Ohtsuki Tomi

Abstract When the electric conductance of a nano-sized metal is measured at low temperatures, it often exhibits complex but reproducible patterns as a function of external magnetic fields called quantum fingerprints in electric conductance. Such complex patterns are due to quantum–mechanical interference of conduction electrons; when thermal disturbance is feeble and coherence of the electrons extends all over the sample, the quantum interference pattern reflects microscopic structures, such as crystalline defects and the shape of the sample, giving rise to complicated interference. Although the interference pattern carries such microscopic information, it looks so random that it has not been analysed. Here we show that machine learning allows us to decipher quantum fingerprints; fingerprint patterns in magneto-conductance are shown to be transcribed into spatial images of electron wave function intensities (WIs) in a sample by using generative machine learning. The output WIs reveal quantum interference states of conduction electrons, as well as sample shapes. The present result augments the human ability to identify quantum states, and it should allow microscopy of quantum nanostructures in materials by making use of quantum fingerprints.

The interplay between non-Hermiticity and disorder plays an important role in condensed matter physics. Here, we report the universal critical behaviors of the Anderson transitions driven by non-Hermitian disorders for a three-dimensional (3D) Anderson model and 3D U(1) model, which belong to 3D class AI† and 3D class A in the classification of non-Hermitian systems, respectively. Based on level statistics and finite-size scaling analysis, the critical exponent for the length scale is estimated as ν=0.99±0.05 for class AI†, and ν=1.09±0.05 for class A, both of which are clearly distinct from the critical exponents for 3D orthogonal and 3D unitary classes, respectively. In addition, spectral rigidity, level spacing distribution, and level spacing ratio distribution are studied. These critical behaviors strongly support that the non-Hermiticity changes the universality classes of the Anderson transitions.

We study the dynamics of Dirac and Weyl electrons in disordered point-node semimetals. The ballistic feature of the transport is demonstrated by simulating the wave-packet dynamics on lattice models. We show that the ballistic transport survives under a considerable strength of disorder up to the semimetal-metal transition point, which indicates the robustness of point-node semimetals against disorder. We also visualize the robustness of the nodal points and linear dispersion under broken translational symmetry. The speed of the wave packets slows down with increasing disorder strength, and vanishes toward the critical strength of disorder, hence becoming the order parameter. The obtained critical behavior of the speed of the wave packets is consistent with that predicted by the scaling conjecture.

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.

Identifying unconventional quantum phase transitions is one of the most fundamental subjects in quantum physics. To this end, critical exponents in disorder-driven quantum phase transitions in Weyl semimetals and symmetry-protected topological phases have been extensively studied in recent years. In this Rapid Communication, we provide precise critical exponent of the Anderson metal-insulator transition in three-dimensional (3D) orthogonal class with particle-hole symmetry, class CI, as ν = 1.16 ± 0.02 . We further study disorder-driven quantum phase transitions in the 3D nodal line Dirac semimetal model, which belongs to class BDI, and estimate the critical exponent as ν = 0.80 ± 0.02 . From a comparison of the exponents, we conclude that a disorder-driven reentrant insulator-metal transition from the topological insulator phase in the class BDI to the metal phase belongs to the same universality class as the Anderson transition in the 3D class BDI. We also argue that small disorder drives the nodal line Dirac semimetal in the clean limit to the metal.

Using numerical simulations, we investigate the distribution of Kondo temperatures at the Anderson transition. In agreement with previous work, we find that the distribution has a long tail at small Kondo temperatures. Recently, an approximation for the tail of the distribution was derived analytically. This approximation takes into account the multifractal distribution of the wavefunction amplitudes (in the parabolic approximation), and power law correlations between wave function intensities, at the Anderson transition. It was predicted that the distribution of Kondo temperatures has a power law tail with a universal exponent. Here, we attempt to check that this prediction holds in a numerical simulation of Anderson’s model of localisation in three dimensions.

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.

The transfer matrix method is inherently serial and is not well suited to modern massively parallel supercomputers. The obvious alternative is to simulate a large ensemble of hypercubic systems and average. While this permits taking full advantage of both OpenMP and MPI on massively parallel supercomputers, a straight forward implementation results in data that does not scale. We show that this problem can be avoided by generating random sets of orthogonal initial vectors with an appropriate stationary probability distribution. We have applied this method to the Anderson transition in the three-dimensional orthogonal universality class and been able to increase the largest L × L cross section simulated from L = 24 [New J. Phys. 16, 015012 (2014)] to L = 64 here.

機械学習，特にニューラルネットワークを利用した固体物理の研究が最近，活発に行われている。これらの研究の現状を，画像解析と強化学習の方法に分けて解説し，従来の方法との比較を行い，今後の展望について考える。

機械学習を利用した画像解析が最近，格段の進歩を遂げた。こうした手法を固体物理に応用する試みが始まっている。

<p>We report the results of a numerical study of the Anderson transition in a model including classical magnetic impurities, which we call the Anderson-Heisenberg model. In this model randomly oriented Heisenberg like magnetic impurities are distributed randomly on a small percentage of lattice sites. These couple locally to the electron with exchange coupling strength J. One of the motivations for studying this model is to better understand the metal insulator transition in doped semiconductors.</p>