Topological insulator (TI) thin films with surface magnetism are expected to exhibit a quantized anomalous Hall effect (QAHE) when the magnetizations on the top and bottom surfaces are parallel, and a quantized topological magnetoelectric (QTME) response when the magnetizations have opposing orientations (axion insulator phase) and the films are sufficiently thick. We present a unified picture of both effects that associates deviations from exact quantization of the QTME caused by finite thickness with non-locality in the side-wall current response function. Using realistic tight-binding model calculations, we show that in $Bi_2Se_3$ TI thin films deviations from quantization in the axion insulator-phase are reduced in size when the exchange coupling of tight-binding model basis states to the local magnetization near the surface is strengthened. Stronger exchange coupling also reduces the effect of potential disorder, which is unimportant for the QAHE but detrimental for the QTME, which requires that the Fermi energy lie inside the gap at all positions.
Axion insulator is an exotic magnetic topological insulator with zero Chern number but a nonzero quantized Chern-Simons magnetoelectric coupling. A conclusive experimental evidence for axion insulators is still lacking due to the small signal of topological magnetoelectric effect (TME). Here we show that the dynamical magnetoelectric coupling can be induced by the \emph{out-of-plane} surface magnetization dynamics in axion insulator thin films, which further generates a polarization current in the presence of an external magnetic field in the same direction. Such a current is finite in the bul…
Three-dimensional topological insulators or axion insulators exhibit the topological magnetoelectric effect, which is isotropic with a universal coefficient of proportionality quantized in units of $e^2/2h$. Here we study the finite-size effect of topological magnetoelectric effect, and find the magnetoelectric coefficients are anisotropic, namely $α_{xx}\neqα_{zz}$. Both of them are shown to converge to a quantized value when the thickness of topological insulator film $d$ increases reaching the three-dimensional bulk limit. The nonzero value of $(α_{xx}-α_{zz})\propto1/d$ could be measured b…
Chromia is a magnetoelectric insulator whose bulk magnetoelectric susceptibility contains a non-zero pseudoscalar component, like that present in magnetized topological insulators. We reveal the dynamic axion field of chromia by measuring the temperature dependence of its non-linear magnetoelectric response using a lock-in technique with an AC electric field stimulus. The electric field dependence, magnetic field dependence, and temperature behavior of the second harmonic lock-in signal are all in agreement with theoretical expectations. Our results demonstrate that chromia magnetoelectric res…
More than forty years ago, axion was postulated as an elementary particle with a low mass and weak interaction in particle physics to solve the strong $\mathcal{CP}$ (charge conjugation and parity) puzzle. Axions are also considered as a possible component of dark matter of the universe. However, the existence of axions in nature has not been confirmed. Interestingly, axions arise as pseudoscalar fields derived from the Chern-Simons theory in condensed matter physics. In antiferromagnetic insulators, the axion field can become dynamical induced by spin-wave excitations and exhibits rich exotic…
Intrinsic magnetic topological insulator ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ is believed to be an axion insulator in its antiferromagnetic ground state. However, direct identification of axion insulators remains experimentally elusive because the observed vanishing Hall resistance, while indicating the onset of the axion field, is inadequate to distinguish the system from a trivial normal insulator. Using numerical Green's functions, we theoretically demonstrate the quantized magnetoelectric current in a tunnel junction of atomically thin ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ sandwiched be…
The orbital motion of electrons in a three-dimensional solid can generate a pseudoscalar magnetoelectric coupling $θ$, a fact we derive for the single-particle case using a recent theory of polarization in weakly inhomogeneous materials. This polarizability $θ$ is the same parameter that appears in the "axion electrodynamics" Lagrangian $Δ{\cal L}_{EM} = (θe^2 / 2 πh) {\bf E} \cdot {\bf B}$, which is known to describe the unusual magnetoelectric properties of the three-dimensional topological insulator ($θ=π$). We compute $θ$ for a simple model that accesses the topological insulator and discu…
Topological magnetoelectric effect (TME) is a hallmark response of the topological field theory, which provides a paradigm shift in the study of emergent topological phenomena. However, its direct observation is yet to be realized due to the demanding magnetic configuration required to gap all the surface states. Here, we theoretically propose that the axion insulators with a simple ferromagnetic configuration, such as MnBi2Te4/(Bi2Te3)n family, provide an ideal playground to realize TME. In a designed triangular prism geometry, all the surface states are magnetically gapped. Under a vertical …
A quantized version of the magnetoelectric effect, known as the topological magnetoelectric effect, can exist in a time-reversal invariant topological insulator with all its surface states gapped out by magnetism. This topological phase, called the axion insulator phase, has been theoretically proposed but is still lack of conclusive experimental evidence due to the small signal of topological magnetoelectric effect. In this work, we propose that the dynamical in-plane magnetization in an axion insulator can generate a "pseudo-electric field", which acts on the surface state of topological ins…
Topological states of quantum matter have attracted great attention in condensed matter physics and materials science. The study of time-reversal-invariant (TRI) topological states in quantum materials has made tremendous progress in both theories and experiments. As a great success, thousands of TRI topological materials are predicted through sweeping search. Richer exotic phenomena are expected to appear in magnetic topological materials because of varied magnetic configurations, but this study falls much behind due to the complex magnetic structures and transitions. Here, we predict the tet…
A graphene bilayer shows an unusual magnetoelectric response whose magnitude is controlled by the valley-isospin density, making it possible to link magnetoelectric behavior to valleytronics. Complementary to previous studies, we consider the effect of static homogeneous electric and magnetic fields that are oriented parallel to the bilayer's plane. Starting from a tight-binding description and using quasi-degenerate perturbation theory, the low-energy Hamiltonian is derived including all relevant magnetoelectric terms whose prefactors are expressed in terms of tight-binding parameters. We con…
One of the intriguing properties characteristic to three-dimensional topological materials is the topological magnetoelectric phenomena arising from a topological term called the $θ$ term. Such magnetoelectric phenomena are often termed the axion electrodynamics, since the $θ$ term has exactly the same form as the action describing the coupling between a hypothetical elementary particle, axion, and a photon. The axion was proposed about forty years ago to solve the so-called strong CP problem in quantum chromodynamics, and is now considered as a candidate for dark matter. In this tutorial, we …
One of the hallmarks of time-reversal-symmetric topological insulators in three dimensions is the topological magnetoelectric effect (TME). So far, a time-reversal breaking variant of this effect has attracted much attention, in the sense that the induced electric charge changes sign when the direction of an externally applied magnetic field is reversed. Theoretically, this effect is described by the so-called axion term. Here, we discuss a time-reversal-symmetric TME, where the electric charge depends only on the magnitude of the magnetic field but is independent of its sign. We obtain this n…
The dynamical axion field is a new state of quantum matter where the magnetoelectric response couples strongly to its low-energy magnetic fluctuations. It is fundamentally different from an axion insulator with a static quantized magnetoelectric response. The dynamical axion field exhibits many exotic phenomena such as axionic polariton and axion instability. However, these effects have not been experimentally confirmed due to the lack of proper topological magnetic materials. Here by combining analytic models and first-principles calculations, we predict a series of van der Waal layered Mn$_2…
We study theoretically a three-dimensional correlated and spin-orbit coupled system, the half-filled extended Fu-Kane-Mele-Hubbard model on a diamond lattice, focusing on the topological magnetoelectric response of the antiferromagnetic insulator phase. In the antiferromagnetic insulator phase, the Dirac-like low-energy effective Hamiltonian is obtained. Then the theta term, which results in the magnetoelectric response, is derived as a consequence of the chiral anomaly. The realization of the dynamical axion field in our model is discussed. The relation with a symmetry broken phase induced by…
Based on a first-principles approach, we show that in a single crystal of a prototypical topological insulator such as Bi$_2$Se$_3$ the difference in the work function between adjacent surfaces with different crystal-face orientations generates a built-in electric field around facet edges. Owing to the topological magnetoelectric coupling for a given broken time-reversal symmetry in the crystal, the electric field, in turn, forces effective magnetic dipoles to accumulate along the edges, realizing the facet-edge magnetic ordering. We demonstrate that the predicted magnetic ordering which depen…
The surface of a 3D topological insulator is a new type of two dimensional electron gas (2DEG). The exactly quantized magneto-optical or axionic response of a 3D topological insulator is a direct result of the Berry's phase protected spin-momentum locked states on its surface. (PRL Viewpoint http://physics.aps.org/node/3810).
We consider two semi-infinite magnetoelectric media with constant dielectric permittivity separated by a planar interface, whose electromagnetic response is described by non-dynamical axion electrodynamics and investigate the radiation of a point-like electric dipole located perpendicularly to the interface. We start from the exact Green's function for the electromagnetic potential, whose far-field approximation is obtained using a modified steepest descent approximation. This procedure yields the standard spherical waves as well as axially symmetric cylindrical superficial waves, which nevert…
Exploration of novel electromagnetic phenomena is a subject of great interest in topological quantum materials. One of the unprecedented effects to be experimentally verified is topological magnetoelectric (TME) effect originating from an unusual coupling of electric and magnetic fields in materials. A magnetic heterostructure of topological insulator (TI) hosts such an exotic magnetoelectric coupling and can be expected to realize the TME effect as an axion insulator. Here we designed a magnetic TI with tricolor structure where a non-magnetic layer of (Bi, Sb)2Te3 is sandwiched by a soft ferr…
Axion Insulator State in Hundred-Nanometer-Thick Magnetic Topological Insulator Sandwich Heterostructures
Department of Physics, The Pennsylvania State University（宾夕法尼亚州立大学物理系）；Department of Physics, Hong Kong University of Science and Technology（香港科技大学物理系）；Materials Research Institute, The Pennsylvania State University（宾夕法尼亚州立大学材料研究所）
Topological insulators have been proposed to be best characterized as bulk magnetoelectric materials that show response functions quantized in terms of fundamental physical constants. Here we lower the chemical potential of three-dimensional (3D) Bi$_2$Se$_3$ films to $\sim$ 30 meV above the Dirac point, and probe their low-energy electrodynamic response in the presence of magnetic fields with high-precision time-domain terahertz polarimetry. For fields higher than 5 T, we observed quantized Faraday and Kerr rotations, whereas the DC transport is still semi-classical. A non-trivial Berry phase…
We explore a model akin to axion electrodynamics in which the axion field $θ(t,\mathbf{x})$ rather than being dynamical is a piecewise constant effective parameter $θ$ encoding the microscopic properties of the medium inasmuch as its permittivity or permeability, defining what we call a $θ$-medium. This model describes a large class of phenomena, among which we highlight the electromagnetic response of materials with topological order, like topological insulators for example. We pursue a Green's function formulation of what amounts to typical boundary-value problems of $θ$-media, when external…
The 'magnetoelectric effect' arises from the coupling between magnetic and electric properties in materials. The Z2 invariant of topological insulators (TIs) leads to a quantized version of this phenomenon, known as the topological magnetoelectric (TME) effect. This effect can be realized in a new topological phase called an 'axion insulator' whose surface states are all gapped but the interior still obeys time reversal symmetry. We demonstrate such a phase using electrical transport measurements in a quantum anomalous Hall (QAH) sandwich heterostructure, in which two compositionally different…
The magnetoelectric coupling of electrons in a three-dimensional solid can be effectively described by axion electrodynamics. Here we report the discovery of the fractional magnetoelectric effect in chiral anomalous semimetals of the three-dimensional massless Wilson fermions, which have linear dispersions at the energy crossing point, and break the chiral symmetry at generic momenta. In the presence of electric and magnetic fields, the time-reversal and parity symmetry breaking give rise to the quarter-quantized topological magnetoelectric effect, which is directly related to the winding numb…
Axionic tunneling from a topological Kondo insulator
Rutgers University（罗格斯大学）；University of Greifswald（格赖夫斯瓦尔德大学）；Harvard University（哈佛大学）；Sun Yat-sen University（中山大学）；Zhejiang University（浙江大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Pennsylvania（宾夕法尼亚大学）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）
The interplay of non-trivial band topology and magnetism gives rise to a series of exotic quantum phenomena, such as the emergent quantum anomalous Hall (QAH) effect and topological magnetoelectric effect. Many of these quantum phenomena have local manifestations when the global symmetry is broken. Here, we report local signatures of the thickness dependent topology in intrinsic magnetic topological insulator MnBi$_2$Te$_4$(MBT), using scanning tunneling microscopy and spectroscopy on molecular beam epitaxy grown MBT thin films. A thickness-dependent band gap with an oscillatory feature is rev…
When time-reversal symmetry is broken on its surface, topological insulators exhibit a magnetoelectric response which is described by axion electrodynamics. A direct consequence of this theory is the appearance of a magnetic field that resembles the one produced by a magnetic image monopole when a point-like electric charge is located near the surface of the material. In this paper we investigate the more realistic problem when the point-like charge is replaced by a finite size sphere at constant potential. We calculate the electromagnetic fields using the potential formulation in a particular…
Axion, first postulated as a hypothetical particle in high-energy physics, is now extended to describe a novel topological magnetoelectric effect derived from the Chern-Simons theory in condensed matter systems. The recent discovered intrinsic magnetic topological insulators MnBi2Te4 and its derivatives have attracted great attention because of their potential as a material platform to realize such a quantized axion field. Since the magnetic exchange gap can bring the half-quantized anomalous Hall effect at the surface, an axion insulator manifests as quantum anomalous Hall and zero Hall plate…
The electrodynamics of topological insulators (TIs) is described by modified Maxwell's equations, which contain additional terms that couple an electric field to a magnetization and a magnetic field to a polarization of the medium, such that the coupling coefficient is quantized in odd multiples of $e^2 / 2 h c $ per surface. Here, we report on the observation of this so-called topological magnetoelectric (TME) effect. We use monochromatic terahertz (THz) spectroscopy of TI structures equipped with a semi-transparent gate to selectively address surface states. In high external magnetic fields,…
We establish the correspondence between the fractional charge bound to a vortex in a textured lattice and the relevant bulk band topology in two-dimensional (2D) topological crystalline insulators. As a representative example, we consider the Kekule textured graphene whose bulk band topology is characterized by a 2D $\mathbb{Z}_{2}$ topological invariant $ν_{\rm 2D}$ protected by inversion symmetry. The fractional charge localized at a vortex in the Kekule texture is shown to be related to the change in the bulk topological invariant $ν_{\rm 2D}$ around the vortex, as in the case of the Su-Sch…
Axion electrodynamics, first proposed in the context of particle physics, manifests itself in condensed matter physics in the topological field theory description of 3d topological insulators and gives rise to magnetoelectric effect, where applying magnetic (electric) field $\mathbf{B}(\mathbf{E})$ induces polarization (magnetization) $\mathbf{p}(\mathbf{m})$. We use linear response theory to study the associated topological current using the Fu-Kane-Mele model of 3d topological insulators in the presence of time-dependent uniform weak magnetic field. By computing the dynamical current suscept…
Many magnetic point-group symmetries induce a topological classification on crystalline insulators, dividing them into those that have a nonzero quantized Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those that do not ("axion-even" or "trivial"). For time-reversal or inversion symmetries, the resulting topological state is usually denoted as a "strong topological insulator" or an "axion insulator" respectively, but many other symmetries can also protect this "axion $Z_2$" index. Topological states are often insightfully characterized by choosing one crystallographi…
The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $θ=π$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of b…
The intricate interplay between nontrivial topology and magnetism in two-dimensional (2D) materials has led to the emergence of many novel phenomena and functionalities. An outstanding example is the quantum anomalous Hall (QAH) effect, which was realized in magnetically doped topological insulators (TIs) in the absence of magnetic field. Recently, the layered van der Waals compound MnBi2Te4 has been theoretically predicted and experimentally verified to be a TI with interlayer antiferromagnetic (AFM) order. It is a rare stoichiometric material with coexisting topology and magnetism, thus repr…
Floquet engineering is an important way to manipulate the electronic states of condensed matter physics. Recently, the discovery of the magnetic topological insulator MnBi$_2$Te$_4$ and its family provided a valuable platform to study magnetic topological phenomena, such as, the quantum anomalous Hall effect, the axion insulator state and the topological magnetoelectric effect. In this work, based on the effective model and first-principles calculations in combination with the Floquet theory, we reveal that the circularly polarized light (CPL) induces the sign reversal of the Chern number of o…
Half-quantized anomalous Hall conductance in topological insulator/ferromagnet van der Waals heterostructures
Linnaeus University（林奈大学）；Nordita, KTH Royal Institute of Technology and Stockholm University（北欧原子理论中心，皇家理工学院和斯德哥尔摩大学）；University of Connecticut（康涅狄格大学）；University of Texas at El Paso（德克萨斯大学埃尔帕索分校）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We study the electromagnetic properties of dense QCD in the so-called Magnetic Dual Chiral Density Wave phase. This inhomogeneous phase exhibits a nontrivial topology that comes from the fermion sector due to the asymmetry of the lowest Landau level modes. The nontrivial topology manifests in the electromagnetic effective action via a chiral anomaly term $~θF^{μν}\tilde{F}_{μν}$, with a dynamic axion field $θ$ given by the phase of the dual chiral density wave condensate. The coupling of the axion with the electromagnetic field leads to several macroscopic effects that include, among others, a…
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）
We search for dynamical magnetoelectric phenomena in three-dimensional correlated systems with spin-orbit coupling. We focus on the antiferromagnetic insulator phases where the dynamical axion field is realized by the fluctuation of the antiferromagnetic order parameter. It is shown that the dynamical chiral magnetic effect, an alternating current generation by magnetic fields, emerges due to the time dependences of the order parameter such as antiferromagnetic resonance. It is also shown that the anomalous Hall effect arises due to the spatial variations of the order parameter such as antifer…
We present a magneto-optical study of the three-dimensional topological insulator, strained HgTe using a technique which capitalizes on advantages of time-domain spectroscopy to amplify the signal from the surface states. This measurement delivers valuable and precise information regarding the surface state dispersion within <1 meV of the Fermi level. The technique is highly suitable for the pursuit of the topological magnetoelectric effect and axion electrodynamics.
We survey various quantized bulk physical observables in two- and three-dimensional topological band insulators invariant under translational symmetry and crystallographic point group symmetries (PGS). In two-dimensional insulators, we show that: (i) the Chern number of a $C_n$-invariant insulator can be determined, up to a multiple of $n$, by evaluating the eigenvalues of symmetry operators at high-symmetry points in the Brillouin zone; (ii) the Chern number of a $C_n$-invariant insulator is also determined, up to a multiple of $n$, by the $C_n$ eigenvalue of the Slater determinant of a nonin…
We study the interaction between a topological insulator nanoparticle and a quantum dot subject to an applied electric field. The electromagnetic response of the topological insulator is derived from axion electrodynamics in the quasistatic approximation. Localized modes are quantized in terms of dipolar bosonic modes, which couples dipolarly to the quantum dot. Hence, we treat the hybrid as a two-level system interacting with a single bosonic mode, where the coupling strength encodes the information concerning the nontrivial topology of the nanoparticle. The interaction of the hybrid with the…
Weyl semimetals (WSM) are a new class of topological materials that exhibit a bulk Hall effect due to time-reversal symmetry breaking, as well as a chiral magnetic effect due to inversion symmetry breaking. These unusual electromagnetic responses can be characterized by an axion term $θ\textbf{E} \cdot \textbf{B}$ with space and time dependent axion angle $θ(\textbf{r} ,t)$. In this paper we compute the electromagnetic fields produced by an electric charge near to a topological Weyl semimetal with two Weyl nodes in the bulk Brillouin zone. We find that, as in ordinary metals and dielectrics, o…
