Dynamical Axion Field in Topological Magnetic Insulators

Dynamical Axion Field in Topological Magnetic Insulators

Rundong Li, Jing Wang, Xiaoliang Qi & Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305-4045 Department of Physics, Tsinghua University, Beijing 100084, China
July 15, 2019

Axions are very light, very weakly interacting particles postulated more than 30 years ago in the context of the Standard Model of particle physics. Their existence could explain the missing dark matter of the universe. However, despite intensive searches, they have yet to be detected. In this work, we show that magnetic fluctuations of topological insulators couple to the electromagnetic fields exactly like the axions, and propose several experiments to detect this dynamical axion field. In particular, we show that the axion coupling enables a nonlinear modulation of the electromagnetic field, leading to attenuated total reflection. We propose a novel optical modulators device based on this principle.

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The electromagnetic response of a three dimensional insulators is described by the Maxwells action , with material-dependent dielectric constant and magnetic permeability , where and are the electromagnetic fields inside the insulator. However, generally, it is possible to include another quadratic term in the effective action , where is the fine structure constant, and is a parameter describing the insulator in question. In the field theory literature this effective action is known as the axion electrodynamics wilczek1987 (), where plays the role of the axion field. Under the periodic boundary condition the partition function and all physical quantities are invariant if is shifted by integer multiples of . Therefore all time reversal invariant insulators fall into two distinct classes described by either or  Qi2008 (). Topological insulators are defined by and can only be connected continuously by time reversal breaking perturbations to trivial insulators defined by . The form of the effective action implies that an electric field can induce a magnetic polarization, whereas a magnetic field can induce an electric polarization. This effect is known as the topological magneto-electric effect (TME) and has the meaning of the magneto-electric polarization . Physically the parameter depends on the band structure of the insulator and has a microscopic expression of the momentum space Chern-Simons form Qi2008 ()


where is the momentum space non-abelian gauge field, with indices referring to the occupied bands. The parameter has been calculated explicitly for several basic models of topological insulatorsQi2008 (); Essin2009 (). In a topological insulator the axion field gives rise to novel physical effects such as the image monopole and anyonic statistics qi2009 (). This field, however, is static in a time-reversal invariant topological insulator. In this work, we consider the anti-ferromagnetic long range order in a topological insulator, which breaks time-reversal symmetry spontaneously, so that becomes a dynamical axion field taking continuous values from to . In the following we will refer to such an antiferromagnetic insulator as a “topological magnetic insulator”. We propose a minimal model in which the antiferromagnetic order break the time reversal symmetry spontaneously and the magnetic fluctuations couple linearly to the axion field, thus realizing the dynamic axion field in condensed matter systems. Compared to its high energy version, the axion proposed here has the advantage that it can be observed in controlled experimental settings Wilczek2009 (). With an externally applied magnetic field, the axion field couples linearly to light, resulting in the axionic polariton. By measuring the attenuated total reflection, the gap in the axionic polariton dispersion can be observed. An attractive feature is that the axionic polariton gap is tunable by changing the external electric or magnetic fields. The control of the light transmission through the material enables a novel type of optical modulator. We also propose another experiment to detect the dynamic axion by microcantilever torque magnetometry, where the double frequency response of the cantilever is a unique signature of the dynamic axion field.

We propose several materials that may realize the topological magnetic insulator with dynamic axion field. One possibility is the topological insulator , , doped with transition metal elements such as  Zhou2006 (); Larson2008 (). Another possible class of material is the 5d transition metal compound with and standing for some 5d transition metal and some alkali metal, respectively. Electrons in 5-orbital can have both strong spin orbital coupling and strong interaction, which is ideal for the realization of the topological magnetic insulatorNagaosa2009 (). We show that such a compound with the corundum structure may have a topological magnetic insulator phase if the states closed to fermi level are formed by orbitals with total angular momentum Kim2009 (). The detail of this proposal is beyond the goal of the present work, and will be presented in a separate paper.wang2009 () We also noticed two very recent works on 5d transition metal compounds with pyrochlore structure, which may also realize the topological magnetic insulator phase.pesin2009 (); guo2009 ()

Effective model for the 3D topological insulator

Although all the physical effects discussed in this paper are generic for any system supporting axionic excitation and do not rely on a specific model, we would like to start from a simple model for concreteness. We adopt the effective model proposed by Zhang et al in Ref. Zhanghaijun2009 () to describe topological insulators , and . The low energy bands of these materials consist of a bonding and an anti-bonding state of orbitals, labeled by and , respectively. The generic form of the effective Hamiltonian describing these four bands is obtained up to quadratic order of momentum in Ref. Zhanghaijun2009 (). Since a lattice regularization is necessary for computing axion field , in the present paper we start from a lattice version of this model, with the Hamiltonian


where , , and the Dirac matrices have the representation in the basis of .

breaking terms.

The above Hamiltonian preserves both time reversal symmetry and parity .

Figure 1: [Color Online] Crystal structure of with three primitive lattice vectors denoted as . A quintuple layer with is indicated in the orange box. The spin ordering configuration giving rises to the mass is indicated by the black arrow, which is antiferromagnetic along direction and ferromagnetic within plane.

In the effective model (2), the time-reversal and spatial inversion transformation are defined as (with the complex conjugation operator) and , respectively. The axion field can be calculated by formula (1) which gives or depending on the value of parametersQi2008 (), as expected from time reversal symmetry. Now we consider a perturbation to the Hamiltonian (2) which can lead to deviation of from or . Since is odd under time reversal and parity operation, only time reversal and parity breaking perturbations can induce a change of . By simple algebra one can show that to the leading order the breaking perturbation must have the form . Thus the perturbed Hamiltonian can still be written as with . For this model, Eq. (1) can be reduced to an explicit expression for :


Although there are four independent parameters in breaking term, only leads to a correction to to the linear order. Thus in the following we will take without the loss of generality, and leave only the term. To see the physical meaning of the term we change the basis to with and . We see that . Physically, and stand for states of the two lattice sites in each unit cell, e.g. the two atom sites in , which are shifted away from the inversion center along direction. By transforming to the new basis, we see that it represents a staggered Zeeman field that points in direction on the sublattice. This staggered Zeeman field can be generated by an antiferromagnetic order, where electron spins point along opposite directions on two sublattices, as shown in Fig. 1. Without the term, topological insulators have protected surface states consisting of odd number of massless Dirac cones. With broken time reversal symmetry, a direct consequence of the term is that it opens a gap in the surface state spectrum. The surface state gap is equal to , independent on the orientation of the surface.

Dynamical axion in the anti-ferromagnetic phase

We have seen above that the axion field can deviate from or if the electrons are coupled to the antiferromagnetic order parameter. Such a staggered field can be induced if an antiferromagnetic long range order is established in the system. Physically, an antiferromagnetic ordered phase can be obtained naturally if the screened Coulomb interaction of electrons become strong. For example, in the family doped with magnetic impurities such as , the substitution of -electrons of or by -electrons of the magnetic impurities effectively enhanced the on-site repulsion of electrons. In the basis the Hamiltonian with interaction can be written as


in which the first term is the kinetic energy given by Eq. (2) and the rest two terms represent the on-site repulsion and the inter-site repulsion between and sites. Possible ordered phase resulting from the interaction include the ferromagnetic phase where the spin on two sublattices and point in the same direction, the antiferromagnetic phase where the spin on two sublattices point in the opposite direction, and the charge density wave (CDW). Correspondingly the order parameters are taken as the ferromagnetic order parameter , the antiferromagnetic order parameter and the CDW order parameter . It is assumed that translational symmetry is preserved and all the order parameters are uniform in space. In the mean field approximation, we find that for a wide range of value for band structure parameters , and , the system develops antiferromagnetic order pointing in the direction if the effect of dominates that of , which thus leads to and axion field .

Axion electrodynamics.

In the mean-field approximation, the antiferromagnetic phase has a static axion field . However, the antiferromagnetic phase also has amplitude and spin wave excitations, which can induce fluctuations of the axion field. The fluctuation of the Neel vector can be generally written as . To the linear order, it can be shown from symmetry analysis that the fluctuation of axion field only depends on , since is a pseudo-scalar. In other words, we have where the coefficient can be determined from Eq. (3). The dispersion of the amplitude mode can be obtained in standard RPA approximation, leading to a massive axion field . Considering the coupling term of axion with electromagnetic field, the effective action describing the axion-photon coupled system is given by


where and are the stiffness, velocity and mass of the spin wave mode , and are the electric field and the magnetic field respectively, and are the dielectric constant and magnetic permeability respectively. The second term describes the topological coupling between the axion and the electromagnetic field, with the fine-structure constant. The third term describes the dynamics of the massive axion. Within the model we have adopted, the parameters and are given by


where and the repeated index indicates summation with .

The axionic polariton.

The dynamic axion field couples nonlinearly to the external electromagnetic field combination . When there is an externally applied static and uniform magnetic field parallel to the electric field of the photon, will couple linearly to  Maiani1986 (). In condensed matter systems, when a collective mode is coupled linearly to photons, hybridized propagating modes called polariton emergemills1974 (). The polaritons can be coupled modes of optical phonon and light through the electric dipole interaction, or coupled modes of magnon and light through the magnetic dipole interaction. Here we propose a novel type of polariton—axionic polariton which is the coupled mode of light and the axionic mode of an antiferromagnet. The dispersion of the axionic polariton can be obtained from the effective action (5) which leads to the following linearized equation of motion Raffelt1988 (); Cameron1993 ()

Figure 2: [Color Online] Axionic polariton and ATR experiment. a. The dispersion of the axionic polariton. The grey area indicates the forbidden band between frequencies and (see text), within which light can not propagate in the sample. The red dashed line shows the bare photon dispersion . b. Setup for the attenuated total reflection (ATR) experiment. Without external magnetic field, the incident light can transmit through the sample. c. When an external magnetic field is applied parallel to the electric field of light, the incident light will be totally reflected if its frequency lies within the forbidden band.

where is the speed of light in the media and is the dielectric constant. Compared to photon, the dispersion of axion can be neglected, in which case the axionic polaritons have the dispersion


with . As shown in Fig. 2 a, this dispersion spectrum consists of two branches separated by a gap between and . The quantity measures the coupling strength between the axion field and the electric field and is proportional to the external magnetic field . Upon turing on , the axionic mode at changes its frequency from to , due to the linear mixing between the axion and the photon field. Physically, the axionic polariton is very similar to the transverse optical phonon polariton, since the axion also leads to an additional contribution to the charge polarization due to the topological magneto-electric effectQi2008 (), . The optical phonon polariton has the same dispersion as Eq. (8), with the parameter replaced by the lattice unscreened plasmon frequency . The key difference between axion and optical phonon is that the coupling between axion and electric field is determined by the external magnetic field , which is thus tunable.

The gap in the axionic polariton spectrum may be experimentally observed using the attenuated total reflection (ATR) method. The geometry is arranged such that the incident light is perpendicular to the surface of the sample and the static magnetic field is parallel to the electric field of light, as shown in Fig. 2 b and c. Since the light can only propagate through the media in the form of the axionic polariton, when the frequency of the incident light is within the gap of the axionic polariton spectrum, a significant increase of the reflectivity will be observed. To estimate the gap we adopt the tight-binding parameters obtained for Zhanghaijun2009 () in Hamiltonian (2). We take a typical exchange splitting for an antiferromagnet , and an estimated dielectric constant . With a magnetic field , we obtain the axion mass and . The gap is approximately , which can be observed experimentally. One unique signature of the axionic polariton is the dependence of the gap on , which can be used to distinguish from usual magnetic polaritons.

Figure 3: [Color Online] Cantilever Torque Magnetometry measurement of axion. A cantilever experiences a torque when the magnetization of the deposited topological magnetic insulator induced by the AC electric field is placed in the applied magnetic field . The and frequency response comes from normal contribution and dynamical axion contribution, respectively. “iOSCAR” is so called interrupted oscillating cantilever-driven adiabatic reversal protocol which can be used to measure the frequency change of cantileverBudakian2003 ().

By changing the magnitude of we can selectively determine the frequency band within which the light is totally reflected. This principle may find application as an amplitude optical modulator working at far-infrared frequency.

Measuring the axion by microcantilever.

As discussed in Ref. Qi2008 (), a static leads to a topological magneto-electric effect, in which a magnetization is induced by electric field. When the dynamics of axion field is considered, the behavior of the magnetization is modified, which can be detected by microcantilever torque magnetometry (MTM). Lohndorf2000 (); Budakian2003 (); Budakian2005 () As shown in Fig. 3, a DC magnetic field and an AC electric field are applied to the topological insulator attached to the tip of the cantilever, with an angle between them. A magnetization will be induced along the direction of the electric field, so that a torque acting on the cantilever is generated. This magnetic force on the cantilever mimics a change in cantilever stiffness, in turn, shifts the cantilever frequency by  Budakian2003 (); Budakian2005 (), Here is the cantilever stiffness, is the cantilever natural frequency, is the effective length of cantilever. With the electric field , the time dependent frequency change is,




It should be noticed that the response with doubled frequency is a unique signature of the axion dynamics, since all the response should have been linear and thus in frequency of if the axion field is static. With typical cantilever parameters in present experiments Budakian2003 (); Budakian2005 () kHZ, N/m, m, G, V/m, Hz and , we obtain the frequency shift Hz and Hz, which are easily detectable in current experimental techniques.

In conclusion, we have proposed the existence of dynamical axions in topological magnetic insulators. We have proposed two experiments in which axions can be detected by its unique coupling to the electromagnetic field. The coupling between axion and photon leads to an axionic polariton which has a polariton gap tunable by magnetic field and thus may be used as a tunable optical modulator.

We wish to thank T. L. Hughes, S. B. Chung, S. Raghu, J. Maciejko, R. B. Liu and B. F. Zhu for insightful discussions. This work is supported by the US Department of Energy, Office of Basic Energy Sciences under contract DE-AC03-76SF00515. JW acknowledges the support of China Scholarship Council, NSF of China (Grant No.10774086), and the Program of Basic Research Development of China (Grant No. 2006CB921500).


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