Daniel C. Ralph
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.
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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.
We derive a semiclassical expression for the Green's function in graphene, in which the presence of a semiclassical phase is made apparent. The relationship between this semiclassical phase and the adiabatic Berry phase, usually referred to in this context, is discussed. These phases coincide for the perfectly linear Dirac dispersion relation. They differ however when a gap is opened at the Dirac point. We furthermore present several applications of our semiclassical formalism. In particular we provide, for various configurations, a semiclassical derivation of the electron's Landau levels, ill…
Lecture notes published in ''Magnetism goes nano'', Lecture Manuscripts of the 36th Spring School of the Institute of Solid State Research, edited by Stefan Bluegel, Thomas Brueckel, and Claus M. Schneider (Forschungszentrum Juelich, 2005).
We propose the semiclassical quantization for complicated electron systems governed by a many-band Hamiltonian. An explicit analytical expression of the corresponding Berry phase is derived. This impact allows us to evaluate the Landau magnetic levels when the rigorous quantization fails, for instance, for bilayer graphene and graphite with the trigonal warping. We find that the magnetic breakdown can be observed for the certain type of classical electron orbits.
Berry-Phase Breakdown and Semiclassical Reconciliation in Topological Dirac Fock-Darwin states
研究拓扑绝缘体表面圆环n-junction中Dirac费米子类人工原子（Fock-Darwin态）的Berry相切换失效现象，揭示电子-空穴核心壳层结构在临界点附近演变，通过半经典方法调和理论与实验结果。
In topological semimetals and insulators, negative longitudinal magnetoresistance and angle-dependent planar Hall effect have been reported arising from the Berry curvature. Using the Boltzmann transport theory, we present a closed-form expression for the nonequilibrium distribution function which includes both the effects of the Berry curvature and Lorentz force. Using this formulation, we obtain analytical expressions for conductivity and resistivity tensors in Weyl semimetals demonstrating a non-monotonic field dependence arising from the competition between the two effects.
Understanding the transport behavior of an electronic system under the influence of a magnetic field remains a key subject in condensed matter physics. Particularly in topological materials, their nonvanishing Berry curvature can lead to many interesting phenomena in magnetotransport owing to the coupling between the magnetic field and Berry curvature. By fully incorporating both the field-driven anisotropy and inherent anisotropy in the band dispersion, we study the semiclassical Boltzmann magnetotransport theory in topological materials with a nonvanishing Berry curvature. We show that as a …
We consider the semiclassical equations of motion of a particle when both an external electromagnetic field and the Berry gauge field in the momentum space are present. It is shown that these equations are Hamiltonian and relations between the canonical and covariant variables are determined through a consistent account of all components of the Berry connection. The Jacobian of the canonical-to-covariant-variables transformation describes the nonconservation of the 'naive' phase space volume in the covariant coordinates (D.Xiao, J.Shi, and Q.Niu, Phys. Rev. Lett. 95, 137204 (2005)).
The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects -- such as Berry phase and Maslov index -- occurring at order $\hbar$ in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase -- which is basica…
We theoretically study the role of the Berry curvature on neutral and charged excitons in two-dimensional transition-metal dichalcogenides. The Berry curvature arises due to a strong coupling between the conduction and valence bands in these materials that can to great extent be described within the model of massive Dirac fermions. The Berry curvature lifts the degeneracy of exciton states with opposite angular momentum. Using an electronic interaction that accounts for non-local screening effects, we find a Berry-curvature induced splitting of $\sim 17$ meV between the 2$p_{-}$ and 2$p_{+}$ e…
The dynamics of observables which are matrices depending on \hbar and taking values in classical phase space is defined retaining the terms up to the first order in \hbar of the Moyal bracket. Within this semiclassical approach a first order lagrangian involving gauge fields is studied as a constrained hamiltonian system. This provides a systematic study of spin dynamics in the presence of non-abelian Berry gauge fields. We applied the method to various types of dynamical spin systems and clarified some persisting discussions. In particular employing the Berry gauge field which generates the T…
Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in…
Starting with general semiclassical equations of motion for electrons in the presence of electric and magnetic fields, we extend the Chambers formula to include in addition to a magnetic field, time-dependent electric fields and bands with Berry curvature. We thereby compute the conductivity tensor $σ_{αβ}(B,ω)$ in the presence of magnetic field for bands in two (2D) and three (3D) dimensions with Berry curvature. We focus then on several applications to magnetotransport for metals with Fermi surface topological transitions in 2D. In particular, we consider a rectangular lattice and a model re…
Using a microscopic theory for the magnetoconductivity at low magnetic fields we show how the Hall and longitudinal conductivity can be calculated in the low scattering rate limit. In the lowest order of the scattering rate, we recover the result of the semiclassical Boltzmann transport theory. At higher order, we get corrections containing the Berry curvature and the orbital magnetic moment. We use this formalism to study the linear longitudinal magnetoconductivity in tilted Weyl semimetals. We discuss how our result is related to the semiclassical Boltzmann approach and show the differences …
The weak field magnetoresistance has seen a revived interest due to the distinct role played by the momentum-space Berry curvature of Bloch electrons. While most previous studies in this regard focus on the inter-scattering motion of semiclassical Bloch electrons in electromagnetic fields, the intra-scattering effects of the semiclassical dynamics augmented by the Berry curvature, magnetic moment and shift vector on the magnetoresistance have been largely overlooked. Here we uncover that these intra-scattering effects, which are neglected in the field-independent relaxation time approximation …
By quantizing the semiclassical motion of excitons, we show that the Berry curvature can cause an energy splitting between exciton states with opposite angular momentum. This splitting is determined by the Berry curvature flux through the $\bm k$-space area spanned by the relative motion of the electron-hole pair in the exciton wave function. Using the gapped two-dimensional Dirac equation as a model, we show that this splitting can be understood as an effective spin-orbit coupling effect. In addition, there is also an energy shift caused by other "relativistic" terms. Our result reveals the l…
We develop a semiclassical theory of nonlinear transport and the photogalvanic effect in non-centrosymmetric media. We show that terms in semiclassical kinetic equations for electron motion which are associated with the Berry curvature and side jumps give rise to a dc current quadratic in the amplitude of the ac electric field. We demonstrate that the circular photogalvanic effect is governed by these terms in contrast to the linear photogalvanic effect and nonlinear I-V characteristics which are governed mainly by the skew scattering mechanism. In addition, the Berry curvature contribution to…
Transverse current due to Berry curvature in phase space is formulated based on the Boltzmann equations with the semiclassical equations of motion for an electron wave packet. It is shown that the Hall effect due to the phase space Berry curvature is absent because the contributions from "anomalous velocity" and "effective Lorentz force" are completely cancelled out.
We study nonlinear magneto-optical responses of metals by a semiclassical Boltzmann equation approach. We derive general formulas for linear and second order nonlinear optical effects in the presence of magnetic fields that include both Berry curvature and orbital magnetic moment. Applied to Weyl fermions, the semiclassical approach (i) captures the directional anisotropy of linear conductivity under magnetic field as a consequence of an anisotropic $B^2$ contribution, which may explain the low-field regime of recent experiments; (ii) predicts strong second harmonic generation proportional to …
Noncentrosymmetric metals are anticipated to exhibit a $dc$ photocurrent in the nonlinear optical response caused by the Berry curvature dipole in momentum space. Weyl semimetals (WSMs) are expected to be excellent candidates for observing these nonlinear effects because they carry a large Berry curvature concentrated in small regions, i.e., near the Weyl points. We have implemented the semiclassical Berry curvature dipole formalism into an $ab~initio$ scheme and investigated the second-order nonlinear response for two representative groups of materials: the TaAs-family type-I WSMs and MoTe$_2…
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…
Weyl semimetals possess low energy excitations which act as monopoles of Berry curvature in momentum space. These emergent monopoles are at the heart of the extensive novel transport properties that Weyl semimetals exhibit. The singular nature of the Berry curvature around the nodal points in Weyl semimetals allows for the possibility of large anomalous transport coefficients in zero applied magnetic field. Recently a new class, termed type-II Weyl semimetals, has been demonstrated in a variety of materials, where the Weyl nodes are tilted. We present here a study of anomalous transport in thi…
Topological linear magnetoresistivity and thermoconductivity induced by noncentrosymmetric Berry curvature
本文基于半经典玻尔兹曼形式和对称性分析，证明非中心对称贝里曲率分布可普遍诱导具有B标度行为的线性磁阻率和热电导率，并在MnBi2Te4等材料中与实验吻合，为拓扑线性磁输运提供了统一理论解释。
Kinetic theory of Dirac fermions is studied within the matrix valued differential forms method. It is based on the symplectic form derived by employing the semiclassical wave packet build of the positive energy solutions of the Dirac equation. A satisfactory definition of the distribution matrix elements imposes to work in the basis where the helicity is diagonal which is also needed to attain the massless limit. We show that the kinematic Thomas precession correction can be studied straightforwardly within this approach. It contributes on an equal footing with the Berry gauge fields. In fact …
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。
Composite Quantum Geometry and Semiclassical Dynamics
推导绝缘体和半导体中一般复合束缚态（如激子和三激子）的半经典运动方程，发现中性复合物中均匀外电场不与贝里曲率项耦合，而出现一种新的量子几何量——复合物不等价贝里连接之差，并讨论了带电复合物的贝里曲率贡献及在魔角扭曲双层石墨烯中的应用。
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…
Within the semiclassical Boltzmann transport theory, the formula for Seebeck coefficient $S$ is derived for an isotropic two-dimensional electron gas (2DEG) system that exhibits anomalous Hall effect (AHE) and anomalous Nernst effect (ANE) originating from Berry curvature on their bands. Deviation of $S$ from the value $S_0$ estimated neglecting Berry curvarture is computed for a special case of 2DEG with Zeeman and Rashba terms. The result shows that, under certain conditions the contribution of Berry curvature to Seebeck effect could be non-negligible. Further study is needed to clarify the …
We develop a theory of circular photogalvanic effect (CPGE) for classically high photon energies which exceed the electron scattering rate but are small compared to the average electron kinetic energy. In this frequency range one can calculate the CPGE by using two different approaches. In the fully quantum-mechanical approach we find the photocurrent density by applying Fermi's golden rule for indirect intraband optical transitions with virtual intermediate states both in the conduction and valence bands. In the framework of the semiclassical approach, we apply a generalized Boltzmann equatio…
Liouville's theorem on the conservation of phase space volume is violated by Berry phase in the semiclassical dynamics of Bloch electrons. This leads to a modification of the phase space density of states, whose significance is discussed in a number of examples: field modification of the Fermi-sea volume, connection to the anomalous Hall effect, and a general formula for orbital magnetization. The effective quantum mechanics of Bloch electrons is also sketched, where the modified density of states plays an essential role.
The magnetization and the de Haas-van Alphen oscillations of Bloch electrons are calculated near commensurate magnetic fluxes. Two phases that appear in the quantization of mixed systems--the Berry's phase and a phase first discovered by Wilkinson--play a key role in the theory.
Berry curvature acts analogous to a magnetic field in the momentum-space, and it modifies the flow of charge carriers and entropy. This induces several intriguing magnetoelectric and magnetothermal transport phenomena in Weyl semimetals. %, including the planar Hall and planar Nernst effect. Here, we explore the impact of the Berry curvature and orbital magnetization on the thermopower in tilted type-I and type-II Weyl semimetals, using semiclassical Boltzmann transport formalism. We analytically calculate the full magnetoconductivity matrix and use it to obtain the thermopower matrix for diff…
We consider semiclassical higher-order wave packet solutions of the Schrodinger equation with phase vortices. The vortex line is aligned with the propagation direction, and the wave packet carries a well-defined orbital angular momentum (OAM) $\hbar l$ ($l$ is the vortex strength) along its main linear momentum. The probability current coils around momentum in such OAM states of electrons. In an electric field, these states evolve like massless particles with spin $l$. The magnetic-monopole Berry curvature appears in momentum space, which results in a spin-orbit-type interaction and a Berry/Ma…
A generalized Peierls substitution which takes into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be an essential element for the correct determination of the semiclassical equations of motion as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expression for the cross-sectional area is derived and used as an illustration for the calculation of the energy levels of Bloch and Dirac electrons.
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。
The geometric or Berry phase, a characteristic of quasiparticles, is fundamental to the underlying quantum materials. The discoveries of new materials at a rapid pace nowadays call for efficient detection of the Berry phase. Utilizing $α$-T$_3$ lattice as a paradigm, we find that, in the Dirac electron optics regime, the semiclassical decay of the quasiparticles from a chaotic cavity can be effectively exploited for detecting the Berry phase. In particular, we demonstrate a one-to-one correspondence between the exponential decay rate and the geometric phase for the entire family of $α$-T$_3$ m…
Within the semiclassical approach, we calculate contributions of the Berry phase and of the Zeeman coupling of the electron moment with the magnetic field to the phase of the Shubnikov - de Haas oscillations for the surface electrons in the Bi$_2$X$_3$ family of three-dimensional topological insulators (X stands for Te or Se). We also discuss a relation of the obtained results with published experimental data on the Shubnikov-de Haas oscillations for this family of topological insulators.
Semiclassical magnetotransport in Weyl semimetals with skyrmion-induced emergent magnetic fields
研究了在现实空间和动量空间拓扑共同作用下，磁场所致输运现象的特性，通过 semiclassical Boltzmann 方法分析纵向磁导率和横向霍尔导电率，揭示了间隔谷散射与现实空间拓扑对响应几何特征的不同控制作用。
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.
We construct a theory for the semiclassical dynamics of superconducting quasiparticles by following their wave-packet motion and reveal rich contents of Berry curvature effects in the phase-space spanned by position and momentum. These Berry curvatures are traced back to the characteristics of superconductivity, including the nontrivial momentum-space geometry of superconducting pairing, the real-space supercurrent, and the charge dipole of quasiparticles. The Berry-curvature effects strongly influence the spectroscopic and transport properties of superconductors, such as the local density of …
Recent results on the semiclassical dynamics of an electron in a solid are explained using techniques developed for ``exotic'' Galilean dynamics. The system is indeed Hamiltonian and Liouville's theorem holds for the symplectic volume form. Suitably defined quantities satisfy hydrodynamic equations.
Linear thermal noise induced by Berry curvature dipole in a four-terminal system
本文研究了四端系统中有限 Berry 曲率偶极子引起的线性热噪声，通过非平衡 Green 函数方法与 semiclassical 结果对比，建立了多端系统终端分辨噪声与 bulk 运输方向分辨噪声的一一对应关系，揭示了量子输运中的对称性选择规则。
We describe a method for measuring the Berry curvature from the wave-packet dynamics in perturbed arrays of evanescently coupled optical waveguides with honeycomb lattice structure. To disentangle the effects of the Berry curvature and the energy dispersion we utilize a difference measurement by propagating the wave packet under the influence of a constant external force back and forth. In this way a non-vanishing Berry curvature is obtained for photonic graphene with small sublattice bias or strain, where the relative error between the exact Berry curvature and the one derived from the semicl…
Theory of Topological Nernst and Thermoelectric Transport in Chiral Magnets
本文用含全相空间 Berry 曲率的受控半经典方法研究手性磁体中拓扑自旋织构下的热电输运，分解能斯特效应并推导广义 Mott 关系。
A physically transparent and mathematically simple semiclassical model is employed to examine dynamics in the central-spin problem. The results reproduce a number of previous findings obtained by various quantum approaches and, at the same time, provide information on the electron spin dynamics and Berry's phase effects over a wider range of experimentally relevant parameters than available previously. This development is relevant to dynamics of bound magnetic polarons and spin dephasing of an electron trapped by an impurity or a quantum dot, and coupled by a contact interaction to neighboring…
We study the dynamics of semiclassical electrons in (gapped) graphene in two complementary limits, i.e. in the Klein tunneling and valley Hall effect regimes, by scattering wave packets off armchair step potentials and by exposing wave packets to a uniform electric field, respectively. Our numerical wave packet simulation goes beyond semiclassical analytical approximations and standard Klein tunneling treatments and allows to study intra- and intervalley scattering processes. We find distinct Klein tunneling characteristics for low and tall steps, which include unusual Berry curvature induced …
Berry Curvature Spectroscopy from Bloch Oscillations
本文提出利用静态电场缀饰下人工晶体的布洛赫振荡共振，通过半经典理论证明缀饰霍尔电导率共振与贝里曲率晶格傅里叶分量成正比，从而实现太赫兹频段的贝里曲率光谱学测量。
The winding numbers for the even d+1 spacetime dimensional Weyl Hamiltonians are calculated in terms of the related Green's functions. It is shown that these winding numbers result in the divergence of the Dirac monopole fields, hence they are equal to the unit topological charge. It is demonstrated that the winding numbers are also equal to the Chern numbers which are expressed as the integral of the Berry field strength. Explicit calculations are presented for the 3+1 and 5+1 dimensional cases. Relevance of these topological invariants for the physical systems like the semiclassical chiral k…
Nonlinear anomalous transverse responses induced by Berry curvature quadrupole in systems with broken time-reversal symmetry
本文基于半经典玻尔兹曼形式，计算了破坏时间反演对称性系统中Berry曲率四极矩诱导的非线性反常热霍尔与能斯特系数，预测了与温度梯度三次方成正比的响应，并指出该效应在已有实验系统中必然存在。
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…
We study the magnetoelectric and magnetothermal transport properties of noncentrosymmetric metals using semiclassical Boltzmann transport formalism by incorporating the effects of Berry curvature and orbital magnetic moment. These effects impart quadratic-B dependence to the magnetoelectric and magnetothermal conductivities, leading to intriguing phenomena such as planar Hall effect, negative magnetoresistance, planar Nernst effect and negative Seebeck effect. The transport coefficients associated with these effects show the usual oscillatory behavior with respect to the angle between the appl…
Anomalous and Topological Hall Effects with Phase-Space Berry Curvatures: Electric, Thermal, and Thermoelectric Transport in Magnets
该研究建立含相空间贝里曲率的半经典理论，统一解释磁体中反常与拓扑霍尔等横向输运效应，推导低温普适关系并揭示自旋轨道耦合与交换耦合相对强度对输运行为的调控作用。
Planar Hall effect in topological Weyl and nodal line semimetals
本研究通过对称性分析与半经典玻尔兹曼方程，探究三维材料中平面霍尔效应的普遍起源，明确洛伦兹力与贝里曲率的不同贡献及对称性特征，揭示其在外尔、节线半金属中的差异，为实验区分两类贡献提供依据。
We argue that ferromagnetic transition metal nanoparticles with fewer than approximately 100 atoms can be described by an effective Hamiltonian with a single giant spin degree of freedom. The total spin $S$ of the effective Hamiltonian is specified by a Berry curvature Chern number that characterizes the topologically non-trivial dependence of a nanoparticle's many-electron wavefunction on magnetization orientation. The Berry curvatures and associated Chern numbers have a complex dependence on spin-orbit coupling in the nanoparticle and influence the semiclassical Landau-Liftshitz equations th…
Third-order Hall effect in the surface states of a topological insulator
本文利用半经典Boltzmann形式体系研究了六角翘曲拓扑绝缘体表面态中的三阶霍尔效应，发现倾斜和翘曲程度能显著增强并调控该响应，且其可作为系统主导阶响应直接在实验中验证。
Motivated by the recent prediction of the Magnus Hall effect in systems with broken inversion symmetry, in this paper we study the Magnus Nernst effect and the Magnus thermal Hall effect. In presence of an in-built electric field, the self rotating wave-packets of electrons with finite Berry curvature generate a Magnus velocity perpendicular to both. This anomalous Magnus velocity gives rise to the Magnus Hall transport which manifests in all four electro-thermal transport coefficients. In this paper, we demonstrate the existence of the Magnus Nernst and Magnus thermal Hall effect in monolayer…
Nonlinear optical response is well studied in the context of semiconductors and has gained a renaissance in studies of topological materials in the recent decade. So far it mainly deals with non-magnetic materials and it is believed to root in the Berry curvature of the material band structure. In this work, we revisit the general formalism for the second-order optical response and focus on the consequences of the time-reversal-symmetry ($\mathcal{T}$) breaking, by a diagrammatic approach. We have identified three physical mechanisms to generate a dc photocurrent, i.e. the Berry curvature, the…
Linear Magnetoelectric Electro-Optic Effect
本文用半经典Boltzmann方法推导电偏置量子材料的广义本构关系，发现Berry曲率与磁矩共同产生的线性磁电电光效应，并给出空间群152和198手性材料的响应张量。
Berry curvature-induced transport signature for altermagnetic order
针对交变磁序探测需复杂自旋敏感装置的问题，提出一种无需自旋选择设备的简单输运测量方法，通过磁导率各向异性探测d波交变磁序参量，为交变磁相变研究提供便捷手段。
An extension of the Hellmann-Feynman theorem to one employing dynamical parameters that vary with time according to quantum dynamics is rigorously derived, avoiding any linear response or other approximations. The resulting theorem for the dynamics of observables, valid to all orders in external fields, is found to contain generalized Berry curvature type of quantities that incorporate the dynamics through explicit and nontrivial time-dependence; these Berry quantities resemble the so called anomalous terms (in semiclassical equations of motion in solids) but are both of magnetic and electric …
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