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%%   A Remark on  the Decay of Superconducting and Magnetic Correlations 
%%             in One-and-Two-Dimensional Hubbard Models       
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%%      by: Nicolas Macris, Jean Ruiz
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{\Large \bf A Remark on the Decay of Superconducting \\
  Correlations in One and Two Dimensional \\[2mm]
  Hubbard Models}
\\[5mm]
Nicolas  MACRIS\footnote[1]{Institut de Physique Th\'eorique,
                  Ecole Polytechnique F\'ed\'erale de Lausanne,
                 CH-1015 Lausanne.\\
{\it E-mail address\/}: macris@eldp.epf{l}.ch} 
              and 
        Jean RUIZ\footnote[2]{CPT-CNRS, 
                   Luminy case 907, F-13288 Marseille Cedex 9.\\
{\it E-mail address\/}: ruiz@cptsu2.univ-mrs.fr}
\\[5mm]
\end{center}

\begin{abstract}
Upper bounds on the decay of various correlation functions
are derived for a general class of lattice fermion models 
with long range hopping matrix.
These bounds extend previous results of Koma and Tasaki~\cite{KT}
and rule out the  possibility of magnetic ordering 
and condensation of superconducting electron
pairs in one and two dimensions for finite temperatures. 
\newline
\vbox{\vspace{2mm}}
\noindent
{\sc key words}: Hubbard model, ODLRO, superconducting correlations, gauge
symmetry.
\end{abstract}

\vbox{\vspace{5mm}}

\indent 
The absence of spontaneous breakdown of continuous symmetry in one and two 
dimensional classical and quantum statistical mechanical systems 
is a well known phenomenon
(see e.g.\t\cite{FP}, [8] and references therein).
For the Hubbard model, which has a global 
$SU(2)$ symmetry, Ghosh~\cite{G} proved  the absence of magnetic
ordering at finite temperatures, using the Bogoliubov inequality.
More recently, Koma and Tasaki~\cite{KT} extended the Mc Bryan and
Spencer method~\cite{MS}, to a general class of Hubbard models
with {\it finite range hoppings.\/} 
Making use of the 
global $U(1)$ gauge symmetry of any quantum system conserving the particle
number,
they proved  the absence of off diagonal long range order (ODLRO)
corresponding to the
condensation of superconducting electron
pairs (such as Cooper pairs).

In this letter, we report on a number of upper bounds for off diagonal
correlation functions in the case of {\it long range hopping matrix.\/}
They are based on the Mc Bryan and Spencer bound proved in
Ref.\t\cite{KT} to which we apply the estimates of 
Messager {\it et al.\/}\t\cite{MMR}.
The case of random hopping is also considered.

We consider on the $(d \leq 2)$-dimensional lattice 
$\relatif^d$, an itinerant electron model with Hamiltonian
\begin{equation}
H = -\sum_{x,y \in \relatif^d}
\sum_{\sigma=\uparrow,\downarrow}
t_{xy} c_{x,\sigma}^{+} c_{y,\sigma}
+V(\{ n_{x,\sigma} \} )
\end{equation}
Here $(t_{xy})$ is the  hopping matrix,
$c_{x,\sigma}^{+}$ and  $c_{x,\sigma}$
are the creation and annihilation operators 
with Fermi statistics
and the interaction
$V(\{ n_{x,\sigma} \} )$ 
is an arbitrary function of the number operators
$n_{x,\sigma}=c_{x,\sigma}^{+}c_{x,\sigma}$.
We introduce the expectation of an arbitrary observable $A$ as
$\langle A \rangle
= \hbox{Tr}\, A e^{-\beta (H -\mu N)} / \hbox{Tr}\,  e^{-\beta( H -\mu N)}$
where $\mu$ is the chemical potential and 
$N=\sum_{x,\sigma} n_{x,\sigma}$ 
is the total number operator.
$\langle A \rangle$
is to be interpreted as the thermodynamic limit
$\Lambda \uparrow \relatif^d$ of the corresponding finite volume expression
$\langle A \rangle_{\Lambda}$
where the sites are restricted to a finite box $\Lambda$.
The following theorem sets bounds on the correlations between the 
superconducting order parameter
$\Delta_x^+ = c_{x,\uparrow}^{+} c_{x,\downarrow}^{+}$,
$\Delta_y = c_{y,\downarrow} c_{y,\uparrow}$.


\newtheorem{theorem}{Theorem}

\begin{theorem}
\begin{description}
\item[\rm (a)]
If $d=1$ and 
$\vert t_{uv} \vert 
\leq 
\frac{t}{ \vert u-v \vert^{\alpha} }$
with
$\alpha >2$,
then
\begin{equation}
\langle \Delta_x^+  \Delta_y  + h.c. \rangle
\leq
\frac{C_0}{ \vert x-y \vert^{2(\alpha -1)} }
\end{equation}
\item[\rm (b)]
If $d=1$
and 
$\vert t_{uv} \vert 
\leq 
\frac{t}{ \vert u-v \vert^2 \log^{(p)} \vert u-v \vert  }$,
then 
\begin{equation}
\langle \Delta_x^+  \Delta_y  + h.c. \rangle
\leq
\frac{C_1}{\left( \log^{(p)} \vert x-y \vert \right)^{\lambda_1(\beta)} }
\label{b}
\end{equation}
\item[\rm (c)]
If $d=2$
and
$\vert t_{uv} \vert 
\leq 
\frac{t}{ \vert u-v \vert^{\alpha} }$
with
$\alpha >4$,
then
\begin{equation}
\langle \Delta_x^+  \Delta_y  + h.c. \rangle
\leq
\frac{C_2}{ \vert x-y \vert^{\lambda_2(\beta)} }
\end{equation}
\item[\rm (d)]
If $d=2$
and 
$\vert t_{uv} \vert 
\leq 
\frac{t}{ \vert u-v \vert^{4} }$,
then
\begin{equation}
\langle \Delta_x^+  \Delta_y  + h.c. \rangle
\leq
\frac{C_3}{\left( \log  \vert x-y \vert \right)^{\lambda_3(\beta)} }
\end{equation}
\item[\rm (e)]
If $d=2$
and 
$
\vert t_{uv} \vert 
\leq t 
\frac{\log^{(2)} \vert u-v \vert \cdots \log^{(p)} \vert u-v \vert }{ \vert
u-v \vert^{4}}
$,
then 
\begin{equation}
\langle \Delta_x^+  \Delta_y  + h.c. \rangle
\leq
\frac{C_4}{\left( \log^{(p)} \vert x-y \vert \right)^{\lambda_4(\beta)} }
\label{e}
\end{equation}
\end{description}
\noindent
where 
$t$, $C_0$, $C_1$, $C_2$, $C_3$, $C_4$ are positive constants and
$\lambda_1$, $\lambda_2$, $\lambda_3$, $\lambda_4$
are non increasing functions of $\beta$ proportional to
$\beta^{-1}$ for large $\beta$.
In (\ref{b}) and (\ref{e}),
$\log^{(p)}$ denotes the $p$-times iterated logarithm.
\end{theorem}
\noindent
{\it Proof.\/}
Formula (6), (10) and the first inequality of (11) in
Ref.\t\cite{KT} give
\begin{equation}
\langle
\Delta_x^+  \Delta_y  + h.c. 
\rangle
\leq
\exp 
\left\{
- 2(\varphi_x - \varphi_y)
+ \beta \sum_{u,v}
| t_{uv}| [ \cosh ( \varphi_u - \varphi_v ) - 1 ] 
\right\}
\label{4p}
\end{equation}
where $\{ \varphi_u \}$ is an arbitray family of real numbers.
They are related to the gauge transformation
\begin{equation}
A \longrightarrow 
e^{-\sum_{u\sigma} \varphi_u c_{u,\sigma}^{+}c_{u,\sigma}}
A
\: e^{\sum_{u\sigma} \varphi_u c_{u,\sigma}^{+}c_{u,\sigma}}
\end{equation}
that plays, in the case under consideration,
the role of the complex translation of Ref.\t\cite{MS}. 
The right hand side of (\ref{4p}) 
has been estimated by a suitable choice of the variables
$\varphi_u$
according to the different hypotheses on 
$t_{uv}$, to obtain upper bounds on the decay of 
two-points correlation functions of 
$SO(N)$-symmetric spins systems \cite{MMR}.
We thus refer to \t\cite{MMR} (see Section 2 and Section 3) 
to conclude the proof.
$\diamondsuit$

\newpage
\noindent
{\it Remarks.\/}
\begin{itemize}
\item
Koma and Tasaki proved the decay given in Statements (a) and (c) of 
Theorem 1  in the case of finite range hopping
matrix:
$\vert t_{xy} \vert =0$
if
$ \vert x-y \vert > R$
where $R$ is some constant.
They already mentioned that the case of long range hopping could
be treated by their techniques using the result of Ito~\cite{I}.
However the treatment of the Mc Bryan Spencer bound given by Ito 
leads to much slower decay than those obtained 
from the estimates of Messager {\it et al.\/}\t\cite{MMR} proposed here.
\item
In dimension $d=1$,
it can also be shown that
power law decay holds for small $\beta$
when  
$\vert t_{uv} \vert \leq t \vert u-v \vert^{-2}$
and exponential decay holds for all $\beta$ when  
$t_{uv}$ decays exponentially.
\item 
The bounds of the theorem hold also for the correlations
$\langle c_{x,\sigma}^{+}c_{y,\sigma}+ h.c. \rangle$,
except in case (a) where the decay goes like
$\vert x-y \vert^{-(\alpha -1)}$. 
\item
The infinite volume limits of  
the free energy and the pressure exist 
for the models under consideration if $\vert t_{uv}\vert\leq t\vert
u-v\vert^{-\alpha}$, $\alpha>d$.
This follows from standard arguments in the theory of the  
thermodynamic limit.
Here we tacitly assume that it is also the case for the correlation 
functions. 
\end{itemize}


We now consider the case of random hopping. 
Obviously the bounds given in Theorem 1 
immediatly extend to this case, but  we can obtain decaying bounds 
even if $d<\alpha<2d$,
thanks to methods used for classical spin-glasses~\cite{VE}.


\begin{theorem}
Whenever
$t_{uv} = \tilde{t}_{uv} |u-v|^{-\alpha} $,
where 
$\tilde t_{uv}$
are bounded i.i.d. random variables with zero mean, then 
\begin{description}
\item[\rm (a)]
if 
$d=1$ and 
$\alpha >1$,
there is a positive random variable
$f( \{ \tilde t_{uv} \} )$
which is finite almost surely such that
\begin{equation}
\langle \Delta_x^+  \Delta_y  + h.c. \rangle
( \{ \tilde t_{uv} \} )
\leq
f( \{ \tilde t_{uv} \} )
\, \frac{1}{ \vert x-y \vert^{(2\alpha -1)} }
\nonumber
\end{equation}
\item[\rm (b)]
if 
$d=2$ and 
$\alpha >2$, 
then for all $0<K<\alpha-1$ and $\gamma <1$, there exists a sequence of
random variables 
$F_N$ such that for $\vert x-y \vert =N$:
\begin{equation}
- \log \vert \langle \Delta_x^+  \Delta_y  + h.c. \rangle \vert 
\geq 
F_N
\nonumber
\end{equation}
with
\begin{equation}
\lim_{N \to \infty}
\frac{F_N}{\left( \log N \right)^{\gamma} }
=K \quad {\rm in } \: L^2{\rm -sense}
\nonumber
\end{equation}
\end{description}
\end{theorem}

\noindent
{\it Proof.}
>From the second inequality of formula~(10) in Ref.\t\cite{KT}
and the variational principle (or Peierls inequality~\cite{R}),
we get
\begin{eqnarray}
\langle A \rangle
&\leq&
e^{- 2(\varphi_x - \varphi_y)}
\: \frac{ {\rm Tr}\, e^{-\beta (H'-\mu N)} }{ {\rm Tr}\, e^{-\beta (H- \mu
N)} }
\\
&\leq&
\exp 
\left\{
- 2(\varphi_x - \varphi_y)
+ \beta \sum_{u,v}
 t_{uv} [ \cosh ( \varphi_u - \varphi_v ) - 1 ]
\langle c^{+}_{u,\sigma} c_{v,\sigma} \rangle'
\right\}
\nonumber
\end{eqnarray}
where $H'$ is the Hamiltonian with
modified hoppings 
$t'_{uv} = t_{uv} \cosh (\varphi_u - \varphi_v )$
and 
$\langle \ \rangle'$
is the expectation with respect to this modified hopping.
As in Ref.\t\cite{VE}, it is sufficient, for $d=1$, to prove that
\begin{equation}
\esp \left\{ \vert \sum_{uv} X_{uv} \vert  \right\}
\equiv
\esp \left\{ \vert \sum_{uv} t_{uv} [ \cosh ( \varphi_u - \varphi_v ) - 1 ]
\vert  \right\}
< \infty
\end{equation}
where $\esp$ is the expectation with respect to the hoppings
$\tilde{t}_{uv}$.

Since, by Schwartz inequality, 
$
\esp \left\{ \vert \sum_{uv} X_{uv} \vert  \right\}
\leq
\left( \esp \left\{ \left( \sum_{uv} X_{uv} \right)^2   \right\}
\right)^{1/2}
$,
one needs to 
bound a diagonal part
$
\sum_{uv} \esp \left\{  X_{uv}^2 \right\}
$
and a non-diagonal part
$
\sum_{ij,kl \atop ij \not=kl} 
\esp \left\{  X_{ij} X_{kl} \right\}
$.
For the diagonal part each term is bounded as in Ref.\t\cite{VE} by 
\begin{equation}
\frac{ \esp \left\{  \tilde{t}_{uv}^2 \right\} }{\vert u-v \vert^{2
\alpha}}
\; [ \cosh ( \varphi_u - \varphi_v ) - 1 ]^2
\end{equation}
because of 
$\vert\langle  c_{u,\sigma}^{+} c_{v,\sigma}\rangle'\vert \leq 1$.
Indeed (for finite volume)
\begin{eqnarray}
\vert {\rm Tr}\,  c_{u,\sigma}^{+} c_{v,\sigma} 
e^{-\beta (H'-\mu N)} \vert
&\leq&
\parallel c_{u,\sigma}^{+} c_{v,\sigma} 
e^{-\beta (H'-\mu N)} \parallel_1
\nonumber
\\
&\leq&
\parallel c_{u,\sigma}^{+} c_{v,\sigma} \parallel_{\infty}\cdot 
\parallel e^{-\beta (H'-\mu N)} \parallel_1
\nonumber
\\
&\leq &
{\rm Tr}\,   
e^{-\beta (H'-\mu N)}
\nonumber
\end{eqnarray}
where 
$\parallel \: . \parallel_1$
is the trace class norm and
$\parallel \: . \parallel_{\infty}$  the operator norm (c.f.
Ref.\t\cite{K}).

For the non diagonal part we use the Dyson expansion
\begin{eqnarray}
\lefteqn{  
e^{-\beta(H_0+H_1)} = e^{-\beta H_0}+  } 
\\
&&
\sum_{n\geq 1} \int_0^{\beta} ds_1
\int_0^{s_1} ds_2 \cdots
\int_0^{s_{n-1}} ds_n 
e^{-s_n H_0} H_1 e^{-(s_{n-1}-s_{n}) H_0} H_1
\cdots
H_1 e^{-(\beta-s_1) H_0}
\nonumber
\end{eqnarray}
where
\begin{eqnarray}
H_0
&=& 
-\sum_{x,y \atop \not= ij,kl}
\sum_{\sigma=\uparrow,\downarrow}
t_{xy}' c_{x,\sigma}^{+} c_{y,\sigma}
+V(\{ n_{x,\sigma} \} )
\\
H_1
&=&
t_{ij}' c_{i,\sigma}^{+} c_{j,\sigma}
+t_{kl}' c_{k,\sigma}^{+} c_{l,\sigma}
\end{eqnarray}
This leads to a convergent series for
$
\langle  c_{i,\sigma}^{+} c_{j,\sigma}\rangle'
\langle  c_{k,\sigma}^{+} c_{l,\sigma}\rangle'
$ in powers of
$\tilde t_{ij}$ 
and 
$\tilde t_{kl}$. 
Since 
$
\esp\{\tilde{t}_{ij} \tilde{t}_{kl}^n\}
=
\esp\{\tilde{t}_{ij}\} \esp\{\tilde{t}_{kl}^n\}
=0
$, 
for any integer $n$, all the first order terms disappear and 
each term of the non diagonal part can be bounded by
\begin{eqnarray}
 C \; \frac{ \esp \left\{  \tilde{t}_{ij}^2 \right\}
\esp \left\{  \tilde{t}_{kl}^2 \right\} }{\vert i-j \vert^{2 \alpha}
\vert k-l \vert^{2 \alpha} }
&&  
[ \cosh^2 ( \varphi_i - \varphi_j ) - \cosh ( \varphi_i - \varphi_j ) ]  
\\
&& \phantom{xxxx} \times \
[ \cosh^2 ( \varphi_k - \varphi_l ) - \cosh ( \varphi_k - \varphi_l ) ]
\nonumber
\end{eqnarray}
We choose 
$\varphi_z - \varphi_x = (\alpha - 1)
\sum_{j=1}^n j^{-1}$
when
$\vert z - x \vert =n$ 
and conclude the proof of the one dimensional case 
by the estimates of Ref.\t\cite{MMR}.

For the two dimensional case 
we use the choice of Picco \cite{P},
$\varphi_z - \varphi_x = K
\sum_{j=1}^n \left( j [ \max \{ 1, \log j\} ]^{1-\gamma} \right)^{-1}$
when $z$ belongs to the square with sides of length $2n+1$
centered at $x$,
and  apply the estimates of Ref.\cite{MMR}  
to (14) and (18), which yields instead of (13)
\begin{equation}
\esp \left\{ \vert \sum_{uv} X_{uv} \vert  \right\}
\leq
 O((\log N)^{2\gamma-1}) + {\rm Cte}
\end{equation}
for any $\gamma < 1$,  and  conclude the proof as in Van Enter \cite{VE}.
$\diamondsuit$


Part of this work was done when the authors where at Rutgers University.
They thank J.L.\t Lebowitz for an interesting discussion. J.R.\t is 
grateful to Courant Institute for Mathematical Science and
Department of Mathematics of Rutgers University for warm hospitality
and acknowledges MRI-CNRS for financial support.   
N.M.\t acknowledges financial support from the  Swiss National
Foundation for Science. 



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\end{thebibliography}
\end{document}
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