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Complex Zeroes of the d = 3 Ising Model: Finite-Size Scaling and Critical Amplitudes
Nuclear Physics B235[FS11] (1984) 123-134 © North-Holland Publishing Company
COMPLEX ZEROES OF THE d = 3 ISING MODEL: FINITE-SIZE SCALING AND CRITICAL AMPLITUDES Enzo MARINARI Service de Physique Theorique, Centre d'Etudes Nucleaires de Saclay, 91 191 Gif-sur- Yvette, Cedex, France Received 15 November 1983
By means of an MC simulation of lattices of size 4 3 to 8 3 , we study the distribution of the complex zeroes closest to the real ß axis of the d = 3 Ising model. We observe they do scale and we measure in this way v and A +/A _. We obtain the prediction A +/A _= 0.45 ± 0.07.
1. Introduction The problem of zeroes in the partition function of a statistical system on a finite lattice is receiving a lot of attention [3,2]. Both exact studies of small systems [2], and the use of methods like the Bethe approximation [3], are suggesting that a lot of physical information can be extracted from the distribution of zeroes; we will specialize here to the d = 3 (simple cubic) Ising model. In this case, for a system of size TV3, if N ^ 5 an exact resolution of the model seems impossible [2]. We will apply an MC integration method trying to follow the line of zeroes that, in the N -» oo limit, are pinching the real ß axis at the critical inverse temperature ßc. In ref. [1] the SU(2) lattice gauge theory was analysed, and the location of the zero closest to the real coupling constant axis was found for a 44 system. We show here that this method is very efficient in order to study the scaling behaviour of different zeroes, close enough to the real ß axis, for increasing lattice size. The main observation is that, if the complex zeroes are generating a real singularity in the infinite-volume limit, the oscillating factor one has to face when moving in the complex plane will not behave as Nd, but (for example the d = 3 Ising model) as N(d~l/p)~ NlA. This claim is supported by results for the first two zeroes on N = 5-S lattices; the agreement between these results and the scaling behaviour hypothesized in ref. [3] is remarkable. We find, for the critical amplitudes ratio, A+/A_=0A5 ± 0.07. The total CPU time used for obtaining these results is - 5 hours of CDC 7600. In sect. 2 we describe the MC procedure [1] we used in order to compute the partition function for complex ß. In sect. 3 we give our numerical results, and in sect. 4 we work out their physical implications. 123
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E. Marinari / Finite-size scaling
2. Monte Carlo integration for complex ß Our goal is to compute, by Monte Carlo integration, the zeroes of the partition function Z(/J), for ß complex, having a small Im/i. These are the zeroes that are pinching the real ß axis in the infinite-volume limit, giving rise to the critical behaviour of the system. We will apply to the d = 3 Ising model the method used in ref. [1] for the SU(2) lattice gauge theory. Our action will be
s--E'(^-i).
(2·ΐ)
where σ, = ± 1 , and the sum runs over first neighbours. The partition function is defined by Z(/?) = E e - ^ ( C ) , c
(2.2)
where the sum runs over all possible configurations the system can assume. An exact computation of Z{ß) was performed in ref. [2] for a 43 system; in this case the number of configurations one has to consider is ~ 1019, and only by exploiting all the symmetries of the system it was possible to solve the 43 system in a realistic computer time. But already a 53 system is absolutely impossible to solve in an exact way (one should consider - 1037 configurations) and the number of states of an 83 system is - 10154*. That is why, to go to larger lattices, we have to use a random sampling. The partition function (2.2) can be expressed as a sum over the energies W the system can reach, where the factor t~ßw is weighted by the density of states with energy W, Nw\ Ζ(β) = ΣΝ**-βιν· w
(2.3)
The energy W can assume values ranging from 0 (for an ordered configuration) up to 2dN3. We explore now, by means of a probabilistic algorithm, the phase space of the system at a given real β (we assume that we start from a configuration that has already been led to thermal equilibrium), and record the energy states assumed by the system (properly normalising the distribution function obtained in this way). We are computing in this way the normalised energy distribution function FAß)
-^(ß)-^(ßT'
r
(2 4)
·
Also using the transfer matrix formalism one would count an unrealistic number of configurations (~10 20 ).
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125
such that
Σ Μ / 0 = ΐ·
(2.5)
W
Let us now compare the two expressions Fw{ß)Z{ß)^
=
Fw(ß')Z(ß')^'w,
that is Fw(ß)-Fw(ß')*-*-n^L,
(2.6)
and integrate this relation over W, by applying the normalization condition (2.5):
^
= LFAß')e-(ß-ß'W-
(2-7)
IP ) w This is the relation that will be useful in the following. We will consider now the case ] 8 ' E R , and ß will be made complex. The sense of this analytic continuation is clear in (2.7); Fw(ß') will be the output of MC runs, done at some fixed β'. Moreover, if we define β = η + /£; η9 ξ e R, we see that in the RHS of (2.7) we have an oscillating factor [cos(£W) +isin(i;W)] and a damping contribution e~ (T,_/r)H/ . The mean value of W is a number of order \ times the volume of the system; this makes apparent the difficulty of going far from the real axis (increasing ζ implies faster oscillations), and of going to larger lattices. But also suggests that, in order to take care of the damping factor, we compute (since our goal is determining the zeroes of Ζ(β) in a situation in which we know there are no zeroes for ß e R), instead of (2.7), the quantity Z
Z(ß) _ Z(ß) fZ(Reß)^1 Z(Re/?) z(jS') I Z(j8') ZFw(ß')t-to-ß'W[co$(tW)-ism(tW)] w
(2.8)
ZFw{ß')t-<*-™ w
We will eventually determine the zeroes by looking at the minima of
Aß)
2
+(Liv(/r)e-(*-/iWsin(|W'))
IZFAß'^-^cosUW))
\2
Z(Reß)
')W '
\ w
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3. Numerical analysis We studied the d= 3 Ising model on lattices of size 43, in order to compare our results with the exact ones of ref. [2], and of sizes going from 53 to 83. We used a heat-bath algorithm to flip the spins of our system (for a description of the method see ref. [5]). We chose randomly the spin to be updated; we recorded the energy of the system after any 57V3 updates of spins. For any lattice size we performed 10 5 N 3 updates (not recording the energies) to bring the system to thermal equilibrium; eventually we performed 5 · lO^/V3 updates at two β' values (0.230 and 0.235) for the 43 lattice, and lO 6 ^ 3 updates at β' = 0.230 for the sizes TV = 5-8. In table 1 we give our average energies for the different lattice sizes. We start by discussing our tests on a 43 lattice: see figs. 1 and 2. In this case the exact result is known [2], and the zeroes of Z have been computed with high precision [4]. The first check concerns the capabihty of our procedure to reproduce complex zeroes when the exact energy distribution function Fw(ß) is used. In ref. [2] the coefficients Cn of the polynomial 96
Z(«)=EQ«»,
(3.1)
/7 = 0
where W
= e"4^
are given; we can now easily compute F$(ß\ inserting it in eq. (2.8), and using our numerical procedure to determine the zeroes of Z. Secondly we performed 5 · 10 5 # 3 updates at ß'= 0.230 and 0.235 (after 105N3 equilibrating flips) and we computed the corresponding zeroes. We want to em phasize also that if we had wanted to use the improved /?'s available (so as to minimize the damping factor of (2.8)), the numerical integration we performed was
TABLE 1
(S) (see (2.1)) at ß = 0.23 for the different lattices sizes N
(S)
5 6 7 8
0.5281(9) 0.5360(7) 0.5388(5) 0.5411(7)
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Fig. 1. Im u$o versus Re utyo for the first seven zeroes on the 43 lattice: comparison between the exact result [4] (dots), the numerical integration of the exact energy distribution (crosses), and MC results for 105 energy values (open circles), ß' = 0.230.
supposed to converge at the correct result independently of β'. That is why we did our check at two different /Ts. We looked at the first seven zeroes w^L4j0> / = 1,... ,7; for all of them the exact result was very close to the one obtained from O0O((|w ( A e x a c t | " l"i0o^hD/l"(4,oexactl < 1%, for / = 1,...,7), both for £ ' = 0.230 and for β'= 0.235. As far as the MC determination of the zeroes is concerned, it is apparent from figs. 1, 2 that the first two zeroes (w%, M^O) do coincide, for both /Ts, inside our error, with the exact ones. An error of the order of one percent is observed for the third zero, while precision is lost for the farther ones. Now when studying larger lattices, we have reasons to be worried: the oscillating factor we are trying to integrate, - cos(£fF), can be written as cos(£vF£), where V is the volume of the system, E is the integration variable, going from zero to one,
316
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1
1
I
Im u too
•X
.300
o
o
*
X
o .200
Γ
o
— X
.100
.100
_i_
.200
Reu .300
.WO
Fig. 2. Same as in fig. 1, but ß ' = 0.235.
and £v is the distance of the interesting zero from the real u axis in the considered volume. The magnitude of the oscillations will dramatically increase with the volume of the system. The situation is in our case nicer than that just because of the presence, in the infinite-volume limit, of a zero on the real U axis; since ξΝ is supposed to behave as [3] N~l/v, the amplitude of the oscillations will not increase as TV3, but as ~iV 14 . That means, for example, that when going from iV = 4 to TV = 8, it will not increase by a factor 8, but just by a factor of ~ 2.5. In tables 2 and 4 we give our results for the first and the second zero, for N = 5-8. We defined our error in the following way: we compute the zero in two clusters of energy values, each one given from 4· 105iV3 spin updates, and separate them by 10 5N3 updates. We assume our statistical precision is one in which these two results coincide and also coincide with the one we get from considering all the 106N3 spin updates. We consider our statistics not large enough to obtain reliable information on zeroes farther than the second one.
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Fig. 3. - In Im u$\0) versus In N, for N = 4-8.
4. Interpretation of results The physical conclusions we will draw here will follow the analysis performed in ref. [3]. The scaling picture [6] is applied to the location of the zeroes of Z(ß) in the complex ß plane. Our first result follows from the relation
r~CN-l/\
(4.1)
where τ is the distance of the closest zero from the physical singularity at N = oo, r= uc(N)-uc(oo). In fig. 3 we plot -Inilmu^) versus InN: the linearity of the results is quite impressive. In table 3 we give the values of v(N9 N') for N = 4 , . . . ,7, N' > N, when assuming τ = Im u(1\ In table 4 we use τ = \u$0 - uc\ with uc = 0.4120 [11]: in the
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TABLE 2
First zero: u = e ~ 4^ N
Re/^o
Im/^o
Re «# 0
Im !$>0
4 5 6 7 8
0.2327 0.2307 0.2289 0.2277 0.2270
0.0561 0.0384 0.0283 0.0220 0.0175
0.3844 0.3927 0.3977 0.4006 0.4023
0.0877 0.0608 0.0452 0.0353 0.0282
Error is 1 on the last digit in columns one and two and 2 on the last digit in columns 3 and 4.
TABLE 3
v(N, N') = (In N' - In tf)/(lnlm w^o ~ hi Im i/JJ) 0 ) N' Ν
4 4
5
6
7
8
0.610(9)
0.612(6) 0.614(4)
0.615(4) 0.619(9) 0.624(20)
0.611(5) 0.612(9) 0.611(14) 0.596(31)
5 6 7 8
TABLE 4
As in table 3, but lnlm u% - ln(Im κ#>ο + (Re u$0 - w c ) 2 ) 1/2 4
Ν
Ν
4 4 5 6 7
5
6
7
8
0.610(6)
0.612(4) 0.614(10)
0.617(3) 0.621(6) 0.628(16)
0.616(3) 0.618(5) 0.621(9) 0.612(22)
8
scaling limit the two definitions are equivalent. We obtain that v from table 4 is closer to the true value (v = 0.631 ± 0.001, from (9)) than the one from table 3; this is for all values of the lattice size. More information is contained in table 4 than in table 3, which can help explain this phenomenon. A systematic smallness of v computed from our small lattices (~ 1%) is apparent; given our error bars we cannot try to study the way in which p(N, N') -> p(oo).
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TABLE 5
The second zero
N
t^c uN o
Im u%
4
0.3444 0.366(3) 0.379 0.386 0.390
0.1433 0.101(2) 0.076(3) 0.054 0.045
5 6 7 8
Error is 1 on the last digit, if not indicated in parenthesis.
A last remark on this point is in order: it is clear that, for increasing N, all the zeroes close enough to the real ß axis should scale according to (4.1). In fig. 4 we plot - In Im u$0 for / = 1,2; also the second zero has the expected behaviour. The following result is based on the relation , cos πα - A _/A + r, tan[(2 - α)φ] = -r'—*- , sin7ra
1.2
1.4
1.6
1.8
(4.2)
2.0
Fig. 4. As in fig. 3, but dots for u$\Q) and crosses for MJ^ 0 , N = 5-8.
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E. Marinari / Finite-size scaling TABLE 6
φ„ ^ tan' HCIm u%0 - Im ii# 0 )/(Re «$> - Re u$0)}
N
Φ
5 6 7 8
56.5ΐί 3 5 58.7 + i f 52.0±33;87
53.8+ti
obtained in ref. [3]. Here a is the specific heat exponent (we will use as input the value a = 2 - dv = 0.107 ± 0.003), A _/A + the critical amplitude ratio for the specific heat, and φ is the angle at which the complex zeroes deviate from the real u axis (in the complex u plane) in the infinite-volume limit (at zero magnetic field). We give in table 6 the φ^ we get for our lattice sizes, from the first and second zeroes. The errors are quite large (obviously enough the precision of the order of one percent in the second zero is reflected by a large indetermination of the slope, see fig. 5). From our data we feel quite comfortable with the assumption that the N dependence of the slope is by far considerably smaller than our statistical indetermination; so we compute φ5,8 = (55.3 ±1.5)°,
(4.3)
and average φ on the lattices of sizes from 5 to 8. We get in this way, from (4), the estimate ^ 7 = 0.45(7),
(4.4)
where the error has a statistical meaning, and the systematic error coming from the finiteness of the lattice is not taken into account (but seems to be much smaller than the statistical one). For a comparison with experimental data, see for example ref. [8] (in particular table 3 and fig. 2 therein); the experimental values range from 0.36 and 0.63, and the comparison with our value is fair enough. On the theoretical ground a computation [7] at the order e2 gives for A +/A _ a value of 0.48 (lowering the value [10] at the order ε, 0.55); also in this case the comparison with our value is more than reasonable. We wish to thank C. Itzykson and J.B. Zuber for their enthusiastic, generous and continuous help. Interesting discussions with D. Beysens, H. Flyberg, R. Pearson and J. Zinn-Justin are gratefully acknowledged. We thank C. Itzykson and G. Parisi for a critical reading of the manuscript.
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Fig. 5. Determination of φ^.
References [1] M. Falcioni, E. Marinari, M.L. Paciello, G. Parisi and B. Taghenti, Phys. Lett. 108B (1982) 331 [2] R.B. Pearson, Phys. Rev. B26 (1982) 6285 [3] C. Itzykson, R.B. Pearson and J.B. Zuber, Nucl. Phys. B220[FS8] (1983) 415; J.B. Zuber, Talk at Workshop on nonperturbative field theory and QCD, Trieste, December 1982, Saclay preprint, SPhT/83/004 (January 1983) [4] R.B. Pearson, private communication [5] E. Marinari, G. Paladin, G. Parisi and A. Vulpiani, 1 / / noise, disorder and dimensionality, Saclay preprint, SPhT/83/96 (July 1983), J. de Phys., to be published [6] M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516 [7] C. Bervillier, Phys. Rev. B14 (1976) 4964
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[8] D. Beysens, A. Bourgou and P. Calmettes, Phys. Rev. A26 (1982) 3589 [9] J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. B21 (1980) 3976; B.G. Nickel, in Phase transitions, status of the experimental and theoretical situation, Cargese 1980, eds. M. Levy, J.C. Le Guillou and J. Zinn-Justin (Plenum, New York, 1981); J. Zinn-Justin, J. de Phys. 42 (1981) 783 [10] P.C. Hohenberg, A. Aharony, B.I. Halperin and E.D. Siggia, Phys. Rev. B13 (1976) 2896 [11] C. Domb, in Phase transitions and critical phenomena, vol. 3, eds. C. Domb and M.S. Green (Academic Press, 1974); J. Zinn-Justin, private communication
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COMPLEX ZEROES OF THE d = 3 ISING MODEL: FINITE-SIZE SCALING AND CRITICAL AMPLITUDES Enzo MARINARI Service de Physique Theorique, Centre d'Etudes Nucleaires de Saclay, 91 191 Gif-sur- Yvette, Cedex, France Received 15 November 1983
By means of an MC simulation of lattices of size 4 3 to 8 3 , we study the distribution of the complex zeroes closest to the real ß axis of the d = 3 Ising model. We observe they do scale and we measure in this way v and A +/A _. We obtain the prediction A +/A _= 0.45 ± 0.07.
1. Introduction The problem of zeroes in the partition function of a statistical system on a finite lattice is receiving a lot of attention [3,2]. Both exact studies of small systems [2], and the use of methods like the Bethe approximation [3], are suggesting that a lot of physical information can be extracted from the distribution of zeroes; we will specialize here to the d = 3 (simple cubic) Ising model. In this case, for a system of size TV3, if N ^ 5 an exact resolution of the model seems impossible [2]. We will apply an MC integration method trying to follow the line of zeroes that, in the N -» oo limit, are pinching the real ß axis at the critical inverse temperature ßc. In ref. [1] the SU(2) lattice gauge theory was analysed, and the location of the zero closest to the real coupling constant axis was found for a 44 system. We show here that this method is very efficient in order to study the scaling behaviour of different zeroes, close enough to the real ß axis, for increasing lattice size. The main observation is that, if the complex zeroes are generating a real singularity in the infinite-volume limit, the oscillating factor one has to face when moving in the complex plane will not behave as Nd, but (for example the d = 3 Ising model) as N(d~l/p)~ NlA. This claim is supported by results for the first two zeroes on N = 5-S lattices; the agreement between these results and the scaling behaviour hypothesized in ref. [3] is remarkable. We find, for the critical amplitudes ratio, A+/A_=0A5 ± 0.07. The total CPU time used for obtaining these results is - 5 hours of CDC 7600. In sect. 2 we describe the MC procedure [1] we used in order to compute the partition function for complex ß. In sect. 3 we give our numerical results, and in sect. 4 we work out their physical implications. 123
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E. Marinari / Finite-size scaling
2. Monte Carlo integration for complex ß Our goal is to compute, by Monte Carlo integration, the zeroes of the partition function Z(/J), for ß complex, having a small Im/i. These are the zeroes that are pinching the real ß axis in the infinite-volume limit, giving rise to the critical behaviour of the system. We will apply to the d = 3 Ising model the method used in ref. [1] for the SU(2) lattice gauge theory. Our action will be
s--E'(^-i).
(2·ΐ)
where σ, = ± 1 , and the sum runs over first neighbours. The partition function is defined by Z(/?) = E e - ^ ( C ) , c
(2.2)
where the sum runs over all possible configurations the system can assume. An exact computation of Z{ß) was performed in ref. [2] for a 43 system; in this case the number of configurations one has to consider is ~ 1019, and only by exploiting all the symmetries of the system it was possible to solve the 43 system in a realistic computer time. But already a 53 system is absolutely impossible to solve in an exact way (one should consider - 1037 configurations) and the number of states of an 83 system is - 10154*. That is why, to go to larger lattices, we have to use a random sampling. The partition function (2.2) can be expressed as a sum over the energies W the system can reach, where the factor t~ßw is weighted by the density of states with energy W, Nw\ Ζ(β) = ΣΝ**-βιν· w
(2.3)
The energy W can assume values ranging from 0 (for an ordered configuration) up to 2dN3. We explore now, by means of a probabilistic algorithm, the phase space of the system at a given real β (we assume that we start from a configuration that has already been led to thermal equilibrium), and record the energy states assumed by the system (properly normalising the distribution function obtained in this way). We are computing in this way the normalised energy distribution function FAß)
-^(ß)-^(ßT'
r
(2 4)
·
Also using the transfer matrix formalism one would count an unrealistic number of configurations (~10 20 ).
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125
such that
Σ Μ / 0 = ΐ·
(2.5)
W
Let us now compare the two expressions Fw{ß)Z{ß)^
=
Fw(ß')Z(ß')^'w,
that is Fw(ß)-Fw(ß')*-*-n^L,
(2.6)
and integrate this relation over W, by applying the normalization condition (2.5):
^
= LFAß')e-(ß-ß'W-
(2-7)
IP ) w This is the relation that will be useful in the following. We will consider now the case ] 8 ' E R , and ß will be made complex. The sense of this analytic continuation is clear in (2.7); Fw(ß') will be the output of MC runs, done at some fixed β'. Moreover, if we define β = η + /£; η9 ξ e R, we see that in the RHS of (2.7) we have an oscillating factor [cos(£W) +isin(i;W)] and a damping contribution e~ (T,_/r)H/ . The mean value of W is a number of order \ times the volume of the system; this makes apparent the difficulty of going far from the real axis (increasing ζ implies faster oscillations), and of going to larger lattices. But also suggests that, in order to take care of the damping factor, we compute (since our goal is determining the zeroes of Ζ(β) in a situation in which we know there are no zeroes for ß e R), instead of (2.7), the quantity Z
Z(ß) _ Z(ß) fZ(Reß)^1 Z(Re/?) z(jS') I Z(j8') ZFw(ß')t-to-ß'W[co$(tW)-ism(tW)] w
(2.8)
ZFw{ß')t-<*-™ w
We will eventually determine the zeroes by looking at the minima of
Aß)
2
+(Liv(/r)e-(*-/iWsin(|W'))
IZFAß'^-^cosUW))
\2
Z(Reß)
')W '
\ w
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3. Numerical analysis We studied the d= 3 Ising model on lattices of size 43, in order to compare our results with the exact ones of ref. [2], and of sizes going from 53 to 83. We used a heat-bath algorithm to flip the spins of our system (for a description of the method see ref. [5]). We chose randomly the spin to be updated; we recorded the energy of the system after any 57V3 updates of spins. For any lattice size we performed 10 5 N 3 updates (not recording the energies) to bring the system to thermal equilibrium; eventually we performed 5 · lO^/V3 updates at two β' values (0.230 and 0.235) for the 43 lattice, and lO 6 ^ 3 updates at β' = 0.230 for the sizes TV = 5-8. In table 1 we give our average energies for the different lattice sizes. We start by discussing our tests on a 43 lattice: see figs. 1 and 2. In this case the exact result is known [2], and the zeroes of Z have been computed with high precision [4]. The first check concerns the capabihty of our procedure to reproduce complex zeroes when the exact energy distribution function Fw(ß) is used. In ref. [2] the coefficients Cn of the polynomial 96
Z(«)=EQ«»,
(3.1)
/7 = 0
where W
= e"4^
are given; we can now easily compute F$(ß\ inserting it in eq. (2.8), and using our numerical procedure to determine the zeroes of Z. Secondly we performed 5 · 10 5 # 3 updates at ß'= 0.230 and 0.235 (after 105N3 equilibrating flips) and we computed the corresponding zeroes. We want to em phasize also that if we had wanted to use the improved /?'s available (so as to minimize the damping factor of (2.8)), the numerical integration we performed was
TABLE 1
(S) (see (2.1)) at ß = 0.23 for the different lattices sizes N
(S)
5 6 7 8
0.5281(9) 0.5360(7) 0.5388(5) 0.5411(7)
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Fig. 1. Im u$o versus Re utyo for the first seven zeroes on the 43 lattice: comparison between the exact result [4] (dots), the numerical integration of the exact energy distribution (crosses), and MC results for 105 energy values (open circles), ß' = 0.230.
supposed to converge at the correct result independently of β'. That is why we did our check at two different /Ts. We looked at the first seven zeroes w^L4j0> / = 1,... ,7; for all of them the exact result was very close to the one obtained from O0O((|w ( A e x a c t | " l"i0o^hD/l"(4,oexactl < 1%, for / = 1,...,7), both for £ ' = 0.230 and for β'= 0.235. As far as the MC determination of the zeroes is concerned, it is apparent from figs. 1, 2 that the first two zeroes (w%, M^O) do coincide, for both /Ts, inside our error, with the exact ones. An error of the order of one percent is observed for the third zero, while precision is lost for the farther ones. Now when studying larger lattices, we have reasons to be worried: the oscillating factor we are trying to integrate, - cos(£fF), can be written as cos(£vF£), where V is the volume of the system, E is the integration variable, going from zero to one,
316
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1
1
I
Im u too
•X
.300
o
o
*
X
o .200
Γ
o
— X
.100
.100
_i_
.200
Reu .300
.WO
Fig. 2. Same as in fig. 1, but ß ' = 0.235.
and £v is the distance of the interesting zero from the real u axis in the considered volume. The magnitude of the oscillations will dramatically increase with the volume of the system. The situation is in our case nicer than that just because of the presence, in the infinite-volume limit, of a zero on the real U axis; since ξΝ is supposed to behave as [3] N~l/v, the amplitude of the oscillations will not increase as TV3, but as ~iV 14 . That means, for example, that when going from iV = 4 to TV = 8, it will not increase by a factor 8, but just by a factor of ~ 2.5. In tables 2 and 4 we give our results for the first and the second zero, for N = 5-8. We defined our error in the following way: we compute the zero in two clusters of energy values, each one given from 4· 105iV3 spin updates, and separate them by 10 5N3 updates. We assume our statistical precision is one in which these two results coincide and also coincide with the one we get from considering all the 106N3 spin updates. We consider our statistics not large enough to obtain reliable information on zeroes farther than the second one.
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Fig. 3. - In Im u$\0) versus In N, for N = 4-8.
4. Interpretation of results The physical conclusions we will draw here will follow the analysis performed in ref. [3]. The scaling picture [6] is applied to the location of the zeroes of Z(ß) in the complex ß plane. Our first result follows from the relation
r~CN-l/\
(4.1)
where τ is the distance of the closest zero from the physical singularity at N = oo, r= uc(N)-uc(oo). In fig. 3 we plot -Inilmu^) versus InN: the linearity of the results is quite impressive. In table 3 we give the values of v(N9 N') for N = 4 , . . . ,7, N' > N, when assuming τ = Im u(1\ In table 4 we use τ = \u$0 - uc\ with uc = 0.4120 [11]: in the
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TABLE 2
First zero: u = e ~ 4^ N
Re/^o
Im/^o
Re «# 0
Im !$>0
4 5 6 7 8
0.2327 0.2307 0.2289 0.2277 0.2270
0.0561 0.0384 0.0283 0.0220 0.0175
0.3844 0.3927 0.3977 0.4006 0.4023
0.0877 0.0608 0.0452 0.0353 0.0282
Error is 1 on the last digit in columns one and two and 2 on the last digit in columns 3 and 4.
TABLE 3
v(N, N') = (In N' - In tf)/(lnlm w^o ~ hi Im i/JJ) 0 ) N' Ν
4 4
5
6
7
8
0.610(9)
0.612(6) 0.614(4)
0.615(4) 0.619(9) 0.624(20)
0.611(5) 0.612(9) 0.611(14) 0.596(31)
5 6 7 8
TABLE 4
As in table 3, but lnlm u% - ln(Im κ#>ο + (Re u$0 - w c ) 2 ) 1/2 4
Ν
Ν
4 4 5 6 7
5
6
7
8
0.610(6)
0.612(4) 0.614(10)
0.617(3) 0.621(6) 0.628(16)
0.616(3) 0.618(5) 0.621(9) 0.612(22)
8
scaling limit the two definitions are equivalent. We obtain that v from table 4 is closer to the true value (v = 0.631 ± 0.001, from (9)) than the one from table 3; this is for all values of the lattice size. More information is contained in table 4 than in table 3, which can help explain this phenomenon. A systematic smallness of v computed from our small lattices (~ 1%) is apparent; given our error bars we cannot try to study the way in which p(N, N') -> p(oo).
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TABLE 5
The second zero
N
t^c uN o
Im u%
4
0.3444 0.366(3) 0.379 0.386 0.390
0.1433 0.101(2) 0.076(3) 0.054 0.045
5 6 7 8
Error is 1 on the last digit, if not indicated in parenthesis.
A last remark on this point is in order: it is clear that, for increasing N, all the zeroes close enough to the real ß axis should scale according to (4.1). In fig. 4 we plot - In Im u$0 for / = 1,2; also the second zero has the expected behaviour. The following result is based on the relation , cos πα - A _/A + r, tan[(2 - α)φ] = -r'—*- , sin7ra
1.2
1.4
1.6
1.8
(4.2)
2.0
Fig. 4. As in fig. 3, but dots for u$\Q) and crosses for MJ^ 0 , N = 5-8.
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E. Marinari / Finite-size scaling TABLE 6
φ„ ^ tan' HCIm u%0 - Im ii# 0 )/(Re «$> - Re u$0)}
N
Φ
5 6 7 8
56.5ΐί 3 5 58.7 + i f 52.0±33;87
53.8+ti
obtained in ref. [3]. Here a is the specific heat exponent (we will use as input the value a = 2 - dv = 0.107 ± 0.003), A _/A + the critical amplitude ratio for the specific heat, and φ is the angle at which the complex zeroes deviate from the real u axis (in the complex u plane) in the infinite-volume limit (at zero magnetic field). We give in table 6 the φ^ we get for our lattice sizes, from the first and second zeroes. The errors are quite large (obviously enough the precision of the order of one percent in the second zero is reflected by a large indetermination of the slope, see fig. 5). From our data we feel quite comfortable with the assumption that the N dependence of the slope is by far considerably smaller than our statistical indetermination; so we compute φ5,8 = (55.3 ±1.5)°,
(4.3)
and average φ on the lattices of sizes from 5 to 8. We get in this way, from (4), the estimate ^ 7 = 0.45(7),
(4.4)
where the error has a statistical meaning, and the systematic error coming from the finiteness of the lattice is not taken into account (but seems to be much smaller than the statistical one). For a comparison with experimental data, see for example ref. [8] (in particular table 3 and fig. 2 therein); the experimental values range from 0.36 and 0.63, and the comparison with our value is fair enough. On the theoretical ground a computation [7] at the order e2 gives for A +/A _ a value of 0.48 (lowering the value [10] at the order ε, 0.55); also in this case the comparison with our value is more than reasonable. We wish to thank C. Itzykson and J.B. Zuber for their enthusiastic, generous and continuous help. Interesting discussions with D. Beysens, H. Flyberg, R. Pearson and J. Zinn-Justin are gratefully acknowledged. We thank C. Itzykson and G. Parisi for a critical reading of the manuscript.
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Fig. 5. Determination of φ^.
References [1] M. Falcioni, E. Marinari, M.L. Paciello, G. Parisi and B. Taghenti, Phys. Lett. 108B (1982) 331 [2] R.B. Pearson, Phys. Rev. B26 (1982) 6285 [3] C. Itzykson, R.B. Pearson and J.B. Zuber, Nucl. Phys. B220[FS8] (1983) 415; J.B. Zuber, Talk at Workshop on nonperturbative field theory and QCD, Trieste, December 1982, Saclay preprint, SPhT/83/004 (January 1983) [4] R.B. Pearson, private communication [5] E. Marinari, G. Paladin, G. Parisi and A. Vulpiani, 1 / / noise, disorder and dimensionality, Saclay preprint, SPhT/83/96 (July 1983), J. de Phys., to be published [6] M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516 [7] C. Bervillier, Phys. Rev. B14 (1976) 4964
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[8] D. Beysens, A. Bourgou and P. Calmettes, Phys. Rev. A26 (1982) 3589 [9] J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. B21 (1980) 3976; B.G. Nickel, in Phase transitions, status of the experimental and theoretical situation, Cargese 1980, eds. M. Levy, J.C. Le Guillou and J. Zinn-Justin (Plenum, New York, 1981); J. Zinn-Justin, J. de Phys. 42 (1981) 783 [10] P.C. Hohenberg, A. Aharony, B.I. Halperin and E.D. Siggia, Phys. Rev. B13 (1976) 2896 [11] C. Domb, in Phase transitions and critical phenomena, vol. 3, eds. C. Domb and M.S. Green (Academic Press, 1974); J. Zinn-Justin, private communication
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