The optical Hall response is theoretically studied for spin-orbit coupled metals with ferroic orders of cluster-type magnetic multipoles. We find that different magnetic multipoles give rise to distinct spectra in the optical Hall conductivity. In the cases of monopole and quadrupole orders, the optical Hall response appears predominantly in high- and low-energy regions, which correspond to the energy scales of electron correlation and kinetic energy, respectively, while the response is dispersed and rather weak in the case of toroidal order. By decomposing the spectra into different interband contributions, we reveal selection rules stemming from the interplay between the antisymmetric spin-orbit coupling and the underlying multipoles. Our results suggest that the optical Hall measurement is useful to detect and distinguish the cluster-type magnetic multipole orders.
Magnetic monopoles and toroidal order are compelling features that have long been theorized but remain elusive in real materials. Multiferroic hexagonal ferrites are an interesting realization of frustrated triangular lattice, where magnetic order is coupled to ferroelectricity and trimerization. Here we propose a mechanism, through which magnetic monopolar and toroidal orders emerge from the combination of 120$^\circ$ antiferromagnetism and trimerization, present in hexagonal manganites and ferrites. The experimentally observable signatures of magnetic monopolar and toroidal orders are identi…
We report our theoretical results on spin conductivity in antiferromagnets by focusing on the role of the magnetic toroidal quadrupole (MTQ) in electron systems. The MTQ is characterized as a time-reversal-odd rank-2 polar tensor degree of freedom in electrons, which is distinct from conventional rank-1 magnetic and magnetic toroidal dipoles. Based on a microscopic $sd$ model analysis for a tetragonal system under both collinear and noncollinear antiferromagnetic orderings, we clarify that the MTQ becomes a source of an extrinsic spin conductivity even with neither a uniform magnetization nor …
We report our theoretical results on the order parameters for the pyrochlore metal Cd$_2$Re$_2$O$_7$, which undergoes enigmatic phase transitions with inversion symmetry breaking. By carefully examining active electronic degrees of freedom based on the lattice symmetry, we propose that two parity-breaking phases at ambient pressure are described by unconventional multipoles, electric toroidal quadrupoles (ETQs) with different components, $x^2-y^2$ and $3z^2-r^2$, in the pyrochlore tetrahedral unit. We elucidate that the ETQs are activated by bond or spin-current order on Re-Re bonds. Our ETQ s…
We show that orbital currents in a CuO2 plane, if present, should be described by two independent parity and time-reversal odd order parameters, a toroidal dipole (anapole) and a magnetic quadrupole. Based on this, we derive the resonant X-ray diffraction cross-section for monoclinic CuO at the antiferromagnetic wavevector and show that the two order parameters can be disentangled. From our analysis, we examine a recent claim of detecting anapoles in CuO.
In vortex-like spin arrangements, multiple spins can combine into emergent multipole moments. Such multipole moments have broken space-inversion and time-reversal symmetries, and can therefore exhibit linear magnetoelectric (ME) activity. Three types of such multipole moments are known: toroidal, monopole, and quadrupole moments. So far, however, the ME-activity of these multipole moments has only been established experimentally for the toroidal moment. Here, we propose a magnetic square cupola cluster, in which four corner-sharing square-coordinated metal-ligand fragments form a noncoplanar b…
We present a general formalism of multipole descriptions under the space-time inversion group. We elucidate that two types of atomic toroidal multipoles, i.e., electric and magnetic, are fundamental pieces to express electronic order parameters in addition to ordinary electric and magnetic multipoles. By deriving quantum-mechanical operators for both toroidal multipoles, we show that electric (magnetic) toroidal multipole higher than dipole (monopole) can become a primary order parameter in a hybridized-orbital system. We also demonstrate emergent cross-correlated couplings between electric, m…
We discuss possible approaches to the problem of the URu2Si2 "Hidden Order" (HO) which remains unsolved after tremendous efforts of researches. Suppose there is no spatial symmetry breaking at the HO transition temperature and solely the time-reversal symmetry breaking emerges owing to some sort of magnetic order. As a result of its 4/mmm symmetry, each uranium atom is a three-dimensional magnetic vortex; its intra-atomic magnetization M(r) is intrinsically non-collinear, so that its dipole, quadrupole and toroidal moments vanish, thus making the vortex "hidden". The first non-zero magnetic mu…
We investigate a possibility of odd-parity multipole orderings in a locally noncentrosymmetric tetragonal compound CeCoSi. By performing symmetrical and microscopic mean-field analyses on a two-orbital tight-binding model, we propose potential odd-parity multipoles hidden in staggered antiferromagnetic and antiferroquadrupole orderings in CeCoSi. We show that $3z^2-r^2$ type of the magnetic quadrupole is induced by the staggered magnetic dipole ordering for a large crystal-field splitting between two orbitals, while $xy$ type of the electric toroidal quadrupole is emergent by the staggered ele…
Here we report a resonant x-ray diffraction (RXD) study at the Cu L3 edge on the multi-chiral system CsCuCl3, exhibiting helical magnetic order in a chiral crystal structure. RXD is a powerful technique to disentangle electronic degrees of freedom due to its sensitivity to electric monopoles (charge), magnetic dipoles (spin), and electric quadrupoles (orbital). We characterize electric quadrupole moments around Cu ascribed to the unoccupied Cu 3d orbital, whose quantization axis is off the basal plane. Detailed investigation of magnetic reflections reveals additional sinusoidal modulations alo…
We report our theoretical study on nucleation of a magnetic vortex around a nonmagnetic impurity doped into ferromagnets. The mechanism lies in the asymmetric Dzyaloshinskii-Moriya interaction arising from the breaking of spatial inversion symmetry by the impurity. By using the spin-wave analysis and Monte Carlo simulations, we show that the asymmetric interaction induces a magnetic vortex with nonzero vorticity $l=+1$. The vortex is stabilized even for less frustration in exchange interactions and in the absence of an external magnetic field. We also find that the magnetic vortex is character…
We theoretically investigate possible symmetry-broken states in ${\rm UNi_4B}$, constructing a localized pseudo triplet crystalline-electric field model. For a long time, its low-temperature symmetry-broken phase in ${\rm UNi_4B}$ has been considered to be a magnetic toroidal order forming atomic-scale vortices lattice with disordered sites at each center of the vortices. However, recent observation of current-induced magnetizations offers a reinvestigation about the validity of this order parameter because of the contradiction in their anisotropy. Our model takes into account the quadrupole d…
We theoretically study NQR and NMR spectra in the presence of odd-parity multipoles originating from staggered antiferromagnetic and antiferroquadrupole orderings. For the $f$-electron metal, CeCoSi, which is a candidate hosting odd-parity multipoles, we derive an effective hyperfine field acting on Co nucleus generated from electronic origin multipole moments at Ce ion under zero and nonzero magnetic fields. We elucidate that emergent odd-parity multipoles give rise to sublattice-dependent spectral splittings in NQR and NMR through the effective hyperfine coupling in the absence of the global…
Odd-parity magnetic and magnetic toroidal multipoles in the absence of both spatial-inversion and time-reversal symmetries are sources of multiferroic and nonreciprocal optical phenomena. We investigate electronic states caused by an emergent odd-parity magnetic quadrupole (MQ) as a representative example of magnetic odd-parity multipoles. It is shown that spontaneous ordering of the MQ leads to an antisymmetric spin-orbital polarization in momentum space, which corresponds to a spin-orbital momentum locking at each wave vector. By symmetry argument, we show that the orbital or sublattice degr…
A whole series of expressions for four species of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal) is provided as a complete basis set to describe arbitrary single-centered spinful electron systems. A compact formula to calculate matrix elements of these multipoles is also derived. A visualization method of an electronic state characterized in terms of multipoles is proposed. The complete basis set is useful to narrow down a candidate order parameter of electron systems in phase transition, to describe a property of cross-correlated phenomena, to analyze spectra of x-r…
We submit that the magnetic space-group Cac (#9.41) is consistent with the established magnetic structure of BaFe2Se3, with magnetic dipole moments in a motif that uses two ladders [Caron J M et al 2011 Phys. Rev. B 84 180409(R)]. The corresponding crystal class m1' allows axial and polar dipoles and forbids bulk ferromagnetism. The compound supports magneto-electric multipoles, including a magnetic charge (monopole) and an anapole (magnetic toroidal dipole) visible in the Bragg diffraction of x-rays and neutrons. Our comprehensive simulation of neutron Bragg diffraction by BaFe2Se3 exploits e…
