A general expression for the orbital magnetization of a Floquet system is derived. The expression holds for a clean system, and is valid for any driving protocol, and arbitrary occupation of the bands. The orbital magnetization is shown to be large not only for Chern insulators, but also for anomalous phases where the Chern number does not fully account for the topology. In addition, the orbital magnetization is shown to take significant values both for a thermal equilibrium occupation of the Floquet bands, and for occupations determined by a quantum quench from an initial state with zero orbital magnetization. For the latter case, the orbital magnetization is shown to be highly sensitive to van Hove singularities of the Floquet bands.
Results are presented for Floquet systems in two spatial dimensions where the Floquet driving breaks an effective time reversal symmetry. The driving protocol also induces flat bands that correspond to anomalous Floquet phases where the Chern number is zero and yet chiral edge modes exist. Analytic expressions for the edge modes, Berry curvature, and the orbital magnetization are derived for the flat bands. Results are also presented for the static Haldane model for parameters when the bands are flat. Floquet driving of the same model is shown to give rise to Chern insulators as well as anomal…
We uncover a new family of few-body topological phases in periodically driven fermionic systems in two dimensions. These phases, which we term correlation-induced anomalous Floquet insulators (CIAFIs), are characterized by quantized contributions to the bulk magnetization from multi-particle correlations, and are classified by a family of integer-valued topological invariants. The CIAFI phases do not require many-body localization, but arise in the generic situation of k-particle localization, where the system is localized (due to disorder) for any finite number of particles up to a maximum nu…
Floquet states have been subject of great research interest since Zel'dovich's pioneering work on the quasienergy of a quantum system subject to a temporally periodic action. Nowadays periodic modulation of the system Hamiltonian is mostly achieved by microwaves leading to novel exciting phenomena in condensed matter physics: Floquet topological insulators, chiral edge states etc. On the other hand, nonthermal optical control of magnetization at picosecond time scales is currently a highly appealing research topic for potential applications in magnetic data storage. Here we combine these two c…
We study the response of ideal spin systems which are interacting with both a strong oscillating magnetic field, and a thermal environment, to a weak probing magnetic field. We demonstrate that even the sign of the resulting mean magnetization depends on the amplitude of the driving field, and that its absolute value can be significantly larger than the equilibrium magnetization in the absence of time-periodic forcing. Since the underlying Floquet-state occupation probabilities are determined by the precise form of the system-bath coupling, future measurements of such effects have the potentia…
We propose a scheme to create an electronic Floquet vortex state by irradiating a two-dimensional semiconductor with the laser light carrying non-zero orbital angular momentum. We analytically and numerically study the properties of the Floquet vortex states, with the methods analogous to the ones previously applied to the analysis of superconducting vortex states. We show that such Floquet vortex states are similar to the superconducting vortex states, and they exhibit a wide range of tunability. To illustrate the potential utility of such tunability, we show how such states could be used for…
We present a new scheme to control the spin exchange interactions between two magnetic ions by manipulating the orbital degrees of freedom using a periodic drive. We discuss two different protocols for orbital Floquet engineering. In one case, we modify the properties of the ligand orbitals which mediate magnetic interactions between two transition metal ions. While in the other case, we mix the d orbitals on each magnetic ion. In contrast to previous works on Floquet engineering of magnetic properties, the present scheme makes use of the AC Stark shift of the states involved in the exchange p…
Magnetism and Topology from Circularly Polarized Phonon Floquet Engineering
RIKEN Center for Emergent Matter Science (CEMS)（RIKEN 涌现物质科学中心）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；The Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）
A semiclassical theory for the orbital magnetization due to adiabatic evolutions of Bloch electronic states is proposed. It renders a unified theory for the periodic-evolution pumped orbital magnetization and the orbital magnetoelectric response in insulators by revealing that these two phenomena are the only instances where the induced magnetization is gauge invariant. This theory also accounts for the electric-field induced intrinsic orbital magnetization in two-dimensional metals and Chern insulators. We illustrate the orbital magnetization pumped by microscopic local rotations of atoms, wh…
Circularly polarised light opens a gap in the Dirac spectrum of graphene and topological insulator (TI) surfaces, thereby inducing a quantum Hall-like phase. We propose to detect the accompanying edge states and their current by the magnetic field they produce. The topological nature of the edge states is reflected in the mean orbital magnetization of the sample, which shows a universal linear dependence as a function of a generalized chemical potential - independent of the driving details and the properties of the material. The proposed protocol overcomes several typically encountered problem…
A space-time crystal is defined as a quantum mechanical system with both spatial and temporal periodicity. Such a system can be described by the Floquet-Bloch (FB) theory. We first formulate a semiclassical theory by constructing a wave-packet through the superposition of the FB wave functions and derive the equations of motion of FB electrons subjected to slowly varying external fields (not to be confused with the fast-changing Floquet drive), revealing behaviors similar to ordinary Bloch electrons but with quantities modified in the Floquet context. Specifically, we study local magnetic mome…
We study the anomalous Hall effect and orbital magnetization in chiral antiferromagnets, constructing a simple tight-binding model on a stacked Kagome lattice structure with spin-orbit coupling and the exchange interaction between the localized spins and itinerant electrons. It is shown that the ground state is an orbital ferromagnet with the easy-plane anisotropy. The relation between the orbital magnetization and the configuration of the Weyl points are discussed.
We calculate orbital spin-dependent magnetization in a two-dimensional electron gas with spin-orbit interaction of Rashba type. Such an orbital magnetization is admitted by the time-reversal symmetry of the system, and gives rise to spin currents when the system is not in thermal equilibrium. The theoretical approach is based on the linear response theory and the Matsubara Green's function formalism. To account for the spin-resolved orbital magnetization a spin-dependent vector potential has been introduced. The spin currents which appear in thermal nonequilibrium due to the spin-resolved orbi…
We investigate the induced orbital magnetization density in a Rashba electron gas with magnetic impurities. Relying on classical electrodynamics, we obtain this quantity through the bound currents composed of a paramagnetic and a diamagnetic-like contribution which emerge from the spin-orbit interaction. Similar to Friedel charge ripples, the bound currents and the orbital magnetization density oscillate as function of distance away from the impurity with characteristic wavelengths defined by the Fermi energy and the strength of the Rashba spin-orbit interaction. The net induced orbital magnet…
We consider periodic adiabatic processes of gapped many-body spinless electrons. We find an additional contribution to the orbital magnetization due to the adiabatic time evolution, dubbed \textit{geometric} orbital magnetization, which can be expressed as derivative of the many-body Berry phase with respect to an external magnetic field. For two-dimensional band insulators, we show that the geometric orbital magnetization generally consists of two pieces, the topological piece that is expressed as third Chern-Simons form in $(t,k_x,k_y)$ space, and the non-topological piece that depends on Bl…
Recently, signatures of quantum anomalous Hall states with spontaneous ferromagnetism were observed in twisted bilayer graphenes (TBGs) near 3/4 filling [1, 2]. Importantly, it was demon-strated that an extremely small current can switch the direction of the magnetization. This offers the prospect of realizing low energy dissipation magnetic memories. However, the mechanism of the current-driven magnetization switching is poorly understood as the charge currents in graphene layers are generally believed to be non-magnetic. In this work, we demonstrate that, in TBGs, the twist-induced reduction…
We theoretically show that in a crystal with a helical lattice structure, orbital and spin magnetizations along a helical axis are induced by an electric current along the helical axis. We propose a simple tight-binding model for calculations, and the results can be generalized to any helical crystals. The induced magnetizations are opposite for right-handed and left-handed helices. The current-induced spin magnetization along the helical axis comes from a radial spin texture on the Fermi surface. This is in sharp contrast to Rashba systems where the induced spin magnetization is perpendicular…
Both spin and orbital degrees of freedom contribute to the magnetic moment of isolated atoms. However, when inserted in crystals, atomic orbital moments are quenched because of the lack of rotational symmetry that protects them when isolated. Thus, the dominant contribution to the magnetization of magnetic materials comes from electronic spin. Here we show that nanoislands of quantum spin Hall insulators can host robust orbital edge magnetism whenever their highest occupied Kramers doublet is singly occupied, upgrading the spin edge current into a charge current. The resulting orbital magnetiz…
The intrinsic orbital magnetization of a TMD monolayer is usually calculated for a plane unbounded system without mentioning the geometrical shape of samples and boundary conditions (BCs) for electron wave functions. The method of calculations includes allowing for the Berry curvature contribution also in the case when the system is described by the two-band minimal model [9]. In the present paper, we show that the geometrical and topological properties of the specimen, as well as the BCs, play an important role in the problem of magnetization even for a macroscopic specimen.
Using Floquet-Bloch theory we study the effect of circularly and linearly polarized light on the electronic structure of (LaNiO$_3$)$_2$/(LaAlO$_3$)$_N$ heterostructure grown along the (111) direction. In equilibrium, a tight-binding fit to the first principles band structure shows that nearest-neighbor hopping plays a dominant role while second-neighbor hopping breaks the particle-hole symmetry and determines the finer band features. The four bands of the LaNiO$_3$ bilayer exhibit both quadratic band touching points and Dirac points. By varying the amplitude of the incident light, one can ind…
The orbital magnetic moment of a finite piece of matter is expressed in terms of the one-body density matrix as a simple trace. We address a macroscopic system, insulating in the bulk, and we show that its orbital moment is the sum of a bulk term and a surface term, both extensive. The latter only occurs when the transverse conductivity is nonzero and owes to conducting surface states. Simulations on a model Hamiltonian validate our theory.
The modern expressions for polarization $¶$ and orbital magnetization $\M$ are $\k$-space integrals. But a genuine bulk property should also be expressible in $\r$-space, as unambiguous function of the ground-state density matrix, "nearsighted" in insulators, independently of the boundary conditions---either periodic or open. While $¶$---owing to its "quantum" indeterminacy---is {\it not} a bulk property in this sense, $\M$ is. We provide its $\r$-space expression for any insulator, even with nonzero Chern invariant. Simulations on a model Hamiltonian validate our theory.
Quantized Chern-Simons Axion Coupling in Anomalous Floquet Systems
Université Libre de Bruxelles（布鲁塞尔自由大学）；International Solvay Institutes（国际索尔维研究所）；Collège de France（法兰西公学院）；CNRS（法国国家科学研究中心）；ENS-Université PSL（巴黎高等师范学院-巴黎文理研究大学）；Sorbonne Université（索邦大学）；Centro Atómico Bariloche（巴里洛切原子能中心）；Instituto Balseiro（巴尔塞罗研究所）；Comisión Nacional de Energía Atomica (CNEA)（国家原子能委员会）；Universidad Nacional de Cuyo (UNCUYO)（门多萨国立大学）；Instituto de Nanociencia y Nanotecnología (INN-Bariloche)（纳米科学与技术研究所）；Consejo Nacional de Investigaciones Científicas y Tecnicas (CONICET)（国家科学研究委员会）
The theory of orbital magnetization is reconsidered by defining additional quantities that incorporate a non-Hermitian effect due to anomalous operators that break the domain of definition of the Hermitian Hamiltonian. As a result, boundary contributions to the observable are rigorously and analytically taken into account. In this framework, we extend the standard velocity operator definition in order to incorporate an anomaly of the position operator that is inherent in band theory, which results in an explicit boundary velocity contribution. Using the extended velocity, we define the electro…
The control of physical properties of solids with short laser pulses is an intriguing prospect of ultrafast materials science. Continuous-wave high-frequency laser driving with circular polarization was predicted to induce a light-matter coupled new state possessing a quasi-static band structure with an energy gap and a quantum Hall effect, coined "Floquet topological insulator". Whereas the envisioned Floquet topological insulator requires well separated Floquet bands and therefore high-frequency pumping, a natural follow-up question regards the creation of Floquet-like states in graphene wit…
Dichroism from Chiral Thermoelectric Probes: Generalized Sum Rules for Orbital and Heat Magnetizations
Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-Université PSL, Sorbonne Université（卡斯勒布罗塞尔实验室，法兰西学院，法国国家科学研究中心，巴黎高等师范学院-巴黎文理研究大学，索邦大学）；International Solvay Institutes（国际索尔维研究所）；Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles（非线性现象与复杂系统中心，布鲁塞尔自由大学）
The observed magnetization switching by circularly polarized ultrafast laser pulses has been attributed to the inverse Faraday effect in which the induced non-equilibrium orbital momentum serves as an effective magnetic filed via spin-orbit coupling for magnetization rotation and switching. We critically examine this scenario by explicitly calculating the magnitude of the induced orbital momentum for generic itinerant band. We show that the calculated induced angular momentum is not large enough for reversing the magnetization in one laser pulse with the order of 100 femtosecond duration. Inst…
Staggered orbital magnetization from itinerant electrons: orbital antiferro- and ferrimagnetic phases
Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）；Centro Brasileiro de Pesquisas Físicas (CBPF)（巴西物理研究中心）；Physics Center of Minho and Porto Universities (CF-UM-UP)（米尼奥与波尔图大学物理中心）
Theory of In-Plane Orbital Magnetization with Layer Hybridization
The Hong Kong University of Science and Technology（香港科技大学）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；Beijing Institute of Technology（北京理工大学）
We propose an orbital magnetothermal effect wherein a temperature gradient generates an orbital magnetization (OM) for Bloch electrons, and we present a unified theory for electrically and thermally induced OM, valid for both metals and insulators. We reveal that there exists an intrinsic response of OM, for which the susceptibilities are completely determined by the band geometric quantities such as interband Berry connections, interband orbital moments, and the quantum metric. The theory can be readily combined with first-principles calculations to study real materials. As an example, we cal…
Real-Space Spectral Approach to Orbital Magnetization
International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）；Centro de Física das Universidades do Minho e Porto（米尼奥和波尔图大学物理中心）；Universidade do Porto（波尔图大学）；University of York（约克大学）；Centro Brasileiro de Pesquisas Físicas (CBPF)（巴西物理研究中心）；Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Physics Center of Minho and Porto Universities (CF-UM-UP)（米尼奥和波尔图大学物理中心）
The modern theory of orbital magnetization addresses crystalline materials at the noninteracting level: therein the observable is the k-space integral of a geometrical integrand. Alternatively, magnetization admits a local representation in r space, i.e. a "density" which may address noncrystalline and/or inhomogeneous materials as well; the Chern number admits an analogous density. Here we provide the formulation for ribbons, polymers, and layered materials, where both k-space and r-space integrations enter the definition of the two observables.
A valley-contrasting Berry curvature in bilayer transition metal dichalcogenides with spin-orbit coupling can generate valley magnetization when the inversion symmetry is broken, for example, by an electric field, regardless of time-reversal symmetry. A nontrivial Berry curvature can also lead to anomalous transport responses, such as the anomalous Hall effect and the anomalous Nernst effect. Applied to a bilayer WSe$_2$, an electric field can tune the Berry curvature and orbital magnetic moment, which has important consequences for the orbital magnetization and the anomalous Nernst responses.…
The orbital magnetization of the electron gas on a two-dimensional kagome lattice under a perpendicular magnetic field is theoretically investigated. The interplay between the lattice geometry and magnetic field induce nontrivial $k$-space Chern invariant in the magnetic Brillouin zone, which turns to result in profound effects on the magnetization properties. We show that the Berry-phase term in the magnetization gives a paramagnetic contribution, while the conventional term brought about by the magnetic response of the magnetic Bloch bands produces a diamagnetic contribution. As a result, th…
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；Hefei National Laboratory（合肥国家实验室）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）
We theoretically demonstrate that 100\% valley-polarized transport in monolayers of MoS$_{2}$ and other group-VI dichalcogenides can be obtained using off-resonant circularly polarized light. By tuning the intensity of the off-resonant light the intrinsic band gap in one valley is reduced, while it is enhanced in the other valley, enabling single valley quantum transport. As a consequence, we predict (i) enhancement of the longitudinal electrical conductivity, accompanied by an increase in the spin-polarization of the flowing electrons, (ii) enhancement of the intrinsic spin Hall effect, toget…
Ensembles of gold nanoparticles present a magnetic behavior which is at odds with the weakly diamagnetic response of bulk gold. In particular, an unusual ferromagnetic order has been unveiled by several experiments. Here we investigate if the combined effect of orbital magnetism of conduction electrons and interparticle dipolar interaction can lead to magnetic ordering. Using different model systems of interacting mesoscopic magnetic dipoles, together with a microscopic description of the electron dynamics within the nanoparticles, we find that a spontaneous magnetic moment may arise in dimers…
Theory of local orbital magnetization: local Berry curvature
CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris（法国国家科学研究中心，巴黎综合理工学院，巴黎理工学院）；Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS（巴黎-萨克雷大学，法国国家科学研究中心）
