Unusual ordered phases of highly frustrated magnets: a review

A Triangular Frustrated Eu(II)-Organic Framework for Sub-Kelvin Magnetic Refrigeration

Attaining sub-Kelvin temperatures remains technologically challenging and often relies on the scarce resource 3He, unless employing adiabatic demagnetization refrigeration. Herein, the active coolant typically consists of weakly coupled paramagnetic ions, whose magnetic interaction strengths are comparable in energy to the relevant temperature regime of cooling. Such interactions depend strongly on inter-ion distances, fundamentally hindering the realization of dense coolants for sub-Kelvin refrigeration. We present a magnetically concentrated triangular coordination network, Eu0.9Ba0.1I2(pyrazine)3, featuring the large s = 7/2 moment of Eu(II). Electron paramagnetic resonance, magnetization, and heat capacity measurements reflect antiferromagnetic correlations between the Eu(II) ions and a dominant easy-plane magnetic anisotropy. The ensuing geometric frustration prevents entropy-annihilating magnetic order down to at least 0.17 K. This remarkably low operational temperature for a low-dimensional, molecule-based magnetic refrigerant opens possibilities for on-chip cryogenic refrigeration, especially in scenarios where established inorganic refrigerants cannot be utilized.

Effect of Fe doping on the electronic properties of CoSn Kagome semimetal

Quantum phenomena in two-dimensional Kagome materials lead to exotic topological states and complex magnetism. Here, we have investigated the detailed electronic properties of Co1-xFexSn as a function of composition (x) to explore the competing electronic interactions for the origin of complex magnetism and topological properties. We find that the screening effect in the valence electrons increases while the correlation effect decreases with an increase in the Fe doping. Valence fluctuations observed at Co and FeL2,3edges showed systematic changes in the magnitude of divalent and trivalent states with the increase inx. Fe 3dstates are found to be more screened by the conduction electrons than the Co 3dstates. A comparison of the theoretical and experimental density of states showed different natures of localized states with strong screening effects on the surface and dominating correlation effects in the bulk forx>0. We have observed localized flat bands on the CoSn (001) surface while quasi-localized flat bands on the Co0.94Fe0.06Sn (001) surface. The distinct character of the bulk and surface band structure is confirmed in the Fe-doped composition. Hence, the bulk-surface interaction present in Co1-xFexSn gives rise to the origin of valence fluctuation, complex magnetism, and topological properties.

Continuum Excitations in a Spin Supersolid on a Triangular Lattice

Magnetic, thermodynamic, neutron diffraction and inelastic neutron scattering are used to study spin correlations in the easy-axis XXZ triangular lattice magnet K_{2}Co(SeO_{3})_{2}. Despite the presence of quasi-2D "supersolid" magnetic order, the low-energy excitation spectrum contains no sharp modes and is instead a broad and structured multiparticle continuum. Applying a weak magnetic field drives the system into an m=1/3 fractional magnetization plateau phase and restores sharp spin wave modes. To some extent, the behavior at zero field can be understood in terms of spin wave decay. However, the presence of clear excitation minima at the M points of the Brillouin zone suggest that the spinon language may provide a more adequate description, and signals a possible proximity to a Dirac spin liquid state.

Frustrated Extended Bose-Hubbard Model and Deconfined Quantum Critical Points with Optical Lattices at the Antimagic Wavelength

The study of geometrically frustrated many-body quantum systems is of central importance to uncover novel quantum mechanical effects. We design a scheme where ultracold bosons trapped in a one-dimensional state-dependent optical lattice are modeled by a frustrated Bose-Hubbard Hamiltonian. A derivation of the Hamiltonian parameters based on Cesium atoms, further show large tunability of contact and nearest-neighbor interactions. For pure contact repulsion, we discover the presence of two phases peculiar to frustrated quantum magnets: the bond-order-wave insulator with broken inversion symmetry and a chiral superfluid. When the nearest-neighbor repulsion becomes sizable, a further density-wave insulator with broken translational symmetry can appear. We show that the phase transition between the two spontaneously symmetry-broken phases is continuous, thus representing a one-dimensional deconfined quantum critical point not captured by the Landau-Ginzburg-Wilson symmetry-breaking paradigm. Our results provide a solid ground to unveil the novel quantum physics induced by the interplay of nonlocal interactions, geometrical frustration, and quantum fluctuations.

Field-Induced Non-BEC Transitions in Frustrated Magnets

Frustrated spin systems have traditionally proven challenging to understand, owing to a scarcity of controlled methods for their analyses. By contrast, under strong magnetic fields, certain aspects of spin systems admit simpler and universal description in terms of hardcore bosons. The bosonic formalism is anchored by the phenomenon of Bose-Einstein condensation (BEC), which has helped explain the behaviors of a wide range of magnetic compounds under applied magnetic fields. Here, we focus on the interplay between frustration and externally applied magnetic field to identify instances where the BEC paradigm is no longer applicable. As a representative example, we consider the antiferromagnetic J_{1}-J_{2}-J_{3} model on the square lattice in the presence of a uniform external magnetic field, and demonstrate that the frustration-driven suppression of the Néel order leads to a Lifshitz transition for the hardcore bosons. In the vicinity of the Lifshitz point, the physics becomes unmoored from the BEC paradigm, and the behavior of the system, both at and below the saturation field, is controlled by a Lifshitz multicritical point. We obtain the resultant universal scaling behaviors, and provide strong evidence for the existence of a frustration and magnetic-field driven correlated bosonic liquid state along the entire phase boundary separating the Néel phase from other magnetically ordered states.

Giant magnetocaloric effect in spin supersolid candidate Na2BaCo(PO4)2

Supersolid, an exotic quantum state of matter that consists of particles forming an incompressible solid structure while simultaneously showing superfluidity of zero viscosity1, is one of the long-standing pursuits in fundamental research2,3. Although the initial report of 4He supersolid turned out to be an artefact4, this intriguing quantum matter has inspired enthusiastic investigations into ultracold quantum gases5-8. Nevertheless, the realization of supersolidity in condensed matter remains elusive. Here we find evidence for a quantum magnetic analogue of supersolid-the spin supersolid-in the recently synthesized triangular-lattice antiferromagnet Na2BaCo(PO4)2 (ref. 9). Notably, a giant magnetocaloric effect related to the spin supersolidity is observed in the demagnetization cooling process, manifesting itself as two prominent valley-like regimes, with the lowest temperature attaining below 100 mK. Not only is there an experimentally determined series of critical fields but the demagnetization cooling profile also shows excellent agreement with the theoretical simulations with an easy-axis Heisenberg model. Neutron diffractions also successfully locate the proposed spin supersolid phases by revealing the coexistence of three-sublattice spin solid order and interlayer incommensurability indicative of the spin superfluidity. Thus, our results reveal a strong entropic effect of the spin supersolid phase in a frustrated quantum magnet and open up a viable and promising avenue for applications in sub-kelvin refrigeration, especially in the context of persistent concerns about helium shortages10,11.

Complete field-induced spectral response of the spin-1/2 triangular-lattice antiferromagnet CsYbSe2

Fifty years after Anderson's resonating valence-bond proposal, the spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLHAF) remains the ultimate platform to explore highly entangled quantum spin states in proximity to magnetic order. Yb-based delafossites are ideal candidate TLHAF materials, which allow experimental access to the full range of applied in-plane magnetic fields. We perform a systematic neutron scattering study of CsYbSe2, first proving the Heisenberg character of the interactions and quantifying the second-neighbor coupling. We then measure the complex evolution of the excitation spectrum, finding extensive continuum features near the 120°-ordered state, throughout the 1/3-magnetization plateau and beyond this up to saturation. We perform cylinder matrix-product-state (MPS) calculations to obtain an unbiased numerical benchmark for the TLHAF and spectacular agreement with the experimental spectra. The measured and calculated longitudinal spectral functions reflect the role of multi-magnon bound and scattering states. These results provide valuable insight into unconventional field-induced spin excitations in frustrated quantum materials.

Emergent Spin Frustration in Neutral Mixed-Valence 2D Conjugated Polymers: A Potential Quantum Materials Platform

Two-dimensional conjugated polymers (2DCPs)─organic 2D materials composed of arrays of carbon sp² centers connected by π-conjugated linkers─are attracting increasing attention due to their potential applications in device technologies. This interest stems from the ability of 2DCPs to host a range of correlated electronic and magnetic states (e.g., Mott insulators). Substitution of all carbon sp² centers in 2DCPs by nitrogen or boron results in diamagnetic insulating states. Partial substitution of C sp² centers by B or N atoms has not yet been considered for extended 2DCPs but has been extensively studied in the analogous neutral mixed-valence molecular systems. Here, we employ accurate first-principles calculations to predict the electronic and magnetic properties of a new class of hexagonally connected neutral mixed-valence 2DCPs in which every other C sp² nodal center is substituted by either a N or B atom. We show that these neutral mixed-valence 2DCPs significantly energetically favor a state with emergent superexchange-mediated antiferromagnetic (AFM) interactions between C-based spin-¹/₂ centers on a triangular sublattice. These AFM interactions are surprisingly strong and comparable to those in the parent compounds of cuprate superconductors. The rigid and covalently linked symmetric triangular AFM lattice in these materials thus provides a highly promising and robust basis for 2D spin frustration. As such, extended mixed-valence 2DCPs are a highly attractive platform for the future bottom-up realization of a new class of all-organic quantum materials, which could host exotic correlated electronic states (e.g., unusual magnetic ordering, quantum spin liquids).

Magnetic phase diagram of a 2-dimensional triangular lattice antiferromagnet Na2BaMn(PO4)2

We report the magnetic phase transitions of a spin-5/2, 2-dimensional triangular lattice antiferromagnet (AFM) Na₂BaMn(PO₄)₂. From specific heat measurements, we observe two magnetic transitions at temperatures 1.15 and 1.30 K at zero magnetic field. Detailed AC magnetic susceptibility measurements reveal multiple phases including the↑↑↓(up-up-down)-phase between 1.9 and 2.9 T at 47 mK when magnetic field is applied along thecaxis, implying that Na₂BaMn(PO₄)₂is a classical 2dTL Heisenberg AFM with easy-axis anisotropy. However, it deviates from an ideal model as evidenced by a hump region with hysteresis between the↑↑↓andV-phases and weak phase transitions. Our work provides another experimental example to study frustrated magnetism in 2dTL AFM which also serves as a reference to understand the possible quantum spin liquid behavior and anomalous phase diagrams observed in sibling systems Na₂BaM(PO₄)₂(M= Co, Ni).

Spin Density Wave versus Fractional Magnetization Plateau in a Triangular Antiferromagnet

We report an excellent realization of the highly nonclassical incommensurate spin-density wave (SDW) state in the quantum frustrated antiferromagnetic insulator Cs_{2}CoBr_{4}. In contrast to the well-known Ising spin chain case, here the SDW is stabilized by virtue of competing planar in-chain anisotropies and frustrated interchain exchange. Adjacent to the SDW phase is a broad m=1/3 magnetization plateau that can be seen as a commensurate locking of the SDW state into the up-up-down (UUD) spin structure. This represents the first example of the long-sought SDW-UUD transition in triangular-type quantum magnets.

Geometric Frustration on the Trillium Lattice in a Magnetic Metal-Organic Framework

In the dense metal-organic framework Na[Mn(HCOO)_{3}], Mn^{2+} ions (S=5/2) occupy the nodes of a "trillium" net. We show that the system is strongly magnetically frustrated: the Néel transition is suppressed well below the characteristic magnetic interaction strength; short-range magnetic order persists far above the Néel temperature; and the magnetic susceptibility exhibits a pseudo-plateau at 1/3-saturation magnetization. A simple model of nearest-neighbor Heisenberg antiferromagnetic and dipolar interactions accounts quantitatively for all observations, including an unusual 2-k magnetic ground state. We show that the relative strength of dipolar interactions is crucial to selecting this particular ground state. Geometric frustration within the classical spin liquid regime gives rise to a large magnetocaloric response at low applied fields that is degraded in powder samples as a consequence of the anisotropy of dipolar interactions.

Mutual information and breakdown of the Perron-Frobenius scenario in zero-temperature triangular Ising antiferromagnets on cylinders

A nominally two-dimensional spin model wrapped onto a cylinder can profitably be viewed, especially for long cylinders, as a one-dimensional chain. Each site of such a chain is a ring of spins with a complex state space. Traditional correlation functions are inadequate for the study of correlations in such a system and need to be replaced with something like mutual information. Being induced purely by frustration, the disorder of a cylindrical zero-temperature triangular Ising antiferromagnet (TIAFM) and attendant correlations have a chance of evading the consequences of the Perron-Frobenius theorem which describes and constrains correlations in thermally disordered one-dimensional systems. Correlations in such TIAFM systems and the aforementioned evasion are studied here through a fermionic representation. For cylindrical TIAFM models with open boundary conditions, we explain and derive the following characteristics of end-to-end mutual information: period-three oscillation of the decay length, halving of the decay length compared to what Perron-Frobenius predicts on the basis of transfer matrix eigenvalues, and subexponential decay-inverse square in the length-for certain systems.

Inq, a Modern GPU-Accelerated Computational Framework for (Time-Dependent) Density Functional Theory

We present inq, a new implementation of density functional theory (DFT) and time-dependent DFT (TDDFT) written from scratch to work on graphic processing units (GPUs). Besides GPU support, inq makes use of modern code design features and takes advantage of newly available hardware. By designing the code around algorithms, rather than against specific implementations and numerical libraries, we aim to provide a concise and modular code. The result is a fairly complete DFT/TDDFT implementation in roughly 12 000 lines of open-source C++ code representing a modular platform for community-driven application development on emerging high-performance computing architectures.

Non-Fermi liquid behavior below the Néel temperature in the frustrated heavy fermion magnet UAu2

The term Fermi liquid is almost synonymous with the metallic state. The association is known to break down at quantum critical points (QCPs), but these require precise values of tuning parameters, such as pressure and applied magnetic field, to exactly suppress a continuous phase transition temperature to the absolute zero. Three-dimensional non-Fermi liquid states, apart from superconductivity, that are unshackled from a QCP are much rarer and are not currently well understood. Here, we report that the triangular lattice system uranium diauride (UAu₂) forms such a state with a non-Fermi liquid low-temperature heat capacity [Formula: see text] and electrical resistivity [Formula: see text] far below its Néel temperature. The magnetic order itself has a novel structure and is accompanied by weak charge modulation that is not simply due to magnetostriction. The charge modulation continues to grow in amplitude with decreasing temperature, suggesting that charge degrees of freedom play an important role in the non-Fermi liquid behavior. In contrast with QCPs, the heat capacity and resistivity we find are unusually resilient in magnetic field. Our results suggest that a combination of magnetic frustration and Kondo physics may result in the emergence of this novel state.

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.

Continuous control of classical-quantum crossover by external high pressure in the coupled chain compound CsCuCl3

In solid materials, the parameters relevant to quantum effects, such as the spin quantum number, are basically determined and fixed at the chemical synthesis, which makes it challenging to control the amount of quantum correlations. We propose and demonstrate a method for active control of the classical-quantum crossover in magnetic insulators by applying external pressure. As a concrete example, we perform high-field, high-pressure measurements on CsCuCl₃, which has the structure of weakly-coupled spin chains. The magnetization process experiences a continuous evolution from the semi-classical realm to the highly-quantum regime with increasing pressure. Based on the idea of "squashing” the spin chains onto a plane, we characterize the change in the quantum correlations by the change in the value of the local spin quantum number of an effective two-dimensional model. This opens a way to access the tunable classical-quantum crossover of two-dimensional spin systems by using alternative systems of coupled-chain compounds.

In real materials, a spin quantum number assumes a fixed value, which makes it challenging to realize a crossover between quantum and classical spin regimes. Here the authors demonstrate such a crossover in a weakly coupled chain compound by controlling the amount of quantum correlations, in the form of the inverse spin quantum number, with external pressure.

Magnetization Process of Atacamite: A Case of Weakly Coupled S=1/2 Sawtooth Chains

We present a combined experimental and theoretical study of the mineral atacamite Cu_{2}Cl(OH)_{3}. Density-functional theory yields a Hamiltonian describing anisotropic sawtooth chains with weak 3D connections. Experimentally, we fully characterize the antiferromagnetically ordered state. Magnetic order shows a complex evolution with the magnetic field, while, starting at 31.5 T, we observe a plateaulike magnetization at about M_{sat}/2. Based on complementary theoretical approaches, we show that the latter is unrelated to the known magnetization plateau of a sawtooth chain. Instead, we provide evidence that the magnetization process in atacamite is a field-driven canting of a 3D network of weakly coupled sawtooth chains that form giant moments.

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.

Proximate Quantum Spin Liquid on Designer Lattice

Complementary to bulk synthesis, here we propose a designer lattice with extremely high magnetic frustration and demonstrate the possible realization of a quantum spin liquid state from both experiments and theoretical calculations. In an ultrathin (111) CoCr₂O₄ slice composed of three triangular and one kagome cation planes, the absence of a spin ordering or freezing transition is demonstrated down to 0.03 K, in the presence of strong antiferromagnetic correlations in the energy scale of 30 K between Co and Cr sublattices, leading to the frustration factor of ∼1000. Persisting spin fluctuations are observed at low temperatures via low-energy muon spin relaxation. Our calculations further demonstrate the emergence of highly degenerate magnetic ground states at the 0 K limit, due to the competition among multiply altered exchange interactions. These results collectively indicate the realization of a proximate quantum spin liquid state on the synthetic lattice.

Impact of the Lattice on Magnetic Properties and Possible Spin Nematicity in the S=1 Triangular Antiferromagnet NiGa_{2}S_{4}

NiGa_{2}S_{4} is a triangular lattice S=1 system with strong two dimensionality of the lattice, actively discussed as a candidate to host spin-nematic order brought about by strong quadrupole coupling. Using Raman scattering spectroscopy we identify a phonon of E_{g} symmetry which can modulate magnetic exchange J_{1} and produce quadrupole coupling. Additionally, our Raman scattering results demonstrate a loss of local inversion symmetry on cooling, which we associate with sulfur vacancies. This will lead to disordered Dzyaloshinskii-Moriya interactions, which can prevent long-range magnetic order. Using magnetic Raman scattering response we identify 160 K as a temperature of an upturn of magnetic correlations. The temperature range below 160 K, but above 50 K where antiferromagnetic correlations start to increase, is a candidate for spin-nematic regime.

Magnon Crystallization in the Kagome Lattice Antiferromagnet

We present numerical evidence for the crystallization of magnons below the saturation field at nonzero temperatures for the highly frustrated spin-half kagome Heisenberg antiferromagnet. This phenomenon can be traced back to the existence of independent localized magnons or, equivalently, flatband multimagnon states. We present a loop-gas description of these localized magnons and a phase diagram of this transition, thus providing information for which magnetic fields and temperatures magnon crystallization can be observed experimentally. The emergence of a finite-temperature continuous transition to a magnon crystal is expected to be generic for spin models in dimension D>1 where flatband multimagnon ground states break translational symmetry.

Interplay of Magnetism and Topological Superconductivity in Bilayer Kagome Metals

The binary intermetallic materials, M_{3}Sn_{2} (M=3d transition metal) present a new class of strongly correlated systems that naturally allows for the interplay of magnetism and metallicity. Using first principles calculations we confirm that bulk Fe_{3}Sn_{2} is a ferromagnetic metal, and show that M=Ni and Cu are paramagnetic metals with nontrivial band structures. Focusing on Fe_{3}Sn_{2} to understand the effect of enhanced correlations in an experimentally relevant atomistically thin single kagome bilayer, our ab initio results show that dimensional confinement naturally exposes the flatness of band structure associated with the bilayer kagome geometry in a resultant ferromagnetic Chern metal. We use a multistage minimal modeling of the magnetic bands progressively closer to the Fermi energy. This effectively captures the physics of the Chern metal with a nonzero anomalous Hall response over a material relevant parameter regime along with a possible superconducting instability of the spin-polarized band resulting in a topological superconductor.

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.

Frustrated Magnetism in Triangular Lattice TlYbS2 Crystals Grown via Molten Flux

The triangular lattice compound TlYbS₂ was prepared as large single crystals via a molten flux growth technique using sodium chloride. Anisotropic magnetic susceptibility measurements down to 0.4 K indicate a complete absence of long-range magnetic order. Despite this lack of long-range order, short-range antiferromagnetic interactions are evidenced through broad transitions, suggesting frustrated behavior. Variable magnetic field measurements reveal metamagnetic behavior at temperatures ≤2 K. Complex low temperature field-tunable magnetic behavior, in addition to no observable long-range order down to 0.4 K, suggest that TlYbS₂ is a frustrated magnet and a possible quantum spin liquid candidate.

Emergent Magnetic State in (111)-Oriented Quasi-Two-Dimensional Spinel Oxides

We report on the emergent magnetic state of (111)-oriented CoCr₂O₄ ultrathin films sandwiched between Al₂O₃ spacer layers in a quantum confined geometry. At the two-dimensional crossover, polarized neutron reflectometry reveals an anomalous enhancement of the total magnetization compared to the bulk value. Synchrotron X-ray magnetic circular dichroism measurements demonstrate the appearance of a long-range ferromagnetic ordering of spins on both Co and Cr sublattices. Brillouin function analyses and ab-initio density functional theory calculations further corroborate that the observed phenomena are due to the strongly altered magnetic frustration invoked by quantum confinement effects, manifested by the onset of a Yafet-Kittel-type ordering as the magnetic ground state in the ultrathin limit, which is unattainable in the bulk.

Heisenberg-Kitaev physics in magnetic fields

Magnetic insulators in the regime of strong spin-orbit coupling exhibit intriguing behaviors in external magnetic fields, reflecting the frustrated nature of their effective interactions. We review the recent advances in understanding the field responses of materials that are described by models with strongly bond-dependent spin exchange interactions, such as Kitaev's celebrated honeycomb model and its extensions. We discuss the field-induced phases and the complex magnetization processes found in these theories and compare with experimental results in the layered Mott insulators [Formula: see text]-RuCl₃ and Na₂IrO₃, which are believed to realize this fascinating physics.

Magnetism study on a triangular lattice antiferromagnet Cu2(OH)3Br

Magnetism of Cu₂(OH)₃Br single crystals based on a triangular lattice is studied by means of magnetic susceptibility, pulsed-field magnetization, and specific heat measurements. There are two inequivalent Cu²⁺ sites in an asymmetric unit. Both Cu²⁺ sublattices undergo a long-range antiferromagnetic (AFM) order at [Formula: see text] K. Upon cooling, an anisotropy crossover from Heisenberg to XY behavior is observed below 7.5 K from the anisotropic magnetic susceptibility. The magnetic field applied within the XY plane induces a spin-flop transition of Cu²⁺ ions between 4.9 T and 5.3 T. With further increasing fields, the magnetic moment is gradually increased but is only about half of the saturation of a Cu²⁺ ion even in 30 T. The individual reorientation of the inequivalent Cu²⁺ spins under field is proposed to account for the magnetization behavior. The observed spin-flop transition is likely related to one Cu site, and the AFM coupling among the rest Cu spins is so strong that the 30 T field cannot overcome the anisotropy. The temperature dependence of the magnetic specific heat, which is well described by a sum of two gapped AFM contributions, is a further support for the proposed scenario.

Pressure-tuning the quantum spin Hamiltonian of the triangular lattice antiferromagnet Cs2CuCl4

Quantum triangular-lattice antiferromagnets are important prototype systems to investigate numerous phenomena of the geometrical frustration in condensed matter. Apart from highly unusual magnetic properties, they possess a rich phase diagram (ranging from an unfrustrated square lattice to a quantum spin liquid), yet to be confirmed experimentally. One major obstacle in this area of research is the lack of materials with appropriate (ideally tuned) magnetic parameters. Using Cs₂CuCl₄ as a model system, we demonstrate an alternative approach, where, instead of the chemical composition, the spin Hamiltonian is altered by hydrostatic pressure. The approach combines high-pressure electron spin resonance and r.f. susceptibility measurements, allowing us not only to quasi-continuously tune the exchange parameters, but also to accurately monitor them. Our experiments indicate a substantial increase of the exchange coupling ratio from 0.3 to 0.42 at a pressure of 1.8 GPa, revealing a number of emergent field-induced phases.

Theoretical studies of quantum magnetism typically assume idealised lattices with freely tunable parameters, which are difficult to realise experimentally. Zvyagin et al. perform challenging measurements at high pressures to tune and to accurately monitor the exchange parameters of a triangular lattice antiferromagnet.

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.

Magnetic structure and spin waves in the frustrated ferro-antiferromagnet Pb2VO(PO4)2

Single crystal neutron diffraction, inelastic neutron scattering, and electron spin resonance experiments are used to study the magnetic structure and spin waves in Pb2VO(PO4)2, a prototypical layered S = 1/2 ferromagnet with frustrating next-nearest neighbor antiferromagnetic interactions. The observed excitation spectrum is found to be inconsistent with a simple square lattice model previously proposed for this material. At least four distinct exchange coupling constants are required to reproduce the measured spin wave dispersion. The degree of magnetic frustration is correspondingly revised and found to be substantially smaller than in all previous estimates.

Competing spin fluctuations and trace of vortex dynamics in the two-dimensional triangular-lattice antiferromagnet AgCrS2

The spin dynamics of the two-dimensional triangular-lattice antiferromagnet AgCrS₂ is investigated by electron spin resonance (ESR) spectroscopy. The g-factor is found to show an unusual non-monotonously temperature dependent behavior, which, along with the super-Curie behavior observed in the ESR intensity data, provides clear evidence for the competition between ferromagnetic and antiferromagnetic fluctuations at temperatures well above T _N. On approaching the Néel temperature T _N from above, the linewidth is found to diverge. Such a divergent behavior could be well described by the Kawamura-Miyashita model due to Z2 type magnetic vortex-antivortex pairing, which is consistent with the expectation for a 2D Heisenberg magnetic system.

Electrodynamics of quantum spin liquids

Quantum spin liquids attract great interest due to their exceptional magnetic properties characterized by the absence of long-range order down to low temperatures despite the strong magnetic interaction. Commonly, these compounds are strongly correlated electron systems, and their electrodynamic response is governed by the Mott gap in the excitation spectrum. Here we summarize and discuss the optical properties of several two-dimensional quantum spin liquid candidates. First we consider the inorganic material herbertsmithite ZnCu₃(OH)₆Cl₂ and related compounds, which crystallize in a kagome lattice. Then we turn to the organic compounds [Formula: see text]-EtMe₃Sb[Pd(dmit)₂]₂, κ-(BEDT-TTF)₂Ag₂(CN)₃ and κ-(BEDT-TTF)₂Cu₂(CN)₃, where the spins are arranged in an almost perfect triangular lattice, leading to strong frustration. Due to differences in bandwidth, the effective correlation strength varies over a wide range, leading to a rather distinct behavior as far as the electrodynamic properties are concerned. We discuss the spinon contributions to the optical conductivity in comparison to metallic quantum fluctuations in the vicinity of the Mott transition.

Field-induced States and Excitations in the Quasicritical Spin-1/2 Chain Linarite

The mineral linarite, PbCuSO_{4}(OH)_{2}, is a spin-1/2 chain with frustrating nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic exchange interactions. Our inelastic neutron scattering experiments performed above the saturation field establish that the ratio between these exchanges is such that linarite is extremely close to the quantum critical point between spin-multipolar phases and the ferromagnetic state. We show that the predicted quantum multipolar phases are fragile and actually suppressed by a tiny orthorhombic exchange anisotropy and weak interchain interactions in favor of a dipolar fan phase. Including this anisotropy in classical simulations of a nearly critical model explains the field-dependent phase sequence of the phase diagram of linarite, its strong dependence of the magnetic field direction, and the measured variations of the wave vector as well as the staggered and the uniform magnetizations in an applied field.

New Description of Evolution of Magnetic Phases in Artificial Honeycomb Lattice

Artificial magnetic honeycomb lattice provides a two-dimensional archetypal system to explore novel phenomena of geometrically frustrated magnets. According to theoretical reports, an artificial magnetic honeycomb lattice is expected to exhibit several phase transitions to unique magnetic states as a function of reducing temperature. Experimental investigations of permalloy artificial honeycomb lattice of connected ultra-small elements, [Formula: see text] 12 nm, reveal a more complicated behavior. First, upon cooling the sample to intermediate temperature, [Formula: see text] 175 K, the system manifests a non-unique state where the long range order co-exists with short-range magnetic charge order and weak spin ice state. Second, at much lower temperature, [Formula: see text] 6 K, the long-range spin solid state exhibits a re-entrant behavior. Both observations are in direct contrast to the present understanding of this system. New theoretical approaches are needed to develop a comprehensive formulation of this two dimensional magnet.

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.

Nuclear Magnetic Resonance Signature of the Spin-Nematic Phase in LiCuVO_{4} at High Magnetic Fields

We report a ^{51}V nuclear magnetic resonance investigation of the frustrated spin-1/2 chain compound LiCuVO_{4}, performed in pulsed magnetic fields and focused on high-field phases up to 56 T. For the crystal orientations H∥c and H∥b, we find a narrow field region just below the magnetic saturation where the local magnetization remains uniform and homogeneous, while its value is field dependent. This behavior is the first microscopic signature of the spin-nematic state, breaking spin-rotation symmetry without generating any transverse dipolar order, and is consistent with theoretical predictions for the LiCuVO_{4} compound.

Scanning tunnelling spectroscopy as a probe of multi-Q magnetic states of itinerant magnets

The combination of electronic correlations and Fermi surfaces with multiple nesting vectors can lead to the appearance of complex multi-Q magnetic ground states, hosting unusual states such as chiral density waves and quantum Hall insulators. Distinguishing single-Q and multi-Q magnetic phases is however a notoriously difficult experimental problem. Here we propose theoretically that the local density of states (LDOS) near a magnetic impurity, whose orientation may be controlled by an external magnetic field, can be used to map out the detailed magnetic configuration of an itinerant system and distinguish unambiguously between single-Q and multi-Q phases. We demonstrate this concept by computing and contrasting the LDOS near a magnetic impurity embedded in three different magnetic ground states relevant to iron-based superconductors—one single-Q and two double-Q phases. Our results open a promising avenue to investigate the complex magnetic configurations in itinerant systems via standard scanning tunnelling spectroscopy, without requiring spin-resolved capability.

It remains difficult to distinguish single-Q and multi-Q magnetic states experimentally. Here, Gastiasoro et al. show that the magnetic configuration of an itinerant system can be mapped out to the local density of states near a magnetic impurity, distinguishing unambiguously between single-Q and multi-Q phases.

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.

Self-Induced Glassiness and Pattern Formation in Spin Systems Subject to Long-Range Interactions

We study the glass formation in two- and three-dimensional Ising and Heisenberg spin systems subject to competing interactions and uniaxial anisotropy with a mean-field approach. In three dimensions, for sufficiently strong anisotropy the systems always modulate in a striped phase. Below a critical strength of the anisotropy, a glassy phase exists in a finite range of temperature, and it becomes more stable as the system becomes more isotropic. In two dimensions the criticality is always avoided and the glassy phase always exists.

Effect of further-neighbor interactions on the magnetization behaviors of the Ising model on a triangular lattice

In this work, we study the magnetization behaviors of the classical Ising model on the triangular lattice using Monte Carlo simulations, and pay particular attention to the effect of further-neighbor interactions. Several fascinating spin states are identified to be stabilized in certain magnetic field regions, respectively, resulting in the magnetization plateaus at 2/3, 5/7, 7/9 and 5/6 of the saturation magnetization M S, in addition to the well-known plateaus at 0, 1/3 and 1/2 of M S. The stabilization of these interesting orders can be understood as the consequence of the competition between Zeeman energy and exchange energy.

Frustrated magnets in high magnetic fields-selected examples

An indispensable parameter to study strongly correlated electron systems is the magnetic field. Application of high magnetic fields allows the investigation, modification and control of different states of matter. Specifically for magnetic materials experimental tools applied in such fields are essential for understanding their fundamental properties. Here, we focus on selected high-field studies of frustrated magnetic materials that have been shown to host a broad range of fascinating new and exotic phases. We will give brief insights into the influence of geometrical frustration on the critical behavior of triangular-lattice antiferromagnets, the accurate determination of exchange constants in the high-field saturated state by use of electron spin resonance measurements, and the coupling of magnetic degrees of freedom to the lattice evidenced by ultrasound experiments. The latter technique as well allowed new, partially metastable phases in strong magnetic fields to be revealed.

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

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

Anomalous Magnetothermopower in a Metallic Frustrated Antiferromagnet

We report the temperature T and magnetic field H dependence of the thermopower S of an itinerant triangular antiferromagnet PdCrO_{2} in high magnetic fields up to 32 T. In the paramagnetic phase, the zero-field thermopower is positive with a value typical of good metals with a high carrier density. In marked contrast to typical metals, however, S decreases rapidly with increasing magnetic field, approaching zero at the maximum field scale for T>70  K. We argue here that this profound change in the thermoelectric response derives from the strong interaction of the 4d correlated electrons of the Pd ions with the short-range spin correlations of the Cr^{3+} spins that persist beyond the Néel ordering temperature due to the combined effects of geometrical frustration and low dimensionality.

Complex Field-Induced States in Linarite PbCuSO4(OH)2 with a Variety of High-Order Exotic Spin-Density Wave States

Low-temperature neutron diffraction and NMR studies of field-induced phases in linarite are presented for magnetic fields H∥b axis. A two-step spin-flop transition is observed, as well as a transition transforming a helical magnetic ground state into an unusual magnetic phase with sine-wave-modulated moments ∥H. An effective J[over ˜]_{1}-J[over ˜]_{2} single-chain model with a magnetization-dependent frustration ratio α_{eff}=-J[over ˜]_{2}/J[over ˜]_{1} is proposed. The latter is governed by skew interchain couplings and shifted to the vicinity of the ferromagnetic critical point. It explains qualitatively the observation of a rich variety of exotic longitudinal collinear spin-density wave, SDW_{p}, states (9≥p≥2).