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\centerline{\bf Series Expansions versus Simulations}
\centerline {Joan Adler}
\centerline{ Department of Physics }
\centerline {Technion-IIT, 32000, Haifa, Israel.}
\bigskip
\centerline {\bf Abstract}
\smallskip
In this chapter we describe the series expansion method for the study of
critical behaviour, and compare it with simulation methods for some
simple models. Series methods have been applied for more than fifty years,
but were eclipsed in popularity by renormalization group and then
simulation methods in the late seventies and eighties and only recently have
experienced a resurge in popularity.
They have in
common with Monte Carlo simulations heavy computational demands
for quality results, and the need for sophisticated analysis in order to
obtain correct critical exponents and temperatures/thresholds.
The series method involves exact enumeration rather
than random samples and therefore does not use random numbers.
However, the startup time can be considerable, and as such it is a method for
precise calculation, and is especially suited to
problems in general dimension or with a parameter such as an external field,
but is less suited than simulations to quick explorations.
A discussion of recent results
and comparisons between series
and simulation studies of several systems of current interest as well
as tabulations of results from both series and simulations for Ising models
and percolation
in general dimension are given.
%change
The overall agreement between the two approaches is very pleasing.
\vfill
\eject
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{ \bf Introduction: }
The transition from ordinary (paramagnetic) iron to the magnetic phase
has been a source of wonder for many centuries.
Other phase transitions are equally fascinating in themselves, and
even more so for the analogies that can be drawn between
transitions in diverse physical systems. For example,
the divergences
at the critical temperature in the superfluid-normal
transition of helium and in certain magnetic transitions
are characterised
by the same critical exponents.
This notion of universality, together with the idea that fluctuations
near critical points occur over many length scales,
is central to the current understanding
of the area of phase
transitions and critical phenomena.
Despite the crucial breakthroughs
of recent years, including the advent of the renormalization group,
many questions about critical phenomena remain unanswered.
Although certain model systems are now well
understood, many real materials of significant technological and
scientific interest exhibit critical phenomena that are still not
well characterized. In particular, materials such as iron, which have longer
ranged interactions, and most disordered alloyed systems such as spin
glasses, ferromagnets in random fields,
or the new superconductors, pose an exciting challenge to both theorists
and experimentalists. In the absence of a thermodynamic framework for the
development of scaling theories, disordered
systems will require high quality experimental and
numerical work for the elucidation of their
properties.
The series expansion method, which predates the advances of the
renormalization group, was then and remains today
an essential numerical tool
for the study
of critical phenomena.
In fact, much of the numerical information on which the early observations
of scaling and universality were based was gathered by this method.
There have been recent significant algorithmic breakthroughs for
both series generation and the other common numerical technique, namely
direct simulation.
Both series expansions and direct simulations are
numerical approaches that can be applied to other
aspects of interest
of models
that undergo phase transitions, such as zero temperature
phase diagrams.
They have advantages in common that in contrast to approximations,
such as
mean-field theory and real space
renormalization group techniques, which are often
unreliable in two and three dimensions and are difficult to
extend in a controlled manner, both series and simulations
are good in dimensions of physical interest
and, at least in principle, are easy to extend
in a controlled manner. Common limitations on both
series and simulations are computer memory, time, and the need for efficient
algorithms.
In general, the best way to study a model that cannot be solved
exactly is to compare the results of simulations, series expansions,
and other complementary approaches if available.
Such comparisons have been made
for many systems that undergo phase transitions and consistent
results have been obtained from series expansions,
$\epsilon$-expansions (generated by the momentum-space
renormalization group, expanding from
a dimension where the model's critical behaviour is known
exactly) and mean-field theory in higher dimensions,
and exact results and simulations in lower dimensions. Systematic presentation
of such comparisons is the aim of this chapter,
which
is arranged as follows. First some exactly solved
models will be defined and their series presented as an introduction
to the major issue that the series analysis must involve fitting to the
correct functional form. This section will assume very little prior
knowledge.
While simulations hardly need further introduction
to the readers of this volume, some aspects of the analysis
of these will also be
reviewed. Next, although generation of series expansions will not be
discussed in depth here, pertinent aspects of utilization of
computer resources for series generation
will be presented for contrast with simulations.
The
common aspects of
and contrasts between
series and simulations will be discussed in the
context of historical developments in statistical
mechanics and in computing that have influenced the
development of both.
This will be followed by some comparisons between series and simulation
studies, and an introduction to the literature where such comparisons
are presented.
A summary and discussion of controversies and points of
current interest will follow.
For those unfamiliar with technical aspects of
critical phenomena, the early chapters of either
Yeomans [1] or Stanley [2] are recommended.
The discussion of the renormalization group in chapter 8 of [1]
is particularly useful as background for comparisons below.
An introduction to
series expansions
can be found in chapters 6 and 8 respectively of these books. This author
has recently presented [3] an introduction to both generation and analysis
of series expansions in a percolation context
for a simulation audience, including sample programs.
Other introductory references are [4-8], with some details of algorithms
for Pad\'e approximants in
[9,10].
The present chapter complements these by presenting
some general computational aspects in detail and providing a comparison of
resources needed and of results
obtained, from series and simulations. Since comparisons
are best made with exactly solved models we will
concentrate on Ising and related models.
Overall, it should be kept in mind that
this chapter is
aimed at students and researchers
who, while interested in results obtained from series expansions,
have not necessarily applied the method themselves.
{ \bf Application of series and simulations to two exactly solved models:}
In this section the results obtained from series
and simulations will be compared using examples of
two exactly solved two-dimensional models. One is the
two-dimensional Ising model, the other the Baxter-Wu (BW) model [11,12]. Their
Hamiltonians are
$$-\beta{\cal H}=J_{ij}\sum_{ij}s_is_j\eqno(1) $$
and $$-\beta{\cal H}=J_{ijk}\sum_{ijk}s_is_js_k,\eqno(2) $$
respectively. For the purposes of
this section the former (which could be defined on any lattice, with ${ij}$
being adjacent pairs of sites) will be studied on the square lattice,
the latter is specific to the
triangular lattice where ${ijk}$ are triplets of sites around the
triangles of the lattice.
In both cases, series for the magnetization $M=$
can be found to arbitrary length by expanding the exact solution.
(The expansion can easily be done by computer algebra; the Ising series was
obtained by expanding the expression for $M(u)=(1+u)^{1/4}(1-u)^{-1/2}
(1-6u+u^2)^{1/8}$ directly
and the BW series is taken from [13] based on the conjectured exact form
of [12].)
We quote these series in Table 1 to order 40.
%change2
The expansion variables are $u=\exp(-4\beta J)$,
known as the ``low-temperature'' variable, and coincidentally the same
for both models. Like all coincidences in physics, there is a deep
symmetry here, and in fact the solutions of the two models are very
similar [12].
Of course, we are sidestepping the issue of generation
(generation is often
the hardest part, see e.g. [5] for details for Ising models)
by using the expansions of the solution
here, but it is in a good pedagogical cause so onward
to the main point of this section.
The main point is that one must have some understanding of the critical
behaviour and make suitable hypotheses about its form, in order to analyse
the series.
The exact solutions to the Ising and BW
models show that their critical behaviour
is different in one very significant way. For the Ising model
$$M\propto(T_c-T)^{\beta}(1+A(T_c-T)+.......),\eqno(3)$$
where $\beta=1/8$,
whereas for the BW [14]
$$M\propto(T_c-T)^{\beta}(1+A(T_c-T)^{\Delta_1}+.......)\eqno(4)$$
where $\beta=1/12$ and $\Delta_1=2/3$. $T_c$ is the critical temperature,
$\beta$ is a dominant critical exponent and $\Delta_1$ a
``correction-to scaling'' exponent. In the two-dimensional Ising case
$\Delta_1$ is replaced by the analytic correction, taking the value 1.
(This analyticity is related to non-linear scaling
fields [15]).
This is however the exception and
a non-analytic ``correction-to-scaling'' such as observed in
the BW model is typical of
most models of physical interest (including the
three-dimensional Ising case). Its origin within the renormalization
group is
usually additional operators that are irrelevant in the renormalization
group sense.
For a general discussion of irrelevant operators see p. 110ff of [1];
an introduction to these can also be found in Ref. 7
in a percolation series context. As will be seen 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 (see below and [16]).
How do we turn some 20-40 numbers into accurate estimates for
critical parameters?
(For realistic three-dimensional problems 15-20 terms is typical today, and
40 or more for some ideal two-dimensional models. For some problems
such as quantum models interesting
results with 8-12 terms can also be obtained.)
If the series were infinite then simply summing
the series would give
the
exact solution of the magnetization. When it is
finite we do have the first $n$ terms of the exact expression, but
convergence is slow
so simply calculating the magnetization by summing a finite number of
terms in the series does not give a very accurate fit to the exact result
in the critical region. Outside the critical region a few terms of the series
are sufficient to enable matching. In fact, since in general the first
few terms of
a series are quite easy to calculate, it is often good to compare a
new simulation
program with the series estimate in the high or low temperature region.
Far from the critical region a few terms of the
series usually agree with the exact solution.
In the critical region simulations usually do much better job than
the summed-up series
on the level of simply graphing the magnetization.
Accurate estimates of the
critical parameters is another question that will be much discussed below,
but first we illustrate the simple approach.
Using one hour of workstation time D. Stauffer has provided data from
some simulations of the
two-dimensional Ising model specifically for this comparison.
These were made with the Glauber algorithm for
size 4800$\times$4800 and 100 timesteps and 960$\times$960 and
1000 timesteps.
In Figure 1 we present the exact solution, results of summing 20 terms of
the series and Stauffer's Monte Carlo
data on a single graph for the two dimensional Ising case.
At low
temperatures the series and exact lines overlap. Simulation data gives
similar results.
For clarity only the exact solution is shown below $u_c=0.14$.
Deviations begin at about 90\% of $u_c$. The simulations of the 960 size
sample for 1000 timesteps
actually follow the exact line for longer than the series.
It is instructive to compare the effect of increasing the series length
and we have chosen to do this at $T_c$, or $u_c=0.17157...$.
We find that while the exact
value here is 0 and the simulation gives 0.705 for the larger sample
and short times and
0.626 for the smaller one at longer times, the 20-term series give
about 0.72. For the Baxter Wu model the series gives
0.93, 0.88, 0.84 and 0.75
respectively at 10,20,40, and 100 terms, showing that increasing the number
of terms is not very helpful here.
Since 20 is anyway realistic we must conclude that the simulation method is
definitely superior in the region
close to criticality case. However no-one attempts to determine critical
behaviour (which is our primary objective)
directly from summing the series so we now turn to more sophisticated
series analysis
methods and comparison with simulation for determining critical parameters.
The basic idea of series analysis is to make an hypothesis for
the critical behaviour (Eqn. 3, 4, etc) and then to fit this to the $n$ terms
of the series. Sometimes several hypotheses are tried and the best
fit is selected.
The series are usually an expansion from some region,
in this case low temperature, towards the critical region.
We must extend the series beyond the low temperature region
and identify the critical point with the
radius of convergence of the series.
Several approaches have been developed for such extrapolation.
An introduction to the older ratio methods can be found in [4,5].
Here we shall discuss several variations based on
the Pad\'e approximant approach, which
works better than it has any right to.
Pad\'e approximants were introduced into critical phenomena by G. Baker
[8].
Variations of this and of
other more general approximant schemes
have been used to obtain most of the results of
physical interest that will be presented below.
One way to
determine the location and nature of the critical point uses the
logarithmic derivative (dlog) of the series. It is based on the idea
that if we take the dlog of a function, say $M(u)=\sum_na_nu^n$
where the critical
behaviour is of the Eqn. 3 type,
$${d\over
du}(\ln \sum_n a_n u^n) = {{ \beta}\over {u_c-u }} +.....,
\eqno(5)$$ then the R.H.S. of (5) will have a pole at
$u=u_c$ with a residue of $\beta$.
Because calculating the pole and the residue of the logarithmic
derivative of the series directly
is not successful in general, Pad\'e approximants are taken and then
their poles and residues are calculated [4,5]. Pad\'e approximants are
polynomial ratios;
the [$L,M$] Pad\'e
approximant ($L+M
\le N$) to a series of order $N$ is the ratio of a polynomial of
order $L$ to a polynomial of order
$M$, such that when the denominator of the approximant is divided
into the numerator, the first $L+M+1$ terms of the expansion of the
Pad\'e approximant match the first
$L+M+1$ terms of the series. The approximants can be calculated
using standard computer languages such as FORTRAN or using computer
algebra packages. MATHEMATICA and MAPLE examples were given in [3].
The FORTRAN approach
is more efficient, but requires many more lines of code. Programs in all
three languages were presented in Ref. 3. The FORTRAN version is preferable
(since it is many times faster),
if one is engaged in many analyses of multiple series but the other versions
are much easier to implement for quick investigations.
The basic idea of all the routines is to first obtain the Pad\'e
approximant and then calculate its pole and residue.
Approximants are usually selected to have high
$L$ and $M$ values, with $L \approx M$. An introduction to numerical
aspects of determining Pad\'e approximants is given in [9] for compiled
languages and in [10] for MATHEMATICA.
A large number of poles and residues must be tabulated and
sorted so that the physical root can be selected and then plotted.
A variant of the MATHEMATICA program (prepared by C. Goldenberg)
from [3] that does this is given
in Table~2 with 20 terms of the Baxter-Wu magnetization [17].
Efficient FORTRAN programs for series analysis are available
from several sources. These programs include subroutines for the
input of the terms of the series, the calculation of the desired
Pad\'e approximants, subroutines for the calculation of the poles and
residues, and subroutines for tabular and graphical output. A good
general reference to series analysis (with programs available on
diskette) is by Guttman[4]. If there are complications such as
more than one expansion variable or significant analytic additions,
more sophisticated approximants
must be considered. Differential and partial differential
approximants are two such methods. Some excellent partial
differential approximant routines are described in [18].
Once the individual poles and residues have been obtained, we must
decide how to deduce the overall estimate for $u_c$ and the
exponents. Simple averaging is not always desirable and the usual
first step is to graph all the residues as functions of the pole
location for a range of physical poles that fall reasonably close
together. Examples of such pole-residue plots for both Ising and BW models
are given
in Figure 2, and estimates taken from these as well as
the exact results are given in Table 3. The exact threshold for both models
is $u_c=0.17157..$;
one set of estimates is taken from reading off the exponent
at this value, another taken by noting where the density of poles is highest.
Another way to obtain exponent estimates is to multiply
the R.H.S. by $u_c-u$ and then to take Pad\'e approximants to the R.H.S.
which will give an estimate of $\beta$. Such estimates are also
given in Table 3.
For the Ising case agreement between different methods
of analysis of the 20 term
series is very good.
For the BW there are deviations. These occur because the simple approximation
of Eqn. 5 takes into account only
the first term on the R.H. sides of Eqns. 3 and 4. An analytic correction
does not invalidate this but there is an
implicit assumption that there is no non-analytic correction.
In order to take this higher order term
into account we must extend the
analysis techniques. One way of doing this is to transform the series
to minimize the interference of the
correction terms prior to calculating the approximants. One such method,
which is a variant of the threshold-biased dlog
Pad\'e, has been
is based
on a generalization of the Roskies transform. Here the
series in
$u$ is transformed to one in the variable
$y=1-(1-u/u_c)^{\Delta_1}$. We obtain Pad\'e approximants to the
series $G(y)=\Delta_1(y-1){\rm d}(\ln M(u))/{\rm d}u$. 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
$\beta$. It can be shown [6] that deviations from the correct
value will be seen as changes in the slope of the $\gamma$ estimates
plotted as a function of the input value of $\Delta_1$. Such a plot
leads to an intersection region near the correct $(\Delta_1,\beta)$
point. A related method involves calculating the logarithmic
derivative of $B(u)=\gamma M(u)-(u_c-u)\,d(M(u))/du$
which has a pole at $p_c$ 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]. These methods also work for cases such as the Ising where there is no
non-analytic correction and we show such graphs for both the Ising (20 terms)
and BW (40 terms)
models in Figure 3.
If the value of $u_c$ is
not known, different trial temperature
values are tested and convergence in the
three-dimensional $u_c, \beta, \Delta_1$ space is sought. If we attempt to
draw graphs at trial $u_c$ values that differ from the exact ones then
we can clearly illustrate the optimal convergence in the correct $u_c$ plane.
Results of
the full series analysis of the 20 and 40 term BW model were given in [13],
and 3D color pictures are given in Figure 1 of Ref. 6,
and we summarise our numerical results from this calculation in
Table 3. 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.
Now let us consider exponent and temperature values from the simulation.
Almost every new simulation technique is tested by application
to the two-dimensional Ising model[19].
All the data published for this case are very good; however it would be
premature to draw conclusions before making a similar comparison for
the BW model. Simulation results for this model have been published
by
two groups. Both our results from [13] and those of
Novotny and Evertz [20] are given in Table 3.
In this case the series are clearly better
behaved and since the behaviour of Eqn. 4 type is more typical of interesting
three-dimensional models this is an interesting conclusion. Of course,
other simulations of the BW with different techniques might give different
results and the reader is welcome to try his/her favourite method.
It would also be of interest to attempt to reanalyse the BW series
with different methods.
One point that must be made is that reanalysis of the series simply requires
transferring some 20 odd numbers whereas reanalysis of existing simulation
data can be very cumbersome.
We conclude this section with a warning.
There are many systems of physical interest which have critical behaviour
that is not of the form of either Eqn 3 or 4. For example the singularity
could be exponential (XY model in two dimensions), have logarithmic
corrections (at the upper critical dimension $d_u$ such as the four-dimensional
Ising model), or be first-order.
Here different analysis techniques must be used, these will be briefly
discussed in
later sections of this chapter when physical realizations are discussed.
More details can be found in [4].
{\bf Comparison of use of computer resources:}
Simulations began slowly in the fifties and sixties and have continued to gain
momentum ever since as computer technology developed.
Since they
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, in practice the samples are never large enough nor
are there enough of them.
In addition, it can be
difficult to determine that the simulated
system has reached equilibrium and to
extrapolate the results to the thermodynamic limit.
Thus there is never enough computer time for any ideal simulation but
any time available can be always be utilised to obtain a few more samples
or a better check on equilibration.
Of course, a simulation is restricted to a particular model in a
given spatial dimension at one time so in order to study
the effect of changing temperature or disorder many different calculations
need to be made.
In response to problems of insufficient time and memory 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. At least prior to the development of more
sophisticated analysis routines of
the histogram weighting type [21], 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. Simulations then were really
the dominant numerical technique
for critical phenomena, but as we shall now argue
their relative advantage to
series is diminishing as simulations
become more complex and series generation and analysis easier.
In contrast to the smooth development of simulations, the development of
series expansions has been far more erratic. The
series method flowered from 1960 on and provided important input
for the ideas of scaling and universality that underlie the
renormalization group.
In addition to problems adapting series methods to remote-site vector
supercomputers in the eighties (problems with both vectorization and with
data transmission were relevant here),
some unpopularity resulted from apparent
violations of hyperscaling that were caused by inappropriate
extrapolation techniques that neglected corrections-to scaling.
This led to a definite downturn in activity relative to the growth
of simulation methods at this time,
although some groups remained active and laid the foundation for
the current revival. 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 this is irrelevant as we shall now show.
Four recent computing advances have been most helpful
to series expansion studies.
The first computer advance is fast communications and networking which
have meant that
transmission of the large data files needed for series generation
is now practical. The second is widespread use of
workstations with large disks and memories
which are usually adequate for much series expansion development.
The third is parallel machines which are suited for certain series generation
methods and the fourth is widespread availability of
computer algebra packages
which are useful both for some aspects of generation and for analysis.
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.
Let us now look at some specific points.
In series expansion studies the major share of the computer resources
is usually spent on the generation of the series.
Series
can usually be generated in more than one way;
some generation methods being more efficient than others.
Algorithmic developments from the eighties include,
Nickel's 1980 star graph formulation [22] for the Ising model and
the 1986 Singh-Chakravarty extension [23] to the
Ising spin-glass Edwards-Anderson susceptibilities, the 1982
Harris [24]
no-free-end (NFE) formalism.
The
NFE
approach can
be
applied to many problems
ranging from low concentration geometric models to magnetic
systems. These were reviewed in [25]. Very recent developments include
significant
extensions to
the finite-lattice method
by Conway and Guttmann [26] and popularization from the studies
of Bhanot and Creutz et al [27],
and the Comi and Butera [28]
method for n-vector models.
The finite lattice methods have been extensively implemented on different types
of parallel machines. Discussion of series generation on parallel machines
can be found in Duarte [29] and Conway [30].
It should be noted that all but the finite
lattice method are in principle as well as practice ``parallel algorithms''
in the sense that series for all dimensions and all values of some
additional parameter such as disorder or field can be generated at once,
not to speak of the fact that all series give results for all temperatures on
one side of the transition
in a single expression.
The reliablity 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 (despite impressions to the contrary
as mistakes can be made, it is true that exactly one correct series exists).
A basic indication
that all the generation
methods are reliable is that spinglass series and percolation
thru to 15th order were obtained by star graph and NFE in the former case
[23, 31]
and NFE [32] and by Sykes et al [33] in the latter case.
All gave exactly the
same series terms. This kind of confirmation cannot exist in
the simulation world, where if nothing else random number
generation seems to be an endless source
of worry for Monte Carlo studies.
The star, NFE and other graph-based generation schemes have two possible
bottlenecks in computer resources.
One is the generation of the graphs themselves;
since this is usually
not done afresh for each calculation, it is
in practice (even though it should not be in principle)
a significant barrier. The other is the calculation of
contributions to a particular series from a given graph. For quantum and
disordered models the latter can be a real problem.
The demands on disk space are usually very large
in the
intermediate stages
(if graph listing are used) but the final series are usually compact.
This compactness of final results mean that several analyses can be made on
each series (often many years after the generation in the event that
physical intuition advances later). Although there are some signs that
simulation data is now sometimes stored for later reanalysis, this requires
so much space that this is unlikely to become commonplace.
In practice the limitations on length of series are usually computer memory
and less often time but these demands are far less than
required for a simulation of equivalent quality. It should however be
noted that while any time available can usually be used to slightly
improve simulation quality, for series there must be enough time
(and memory) to
obtain another term in the series and often the next term will take as much
time as all the previous ones together did.
The last point in this discussion of comparative
computer resources is the far from minor point of comparing
the literature about series with that about simulations.
There is no question that today there is far more good published instructional
material on how to simulate
(for example [34]) than there is about series expansions, and material
with hands-on instructions for series generation is particularly
hard to obtain.
Several educational references to series analysis do exist and
were mentioned in the previous section;
and a
program has been published by Mertens [35] for enumerating lattice
animals.
Introductory
material pertaining to recent advances
in the graphical generation methods has not yet been published,
but this is a situation that will hopefully be remedied in the near
future [36],
following renewed interest in series and the Haifa workshop of June, 1994.
Series generation for the different methods has been discussed in
a more technical way in some of the papers cited above, and
an elementary discussion of series generation for percolation
can be found in [3].
In summary, at this point in time
the gap between series and simulation applications is narrowing owing to
developments in computation.
Series methods for a given expenditure of computer time nearly always give
more precise results.
Reanalysis of series is easier and more physical
cases can be studied simultaneously.
On the other hand startup time for series is usually
longer, and the situation concerning introductory literature for series
generation leaves a lot to be desired.
We now look at some comparative results for physical models.
\vfill
\eject
{\bf Comparisons for physical models:}
A diagram relating some of the different models discussed below
is given in Figure 4, this type of diagram was originated by Stanley
with the epithet of a ``metro map''.
In an $n-$vector magnet[2] the spin has $n$ equivalent components; in the $q-$
state Potts model[37] there are $q$ possible real values.
For the n-vector models
$d_u=4$ and there is no
finite-temperature transition in $d\le2$ for $n\ge3$. For the Potts models
$d_u=6$ except for the Ising case.
A partial report (on critical temperatures only) for certain
cases of critical behaviour of the form of
Eqn 4 has been given in
[38].
This paper, based on a conference
talk, by the author complements this present rsection. Following a
discussion of some details of specific interest we conclude with two
tables summarising numerical results from series, simulations and
renormalization group calculations.
Table 4 is for percolation critical exponents and
Table 5
for selected Ising critical temperatures and percolation thresholds.
The Ising model belongs to both families ($n=1$, $q=2$) so we will
discuss it first. Four aspects of interest are the three-dimensional
case which was at the centre of the hyperscaling controversy,
the dynamical exponent (especially in two dimensions),
the logarithmic corrections at the upper critical dimension
and the roughening transition that occurs at the interface of
an Ising model with opposing boundary conditions.
Although problems with apparent hyperscaling violations caused by
neglect of corrections-to-scaling occurred
in quite a few systems in certain analyses, the big excitement
was over the three-dimensional Ising model. This happened when it became
clear that the older series analyses gave values for $\gamma$,
the critical exponent of the susceptibility, and $\nu$ of the
correlation length that disagreed with renormalization group estimates
and violated hyperscaling. Also, critical temperature estimates
disagreed with some early simulations. Resolution of the controversy
happened when Nickel [22] developed the star-graph expansions
leading to 21-term series on the bcc lattice
and showed that the exponents indeed obeyed hyperscaling.
A little later the groups at Santa Barbara (with a special purpose
machine) and Edinburgh (on the DAP array processor) [39] found precise
estimates of $T_c=0.221650\pm0.000005$
and $T_c=0.221655\pm0.000005$ repectively
from the simple cubic lattice, and independently
Adler [40] found $T_c=0.221655\pm0.000005$
from a reanalysis of the older series.
Since then there have been other series analyses, we mention Fisher and
Liu [41] who calculated many critical amplitude values and a slightly
lower $K_c$ and many other
simulations, including Blote et al [42] ($T_c= 0.2216546\pm 0.0000010$) and
Landau et al [43]
$T_c=0.2216593\pm0.0000022$.
This precision is substantially beyond the capability of the
current series, but it is rewarding to see that the much older
series estimates
fall well within the range. One should note that there is a conjecture
by Rosengren [44] ($T_c=0.22165863$)
that stimulated several further calculations, before it
was shown
by Fisher [45],
that the Rosengren conjecture is not unique and that there are
other conjectures of the same type that give different thresholds.
In the meantime
Baker,
gave decisive numerical proof of hyperscaling [46], via a calculation of
the coupling constant $g*$. Although this proof does not depend directly on
the exact value of [44], it doesnt depend on the critical point falling close
to it or to the values of [40,42,43]. Recently Salman and Adler [47] have
reanalysed the low-temperature series of Creutz et al [27]
and find values that fall at the lower end of the above studies
with $K_c=0.221653$.
Higher dimension Ising values from both series and simulations
were discussed in detail in [38],
with exception of four dimensions where logarithmic corrections
complicate matters. Adler and Stauffer [48] have recently
looked again at four dimensions and find that it is possible to reconcile
both series and simulation results with $K_c=0.14970$, a little above older
series values.
All the results described above are summarised in Table 5, which we
hope gives a useful ready reference, but does not claim to include
all estimates calculated.
We apologise to authors whose values were not
cited, we have given a subjective selection as to attempt to include all
estimates would be impractical.
Dynamical exponents for the Ising model
have been of considerable interest during the last two decades.
Defining one such exponent, $z$,
from the decay of the magnetization at $T_c$ with time $t$
$$M\propto t^{-\beta/(\nu z)} \eqno (5) $$
a large number of simulation
studies have attempted to calculate $z$, with varying degrees of success.
We refer the reader to [49] for a review of the simulation numbers for $d=3$.
We quote here some of the more recent values for $d=2$
$2.172\pm 0.006$ [50], $2.18\pm0.02$
[51], and $2.165\pm 0.010$ and more recently by the same method
$2.160\pm0.005$ [52].
It is not clear to us what the nature of the corrections-to-scaling should
be in this case. Both logarithmic corrections and a confluent
non-analytic correction have been informally proposed.
There are several series estimates for $z$
and the most recent series generation
by Damman and Reger [53] was a real ``tour-de-force'' using an original
approach that enabling significant extension of the series, so these are
the only two-dimensional
series we will discuss here.
Damman and Reger have carried out a comprehensive analysis
including a search for corrections (which they did not find) and quoted
$z=2.183\pm 0.005$. Reanalysis of the series by Adler found a correction at
$\Delta_1>1.0$, and this leads to $z=2.165\pm 0.015$, as quoted in [52]
in good agreement with the Ito
value [52]. A
full description
including analysis of some extended series is being prepared by Damman
and Adler at present. Attempts to impose logarithmic corrections show
that these are not present and return a $z$ consistent with the above.
In three-dimensions the series are not very long, and the simulations results
are far more precise still.
Percolation ($q\rightarrow 1$)
series have been discussed in [3] for simple two
dimensional cases. In higher dimensions the situation was summarized in [31].
We reprint part of the Table from [31] in Table 4, (omitting some
of the oldest values, when newer results have been obtained
by the same techniques)
noting that there have been no real changes in exponent values
in the time since this appeared.
Two-dimensional Potts models have exact solutions for the critical
points and for the exponents for ferromagnets with $q\le4$.
The very earliest series obtained good exponents [54], then some longer
series were analysed neglecting rather large corrections to scaling,
which were included in a set of papers by Adler, Privman and Enting [55].
Long series have recently been generated by finite lattice methods in two
and three dimensions.
For $q>4$ and for the $q=3$ case on the triangular lattice (3PAFT)
the transition is first order; at present there is no universally accepted
method for
analyzing series with first order transitions [56],
in cases where matching cannot be done between low and high temperature series,
but using simple methods and
allowing for non-analytic corrections to scaling gives good numerical agreement
with exact values for the Potts ferromagnets [57] and for the 3PAFT [58].
%change
The 3PAFT is the only two-dimensional Potts model for which there is no
exact value for the critical point, the exact result being limited to the
observation that there is no finite temperature second order transition.
The series analysis gives $T_t=0.628\pm0.004$ and the simulations
$T_t=0.62731\pm 0.00006$, and clear evidence of the first order transition.
XY and Heisenberg models ($n=2$ and 3) are harder to study with both series
and simulations than the Ising case, but nevertheless much progress has
been made.
In two dimensions the XY has an essential singularity, and the Heisenberg
model has no ordered phase.
The longest series here have recently been obtained by Butera and Comi
and analysed by Guttmann [59].
Three dimensional critical points were reviewed in [38] and
%change
there is also a summary in Butera and Comi [59].
%change
The final topics to be discussed here are two variants of the Ising model.
One is the question of roughening transitions in the interface between
a domain of up and a domain of down spins,
where the related solid-on-solid model is also often studied.
The older series analyses of Adler [60] ten years ago
gave critical temperatures of
$0.404(12)$ and $0.787(24)$ for these two systems respectively.
Many simulations were carried out in the meantime,
%change3
giving numbers that oscillated above and below these,
but the most recent simulations [61] give 0.40754(5) and 0.8061(3)
are far more precise, and well within the range of the older series values.
Higher precision from the series would require the generation of
substantially more terms
in the expansion.
The second topic is the Ising model in a random field. After a long
hiatus in activity here, a recent calculation by Gofman et al [62]
doubled the
length of the series and conclusively showed that there are two independent
exponents. More recent simulations [63] confirm some of this in
three dimensions, and it would
be of interest to have some more of these examined by simulation.
{\bf In Summary:} A comparison of series and simulation methods for
studying critical phenomena for exactly solved and other systems
has been given. Overall the results are in pleasing agreement.
%change4
If we return to the discussion at the beginning of this chapter, where the
claim was made that the best way to study a model that cannot be solved exactly
is to use different approaches, we can see
from Tables 4 and 5 and other models as discussed in the text
that there is a clear consensus on critical points for general dimensional
percolation and Ising models, as well as many other systems.
The general trend is that for certain special cases in two or three dimensions,
simulation values are more precise but are included in the series estimates.
For higher dimensions the series are usually more precise.
Open problems of current interest for series enthusiasts include extended
series for dynamic Ising exponents in
dimensions other than two, extended series for the roughening transition
and the development of methods to analyse
series with first order transitions and tricritical points.
In addition the entire area of series for quantum problems is
developing quickly, and early results here eg [64.65] are most interesting.
{\bf Acknowledgements:}
Support from the GIF (German Israel Foundation) to enable the author
to travel to Cologne where much of this manuscript was written
is acknowledged with thanks.
The constant support of D. Stauffer throughout the preparation of the
manuscript was essential for its completion.
The calculations of the author reported herein
were carried out with support from the
GIF (XY, Heisenberg and Potts models), from the
US-Israel Binational Science Foundation (development of analysis methods)
and from the
Israel Academy of Science (roughening transition results.)
Hospitality from R. Palmer at Duke where the manuscript was completed
was also very important.
\vfill\eject
\baselineskip 13pt
\medskip \noindent {\bf References} \smallskip \frenchspacing
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%change
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%change
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R. R. P. Singh and J. Adler.
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\item{38.}
J. Adler in
``Recent Developments in Computer Simulation Studies
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\item{47.}Z. Salman and J. Adler, in preparation.
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%change2
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J. Phys. A ,
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\vfill
\eject
\baselineskip=20pt
\noindent{\bf FIGURE CAPTIONS}
\noindent{\bf FIGURE 1:} Graphs of
the magnetization of the
Ising model as a function of temperature
from direct substitution of the series, from
simulations and from the exact expression.
\noindent{\bf FIGURE 2:} Pole-residue (temperature $u_c$-exponent $\beta$)
plots for (a) the 2D Ising model and (b) the Baxter-Wu model, from
the Dlog Pad\'e analysis.
\noindent{\bf FIGURE 3:} Graphs of Pad\'e approximants to the critical exponent
$\beta$ of the magnetization as functions of the trial
correction-to-scaling value for
(a) a 40-term Baxter-Wu series and (b) a 20-term Ising series.
Note the interesctions of the approximants
near $\Delta_1=0.66$ in (a) and $\Delta_1=1$ in ((b),
at the exact values of $\beta$.
\noindent{\bf FIGURE 4:} Diagram of the relationship between some popular
$n-$ vector
and $q-$ state Potts models. Any resemblance to the Physics building
at the Technion is purely imaginary.
\vfill
\eject
\baselineskip 12pt
\settabs 5 \columns
\noindent {\bf TABLE 1.} The first 40 terms of the
Baxter-Wu model magnetization series, $M(u)=
\sum_{i=0}^{40}a(i) u^i.$
\smallskip
\hrule
\smallskip
\+ $i$ &$a(i)$\cr
\smallskip
\hrule
\smallskip
\+ 0 & 1.0e+0
\cr \+ 1 & 0.0e+0
\cr \+ 2 & 0.0e+0
\cr \+ 3 & -2.e+0
\cr \+ 4 & -1.2e+01
\cr \+ 5 & -6.6e+01
\cr \+ 6 & -3.5e+02
\cr \+ 7 & -1.848e+03
\cr \+ 8 & -9.78e+03
\cr \+ 9 & -5.2012e+04
\cr \+ 10 & -2.78118e+05
\cr \+ 11 & -1.495092e+06
\cr \+ 12 & -8.077274e+06
\cr \+ 13 & -4.383648e+07
\cr \+ 14 & -2.38889424e+08
\cr \+ 15 & -1.306708196e+09
\cr \+ 16 & -7.171779996e+09
\cr \+ 17 & -3.948255072e+10
\cr \+ 18 & -2.17967820876e+11
\cr \+ 19 & -1.206373261572e+12
\cr \+ 20 & -6.692352865914e+12
\cr \+ 21 & -3.7204913820578e+13
\cr \+ 22 & -2.0723928199248e+14
\cr \+ 23 & -1.15645382553165e+15
\cr \+ 24 & -6.46413153590575e+15
\cr \+ 25 & -3.618804588707499e+16
\cr \+ 26 & -2.02883110713227064e+17
\cr \+ 27 & -1.138963792045563204e+18
\cr \+ 28 & -6.402045451452033852e+18
\cr \+ 29 & -3.6027785027824272156e+19
\cr \+ 30 & -2.02971327340540689416e+20
\cr \+ 31 & -1.144672634469195243774e+21
\cr \+ 32 & -6.461763298017490096236e+21
\cr \+ 33 & -3.6510670236348960220518e+22
\cr \+ 34 & -2.06474033826802954554852e+23
\cr \+ 35 & -1.168604601894680208737028e+24
\cr \+ 36 & -6.619230155887851818416352e+24
\cr \+ 37 & -3.7520426992289993967183642e+25
\cr \+ 38 & -2.12829788606061196105748508e+26
\cr \+ 39 & -1.208054495766604253678875824e+27
\cr \+ 40 & -6.861456053144684396262799662e+27
\cr
\smallskip
\hrule
\vfill
\eject
\baselineskip=13pt
\def\uncatcodespecials{\def\do##1{\catcode`##1=12 } \dospecials}
\def\setupverbatim{%
\par \tt \spaceskip=0pt % Make sure we get fixed tt spacing
\obeylines\uncatcodespecials\obeyspaces\verbatimdefs}
% This macro turns on verbatim mode until @endverbatim is seen.
\def\verbatim{\begingroup \setupverbatim
\parskip=0pt plus .05\baselineskip \parindent=0pt
\catcode`\ =13 \catcode`\^^M=13 \catcode`\@=0
\verbatimgobble}
{\catcode`\^^M=13{\catcode`\ =13\gdef\verbatimdefs{\def^^M{\
\par}\let =\ }} \gdef\verbatimgobble#1^^M{}}
% This defines @endverbatim to end the group which begins with \verbatim
\let\endverbatim=\endgroup
\noindent{\bf TABLE 2.} Mathematica program for the evaluation of Pad\'e
approximants for the 20 term Baxter-Wu series.
\medskip
\verbatim
t = 1.0 -2.*10^0*K^3 -1.2*10^01*K^4 -6.6*10^01*K^5 -
3.5*10^02*K^6 -1.848*10^03*K^7 -9.78*10^03*K^8 -
5.2012*10^04*K^9 -2.78118*10^05*K^10 -1.495092*10^06*K^11 -
8.077274*10^06*K^12 -4.383648*10^07*K^13 -2.38889424*10^08*K^14 -
1.306708196*10^09*K^15 -7.171779996*10^09*K^16 -
3.948255072*10^10*K^17 -2.17967820876*10^11*K^18 -
1.206373261572*10^12*K^19 -6.692352865914*10^12*K^20
Print[t]
<<"Calculus`Pade`"
order=20;
ft=N[Collect[D[Normal[Series[Log[t],{K,0,order}]],K],K],60];
pp[l_,m_]:=Roots[Denominator[Pade[ft,{K,0,l,m}]]==0,K];
qq[l_,m_]:=Numerator[Pade[ft,{K,0,l,m}]];
rr[l_,m_]:=D[Denominator[Pade[ft,{K,0,l,m}]],K];
(* Definition of a function which finds the minimum real pole in the *)
(* [L,M] Pade approximant: *)
GetPole[l_,m_]:=Module[{temp},temp=pp[l,m];
If[Depth[temp]==2,temp[[2]],
Min[Cases[Table[temp[[n,2]],{n,1,Length[temp]}]
,x_Real/;x>0]]]]
(* Definition of a function which finds the residue corresponding to *)
(* the minimum real pole in the [L,M] Pade approximant: *)
GetResidue[l_, m_] := qq[l, m]/rr[l, m] /. K -> GetPole[l, m]
prlist=Flatten[Table[{GetPole[l,m],GetResidue[l,m]},{l,6,14},{m,6,14}],1]
ListPlot[prlist]
ListPlot[prlist,PlotRange->{{0.17,0.172},Automatic}]
@endverbatim
\vfill
\eject
\settabs 6 \columns
\noindent{\bf TABLE 3.} Comparison of results for different analyses of the
magnetization of the Ising and Baxter-Wu models.
\bigskip
\hrule
\smallskip
\+ method &Ising $\beta$ &Ising $u_c$& BW $\beta$ & BW $u_c$ &BW $\Delta_1$
\cr
\smallskip
\hrule
\bigskip
\+exact &1/8 &0.17157&0.17157&1/12&2/3\cr
\+MC [13]&&&input&input&$0.6\pm0.2$\cr
\+MC [20]&&&&&$<1.0$\cr
\+20 term series\cr\+ (Dlog Pade)
&0.125&input&$0.078(2)
$&input&1 (input)\cr
\+ 20 terms(M2)&$0.1250(5)$
&input&$0.0834(2)$&input&$0.660(6)$\cr
\+ 40 terms(M2)&&&
$0.0833(1)$&$0.17157(10)$
&$0.665(5)$\cr
\smallskip
\hrule
\bigskip
\vfill \eject
\settabs 7 \columns
\noindent{\bf TABLE 4.} Percolation critical exponents.
%seperate file owing to magnification trouble
\vfill
\eject
\settabs 4\columns
\noindent {\bf TABLE 5.} Bond percolation thresholds and Ising
model critical temperatures
for hypercubic lattices.
\bigskip
\hrule
\smallskip
\+ Method &Percolation ($p_c$) &Method& Ising($J/{k_BT_c}$)
\cr
\smallskip
\hrule
\bigskip
D=2
\smallskip
\+ Exact &1/2 &Exact &$(1/2)\ln(1+\sqrt(2))$
\cr
\bigskip
D=3
\smallskip
\+ MC [Ziff]$^*$ &$0.248810\pm0.000005$& MC [42]&$0.2216546\pm 0.0000010$ \cr
\+ MC [68]&$0.24883\pm 0.00005$&MC [43]&$0.2216593\pm0.0000022$ \cr
\+ Series [31] &$0.2488\pm0.0002$& Series [40]&$0.221655\pm 0.000005$ \cr
\bigskip
D=4
\smallskip
\+ MC [66] &$0.16013\pm0.00012$&MC [67]&$0.14970\pm0.00001$ \cr
\+ Series [31] &$0.16005\pm0.00015$&Series [48]&$0.14968\pm0.00003$ \cr
\bigskip
D=5
\smallskip
\+ &&MC [Stauffer, 69]&$0.11390\pm0.00002$\cr
\+ &&MC [70]&$0.113929\pm0.000045$\cr
\+ Series [31] &$0.11819\pm0.00004$&Series [69]&$0.113935\pm0.000015$ \cr
\bigskip
D=6
\smallskip
\+ &&MC [71]&$0.09230\pm 0.00005$\cr
\+ Series [31] &$0.09420\pm0.0001$&Series [71]&$0.092295\pm 0.00003$ \cr
\bigskip
D=7
\smallskip
\+ &&MC [71]&$0.07772\pm0.0003$\cr
\+ Series [31] &$0.078685\pm0.00003$&Series [71] &$0.077706\pm 0.000002$ \cr
\bigskip
D=8
\smallskip
\+ Series [31] &$0.06770\pm0.00005$ \cr
\bigskip
D=9
\smallskip
\+ Series [31] &$0.059500\pm0.00005$ \cr
\bigskip
\hrule
\bigskip
$^*$ unpublished
\end
\end
\end
\bye
\end