_{j}^{+} &= c_{j\uparrow}^{\dagger} c_{j\downarrow}, \hspace{5mm}
S_{j}^{-} = c_{j\downarrow}^{\dagger} c_{j\uparrow}, \hspace{5mm}
S_{j}^{z} = \frac{1}{2}(n_{j\uparrow} - n_{j\downarrow}), \nonumber \\
\eta_{j}^{+} &= c_{j\uparrow}^{\dagger} c_{j\downarrow}^{\dagger}, \hspace{5mm}
\eta_{j}^{-}  = c_{j\downarrow} c_{j\uparrow}, \hspace{5mm}
\eta_{j}^{z}  = \frac{1}{2}(n_{j\uparrow}+n_{j\downarrow}-1).
\end{align}
The ${SU(2)}$ symmetry of the fermionic ${R}$-operator for
the 1D Hubbard model (\ref{Rhubbard}) is represented
by the following commutation relations,
\begin{align}
&\Big[ \mathcal{R}_{jk}^{\rm h}(u,v), S_{j}^{+} + S_{k}^{+} \Big]=0, \hspace{5mm}
\Big[ \mathcal{R}_{jk}^{\rm h}(u,v), S_{j}^{-} + S_{k}^{-} \Big]=0, \nonumber \\
&\Big[ \mathcal{R}_{jk}^{\rm h}(u,v), S_{j}^{z} + S_{k}^{z} \Big]=0.
\label{spinR}
\end{align}
It is easy to see that the relations (\ref{spinR}) extend to the ${SU(2)}$
symmetry of the monodromy operator
\begin{align}
\mathcal{T}_{a}(u,u_0) = \mathcal{R}_{aN}^{\rm h}(u,u_0)
\ldots \mathcal{R}_{a1}^{\rm h}(u,u_0) \nonumber
\end{align}
as
\begin{align}
\Big[ \mathcal{T}_{a}(u,u_0), S_{a}^{\alpha} + S^{\alpha} \Big]=0,
\hspace{3cm} \alpha= \pm, z.
\label{spinmonodromy}
\end{align}
Taking the supertrace ${{\rm Str}_{a}}$ of (\ref{spinmonodromy}), we obtain
\begin{align}
\Big[ \tau(u,u_0), S^{\alpha} \Big]=0,
\hspace{4cm} \alpha= \pm, z,
\label{spintransfer}
\end{align}
which shows that the transfer operator ${\tau(u,u_0)}$ is the spin ${SU(2)}$
invariant.

Next we consider the charge ${SU(2)}$ symmetry. For the generator
${\eta_{j}^{z}}$, the same relation as (\ref{spinR}) holds,
\begin{align}
\Big[ \mathcal{R}_{jk}^{\rm h}(u,v), \eta_{j}^{z} + \eta_{k}^{z} \Big]=0,
\end{align}
which extends to the symmetry of the monodromy operator and the transfer
operator as,
\begin{align}
\Big[ \mathcal{T}_{a}(u,u_0), \eta_{a}^{z} + \eta^{z} \Big]=0,
\hspace{5mm}
\Big[ \tau(u,u_0), \eta^{z} \Big]=0.
\end{align}
As for the other generators ${\eta_{j}^{\pm}}$, we do not have the relations
like (\ref{spinR}). Instead, we have discovered the following anti-commuting
relation for ${\eta_{j}^{\pm}}$,
\begin{align}
\Big\{ \mathcal{R}_{jk}^{\rm h}(u,v), \eta_{j}^{\pm} - \eta_{k}^{\pm} \Big\} = 0.
\label{chargeR}
\end{align}

The local relation (\ref{chargeR}) extends to the identities for
the monodromy operator,
\begin{align}
\Big[ \mathcal{T}_{a}(u,u_0), \eta_{a}^{\pm} \Big]
+ \Big\{ \mathcal{T}_{a}(u,u_0), \eta^{\pm} \Big\} = 0,
\label{chargepm}
\end{align}
which can be proved as follows,
\begin{align}
&\mathcal{T}_{a}(u,u_0) \left( \eta_{a}^{\pm} + \eta^{\pm} \right)
\nonumber\\
&=\mathcal{R}_{aN}^{\rm h}(u,u_0) \ldots \mathcal{R}_{a2}(u,u_0)
\mathcal{R}_{a1}^{\rm h}(u,u_0)
(\eta_a^{\pm} - \eta_1^{\pm} + \eta_2^{\pm} \ldots + \eta_N^{\pm})
\nonumber\\
&= \mathcal{R}_{aN}^{\rm h}(u,u_0) \ldots \mathcal{R}_{a2}^{\rm h}(u,u_0)
(-\eta_a^{\pm} + \eta_1^{\pm} + \eta_2^{\pm} \ldots + \eta_N^{\pm})
\mathcal{R}_{a1}^{\rm h}(u,u_0)
\nonumber\\
&= \ldots
\nonumber\\
&= (\eta_a^{\pm} + \eta_1^{\pm} - \eta_2^{\pm} \ldots - \eta_N^{\pm})
\mathcal{R}_{aN}^{\rm h} (u,u_0) \ldots \mathcal{R}_{a2}^{\rm h} (u,u_0)
\mathcal{R}_{a1}^{\rm h} (u,u_0) \nonumber \\
&= (\eta_{a}^{\pm} - \eta^{\pm}) \mathcal{T}_{a}(u,u_0).
\end{align}
Here we have used the fact that ${N}$ is even.

Taking the supertrace of (\ref{chargepm}), we obtain
\begin{align}
\Big\{ \tau(u,u_0), \eta^{\pm} \Big\} = 0.
\label{anticummute}
\end{align}
Namely the transfer operator anti-commutes with the generators ${\eta^{\pm}}$.
Thus contrary to the naive anticipation, the transfer operator itself does not
commute with the generators of the charge ${SU(2)}$ symmetry.
However a combination ${ \tau(u_0,u_0)^{-1} \tau(u,u_0)}$
commutes with ${\eta^{\pm}}$
\begin{align}
\Big[   \tau(u_0,u_0)^{-1} \tau(u,u_0), \eta^{\pm} \Big] =0,
\end{align}
due to the relation
\begin{align}
\Big\{ \tau(u_0,u_0), \eta^{\pm} \Big\} = 0.
\label{shifteta}
\end{align}

Of course, ${\tau(u_0,u_0)^{-1} \tau(u,u_0) }$ commutes with
the other generators of the ${SO(4)}$ algebra.
Since we have defined the local conserved operators by eq. (\ref{conserved}),
\begin{align}
I_n = \frac{{\rm d}^n}{{\rm d} u^n} \ln
\left( \tau(u_0,u_0)^{-1} \tau(u,u_0)  \right) \Big|_{u=u_0}, \hspace{2cm}
n=1,2, \ldots
\end{align}
it follows that they are all ${SO(4)}$ invariant.
In particular, we have proved that
a generalized Hubbard model (\ref{modifiedhub}) has the
${SO(4)}$ symmetry. It is also possible
to prove the ${SO(4)}$ invariance of
the Hamiltonian (\ref{modifiedhub}) directly
using the expression (\ref{hamiltonianRcheck}) and the commutation relations
\begin{align}
&\Big[ \mathcal{\check{R}}_{jk}^{\rm h}(u,v), S_{j}^{\alpha} + S_{k}^{\alpha} \Big]=0, \hspace{4cm} \alpha = \pm,z, \nonumber \\
&\Big[ \mathcal{\check{R}}_{jk}^{\rm h}(u,v),
\eta_{j}^{z} + \eta_{k}^{z} \Big]
=0, \hspace{5mm}
\Big[ \mathcal{\check{R}}_{jk}^{\rm h}(u,v), \eta_{j}^{\pm} - \eta_{k}^{\pm} \Big]
=0.
\end{align}
where ${\check{\mathcal{R}}_{jk}^{\rm h}(u,v) = \mathcal{P}_{jk} \mathcal{R}_{jk}^{\rm h}(u,v)}$ as before.

The relation (\ref{shifteta}) means that the shift
 operator ${\hat{U}=\tau(u_0,u_0)}$  anticummutes with ${\eta^{\pm}}$
\begin{align}
\hat{U} \eta^{\pm} = - \eta^{\pm} \hat{U}.
\label{shift2}
\end{align}
If we define the lattice momentum operator ${\hat{\Pi}}$
\cite{takhtajan,murakami} by
\begin{align}
\hat{U} = \exp {\rm i} \hat{\Pi},
\label{momentum}
\end{align}
the relation (\ref{shift2}) implies that
\begin{align}
\hat{\Pi} \eta^{\pm} - \eta^{\pm} \hat{\Pi} =  \pi \eta^{\pm}.
\label{pichange}
\end{align}
This indicates that the generators ${\eta^{\pm}}$ changes the momentum
eigenvalue by ${\pi}$. \cite{yang1,murakami} Note that the lattice
momentum operator ${\hat{\Pi}}$ is defined
in (\ref{momentum}) mod ${2 \pi}$.

The lattice momentum operator ${\hat{\Pi}}$ is also a conserved
operator
\begin{align}
\left[ \hat{\Pi}, I_n \right] = 0,
\end{align}
but it is not contained in ${\{I_{n}\}}$. Actually ${\hat{\Pi}}$
is a non-local operator (we refer ref. \cite{murakami} for
the detailed discussions on ${\hat{\Pi}}$). Furthermore the
relation (\ref{pichange}) suggests that it does not have
the ${SO(4)}$ symmetry.

\section{Concluding Remarks}
\setcounter{equation}{0}
\renewcommand{\theequation}{7.\arabic{equation}}
In this paper, we have presented a new fermionic approach
for the integrability of the 1D Hubbard model. The central
object in our approach is the fermionic ${R}$-operator, which
 we introduced recently to discuss the integrability of the ${XXZ}$
fermion model. We have shown that the fermionic ${R}$-
operator can be generalized to the ${XYZ}$ fermion model.
It gives an operator solution of the Yang-Baxter equation (YBE).
We also considered the decorated Yang-Baxter equation (DYBE) for
the fermionic ${R}$-operator and found that the free-fermion condition
is necessary. In other words, the fermionic ${R}$-operator related
to the ${XY}$ fermion model fulfills the YBE and the DYBE.

The 1D Hubbard model can be regarded as two ${XX}$ fermion models
with an interaction term.  Using the YBE and the DYBE
for the ${XX}$ fermion model, we have established
a new kind of the Yang-Baxter relation for the 1D Hubbard model.
This gives an alternative proof for the integrability of the 1D Hubbard model.
We have obtained the fermionic ${R}$-operator
for the 1D Hubbard model and confirmed that it satisfies the YBE.
The connection between the fermionic ${R}$-operator and the
graded Yang-Baxter relation for the 1D Hubbard model
\cite{wadati2} will be discussed in a separate paper. \cite{shiroishi5}

From the fermionic ${R}$-operator of the 1D Hubbard model, we have
constructed a new fermionic transfer operator. Because of the non-additive
property of the ${R}$-operator, the transfer operator generates
a fermionic integrable Hamiltonian (\ref{modifiedhub})
which generalize the 1D Hubbard model.

One of the advantages of our fermionic approach is that we can
discuss the ${SO(4)}$ symmetry at the level of the (local) fermionic
${R}$-operator, which immediately leads to the ${SO(4)}$ symmetry of
all the local conserved operators derived from the transfer operator.
It is known that the ${SO(4)}$ symmetry of the 1D Hubbard
model is extended to the Yangian symmetry on the infinite lattice
\cite{uglov,murakami2}. It is interesting to investigate
the relation between the fermionic
${R}$-operator and the Yangian symmetry.

\vspace{5mm}
\begin{center}
{\bf Acknowledgment}
\end{center}
The authors are grateful to Y. Komori
for useful discussions and careful reading of the manuscript.
This work
is in part supported by Grant-in-Aid for
JSPJ Fellows from the
Ministry of Education, Science, Sports and Culture of Japan.

\vspace{30pt}

\begin{flushleft}
APPENDIX A: A Simple Derivation of the Fermionic ${R}$-Operator
\end{flushleft}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
The ${R}$-matrix of the ${XYZ}$ spin models in terms of the Pauli
spin matrix ($\sigma^{+}_{j}, \sigma^{-}_{j}$ and
$\sigma_{j}^{z}$) is given by
\begin{align}
R_{jk}(u) =& a(u) \dfrac{1 + \sigma^z_j \sigma^z_k}{2}
            + b(u) \dfrac{1 - \sigma^z_j \sigma^z_k}{2}
            + c(u) (\sigma^+_j \sigma^-_k + \sigma^+_k \sigma^-_j)
            \nonumber\\
           & + d(u) (\sigma^+_j \sigma^+_k + \sigma^-_j \sigma^-_k),
\label{8V}
\end{align}
where the Boltzmann weights ${a(u), b(u), c(u)}$ and ${d(u)}$ are the same as
 (\ref{baxterpara}). The ${R}$-matrix (\ref{8V}) satisfies the Yang-Baxter equation
\begin{align}
R_{12}(u-v) R_{13}(u) R_{23}(v) = R_{23}(v) R_{13}(u) R_{12}(u-v).
\label{YBERmatrix}
\end{align}
We like to obtain the fermionic ${R}$-operator (\ref{R})
from the ${R}$-matrix (\ref{8V}) by means of
the Jordan-Wigner transformation \cite{lieb2}
\begin{align}
\sigma_{j}^{-} &= \exp \left\{ {\rm i} \pi
\sum_{k=1}^{j-1} n_{k} \right\} c_{j}, \nonumber \\
\sigma_{j}^{+} &= \exp \left\{ -{\rm i} \pi
\sum_{k=1}^{j-1} n_{k} \right\} c_{j}^{\dagger},
 \label{JWT}
\end{align}
However it is not possible to apply the Jordan-Wigner transformation directly
to (\ref{8V}) because of the non-local term ${R_{13}(u)}$ in the Yang-Baxter equation (\ref{YBERmatrix}). For this reason, we consider an equivalent ${R}$-matrix ${\check{R}_{jk}(u)}$
\begin{align}
\check{R}_{jk}(u) &  \equiv P_{jk} R_{jk}(u)  \nonumber \\
= & a(u) \dfrac{1 + \sigma^z_j \sigma^z_k}{2}
           + c(u) \dfrac{1 - \sigma^z_j \sigma^z_k}{2}
           + b(u) (\sigma^+_j \sigma^-_k
                     + \sigma^-_j \sigma^+_k)
\nonumber\\
           & + d(u) (\sigma^+_j \sigma^+_k
                      + \sigma^-_j \sigma^-_k),
\label{checkRmatrix}
\end{align}
which satisfies the Yang-Baxter equation in the form
\begin{align}
\check{R}_{12}(u-v) \check{R}_{23}(u) \check{R}_{12}(v)
= \check{R}_{23}(v) \check{R}_{12}(u) \check{R}_{23}(u-v).
\label{YBEcheckRmatrix}
\end{align}
Here ${P_{jk}}$ is the permutation matrix
\begin{align}
P_{jk} = \dfrac{1 + \sigma^z_j \sigma^z_k}{2}
                  + \sigma^+_j \sigma^-_k
                  + \sigma^-_j \sigma^+_k.
\end{align}

Now we can apply the Jordan-Wigner transformation (\ref{JWT}) to
(\ref{checkRmatrix}) and (\ref{YBEcheckRmatrix}). Then we obtain
a fermionic solution
\begin{align}
       \check{\mathcal{R}}_{jk}(u) =& a(u) \{ n_j n_k +
(1-n_j)(1-n_k)\}
           + c(u) \{n_j (1-n_k) + (1-n_j) n_k \}
\nonumber\\
           +  & b(u) (c_j^{\dagger} c_k + c_k^{\dagger} c_j)
           + d(u) (c_j^{\dagger} c_k^{\dagger} - c_j c_k),
\label{fermioncheckR}
\end{align}
satisfying
\begin{align}
\check{\mathcal{R}}_{12}(u-v) \check{\mathcal{R}}_{23}(u)
\check{\mathcal{R}}_{12}(v)
= \check{\mathcal{R}}_{23}(v) \check{\mathcal{R}}_{12}(u)
\check{\mathcal{R}}_{23}(u-v).
\label{fermionYBEcheckRmatrix}
\end{align}
In the derivation, the following transformation laws are
of particular importance,
\begin{align}
&\sigma^{+}_{j} \sigma^{+}_{j+1} \longrightarrow c_{j}^{\dagger} c_{j+1}^{\dagger}, \nonumber \\
&\sigma^{-}_{j} \sigma^{-}_{j+1} \longrightarrow - c_{j} c_{j+1}.
\end{align}
Strictly speaking, ${\check{\mathcal{R}}_{jk}(u)}$ is defined in
(\ref{fermioncheckR}) only for ${k=j+1}$. However we extends its definition to arbitrary ${j}$ and ${k}$ by the formula
(\ref{fermioncheckR}).

Finally multiplying the products of the fermionic permutation operators
(\ref{fermionicpermutation})
\begin{align}
\mathcal{P}_{12} \mathcal{P}_{13} \mathcal{P}_{23}
= \mathcal{P}_{23} \mathcal{P}_{13} \mathcal{P}_{12},
\label{permutatorYBE}
\end{align}
to (\ref{fermioncheckR}) from the left,
we obtain
\begin{align}
\mathcal{R}_{12}(u-v) \mathcal{R}_{13}(u)
\mathcal{R}_{23}(v)
= \mathcal{R}_{23}(v) \mathcal{R}_{13}(u)
\mathcal{R}_{12}(u-v),
\end{align}
with
\begin{align}
\mathcal{R}_{jk}(u) &= \mathcal{P}_{jk} \check{\mathcal{R}}_{jk}(u)  \nonumber \\
                    &= a(u) \{ -n_j n_{k} + (1-n_j)(1-n_{k})\}
           + b(u) \{n_j (1-n_k) + (1-n_j) n_k\}
\nonumber\\
            & +c(u) (c_j^{\dagger} c_k + c_k^{\dagger} c_j)
           - d(u) (c_j^{\dagger} c_k^{\dagger} + c_j c_k).
\end{align}
This is identical to the fermionic ${R}$-operator (\ref{R}).
In this way we can obtain the fermionic ${R}$-operator (\ref{R}) from the ${R}$-matrix of the ${XYZ}$ spin model.

Just in the same way, the fermionic ${R}$-operator for the 1D
Hubbard model can be obtained from the ${R}$-matrix of the coupled spin model (\ref{coupledspin}).

redpajama_set_name: RedPajamaArXiv


text:
River Regulation, Flood Control, and Water Conservation and Utilization: Hearings Before the Committee on Commerce, United States Senate, Sixty-fourth Congress, Second Session, on H.R. 14777, an Act to Provide for the Control of Floods of the Mississippi River and of the Sacramento River, California, and for Other Purposes, and S. 5736, a Bill to Promote Interstate Commerce, Agriculture, and the General Welfare by Providing for the Development and Control of Waterways and Water Resources, for Water Conservation, for Flood Control, Prevention, and Protection; for the Application of Flood Waters to Beneficial Uses; and for Cooperation in Such Work with States and Other Agencies, and for Other Purposes, Together with Hearings, Reports, and Documents of Preceding Congresses .... Dec. 19-22, 1916
United States. Congress. Senate. Committee on Commerce
U.S. Government Printing Office, 1917 - 430 pagini
acres Agriculture amendment amount annually appropriation authorized banks basin benefits bill building bureaus called Canal carry CHAIRMAN channel Chief commerce commission committee complete Congress connection conservation consideration considered construction continue cooperation coordination cost course Department drainage effect engineers entire expended fact flood flood control flood waters flow follows forests fund Government House important improvement increase interests irrigation lands levee lower matter means ment miles Mississippi River Missouri natural navigation necessary needed Ohio organization overflow Panama passed plans possible practically President prevention problem proposed protection question reclamation recommend reference regard region regulation relating Representatives reservoirs result river and harbor river regulation Secretary secure Senator NEWLANDS soil storage streams Survey thing tion transportation tributaries United Valley various water power watersheds waterways
Pagina 427 - Georgia, Idaho, Illinois, Indiana, Iowa, Kansas, Kentucky, Louisiana, Maine; Maryland, Massachusetts, Michigan, Minnesota, Mississippi, Missouri, Montana, Nebraska, New Hampshire, New Jersey, New York, North Carolina, North Dakota, Ohio, Oklahoma, Oregon, Pennsylvania, Rhode...
Pagina 83 - Be it further resolved that the Secretary be instructed to send a copy of this resolution to the President of the United States.
Pagina 64 - reclamation fund," to be used in the examination and survey for and the construction and maintenance of irrigation works for the storage, diversion, and development of waters for the reclamation of arid and semiarid lands in the said States and Territories...
Pagina 71 - We declare our firm conviction that this conservation of our natural resources is a subject of transcendent importance which should engage unremittingly the attention of the nation, the States, and the people in earnest cooperation.
Pagina 255 - An Act to provide for the control of the floods of the Mississippi River and of the Sacramento River, California, and for other purposes...
Pagina 181 - That a Council of National Defense is hereby established for the coordination of industries and resources for the national security and welfare, to consist of the Secretary of War, the Secretary of the Navy, the Secretary of the Interior, the Secretary of Agriculture, the Secretary of Commerce, and the Secretary of Labor.
Pagina 181 - Commission, consisting of the Secretary of War, the Secretary of the Interior, the Secretary of Agriculture, and two members of the Senate, to be selected by the President of the Senate, and two members of the House of Representatives, to be selected by the Speaker...
Pagina 115 - ... correct, permanently locate, and deepen the channel and protect the banks of the Mississippi River; improve and give safety and ease to the navigation thereof ; prevent destructive floods ; promote and facilitate commerce, trade, and the postal service...
Pagina 49 - Agriculture is hereby authorized, and on such conditions as he deems wise, to stipulate and agree with any State or group of States to cooperate in the organization and maintenance of a system of fire protection on any private or State forest lands within such State or States and situated upon the watershed of a navigable river...
Pagina 80 - Vice President of the United States, to the Speaker of the House of Representatives, and to each Senator and Representative from California in the Congress of the United States.
Titlu River Regulation, Flood Control, and Water Conservation and Utilization: Hearings Before the Committee on Commerce, United States Senate, Sixty-fourth Congress, Second Session, on H.R. 14777, an Act to Provide for the Control of Floods of the Mississippi River and of the Sacramento River ...
Autor United States. Congress. Senate. Committee on Commerce
Editor U.S. Government Printing Office, 1917
Exemplar original de la Biblioteca Congresului SUA
Scanată la 8 aug. 2006

redpajama_set_name: RedPajamaCommonCrawl


text:
OutKast, Beck, and Jack White Lead Forecastle Festival 2014
Written By Cynthia Orgel
| March 4, 2014 - 4:28 pm | Updated 8 years ago
Home » News » OutKast, Beck, and Jack White Lead Forecastle Festival 2014
Last summer at the 11th annual Forecastle Festival in Louisville, Kentucky, Big Boi spit rhymes and captivated a sea of fans, all while sporting a leg brace. This year, he returns to the Waterfront Park the weekend of July 18 through 20, and will not only be more mobile, but his other (hip-hop) half, André 3000, will share the stage with him. That's right: OutKast are headlining another music festival. The ATLiens are topping the bill at the 12th annual Forecastle Fest, alongside Beck and Jack White. Louisville's own legendary post-rock group, Slint, will reunite at the three-day showcase for their first hometown show in almost a decade.
Here's the rest of the roster: the Replacements, Spoon, Against Me!, Charli XCX, Band of Horses, Jenny Lewis, tUnE-yArDs, Sharon Van Etten, Action Bronson, Sun Kil Moon, the Black Lips, Ray Lamontagne, Nickel Creek, Dwight Yoakam, Sharon Jones & the Dap-Kings, Gary Clark Jr., St. Lucia, Flume, Local Natives, Hurray for the Riff Raff, Avey Tare's Slasher Flicks, Twenty One Pilots, Jason Isbell & the 400 Unit, Trampled By Turtles, Lord Huron, JJ Grey & Mofro, Brett Dennen, Tourist, Lucius, Nightmares on Wax (live), Claude VonStroke, Spanish Gold, Drew Holcomb & the Neighbors, Reignwolf, Waxahatchee, Chrome Sparks, Hayes Carll, Willie Watson, Foy Vance, the Districts, Benjamin Booker, Johnnyswim, Leagues, Old Baby, Blue Sky Black Death, Matrimony, Mount Moriah, Seluah, the Weeks, JaLin Roze, Young & Sick, Jill Andrews, the Wans, and Goodbye June.
GA and VIP weekend passes go on sale this Friday, March 7, at noon EST. Information on single-day passes and late-night show tickets will be made available soon. Stay tuned for more details.
Beck, Phoenix Join Forces for 'Summer Odyssey' Tour
Cynthia Orgel
Beck, Forecastle Festival, jack white, Music News, Outkast
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Elton John Lyricist Bernie Taupin's Memoir, Scattershot Due This Fall