Odd-Parity Multipoles by Staggered Magnetic Dipole and Electric Quadrupole Orderings in CeCoSi

Abstract

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 electric quadrupole ordering for a small crystal-field splitting. Furthermore, we discuss a magneto-electric effect and elastic-electric effect due to the odd-parity multipoles, which will be useful to identify order parameters in CeCoSi.The breaking of the spatial inversion symmetry has been attracting attention in condensed matter physics. The effect of the inversion symmetry breaking in crystals is described by an antisymmetric spin-orbit interaction (ASOI) in the form of g(k)·σ where g(k) is an odd function with respect to the wave vector k and σ is the spin. The ASOI leads to unconventional physical phenomena, such as a current-induced magnetization which is the so-called Edelstein effect, 1, 2) noncentrosymmetric superconductivity, 3) and spin Hall effect. 4,5) Similar noncentrosymmetric physics arises in a locally noncentrosymmetric system with the staggered-type ASOI once a spontaneous electronic ordering breaks the global inversion symmetry. For example, staggered magnetic and/or orbital orderings on a zigzag chain, 6-9) honeycomb, 10-14) diamond, 15,16) and bi-layer structures [17][18][19] give rise to clustertype odd-parity multipoles, such as the magnetic toroidal dipole and electric octupole. [20][21][22] The f -electron metallic compound CeCoSi is a candidate for such cluster-type oddparity multipoles. The crystal structure is a centrosymmetric tetragonal CeFeSi-type structure (P4/nmm, D 7 4h , No. 129) and there are two Ce sites (referred as Ce A and Ce B ) connected by the inversion operation, 23) as shown in Fig. 1(a). While changing temperature and pressure, CeCoSi undergoes two phase transitions: the antiferromagnetic (AFM) order at T N = 8.8 K at ambient pressure 24,25) and the hidden order, the latter of which dominantly appears under pressure. [26][27][28] Recently, the experiment implies that the hidden order under pressure corresponds to the antiferroquadrupole (AFQ) order, 27, 28) since it shows similar behavior to the AFQ phase observed in CeB 6 and CeTe. [29][30][31][32][33][34][35] Interestingly, the unit of the staggered AFM and AFQ orders in CeCoSi accompanies the cluster-type odd-parity multipoles, as the staggered alignment of the even-parity multipoles at two Ce sites breaks the global inversion symmetry. However, it has not been clarified what types of odd-parity multipoles can be active in the AFM and AFQ phases. The theoretical identifications are helpful not only to determine order parameters ...

NQR and NMR spectra in the odd-parity multipole material CeCoSi

“…In the present discussion, we consider the local multipole degrees of freedom at a Ce ion described by the low-energy crystal-field levels up to the firstexcited level. We assume that the low-energy levels consist of the ground-state Γ 7 doublet and the first-excited Γ 6 doublet in C 4v site symmetry [60]. Within these low-energy levels, even-parity electric and magnetic multipoles with rank l ≤ 5 are activated, as discussed below [60,[62][63][64][65].…”

Spin-orbital-momentum locking under odd-parity magnetic quadrupole ordering

“…We here consider the piezo-electric effect where the symmetric distortion ζ is induced by the electric field E ν , i.e., ζ = ν Λ ζν E ν for ζ = u, v, yz, zx, xy. The current-induced distortion tensor Λ ζν is calculated by the linear response theory as [9,79,102]…”

Multipole classification in 122 magnetic point groups for unified understanding of multiferroic responses and transport phenomena

“…Meanwhile, the concept of multipole covers not only the atomic electronic degrees of freedom but also ones over multi sites and orbitals, such as the hybrid multipole for the interorbital degrees of freedom between different orbitals [36,37,51], the cluster multipole for the on-site degrees of freedom in a cluster [16,52], the bond multipole for the off-site bond degrees of freedom in a cluster [53,54], and the k multipole for the band modulations and spin splittings in the electronic band structures [33,34]. The microscopic description in terms of the electronic multipole degrees of freedom is useful to understand the macroscopic responses in accordance with the crystallographic symmetry [33,34,[55][56][57][58][59][60], such as the magnetoelectric effect in the magnetic toroidal dipole orderings, e.g., Cr 2 O 3 [61], LiCoPO 4 [62,63], and UNi 4 B [64,65] and in the magnetic quadrupole orderings, e.g., Co 4 Nb 2 O 9 [66][67][68][69], anomalous Hall effect in the magnetic octupole orderings, e.g., Mn 3 Sn [15,16], magnetopiezoelectric effect in the magnetic quadrupole/hexadecapole orderings, e.g., Ba 1−x K x Mn 2 As 2 [70] and EuMn 2 Bi 2 [71], and magnetostriction effect in the magnetic octupole orderings [72,73]. Recent studies also clarified that the multipole description gives the systematic microscopic understanding of the band modulation in the AFM orderings [54,[74][75][76], e.g., κ-(BETD-TTF) 2 Cu[N(CN) 2 ]Cl…”