%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LaTeX file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Uniqueness of Ground State
% in Exactly Solvable
% Hubbard, Perodic Anderson, and Emery Models
%
% LJ5041
% Hal Tasaki
%
% Department of Physics, Gakushin University,
% Mejiro, Toshima-ku, Tokyo 171, Japan
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{article}
\oddsidemargin -0mm\evensidemargin -0mm\topmargin -12mm
\textheight 654pt\textwidth 458pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{psTexture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%\begin{center}
\noindent
{\Large\bf
Uniqueness of Ground State in
Exactly Solvable \\Hubbard, Periodic Anderson, and Emery Models\\}
\par\noindent
{\large Hal Tasaki}\par\noindent
{\small\em Department of Physics, Gakushuin University,
Mejiro, Toshima-ku, Tokyo 171, JAPAN}
%\end{center}
\par\bigskip\noindent
We study the exactly solvable strongly interacting electron
models recently introduced by Brandt and Giesekus, and further
generalized by other authors.
For a very general class of models, including the Hubbard, the
periodic Anderson, and
the Emery models with certain hopping matrices and infinitely large
on-site
Coulomb repulsion on d-sites, we prove that the known exact ground
sate is
indeed the unique ground state
for a certain electron number.
The uniqueness guarantees that one can discuss physics of various
strongly
interacting electron systems by analyzing the exact ground states.
%\par\noindent PACS Numbers: 71.28+d, 75.10.Lp, 02.90+p
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par\bigskip\bigskip\noindent
In spite of considerable interest, various aspects of strongly interacting
electron systems remain to be understood.
Recently Brandt and Giesekus \cite{BG} introduced models of tight
binding electrons with infinitely large on-site Coulomb repulsion,
in which they were able to write down the exact ground states.
Some generalization of the models were found by Mielke \cite{M}, by Strack
\cite{S}, and by Tasaki \cite{T}.
In particular the cell construction in \cite{T}, which we shall use in the
present Letter as well, provides the most general treatment of the class of
the solvable models \cite{FNT}.
The class of models now includes various versions of the
Hubbard, the periodic
Anderson, and the
Emery models with specific hopping matrices and $U=\infty$ on
d-sites.
Unlike in many solvable models, the ground states of Brandt and Giesekus have
nontrivial structure, and are expected to contain rich physics.
Although the solvable models are in some sense artificial,
it is expected that the models provide typical examples which exhibit
various interesting phenomena generated by interplay between strong
Coulomb interaction and kinetic motion of electrons.
In \cite{T} it was pointed out that the exact ground states have the
so-called RVB (resonating-valence-bond) structure, and was speculated
that some of them exhibit superconductivity \cite{Tnote}.
In \cite{BL1}, Bares and Lee performed a detailed analysis of the
solvable Emery model in one dimension, and discussed its relevance to
the physics of the Kondo insulator.
In \cite{BG,M,T}, the exact ground state was speculated to be the unique
ground state of each model, but no proof was given.
This has been a serious disadvantage when one wishes to draw physical
conclusions by analyzing the exact ground states.
Recently Bares and Lee \cite{BL1,BL2} announced that they obtained a
proof of uniqueness of the ground state for some one-dimensional
models.
In the present Letter,
we prove that the exact ground state is nonvanishing and is indeed the
unique ground state (in a finite volume) for a quite general class of
models with the number of electrons fixed to the twice of the number
of ``cells'' in the lattice.
(Our argument is different from that of Bares and Lee \cite{BL2}.)
The class includes {\em all\/} the concrete models (of
Brandt-Giesekus type) considered in literature, among which are the
two and three dimensional Hubbard and periodic Anderson models
introduced in
\cite{BG,M}, the two dimensional Emery \cite{S} and Hubbard \cite{T}
models which mimic the CuO$_2$
structure, the one dimensional Emery and
the periodic Anderson
models \cite{S,BL1}.
(See Fig.1.)
We believe that the present results provide a basis for future studies, in
which one extracts various physics out of the exact ground states
of the Brandt-Giesekus type.
\par\bigskip\noindent
{\em The models and main results:}
We shall describe the solvable models in their most general forms.
We first construct the lattice to work with.
A cell $C$ is a finite set of sites, where each site $x\in C$ is
classified
either as a $U=\infty$ site (or a d-site) which can carry at most one
electron, or a
$U=0$ site (or a p-site) which can carry at most two electrons.
The lattice $\Lambda_N$ is constructed by starting from the empty set
$\Lambda_0=\phi$, and successively adding cells $C_1, C_2,
\ldots, C_N$,
where the cells need not be identical.
When adding a new cell $C_i$ to the lattice $\Lambda_{i-1}$ (which
consists of
$C_1,\ldots, C_{i-1}$), we identify some (including none) of the sites
in $C_i$
with sites in $\Lambda_{i-1}$ in a one-to-one fashion, noting that sites
of the
same type should be identified.
(See Fig.1.)
The only nontrivial requirement (which is introduced in
the present Letter) in the construction is the following
``three electrons condition''.
{\em When a new cell $C_i$ is added to $\Lambda_{i-1}$ to form
$\Lambda_i$, we have either;
1) the sites
in $C_i$ which are not identified with sites in $\Lambda_{i-1}$ can
carry at least three electrons,
or 2) the unidentified site in $C_i$ is a single p-site.}
This is quite a reasonable requirement \cite{FNthree}, which is satisfied
in all the concrete models (of Brandt-Giesekus type) studied in
literature \cite{FNspin}.
We shall consider an electron system on the resulting lattice
\cite{FNunion}
$\Lambda_N=C_1\cup\cdots\cup C_N$.
For a site $x\in\Lambda_N$, $c_{x\sigma}$, $c^\dagger_{x\sigma}$, and
$n_{x\sigma}=c^\dagger_{x\sigma}c_{x\sigma}$ denote the
annihilation, the
creation, and the number operators, respectively, of an electron at
site $x$
with spin $\sigma=\uparrow,\downarrow$.
The states are constructed by operating $c^\dagger_{x\sigma}$ with
various
$x$ and $\sigma$ to the vacuum state $\Phi_0$, but we only allow the
states
which satisfy $n_{x\uparrow}n_{x\downarrow}\Phi=0$ for any $U=\infty$
site $x$.
This restriction effectively takes into account infinitely large
on-site Coulomb repulsion on d-sites.
With a cell $C$, we associate the Hamiltonian
\begin{equation}
H(C)=\sum_{\sigma=\uparrow,\downarrow}
\alpha_\sigma(C){\cal P}(C)\alpha^\dagger_\sigma(C),
\label{HC}
\end{equation}
with
\begin{equation}
\alpha_\sigma(C)=\sum_{x\in C} \lambda^{(C)}_x c_{x\sigma},
\label{alpha}
\end{equation}
where $ \lambda^{(C)}_x$ are nonvanishing real coefficients \cite{FNcomplex}.
The projection operator onto the space of the allowed sates is
\begin{equation}
{\cal P}(C)=\prod_{x\in C_{U=\infty}}
(1-n_{x\uparrow}n_{x\downarrow}),
\label{PC}
\end{equation}
where $ C_{U=\infty}$ is the set of $U=\infty$ sites in $C$.
Then the Hamiltonian for the whole lattice $\Lambda_N$ is
\begin{equation}
H_N=\sum_{i=1}^NH(C_i),
\label{HN}
\end{equation}
where the coefficients $\lambda_x^{(C_i)}$ are chosen and fixed
independently in each cell.
Before discussing the ground state of the model, we shall rewrite the
Hamiltonian (\ref{HN}) into the ``standard form'' (\ref{HN2}).
(This rewriting is not necessary for the uniqueness proof.)
Note that there are operator identities \cite{FNopid}
$ c_{y\sigma}{\cal P}(C) c^\dagger_{x\sigma}
=-{\cal P}(C) c^\dagger_{x\sigma} c_{y\sigma}{\cal P}(C)$
for $x\ne y\in C$,
$ c_{x\sigma}{\cal P}(C) c^\dagger_{x\sigma}
={\cal P}(C)(1-n_{x\uparrow}-n_{x\downarrow}){\cal P}(C)$
for $x\in C_{U=\infty}$, and
(trivially)
$ c_{x\sigma}{\cal P}(C) c^\dagger_{x\sigma}
={\cal P}(C)(1-n_{x\sigma}){\cal P}(C)$
for $x\not\in C_{U=\infty}$.
By using these identities we find
\begin{equation}
H_C=\sum_{\sigma=\uparrow,\downarrow} \sum_{x,y\in C}
\lambda^{(C)}_x \lambda^{(C)}_y c_{y\sigma}{\cal P}(C) c^\dagger_{x\sigma}
={\cal P}(C)\left\{\epsilon(C)-\sum_{\sigma=\uparrow,\downarrow}
\sum_{x,y\in C}t^{(C)}_{xy} c^\dagger_{x\sigma} c_{y\sigma}
\right\}{\cal P}(C),
\label{HC2}
\end{equation}
where $t^{(C)}_{xy}= \lambda^{(C)}_x \lambda^{(C)}_y$ for $x\ne y$,
$t^{(C)}_{xx}=2( \lambda^{(C)}_x)^2$ if $x\in C_{U=\infty}$,
$t^{(C)}_{xx}=( \lambda^{(C)}_x)^2$ if $x\not\in C_{U=\infty}$, and
$\epsilon(C)=\sum_{x\in C}( \lambda^{(C)}_x)^2$.
By summing up this expression, we get
\begin{equation}
H_N=-E_0-{\cal P}_N\sum_{\sigma=\uparrow,\downarrow}
\sum_{x,y\in\Lambda_N}t_{xy} c^\dagger_{x\sigma} c_{y\sigma},
\label{HN2}
\end{equation}
where $t_{xy}=\sum_{i=1}^Nt^{(C_i)}_{xy}$,
$E_0=-\sum_{i=1}^N\epsilon(C_i)$,
and
\begin{equation}
{\cal P}_N=\prod_{i=1}^N {\cal P}(C_i).
\label{PN}
\end{equation}
To derive (\ref{HN2}),
we have used that fact that $H_N$ operates only on the
allowed states, which are now characterized as ${\cal P}_N\Phi=\Phi$.
The main result of the present Letter is the following.
\bigskip\noindent
{\bf Theorem:}
Consider the model on $\Lambda_N$.
Then we have;
\par\noindent
i) Let $N_\uparrow,N_\downarrow$ be nonnegative integers with
$N_\uparrowFrom the simple operator identity
${\cal P}(C_i)\alpha^\dagger_\sigma(C_i)
{\cal P}_N\alpha^\dagger_\sigma(C_i)
={\cal P}_N(\alpha^\dagger_\sigma)^2=0$,
and the fact that the Hamiltonian
(\ref{HN}) is nonnegative, it obviously follows that the state
(\ref{Phi0}), if nonvanishing, is an exact ground sate of the model.
More delicate issues, which are solved in the present Letter, have been
to show the
state (\ref{Phi0}) is nonvanishing and is the unique ground state.
By using the standard argument based on the $SU(2)$ invariance of the model,
the above theorem implies the following.
\par\bigskip\noindent
{\bf Corollary:}
Consider the model on $\Lambda_N$.
In the sector with the total electron number $2N$,
the ground state of the
Hamiltonian (\ref{HN}) is unique, and is given by (\ref{Phi0}).
\par\bigskip
For the electron number strictly greater than $2N$, one can easily see (by
construction) that the ground sates are degenerate \cite{BG,M}.
Although we do not study such cases in detail, we remark that i) of the above
theorem has an immediate consequence that the ground states for $2N+n$
electrons has total spin not greater than $n/2$.
\par\bigskip\noindent
{\em Proof:}
We shall prove the theorem by induction.
We first set $N=0$ and $\Lambda_0=\phi$.
Then both i) and ii) are trivial, where in the latter we let
$A^\dagger_0=1$.
We assume the statements of the theorem for
$\Lambda_N=C_1\cup\cdots\cup C_N$,
and
prove them for $\Lambda_{N+1}=C_1\cup\cdots\cup
C_{N+1}=\Lambda_N\cup
C_{N+1}$.
In what follows, we abbreviate $\Lambda_N$ and $C_N$ as
$\Lambda$ and $C$,
respectively.
We start from ii).
Let $\Phi$ be a state on $\Lambda_{N+1}=\Lambda\cup C$ with
$(N+1)$ up
electrons and
$(N+1)$ down electrons, and assume that
\begin{equation}
H_{N+1}\Phi=0.
\label{HPhi=0}
\end{equation}
Since $H_{N+1}$ is a sum of nonnegative operators, we find that
(\ref{HPhi=0}) is equivalent to the condition that both
\begin{equation}
H_N\Phi=0,
\label{HNPhi=0}
\end{equation}
and
\begin{equation}
{\cal P}(C)\alpha^\dagger_\sigma(C)\Phi=0,
\label{Palpha}
\end{equation}
for $\sigma=\uparrow,\downarrow$ hold.
Let $\tilde{C}$ be the set of sites in $C$ which are not identified with
sites in
$\Lambda$ when one constructs $\Lambda_{N+1}=\Lambda\cup C$.
(We then have $\Lambda\cap\tilde{C}=\phi$ and
$\Lambda\cup\tilde{C}=\Lambda_{N+1}$,
where
$\cup$ means the simple union.)
We can decompose $\Phi$ according to the numbers of up and down
electrons
(denoted as $ n_\uparrow$ and $n_\downarrow$, respectively)
contained in $\tilde{C}$ as
\begin{equation}
\Phi=\sum_{ n_\uparrow=0,1,2,\ldots}
\sum_{n_\downarrow=0,1,2,\ldots}
\Phi_{ n_\uparrow,n_\downarrow}.
\label{dec1}
\end{equation}
In the state $\Phi_{ n_\uparrow,n_\downarrow}$ with $n_\sigma>1$ for
$\sigma=\uparrow$ or $\downarrow$, the number of
spin-$\sigma$ electrons in $\Lambda$ is
strictly less than
$N$.
Then (\ref{HNPhi=0}) and the assumed i) for $\Lambda$ imply
$\Phi_{ n_\uparrow,n_\downarrow}=0$.
Thus the decomposition (\ref{dec1}) becomes
\begin{equation}
\Phi=\Phi_{1,1}+\Phi_{1,0}+\Phi_{0,1}+\Psi_{0,0},
\label{dec2}
\end{equation}
with
\begin{equation}
\Phi_{1,1}=\sum_{x,y\in\tilde{C}}
c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
\Psi_{1,1}(x,y),
\label{Phi11}
\end{equation}
and
\begin{equation}
\Phi_{1,0}=\sum_{x\in\tilde{C}}
c^\dagger_{x\uparrow}\Psi_{1,0}(x),\quad
\Phi_{0,1}=\sum_{y\in\tilde{C}}
c^\dagger_{y\downarrow}\Psi_{0,1}(y),
\label{Phi1001}
\end{equation}
where various $\Psi$ denote states in which electrons live only on
$\Lambda$.
Since $H_N$ acts only on $\Lambda$, the condition (\ref{HNPhi=0})
implies (among
other relations) $H_N\Phi_{1,1}=0$, which further reduces to
$H_N\Psi_{1,1}(x,y)=0$ for each $x,y\in\tilde{C}$.
Noting that the state $\Psi_{1,1}(x,y)$ has $N$ up electrons and $N$
down
electrons, the assumed ii) for $\Lambda$ implies
\begin{equation}
\Psi_{1,1}(x,y)=\psi_{x,y}{\cal P}_NA^\dagger_N\Phi_0,
\label{Psi11=}
\end{equation}
where $\psi_{x,y}$ are undetermined coefficients.
Next we examine the condition (\ref{Palpha}), which, with the
decomposition
(\ref{dec2}), now reads
\begin{equation}
{\cal P}(C)\sum_{z\in C}
\lambda^{(C)}_z c^\dagger_{z\sigma}
(\Phi_{1,1}+\Phi_{1,0}+\Phi_{0,1}+\Psi_{0,0})=0,
\label{Palpha2}
\end{equation}
for $\sigma=\uparrow,\downarrow$, where we used the definition
(\ref{alpha}) of $\alpha_\sigma(C)$.
We shall again decompose the left-hand side of (\ref{Palpha2})
according to
the numbers of up and down electrons in $\tilde{C}$.
Clearly each state in the decomposition must vanish independently.
Let us suppose that the cell $C$ satisfies 1) of the ``three electrons
condition'', and start from the sector with three electrons in
$\tilde{C}$.
The contribution comes only from $\Phi_{1,1}$, and we get
from (\ref{Palpha2}) that
\begin{eqnarray}
&&0={\cal P}(C)\sum_{z\in\tilde{C}}
\lambda^{(C)}_z c^\dagger_{z\sigma}
\sum_{x,y\in\tilde{C}}\psi_{x,y}
c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
{\cal P}_N A^\dagger_N\Phi_0
\nonumber \\
&&=\sum_{x,y,z\in\tilde{C}}\psi_{x,y}
\lambda^{(C)}_z{\cal P}(C)
c^\dagger_{z\sigma} c^\dagger_{x\uparrow}
c^\dagger_{y\downarrow}{\cal P}_N
A^\dagger_N\Phi_0,
\label{eq0}
\end{eqnarray}
where we used explicit form of $\Psi_{1,1}(x,y)$ in (\ref{Psi11=}).
The equation is easy to analyze since the right-hand side factorizes
into the
states on $\tilde{C}$ and on $\Lambda$.
It is also essential that ${\cal P}_N A^\dagger_N\Phi_0$ is
nonvanishing from the
assumed ii)
for $\Lambda$.
By setting $\sigma=\uparrow$, (\ref{eq0}) yields
\begin{equation}
\psi_{x,y} \lambda^{(C)}_z=\psi_{z,y} \lambda^{(C)}_x,
\label{eq1}
\end{equation}
for the compatible combinations of $x,y,z\in\tilde{C}$, i.e., those satisfy
$x\ne y$, and $z\ne x,y$ if $z\in C_{U=\infty}$.
By setting $\sigma=\downarrow$ in (\ref{eq0}), we get
\begin{equation}
\psi_{x,y} \lambda^{(C)}_z=\psi_{x,z} \lambda^{(C)}_y,
\label{eq2}
\end{equation}
again for $x,y,z\in\tilde{C}$ with similar compatible conditions.
Because of 1) of the ``three electrons condition'', we see that the sets of
compatible $(x,y,z)$ in (\ref{eq1}), (\ref{eq2}) are not empty.
Then the equations (\ref{eq1}), (\ref{eq2}) are easily found to
possess the
unique (apart from multiplication by a constant) solution
\begin{equation}
\psi_{x,y}= \lambda^{(C)}_x \lambda^{(C)}_y.
\label{psi}
\end{equation}
When the cell $C$ satisfies 2) of the ``three electrons condition'',
the analysis is trivial.
Since we must have $x=y$ in (\ref{Phi11}), we can set
$\psi_{xx} = (\lambda^{(C)}_x)^2$ to be consistent with
(\ref{psi}).
In the following, we do not have to distinguish between the cases 1)
and 2).
Next we set $\sigma=\downarrow$ in (\ref{Palpha2}) and consider the
sector with
one up electron and one down electron in $\tilde{C}$.
Both $\Phi_{1,1}$ and $\Phi_{1,0}$ contribute, and we get
\begin{equation}
\sum_{x,y\in\tilde{C}}\sum_{z\in\delta C} \lambda^{(C)}_x
\lambda^{(C)}_y \lambda^{(C)}_z{\cal P}(C) c^\dagger_{z\downarrow}
c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
{\cal P}_NA^\dagger_N\Phi_0
+\sum_{x,z\in\tilde{C}} \lambda^{(C)}_z{\cal P}(C)
c^\dagger_{z\downarrow} c^\dagger_{x\uparrow}\Psi_{1,0}(x)=0,
\label{eq3}
\end{equation}
where $\delta C=C\backslash\tilde{C}$ (i.e., the set of sites in $C$
identified
with those in
$\Lambda$).
We also used the solution (\ref{psi}).
By using the operator identities in \cite{FNopid}, we find that
${\cal P}(C) c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
c^\dagger_{z\downarrow}{\cal P}_N={\cal P}(C){\cal P}_z
c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
c^\dagger_{z\downarrow}
{\cal P}_z{\cal P}(\Lambda\backslash\{z\})
={\cal P}(C) c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
{\cal P}_z c^\dagger_{z\downarrow}{\cal P}_z
{\cal P}(\Lambda\backslash\{z\})
={\cal P}(C) c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
{\cal P}_N c^\dagger_{z\downarrow}$, where
${\cal P}(\Lambda\backslash\{z\})
=\prod_{x\in(\Lambda\backslash\{z\})_{U=\infty}}
(1-n_{x\uparrow}n_{x\downarrow})$ commutes with
$ c^\dagger_{z\downarrow}$.
Then the equation (\ref{eq3}) can be rewritten as
\begin{equation}
\sum_{x,y\in\tilde{C}} \lambda^{(C)}_x \lambda^{(C)}_y
{\cal P}(C) c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
\{{\cal P}_N\sum_{z\in\delta C}
\lambda^{(C)}_z c^\dagger_{z\downarrow}
A^\dagger_N\Phi_0-( \lambda^{(C)}_x)^{-1}\Psi_{1,0}(x)\}=0,
\end{equation}
which is again factorized, and yields
\begin{equation}
\Psi_{1,0}(x)= \lambda^{(C)}_x{\cal P}_N
\sum_{y\in\delta C} \lambda^{(C)}_y
c^\dagger_{y\downarrow} A^\dagger_N\Phi_0.
\label{Psi10}
\end{equation}
By setting $\sigma=\downarrow$ in (\ref{Palpha2}) and looking at the
same sector, we get
\begin{equation}
\Psi_{0,1}(y)=- \lambda^{(C)}_y{\cal P}_N\sum_{x\in\delta C}
\lambda^{(C)}_x c^\dagger_{x\uparrow} A^\dagger_N\Phi_0.
\label{Psi01}
\end{equation}
Finally we set $\sigma=\uparrow$ in (\ref{Palpha2}) and consider the
sector with
only one up electron in $\tilde{C}$.
We see that
$\Phi_{1,0}$ and $\Psi_{0,0}$ contribute, and by using (\ref{Psi10}),
we get
\begin{equation}
-\sum_{x\in\tilde{C}}\sum_{y,z\in\delta C}
\lambda^{(C)}_x \lambda^{(C)}_y \lambda^{(C)}_z
c^\dagger_{x\uparrow}{\cal P}_N c^\dagger_{z\uparrow}
c^\dagger_{y\downarrow} A^\dagger_N\Phi_0
+\sum_{z\in\tilde{C}} \lambda^{(C)}_z
c^\dagger_{z\uparrow}\Psi_{0,0}=0,
\end{equation}
which imply
\begin{equation}
\Psi_{0,0}={\cal P}_N\sum_{x,y\in\delta C} \lambda^{(C)}_x
\lambda^{(C)}_y c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}
A^\dagger_N\Phi_0.
\label{Psi00}
\end{equation}
By combining the solutions (\ref{Psi11=}), (\ref{psi}), (\ref{Psi10}),
(\ref{Psi01}), and (\ref{Psi00}) with the decomposition (\ref{dec2}),
(\ref{Phi11}), and (\ref{Phi1001}),
and noting that ${\cal P}(C){\cal P}_N={\cal P}_{N+1}$, we find that,
only by using
necessary
conditions for (\ref{HPhi=0}), the state $\Phi$ has been uniquely
determined
to have the desired form
\begin{equation}
\Phi={\cal P}_{N+1}\left(\prod_{\sigma=\uparrow,\downarrow}
\alpha^\dagger_\sigma(C)\right) A^\dagger_N\Phi_0
={\cal P}_{N+1}A^\dagger_{N+1}\Phi_0,
\label{Phi}
\end{equation}
which clearly satisfies (\ref{HPhi=0}).
It only remains to show $\Phi$ is nonvanishing.
>From (\ref{Phi}), we find
\begin{equation}
\Phi_{1,1}=\left({\cal P}(C)\sum_{x,y\in\tilde{C}}
\lambda^{(C)}_x \lambda^{(C)}_y c^\dagger_{x\uparrow}
c^\dagger_{y\downarrow}\right){\cal P}_N
A^\dagger_N\Phi_0.
\end{equation}
Noting that
${\cal P}(C)\sum_{x,y\in\tilde{C}} \lambda^{(C)}_x \lambda^{(C)}_y
c^\dagger_{x\uparrow} c^\dagger_{y\downarrow}\Phi_0$ is
nonvanishing due to
the ``three electrons condition'', and ${\cal P}_N A^\dagger_N\Phi_0$
is nonvanishing
due to the
assumed ii)
for $\Lambda$, we find that $\Phi_{1,1}$ (which may be regarded as
the direct product
of these sates) is nonvanishing as well.
Since the states in the decomposition (\ref{dec2}) are mutually
orthogonal,
we have shown that $\Phi$ is nonvanishing.
The desired ii) in the theorem has been proved.
To prove i), we assume that
$\Phi$ is a state on $\Lambda_{N+1}=\Lambda\cup C$ with the
number of up or down electrons strictly less than $N+1$, and
that $\Phi$ satisfies
(\ref{HPhi=0}).
Again we decompose $\Phi$ as in (\ref{dec1}).
The condition (\ref{Palpha}) implies that there is at least one
combination
$( n_\uparrow,n_\downarrow)$ with $ n_\uparrow\ge1$,
$n_\downarrow\ge1$ such that
$\Phi_{ n_\uparrow,n_\downarrow}\ne0$.
This, however, contradicts with the conclusion from (\ref{HNPhi=0})
and the
assumed i) for $\Lambda$ that
$\Phi_{ n_\uparrow,n_\downarrow}=0$ whenever
$ n_\uparrow\ge1$,
$n_\downarrow\ge1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par\bigskip\bigskip
I wish to thank Pere-Anton Bares and Patrick Lee for stimulating
correspondence.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{thebibliography}{99}
\bibitem{BG}
U. Brandt and A. Giesekus, Phys. Rev. Lett. {\bf 68}, 2648 (1992).
\bibitem{M}
A. Mielke, J. Phys. {\bf A25}, 6507 (1992).
\bibitem{S}
R. Strack, Phys. Rev. Lett. {\bf 70}, 833 (1993).
\bibitem{T}
H. Tasaki, Phys. Rev. Lett. {\bf 70}, 3303 (1993).
\bibitem{FNT}
In \cite{T}, models only with d-sites (see below) was considered.
By relaxing this condition (as we do here), one gets literally the most
general description of the solvable models.
\bibitem{Tnote}
In \cite{T} it was claimed, directly below eq. (8), that a loop
configuration with nonvanishing contribution must contain a loop
which has the two source bonds in it.
This claim is incorrect in general since there is a contribution
from a configuration with a chain of ``connected'' loops
containing the source bonds.
The calculation in \cite{T} for the tree models is not
affected by this error.
\bibitem{BL1}
P.-A. Bares and P.A. Lee, preprint.
\bibitem{BL2}
P.-A. Bares and P.A. Lee, private communication.
\bibitem{FNthree}
In some models, we have to impose open boundary conditions to fulfill
the ``three electrons condition''.
Although one can construct models in which the ``three electrons
condition'' is violated, no such model of physical interest has been
proposed.
We note, in particular, that if the condition is violated for all
$i=1,\ldots,N$, then the resulting lattice can contain at most $2N$
electrons.
Therefore the only possibilities are that there is no state with $2N$
electrons, or that the lattice is completely filled by electrons and the
ground states are degenerate.
(Note that we always have some d-sites, on which we can put either up
or down electron.)
\bibitem{FNspin}
In a model of ``electrons'' with spin $S$ as in \cite{S},
we replace ``three'' in 1) by $2(S+1)$.
Then the uniqueness theorem can be extended in a trivial manner.
\bibitem{FNunion}
Throughout the present Letter, the symbol $\cup$
implicitly means
that proper
identifications of sites are made.
\bibitem{FNcomplex}
By letting the coefficients $ \lambda^{(C)}_x$ complex,
we get models which cannot be mapped to the models with real
$ \lambda^{(C)}_x$ by
gauge transformations.
Here we do not go into details of such models with complex
hoppings.
\bibitem{FNopid}
Let ${\cal P}_x=1-n_{x\uparrow}n_{x\downarrow}$.
It is easy to verify that
${\cal P}_x c^\dagger_{x\sigma}{\cal P}_x={\cal P}_x
c^\dagger_{x\sigma}$, and $[{\cal P}_x, c^\dagger_{y\sigma}]=0$,
$[{\cal P}_x,{\cal P}_y]=0$ for $x\ne y$.
Then
$ c_{y\sigma}{\cal P}_y{\cal P}_x c^\dagger_{x\sigma}
={\cal P}_y c_{y\sigma}{\cal P}_y{\cal P}_x
c^\dagger_{x\sigma}{\cal P}_x
={\cal P}_x{\cal P}_y c_{y\sigma}
c^\dagger_{x\sigma}{\cal P}_x{\cal P}_y$,
which proves the first identity.
The second identity follows from
$n_{x\sigma} c^\dagger_{x\sigma}= c^\dagger_{x\sigma}$ and
$(n_{x\sigma})^2=n_{x\sigma}$.
\end{thebibliography}
%\newpage\centerline{\psfigs{lattice.ps}{750}}
\par\bigskip\bigskip\noindent
{\bf Fig. 1.}
Examples of models (lattice structures) to which our uniqueness
theorem apply.
A d-site with $U=\infty$ is denoted by $\bullet$, and a p-site with
$U=0$ by $\circ$.
On the left of each lattice is the corresponding unit cell.
The models are; (a) the one-dimensional periodic Anderson model
\cite{S}, (b) the two-dimensional CuO$_2$-like model with extra
hopping between O-sites \cite{BG}, and (c) the two-dimensional
extended Emery
model \cite{S}.
These models become solvable by choosing appropriate hopping
matrices and filling factor.
We fix the electron number equal to the twice
of the number of cells in the lattice. The ``three electrons condition'' is
satisfied in the models (a) and (b) with open boundary conditions, and
in the model (c) with open or periodic boundary conditions.
\end{document}