Strong Frustration Research Articles

(Cs X)Cu5O2(PO4)2 ( X = Cl, Br, I): A Family of Cu2+ S = 1/2 Compounds with Capped-Kagomé Networks Composed of OCu4 Units.

Three new salt inclusion compounds (Cs X)Cu5O2(PO4)2 ( X = Cl, Br, I), phosphate analogues of the kagomé mineral averievite, are reported. Their crystal structures are composed of trigonal networks of corner-sharing OCu4 anion-centered tetrahedra, forming capped-kagomé planes, which can also be regarded as two-dimensional slices along the [111] direction of a pyrochlore lattice. Magnetization and heat capacity measurements reveal strong geometric frustration of this network and complex magnetic behavior. X-ray and neutron diffraction studies show that all three compounds undergo a trigonal-to-monoclinic phase transition upon cooling, with a first-order phase transition seen in CsBr and CsI analogues. Along with the previously reported (CsCl)Cu5O2(VO4)2, these three new compounds belong to a large family of OCu4-based networks, which are a playground for studying frustrated quantum magnetism.

Magnetic and transport properties of Mn2FeAl

Experimental studies of polycrystalline Mn2FeAl revealed that it crystallizes in a primitive cubic structure of β-Mn type (space group P4132) and is an antiferromagnet with Néel temperature TN = 40 K. The alloy exhibits temperature dependence of transport properties typical for semiconductors and has abnormally high electrical resistivity (ρ ∼ 238 μΩ·cm at room temperature). High Curie-Weiss temperature ΘCW ∼ –905 K and a large value of the frustration parameter ƒ ≈ 23 points to a strong frustration of the magnetic subsystem. These experimental findings are in contrast with reported results of first-principles calculations which predicted Mn2FeAl to be half metallic ferrimagnet crystallizing in the Heusler crystal structure.

Fullerene and endometallofullerene Kagome lattices with symmetry-forced spin frustration.

Salts of fullerene C60˙- and endometallofullerene Sc3N@Ih-C80˙- radical anions with Bu3MeP+ cations ((Bu3MeP+)3(C60˙-)3·C6H4Cl2 (1) and (Bu3MeP+)3(Sc3N@Ih-C80˙-)3·C6H4Cl2 (2)) have been obtained. The C3 symmetry of the Bu3MeP+ cation provides 2D Kagome lattices with an equilateral triangle arrangement of fullerenes in accordance with trigonal crystal symmetry P31m. The C60˙- and Sc3N@Ih-C80˙- radical anions preserve their monomeric forms in 1 and 2 with the S = 1/2 spin state down to 1.9 K. The close packing of the fullerene radical anions results in strong antiferromagnetic coupling of the spins with Weiss temperatures of -108 K for 1 and -43 K for 2. Compound 1 is a rare example of a magnetic system in which in spite of the strong magnetic coupling of spins no long-range ordering is observed down to 1.9 K. The 13C NMR spectra of the 13C enriched sample of 1 support the absence of the antiferromagnetic ordering of spins down to 1.5 K. Thus, the crystals of 1 preserve large spin frustration forced by the trigonal symmetry. Therefore, compound 1 is a promising candidate for the first observation of a quantum spin liquid (QSL) state in a fullerene-based system. Isostructural salt 2 is the first compound that contains monomeric paramagnetic Sc3N@Ih-C80˙- radical anions stable down to 1.9 K, which show strong spin frustration. These data indicate the ability of endometallofullerenes to give exotic magnetic systems such as QSLs.

A proof of concept: Thermodynamics of aluminum – transition metal highly concentrated alloys

In a previous work by the present authors (Acta Mater. 148, 2018, 263–279), the thermodynamics of V–Nb–Ta–Mo–W highly concentrated alloy at equiatomic concentration was investigated using the cluster variation method in the irregular tetrahedron approximation. It was shown that the disordered solid solution is stabilized in this highly concentrated area of the system as prescribed by the high entropy alloy (HEA) paradigm. The configurational entropy, however, has only a minor (and sometimes contrary) influence in this stabilization. It was then postulated that configurational frustration, i.e. the competition between conflicting interactions, is the more relevant factor in the stabilization of the highly-concentrated alloys.The present work elaborates this idea by further considering the cases of V–Nb–Mo–Al and V–Nb–W–Al quaternary systemas and the corresponding quinary systems with Ta addition. The quaternary system are characterized by strong Al – M pair interactions that could enhance configurational frustration. The introduction of Ta, with weaker Ta – Al and practically non-existing Ta – Nb interactions, is meant to disrupt the frustration, hence leading to a decrease of the stability of the disordered alloys. It was shown that V–Nb–Mo–Al and its quinary do not exhibit configurational frustration as expected, since a stable compound with stoichiometry VMoNbAl is formed, and the stability of their disordered alloys (A2) is low. On the contrary, V–Nb–W–Al and its quinary demonstrated strong configurational frustration and correspondingly a stronger A2 stabilization. Deviations from equiatomic composition were also investigated and even stronger A2 stabilization was observed in this case. These results show that the stabilization of the disordered phase can only be attributed to configurational frustration, corroborating the previous work's conclusion that the effect of configurational entropy is only marginal in this stabilization.

Quasi-degenerate magnetic states in α-RuCl3

Exploring quantum spin liquid (QSL) state has both fundamental scientific value and realistic application potential. Recently, α-RuCl3 was experimentally observed to hold in-plane zigzag antiferromagnetic (AFM) order at low temperature, which was further proposed to be proximate to a Kitaev QSL ground state. We have studied the magnetic properties of α-RuCl3 in the framework of electronic structure calculation based on density functional theory (DFT) with Hubbard U correction (DFT+U) and spin–orbit coupling. When the intra-orbital Hubbard interaction U and the inter-orbital Hund’s coupling J adopt the commonly accepted values of U = 2.0 eV and J = 0.4 eV, the zigzag AFM order indeed owns the minimum energy, consistent with the experimental observation. More importantly, we find that compared with the ferromagnetic order in the previous theoretical studies, there exist a series of magnetic configurations energetically even closer to the zigzag AFM ground state. The further calculations and analysis indicate that these low-energy magnetic states are closely related to the electronic correlation effect of Ru 4d orbitals. By decreasing U and increasing J with just about 0.2 eV, they become energetically degenerate with the zigzag AFM order, inducing strong magnetic frustration and then yielding a state without long-range magnetic order but with nonzero local moments. Considering the facts that theoretically the pressure usually reduces the intra-orbital Hubbard interaction and meanwhile enhances the inter-orbital Hund’s coupling, while experimentally the pressure drives α-RuCl3 into a quantum disordered phase, our results provide a perspective to understand the exotic magnetic behaviors of α-RuCl3.

Effect of Zn doping on the antiferromagnetism in kagomeCu4−xZnx(OH)6FBr

Barlowite Cu$_4$(OH)$_6$FBr shows three-dimensional (3D) long-range antiferromagnetism, which is fully suppressed in Cu$_3$Zn(OH)$_6$FBr with a kagome quantum spin liquid ground state. Here we report systematic studies on the evolution of magnetism in the Cu$_{4-x}$Zn$_x$(OH)$_{6}$FBr system as a function of $x$ to bridge the two limits of Cu$_4$(OH)$_6$FBr ($x$=0) and Cu$_3$Zn(OH)$_6$FBr ($x$=1). Neutron-diffraction measurements reveal a hexagonal-to-orthorhombic structural change with decreasing temperature in the $x$ = 0 sample. While confirming the 3D antiferromagnetic nature of low-temperature magnetism, the magnetic moments on some Cu$^{2+}$ sites on the kagome planes are found to be vanishingly small, suggesting strong frustration already exists in barlowite. Substitution of interlayer Cu$^{2+}$ with Zn$^{2+}$ with gradually increasing $x$ completely suppresses the bulk magnetic order at around $x$ = 0.4, but leaves a local secondary magnetic order up to $x\sim 0.8$ with a slight decrease in its transition temperature. The high-temperature magnetic susceptibility and specific heat measurements further suggest that the intrinsic magnetic properties of kagome spin liquid planes may already appear from $x>0.3$ samples. Our results reveal that the Cu$_{4-x}$Zn$_x$(OH)$_6$FBr may be the long-thought experimental playground for the systematic investigations of the quantum phase transition from a long-range antiferromagnet to a topologically ordered quantum spin liquid.

Efficient Monte Carlo simulation of a dissipative Ising chain

We numerically study the dissipative transverse field Ising model in a bosonic bath with Ohmic spectral density. We present Monte Carlo techniques for studying this model in previously inaccessible regimes of strong frustration. We then consider a well-studied limit of this model, infinite-separation, and further show that even for finite separations there is no magnetic ordering associated with the case of an infinite bath cutoff frequency. We discuss future applications for the Monte Carlo method.

Effect of chemical and hydrostatic pressure on the cubic pyrochlore Cd2Ru2O7

The cubic pyrochlore $\mathrm{C}{\mathrm{d}}_{2}\mathrm{R}{\mathrm{u}}_{2}{\mathrm{O}}_{7}$ develops a metalliclike state below the antiferromagnetic transition at ${T}_{\mathrm{N}}\ensuremath{\approx}90\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, in contrast to the sharp transition to an insulating state below ${T}_{\mathrm{N}}$ of the $5d$-analog ${\mathrm{Cd}}_{2}{\mathrm{Os}}_{2}{\mathrm{O}}_{7}$. We have synthesized polycrystalline $\mathrm{C}{\mathrm{d}}_{2}\mathrm{R}{\mathrm{u}}_{2}{\mathrm{O}}_{7}$ under high pressure, and investigated the responses of its electronic behavior to hydrostatic pressure and isovalent substitutions of ${\mathrm{Ca}}^{2+}$ for ${\mathrm{Cd}}^{2+}$. Three characteristic anomalies are identified in the electrical transport and magnetic properties of $\mathrm{C}{\mathrm{d}}_{2}\mathrm{R}{\mathrm{u}}_{2}{\mathrm{O}}_{7}$, signaling an intimate correlation between charge and spin degrees of freedom. Interestingly, we found that the metalliclike state of $\mathrm{C}{\mathrm{d}}_{2}\mathrm{R}{\mathrm{u}}_{2}{\mathrm{O}}_{7}$ below ${T}_{\mathrm{N}}$ is very fragile and can be suppressed by a small hydrostatic pressure of $\ensuremath{\le}1\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ or substitution of 5--10% ${\mathrm{Ca}}^{2+}$ for ${\mathrm{Cd}}^{2+}$, resulting in a resistivity behavior similar to that of ${\mathrm{Cd}}_{2}{\mathrm{Os}}_{2}{\mathrm{O}}_{7}$. In addition, the resultant insulating state below ${T}_{\mathrm{N}}$ is very robust against pressure, and the resistivity evolves gradually into two distinct activated regions under higher pressures. We constructed a temperature-pressure phase diagram for $\mathrm{C}{\mathrm{d}}_{2}\mathrm{R}{\mathrm{u}}_{2}{\mathrm{O}}_{7}$ and discussed its peculiar metalliclike state in terms of the electronic itinerancy/localization dichotomy via side-by-side comparisons with the related compounds ${A}_{2}\mathrm{R}{\mathrm{u}}_{2}{\mathrm{O}}_{7}(A=\mathrm{Ca},\mathrm{Hg})$ and ${\mathrm{Cd}}_{2}{\mathrm{Os}}_{2}{\mathrm{O}}_{7}$. Our results demonstrate that these cubic $4d/5d$ pyrochlore oxides offer an important paradigm for studying the exotic physics of correlated electrons on the border of (de)localization in the presence of strong geometrical frustration.

Chern-Simons fermionization approach to two-dimensional quantum magnets: Implications for antiferromagnetic magnons and unconventional quantum phase transitions

We develop an approach to describe antiferromagnetic magnons on a bipartite lattice supporting the N\'{e}el state using fractionalized degrees of freedom typically inherent to quantum spin liquids. In particular we consider a long-range magnetically ordered state of interacting two-dimensional quantum spin$-1/2$ models using the Chern-Simons (CS) fermion representation of interacting spins. The interaction leads to Cooper instability and pairing of CS fermions, and to CS superconductivity which spontaneously violates the continuous $\mathrm{U}(1)$ symmetry generating a linearly-dispersing gapless Nambu-Goldstone mode due to phase fluctuations. We evaluate this mode and show that it is in high-precision agreement with magnons of the corresponding N\'{e}el antiferromagnet irrespective to the lattice symmetry. Using the fermion formulation of a system with competing interactions, we show that the frustration gives raise to nontrivial long-range four, six, and higher-leg interaction vertices mediated by the CS gauge field, which are responsible for restoring of the continuous symmetry at sufficiently strong frustration. We identify these new interaction vertices and discuss their implications to unconventional phase transitions. We also apply the proposed theory to a model of anyons that can be tuned continuously from fermions to bosons.

Robust non-Abelian spin liquid and a possible intermediate phase in the antiferromagnetic Kitaev model with magnetic field

We investigate the non-Abelian topological chiral spin liquid phase in the two-dimensional (2D) Kitaev honeycomb model subject to a magnetic field. By combining density matrix renormalization group (DMRG) and exact diagonalization (ED) we study the energy spectra, entanglement, topological degeneracy, and expectation values of Wilson loop operators, allowing for robust characterization. While the ferromagnetic (FM) Kitaev spin liquid is already destroyed by a weak magnetic field with Zeeman energy $H_*^\text{FM} \approx 0.02$, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, $H_*^\text{AFM} \approx 0.2$. Interestingly, for larger fields $H_*^\text{AFM} < H < H_{**}^\text{AFM}$, an intermediate gapless phase is observed, before a second transition to the high-field partially-polarized paramagnet. We attribute this rich phase diagram, and the remarkable stability of the chiral topological phase in the AFM Kitaev model, to the interplay of strong spin-orbit coupling and frustration enhanced by the magnetic field. Our findings suggest relevance to recent experiments on RuCl$_3$ under magnetic fields.

Frustration of square cupola in Sr(TiO) Cu4(PO4)4

The structural and magnetic properties of the square-cupola antiferromagnet Sr(TiO)Cu$_{4}$(PO$_{4}$)$_{4}$ are investigated via x-ray diffraction, magnetization, heat capacity, and $^{31}$P nuclear magnetic resonance experiments on polycrystalline samples, as well as density-functional band-structure calculations. The temperature-dependent unit cell volume could be described well using the Debye approximation with the Debye temperature of $\theta_{\rm D} \simeq $ 550~K. Magnetic response reveals a pronounced two-dimensionality with a magnetic long-range-order below $T_{\rm N} \simeq 6.2$~K. High-field magnetization exhibits a kink at $1/3$ of the saturation magnetization. Asymmetric $^{31}$P NMR spectra clearly suggest strong in-plane anisotropy in the magnetic susceptibility, as anticipated from the crystal structure. From the $^{31}$P NMR shift vs bulk susceptibility plot, the isotropic and axial parts of the hyperfine coupling between $^{31}$P nuclei and the Cu$^{2+}$ spins are calculated to be $A_{\rm hf}^{\rm iso} \simeq 6539$ and $A_{\rm hf}^{\rm ax} \simeq 952$~Oe/$\mu_{\rm B}$, respectively. The low-temperature and low-field $^{31}$P NMR spectra indicate a commensurate antiferromagnetic ordering. Frustrated nature of the compound is inferred from the temperature-dependent $^{31}$P NMR spin-lattice relaxation rate and confirmed by our microscopic analysis that reveals strong frustration of the square cupola by next-nearest-neighbor exchange couplings.

Physical properties of MgAl2O4, CoAl2O4, NiAl2O4, CuAl2O4, and ZnAl2O4 spinels synthesized by a solution combustion method

MgAl2O4, CoAl2O4, NiAl2O4, CuAl2O4, and ZnAl2O4 spinels have been synthesized by a solution combustion method following with secondary calcination. These aluminates were characterized by X-ray diffraction, X-ray photoelectron spectroscopy, ultraviolet–visible spectroscopy, and scanning electron microscope. As-prepared samples show pure spinel phase with good crystalline, vivid colour, fine particle size, and porous structure. Phenomenon of anti-site disorder in these samples has been investigated, which shows dependence on the radius of the A-site ions. Magnetic properties and heat capacities of these samples were further characterized by magnetic property measurement system and physical property measurement system. Among the magnetic compounds, NiAl2O4 is typical antiferromagnetic, CoAl2O4 shows interesting spin-glass-like ground state, and CuAl2O4 exhibits characteristic of spin liquid due to strong spin frustration and site inversion effects in the system. Heat capacities of these compounds are further studied. The values of the Debye and Einstein temperatures are given through fitting the data. Furthermore, the magnetic contribution ΔCm and magnetic entropies ΔSm of the magnetic samples are obtained after subtraction of the phonon contribution.

Anomalous double-stripe charge ordering in β−NaFe2O3 with double triangular layers consisting of almost perfect regular Fe4 tetrahedra

The physical properties of the mixed-valent iron oxide $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{NaFe}}_{2}{\mathrm{O}}_{3}$ were investigated by means of synchrotron radiation x-ray diffraction, magnetization, electrical resistivity, differential scanning calorimetry, $^{23}\mathrm{Na}$ NMR, and $^{57}\mathrm{Fe}\phantom{\rule{4pt}{0ex}}\mathrm{M}\stackrel{\ifmmode \ddot{}\else \&#34;{}\fi{}}{\mathrm{o}}\mathrm{ssbauer}$ measurements. This compound has double triangular layers consisting of almost perfect regular ${\mathrm{Fe}}_{4}$ tetrahedra, which suggests geometrical frustration. We found that this compound exhibits an electrostatically unstable double-stripe-type charge ordering, which is stabilized by the cooperative compression of ${\mathrm{Fe}}^{3+}{\mathrm{O}}_{6}$ octahedra, owing to a valence change and ${\mathrm{Fe}}^{2+}{\mathrm{O}}_{6}$ octahedra due to Jahn-Teller distortion. Our results indicate the importance of electron-phonon coupling for charge ordering in the region of strong charge frustration.

Origin of Ferrimagnetism and Ferroelectricity in Room-Temperature Multiferroic ϵ−Fe2O3

In devices, being able to control magnetism with electric rather than magnetic fields is very appealing, and magnetoelectric multiferroic materials are key to this functionality. \ensuremath{\epsilon}-FeO is a room-temperature multiferroic featuring both strong magnetization and strong ferroelectricity simultaneously, but the origin of this multiferroicity is unclear. This work shows that the compound's ferrimagnetism arises from strong spin frustration, and that a low-energy transition state is responsible for switchable ferroelectricity. At a higher level, these results also suggest how to search for other high-performance multiferroics, for use in next-generation data storage.

Suppressed density of states in self-generated Coulomb glasses

We investigate the structure of metastable states in self-generated Coulomb glasses. In dramatic contrast to disordered electron glasses, we find that these states lack marginal stability. Such absence of marginal stability is reflected by the suppression of the single-particle density of states into an exponentially soft gap of the form . To analytically explain this behavior, we extend the stability criterion of Efros and Shklovskii to incorporate local charge correlations, in qualitative agreement with our numerical findings. Our work suggests the existence of a new class of self-generated glasses dominated by strong geometric frustration.

Low-Temperature Lattice Effects in the Spin-Liquid Candidate κ-(BEDT-TTF)2Cu2(CN)3

The quasi-two-dimensional organic charge-transfer salt κ -(BEDT-TTF) 2 Cu 2 (CN) 3 is one of the prime candidates for a quantum spin-liquid due the strong spin frustration of its anisotropic triangular lattice in combination with its proximity to the Mott transition. Despite intensive investigations of the material’s low-temperature properties, several important questions remain to be answered. Particularly puzzling are the 6 K anomaly and the enigmatic effects observed in magnetic fields. Here we report on low-temperature measurements of lattice effects which were shown to be particularly strongly pronounced in this material (R. S. Manna et al., Phys. Rev. Lett. 2010, 104, 016403)). A special focus of our study lies on sample-to-sample variations of these effects and their implications on the interpretation of experimental data. By investigating overall nine single crystals from two different batches, we can state that there are considerable differences in the size of the second-order phase transition anomaly around 6 K, varying within a factor of 3. In addition, we find field-induced anomalies giving rise to pronounced features in the sample length for two out of these nine crystals for temperatures T &lt; 9 K. We tentatively assign the latter effects to B-induced magnetic clusters suspected to nucleate around crystal imperfections. These B-induced effects are absent for the crystals where the 6 K anomaly is most strongly pronounced. The large lattice effects observed at 6 K are consistent with proposed pairing instabilities of fermionic excitations breaking the lattice symmetry. The strong sample-to-sample variation in the size of the phase transition anomaly suggests that the conversion of the fermions to bosons at the instability is only partial and to some extent influenced by not yet identified sample-specific parameters.

Effect of Pb doping on metallic state of cubic pyrochlore Cd2Ru2O7

Many exotic phenomena in strongly correlated electron systems, such as unconventional superconductivity, metal-insulator transition, and quantum criticality, take place in the intermediate regime between localized and itinerant electronic state. To understand the electronic behaviors near the localized-to-itinerant crossover remains a challenging problem in condensed matter physics. The Ru5+ cubic pyrochlores A2Ru2O7 (A=Cd, Cd, Hg) constitute such a system that the Ru-4d electrons acquire characters of both itinerancy and localization. In addition, the magnetic Ru5+ ions that are situated on the vertices of corner-shared tetrahedral lattice are expected to experience strong geometrical frustration given an antiferromagnetic (AF) arrangement. In this work, we investigate the cubic pyrochlore Cd2Ru2O7, which develops a peculiar metallic state below the AF transition. We synthesize a series of Pb-doped Cd2-xPbxRu2O7 (0 x 2) polycrystalline samples under high-pressure condition, and study the effects of Pb doping on their crystal structure and physical properties. Although the Pb2Ru2O7 pyrochlore is a Pauli paramagnetic metal, we find that the substitution of 10% Pb2+ for Cd2+ in Cd1.8Pb0.2Ru2O7 converts the metallic state of Cd2Ru2O7 into an insulating ground state, in a manner similar to the consequence of exerting hydrostatic pressure or substituting 10% Ca2+ for Cd2+ ions as we found recently. We propose that the electronic state of Cd2Ru2O7 be located at the itinerancy to localization crossover, and the introduction of chemical disorder via Pb2+ substitution may enhance the localized character and induce the metal-to-insulator transition. Our results further demonstrate that the cubic Ru5+ pyrochlore oxides offer an important paradigm for studying the exotic physics of correlated electrons on the border of (de)localization in the presence of strong geometrical frustration.

Synthesis, structure and magnetism of the new S = 1 kagome magnet NH4Ni2.5V2O7(OH)2⋅H2O

Kagome antiferromagnets (KAFMs) have long been known to host exotic electronic states due to their strong geometric frustration, including the quantum spin liquid state in systems. Away from that limit, S = 1 KAFMs are also predicted to host unconventional ground states such as spin nematic phases, but a paucity of studies on known model materials has restricted progress.Here, we present the crystal structure and preliminary magnetization measurements on the newly synthesized S = 1 KAFM, NH4Ni2.5V2O7(OH)H2O, which has the three-fold symmetry of the kagome lattice but significant site depletion, with ∼ site occupancy. Bulk magnetic data show clear evidence of frustration and competition between ferromagnetic and antiferromagnetic interactions. We propose that the magnetic Hamiltonian is frustrated and that anisotropic terms cause the formation of an unconventional ground state.

Structural and Magnetic Properties of Fe-Doped Mn2O3 Orthorhombic Bixbyite

There is renewed interest in non-perovskite AB O3-type compounds owing to various functional properties such as catalysts, cathode material, and luminescence. The bixbyite compounds have been the subject of intense study in the recent literature owing to their fascinating structural and magnetic properties. In the present work, we have prepared a Fe-doped Mn2O3 bixbyite compound by a mechanochemical synthesis method and studied structural and magnetic properties. The compound is found to be orthorhombic at room temperature and exhibits two antiferromagnetic transitions at 80 and 30 K. Signatures of strong magnetic frustration coexisting with the long-range magnetic order are observed in this sample from the magnetization measurements. The results of our preliminary studies are presented here in light of the current interest in search for novel physical properties in non-perovskite AB O3 compounds.

Perpendicular susceptibility and geometrical frustration in two-dimensional Ising antiferromagnets: Exact solutions

Discovery of new materials and improved experimental as well as numerical techniques have led to a renewed interest in geometrically frustrated spin systems. However, there are very few exact results available that can provide a benchmark for comparison. In this work, we calculate exactly the perpendicular susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$ for an Ising antiferromagnet with (i) nearest-neighbor pair interaction on a kagome lattice where strong frustration prevents long-range ordering and (ii) elementary triplet interactions on a kagome lattice which has no frustration but the system remains disordered down to zero temperature. By comparing with other known exact results with and without frustration, we propose that an appropriately temperature-scaled ${\ensuremath{\chi}}_{\ensuremath{\perp}}$ can be used as a quantitative measure of the degree of frustration in Ising spin systems.