根据布洛赫定理的基本假设，当费米能级处于导带或者价带中时，电子波函数在有序晶体中是扩展的，这意味着在整个晶体中找到一个电子的概率是相同的，这种扩展态具有金属性。一旦在有序晶体中引入无序杂质，晶格的周期性将会被破坏，导致电子波函数将呈指数衰减。人们认识到在无序系统中，传统的能带理论失效了，取而代之的是安德森局域化理论。随着无序的增强，平均自由程 l 变短，电导率下降，电子的波函数呈现指数衰减趋势。电子将被局域在杂质的周围，体现出绝缘体的性质。系统将经历一个金属-绝缘体转变，这种由无序导致的局域化过程，称为安德森局域化。对于三维系统，无序杂质的引入会导致局域化－非局域化相变，并且存在一个阈值，临界杂质浓度Wc  = 16.5，也称为迁移率边。当杂质浓度低于这个阈值时，即 W < Wc，电子将以扩展态的形式存在，系统表现出金属性，而当杂质浓度高于这个阈值时，即 W > Wc，电子将以局域态的形式存在，系统表现为绝缘体。进一步地，根据有限尺寸标度分析的观点，在一维和二维无序非关联无序的电子系统中，任意小浓度的杂质都会导致体系中的电子局限在一个小尺度的范围内。然而，有研究者指出在一维系统存在长程关联的无序时，也可能会出现扩展态。这要取决于无序强度和长程关联之间的竞争关系。例如，如果存在长程关联的无序杂质，在一维系统的能带中心就可以发现一个连续的扩展态能谱。
近年来，机器学习领域以及处理器GPU的迅猛发展，为我们带来了极大的便利。机器学习在许多不同的领域中都很流行，不仅仅是图像识别和语音转录，同时在协助学术研究方面也很高效。除了能够分析物理学和天体物理学中高能量下的实验数据，还可以用来识别物质的相变。我们利用机器学习技术研究了低维无序系统中的量子相变问题。相比传统计算方法，机器学习的优势在于：只需要输入最原始的波函数，通过神经网络的学习和训练，并不需要先验的物理学的知识就可以得到清晰的相图。本文就是利用机器学习的优势帮助人们识别人眼难以识别的扩展态——局域态相变图，只需要输入格点波函数的信息，就可以输出可靠的判断。
本文总共有四章。在第一章介绍了无序系统的基本理论知识，包括安德森模型、局域化理论等。在第二章介绍了机器学习的基础理论，包括机器学习的回归分类算法以及神经网络中的全连接网络和卷积神经网络。第三章将机器学习应用到低维无序系统中，包括一维的单链模型和双链模型并得到了和传统理论计算类似的相图。我们还发现这两种模型的W - p 相图在总体趋势上是一致的。唯一不同的是，在双链模型中链间短程关联耦合的存在，它为电子提供了一个额外的通道，导致传输特性上略有不同。对于单链模型的扩展状态只存在于中心能带，而边缘只存在于局域状态。而对于双链模型，随着链间跳跃积分的增加，扩展状态不仅存在于能带中心，而且在能带边缘也存在大量的扩展状态。我们还研究了二维电子气存在自旋轨道耦合时的安德森转变，我们同样得到了近似的相图和临界杂质浓度。在第四章中我们做了简要的总结，并对接下来的研究工作做了展望。

According to the basic assumption of Bloch’s theorem, when the Fermi energy is in the conduction band or valence band, the electron wave function is extended in the ordered crystal, which means that the probability of finding an electron in the whole crystal is the same. Once disordered impurities are introduced into the ordered crystal, the periodicity of the lattice will be destroyed, resulting in the exponential decay of the electron wave function. It was recognized that in disordered systems, the traditional band theory failed and was replaced by Anderson’s localized theory. With the increase of disorder, the average free path l becomes shorter, the conductivity decreases, and the electron’s wave function declines exponentially. The electrons will be localized around the impurity, reflecting the nature of the insulator. The system will undergo a metal-insulator transition, a process of localization caused by disorder known as Anderson localization. For a three-dimensional system, the introduction of disordered impurities leads to localized-delocalized phase transitions, and there is a threshold value, the critical impurity concentration Wc  = 16.5, also known as the mobility edge. When the impurity concentration is lower than this threshold value, that is, W < Wc, the electrons will exist in the form of an extended state and the system will exhibit the metallic property. When the impurity intensity is higher than this threshold value, that is, W > Wc, the electrons will exist in the form of a localized state and the system willbehave as an insulator. Furthermore, from the point of view of finite size scale analysis, in one and two-dimensional disordered electronic systems, any uncorrelated disorder of strength causes all electronic states to be confined to a small scale. However, some researchers point out that the extended state may also occur in one-dimensional systems when there is a disorder of long range correlation. This depends on the competitive relationship between the disorder intensity and the long range association. For example, if there are disordered impurities with long range correlation, a continuous extended state energy spectrum can be found in the energy band center of a one-dimensional system. 

In recent years, the rapid development of machine learning and GPU has brought us greatconvenience. Machine learning is popular in many different fields, not only in image recognition and speech transcription, but also in assisting academic research. In addition to being able to analyze experimental data at high energies in physics and astrophysics, it can also be used to identify phase transitions in matter. We study the quantum phase transition in low-dimensional disordered systems using machine learning techniques. Compared with traditional calculation methods, machine learning has the advantage of only inputting the most primitive wave function, learning and training through neural network, and obtaining clear phase diagram without prior knowledge of physics. In this paper, the advantage of machine learning is used to help people to recognize the local-delocalization transition, which is difficult to be recognized by human eyes. Reliable judgment can be output only by inputting the information of lattice point wave function.
There are four chapters in this paper. In the first chapter, the basic theoretical knowledge of disordered system is introduced, including Anderson model and localization theory. In the second chapter, the basic theory of machine learning is introduced, including the regression classification algorithm of machine learning, the fully connected network and the convolutional neural network. In chapter 3, machine learning is applied to low-dimensional disordered systems, including onedimensional single-chain model and double-chain model. We find that the w - p phase diagram in these two models is consistent in the overall trend. The only difference is the existence of short-range association coupling between chains in the double-chain model, which provides an additional channel for electrons, resulting in slightly different transport characteristics. For the extended state of the single-chain model, only the central band exists, and the edge only exists in the localized state. For the double-chain model, with the increase of inter-chain jump integral, the extended states exist not only in the band center, but also in the band edge. We also study the Anderson transition of two-dimensional electron gas with spin orbit coupling. We also obtain approximate phase diagram and critical impurity concentration. In the fourth chapter, we made a brief summary and made a prospect for the following research work.