The massive schwinger model on the lattice studied via a local hamiltonian Monte Carlo method

Nuclear Physics B225 [FS9] (1983) 204-220 © North-Holland Publishing Company

T H E MASSIVE SCHWINGER MODEL O N T H E L A T T I C E STUDIED VIA A L O C A L H A M I L T O N I A N M O N T E C A R L O M E T H O D A. SCHILLER and J. RANFT Sektion Physik, KarI-Marx-Universitiit Leipzig, DDR

Received 8 April 1983 A local hamiltonian Monte Carlo method is used to study the massive Schwinger model. A non-vanishing quark condensate is found and the dependence of the condensate and the string tension on the background field is calculated. These results reproduce well the expected continuum results. We study also the first order phase transition which separates the weak and strong coupling regimes and find evidence for the behaviour conjectured by Coleman.

1. Introduction T h e r e are two outstanding problems in lattice gauge theory: (i) the construction of methods, which allow efficient and accurate Monte Carlo calculations in the presence of fermions and (ii) the question of universality of the lattice action and the existence of the continuum limit of lattice gauge theory. The Schwinger model [1], that is quantum electrodynamics of one Dirac particle of mass m and charge g in 1 + 1 dimension is a very interesting model to study properties which are expected to be also properties of Q C D - l i k e quark confinement and chirai symmetry breaking. At the same time the lattice Schwinger model is also a good laboratory to study the two problems in lattice gauge theory mentioned above. The most important question to be studied using lattice gauge theory is to solve Q C D in the large distance, strong coupling regime. There has been much progress along these lines during the last years. Several methods are available now to include fermions into Monte Carlo calculations [2] and there are first results regarding the masses and magnetic moments of mesonic and baryonic bound states of quarks [3]. At the same time however, it has been realized [4] that there are very large fluctuations in these parameters as calculated on lattices of rather limited size. Also the problem of universality and of a proper continuum limit of lattice gauge theories are unsolved [5]. Studying lattice gauge theories including fermions in 1 + 1 dimension is more easy than in higher dimensions due to the existence of a local hamiltonian Monte Carlo method [6] which, so far could not be generalized to higher dimensions. This method is competitive in computational speed to the usual lagrangian methods to 204

A. Schiller, J. Rand / Massive $chwinger model

205

study pure gauge theories without fermions on the lattice. This method was used so far to study simple fermion models and coupled fermion-boson models [7]. The application to the massless Schwinger model [8] gave an impressive agreement of the mass of the bosonic bound state calculated numerically with the exact result of the continuum model. Here we want to report on the application of this method to the massive Schwinger model. First results of our study were already published elsewhere [9]. They confirm that it is possible, using this Monte Carlo method, to obtain results which agree with exact results of the continuum theory. Our paper is organized as follows. In order to make the paper self-contained we discuss in sect. 2 the well-known properties of the Schwinger model [10] and present the hamiltonian lattice formulation. In sect. 3 we apply the local hamiltonian method of Hirsch et al. [7] and of Martin and Otto [8] to the massive Schwinger model. In sect. 4 we present the Monte Carlo calculation and discuss the results regarding the fermion condensate and string tension (change in average energy with background field 0). In sect. 5 we present our results regarding the first order phase transition between weak and strong coupling for the model with the background field 0 = lr.

2. Properties and hamiitonian lattice formulation of the massive Schwinger model The massive Schwinger model is defined by the lagrangian density .LP= - ~ ' . , . F ~'' + ~(i/~ -m)~k

(1)

with t ~ = ¢'O~, = ¢'(0~, + igA~,) , F,,v = O,,Av - O,.A,,.

(2)

Throughout the paper we use the notations goo = - g l l = 1,

e ° l = - e 1°= 1 .

(3)

As found by Coleman et al. [11] the solution of the model involves an arbitrary parameter 8 totally independent of g and m. This 8 parameter has been interpreted by Coleman [10] by introducing a constant electric background field into the model by adding the term

1 0 2 2zr ge""F~'~

(4)

to the lagrangian. The origin of this background field are external test charges +q and - q at distance +L(ILI-~oo) which are connected to the angle 0 by O q=~g

=ag.

(5)

A. Schiller, J. Ran[t / Massive Schwinger model

206

For later use also the variable a is introduced. In hamiltonian language this model has been studied by Coleman [10] bosonizing the system and integrating out redundant gauge field variables (in Coulomb gauge) ff~a=l

1 2,1 ~H,2 + ~(0~o') -,~tx 2cr2 - c r n ~ cos ( 2 4 ~ r - 0 ) .

(6)

Here normal ordering with respect to the mass t~ is understood. The constants 0z and c are defined as follows

tz = g/'Vr~,

c = exp (y)/2~',

y = 0.5¢74 •. •.

(7)

H~ is the canonical momentum of the real field or which was introduced to eliminate the fermions i ~ t b ~ ½(O~,o')(O~'o'),

~$ ~ - c ~ cos (2,/~ o-).

(8)

In the limit of massless fermions (m = 0) the model with hamiltonian density (6) is exactly soluble and describes a massive boson of mass t~ which can be interpreted as a fermion-antifermion bound state. The chiral symmetry is spontaneously broken, the quarks (fermions) are confined. For m = 0 also the dependence of ~ on the background field variable 0 is removed and the external test charge are completely screened. The quark condensate is expected to be (Eo is the ground state energy)

(0~) = d~-~Eo(m, O)l,,=o = -ctz cos 0.

(9)

In the case of massive fermions the model of (6) has been studied in the strong (m <>g) coupling regions. Defining a string tension as increase in the ground state energy per unit length due to external sources 1 T = -f~ (E(O) - E(O)),

(10)

one can study whether besides of the quark charges the test charges are confined too. If q is an integer multiple of the quark charge g the test charges are screened in the massive Schwinger model (T = 0). For any other value of q/g the test charges are confined and a non-zero string tension occurs. Special attention has been given to the case of the background field 101 = w (lal = ½). As pointed out by Coleman, a first order phase transition appears between strong coupling and weak coupling regions. At strong coupling, at small values of m/g, the model has only a single vacuum. Above some critical value of m / g there are two degenerate vacuum states breaking the charge conjugation symmetry spontaneously. Half-asymptotic states appear corresponding to a pair of widely separated unit charges provided that they are in a certain order. Below the critical value no liberation of these states occurs.

A. Schiller, I. Ranft / Massive Schwinger model

207

The lattice formulation of the massive Schwinger model is well-known [12] and we sketch here only briefly its derivation. Using the axial gauge (A0 = 0) we obtain from (1) and (4) the continuum hamiltonian density

~( = ~([[A, -- got )2 + ~(i3"1(0, + igA ~) + m )d/

( 11 )

[[IAI(X), A,(y)] = -iS(x - y ) .

(12)

with

Introducing a spacelike lattice with lattice spacing a, setting the fermion spinor components on neighbouring sites in the manner of Susskind [13] and the gauge fields on links between the sites one obtains

H = ½ a g 2 Y . ( L . - a ) 2 + - - iY . ( ~ .+U . . . +14t. +1 - h.c.) + m Y. ( - 1)"dr.+dr. .

2a

,.

.

[L,,, U,..,.+I]=6.mU . . . . i .

(13) (14)

We have used the following representation for the Dirac matrices 3'°=(10

?1)'

3''=(71

10)"

(15)

The sum in (13) runs over all sites at a given time. The link operator is connected to the gauge field A,(n) as usual U... +x = exp (igaA ,(n)) --- exp iO..

(16)

The operator L . = lla,(n)/g generates rotation in 0. with an integer number allowed eigenvalue spectrum. With

{x+.,x.}=a,,,,,,

)t'. = (i)"0.,

{x.,xm} = 0

(17)

and introducing the boson number representation e i°" = a.+, 4-

+

[L., a,.] = 6 . , . a , . ,

[L., a,.] = - 6 . , . a , . ,

(18)

we obtain the lattice hamiltonian used in our calculations 2

1

+

+

H=½g2a ~ (L,,-o~) +~a ~ (X,,a,,x.4-,+h.c.)+m ~(,-1)"X~X.

(19)

The three pices of H can be recognized as the gauge or electric field strength term H o , the fermion-gauge field interaction term Hr-G and the quark mass term HM. If we denote the eigenstates of L . by II) then a.+ (a.) increases (decreases) the boson number by one

a+.ll)=]l+l),

a.ll)=ll-1).

(20)

208

A. Schiller, J. Ranft / Massive Schwinger model

Boson and fermion operators commute. Therefore both fermions and bosons are described by occupation numbers. At a given time t~ the site n for one fermionic degree of freedom can be either empty or full whereas the link between the sites n and n + 1 is occupied by positive or negative integers for the bosons. Denoting the corresponding fermion and boson numbers by it,. (=0, 1) and k t,. the state at time t~ is given by Jr1) = H(x.+ Y~..(a.+)k~,"[o) n

(21)

x.10) = 0 ,

(22)

with L.10) = 0 .

3. Application of the local hamiltonian method We use a method, which was proposed and applied to solve non-relativistic 1 + 1 dimensional models by Hirsch et al, [7]. Martin and Otto [8] applied the method first to a lattice gauge model, the massless Schwinger model. Given the lattice hamiltonian (19), all properties of the model are determined from expectation values ( 0 ) = Tr (O e - t m ) / T r (e -oM)

(23)

where/3 is the inverse temperature. The traces in (23) are calculated by inserting a complete set of states at each interval ( n A t = [3) T r ( e -all) =

Y. (ttle , -a.t-t Itn) , '" • " ( t'31 e -a*u [2)( t' t'21 e -a~t4 [t't). t;,t~,...,t~

(24)

However the matrix elements of e =am are not local, Because matrix elements of local operators are more easy to evaluate, the hamiltonian is split into two pieces H = Ht + H2,

(25)

where the H~ consist of sums of commuting terms which couple only nearest neighbour sites. Inserting further complete sets of states we obtain finally Tr (e - ° H ) =

E (tlle-a'H21t2.)(te.le-a'n'lt2.-t) fl*/2,...)/2n • " (t3le-a*u2lt2)(t2[e-Z~"l[q) + O(Ar3).

(26)

In order to use this method we split our hamiltonian H (19) into two pieces acting on variables attached to even (odd) links. A link between sites n and n + 1 is even (odd) if n is even (odd). We obtain

I"t,= £ H~"> n odd

=~g

t2a

~ (L,,-ot)2+ n odd

2 - ~n

(x.a.x.+a+h.c.)-m d

++

~, X . X . n odd

+

,

(27a)

209

A. Schiller, J. Ranft / Massive Schwinger model

H2 = E H~~' tl even

1

1

2

= ~g a

Y. ( L . - a ) 2+ .

even

__~_x

2a

,r

+

L

+

O¢.a.x.÷l+h.c.)+m

n even

~.

+

X nX,, •

(27b)

n even

Note that HI and H2 consist each of sums of commuting terms. Each term in H1,2 only changes quantum numbers on a link and the adjoint sites. Therefore e -a'u'=

]-I e -a'uW.

(28)

n odd

The matrix elements needed to calculate have the following structure (tele-a'H~]t0 = l-I (01(X,,+0'2'"'~(x.)i~'"(a.) k''" n odd

×e

-- A r H ° ~

)t

+ \i I

~.)

/

+

"

'"~.+l)'""'l(a.+)k""[0) l-I 6k,..k2...

(29)

n even

They can be easily calculated to order 0(AT 3) using the relation e

-

= e -~a'/2)t't~' e -ta*/2)t't'~' e -a'u¢"-L e-ta'/2~n'g~'e-ta'/2)n ~, +O(AT a).

(30)

Specifying the numbers i and k all non-vanishing local matrix elements between two neighbouring times slices tt and tt+l are given in table 1. Full (open) circles denote occupied (empty) sites for the fermion components. The links between sites n and n + 1 have boson occupation numbers kt at time tt and kt+l at tt+l. The splitting procedure for the hamiitionian allows the introduction of "strings" connecting the occupied fermion sites in time. As for the massless model [8], besides of the obvious periodic boundary conditions in time also periodic boundary conditions in space for each of the strings are imposed. For updating the lattice at one link and the neighbouring sites, one has to consider a block of four elementary matrix elements. In fig. 1 we give the four possible transitions. They occur between the configurations given by the solid and dashed fermion strings and vice versa. The matrix elements given in the table sit on the shadowed squares of the lattice where the fermions can hop and interact. The updating is most conveniently done using importance sampling. 4. The Monte Carlo calculation and results on fermion condensate and string tension

Monte Carlo calculations for the Schwinger model were so far reported in refs. [8, 14, 15]. Our calculations have been performed on ESER and IBM computers.

A. Schiller, J. Ran[t /

210

Massive Schwinger model

TABLE 1 Matrix elements - (-l)nm,a~ ")

8k,.k,+, exp (-½gZaAr(kt v

n

? ;

v

t~).l

~ 8k, kt.l exp (-½gEaAl"(kt - a ) 2) .

2

X

n

AT

, , . , . °xo( .','a,.,(,-o,'.(,., o>'> ,',o,.> ,,°, 2

,1

/Ark

-Sk,+ ~.k,.,exp (-~g2aAr((k, - a )2 + (k,+ ~ - a) ) - (- I) ~mzlr) sinh ~a)

Space-time lattices of size 40 x 100, 20 x 50 and 40 x 50 have been used to study some of the interesting properties of the massive Schwinger model mentioned in sect. 2. As the initial lattice, we have chosen the ground state for the massless model in a vanishing external background field. The particle configuration is characterized by setting the boson numbers on the links equal to zero and by equally spacing

Fig. 1. Possible elementary transitions.

A. Schiller,J. Ranft / Massive Schwingermodel

211

the fermion occupation numbers zero and one on the sites. In this massless lattice model two degenerate states exist: occupied even sites and unoccupied odd sites or vice versa. One state can be transformed into the other by shifting the lattice by an odd number of lattice spacings. This corresponds to a discrete chiral transformation. Therefore, the discrete chiral symmetry is spontaneously broken. Studying the massive model we have to choose that ground state which smoothly changes when the quark mass is increased from zero. Within our convention the quark mass term has the form HM = m Y~( - 1 ) "X ,+X , ,

(31)

n

from what follows that the odd sites have to be occupied in the correct ground state for the massive Schwinger model on the lattice. In the massless limit we start always from this ground state as well. We have fixed the asymmetry A~-/a to 0.2 to minimize the neglected contributions in the trace (26) and in the matrix elements. For the observables studied we have checked that a change to zir/a = 0.1 does not change the results significantly. In the following we set the lattice spacing a equal to 1. Starting with the initial lattice and choosing m and a values (la] <~0.5) we have usually heated the lattice in 600 sweeps. This number in general turns out to be sufficient to stabilize the observables being measured. For following small changes in the model parameters typically 200 iterations have been used to thermalize the system again. For most runs we use relatively large couplings g where we reach rapidly the equilibrium. The measurements have been done always after five sweeps through the lattice to decrease correlation effects between the measured variables. All errors reported in the figures are statistical ones. They may underestimate the real errors since, e.g, finite lattice size effects cannot be neglected a priori and correlation effects are not completely excluded. The continuum Schwinger model predicts a non-zero fermion condensate for vanishing quark masses as a result of the spontaneously broken chiral symmetry. Marinari et al. [ 15], using euclidean Monte Carlo methods with fermions, reproduce thi~ continuum prediction approximately as an extrapolation from the massive model. In our local hamiltonian method we obtain a non-zero condensate defined as

for zero quark masses consistent with eq. (9). As can be seen in fig. 2 the quantity ( ~ b ) / g with increasing m / g ratio is nearly independent of the different couplings used. The lattice prediction for the condensate as a function of the background field value a for zero quark masses is compared with the exact continuum result in

212

A. Schiller, J. Ranft / Massive Sch winger model 40 x 100 ,, g : 0 2

I

• g :04

o g:O.?

-0.2@

÷

-0.4 -

K> -06 -

-0.8

÷ +

-

0

I

I

I

I

I

I

0.2

0,4

0.6

0.8

1.0

1.2

m

g

Fig. 2. Quark condensate over coupling as function of the ratio m/g for different couplings.

i

0.08

i

40xi00 m=O0 g-04

I

~,

Om 004 002 0 -OOZ -0.04 -006 -008 I

0.5

I

1.0

Cl

Fig. 3. Q u a r k condenstate at m = 0.0 and g = 0.4 versus background field compared to the expected c o n t i n u u m behaviour of eq. (9).

A. Schiller, Z Ranft / Massive Schwinger model E

213

20 xSO 9=.~ m-O.

-.29

++++++

-.30

+

Fig. 4. Average energy at m = 0.0 and g = 0.4 versus background field.

fig. 3. A coupling g = 0.4 is used. The Monte Carlo data of the condensate have been obtained using 120 and partly 240 m e a s u r e m e n t s per point. The normalization is in rough agreement with the continuum expectation, especially also for a ~>0.5. It is encouraging that the lattice calculation within this approach reproduces both magnitude and t~-dependence of the condensate whereas the continuous y5 symm e t r y responsible for this behaviour is recovered only in the limit a -~ 0. We have checked that with increasing mass at fixed coupling the sign changing effect of fig. 3 disappears. For large mass and outside the phase transition at a = 0 . 5 the condensate is roughly independent of o~. Now we study the string tension defined in eq. (10) as change in the average energy when varying the background field. This string tension characterizes the behaviour of the test charges responsible for the constant electric field. For this analysis we use a lattice of size 20 x 50. For vanishing quark masses the test charges in the continuum model are screened. We expect no dependence of the ground state energy on the background field p a r a m e t e r o~. In fig. 4 we show the average energy versus a for zero quark masses and coupling g = 0.4. 120 m e a s u r e m e n t s per data point have been used. Clearly the string tension vanishes within errors. In the massive model for a # n, where n is an integer, the string tension is to be expected different from zero indicating confined test charges. For small masses one can extract the string tension T ( a ) (10) from the hamiitonian density (11) T ( a ) cc I - cos ( 2 I r a ) .

20x50

g=l. m-,1

+

-20

+÷ +

(33)

++ +

+ +

-.25 i

J

+ $

O. .2 Fig. 5. Average energy at m = 0.2 and g = 1.0 versus background field.

A. Schiller, J. Ranft / Massive Schwinger model

214

20x50 g=.7 m - .S

,÷~-÷, ~

-.30 • ~, i n c r e a s i n g

,~

~.d e c r e a s i n g

-,

-,,

~

,~,

,

,+"

',

+ ,

-.40 A

0

.~

~

i

.~

~

÷ ~.+..~ |

.~,

Fig. 6. H y s t e r e s i s b e h a v i o u r of the a v e r a g e e n e r g y at m = 0.5 a n d g = 0.7 as function of the b a c k g r o u n d field.

With g = 1.0, m = 0.1 and 45 measurements per point we find the behaviour as shown in fig. 5. We observe the increase in the average energy (string tension non-zero) with a symmetric m a x i m u m around a = 0.5. For a = 0.0 and 1.0 the energies are equal within statistical errors indicating the described screening of the external test charges. We note that the ratio m/g = 0.1 in this study is rather small. Larger m/g values are studied in the next section.

5. Phase transition studies in the massive Schwinger model at eL= 0.5

As noticed in the last section, for small m/g the string tension is almost symmetric around a = 0.5. Fixing the coupling to g = 0.7 and m = 0.5 we obtain the average energy as function of the background field shown in fig. 6. Our starting configuration is that for a = 0. In steps of 0.05 in a we run to a = 1 and come back to the initial value. 120 (240) measurements have been used to obtain the data in the region 0~ a >10.5 (0.5 > a ~ 0 ) . The lines in the figures are drawn to guide the eyes. Different from fig. 5 we find a non-symmetric behaviour around a = 0 . 5 , A p a r t from the confinement and screening mechanisms mentioned before we observe a typical hysteresis behaviour for the energy as function of ~. Crossing a = 0.5 the system first runs into an energetically unfavoured metastable phase followed by a decay of this phase into the stable vacuum. Below a = 0.5 in our boson n u m b e r representation used the stable ground state is characterized by a configuration where most of the links are empty. Such a configuration minimizes the energy obtained from the hamiltonian (19). A b o v e a = 0.5 the energetically favoured state is that of an average occupation number one accompanied by some fluctuations. At ct = 0 . 5 both phases are characterized by the same average energy (see fig. 6), therefore two degenerate stable vacua exist.

,~-~

~

~~

g~ ~

e~ e

~"

, g=~"

~

E8

~'~

m

o o o o o o o o o o o o o o o o o o o o o o IJ o o o o o o ~ o o o

o o o o o o o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o o o o o m o

o o o o o o o o o o m o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o o o o o o o o o o o o o o o o o

o o o o o o o o o

i

o o o ~ o ~ o

o

o o m

o o o a o o ~ o o o o

o

o o o o o o o o o o o

o o o o o o ~ o o o m

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o ~ o o o o

o o o o o o o o o o o

o o o o o o o ~ o o o

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o o o o m o o o o o o

o o o o o o o o o o o

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o o o o o o o o o o o

o o o o

o o o o o o o o o o o ~

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o o o o o o

o o o

o

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o o o o o

o

o

~ o o o o o o o o

o

o o o o o o o o o o o

o o o o o o o o o o o

o~

~

o o o o o o o o o o o

o o o o o o o o o o o

o o e o o o o o o o o

o o o o o o o

o

o ~ o o o o o o o o ~

~ o o o o o o o o o o

o o o o o o o o o ~ o

o o o o o o o o o o o

o o o o ~ o o o o o o

o o o o o o o o o o o

o o o o a o o o o o o

e

o w w o o o o o o o o

o o o o o o o o o o o

o

o o o o o

o o o o o

o o o o o o o o o o o

o o o o o o o o o o o

o o o o o ~

o o o o o o o o o o o

o o o o

L'

o o o o a o o o o o o

o o o o o o o o o o o

~ o o o o o o o o o o

c:

,--, n

N

S"

~.. ,~} C~.

~.~

~-i~g

_~.

2.~.~ ~ ~

:::)

216

A. Schiller, J. Ranft / Massive Schwinger model

-.25

~

20x50 9-.7 rn .5

,7

=

-.55 -.30

-.:15

"

500

1000

1500

iterations

Fig. 8. Energy versus number of iterations in the mixed phase run o f fig. 7 for (~ = 0.45 and a = 0.55.

The digits give the boson numbers on the links. The vertical direction is a part of the time axis. Setting a = 0.45 we show this part of the lattice after about 500, 1000 and 1500 iterations in figs. 7b, c and d. Clearly the rise of the correct "zero phase" is visible, similar to the growth of a crystal. Ditierent to this we see in fig. 7e the coexistence of both phases after 1500 iterations fixing a to 0.50. In Fig. 8 the average energy is studied as a function of the n u m b e r of iterations for a = 0.45 and a = 0.55 indicating the hopping of the energy into the stable ground state value after a certain "time". The Monte Carlo data are averaged over ten measurements to decrease somewhat the fluctuations. The observed first-order phase transition at a = 0.5 corresponds to the existence of two degenerate vacuum states breaking the charge conjugation symmetry spontaneously for relatively large m/g values. The non-symmetric behaviour of the energy around (~ = 0.5 allows us to find the critical value of the mass over coupling ratio (m/g)c above which this transition should start. Below (rn/g)c we expect a rather symmetric energy distribution. In figs, 9a, b the average energy is studied as a function over the background field values at fixed coupling constants g = 1.0 and g = 0.7, varying m/g. Again we have used a lattice of size 20 x 50 and have accumulated 120 measurements in 600 iterations per data point. From these figures we extract the critical ratios (m/g)c = 0 . 3 0 + 0 . 0 1 , g = 1.0, (34a)

(m/g)¢ =

0.31 + 0.01,

g = 0.7,

(34b)

in agreement with the numbers reported by H a m e r et al. [16] using finite-size scaling methods. A hysteresis curve near the critical coupling is shown in fig. 10 at coupling g = 0.7 and m/g = 0.32. Crossing a = 0.5 from below the system runs first into the stable phase (decreasing energy) and hopping then into the metastable one. This indicates the large tunneling probability between the two phases near the endpoint of the

A. Schiller, J. Ran[t / Massive $chwinger model E

20x50 g-1.

-.15

,"

+ .29

o

217

,~,

.

'

i

/r:,\

.:.20

/

.

\

f

~5

|

i

i

.60 E

i

.SO

i

i

.60

20xS0 g'7

-.23 +.30

-.27

i

i

i

./.0

/

.SO

i

.60

Fig. 9. S t u d y of the a v e r a g e e n e r g y a r o u n d a = 0 . 5 0 s t a r t i n g w i t h a = 0.40 for different r a t i o s g = 1.0; (b) g = 0.7.

E20'

20;50 g-.7 m/g-.32

÷ increasing

~-

+ decreasing

a.

m/g.

,~, /

,'

i /

-.2S

f

/ "\

ii

~,.

-3o

"f .'~0

'

•s'o

Fig. 10. H y s t e r e s i s run v e r s u s a n e a r the critical r a t i o

'

(re~g) for

.70'

-"

g = 0.7 a n d

m/g

= 0.32.

(a)

A. Schiller, J. Ranft / Massive Schwinger model

218

first order phase transition line at a = 0.5. The observed behaviour is closely related to the phase transition for the massive Schwinger model at a = 0.5 (0 = ~r) conjectured by Coleman when going from the strong coupling to the weak coupling regime. The results found are in good agreement with the expected transition from a single vacuum at small m/g to two degenerate vacuum states at large m/g corresponding to the liberation of Coleman's half-asymptotic states. Fixing the background field value to 0.5 we have studied the average boson occupation number L as function of the iteration number in the weak and strong coupling region. A lattice of size 40 x 50 and a coupling g = 1.0 have been used. Starting with the initial ground state where all links are empty we obtain the results as shown in fig. 11. Every tenth value L is given in the figures. The averaged values reported have been obtained excluding the first 500 iterations used for heating. Above the critical value (rn/g)c=0.30 the system remains in the initially chosen ground state characterized by the zero occupation number for the bosons (fig. 1 la). In the strong coupling region the ground state is not specified by a selected average boson number. The number L can fluctuate between zero and one as shown in fig. l l b . For a large enough number of iterations we expect an ( L ) = 0 . 5 0 . Nearer to

Z,O=50 g=l. v.=.5 • nn/g - . 3 5 + rn/g =.40


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Fig. 11. A v e r a g e boson occupation n u m b e r v e n u s n u m b e r of iterations at ~ = 0 . 5 0 and g = 1.0 (a)

m/g = 0.35 and 0.40 (b) m/g = 0.10 and 0.25.

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the phase transition point the fluctuations are of large "time" order. This explains the still rather small (but increasing) average number for m/g = 0.25.

7. Summary We find the local hamiltonian Monte Carlo method [6] for lattice gauge theories including fermions in two dimensions to work very efficient. The known continuum results for the Schwinger model are reproduced without difficulty. We study here especially the fermion condensate and the string tension and their dependence on m/g and the background field parameter a. At a = 0.5 we find the phase transition conjectured by Coleman and determine the critical (m/g)~0.30. At weak coupling, above (m/g)c=0.30 the model is characterized by the existence of two degenerate vacua. These vacua can be well identified by our average boson occupation numbers as well as by direct inspection of the boson number distribution. According to Coleman this phase transition should be connected with the existence of semi-asymptotic states in the weak coupling phase. So far we were not able to identify order parameters related to these states. We acknowledge discussions with G. Bhanot, J. Kripfganz and H.-J. M6hring and help in running computer programs by S. Henger. It is a pleasure of A.S. to thank the CERN Theory Division where part of this work was done for its hospitality.

References [1] J. Schwinger, Phys. Rev. 128 (1962) 2425; J. Lowenstein and J. Swieca, Ann. of Phys. 68 (1971) 172 [2] D.J. Scalapino and R.L. Sugar, Phys. Rev. Lett. 46 (1981) 519; D. Weingarten and D. Petcher, Phys. Lett. 99B (1981) 333; F. Fucito, E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B180 [FS2] (1981) 369; A. Hasenfratz and P. Hasenfratz, Phys. Lett. 104B (1981) 489; C.B. Lang and H. Nicolai, Nucl. Phys. B200 [FS4] (1982) 135; J. Kuti, Phys. Rev. Lett. 49 (1982) 183; G. Parisi, Nucl. Phys. B205 [FSS] (1982) 337 [3] For a summary see Proc. of the XXI Int. Conf. on High Energy Physics, Paris (1982); C. Rebbi, p. C3-723 and G. Martinelli, p. C3-278; Proc. of the John Hopkins Workshop, Florence (1982), P. Hasenfratz (p. 1), H. Hamber (p. 71), G. Martinelli (p. 121) [4] P. Hasenfratz and I. Montvay, Phys. Rev. Lett. 50 (1983) 309; G. Martinelli, G. Parisi, R. Petronzio and F. Rapuano, Phys. Lett. 122B (1983) 283 [5] G. Bhanot and R. Dashen, Phys. Lett. 113B (1982) 299 [6] I.E. Hirsch et al., Phys. Rev. Lett. 47 (1981) 1628 [7] I.E. Hirsch, R.L. Sugar. D.J. Scalapino and R. Blankenbecler, Phys. Rev. B26 (1982) 1304 [8] O. Martin and S. Otto, Nucl. Phys. B203 (1982) 193 [9] J. Ranft and A. Schiller, Phys. Lett. 122B (1983) 403

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