一维系统中的Majorana费米子和分数费米子

发布时间：2018-11-25 12:03

【摘要】：在低能级性质可以近似用Dirac电子来描述的系统中,Majorana费米子以及分数费米子会以零模束缚态的形式出现在拓扑非平庸区域与平庸区域的边界。其根本原因是在边界附近与Dirac旋量耦合的标量场呈现出了扭结型的分布。该扭结型的标量场会打开系统的能隙,并且它在拓扑上不同于均匀分布的标量场。Majorana费米子和分数费米子正是在这种拓扑非平庸的标量场下出现的。我们在第一章介绍了几个相关的比较著名的模型,同时也为之后几章提供了部分理论支撑。在第二章,我们研究了一个一维Rashba系统,发现一般形式的周期性磁场可以在其边界诱导出Majorana费米子和分数费米子。在该系统中Majorana费米子依赖于邻近效应引入的超导配对势,而分数费米子的出现则不需要。与均匀磁场不同的是,周期性磁场可以在系统的能谱中诱导出很多的能隙。当化学势穿过其中任一能隙之中时,超导配对势的引入便可以使得Majorana费米子出现在系统边界。分数费米子则出现在两个能隙的交叠部分,并且要求磁场的周期必须是Rashba自旋轨道耦合长度的整数倍。两种零模束缚态的出现对磁场的具体形式则没有特定的要求。第三章中,我们研究了一个由Majorana费米子组成的一维单链格点系统,发现当系统的相互作用足够强时会有tricritical Ising (TCI)相变发生。该类型的相变处于Ising相变与一级相变的交点处,且会表现出超对称的性质。我们利用密度矩阵重整化群数值方法对该系统进行了详细的研究,并且得到了TCI相变点处系统的参数。我们发现该临界点处有很强的相互作用,一般的系统很难达到这种强度,但我们发现在Majorana格点系统中,原则上只需调节化学势便可以任意改变相互作用与跃迁项的比值,从而使其到达TCI相变点附近。第四章中我们同样考虑了Majorana格点系统,只是这次是一个梯子形状的模型,它可以在拓扑绝缘体表面实现。与上一章类似,我们通过详尽的解析与数值分析发现了TCI相变点,它的两侧分别是Ising相变与一级相变。该模型的优势在于它不要求系统具有很强的相互作用。最后我们讨论了实验上如何调节相关参量使得系统到达TCI相变点附近,以及超对称所表现出的特征。最后一章我们简单总结了本文的主要发现并且对未来可能的研究方向进行了简短的讨论。
[Abstract]:In a system where the properties of low energy levels can be approximately described by Dirac electrons, Majorana fermions and fractional fermions appear in the form of zero-mode bound states at the boundary between the topological non-mediocre region and the mediocre region. The fundamental reason is that the scalar field coupled with Dirac spinor near the boundary presents a kink distribution. The scalar field of the kink type will open the energy gap of the system, and it is different from the scalar field of uniform distribution in topology. The Majorana fermion and fractional fermion appear under the nonmediocre scalar field of this topology. In the first chapter, we introduce several famous models and provide some theoretical support for the later chapters. In chapter 2, we study a one-dimensional Rashba system and find that the general periodic magnetic field can induce Majorana fermion and fractional fermion at its boundary. In this system, the Majorana fermion depends on the superconducting pairing potential introduced by the proximity effect, but the presence of fractional fermion is not necessary. Different from the uniform magnetic field, the periodic magnetic field can induce a lot of energy gaps in the energy spectrum of the system. When the chemical potential passes through any of the energy gaps, the introduction of the superconducting pairing potential makes the Majorana fermion appear at the system boundary. The fractional fermion appears in the overlapping part of the two energy gaps, and the period of the magnetic field must be an integer multiple of the coupling length of the Rashba spin orbit. The appearance of the two zero-mode bound states has no specific requirements for the specific form of magnetic field. In chapter 3, we study a one-dimensional single-chain lattice system composed of Majorana fermions. It is found that the tricritical Ising (TCI) phase transition occurs when the interaction of the system is strong enough. This type of phase transition is located at the intersection of the Ising phase transition and the first order phase transition and shows supersymmetry. We use the density matrix renormalization group method to study the system in detail and obtain the system parameters at the TCI phase transition point. We find that there is a strong interaction at this critical point, and it is difficult for a general system to achieve this intensity. However, we find that in a Majorana lattice system, in principle, the ratio of interaction to transition term can be arbitrarily changed by simply adjusting the chemical potential. So that it can reach the TCI phase transition point. In Chapter 4 we also consider the Majorana lattice system but this time it is a ladder shaped model which can be implemented on the surface of a topological insulator. Similar to the previous chapter, we find out the TCI phase transition point through detailed analytical and numerical analysis, the two sides of which are the Ising phase transition and the first order phase transition, respectively. The advantage of the model is that it does not require the system to have strong interaction. Finally, we discuss how to adjust the relevant parameters to make the system reach the TCI phase transition point and the characteristics of supersymmetry. In the last chapter, we briefly summarize the main findings of this paper and briefly discuss the possible research directions in the future.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O572.2

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