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%
\preprint{NYU--TH--94/07/01}
\title{ Low--Temperature Series for Renormalized Operators: the
Ferromagnetic Square--Lattice Ising Model }
\author{
{\bf J. Salas} \thanks{E-mail: salas@mafalda.physics.nyu.edu} \\
{\em Department of Physics} \\
{\em New York University} \\
{\em 4 Washington Place} \\
{\em New York, NY 10003, USA} }
\begin{document}
\setcounter{page}{0}
\maketitle
\thispagestyle{empty}
\begin{abstract}
A method for computing low--temperature series for renormalized operators
in the two--dimensional Ising model is proposed.
Series for the renormalized magnetization and
nearest--neighbor correlation function are given for the majority rule
transformation on $2 \times 2$ blocks and random tie--breaker.
These series are applied to the study at very low temperature
of the first--order phase transition
undergone by this model.
We analyze how truncation in the renormalized Hamiltonian leads to
spurious discontinuities of the Renormalization Group transformation.
\end{abstract}
\vspace*{1cm}
\noindent
{\bf Keywords}: Renormalization group, position--space renormalization--group
transformations, Ising model, low--temperature expansions.
\newpage
\section{Introduction}
The behavior of the Renormalization Group (RG) in the vicinity of
first order phase
transitions has been a very controversial matter for the last 20 years.
In 1975 Nienhuis and Nauenberg \cite{nienhuis_nauenberg} proposed that
the RG transformations behave near first--order transition
points in a similar fashion
as near standard critical points.
Each RG step is smooth (i.e. the renormalized couplings
are analytic functions of the original ones, even at the transition points).
Singular behavior is recoved as we infinitely iterate this transformation
near a fixed point.
Moreover, first--order transition points are governed by a so--called
``discontinuity
fixed point'' (DFP), characterized by
i) A domain of attraction which includes the transition surface.
ii) Zero correlation length (In most systems, first--order transition points
possess a finite correlation length. See \cite{clocks} for a counterexample).
iii) A relevant operator whose critical exponent is given by the dimensionality
of the system $y=d$.
As a matter of fact, there are as many exponents $y=d$ as phases coexist at
the transition line
\footnote{Here we take into account the (trivial) critical
exponent associated with the renormalization
of the identity operator in the Hamiltonian.} \cite{fisher_berker}.
In the Ising model it is believed that the DFP is located at
zero temperature \cite{klein_wallace_zia}.
This picture was criticized by some authors
\cite{blote_swendsen,lang,toni_okawa_shimizu,decker_hasenfratz,hasenfratz_2}
who claimed that the RG flow is itself discontinuous
at the transition line.
That is, they claimed that
the renormalized Hamiltonian has different limiting values depending on
how the original Hamiltonian approaches the transition line.
As a result, they doubted whether the DFP would exist at all.
Most of these claims were based on Monte Carlo Renormalization Group (MCRG)
computations.
In ref.~\cite{hasenfratz_2} non--rigorous analytical arguments were
given to support the same conclusion.
In ref.~\cite{toni_salas2} it was argued that the observed discontinuities
are artifacts due to the truncation of the Hamiltonian space inherent in the
MCRG approach.
In fact, for the two--dimensional Ising model and majority rule with
$2 \times 2$ blocks it was found that the discontinuity in the magnetic
field was of the same order as the truncation error.
Moreover, as the number of operators included in the computation was increased,
the size of this discontinuity decreased.
This puzzle was solved partially by van Enter--Fern\'andez--Sokal
\cite{enter_fernandez_sokal}, who showed
that for systems with bounded dynamical variables and interacting through a
Hamiltonian belonging to the space ${\cal B}^1$ (i.e. the space of
real, absolutely summable and translation--invariant interactions) the
RG flow is always continuous and single--valued, {\em whenever
it exists at all}
(subject to some very mild locality conditions on the RG transformation).
For finite systems the existence of the transformation (i.e. of the
renormalized Hamiltonian) is trivial.
In the thermodynamic limit, however, this is a very subtle problem.
As a matter of fact, these authors proved that the renormalized Hamiltonian
does {\em not} exist in the two--dimensional Ising model when the temperature
is low enough, for the Kadanoff
transformation, decimation, block average and some particular cases of
majority rule.
On the other hand the majority rule with blocks of size $b = 2$
(the case most considered in the literature) is still an
open problem.
Notice that the pathologies always occur at low temperatures.
In such a regime there is an alternative to MCRG computations: the
low--temperature (low--$T$) expansions
\cite{domb,creutz,bhanot_creutz_lacki,guttmann,briggs_enting_guttmannII}.
In this paper we propose to study the behavior of several RG
transformations using low--$T$ expansions.
This approach has several advantages over MCRG computations.
MCRG methods have three sources of
errors: statistical errors, finite--size effects and truncation errors.
Series expansions do not suffer from the first two, as the
observable quantities are obtained directly in the thermodynamic
limit and no stochastic process is involved
\footnote{ Note that unlike many applications of series expansions, here we
are really interested in the behavior at low temperature and {\em not} in the
critical region $T \approx T_c$. Therefore, no extrapolation procedure is
involved.}.
If we wished to obtain a renormalized Hamiltonian from the renormalized
expectation values, then a truncation scheme would be involved.
However, in this paper we will use our results to study the truncation
procedure itself and learn why it works or does not work.
If we truncate the renormalized Hamiltonian (i.e. we allow only a finite
number of renormalized interactions), we can obtain estimates for those
couplings by solving a highly non--linear set of equations,
which involve expectation values of operators computed in the
renormalized measure.
In this paper we develop a procedure to compute series expansions for these
expectation values, which
have not been computed previously (to our knowledge) in the
literature.
For real--space RG transformations the expectation value of an operator $O$
with respect the renormalized measure
can be written as an expectation value in the original measure of a
certain composite operator $\tilde{O}$.
This composite operator $\tilde{O}$ is equal to the original operator $O$
acted upon by a probability kernel (which is the mathematical object
representing the RG transformation).
Thus, if we know how to obtain the low--$T$ expansions in the original
(or unrenormalized) measure, then we can compute any expectation value
by doing the corresponding integral.
These series can be useful in two other ways:
i) They provide a real check for MCRG computations at low temperature.
Expectation values coming from the Monte Carlo simulations can be compared
with the low--$T$ predictions.
ii) When performing a RG transformation the system is viewed at a
larger spatial scale.
For that reason we believe that the low--$T$ series for the
renormalized magnetization, susceptibility and specific heat
could be used to extract the critical exponents (using standard
series--extrapolation techniques).
In fact, a better convergence could be expected for these ``improved'' series.
It would be interesting to devise a computational procedure
to generate these series to an arbitrary order.
On the other hand, the main goal of this paper is to analyze the
truncation issue in the Ising model.
Starting at the first order transition line and at very low temperature,
we would like to know whether it is possible to obtain estimates for the
renormalized couplings in such a
way that the truncated interaction does not contain any odd term.
An affirmative answer
would imply that the approximate RG transformation, restricted to some
finite--dimensional subspace of ${\cal B}^{1}$,
is continuous at the transition line.
We find that this situation occurs for the majority rule transformation (on
$2 \times 2$ blocks) when restricted to a subspace containing a magnetic field
and a nearest--neighbor interaction.
On the other hand, we find that this is not the case
for the decimation and large--$p$ Kadanoff
transformations restricted to the latter two--dimensional subspace or for the
majority rule transformation when restricted to the three--dimensional subspace
containing magnetic field, nearest--neighbor and next--to--nearest--neighbor
interactions.
In all of these cases, the renormalized magnetic field is {\em non--zero}
implying that the approximate RG map is {\em discontinuous}.
Thus, the typical situation seems to be that truncation induces discontinuities
in the RG transformation when restricted to some finite--dimensional
subspace of the interaction space.
However, the relation between these results on truncation and
the results of \cite{enter_fernandez_sokal} on non--Gibbsianness
is far from clear.
This paper is organized as follows. In Section~2 we describe the way the
low--$T$ expansions for renormalized observables can be obtained.
We give three examples for the two--dimensional Ising case: decimation,
Kadanoff transformation and majority rule, all of them with block size $b=2$.
In Section~3 we explain how to generate the low-$T$ series for
the latter example using a computer algorithm.
To show the performance of the method, we
construct the series for the magnetization and energy density
up to 15 terms.
In Section~4 the study of those RG transformations near the Ising first--order
phase transition line is considered.
Finally in Section~5 we present our conclusions.
\section{Series Expansions for Renormalized Operators}
\subsection{ Review of Low-$T$ Expansions}
Let us consider for simplicity a ferromagnetic Ising model on a
two--dimensional square lattice.
The spins
take the values $\pm 1$ and interact through the Hamiltonian
%
\be
\label{ising_hamiltonian}
{\cal H} = - K \sum_{\langle i,j \rangle} (\sigma_i \sigma_j - 1)
- H \sum_{i} (\sigma_i - 1)
\ee
%
where the first sum is over all the nearest--neighbor pairs of spins, and the
second one over every point $i = (i_x,i_y)$ of the lattice.
The partition function for a system of $N$ spins with periodic boundary
conditions is then
%
\be
\label{ising_z}
Z_N = \sum_{ \{ \sigma = \pm 1 \} } e^{ K \sum_{\langle i,j \rangle}
(\sigma_i \sigma_j - 1) + H \sum_{i} (\sigma_i - 1) }
\ee
%
We have absorbed the term $\beta = 1/kT$ in the definition of
the coupling constants $K \ge 0$ and $H$.
We are mainly interested in the zero--field case ($H=0$), but for future
convenience we keep the second term of the Hamiltonian
\reff{ising_hamiltonian}.
This term will be necessary to obtain the zero--field susceptibility
(see below).
The first step to compute low--$T$ expansions is to find out the ground states
of the system at $T=0$.
In our case it is easy to realize that when $H=0$ there are only
two translation--invariant ground states.
Both of them are completely ordered configurations with magnetization
$+1$ and $-1$ respectively.
When $H \neq 0$ then there is only one ground state whose
magnetization is parallel to the magnetic field $H$.
We will choose hereafter the ($+1$)--state as our ground state.
This implies that the magnetic field should be always non--negative
($H \geq 0$).
Furthermore, we have normalized the Hamiltonian \reff{ising_hamiltonian}
in such a way that ${\cal H}(+1) = 0$.
Looking at eq. \reff{ising_z} it is easy to realize that each flipped spin
is penalized by a factor $\lambda = \exp(-2H)$ in the partition function.
And each unsatisfied bond (i.e. a bond with both spins in opposite states)
is suppressed by a factor $\mu = \exp(-2K)$.
All the spin configurations with $n$ flipped
spins and $m$ unsatisfied bonds give the same contribution to the partition
function \reff{ising_z} and equal to $\mu^m \lambda^n$.
So we can group these
configurations together and express the partition function as
%
\be
\label{ising_partition}
Z_N(\mu,\lambda) = \sum_{m,n} Z^{(N)}_{m,n} \mu^m \lambda^n
\ee
%
where $Z^{(N)}_{m,n}$ is the number of configurations with $m$ unsatisfied
bonds and $n$ flipped spins that occur in the system.
These numbers depend explicitly
on the size of the system, as well as on the boundary conditions.
The first term of the expansion corresponds to
the ground state, the second to one flipped spin ($n=1$, $m=4$), the third to
two nearest--neighbor flipped spins ($n=2$, $m=6$), and so on.
With this choice of boundary conditions, $Z^{(N)}_{m,n} = 0$
for odd values of $m$.
This expansion is exact for finite $N$ if all the $2^N$ possible configurations
are taken into account.
The low-$T$ expansion of the partition function \reff{ising_partition} contains
the most relevant terms when the temperature goes to zero.
It can also be viewed as an enumeration of the low--energy excitations of the
system.
Here we are interested in developing an expansion valid as
$K \rightarrow \infty$
with $H$ bounded (i.e. an expansion in powers of $\mu$ ($\ll 1$) whose
coefficients are functions of $\lambda$)
\footnote{Different expansions are obtained when $H \rightarrow \infty$ and
$K$ remains bounded or when both $K$ and $H$ diverge with $K/H \rightarrow$
constant.}.
Thus, the dominant terms are those with the smallest values of $m$.
For a given value of $m$ the possible values of $n$ are finite. For
excitations which do not see the boundary of the system the allowed
values of $n$ are given by
$n \in [m/4, m^2/16] \cup [N-m^2/16, N-m/4]$
(resp. $[(m+2)/4, (m^2-4)/16] \cup [N-(m^2-4)/16,N-(m+2)/4]$)
when $m/2$ is even (resp. odd).
All the terms with the same $m$, irrespective of $n$, are considered to
contribute at the same order (i.e. $\lambda$ is considered to be of order 1).
This feature implies that we can compute derivatives of the series expansions
with respect to the magnetic field $H$.
When the temperature is very close to zero only a few terms are needed to
provide an accurate description of the system.
However, as the temperature increases we have to
include more and more terms in the expansion to attain a similar accuracy.
Actually, the partition function expansion is a technical tool to compute
the expectation values of some local operators: the energy density
$E = \langle \sigma_{(0,0)} \sigma_{(1,0)} \rangle$ and the magnetization
$M= \langle \sigma_{(0,0)} \rangle $.
The relations for a finite system are the following
%
\begin{subeqnarray}
\slabel{def_energy_N}
E_N(\mu,\lambda)
&=& 1 + { 1 \over 2N}
{1 \over Z_N} {\partial Z_N \over \partial K} =
1 - { 1 \over N}
{\mu \over Z_N} {\partial Z_N \over \partial \mu}
= \sum_{m,n} E_{m,n}^{(N)} \mu^m \lambda^n \\
\slabel{def_magnetization_N}
M_N(\mu,\lambda)
&=& 1 + {1 \over N}
{1 \over Z_N} {\partial Z_N \over \partial H} =
1 - {1 \over N}
{2\lambda \over Z_N} {\partial Z_N \over \partial \lambda}
= \sum_{m,n} M_{m,n}^{(N)} \mu^m \lambda^n
\end{subeqnarray}
As before, the coefficients $\{ E_{m,n}^{(N)},M_{m,n}^{(N)} \}$ do
depend on the lattice size and, in general, on the boundary conditions.
Let us discuss now the thermodynamic limit ($N \rightarrow \infty$)
of these expansions.
In this limit, the contribution of all the terms with the same $m$ is not
in general of the same order. In particular, for $H > 0$ the configurations
with $n$ near $N$ (for instance, $n \in [N-m^2/16, N-m/4]$ for $m/2$ even) are
exponentially suppressed, and can therefore be dropped.
Moreover, for $H=0$ the $\sigma \rightarrow - \sigma$ symmetry implies that
the contribution of the terms with $n$ near zero is
equal to the one of those with $n$ near $N$. However, at $H=0^+$ only the
first set is selected. Therefore, for $H > 0$ or $H = 0^+$ the correct
expansion is obtained by taking all the terms with $n$ near zero.
On the other hand, the series corresponding to the partition function
\reff{ising_partition} are meaningless when $N \rightarrow \infty$,
as all the coefficients
$Z^{(N)}_{m,n}$ (except for $Z^{(N)}_{0,0} = 1$) diverge in that limit.
This is not true for the series (\ref{def_energy_N},\ref{def_magnetization_N})
whose coefficients have a well-defined limit
\begin{subeqnarray}
\slabel{def_energy}
E(\mu,\lambda)
&=& \sum_{m,n} E_{m,n} \mu^m \lambda^n \quad ; \qquad
E_{m,n} = \lim_{N \rightarrow \infty} E_{m,n}^{(N)} \\
\slabel{def_magnetization}
M(\mu,\lambda)
&=& \sum_{m,n} M_{m,n} \mu^m \lambda^n \quad ; \qquad
M_{m,n} = \lim_{N \rightarrow \infty} M_{m,n}^{(N)}
\end{subeqnarray}
%
Here it is assumed that the limit $N \rightarrow \infty$ commutes
(for both quantities) with the expansion in $\mu$ and $\lambda$.
This fact is necessary to identify the limiting series with the thermodynamic
limits of the energy density and magnetization.
The coefficients $\{ E_{m,n}, M_{m,n} \}$ do not depend on the
boundary conditions of the finite systems.
Finally, the specific heat $C_v$ and the susceptibility $\chi$ are
defined as follows
\begin{subeqnarray}
\slabel{def_cv}
C_v(\mu,\lambda) &=& \sum_{\langle x,y \rangle}
\left[ \langle \sigma_x \sigma_y \sigma_{(0,0)}
\sigma_{(1,0)} \rangle - E^2 \right] =
{\partial E \over \partial K} =
-2\mu {\partial E \over \partial \mu} \\
\slabel{def_chi}
\chi(\mu,\lambda) &=& \sum_x \left[ \langle \sigma_x \sigma_{(0,0)}
\rangle - M^2 \right] =
{\partial M \over \partial H} =
-2\lambda {\partial M \over \partial \lambda}
\end{subeqnarray}
%
where the sum $\sum_{\langle x,y \rangle}$ is over all nearest--neighbor pairs
of spins.
The series expansions for the zero--field case ($H=0^+$ or $\lambda=1^-$)
can be easily obtained from the previous ones by summing over the index $n$.
For example, $M(\mu) = \sum_m M_m \mu^m$ where $M_m = \sum_n M_{m,n}$.
For the two--dimensional Ising model we can easily compute the corresponding
zero--field expansions for the energy density, specific heat
and magnetizations from the known exact solutions
\cite{onsager,kaufman_onsager,yang}
and the aid of an algebraic manipulator such as Mathematica.
However, the zero--field susceptibility is not exactly known.
Series are available up to order ${\cal O}(\mu^{56})$
\cite{briggs_enting_guttmannII}.
In this paper we are mainly concerned about the computation of expectation
values of more complicated local observables $O$.
By local operator we mean an operator which only depends on a finite number
of spins.
Our definitions of the energy density and the magnetization do satisfy this
property.
The previous procedure can be generalized to include also this case by
adding to the Hamiltonian \reff{ising_hamiltonian} a new term proportional
to a translation--invariant version of the operator $O$.
However, this method is not feasible for very complicated operators, such as
the ones considered in the next Section.
In this paper we propose to use the following identity
%
\be
\label{value_o}
\langle O \rangle = \lim_{N \rightarrow \infty}
{1 \over Z_N} \sum_{ \{ \sigma \pm 1\} } O(\sigma)
e^{-{\cal H}(\sigma)}
\ee
%
to overcome this problem.
The term $\exp(-{\cal H})$ can be expanded
in terms of configurations with $m$ unsatisfied bonds and $n$ flipped spins
as we did in \reff{ising_partition}.
In this case not all the configurations with the same values of $m$ and $n$
give the same contribution to the numerator of \reff{value_o}.
This contribution is equal to $\mu^m \lambda^n$ times the value of the
operator $O(\sigma)$ at the configuration.
Let us consider a simple example.
To compute the magnetization series one has to consider, for instance,
the operator $O = \sigma_{(0,0)}$ (translation invariance assures that the
mean value of this operator will coincide with the magnetization
\reff{def_magnetization}).
For instance, the contribution of the one--flip configurations
is different depending
on whether the flipped spin coincides or not with $\sigma_{(0,0)}$.
In the first case it is equal to $ - \mu^4\lambda$ and in the second one to
$+ \mu^4\lambda$.
The same occurs for more complicated configurations (and operators).
For a finite volume we obtain in this way
an expansion similar to (\ref{def_energy_N},
\ref{def_magnetization_N}).
The final result $\langle O \rangle = \sum_{m,n} O_{m,n} \mu^m \lambda^n$
is obtained after performing the thermodynamic limit.
The main advantage of this method is that it allows the computation of
low--$T$ series for arbitrary operators.
Its main drawback is that we need to compute two series for each observable,
not one as in the former method.
Furthermore, in Section~3 it is shown that its implementation on a computer is
much less efficient than the corresponding to the first procedure.
Its interest relies on the fact that this method could
be used to compute the expectation values of any renormalized operator.
\subsection{ Renormalization Group Transformations}
RG transformations are usually viewed as a map in a
certain space of Hamiltonians (i.e. ${\cal B}^1$).
This approach has a main drawback: for some commonly used RG transformations
the image Hamiltonian does not belong to the space
${\cal B}^1$ when the original interaction is located in the vicinity of
the Ising first order phase transition at low enough temperature.
On the other hand, strictly local RG transformations do always exist as a
map in the space of translation--invariant measures
\cite{enter_fernandez_sokal}.
Let us consider the RG transformations from this alternative point of view.
The original Ising system can be completely described by means of a
probability distribution $\nu$ over its configuration space.
Later on, the relationship between this measure and the Hamiltonian
\reff{ising_hamiltonian} will be discussed.
The next step is to define the renormalized spins.
First we divide the whole lattice into blocks.
(for simplicity we will assume here that these are $2 \times 2$ blocks).
To each block $B_i$ we associate a new (renormalized) spin
$\sigma^\prime_i$ \footnote{We will denote renormalized quantities with a
prime}.
The RG transformation is the rule which gives the
$\{ \sigma^\prime \}$ configuration from the original one $\{ \sigma \}$.
This rule could be either stochastic or deterministic,
but in any case the renormalized spin should only depend on the
spins belonging to the corresponding block (strict locality condition).
Mathematically speaking we give a probability kernel
$T(\sigma, d\sigma^\prime)$.
For each configuration of the original spins $\{ \sigma \}$, $T(\sigma,\cdot)$
is a probability distribution for the $\{ \sigma^\prime \}$ spins and
furthermore, it satisfies the property
$\int T(\sigma, d\sigma^\prime) = 1$.
On the other hand, it is usually assumed that $T$ is strictly local in
position space and that it maps translation--invariant measures into
translation--invariant ones.
The probability distribution $\nu^\prime$ of the image system is given by
%
\be
\nu^\prime = \nu T = \int d\nu(\sigma) T(\sigma, \cdot)
\ee
%
and the expectation value of any local observable in this renormalized measure
can be written as
%
\be
\label{def_mean_renor}
\langle O \rangle_{\nu^\prime}
= \int d\nu(\sigma) \left[ \int T(\sigma, d\sigma^\prime)
O(\sigma^\prime) \right]
= \langle \tilde{O} (\sigma) \rangle_\nu
\ee
%
The probability kernel $T(\sigma,\cdot)$ when acting on the measure
$d\nu(\sigma)$ gives a probability distribution on the new spins
$\{ \sigma^\prime \}$ (i.e. a renormalized measure $\nu^\prime$).
On the other hand, we can consider its action on the operator
$O(\sigma^\prime)$.
In this case the results is a composite operator $\tilde{O}(\sigma) =
(T \cdot O) (\sigma)$ which depends only on the original spins.
Thus, the expectation value of any local renormalized operator is equal
to the mean value of a certain composite operator in the original measure.
This discussion is general: the conclusions hold whether the systems can be
described or not by a Hamiltonian ${\cal H} \in {\cal B}^1$.
Now we take into account the role of the Hamiltonians.
Given an interaction ${\cal H} \in {\cal B}^1$ we can construct a measure
over the spin configuration space using the Gibbs prescription
%
\be
\label{gibbs_prescription}
d\nu (\sigma) = d\nu^0(\sigma) { 1 \over Z } e^{-{\cal H} (\sigma)}
\ee
%
For finite systems this formula gives the correct answer, but for infinite
systems one has to be more careful and consider the limit of the measures
for finite systems and given boundary conditions as the size of the systems
goes to infinite in a given sense.
In \reff{gibbs_prescription} $d\nu^0(\sigma)$
is the a--priori measure we assign to the space of configurations of a single
spin (in our case it is just the counting measure which gives to each state a
probability 1/2).
For finite systems the relation between Hamiltonians and
measures is one--to--one. However, in the thermodynamic limit that is not the
case: one Hamiltonian can be associated to several measures (i.e. at first
order phase transitions) or there are perfectly sound measures which cannot be
constructed via the Gibbs prescription from any sensible Hamiltonian
\cite{enter_fernandez_sokal}.
The Hamiltonian \reff{ising_hamiltonian} does belong obviously to the set
${\cal B}^1$, so we can construct the measure $\nu$ using
\reff{gibbs_prescription}.
Then the expectation value \reff{def_mean_renor} of any local
renormalized operator can be written as
%
\be
\label{def_final}
\langle O \rangle_{\nu^\prime} = \langle \tilde{O} \rangle_\nu =
\lim_{N \rightarrow \infty} {1 \over Z_N} \sum_{ \{ \sigma = \pm 1 \} }
\tilde{O}(\sigma) e^{-{\cal H}(\sigma)}
\ee
%
where the definition of $d\nu^0$ has been taken into account.
In Section~2.1 we showed how to obtain low--$T$ expansions for a general
mean value $\langle O \rangle_\nu$.
Thus, the same procedure can be applied to \reff{def_final}, and
series of the type
$\langle O \rangle_{\nu^\prime} = \sum_{m,n} O^\prime_{m,n} \mu^m \lambda^n$
are obtained.
The practical applicability of this method relies heavily on the actual
form of the kernel $T$ as it is shown below.
This procedure can also be easily generalized to several RG steps.
It is important to remark that
this method does not suffer from any of the pathologies which are exhibited
by the RG when we try to define it
as a map from a Hamiltonian space into a Hamiltonian space.
Here we have not tried to define any renormalized interaction
${\cal H}^\prime$ related with the renormalized measure $\nu^\prime$ via
the Gibbs prescription \reff{gibbs_prescription}.
Our results are independent of the Gibbsian or non--Gibbsian nature of the
renormalized measure.
Let us illustrate this method with three examples:
\subsubsection*{Example 1: Decimation}
This case is really simple because this transformation fixes one spin of
the block to be the renormalized one.
In particular, the (deterministic) kernel $T$ takes the form
%
\be
T(\sigma,\sigma^\prime) = \prod_i
\delta( \sigma^\prime_i , \sigma_{2i} )
\ee
%
where the product is over all sites $i$ of the renormalized system.
We are only interested in computing observables that are monomials of the
spins ($O = \{ \sigma_{(0,0)}, \sigma_{(0,0)}\sigma_{(1,0)} \}$).
So it is enough to compute for each RG transformation the composite
operator $\tilde{\sigma}_i$.
In this case this is equal to
%
\be
\tilde{\sigma}_i = \int T(\sigma,d\sigma^\prime) \sigma^\prime_i
= \sigma_{2i}
\ee
%
This implies that the zero--field quantities are given by
%
\begin{subeqnarray}
M^\prime(\mu,1^-) &=& \langle \tilde{\sigma}_{(0,0)} \rangle =
M(\mu,1^-) \\
E^\prime(\mu,1) &=&
\langle \tilde{\sigma}_{(0,0)} \tilde{\sigma}_{(1,0)} \rangle
= \langle \sigma_{(0,0)} \sigma_{(2,0)} \rangle (\mu,1)
\end{subeqnarray}
%
where the r.h.s. of the second equation is just the unrenormalized
third neighbor correlation function.
This case is trivial: the
renormalized correlation functions are equal to the unrenormalized ones at
twice the distance.
And these functions can be obtained
in the two--dimensional Ising model from the
exact solution \cite{onsager,kaufman_onsager,yang}.
On the other hand, this method also allows to obtain the renormalized
susceptibility and specific heat.
However, they cannot be computed by using derivatives as in the usual Ising
model (that is because we do not know the renormalized coupling constants
$H^\prime$ and $K^\prime$, if they exist).
One is forced to use their definitions (\ref{def_cv},\ref{def_chi})
in terms of correlation functions.
It would be very interesting to devise an algorithm to build the low-$T$ series
for such quantities to an arbitrary order.
\subsubsection*{Example 2: Kadanoff Transformation}
This is given by the following (stochastic) probability kernel
%
\be
\label{kadanoff}
T(\sigma, \sigma^\prime) = \prod_i
{ e^{ p\sigma^\prime_i \sum_{j \in B_i} \sigma_j }
\over
2 \cosh (p \sum_{j \in B_i} \sigma_j) }
\ee
%
where $p$ is a free real parameter. Then,
%
\be
\tilde{\sigma}_i = \tanh \left( p \sum_{k \in B_i } \sigma_k
\right)
\ee
%
The first terms can be computed by hand
%
\begin{subeqnarray}
M^\prime(\mu,1^-) &=& \tanh 4p - 4(\tanh 4p - \tanh 2p) \mu^4 -
4 (3 \tanh 4p - 2 \tanh 2p) \mu^6
\nonumber \\
&-& (36 \tanh 4p - 4 \tanh 2p) \mu^8 + {\cal O}(\mu^{10}) \\
E^\prime(\mu,1 ) &=& \tanh^2 4p
- 8 (\tanh^2 4p - \tanh 4p \tanh 2p) \mu^4
\nonumber \\
&-& 2 (11 \tanh^2 4p - 6 \tanh 2p \tanh 4p - \tanh^2 2p ) \mu^6
\nonumber \\
&-& (43 \tanh^2 4p + 40 \tanh 2p \tanh 4p - 20 \tanh^2 2p) \mu^8
+ {\cal O}(\mu^{10})
\end{subeqnarray}
%
The limit $p \rightarrow 0$ corresponds to the case in which the
$\sigma^\prime$ are not correlated with the original spins and thus,
the renormalized spins do not interact among them.
For this reason both quantities are zero.
The limit $p \rightarrow \infty$
corresponds to the majority rule with equally--probable tie--breaker.
This case will be treated in the next section.
\subsubsection*{Example 3: Majority Rule}
In this case
%
\be
\label{majority}
T(\sigma^\prime, \sigma) = \prod_i
\delta \left(
\sigma^\prime_i - {\rm sign} \left( \sum_{j \in B_i} \sigma_j
\right) \right)
\ee
When ${\rm sign}(\cdot) = 0$ we choose $\sigma^\prime = -1$ or $+1$ with
probabilities $q \in [0,1]$ and $1-q$ respectively.
The composite operator $\tilde{\sigma}$ takes the form
%
\be
\tilde{\sigma}_i = {\rm sign} \left( \sum_{k \in B_i} \sigma_k
\right)
\ee
%
The first terms for general $q$ are:
%
\begin{subeqnarray}
M^\prime(\mu,1^-) &=& 1 - 8q\mu^6 - (10 + 44q) \mu^8 + {\cal O}(\mu^{10}) \\
E^\prime(\mu,1 ) &=& 1 - 16q\mu^6 - (20 + 88q - 4q^2)\mu^8
+ {\cal O}(\mu^{10})
\end{subeqnarray}
%
The result with $q=1/2$ was first reported in
ref.~\cite{enter_fernandez_sokal}.
Notice that the ${\cal O}(\mu^{4})$ term vanishes.
This is due to the fact that one--spin excitations cannot produce any
flipped renormalized spin $\sigma^\prime = -1$.
\section{Series for the Majority Rule and $q= 1/2$ }
The low--$T$ series for this particular transformation can be improved
systematically with the aid of a computer algorithm.
The one used here is inspired on the Recursive Counting Method (RCM) of
refs.~\cite{creutz,bhanot_creutz_lacki} where details can be found.
This one consists essentially on a recursive enumeration of the most relevant
configurations of the system and
can be easily implemented on a computer.
However, there are several differences which should be noticed.
We place the spins on a $L_x \times L_y$ square lattice with periodic
boundary conditions in the $x$--direction and fixed on the other one.
In particular we put cold walls of $+1$ spins at both vertical ends of
our system.
This fact automatically selects the $(+1)$ configuration as our
ground state.
The desired series for renormalized operators cannot be related in a simple
way to derivates of the partition function.
For our purposes it is rather useful to write \reff{def_final} in the
following equivalent form (for the magnetization)
valid only for this RG transformation
%
\be
M^\prime =
\langle \tilde{\sigma}_{(0,0)} \rangle =
\left\langle {\rm sign} \left( \sum_{k \in B_{(0,0)}} \sigma_k \right)
\right\rangle =
\sum_{ \{ s_i \pm 1\} } {\rm sign} \left( \sum_{k \in B_{(0,0)}} s_k \right)
\left\langle \prod_{j \in B_{(0,0)} } \delta_{\sigma_j,s_j} \right\rangle
\label{final}
\ee
%
The procedure is simple: i) Decide where to place the renormalized spin
on the lattice.
ii) For each configuration $\{ s_i \}$ of the original spins belonging to
the block $B_{(0,0)}$, compute the expectation
value $\langle \prod_{j \in B_{(0,0)} } \delta_{\sigma_j,s_j} \rangle$.
Notice that this expectation value should be calculated with the
unrenormalized measure.
iii) Finally we obtain $M^\prime$ using the later formula.
For the renormalized energy
$E^\prime = \langle \tilde{\sigma}^\prime_{(0,0)}
\tilde{\sigma}^\prime_{(1,0)} \rangle$
the formula is very similar, although there are two renormalized spins
involved (and two blocks).
The expectation value
$\langle \prod_{j \in B_{(0,0)} } \delta_{\sigma_j,s_j} \rangle$
can be obtained using the RCM.
The only difference is
that when we arrive at any of the spins $\sigma_j \in B_{(0,0)}$ we have to
fix its value to $s_j$.
The sum in eq.~\reff{final} contains in general $2^{4b}$ terms, where $b$ is
the number of blocks involved in the computation ($b=1$ for the magnetization
and $b=2$ for the energy).
This feature makes this method much slower than the pure RCM.
However there is a trick which allows us to save a factor of 1.6 in CPU time.
When $q = 1/2$, configurations with ${\rm sign}(\cdot) = 0$ do not have a net
contribution to \reff{final}: half of the times they give some contribution
and the other half, minus this one.
Another disadvantage of our procedure is that it breaks the homogeneity
of the lattice.
There are some special blocks ($B_{(0,0)}$ for the magnetization, and
$B_{(0,0)}$ and $B_{(1,0)}$ for the energy density)
which are clearly different from the rest.
This feature implies that, for a given order, our method needs a larger
lattice than the RCM.
Here the length of the series is mainly limited
by $L_x$: the result is exact up to order ${\cal O}(\mu^{L_x-2})$ whenever
$L_y \geq L_x$.
In our case the first statement is true, but the order ${\cal O}(\mu^{L_x-2})$
is achieved only if $L_y \geq 2L_x -4$ (Here we assume that the
renormalized spins are placed in the middle of the lattice\footnote{In this
way we minimize the border effects due to the cold walls} and, for
$E^\prime$ the bond which joins both spins is parallel to the $x$--axis).
As in refs.~\cite{creutz,bhanot_creutz_lacki}
we can improve the performance of the algorithm by
introducing different couplings ($K_x$ and $K_y$) for horizontal and
vertical bonds.
If we want to compute $M^\prime$ to order ${\cal O}(\mu^{2L})$
we need a lattice of size $L_y = 2L-3$ and $L_x = (L+2)/2$ ($L_x= (L+1)/2$)
if $L$ is even (odd).
In this way we obtain half of the terms which contribute to
${\cal O}(\mu^{2L})$ and the rest can be recovered using the symmetry of the
result under $K_x \leftrightarrow K_y$.
This is not longer true for $E^\prime$ as the bond joining the renormalized
spins distinguishes one axis from the other.
To overcome this difficulty we have to run the program twice: the first time
that bond is horizontal and the second one vertical.
To obtain the same precision we have to use different lattice sizes.
When the bond is horizontal we need a lattice with $L_y = 2L-4$ and
$L_x = (L+5)/2$ ($L_x = (L+6)/2$) when $L$ is odd (even).
And if it is vertical, $L_y = 2L-2$ and $L_x = (L+1)/2$ ($L_x = (L+2)/2)$.
In this way we have been able to obtain the series
(\ref{def_energy}, \ref{def_magnetization}) up to order ${\cal O}(\mu^{30})$.
The result is displayed in Table~1.
In this algorithm we need to deal with
very large numbers, much larger than the precision of the computer (32 bits
in our case).
For that reason, we used modular arithmetic in the FORTRAN code to obtain
all the coefficients.
And all the series manipulation was done using Mathematica, which allows
infinite--precision integer arithmetic.
We checked the algorithm by reproducing the known series for the unrenormalized
observables $M$, $E$ and $\chi$.
With the use of more sophisticated tricks to save memory these series could
be extended a lot more.
\section{Study of the First--Order Phase Transition at Very
Low Temperatures}
For the two--dimensional Ising model some rigorous results are known
about the behavior of the RG at the first--order
phase transition.
The authors of ref.~\cite{enter_fernandez_sokal} found that the renormalized
measure is not Gibbsian for some particular RG
transformations {\em at} the transition line.
These are the following
\begin{itemize}
\item
Decimation for blocks of size $b = 2$ and
$K > (1/2) \cosh^{-1}(1 + \sqrt{2})$.
For $b \geq 3$ they only could prove this statement for large enough $K$.
\item
Kadanoff transformation with $0 < p < \infty$, block size $b \geq 1$,
and sufficiently large $K$.
\item
Majority rule for blocks of size $b = 7,41,\ldots$ and $K$ large enough.
\item
Block--averaging transformation for even $b \geq 2$ and sufficiently large
$K$.
In this case they were also able to prove that the same conclusion
is true for arbitrary magnetic field $H$ provided $K$ is large enough.
\end{itemize}
In actual MCRG calculations one chooses by hand a linear
subspace $V_n \in {\cal B}^1$ of the space of sensible Hamiltonians.
Then, given certain renormalized expectation values, one tries to
obtain a renormalized Hamiltonian ${\cal H}^{\prime}_n \in V_n$ in such a
way that the measure constructed from ${\cal H}^{\prime}_n$ is similar in
some sense to the true renormalized measure $\nu^\prime$.
Most ``reconstruction'' methods are based in Schwinger--Dyson equations
\cite{swendsen,toni_okawa,toni_salas1}.
The idea is simple: minimize a certain functional (which depends
on the method) involving both renormalized expectation values (the input) and
renormalized couplings (the output).
It can be shown \cite{toni_salas1} that these methods provide a unique
solution ${\cal H}^\prime_n$, which coincides with the true one
${\cal H}^\prime$ if this latter
interaction belongs to the trial subspace $V_n$.
The key property of these functionals is that they are strictly convex.
Here we will consider the procedure given in ref.~\cite{enter_fernandez_sokal}.
It is based on the minimization of
the relative density entropy with respect to the true renormalized measure
$\nu^\prime$.
This functional in also strictly convex and thus, the solution is
unique in each $V_n$.
They also proved that the solution ${\cal H}^{\prime}_n$
should satisfy the following conditions
%
\be
\label{matching_eqs}
\langle O_i \rangle_{\nu^\prime} =
\langle O_i \rangle_{\nu_n^\prime}; \qquad \forall O_i \in V_n
\ee
%
where $\nu_n^\prime$ is one Gibbs measure constructed from the Hamiltonian
${\cal H}^{\prime}_n$.
In this case we have the same number of equations than unknown parameters.
However, when we restrict these equations to a zero--field subspace it is
not always possible to find a solution.
If the measure $\nu^\prime$ is Gibbsian we expect that the sequence of
solutions ${\cal H}_n^\prime$ will converge to the true (and existing) solution
${\cal H}^\prime$. However, if the measure is non--Gibbsian the situation is
less clear. It could happen that the norm in ${\cal B}^1$ of the
solutions ${\cal H}_n^\prime$ will diverge as $n \rightarrow \infty$.
\noindent
Remark: More generally one could choose to look for a renormalized Hamiltonian
in some {\em affine} subspace $A_n = V_n + {\cal H}_0$, where ${\cal H}_0$
is some fixed element of ${\cal B}^1$. This will be relevant for Case I below.
Using low--$T$ expansions we can study this procedure with no much difficulty
and no statistical errors.
In this section we will mainly treat the majority rule transformation with
$q=1/2$ and block size $b=2$.
\subsection*{Case I: $V_1 = \{ H \}$ }
Here the subspace $V_1$ contains only the magnetization; so we should
solve the following equation:
%
\be
\label{matching_1}
M(K,H^\prime) = M^\prime(K,H)
\ee
%
In this case,
${\cal H}^\prime_1 = (K,H^\prime)$ is the approximate renormalized
Hamiltonian chosen within the affine subspace $A_1$ and
the element ${\cal H}_0$ is equal to the nearest--neighbor
interaction ${\cal H}_0 = (K,0)$.
Notice that any RG transformation satisfying $M(K,0^+) \neq M^\prime(K,0^+)$
is discontinuous at $H=0$ when restricted to this affine subspace $A_1$.
The main interest of this case relies on its connection with
ref.~\cite{decker_hasenfratz}, where it was claimed that eq.~\reff{matching_1}
could be used to compute numerically the leading critical exponent of the Ising
DFP.
They considered the Kadanoff transformation with $p=2.5$, which we now
know that it does not lead to any renormalized Gibbsian measure.
Actually, they used a method due to Wilson \cite{wilson} which allows to
linearize a RG transformation near a fixed point without
suffering from truncation errors.
However, it is required that this fixed point possesses only one relevant
operator, and in the present case there are two relevant operators at the
DFP: the magnetic field and the temperature $\sim 1/K$
\cite{klein_wallace_zia,toni_salas2}.
We can repeat the same calculation using the low--$T$ series obtained
in Section~2 by generalizing them to $H \neq 0$.
As a matter of fact, it is not very
difficult to notice that the leading term in $1 - M(\mu,\lambda)$ comes from
one--spin flips, so it is proportional to $\lambda$.
On the other hand,
the leading term in $1 - M^\prime(\mu,\lambda)$ is due to two--spin flips,
and thus, it is proportional to $\lambda^2$.
The final result is
%
\be
H^\prime = 2 H + 2K - {1 \over 2} \log 2
\ee
%
This means that there is a jump ($= 2K - (1/2)\log 2$) at the transition
line as Decker {\em et al} obtained.
Notice that the size of the discontinuity decreases as $K$ does.
However, the slope is different from theirs.
The critical exponent would be $y = 1$ contrary to their result
and the DFP prediction ($y=2$).
The same can be done for the decimation transformation with $b=2$.
In this case everything is much simpler because
%
\be
M(K,H^\prime) = M^\prime(K,H) = M(K,H)
\ee
%
This implies that $H^\prime = H$ and there is no jump at $H=0$.
The most relevant exponent is not longer relevant, but marginal ($y = 0$),
contrary to the previous results.
In summary,
we have obtained very different results for the critical exponent $y$
depending on the used RG transformation. The critical exponents do not
depend on the RG transformation, so these results are a signal that
this matching method cannot be applied to this particular case.
On the other hand, only decimation is continuous at $H=0$, although the
relation between $H$ and $H^\prime$ is trivial.
\subsection*{Case II. $V_2 = \{ H, K \}$ }
Now our subspace contains the original interaction (${\cal H} \in V_2$).
We will try to match both the energy density and the zero field magnetization
with a different zero--field Hamiltonian.
If this matching can be performed, it would mean that the RG transformation
is not discontinuous at the transition line (when restricted to this
coupling subspace $V_2$).
However, this does not mean that that the renormalized measure is Gibbsian.
On the other hand, if the renormalized Hamiltonian ${\cal H}^\prime$ exists,
it is not guarantied that the approximants ${\cal H}_n^\prime$ do not
contain any odd coupling. However, as $n \rightarrow \infty$ these odd
couplings should vanish because the exact RG transformation is continuous
(assuming its existence).
First we define $K^\prime$ as the nearest--neighbor coupling such that
%
\be
\label{matching_energy}
E^\prime(K,0) = E(K^\prime,0)
\ee
%
Using the result given in Section~2 and the well--known expansion of the
Onsager solution
%
\begin{subeqnarray}
E(\mu,1) &=& 1 - 4\mu^4 -12 \mu^6 - 36 \mu^8 + {\cal O}(\mu^{10}) \\
M(\mu,1^-)&=& 1 - 2\mu^4 - \phantom{1}8 \mu^6 - 34 \mu^8
+ {\cal O}(\mu^{10})
\end{subeqnarray}
%
we find that
%
\be
\mu^\prime = \sqrt{2} \mu^3 + {63 \over 8\sqrt{2} } \mu^5 + {\cal O}(\mu^6)
\ee
%
The magnetization $M$ at this particular value of $\mu = \mu^\prime$
is equal to
%
\be
M(\mu^\prime,1^-) = 1 - 4 \mu^6 - {63 \over 4 } \mu^8 + {\cal O}(\mu^9)
\ee
%
and this expansion should be compared with the renormalized magnetization
$M^\prime(\mu,\lambda)$ given in Section~2.
We find that
%
\be
\label{matching_condition}
M(\mu^\prime,1^-) > M^\prime(\mu,1^-)
\ee
%
This equation means that we can give account of the observed renormalized
magnetization with a system with zero field and
$K^\prime = -(1/2)\log \mu^\prime \approx 3 K - (1/4) \log 2$.
This system is not in a pure phase,
but in a mixed phase because the renormalized
magnetization $M^\prime(\mu,1^-)$ is strictly smaller than
$M(\mu^\prime,1^-)$.
Thus, eq.~\reff{matching_eqs} is satisfied by a measure $\nu_2^\prime$
which is a convex linear combination of the two pure phases $\nu^\pm$
characterizing the two--dimensional Ising model at low temperature
and $H = 0^\pm$ (i.e. $\nu_2^\prime = \alpha \nu^+ + (1-\alpha) \nu^-$ for
some $\alpha \in (0,1)$).
The same game can be played with the other two RG
transformations considered in Section~2.
The easiest case is the decimation transformation, where conclusions can be
drawn for every $K > K_c$.
In the two--dimensional Ising model it is well--known that
$\langle \sigma_{(0,0)} \sigma_{(1,0)} \rangle >
\langle \sigma_{(0,0)} \sigma_{(2,0)} \rangle $ for $0 < K < \infty$.
This implies immediately that $E(K,0) > E^\prime(K,0)$ and
$K^\prime < K$ if we take into account that $E(K,0)$ is a strictly increasing
function of $K$.
On the other hand, the renormalized magnetization coincides with the
unrenormalized one
(i.e. the RG flow follows the lines of constant magnetization).
And $M(K,0^+)$ is also a strictly increasing function of $K$ for
$K > K_c$.
Combining both pieces we obtain that $M(K^\prime,0^+) < M(K,0^+)$ for all
$K > K_c$.
This is so because the direction of the RG flow is reversed: it goes from
low--temperature to high--temperature ($K^\prime < K$).
So, we have to increase the magnetic field to keep the magnetization constant,
unless the magnetization at the starting point is zero.
This condition is only held above the critical temperature.
In summary, we cannot match the renormalized observables using a zero--field
Hamiltonian along the whole first--order transition line for this RG
transformation.
For the Kadanoff transformation and large (but {\em finite}) $p$ the same
result holds: one cannot match the energy densities and the magnetizations
with a zero--field nearest--neighbor interaction.
This can only be proved when $p$ is large enough.
The reason is clear: the leading term of $E^\prime$ is $\tanh^2 4p$ and if
$p$ is not large, then the solution of \reff{matching_energy} does not
satisfy $\mu^\prime \ll 1$ and the low--$T$ series
for $\mu^\prime$ are then meaningless.
For finite $p$ we can always choose $\mu_0$ such that for $\mu < \mu_0$
the leading term of $E^\prime(\mu,1)$ is dominated by a term which does not
depend on $\mu$. Then
%
\be
E^\prime(\mu,1) = 1 - 4 e^{-8p} + {\cal O}(e^{-16p})
\ee
%
if we choose $\mu_0 \sim \exp(-3p)$.
The solution of eq.~\reff{matching_energy} is then
%
\be
\mu^\prime = e^{-2p} - {3 \over 4} e^{-6p} + {\cal O}(e^{-10p})
\ee
%
and
%
\be
M(\mu^\prime,1^-) = 1 - 2e^{-8p} - 3 e^{-12p} + {\cal O}(e^{-16p})
\ee
%
which should be compared with the expansion of the renormalized magnetization
for $p$ very large and $\mu < \mu_0$
%
\be
M^\prime(\mu,1^-) = 1 - 2 e^{-8p} + {\cal O}(e^{-16p})
\ee
%
We find that at leading term both quantities are the same, but the
next--to--leading term is different.
In particular we find that $M^\prime(\mu,1^-) > M(\mu^\prime,1^-)$, so
we cannot match both $E^\prime$ and $M^\prime$ with a zero--field
Ising interaction.
This discussion is valid as long as $p$ is large but {\em finite}.
When $p$ diverges the leading term of $1 - E^\prime(\mu,1)$ is proportional
to $\mu^6$ and we re--obtain the result for the majority rule transformation
with $q=1/2$.
\subsection*{Case III. $V_3 = \{ H, K, L \}$ }
Now we are considering a Hamiltonian with an additional next--to--nearest
neighbor term $L \sum \sigma_i \sigma_k$.
First of all we have to compute the renormalized mean value of the
next--to--nearest neighbor correlation function. The result for the
majority rule with random tie--breaker is
%
\be
F^\prime(\mu, 1) =
\langle \tilde{\sigma}_{(0,0)} \tilde{\sigma}_{(1,1)} \rangle =
1 - 4\mu^6 - 64 \mu^8 -336 \mu^{10} - 1578 \mu^{12} + {\cal O}(\mu^{14})
\ee
%
The second step is to write down the expressions for
$\langle O_i \rangle_{\nu_n}$, $\forall O_i \in V_3$.
The result for zero magnetic field is
%
\begin{subeqnarray}
\slabel{e_v3}
E(\mu,\gamma,1) &=& 1 - 4\mu^4\gamma^4 - 12\mu^6\gamma^8
- 24 \mu^8\lambda^{12} - 32 \mu^8\lambda^{10} +
36 \mu^8\lambda^8 \nonumber \\
&-& 40 \mu^{10}\lambda^{16}
+ {\cal O}(\mu^8\lambda^6) \\
\slabel{f_v3}
F(\mu,\gamma,1) &=& 1 - 4\mu^4\gamma^4 - 16\mu^6\gamma^8
-36 \mu^8\lambda^{12} - 40 \mu^8\lambda^{10} +
36 \mu^8\lambda^8 \nonumber \\
&-& 64 \mu^{10}\lambda^{16}
+ {\cal O}(\mu^8\lambda^6) \\
\slabel{m_v3}
M(\mu,\gamma,1^-) &=& 1 - 2\mu^4\gamma^4 - 8\mu^6\gamma^8
- 20 \mu^8\lambda^{12} - 24 \mu^8\lambda^{10} +
18 \mu^8\lambda^8 \nonumber \\
&-& 40 \mu^{10}\lambda^{16}
+ {\cal O}(\mu^8\lambda^6)
\end{subeqnarray}
%
where $\gamma = \exp(-2L)$.
Now we have to find out a pair $(\mu^\prime,\gamma^\prime)$ such that
%
\be
E(\mu^\prime,\gamma^\prime,1) = E^\prime(\mu,1); \qquad
F(\mu^\prime,\gamma^\prime,1) = F^\prime(\mu,1)
\ee
%
The solution to leading term is
%
\be
\label{leading_term}
\mu^\prime = 4 \mu^2 + {\cal O}(\mu^4); \qquad
\gamma^\prime = \left({1 \over 32 \mu^\prime }\right)^{1/4}
( 1 + {\cal O}(\mu^\prime))
\ee
%
This implies that $K^\prime \approx 2K - \log 2 > 0$ and
$L^\prime \approx (5/8) \log 2 - K^\prime/4 \approx (7/8) \log 2 - K/2 < 0$.
So, as $K \rightarrow \infty$, $K^\prime$ and $-L^\prime$ also diverge.
The latter relation \reff{leading_term} between $\mu^\prime$ and
$\gamma^\prime$ should be taken into account when computing the actual order of
a given term in the expansion of the partition function
$Z_N(\mu^\prime,\gamma^\prime,1^-)$ and its derivatives.
In our case, this implies that the first two excitations to the ground state
are of order $\mu^{\prime 3}$ and $\mu^{\prime 4}$ respectively.
We have considered here all the excitations up to order
${\cal O}(\mu^{\prime 6})$.
A straightforward computation leads to the next--to--leading terms of
eq. \reff{leading_term}.
\begin{subeqnarray}
\mu^\prime &=& 4 \mu^2 \left[ 1 - {69 \over 16} \mu^2 + \sqrt{2} \mu^3 +
{17027 \over 512} \mu^4 + {\cal O}(\mu^5) \right] \\
\gamma^\prime &=& \left({1 \over 32 \mu^\prime}\right)^{1/4} \left[ 1 +
{327 \over 256} \mu^\prime - {3 \over 16 \sqrt{2} } \mu^{\prime 3/2} +
{144177 \over 131072} \mu^{\prime 2} + {\cal O}(\mu^{\prime 5/2}) \right]
\end{subeqnarray}
%
The magnetization \reff{m_v3} computed at the latter solution is equal to
%
\be
M(\mu^\prime,\gamma^\prime,1^-)
= 1 - 4\mu^4 - 32 \mu^8 - {2689 \over 16} \mu^{10} +
{\cal O}(\mu^{11}) <
M^\prime(\mu,1^-)
\ee
%
This implies that we cannot match the renormalized expectation values with
a zero--field interaction belonging to $V_3$.
\section{Conclusions}
In this note we have shown how to compute low--temperature expansions for
the expectation values of local operators computed in the renormalized
measure. In particular we have analyzed three RG transformations: decimation,
Kadanoff transformation with large but finite parameter $p$ and majority
rule with random tie--breaker. All of them are defined on $2 \times 2$
blocks. We have been able to compute the first terms of the series
corresponding to the
renormalized magnetization and nearest--neighbor two--point correlation
function for all these transformations.
For the majority rule case, a computer algorithm has been devised to
provide those series to an arbitrary high order. The main limitation
of this computational method is the huge memory needed. With the use of
more sophisticated programming tricks we expect to increase the order of
both series. Here they are reported up to order ${\cal O}(\mu^{30})$.
These results are useful as checks for MCRG computations.
Another interesting point would be to devise a new algorithm to obtain
the series for the renormalized susceptibility and specific heat to
an arbitrary order.
As explained in Section~2, these quantities are not related by simple
derivatives to the partition function, and we need to use their definition
in terms of sum over connected correlation functions. This feature makes
their computation a more involved matter.
The main goal of this note was the analysis of the truncation issue in the
Ising model.
The unrenormalized system is located at the Ising first--order transition
line and very low temperature ($H = 0, K \gg K_c$).
For the three transformations considered we have found that we need a
magnetic field to solve the matching equations \reff{matching_eqs}
when we restrict our estimated Hamiltonian to belong to a certain
finite--dimensional subspace of ${\cal B}^1$.
In particular, for the decimation and Kadanoff transformations this
matching cannot be performed when restricting the equations to $V_2$.
For majority rule, in this case the equations admit a zero--field solution
but when we consider the (larger) subspace $V_3$ we also need a magnetic field.
So its seems that truncation in the renormalized Hamiltonian induces some
spurious odd operators (we have only found non--zero magnetic fields, but
there is no reason why more complicated odd operators should not appear
for larger subspaces $V_n$). So, the RG transformations are discontinuous
at the Ising transition line when restricted to some finite--dimensional
subspace of the interaction space ${\cal B}^1$.
However, these results do not clarify the interplay between truncation
and non--Gibbsianness. It is known \cite{enter_fernandez_sokal}
that the decimation and Kadanoff
transformations lead to non--Gibbsian renormalized measures when we start
at low enough temperature; and in these cases we have shown that the
approximate RG transformation is discontinuous.
For the majority rule the situation is
less clear, as it is not known the nature of the renormalized measure.
The authors of ref.~\cite{enter_fernandez_sokal} conjectured that in this case
the renormalized measure is also non--Gibbsian, but they were able to prove it
only for certain special block sizes ($7 \times 7$, $41 \times 41$, $\ldots$).
In any case, this model leads to a continuous approximate RG transformation
for the subspace $V_2$, but a discontinuous one for $V_3$. It is an open
question what happens for larger subspaces $V_n$.
It would be very interesting to find a transformation
which leads to a Gibbsian measure at low temperatures. In this case we
could isolate the effect of truncation from non--Gibbsianness.
A systematic study of the behavior of the estimates ${\cal H}_n^\prime$
could also be useful. When the renormalized measure is Gibbsian, the odd
couplings should go to zero because the transformation is in this case
continuous and single--valued. If the renormalized measure is non--Gibbsian
then it is not known what could happen.
\subsection*{Acknowledgements}
We would like to thank A. Sokal for his encouragement and for
illuminating discussions. We also acknowledge helpful comments by
J.L.F.~Barb\'on, R.~Fern\'andez and M.~Garc\'{\i}a P\'erez.
This research has been supported by a MEC(Spain)/Fulbright grant.
\newpage
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\newpage
\begin{table}
\centering
\begin{tabular}{|r|r|r|} \hline
m &$M^\prime_m $ & $E^\prime_m $ \\ \hline
0 & 1& 1 \\
6 & -4& -8 \\
8 & -32& -63 \\
10 & -168& -312 \\
12 & -816& -1328 \\
14 & -3964& -5318 \\
16 & -19628& -21389 \\
18 & -99120& -89806 \\
20 & -508848& -396826 \\
22 & -2647012& -1828884 \\
24 & -13917848& -8690181 \\
26 & -73827576& -42212476 \\
28 & -394527840& -208509354 \\
30 & -2121643804& -1043875370 \\ \hline
\end{tabular}
\caption{ Series expansions for the zero--field renormalized magnetization
$M^\prime = \sum_m M^\prime_m \mu^m $
and two--point correlation function
$E^\prime = \sum_m E^\prime_m \mu^m $.
Only the non--zero contributions are displayed. }
\end{table}
\end{document}