## トピック

### Magnetic topological degeneracy and transport phenomena

- Spin and anomalous Hall effects emerging from topological degeneracy in Dirac fermion system CuMnAs
,*Vu Thi Ngoc Huyen*,*Yuki Yanagi**Michi-To Suzuki*

Phys. Rev. B**104**, 035110/1-12 (2021)- Orthorhombic CuMnAs has been proposed as an antiferromagnetic semimetal hosting nodal line and Dirac points around the Fermi level. We investigate relations between the topological bands and transport phenomena, i.e., the spin Hall effect and anomalous Hall effect, in orthorhombic CuMnAs with first-principles calculations combined with a symmetry analysis of the magnetic structures and of the (spin) Berry curvature. We show the nodal line gapped with spin-orbit coupling in CuMnAs dominantly generates large spin Hall conductivity in the ground state. Although the magnetic symmetry in the ground state of CuMnAs forbids the finite anomalous Hall effect, applied magnetic fields produce a significant anomalous component of the Hall conductivity with the magnetic symmetry breaking. We identify that the dominant contribution to anomalous Hall components comes from the further lifting of band degeneracy under external magnetic fields for the Bloch states generated with the splitting of nodal lines by spin-orbit coupling near the Fermi energy.

- First-principles investigation of magnetic and transport properties in hole-doped

shandite compounds Co_{3}In_{x}Sn_{2-x}S_{2} , Junya Ikeda, Kohei Fujiwara, Kentaro Nomura, Atsushi Tsukazaki, and*Yuki Yanagi**Michi-To Suzuki*

Phys. Rev. B**103**, 205112/1-8 (2021)-
Co-based shandite Co
_{3}Sn_{2}S_{2}is a representative example of magnetic Weyl semimetals showing rich transport phenomena. We thoroughly investigate magnetic and transport properties of hole-doped shandites Co_{3}In_{x}Sn_{2-x}S_{2}by first-principles calculations. The calculations reproduce nonlinear reduction of anomalous Hall conductivity with doping In for Co_{3}Sn_{2}S_{2}as reported in experiments against the linearly decreased ferromagnetic moment within virtual crystal approximation. We show that a drastic change in the band parity character of Fermi surfaces, attributed to the nodal rings lifted energetically with In doping, leads to strong enhancement of anomalous Nernst conductivity with reversing its sign in Co_{3}In_{x}Sn_{2-x}S_{2}.

- Topology analysis for anomalous Hall effect in the noncollinear antiferromagnetic states of Mn
_{3}*A*N (*A*= Ni, Cu, Zn, Ga, Ge, Pd, In, Sn, Ir, Pt) - Vu Thi Ngoc Huyen,
, Kunihiko Yamauchi, and Tamio Oguchi*Michi-To Suzuki*

Phys. Rev. B**100**, 094426/1-9 (2019) -
We investigate topological features of electronic structures which produce large anomalous Hall effect in the noncollinear antiferromagnetic metallic states of antiperovskite manganese nitrides by first-principles calculations. We first predict the stable magnetic structures of these compounds to be noncollinear antiferromagnetic structures characterized by either T
_{1g}or T_{2g}irreducible representation by evaluating the total energy for all of the magnetic structures classified according to the symmetry and multipole moments. The topology analysis is next performed for the Wannier tight-binding models obtained from the first-principles band structures. Our results reveal the small Berry curvature induced through the coupling between occupied and unoccupied states with the spin-orbit coupling, which is widely spread around the Fermi surface in the Brillouin zone, dominantly contributes after the-space integration to the anomalous Hall conductivity, while the local divergent Berry curvature around Weyl points has a rather small contribution to the anomalous Hall conductivity.**k**

### Cluster multipole theory

- Multipole expansion for magnetic structures: A generation scheme for symmetry-adapted orthonormal basis set in crystallographic point group
, T. Nomoto, R. Arita,*Michi-To Suzuki*, S. Hayami, H. Kusunose*Y. Yanagi*

Phys. Rev. B**99**, 174407/1-10 (2019) Editors' suggestion-
We propose a systematic method to generate a complete orthonormal basis set of multipole expansion for magnetic structures in arbitrary crystal structure. The key idea is the introduction of a virtual atomic cluster of a target crystal on which we can clearly define the magnetic configurations corresponding to symmetry-adapted multipole moments. The magnetic configurations are then mapped onto the crystal so as to preserve the magnetic point group of the multipole moments, leading to the magnetic structures classified according to the irreducible representations of the crystallographic point group. We apply the present scheme to pyrochlore and hexagonal
*AB*O_{3}crystal structures and demonstrate that the multipole expansion is useful to investigate the macroscopic responses of antiferromagnets.

### First-principles study for multipole order phase

- First-principles theory of magnetic multipoles in condensed matter systems
, Hiroaki Ikeda, Peter M. Oppeneer*Michi-To Suzuki*

J. Phys. Soc. Jpn.**87**, 041008/1-24 (2018)- The multipole concept, which characterizes the spacial distribution of scalar and vector objects by their angular dependence, has already become widely used in various areas of physics. In recent years it has become employed to systematically classify the anisotropic distribution of electrons and magnetization around atoms in solid state materials. This has been fuelled by the discovery of several physical phenomena that exhibit unusual higher rank multipole moments, beyond that of the conventional degrees of freedom as charge and magnetic dipole moment. Moreover, the higher rank electric/magnetic multipole moments have been suggested as promising order parameters in exotic hidden order phases. While the experimental investigations of such anomalous phases have provided encouraging observations of multipolar order, theoretical approaches have developed at a slower pace. In particular, a materials’ specific theory has been missing. The multipole concept has furthermore been recognized as the key quantity which characterizes the resultant configuration of magnetic moments in a cluster of atomic moments. This cluster multipole moment has then been introduced as macroscopic order parameter for a noncollinear antiferromagnetic structure in crystals that can explain unusual physical phenomena whose appearance is determined by the magnetic point group symmetry. It is the purpose of this review to discuss the recent developments in the first-principles theory investigating multipolar degrees of freedom in condensed matter systems. These recent developments exemplify that ab initio electronic structure calculations can unveil detailed insight in the mechanism of physical phenomena caused by the unconventional, multipole degree of freedom.

### Anomalous Hall effect in antiferromagnets

- Cluster multipole theory for anomalous Hall effect in antiferromagnets
, T. Koretsune, M. Ochi, R. Arita*Michi-To Suzuki*

Phys. Rev. B**95**, 094406/1-11 (2017) Editors' suggestion-
We introduce a cluster extension of multipole moments to discuss the anomalous Hall effect (AHE) in both ferromagnetic (FM) and antiferromagnetic (AFM) states in a unified framework. We first derive general symmetry requirements for the AHE in the presence or absence of the spin-orbit coupling, by considering the symmetry of the Berry curvature in
space. The cluster multipole (CMP) moments are then defined to quantify the macroscopic magnetization in non-collinear AFM states, as a natural generalization of the magnetization in FM states. We identify the macroscopic CMP order which induces the AHE. The theoretical framework is applied to the non-collinear AFM states of Mn**k**_{3}Ir, for which an AHE was predicted in a first-principles calculation, and Mn_{3}Z (Z=Sn, Ge), for which a large AHE was recently discovered experimentally. We further compare the AHE in Mn_{3}Z and bcc Fe in terms of the CMP. We show that the AHE in Mn_{3}Z is characterized with the magnetization of a cluster octupole moment in the same manner as that in bcc Fe characterized with the magnetization of the dipole moment.

### First-principles study for heavy fermion systems

- Change of Fermi surface topology in CeRu
_{2}Si_{2}studied by LSDA+*U*method and Hisatomo HARIMA*Michi-To SUZUKI*

J. Phys. Soc. Jpn.**79**(2010) 024705/1-5.-
The detailed electronic structures of CeRu
_{2}Si_{2}have been investigated for a nonmagnetic state and a magnetic state under magnetic fields by an LSDA+*U*method. It is found that the Si position in the crystal structure is an essential parameter to reproduce precise topology of the experimentally determined Fermi surfaces of non-*f*reference LaRu_{2}Si_{2}. As for CeRu_{2}Si_{2}, the LDA calculation fails to predict the nonmagnetic ground state, which is experimentally reported to be mainly occupied by |*j*=5/2,*j*=±5/2> orbitals, and the Fermi surfaces. The nonmagnetic case solution of LSDA+_{z}*U*method greatly improves the electronic structure in the nonmagnetic CeRu_{2}Si_{2}. An LSDA+*U*method is also applied to investigate the electronic structure under applied magnetic fields, then the change of the so-called large Fermi surfaces to small Fermi surfaces is successfully described.

- Emergent Loop-Nodal
*s*_{±}-Wave Superconductivity in CeCu_{2}Si_{2}: Similarities to the Iron-Based Superconductors - Hiroaki Ikeda,
, and Ryotaro Arita*Michi-To Suzuki*

Phys. Rev. Lett.**114**, 147003/1-5 (2015) -
Heavy-fermion superconductors are prime candidates for novel electron-pairing states due to the spin-orbital coupled degrees of freedom and electron correlations. Superconductivity in CeCu
_{2}Si_{2}discovered in 1979, which is a prototype of unconventional (non-BCS) superconductors in strongly correlated electron systems, still remains unsolved. Here we provide the first report of superconductivity based on the advanced first-principles theoretical approach. We find that the promising candidate is an*s*_{±}-wave state with loop-shaped nodes on the Fermi surface, different from the widely expected line-nodal*d*-wave state. The dominant pairing glue is magnetic but high-rank octupole fluctuations. This system shares the importance of multiorbital degrees of freedom with the iron-based superconductors. Our findings reveal not only the long-standing puzzle in this material, but also urge us to reconsider the pairing states and mechanisms in all heavy-fermion superconductors.

### Iron-based superconductors

- First-principles study of magnetic properties in Fe-ladder compound BaFe
_{2}S_{3} , Ryotaro Arita, and Hiroaki Ikeda*Michi-To Suzuki*

Phys. Rev. B**92**, 085116/1-6 (2015)-
We study the magnetic, structural, and electronic properties of the recently discovered iron-based superconductor BaFe
_{2}S_{3}based on density functional theory with the generalized gradient approximation. The calculations show that the magnetic alignment in which the spins are coupled ferromagnetically along the rung and antiferromagnetically along the leg is the most stable in the possible magnetic structure within an Fe ladder and is further stabilized with the periodicity characterized by the wave vector**Q**=(π,π,0), leading to the experimentally observed magnetic ground state. The magnetic exchange interaction between the Fe ladders creates a tiny energy gap, the size of which is in excellent agreement with the experiments. Applied pressure suppresses the energy gap and leads to an insulator-metal transition. Finally, we also discuss what type of orbitals can play crucial roles on the magnetic and insulator-metal transition.

### Hidden order

- Multipole order and global/site symmetry in the hidden order phase of URu
_{2}Si_{2} and Hiroaki Ikeda*Michi-To Suzuki*

Phys. Rev. B**90**,184407/1-8 (2014)-
On the basis of group theory and first-principles calculations, we investigate high-rank multipole orderings in URu
_{2}Si_{2}, which have been proposed as a genuine primary order parameter in the hidden-order phase below 17.5 K. We apply Shubnikov group theory to the multipole ordered states characterized by the wave vector**Q**_{0}=(0,0,1) and specify the global/site symmetry and the secondary order parameters, such as induced dipole moments and change in charge distribution. We find that such antiferroic magnetic multipole orderings are particularly advantageous to conceal the primary order parameter due to preserving high symmetry in charge distribution. Experimental observations of the induced low-rank multipoles, which are explicitly classified in this paper, will be key pieces to understand the puzzling hidden-order phase.