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\noindent {\bf Modern Series Analysis Techniques and the Relation to Monte
Carlo Results on Similar Models.}
\vskip 0.5cm
\noindent {Joan Adler}
\vskip 0.5cm
\noindent { Department of Physics }
\noindent {Technion-IIT, 32000, Haifa, Israel.}
\vfill
\noindent {\bf Abstract.} In this paper an introduction to modern techniques
of series analysis and a comparison of results with those of Monte Carlo
simulations is given. Discussions of the relative advantages of and
resources required for
both methods and the relative precision of the results
are presented.
\vskip 0.75cm
\noindent {\bf 1. Introduction }
\vskip 0.25cm
\noindent This article complements two recent reviews by the author:
an introduction [1] to both generation and analysis
of series expansions in a percolation context for a student audience,
and a comparison [2] of results from the analysis of series with
simulations
for two exactly
solved models (Ising and Baxter-Wu). Both
are aimed at a simulation audience and the former includes sample programs.
General introductory references to different
aspects of series expansion methods
can be found in [3-12].
In the present article a comparison of results
from series, simulations and some other techniques for several
systems will be given. A comparative
discussion of the
two methods in Section 2, will include a comparison of the relative
resources needed, advantages and disadvantages of both
and some historical developments.
The clear division between series generation and analysis will be described in
Section 3, together with an introduction to some methods of analysis
that allow for the effects of non-analytic corrections-to-scaling.
In Section 4, results for several systems from both simulations and series
will be presented.
Conclusions
pertaining to the
relative merits of series and simulations will be given in Section 5.
\vskip 0.5cm
\noindent {\bf 2. A General Comparison of Series and Simulations}
\vskip 0.25cm
\noindent Since the following comparison is written from a series-expansion
point-of-view some simulation proponents amongst the
readers may find it a little biased. However,
even if you do think there is bias in favour of series
please bear
with in order to appreciate the reasons for the
resurgence of interest in series
expansions that has begun
in the last year or so. Since the following is also written from the viewpoint
of Computational Physics (not all supporters of series methods really like
to see them categorised as Computational Physics) others may find other biases.
Simulation methods began slowly in the
fifties and sixties and have continued to gain
momentum ever since as computer technology developed.
Simulations
are based on studying a selection of
realizations of finite systems,
and
would give exact results if the system were
infinite and if a sufficient number of independent configurations
could be generated.
Overall, the total collection of results from computer
simulations is impressive.
While there is never enough computer time for any quality simulation,
(just ask the committees dividing computer time)
any time available can be always be utilised to obtain a few more samples
or a better check on equilibration. Simulations are in fact extremely
efficient in terms of their utilization of computer resources.
However there are some problems with simulations. For example for
some
systems of interest, such as spin glasses,
it can be
difficult to determine that the simulated
system has reached equilibrium and to
extrapolate the results to the thermodynamic limit.
Also for problems of current concern to the general condensed
matter community,
such as groundstates of quantum systems, simulations seem
to have problems of principle as to how to simulate to obtain answers to
some questions of interest.
In addition to these examples of specific problems there is the general
fact that
one must admit that
a simulation is always restricted to a particular model in a
given spatial dimension at one time. Thus in order to study
the effect of changing temperature more than a little
or changing disorder even a little many different calculations
need to be made.
Simulations tend to be difficult to do in high dimensions and at very
low temperatures where there can be long relaxation times.
Thus there is certainly room in modern Computational
Physics for alternative numerical techniques.
The most developed alternative
is the series expansion method, which is especially good for low-temperatures,
and high dimensions and has no involvement
(and therefore no problems) with random number generators.
In contrast to the smooth development of simulations
and the consistent success of simulations in
achieving efficient
use of avaliable computer resources, the development of
series expansions has been more erratic,
and incompatible at times with state-of-the art hardware. The
series method flowered from 1960 on and provided important input
for the ideas of scaling and universality that underlie the
renormalization group. However certain problems, that will be
discussed in detail in the next paragraph,
led to the situation where even the biggest fans of the series approach
were forced to grudgingley admit that
simulations really were the dominant numerical technique
for critical phenomena in the eighties.
After presenting the solutions to these problems with the series methods
I shall argue
that the relative advantage of simulations to
series is diminishing as
simulations become more complex and series generation and analysis easier.
Series methods have traditionally used less computer resources than
simulations.
The ideal series machine of the seventies was the mainframe, situated locally
with access to a tape drive to read in graph lists and read out
components of the series. Preferably, this machine belonged to a small research
group. As long as such machines were all one had, and series could use
these old mainframes exactly like simulations, the intrinsic efficiency
of the series methods led to superior results.
However with the move
to remote-site vector
supercomputers in the eighties insurmountable
problems with both vectorization and
data transmission of graph lists led to effective limits on resources
avaliable for series, and as everyone was jumping on the vectorization
bandwagon, series were left behind.
Coinciding with
this resource question
some justified unpopularity for series resulted from apparent
violations of hyperscaling, that were caused by inappropriate
extrapolation techniques that neglected corrections-to-scaling.
These two effects together
led to a definite downturn in activity relative to the growth
of simulation methods at this time,
although some groups remained active in series expansions
and laid the foundation for
the current revivial. The analysis question is now well under control and
while
series methods have not been adapted for vector computers in the meantime,
other
recent developments in computing have been so favourable to series
expansions calculations that the matter of
vectorisability is largely irrelevant as we shall now show.
Despite my biased(?) view that simulation proponents reponded very well
to changes in computer hardware, many simulation experts think that
they have always experienced
serious problems
of insufficient time and memory. As a result of this
there has been an
incredible advance in development of algorithms for simulations in the last
fifteen years. This push began with development of algorithms suited to
special purpose and
vector computers of the last decade and has continued on the parallel
machines more recently.
This increased sophistication has a downside because
prior to the development of
analysis routines of
the histogram weighting type, simulations
tended to demand very little disk storage or communication
overhead and so were ideally suited to the remote site vector supercomputers
of the eighties. Now they would experience some of the same limitations in data
transmission and need for disk storage as do series methods, if these needs
were not now being met by recent advances in computers and communications.
Four recent computing advances that have been most helpful
to series expansion studies are:
\item {1.}
fast communications and networking,
\item{2.} widespread use of
workstations with large disks and memories,
\item{3.}
parallel machines, and
\item{4.}
MATHEMATICA and MAPLE computer algebra packages.
\noindent The first three of these have also helped simulations, but
relatively speaking
improved data transmissions have meant that for series remote-site work has
moved from almost impossible unless tapes were sent,
to commonplace whereas for simulations it has been a gradual improvement.
Computer algebra on the other hand has not had much impact on simulations
but
has played an enormous r\^ole in series development for both analysis and
generation.
At least until recently the time overhead on starting a new series
project was far greater than for a simulation, since series algorithms were
relatively more complex. This is no longer the case as, at least for startup,
computer algebra can simplify the complexity (a few terms can be easily
generated with computer algebra, even if it is too slow for
an entire project), and simulation algorithms
have become more complex.
\vskip 0.5cm
\noindent {\bf 3. Generation and Analysis of Series}
\vskip 0.25cm
\noindent Series expansions are one member of the general family of
perturbation expansions around some limit of the model
for which an exact solution can be found. In all of these extrapolation
must be made to the region of the
model that is of interest, and this is not always reliable.
For example, in some field-theoretic methods one expands around
the upper critical dimension. This can be as high as six or even
eight for some interesting problems
and hence extrapolation to physical dimensions may be questionable.
One perturbation method that has proved extremely successful
in general dimension is the method of exact series expansions.
Here the exact expression for
the order parameter, susceptibility
or other desired
quantity
is obtained as a series in increasing powers
of some variable such as inverse temperature, $T^{-1}$,
for magnets, or concentration, $ p$, of conducting particles in a mixture
of insulating and conducting particles
(the percolation problem). This expansion is calculated
on a term-by-term basis, and usually uses graph enumerations. The expansion
would give us the exact solution if it could be carried to
infinite order, but near the limit of low (high) concentration
an expansion in $p\ (q=1-p)$ to a finite order will be quite reliable.
Extrapolation is made towards the region where the expansion variable reaches
a critical value and the quantity being expanded exhibits a singularity.
Expansions from the disordered phase are especially well behaved;
for magnetic systems this is the high temperature, and for percolation
the low-concentration limit. The best results are obtained when
extrapolations can be made from both extremes and the results shown to be
consistent.
We note that
series expansion data are extremely compact
for storage,
and hence many
analyses can easily be made on the same series data.
Secondly, the
series contain information on all possible thresholds,
unlike simulations which, in many
cases, must be carried out separately
for each trial temperature or threshold value.
Thus, if there appears a reason to investigate the behaviour
in another location at a later date, it is possible
to obtain a new prediction from the series by recalling
a very few coefficients
rather than repeating the entire calculation.
Series expansions studies are always divided into two parts;
generation and analysis. See [1-2] for a recent discussion of
this question.
The reliability of series generation has an impressive record.
While series can give different critical parameters based
on the analysis method there is only one correct series for
each quantity for each model.
Generation with different algorithms has consistently
led to identical terms for a wide variety of problems.
This is in contrast to
the simulation world, where if nothing else random number
generation seems to be an endless source of worry.
A summary of the current state of algorithms for the generation of
series is given in [2] together with a comphrensive list of references.
Several of the algorithms (especially those for high temperature/low
density series) enable simultaneous generation of series for general dimension
and disorder or field parameters in a single calculation.
To analyse a series (or a simulation)
one must have some understanding of the critical
behaviour and make suitable hypotheses about its form, in order to analyse
the series. Analysis of series does seem to be more complex than analysis
of simulations (at least prior to the advent of the histogram methods.)
For series not at critical dimensions where logarithms complicate matters,
critical behaviour is usally of the form (on the low temperature side for the
magnetization, $M$)
$$M\sim(T_c-T)^{\beta}(1+A(T_c-T)^{\Delta_1}+.......)\eqno(4)$$
Here $T_c$ is the critical temperature,
$\beta$ is a dominant critical exponent and $\Delta_1$ a
``correction-to-scaling'' exponent.
This is also the exact form for the Baxter-Wu model.
In the two-dimensional Ising case
$\Delta_1$ is simply analytic (this is related to non-linear scaling
fields), and so
$$M\sim(T_c-T)^{\beta}(1+A(T_c-T)+.......),\eqno(3)$$
The two-dimensional Ising is
the exception and
a non-analytic ``correction-to-scaling'' such as observed in
the Baxter-Wu model is typical of
most models of physical interest (including the
three-dimensional Ising). Its origin is
usually additional operators that are irrelevant in the renormalization
group sense.
These ``corrections-to-scaling''
are of paramount importance to series analysis. Failure to allow
for their effect causes exponent estimates that violate
thermodynamic relations known as hyperscaling relations.
Many of the results presented in the next section are based on
two methods developed by the author in collaboration with M. Moshe and
V. Privman [7,8] in the early eighties that explicitly
allow for corrections-to scaling. Other good methods are discussed in [5].
Our methods
are based on
transforming the series
to minimize the interference of the
correction terms prior to calculating the approximants. They are
variants of the threshold-biased dlog
Pad\'e, and one has been
based
on a generalization of the Roskies transform. Here a series
series in the variable $T$ (which could stand for $p$ too)
is transformed to one in the variable
$y=1-(1-T/T_c)^{\Delta_1}$. We obtain Pad\'e approximants to the
series $G(y)=\Delta_1(y-1){\rm d}(\ln M(T))/{\rm d}T$. At the
correct value of $\Delta_1$, a set of different high and
near-diagonal threshold-biased approximants of $G(y)$ all should
give the correct dominant exponent, for example
$\beta$ if we study magnetization.
It can be shown [8] that deviations from the correct
value will be seen as changes in the slope of the dominat estimates
plotted as a function of the input value of $\Delta_1$. Such a plot will have
an intersection region near the correct $(\Delta_1,\beta)$
point. A related method involves calculating the logarithmic
derivative of $B(T)=\gamma M(T)-(T_c-T)\,d(M(T))/dT$
which has a pole at $T_c$ with residue
$\beta-\Delta_1$. Again threshold-biased approximants are
calculated and graphed and an intersection region is found near the
correct $(\Delta_1,\beta)$ point. In many papers these methods are known as
M2 and M1, respectively.
Interactive
graphics subroutines are very useful for these methods and were described in
Ref. [7].
If the value of $T_c$ is
not known, different trial temperature
values are tested and convergence in the
three-dimensional $T_c, \beta, \Delta_1$ space is sought. If we attempt to
draw graphs at trial $T_c$ values that differ from the exact ones then
we can clearly illustrate the optimal convergence in the correct $T_c$ plane.
An example of such a graph is shown for the susceptibilty of
five-dimensional bond
percolation for the M2 analysis in Section 4.4.
We must emphasise that the nature of the
convergence
with the clearly defined optimal intersection regions
is such that all three parameters can be optimally determined.
We are not left with a set of alternate fits, as is often the case with
a pole-residue plot or with some methods of analysing simulations.
\vskip 0.5cm
\noindent {\bf 4. Results for Several Systems}
\vskip 0.25cm
\noindent In this section
results
for some of the standard
systems of critical phenomena will be summarised.
Series analysis for systems with exact
solutions is important for benchmaking, see [2] for a review
of the 2D Ising and Baxter-Wu cases, but
emphasis will be placed below on results for systems that do
not have exact solutions.
\vskip 0.5cm
\noindent {\bf 4.1 Ising models}
\vskip 0.25cm
\noindent The three dimensional Ising model has been the main
``realistic'' test for
new methods of series analysis and simulations.
There is not space here to quote all the results obtained from
series and simulations for this model so a biased selection is given.
Early results for the simple-cubic critical point include
$K_c=k_B T_c/J= 0.221654\pm 0.000005$ from the Monte Carlo
renomalization group simulation of
Pawley {\it et al.} [13], and
$0.221655\pm 0.000005$ from a graphical
Roskies-transform analysis
by Adler [14].
More recently
Liu and Fisher [15], (using inhomogeneous
differential approximants), found a central
threshold of $0.221630$, and Landau {\it et al.} [16]
found $0.2216593\pm 0.0000022$ from simulations.
This latest value has a precision that is considerably beyond that obtainable
with current lengths of series, but overlaps with Adler's result.
In higher dimensions, the series are
more than competitive with results obtained via simulations.
Series to 15th order for the pure system
were obtained as a by-product of the generation of
new series for the Ising model in a random field (Gofman {\it et al.}[17]),
and were analysed in [18,19] in parallel with simulations.
The simulations in [18,19] were of a slightly unsusal kind, in that critical
temperatures were dertermined from relaxation of the magnetization.
In five dimensions $K_c=0.113935\pm 0.000015$
was found from these series (the best fit was obtained
for $\gamma=1.0005\pm0.005$ and $\Delta_1=0.4\pm0.1$, the exact values being 1
and 1/2 respectively)
to be compared with $0.11391\pm 0.00001$ from the simulations [18].
Older 11 term series analysed by Guttmann [20] based on imposing $\gamma=1.0$,
gave $0.11392$ and $\Delta_1=0.5\pm 0.05$ compatible
with the simulations of [18].
This led us to be a little concerned about the series, since longer series
should give better results,
but the recent simulations by Rickwardt {\it et al.} [21] give
$0.113935\pm 0.000015$
confirming the series value of [18].
In six dimensions our series gave $0.092295\pm0.000003 $ whereas simulations
suggest $0.09230\pm0.00005 $
and in seven dimensions the results are $0.077706\pm0.000002 $ and
$0.07772\pm0.00003 $ respectively [19].
The simulations of [18,19],
which when made two years ago
were the largest ever simulated in those dimensions,
used a parallel
machine with 32 Intel i860 processors whereas the series were obtained as a
byproduct of another study. The longer series for the
the four dimensional Ising case requires analysis that allows for logarithmic
corrections which is being undertaken at the moment.
\vskip 0.5cm
\noindent {\bf 4.2 Heisenberg models}
\vskip 0.25cm
\noindent
Prior to 1993
published results for the critical temperature
from series expansions up to 12th order
for the three-dimensional
classical Heisenberg and XY models did not agree
very well with high-precision Monte Carlo estimates. The series result
was $K_c=0.6924(2)$ [22].
The simulation estimates were $0.6929$ from the Metropolis method
[23], $0.6930(1)$ from a cluster method [24] and $0.693035(37)$
from a multiple cluster method [25].
In order to clarify this discrepancy Adler {\it et al.} [26] analyzed extended
high-temperature series expansions of the susceptibility, the second
correlation moment, and the second field derivative of the susceptibility,
which were derived by L\"uscher and Weisz [27] and Butera {\it et al.} [28]
for general $O(n)$ vector spin models on $D$-dimensional hypercubic lattices
up to 14{th} order in $K \equiv J/k_B T$. We found $
K_c = 0.6929 \pm 0.0001 $, in
good agreement with the simulations, although substantially less precise.
Exponents were in good agreement
with standard field theory exponent estimates.
\vskip 0.5cm
\noindent {\bf 4.3 XY models}
\vskip 0.25cm
\noindent
In [26]
we also reanalysed the XY model series to find $K_c=0.45414\pm0.00007$.
More recently,
Butera, Comi and Guttmann [29]
have extended the XY series to 17 terms and found exponent values
in good agreement with the field theory and simulation ones.
The simulation temperatures are again more precise and we quote
$0.45421(8) $ [30], and $0.4542(1) $ [31].
The
first-ever experimental
observation of a non-analytic correction-to-scaling was
made in superfluid $^4$He,
which is in the same universality class as the XY model.
from the shorter 14-term series we were unable to observe any clear
interection indicating the non-analytic correction,
but a a reanalysis of the series of
[29] with M2 clearly shows that near $K_c=04541$ there is
a clear intersection near $\Delta_1=0.5$, in good agreement with the
renormalization group and experimental result.
\vskip 0.5cm
\noindent {\bf 4.3 Percolation}
\vskip 0.25cm
\noindent
Series expansions for general moments of the bond percolation cluster size
distribution on hypercubic lattices to 15th order
in the concentration were obtained by Adler {\it et \ al} [32] in 1990.
This was one more term
than the previously published series for the mean cluster
size
in 3D and four
terms more for higher moments and higher dimensions.
Critical exponents, amplitude ratios and thresholds were calculated
from these and other series by a variety of independent analysis
techniques, and compared with other results.
In Figure 1
an example of the M2 analysis is shown for five-dimensional bond percolation.
Here optimal convergence occurs for $p_c=0.11819\pm0.00004$.
\vfill
\eject
.
\vskip8.0cm
\vfill
{\bf Figure 1.} Plot of approximants to the dominant exponent $\gamma$
as a fuction of correction-to-scaling exponent $\Delta_1$, from the M2 analysis
at five trial thresholds for the susceptibilty of five dimensional
percolation.
\vskip 0.5cm
\noindent This theshold was deduced from several measurements
on different moments with both M1 and M2 and
remains the most accurate five-dimensional threshold to the best of
my knowledge. The exponent of the mean-cluster size can be seen to be
$\gamma=1.185\pm0.005$ and the correction exponent $0.55\pm0.15$
in excellent agreement with
$\epsilon$-expansion values [33] of 1.18 and 0.45, respectively.
Older simulation results [34] gave $1.3\pm0.1$
Accurate exponents and thresholds were found in dimensions 3-9,
and in the most interesting case of three dimensions the series gave
$p_c=0.2488\pm0.0002$, which was in contradiction with the then-accepted
$0.2493$ [35]. This caused us a delay in submission while we tried to
understand
the series and simulation values
could be so far off, but we later heard
that Ziff and Stell found $0.248810\pm0.00005$. The exponents
agree with $\epsilon$-expansion values in the high dimensions and Ziff and
Stell in three dimensions.
\eject
\noindent {\bf 5. Conclusions}
\vskip 0.25cm
\noindent
There is a consistent pattern in these models of more precision from recent
simulations in two or three dimensions, with the simulation estimates
falling well within the series error bounds. There is as yet no case
even amongst more complex systems where a series analysis that allowed for
corrections-to-scaling failed to include later precise simulation results.
In higher dimensions, the situation is reversed,
with higher precision being found for the series.
In every case
the series results are obtained with orders of magnitude less computer
resources, (but not always less manpower.)
It is comforting that as things stand at present there remain no
serious discrepancies between series and simulations in the simple models
discussed herein.
\noindent{\bf Acknowledgements:} I thank the BSF and GIF for support
during many of the calculations described above. Discussions with
all my co-authors of the results reviewed here, and many others active
in series and simulations were very helpful.
\vskip 1.0cm
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\end