Phase structure of compact lattice QED with dynamical fermions

Nuclear Physics B (Proc. Suppl.) 9 (1989) 114-118

114

North-Holland, Amsterdam

PHASE STRUCTURE

OF COMPACT LATTICE

QED WITH DYNAMICAL FERMIONS

H.C. Hege and A. Nakamura FB Physlk,

Freie Universit~t

Berlin, D-]O00 Berlin 33, FRG

The phase structure of compact lattice QED with dynamical Wilson fermions is studied by Monte Carlo simulation. The fermion contribution is calculated using an exact algorithm, capable of large Markov steps in configuration space. A first order phase transition line in the (~,~) plane, ending at the quenched point (~I.0,<=0) has been found. The phases are explored by measuring gauge dependent quantities, photon and fermion propagators, besides the more conventional gauge invariant observables. For this we applied a non-iterative gauge fixing algorithm.

From the phenomenological

point of view weak

coupling QED is the most successful theory due to the excellent rimental

results

perturbative

physical

agreement

of expe-

and those obtained within the

approach.

However,

to understand

For the pure gauge part of lattice QED there are two formulations, odic) and the non-compact non-compact

the compact

version models conventional

coupling continuum

(peri-

one. Whi~e the

theory directly,

weak

the com-

QED as a complete field theory it is necessary

pact theory contains additional

to get insight into the theory at strong coup-

excitations

ling, too. Recently it has been suggested,

exponentially

based upon Schwinger-Dyson

but fits into the general lattice gauge theory

calculations

quenched ladder approximation,

that in the

strong coupling regime a nontrivial beta function

fermions,

cal consequences. such an unexpected

in the presence of

it may have phenomenologi-

It has been pointed out, that behaviour

of strong coupling

QED may revive certain technicolor suppression neutral

zero of the

exists ] . If the fixed point

really exists and survives dynamical

in the

models by

of undesired flavour-changing

currents 2. The speculation

sharp lines observed

that the

in positron and electron

spectra in heavy ion collisions

may be caused

by a new phase of QED 3 (but see i.e. Ref.4), raised the interest in non-perturbative tigations

inves-

of QED, too. The lattice formulation

of QED allows non-perturbative

calculations

scheme.

topological

whose density vanishes

in the weak coupling region),

Since gauge invariant lattice Dirac

operators compact

(monopoles,

are customarily

defined in terms of

gauge field variables Un, =exp(i@n,u) ,

the compact lattice version looks more natural from a conceptual

point of view.

Up to now, only staggered fermions

have

been employed in Monte Carlo simulations

of

lattice QED. Recently Dagotto and Kogut found phase transitions QED 5

of first order in compact

and of second order in noncompact

with staggered fermions

QED 6

coupled to compact

gauge field variables. Here we report the first results

of a

compact lattice QED Monte Carlo simulation with dynamical

Wilson fermions,

that is with

action

which may be more reliable than the ladder approximation. transition

The existence

of a second order

S=B~Re Un,~Un+~,vU~+v,uU~,v+~n(1-
(])

in strong coupling lattice QED,

which might allow a non-trivial

continuum limit

to exist, would be of great interest.

0920-5632/89/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Qnn'=[( ( 1-~U)Un,u~n+~,n "+ ( 1+Y~)U~.,~n_u,n. ]

H.C. Hege, A. Nakamura/ Phase structure of compact lattice QED

115

where n denotes sites and ~,~ directions in the

discussed in Ref.9. Configurations

four dimensional hypercubic lattice. We always

this algorithm,

used one flavour and antiperiodic boundary

ted, even near first order phase transitions,

conditions for fermions.

as was observed in a QCD simulation I0.

The four dimensional ~ure U(1) lattice gauge theory,

proved to be weakly correla-

In fig.1 we show the value of the plaquette

i.e. compact lattice QED, already

exhibits an interesting phase structure:

produced by

energy as function of Monte-Carlo sweeps at a

B=0.9 and ~:0.175

and 43*8 lattices each with

strong-coupling confining phase and a weak-

ordered and disordered start. These histories

coupling deconfined Coulombic phase. The dyna-

display two co-existing phases and therefore

mics of the phase transition is well under-

suggest strongly the oceurence of a first

stood:

order phase transition.

in the confinement

phase the presence of

a gas of monopoles and antimonopoles

causes the

narrow electric flux tube producing a linear potential.

In the weak coupling phase the

density of monopoles

.7 ~

E

vanishes exponentially,

the static potential is Coulombic and the model

.6

has a line of fixed points, each assigned to a trivial model with another renormalized fine

.5

structure constant. The order of the phase transition point

10

(which extends to a phase

20

transition line in the fundamental-adjoint

30

40 50 number

60 70 of sweeps

80

parameter space) is still controversial. Most recent MCRG calculations 7 indicate a discontinous phase transition. On finite lattices with periodic boundary conditions

FIGURE I Plaquette energy as function of sweeps (each involving 10 hits per link and 10 updates per hypercube) starting with an ordered/disordered configuration on a 43*8 lattice.

very stable topological excitations

divide the configuration space in different

We found jumps in many observables for instance

sectors, which local Monte Carlo algorithms

in <~@>, all of them strictly correlated with

have problems to cross8.

the jumps in the plaquette action. The eigen-

Simulating compact QED

in the strong coupling region it is therefore

value distribution of the fermion matrix is

important to use an algorithm capable of large

quite different for each phase, see fig.2 and

Markov steps in configuration space.

may be used for discriminating them.

We use a Metropolis-type

algorithm without

The "positronium" masses also jump: from

any systematic bias and allowing large Markov

mpara=1.37±O.O8 and mortho=1.67±O.07

steps with customary acceptance rates.

disordered phase, to mpara=1.84±O.07 and

It

updates the links on a hypercube several times

mortho=2.01±O.06

(here 10 times) before proceeding to the next

of the inverse lattice spacing).

one, and thereby decorrelates successive configurations.

The columns of (I-~Q) -I, required

in the

in the ordered one (in units

In fig.3a,b and c we plot the plaquette energy as function of ~ for 2=0.90,

0.95 and

when stepping from one hypercube to the next,

1.0 on 44 and 43*8 lattices.

are calculated by a preconditioned CR/CG sol-

phase transition observed at B=0.90 persists

ver, starting from a 4th order hopping parame-

and moves to larger fermion mass with increa-

ter expansion. Details of this algorithm are

sing B.

The discontinous

H.C. Hege, A. Nakamura / Phase structure of compact lattice QED

116

At 8=1.0 the observables

display for all

values the ordered phase as expected, ,<

,5-

since

the quenched phase transition point for our

':.

". '.- :' ~: .:".:".:i-~':i:."i i'": : "'" ." •

• . . , . .

,

.

•

'.

.

.

.

.

.

.

•

. . . .

,

-"

lattice sizes lies at 6~0.995.

The phase tran-

sition continues to small ~ (fig.3d), and rea.0-

ches the quenched . ,

. , . . .

. . . .

,

.

:

.

.

•

v :::::

.:,"

2....,.'v',.',:'.~

~.';,.;.

'.Z."

'~'A

'.

!,':.',.':'.'..'.'.

(probably weakly first

order) phase transition point. At ~=0.I the

, ..

:Zii:i: :i::: i : .... : i~:':!!~:7 i i'::~i~::IZ:7:!!: :: :i(: ;: •

-.5-

"

. ' . . . .

fluctuation of the fermion determinant ratio around I is within a few percent, nevertheless

'.,"

the phase transition is most probably of first order. Dynamical fermions seem to strengthen ,<

.5.

this phase transition. We repeated this kind of analysis and observed several phase transition points, shown in fig.4. One is tempting to relate the

.0.

transition with the zero mode due to the small fermion mass;

we investigated the eigenvalue

distribution but found no eigenvalue very

-.5.

close to zero. The phase diagram of lattice o'.o ....

o'.5 ....

1'.o ....

1'.5 ....

QED with staggered fermions is pictured in

40

Re(X )

Ref.4 as follows:

also running with decreasing fermion mass to

FIGURE 2 Typical eigenvalue distribution of the matrix I-~Q in the disordered phase (above) and ordered phase (below) at ~=0.9 and K:0.175.

lower 8, is broken at intermediate fermion mass and reappears for small fermion mass as a

(a)

(b)

(c)

(d)

p:o.9o

p=o.95

p=too

~:0.10

0.8 E#aq

the phase transition line,

o.7

o.6

O

"L'L---~

4~ 0.4 -~ I

0

0.1/.

I 0.17 K

I

0.20

zt

I

0.1/. K

I

0.17 0

tz I

I

0.1/.

0.17 0.96

I

0 '-~.e

1.00

~

I 1.10

K

FIGURE 3 Plaquette energ~ as function of ~ for B:O.90 (a), 0.95 (b) and 1.O (c) and as function of B for ~=0.I0 (d) on 4" (O) and 43*8 (o) lattices (empty: disordered start, filled: ordered start, halffilled: ordered and disordered start).

H.C. Hege, A. Nakamura / Phase structure of compact lattice QED

chiral symmetry breaking transition the SU(3)

(similar to

method in Ref. 13.

case, compare Ref.5 and e.g. Ref.12).

The phase structure that in SU(3)

in fig.4 is also similar to

with Wilson fermions on finite

In Tab. I we show the photon and fermion masses at various

(B,~) pairs on a 43*8 lat-

tice. Photon propagators

lattices 11

1l?

Ai(O)>

<~sin Ai(~) sin

(i=1,2 and 3) are f~tted with the usual

cosh form, while fermion propagators K

r

r

G(~)=<~(~)~(O)>

i

I

are fitted with

IC*sinh(Nt/2-~)l.

0.2

=

The basis of this form will

be discussed in detail elsewhere. disordered

In the

phase the photon mass is ~ I. In

the quenched

0.1

IG(~)l

i

case the photon becomes imme-

diately massless transition

when moving through the phase

point into the deconfined

phase

(this was previously observed in Ref.13). With dynamical

fermions,

however,

the photon re-

mains massive even in the ordered phase. This is probably

FIGURE 4 Phase diagram of compact lattice QED with Wilson fermions in the (B,~) plane.

because the photon mass renorma-

lization due to fermion loops produces "plasmon" mass on a finite lattice.

In order to clarify the physical meaning of each phase, we fix now the gauge to the covariant Landau gauge. This opens the possibility calculate

gauge dependent,

quantities,

familiar

in perturbation

theory.

In case of U(1), unlike non-Abelian lattice gauge fields,

TABLE I Photon and fermion mass (in units of the inverse lattice spacing) extracted from propagator fits of about 100-200 equilibrated configurations (1000 in the quenched case) on a 43*8 lattice.

the gauge transformation

A'~(x)=Ap(x)+

~X(x)

configuration

exactly to Landau gauge is easily

which fixes

the whole B

found.

Introducing forward/backward

derivatives,

~+)f(x)=f(x+~)-f(x)

~-)f(x)=f(x)-f(x-~),

lattice

and

we set

×(x) = ~ ( X - F ) 3 ~ - ) A

(y)

(2a)

Y where the lattice Green function

I

~(X) = ~ ~e ipx / 4~sin2p Vol p U

is defined as

/2.

(2b)

~

~-)A~(x)

satisfies

the Landau gauge condition

= 0 exactly,

unlike the iteration

phase

mphoton

0.97 1.00 1.05

0 0 0

dis. ord. ord.

0.86 ( 5 ) 0.25 ( 3 ) 0.16 ( 2 )

1.01

0.100

ord.

1.08 (13)

0.9O 0.90 0.90 0.90

0.160 0.175 0.175 0.180

dis. dis. ord. ord.

1.05 0.90 0.99 0.56

The renormalized propagators

Then A '

to see the photon mass with

staggered fermions.

to

but fundamental

such as the photon and electron

propagators,

be interesting

a

It would

(19) (45) (03) (08)

mfermio n

0.87 0.78 0.69 0.60

(5) (10) (6) (12)

fermion mass measured

decreases

as ~ increases

by

until

~=0.18. At ~=O.175, where we observe the phase transition,

the fermion mass is far from zero.

B.C. liege, A. Nakamura / Phase structure of compact lattice QED

118

This is consistent

with the behaviour

£ermion matrix eigenvalues. transition

Therefore

of the

this

cannot originate from the chirality

~f fermions.

Above K~0.18 the fermion mass

suddenly takes large values,

probably we are

RE FE RE NCES I°V.P° 191 V.A. 149; C.N. Nucl.

Gusynin and V.A. ~iransky, Phys.Lett.B (1987) 141 ; Miransky, Nouvo Cimento 90A (1985) Leung, S.T. Love, and W.A. Bardeen, Phys. B 273 (1986) 649.

beyond ~c" In conclusion,

using an exact fermion algo-

rithm, we have found a phase transition line in compact lattice QED with Wilson fermions, seems to continue up to the quenched transition

phase

point. Since this point is most

probably first order and the transition, playing coexisting fermion mass, fermions,

dis-

states at medium and small

is strengthened

we conclude

the lattices observed

which

by dynamical

despite the smallness

of

(44 to 63x8) that the whole

phase transition line is first order.

In order to characterize gauge dependent

the two phases also

quantities

have been measured

2.M. Bando, T. Morozumi, H.So, and K.Yamayaki, Phys.Rev.Lett. 59 (1987) 384, and references therein. 3.L.S. Celenza, V.K. Mishra, C.M. Shakin, and K.F. Liu, Phys.Rev.Lett. 57 (1986) 55, and references in 4. 4.R.D. Peccei, J. Sola, and C. Wetterich, Phys.Rev. D 37 (1988) 2492. 5.J.B. Kogut 59 (1987) E. Dagatto 295 (1988)

and E. Dagatto, Phys.Rev.Lett. 617; and J.B. Kogut, Nucl.Phys., B 123.

6.J.B. Kogut, E. Dagatto, and A. Kocic, Phys.Rev.Lett. 60 (1988) 772. 7.A. Hasenfratz,

Phys.Lett.

B 201

(1988)

492

using a method which allows to fix a configuration to Landau gauge in one step. These

quanti-

ties behave as expected within the disordered phase.

In the quenched

suddenly massless phase, however,

with dynamical

small K it acquires structure

fermions even at

a heavy mass. The phase

becomes much more complex with dyna-

mical fermions analyses.

case the photon becomes

when entering the ordered

and requires

further

finite size effects

and the behaviour

and monopol@s

of the

ACKNOWLEDGEMENTS

cussions

with G.Feuer,

Stamatescu,

been performed the computer X-MP/18).

K.Kanaya,

and W.Theis.

valuable disV.Linke,

I.O.

The calculations

have

at ZIB Berlin

10.Ph. de Forcrand, M. Haraguchi, H.C. Hege, V. ~inke, A. Nakamura, and I.O. Stamatescu, Phys.Rev.Lett. 58 (1987) 2011. 11.Ph. de Forcrand and I.O. Stamatescu, Nucl.Phys.B 304 (1988) 628.

the

in the (B,~) plane 14

We would like to acknowledge

9.A. Nakamura, G. Feuer, H.C. Hege, V. Linke, and M. Haraguchi, to appear in Comp. Phys .Comm.

extensive

Currently we are investigating

eigenvalues

8.V. Gr6sch, K. Jansen, J. Jersak, C.B. Lang, T. Neuhaus, and C. Rebbi, Phys.Lett. B 162 (1985) 171.

(CRAY X-MP/24)

center of university Kiel

(CRAY

and

12.J.B. Kogut, E.V.E. Kovacs, and D.K.Sinclair, Nucl.Phys. 231.

B 290 (1987)

13.P. Coddington, A. Hey, J. Mandula, and M. Ogilvie, Phys.Lett. B 197 (1987) 197. 14.H.C. Hege, V. Linke and A. Nakamura (in preparation).