Extended scaling functions for Ising systems with dimensionality 1⩽d⩽4

Physica A 267 (1999) 165–172

Extended scaling functions for Ising systems with dimensionality 16d64 Manuel I. Marques, Julio A. Gonzalo ∗ Departamento de Fsica  de Materiales, C-IV, Universidad AutÃonoma de Madrid, 28049 Madrid, Spain Received 12 July 1998

Abstract Finite-size lattice Monte Carlo simulations of phase transitions in Ising systems (16d64) allow the determination of simple binomial scaling functions for T ¡ Tc and T ¿ Tc , consistent with the asymptotic behavior for the critical isotherm, the spontaneous magnetization and the zero eld susceptibility, which describe quite well the scaling behavior in an extended range of c 1999 Elsevier the scaling variable, 0:016|(T − Tc )=Tc |L1= 620, for all four dimensionalities. Science B.V. All rights reserved. PACS: 75.10.H; 11.10.J; 75.30.K; 02.50.N Keywords: Scaling functions; Ising model; Monte Carlo simulations

Since the homogeneity assumption [1–5] was formulated for the free energy of cooperative systems undergoing phase transitions, eorts were made [6 –15] to cast the scaling function, e.g. h(x) in H=M  = h(||=M 1= ), in a simple form. Further developments [16,17] lead to the renormalization group approach to phase transitions which provided the subsequent unifying picture for cooperative phenomena at phase transitions. Nevertheless the renormalization group approach does not give direct hints regarding scaling functions, and has not played a prominent role in the search for constructing explicit scaling functions. A well known and useful method [18–22] to study systems with many degrees of freedom undergoing phase transitions is the Monte Carlo method. In particular, phase transitions in Ising systems of increasing dimensionality are especially favorable to carry out numerical simulations in the vicinity of the transition temperature, which can provide a good empirical basis to investigate the asymptotic behavior of the phase transition. Of course a very substantial amount of work using Monte Carlo simulations ∗

Corresponding author. Fax: +34-91-397-85-79; e-mail: [email protected].

c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 7 5 - X

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has been devoted in the last two decades to investigate Ising, X −Y , Heisenberg and other systems undergoing phase transitions [23–32], but most of it has not been focused on the scaling functions as such and their dependence on the dimensionality. In this work we will use numerical nite-size lattice Monte Carlo simulations of phase transitions in Ising systems with increasing dimensionality in an eort to characterize the respective scaling functions in a compact and selfconsistent form. Our aim is to get a simple functional expression common to all dimensionalities diering only in the respective critical exponents and a few numerical factors of order one. We consider calculations until the marginal dimensionality d = 4. According to nite-size scaling theory [4] the singular part of the free energy for a spin 12 Ising ferromagnetic system of linear dimension L can be written as ˜ − Tc )=Tc |L1= ; HL = ) ; Fs (T; H; L) = L−(2−)= F(|(T

(1)

where T is the temperature, H the magnetic eld and ; ;  and  are the usual critical ˜ y), with x = |(T − Tc )=Tc |L1= and ˜ exponents, and F(|(T − Tc )=Tc |L1= ; HL = ) = F(x; y = HL = , is the free energy in scaled form. Taking the partial derivative of Fs with respect to H , and making use of the relationships among critical exponents we get 1= ˜ ˜ y)=@y) = f(||L ; 1) ML = = (@F(x;

(2)

which de nes a scaling function 1= ˜ ; 1) M=H 1= = f(||=M

(3) − =

− =

and M ∼ L on the left-hand side identifying formally H and M with H ∼ L and the right-hand side, respectively. This scaling function must conform to the expected behavior at (a) the critical isotherm ( = 0), M=H 1= ∼ const: → ML = ∼ const ;

(4)

obtained using the formal relation H ∼ L− = , (b) the spontaneous magnetization ( ¡ 0; H = 0), ||=M 1= ∼ const: → ML = ∼ (||L1= ) ;

(5)

obtained eliminating M and multiplying both terms of the equation by L = , and (c) the zero eld susceptibility ( ¿ 0; H → 0), M=H ∼ const:||− (−1) → ML = ∼ (||L1= )− (−1)=2 ;

(6)

where ( − 1) = is the zero eld susceptibility exponent. This last expression can be obtained using the formal identi cations H ∼ L− = and M ∼ L− = . This behavior may be interpreted in the following way: in any nite-size lattice with periodic boundary conditions there is a residual eld (H ) ∼ L− = and a residual magnetization (M ) ∼ L− = which tend to zero as L → ∞. As mentioned previously our aim is to construct compact scaling functions compat1= ˜ ; 1), ible with the asymptotic behavior speci ed by (a) – (c) above. We try for f(||L 1= simple binomial forms compatible with (a) – (c) asymptotically, i.e. for ||L → 0 and

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Table 1 Critical temperature and critical exponents for Ising systems with dimensionalities d = 1; 2; 3; 4 Dimension, d

Tc









1 2 3 4

0 2.2692 4.5116 6.6879

0 1=8 0.33 1=2

∞ 15 4.8 3

1=2 7=4 1.24 1

3=2 0 −0:0=0 0

1=2 1 0.63 1=2

for ||L1= 1, but make room, within the freedom granted by the homogeneity assumption, for modi cations capable of describing well also the Monte Carlo simulation data for ||L1= ∼ 1. First, we try m¡ = f˜d¡ (1; x) = Ad (1 + Bd¡ (x)|x|zd¡ ) =zd¡

for  ¡ 0 ;

m¿ = f˜d¿ (1; x) = Ad (1 + Bd¿ (x)|x|zd¿ )− (−1)=2zd¿

for  ¿ 0 :

(7) (8)

Here m = ML = ∼ M=H 1= ; x = L1= ∼ =M 1= , and Ad ; Bd (∞) (i.e. Bd (x) for x → ∞) are numerical constants of order unity which may take dierent values for dierent dimensionalities 16d64. The auxiliary exponents z (z ¡ for  ¡ 0, z¿ for  ¿ 0) are chosen for a best t to the numerical simulation data corresponding to each dimensionality in the intervening region ||L1= ∼ 1, leaving invariant automatically the asymptotic behavior for ||L1= → ∞. We will see below that these auxiliary exponents z come out to be very close or identical to the corresponding combinations of exponents z¡ = = and z ¿ =  for each dimensionality. Then a closer look at the behavior of the scaled data shown below, clearly indicates that in order to reproduce the observed behavior at  → 0, where clearly @m=@ 6= 0, it is required for B(x)x z to go from ∼ (x=x0 ) at x=x0 1 to ∼ x z at x=x0  1, which can be achieved writing B(x) ∼ B(∞)=(|tanh(x=x0 )|) z−1 . Table 1 gives the critical temperature and the critical exponents for Ising systems with dimensionalities d = 1; 2; 3; 4. These numbers will be used below to display the raw data for M and T from the numerical simulation in scaled form. Standard Metropolis Monte Carlo simulations for lattices of increasing linear dimension L were performed with a desk top computer (Pentium at 266 MHz) in the usual way [22]. We are aware that more sophisticated calculations have been performed to study Ising systems with longer linear dimension L and greater dimensionalities [33–35], but, for our propose, they are not absolutely required. Our Monte Carlo calculations involved about 20 000 Monte Carlo steps per spin for each data point. Initial conditions for each temperature were taken as the equilibrium conditions for the previous temperature, starting with M ≈ 1 for the lowest temperature and thermalizing. Fig. 1(a) – (c) shows the raw Monte Carlo data for M (T ) used to construct our scaling function, with d = 2; 3; 4 using Ld spins with 306L690 for d = 2; 56L630 for d = 3 and 76L613 for d = 4. The data for d = 1 are not shown, but using a small constant le H ∼ 0:001 to avoid peculiar complications due to the critical point

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Fig. 1. Magnetization versus temperature data for Ising systems from Monte Carlo simulation with dierent values of the linear dimension L. (a) d = 2, (b) d = 3 and (c) d = 4.

at Tc = 0, data were obtained and they showed, as expected, scaling behavior at ||L1= values substantially larger than for higher dimensionalities. Fig. 2(a) – (c) presents in log–log form the scaled data (ML = versus ||L1= ) corresponding to  ¡ 0 and  ¿ 0 for d = 2; 3; 4, using the data in Fig. 1(a) – (c) and the critical temperature and critical exponents given in Table 1. It can be seen, as expected, that the data scale well, and that small systematic deviations from perfect scaling are more noticeable at the larger values of ||L1= for the smaller L values in each case due probably to magnetization saturation eects. Also, for the smaller L at low ||L1= values, deviations take place earlier due to more pronounced nite-size eects, as may be expected. The scaling function (m versus |x|) given by Eq. (7) ( ¡ 0) and Eq. (8) ( ¿ 0) are shown (full lines), along with the starting scaling functions (with zd¡ = zd¿ = 1, Bd¡ (x) = Bd¡ (∞), and Bd¿ (x) = Bd¿ (∞)). These starting scaling functions give good ttings to the data only away from the critical point (dashed lines). We note that Eqs. (7) and (8) describe quite well the observed behavior in a very large interval of the scaling variable, 0:016||L1= 620.

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Fig. 2. Log–Log plots of the scaled data ML = versus ||L1= for Ising systems. Figure (a) corresponds to dimension d = 2, (b) to dimension d = 3 and (c) to dimension d = 4. Scaling functions are plotted. Full lines are for m = f˜ d (1; x), in both branches (Eq. (7) for  ¡ 0 and Eq. (8) for  ¿ 0), taking zd¡ as = and zd¿ as . Dashed lines correspond to the starting scaling functions (zd¡ = zd¿ = 1, Bd¡ (x) = Bd¡ (∞), and Bd¿ (x) = Bd¿ (∞)).

Fig. 3(a) – (c) presents a close look of (ML = versus L1= ) for small . It gives, in linear form, the scaled data, the full scaling functions (m versus x), that is Eqs. (7) and (8) with the parameters zd¡ = =, zd¿ = , and Bd¡ (x) = Bd¡ (∞)=[|tanh(x=x0¡ )|]zd¡ −1 and Bd¿ (x)=Bd¿ (∞)=[|tanh(x=x0¿ )|]zd¿ −1 , together with the starting scaling functions (with zd¡ = zd¿ = 1, Bd¡ (x) = Bd¡ (∞), and Bd¿ (x) = Bd¿ (∞)). It is clear that in the close vicinity of the critical point, || = 0, the t to the data is substantially improved with the full scaling function. Since the critical exponents for each dimensionality are xed, the best t to the data involves optimizing only three parameters Ad ; Bd (∞) and x0d , for each phase, the ferromagnetic phase ( ¡ 0) and the paramagnetic phase ( ¿ 0). Note also that the scaling data, and the scaling functions, are highly asymmetric with respect to ||=0, for d=2 and 3, and that they become much more symmetric for d=4 (margined dimensionality), where the mean eld exponents = 12 , ( − 1)=2 = =2 = 12 ,

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Fig. 3. Linear plot of the scaled data ML = versus  L1= for Ising systems. Figure (a) corresponds to dimension d = 2, (b) to dimension d = 3 and (c) to dimension d = 4. Scaling functions are plotted. Full lines for m = f˜ d (1; x), in both branches (Eq. (7) for  ¡ 0 and Eq. (8) for  ¿ 0), taking zd¡ as = and zd¿ as . Dashed lines correspond to the starting scaling functions (zd¡ = zd¿ = 1, Bd¡ (x) = Bd¡ (∞), and Bd¿ (x) = Bd¿ (∞)).

which determine the asymptotic behavior at  ¡ 0 and  ¿ 0, respectively, become equal. Table 2 summarizes the scaling function coecients and exponents for Ising systems with dimensionalities 26d64, used to t the scaled Monte Carlo data. We have not used any special criteria in order to choose the tting parameters except looking for a sucient good visual tting to the Monte Carlo data in each case. The empirical determination of the exponents zd¡ and zd¿ has also been done in the same way at || ∼ 1. We note, however, how values of = and  for zd¡ and zd¿ , respectively, give excellent ts in all cases. We have compared our scaling functions for  ¿ 0 with other scaling functions previously using [6 –11] for Ising data with d = 3 and we have found that the former

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Table 2 Scaling function coecients Ad ; Bd (∞); x0 and exponents zd for Ising systems with dimensionalities d = 1; 2; 3; 4: The last columns show approximate values for zd¡ and zd¿ in terms of the critical exponents ¡0 Dimension, d

Ad

Bd¡ (∞)

x0¡

zd¡

=

1 2 3 4

– 1 1.10 1.325

– 15.52 21.30 20.47

– 0.373 0.192 0.566

– 1:9 ± 0:3 2:5 ± 0:5 2:5 ± 1:0

– 1.875 2.514 3

Dimension, d

Ad

Bd¿ (∞)

x0¿

zd¿



1 2 3 4

(1) 1 1.10 1.325

– 1.21 1.65 2.17

– 0.454 0.427 0.512

– 1:9 ± 0:3 1:6 ± 0:3 1:5 ± 0:3

0.5 1.875 1.584 1.5

¿0

Fig. 4. Log–Log plot of the scaled data ML = versus L1= for Ising systems with d = 3 and  ¿ 0. Previous scaling functions of Arrot and Noakes, Ho and Lister, Vicentini-Missoni et al. and Milosevic and Stanley (by series extrapolation methods) are compared with our scaling function, Eq. (8) (full line). An arrow indicates the end of the numerical data by Milosevic and Stanley method. Gaunt and Domb equation obtained by Pade approximants behaves similar to the functions plotted by means of broken lines.

(our scaling function) ts better the Monte Carlo data in a more extended range of the variables (see Fig. 4). It may be concluded that: (a) simple and accurate scaling functions valid for the Ising systems with dimensionalities 16d64 have been constructed, based solely on the known asymptotic behavior of the critical isotherm, the spontaneous polarization temperature dependence ( ¡ 0) and the zero eld susceptibility ( ¿ 0); (b) these scaling functions are asymmetric with respect to the critical point ( = 0, H = 0) for d ¡ 4 and have the same functional expression for all four dimensionalities examined provided

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that the respective critical exponents are used, and (c) in spite of their simplicity, they are valid in an extraordinary wide range. We acknowledge gratefully the nancial support from CICyT through grant PB96-0037. References [1] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York, Oxford, 1971. [2] C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, Academic Press, New York, 1972. [3] B. Widom, J. Chem. Phys. 43 (1965) 3898. [4] M.E. Fisher, Rep. Prog. Phys. 30 (1967) 615. [5] R.B. Griths, Phys. Rev. 1958 (1967) 176. [6] A. Arrot, J.E. Noakes, Phys. Rev. Lett. 19 (1967) 786. [7] J.T. Ho, J.D. Litser, Phys. Rev. Lett. 22 (1969) 603. [8] M. Vicentini-Missoni, J.M. Levelt Sengers, M.S. Green, Phys. Rev. Lett. 22 (1969) 390. [9] D.S. Gaunt, C. Domb, J. Phys. C 3 (1970) 1442. [10] S. Milosevic, H.E. Stanley, Phys. Rev. B 6 (1972) 986. [11] S. Milosevic, H.E. Stanley, Phys. Rev. B 6 (1972) 1002. [12] P. Scho eld, Phys. Rev. Lett. 22 (1969) 606. [13] J.A. Gonzalo, Phys. Rev. B 1 (1970) 3125. [14] J. Ho, Phys. Rev. Lett. 26 (1971) 1485. [15] J.A. Gonzalo, J. Phys. C 13 (1980) 241. [16] L.P. Kadano, Physics (N.Y.) 2 (1966) 263. [17] K. Wilson, J.B. Kogut, Phys. Rep. C 12 (1975) 75. [18] K. Binder, Introduction: theory and “technical” aspects of Monte Carlo simulations, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Springer, New York, 1979, pp. 1– 43. [19] D.P. Landau, Phase diagrams of mixtures and magnetic systems, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Springer, New York, 1979, pp. 121–141. [20] K. Binder (Ed.), Applications of Monte Carlo Method in Statistical Physics, second ed., Springer, New York, 1987. [21] J.M. Yeomans, Statistical Mechanics of Phase Transitions, Clarendon Press, Oxford, 1992. [22] P. Back, Phys. Today (1983) 25. [23] D.P. Landau, M. Blume, Phys. Rev. B 13 (1976) 287. [24] A.M. Ferrenberg, D.P. Landau, Phys. Rev. B 44 (1991) 5081. [25] A.M. Ferrenberg, D.P. Landau, J. Appl. Phys. 70 (1991) 6215. [26] K. Binder, K. Vollmayr, H.-P. Deutsh, J.D. Reger, M. Scheucherand, D.P. Landau, Int. J. Mod. Phys. C 3 (1992) 1025. [27] K. Chen, A.M. Ferenberg, D.P. Landau, J. Appl. Phys. 73 (1993) 5488. [28] H.W.J. Blote, E. Luijtenand, J. Heringa, J. Phys. A: Math. Gen. 28 (1995) 6289. [29] G. Kamieniarz, F. Mallezie, R. Dekeyser, Phys. Rev. B 38 (1988) 6941. [30] H.W.J. Blote, G. Kamieniarz, Physica A 196 (1993) 455. [31] P. Pawlichi, G. Kamieniarz, L. Debski, Physica A 242 (1997) 290. [32] U. Nowak, A. Hucht, J. Appl. Phys. 76 (1994) 6341. [33] R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 58 (1987) 86. [34] U. Wol, Phys. Lett. B 228 (1989) 379. [35] G. Parisi, J.J. Ruiz-Lorenzo, Phys. Rev. B 54 (1996) 3698.