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Abstract Here we review scanning tunneling microscopy research on the surface determination for various types of kagome materials, including 11-type (CoSn, FeSn, FeGe), 32-type (Fe3Sn2), 13-type (Mn3Sn), 135-type (AV3Sb5, A = K, Rb, Cs), 166-type (TbMn6Sn6, YMn6Sn6 and ScV6Sn6), and 322-type (Co3Sn2S2 and Ni3In2Se2). We first demonstrate that the measured step height between different surfaces typically deviates from the expected value of ±0.4 ∼0.8Å, which is owing to the tunneling convolution effect with electronic states and becomes a serious issue for Co3Sn2S2 where the expected Sn-S interlayer distance is 0.6Å. Hence, we put forward a general methodology for surface determination as atomic step geometry imaging, which is fundamental but also experimentally challenging to locate the step and to image with atomic precision. We discuss how this method can be used to resolve the surface termination puzzle in Co3Sn2S2. This method provides a natural explanation for the existence of adatoms and vacancies, and beyond using unknown impurity states, we propose and use designer layer-selective substitutional chemical markers to confirm the validity of this method. Finally, we apply this method to determine the surface of a new kagome material Ni3In2Se2, as a cousin of Co3Sn2S2, and we image the underlying kagome geometry on the determined Se surface above the kagome layer, which directly visualizes the p-d hybridization physics. We emphasize that this general method does not rely on theory, but the determined surface identity can provide guidelines for first-principles calculations with adjustable parameters on the surface-dependent local density of states and quasi-particle interference patterns.

Abstract The appearance of van Hove singularities near the Fermi level leads to prominent phenomena, including superconductivity, charge density wave, and ferromagnetism. Here a bilayer Kagome lattice with multiple van Hove singularities is designed and a novel borophene with such lattice (BK‐borophene) is proposed by the first‐principles calculations. BK‐borophene, which is formed via three‐center two‐electron (3c–2e) σ‐type bonds, is predicted to be energetically, dynamically, thermodynamically, and mechanically stable. The electronic structure hosts both conventional and high‐order van Hove singularities in one band. The conventional van Hove singularity resulting from the horse saddle is 0.065 eV lower than the Fermi level, while the high‐order one resulting from the monkey saddle is 0.385 eV below the Fermi level. Both the singularities lead to the divergence of electronic density of states. Besides, the high‐order singularity is just intersected to a Dirac‐like cone, where the Fermi velocity can reach 1.34 × 106 m s−1. The interaction between the two Kagome lattices is critical for the appearance of high‐order van Hove singularities. The novel bilayer Kagome borophene with rich and intriguing electronic structure offers an unprecedented platform for studying correlation phenomena in quantum material systems and beyond.

Abstract Geometrically perfect S = ½ kagome lattices with frustrated magnetism are typically electrical insulators. Electron or hole doping is predicted to induce an exotic conducting state including superconductivity. Herein, an unconventional strategy of doping an S = ½ kagome lattice CoCu3(OH)6Cl2 is adopted – a structural analogue of a well‐known quantum spin liquid (QSL) candidate herbertsmithite (ZnCu3(OH)6Cl2) – by integrating it with reduced graphene oxide (rGO) via in situ redox chemistry. Such an integration drastically enhances the electrical conductivity, resulting in the transformation of an insulator to a semiconductor, corroborating the respective density of states obtained from the density functional theory calculations. Estimation of the magnetic moments, data on the Hall‐effect measurements, Bader charge analysis, and photoemission signals, altogether provide a bold signature of remote hole doping in CoCu3(OH)6Cl2 by rGO. The remote doping provides an alternative to the site doping approach to impart exotic electronic properties in spin liquid candidates, specifically, the generation of topological states like Dirac metal is envisioned.

In this work we studied the vortex-configuration in a mesoscopic superconducting square in presence of an external magnetic field H applied parallel to its surface vector. We study the magnetization curves in a complete loop, the magnretic induction and the superconducting-electron density for the sample considering Neumann boundary conditions for the order parameter, via length Gennes extrapolation . The sample presents a pinnin center as a Kagome-type laticce at different critical temperature. We solve the generalized time-dependent Ginzburg-Landau equations for a two-condensate system using the Link-variable method considering a Field-Cooling process. Our results show that the vortices are always located at the pinning centers in non-conventional configurations, due to cuopling used.

Abstract Controllably fabricating low-dimensional systems and unraveling their exotic states at the atomic scale is a pivotal step for the construction of quantum functional materials with emergent states. Here, by utilizing the elaborated molecular beam epitaxy growth, we obtain various FexSey phases beyond the single-layer FeSe/SrTiO3 films. A synthetic strategy of lowering substrate temperature with superfluous Se annealing is implemented to achieve various stoichiometric FeSe-derived phases, ranging from 1:1 to 5:8. The phase transitions and electronic structure of these FexSey phases are systematically characterized by atomic resolution scanning tunneling microscopy measurements. We observe the long-ranged antiferromagnetic order of the Fe4Se5 phase by spin-polarized signals with striped patterns, which is also verified by their magnetic response of phase shift between adjacent domains. The electronic doping effect in insulating Fe4Se5 and the kagome effect in metallic Fe5Se8 are also discussed, where the kagome lattice is a promising structure to manifest both spin frustration of d electrons in a quantum-spin-liquid phase and correlated topological states with flat-band physics. Our study provides promising opportunities for constructing artificial superstructures with tunable building blocks, which is helpful for understanding the emergent quantum states and their correlation with competing orders in the FeSe-based family.

Topological photonic insulators provide a novel approach to achieve high-efficiency terahertz transmissions due to the presence of the topological edge states. In this work, the frequency ranges of the topological edge states can be controlled by varying the radius of rods in the kagome lattice and the distance between them. In addition, a gap can be generated between two different frequency ranges and the frequency selecting functions can be realized by constructing different types of domains. Similarly, sandwich a non-trivial photonic crystal between two trivial ones also capable of frequency selection for its topological edge states. Our findings suggest that this work has significant potential for applications in photonic integrated devices.

In this work, we study the effect of electron doping on the kagome superconductor CsV3Sb5. Single crystals and powders of CsV3Sb5−xTex are synthesized and characterized via magnetic susceptibility, nuclear quadrupole resonance, and x-ray diffraction measurements, where we observe a slight suppression of the charge density wave transition temperature and superconducting temperature with the introduction of electron dopants. In contrast to hole doping, both transitions survive relatively unperturbed up to the solubility limit of Te within the lattice. A comparison is presented between the electronic phase diagrams of electron- and hole-tuned CsV3Sb5.

Compounds with a kagome lattice exhibit intriguing properties and the charge density wave (CDW) adds an additional layer of interest to research on them. In this study, we investigate the temperature and magnetic field dependent electrical properties under a chemical substitution and hydrostatic pressure of ScV _6 Sn _6 , a non-magnetic CDW compound. Substituting 5% Cr at the V site or applying 1.5 GPa of pressure shifts the CDW from 92 K to ∼ 50 K. This shift is attributed to the movement of the imaginary phonon band, as revealed by the phonon dispersion relation. The longitudinal and Hall resistivities respond differently under these stimuli. The magnetoresistance (MR) retains its quasilinear behavior under pressure, but it becomes quadratic after Cr substitution. The anomalous Hall-like behavior of the parent compound persists up to the respective CDW transition under pressure, after which it decreases sharply. In contrast, the longitudinal and Hall resistivities of Cr substituted compounds follow a two-band model and originate from the multi carrier effect. These results clearly highlight the role of phonon contributions in the CDW transition and call for further investigation into the origin of the anomalous Hall-like behavior in the parent compound.

A wider scope in topological semimetals accelerates various attractive quantum phenomena for the last two decades. We here report the detailed investigation of magnetic Co3Sn2S2 Weyl semimetal based on the cobalt accompanied kagome lattice for the first time. We emphasize the highly motivated theoretical and experimental analyses in each topological terms of Co3Sn2S2 that led to intensifying emergent topological materials with inherent properties, particularly in the pursuance of magnetic rich quantum anomalous Hall effect. We additionally highlight the Co3Sn2S2 surface band structure, recognized in the diverse fields of study from the perception of topological properties to modern electrochemistry approaches. From materials perspective, this typical kagome Co3Sn2S2 Weyl semimetal alone has extensively covered more enriching nontrivial surface states than topological insulators and nonmagnetic semimetals even in the absence of magnetic field and external strain. Some specific examples include Weyl nodes linked flat band magnetism, a large anomalous Hall effect, anomalous Hall angle, giant magnetooptical effect, and thermomagnetic Nernst effect. In addition, we selectively extend the review on the undisputed surface band characteristics of Co3Sn2S2, by which a major concept of magnetic Weyl semimetal could hierarchically meet the contemporary quantum phenomena. This strongly led to more focus on nontrivial magnetic topological studies for gaining insightful perspectives.

The spin–orbit coupling (SOC) effect is the dominant origin of the topological properties in a topological insulator which can be induced and enhanced through various methods. Here we theoretically study the effects of varying SOC on the topological properties of two-dimensional kagome lattice based on the tight-binding approximation. We find that the system can undergo a transition between two non-trivial topological states by tuning the SOC strength. In addition, the topological phase transition can be achieved with a weaker SOC under a uniform tensile strain, which suggests that strain engineering can be used to regulate topological states. Besides, the characteristics of optical spin Hall conductivity are also studied, which supplies an experimental determination to reveal the different topological states. Furthermore, we propose an external mechanical strain modulated spin-switch device, which can generate opposite spin currents under stretch and compressed strain. Our results indicate that with the combination of SOC and strain, kagome lattice materials have potential applications on tunable spintronics and optoelectronic devices.

Recently the Rydberg blockade effect has been utilized to realize quantum spin liquid (QSL) on a kagome lattice. Evidence of QSL has been obtained experimentally by directly measuring non-local string order. In this paper, we report a Bardeen–Cooper–Schrieffer (BCS)-type variational wave function study of the spin liquid state in this model. This wave function is motivated by mapping the Rydberg blockade model to a lattice gauge theory, where the local gauge conservations replace the role of constraints from the Rydberg blockade. We determine the variational parameter from the experimental measurement of the Rydberg atom population. Then we compare the predictions of this deterministic wave function with the experimental measurements of non-local string order. Combining the measurements on both open and closed strings, we extract the fluctuations only associated with the closed-loop as an indicator of the topological order. The prediction from our wave function agrees reasonably well with the experimental data, with only one fitting parameter determined by measurement of Rydberg atom population. Our variational wave function provides a simple and intuitive picture of the QSL in this system that can be generalized to similar spin liquid phases in other lattice geometry.

The magnon band topology due to the Dzyaloshinskii–Moriya interaction (DMI) and its relevant topological thermal Hall effect has been extensively studied in kagome lattice magnets. In this theoretical investigation, we report a new mechanism for phase transitions between topological phases of magnons in kagome ferromagnets by tuning the anisotropic nearest-neighbor ferromagnetic interaction and DMI. Using the linear spin-wave theory, we calculate the Chern number and thermal Hall conductivity of magnons in low temperature regime. We show the magnon band structures and magnonic edge states in each topological phase. From the topological phase diagram, we find a sign reversal of the thermal Hall conductivity upon tuning the modulation factors. We explicitly demonstrate the correspondence of thermal Hall conductivity with the propagation direction of the magnonic edge states. Finally, we discuss candidate materials as experimental realizations of our theoretical model.

We report an ultra-high vacuum low-temperature scanning tunneling microscopy (STM) study of the C60 monolayer grown on Cd(0001). Individual C60 molecules adsorbed on Cd(0001) may exhibit a bright or dim contrast in STM images. When deposited at low temperatures close to 100 K, C60 thin films present a curved structure to release strain due to dominant molecule–substrate interactions. Moreover, edge dislocation appears when two different wavy structures encounter each other, which has seldomly been observed in molecular self-assembly. When growth temperature rose, we found two forms of symmetric kagome lattice superstructures, 2 × 2 and 4 × 4, at room temperature (RT) and 310 K, respectively. The results provide new insight into the growth behavior of C60 films.

Averievite-type compounds with the general formula (MX)[Cu5O2(TO4)], where M = alkali metal, X = halogen and T = P, V, have been synthesized by crystallization from gases and structurally characterized for six different compositions: 1 (M = Cs; X = Cl; T = P), 2 (M = Cs; X = Cl; T = V), 3 (M = Rb; X = Cl; T = P), 4 (M = K; X = Br; T = P), 5 (M = K; X = Cl; T = P) and 6 (M = Cu; X = Cl; T = V). The crystal structures of the compounds are based upon the same structural unit, the layer consisting of a kagome lattice of Cu2+ ions and are composed from corner-sharing (OCu4) anion-centered tetrahedra. Each tetrahedron shares common corners with three neighboring tetrahedra, forming hexagonal rings, linked into the two-dimensional [O2Cu5]6+ sheets parallel to (001). The layers are interlinked by (T5+O4) tetrahedra (T5+ = V, P) attached to the bases of the oxocentered tetrahedra in a “face-to-face” manner. The resulting electroneutral 3D framework {[O2Cu5](T5+O4)2}0 possesses channels occupied by monovalent metal cations M+ and halide ions X−. The halide ions are located at the centers of the hexagonal rings of the kagome nets, whereas the metal cations are in the interlayer space. There are at least four different structure types of the averievite-type compounds: the P-3m1 archetype, the 2 × 2 × 1 superstructure with the P-3 space group, the monoclinically distorted 1 × 1 × 2 superstructure with the C2/c symmetry and the low-temperature P21/c superstructure with a doubled unit cell relative to the high-temperature archetype. The formation of a particular structure type is controlled by the interplay of the chemical composition and temperature. Changing the chemical composition may lead to modification of the structure type, which opens up the possibility to tune the geometrical parameters of the kagome net of Cu2+ ions.

Magnetic competition in topological kagome magnets is studied by incorporating the spin–orbit coupling, anisotropic Hund coupling and spin exchange into a tight-binding electron dynamics in the kagome lattice. Using the Bogoliubov variational principle we find the stable phases at zero and finite temperatures. At zero temperature and in the strong Ising-Hund coupling regime, a magnetic tunability from the out-of-plane ferromagnetism to the in-plane antiferromagnetism is achieved through a universal property of the critical in-plane Hund coupling. At low temperature the out-of-plane ferromagnetism is stable until a finite crossing temperature. Above the crossing temperature the in-plane antiferromagnetism is stable, but the magnetization of the out-of-plane ferromagnetism still survives. This suggests a metastable coexistence of these magnetic phases in a finite temperature range. A large anomalous Hall conductance is observed in the Ising-Hund coupling limit.

Spin and charge are two interrelated properties of electrons. Most of previous works on topological phases of matter tend to study the topological electronic bands and topological magnon excitations separately. In this work, by combining density functional theory and the Schwinger boson method, we determine the topological electronic band structure and topological magnon spectrum simultaneously in the ferromagnetic phase of CoCu _3 (OH) _6 Cl _2 . This Herbertsmithite material has an ABC-stacking Kagome-lattice structure with fractional occupancies on the Cu sites. We find that the hopping parameters of the electronic and magnonic bands can mutually influence each other and predict in further model studies the possibility of magnetically driven topological phase transitions in the electronic bands. Our study also provides a useful method to deal with correlated topological magnetic systems with narrow band gaps self-consistently.

The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them. Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice. We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations. The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field. We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM). For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures.

A new mineral, dioskouriite, CaCu4Cl6(OH)4∙4H2O, represented by two polytypes, monoclinic (2M) and orthorhombic (2O), which occur together, was found in moderately hot zones of two active fumaroles, Glavnaya Tenoritovaya and Arsenatnaya, at the Second scoria cone of the Northern Breakthrough of the Great Tolbachik Fissure Eruption, Tolbachik volcano, Kamchatka, Russia. Dioskouriite seems to be a product of the interactions involving high-temperature sublimate minerals, fumarolic gas and atmospheric water vapor at temperatures not higher than 150 °C. It is associated with avdoninite, belloite, chlorothionite, eriochalcite, sylvite, halite, carnallite, mitscherlichite, chrysothallite, sanguite, romanorlovite, feodosiyite, mellizinkalite, flinteite, kainite, gypsum, sellaite and earlier hematite, tenorite and chalcocyanite in Glavnaya Tenoritovaya and with avdoninite and earlier hematite, tenorite, fluorophlogopite, diopside, clinoenstatite, sanidine, halite, aphthitalite-group sulfates, anhydrite, pseudobrookite, powellite and baryte in Arsenatnaya. Dioskouriite forms tabular, lamellar or flattened prismatic, typically sword-like crystals up to 0.01 mm × 0.04 mm × 0.1 mm combined in groups or crusts up to 1 × 2 mm2 in area. The mineral is transparent, bright green with vitreous luster. It is brittle; cleavage is distinct. The Mohs hardness is ca. 3. Dmeas is 2.75(1) and Dcalc is 2.765 for dioskouriite-2O and 2.820 g cm−3 for dioskouriite-2M. Dioskouriite-2O is optically biaxial (+), α = 1.695(4), β = 1.715(8), γ = 1.750(6) and 2Vmeas. = 70(10)°. The Raman spectrum is reported. The chemical composition (wt%, electron microprobe data, H2O calculated by total difference; dioskouriite-2O/dioskouriite-2M) is: K2O 0.03/0.21; MgO 0.08/0.47; CaO 8.99/8.60; CuO 49.24/49.06; Cl 32.53/32.66; H2O(calc.) 16.48/16.38; -O=Cl −7.35/−7.38; total 100/100. The empirical formulae based on 14 O + Cl apfu are: dioskouriite-2O: Ca1.04(Cu4.02Mg0.01)Σ4.03[Cl5.96(OH)3.90O0.14]Σ10∙4H2O; dioskouriite-2M: (Ca1.00K0.03)Σ4.03(Cu4.01Mg0.08)Σ4.09[Cl5.99(OH)3.83O0.18]Σ10∙4H2O. Dioskouriite-2M has the space group P21/c, a = 7.2792(8), b = 10.3000(7), c = 20.758(2) Å, β = 100.238(11)°, V = 1531.6(2) Å3 and Z = 4; dioskouriite-2O: P212121, a = 7.3193(7), b = 10.3710(10), c = 20.560(3) Å, V = 1560.6(3) Å3 and Z = 4. The crystal structure (solved from single-crystal XRD data, R = 0.104 and 0.081 for dioskouriite-2M and -2O, respectively) is unique. The structures of both polytypes are based upon identical BAB layers parallel to (001) and composed from Cu2+-centered polyhedra. The core of each layer is formed by a sheet A of edge-sharing mixed-ligand octahedra centered by Cu(1), Cu(2), Cu(3), Cu(5) and Cu(6) atoms, whereas distorted Cu(4)(OH)2Cl3 tetragonal pyramids are attached to the A sheet on both sides, along with the Ca(OH)2(H2O)4Cl2 eight-cornered polyhedra, which provide the linkage of the two adjacent layers via long Ca−Cl bonds. The Cu(4) and Ca polyhedra form the B sheet. The difference between the 2M and 2O polytypes arises as a result of different stacking of layers along the c axis. The cation array of the layer corresponds to the capped kagomé lattice that is also observed in several other natural Cu hydroxychlorides: atacamite, clinoatacamite, bobkingite and avdoninite. The mineral is named after Dioskouri, the famous inseparable twin brothers of ancient Greek mythology, Castor and Polydeuces, the same in face but different in exercises and achievements; the name is given in allusion to the existence of two polytypes that are indistinguishable in appearance but different in symmetry, unit cell configuration and XRD pattern.

High pressure electric transport and synchrotron x-ray diffraction (XRD) measurements together with the first-principles calculations are performed on a shandite compound Pd _3 Pb _2 Se _2 which contains the Kagome lattice of the transition metal Pd. A pressure-induced superconducting transition is observed above 25 GPa, for the first time in the shandite compounds, although the crystal structure of the compound seems to be very robust and persists up to the highest pressure in the XRD study (76.3 GPa). The superconducting transition temperature is about 2.2 K and almost does not change with pressure. The carrier density suddenly increases around 20 GPa possibly due to the emergence of two electron pockets at the Γ point. Our work indicates that the superconductivity in Pd _3 Pb _2 Se _2 is strongly correlated to its electronic structure.

Two new members of the A2B′Cu3F12 family of kagome-related materials have been prepared, in order to further understand the crystal-chemical relationships, phase transitions and magnetic behaviour within this family of potentially frustrated S = ½ two-dimensional quantum magnets. Cs2TiCu3F12 adopts a crystal structure with the ideal kagome lattice topology (space group R m) at ambient temperature. Diffraction studies reveal different symmetry-lowering structural phase transitions in single crystal and polycrystalline forms at sub-ambient temperatures, with the single crystal form retaining rhombohedral symmetry and the powder form being monoclinic. In both cases, long-range antiferromagnetic order occurs in the region 16–20 K. Rb2TiCu3F12 adopts a distorted triclinic structure even at ambient temperatures.

We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.

In the present work, we directly visualize the multi-quanta cages (MQCs) consisting of the giant vortices pinned by the elongated antidots using low-temperature scanning Hall probe microscopy. The periodic but sufficiently isolated MQCs, observed at various magnetic fields, are in a good agreement with the simulated vortex states based on the time-dependent Ginzburg–Landau (tdGL) equations. Due to the competition between the interstitial vortices and the pinned giant vortices, the formation and collapse of the MQCs can be tuned by varying magnetic field. The experimental statistics of the interstitial vortices confined in the MQCs show that the interstitial vortex patterns become more disordered at higher magnetic fields. The stability of the degenerate vortex states and the multi-quanta confinement effects under an external current are also investigated by using the tdGL simulations. The splitting of the free energy of the degenerate vortex states indicates that applying external current can eliminate parts of the degenerate vortex states.

In this work we investigated the critical properties of the anti-ferromagnetic XY model on a two dimensional Kagome lattice under single-ion easy-axes anisotropy. Employing the mean field theory, we found that this model shows a second order phase transition from disordered to all-in all-out state for any value of anisotropy.

It is noted that the pair correlation matrix χ of the nearest neighbor Ising model on periodic three-dimensional (d=3) kagome-like lattices of corner-sharing triangles can be calculated partially exactly. Specifically, a macroscopic number 1/3N+1 out of N eigenvalues of χ are degenerate at all temperatures T, and correspond to an eigenspace L_ of χ, independent of T. Degeneracy of the eigenvalues, and L_ are an exact result for a complex d=3 statistical physical model. It is further noted that the eigenvalue degeneracy describing the same L_ is exact at all T in an infinite spin dimensionality m limit of the isotropic m-vector approximation to the Ising models. A peculiar match of the opposite m=1 and m→ ∞ limits can be interpreted that the m→ ∞ considerations are exact for m=1. It is not clear whether the match is coincidental. It is then speculated that the exact eigenvalues degeneracy in L_ in the opposite limits of m can imply their quasi-degeneracy for intermediate 1≤m

We demonstrate that an acoustic Kagome lattice formed by an array of interconnected resonant cavities exhibits a new class of topological states protected by C _3 symmetry, and it is characterised by a topological invariant in the form of a winding number in Pauli vector space. This acoustic topological metamaterial can be considered as the two-dimensional analogue of the Su–Schrieffer–Heeger model, exhibiting a topological transition when a detuning is introduced between the inter-cell and intra-cell hopping amplitudes. The topological transition caused by such detuning is accompanied by the opening of a complete topological band gap, which may host edge states. The edge states emerge on either truncated ends of the lattice terminated by a cladding layer or at the domain walls between topologically nontrivial and trivial domains. First-principles simulations based on full-wave finite element method are used to design the lattice and confirm our analytical predictions.