Quantum percolation transition in three dimensions: Density of states, finite-size scaling, and multifractality

Ujfalusi, László; Varga, Imre

doi:10.1103/physrevb.90.174203

Cited by 9 publications

(17 citation statements)

References 34 publications

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“…Here, the critical disorder W c is estimated to be 16.54 ± 0.01 by the finite size scaling analysis of the Lyapunov exponent calculated by the transfer matrix method. 267,268) 3D quantum percolation model.-We next consider the 3D quantum site percolation model described by the following Hamiltonian, [269][270][271][272]…”

Section: D Anderson Model and Quantum Percolationmentioning

confidence: 99%

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Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions

Ohtsuki

Mano

2020

J. Phys. Soc. Jpn.

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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

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Section: D Anderson Model and Quantum Percolationmentioning

confidence: 99%

“…According to the study of quantum percolation, 272,273) the strongly localized states, socalled molecular states, exist at some energies, such as E = 0, ±1, ± √ 2, ± √ 3. These states are peculiar to the quantum percolation model and are degenerate, resulting in the strong peaks in the density of states.…”

Section: D Anderson Model and Quantum Percolationmentioning

confidence: 99%

Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions

Ohtsuki

Mano

2020

J. Phys. Soc. Jpn.

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“…In addition, we can draw the phase diagram in the W-E plane easily using the present method, while the transfer matrix method , 48) while that in (b) is the spline interpolation from the data in ref. 27) 8/11…”

Section: Lettersmentioning

confidence: 99%

“…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]…”

mentioning

confidence: 99%

“…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 According to the studies on quantum percolation, 27,[31][32][33] many strongly localized states, so-called molecular states, appear at energies, e.g., E = 0, ±1, ± √ 2, when we set v x = 0.…”

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

Mano

Ohtsuki

2017

J. Phys. Soc. Jpn.

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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

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Machine learning wave functions to identify fractal phases

Čadež,

Dietz,

Rosa

et al. 2023

Phys. Rev. B

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Quantum percolation transition in three dimensions: Density of states, finite-size scaling, and multifractality

Cited by 9 publications

References 34 publications

Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions

Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions

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

Machine learning wave functions to identify fractal phases

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