Finite-size scaling analysis of the critical behavior of the Baxter–Wu model

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Physica A 352 (2005) 447–458 www.elsevier.com/locate/physa

Finite-size scaling analysis of the critical behavior of the Baxter–Wu model S.S. Martinos, A. Malakis, I. Hadjiagapiou Department of Physics, University of Athens, GR 15784 Athens, Greece Received 20 May 2004 Available online 2 February 2005

Abstract We use the recently developed critical minimum energy subspace (CrMES) approximation scheme to study the critical behavior of the Baxter–Wu model. This scheme uses only a properly determined part of the energy spectrum and allows us to obtain high accuracy for relatively large systems with considerably reduced computational effort. The density of states is constructed by a multi-range Wang–Landau sampling and from this we obtain the critical properties of specific heat and of the ‘‘reduced Binder cummulant’’. The good agreement of our results with existing exact solutions demonstrates the accuracy of our approximation technique. r 2005 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 64.60.Cn; 75.10.Hk Keywords: Ising model; Triangular lattice; Baxter–Wu model; Wang–Landau method

1. Introduction Numerical simulations are widely used for the study of macroscopic systems [1,2]. But even though they can be performed on relatively large systems numerical simulations cannot give the true behavior in the critical region. A system exhibits Corresponding author.

E-mail address: [email protected] (S.S. Martinos). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.12.062

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true critical behavior only in the limit in which its size becomes infinite (thermodynamic limit). There are several techniques for extracting bulk properties of thermodynamic systems from the data obtained from simulations in finite systems. One of the most widely used such technique is finite-size scaling (FSS), especially for extracting the values of critical exponents [3–6]. By this technique we essentially study how thermodynamic quantities vary with the size of the system and we put them into relations that include ‘‘universal’’ functions, i.e., functions not dependent on the size of the system. FSS is a theory for the properties of a system in the critical region and thus it has been originally developed for the study of second-order phase transitions. However it has been also successfully applied in the case of first-order transitions [7–9]. For the last case, the parameters that are introduced in relations expressing FSS are determined by the dimensionality of the lattice and not by the critical exponents. According to FSS the free energy of an L
L lattice is given by the scaling ansatz F ðL; TÞ ¼ Lc F~ ðtLy Þ ,

(1)

log(DE2/L2)

5.95 5.90

Data: SCALTRIANGDE_change Model: LinFunct

5.85

Chi^2 = 7.9545E-7 R^2 = 0.99996

5.80

a b

1.08063 ± 0.00341 0.94983 ± 0.015

5.75 5.70 5.65 5.60 5.55 5.50 4.20

4.25

4.30

4.35

4.40

4.45

4.50

4.55

4.60

logL Fig. 1. Variation of the width DE of the critical minimum energy subspace with lattice size (DE is the width of the minimum energy range that is needed to obtain an accuracy r ¼ 106 for the specific heat at the temperature of its maximum). The six points in the diagram are for lattice sizes L ¼ 69; 75, 78, 84, 87, 96. The parameter a gives the slope of the log–log fit and estimates the ratio a=n:

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where c ¼ ð2  aÞ=n and t ¼ j1  T=T c j with T c the critical temperature of the infinite lattice and a; n the specific heat and correlation length critical exponents, respectively. The critical behavior of the correlation length ðx ¼ x0 tn Þ suggests that y ¼ 1=n: The function F~ ðwÞ is ‘‘universal’’ in the sense that it is independent from the lattice size. For the specific heat (per lattice site), the study of which is the main purpose of the present work, FSS predicts the scaling relation cL La=n ¼ c~ðtL1=n Þ ,

(2)

where the function c~ðwÞ is again independent from lattice size. We mention that relation (2) is valid if the specific heat has a power-law singularity at the critical point and not a logarithmic one. Moreover it is valid in this simple form if no logarithmic corrections in the power law are needed. For the Baxter–Wu model, however, specific heat exhibits a simple power-law singularity and we do not need any correction to expression (2). (See Ref. [5] p. 237 and references therein.) In the present work, we study the scaling properties of the specific heat and the socalled Binder cummulant for the Baxter–Wu model. The Baxter–Wu model is a triangular lattice with the Hamiltonian, in the absence of external magnetic field, X H ¼ J si sj sk , (3)

CL

ijk

38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

L=30 L=51 L=78 L=96

2.22

2.24

2.26

2.28

2.30

2.32

T Fig. 2. Dependence of the specific heat C L on the temperature for various lattice sizes. As L becomes large the ‘‘pseudocritical’’ temperature approaches the bulk critical temperature.

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where the sum is taken over all triangular faces of the lattice and the spins si , sj ; sk take on the values 1 . The model is a special case of a more general one introduced by Wood and Griffiths [10], in which the Hamiltonian takes into account not only the triplet interaction (3) but also nearest-neighbor pair interaction of the usual Ising type. The triangular lattice can be decomposed into three interpenetrating sublattices [11] so that any triangular face contains one spin from each of them. As we see from (3) the Hamiltonian of Baxter–Wu model is invariant if the spins of any two sublattices are reversed and the resulting state has equal probability with the initial one (this ‘‘symmetry’’ of the model corresponds, in some way, to the known up–down spin-reversal symmetry of the usual Ising model). As a consequence the ground state of the Baxter–Wu model is four-fold degenerate: there is one ferromagnetic state with all the spins up, while the three ferrimagnetic ground states have the spins in one of the three sublattices up and the spins in the other two down. An exact solution by Baxter and Wu a second-order phase transition at pffiffigives ffi the critical temperature kT c ¼ 2= lnð1 þ 2Þ with critical exponents a ¼ a0 ¼ n ¼ n0 ¼ 23: In Refs. [12,13] results are given for the magnetic susceptibility and spontaneous magnetization critical exponents. The Baxter–Wu model is believed to belong to the same universality class as the four-state Potts model [14,15]. However, while for the Potts model a logarithmic correction to scaling is needed 0.40 0.35 0.30 L=30

CL L-1

0.25

L=51 L=78

0.20

L=96

0.15 0.10 0.05 0.00 -60

-40

-20

0

20

40

60

3/2

(T-TC )L

Fig. 3. Scaled specific heat cL La=n vs. scaled reduced temperature ðT  T c ÞL1=n : The four curves for different L falls into one in agreement with scaling relation (2).

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there is no evidence for such a correction in the case of the Baxter–Wu model [16–18]. In addition to the exact solutions given in Refs. [12,13] and the above-mentioned works several other papers have been published for the Baxter–Wu model. Phase diagrams in the presence of both pair and triplet interactions have been derived by Malakis [19,20], by using variational approximations. In our recent paper [21], we investigate the scaling properties of the distribution of magnetization, while in Ref. [22] we compare FSS behavior for first- and second-order phase transitions. Also, the so-called short time dynamics of the model has recently attracted the interest of several authors [23–25]. In Section 2 we present a brief outline of the ‘‘critical minimum energy subspace technique’’ (CrMES) [26], which will be used for speeding up our simulations. Numerical results are presented in Section 3 by combining the CrMES scheme with the Wang–Landau algorithm. Finally, we present our conclusions in Section 4.

4

-2 Data: SCALTRIANGTLC_SM Model: LinFunct

Data: SCALTRIANGTLC_change Model: LinFunct

log(TLmax-TC)

Chi^2 = 0.00003 R^2 = 0.99984

logCLmax

a 1.01941 ± 0.00407 b -1.08987 ± 0.01597

2

a -1.50283 ± 0.00774 b 0.95567 ± 0.02767

-4

-6 3.2

(a)

Chi^2 = 0.00019 R^2 = 0.99968

3.4

3.6

3.8 4.0 logL

4.2

4.4

4.6

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 logL

(b)

2.32 2.31

Data: Data1_B Model: extrapol Chi^2 = 2.2379E-8 R^2 = 0.99987

TLmax

2.30

a 2.62226 ± 0.11296 w 1.50722 ± 0.01679 b 2.26928 ± 0.00018

2.29 2.28 2.27 10

(c)

20

30

40

50

60

70

80

L

Fig. 4. (a) Estimation of the ratio a=n from the slope (parameter a) of the curve log C max vs. log L: The L points in the diagram are for lattice sizes L ¼ 27 to 96. (b) Estimation of 1=n from the slope (parameter a) of the curve logðT max  T c Þ vs. log L: (c) Simultaneous estimation of 1=n (parameter w) and T c (parameter L b) from extrapolation of T max vs. L: The points in diagrams (b) and (c) are for lattice sizes L ¼ 15 to 78. L

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2. Numerical technique The CrMES technique is based on the following simple idea. Let us suppose that we know the number of states GðEÞ for a finite system of linear size L and dimension d or, equivalently, the microcanonical entropy SðEÞ ¼ ln GðEÞ: Then its specific heat, at any temperature, is expressed by the usual statistical sums ðkB ¼ 1Þ: ( E max X d 2 C L ðTÞ ¼ L T Z1 E 2 exp½SðEÞ  bE E min

 Z 1

E max X

E exp½SðEÞ  bE

!2 9 =

ð4Þ

;

E min

with the partition function Z given by Z¼

E max X

exp½SðEÞ  bE .

(5)

E min

In practice, of course, the entropy SðEÞ is not exactly known. Instead we have only an approximation obtained by a numerical simulation such as entropic sampling [27] 0.67

0.66

VL

0.65

0.64 L=30 L=51

0.63

L=78 L=96

0.62

2.22

2.24

2.26

2.28

2.30

2.32

T Fig. 5. Variation of reduced cummulant V L with temperature. In the critical region V L deviates from the ‘‘Gaussian’’ value 23 and it exhibits a minimum for temperatures that seem to approach T c as the lattice size becomes large.

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or the more efficient Wang–Landau method [28,29] that we also adopt in the present work. Thus, expressions (4) and (5) give approximate results with accuracy depending on the accuracy of the numerical simulation. Now let E~ be the energy that corresponds to the maximum term exp½SðEÞ  bE of the partition function at the temperature of interest. Because of the sharpness of this maximum, we can restrict the summations in Eq. (4) in a subrange ðE~  ; E~ þ Þ around E instead of the total energy range ðE min ; E max Þ and obtain an approximation to the specific heat. As our investigations show [26], the errors introduced by this approximation are much smaller than the already existing errors from the approximate computation of SðEÞ by numerical simulations. Consequently, by this scheme we can have an efficient estimation of critical behavior using a small part of the energy space. We need SðEÞ only in this energy subspace and this restriction greatly facilitates the Wang–Landau algorithm for the computation of SðEÞ: Details for the implementation of the CrMES scheme can been found in Ref. [26], where the definition of an appropriate sequence of energy subranges has been shown to produce the proper energy range ðE~  ; E~ þ Þ for any desired accuracy of the specific heat at a given temperature. Note that, by the term ‘‘accuracy’’ we mean that the (relative) error j½cL ðE~  ; E~ þ Þ=cL ðE min ; E max Þ  1j; in the computation of specific heat from Eq. (4) using the restricted energy range relative to the value obtained using the

2.38 Data: LTLV_B Model: extrapol

2.36

Chi^2 = 6.2949E-8 R^2 = 0.99992 a w b

TLV

2.34

14.40416 ± 0.44908 1.84834 ± 0.0117 2.27028 ± 0.00018

2.32

2.30

2.28

2.26 0

20

40

60

80

100

L T min L

Fig. 6. Extrapolation of temperature of V L minimum vs. L (same as in Fig. 4c). The value of parameter b was expected to be the bulk critical temperature.

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total energy range, satisfies a predefined criterion, the same for all lattice sizes: j½cL ðE~  ; E~ þ Þ=cL ðE min ; E max Þ  1jpr: We shall use, in this paper, the accuracy criterion r ¼ 106 ; which is very strict compared to the relative errors produced in the specific heat by the available methods for determining the density of states (DOS), for instance by the Wang–Landau method. An important feature of the CrMES scheme must be mentioned here. At the ‘‘pseudocritical’’ temperature T L (i.e., the temperature of maximum of specific heat) the width DE ¼ E~ þ  E~  of the energy range for a given accuracy is expected to obey a scaling law of the form [26] DE  La=2n , Ld=2

(6)

where a and n are the specific heat and correlation length critical exponents, respectively. The same scaling law is valid for the width DE c of the energy range at the true critical temperature, as verified for the two- and three-dimensional Ising model in Ref. [26]. Fig. 1 refers to Baxter–Wu lattices of linear sizes L ¼ 69 to L ¼ 96: For this model a ¼ n ¼ 23 and the plot of logðDE 2 =L2 Þ vs. log L must have a slope a=n ¼ 1: For each lattice size ðLÞ; DE is the width of energy range determining the specific heat with a relative accuracy r ¼ 106 at the corresponding pseudocritical temperature. The value of the slope, a; in this figure is close to the expected value 2.38

Data: LTLV_B Model: extrapolB

2.36

Chi^2 = 2.6273E-8 R2 = 0.99997

TLmin

2.34

Tc a w b

2.32

2.26904 ± 0.00049 3.62033 ± 2.38925 1.50471 ± 0.15693 8.90714 ± 5.8544

2.30

2.28

2.26 0

20

40

60

80

100

L Fig. 7. Fitting of the values of to the function T c þ aLw ð1 þ bL1 Þ: By this fitting we obtain estimates for the critical temperature T c and for the correlation length critical exponent. T min L

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ða=n ¼ 1Þ and, of course, this fact demonstrates that scaling law (6) is valid, and provides additional support to the CrMES scheme.

3. Finite-size scaling behavior We proceed in this section to apply Eq. (4) of Section 2 for the computation of the specific heat C L ðTÞ . As it has been mentioned above, we shall use the minimum energy subspace to obtain an ‘‘accuracy’’ r ¼ 106 in the estimation of specific heat at the pseudocritical temperature, i.e., the temperature of maximum specific heat. We compute the density of states in this energy range by using the Wang–Landau algorithm [28]. We begin with an initialp modification factor f 0 ¼ e1 and we perform ffiffiffiffi 25 Wang–Landau iterations: f jþ1 ! f j : To complete iteration we apply the following histogram flatness criterion: max HðEÞ  min HðEÞ p0:05 . max HðEÞ In Fig. 2 the dependence of the specific heat C L on temperature near T c is shown, while in Fig. 3 the coincidence of the four curves demonstrates the validity of the scaling relation (2) with the known values of the critical exponents (i.e., a ¼ n ¼ 23).

log(2/3 - VLmin)

-2.6 -2.8

Data: LOGLLOGVLMIN_SM Model: LinFunct

-3.0

Chi^2 = 0.00004 R^2 = 0.99985

-3.2

a b

-1.06267 ± 0.00493 0.56848 ± 0.01989

-3.4 -3.6 -3.8 -4.0 -4.2 -4.4 3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

logL Fig. 8. Estimation of the ratio a=n from the slope (parameter a) of the curve logð23  V min L Þ vs. log L: The points in the diagram are for lattice sizes L ¼ 24 to 96.

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Estimates for the critical exponents are obtained from the slopes of the curves log C max vs. log L and logðT max  T c Þ vs. log L: C max is the maximum value of the L L L specific heat for lattice size L and T max is the corresponding temperature L (pseudocritical temperature). The slopes of these two curves must be a=n ¼ 1 and 1=n ¼ 1:5; respectively. In Figs. 4a and b, we present the corresponding linear fittings and the expected slopes are the parameters a in the insets of the figures. In Fig. 4c, we show the fitting of the values of the pseudocritical temperature T max L for various L to the function aLw þ b letting a; w and b as free parameters. From this we obtain a good estimate not only for the correlation length critical exponent (the inverse of parameter w), but also for the bulk critical temperature T c (parameter b). A further point of interest in our consideration is the reduced Binder cummulant V L defined by [9]

4 E (7) V L ¼ 1  L2 . 3 E2 L An analogous cummulant has been introduced for the distribution of magnetization instead of energy [8,30]. Although V L is not a measurable physical quantity it has been found that it is a characteristic feature of a distribution and very useful in simulations. Fig. 5 shows the variation of V L with temperature for different lattice sizes. As we see, far from the critical temperature and for increasing L the cummulant approaches the expected value 23: For these temperatures

 and 2for large L the distribution of energy is almost a Gaussian and thus E 4 ¼ E 2 : Near the critical temperature V L exhibits a characteristic minimum increasing with L: The temperature T min L ; where this minimum occurs, is not the same with the temperature of specific heat maximum but it should approach the bulk critical temperature as L becomes large. From the discussion in Ref. [9] we deduce that the difference of the minimum value V min of the energy cummulant (7) from 23 must scale as L 2 3

dþa=n ~ 1=n V ððT min  V min Þ L L L  T c ÞL

(8)

1=n and thus T min : In a similar way as in Fig. 4c, we show in L must approach T c as L min Fig. 6 the fitting of T L for various L to the function aLw þ b: We observe that T min approaches the bulk critical temperature as in the case of the pseudocritical L temperature (Fig. 4c). The resulting value of the exponent w on the other hand is not close to the inverse of correlation length critical exponent. In Fig. 7, we show a further attempt to obtain a correction to scaling of V L : we fit the values of T min to L the function T c þ aLw ð1 þ bL1 Þ: In this way we obtain a better estimate for T c and an estimate for the exponent w close to the value 1=n ¼ 1:5; but the fit is not as accurate as the corresponding fitting for the specific heat. In Fig. 8, we show a linear fitting of logð23  V min L Þ vs. log L: According to Eq. (8) the slope of the line must be d þ a=n ¼ 1: As we see, the slope (parameter a) has this value with good accuracy. However, the conclusion is that the specific heat scaling is more simple than that of the energy cummulant.

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4. Conclusions Our numerical results for the Baxter–Wu model are in complete agreement with theoretical predictions that arise from existing analytic solutions for this model. Especially our numerical simulations demonstrate the fact that the singularity of specific heat at the critical temperature obeys a simple power-law and no logarithmic corrections to finite-size scaling are needed. The use of Wang–Landau sampling combined to our minimum energy subspace technique gives numerical results much more accurate than those obtained by the usual Metropolis algorithm at least for the energy distribution and the specific heat. This accuracy makes possible the simultaneous estimation of critical temperature and critical exponents. For the reduced Binder cummulant our results are in accordance with the deviation of energy distribution from the Gaussian form in the critical region. The finite-size scaling laws for the behavior of this quantity seems to be more sophisticated and it appears that, at the moment, this is not the best route by which we could determine critical exponents or critical temperatures. However, Binder cummulant is a very good criterion that, in some cases, helps us to distinguish between first- and second-order phase transitions. In conclusion, the present work provides additional support for the efficiency of our approximation technique of minimum energy subspace. Since the restriction of the energy space is speeding up our simulations, we can considerably improve the accuracy of our estimates for the critical parameters by carrying out more independent (Wang–Landau) random walks in the appropriate energy subspace, thus reducing statistical errors. References [1] D.P. Landau, K. Binder, A guide to Monte Carlo Simulations in Statistical Physics, University Press, Cambridge, 2000. [2] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Clarendon Press, Oxford, 1999. [3] M.E. Fisher, in: M.S. Green (Ed.), Critical Phenomena, Academic Press, New York, 1971. [4] M.E. Fisher, M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516. [5] M.N. Barber, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Academic Press, New York, 1983. [6] V. Privman (Ed.), Finite-Size Scaling and Numerical Simulations of Statistical Systems, World Scientific, Singapore, 1990. [7] M.E. Fisher, A.N. Berker, Phys. Rev. B 26 (1982) 2507. [8] K. Binder, D.P. Landau, Phys. Rev. B 30 (1984) 1477. [9] M.S.S. Challa, D.P. Landau, K. Binder, Phys. Rev. B 34 (1986) 1841. [10] D.W. Wood, H.P. Griffiths, J. Phys. C 5 (1972) L253. [11] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, third ed., Academic Press, Tunbridge Wells, UK, 1989, p. 314. [12] R.J. Baxter, et al., J. Phys. A 8 (1975) 245. [13] R.J. Baxter, I.G. Enting, J. Phys. A 9 (1976) L149. [14] M.A. Novotny, D.P. Landau, Phys. Rev. B 24 (1981) 1468. [15] F.C. Alcaraz, J.C. Xavier, J. Phys. A 30 (1997) L203; F.C. Alcaraz, J.C. Xavier, J. Phys. A 32 (1999) 2041.

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