Gapless quantum spin liquid ground state in the two-dimensional spin-1/2 triangular antiferromagnet YbMgGaO4

Evidence of Dirac Quantum Spin Liquid in YbZn_{2}GaO_{5}

The emergence of a quantum spin liquid (QSL), a state of matter that can result when electron spins are highly correlated but do not become ordered, has been the subject of a considerable body of research in condensed matter physics [1,2]. Spin liquid states have been proposed as hosts for high-temperature superconductivity [3] and can host topological properties with potential applications in quantum information science [4]. The excitations of most quantum spin liquids are not conventional spin waves but rather quasiparticles known as spinons, whose existence is well established experimentally only in one-dimensional systems; the unambiguous experimental realization of QSL behavior in higher dimensions remains challenging. Here, we investigate the novel compound YbZn_{2}GaO_{5}, which hosts an ideal triangular lattice of effective spin-1/2 moments with no detectable inherent chemical disorder. Thermodynamic and inelastic neutron scattering measurements performed on high-quality single crystal samples of YbZn_{2}GaO_{5} exclude the possibility of long-range magnetic ordering down to 0.06 K, demonstrate a quadratic power law for the specific heat and reveal a continuum of magnetic excitations in parts of the Brillouin zone. Both low-temperature thermodynamics and inelastic neutron scattering spectra suggest that YbZn_{2}GaO_{5} is a U(1) Dirac QSL with spinon excitations concentrated at certain points in the Brillouin zone. We advanced these results by performing additional specific heat measurements under finite fields, further confirming the theoretical expectations for a Dirac QSL on the triangular lattice of YbZn_{2}GaO_{5}.

Spiral Spin Liquid in a Frustrated Honeycomb Antiferromagnet: A Single-Crystal Study of GdZnPO

The frustrated honeycomb spin model can stabilize a subextensively degenerate spiral spin liquid with nontrivial topological excitations and defects, but its material realization remains rare. Here, we report the experimental realization of this model in the structurally disorder-free compound GdZnPO. Using a single-crystal sample, we find that spin-7/2 rare-earth Gd^{3+} ions form a honeycomb lattice with dominant second-nearest-neighbor antiferromagnetic and first-nearest-neighbor ferromagnetic couplings, along with easy-plane single-site anisotropy. This frustrated model stabilizes a unique spiral spin liquid with a degenerate contour around the K{1/3,1/3} point in reciprocal space, consistent with our experiments down to 30 mK, including the observation of a giant residual specific heat. Our results establish GdZnPO as an ideal platform for exploring the stability of spiral spin liquids and their novel properties, such as the emergence of low-energy topological defects on the sublattices.

Crystal Growth, Structure, and Diverse Magnetic Behaviors in Frustrated Triangular Lattice REBO3 (RE = Tb-Yb)

Triangular lattice (TL) materials are a rich playground for investigating exotic quantum spin states and related applications in quantum computing and quantum information. Millimeter-level single crystals of REBO3 (RE = Tb-Yb) with a nearly perfect RE-based TL have been successfully grown via a high-temperature flux method and structurally characterized via single-crystal X-ray diffraction. These 113-type materials crystallize in a monoclinic crystal system with a C2/c space group. Anisotropic magnetism and dominant antiferromagnetic interactions are found for the above materials based on DC magnetic susceptibility measurements. The comprehensive low-temperature specific heat data of REBO3 (RE = Tb-Tm) are characterized on single crystals for the first time, which exhibit diverse magnetic behaviors. Specifically, two weak-field-induced transitions could be found in the case of DyBO3 based on the specific heat measurements. Our results suggest that REBO3 (RE = Tb-Yb) is a TL magnetic system for investigating potential quantum magnetism.

Two-Dimensional Cobalt(II) Benzoquinone Frameworks for Putative Kitaev Quantum Spin Liquid Candidates

The realization and discovery of quantum spin liquid (QSL) candidate materials are crucial for exploring exotic quantum phenomena and applications associated with QSLs. Most existing metal-organic two-dimensional (2D) quantum spin liquid candidates have structures with spins arranged on the triangular or kagome lattices, whereas honeycomb-structured metal-organic compounds with QSL characteristics are rare. Here, we report the use of 2,5-dihydroxy-1,4-benzoquinone (X2dhbq, X = Cl, Br, H) as the linkers to construct cobalt(II) honeycomb lattices (NEt4)2[Co2(X2dhbq)3] as promising Kitaev-type QSL candidate materials. The high-spin d7 Co2+ has pseudospin-1/2 ground-state doublets, and benzoquinone-based linkers not only provide two separate superexchange pathways that create bond-dependent frustrated interactions but also allow for chemical tunability to mediate magnetic coupling. Our magnetization data show antiferromagnetic interactions between neighboring metal centers with Weiss constants from -5.1 to -8.5 K depending on the X functional group in X2dhbq linkers (X = Cl, Br, H). No magnetic transition or spin freezing could be observed down to 2 K. Low-temperature susceptibility (down to 0.3 K) and specific heat (down to 0.055 K) of (NEt4)2[Co2(H2dhbq)3] were further analyzed. Heat capacity measurements confirmed no long-range order down to 0.055 K, evidenced by the broad peak instead of the λ-like anomaly. Our results indicate that these 2D cobalt benzoquinone frameworks are promising Kitaev QSL candidates with chemical tunability through ligands that can vary the magnetic coupling and frustration.

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.

Erbium-Excess Gallium Garnets

A series of garnets of formula Er3+xGa5-xO12 are described, for which we report the crystal structures for both polycrystalline and single-crystal samples. The x limit in the garnet phase is between 0.5 and 0.6 under our conditions, with the Er fully occupying the dodecahedral (24c) garnet site plus some of the octahedral site (16a) in place of the Ga normally present. Long-range antiferromagnetic order with spin-ice-like frustration is suggested by the transition temperature (TN ≈ 0.8 K) being lower than the Curie-Weiss theta. The magnetic ordering temperature does not depend on the Er excess, but there is increasing residual entropy as the Er excess is increased, highlighting the potential for unusual magnetic behavior in this system. The field-dependent magnetic entropy trend is consistent with the reported behavior for frustrated triangular magnetic systems: an increasing transition temperature with a broader hump as the applied field increases [Xing, J.; Phys. Rev. Mater. 2019, 3(11), 114413;Filippi, J.; Solid State Commun. 1977, 23(9), 613-616; Bloxsom, J. A. Thermal and Magnetic Studies of Spin Ice Compounds. University College London, 2016].

Possible realization of three-dimensional quantum spin liquid behavior in HoVO4

The study of geometrically frustrated magnetic systems with unusual crystal field ground states offers a possibility of realizing the new aspects of physics of disordered systems. In this study, we report our results of structural, magnetic susceptibility, heat capacity measurements, along with density functional theory (DFT) calculations on HoVO₄; a compound in which the presence of a distorted kind of HoO₈polyhedral leads to multiple magnetic interaction paths. The observed broad maximum below 10 K in the temperature response of DC susceptibility curves implies the presence of short-range correlations. AC susceptibility rules out the possibility of any kind of spin freezing. Temperature dependent heat capacity measurement at zero field indicate towards the absence of long-range ordering, along with the presence of a broad maximum centered around 14 K. The residual heat capacity exhibits a characteristic power-law (T^α) behavior with the exponentαnearly equal to 2, which is analogous to that observed for other three-dimensional (3D) quantum spin liquid (QSL) systems. The DFT calculations signify the presence of dominant second and third nearest neighbor interactions, which in turn lead to magnetic frustration in our system. Our investigations suggest that HoVO₄can be a candidate for realizing a 3D QSL state.

Crystal structure and specific heat of calcium lanthanide oxyborates Ca4 LnO(BO3)3

Calcium lanthanide oxyborates Ca4 LnO(BO3)3 are of interest for their optical and electromechanical properties. Their crystal structure has been well characterised using powder and single-crystal X-ray diffraction but there remains some disagreement regarding cation ordering in these compounds. In this study, combined X-ray and neutron powder diffraction was employed to study the cation distribution and obtain accurate boron and oxygen atomic coordinates for six Ca4 LnO(BO3)3 compounds (Ln = Pr, Nd, Tb, Ho, Er, Yb) at room temperature and one (Ln = Tb) at 50 and 1.5 K. All compounds adopt the previously reported monoclinic structure with space group Cm. The Ln 3+ ions are disordered over two of the three metal sites, with the extent of disorder increasing across the lanthanide series with decreasing ionic radius. Low-temperature neutron data for Ca4TbO(BO3)3 showed a decrease in paramagnetic scattering on cooling but no obvious magnetic Bragg or diffuse scattering at the lowest temperature of 1.5 K. We report specific heat data at cryogenic temperatures for eight Ca4 LnO(BO3)3 compounds and relate the magnetic properties of these compounds to their structural behaviour.

LiYbSe2: Frustrated Magnetism in the Pyrochlore Lattice

Three-dimensionally (3D) frustrated magnets generally exist in the magnetic diamond and pyrochlore lattices, in which quantum fluctuations suppress magnetic orders and generate highly entangled ground states. LiYbSe₂ in a previously unreported pyrochlore lattice was discovered from LiCl flux growth. Distinct from the quantum spin liquid (QSL) candidate NaYbSe₂ hosting a perfect triangular lattice of Yb³⁺, LiYbSe₂ crystallizes in the cubic pyrochlore structure with space group Fd3m (No. 227). The Yb³⁺ ions in LiYbSe₂ are arranged on a network of corner-sharing tetrahedra, which is particularly susceptible to geometrical frustration. According to our temperature-dependent magnetic susceptibility measurements, the dominant antiferromagnetic interaction in LiYbSe₂ is expected to appear around 8 K. However, no long-range magnetic order is detected in thermomagnetic measurements above 70 mK. Specific heat measurements also show magnetic correlations shifting with applied magnetic field with a degree of missing entropy that may be related to the slight mixture of Yb³⁺ on the Li site. Such magnetic frustration of Yb³⁺ is rare in pyrochlore structures. Thus, LiYbSe₂ shows promise in intrinsically realizing disordered quantum states like QSL in pyrochlore structures.

Non-magnetic ion site disorder effects on the quantum magnetism of a spin-1/2 equilateral triangular lattice antiferromagnet

With the motivation to study how non-magnetic ion site disorder affects the quantum magnetism of Ba₃CoSb₂O₉, a spin-1/2 equilateral triangular lattice antiferromagnet, we performed DC and AC susceptibility, specific heat, elastic and inelastic neutron scattering measurements on single crystalline samples of Ba_2.87Sr_0.13CoSb₂O₉with Sr doping on non-magnetic Ba²⁺ion sites. The results show that Ba_2.87Sr_0.13CoSb₂O₉exhibits (i) a two-step magnetic transition at 2.7 K and 3.3 K, respectively; (ii) a possible canted 120 degree spin structure at zero field with reduced ordered moment as 1.24μ_B/Co; (iii) a series of spin state transitions for bothH∥ab-plane andH∥c-axis. ForH∥ab-plane, the magnetization plateau feature related to the up-up-down phase is significantly suppressed; (iv) an inelastic neutron scattering spectrum with only one gapped mode at zero field, which splits to one gapless and one gapped mode at 9 T. All these features are distinctly different from those observed for the parent compound Ba₃CoSb₂O₉, which demonstrates that the non-magnetic ion site disorder (the Sr doping) plays a complex role on the magnetic properties beyond the conventionally expected randomization of the exchange interactions. We propose the additional effects including the enhancement of quantum spin fluctuations and introduction of a possible spatial anisotropy through the local structural distortions.

NaCo2(SeO3)2(OH): competing magnetic ground states of a new sawtooth structure with 3d7 Co2+ ions

We report the high-pressure synthesis and a comprehensive study of NaCo2(SeO3)2(OH) sawtooth chain structure using bulk magnetic properties and neutron scattering.

Magnetic properties and signatures of ordering in triangular lattice antiferromagnet KCeO2

The magnetic ground state and the crystalline electric field level scheme of the triangular lattice antiferromagnet KCeO 2 are investigated. Below T N = 300 mK, KCeO 2 develops signatures of magnetic order in specific heat measurements and low energy inelastic neutron scattering data. Trivalent Ce 3 + ions in the D 3 d local environment of this compound exhibit large splittings among the lowest three 4 f 1 Kramers doublets defining for the free ion the J = 5 / 2 sextet and a ground state doublet with dipole character, consistent with recent theoretical predictions in M. S. Eldeeb et al. Phys. Rev. Materials 4, 124001 (2020). An unexplained, additional local mode appears, and potential origins of this anomalous mode are discussed.

Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice

At strong repulsion, the triangular-lattice Hubbard model is described by s=1/2 spins with nearest-neighbor antiferromagnetic Heisenberg interactions and exhibits conventional 120° order. Using the infinite density matrix renormalization group and exact diagonalization, we study the effect of the additional four-spin interactions naturally generated from the underlying Mott-insulator physics of electrons as the repulsion decreases. Although these interactions have historically been connected with a gapless ground state with emergent spinon Fermi surface, we find that, at physically relevant parameters, they stabilize a chiral spin liquid (CSL) of Kalmeyer-Laughlin (KL) type, clarifying observations in recent studies of the Hubbard model. We then present a self-consistent solution based on a mean-field rewriting of the interaction to obtain a Hamiltonian with similarities to the parent Hamiltonian of the KL state, providing a physical understanding for the origin of the CSL.

Survival of itinerant excitations and quantum spin state transitions in YbMgGaO4 with chemical disorder

A recent focus of quantum spin liquid (QSL) studies is how disorder/randomness in a QSL candidate affects its true magnetic ground state. The ultimate question is whether the QSL survives disorder or the disorder leads to a “spin-liquid-like” state, such as the proposed random-singlet (RS) state. Since disorder is a standard feature of most QSL candidates, this question represents a major challenge for QSL candidates. YbMgGaO₄, a triangular lattice antiferromagnet with effective spin-1/2 Yb³⁺ions, is an ideal system to address this question, since it shows no long-range magnetic ordering with Mg/Ga site disorder. Despite the intensive study, it remains unresolved as to whether YbMgGaO₄ is a QSL or in the RS state. Here, through ultralow-temperature thermal conductivity and magnetic torque measurements, plus specific heat and DC magnetization data, we observed a residual κ₀/T term and series of quantum spin state transitions in the zero temperature limit for YbMgGaO₄. These observations strongly suggest that a QSL state with itinerant excitations and quantum spin fluctuations survives disorder in YbMgGaO₄.

It remains an open question as to whether the quantum spin liquid state survives material disorder, or is replaced by some spin-liquid like state. Here, Rao et al succeed in resolving a resolving a κ₀/T residual in the thermal conductivity of YbMgGaO₄ strongly suggesting the survival of the quantum spin liquid state.

Stripe-yzmagnetic order in the triangular-lattice antiferromagnet KCeS2

Yb- and Ce-based delafossites were recently identified as effective spin-1/2 antiferromagnets on the triangular lattice. Several Yb-based systems, such as NaYbO2, NaYbS2, and NaYbSe2, exhibit no long-range order down to the lowest measured temperatures and therefore serve as putative candidates for the realization of a quantum spin liquid. However, their isostructural Ce-based counterpart KCeS2exhibits magnetic order belowTN= 400 mK, which was so far identified only in thermodynamic measurements. Here we reveal the magnetic structure of this long-range ordered phase using magnetic neutron diffraction. We show that it represents the so-called 'stripe-yz' type of antiferromagnetic order with spins lying approximately in the triangular-lattice planes orthogonal to the nearest-neighbor Ce-Ce bonds. No structural lattice distortions are revealed belowTN, indicating that the triangular lattice of Ce3+ions remains geometrically perfect down to the lowest temperatures. We propose an effective Hamiltonian for KCeS2, based on a fit to the results ofab initiocalculations, and demonstrate that its magnetic ground state matches the experimental spin structure.

Development of short and long-range magnetic order in the double perovskite based frustrated triangular lattice antiferromagnet Ba[Formula: see text]MnTeO[Formula: see text]

Frustrated magnets based on oxide double perovskites offer a viable ground wherein competing magnetic interactions, macroscopic ground state degeneracy and complex interplay between emergent degrees of freedom can lead to correlated quantum phenomena with exotic excitations highly relevant for potential technological applications. By local-probe muon spin relaxation ([Formula: see text]SR) and complementary thermodynamic measurements accompanied by first-principles calculations, we here demonstrate novel electronic structure and magnetic phases of Ba[Formula: see text]MnTeO[Formula: see text], where Mn[Formula: see text] ions with S = 5/2 spins constitute a perfect triangular lattice. Magnetization results evidence the presence of strong antiferromagnetic interactions between Mn[Formula: see text] spins and a phase transition at [Formula: see text] = 20 K. Below [Formula: see text], the specific heat data show antiferromagnetic magnon excitations with a gap of 1.4 K, which is due to magnetic anisotropy. [Formula: see text]SR reveals the presence of static internal fields in the ordered state and short-range spin correlations high above [Formula: see text]. It further unveils critical slowing-down of spin dynamics at [Formula: see text] and the persistence of spin dynamics even in the magnetically ordered state. Theoretical studies infer that Heisenberg interactions govern the inter- and intra-layer spin-frustration in this compound. Our results establish that the combined effect of a weak third-nearest-neighbour ferromagnetic inter-layer interaction (owing to double-exchange) and intra-layer interactions stabilizes a three-dimensional magnetic ordering in this frustrated magnet.

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.

Structure and Magnetic Properties of Melilite-Type Compounds RE2Be2GeO7 (RE = Pr, Nd, Gd-Yb) with Rare-Earth Ions on Shastry-Sutherland Lattice

Rare-earth (RE)-based frustrated magnets, such as typical systems of combining strong spin-orbit coupling (SOC), geometric frustration, and anisotropic exchange interaction, can give rise to diverse exotic magnetic ground states such as quantum spin liquid. The discovery of new RE-based frustrated materials is crucial for exploring the exotic magnetic phases. Herein, we report the synthesis, structure, and magnetic properties of a family of melilite-type RE₂Be₂GeO₇ (RE = Pr, Nd, and Gd-Yb) compounds crystallized in a tetragonal P4̅2₁m structure, where magnetic RE³⁺ ions lay out on the Shastry-Sutherland lattice (SSL) within the ab plane and are well separated by nonmagnetic [GeBe₂O₇]^6- polyhedrons along the c-axis. Temperature (T)-dependent susceptibilities χ(T) and isothermal magnetization M(H) measurements reveal that most RE₂Be₂GeO₇ compounds except RE = Tb show no magnetic ordering down to 2 K despite the dominant antiferromagnetic (AFM) interactions, where Tb₂Be₂GeO₇ undergoes AFM transition with Néel temperature T_N ∼ 2.5 K and field-induced spin flop behaviors (T < T_N). In addition, the calculated magnetic entropy change ΔS_m from the isothermal M(H) curves reveals viable magnetocaloric effect for RE₂Be₂GeO₇ (RE = Gd and Dy) in liquid helium temperature regimes; Gd₂Be₂GeO₇ shows the maximum ΔS_m up to 54.8 J K^-1 kg^-1 at ΔH = 7 T and Dy₂Be₂GeO₇ has the largest value ΔS_m = 16.1 J K^-1 kg^-1 at ΔH = 2 T in this family. More excitingly, the rich diversity of RE ions in this family enables an archetype for exploring exotic quantum magnetic phenomena with large variability of spin located on the SSL lattice.

Frustrated Heisenberg J1-J2 model within the stretched diamond lattice of LiYbO2

We investigate the magnetic properties of LiYbO2, containing a three-dimensionally frustrated, diamond-like lattice via neutron scattering, magnetization, and heat capacity measurements. The stretched diamond network of Yb3+ ions in LiYbO2 enters a long-range incommensurate, helical state with an ordering wave vector k = ( 0.384 , ± 0.384 , 0 ) that "locks-in" to a commensurate k = ( 1 / 3 , ± 1 / 3,0 ) phase under the application of a magnetic field. The spiral magnetic ground state of LiYbO2 can be understood in the framework of a Heisenberg J 1 - J 2 Hamiltonian on a stretched diamond lattice, where the propagation vector of the spiral is uniquely determined by the ratio of J 2 / J 1 . The pure Heisenberg model, however, fails to account for the relative phasing between the Yb moments on the two sites of the bipartite lattice, and this detail as well as the presence of an intermediate, partially disordered, magnetic state below 1 K suggests interactions beyond the classical Heisenberg description of this material.

Signatures of a Spin-1/2 Cooperative Paramagnet in the Diluted Triangular Lattice of Y_{2}CuTiO_{6}

We present a combination of thermodynamic and dynamic experimental signatures of a disorder driven dynamic cooperative paramagnet in a 50% site diluted triangular lattice spin-1/2 system: Y_{2}CuTiO_{6}. Magnetic ordering and spin freezing are absent down to 50 mK, far below the Curie-Weiss scale (-θ_{CW}) of ∼134  K. We observe scaling collapses of the magnetic field and temperature dependent magnetic heat capacity and magnetization data, respectively, in conformity with expectations from the random singlet physics. Our experiments establish the suppression of any freezing scale, if at all present, by more than 3 orders of magnitude, opening a plethora of interesting possibilities such as disorder stabilized long range quantum entangled ground states.

Possible itinerant excitations and quantum spin state transitions in the effective spin-1/2 triangular-lattice antiferromagnet Na2BaCo(PO4)2

The most fascinating feature of certain two-dimensional (2D) gapless quantum spin liquid (QSL) is that their spinon excitations behave like the fermionic carriers of a paramagnetic metal. The spinon Fermi surface is then expected to produce a linear increase of the thermal conductivity with temperature that should manifest via a residual value (κ₀/T) in the zero-temperature limit. However, this linear in T behavior has been reported for very few QSL candidates. Here, we studied the ultralow-temperature thermal conductivity of an effective spin-1/2 triangular QSL candidate Na₂BaCo(PO₄)₂, which has an antiferromagnetic order at very low temperature (T_N ~ 148 mK), and observed a finite κ₀/T extrapolated from the data above T_N. Moreover, while approaching zero temperature, it exhibits series of quantum spin state transitions with applied field along the c axis. These observations indicate that Na₂BaCo(PO₄)₂ possibly behaves as a gapless QSL with itinerant spin excitations above T_N and its strong quantum spin fluctuations persist below T_N.

Fractionalized Excitations Revealed by Entanglement Entropy

Fractionalized excitations develop in many unusual many-body states such as quantum spin liquids, disordered phases that cannot be described using any local order parameter. Because these exotic excitations correspond to emergent degrees of freedom, how to probe them and establish their existence is a long-standing challenge. We present a general procedure to reveal the fractionalized excitations using real-space entanglement entropy in critical spin liquids that are particularly relevant to experiments. Moreover, we show how to use the entanglement entropy to construct the corresponding spinon Fermi surface. Our work defines a new pathway to establish and characterize exotic excitations in novel quantum phases of matter.

Magnetic Properties of Quasi-One-Dimensional Lanthanide Calcium Oxyborates Ca4LnO(BO3)3

This study examines the lanthanide calcium oxyborates Ca₄LnO(BO₃)₃ (Ln = La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Y, Er, Yb). The reported monoclinic structure (space group Cm) was confirmed using powder X-ray diffraction. The magnetic Ln³⁺ ions are situated in well-separated chains parallel to the c axis in a quasi-one-dimensional array. Here we report the first bulk magnetic characterization of Ca₄LnO(BO₃)₃ using magnetic susceptibility χ(T) and isothermal magnetization M(H) measurements at T ≥ 2 K. With the sole exception of Ca₄TbO(BO₃)₃, which displays a transition at T = 3.6 K, no magnetic transitions occur above 2 K, and Curie–Weiss analysis indicates antiferromagnetic nearest-neighbor interactions for all samples. Calculation of the magnetic entropy change ΔS_m indicates that Ca₄GdO(BO₃)₃ and Ca₄HoO(BO₃)₃ are viable magnetocaloric materials at liquid helium temperatures in the high-field and low-field regimes, respectively.

The monoclinic lanthanide calcium oxyborates Ca₄LnO(BO₃)₃ contain well-separated chains of magnetic Ln³⁺ ions. Bulk magnetic characterization suggests quasi-one-dimensional behavior with no magnetic ordering above 2 K except in Ca₄TbO(BO₃)₃ (T_tr = 3.6 K). Ca₄GdO(BO₃)₃ and Ca₄HoO(BO₃)₃ are viable magnetocaloric materials at liquid helium temperatures in the high-field and low-field regimes, respectively.

Spin liquids in geometrically perfect triangular antiferromagnets

The cradle of quantum spin liquids, triangular antiferromagnets show strong proclivity to magnetic order and require deliberate tuning to stabilize a spin-liquid state. In this brief review, we juxtapose recent theoretical developments that trace the parameter regime of the spin-liquid phase, with experimental results for Co-based and Yb-based triangular antiferromagnets. Unconventional spin dynamics arising from both ordered and disordered ground states are discussed, and the notion of a geometrically perfect triangular system is scrutinized to demonstrate non-trivial imperfections that may assist magnetic frustration in stabilizing dynamic spin states with peculiar excitations.

Magnetism and spin-gap behaviour in the layered Pr3 T 4Al12 (T = Fe, Ru, Os) compounds with the distorted Kagomé lattice

We have investigated the physical and magnetic properties of the distorted Kagomé lattice structure compounds; Pr[Formula: see text]Al₁₂ (T  =  Fe, Ru, Os) which crystallize in the hexagonal Gd₃Ru₄Al₁₂-type structure with space group P6₃/mmc (No. 194). The three compounds show long-range magnetic orderings of FM type at T _C  =  36.5 K (Fe), 39 K (Ru) and 37 K (Os) as indicated by the temperature dependences of magnetic susceptibility, [Formula: see text], specific heat, C _p(T) and electrical resistivity, [Formula: see text]. Pr₃Ru₄Al₁₂ shows an additional anomaly at 7 K which is associated to possible spin reorientation of some of the Pr moments. We also observed an additional anomaly in Pr₃Fe₄Al₁₂ at 132 K which might be associated to the formation of a spin density wave. The Sommerfeld coefficient, [Formula: see text] extracted from C _p(T) in the temperature range immediately above T _C indicate heavy fermion behaviours in Pr[Formula: see text]Al₁₂ (T  =  Fe, Ru, Os). [Formula: see text] of the three compounds show spin-gap behaviours below T _C of comparable energy gaps magnitude.

Randomly Hopping Majorana Fermions in the Diluted Kitaev System α-Ru_{0.8}Ir_{0.2}Cl_{3}

dc and ac magnetic susceptibility, magnetization, specific heat, and Raman scattering measurements are combined to probe low-lying spin excitations in α-Ru_{1-x}Ir_{x}Cl_{3} (x≈0.2), which realizes a disordered spin liquid. At intermediate energies (ℏω>3  meV), Raman spectroscopy evidences linearly ω-dependent Majorana-like excitations, obeying Fermi statistics. This points to robustness of a Kitaev paramagnetic state under spin vacancies. At low energies below 3 meV, we observe power-law dependences and quantum-critical-like scalings of the thermodynamic quantities, implying the presence of a weakly divergent low-energy density of states. This scaling phenomenology is interpreted in terms of the random hoppings of Majorana fermions. Our results demonstrate an emergent hierarchy of spin excitations in a diluted Kitaev honeycomb system subject to spin vacancies and bond randomness.

Spatial anisotropy of the quantum spin liquid system YbMgGaO4 revealed by ab initio calculations

YbMgGaO₄ was recently proposed as a promising quantum-spin-liquid candidate material. However, some details of its structure, such as those related to a spatial anisotropy, were not completely understood. In this work, we perform ab initio calculations based on density-functional-theory to investigate the structural, the electronic and the magnetic properties of YbMgGaO₄. The geometrical model was constructed to take into account disorder effects produced by the random distribution of Ga and Mg along the lattice. We found a substantial spatial anisotropy revealed by variations up to 8% in the Mg-O and Ga-O bond lengths, which results in variations up to 3% in the Yb-Yb distances along its triangular lattice. Thus, the Yb lattice was not perfectly triangular. Furthermore, we demonstrate an out-of-plane magnetization at the Yb atoms with magnetic anisotropy energy of [Formula: see text] eV/Yb and a small interlayer exchange of [Formula: see text] eV/Yb, demonstrating that the system is only approximately two-dimensional. The presented results provide insights for an atomic-scale understanding of YbMgGaO₄ with density-functional-theory calculations.

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.

Intertwined dipolar and multipolar order in the triangular-lattice magnet TmMgGaO4

A phase transition is often accompanied by the appearance of an order parameter and symmetry breaking. Certain magnetic materials exhibit exotic hidden-order phases, in which the order parameters are not directly accessible to conventional magnetic measurements. Thus, experimental identification and theoretical understanding of a hidden order are difficult. Here we combine neutron scattering and thermodynamic probes to study the newly discovered rare-earth triangular-lattice magnet TmMgGaO₄. Clear magnetic Bragg peaks at K points are observed in the elastic neutron diffraction measurements. More interesting, however, is the observation of sharp and highly dispersive spin excitations that cannot be explained by a magnetic dipolar order, but instead is the direct consequence of the underlying multipolar order that is "hidden" in the neutron diffraction experiments. We demonstrate that the observed unusual spin correlations and thermodynamics can be accurately described by a transverse field Ising model on the triangular lattice with an intertwined dipolar and ferro-multipolar order.

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.

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.

Electron spin resonance on the spin-1/2 triangular magnet NaYbS2

The delafossite structure of NaYbS₂ contains a planar spin-1/2 triangular lattice of Yb³⁺ ions and features a possible realisation of a quantum spin-liquid state. We investigated the Yb³⁺ spin dynamics by electron spin resonance (ESR) in single-crystalline samples of NaYbS₂. Very clear spectra with a well-resolved and large anisotropy could be observed down to the lowest accessible temperature of 2.7 K. In contrast to the ESR properties of other known spin-liquid candidate systems, the resonance seen in NaYbS₂ is accessible at low fields (<1 T) and is narrow enough for accurate characterisation of the relaxation rate as well as the g factor of the Yb³⁺ spins.

Rearrangement of Uncorrelated Valence Bonds Evidenced by Low-Energy Spin Excitations in YbMgGaO_{4}

dc-magnetization data measured down to 40 mK speak against conventional freezing and reinstate YbMgGaO_{4} as a triangular spin-liquid candidate. Magnetic susceptibility measured parallel and perpendicular to the c axis reaches constant values below 0.1 and 0.2 K, respectively, thus indicating the presence of gapless low-energy spin excitations. We elucidate their nature in the triple-axis inelastic neutron scattering experiment that pinpoints the low-energy (E≤J_{0}∼0.2  meV) part of the excitation continuum present at low temperatures (T<J_{0}/k_{B}), but completely disappearing upon warming the system above T≫J_{0}/k_{B}. In contrast to the high-energy part at E>J_{0} that is rooted in the breaking of nearest-neighbor valence bonds and persists to temperatures well above J_{0}/k_{B}, the low-energy one originates from the rearrangement of the valence bonds and thus from the propagation of unpaired spins. We further extend this picture to herbertsmithite, the spin-liquid candidate on the kagome lattice, and argue that such a hierarchy of magnetic excitations may be a universal feature of quantum spin liquids.

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.

Field-tunable quantum disordered ground state in the triangular-lattice antiferromagnet NaYbO2

Antiferromagnetically coupled S = 1 / 2 spins on an isotropic triangular lattice are the paradigm of frustrated quantum magnetism, but structurally ideal realizations are rare. Here, we investigate NaYbO2, which hosts an ideal triangular lattice of effectiveJ eff = 1 / 2 moments with no inherent site disorder. No signatures of conventional magnetic order appear down to 50 mK, strongly suggesting a quantum spin liquid ground state. We observe a two-peak specific heat and a nearly quadratic temperature dependence, in agreement with expectations for a two-dimensional Dirac spin liquid. Application of a magnetic field strongly perturbs the quantum disordered ground state and induces a clear transition into a collinear ordered state, consistent with a long-predicted up-up-down structure for a triangular-lattice XXZ Hamiltonian driven by quantum fluctuations. The observation of spin liquid signatures in zero field and quantum-induced ordering in intermediate fields in the same compound demonstrates an intrinsically quantum disordered ground state. We conclude that NaYbO2 is a model, versatile platform for exploring spin liquid physics with full tunability of field and temperature.

Scaling and data collapse from local moments in frustrated disordered quantum spin systems

Recently measurements on various spin–1/2 quantum magnets such as H₃LiIr₂O₆, LiZn₂Mo₃O₈, ZnCu₃(OH)₆Cl₂ and 1T-TaS₂—all described by magnetic frustration and quenched disorder but with no other common relation—nevertheless showed apparently universal scaling features at low temperature. In particular the heat capacity C[H, T] in temperature T and magnetic field H exhibits T/H data collapse reminiscent of scaling near a critical point. Here we propose a theory for this scaling collapse based on an emergent random-singlet regime extended to include spin-orbit coupling and antisymmetric Dzyaloshinskii-Moriya (DM) interactions. We derive the scaling C[H, T]/T ~ H^−γF_q[T/H] with F_q[x] = x^q at small x, with q ∈ {0, 1, 2} an integer exponent whose value depends on spatial symmetries. The agreement with experiments indicates that a fraction of spins form random valence bonds and that these are surrounded by a quantum paramagnetic phase. We also discuss distinct scaling for magnetization with a q-dependent subdominant term enforced by Maxwell’s relations.

There are many proposals for new forms of quantum matter in frustrated magnets but in practice disorder prevents the realisation of theoretically-tractable idealised models. Kimchi et al. show that recently observed scaling behavior common to several disordered quantum magnets can be understood as the emergence of a universal random-singlet regime.

Fractionalized excitations in the partially magnetized spin liquid candidate YbMgGaO4

Quantum spin liquids (QSLs) are exotic states of matter characterized by emergent gauge structures and fractionalized elementary excitations. The recently discovered triangular lattice antiferromagnet YbMgGaO₄ is a promising QSL candidate, and the nature of its ground state is still under debate. Here we use neutron scattering to study the spin excitations in YbMgGaO₄ under various magnetic fields. Our data reveal a dispersive spin excitation continuum with clear upper and lower excitation edges under a weak magnetic field (H = 2.5 T). Moreover, a spectral crossing emerges at the Γ point at the Zeeman-split energy. The corresponding redistribution of the spectral weight and its field-dependent evolution are consistent with the theoretical prediction based on the inter-band and intra-band spinon particle-hole excitations associated with the Zeeman-split spinon bands, implying the presence of fractionalized excitations and spinon Fermi surfaces in the partially magnetized QSL state in YbMgGaO₄.

Recent experiments have indicated that YbMgGaO₄ may be a quantum spin liquid with spinon Fermi surfaces but additional evidence is needed to support this interpretation. Shen et al. show weak magnetic fields cause changes in the excitation continuum that are consistent with spin liquid predictions.

Spinon Fermi Surface in a Cluster Mott Insulator Model on a Triangular Lattice and Possible Application to 1T-TaS_{2}

1T-TaS_{2} is a cluster Mott insulator on the triangular lattice with 13 Ta atoms forming a star of David cluster as the unit cell. We derive a two-dimensional XXZ spin-1/2 model with a four-spin ring exchange term to describe the effective low energy physics of a monolayer 1T-TaS_{2}, where the effective spin-1/2 degrees of freedom arises from the Kramers degenerate spin-orbital states on each star of David. A large scale density matrix renormalization group simulation is further performed on this effective model and we find a gapless spin liquid phase with a spinon Fermi surface at a moderate to large strength region of the four-spin ring exchange term. All peaks in the static spin structure factor are found to be located on the "2k_{F}" surface of a half-filled spinon on the triangular lattice. Experiments to detect the spinon Fermi surface phase in 1T-TaS_{2} are discussed.

Topography of Spin Liquids on a Triangular Lattice

Spin systems with frustrated anisotropic interactions are of significant interest due to possible exotic ground states. We have explored their phase diagram on a nearest-neighbor triangular lattice using the density-matrix renormalization group and mapped out the topography of the region that can harbor a spin liquid. We find that this spin-liquid phase is continuously connected to a previously discovered spin-liquid phase of the isotropic J_{1}-J_{2} model. The two limits show nearly identical spin correlations, making the case that their respective spin liquids are isomorphic to each other.

Spin-Glass Ground State in a Triangular-Lattice Compound YbZnGaO_{4}

We report on comprehensive results identifying the ground state of a triangular-lattice structured YbZnGaO_{4} as a spin glass, including no long-range magnetic order, prominent broad excitation continua, and the absence of magnetic thermal conductivity. More crucially, from the ultralow-temperature ac susceptibility measurements, we unambiguously observe frequency-dependent peaks around 0.1 K, indicating the spin-glass ground state. We suggest this conclusion holds also for its sister compound YbMgGaO_{4}, which is confirmed by the observation of spin freezing at low temperatures. We consider disorder and frustration to be the main driving force for the spin-glass phase.

Spinon Magnetic Resonance of Quantum Spin Liquids

We describe electron spin resonance in a quantum spin liquid with significant spin-orbit coupling. We find that the resonance directly probes spinon continuum, which makes it an efficient and informative probe of exotic excitations of the spin liquid. Specifically, we consider spinon resonance of three different spinon mean-field Hamiltonians, obtained with the help of projective symmetry group analysis, which model a putative quantum spin liquid state of the triangular rare-earth antiferromagnet YbMgGaO_{4}. The band of absorption is found to be very broad and exhibit strong van Hove singularities of single spinon spectrum as well as pronounced polarization dependence.

Disorder-Induced Mimicry of a Spin Liquid in YbMgGaO_{4}

We suggest that a randomization of the pseudodipolar interaction in the spin-orbit-generated low-energy Hamiltonian of YbMgGaO_{4} due to an inhomogeneous charge environment from a natural mixing of Mg^{2+} and Ga^{3+} can give rise to orientational spin disorder and mimic a spin-liquid-like state. In the absence of such quenched disorder, 1/S and density matrix renormalization group calculations both show robust ordered states for the physically relevant phases of the model. Our scenario is consistent with the available experimental data, and further experiments are proposed to support it.