Nuclear Physics B (Proc. Suppl.) 4 (1988) 585-589 585 North-Holland, Amsterdam

EXACT TREATMENT OF WILSON FERMIONS IN MONTE CARLO SIMULATIONS*

Atsushi NAKAMURA

FB Physik, Freie Univ. Berlin, Arnimallee 14, D-1000 Berlin 33

A project to treat dynamical Wilson fermion in an exact manner is reported. An algorithm for this purpose is briefly described and numerical results for SU(3) and U(1) on 44 are presented. Comparison with the bush-factorized algorithm is done and they agree reasonably with each other. Indications of first order phase transtions are observed near 8=5.3 and k=0.14 with 3 flavors for SU(3) and near 8=0.9 and k:0.17 with 1 flavor for U(1).

I. EXACT ALGORITHM, WHAT, WHY AND HOW

I.i. What is an "exact algorithm"

After integrating over fermion varia-

bles, the partition function has a form,

Z : ~[dU] (detW) f e-SyM, (i)

where W is a fermion matrix. During the

updating process of U, we need the ratio

of the fermion determinant,

= detW[U+6U]/detW[U]

= det(I+w-l~u), (2)

where 6W : W[U+4U]-W[U] and W -I :

(W[U]) -I .

We have eliminated fermionic fields,

but, as a price, we have got a non-

local object. After we change one link

variable, all elements of W -I, the fermi-

on propagators, become different from

the previous ones. In the exact algo-

rithm we compute the ratio of the

fermion determinant evez~ time we update

one link variable.

1.2. Why we need "exact algorithms"

Since the first trials to include the

dynamical fermion loops in Monte

Carlo calculations of lattice gauge the-

oriesl, considerable progress has been

made in this field 2. Most of them, how-

ever, use so-called small step algo-

rithms. One likes to reduce the calcula-

tion of W -I which costs a lot of CPU

time, and usually W -I is refreshed only

once or several times during one updates

all U's on a lattice. In compensation

for this merit one has to accept small

Markov steps in Langevin-type algo-

rithms. In the bush-factorized algo-

rithm 3 which we will discuss later we

allow large Markov steps in the config-

uration space, but we miss long range

correlations due to quark propagators.

The bias in these approximate methods

may become severer when we arrive

the very small fermion mass region.

We have been pursuing a project of

Monte Carlo calculations by an exact

algorithm 4. Our aim is, first of all,

to establish standards that are free

from systematic errors except the fi-

nite size effects. If we could be al-

lowed to be very optimistic, we ex-

pect that the size becomes bigger in

* This is based on the collaboration with Ph.de Forcrand (Cray Research, Wisconsin), M.Haraguchi (Fujitsu• Tokyo) and G.Feuer, H.C.Hege, V.Linke and I.O.Stamatescu (Freie Univ. Berlin).

0920 5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

5 8 6 A. Nakamura / Exact treatment of Wilson fermions

future like in the quenched and approxl-

mate dynamical fermion cases. Secondly,

we hope that the exact algorithm will

be a starting point of developping new

approximate methods with large Markov

steps. One possibility is to employ

cheaper methods for the calculation

of W -I, such as the blocking method. 5

1.3. How the "exact algorithm" works

We summarize in the following flow

chart how a gauge configuration is up-

dated. First we divide a lattice into

hyper-cubes whose ]inks do not overlap

with each other. There are eight pos-

sible partitions in 4-dimensions. All

eight partitions are taken one after

another in a random order. Every hyper-

cube of the chosen partition is updated.

W -I is calculated by solving

Wx = e (i) (3)

where e (i) is a unit vector the i-th

component of which is non-zero. We

get one column of W -I everytime we

solve eq.(3). The index i of e (i) runs

over the 16 sites of the hyper-cube.

Therefore we solve eq.(3) 4x3x16 (4x16)

times for SU(3) ( for U(1) ) on a hyper-

cube. We call the obtained (4(x3)x16)x

(4(x3)x16) submatrix by H,

H C W -I (4)

6

7

8

9

i0

ii

12

13

14

]5

16

17

18

19

20

21

22

23

24

25

26

Flow C h a r t ]

27

28

29

30

31

32

1 Run over hypercubes on the lattice

Calculate H by solving Wx:e ([)

Re)eat several times

Run over all 32 ]inks U on

a hypercube

U 0 : U

gO : 1

Reneat several times

Propose a new link

value U'

C d l c u l a t e g:

g ,

detH(U' )/det H(U 0)

g ~' /9 0

Metropolis:

If U' is accepted,

then, U-U' and

g0 g'

Calculate H by

Woodbury's formula

. . . . . . . . . . . . . . . . . . . . . . +

+ . . . . . . . . . . . . . . . . . . . . . . . . . . . +

3 3 + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +

We use here the conjugate gradient

(CG) and conjugate residual (CR) methods

with a hopping parameter expansion pre-

condition 6 i.e. we solve i

VWx = Ve (i) (51

A. Nakamura / Exact treatment of Wilson fermions 587

in stead of eq.(3). Here a matrix V is

a first order hopping parameter expan-

sion approxlmation of W -I For a

review of CG/CR see for example Ref.(7).

The number of operations CR requires is

roughly half of that CG needs, but the

hermlte part of VW should be positive

definite. Of course, when we employ a

precondition, we expect that VW = I,

and that the eigen-values distribute

around one. But this is not guuanteed.

Therefore we start with CR and, if it

does not reach the required precision

after some iterations, we switch to CG.

In our experience so far this procedure

becomes necessary for k=k c in U(]) and

for an asymmetric SU(3) lattice. 8

Now we update 32 link variables on a

hyper-cube according to the Metropolis

algorithm. During• the update of a link,

we can calculate ~ from H with little

CPU-time. But after U 0 is changed we

have to refresh H. Here we use Woodbury's

formula,

Hne w : H - H6W(I+H6W)-IH. (6)

This formula requires a relatively small

number of operations in order to refresh

a matrix inverse. 9 If we apply it to a

whole matrix W, however, we need a very

large memory and may have rounding-error

problems.

With help of Woodbury's formula, we

can update all 32 links without solving

eq. (5). But we gain more; we can update

thts hyper-cube as many times as we

wdnt (line 5 in the flow chart). We

update o r l e hyper-cube typically =i0 times

before" going to the next hyper-cube.

The performance and the choice of para-

metors are discussed in Ref.(6).

2. Numerical results

First let us discuss data on SU(3).

Fig.la shows the behaviour of the

plaquette energy at fl-5.3 and k:0.14

with 3-flavors as a function of the

number of sweeps. The points fluctuate

strongly and suggest a two-phase

structure.

055 Eplaq

o5o

. , , , • , .... (a)

K.O13

045 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

' 8b ' , ~ o ' 2 ~ o ' a ~ o ' . 6 o o ' 8'o ' ,~o

number Of sweeps

/ K.OI5

FIG.Ia

• ~ (l~) Eplaq

.5;

.5(

. 4 ~ ~

.46 i 4o0o 6 o ; 0 ' ~ i ~ o ' , o 6 o o

number of sweeps

J ~8o6--

FIG.Ib

In order to see the character of

each phase, we start from the 240-th con-

figuration and run the simulatlons with

k-0.13 and 0.15. The runs seem to con-

verge to the values near the lower and

upper levels at k-0.14, respective-

ly. In Fig.2, we present the value

from the Polyakov line. These data sug-

gest that, at 8 5.3 and k=0.14 with

588 A. Nakamura / Exact treatment o/Wilson fermions

3-flavors on 44 lattice, we are very

near to a phase transition line, and

when we go to lighter (heavier) quark

mass region, we enter the deconfining

(confining) phase.

Q2(

L QI [

0 . 1 (

0 . 0 ~.

I

tl, i, '

tOO 200 300

numbe r of sweeps

FIG.2

400

In Fig.lb we give the corresponding

results for 8:5.3 and k=0.14 from the

bush-factorized noisy calculation. Here

again we see the two-phase structure

and the values obtained by two algo-

rithms are consistent. This is good

news for the bush-factorized method.

We see the difference rather in the

"speed" of the iteration. In the first

4000 iterations we see only monotonic

behavior in the bush-factorized method.

The period of one phase is O(1000) sweeps

in the bush-factorized algorithm and

O(100) sweeps in the exact one. This

is good news for the exact algorithm,

i.e., it sweeps very quickly the con-

figuration space.

Next we report preliminary results

on U(1), where we work with one flavor.

In Fig.3a we plot the plaquette energies

at 8:0.9 and k=0.16 as a function of the

5, . 5 5

sweeps with hot and cold starts. After

some sweeps both converge to th~ same

value which is a little bit highest than

the quenched value. In Fig.3b, where

we give the result at k 0.17, both be-

have violently. Looking carefully

each line, we see jumps between two

phases. Although further investiga-

tions on bigger lattices are necessary,

these results may suggest a first {Jrder

phase transition in (compact) QED, as

claimed by Dagotto and Kogut for the

staggerd fermion case. I0 U(1) gauge

theory, if light fermions are included,

could have a more complicated phase

structure than expected.

. . . . . . . . i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . t . . . . . . . . . , . . . . . . . . .

. 5 0

Hot Start k=O.t 6

.40 . . . . . . . . . i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . ] . . . . . . . . . J . . . . . . . . . i . . . . . . . . . i . . . . . . . . .

10 20 30 40 50 60 70 80 # of Sweeps

FIG.3a

. 6 5

6, .60

. 5 5

. 5 0

. 45

. 40

Cold Start W

Hot Start

5'0

FIC.3b

I ,

k=0.17

# of Sweeps

A. Nakamura / Exact treatment of Wilson fermions 589

Flnal]y let us mention the CPU time.

There is some misunderstanding that our

group can play with the exact algorithm

because we are very rich in computer

time. This is in any sense not true.

Fig.]a, for example, costs roughly 60

hours on a VP200 and Fig.3a costs less

than ]0 hours on a Cray-XMP. Our program

is not yet fully optimized and we

believe that we can reduce the CPU-time

still by a factor two.

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