Four types of atomic-scale multipoles, electric, magnetic, magnetic toroidal, and electric toroidal multipoles, give a complete set to describe arbitrary degrees of freedom for coupled charge, spin, and orbital of electrons. We here present a systematic classification of these multipole degrees of freedom towards the application in condensed matter physics. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structures and linear responses, such as the magneto-electric effect, magneto-current (magneto-gyrotropic) effect, spin conductivity, Piezo-electric effect, and so on. Moreover, we exhibit a complete table to represent the active multipoles under 32 crystallographic point groups. Our comprehensive and systematic analyses will give a foundation to identify enigmatic electronic order parameters and a guide to evaluate peculiar cross-correlated phenomena in condensed matter physics from microscopic point of view.

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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 ...

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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 inversion symmetry. We mainly examine behaviors of the NQR and NMR spectra in three odd-parity multipole ordered states: a y-type magnetic toroidal dipole order with a staggered x-type antiferromagnetic structure, an xy-type electric toroidal quadrupole order with a staggered x 2 − y 2-type antiferroquadrupole structure, and a z-type electric dipole order with a staggered 3z 2 − r 2type antiferroquadrupole structure. We show that different odd-parity multipole orders lead to different fielddependent spectral splittings in NMR, while only the xy-type electric toroidal quadrupole order exhibits the NQR spectral splitting. We also present possible sublattice-dependent spectral splittings for all the odd-parity multipole orders potentially activated in low-energy crystal-field levels, which will be useful to identify oddparity order parameters in CeCoSi by NQR and NMR measurements.

Magnetic toroidal dipole (MTD) is one of a fundamental constituent to induce magneto-electric effects in the absence of both spatial inversion and time-reversal symmetries. We report on a microscopic investigation of the atomic-scale MTD in solids by taking into account the orbital degree of freedom with a different parity. We construct an effective twoorbital d-f tight-binding model on a polar tetragonal system for describing the atomic-scale MTD, which are obtained by incorporating the atomic spin-orbit coupling and odd-parity hybridization. The effective model exhibits two types of the MTDs: in-plane x, y components activated through spontaneous ferromagnetic ordering or external magnetic field and an out-of-plane z component by a spontaneous odd-parity hybridization without spin moments. We show that the intraorbital (inter-orbital) Coulomb interaction in multi-orbital systems plays an important role in stabilizing the in-plane (out-of-plane) MTD orderings. We also examine the magneto-electric effect under each MTD ordering by calculating a linear response tensor. We show that the odd-parity hybridization enhances the magneto-electric effect for the in-plane MTDs, while it suppresses that for the out-of-plane MTD.

We theoretically propose a realization of a nonlinear nonreciprocal transport in antiferromagnets without relying on the relativistic spin-orbit coupling. Through the symmetry and microscopic model analyses, we show that a local spin scalar chirality to induce an asymmetric band modulation becomes a source of a Drude-type nonlinear transport, while an electric polarization induced by a collinear spin configuration in a triangle unit leads to a Berry-curvature-dipole-type nonlinear transport. We demonstrate that 120 • antiferromagnetic ordering on a triangular lattice and a breathing kagome lattice in an external magnetic field are typical examples. Our results open a new direction of designing and engineering functional materials with showing rich parity-violating transport phenomena by spontaneous magnetic phase transitions.

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