Field theory of self-avoiding random chains

Volume 103A, number 1,2

PHYSICS LETTERS

18 June 1984

FIELD THEORY OF SELF-AVOIDING RANDOM CHAINS T. HOFSASS and H. KLEINERT Freie Universitiit Berlin, Institut far Theorie der Elementarteilchen, Arnimallee 14, 1000 Berlin 33, Germany Received 28 November 1983 Revised manuscript received 20 March 1984

We present a new lattice model whose partition function is equal to the sum over all self-avoiding closed random chains of m colors. The fluctuating variables are pure phases similar to an X Y model and, contrary to previous proposals, no awkward n ~ 0 limits are involved. The model can be transformed to a real O(m) invariant field theory, which shows that the critical indices are O (rn) like. There exists a simple relation to O (m) spin models which serves to estimate the critical temperatures.

Self-avoiding random chains play an important role in polymer physics [ 1 ]. It is therefore desirable to possess a simple model which permits a complete study o f the statistical mechanics of such chains. Guided by the knowledge that classical planar spin models are dually equivalent to non-backtracking oriented random chains [2] and that a model involving an n-dimensional spin vector S a o f length S 2 = n contains, in the strong coupling expansion and the limit n ~ 0, all configurations o f a single self-avoiding random chain [3,4], it has been suggested [5] that the partition function m

Z--I-I

(1)

f d f 2 ( x ) eXP(a~=l [3a ~ . S a ( X ) S a ( x + i ) ) ,

X

=

X, I

,

with n

S2 = n -+ O ,

(2)

a=l

should be used to study grand canonical ensembles of self-avoiding polymer chains with m colors. The measure o f integration dg2 covers the surface o f the n-dimensional sphere and is normalized to unity, and the vectors i run over all l q positively oriented next neighbors. The parameters/3 a are the Boltzmann factors e x p ( - e a / T ) = exp(-/3aP°l) where ea is the energy per link o f the polymer chain. Unfortunately, this model does not really fulfill its purpose. When performing a low/3a (i.e. low T) expansion, the partition functions contain contributions of the form • 1 a~ 2 2 2 1 f da(x)f da(x +,) 2 S (X)Sa(X +i)=gm

~a[3 2 .

(3)

a

These correspond to chains running back and forth on the same link, which a self-avoiding chain cannot do. In addition, when allowing for a break-up of chains by adding to the exponent in (1) an external field term Zx, a ha(x)Sa(X ) with a Boltzmann factor h a = exp(--ebr/T), there are terms ~ Z a h 2 which correspond to spurious "zero-link" objects [6,7]. 67

Volume 103A, number 1,2

PHYSICS LETTERS

18 June 1984

The purpose of this note is to remedy such difficulties by setting up a new model which has the additional merit of being much simpler than (1). If {L} denotes all self-avoiding closed random chain configurations we want to calculate, Z = ~

{L)

exp[-

(e/T)l] = ~/3l,

(4)

{L)

where l denotes the total number of link vectors i occupied by the chains. We may assign to each link vector i an occupation number ni(x ) whose value can be zero or one. The property of being self-avoiding means that whenever one looks at all occupation numbers around each site x, the numbers ni have to be either all zero, or two of them can be unity which means that q/2

i=1

ni(x - i) +ni(x )=0

or

2.

This constraint can be written as follows: 7r

l-I ~ x z(x)=0,2

82ini(x-O+ni(x)'z(x) = x[I f-~ d0(x) 2~v z(x)=0,2

=x[I/d0(x) 2n --Tl"

z (x)=0,2

exp(-i

~x O(X)Z(x))exp(i ~ [O(x) + O(x +i)]ni(x)) .

Introducing the complex pure phase variables

(5)

U(x) = e i°(x), this becomes

x['I(/dO(x) (1+ [U*(x)]2})[-I [U(x)U(x+i)] hi(x)

(6)

xd

--/1

Multiplying this with the Boltzmann factor/3 hi(x) and summing over all Z--

ni(x ) gives the

partition function

H(f dO(x) 2--~---{I + [V*(x)] 2})HI1 + /3U(x)U(x+i)] .

(7)

It can easily be checked that an expansion in powers of/3 does indeed reproduce all self-avoiding chains. The factor 1 + (U*) 2 in the measure of integration makes sure that each site is either empty or touched by two occupied links, while the factors U(x)U(x +i)guarantee the connectedness of the chain. Notice that Z is real since it is invariant under the exchange U-+ U*, although it does not have a conventional Boltzmann form. The calculation of the thermodynamic properties is most convenient by transforming Z into a theory of real fields. For this we simply note that

Z = ~(/

du(x) 6(u(x)){1 --~

+ ½[d/du(x)]2}) 1-I [1 +/3u(x)u(x+i)],

coincides with (7), since a (d/du) 2 has the same effect upon u 2, in the integral in the integral f~_~vdO/2n.Alternatively, we can write Z = U (/

du(x) / . _oo

68

(8)

x,i

_i

da(x)~[1 + la2(x)l ~

) exp (, ~a(x)u(X)x

f~_=du 6(u), as ( U * ) 2 has upon

x,iFI[l +/3u(x)u(x+i)]

U2

(9)

Volume 103A, number 1,2

PHYSICS LETTERS

18 June 1984

The model can easily be extended to m colors, by using the constraint m

tin

a=l

G aZit'a(x--i)+na(x)]'za(x)~X z(x)=0,2 ~ a~2aza(x)'z(x)' x za(x)=0,2

(lO)

where the na(x) can take on the values zero or one, corresponding to the link x, x + i being empty or carrying color a. The first Kronecker 6 ensures self-avoidance within each color a [see (5)]. To have avoidance between different colors as well, we have to make sure that whenever a line of color a passes through a site x[z17(x)= 2], no line of a different color can pass through the same site. This is guaranteed by the second Kronecker 8. In order to exhibit the underlying O(m) symmetry, we also introduce the superfluous constraint

11 ~ 8Nan~.(x)zi(x). x,i zi---0,1

(11)

This gives

Z= x,a[-If ~--~

~ ~

--TT

E j • ~d6/(x) x,

--Tr

X H (1 + [V*(x)l 2} [1 [1 + V/*(x)l [ I

x

xi

where V - exp(i6),

Vi -

z=Hf x,a ~d°17(x) ~x

I-I {1+

[Ua(x)]2V2(x)}

--Tf

x,i,a

[1

+~aU17(x)U17(x+i)Vi(x)],

exp 08i). The integrals over V and

(1+

a~ [Ua(x)] 2 ) xH,t. (1 +

Vi can be done and

~a ~aUa(x)Ua(x+i))

.

(12) we arrive at (13)

This is the direct generalization of (7) to m colors. The same generalization takes place in the real field representations (8) and (9) with u and a becoming real O(m) vector fields. This symmetry determines the universality class of the critical indices. There is no problem of allowing for the possibility of break-up of chains.by inserting magnetic fields - Zx,17h17(x)U17(x),such that the complete partition function of self-avoiding random chains with m colors is

Z =~ x,a

? du17 i?

_oo

d%

(1

exp(-

× ~I(l+~aU17(x)u17(x+i). X,I

+h17(x)]

_i~.

(14)

17

It is useful to compare this with the O(m) spin model m

Zo(m)= I-I H fda(x)exp(~m ~ Sa(x)S17(x+i)), 17=1 x ~ x,i,a

(15)

in which case there is a well-known similar field theory

69

Volume 103A, n u m b e r 1,2 oo

ZOm,=g f X exp(- ~

PHYSICS LETTERS

18 June 1984

i,~ daa f duo ~i ~x (½m- 1)!(½[al)-(m/2-1)Im/2_l(la[) [%(x)+ha(x)]Ua(X))~

exp( a~

~aUa(X)Ua(X+i)).

(16)

Here, the expansion of the Bessel factors 2

x[-I[ ( m / 2 - 1 ) ! a ~ l 2 (m/2--1)! ( a ~ l 2 ) + 1 + -(-~-/T)T ~aa + (m--7-Sr+T)'TT! ~%

"

]

(17)

gives rise to all possible multiple occupancies of sites. The case m = 1 reduces to the Ising model Zo(1)~-zIsing=

x~I

f

da ~x

du f _i

_oo

(

ch a exp -

~

~x [o~(x)+h(x)]u(x))11X,! exp[,lU(X)U(x+i)].

(18)

Since the integration of the a variables restricts u to the values -+1, we can write n exp [/3lu(x)u(x X,t

+i)]

= (ch/31)Nq/2

[-I [1 + th/31 u(x)u(x +i)]

.

(19)

X,i

Thus, apart from the trivial factor (ch/3)Nq/2, the partition function of self-avoiding random chains with m = 1 can be obtained from that of the Ising model by identifying th/31 =/3 and performing a perturbation expansion in *l 1 2 AV = log(l + ga ) - log ch a = - AC~4 + ....

(20)

The lowest contribution of AV is suppressed by a Boltzmann factor/38, such that there are practically no corrections close to the critical point. In table 1 we compare the critical values obtained from th/31 with recent Monte Carlo data on random chains by Helfrich's group [8] and see that the agreement is indeed excellent ,2 For general m, there is a similar relation between (14) and (19). Expanding the exponential into Gegenbauer polynomials ~o

exp@ m

~a.Sa(x)Sa(x+ i))

= n=~Odn(/3m)C(nm/2-1)(EaSa(X)Sa('X+i)),

(21)

with P(m/2) do(~3)~_(/3~11m/2-1(/3)

dl(/3)

_ P(m/2)

(/3/2)rn/2-1m

m 21ml2(/3)'

we find the low temperature series

Zo(m)=[do(/3m)]Nql 2~Ifd~2II+ E (-~ lml2(/3m)~ni(X)cm/2-1(EaSa(X)Sa(X+i (22) ))} x,i {ni(x)=l,2....} 2 1 ~ ) ! ni(x) " 4:1 Notice that the partition function of the lsing model would be obtained from the original Ansatz by allowing in the constraint (5), for all multiple occupancies of a site, i.e. by summing Z over 0, 2, 4 . . . . . This replaces in (7) 1 + (U*) 2 by 1 + (U*) 2 + (U*) 4 + ... and in (9) 1 + c~2/2 by 1 + c~2/2 +c~4/4! + ... = ch a. . 2 Apart from their five-dimensional which m u s t be wrong.

70

Volume 103A, number 1,2

PHYSICS LETTERS

18 June 1984

Table 1 Transition temperatures of self-avoiding random chains of rn colors on a s.c. lattice as estimated from those of the O(m) spin " C C model via the relaUon t3C ~ , Im/203m)/Im/2_l(13 m) = th ~31,I1(~)/I003~) , cth 13~ - 1/t3~.... for m = 1, 2, 3. . . . . The third row gives the Monte Carlo data of ref. [8] with dubious results put in parentheses. The last row contains the mean field estimates.

m

q/2

1

~ th ~] 13~c

2

/3~ C e 11 (132)/I0~ 2) 13~c

3

#~ cth t3~ - 1/#~ 13~c

all

~MF

2

3

4

5

...

~1

0.4407 0.4142 0.42

0.2217 0.2181 0.22

0.1499 0.1488 0.15

0.1140 0.1135 (0.14)

... ... ...

1/q 1/q

0.439 0.24

0.298 0.15

0.227 0.11

... ...

2/q

0.694 0.224

0.457 0.150

0.341 0.113

... ... ...

3/q 1/q

0.167

0.125

0.1

...

1/q

0.75 0.35 (0.45)

0.25

For/3 ~ ~c, the diagrams are d o m i n a t e d by produce all self-avoiding loops o f m colors.

ni(x)

= 1 loops. In this case since

Zo(m) ~ [do(flm ) ] Nq/2 ( {~L) mn[L ] [Irn/2(~m)/Im/2-1

C 1 (z) = (m - z)z,

1/q

the dr2 integrals

(~m)] l ) ,

where n [L] is the n u m b e r o f closed loops in the ensemble {L). Corrections arise only to order (Im/2/Im/2_ Thus we may use the critical temperatures o f O ( m ) spin models and estimate ~c f r o m the relation

(23)

1)8.

[3c =Im/2 ( [3c )/lm/2 - 1 (~Cm) " This gives the numbers shown in table 1. Our m o d e l can easily be studied by mean field m e t h o d s . This gwes a critical point at ~MF = 1/q as c o m p a r e d w i t h the O ( m ) value m/q. For large t3, the total l o o p length approaches N , which is in contrast to the O(1) m o d e l where it is Nq/2 since then, multiple occupancies o f each site allow to fill each link. F l u c t u a t i o n corrections to the m e a n field solution will be given elsewhere. The authors are grateful to W. Helfrich for interesting discussions. [1] P.J. Flory, Principles of polymer chemistry (Cornell Univ. Press, Ithaca, NY, 1953); P.G. de Gennes, Scaling concepts in polymer physics (Cornell Univ. Press, Ithaca, NY). [2] T. Banks, R. Myerson and J. Kogut, Nucl. Phys. B129 (1977) 493. [3] P.G. de Gennes, Phys. Lett. 38A (1972) 339; J. De Cloiseaux, J. Phys. (Paris) 36 (1975) 281. [4] R.G. Bowers and A. McKerrel, J. Phys. C6 (1975) 2721. [5] W. Helfrich and F. Rys, J. Phys. A15 (1982) 599; P. Pfeuty and J.C. Wheeler, Phys. Lett. 84A (1981) 493. [6] J.C. Wheeler and P. Pfeuty, Phys. Rev. A24 (1981) 1050. [7] P.D. Gujrati, Chicago preprint (1982). [8] M. Karowski, H.-J. Thun, W. Helfrich and F. Rys, Berlin preprint (1983). 71