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\begin{document}
\author{{\bf J. Rodrigo Parreira}\thanks{%
Supported by CNPq} \\
%EndAName
Department of Physics \\
Princeton University\\
P.O. Box 708\\
08544-0708 Princeton USA\\
{\bf e-mail}: parreira@math.princeton.edu}
\title{{\bf Phase Transitions in Hierarchical Models: The Role of
Coupling Coefficients }} \maketitle
\begin{abstract}
We investigate the role of coupling coefficients in determining the
existence of
phase transitions for Hierarquical models of Dyson type in arbitrary
dimensions. In
particular, we show that, when the random spin variables display discrete
symmetry, and the coupling constants are chosen so that asymptotically
they match the behavior of the usual Laplacean, a phase transition is not
expected to occur in 2 dimensions. We show this fact in two different ways.
First we
develop a heuristic argument based on the balance of energy and entropy
of the model. The rigorous proof is then obtained by the generalization to
several
dimensions of a theorem due to Dyson that states the non-existence of
phase-transitions when the coupling constants of the model are bounded.
\end{abstract}
{\bf Key Words}: Classical spin systems, Hierarchical models, Ising
model, Phase Transitions
\section{Introduction.}
Hierarchical models (HM) were introduced in physics literature by Dyson \cite
{D} for the study of ferromagnetic spin systems with long range
interactions in one dimension. A few years latter, Baker \cite{B}
rediscovered them and pointed out that, for this kind of system, the
Renormalization Group transformations are exact.
The basic idea to construct HM is to integrate
the short distance
interactions of the Hamiltonian in order to extract the long distance
asymptotic behavior of the model.
This is done by splitting the spin variables into two parts. The first one,
generated by the so-called block-spin
transformation, will describe the effective interaction between mean-spins
defined over blocks
of a predefined size. The other one, orthogonal to the first, is produced by
the fluctuation transformation and will be used to describe the
interactions inside each block.
However being a strong simplification of the original model, and even
not displaying some important symmetry characteristics, such as the
invariance under translations of the Hamiltonian, HM are
believed to preserve all relevant information concerning the infrared
behavior of the original models. In this sense, they are frequently used to
describe
situations close to the critical temperature, in various statistical
mechanics models, where this kind of effects dominate the global system.
More recently, Paiva and Perez \cite{PP} developed a Hierarchical
Laplacean which generalizes Dyson's HM for any dimension and
block size. Although their main concern was the study of random walks in
a random environment, their formulation, being general, permits one to
build any of the usual statistical mechanics spin systems, in a
hierarchical version. The coupling parameters in their Hierarchical
Laplacean are chosen so that its large distance behavior is equal to the
one showed by the usual lattice Laplacean.
We are going to show in this paper that, if the coupling parameters are
chosen in such a way, the Ising Hierarchical Model does not have a phase
transition in two dimensions. We do this in two different ways. The first
one is heuristic but in some sense clarifies the point we have in mind.
We develop an energy-entropy argument for this model, calculating the
energy cost of inserting, in a system that finds itself in a pure phase,
a region of reversed magnetization. It turns out that, for one and two
dimenions, this energy cost is finite, so that the pure phase is unstable
and long range order is not expected to occur for any positive
temperature (for details about this kind of argument, the reader is
refered to \cite{G}). The second way of proving this fact is rigorous,
since we extend a theorem by Dyson \cite{D}, that asserts the absence of a
phase transition when the coupling parameters are bounded, to the case of
any dimension and block size (Dyson's result is restricted to one
dimension and block size equals to $2$). We then find that, for the
particular choice of coupling coefficients of Paiva and Perez, $d=2$ is
exactly the limiting dimension, so that for $d \leq 2$ a phase transition is
not
expected. It is important to remark that our proof lies strongly on the
use of Griffiths inequalities \cite{G2} which are not available for
models in which the spin variables display continuous symmetry. It would
be very interesting to obtain similar results in such a case.
\section{The Hierarchical Laplacean}
Consider a $d-$dimensional region $\Lambda \subset {\bf Z}^d$. Without
loss of generality we assume the number of point inside $\Lambda$ to
be $L^{dN}$. $\Lambda$ is now divided
in blocks of volume $L^d$. A block of first hierarchy is defined by:
$$
B_x^{(1)}=L^{-\frac d2}\sum_{y\in B(Lx,L)}\delta _y,
$$
where $B(Lx,L)$ means the block of center $Lx$ and side $L$. A second
hierarchy block by:
$$
B_x^{(2)}=L^{-\frac d2}\sum_{y\in B(Lx,L)}B_y^{(1)},
$$
and so on.
The Hierarchical Laplacean is given by:
$$
-\Delta _h=\sum_{x\in {\bf Z}^d}\sum_{n\geq 1}L^{-2n}Q_x^{(n)},
$$
where:
$$
Q_x^{(n)}=\sum_{y\in B(Lx,L)}\mid B_y^{(n-1)}\rangle \langle B_y^{(n-1)}\mid
-\mid B_x^{(n)}\rangle \langle B_x^{(n)}\mid .
$$
The first term of the difference is the block operator, that projects on
functions which have support on $B(L^nx,L^n)$. The $Q_x^{(n)}$ is the
fluctuation operator.
The $L^{-2n}$ prefactor is introduced to obtain the standard Green's function
decay with distance, as we are going to see. Using the fact that:
$$
\sum_{x\in {\bf Z}^d}\sum_{n\geq 1}Q_x^{(n)}={\bf 1},
$$
we can immediatly show that the associated Green's function for this
operator is given by:
$$
G(x,y;z)=(-\Delta _h-z)^{-1}(x,y)=
$$
$$
=\left( -\delta _x,\sum_{n\geq 1}\sum_{w\in {\bf Z}^d}\frac{Q_w^{(n)}}{%
L^{-2n}-z}\delta _y\right) .
$$
Developing this expression, we have:
$$
G(0,x;z)=(L^d-1)\sum_{n\geq N(0,x)}L^{-dn}(L^{-2n}-z)^{-1}+\delta
_{0,x}(L^{-2}-z)^{-1},
$$
where $N(0,x)$ is the smallest hierarchy that contains both $0$ and $x$.
If we now introduce the hierarchial distance $\mid x-y\mid _h$
defined by:
$$
\mid x-y\mid _h=
\left
\{
\begin{array}{c}
L^{N(x,y)}\,,if\;\;\;x\neq y \\
\\
0\;\;\;\;\;\;\;\;,if\;\;\;x=y
\end{array}
\right.
$$
and make the remark that $\mid x-y\mid \rightarrow \infty $ ($\mid .\mid $is
the usual
euclidean distance) implies $\mid x-y\mid _h\rightarrow \infty $, we are
lead to, for $d>2$, taking the limit $z\rightarrow 0$:
$$
-\Delta _h^{-1}=\frac{1-L^{-2}}{L^{-2}-L^{-d}}L^{-2N}(\mid x\mid
_h)^{-d}+\delta _{0,x}L^2.
$$
For $d=1$, we have:
$$
-\Delta _h^{-1}=(L-1)\mid x\mid _h,
$$
and, for $d=2$:
$$
-\Delta _h^{-1}=(L^2-1)\log {}_L\mid x\mid _h,
$$
so that in every case the asymptotic decay of the hierarchical
Laplacean is equal to the one of the usual Laplacean, namely $|x|^{2-d}$.
This Hierarchical Laplacean is an extension of Dyson's first Hierarchical
Model in the sense that taking one-dimensional spin arrays with $L=2$ we
recover the energy expression in \cite{D}.
\section{Energy-Entropy Argument for The Hierarquical Model}
Suppose now that, at each point $x \in
\Lambda$, is defined an Ising random variable $\varphi(x)$ that can assume
values on the set $\{-1,+1\}$.
To develop our energy-entropy argument, we need to study the energy
cost of inserting a `bubble' of reversed spins in a purely magnetized phase.
The quadratic form corresponding to the energy of this system can be easily
calculated:
$$
H_{N}=(\varphi ,-\Delta _h\varphi )=
$$
\beq
=\sum_{n=0}^N L^{-n(d+2)}(L^{-2}-1+\delta _{n,0})\sum_{x\in Z^d}[\varphi
_B^{(n)}(x)]^2. \label{H}
\eeq
The block-spin variables $\varphi
_B^{(n)}(x)$ are defined as follows:
$$
\varphi _B^{(0)}(x)=\varphi (x),
$$
$$
\varphi _B^{(n)}(x)= \sum_{y\in B(Lx,L)}\varphi (y).
$$
For the totally magnetized system, the energy will be, when $N
\rightarrow \infty$:
$$
(\varphi ,-\Delta _h\varphi )_0=\sum_{n=0}^\infty L^{-n(d+2)}(L^{-2}-1+\delta
_{n,0})\sum_{x\in Z^d}L^{2dn}
$$
The energy cost for the introduction of a whole $k$ hierarchy of reversed
spins will be given by:
$$
\Delta E=(\varphi ,-\Delta _h\varphi )_{+}-(\varphi ,-\Delta _h\varphi )_0,
$$
where $(\varphi ,-\Delta _h\varphi )_{+}$ is the model energy with the
reversed hierarchy. We have also:
$$
(\varphi ,-\Delta _h\varphi )_{+}=(\varphi ,-\Delta _h\varphi )_{+}^{n\leq
k}+(\varphi ,-\Delta _h\varphi )_{+}^{n>k},
$$
$$
(\varphi ,-\Delta _h\varphi )_0=(\varphi ,-\Delta _h\varphi )_0^{n\leq
k}+(\varphi ,-\Delta _h\varphi )_0^{n>k},
$$
but $(\varphi ,-\Delta _h\varphi )_{+}^{n\leq k}=(\varphi ,-\Delta _h\varphi
)_0^{n\leq k}$. In view of these properties, we obtain:
$$
\Delta E=\sum_{n>k}^\infty \frac{L^{-2n}L^{-2}}{L^{dn}}\left\{ \left[
L^{dn}-2L^{dk}\right] ^2-L^{2dn}\right\} .
$$
Changing variables, $n-k=j$:
$$
\Delta E=\sum_{j=0}^\infty \frac{L^{-2(j+k)}L^{-2}}{L^{d(j+k)}}\left\{
\left[ L^{d(j+k)}-2L^{dk}\right] ^2-L^{2d(j+k)}\right\} =
$$
$$
=L^{k(d-2)}\sum_{j=0}^{\infty}4L^{-j(d+2)}L^{-2}\{1-L^{dj}\}=L^{k(d-2)}%
\Gamma (d),
$$
where $L^{k(d-2)}$ is the volume of the reversed block in $d-2$ dimensions.
The conclusion
is that the Hierarchical Model defined by (\ref{H}) seems not to have a phase
transition for dimension less than three, since for one and two
dimensions the energy cost of inserting a region of reversed
magnetization in a pure phase is finite. This fact is related to the chosen
decay of the Hierarchical Laplacean with distance, as we are going to see
next.
\section{Absence of Phase Transitions for Bounded Coupling Parameters}
We begin by rewriting the expression of the Hamiltonian (\ref{H}) as
\beq
H_{N}=\sum_{n=0}^{N}L^{-2dn}J_{n} \sum_{x \in \Lambda}
[\varphi_{B}^{(n)}(x)]^{2}, \label{modH}
\eeq
where
\beq
J_{n}=L^{n(d-2)}(L^{-2}-1+\delta_{n,0}) \label{J}
\eeq
defines the coupling parameters of the model.
The interaction of a spin with all other spins in $\Lambda$ is given by
\beq
I_{N}=\sum_{n=1}^{N}2J_{n}L^{-2dn}(L^{dn}-1). \label{I}
\eeq
It is also convenient to define the function
\beq
f_{N}(n) = L^{-2dn} \langle \; [\varphi_{B}^{(n)}]^{2} \; \rangle_{N}.
\label{f} \eeq
so that $f_{N}(0)=1$. Here, the expectation value is taken over the system
confined in a finite volume.
The spontaneous magnetization in the thermodynamical limit can then be
represented as
\beq
m^{2} = \lim_{n \rightarrow \infty} f(n), \label{m}
\eeq
where $f(n)=\lim_{N \rightarrow \infty} f_{N}(n)$.
{\bf Theorem}: {\it If the $J_{n}$ are bounded, $m^2 =0$ in the limit $N
\rightarrow \infty$.}
{\bf Proof}
As in \cite{D}, we need only prove the theorem for $J_{n}=1$. Let now
$p_{N}$ be the probability, for finite $N$, that two spins,
$\varphi(x)$ and $\varphi(x')$, such that both $x$ and $x'$ lie in the same
first hierarchy block, are parallel. By Griffiths' inequalities, this
probability does not decrease if all spins in the system, except
$\varphi(x)$ and $\varphi(x')$ are locked parallel by infinite couplings.
We have then
$$
p_{N} \leq \frac{1}{1+ e^{-2 \beta I_{N}}}.
$$
Using the fact that $p_{N} < \lim_{n \rightarrow \infty} p_{N}$ (since
$I_{N}$ is monotonically increasing in $N$), we have
\beq
p_{N} < p = \frac{1}{1+ e^{-2 \beta I}} = g <1, \label{p}
\eeq
where, since $J_{n}=1$, we have
$$
I=\lim_{N\rightarrow \infty} I_{N}=2 L^{-2} \left( \frac{1}{L^{d}-1}-
\frac{1}{L^{2d}-1} \right).
$$
Since all the spins contained in a given block are equivalent, we have
$$
f_{N}(n)=L^{-2dn}L^{d(n-1)}\frac{L^{d}(L^{d}-1)}{2}\langle
\; [\varphi(x)+\varphi(x')]\varphi_{B}^{(n)}(x) \; \rangle_{N}
$$
so that
$$
f_{N}(n)=L^{-dn}(L^{d}-1)p_{N} \langle \; \varphi_{B}^{(n)}(x) \;
\rangle_{N,lock} $$
We now stress that the above average is taken over the model where the two
spins $\varphi(x)$ and $\varphi(x')$ are locked parallel.
Again, by Griffiths inequalities, the above average will not decrease if
the
spins contained in each first hierarchy block are all locked parallel. This
is equivalent to a hierarchical model of the same kind, with the same
previous coupling parameters but with a number of spins $ L^{d(N-1)}$. We
have then
\beq
f_{N}(n) \leq L^{-d(n-1)}(L^{d}-1)p_{N} \langle \; \varphi_{B}^{(n-1)}(x) \;
\rangle_{N-1}=p_{N}f_{N-1}(n-1). \label{recur}
\eeq
Iterating (\ref{recur}), we obtain
$$
f_{N}(n) < gf_{N-1}(n-1) < g^{n}f_{N-n}(0) = g^{n},
$$
so that we have, in the thermodynamical limit
$$
f(n) \leq g^{n},
$$
where, by (\ref{p}), $g<1$. Using then the definition of the spontaneous
magnetization (\ref{m}), we finally get
$$
m^{2}=\lim_{n \rightarrow \infty} f(n) \leq \lim_{n \rightarrow \infty}
g^{n} =0.
$$
{\bf Corollary}: {\it The model defined by} (\ref{modH}) {\it has zero
spontaneous magnetization at any finite temperature in two dimensions}
{\bf Proof}
Imediate, since we see that, for $d=2$, the coupling coefficients $J_{n}$,
defined by (\ref{J}), are bounded.
{\bf Acknowledgements}: We would like to thank Professor J. Fernando
Perez for very important
discussions during the first steps of this work.
\begin{thebibliography}{9}
\bibitem{D} Dyson, F. J.: {\it Commun. Math. Phys.}, {\bf 12}, 91 (1969)
\bibitem{B} Baker Jr., G. A.: {\it Phys. Rev. B}, {\bf 5}, 2622 (1972)
\bibitem{PP} Paiva, C.; Perez, J. F.: {\it J. Stat. Phys.}, {\bf 71}, 435
(1993)
\bibitem{G} Griffiths, R. B. in {\it Statistical Mechanics and
Quantum Field Theory}, C. de Witt, R. Stora eds. (Gordon and Breach, New
York, 1971)
\bibitem{G2} Griffiths, R. B: {\it J. Math. Phys.}, {\bf 8}, 478 (1967)
\end{thebibliography}
\end{document}