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Critical behaviour of compact electrodynamics in 4−ϵ dimensions
Volume 242, number 1
PHYSICS LETTERS B
31 May 1990
CRITICAL BEHAVIOUR OF COMPACT ELECTRODYNAMICS I N 4 - ~ D I M E N S I O N S E.-M. I L G E N F R I T Z and A. S C H I L L E R Sektion Physik, Karl-Marx-Universit~it Leipzig, DDR-701 Leipzig, GDR
Received 12 February 1990
Using truncated generalized Sierpinski carpets constructed from embedding hypercubic lattices we investigate the phase structure of the U ( 1) gauge theory for 4 - e dimensions. Slightly below d= 4 the first order phase transition disappears, d = 4 is conjectured to be the upper critical dimension.
The phase structure o f c o m p a c t U ( 1 ) lattice gauge theory in the charge one (coupling r ) and two (fla) m i x e d action plane r e m a i n s interesting, also from the p o i n t o f view o f testing new tools a n d approaches. Evidence as accumulating that puts the c o n t i n u u m limit o f this theory, restricted to W i l s o n ' s action, into question. There is a first order transition line [1 ] which extends into the region o f fla < 0. In o r d e r to understand better the critical b e h a v i o u r it is worthwhile to investigate the phase structure at dimension d e 4. The n o n - p e r t u r b a t i v e m e t h o d o f lattice simulation o f field theories can be c o n t i n u e d to d = 4 - e, using the construction o f fractal lattices [2]. A p a r t from this, there are a d d i t i o n a l reasons to study field theoretical models on non-standard geometries, near to the a p p r o p r i a t e c o n t i n u u m limit. The concept o f s m o o t h s p a c e - t i m e is likely to b r e a k down at the Planck scale a n d to be replaced by irregular metrics a n d / o r d y n a m i c a l dimensionality. N o n s t a n d a r d geometries can be i m p l e m e n t e d using fractal lattices [8] a p p r o a c h i n g flat four-dimensional space at large scales, Critical b e h a v i o u r on fractals has been investigated within the last few years m a i n l y for spin systems [ 4 ]. The first investigations o f some models, placed on particular fractal structures, have pointed out that only fractals having infinite order o f ramification allow for a continuous change o f phase structure with the fractional dimension, and the requirement o f translational invariance corresponds to low lacunarity [5 ]. All this suggests to use Sierpinski carpets.
The first papers concerning gauge field theories on fractal lattices were devoted to the study o f the phase structure o f the Z ( 2 ) gauge m o d e l in 4 - e [2 ] and Z ( 3 ) [6] in 3 - e d i m e n s i o n s a n d between 3 a n d 4 dimensions, supporting the conjecture to have an upper critical dimension 4 and 3, respectively. There was found a specific dimensionality ( " c o u p l i n g " d i m e n sion), suggested b y the plaquette structure o f the action, to give a smooth interpolation o f the critical coupling tic between integer dimensions, in contrast to the H a u s d o r f f dimension. In the present p a p e r we investigate what happens to the first order transition o f U ( 1 ) lattice gauge theory with Wilson action (fla= 0) below four d i m e n sions. Since the latent heat o f this transition is very small, we need m o r e elaborate tools to compare the four-dimensional t h e o r y with its lower-dimensional counterparts. In our simulations we used non-perfect lattices constructed by damaging the e m b e d d i n g D-dimensional hypercubic lattice ( D = 4) in a way that higherdimensional Sierpinski "carpets" suggest. Local gauge invariance ought to be kept on fractal lattices, too. This dictates to delete plaquettes from the hypercubic lattice. The notion o f a regular self-similar Sierpinski carpet is well defined in any e m b e d d i n g d i m e n s i o n D (integer). On a lattice this means to cut out blocks o f fundamental hypercubes o f different sizes (2n) 9, n = 0, 1..... n . . . . We decided to do this in a stochastic way, picking the blocks r a n d o m l y instead o f regularly. Deleting a block means to cancel all elementary
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
89
Volume 242, number 1
PHYSICS LETTERS B
hypercubes belonging to the block. Our rule for deleting the latter ones means omitting the forward plaquettes emanating from "leftmost" corners of the hypercubes (see fig. 1 ). These Sierpinski lattices are necessarily truncated. The minimal blocks are obviously the elementary hypercubes themselves (n = 0). The maximal blocks define the n u m b e r of levels of the fractal construction, K = F/max "[- l, which is limited by the size of the embedding lattice. All plaquettes not deleted are kept "active" in the action with the same coupling //. The links of the embedding lattice, which are not members of at least one active plaquette, have to be cancelled. In a block of size (2") ° the number of eliminated (unorientared) plaquettes is given by ½D(D- 1 ) (2") D, the number of links to be deleted is D. 2" ( 2 " - 1 ) o - 1. The same fraction c determines how many of the blocks of each size has to be deleted from the embedding lattice. Choosing c one determines the Hausdorff dimension
have to check whether all active plaquettes form a connected structure. Different random constructions of a fractal lattice of given dH will lead to different values of d. Strictly speaking, an ideal fractal structure could be represented only by the quenched average over a large number of fractal realizations, moreover in the limit of K-,c~. Apart from a few cases, however, our Monte Carlo simulations are based on only one single realization of the fractal lattice. For a given fractal lattice with Wilson action the partition function is Z=
dURexp
(
•
f l ~ Sw(Up) p
')
,
(3)
."
where ~R and 2p are restricted to active links and plaquettes, respectively. We approximate the U (1) gauge theory by a Z (N) one (with N = 16). The group elements on the links ~ are given by UR= e x p ( i 0 0 ,
dH = D - - l n ( 1 - c ) / l n 2.
(1)
The "coupling" dimension mentioned above is abstracted from the regular lattice
d=I+Np/NR,
(2)
where Np is the number of active plaquettes and NR the number of retained links. It corresponds to the "nearest neighbour" dimension dn, = number of bonds/number of sites, in the case of spin systems [ 4 ]. Before accepting such a randomly broken lattice as a Sierpinski lattice, we
/1 //
Fig. 1. Plaquettes to be deleted in a l ° block ( D = 3 shown for obvious reason), the "leftmost" corner is indicated.
27~jR N ' JR=0,1 ..... N - 1 .
0R
(4)
The group integration is understood as f dU~.... ~N-- 1 j~=o ..., and the action is
Sw(Up)=cosOp,
0p= Z r/(~,p)OR,
(5)
~p
t/(t~, p) = + 1 accounts for the relative orientation of plaquettes and links. Our Monte Carlo simulations for Z (16) have been performed with a five-hit Metropolis algorithm (selfcontrolling for 50% acceptance rate) on 44 and 64 embedding lattices in four dimensions. Our data consists of 2000-6000 measurements which are taken after every fifth sweep. The observables measured are the mean plaquette (spatially averaged for the given configuration) e= ~
90
31 May 1990
1
~ Sw(Up) ,
(6)
and the specific heat defined similarly to Lautrup and Nauenberg [ 7 ] (i.e. normalized by the dispersion of an individual plaquette)
Volume 242, number 1
PHYSICS LETTERS B
p'~p
×(~
[(Sw(Uo)2)-(Sw(Up))2])
.
(7)
. P
This specific heat has been calculated from the indicated averages for the whole sample, whose number o f configurations corresponds to the requirement that Cv did not change within 5% after 500 (1000 near the transition) measurements. In order to estimate the relative error o f the normalized specific heat we repeated the measurement near the phase transition on a 4 4 lattice over 5 more samples and found it o f the same size as the error of the standard specific heat. This amounts to a m a x i m u m of 5% for the other samples consisting of 2000-6000 measurements. In fig. 2 we show the normalized specific heat Cv versus fl for the smallest embedding and fractal lattices available. The non-integer dimensions d = 3.96 and d = 3.93 are created by deleting 3 and 6 elementary hypercubes. In both cases the Hausdorff dimension keeps much nearer to D = 4 than the "coupling" dimension d. We prevented from cutting out 2 4 blocks from a 44 lattice in order to stay near to four dimensions. The critical tic determined from the peak of Cv is shifted in qualitative agreement with mean field arguments. The peak value of Cv decreases indicating a toss o f coherence over the damaged lattice. At
this level nothing can be said about the presence o f a genuine phase transition and its eventual order. A finite size analysis could help to reach more definite conclusions. We compare in fig. 3 the specific heat measurements for 64 and 4 4 embedding lattices. Unfortunately, these lattice sizes are too small to extract any asymptotic scaling behaviour, and 6 4 also did not allow to construct a second level of our fractal construction. The dimension d = 3.93 is reached by deleting 31 elementary hypercubes from the 64 host lattice. We notice that the peak value of Cv rises clearly weaker than proportional to the volume, which would be expected for a first order phase transition. One must say, however, that this does not happen on regular four-dimensional lattices of that size either [7]. The so-called spectral function method [ 8 ] offers itself as an alternative possibility to detect first order transitions even on small lattices. One expects to observe different phases on a finite lattice, coexisting and tunneling from one phase to the other. This can be seen in a histogram, in our case of the mean plaquette e, showing a clear double-bump structure. At the critical coupling fl~ each b u m p carries the same weight. While the maxima should stay at fixed values o f the order parameter with increasing volume, they will become more and more pronounced due to the increasing tunneling barrier. In fig. 4 the spectral density of the mean plaquette at the respective critical coupling is shown for the same fractal lattices as in fig. 3 (both normalized arbitrarily to the same area). The width of the distri-
20-
25-
4 4
~ d=4.00
d =3.93
a=3.96 r_~
16-
'~ o
12
31 May 1990
or L\T
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ID 54 0.940
I 0.960
i
0,980
I
I
1,OO0
1.020
-
1.040
Fig. 2. Normalized specific heat Cv versus fl for a 4 4 embedding lattice and several "coupling" dimensions d
0, 0.980
I
I
I
1.000
1.029
1.040
Fig. 3. Normalized specific heat Cv versus fl for 4 4 and 64 embedding lattices at d= 3.93. 91
Volume 242, number 1
PHYSICS LETTERS B
800
0.750
64
•
44
0.650-
[] []
oo
400-
(a i
0.700-
[]
[]
d=4.00 .e=LOO14
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[]
d=3.93 600
31 May 1990
0.600~ "'"
41, o
":" ' "
......
o~
...t
0.550
200-
""
"
"
I" '~ " ' " - "
"
~1
'}'~,
....
,'1~
",,-' " , : ":
-.'.~
. ,:.,:...
[]o 0.500
I1450
- 'q.~ ~ g ~ ---. 0.650 0.750 mean plaquette
0.550
Fig. 4. Spectral density N(e) of mean plaquette at critical couplings as obtained from fig. 3 for 4~ and 6~ lattices at d= 3.93 ( normalized to same area ).
500
0
1000
1500
800
~ []
3000
0.750
64
.e=l.O~O
d=a.ga
(b)
O. 7 0 0
:~" :~ -.,,
~$ 0.650
0.600
64
2500 measurements
,~.,~ ,"~.'~ "~:
.':~;:'-.
• ~.%'...'--. ~.,..~- : - ~ > ' ~ , ' a t , , ~
N(e)
2000
~
[]
~$,..-..... •" . ,
"
-
".'':
.
r - ~ z . . . : , . : -~..X, ~.'-~.: ~ ,
"
' J
. '
.,
.,..,
"
"s"
~ ."" :
.I 0 . 5 5 0
600
~
d=4.00
[] 0.500 0
3000
40'00
5000
6000
Fig. 6. Mean plaquette versus Monte Carlo time at the respective critical/Y (for a 64 lattice} in dimension (a) d=4.00 (measurements after each fifth iteration), (b) d= 3.93 (measurements after each 25th iteration ).
.,/ 0.550
2000
measurements
/o 0.500
1000
0.600
0.650
0.700
n.~e a n p l a q u e t t ; e
Fig. 5. Spectral density N(e) at fl= 1.0014 and 1.0380 for lattice at d-4.00 and d= 3.93.
a 64
bution shrinks with increasing volume and does not show a pronounced double-bump structure. On the other hand, we compare in fig. 5 the spectral densities for the four-dimensional and fractal lattices (at the respective critical coupling and for the larger volu m e ) . Contrary to the fractal case, the four-dimensional lattice system shows two clearly separated m a x i m a of the spectral density, signalling the first order nature of the phase transition. Taken together, figs. 4 and 5 strongly suggest that the U ( 1 )-gauge theory with Wilson action on a fractal lattice of dimension d = 4 - e has a second order phase transition. The lesson of fig. 5 is supported by comparing figs. 6a with 6b which show the Monte Carlo t i m e history of the order parameter, the mean plaquette for the four-dimensional and fractal lattices, respectively. Recently, a new m e t h o d has been proposed [ 9 ] to 92
find the location o f critical couplings more precisely from the fourth order cumulant of the spectral density versus//. It has been~argued, moreover, that a finite size analysis [9,10] o f the cumulant is able to identify even the nature of the phase transition. The fourth-order cumulant BL (where L v is the size o f the embedding lattice) is defined as follows (e 4 )
BL(fl) = 1
3(e2)2
.
(8)
Here the averages are understood according to the spectral density at the given ft. In principle, the spectral functions for several flvalues can be determined from a single one at once using the idea ofreweighting [ 10,11 ]. In fact, we have analyzed a limited amount of data taken at s o m e fl (near to the proposed transition point ) by calculating the cumulant function. The m i n i m u m o f the cumulant is pointing to the more precise critical coupling. At this/Y-value a higher statistics Monte Carlo run has been performed. After one or more iterations of this procedure these data have been used to con-
Volume 242, number 1
PHYSICS LETTERSB
0.668
31 May 1990
~-BL(fl¢(L)),~ 4 CL(fl¢(L)) 3 Np[Eo(L,L(L))pc(L)]2,
n~.(t~) 0.665 0.602
'",..,
/,/'
""-,.,.
////
', f "" 0.656
,
0.960
= .
...... 44 d=4.00I ............ 44 d=a.gal
0.980
1.000
1.020
1.040
Fig. 7. Fourth cumulant BL v e r s u s fl from the spectral densities taken atfl=0.980 and 1.012 for the 44 andfl= 1.0014 and 1.0380 for the 64 lattices at d= 4.00 and 3.93, respectively. struct the respective curves in fig. 7. Comparing the geometrical shape of the c u m u l a n t curves, fitted as an inverted gaussian, in terms of the ratio R = w i d t h / height, one convinces oneself that the curve shrinks in four d i m e n s i o n s a n d widens in 4 - e dimensions with increasing volume. This is the expected behaviour for a first- a n d second-order transition, respectively, a n d can be understood within a p h e n o m e n o logical description of the spectral density N ( e ) . I n this framework the first order transition is described by a superposition of two equal weight gaussians at//o which should be extrapolated to n e a r b y / / by reweighting. The gaussians are centered at the averages eo+ a n d eo_, referring to the respective phases, and shrink proportionally to the volume. In the infinite volume limit a sharp m i n i m u m remains: 2(Eg+ + e ~ _ ) 3-f2 -- 2 2" t o+ ± C o - )
B rain = 1
(9)
Alternatively, a second order phase transition can be modelled by one single gaussian at//o extended as above to n e a r b y / / NL(e,//) = ~
const
exp [ (//~ --//) eNp]
/
×exp(,
[e-eo(L://¢)]:_fl~Np~ 2CL (//~)
(10)
] "
The specific heat m a x i m u m CL (//o(L)) rises ocL c"~, where ot is the thermal and v the correlation length critical exponent with ot/v
(11)
a n d vanishes in the finite volume limit. The gaussian ansatz is too crude to describe the global shape of the c u m u l a n t as function of//. In conclusion, we have d e m o n s t r a t e d that the firstorder phase transition of the U ( 1 ) gauge theory with Wilson action becomes lost when the embedding lattice is made irregular by r a n d o m l y deleting a small fraction ofplaquettes. We have to a d m i t that, due to the small embedding lattices we could afford, this did not really produce a fractal structure with self-similar defects over some different scales. On the other hand, from our earlier studies on Z (3) we know that more iterations of the fractal construction (Up to some scale of translational invariance) render phase transitions even softer. Further studies, both including more complicated actions (with admixtures of multiple charge loops) a n d on larger lattices allowing for more levels of damaging the regular lattice (looking more fractaMike) would be desirable.
References
[ 1] A. Hasenfratz, Phys. Lett. B 201 (1988) 492, and references therein. [2] G. Bhanot and R. Salvador,Phys. Lett. B 167 (1986) 343. [ 3 ] F. Englert, J.-M. Fr6re, M. Rooman and P. Spindel, Nucl. Phys. B 280 [FS18] (1987) 147. [4] G. Bhanot. H. Neuberger and J. Shapiro, Phys. Rev. Lett. 53 (1984) 2277; G. Bhanot, D. Duke and R. Salvador, Phys. Lett. B 165 (1985) 355. [5 ] Y. Gefen, B.B. Mandelbrotand A. Aharony,Phys. Rev. Lett. 45 (1980) 855; J. Phys. A 17 (1984) 1277; Y. Gefen, Y. Meir, B.B. Mandelbrot and A. Aharony, Phys. Rev. Lett. 50 (1983) 145. [6 ] E.M. Ilgenfritz,J. Ranft and A. Schiller. LeipzigUniversity preprint KMU-HEP 88-02 ( 1988). [ 7 ] B. Lautrup and M. Nauenberg, Phys. Lett. B 95 (1980) 63. [8] G. Bhanot, K. Bitar and R. Salvador, Phys. Lett. B 188 (1987) 246; S.R. Sharpe, talk Intern. Symp. Lattice '88 (Batavia, 1988). [9] K. Binder and D.P. Landau, Phys. Rev. B 30 (1984) 1477; M.S.S. Challa, D.P. Landau and K. Binder, Phys. Rev. B 34 (19861 1841. [ 10] G.V. Bhanot and S. Sanielevici, Phys. Rev. D 40 ( 1989) 3454. [ 11] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 63 (1989) 1195.
93
PHYSICS LETTERS B
31 May 1990
CRITICAL BEHAVIOUR OF COMPACT ELECTRODYNAMICS I N 4 - ~ D I M E N S I O N S E.-M. I L G E N F R I T Z and A. S C H I L L E R Sektion Physik, Karl-Marx-Universit~it Leipzig, DDR-701 Leipzig, GDR
Received 12 February 1990
Using truncated generalized Sierpinski carpets constructed from embedding hypercubic lattices we investigate the phase structure of the U ( 1) gauge theory for 4 - e dimensions. Slightly below d= 4 the first order phase transition disappears, d = 4 is conjectured to be the upper critical dimension.
The phase structure o f c o m p a c t U ( 1 ) lattice gauge theory in the charge one (coupling r ) and two (fla) m i x e d action plane r e m a i n s interesting, also from the p o i n t o f view o f testing new tools a n d approaches. Evidence as accumulating that puts the c o n t i n u u m limit o f this theory, restricted to W i l s o n ' s action, into question. There is a first order transition line [1 ] which extends into the region o f fla < 0. In o r d e r to understand better the critical b e h a v i o u r it is worthwhile to investigate the phase structure at dimension d e 4. The n o n - p e r t u r b a t i v e m e t h o d o f lattice simulation o f field theories can be c o n t i n u e d to d = 4 - e, using the construction o f fractal lattices [2]. A p a r t from this, there are a d d i t i o n a l reasons to study field theoretical models on non-standard geometries, near to the a p p r o p r i a t e c o n t i n u u m limit. The concept o f s m o o t h s p a c e - t i m e is likely to b r e a k down at the Planck scale a n d to be replaced by irregular metrics a n d / o r d y n a m i c a l dimensionality. N o n s t a n d a r d geometries can be i m p l e m e n t e d using fractal lattices [8] a p p r o a c h i n g flat four-dimensional space at large scales, Critical b e h a v i o u r on fractals has been investigated within the last few years m a i n l y for spin systems [ 4 ]. The first investigations o f some models, placed on particular fractal structures, have pointed out that only fractals having infinite order o f ramification allow for a continuous change o f phase structure with the fractional dimension, and the requirement o f translational invariance corresponds to low lacunarity [5 ]. All this suggests to use Sierpinski carpets.
The first papers concerning gauge field theories on fractal lattices were devoted to the study o f the phase structure o f the Z ( 2 ) gauge m o d e l in 4 - e [2 ] and Z ( 3 ) [6] in 3 - e d i m e n s i o n s a n d between 3 a n d 4 dimensions, supporting the conjecture to have an upper critical dimension 4 and 3, respectively. There was found a specific dimensionality ( " c o u p l i n g " d i m e n sion), suggested b y the plaquette structure o f the action, to give a smooth interpolation o f the critical coupling tic between integer dimensions, in contrast to the H a u s d o r f f dimension. In the present p a p e r we investigate what happens to the first order transition o f U ( 1 ) lattice gauge theory with Wilson action (fla= 0) below four d i m e n sions. Since the latent heat o f this transition is very small, we need m o r e elaborate tools to compare the four-dimensional t h e o r y with its lower-dimensional counterparts. In our simulations we used non-perfect lattices constructed by damaging the e m b e d d i n g D-dimensional hypercubic lattice ( D = 4) in a way that higherdimensional Sierpinski "carpets" suggest. Local gauge invariance ought to be kept on fractal lattices, too. This dictates to delete plaquettes from the hypercubic lattice. The notion o f a regular self-similar Sierpinski carpet is well defined in any e m b e d d i n g d i m e n s i o n D (integer). On a lattice this means to cut out blocks o f fundamental hypercubes o f different sizes (2n) 9, n = 0, 1..... n . . . . We decided to do this in a stochastic way, picking the blocks r a n d o m l y instead o f regularly. Deleting a block means to cancel all elementary
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
89
Volume 242, number 1
PHYSICS LETTERS B
hypercubes belonging to the block. Our rule for deleting the latter ones means omitting the forward plaquettes emanating from "leftmost" corners of the hypercubes (see fig. 1 ). These Sierpinski lattices are necessarily truncated. The minimal blocks are obviously the elementary hypercubes themselves (n = 0). The maximal blocks define the n u m b e r of levels of the fractal construction, K = F/max "[- l, which is limited by the size of the embedding lattice. All plaquettes not deleted are kept "active" in the action with the same coupling //. The links of the embedding lattice, which are not members of at least one active plaquette, have to be cancelled. In a block of size (2") ° the number of eliminated (unorientared) plaquettes is given by ½D(D- 1 ) (2") D, the number of links to be deleted is D. 2" ( 2 " - 1 ) o - 1. The same fraction c determines how many of the blocks of each size has to be deleted from the embedding lattice. Choosing c one determines the Hausdorff dimension
have to check whether all active plaquettes form a connected structure. Different random constructions of a fractal lattice of given dH will lead to different values of d. Strictly speaking, an ideal fractal structure could be represented only by the quenched average over a large number of fractal realizations, moreover in the limit of K-,c~. Apart from a few cases, however, our Monte Carlo simulations are based on only one single realization of the fractal lattice. For a given fractal lattice with Wilson action the partition function is Z=
dURexp
(
•
f l ~ Sw(Up) p
')
,
(3)
."
where ~R and 2p are restricted to active links and plaquettes, respectively. We approximate the U (1) gauge theory by a Z (N) one (with N = 16). The group elements on the links ~ are given by UR= e x p ( i 0 0 ,
dH = D - - l n ( 1 - c ) / l n 2.
(1)
The "coupling" dimension mentioned above is abstracted from the regular lattice
d=I+Np/NR,
(2)
where Np is the number of active plaquettes and NR the number of retained links. It corresponds to the "nearest neighbour" dimension dn, = number of bonds/number of sites, in the case of spin systems [ 4 ]. Before accepting such a randomly broken lattice as a Sierpinski lattice, we
/1 //
Fig. 1. Plaquettes to be deleted in a l ° block ( D = 3 shown for obvious reason), the "leftmost" corner is indicated.
27~jR N ' JR=0,1 ..... N - 1 .
0R
(4)
The group integration is understood as f dU~.... ~N-- 1 j~=o ..., and the action is
Sw(Up)=cosOp,
0p= Z r/(~,p)OR,
(5)
~p
t/(t~, p) = + 1 accounts for the relative orientation of plaquettes and links. Our Monte Carlo simulations for Z (16) have been performed with a five-hit Metropolis algorithm (selfcontrolling for 50% acceptance rate) on 44 and 64 embedding lattices in four dimensions. Our data consists of 2000-6000 measurements which are taken after every fifth sweep. The observables measured are the mean plaquette (spatially averaged for the given configuration) e= ~
90
31 May 1990
1
~ Sw(Up) ,
(6)
and the specific heat defined similarly to Lautrup and Nauenberg [ 7 ] (i.e. normalized by the dispersion of an individual plaquette)
Volume 242, number 1
PHYSICS LETTERS B
p'~p
×(~
[(Sw(Uo)2)-(Sw(Up))2])
.
(7)
. P
This specific heat has been calculated from the indicated averages for the whole sample, whose number o f configurations corresponds to the requirement that Cv did not change within 5% after 500 (1000 near the transition) measurements. In order to estimate the relative error o f the normalized specific heat we repeated the measurement near the phase transition on a 4 4 lattice over 5 more samples and found it o f the same size as the error of the standard specific heat. This amounts to a m a x i m u m of 5% for the other samples consisting of 2000-6000 measurements. In fig. 2 we show the normalized specific heat Cv versus fl for the smallest embedding and fractal lattices available. The non-integer dimensions d = 3.96 and d = 3.93 are created by deleting 3 and 6 elementary hypercubes. In both cases the Hausdorff dimension keeps much nearer to D = 4 than the "coupling" dimension d. We prevented from cutting out 2 4 blocks from a 44 lattice in order to stay near to four dimensions. The critical tic determined from the peak of Cv is shifted in qualitative agreement with mean field arguments. The peak value of Cv decreases indicating a toss o f coherence over the damaged lattice. At
this level nothing can be said about the presence o f a genuine phase transition and its eventual order. A finite size analysis could help to reach more definite conclusions. We compare in fig. 3 the specific heat measurements for 64 and 4 4 embedding lattices. Unfortunately, these lattice sizes are too small to extract any asymptotic scaling behaviour, and 6 4 also did not allow to construct a second level of our fractal construction. The dimension d = 3.93 is reached by deleting 31 elementary hypercubes from the 64 host lattice. We notice that the peak value of Cv rises clearly weaker than proportional to the volume, which would be expected for a first order phase transition. One must say, however, that this does not happen on regular four-dimensional lattices of that size either [7]. The so-called spectral function method [ 8 ] offers itself as an alternative possibility to detect first order transitions even on small lattices. One expects to observe different phases on a finite lattice, coexisting and tunneling from one phase to the other. This can be seen in a histogram, in our case of the mean plaquette e, showing a clear double-bump structure. At the critical coupling fl~ each b u m p carries the same weight. While the maxima should stay at fixed values o f the order parameter with increasing volume, they will become more and more pronounced due to the increasing tunneling barrier. In fig. 4 the spectral density of the mean plaquette at the respective critical coupling is shown for the same fractal lattices as in fig. 3 (both normalized arbitrarily to the same area). The width of the distri-
20-
25-
4 4
~ d=4.00
d =3.93
a=3.96 r_~
16-
'~ o
12
31 May 1990
or L\T
1> r,..)
~, d = 3 . 9 3
r f
20-
15-
÷°"
/
\
.o-o~ 10-
o
ID 54 0.940
I 0.960
i
0,980
I
I
1,OO0
1.020
-
1.040
Fig. 2. Normalized specific heat Cv versus fl for a 4 4 embedding lattice and several "coupling" dimensions d
0, 0.980
I
I
I
1.000
1.029
1.040
Fig. 3. Normalized specific heat Cv versus fl for 4 4 and 64 embedding lattices at d= 3.93. 91
Volume 242, number 1
PHYSICS LETTERS B
800
0.750
64
•
44
0.650-
[] []
oo
400-
(a i
0.700-
[]
[]
d=4.00 .e=LOO14
64
[]
d=3.93 600
31 May 1990
0.600~ "'"
41, o
":" ' "
......
o~
...t
0.550
200-
""
"
"
I" '~ " ' " - "
"
~1
'}'~,
....
,'1~
",,-' " , : ":
-.'.~
. ,:.,:...
[]o 0.500
I1450
- 'q.~ ~ g ~ ---. 0.650 0.750 mean plaquette
0.550
Fig. 4. Spectral density N(e) of mean plaquette at critical couplings as obtained from fig. 3 for 4~ and 6~ lattices at d= 3.93 ( normalized to same area ).
500
0
1000
1500
800
~ []
3000
0.750
64
.e=l.O~O
d=a.ga
(b)
O. 7 0 0
:~" :~ -.,,
~$ 0.650
0.600
64
2500 measurements
,~.,~ ,"~.'~ "~:
.':~;:'-.
• ~.%'...'--. ~.,..~- : - ~ > ' ~ , ' a t , , ~
N(e)
2000
~
[]
~$,..-..... •" . ,
"
-
".'':
.
r - ~ z . . . : , . : -~..X, ~.'-~.: ~ ,
"
' J
. '
.,
.,..,
"
"s"
~ ."" :
.I 0 . 5 5 0
600
~
d=4.00
[] 0.500 0
3000
40'00
5000
6000
Fig. 6. Mean plaquette versus Monte Carlo time at the respective critical/Y (for a 64 lattice} in dimension (a) d=4.00 (measurements after each fifth iteration), (b) d= 3.93 (measurements after each 25th iteration ).
.,/ 0.550
2000
measurements
/o 0.500
1000
0.600
0.650
0.700
n.~e a n p l a q u e t t ; e
Fig. 5. Spectral density N(e) at fl= 1.0014 and 1.0380 for lattice at d-4.00 and d= 3.93.
a 64
bution shrinks with increasing volume and does not show a pronounced double-bump structure. On the other hand, we compare in fig. 5 the spectral densities for the four-dimensional and fractal lattices (at the respective critical coupling and for the larger volu m e ) . Contrary to the fractal case, the four-dimensional lattice system shows two clearly separated m a x i m a of the spectral density, signalling the first order nature of the phase transition. Taken together, figs. 4 and 5 strongly suggest that the U ( 1 )-gauge theory with Wilson action on a fractal lattice of dimension d = 4 - e has a second order phase transition. The lesson of fig. 5 is supported by comparing figs. 6a with 6b which show the Monte Carlo t i m e history of the order parameter, the mean plaquette for the four-dimensional and fractal lattices, respectively. Recently, a new m e t h o d has been proposed [ 9 ] to 92
find the location o f critical couplings more precisely from the fourth order cumulant of the spectral density versus//. It has been~argued, moreover, that a finite size analysis [9,10] o f the cumulant is able to identify even the nature of the phase transition. The fourth-order cumulant BL (where L v is the size o f the embedding lattice) is defined as follows (e 4 )
BL(fl) = 1
3(e2)2
.
(8)
Here the averages are understood according to the spectral density at the given ft. In principle, the spectral functions for several flvalues can be determined from a single one at once using the idea ofreweighting [ 10,11 ]. In fact, we have analyzed a limited amount of data taken at s o m e fl (near to the proposed transition point ) by calculating the cumulant function. The m i n i m u m o f the cumulant is pointing to the more precise critical coupling. At this/Y-value a higher statistics Monte Carlo run has been performed. After one or more iterations of this procedure these data have been used to con-
Volume 242, number 1
PHYSICS LETTERSB
0.668
31 May 1990
~-BL(fl¢(L)),~ 4 CL(fl¢(L)) 3 Np[Eo(L,L(L))pc(L)]2,
n~.(t~) 0.665 0.602
'",..,
/,/'
""-,.,.
////
', f "" 0.656
,
0.960
= .
...... 44 d=4.00I ............ 44 d=a.gal
0.980
1.000
1.020
1.040
Fig. 7. Fourth cumulant BL v e r s u s fl from the spectral densities taken atfl=0.980 and 1.012 for the 44 andfl= 1.0014 and 1.0380 for the 64 lattices at d= 4.00 and 3.93, respectively. struct the respective curves in fig. 7. Comparing the geometrical shape of the c u m u l a n t curves, fitted as an inverted gaussian, in terms of the ratio R = w i d t h / height, one convinces oneself that the curve shrinks in four d i m e n s i o n s a n d widens in 4 - e dimensions with increasing volume. This is the expected behaviour for a first- a n d second-order transition, respectively, a n d can be understood within a p h e n o m e n o logical description of the spectral density N ( e ) . I n this framework the first order transition is described by a superposition of two equal weight gaussians at//o which should be extrapolated to n e a r b y / / by reweighting. The gaussians are centered at the averages eo+ a n d eo_, referring to the respective phases, and shrink proportionally to the volume. In the infinite volume limit a sharp m i n i m u m remains: 2(Eg+ + e ~ _ ) 3-f2 -- 2 2" t o+ ± C o - )
B rain = 1
(9)
Alternatively, a second order phase transition can be modelled by one single gaussian at//o extended as above to n e a r b y / / NL(e,//) = ~
const
exp [ (//~ --//) eNp]
/
×exp(,
[e-eo(L://¢)]:_fl~Np~ 2CL (//~)
(10)
] "
The specific heat m a x i m u m CL (//o(L)) rises ocL c"~, where ot is the thermal and v the correlation length critical exponent with ot/v
(11)
a n d vanishes in the finite volume limit. The gaussian ansatz is too crude to describe the global shape of the c u m u l a n t as function of//. In conclusion, we have d e m o n s t r a t e d that the firstorder phase transition of the U ( 1 ) gauge theory with Wilson action becomes lost when the embedding lattice is made irregular by r a n d o m l y deleting a small fraction ofplaquettes. We have to a d m i t that, due to the small embedding lattices we could afford, this did not really produce a fractal structure with self-similar defects over some different scales. On the other hand, from our earlier studies on Z (3) we know that more iterations of the fractal construction (Up to some scale of translational invariance) render phase transitions even softer. Further studies, both including more complicated actions (with admixtures of multiple charge loops) a n d on larger lattices allowing for more levels of damaging the regular lattice (looking more fractaMike) would be desirable.
References
[ 1] A. Hasenfratz, Phys. Lett. B 201 (1988) 492, and references therein. [2] G. Bhanot and R. Salvador,Phys. Lett. B 167 (1986) 343. [ 3 ] F. Englert, J.-M. Fr6re, M. Rooman and P. Spindel, Nucl. Phys. B 280 [FS18] (1987) 147. [4] G. Bhanot. H. Neuberger and J. Shapiro, Phys. Rev. Lett. 53 (1984) 2277; G. Bhanot, D. Duke and R. Salvador, Phys. Lett. B 165 (1985) 355. [5 ] Y. Gefen, B.B. Mandelbrotand A. Aharony,Phys. Rev. Lett. 45 (1980) 855; J. Phys. A 17 (1984) 1277; Y. Gefen, Y. Meir, B.B. Mandelbrot and A. Aharony, Phys. Rev. Lett. 50 (1983) 145. [6 ] E.M. Ilgenfritz,J. Ranft and A. Schiller. LeipzigUniversity preprint KMU-HEP 88-02 ( 1988). [ 7 ] B. Lautrup and M. Nauenberg, Phys. Lett. B 95 (1980) 63. [8] G. Bhanot, K. Bitar and R. Salvador, Phys. Lett. B 188 (1987) 246; S.R. Sharpe, talk Intern. Symp. Lattice '88 (Batavia, 1988). [9] K. Binder and D.P. Landau, Phys. Rev. B 30 (1984) 1477; M.S.S. Challa, D.P. Landau and K. Binder, Phys. Rev. B 34 (19861 1841. [ 10] G.V. Bhanot and S. Sanielevici, Phys. Rev. D 40 ( 1989) 3454. [ 11] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 63 (1989) 1195.
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