PHYSICAL REVIEW E 92, 042167 (2015)
Proper encoding for snapshot-entropy scaling in two-dimensional classical spin models Hiroaki Matsueda and Dai Ozaki Advanced Course of Information and Electronic System Engineering, Sendai National College of Technology, Sendai 989-3128, Japan (Received 12 May 2014; revised manuscript received 30 June 2015; published 30 October 2015) We reexamine the snapshot entropy of the Ising and three-states Potts models on the L × L square lattice. Focusing on how to encode the spin snapshot, we find that the entropy at Tc scales asymptotically as S ∼ (1/3) ln L that strongly reminds us of the entanglement entropy in one-dimensional quantum critical systems. This finding seems to support that the snapshot entropy after the proper encoding is related to the holographic entanglement entropy. On the other hand, the anomalous scaling Sχ ∼ χ η ln χ for the coarse-grained snapshot entropy holds even for the proper encoding. These features originate in the fact that the largest singular value of the snapshot matrix is regulated by the proper encoding. DOI: 10.1103/PhysRevE.92.042167
PACS number(s): 05.50.+q, 05.10.Cc, 89.70.Cf, 11.25.Hf
I. INTRODUCTION
Quantum entanglement and holography are two fundamental concepts in current theoretical physics. The entropy of the entanglement has information similar to the two-point correlation function of scaling operators and is recognized to be a new-order parameter efficient for detecting quantum-ordered states. On the other side, the holography provides us with a method of how to transform the quantum entanglement into classical quantities. A representative holographic connection is the so-called Ryu-Takayanagi formula in the string theory that transforms the entanglement entropy into the minimal surface area on the classical side with negative curvature [1]. The negative curvature space is characterized by the presence of various length scales, and in other words the holography seems to be a method of how to efficiently embed the quantum data into the classical memory space. Although the holography has been mainly examined in string-theory community, one of the authors (HM) has also proposed a possible alternative holographic theory appearing in the statistical physics [2]. Interestingly, the theory is based on the image processing of spin snapshot data in the classical two-dimensional (2D) Ising and three-states Potts models. The image processing is done by the singular value decomposition (SVD), and because of the universal nature of the singular values, the information entropy defined from the singular values shows a universal feature that reminds us of the entanglement entropy in 1D quantum critical systems. We call this entropy as snapshot entropy. In the previous paper, this correspondence was considered to originate from the Suzuki-Trotter decomposition of the corresponding 1D transverse-field quantum Ising model [3]. SVD can also be applied to both of classical and quantum systems, and the function of SVD as length-scale decomposition works in both systems. Thus, we think that this common feature can detect the quantum-classical correspondence. However, there are still many questions for the physical meaning of the snapshot entropy, although related works have recently appeared [4–6]. One of mysteries is why the snapshot entropy at Tc behaves as S ∼ ln L, where L is the linear system size of 2D snapshots. According to the conformal field theory (CFT), the entanglement entropy for subsystem size L in 1D critical systems is obtained as L c SEE = ln + c1 , (1) 3 a 1539-3755/2015/92(4)/042167(10)
where a is a lattice cut-off (we take a = 1), c is the central charge of corresponding CFT, and c1 is a nonuniversal positive constant [7,8]. If the snapshot entropy is a key quantity of holography, we expect that the amount of information should conserve after the quantum-classical correspondence. Thus, it is necessary to do more sophisticated analysis for the scaling formula of the snapshot entropy up to the coefficient. A hint to resolve this problem appears in the author’s recent work on the snapshot entropy of fractal images [5]. Fractal images can be regarded as ideal Ising snapshots at Tc owing to their self-similarities. Here, a fractal is defined by the tensor product of the h × h unit cell matrix H . Then, the tensor product of N copies of H , M = H ⊗ H ⊗ · · · ⊗ H ⊗ H , represents the fractal with N different length scales. An important point is that the elements of the unit cell matrix H should be taken so that the tensor product properly creates the desired fractal image. We call this “proper encoding.” If this point is satisfied, the entropy is given by S ∝ N ∝ ln L for the total linear system size L = hN , and we know that the snapshot entropy counts the number of the different cluster sizes. Furthermore, this construction of the snapshot entropy is equivalent with the entanglement entropy in 1D quantum critical systems, since their spectra have the same mathematical structure. At least for the exact fractal cases, the snapshot entropy does represent the holography of quantum entanglement. Therefore, the proper encoding is quite important for obtaining the proper scaling formula. For example, let us consider the unit cell matrix of the Sierpinski carpet given by ⎛ ⎞ 1 1 1 H = ⎝1 0 1⎠. (2) 1 1 1 The tensor product of two copies of H is given by ⎛ 1 1 1 1 1 1 1 1 ⎜ 1 0 1 1 0 1 1 0 ⎜ ⎜ 1 1 1 1 1 1 1 1 ⎜ ⎜ 1 1 1 0 0 0 1 1 ⎜ H ⊗H =⎜ 1 0 1 0 0 0 1 0 ⎜ 1 1 1 0 0 0 1 1 ⎜ ⎜ 1 1 1 1 1 1 1 1 ⎜ ⎝ 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, (3) ⎟ ⎟ ⎟ ⎟ ⎠
©2015 American Physical Society
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and clearly the level-2 fractal structure emerges. Here, the proper encoding is (0,1). However, we cannot exchange 0 and 1, since the matrix elements M are all zero except for the central pixel. In this viewpoint, the previous works did not take care about the proper encoding. Therein, the up and down spins of the Ising model were taken to be +1 and −1. We call this “normal encoding” in Ising models in comparison with the proper encoding. We need to reconsider what is the proper encoding of the snapshot matrix. The purpose of this paper is to confirm the role of the proper encoding on the snapshot entropy near Tc . Then, the proper encoding will make it clear to know the close connection among the central charge, hierarchical cluster spins, and the entropy formula. We will actually find that the proper encoding is crucial for finding the scaling relation of the snapshot entropy at Tc . The proper selection of the matrix elements would be 0 and 1 for up and down spins of the Ising model (it is better to slightly change 0 with a small positive number ). This is actually a direct extension of the tensor product construction of the Sierpinski carpet. The selection for the Potts model would be 0, 1, and 2. In both cases, we will find by the precise numerical simulations that the entropy scaling is given by S ∼ (1/3) ln L, and this is similar to Eq. (1) except for the central charge c. We will conclude that the snapshot entropy detects modelindependent universal features, and one of the features is the presence of the various-size clusters. We will also mention the largest eigenvalue spectrum, and this also shows similar scaling behavior. We will point out that the largest singular value should be regulated by the proper encoding. The organization of this paper is as follows. In Sec. II, we discuss the role of the encoding on the SVD spectrum. In Sec. III, the numerical data and the scaling analysis for the 2D Ising model are presented, and we will give a theoretical interpretation of the scaling based on CFT. In Sec. IV, we examine close connection between the tensor product of the properly encoded snapshot matrix and the cluster distribution. We also present the finite-χ scaling in Sec. V. The reliability of our scaling is examined in Sec. VI by taking the three-states Potts model with the different central charge. In Sec. VII, we discuss why the central charge is lost in the entropy formula. The final section is devoted to the summary. II. PROPER ENCODING OF SNAPSHOT DATA A. Role of the largest singular value on the snapshot entropy
The Hamiltonian of the Ising model is defined by
H = −J σi σj ,
(4)
i,j
where σi = ±1 and the sum runs over the nearest-neighbor lattice sites i,j , and J (> 0) is the exchange interaction. We consider the square lattice under the periodic boundary condition, and the system size is taken to be L ×√L. The critical temperature is known to be Tc /J = 2/ ln(1 + 2) = 2.2692. The central charge of CFT is c = 1/2. We regard a snapshot (a spin configuration) as the following matrix: Mn (x,y) = σi ,
(5)
FIG. 1. (Color online) Temperature and system-size dependence on S: L = 32 = 25 (open purple triangles), L = 64 = 26 (filled red triangles), L = 128 = 27 (open blue circles), and L = 256 = 28 (filled black circles). A dashed vertical line is a guide to Tc . The upper panel (a) is numerical result for Mp (x,y) = (σi + 1)/2 with i = (x,y), and the lower panel (b) is for Mn (x,y) = σi .
with i = (x,y). The index n means the normal encoding. To obtain the snapshot, we basically perform Monte Carlo (MC) simulation by the Swendsen-Wang algorithm (note that Figs. 1 and 6 are taken by the simple Metropolis algorithm). Then, we typically take 104 MC steps. Here, one MC step corresponds to all possible cluster flips for a given spin configuration. At each T , we calculate the snapshot entropy with the help of SVD. The proper encoding of Eq. (5) is defined by Mp (x,y) = 12 (σi + 1).
(6)
In the ferromagnetic phase, we need to take care about many zeros of matrix elements, since in such a case, SVD and entropy calculation become unstable. If the total magnetization σ is positive, we replace M(x,y) = −1 into 0. If the i i magnetization is negative, we take M(x,y) = 1 → 0 and
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M(x,y) = −1 → 1. We are particularly interested in the case of Eq. (6). As already mentioned, the snapshot data M(x,y) can be regarded as a matrix. To extract the universal information of the matrix data, we apply SVD to M(x,y). The SVD is defined by M(x,y) =
L
Un (x) n Vn (y),
(7)
n=1
√ where n is the nth singular value, and Un (x) and Vn (y) are nth column unitary matrices. The square of the singular value, n , is equal to the eigenvalue of the density matrix defined by ρ = MM † .
(8)
We align the eigenvalues so that 1 2 · · · L . Each eigenvalue n is normalized to be n λ n = L n=1
n
(9)
.
Then, the snapshot entropy is defined by S=−
L
λn ln λn .
(10)
n=1
In the following, we examine the T and L dependencies of this entropy S. Before going into details of numerical data, we mention the density matrices for both normally and properly encoded snapshot data, Mp and Mn , respectively. These data are related with Mp = 12 (Mn + B),
(11)
where the L × L background matrix B is defined by ⎛ ⎞ 1 ··· 1 ⎜ ⎟ B = ⎝ ... . . . ... ⎠. 1
···
(12)
1
Then the density matrix for Mp is given by ρp = 14 ρn + 14 (BMn† + Mn B) + 14 LB,
(13)
where we have used B 2 = LB. The second term, † (BMn + Mn B)/4, is very small for paramagnetic states and near Tc , since Mn contains both of ±1 components and the matrix product almost cancels out. Therefore, the eigenvalues of ρp are quite different from those of ρn as long as the background matrix B is not the unit matrix. This means that the proper encoding is characterized by B and in the present case, B in Eq. (12) affects the final result very much. According to the recent paper [4], the normally encoded density matrix ρn corresponds to the spin-spin correlator −η
ρn (x,y) ∼ G(x − y) ∝ |x − y| ,
(14)
with the anomalous dimension η = 1/4, and is diagonalized by the Fourier transformation. The background matrix B is also diagonalized by the Fourier transformation, and the third term in Eq. (13) contributes to the uniform (k = 0) component. † Thus, if the second term in Eq. (13), (BMn + Mn B)/4, does not affect so much for the final result, the proper
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encoding makes the maximum eigenvalue increased. After the normalization of a set of all the eigenvalues, the snapshot entropy decreases, since the probability of only the largest singular-value state increases. As we have already mentioned in the introduction, the entropy with the normal encoding is given by S ∼ ln L. Thus, it is natural that a factor less than unity appears in front of the logarithmic term. The above statement only guarantees the presence of the small factor, and we do not mention anymore about its physical origin. However, it is found in the entropy calculation by SVD that the magnitude of the largest eigenvalue λ1 is crucial for determining the total entropy value. This point has been already clarified in the quantum side by the replica method [9]. According to CFT, the logarithm of the largest eigenvalue is a half of the total entanglement entropy: SEE . (15) 2 It is interesting if this property is also reflected to the classical side. If λ1 is the uniform component of the Fourier-transformed correlator G(k = 0), −ln λ1 is almost equal to the definition of the entanglement entropy in terms of Calabrese-Cardy’s approach [8]. We guess that the proper encoding facilitates to pick up the uniform component, although in the present stage a role of additional constant factor by the proper encoding on λ1 is still missing. We will examine whether this scaling equation is also satisfied on the classical side as well as the entanglement entropy formula. −ln λ1 =
B. Representation of snapshot matrix and nonzero singular values
Let us further refer to the representation dependence on the SVD spectra. We focus on whether a particular choice of representation can detect fine structures of the spin configuration. Then, we would like to confirm that the background matrix is necessary for the representation. If this is correct, the background matrix leads to the important result that the choice is related to a proper regulation of the largest eigenvalue. For these purposes, we consider the following examples (α = 1): 1 1 1 1 1 1 α α M1 (α) = , M2 (α) = . α α α α α α 1 1 (16) Here, we regard an alignment of 1 (or α) as a spin string, and in the case of M2 the strings of 1 and α are classically entangled. At first let us take α = 0. Then, their partial density matrices are given by 4 0 2 0 M1 (0)M1 (0)† = , M2 (0)M2 (0)† = . (17) 0 0 0 2 Thus, the matrix M2 (0) has larger entropy than that for the matrix M1 (0). This means that the complexity of string configuration affects the entropy value. However, if we replace 0 to −1, the situation changes. Actually, 4 −4 M1 (−1)M1 (−1)† = M2 (−1)M2 (−1)† = ,(18) −4 4 and then we cannot identify the essential difference between M1 (−1) and M2 (−1) by the snapshot entropy that is only
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dependent to the eigenvalues of this matrix. We think that the excess entropy induced by a larger-entropy representation is an artifact, and we should determine the minimum-entropy state. In this strategy, negative components are not willing, since the effect of the negative sign on the partial density matrix sometimes vanishes. The uniform shift of the matrix elements, like the background matrix B in the previous subsection, is an efficient way of taking account of the above strategy. We introduce the shifted matrix for M1 as M1 (α; ) = M1 (α) − B 1− 1−
= α− α−
1−
α−
1−
. (19) α−
Then, the partial density matrix is given by M1 (α; )M1 (α; )† 4(1 − )2 = 4(1 − )(α − )
4(1 − )(α − ) , 4(α − )2
(20)
and we should take = α for the minimum entropy state. Finally, we obtain M1 (α; α) = (1 − α)M1 (0),
(21)
and taking α = 0 is the simplest selection of the matrix elements. This is one interpretation of taking 0 matrix elements. III. NUMERICAL RESULTS FOR SNAPSHOT ENTROPY WITH NORMAL AND PROPER ENCODING A. Single snapshot entropy
Let us examine the numerical results for the Ising model. Figure 1(a) shows S as functions of T and L. We have used single snapshot for each T , and do not take statistical average. The statistical error becomes smaller with increasing L owing to the self-averaging feature of the snapshot entropy, but at the same time we would like to find the temperature range in which critical fluctuation is mostly enhanced. We will later also present the sample average. In Fig. 1(a), we have used Eq. (6). On the other hand, Fig. 1(b) is equal to the previous data with use of Eq. (5). The most important contrasts between them are the line shape and the temperature range of critical fluctuation, although both of them become efficient criteria for the ferromagnetic phase transition. The difference between ρp and ρn by B clearly appears. We find that the change in S near Tc is abrupt in Fig. 1(a). This feature is somehow different from that in Fig. 1(b), where we have observed that S only weakly changes far above Tc and starts to decrease slightly above Tc , followed by a quite asymmetric tail below Tc . Because of these features, we could not exactly separate the critical behavior from the high-T ln L feature originating from the random matrix theory in the previous work [2]. In the present case, the high-T and near-Tc features are different and they can be identified independently. Furthermore, the data in Fig. 1(a) are more symmetric at around Tc than those in Fig. 1(b). This may indicate the fundamental nature of the exact solution in which the divergence of the specific heat is symmetric near Tc . The
critical fluctuation above Tc may also suggest that the entropy detects the violation of the ferromagnetic order. B. Average snapshot entropy
Let us start examination of the snapshot entropy with the proper encoding. For sophisticated analysis, we take the sample average as S =
Nav 1
S(Ml ), Nav l=1
and also we calculate the variance as
Nav 1
σ = {S(Ml ) − S}2 , Nav l=1
(22)
(23)
where S(Ml ) is equal to Eq. (10) for the single snapshot Ml . The index l identifies the lth snapshot. We typically take Nav = 103 ∼ 104 after MC steps at each T . At first, we focus on S at T = Tc as a function of ln L in Fig. 2. Although there is relatively large variance σ from S, we find that the gradient is roughly 1/3. It seems to be twice of the coefficient c/3 = 1/6 of Eq. (1) for the transverse-field 1D Ising model (c = 1/2), into which the 2D classical Ising model is transformed. Otherwise, we may be looking at just 1/3 in the coefficient. In the present stage, it is not obvious to determine which one is the better interpretation. However, the presence of the nontrivial factor 1/3 strongly reminds us of the coefficient of Eq. (1). In Fig. 2, we also plot S for the snapshot data in which we encode up and down spins as (=0.1) and 1 instead of taking = 0. The gradient of this entropy is much closer to 1/3. According to Ref. [5] in which we have examined the entropy for the fractal images, if the tensor-product factorization is satisfied for a given image, the SVD spectra are degenerate. If we introduce a finite amount of a perturbation by , then the degeneracy is lifted. In this case, we expect a power-law decay of the spectra. Since the power-law decay of the SVD spectra is related to the criticality of the Ising model [4,6], the introduction of the small perturbation may be sometimes
FIG. 2. (Color online) Black solid line with open circles: S for the Ising model at T = Tc as a function of ln L. The gradient of the dashed guideline is 1/3. The error bar represents the variance σ . The red fine-solid line with filled circles is obtained when we encode up and down spins as (=0.1) and 1 instead of using 0 and 1.
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since Eq. (24) would be the asymptotic one in the large-L limit. It is necessary to confirm why the gradient is close to 1/3 by different approaches. For this purpose, we estimate the gradient of the logarithmic term by the tensor product form used in the previous work [5]. Since the Ising spin snapshot is a fractal-like one, we can phenomenologically construct a large spin configuration by starting with a smaller snapshot and then by taking the tensor product of the copies of the smaller snapshot. According to our exact construction of the snapshot entropy in the fractal case [5], this procedure does produce the logarithmic term compatible to Eq. (1). Therefore, this is a nice phenomenological method to extract the scaling. One comment is that the image constructed by the tensor product is in most cases not so similar to the real spin configuration obtained by the large-scale Monte Carlo simulation. We should be careful for the fact that this is just a phenomenological method. Let us consider the case in which H is a unit cell matrix representing a particular snapshot of the Ising model. We assume that the size of the matrix is h × h. Then, the snapshot entropy is given by Sph = SH N =
FIG. 3. (Color online) (a) Temperature and system-size dependence on S for Mp (x,y) = (σi + 1)/2 with i = (x,y): L = 32 = 25 (fine-dashed purple line), L = 64 = 26 (dashed red line), L = 128 = 27 (bold-solid blue line), L = 256 = 28 (solid black line), and L = 512 = 29 (fine-solid black line). A dashed line is a guide to Tc . (b) We also plot S − (1/3) ln L in order to see Binder-parameter-like feature. The inset estimates the coefficient of the logarithmic scaling with use of the phenomenological model.
necessary for self-similar-type images. This is the reason why we have checked the role of the perturbation on the magnitude of the gradient. In Fig. 3, we plot S − (1/3) ln L as well as S. This is Binder-parameter-like representation, and actually all the data intersects with each other at T = Tc . Thus, the numerical data suggest the scaling given by S ∼
1 3
ln L + α,
(24)
with a negative constant α slightly larger than −0.5. The negative constant is quite strange, since the positivity of the entropy is violated in the small-L limit. Thus, the above scaling is only an asymptotic relation for large-L region. Going back to Fig. 2, we see that the slight deviation of the gradient from 1/3 is remarkable at smaller-L region. This is natural,
SH ln L, ln h
(25)
where L = hN and SH is the snapshot entropy for the unit cell H . We can extract the coefficient SH / ln h from the main panel in Fig. 3 by identifying h with L. The numerical result is shown in the inset of Fig. 3, and we can see a rough tendency that the coefficient is naturally extrapolated to 1/3 in the large-h limit. Thus it is reasonable for us to rely on Eq. (24). To examine the physical origin of the coefficient 1/3, it is important to calculate the snapshot entropy for alternative spin models with different central charge. Later we will mention this point. C. The largest SVD spectrum
Let us next examine whether the feature characterized by Eq. (15) appears in the snapshot spectrum. For this purpose, we examine −ln λ1 . The result is shown in Fig. 4. Here we take the sample average after taking the logarithm of λ1 . This may generate large sample deviation, since the logarithm for λ1 < 1 changes quite rapidly. On the other hand, the obtained numerical data are not so seriously fluctuating. Therefore, we can say that the data quality is retained. We find in Fig. 4(a) that the overall feature is similar to S, and this is also a good signal for the phase transition. We thus think that there is certain correspondence between the largest SVD spectrum and the full snapshot entropy. Actually, at T = Tc in Fig. 4(b) and the inset, we find the scaling given by −ln λ1 ∼
1 24
ln L + α ∼ 18 S,
(26)
where the additional constant α is almost zero in this case. The coefficient of this equation is different from Eq. (15), but it is still depending on the value which strongly reminds us of the CFT description of the entanglement entropy.
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FIG. 4. (Color online) (a) Temperature and system-size dependence on −ln λ1 for Mp (x,y) = (σi + 1)/2 with i = (x,y): L = 32 = 25 (fine-dashed purple line), L = 64 = 26 (dashed red line), L = 128 = 27 (bold-solid blue line), L = 256 = 28 (solid black line), and L = 512 = 29 (fine-solid black line). (b) We also plot −ln λ1 − (1/24) ln L in order to see the Binder-parameter-like feature. The inset shows the gradient of the logarithmic term at Tc . The gradient of the guideline is 1/24.
FIG. 5. (Color online) Temperature dependence on the SVD spectra for L = 128 = 27 . The bold-solid blue line represents the largest spectrum −ln λ1 . The other data above the gap are −ln λi for i = 2,3,...,20 from the bottom to the top, respectively. (a) Mp (x,y) = (σi + 1)/2 and (b) Mn (x,y) = σi . The red data with open circles in the inset of (a) are obtained by taking L = 256 = 28 .
To show peculiarity of the largest SVD spectrum, we plot the average spectra −ln λi (λ1 > λ2 > · · · > λ20 ) for L = 128 in Fig. 5(a). An important point is that the temperature dependence on −ln λ1 is different from others and S is dominated by only −ln λ1 . We observe that there is a large spectral gap between −ln λ1 and others (a band above the gap) in the ferromagnetic phase. We also observe that the gap tends to decrease toward Tc , and takes the minimum value. In Fig. 5(a), the minimum gap is seen slightly above Tc . In the inset of the same figure, we plot the lowest and second-lowest spectra for L = 128 and L = 256. The cusp structure of the second-lowest spectrum near Tc becomes sharper with increasing L, and the cusp point seems to slightly shift toward Tc . The gap size does not strongly depend on L. In the paramagnetic phase, the gap size shows weaker
temperature dependence in comparison with the ferromagnetic phase. As we have already mentioned in Sec. II, λ1 increases by the proper encoding (0,1), and then the average spectrum −ln λ1 decreases. As a result, this gap structure appears. Figure 5(b) is the SVD spectrum with the normal encoding (−1,1). Therein, the lift of the degeneracy between −ln λ1 and the continuous band is not complete above Tc . The mixture among these spectra blurs the essential properties of −ln λ1 , when we calculate the snapshot entropy. The gap protects the critical behavior against the temperature fluctuation. Therefore, the proper encoding plays crucial roles on the formation of the lowest spectrum separated from the continuous band. This kind of protection mechanism of the lowest SVD spectrum has been also discussed in the context of edge modes of the topological insulators.
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following unit cell matrix: 1 H = 1
IV. TENSOR PRODUCT OF PROPERLY ENCODED MATRIX AND CLUSTER DISTRIBUTION A. Encoding and cluster distribution
Let us consider the meaning of the coefficient 1/3 of the logarithmic entropy scaling in more physical standpoint. We first examine that the encoding affects the distribution of the number of the ferromagnetic cluster sizes in the snapshot. This is crucial for determining the proper spin-spin correlator form. To see this, we introduce an effective model defined by the ⎛ 1 ⎜1 H ⊗H =⎝ 1 1
1 1 1 1
0 0 0 0
1 0 1 0
1 1 1 1
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1 1 1 1
0 0 0 0
1 1
0 0
1 , 0
(27)
where the left half block (4 spins) represents a large spin cluster, while the right upper site represents a small cluster (1 spin). As already mentioned in the Introduction, the fractallike spin configuration can be made by the multiple tensor product of the unit cell H . Taking the tensor product between two copies of H , we obtain 1 0 1 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 1 0 0
1 1 0 0
0 0 0 0
⎞ 1 0⎟ . 0⎠ 0
(28)
To clearly look at cluster distribution, we omit zero components from H ⊗ H (it is also possible to omit 1). The result is the following: ⎛ ⎞ 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎜1 1 ⎟ H ⊗H →⎝ (29) ⎠. 1 1 1 1 1 1 1 1 1 1 Then, we find certain distribution of various-size clusters. Here, we define a set of up spins (labeled by 1) connected along bond direction as a cluster. The largest cluster size is 10. We observe a rough tendency that the number of the clusters increases with decreasing the cluster size. This tendency matches with the spin configuration realized by the MC simulation at Tc . On the other hand, if we consider the normal encoding (−1,1), the situation changes. Let us next consider the unit cell matrix 1 1 −1 1 H = , (30) 1 1 −1 −1 and calculate the tensor product. Then we obtain
⎛ 1 ⎜1 H ⊗H →⎝ 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
⎞ 1 1 1 1 1 1 1 1 1 ⎟ , 1 1 1 ⎠ 1 1 1 1
(31)
where we have omitted −1. We clearly see that there are many large spin clusters and the distribution changes. B. Cluster distribution and coefficient of logarithmic entropy scaling
Let us next discuss a close relationship between the cluster distribution and the coefficient of the logarithmic term in the snapshot entropy formula. According to Ref. [5], the snapshot entropy is given by S=N
r
(−γj ln γj ) = A ln L,
A = 0.5408 for Eq. (30). The former is relatively close to 1/3. This suggests that a better cluster configuration tends to provides us with a A value close to 1/3. For comparison, we calculate A of the √ Sierpinski carpet. This case provides us with γ± = (1/2) ± ( 3/4) and A = 0.2237. Alternatively, we can set up a unit cell matrix that represents a slightly ordered but still critically fluctuating spin configuration. For instance, we have
(32)
⎛ 1 ⎜1 ⎜ H =⎜ ⎝1 1
j =1
where r = rank(H H † ) = 2 (we denote this as j = ±), and γj are the normalized eigenvalues of H H † . This equation tells us that there exist unit cell matrices with a particular set of γj values that leads to a proper gradient A. We compare the A value for √ Eq. (27) with that for Eq. (30). We find γ± = (1/2) ± ( 17/10) for Eq. (27), and γ± = (1/2) ± (1/4) for Eq. (30). Now the system after N tensor products is a rectangular shape 2N × 4N , and thus we identify a typical linear size to be L2 = 23N and N = (2/3) log2 L. We then obtain A = 0.2858 for Eq. (27) and
1 0 1 1
1 1 1 1
⎞ 1 1⎟ ⎟ ⎟. 1⎠ 1
(33)
2 The eigenvalues γ± = (15 ± √ matrix H has two nonzero 189)/2. In this case, L = 4N and we obtain A = 0.1251 much smaller than 1/3. Summarizing this section, we think that the cluster distribution is related to the magnitude of the coefficient of the logarithmic entropy scaling. This is a suggestive statement,
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since in our usual experience of MC simulation the size distribution of various clusters looks common in both of the Ising and three-states Potts snapshots at Tc . Then, the entropy may not be observing model-dependent parameters such as the central charge, even if the presence of the logarithmic term itself is related to criticality of our target models. We will refer to this point in the numerical simulation of the Potts model. V. FINITE-χ SCALING
Up to now we have confirmed that the encoding affects the entropy scaling very much. Thus, we are also interested in whether the coarse-grained snapshot entropy formula is also deformed or not. The coarse-grained entropy is defined by
χ
Proper encoding for snapshot-entropy scaling in two-dimensional classical spin models Hiroaki Matsueda and Dai Ozaki Advanced Course of Information and Electronic System Engineering, Sendai National College of Technology, Sendai 989-3128, Japan (Received 12 May 2014; revised manuscript received 30 June 2015; published 30 October 2015) We reexamine the snapshot entropy of the Ising and three-states Potts models on the L × L square lattice. Focusing on how to encode the spin snapshot, we find that the entropy at Tc scales asymptotically as S ∼ (1/3) ln L that strongly reminds us of the entanglement entropy in one-dimensional quantum critical systems. This finding seems to support that the snapshot entropy after the proper encoding is related to the holographic entanglement entropy. On the other hand, the anomalous scaling Sχ ∼ χ η ln χ for the coarse-grained snapshot entropy holds even for the proper encoding. These features originate in the fact that the largest singular value of the snapshot matrix is regulated by the proper encoding. DOI: 10.1103/PhysRevE.92.042167
PACS number(s): 05.50.+q, 05.10.Cc, 89.70.Cf, 11.25.Hf
I. INTRODUCTION
Quantum entanglement and holography are two fundamental concepts in current theoretical physics. The entropy of the entanglement has information similar to the two-point correlation function of scaling operators and is recognized to be a new-order parameter efficient for detecting quantum-ordered states. On the other side, the holography provides us with a method of how to transform the quantum entanglement into classical quantities. A representative holographic connection is the so-called Ryu-Takayanagi formula in the string theory that transforms the entanglement entropy into the minimal surface area on the classical side with negative curvature [1]. The negative curvature space is characterized by the presence of various length scales, and in other words the holography seems to be a method of how to efficiently embed the quantum data into the classical memory space. Although the holography has been mainly examined in string-theory community, one of the authors (HM) has also proposed a possible alternative holographic theory appearing in the statistical physics [2]. Interestingly, the theory is based on the image processing of spin snapshot data in the classical two-dimensional (2D) Ising and three-states Potts models. The image processing is done by the singular value decomposition (SVD), and because of the universal nature of the singular values, the information entropy defined from the singular values shows a universal feature that reminds us of the entanglement entropy in 1D quantum critical systems. We call this entropy as snapshot entropy. In the previous paper, this correspondence was considered to originate from the Suzuki-Trotter decomposition of the corresponding 1D transverse-field quantum Ising model [3]. SVD can also be applied to both of classical and quantum systems, and the function of SVD as length-scale decomposition works in both systems. Thus, we think that this common feature can detect the quantum-classical correspondence. However, there are still many questions for the physical meaning of the snapshot entropy, although related works have recently appeared [4–6]. One of mysteries is why the snapshot entropy at Tc behaves as S ∼ ln L, where L is the linear system size of 2D snapshots. According to the conformal field theory (CFT), the entanglement entropy for subsystem size L in 1D critical systems is obtained as L c SEE = ln + c1 , (1) 3 a 1539-3755/2015/92(4)/042167(10)
where a is a lattice cut-off (we take a = 1), c is the central charge of corresponding CFT, and c1 is a nonuniversal positive constant [7,8]. If the snapshot entropy is a key quantity of holography, we expect that the amount of information should conserve after the quantum-classical correspondence. Thus, it is necessary to do more sophisticated analysis for the scaling formula of the snapshot entropy up to the coefficient. A hint to resolve this problem appears in the author’s recent work on the snapshot entropy of fractal images [5]. Fractal images can be regarded as ideal Ising snapshots at Tc owing to their self-similarities. Here, a fractal is defined by the tensor product of the h × h unit cell matrix H . Then, the tensor product of N copies of H , M = H ⊗ H ⊗ · · · ⊗ H ⊗ H , represents the fractal with N different length scales. An important point is that the elements of the unit cell matrix H should be taken so that the tensor product properly creates the desired fractal image. We call this “proper encoding.” If this point is satisfied, the entropy is given by S ∝ N ∝ ln L for the total linear system size L = hN , and we know that the snapshot entropy counts the number of the different cluster sizes. Furthermore, this construction of the snapshot entropy is equivalent with the entanglement entropy in 1D quantum critical systems, since their spectra have the same mathematical structure. At least for the exact fractal cases, the snapshot entropy does represent the holography of quantum entanglement. Therefore, the proper encoding is quite important for obtaining the proper scaling formula. For example, let us consider the unit cell matrix of the Sierpinski carpet given by ⎛ ⎞ 1 1 1 H = ⎝1 0 1⎠. (2) 1 1 1 The tensor product of two copies of H is given by ⎛ 1 1 1 1 1 1 1 1 ⎜ 1 0 1 1 0 1 1 0 ⎜ ⎜ 1 1 1 1 1 1 1 1 ⎜ ⎜ 1 1 1 0 0 0 1 1 ⎜ H ⊗H =⎜ 1 0 1 0 0 0 1 0 ⎜ 1 1 1 0 0 0 1 1 ⎜ ⎜ 1 1 1 1 1 1 1 1 ⎜ ⎝ 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, (3) ⎟ ⎟ ⎟ ⎟ ⎠
©2015 American Physical Society
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and clearly the level-2 fractal structure emerges. Here, the proper encoding is (0,1). However, we cannot exchange 0 and 1, since the matrix elements M are all zero except for the central pixel. In this viewpoint, the previous works did not take care about the proper encoding. Therein, the up and down spins of the Ising model were taken to be +1 and −1. We call this “normal encoding” in Ising models in comparison with the proper encoding. We need to reconsider what is the proper encoding of the snapshot matrix. The purpose of this paper is to confirm the role of the proper encoding on the snapshot entropy near Tc . Then, the proper encoding will make it clear to know the close connection among the central charge, hierarchical cluster spins, and the entropy formula. We will actually find that the proper encoding is crucial for finding the scaling relation of the snapshot entropy at Tc . The proper selection of the matrix elements would be 0 and 1 for up and down spins of the Ising model (it is better to slightly change 0 with a small positive number ). This is actually a direct extension of the tensor product construction of the Sierpinski carpet. The selection for the Potts model would be 0, 1, and 2. In both cases, we will find by the precise numerical simulations that the entropy scaling is given by S ∼ (1/3) ln L, and this is similar to Eq. (1) except for the central charge c. We will conclude that the snapshot entropy detects modelindependent universal features, and one of the features is the presence of the various-size clusters. We will also mention the largest eigenvalue spectrum, and this also shows similar scaling behavior. We will point out that the largest singular value should be regulated by the proper encoding. The organization of this paper is as follows. In Sec. II, we discuss the role of the encoding on the SVD spectrum. In Sec. III, the numerical data and the scaling analysis for the 2D Ising model are presented, and we will give a theoretical interpretation of the scaling based on CFT. In Sec. IV, we examine close connection between the tensor product of the properly encoded snapshot matrix and the cluster distribution. We also present the finite-χ scaling in Sec. V. The reliability of our scaling is examined in Sec. VI by taking the three-states Potts model with the different central charge. In Sec. VII, we discuss why the central charge is lost in the entropy formula. The final section is devoted to the summary. II. PROPER ENCODING OF SNAPSHOT DATA A. Role of the largest singular value on the snapshot entropy
The Hamiltonian of the Ising model is defined by
H = −J σi σj ,
(4)
i,j
where σi = ±1 and the sum runs over the nearest-neighbor lattice sites i,j , and J (> 0) is the exchange interaction. We consider the square lattice under the periodic boundary condition, and the system size is taken to be L ×√L. The critical temperature is known to be Tc /J = 2/ ln(1 + 2) = 2.2692. The central charge of CFT is c = 1/2. We regard a snapshot (a spin configuration) as the following matrix: Mn (x,y) = σi ,
(5)
FIG. 1. (Color online) Temperature and system-size dependence on S: L = 32 = 25 (open purple triangles), L = 64 = 26 (filled red triangles), L = 128 = 27 (open blue circles), and L = 256 = 28 (filled black circles). A dashed vertical line is a guide to Tc . The upper panel (a) is numerical result for Mp (x,y) = (σi + 1)/2 with i = (x,y), and the lower panel (b) is for Mn (x,y) = σi .
with i = (x,y). The index n means the normal encoding. To obtain the snapshot, we basically perform Monte Carlo (MC) simulation by the Swendsen-Wang algorithm (note that Figs. 1 and 6 are taken by the simple Metropolis algorithm). Then, we typically take 104 MC steps. Here, one MC step corresponds to all possible cluster flips for a given spin configuration. At each T , we calculate the snapshot entropy with the help of SVD. The proper encoding of Eq. (5) is defined by Mp (x,y) = 12 (σi + 1).
(6)
In the ferromagnetic phase, we need to take care about many zeros of matrix elements, since in such a case, SVD and entropy calculation become unstable. If the total magnetization σ is positive, we replace M(x,y) = −1 into 0. If the i i magnetization is negative, we take M(x,y) = 1 → 0 and
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M(x,y) = −1 → 1. We are particularly interested in the case of Eq. (6). As already mentioned, the snapshot data M(x,y) can be regarded as a matrix. To extract the universal information of the matrix data, we apply SVD to M(x,y). The SVD is defined by M(x,y) =
L
Un (x) n Vn (y),
(7)
n=1
√ where n is the nth singular value, and Un (x) and Vn (y) are nth column unitary matrices. The square of the singular value, n , is equal to the eigenvalue of the density matrix defined by ρ = MM † .
(8)
We align the eigenvalues so that 1 2 · · · L . Each eigenvalue n is normalized to be n λ n = L n=1
n
(9)
.
Then, the snapshot entropy is defined by S=−
L
λn ln λn .
(10)
n=1
In the following, we examine the T and L dependencies of this entropy S. Before going into details of numerical data, we mention the density matrices for both normally and properly encoded snapshot data, Mp and Mn , respectively. These data are related with Mp = 12 (Mn + B),
(11)
where the L × L background matrix B is defined by ⎛ ⎞ 1 ··· 1 ⎜ ⎟ B = ⎝ ... . . . ... ⎠. 1
···
(12)
1
Then the density matrix for Mp is given by ρp = 14 ρn + 14 (BMn† + Mn B) + 14 LB,
(13)
where we have used B 2 = LB. The second term, † (BMn + Mn B)/4, is very small for paramagnetic states and near Tc , since Mn contains both of ±1 components and the matrix product almost cancels out. Therefore, the eigenvalues of ρp are quite different from those of ρn as long as the background matrix B is not the unit matrix. This means that the proper encoding is characterized by B and in the present case, B in Eq. (12) affects the final result very much. According to the recent paper [4], the normally encoded density matrix ρn corresponds to the spin-spin correlator −η
ρn (x,y) ∼ G(x − y) ∝ |x − y| ,
(14)
with the anomalous dimension η = 1/4, and is diagonalized by the Fourier transformation. The background matrix B is also diagonalized by the Fourier transformation, and the third term in Eq. (13) contributes to the uniform (k = 0) component. † Thus, if the second term in Eq. (13), (BMn + Mn B)/4, does not affect so much for the final result, the proper
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encoding makes the maximum eigenvalue increased. After the normalization of a set of all the eigenvalues, the snapshot entropy decreases, since the probability of only the largest singular-value state increases. As we have already mentioned in the introduction, the entropy with the normal encoding is given by S ∼ ln L. Thus, it is natural that a factor less than unity appears in front of the logarithmic term. The above statement only guarantees the presence of the small factor, and we do not mention anymore about its physical origin. However, it is found in the entropy calculation by SVD that the magnitude of the largest eigenvalue λ1 is crucial for determining the total entropy value. This point has been already clarified in the quantum side by the replica method [9]. According to CFT, the logarithm of the largest eigenvalue is a half of the total entanglement entropy: SEE . (15) 2 It is interesting if this property is also reflected to the classical side. If λ1 is the uniform component of the Fourier-transformed correlator G(k = 0), −ln λ1 is almost equal to the definition of the entanglement entropy in terms of Calabrese-Cardy’s approach [8]. We guess that the proper encoding facilitates to pick up the uniform component, although in the present stage a role of additional constant factor by the proper encoding on λ1 is still missing. We will examine whether this scaling equation is also satisfied on the classical side as well as the entanglement entropy formula. −ln λ1 =
B. Representation of snapshot matrix and nonzero singular values
Let us further refer to the representation dependence on the SVD spectra. We focus on whether a particular choice of representation can detect fine structures of the spin configuration. Then, we would like to confirm that the background matrix is necessary for the representation. If this is correct, the background matrix leads to the important result that the choice is related to a proper regulation of the largest eigenvalue. For these purposes, we consider the following examples (α = 1): 1 1 1 1 1 1 α α M1 (α) = , M2 (α) = . α α α α α α 1 1 (16) Here, we regard an alignment of 1 (or α) as a spin string, and in the case of M2 the strings of 1 and α are classically entangled. At first let us take α = 0. Then, their partial density matrices are given by 4 0 2 0 M1 (0)M1 (0)† = , M2 (0)M2 (0)† = . (17) 0 0 0 2 Thus, the matrix M2 (0) has larger entropy than that for the matrix M1 (0). This means that the complexity of string configuration affects the entropy value. However, if we replace 0 to −1, the situation changes. Actually, 4 −4 M1 (−1)M1 (−1)† = M2 (−1)M2 (−1)† = ,(18) −4 4 and then we cannot identify the essential difference between M1 (−1) and M2 (−1) by the snapshot entropy that is only
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dependent to the eigenvalues of this matrix. We think that the excess entropy induced by a larger-entropy representation is an artifact, and we should determine the minimum-entropy state. In this strategy, negative components are not willing, since the effect of the negative sign on the partial density matrix sometimes vanishes. The uniform shift of the matrix elements, like the background matrix B in the previous subsection, is an efficient way of taking account of the above strategy. We introduce the shifted matrix for M1 as M1 (α; ) = M1 (α) − B 1− 1−
= α− α−
1−
α−
1−
. (19) α−
Then, the partial density matrix is given by M1 (α; )M1 (α; )† 4(1 − )2 = 4(1 − )(α − )
4(1 − )(α − ) , 4(α − )2
(20)
and we should take = α for the minimum entropy state. Finally, we obtain M1 (α; α) = (1 − α)M1 (0),
(21)
and taking α = 0 is the simplest selection of the matrix elements. This is one interpretation of taking 0 matrix elements. III. NUMERICAL RESULTS FOR SNAPSHOT ENTROPY WITH NORMAL AND PROPER ENCODING A. Single snapshot entropy
Let us examine the numerical results for the Ising model. Figure 1(a) shows S as functions of T and L. We have used single snapshot for each T , and do not take statistical average. The statistical error becomes smaller with increasing L owing to the self-averaging feature of the snapshot entropy, but at the same time we would like to find the temperature range in which critical fluctuation is mostly enhanced. We will later also present the sample average. In Fig. 1(a), we have used Eq. (6). On the other hand, Fig. 1(b) is equal to the previous data with use of Eq. (5). The most important contrasts between them are the line shape and the temperature range of critical fluctuation, although both of them become efficient criteria for the ferromagnetic phase transition. The difference between ρp and ρn by B clearly appears. We find that the change in S near Tc is abrupt in Fig. 1(a). This feature is somehow different from that in Fig. 1(b), where we have observed that S only weakly changes far above Tc and starts to decrease slightly above Tc , followed by a quite asymmetric tail below Tc . Because of these features, we could not exactly separate the critical behavior from the high-T ln L feature originating from the random matrix theory in the previous work [2]. In the present case, the high-T and near-Tc features are different and they can be identified independently. Furthermore, the data in Fig. 1(a) are more symmetric at around Tc than those in Fig. 1(b). This may indicate the fundamental nature of the exact solution in which the divergence of the specific heat is symmetric near Tc . The
critical fluctuation above Tc may also suggest that the entropy detects the violation of the ferromagnetic order. B. Average snapshot entropy
Let us start examination of the snapshot entropy with the proper encoding. For sophisticated analysis, we take the sample average as S =
Nav 1
S(Ml ), Nav l=1
and also we calculate the variance as
Nav 1
σ = {S(Ml ) − S}2 , Nav l=1
(22)
(23)
where S(Ml ) is equal to Eq. (10) for the single snapshot Ml . The index l identifies the lth snapshot. We typically take Nav = 103 ∼ 104 after MC steps at each T . At first, we focus on S at T = Tc as a function of ln L in Fig. 2. Although there is relatively large variance σ from S, we find that the gradient is roughly 1/3. It seems to be twice of the coefficient c/3 = 1/6 of Eq. (1) for the transverse-field 1D Ising model (c = 1/2), into which the 2D classical Ising model is transformed. Otherwise, we may be looking at just 1/3 in the coefficient. In the present stage, it is not obvious to determine which one is the better interpretation. However, the presence of the nontrivial factor 1/3 strongly reminds us of the coefficient of Eq. (1). In Fig. 2, we also plot S for the snapshot data in which we encode up and down spins as (=0.1) and 1 instead of taking = 0. The gradient of this entropy is much closer to 1/3. According to Ref. [5] in which we have examined the entropy for the fractal images, if the tensor-product factorization is satisfied for a given image, the SVD spectra are degenerate. If we introduce a finite amount of a perturbation by , then the degeneracy is lifted. In this case, we expect a power-law decay of the spectra. Since the power-law decay of the SVD spectra is related to the criticality of the Ising model [4,6], the introduction of the small perturbation may be sometimes
FIG. 2. (Color online) Black solid line with open circles: S for the Ising model at T = Tc as a function of ln L. The gradient of the dashed guideline is 1/3. The error bar represents the variance σ . The red fine-solid line with filled circles is obtained when we encode up and down spins as (=0.1) and 1 instead of using 0 and 1.
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since Eq. (24) would be the asymptotic one in the large-L limit. It is necessary to confirm why the gradient is close to 1/3 by different approaches. For this purpose, we estimate the gradient of the logarithmic term by the tensor product form used in the previous work [5]. Since the Ising spin snapshot is a fractal-like one, we can phenomenologically construct a large spin configuration by starting with a smaller snapshot and then by taking the tensor product of the copies of the smaller snapshot. According to our exact construction of the snapshot entropy in the fractal case [5], this procedure does produce the logarithmic term compatible to Eq. (1). Therefore, this is a nice phenomenological method to extract the scaling. One comment is that the image constructed by the tensor product is in most cases not so similar to the real spin configuration obtained by the large-scale Monte Carlo simulation. We should be careful for the fact that this is just a phenomenological method. Let us consider the case in which H is a unit cell matrix representing a particular snapshot of the Ising model. We assume that the size of the matrix is h × h. Then, the snapshot entropy is given by Sph = SH N =
FIG. 3. (Color online) (a) Temperature and system-size dependence on S for Mp (x,y) = (σi + 1)/2 with i = (x,y): L = 32 = 25 (fine-dashed purple line), L = 64 = 26 (dashed red line), L = 128 = 27 (bold-solid blue line), L = 256 = 28 (solid black line), and L = 512 = 29 (fine-solid black line). A dashed line is a guide to Tc . (b) We also plot S − (1/3) ln L in order to see Binder-parameter-like feature. The inset estimates the coefficient of the logarithmic scaling with use of the phenomenological model.
necessary for self-similar-type images. This is the reason why we have checked the role of the perturbation on the magnitude of the gradient. In Fig. 3, we plot S − (1/3) ln L as well as S. This is Binder-parameter-like representation, and actually all the data intersects with each other at T = Tc . Thus, the numerical data suggest the scaling given by S ∼
1 3
ln L + α,
(24)
with a negative constant α slightly larger than −0.5. The negative constant is quite strange, since the positivity of the entropy is violated in the small-L limit. Thus, the above scaling is only an asymptotic relation for large-L region. Going back to Fig. 2, we see that the slight deviation of the gradient from 1/3 is remarkable at smaller-L region. This is natural,
SH ln L, ln h
(25)
where L = hN and SH is the snapshot entropy for the unit cell H . We can extract the coefficient SH / ln h from the main panel in Fig. 3 by identifying h with L. The numerical result is shown in the inset of Fig. 3, and we can see a rough tendency that the coefficient is naturally extrapolated to 1/3 in the large-h limit. Thus it is reasonable for us to rely on Eq. (24). To examine the physical origin of the coefficient 1/3, it is important to calculate the snapshot entropy for alternative spin models with different central charge. Later we will mention this point. C. The largest SVD spectrum
Let us next examine whether the feature characterized by Eq. (15) appears in the snapshot spectrum. For this purpose, we examine −ln λ1 . The result is shown in Fig. 4. Here we take the sample average after taking the logarithm of λ1 . This may generate large sample deviation, since the logarithm for λ1 < 1 changes quite rapidly. On the other hand, the obtained numerical data are not so seriously fluctuating. Therefore, we can say that the data quality is retained. We find in Fig. 4(a) that the overall feature is similar to S, and this is also a good signal for the phase transition. We thus think that there is certain correspondence between the largest SVD spectrum and the full snapshot entropy. Actually, at T = Tc in Fig. 4(b) and the inset, we find the scaling given by −ln λ1 ∼
1 24
ln L + α ∼ 18 S,
(26)
where the additional constant α is almost zero in this case. The coefficient of this equation is different from Eq. (15), but it is still depending on the value which strongly reminds us of the CFT description of the entanglement entropy.
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FIG. 4. (Color online) (a) Temperature and system-size dependence on −ln λ1 for Mp (x,y) = (σi + 1)/2 with i = (x,y): L = 32 = 25 (fine-dashed purple line), L = 64 = 26 (dashed red line), L = 128 = 27 (bold-solid blue line), L = 256 = 28 (solid black line), and L = 512 = 29 (fine-solid black line). (b) We also plot −ln λ1 − (1/24) ln L in order to see the Binder-parameter-like feature. The inset shows the gradient of the logarithmic term at Tc . The gradient of the guideline is 1/24.
FIG. 5. (Color online) Temperature dependence on the SVD spectra for L = 128 = 27 . The bold-solid blue line represents the largest spectrum −ln λ1 . The other data above the gap are −ln λi for i = 2,3,...,20 from the bottom to the top, respectively. (a) Mp (x,y) = (σi + 1)/2 and (b) Mn (x,y) = σi . The red data with open circles in the inset of (a) are obtained by taking L = 256 = 28 .
To show peculiarity of the largest SVD spectrum, we plot the average spectra −ln λi (λ1 > λ2 > · · · > λ20 ) for L = 128 in Fig. 5(a). An important point is that the temperature dependence on −ln λ1 is different from others and S is dominated by only −ln λ1 . We observe that there is a large spectral gap between −ln λ1 and others (a band above the gap) in the ferromagnetic phase. We also observe that the gap tends to decrease toward Tc , and takes the minimum value. In Fig. 5(a), the minimum gap is seen slightly above Tc . In the inset of the same figure, we plot the lowest and second-lowest spectra for L = 128 and L = 256. The cusp structure of the second-lowest spectrum near Tc becomes sharper with increasing L, and the cusp point seems to slightly shift toward Tc . The gap size does not strongly depend on L. In the paramagnetic phase, the gap size shows weaker
temperature dependence in comparison with the ferromagnetic phase. As we have already mentioned in Sec. II, λ1 increases by the proper encoding (0,1), and then the average spectrum −ln λ1 decreases. As a result, this gap structure appears. Figure 5(b) is the SVD spectrum with the normal encoding (−1,1). Therein, the lift of the degeneracy between −ln λ1 and the continuous band is not complete above Tc . The mixture among these spectra blurs the essential properties of −ln λ1 , when we calculate the snapshot entropy. The gap protects the critical behavior against the temperature fluctuation. Therefore, the proper encoding plays crucial roles on the formation of the lowest spectrum separated from the continuous band. This kind of protection mechanism of the lowest SVD spectrum has been also discussed in the context of edge modes of the topological insulators.
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following unit cell matrix: 1 H = 1
IV. TENSOR PRODUCT OF PROPERLY ENCODED MATRIX AND CLUSTER DISTRIBUTION A. Encoding and cluster distribution
Let us consider the meaning of the coefficient 1/3 of the logarithmic entropy scaling in more physical standpoint. We first examine that the encoding affects the distribution of the number of the ferromagnetic cluster sizes in the snapshot. This is crucial for determining the proper spin-spin correlator form. To see this, we introduce an effective model defined by the ⎛ 1 ⎜1 H ⊗H =⎝ 1 1
1 1 1 1
0 0 0 0
1 0 1 0
1 1 1 1
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1 1 1 1
0 0 0 0
1 1
0 0
1 , 0
(27)
where the left half block (4 spins) represents a large spin cluster, while the right upper site represents a small cluster (1 spin). As already mentioned in the Introduction, the fractallike spin configuration can be made by the multiple tensor product of the unit cell H . Taking the tensor product between two copies of H , we obtain 1 0 1 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 1 0 0
1 1 0 0
0 0 0 0
⎞ 1 0⎟ . 0⎠ 0
(28)
To clearly look at cluster distribution, we omit zero components from H ⊗ H (it is also possible to omit 1). The result is the following: ⎛ ⎞ 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎜1 1 ⎟ H ⊗H →⎝ (29) ⎠. 1 1 1 1 1 1 1 1 1 1 Then, we find certain distribution of various-size clusters. Here, we define a set of up spins (labeled by 1) connected along bond direction as a cluster. The largest cluster size is 10. We observe a rough tendency that the number of the clusters increases with decreasing the cluster size. This tendency matches with the spin configuration realized by the MC simulation at Tc . On the other hand, if we consider the normal encoding (−1,1), the situation changes. Let us next consider the unit cell matrix 1 1 −1 1 H = , (30) 1 1 −1 −1 and calculate the tensor product. Then we obtain
⎛ 1 ⎜1 H ⊗H →⎝ 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
⎞ 1 1 1 1 1 1 1 1 1 ⎟ , 1 1 1 ⎠ 1 1 1 1
(31)
where we have omitted −1. We clearly see that there are many large spin clusters and the distribution changes. B. Cluster distribution and coefficient of logarithmic entropy scaling
Let us next discuss a close relationship between the cluster distribution and the coefficient of the logarithmic term in the snapshot entropy formula. According to Ref. [5], the snapshot entropy is given by S=N
r
(−γj ln γj ) = A ln L,
A = 0.5408 for Eq. (30). The former is relatively close to 1/3. This suggests that a better cluster configuration tends to provides us with a A value close to 1/3. For comparison, we calculate A of the √ Sierpinski carpet. This case provides us with γ± = (1/2) ± ( 3/4) and A = 0.2237. Alternatively, we can set up a unit cell matrix that represents a slightly ordered but still critically fluctuating spin configuration. For instance, we have
(32)
⎛ 1 ⎜1 ⎜ H =⎜ ⎝1 1
j =1
where r = rank(H H † ) = 2 (we denote this as j = ±), and γj are the normalized eigenvalues of H H † . This equation tells us that there exist unit cell matrices with a particular set of γj values that leads to a proper gradient A. We compare the A value for √ Eq. (27) with that for Eq. (30). We find γ± = (1/2) ± ( 17/10) for Eq. (27), and γ± = (1/2) ± (1/4) for Eq. (30). Now the system after N tensor products is a rectangular shape 2N × 4N , and thus we identify a typical linear size to be L2 = 23N and N = (2/3) log2 L. We then obtain A = 0.2858 for Eq. (27) and
1 0 1 1
1 1 1 1
⎞ 1 1⎟ ⎟ ⎟. 1⎠ 1
(33)
2 The eigenvalues γ± = (15 ± √ matrix H has two nonzero 189)/2. In this case, L = 4N and we obtain A = 0.1251 much smaller than 1/3. Summarizing this section, we think that the cluster distribution is related to the magnitude of the coefficient of the logarithmic entropy scaling. This is a suggestive statement,
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PHYSICAL REVIEW E 92, 042167 (2015)
since in our usual experience of MC simulation the size distribution of various clusters looks common in both of the Ising and three-states Potts snapshots at Tc . Then, the entropy may not be observing model-dependent parameters such as the central charge, even if the presence of the logarithmic term itself is related to criticality of our target models. We will refer to this point in the numerical simulation of the Potts model. V. FINITE-χ SCALING
Up to now we have confirmed that the encoding affects the entropy scaling very much. Thus, we are also interested in whether the coarse-grained snapshot entropy formula is also deformed or not. The coarse-grained entropy is defined by
χ
