Critical Exponents for the n-Vector Model in Three Dimensions from Field Theory

PHYSICAL

V O L U M E 39, N U M B E R 2

REVIEW

LETTERS

11 J U L Y

1977

Critical Exponents for the /i-Vector Model in Three Dimensions from Field Theory J. C. L e Guillou Laboratoire

de Physique

Theorique

et Hautes Energies,

Universite

Paris

VI, 75230 Paris

Cedex 05,

France

and J. Zinn-Justin Service

de Physique

Theorique,

Centre

d* Etudes Nucleaires de Saclay, (Received 14 April 1977)

91190 Gif-sur-Yvette,

France

We present a new calculation of the critical exponents of the η-vector model through field-theoretical methods. The coefficients of the renormalization functions of the (φ ) theory are expanded in powers of the coupling constant. Asymptotic estimates of large order of perturbation series are used to transform the divergent perturbation series into a convergent one. As a consequence, new and more precise values of critical exponents are obtained. 2 2

Initiated by Wilson, the field-theoretical ap­ proach to critical phenomena has been e x t r e m e ­ ly successful in the domain of phase transitions. In particular, it has been possible to calculate physical quantities such as critical exponents through the famous Wilson- Fisher e = 4 - d ex­ pansion. M o r e recently, use has been made of perturbation s e r i e s for the g

2

T A B L E I. Perturbative expansion (Ref. 8) of renor­ malization-group functions of the gUp ) field theory with Oin) symmetry in three dimensions. 2 2

η - 1 W(g)

= - g + g -

3

Y

_ ,

(g)

0.376526828 g

-

1 ~ I

A

- 0.4224965707 g

2

-

g + 27"

0.02245952

•

0.3510695978 g

+ 0.49554751 g

5

g

3

'

2

4

- 0.749689 g

6

7

0.0230696213 g +0.0198868203 g 3

g

5

+ 0.0303679

0.0109739369 g

2

+ 0.0009142223 g *0.0017962229 g

g

4

6

4,5

6

2

N(g)

-

W(g)

0.00065370 g

- -

g + g

3

+ 0.0013878 η « 2

5

- 0.4029629630 g

2

g

6

• 0.3149169420 g

3

4

4

7

Y

_ 1

(g)

0.317928484 g

-

1 - { -

2

N(g)

-

g + 2l g

0.02219865

+ 0.39110247 g

5

g

2

7

~ 0.0259419075 g +0.0205323538 g 3

+ 0.0279829

5

- 0.552448 g

6

g

0.01185l8519g + 0.0009873601 g 2

4

6

+ 0.0018368107 g

3

4

8

-

W(g)

0.00058633 g

= - g + g

+ 0.0012514 η = 3

5

- 0.3832262015 g

2

g

6

+ 0.2829466813 g

3

4

9

The main ingredient that w e shall use in the analysis of the perturbation s e r i e s i s the follow­ ing: It can be shown ' that in the g(

7

A(g)=EKA g\ K

Y~'(g)

N(g)

K

+ 0.31255586 g

5

ΤΤΓ

+

g

G

0.0122436486 g - 0.00050410 g

" °-

0 2 7 6 6 7 3 0 1

2

7

9g *0.0201190591 g 3

g

3

g

4

6

+ 0.0010200001 g «0.0017919258 g

+ 0.0010883

5

- 0.414861 g

6

4

6

η - ρ -

-

(2)

2

·»• 0 . 0 2 4 7 4 9 7

5

(1)

where a and b have been calculated in three di­ mensions for the various renormalization-group 7

8

0.02101293

-

behave f o r large Κ (K-~+<») as K

1 " j2

-

2 2

W(g)

A ~K\{-a) K\

0.270333298 g

γ"'( ) 8

•

2 g

- 0.4398148149 g

0.447316097 g 1 - 1 g + JL

= -

H(g)

g

-

0.02062072

0.00066062 g

5

-

6

1.034928

4

g

7

- 0.0184623402 3+0.0172838882 g

g

4

g

+ 0.0299914

5

0.0092592593 g -

• 0.3899226895 g

+ 0.63385550 g

5

2

g

3

2

g

6

+ 0.0007713750 g +0.0015898706 g 3

• 0.0014103

g

4

6

95 CPSC

7 - paper 61

527

PHYSICAL

V O L U M E 39, N U M B E R 2

functions

etc.].

v(g),

REVIEW

T h e p r e s e n c e of a A !

b e h a v i o r had b e e n anticipated b y B a k e r et a/.,

LETTERS

11 J U L Y 1977

the structure of the n e a r e s t singularity of B(g). Under the assumption that B{g) i s analytic in a

5

w h o used the P a d e - B o r e l method to sum the s e r ­

cut p l a n e , it i s p o s s i b l e to map the cut plane on

i e s and obtain v a l u e s f o r the c r i t i c a l exponents

a c i r c l e and obtain a convergent e x p a n s i o n

for I s i n g - l i k e s y s t e m s (n = 1 ) .

B(g)—and

T h e m o r e complete

information now a v a i l a b l e a l l o w s us to apply a m o r e p r e c i s e method.

t h e r e f o r e A (g)—so

10

for

that

A&^f^e-'B^dt.

Defining the B o r e l t r a n s ­

f o r m of A(g) by

(4)

A s shall b e explained b e l o w , this method can b e

Β®=Σκ(Ακ/ΚΙ)Ζ ,

(3)

Κ

w e can determine f r o m E q . (2) the position and

Δ

further refined with u s e of the known coefficient b of E q . ( 2 ) , |

O u r b e s t e s t i m a t e s f o r the c r i t i c a l

exponents a r e p r e s e n t e d in the following t a b l e :

η

1

2

3

0

g* γ η ν ω

1.414±0.003 1.2402± 0.0009 0.0315± 0.0025 0.6300± 0.0008 0.782± 0.010 0.493± 0.007 0.325± 0.001

1.405±0.002 1.3160± 0.0012 0.0335± 0.0025 0.6693± 0.0010 0.778± 0.008 0.521±0.006 0.346± 0.001

1.391±0.001 1.3866± 0.0012 0.0340± 0.0025 0.7054± 0.0011 0.779± 0.006 0.550±0.005 0.3647± 0.0012

1.4170 ± 0.0045 1.1615± 0.0011 0.0260± 0.0030 0.5880*0.0010 0.790± 0.015 0.465±0.010 0.3020± 0.0013

ων

1 }

β

H e r e , g* i s the fixed-point value of the r e n o r m a l ­ i z e d coupling constant, n o r m a l i z e d in such a w a y that W(g) = -g+g +0(g ), 2

f o r the field and Ζ (g) W(g)

tical exponents w h i c h g o v e r n the b e h a v i o r n e a r the critical t e m p e r a t u r e T

f o r the v e r t e x b y

(4)

and γ and ν a r e the c r i ­

3

= (d-4)[dlng /dg]'\

g =gZ /Z . 0

0

of the magnetic s u s ­

Z

ceptibility χ and of the c o r r e l a t i o n length ξ , r e ­

φ

spectively, such that

a r e d e t e r m i n e d b y the functions

c

( 2 ) 2

b e i n g the r e n o r m a l i z a t i o n constant f o r the i n s e r t i o n , the c r i t i c a l exponents η, ν, and γ

(5)

v(g)

=

The exponent 77 g i v e s the l a r g e - d i s t a n c e b e h a v i o r

v{g)

= [2 + W(g)d lnZ /dg

Y(g)

=

k~\T-T \'\

x~|t-T |-7,

c

c

at T

of the s p i n - s p i n c o r r e l a t i o n function G(*r)

c

~χ ~ ~ , 2

α

η

and ω g o v e r n s the leading c o r r e c t i o n s

(6)

2

{4)

2

W(g)[dlv>Z(g)/dg], (2)

- v(g)],

(7)

v(g)[2-V(g)],

evaluated a t g =g*, i . e . ,

to scaling. O u r r e s u l t s f o r the c r i t i c a l exponents in the c a s e with n = 1 a r e c o m p a t i b l e with, but m o r e

*7 =*?(£·*),

v = v(g*)>

y=r(^*)-

(8)

p r e c i s e than, those obtained in Ref. 5: γ = 1.2410

F i n a l l y , the leading c o r r e c t i o n s to the scal i n g

± 0 . 0 0 2 ; 77 = 0 . 0 2 1 ± 0 . 0 2 ; ν = 0.627± 0.01; u> = 0.78

l a w s a r e g o v e r n e d b y the exponent

± 0 . 0 1 ; and Δ = 0.49± 0.01. Α

O u r results give also

m o r e p r e c i s e v a l u e s than the most r e c e n t h i g h temperature-series estimates

11

and a r e in a g r e e ­

L e t u s denote b y A^Hg),

ment with them, even now f o r the exponent γ a l ­

tions W(g),

though the v a l u e i s definitively l o w e r than the c e n ­

tively.

t r a l value given b y h i g h - t e m p e r a t u r e s e r i e s . O u r r e s u l t s a r e a l s o in v e r y good a g r e e m e n t with the most r e c e n t e x p e r i m e n t a l r e s u l t s

1 2

(see Table II).

(9)

u>=W'(g*).

W'(g),

U)

K

K

w h e r e the coefficients A the other functions.

2

z e r o g* of the r e n o r m a l i z a t i o n - g r o u p function W(g)

of the gfcp (x)] 2

2

field theory, which i s d e ­

fined through the r e n o r m a l i z a t i o n constants Ζ (g) 96 528

CPSC

7 - paper 61

^ have b e e n calculated

u p to Κ = 7 f o r W(g) and u p to Κ = 6 f o r

to d e r i v e these r e s u l t s .

The c r i t i c a l b e h a v i o r of a s e c o n d - o r d e r p h a s e

respec­

(10)

A (g)=B A ^g\ by N i c k e l

transition i s g o v e r n e d b y the i n f r a r e d - s t a b l e

the f u n c ­

andy(^),

T h e i r perturbation expansion i s

W e shall b r i e f l y sketch the method which w e u s e d 1 3

for i = 1-5,

η(&), v(g),

8

It has recently b e e n s h o w n

that the asymptotic b e h a v i o r of A ^ K

l)

7

for large Κ

i s given b y A^~K\(-a) K (i)C [l+0(l/K)], K

b

(i)

(11)

w h e r e the n u m e r i c a l v a l u e s of a and b a r e a s f o l -

PHYSICAL

V O L U M E 39, N U M B E R 2

REVIEW

T A B L E Π. Comparison of our results (first column) with those of Baker et al. (second column; Ref. 5 ) , ex­ perimental data (third column; Ref. 1 2 ) , and high-tempperature-series results (fourth and rightmost column; Ref. 1 1 ) .

LETTERS

B (gt)

11 J U L Y 1977

of the function A (g)

{i)

(i)

/ °°

Ξ

π

1 240±0.007

1.24110.002

1.2402±0.0009

Y

» 1

1 .23

1.24

1 .27

1.28

a

o

by writing

e'H 'B^{U)dt,

(13)

b

where b' will be varied in the neighborhood of b and where U=gt The important point is that the behavior (11) for large Κ gives the location and the nature of the nearest singularity of the Borel transform B (U). This singularity is located at £ / = - l / a , where B (U) behaves as (1 + aU) ' "\ or as l n ( l + all) for b' = b + 1 . Assuming that B (U) is analytic in the i/-plane cut f r o m - 1/a to - o o , we can now map this cut plane into a c i r ­ cle in the χ plane with 0

1

e

250 °· *^ -0.007 +

1

0 0 3

υ

(i)

0.021±0.02

η 0.0315±0.0025

0. 016±0.007 0 016±0.014

ν 0.6300±0.0008

0.627±0.01

0. 625±0.003 0. 625±0.005

0.320±0.016

6 0.325 ±0.001

0 . 321

0.323

(i)

a

°·

a

b

e

U=x(l-iax)' =

328±0.004

n=

c

U x",

(14)

n

1

the integration in (13) running f r o m x = 0 to χ = 4/α. The B o r e l transform is now given in (13) through a convergent s e r i e s in x, which leads to

n = 2

0. 675±0.001

Σ

2

316±0.008

0.

ν 0.6693±0.0010

(i)

f

0.312±0.005

0.329 0.

..Ο+0.002 -0.008

6 3 8

b mb

d

Α \)=Σ„ΰ (

n = 3

η

{ f~e'

Ψ\x(t)]»

dt}.

(15)

c

We have first used this result to calculate the z e r o g* of W(g). We have done this by several methods. One method consists of calculating the z e r o g* of W(g) for different number of terms of the new expansion (15). Because the successive values of g* seemed to converge v e r y smoothly, we extrapolated them with use of Pad6 approximants. Another approach used was to write W{g) as -gF(g) o r -g+g F(g), and then apply our method on the F(g) This latter method seems to converge faster than the f o r m e r one and, without the use of any Pad§ extrapolation, yields precise values of g* which are consistent with those ob­ tained from the other methods. In each case, we have varied b' in a neighborhood of b in order to generate a function B(U) with a weak singularity

f

[(l + aU) ,

1

375 °· -0.01 +

g

0 2

, , J / ; >

Υ

1.3866±0.0012

η

0.0340±0.0025

1.405±0.02

h

i

1 42 ° · ^ -0.01 +

1 ,

ν 0.7054±0.001 1

2

0.043±0.0l4

g

0.040±0.008

h

0 U

0

Ζ

'

7025 °· -0.005 +

g

0 1 0

/ U O

2

a

b

c

d

0.717±0.007

h

0.725±0.015

1

0

Chang et al., Ref. 12. Hocken and Moldover, Ref. 12. G r e e r , Ref. 12. Mueller et al., Ref. 1 2 ; Grey wall and Ahlers, Ref.

12.

C a m p et al., Ref. 1 1 . Domb, Ref. 1 1 . ^Ritchie and Fischer, Ref. 1 1 . F e r e r et al. Ref. 1 1 . ^Camp and Van Dyke, Ref. 1 1 .

h

1/2

t

« = 0.147 774 2 2 . . . ,

b

(i)

w(g);

= j 3 + i w f o r W{g)Mg)Mg)\ ( 2 + i w for

(1 + aU)' ] l/2

at U= - 1/a.

{i)

lows:

{ 5 + i n for

ln(l+fl£/),

This has given us a range of values for g*, for which we have calculated the other quantities A (g), again with application of the same meth­ ods as for Wig).

(12)

T){g).

We introduce the generalized B o r e l transform

We have made other checks like calculating ex­ ponents f r o m the direct s e r i e s and its inverse, and verified the scaling laws 2/9 = 3i/ - y and γ = i/(2 - η) between the independently computed e x ­ ponents. A l l the results that we have obtained are consistent with one another. Finally, several advantages of our approach with respect to the Pad6-Borel method are to be emphasized. F i r s t , it gives more stable and a c 97 CPSC

7 - paper 61

529

P H Y S I C A L

V O L U M E 39, N U M B E R 2

R E V I E W

curate v a l u e s , as w e have v e r i f i e d , f o r instance,

diate coupling, the a c c u r a c y i s , f o r s i x t e r m s of the p e r t u r b a t i o n s e r i e s , of the o r d e r of 1 0 * f o r

11 J U L Y 1977

don and B r e a c h , N e w Y o r k , to b e p u b l i s h e d ) .

on the calculation of the g r o u n d - s t a t e e n e r g y of the anharmonic o s c i l l a t o r w h e r e , f o r an i n t e r m e ­

L E T T E R S

G. A. Baker,

5

Meiron, G

B. G. Nickel,

M . S. G r e e n ,

and P . I.

P h y s . R e v . L e t t . 36, 1351 ( 1 9 7 6 ) .

L . N. Lipatov,

P i s m a Z h . E k s p . T e o r , F i z . 2 5 , 116

(1977) [ J E T P L e t t , (to b e p u b l i s h e d ) ] , L e n i n g r a d I n s t i ­

2

the Pade" method, 1 0 " f o r the P a d 6 - B o r e l m e t h ­ 3

od,

and 3 x l 0 "

4

tute of N u c l e a r P h y s i c s R e p o r t s N o . 253 and N o . 255, 1976 (to b e p u b l i s h e d ) .

f o r our method. Second, to i m ­

p r o v e efficiently the rate of c o n v e r g e n c e of a s e r ­

7

i e s by m e a n s of the P a d e * - B o r e l method, the p e r ­ This

condition is not c r u c i a l in o u r a p p r o a c h .

This is

p a r t i c u l a r l y important in the calculation of c r i t i ­ cal exponents f r o m φ field theory in t h r e e d i ­ 4

5

and v (g)=W(g)(d/dg)ln[z (g)], 2

only u s e d 1/γ(&) whose series a l ­

{2)

8

B . G . N i c k e l , to b e p u b l i s h e d .

Our Table I gives

numbers used h e r e .

turbation s e r i e s has to alternate in s i g n .

mensions, w h e r e B a k e r et al.

E . B r e z i n , J . C . L e G u i l l o u , and J . Z i n n - J u s t i n ,

P h y s . R e v . D 15, 1544, 1558 ( 1 9 7 7 ) .

ternate in sign, the other c r i t i c a l exponents being

9

P . G . de G e n n e s ,

Cloizeaux,

P h y s . L e t t . 4 4 A , 271 ( 1 9 7 3 ) ; J . d e s

P h y s . R e v . A 10, 1665 ( 1 9 7 4 ) , and J . P h y s .

( P a r i s ) 36, 281 ( 1 9 7 5 ) . Rep. 1 0

S e e a l s o D . S. M c K e n z i e ,

J. J. Loeffel,

C e n t r e d ' E t u d e s N u c l e a i r e s de S a c l a y

Report No. S A C L A Y - D P h - T / 7 6 - 2 0 ! 1

Phys.

2 7 C , 35 ( 1 9 7 6 ) . (unpublished).

W . J. C a m p , D. M . Saul, J. P. Van Dyke,

and

obtained by s c a l i n g r e l a t i o n s . With o u r method,

M. Wortis,

w e could calculate a l l exponents independently.

and J . P . V a n D y k e , J . P h y s . A 9 , 731 ( 1 9 7 6 ) ; C . D o m b ,

W e a r e e x t r e m e l y grateful to G. A . B a k e r , B . G. N i c k e l , and D. I. M e i r o n f o r communicating to us,

p r i o r to publication, the T a y l o r s e r i e s listed

in T a b l e I.

W e e s p e c i a l l y thank B . G. N i c k e l f o r

c o r r e s p o n d e n c e concerning these n u m b e r s .

P h y s . R e v . Β 14, 3990 ( 1 9 7 6 ) ; W . J . C a m p

in Phase Transitions

1 9 7 4 ) , V o l . 3; M . F e r e r , Fisher, l 2

E . B r e z i n , J . C . L e G u i l l o u , and J . Z i n n - J u s t i n ,

Phase Transitions

P h y s . R e v . Β 4 , 3184 ( 1 9 7 1 ) .

and Critical

Phenomena,

C . D o m b and M . S. G r e e n ( A c a d e m i c ,

edited b y

New York,

1976),

Vol. VI. 3

K . G . W i l s o n and Μ . E . F i s h e r , G. Parisi,

P h y s . R e v . Lett. 28,

P h y s . R e v . L e t t . 3 7 , 29 ( 1 9 7 5 ) ; S. C .

P h y s . R e v . A 14, 1770 ( 1 9 7 6 ) ; Κ. H . M u e l l e r , P h y s . R e v . L e t t . 34, 513

( 1 9 7 5 ) ; D . S. G r e y w a l l and G . A h l e r s ,

P h y s . R e v . A 7,

2145 ( 1 9 7 3 ) . 1 3

A p p l y i n g the s a m e m e t h o d f o r d = 2 and η = 1 ( I s i n g

η = 0.19±0.07.

y = 0 . 9 8 ± 0.07, and ω = 1.1 ± 0 . 3 .

1

In t h i s

c a s e the p e r t u r b a t i o n s e r i e s h a s o n l y b e e n c a l c u l a t e d in P r o c e e d i n g s of the 1973 C a r g e s e

m e r S c h o o l , edited b y E . B r e z i n and J . M . C h a r a p

98 530

E.

m o d e l ) w e h a v e found g* = 1.85 ± 0.07, y = 1.79 ± 0 . 0 7

240 ( 1 9 7 2 ) . 4

M . A . M o o r e , and M . W o r t i s ,

P h y s . R e v . L e t t . 37, 1481 ( 1 9 7 6 ) ; R . H o c k e n and

F . P o b e l l , and G . A h l e r s , in

edited York,

R . F . C h a n g , H . B u r s t y n , J . V . S e n g e r s , and A . J .

Greer, K . G. Wilson,

r

New

P h y s . R e v . Β 5, 2668 ( 1 9 7 2 ) .

M . R. Moldover, !

Phenomena

P h y s . R e v . Β 4 , 3954 ( 1 9 7 1 ) ; D . S . R i t c h i e and Μ .

Bray,

2

and Critical

b y C . D o m b and M . S. G r e e n ( A c a d e m i c ,

CPSC

7 - paper 61

Sum­ (Gor­

( R e f . 8) up to o r d e r 5 f o r W(g) other functions.

and to o r d e r 4 f o r the