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\begin{document}
\begin{center}
{\Large\bf Quantum fluctuations in quantum lattice-systems}
\smallskip\\
{\Large\bf with continuous symmetry\footnote{
To appear in J.~Stat.~Phys. (1996)}}
\bigskip
\bigskip
{\large Tsutomu Momoi\footnote{E-mail: momoi@cm.ph.tsukuba.ac.jp.}}
\bigskip
\smallskip
{\it Institute of Physics, University of Tsukuba,
Tsukuba, Ibaraki 305, Japan.\/}
\bigskip
\smallskip
(\hspace{5cm})\\
\end{center}
%\bigskip
\begin{abstract}
\begin{normalsize}
\noindent
We discuss conditions for the absence of spontaneous breakdown of
continuous symmetries in quantum lattice systems at $T=0$. Our analysis
is based on Pitaevskii and Stringari's idea that the uncertainty relation
can be employed to show quantum fluctuations. For the one-dimensional
systems, it is shown that the ground state is invariant under the continuous
transformation if a certain uniform susceptibility is finite.
For the two- and three-dimensional systems, it is shown that
truncated correlation functions cannot decay
any more rapidly than $|r|^{-d+1}$
whenever the continuous symmetry is spontaneously broken.
Both of these phenomena occur owing to quantum fluctuations.
Our theorems cover a wide class of quantum lattice-systems having
not-too-long-range
interactions.
\end{normalsize}
\end{abstract}
\smallskip
\noindent
{\bf KEY WORDS:} Quantum fluctuations; ground states; symmetry breaking;\\
uncertainty relation; clustering.
\smallskip
\section{Introduction}
\hspace*{\parindent}
It is well known that continuous symmetries
cannot be spontaneously broken in one- and two-dimensional systems
at nonzero temperatures if the interactions are short range.
Since Mermin and Wagner,\upcite{MerminW} and
Hohenberg\upcite{Hohenberg} showed rigorous proofs, several
papers have appeared, proving the invariance of the
state under the continuous transformation.\uprcite{DobrushinS}{BonatFPK}
These arguments, however, work only at finite temperatures.
Absence of symmetry breaking in the ground state of the
one-dimensional quantum systems has been a long-standing question. This
problem was discussed by using an extension of the Bogoliubov
inequality\upcite{Takada} and using the uncertainty
relation.\upcite{PitaevskiiS,Shastry} Takada\upcite{Takada}
argued the relation between the absence of long-range order and the
dispersion form of the excitation
spectrum, and thereby showed that,
if the lowest excitation frequency has a gapless
$k$-linear form, the ground state cannot show symmetry breaking.
Pitaevskii and Stringari\upcite{PitaevskiiS}
proposed a zero-temperature analogue of the Bogoliubov inequality,
using the uncertainty relation
of the quantum mechanics. They presented a method for
showing the absence of breakdown of continuous symmetry in the ground
state. After that,
Shastry\upcite{Shastry} pointed out that one can complete the proof
for the one-dimensional Heisenberg antiferromagnet
combining their method and the infrared bound given by
Dyson, Lieb and Simon.\upcite{DysonLS}
The method proposed by Pitaevskii and Stringari\upcite{PitaevskiiS}
can be successfully applied only when we have a rigorous upper bound of
the susceptibility at the whole momentum space. It is however
difficult (to date) to obtain upper bounds of the momentum-dependent
susceptibility in general quantum systems.
Another well-known theorem for short-range systems
with continuous symmetry is the
Nambu-Goldstone theorem, which states that
there exist gapless elementary excitations whenever any continuous symmetry
is spontaneously broken.
This theorem was also proved for the lattice systems.\upcite{LandauFPW}
Furthermore, Martin\upcite{Martin} showed that some truncated correlation
functions at finite temperatures have power-decay behavior
slower than or equal to $|r|^{-1}$
in three-dimensional systems if continuous symmetry is spontaneously
broken.
In the present paper, we extend the method by Pitaevskii and
Stringari,\upcite{PitaevskiiS} using the technique developed
by Martin,\upcite{Martin}
and thereby show conditions for the ground state of quantum systems
being invariant under the continuous transformation.
We obtain the following results on the continuous-symmetry breaking
in the ground states.
\begin{itemize}
\item[1.] In the one-dimensional system, if a certain uniform
susceptibility is finite, the ground state has continuous symmetry, i.e.,
\begin{equation}
\omega ( \sigma_\theta (A)) = \omega ( A )
\end{equation}
for any local observable $A$. Here $\omega ( \cdots )$ denotes the
ground state and $\sigma_\theta$ denotes
the continuous transformation under which
the interactions of the Hamiltonian are invariant. (See theorem 1.)
\item[2.] In more-than-one-dimensional ($d>1$) systems, if any continuous
symmetry is spontaneously broken in the ground state as
${d\over d\theta} \omega ( \sigma_\theta (A) ) |_{\theta=0} \ne 0$
with a local observable $A$
and if a certain uniform susceptibility is finite, the truncated two-point
correlation function
of $A$ shows a power-decay slower than or equal
to $O(1/r^{d-1})$. Here we denote the
dimensionality of the system by $d$. (See theorem 2.)
\end{itemize}
Both of these phenomena occur as a consequence of quantum fluctuations.
In our discussion, we define the ground state applying an infinitesimally
small field.
We derive these results, using rigorous inequalities and
assuming the clustering property of this ground state.
(Note that this assumption is quite reasonable, though it cannot be verified
within the presently available techniques in mathematical physics.)
These theorems are applicable to a wide class of quantum lattice systems
having
not-too-long-range interactions and continuous symmetries. Quantum spin
systems, lattice fermion-systems and hard-core bose systems
are included, for example.
\section{Theorems and physical consequences}
\subsection{Preliminaries}
We first give some notations. We denote the $d$-dimensional lattice by
$\L$, which
is taken as ${\bf Z}^d$. For each lattice point $x\in\L$,
there are the algebra
$\A_x$ of operators and the finite-dimensional Hilbert space $\h_x$. For any
bounded subset $\Lambda\subset\L$, local operators which are defined
on $\Lambda$ generate the local algebra $\A_\Lambda$ of
observables and the Hilbert space is given
by $\h_\Lambda = \bigotimes_{x\in\Lambda} \h_x$.
For simplicity, we present arguments for quantum systems with two-body
interactions. We can easily extend the following arguments to models with
more-than-two-body interactions.
Let $\L$ be the translationally invariant lattice and
$H_\Lambda$ be the Hamiltonian in the finite-volume lattice
$\Lambda\subset\L$, which is given by
\begin{equation}
H_\Lambda = \sum_{x,y\in \Lambda} \phi(x,y).
\end{equation}
Here $\phi(x,y)$ denotes the translationally invariant interaction
defined on ${\h}_x \otimes {\h}_y$
with the norm $\Vert \phi(x,y)\Vert = \psi (x-y)$. We restrict our discussions
to the models that have not-too-long-range interactions satisfying
\begin{equation}\label{eq:range}
\sum_{x\in\L} |x|^2 \psi (x) <\infty
\end{equation}
and that have, at least,
the $U(1)$-continuous symmetry, i.e.,
\begin{equation}\label{eq:symmetry}
[\phi (x,y), J_\Omega ]=0
\end{equation}
for any $x$, $y\in \Omega$ and local subset $\Omega\subset\L$.
Here $J_\Omega$ denotes the generator
of the (global) symmetry-transformations of operators in $\A_\Omega$.
The continuous symmetry-transformation is given by
\begin{equation}
\sigma_\theta (A) = \exp (i\theta J_\Omega)A\exp (-i\theta J_\Omega)
\end{equation}
for any $A\in \A_\Omega$.
To define the ground state, we select
a proper order parameter and then apply the corresponding symmetry-breaking
field.
Let us define the ground state in the form
\begin{equation}\label{eq:GS}
\omega ( \cdots )
= \lim_{B\downarrow 0} \lim_{\Lambda \uparrow \L}
\lim_{\beta\uparrow\infty}
{{\rm Tr} \cdots \exp \{-\beta (H_\Lambda -BO_\Lambda) \}
\over {\rm Tr}\exp \{-\beta (H_\Lambda -BO_\Lambda) \}},
\end{equation}
where $O_\Lambda$ denotes the order-parameter operator and $B$ is the
real-valued symmetry breaking field. It is known that the limits are
well-defined by choosing suitable sequences of $\Lambda$ and $B$.
(See Appendix A of ref.~\cite{KomaT1994}, for example.)
We restrict our discussions to the case that the order-parameter operator has
a sublattice-translational invariance. Hence the ground state defined by
(\ref{eq:GS}) has the following sublattice-translational invariance
\begin{equation}\label{cond:translation}
\omega (A) = \omega (\tau_x (A))
\end{equation}
for any $x\in \L_{\rm s}$ and $A\in \A_\Omega$ on a local subset $\Omega$.
Here $\tau_x$ denotes the space translation by $x$ and
$\L_{\rm s}$ denotes a set of sites in a sublattice.
If we consider antiferromagnets on a bipartite lattice, for example,
the order parameter
is set as the staggered magnetization and $\L_{\rm s}$ is one of two
sublattices. In ordinary ferromagnets, the ground state
has the full lattice-translational invariance and hence $\L_{\rm s}=\L$.
In the following discussions, we assume the clustering property of the state,
\begin{equation}\label{cond:cluster}
|\omega ( \tau_x (A) B )
- \omega ( \tau_x (A) ) \omega ( B ) |
\le O \left( {1 \over |x|^\delta} \right)
\end{equation}
with $\delta>0$ for sufficiently large $|x|$ and any $A$, $B\in\A_\Omega$
on a local subset
$\Omega$.
This property means that observations at two points separating far away from
one another do not affect each other. Note that this is a quite
natural assumption. It is believed that,
by selecting a proper order parameter, the state $\omega ( \cdots )$
becomes an pure state, i.e., it has the clustering property.
{\bf Remark:} It is widely believed that any physically
natural equilibrium state has the clustering property.
In studies on finite systems, we sometimes encounter states
which do not have the cluster property. For example, consider the
ground state of the three-dimensional Heisenberg antiferromagnet.
It is shown that the ground state
of finite-volume systems is invariant under the global spin
rotation\upcite{Marshall,LiebM} and it has
a long-range order in the infinite-volume limit.\upcite{KennedyLS}
Taking the infinite-volume limit of the ground state of finite systems, one
can define a ground state that does not have the clustering property.
However, as discussed in ref.~\cite{KomaT1994}, this
symmetric ground state is unphysical and only a mathematical object.
It is believed that in the thermodynamic limit
this state is decomposed into pure states
and one of the pure states appears as a natural state in the real
system.\upcite{Ruelle,BratteliR}
%\newpage
\subsection{Main Theorems}
In this section, we show our theorems. Physical
consequences of the theorems will be discussed in sections~\ref{sec:one-d}
and \ref{sec:two-three-d}, and proofs are given in section~\ref{sec:proof}.
The statement that the state $\omega ( \cdots )$
has the continuous symmetry is equivalent to
\begin{equation}
{d\over d\theta} \omega ( \sigma_\theta (A) ) \biggr|_{\theta =0} = 0
\end{equation}
for any $A\in \A_\Lambda$ on any subset $\Lambda \subset \L$.
We consider the transformation $\sigma_\theta$ in which $J_\Omega$ is
given by $J_\Omega = \sum_{x\in \Omega}\tau_x (J_0)$ with a bounded
self-adjoint operator $J_0\in\A_0$.
In this case, we have
\begin{equation}\label{eq:deriv}
{d\over d\theta} \omega ( \sigma_\theta (A) ) \biggr|_{\theta =0}
= i\omega ( [J_\Lambda,A] )
\end{equation}
for any $A\in\A_\Lambda$ and any subset $\Lambda\subset\L$.
Without loss of generality, we consider the operator $A$ on the subset
$\Lambda = \{\mbox{$x\in\L:$ $|x|\le r$}\}$, where $r$ is a finite constant.
To discuss the quantity $\omega ( [J_\Lambda,A] )$,
we use the sublattice-translational invariance (\ref{cond:translation})
and hence we have
\begin{equation}\label{eq:translation}
\omega ( [J_\Lambda,A] )
= {1\over |\Omega_{\rm S}|}\sum_{x\in \Omega_{\rm S}}
\omega ( [J_\Omega,\tau_x (A)] )
\end{equation}
for any $A\in\A_\Lambda$,
where $\Omega_{\rm S} = \{\mbox{$x\in\L_{\rm S}:$ $|x|\le R$} \}$ and
$\Omega = \{\mbox{$x\in\L:$ $|x_i|\le R_0$ for $i=1,\dots ,d$} \}$ with
$R_0 = R+r$. [Though equation (\ref{eq:translation}) holds by
setting $\Omega$ as $\{ x\in \L$~:~$|x|\le R+r \}$, we have taken $\Omega$
as the hyper-cubic lattice for convenience in later discussions.]
Bounding the absolute value of the right-hand side of~(\ref{eq:translation})
with the uncertainty relation and the Kennedy-Lieb-Shastry
inequality\upcite{KennedyLS}, and
estimating the $R$ dependence of the upper bound,
we obtain the following lemma.
%\vspace{12mm}
\vspace{6mm}
{\bf Lemma.} Let the interaction satisfy~(\ref{eq:range})
and (\ref{eq:symmetry}), and assume that the ground state (\ref{eq:GS})
satisfies (\ref{cond:translation}) and (\ref{cond:cluster}).
Consider $\A_\Lambda$ on the subset $\Lambda=\{ x\in \L : |x| \le r \}$,
where $r$ is a finite constant, and let
$\Omega_{\rm S} = \{\mbox{$x\in\L_{\rm S}:$ $|x|\le R$} \}$ and
$\Omega = \{\mbox{$x\in\L:$ $|x_i|\le R_0$ for $i=1,\dots,d$} \}$ with
$R_0=R+r$.
Furthermore, define the uniform susceptibility of $J$ by
\begin{equation}\label{def:suscep2}
\chi_J = \lim_{\Omega \uparrow \L} {2 \over |\Omega|}
\int^\infty_0 d\lambda
\{ \omega( J_\Omega J_\Omega(i\lambda) ) - \omega^2( J_\Omega ) \}
\ge 0
\end{equation}
assuming existence of the limit,\footnote{
Mathematically speaking, existence of the limit in (\ref{def:suscep2})
may be nontrivial. It should
be remarked that this definition of the uniform susceptibility is
equivalent to the standard one, which has been used in many papers
in physics. See Appendix.}
where $J_\Omega (t)$ is the time-evolved operator of $J_\Omega$.
Then, the right-hand side of (\ref{eq:translation}) is bounded as
\begin{equation}\label{Lemma}
\left|{1 \over |\Omega_{\rm s}|} \sum_{x\in \Omega_{\rm s}}
\omega ([J_\Omega, \tau_x (A)] ) \right|^2
\le \left\{
\begin{array}{ll}
O(R^{d-1-\delta})\cdot\sqrt{\chi_{J}} &
\mbox{\hspace{1cm}} (0<\deltad)
\end{array} \right.
\end{equation}
for sufficiently large $R$ and any $A\in\A_\Lambda$.
%\vspace{12mm}
\vspace{6mm}
We will give a proof in section~\ref{sec:proof}. As shown in the proof,
this lemma comes from the uncertainty relation of the quantum mechanics.
Hence the inequality (\ref{Lemma}) can show purely quantum effects.
In the following,
we discuss physical consequences of the bound in each dimension.
It should be remarked that these results are applicable to various
models on arbitrary lattices that have the translation invariance.
Selecting bonds of the non-vanishing interactions $\phi (x,y)$,
we can define various lattices on ${\bf Z}^d$.
The results depend only on the dimensionality $d$ of the lattice.
First, we discuss one-dimensional systems, in which $\L={\bf Z}$.
By taking the $R\rightarrow\infty$ limit of (\ref{Lemma}),
the above lemma shows conditions for the absence
of continuous-symmetry breaking in one-dimensional systems.
%\vspace{12mm}
\vspace{6mm}
{\bf Theorem 1.} Let $\L$ be a one-dimensional lattice and the
interaction $\phi (x,y)$
satisfy~(\ref{eq:range}) and (\ref{eq:symmetry}). Assume the ground
state (\ref{eq:GS}) satisfies the properties (\ref{cond:translation})
and (\ref{cond:cluster}). If the infinite volume limit in the
definition (\ref{def:suscep2}) of the uniform susceptibility exists and if
this
susceptibility $\chi_J$ is not diverging, the ground state (\ref{eq:GS})
is invariant under the continuous transformation $\sigma_\theta$, i.e.,
\begin{equation}\label{eq:Theorem1}
{d \over d\theta}\omega ( \sigma_\theta (A) ) \biggr|_{\theta=0} = 0
\end{equation}
for any $A\in\A_\Lambda$ on any finite subset $\Lambda$.
%\vspace{12mm}
\vspace{6mm}
Physical meanings of this theorem are discussed in
section~\ref{sec:one-d}.
An advantageous point of this theorem is that the results depend only on the
``uniform'' susceptibility, not on other momentum-dependent
susceptibilities.
The condition that the uniform susceptibility is
finite (or vanishing) is physically important. (See examples in the
next section.) We cannot improve the condition
without further detailed properties of the model,
since the uniform susceptibility is finite or
diverging, depending on each model.
Next, we discuss two- and three-dimensional systems.
For these systems, we consider the case that the continuous symmetry is
spontaneously broken. Slight modifications of the lemma
give the following bound for a truncated two-point correlation function.
%\vspace{12mm}
\vspace{6mm}
{\bf Theorem 2.} Let $\L$ be a more-than-one-dimensional ($d>1$) lattice,
and $\phi (x,y)$ satisfy~(\ref{eq:range}) and (\ref{eq:symmetry}).
Assume that the ground state (\ref{eq:GS}) satisfies the conditions
(\ref{cond:translation}) and (\ref{cond:cluster}).
If continuous symmetry is spontaneously broken in the ground
state (\ref{eq:GS}), i.e., $\omega ( [J_\Lambda,A] ) \ne 0$
with an operator $A\in \A_\Lambda$ on an arbitrary subset
$\Lambda \subset \L$, and if the infinite volume limit in
(\ref{def:suscep2}) exists and $\chi_J <\infty$,
the truncated two-point correlation function of $A$ shows the slow
clustering as
\begin{equation}\label{eq:Theorem2}
|\omega ( A^* \tau_x (A) )
- \omega ( A^* ) \omega ( \tau_x (A) ) |
\ge O\left( {1\over |x|^{d-1}} \right)
\end{equation}
for sufficiently large $|x|$ with $x\in\L_{\rm S}$.
%\vspace{12mm}
\vspace{6mm}
We discuss the meaning of this theorem in section~\ref{sec:two-three-d}
and give a proof in section~\ref{sec:proof}.
Under some conditions, this theorem states that the truncated
correlation function of $A$ cannot show
any exponential decay in the ordered ground state.
This result hence corresponds to an extension
of the Nambu-Goldstone theorem.
This theorem shows the conditions for existence of quantum fluctuations
and shows strong correlation between the fluctuations.
(Remember that in the classical model there is no fluctuation in the
ground state and hence the truncated two-point correlation function
vanishes.)
The condition for the uniform susceptibility appears in
the theorem again and it is important in this case, as well. (See
examples in section~\ref{sec:two-three-d}.)
%\newpage
\subsection{One-dimensional systems}
\label{sec:one-d}
First we discuss one-dimensional systems, whose lattice is set as
${\bf Z}$. Among the assumptions of Theorem~1, the finiteness of $\chi_J$
is physically important. It determines whether the ground state shows
symmetry breaking or not.
To clarify the meaning of Theorem~1, we display three examples.
{\bf Example 1.} {\em Spin SU(2) symmetry.}
Let us first consider the one-dimensional spin-$S$ Heisenberg antiferromagnet
on the lattice $\L(={\bf Z})$. The Hamiltonian in $\Lambda\subset\L$
is given by
\begin{equation}\label{eq:Heisenberg}
H_\Lambda = \sum_{\langle i,j \rangle \in \Lambda}
(S_i^x S_j^x + S_i^y S_j^y + S_i^z S_j^z),
\end{equation}
where $S^\alpha_i$ ($\alpha=x$, $y$, $z$) denote the spin operators on the
site $i$ satisfying
$[S^\alpha_j,S^\beta_k]=i\delta_{jk}\epsilon_{\alpha\beta\gamma}S^\gamma_j$
with ${\bf S}^2=S(S+1)$.
The summation runs over all the nearest-neighbor sites.
As a generator of the $U(1)$ rotation, we take
\begin{equation}
J_\Lambda = \sum_{i\in\Lambda} S^z_i.
\end{equation}
This model clearly satisfies the conditions~(\ref{eq:range})
and (\ref{eq:symmetry}).
Setting the order-parameter operator of the antiferromagnetism as
\begin{equation}
O_\Lambda = \sum_{i\in\Lambda} (-1)^i S^x_i,
\end{equation}
we define the ground state $\omega(\cdots)$ by (\ref{eq:GS}).
By definition, the ground state satisfies the sublattice-translation
invariance (\ref{cond:translation}).
In this model, the quantity $\chi_J$ is the uniform magnetic
susceptibility of the ground state $\omega(\cdots)$.
It has been proved
in refs.~\cite{DysonLS} and \cite{KennedyLS} that $\chi_J$ is bounded from
above by a finite constant for the Heisenberg antiferromagnets
on hyper-cubic lattices.\footnote{\label{footnote}
Though, in refs.~\cite{DysonLS} and \cite{KennedyLS}, they discussed only
antiferromagnets without any magnetic field, their arguments can be easily
extended to the Hamiltonian with the staggered magnetic field,
$H_\Lambda - B O_\Lambda$, and hence we can show that their bound on the
susceptibility holds for this system as well.}
Finally we assume that $\omega(\cdots)$ satisfies the clustering property.
Under this assumption, Theorem~1 hence states that
the ground state $\omega(\cdots)$ has the spin-rotational symmetry.
For the system whose uniform susceptibility is not diverging,
Theorem 1 states that
quantum fluctuations suppress spin ordering, even if some
momentum-dependent susceptibility is diverging.
The correlation of $k=0$ is however special. Theorem~1 does not
exclude the possibility of ferromagnetism, since in the ferromagnets
the uniform transverse susceptibility, which is nothing but $\chi_J$,
diverges. As is well known,
the one-dimensional Heisenberg ferromagnet has the fully ordered ground
state. Thus the spin long-range correlation with the zero momentum can survive
quantum fluctuations. Furthermore Theorem 1 says that
ferrimagnetism can occur as well.
Some models indeed show the ferrimagnetic order even
in the one-dimensional system.\upcite{ShenQ}
In the ferrimagnetism, antiferromagnetic long-range
order coexists with ferromagnetic order. From
the theorem we learn that this antiferromagnetic order
can appear owing to the existence of ferromagnetic order.
{\bf Example 2.} {\em Spin O(2) symmetry.}
Next we consider the one-dimensional spin-$S$ $XY$ ferromagnet,
whose Hamiltonian is
\begin{equation}
H=-\sum_{\langle i,j \rangle\in\Lambda} (S^x_i S^x_j + S^y_i S^y_j).
\end{equation}
The summation runs over all nearest-neighbor sites. This model is
invariant under the $O(2)$ rotation, whose generator is
$J_\Lambda=\sum_{i\in\Lambda} S^z_i$.
This model is expected to have strong correlation at $k=0$.
We hence set the order parameter as $O_{\Lambda}=\sum_{i\in\Lambda}S^x_i$,
thereby defining the ground state by (\ref{eq:GS}).
The Hamiltonian and the ground state clearly satisfy the conditions
(\ref{eq:range}), (\ref{eq:symmetry}) and (\ref{cond:translation}).
Since this model has only the $O(2)$ symmetry and may have
weak $S^z$-correlation, situations are different from the
ferromagnets in the above example.
We expect $\chi_J$ is not diverging in the $XY$ model and hence, from
Theorem~1, the ground state has the $O(2)$ rotational invariance.
{\bf Example 3.} {\em U(1)-gauge symmetry of fermions.}
Let us consider the breakdown of the $U(1)$-gauge
symmetry of fermions. (The Hilbert space of fermion systems is not
a simple tensor product of the local Hilbert spaces and hence
some modifications
to the notations are needed. Furthermore, each observable in the algebra
$\A_\Omega$ should contain multiplets of an even
number of fermion operators, so that $[A,B]=0$ for
any $A\in\A_{\Lambda_1}$ and
$B\in\A_{\Lambda_2}$ with $\Lambda_1 \cap \Lambda_2 =\emptyset$. Thereby
our theorems still work for the fermion systems,
as well.)
As an example of correlated lattice-fermions,
we consider the one-dimensional Hubbard model, whose Hamiltonian is given by
\begin{equation}\label{eq:Hubbard}
H_\Lambda
= - t \sum_{\langle i,j \rangle \in \Lambda}
\sum_{\sigma=\uparrow,\downarrow}
(c_{i\sigma}^* c_{j\sigma} + c_{j\sigma}^* c_{i\sigma})
+ U\sum_{i\in\Lambda} n_{i\uparrow}n_{i\downarrow}
-\mu \sum_{i\in\Lambda} (n_{i\uparrow} + n_{i\downarrow}).
\end{equation}
The summation of the hopping term runs over all the neatest-neighbor sites.
We denote the creation operator of the fermion at site $i$
with spin $\sigma$ by
$c_{i\sigma}^*$ and the number operator of the fermion by $n_{i\sigma}$.
The generator of the gauge transformation is given by
$J_\Lambda = \sum_{i\in\Lambda} (n_{i\uparrow}+n_{i\downarrow})$
and hence $\chi_J$ is the uniform charge
susceptibility, or the compressibility. This model satisfies the conditions
(\ref{eq:range}) and (\ref{eq:symmetry}). Under the assumption of the
clustering property, Theorem~1 states for this model that, if the
compressibility is finite, there is no breakdown of the
$U(1)$-gauge symmetry.
Here we mention about the model proposed by Essler et
al.\upcite{EsslerKS} In their model the
ground state has superconductivity even in the one-dimensional
system. It should be noted that the compressibility is
diverging in the ground state of their model,
and hence Theorem~1 is not applicable to their model.
%\newpage
\subsection{Two- and three-dimensional systems}
\label{sec:two-three-d}
In this section we discuss two- and three-dimensional systems, whose
lattice is taken as ${\bf Z}^2$ or ${\bf Z}^3$.
To clarify the meaning of Theorem~2, let us consider two examples.
{\bf Example 4.} We again discuss the spin-symmetry breaking of the Heisenberg
antiferromagnet (\ref{eq:Heisenberg}).
Here we take the lattice $\L$ as ${\bf Z}^2$ or
${\bf Z}^3$. We set the order-parameter operator as
$O_\Lambda = \sum_{r\in\Lambda} S^x_r
\exp (i{\bf q}\cdot {\bf r})$
with ${\bf q}=(\pi,\ldots,\pi)$ and the generator of rotation as
$J_\Lambda = \sum_{r\in\Lambda} S^z_r$.
This model hence satisfies the conditions (\ref{eq:range})
and (\ref{eq:symmetry}), and the ground state $\omega (\cdots)$ defined by
(\ref{eq:GS}) satisfies (\ref{cond:translation}).
The occurrence of symmetry breaking in $\omega (\cdots)$
is proved for the two-dimensional $S\ge 1$
models\upcite{NevesP,KomaT1993}
and for the three-dimensional arbitrary-$S$
models.\upcite{KennedyLS,KomaT1993}
Existence of long-range order is also proved for anisotropic Heisenberg
antiferromagnets.\uprcite{KennedyLS2}{NishimoriO}
In these models, the ground state hence shows
\begin{equation}
\omega ( [J_\Lambda , S^y_r] ) = -i \omega ( S^x_r )
\ne 0.
\end{equation}
Furthermore, the finiteness of $\chi_J$ is proved in
refs.~\cite{DysonLS} and \cite{KennedyLS}.
(See also the footnote~3 on page~\pageref{footnote}.)
Using these results and assuming the clustering property of
$\omega(\cdots)$, we find from Theorem~2 that
the transverse-spin correlation shows the slow clustering as
\begin{equation}\label{eq:spin_correlation}
|\omega ( S^y_0 S^y_r )| \ge O\left( {1\over |r|^{d-1}} \right)
\end{equation}
for $0$,~$r\in\L_{\rm S}$ and for sufficiently large $|r|$.
Note that $\omega(S^y_r)=0$ by definition of the ground state.
Hence (\ref{eq:spin_correlation}) shows
that there are quantum fluctuations in the ground state
and they are strongly correlated.
Shastry\upcite{Shastry} showed that the transverse-structure factor
diverges as $\omega(S^y_k S^y_{-k})\sim 1/|{\bf k}-{\bf q}|$ at
${\bf k}\simeq {\bf q}$ in the ground state with N\'eel order.
This indicates that the transverse-correlation function decays as
$\omega(S^y_0 S^y_r)\sim (-1)^r /|r|^{d-1}$. Thus this example shows that
our bound (\ref{eq:Theorem2}) is optimal.
It may be worth mentioning about another Nambu-Goldstone-type theorem for
the excitation spectrum of the Heisenberg
antiferromagnets.\uprcite{Momoi1994}{Momoi1995}
It states that the
N\'eel-ordered ground state has a gapless excitation spectrum and the
lowest frequency of excitations is bounded from above by a gapless $k$-linear
form around ${\bf k}\simeq {\bf 0}$ and {\bf q}.
These two Nambu-Goldstone-type theorems may closely relate to
each other.
Furthermore we discuss
the ferromagnetic Heisenberg model, in which $\chi_J$ is diverging,
to clarify the significance of the condition on $\chi_J$.
The ground state of the ferromagnet can be written as a direct product
of local spins and it does not fluctuate. Hence the truncated two-point
correlation function is always vanishing.
Thus the Heisenberg ferromagnet
is a special model, which does not contain quantum fluctuations in the
ground state. Our theorem successfully excludes this special case.
{\bf Example 5.} Finally we consider lattice fermion-systems, e.g.,
the Hubbard model (\ref{eq:Hubbard}) on ${\bf Z}^2$ or ${\bf Z}^3$,
and consider the spontaneous breakdown of the $U(1)$-gauge symmetry.
As the order parameter, we take, for example,
\begin{equation}
O_\Lambda = \sum_{i\in\Lambda}O^+_i
= \sum_{i\in\Lambda} (c_{i\uparrow}^* c_{i\downarrow}^*
+ c_{i\downarrow} c_{i\uparrow}).
\end{equation}
One can take other types of order parameters, as well.
The generator of gauge transformation is given by
$J_\Lambda=\sum_{i\in\Lambda} (n_{i\uparrow} + n_{i\downarrow})$ and hence
$\chi_J$ denotes the charge susceptibility. Assume that the ground state
defined by (\ref{eq:GS}) shows superconductivity and satisfies
\begin{equation}
\omega ( [J_\Lambda , O^-_j] ) = 2 i \omega ( O^+_j ) \ne 0,
\end{equation}
where $O^-_j = i c_{j\uparrow}^* c_{j\downarrow}^*
- i c_{j\downarrow} c_{j\uparrow}$.
For this system, Theorem~2 states that, if the compressibility is
finite, we have
\begin{equation}
|\omega ( O^-_0 O^-_r )| \ge O\left( {1\over |r|^{d-1}} \right)
\end{equation}
for sufficiently large $|r|$.
It should be remarked that the Coulomb interaction does not satisfy
the condition (\ref{eq:range})
and hence Theorem~2 is not applicable to the
models that contain the Coulomb interactions. Decay of correlation
functions in these systems may closely relate to the
Anderson-Higgs phenomena and it is out of scope of this paper.
%\newpage
\subsection{Proof of Theorems}
\label{sec:proof}
In this section, we shall show proofs of Lemma, Theorem~1 and Theorem~2.
%\vspace{12mm}
\vspace{6mm}
{\it Proof of Lemma.}
As in ref.~\cite{PitaevskiiS}, we use the following two
inequalities; one is the uncertainty relation,\upcite{PitaevskiiS}
\begin{equation}\label{ineq:Uncertainty}
|\omega ( [C,A])|^2
\le \omega ( \{ \Delta C^*,\Delta C \} )
\omega ( \{ \Delta A^*,\Delta A \} )
\end{equation}
for any $A$, $C\in\A_\Omega$ with $\Delta C = C - \omega ( C )$
and $\Delta A = A - \omega ( A )$, and
the other is Kennedy, Lieb and Shastry's
inequality,\upcite{KennedyLS}
\begin{equation}\label{ineq:KLS}
\omega ( \{\Delta C^*, \Delta C\})^2
\le D(C) \omega ( [ [ C^*, H_\Omega ], C ] )
\end{equation}
for any $C\in\A_\Omega$.
Here $H_\Omega$ denotes the Hamiltonian on $\Omega$ and $D(C)$ denotes the
Duhamel two-point function of $C$,
\begin{equation}
D(C) = \lim_{B\downarrow 0}\lim_{\Lambda\uparrow\L}
\lim_{\beta\uparrow\infty}
\int^\beta_0 d\lambda
\{ \omega_{\Lambda,B} ( C^* C( i\lambda ) )
- \omega_{\Lambda,B} ( C^* ) \omega_{\Lambda,B} ( C ) \} ,
\end{equation}
where
\begin{equation}
\omega_{\Lambda,B} ( \cdots )
= \frac{ {\rm Tr} [\cdots \exp \{-\beta(H_\Lambda -BO_\Lambda)\}] }
{ {\rm Tr} [\exp \{-\beta(H_\Lambda -BO_\Lambda)\}] }
\end{equation}
and
\begin{equation}\label{eq:time_evolution}
C(t) = \exp \{ it (H_\Lambda -BO_\Lambda)\}
C \exp \{ -it (H_\Lambda -BO_\Lambda)\}.
\end{equation}
Both inequalities (\ref{ineq:Uncertainty}) and (\ref{ineq:KLS})
were first obtained for finite-volume systems.
Taking the thermodynamic limit of the inequalities, one obtains
(\ref{ineq:Uncertainty}) and (\ref{ineq:KLS}). Combining
(\ref{ineq:Uncertainty}) and (\ref{ineq:KLS}), we have
\begin{equation}
\label{ineq:PitaevskiiS}
|\omega ( [C,A])|^2
\le \biggl\{ D(C) \omega ( [ [ C^*, H_\Omega ], C ] )\biggr\}^{1/2}
\omega ( \{ \Delta A^*, \Delta A\} )
\end{equation}
for any $A$, $C\in \A_\Omega$, where $\Delta A = A - \omega ( A )$.
Setting $A$ as $A_{\Omega_{\rm S}} = |\Omega_{\rm S}|^{-1}
\sum_{x\in\Omega_{\rm S}} \tau_x (A)$ with $A\in\A_{\Lambda}$ and
$C=J_\Omega$ in (\ref{ineq:PitaevskiiS}), we obtain an upper
bound of~(\ref{eq:translation}).
To estimate properly the $R$ dependence of the right-hand side of
(\ref{ineq:PitaevskiiS}), we use the smooth
action\upcite{FrohlichP,KleinLS} of $J_{\Omega}$.
We set the operator $C$ as
\begin{equation}
C = J_f = \sum_{x\in\L} f(x) J_x,
\end{equation}
where $f(x)=1$ for $x\in\Omega$, and $f(x)\rightarrow 0$ as
$|x|\rightarrow\infty$.
Defining $x_{\rm max}$ by $x_{\rm max}=\max_i |x_i|$,
we set the function $f(x)$ in the form
\begin{equation}
f(x) = \left\{
\begin{array}{ll}
1 & \mbox{\hspace{1cm}} (x_{\rm max}d),
\end{array} \right.
\end{equation}
where we have used the clustering property~(\ref{cond:cluster}).
Calculations of $\omega ( [[J_f,H],J_f] )$ are also given
in ref.~\cite{Martin}. Though the definition of the smooth function
$f(x)$ is different from ours, the derivations and results of
ref.~\cite{Martin} still hold only by changing the spherical supports
to the hyper-cubic ones. Thus we have
\begin{equation}\label{ineq:DoubleC}
\omega ( [[J_f,H],J_f] ) \le M \Vert J_0 \Vert^2 R^{d-2}
\sum_x |x|^2 \psi (x),
\end{equation}
where $M$ is a positive finite constant.
If we use $J_\Omega$ instead of $J_f$ in~(\ref{ineq:DoubleC}),
$\omega ([[J_\Omega,H],J_\Omega])$ can be bounded by the form $R^{d-1}$.
Thus in (\ref{ineq:DoubleC}) the double commutator is
better estimated due to the smooth action.
Inserting~(\ref{ineq:DTF3})--(\ref{ineq:DoubleC})
into~(\ref{ineq:1}), we obtain~(\ref{Lemma}).~\qed
%\vspace{12mm}
\vspace{6mm}
{\it Proof of Theorem~1.}
Setting $d=1$ in Lemma, taking the $R\rightarrow\infty$ limit, and
using (\ref{eq:deriv}) and (\ref{eq:translation}),
one obtains (\ref{eq:Theorem1}) for any $\delta >0$,
if $\chi_J <\infty$.~\qed
%\vspace{12mm}
\vspace{6mm}
{\it Proof of Theorem~2.}
Consider the case that all hypotheses of this theorem are satisfied
and furthermore assume that the truncated two-point
correlation function of $A$ decays faster than $1/|x|^{d-1}$, i.e.,
\begin{equation}\label{eq:assumption}
|\omega (A^* \tau_x (A)) - \omega (A^*)\omega (\tau_x(A))| \le
o \left( {1 \over |x|^{d-1}} \right).
\end{equation}
Here $o(|x|^{-d+1})$ denotes a number that is lower order than $|x|^{-d+1}$.
Using (\ref{eq:assumption}) instead of the clustering
property (\ref{cond:cluster}), one can obtain
\begin{equation}
\omega(\{ \Delta A^*_{\Omega_{\rm S}}, \Delta A_{\Omega_{\rm S}} \})
\le o(R^{-d+1})
\end{equation}
instead of (\ref{ineq:ACR}). Thus, slightly modifying the proof of Lemma,
we obtain
\begin{equation}\label{contradict}
\left| {1\over |\Omega_{\rm s}|} \sum_{x\in\Omega_{\rm s}}
\omega ([J_\Omega,\tau_x(A)]) \right|^2
\le o(R^0),
\end{equation}
where $o(R^0)$ denotes a number that vanishes in the $R\rightarrow \infty$
limit.
(Remember that we are in the condition $\chi<\infty$.)
In the $R\rightarrow\infty$ limit, (\ref{contradict}) shows
$\omega ([J_\Lambda,A])=0$. This clearly contradicts with the
condition $\omega([J_\Lambda,A]) \ne 0$ and hence, by contradiction,
we arrive at (\ref{eq:Theorem2}).~\qed
\section*{Acknowledgments}
The author would like to thank Professor~K.~Kubo for critically reading
this manuscript and useful comments, and
Professor~S.~Takada and Dr.~T.~Koma for stimulating discussions.
He also acknowledges the financial support by the Japan Society
for the Promotion of Science (JSPS).
%\newpage
\renewcommand{\thesection}{\Alph{section}}
\setcounter{section}{0}
\section{Definitions of the uniform susceptibility}
We give a comment on the definition of the uniform susceptibility
(\ref{def:suscep2}).
In the literature, the susceptibility is usually defined by
\begin{equation}\label{def:suscep1}
X_J\equiv\lim_{B\downarrow 0} \lim_{\Lambda\uparrow \L}
{1\over |\Lambda|} D_{\Lambda,B}(J_\Lambda),
\end{equation}
where
\begin{equation}
D_{\Lambda,B} (A) = \lim_{\beta\uparrow\infty}
\int^\beta_0 d\lambda
\{ \omega_{\Lambda,B}^\beta (A^* A(i\lambda))
- \omega_{\Lambda,B}^\beta (A^*) \omega_{\Lambda,B}^\beta (A) \}
\end{equation}
with
\begin{equation}
\omega_{\Lambda,B}^\beta (\cdots)
= { {\rm Tr} [\cdots \exp\{-\beta(H_\Lambda - B O_\Lambda) \}] \over
{\rm Tr} [\exp\{-\beta(H_\Lambda - B O_\Lambda) \}] }.
\end{equation}
For an arbitrary self-adjoint operator $A\in\A_\Lambda$, $D_{\Lambda,B} (A)$
can be written as
\begin{equation}
D_{\Lambda,B} (A) =
2 \int^\infty_0 d\lambda
\{ \omega^{\beta=\infty}_{\Lambda,B} (A A(i\lambda))
- \omega^{\beta=\infty}_{\Lambda,B} (A)
\omega^{\beta=\infty}_{\Lambda,B} (A) \}.
\end{equation}
In (\ref{def:suscep1}), the limits are taken so that the state
$\omega^\beta_{\Lambda,B}(\cdots)$ converges.
Here we assume that the limits of the quantity in (\ref{def:suscep1})
exist and that $X_J$ is well-defined.
Our definition of the uniform susceptibility is however different from
(\ref{def:suscep1}). In this paper, we have defined the uniform susceptibility
as follows
\begin{eqnarray}
\chi_J &\equiv& \lim_{\Omega\uparrow\L} \lim_{B\downarrow 0}
\lim_{\Lambda\uparrow \L}
{1\over |\Omega|} D_{\Lambda,B}(J_\Omega) \nonumber\\
&=& \lim_{\Omega\uparrow\L} {2 \over |\Omega|}
\int^\infty_0 d\lambda \{ \omega (J_\Omega J_\Omega(i\lambda))
- \omega (J_\Omega) \omega (J_\Omega) \}
\end{eqnarray}
taking suitable subsequences of $\Lambda$ and $B$, where $\Omega$ is set
as the hyper-cubic subsets $\{ x\in\L$~:~$|x_i|\le R_0$~for~$i=1,\dots,d \}$
and
\begin{equation}
\omega (\cdots)
= \lim_{B\downarrow 0} \lim_{\Lambda\uparrow\L}
\lim_{\beta\uparrow\infty} \omega_{\Lambda,B}^\beta (\cdots).
\end{equation}
In this Appendix, we shall show that these two definitions are equivalent and
hence $\chi_J$ converges to $X_J$.
Consider a finite subset $\Lambda(\supset\Omega)$ and a function $g(x)$
defined by
\begin{equation}
g(x) = \left\{
\begin{array}{ll}
1 & \mbox{\hspace{1cm}} (x\in\Omega) \\%[0.5cm]
0 & \mbox{\hspace{1cm}} (x\notin\Omega),
\end{array} \right.
\end{equation}
then we have $J_\Omega = \sum_{x\in\Lambda} g(x) J_x$ and
\begin{eqnarray}
{1\over|\Omega|} D_{\Lambda,B} (J_\Omega)
&=& {1\over|\Omega|} {1\over|\Lambda|} D_{\Lambda,B} (\sum_k g_{-k} J_k)
\nonumber\\
&=& {1\over|\Omega|} {1\over|\Lambda|} \sum_k |g_k|^2 D_{\Lambda,B} (J_k),
\label{eq:relation}
\end{eqnarray}
where
$J_k = |\Lambda|^{-1/2} \sum_{x\in\Lambda} J_x \exp(ikx)$ and
$g_k = \sum_{x\in\Lambda} g(x) \exp(ikx)$.
In the thermodynamic limit, (\ref{eq:relation}) can be written as
\begin{equation}\label{eq:relation2}
\lim_{B\downarrow 0} \lim_{\Lambda \uparrow \L}
{1\over|\Omega|} D_{\Lambda,B}(J_\Omega)
= {1\over|\Omega|} \int_{|k_i|\le \pi} {d^d k \over (2\pi)^d} |g_k|^2 X_J (k),
\end{equation}
where
\begin{equation}
X_J (k) = \lim_{B\downarrow 0} \lim_{\Lambda\uparrow \L} D_{\Lambda,B} (J_k).
\end{equation}
The function $|\Omega|^{-1} |g_k|^2$ has the following two properties;
\begin{equation}
\int_{|k_i|\le \pi} {d^d k \over (2\pi)^d} {1\over |\Omega|} |g_k|^2 = 1
\end{equation}
and
\begin{eqnarray}
\lim_{\Omega\uparrow\L} {1\over |\Omega|} |g_k|^2
&=& \lim_{R_0 \uparrow \infty} {1\over (2R_0+1)^d}
\left\{ \prod^d_{i=1} {\sin k_i ( R_0+1/2 ) \over \sin k_i/2} \right\}^2
\nonumber\\
&=& 0
\end{eqnarray}
for any $k$ satisfying $k\ne 0$ and $|k_i|\le\pi$.
Hence it converges to the Dirac's delta function,
\begin{equation}\label{eq:delta}
\lim_{\Omega\uparrow\L} {1\over|\Omega|} |g_k|^2 = (2\pi)^d \delta (k)
\end{equation}
for $|k_i|\le\pi$. Inserting (\ref{eq:delta}) into (\ref{eq:relation2}) and
using $X_J (k=0) = X_J$, we thus find that $\chi_J = X_J$.
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