Spectral Indicators of Piezomagnetically Induced Symmetry Breaking in Altermagnets

Abstract

Recent developments in the multipole reformulation of X-ray absorption spectroscopy (XAS) have provided a unified framework to describe magnetic and orbital responses in terms of ferroic multipole order parameters. X-ray magnetic circular dichroism (XMCD) is known to probe spin, orbital, and anisotropic magnetic dipole (AMD) moments. Its applications to altermagnets and noncollinear antiferromagnets have revealed that the XMCD response is often governed by the ferroic states of the AMD in the photo-excited states rather than by conventional magnetic dipoles in the ground states. In this work, we extend the multipole-based analysis to X-ray magnetic linear dichroism (XMLD) and demonstrate that XMLD in altermagnets can be understood as a manifestation of piezomagnetic effects: linear couplings between magnetic dipole and electric quadrupole moments. Using symmetry analysis combined with exact diagonalization calculations of $L_{2,3}$-edge XAS, we systematically investigate representative altermagnets, including $α$-MnTe, MnF$_2$, and CrSb. We show that the ferroic ordering of higher-rank magnetic multipoles, particularly spinful magnetic octupoles, gives rise to characteristic field-odd XMLD signals that directly reflect the underlying piezomagnetic response tensors allowed by magnetic point-group symmetry. Furthermore, we discuss XMCD signals induced by piezomagnetic effects, in which strain generates magnetic dipole moments. Our results establish XMLD and XMCD as element-specific probes of magnetoelastic multipole order in altermagnets and provide a general symmetry-based pathway to identify hidden ferroic multipoles and strain-controllable spin phenomena beyond conventional ferromagnetism.

Figures (4)

• Figure 1: Schematic illustration of electronic and magnetic response functions in altermagnetic systems. (Left) Response of the electric quadrupole moment $Q_{x^2-y^2}$ to an external magnetic field $H$, where brown arrows indicate the direction of the applied magnetic field. In an altermagnet possessing magnetic dipole symmetry $M_z$, the application of a magnetic field induces an electric quadrupole moment through the coupling between $M_z$ and the field. At sufficiently high magnetic fields, however, the magnetic dipole $M_z$ can reverse its orientation. As a result, the induced quadrupole response becomes symmetric (even) with respect to the magnetic field $H_z$. In contrast, if magnetic octupoles, $M_{xyz}$ and $M_{x(x^2-y^2)}$, do not directly flip with the magnetic field, they lead to antisymmetric (odd) linear responses in $Q_{x^2-y^2}$ as a function of $H_z$ or $H_x$, respectively. (Right) Response of the magnetic dipole to external symmetric strain $\sigma_{x^2-y^2}$, with green arrows indicating the direction of the applied strain. Each panel illustrates the emergence of magnetic moments under corresponding multipolar orderings. Similar to the magnetic field response, the induced magnetic dipole is governed by the linear coupling between the applied strain and the pre-existing multipolar symmetry of the $d$-wave and $g$-wave altermagnetic states.
• Figure 2: Magnetic multipole structure and X-ray absorption spectra in $\alpha$-MnTe. (a) Crystal and magnetic structure of $\alpha$-MnTe with the NiAs-type lattice. The in-plane Néel vector is oriented along the $[1\bar{1}00]$ direction, described by the magnetic point group $mm'm'$. The green arrows represent the spin direction. This symmetry allows (b) a magnetic dipole $M_{10}$ along the $c$-axis and (c) a spinful magnetic dipole $M_{10}^{(1,1)}$, whose microscopic distribution originates from the $yz$-type quadrupole moment associated with $s_y$ spin component, where $x \parallel a$, $y \parallel b^{*}$, and $z \parallel c$. In the spherical-harmonic representation, red and blue colors denote the sign of the magnetic monopole charge density defined by $\mathbf{m}\cdot\mathbf{r}$, while green and yellow indicate the sign of the spin operator. (d) X-ray absorption spectrum (XAS) and X-ray magnetic circular dichroism (XMCD) spectra around the Mn $L$ edge for magnetic fields applied parallel and antiparallel to the $c$-axis. (e) X-ray magnetic linear dichroism (xyXMLD) spectra measured between orthogonal linear polarizations $E\parallel a$ and $E\parallel b$ under opposite magnetic fields. (f) Magnetic-field dependence of the XMLD intensity at $E = 638$ eV. (g) XMCD spectra under uniaxial stress ($P_{\sigma}=\pm0.3$ GPa) applied perpendicular to the mirror plane. (h) Comparison between magnetic-field-induced and uniaxial-XMCD spectra. (i) Uniaxial-stress dependence of the XMCD intensity at $E = 638$ eV.
• Figure 3: Magnetic multipole structure and X-ray absorption spectra in MnF$_2$. (a) Crystal and magnetic structure of MnF$_2$ with the rutile structure. The antiferromagnetic spins are aligned along the $c$-axis, reducing the symmetry to the magnetic point group $4'/mmm'$. This symmetry allows (b) the $xyz$-type magnetic octupole moments ($M_{3,\pm2}$), and (c) their spinful counterpart of the $xy$-type quadrupole associated with the $s_z$ spin component, $M_{3,\pm2}^{(1,-1)}$, where $x \parallel a$, $y \parallel b$, and $z \parallel c$. (d) X-ray absorption spectrum (XAS) and X-ray magnetic circular dichroism (XMCD) spectra around the Mn $L$ edge for magnetic fields applied parallel and antiparallel to the $c$-axis. (e) X-ray magnetic linear dichroism (xyXMLD) spectra measured between orthogonal linear polarizations ($E\parallel x$ and $E\parallel y$) under opposite magnetic fields. (f) Magnetic-field dependence of the XMLD intensity at $E = 638$ eV. The xyXMLD signal exhibits a linear dependence on the applied magnetic field and reverses sign depending on the Néel vector orientation ($\mathbf{N}\parallel[001]$ and $[00\bar{1}]$). (g) XMCD spectra under uniaxial pressure ($P_{\sigma}=\pm0.3$ GPa). (h) Comparison between magnetic-field-induced and uniaxial-pressure-induced XMCD spectra. (i) Uniaxial-pressure dependence of the XMCD intensity at $E = 638$ eV.
• Figure 4: Magnetic multipole structure and calculated X-ray absorption spectra in CrSb. (a) Crystal and magnetic structure of CrSb with the magnetic point group $6'/m'mm'$. This symmetry permits (b) the $(x^3-3xy^2)$-like magnetic octupole moments ($M_{3,\pm3}$) and (c) their spinful counterpart of the $z(x^3-3xy^2)$-type hexadecapole associated with the $s_z$ spin component, $M_{3,\pm3}^{(1,1)}$, where $x \parallel a$, $y \parallel b^{*}$, and $z \parallel c$. (d) X-ray absorption spectrum (XAS) and X-ray magnetic circular dichroism (XMCD) spectra around the Cr $L$ edge for magnetic fields applied parallel and antiparallel to the $b^{*}$ direction. (e) X-ray magnetic linear dichroism (xyXMLD) spectra measured between orthogonal linear polarizations ($E\parallel a$ and $E\parallel b^{*}$) under opposite magnetic fields. The XMLD signal reverses its sign upon inversion of the magnetic multipole domain, demonstrating its odd response with respect to the external magnetic field. (f) Magnetic-field dependence of the xyXMLD intensity at $E = 582$ eV. The signal exhibits a linear dependence on the applied magnetic field and changes sign depending on the Néel vector orientation ($\mathbf{N}\parallel[0001]$ and $[000\bar{1}]$). (g) XMCD ($k||b^*$) spectra under uniaxial pressure ($P_{a}=\pm0.3$ GPa). The stress-induced XMCD emerges through piezomagnetic symmetry breaking even in the absence of net magnetization. (h) Comparison between magnetic-field-induced and uniaxial-pressure-induced XMCD spectra. (i) Uniaxial-pressure dependence of the XMCD intensity at $E = 574$ eV.