Phase Diagrams of Three-Dimensional Anderson and Quantum Percolation Models using Deep Three-Dimensional Convolutional Neural Network

The three-dimensional Anderson model is a well-studied model of disordered electron systems that shows the delocalization--localization transition. As in our previous papers on two- and three-dimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. {\bf 85}, 123706 (2016), {\bf 86}, 044708 (2017)], we used an image recognition algorithm based on a multilayered convolutional neural network. However, in contrast to previous papers in which 2D image recognition was used, we applied 3D image recognition to analyze entire 3D wave functions. We show that a full phase diagram of the disorder-energy plane is obtained once the 3D convolutional neural network has been trained at the band center. We further demonstrate that the full phase diagram for 3D quantum bond and site percolations can be drawn by training the 3D Anderson model at the band center.

density from 3D to 2D is that we can then use standard 2D image recognition. The drawback is that we lose the information of electron density distribution along one direction.
In this Letter, we apply 3D image recognition via deep 3D CNN to analyze a random 3D electron system. The Anderson model for delocalization-localization transition 18) and the 3D quantum bond and site percolation models were studied. We show that the full phase diagram of the disorder-energy (W-E) plane is available once the 3D CNN has been trained at the band center, i.e., along E = 0 (see arrows in Fig. 2(b)). We further demonstrate that the twodimensional phase diagrams (see Fig. 3) for the 3D quantum percolation model, both bond and site types, can be drawn by training the 3D Anderson model at the band center, without learning the features of quantum percolations.
Models and method.-We consider the following 3D Anderson Hamiltonian, 18) where c † x (c x ) denotes the creation (annihilation) operator of an electron at a site x = (x, y, z) that forms a simple cubic lattice, and v x denotes the random potential at x. · · · indicates the nearest-neighbor hopping. A box distribution is assumed with each v x uniformly and independently distributed at the interval [−W/2, W/2]. At energy E = 0, i.e., at the center of the band, the wave functions are delocalized when W < W c and the system is a diffusive metal. For W > W c , the wave functions are exponentially localized and the system is an Anderson insulator. Here, the critical disorder W c at E = 0 is estimated to be 16.54 ± 0.01 19) by the finite size scaling analysis of the Lyapunov exponent calculated from the transfer matrix method. [20][21][22][23] Next, we consider the 3D quantum bond and site percolation models described by the following Hamiltonian, [24][25][26][27] where electrons are allowed to move on connected sites of a classical percolation, [28][29][30] The hopping parameter t xx ′ is defined as where the energy unit is the transfer energy between connected sites. In the case of bond percolation, bonds are randomly connected with a probability p B . In the case of site percolation, each site is filled with a probability p S , and bonds are connected only if the sites on both 2/11 sides of the bond are filled. For each realization of bond and site percolations, we identify the maximally connected cluster with a depth-first search, and analyze only the states in this cluster.
These states are peculiar to the quantum percolation model, and are degenerate, resulting in strong peaks in the density of states. Due to this degeneracy, any linear combination is possible, which results in difficulty in judging the delocalized/localized phases. We therefore assume a very weak site random potential of the order of 10 −3 , namely, v x ∈ [−10 −3 /2, 10 −3 /2], and thereby lift the degeneracy.
Random numbers were generated by the Mersenne Twister algorithm, 34) MT2203. We imposed the periodic boundary condition, and the maximum modulus of the eigenfunction was shifted to the center of the system to improve the accuracy of the 3D image recognition.
For all calculations, we considered a 40 × 40 × 40 lattice, and diagonalized the Hamiltonian.
For training data and the data for Fig. 2(a), we used the sparse matrix diagonalization Intel MKL/FEAST, 35) since we restricted the energy range to around E = 0. For drawing the phase diagrams, we used the standard Linear Algebra package LAPACK 36) for diagonalization, and obtained all the eigenvalues and eigenvectors.
To calculate the probabilities of delocalized and localized phases, P deloc and P loc (= 1 − P deloc ) for a given eigenfunction, we adopted a 3D CNN using Keras 37)  Through the hidden layers that consist of a convolution layer, max pooling layer, and denselyconnected layer, our neural network outputs two real numbers, i.e., the probability of a delocalized (localized) phase P deloc (P loc ) in the case of the Anderson and quantum percolation models. The network utilizes the AdaDelta solver 38) as the stochastic gradient decent solver and rectified linear unit (ReLU) as its activation function except for the last layer, which is activated by the softmax function. 39) The loss function is defined by the cross entropy (which is categorical crossentropy in Keras), which was minimized during the training. To avoid overfitting, dropout processes, which randomly drop half of the inputs, were implemented in some layers (see Fig. 1). Before training, we randomly chose 10% of the training data as a validation data set. 3/11 determined the network weight parameters, which captured the features of the delocalized/localized phase. For training data, we restricted the energy region to the vicinity of the band center, E = 0, in the Anderson model. The main purpose of this work was to obtain global delocalization-localization phase diagrams of 3D random electron systems, and we assumed that the critical disorder at E = 0, W c = 16.54, was known in advance.
As the training data of the delocalized phase, we prepared 4000 eigenstates in the range,  Fig. 2(b).
To confirm that the network sufficiently captures the features of the eigenstates, we prepared test data from the Anderson model for various W and E, and determined the phase diagram in W-E parameter space and then compared it with those obtained by other researchers. [40][41][42] Once the results were confirmed, we applied this trained CNN to bond and site quantum percolation models, the phase diagrams of which are less well-known. 5 independent samples were prepared for the test data from the Anderson and quantum percolation models, and the output probabilities are averaged over the 5 samples.
Result for Anderson model.-In Fig. 2(a), we show the probabilities P deloc and P loc at We then prepared eigenfunctions all over the energy spectra with varying W, and let the CNN determine the phase. The result is shown in Fig. 2(b) where the intensity P deloc is plotted as a color map, which is in good agreement with the previous results from the transfer matrix. 40,41) The flat phase boundary around W c , |E| 6, together with the rapidly changing critical disorder outside the band edge |E| = 6 in the absence of randomness, are reproduced.
For relatively small disorder, the CNN judges that eigenstates near the band edge are localized. With increasing W but fixed E around E = 7, the localized states near the band edge are delocalized and then localized again, a re-entrant phenomenon known for the Anderson model with the box distributed site energies. [40][41][42] Note that the states near the band edge with small disorders look rather like bound states trapped by the potential fluctuation, and smoothly change in space. It is interesting that the CNN trained only around E = 0, where the random quantum interference was strong and the 4/11 wave function amplitudes were rapidly fluctuating, correctly judged the smooth bound states as localized.
Application to quantum percolation. -Figs. 3(a) and (b) show the phase diagrams of the bond and site quantum percolation models, respectively. Eigen energies of the maximal clusters are plotted on the horizontal axis, while the probability p B or p S is plotted on the vertical axis. All over the energy spectra, the quantum percolation transition occurs well above the classical percolation threshold, p classical B ≈ 0.2488 and p classical S ≈ 0.3116. 43,44) The energy dependence of the quantum site percolation threshold is consistent with the finite size scaling analysis of multifractality [45][46][47] by Ujfalusi and Varga,27) where a strong energy dependence is observed near E = 0 and near the band edge. [48][49][50] In the quantum percolation models, even on the maximal cluster, there appear energetically degenerate molecular states, the amplitudes of which are localized on finite sites, while being exactly zero on other sites. 31,32) As stated while explaining the model, we lifted the degeneracy by introducing a very weak site randomness. The molecular states can coexist with other states, but their numbers are much smaller than other non-molecular states except at specific energies E = 0, ±1, ± √ 2, · · · where the shapes of molecular states are relatively simple. Because we average the probability P deloc over a finite energy region, the phase is judged to be delocalized (metal) if the other non-molecular states are delocalized.
Summary and concluding remarks.-In this paper, we applied 3D image recognition to draw the global phase diagrams of 3D random electron systems. We have shown that once the 3D CNN is trained in a small region of the Anderson transition, the global phase diagram can be obtained by using this training. Furthermore, the CNN trained for the Anderson model at E = 0 allows us to draw the phase diagrams of the quantum bond and site percolation models. Thus, the present work has opened a way to use 3D deep CNN for condensed matter physics. Application to strongly correlated 3D electron systems with disorder [51][52][53] is one of the interesting problems that could benefit from this approach. Another interesting application is to distinguish rare event states in Weyl/Dirac semimetals 54,55) from other types of states, which is especially important in discussing the Weyl/Dirac semimetal to metal transition. [56][57][58][59] The present method can be extended easily to analyze other quantum percolation problems, such as systems with magnetic fields or spin-orbit interactions, i.e., systems belonging to different symmetry classes. [60][61][62][63] In these systems, the quantum percolation threshold is expected to be lowered due to the breaking of the time reversal or spin-rotational symmetries. 26) How the energy-dependent quantum percolation threshold is changed by the breaking of the symmetries is an interesting problem that remains to be studied in the future.

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Let us conclude this Letter by comparing the present method with the transfer matrix method, [19][20][21][22][23] and stating that the present method and the transfer matrix method are complementary. Due to the zero connections between sites that appear randomly, the transfer matrix analysis is not applicable to the quantum percolation problem. This situation is similar to topological insulators with a specific set of parameters. 16) The method presented here is, in this sense, more general than the transfer matrix method. In addition, we can draw the phase diagram in the W-E plane easily using the present method, while the transfer matrix method requires a one-dimensional parameter change, hence drawing the global W-E phase diagram is more numerically demanding. On the other hand, the transfer matrix method gives a more precise estimate of the critical point than the present method. The estimate of the critical exponent is also possible in the finite size scaling analysis of the Lyapunov exponent obtained by the transfer matrix method, which is so far impossible using the present method. A possible way to estimate the critical exponent by the present method is to change the system size L and observe how the quantities, such as probabilities of delocalized/localized states and/or convergence time vary with L. The independent estimate of critical disorder W c , with the aid of unsupervised training, is also an important topic for the future.   48) while that in (b) is the spline interpolation from the data in ref. 27) 8/11