# Theory of NMR and NQR spectra under odd-parity multipole orderings

The breaking of the spatial inversion symmetry in solids has been attracting much interest, since it causes a variety of interesting physical phenomena, such as the magneto-electric effect and nonreciprocal transport. The electronic ordered parameters which break the spatial inversion symmetry are expressed as "odd-parity multipoles", e.g., magnetic toroidal dipole and magnetic quadrupole. These odd-parity multipoles have been observed by using the second harmonic generation and x-ray resonant scalttering experiments.

In the present study, we investigate a further possibility of the direct observation for odd-parity multipoles via NMR and NQR measurements.
We consider the f-electron compound CeCoSi, which is a candidate for odd-parity multipoles, as a target material.
In CeCoSi, it is suggested that the staggered antiferromagnetic and antiferroquadrupole ordered states are realized in experiments, which are regarded as the magnetic toroidal dipole and electric toroidal quadrupole from the viewpoint of the odd-parity multipoles.
We examine how odd-parity multipoles modulate the NMR and NQR spectra by analyzing the theoretical model.
By deriving an effective hyperfine coupling between ^{59}Co nulear moments and Ce f-electron multipoles in each ordered state, we find that resonant spectral peaks are split once the odd-parity multipole are activated.
Moreover, we show that the way of splittings in NMR and NQR strongly depends on the types of odd-parity multipoles, which can be identified through the measurements.

M. Yatsushiro and S. Hayami,
Phys. Rev. B 102, 195147 (2020)

# Microscopic theory of spin-split and reshaped electronic band structures without relying on spin-orbit coupling based on augmented multipoles

Spin-orbit coupling (SOC) is one of the most important elements in condensed matter physics, which is a source of various fascinating phenomena, such as magneto-electric effect, anomalous Hall effect, and nonreciprocal transport. Meanwhile, the recent studies have revealed that a specific antiferromagnetic phase transition leads to a momentum-dependent spin splitting in the band structure, which results in similar SOC-related physics even without relying on the SOC. This provides potential features of antiferromagnetic spintronics with considerably weak SOC in light-element materials.

Motivated by these studies, we theoretically developed a formalism and established an efficient microscopic bottom-up design procedure of electronic band structures in SOC free antiferromagnets. By introducing the concept of augmented multipoles, we demonstrated that the momentum-dependent spin splittings are caused by the effective multipole couplings. Our general scheme can be ubiquitously applied to any types of magnetic orderings with collinear, coplanar, and noncoplanar spin structures and any lattice systems, and predict cross-correlated responses in detail through analysis of the effective multipole couplings. To promote further experimental and theoretical developments, we list up various candidate materials showing intrinsic band deformations in accordance with MAGNDATA, magnetic structures database.

S. Hayami, Y. Yanagi, and H. Kusunose,
Phys. Rev. B 101, 220403(R) (2020),

S. Hayami, Y. Yanagi, and H. Kusunose,
Phys. Rev. B 102, 144441 (2020)

# Nonreciprocal magnon excitations by anisotropic bond interactions

Asymmetric magnon excitations with respect to the wave number in noncentrosymmetric magnets has drawn considerable interest in condensed matter physics, since it leads to intriguing phenomena, such as nonreciprocal directional optics. On the basis of the microscopic theory, the mechanisms of such nonreciprocal magnon excitations are limited in the noncentrosymetric systems with the Dzyaloshinskii-Moriya interaction and the frustrated systems with the competing exchange interactions.

To open up a further mechanism to induce nonreciprocal magnons, we focus on the role of magnetic anisotropy in the lattice structure.
We consider the staggered antiferromagnetic ordering in the honeybomc lattice structure, which breaks the global inversion symmetry.
By performing a linear-spin wave claculation using the Holstein-Primakoff transformation, we theoretically elucidate that the symmetric anisotropic exchange intearction on the bond becomes the microscopic origin of the nonreciprocal magnon exciations.
We also show that the direction of magnon nonreciprocity is controlled by applying the in-plane magnetic field.

T. Matsumoto and S. Hayami,
Phys. Rev. B 101, 224419 (2020)

# Chiral stripe phase by the d-p hybridization under the strong spin-charge coupling

In condensed matter physics, mutual interplay between electronic degrees of freedom, i.e., charge, spin, and orbital, gives rise to various fascinating phenomena, such as the colossal magnetoresistance and multiferroics. One of the typical materials showing such entanglement physics is a 3d transition metal oxide SrFeO3. The recent experiment shows that SrFeO3 exhibits a complex magnetic phase diagram, which comprises a plethora of multiple-Q spiral states, such as double-Q and quadruple-Q states. Although it has been known that such a multiple-Q spiral state is stabilized by the Dzyaloshinskii-Moriya interaction originating from the spin-orbit coupling, the spin-orbit coupling in SrFeO3 has been considered to be negligibly small. Thus, the mechanism of multiple-Q states in SrFeO3 is still an open problem. Recently, theoretical studies have shown that the spin-charge coupling between itinerant electron spin and localized spin gives another route to stabilize the multiple-Q states even without the spin-orbit coupling. There, the effective magnetic interactions between localized spins mediated by itinerant electrons, which include the Ruderman-Kittel-Kasuya-Yosida interaction, play an important role inducing the spiral state and multiple-Q spiral states. This type of mechanism works well for the materials with the weak spin-charge coupling, such as the f-electron compounds. Meanwhile, a possibility of multiple-Q states in the strong-coupling regime, which might be appropriate for SrFeO3, remains unclear.

In the present study, we investigate a possible realization of multiple-Q states in the d-p model on a square lattice with the d-p hybridisation and strong spin-charge coupling.
We examine what types of magnetic states are realized in the model by variational calculations.
We determine the optimal magnetic states in the ground state by comparing the internal energies for various magnetic states: ferromagnetic, staggered antiferromagnetic, single-Q spiral, and double-Q spiral states.
As a result, we find that the double-Q state becomes the ground-state for the strong d-p hybridaisation regime.
The result indicates the importance of the d-p hybridization to realize multiple-Q states in the strong-coupling system such as transition metal oxide systems.

R. Yambe and S. Hayami,
J. Phys. Soc. Jpn. 89, 013702 (2020)

# Cluster odd-parity multipoles in CeCoSi

In a solid, electrons with the charge, spin, and orbital degrees of freedom, give rise to intriguing quantum states due to the synergetic effect between the spin-orbit coupling and crystal field. Such quantum states have been well described by introducing microscopic multipoles, whose concept has been mainly developed in localized f-electron systems. In fact, the conventional multipole orderings, such as electric quadrupole and magnetic octupole orderings, have been discovered in the f-electron systems. On the other hand, odd-parity multipoles, which are activated in the absence of the spatial inversion symmetry, have recently been attracted. For instance, magnetic toroidal dipole and electric octupole belong to this category. One of the main characters under odd-parity multipoles is the cross-correlation phenomena including the magneto-electric (ME) effect and piezo-electric effect. The f-electron compound CeCoSi is a candidate to exhibit odd-parity multipoles. Although the crystal structure of CeCoSi is cnetrosymmmetric, the spontaneous staggered electronic orderings with the ordering vector q=0 break the spatial-inversion symmetry, which lead to the emergence of odd-parity multipoles. Recently, the experiments indicated the antiferromagnetic (AFM) order at ambient pressure and the antiferroquadrupole (AFQ) order under high pressure, although their order parameters have not been identified yet.

In the present study, we investigate what types of odd-parity multipoles are activated and how cross-correlation phenomena are induced in CeCoSi. First, we investigate potential candidates of odd-parity multipoles by taking into account the crystal field splitting of the 4f electron at Ce ion and the hybridization with the Co 3d electrons. Next, we construct an effective two-orbital model and examine stable multipole orderings within the mean-field calculations. As a result, we find that staggered AFM ordering along the z direction, which is accompanied with odd-parity 3z2-r2-type magnetic quadrupole, is dominantly stabilized for a large crystal field splitting. On the other hand, when the crystal field splitting becomes relatively small under pressure, the x2-y2 type AFQ ordering with odd-parity xy-type electric toroidal quadrupole is stabilized.
As odd-parity multipoles are expected to be identified by detecting a current-induced magnetization and current-induced distortion, our results will provide a deep understanding of electronic orderings observed by the experiments.

M. Yatsushiro and S. Hayami,
J. Phys. Soc. Jpn. 89, 013703 (2020)

# Spin-split band structure induced by collinear magnetic orderings

The atomic spin-orbit coupling, which couples the orbit and spin degrees of freedom in electrons, has been attracting much interest, as it leads to unusual topological properties and multiferroics phenomena. Especially, in the noncentrosymmetric cyrstals, an effective magnetic field depending on the asymmetric potential gradient in the crystal structure couples with the electron spin, which results in the spin conductive phenomena, such as the spin-current generation and spin-Hall effect. Such spin-dependent conductive properties are well understood from the emergent antisymmetric spin-orbit coupling which arises from the coupling between electron momentum and spin in wave-number space. Meanwhile, the recent studies show that the spontaneous symmetry breaking of the crystal symmetry provides another spin-orbit coupling physics. Such an emergent spin-orbit interaction is caused by the antiferromagnetic ordering even without the atomic spin-orbit coupling, which gives alternative way to engineer the spin-orbit coupling physics for the materials with the small atomic spin-orbit coupling, such as organic conductors and 3d transition metal oxides. In fact, an antiferromagnetic organic conductor, κ-(BEDT-TTF)2Cu[N(CN)2]Cl, exhibits the symmetric spin splitting in the electronic band structure and spin-current generation owing to the glide symmetry breaking. However, the microscopic conditions for emergent spin-orbit physics are still unclear.

We here theoretically investigate the symmetry and microscopic conditions for the spin-split band structure by focusing on the collinear antiferromagnet in the absence of the atomic spin-orbit coupling.
As a result, we clarify that anisotropic kinetic motions of electrons in a collinear antiferromagnet gives rise to an effective spin–orbit interaction in momentum space on the basis of a microscopic multipole description.
We present a systematic classification of the spin splitting in terms of specific antiferomagnetic ordered patterns under 32 point groups, which provides a reference to explore physical phenomena driven by the spin-split band structures, such as a spin-current generation by electric (thermal) current and a uniform magnetization by a strain field.

S. Hayami, Y. Yanagi, and H. Kusunose,
J. Phys. Soc. Jpn. 88, 123702 (2019)

# Electric toroidal quadrupole in spin-orbit-coupled metal Cd2Re2O7

Spatial inversion symmetry in crystals is one of the most important elements to characterize physical phenomena. Among them, the atomic spin-orbit coupling without the spatial inversion symmetry has been extensively studied, as it becomes a source to induce interesting phenomena, such as multiferroics and spin-Hall effect. Recently, such phenomena caused by the spatial inversion symmetry breaking is understood by using a concept of "odd-parity multipoles" that can express the microscopic degrees of freedom in electron systems. For example, odd-parity multipoles in the presence of time-reversal symmetry consist of odd-rank multipoles, such as electric dipole and octupole, and even-rank multipoles, such as electric toroidal monopole and quadrupole. In particular, spontaneous electronic orderings of electric dipole and electric toroidal monoopole and quadrupole moments give rise to the antisymmetric spin splitting in the band structure and linear magneto-electric effect. However, it remains elusive when such odd-parity multipoles emerge and how they are detected in experiments.

We here investigate the pyrochlore compound Cd2Re2O7 in 5d electron systems as a prototype of odd-parity multipole orderings.
In particular, we focus on a possibility that the electric toroidal quadrupole corresponds to the order parameters for the two structural phase transitions observed around 200K and 120K.
By analyzing a tight-binding model which incorpolartes the effect of the atomic spin-orbit coupling, we clarify the microscopic expressions of the electric toroidal quadrupoles hidden in the tetragaonal unit.
Moreover, we find that the orderings of the electroic toroidal quadrapoles induce the antisymmetric spin splitting in the electronic band structure, cross-correlated phenomena, and nonreciprocal transport.

S. Hayami, Y. Yanagi, H. Kusunose, and Y. Motome,
Phys. Rev. Lett. 122, 147602 (2019)

# Classification of multipoles in solids

Interplay between fundamental electronic degrees of freedom in solids, such as charge, spin, and orbital, has attracted growing interest in various fields of condensed matter physics. The concept of multipoles, which are used to characterize electric charge and current distributions, has been widely developed to describe such multiple degrees of freedom in a unified way. Under the space-time inversion group, there are four types of multipoles according to their spatial inversion and time-reversal properties: electric (E: polar tensor with time-reversal even), magnetic (M: axial tensor with time-reversal odd), magnetic toroidal (MT: polar tensor with time-reversal odd), and electric toroidal (ET: axial tensor with time-reversal even) multipoles. The multipole description has been extensively studied in f-electron systems as an atomic object, while it is extended to a cluster consisting of several atomic sites, multiple hybrid orbitals, and molecular orbitals. These studies of multipoles are useful to cover various unconventional order parameters in a systematic manner and expect physical phenomena from the symmetry and microscopic viewpoints.

In the present study, we discuss a general microscopic formalism for four types of multipoles and its application to solids~.
Starting from the classical multipole expansion of electromagnetic scalar and vector potentials, and using the mutual relationship among four multipoles, we derive the quantum-mechanical operator expressions of MT and ET multipoles in addition to conventional E and M multipoles.
We demonstrate that these multipole degrees of freedom can be active in the Hilbert space spanned by orbitals with different azimuthal quantum number, e.g., s-d, p-d, and d-f hybrid orbitals.
We also discuss emergent cross-correlated couplings, such as magneto-electric and magneto(electro)-elastic couplings under multipole orderings.
Moreover, we present a comprehensive classification of multipoles under 32 crystallographic point groups and what physical phenomena are expected in unconventional MT and ET multipoles.

S. Hayami, M. Yatsushiro, Y. Yanagi, and H. Kusunose,
Phys. Rev. B 98, 165110 (2018)** [selected in Editors' Suggestion]**

# Neel- and Bloch-type skyrmions in spin-orbit coupled metals

Noncoplanar spin textures in itinerant magnets have been attracting much interest in condensed matter physics, since they act as a huge effective magnetic field for itinerant electrons through the spin Berry phase mechanism and bring about unusual quantum transport phenomena. Chiral magnets with the spin-orbit coupling (SOC) are good platforms of stabilizing noncoplanar spin configurations, such as skyrmion crystals. Besides the SOC, recent theoretical studies demonstrated different origins of similar noncoplanar spin structures by the spin-charge coupling in itinerant magnets.

In the present study, we push forward these theoretical studies to a more realistic situation by considering both the SOC and the spin-charge coupling in magnetic conductors. Specifically, we consider the Kondo lattice model with the Rashba-type SOC on a polar square system. We derive an effective spin model with generalized Ruderman-Kittel-Kasuya-Yosida interactions including the anisotropic and antisymmetric exchange interactions. By performing numerical simulated annealing, we find that the model exhibits vortex crystals of both Neel and Bloch type even in the absence of an external magnetic field. Moreover, we show that a magnetic field turns the vortex crystals into Neel- and Bloch-type skyrmion crystals.

S. Hayami and Y. Motome,
Phys. Rev. Lett. 121, 137202 (2018)** [selected as Cover image]**

S. Hayami and Y. Motome,
IEEE Transactions on Magnetics 55, 0018-9464 (2018)

# Derivation of quantum-mechanical operators for toroidal multipoles

Mutual interplay between fundamental degrees of freedom of electrons in solids, i.e., charge, spin, and orbital, has attracted growing interest in various context. The concept of multipole has been developed to describe such complex electronic degrees of freedom, such as magnetic monopole and electric quadrupole, in a unified manner. Under the space-time inversion group, there are four types of multipoles according to their presence/absence of spatial inversion and time-reversal symmetries: electric (E) multipole, magnetic (M) multipole, magnetic toroidal (MT) multipole, and electric toroidal (ET) multipole. Among the toroidal multipoles, the MT dipole has been extensively investigated due to its potential role for exotic phenomena, such as magneto-electric effect and nonreciprocal directional dichroism. Such a MT dipole is often identified with a vortex-type magnetic orderings over several atomic sites as a classical object. However, MT multipole can arise even at each atomic site as a quantum object. Thus, it is desirable to obtain quantum-mechanical operator expressions of toroidal multipoles, in contrast to previous discussions as classical electromagnetic quantities. Once we obtain such expressions, we can clarify when it can be a primary order parameter characterizing thermodynamic phases in condensed matter.

In the present study, we discuss a general microscopic formalism to describe not only MT but also ET multipoles. By summarizing the classical description in the expansion of electromagnetic potentials, and using the mutual correspondence among four fundamental multipoles, we derive the quantum-mechanical operator expressions of both ET and MT multipoles. We demonstrate that the atomic ET and MT multipoles can be activated in the Hilbert space spanned by orbitals with different azimuthal quantum number, e.g., s-d, p-d, and d-f hybrid orbitals, as shown in Fig. 1. We find that the atomic ET and MT multipoles in addition to ordinary E and M multipoles constitute a complete set to express an arbitrary degree of freedom in the hybrid orbitals. We also demonstrate emergent cross-correlated couplings, such as magneto-electric and magneto(electro)-elastic couplings, in the presence of an ET or MT multipole ordering.

S. Hayami and H. Kusunose,
J. Phys. Soc. Jpn. 87, 033709 (2018)

# Stabilization mechanism of noncoplanar magnetic orderings in itinerant magnets

Noncoplanar spin textures in itinerant magnets have been attracting much interest in condensed matter physics, since they act as a huge effective magnetic field for itinerant electrons through the spin Berry phase mechanism and bring about unusual quantum transport phenomena. Chiral magnets with the spin-orbit coupling (SOC) are good platforms of stabilizing noncoplanar spin configurations, such as skyrmion crystals. Recently, however, another origin of noncoplanar spin configuration has been explored in itinerant magnets. The key ingredient is the instability of the Fermi surfaces at particular electron fillings, which does not need the presence of the SOC. Here, we investigate a further possibility of noncoplanar spin configurations in itinerant magnets.

For this purpose, we consider the Kondo lattice model on two-dimensional square and triangular lattices. By performing a large-scale simulation based on Langevin dynamics, we find double-Q vortex crystals on the square lattice, while we find triple-Q skyrmion crystals with an unusual topological number of two on the triangular lattice at zero magnetic field. We examine the stabilization mechanism by two methods in a complimentary way: perturbation expansion with respect to the spin-charge coupling and variational calculations. All these studies give consistent results and shows that noncoplanar multiple-Q states become a new ground state in itinerant magnets. We also derive the effective spin model to capture the underlying physics of the instability toward noncoplanar multiple-Q states in itinerant magnets, which will be helpful to avoid laborious calculations for the itinerant electron systems.

S. Hayami, R. Ozawa, and Y. Motome,
Phys. Rev. B 95, 224424 (2017),
erratum ** [selected in Kaleidoscopes]**

R. Ozawa, S. Hayami, and Y. Motome,
Phys. Rev. Lett. 118, 147205 (2017)

R. Ozawa, S. Hayami, K. Barros, G. W. Chern, Y. Motome, and C. D. Batista,
J. Phys. Soc. Jpn. 85, 103703 (2016)

S. Hayami and Y. Motome,
Phys. Rev. B 90, 060402(R) (2014)

# Skyrmion crystals in frustrated magnets

Magnetic skyrmions, which are characterized as topologically-nontrivial magnetic textures, have been extensively studied since their discovery in B20 compound without the inversion symmetry, since they lead to unconventional topological Hall effect and magneto-electric effect. It is known that the Dzyaloshinskii-Moriya interaction originating from the atomic spin-orbit coupling without the inversion symmetry plays an important role in realizing magnetic skyrmions. Meanwhile, recent theoretical studies exhibit that the skyrmions are stabilized in frustrated magnets.

In the present study, we investigate a classical Heisenberg model with the frustrated exchange interactions and the single-ion anisotropy on a triangular lattice in order to clarify the stability of skyrmion crystals without the atomic spin-orbit coupling.
By using analytical calculations, variational calculations, and Monte Carlo calculations, we find the conditions for the emergence of skyrmion structures.
We elucidate that the following three ingredients are enough to obtain field-induced skyrmion crystals: (1) C6 symmetry, (2) finite Q ordering due to competing interactions, and (3) easy-axis anisotropy.

S. Hayami, S.-Z. Lin, and C. D. Batista,
Phys. Rev. B 93, 184413 (2016)

C. D. Batista, S.-Z. Lin, S. Hayami, and Y. Kamiya,
Rep. Prog. Phys. 79, 084504 (2016)

S.-Z. Lin and S. Hayami,
Phys. Rev. B 93, 064430 (2016)

# Unconventional multipole orders and off-diagonal responses induced by hidden antisymmetric spin-orbit coupling

The relativistic spin-orbit coupling in the absence of spatial inversion symmetry has been extensively studied because it leads to various fascinating phenomena, such as unconventional superconductivity and multiferroics. A key concept in these phenomena is the antisymmetric spin-orbit coupling under the inversion symmetry breaking. Recently, it is recognized that a minimal ingredient for the antisymmetric spin-orbit coupling is local parity mixing originating from the inversion symmetry breaking at the lattice sites. phenomena induced by local parity mixing.

We here investigate the influence of the local parity mixing with focusing on itinerant electron systems. As a result, we find that a toroidal ordering, which has been ever discussed only for magnetic insulators, is realized in metallic systems, and induces novel magnetic transport and magnetoelectric effects. Furthermore, we clarify that a spontaneous parity breaking by charge, spin, and orbital ordering activates locally an antisymmetric spin-orbit coupling in the site-dependent form and results in the spin splitting in the band structure and magnetoelectric effect. Our results pave the way for novel electronic ordering, transport, and magnetoelectric

S. Hayami, H. Kusunose, and Y. Motome,
Phys. Rev. B 97, 024414 (2018)

Y. Yanagi, S. Hayami, and H. Kusunose,
Phys. Rev. B 97, 020404 (2018)

Y. Yanagi, S. Hayami, and H. Kusunose,
Physica B: Condensed Matter 536, 107-110 (2018)

S. Hayami, H. Kusunose, and Y. Motome,
Physica B: Condensed Matter 536, 649-653 (2018)

S. Hayami, H. Kusunose, and Y. Motome,
J. Phys.: Condens. Matter 28, 395601 (2016)

S. Hayami, H. Kusunose, and Y. Motome,
J. Phys. Soc. Jpn. 85, 053705 (2016)

S. Hayami, H. Kusunose, and Y. Motome,
J. Phys.: Conf. Ser. 592, 012101 (2015)

S. Hayami, H. Kusunose, and Y. Motome,
J. Phys.: Conf. Ser. 592, 012131 (2015)

S. Hayami, H. Kusunose, and Y. Motome,
J. Phys. Soc. Jpn. 84, 064717 (2015)

S. Hayami, H. Kusunose, and Y. Motome,
Phys. Rev. B 90, 081115(R) (2014)

S. Hayami, H. Kusunose, and Y. Motome,
Phys. Rev. B 90, 024432 (2014) ** [selected in Editors' Suggestion]**

# Three-dimensional Dirac electrons with noncoplanar multiple-Q order

Noncollinear and noncoplanar spin textures in solids manifest themselves not only in their peculiar magnetism but also in unusual electronic and transport properties. We here study a noncoplanar order on a simple cubic lattice and its influence on the electronic structure. We show that a four-sublattice triple-Q order induces three-dimensional massless Dirac electrons at commensurate electron fillings. The Dirac state is doubly degenerate, while it splits into a pair of Weyl nodes by lifting the degeneracy by an external magnetic field; the system is turned into a Weyl semimetal in an applied field. In addition, we point out the triple-Q Hamiltonian in the strong coupling limit is equivalent to the 3D π-flux model relevant to an AIII topological insulator. We examine the stability of such a triple-Q order in two fundamental models for correlated electron systems: a Kondo lattice model with classical localized spins and a periodic Anderson model. For the Kondo lattice model, performing a variational calculation and Monte Carlo simulation, we show that the triple-Q order is widely stabilized around 1/4 filling. For the periodic Anderson model, we also show the stability of the same triple-Q state by using the mean-field approximation. For both models, the triple-Q order is widely stabilized via the couplings between conduction electrons and localized electrons even without any explicit competing magnetic interactions and geometrical frustration. We also show that the Dirac electrons induce peculiar surface states: Fermi “arcs” connecting the projected Dirac points, similar to Weyl semimetals.

S. Hayami, T. Misawa, Y. Yamaji, and Y. Motome,
Phys. Rev. B 89, 085124 (2014)

# Charge order in the Kondo systems on a cubic lattice

In heavy-fermion systems, two competing interactions originating from the interplay be- tween localized electrons and conduction electrons play an important role; one is the Ruderman-Kittel-Kasuya-Yosida interaction and the other is the Kondo coupling. Their competition leads to a quantum critical point between a magnetically-ordered state and a Fermi liquid state, and quantum fluctuations enhanced near the quantum critical point are the source of interesting phenomena. In the present study, we focus on a possibility of charge ordered states in such situations. The possibility of charge order in Kondo systems has long been studied, but the realization in realistic three-dimensional systems has not been studied systematically.

We explore charge ordered states on a three-dimensional cubic lattice. Especially, we focus on competition and cooperation between charge orders and magnetic orders. By investigating the ground state of the periodic Anderson model at 3/2 filling by the mean-field approximation, we find that the model exhibits three different charge ordered states. Despite the absence of apparent frustration in the cubic lattice, one of the charge ordered states shows a noncoplanar triple-Q magnetic order. We show that the origin is likely to be the effective geometrical frustration induced by charge order with the ordering wave vector (π, π, π) on the cubic lattice. We also examine the electronic structure in the charge ordered state with the noncoplanar magnetic order.

S. Hayami, T. Misawa, and Y. Motome,
JPS Conf. Proc. 3, 016016 (2014)

# Partial disorder in the Kondo lattice systems

We investigate the effect of geometrical frustration in the quantum critical region, with emphasis on the emergence of a partially disordered state. The partially disordered state is characterized by the coexistence of magnetic order and a nonmagnetic Kondo singlet, which gives an intriguing example of self-organization to relieve geometrical frustration.

By using the mean-field calculation, we examine the nature of the partially disordered state in the periodic Anderson model on a triangular lattice.
As a result, we find a collection of different types of the partially disordered states, both insulating and metallic, depending on the model parameters. At half-filling, we obtain a partially-disordered insulating state between a noncollinear antiferromagnetic metal and a Kondo insulator. The partially disordered state is stabilized by releasing the frustration with self-organizing the system into the coexistence of collinear antiferromagnetic order on unfrustrated honeycomb subnetwork and nonmagnetic state at the remaining sites.
Furthermore, we find another type of partially-disordered insulating states at two commensurate fillings, 2/3 and 8/3 fillings. These states show different nature from that at half-filling: The nonmagnetic sites appear to be simply paramagnetic rather than the singlet state due to the hybridization; they are not separated from but rather connected with the magnetic sites. Reflecting the difference, the partially disordered states at 2/3 and 8/3 fillings exhibit distinct responses to the spin anisotropy and magnetic field compared to the half-filling case.
As a particularly interesting point, we find that the partially-disordered insulating state at 2/3 filling is changed into a partially-disordered metallic state by hole doping without showing a phase separation.

S. Hayami, M. Udagawa, and Y. Motome,
J. Phys. Soc. Jpn. 81,103707 (2012)

S. Hayami, M. Udagawa, and Y. Motome,
J. Phys.: Conf. Ser. 400, 032018 (2012)

S. Hayami, M. Udagawa, and Y. Motome,
J. Phys. Soc. Jpn. 80, 073704 (2011)