\documentstyle[12pt]{article}
\begin{document}
\author{{\bf J. Rodrigo Parreira}\thanks{\hspace{0.1in}Supported by CAPES.} \and
{\bf O. Bolina}\thanks{\hspace{0.1in}Supported by FAPESP.} \and {\bf J.
Fernando Perez}\thanks{\hspace{0.1in}Partially supported by CNPq.} \\
%EndAName
Instituto de F\'{\i}sica\\
Universidade de S\~ao Paulo\\
P.O. Box 66318\\
05389-970 S\~ao Paulo\\
Brazil}
\title{{\bf N\'eel Order in the Ground State of Spin-1/2 Heisenberg Antiferromagnetic
Multilayer Systems} }
\date{July, 1996}
\maketitle
\begin{abstract}
We show existence of N\'eel order for the ground state of a system with $M$
two-dimensional layers with spin-1/2 and Heisenberg antiferromagnetic
coupling, provided $M\geq 8$. The method uses the infrared bounds for the
ground state combined with the ideas introduced by Kennedy, Lieb and Shastry.
\end{abstract}
{\bf in:} {\it Journal of Statistical Physics} (to appear)
\section{Introduction}
The existence of N\'eel order in the ground state of two-dimensional
spin-1/2 Heisenberg antiferromagnets remains an open problem \cite{KLS88}.
For spin $S\geq 1$, Jord\~ao Neves and Perez \cite{NP86} adapted the
techniques of Dyson, Lieb and Simon \cite{DLS78} to show long range order
(LRO) in the ground state (actually, due to a numerical oversight corrected
in \cite{AKLT88}, the result was claimed to be valid only for $S\geq 3/2$ ).
Nevertheless, the method is not sharp enough to obtain order for the $S=1/2$
case even for the ground state of the three-dimensional model \cite{Parr95}
. Later, Kennedy, Lieb and Shastry \cite{KLS88} improved these techniques to
show N\'eel order for $S\geq 1/2$ and $d\geq 3$ .
In this note we consider the ground-state of a system composed of an even
number $M$ of two-dimensional layers with $S\geq 1/2$. Each layer is an
infinite two-dimensional square lattice so that the finite volume
Hamiltonian is given by:
\[
{\cal H}_{M,N}=\sum_{x\in \Lambda }\sum_{i=1}^3{\bf S}_x.{\bf S}_{x+{\bf l}%
_i},
\]
where $\Lambda =\{1,...,N\}^2\times \{1,...,M\}$, $N\,$ is even and ${\bf l}%
_i$ is the unit vector in the $i$-direction. We are interested in the
thermodynamical limit $N\rightarrow \infty $, with fixed $M$. For $M=1$ we
would have the two-dimensional model and in the limit $M\rightarrow \infty $
we obtain the $d=3$ system so that our Hamiltonian interpolates between two
and three dimensions.
We shall take periodic boundary conditions in all directions. The model with
open boundary conditions in the third direction, the relevant one when
dealing with real systems of layers, cannot be controlled with our methods,
since the use of periodic boundary conditions is essential the derivation of
the crucial infrared bound. On intuitive grounds one would expect to need a
bigger number of layers for N\'eel order to appear.
We show the existence of N\'eel order provided $M\geq 8$, for $S\geq \frac
12 $.
It should be stressed that, to the system under consideration, the
Mermin-Wagner argument applies, so that N\'eel order is absent for any
positive temperature. This is in contrast to the three-dimensional model
proposed and discussed in \cite{KLS88} where the coupling constant in the
third dimension was affected by a factor $r$, with $0\leq r\leq 1$, thus
simulating an interpolation between the 2- and 3-dimensional models ($r=0,1$
corresponding respectively to 2 and 3 dimensions). For the latter model
N\'eel order was shown to hold, both for the ground state and sufficiently
low temperatures, if $r\geq 0.16$.
\section{The technique}
Our proof is carried out along the same lines of \cite{KLS88}. After
defining
\[
S_q=\frac 1{\left| \Lambda \right| ^{1/2}}\sum_{x\in \Lambda }S_x^3\exp
\{-iqx\}
\]
and $g_q=\left\langle S_{-q}S_q\right\rangle $, where by $\left\langle
.\right\rangle $ we mean the expectation value in the ground state, we have
the sum rule
\begin{equation}
I\equiv \frac 1M\sum_{q_3\in B_M}\frac 1{(2\pi )^2}\int g_qd^2q=\frac{S(S+1)}%
3. \label{I}
\end{equation}
Here, the summation on $q_{3}$is taken over $B_M=\left\{ \frac{2\pi }%
Mk,k=1,2,...,M\right\} $. We also have the infrared bound \cite{NP86},
\begin{equation}
0\leq g_q\leq f_q=\left( \frac{\left\langle [[S_q,{\cal H}%
_M],S_{-q}]\right\rangle }{4E_{q-Q}}\right) ^{1/2}, \label{irb}
\end{equation}
where
\begin{equation}
E_q=\sum_{i=1}^3(1-\cos q_i), \label{f:M1}
\end{equation}
and $Q$ is the point $(\pi ,\pi ,\pi )$.
Under the assumption of absence of LRO we have the following formula:
\begin{equation}
\frac 1M\sum_{q_3\in B_M}\frac 1{(2\pi )^2}\int g_q(\cos q_1+\cos q_2+\cos
q_3)d^2q=-\frac{e_0}3, \label{c}
\end{equation}
where $-e_0$ is the ground state energy per site of the system.
We then maximize the integral $I$, over all functions $g_q$ satisfying both
the infrared bound (\ref{irb}) and the constraint (\ref{c}). If $I_{\max }$,
the maximum value so obtained, is less than $\frac{S\left( S+1\right) }3$,
we have a contradiction, implying the existence of LRO.
\section{The results}
The expectation value of the double commutator in (\ref{irb}) can be
calculated explicitly and equals
\[
\frac 23[(2-\cos q_1-\cos q_2)\rho _1+(1-\cos q_3)\rho _3],
\]
where
\[
\rho _1=-\left\langle {\bf S}_x.{\bf S}_{x+l_1}\right\rangle \; {\rm and }%
\; \rho _3=-\left\langle {\bf S}_x.{\bf S}_{x+l_3}\right\rangle.
\]
It should be first remarked that $\rho _1$ and $\rho _3$ are non negative.
This result is a Griffiths' inequality of the first type for the system. Its
proof can be reduced to the proof, given by Ginibre \cite{Gi}, of a class of
such inequalities for quantum ferromagnetic spin systems. This is achieved
by performing on the system a rotation by an angle $\pi $ around the 3rd
spin direction, on the even sublattice. This transforms the Hamiltonian of
the system into:
\[
{\cal H}_{M,N}^{\prime }=\sum_{x\in \Lambda }\sum_{i=1}^3\left( -S_x^1.S_{x+%
{\bf l}_i}^1-S_x^2.S_{x+{\bf l}_i}^2+S_x^3.S_{x+{\bf l}_i}^3\right) ,
\]
To the expectation values in the ground state of the modified system, the
method used in the proof of the Corollary of Theorem 5, page 110 of \cite{Gi}
applies (as the sign of the coupling in the third direction is irrelevant),
so that for any pair of sites $x$ and $y$:
\[
\left\langle S_x^1S_y^1+S_x^2S_y^2\right\rangle ^{\prime }\geq 0.
\]
{\bf Remark: }Actually Ginibre's proof is given for a positive temperature
state, but the inequality clearly survives the limit $\beta \rightarrow
\infty $ .
Undoing the transformation, we obtain for the original system and for any
pair of nearest neighbor sites $x$ and $y$:
\[
\left\langle S_x^1S_y^1+S_x^2S_y^2\right\rangle \leq 0.
\]
Using the isotropy in the spin variable, we finally get, for any pair of
nearest neighbor sites $x$ and $y$
\[
\left\langle {\bf S}_x.{\bf S}_y\right\rangle =\left\langle
S_x^1S_y^1+S_x^2S_y^2+S_x^3S_y^3\right\rangle \leq 0.
\]
At this point our method departs from the one used in \cite{KLS88} as we
have not been able to estimate $\rho _1$ and $\rho _3$ in terms of the
ground state energy density $e_0$. So, we consider separately two
situations:
\[
{\rm Case \; 1: } \; \rho _1\leq \frac{e_0}4,\rho _3\leq e_0
\]
\[
{\rm Case \; 2: } \; \rho _1\leq \frac{e_0}2,\rho _3\leq \frac{e_0}2
.
\]
One of these two cases always hold, since $e_0=2\rho _1+\rho _3$, $\rho
_1\geq 0$ and $\rho _3\geq 0$. We shall compute $I_{\max }$ for each case at
a time. The conclusion follows if we get $I_{\max }<\frac{S(S+1)}3$ in both
cases.
As in \cite{KLS88}, the maximum of $I$, for all $g_q$ subjected to (\ref{irb}%
), is attained by
\[
g_q=f_q\chi (\cos q_1+\cos q_2+\cos q_3<\alpha )
\]
where $\alpha $ is a positive constant and $\chi (.)=1$ when condition $%
\left( .\right) $ is true and zero otherwise. We perform a numerical
calculation in order to determine the value of $\alpha $ so that the
constraint (\ref{c}) is verified. A lower bound for $e_0$ can be obtained by
using the N\'eel state as a variational trial ground state. For our system
it is given by $e_0\geq e_0^{\left( N\right) }=0.75$. Replacing $e_0$ by $%
e_0^{\left( N\right) }$, we compute the value of $\alpha ^{\left( N\right) }$%
, for $M=8$ :
\[
{\rm Case \; 1: } \; \alpha _1^{\left( N\right) }=0.71160
\]
\[
{\rm Case \; 2: } \; \alpha _2^{\left( N\right) }=0.72056
\]
and for both of these values the integral $I_{\max }$ is calculated and
shown to be less than $\frac{S\left( S+1\right) }3$:
\[
{\rm Case \; 1: } \; I_{\max ,1}^{\left( N\right) }=0.24388,
\]
\[
{\rm Case \; 2: } \; I_{\max ,2}^{\left( N\right) }=0.24650.
\]
By monotonicity we have that $I_{\max ,1}\leq I_{\max ,1}^{\left( N\right) }$
and $I_{\max ,2}\leq I_{\max ,2}^{\left( N\right) }$, implying $I_{\max }<%
\frac{S\left( S+1\right) }3$, which concludes the proof.
\begin{thebibliography}{9}
\bibitem{KLS88} T. Kennedy, E. H. Lieb and B. S. Shastry; {\it J. Stat.
Phys.}, {\bf 53}:1019-1030 (1988).
\bibitem{NP86} E. Jord\~ao Neves and J. Fernando Perez; {\it Phys. Lett.},%
{\it \ }{\bf 114A}: 331-333 (1986).
\bibitem{AKLT88} I. Affleck, T. Kennedy, E.H.Lieb and H. Tasaki; {\it %
Commun. Math. Phys.}, {\bf 115}: 477-528 (1988).
\bibitem{DLS78} F. J. Dyson, E. H. Lieb and B. Simon; {\it J. Stat. Phys.}%
{\bf \ 18}: 335-383(1978).
\bibitem{Parr95} J. R. Parreira; Ph.D. Thesis, Instituto de F\'\i sica,
Universidade de S\~ao Paulo (1995)
\bibitem{Gi} J. Ginibre; Correlations in Ising Ferromagnets, Carg\`ese
Lectures in Physics vol.4, 95-112, Gordon and Breach, (1970)
\end{thebibliography}
\end{document}