Abstract
The inverse renormalization group is studied based on the image superresolution using the deep convolutional neural networks. We consider the improved correlation configuration instead of spin configuration for the spin models, such as the twodimensional Ising and threestate Potts models. We propose a blockcluster transformation as an alternative to the blockspin transformation in dealing with the improved estimators. In the framework of the dual Monte Carlo algorithm, the blockcluster transformation is regarded as a transformation in the graph degrees of freedom, whereas the blockspin transformation is that in the spin degrees of freedom. We demonstrate that the renormalized improved correlation configuration successfully reproduces the original configuration at all the temperatures by the superresolution scheme. Using the rule of enlargement, we repeatedly make inverse renormalization procedure to generate larger correlation configurations. To connect thermodynamics, an approximate temperature rescaling is discussed. The enlarged systems generated using the superresolution satisfy the finitesize scaling.
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Introduction
Wilson revealed that the renormalization group (RG) is a key concept in understanding critical phenomena of phase transitions^{1,2}. Kadanoff’s blockspin transformation is an idea to realize renormalization in real space^{3}. The combination of the RG with Monte Carlo simulation has been successfully used as the Monte Carlo RG^{4,5,6,7}. The inverse operation to generate a large system, which is called an inverse RG, was proposed by Ron, Swendsen, and Brandt^{8}. This approach is free of critical slowing down for large systems.
Recent developments of machinelearningbased techniques have been applied to fundamental research, such as statistical physics^{9}. A technique of supervised learning for image classification was used by Carrasquilla and Melko^{10} to propose a paradigm that is complementary to the conventional approach of studying interacting spin systems. By using large datasets of spin configurations, they classified and identified a hightemperature paramagnetic phase and a lowtemperature ferromagnetic phase of the twodimensional (2D) Ising model. Shiina et al.^{11} extended and generalized this idea so as to treat various spin models including the multicomponent systems and the systems with a vector order parameter. The configuration of a longrange spatial correlation was considered instead of the spin configuration itself. Not only the secondorder and the firstorder transitions but also the BerezinskiiKosterlitzThouless (BKT) transition^{12,13,14,15} was studied.
Tomita et al.^{16} have made further progress in this approach. In the machinelearning study of the phase classification of spin models, the Fortuin–Kasteleyn (FK)^{17,18} representationbased improved estimators^{19,20} of the correlation configuration were employed as an alternative to the ordinary correlation configuration. This method of improved estimators was applied not only to the classical spin models but also to the quantum Monte Carlo simulation using the loop algorithm. They analyzed the BKT transition of the spin1/2 quantum XY model on the square lattice.
Another application of machinelearning study for image processing is superresolution (SR), which is a class of techniques that enhance the resolution of an imaging system. Efthymiou et al.^{21} proposed a method to increase the size of lattice spin configuration using SR, deep convolutional neural networks^{22}. This study is related to the inverse RG approach^{8}. At high temperatures, however, there is a problem that the noise is largely random and difficult to learn. Because a significant reduction of variance is obtained for improved estimators at high temperatures of a disordered phase^{23}, the improved correlation configuration could reduce the difficulty of SR.
There have been some other proposals for the combination of neural network and RG^{24,25,26,27,28}. The work by Efthymiou et al.^{21} is an exceptional one to investigate an inverse RG.
In this paper, we study the inverse RG of spin models based on the SR. We consider the improved estimator of the correlation configuration instead of the spin configuration. As for the renormalization process, we propose a blockcluster transformation as an alternative to a blockspin transformation. Then, we can set up an inverse RG procedure using the SR technique. The resolution of the enhanced configuration at high temperatures is much improved compared to the SR using the spin configuration. We make inverse renormalization procedure repeatedly to generate larger correlation configurations. Introducing an approximate temperature rescaling, we show the finitesize scaling (FSS)^{29,30,31} for the enlarged systems. For the spin models, we treat the 2D Ising model and the 2D threestate Potts model.
Results
Monte Carlo renormalization group
Blockspin transformation
We start with the RG process. To realize the RG in real space Monte Carlo simulation, Kadanoff’s blockspin transformation^{3} is conventionally used. A majority rule is employed to determine a block spin from \(2 \times 2\) spins in a block, for the squarelattice Ising model, for example.
We do not consider the transformation of the Hamiltonian. Instead, we study the transformation property of the correlation. We calculate the correlation with a distance of L/2 as in the machinelearning study of the phase classification of spin models^{11,16}, where L is a linear system size. This type of correlation function was used along with the generalized scheme for the probabilitychanging cluster algorithm^{32}. For actual calculation, we treat the average value of the xdirection and the ydirection for the sitedependent correlation, that is,
where \(g[s, s']\) denotes a spin–spin correlation between a spin pair s and \(s'\).
We performed the Monte Carlo simulation of the Ising model (2state Potts model) on the \(64 \times 64\) square lattice using the SwendsenWang cluster update^{33}, and made blockspin transformations repeatedly. When a blockspin transformation is made one time, the linear system size becomes a half of the original size. We measured the spaceaveraged value of \(g_i(L/2)\) for the original Ising spins and also for the block spins. We plot the temperature (T) dependence of the total correlation,
by circles in Fig. 1a, where \(N (=L \times L)\) is the system size, and the angular brackets denote the Monte Carlo average. The temperature is measured in units of the interaction J (in terms of the Potts model). We also plot the temperature dependence g(T) of the Ising model with \(L=32\) and \(L=16\) by solid curves. We observe that the blockspin correlation g(T) of \(L=32\) produced from \(L=64\) system and the correlation g(T) of the true \(L=32\) system cross at the exact \(T_c = 1/\ln (1+\sqrt{2}) = 1.1346 \cdots \), which is shown by vertical dashed line. The twotime blockspin correlation g(T) of \(L=16\) and the correlation of the true \(L=16\) system cross at the exact \(T_c\). It is noteworthy that there are some corrections to scaling.
As another example, we treated the threestate Potts model on the square lattice. For the blockspin transformation, we use a majority rule. We plot the temperature dependence g(T) of the threestate Potts model on the square lattice with \(L=64\) in Fig. 1b. The blockspin correlations are compared with the correlations of the true \(L=32\) and \(L=16\) systems as in the case of the Ising model, Fig. 1a. We again observe the crossing at the exact \(T_c\), which is \(T_c = 1/\ln (1+\sqrt{3}) = 0.9950 \cdots \) for the threestate Potts model.
Blockcluster transformation
In the development of the cluster update of the Monte Carlo simulation, the socalled improved estimators^{19,20} were proposed for the measurement of the correlation. In calculating spin correlations, only the spin pair belonging to the same FK cluster should be considered. In the improved estimator for the cluster representation of the qstate Potts model (including the Ising model), the correlation becomes 1 for the spin pair belonging to the same FK cluster, whereas it becomes 0 for the spins of different clusters. In the framework of the dual Monte Carlo algorithm^{34,35,36}, the Markov process in the cluster update alternates between the original spin configurations (spin) and the space of the configurations of auxiliary variables (graph). Then, an improved estimator is considered to be an estimator defined in terms of the graph degrees of freedom rather than the original spin degrees of freedom.
In the proposal of using SR by Efthymiou et al.^{21}, there is a problem that the noise is largely random and difficult to learn at high temperatures. We will discuss in the present paper that the improved correlation configuration solves this difficulty. However, we cannot directly apply the improved estimator for the blockspin transformation; we cannot specify the FK cluster to which a selected block spin belongs. Thus, we here propose another renormalization procedure, a blockcluster transformation. The detailed procedure of a blockcluster transformation is described in the section of Methods.
We performed the blockcluster transformation for the 2D Ising model and the threestate Potts model. The plots of the temperature dependence of the correlation g(T) are given by crosses in Fig. 1a,b. We may compare the results of the blockspin transformation (circles) and those of the blockcluster transformation (crosses). We observe that the renormalized values of g(T) are almost the same for both the Ising model and the threestate Potts model. At high temperatures fluctuations of the blockcluster transformation are smaller than those of the blockspin transformation because the improved estimators are used in the blockcluster transformation. There are very small deviations at low temperatures, which depend on the renormalization scheme. At \(T_c\) of the Ising model, the blockcluster value of \(g(T_c)\) for \(L=32\) (purple cross) is 1.6% larger than the blockspin value (purple circle), whereas the true value of \(L=32\) (purple curve) is between, and close to the value of the blockcluster transformation. From the viewpoint of transformation property of renormalization, the blockcluster transformation could be better.
Inverse renormalization group based on superresolution
Extension of the method of Efthymiou et al
We perform the inverse operation of RG, extending the method of SR with a deep convolutional neural network (CNN)^{22} due to Efthymiou et al.^{21}. We use the improved estimator of the correlation configuration instead of the spin configuration. We give the detailed description of SR in the section of “Methods”. We emphasize that the present method can be applied to any qstate Potts model because we deal with the correlation configuration. In the case of the spin configuration, only the Ising model can be treated.
We performed the procedure of SR for the 2D Ising model and threestate Potts model. Simulating \(32 \times 32\) systems, we obtain sets of original improved correlation configuration (\(32 \times 32\)) and the truncated improved correlation configuration (\(16 \times 16\)) using blockcluster transformation. For training data we use 8000 sets of configurations for each temperature. In Fig. 2, we plot the sample average of \(\sum _i \xi _i\) and \(\sum _i g_i\); that is, g(T) for \(L=32\) (brown pluses) and \(L=16\) (blue pluses). Here, \(\xi _i\) and \(g_i\) are original and truncated improved correlation configurations, respectively. Using the SR technique, the parameters (\(\theta = (W,b)\)) are tuned for each temperature. For test data we use other independent 6000 sets of original and truncated configurations. The improved correlation configuration of \(L=32\) is reproduced from the \(L=16\) truncated configuration using the optimized parameters \(\theta \). We compare the original \(\langle \sum _i \xi _i \rangle \) (brown pluses) and the reproduced \(\langle \sum _i \xi _i' \rangle \) (red circles) in Fig. 2. We do not observe appreciable differences; it means that the reproduction is almost perfect for all the temperatures. In Fig. 2, for comparison, we also plot the results of SR for the ordinary correlation configuration produced from the spin configuration (lime circles). At high temperatures, there is a deviation from the true values of \(L=32\) (brown pluses). To avoid this difficulty, an additional term was added in the regularization term in the loss function in Ref. ^{21}. We do not need to add such an additional term for the improved estimator. The relative deviation of reproduction \(\langle \sum _i \xi _i \rangle \) \(\rightarrow \) \(\langle \sum _i \xi _i' \rangle \) is smaller than 0.4% at all the temperatures.
Next consider the further increase of the system size. From the improved correlation configuration of \(32 \times 32\) with tuned parameters \(\theta \), we generate the SR configuration of \(64 \times 64\). The \(2L \times 2L\) output of the first SR is regarded as the input of a new network \(2L \times 2L \rightarrow 4L \times 4L\). Following Efthymiou et al.^{21}, we assume that the parameters (the weight matrix W and the bias vector b) are independent of the system size. In the same way, we generate \(128 \times 128\) enlarged image, \(256 \times 256\) enlarged image, repeatedly. In Fig. 3, we plot the iterative SR for both 2D Ising model and 2D threestate Potts model. It is noteworthy that we make Monte Carlo simulations only for the system size of \(32 \times 32\). This SR procedure is a geometric procedure, and temperature has its own meaning only for \(32 \times 32\) system.
Temperature rescaling
For the onedimensional Ising model, selfsimilar transformation of the Hamiltonian using the decimation scheme is possible. However, the nearestneighbor 2D Ising model will be mapped to the model with complex interactions, such as the nextnearestneighbor interaction, fourspin interaction, etc. Thus, for the 2D models, the transformation of Hamiltonian is not selfsimilar after a blockspin RG transformation, and therefore temperature alone is not sufficient to describe the coupling space of the RG configuration. We follow Efthymiou et al.^{21} to describe a method to approximate the rescaling of temperature numerically. We transform temperature such as \({\tilde{T}} = F(T)\) with the RG transformation. To avoid confusion with the activation function f, we here use F instead of f. In opposite direction of inverse RG transformation, transformation such as \(T = F^{1}({\tilde{T}})\) is expected. To find the rescaling, we compare the correlation g(T) calculated for enlarged system of \(64 \times 64\) with the true g(T) of \(64 \times 64\) system. We find the transformation \({\tilde{T}} \rightarrow T\) by requiring the corresponding g(T) curves to collapse.
Once the transformation \(T = F^{1}({\tilde{T}})\) from \(32 \times 32\) to \(64 \times 64\) systems is established, we use the same transformation to the inverse RG procedure of \(64 \rightarrow 128\) and \(128 \rightarrow 256\). In Fig. 4, we show the temperature rescaling results. The temperatures of the output of SR for \(64 \times 64\) (orange crosses) are rescaled such that they collapse with the true g(T) obtained by the Monte Carlo simulation of \(64 \times 64\) (orange curve), which is shown in the inset. The outputs of \(128 \times 128\) and \(256 \times 256\) are rescaled using the transformation \(F^{1}\) obtained by \(32 \rightarrow 64\) transformation. We observe that the range of the rescaled temperature shrinks. Monte Carlo results of \(128 \times 128\) and \(256 \times 256\) are also given in Fig. 4 for the sake of comparison. For high temperature sides, temperature rescaling works quite well. However, for low temperature side, deviation becomes appreciable when the temperature becomes away from \(T_c\). It comes from the saturation effects of g(T), that is, it approaches 1 as \(T \rightarrow 0\). This phenomena often appear in FSS analysis of magnetization, for example.
For geometric inverse RG, only the Monte Carlo simulation of \(32 \times 32\) system is enough. For temperature rescaling, Monte Carlo simulation of \(64 \times 64\) should be added. With these small sizes of simulation, we can obtain the information on larger system sizes.
Finitesize scaling
Now that we have obtained the temperature dependence g(T) for various sizes from small sizes, we try a FSS analysis^{29,30,31} to examine critical phenomena. The FSS function for the equation of state can be written as
where \(t=TT_c\) with the critical temperature \(T_c\); \(\nu \) and \(\beta \) are the correlationlength and magnetization critical exponents, respectively. In Fig. 5, we show the FSS plots of the Ising model and the threestate Potts model. That is, \(g(T) L^{2\beta /\nu }\) is plotted as a function of \(t L^{1/\nu }\). As for the critical temperature \(T_c\), we use the exact values because in the temperature rescaling transformation shown in the inset of Fig. 4, the fixed points, \(T^* = F^{1}(T^*)\), are very close to the exact values. For the critical exponents, \(2\beta /\nu (= \eta )\) and \(1/\nu \), bestfitted values for data collapsing up to \(L=128\) were used. The chosen \(2\beta /\nu \) and \(1/\nu \) are 0.240 (1/4) and 1.010 (1) for the Ising model, and 0.255 (4/15) and 1.215 (6/5) for the threestate Potts model. In the parentheses, the exact exponents are given. Estimated critical exponents are \(1 \sim 4 \%\) accuracy. We obtained good FSS, although the original system of small size of \(L=32\) has larger corrections to FSS.
Discussion
We have successfully realized the inverse RG based on SR approach. We have proposed the blockcluster transformation as an alternative to the blockspin transformation to use the improved estimators. We have made inverse renormalization procedure repeatedly to generate larger correlation configurations. We here make remarks on the advantage of the present method and the future directions of the research.
The advantage of using cluster representation of spin models is remarkable, and the improved estimator is quite useful. In doing so, we have introduced a blockcluster transformation, and we investigate the improved correlation configuration. The statistical advantage of improved estimator is well known^{23}; at high temperatures above \(T_c\), the errors for the spatial average of correlation is drastically reduced. In addition, this study elucidated the advantage in improved correlation configuration itself.
We make comments on the image as an object of image processing. Images usually have some smooth parts together with some edges. Abstract painting is not an object of image processing. A similar example is found in the case of text compression; we discuss the compression of text in natural language, while the compression of random sequences is impossible. The spin configurations have particular characteristics. There is a longrange correlation up to the correlation length. Moreover, there are some symmetries. The orderedstate spin configurations at low temperatures have some smooth parts. However, hightemperature spin configurations are random. On the other hand, improved correlation configurations have cluster structures even at high temperatures. The choice of proper “image” is important in applying the technique of image processing to the spin configuration problems.
The RG transformation is a coarsegrained procedure; thus, the truncated systems (blockspin or blockcluster transformation) have less information. Practically, it is difficult to find appropriate procedure of the inverse RG. The advantage of using the machinelearning is to find a rule to connect the renormalized configuration and the original configuration by searching for large amounts of datasets. This is the idea of the machinelearning. A single realization of \(32 \times 32\) system in the present study does not have the information of larger systems. With the help of SR technique of machinelearning, we can obtain the information of larger systems which include the scaling properties.
As a preliminary study, we are also considering the blockcluster transformation for the threedimensional (3D) systems. In the case of the simple cubic lattice, we determine a label of the block cluster from the labels of \(2 \times 2 \times 2\) sites using a majority rule for the labels. The blockcluster transformation works very well for the 3D Ising model. For the 3D Ising model, a trial to improve convergence using the modified blockspin transformation^{38} in the Monte Carlo RG calculation was reported recently^{39}. In the Monte Carlo RG studies^{6,7,39}, the correlations between different blocking levels (m), (n), \(\langle s_{\gamma }^{(n)} s_{\beta }^{(m)} \rangle  \langle s_{\gamma }^{(n)} \rangle \langle s_{\beta }^{(m)}\rangle \), are calculated. In the present formalism of blockcluster transformation, such correlations can be calculated. It will be interesting to apply the blockcluster transformation, where firstorder moments \(\langle s_{\beta } \rangle \) are automatically zero because of the improved estimator.
It is wellknown that the wavelet transformation is a highly efficient representation of images by decomposing the image signal into highfrequency and lowfrequency subbands. Guo et al.^{40} proposed a deep wavelet SR method to recover missing details of lowresolution images. Tomita^{41} performed the wavelet analysis of a configuration of FK clusters. The SR study on spin models using wavelet transformation will be informative.
For future studies, it will be interesting to treat continuous spin systems, such as the XY (clock) model, along the present scheme. The application of RG and inverse RG analyses to quantum systems^{42} is also challenging.
Methods
Detailed procedure of Blockcluster transformation
In the cluster update of Monte Carlo simulation, the spins are classified by the FK clusters^{17,18}, and we assign a label to each FK cluster. We note that the connection between the magnetization of the qstate Potts model and the percolation probability of the cluster model was discussed by Hu^{43,44}. We employ a majority rule to determine a label of the block cluster from the labels of \(2 \times 2\) sites. The procedure of the blockcluster transformation is schematically illustrated in Fig. 6, where different colors are used to assign the labels of clusters, not the spins. Even if adjacent sites have the same spins, these sites may have different colors because of the FK cluster. If the labels of two sites coincide, we take this label as the label of the block. The choice of two sites is sixfold. If there is no pair to coincide, we choose one label from four sites with 25% probability. When two sites have one label and the other two sites have another label, one may pick up one with 50% probability. However, we can take the label of the pair, which was first picked up, deterministically; it is the same situation as the twoup twodown case of the Isingmodel blockspin transformation^{21}. It is noteworthy that the procedure of blockcluster transformation is the same for qstate models irrespective of q. In the framework of the dual Monte Carlo algorithm^{34,35,36}, the blockcluster transformation is a transformation in the graph degrees of freedom, whereas the blockspin transformation is that in the spin degrees of freedom. It is instructive to compare the program code for the blockcluster transformation with that for the blockspin transformation. The essential part to select the label of the block cluster is
which is in parallel to the blockspin transformation:
In the program code, \(\texttt {cluster[]}\) stands for the label of the original cluster, whereas \(\texttt {cluster\_blk[]}\) stands for that of the block cluster. The command means that if the two labels in the block coincide, we choose this label as the label of the block cluster.
Detailed description of SR
We describe the detailed procedure of the SR with a deep CNN. It is an extension of the method by Efthymiou et al.^{21}. The improved estimator of the correlation configuration is used instead of the spin configuration. Starting from \(2L \times 2L\) spin system of improved correlation configuration, \(\{ \xi _i \}\), we produce \(L \times L\) system of improved correlation configuration, \(\{ g_i \}\), using the blockcluster transformation. We aim to reproduce the \(2L \times 2L\) target improved correlation configuration using the SR mapping of a supervised learning approach. This procedure is also illustrated in Fig. 6. The improved correlation configurations take a value of 1, 1/2, or 0 because we consider the average value of x and ydirections. According to Ref. ^{16}, they are represented in white, light gray or dark gray.
We consider three layers for the SR CNN, patch extraction and representation, nonlinear mapping, and reconstruction^{22}. The first layer is an upsampling layer by copying each configuration to \(2 \times 2\) block. This upsampling procedure is appropriate both for low temperature and high temperature. It is in contrast to the situation of using spin configuration^{21}, where a simple upsampling procedure is insufficient for random configuration of hightemperature side. For the improved correlation configuration, it takes \(+1\) when the correlation length is as large as half of the system size, L/2, and the sites with \(+1\) correlation form a small cluster (see Fig. 1f of Ref. ^{16}). For a convolution layer, as the second layer, the transformation \(f(W * x + b)\) is applied to the input x, a \(2 \times 2\) improved correlation configuration. Here, W (\(2 \times 2\)) is the filter, b being the bias vector, and \(*\) being the convolution operation. To avoid truncating the image edge, we add the periodic boundary padding. For activation function f, we use a sigmoid function, which gives the probability of each site For a loss function, we use the crossentropy loss function between \(\{ \xi _i \}\) and \(\{ p_i \}\) (continuous variable);
where \(\cdot \) denotes the elementwise product between matrices. As a library we use “BCEWithLogitsLoss”, where BCE stands for binary cross entropy. Parameters (\(\theta =(W,b)\)) are tuned to minimize a loss function. We employ the Adam method^{37} as an optimizer. As the third layer, using the optimized parameters, we calculate each \(\{ p_i \}\), and determine \(+1\) or 0 depending upon this probability. Repeating this process two times, we emulate a configuration as the sum of x and ydirections.
Thus, we can reproduce \(\{ \xi _i' \}\) of \(2L \times 2L\) size. Because we deal with the correlation, we can treat the threestate Potts model; the permutational symmetry is taken into account. When we consider the spin configuration, we cannot follow the present superresolution procedure. We emphasize that the same procedure can be used for any qstate Potts model.
References
Wilson, K. G. Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B 4, 3174–3183 (1971).
Wilson, K. G. Renormalization group and critical phenomena. II. Phasespace cell analysis of critical behavior. Phys. Rev. B 4, 3184–3205 (1971).
Kadanoff, L. P. Scaling laws for ising models near \(T_c\). Physics 2, 263–272 (1966).
Ma, S.K. Renormalization group by Monte Carlo methods. Phys. Rev. Lett. 37, 461–464 (1976).
Swendsen, R. H. Monte Carlo renormalization group. Phys. Rev. Lett. 42, 859–861 (1979).
Pawley, G. S., Swendsen, R. H., Wallace, D. J. & Wilson, K. G. Monte Carlo renormalizationgroup calculations of critical behavior in the simplecubic Ising model. Phys. Rev. B 29, 4030–4040 (1984).
Baillie, C. F., Gupta, R., Hawick, K. A. & Pawley, G. S. Monte Carlo renormalizationgroup study of the threedimensional Ising model. Phys. Rev. B 45, 10438–10453 (1992).
Ron, D., Swendsen, R. H. & Brandt, A. Inverse Monte Carlo renormalization group transformations for critical phenomena. Phys. Rev. Lett. 89, 275701 (2002).
Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).
Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017).
Shiina, K., Mori, H., Okabe, Y. & Lee, H. K. Machinelearning studies on spin models. Sci. Rep. 10, 2177 (2020).
Berezinskii, V. L. Destruction of longrange order in onedimensional and twodimensional systems having a continuous symmetry Group I. Classical systems. Sov. Phys. JETP 32, 493–500 (1970).
Berezinskii, V. L. Destruction of longrange order in onedimensional and twodimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. Phys. JETP 34, 610–616 (1972).
Kosterlitz, J. M. & Thouless, D. Ordering, metastability and phase transitions in twodimensional systems. J. Phys. C Solid State Phys. 6, 1181–1203 (1973).
Kosterlitz, J. M. The critical properties of the twodimensional xy model. J. Phys. C Solid State Phys. 7, 1046–1060 (1974).
Tomita, Y., Shiina, K., Okabe, Y. & Lee, H. K. Machinelearning study using improved correlation configuration and application to quantum Monte Carlo simulation. Phys. Rev. E 102, 021302(R) (2020).
Kasteleyn, P. W. & Fortuin, C. M. Phase transitions in lattice systems with random local properties. J. Phys. Soc. Jpn. Suppl. 26, 11 (1969).
Fortuin, C. M. & Kasteleyn, P. W. On the randomcluster model: I. Introduction and relation to other models. Physica 57, 536–564 (1972).
Wolff, U. Asymptotic freedom and mass generation in the O(3) nonlinear \(\sigma \)model. Nucl. Phys. B 334, 581–610 (1990).
Evertz, H. G., Lana, G. & Marcu, M. Cluster algorithm for vertex models. Phys. Rev. Lett. 70, 875–879 (1993).
Efthymiou, S., Beach, M. J. S. & Melko, R. G. Superresolving the Ising model with convolutional neural networks. Phys. Rev. B 99, 075113 (2019).
Dong, C., Loy, C. C., He, K. & Tang, X. Image superresolution using deep convolutional networks. IEEE Trans. Pattern Anal. Mach. Intell. 38, 295 (2016).
Holm, C. & Janke, W. Critical exponents of the classical threedimensional Heisenberg model: A singlecluster Monte Carlo study. Phys. Rev. B 48, 936–950 (1993).
Bény, C. Deep learning and the renormalization group. arXiv:1301.3124 (2013).
Mehta, P., & Schwab, D. J. An exact mapping between the variational renormalization group and deep learning. arXiv:1410.3831 (2014).
Iso, S., Shiba, S. & Yokoo, S. Scaleinvariant feature extraction of neural network and renormalization group flow. Phys. Rev. E 97, 053304 (2018).
KochJanusz, M. & Ringel, Z. Mutual information, neural networks and the renormalization group. Nat. Phys. 14, 578–582 (2018).
Li, S.H. & Wang, L. Neural network renormalization group. Phys. Rev. Lett. 121, 260601 (2018).
Fisher, M. E. Critical Phenomena. In Proc. Enrico Fermi Int. School of Physics Vol. 51. (ed. Green, M. S.) (Academic Press, 1971).
Fisher, M. E. & Barber, M. N. Scaling theory for finitesize effects in the critical region. Phys. Rev. Lett. 28, 1516–1519 (1972).
Binder, K. Finite size scaling analysis of ising model block distribution functions. Z. Phys. B 43, 119–140 (1981).
Tomita, Y. & Okabe, Y. Finitesize scaling of correlation ratio and generalized scheme for the probabilitychanging cluster algorithm. Phys. Rev. B 66, 180401(R) (2002).
Swendsen, R. H. & Wang, J. S. Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987).
Kandel, D. & Domany, E. General cluster Monte Carlo dynamics. Phys. Rev. B 43, 8539–8548 (1991).
Kawashima, N. & Gubernatis, J. E. Generalization of the Fortuin–Kasteleyn transformation and its application to quantum spin simulations. J. Stat. Phys. 80, 169–221 (1995).
Kawashima, N. & Harada, K. Recent developments of worldLine Monte Carlo methods. J. Phys. Soc. Jpn. 73, 1379–1414 (2004).
Kingma, D. P., Ba, J., Adam: A method for stochastic optimization. arXiv:1412.6980 (2014).
Blöte, H. W. J., Heringa, J. R., Hoogland, A., Meyer, E. W. & Smit, T. S. Monte Carlo renormalization of the 3D Ising model: Analyticity and convergence. Phys. Rev. Lett. 76, 2613–2616 (1996).
Ron, D., Brandt, A. & Swendsen, R. H. Surprising convergence of the Monte Carlo renormalization group for the threedimensional Ising model. Phys. Rev. E 95, 053305 (2017).
Guo, T., Mousavi, H. S., Vu, T. H. & Monga, V. Deep wavelet prediction for image superresolution. In IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) 104–113 (2017).
Tomita, Y. Measurement of entanglement entropy in the twodimensional Potts model using wavelet analysis. Phys. Rev. E 97, 052128 (2018).
Wu, Y. & Car, R. Continuoustime Monte Carlo renormalization group. Phys. Rev. B 102, 014456 (2020).
Hu, C.K. Percolation, clusters, and phase transitions in spin models. Phys. Rev. B 29, 5103–5018 (1984).
Hu, C.K. Sitebondcorrelated percolation and a sublattice dilute Potts model at finite temperatures. Phys. Rev. B 29, 5109–5116 (1984).
Acknowledgements
This work was supported by a GrantinAid for Scientific Research from the Japan Society for the Promotion of Science, Grant Number, JP19K03657. It was also supported by a Research Fellowships of Japan Society for the Promotion of Science for Young Scientists, Grant Number 20J12472. K. S. is grateful to the A*STAR (Agency for Science, Technology and Research) Research Attachment Programme of Singapore for financial support. We generated eps files of figures using python library of matplotlib.
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K.S. and Y.O. designed the study and performed computer calculations. All authors analyzed the results, and approved the final manuscript.
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Shiina, K., Mori, H., Tomita, Y. et al. Inverse renormalization group based on image superresolution using deep convolutional networks. Sci Rep 11, 9617 (2021). https://doi.org/10.1038/s4159802188605w
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DOI: https://doi.org/10.1038/s4159802188605w
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