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Axial Magnetoelectric Effect in Dirac Semimetals

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Axial Magnetoelectric Effect in Dirac Semimetals PHYSICAL REVIEW LETTERS 126, 247202 (2021) Axial Magnetoelectric Effect in Dirac Semimetals Long Liang ,1 P. O. Sukhachov ,2 and A. V. Balatsky1,3 1Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 2Department of Physics, Yale University, New Haven, Connecticut 06520, USA 3Department of Physics and Institute for Materials Science, University of Connecticut, Storrs, Connecticut 06269, USA (Received 13 December 2020; revised 1 March 2021; accepted 18 May 2021; published 17 June 2021) We propose a mechanism to generate a static magnetization via the “axial magnetoelectric effect” à (AMEE). Magnetization M ∼ E5ðωÞ × E5ðωÞ appears as a result of the transfer of the angular momentum of the axial electric field E5ðtÞ into the magnetic moment in Dirac and Weyl semimetals. We point out similarities and differences between the proposed AMEE and a conventional inverse Faraday effect. As an example, we estimated the AMEE generated by circularly polarized acoustic waves and find it to be on the scale of microgauss for gigahertz frequency sound. In contrast to a conventional inverse Faraday effect, magnetization rises linearly at small frequencies and fixed sound intensity as well as demonstrates a nonmonotonic peak behavior for the AMEE. The effect provides a way to investigate unusual axial electromagnetic fields via conventional magnetometry techniques. DOI: 10.1103/PhysRevLett.126.247202 — à Introduction. We are now witnessing a surge of interest M ∼ E5ðωÞ × E5ðωÞ: ð1Þ in quantum matter with an unusual relativisticlike dis- persion relation [1–4]. We mention bosonic (acoustic Therefore, we call this phenomenon the “axial magneto- waves, photonic crystals, and magnons) and fermionic electric effect” (AMEE). The AMEE is qualitatively differ- (A phase of superfluid helium-3, d-wave superconductors, ent from the IFE. The synthetic nature of the pseudoelectric graphene, topological insulators) Dirac and Weyl materials. field in Dirac and Weyl semimetals means that one can The key topological features for all of them are the linear induce magnetization using phonons without any electro- energy spectrum and the presence of several gapless nodes magnetic fields. This situation reminds us of the difference (valleys) that enable exotic artificial fields, which is unique between the Aharonov-Bohm [15] and Aharonov-Casher for Dirac materials. [16] effects, where the gauge potentials, while similar, have The dynamics of excitations in Dirac and Weyl semi- a different origin. metals and their interplay with electromagnetic fields is a The origin of the pseudoelectromagnetic fields is rooted rapidly developing research area. We focus on the inverse in the relativisticlike energy spectrum and topology of Faraday effect (IFE) [5–9] that is a well-known example of Dirac matter. In the case of Dirac semimetals, axial gauge light-matter interaction. In this case, the angular momen- fields could be generated via strains [17–23] as well as tum from a dynamical electromagnetic (photon) field EðtÞ magnetization textures [24,25]. In graphene, pseudomag- is transferred into the magnetic moment of electrons, netic fields were observed experimentally in Refs. [26–29]. leading to a static magnetization M ∼ EðωÞ × EÃðωÞ. Previous work focuses mostly on static axial fields. The This effect was recently investigated for Dirac and Weyl studies related to the dynamics of these fields have quickly semimetals in Refs. [10–14]. become an active research area. In particular, we mention In this Letter, we propose a mechanism to generate a ultrasonic attenuation [21,30,31], the acoustogalvanic static magnetization in Dirac and Weyl semimetals through effect [32], and torsion-induced chiral magnetic current dynamical axial gauge fields, whereby the angular momen- [33]. The circularly polarized sound can be generated by tum is transferred from the axial or pseudoelectric field acoustic transducers coupled to acoustic quarter-wave E5ðtÞ into the magnetic moment of electrons plates [34]; see also Refs. [35–37]. In addition to the acoustic method, we also point out recent work on gen- eration of circularly polarized phonons [38]. It has been shown that such phonons can be induced optically [39] in Cd3As2 [40]. Therefore, the proposed AMEE is timely and Published by the American Physical Society under the terms of instrumental in the elucidation of the axial fields in three- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to dimensional semimetals. the author(s) and the published article’s title, journal citation, The AMEE is universal in the sense of measuring the and DOI. Funded by Bibsam. response of nodal states to dynamic axial fields regardless 0031-9007=21=126(24)=247202(6) 247202-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 126, 247202 (2021) FIG. 2. The Feynman diagram used to calculate the AMEE. The cross denotes an external magnetic field. The wavy lines represent the axial gauge fields. General theory.—For weak deviations from equilibrium, the magnetization is obtained as a functional derivative of FIG. 1. Schematic model setup for the axial magnetoelectric effect. Circularly polarized axial electric fields lead to the static the effective action S with respect to a magnetic field à M − →0δ δB magnetization M ∼ E5ðωÞ × E ðωÞ of Dirac semimetals. The ¼ limB S= . (See Fig. 2 for the corresponding 5 −A resulting magnetization is picked up by a magnetometer Feynman diagram.) Here, A5;μ ¼fA5;0; 5g is the axial such as SQUID. gauge field that gives rise to a time-dependent axial electric field E5 ¼ −∂tA5 − ∇A5;0. While the diagram in Fig. 2 resembles that for the chiral anomaly, it has a different of their origin. Therefore, it could be realized in various structure, where the external lines correspond to two axial experimental setups where the position of the Weyl nodes and one vector gauge fields. Therefore, the chiral anomaly can be manipulated (for a recent study of the manipulation action does not contribute to the magnetization. This allows of the Weyl nodes via light pulses, see Ref. [41]). us to calculate the effect separately for each Weyl node. We By using a dynamical deformation as a characteristic further assume that the wave vector of the chiral electric example, we found that the induced magnetization scales as field is small enough to ignore the internode transitions. ∝ μω ω M for small frequencies at fixed sound intensity. In general, there could be first-order effects, i.e., M ∼ E5. This is in drastic contrast to the conventional IFE, where They, however, produce an oscillating magnetization that ∝ 1 ω the magnetization scales as M = in the dirty limit. The averages to zero for dynamical axial electric fields as well explanation of this scaling is simple and follows from the as requires an explicit breakdown of the time-reversal nature of the axial electric field. As with conventional IFEs, symmetry. Therefore, we focus on a second-order rectifi- the induced magnetization scales as the second power of cation effect, which gives a static magnetization even in a E E the effective axial electric field 5. Because 5 is propor- time-reversal symmetric system. This fact makes the tional to the time derivative of the gauge potentials, which AMEE universal and attractive from a practical point of ∼ ω in turn are determined by the strain tensor uij , it results view. For example, while the first-order effects cancel in in a two powers of frequency difference when the sound Dirac semimetals, the AMEE survives. 2 intensity I ∼ ω is fixed. Hence, we expect that the AMEE Without the loss of generality, we assume that E5 is has the overall scaling ∼ω2 relative to the IFE for conven- polarized in the x–y plane. Then, the magnetization for a tional metals. Furthermore, our estimates show that the single node is magnetization induced via the AMEE is within the reach of − χ ω q E E modern magnetometry technology such as the supercon- Mz ¼ i ð ; Þð 5;ω;q × 5;−ω;−qÞz; ð2Þ ducting quantum interference device (SQUID). The corre- sponding model setup is schematically shown in Fig. 1. with the response function χðω; qÞ given by Z e3 X χðω; qÞ¼ d k I ðω; qÞðvmvm∂ vm − vmvm∂ vmÞ 4ω2 mmm y x kx y y y kx x m X Z 2 m 2 m m m m m m þ d k Immlðω; qÞðϵl − ϵmÞ½ðϵl − ϵmÞ ðΩxyÞ þ vx vx gyy − vx vy gxyþðx ↔ yÞ : ð3Þ l≠m ω q Ωm m Here, e is the electric charge, the expression for Imnlð ; Þ and xy and gij are the Berry curvature [48] and the is given in the Supplemental Material (SM) p[42]ffiffiffiffiffiffiffiffiffi, m ¼ quantum metric [49] of the band, respectively (see the ϵ 2 2 denotes the electron and hole bands, Æ ¼ vi ki is the SM [42] for the explicit expressions). The first term in m ∂ ϵ the curly brackets is the intraband contribution, which energy dispersion, vi ¼ ki m is the quasiparticle velocity, 247202-2 PHYSICAL REVIEW LETTERS 126, 247202 (2021) 10 2 depends only on the dispersion. The second term originates 10 from interband processes. In this Letter, we focus on the 20 orbital contribution to the magnetization. The spin con- 8 tribution is estimated to be weak for intraband processes: 10 ω ≪ 2 μ 6 j j (see also Ref. [14] for the IFE results). 0 The minimal Weyl Hamiltonian X 4 -10 H ¼ λ viσipi ð4Þ -20 2 1.5 2 2.5 i was used in the derivation of the response function χðω; qÞ. 0 Here, λ ¼denotes the chirality of Weyl nodes, σ are the i -2 Pauli matrices, p ¼ −i∇ is the momentum operator, and ℏ 0.1 0.15 0.2 0.25 0.3 and the light velocity c are taken to be unity throughout this Letter.
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