大槻 東巳

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.

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。

We clarify unconventional forms of the scaling functions of conductance, critical conductance distribution, and localization length in a disorder-driven quantum phase transition between band insulator and Weyl semimetal phases. Quantum criticality of the phase transition is controlled by a clean-limit fixed point with spatially anisotropic scale invariance. We argue that the anisotropic scale invariance is reflected on the scaling function forms in the quantum phase transition. We verify the proposed scaling function forms in terms of transfer-matrix calculations of conductance and localization length in a tight-binding model.

In electronic band structure of solid-state material, two band-touching points with linear dispersion appear in pairs in the momentum space. When they annihilate each other, the system undergoes a quantum phase transition from a three-dimensional (3D) Weyl semimetal (WSM) phase to a band insulator phase such as a Chern band insulator (CI) phase. The phase transition is described by a new critical theory with a "magnetic dipole"-like object in the momentum space. In this paper, we reveal that the critical theory hosts a novel disorder-driven quantum multicritical point, which is encompassed by three quantum phases: a renormalized WSM phase, a CI phase, and a diffusive metal (DM) phase. Based on the renormalization group argument, we first clarify scaling properties around the band-touching points at the quantum multicritical point as well as all phase boundaries among these three phases. Based on numerical calculations of localization length, density of states, and critical conductance distribution, we next prove that a localization-delocalization transition between the CI phase with a finite zero-energy density of states (zDOS) and DM phase belongs to an ordinary 3D unitary class. Meanwhile, a localization-delocalization transition between the Chern insulator phase with zero zDOS and a renormalized WSM phase turns out to be a direct phase transition whose critical exponent ν=0.80±0.01. We interpret these numerical results by a renormalization group analysis on the critical theory.

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. 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.

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. 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.

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。

Three-dimensional random electron systems undergo quantum phase transitions and show rich phase diagrams. Examples of the phases are the band gap insulator, Anderson insulator, strong and weak topological insulators, Weyl semimetal, and diffusive metal. As in the previous paper on two-dimensional quantum phase transitions [J. Phys. Soc. Jpn. 85, 123706 (2016)], we use an image recognition algorithm based on a multilayered convolutional neural network to identify which phase the eigenfunction belongs to. The Anderson model for localization-delocalization transition, the Wilson-Dirac model for topological insulators, and the layered Chern insulator model for Weyl semimetal are studied. The situation where the standard transfer matrix approach is not applicable is also treated by this method.

Random electron systems show rich phases such as Anderson insulator, diffusive metal, quantum Hall and quantum anomalous Hall insulators, Weyl semimetal, as well as strong/weak topological insulators. Eigenfunctions of each matter phase have specific features, but owing to the random nature of systems, determining the matter phase from eigenfunctions is difficult. Here, we propose the deep learning algorithm to capture the features of eigenfunctions. Localization-delocalization transition, as well as disordered Chern insulator-Anderson insulator transition, is discussed.

Regarding three-dimensional (3D) topological insulators and semimetals as a stack of constituent twodimensional (2D) topological (or sometimes nontopological) systems is a useful viewpoint. Here, we perform a comparative study of the paradigmatic 3D topological phases: Weyl semimetal (WSM), strong and weak topological insulators (STI/WTI), and Chern insulator (CI). By calculating the Z and Z(2) indices for the thin films of such 3D topological phases, we follow dimensional evolution of topological properties from 2D to 3D. It is shown that the counterparts of STI and WTI in the time-reversal symmetry broken CI system are, respectively, WSM and CI phases. The number N-D of helical Dirac cones emergent on the surface of a topological insulator is shown to be identical to the number N-W of the pairs of Weyl cones in the corresponding WSM phase: N-D = N-W. To test the robustness of this scenario against disorder, we have studied the transport property of disordered WSM thin films, taking into account both the bulk and surface contributions.

Low-energy magnon bands in a two-dimensional spin-ice model become integer quantum magnon Hall bands under an out-of-plane field. By calculating the localization length and the two-terminal conductance of magnon transport, we show that the magnon bands with disorders undergo a quantum phase transition from an integer quantum magnon Hall regime to a conventional magnon localized regime. Finite size scaling analysis as well as a critical conductance distribution shows that the quantum critical point belongs to the same universality class as that in the quantum Hall transition. We characterize thermal magnon Hall conductivity in a disordered quantum magnon Hall system in terms of robust chiral edge magnon transport.

Disordered non-interacting systems are classified into ten symmetry classes, with the unitary class being the most fundamental. The three and four-dimensional unitary universality classes are attracting renewed interest because of their relation to three-dimensional Weyl semi-metals and four-dimensional topological insulators. Determining the critical exponent of the correlation/localisation length for the Anderson transition in these classes is important both theoretically and experimentally. Using the transfer matrix technique, we report numerical estimations of the critical exponent in a U(1) model in three and four dimensions.

We studied the effects of disorder in a three-dimensional layered Chern insulator, which, in the clean limit, is either a Chern insulator or a Weyl semimetal depending on an interlayer coupling strength. By calculating the localization length by the transfer matrix method, we found two distinct types of metallic phases between the Anderson insulator and the Chern insulator: one is a diffusive metallic phase and the other is a renormalized Weyl semimetal phase. By calculating the conductance and density of states, we characterize these two metallic phases and reveal a critical nature of a quantum critical line between these two metallic phases.

<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>

<p>トポロジカル絶縁体とワイル半金属それぞれのナノ薄膜の2次元的トポロジカル特性に着目し，膜厚を制御した時の性質の変化を比較した．それぞれ時間反転に対する対称性が異なるため，薄膜に加工した時の2次元的トポロジカル特性は，前者は量子スピンホール効果，後者はいわゆる量子異常ホール効果に類似したものとなるが，膜厚を制御したときの2次元的トポロジカル特性の現れ方に完全な類似性を見いだす事ができた．</p>

<p>乱れたワイル半金属表面に電子波束を注入した際のダイナミクスを報告する。チェビシェフ展開を用いた高精度の時間発展シミュレーションにより、ワイル半金属における量子拡散現象を明らかにする。</p>

<p>前回Lifshitz tailの原因となる局在電子をワイル半金属において存在することを数値的に確認し報告した。引き続きワイル半金属をモデルに今回は定量評価として局在電子の系のサイズ依存性について検証した。局在電子数とサイズの関係から現実的なサイズでLifshitz tailが半金属-金属相転移に影響を及ぼすのか議論する。</p>

We show how the two-dimensional (2D) topological insulator evolves, by stacking, into a strong or weak topological insulator with different topological indices, proposing a new conjecture that goes beyond an intuitive picture of the crossover from quantum spin Hall to weak topological insulator. Studying the conductance under different boundary conditions, we demonstrate the existence of two conduction regimes in which conduction happens through either surface or edge conduction channels. We show that the two conduction regimes are complementary and exclusive. Conductance maps in the presence and absence of disorder are introduced, together with 2D Z2-index maps, describing the dimensional crossover of the conductance from the 2D to the 3D limit. Stacking layers is an effective way to invert the gap, an alternative to controlling the strength of spin-orbit coupling. The emerging quantum spin Hall insulator phase is not restricted to the case of odd numbers of layers.

We show how the two-dimensional (2D) topological insulator evolves, by stacking, into a strong or weak topological insulator with different topological indices, proposing a new conjecture that goes beyond an intuitive picture of the crossover from quantum spin Hall to weak topological insulator. Studying the conductance under different boundary conditions, we demonstrate the existence of two conduction regimes in which conduction happens through either surface or edge conduction channels. We show that the two conduction regimes are complementary and exclusive. Conductance maps in the presence and absence of disorder are introduced, together with 2D Z(2)-index maps, describing the dimensional crossover of the conductance from the 2D to the 3D limit. Stacking layers is an effective way to invert the gap, an alternative to controlling the strength of spin-orbit coupling. The emerging quantum spin Hall insulator phase is not restricted to the case of odd numbers of layers.

We numerically demonstrate a practical means of systematically controlling topological transport on the surface of a three-dimensional topological insulator, by introducing strong disorder in a layer of depth d extending inward from the surface of the topological insulator. The dependence on d of the density of states, conductance, scattering time, scattering length, diffusion constant, and mean Fermi velocity are investigated. The proposed control via disorder depth d requires that the disorder strength be near the large value which is necessary to drive the topological insulator into the nontopological phase. If d is patterned using masks, gates, ion implantation, etc., then integrated circuits may be fabricated. This technique will be useful for experiments and for device engineering.