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Does the topological charge scale in the lattice CP3-model?
Physics Letters B 299 (1993) 293-298 North-Holland
P IqYSIC$ I_ETTER$ g
Does the topological charge scale in the lattice Cp3-model? ¢r M a r t i n H a s e n b u s c h a n d Steffen M e y e r Fachbereich Physik, Universitdt Kaiserslautern, W-6 750 Kaiserslautern, FRG
Received 30 September 1992; revised manuscript received 19 November 1992
Using a multigrid Monte Carlo algorithm with piecewise-linear interpolation operator for the two dimensional lattice CPa model with the standard quadratic action, we compare our results for the correlation length and the topological susceptibility with recent overrelaxation data. We do not confirm the scaling for the universal topological susceptibility found for smaller correlation length in the overrelaxation simulation. We find clear evidence that a single state saturates the 2-point correlation function at large distances.
I. Introduction
The development of efficient Monte Carlo algorithms which avoid critical slowing-down has opened a new field o f interest in computational physics. Nearly critical systems in statistical mechanics and lattice field theories close to the continuum limit can now be studied with nonlocal Monte Carlo algorithms. For two dimensional asymptotically free theories the connection between nonperturbative lattice studies and the perturbative continuum behaviour can therefore be investigated much more efficiently than previously with local Monte Carlo algorithms. A particular promising approach to overcoming critical slowing-down is the implementation o f multigrid ideas for Monte Carlo simulations [ 1-3 ]. It has recently been demonstrated by us, that a multigrid algorithm with higher interpolation operator has a very small dynamical critical exponent for some two dimensional nonlinear a models including the O ( 3 )vector-model [4], The Cp3-model and the chiral S U ( 3 ) × S U ( 3 ) spin model [5]. Besides the O ( N ) invariant vector models, the CP N- l-models have been studied on the lattice for the last decade because of the close resemblance between many o f their dynamical features and those o f QCD. In particular, Monte Carlo studies o f the CP 3 model have been pursued -a- Supported by Deutsche Forschungsgemeinschaftthrough grant
#Me 567/5-3.
extensively, mainly to study the spectrum, topology and universality on the lattice and to match analytic results like the 1 / N expansion. In a comparison o f the dynamical critical behaviour o f the multigrid algorithm with the local overrelaxation technique for the Cp3-model, we found for our particular implementation the multigrid algorithm superior for correlation length larger than 25 in lattice units [ 6 ]. In these numerical studies the accurate extraction o f a unique correlation length is of quite some importance. In particular the assumption that a single state saturates the 2-point correlation function at large distances has to be verified. In this note we compare our results with recent overrelaxation data [ 7 ], where the asymptotic scaling violations in the Cp3-model with the standard quadratic action first found in ref. [5 ] are confirmed, but scaling for the topological susceptibility was observed and a value for the dimensionless combination Z,~ 2 = 0.156 (2) has been obtained in the continu u m limit. Also some discrepancies for nonlocal observables at large correlation length where reported in ref. [ 7 ].
2. T h e lattice C P N-1 m o d e l s
A class o f nonlinear a models o f great interest are the CP s - 1 models where the manifold is a ( 2 N - 1 ) dimensional sphere, where points related by a U ( 1 )
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
293
Volume 299, number 3,4
PHYSICS LETTERSB
transformation are identified. A simple choice of the lattice action for the CP N- 1 model [ 8 ], which preserves this local U ( 1 ) gauge invariance of the classical action and is invariant under global U ( N ) transformations, is given by
S=2fl Y, ( 1 - l ~ z j l 2 ) ,
(1)
( i,j )
where z~ is an N-component complex unit vector and fl is the coupling constant. Different lattice regularizations of the CP N- 1 models have been discussed in the past [ 8 ]. The correlation length ~ is obtained from the exponential decay of the invariant connected 2-point function
G(i,j)
1 = (Iz~zjl2) - ~ .
(2)
From the energy eigenstate expansion of G(i, j) one can obtain for large euclidean time the mass gap m = ~-1 under the assumption that m is an isolated point in the spectrum, while the contributions from higher states are exponentially damped relative to the first term. States directly above the mass gap corresponding to nonzero momenta do not contribute, if one takes the zero momentum operator in eq. (2). The magnetic susceptibility Z is given by 1
Z=-~ ~ G(i,j).
28 January 1993
simple update procedure for the C P N - l model in terms of an embedded XY-model with complex, sitedependent couplings, which can be applied to general nonlinear a models [ 5 ]. In the following we first give a prescription of the embedding and then explain the multigrid scheme for the embedded XY model. In order to reduce the computational complexity of the algorithm we use a fixed generator 2~ of the U (N) group for one cycle of the multigrid algorithm. The updates with this single generator can be translated to updates of an embedded two dimensional XY model with the action
S=2fl ~ [ 1 - l e x p ( - i ~ 2 l ) z ~ e x p ( - i ( ~ j 2 t ) z j l 2] , ( i,j )
(5) where we set the real variables q~= 0 at the beginning of each iteration. The original fields of the action eq. ( 1 ) are then updated according
More explicitly, for the CP 3 model we take four of the generators of the group U (4), which are given by ();t)t,t= 1 for l = 1, 2, 3, 4 while all other coefficients of the matrix are zero. With these generators the action given in eq. (5) takes a particular simple form. For example we obtain for l = 1
(3) exp( - iq~i2t) =
Of particular interest in the CpN-m-model is the topological susceptibility Zt = ( 1/L 2) ( Q2 ), where Qt is the topological charge, which classifies the various topological sectors of the classical theory. On the lattice, the topological charge is defined in terms of the sum over phase angles O~ around the elementary plaquettes p 1
at= ~-~n~ Op.
(6)
z; = exp ( - i¢i2t) Z i .
'exp-iOi001)10 O 0 0
0 0
1 0
"
If we now consider
si = exp ( - i¢i) z as embedded
(7)
XY spins, we get the action
S x y = - ~ Re(sic~),
(8)
( i,j)
(4) where the complex couplings are given by
The dimensionless combination Zt~2 has recently been computed in the 1/N expansion to order O( I / N 2) [9].
c=4fl ~ zkizk,
3. Multigrid Monte Carlo algorithm
with the property that c<~,~>=c<~j>. In our simulations we used subsequently the generators 1, 2, 3 and 4 for the embeddings. At the end of this embedding cycle we multiplied all the spins of the lattice with a U (4) matrix, which
In refs. [ 5,6 ] we gave a description of a particular 294
4
__
(9)
k=2
Volume 299, number 3,4
PHYSICS LETTERSB
was choosen with the probability according to the Haar measure of the U (4) group in order to restore ergodicity. We now describe the multigrid updating scheme. Updates on the various levels of a multigrid system can be viewed as nonlocal updates on the corresponding fundamental system. This unigrid point of view needs less formalism to describe the algorithm, and furthermore it is more general in the sence, that not all nonlocal changes of the fundamental field configuration can be interpreted as single site changes on a multigrid system. We use a Metropolis algorithm to update the system. But the proposals for a new field configuration are in general nonlocal. They are given in terms of the embedded XYspins, see eq. (7) by s~ = e x p ( - i T t A : ) s j ,
(10)
where ~ is a random number with an even probability distribution. The kernels Aj determined the relative amplitude of the change of the fields at the sites j. The Aj are chosen such that they are only nonzero within LB × LB blocks and the average over the block is normalized to 1. In two dimensions for at least pieeewise linear kernels the mean step size ( ~ ) is constant for increasing block sizes LB, while it decreases like 1/ v/La for kernel Aj= const, within a block. In our numerical work we used kernels with a pyramidal shape. These elementary updates build whole cycles. First we sweep through the fundamental lattice with a local Metropolis update, next over all disjoint blocks LB=2, 4 .... and so on up to the maximum block size LBm~,= ½L, and then start again the new cycle with a local Metropolis sweep. In ref. [ 5 ] we found that the block updates need overlap. We satisfied this demand by translating the fields after each cycle by a randomly chosen distance. In order to reduce the dependence of the Markov chain on the special properties of our multigrid Metropolis implementation, we use a multi-hit Metropolis update for the simulations discussed in the following. In particular, we made 5 hits in the local Metropolis update used for the fundamental and first coarsened lattice of the multigrid, while for all other levels 10 hits are made. From a finite size scaling analysis of the integrated autocorrelation times o f x we determined the dynam-
28 January 1993
ical critical exponent z = 0.2( 1 ) of our multigrid algorithm [ 5,6 ].
4. Numerical results for the Cp3-model Apart from refs. [5-7] several other numerical studies of the CPa-model with the standard action eq. ( 1 ), but at smaller correlation length and with larger errors due to critical slowing down of the cluster algorithm have been performed lately [ 10,11 ]. In refs. [ 5,6 ] we determined the correlation length by fitting the invariant zero momentum correlation function to G(xo, ~ e x p ( -
~)+exp(L-~xo)
(11)
in the interval ~ to 3 ~ - for an example see fig. 1 of ref. [ 5 ] - and checked the stability and significance of these fits by comparing with further fits in the intervals ½~to 3~, ~to 2~up to as large a distance as a fit can be obtained. The error is estimated by a binning procedure with various bin lengths. In ref. [ 7 ] it has been argued, that in our analysis there may be a problem with higher states and a real discrepancy beyond errors was found for ~ at fl= 3.1. In our simulation we have stored all estimates of the 2-point function at separations Xo= 1, 2, 3 .... ,½L to verify the assumption of the fit eq. (5), that a single state dominates the large distance behaviour of the correlation function. In fig. 1 we show a plot of the effective correlation length ~rf(Xo), which is obtained from eq. (5) by inserting Xo and Xo+ 1, as a function of separation Xo for fl= 3.1 on a lattice with lattice size L = 2 56 (upper part) and L = 512 (lower part). Within the accuracy of our simulation we do not see finite size effects for the correlation length and for other static quantities. in fig. 2 the effective correlation length for weaker coupling fl= 3.2 on an L = 512 lattice is plotted. The error bars shown on all these plots have been determined by a binning procedure with various bin lengths. For both lattice sizes a single state dominates the behaviour of the 2-point correlation function from distance x0 = ~ and extends to Xo~ 3~. To allow for a direct comparison with the procedure proposed in ref. [7 ] we have also quoted the values of the effective correlation length ~ff(Xo) at separation j~ with j = 1, 295
Volume 299, n u m b e r 3,4
P H Y S I C S LETTERS B
4O 38 36 m
! i
34 32 30 28
i
0
20
40
i
60
80
100
120
100
120
x
40 38
34
~
•
i 30
i
28
i 20
. . . . . . . . . ~ 40
~ 60
~ 80
x
Fig. 1. The effective correlation length ~fr(Xo) as a function of separation for fl= 3.1 and lattice sizes L = 25 6 ( u p p e r part) and L = 5 12 (lower part).
56
5, .I 8
!
i
T
48 46 44 42 40 38
20
40
60
i
I
i
i
80
1~
120
140
160
x
Fig. 2. The effective correlation length d-.~r(Xo) as a function of separation f o r / / = 3.2 and lattice size L = 5 12.
2 and label them ~, and ~2 respectively in table 1. Further evidence that our estimates of the correlation lengths are reliable, is obtained from a refined analysis of the short distance behaviour of the 2-point 296
28 January 1993
correlation function in the range 0~7 [5 ]. We obtain ~2Zt= 0.138 (2), which is consistent with the dropping of ~2Z~ at weaker coupling as obtained previously by us [6 ]. The value is more than 10% below the continuum value of ref. [ 7 ]. To leading order the large N-expansion predicts ~2Z~=0.05968... for N = 4 [ 14 ]. In fig. 3 we summarize our results for the nonscaling behaviour of the topological susceptibility and the asymptotic scaling violations of the correlation length.
Volume 299, number 3,4
PHYSICS LETTERS B
28 January 1993
Table 1 Results for the correlation length ~, the effectivecorrelation lengths ~t and ~2and the topologicalsusceptibilityXtfor different lattice sizes L and couplingsft. L
fl
~
~l
~2
Zt103
128 256 512 512 512
2.9 3.1 3.1 3.2 3.3
18.5(1) 38.0(2) 37.8(3) 53.2(3) 77.5(1.1)
18.53(8) 37.96(10) 37.63(18) 53.18(16) 76.4(9)
18.70(12) 38.12(26) 38.07(44) 53.38(35) 77.7(1.6)
0.443(3) 0.100(1) 0.101(2) 0.0487(8) 0.0204(4)
required to study the lattice model in the c o n t i n u u m limit.
.15 t ]E
~'~. 10
j
200
150 f
100
t
t
i
t
~
~
2.5
,
,
~
,
~
~
,
,
t
l
I
i
i
j
L
J
i
3. o
Fig. 3. The universal topologicalsusceptibility and the massgap defect J,, = rn (n/~)- t/2exp (nil) as a function of the coupling P. ~t~
2
6. Acknowledgement It is a pleasure to thank Gerhard Mack, and Klaus P i n n for m a n y suggestions. We would like to thank U. Wolff for helpful correspondence. This work is supported by Deutsche Forschungsgemeinschaft through grant Me 567/5-3 and the G e r m a n - I s r a e l i F o u n d a t i o n for Scientific Research, We also acknowledge partial support by the H L R Z at JiJlich, where some of the computations have been performed. It is a pleasure to thank the R e c h e n z e n t r u m at the University of Karlsruhe for allocating time on their VP $ 6 0 0 / 2 0 computer.
5. Conclusions References We have demonstrated, that using the multigrid Monte Carlo algorithm with piecewise linear interpolation operator for the CP 3 model, consistent correlation lengths with a one percent accuracy can be obtained on lattices up to size L = 512. In addition, we do not find any unexpected large time separation behaviour of the effective correlation length as has been observed with an overrelaxation algorithm [ 7 ]. We also confirm the scaling violations of the universal topological susceptibility found earlier by us. The value o f z t ( 2 f o r f l = 3.2 is not compatible with the c o n t i n u u m estimate given in ref. [7]. The substantial i m p r o v e m e n t in efficiency allows us to study the CP 3 model much closer to criticality than previous studies, but further i m p r o v e m e n t s are
[ 1] G. Parisi, in: Progress in gauge field theory, eds. G. 't Hooft et al. (Plenum, New York, 1984) p. 531. [2] G. Mack and A. Pordt, Commun. Math. Phys. 97 (1985) 267; G. Mack, in: Nonperturbative quantum field theory, eds. G. 't Hoofl et al. (Plenum, New York, 1988) p. 309. [3] J. Goodman and A.D. Sokal, Phys. Rev. Lett. 56 (1986) 1015; Phys. Rev. D 40 (1989) 2035. [4] M. Hasenbusch,S. Meyerand G. Mack, Nucl.Phys. B (Proc. Suppl.) 20 (1991) 110. [5] M. Hasenbusch and S. Meyer, Phys. Rev. Lett. 68 (1992) 435. [6] M. Hasenbusch and S. Meyer, Phys. Rev. D 45 (1992) R4376. [7] U. Wolff, Phys. Lett. B 284 (1992) 94. [ 8 ] M. Stone, Nucl. Phys. B 152 (1979) 97, E. Gava, R. Jengo and C. Omero, Nucl. Phys. B 168 (1980) 465; 297
Volume 299, number 3,4
PHYSICS LETTERS B
P. Di Vecchia, A. Holtkamp, R. Musto, F. Nicodemi and R. Pettorino, Nucl. Phys. B 190 ( 1981 ) 719. [ 9 ] M. Campostrini and P. Rossi, Phys. Lett. B 272 ( 1991 ) 305. [ 10] K. Jansen and U.-J. Wiese, Nucl. Phys. B 370 (1992) 762.
298
28 January 1993
[ 11 ] A.C. Irving and C. Michael, Nucl. Phys. B 371 (1992) 521. [ 12 ] M. Hasenbusch and S. Meyer, in preparation. [ 13 ] R. Gupta and C.F. Baillie, Phys. Rev. B 45 (1992) 2883. [ 14] M. LiJscher, Phys. Lett. B 78 (1978) 465.
P IqYSIC$ I_ETTER$ g
Does the topological charge scale in the lattice Cp3-model? ¢r M a r t i n H a s e n b u s c h a n d Steffen M e y e r Fachbereich Physik, Universitdt Kaiserslautern, W-6 750 Kaiserslautern, FRG
Received 30 September 1992; revised manuscript received 19 November 1992
Using a multigrid Monte Carlo algorithm with piecewise-linear interpolation operator for the two dimensional lattice CPa model with the standard quadratic action, we compare our results for the correlation length and the topological susceptibility with recent overrelaxation data. We do not confirm the scaling for the universal topological susceptibility found for smaller correlation length in the overrelaxation simulation. We find clear evidence that a single state saturates the 2-point correlation function at large distances.
I. Introduction
The development of efficient Monte Carlo algorithms which avoid critical slowing-down has opened a new field o f interest in computational physics. Nearly critical systems in statistical mechanics and lattice field theories close to the continuum limit can now be studied with nonlocal Monte Carlo algorithms. For two dimensional asymptotically free theories the connection between nonperturbative lattice studies and the perturbative continuum behaviour can therefore be investigated much more efficiently than previously with local Monte Carlo algorithms. A particular promising approach to overcoming critical slowing-down is the implementation o f multigrid ideas for Monte Carlo simulations [ 1-3 ]. It has recently been demonstrated by us, that a multigrid algorithm with higher interpolation operator has a very small dynamical critical exponent for some two dimensional nonlinear a models including the O ( 3 )vector-model [4], The Cp3-model and the chiral S U ( 3 ) × S U ( 3 ) spin model [5]. Besides the O ( N ) invariant vector models, the CP N- l-models have been studied on the lattice for the last decade because of the close resemblance between many o f their dynamical features and those o f QCD. In particular, Monte Carlo studies o f the CP 3 model have been pursued -a- Supported by Deutsche Forschungsgemeinschaftthrough grant
#Me 567/5-3.
extensively, mainly to study the spectrum, topology and universality on the lattice and to match analytic results like the 1 / N expansion. In a comparison o f the dynamical critical behaviour o f the multigrid algorithm with the local overrelaxation technique for the Cp3-model, we found for our particular implementation the multigrid algorithm superior for correlation length larger than 25 in lattice units [ 6 ]. In these numerical studies the accurate extraction o f a unique correlation length is of quite some importance. In particular the assumption that a single state saturates the 2-point correlation function at large distances has to be verified. In this note we compare our results with recent overrelaxation data [ 7 ], where the asymptotic scaling violations in the Cp3-model with the standard quadratic action first found in ref. [5 ] are confirmed, but scaling for the topological susceptibility was observed and a value for the dimensionless combination Z,~ 2 = 0.156 (2) has been obtained in the continu u m limit. Also some discrepancies for nonlocal observables at large correlation length where reported in ref. [ 7 ].
2. T h e lattice C P N-1 m o d e l s
A class o f nonlinear a models o f great interest are the CP s - 1 models where the manifold is a ( 2 N - 1 ) dimensional sphere, where points related by a U ( 1 )
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
293
Volume 299, number 3,4
PHYSICS LETTERSB
transformation are identified. A simple choice of the lattice action for the CP N- 1 model [ 8 ], which preserves this local U ( 1 ) gauge invariance of the classical action and is invariant under global U ( N ) transformations, is given by
S=2fl Y, ( 1 - l ~ z j l 2 ) ,
(1)
( i,j )
where z~ is an N-component complex unit vector and fl is the coupling constant. Different lattice regularizations of the CP N- 1 models have been discussed in the past [ 8 ]. The correlation length ~ is obtained from the exponential decay of the invariant connected 2-point function
G(i,j)
1 = (Iz~zjl2) - ~ .
(2)
From the energy eigenstate expansion of G(i, j) one can obtain for large euclidean time the mass gap m = ~-1 under the assumption that m is an isolated point in the spectrum, while the contributions from higher states are exponentially damped relative to the first term. States directly above the mass gap corresponding to nonzero momenta do not contribute, if one takes the zero momentum operator in eq. (2). The magnetic susceptibility Z is given by 1
Z=-~ ~ G(i,j).
28 January 1993
simple update procedure for the C P N - l model in terms of an embedded XY-model with complex, sitedependent couplings, which can be applied to general nonlinear a models [ 5 ]. In the following we first give a prescription of the embedding and then explain the multigrid scheme for the embedded XY model. In order to reduce the computational complexity of the algorithm we use a fixed generator 2~ of the U (N) group for one cycle of the multigrid algorithm. The updates with this single generator can be translated to updates of an embedded two dimensional XY model with the action
S=2fl ~ [ 1 - l e x p ( - i ~ 2 l ) z ~ e x p ( - i ( ~ j 2 t ) z j l 2] , ( i,j )
(5) where we set the real variables q~= 0 at the beginning of each iteration. The original fields of the action eq. ( 1 ) are then updated according
More explicitly, for the CP 3 model we take four of the generators of the group U (4), which are given by ();t)t,t= 1 for l = 1, 2, 3, 4 while all other coefficients of the matrix are zero. With these generators the action given in eq. (5) takes a particular simple form. For example we obtain for l = 1
(3) exp( - iq~i2t) =
Of particular interest in the CpN-m-model is the topological susceptibility Zt = ( 1/L 2) ( Q2 ), where Qt is the topological charge, which classifies the various topological sectors of the classical theory. On the lattice, the topological charge is defined in terms of the sum over phase angles O~ around the elementary plaquettes p 1
at= ~-~n~ Op.
(6)
z; = exp ( - i¢i2t) Z i .
'exp-iOi001)10 O 0 0
0 0
1 0
"
If we now consider
si = exp ( - i¢i) z as embedded
(7)
XY spins, we get the action
S x y = - ~ Re(sic
(8)
( i,j)
(4) where the complex couplings are given by
The dimensionless combination Zt~2 has recently been computed in the 1/N expansion to order O( I / N 2) [9].
c
3. Multigrid Monte Carlo algorithm
with the property that c<~,~>=c<~j>. In our simulations we used subsequently the generators 1, 2, 3 and 4 for the embeddings. At the end of this embedding cycle we multiplied all the spins of the lattice with a U (4) matrix, which
In refs. [ 5,6 ] we gave a description of a particular 294
4
__
(9)
k=2
Volume 299, number 3,4
PHYSICS LETTERSB
was choosen with the probability according to the Haar measure of the U (4) group in order to restore ergodicity. We now describe the multigrid updating scheme. Updates on the various levels of a multigrid system can be viewed as nonlocal updates on the corresponding fundamental system. This unigrid point of view needs less formalism to describe the algorithm, and furthermore it is more general in the sence, that not all nonlocal changes of the fundamental field configuration can be interpreted as single site changes on a multigrid system. We use a Metropolis algorithm to update the system. But the proposals for a new field configuration are in general nonlocal. They are given in terms of the embedded XYspins, see eq. (7) by s~ = e x p ( - i T t A : ) s j ,
(10)
where ~ is a random number with an even probability distribution. The kernels Aj determined the relative amplitude of the change of the fields at the sites j. The Aj are chosen such that they are only nonzero within LB × LB blocks and the average over the block is normalized to 1. In two dimensions for at least pieeewise linear kernels the mean step size ( ~ ) is constant for increasing block sizes LB, while it decreases like 1/ v/La for kernel Aj= const, within a block. In our numerical work we used kernels with a pyramidal shape. These elementary updates build whole cycles. First we sweep through the fundamental lattice with a local Metropolis update, next over all disjoint blocks LB=2, 4 .... and so on up to the maximum block size LBm~,= ½L, and then start again the new cycle with a local Metropolis sweep. In ref. [ 5 ] we found that the block updates need overlap. We satisfied this demand by translating the fields after each cycle by a randomly chosen distance. In order to reduce the dependence of the Markov chain on the special properties of our multigrid Metropolis implementation, we use a multi-hit Metropolis update for the simulations discussed in the following. In particular, we made 5 hits in the local Metropolis update used for the fundamental and first coarsened lattice of the multigrid, while for all other levels 10 hits are made. From a finite size scaling analysis of the integrated autocorrelation times o f x we determined the dynam-
28 January 1993
ical critical exponent z = 0.2( 1 ) of our multigrid algorithm [ 5,6 ].
4. Numerical results for the Cp3-model Apart from refs. [5-7] several other numerical studies of the CPa-model with the standard action eq. ( 1 ), but at smaller correlation length and with larger errors due to critical slowing down of the cluster algorithm have been performed lately [ 10,11 ]. In refs. [ 5,6 ] we determined the correlation length by fitting the invariant zero momentum correlation function to G(xo, ~ e x p ( -
~)+exp(L-~xo)
(11)
in the interval ~ to 3 ~ - for an example see fig. 1 of ref. [ 5 ] - and checked the stability and significance of these fits by comparing with further fits in the intervals ½~to 3~, ~to 2~up to as large a distance as a fit can be obtained. The error is estimated by a binning procedure with various bin lengths. In ref. [ 7 ] it has been argued, that in our analysis there may be a problem with higher states and a real discrepancy beyond errors was found for ~ at fl= 3.1. In our simulation we have stored all estimates of the 2-point function at separations Xo= 1, 2, 3 .... ,½L to verify the assumption of the fit eq. (5), that a single state dominates the large distance behaviour of the correlation function. In fig. 1 we show a plot of the effective correlation length ~rf(Xo), which is obtained from eq. (5) by inserting Xo and Xo+ 1, as a function of separation Xo for fl= 3.1 on a lattice with lattice size L = 2 56 (upper part) and L = 512 (lower part). Within the accuracy of our simulation we do not see finite size effects for the correlation length and for other static quantities. in fig. 2 the effective correlation length for weaker coupling fl= 3.2 on an L = 512 lattice is plotted. The error bars shown on all these plots have been determined by a binning procedure with various bin lengths. For both lattice sizes a single state dominates the behaviour of the 2-point correlation function from distance x0 = ~ and extends to Xo~ 3~. To allow for a direct comparison with the procedure proposed in ref. [7 ] we have also quoted the values of the effective correlation length ~ff(Xo) at separation j~ with j = 1, 295
Volume 299, n u m b e r 3,4
P H Y S I C S LETTERS B
4O 38 36 m
! i
34 32 30 28
i
0
20
40
i
60
80
100
120
100
120
x
40 38
34
~
•
i 30
i
28
i 20
. . . . . . . . . ~ 40
~ 60
~ 80
x
Fig. 1. The effective correlation length ~fr(Xo) as a function of separation for fl= 3.1 and lattice sizes L = 25 6 ( u p p e r part) and L = 5 12 (lower part).
56
5, .I 8
!
i
T
48 46 44 42 40 38
20
40
60
i
I
i
i
80
1~
120
140
160
x
Fig. 2. The effective correlation length d-.~r(Xo) as a function of separation f o r / / = 3.2 and lattice size L = 5 12.
2 and label them ~, and ~2 respectively in table 1. Further evidence that our estimates of the correlation lengths are reliable, is obtained from a refined analysis of the short distance behaviour of the 2-point 296
28 January 1993
correlation function in the range 0
Volume 299, number 3,4
PHYSICS LETTERS B
28 January 1993
Table 1 Results for the correlation length ~, the effectivecorrelation lengths ~t and ~2and the topologicalsusceptibilityXtfor different lattice sizes L and couplingsft. L
fl
~
~l
~2
Zt103
128 256 512 512 512
2.9 3.1 3.1 3.2 3.3
18.5(1) 38.0(2) 37.8(3) 53.2(3) 77.5(1.1)
18.53(8) 37.96(10) 37.63(18) 53.18(16) 76.4(9)
18.70(12) 38.12(26) 38.07(44) 53.38(35) 77.7(1.6)
0.443(3) 0.100(1) 0.101(2) 0.0487(8) 0.0204(4)
required to study the lattice model in the c o n t i n u u m limit.
.15 t ]E
~'~. 10
j
200
150 f
100
t
t
i
t
~
~
2.5
,
,
~
,
~
~
,
,
t
l
I
i
i
j
L
J
i
3. o
Fig. 3. The universal topologicalsusceptibility and the massgap defect J,, = rn (n/~)- t/2exp (nil) as a function of the coupling P. ~t~
2
6. Acknowledgement It is a pleasure to thank Gerhard Mack, and Klaus P i n n for m a n y suggestions. We would like to thank U. Wolff for helpful correspondence. This work is supported by Deutsche Forschungsgemeinschaft through grant Me 567/5-3 and the G e r m a n - I s r a e l i F o u n d a t i o n for Scientific Research, We also acknowledge partial support by the H L R Z at JiJlich, where some of the computations have been performed. It is a pleasure to thank the R e c h e n z e n t r u m at the University of Karlsruhe for allocating time on their VP $ 6 0 0 / 2 0 computer.
5. Conclusions References We have demonstrated, that using the multigrid Monte Carlo algorithm with piecewise linear interpolation operator for the CP 3 model, consistent correlation lengths with a one percent accuracy can be obtained on lattices up to size L = 512. In addition, we do not find any unexpected large time separation behaviour of the effective correlation length as has been observed with an overrelaxation algorithm [ 7 ]. We also confirm the scaling violations of the universal topological susceptibility found earlier by us. The value o f z t ( 2 f o r f l = 3.2 is not compatible with the c o n t i n u u m estimate given in ref. [7]. The substantial i m p r o v e m e n t in efficiency allows us to study the CP 3 model much closer to criticality than previous studies, but further i m p r o v e m e n t s are
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