Nota Científica 24/79
A DIRECT RENORMALIZATION GROUP APPROACH FOR THE EXCLUDED VOLUME PROBLEM
S. L. A. de Queiroz and C. M. Chaves
DEPARTAMENTO ÜE FÍSICA Outubro 1979
A DIRECT RENORMALIZATION GROUP APPROACH
FOR THE EXCLUDED VOLUME PROBLEM*
S . L . A. d e Q u e i r o z and C . M. Chav.es
Departamento de F í s i c a , P o n t i f í c i a Universidade Católica Cx.P. 38071, Rio de Janeiro, RJ, B r a s i l
O c t o b e r 1 9 7 9
l
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:. rr'-F'' '
ABSTRACT. We—propose 9 direct renormalization group approach/to the excluded volume problem in a square lattice by considering percolating self-avoiding paths in a b x b cell, where b = 2,3. Two ways of counting these paths are presented. In the first one, we get the exponent v * » 0.715 for b • 2 and v • 0.719 for b • 3, whereas in the second one v • 0.771 for b • 2 and v *- 0.748 for b - 3. Comments are made on the extrapolation to b + ».
RESUMO. Um formalism© usando o grupo de renormalização no espaço real e proposto para o problema do volume excluído nuia rede quadrada. Considera-t e caminhos que percolam numa célula b x b, o*ide b • 2,3, e que se; am
também caminhos aleatórios sem repetição. Duas maneiras de contagem são apresentadas. Pa primeira obtém-se v - 0.715 para b » 2 e v • 0.719 para b • 3, enquanto que na segunda maneiva v * 0.771 para b - 2 e v • 0.748 para b • 3. Comenta-se a extrapolação desses resultados para b •* ».
2
Let R denote a site on a lattice and let M ($) be the number of self-avoiding random walks (SAW'S) of n steps beginning at the origin and finishing at site R. The mean square end-to-end distance <Rn 2 > of an n-step walk is given
1.2 by '
< R 2> =
I R2 M .($)
ft
n 11I
Mn<*>
a
nIf we introduce a parameter P, such that to any n-step SAW corresponds a weight Pn, the mean square end-to-end
2
distance is given by
+I R2 Pn M (R)
ç 2(p) =R d 2 (!) ín P M«( S )
R,n n
where now the average is computed over all possible SAW'S. When P approaches a critical value P from below, Ç(P)
—v diverges as ( P ~ P )
The excluded volume problem is related to statistics
1 3 **
of polymers ' , and has also been shown to be connected with the n •* 0 limit of the n-component spin model. Here we propose a position-space renormalization group approach to this problem, which allows us to calculate directly the exponent v and P . We make use of the relation between percolation and SAW in a lattice5. An alternative approach has been given recently by Shapiro2.
Consider the cell in a square lattice shown in Fig. 1, for which the scaling factor is b = 2. We count all the vertical (or horizontal, which is equivalent) percolating
paths6 through the cell, starting at the origin-taken as
point 1 in Fig. 1 - and ending at points labelled 3 or 6. Paths which are not SAW'S are not counted and a path with n steps has a weight p . Thus the renormalized weight p* is given by
p» = p2 + 2p3 + ph (2)
since we have one two-step walk(123), two three-step walks (1456 and 1256) and one four-step walk(14523) . The recursion relation (2) has two fixed points, the trivial one at p* = 0 and p = 0.4656. At this point A = |^- = 2.636 and v = ~ 7 =
*c r 3p £n>
0.715, in excellent agreement with other results2'7*8.
Por a b = 3 cell the same procedure -jives
p» = 4p9 + 2p8 + 9p7 + 5p6 + 9p5 + 3P4 + p3 (3)
from which we get p = 0.4468 and v = 0.719. c
We can also count all the vertical and horizontal SAW'S; clearly, this has the effect of doubling the coefficients of the recursion relations of the first case. We then get
p» . 2p2 + 4p3 + 2P1* (b = 2) (4)
and analogously for b - 3.
Our results, for both procedures, are displayed in table 1. The fact that the values of p„ in column I (the
c
first definition), are greater than those obtained in II was to be expected, since the definition of an allowed SAW in the former scheme is more restrictive than in the latter one. Although the second procedure seems logically more justified
4
b = 3, for example, the two val>*3s obtained for the exponent v differ by less than 4%. Besides, the two series seem to be converging approximately to the same value when b increases.
We have extrapolated6 our results to b •> «, by plotting v against 1/b2 (see table 1). The two series
extrapolate practically to the same value, within less than 1%.
Of course such extrapolation may be criticized because it is not certain that we are in an asymptotical region neither it is clear that an extrapolation of the results against 1/b2 in a SAW problem can be supported by the same arguments as in a pure percolation problem6. Nevertheless, the fact that the values obtained are very close to each other and with the values quoted in the literature2»7/8 indicates that our definition and our method allows the calculation of the
REFERENCES
* Work supported by Brazilian agencies FINEP and CNPq. 1. D. S. McKenzie, Polymers and Scaling, Physics Reports
22, 35 (1976).
2. B. Shapiro, J. Phys. Cll, 2829 (1978).
2. T. C. Lubensky, C. Dasgupta and C. M. Chaves, J. Phys. All, 2219 (1978).
4. P. G. de Gennes, Phys. Lett. 38A, 339 (1972). 5. H. E. Stanley, J. Phys. Al£, L211 (1977).
6. P. M. C. de Oliveira, S. L. A. de Queiroz, R. Riera and C. M. Chaves, submitted to J. Phys. A.
7. H. J. Hilhorst, Phy-s. Lett. 56A, 153 (1976). 8. H. J. Hilhorst, Phys. Rev. B16, 1253 (1977).
TABLE CAPTION
Table 1 - Values obtained for p . X and v for b = 2 and b = 3, c
for both definitions of percolating paths. Extrapolated values of p and v are also given.
Pc( b « 2 ) •
Pc( b « 3 )
Pc<b"~>
A ( b - 2 ) X(b-3) v(b=2) v(b»3) v(b-**»)
I
0 . 4 6 5 6
0 . 4 4 6 8
0 . 4 2 8
2 . 6 3 6 4 . 6 1 2 0 . 7 1 5 0 . 7 1 9 0 . 7 2 3
I I
0 . 2 9 7 1
0 . 3 6 6 0
0 . 4 2 2
2 . 4 5 7 4 . 3 4 4 0 . 7 7 1 0 . 7 4 8 0 . 7 3 0
FIGURE CAPTION
Figure 1 - A b x b cell for a square lattice. Here b = 2. The origin is taken at point 1.
