selected topics from our recent research

**Multiple Dirac cones in atomically-thin honeycomb materials**

The discovery of monolayer graphen has initiated two important fields, science of atomically-thin materials and Dirac semimetals. They have been rapidly growing, and recently, topological states of matter have been intensively studied for atomically-thin transition metal compounds. In these systems, novel topological states may be stabilized by the interplay between charge, spin, and orbital degrees of freedom under strong electron correlations. In the present study, we theoretically explore such novel properties in

*e*

_{g}electron states of the transition metals, by using

*ab initio*calculations and model analyses. We find that monolayer transition-metal trichalcogenides with a honeycomb structure exhibit eight independent Dirac cones, and the system can turn into a topological ferromagnetic state with a high Chern number under the spin-orbit coupling and electron correlations. Furthermore, we elucidate that various types of topological phases and transitions between them are realized in a wide range of atomically-thin honeycomb materials with

*e*

_{g}electrons.

Y. Sugita and Y. Motome, preprint (arXiv:1805.07068)

Y. Sugita, T. Miyake, and Y. Motome, Physica B: Condensed Matter

Y. Sugita, T. Miyake, and Y. Motome, Phys. Rev. B

Y. Sugita, T. Miyake, and Y. Motome, Physica B: Condensed Matter

**536**, 48 (2018)Y. Sugita, T. Miyake, and Y. Motome, Phys. Rev. B

**97**, 035125 (2018) [selected in Kaleidoscope]**Soliton lattice in chiral magnets**

In monoaxial chiral magnets, which has left- and right-handed degrees of freedom along a specific axis in the crystals, a helimagnetic state is partially relaxed by applying a magnetic field perpendicular to the chiral axis, and it can turn into the so-called chiral soliton lattice with a periodic array of soliton-like spin helices. Since the pioneering works by Dzyaloshinskii in 1960's, extensive studies have been done for a long time, and recently, the chiral soliton lattice was observed by using real-space probes, which has accelerated the researches from both theories and experiments. Theoretically, however, there are less studies including the lattice discreteness and itinerant electron degrees of freedom. Here, we consider a model including the itinerant electrons for such monoaxial chiral magnets, and study the ground-state and finite-temperature properties by using variational calculations and Monte Carlo simulations, respectively. We successfully explain the experimental results of the nonlinear negative magnetoresistance and the lock-in of the period of chiral soliton lattices in an applied magnetic field. These results would contribute to further development of the cooperative studies between theories and experiments on this rapidly growing topic.

S. Okumura, Y. Kato, and Y. Motome, J. Phys. Soc. Jpn.

S. Okumura, Y. Kato, and Y. Motome, Physica B: Condensed Matter

S. Okumura, Y. Kato, and Y. Motome, J. Phys. Soc. Jpn.

**87**, 033708 (2018)S. Okumura, Y. Kato, and Y. Motome, Physica B: Condensed Matter

**536**, 223 (2018)S. Okumura, Y. Kato, and Y. Motome, J. Phys. Soc. Jpn.

**86**, 063701 (2017)**Kitav-Heisenberg model in**

*d*^{7}high-spin systemsThe Kitaev model proposed by A. Kitaev in 2006 is a localized spin model that realizes an exact spin liquid ground state. Since Jackeli and Khaliullin pointed out that the model could give a good description of some spin-orbit coupled Mott insulators in the

*d*

^{5}low-spin state, 5

*d*- and 4

*d*-electron systems, such as Ir and Ru compounds, have been intensively studied as the Kitaev candidates. However, the researches of the Kitaev spin liquids have been mostly limited to such

*d*

^{5}low-spin systems thus far, and the candidate materials are still limited. In order to explore a new platform of the Kitaev spin liquids, we here investigate the possibility of realizing the Kitaev model in the

*d*

^{7}high-spin state. We find that, similar to the

*d*

^{5}low-spin case, the

*d*

^{7}high-spin systems can realize spin-orbit coupled Mott insulators with the effective magnetic moment 1/2, and the low-energy effective model is given by the Kitaev-Heisenberg model with the anisotropic Kitaev-type interactions and the isotropic Heisenberg exchange interactions. Our results would contribute to provide a new playground for the Kitaev spin liquids beyond the

*d*

^{5}low-spin systems.

R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B

**97**, 014408 (2018)**Three-dimensional chiral spin liquid transition**

The chiral spin liquid is a categoly of quantum spin liquids where the time reversal symmetry is spontaneously broken. This exotic magnetic state has been extensively studied, especially in two dimensions, in the context of the high

*T*

_{C}superconductivity, the fractional quantum Hall effect, frustrated quantum magnets, etc. However, much less is known for three dimensions (3D). Here, we consider the chiral spin liquids in an extension of the exactly-soluble Kitaev model to 3D. In particular, we study finite-temperature properties of the model on a hypernonagon lattice with nine-site elementary loops. By performing Monte Carlo simulations for the effective models associated with

*Z*

_{2}fluxes defined on each loop, we find a finite-temperature phase transition between the paramagnetic state and the chiral spin liquid. This is the first unbiased results on finite-temperature properties of the 3D chiral spin liquids to our knowledge, which would significantly contribute to the future study of the exotic states.

Y. Kato, Y. Kamiya, J. Nasu, and Y. Motome, Phys. Rev. B

**96**, 174409 (2017)**Magnetic skyrmion with a high topological number**

Magnetic skyrmions are microscopic swirling textures composed of electron spins in solids. Owing to the topologically protected nature and a variety of responses to electric and magnetic fields, they have attracted much attention for potential applications to new magnetic devices. Thus far, magnetic skyrmions with topological number of unity, which are stable in an applied magnetic field, have been extensively studied, but it has been desired to realize new types of skyrmions for further extending the possibility of applications. We here show that a skyrmion crystal with unusually high topological number of two is stabilized in itinerant magnets at a zero magnetic field. The results are obtained for a minimal Kondo lattice model by an unrestricted large-scale numerical simulation. Furthermore, we find that the topological number can be switched successively by a magnetic field as 2<=>1<=>0. Our results will stimulate new fundamental scientific principles for potential applications, such as multiple digital switching memory by a weak magnetic field.

**Magnetization curve and magnetoelectric behavior from antiferromagnetic square cupolas**

Systems without spatial inversion and time reversal symmetry often exhibit cross correlations between magnetism and electricity (linear magnetoelectric effects). As a new playground of such interesting phenomena, a recently synthesized Cu oxide Ba(TiO)Cu

_{4}(PO

_{4})

_{4}has attracted much attention. The lattice structure of this compound is composed of an assembly of low-symmetric units Cu

_{4}O

_{12}forming square cupolas, where spatial inversion symmetry is broken locally. In addition, localized magnetic moments at Cu sites exhibit a magnetic long-range order at low temperature, which breaks time reversal symmetry as well. These lead to a magnetoelectric effect of quadrupole type, and indeed, a dielectric anomaly was observed. We here construct a minimal theoretical model, which successfully reproduces the full magnetization curves measured at the Institute for Solid State Physics, University of Tokyo. Our model points to an importance of the antisymmetric exchange interaction called the Dzyaloshinskii-Moriya interaction arising from the low symmetry of square cupolas. Furthermore, elaborating the ground-state and finite-temperature phase diagrams of the model, we predict five different antiferromagnetic phases. As different types of magnetoelectric behaviors are expected in each phase, further experiments in the high magnetic field regime are currently in progress.

Y. Kato, K. Kimura, A. Miyake, M. Tokunaga, A. Matsuo, K. Kindo, M. Akaki, M. Hagiwara, M. Sera, T. Kimura, and Y. Motome, Phys. Rev. Lett.

**118**, 107601 (2017)**Fractional spin fluctuations**

Fractionalization of quantum spins is a prominent feature of quantum spin liquids. Fractional excitations emergent from the fractionalization have their own energy scales depending on types of the quasiparticles. Hence, they are expected to affect the spin dynamics not only in quantum spin liquid states but also paramagnetic states in proximity to the quantum spin liquids. We here compute the spin dynamics at finite temperatures for the Kitaev model, whose ground state is exactly given by quantum spin liquids. Developing and applying new numerical techniques based on a Majorana fermion representation, the cluster extension of dymanical mean-field theory and the continuous-time quantum Monte Carlo method, we clarify the temperature dependences of the dynamical spin structure factor, the NMR relaxation rate, and the magnetic susceptibility. We find unusual behavior never seen in conventional magnets: a dichotomy between static and dynamical spin correlations. It was thus far hard to compute temperature dependences of static quantities. This is the first attempt to our knowledge to systematically study the finite-temperature dynamics of quantum spin liquids.

J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett.

**117**, 157203 (2016)**Fermionic excitations from fractionalization of quantum spins**

The quantum spin liquid is an exotic quantum disordered state driven by strong quantum fluctuations and many-body effects in insulating magnets. While many candidate materials of this state have been discovered, the theoretical treatment is still controversial; in particular, less is known thus far about finite-temperature properties, which are crucial for critical comparison with experiments. We here study the magnetic Raman scattering spectrum at finite temperatures for the Kitaev model which possesses the exact quantum spin liquid ground state, and compare the results with experiments for a candidate Ru compound. We find that the temperature dependence of the Raman intensity is well described by the Fermi distribution function, reflecting the fact that the fundamental excitations in this system are governed by Majorana fermions emergent from the fractionalization of quantum spins. We show that our result quantitatively agrees with experimental results from about 10K to room temperature, which strongly suggests the existence of emergent Majorana fermions in the real material in this wide temperature range. As the comparison of the Raman intensity is applicable to other candidate materials, it will be widely used as a experimental hallmark of quantum spin liquids.

Y. Motome, Press Release [UTokyo Research]

J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R. Moessner, Nature Physics

J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R. Moessner, Nature Physics

**12**, 912 (2016)**Thermal fractionalization of quantum spins**

In general, it is hard to prove that a magnet is in a quantum spin liquid state. This is because we need to show the alibi (so-called probatio diabolica), as the quantum spin liquids do not show any symmetry breaking and magnetic ordering down to zero temperature. To avoid the difficulty, a lot of efforts have been made to capture one of prominent features of quantum spin liquids, fractionalization of quantum spins. The spin fractionalization is a phenomenon in which a fundamental degree of freedom, electron spin, is fractionalized into several degrees of freedom. This is regarded as a spin version of charge fractionalization in fractional quantum Hall systems. We here clarify how the spin fractionalization affects thermodynamics, by applying quantum Monte Carlo simulation to the Kitaev model, whose ground state is exactly given by quantum spin liquids. We find that the fractionalized spins release their entropy successively at largely different temperature scales. Our results will be useful for the experimental identification of quantum spin liquids.

J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. B

**92**, 115122 (2015)**Finite-temperature transition in chiral spin liquids**

Chiral spin liquids, in which time-reversal symmetry is broken but a long-range magnetic order is absent, have long been studied as a new quantum state of matter. In particular, recently, it has attracted growing interest as the realization of non-Abelian anyons that are useful in fault-tolerant quantum computations. However, less is know how it behaves at finite temperature under thermal fluctuations, although it is important for the quantum computations. By the quantum Monte Carlo simulation, we study the finite-temperature properties of a Kitaev model defined on a decorated-honeycomb lattice, which is know to be a chiral spin liquid at zero temperature. We found that the chiral spin liquid remains robust against thermal fluctuations and it is distinguished from the high-temperature paramagnet by a phase transition. We also clarified that the nature of the phase transition depends on the statistical property of elementary excitations in the chiral spin liquids, anyons. The results pave the way for further understanding of not only quantum magnetism but also quantum computations using non-Abelian anyons.

**Vaporization in quantum spin liquids**

Insulating magnets exhibit paramagnetic behavior with disordered spin directions at high temperature, while magnetic orders with spontaneous symmetry breaking below some critical temperature. The former corresponds to ggash of quantum spins, and the latter gsolidh among the states of matter. In 1973, P. W. Anderson proposed a new quantum state, "quantum spin liquid (QSL)", in which quantum spins are disordered but strongly correlated with each other. This state has been investigated experimentally and theoretically thus far, but its thermodynamic properties have not been revealed yet. In order to solve this problem, we investigate the thermodynamics of a three-dimensional Kitaev model, whose ground state is exactly shown to be a QSL, by newly developing a quantum Monte Carlo simulation on the basis of the Majorana fermion representation. As a result, we discover that a finite-temperature phase transition takes place between the low-temperature QSL and high-temperature paramagnet. We also show that this phase transition is characterized by the change of the topology in the excited states. Our results create a stir in the experimental studies where the adiabatic connection between QSL and paramagnet is implicitly assumed.

Y. Motome, Parity

J. Nasu, M. Udagawa, and Y. Motome, BUTSURI

Y. Motome, Parity

J. Nasu, M. Udagawa, and Y. Motome, J. Phys.: Conf. Ser.

Y. Motome, Press Release [UTokyo Research]

J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. Lett.

J. Nasu, T. Kaji, K. Matsuura, M. Udagawa, and Y. Motome, Phys. Rev. B

**31**(1), 22 (2016)J. Nasu, M. Udagawa, and Y. Motome, BUTSURI

**70**, 776 (2015)Y. Motome, Parity

**30**(10), 36 (2015)J. Nasu, M. Udagawa, and Y. Motome, J. Phys.: Conf. Ser.

**592**, 012115 (2015)Y. Motome, Press Release [UTokyo Research]

J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. Lett.

**113**, 197205 (2014)J. Nasu, T. Kaji, K. Matsuura, M. Udagawa, and Y. Motome, Phys. Rev. B

**89**, 115125 (2014)**Unconventional multipole orders and off-diagonal responces 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. 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 phenomena induced by local parity mixing.

S. Hayami, H. Kusunose, and Y. Motome, Kotai Butsuri 50, 217 (2015)

S. Hayami, H. Kusunose, and Y. Motome, J. Phys. Soc. Jpn.

S. Hayami, H. Kusunose, and Y. Motome, J. Phys.: Conf. Ser.

S. Hayami, H. Kusunose, and Y. Motome, J. Phys.: Conf. Ser.

S. Hayami, H. Kusunose, and Y. Motome, Phys. Rev. B

S. Hayami, H. Kusunose, and Y. Motome, Phys. Rev. B

S. Hayami, H. Kusunose, and Y. Motome, J. Phys. Soc. Jpn.

**84**, 064717 (2015)S. Hayami, H. Kusunose, and Y. Motome, J. Phys.: Conf. Ser.

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

**592**, 012101 (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) [Editors' Suggestion]**Noncoplanar magnetic order and 3D massless Dirac electrons on a cubic lattice**

Noncoplanar multiple-Q orders, which are characterized by more than a single ordering wave vector, often lead to topologically nontrivial states and new low-energy excitations. To explore the possibility of such multiple-Q orders on unfrustrated lattices, we investigate noncoplanar magnetic orders on a simple cubic lattice. As a result, we find that a triple-Q magnetic order on the cubic lattice significantly affects the low-energy single-particle spectrum, resulting in the three-dimensional massless Dirac electrons. We also show that such magnetic ordering possesses gapless surface states, becomes a Weyl semimetal in an applied magnetic field, and changes into a topological insulator by an appropriate perturbation. Moreover, we examine the stability of the triple-Q state by mean-field calculations and Monte Carlo simulation for itinerant electron models, such as the Kondo lattice model and the periodic Anderson model. Our results indicate that the itinerant nature of electrons, rather than the geometrical frustration, plays an important role in realizing noncoplanar magnetic orders.

S. Hayami, T. Misawa, Y. Yamaji, and Y. Motome, Phys. Rev. B

S. Hayami, T. Misawa, and Y. Motome, JPS Conf. Proc.

**89**, 085124 (2014)S. Hayami, T. Misawa, and Y. Motome, JPS Conf. Proc.

**3**, 016016 (2014)**Spin-orbital frustration in pyrochlore molybdenum oxides**

Pyrochlore molybdenum oxides

*A*

_{2}Mo

_{2}O

_{7}provide a fertile ground for studying metal-insulator transitions and associated unconventional magnetic phenomena. In the insulating materials, a long-range magnetic order is suppressed, and at low temperatures a spin glass behavior appears with frozen magnetic moments in random directions. It has long been believed that the absence of long-range ordering is due to the geometrical frustration between isotropic antiferromgnetic exchnge couplings on the pyrochlore lattice. Recently, however, the neutron scattering experiment for a single crystal of Y

_{2}Mo

_{2}O

_{7}has revealed an unexpected behavior, that is, the existence of weak ferromagnetic spin fluctuations. This urges reconsideration of the microscopic origin of the spin glass behavior. Performing the state-of-the-art first-principles calculations including the relativistic spin-orbit coupling, we clarified that the ground state suffers from keen competition between antiferromagnetic and ferromagnetic states. Furthermore, through the analyses of localized spin model and multi-orbital Hubbard model, we found that the magnetic competition couples with the competition between different orbital orders via the spin-orbit coupling. The new picture of magnetic frustration originating from the orbital degrees of freedom well explains the experimental results.

H. Shinaoka, Y. Motome, T. Miyake, and S. Ishibashi, Phys. Rev. B

**88**, 174422 (2013) [selected in Kaleidoscope]**Charge order in Kondo lattice systems**

The Kondo lattice model, in which conduction electrons couple with localized quantum spins, is one of the most fundamental models for heavy fermion systems. The model has been intensively studied for searching for novel quantum phases. In particular, the possibility of a charge ordered phase has been examined for more than 30 years, but it was not clarified except for the special cases in one and infinite dimensions. Utilizing two sophisticated numerical methods, a cluster extension of the dynamical mean-field theory and a multi-variables variational Monte Carlo method, we solve this problem and show the evidence of charge ordering in two dimensions. Furthermore, we find that the local Kondo singlet formation plays a key role in stabilizing the charge order. Our results indicate that the charge order in the Kondo lattice systems is qualitatively different from those by intersite Coulomb repulsion. We extend the study to three-dimensional systems, and find that the charge order appears with a noncoplanar magnetic ordering due to the effective frustration under the charge order.

T. Misawa, J. Yoshitake, and Y. Motome, Phys. Rev. Lett.

S. Hayami, T. Misawa, and Y. Motome, JPS Conf. Proc.

**110**, 246401 (2013)S. Hayami, T. Misawa, and Y. Motome, JPS Conf. Proc.

**3**, 016016 (2014)**Quantum anomalous Hall effect in kagome ice**

Spin ice exhibits a peculiar magnetization plateau under the [111] magnetic field. It is considered to be realized by forming a spin liquid state called kagome ice on the [111] kagome planes. For clarifying the effect of the locally-correlated kagome ice on itinerant electrons, we investigate numerically the electronic and transport properties of the spin-ice type Kondo lattice model on a kagome lattice. As a result, we find that the kagome ice spin correlation opens a charge gap in the electronic state in spite of absence of magnetic long-range order. Moreover, the insulating state is a quantum anomalous Hall insulator with a quantization of the Hall conductivity. As increasing magnetic field, the charge gap closes, but opens again in the fully-polarized insulating state, in which the Hall conductivity is quantized at a different value. We show that this is considered as a transition between different topological insulators.

H. Ishizuka and Y. Motome, Phys. Rev. B

**87**, 081105(R) (2013)**Dirac half-metal in a triangular ferrimagnet**

A monolayer of carbon, graphen, has attracted much interest because of the Dirac electrons with a peculiar linear dispersion in the electronic state. However, it has some difficulty for application to spintronics as the spin-orbit coupling is very weak. We here theoretically propose a possibility of the Dirac electrons from a different point of view. Considering a triangular magnet with itinerant electrons, we show that the Dirac electronic state with a linear dispersion emerges in underlying three-sublattice ferrimagnetic order. The Dirac electrons are perfectly spin-polarized, i.e., the system is half metallic. Furthermore, by Monte Carlo simulation, we clarify the parameter region where the Dirac half-metallic phase is stabilized. The discovery of such new Dirac electrons will stimulate further development of spintronics.

H. Ishizuka and Y. Motome, Phys. Rev. Lett.

**109**, 237207 (2012)**Metallic partial disorder in a periodic Anderson model on a triangular lattice**

In our previous studies, we found the partially disordered states in the Kondo lattice model, periodic Anderson model, and Ising Kondo lattice model on a triangular lattice. The partially disordered states, however, are all insulating with a finite charge gap. Meanwhile, the partially disordered states in experiments are metallic. It is therefore important to find a metallic partial disorder theoretically for providing understanding of the experiments. Here, we extend the mean-field analysis of the periodic Anderson model to other commensurate fillings and carrier-doped regions around them. As a result, we find that a different type of partial disorder emerges at other commensurate fillings, and that hole doping induces a metallic partially disordered state.

S. Hayami, M. Udagawa, and Y. Motome, J. Phys. Soc. Jpn.

**81**, 103707 (2012)**Partial disorder in an Ising-spin Kondo lattice model on a triangular lattice**

The triangular-lattice Ising model, a fundamental model for geometrically frustrated systems, exhibits macroscopic degeneracy in the ground state when the interactions are restricted to nearest neighbors and antiferromagnetic. A nontrivial emergent state from this degeneracy is a partial disorder, which is coexistence of magnetic order and paramagnetic moments. In two dimensions, however, the partial disorder is fragile against thermal fluctuations, and does not form a long-range order. Stimulated by recent experimental findings of a partial disorder in quasi-two-dimensional conducting systems, we explore the possibility of partial disorder by the interaction between localized moments and itinerant electrons. By studying an Ising-spin Kondo lattice model on a triangular lattice by Monte Carlo simulation, we found that a partial disorder is stabilized even in two dimensions in the spin-charge coupled system. We clarified that the charge degree of freedom plays a crucial role in the emergence of partial disorder through an electronic phase separation, charge ordering, and opening of the charge gap.

H. Ishizuka and Y. Motome, Phys. Rev. B

H. Ishizuka and Y. Motome, Phys. Rev. Lett.

**87**, 155156 (2013)H. Ishizuka and Y. Motome, Phys. Rev. Lett.

**108**, 257205 (2012)**Hidden multiple-spin interactions in frustrated itinerant-electron systems**

Anomalous Hall effect by spin scalar chiral ordering has attracted much attention. Recently, we found that a four-sublattice noncoplanar chiral order appears near 1/4 filling in one of the fundamental models of frustrated itinerant-electron systems, a Kondo lattice model on a triangular lattice. This state, however, is not explained by the previously-known scenario of the nesting of Fermi surface, and hence, the origin was unclear. We here investigate the stabilization mechanism by carefully examining the perturbation in terms of the spin-charge coupling up to fourth order. As a result, we found that the effective exchange interactions in the second order (RKKY interactions) are degenerate due to the frustration, and that the higher fourth-order contributions play a decisive role. Among many effective multiple-spin interactions in the fourth order, the biquadratic interaction is critically enhanced with a positive coefficient. This is not due to the nesting but a Fermi surface connection, which we call the generalized Kohn anomaly. Our results suggest that the nontrivial stabilization mechanism is hidden in the wide range of frustrated itinerant electron systems.

Y. Akagi, M. Udagawa, and Y. Motome, Phys. Rev. Lett.

**108**, 096401 (2012)**Resistivity minimum in spin-ice conduction systems**

In frustrated magnets called spin ice, a long-range magnetic order is suppressed and a spin-liquid like state can emerge with showing only local spin correlations obeying the so-called ice rule. When such peculiar spatial magnetic texture interacts with itinerant electrons, one can expect some new transport phenomena. We here consider this problem in a model in which spin-ice type Ising moments are coupled with itinerant electrons on a pyrochlore lattice, by employing a cluster extension of the dynamical mean-field theory. As a result, we found that, in low electron density region, the system exhibits a spin-ice like liquid state, and more importantly, the electrical resistivity shows a minimum corresponding to the development of local spin correlations. This clearly shows that the special ice-rule correlation becomes a strong scatterer of electrons, giving a completely new mechanism of resistivity minimum distinct from the conventional Kondo effect. The results suggest the possibility of understanding the resistivity minimum recently observed in some Ir pyrochlore oxides.

M. Udagawa, H. Ishizuka, and Y. Motome, Phys. Rev. Lett.

**108**, 066406 (2012)**Partial disorder in the periodic Anderson model on a triangular lattice**

In Kondo lattice systems, geometrical frustration has recently attracted much interest as a new control parameter to induce novel quantum states in the competition between the RKKY interaction and Kondo coupling. In particular, a partial disordered (PD) state was suggested both experimentally and theoretically, in which a magnetic order appears only on a subset of sublattices so as to relieve the frustration; however, its basic properties as well as the stabilization mechanism remain unclear. We here investigate the ground state of a fundamental model for the Kondo lattice systems, i.e., the periodic Anderson model, on a frustrated triangular lattice at half filling by using the mean field approximation. We found that a PD state is stabilized concomitant with charge ordering between a 120 degree AF metal and a Kondo insulator in the region from weak to intermediate correlation, and that the PD state is insulating and shows a characteristic crossover in the electronic structure. From a comparison to the previous results for the Kondo lattice models, it is clarified that the charge degree of freedom of localized electrons plays a crucial role in stabilizing PD.

S. Hayami, M. Udagawa, and Y. Motome, J. Phys. Soc. Jpn.

**80**, 073704 (2011)**Metal-insulator transition in charge-frustrated systems under ice-rule local constraint**

Ice rule, which is a local constraint named after the configuration of protons in water ice, appears also in the charge configuration in frustrated charge-ordering systems. Transport properties in such charge-frustrated systems have attracted attention both experimentally and theoretically, but detailed understanding is not obtained so far. We here examine this problem by investigating electronic and transport properties for an extended Falicov-Kimball model on various frustrated lattice structures, in which the configuration of localized particles obeys the ice rule. As a result, we find that a considerably small interaction compared to the bandwidth opens a charge gap and the system becomes `charge-ice insulator' in which itinerant electrons are localized in ice-rule configurations. In particular, in the model on the pyrochlore lattice, the system shows a phase transition from a metal to the charge-ice insulator without entering into the Anderson insulator that is seen in the case with random configuration. These novel behaviors are considered to originate from the particular spatial correlation due to the gauge structure hidden in the ice-rule systems.

H. Ishizuka, M. Udagawa, and Y. Motome, Phys. Rev. B

**83**, 125101 (2011)**Spin glass transition and spin-lattice coupling in pyrochlore antiferromagnets**

Many frustrated magnets exhibit a spin glass state at low temperatures. Usually the spin glass is induced by randomness, however, in several pyrochlore magnets, a spin glass transition takes place even in a best-quality sample with virtually disorder free, and furthermore, the transition temperature is almost independent of the strength of disorder. To explore the origin of these peculiar behaviors, we investigate the effect of spin-lattice coupling in a bond-disordered pyrochlore Heisenberg antiferromagnet by extensive Monte Carlo simulations. As a result, we find that the spin glass transition temperature is strongly enhanced by the spin-lattice coupling, and remarkably, becomes almost constant in a wide range of the strength of disorder. This is presumably because the spin-lattice coupling enhances the spin collinearity and suppresses thermal fluctuations. The spin glass transition is of second order and its critical properties are compatible with the conventional ones. Our results well account for the puzzling behaviors observed in experiments.

H. Shinaoka, Y. Tomita, and Y. Motome, Phys. Rev. B

H. Shinaoka, Y. Tomita, and Y. Motome, J. Phys.: Conf. Ser.

H. Shinaoka, Y. Tomita, and Y. Motome, Phys. Rev. Lett.

H. Shinaoka and Y. Motome, Phys. Rev. B

H. Shinaoka, Y. Tomita, and Y. Motome, J. Phys.: Conf. Ser.

**90**, 165119 (2014)H. Shinaoka, Y. Tomita, and Y. Motome, J. Phys.: Conf. Ser.

**400**, 032087 (2012)H. Shinaoka, Y. Tomita, and Y. Motome, Phys. Rev. Lett.

**107**, 047204 (2011)H. Shinaoka and Y. Motome, Phys. Rev. B

**82**, 134420 (2010)H. Shinaoka, Y. Tomita, and Y. Motome, J. Phys.: Conf. Ser.

**320**, 012009 (2011)**Spin scalar chirality ordering and anomalous Hall effect in triangular-lattice ferromagnetic Kondo models**

Recently, spin scalar chirality has attracted much attention as a novel origin of the anomalous Hall effect (AHE), independent of the relativistic spin-orbit coupling. In fact, it was pointed out that AHE is induced by noncoplanar magnetic orders on the kagome or triangluar lattice. However, in the previous studies, such magnetic orders were given by hand with neglecting effects of itinerant electrons, and their stability relative to other orders was not examined. In the present study, we investigate the ground state of the ferromagnetic Kondo model on the triangular lattice by variational calculations for various magnetic states up to four-sublattice orders. As a result, we find that a four-sublattice scalar chiral order emerges around 1/4 filling, in addition to 3/4 filling which was predicted previously. The 1/4 filling phase is stable in a wider range of parameters than the nesting-driven 3/4 filling one. We also compute the Hall conductivity in these chiral phases, which is quantized according to the Chern number in gapped insulating states.

Y. Akagi and Y. Motome, J. Phys. Soc. Jpn.

Y. Akagi and Y. Motome, J. Phys.: Conf. Ser.

**79**, 083711 (2010)Y. Akagi and Y. Motome, J. Phys.: Conf. Ser.

**320**, 012059 (2011)**Partial Kondo screening in frustrated Kondo lattice systems**

One of the most important concepts in Kondo lattice systems is competition between the Kondo coupling and the RKKY interaction. The competition leads to a quantum critical point (QCP) between a magnetically-ordered state and a Fermi liquid state, and furthermore, it is the origin of novel phenomena around the QCP, such as a non-Fermi liquid behavior and a superconductivity. To explore a new quantum phase related to the competition, we investigate the ground state of geometrically-frustrated Kondo lattice systems by employing a high-precision variational Monte Carlo simulation. We find that a partially-ordered state, in which a magnetic order and a Kondo spin singlet coexists, emerges between a magnetically-ordered state stabilized by the RKKY interaction and a Kondo spin liquid state stabilized by the Kondo coupling. We clarified that this new quantum phase is stabilized by quantum fluctuations as well as magnetic anisotropy, and that it is accompanied by a charge disproportionation.

Y. Motome, K. Nakamikawa, Y. Yamaji, and M. Udagawa, Phys. Rev. Lett.

Y. Motome, Y. Yamaji, and M. Udagawa, J. Phys.: Conf. Ser.

Y. Motome, K. Nakamikawa, Y. Yamaji, and M. Udagawa, J. Phys. Soc. Jpn.

**105**, 036403 (2010)Y. Motome, Y. Yamaji, and M. Udagawa, J. Phys.: Conf. Ser.

**145**, 012068 (2009)Y. Motome, K. Nakamikawa, Y. Yamaji, and M. Udagawa, J. Phys. Soc. Jpn.

**80**, Suppl. A, SA133 (2011)**Quantum melting of charge-ice insulator and non-Fermi-liquid behavior**

Ice rule is a local constraint observed in a broad range of condensed matter, such as the proton position in water ice and the Ising spin configuration in spin ice. This local constraint is not enough to let the system ordered, and the ground state often retains macroscopic degeneracy. However, the disordered state is not completely random but has a specific spatial correlation due to a hidden gauge structure. To clarify the effect of special correlations on itinerant electrons, we consider an extended Falicov-Kimball model which describes a correlation between ice-rule localized particles and itinerant electrons, and find an exact solution of this model on a loopless tetrahedron Husimi cactus. We show that the system becomes a charge-ice insulator with 1D-like electronic structure in large correlation regime, and that non-Fermi-liquid behavior appears at the quantum critical point where the charge ice melts.

M. Udagawa, H. Ishizuka, and Y. Motome, Phys. Rev. Lett.

**104**, 226405 (2010)**Chirality-driven heavy-mass behavior**

Heavy-mass behavior in transition metal compounds, such as spinel oxide LiV

_{2}O

_{4}, has attracted much attention, but its origin still remains unclear. Focusing on the synergy between strong electron correlation and geometrical frustration, we investigate the kagome Hubbard model by employing a cluster extension of the dynamical mean-field theory and the continuous-time quantum Monte Carlo simulation. From the detailed analysis of the cluster density matrix, we find that the system exhibits a hierarchy in the energy scale among charge, spin, and chirality degrees of freedom, and the chirality becomes dominant at the lowest temperature. Furthermore, by calculating the specific heat and entropy, we clarify that heavy-mass behavior emerges from a large amount of entropy associated with the chirality. These results indicate that the frustration plays not only a secondary role just to suppress long-range ordering but also an intensive role to invoke a novel electronic state through a formation of multiple degree of freedom such as chirality.

M. Udagawa and Y. Motome, Phys. Rev. Lett.

M. Udagawa and Y. Motome, J. Phys.: Conf. Ser.

**104**, 106409 (2010)M. Udagawa and Y. Motome, J. Phys.: Conf. Ser.

**200**, 012214 (2010)**Phase competition and phase separation in the pyrochlore double-exchange model**

Double-exchange model is one of the most fundamental models describing the interplay between itinerant electrons and localized moments, and has been extensively studied in the research of CMR manganites. It has been clarified that the model exhibits bicritical behavior and phase separation (PS) between a ferromagnetic metal (FM) and an antiferromagnetic (AF) insulator. However, surprisingly less is known for the effect of frustration on the model. We here investigate finite-temperature properties of the model on the 3D frustrated pyrochlore lattice by Monte Carlo simulation. As a result, we find that FM becomes unstable gradually with increasing AF super-exchange interaction and is finally taken over by a peculiar paramagnetic metal (PM) which shows an almost temperature-independent behavior with incoherent transport, and that PS takes place between FM and PM. These are specific features to the frustrated systems, not seen in the absence of frustration. The results well explain the novel properties recently observed in Mo pyrochlore oxides under external pressure.

Y. Motome and N. Furukawa, Phys. Rev. Lett.

Y. Motome and N. Furukawa, Phys. Rev. B

Y. Motome and N. Furukawa, J. Phys.: Conf. Ser.

**104**, 106407 (2010)Y. Motome and N. Furukawa, Phys. Rev. B

**82**, 060407(R) (2010)Y. Motome and N. Furukawa, J. Phys.: Conf. Ser.

**200**, 012131 (2010)