Kapellasite: a kagome quantum spin liquid with competing interactions

Low-temperature behaviors of the dipolar magnet Dy3Sb3Zn2O14with a strongly site-mixing disordered kagome lattice

Interesting behaviors may emerge in the magnetic frustrated materials with significant site-mixing disorder. We present the results of the structural, magnetic susceptibility, and specific heat measurements of Dy3Sb3Zn2O14with ∼20%Dy/Zn site-mixing disorder, which results in either a diluted 2D triangular lattice, or an intermediate structure between the kagome and pyrochlore lattice. In addition to the sharp anomaly of the temperature dependence of specific heat atT∼0.35 K, which was attributed to the emergent charge order state for the sample with less disorder, a broad peak atT∼1.5 K, and a small hump belowT∼0.1 K are observed. The measured temperature dependence of specific heat and the Monte Carlo simulation suggest that the magnetic frustration persists despite of a strong site-mixing disorder.

Structural and magnetic studies of the frustratedS = 1 kagome magnet NH4Ni2Mo2O10H3

The strong geometric frustration of the kagome antiferromagnets (KAFMs) can destabilise conventional magnetic order and lead to exotic electronic states, such as the quantum spin-liquid state observed in someS=12KAFM materials. However, the ground state ofS = 1 KAFM systems are less well understood. Spin nematic phases and valence bond solid ground states have been predicted to form but a paucity of experimental realisations restricts understanding. Here, theS = 1 KAFM NH4Ni2Mo2O10H3is presented, which has the 3-fold symmetry of the kagome lattice but significant site depletion, with∼64%site occupancy. Frustration and a competition between exchange interactions are evidenced through the suppression of order below the Weiss temperature|θW|and observation of ferromagnetic and antiferromagnetic characteristics in the magnetisation data. A semi spin glass ground state is predicted based on the ac-field frequency dependence of the magnetic transition and ferromagnetic signal.

Thermodynamic Properties and DFT Study on Highly Frustrated Cr3BO6: Coexistence of Spin-Singlets with Long-Range Magnetic Order

The triangle-based magnetic subsystem of borates with the mineral norbergite structure M3BO6 (M = Fe, Cr, V) makes these compounds unique to investigate rare quantum ground states influenced by strong magnetic frustration. In this work, we investigated the thermal and magnetic properties of Cr3BO6 to find that despite very large negative Weiss temperature Θ = -160.7 K, it orders only at TN = 4.5 K and experiences a spin-flop transition at µ0H = 5 T. Density functional theory (DFT) calculations of exchange interaction parameters allow for suggesting the model of magnetic subsystem in chromium borate Cr3BO6. The results prove the decisive role of magnetic frustration on the formation of long-range order, providing therefore a basis for future study. Both experimental data and first-principles calculations point to the coexistence of chromium spin-singlets with long-range antiferromagnetic order.

Unusual Exchange Couplings and Intermediate Temperature Weyl State in Co_{3}Sn_{2}S_{2}

Understanding magnetism and its possible correlations to topological properties has emerged to the forefront as a difficult topic in studying magnetic Weyl semimetals. Co_{3}Sn_{2}S_{2} is a newly discovered magnetic Weyl semimetal with a kagome lattice of cobalt ions and has triggered intense interest for rich fantastic phenomena. Here, we report the magnetic exchange couplings of Co_{3}Sn_{2}S_{2} using inelastic neutron scattering and two density functional theory (DFT) based methods: constrained magnetism and multiple-scattering Green's function methods. Co_{3}Sn_{2}S_{2} exhibits highly anisotropic magnon dispersions and linewidths below T_{C}, and paramagnetic excitations above T_{C}. The spin-wave spectra in the ferromagnetic ground state is well described by the dominant third-neighbor "across-hexagon" J_{d} model. Our density functional theory calculations reveal that both the symmetry-allowed 120° antiferromagnetic orders support Weyl points in the intermediate temperature region, with distinct numbers and the locations of Weyl points. Our study highlights the important role Co_{3}Sn_{2}S_{2} can play in advancing our understanding of kagome physics and exploring the interplay between magnetism and band topology.

CoMOF5(pyrazine)(H2O)2 (M = Nb, Ta): Two-Layered Cobalt Oxyfluoride Antiferromagnets with Spin Flop Transitions

Two cobalt oxyfluoride antiferromagnets CoMOF₅(pyz)(H₂O)₂ (M = Nb 1, Ta 2; pyz = pyrazine) have been synthesized via conventional hydrothermal methods and characterized by thermogravimetric (TGA) analysis, FTIR spectroscopy, electron spin resonance (ESR), magnetic susceptibility, and magnetization measurements at both static low field and pulsed high field. The single-crystal X-ray diffraction indicates both compounds 1 and 2 are isostructural and crystallize in the monoclinic space group C2/m with a two-dimensional Co²⁺ triangular lattice in the ab plane, separated by the nonmagnetic MOF₅ (M = Nb 1, Ta 2) octahedra along the c-axis with large intertriangular-lattice Co···Co distance. Because of low dimensionality together with frustrated triangular lattice, compounds 1 and 2 exhibit no long-range antiferromagnetic order until ∼3.7 K. Moreover, a spin flop transition is observed in the magnetization curves at 2 K for both compounds, which is further confirmed by ESR spectra. In addition, the ESR spectra suggest the presence of a zero-field spin gap in both compounds. The high field magnetization measured at 2 K saturates at ∼7 T with M_s = 1.55 μ_B for 1 and 1.71 μ_B for 2, respectively, after subtracting the Van Vleck paramagnetic contribution, which is usually observed for Co²⁺ ions with pseudospin spin of 1/2 at low temperature. Powder-averaged magnetic anisotropy of g = 3.10 for 1 (3.42 for 2) and magnetic superexchange interaction J/k_B = -3.2 K for 1 (-3.6 K for 2) are obtained.

Chemistry of Quantum Spin Liquids

Quantum spin liquids are an exciting playground for exotic physical phenomena and emergent many-body quantum states. The realization and discovery of quantum spin liquid candidate materials and associated phenomena lie at the intersection of solid-state chemistry, condensed matter physics, and materials science and engineering. In this review, we provide the current status of the crystal chemistry, synthetic techniques, physical properties, and research methods in the field of quantum spin liquids. We highlight a number of specific quantum spin liquid candidate materials and their structure-property relationships, elucidating their fascinating behavior and connecting it to the intricacies of their structures. Furthermore, we share our thoughts on defects and their inevitable presence in materials, of which quantum spin liquids are no exception, which can complicate the interpretation of characterization of these materials, and urge the community to extend their attention to materials preparation and data analysis, cognizant of the impact of defects. This review was written with the intention of providing guidance on improving the materials design and growth of quantum spin liquids, and to paint a picture of the beauty of the underlying chemistry of this exciting class of materials.

Quantum spin-liquid states in an organic magnetic layer and molecular rotor hybrid

The exotic properties of quantum spin liquids (QSLs) have continually been of interest since Anderson's 1973 ground-breaking idea. Geometrical frustration, quantum fluctuations, and low dimensionality are the most often evoked material's characteristics that favor the long-range fluctuating spin state without freezing into an ordered magnet or a spin glass at low temperatures. Among the few known QSL candidates, organic crystals have the advantage of having rich chemistry capable of finely tuning their microscopic parameters. Here, we demonstrate the emergence of a QSL state in [EDT-TTF-CONH₂]₂ ⁺[[Formula: see text]] (EDT-BCO), where the EDT molecules with spin-1/2 on a triangular lattice form layers which are separated by a sublattice of BCO molecular rotors. By several magnetic measurements, we show that the subtle random potential of frozen BCO Brownian rotors suppresses magnetic order down to the lowest temperatures. Our study identifies the relevance of disorder in the stabilization of QSLs.

Pb(OF)Cu3(SeO3)2(NO3): a selenite fluoride nitrate with a breathing kagomé lattice

The Cu²⁺-based breathing kagomé material Pb(OF)Cu₃(SeO₃)₂(NO₃) with both corner-sharing and edge-sharing has been synthesized. Magnetic measurements suggest ferromagnetic interactions inside the layers and antiferromagnetic interactions between the neighboring layers, leading to an antiferromagnetic ground state with a field-induced spin-flip transition at low temperature.

Gapless spin liquid in a square-kagome lattice antiferromagnet

Observation of a quantum spin liquid (QSL) state is one of the most important goals in condensed-matter physics, as well as the development of new spintronic devices that support next-generation industries. The QSL in two dimensional quantum spin systems is expected to be due to geometrical magnetic frustration, and thus a kagome-based lattice is the most probable playground for QSL. Here, we report the first experimental results of the QSL state on a square-kagome quantum antiferromagnet, KCu6AlBiO4(SO4)5Cl. Comprehensive experimental studies via magnetic susceptibility, magnetisation, heat capacity, muon spin relaxation (μSR), and inelastic neutron scattering (INS) measurements reveal the formation of a gapless QSL at very low temperatures close to the ground state. The QSL behavior cannot be explained fully by a frustrated Heisenberg model with nearest-neighbor exchange interactions, providing a theoretical challenge to unveil the nature of the QSL state.

Synthetic investigation of competing magnetic interactions in 2D metal-chloranilate radical frameworks

The discovery of emergent materials lies at the intersection of chemistry and condensed matter physics. Synthetic chemistry offers a pathway to create materials with the desired physical and electronic structures that support fundamentally new properties. Metal–organic frameworks are a promising platform for bottom-up chemical design of new materials, owing to their inherent chemical predictability and tunability relative to traditional solid-state materials. Herein, we describe the synthesis and magnetic characterization of a new 2,5-dihydroxy-1,4-benzoquinone based material, (NMe₂H₂)_3.5Ga₂(C₆O₄Cl₂)₃ (1), which features radical-based electronic spins on the sites of a kagomé lattice, a geometric lattice known to engender exotic electronic properties. Vibrational and electronic spectroscopies, in combination with magnetic susceptibility measurements, revealed 1 exhibits mixed valency between the radical-bearing trianionic and diamagnetic tetraanionic oxidation states of the ligand. This unpaired electron density on the ligand forms a partially occupied kagomé lattice where approximately 85% of the lattice sites are occupied with an S = ½ spin. We found that gallium mediates ferromagnetic coupling between ligand spins, creating a ferromagnetic kagomé lattice. By modulation of the interlayer spacing via post-synthetic cation metathesis of 1 to (NMe₄)_3.5Ga₂(C₆O₄Cl₂)₃ (2) and (NEt₄)₂(NMe₄)_1.5Ga₂(C₆O₄Cl₂)₃ (3), we determined the nature of the magnetic coupling between neighboring planes is antiferromagnetic. Additionally, we determined the role of the metal in mediating this magnetic coupling by comparison of 2 with the In³⁺ analogue, (NMe₄)_3.5In₂(C₆O₄Cl₂)₃ (4), and we found that Ga³⁺ supports stronger superexchange coupling between ligand-based spins than In³⁺. The combination of intraplanar ferromagnetic coupling and interplanar antiferromagnetic coupling exchange interactions suggests these are promising materials to host topological phenomena.

2D metal–organic frameworks provide insight into kagomé spin physics.

Spin liquid mediated RKKY interaction

We propose an RKKY-type interaction that is mediated by a spin liquid. If a spin liquid exists such an interaction could leave a fingerprint by ordering underlying localised moments such as nuclear spins. This interaction has a unique phenomenology that is distinct from the RKKY interaction found in fermionic systems; most notably the lack of a Fermi surface and absence of the requirement for itinerant electrons, since most spin liquids are insulators. We demonstrate that the interaction is predominately shaped by the lattice symmetries of the underlying spin liquid. As a working example we investigate the possible ordering of nuclear spins that interact through an underlying lattice of the two-dimensional spin-1/2 kagome antiferromagnet (KHAF), although the treatment remains general and can be extended to other spin liquids and dimensions. We find that several different nuclear spin orderings minimise the RKKY-type energy induced by the KHAF but are unstable due to a zero-energy flat magnon band in linear spin-wave theory. Despite this we show that a small magnetic field is able to gap out this magnon spectrum resulting in an intricate nuclear magnetism.

Novel excitations near quantum criticality in geometrically frustrated antiferromagnet CsFeCl3

The investigation of materials that exhibit quantum phase transition provides valuable insights into fundamental problems in physics. We present neutron scattering under pressure in a triangular-lattice antiferromagnet that has a quantum disorder in the low-pressure phase and a noncollinear structure in the high-pressure phase. The neutron spectrum continuously evolves through critical pressure; a single mode in the disordered state becomes soft with the pressure and it splits into gapless and gapped modes in the ordered phase. Extended spin-wave theory reveals that the longitudinal and transverse fluctuations of spins are hybridized in the modes because of noncollinearity, and previously unidentified magnetic excitations are formed. We report a new hybridization of the phase and amplitude fluctuations of the order parameter near a quantum critical point in a spontaneously symmetry-broken state.

Spin Chain Network Construction of Chiral Spin Liquids

We show that a honeycomb lattice of Heisenberg spin-1/2 chains with three-spin junction interactions allows for controlled analytical studies of chiral spin liquids (CSLs). Tuning these interactions to a chiral fixed point, we find a Kalmeyer-Laughlin CSL phase which here is connected to the critical point of a boundary conformal field theory. Our construction directly yields a quantized spin Hall conductance and localized spinons with semionic statistics as elementary excitations. We also outline the phase diagram away from the chiral point where spinons may condense. Generalizations of our approach can provide microscopic realizations for many other CSLs.

Orbital Transitions and Frustrated Magnetism in the Kagome-Type Copper Mineral Volborthite

Volborthite Cu₃V₂O₇(OH)₂·2H₂O is a copper mineral that materializes a two-dimensional quantum magnet comprising a kagome net of spin-¹/₂ Cu²⁺ ions. We prepared single crystals of volborthite using hydrothermal conditions and investigated their crystal structures and magnetic properties. Unusual orbital "switching" and "flipping" transitions were observed: in the former type of transition (switching), the Cu 3d orbital occupied by an unpaired electron changes between the d(3z²-r²) and d(x²-y²) types, and in the latter type of transition (flipping), the d(x²-y²)-type orbitals change their directions. Their origin is ascribed to variations in the orientation of water molecules in the gap between the kagome layers and the accompanying changes of hydrogen bonding. These orbital transitions dramatically modify the magnetic interactions between Cu²⁺ spins, from the anisotropic kagome type to the formation of spin trimers over the kagome net. The effective spin ¹/₂ generated on the trimers exhibits a frustrated magnetism, resulting in a rich phase diagram in the magnetic fields. Volborthite is a unique compound showing an exceptional interplay between the orbital and spin degrees of freedom.

The half magnetization plateau in Ni3V2O8 studied by electron spin resonance

Ni₃V₂O₈, regarded as an S  =  1 kagome staircase lattice antiferromagnet, possesses a novel magnetic field-temperature phase diagram. Specifically, a half plateau region is observed in the high field magnetization curve for magnetic fields in the range of 11-19 T. This experimental observation is theoretically unexpected for a standard kagome lattice antiferromagnet, and consequently, the underlying magnetic structure is still unclear. Multi-frequency electron spin resonance results in this study strongly support a collinear magnetic arrangement at the half plateau region. The resonant modes can be well fit by only considering the antiferromagnetic interactions on a four-spin sublattice of the spine sites.

A series of magnon crystals appearing under ultrahigh magnetic fields in a kagomé antiferromagnet

Geometrical frustration and a high magnetic field are two key factors for realizing unconventional quantum states in magnetic materials. Specifically, conventional magnetic order can potentially be destroyed by competing interactions and may be replaced by an exotic state that is characterized in terms of quasiparticles called magnons, the density and chemical potential of which are controlled by the magnetic field. Here we show that a synthetic copper mineral, Cd-kapellasite, which comprises a kagomé lattice consisting of corner-sharing triangles of spin-1/2 Cu2+ ions, exhibits an unprecedented series of fractional magnetization plateaus in ultrahigh magnetic fields of up to 160 T. We propose that these quantum states can be interpreted as crystallizations of emergent magnons localized on the hexagon of the kagomé lattice.

Quantum magnetisms in uniform triangular lattices Li2AMo3O8 (A = In, Sc)

Molecular based spin-1/2 triangular lattice systems such as LiZn₂Mo₃O₈ have attracted research interest. Distortions, defects, and intersite disorder are suppressed in such molecular-based magnets, and intrinsic geometrical frustration gives rise to unconventional and unexpected ground states. Li₂AMo₃O₈ (A = In or Sc) is such a compound where spin-1/2 Mo₃O₁₃ clusters in place of Mo ions form the uniform triangular lattice. Their ground states are different according to the A site. Li₂InMo₃O₈ undergoes conventional 120° long-range magnetic order below T_N = 12 K whereas isomorphic Li₂ScMo₃O₈ exhibits no long-range magnetic order down to 0.5 K. Here, we report exotic magnetisms in Li₂InMo₃O₈ and Li₂ScMo₃O₈ investigated by muon spin rotation (μSR) and inelastic neutron scattering (INS) spectroscopies using polycrystalline samples. Li₂InMo₃O₈ and Li₂ScMo₃O₈ show completely different behaviors observed in both μSR and INS measurements, representing their different ground states. Li₂InMo₃O₈ exhibits spin wave excitation which is quantitatively described by the nearest neighbor anisotropic Heisenberg model based on the 120° spin structure. In contrast, Li₂ScMo₃O₈ undergoes short-range magnetic order below 4 K with quantum-spin-liquid-like magnetic fluctuations down to the base temperature. Origin of the different ground states is discussed in terms of anisotropies of crystal structures and magnetic interactions.

Vesignieite: An S=1/2 Kagome Antiferromagnet with Dominant Third-Neighbor Exchange

The spin-1/2 kagome antiferromagnet is an archetypal frustrated system predicted to host a variety of exotic magnetic states. We show using neutron scattering measurements that deuterated vesignieite BaCu_{3}V_{2}O_{8}(OD)_{2}, a fully stoichiometric S=1/2 kagome magnet with <1% lattice distortion, orders magnetically at T_{N}=9  K into a multi-k coplanar variant of the predicted triple-k octahedral structure. We find that this structure is stabilized by a dominant antiferromagnetic third-neighbor exchange J_{3} with minor first- or second-neighbor exchanges. The spin-wave spectrum is well described by a J_{3}-only model including a tiny symmetric exchange anisotropy.

Special temperatures in frustrated ferromagnets

The description and detection of unconventional magnetic states, such as spin liquids, is a recurring topic in condensed matter physics. While much of the efforts have traditionally been directed at geometrically frustrated antiferromagnets, recent studies reveal that systems featuring competing antiferromagnetic and ferromagnetic interactions are also promising candidate materials. We find that this competition leads to the notion of special temperatures, analogous to those of gases, at which the competing interactions balance, and the system is quasi-ideal. Although induced by weak perturbing interactions, these special temperatures are surprisingly high and constitute an accessible experimental diagnostic of eventual order or spin-liquid properties. The well characterised Hamiltonian and extended low-temperature susceptibility measurement of the canonical frustrated ferromagnet Dy₂Ti₂O₇ enables us to formulate both a phenomenological and microscopic theory of special temperatures for magnets. Other members of this class of magnets include kapellasite Cu₃Zn(OH)₆Cl₂ and the spinel GeCo₂O₄.

Competing interactions in frustrated magnets give rise to complex emergent phenomena, which challenge a full microscopic understanding but invite comparison to other systems. Bovo et al. find an analogy to classical gases and identify special temperatures that reveal fine details of the microscopic Hamiltonian.

Emergent Chiral Spin State in the Mott Phase of a Bosonic Kane-Mele-Hubbard Model

Recently, the frustrated XY model for spins 1/2 on the honeycomb lattice has attracted a lot of attention in relation with the possibility to realize a chiral spin liquid state. This model is relevant to the physics of some quantum magnets. Using the flexibility of ultracold atom setups, we propose an alternative way to realize this model through the Mott regime of the bosonic Kane-Mele-Hubbard model. The phase diagram of this model is derived using bosonic dynamical mean-field theory. Focusing on the Mott phase, we investigate its magnetic properties as a function of frustration. We do find an emergent chiral spin state in the intermediate frustration regime. Using exact diagonalization we study more closely the physics of the effective frustrated XY model and the properties of the chiral spin state. This gapped phase displays a chiral order, breaking time-reversal and parity symmetry, but is not topologically ordered (the Chern number is zero).

Spin-liquid-like state in a spin-1/2 square-lattice antiferromagnet perovskite induced by d10-d0 cation mixing

A quantum spin liquid state has long been predicted to arise in spin-1/2 Heisenberg square-lattice antiferromagnets at the boundary region between Néel (nearest-neighbor interaction dominates) and columnar (next-nearest-neighbor interaction dominates) antiferromagnetic order. However, there are no known compounds in this region. Here we use d¹⁰-d⁰ cation mixing to tune the magnetic interactions on the square lattice while simultaneously introducing disorder. We find spin-liquid-like behavior in the double perovskite Sr₂Cu(Te_0.5W_0.5)O₆, where the isostructural end phases Sr₂CuTeO₆ and Sr₂CuWO₆ are Néel and columnar type antiferromagnets, respectively. We show that magnetism in Sr₂Cu(Te_0.5W_0.5)O₆ is entirely dynamic down to 19 mK. Additionally, we observe at low temperatures for Sr₂Cu(Te_0.5W_0.5)O₆-similar to several spin liquid candidates-a plateau in muon spin relaxation rate and a strong T-linear dependence in specific heat. Our observations for Sr₂Cu(Te_0.5W_0.5)O₆ highlight the role of disorder in addition to magnetic frustration in spin liquid physics.

Strong spin frustration and negative magnetization in LnCu3(OH)6Cl3 (Ln = Nd and Sm) with triangular lattices: the effects of lanthanides

Herbertsmithite- and kapellasite-type compounds with triangular lattices (i.e. Kagomé) as the most promising candidates for realizing the exotic quantum spin liquid (QSL) state have recently attracted significant attention in condensed matter physics and materials science but are often adversely affected by dimensional imperfections arising from significant cation mixing. Also, interaction mechanisms between the Kagomé lattices and ionic impurities remain unclear. Herein we report on the synthesis, crystal structures and magnetic properties of a new class of kapellasite-type compounds LnCu₃(OH)₆Cl₃ (Ln = Nd and Sm) with two overlapped triangular lattices. These compounds are characterized by the triangular lattices of Cu²⁺ superimposed by another triangular lattice of paramagnetic Ln³⁺. The magnetic properties of LnCu₃(OH)₆Cl₃ feature strong spin frustrations as well as antisymmetrical Dzyaloshinskii-Moriya interactions resulting in canted antiferromagnetic ordering with the Néel temperature (T_N) of ∼20 K and ∼18 K for NdCu₃(OH)₆Cl₃ and SmCu₃(OH)₆Cl₃, respectively. Moreover, negative magnetization at low temperatures was firstly observed in Kagomé lattice compounds, arising from geometrical spin frustration and competing interactions within two overlapped triangular lattices.

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 [Formula: see text] 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, NH₄Ni_2.5V₂O₇(OH)[Formula: see text]H₂O, which has the three-fold symmetry of the kagome lattice but significant site depletion, with  ∼[Formula: see text] 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.

Electronic and magnetic properties of low-dimensional system Co2TeO3Cl2

The electronic and magnetic properties of transition metal oxyhalide compound Co₂TeO₃Cl₂ are investigated using first principle calculations within the framework of density functional theory. To find the underlying spin-lattice of this compound, various hopping integrals and exchange interactions are calculated. The calculations reveal that the dominant inter-chain and intra-chain interactions are in the ab plane. The exchange path is visualized by Wannier function plotting. The nearest neighbor and next nearest neighbor exchange interactions are antiferromagnetic, making the system frustrated in low dimension. Calculations are also done with spin-orbit coupling (SOC) to find out the effect of SOC on this compound. Calculation of magnetocrystalline anisotropy suggests that the easy axis is along the crystallographic b direction.

Clustering of Topological Charges in a Kagome Classical Spin Liquid

Fractionalization is a ubiquitous phenomenon in topological states of matter. In this work, we study the collective behavior of fractionalized topological charges and their instabilities, through the J_{1}-J_{2}-J_{3} Ising model on a kagome lattice. This model can be mapped onto a Hamiltonian of interacting topological charges under the constraint of Gauss' law. We find that the recombination of topological charges gives rise to a yet unexplored classical spin liquid. This spin liquid is characterized by an extensive residual entropy, as well as the formation of hexamers of same-sign topological charges. The emergence of hexamers is reflected by a half-moon signal in the magnetic structure factor, which provides a signature of this new spin liquid in elastic neutron-scattering experiments. To study this phase, a worm algorithm has been developed which does not require the usual divergence-free condition.

Crystalline Electric-Field Randomness in the Triangular Lattice Spin-Liquid YbMgGaO_{4}

We apply moderate-high-energy inelastic neutron scattering (INS) measurements to investigate Yb^{3+} crystalline electric field (CEF) levels in the triangular spin-liquid candidate YbMgGaO_{4}. Three CEF excitations from the ground-state Kramers doublet are centered at the energies ℏω=39, 61, and 97 meV in agreement with the effective spin-1/2 g factors and experimental heat capacity, but reveal sizable broadening. We argue that this broadening originates from the site mixing between Mg^{2+} and Ga^{3+} giving rise to a distribution of Yb-O distances and orientations and, thus, of CEF parameters that account for the peculiar energy profile of the CEF excitations. The CEF randomness gives rise to a distribution of the effective spin-1/2 g factors and explains the unprecedented broadening of low-energy magnetic excitations in the fully polarized ferromagnetic phase of YbMgGaO_{4}, although a distribution of magnetic couplings due to the Mg/Ga disorder may be important as well.

Infrared phonons as a probe of spin-liquid states in herbertsmithite ZnCu3(OH)6Cl2

We report on temperature dependence of the infrared reflectivity spectra of a single crystalline herbertsmithite in two polarizations-parallel and perpendicular to the kagome plane of Cu atoms. We observe anomalous broadening of the low frequency phonons possibly caused by fluctuations in the exotic dynamical magnetic order of the spin liquid.

Quantum spin liquids: a review

Quantum spin liquids may be considered 'quantum disordered' ground states of spin systems, in which zero-point fluctuations are so strong that they prevent conventional magnetic long-range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, which is of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments in relation to study quantum spin liquids, and to the diverse probes used therein.

Emergent order in the kagome Ising magnet Dy3Mg2Sb3O14

The Ising model—in which degrees of freedom (spins) are binary valued (up/down)—is a cornerstone of statistical physics that shows rich behaviour when spins occupy a highly frustrated lattice such as kagome. Here we show that the layered Ising magnet Dy₃Mg₂Sb₃O₁₄ hosts an emergent order predicted theoretically for individual kagome layers of in-plane Ising spins. Neutron-scattering and bulk thermomagnetic measurements reveal a phase transition at ∼0.3 K from a disordered spin-ice-like regime to an emergent charge ordered state, in which emergent magnetic charge degrees of freedom exhibit three-dimensional order while spins remain partially disordered. Monte Carlo simulations show that an interplay of inter-layer interactions, spin canting and chemical disorder stabilizes this state. Our results establish Dy₃Mg₂Sb₃O₁₄ as a tuneable system to study interacting emergent charges arising from kagome Ising frustration.

Frustration in lattices of interacting spins can lead to rich and exotic physics, such as fractionalized excitations and emergent order. Here, the authors demonstrate a low-temperature transition from a disordered spin-ice-like phase to an emergent charge ordered phase in the bulk kagome Ising magnet Dy₃Mg₂Sb₃O₁₄.

Muon Spin Relaxation Evidence for the U(1) Quantum Spin-Liquid Ground State in the Triangular Antiferromagnet YbMgGaO_{4}

Muon spin relaxation (μSR) experiments on single crystals of the structurally perfect triangular antiferromagnet YbMgGaO_{4} indicate the absence of both static long-range magnetic order and spin freezing down to 0.048 K in a zero field. Below 0.4 K, the μ^{+} spin relaxation rates, which are proportional to the dynamic correlation function of the Yb^{3+} spins, exhibit temperature-independent plateaus. All these μSR results unequivocally support the formation of a gapless U(1) quantum spin liquid ground state in the triangular antiferromagnet YbMgGaO_{4}.

Magnetic Behavior of Volborthite Cu_{3}V_{2}O_{7}(OH)_{2}·2H_{2}O Determined by Coupled Trimers Rather than Frustrated Chains

Motivated by recent experiments on volborthite single crystals showing a wide 1/3-magnetization plateau, we perform microscopic modeling by means of density functional theory (DFT) with the single-crystal structural data as a starting point. Using DFT+U, we find four leading magnetic exchanges: antiferromagnetic J and J_{2}, as well as ferromagnetic J^{'} and J_{1}. Simulations of the derived spin Hamiltonian show good agreement with the experimental low-field magnetic susceptibility and high-field magnetization data. The 1/3-plateau phase pertains to polarized magnetic trimers formed by strong J bonds. An effective J→∞ model shows a tendency towards condensation of magnon bound states preceding the plateau phase.

Anisotropic Exchange within Decoupled Tetrahedra in the Quantum Breathing Pyrochlore Ba_{3}Yb_{2}Zn_{5}O_{11}

The low energy spin excitation spectrum of the breathing pyrochlore Ba_{3}Yb_{2}Zn_{5}O_{11} has been investigated with inelastic neutron scattering. Several nearly resolution limited modes with no observable dispersion are observed at 250 mK while, at elevated temperatures, transitions between excited levels become visible. To gain deeper insight, a theoretical model of isolated Yb^{3+} tetrahedra parametrized by four anisotropic exchange constants is constructed. The model reproduces the inelastic neutron scattering data, specific heat, and magnetic susceptibility with high fidelity. The fitted exchange parameters reveal a Heisenberg antiferromagnet with a very large Dzyaloshinskii-Moriya interaction. Using this model, we predict the appearance of an unusual octupolar paramagnet at low temperatures and speculate on the development of intertetrahedron correlations.

Na2Cu7(SeO3)4O2Cl4: a selenite chloride compound with Cu7 units showing spin-frustration and a magnetization plateau

A selenite chloride, Na2Cu7(SeO3)4O2Cl4 (), was prepared via a conventional hydrothermal method. Na2Cu7(SeO3)4O2Cl4 crystallizes in the triclinic space group P1[combining macron] and features an isolated reverse triangular dipyramid Cu7, which is assembled with two corner-shared Cu4 tetrahedral units. Magnetic measurements suggest that shows the spin-frustration effect with antiferromagnetic ordering at ∼5 K, while an unusual magnetization plateau is observed at an applied field of >4 T.

NaKV4O9·2H2O: a new 2D magnetic compound with a 1/5-depleted square lattice

A new vanadate compound NaKV4O9·2H2O is successfully synthesized by a conventional hydrothermal method. This compound crystallizes in the monoclinic system with the space group C2/c, showing a typical 2D layered structure built from VO5 pyramids, in which the layers are separated by Na(+), K(+), and H2O. The topology structure of magnetic V(4+) ions shows a quite interesting 1/5-depleted square lattice, which is quite similar to that of a famous low-dimensional quantum spin system CaV4O9. A structural and magnetic comparison confirmed that the title compound may exhibit a more pronounced 2D character with a large spin gap.

Spin Liquid State in the 3D Frustrated Antiferromagnet PbCuTe_{2}O_{6}: NMR and Muon Spin Relaxation Studies

PbCuTe_{2}O_{6} is a rare example of a spin liquid candidate featuring a three-dimensional magnetic lattice. Strong geometric frustration arises from the dominant antiferromagnetic interaction that generates a hyperkagome network of Cu^{2+} ions although additional interactions enhance the magnetic lattice connectivity. Through a combination of magnetization measurements and local probe investigations by NMR and muon spin relaxation down to 20 mK, we provide robust evidence for the absence of magnetic freezing in the ground state. The local spin susceptibility probed by the NMR shift hardly deviates from the macroscopic one down to 1 K pointing to a homogeneous magnetic system with a low defect concentration. The saturation of the NMR shift and the sublinear power law temperature (T) evolution of the 1/T_{1} NMR relaxation rate at low T point to a nonsinglet ground state favoring a gapless fermionic description of the magnetic excitations. Below 1 K a pronounced slowing down of the spin dynamics is witnessed, which may signal a reconstruction of spinon Fermi surface. Nonetheless, the compound remains in a fluctuating spin liquid state down to the lowest temperature of the present investigation.

Extending the Family of V(4+) S=(1/2) Kagome Antiferromagnets

The ionothermal synthesis, structure, and magnetic susceptibility of a novel inorganic-organic hybrid material, imidazolium vanadium(III,IV) oxyfluoride [C3 H5 N2 ][V9 O6 F24 (H2 O)2 ] (ImVOF) are presented. The structure consists of inorganic vanadium oxyfluoride slabs with kagome layers of V(4+) S=${{ 1/2 }}$ ions separated by a mixed valence layer. These inorganic slabs are intercalated with imidazolium cations. Quinuclidinium (Q) and pyrazinium (Pyz) cations can also be incorporated into the hybrid structure type to give QVOF and PyzVOF analogues, respectively. The highly frustrated topology of the inorganic slabs, along with the quantum nature of the magnetism associated with V(4+) , means that these materials are excellent candidates to host exotic magnetic ground states, such as the highly sought quantum spin liquid. Magnetic susceptibility measurements of all samples suggest an absence of conventional long-range magnetic order down to 2 K despite considerable antiferromagnetic exchange.

Resonating-Valence-Bond Physics Is Not Always Governed by the Shortest Tunneling Loops

It is well known that the low-energy sector of quantum spin liquids and other magnetically disordered systems is governed by short-ranged resonating-valence bonds. Here we show that the standard minimal truncation to the nearest-neighbor valence-bond basis fails completely even for systems where it should work the most, according to received wisdom. This paradigm shift is demonstrated for the quantum spin-1/2 square kagome, where strong geometric frustration, similar to the kagome, prevents magnetic ordering down to zero temperature. The shortest tunneling events bear the strongest longer-range singlet fluctuations, leading to amplitudes that do not drop exponentially with the length of the loop L, and to an unexpected loop-six valence-bond crystal, which is otherwise very high in energy at the minimal truncation level. The low-energy effective description gives in addition a clear example of correlated loop processes that depend not only on the type of the loop but also on its lattice embedding, a direct manifestation of the long-range nature of the virtual singlets.

Rare-Earth Triangular Lattice Spin Liquid: A Single-Crystal Study of YbMgGaO4

YbMgGaO4, a structurally perfect two-dimensional triangular lattice with an odd number of electrons per unit cell and spin-orbit entangled effective spin-1/2 local moments for the Yb(3+) ions, is likely to experimentally realize the quantum spin liquid ground state. We report the first experimental characterization of single-crystal YbMgGaO4 samples. Because of the spin-orbit entanglement, the interaction between the neighboring Yb(3+) moments depends on the bond orientations and is highly anisotropic in the spin space. We carry out thermodynamic and the electron spin resonance measurements to confirm the anisotropic nature of the spin interaction as well as to quantitatively determine the couplings. Our result is a first step towards the theoretical understanding of the possible quantum spin liquid ground state in this system and sheds new light on the search for quantum spin liquids in strong spin-orbit coupled insulators.

Frozen State and Spin Liquid Physics in Na_{4}Ir_{3}O_{8}: An NMR Study

Na_{4}Ir_{3}O_{8} is a unique case of a hyperkagome 3D corner sharing triangular lattice that can be decorated with quantum spins. It has spurred a lot of theoretical interest as a spin liquid candidate. We present a comprehensive set of NMR data taken on both the ^{23}Na and ^{17}O sites. We show that disordered magnetic freezing of all Ir sites sets in below T_{f}~7 K, well below J=300 K, with a drastic slowing down of fluctuations to a static state revealed by our T_{1} measurements. Above typically 2T_{f}, physical properties are relevant to the spin liquid state induced by this exotic geometry. While the shift data show that the susceptibility levels off below 80 K, 1/T_{1} has little variation from 300 K to 2T_{f}. We discuss the implication of our results in the context of published experimental and theoretical work.

Spin susceptibility of quantum magnets from high to low temperatures

We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.

From spin glass to quantum spin liquid ground states in molybdate pyrochlores

We present new magnetic heat capacity and neutron scattering results for two magnetically frustrated molybdate pyrochlores: S=1 oxide Lu_{2}Mo_{2}O_{7} and S=1/2 oxynitride Lu_{2}Mo_{2}O_{5}N_{2}. Lu_{2}Mo_{2}O_{7} undergoes a transition to an unconventional spin glass ground state at T_{f}∼16  K. However, the preparation of the corresponding oxynitride tunes the nature of the ground state from spin glass to quantum spin liquid. The comparison of the static and dynamic spin correlations within the oxide and oxynitride phases presented here reveals the crucial role played by quantum fluctuations in the selection of a ground state. Furthermore, we estimate an upper limit for a gap in the spin excitation spectrum of the quantum spin liquid state of the oxynitride of Δ∼0.05  meV or Δ/|θ|∼0.004, in units of its antiferromagnetic Weiss constant θ∼-121  K.

Emergent chiral spin liquid: fractional quantum Hall effect in a kagome Heisenberg model

The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a spin-½ isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.

Molecular quantum magnetism in LiZn2Mo3O8

Inelastic neutron scattering at low temperatures T≤30  K from a powder of LiZn2Mo3O8 demonstrates this triangular-lattice antiferromagnet hosts collective magnetic excitations from spin-1/2 Mo3O13 molecules. Apparently gapless (Δ<0.2  meV) and extending at least up to 2.5 meV, the low-energy magnetic scattering cross section is surprisingly broad in momentum space and involves one-third of the spins present above 100 K. The data are compatible with the presence of valence bonds involving nearest-neighbor and next-nearest-neighbor spins forming a disordered or dynamic state.

de Vries M, Lightfoot P. A hybrid vanadium fluoride with structurally isolated S = 1 kagome layers

A novel hybrid vanadium fluoride displaying isolated S = 1 kagome layers separated by ammonium and ethylammonium cations is prepared; this compound shows interesting magnetic properties.