CHINESE JOURNAL OF PHYSICS
VOL. 15, NO. 3
AUTUMN, 1977
On the Scaling Factor in the Renormalization Group
R
INSE
R
ESEARCH
G
R O U P*
P. 0. Box l-6. Tamsui, Taiwan 251
(Received August 27, 1977)
The importance of the role of the scaling factor in the renormalization group
transformation is discussed. The variational calculations on 2-dimensional Ising
model by Kadanoff et al is used as an example to display the idea. Numerical
results of Kadanoff et al. are found to be in good agreement with our picture.
The internal contradiction mentioned by them is not a contradiction according to
our picture.
-__ __
__-
I. INTRODUCTION
m ECENTLY, the renormalization group has been well developed in solving the critical phenomena.
AxThe main concepts of the renormalization group had been interpreted clearly by K. G. Wilsoncl)
and many author&. But they often ignored the role of the scaling factor. In fact, the physical
interpretation of the scaling factor can help the understanding ,of the renormalization group.
In this paper we will describe the role of the scaling factor in the renormalization group. In
order to show our idea clearly we use the variational approximate method constructed by Kadanoff
et al.@) as an example. They built a variational scheme to carry out the renormalization group
transformation process for 2-dimensional Ising model. Their results are in good agreement with the
exact solution. But, on the other hand, they thought there exists an internal contradiction in their
procedure of calculation. In our opinion, there is no contradiction at all. We will explain this point
in terms of the scaling idea.
II. THE RENORMALIZATION CROUP AND THE SCALING FACTOR
The purpose of the renormalization group approach is to understand the observed large correlation length produced primarily by some short-range interaction. For example, the Ising model with
nearest neighbor coupling exists a critical temperature where the correlation length goes to infinite.
If we average the Hamiltonian over a local region with size larger than the original coupling range,
the coupling range of the resultant Hamiltonian is larger than or of the same order as linear dimension that local region, which is presumably larger than the original interaction range. The successive
average processes will result in a cascade effect in whole system and display how the correlation
* Research in Natural Sciences and Engineering.
~~:6~~~~~Jll~~~~~~~~:~~~~~~
(1) K.G. Wilson, Phys. Rev. B4, 3174 (1971), B4, 3184 (1971).
(2) K. G. Wilson and J. Kogut, Physics Reports 12C, 76 (1974); K. G. Wilson, Rev. of Mod. Phys. 47, 1,
i3)
(1975); Shang-Keng Ma, Rev. of Mod. Phys. 45, 589, (1973).
L. P. Kadanoff, A Houghton, and M. C. Yulabik, Journal of Statistical Physics, 14, (1976).
236
RINSE RESEARCH GROUP
237
propagates. In order to realize renormalization group approach to the determination of the singular
behavior of a physical system near its critical point, it requires the construction of a renormalization
group transformation appropriate to the problem, and the location of a fixed point of the transformation related to the original Hamiltonian of the physics system. In usual formulation of renormalization groupc1y2), the transformation must be introduced with a parameter and its value must be
properlf chosen in order to find an interesting fixed point for the transformation. Why a chosen
parameter is necessary for finding the fixed point? If the Hamiltonian is not at the fixed point, how
is the parameter chosen?
To answer the above questions we begin with brief description of the methodology of renormalization group. For conveniently, wk speak in terms of a specific physical system, a system of
interacting spins u on a lattice which.undergoes a ferromagnetic transition at some critical temperature.
The transformation operator R transforms the old Hamiltonian of the system, H(o) (assume including
the factor l/K,Z’), into a new Hamiltonian W(p),
fl(~)=RV(u))
(1)
where H(D) represents the effective interactions between blocks of old spins u, which is treated as
single new spins ,u. For maintaining the same thermodynamic behavior the transformation must insure
the invariant of free energy, i.e.,
F- -In x e-@(P) _ -In x CRC*)
(2)
(Pu)
t(r)
where the free energy F includes the factor l/K,T, and the sum x sums over all configurations of
{XI
all lattice spin variables x on the lattice.
Now, following Kadanoff et al., the transformation can be constructed as follows:
H’(P) = R {H(u)) = -In {F exp( --Ei(,u, a>>
D
(3)
ti(,U, u) = H(u) - ZQf, 6) -U?(U)
(4)
with
Here T is arbitrary, but U is defined by the relation
U#) - -In IF1 exp +(A u)
(5)
It is easy to prove that the transformation (3) satisfies Eq. (2). For the new Hamiltonian H’(p), to
have the similar symmetry properties as that of H(u), one must do two things. The first is to embed
the new lattice ‘spins ,o in proper position relative to the old u ones. One example of such a pair
of lattice is shown in Fig. 1. The second is T has to be chosen to satisfy the symmetry relation
f’(Gj(Pu)* Gj(u>)=p(Pv
0)
for all i
where Gj describe all possible lattice transilation and rotation operators on the p-lattice.
one can choose
5$.f, U) = z In (
I+ pp;un(,) )
(6)
For example,
(7)
for the lattice in Fig. la, where p .is the scaling parameter; the function n(m) gives the label
n of the spin u, in terms of the label m of the new spin p,,, to which u corresponds.
Notice that athough the old spins u and new spins p are all taken values f 1, ‘an average of an
arbitrary function f(u) from the Hamiltonian H(u) will generally be different from the average of the
corresponding function fJ,o) fro-m the Hamiltonian H’(p). Since H(o) and H’(p) are Hamiltonians
describing the same system, u and y sohuld describe the spin variable with different scale. Furthermore, the transformation from u to ,u is somewhat arbitary. Different choice of transformations
result to different F(,o)‘s. Obviously, the average of a function f,(,o) from different H’(,u)‘s will be
238
ON THE SCALING FACTOR IN THE RENORMALIZATION GROUP
L
x:
J
.
x
.
.
.
.
.
l
I
*
.
x’
%
I(
.
,
(lh
.
(Ia)
=(r and X=,U (la) the spins ,U system have the same lattice
Fig. 1. The pairs of lattices.
form as 0 system with the lattice constant /r a and (lb) 2a.
l
different. The difference should be compensated by the different scaling (of y to a) which is determined by that particular transformation. The arbitrariness makes the scaling not very useful unless
there is some restriction. The point will be more clear if we look at the critical point. +4t this
particular temperature we can find a renormalization group transformation such that the Hamiltonian
is unchanged under this transformation, i.e.,
H*(P)-R{H*(~)
.(g)
which is called the “fixed point”. This is a rather rigid restriction. Since the Hamiltonian is exactly
the same as before. In this case the scaling factor will show how the spin-spin correlation falls
down. Let us take the 2-dimensional lattice shown in Fig. la as an example. Since H’(U)--H(U) at
critical point we have
<cI~OIu~~O> =<%%)>
(9)
If the-scaling of ,u to u is cz, then
~e<%YJCdTr)> =<Q%>
(10)
where a! is the scaling factor. Because
<~UW>= 1 P
hence &‘=(/ 2-)“.
(::)
More specifically, for 2-dimensional Ising model n-f, the scaling factor is then
a2=21/8
(12)
OI
This calculation shows that you have to scale the spin variable 2t/le every times to make the Ha
miltonian “f ix”. The transformation does not automatical “fix” at the critical poin. They are fixed
only if you properly scaled the spin variable. We can look at this fact at different point of view.
At the critical point the spin-spin correlation is not constant, it falls off at a certain power. Hence,
the average spin values over larger region is smaller than the over smaller region. In order to let
Hamiltonian look exact the same, the spins should be properly scaled.
In the case TfT,, the interpretation of the scaling factor is not so clear due to undefiniteness
in the form of Hamiltonian. However, if there is some resonable restriction on the form of Hamiltonian the transformation is no more arbitrary. Since the interaction strength of new Hamiltonian is
RINSE RESEARCH GROUP
239
not the same as that of old one, the scaling factor can not exactly describe the behavior of the spinspin correlation. But in some sense it is the proper scale of the spin variable at that stage, i.e.,
at that characteristic basic length, for that form of Hamiltonian to describle the system. As T#T,,
after infinite times transformations the Hamiltonian should go to another kinds of “fixed” point. For
T>T,, at sufficient large characteristic length the coupling will be very weak and eventually goes to
zero. Naturally, the proper scale of the new spin variable relative to the old one will be zero. also.
For T<T,, there exists a finite magnetization. Hence the proper spin scale after sufficient large size
wiJ1 be in such a way that the magnetization remains the same magnitude. This corresponds to the
coupling strength goes to infinite and scaling factor goes to 1: In next section we will display this
picture by KadanofYs variational calculation on 2-dimensional Ising model.
III. CALCULZTION O F T H E S C A L I N G F A C T O R
How can we know the scaling factor for a particurar transformation?
The scaling factor is related to the scaling parameter introduced into the transformation. For
example, when ? is chosen by Eq. (7), one can derive
(13)
Here the scaling factor is just the scaling parameter p, which must be chosen 2’/18 to get the “fixed”
Hamiltonian at the critical point. But for some other ?‘(,Gu), such as that given by Nicmeizer and
Leeuwen@ and Kadanoff et a l .($I the relationship between the scaling factor and the scaling parameter is not explicit. The calculation of the spin-spin correlation for old and .new spins will automaticatty give us the scaling factor. Unfortunately, the. correlation is often not known before the
whole problem is solved. However, an external probe m&y help us to overcome this difficulty.
Consider the system probed by an external field h. The field couples the spin variables with
the form hb. When u-lattice is transformed to p-lattice, the field coupling energy term becomes
h’p, where ~5’ can be interpreted as new field strength. If there is no scaling between u and p, h’
should equal to zh where z is the number of spin a unit block. Since p is scaled, the strength
is the scaling factor.
coupled to the external field is also reduced. Accordingly, K
zh k-.0
One complexity occurres. There might be more than one terms coupled to the externsl field.
Although in original Hamiltonian, the spin is often assumed to couple magnetic field with the form
hu, after transformation there apper other terms. For example, in Kadanoff’s variational calculation
on 2-dimensional Ising model the term krUiUjUl will appear. Without external field there are only
ktUiUi and krUiUjU,Ul terms in Hamiltonian. The field will introduce terms klui and k,UiUjUhe
Hence, kl and k, play the role of external field. Now, the scaling is actually a matrix instread of
a pure number.
(14)
where the prime represents the new strength after the transformation. The zero field limit is defined
to avoid the disturbance of the system by the field.
Often the largest eigenvalue play the role of the scaling factor. It is particularly true at critical
point. Since after infinite numbers of same transformations, the ‘matrix will become a constant number
an where a! is the largest eigenvalue and n is the times of transformation.
(4) Th. Niemeijer and J.M. J. Van Leeuwen, Physica 71, 17 (1974).
240
ON THE SCALING FACTOR IN THE RENORMALIZATION GROUP
Now, we will show that the variational calculation on 2-dimensional Ising model of KadanolPs
et al. is coincident to our picture. We begin with a brief description of their method. They wrote
the potential in terms of the sum of basic blocks (4 spins). In that case, it is easy to prove that
the potential in a block can be written as a function of the total spin s~=u~+u~+u~+u,. Kadonoff
et al. construct a transformation relates potential V(m,) for p-lattice (rnI=ul +ue+u8+u,) and V(s,)
for u-lattice.
‘&%) (Cos h p)z(tanh @_4,,
exp CWM- q S,(Z)
with
and
z&) = In [2 cash (ps,)]
where S.(s,) and W(s,) are defined as follows:
S,=l
&=Z,Z-2,...,
-z
S,+,(S,)=[S,S,(s,)-(z-n+1)~,-,(s,)l/(n+1)
\
w(s,)=z!/[(z-s,)/2~![(z+~,)/21!
!
(16)
The potential can be written in a general form
(17)
Here the free energy does not invariant under this transformation. However, it can be proved that
for the transformation of eq. (15) with any choice of the free energy calculated from v’(m,) will have
lower value than original one. The whole problem then solved in a variational way by Kadanoff et
al.. For our propose, we are going to calculate the scaling factor for any arbitrary parameter P.
From Eq. (15) we can derive the relation ’
(18)
with
and A. has been defined in Eq. (15). At nontrivial tixed point in 2-dimentsional Ising model P0.766 and K,=O.1392, K,=-0.006865, we have
with z=4. The scaling factor LX is then
1.003 -0.104
$ =largest eigenvalue of
0.153
0.632
(
2: 1/2”3
At ke=k,=O and P=O it is easy to obtain
(20)
241
RINSE RESEARCH GROUP
a=0
(21)
and at k,=k,=oo and P=;co
A,=
(
2
2
2
2
1
Therefore, we obtain a: = 1.
IV. COMMENT ON THE VARIATIONAL APPROXIMATION
t
Although the scaling idea is true for all case, our discussions base entirely on the Kadanoff’s
variational approximation. There are two reasons. First, the variational approximation sounds promising. The method looks general enough for the various models. We intend to apply it to various
problems. Second, the process af the calculation of this method seems to be a very nice display of
the scaling idea. Besides, the internal constradiction mentioned by Kadanoff et al. is not a contradiction from our point of view.
Since the variational parameter P of the transformation is related to the scaling factor of the
transformation. A choice of P is a choice of the scaling factor. Hence the base value P gives a
proper scale of the new spin. Since the new spin actually stand for average variable over a larger
region, the scaling factor indicates how the new spin behaves at every new stage. As we explained
in the content, for T>T, the scaling factor will gradually goes to zero stage by stage. This corresponds to the parameter P approach 0. No matter how close to T, P should end up with 0 for
sufficient large scale. On the other hand, for T<T,, there is a finite magnetization, the spin can
not be scaled down without an end. The scaling factor should go to 1 for sufficient large scale.
This corresponds to p+co. The numerical results of Kadanoff et al. proved this picture. Hence, the
variation of P in every stage is a nature way.
Then, why do the eigenvalues at the fixed point (at which P is fixed) describes the critical exponent so accurate? It is not surprising for the critical exponent that describes the behavior exact
‘at critical point (for example, the power of spin-spin correlation at critical tenperature). What we
should discuss is the critical behavior with a small deviation from T,: For example, we concern
with the correlation length E as a function of (T-TT,). In this case, the critical behavior is governed
by the transformation of interaction strength i. e., the transformation for k, and k, to k; and k:. The
important quantity here is the eigenvalue of matrix
(23)
This value indicates how the deviation of interaction strength from this “fixed point” glows. It is
definitely not right in the limit of infinite numbers of transformations, since the strength would go
to infinite instead of zero (for T<T,). However, when the transformation stage is well inside the
correlation length this description is eligible. The critical exponent actually does not require the whole
range of transformation properties. For simplicity, let us consider a certain amount of the interaction
strength as a standard (k,). For different temperature T, and Te the numbers of transformations will
be different for their effective strength to reach this value k,. But after they reach k,, they have
same behavior for suceed transformation. Hence, the ratio of basic length to have same behavior for
systems with T, and Te can be obtained from the numbers of stages before they reach k,. This ratio
is just how correlation length E goes with temperature.
Thus, there are two ways to get the information of critical exponent. We can solve the complete statistical problem and get the partition function or free energy. The critical behavior is then
automatically obtained. Or we can look at the transformation scaling near the tied point. In the
former case, we have to take care of the whole behavior of the system. In the latter one, we notice
242
ON THE SCALING FACTOR IN THE RENORMALIZATION GROUP
only the scaling near fixed point and extract out the critical behavior. There are not contradiction
between these two.
There is one more comment. To fix the parameter P at every stage does not imply the same
scaling of the interaction strength. The matrix
(24)
is not the same for the same P and different k ’ s . Hence, keeping the parameter p fixed does not even
mean taking the tied scaling for approximation.
.f