Motivated by recent experiments in Gd$_3$Ru$_4$Al$_{12}$, we demonstrate the emergence of atomic scale Skyrmions in interacting spins on a breathing kagome lattice with competing nearest neighbor ferromagnetic and next nearest neighbor antiferromagnetic exchange interactions. In the presence of an applied longitudinal magnetic field, the ground state magnetic order evolves from a helical phase at low fields to a Skyrmion phase at intermediate fields before finally entering a polarized phase at high fields. The size of each Skyrmion spans only two unit cells of the lattice, in contrast to tens to hundreds of unit cells in most chiral magnets. Furthermore, the Skyrmions are driven not by chiral interactions but by the interplay between competing exchange interactions and geometric frustration, just as in Gd$_3$Ru$_4$Al$_{12}$ . When itinerant electrons are coupled to the localized moments, they exhibit the usual Skyrmion-driven topological Hall effect (THE) arising from the real space Berry curvature of the the Skyrmion texture. The small size of the Skyrmions in this system yield a strong local Berry curvature that results in an enhanced THE, which is investigated using a strong coupling approximation between the spins of the itinerant electrons and the localized moments. Our results will be crucial in understanding the experiments in Gd$_3$Ru$_4$Al$_{12}$ and other members of the same family of metallic frustrated magnets.
We numerically study the spin-$1/2$ breathing kagome lattice. In this variation of the kagome Heisenberg antiferromagnet, the spins belonging to upward and downward facing triangles have different coupling strengths. Using the density matrix renormalization group (DMRG) method and exact diagonalization, we show that the kagome antiferromagnet spin liquid is extremely robust to this anisotropy. Materials featuring this anisotropy -- and especially the recently studied vanadium compound $[{\mathrm{NH}}_{4}{]}_{2}[{\mathbf{C}}_{7}{\mathbf{H}}_{14}\mathbf{N}][{\mathbf{V}}_{7}{\mathbf{O}}_{6}{\math…
In the vanadium oxyfluoride compound (NH$_4$)$_2$[C$_7$H$_{14}$N][V$_7$O$_6$F$_{18}$] (DQVOF), the V$^{4+}$ (3d$^1$, $S=1/2$) ions realize a unique, highly frustrated breathing kagome lattice composed of alternately-sized, corner-sharing equilateral triangles. Here we present an $^{17}$O NMR study of DQVOF, which isolates the local susceptibility of the breathing kagome network. By a fit to series expansion we extract the ratio of the interactions within the breathing kagome plane, $J_\triangledown / J_\vartriangle = 0.55(4)$, and the mean antiferromagnetic interaction $\bar{J}=60(7)$~K. Spin …
We study the zero-temperature phase diagram of the spin-$\frac{1}{2}$ Heisenberg model with breathing anisotropy (i.e., with different coupling strength on the upward and downward triangles) on the kagome lattice. Our study relies on large scale tensor network simulations based on infinite projected entangled-pair state and infinite projected entangled-simplex state methods adapted to the kagome lattice. Our energy analysis suggests that the U(1) algebraic quantum spin-liquid (QSL) ground-state of the isotropic Heisenberg model is stable up to very large breathing anisotropy until it breaks do…
Motivated by recent experiments on the vanadium oxyfluoride material DQVOF, we examine possible spin liquid phases on a breathing kagome lattice of S=1/2 spins. By performing a projective symmetry group analysis, we determine the possible phases for both fermionic and bosonic $\mathbb{Z}_2$ spin liquids on this lattice, and establish the correspondence between the two. The nature of the ground state of the Heisenberg model on the isotropic kagome lattice is a hotly debated topic, with both $\mathbb{Z}_2$ and U(1) spin liquids argued to be plausible ground states. Using variational Monte Carlo …
We present measurements on a series of materials, Li$_2$In$_{1-x}$Sc$_x$Mo$_3$O$_8$, that can be described as a 1/6th-filled breathing kagome lattice. Substituting Sc for In generates chemical pressure which alters the breathing parameter non-monotonically. $μ$SR experiments show that this chemical pressure tunes the system from antiferromagnetic long range order to a quantum spin liquid phase. A strong correlation with the breathing parameter implies that it is the dominant parameter controlling the level of magnetic frustration, with increased kagome symmetry generating the quantum spin liqu…
High-pressure modulation of breathing kagome lattice: Cascade of Lifshitz transitions and evolution of the electronic structure
Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿罗森多夫赫尔姆霍兹中心）；TUD Dresden University of Technology（德累斯顿工业大学）；Universität Stuttgart（斯图加特大学）；Synchrotron SOLEIL（索雷尔同步辐射中心）；ESRF（欧洲同步辐射装置）；ShanghaiTech University（上海科技大学）；Renmin University of China（中国人民大学）
In this work, topological properties of a ferromagnetic Heisenberg model on a breathing kagome lattice are investigated extensively in the presence of Dzyaloshinskii-Moriya interaction. While the kagome ferromagnet hosts only a single first order topological phase, the breathing kagome system exhibits multiple first and second order topological phases along with their coexistence. Magnon dispersion relation is obtained by using linear spin wave theory. Flat band and Dirac cones are obtained in the absence of Dzyaloshinskii-Moriya interaction. A topological phase diagram is presented where seve…
We study the impact of trimerization (breathing) of the nearest-neighbor (NN) exchange on the planar XY spin-1/2 ferromagnet on the kagome lattice, including additional four-site ring exchange. For uniform NN exchange, this model has previously been shown to transit from a long-range ordered ferromagnet into a Z2 quantum spin liquid by virtue of the ring exchange. Using quantum Monte-Carlo calculations, based on the stochastic series expansion, we present results for the spin stiffness, the quantum phase diagram, the longitudinal static, as well as the transverse dynamic structure factor. Our …
We study the effects of an in-plane Dzyaloshinskii-Moriya interaction under an external magnetic field in the highly frustrated kagome antiferromagnet. We focus on the low-temperature phase diagram, which we obtain through extensive Monte-Carlo simulations. We show that, given the geometric frustration of the lattice, highly non trivial phases emerge. At low fields, lowering the temperature from a cooperative paramagnet phase, the kagome elementary plaquettes form non-coplanar arrangements with non-zero chirality, retaining a partial degeneracy. As the field increases, there is a transition fr…
Excitations of ordered insulating magnets gain renewed interest due to their potential topological properties and the natural realization of magnetic analogues of the celebrated topological models. In this paper we go beyond these parallels and explore what else is there in the unconventional excitations of quantum magnets. We study the topologically nontrivial multiplet excitations of the antiferromagnetic spin-1/2 kagome system with strong breathing anisotropy and Dzyaloshinskii-Moriya interaction. We show that in the chiral magnetic ground state the excitations can be characterized by a spi…
We investigate the spin-1/2 Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e. with weak and strong triangular units), constructing an improved simplex Resonating Valence Bond (RVB) ansatz by successive applications (up to three times) of local quantum gates which implement a filtering operation on the bare nearest-neighbor RVB state. The resulting Projected Entangled Pair State involves a small number of variational parameters (only one at each level of application) and preserves full lattice and spin-rotation symmetries. Despite its simple analytic form, the sim…
A second-order topological insulator in $d$ dimensions is an insulator which has no $d-1$ dimensional topological boundary states but has $d-2$ dimensional topological boundary states. It is an extended notion of the conventional topological insulator. Higher-order topological insulators have been investigated in square and cubic lattices. In this paper, we generalize them to breathing Kagome and pyrochlore lattices. First, we construct a second-order topological insulator on the breathing Kagome lattice. Three topological boundary states emerge at the corner of the triangle, realizing a 1/3 f…
Ca$_{10}$Cr$_7$O$_{28}$ is a novel spin-$1/2$ magnet exhibiting spin liquid behaviour which sets it apart from any previously studied model or material. However, understanding Ca$_{10}$Cr$_7$O$_{28}$ presents a significant challenge, because the low symmetry of the crystal structure leads to very complex interactions, with up to seven inequivalent coupling parameters in the unit cell. Here we explore the origin of the spin-liquid behaviour in Ca$_{10}$Cr$_7$O$_{28}$, starting from the simplest microscopic model consistent with experiment - a Heisenberg model on a single bilayer of the breathin…
Motivated by recent experimental progress on various cluster Mott insulators, we study an extended Hubbard model on a breathing Kagomé lattice with a single electron orbital and $1/6$ electron filling. Two distinct types of cluster localization are found in the cluster Mott regime due to the presence of the electron repulsion between neighboring sites, rather than from the on-site Hubbard interaction in the conventional Mott insulators. We introduce a unified parton construction framework to accommodate both type of cluster Mott insulating phase as well as a trivial Ferm liquid metal and discu…
Engineering non trivial topological phases in materials, specially skyrmion-like arrangements, has been of great interest in the last decade due to its potential technological applications. In this work, we study a model of electrons coupled to a magnetic texture in the kagome lattice interacting with magnetic moments via Rashba spin orbit coupling in the large Hund interaction limit. We obtain the effective spin Hamiltonian and study the emergent low temperature phases under an external magnetic field using large scale Monte Carlo simulations. We show that strong geometric frustration, charac…
The nature of the ground state of the spin $S=1/2$ Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e., with different superexchange couplings $J_{\vartriangle}$ and $J_{\triangledown}$ within elementary up- and down-pointing triangles) is investigated within the framework of Gutzwiller projected fermionic wave functions and Monte Carlo methods. We analyze the stability of the U(1) Dirac spin liquid with respect to the presence of fermionic pairing that leads to a gapped $\mathbb{Z}_{2}$ spin liquid. For several values of the ratio $J_{\triangledown}/J_{\vartriangl…
We show that classical Heisenberg antiferromagnets on the breathing kagome lattice can be a platform to realize a zero-field topological order of the scalar spin chirality which can be viewed as a miniature skyrmion crystal (SkX) of discrete form with a small number of spins in its magnetic unit cell. In the model, a third nearest-neighbor (NN) antiferromagnetic interaction along the bond direction $J_3$ and the breathing bond-alternation characterized by the ratio of the NN interaction for large triangles to that for small ones, $J_1'/J_1$, are essential. It is found by means of Monte Carlo s…
In breathing pyrochlores and kagomes, couplings between neighbouring tetrahedra and triangles are free to differ. Breathing lattices thus offer the possibility to explore a different facet of the rich physics of these systems. Here we consider nearest-neighbour classical Heisenberg interactions, both ferromagnetic and antiferromagnetic, and study how the anisotropy of breathing lattices modifies the mode spectrum of pyrochlore and kagome systems. The nature and degeneracy of the flat bands are shown to be preserved for any value of the anisotropy. These flat bands can coexist with Dirac nodes …
Precompression engineering of metal-insulator transition and magnetism in designed breathing kagome systems
School of Physics and Astronomy, and Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University（北京师范大学物理与天文学院，多尺度自旋物理教育部重点实验室）；College of Physics, Hebei Normal University, and Hebei Advanced Thin Films Laboratory（河北师范大学物理学院，河北省先进薄膜实验室）
We theoretically study the topological properties of the tight-binding model on the breathing kagome lattice with antisymmetric spin-orbit coupling (SOC) between nearest neighbors. We show that the system hosts nontrivial topological phases even without second-nearest-neighbor hopping, and that the weakly dispersing band of the kagome lattice can become topological. The main results are presented in the form of phase diagrams, where the $\mathbb{Z}_2$ topological index is shown as a function of SOC (intrinsically allowed and Rashba) and lattice trimerization. In addition, exact diagonalization…
The stability of a miniature skyrmion crystal (SkX) with only a small number of spins in the magnetic unit cell has been theoretically investigated in $J_1$-$J_3$ antiferromagnets on the breathing kagome lattice with a single-ion anisotropy $D$ at zero field. It is found by means of Monte Carlo simulations that due to the breathing bond-alternation, a zero-field triple-${\bf Q}$ miniature SkX can be stabilized not only in the specific case of $D=0$ [K. Aoyama and H. Kawamura, Phys. Rev. B 105, L100407 (2022)] but also in more general situations with easy-axis ($D<0$) and easy-plane ($D>0$) ani…
Magnetic skyrmion textures are realized mainly in non-centrosymmetric, e.g. chiral or polar, magnets. Extending the field to centrosymmetric bulk materials is a rewarding challenge, where the released helicity / vorticity degree of freedom and higher skyrmion density result in intriguing new properties and enhanced functionality. We report here on the experimental observation of a skyrmion lattice (SkL) phase with large topological Hall effect and an incommensurate helical pitch as small as 2.8 nm in metallic Gd3Ru4Al12, which materializes a breathing kagomé lattice of Gadolinium moments. The …
Skyrmions are of interest both from a fundamental and technological point of view, due to their potential to act as information carriers. But one challenge concerns their manipulation, especially at high temperature where thermal fluctuations eventually disintegrate them. Here we study the competition between skyrmions and a chiral spin liquid, using the latter as an entropic buffer to impose a quasi-vacuum of skyrmions. As a result, the temperature becomes a knob to tune the skyrmion density from a dense liquid to a diluted gas, protecting the integrity of each skyrmion from paramagnetic disi…
It is shown here that a Kagomé magnet, with Heisenberg and Dzyaloshinskii-Moriya interactions causes non trivial topological and chiral magnetic properties. Chirality---that is, left or right handedness---is a very important concept in a broad range of scientific areas, and particularly, in condensed matter physics. Inversion symmetry breaking relates chirality with skyrmions, that are protected field configurations with particle-like and topological properties. Here, the reported numerical simulations and theoretical considerations reveal that the magnetic excitations of the Kagomé magnet can…
Magnetic Anisotropies and Skyrmion Lattice Related to Magnetic Quadrupole Interactions of the RKKY Mechanism in Frustrated Spin-Trimer System Gd$_{3}$Ru$_{4}$Al$_{12}$ with a Breathing Kagome Structure
Motivated by the recent developments on cluster Mott insulating materials such as the cluster magnet LiZn$_2$Mo$_3$O$_8$, we consider the strong plaquette charge ordered regime of the extended Hubbard model on a breathing Kagome lattice and reveal the properties of the cluster Mottness. The plaquette charge order arises from the inter-site charge interaction and the collective motion of three localized electrons on the hexagon plaquettes. This model leads naturally to a reduction of the local moments by 2/3 as observed in LiZn$_2$Mo$_3$O$_8$. Furthermore, at low temperatures each hexagon plaqu…
The geometrical frustration nature of the kagome lattice makes it a great host to flat electronic band, non-trivial topological properties, and novel magnetisms. Metallic kagome antiferromagnet YMn$_{6}$Sn$_{6}$ exhibits the topological Hall effect (THE) when an in-plane magnetic field is applied. THE is typically associated with the nanometer-sized non-coplanar spin structure of skyrmions in non-centrosymmetric magnets with large Dzyaloshinskii-Moriya interaction. Here we use single crystal neutron diffraction to determine the field/temperature dependence of the magnetic structure in YMn$_{6}…
Anomalous Hall Response Induced by Correlated Disorder in the Breathing Kagome Lattice Mn$_{3}$Sn
Oak Ridge National Laboratory（橡树岭国家实验室）；The Pennsylvania State University（宾夕法尼亚州立大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Brigham Young University（杨百翰大学）；Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Brookhaven National Laboratory（布鲁克海文国家实验室）
Tuneable skyrmion and anti-skyrmion fluids via mechanical strain in chiral kagome lattice
CONICET and Facultad de Ingeniería, Universidad de Mendoza（门多萨大学工程学院）；Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB), UNLP-CONICET, Facultad de Ciencias Exactas（拉普拉塔国立大学理学院液体与生物系统物理研究所）；Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata（拉普拉塔国立大学工程学院基础科学系）；CNRS, Université de Bordeaux, LOMA, UMR 5798（波尔多大学）；Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS（图卢兹大学）；Centro de Nanotecnología Aplicada, Facultad de Ciencias, Universidad Mayor（马约尔大学科学学院应用纳米技术中心）
Properties of magnon modes localized on a ferromagnetic skyrmion are studied. Three types of possible asymptotic behavior of the modes eigenfrequencies are found for the case of large skyrmion radius $R_s$, namely $ω_0\sim R_s^{-2}$ for the breathing mode, $ω_{-|μ|}\sim R_s^{-1}$ and $ω_{|μ|}\sim R_s^{-3}$ for modes with negative and positive azimuthal quantum numbers, respectively. A number of properties of the magnon eigenfunctions are determined. This enables us to demonstrate that the skyrmion dynamics based on the traveling wave model is described by the massless Thiele equation.
Observation of Magnetic-Anisotropy Crossover and High-Temperature Skyrmions in the Dirac Magnet Fe3Ge with a Distorted Kagome Lattice
Zhejiang University of Technology（浙江工业大学）；Fudan University（复旦大学）；Nanjing University of Posts and Telecommunications（南京邮电大学）；Ningbo University（宁波大学）；Academia Sinica（中央研究院）；Shanghai University（上海大学）；Suzhou University of Technology（苏州科技大学）；Qufu Normal University（曲阜师范大学）；National Taiwan Normal University（台湾师范大学）；Zhejiang University（浙江大学）；Shanghai Jiao Tong University（上海交通大学）
From strong to weak correlations in breathing-mode kagome van der Waals materials: Nb$_3$(F,Cl,Br,I)$_8$ as a robust and versatile platform for many-body engineering
We theoretically predict and classify the localized modes of a skyrmion in a collinear uniaxial antiferromagnet and discuss how they can be excited. As a central result, we find two branches of skyrmion eigenmodes with distinct physical properties characterized by being low or high energy excitations. The frequency dependence of the low-energy modes scales as $R_0^{-2}$ for skyrmions with large radius $R_0$. Furthermore, we predict localized high-energy eigenmodes, which have no direct ferromagnetic counterpart. Except for the breathing mode, we find that all localized antiferromagnet skyrmion…
Magnetic materials with competing magnetocrystalline anisotropy and dipolar energies can develop a wide range of domain patterns, including classical stripe domains, domain branching, as well as topologically trivial and non-trivial (skyrmionic) bubbles. We image the magnetic domain pattern of Fe$_3$Sn$_2$ by magnetic force microscopy (MFM) and study its evolution due to geometric confinement, magnetic fields, and their combination. In Fe$_3$Sn$_2$ lamellae thinner than 3 $μ$m, we observe stripe domains whose size scales with the square root of the lamella thickness, exhibiting classical Kitte…
The higher-order topological insulator (HOTI) protected by spacial symmetry has been studied in-depth on models with square lattice. Our work, based on an alternative model on the breathing Kagome lattice, revealed that the different types of corners in the lattice could actually be conditionally gapless, or always gapped. Using the Wilson loop formalism, we argue that these corner states occur when the eigenvalues of the Wannier Hamiltonian cross through a certain reference point during the conceptual "pumping" procedure. The results demonstrate the corner of the Kagome lattice based HOTI is …
We report results from spin trimer-based cluster quantum Monte Carlo simulations for the thermodynamic properties of two-dimensional frustrated quantum antiferromagnets that are composed of weakly-coupled three-spin (trimer) clusters. In particular, we consider the spin-1/2 kagome lattice with a strong breathing distortion, and the triangle-square lattice model proposed previously for the cuprate La${}_4$Cu${}_3$MoO${}_{12}$. For both cases, we demonstrate that an appropriately chosen trimer-based computational basis allows us to significantly reduce the quantum Monte Carlo sign problem down t…
Chiral magnets support topological skyrmion textures due to the Dzyaloshinskii-Moriya spin-orbit interaction. In the presence of a sufficiently large applied magnetic field, such skyrmions are large amplitude excitations of the field-polarized magnetic state. We investigate analytically the interaction between such a skyrmion excitation and its small amplitude fluctuations, i.e., the magnons in a clean two-dimensional chiral magnet. The magnon spectrum is found to include two magnon-skyrmion bound states corresponding to a breathing mode and, for intermediate fields, a quadrupolar mode, which …
The elementary excitations of skyrmion crystals experience both emergent magnetic fields and anharmonic interactions brought about by the topologically nontrivial noncollinear texture. The resulting flat bands cause strong spontaneous quasiparticle decay, dressing the eigenmodes of skyrmion crystals with a finite zero-temperature quantum lifetime. Sweeping the flat bands through the spectrum by changing the magnetic field leads to an externally controllable energy-selective magnon breakdown. In particular, we uncover that the three fundamental modes, i.e., the anticlockwise, breathing, and clo…
We theoretically study spin-wave modes and their intense excitations activated by microwave magnetic fields in the skyrmion-crystal phase of insulating magnets by numerically analyzing a two-dimensional spin model using the Landau-Lifshitz-Gilbert equation. Two peaks of spin-wave resonances with frequencies of ~1 GHz are found for in-plane a.c. magnetic field where distribution of the out-of-plane spin components circulates around each skyrmion core. Directions of the circulations are opposite between these two modes, and hence the spectra exhibit salient dependence on the circular polarizatio…
Spectrum of spin eigenmodes localized on a ferromagnetic skyrmion pinned by a geometrical defect (bump) of magnetic films is studied theoretically. By means of direct numerical solution of the corresponding eigenvalue problem and finite element micromagnetic simulations we demonstrate, that the curvature can induce localized modes with higher azimuthal and radial quantum numbers, which are absent for planar skyrmions (for the same parameters). The eigenfrequencies of all modes, except the breathing and gyromodes decreases with increasing curvature. Due to the translational symmetry break, the …
The skyrmion is a topologically stable spin texture, in which the spin direction wraps a sphere. The topological nature gives rise to emergent electromagnetic phenomena such as topological Hall effect. Recently, the crystallization of nanoscale skyrmions was observed in chiral helimagnets with use of the neutron diffraction and Lorentz transmission electron microscopy. The skyrmions are quite mobile under electric current density as low as $\sim$ 10$^6$ A/m$^2$, and, in some cases, stable up to near room temperature. These features suggest that the skyrmions may work as a magnetic stable-varia…
We report on the observation of a large topological Hall effect (THE) over a wide temperature region in a geometrically frustrated Fe3Sn2 magnet with a kagome-bilayer structure. We found that the magnitude of the THE resistivity increases with temperature and reaches -0.875 μΩ.cm at 380 K. Moreover, the critical magnetic fields with the change of THE are consistent with the magnetic structure transformation, which indicates that the real-space fictitious magnetic field is proportional to the formation of magnetic skyrmions in Fe3Sn2. The results strongly suggest that the large THE originates f…
The kagome lattice, consisting of interconnected triangles and hexagons uniquely, is an excellent model system for study frustrated magnetism, electronic correlation and topological electronic structure. After an intensive investigation on frustrated magnetism in insulating magnetic kagome lattices, the interplay between charge and spin degrees of freedom via spin-orbital coupling in metallic systems (kagome metals) has become an attractive topic recently. Here, we study centrosymmetric YMn6Sn6 with Mn kagome lattice. We discover that it exhibits giant topological Hall effect near room tempera…
Anomalous Hall and Nernst effects driven by static and fluctuating spin chiralities on Kagome lattice
School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences（中国科学院大学电子电气与通信工程学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学科维理理论科学研究所）
Spin-1 antiferromagnets are abundant in nature, but few theories or results exist to understand their general properties and behavior, particularly in situations when geometric frustration is present. Here we study the $S=1$ Kagome compound Na$_2$Ti$_3$Cl$_8$ using a combination of Density Functional Theory, Exact Diagonalization, and Density Matrix Renormalization Group methods to achieve a first principles supported explanation of exotic magnetic phases in this compound. We find that the effective magnetic Hamiltonian includes essential non-Heisenberg terms that do not stem from spin-orbit c…
