% PRL on `Electronic model for superconductivity'
% draft for the revised version Sept. 1992
% Fabian Essler, Vladimir Korepin, Kareljan Schoutens
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\begin{document}
\baselineskip=15pt
\null\vskip -2cm
\hfill {ITP-SB-92-20, May 1992}
\vskip 1.5cm
\begin{center}
{\Large Electronic model for superconductivity%
\footnote{Work supported in part by NSF grant PHY-9107261}}\\
\vskip 2.0cm
{\large Fabian H.L. E\char'31 ler}\\
\vspace{2mm}
{\large Vladimir E. Korepin}\\
\vspace{2mm}
{\large and}\\
\vspace{2mm}
{\large Kareljan Schoutens}\\
\vskip .5cm
{\sl Institute for Theoretical Physics\\
State University of New York at Stony Brook\\
Stony Brook, NY 11794-3840, U.S.A.}
\vskip 1.5cm
{\bf Abstract}
\end{center}
\vspace{.3cm}
%\baselineskip=18pt
\baselineskip=32pt
\noindent
We consider a model of strongly correlated electrons that exhibits
superconductivity. It differs from the Hubbard model by nearest
neighbour interactions. We find the ground state wave function
(in one, two or three dimensions) and show it to be superconducting for
attractive and moderately repulsive on-site interaction.
\vfill
\noindent PACS \ \
71.20.Ad\ \ % Electron states: developments in mathematical
% and computational techniques
74.20-Z\ \ %Theory of Superconductivity
75.10.Jm\ \ % Magnetic properties and materials: Quantised spin models
\newpage
%\baselineskip=18pt
\baselineskip=30pt
The phenomenon of high-$T_c$ superconductivity has led to
an increased interest in theoretical models for superconductivity
other than the BCS theory.
In \cite{ARZ} P.W. Anderson and F.C. Zhang and T.M. Rice proposed that
superconductivity can occur in models based on purely electronic
interactions.
In this letter we discuss the electronic model that we introduced in
\cite{EKS} and show that it is superconducting.
We determine the ground state structure at zero temperature and
investigate under what conditions it exhibits superconductivity.
We will prove that for negative $U$ (attractive case) the model has
a unique ground state which has ODLRO and is thus superconducting.
This result holds for lattices of arbitrary dimension. We will
also discuss the phase diagram for positive $U$,
where we will find that superconductivity persists if the repulsion
$U$ is smaller than a certain critical value $U_c$. The neighbouring
phases are a phase of the supersymmetric $t$-$J$ model, and an
insulator phase.
\vskip .3cm
Electrons on a lattice are described by operators $c_{j,\sigma}\ $,
$j=1,\ldots,L$, $\sigma=\up,\down$, where $L$ is the total number of
lattice sites.
These are canonical Fermi operators satisfying
$\{ c^\dagger_{i,\sigma} , c_{j,\tau} \} = \delta_{i,j}
\delta_{\sigma,\tau}$. The state $\vac $ (the Fock vacuum) satisfies
$c_{i,\sigma} \vac = 0$. By $n_{i,\si}= c^\dagger_{i,\si} c_{i,\si}$
we denote the number operator for electrons with spin $\si$ on site
$i$ and we write $n_i=n_{i,\up} + n_{i,\down}$.
The hamiltonian of the new model on a general
$d$-dimensional lattice can be written as
\be
H = H^0
+ U \, \sum_{j=1}^L (n_{j,\up}-\half)(n_{j,\down}-\half)
- \mu \, \sum_{j=1}^L n_j\ ,
\label{hamil}
\ee
where $H^0 = - \sumnn \, H^0_{j,k} \ \ (\langle j,k \rangle\ {\rm
are}\ {\rm nearest\ neighbours})$, and
\bea
H^0_{j,k} &=&
\cd_{k,\up} c_{j,\up}(1-n_{j,\down}-n_{k,\down})
+ \cd_{j,\up} c_{k,\up}(1-n_{j,\down}-n_{k,\down})
\nonu
&& + \cd_{k,\down} c_{j,\down}(1-n_{j,\up}-n_{k,\up})
+ \cd_{j,\down} c_{k,\down}(1-n_{j,\up}-n_{k,\up})
\nonu
&& + \half (n_j - 1)(n_k - 1)
+ \cd_{j,\up} \cd_{j,\down} c_{k,\down} c_{k,\up}
+ c_{j,\down} c_{j,\up} \cd_{k,\up} \cd_{k,\down}
\nonu
&& - \half (n_{j,\up}-n_{j,\down})(n_{k,\up}-n_{k,\down})
- \cd_{j,\down} c_{j,\up} \cd_{k,\up} c_{k,\down}
- \cd_{j,\up} c_{j,\down} \cd_{k,\down} c_{k,\up}
\nonu
&& + (n_{j,\up}-\half)(n_{j,\down}-\half)
+ (n_{k,\up}-\half)(n_{k,\down}-\half) \ .
\label{hamil0jk}
\eea
This hamiltonian contains kinetic terms and interaction terms that
combine those of the Hubbard model and of the $t$-$J$ model.
It also contains a hopping term for local electron pairs (spin-down and
spin-up electrons occupying the same site).
The interaction terms are very similar to the ones proposed by J.E.
Hirsch in his model of superconductivity, which was derived by a
tight-binding analysis \cite{hirsch}.
The second term in (\ref{hamil}) is the on-site Hubbard interaction
term and $\mu$ is the chemical potential. The Hubbard coupling $U$
will determine the ratio of local electron pairs to single (unpaired)
electrons in the ground state.\\
The state $\sum_{\vec{x}}\exp{(i\ {\vec k}\cdot{\vec x})}\
\cd_{{\vec x},\down}\cd_{{\vec x},\up}\ \vac$ (where $\vec x$ runs
over all sites of a $d$-dimensional lattice) is an eigenstate of the
hamiltonian (\ref{hamil}).
We call this state a {\it localon} state of momentum $\vec k$.
The hamiltonian $H^0$ has a rich symmetry structure. It is invariant
under two independent $SU(2)$ symmetries and under eight
supersymmetries. Together with the number operator for local electron
pairs and with the identity operator these symmetries generate the
superalgebra $U(2|2)$ (see \cite{EKS} for more details). The first of
the two $SU(2)$ algebras corresponds to ordinary spin; the generators
of the second $SU(2)$ algebra are
\be
\eta = \sum_{j=1}^L c_{j,\up} c_{j,\down}\ , \qquad
\etad = \sum_{j=1}^L \cd_{j,\down} \cd_{j,\up}\ , \qquad
\etaz = \sum_{j=1}^L \half (1-n_j)\ .
\label{su2pair}
\ee
It can be seen that the operator $\etad$ creates a localon of momentum
zero. The fact that it commutes with the hamiltonian $H^0$ and with
the $U$-term in (\ref{hamil}) makes
it possible to construct eigenstates of the full hamiltonian $H$
that contain a large number of zero momentum local electron pairs. Such states
will play a crucial role in our discussion below.
The hamiltonian density $H^0_{j,k}$ acts as a {\it graded
permutation} $\Pi^g_{j,k}$ of the electronic states at sites $j$ and
$k$. By `graded' we mean that there is an extra minus sign if the two
states that are permuted are both single electron states.
For example,
\be
H^0_{j,k} \, \cd_{j,\up} \vac
= \cd_{k,\up} \vac \, , \quad
H^0_{j,k} \, \cd_{j,\up} \cd_{k,\down} \vac
= - \cd_{j,\down} \cd_{k,\up} \vac \, , \quad
{\rm etc.}
\label{perm}
\ee
This property implies that the number operators ${\hat N}_\up$, ${\hat
N}_\down$ (the number operators of {\it single} electrons with given
spin) and ${\hat N}_l$ (the number operator of local electron pairs), defined by
\be
{\hat N}_\up +{\hat N}_l = \sum_{j=1}^L n_{j,\up}\ , \qquad
{\hat N}_\down + {\hat N}_l = \sum_{j=1}^L n_{j,\down}\ , \qquad
{\hat N}_l = \sum_{j=1}^L n_{j,\up}\ n_{j,\down}\ ,
\label{nums}
\ee
all commute with $H^0$, so that $H^0$ can be diagonalised within a sector
with given numbers $N_{\up}$, $N_{\down}$ and $N_l$. This implies that
our hamiltonian does not allow for decay of local electron pairs into two single
electrons.
In the sectors without local electron pairs $H^0$ reduces to the hamiltonian
of the supersymmetric $t$-$J$ model with $t=1$, $J=2$ (in our
discussion below we always consider this special case), and the model
is isomorphic to the spin-$\half$ $XXX$ model in the sector with only
vacancies and local electron pairs.
Abstract graded permutations of $2$ species of bosons and $2$ species
of fermions were first considered as a dynamical hamiltonian by
B.Sutherland in \cite{suth}, where the ground state energy for the
one-dimensional model was computed.
\vskip 4mm
Let us now discuss physical aspects of the new model, which hold in
arbitrary dimensions.
We will first establish that certain eigenstates of $H^0$ have
the property of off-diagonal long range order (ODLRO), which
is characteristic of superconductivity \cite{odlro}. Let us consider
an eigenstate $|\psi\rangle$ of $H^0$ which is a highest weight
state of the $\eta$-pairing $SU(2)$ algebra (\ref{su2pair}). It has
the properties
\be
\eta |\psi\rangle = 0, \qquad
\etaz |\psi\rangle = \half(L-N_c) |\psi\rangle \ ,
\ee
where $N_c = N_\up + N_\down + 2 N_l$ is the number of electrons in
$|\psi\rangle$. We can then construct additional eigenstates of the
form
\be
|\psi_n\rangle = (\etad)^n |\psi\rangle , \qquad n=0,1,2,\ldots, L-N_c \ .
\label{psin}
\ee
Following \cite{pairing} we consider the following off-diagonal
matrix element ($k\neq l$) of the reduced density matrix $\rho_2$
for the state $|\psi_n\rangle$
\be
(\rho_2)_{kl}
= \langle (k,\down) (k,\up) | \, \rho_2 \, | (l,\up) (l,\down) \rangle
= \frac{ \strutje \langle \psi | \eta^n \cd_{k,\down} \cd_{k,\up}
c_{l,\up} c_{l,\down} (\etad)^n | \psi \rangle}{ \str
\langle \psi | \eta^n (\etad)^n |\psi\rangle} \ .
\label{r2}
\ee
We consider the thermodynamic limit $L\longrightarrow\infty$,
$N\longrightarrow\infty$, $n\longrightarrow\infty$, where the ratios
$\frac{n}{L}$ (superconducting density) and $\frac{N_c}{L}$ (normal
state density) are kept fixed. If the matrix element
(\ref{r2}) approaches a nonzero value $A$ asymptotically at large
distances ($1\ll |k-l|\ll L$), then the state $|\psi_n\rangle$
exhibits ODLRO and is superconducting. The value $A$ can be found by
averaging over $k$ and $l$ and then using the SU(2) structure of the
generators (\ref{su2pair})
\vskip-.3mm
\be
A =\lim_{L\to\infty}\frac{1}{L^2} \sum_{k,l} (\rho_2)_{kl}
= \frac{n}{L} (1-\frac{N_c}{L}-\frac{n}{L}) \ .
\ee
The result establishes the property of ODLRO for the states $|\psi_n\rangle$.
We now consider the phase diagram of the hamiltonian $H$ at zero
temperature. Equation (\ref{hamil}) defines the model in the
framework of the grand canonical ensemble. We now change our point of
view to the canonical ensemble, dropping the chemical potential
$\mu$ from (\ref{hamil}), fixing the magnetisation to zero and the
density $D=\frac{N}{L}$ ($N=N_c+2n$ is the complete number of
electrons) to a value in the interval $0\le D\le 2$.
On the basis of the analysis below, we propose the phase diagram shown
in figure 1.
\null
\vskip 7cm
\special{psfile=fig1.ps hoffset= -40 voffset=-180 vscale=70 hscale=70}
\vskip .6cm
\centerline{Fig.1 : Ground states in the canonical ensemble}
We claim that the
ground state in the areas I and II is of the form $|\psi_n\rangle$
as in (\ref{psin}), implying that in these regions the model
is superconducting. For the region I, corresponding to negative
$U$, this is rigourously established by the following theorem.
\vskip 4mm
\noindent {\bf Theorem.} The ground state of the hamiltonian
(\ref{hamil}), with $\mu=0$ and with $U<0$, in the sector
with an even number $N$ of electrons and zero magnetisation
is unique and is given by
$ |\Psi_{N\over 2}\rangle = (\etad)^{N\over 2} \vac\ $. The ground
state energy is $E_g = U {L \over 4} - M$, where $M$ is the number
of nearest neighbour links on the lattice.
\vskip 4mm
\noindent {\bf Proof.} In the sector with $N$ electrons the
hamiltonian reads
\be
H_{(N)} = H^0 + U \sum_{j=1}^L (n_{j,\up} n_{j,\down})
+ ({L \over 4}- {N\over 2}) U .
\ee
We first consider the term $H^0$. The fact that $H^0$ is equal
to minus the sum of graded permutations shows that the energy
$E^0$ is bounded from below by $-M$. One state which saturates
this value is the empty state $\vac$. Using the $U(2|2)$ symmetry of
$H^0$ we can construct the $N$-particle state $|\Psi_{N\over 2}\rangle
= (\etad)^{N\over 2} \vac $, which by construction has the same energy
$E^0$.
%If we then use the continuous
%symmetries of the hamiltonian $H^0$, which we listed above and which
%form the superalgebra $U(2|2)$, we can generate a total of $4L$ states
%of $H^0$ energy $-M$
%\bea
%(\etad)^n \vac\ ,&& \qquad n=0,1,\ldots,L
%\nonu
%(\etad)^n \Qd_\sigma \vac\ ,&& \qquad \sigma=\up,\down,
%\quad n=0,1,\ldots,L-1
%\nonu
%(\etad)^n \Qd_\up \Qd_\down \vac\ ,&& \qquad n=0,1,\ldots,L-2 \ ,
%\label{ground states}
%\eea
%where $Q^\dagger_\sigma=\sum_{j=1}^L\cd_{j,\sigma}(1-n_{j,-\sigma})$.
%[We remark that these states form a so-called a-typical representation
%of $U(2|2)$.]
If we now take into account the remaining terms in
$H_{(N)}$, we find that they are bounded from below by the value
$U{L \over 4}$ ($U<0$), where the minimum is reached for states that
have $N\over 2$ local electron pairs. This is precisely the case for the state
$|\Psi_{N\over 2}\rangle$. This shows that $|\Psi_{N\over
2}\rangle$ is a ground state of $H$. To show that it is the {\it
unique} ground state, we should check that there are no other states
with ${N\over 2}$ local electron pairs (and no single electrons) that
saturate the
lower bound $-M$ of $H^0$.
%\footnote{It is actually possible to prove a stronger statement, which is
%that there are no states other than the $4L$ states in
%(\ref{ground states}) which saturate
%the lower bound $H^0=-M$.}
This can be proved by an elementary application of the Perron-Frobenius
theorem as follows. We consider the action of
$1-H^{(0)}=1+\sumnn \Pi^g_{j,k}$, on the space of
all states with ${N\over 2}$ local pairs and no single electrons,
which can be represented as a square matrix of size
$\left( \begin{array}{c} L \\ {N\over 2} \end{array}\right)$.
It is clear that this matrix is symmetric, that its entries
are non-negative and that there exists a positive power of the matrix
which is such that all entries are positive.
% This is because $(1+\sumnn \Pi^g_{j,k})^M$
% can be expressed as a sum over strings of permutation operators
% between neighbouring sites with positive coefficients. For
% sufficiently large $M$, any two
% sites on the lattice will be ``connected'' by such a string of
% permutation operators.
% Now any two localonic states $1$ and $2$ in the sector under
% discussion can be obtained by permutations of local electron pairs on certain
% sites. For sufficiently large $M$ $(1+\sumnn \Pi^g_{j,k})^M$ will
% contain terms wich carry $1$ into $2$, and thus all matrix elements
% of $(1+\sumnn \Pi^g_{j,k})^M$ will be greater than zero.
For this situation the Perron-Frobenius theorem guarantees that there
is a unique state with a maximal eigenvalue for $1-H^{0}$. This state,
which we already identified as $|\Psi_{N\over 2}\rangle$, is the
unique ground state of $H^0$ in the sector with ${N\over 2}$ local
pairs and thus the unique ground state of $H$ in the sector with a
total number of $N$ electrons.\ \ \qed
%\ \footnote{Uniqueness of the
%ground state for the attractive Hubbard model was proven in \cite{lieb2}.}
\vskip .3cm
To establish the phase diagram in figure $1$ for positive coupling $U$
it is more convenient to work in the grand canonical ensemble first,
and then translate the results to the $D$-$U$ plane in the canonical
ensemble. The phase diagram at zero temperature in the grand canonical
ensemble is given in figure 2.
In order to derive this diagram we rewrite the hamiltonian
(\ref{hamil}) as a function of $\mu$ and $U$ as
\be
H(\mu ,U) = H^0 - (\mu + \half U) \, (N_\up + N_\down)
- 2 \mu \, N_l +U{L\over 4}\ .
\label{ham}
\ee
We first note that (up to a constant) under the particle-hole
transformation $c^\dagger_{j,\sigma}\leftrightarrow c_{j,\sigma}$
the hamiltonian transforms according to $H(\mu, U)\longrightarrow
H(-\mu, U) -2\mu\ L$. Therefore it is sufficient to determine the
ground states for $\mu\le 0$; the ones for $\mu > 0$ can then be
obtained by a particle-hole transformation.
We also note that all eigenstates of the (supersymmetric) $t$-$J$
model are eigenstates of the hamiltonian (\ref{hamil}) as well.
Using this fact we will be able to express the ground state wave
function $|\Psi_g\rangle$ of the hamiltonian (\ref{ham}) in terms of
the ground state wave function $|t-J\rangle$ of the $t$-$J$ submodel
(in the grand canonical ensemble) and the $U(2|2)$ generators. In one
dimension the $t$-$J$ ground state wave function is known exactly
\cite{tJ} and our results become explicit.
\null
\vskip 7cm
\special{psfile=fig2.ps hoffset=-40 voffset=-180 vscale=70 hscale=70}
\vskip .5cm
\centerline{Fig.2 : Ground states in the grand canonical ensemble}
\vskip .3cm
It was proved by B. Sutherland in \cite{suth} that the
ground state energy of $H^0$, which is minus the sum over graded
permutations on four states (two bosons and two fermions), will be
equal to that of the $t$-$J$ submodel, which is minus the sum over
graded permutations on three states. Here the numbers of the two
species of fermions and the total number of bosons is fixed but
arbitrary. This theorem can be used as follows :
If no local electron pairs are present in the ground states, the hamiltonian
reduces to the $t$-$J$ hamiltonian with effective chemical potential
$\mu_{tJ} = \mu + \half U$. Whether it is energetically favourable to
have local electron pairs in the ground state is then determined exclusively by
the term $-2\mu\ N_l$ in the hamiltonian.
We recall the following results for the ground-state
of the $t$-$J$ model at zero temperature as a function of $\mu_{tJ}$
(see, for example \cite{tJ}).
For $\mu_{tJ}<0$ the ground state is the empty state. Once
$\mu_{tJ}$ becomes positive the ground state starts filling up,
with the density monotonically increasing as a function of $\mu_{tJ}$
until it reaches half-filling for a finite value $\mu_{tJ}=\mu_c$.
In one dimension the Bethe Ansatz solution gives $\mu_c=2 \log 2$.
The equations $\mu + \half U = 0$ and $\mu + \half U = \mu_c$ define
two critical lines in the $\mu$-$U$ plane for our model (see figure 2).
Let us now consider the regions $II$, $III$, $IV$ and $V$ in figure 2.
The ground states in the regions $III'$ and $V'$ will follow from the
ones in $III$ and $V$ by particle-hole correspondence. Note that below
we use the term ``$t$-$J$ submodel'' in the context of the grand
canonical ensemble ({\sl i.e.} including the $\mu_{tJ}(N_\up+N_\down)$
-term).
In region $V$ the ground state of the $t$-$J$ submodel is empty.
Furthermore, since $\mu < 0$ Sutherland's result tells us that
it is not favourable for local electron pairs to enter the ground state.
We conclude that the ground state in region $V$ is the empty state,
{\sl i.e.} $|\Psi_g\rangle =\vac$.
If we now cross the line $\mu+\half U=0$ to enter the region $III$
($-2\mu\le U\le 2(\mu_c-\mu),\ \mu <0$), there are single electrons
present in the ground state of the $t$-$J$ model.
Since $\mu$ is still negative, it is not energetically
favourable to have local electron pairs in the ground state. We conclude that in
region $III$ the ground state is that of the $t$-$J$ model (which is
presumably metallic) without
any local electron pairs, {\sl i.e.} $|\Psi_g\rangle =|t-J\rangle$.
At the line $\mu+\half U = \half U_c$ the $t$-$J$ ground state reaches
half filling and in the entire region $IV$ the $t$-$J$ half-filled
ground state is the ground state of our model.
Let us finally look at region $II$, where $0\le U\le U_c=2\mu_c$ and
$\mu=0$ ($U_c=4\log 2$ in one dimension).
Let us first consider the ground state $|t-J\rangle$ of the $t$-$J$
submodel, which is also an eigenstate of the hamiltonian
(\ref{hamil}) of energy $E$. It has a certain
filling, which varies from zero for $U=0$ to one for $U=U_c$.
As $\mu =0$ it follows from Sutherland's theorem that the ground state
energy of the hamiltonian (\ref{ham}) is equal to $E$, and thus that
$|t-J\rangle$ is a state of lowest energy of $H(0 , U)$.
Using the fact that $[\eta^\dagger, H_0]=0$, we can construct
additional states with energy $E$ of the form
$(\eta^\dagger)^{n}|t-J\rangle$. In figure 2 the segment under
consideration is thus singular, representing a large number of
possible ground states. However, if we pass to the $D$-$U$ plane in
figure 1 this singularity is resolved because the number $n$ of
local electron pairs adjusts itself to the density that is imposed. Thus a
non-zero number of local electron pairs will enter the ground state in
the region II in figure 1. The resulting ground state wave function is
of the form $|\Psi_g\rangle =(\eta^\dagger)^{n}|t-J\rangle$ and we
already showed that it is superconducting (note that $\eta
|t-J\rangle=0$).
The ground state consists of a condensate of zero-momentum local
pairs, and single non-paired electrons in a Fermi sphere.
Due to the (superconducting) condensation, the volume of the Fermi
sphere of the unpaired electrons is smaller than the volume of the
Fermi sphere for a free electron gas (Luttinger's theorem
is not applicable \cite{don}).
In sector $III$ in figure 1 the ground state is that of the $t$-$J$
submodel (no local electron pairs). In sector $III'$ the ground state is that of
the particle-hole transformed $t$-$J$ model. It can be shown to be of
the form $(\eta^\dagger)^{L-N} |t-J\rangle$, where $|t-J\rangle$ is
the ground state of the $t$-$J$ submodel for the opposite value $\mu$
(in sector $III$). Sectors $III$ and $III'$ are
separated by an energy gap of $H$ that exists at half
filling, $D=1$, for $U>U_c$. This situation is similar to the the one
in the repulsive 1-d Hubbard model, where a gap arises at half filling
\cite{liebwu}. In regions $I$ and $II$ (dotted line in fig 2) the
compressibility is infinite, which is intimately connected to the
presence of the superconducting condensate. The situation is quite
similar to Bose condensation in a free Bose gas, and can be
`regularised' by a perturbation of the hamiltonian.
For the attractive case we also determined the wavefunction of the
supercurrent in a circular wire, threaded by a magnetic field
\cite{tbp} (see also \cite{flux}). We found it to be a bound state of
all local electron pairs . Its wave function is equal to the one of a string
solution in the one dimensional spin-${1\over 2}$ Heisenberg XXX
ferromagnet.
\vskip 6mm
In conclusion, we have shown that the model that we introduced in
\cite{EKS} provides a particularly simple example of a superconducting
system. The mechanism of superconductivity is similar to the one of
the strongly attractive Hubbard model.
At zero temperature, the superconductivity already exists in one
dimension and we have seen that it persists if the on-site interaction
becomes weakly repulsive. We expect that in three dimensions
the superconductivity will persist at finite temperature. In a future
publication we will further clarify the physics of the model in one
dimension by using the Bethe Ansatz solution \cite{tbp}.\\
Due to the experimentally established fact that Cooper pairs in
high-$T_c$ superconductors are rather small, our model (with
``zero-size'' pairs) might have applications in this field.
In order to make contact with experiment it is necessary to perturb the
model (change the coefficients of the various interactions in our
hamiltonian). If this is done the most important newly occuring
phenomenon is the decay of local electron pairs into two single electrons. The
physics of this process has been studied by T.D.Lee and R.Friedberg in
their field theoretical model of superconductivity \cite{leefr}. In
their model electron pairs are described by a scalar Bose field
interacting with fermionic fields (representing the single electrons)
{\sl via} the above mentioned decay process. Their results (obtained
in perturbation theory) show the existence of superconductivity and
thus indicate that the decay of local electron pairs would not destroy
the superconducting properties of our model.
\vskip 6mm
It is a pleasure to thank M. Fowler, V.L. Ginzburg, B. Sutherland and
C.N. Yang for interesting discussions.
\vskip 6mm
\frenchspacing
\begin{thebibliography}{11}
%\bibitem{locpair}
% R. Micnas, J. Ranninger and S. Robaszkiewicz, Rev. Mod. Phys.
% {\bf 62} (1990) 113; \\
% V.L. Ginzburg, ``Once again about High Temperature
% Superconductivity'', preprint Lebedev Phys. Inst., Moscow 1992
\bibitem{ARZ}
P.W.Anderson,\ Science {\bf 235} (1987) 1196\\
F.C.Zhang, T.M.Rice,\ Phys.Rev.{\bf B 37} (1988) 3759
\bibitem{EKS}
F.H.L. E\char'31 ler, V.E. Korepin and K. Schoutens, \prl {\bf 68}
(1992) 2960
\bibitem{hirsch}
J.E. Hirsch, Physica {\bf C158} (1990) 326
\bibitem{suth}
B. Sutherland, \pr {\bf B12} (1975) 3795
\bibitem{odlro}
C.N. Yang, Rev.Mod.Phys. {\bf 34} (1962) 694
\bibitem{pairing}
C.N. Yang, \prl {\bf 63} (1989) 2144; \\
C.N. Yang and S. Zhang, Mod.Phys.Lett. {\bf B4} (1990) 759
%\bibitem{lieb2}
% E. Lieb, \prl {\bf 62} (1989) 1201
\bibitem{tJ}
P. Schlottmann, \pr {\bf B36} (1987) 5177; \\
P.-A. Bares, G. Blatter and M. Ogata, \pr {\bf B44} (1991) 130
\bibitem{liebwu}
E. Lieb and F.Y. Wu, Phys.Rev.Lett. {\bf 20} (1968) 1445
\bibitem{tbp}
F.H.L. E\char'31 ler, V.E. Korepin and K. Schoutens,
in preparation
\bibitem{flux}
B. Sutherland and S. Shastry, \prl {\bf 65} (1990) 243, 1833\\
M. Fowler and N. Yu, preprint UVA, 1992
\bibitem{leefr}
R.Friedberg, T.D.Lee and H.C.Ren, Phys.Lett {\bf A152} (1991) 417
and references therein
\bibitem{don}
S. Doniach and E.H. Sondheimer, {\sl Green's functions for Solid
State Physicists, p.124}, Addison Wesley 1974
%\bibitem{lutt}
% J.M. Luttinger, Phys.Rev {\bf 119} (1960) 1153
\end{thebibliography}
\end{document}