Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems

Transcription

1 Commun. Theor. Phys. 66 (2016) Vol. 66, No. 3, September 1, 2016 Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems Xiao-Teng Li ( 李晓腾 ) and Xiao-Song Chen ( 陈晓松 ) Institute of Theoretical Physics, Key Laboratory of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing , China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing , China (Received May 23, 2016; revised manuscript received June 3, 2016) Abstract Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λ (L, t) = L γ/ν f (tl 1/ν ), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ 1, λ 2 and λ 6, we get the finite-size scaling form of the second moment correlation length ξ(l, t) = L ξ(tl 1/ν ). It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic. PACS numbers: an, De, Da, Kd Key words: critical phenomena, finite-size scaling, principal fluctuation modes 1 Introduction Complex system refers to systems with enormous agents interacting with each other. From microscopic interactions under different external conditions, emergent phenomena and collective behaviors appear in a macroscopic scale. The interactions in complex systems have often multiple scales of length and time and are of character of complexity. In recent decades, datasets of a variety of complex systems become available. Data analysis and techniques for data analysis have aroused broad interests of scientists. [1 2] Among all data analysis techniques, principal component analysis is the most fundamental one with various applications, such as dimension reducing, [3 4] clustering [5] and eigen-mode extraction. [6 12] From data of a complex system consisting of N agents, the correlation between any two agents and therefore an N N correlation matrix can be obtained. Using the pricipal component analysis, we can get N independent principal fluctuation modes of agents from the N N correlation matrix. The fluctuations of whole system are dominated usually by a few of principla fluctuation modes. [6] From the characters of these principal fluctuation modes, we can have a good understanding of global properties. For thermodynamic systems with finite size, their thermodynamic functions satisfy the finite-size scaling form in the neighborhood of a critical point. [13 14] This can be used to identify the continuous phase transition in a finite system. If thermodynamic functions of a finite system follow finite-size scaling laws, there is a continuous phase transition in the system. With the critical exponents determined from finite-size scaling forms, the universality class of the continuous phase transition can be confirmed. For many complex systems, their principal fluctuation modes instead of thermodynamic functions can be investigated usually. The studies of the critical behaviors of principal fluctuation modes are of great interest. We like to know the finite-size behaviors of principal fluctuation modes near a critical point. In this article, we propose a finite-size scaling form of principal fluctuation modes. We can use this scaling form to study the critical phenomena of complex systems. Using the Ising model on a two-dimensional simple square lattice [15] as an example, we will investigate the critical behaviors of principal fluctuation modes. It is found that the principal fluctuation modes near critical point show critical behaviors and satisfy the finite-size scaling we proposed. From eigen values of principal fluctuation modes, we can calculate the second moment correlation length which follows a finite-size scaling form. Our paper is organized as follows. In Sec. 2, we introduce the principla fluctuation modes of complex systems and propose a finite-size scaling form for them. Taking the Ising model on a two-dimensional square lattice as an example, we investigate principal fluctuation modes and their finite-size scaling behaviors near the critical point in Sec. 3. The second moment correlation length is calculated from the eigen values of principal fluctuation modes Supported by the National Natural Science Foundation of China under Grant Nos and chenxs@itp.ac.cn c 2016 Chinese Physical Society and IOP Publishing Ltd

2 356 Communications in Theoretical Physics Vol. 66 in Sec. 4. Finally, we make conclusions in Sec Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems 2.1 Principla Fluctuation Modes of Complex Systems In a complex system consisting of N agents, agents interact with each other and they are correlated. Define a snapshot I of the system as configuration I, the state of an agent i is characterized by S i (I). From all configurations of the system, the average state of agent i is calculated as S i = 1 R S i (I), (1) R I=1 where R is the number of the configurations. The agent i has a fluctuation δs i (I) S i (I) S i in the configuration I. The correlation between agents i and j is defined as C ij = δs i δs j = 1 R δs i (I)δS j (I). (2) R I=1 With c ij as its elements, an N N correlation matrix C is introduced. There are N eigenvectors and eigenvalues for the correlation matrix C. The eigenvector corresponding an eigenvalue λ n is written as b n = b 1n b 2n b Nn, (3) which satisfies the equation Cb n = λ n b n, n = 1, 2,..., N. (4) All eigenvectors are normalized and orthogonal to each other. Any eigenvectors b n and b l follow the condition b n b l = b jn b jl = δ nl, (5) j where δ nl = 0 when n l and δ nl = 1 if n = l. From an eigenvector b n, we can define a principal fluctuation mode δ S n = δs j b jn. (6) j=1 For a transform matrix B defined by elements B in = b in, there are the relations B B T = I and B T = B 1. Equation (4) can be rewritten as C B = B Λ, where Λ is a diagonal matrix with elements Λ nl = λ n δ nl. Using the orthogonal condition of Eq. (5), we get the correlation between principal fluctuation modes C nl δ S n δ S l = λ n δ nl, (7) There is no correlation between different principal fluctuation modes and the mean square of a principal fluctuation mode δ S n is equal to λ n. From the N eigenvalues of correlation matrix C and their eigenvectors, we can calculate the correlation between agent i and j as C ij = b in b jn λ n = Q ij n λ n, (8) n=1 n=1 where Q ij n = b in b jn is the link strength between agent i and j of n-th principla fluctuation mode. We can expect that the correlation length of a complex system is related to the eigenvalues of its principal fluctuation modes. We define the state of system as M = S j. as j=1 The total correlation of an agent i can be calculated ( N D i = δs i δm = C ij = Q ij n )λ n, (9) j=1 k=1 j=1 where N j=1 Q ij n is the total link of the agent i in n-th principla fluctuation mode. The average of D i gives the susceptibility χ = 1 D i = 1 N N δm 2 = Q n λ n, (10) i=1 n=1 where Q n = ij Q ij n/n. is the average link of n-th principal fluctuation mode. We can consider the eigen value λ n as the susceptibilty of n-th principal fluctuation mode. The susceptibility of system is the sum of all eigen value λ n with a weight factor Q n. Corresponding to N interacting agents in the system, there are N independent principla fluctuation modes. In some cases, the susceptibility is dominated just by a few of principla fluctuation modes. From the investigations of several principla fluctuation modes, we can catch the global behaviors of system. 2.2 Finite-Size Scaling of Principal Fluctuation Modes According to the finite-size scaling theory of critical phenomena, [13 14] the susceptibility of a finie system with size L has the following finite size scaling form χ(l, t) = L γ/ν F χ (tl 1/ν ), (11) where t = (T T c )/T c is the reduced temperature and T c is the critical temperature. Because of the relation between susceptibility and eigen values λ n of principal fluctuation modes in Eq. (10), we suppose that λ n follow a finite-size scaling form as λ n (L, t) = L ζ n λn (tl 1/ν ), (12) for a few of dominant principal fluctuation modes. We anticipate that the exponent ζ n is equal to the ratio of critical exponent γ/ν and is independent of n. The finite-size scaling form in Eq. (12) can be used to investigate the critical behaviors of complex systems. 3 Finite-Size Scaling of Principal Fluctuation Modes in Two-Dimensional Ising Model Here we use the Ising model on a two-dimensional simple square lattice with zero external field to study its principal fluctuation modes. For the two-dimensional square lattice, periodic boundary conditions are taken. There are

3 No. 3 Communications in Theoretical Physics 357 N spins, which interact each other and have the Hamiltonian H = J i,j S i S j, (13) where interactions are restricted to the nearest neighbors. The spin S i at site i can point up or down and has S i = ±1 respectively. A configuration with {S i } = (S 1, S 2,..., S N ) appears with a probability p({s i }) = 1 Z e βh, (14) Z = e βh, (15) {S i} where β = 1/(k B T ) and k B is the Boltzmann s constant. The statistical average of any observable A({S i }) is calculated as A({S i }) = 1 A({S i }) e βh, (16) Z {S i} where the summation can be done for the sampled configurations {S i } simulated by the Wolff algorithm. In a finite Ising model, there is no symmetry breaking. We have always S i = 0 if all configurations are considered in the average. Correspondingly, the average of total magnetization M = i S i = 0. To characterize the appearance of ferromagnetic phase, we restrict the statistical average to the configurations with positive total magnetization. In this case, the averages s i = S i and m = M /N are nonzero. In the bulk limit N, m = 0 for temperature T > T c and m > 0 for temperature T < T c. The nonzero magnetization m > 0 indicate the appearance of ferromagnetic phase. If the total magnetization M is negative after a Monte Carlo step, we make a flip S i S i to all spins so that M become positive again. For this flip, the total energy of the Ising model is unchanged. Using only the configurations with positive total magnetization, we can define an N N correlation matrix C with elements C ij = S i S j S i S j. (17) For the correlation matrix C, there are N eigenvalues λ i and N corresponding eigenvectors b i., where i = 1, 2,..., N. An eigen vector b i is the field defined on a two-dimensional simple square lattice. The principal fluctuation mode δ S n is the summation of all fluctuations δs j at the site j with coefficent b jn. 3.1 Finite-Size Scaling at the Critical Point At the critical point T = T c of two-dimensional Ising model, the finite-size scaling form of principal fluctuation modes becomes λ n (L, 0) = L ζn λ n (0). (18) The logarithm of this equation gives ln λ n (L, 0) = ζ n ln L + ln λ n (0), (19) so that the log-log plot of λ versus L at T = T c is a straight line with slope equat to the exponent ζ. We find that the eigen values λ n have degeneracy. These results are shown in Fig. 1. In the first degerate group, eigen values λ 2, λ 3, λ 4, and λ 5 are equal. The eigen values λ 6, λ 7, λ 8 and λ 9 in the second degerate group have the same results within the range of error. The degeneracy here is the consequence of the symmetry in simple square lattice, which will be discussed in Subsec From the slopes of straight lines in Fig. 1, we can get the exponent ζ n. The results are summarized in Table 1. As we suspected in the last section, the exponent ζ n is independent of n and equal to the the exponent ratio γ/ν = 7/4 of two-dimensional Ising model. Fig. 1 Log-Log plot of λ versus L at T = T ك for n = 1, 2,..., 9. The critical exponents ζ are given by the slopes of the linear lines. Table 1 Critical exponent of n-th eigenvalue λ. ζ (6) ζ (3) ζ (3) ζ (4) ζ (4) ζ (2) ζ (3) ζ (3) ζ (3) 3.2 Finite-Size Scaling Functions of Principal Fluctuation Modes At temperatures around the critical point T c, the largest eigen values λ 1 (L, t) simulated for system sizes L = 16, 32, 64 are shown in Fig. 2(a). According to the finite-size scaling form of Eq. (12), the Monte Carlo data of λ 1 (L, t) should collapse into one curve of scaling variable tl 1/ν after multiplying L ζ 1, which is demonstrated in Fig. 2(b). In Fig. 3(a), the degenerate eigen values λ 2 (L, t), λ 3 (L, t), λ 4 (L, t) and λ 5 (L, t) are shown with respect to temperaure T for system sizes L = 16, 32, 64. The finite-size scaling functions λ n (tl 1/ν ) = λ n (L, t)l ζn of n = 2, 3, 4, 5 are presented in Fig. 3(b). In Fig. 4, we present the degenerate eigen values λ 6 (L, t), λ 7 (L, t), λ 8 (L, t), and λ 9 (L, t) on the left and their finite-size scaling functions on the right.

4 358 Communications in Theoretical Physics Vol. 66 Fig. 2 Eigenvalue λ 1 (L, t) is shown as a function of temperature T for system sizes L = 16, 32, 64 in the left. Using λ 1 (L, t)l ζ 1 = λ 1 (tl 1/ν ), different curves in the left collapse into one curve in the right. be used to determine the critical point from its fixed point. As analogous to the cumulant ratio of magnetization, we anticipate that the finite-size scaling function f n/l (tl 1/ν ) is universal. The eigenvalue ratio f n/l (0) at the critical point is a universal constant. In Fig. 5, the eigenvalue ratios R 1/l (L, t) of λ 1 to λ l for l = 2, 3, 4, 5 are shown with respect to temperature T and the scaling variable tl 1/ν in the left and right respectively. We find a perfect finite-size scaling for the Monte Carlo data of different sytem sizes L = 16, 32, 64. The eigenvalue ratios R 1/l (L, t) of n = 6, 7, 8, 9 are presented in Fig. 6. With temperature T as variable, the curves R 1/l of system sizes L = 16, 32, 64 differ. After using the scaling variable tl 1/ν, the different curves of R 1/l in the left collapse into one curve in the right. Fig. 3 (a) Degenerate eigen values λ 2(L, t), λ 3(L, t), λ 4 (L, t) and λ 5 (L, t) versus temperature T for system sizes L = 16, 32, 64. (b) finite-size scaling functions λ (L, t)l ζn = λ (tl 1/ν ) of n = 2, 3, 4, 5. Fig. 5 Eigenvalue ratio λ 1 /λ of l = 2, 3, 4, 5 versus temperature T and the scaling variable tl 1/ν for system sizes L = 16, 32, 64. Monte Carlo data of different L demonstrate a fixed point at the critical point. Fig. 4 (a) Degenerate eigen values λ 6(L, t), λ 7(L, t), λ 8 (L, t) and λ 9 (L, t) versus temperature T for system sizes L = 16, 32, 64. (b) Finite-size scaling functions λ (L, t)l ζn = λ (tl 1/ν ) of n = 6, 7, 8, Eigenvalue Ratios of Principal Fluctuation Modes Since the eigenvalues λ n (L, t) of different principal fluctuation modes follow the finite-size scaling form Eq. (12) with the same exponent ζ n, the eigenvalue ratio R n/l (L, t) λ n (L, t)/λ l (L, t) has the finite-size scaling form R n/l (L, t) = λ n (tl 1/ν ) λ l (tl 1/ν ) = f n/l(tl 1/ν ). (20) At critical point with t = 0, the ratio R n/l (L, 0) = f n/l (0) is independent of system size L. This property of R n/l can Fig. 6 Eigenvalue ratio λ 1 /λ for n = 6, 7, 8, 9 versus temperature T and the scaling variable tl 1/ν for system sizes L = 16, 32, 64. Monte Carlo data of different system sizes have a fixed point at T ك. 3.4 Space Distribution of Principal Fluctuation Modes From the N-dimensional eigen vector b n, we can get the space distribution b n (r) of n-th principal fluctuation mode. The space distribution function b n (r) satisfies the normalization condition b n (r) 2 = 1. (21) r

5 No. 3 Communications in Theoretical Physics To characterize the space distribution bn (r), we make the following Fourier analysis 1 b n (k) exp(ikr), (22) bn (r) = N k where 1 b n (k) = bn (r) exp( ikr). (23) N r 359 The summation over vector k in Eq. (22) is done for k = (kx, ky ) = ((2π/L)nx, (2π/L)ny ) and π < kx, ky π. For Fourier components b n k, there is also a normalization condition b n (k) 2 = 1. (24) k Fig. 7 Rescaled space distributions of principla fluctuation modes b n (r) = L bn (r) of groups n = 1, n = 2, 3, 4, 5 and n = 6, 7, 8, 9 at the critical point T = Tc for system size L = 128.

6 360 Communications in Theoretical Physics Vol. 66 In Fig. 7, the rescaled space distribution b n (r) = L b n (r) is presented in three groups of n = 1, n = 2, 3, 4, 5 and n = 6, 7, 8, 9. For the first group n = 1, the space distribution is very flat and all spins of the system fluctuate synchronously. The space distributions of the second group with n = 2, 3, 4, 5 have one peak and one valley. In the third group with n = 6, 7, 8, 9, the space distribution functions of principal fluctuation modes have two peaks and two valleys. Before we make the Fourier analyses of the space distribution of principal fluctuation modes, we show the Fourier space of two-dimensional square lattice with periodic boundary conditions in Fig. 8. Table 2 Fourier components of principal fluctuation modes in the second and the third group at the critical point T ك and system size L = 128. )) 2 لج ˆb (±(0, 2 2 0)), لج ˆb (±( 2 Group II b 2 (r) b 3 (r) b 4 (r) b 5 (r) )) 2 لج 2, لج ˆb (±( 2 2 )) لج, 2 لج ˆb (±( 2 Group III b 6 (r) b 7 (r) b 8 (r) b 9 (r) , Fig. 8 (Color online) Fourier space ((2π/L)n (2π/L)n ) of two-dimensional square lattice with periodic boundary conditions. The sites with the same color have equal k. The first principal fluctuation mode b 1 (r) has only (0, 0) componen so that ˆb 1 (0) 2 = The principal fluctuation modes of the second group consist of four components with k = ±(0, (2π/L)) and k = ±(2π/L, 0). We present the Fourier components ˆb n (±(2π/L, 0)) 2 and ˆb n (±(0, 2π/L)) 2 of n = 2, 3, 4, 5 in Table 2. Within the error range of Monte Carlo data, the normalization condition in Eq. (24) is satisfied for n = 2, 3, 4, 5. In the third group, principal fluctuation modes consist of four components of k = ±(2π/L, 2π/L) and k = ±(2π/L, 2π/L). The Fourier components of principal fluctuation mode b n (r) are given in Table 2 for n = 6, 7, 8, 9 and satisfy the normalization condition of Eq. (24). 4 The Second Moment Correlation Lenght and Principal Fluctuation Modes From the correlation matrix C ij, we can get the correlation function G(r i, r j ) = C ij = δs i δs j = b n (r i )b n (r i )λ n, (25) n=1 which covers the contributions of all principal fluctuation modes. With the correlation function, the second moment correlation length squared can be calculated as ξ 2 = 1 2d r i,r j r j r i 2 G(r i, r j ) r i,r j G(r i, r j ). (26) In the bulk limit L, vector k of the Fourier space is continuous and the second moment correlation can be written as [16] ξ 2 Ĝ(0) Ĝ 1 (k) k=0 = k 2, (27) where the Fourier coefficien of the correlation function Ĝ(k) = 1 G(r i, r j ) e ik (rj ri), (28) N r i,r j and can be calculated as Ĝ(k) = λ n ˆb n (k) 2. (29) n For finite system, vector k of the Fourier space is discrete and the definition of the second moment correlation lenght in Eq. (27) is replaced by ξ 2 = 1 [ Ĝ(0) ] k 2 Ĝ(k) 1. (30) The second moment correlation length in the x- direction is calculated at k = ±(2π/L, 0) where ( ( 2π )) 5 Ĝ ± L, 0 = λ 2 ˆb ( ( 2π )) n ± L, 0 2 = λ2. (31) n=2 Therefore, we get the second moment correlation length of the x-direction ξ 10 (L, t) = L 1 2π [f 1/2(tL 1/ν ) 1] 1/2, (32)

7 No. 3 Communications in Theoretical Physics 361 which follows the finite-size scaling form with ξ(l, t) = L ξ(tl 1/ν ), (33) ξ 10 (tl 1/ν ) = 1 2π [f 1/2(tL 1/ν ) 1] 1/2. (34) Similarly, the Fourier coefficient ( ( Ĝ ± 0, 2π )) 5 = λ 2 L ˆb ( ( n ± 0, 2π )) 2 = λ2. (35) L 2 Therefore, the second moment correlation length in the y-direction ξ 01 = ξ 10. (36) Our Monte Carlo simulation results of ξ 2 10 are shown versus temperature T and for different system sizes in Fig. 9(a). The second moment correlation length scaled is presented with respect to the scaling variable tl 1/ν in Fig. 9(b). The different curves of L = 16, 32, 64 in the left collapse into one curve in the right. Fig. 9 Second moment correlation length squared ξ 2 10 of the x-direction. (a) ξ 2 10 as function of temperature for L = 16, 32, 64. (b) ξ 10 /L 2 as function of the scaling variable tl 1/ν. At k = ±(2π/L, 2π/L), we have ( ( 2π Ĝ ± L, 2π 9 ( ( 2π ))=λ 6 ˆb n ± L L, 2π )) 2 = λ6. (37) L n=6 According to Eq. (30), the second moment correlation length of the (1, 1)-direction can be calculated as ξ 11 (L, t) = L ξ 11 (tl 1/ν ), (38) ξ 11 (tl 1/ν ) = 1 2 2π [f 1/6(tL 1/ν ) 1] 1/2. (39) The Monte Carlo results of ξ 2 11 and its scaling function ξ 11 /L 2 are given in Fig. 10. Similarly, the second moment correlation length of the (1, 1)-direction is equal to ξ 11. After making a comparison of Fig. 10 with Fig. 9, we can conclude that the second moment correlation length of the (1, 1)-direction is different from that of the (1,0)-direction. Therefore, the second moment correlation length in the two-dimensional square lattice is anisotropic. This is in agreement with the anisotropy of the exponential correlation length. Fig. 10 Second moment correlation length squared ξ 2 11 of the (1, 1)-direction. (a) ξ 2 11 as function of temperature for L = 16, 32, 64. (b) ξ 11 /L 2 as function of the scaling variable tl 1/ν. 5 Conclusions For the data of a complex system consisting of N agents, the correlations between all agents can be calculated. With the correlations as elememnts, an N N correlation matrix C of the complex system can be obtained. The N eigenvectors of C define the N principal fluctuation modes of the complex system. The mean square of a principal fluctuation mode is equal to its corresponding eigenvalue. It is observed often that the fluctuations of complex system are dominated just by a few of principal fluctuation modes with larger eigenvalues. In this case, the complex system can be studied by investigating some of the N principal fluctuation modes. From the dominant principal fluctuation modes, the global properties of complex systems, such as susceptibility, can also be calculated. Near the critical point of a complex system, the mean squares of dominant principal fluctuation modes are anticipated to have critical behaviors similar to that of susceptibility. For a finite complex system near its critical point with small reduced temperature t = (T T c )/T c, the eigenvalues of the dominant principal fluctuation modes follow the finite-size scaling form λ n (L, t) = L ζn f n (tl 1/ν ), where ν is the critical exponent of correlation length. In comparison with thermodynamic functions which characterize global properties of system, principal fluctuation modes are related to the length scales from microscopic to macroscopic. More informations of critical behaviors are exist in principal fluctuation modes and these could be studied in the future investigations. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes are investigated. The first 9 prinicipal fluctuation modes are divided into three groups. The largest eigenvalue is λ 1. In the second group, the eigenvalues λ 2, λ 3, λ 4, and λ 5 are equal and they are degenerate. The eigenvalues λ 6, λ 7, λ 8 and λ 9 of the third group are the same. At the critical point T = T c, we find that the principal eigenvalues follow a power law λ n (L, 0) L ζn. We find that ζ n is independent of n and ζ n = γ/ν for twodimensional Ising model, where γ is the critical exponent

8 362 Communications in Theoretical Physics Vol. 66 of susceptibilty. In Ref. [17], two small correlation matrices are considered and it was found that the leading and subleading eigenvalues are governed by different exponents. Therefore, further investigations are needed to clarify if the independence of ζ n on n exists in general. Around the critical point, our Monte Carlo data of L = 16, 32, 64 demonstrate that the eigenvalues λ n with n from 1 to 9 satisfy its finite-size scaling form given above. Correspondingly, the eigenvalue ratios R n/l (L, t) = λ n /λ l are presented and they follow the finite-size scaling form R n/l (L, t) = f n/l (tl 1/ν ). For finite systems, the second moment correlation length is defined as ξ = [Ĝ(0)/Ĝ(k) 1]/ k 2, where Ĝ(k) is the Fourier component of the correlation function. At k = 0, we have Ĝ(0) = λ 1. The Fourier component Ĝ(k) at k = (2π/L, 0) consists of contributions of λ 2, λ 3, λ 4, λ 5 and we get Ĝ(k) = λ 2. Using the finite-size scaling behaviors of eigenvalues, we can obtain the finitesize scaling form of the second moment correlation length in the x-direction ξ 10 = L ξ 10 (tl 1/ν ). It can be shown that the second moment correlation lenghts in y direction is equal to that of x direction. At k = (2π/L, 2π/L), Ĝ(k) = λ 6 can be got. Therefore, the second moment correlation length in the (1, 1)-direction follows the scaling form ξ 11 = L ξ 11 (tl 1/ν ) also. It can be demonstrated that ξ 11 is equal to the second moment correlation length in the (1, 1) direction. However, ξ 11 and ξ 10 are different. Therefore, the second moment correlation length of the Ising model on the two-dimensional square lattice is anisotropic and has the similar anisotropy as the exponential correlation length. [16] Our investigations of principal fluctuation modes in the two-dimensional Ising mode can be extended to other complex systems. It is very interesting to study the effects of boundary conditions, dimensionality of systems and types of order parameters on princopal fluctuation modes. Nowdays, more and more data of the earth system and the human socities become available. We can investigate these systems from the aspect of principal fluctuation modes. References [1] M.E.J. Newman, Contemporary Physics 46 (2005) 323. [2] J. Kwapien, S. Drozdz, J. Kwapien, and S. Drozdz, Phys. Rep. 515 (2012) 115. [3] C. Kamath, Int. J. Uncertain. Quantif. 2 (2012) 73. [4] P. Bect, Z. Simeu-Abazi, and P.L. Maisonneuve, Computers in Industry 68 (2015) 78. [5] K.Y. Yeung and W.L. Ruzzo, Bioinformatics 17 (2001) 763. [6] Y. Yan, M.X. Liu, X.W. Zhu, and X.S. Chen, Chin. Phys. Lett. 29 (2012) [7] Robert Cukier, J. Chem. Phys. 135 (2011) [8] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A. Nunes Amaral, and H. Eugene Stanley, Phys. Rev. Lett. 83 (1999) [9] D.J. Fenn, M.A. Porter, S. Williams, M. McDonald, N.F. Johnson, and N.S. Jones, Phys. Rev. E 84 (2011) [10] W.J. Ma, C.K. Hu, and R. Amritkar, Phys. Rev. E 70 (2004) [11] A. Sensoy, S. Yuksel, and M. Erturk, Physica A 392 (2013) [12] M. MacMahon and D. Garlaschelli, Phys. Rev. X 5 (2015) [13] V. Privman and M.E. Fisher, Phys. Rev. B 30 (1984) 322. [14] V. Privman, Finite Size Scaling and Numerical Simulation of Statistical Systems, World Scientific, Singapore (1990). [15] H. Nishimori and G. Ortiz, Elements of Phase Transition and Critical Phenomena, Oxford University Press, Oxford (2011). [16] X.S. Chen and V. Dohm, Eur. Phys. J. B 15 (2000) 283. [17] Y. Deng, Y. Huang, J.L. Jacobsen, J. Salas, and A.D. Sokal, Phys. Rev. Lett. 107 (2011)