Estimate of the lattice QED phase transition from hamiltonian strong coupling expansions

Volume 198, number 4

PHYSICS LETTERS B

3 December 1987

ESTIMATE OF THE LATTICE QED PHASE TRANSITION FROM HAMILTONIAN STRONG COUPLING EXPANSIONS C J BURDEN Department of Theoreucal Physics, Research School of Ph vswal Sciences, The A ustrahan National Um verslly, Canberra ACT 2601, 4ustraha Received 1 September 1987

Hamlltonlan strong couphng expansmns are evaluated for latnce quantum elecirodynamlcs m (3+ 1) &menslons using the hnked cluster algortlhm Pade analysis of the chlral condensate series (to order g-iS) shows evidence for a possible continuous transmon at 1/g~= 0 75 _+0 03 when the bare fermmn mass is zero The transmon is signalled by a vamshmg ehlral condensate with crmcal index 0 22 _+0 02 A discontinuous phase transition is by no means ruled out by the calculanon, m which case the value quoted above is an upper bound on I/~ Strong couphng expansmns for the vacuum energy to order g-Z8 and scalar photonball mass gap to order g - TM are also given

Lattice q u a n t u m electrodynamlcs is expected to undergo a phase transition between the strong coupling (confining) and weak coupling ( u n c o n f i n i n g ) limits of the theory This transition is well established for the pure gauge theory both analytically [ 1 ] and numerically [2], while recent r e n o r m a h z a t i o n group analyses [ 3 ] favour a c o n t m u o u s transition for the Wilson action A M o n t e Carlo study of full QED with dynamical staggered fermions by Azcoltl et al [4] has also produced a phase transition. Applying a pseudo-fermion algorithm to four-dimensional euclidean Q E D AzCOltl et al. measure a transition in the vicimty of 1/g 2 = 1 0 for small ferm i o n masses The transition ts signalled by a simultaneous restoration of chiral symmetry and onset of deconfinement. A more recent Monte Carlo analysis by Kogut a n d Dagotto [5] using hybrid stochastic differential equations favours a first-order transition for QED when four flavours of very light fermions are present In this letter we apply the linked cluster method [6] to obtain strong couphng expansions for hamiltoman Q E D in (3 + 1) dtmenslons. The method has previously been applied to the pure gauge case [ 7], where m e a s u r e m e n t s of the string tension a n d specific heat were used to estimate the transition at 1/g2=O 85 The presence of hght dynamical fer-

m i o n s makes the string tension an unsuitable order parameter for the present case On the other hand, fermlon d o u b h n g provides us with a chlral symmetry which is spontaneously broken in the strong couphng limit [8] Using Pad6 analysis techniques we obtain an estimate of the ehiral symmetry restormg t r a n s m o n for zero bare fermlons mass, assummg a c o n t i n u o u s transition Series expansions for the specific heat a n d scalar " p h o t o n b a l l " mass gap are also given, but prove to be considerably less useful for determining the phase transition We use the K o g u t - S u s s k i n d Q E D h a m i l t o n i a n [8,9]

H=(g2/2a)W,

(la)

W=Wo+yW~+y2V~

(y=I/g2),

(Ib)

Wo= W~ + v¢,,

=Z E~+/z Z (-1)~'+'-*'~zt(r)z(r), (~

(lc)

r

W, = f. q(r,) [ z t ( r ) U , ( r ) z ( r + a h + h . c

],

fl

rlt(r)=(_l)r,+

+r,

I4~ = - Z ( U . . + g~..), p

,,

(ld)

(le)

where r = (r~, r~, r3) labels the sites, £ the links, p the 525

Volume 198, number 4

PHYSICS LETTERS B

plaquettes and t = 1, 2, 3 the &rectxons on a square three-&mensmnal spatml lattice with spacing a The d~menslonless fermxon mass/z ~s related to its contlnuum counterpart by

l~= 2am/g 2

(2)

The bosonlc operators E, U and single-componentper-site fermmn operators Z satisfy [E~,U~]=U~d~,

[EQ, U ~ ] = - U ~ 6 ~ ,

{Z* (r), z ( r ) } = d , r , [E~, z ( r ) ] = [E~, z*(r)] = [ U~, g(r)] = [ U~, x*(r)] = 0

(3)

means the product UI (r) U2( r+ ai ) x U](r+a2)U~(r) around the plaquette p In the

Uop

nmve continuum limit, the hamlltonmn (1) becomes the standard continuum Q E D h a m d t o m a n with couphng g and two flavours spin 1/2 fermlons of mass /t The mechamcs of the hamfltonlan strong couphng expansmn for gauge fields with fermmns are well known [ 10] We work m a strong couphng basis wxth an integer flux defined on each link and a f e r m m m c state I + ) on each site satisfying Z*l-)=l+), xl-)=10),

z*l+)=0,

The ground state of the unperturbed hamlltonian Wo has zero flux on each hnk, while the arbitrariness m the fermmn sector ~s broken at first order of perturbation to the fermlon state [ A ) defined by IA ) = l + ) =1-)

on oddmtes, on even sites

Since continuum chiral transformatmns correspond to shifts of one spacing on the lattice [ 8], the unperturbed ground state is chlrally asymmetric Restoratmn of chiral symmetry as signalled by a zero value o f the average chiral condensate per site" ( @ ~ / / ) lamce =

1

2N~

~ (_l)r,+r2+r~[z(rt),z(r)], T

(4) where N~ is the n u m b e r o f lattice sites Details o f the hnked cluster method applied to 526

gauge theories with fermions are set out m detail m ref [ 11 ], where (2 + 1)-dimensional Q E D is studied Strong couphng expansions for quantmes such as the v a c u u m energy or choral condensate per site are evaluated numerically as a sum of contributions from fimte sublatt~ces or clusters, the contribution from each cluster representing the strong couphng diagrams spanning that cluster [ 5 ] The required set of clusters up to a particular order of perturbatmn theory ~s first generated m the computer, and series expansmns are then evaluated for the hamlltoman restricted to each cluster An ~terat~ve procedure then gives the bulk series expansion The only sxgmficant comphcatlon involved m going from ( 2 + 1 ) to ( 3 + 1 ) dimensmns ~s the inclusion of closed clusters (such as the cube at order y6) [7] in a d d m o n to the usual open clusters [ 11 ] occurring at each other In table 1 we hst series expansion coefficients to order y14 (or g-28) for the ground state energy COo of the dimensionless h a m d t o m a n W, and for the chlral condensate (qT~u) lat't,ce at several values of the dimensionless mass # Generating the required clusters to thxs order took approximately 20 h C P U time on a Mxcrovax II computer, and the subsequent generation of series expansions took less than 20 min for each value of p Table 2 hsts estimates of the choral condensate and specific heat [ 12 ]

C= - 2y~-d2ooo/dy 2,

xJ+)=J-)

3 December 1987

(5)

obtained by fitting Pad6 and Shafer approxlmants to the strong couphng series Plots of these q u a n t m e s are shown in figs 1 and 2 for # = 0 , 0 and 0 5 , respectively If a second-order transition is assumed, the specific heat should diverge m the vicinity of the transition a s C~ (Yc-Y)-",

(6)

whale the choral condensate, for the massless case, is expected to vanish as ( ~ / / ) lattice ~ (Yc -Y)~

(7)

Singularities such as (6) or (7) will show up xn the logarithmic derivatives of functions as simple poles, the residue gxvmg the value of the critical exponent Since Padd approxlmants are able to represent poles accurately [ 13 ], we apphed Padd analysis to series

Volume 198, number 4

¢-q

I 1 [

PHYSICS LETTERS B

[

I

I

I

~ 7 ~

expansions o f the loganthmxc derivatives o f C and (ff'~') ~t~'~ A t / z = 0 the p r o c e d u r e exposed singularities m both quantities at the unphys~cal point Yo ~ - 0 0162 These singularities move out along the negative y2 axis as/z increases, and a p p r o a c h the origin a t / z - - , - 1/2, at which p o i n t the hghtest unperturbed meson-hke state becomes mass degenerate w~th the u n p e r t u r b e d v a c u u m A s~mllar p h e n o m e non has been seen m the ( 2 + l ) - & m e n s l o n a l case [ 11 ] To r e m o v e this unphyslcal singularity we used the Euler transform [ 13 ]

I

~ -

t=x/(1-X/Xo),

A

where x = y ~-to shift the u n w a n t e d singularity to infinity, a n d Pad6 analysed the logarithmic derivative o f the resultant series F o r the specific heat, new (unphysical) s m g u l a r m e s In the complex x-plane were obtained, but no clear evidence for a stable singularity on the positive y-ax~s was seen On the other hand, evidence for a stable algebrazc zero m the choral condensate d i d emerge F r o m the results o f our Pad6 analys~s shown in table 3 we estimate

v

I

o 7~ e~

y~ ~ 0 75_+0 03, I

o

I

o

-~ e~

~g

3 December 1987

~

~

y ~ 0 22_+0 02

(8)

f o r / z = 0. The strong couphng expansion ~s l n v a h d m the weak couphng regime Y>Yc, where M o n t e Carlo results [4] suggest ( ~ / ) ~ a u ' c e - - 0 for the massless theory In fig I we have also plotted ( ~ / ~ f f ) l a t t , c e = K ( y c - v ) ' m the VlClmty o f y ~ = 0 75, with K c h o sen to be 0 3 to match the general b e h a v l o u r o f series a p p r o x l m a n t s T h e sharp d r o p m ( f f ' ~ ) ~a,t,ce near yc ~s consistent with the r a p i d j u m p m o r d e r p a r a m e ters seen m the M o n t e Carlo analysis o f ref [4] Turning now to the specific heat, we see no evidence to confirm or & s p r o v e the expected singularity at Yc in the plot o f C in fig. 1. Using the chlral condensate plot as a grade, one mzght expect the a s y m p t o t i c beh a v l o u r (6) to set in rapidly near y ~ 7 , though the series a p p r o x l m a n t s clearly do not converge well enough to exhibit this Finally, we note that if the transition were m fact first order, and not continuous, the strong couphng ser~es expansions wall g~ve correct values only out to the trans~tmn At this point, we expect the choral condensate curve to d r o p suddenly to zero, the value o f y~ given by (8) being an 527

Volume 198, number 4

PHYSICS LETTERS B

3 December 1987

Table 2 Estimates of the speofic heat t~and chlral condensate
/t 00

specific heat

choral condensate

03

02 04

0 1883(9) 1 0(2)

06 08

4(1)

00 02 04 06 08

05 03847(1) 0288(6) 021(2)

05

0 20719(3) I 20(2)

0 20400(I) 1 25(I)

5 1(4) 16(2)

u p p e r b o u n d o n t h e actual t r a n s l U o n A s~mdar P a d 6 analys~s was c a r r i e d o u t on the series expansxons for C a n d ( V)~u) ~ t , ~ at/~ = 0 5. O n c e a g a m the C series was p l a g u e d w i t h unphys~cal singularities a n d n o result for the p h a s e t r a n s m o n was

20

0 14474 1 204(I)

5 30(5) 16 5(7)

05 043571 03518(6) 028(1)

5 20(3) 15 6(5)

05 045391 03815(1) 0317(3) 0 26(2)

05 049087 046769 043711(1) 0 4015(5)

seen F o r the chlral c o n d e n s a t e , a v e r y clear algebraic z e r o s h o w e d up f r o m a P a d 6 analysis o f the loga r i t h m i c d e r i v a t i v e after E u l e r t r a n s f o r m i n g a w a y a singularity at y2 ~ _ 0 0614 F r o m table 3 we e s t i m a t e !

I

I

I

I

I

!

I

/t[2/3

]

[2/2/1]~/tI1/2/2]

I

200

300

~: loo

200

oo

100

o5

00

j

~

[

3

/

3

1

-

05

<@~'>o3

13]

[t./3]

I Or,

I

y

', ~ 08

(ff ¢) -

l

_ 1.2

Fig 1 Series approx]mants to the specxfic heat ~ and chlral condensate (~q/)~u,~ at/~=0 The dashed curve is the asymptotxc behavlour (~q/) ~"'~ =K( Vo-Y)' with Vo=075, 7 = 0 2 2 from the Pade result (8) and K= 03 chosen to match the series approxlmants Pad6 approxlmants (calculated as funcUons of v-') are labelled [p/q] and Shafer approxlmants [p/q/r] 528

~

~

,[3131

03 -

,

', I

0.0

-

"~-~[1/2/3]

0.1 0.0

0/~ -

~[2/2/2] [3/2; "~.~[3#4

02

\~L

J'[3fl,]

t2/2/2] ~ / / [ 2 1 2 / 2 1 [4/31/

",

t112131J

02 -

, t I

01-

'

I I i

l

oo

0.0

-

1

04

i

y

l

08

-

i i

2

Fxg 2 The same as m fig 1 for #=0 5 The asymptotic fit (q/~')~a"'C~=K(yo-V)' has v,~=0 914, y=0 196 fromeq (9)and K chosen to be 0 4

Volume 198 number 4

PHYSICS LETTERS B

3 December 1987

Table 3 Estimates of v~ and Y (in brackets) from poles m the [&IM] Pade approxlmants to the logarithmic derivative of the Euler transformed series expansion of ( qty.,~ J'"'" Esnmates are gr~ en for p = 0 and ~z= 0 5

00

05

N

[N/N-l]

[N/N]

[N/N+ II

1 2 3

0 6175(0 20531 0 7609(0 2197)

0 5057(0 1935) 0 7409(0 2176) 0 7658(0 2203)

0 7330(0 2167) 0 7184(0 2161)

1 2 3

0 8537(0 1870) 0 9148(0 1958)

0 7648(0 1767) 0 9145(0 1957) 0 9145(0 1957) "~

0 9140(0 1957) 0 9162(0 1962)

" Approxlmant with an intervening pole

y~=0.914_+0002,

),=0196+0005

(9)

Series a p p r o x l m a n t s to tff a n d (~7~') lalt,ce and the asymptotic behavlour (g)~,} la.,ce = K ( y c _ y ) ) ( c h o o s i n g K = 0 4) are p l o t t e d in fig 2 F o r n o n - z e r o b a r e f e r m l o n mass. chlral s y m m e t r y is explicitly broken, a n d a n o n - z e r o chlral c o n d e n s a t e is e x p e c t e d for all c o u p l i n g s I f this is the case, the b e h a v l o u r a r o u n d Yc q u o t e d in (9) r e p r e s e n t s an a n a l y t i c c o n t i n u a t i o n o f (v2g/)lall,ct. past the actual t r a n s i t i o n p o i n t T h e true c u r v e will f o l l o w the strong c o u p h n g c u r v e to the p h a s e t r a n s i t i o n , f r o m w h i c h p o i n t the w e a k coup h n g b e h a v m u r takes o v e r T h i s i n t e r p r e t a t i o n is c o n s i s t e n t w i t h an a s s u m p t m n t h a t t h e critical m verse c o u p l i n g 3'~ increases s m o o t h l y f r o m y ~ 0 75 a t / z = 0 o u t to the p u r e gauge d e c o n f i n l n g t r a n s i t i o n at yc-~O 85 as t t - - , ~ F o r c o m p l e t e n e s s we h a v e also a t t e m p t e d a strong c o u p l i n g series e x p a n s i o n a b o u t the u n p e r t u r b e d scalar p h o t o n b a l l state

1 v-~ ~ (uo~+g~) [o),

using a s i m i l a r a l g o r i t h m to the o n e d e s c r i b e d for the ( 2 + 1 ) - d i m e n s i o n a l case m r e f [ 11 ] E x p a n s i o n coefficients to o r d e r g - 1 6 o f the scalar state m a s s gap oJ~ are h s t e d m table 4 for v i n o u s v a l u e s o f the dim e n s i o n l e s s bare f e r m l o n m a s s / t Fig 3 shows plots o f Pad6 a n d S h a f e r a p p r o x l m a n t s to these series As in the ( 2 + 1 ) - d i m e n s i o n a l case, the b e h a v l o u r o f these c u r v e s is d o m i n a t e d by an a c c i d e n t a l m a s s deg e n e r a c y b e t w e e n the scalar p u r e gauge state a n d a m e s o n - h k e state at p = 1/2 m the strong c o u p h n g l i m i t In fact, plots o f the m a s s gap m (2 + 1) a n d (3 + 1 ) d i m e n s i o n s are a l m o s t i d e n t i c a l to this o r d e r [ 11 ] T h e series e x p a n s i o n s are t o o short to g w e any reliable i n d i c a t i o n o f a p h a s e t r a n s m o n , e i t h e r by visual m s p e c t m n o f fig 3 or by P a d 6 analysis o f the l o g a r i t h m i c d e n v a n v e o f o) 1 T o s u m m a r i z e , by Pad6 a n a l y s i n g strong c o u p h n g e x p a n s i o n s m massless ( 3 + 1 ) - & m e n s l o n a l q u a n t u m e l e c t r o d y n a m l c s , we find an algebraic z e r o m the chiral c o n d e n s a t e at the v a l u e q u o t e d m e q (8) I f the t r a n s i t i o n is c o n t i n u o u s , this z e r o gives the loc a t i o n o f the p h a s e t r a n s m o n I f ~t is first order, the

Table 4 Coefficients of v; ~ m the strong couphng series expansmns of the scalar photonball mass gap o) N

/.t 001

0 1 2 3 4

4 5 30013329835245 - 68 367887016299 2361 173364438 -99837 4927470

04 4 11 6959064327485 -41 701703638111 -295 3730919176 - 35498 2701007

08 4 - 2 22965440356744 26 447064206759 -431 8178508817 7259 59912810

12 4 - 0 622471210706505 7 78028517847E-3 - 12 21050872990 101 724446048

529

Volume 198, number 4 50

,

PHYSICS LETTERS B

i

c/

I

=0t~

I

i

I

~t =oo~

~-.0 -

2

/.t =0B 35 t

t

t

t

I

t

I

05

O0

3 December 1987

e m e r g e as m o r e t e r m s o f the strong c o u p l i n g series b e c o m e available, it will p r o v i d e c o m p e l l i n g evid e n c e for a s e c o n d - o r d e r t r a n s i t i o n , a n d h e n c e an ult r a v i o l e t fixed p o i n t for Q E D As the d i m e n s i o n l e s s m a s s # is increased, the zero in ( V V ) ,a.lce m o v e s to h i g h e r v a l u e s o f the i n v e r s e c o u p l i n g Since we e x p e c t ( C V ) ,au,ce #: 0 In the massive theory, these zeros give an u p p e r b o u n d on the actual t r a n s i t i o n Series e x p a n s i o n s for the s y m m e t ric scalar p h o t o n b a l l m a s s gap h a v e also b e e n determ i n e d , b u t are t o o short to shed f u r t h e r light on the n a t u r e o f the phase t r a n s i t i o n I t h a n k C J H a m e r for useful discussions a n d for r e a d i n g the m a n u s c r i p t , a n d also T C C h o y a n d R J B a x t e r for useful I n f o r m a t i o n

Y F~g 3 Graphs of the scalar photonball mass gap oJ t for various values of/~ The shaded areas exhlb~t the spread of the [ 1/2], [2/1 ] and [2/2] Pade and [ 1/1/1 ] Shafer approxlmants excluding obvious outhers

q u o t e d v a l u e o f Yc is an u p p e r b o u n d on t h e actual critical i n v e r s e c o u p h n g T h e o b v i o u s d i s c r e p a n c y w i t h the v a l u e s o b t a i n e d in the e u c l i d e a n M o n t e C a r l o analyses o f refs [4,5] is n o t p a r t i c u l a r l y significant, for two reasons Firstly, t h e r e is n o r e a s o n to e x p e c t an e x a c t m a t c h i n g b e t w e e n the l o c a t i o n o f the p h a s e t r a n s i t i o n s in the e u c l i d e a n l a g r a n g l a n a n d c o n t i n u o u s - t i m e h a m l l t o m a n f o r m a l i s m , t h o u g h any critical e x p o n e n t s are generally e x p e c t e d to be universal Secondly, b e c a u s e o f f e r m i o n d o u b l i n g in the dlscretazed t i m e d i r e c t i o n , the e u c l i d e a n f o r m a l i s m has f o u r f l a v o u r s o f f e r m i o n s in the n a i v e c o n t i n u u m limit, as o p p o s e d to t w o in o u r case. O n e can therefore argue that we are c o n s i d e r i n g a d i f f e r e n t c o n t m u u m t h e o r y with a d i f f e r e n t chlral s y m m e t r y group T h e o r d e r o f the phase t r a n s i t i o n w h e n two f l a v o u r s o f f e r m l o n s are p r e s e n t is still an o p e n quest i o n Pad6 analysis o f the specific h e a t series to O ( y ~4) has failed to find a stable pole c o i n c i d e n t w i t h the z e r o in the chxral c o n d e n s a t e I f such a p o l e d o e s

530

References [ 1] A Guth, Phys Rev D 21 (1980) 2291 [21 M Creutz, L Jacobs and C Rebbl, Phys Rev D 20 (1979) 1915, B LautrupandM Nauenberg, Phys Lett B95 (1980)63 [3] R Gupta, M A NovomyandR Cordery, Phys Lett B 172 (1986) 86, C B Lang, Phys Rev Lett 57 (1986) 1828, Nucl Phys B 280 (1987) 255 [4] V Azcoitl et al, Phys Lett B 175 (1986) 202 [5] J B Kogut and E Dagotto, Phys Rev Lett 59 (1987) 617 [ 6 ] A C Irving and C J Hamer, Nucl Phys B 230 IFSI0] (1984) 361 [ 7 ] A C Irving and C J Hamer, Nucl Pbys B 235 [FSII] (1984) 358 [8] L Susskmd, Plays Rev D 16 (1977) 3031 [9] J Kogut and L Susskmd, Phys Rev D 11 (1975) 395 [10] T Banks, L SusskmdandJ Kogut, Ph~s Rev D 13 (19761 1043 [ 11 ] C J Burden and C J Hamer, Hamfltoman strong couphng expansions for (2+ l)-dlmenslonal quantum electrodynamlcs, ANU preprmt (1987) [12] CJ H a m e r a n d M N Barber, J Phys A 14 (1981) 241 [ 13 ] D S Gaunt and A J Guttmann, Phase transitions and critical phenomena, eds C Domb and M S Green, Vol 3 (Academic Press, London, 1974) p 181