Accelerating Abelian gauge dynamics

Nuclear Physics B (Proc. Supp]_) '20 (19'9~) ll,.!-_-ll7

114

ACCELERATING ABELIAN GAUGE DYNAMICS Stephen L. ADLER Institute for Advanced Study, Princeton, NJ 08540, and Gyan BHANOT* Thinking Machines Corporation, 245 First Street,, Cambridge, MA 02142 and Institute for Advanced Study, Princeton, NJ ~540 In this paper, we suggest a new acceleration method Tor Abe~an gauge theories based on 5near transformar'runs to vafmbles which weight a;J ]emgth scales equa|Jy. We measure the autocorre~t~m IL;mefor 1~e Polyakov loop and the p|aquette at ~ = LO ]n the U(]) gauge theory in four dimen~,~s, for the ne~ method and for standard Metropolis update~ We find a dramat~ improvement [or the new method over the Metropolis method. Computing the c~t;cal e~onent z for the new method remains an important open issue. Critical slov~ng down poses a sedous problem, in the study of systems near cr~cafity. For spin modds, there are new many successful techn~quns ava;laMe to reduceor e~minate crit~a] sSow;ngdown. In the caseof gauge the~es, and in part~ular noa-Abe~an gauge theories, the~e has bee. less pro~-e.~. Wh~e overrelasadon has proved asdul, t ~ e are no know. methodswh~.h succeedin e~m~a~ng c~_.~ I dcm~g do~n. To some extent the Im~blem ~sone of identify" g relevant ~ cenfigumt~ns c o n t n ~ n g to the dynamics. However, even when the operators are known (such as the monopole configurations for the compact/7(1) gauge theory), it has not been fea~ble to accelerate the dynamic~ A generagy api~ca~.method that has been known for some time is F o ~ r ~ acceleration. The basic idea here is to update the dynamics at the same rate at all length scales, wh~:h can he cloneby going to momentum space variables. However, in the case of gauge fields, the me*hod as usually implemented requires accurate gauge fixing, which turns out to be too expensive in computer time for the method to be competitive with usual techniques. Here we suggest a method based on simple changes of variables that weight all length scales equally when the linear dimensions of the lattice are powers of two. To be specific, consider the U(1)

gau~e tim~ry on an N= x N , x N~ x N t = 2 ' lat-

with per~f~ boundary cenKe~s_ Let ~ and be the angles as~:)c:~temlwith the gauge Enks ~-~ and Us in the dine direction (/71 = e~ etc.). Define the change of va~uMes

In thk simple example, there are only two momenta p0ssib~ealong the time direction and 01 and 02 are indeed the ze~ mode and the mode corresponding to momentum = T . In terms of the fields /71 and Uz this change of variables is given by

u, = ~ , where

~

=

e ~,

~, = ~ - ~ ,

(2)

etc.

We consider two general-uzationsof this change of variables to any value of the lattice size which is a p o w ~ of two. For e~ample, for Nt = 4 , in method 1, we have new va~ables defined by

4~=

0~ ÷ 0 . - - 0 ~ - 0 ,

0~-02-0s+0,

~=

01 - - 02 "~L Oa - - 04 :D

~4 =

(3)

and in method 2, the new variables X are defined by

*Speaker at the conference. 0920-5632/91/$3.50 @ Elsevier Science Publishers B.V. (North-Holland)

=

X1 -- X~ -- Xa

S.L,. Adler. G. Bha~zo~,/A cceteral:ing abe~an gauge ~'namics .~.- =

~,=

Xz -- X~ ÷ Xz

x~ + x~ + x , .

(4)

~ + ez + e= + e ~ - e = - e , - e r - an

~=

e~ - e , - 8 = - e~ - e, + e, + ~ + e=

~=

e, + ~ -

e = - e~ + e= + e n - e ~ - Pn

~=

e, - ~ -

e= + e, + ~,= - e , - ~ + e,

..~= ~ - e= + e = - e~ + e = - e, + ~ -

noA-10c=i a=l c|mse~ io s=,d~a ~

tl~ ~

mvde=gtk m=l~ to rd== =t t~e ~ = ¢

x~-x=+x=-x,

sm=ll~ for tke = m ~

/~=

xz+x~+x~+x=.

....

~;~e= ....

as c=mlmml = q ~ a ~

(6)

where the ~'s are tee Enk an&les al0q~ t ~ t t ~ , e direction. Hemls~:ally. l~hetramdrmmallJ~ of metlmd I is modvated by Fourier a.alysisz aria that of mdlxxl 2 by mldt~flN~, z Method 1is ebtalm~ by malSmg tee a ~ i , the Fo=rie~ ~ Eeld mmlables that when the Fourie~ coelEcient is pmidve, it is rel~aced by plus one and if it is negall~e, by m~n~ one. MetSod 2 ~s o S r : a ~ l 5y P:,eraSngt S e ~ ~ of Eq. (Z) ~,e," ~ ,=,~-~" h, ~=,=,=,~~ 2. Since these are both h e a r t m m S ~ m a t ~ on the ~'s, the measure is imraria~ up t o a constant ( ~ =

tk~t t j ~ / t n ~ t J

e , . (s)

wEile method 2 becomes

#==

the O's or X's appearing at mo~ once in each term. This means that the computationa| complexity doe~ not ~ncrease ~n making these changes of variab~. One now does normal checkerboard MetroT)o~ up~afin 8 but on the 0 or X var'~ab~es;nstead of the ¢~ va(~bles. The me.hod is appr~cl in successk~ in of the four co~£mat~ £~r-~'fions for a cornpr~.e ~pclate. The Gau~sian w~dth of the random numbers . ~ d in the Me~op0r~s = p d a ~ were tu,e~ to ensure a 50 pe~,.~=t i~.¢pta~ae rate for tee tkree m~Eo¢~

![ A~ = $. t~--n m~hocl 1 reads

~=

:15

~x~=~x=._~ T~is e o u s t a . t

;s in fact unity when the t r a n s 4 r ~ are appEecl to an integrand which is p~rlocEc in the ~'s with period 2~r. with the ~, 8 and X integ~t~ms all extending from 0 to 2~r. Also, since no two of the ~'s arein the same plaquette, for actions which are only functions of the plaquette (as is usually the case), the action can still be written as a sum of local terms with

=tmcmred i~!= tli=t = ¢ f=~4 i= I=~i¢= ~

t]ke-

I~¢¢ i¢,a~¢=i~ =Iko m~== t l ~ tlr=epmlga=s ==e mites m= ~ r y f a ~ ~ = = l l y a= s ~ e ~ =f a i m t I -

t o ka~e a ~ e a k f . , ~ o r d e r ~

K a r ~ ---- 1

~id= r a t l ~ -'~= am~omTek~m trees for s~mlocal , l ~ l a t ~ I m •

~Ec=lt time l ~ = e r a t ~ the

We ~ouc=.~.z~l o=r ~ o= oue ~4,e of p. ,amety p = ~0. TI~ q=a=~t~ t~at =e =s=l to monitor the cemerSm=e rate is deEm~l as foBomsc If @(/=) ;~ the es~d=-~m val,e of a, ob~m=~l~ Q at t h e J~tE ~

,

and i f one does a m e a ~ a r e n ' ~ l

o f th;s observable over ~

time steps of which ~z

S.L. Adler, "q. Bhanot /Acceteratiug abelian gauge dynamics

116

Moq.PoL Loop. . . . .

I

. . . .

I ....

S~gn oF PoJ. Loop.

I ....

I'"

'"

LOO

O75

O.5O

o~ _

,..I

....

I ....

I ....

I

....

k

Figure 1: Autocorrelation function of Eq. (7) for the magnitude of the Polyakov loop for ~n 84 laR~ce at fl = 1. The top graph is for normal updating of the link variables and the bottom ~ graphs are for methods 1 and 2. measurements are discarded, then we define the autocorre]ation function, C(i) =

,~._,~+~_~

-t

~--~+z j

F~ure 2: The same as Figure 1 but for the sign of the Polyako: Ionp. both methods as well as for direct update o f the ~'s. In all cases, there was a dramatic improvement in the autocorrdation t i m e for both method 1 as as method 2 over direct updating o f the ~ ' ~ As one

would expect, the improvernent was proportionately greater for larger systonm since the correlation length a t f l = 1 is certainly greater than 8. T h e ratio C(k)/@(O) for an 84 lardce is shown in Figures 1

"

(7)

There are two types of critical s i t i n g down assoc~ ated with any updating procedure. One is the rate at which the system approaches the eqnii~brlum ~mtributlon. The other is the rate at which statistically independent configurations are generated. Clearly, it is the sucond of these that is more relevant in a large numerical simulat'mn. One studies the second type of autocorrelation by taking a su~c~ently large value of ~zz in Eq. (7) so that one starts the measurements already in the equilibrium distribution. This can be monitored by computing the expectation values of the plaquette and the Polyakov loop (magnitude and sign). We found that nz ---- 5000 was adequate for these lattice s'~es at fl ~ 1. We chose r~2 = 10000 in our study, although this was marginal for 84 lattices for the usual method of updating the q?s. We always used a one hit Metropolis method with a checkerboard update. We have studied 24, 44 and 84 systems. We measured the plaquette and the Polyakov loop for

to 3 for the magnitude of the Polyakov loop and its s ~ , and the plaquette, respectively. PJaquet~e "''1

....

I ....

I . . . .

I ....

I

i.00

0.U( n

2a

~

~

90

100

k

Figure 3: Same as Figure 1 hut for the plaquette. The top graph again is the usual link update, the middle graph is for method 2 and the bottom graph is method 1.

S.L. Adler~ G. Bhz~ot /Accelerati~g abe//aa gauge dynamics Note that in al~ three cases,the autocorre~on t~me, particularly for the P~t'ak~v loop, is dramat~..a|ly smaSef for the new methods as compared to standard link update. In Summary, ot~r meth¢~ are a definite improve,umt over the MetropoEs update of tee ~'s directly. However. in terms of tee f~mela corre/e~ ~ ~ eL% our study ~ not accurate enough to estabEsb whether the impmveme~ ;sin t h e ~ t c o r t l ~ e x p c x u m t z . TbSs is aa important open msueCae this method be appEed to nomAbeftan gauge ~lN~r~s? An apim~cb wh~:h ~ are angular varmbles (~u4e~ angles or v a t ~ m thereof). The t r ~ a s f o n n . - ~ of Eq~ (S) an~ (6) cas be applied ~ r e ~ to the azimuthal angb~s,amJ c~a be

117

applied to the polar an~Jeas well when the integrand is extended to be periodic (although not anaJyt~) in the polar angle variable. Resu|tsof th~s study w[l| be reported elsewhere.

AOKNOWLEDG~MEI~S This work was partially supported by the U~5. Departnumt of Energy through grant n.mber DEFGO2-90ER405,4Z. REFEREIICES 1. O. E. La~ord ill in 'C~#,/~adP ~ , P,a~dom Systems, Gnse Theodes'. Les H e e d ~ lg84. K- ~ ~ I I~ S*.~ eds. (Pk~h

PI~,~.~.

lgST.