Barry Bradlyn, Mikel Iraola
In these notes, we review the role of Berry phases and topology in noninteracting electron systems. Topics including the adiabatic theorem, parallel transport, and Wannier functions are reviewed, with a focus on the connection to topological insulators.
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Berry phase effects in magnetism
讲座笔记发表在“磁性纳米化”，固体研究所第36届春季学校讲座手册，由Stefan Bluegel，Thomas Brueckel和Claus M. Schneider（Forschungszentrum Juelich，2005）.
Lecture notes used in a graduate-level Introduction to Solid State Physics course at Cornell University, to serve as a supplement to textbooks at the level of Ashcroft & Mermin.
Tutorial: From Topology to Hall Effects -- Implications of Berry Phase Physics
本教程全面介绍Berry相位的数学框架及其与量子系统拓扑性质的联系，深入探讨量子霍尔效应等拓扑效应及拓扑绝缘体的输运性质，旨在帮助读者掌握Berry相位在拓扑量子现象中的核心作用。
Berry phase, topology, and diabolicity in quantum nano-magnets
本文提出了量子磁体魔点（简并）的拓扑理论。用恶魔指数刻画恶魔点，导出了恶魔点的拓扑求和规则。澄清了Fe_8分子磁体的魔点缺失悖论。一种新的方法也被开发，以提供一个简单的解释，在相消干涉由于Berry相位，在双轴系统，如Fe 8中发现的恶魔点的完整集合。
These lecture notes cover undergraduate textbook topics (e.g. as in Sakurai), and also additional advanced topics at the same level of presentation. In particular: EPR and Bell; Basic postulates; The probability matrix; Measurement theory; Entanglement; Quantum computation; Wigner-Weyl formalism; The adiabatic picture; Berry phase; Linear response theory; Kubo formula; Modern approach to scattering theory with mesoscopic orientation; Theory of the resolvent and the Green function; Gauge and Galilei Symmetries; Motion in magnetic field; Quantum Hall effect; Quantization of the electromagnetic f…
Electron motion in crystals is governed by the coupling between crystal momentum and internal degrees of freedom such as spin implicit in the band structure. The description of this coupling in terms of a momentum-dependent effective field and the resultant Berry phase has greatly advanced our understanding of diverse phenomena including various Hall effects and has lead to the discovery of new states of matter exemplified by topological insulators. While experimental studies on topological systems have focused on the gapless states that emerge at the surfaces or edges, the underlying nontrivi…
Lecture Notes: The two-dimensional electron Wigner crystal -- What's old and what's new?
该讲座笔记以对话式风格，在近期实验背景下介绍回顾二维电子维格纳晶体的新旧观点，着重讲述其半经典描述、量子熔化转变、自旋序及受贝里曲率影响的方式。
Emergent Topology in Kagome Ferromagnets
研究二维kagome铁磁体中拓扑磁振子相，通过引入手性相互作用项，利用动量空间表示和线性自旋波理论，探讨DMI与拓扑轨道耦合的相互作用，揭示其对拓扑性质的影响，建立晶格几何、手性和磁振子拓扑的对应关系。
How pairing mechanism dictates topology in valley-polarized superconductors with Berry curvature
研究配对机制对具有Berry曲率的谷极化系统拓扑超导性的影响，发现短程吸引与过屏蔽排斥相互作用分别使BdG Chern数与Berry曲率符号相同或相反，为区分配对机制提供了实验判据。
We investigate both first-order topology, as realized through Haldane's model, and second-order topology, implemented through an additional Kekulé-distortion, on the honeycomb lattice. The interplay and competition of both terms result in a phase diagram at half-filling which contains twelve distinct phases. All phases can be characterized by the first Chern number or by a quantized $\mathbb{Z}_Q$ Berry phase. Highlights include phases with high Chern numbers, a novel $\mathbb{Z}_6$ topological phase, but also coupled kagome-lattice Chern insulators. Furthermore, we explore the insulating phas…
Recently, optical lattices with non-zero Berry's phases of Bloch bands have been realized. New approaches for measuring Berry's phases and topological properties of bands with experimental tools appropriate for ultracold atoms need to be developed. In this paper, we propose an interferometric method for measuring Berry's phases of two dimensional Bloch bands. The key idea is to use a combination of Ramsey interference and Bloch oscillations to measure Zak phases, i.e. Berry's phases for closed trajectories corresponding to reciprocal lattice vectors. We demonstrate that this technique can be u…
The smooth topology change of Berry's phase from a Dirac monopole-like configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection ${\cal A}_{k}$ is that the geometrical center of the monopole-like configuration and the origin of the Dirac string are displaced in the parameter space. Gauss' theorem $\int_{S}(\nabla\times {\cal A})\cdot d\vec{S}=\int_{V} \nabla\cdot (\nabla\times {\cal A}) dV=0$ for a volume $V$ which is free of singularities shows that a combination of …
Berry Phase and Topology in Ultrastrongly Coupled Quantum Light-Matter Systems
本文提出一个理论框架，用于分析二维材料超强耦合到腔电磁场的量子几何与拓扑，并在深度强耦合区域发现向拓扑平凡相的再入转变，通过幺正映射与 Haldane 模型建立联系。
Tunable Topology and Berry Curvature Dipole in Transition Metal Dichalcogenide Janus Monolayers
本文通过第一性原理与紧束缚计算研究 WTe₂ 衍生 Janus 单层，揭示其拓扑平庸却具有巨大 Berry 曲率偶极矩，并用低能有质量 Dirac 模型解释其非线性霍尔效应潜力。
This lecture note adresses the correspondence between spectral flows, often associated to unidirectional modes, and Chern numbers associated to degeneracy points. The notions of topological indices (Chern numbers, analytical indices) are introduced for non specialists with a wave physics or condensed matter background. The correspondence is detailed with several examples, including the Dirac equations in two dimensions, Weyl fermions in three dimensions, the shallow water model and other generalizations.
Tunable moiré materials for probing Berry physics and topology
本综述介绍可调谐莫尔材料作为研究贝里物理与拓扑的通用平台，梳理其能带调控优势、研究工具和最新实验突破，指出当前挑战并展望未来研究方向。
We consider the extent to which symmetry eigenvalues reveal the topological character of bands. Specifically, we compare distinct atomic limit phases (band representations) that share the same irreducible representations (irreps) at all points in the Brillouin zone and, therefore, appear equivalent in a classification based on eigenvalues. We derive examples where such "irrep-equivalent" phases can be distinguished by a quantized Berry phase or generalization thereof. These examples constitute a generalization of the Su-Schrieffer-Heeger chain: neither phase is topological, in the sense that l…
Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a $π$-flux of Berry phase. Nodal-line rings in $\mathcal{PT}$-symmetric systems with negligible spin-orbit coupling (here described as "nodal class AI") can carry an additional "monopole charge", which further enhances their stability. Here, we relate two recent theoretical advancements in the study of band topology in nodal class AI. On the one hand, cohomology classes of real vector bundles were used to relate the monopole charge of nodal-line rings to their linking with nodal lines …
A Group-Theoretical Framework for Local k-Space Topology and Berry Phase in 2D Photonic Systems
本文构建了融入光子系统关键特征的群论框架，研究二维光子系统中局域k空间拓扑与贝里相位的对称性起源，分析结构设计与对称性扰动对拓扑特征的操控，相关分析可扩展至多类人工波晶体。
Relative hybridization textures as local coordinates for band geometry and topology
研究提出用相对杂化坐标\(Z\)连接能带全局量与局部自由度，它能重构局部投影算符。在有效坐标片上其纹理编码多种物理量，坐标片障碍对应秩降缺陷。通过QWZ和晶格BHZ模型验证，为能带几何拓扑与微观结构联系提供扇区分辨框架。
Dirac zeros in an orbital selective Mott phase: Green's function Berry curvature and flux quantization
本文在轨道选择性莫特相中证明对称性保护的格林函数零点交叉具有量子化贝里磁通，提出狄拉克零点概念，为诊断强关联金属体系中的隐藏拓扑提供了新途径。
Topology ultimately unveils the roots of the perfect quantization observed in complex systems. The 2D quantum Hall effect is the celebrated archetype. Remarkably, topology can manifest itself even in higher-dimensional spaces in which control parameters play the role of extra, synthetic dimensions. However, so far, a very limited number of implementations of higher-dimensional topological systems have been proposed, a notable example being the so-called 4D quantum Hall effect. Here we show that mesoscopic superconducting systems can implement higher-dimensional topology and represent a formida…
We study the weak antilocalization (WAL) effect in the magnetoresistance of narrow HgTe wires fabricated in quantum wells (QWs) with normal and inverted band ordering. Measurements at different gate voltages indicate that the WAL is only weakly affected by Rashba spin-orbit splitting and persists when the Rashba splitting is about zero. The WAL signal in wires with normal band ordering is an order of magnitude smaller than for inverted ones. These observations are attributed to a Dirac-like topology of the energy bands in HgTe QWs. From the magnetic-field and temperature dependencies we extrac…
The Hodge structure of Berry-phase transport: topology, geometry, and noise
该研究通过霍奇-德拉姆分解将贝里曲率扇区与输运、涨落关联，证明电流噪声谱可双向分离全局带拓扑与局域带几何。
We outline an approach to endow a plain vanilla material with topological properties by creating topological bands in stacks of manifestly nontopological atomically thin materials. The approach is illustrated with a model system comprised of graphene stacked atop hexagonal-boron-nitride. In this case, the Berry curvature of the electron Bloch bands is highly sensitive to the stacking configuration. As a result, electron topology can be controlled by crystal axes alignment, granting a practical route to designer topological materials. Berry curvature manifests itself in transport via the valley…
Quantum oscillations can be used to determine properties of the Fermi surface of metals by varying the magnitude and orientation of an external magnetic field. Topological insulator surface states are an unusual mix of normal and Dirac fermions. Unlike in graphene and simple metals, Berry's geometric phase in topological insulator surface states is not necessarily quantised. We show that reliably extracting this geometric phase from the phase offset associated with the quantum oscillations is subtle. This is especially so in the presence of a Dirac gap such as that associated with the Zeeman s…
Phonons as collective excitations of lattice vibrations are the main heat carriers in solids. Tremendous effort has been devoted to investigate phonons and related properties, giving rise to an intriguing field of phononics, which is of great importance to many practical applications, including heat dissipation, thermal barrier coating, thermoelectrics and thermal control devices. Meanwhile, the research of topology-related physics, awarded the 2016 Nobel Prize in Physics, has led to discoveries of various exotic quantum states of matter, including the quantum (anomalous/spin) Hall [Q(A/S)H] e…
We examine excitons formed in the bulk of a topological insulator as the system is tuned via a parameter between topological and trivial insulating phases, arguing that nontrivial topology has fingerprints in the spectrum of these excitons. The closely related hydrogen atom problem is well known to have a degeneracy due to a hidden symmetry, and the changes to the excitonic spectrum that we find can be understood as a result of breaking of this underlying symmetry due to the Berry phase. Furthermore, this phase is found to affect the spectrum in the topological parameter regime much more stron…
Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and by the absence of non-Hermitian skin effects. Here, we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach we present and study the pseudo-Hermitian version of well known Hermitian topological phases, raging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to …
Quantization of intra- and inter-band Berry phases in the shift current
本文提出一类仅与非中心对称空间群相容的新型拓扑绝缘体，证明其位移矢量被量子化为Bravais晶格矢量的有理倍数，并揭示带内与带间Berry相的量子化相互作用可显著增强位移电流，电导率远超传统铁电材料。
Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
本文在二维非互易跳跃Su-Schrieffer-Heeger模型中实现了边缘态与皮肤效应共存的角落态，通过$\mathbb{Z}_4$ Berry相位和绕数表征了拓扑不变量，并确定了拓扑保护局域态的位置。
Topology in the space-time scaling limit of quantum dynamics
本研究探究量子猝灭动力学时空标度极限下的拓扑效应，提出基于动力学缠绕数的拓扑表征，发现贝里相的阶梯行为可揭示局域不可见磁通量，并给出有限系统的实验观测方案。
An effective Hamiltonian describing the surface states of a toroidal topological insulator is obtained, and it is shown to support both bound-states and charged zero-modes. Actually, the spin connection induced by the toroidal curvature can be viewed as an position-dependent effective vector potential, which ultimately yields the zero-modes whose wave-functions harmonically oscillate around the toroidal surface. In addition, two distinct Berry phases are predicted to take place by the virtue of the toroidal topology.
Berry phase and Berry curvature play a key role in the development of topology in physics and do contribute to the transport properties in solid state systems. In this paper, we report the finding of novel nonzero Hall effect in topological material ZrTe5 flakes when in-plane magnetic field is parallel and perpendicular to the current. Surprisingly, both symmetric and antisymmetric components with respect to magnetic field are detected in the in-plane Hall resistivity. Further theoretical analysis suggests that the magnetotransport properties originate from the anomalous velocity induced by Be…
Adjoint computation of Berry phase gradients
针对Berry相位因依赖全局积分而难以求导的问题，提出一种仅需一次正向和一次伴随计算即可高效计算Berry相位梯度的伴随方法，以支持拓扑优化设计具有定制拓扑波性质的新材料。
Band topology of anomalous quantum Hall insulators can be precisely addressed by computing Chern numbers of constituent non-degenerate bands that describe quantized, Abelian Berry flux through two-dimensional Brillouin zone. Can Chern numbers be defined for $SU(2)$ Berry connection of two-fold degenerate bands of materials preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries or combined $\mathcal{PT}$ symmetry, without detailed knowledge of underlying basis? We affirmatively answer this question by employing a non-Abelian generalization of Stokes' theorem and…
When Adiabaticity Is Not Enough to Study Topological Phases in Solid-State Physics: Comparing the Berry and Aharonov-Anandan Phases in 2D Materials
针对无能隙2D材料中绝热近似失效的问题，提出利用Aharonov-Anandan相位等时间依赖拓扑量，结合Floquet理论分析电磁辐射下graphene的拓扑、带间跃迁与电流涡旋，为Berry相位提供互补见解。
Berry Curvature Signatures in Chiroptical Excitonic Transitions
本文提出并实验验证了一种基于能量和动量分辨光学跃迁速率测量的通用方法，通过偏振调制激发的角分辨光电子能谱在WSe₂中利用激子提取贝里曲率纹理，并验证了该方法对超快散射过程的鲁棒性。
Bi$_{2}$Se$_{3}$ is a well known 3D-topological insulators(TI) with a non-trivial Berry phase of $ \left(2n+1\right)π$ attributed to the topology of the band structure. The Berry phase shows non-topological deviations from $ \left(2n+1\right)π$ in presence of a perturbation that destroys time reversal symmetry and gives rise to a quantum system with massive Dirac fermions and finite band gap. Such a band gap opening is achieved on account of the exchange field of a ferromagnet or the intrinsic energy gap of a superconductor that influences the topological insulator surface states by virtue of …
Ideal Chern insulating phases arise in two-dimensional systems with broken time-reversal symmetry. They are characterized by having nearly-flat bands, and a uniform quantum geometry -- which combines the Berry curvature and quantum metric -- and by being incompressible. In this work, we analyze the role of the quantum geometry in ideal Chern insulators aiming to describe transport in presence of external out-of-plane magnetic and electric fields. We firstly show that in the absence of external perturbations, novel Berry connections appear in ideal Chern insulating phases. Secondly, we provide …
Switchable band topology and geometric current in sliding bilayer elemental ferroelectric
本文发现双层铋单层的层间滑动可诱导拓扑平庸态与非平庸态交替相变，并调控Berry曲率偶极子及非线性反常霍尔电流，为拓扑存储应用提供了高能效机械控制平台。
Topological anisotropic non-Fermi liquid from a Berry-dipole semimetal
研究拓扑量子临界点处伯里二极子半金属在长程库仑相互作用下的行为，预测出各向异性非费米液体及其拓扑特征的观测测试。
When electrons are confined in two-dimensional (2D) materials, quantum mechanically enhanced transport phenomena, as exemplified by the quantum Hall effects (QHE), can be observed. Graphene, an isolated single atomic layer of graphite, is an ideal realization of such a 2D system. Here, we report an experimental investigation of magneto transport in a high mobility single layer of graphene. Adjusting the chemical potential using the electric field effect, we observe an unusual half integer QHE for both electron and hole carriers in graphene. Vanishing effective carrier masses is observed at Dir…
We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to remove EPs fr…
The role of Berry curvature derivatives in the optical activity of time-invariant crystals
本文建立时间反演不变晶体中光学活性与Berry曲率k空间导数的关系，揭示手性晶类圆二色性的量子几何起源。
Hodge Topology of Semiclassical Transport: A Coordinate-Free Geometric Framework for the Anomalous Hall Effect and Non-Linear Berry Dipole
利用Brillouin区的Hodge-de Rham分解建立拓扑带中反常输运的坐标无关微分几何框架，通过正则化Berry曲率2-形式得到全局光滑的代理势，统一线性与非线性输运的几何描述，并证明Hodge势与最大局域Wannier函数规范的联系。
Electronic band structures dictate the mechanical, optical and electrical properties of crystalline solids. Their experimental determination is therefore of crucial importance for technological applications. While the spectral distribution in energy bands is routinely measured by various techniques, it is more difficult to access the topological properties of band structures such as the Berry phase γ. It is usually thought that measuring the Berry phase requires applying external electromagnetic forces because these allow realizing the adiabatic transport on closed trajectories along which qua…
Moiré Topology in Twisted Structures with Noncollinear Spin-Orbit Coupling
提出基于非共线自旋轨道耦合实现拓扑莫尔微带的新途径，在HgI2中验证机制，并通过机器学习代理优化为PbI2，获得更窄更隔离的拓扑平带，支持关联磁性及量子自旋霍尔和陈绝缘体相。
Topological Landau Theory
本文扩展朗道相变理论，引入序参量拓扑性，证明超导系统中序参量在参数循环演化下获得贝里相位，并识别约瑟夫森结中的实验特征。
Strain-tunable Berry curvature in quasi-two-dimensional chromium telluride
研究通过第一性原理模拟与界面表征，揭示准二维 Cr2Te3 薄膜中反常霍尔符号反转及驼峰峰源于应变可调的 Berry 曲率和磁畴，为拓扑电子学提供新途径。
Berry phases strongly affect the properties of crystalline materials, giving rise to modifications of the semiclassical equations of motion that govern wave-packet dynamics. In non-Hermitian systems, generalizations of the Berry connection have been analyzed to characterize the topology of these systems. While the topological classification of non-Hermitian systems is being developed, little attention has been paid to the impact of the new geometric phases on dynamics and transport. In this work, we derive the full set of semiclassical equations of motion for wave-packet dynamics in a system g…
Topological Diagnosis of Strongly Correlated Electron Systems
针对强关联系统拓扑诊断难题，提出基于Green函数Berry曲率的通用框架，发现零点通量量子化可直接探测Mott绝缘体拓扑，为搜索强关联拓扑材料提供新方法。
Harnessing Nonlinear Dynamics for Time-Driven Berry Phase in Classical Systems
本文研究了经典非线性系统中时间驱动 Berry 阶段的积累，提出了一种无需改变内部或外部条件即可使 gauge 变量随时间自然演化的机制，并通过实验验证了频率依赖的 Berry 阶段理论预测。
Strain-induced Berry phase in chiral superconductors
应变诱导手性超导体中 Berry 阶段的拓扑效应通过狄拉克锥结构和超流刚性张量的观测得以实现。
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