Gapless spin liquid of an organic triangular compound evidenced by thermodynamic measurements

Spin Frustration in Organic Radicals

Spin frustration, which results from geometric frustration and a systematical inability to satisfy all antiferromagnetic (AF) interactions between unpaired spins simultaneously, is under the spotlight for its importance in physics and materials science. Spin frustration is treated as the structural basis of quantum spin liquids (QSLs). Featuring flexible chemical structures, organic radical species exhibit great potential in building spin-frustrated molecules and lattices. So far, the reported examples of spin-frustrated organic radical compounds include triradicals, tetrathiafulvalene (TTF) radicals and derivatives, [Pd(dmit)2 ] compounds (dmit=1,3-dithiol-2-thione-4,5-dithiolate), nitronyl nitroxides, fullerenes, polycyclic aromatic hydrocarbons (PAHs), and other heterocyclic compounds where the spin frustration is generated intra- or intermolecularly. In this Minireview, we provide a brief summary of the reported radical compounds that possess spin frustration. The related data, including magnetic exchange coupling parameters, spin models, frustration parameters, and crystal lattices, are summarized and discussed.

Fluctuating magnetic droplets immersed in a sea of quantum spin liquid

The search of quantum spin liquid (QSL), an exotic magnetic state with strongly fluctuating and highly entangled spins down to zero temperature, is a main theme in current condensed matter physics. However, there is no smoking gun evidence for deconfined spinons in any QSL candidate so far. The disorders and competing exchange interactions may prevent the formation of an ideal QSL state on frustrated spin lattices. Here we report comprehensive and systematic measurements of the magnetic susceptibility, ultralow-temperature specific heat, muon spin relaxation (μSR), nuclear magnetic resonance (NMR), and thermal conductivity for NaYbSe2 single crystals, in which Yb3+ ions with effective spin-1/2 form a perfect triangular lattice. All these complementary techniques find no evidence of long-range magnetic order down to their respective base temperatures. Instead, specific heat, μSR, and NMR measurements suggest the coexistence of quasi-static and dynamic spins in NaYbSe2. The scattering from these quasi-static spins may cause the absence of magnetic thermal conductivity. Thus, we propose a scenario of fluctuating ferrimagnetic droplets immersed in a sea of QSL. This may be quite common on the way pursuing an ideal QSL, and provides a brand new platform to study how a QSL state survives impurities and coexists with other magnetically ordered states.

Coexistence of spin liquid state and magnetic correlations in 3d-5dbased triangular-lattice antiferromagnet Sr3CuIr2O9

Here, we report detailed lattice structure, magnetization (dc and ac) and specific heat measurements on a 3d-5dbased new triple-perovskite material Sr3CuIr2O9. The Sr/Cu forms a layered structure of triangular-lattice while the Ir forms Ir2O9dimers which lie in chain as well as simultaneously makes layered triangular-lattice with neighboring atoms. Due to random site-sharing with Sr2+, the Cu2+(3d9, spin-1/2) forms a diluted magnetic lattice, thus giving a disordered in-plane exchange interaction. Opposed to conventionalJeffmodel, the Ir5+(5d4,Jeff= 0) is believed to be magnetic here which participates both in-chain and in-plane magnetic interactions. This complex lattice structure driven competing exchange interaction leads the ground state to a gapless quantum-spin-liquid state which coexists with (weak) ferromagnetic spin correlations. While underling the importance of spin state (spin-1/2), we believe that the combined effect of lattice structure, geometric frustration, spin-orbit coupling and spin state has given rise this interesting ground state in this material.

Spin-Liquid Insulators Can Be Landau's Fermi Liquids

The long search for insulating materials that possess low-energy quasiparticles carrying electron's quantum numbers except charge-inspired by the neutral spin-1/2 excitations, the so-called spinons, exhibited by Anderson's resonating-valence-bond state-seems to have reached a turning point after the discovery of several Mott insulators displaying the same thermal and magnetic properties as metals, including quantum oscillations in a magnetic field. Here, we show that such anomalous behavior is not inconsistent with Landau's Fermi liquid theory of quasiparticles at a Luttinger surface. That is the manifold of zeros within the Brillouin zone of the single-particle Green's function at zero frequency, and which thus defines the spinon Fermi surface conjectured by Anderson.

Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on theT_cdome. There, a plausiblesx2+y2-dx2-y2-dxy-wave symmetry, in place of the conventionaldx2-y2-wave, governs the nematicity-enhanced pairing with a sharp rise in theT_cdome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.

Mixed-Valence Conductors from Ni Bis(diselenolene) Complexes with a Thiazoline Backbone

Highly conducting, mixed-valence, multi-component nickel bis(diselenolene) salts were obtained by electrocrystallization of the monoanionic species [Ni(Me-thiazds)₂]^-1 (Me-thiazds: N-methyl-1,3-thiazoline-2-thione-4,5-diselenolate), with 1:2 and 1:3 stoichiometries depending of the counter ion used (Et₄N⁺ and nBu₄N⁺ vs Ph₄P⁺, respectively). This behavior strongly differs from that of the corresponding monoanionic dithiolene complexes whose oxidation afforded the single component neutral species. This provides additional rare examples of mixed-valence conducting salts of nickel diselenolene complexes, only known in two examples with the dsit (1,3-dithiole-2-thione-4,5-diselenolate) and dsise (1,3-dithiole-2-selone-4,5-diselenolate) ligands. The mixed-valence salts form highly dimerized or trimerized bi- and trimetallic units, rarely seen with such nickel complexes. Transport measurements under a high pressure (up to 10 GPa) and band structure calculations confirm the semiconducting character of [Ph₄P][Ni(Me-thiazds)₂]₃ and the quasi metallic character of [Et₄N][Ni(Me-thiazds)₂]₂ and [NBu₄]_x[Ni(Me-thiazds)₂]₂ salts (0 < x < 1).

Emergence of Spinons in Layered Trimer Iridate Ba_{4}Ir_{3}O_{10}

Spinons are well known as the elementary excitations of one-dimensional antiferromagnetic chains, but means to realize spinons in higher dimensions is the subject of intense research. Here, we use resonant x-ray scattering to study the layered trimer iridate Ba_{4}Ir_{3}O_{10}, which shows no magnetic order down to 0.2 K. An emergent one-dimensional spinon continuum is observed that can be well described by XXZ spin-1/2 chains with a magnetic exchange of ∼55  meV and a small Ising-like anisotropy. With 2% isovalent Sr doping, magnetic order appears below T_{N}=130  K along with sharper excitations in (Ba_{1-x}Sr_{x})_{4}Ir_{3}O_{10}. Combining our data with exact diagonalization calculations, we find that the frustrated intratrimer interactions effectively reduce the system into decoupled spin chains, the subtle balance of which can be easily tipped by perturbations such as chemical doping. Our results put Ba_{4}Ir_{3}O_{10} between the one-dimensional chain and two-dimensional quantum spin liquid scenarios, illustrating a new way to suppress magnetic order and realize fractional spinons.

Resistivity and thermal conductivity of an organic insulator β'-EtMe3Sb[Pd(dmit)2]2

A finite residual linear term in the thermal conductivity at zero temperature in insulating magnets indicates the presence of gapless excitations of itinerant quasiparticles, which has been observed in some candidate materials of quantum spin liquids (QSLs). In the organic triangular insulator β′–EtMe₃Sb[Pd(dmit)₂]₂, a QSL candidate material, the low-temperature thermal conductivity depends on the cooling process and the finite residual term is observed only in samples with large thermal conductivity. Moreover, the cooling rate dependence is largely sample dependent. Here we find that, while the low-temperature thermal conductivity significantly depends on the cooling rate, the high-temperature resistivity is almost perfectly independent of the cooling rate. These results indicate that in the samples with the finite residual term, the mean free path of the quasiparticles that carry the heat at low temperatures is governed by disorders, whose characteristic length scale of the distribution is much longer than the electron mean free path that determines the high-temperature resistivity. This explains why recent X-ray diffraction and nuclear magnetic resonance measurements show no cooling rate dependence. Naturally, these measurements are unsuitable for detecting disorders of the length scale relevant for the thermal conductivity, just as they cannot determine the residual resistivity of metals. Present results indicate that very careful experiments are needed when discussing itinerant spin excitations in β′–EtMe₃Sb[Pd(dmit)₂]₂.

Frustration, strain and phase co-existence in the mixed valent hexagonal iridate Ba3NaIr2O9

Using detailed synchrotron diffraction, magnetization, thermodynamic and transport measurements, we investigate the relationship between the mixed valence of Ir, lattice strain and the resultant structural and magnetic ground states in the geometrically frustrated triple perovskite iridate Ba₃NaIr₂O₉. We observe a complex interplay between lattice strain and structural phase co-existence, which is typically not observed in this family of compounds. The low temperature magnetic ground state is characterized by the absence of long-range magnetic order, and points towards the condensation of a cluster glass state from an extended regime of short range magnetic correlations.

A Discrepancy in Thermal Conductivity Measurement Data of Quantum Spin Liquid β′-EtMe3Sb[Pd(dmit)2]2 (dmit = 1,3-Dithiol-2-thione-4,5-dithiolate)

A molecular Mott insulator β′-EtMe3Sb[Pd(dmit)2]2 is a quantum spin liquid candidate. In 2010, it was reported that thermal conductivity of β′-EtMe3Sb[Pd(dmit)2]2 is characterized by its large value and gapless behavior (a finite temperature-linear term). In 2019, however, two other research groups reported opposite data (much smaller value and a vanishingly small temperature-linear term) and the discrepancy in the thermal conductivity measurement data emerges as a serious problem concerning the ground state of the quantum spin liquid. Recently, the cooling rate was proposed to be an origin of the discrepancy. We examined effects of the cooling rate on electrical resistivity, low-temperature crystal structure, and 13C-NMR measurements and could not find any significant cooling rate dependence.

Thermal Properties and Instability of a U(1) Spin Liquid on the Triangular Lattice

We study the effect of Dzyaloshinskii-Moriya (DM) interaction on the triangular lattice U(1) quantum spin liquid (QSL) which is stabilized by ring-exchange interactions. A weak DM interaction introduces a staggered flux to the U(1) QSL state and changes the density of states at the spinon Fermi surface. If the DM vector contains in-plane components, then the spinons gain nonzero Berry phase. The resultant thermal conductances κ_{xx} and κ_{xy} qualitatively agree with the experimental results on the material EtMe_{3}Sb[Pd(dmit)_{2}]_{2}. Furthermore, owing to perfect nesting of the Fermi surface, a spin density wave state is triggered by larger DM interactions. On the other hand, when the ring-exchange interaction decreases, another antiferromagnetic (AFM) phase with 120° order shows up which is proximate to a U(1) Dirac QSL. We discuss the difference of the two AFM phases from their static structure factors and excitation spectra.

Quantum Disordering of an Antiferromagnetic Order by Quenched Randomness in an Organic Mott Insulator

The behavior of interacting spins subject to randomness is a longstanding issue and the emergence of exotic quantum states is among intriguing theoretical predictions. We show how a quantum-disordered phase emerges from a classical antiferromagnet by controlled randomness. ^{1}H NMR of a successively x-ray-irradiated organic Mott insulator finds that the magnetic order collapses into a spin-glass-like state, immediately after a slight amount of disorder centers are created, and evolves to a gapless quantum-disordered state without spin freezing, spin gap, or critical slowing down, as reported by T. Furukawa et al. [Phys. Rev. Lett. 115, 077001 (2015)]PRLTAO0031-900710.1103/PhysRevLett.115.077001 through sequential reductions in the spin freezing temperature and moment.

Nature of the randomness-induced quantum spin liquids in two dimensions

The nature of the randomness-induced quantum spin liquid state, the random-singlet state, is investigated in two dimensions (2D) by means of the exact-diagonalization and the Hams-de Raedt methods for several frustrated lattices, e.g. the triangular, the kagome and the J ₁-J ₂ square lattices. Properties of the ground state, the low-energy excitations and the finite-temperature thermodynamic quantities are investigated. The ground state and the low-lying excited states consist of nearly isolated singlet-dimers, clusters of resonating singlet-dimers, and orphan spins. Low-energy excitations are either singlet-to-triplet excitations, diffusion of orphan spins accompanied by the recombination of nearby singlet-dimers, creation or destruction of resonating singlet-dimers clusters. The latter two excitations give enhanced dynamical 'liquid-like' features to the 2D random-singlet state. Comparison is made with the random-singlet state in a 1D chain without frustration, the similarity and the difference between in 1D and in 2D being highlighted. Frustration in a wide sense, not only the geometrical one but also including the one arising from the competition between distinct types of interactions, play an essential role in stabilizing this frustrated random singlet state. Recent experimental situations on both organic and inorganic materials are reviewed and discussed.

Absence of Magnetic Thermal Conductivity in the Quantum Spin Liquid Candidate EtMe_{3}Sb[Pd(dmit)_{2}]_{2}

We present the ultralow-temperature specific heat and thermal conductivity measurements on single crystals of triangular-lattice compound EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, which has long been considered as a gapless quantum spin liquid candidate. In specific heat measurements, a finite linear term is observed, consistent with the previous work [S. Yamashita et al., Nat. Commun. 2, 275 (2011)NCAOBW2041-172310.1038/ncomms1274]. However, we do not observe a finite residual linear term in the thermal conductivity measurements, and the thermal conductivity does not change in a magnetic field of 6 T. These results are in sharp contrast to previous thermal conductivity measurements on EtMe_{3}Sb[Pd(dmit)_{2}]_{2} [M. Yamashita et al., Science 328, 1246 (2010)SCIEAS0036-807510.1126/science.1188200], in which a huge residual linear term was observed and attributed to highly mobile gapless excitations, likely the spinons of a quantum spin liquid. In this context, the true ground state of EtMe_{3}Sb[Pd(dmit)_{2}]_{2} has to be reconsidered.

Specific Heat Study of 1D and 2D Excitations in the Layered Frustrated Quantum Antiferromagnets Cs_{2}CuCl_{4-x}Br_{x}

We report an experimental and theoretical study of the low-temperature specific heat C and magnetic susceptibility χ of the layered anisotropic triangular-lattice spin-1/2 Heisenberg antiferromagnets Cs_{2}CuCl_{4-x}Br_{x} with x=0, 1, 2, and 4. We find that the ratio J^{'}/J of the exchange couplings ranges from 0.32 to ≈0.78, implying a change (crossover or quantum phase transition) in the materials' magnetic properties from one-dimensional (1D) behavior for J^{'}/J<0.6 to two-dimensional (2D) behavior for J^{'}/J≈0.78. For J^{'}/J<0.6, realized for x=0, 1, and 4, we find a magnetic contribution to the low-temperature specific heat, C_{m}∝T, consistent with spinon excitations in 1D spin-1/2 Heisenberg antiferromagnets. Remarkably, for x=2, where J^{'}/J≈0.78 implies a 2D magnetic character, we also observe C_{m}∝T. This finding, which contrasts the prediction of C_{m}∝T^{2} made by standard spin-wave theories, shows that Fermi-like statistics also plays a significant role for the magnetic excitations in spin-1/2 frustrated 2D antiferromagnets.

Randomness-Induced Quantum Spin Liquid Behavior in the s=1/2 Random-Bond Heisenberg Antiferromagnet on the Pyrochlore Lattice

We investigate the zero- and finite-temperature properties of the random-bond s=1/2 Heisenberg antiferromagnet on the pyrochlore lattice by the exact diagonalization and the Hams-de Raedt methods. We find that the randomness induces the gapless quantum spin liquid (QSL) state, the random-singlet state. Implications to recent experiments on the mixed-anion pyrochlore-lattice antiferromagnet Lu_{2}Mo_{2}O_{5}N_{2} exhibiting gapless QSL behaviors are discussed.

Strong quantum fluctuations in a quantum spin liquid candidate with a Co-based triangular lattice

Currently under active study in condensed matter physics, both theoretically and experimentally, are quantum spin liquid (QSL) states, in which no long-range magnetic ordering appears at low temperatures due to strong quantum fluctuations of the magnetic moments. The existing QSL candidates all have their intrinsic disadvantages, however, and solid evidence for quantum fluctuations is scarce. Here, we report a previously unreported compound, [Formula: see text], a geometrically frustrated system with effective spin-1/2 local moments for Co²⁺ ions on an isotropic 2-dimensional (2D) triangular lattice. Magnetic susceptibility and neutron scattering experiments show no magnetic ordering down to 0.05 K. Thermodynamic measurements show that there is a tremendous amount of magnetic entropy present below 1 K in 0-applied magnetic field. The presence of localized low-energy spin fluctuations is revealed by inelastic neutron measurements. At low applied fields, these spin excitations are confined to low energy and contribute to the anomalously large specific heat. In larger applied fields, the system reverts to normal behavior as evident by both neutron and thermodynamic results. Our experimental characterization thus reveals that this material is an excellent candidate for the experimental realization of a QSL state.

Fragmented Electronic Spins with Quantum Fluctuations in Organic Mott Insulators Near a Quantum Spin Liquid

Magnetic structures of organic Mott insulators X[Pd(dmit)_{2}]_{2} (X=Me_{4}P, Me_{4}Sb), of which electronic states are located near a quantum spin liquid (X=EtMe_{3}Sb), are demonstrated by ^{13}C nuclear magnetic resonance. Antiferromagnetic spectra and nuclear relaxations show two distinct magnetic moments within each Pd(dmit)_{2} molecule, which cannot be described by single band dimer-Mott model and requires intramolecular electronic correlation. This unconventional fragmentation of S=1/2 electron spin with strong quantum fluctuation is presumably caused by nearly degenerated intramolecular multiple orbitals, and shares a notion of quantum liquids where electronic excitations are fractionalized and S=1/2 spin is no longer an elementary particle.

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 TN = 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.

Spin-orbital entangled liquid state in the copper oxide Ba3CuSb2O9

Structure with orbital degeneracy is unstable toward spontaneous distortion. Such orbital correlation usually has a much higher energy scale than spins, and therefore, magnetic transition takes place at a much lower temperature, almost independently from orbital ordering. However, when the energy scales of orbitals and spins meet, there is a possibility of spin-orbital entanglement that would stabilize novel ground state such as spin-orbital liquid and random singlet state. Here we review on such a novel spin-orbital magnetism found in the hexagonal perovskite oxide Ba₃CuSb₂O₉, which hosts a self-organized honeycomblike short-range order of a strong Jahn-Teller ion Cu²⁺. Comprehensive structural and magnetic measurements have revealed that the system has neither magnetic nor Jahn-Teller transition down to the lowest temperatures, and Cu spins and orbitals retain the hexagonal symmetry and paramagnetic state. Various macroscopic and microscopic measurements all indicate that spins and orbitals remain fluctuating down to low temperatures without freezing, forming a spin-orbital entangled liquid state.

Low-Energy Excitations in Quantum Spin Liquids Identified by Optical Spectroscopy

The electrodynamic response of organic spin liquids with highly frustrated triangular lattices has been measured in a wide energy range. While the overall optical spectra of these Mott insulators are governed by transitions between the Hubbard bands, distinct in-gap excitations can be identified at low temperatures and frequencies, which we attribute to the quantum-spin-liquid state. For the strongly correlated β^{'}-EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, we discover enhanced conductivity below 175  cm^{-1}, comparable to the energy of the magnetic coupling J≈250  K. For ω→0, these low-frequency excitations vanish faster than the charge-carrier response subject to Mott-Hubbard correlations, resulting in a dome-shaped band peaked at 100  cm^{-1}. Possible relations to spinons, magnons, and disorder are discussed.

Tunable Quantum Spin Liquidity in the 1/6th-Filled Breathing Kagome Lattice

We present measurements on a series of materials, Li_{2}In_{1-x}Sc_{x}Mo_{3}O_{8}, that can be described as a 1/6th-filled breathing kagome lattice. Substituting Sc for In generates chemical pressure which alters the breathing parameter nonmonotonically. Muon spin rotation experiments show that this chemical pressure tunes the system from antiferromagnetic long range order to a quantum spin liquid phase. A strong correlation with the breathing parameter implies that it is the dominant parameter controlling the level of magnetic frustration, with increased kagome symmetry generating the quantum spin liquid phase. Magnetic susceptibility measurements suggest that this is related to distinct types of charge order induced by changes in lattice symmetry, in line with the theory of Chen et al. [Phys. Rev. B 93, 245134 (2016)PRBMDO2469-995010.1103/PhysRevB.93.245134]. The specific heat for samples at intermediate Sc concentration, which have the minimum breathing parameter, show consistency with the predicted U(1) quantum spin liquid.

Mixed-valence insulators with neutral Fermi surfaces

Samarium hexaboride is a classic three-dimensional mixed valence system with a high-temperature metallic phase that evolves into a paramagnetic charge insulator below 40 K. A number of recent experiments have suggested the possibility that the low-temperature insulating bulk hosts electrically neutral gapless fermionic excitations. Here we show that a possible ground state of strongly correlated mixed valence insulators—a composite exciton Fermi liquid—hosts a three dimensional Fermi surface of a neutral fermion, that we name the “composite exciton.” We describe the mechanism responsible for the formation of such excitons, discuss the phenomenology of the composite exciton Fermi liquids and make comparison to experiments in SmB₆.

Samarium hexaboride is a candidate topological insulator but recent experiments have found behaviour indicative of a metallic Fermi liquid phase. Here the authors show that the conflicting observations can be accommodated by a model where strong interactions drive the formation of exotic neutral quasiparticles.

Randomness-induced quantum spin liquid on honeycomb lattice

Quantum entanglement in magnetic materials is expected to yield a quantum spin liquid (QSL), in which strong quantum fluctuations prevent magnetic ordering even at zero temperature. This topic has been one of the primary focuses of condensed-matter science since Anderson first proposed the resonating valence bond state in a certain spin-1/2 frustrated magnet in 1973. Since then, several candidate materials featuring frustration, such as triangular and kagome lattices, have been reported to exhibit liquid-like behavior. However, the mechanisms that stabilize the liquid-like states have remained elusive. Here, we present a QSL state in a spin-1/2 honeycomb lattice with randomness in the exchange interaction. That is, we successfully introduce randomness into the organic radial-based complex and realize a random-singlet (RS) state (or valence bond glass). All magnetic and thermodynamic experimental results indicate the liquid-like behaviors, which are consistent with those expected in the RS state. Our results suggest that the randomness or inhomogeneity in the actual systems stabilize the RS state and yield liquid-like behavior.

Highly Spin-Frustrated Magnetism in the Topochemically Prepared Triangular Lattice Cluster Magnets Na3 A2 (MoO4 )2 Mo3 O8 (A=In, Sc)

The physical properties of novel cluster-based triangular lattice antiferromagnets Na₃ A₂ (MoO₄ )₂ Mo₃ O₈ (A=In, Sc), synthesized through a topochemical Na-intercalation to nonmagnetic Na₂ A₂ (MoO₄ )₂ Mo₃ O₈ , are reported. The S=1/2 [Mo₃ ]¹¹⁺ clusters form a regular triangular lattice, which gives the magnetic system a strong geometrical spin frustration effect. Despite the strong antiferromagnetic couplings among [Mo₃ ]¹¹⁺ clusters, they show no long-range magnetic orderings down to 0.5 K with the finite residual magnetic entropy. The ground states of Na₃ A₂ (MoO₄ )₂ Mo₃ O₈ have been characterized as a quantum spin liquid, owing to the strong spin frustration of cluster spins on the triangular lattice.

Charge and Lattice Fluctuations in Molecule-Based Spin Liquids

Spin liquid (SL) systems have been the subject of much attention recently, as they have been theoretically predicted to not freeze, even at 0 K. Despite extensive searches being made for such a system, only a few candidates have been found. All of these candidates share geometrical frustrations that are based on triangular lattices. We applied vibrational spectroscopy to one of the candidates of a molecule-based SL system, and we compared its results against three antiferromagnetic compounds and four charge-ordered compounds. All of their structural motifs belong to triangular lattices. The C=C stretching modes in the SL state indicated that there were charge and lattice fluctuations. These fluctuations were suppressed but non-negligible in the AF compounds. This finding is potentially significant, as it indicates that a hidden lattice and charge fluctuation are the driving force of a geometrical frustration, which eventually leads to a SL state.

3D Spin-Liquid State in an Organic Hyperkagome Lattice of Mott Dimers

We report the first 3D spin liquid state of isotropic organic spins. Structural analysis, and magnetic and heat-capacity measurements were carried out for a chiral organic radical salt, (TBA)_{1.5}[(-)-NDI-Δ] (TBA denotes tetrabutylammonium and NDI denotes naphthalene diimide), in which (-)-NDI-Δ forms a K_{4} structure due to its triangular molecular structure and an intermolecular π-π overlap between the NDI moieties. This lattice was identical to the hyperkagome lattice of S=1/2 Mott dimers, and should exhibit 3D spin frustration. In fact, even though the high-temperature magnetic susceptibility followed the Curie-Weiss law with a negative Weiss constant of θ=-15  K, the low-temperature magnetic measurements revealed no long-range magnetic ordering down to 70 mK, and suggested the presence of a spin liquid state with a large residual paramagnetism χ_{0} of 8.5×10^{-6}  emu g^{-1} at the absolute zero temperature. This was supported by the ^{14}N NMR measurements down to 0.38 K. Further, the low-temperature heat capacities c_{p} down to 68 mK clearly indicated the presence of c_{p} for the spin liquid state, which can be fitted to the power law of T^{0.62} in the wide temperature range 0.07-4.5 K.

Slow dynamics of electrons at a metal-Mott insulator boundary in an organic system with disorder

The Mott transition-a metal-insulator transition caused by repulsive Coulomb interactions between electrons-is a central issue in condensed matter physics because it is the mother earth of various attractive phenomena. Outstanding examples are high-Tc (critical temperature) cuprates and manganites exhibiting colossal magnetoresistance. Furthermore, spin liquid states, which are quantum-fluctuation-driven disordered ground states in antiferromagnets, have recently been found in magnetic systems very near the Mott transition. To date, intensive studies on the Mott transition have been conducted and appear to have established a nearly complete framework for understanding the Mott transition. We found an unknown type of Mott transition in an organic spin liquid material with a slightly disordered lattice. Around the Mott transition region of this material under pressure, nuclear magnetic resonance experiments capture the emergence of slow electronic fluctuations of the order of kilohertz or lower, which is not expected in the conventional Mott transition that appears as a clear first-order transition at low temperatures. We suggest that they are due to the unconventional metal-insulator fluctuations emerging around the disordered Mott transition in analogy to the slowly fluctuating spin phase, or Griffiths phase, realized in Ising spin systems with disordered lattices.

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.

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.

Absence of Magnetic Thermal Conductivity in the Quantum Spin-Liquid Candidate YbMgGaO_{4}

We present the ultralow-temperature specific heat and thermal conductivity measurements on single crystals of YbMgGaO_{4}, which was recently argued to be a promising candidate for a quantum spin liquid (QSL). In a zero magnetic field, a large magnetic contribution of specific heat is observed, and exhibits a power-law temperature dependence (C_{m}∼T^{0.74}). On the contrary, we do not observe any significant contribution of thermal conductivity from magnetic excitations. In magnetic fields H≥6  T, the exponential T dependence of C_{m} and the enhanced thermal conductivity indicate a magnon gap of the fully polarized state. The absence of magnetic thermal conductivity at the zero field in this QSL candidate puts a strong constraint on the theories of its ground state.

Fate of Spinons at the Mott Point

Gapless spin liquids have recently been observed in several frustrated Mott insulators, with elementary spin excitations-"spinons"-reminiscent of degenerate Fermi systems. However, their precise role at the Mott point, where charge fluctuations begin to proliferate, remains controversial and ill understood. Here we present the simplest theoretical framework that treats the dynamics of emergent spin and charge excitations on the same footing, providing a new physical picture of the Mott metal-to-insulator transition at half filing. We identify a generic orthogonality mechanism leading to strong damping of spinons, arising as soon as the Mott gap closes. Our results indicate that spinons should not play a significant role within the high-temperature quantum critical regime above the Mott point-in striking agreement with all available experiments.

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}.

Anomalous vacuum energy and stability of a quantum liquid

We show that the vacuum (zero-point) energy of a low-temperature quantum liquid is a variable property which changes with the state of the system, in notable contrast to the static vacuum energy in solids commonly considered. We further show that this energy is inherently anomalous: it decreases with temperature and gives a negative contribution to a system's heat capacity. This effect operates in an equilibrium and macroscopic system, in marked contrast to small or out-of-equilibrium configurations discussed previously. We find that the negative contribution is over-compensated by the positive term from the excitation of longitudinal fluctuations and demonstrate how the overall positive heat capacity is related to the stability of a condensed phase at the microscopic level.

Origin of the Spin-Orbital Liquid State in a Nearly J=0 Iridate Ba_{3}ZnIr_{2}O_{9}

We show using detailed magnetic and thermodynamic studies and theoretical calculations that the ground state of Ba_{3}ZnIr_{2}O_{9} is a realization of a novel spin-orbital liquid state. Our results reveal that Ba_{3}ZnIr_{2}O_{9} with Ir^{5+} (5d^{4}) ions and strong spin-orbit coupling (SOC) arrives very close to the elusive J=0 state but each Ir ion still possesses a weak moment. Ab initio density functional calculations indicate that this moment is developed due to superexchange, mediated by a strong intradimer hopping mechanism. While the Ir spins within the structural Ir_{2}O_{9} dimer are expected to form a spin-orbit singlet state (SOS) with no resultant moment, substantial frustration arising from interdimer exchange interactions induce quantum fluctuations in these possible SOS states favoring a spin-orbital liquid phase down to at least 100 mK.

Effective magnetic coupling with strong spin frustration in (Ph3MeP+)(C60˙−) and reversible C60˙− dimerization in (Ph3MeP+)(C60˙−)·C6H5CN. Effect of solvent on structure and properties

Antiferromagnetic coupling of spins with spin frustration and fullerene dimerization are observed for (Ph₃MeP⁺)(C₆₀˙⁻) (left) and (Ph₃MeP⁺)(C₆₀˙⁻)·C₆H₅CN (right).

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.

Gapless quantum spin liquid in an organic spin-1/2 triangular-lattice κ-H3(Cat-EDT-TTF)2

We report the results of SQUID and torque magnetometry of an organic spin-1/2 triangular-lattice κ-H(3)(Cat-EDT-TTF)(2). Despite antiferromagnetic exchange coupling at 80-100 K, we observed no sign of antiferromagnetic order down to 50 mK owing to spin frustration on the triangular lattice. In addition, we found nearly temperature-independent susceptibility below 3 K associated with Pauli paramagnetism. These observations suggest the development of gapless quantum spin liquid as the ground state. On the basis of a comparative discussion, we point out that the gapless quantum spin liquid states in organic systems share a possible mechanism, namely the formation of a band with a Fermi surface possibly attributed to spinons.

Gapless spin liquid ground state in the S = 1/2 vanadium oxyfluoride kagome antiferromagnet [NH4]2[C7H14N][V7O6F18]

The vanadium oxyfluoride [NH(4)](2)[C(7)H(14)N][V(7)O(6)F(18)] (DQVOF) is a geometrically frustrated magnetic bilayer material. The structure consists of S = 1/2 kagome planes of V(4+) d(1) ions with S = 1 V(3+) d(2) ions located between the kagome layers. Muon spin relaxation measurements demonstrate the absence of spin freezing down to 40 mK despite an energy scale of 60 K for antiferromagnetic exchange interactions. From magnetization and heat capacity measurements we conclude that the S = 1 spins of the interplane V(3+) ions are weakly coupled to the kagome layers, such that DQVOF can be viewed as an experimental model for S = 1/2 kagome physics, and that it displays a gapless spin liquid ground state.

Novel Pauli-paramagnetic quantum phase in a Mott insulator

In Mott insulators, the strong electron-electron Coulomb repulsion localizes electrons. In dimensions greater than one, their spins are usually ordered antiferromagnetically at low temperatures. Geometrical frustrations can destroy this long-range order, leading to exotic quantum spin liquid states. However, their magnetic ground states have been a long-standing mystery. Here we show that a quantum spin liquid state in the organic Mott insulator EtMe(3)Sb[Pd(dmit)(2)](2) (where Et is C(2)H(5)-, Me is CH(3)-, and dmit is 1,3-dithiole-2-thione-4,5-dithiolate) with two-dimensional triangular lattice has Pauli-paramagnetic-like low-energy excitations, which are a hallmark of itinerant fermions. Our torque magnetometry down to low temperatures (30 mK) up to high fields (32 T) reveals distinct residual paramagnetic susceptibility comparable to that in a half-filled two-dimensional metal, demonstrating the magnetically gapless nature of the ground state. Moreover, our results are robust against deuteration, pointing toward the emergence of an extended 'quantum critical phase', in which low-energy spin excitations behave as in paramagnetic metals with Fermi surface, despite the frozen charge degree of freedom.