Using a renormalization group method, we calculate 800 high-temperature
coefficients of the magnetic susceptibility of the hierarchical Ising model.
The conventional quantities obtained from differences of ratios of coefficients
show unexpected smooth oscillations with a period growing logarithmically and
can be fitted assuming corrections to the scaling laws with complex exponents.Comment: 10 pages, Latex , uses revtex. 2 figures not included (hard copies
  available on request

A. J. Guttmann

A. Zamolodchikov

B. Nickel

D. S. Gaunt

E. Lerner

F. Dyson

G. Baker

G. Bhanot

G. Ordaz

G. Parisi

H. Knops

H. Koch

K. Wilson

P. Bleher

P. Collet

P. Hazenfratz

Th. Niemeijer

V. G. J. Rodgers

Y. Meurice

English

arXiv

Using a renormalization group method, we calculate 800 high-temperature coefficients of the magnetic susceptibility of the hierarchical Ising model. The conventional quantities obtained from differences of ratios of coefficients show unexpected smooth oscillations with a period growing logarithmically and can be fitted assuming corrections to the scaling laws with complex exponents

Meurice, Y

Ordaz, G

Rodgers, V G J

CERN Document Server

U. of Iowa preprint
Evidence for Complex Subleading Exponents from the
High-Temperature Expansion of the Hierarchical Ising Model
Y. Meurice, G. Ordaz and V. G. J. Rodgers
Department of Physics and Astronomy
University of Iowa, Iowa City, Iowa 52246, USA
Abstract
Using a renormalization group method, we calculate 800 high-temperature
coecients of the magnetic susceptibility of the hierarchical Ising model. The
conventional quantities obtained from dierences of ratios of coecients show
unexpected smooth oscillations with a period growing logarithmically and can
be tted assuming corrections to the scaling laws with complex exponents.
1
The renormalization group method [1] has enhanced considerably our understanding of
elementary processes and critical phenomena. In particular, it has allowed the computation
of the critical exponents of lattice models in various dimensions. On the other hand, the
critical exponents can be estimated from the analysis of high-temperature series [2]. Showing
that the two methods give precisely the same answers is a challenging problem [3]. More
generally, much could be gained if we could combine these two approaches, in particular, in
the context of lattice gauge theories.
As far as the numerical values of the critical exponents are concerned, there are two
diculties. The rst one [4] is that one needs much longer high-temperature series than the
ones available [5] (which do not go beyond order 25 in most of the cases) in order to make
precise estimates. The second is that the practical implementation of the renormalization
group usually requires projections into a manageable subset of parameters characterizing
the interactions. It is nevertheless possible to design lattice models [6], called hierarchical
models, which can be seen as approximate version of nearest neighbor models, for which such
projections are unnecessary. For the hierarchical models, the renormalization group transfor-
mation reduces to a recursion formula which is a simple integral equation involving only the
local measure. This simplicity allows one to control rigorously [7] the renormalization group
transformation and to obtain accurate estimates of the critical exponents [8]. As explained
in the next paragraph, the recursion formula also allows to calculate the high-temperature
expansion to very high order. Consequently, the hierarchical model is well suited to study
the questions addressed above.
In this letter, we report the results of a large scale calculation which performs the high-
temperature expansion of the magnetic susceptibility of the hierarchical Ising model up to
order 800. The method of calculation uses directly the renormalization group transformation
with a rescaling of the spin variable appropriate to the study of the high-temperature xed
point. This method has been presented in Ref. [9] and checked using results obtained with
conventional graphical methods [10]. For the sake of briefness, we shall follow exactly the
set-up and the notations of Refs. [9,11] where the basic facts concerning the hierarchical
2
model were reviewed. The hierarchical models considered here have 2
n
sites, and the free
parameter c which controls the strength of the interactions is set equal to 2
1 2=D
in order
to approximate a nearest neighbor model in D-dimensions. We dene the nite volume
magnetic susceptibility as

n
() = 1 + b
1;n
 + b
2;n

2
+ ::: (1)
In most of the calculations presented below, we have used n=100, which corresponds to a
number of sites larger than 10
30
. From a mathematical point of view, the calculation of 
n
amounts to repeating n times the Fourier transform of the recursion formula [9]
R
n+1
(k) = C
n+1
exp( 
1
2
(
c
2
)
n+1
@
2
@k
2
)(R
n
(
k
p
2
))
2
; (2)
expanded to the desired order in , with the initial condition R
0
= cos(k). The constant C
n+1
can be adjusted in such a way that the Taylor expansion ofR
n+1
(k) reads 1 (1=2)k
2

n+1
+:::.
This calculation has been implemented with a C program and ran for 6 weeks on a DEC-
alpha 3000 in order to obtain 800 coecients for D = 3.
For the discussion which follows, it is crucial to estimate precisely the errors made in
the calculation of the coecients. There are two sources of errors: the numerical round-os
and the nite number of sites. We claim that with 2
100
sites and 3  D  4, the nite
volume eects are several order of magnitude smaller than the round-o errors. From Eq.
(2), one sees that the leading volume dependence will decay like (c=2)
n
. This observation
can be substantiated by using exact results at nite volume [10] for low order coecients,
or by displaying the values of higher order coecients at successive iterations as in Figure
1 of Ref. [9]. In both cases, we observed that the (c=2)
n
law worked remarkably well. For
the main calculation presented below, we have used c = 2
1
3
(i.e. D = 3) and n = 100,
which gives volume eects of the order of 10
 20
. On the other hand, the round-o errors are
expected to grow like the square root of the number of arithmetical operations. In Ref. [9],
we have estimated that this number was approximately n:m
2
for a calculation up to order m
in the high-temperature expansion with 2
n
sites. Putting this together, we estimated that
3
for n = 100, the error on them-th coecient will be of order m10
 16
. We have veried this
approximate law by calculating the coecients using a rescaled temperature and undoing
this rescaling after the calculation. We have chosen the rescaling factor to be 0.8482 and
the rescaled critical temperature is then approximately 1. This prevents the appearance
of small numbers in the calculation. If all the calculations could be performed exactly,
we would obtain the same results as with the original method. However, for calculations
with nite precision, the two calculations have independent round-o errors. Comparing
the results obtained with the two methods for the coecients up to order 200 shows that
the numerical uctuations of b
m
grow approximately like m  10
 16
. More conservatively,
we can say that the numerical errors are bounded by m 10
 15
. We conclude that for the
calculations reported below, the errors on the coecients are dominated by the numerical
round-os and we estimate that they do not exceed 10
 12
.
In order to estimate , we used standard methods described in Refs. [2,4]. For the sake
of deniteness, we recall a few denitions. First, we dene r
m
= b
m
=b
m 1
, the ratio of two
successive coecients. We then dene the normalized slope S
m
and the extrapolated slope
b
S
m
as
S
m
=  m(m  1)(r
m
  r
m 1
)=(mr
m
  (m  1)r
m 1
) ;
(3)
b
S
m
= mS
m
  (m  1)S
m 1
:
The extrapolated slope, which is free of order n
 1
corrections [4], is displayed in Fig.1 for
m  200. For comparison, we have also displayed the results for D = 3:5 and 4. A surprising
feature is the clear appearance of large oscillations for D = 3. When D is increased, the
amplitude of these oscillations diminishes. They are still present at D = 4 and can be seen
better by plotting
b
S
m+1
 
b
S
m
. One important point of this letter is to establish that these
oscillations are not due to the errors discussed above. As a consequence of the multiplications
by m appearing in the denition of the the extrapolated slope, the errors are amplied by a
factor which can be as large as 10
5
for m near 100 and 10
7
for m near 500. However, even
4
when multiplied by such a large factors our most conservative estimate of the numerical
errors gives errors on the extrapolated slope which are several order of magnitude smaller
than the amplitude of the oscillations. We have made independent checks of this statement
for D = 3 by calculating directly the extrapolated slope for n =100 and 200 and by using an
intermediate temperature rescaling as explained above. The smoothness of the oscillations
appears clearly in Fig. 2, where
b
S
m
is displayed for 50  m  800. This smoothness rules
out large numerical uctuations. In conclusion, we have established that the oscillations in
the extrapolated slope are a genuine feature of the model considered. Fig. 2 also shows
that the extrema are not equally spaced. Instead, the location of one extremum can be
approximately found by multiplying the location of the previous extremum by 1.19. In
other words, the extrema of Fig. 2 would look equally spaced if the abscissa variable had
been ln(m) instead of m. This of course suggests the use of a complex exponents since
Re(m
i
) = cos(ln(m)).
In the conventional description [12] of the renormalization group ow near a xed point
with only one eigenvalue 
1
> 1, one expects that the magnetic susceptibility can be ex-
pressed as
 = (
c
  )
 
(A
0
+A
1
(
c
  )

+ ::::) ; (4)
with  = jln(
2
)j=ln(
1
) and 
2
being the largest of the remaining eigenvalues. It is usually
assumed that these eigenvalues are real. This implies [4] that
b
S
m
=    1 +Bm
 
+O(m
 2
) : (5)
If  is real, there is no room for the oscillations in this description. Nevertheless, the
fact that the period of oscillation increases logarithmically with m suggests that one could
modify slightly Eq. (5) by allowing B and  to be complex and selecting the real part of
the modied expression. This introduces two new parameters and we have chosen to use
the following modied parametrization of the extrapolated slope:
b
S
m
=    1 +Km
 
cos(2
ln(m=m
0
)
ln()
) +O(m
 2
) : (6)
5
This parametric expression allows good quality ts for m large enough. For instance, a
least square t for the m  300 data, yields  = 1:412, m
0
= 512,  = 0:67,  = 1.310
and K = 2:53. The t is displayed on Fig. 2. More accurate results could presumably
be obtained if we had a consistent description of the oscillations involving denite relations
among the parameters of Eq. (6). The value of  is in good agreement with the result [7,8]
obtained with the -expansion, namely 1.300. The value of  is not far from jln(
2
)j=ln(
1
)
which is approximately 0.46 according to Ref. [7,8].
We have considered two possible explanations of the oscillatory behavior. The rst one
is that one could replace 
2
by a couple of complex conjugated eigenvalues. This possibility
is not realized in any perturbative calculation we know. For instance, the gaussian spectrum
is real and its largest eigenvalues are widely separated. One could imagine that when D
is continuously evolved from 4 to 3 for the hierarchical model, two real eigenvalues merge
into each other and subsequently evolve as complex conjugate of each other. However,
the fact that the oscillations persists at D = 4, as indicated in Fig. 1, goes against such
an explanation. Conformal theories in two dimensions provide examples of calculations of
the renormalization group eigenvalues as the eigenvalues of the matrix of derivative of the
beta functions. For low order calculations, this matrix is symmetric [13] which implies real
eigenvalues. A more attractive possibility, is that the susceptibility of the hierarchical model
satises a renormalization group equation of the type discussed in section II of Ref. [14]. This
equation allows corrections to the scaling law of the type (1+(A
1
(
c
 )
i2
ln()
+c:c)+::), in the
simplied case where only one eigenvalue  is considered. Considering the values obtained
with the t we see that  = 1:412 is close to the the value of the largest eigenvalue [7,8]

1
=1.427. This result suggests that one should try to nd an equation for the susceptibility
of the hierarchical model related to the one discussed in Ref. [14].
In conclusion, we have shown that a calculational method of the high-temperature ex-
pansion based on the renormalization group method can be a very powerful tool when the
hierarchical approximation is used. It would be worth trying to improve this method beyond
this approximation. Our analysis of the magnetic susceptibility has shown that unexpected
6
oscillations appear in the extrapolated slope. A detailed understanding of these oscillations
is required in order to allow a precise comparison between the results obtained from the
high-temperature expansion and the -expansion .
One of us (Y.M.), stayed at the Aspen Center for Physics during the last stage of this
work and beneted from stimulating conversations with the participants, especially with N.
Warner.
7
REFERENCES
[1] K. Wilson, Phys. Rev. B. 4 3185 (1971) ; K. Wilson and J. Kogut, Phys. Rep. 12 75
(1974).
[2] D. S. Gaunt and A. J. Guttmann, in Phase Transitions and Critical Phenomena vol. 3,
C. Domb and M. S. Green, eds., (Academic Press, New York, 1974) ; A. J. Guttmann,
in Phase Transitions and Critical Phenomena vol. 13, C. Domb and Lebowitz, eds.,
(Academic Press, New York, 1989).
[3] There is a large amount of literature on this subject. References can be found, e.g.,
in Phase Transitions, Cargese 1980, M. Levy, J.C. Le Guillou and J. Zinn-Justin eds.,
(Plenum Press, New York, 1982).
[4] B. Nickel, Lecture Notes published in Ref. [3].
[5] For recent calculations see e.g: G. Bhanot, M. Creutz, U. Glasser and K. Schilling,
Phys. Rev. B 49 , 12909 (1994) .
[6] F. Dyson, Comm. Math. Phys. 12, 91 (1969) ; G. Baker, Phys. Rev. B5, 2622 (1972).
[7] P. Bleher and Y. Sinai, Comm. Math. Phys. 45, 247 (1975) ; P. Collet and J. P.
Eckmann, Comm. Math. Phys. 55, 67 (1977) and Lecture Notes in Physics 74 (1978) ;
H. Koch and P. Wittwer, Comm. Math. Phys. 106 495 (1986) , 138 (1991) 537 , 164
(1994) 627 .
[8] P. Collet, J.-P. Eckmann, and B. Hirsbrunner, Phys. Lett. 71B, 385 (1977).
[9] Y. Meurice and G. Ordaz, J. Stat. Phys. 80 (in press), hep-lat 9409021.
[10] Y. Meurice, J. Math. Phys. 36 1812 (1995).
[11] Y. Meurice, G. Ordaz and V. G. J. Rodgers, J. Stat. Phys. 77, 607 (1994).
[12] G. Parisi, Statistical Field Theory (Addison Wesley, New-York, 1988).
8
[13] A. Zamolodchikov, Sov. J. Nucl. Phys. 46, 1090 (1987).
[14] Th. Niemeijer and J. van Leeuwen, in Phase Transitions and Critical Phenomena, vol.
6, C. Domb and M. S. Green, eds., (Academic Press, New York, 1976).
9
Figure Captions
Fig. 1: The extrapolated slope
b
S
m
for m  200 and D = 3, 3.5 and 4.
Fig.2: The dots are the extrapolated slope
b
S
m
for 50  m  800 and D = 3. The continuous
curve is the t described in the text.
10


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