Topological flat bands in frustrated kagome lattice CoSn
Mingu Kang, Shiang Fang, Linda Ye, Hoi Chun Po, Jonathan Denlinger, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Efthimios Kaxiras, Joseph G. Checkelsky, Riccardo Comin
Electronic flat bands in momentum space, arising from strong localization of electrons in real space, are an ideal stage to realize strong correlation phenomena. In certain lattices with built-in geometrical frustration, electronic confinement and flat bands can naturally arise from the destructive interference of electronic hopping pathways. Such lattice-borne flat bands are often endowed with nontrivial topology if combined with spin-orbit coupling, while their experimental realization in condensed matter system has been elusive so far. Here, we report the direct observation of topological flat bands in the vicinity of the Fermi level in frustrated kagome system CoSn, using angle-resolved photoemission spectroscopy and band structure calculations. The flat band manifests itself as a dispersionless electronic excitation along the G-M high symmetry direction, with an order of magnitude lower bandwidth (below 150 meV) compared to the Dirac bands originating from the same orbitals. The frustration-driven nature of the flat band is directly confirmed by the real-space chiral d-orbital texture of the corresponding effective Wannier wave functions. Spin-orbit coupling opens a large gap of 80 meV at the quadratic band touching point between the Dirac and flat bands, endowing a nonzero Z2 topological invariant to the flat band in the two-dimensional Brillouin zone. Our observation of lattice-driven topological flat band opens a promising route to engineer novel emergent phases of matter at the crossroad between strong correlation physics and electronic topology.
Layered kagome-lattice 3d transition metals are emerging as an exciting platform to explore the frustrated lattice geometry and quantum topology. However, the typical kagome electronic bands, characterized by sets of the Dirac-like band capped by a phase-destructive flat band, have not been clearly observed, and their orbital physics are even less well investigated. Here, we present close-to-textbook kagome bands with orbital differentiation physics in CoSn, which can be well described by a minimal tight-binding model with single-orbital hopping in Co kagome lattice. The capping flat bands wit…
CoSn is a Pauli paramagnet with relatively flat d-bands centered about 100 meV below the Fermi energy Ef. Single crystals of CoSn lightly doped with Fe, In, or Ni are investigated using x-ray and neutron scattering, magnetic susceptibility and magnetization, ac susceptibility, specific heat and resistivity measurements. Within the rigid band approximation, hole doping with a few percent of Fe or In should move the flat bands closer to Ef, whereas electron doping with Ni should move the flat bands further away from Ef. We provide evidence that this indeed occurs. Fe and In doping drive CoSn tow…
Quantum interference on the kagome lattice generates electronic bands with narrow bandwidth, called flat bands. Crystal structures incorporating this lattice can host strong electron correlations with non-standard ingredients, but only if these bands lie at the Fermi level. In the six compounds with the CoSn structure type (FeGe, FeSn, CoSn, NiIn, RhPb, and PtTl) the transition metals form a kagome lattice. The two iron variants are robust antiferromagnets so we focus on the latter four and investigate their thermodynamic and transport properties. We consider these results and calculated band …
In two-dimensional (2D) metallic kagome lattice materials, destructive interference of electronic hopping pathways around the kagome bracket can produce nearly localized electrons, and thus electronic bands that are flat in momentum space. When ferromagnetic order breaks the degeneracy of the electronic bands and splits them into the spin-up majority and spin-down minority electronic bands, quasiparticle excitations between the spin-up and spin-down flat bands should form a narrow localized spin-excitation Stoner continuum coexisting with well-defined spin waves in the long wavelengths. Here w…
We investigate the connection between highly frustrated kagome based Hamiltonians and a recently synthesized family of materials containing Ti3+ S=1/2 ions. Employing a combination of all electron density functional theory and numerical diagonalization techniques, we establish the Heisenberg Hamiltonians for the distorted kagome antiferromagnets Rb2NaTi3F12, Cs2NaTi3F12 and Cs2KTi3F12. We determine magnetization curves in excellent agreement with experimental observations. Our calculations successfully clarify the relationship between the experimental observations and the magnetization-plateau…
Kagome-net, appearing in areas of fundamental physics, materials, photonic and cold-atom systems, hosts frustrated fermionic and bosonic excitations. However, it is extremely rare to find a system to study both fermionic and bosonic modes to gain insights into their many-body interplay. Here we use state-of-the-art scanning tunneling microscopy and spectroscopy to discover unusual electronic coupling to flat-band phonons in a layered kagome paramagnet. Our results reveal the kagome structure with unprecedented atomic resolution and observe the striking bosonic mode interacting with dispersive …
Spin ices are exotic phases of matter characterized by frustrated spins obeying local ice rules, in analogy with the electric dipoles in water ice. In two dimensions, one can similarly define ice rules for in-plane Ising-like spins arranged on a kagome lattice. These ice rules require each triangle plaquette to have a single monopole, and can lead to various unique orders and excitations. Using experimental and theoretical approaches including magnetometry, thermodynamic measurements, neutron scattering and Monte Carlo simulations, we establish HoAgGe as a crystalline (i.e. non-artificial) sys…
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…
Electronic flat band systems are a fertile platform to host correlation-induced quantum phenomena such as unconventional superconductivity, magnetism and topological orders. While flat band has been established in geometrically frustrated structures, such as the kagome lattice, flat band-induced correlation effects especially in those multi-orbital bulk systems are rarely seen. Here we report negative magnetoresistance and signature of ferromagnetic fluctuations in a prototypical kagome metal CoSn, which features a flat band in proximity to the Fermi level. We find that the magnetoresistance i…
The anomalous Hall effect is investigated for ferromagnetic Fe3Sn2 single crystal with geometrically frustrated kagome bilayer of Fe. The scaling behavior between anomalous Hall resistivity rho_{xy}^{A} and longitudinal resistivity rho_{xx} is quadratic and further analysis implies that the AHE in Fe3Sn2 single crystal should be dominated by the intrinsic Karplus-Luttinger mechanism rather than extrinsic skew-scattering or side-jump mechanisms. Moreover, there is a sudden jump of anomalous Hall conductivity sigma_{xy}^{A} appearing at about 100 K where the spin-reorientation transition from th…
Quantum spin-1/2 kagome Heisenberg antiferromagnet is the representative frustrated system possibly hosting a spin liquid. Clarifying the nature of this elusive topological phase is a key challenge in condensed matter, however, even identifying it still remains unsettled. Here, we apply a magnetic field and discover a series of spin gapped phases appearing at five different fractions of magnetization by means of grand canonical density matrix renormalization group, an unbiased state-of-art numerical technique. The magnetic field dopes magnons and first gives rise to a possible Z3 spin liquid p…
We investigate analytically and numerically the classical ground states of frustrated Heisenberg models on pyrochlore and kagome lattices in zero and finite magnetic fields. Each model has a wide region in the microscopic parameter space, where the propagation vector is turned to a commensurate position equal to a half of the reciprocal lattice vector with a nontrivial star. Within these regions the zero-field ground states for both models correspond to noncoplanar triple-k spin configurations. A universal appearance of the 3-k states can be related to the spin-space dimensionality. A strong m…
We investigate the lattice dynamics in the spin-1/2 frustrated kagome compound herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$ by a combination of infrared spectroscopy measurements and ab initio density functional theory calculations, and provide an unambiguous assignment of infrared-active lattice vibrations involving in-plane and out-of-plane atom displacements in the kagome layers. Upon cooling, non-thermal red-shifts and broadening appear specifically for modes that deform the kagome layer or affect the Cu-O-Cu bond angles, thus creating pronounced modifications of the antiferromagnetic exchange c…
The objective of the present work was to analyze the possibility of realization of quantum spin liquid in three volcanic minerals - averievite (Cu5O2(VO4)2(CuCl)), ilinskite (NaCu5O2(SeO3)2Cl3), and avdononite (K2Cu5Cl8(OH)4 2H2O - from the crystal chemistry point of view. Based on the structural data, the sign and strength of magnetic interactions have been calculated and the geometric frustrations serving as the main reason of the existence of spin liquids have been investigated. According to our calculations, the magnetic structures of averievite and ilinskite are composed of antiferromagne…
In frustrated magnets the Dzyaloshinsky-Moriya interaction arising from spin-orbit coupling can induce a magnetic long-range order. Here, we report a theoretical prediction of thermal Hall effect in frustrated kagomé magnets such as KCr$_3$(OH)$_6$(SO4)$_2$ and KFe$_3$(OH)$_{6}$(SO$_{4}$)$_2$. The thermal Hall effects in these materials are induced by scalar spin chirality as opposed to DMI in previous studies. The scalar spin chirality originates from magnetic-field-induced chiral spin configuration due to non-coplanar spin textures, but in general it can be spontaneously developed as a macro…
Rb2SnCu3F12 has been studied using synchrotron powder X-ray diffraction, powder neutron diffraction and single crystal X-ray diffraction at a range of temperatures (100 to 500 K). A broad but clear phase transition, possibly of re-entrant character, is found to occur in the powder form, whereas the corresponding transition is not seen in the single crystal.
Spin dynamics of the spin-1/2 kagome lattice antiferromagnet Cs$_2$Cu$_3$SnF$_{12}$ was studied using high-resolution, time-of-flight inelastic neutron scattering. The flat mode, a characteristic of the frustrated kagome antiferromagnet, and the low-energy dispersive mode, which is dominated by magnons, can be well described by the linear spin-wave theory. However, the theory fails to describe three weakly dispersive modes between 9 and 14 meV. These modes could be attributed to two-spinon bound states, which decay into free spinons away from the zone center and at a high temperature, giving r…
The spin wave excitations of the ideal S=5/2 Kagome lattice antiferromagnet KFe3(OH)6(SO4)2 have been measured using high-resolution inelastic neutron scattering. We directly observe a flat mode which corresponds to a lifted "zero energy mode," verifying a fundamental prediction for the Kagome lattice. A simple Heisenberg spin Hamiltonian provides an excellent fit to our spin wave data. The antisymmetric Dzyaloshinskii-Moriya interaction is the primary source of anisotropy and explains the low temperature magnetization and spin structure.
Motivated by recent experimental work on FeCrAs, we study the magnetic properties of classical spin models on the hexagonal Fe$_2$P/ZrNiAl structure (space group $P{\bar{6}}2m$). When both transition metal/rare earth sites carry magnetic moments, one has alternating distorted kagome and triangular lattice layers. Each point of the triangular lattice lies at the centre of a distorted hexagon and actually corresponds to three isolated magnetic atoms: a "trimer". We show that a simple model consisting of antiferromagnetically correlated Heisenberg spins on the frustrated distorted kagome lattice …
The magnetization and specific heat measurements have been performed on single-crystalline Gd_3_Ru_4_Al_12_ with a distorted Kagome lattice structure. This spin system is regarded as an antiferromagnetic triangular lattice of XY like Heisenberg model at low temperatures. The magnetic phase diagrams indicate the existence of frustration and Z_2_ degeneracy. The magnetization and specific heat imply the successive phase transitions with partial disorder and a T-shaped spin structure in the ground state.
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…
We have studied the Heisenberg antiferromagnets on two-dimensional frustrated lattices, triangular and kagome lattices using linear spin-wave theory. A collinear ground state ordering is possible if one of the three bonds in each triangular plaquette of the lattice becomes weaker or frustrated. We study spiral order in the Heisenberg model along with Dzyaloshinskii-Moriya (DM) interaction and in the presence of a magnetic field. The quantum corrections to the ground state energy and sublattice magnetization are calculated analytically in the case of triangular lattice with nearesr-neighbour in…
We explore the phase diagram and the low-energy physics of three Heisenberg antiferromagnets which, like the kagome lattice, are networks of corner-sharing triangles but contain two sets of inequivalent short-distance resonance loops. We use a combination of exact diagonalization, analytical strong-coupling theories, and resonating valence bond approaches, and scan through the ratio of the two inequivalent exchange couplings. In one limit, the lattices effectively become bipartite, while at the opposite limit heavily frustrated nets emerge. In between, competing tunneling processes result in s…
We propose a general non-perturbative scheme that quantitatively maps the low-energy sector of spin-1/2 frustrated Heisenberg antiferromagnets to effective Generalized Quantum Dimer Models. We develop the formal lattice independent frame and establish some important results on (i) the locality of the generated Hamiltonians (ii) how full resummations can be performed in this renormalization scheme. The method is then applied to the much debated kagome antiferromagnet for which a fully resummed effective Hamiltonian - shown to capture the essential properties and provide deep insights on the mic…
We report muSR experiments on Mg{x}Cu{4-x}(OH)6Cl2 with x \sim 1, a new material isostructural to Herbertsmithite exhibiting regular kagome planes of spin 1/2 (Cu^{2+}), and therefore a candidate for a spin liquid ground state. We evidence the absence of any magnetic ordering down to 20 mK (\sim J/10^4). We investigate in detail the spin dynamics on well characterized samples in zero and applied longitudinal fields and propose a low T defect based interpretation to explain the unconventional dynamics observed in the quantum spin liquid phase.
Entanglement renormalization techniques are applied to numerically investigate the ground state of the spin-1/2 Heisenberg model on a kagome lattice. Lattices of N={36,144,inf} sites with periodic boundary conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be a valence bond crystal (VBC) with a 36-site unit cell, compatible with a previous proposal. Its energy per site, E=-0.43221, is an exact upper bound and is lower than the energy of any previous (gapped or algebraic) spin liquid candidate for the ground state.
Strain applied to a condensed-matter system can be used to engineer its excitation spectrum via artificial gauge fields, or it may tune the system through transitions between different phases. Here we demonstrate that strain tuning of the ground state of otherwise highly degenerate frustrated systems can induce novel phases, both ordered and disordered. For the classical Heisenberg antiferromagnet on the kagome lattice, we show that weak triaxial strain reduces the degeneracies of the system, leading to a classical spin liquid with non-coplanar configurations, while stronger strain drives the …
We study the XXZ spin-one quantum magnet on the kagome lattice as an example where quantum fluctuations on highly degenerate classical ground states lead to various exotic quantum ground states. Previous studies have predicted several quantum phases, but different analytical approaches do not necessarily lead to the same physical picture. In this work, we use Quantum Monte Carlo computations to critically examine some of the predictions made in the string-net mean-field theory and the degenerate perturbation theory combined with duality analysis and effective field theory. It is found that the…
Time-domain magneto-THz spectroscopy is utilized to study the frustrated magnet averievite Cu$_{5-x}$Zn$_x$V$_2$O$_{10}$(CsCl). Pronounced THz resonances are observed in unsubstituted samples ($x=0$) when cooling below the onset of short-range magnetic correlations. The influence of external magnetic effects confirms the magnetic origin of these resonances. Increasing Zn substitution suppresses the resonances, as frustration effects dominate, reflecting the non-magnetic phases for $x> 0.25$ compounds. The temperature evolution of the THz spectra is complemented with electron spin resonance spe…
Spin correlations in the geometrically frustrated RBaCo4O7 compounds, usually described as an alternating stacking of Kagome and triangular layers on a hexagonal lattice, have been studied by mean-field approach and by Monte-Carlo simulations. The behavior of the system was modelled with an isotropic Heisenberg Hamiltonian as a function of the relevant parameter Jout/Jin, representing the ratio between exchange integrals inside the Kagome layers, Jin, and between Kagome and triangular layers, Jout. This ratio can be varied in real systems by appropriate chemical substitutions. At the mean-fiel…
Topological magnon insulators in insulating Kagomé ferromagnets have been extensively studied in a series of papers. It has been established that Dzyaloshinskii-Moriya interaction (DMI) is the key ingredient to observe a nontrivial topological magnon with edge modes. However, insulating antiferromagnets on the Kagomé lattice are frustrated systems considered as a playground for studying quantum spin liquid physics. In these systems the DMI can induce a coplanar but noncollinear magnetic orders with a $\mathbf{q}=0$ propagating wavevector. We show that topological magnon bands are absent in thi…
We find that the antiferromagnetic Heisenberg model on the Kagome lattice with nearest neighboring exchange coupling(NN-KAFH) belongs to a continuous family of fully-frustrated Heisenberg model on the Kagome lattice, which has no preferred classical ordering pattern. The model within this family consists of the first, second and the third neighboring exchange coupling $J_{1}$, $J_{2}$, and $J_{3}$, with $J_{2}=J_{3}$. We find that when $-J_{1}\leq J_{2}=J_{3}\leq 0.2J_{1}$, the lowest band of $J(\mathbf{q})$, namely, the Fourier transform of the exchange coupling, is totally non-dispersive. Ex…
We present an extensive gallium NMR study of the geometrically frustrated kagome bi-layer compound SrCr_(9p)Ga_(12-9p)O_(19) (Cr^3+, S=3/2) over a broad Cr-concentration range (.72<p<.95). This allows us to probe locally the kagome bi-layer susceptibility and separate the intrinsic properties due to the geometric frustration from those related to the site dilution. Our major findings are: 1) The intrinsic kagome bi-layer susceptibility exhibits a maximum in temperature at 40-50 K and is robust to a dilution as high as ~20%. The maximum reveals the development of short range antiferromagnetic c…
Kalmeyer-Laughlin (KL) chiral spin liquid (CSL) is a type of quantum spin liquid without time-reversal symmetry, and it is considered as the parent state of an exotic type of superconductivity--anyon superconductor. Such exotic state has been sought for more than twenty years, however it remains unclear whether it can exist in realistic system where time-reversal symmetry is breaking (T-breaking) spontaneously. By using the density matrix renormalization group, we show that KL CSL exists in a frustrated anisotropic kagome Heisenberg model (KHM), which has spontaneously T-breaking. We find that…
The competition between interactions in frustrated magnets allows a wide variety of new ground states, often exhibiting emergent physics and unique excitations. Expanding the suite of lattices available for study enhances our chances of finding exotic physics. Mn$_2$Sb$_2$O$_7$ forms in a chiral, kagome-based structure in which a fourth member is added to the kagome-plane triangles to form an armchair unit and link adjacent kagome planes. This structural motif may be viewed as intermediate between the triangles of the kagome network and the tetrahedra in the pyrochlore lattice. Mn$_2$Sb$_2$O$_…
Optical conductivity measurements are combined with density functional theory calculations in order to understand the electrodynamic response of the frustrated Mott insulators Herbertsmithite $\mathrm{ZnCu_{3}(OH)_{6}Cl_{2}}$ and the closely-related kagome-lattice compound $\mathrm{Y_{3}Cu_{9}(OH)_{19}Cl_{8}}$. We identify these materials as charge-transfer rather than Mott-Hubbard insulators, similar to the high-$T_c$ cuprate parent compounds. The band edge is at 3.3 and 3.6 eV, respectively, establishing the insulating nature of these compounds. Inside the gap, we observe dipole-forbidden lo…
Magnetization, specific heat, and neutron scattering measurements were performed to study a magnetic transition in jarosite, a spin-5/2 kagome lattice antiferromagnet. When a magnetic field is applied perpendicular to the kagome plane, magnetizations in the ordered state show a sudden increase at a critical field H_c, indicative of the transition from antiferromagnetic to ferromagnetic states. This sudden increase arises as the spins on alternate kagome planes rotate 180 degrees to ferromagnetically align the canted moments along the field direction. The canted moment on a single kagome plane …
We present the crystal structure and magnetic properties of Y$_{3}$Cu$_{9}$(OH)$_{19}$Cl$_{8}$, a stoichiometric frustrated quantum spin system with slightly distorted kagome layers. Single crystals of Y$_{3}$Cu$_{9}$(OH)$_{19}$Cl$_{8}$ were grown under hydrothermal conditions. The structure was determined from single crystal X-ray diffraction and confirmed by neutron powder diffraction. The observed structure reveals two different Cu-positions leading to a slightly distored kagome layer in contrast to the closely related YCu$_{3}$(OH)$_{6}$Cl$_{3}$. Curie-Weiss behavior at high-temperatures w…
Transverse field Ising model is a common model in quantum magnetism and is often illustrated as an example for quantum phase transition. Its physical origin in quantum magnets, however, is actually not quite well-understood. The quantum mechanical properties of this model on frustrated systems are not well-understood either. We here clarify the physical origin, both extrinsic one and intrinsic one, for the transverse field of the quantum Ising model, and then explain the quantum effects in the Kagome system. We discuss the quantum plaquette order and the quantum phase transition out of this or…
The structural-magnetic models of 25 AFM kagome cuprates similar to herbertsmithite (ZnCu3(OH)6Cl2) - a perspective spin liquid - have been calculated and analyzed. Main correlations between the structure and magnetic properties of these compounds were revealed. It has been demonstrated that, in all AFM kagome cuprates, including herbertsmithite, there exists the competition between the exchange interaction and the antisymmetric anisotropic exchange one (the Dzyaloshinskii-Moriya interaction), as magnetic ions are not linked to the center of inversion in the kagome lattice. This competition is…
