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Comparison between finite-size field theory and Monte Carlo simulations for the three-dimensional Ising model
PHYSICA ELSEVIER
Physica A 232 (1996) 375-396
Comparison between finite-size field theory and Monte Carlo simulations for the three-dimensional Ising model X . S . C h e n a'*, V . D o h m a, A . L . T a l a p o v b a lnstitut fffr Theoretische Physik, Technische Hochschule Aachen, D-52 056 Aachen, Germany b Landau Institute for Theoretical Physics, GSP-1, 117940 Moscow V-334, Russia
Received 7 February 1996
Abstract
Recent field-theoretic predictions of finite-size effects for various thermodynamic quantities near the critical point are compared with new accurate Monte Carlo (MC) data for the L x L x L Ising model obtained from a cluster-algorithm special-purpose computer. Owing to the large size (L = 32,64, 128,256) of the Ising models, these MC data permit to test the finite-size field theory in the asymptotic scaling region above and below To. The results constitute the most accurate confirmation of finite-size scaling in three dimensions obtained to date in a wide range of temperatures. Agreement with the predicted shape of the finite-size scaling functions is found on a quantitative level. P A C S : 05.50.+q;05.70.JK; 64.60.Ak; 75.40.Mg
1. Introduction
Recently [1] quantitative field-theoretic calculations have been performed to describe finite-size effects on various thermodynamic quantities near the critical point o f the (~4 model with a one-component field tp in three dimensions. This model is believed to belong to the same universality class as the three-dimensional Ising model. A comparison with available Monte Carlo (MC) data for the L x L x L Ising model with periodic boundary conditions (b.c.) was carried out, and good agreement with the field-theoretic results for the same geometry and b.c. was found in most cases. On a fully quantitative level, however, a number of questions remained unanswered partly because o f the limited accuracy o f the MC data but primarily because o f the limited size L (in most cases * Corresponding author. 0378-4371/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PI1 S0378-4371 (96)00 130-6
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X..S. Chen et al./Physica A 232 (1996) 375-396
L ~<38) of the Ising models for which MC data were available over the full range of the relevant temperature region above and below To. This limitation prevented a conclusive comparison with the field-theoretic predictions which, in their present asymptotic form [1], are applicable only to the asymptotic (large L, small IT - TcI/T~) finite-size scaling region. The advent of the first cluster-algorithm special-purpose computer able to simulate three-dimensional Ising models [2] has significantly changed the situation. Accurate Monte Carlo data for the L × L x L Ising model have been obtained from this computer for L = 64, 128,256 not only at T~ [2] but also above and below T~. This should permit to perform a quantitative test of the finite-size field theory in the asymptotic scaling region. In this paper we shall carry out this comparison and answer a number of questions that were left open previously. We shall also take into account new MC data for L = 32 (even if they are not entirely in the asymptotic region) because they are considerably more accurate than the previous ones [1]. Specifically we shall study the specific heat C, two versions of the finite-size magnetization m (1) and m (2), the susceptibilities ~+ and Z- above and below T~ and a cumulant ratio U. A large number of numerical studies of these quantities have already been performed in the past as reviewed in [3-7]. The main objective of our present analysis is to identify the topology of the temperature dependence of the finite-size quantities relative to the bulk quantities, to perform an accurate test of the finite-size scaling structure, and to determine the shape of the finite-size scaling functions. We shall find quantitative agreement between the field-theoretic predictions and the MC data in most cases. We shall compare some of the new MC data also with the earlier finite-size theories [8, 9]. The results of our analysis constitute the most accurate confirmation of finite-size scaling [10] in three dimensions to date and provide further support for the kind of improved finite-size perturbation approach introduced recently [1, 1 I]. Remaining disagreements between a few of the MC data and the asymptotic theoretical results can be attributed either to inaccuracies of the one-loop approximation for the asymptotic scaling functions or to nonasymptotic effects contained in the MC data.
2. Special-purpose computer simulations Simulations of the L x L × L simple cubic Ising model with periodic b.c. were performed on the special-purpose computer (SPC) which implements in hardware the efficient cluster-flip Wolff algorithm [12]. At all temperature points ten runs with different sequences of random numbers were performed. The obtained ten sets of data were used to calculate mean values and standard deviations. Each run was organized in the usual way: initially nt clusters were flipped to reach thermal equilibrium. After that measurements were carried out nm times, nf clusters were flipped between any two measurements. The mean number of spins in a cluster
X.X Chen et aL / Physica A 232 (1996) 375~396
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depends on temperature. At low temperatures the mean cluster includes almost all lattice spins, at infinitely high temperature each cluster consists of only one spin. Near Tc the mean number of spins in a cluster changes drastically. The value of n f in most cases was chosen from the condition that approximately all the lattice spins should be flipped between two measurements. During each measurement two quantities were calculated by the SPC: the total magnetization Mtot = ~ S, which is the sum of all lattice spins S, and the dimensionless total energy (in units of coupling constant J ) Etot = - ~ SS', where the sum includes all pairs of nearest neighbor spins S and S'. The spins S are supposed to take the values -4-1. Using Mtot and Eto t the host computer, which controls the SPC, calculated the (dimensionless) thermodynamic quantities C, M O ) , M (2), Z + and Z- given below for L = 32,64, 128, and 256. In this paper we use these thermodynamic quantities normalized to one spin. The specific heat (divided by ks) was calculated as C = K2((E2ot ) -
(Etot)2)/L 3,
(1)
where K = J / k , T is the dimensionless inverse temperature, and L is in units of the lattice constant. We also obtained two versions of the finite-size magnetization M O ) = ([Mtotl)/Z 3, M(2)
=
[ M 2 \1/2/13 \"*tot/ /~ ,
(2)
(3)
and the reduced susceptibilities Z + ..=_(M2t)/L 3, Z-
:
((M?ot) -
(4)
(IM, o,I)2)/L 3.
(5)
For each of the ten runs the host computer also calculated (Mt4ot) which was used to get the cumulant ratio U : 1 -- ½(M~ot)/(Miot). 4 2 2
(6)
The MC data of these quantities will be presented in Section 4.
3. Summary of field-theoretic results The field-theoretic results for the various thermodynamic quantities defined in Section 2 have been shown [1] to attain a finite-size scaling form [10] in the limit of large L and small It[, t = ( T - T¢)/T~ where Tc is the bulk critical temperature. The scaling form for these (dimensionless) quantities reads C ( t , L ) = L~/Vpc(x) + CB,
(7)
M O ) ( t , L ) = L-~/vPO)(x),
(8)
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X.S. Chen et al. IPhysica A 232 (1996) 375-396
M(2)(t,L) = L-~/~p(~)(x),
(9)
Z+(t,L) = L~/Vp+(x), Z-(t,L)
=
(10)
L~/~P~(x),
(11)
U(t,L) = U(x),
(12)
where x = tZ 1Iv (L in units of the lattice constant). The field-theoretic calculations of the various scaling functions Pc,P~),P(~),P+,P~ and U have been performed in three dimensions by means of an improved perturbation approach [1, 11] that is well applicable both to T/> Tc and T < To. The results are approximate with regard to the form of the finite-size scaling functions Pi(x) given below but they fully incorporate the Liu-Fisher results for the bulk critical exponents [13] = 0.100,
fl = 0.3305,
7 = 1.2395,
v = 0.6335.
Using slightly different values for the exponents given in the more recent literature [14-19] would not change the results of the present paper. The analytic expressions for the scaling functions, for the case of a three-dimensional cube with periodic boundary conditions, read
Pc(x) = Ac ~-~/v [-2(1 + 12u*Rz)u*-10~(Y) + 2R2 + 2v/a],
(13)
P(M)(X) = AM ?(fl/v)-3/4 /~-1/2 01(g),
(14)
P(2M)(X) = .4M ?(fl/v)--3/4 R-1/2 [02( y)] 1/2,
( 15 )
P+(x) = --z-J+?(3/z)-~/v~---1 02(r),
(16)
P ; ( x ) = j;?(3/2)-~/v/~-1 [02(Y) - 01(Y)2],
(17)
U(x) = 1 -- ½04(Y)/O2(Y) 2,
(18)
where
fods
Sm exp(- 1 ys 2 _ S4 )
(19)
and e ( x ) = 1 - 18u*R2(x )
(20)
with u* = 0.0412 being the fixed-point value of the renormalized coupling [1]. The expressions for the auxiliary scaling functions ~(x),R2(x), and Y(x) are given in the Appendix. These functions depend on the ratio ~0fii between the amplitude 40 of the
X.S. Chen et al. IPhysica A 232 (1996) 375-396
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asymptotic bulk correlation length ¢0t -v above Tc and the lattice constant ti. For the sc Ising model this ratio is taken from [13] as ~0/a = 0.495. As noted in [1], the structure of the d - - 3 perturbative expressions (13)-(20) differs fundamentally from the structure that would be obtained within the e-expansion approach of Refs. [8] and [9]. In the latter approach the quantities u* and Y (as well as R and the amplitudes Ac,-~i) would be expanded in powers of ~ = 4 - d or ~:1/2 and subsequently an e-expansion would be performed for the functions zgm(Y). This would destroy the exponential structure of the integrands of the functions O,,(Y) in (19). This would be inadequate since the exponent in (19) should be interpreted as a physical quantity in its own right, namely as a constrained free energy [1]. Keeping the exponential structure of Om(Y) has been demonstrated to yield significantly better agreement with the MC data for the specific heat below T~ than that obtained from the e-expansion approach (see also Section 4.1). In the bulk limit (L ~ e~) the usual power laws are reproduced from (13)-(17) as C(t, ~ ) = (A±/~)ltl -~ + C8,
(21)
M(1)(t, cx~) =M(2)(t,c~) =AMIt] ~,
(22)
z+(t,~)- -a
z+ It I-~ ,
(23)
Z-(t, c~) = A; Itl-~.
(24)
The relation between the amplitudes Ac, -~M, A~ 5z: and the bulk amplitudes A-, AM, Af are given in the Appendix. Only these bulk amplitudes and the bulk constant C8 are treated as adjustable parameters. More specifically, for the quantities C,M(i),z+ and Zthe bulk amplitudes A - and CB,AM,A +, and A~-, respectively, have been independently adjusted to the corresponding quantities obtained from the series extrapolation for the bulk Ising model by Liu and Fisher [13] as given in the Appendix. (For a discussion + -of the ensuing universal bulk amplitude ratios such as A+/A - or A z/A z see Ref. [1].) Thus the expressions (13)-(18) are to be considered as quantitative predictions of finite-size effects without any adjustments to finite-size data. This provides the possibility of conclusive tests of the finite-size field theory in the asymptotic scaling region above, at and below T~.
4. Comparison between field theory and MC simulations The main purpose of this section is to compare the new MC data for L -----32,64, 128, 256 with the field-theoretic results in order to test the finite-size scaling structure and in order to answer the questions raised previously [1] with regard to the topology of the finite-size curves relative to the bulk curves and with regard to the shape of the finite-size scaling functions. By including the new more accurate L = 32 data we are
X..S. Chen et al./Physica A 232 (1996) 375-396
380
Table 1 Selected MC data from [19] (the standard errors are given in the usual way: for example, 262.26(7) means 262.26 4- 0.07) L
K
M(U
Z-
C • K -2
32 32 32 32 32
0.2205 0.221653 0.2218 0.2225 0.224
0.11143(9) 0.18471(3) 0.1980(3) 0.2645(3) 0.3719(2)
171.5(2) 262.26(7) 264.7(5) 217(2) 64.7(8)
25.07(6) 45.55(2) 48.9(1) 59.1(4) 50.8(4)
64 64 64 64 64
0.221 0.221653 0.2217 0.222 0.223
0.0597(2) 0.1288(2) 0.1379(3) 0.1979(3) 0.3129(1)
459(2) 1038(2) 1045(5) 774(5) 126(1)
24.9(1) 55.1(2) 58.6(3) 72.5(5) 57.0(3)
128 128 128 128 128
0.2215 0.221653 0.2217 0.2218 0.222
0.0500(1) 0.0898(3) 0.1092(5) 0.1512(3) 0.20259(8)
2344(9) 4057(14) 4037(39) 2352(30) 704(4)
36.6(4) 66.3(5) 78(1) 83.9(7) 72.7(5)
256 256 256
0.221653 0.22186 0.22198
0.0619(2) 15912(70) 0.17144(5) 1326(10) 0.19885(4) 739(6)
77.6(7) 78.9(5) 71.8(7)
able to demonstrate the onset o f nonasymptotic deviations from the finite-size scaling structure. In most cases the statistical errors o f the MC data shown in the figures are smaller than the size o f the symbols used. A selected set o f MC data is given in Table 1. For the critical value o f K we take Kc = J/ksTc = 0.221653 which is supposed to be close to the exact value [2]. A complete documentation o f the MC data will be given in [19]. For general remarks on the strategy o f the comparison in this section and for comments on the various thermodynamic quantities we refer to Section 7 of [1].
4.1. Specific heat The previous analysis [1] of finite-size effects on the specific heat C(t,L) suffered from two problems: (i) In a scaling plot, the MC data for 8~
381
X..S. Chen et al. IPhysica A 232 (1996) 375-396 3.5 ///
-,
o MC "\
'
3.0
,
j/
*
Data: L = 128
•
64
~
32
xx +~
2.5
Pc 2.0
1.5
1.0
,
-4.0
I
-2.0
i
I
0.0 X
i
I
2.0
i
4.0
Fig. 1. Scaling plot of the MC data for the specific heat C versus x = tL 1/" as defined in (1) and (7). Solid line is the theoretical prediction according to (13). Dashed line is the result of the method of Ref. [9] corresponding to the dashed lines in Figs. 18 and 19 of Ref. [1]. theory and because o f the limited accuracy o f the MC data, the conclusions drawn from this analysis ((i) with regard to the scaling structure and (ii) with regard to the topology relative to the bulk curve) were not definitive in a fully quantitative sense. Problem (i) is significantly reduced owing to the larger values of L in the range 32~ 1.0) is a trivial bulk effect because of the one-loop inaccuracy for the asymptotic amplitude ratio A+/A - [1]. In Fig. 1 also the new more accurate data for L = 32 (crosses) are included in order to demonstrate the nonasymptotic effect on these data below Tc where they do not collapse with the L = 64 and L = 128 data. For comparison we also show the dashed line which represents the result of the earlier perturbation approach o f [9] where the exponential structure o f the functions ~ m ( Y ) , (19), is not maintained. Problem (ii) has become less severe both because of the larger values o f L and because o f the improved accuracy o f the new MC data. The finite-size effect on the MC data relative to the bulk specific heat (solid line) is demonstrated in Fig. 2 for L = 32, L = 64 and L = 128. Owing to the small error bars we now see unambiguously that C ( t , L ) - C(t, cx~) is positive slightly below tmax. This topology agrees with the
X.S. Chen et aL /Physica A 232 (1996) 375-396
382
4.0
r
• MC Data: L = 3 2
3.0
!1 !
C
Ill
2.0
1.0
0.0 -0.02
I
I
-0.01
t
I
0.00 (T-T¢)/To
(a)
I
0.01
I
0.02
5.0 • MC Data: L = 64 4.0
3.0 J
C
J
Y
2.0
1.0
I
0'0.008 (b)
I
-0.004
I
I
0.000 (T-To)/T~
I
I
0.004
0.008
Fig. 2. MC data (with error bars) for the specific heat C versus reduced temperature for L = 32 (a), L = 64 (b) and L = 128 (c). Solid lines are the bulk specific heat of Liu and Fisher [13].
X..S. Chen et aL I Physica A 232 (1996) 375-396 5.0
383
I
• MC Data:
L = 128
4.0
/
3.0
C 2.0
1.0
0.0 -0.004
(c)
I
I
-0.002
0.000
,
I
0.002
0.004
(T-Tc)/T c Fig. 2. Continued.
(asymptotic) theoretical predictions shown in Fig. 3 for L = 32, L = 64 and L = 128 relative to the dashed line representing the (asymptotic) bulk specific heat. The previous analysis of the L = 32 [1,11] and L -- 50 data [20] in Fig. 13 of [1] is consistent with this result. We note that this topology of the specific heat of the Ising model is distinctly different from that of the XY model with a two-component order parameter [21,22]. It is remarkable that this nontrivial dependence on the number of components of the order parameter is anticipated already in lowest order of bare field theory as demonstrated in Fig. 2 of [1]. We conclude that the quantitative agreement of the predicted specific heat with the new MC data provides further support for the improved perturbation approach [ 1, 11] of renormalized finite-size field theory.
4.2. Magnetization Similar to the specific heat, the scaling plots for the magnetization did not yield perfect data collapsing in the previous analysis below Tc (see Figs. 10 and 12 of [1]). Correspondingly there existed systematic deviations from the asymptotic theoretical predictions for P(~)(x) and P~)(x) in the range x~< - 1 where the MC data did not collapse. By contrast, the new MC data with 64~
X..S. Chen et al. I Physica A 232 (1996.) 375-396
384
4.0
I
i
I
i t
/1
r
/
///
3.0
C
2.0
1.o
o.o -0.02
I
i
I
-0.01
,
I
0.00
(a)
0.01
0.02
(T-Tc)/T c
5.0
I
i
I
l
I
I
l t
/ /
L = 128 p
4.0
3.0
C 2.0
1.0
0.0
I
-0.004
(b)
I
-0.002
t
I
0.000
i
0.002
0.004
(T-Tc)/T ~
Fig. 3. Theoretical predictions for the asymptotic specific heat C (solid lines) versus reduced temperature according to (7) and (13) for L = 32 (a) and for L -----64, L = 128 (b). Dashed lines are the bulk limits of the solid lines.
X.S. Chen et al./Physica A 232 (1996) 375-396
385
3.5
• MC Data:
3.0
L =
256
2.5 2.0 p M (1) 1.5
1.0
0.5
0.0
I
-5.0
I
-3.0
I
I
I
-1.0
I
I
1.0
I
3.0
I
5.0
X
Fig. 4. Scaling plot of the MC data for the magnetization M O) versus x tZ l/v aS defined in (2) and (8). Solid line is the theoretical prediction according to (14). Dashed line is the scaling function "p(2) shown in M Fig. 5 as solid line. --
from the scaling structure of both M O) and M (2) are exhibited by the L = 32 MC data for x ~< - 2.0 in Figs. 4 and 5. In the previous analysis [1] it was predicted that in some range below Tc the deviation M (1) --Mbulk should be negative and that M(1)(t,L) should cross the bulk curve Mbulk(t) slightly below Tc whereas M(2)--Mbulk should be positive for all t, i.e., M(z)(t,L) should not cross Mbulk(t). The available MC data were consistent with this prediction but no definitive confirmation was possible because o f the nonasymptotic size o f L and because o f the limited accuracy of the MC data. In Figs. 6 and 7 the new MC data for M (l) and M (2) are shown together with the bulk curve (dashed lines). The enlarged plots in the insets in Fig. 6 demonstrate that the M (1) data are indeed below the bulk curve in some range below Tc. The corresponding (asymptotic) theoretical predictions for M (1) and M (2) are shown in Figs. 8 and 9 together with the (asymptotic) bulk curve. We see that the predicted topological differences between m (1) and M (2) are further supported by the new MC data. They also agree in magnitude with the theoretical predictions o f these subtle finite-size effects. We note that these results are consistent with the positivity of Z- which, according to (5), is proportional to the difference [ m ( 2 ) ] 2 - [M(I)] 2.
4.3. Susceptibility The scaling plots o f the new MC data for Z+ and Z - are shown in Figs. 10 and 11 together with the theoretical predictions. While the MC data scale significantly better
X.S. Chen et al. / Physica .4 232 (1996) 375-396
386
3.5
• MC Data: L = 256 o 128
3.0 • " +
•
364
2.5
2.0
pM (2) 1.5
1.0
0.5
I
I
0"~5.0
i
I
-3.0
I
-1.0
i
I
I
10
I
3.0
5.0
X Fig. 5. Scaling plot of the MC data for the magnetization M (2) versus x = tL llv as defined in (3) and (9). Solid line is the theoretical prediction according to (15). Dashed line is the scaling function P ~ ) shown in Fig. 4 as solid line. 0.40
i
I
i
r
i
f
i
• MC Data: L = 32
0.30
• :.
x
M (1)
0.20
°o
0.10
0.00 -0.010
(a)
,
I -0.005
i
I
0.000
1
0.005
J
0.010
(T-Tc)/l'~
Fig. 6. MC data for the magnetization M O) versus reduced temperature for L = 32 (a), L = 64 (b) and L = 128 (c). Dashed lines are the bulk magnetization of Liu and Fisher [13]. The insets are enlargements of those MC data (with error bars) which arc below the bulk curve.
X.S. Chen et aL I Physica A 232 (1996,) 375-396
387
I
i
• MC Data: L = 64 0.25 t~
0.20
M (1)
~e
0.15 k
\ k
0.10
0.05
0.00
-0.004
I
I
0.000
-0.002
I
0.004
0.002
(T-T~)/T~
(b)
0.25
I
I
• MCData:
i
L=128
0.20 \
0,15 \
M (1)
'b \ \
0.10
q
0.05 i i i
0.00
-0.002
(c)
I -0.001
I
0.0O0
(T-Tc)FF , Fig. 6. Continued.
L
I 0.001
0.002
X.S. Chen et al./ Physica A 232 (1996) 375-396
388 0.40
• MC Data: L = 32
~',°
0.30 \° \\°
M (2)
•
"°
0.20
\ \\
0.10
0.00 -0.010
I
-0.005
0.000
(a)
0.005
0.010
(T-Tc)/T c
I
• MC Data: L = 64 0.25 ~
°
0.20 o
M(2)
o.15
\
\
0.10 tt I = I = = I I I
0.05
0.00 -0.004
(b)
,
I -0.002
I
I
0,000
0.002
I
0.004
(T-Tc)/T c
Fig. 7. MC data for the magnetization M (2) versus reduced temperature for L = 32 (a), L = 64 (b) and L = 128 (e). Dashed lines are the bulk magnetization o f Liu and Fisher [13].
389
X.S. Chen et al. IPhysica A 232 (1996) 375-396 0.25
I
• MC Data: L = 128 0.20
0.15 M (2) ° \
0.10
0.05
0.00
i
-0.002 (C)
I
i
-O.001
i
0.000 (T-Tc)/T c
I
0.001
i
0.002
Fig. 7. Continued. than the previous ones in the range x ~<0 (compare Figs. 6 and 8 of [1]), the asymptotic theoretical results for P+ and P~- (solid lines) agree well with the data only for x > 0. Systematic deviations of 0(10%) exist for P+ below Tc and, like previously [1], for P~- near the maximum around x ~<0. As far as P+ is concerned, the origin of these deviations is presumably a trivial bulk effect due to the theoretical one-loop inaccuracy implied by the finite-size calculation for the ratio of b u l k amplitudes above and below Tc (only A+ was adjusted [1]). As far as P~- is concerned, the deviation of the solid line from the MC data near the maximum is presumably a true inaccuracy of the finite-size calculation. On the other hand, we infer from the MC data in Fig. 11 that even for L = 128 there still exist nonasymptotic deviations near the maximum and that the true asymptotic maximum of P~- is not yet accurately determined by the presently available MC data for L = 256. This demonstrates that, near the maximum, Z- is an extremely sensitive quantity resulting from the difference of the two quantities [M(2)] 2 and [M(I)] 2 which, according to Figs. 6 and 7, are of comparable size. Therefore the deviation of the theoretical prediction from the data near the maximum is not surprising and well within the expected inaccuracy implied by the low-order finite-size perturbation theory.
X..S. Chen et al./ Physica A 232 (1996) 375-396
390 0.4
I
I
I
I
=
I
I
I
t
0.3
L= 32
"",
M (1)
k
0.2
0.1
0.0 -0.010
t
I
-0.005
(a)
I
i
0.000
0.005
0.010
(T-Tc)/T c
0.25
0.20
0.15
M (1) 0.10
0.05
0"-000.002
(b)
-0.001
0.000
0.001
0.002
(T-Tc)/T c
Fig. 8. Theoretical predictions for the asymptotic magnetization M (]) versus reduced temperature according to (8) and (14) for L = 32 (a) and L = 128 (b). Dashed lines are the bulk power law (22).
X.S. Chen et al./Physica A 232 (1996) 375-396 0.4
i
1
i
I
i
I
391
i
0.3
M (2)
0.2
0.1
I
0.0
-0.010
i
-0.005
I
0.000
(a)
i
0.005
0.010
(T-T~)/I" c
0.25
••//•
0.15
M(2) 0.10
128
\
0.05
0.00
=
-0.002
(b)
J
-0.001
I
L
0.000
I
0.001
I
0,002
(T-Tc)/T c
Fig. 9. Theoretical predictions for the asymptotic magnetization M (2) versus reduced temperature according to (9) and (15) for L = 32 (a) and L = 128 (b). Dashed lines are the bulk power law (22).
392
,Y.S. Chen et a l . / P h y s i c a A 232 (1996) 3 7 5 - 3 9 6
5.0 • MCData: L = 2 5 6 128 64 32
4.0
p
+
3.0 L
2.0
1.0
0.0
I
-2.0
I
I
0.0
I
I
2.0
I
4.0
=
6.0
X Fig. 10. Scaling plot of the MC data for the susceptibility X+ versus x = tL l/~ as defined in (4) and (10). Solid line is the theoretical prediction according to (16).
0.4 L=256
• MCData:
0.3
Pz- 0.2
0.1
0.0
I
-5.0
I
-3.0
i
I
-1,0
~
I
1.0
3.0
X Fig. 11. Scaling plot of the MC data for the susceptibility Z - versus x ~ tL J/v as defined in (5) and (11). Solid line is the theoretical prediction according to (17).
X..S. Chen et al. IPhysica A 232 (1996) 375-396 i
I
i
393
i
I
I
i
• M C Data: L = 256 128 64 32
0.60 0.50
0.40
U
0.30
\
0.20
0.10
0.00 -0.10 -10.0
i
I -5.0
i
I 0.0
i
I 5.0
l 0.0
X Fig. 12. Scaling plot of the MC data for the cumulant ratio U versus x = tLl/v as defined in (6) and (12). Solid line is the theoretical prediction according to (18).
4.4. Curnulant ratio The new MC data for the cumulant ratio U(t,L) are shown in the scaling plot of Fig. 12. The data scale quite well and have less scatter than the earlier MC data. As previously [1], the agreement with the theoretical curve is satisfactory but not perfect, presumably because of the one-loop approximation of the theory. A closer inspection of the value of U(O,L) at Tc = J/ksKc in Fig. 12 shows that the new MC data still exhibit a systematic L dependence as substantiated by the numerical results
U(O,L) = 0.46992(8)
for L = 32,
U(O,L) = 0.4664(7)
for L = 64,
U(O,L) = 0.464(1)
for L = 128,
U(O,L) = 0.459(1)
for L = 256.
Thus the universal value U* = l i m t - - . ~ U(O,L) could be slightly smaller than 0.459. This would be in reasonable agreement with the (one-loop) d = 3 value U* = 0.417
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X..S. Chen et al. IPhysica A 232 (1996) 375-396
predicted in [1] which is slightly better than the (one-loop) result U* = 0.405 of the e-expansion [8]. It should be noted, however, that the true origin of the apparent L dependence of U at K = Kc as given above is not yet clear at present. One would have expected a decreasing change of U(O,L) as L is increased by factors of 2. It remains to be seen whether a more precise determination of Kc will reduce this apparent L dependence to that of the pure nonasymptotic (Wegner) corrections at the exact critical temperature.
Acknowledgements This comparison between field theory and Monte Carlo simulations would not have been carried out without the stimulating and continued interest by W. Selke. We also appreciate useful discussions with H.W.J. B16te. Support by the Sonderforschungsbereich 341 "Physik mesoskopischer und niedrigdimensionaler metallischer Systeme" der Deutschen Forschungsgemeinschaft is acknowledged. MC simulations were supported by grants 07-13-210 of NWO, the Dutch Organization of Scientific Research; INTAS-93-0211; MOQ300 of ISF, the International Science Foundation and 93-02-2018 of RFFR, the Russian Foundation for Fundamental Research.
Appendix The scaling function 7(x) appearing in (13)-(17) is determined implicitly by
7(x) 3/2 = (47ru*)l/2[fi(x) + 1202(fi(x))],
(A.1)
.~(x) = Q*(5/~o)l/v(4Tcu*)-l/27(x)(3/2)-l/vx
(A.2)
with
where Q* = 0.945 [I]. The scaling function Y(x) is given by
Y(x) = (47ru*)-1/27(x)3/2 × [Q*(5/~o)VV7(x)-l/Vx(1 + 6u'R2) + 12u*(R1 + R2)],
(A.3)
where Rl(X) = ~z-17(x)-lll(72), g2(x) = (47r3) -1 7( x )I2(72 )
(A.4) 1 2"
(A.5)
X.S. Chen et al./Physica A 232 (1996) 375-396
395
~2
The expression for Im(f ), m = 1,2, reads OG
im(~2 ) =
f dzz m- 1 exp(-~2z/4g2 )[K(z) 3 - (rc/z) 3/2 - 1],
(A.6)
o OQ
K(z) = Z
exp(-J2z)"
(A.7)
j=--O~
The amplitudes in (13)-(17) are related to the corresponding bulk amplitudes in (21)-(24) according to
Ac = (A-/~)(2Q*)~(gl/~o)~/~[1/2u * - 4 + 2v/e] -1,
(A.8)
A~t = AM(2Q*)-#(~/¢0)-#/v(4ztu * )1/4V/8,
(A.9)
~+
+
A X = Az Q*~(gt/~o)~/~(4rtu* ) - 1/2(1 - 81 u .2),
(A.10)
f~; = A ; (2Q*)r(~/~o)~/v(4ztu*)-I/~(1 + 18u* + 81u'2).
(A.11)
The bulk amplitudes A - , AM, A f and CB are adjusted to those of Liu und Fisher [13], i.e., A - = 0.295, A2 = 2.92, A + = 1.0928, A z = 0.220 and CB = B - = -1.965 (in units of the lattice constant). The expressions (A.1)-(A.11) have also been used in [1]. Note that the O(u .2) terms in (A. 10) and (A. 11 ) result from the product of the one-loop finite-size expressions for/~-1 and 02(Y) and 02(Y) - 01(Y) 2 in the bulk limit. They must not be dropped in order to ensure the proper bulk amplitudes in (23) and (24).
Note added in proof In a recent paper by A.L. Talapov and H.W.J. B15te (J. Phys. A: Math. Gen., in print) the more precise value Kc = 0.2216544(3) is presented which may reduce the apparent L dependence of U(0, L) in Section 4.4.
References [1] A. Esser, V. Dohm and X.S. Chen, Physica A 222 (1995) 355. [2] A.L. Talapov, L.N. Shchur and H.W.J. Blrte, JETP Lett. 62 (1995) 174. [3] K. Binder, in: Phase Transitions and Critical Phenomena, Vol. 5b, eds. C. Domb and M.S. Green (Academic Press, London, 1976) p. 1. [4] M.N. Barber, in: Phase Transitions and Critical Phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983) p. 145. [5] V. Privman, in: Finite Size Scaling and Numerical Simulation of Statistical Systems, ed. V. Privman (World Scientific, Singapore, 1990) p. 1. [6] K. Binder, Annu. Rev. Phys. Chem. 43 (1992) 33. [7] K. Binder, in: Computational Methods in Field Theory, eds. H. Gausterer and C.B. Lang (Springer, Berlin, 1992) p. 59.
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[8] E. Br6zin and J. Zinn-Justin, Nucl. Phys. B 257 (1985) 867. [9] J. Rudnick, H. Guo and D. Jasnow, J. Stat. Phys. 41 (1985) 353. [10] M.E. Fisher, in: Critical Phenomena, International School of Physics "Enrico Fermi", Course 51, ed. M.S. Green (Academic Press, New York, 1971). [11] A. Esser, V. Dohm, M. Hermes and J.S. Wang, Z. Phys. B 97 (1995) 205. [12] U. Wolff, Phys. Rev. Lett. 62 (1989) 361 [13] A.J. Liu and M.E. Fisher, Physica A 156 (1989) 35. [14] A.M. Ferrcnberg and D.P. Landau, Phys. Rev. B 44 (1991) 5081. [15] C.F. Baillie, R. Gupta, K.A. Hawick and G.S. Pawley, Phys. Rev. B 45 (1992) 10438. [16] A.J. Guttmann and I.G. Enting, J. Phys. A: Math. Gen. 27 (1994) 8007. [17] G. Bhanot, M. Creutz, U. Glfissner and K. Schilling, Phys. Rev. B 49 (1994) 12909. [18] H.W.J. Bl6te, E. Luijten and J.R. Heringa, J. Phys. A: Math. Gen. 28 (1995) 6289. [19] H.W.J. Blfte and A.L. Talapov, to be published. [20] S. Dasgupta, D. Stauffer and V. Dohm, Physica A 213 (1995) 368. [21] X.S. Chen, V. Dohm and A. Esser, J. Phys. I France 5 (1995) 205. [22] X.S. Chen and V. Dohm, to be published.
Physica A 232 (1996) 375-396
Comparison between finite-size field theory and Monte Carlo simulations for the three-dimensional Ising model X . S . C h e n a'*, V . D o h m a, A . L . T a l a p o v b a lnstitut fffr Theoretische Physik, Technische Hochschule Aachen, D-52 056 Aachen, Germany b Landau Institute for Theoretical Physics, GSP-1, 117940 Moscow V-334, Russia
Received 7 February 1996
Abstract
Recent field-theoretic predictions of finite-size effects for various thermodynamic quantities near the critical point are compared with new accurate Monte Carlo (MC) data for the L x L x L Ising model obtained from a cluster-algorithm special-purpose computer. Owing to the large size (L = 32,64, 128,256) of the Ising models, these MC data permit to test the finite-size field theory in the asymptotic scaling region above and below To. The results constitute the most accurate confirmation of finite-size scaling in three dimensions obtained to date in a wide range of temperatures. Agreement with the predicted shape of the finite-size scaling functions is found on a quantitative level. P A C S : 05.50.+q;05.70.JK; 64.60.Ak; 75.40.Mg
1. Introduction
Recently [1] quantitative field-theoretic calculations have been performed to describe finite-size effects on various thermodynamic quantities near the critical point o f the (~4 model with a one-component field tp in three dimensions. This model is believed to belong to the same universality class as the three-dimensional Ising model. A comparison with available Monte Carlo (MC) data for the L x L x L Ising model with periodic boundary conditions (b.c.) was carried out, and good agreement with the field-theoretic results for the same geometry and b.c. was found in most cases. On a fully quantitative level, however, a number of questions remained unanswered partly because o f the limited accuracy o f the MC data but primarily because o f the limited size L (in most cases * Corresponding author. 0378-4371/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PI1 S0378-4371 (96)00 130-6
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L ~<38) of the Ising models for which MC data were available over the full range of the relevant temperature region above and below To. This limitation prevented a conclusive comparison with the field-theoretic predictions which, in their present asymptotic form [1], are applicable only to the asymptotic (large L, small IT - TcI/T~) finite-size scaling region. The advent of the first cluster-algorithm special-purpose computer able to simulate three-dimensional Ising models [2] has significantly changed the situation. Accurate Monte Carlo data for the L × L x L Ising model have been obtained from this computer for L = 64, 128,256 not only at T~ [2] but also above and below T~. This should permit to perform a quantitative test of the finite-size field theory in the asymptotic scaling region. In this paper we shall carry out this comparison and answer a number of questions that were left open previously. We shall also take into account new MC data for L = 32 (even if they are not entirely in the asymptotic region) because they are considerably more accurate than the previous ones [1]. Specifically we shall study the specific heat C, two versions of the finite-size magnetization m (1) and m (2), the susceptibilities ~+ and Z- above and below T~ and a cumulant ratio U. A large number of numerical studies of these quantities have already been performed in the past as reviewed in [3-7]. The main objective of our present analysis is to identify the topology of the temperature dependence of the finite-size quantities relative to the bulk quantities, to perform an accurate test of the finite-size scaling structure, and to determine the shape of the finite-size scaling functions. We shall find quantitative agreement between the field-theoretic predictions and the MC data in most cases. We shall compare some of the new MC data also with the earlier finite-size theories [8, 9]. The results of our analysis constitute the most accurate confirmation of finite-size scaling [10] in three dimensions to date and provide further support for the kind of improved finite-size perturbation approach introduced recently [1, 1 I]. Remaining disagreements between a few of the MC data and the asymptotic theoretical results can be attributed either to inaccuracies of the one-loop approximation for the asymptotic scaling functions or to nonasymptotic effects contained in the MC data.
2. Special-purpose computer simulations Simulations of the L x L × L simple cubic Ising model with periodic b.c. were performed on the special-purpose computer (SPC) which implements in hardware the efficient cluster-flip Wolff algorithm [12]. At all temperature points ten runs with different sequences of random numbers were performed. The obtained ten sets of data were used to calculate mean values and standard deviations. Each run was organized in the usual way: initially nt clusters were flipped to reach thermal equilibrium. After that measurements were carried out nm times, nf clusters were flipped between any two measurements. The mean number of spins in a cluster
X.X Chen et aL / Physica A 232 (1996) 375~396
377
depends on temperature. At low temperatures the mean cluster includes almost all lattice spins, at infinitely high temperature each cluster consists of only one spin. Near Tc the mean number of spins in a cluster changes drastically. The value of n f in most cases was chosen from the condition that approximately all the lattice spins should be flipped between two measurements. During each measurement two quantities were calculated by the SPC: the total magnetization Mtot = ~ S, which is the sum of all lattice spins S, and the dimensionless total energy (in units of coupling constant J ) Etot = - ~ SS', where the sum includes all pairs of nearest neighbor spins S and S'. The spins S are supposed to take the values -4-1. Using Mtot and Eto t the host computer, which controls the SPC, calculated the (dimensionless) thermodynamic quantities C, M O ) , M (2), Z + and Z- given below for L = 32,64, 128, and 256. In this paper we use these thermodynamic quantities normalized to one spin. The specific heat (divided by ks) was calculated as C = K2((E2ot ) -
(Etot)2)/L 3,
(1)
where K = J / k , T is the dimensionless inverse temperature, and L is in units of the lattice constant. We also obtained two versions of the finite-size magnetization M O ) = ([Mtotl)/Z 3, M(2)
=
[ M 2 \1/2/13 \"*tot/ /~ ,
(2)
(3)
and the reduced susceptibilities Z + ..=_(M2t)/L 3, Z-
:
((M?ot) -
(4)
(IM, o,I)2)/L 3.
(5)
For each of the ten runs the host computer also calculated (Mt4ot) which was used to get the cumulant ratio U : 1 -- ½(M~ot)/(Miot). 4 2 2
(6)
The MC data of these quantities will be presented in Section 4.
3. Summary of field-theoretic results The field-theoretic results for the various thermodynamic quantities defined in Section 2 have been shown [1] to attain a finite-size scaling form [10] in the limit of large L and small It[, t = ( T - T¢)/T~ where Tc is the bulk critical temperature. The scaling form for these (dimensionless) quantities reads C ( t , L ) = L~/Vpc(x) + CB,
(7)
M O ) ( t , L ) = L-~/vPO)(x),
(8)
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X.S. Chen et al. IPhysica A 232 (1996) 375-396
M(2)(t,L) = L-~/~p(~)(x),
(9)
Z+(t,L) = L~/Vp+(x), Z-(t,L)
=
(10)
L~/~P~(x),
(11)
U(t,L) = U(x),
(12)
where x = tZ 1Iv (L in units of the lattice constant). The field-theoretic calculations of the various scaling functions Pc,P~),P(~),P+,P~ and U have been performed in three dimensions by means of an improved perturbation approach [1, 11] that is well applicable both to T/> Tc and T < To. The results are approximate with regard to the form of the finite-size scaling functions Pi(x) given below but they fully incorporate the Liu-Fisher results for the bulk critical exponents [13] = 0.100,
fl = 0.3305,
7 = 1.2395,
v = 0.6335.
Using slightly different values for the exponents given in the more recent literature [14-19] would not change the results of the present paper. The analytic expressions for the scaling functions, for the case of a three-dimensional cube with periodic boundary conditions, read
Pc(x) = Ac ~-~/v [-2(1 + 12u*Rz)u*-10~(Y) + 2R2 + 2v/a],
(13)
P(M)(X) = AM ?(fl/v)-3/4 /~-1/2 01(g),
(14)
P(2M)(X) = .4M ?(fl/v)--3/4 R-1/2 [02( y)] 1/2,
( 15 )
P+(x) = --z-J+?(3/z)-~/v~---1 02(r),
(16)
P ; ( x ) = j;?(3/2)-~/v/~-1 [02(Y) - 01(Y)2],
(17)
U(x) = 1 -- ½04(Y)/O2(Y) 2,
(18)
where
fods
Sm exp(- 1 ys 2 _ S4 )
(19)
and e ( x ) = 1 - 18u*R2(x )
(20)
with u* = 0.0412 being the fixed-point value of the renormalized coupling [1]. The expressions for the auxiliary scaling functions ~(x),R2(x), and Y(x) are given in the Appendix. These functions depend on the ratio ~0fii between the amplitude 40 of the
X.S. Chen et al. IPhysica A 232 (1996) 375-396
379
asymptotic bulk correlation length ¢0t -v above Tc and the lattice constant ti. For the sc Ising model this ratio is taken from [13] as ~0/a = 0.495. As noted in [1], the structure of the d - - 3 perturbative expressions (13)-(20) differs fundamentally from the structure that would be obtained within the e-expansion approach of Refs. [8] and [9]. In the latter approach the quantities u* and Y (as well as R and the amplitudes Ac,-~i) would be expanded in powers of ~ = 4 - d or ~:1/2 and subsequently an e-expansion would be performed for the functions zgm(Y). This would destroy the exponential structure of the integrands of the functions O,,(Y) in (19). This would be inadequate since the exponent in (19) should be interpreted as a physical quantity in its own right, namely as a constrained free energy [1]. Keeping the exponential structure of Om(Y) has been demonstrated to yield significantly better agreement with the MC data for the specific heat below T~ than that obtained from the e-expansion approach (see also Section 4.1). In the bulk limit (L ~ e~) the usual power laws are reproduced from (13)-(17) as C(t, ~ ) = (A±/~)ltl -~ + C8,
(21)
M(1)(t, cx~) =M(2)(t,c~) =AMIt] ~,
(22)
z+(t,~)- -a
z+ It I-~ ,
(23)
Z-(t, c~) = A; Itl-~.
(24)
The relation between the amplitudes Ac, -~M, A~ 5z: and the bulk amplitudes A-, AM, Af are given in the Appendix. Only these bulk amplitudes and the bulk constant C8 are treated as adjustable parameters. More specifically, for the quantities C,M(i),z+ and Zthe bulk amplitudes A - and CB,AM,A +, and A~-, respectively, have been independently adjusted to the corresponding quantities obtained from the series extrapolation for the bulk Ising model by Liu and Fisher [13] as given in the Appendix. (For a discussion + -of the ensuing universal bulk amplitude ratios such as A+/A - or A z/A z see Ref. [1].) Thus the expressions (13)-(18) are to be considered as quantitative predictions of finite-size effects without any adjustments to finite-size data. This provides the possibility of conclusive tests of the finite-size field theory in the asymptotic scaling region above, at and below T~.
4. Comparison between field theory and MC simulations The main purpose of this section is to compare the new MC data for L -----32,64, 128, 256 with the field-theoretic results in order to test the finite-size scaling structure and in order to answer the questions raised previously [1] with regard to the topology of the finite-size curves relative to the bulk curves and with regard to the shape of the finite-size scaling functions. By including the new more accurate L = 32 data we are
X..S. Chen et al./Physica A 232 (1996) 375-396
380
Table 1 Selected MC data from [19] (the standard errors are given in the usual way: for example, 262.26(7) means 262.26 4- 0.07) L
K
M(U
Z-
C • K -2
32 32 32 32 32
0.2205 0.221653 0.2218 0.2225 0.224
0.11143(9) 0.18471(3) 0.1980(3) 0.2645(3) 0.3719(2)
171.5(2) 262.26(7) 264.7(5) 217(2) 64.7(8)
25.07(6) 45.55(2) 48.9(1) 59.1(4) 50.8(4)
64 64 64 64 64
0.221 0.221653 0.2217 0.222 0.223
0.0597(2) 0.1288(2) 0.1379(3) 0.1979(3) 0.3129(1)
459(2) 1038(2) 1045(5) 774(5) 126(1)
24.9(1) 55.1(2) 58.6(3) 72.5(5) 57.0(3)
128 128 128 128 128
0.2215 0.221653 0.2217 0.2218 0.222
0.0500(1) 0.0898(3) 0.1092(5) 0.1512(3) 0.20259(8)
2344(9) 4057(14) 4037(39) 2352(30) 704(4)
36.6(4) 66.3(5) 78(1) 83.9(7) 72.7(5)
256 256 256
0.221653 0.22186 0.22198
0.0619(2) 15912(70) 0.17144(5) 1326(10) 0.19885(4) 739(6)
77.6(7) 78.9(5) 71.8(7)
able to demonstrate the onset o f nonasymptotic deviations from the finite-size scaling structure. In most cases the statistical errors o f the MC data shown in the figures are smaller than the size o f the symbols used. A selected set o f MC data is given in Table 1. For the critical value o f K we take Kc = J/ksTc = 0.221653 which is supposed to be close to the exact value [2]. A complete documentation o f the MC data will be given in [19]. For general remarks on the strategy o f the comparison in this section and for comments on the various thermodynamic quantities we refer to Section 7 of [1].
4.1. Specific heat The previous analysis [1] of finite-size effects on the specific heat C(t,L) suffered from two problems: (i) In a scaling plot, the MC data for 8~
381
X..S. Chen et al. IPhysica A 232 (1996) 375-396 3.5 ///
-,
o MC "\
'
3.0
,
j/
*
Data: L = 128
•
64
~
32
xx +~
2.5
Pc 2.0
1.5
1.0
,
-4.0
I
-2.0
i
I
0.0 X
i
I
2.0
i
4.0
Fig. 1. Scaling plot of the MC data for the specific heat C versus x = tL 1/" as defined in (1) and (7). Solid line is the theoretical prediction according to (13). Dashed line is the result of the method of Ref. [9] corresponding to the dashed lines in Figs. 18 and 19 of Ref. [1]. theory and because o f the limited accuracy o f the MC data, the conclusions drawn from this analysis ((i) with regard to the scaling structure and (ii) with regard to the topology relative to the bulk curve) were not definitive in a fully quantitative sense. Problem (i) is significantly reduced owing to the larger values of L in the range 32~
X.S. Chen et aL /Physica A 232 (1996) 375-396
382
4.0
r
• MC Data: L = 3 2
3.0
!1 !
C
Ill
2.0
1.0
0.0 -0.02
I
I
-0.01
t
I
0.00 (T-T¢)/To
(a)
I
0.01
I
0.02
5.0 • MC Data: L = 64 4.0
3.0 J
C
J
Y
2.0
1.0
I
0'0.008 (b)
I
-0.004
I
I
0.000 (T-To)/T~
I
I
0.004
0.008
Fig. 2. MC data (with error bars) for the specific heat C versus reduced temperature for L = 32 (a), L = 64 (b) and L = 128 (c). Solid lines are the bulk specific heat of Liu and Fisher [13].
X..S. Chen et aL I Physica A 232 (1996) 375-396 5.0
383
I
• MC Data:
L = 128
4.0
/
3.0
C 2.0
1.0
0.0 -0.004
(c)
I
I
-0.002
0.000
,
I
0.002
0.004
(T-Tc)/T c Fig. 2. Continued.
(asymptotic) theoretical predictions shown in Fig. 3 for L = 32, L = 64 and L = 128 relative to the dashed line representing the (asymptotic) bulk specific heat. The previous analysis of the L = 32 [1,11] and L -- 50 data [20] in Fig. 13 of [1] is consistent with this result. We note that this topology of the specific heat of the Ising model is distinctly different from that of the XY model with a two-component order parameter [21,22]. It is remarkable that this nontrivial dependence on the number of components of the order parameter is anticipated already in lowest order of bare field theory as demonstrated in Fig. 2 of [1]. We conclude that the quantitative agreement of the predicted specific heat with the new MC data provides further support for the improved perturbation approach [ 1, 11] of renormalized finite-size field theory.
4.2. Magnetization Similar to the specific heat, the scaling plots for the magnetization did not yield perfect data collapsing in the previous analysis below Tc (see Figs. 10 and 12 of [1]). Correspondingly there existed systematic deviations from the asymptotic theoretical predictions for P(~)(x) and P~)(x) in the range x~< - 1 where the MC data did not collapse. By contrast, the new MC data with 64~
X..S. Chen et al. I Physica A 232 (1996.) 375-396
384
4.0
I
i
I
i t
/1
r
/
///
3.0
C
2.0
1.o
o.o -0.02
I
i
I
-0.01
,
I
0.00
(a)
0.01
0.02
(T-Tc)/T c
5.0
I
i
I
l
I
I
l t
/ /
L = 128 p
4.0
3.0
C 2.0
1.0
0.0
I
-0.004
(b)
I
-0.002
t
I
0.000
i
0.002
0.004
(T-Tc)/T ~
Fig. 3. Theoretical predictions for the asymptotic specific heat C (solid lines) versus reduced temperature according to (7) and (13) for L = 32 (a) and for L -----64, L = 128 (b). Dashed lines are the bulk limits of the solid lines.
X.S. Chen et al./Physica A 232 (1996) 375-396
385
3.5
• MC Data:
3.0
L =
256
2.5 2.0 p M (1) 1.5
1.0
0.5
0.0
I
-5.0
I
-3.0
I
I
I
-1.0
I
I
1.0
I
3.0
I
5.0
X
Fig. 4. Scaling plot of the MC data for the magnetization M O) versus x tZ l/v aS defined in (2) and (8). Solid line is the theoretical prediction according to (14). Dashed line is the scaling function "p(2) shown in M Fig. 5 as solid line. --
from the scaling structure of both M O) and M (2) are exhibited by the L = 32 MC data for x ~< - 2.0 in Figs. 4 and 5. In the previous analysis [1] it was predicted that in some range below Tc the deviation M (1) --Mbulk should be negative and that M(1)(t,L) should cross the bulk curve Mbulk(t) slightly below Tc whereas M(2)--Mbulk should be positive for all t, i.e., M(z)(t,L) should not cross Mbulk(t). The available MC data were consistent with this prediction but no definitive confirmation was possible because o f the nonasymptotic size o f L and because o f the limited accuracy of the MC data. In Figs. 6 and 7 the new MC data for M (l) and M (2) are shown together with the bulk curve (dashed lines). The enlarged plots in the insets in Fig. 6 demonstrate that the M (1) data are indeed below the bulk curve in some range below Tc. The corresponding (asymptotic) theoretical predictions for M (1) and M (2) are shown in Figs. 8 and 9 together with the (asymptotic) bulk curve. We see that the predicted topological differences between m (1) and M (2) are further supported by the new MC data. They also agree in magnitude with the theoretical predictions o f these subtle finite-size effects. We note that these results are consistent with the positivity of Z- which, according to (5), is proportional to the difference [ m ( 2 ) ] 2 - [M(I)] 2.
4.3. Susceptibility The scaling plots o f the new MC data for Z+ and Z - are shown in Figs. 10 and 11 together with the theoretical predictions. While the MC data scale significantly better
X.S. Chen et al. / Physica .4 232 (1996) 375-396
386
3.5
• MC Data: L = 256 o 128
3.0 • " +
•
364
2.5
2.0
pM (2) 1.5
1.0
0.5
I
I
0"~5.0
i
I
-3.0
I
-1.0
i
I
I
10
I
3.0
5.0
X Fig. 5. Scaling plot of the MC data for the magnetization M (2) versus x = tL llv as defined in (3) and (9). Solid line is the theoretical prediction according to (15). Dashed line is the scaling function P ~ ) shown in Fig. 4 as solid line. 0.40
i
I
i
r
i
f
i
• MC Data: L = 32
0.30
• :.
x
M (1)
0.20
°o
0.10
0.00 -0.010
(a)
,
I -0.005
i
I
0.000
1
0.005
J
0.010
(T-Tc)/l'~
Fig. 6. MC data for the magnetization M O) versus reduced temperature for L = 32 (a), L = 64 (b) and L = 128 (c). Dashed lines are the bulk magnetization of Liu and Fisher [13]. The insets are enlargements of those MC data (with error bars) which arc below the bulk curve.
X.S. Chen et aL I Physica A 232 (1996,) 375-396
387
I
i
• MC Data: L = 64 0.25 t~
0.20
M (1)
~e
0.15 k
\ k
0.10
0.05
0.00
-0.004
I
I
0.000
-0.002
I
0.004
0.002
(T-T~)/T~
(b)
0.25
I
I
• MCData:
i
L=128
0.20 \
0,15 \
M (1)
'b \ \
0.10
q
0.05 i i i
0.00
-0.002
(c)
I -0.001
I
0.0O0
(T-Tc)FF , Fig. 6. Continued.
L
I 0.001
0.002
X.S. Chen et al./ Physica A 232 (1996) 375-396
388 0.40
• MC Data: L = 32
~',°
0.30 \° \\°
M (2)
•
"°
0.20
\ \\
0.10
0.00 -0.010
I
-0.005
0.000
(a)
0.005
0.010
(T-Tc)/T c
I
• MC Data: L = 64 0.25 ~
°
0.20 o
M(2)
o.15
\
\
0.10 tt I = I = = I I I
0.05
0.00 -0.004
(b)
,
I -0.002
I
I
0,000
0.002
I
0.004
(T-Tc)/T c
Fig. 7. MC data for the magnetization M (2) versus reduced temperature for L = 32 (a), L = 64 (b) and L = 128 (e). Dashed lines are the bulk magnetization o f Liu and Fisher [13].
389
X.S. Chen et al. IPhysica A 232 (1996) 375-396 0.25
I
• MC Data: L = 128 0.20
0.15 M (2) ° \
0.10
0.05
0.00
i
-0.002 (C)
I
i
-O.001
i
0.000 (T-Tc)/T c
I
0.001
i
0.002
Fig. 7. Continued. than the previous ones in the range x ~<0 (compare Figs. 6 and 8 of [1]), the asymptotic theoretical results for P+ and P~- (solid lines) agree well with the data only for x > 0. Systematic deviations of 0(10%) exist for P+ below Tc and, like previously [1], for P~- near the maximum around x ~<0. As far as P+ is concerned, the origin of these deviations is presumably a trivial bulk effect due to the theoretical one-loop inaccuracy implied by the finite-size calculation for the ratio of b u l k amplitudes above and below Tc (only A+ was adjusted [1]). As far as P~- is concerned, the deviation of the solid line from the MC data near the maximum is presumably a true inaccuracy of the finite-size calculation. On the other hand, we infer from the MC data in Fig. 11 that even for L = 128 there still exist nonasymptotic deviations near the maximum and that the true asymptotic maximum of P~- is not yet accurately determined by the presently available MC data for L = 256. This demonstrates that, near the maximum, Z- is an extremely sensitive quantity resulting from the difference of the two quantities [M(2)] 2 and [M(I)] 2 which, according to Figs. 6 and 7, are of comparable size. Therefore the deviation of the theoretical prediction from the data near the maximum is not surprising and well within the expected inaccuracy implied by the low-order finite-size perturbation theory.
X..S. Chen et al./ Physica A 232 (1996) 375-396
390 0.4
I
I
I
I
=
I
I
I
t
0.3
L= 32
"",
M (1)
k
0.2
0.1
0.0 -0.010
t
I
-0.005
(a)
I
i
0.000
0.005
0.010
(T-Tc)/T c
0.25
0.20
0.15
M (1) 0.10
0.05
0"-000.002
(b)
-0.001
0.000
0.001
0.002
(T-Tc)/T c
Fig. 8. Theoretical predictions for the asymptotic magnetization M (]) versus reduced temperature according to (8) and (14) for L = 32 (a) and L = 128 (b). Dashed lines are the bulk power law (22).
X.S. Chen et al./Physica A 232 (1996) 375-396 0.4
i
1
i
I
i
I
391
i
0.3
M (2)
0.2
0.1
I
0.0
-0.010
i
-0.005
I
0.000
(a)
i
0.005
0.010
(T-T~)/I" c
0.25
••//•
0.15
M(2) 0.10
128
\
0.05
0.00
=
-0.002
(b)
J
-0.001
I
L
0.000
I
0.001
I
0,002
(T-Tc)/T c
Fig. 9. Theoretical predictions for the asymptotic magnetization M (2) versus reduced temperature according to (9) and (15) for L = 32 (a) and L = 128 (b). Dashed lines are the bulk power law (22).
392
,Y.S. Chen et a l . / P h y s i c a A 232 (1996) 3 7 5 - 3 9 6
5.0 • MCData: L = 2 5 6 128 64 32
4.0
p
+
3.0 L
2.0
1.0
0.0
I
-2.0
I
I
0.0
I
I
2.0
I
4.0
=
6.0
X Fig. 10. Scaling plot of the MC data for the susceptibility X+ versus x = tL l/~ as defined in (4) and (10). Solid line is the theoretical prediction according to (16).
0.4 L=256
• MCData:
0.3
Pz- 0.2
0.1
0.0
I
-5.0
I
-3.0
i
I
-1,0
~
I
1.0
3.0
X Fig. 11. Scaling plot of the MC data for the susceptibility Z - versus x ~ tL J/v as defined in (5) and (11). Solid line is the theoretical prediction according to (17).
X..S. Chen et al. IPhysica A 232 (1996) 375-396 i
I
i
393
i
I
I
i
• M C Data: L = 256 128 64 32
0.60 0.50
0.40
U
0.30
\
0.20
0.10
0.00 -0.10 -10.0
i
I -5.0
i
I 0.0
i
I 5.0
l 0.0
X Fig. 12. Scaling plot of the MC data for the cumulant ratio U versus x = tLl/v as defined in (6) and (12). Solid line is the theoretical prediction according to (18).
4.4. Curnulant ratio The new MC data for the cumulant ratio U(t,L) are shown in the scaling plot of Fig. 12. The data scale quite well and have less scatter than the earlier MC data. As previously [1], the agreement with the theoretical curve is satisfactory but not perfect, presumably because of the one-loop approximation of the theory. A closer inspection of the value of U(O,L) at Tc = J/ksKc in Fig. 12 shows that the new MC data still exhibit a systematic L dependence as substantiated by the numerical results
U(O,L) = 0.46992(8)
for L = 32,
U(O,L) = 0.4664(7)
for L = 64,
U(O,L) = 0.464(1)
for L = 128,
U(O,L) = 0.459(1)
for L = 256.
Thus the universal value U* = l i m t - - . ~ U(O,L) could be slightly smaller than 0.459. This would be in reasonable agreement with the (one-loop) d = 3 value U* = 0.417
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X..S. Chen et al. IPhysica A 232 (1996) 375-396
predicted in [1] which is slightly better than the (one-loop) result U* = 0.405 of the e-expansion [8]. It should be noted, however, that the true origin of the apparent L dependence of U at K = Kc as given above is not yet clear at present. One would have expected a decreasing change of U(O,L) as L is increased by factors of 2. It remains to be seen whether a more precise determination of Kc will reduce this apparent L dependence to that of the pure nonasymptotic (Wegner) corrections at the exact critical temperature.
Acknowledgements This comparison between field theory and Monte Carlo simulations would not have been carried out without the stimulating and continued interest by W. Selke. We also appreciate useful discussions with H.W.J. B16te. Support by the Sonderforschungsbereich 341 "Physik mesoskopischer und niedrigdimensionaler metallischer Systeme" der Deutschen Forschungsgemeinschaft is acknowledged. MC simulations were supported by grants 07-13-210 of NWO, the Dutch Organization of Scientific Research; INTAS-93-0211; MOQ300 of ISF, the International Science Foundation and 93-02-2018 of RFFR, the Russian Foundation for Fundamental Research.
Appendix The scaling function 7(x) appearing in (13)-(17) is determined implicitly by
7(x) 3/2 = (47ru*)l/2[fi(x) + 1202(fi(x))],
(A.1)
.~(x) = Q*(5/~o)l/v(4Tcu*)-l/27(x)(3/2)-l/vx
(A.2)
with
where Q* = 0.945 [I]. The scaling function Y(x) is given by
Y(x) = (47ru*)-1/27(x)3/2 × [Q*(5/~o)VV7(x)-l/Vx(1 + 6u'R2) + 12u*(R1 + R2)],
(A.3)
where Rl(X) = ~z-17(x)-lll(72), g2(x) = (47r3) -1 7( x )I2(72 )
(A.4) 1 2"
(A.5)
X.S. Chen et al./Physica A 232 (1996) 375-396
395
~2
The expression for Im(f ), m = 1,2, reads OG
im(~2 ) =
f dzz m- 1 exp(-~2z/4g2 )[K(z) 3 - (rc/z) 3/2 - 1],
(A.6)
o OQ
K(z) = Z
exp(-J2z)"
(A.7)
j=--O~
The amplitudes in (13)-(17) are related to the corresponding bulk amplitudes in (21)-(24) according to
Ac = (A-/~)(2Q*)~(gl/~o)~/~[1/2u * - 4 + 2v/e] -1,
(A.8)
A~t = AM(2Q*)-#(~/¢0)-#/v(4ztu * )1/4V/8,
(A.9)
~+
+
A X = Az Q*~(gt/~o)~/~(4rtu* ) - 1/2(1 - 81 u .2),
(A.10)
f~; = A ; (2Q*)r(~/~o)~/v(4ztu*)-I/~(1 + 18u* + 81u'2).
(A.11)
The bulk amplitudes A - , AM, A f and CB are adjusted to those of Liu und Fisher [13], i.e., A - = 0.295, A2 = 2.92, A + = 1.0928, A z = 0.220 and CB = B - = -1.965 (in units of the lattice constant). The expressions (A.1)-(A.11) have also been used in [1]. Note that the O(u .2) terms in (A. 10) and (A. 11 ) result from the product of the one-loop finite-size expressions for/~-1 and 02(Y) and 02(Y) - 01(Y) 2 in the bulk limit. They must not be dropped in order to ensure the proper bulk amplitudes in (23) and (24).
Note added in proof In a recent paper by A.L. Talapov and H.W.J. B15te (J. Phys. A: Math. Gen., in print) the more precise value Kc = 0.2216544(3) is presented which may reduce the apparent L dependence of U(0, L) in Section 4.4.
References [1] A. Esser, V. Dohm and X.S. Chen, Physica A 222 (1995) 355. [2] A.L. Talapov, L.N. Shchur and H.W.J. Blrte, JETP Lett. 62 (1995) 174. [3] K. Binder, in: Phase Transitions and Critical Phenomena, Vol. 5b, eds. C. Domb and M.S. Green (Academic Press, London, 1976) p. 1. [4] M.N. Barber, in: Phase Transitions and Critical Phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983) p. 145. [5] V. Privman, in: Finite Size Scaling and Numerical Simulation of Statistical Systems, ed. V. Privman (World Scientific, Singapore, 1990) p. 1. [6] K. Binder, Annu. Rev. Phys. Chem. 43 (1992) 33. [7] K. Binder, in: Computational Methods in Field Theory, eds. H. Gausterer and C.B. Lang (Springer, Berlin, 1992) p. 59.
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[8] E. Br6zin and J. Zinn-Justin, Nucl. Phys. B 257 (1985) 867. [9] J. Rudnick, H. Guo and D. Jasnow, J. Stat. Phys. 41 (1985) 353. [10] M.E. Fisher, in: Critical Phenomena, International School of Physics "Enrico Fermi", Course 51, ed. M.S. Green (Academic Press, New York, 1971). [11] A. Esser, V. Dohm, M. Hermes and J.S. Wang, Z. Phys. B 97 (1995) 205. [12] U. Wolff, Phys. Rev. Lett. 62 (1989) 361 [13] A.J. Liu and M.E. Fisher, Physica A 156 (1989) 35. [14] A.M. Ferrcnberg and D.P. Landau, Phys. Rev. B 44 (1991) 5081. [15] C.F. Baillie, R. Gupta, K.A. Hawick and G.S. Pawley, Phys. Rev. B 45 (1992) 10438. [16] A.J. Guttmann and I.G. Enting, J. Phys. A: Math. Gen. 27 (1994) 8007. [17] G. Bhanot, M. Creutz, U. Glfissner and K. Schilling, Phys. Rev. B 49 (1994) 12909. [18] H.W.J. Bl6te, E. Luijten and J.R. Heringa, J. Phys. A: Math. Gen. 28 (1995) 6289. [19] H.W.J. Blfte and A.L. Talapov, to be published. [20] S. Dasgupta, D. Stauffer and V. Dohm, Physica A 213 (1995) 368. [21] X.S. Chen, V. Dohm and A. Esser, J. Phys. I France 5 (1995) 205. [22] X.S. Chen and V. Dohm, to be published.