Comparison of the simplicial method with Wilson's lattice gauge theory for U(1)3

Volume 168B, number 4 PHYSICS LETTERS 13 March 1986

C O M P A R I S O N OF T H E SIMPLICIAL M E T H O D W I T H W I L S O N ' S LAqIq'ICE GAUGE T H E O R Y FOR U ( | ) 3 :"

Kevin C A H I L L and Randolph R E E D E R ~

Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA

Received 20 December 1985

In the simplicial version of lattice gauge theory, euclidean path integrals are approximated by tiling spacetime with simplexes and by linearly interpolating the fields throughout each simplex from their values at the vertices. This method is compared with Wilson's lattice gauge theory for U(I) in three dimensions. As a standard of comparison, the exact values of Creutz ratios of Wilson loops in the cont inuum theory are computed. Monte Carlo computat ions using the simplicial method give Creutz ratios within a few percent of the exact values for reasonably sized loops at /3 = 1, 2, and 10. Similar computations usingWilson's method give ratios that typically differ from the exact values by factors of two or more for 1 ~</3 ~< 3.5 and that have the wrong /3 dependence. The better accuracy of the simplicial method is due to its use of the action and domain of integration of the exact theory, unaltered apart from the granularity of the simplicial lattice. Data on the action density and the mass gap are also presented.

Creutz and others [1,2] have shown Wilson's lattice gauge theory [3] to be a practical way to study gauge theories nonperturbatively. However, the action and domain of integration of Wilson's method resemble those of the exact theory only when the group dements are near the identity, i.e. only for weak coupling. In a theory with a running coupling constant, one may try to reduce the resultant errors by working at weak coupling. But in SU(3) the physical size of the lattice spacing shrinks with the coupling g like exp[-1/(2b0g2)] with b 0 = 11/(16rr 2) [4]. If one reduces the coupling, then one must compute on a larger lattice in order to encompass the same physical phenomenon. A reduc- tion o fg from 1.0 to 0.5 requires an increase of a 104 lattice to one of size (1010) 4 , which is inconveniently large.

In an earlier paper [5], we proposed and tested a method for approximating euclidean path integrals at arbitrary coupling. In this method one tiles spacetime with simplexes and linearly interpolates the fields

~ Research supported by the US Department of Energy under grant DE-FG04-84ER40166.

1 Present address: AT-2, MS-H821, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

throughout each simplex from their values at the vertices. The fields are defined throughout spacetime. The method uses the action and domain of integration of the exact theory, unaltered apart from the granularity of the simplicial lattice.

Here we describe an application of this simpiicial method to U(1) in three dimensions. By using a heat- bath Monte Carlo [6], we computed the mean value of the action density, the (vanishing) mass gap, and the Creutz ratios of various quartets of Wilson loops. We derived an exact formula for the Creutz ratios x(L,J) of U(1)3 and used it as a standard to compare the accuracy of the simpiicial method with that of Wilson's lattice gauge theory. We made this comparison at three values of the dimensionless inverse temperature/3 = 1 lag 2, where a is the lattice spacing. For loops that have a spatial extent La of between 4 and 7 lattice spac ings and a temporal extent Ja of between 2 and 8 spac- ings, the simplicial method on a 162-by-64 lattice gave Creutz ratios x(L,J) within 20% of the exact values for strong coupling/3 = 1, intermediate coupling 13 = 2, and weak coupling/3 = 10. For these loops and these /3's, the average error of our Creutz ratios is 5.9%. In contrast, Wilson's lattice gauge theory, to judge from results reported by Bhanot and Creutz [7] and by

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Ambj~brn, Hey, and Otto [8], gives Creutz ratios that typically differ from the exact values by factors of two or more for 1 ~</3 ~< 3.5 and that have the wrong/3 dependence. But according to the weak-coupling expansion of Miiller and Riahl [9], Wilson's method on an infinite lattice should display acceptable errors at ~ = 1 0 .

For U(1)3 at intermediate and strong coupling, Wilson's method is less accurate than the simplicial method because it does not use the action or the domain of integration of the continuum theory. The basic variables of the Wilson method are the group elements U n = exp(igaA n a~,~)which for U(1) are simply the phases U n = exp(igaAn) where A n is the gauge field. The Wilson action and domain of integration resemble those of the continuum theory when the dimensionless angles IgaA nal or IgaA n I are much less than unity, which is true only for weak coupling. At stronger coupling, the extra terms in Wilson's action and the to- pology and curvature of his domain of integration be- come important.

In the simplicial method, spacetime is filled with a cubic lattice each cube of which is tiled with six (te- trahedral) simplexes. Each spacetirne point x lying in a simplex with vertices v i can be uniquely expressed in the form x = ~,iPioi in which the four nonnegative weights Pi sum to one. We use this formula to linearly interpolate the fieldAn(X) at the point x from its values A(n, vi) at the vertices vi: An(x ) = ZiP.r4(n, oi). Since the interpolated fields are defmed throughout spacetime, we use the euclidean action of the continuum theory,

where Q(A) is obtained from Q(A) by replacing the operator A n (x) with the interpolated field A n (x).

Because we used the temporal gauge and a 162-by - 64 lattice, with 16 384 vertices, our action is a quartic polynomial in 32 768 variables, the A(n,i ,],k). How- ever since each field variable A(n,i ,] ,k) influences the action in only 24 simplexes, it is coupled to only 30 of these 32 768 variables. We used the fact that the action is quadratic to write a heat-bath algorithm [6] that only requires knowledge of the first and second deri- vatives of the action with respect to each field variable. The algorithm constructs the parabola that describes the dependence of the action on each A, goes to the minimum of the parabola, and then adds noise at the inverse temperature 13.

The most accessible of the physical quantities we measured is the mean value of the action per cube, (S> e. The action is a diagonalizable quadratic form, S = [3Xicoiq 2 , and its mean value per cube can be computed exactly: (S) c = (N - NO)/2N c where N i s the number of A, N O is the number of q that have vanishing ~o, i.e. the number of zero modes, and N c = N/2 is the number of cubes in the lattice. For an Ns2-by-Nt lat-

t ice ,N o = 5 + 3(N s - 2). For our lattice N = 32 768, N O = 47, and (S) e = 0.9986. A run from a cold start at 13 = 1 gave (S> c = 0.9990 after 1650 sweeps of which the last 550 were averaged. Similar runs at 13 = 2 and /3 = 10 gave (S) e = 0.9981 and (S) e = 0.9982.

There is no mass gap in U(1). We verified that our Monte carlo gave a vanishing mass gap by using it to compute the product o f the mass gap E times the lattice spacing a from a ratio of two-point functions,

S(A) = f d3x ¼Fran(X) 2 . . . [(A(n,i,],K + 1 +k)A(n,i ,],k)) \ Lrn In - . .---:v--~--~.. - - - ~ aE~K-~.. ~ (A(n, t , l ,K+k)A(n, t , l ,k)) ) ' (2)

Because the interpolation is linear in the field variables A, the jacobian det[bAn(X)/aA(m,i,],k)] cancels in ratios of path integrals. By restricting spacetime to a finite volume, we approximate the mean value in the vacuum of a euclidean-time-ordered operator, Q(A) by a multiple integral over the A(n,i,],k):

(0 tTQ(A)IO)

a sum being understood over i,], and k in both the numerator and the denominator. A run of 200 sweeps and 100 measurements, starting from thermalized fields at/5 = 1, gave for K = 0 - 6 the values aE = 0.0204, 0.0119, 0.0087, 0 .0071,0 .0061,0 .0053 and 0.0049.

In temporal-gauge U(1), the Wilson loop (functional) is the mean value in the vacuum of a euclidean-time- ordered product,

fII dA(n,i , j ,k) exp [---S(A)] Q(A ) f lI dA(n, i,] , k) exp [-S(A)]

(1) W(r,t) = (OlT exp(ig f An(X) dXn ) ]O),

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in which the line integral runs from 0 to r at time 0 and from r to 0 at time t. Wilson loops vanish in more than two dimensions due to a string singularity in the line integral of the connection [10]. Creutz introduced the practice of measuring ratios of products of Wilson loops in a way that separates the physically important area term from this singularity [ 1 ]. The Creutz ratio of a quartet o f Wilson loops is a logarithm of a ratio of their products:

X(L,J)--- x(r/a, t/a)

, [ W ( r , t ) W ( r - a , t - a ) ] = - m [ w ( r - a, t )W(r, t - a ) ] "

(3)

The Creutz ratio is insensitive to terms in ln(W) that are independent of r and t or that depend linearly on r or t. Whenever the loops are dominated by an area law, ×(L , J ) directly measures the string tension, ×(L,J )

o(fl). In Wilson's lattice gauge theory, this occurs when L and J are large or when the coupling is large

[21. With

F(q) = 1 - q - [q + (1 + q2)1/2] -1

+q ln[q +(1 + q2)1/2],

our formula for the exact Creutz ratio in U(1)3 is

x(L , J ) = (2zr/3) -1 [ (3"- 1 ) F ( L / ( J - 1)) - J F ( L / J )

+ (L - 1)F(J[(L - 1)) - L F ( J / L )

+JF( (L - 1 ) / J ) - ( S - 1)F((L - 1 ) / ( J - 1))

+ L F ( ( J - 1)/L) - ( L - 1 ) F ( ( J - 1)/(L - 1))] (4)

in which L = r/a and J = t/a are the extents of the largest loop of the ratio in units of the lattice spacing a. The exact X is symmetric, ×(L,J ) = ×(J ,L) , and depends upon/3 exclusively through the factor 1//3 = ag 2, as expected in a free theory. In the limit of large J,

x (L , J ) -~, (2zr/3) -1 ln[L/ (L - 1)]

which is the difference a[V(La) - V(La - a)] where V(r) = (g2/27r)In(r) is the static potential of two charges g separated by the distance r.

To compute Creutz ratios at/3 = 1, we made 12 000 sweeps with measurements every other sweep, starting

from fields thermalized by 700 sweeps. For L = r/a = 5 - 7 and J = t/a = 2--6, these x ( L , J ) are within 7% of the exact values. For L = 3 and 4 and J = 2 and 3, they have errors of 6% or less. To compute Creutz ratios at/3 = 2, we used 2 750 sweeps with measurements every third sweep, starting from thermalized fields. For L = 6 - 8 and J = 2 - 8 , these x ( L , J ) are within 8% of the exact values. For L = 4 and 5 and J = 2 - 4 , they have errors of less than 6%. To compute Creutz ratios at/3 = 10, we did 1 650 sweeps with measurements every third sweep, starting from thermalized fields. For L = 5 - 8 and J = 2 - 8 , these x ( L , J ) are within 7% of the exact values, with an average error of only 3.8%. For L = 3 and 4 a n d J = 2--4, they have errors running from 1 to 11%.

For all three values of/3, our Creutz ratios ×(L,J ) are within 20% of the exact values for L = r/a = 4 - 7 and J = t/a = 2 - 8 . The average error of these X is 5.9%. For L = 5 - 7 and J = 2--6, the average error of the ×(L,J ) is 3.5% at/3 = 1. For L = 6 - 8 and J = 2 - 8 , the average error of the ×(L,J ) is 4.0% at/3 = 2 and 3.9% at/3= 10.

Using Wilson's method, Bhanot and Creutz [7] and Ambj6rn et al. [8] have reported data on Wilson loops for U(1)3 from runs on a 163 lattice. Both groups fitted square Wilson loops to curves of the form

W(La,La) = exp[-o( /3)L 2 +P(fl)L + B ~ ) ] ,

and reported the values of the string tension o(~) for various/3 between 1 and 3.5. They did not report Creutz ratios or the values of any non-square Wilson loops. In order to extract Creutz ratios ×(L,L) for square loops from their data, we make the reasonable assumption that the Wilson loops W(La, (L - 1)a) and W((L - 1)a,La), had they measured them, would have fitted the same curves as their W(La,La) with L 2 replaced by L(L - 1) and L replaced by L - 1/2. This assumption is implicit in their extraction of string ten- sions from those curves.

From this assumption it follows that the Creutz ratios ×(L,L), had they measured them, would have been given by the relation ×(L,L) ~ 0([3) wi th L taken somewhat beyond the midpoint of the range of L they used to fit their loops. Since Bhanot and Creutz fitted Wilson loops with values of L from 1 - 4 , we identify in fig. 1 their o(~) with ×(3,3). Since AmbjCrn et al. fitted loops with values of L from 4 - 8 , we identify in fig. 2 their o(fl) with ×(7,7). Of the 16× so inferred, 2

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o -

o -

.~o

COo" C_

N

C-o-

o

o -

~b

c~3'. 00 2'. 00 4'. 00 8'. 00 8'. 00 10.00 Beta

Fig. 1. Creutz ratios ×(3,3) obtained from the simplicial method (octagons) and Wilson's method (plusses). The exact ×(3,3)'s are represented by the solid curve.

are within 5% of the exact values; 5 differ by 25 to 59%; and 9 are off by 151% to 1 481%. The errors of these last 9 points cannot be ascribed to the crudity of

o-

o3 o LS~ fQo- C_

N ~ o

C-O" C~

o

o

~0.00

%

2.00 4.00 6'. 00 8'. 00 10.00 Beta

Fig. 2. Creutz ratios x(7,7) obtained from the simplieial method (octagons) and Wilson's method (plusses). The exact x(7,7)'s are represented by the solid curve.

our identification of X with a. Nor are they a finite- tize effect; in U(1)3 with the Wilson action such effects are small on a 163 lattice [8, 11 ], because of the mass gap. These errors reflect the incorrect 13 dependence of their Wilson loops.

For other theories in other dimensions, Wilson's method may be more accurate at intermediate and strong coupling than it is for U(1)3. In the special case of U(1)3, Wilson's formulation predicts behavior that is qualitatively different from that of the continuum theory. The compactification in Wilson's method in- troduces nonlinearities that result in a linear confining potential at long distances. These nonlinear effects are often described in terms of monopole excitations, which are not present in the continuum theory or the simplicial method.

In order to examine the accuracy of Wilson's method for weak coupling, we turn to the weak-coupling expansion of MOiler and Rfihl [9]. For an infinite lattice, the first two terms of their weak-coupling expansion for a U(t)3 Wilson loop are

ln[ W(La,Ja)] .~ - (1/f l + 1/3/32) Wl (La,Ja).

By using the numerical values for Wl(La,Ja ) of Ambj6rn et al. [8], we find that the Wilson method on an infinite lattice should give the following Creutz ratios for/3 = 10: ×(3,3) ~. 0.0099, ×(4,4) ~- 0.0069, X(5,5) ~- 0.0053, X(6,6) ~. 0.0043, X(7,7) ~ 0.0036, and X(8,8) ~- 0.0031. These values differ from the exact X by 8.6%, 6.3%, 5.0%, 4.4%, 4.8%, and 3.1%, errors that are only about twice those of the simplicial method. Two of these values appear in the figures.

On the basis of the data reported here, and illustrated by the figures, we conclude that the simplicial method is more accurate than Wilson;s lattice gauge theory for U(1)3 , particularly for intermediate and strong coupling. The better accuracy of the simplicial method is due to its use of the action and domain of integration of the exact theory, unaltered apart from the granularity of the simplicial lattice. Wilson's action and domain of integration are close to those of the exact theory for weak coupling, but not for intermediate or strong coupling. However the resultant errors may be less severe for other theories in other dimensions.

The action of the simplicial method possesses gauge invariance only up to the granularity of the lattice. For U(1)3 in the temporal gauge, this lack of exact gauge invariance was unimportant; whether it would matter

384


for a nonabelian gauge theory is an open question. We are now investigating this question for SU(2) 3 .

The authors are grateful to Jay Banks, Michael Creutz, Roy Glauber, Sudhakar Prasad, Brent Richert and Paul Smith for useful conversations. This research was supported by the Department of Energy under grant DE- FG04-84ER40166 .

References

[1] M. Creutz, Phys. Rev. Lett. 45 (1980) 313. [2] M. Creutz, Quarks, gluons and lattices (Cambridge U_P.,

London, 1983).

[3] K. Wilson, Phys. Rev. D10 (1974) 244. [4] R.K. Ellis, in: Gauge theory on a lattice (1984) (Depart-

ment of Commerce, Springfield, VA, 1984) p. 191. [5] K. CahiU, S. Prasad and R. Reeder, Phys. Lett. 149B

(1984) 377. [6] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20

(1979) 1915. [7] G. Bhanot and M. Creutz, Phys. Rev. D21 (1980) 2892. [8] J. Ambj¢rn, A.J.G. Hey and S. Otto, Nuel. Phys. B210

(1982) 347. [9] V.F. M/iller and W. Riihl, Z. Phys. C9 (1981) 261 ; in:

Proc. 17th Winter School of Theoretical physics (Karpacz, 1980).

[10] K. CahiU and D. Stump, Phys. Rev. D20 (1979) 2096. [11] R. Horrsley and U. Wolff, Phys. Lett. 105B (1981) 290.

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Comparison of the simplicial method with Wilson's lattice gauge theory for U(1)3