Applications of neural networks to condensed matter physics are becoming popular and beginning to be well accepted. Obtaining and representing the ground and excited state wave functions are examples of such applications. Another application is analyzing the wave functions and determining their quantum phases. Here, we review the recent progress of using the multilayer convolutional neural network, so-called deep learning, to determine the quantum phases in random electron systems. After training the neural network by the supervised learning of wave functions in restricted parameter regions in known phases, the neural networks can determine the phases of the wave functions in wide parameter regions in unknown phases; hence, the phase diagrams are obtained. We demonstrate the validity and generality of this method by drawing the phase diagrams of two-and higher dimensional Anderson metal-insulator transitions and quantum percolations as well as disordered topological systems such as three-dimensional topological insulators and Weyl semimetals. Both real-space and Fourier space wave functions are analyzed. The advantages and disadvantages over conventional methods are discussed. * ohtsuki@sophia.ac.jp

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 twoand three-dimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. 85, 123706 (2016), 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.Introduction.-Applying machine learning methods to solve problems in condensed matter physics has proven to be successful. Ising and spin ice models, 1, 2) low dimensional topological systems, 3, 4) strongly correlated systems, [5][6][7][8][9][10][11][12][13][14] as well as random two-and threedimensional (2D, 3D) topological and non-topological systems, [15][16][17] have been studied using machine learning.In previous papers, 15,16) we presented studies of 2D and 3D random electron systems via a multilayer convolutional neural network (CNN) approach called deep learning, and the features of quantum phase transitions such as delocalization-localization transitions (the Anderson metal-insulator transition) and topological-nontopological insulator transitions were shown to be captured. We diagonalized the Hamiltonian, obtained the eigenfunctions Ψ ν (r), and trained the neural network by feeding the electron density |Ψ ν (r)| 2 for a specific quantum phase. We used 2D image recognition, and for 3D systems, integration over one direction was performed to reduce the 3D electron density to 2D. The advantage of reducing the electron * ohtsuki@sophia.ac.jp 1/11 J. Phys. Soc. Jpn. LETTERS

Quantum material phases such as the Anderson insulator, diffusive metal, and Weyl/Dirac semimetal as well as topological insulators show specific wave functions both in real and Fourier spaces. These features are well captured by convolutional neural networks, and the phase diagrams have been obtained, where standard methods are not applicable. One of these examples is the cases of random lattices such as quantum percolation. Here, we study the topological insulators with random vacancies, namely, the quantum percolation in topological insulators, by analyzing the wave functions via a convolutional neural network. The vacancies in topological insulators are especially interesting since peculiar bound states are formed around the vacancies. We show that only a few percent of vacancies are required for a topological phase transition. The results are confirmed by independent calculations of localization length, density of states, and wave packet dynamics.

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Machine learning has recently been applied to many problems in condensed matter physics. A common point of many proposals is to save computational cost by training the machine with data from a simple example and then using the machine to make predictions for a more complicated example. Convolutional neural networks (CNN), which are one of the tools of machine learning, have proved to work well for assessing eigenfunctions in disordered systems. Here we apply a CNN to assess Kohn-Sham eigenfunctions obtained in density functional theory (DFT) simulations of the metal-insulator transition of a doped semiconductor. We demonstrate that a CNN that has been trained using eigenfunctions from a simulation of a doped semiconductor that neglects electron spin successfully predicts the critical concentration when presented with eigenfunctions from simulations that include spin.