SOLVING ~b4,2 FIELD THEORY WITH MONTE ... - Science Direct

Nuclear Physics B210[FS6] (1982) 210-228 (~) North-Holland Publishing Company

SOLVING ~b4,2 FIELD T H E O R Y WITH M O N T E C A R L O Fred COOPER, B. FREEDMAN ~ and Dean PRESTON Theoretical Division, Los Alamos National Laboratory, University of California, Los Alamos, New Mexico 87545, USA

Received 19 February 1982 We study lattice got~4 field theory for all go and fixed renormalized mass M in one and two dimensions using Monte Carlo techniques. We calculate the dimensionless renormalized coupling constant g a = gR/M4-d, where d is the dimension of space-time, at fixed small values of the lattice spacing a for various go and lattice sizes. Our results are in quantitiative agreement with the analyses of high temperature and strong coupling series which rely on extrapolation from large to small lattice spacing.

1. Introduction R e c e n t l y go¢~4 field t h e o r y has b e e n s t u d i e d in the r e g i m e of i n t e r m e d i a t e a n d large go a n a l y t i c a l l y using t h e p a t h i n t e g r a l r e p r e s e n t a t i o n of t h e g e n e r a t i n g functional of t h e G r e e n functions. T h e two m e t h o d s u s e d h a v e b e e n t h e high t e m p e r a t u r e series a p p r o a c h of B a k e r a n d K i n c a i d [1] a n d t h e r e l a t e d s t r o n g c o u p l i n g series of B e n d e r , C o o p e r , G u r a l n i k , R o s k i e s a n d S h a r p [3, 4]. E a c h of t h e s e m e t h o d s is b a s e d on r e p l a c i n g t h e c o n t i n u u m by a e u c l i d e a n lattice with s p a c i n g a a n d e x p a n d i n g in t h e k i n e t i c energy. In t h e s e m e t h o d s , o n e o b t a i n s series for d i m e n s i o n less s c a t t e r i n g a m p l i t u d e s such as ~g = g R / M 4-d in t h e d i m e n s i o n l e s s c o r r e l a t i o n l e n g t h ~2 = 1 / M 2 a 2 for fixed v a l u e s of goa 4-d (high t e m p e r a t u r e series) or M 4 a a / g o (strong c o u p l i n g series), gR is t h e r e n o r m a l i z e d f o u r - p o i n t c o u p l i n g c o n s t a n t a n d M the r e n o r m a l i z e d m a s s p a r a m e t e r at q 2 = 0. T h e m a j o r u n c e r t a i n t y in t h e s e c a l c u l a t i o n s is t h e e x t r a p o l a t i o n of-the c o r r e l a t i o n l e n g t h series f r o m small to large c2 (a ~ 0). This is a c c o m p l i s h e d by using v a r i a t i o n s of P a d 6 a p p r o x i m a n t t e c h n i q u e s which a r e d i s c u s s e d in d e t a i l in refs. [3, 4]. B e c a u s e of the c o n t r o v e r s i a l conclusions a r r i v e d at by the a b o v e a u t h o r s : p o s s i b l e b r e a k d o w n of h y p e r s c a l i n g in t h r e e d i m e n s i o n s * , the triviality of g04~4 field t h e o r y in d = 4, it is i m p o r t a n t to verify t h e conclusions of t h e s e a u t h o r s using t e c h n i q u e s which d o n o t rely on e x t r a p o l a t i o n in the c o r r e l a t i o n length. T h e M o n t e C a r l o m e t h o d [6], which consists of d i r e c t l y e v a l u a t i n g the lattice p a t h i n t e g r a l for go, m20 a n d a r b i t r a r y lattice s p a c i n g a, d o e s n o t rely on an e x t r a p o l a t i o n t e c h n i q u e of u n k n o w n accuracy. T h e price o n e p a y s in M o n t e C a r l o is t h e r e s t r i c t i o n to a crystal Permanent address: Physics Department, Indiana University Bloomington, Indiana 47405. * The result of Baker et al. that to > 0 for d = 3 has been contested in ref. [2]. 210

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of finite volume. The finite volume effects are minimized by imposing periodic boundary conditions. In principle, Monte Carlo can be used to study both sides of a critical point [7]*. This is especially useful for models with several distinct phases, g o ~ 4 field theory exhibits two phases, a "disordered" phase in which the vacuum expectation value of the field vanishes and an " o r d e r e d " phase which is characterized by spontaneous magnetization. In this p a p e r we use Monte Carlo to study the "disordered" phase of l a t t i c e go~b4 field theory in one and two dimensions. After implementing a mass renormalization, gR is computed as a function of go and the lattice spacing. In sect. 2 we discuss lattice go~b4 field theory, its relationship to the Ising model, and the order parameters we measure by Monte Carlo. Sect. 3 is devoted to a discussion of the Monte Carlo, high temperature and strong coupling approaches for studying g0~b 4 theory. In sect. 4 we describe some of the details of our numerical studies. The gaussian (go = 0) and Ising (go ~ oo) limits provide useful checks on the continuous-spin algorithm used for all go. These limits are discussed in sect. 5. In sect. 6 we present our results for gR(go, a) and briefly summarize our work in sect. 7.

2. The model

The continuum field theory we want to study is defined by the euclidean action

SE = f ddx[~Ogq)O~ -~rnoQ '1 2 _ 2 +~..O , g o _ 4 ] 1.

(2.1)

In order to define the path integral it is necessary to introduce a d-dimensional lattice. We choose a hypercubic lattice with periodic boundary conditions. At each site in the lattice there is a real field tb(i) which takes values from R, - o o ~a .

(4.2)

In a statistical mechanical model eq. (4.2) is replaced by the condition that the number of spins within a correlated block of size ~5d must be large, while the size of a correlated block is much smaller than V. In practice we used Monte Carlo on lattices with Ns~ a ' and M is held constant. Working in finite volume* (L = N a ) , eq. (5.12) is replaced by w = I n (g~/gR)/ln ( N / N ' ) ,

N'>N>

1.

(5.13)

It has been rigorously proved that the "anomalous" ~o is positive semidefinite [14]. m/> 0.

(5.14)

Eq. (5.14) taken as an equality is a "hyperscaling relation." In the limit ~o ~ oo and a ~ 0 the renormalized coupling tends to an asymptotic value denoted by g*. This value is reached in the Ising model as/3 -~/3c. If w > 0 then g* = 0 and hyperscaling fails; the hyperscaling value of o) = 0 implies that g * > 0. Using Monte Carlo we verified that eq. (5.5) holds in the limit ff0~oo. The p a r a m e t e r / x e was found by measuring the asymptotic value of the ratio - m ~ / ~ o keeping M constant. A Monte Carlo simulation of the Ising model was then performed giving us the values for/~ and gR(/3). In all cases, the measured values for/3 and/x 2 satisfied (5.6) within the statistical fluctuations of our measurements. In addition, gR(flc) and the value for gR (go = oO, a) calculated with the continuous spin algorithm for large N were found to be in close agreement (> 1 is shown. The Ising nature of this configuration (small dispersion in spin norm) is quite apparent.

'go : 4 5 xl

Fig. 2. A

sample spin configuration on a 5 x 5 lattice for go = 45 >> 1.

F. Cooper et al. / ~ ~,2 field theory with Monte Carlo

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d=g I

L

I

I

0.8

0.6

/N

0.4

0.2

0.

I

o.

0.2

.

I

I

I

o.,

0.6

0.8

'V(,o+Ll Fig. 3. Average spin n o r m (Is I) versus go in two dimensions, (Is I) ~"/31/2 in the limit go -~ a3,

Fig 3 is a graph of the average spin norm versus ~o. In the limit ~o-+ oo we see that (Isl)~[-m2a2/4~o]a/2=fl 1/2. Fig. 4 shows the dispersion in Isl as a function of go. The dispersion vanishes as ~0 ~ oo. All these results are consistent with the picture that the spins take on discrete values in the limit ~o-~ oo.

6. Results In this section we plot gR for various values of go and N. The choice M = 2 sufficiently reduces finite volume effects on our lattice of length L = 3. In previous sections we have described the method by which ~R is computed. The accuracy of our measurements depended on (1) the size of statistical fluctuations and (2) solving the mass constraint. In a real Monte Carlo experiment these two sources of error are not independent. If we denote the measured mass by M M o then [ M M c - M [ can differ from zero because of random fluctuations and our choice for m 2. Fortunately, ~R was found to be insensitive to small variations in m 2. To estimate the error in gR we performed several independent measurements: gR(1), gR(2) . . . . . gR(M) •

(6.1)

224

F. Cooper et al. / ~ ~,2 field theory with Monte Carlo d:2 I

1

I

I

I

0.2

0.4

0.6

0.6

0.8

0.6 /x,

v 0.4

0.2

O.

O.

o,C,O+ ol F i g . 4. The dispersion in the spin norm as a function of go. The spin is clearly discrete in the limit go ~ oo.

Starting from different initial configurations, each measurement was made with the same values for go, a and mo2. The estimated value for ga was found by averaging: 1

M

~ R = ~ - E= gR(1") •

(6.2)

The relative error, 6ga, was calculated from the variance among the {gR(/)}:

where

(6.3) 1

M

M - 1 j=ZlI R(J)- gRI2 The relative error in ga was largest in regions where ga became small; for example when go approached zero. In the region where the spin density changes from gaussian-like to Ising-like, errors in ga were sufficiently small (roughly 5 %) to make an accurate determination of ga possible. The 5% error in our data is not indicatec with error bars in figs. 5-8.

F. Cooper et al. / 4~.2 [ieM theory with Monte Carlo

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d=l 8

I

1

I

I

I

I

I

I

I

7 6 N=20

!

4

5

I

2 0.0

I 0.2

I

I 0.4

I

I 0.6

I

I 0.8

I

1.0

g o / ( g o + I00) Fig. 5. ~a as a function of go for N = 5, 10 and 20 in one dimension for M = 2.

d = 1. In fig. 5 we plot g a as function of go for N = 5, 10 and 20. The renormalized mass was chosen to be M = 2 giving M V lid = 6. The data show that ~r~ (go --- oo) decreases in the c o n t i n u u m limit t o w a r d the k n o w n value 6.0. T h e m o n o t o n i c i t y of ~R(go) is evident in fig. 5. Fig. 6 is a c o m p a r i s o n of our M o n t e Carlo data for M = 2 and N = 20 (~2 = 11.11) with the short strong coupling series for gR given

d=l I

I

I

I

1

I

I

I

1

1

I

1

1

6

QR t3

4

2

0.0

!

0.2

1

0.4 0.6 go / (%+ I001

0.8

1,0

Fig. 6. Comparison of Monte Carlo data for M = 2, d = 1 and N = 20 with the strong coupling series for gR of ref. [5].

226

F. Cooper et al. / 4~,2 field theory with Monte Carlo d:2

24

I

I

I

I

I

I

I

l

I

l

I 0.2

I

I 0.4

I

[ 0.6

L

l 0.8

I

22

20 18 gA 16 14 12 IO

0.0

1.0

g o / [ ~ o + I00) Fig. 7. ~R as a function of go for N = 4, 8 and 16 in two dimensions for M = 2.

in ref. [5]. A g r e e m e n t is excellent when go is large (go>~ 100). As expected, the strong coupling series shows large deviations from the more reliable Monte Carlo results when go lies outside the strong coupling regime. Comparison of gR at the same go, N for M 2 = 2, 4 revealed that gR was invariably larger for M 2 = 4. For example, with go = 5000, N = 20 we found gR increased from 5.41 at M 2 = 2 to 5.91 at M 2 = 4. The approach of gR at large go toward the limiting value of 6.0 shows that the effect of finite lattice volume disappears as the C o m p t o n wavelength of the particle decreases ( M increases). d = 2 . Fig. 7 shows gR as a function of go for N = 4 , 8 and 16 with M = 2 . In the strong coupling regime, gR ~ 15 for N = 8, 16 in good agreement with the strong coupling [5] (14.95) and high t e m p e r a t u r e [1] (14.5 + 0.2) results in the continuum limit. As for d = 1, gR increases monotonically with increasing go. In fig. 8 we compare our M o n t e Carlo results with the strong coupling series in ref. [5]. Table 1 shows the disappearance of finite volume effects on a 16 x 16 lattice as the renormalized mass is increased. In both one and two dimensions we found that gR tended to a finite nonzero limit as go, ~ 2~oo. This observation verifies the validity of hyperscaling for d = l , 2.

7. Summary In this paper we have shown how to apply Monte Carlo methods to the stud~ of a sclar field theory. A mass renormalization eliminated the bare mass in favo

F. Cooper et aL / ~ 4,2 field theory with Monte Carlo

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d=2 18

I

I

I

I

I

I

16--

14--

~R 12 o

°

I0

q 0

8

I

I

0.0

0.2

0.4

0.6

I 0.8

I

1.0

go/(go+lO0) Fig. 8. Comparison of Monte Carlo data for M = 2, d = 2 and N = 16 with the strong coupling series for gR of ref. [5].

of the physical mass. Using the continuous-spin algorithm we were able to measure the two- and four-point Green functions of the theory. Working always in finite volume, the continuum limit of the field theory was recovered by increasing the number of lattice sites. go~b4 theory exhibited a nonzero scattering amplitude in one and two dimensions. Asymptotic values for the renormalized coupling constant obtained by Monte Carlo were in good agreement with high temperature and strong coupling results. For the lattices we studied, no evidence of hyperscaling violation was seen in our simulations. A question left unanswered is whether the same behavior describes the ordered phase of go~b4 field theory. TABLE

1

~a(go = oo) on a 162 lattice for four values of the renormalized mass

1/M 2

gr~ (go = oo)

0.68 0.50 0.40 0.25

10.8 12.7 13.8 15.0

The proximity of ~R = 15.0 to the continuum results of the high temperature (14.5+0.2) and strong coupling (14.95) series shows that finite volume effects are negligible for M = 2.

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In principle, M o n t e C a r l o s i m u l a t i o n s include all possible field configurations. If a p a r t i c u l a r e x c i t a t i o n of t h e v a c u u m is physically i m p o r t a n t , t h e n it s h o u l d be s e e n u p o n carefu l e x a m i n a t i o n of the s i m u l a t e d configurations. T h u s we saw the p r e s e n c e of Ising-like spin c o n f i g u r a t i o n s in the s t r o n g c o u p l i n g r eg i o n . O n e a d v a n t a g e of t h e M o n t e C a r l o m e t h o d as a p p l i e d to g0~b4 is the a l m o s t trivial g e n e r a l i z a t i o n to m o r e c o m p l i c a t e d field t h e o r i e s 115]* such as t h e ab el i an H ig g s m o d e l or an N - c o m p o n e n t scalar field t h e o r y with an i n t e r n a l s y m m e t r y . U s i n g M o n t e C a r l o we can m e a s u r e the l o w - l y i n g masses, t h e s c a t t e r i n g a m p l i t u d e s , and t h e p h a s e s t r u c t u r e of t h e s e t h e o r i e s .

References [1] G.A. Baker, Jr., and J. Kincaid, J. Stat. Phys. 24 (1981) 469; Phys. Rev. Lett. 42 (1979) 1431 [2] B. Nickel, Carg~se Lectures, 1980, to be published; R. Roskies, in preparation [3] C.M. Bender, F. Cooper, G.S. Guralnik and D.H. Sharp, Phys. Rev. D19 (1979) 1865 [4] C.M. Bender, F. Cooper, G.S. Guralnik, R. Roskies and D.H. Sharp, Phys. Rev. D23 (1981) 2976 [5] G.A. Baker, L.P. Benofy, F. Cooper and D. Preston, Analysis of the lattice strong coupling series for got~4 field theory in d dimensions, Los Ala~mospreprint (1981), Nucl. Phys. B210[FS6] (1982), to appear [6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J, Chem. Phys. 21 (1953) 1087 [7] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20 (1979) 1915 [8] K. Binder, in Phase transitions and critical phenomena, ed. C. Domb and M.S. Grren, vol. 56 (Academic Press, New York, 1976) [9] B. Freedman, P. Smolensky and D.H. Weingarten, A Monte Carlo study of~b 4 field theory, Indiana University preprint IUHET-68 [10] M. Creutz and B. Freedman, Ann. of Phys. 132 (1981) 427 [11] M.E. Fisher, Rep. Prog. Phys, 30 (1976) 615; H. Wilson and J. Kogut, Phys. Reports 12 (1974) 75 [12] (a) B. Lautrup and M. Nauenberg, CERN preprint, TH-2873-CERN, (May, 1980); (b) G. Bhanot and B. Freedman, Nucl. Phys. B190 [FS3] (1981) 357 [13] C. Domb, in Phase transitions and critical phenomena, ed. C. Domb and M.S. Green, vol. 3 (Academic Press, New York, 1976) [14] R. Schrader, Phys. Rev. B14 (1976) 172; G.A. Baker and S. Krinsky, J. Math. Phys. 18 (1977) 590 [15] M. Creutz, Phys. Rev. D21 (1980) 1006; G. Bhanot and B. Freedman, BNL preprint (May, 1980) [16] S. Shinker and J. Tobochnik, M.S.C. report no. 4243 (April, 1980)

* A Monte Carlo renormalization group analysis of the d = 2 Heisenberg model was made in ref. [16].