自旋-轨道耦合材料新奇物性的对称性判定及设计理论

自旋-轨道耦合作用是现代凝聚态物理诸多重要现象的核心，与自旋电子学、低维磁性物态和拓扑物态等前沿领域密切相关。近年来的研究发现自旋-轨道耦合可以诱导不同拓扑物态间的相变，从而产生自旋-轨道耦合驱动的独特输运效应，如陈绝缘体具有量子反常霍尔效应，拓扑绝缘体表现出量子自旋霍尔效应等。对自旋-轨道耦合新奇物相的判定及其独特输运效应的设计是将其应用至自旋电子学、谷电子学等领域中的关键。
正如“凝聚态物理之父”安德森所言：“稍微夸张地说，研究物理就是研究对称性”，群论和对称性分析是处理各种物理问题的有力工具。近年来基于对称性指标等理论系统预言逾8000种拓扑材料，是使用对称性设计和判定新奇拓扑材料的经典成功案例。为高效设计自旋-轨道耦合功能材料，亟需发展新的对称性理论来设计具有稳健量子化输运效应或显著反常输运效应材料，以克服以往根据原子序数设计得到的自旋-轨道耦合材料普遍不稳定的缺点，拓宽可用材料库。本文主要研究自旋-轨道耦合驱动下拓扑物态的对称性判定理论以及具有相关输运效应材料的对称性设计理论，具体将从以下方面展开研究。
探究了对称性保护的轨道简并态可以产生一阶自旋-轨道耦合，提出了部分占据的轨道多重态下电子关联增强自旋-轨道耦合效应的机制，明确轨道极化量是该增强效应的核心。基于以上机制，提出了具有强自旋-轨道耦合效应轻元素材料的设计准则，并在二维材料数据库中筛选得到71 种轻元素候选材料，并系统得到9种高温陈绝缘体材料。以单层Fe₂S₂为例，结合第一性原理计算和紧束缚模型计算定量解释了其628 meV的拓扑带隙。
建立了准对称性与准对称群理论，其中准对称性定义为禁戒一阶微扰效应的隐藏对称性，提供了存在准晶体对称性禁戒一阶微扰效应的晶体点群。基于以上理论，提出准对称性禁戒的轨道角动量矩阵元是设计自旋-轨道耦合导致二阶简并劈裂的关键。以四方AgLa为例，基于第一性原理计算，展示了自旋-轨道耦合驱动的狄拉克节线半金属到准狄拉克半金属相变，证明了准狄拉克点的微小自旋-轨道耦合劈裂受准空间反演对称性的保护。
通过建立本征磁性拓扑绝缘体MnBi₂Te₄/(Bi₂Te₃)_n家族的统一超晶格模型，构建体系在调节磁性交换作用或等效自旋-轨道耦合作用以及层间距下的拓扑相图，使用空间反演对称性和晶体平移对称性联合保护的对称性指标完整刻画了铁磁轴子绝缘体相、三维陈绝缘体相、三维量子自旋霍尔绝缘体相、外尔半金属相之间的相变行为，指出轴子绝缘体相和三维量子自旋霍尔绝缘相具有相同的Z₄不变量，需要利用晶格平移对称性保护的Z₂不变量进行区分。基于第一性原理计算，指出轴子绝缘体相具有表面半整数霍尔电导，并表明手性转角态是铁磁轴子绝缘体相的指纹信号。

Spin-orbit coupling (SOC) is at the core of many important phenomena in modern condensed matter physics, closely related to frontier areas such as spintronics, low-dimensional magnets, and topological states. Recent research has found that SOC can induce phase transitions between different topological states, leading to unique transport effects driven by SOC, such as the quantum anomalous Hall effect in Chern insulators and the quantum spin Hall effect in topological insulators. Diagnosing novel topological phases and designing their unique transport effects are key to applying them in spintronics, valleytronics, and other subfields. 

"It is only slightly overstating the case to say that physics is the study of symmetry.", said by Philip Anderson. Group theory and symmetry analysis are powerful tools for addressing various physical problems. Recently, over 8000 topological materials have been systematically predicted based on symmetry theories like symmetry-based indicator, serving as a huge success of symmetry diagnosis of topological states. For the efficient design of SOC-driven functional materials, there is an urgent need to develop new symmetry theories to design materials with robust quantized transport effects or significant anomalous transport effects. The new symmetry-based theories are expected to overcome the common instability issue of functional materials designed solely based on atomic numbers, and to expand the pool of candidate materials. Therefore, this thesis primarily investigates the symmetry diagnosis of topological states driven by SOC and the symmetry-based design principles of materials with relevant transport effects. Specifically, it will address the following aspects:

As the symmetry-protected orbital multiplets can generate first-order SOC, a mechanism for enhancing the SOC effect through electronic correlations in partially occupied orbital multiplets is proposed, emphasizing that orbital polarization is the core of this enhancement. Based on this mechanism, design principles for materials with strong SOC effects in light elements are proposed, resulting in the screening of 71 candidate materials and 9 high-temperature Chern insulators from two-dimensional materials database. Taking monolayer Fe₂S₂ as an example, a quantitative explanation of its large topological band gap of 628 meV is provided through the first-principles calculations and tight-binding model calculations.

A generic theory on quasi-symmetry, defined as the hidden symmetry that eliminates the first-order perturbation, and quasi-symmetry group is established, providing all the possible crystalline symmetry as quasi-symmetry in crystallographic point groups. Based on this theory, we proposed that the vanish of orbital angular momentum matrix element, ensured by quasi-symmetry, is crucial for designing the tiny energy splitting induced by the second-order SOC effect. Taking tetragonal AgLa as an example and applying first-principles calculations, an SOC-driven transition from a Dirac nodal-line semimetal to a quasi-Dirac semimetal is demonstrated, proving that the tiny SOC splitting of quasi-Dirac points is protected by quasi-inversion symmetry. 

By establishing a unified superlattice model for intrinsic magnetic topological insulators MnBi₂Te₄/(Bi₂Te₃)_n family, a topological phase diagram via tuning the magnetic exchange or SOC and interlayer distance is predicted. Meanwhile, the complete characterization of the ferromagnetic axion insulating phase, the three-dimensional Chern insulating phase, the three-dimensional quantum spin Hall (3DQSH) insulating phase, and the Weyl semimetal phase is provided, based on the symmetry indicators and invariants protected by inversion symmetry and crystal translational symmetry. It is pointed out that the axion insulating phase and the 3DQSH insulating phase share the same Z₄ invariant, requiring the extra Z₂ invariants protected by lattice translation symmetry for distinction. Based on first-principles calculations, it is indicated that the axion insulating phase exhibits surface half-quantized Hall conductivity, and it is demonstrated that the chiral hinge states are the representations of the ferromagnetic axion insulator phase. 

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