几种拓扑声子态的理论计算研究

拓扑材料由于其奇特的物理性质以及在无耗散功能器件上的潜在应用，受到了广泛的关注。最先受到关注的是基于量子霍尔效应的拓扑绝缘体材料，其不同于朗道对称破缺的物态分类机制掀起了凝聚态物理研究的热潮。随着拓扑理论的发展，对拓扑半金属、拓扑超导等新的拓扑物态的研究使得拓扑理论更加的完善。人们认识到这些拓扑物态的存在与时间反演对称性、粒子空穴对称性、晶体结构对称性等等息息相关，并据此建立了丰富的拓扑分类方式。根据拓扑分类和拓扑指标来搜索拓扑材料逐渐成为拓扑材料领域的一个重要方向。过去十几年里，电子拓扑材料已经得到了大量的理论证明和实验验证。而声子，作为另一种重要的元激发，对热输运和声信号等等有着重要的作用，它的拓扑声子物态和对应的拓扑声子材料的研究还处于一个比较基础的阶段。在本学位论文中，我们主要研究了声子系统中几种新奇的拓扑物态以及实际材料中的拓扑性质。具体研究内容如下：
一、根据对称分析，我们研究了230 个空间群中所有可能存在的直线型节线声子。由于极高的对称性要求，这些声子节线都被束缚在布里渊区的高对称线上。通过计算围绕节线的闭合环路的贝利相位，我们将节线分成了两种类型：具有非平庸𝜋 相位的线性色散声子节线和具有平庸的2𝜋 相位的二次色散声子节线，丰富和完善了拓扑分类。线性色散声子节线材料具有非平庸的拓扑表面态，可以作为实验观测的候选对象。二次色散声子节线是声子系统中具有最高阶色散的直线型声子节线，尽管它没有对应的拓扑表面态，但是可以通过对称破缺来实现拓扑非平庸的声子态，是研究拓扑相变的理想平台。我们的工作还给出了所有节线的动量位置和对称要求，对拓扑材料的搜索提供了重要的指导作用。
二、我们研究了高对称性点和高对称性线上charge-2 的狄拉克声子和外尔声子共存的情况，通过对称分析和筛选，我们确定了5 个候选空间群19、90、92、94和96。在这一过程中，我们还发现charge-2 的狄拉克声子不仅源于高对称点上的四维不可约表示，还可源于高对称线上两个二维不可约表示的偶然简并。不同于传统的成对存在的手性准粒子，我们发现charge-2 的狄拉克声子和两个陈数相反的外尔点共存。结合这些准粒子的动量位置，我们考虑了在投影面上可能出现的表面弧形貌。在Na₂Zn₂O₃ 的投影面上，我们发现一个charge-2 狄拉克点投影连接着两个外尔点的投影，形成极长的双表面弧，这与我们的理论预测相一致。我们的工作加深了对拓扑准粒子共存态的理解，并为实现极长的拓扑表面弧提供了一个新的思路。
三、近几年来，晶体固体中具有较高拓扑电荷的非常规外尔点越来越受到人们的广泛关注。在这些高陈数的外尔点中，陈数为±3 的三重外尔点一直没有材料的实现。我们的工作深入研究了受到六重螺旋对称性保护的三重外尔声子，在这个基础上，我们发现了受到6₃螺旋旋转对称性强制保护的沙漏型三重外尔声子。我们的工作证明了沙漏型三重外尔声子只存在于手性空间群173 和182。以173 号空间群的LiIO₃ 为例，我们深入研究了沙漏型三重外尔声子的拓扑性质，发现它在𝑘_𝑧 方向具有线性的色散关系，在𝑘_𝑥-𝑘_𝑦 平面上具有二次色散关系。在(001) 表面上，我们发现两个沙漏型外尔声子的投影叠加在一起，连接着六个等价的单外尔声子的投影，因此，(001) 表面的六螺旋表面态是沙漏型三重外尔声子的一个重要拓扑特征。这个工作提出了一类崭新的对称性强制保护的拓扑准粒子，为三重外尔声子的实验观测提供了一个理想的平台。

Topological materials have attracted extensive attention due to their unique physical properties and potential applications in dissipationless functional devices. Topological insulators based on quantum Hall effect have first attracted attention, whose classification mechanism differs from the Landau symmetry-breaking theory and sparked a wave of research in condensed matter physics. With the development of topology theory, other topological states, such as topological semimetals and topological superconductivity, have come to light, leading to a more comprehensive understanding of these systems. It is recognized that the existence of these topological states of matter is related to time-reversal symmetry, particle-hole symmetry, crystal structure symmetry, etc., and a rich topological classification method has been established accordingly. Searching for topological materials based on topological classification and topological indices has become an important direction in the field of topological materials. Over the past decade, electronic topological materials have received significant theoretical and experimental verification. Phonons, as another important elementary excitation, play a crucial role in heat transport and acoustic signals, and research on their topological phonon states and corresponding topological phonon materials is still in a relatively early stage. In this thesis, we focus on studying several novel topological states in phonon systems and their topological properties in realistic materials. The specific research contents are as follows:

1. According to the symmetry analysis, we study all possible linear nodal line phonons in 230 space groups. Due to the extremely high symmetry requirements, these phonon nodal lines are bound to the high symmetry lines of the Brillouin zone. By calculating the Berry phase around the nodal lines, we have classified them into two types: the linear nodal line with a non-trivial 𝜋 phase and the quadratic nodal line phonons with a trivial 2𝜋 phase, which enriching and refining the topological classification. Linear nodal line phonon materials with nontrivial topological surface states are candidates for experimental observations. The quadratic nodal line is the straight line nodal line with the highest order dispersion in the phonon system. Although it has no corresponding topological surface state, it can realize the topologically non-trivial phonon state through symmetry breaking, which is an ideal platform for studying topological phase transitions. Our work also provides important guidance for searching for topological materials by giving the momentum positions and symmetry requirements of all the nodal lines.

2. We investigate the coexistence of charge-2 Dirac phonons and Weyl phonons at high symmetry points and high symmetry lines. Through symmetry analysis and screening, we identify five candidate space groups 19, 90, 92, 94 and 96. In the process, we also find that the charge-2 Dirac phonon originates not only from the four-dimensional irreducible representation on the high symmetry point, but also from the accidental degeneracy of two two-dimensional irreducible representations on the high symmetry line. Unlike traditional chiral quasiparticles that exist in pairs, we find that the charge-2 Dirac phonon coexists with two Weyl phonons with opposite Chern numbers. Combined with the momentum positions of these quasiparticles, we consider the possible surface Fermi arc on the projection plane. On the projection plane of Na₂Zn₂O₃, we find that the projection of a charge-2 Dirac phonon connects the projections of two Weyl phonons, forming an extremely long double surface arc, which is consistent with theoretical predictions. Our work deepens the understanding of the coexistence of topological quasiparticles and provides a new avenue for realizing extremely long topological surface arcs.

3. In recent years, unconventional Weyl points with higher topological charges in crystalline solids have attracted more and more attention. Among these Weyl points, the triple Weyl point with a Chern number of ±3 has not been realized materially. Our work has deeply studied the triplet Weyl phonon protected by the sixfold screw rotation symmetry. Based on this, we discover the hourglass triple Weyl phonon protected by the screw rotation symmetry of 6₃. Our work proves that the hourglass triple Weyl phonon exists only in chiral space groups 173 and 182. Taking the LiIO₃ of space group 173 as an example, we deeply study the topological properties of the hourglass triple Weyl phonon, and find that it has linear dispersion relation in the 𝑘_𝑧 direction, and quadratic dispersion relation in the 𝑘_𝑥-𝑘_𝑦 plane. On the (001) surface, we find that the projections of two hourglass triple Weyl phonons are superimposed together, connecting the projections of six equivalent single Weyl phonons. Therefore, the six-helical surface states of the (001) surface is an important topological feature of the hourglass triple Weyl phonons. This work proposes a new class of symmetry-enforced topological quasiparticles, which provides a perfect platform for the experimental observation of triple Weyl phonons.

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